PERTURBATION METHODS IN

FLUID MECHANICS

PERTURBATIONMETHODS IN

FLUID MECHANICS

By Milton Van DykeDEPART:\IE:\T OF AERO:\ACTICS A:\D ASTRO:\AUTICS

STA:\FORD c':\IVERSITY

STA:\FORD, CALlFOR:\IA

Annotated Edition

SKY AND WATER I, 1938by l\'1. C. Escher

Courtesy of ,"oqnl ("'111 ,', s r ',.' <..h CIles.. an -raIlClSCO. LaQ:ulla Beach :\C\\" York -\Chlcago. and the Es ~h - F I.' H' . . <inc.. l t'l DUIl( .1tHHL aags (TcIllccntCI1l11SCUIIL The Hague

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f: I ' . en mg 0 one ow into another (P, 8lJ) that is the hcall

o l le method of matched, , , . . .' ,as\ mptotlc expansions chscussed in Chapter J,

p1975

THE PARABOLIC PRESS

Stanford, California

f

PREFACE TO AN:-;OTATED EDITION

PREFACE TO ORIGI;\IAL EDITIO~ .

CONTENTS

XI

Xlll

196-!. 1975 BY ~VIILTO:-': VA:-': D ...·KE. ALL RIGIITS RESERVED.

THE PARABOLIC PRESSPost OfTicc Box 3032StanfOl·d. California 91305

International Standard Book "umber: ISR~ 0-915760-01-0Librarv of Congress Catalog Card l'\umber: 75-15072

Ori gi nally published 196-! bv Academic PressAnnotated Edition 1975

Printed in the United States of America

I. THE NATURE OF PERTURBATION THEORY

1.1. .-\pproximations in fluid mechanics . .1.2. Rational and irrational approximations.1.3. Examples of perturbation expansions1.4. Regular and singular perturbation problems

II. SOME REGULAR PERTURBATION PROBLEMS

2.1. Introduction; basic flow past a circle2.2. Circle in slight shear flow . . . .2.3. Slightly distorted circle . . . . .2.4. Circle in slightly compressible flow2.5. Effect of slight viscosity. . . . .2.6. Boundary layer by coordinate expansion.

Exercises . . . . . . . . .

III. THE TECHNIQUES OF PERTURBATION THEORY

3. I. Introduction; limit processes .3.2. Gauge functions and order symbols. . . . .3.3. Asymptotic representations; asymptotic series3.4. Asymptotic sequences .. . . . . . . . .3.5. Convergence and accuracy of asymptotic series3.6. Propertics of asymptotic expansions.3.7. Successivc approximations. . .3.8. Transfer of boundary conditions3.9. Direct coordinate expansions3.10. Inverse coordinate expansions .3. I I. Change of type and of characteristics

Exercises .

v

I246

9111315161719

212326283032343637414243

vi Contents Contents "ii

IV. SOME SINGULAR PERTURBATION PROBLEMSIN AIRFOIL THEORY

4.1. Introduction .4.2. Formal thin-airfoil expansion4.3. Solution of the thin-airfoil problem4.4. ::\onunifonnity for elliptic airfoil .4.5. ::\onuniqueness; eigensolutions. .4.6. JoukO\\'ski airfoil; leading-edge drag4.7. Bicoll\c:\ airfoil; n:ct<lngular airfoil.4.8. .-\ multiplicati\'e correction for round edges4.9. Local solution ncar a round edge . . . .4.10. :\Iatching \\ith solution ncar round edge4.11. :\Iatching \\ith solution near sharp edge4.12..\ shifting correction for round edges

Exercises . . . . : . . . . . . . .

V. THE METHOD OF MATCHED ASYMPTOTICEXPANSIONS

5.1. Historical introduction . . . . . . . . .5.2. :\onuniformity of straightformlnl expansion5.3. A physical criterion for uniformity . . . .5.4. The role of composite and inner expansions5.5. Choice of inner \'ariables5.6. The role of matching .5.7. :\Iatching principles5.8. Intermediate matching5.9. :\latching order . . .5.10. Construction of composite expansions .

Exercises .

45464850525456596264687173

7778808385888991939497

6.8.

VII.

7.1.7.2.7.3.7.4.7.5.7.6.7.7.7.8.7.9.7.10.7.11.7.12.7.13.7.14.7.15.

VIII.

8.1.8.2.8.3.8.4.8.5.8.6.8.7.

t'tilitv of the method of strained coordinates.Exercises . .

VISCOUS FLOW AT HIGH REYNOLDS NUMBER

Introduction ..\lternative interpretations of flat-plate solution.Outer c:\pansion for flat plate; basic in\'iscid flo\\'Inner expansion; boundary-layer equations; matchingBoundary-layer solution for flat plateL'niqueness of the Blasius solution .1"10\\ due to displacement thickness. . . . . . .Second-order boundary layer for semi-infinite plateSecond-order boundary layer for finite plateI,ocal and integrated skin frictionThird approximation for semi-infinite plateThe effect of changing boundary-layer coordinates.\lternative coordinates for flat phiteDetermination of optimal coordinates . . .Extension of the idea of optimal coordinatesExercises . . . . . . . . . . . .

VISCOUS FLOW AT LOW REYNOLDS NUMBER

Introduction .Stokes' solution for sphere and circle . .The paradoxes of Stokes and WhiteheadThe Oseen approximation. . . . . .Second approximation far from sphere.Second approximation near sphereHigher approximations for circleExercises .

118119

121122124126129131132134136137139140142144145146

149151152153156159161165

VI. THE METHOD OF STRAINED COORDINATES

6.1. Historical introduction . . . . . . . . . . .6.2. A model ordinary differential equation. . . . .6.3. Comparison with method of matched expansions6.4. Nonuniformity in supersonic airfoil theory.6.5. First approximation by strained coordinates6.6. :\Iodifications for corners and shock waves .6.7. First approximation by matched expansions

99101104106109112116

IX. SOME INVISCID SINGULAR PERTURBATIONPROBLEMS

9.1. Introduction .9.2. Lifting wing of high aspect ratio .. .9.3. Lifting-line theory by matched asymptotic expansions.9.4. Summary of third approximation .9.5. Application to elliptic \\'ing . . . . . . . . . . . .

167167170173174

viii

9.6.9.7.9.8.9.9.9.10.9.11.9.129.13.

Contents

Slightly supersonic flow past a slender circular cone .Second approximation and shock position .Third approximation for pressure on cone. .Hypersonic flow past thin blunted wedge . .Small-disturbance solution for blunted wedge~Iiddle expansion for entropy layer. . .Inner expansion for entropy layerComposite expansions for blunted wedgeExercises .

176178180182185186187190192

Contents

Note 13. Lifting wing of high aspect ratioNote 14. The method of multiple scales .Note 15. Analysis and improvement of seriesNote 16. The resolution of paradoxes .

REFERENCES AND AUTHOR INDEX

SUBJECT INDEX .

ix

239241243247

249

263

X. OTHER ASPECTS OF PERTURBATION THEORY

10.1. Introduction .10.2. The method of composite equations10.3. The method of composite expansionslOA. The method of multiple scales . . .10.5. The pre\"alence of logarithms10.6.. Impro\'ement of series; natural coordinates.10.7. Rational fractions . . . . . .10.8. The Euler transformation . . . . . . .10.9. Joining of coordinate expansions ....10.10.Joining of diflerent parameter expansions

Exercises .

NOTES

195195197198200202205207210212213

Note 1.Note 2.Note 3.Note 4.Note 5.Note 6.Note 7.Note 8.Note 9.Note 10.Note 11.Note 12.

IntroductionComputer extension of regular perturbationsComments on the exercises .The asymptotic matching principle .The theory of matching .Alternative rules for composite expansionsUtility of the method of strained coordinatesFlat plate at high Reynolds number; triple decks.Extension of the idea of optimal coordinatesThe sphere and circle at low Reynolds numberTranscendentally small termsViscous flow past paraboloids

215215217220225227228230232234236238

PREFACE TO THE

ANNOTATED EDITION

The technical literature devoted to perturbation methods in me­chanics has burgeoned since this book appeared in 1964. Techniquesfor treating singular perturbation problems, then unfamiliar andesoteric, are now part of the analytical apparatlls of anyone interestedin research. Applications in continuum mechanics, first limitedmainly to classical fluid dynamics and linear elasticity, have spreadover a wide range of fields, and new techniques are being developedwhile older ones are refined. The resulting research papers, even ifrestricted to fluid mechanics, number in the thousands.

Other books have meanwhile appeared to codify and guide thisbody of work. Those of an applied character similar to this one are thebooks of Bellman (1964), Cole (1968), and Nayfeh (1973). Moremathematically oriented treatments include O'Malley (1968, 1974)and Eckhaus (1973).

Despite that competition, and a limited scope, the present book hasapparently remained useful. Its modest sales continued until its evenmore modest price made it unprofitable for the original publisher tokeep it in print. Rather than let the price increase, I have elected torepublish it myself.

In so doing, I have taken the opportunity to correct a number oferror in the text, and to bring it more up to date by adding a section ofNotes at the end. These are keyed to the main text, where an indicatorin the margin of the page, of the form shown here, directs the reader Seeto a discussion of subsequent developments and further references. NoteOf course only a small fraction of the papers published in the pastdecade have been mentioned, even though the list of References at theend has been doubled. However, an attempt has been made to in-clude any reference that provides a significant new idea or result, aswell as any of close relevance to the original text.

The most common criticism from reviewers and readers (aside fromthe restriction to fluid mechanics) is that this book is somewhat tooconcise for self study or use as a text. It has, nevertheless, been usedas the textbook for more college courses than I could have anticipated.For that purpose, the Exercises posed at the end of each chapter are

xi

xii Prcfa'T

~n some cases rather. too demanding (and in a few cases practicallyInsoluble). I have tned to improve this situation in Note 3. On re­ques~, I should be glad to provide colleagues with my versions of thesolutIOns, together \'vith additional exercises I have used in recentyears.

This Annotated Edition. like the original, derives from a one­quarter graduate course that I have continued to teach in the Depart­ment of Aeronautics and Astronautics at Stanford University since1959. Lik~ the original, it draws on my research and that of my stu­dents, \'vhlch has been supported for many years by the Air ForceOffice of Scientific Research.. I am in?ebted to a number of colleagues who ansv,'ered my pleaf?r help with the l\'otes. or otherwise contributed corrections, sugges­tIOn> and references. Of course I could not do justice to all their sug­gestIOns, .but fo~ valuable assistance I thank A. Acrivos, S. A. Berger.P. A. BOIs.,S. :'\. Brown, S. Corrsin. R. T. Davis, C. Domb, J. Ellin­wood, L. E. FraenkeL C. Franc;ois. P. Germain, C. R. Illingworth,K. P. Kerney. P. A. Lagerstrom. R. E. :Melnik, A. :Messiter, R. Medan.J. W. ~liles, X Riley, O. S. Ryzhov, V. V. Sychev, and H. Viviand.

The fraternity of typographers, printers, publishers, and booksellersis a ~riendly one, and I have found them all remarkably generous inhelpIng a neophyte. I could not have undertaken to republish thisbook myself without the advice of Sharon Hawkes. Jack McLean.Eva ~Xq~ist, Dorothy RiedeL and especially my friend's John McNeilan~ ~\ JI!Iam Kauffman at Annual Reviews Inc. Like the original, thisreVISIOn could .not hav: been written without the help and encourage­ment of my \vlfe Sylna, and I rededicate it to her as a gift of love.

MILTON V A:'oI DYKE

Stanford, CaliforniaJune, 1975

PREFACE TO

THE ORIGINAL EDITION

This book is devoted primarily to the treatment of singular perturba­tion problems as they arise in fluid mechanics. In particular, it gives aunified exposition of two rather general techniques that have beendeveloped during the last fifteen years, and which are associatedwith the names of Lagerstrom, Kaplun, and Cole, and of Lighthilland \\'hitham. This emphasis on what might to the uninitiatedappear to be the pathological aspects of perturbation theory is justifiednot so much by the novelty of these techniques as by the fact that singularperturbations seem to be the rule rather than the exception in fluidmechanics, and are being increasingly encountered in current research.However, the book begins with general methods applicable to regularas well as singular perturbations, because no connected account of themis available.

The exposition is largely by means of examples, and these are­except for a few mathematical models-drawn solely from fluid mechanics.It is true that the techniques discussed are rapidly finding applicationin other branches of applied mechanics, and I hope the book will proveuseful to workers in those fields. However, both of the general techniquesmentioned above were invented to handle problems in fluid flow, andhave been largely developed and applied within that field. In fact, theexamples are largely confined to what might at mid-century be charac­terized as classical aerodynamics. It is evident, however, that singularperturbation problems abound in such new subjects as non-equilibriumand radiating flows, magnetohydrodynamics, plasma dynamics, andrarefied-gas dynamics. The techniques discussed will certainly findfruitful application there, as well as in oceanography, meteorology, andother domains of the great world of fluid motion.

This book is the outgrowth of a succession of notes prepared for agraduate course that I have taught since 1959 in the Department ofAeronautics and Astronautics at Stanford University. It naturally drawsheavily on my own research and that of my students, much of whichhas been supported by the Air Force Office of Scientific Research.

The heart of the book is the study, in Chapter IV, of incompressiblepotential flow past a symmetrical thin airfoil. This problem, though

xiii

xiv Preface to Original Edition rconceptually simple and inyolying only the two-dimensional Laplaceequation, embodies most of the features of both regular and singularperturbations. In particular, it seryes to introduce the two standardmethods of treating singular perturbation problems, References to thisbasic problem therefore recur throughout the subsequent chapters.

I would urge the reader not to ignore the exercises. They providein concise form many additional details, further references, and generali­zations and extensions of the material in the text,

:\Iy first debt is to P..-\. Lagerstrom, who has been my teacher,colleague, and friend as well as the co-denJoper of one of the twomain techniques described here for handling singular perturbationproblems. :\Iany of the ideas presented also bear the ilnprint of myyears of collaboration with R. T. Jones, :\1. .-\. Heaslet, and theircolleagues at .-\mes Laboratory. I am indebted to a number of othercolleagues for helpful comments and criticism, including in particularO. Burggraf, I-Dee Chang, G. Emanuel, S, Kaplun, S. ::\adir, B. Perry,and .-\. F. Pillow. This book \\'ould not have been written without thehelp and encouragement of my \yife Sylyia, and I dedicate it to her asa gift of loye.

:\IILTO:\" VA:\" DYKE

Stanford, CaliforniaMay, 1964

Chapter I

THE NATURE

OF PERTURBATION THEORY

1.1. Approximations in Fluid Mechanics

Fluid mechanics has pioneered in the solution of nonlinear partialdifferential equations, In contrast with the basic equations in many otherbranches of mathematical physics, those gO\'erning fluid motion areessentially nonlinear (more precisely, quasi-linear); and this is truewhether or not yiscosity and compressibility are included. The onlyimportant exception is the well explored case of irrotational motion ofan incompressible im'iscid fluid, which leads to Laplace's equation, thenonlinearity then appearing only algebraically in the Bernoulli equation,proyided there are no free boundaries.

Because of this basic nonlinearity, exact solutions are rare in anybranch of fluid mechanics. They are usually self-similar solutions, for\vhich the partial differential equations reduce, by yirtue of a high degreeof symmetry, to ordinary differential equations. So great is the needthat a solution is loosely termed "exact" eyen when an ordinarydifferential equation must be integrated numerically. Lighthill (1948)has giyen a more or less exhaustive list of such solutions for inviscidcompressible flow:

(a) steady supersonic flow past a concaye corner,(b) steady supersonic flow past a conyex corner,(c) steady supersonic flow past an unyawed circular cone,(d) infinite plane wall moved impulsively into still air,(e) infinite plane wall moyed impulsively away from still air,(f) circular cylinder expanding uniformly into still air,(g) sphere expanding uniformly into still air.

Again, from Schlichting (1960) one can construct a partial list forincompressible viscous flow:

(a) steady flow between infinite parallel plates, through a circularpipe, or between concentric circular pipes,

1.2. Rational and Irrational Approximations

:\10st useful approximations are valid when one or more of theparameters or variables in the problem is small (or large). This pertur­bation quantity is often one of the dimensionless parameters:

(b) steady flow bet\\cen a fixed and a sliding parallel plate or concentriccircular pipe,

(c) steady flow between concentric rotating cylinders,(d) plane or axisymmetric flow against an infinite plate,(e) steadv rotation of an infinite flat disk,(f) stead"y plane flow between di\'ergent plates,(g) impulsive or sinusoidal motion of an infinite flat plate in its own

plane.

It is typical of these self-similar flows that they involve idealized geom­etries far from most shapes of practical interest.

To proceed further, one must usually approximate. (A recent alter­native is to launch an electronic computing program!) Approximationis an art, and famous names are usually associated with successfulapproximations:

Prandtl wing theory,Karman-Tsien method for airfoils in subsonic flow,Prandtl-Glauert approximation for subsonic flow,Janzen-Rayleigh expansion for subsonic flow,Stokes and Oseen approximations for viscous flow,Prandtl boundary-layer theory,Karman-Pohlhausen boundary-layer approximation,:\ewton-Busemann theory of hypersonic flow.

In some important fields an altogether successful approximation is yet tobe found. Examples are separated viscous flow, and hypersonic flowpast a blunt body.

I. The Nature of Perturbation Theory 3

Distanee <t: I

Time

1.2. Rational and Irrational Approximations

Blasius series for boundary layer:

Impulsive motion in viscous or

compressible fluid:

In such cases one speaks of a coordinate perturbation.An approximation of this sort becomes increasingly accurate as the

perturbation quantity tends to zero (or infinity). It is therefore anasymptotic solution. In principle, one can improve the result by embeddingit as the first step in a systematic scheme of successive approximations.The resulting series, though not necessarily convergent, is by construc­tion an asymptotic expansion. In practice, one usually calculates onlythe first approximation, sometimes the second. The chief virtue of asecond approximation is often that it helps to understand the first. Onlyrarely does one proceed as far as the fifth or sixth approximation; butthe possibility of continuing indefinitely is of fundamental significance.We shall call an approximation of this sort a rational approximation.

On the other hand, some \"Cry useful approximations do not becomeexact in any known limit. Examples are:

Karman-Tsien method for airfoils in subsonic flow (Liepmann andPuckett, 1947, p. 176),

Shock-expansion theory, and its extension to axisymmetric and three-dimensional flows (Hayes and Probstein, 1959, p. 265),

Spreiter's local linearization in transonic flow (Spreiter, 1959),Chester-Chisnell theory of shock dynamics (Chester, 1960),Mott-Smith theory of shock structure (:\IIott-Smith, 1951).

\Ve shall call such a method an irrational approximation. Unless furtherstudy should reveal its asymptotic nature, an irrational approximationrepresents a dead end. One must accept whatever error it involves, withno possibility of improving the accuracy by successive approximations.

Only rational approximations are considered in this book. Thus weare concerned with asymptotic expansions, for small or large values ofsome parameter or independent variable, of the solutions of the equationsof fluid motion,

We will often find it convenient to denote the perturbation quantitybye, and to define it so that it is small. For example, in boundary-layer

.\n unusual example is Garabedian's (1956) analysis of axisymmetricfree-streamline flow, which is based upon the approximation that thenumber of space dimensions differs only slightly from t\\·o. In all thesecases one speaks of a parameter perturbation. The perturbation quantitymay also be one of the independent variables (in dimensionless form):

'I

,II

Mach number <{; I

Thickness ratio <{; 1

Aspect ratio ~ 1

Reynolds number <{; 1

Reynolds number ~ 1

~\Iach number ~ 1, (y - I) <{; I

Reduced frequency <{; I

Knudsen number ~ I

Janzen-Rayleigh expansion:

Thin-airfoil theorv:

Lifting-line theory:

Stokes, Oseen flow:

Boundary-layer theory:

Newton-Busemann theory:

Quasi-steady theory:

Free-molecule theory:

2

4 I. The Nature of Perturbation Theory 1.3. Examples of Perturbation Expansions 5

(jmax 1 I jl,12 .)-(j = 1 +. ~ e .- 2W (1 T T)e'

In thin-airfoil or slender-body theory, the solution for an elongatedobject is found as the perturbation of a uniform stream. Hantzsche(1943) has calculated the thin-airfoil expansion for an elliptic airfoil ofthickness ratio <: at zero incidence in a subsonic compressible stream.For the maximum speed he finds

(1.2a)

(1.2b)y --,- 1 ]lJ2

T=---4 (32(3 = Vl- :11 2,

I ;1J2 \ 7T [1 ..) 1 3 . I- ~:J (4 I -;- 2' T(l -, T)(8 - M-)] - '2 -- ':!. T - ! T\ e3

M2- 2(32 [I -~. ~ T( I + T)(8 - J12)]e4 Iog e ...!.- O(e4 )

\\'here

theory f can be taken as the reciprocal of the Reynolds number, or ofits square root.

As f tends to zero, the flow must be assumed to approach a limit,which may be termed the basic solution. For example, at high Reynoldsnumbers the viscous flow past most semi-infinite bodies approaches thecorresponding inviscid motion. That this approach is not uniform isclear from the ideas of boundary-layer theory. However, for bodies onwhich the boundary layer separates, we do not yet know the appropriatelimit-or, indeed, whether a limit exists (cf. Section 7.1).

I n parameter perturbations the basic solution is often a uniform parallelstream or other trivial flow. Then one usually regards it as the "zerothapproximation," and calls the leading perturbation therefrom the firstapproximation or first-order solution. In most coordinate perturbations,on the other hand, the basic solution is a nontrivial solution of the fullequations-for example, one of the self-similar flows discussedpreviously-and is itself usually regarded as the first approximation.

1.3. Examples of Perturbation Expansions

These general remarks may be illustrated by exhibiting some typicalperturbation expansions for which a number of terms have beencalculated by assiduous researchers. In the Janzen-Rayleigh or i1J2­expansion method, the effects of compressibility at subsonic speeds arestudied by perturbing the basic solution for incompressible flow. Thefirst-order correction is proportional to the square of the free-stream:\Iach number .\1, and higher approximations proceed by successivepowers of JJ2. Simasaki (1956) has calculated six terms of the series forsubsonic flo\\' past a circular cylinder without circulation. For themaximum speed Qlll3x-which occurs on the surface at the ends of thecross-stream diameter-referred to the free-stream speed U, he finds,with the adiabatic index y equal to 1.405:

SeeNote

2

Here, and throughout this book, log denotes the natural logarithm. In(1.2 a) as in many perturbation problems, logarithms of the perturbationquantity appear unexpectedly. For reasons discussed in Section 10.5,the unknO\\'Il next term, which is indicated to be of order e1, ought tobe included with the term in <:1 log f as the fourth approximation.

.-\t 1m\' Reynolds numbers the viscous flow past a three-dimensionalbody is described by the Stokes approximation. For bodies with fore-and­aft symmetry a better estimate of the drag is given by the Oseenapproximation. Goldstein (1929) has calculated six terms of a series forthe Oseen drag of a sphere. In terms of the Revnolds number R = Ua vbased upon the radius a, his result for the drag coefficient is

c ~ -~- - 6n. (' -'- ~ _ J..?_ 2 .2.!.- 3D - PU 2a2 - R ,I . 8 R 320 R + 2560 R

SeeNote

2

(/1';;"= 2.00000 + 1. I6667J\!I2 --1,2.5812911:/4

T 7.53386J.16 --r- 25.693422\1J8 T 96.79287JVJ10 + '" (1.1)

30,179 R4 ,2,150,400 .

122,519 R5 -+- ... )' (1.3)17,203,200 .

This method is believed to yield a series that converges if the flow ispurely subsonic. (A proof of convergence for sufficiently small 1112 wasreported by Wendt (1948), but never published.) Indeed, it appears fromthe behavior of the above coefficients that the convergence becomessuspect when Al approaches its critical value of about 0.40, at whichqmax becomes sonic (d. Exercise 10.6).

Here the last coefficient has been corrected by Shanks (1955). Thesolution of the full 0J avier-Stokes equations departs from the Oseenapproximation after the second term. According to the work of Proudmanand Pearson (1957) it gives instead

(1.4)

SeeNote

10

6 I. The Nature of Perturbation Theory 1.4. Regular and Singular Perturbation Problems 7

At high Reynolds numbers the viscous flow very close to a body isdescribed by Prandtl's boundary-layer equations. Because they arenonlinear, additional approximations are usually required. A coordinateperturbation for small distances from the stagnation point of any smoothsymmetrical plane shape, initiated by Blasius, has been carried to sixterms by Tifford (1954). For the coefficient of local skin friction on aparabola this gives (Yan Dyke, 1964a)

T 2 [X'X 3 (X 0;cf =,c .. ., ,= - 1.23259 - - 1.931~61_) -- 3.11051 -)~pC~ vi R a \ a I a

-,- 5.02~92 (~')' "7" ~.14109Cf -- 13.18662 (: (;- .,,] (\.5)

Here x is the distance along the surface, and R again the Reynolds numberbased upon the nose radius a.

Incompressible flow past a thin wing of high aspect ratio A is describedapproximately by Prandtl's lifting-line theory. The calculation ofhigher approximations is discussed in Chapter 9. For an elliptical wingthe lift-curve slope is found to be (Yan Dyke, 1964b)

deL = 2rr[1 __ ~ _ ~ log Adcx .--1 rr2 .--1 2

(1.6)

The notoriously difficult problem of hypersonic flow past a bluntbody has been attacked, in the :\ewton-I3usemann approximation, byexpanding in po\\'ers of I .112 and (y - I) (y I). Chester (1956b) hascarried the series farthest for the body of revolution that produces aparaboloidal shock waYe. At .11 ~ x he finds the ratio of the stand-offdistance Ll of the shock wave to its nose radius b to be

In later chapters we shall have something further to say about each ofthese examples.

lA. Regular and Singular Perturbation Problems

Under the happiest circumstances, a perturbation solution leads toaltogether satisfactory results. The series cannot often be presumed to

SeeNote

12

SeeNote

13

converge, particularly for parameter perturbations. :.Jevertheless, itsasymptotic nature means that a few terms may yield ample accuracv,for reasonably small c:, everywhere in the flow field. Such uniform validitvappears to be true, for example, of the Janzen-Rayleigh expansion belo\~'the critical :.vlach number. One speaks then of a regular perturbationproblem.

On the other hand, one may find that the straightforward perturbationsolution is not uniformly valid throughout the flow field. The bestknown example is unseparated viscous flO\V at high Reynolds numbers,where a perturbation of the basic im-iscid motion fails near the surface,and must be supplemented by the boundary-layer approximation. :\otonly does the first approximation break down locally in such cases, butthe difficulty is compounded in higher approximations --if thev can becalculated at all -so that in the region of non-uniformitv the' solutiongrows \\"()rse rather than better. Onethen speaks of a singul;lr perturbationproblem.

Singular perturbation problems arise frequenth- in fluid mechanics.They have been studied increasingly in the recent iiterature as the requi­site mathematical techniques were developed. Fresh insight is gainedinto e\'en some classical problems by recognizing their singular nature.For these reasons this book is largely devoted to singular perturbationproblems.

'1''''0 more or less general methods have recently been developed fortreating singular perturbation problems. One is a generalization of thenotions of boundary-layer theory, which we call the method of matchedCBYlllptotic e.\pallsiolls. The other is the extension of an idea due toPoincare, \\hich we call the method of strained coordillates. :\Iathematicaljustification of these two procedures is in its infancv. Therefore noprecise statements can be made as to when either of thc~ can be applied,as to which is preferable in a given problem, or as to how the twomethods are related. i\evertheless, we will attempt to gain some insightinto these questions by studying a variety of illustrative examples. Insome cases, ordinary differential equations will be adopted as simplemodels to demonstrate the essential points. \Vhenever possible,however, we draw our examples from the recent literature in fluidmechanics, so that partial differential equations are usually encoun­tered.

Because we are concerned with techniques having some generalitv ofapplication, our examples will-as in the results already quoted--,,-betaken from the theories of both inviscid and viscous flows, and for speedsranging from incompressible to hypersonic. The reader is assumed tounderstand the physical bases of those problems, which will often

8 I. The Nature of Perturbation Theory

illuminate the mathematics. He is also expected to be familiar with theelementary processes of analysis, the basic theory of partial differentialequations, and the fundamental notions of complex-variable theory,including in particular the grand concept of the complete description ofan analytic function that results from the process of analytic continuation. Chapter 11

SOME REGULAR

PERTURBATION PROBLEMS

2.1. Introduction; Basic Flow past a Circle

\Ye illustrate some of the techniques of perturbation theory byconsidering first several related regular perturbation problems. Thesetechniques \"ill be systematized in the next chapter. The complicationsassociated \\ith singular perturbations are deferred to later chapters.

Because exact solutions are rare, one cherishes them, and seeks toexploit them as fully as possible. Thus a single exact solution is oftenperturbed in a number of \vays, to explore different ctlects. In the usualaerodynamic problem of unbounded flow past a body, one may perturb

(i) the distant boundary conditions, far upstream,(ii) the near boundary conditions, at the body surface,

(iii) the equations of motion,

and each of these possibilities may in turn be carried out in a \"ariety ofways. For example, the exact inviscid solution for supersonic flow pasta circular cone has been perturbed as in (i) to study the effects of

(a) angle of attack (Stone, 1948),(b) pitching motion (Kawamura and Tsien, 1964),

as in (ii) for the effects of

(c) initial curvature of an ogival tip (Cabannes, 1951),(d) departure from circular cross section (Ferri et al., 1953),(e) slight blunting (Yakura, 1962),

and as in (iii) for the effects of

(f) slight viscosity (Hantzsche and Wendt, 1941).

To illustrate these possibilities without excessive calculation, we adoptas our basic solution the steady plane motion of an incompressible

9

10 II. Some Regular Perturbation Problems 2.2. Circle in Slight Shear Flow 11

(2.4c)

(2.4b)

(2.5a)

Fig. 2.2. Slight shear flow past a circle.

---­U-

The corresponding velocity potential is

and these can of course be combined in the complex expression

2.2. Circle in Slight Shear Flow

Consider first a slight perturbation of the boundarv conditions farupstream. Let the oncoming stream be a parallel flow with small constant\'Orticity (Fig. 2.2), its speed being

(2.1 )

(2.2)\~.p = -w(.p)

dl/J = u dy - v dx

In inviscid motion it satisfies the differential equation

-­U---

Fig. 2.1. Basic incompressible flow past a circle.

which expresses the physical fact that vortiCity is constant alongstreamlines in the absence of viscosity. This is a nonlinear equationunless the vorticity w(t/J) is a linear function of t/J. For uniform flow farupstream the vorticity vanishes, leaving Laplace's equation

imiscid fluid past a circle \\'ithout circulation (Fig. 2.1). It is convenientto work with the stream function as well as the velocity potential, becausewe later consider rotational flow, for which no potential exists.

For plane incompressible flow the stream function is defined in Carte­sian coordinates by

(2.3a)so that its stream function is, to within a constant,

Here, and throughout this book, subscripts are used to indicate partialderivatives. The boundary conditions are .pX) = U(y T~E y~) c= U[r sin 8 -LiE r

2(1 - cas 28)]

a . a(2.5b)

together with some condition to rule out circulation, such as a require­ment that the flow be symmetric about the line e = 0:

.p(r, 8) = -.p(r, -8)

upstream:

surface:

.p(r, 8) -+ Ur sin 8

.p(a,8) = 0

as r -+ 00 (2.3b)

(2.3c)

(2.3d)

with the vorticity

The full problem is therefore

.prr +- .pr +- .pee = E Ur r2 a

(2.5c)

(2.6a)

The solution is given by the uniform stream plus a dipole at the center

of the circle:2

.po = U(r - 7) sin 8 (2.4a)

.p -+ U[r sin 8 +- -iE ~ (1 - cos 28)]

.p(a,8) = 0

as r -+ 00 (2.6b)

(2.6c)

12 II. Some Regular Perturbation Problems 2.3. Slightly Distorted Circle 13

Discoverv of a particular integral has reduced the diHerential equationto homogeneous form-the I,aplace rather than the Poisson equation-sothat the ~olllplelllelltar)'sollition Xl satisfies the problem

(2.13)

(2.12)r = a( I - e sin2 8)

.p(r, 8; e) = .po(r, 8) + e.pl(r, 8) + ...

2.3. Slightly Distorted Circle

Consider next a perturbation of the boundary conditions at the surfaceof the body. This introduces some features of perturbation theory thatdid not appear in the preceding example. Let the body (Fig. 2.3) bedescribed by

This may be regarded either as the first approximation for an ellipse, orthe exact description of a more complicated curve.

As before, we tentatively assume a perturbation expansion in powersof e:

first-order perturbation consists of the rotational part of the free stream,its image in the circle, and a constant to adjust the stream function tozero on the surface,

Ordinarily one would write "-+- ... " or "- i~ 0(:2)" at the end of thisequation, to emphasize that the result is correct only to first order forsmall E. However, in this untypical case the perturbation expansionterminates; the solution is exact [cf. Lamb, 1932, p. 235, Eq. (23)].For any other upstream profile, however, the problem would be non­linear, and thc perturbation solution would be an infinite series inpowcrs of J:: (see Exercise 2.4).

This solution has been considered by Hall (1956) as a model for theef1"ect of shear upon the reading of a pitot tube in a wake or boundaryla,yer. It is easily shown that thc stagnation streamline !/J = 0 originatesupstream at y ,~ ~lca. Hencc the measured stagnation pressure is greaterthan that directly upstream; the impact tube reads too high a value. ThiseHect has been observed experimentally. Hall treats the same problemfor a sphere, which is a great deal more complicated.

(2.9)

(2.7)

(2.8a)

(2.8b)

(2.8e)

(2.lOa)

as l' - 00

, --:- Xl!:.. +' Xlee = 0XUI' I l' 1'2

.pI =1 C 1'''(1 - cos 28) , Xl(r, 8)a

U.pI - -1- 1'''(1 - cos 28), a

In this simple problem a particular integral that accounts for thenonhomogeneous terms on the right-hand side of the diHerentialequation (2.8a) is given by the rotational part of the flow far upstream.Thus it is convenient to express the solution as

To obtain a unique solution we must add, for example, the requirementthat no additional circulation is induced by the body.

If the dimensionless "\"()fticitv number" I:: is small, it seems likelythat the flow will depart only slightly from our previous solution forirrotational motion. On that assumption we tentatively set

where l" is the basic solution (2.4a). Substituting into the full problemro b .(2.6) and equating like powers of J:: yields for the first-order pertur atlOn!/Jl the problem

The basic solution consists of the uniform stream-a dipole at infinity­plus its image in the circle-a dipole at the origin. Similarly, the

This is as readily solved by separation of variables as was the basicproblem (2.3), there being again a unique solution that is free of ci~cula­

tion. Then collecting results gives as the complete first-order solutlOn

h(a, 8) = - i Ua( I - cos 28)

­U--

...... -----

Fig. 2.3. Cniform flow past a slightly distorted circle.

and substitute into the full problem. In the equation of motion (2.3a)

(2.11)

(2.l0b)

(2.10e)

as l' - ,00Xl - canst

a" - ~ 1'2 a3].p = U(r -" -1-') sin 8 ---, }e [j [--;;- (1 - cos 28) -, ----;:2" cos 28 - a

14 II. Some Regular Perturbation Problems 2.4. Circle in Slightly Compressible Flow 15

and upstream boundary condition (2.3b) we can again equate like powersof E to obtain for the first two terms

Howeyer, there is a complication in the condition of tangent flow at thesurface of the body. Written out precisely it is

~Jo[a(1 - C' sin~ 8),8] ...:.... e.pl[a(1 -- e sin2 8),8] _1- ... = 0

\~.po = 0,

\~.pl =. 0,

.po -+ Ur sin 8 as r -+ r:t)

as l' -+ r:t)

(2.14a)

(2.14b)

(2.15a)

2.4. Circle in Slightly Compressible Flow

Consider now a perturbation of the equations of motion. Let the fluidbe slightly compressible, the free-stream :Vlach number being small. It isconvenient to work with the velocity potential, because the connectionbetween the stream function and the velocity is complicated by variationsof density. Let the velocity vector be given by q = C grad <p. Then forplane flow of a perfect gas the full potential equation is (Oswatitsch,1956, p. 328)

1>",: ,- (PUll = .1/1 2 [4>":2"1>",, ---'- 24>~4>y4>xll -1>/4>y"

The last form is obtained bv using the basic solution (2.4a) for tfo·The perturbation probl~m (2.14b), (2.16b) now has the form of the

basic problem, and is just as easily solved. Thus the complete first-orderapproximation is found to be

.p = U(r -- a2

) sin 8 + ~eU(3!!..- sin (j- a: sin 38) + O(e2) (2.17)

I' ' I' I'

Values at the surface of the body, which are usually those of mostinterest, could be obtained simply by setting r equal to its surfacevalue (2.12). However, it is consistent with th,e app~oximation alreadyintroduced to simplify the results by droppmg hlgh~r-order t.erms,which have no significance. This is accomplished by agam expandmg mTavlor series about the basic yalue I' = a. Thus, for example, onefinds for the surface speed

(2.19a)

(2.20a)

(2.20b)

rP -+ I' cos 0 as l' -+ x;

rP,(1, 8) = 0

I. ,2[( (- ,rPo C)( 2 rP/".11 ,rPr -c -;- 0 --;-8 4>, ""7' -2 )- cr 1'- C ' l'

_. (y _ 1)("rP2 - rP02

_ 1)(,4> _ <1>, ...:... <1>8.0...\] (2.19b)I 1'2 ,ir r 1'1 ,

upstream:

surface:

.J. 4>r 4)80'l'rl' - - --2

l' r'

It is convenient to choose the length scale such that the radius of thecircle is unity. Then the boundary conditions are

rPIrl' ""7' -~ + rPl,eo = 2NP[(',J..- - 2-,,) cos {} - _1_ cos 38] (2.21), l' 1'2 1'7 1'" 1'3

plus a requirement of symmetry to rule out circulation.Rather than assume a perturbation expansion as before, we take this

opportunity to illustrate iteration as an alternative way of finding succes­sive approximations. To this end, all terms representing the effects ofcompressibility haH been written on the right-hand side of the differ­ential equation. ~eglecting them altogether leads to the basic incompres­sible solution, given by (2.4b) with U = a = 1. To calculate thefirst-order effects of compressibility, we use that basic solution to evaluatethe nonlinear right-hand side, and solve again. The differential equationbecomes

\\'here .11 is the free-stream :\Iach number. Transforming this to polarcoordinates gins

In iterating it is convenient to calculate the complete solution at eachstage, rather than only a small correction to the previous result. Then the(2.18)

(2.16a)

(2.16b)

(2.15b)

q,= U(2 sin 8 + e sin 38 + ...)

.po(a, 8) =, 0

.pl(a, 8) c= a sin~ (j .por(a, (j) = }Ua(3 sin 8- sin 38)

It is now possible to equate like powers of e, giving

Here the perturbation parameter f' appears implicitly, in the ~rst

argument of the functions, as well as explicitly, so that it is not possibledirectly to equate like powers of I' to zero. This can be achiend only byexpanding the functions to exhibit explicitly their dependence on E.

If we assume that .pI , like .po' is analytic in its dependence upon r, wecan expand in Taylor series about r = a. Keeping only linear terms inE gives

16 II. Some Regular Perturbation Problems 2.6. Boundary Layer by Coordinate Expansion 17

To this must be added a soluti<)n of the homogcneous equation ,-theLaplace equation-that restores the boundary conditions, The finalrcsult is

Calculating from this the ma:\imum surface speed reproduces the firsttwo terms of Simasaki's series (1.1) quoted in Chapter 1.

Higher approximations can be found by repeating the iterationprocedure, the only complication being that the computational laborinvoh'ed gro\vs at a staggering rate. It can be minimized by usingcomplex-nriable theory and by systematizing the procedure.

full boundary conditions (2.20) are to be imposed at each stage. Toindicate this change, \\'e denote the successive approxirnations byRoman-numeral rather than Arabic subscripts. The term in (y I) inthe differential equation is seen to have no effect upon the first approxi­mation 1J, . The first-order effect of compressibility is independent ofthe thermodynamic properties of the gas.

Follo\ving Rayleigh (1916), \ve find by separation of variables a partic­ular integral of the iteration equation (2.21) that vanishes at infinity:

(2.25a)

\anishes, leading to Eq. (2.2) for tf;. Thus the basic inviscid solution(2.4a) becomes exact in that limit. Supposc that we try to iterate to obtain;1 perturbation solution for large Reynolds numbers. The equation shouldht: transformed to polar coordinates, and the condition of no slip at the"urfact: added. Howe\'er, it is clear \vithout carrying out these ddailsthat the iteration attempt \\ill fail. The right-ha~d side nnishes \\'hen>,'\'.lluated in terms of the basic solution, so that the Reynolds numberdocs not enter into the problem. .

The resolution of this dilemma is of course pro\'ided by Prandtl's[loundary-Ia\'er thcory. Despite their superficial resemblance, tnt:i'roblcms of slight cornpressibility and slight viscosity are essentiallyditft:rent. For the first, tht: basi~ solution is a valid' approximatio~"\t:ry,,lwrt: in the flo,,' ficld, \vhereas for the second it is ilwalid in a:wighborhood of the surface no matter how large the Re\nolds number.For this reason the effect of slight compressibility is a regul'ar perturbation,\\'hereas the effect of slight \'iscosity is a singular perturbation.

2.6. Boundary Layer by Coordinate Expansion

Singular perturbation problems and the basic ideas of boundary-layertheory will be discussed in later chapters. For the present, it is i~for'm­

ati\'e simply to recall the well-knO\m results of Prandtl. _-\t high Revnoldsnumbers, viscosity is significant only \vithin a thin laver near the ;urfaceof a body, where Eq. (2.24) for tf; m'ay be approximated by

dq!f;y!f;J:Y - !f;,4l1Y = v!f;YYlI T q d­

x

(2.23)

(2.22)"[ I I 1 I I 1 ]<f, =, .11- (-- ~ -- - -) cos 8- ,- - cos 38Ii> 12 1''' 2 1'3 4 l'

I ,,[ 13 1 1 1 1 1\1' - r I cos 8 - .II- \12 r -- '2 Y3 - 12 rs) cos 8

1 1 1 1 ]- \- - - - - -) cos 3812 1'3 4 l'

'/>1

SeeNote

2

2.5. Effect of Slight Viscosity

One might attempt to treat viscosity in the same way as compressibility.It is convenient to work with the stream function, which in planeincompressible flow satisfies the equation

Here.\' and yare cun-ilinear coordinates along and normal to the bodysurface (Fig. 2.4). Evidently one integration has been performed withrespect to y to reduce the equation from fourth to third order. The lastterm is the function of integration, q being the inviscid surface speed.

(2.24)

Here v is the kinematic viscosity. If we again introduce dimensionlessvariables such that U = a = 1, v can be replaced by R-\ whereR = Ua/v is the Reynolds number based on radius. This equationexpresses the physical fact that vorticity is convected with the localvelocity (the left-hand side) and simultaneously diffused by viscosity(the right-hand side).

At infinite Reynolds number the right-hand side of the equation

x

b 1

Fig. 2.4. Boundary-layer coordinates for a circle.

18 II. Some Regular Perturbation Problems Exercises 19

Two boundary conditions are provided by the requirement of zerovelocity at the surface:

The third condition, that the inviscid surface speed be attained at theouter edge of the boundary layer, may be written

EXERCISES

2.1. /'li/sating circle. Consider uniform plane flo\\' of an incompressiblei1l\iscid liquid past a circular cylinder whose radius \'aries slightly \\·ith time'Il'cording to all .. ol(t)]. Calculate the velocity potential to first order in f.

Find the stream function, and show that it docs not \'anish on the surface .

This yanishes at x a ~= 1.6, suggesting that the boundary layer separatesfrom a circle at about 92) from the stagnation point. Although thisestimate has been refined to J09" by calculating four more terms in theseries (Schlichting, 1960, p. 154), the occurrence of separation im'alidatesthe whole analysis. The resulting broad wake drastically alters the flowO\er the front of the body. Therefore although Prandtl's Eq. (2.25a) is\aEd there, the inviscid surface speed q is not given correctly by (2.27)but is unknown (see Fig. 7.1). Experiments show that separation actuallyoccurs at about 81".

The Blasius series is self-consistent only for bodies where the solutionsho\\':,; no separation. An example is the parabola in a uniform stream,for \\'hich the 6-term counterpart of (2.29) was shO\m in Eq. (1.5) ofChapter I.

(2.25c)

(2.25b)

w7) = V.- 1- (2.26). "y }'a

.jJ(x,O) = .jJl", 0) = 0

with the understanding that here y --4- X! means far from the surfaceonly on the small scale of the boundary-layer thickness.

The previous examples were all parameter perturbations. We nowconsider a coordinate perturbation for the boundary layer on a circle.Within the framework of Prandtl's theory, this is again a regular pertur­bation.

Suppose that the distance .\' from the stagnation point is smallcompared \\·ith the radius a of the circle. Then the stream function canbe expanded in powers of .\' a, and it is clear from symmetry that onlyodd powers will appear. It is convenient to write the expansion as

which is the standard form of the Blasius series (Schlichting, 1960,p. 146). We take the in~'iscid surface speed from the basic solution

and substitute into the boundary-layer problem (2.25). Upon equatinglike powers of x (I, we obtain a sequence of problems involving ordinarydifferential equations:

I~" -i- Af~' - 1/ T I = 0, A(O) = 11 '(0) = 0, 1/(00) = I (2.28a)

I~" T Af~' - 4//13' -i- 3/~'j3 -i- I = 0, 13(0) = 13'(0) = 0, 13'(00) = i(2.28b)

The first of these is Hiemenz' classical problem of viscous flow at a planestagnation point, and the remainder are linear perturbations thereof.

Numerical integration (Tifford, 1954) gives j{'(O) = 1.2325877 andj~'(O) = 0.7244473. Hence the expansion for the coefficient of skinfriction is found to be

\. [V 1\,)3 ]q = 2U sin ~ = 2U ~ - 6(~ -+- ...

T I [V .\:)3]c[ - -- = -. 6.973:'" - 2.732 ('- + ...}pU2 viR a a

(2.27)

(2.29)

2.2. Slightly porolls circle. :\ uniform stream of incompressihle inviscid liquidSee tlm\s past a hollo\\' circular cylinder whose surface is made porous by drillingote m'lI1y tiny holes normal to the surface..-\ssume that the normal velocity at:) the surface is some sm,lll parameter f times the difference in pressure coefficient

het\\'Cen the outer surface and the interior (\\'hich is assumed to he at constantpressure throughout), and that the net flux through the surface is zero. Calculatethe stream function of the external flow approximately to order f, and theinterior pressure. Check the signs of these results by physical reasoning. \\'hatis the corresponding result for a slightly compressible flow, if only linear termsin E and JJ2 are retained?

2.3. Corrugated quasi cylinder. Consider an infinitely long body of reyolutionSec (Fig. 2.5) of radius all E sin (zb)]. Calculate approximately the three-ote dimensional yelocity potential for uniform incompressible flow (\yithout eir-

.:) culation) normal to the axis, keeping only linear terms in E. Csing expansionsfor the Bessel functions, simplify your solution for the case of waYelengthso great that only linear terms in a.b are retained, and interpret the result as aquasi-two-dimensional one. Show that in the opposite extreme of yery largeah the perturbation is a plane harmonic function near the surface, with 8appearing only as a parameter; and justify this by physical arguments.

20 II. Some Regular Perturbation Problems

(3.1)

SeeNote

3

Fig. 2.5. Infinite corrugated cylinder.

2.4. Circle in parabolic shear. A circular cylinder of radius a is symmetricallyplaced in a parallel stream of incompressible inviscid fluid having the paraholicvelocity profile u = C(l·- ~f)·2a2). Find an exact implicit expression for thevorticity w(~J), and expand to give w as a series, keeping terms of order f2.

Carry out a perturbation solution for the flow, showing that a difficulty arisesin the term of order e because no solution can be found for which the velocitydisturbances produced hy the body disappear far upstream.

Chapter III

THE TECHNIQUES

OF PERTURBATION THEORY

3.1. Introduction; Limit Processes

The examples in the preceding chapter have served to introduce\,lrious techniques for handling perturbation problems. \Ve now seekto classify and generalize those that are of common utility . We begin\\'ith some matters of notation, definitions, and relevant processes ofanall·sis.

\Ye are concerned with finding approximate solutions of the equationsof fluid motion that are close to the exact solutions in some useful sense.This iI1\'olves various kinds of equality, which in decreasing degree ofidentification will be expressed by the following symbols:

identical withequal toasymptotically equal to (in some given limit)

'" approximately equal to (in any useful sense)oc proportional to

.-\s discussed in Chapter I, we consider approximations that dependupon a limit process, the result becoming exact as a perturbation quantityapproaches zero or some other critical value. One often encounters adouble or multiple limit process, in \vhich two or more perturbationquantities approach their limits simultaneously. Because the order ofcarrying out several limits cannot in general be interchanged, one mustfrequently specify the relative rates of approach. This specificationprovides a similarity parameter for the problem. The following are somefamiliar examples:

(a) Plane transonic small-disturbance theory for a wing of thicknessratio f (von Karman, 1947):

e -~ 0 IlVI--+l\

lVI-I-----m- = O(1)

e

21

22 III. The Techniques of Perturbation Theory 3.2. Gauge Functions and Order Symbols 23

(d) Hypersonic small-disturbance form of:\"ewton-Buscmann appro:..:i­mation for a slender body of thickness ratio I' (Cole, 1957):

(c) :\"e\yton-Busemann appro:..:imation for hypersonic flow past ablunt body (Cole, 1957):

(b) Hypersonic small-disturbance theory for a body of thicknessratio I (Hayes and Probstein, 1959, p. 36):

(d) (log R 4 + y --~) instead of log R in yiscous flow past a circleat low Reynolds number (y being Euler's constant), becausethe first two terms of the Stokes expansion are thereby combinedinto one (Kaplun, 1957; see Section 8.7),

(e) I-' (I - c) instead of e, e being the density ratio across a normalshock, in the :\"ewton-Busemann approximation for the standoffdistance of the detached shock wave on a blunt body in super­sonic flow, hecause it then becomes infinite for ili ->- 1, asit should (Serbin, 1958),

(f) (.--1 1m- I ~ .. y instead of (A2 -. 2ABR-12 ..,) for the dragof a bluff body in laminar flow at high I{eynolds numbers,\\'hich is suggested hy theory and agrees better with knownresults (Imai, 1957b),

(g) 2" (1 - 2 A .- ...) instead of 2,,( 1 - 2 .--1-- ... ) for the lift-curyeslope of an elliptic wing of high aspect ratio .--1, hecause it thenvanishes for .--1 ------>- 0, as it should (see Chapter IX),

(h) g \/i----=--{~, g being a parabolic coordinate, instead of x in theHlasius series for the boundary layer on a parabola, because theradius of conyergence is thereby e:..:tended to infinity (seeChapter X),

(i) I' (2 -- c) instead of e in free-streamline theory (Garabedian,1956), where 2 .- I' is the number of space dimensions, becausethe radius of conyergence is thereby increased.

3.2. Gauge Functions and Order Symbols

The solution of a probkm in fluid mechanics will depend uponthe coordinates, say x, y, Z, t, and also upon yarious parameters. Oneor more of these quantities may, by appropriate redefinition, be regardedas yanishingly small in a perturbation solution. \Ve consider the behayiorof the solution as it depends upon one such perturbation quantity, withthe other coordinates and parameters fixed. Thus we seek to describe the\\ay in which a function f(e) behaves as e approaches zero. An analogoussituation has already arisen in the upstream boundary conditions (2.3b),(2.6b), etc., where it was necessary to describe the behayior of the solution'far from the body.

There are a number of possible descriptions, of varying degrees ofprecision. \Ve discuss six of them, in increasing order of refinement.First, one may simply state whether or not a limit exists. For example,sin 21' has a limit as e ------>- 0, whereas sin 2 1-' has not. However, we areconcerned only with problems where a limit is be1ieyed to exist.

I----c---:----;; =c= O( I)(y -- IFH-

I--- =c= 0(1)Me

E -~ 0.II -~ .~

y --~

e - -+ 0.II - ~ x

y - I

In the last e:..:ample one might haH: anticipated two similarity parameters,but only one is found to be signiricant.

.-\ pe~turbation quantity is l~eYer uniquely defined. For e:..:ample, thethickness parameter for a slender body may he taken as its thicknessratio, ma:..:imum slope, mean slope, etc. Of course it may also he changedby a constant multiplier, as in referring Reynolds number to radius ratherthan diameter for a sphere. One should be alert to the possibility ofe:..:ploiting this freedom by replacing the olwious choice by an alternatiyethat is superior to it in some respect. The possibilities are too diyerseto be subject to ruks. They can only be suggested here by listing anumher of cases \\here an ingenious choice of the perturbation quantity,usually suggested by extran~ous considerations, leads to simplificationor improyement of the results:

(a) (.1lJ2 - I) instead of (Jl - 1) in transonic small-disturbancetheory, so that the result is nlid also in the adjacent regimes ofsubs~nic and supersonic flow (Spreiter, 1953),

(b) I v/lliz - I instead of I M in hypersonic small-disturbancetheory, so that the result is nlid also in the adjacent regime ofsupersonic flow (\'an Dyke, 1951),

(c) (y- 1)(y...L I) instead of (y- 1) in the Newton-Busemannapproximation for hypersonic flow, because it can be identifiedwith the density ratio across a strong shock wave (Hayes andProbstein, 1959, p. 7),

24 III. The Techniques of Perturbation Theory 3.2. Gauge Functions and Order Sytnbols 25

~econd, one may describe the limiting yalue Ijuulitath;ely. There arcthree possibilities: the function may be

adyantageous. For example, it might under some circumstances proveuseful to replace the first case in (3.4) by the equiyalents

It is a peculiarity of this mode of description that the first case is includedin the second; a function that \'<lI1ishes is also bounded. lIowcyer, onewould naturally usc the first description when possible, because it is

more preCise.Third, one may describe the limiting nlue quallt/tutirely. There arc

again three possibilities, only the second haying been refined:

One ordinarily chooses the real powers of 8 as gauge functions, becausethey haye the most familiar properties. However, this set is not complete.I t fails, for example, to describe log I e, which becomes infinite as E tendsto zero, but more slowly than any power of E. The powers of E musttherefore be supplemented, when necessary, by its logarithm, expo­nential, log log, etc., or their equiyalents. Examples are

etc.

1= O( log log ---;)

o(__e .)1 I ',e

(3.5)

exp(-cosh!) = O(exp[ -}e1 '])e,

O(tan e),sin 2e == O(2e),

1cosh - = 0(e1 /'),

e

as e·-->- 0

as e - .. 0

a constant

f(e) --. 0(e) %

f(e) ----->- cc

(a) limf(e)= 0(b) limf(e) ,= c,(c) limf(e) 0= 'l~

(a) v,wishing:(b) bounded:(c) infinite:

Like the perturbation quantity e itself, the gauge functions arc notunique, and a choice other than the obyious one may occasionally be

(3.6)sin 2e == 0(l), 0(1), 0(e1l2 ) , o(e1:2), etc.

I lo\\-e\'er, we assume that the sharpest possible estimate is always giyen.This means, for example, choosing the largest possible po\\-er of e as thegauge function, and using 0 only when one has insufficient knowledgeto use O. Of course the result may still be only an upper bound for lackof sufficient information.

The mathematical order expressed by the symbols 0 and 0 is formallydistinct from physical order of magnitude, because no account is kept ofconstants of proportionality. Therefore Ke is 0(8) even if K is tenthousand. In physical problems, however, one has at least a mysticalhope, almost inyariably realized, that the two concepts are related. Thusif the error in a physical theory is O(e) and e has been sensibly chosen,one can expect that the numerical error will not exceed some moderatemultiple of e: possibly 2e or eYen 2m, but almost certainly not IOe.

The rules for simple operations with order symbols are -evident fromthis physical connection. For example, the order of a product (or ratio)is the product (or ratio) of the orders; the order of a sum or differenceis that of the dominant term~i.e., the term of order em haying the

Often, as in (1.4), one writes log f: where log 18 would be more appropri­ate.

:\either order symbol necessarily describes the actual rate of approachto the limit, but proyides only an upper bound. Thus it is formallycorrect to replace the first example in (3.4) by

(3.4)

(3.3)

(3.2)

for all III

lim.[(e) == 0£--->0 8(e)

if

if

1 -case == O(e2) = o(e)

sec-1(1 _L e) = O(el!2) = 0(1)

exp(-I ie) = o(e ill)

as e -->- 0f(e) = 0[8(e)]

f(e) 0= 0[8(e)]

sin 2e = O(e),

-Vl-- e2 = 0(1),

1cot e = 0(-),

e

The svmbol 0 is used instead if the ratio tends to zero. One writes

Some examples are

Fourth, one may describe qualitati,'ely the rate at which the limit isapproached. Only casts (a) and (c) abo\'C are thus refined. This can bedone by comparing \vith a set of gauge .timetiolls. These are functionsthat are so familiar that their limiting bchayior can be regarded as kno\\'l1intuitiyely. The comparison is made using the order symbols () ("bigoh") and 0 ("little oh"). They proyide an indispcnsible means for keepingaccount of the degree of appro:-;imation in a perturbation solution.

The symbol 0 is used if comparing f(e) \\ith some gauge function8(1') sho\vs that the ratio f(r) 8(r) remains bounded as e ---+ O. One writes

lim fJtJ, ...0 8(e)

26 III. The Techniques of Perturbation Theory 3.3. Asymptotic Representations; Asymptotic Series 27

smailest value of 1II-etc. Order symbols may be integrated with respectto either f or another variable. It is not in general permissible to differ­entiate order relations. :\ewrtheless, in physical problems one commonlyassumes that differentiation \\'ith respect to another variable is legitimate,so that the derivatin: has the same order as its antecedent. For otherproperties of order symbols see the first chapter of Erdelyi (1956).

:lIHI the error is of still smaller order:

(3.9c)

Fmther terms can be added by repeating this process. Thus one con­structs the m:\'mptotic expansion or asymptotic series to .V terms, written as

3.3. Asymptotic Representations; Asymptotic Series

:\ fifth scheme is to describe qllilntitati7'ely the rate at which a functionapproaches its limit. This constitutes a refinement of the fourth scheme -­the use of order symbols --just as the third scheme does of the second.We simply restore the constant of proportionality, and write

lim f(e) = c,~" 8(e)

If the fu nction f( Ie') \vere known, together \vith the gauge functions,\,( Ie), the coefficients c" of the asymptotic expansion could be computedin succession from

(3.IOa)

(3.lOh)as e ->- ()

as e -~ 0

~-

f(e) = l c",s,,(e) -- o[3,-(e)]n ·,1

and dctlned by

(3.7a)

(3.7b)

as e ----->- 0f(e) - c8(e)

that is, if

if

(3.11)

(3.12)

sin 2e- 2e -- ~ e3 -'- .i. e'; .,-, 3' 15 .

.J e--/ dt ,-, --:-- -- 1- ec- 2e2 - 6e3 -'" -~ l (--I)"lI!e"

• 0 I .- Et ,,~U

Itt hegallge functions are all integral positi \'e pcmers of Ie, one speaks of,Ill lI,'ylllptotic pozcer series, .\s the number .Y of terms increases \\'ithoutlimit, one obtains an infinite asymptotic series, \\-hich may be either con­\ergent or divergent.

Some examples of asymptotic expansions are(3.8)

(3.7c)

(3.9a)as e ----->- 0

2sech- l e -log­

e

f(e) = c3(e) ---:- o[3(e)]

Icot e "'"' -,

e

sin 2e - 2e,

This is the asymptotic form or mymptotic representation of the function,and constitutes the leading term in the asymptotic expansion discussedbelow. Some examples are

Sixth, the preceding description-which is the most preClse onepossible using only one gauge function-can be refined by addingfurther terms. Consider the difference between the function in questionand its asymptotic form as a new function, and determine its asymptoticform. The result can be written

where the second gauge function 82(1') IS necessarily of smaller orderthan the first:

log 11' -- (11 - }) log 11 - 11 -'- log\/21T -

or (3.9b)The first two of these connrge if extended indefinitely; the latter threediverge.

28 III. The Techniques of Perturbation Theory 3.4. Asymptotic Sequences 29

It is proper to regard a distant boundary conditio~ as an asymptoticrelation. For example, (2.6b) will henceforth be rewntten as

Cnseparated :aminar flow over smooth bodies at highReynolds number R, where e = l;R (Yan Dyke,1962a)

Separated Oseen flow at high Reynolds number R,e e== 1jR (Tamada and l\Iiyagi, 1962)

(',amples of Chapter II, it consists of integral powers, E". Fractionalpo\\'ers may also occur, particularly in singular perturbation problems.Examples are:

Logarithms may occur at some stage, examples being

(2.6b')as r -+ 00,p __ U[-l ~ r2(1 -- cos 28) + r sin 8]

It must be understood that this admits the possibility of an error of orderoCr). Actually, in the problem in question, the next term in this asy.mptoticexpansion is 0(1); the stream function must be left unp:escnbed farupstream to within a constant, which corresponds physIcally to thedisplacement of the stagnation streamline.

that are arranged in decreasing order: 0" .1 = 0(0,,)' This i.s the asy.mpt~tic

sequence associated with the problem. It cannot be pr~scnbed ~rbitranly,

because it must be sufficiently complete to descnbe loganthms, forexample, if they appear. On the other hand, there are an unlimited num­ber of alternatives to any particular asymptotic sequence:

(3.15)x> 1x =.C 1

x < 100,

0,I,

Axisymmetric flow at low Reynolds number R,e ~ R (Proudman and Pearson, 1957)

Supersonic axisymmetric slender-body theory,e = thickness parameter (Broderick, 1949)

:'\ewton-Busemann approximation for planehypersonic flow past a blunt body, e =:=

(y - I)/(y _!- I) (Chester, 1956a)Plane viscous flow at 10\\' Reynolds number

R, e :-~ R (Kaplun, 1957; Proudman andPearson, 1957)

Subsonic thin-airfoil theory for round-nosedprofile, e == thickness parameter (Hantzsche,1943)

Laminar flow over flat plate at high Reynoldsnumber R, e = IR (Goldstein, 1956; Imai,1957a)

e-:t:/E

lim--=,,",0 e-1 / E

I, e, e~ log e, e2, e3 log e,'jc', •..

I, e~ log e, e~, e l log~ e,

e l log e, eJ , ...

e log e, e, e" log e, e2 , ...

(log e)-1, (log e)-2,(loge)'3, ...

I, eL2, e, e~'~ log e,3-'>

t: '-, •••

The question naturally anses how one can be sure of guessing the

In the last two examples, earlier im'estigators had obtained erroneoussolutions because they did not suspect the presence of logarithmic terms.Other examples arising in boundary-layer theory have been discussedby Stewartson (1957), who shows that even log log's occur in the asymp­totic solution far downstream on a circular cylinder.

Exponentially small terms are seldom en~ountered, and are difficultto deal with. The following example shows that the estimate O(rlje)has very little practical value:

(3.14)

(3.13)

-.- 21og(1 + e) -L 10g(1 -;- e2 ) - 2 10g(1 + e3) + '"

, e " 756(' e ')';,-.- 6(3 T 2e2) - -5 3 + 2e2 --;-'"

__ 2 tan e - 2 tan3 e - 2 tan5 e -l- ...

The last two forms illustrate the fact that alternative sequences need notbe equivalent: corresponding terms are not of the same order. ~oth

the asymptotic sequence and the asymptotic expansion itself ~re umqueif the perturbation quantity (e.g., E) and the gauge functlOns (e.g.,Ern, log IE, log log IE, etc.) are specified. ,

'rVe have seen that one way of attacking a perturbation problem IS toassume the form of a series ~olution, This requires guessing an appro­priate asymptotic sequence. The simplest possibility is that, as in the

3.4. Asymptotic Sequences

The process just described of constructing an expansion term by termis effectivelv that employed in perturbation solutions, such as those ofChapter 11.< Thus in each problem a perturbation solution generates aspecial sequence of gauge functions

30 III. The Techniques of Perturbation Theory 3.5. Convergence and Accuracy of Asymptotic Series 31

2 4 6 8 10 12n

(b)

2 4 6 8 10 12Number of term, n

(a)

i~ divcrgent for all Ie: no matter how small, but a few terms give goodaccuracy for moderately small E. The first term alone is correct to threesil!niticant fIgures for ([ c:) = 4.

, The utility of an asymptotic expansion lies in the fact that the erroris. In definition, of the order of the first neglected term, and therefore tendsrapt'dly to zero as E is reduced. For a fixed value of f, the error can also bedecreased at first by adding terms; but if the series is divergent, a pointis e\el~tually reached beyond which additional terms increase the error.This beha\ior is indicated in Fig. 3.1. These properties are often ideal

Fig. 3.1. Behavior of terms in series, (a) Slowly cOliH'rgent series. (b) Diwrgenta~~;rllptotIe Serit'S.

for practical purposes, particularly in parameter perturbations of the sortc\l:'lTlplitled by thin-airfoil theory. Then only small \'alues of c' are ofpractical interest; and only a few terms are calculated, so that the pointof inereasing error is not reached.

There are other problems, howc\'er, in which one attempts to make c'

as large as possible. This is true of such parameter perturbations asexpansions for large or small Revnolds numbers. It is almost alwaystrue of coordinate perturbations, b~cause one intends to apply the resuitsas far from the origin as possible. C nder such circumstances, convergenccmay be of considerable practical interest. As discussed in Chapter X,onc can sometimes imnrove the rate and radius of convergence or cvenITnder a divergent ser'ies convergent. '

From a physical point of vie\~, the perturbation quantity E assumesonly positive real values, However, mathematical insight is often gainedhv emisioning its analytic continuation into the complex plane (Fig. 3.2).This is particularly useful \vhen the solution is a power series in theperturbation quantity. Thus one considers the complex NI2-p lane in theJanzen-Rayleigh method, the complex thickness-ratio-plane in thin-

(3.16)

as £ _.~ 0 (3.17)

.' I, !2E [. 9.) ,) ( I IT)T '-..1---- /- /..1.- s- ... cos - - -~ 0\ e "IT 1128 -, E 4

~. (~e -- l~ke3 ~....)sin l~ .- ~)]

for the Bessel function has an infinite radius of convergence, but manyterms are needed for accurate results unless Ie' is small. \Vith I' 4, forexample, the first three terms actually increase i,n magnitude -so ~hat

the series appears to be diverging-and at least ught terms ,are reqU1~ed

for three-figure accuracy. On the other hand, the asymptotIc expansIOn

proper asymptotic sequence..\pparently there are no general rules, hutthe following principles are of some help:

(a) When in doubt, oyerguess. A superfluous term will drop out byproducing for its coefficient a homogeneous problem, whose

solution (if unique!) is zero, _(b) Be ready to suspect the presence of logarithmic terms at the tirst

hint of difficulty.(c) Iteration \vill so~etimes (but not always!) produce the proper

sequence automatically.

One usually has a feeling when the solution is progressing properly; allterms match, and co~)plicated expressions often simplify. 'Vithc:\perience, one learns when the absenc: of suc~ reassurance suggests are-e:\amination of the assumed form of the senes. However, the onlyfool-proof procedure is to lea\'e the asymptotic sequence un~pecifi,e~,

and to determine it term lw term in the course of the solutIOn. 1 hiStechnique will be illustrated'in Chapters VII and VIII.

3.5. Convergence and Accuracy of Asymptotic Series

'Ve have seen that an infinite asymptotic series may either COI1\'ergefor some range of f, or di\'erge for all f, In p~rturbation prol~le,ms o.neoften neither kno\\'s nor cares \yhether the senes com'erges, 1 hIS pOintof \iew has been persuasi \Tly set forth by ] eHreys (1926). It is a fall:lcyto think that cOlwcrgenee is necessarily of practical value. :\Iathematlcalconvergence depends upon the beha\ior of terms of indefinite,ly highorder, \\'hereas in physical problems one can calculate only th~ tirs: :e~\'terms and hope that they rapidly approach the true solutlOn. I hISrequirement may sometimes be better met with a divergent than a con­

vergent series. Thus the expansion

The radius of convergence is thereby extended to the nearest singularityin the complex plane of ie, and the utility of the series often greatly

improved.

3.6. Properties of Asymptotic Expansions

I n substituting an assumed series solution into a perturbation problemone must carry out such operations as addition, multiplication, and

33

(3.19)

(3.20a)

hut not uniformly near.\' = 0, %

but not uniformly near x = 0

3.6. Properties of Asymptotic Expansions

10. /_ = 0(10),vx

10 log x = 0(10),

\Vith respect to the gauge functions en, these two functions have identicalasymptotic expansions to any number of terms. Their difference is sosmall that it would become evident only if the infinite sequence of powersof e were exhausted, for example by summing them. It happens thatthis is easily accomplished in this example by making the Euler trans-

,\ singular perturbation problem is best defined as one in which no singleasymptotic expansion is uniformly valid throughout the field of interest.The nonuniformities illustrated in (3.19) arise in practical singularperturbation problems. For example, the first will be encountered at around leading edge in subsonic thin-airfoil theory, and the second at asharp leading edge or in plane yiscous flow at low Reynolds numbers.

\\'e have obsen'ed that each function has a un'ique asymptoticl'\:pansion if the gauge functions or asymptotic sequence is prescribed.On the other hand, the statement is often made that different functionsmay have the same asymptotic expansion. The extent of this non­uniqueness may be understood by considering the following example:

Jifferentiation. Addition and subtraction are justified in general.:\ Iulti plication is valid if the result is an asymptotic expansion. It isnot in general permissible to differentiate an asymptotic expansion withrespect to either the perturbation quantity or another variable. Theseand other properties of asymptotic expansions are discussed by Erdelyi(1956). However, it appears that results are not available in sufficientgenerality to cover such commonly occuring series as those involvinglogarithmic terms. In practice, therefore, one ordinarily carries outformally such operations as differentiation with respect to another variablevvithout attempting to justify them. When they are not justified, non­IIniformities will arise in the solution.

I n a physical problem, the coefficients in an asymptotic expansion willdqwnd upon space or time variables other than e. The series is said tohe uniformly ~'alid (in space or time) if the error is small uniformly inthose \'ariables. Examples of nonuniformity in x are .

(3.18)_ ee=--

. 1 - e

Imoginary

i'\C,-g:;,c;:r;:nc,

Possible IslfIgulorify \ I--VC:::'======>Reolc

III. The Techniques of Perturbation Theory

Fig. 3.2. COlllpleX plane of perturbation quantity c',

airfoil theon', and ~() on. Then one can take advantage of the p()\\'erfulunifying vie'v\'point of complex-v<uiahle theory.

S(;me knov\ledge of-,and fee-ling for-",the principle of analytic con­tinuation is essen'tial. .-\n analytic function has a power series develop­ment at every regular point. 'l:his cOl1\'erges within a eirele that extendsto the nearest singularity..-\ function defined in any region, or eycn onjust a line segment, is ordinarily defined uniquely in a much larger regionof the complex plane (possibly on sev'eral Riemann sheets), and can becompleted by the process of analytic continuati,on. .

Sometimes the first fevv' terms of a perturhatlOn solutlOn suggest thatthe series cOll\"ergcs, but has its radius of convergence limited for noapparent physical reason. According to the principles Just outlined, thismust result from a singularity in the complex plane ot I' elsewhere thanon the po~itive real a\:is, Sev'eral examples to be discussed in Chapter Xsuggest that in these circumstances the singularity ordinarily lies o,n thenegatiTe real axis, and in fact at I' ~ 1, if the most natural chOlce ofvariables has been made (Fig, 3.2). This artificial limitation can beeliminated by shifting the sing~larity to infinity using a simple conformal

mapping, the Euler trawjormation:

32

SeeNote

IS

34 III. The Techniques of Perturbation Theory 3.7. Successive Approximations 35

(3.20h)

SeeNote

11

formation (3.18). With respect to powers of the new. perturba~ion

quantity t-, the t\yO functions have different asymptotic expansIOns(which happen to terminate):

1 --_._. ......, 1 -- e1 ."-- e

1 -i- e·· I ·, .­----_. 1 .- e' -;- e . e- l/ ' -- eee·- l

"

1 -.- e

One speaks of e L' as being transcendentally small compared with thesequence of powers of 1-, because it is 0(1:"') for ~ny t~ no matter how

I 1.:"1 I.' n a different level of magl1ltude, I- Itself IS transcendent-arge ...Jlmlary,o . ., fally small compared \\ith t he sequence (log e) .11/, which anse~, ..orex;mple, in plane viscous flo\\' at low Reynold~ numb~rs. The pos,slbllItyof dealing with transcendentally small terms WIll be discussed 111 ChapterVIII in connection with that problem.

3.7. Successive Approximations

The perturbation problems encou~tere~ in fll~id mech.anics usuallyinyolye a system of ordinary or partial differential equations. togeth.erwith approj.)riate initial and boundary co~ditio~s. Integro-dlffe:e~tlalequations may also arise, as in problems 1I1yoh'1I1g t~ermal radlatl~n.

There are two systematic procedures for finding a solutIOn by succeSSlyeapproximations: both of which were illustrated in the previous chapter:

(i) substitution of an assumed series, .(ii) iteration upon a basic approximate solutIOn.

In the first method--\yhich is somewhat more common-the guidin.gprinciple is that since the expansion must hold, at lea~t in an asympto.t~c

sense for arbitrary values of the perturbatIOn quantity E, terms of hkeorder'in E must separately satisfy each equality. That is, one can equatelike powers of E, terms in Em log" E having the same values of both 111 and

n, etc. . 'hEach method has its advantages and disadvantages, whlc can

sometimes be exploited by \vorking with a combinati~n of the two.The most important of these differences can be summanzed as follows:

(a) Iteration can be started only if an appropriate initial approximationis known. Series expansion is more automatic, because it ~an

generate the basic approximation if one substitutes a ser:es ~\'lth

the asymptotic sequence left unspecified. An example IS givenin Chapter VII.

(b) Iteration eliminates the need to guess the asymptotic sequence.It is therefore safer than assuming an expansion, unless oneleayes the aSYlnptotic sequence unspecitied. For n:ample, it\vill often though not ah\'ays, in singular perturbationproblems produce logarithn1s in higher-order terms that arcmissed if a power series is assulned.

(c) Beyond the second term the series expansion is more systematicbecause it produces only significant results, \\·hereas iteration\\'ill, in nonlinear problems, generate some higher-order terms,to which no significance can he attached because others of equalorder are missing. For example, in the Janzen-Rayleigh solutionfor a circle (Section 2.4), the next iteration step mndd clearlyproduce not only terms in .114, which are correct and identical\\'ith those giyen hy series expansion, but also some terms in .1J6and JI8 that should be disregarded because they arc incomplete.

(d) Iteration will yield in a single step groups of terms of nearly thesame order that require seyeral steps in a series expansion. Forexample, in axisymmetric slender-body theory, each iterationadds one more of the following groups of terms:

[IJ, [e 2 log e, e2], [e 4 log2 e, e4 log e, elJ, ...

\Yhiche\'er method is used, there are certain features common to allperturbation solutions. The basic solution may be linear or nonlinear,but all higher approximations are governed by linear equations withlinear boundary conditions. An exception arises in the case of transonicand hypersonic small-disturbance theory, where the double limit processis specifically designed to retain in the perturbation the essential non­linearity of the problem. In those special cases only the third and sub­sequent terms in the series satisfy linear problems. Otherwise, becausefirst-order perturbations are linearly independent, they may be super­imposed. For example, the three perturbations studied separately inSections 2.2, 2.3, and 2.4 may be added to give the first-order solutionfor slightly compressible flow past a slightly distorted circle in a slightshear flow. Higher approximations, however, will be coupled throughcross products of the various E'S.

Although the equations governing higher approximations are linear,they ordinarily contain nonconstant coefficients that depend upon theprevious approximations. It is often possible to simplify the computationdramatically by taking advantage of known relations for those earlierresults. A simple example will appear in Chapter IV in thin-airfoiltheory, where both the differential equations and the boundary conditionsare used to simplify their counterparts in subsequent approximations.

36 III. The Techniques of Perturbation Theory 3.9. Direct Coordinate Expansions 37

As in the Janzen-Rayleigh solution of Section 2.4, higher-orderproblems typically differ from one another only in the appearance ofsuccessi\'e!y more complicated nonhomogeneous terms (depending uponprevious approximations) in the differential equations. These arcaccounted for by finding a particular integral. Usually the best way ofseeking a particular integral is to guess it. Only after that attempt hasfailed should one apply more sophisticated processes of analysis.

One should also be on the alert for the occasional problem in whicha general particular integral can be found in terms of previous approxi­mations. We illustrate this possibility by two examples from the small­disturbance theory of axisymmetric compressible flow. First, in anapproximate linearized theory of supersonic propellers the second-ordervelocity potential is found to satisfy the nonhomogeneous wave equation

rP2r rP200~2<b2c,' (I .- .112)4>2JJ' -, rP2,,, -,- r- - IT

d' a solid body (as for the slightly distorted circle of Section 2.3), a free,;treamline, a shock wave, etc. (Fig. 3.3). In order to carrv out a systematicc\.pansion scheme, the boundary condition must in each case b'e expres­sed in terms of quantitites evaluated at the basic position of the surface,corresponding to c = O. Othenvise 8 will appear implicitly as well as

".'

~,,::.' : : : " :..

., .... :.:.,. ',' ".

:',::,'::::',' .... ',.'

,»;;,;;;,;;;";,, ;

Fig. 3.3. Examples of transfer of boundary conditions.

Second, if one seeks to improve the linearized theory of subsonic orsupersonic flo\\' past a body of re\'olution by considering nonlinear terms,the second-order potential satisfies

Particular integrals of this sort are also ordinarily found by trial ratherthan by systematic analysis. Analogous solutions of some homogeneousequati~ns- were discove~ed by Lin and Schaaf (1951) for viscous flow.

~:f<b2 = .11 2 :(2 .1. (y -- I )J12]rP1XrPl.1J' T 2rPl/rPm -';-- rPi,'rPlrr} (3.22a)

where again ='24>1 = O. Van Dyke (1952) has found that to withinthird-order terms a particular integral is given by

3.9. Direct Coordinate Expansions

explicitly in the perturbation expansion, so that the result is unneces­sarily complicated, the series is not an asymptotic expansion, and it isnot possible to equate like functions of 8.

The transfer of a boundary condition is effected by using a kn()\\'ledgeof the way in which the solution varies in the vicinity of the basic surface.Often the solution is known to be analytic in the ~oordinates, in \\hichcase the transfer is accomplished by expanding in Taylor series aboutthe valtles at the basic surface. In the first approximation this usuallvmeans that the condition is simply shifted from the disturbed to th~

hasic surface. I n axisymmetric slender- bodv theorv on the other handthe solution is singul~r on the axis, but th~ transfe~ can be carried ou~lI::;ing the fact that the \'clocity potential varies ncar the axis like log ror the radial velocity like I r.

..\fter the solution is calculated, values of flow quantities are oftenrequired at the body or other surface. These can be found in simplestform by repeating the transfer process, expressing them in terms ofvalues at the basic surface. Both of these transfer processes areillustrated in Chapter I V; see also Exercises 2.1, 2.3, and 3.2.

(3.22b)

(3.2Ia)

(3.2Ib)

= 2.HrPV8

where the first approximation is a solution of the homogeneous equation=24>1 = O. Burns (1951) has noticed that a particular integral is alwaysgiven in terms of the first-order solution by

.11 -I-4>2/) =-= i=- .1/'2 ,\"1'10

3.8. Transfer of Boundary Conditions

Often a boundary condition is imposed at a surface whose positionvaries slightly with the perturbation quantity 1-.'. The surface may be that

Perturbation problems in which the small quantity is a dimensionlesscombination involving the coordinates (space or time) rather than theparameters alone have certain special features. A useful discussion ofthe distinction between parameter and coordinate expansions is given

38 III. The Techniques of Perturbation Theory 3.9. Direct Coordinate Expansions 39

by Chang (1961). The essential point is that no derivatives with respectto a parameter occur, and it is therefore possible to calculate the solutionfor one value of the parameter without considering other values. Ordina­rily one seeks an approximation for either small or large values of one ofthe coordinates. It is useful to designate these respectively as direct andinverse coordinate e.\pansions.

A direct coordinate expansion is natural to a problem governed byparabolic or hyperbolic differential equations. One expands the solutionfor small values of a time-like variable, which can of course be a spacecoordinate rather than actual ti me. The perturbation quantity mustincrease in the positi\'e sense of that variable, following time's arrow.Then there is no back\vard influence, so that each term in the perturbationseries is independent of later ones, and can be calculated in its turn.The result is a perturbation expansion that describes the early stages ofthe evolution of the solution from a known basic initial state.

The following are typical examples of direct coordinate expansions.Goldstein and Rosenhead (1936) have calculated the growth of theboundary layer on a cylinder set impulsinly into motion by expandingin PO\\"Crs of the time, the governing equations being parabolic. :\ear thestagnation point of a circle, for example, they find the skin friction tobe gi\'en by

T ---pvL'~Cr~.\· __I_. [I - 1.42442(U]t) - O.21987(uA2 -: ... ] (3.23)'/7TU l f

where C I is the gradient of im'iscid surface speed at the nose and.\' thedistance along the surface..\s shown in Fig. 3.4, this series obviously

diverges for large time, where it should approach Hiemenz' result forsteady flow. Such nonuniformity usually arises in direct coordinate\:xpansions (but see Sections 10.6 to 10.8).

.\ case \vhere a space coordinate assumes the time-like role is Blasius'expansion of the steady boundary layer on a cylinder in powers of thedistance .~' from th~ stagnation point, the boundary-layer equation(2.25a) bemg paraboltc for steady as well as unsteady motion. The result(1.5) for the parabola is beliend to converge only for.\' a 7T 4 (Yanf)~'ke, 1964~). _:nother such case, iJWol\'ing hyperbolic equations, is the~lxlsymmetncCrocco problem: a perturbation of the self-similar solutionfor sup.erso~ic flo:\' pa~t a circular cone yields the initial flow gradients~lt the tiP 01 an oglve 01 reY<llution (Cabannes, 1951).

F<.)r ~lliptic equations, coordinate expansions usually provide on Ivqllalttatl\e results. One ordinarily encounters a boundarv-\'alue rathe'rthan an initial-value problem. Then because of back\~'ard influenceany local solution depends on remote boundan' conditions and it isnot pos~ible to ~alculate successin terms of ,;n expansion' for smallvalues ?t a coordmate..\11 that can be achieved is to find the form of the\:xpanSlOn, each term being indeterminate bv one or more constants.For example, Carrier and Lin (1948) have exa~ined the nature of viscous~l~l\\'. near. the ,leading edge of a flat plate by expanding for small radius,I heir senes tor the stream function is, after correction

.f. 2 J :3 " ( 8 . 38) B"J (8 5 )'I'~' .'11' .. cos 2: _. cos - .,- ri)· cos J - cos -82 . _ 2

-I"- ~~. 1':3:2 sin~ 8[log r sin 8 -;' (8 - 7T) cos 8J

See~ote

8

(3.24). 8 (. 28 2' 4-'- '5 sIn - S1I1 8) .J.. 3C sin:3 8} ,- ...

\vhere 8 = O. ~n the plate .. The constants A, B, C, '" depend uponboundary conditIOns far outSIde the range of \'alidity of the expansion,and are therefore undeterminable within the framework of theanalysis.

. An exc.ep~ion is t~e unconventional case of an initial-value problemtor an e1hptlc e.9uatlOn. In the inverse problem of supersonic flow pasta blunt body (fig. 3.5), the detached shock wave is prescribed, and oneseeks the body that produces it. Although the flow is subsonic near theaxis, the stream function can be expanded in po\vers of the distancedownstream of the shock wave, and the coefficients found in successionCabannes (1956) has calculated seven terms of the series when Al = 2:

6

Hiemenz----

42OL.- -'- ----L ---.J

o

2rr----------.,.----,..-,/

",'/

.-'.-' 2 terms of (323).-'

,/

, ~--.-'

Fig. 3.4. Growth of skin friction near a stagnation point.

I term

For a paraboloidal shock wan, the stream function is described near theaxis by

41

(3.27)

3.10. Inverse Coordinate Expansions

3.10. Inverse Coordinate Expansions

In contrast to direct expansions, which usually possess a finite radiusof convergence, inverse expansions for large values of a coordinateordinarily appear to be divergent asymptotic series. They also sufferfrom indeterminacy irrespective of the type of the governing equations.For elliptic equations the situation is the same as that discussed abovefor a direct expansion. However, the undetermined constants can some­times be related to simple integral properties of the solution. Thus thefirst few constants in the expansion for subsonic flow far from a finitebody can be identified with its lift, drag, moment, etc. (Imai, 1953;Chang, 1961). For parabolic and hyperbolic equations, indeterminacyarises because the expansion runs contrary to the direction of the time­like variable. Eigensolutions therefore appear at an early stage, whoseconstant multipliers depend upon certain details of the previous history.Occasionally a few of these can be supplied without detailed knowledgeof past events by invoking some global conservation principle (cf. Section4.5). :\lore often a sequence of constants remains undetermined.

The form of an inverse coordinate expansion varies \videly with thetype of the equations, the number of space dimensions, and the extentof the body. In some problems the leading term of the expansion isobvious; for example, it is evidently the undisturbed stream for the steadyflo\\' far from a finite body, the conical motion for flow far downstreamon a blunted cone, and the corresponding steady motion for the flowlong after an impulsive start. Then one perturbs that basic solution tofind how it is approached. The approach is sometimes algebraic, in inversepowers of the large coordinate, as for noncirculatory potential flowfar from a finite body (Imai, 1953). It very often involves logarithmic aswell as algebraic terms, as for circulatory potential motion or viscousflow far from a body (Chang, 1961). In time-dependent problems it isoften exponential, as for unsteady viscous or free-streamline flows(Kelly, 1962; Curle, 1956).

'liVe quote one example. The Blasius series for the boundary layer(Section 2.6) is a direct coordinate expansion for small distances fromthe stagnation point. On a parabolic cylinder we can supplement thatapproximation by an inverse expansion for large distances. It is evidentthat the leading term is the solution for the flat plate, because far down­stream the nose radius is negligible compared with the dimensions ofinterest. Perturbing that basic solution yields as the complement to (1.5)for the coefficient of skin friction

(3.26)

(3.25)

M>I~ x

~ = ~(' Vt ') _ (Il __ 1)2.('J!!..,) 2

-f- ...a 5 _ a - 75 _ a '

~ .- ~ ~ .- ~~(~l~ .- 221;(~)a .- J.~i4~-(~)·

_ 7051(~I:) _ 1,817,9_09 (~)6

243 a 2916 a

Fig. 3.5. Supersonic flow past a blunt body.

III. The Techniques of Perturbation Theory

Here 11 is the number of space dimensions: n = 2 for plane flow andn = 3 for axisymmetric flow. \Ve can expect the result to be the moreaccurate for plane motion, because it is exact for the one-dimensionalpiston problem 11 = I. The series evidently diverges for large time, butCabannes attempts to estimate its limit as the maximum given by thetwo known terms. This yields L1 a = 0.107 for plane flow, comparedwith accurate numerical ~alculations of 0.377 for the circular cylinder.We reconsider this discrepancy in Section 10.7.

.\side from such rare exceptions, direct coordinate expansions can beeffectivelY applied to elliptic equations only by treating them as if theywere par~bolic and truncating the series at a finite number, of te:ms.

The blunt-body problem is so intractable that several Investigatorshave been willing'to introduce another independent nriable in order torender the equations parabolic or hyperbolic. Thus Cabannes (195~)

considers an impulsive start, and expands the unsteady flow field Inpowers of the time. He suggests that the accuracy will in~rease with:Vlach number; at IV1 = oc and for y = 7 5 he finds as the ratIO of shock­\vave standoff distance L1 to nose radius a for any smooth body:

40

42 III. The Techniques of Perturbation Theory Exercises 43

Here x is the abscissa, and C\ is an undetermined constant multiply~ngthe first of an infinite sequence of eigensolutions for the flat-plate solutIOn

(Section 7.6). . 'Sometimes the leading term is by no means ob\'lous. Free-streamlme

motion proyides an example of the complications that may appear. Inplane flow the width of the dead\\'ater .region increase~ far downstreamlike ,\"12, but in axisymmetric flow It has the unhkcly gro\vth of

Xl /2(log X)-I /4 (Leyinson, 1946).

equations by the free-stream :\Iach cones. Again this defect is inconse­quential if the body is sufficiently smooth, but other\\'ise leads to non­uniformities (ef. Chapter YI and Exercise 3.4).

EXERCISES

3.1. .1lodified hypersonic similarity rule. According to hypersonic small­disturbance theory the pressure coefficient on a slender wedge or cone of semi­Ycrtex anglc c has the form

I ---C "l:; ---- F(' /11 z -' Ie)

')1,< .llz ,_ I v.

Calculate c/> using this condition. How smooth must f be? \Vhat happens to thesolution if that restriction is violated?

lim rZq,,. = l3 dd- [ef(t)J3,.,)0 t

cP(O, y) = cP£(O, y) == 0 for y > 0

\vhere f(x) = 0 for x < 0c/>ix,O) = ej'(x),

q,yy - ~xx :::= E~J'J"

3.2. Tran4er of tangency condition. :\ small sphere pulsates in still air, itsradius varying with time as ff(t), and thereby produces \\eak outgoing \\'aves\dlOse velocity potential satisfies the acoustic equation q,u = Cz ,\zq,. Showthat if the function f is sufficiently smooth the tangency condition can approxi­mately be transferred to the origin as

De\'ise an alternative form that can be expected to be superior for thick bodiesin \'ie\\' of :\'e\\'tonian impact theory, according to which the pressure coefficientat any point on a body in hypersonic flight is twice the square of the sine ofthe angle of the surface with the stream. Exhibit the degree of improvementrealized by making numcrical comparison with the full solutions at .11 = x.Ill\'estigate \\hether the result can be extended further to thick bodies at lo\\'erspeeds if one is guided by the supersonic similarity rule

3.3. Estimate of ultimate 'calue from coordinate e,\pansioll. Test Cabannes'idea for estimating the value of (3.26) at infinite time by applying it to (3.23)and (1.5), where the answers are known. Should one choose the value at theinflection point when no extremum exists? Try to devise a better or morerational scheme of this sort.

3.4. Effect of change of characteristics, The following problem is a mathe­matical model of steady supersonic flow over the upper surface of a thin airfoil:

3.11. Change of Type and of Characteristics

A curious feature of perturbation methods is that they may spuri~u.slychange the type of the go\'erning partial different.ial e~uations..\ stnk1l1gexample is Prandtl's boundary-layer approxm1atlOn. ;\lthough t~e~ayier-Stokes equations are elliptic, they are replaced by pa.rabohcequations inside the boundary layer, and by elliptic or. hyperl:oh~ onesoutside, according as the flo\v is subsonic or super.som.c..\gam, In thetheon' of surface \\'a\es the elliptic Laplace equatIOn IS replaced by anonli~ear hyperbolic equation in the. shallc)W.-water .ap~r~ximation(Stoker, 1957). ConYerscly, the hyperbohc equatIOns of .m\'l~Cld super­sonic motion become elliptic in the slender-body approximatIOn (\\ ard,

1955). . .,These changes of type imply significant changes m. t.he reglOn.s ot

influence and dependence, and in the boundary conditIOns reqUlre.d.Thus Prandtl's boundarv-laH'r equations, because they are parabohc,can be integrated stcp-b;'-st~p do\\nstream. The back\:'ard influen~e o~their elliptic antecedents has been suppressed, but \\'111 reassert .Itselfin higher approximations (cf. Chapter YII). Simi~arly,. ~he Kutt:l­Jouko\\'ski condition must be abandoned at a subsomc tralhng edge mthin-wing theory, because its upstream influence has been lost.

Some assumption of smoothness underlies any such change of type.The true type of the equations must be recognized :vherenr thatassumption is yiolated. Otherwise the perturbati.o~ sol~tlOn \\'111 breakdown at least locally. Thus boundary-layer theory IS 1l1yahd near a corner,as are the shallow~water and slender-body approximations. These thusbecome singular perturbation problems as the result of discontinuities

in the boundary conditions,Often the te~dency toward change of type is incomplete; the pertur­

bation equations me~ely become "less hyperbolic" or ,"more elliptic,"For hyperbolic equations this means that the charactenstlc surfaces arechanged. An example is supersonic small-di~turban~e theory, where. ateach stage the true characteristics are approximated 111 the perturbatIOn

44 III. The Techniques of Perturbation Theory

Solve it by first neglecting the right-hand side of the different~al equation,and then iterating to find the second approximation. ~:ompare with the ~xact

solution to deduce under what restrictions on the functIOn f t~c nth approxima­tion is valid near the surface. Discuss what happens at great distances.

Chapter IV

SOME SINGULARPERTURBATION PROBLEMS

IN AIRFOIL THEORY

4.1. Introduction

We proceed now to consider problems in which the straightforwardperturbation schemes used heretofore break down in some region of theflo\v field. There the ratio of successive tcrms is not small, as wasassumed, so that locally the approximation ceases to be an asymptoticexpansion..\s a consequence of this nonuniformity, one may miscal­culate or even lose essential results, such as the skin friction and heattransfer in viscous flow at high Reynolds numbers, or the drag of a thinairfoil. ~Ioreonr, at some early stage it usually becomes impossibleto proceed even formally to higher approximations. These are singularperturbation problems. .\n excellent surny by Friedrichs (1955)discusses their occurrence in branches of mathematical physics otherthan fluid mechanics.

The prototype of a singular perturbation problem is Prandtl's bound­ary-layer theory. However, we introduce the subject instead by studyingthe simpler problem of incompressible flow past a thin airfoil. Becausethe motion is governed by the two-dimensional Laplace equation, whichis linear, it is possible to exhibit the results in analytic form, and sodisplay the main ideas more clearly. This problem is also admirablysuited for introducing two well-known methods that have been devisedfor treating singular perturbation problems. Thus the present chapterforms the cornerstone of the book. The subsequent chapters are simplydevoted to amplifying and extending the ideas set forth here in connectionwith our standard problem of the thin airfoil.

The plan of attack is the following. We first calculate formally thethin-airfoil expansion for a general symmetric profile. Then for severalspecific shapes we exhibit the nonuniformities and nonuniquenesses thatarise because we violate the assumption of small disturbances at stagna­tion points. Finally, we correct those defects, first by using an intuitive

45

46 IV. Some Singular Perturbation Problems in Airfoil Theory4.2. Formal Thin-Airfoil Expansion 47

Fig. 4.1. Symmetric flow past thin airfoil.

physical argument, second by matching with a local solution in the spiritof boundary-layer theory, and third by straining the coordinates in the

case of a round edge.

The flo\v is irrotational, and it is connnient·-with extension to threedimensional flows in view-to work with a nlocity potential cPo Let itbc normalized such that the velocity nctor is given by q = U grad cPwhere U is the free-stream speed. Then the full problem is

Th.is inn(:cIJOl.Is-looki,ng problem, involving only the simplest linearl'!hptlc pa;:lal dIfferentIal equ~tion, is difficult to soh'e \\'ith acceptable:1~cIJracy. 1he standard numencal method of conformal mapping due toI heodorsen (.-\bbott and von Doenhoff, 1959) is tedious and inaccurate.llence .the ~hin-airfoil approximation is of practical utilitv as well astheoretical lI1terest. ~t is even more useful in subsonic ~omprcssibleHO\v, \"herc the equatIon of motion is nonlinear, and in three dimensions\,herc conformal mapping is inapplicable. '

',"c seek thc asymptotic expansion of the solution as the thicknessparameter I: tcnds to zero. In the limit, the airfoil degenerates to a line\\ hlch c~lUses no disturbance of the free stream, so the basic solution isthe Ul1lfOn~1 par~llel fl<!\,·. 'Ve. tentatively assume that the asymptotic~cncs has, for a gIven thIckness function T, the form

<I>(x, y; c) --., x' e<l>tC\', y) - e2q;l\', y). e:l(bix, y) _. ... (4.2)

It is imp~iedr ?e.re that the expansion proceeds indefinitely in integralpO\,H:rs of I'. 1hIS appears to be true for incompressible flo\\' unless 'the<urtoIl IS extremely blunt; see Exercise 4.3. For subsonic compressibleH(m past a round,-nosed profile, however, our example (1.2) illustratestl1<1t loganthms of c appear beginning \vith 1:.'4 log 1'.

. In order to :ubstitute this expansion into the ft;ll problem and equatelIke po\\'ers Of, 1:.', we must transfer the tangency condition (4.1 c) to thea\:lsy - O. It \ve assume that the cP,,(x, .1') are analvtic in \' at \. = 0\'e can expand in Taylor series to obtain ...'

<by(x, =eT) = e<pIY(x, 0 =) - e2[<p2y(x, 0 :-: )

- T(x)<pJ'lix,O __=)] .. e:l [<p3Y{X, 0 =)

'. T<P2Vy(X, 0:::),... }T~(Pl!iVl\',0 =.)] (4.3)

a,~d sim,ilarly for cP.l'. Here .I' =~ O__~__ refers to the top and bottom sidesof the slIt to whIch the airfoil degenerates in the limit as E:: ->- 0 and across\,hich cPy is di.scontinuOl~s .( \\hereas cP.r is continuous). This' expansion,thou!5.h essentIal to obtalJ1lJ1g a senes of the desired form (sec Section3.~), IS or~e. source of the nonuniformities that we encounter later. \VemIght antICIpate difficulties because of the successively higher derivativesthat appear. .

. T~~s we obtain a sequen~e,of pr~)blems that are linear in the tangencycondltlO.n as well as the dIfferentIal equation, the first being that ofconventIOnal linearized thin-airfoil theory:

(4.1a)

(4.1 b)

(4. Ic)

as x2 -+ y2 -+ CfJ

at y ='=:':: eT(x)

cP·",-,x-,o(l)

0/" = _I- eT'(x)0/£ .-tangency condition:

differential equation:

upstream condition:

y

t f:... Y=±f.T(x)

LC:~:~--'xLE TE

4.2. Formal Thin-Airfoil Expansion

\Ve consider uniform plane flow of an incompressible inviscid fluidpast a thin airfoil. For simplicity, we restrict attention to symmetricflow, because it embodies all the essential features. However, asymmetriceffects due to camber and incidence can be treated similarly. For detailsthe reader can see the modern expositions of thin-airfoil theory givenby Lighthill (1951), Jones and Cohen (1960), and Thwaites (1960).

\Ve use Cartesian coordi nates, \\·ith the chord of the airfoil spanningany convenient segment of the x-axis, whose length may be assumed tobe of order unity. Then let the airfoil be described by y c-= =I' T(x),where E is some thickness parameter, and T(x) a function of order unitythat gives the thickness distribution (Fig. 4.1). Varying E:: produces a

family of affinely related profiles.

The second term in the asymptotic condition (4.1 b) merely requiresthat the perturbation potential vanish at infinity, although it is actuallyO( 1r) for a closed body. This requirement ensures a unique solution

by ruling out circulation.

~lJ'J' --1- 4>l!/.IJ =-: 0

<PI = 0(1)

o/JY{x,O ±) = =T'(x)

(4.4a)

(4.4b)

(4.4c)

49

(4J~a)

(4.9a)

(4.8c)

(4.9c)

(4.8b)

(4.9b)Y</>ny(g, 0 +)(x - g)2 + y2

..I. _'..1. = ~ JTE <p"if, 0 +) deIf'nx l'f'nv . C S

~ I.E X + ly - S

I --- I<P ,= - log vix2 + y2 = - Re log z

2~ 2~

I Y<py = -2 0'-2

~ x" T Y

1 .TE

<Pny(x, y) = -- j~ I.E

4.3. Solution of the Thin-Airfoil Problem

In order to calculate surface values, as in (4.7), we need only

representation of the body by a distribution of singularities. Sources,dipoles, and so on, can be placed either on the surface or inside the body.In thin-airfoil theory they are naturally distributed along the axis betweenthe leading and trailing edges. Only sources and sinks are required inour symmetrical problem, an equal quantity of each being needed fora closed body.

A point source of unit strength at the origin gives

For a sufficiently smooth airfoil, considerations of continuity show thatthe local source strength must be twice the airfoil slope. This approxima­tion is, in fact, justified to precisely the same extent as our transfer ofthe tangency condition to the axis. Hence the solution of anyone of thethin-airfoil problems (4.4), (4.5), etc., is giyen by

as can be verified by direct substitution, or in complex form

Here the Cauchy principal value of the divergent integral is indicatedif x lies between the leading and trailing edges. A table of the airfoilintegral (4.10) for a number of thickness functions T(x) is given by VanDyke (1956).

(4.7)

(4.5c)

(4.6c)

(4.6a)

(4.6b)

(4.5a)

(4.5b)

</>3XX 7 </>3YY = 01>3 •.~ 0(1) as x2 T y2 ->- 00

<P3Y(X, 0 :!J = :.: T(X)</>2Ax,0) T T(x)T('\')<Pl"1I(tl:, 0)

:~ T(X)<P2YY(X, 0) - ~ P(X)<Pl!IlI"(X, 0)

= _l: [T(X):<P2J.(X, 0);- ~-T(x)T"(x)}]'

</>2XX --;- </>2VY = 0

<P2 = o( I ) as x2 + y 2 ->- 00

1>2"(X,O I:) = :: T(x)</>v'(x, 0) -i~ T(x)</>m(x,O)

= .L: [T(X)</>lJ(X, 0)]'

IV. Some Singular Perturbation Problems in Airfoil Theory48

Of course this process represents another possible source of nonuni­formities.

The simpler alternatiye forms gi\'en for the second- and third-ordertangency conditions (4.5c) and 4.6c) are obtained by making use of boththe differential equations and the tangency conditions from preyiousapproximations..\s a result of this modification, each problem is formallythe same as the first. Any higher approximation can be regarded as thefirst approximation for some fictitious airfoil whose thickness functionT,,(x) is the last square bracket in the tangency condition.

Flow quantities at the surface of the airfoil can be expressed as powerseries in f by relating them, again through Taylor series expansion, tothe yelocity components on the axis. Because ef;"!1 is giyen there by thetangency condition, one needs only ef;1I .r(x, 0). Thus, for example, thesurface speed q is found to be giyen by

4.3. Solution of the Thin-Airfoil Problem

We have reduced each higher-order thin-airfoil problem to the formof the first. This crucial problem can be solyed in various ways. It isusually simplest to use complex-yariable theory, the best method oftenbeing to guess (ef;.c - ief; y) as a function of the complex variable z _c= x + iythat satisfies the boundary conditions. A useful table of such solutionsis giyen by Jones and Cohen (1960).

Methods based upon the complex variable haye the disadyantage,howeyer, that they cannot be generalized to three-dimensional flows.An alternative method that is capable of such generalization is the

51

o

o

1st approx.

2nd approx. (4.131

Exact (4.1401

Lighthill's rule (4.3Ibl

04 02Ixl

o

u~

0608

4.4. Nonuniformity for Elliptic Airfoil

Fig. 4.3. Surface speed on ellipse with Ii - 0.5.

1.2

0.4

.!.- 08U

The perturbation yelocity is singular like a multiple of p 1'1 !~, where I'

is the distance from the stagnation point. Hence the region of non­uniformity is a circular neighborhood whose radius is of order t~. As

16 .....---------------------,

.. ~.,;..:..'.~••~.'c..-'------;1

might haye been anticipated on physical grounds, this is of the orderof the leading-edge radius.

The failure of the perturbation solution is compounded in higherapproximations. The problem (4.5) for <P~ is seen to be identical withthat for <Pl' Hence the second approximation for surface speed is foundfrom (4.7) as

(4.11 )

(4.12)

,y= ±e ¥/-x2

1,

IV. Some Singular Perturbation Problems in Airfoil Theory50

Substituting into (4.10) yields as the solution of the first-order problem(4.4) for <P1.J: on the axis

4.4. Nonuriiformity for Elliptic Airfoil

:.Y10st subsonic airfoils haye round leading edges. The simplest finiteround-nosed profile is the ellipse, which we will investigate in consider­able detail. Let the thickness function be T(,\') =--= vi - x~, which describesan ellipse of thickness ratio £ extending over - I ~ x ~ 1 (Fig. 4.2).The radius of the leading and trailing edges is £2.

:.YIore generally, the complex perturbation nlocity throughout the flowfield is, from the table of Jones and Cohen:

Fig. 4.2. Elliptic airfoil.

-I

<PlJ'(X, 0) = - ~ f ' gdg7T • -1 V 1 -- e (x - g)

and this is singular like the square of E rl!". The first and second approxi­mations for the case E - 0.5 are compared with the exact result in Fig. 4.3,where the diyergence of the series near the stagnation point is evident.Further terms can be calculated indefinitely, and the 2nth approximationfound to be singular like (c 1'1/2)211, with the signs alternating in a futile

. .

attempt to attain zero speed at the stagnation point by superposingsuccessively stronger singularities. (As in the first approximation, velocitycomponents of odd order are also singular, but contribute only regularterms to the surface speed.) Thus the solution is improved at each stage

\Vith proper attention to principal \"alues at branch cuts, this reproduces(4.11) on the axis. Then (4.7) gins the familiar result of linearized thin­airfoil theory that the surface speed is constant on an ellipse, withq U = 1 --;- c. This \"alue happens to be exact at the middle of theellipse, where the maximum occurs (Fig. 4.3).

It is a misleading circumstance that the first-order surface speed isfinite even at the ends of the airfoil. According to (4.12) the perturbationvelocity itself is singular at the leading and trailing edges. The assumptionof small disturbances has been violated at the stagnation points, and as aconsequence the solution has broken down locally. The thin-airfoilsolution is not uniformly valid near the edges.

q , 1.),\'2- ~ 1 -r- E - - -E- ---e-U 2 I-,\'2

(4.13)

52 IV. Some Singular Perturbation Problems in Airfoil Theory 4.5. Nonuniqueness; Eigensolutions 53

and hence not uniformly near the stagnation points x

which confirms and extends our result (4.13). :\Ioreo\'cr, it makes evidentthe source of the nonuniformity. The singular tcrms arise from expandingthe denominator by the binomial theorem, which is justified only if

This is our first example of an eigensolution. The role of eigensolutionsin inverse coordinate expansions was discussed in Section 3.10. Althoughthe present example is manifestly a parameter expansion, it can-becauseit is a singular perturbation problem-be regarded also as an inversecoordinate expansion. That is, the thin-airfoil solution becomes accuratefor distances from the edges large compared with their radius 1':2. Theeigensolution (4.16) represents ignorance of the details of the flow nearthe edges.

The x-derivative of (4.16) is also an eigensolution for the ellipticairfoil. It consists of dipoles at the leading and trailing edges (whosestrengths need no longer be equal in order to satisfy the upstreamcondition). Similarly, an infinite sequence of successively more singulareigensolutions-consisting of quadrupoles, octupoles, etc.-is formed byrepeated differentiation with respect to x.

In order to make the solution unique, the constant multiplying eachpossible eigensolution must be determined. This can be attempted inthe following successively mOre sophisticated ways:

(i) Principle of minimum singularity. A very reliable bit of mysticismasserts that in any case of nonuniqueness the correct solution involvesthe weakest possible singularity. For the elliptic airfoil, this principleserves to rule out all eigensolutions in the first-order solution, all exceptsources in the second and third approximations, all except sources anddipoles in the fourth and fifth, and so on. A justification of the principle\\'ill appear later in Section 5.6 when we discuss matching with a localsolution of boundary-layer type.

(ii) Global conseY'l.'ation principle. It is sometimes possible to find aconservation law that will serve to determine an eigensolution. For theelliptic airfoil, the principle of conservation of mass serves to rule outthe source eigensolution (4.16). A convenient way of accomplishing thisis to work not with the velocity potential but with the stream function,which assures that mass is conserved globally. Then the weakest admis­sible eigensolution is a dipole rather than a source (see Exercise 4.1).Together with the first method, this defers the indeterminacy to thefourth approximation for the elliptic airfoil.

(iii) iHatching with supplementary expansion, Later in this chapter wegeneralize Prandtl's idea of examining in detail the region of nonuniform­ity. An essential feature of that procedure is the possibility of matchingwith a supplementary asymptotic expansion valid in that region. Thisprocess serves to rule out all eigensolutions in the thin-airfoil expansionfor the ellipse.

(4.15)

(4.16c)

(4.16a)

(4.14a)

(4.14b)

(4.16b)

1 + eq

U

Expanding this formally for small I': yields

It is evident that this leaves each of the problems (4.4), (4,5), etc.,unaffected except at the leading and trailing edges, where the solutionfor the ellipse is singular anyhow.

4.5. Nonuniqueness; Eigensolutions

The verification just given is important, because for a round nose thesolution of the thin-airfoil problem is not mathcmatically unique. Thetangency conditions (4.4c), (4.5c), etc., are singular at a round cdge.Therefore one is entitled to add to the solution any harmonic functionthat is singular there but does not otherwise affect the boundary condi­tions. In the present problem one such function is given by a point sourcelocated at one edge and a sink of equal strength at the other:

except within a distance of order f: 2 from either edge, where it becomesworse.

The formal thin-airfoil expansion can be verificd by comparison withthe full solution, which is readily calculated by conformal mapping orseparation of variables in el1iptic coordinates. The exact result for surfacespeed is

54 IV. Some Singular Perturbation Problems in Airfoil Theory 4.6. Joukowski Airfoil; Leading-Edge Drag 55

The drag coefficient can be calculated by integrating the pressure overthc surface of the airfoil according to

(4.19)

(4.18)~ = I --'-- e(1 - 2x) - ~e2 !..-=- x (I -1- 2x)" ~- ...V' 2 I +x

p P 4x"-3C -~ _---=-...2:.. = --2e(1 -- 2x)- 2e" x --I-.- + ...,,- tp V" --,- x

and, from Bernoulli's equation, for the surface pressure coefficient

By continuing with the second approximation we obtain for the surfacespced

In the preceding section we used none of these methods, but simplycompared with the known exact solution.

All these methods may fail in more complicated problems. An exampleis the third approximation for the boundary layer on a semi-infinite flatplate, which is discussed in Chapter VII. In that case it is perhapsimpossible, eycn in principle, to determine even the first eigensolutionshort of solying the full problem. In such difficult problems, attemptshayc been made to determine the first eigensolution by patching at anarbitrary point with a different approximate solution (Imai, 1957a) orby comparing with a numerical solution (Traugott, 1962).

Howe\-er, substituting the thin-airfoil expansion (4.19) shows that thesecond-order term is dinrgent at the leading edge. Keeping onlyfirst-order tcrms giyes a negatiye drag:

4.6. Joukowski Airfoil; Leading-Edge Drag

The failure of thc thin-airfoil expansion at a round edge means thatone cannot calculate the drag by integrating surface pressures. Indeed,diyergent integrals appear beyond the first approximation. The ellipticairfoil obscures this matter because the effects of the leading and trailingedges cancel by symmetry. \\'e therefore consider instead a simpleprofile with only one round edge.

To second order a symmetrical Joukowski airfoil (Fig. 4.4a) is described

I' f1Cd ,--= ~U" j P dy = e C,,(x)T'(x) dx"p -1

(4.20)

(4.21)

by the thickness function T(x) = (1 -- x)VI - x 2• The thickness

ratio is E at midchord, and (3V3 4)1- ~ 1.30E at the thickest point.For the first-order streamwise velocity perturbation on the axis (4.10) gives

1'\ 1 + t - 2t"<Pllx, 0) == - - J ., dt= 1 - 2x,71' J -1 Vl- t- (x- t)

(4.22)

This result is obyiously wrong, for the drag must yanish according tod' Alembert's principle. Because first-order thin-airfoil theory fails topredict the pressure rise near the stagnation point (Fig. 4.4b), it missesa positi\'e contribution to drag that would exactly cancel the thrust ofthe remainder of the airfoil that we haye just calculated.

Jones (1950) has shown that this leading-edge drag can be recoveredby calculating the drag not from surface pressures but with a momentumcontour that ayoids the region of inyalidity near the leading edge. It canalso be recoyered by correcting the surface pressure near the edge usingmethods discussed later; and only this method is successful in the secondand higher approximations.

Thus one finds that the leading-edge drag to be added to the integralof surface pressure in lincarized theory is the same for any round-nosedairfoil as for our Joukowski profile. It is given in general terms by

where a is the leading-edge radius, equal to 4E2 for the Joukowski airfoilof Fig. 4.4. This is just the drag of an infinite parabola of the same noseradius, defined-following Prandtl and Tietjens (1934)-as the drag ofa nose section separated by a cut into which the ambient pressure

x2 < I (4.17)

( b)

r Exact'" " " "."..~ ,,' ..

6C:

.. xI

y

t

Symmetrical Joukowski airfoil. (a) Geometry. (b) Surface pressure.

130E

Fig. 4.4.

uc::>

56 IV. Some Singular Perturbation Problems in Airfoil Theory 4.7. Biconvex Airfoil; Rectangular Airfoil 57

penetrates, in the limit as the cut moves downstream to infinity (Fig.4.5). That this should be the case is clear from the fact that as its thicknessvanishes any round-nosed airfoil is approximated by a parabola to anincreasing number of radii downstream from the leading edge. This roleof the osculating parabola will be exploited further in Section 4.8.

Fig. 4.5. Scheme defining drag of parahola.

.-\ similar violation of d' .-\Iembert's principle occurs for lifting airfoilsof zero thickness. In that case, however, integration of the surface pres­sures predicted by linearized theory yields a spurious drag rather than athrust. Thus for an inclined flat plate it evidently gives a resultantnormal force, \\'hich has a dO\vnstream component. The resolution ofthis parado\: is again associated \vith a singularity at the leading edge.The additional leading-edge thrust required to restore the drag to zerocan be found by the methods just described for the symmetrical airfoil(von Karman and Burgers, 1935, pp. 51-52).

4.7. Biconvex Airfoil; Rectangular Airfoil

\Ve consider two other shapes, invoh'ing stagnation edges that arerespectivcly sharper and blunter than those of the elliptic airfoil. First,the thickness function T(x) = I __ ,\'2 describes a biconvex airfoilformed of two parabolic arcs (Fig. 4.6).

Fig. 4.6. Biconvex airfoil.

Proceeding as before gives for the first approximation

(4.23)

.-\t the sharp stagnation edges the perturbation velocity is logarithmicallyinfinite, behaving like E log r even on the surface. Again this failure iscompounded in higher approximations. Thus to second order thesurface speed is found to be

~ -- 1 + : e(2 - ,\' log ~! ~,) -i- e2[ ;2 (2 - ,\' log ~! J"

- ~2 log" ~ ~ :: - (1 - ,\'2)] (4.24)

This is singular like (c: log r)2 at the stagnation points, and the nthterm is found to be singular like (c: log r)". Hence the expansion is invalidwithin a distance of either edge of order e-1 ;<. Because this is smallerthan any power of f, it is usually negligible in practice; and at the trailingedge the nonuniformity would be swamped by viscous effects. Ho\vever,it is of fundamental interest to understand and correct this minordefect as \\'ell as the more serious one at a round edge.

The singularities are so weak in this example that no practicaldifficulties arise. :\11 eigensolutions can be excluded by invoking theprinciple of minimum singularity, because even the weakest of them­the source eigensolution (4.16)-is more singular at the stagnation pointsthan any term in the thin-airfoil expansion. Furthermore, the integral(4.20) for the drag coefficient is com-ergent. Thus the formal thin-airfoilsolution can be calculated to indefinitely high order for a sharp-edgedairfoil.

Second, consider the rather extreme case of a thin rectangular airfoil(Fig. 4.7), which can be described by the thickness function T(x) =

H(I -- x2) = H(x --l- I) - H(x - I), Here H is the Heaviside unitstep function, which is zero for negative values of its argument and unity

Y

t (Y:±CHU-X1lr I

'E:TTTTTTTTTTT':~:::::::::~:~:3- -- x

Fig. 4.7. Rectangular airfoil.

60 IV. Some Singular Perturbation Problems in Airfoil Theory 4.8. A Multiplicative Correction for Round Edges 61

Here s is the distance into the edge, and c'; the maximum speed on theparabola. The latter is obviously nearly equal to the free-stream speed[', but its value will not be needed. Expanding this expression formallyfor small values of the nose radius a (more precisely, for small values ofthe ratio a s) yields the series

down within a distance from the leading edge of the order of the noseradius a, which is proportional to 1'2. In that vicinity the airfoil can beapproximated by a parabola ha\-ing the same nose radius. :Yloreover,the approximation becomes better as the thickness is decreased, in thatit holds to a given accuracy over an increasing number of nose radii.

The exact surface speed on the parabola is easily calculated, orextracted through a limit process from the result (4.14a) for the ellipse,as

if we recall that "q"! U is of the form 1 + 0(8), and retain only terms oforder 1'2, we obtain the simpler rule

(4.3Gb)qz ;-s-- ("Q2" aU .. 'V s + ai2 U + 4S)

Here the subscripts indicate that the formal second-order solution"q2" is rendered uniformly valid to second order. This is to be understoodin the sense that both leading and secondary terms-of relative orderI'-are given correctly everywhere on the airfoil for the perturbation inspeed, or the pressure coefficient.

This rule was first deduced by Lighthill (1951) by quite differentreasoning that is described later in Section 4.12. He shows that itapplies also at angle of attack, and to cambered airfoils. In the lattercase, however, the perturbation is correct only to first order near thenose. Henceforth we refer to (4.30b) as Ughthill's rule.

.-\s an example, we apply the rule to the leading edge of the ellipticairfoil, for which "qz" U is given by (4.13). With s = x --:- 1 anda = e2 we find

(4.28)

(4.29)L)I aI -- 4s --~'q" =

The reason is that near the leading edge, where the disturbances arelarge, the exact speed on the airfoil is nearly that on the osculatingparabola. Far from the edge, on the other hand, the correction factorapproaches unity, so that the thin-airfoil solution for the airfoil isrecovered where it is valid.

It is possible to simplify this rule. The denominator has already beenexpanded formally, so that no damage results from further expansion;and the same is true of the thin-airfoil solution"q" for the airfoil. Then

The quotation marks indicate that this is only a formal result. Thismust be the result that ,,-ould be given by thin-airfoil theory. It can infact be extracted from the second-order solution (4.13) for the ellipse.The two terms sho\\-n here constitute the second approximation, becausethe nose radius is of order ,2. There is no term of order E because thefirst-order perturbation happens to vanish at the surface of a parabola.

We now claim that the ratio of these two expressions serves as a multi­plicative correction factor that converts the formal thin-airfoil solution"q" for the speed on any airfoil of nose radius a into a uniformly validapproximation ij:

(4.3Ia)

(4.31b)

q2_~--1-+----X-(1 I 1 121-2x)- T e T -4e ---

U 1 + x + -~e2 1 -.\'

The singularity at the leading edge has been removed, and replaced bythe proper zero. Of course a singularity remains at the trailing edge, sothat the solution is not symmetrical in x; but that too can be eliminatedby applying Lighthill's rule again with s = I-x. This yields, afterfurther simplification, the fully uniform result

The result is shown in Fig. 4.3 in comparison with the exact and thethin-airfoil solutions, which displays its uniform validity.

It is instructive to interpret the above derivation of Lighthill's rule inthe light of conformal mapping. If we seek to map a uniform streaminto the flow past an airfoil, the thin-airfoil approximation cannot copewith the rapid changes near a round nose. However, we can use theproduct of two mappings-from the uniform stream to the osculatingparabola, and from that parabola to the airfoil-and thin-airfoil theoryis adequate for the latter, although the former must be performedexactly. The two terms in (4.30b) correspond to these two mappings.

(4.30a)"q"

! s

-V s + a!2q=---

a1--+ ...4s

62 IV. Some Singular Perturbation Problems in Airfoil Theory 4.9. Local Solution near a Round Edge 63

4.9. Local Solution near a Round Edge

We now reconsider the round-nosed airfoil from the point of view ofPrandtl's boundary-layer theory. That is, we examine the details of theflow in the region of nonuniformity by introducing magnified variables.We thereby obtain a local solution that complements the thin-airfoilseries, and will be matchcd \\-ith it in the next section. The relationshipbetween this formal procedure and the preceding intuitive argumentwill become clear in the course of the analysis.

For definiteness \ve consider the ellipse; it will be evident how theresults apply to other round-nosed airfoils. We study only the non­uniformity at the leading edge; the trailing edge is treated in exactly thesame way. For simplicity we again discuss only the surface speed, butthe method can obviously be applied to the entire flow field. \Ve examinethe second approximation and, in order to exhibit the effecti\-eness ofthe method, suppose that it contains an unknown multiple of the sourceeigensolution (4.16) in each term..-\.ccording to (4.13) and (4.16) theformal thin-airfoil series for the surface speed is then

denoted by capital letters-that are of order unity in the region of non­uniformity. Evidently s is to be magnified by a factor 1 p2, and the otherCartesian coordinate y is treated similarly, because the region of non­uniformity was seen to be circular. Thus we introduce the newcoordi nates

(4.34)

(4.33b)

Ss

=--":2'e

The dependent variable 1J should be correspondingly magnified byintroducing 1J _.~ 1J p2. However, this modification cancels out of thepresent pr~blem because both the differential equation and boundaryconditions are linear in 1J.

We now consider the result of letting 10 tend to zero with the magnifiedvariables fixed. That is, as the thickness ratio vanishes we continuallymagnify the vicinity of the leading edge in such a way that the noseradius remains finite. In the limit the airfoil shape becomes

In these variables the ellipse is described by

(4.32a)C I -) CL - 1 - e(1 -"- __'I ) __ -e21-~"':"" _...::..g_)

U - 1 __ .\"2 2 \ 1 - ,\-2 1 __ .\"2

and the thin-airfoil series (4.32'1) for the surface speed may be written as

Of course conservation of mass can be used, as discussed in Section4.5, to shO\v that C1 ~ C2 = O. Howenr, we want to show that thisfollows from matching with the local solution, as is necessary in moredifficult problems.

Because the nonuniformity under consideration occurs at the leadingedge, it is cOlwenient to shift the origin there (Fig. 4.2) by settings = x + 1. Then the ellipse is described by

(4.33c)

--5

as e -+ 0 with S = 0(1)

which is an infinite parabolic cylinder of unit nose radius. Laplace'sequation is unchanged by our magnification of coordinates. Hence thelocal problem is again that of potential flow past the osculating parabola(Fig. 4.9).

(4.33a)

(4.32b)withas e --->- 0

q ,[. C1 ] 1 0[(1 - S)2 C2 ]U '"'-J 1 --r- e 1 --;- s(2 _ s) - 2.e- s(2----.,) + s(2--0

~ = 0(1)s

This asymptotic expansion is invalid in the region where s = 0(102),

because there the third term is of order unity rather than small aswas assumed.

\Ve aim to introduce new independent and dependent variables-

Fig. 4.9. Local Bow past osculating parabola_

Additional boundarv conditions are required to make the solutionunique. Clearly the flo~v should be symmetrical about the S-axis. On the

64 IV. Some Singular Perturbation Problems in Airfoil Theory 4.10. Matching with Solution near Round Edge 65

other hand, it is not obvious that the flow far upstream should be uniform.However, any other possibility involves a stronger singularity at infinity,and can therefore be ruled out by the principle of minimum singularity(Section 4.5). \Ve cannot conclude, however, that the speed of theoncoming stream is the original free-stream speed U; we therefore denoteit again by C i and determine it by matching with the thin-airfoil solution.Note that the value of Vi was not required for deriving Lighthill's rulein the last section, because it canceled out.

The boundary-value problem for if;(S, Y) can readily be solved forthe surface speed. However, the result is available from (4.28) as

We now apply this principle to find the unknown constants C1 and C2

in the outer expansion (4.32) and Vi in the inner solution (4.35). At theoutset, we know only the first term of the outer expansion, representingthe undisturbed stream, and no terms of the inner expansion. \Ye there­fore choose m = n = I, in order to find a first approximation to theeffective free-stream speed Vi in the inner problem. It is convenient tosystematize the procedure as follows:

as e-O with S = 0(1) (4.35)

I-term outer expansion:

rewritten in inner variables:

expanded for small e:

I-term inner expansion:

q "-' UU

U

U

(4.37a)

(4.37b)

(4.37c)

(4.37d)

By equating (4.37d) and (4.38d) according to the matching principle(4.36) v.-e obtain

u,(l

LT,

Thus we have two different asymptotic approximations to the solutionfor small F. The thin-airfoil expansion (4.32b) is valid except near theleading edge, where the local solution (4.35) applies.

It is helpful to introduce now some terminologv that will be extendedin the next chapter. Following Kaplun (1954) and Lagerstrom and Cole(1955) we call the thin-airfoil series the outer expansion. It is the resultof letting f:: tend to zero with the aliter variables sand y (or x and y)fixed. Similarly, the local solution is the first term of the inner e.\pansion,which is the result of letting f:: tend to zero with the inner z'ariables Sand Yfixed. (The subscript on L'i is now recognized as standing for "inner.")

I-term inner e.\pansion:

rewritten III outer variables:

expanded for small e:

I-term outer expansion:

Ui=U

(4.38a)

(4.38b)

(4.38c)

(4.38d)

(4.39)

That is, to first order the effective free-stream speed for the inner solutionis the actual free-stream speed, which is plausible physically. Exercise4.2 shows that this matching cannot be carried out if the inner solutionis singular at infinity, which justifies our appeal to the principle ofminimum singularity.

\Ye now reverse the process to find the second term in the outerexpansion. Taking m = I and n = 2 we haveSee

Note4

4.10. Matching with Solution near Round Edge

In general, as in this example, the inner and outer expansions comple­ment each other, one being valid in the region where the other fails.An important feature of this complementarity is that one can match thetwo expansions. Matching is essential to the 'success of the method, andwill be discussed in detail in the next chapter. For the present we simplyassert that here, as in many problems, we can apply what we will call(Section 5.7) the asymptotic matching principle:

The m-term inner expansion of (the n-term outer expansion)= the n-term outer expansion of (the m-term inner expansion). (4.36)

He:e. n: and n may be taken as any two integers, equal or unequal. BydefimtIOn, the m-term inner expansion of the n-term outer expansion isfound by rewriting it in inner variables, expanding asymptotically forsmall E, and truncating the result to m terms; and conversely for theright-hand side of (4.36).

I-term inner expansion:

rewritten in outer variables:

5' expanded for small e:

'\ 2-term outer expansion:

(4.40a)

(4.40b)

(4.40c)

(4.40d)

66 IV. Some Singular Perturbation Problems in Airfoil Theory 4.10. Matching with Solution near Round Edge 67

By comparing (4.40d) and (4.4ld), we find that they match only if C1vanishes; there can be no source eigensolution in the first-order thin­airfoil solution.

\Ve now re\'erse the process once more to find the second term in theinner expansion. With m = n = 2 we have

2-term outer e.\pansion: q ....... Ull -t- e[l + -~_]I (4.41a)I 0'(2 - 0') \

rc\uitten in inner vari,lbles: U;1 f- e[1 C\ ] / (4.4lb)T 828(2 _ 828) \

expanded for small 8: qC\ 1 ) (4.41c)= . 288 +. T'"

~~ 'f C1 i: 0II-term inner expansion: L 28S 1 (4.41d)I 1 if C1 = oj

and retaining the required number of terms (Fig. 4.10). The solutioncan be found 'by making a coordinate perturbation of the basic flow past the

order that for the osculating parabola, the eft'ective free-stream speedfor that parabola dift'ers slightly from the actual free-stream speed, It isincreased Iw the effects of relatively remote parts of the airfoil, whichmay bc rcg;uded as formed by adding sinks to the parabola. Thus theparabolic nose is flying in a stream that has been speeded up inorder to flow past the rest of the airfoil.

We could now return again to the outer expansion by choosing Tn ,.,,,2and 11 '---0 3, and show that matching requires C2 to vanish; there are nosource eigcnsolutions in the second-order thin-airfoil solution, Dipoleand higher-order eigensolutions would be ruled out by the same process,

Thi~ process could in principle be continued indefinitely. Beginning\\'ith the next step, ho\\"Cver, it must be recognized that the inner problemis no longer that for a parabola, but for the more complicated shapeobtained by expanding (4.33b):

(4.45)

1- • /-2S'( 1 I 2S' 1 4 S'2 ... )1 ....... :. v' - 4e , _. 32e, -

as e --+ 0 with ,'" = 0(1)(4.42a)

(4.42b)

(4.42c)

(4.42d)

q ....... U(1 - 8)

= C(I + 8)ql _., 8)

-, C(1 -;- 8)

2-term outer e.\panO'ion:

rewrittcn in inner \'ariables:

cxpanded for small e:

2-term inner expansion:

Physically, this means that although the nature of the flow near theleading edge depends upon only the local airfoil shape, and is to second

This last result indicates that the second term in the inner expansionmust be of order f. :\ow it is clear from (4.33b) that the error inapproximating the nose of the ellipse by the osculating parabola (4.33c)is only of order f2. Hence the inner solution is that for a parabola tosecond as well as first order, provided that U i is refined. \Ve find

i,

Ii

II

!I

i

IiI, I

IiI

(4.46)

I -'- 8

(l - 8 25')2+ S(2=- 8 28)

, 3. 8 S2(88 - 23) ,--;-- 4e'J ~-:-2S - 8

4 :f2(T+-'2sfi -- ...]

I

~l

qYf

Fig. 4.10. Successi\'e inner expansions for shape of ellipse.

osculating parabola. Of course the inner expansion (4.45) for the airfoilshape diverges for large S" in fact for S ~> 2 E

2. Hence the inner expansion

will be nonuniform at S' = x. It can be extracted from the knownexact solution by introducing inner variables and expanding, whichyields

(4.44)

(4.43a)

(4.43b)

(4.43c)

(4.43d)

/---­C I 28

q....... 'iy 1 -t- 28

_ U ;-2s=- . iV 20' -'- 8 2

_( I e2 •= Vi I - 4s + ...)

= Vi.#"

Vi = V(1 + 8)

rewritten in outer variables:

2-term inner e.\panO'io/l:

~ expanded for small 8:

\ 2-term outer expansion

and matching shows that to second order

68 IV. SOIne Singular Perturbation ProbleIns in Airfoil Theory 4.11. Matching with Solution near Sharp Edge 69

Singularities at S = 'i) are seen to appear beginning with the fifth termof the inner expansion. This divergence is of no concern, because theinner expansion is used at nowhere near such great distances.

The appropriate inner solution is a symmetric potential flow past thiswedge with the weakest possible singularity at infinity. It can be found,as indicated in Fig. 4.12, by a familiar elementary conformal transforma-

4.11. Matching with Solution near Sharp Edge

We now apply the same technique to a sharp edge. The example of thebiconvex airfoil (Section 4.7) shows that sharp edges lead to logarithmicsingularities in the thin-airfoil expansion. These are so weak that withoutdevoting any special attention to them one can calculate an unlimitednumber of terms in the series. However, it is of interest to understandthe source of the logarithms by examining the flow in the immediatevicinity of the edge.

For simplicity we consider only the leading edge of the biconvexairfoil, the extension to any sharp edge being evident. With the originshifted to the leading edge (Fig. 4.6) the airfoil is described by

Fig. 4.12. Conformal mapping for flow past wedge.

y = ± e(2s - S2) (4.47a)

TJ

.. t-IC=c+iTJI

.. I

y

~IZ=S+iYI

::::%~2t~~~S

The thin-airfoil expansion (4.24) was seen in Section 4.7 to fail within adistance s = 0(e- 1 £) of the leading edge. Hence appropriate magnifiedinner variables are

Y = yelle (4.48)

tion of uniform parallel flow. It can equally well be found by separationof variables in polar coordinates, which is a method that can be extendedto three-dimensional problems. Thus the inner solution for surface speedis found to be

in terms of which airfoil is described byq"""" ['is tan- l 2e/(;'T - tan-l2£) as e ->- 0 with S = 0(1) (4.49)

.\s e tends to zero the second term vanishes faster than any power of e.

Hence whereas a round edge is locally parabolic only to second order, asharp edge is approximated to an)' order by a wedge (Fig. 4.11). Thewedge angle is vanishingly small, our airfoil being approximated by

Fig. 4.11. Inner solution for sharp-edged airfoil.

s' = exp(log s') = exp(e log s) = 1 + e log s + te210g2 s +... (4.50)

Here, in contrast to the round nose, even the first term of the innerexpansion di\"crges at infinity. It gives unbounded velocities as S' -+ x,so that the factor ['j" cannot be interpreted as an equivalent free-streamspeed, but merely as the surface speed at S c-= 1. As for the round nose,Ui can be found by matching with the outer expansion of thin-airfoiltheory.

In order to match, we need the outer expansion of the inner solution(4.49). This requires forcing a small fractional power into a formalseries in powers of the exponent, and is accomplished using the rule that

which is not uniformly valid for small s. It fails where clog s = 0(1), sothat s = 0(r1jE

). This result explains the mysterious occurrence oflogarithms in the thin-airfoil expansion for a sharp edge.

In matching the two expansions we need not, as in the case of the roundnose, work up from one approximation to the next, but can at once treatany desired approximation. Let us consider the second approximation.

(4.47c)

(4.47b)

as e -+ 0 with S = 0(1)y.",,-, == 2eS

70 IV. Some Singular Perturbation Problems in Airfoil Theory 4.12. A Shifting Correction for Round Edges 71

Then the matching principle (4.36) is to be applied with m = n = 2,for which we find

(4.54)

(4.55)

z= x + (1'

.J. '-I., ze'1',' - Z'l"1 = 1-·· .",

. v2(z - ~e2)

4.12. A Shifting Correction for Round Edges

\Ve return to the singularity at a round edge, and describe an alter­native way of correcting it. This serves to introduce a second generalmethod of handling singular perturbation problems, which is discussedin detail in Chapter VI.

Consider the simplest round-nosed shape, the parabola. \Ve muste.\:amine the complete velocity field, because the surface speed suffersdistortion through transfer of the boundary conditions. For the parabolay =c-o :::.: f v2x of nose radius t;2 the first-order complex velocity can beextracted from that (4.12) for the ellipse as

1> ' .- ieP "c I - ~ - O(e2)J, ,!I v2z"

\Ve compare \vith the corresponding exact result \\'hich, by conformalmapping or separation of nriables in parabolic coordinates, is found to be

\re see that the first approximation (4.54) becomes exact if the origin ofcoordinates is simply shifted by !E2. This remO\'es the square-rootsingularity from the nrtex and places it inside the parabola at the focus,\\'hich is the singular point of the conformal mapping. Thus thin-airfoiltheory is seen to give the exact source distribution for a parabola, butdisplaced by half its nose radius.

This remarkable property must hold approximately for any round­nosed airfoil. For that reason :\lunk (1922) ad\'ocated modifying theabscissae to shift the leading edge of any profile by half its radius. It isclear that a simple translation is not correct for a finite airfoil, becausethat would leave the trailing edge misplaced. For the ellipse of Fig. 4.2,a uniform contraction of the x-coordinate e\'idently provides the requiredshift at both ends, as do an unlimited number of more complicatedstrainings. To retain the benefits of complex-variable theory, we straininstead the single nriable z = ,\ -;- (v. .

Lighthill (1949a) has proposed the follO\ving principle for findingstrained coordinates that restore uniform validity in a wide class ofsingular perturbation problems:(4.53)

(4.52d)

(4.52e)

(4.52c)

(4.52a)

(4.5Ie)

(4.5Ic)

(4.5Id)

(4.5Ia)

(4.52b)

." [' [( 2) I 2 ('I S" 2)]= eo"~ i 1 -:;; i:;;e og. -:;;

~= V[(1 -~) --:- ~e(2 + log S - log 2) -;- ... ], IT 1T

= U[I -'- §e(2 - log ~)]

= U[~I-§) --,--§e(2--,--logS-log2)]

2 [ e-l/eS ] I= Vi I --'--;/ 2 --:- (1 - e-l/eS) log2 _ e-l/eS ! (4.5Ib)

q ...., UiS tan-l 2< '(:r - tan- l 2el

2-term outerexpansion:

rC\\Ti ttcn ininner variables:

rewritten inouter variables:

expanded forsmal1 e:

rewritten inouter variables:

rewritten ininner variables:

2-term innerexpansIOn:

expanded forsmal1 e:

The rule (4.50) has been used in obtaining (4.52e). The additional laststep (4.5Ie) or (4.52e) is required because the comparison of the tworesults must be made in terms of either outer or inner variables alone.Then equating (4.51e) and (4.52d) yields (but see exercise 4.5)

2-term innerexpansion:

2-term outer expan- 2 [ s ] Iszon: q ,...., V: I +;/ 2 + (1 - s) log 2 -s \

SeeNote

4

We illustrate this principle by application to the thin-airfoil expansion

This completes the inner expansion (4.49) to second order. The readercan, using (4.24), find the next term in (4.53) by matching withm = 11 = 3.

Higher approximations shall beno more singular than the first. (4.56)

72 IV. Some Singular Perturbation Problems in Airfoil Theory Exercises 73

meter. This approximation is uniformly valid to first order, as can beverified by comparing with the exact solution:

for the ellipse. It is necessary to examine the third approximation,because the second is already no more singular than the first. For thisreason the straining is of order 102 rather than E.

Extending the first approximation (4.12) yields the thin-airfoil expan­sion as

. e (, ~-- zZ--<Do; -_. lrp:~ ,"CO I + -I-- I --- -"--1---;--:;)'

- e z .. - T c..,(4.62)

~ow suppose that, as in the case of the parabola, this result becomesuniformly valid if z is replaced by a slightly strained coordinate ~. Set

(4.57)

Because we have manipulated the solution, rather than introducingthe straining into the original problem, the above procedure is really amodification of Lighthill's technique suggested by Pritulo (1962); seeExercise 6.4.

:\pplying his technique to a general profile led Lighthill (1951) to therule (4.30b) for correcting thin-airfoil theory at round edges. However,in contrast to our previous derivation of that rule, Lighthill's methodis not successful for edges that are sharp, square, or in fact of any shapeother than round.

(4.58)

:\O\\' the third term \yill be no more singular than the second at , --= -+­(and also at ?; = 0) if we set

EXERCISES

4.1. Thill-airfoil theory usillg stream function. Reproduce the analysis ofSections 4.2 and 4.3 using the stream function instead of the velocity potential.Carry out the solution for the elliptic airfoil, and \'erify that indeterminatesource eigensolutions are automatically excluded.

4.2..Hatching and the principle of minimum singularit),. The local problemof symmetric A0\\' past an osculating parabola (Fig. 4.9) has eigensolutionsthat are singular at infinity. Show that the least singular of them adds to thesurface speed a multiple of [.5':3 (I 28)]12. This can be done, for example,by introducing parabolic coordinates g and r; and seeking a solution of thepotential problem as a polynomial in (g ,ir;). Then sho\\' that matching \\'iththe thin-airfoil solution for any round-nosed profile excludes this eigensolution,in accord \\-ith the principle of minimum singularity.

4.3. J-Iatching zfith local solution for square edge. Introduce mner variablesappropriate to the leading edge of the rectangular airfoil of Fig. 4.7. Show thatthe inner solution is that for f10\\- past a semi-infinite rectangular slab of thick­ness 2, on whose sides the surface speed is given in terms of a parameter tC\Iilne-Thompson 1960, Section 10.6) by

(4.59)

(4.60)

(4.6Ia)

Consider for simplicity the first of these possibilities, which providesthe uniform contraction of abscissae already proposed. Then the first­order solution becomes

where the straining function Z2 , which was -1 for the parabola, is to bedetermined. :\Iaking this substitution in (4.57) and expanding again inpowers of 10 yields

where

(4.6Ib)TT It,-- I

q = lJ i\) t=l \vhere 7TS = vtZ - l' - cosh-1 t

Although one can eliminate ~ here to obtain an explicit solution, that isnot possible with any other choice of Z2 . In general, one must be contentwith an implicit representation of the solution in terms of ~ as a para-

:\Iatch with the outer solution (4.27) to shO\v that the coefficient of the eigen­solution is C = J log(47T 1-'), so that the asymptotic sequence for the thin­airfoil expansion (4.2) contains logarithms beginning \\-ith E2 10g E.

74 IV. Some Singular Perturbation Problems in Airfoil Theory Exercises 75

where

Deduce the form of the function f by dimensional reasoning, using the factthat 1:'2 is the only characteristic length in the problem, thus completing thesolution except for an unknown multiple of an eigensolution. Interpret this asthe slender-body form of a particular solution of Laplace's equation. Find itsconstant by comparison \\·ith the exact solution. [See Chang (1961) for adiscussion of the role played here by the artijicial parameter E, and applicationof the method of matched asymptotic expansions. For the paraboloid in sub­sonic compressible flow, sec Van Dyke (1958c).]

4.8. :";mall-disturbance theory for paraboloid of re'wlutiO/l. Consider axi­symmetric incompressible flow past the paraboloid of rC\'olution r = V2t;2X •

Dcri\'e the counterpart of (4.4) for the first perturbation, but do not transferthe tangency condition to the axis. Solve by separating variables in paraboloidalcoordinates, showing that the disturbances are of order t;2 rather than E, and that

\'erify that this result becomes exact if the ongll1 is shifted downstream byhalf the nose radius.

Expand the above approximation formally for r small, of order E, to obtainthe slender-body solution, and verify that it satisfies the approximate equation$rr $,.1''' °of slender-body theory. Improve this result by iterating uponthe full Laplace equation to find the second-order slender-body solution as

4.9• .':;ubsollic thill-airfoil theory. Deduce the first- and second-order problems,corresponding to (4.4) and (4.5), for symmetric subsonic compressible flow,where the Laplace equation (4.la) is replaced by (2.19a). Show that a changein vertical scale (the Prandtl-Glauert transformation) reduces the first to anequivalent incompressible problem. Verify that a general particular integral(Section 3.7) for the second-order equation is given by (Van Dyke, 1952)

4Il = - cot--l e

7T

Compare with the exact result

4.4. Correctioll rule for sharp edges. Devise a multiplicative correction rulefor sharp edges, analogous to Lighthill's rule (4.30b) for round edges. Discussthe prospects for treating square edges similarly, and also sharp tips on slenderaxisymmetric bodies.

qU

4.5. Exact solutioll for bicoll'vex ailfoil. \Iilne-Thompson (1960, Section 6.51)gives the exact solution for symmetrical potential flow past a biconvex profileformed by two circular arcs. To \\·hat order can these be replaced by parabolicarcs in thin-airfoil theory? In the coordi nates of Fig. 4.6 the exact surface speed is

q 4 cosh 1) - cos 1l7T;2

U n2 I ~- cosh 21)/n

Extract from (his the outer and inner expansions, and compare with the resultsof Sections 4.7 and 4.11. Sho\\ that equating (4.5Id) and (4.52e) instead of(4.5le) and (4.52d) does not give (4.53). Explain the reason for the difference,and discuss its implications.

4.6. Block matchillg for e!liptic air(oil. She)\\' that instead of matching stepby step as in Section 4.10, one can at once match the 3-term outer expansion(4.32) \\·ith the 2-term inner expansion (4.35).

4.7 • .Yose-correction rules for bodies o( revolution. Devise a rule analogous to(4.30b) for round-nosed bodies of rC\olution in axisymmetric incompressibleflow. Cse the fact that the surface speed on a paraboloid of revolution is thesame as on a parabolic cylinder. Apply the rule to the ellipsoid of revolutionformed by rotating Fig. 4.2 about the x-axis, for which slender-body theory(Van Dyke, 1959) gives the surface speed as

q ."[ 2 I _ 1 ]U = 1--,-- e- log~ - 2(1 ~ l~ x 2 ) -,- O(eJ log2 e)

SeeNote

4

SeeNote

3

In the same way devise a rule for treating sharp-nosed bodies of revolution,and apply it to the spindle formed by rotating Fig. 4.6, for which slender-body

theory gives

q

U1 ! e2 [2(1 - 3x2 ) log 2 _ ., - 3-;.- Ilx2] + O(eJ log2 e)

evl - x-

Calculate the second-order solution for the wavy wall y ~ E sin x. Calculatethe surface pressure coefficient from

76 IV. Some Singular Perturbation Problems in Airfoil Theory

and check that it satisfies thc second-order similarity rule for airfoils (Hayes,1955):

If at

]v! = 0,

then for

J1(1 ;, 0,

Chapter V

THE METHODOF MATCHED ASYMPTOTIC

EXPANSIONS

5.1. Historical Introduction

\Ye introduced in the preceding chapter a method for treating singularperturbation problems that is a generalization of the boundary-layertheory of Prandtl (1905). This has in the past been called the method of"inner and outer expansions" or of "double asymptotic expansions."\\'e prefer to follow Bretherton (1962) in speaking of the method ofmatched asymptotic expansions.

The ideas underlying the method haye grown through the years. Itwas being used in the 1950's by Friedrichs (1953, p. 126; 1954, p. B- J84)and his students. It was systematically deyeloped and applied to yiscousflows at the California Institute of Technology. Kaplun (1954) introducedthe formal inner and outer limit processes for boundary-layer theory,and the corresponding inner and outer expansions. Later, in studyingflows at low Reynolds number, Kaplun and Lagerstrom (1957) made apenetrating analysis of the matching process (see also Lagerstrom,1957). Kaplun (1957) used those ideas to gain deeper insight into theresolution of the Stokes paradox for plane flow at low Reynolds number.Lagerstrom and Cole (1955) evaluated the method in comparison withnew exact solutions of the N avier-Stokes equations for a sliding andexpanding circular cylinder. Coles (1957) applied it to some specialsolutions for the compressible boundary layer.

Proudman and Pearson (1957) applied this expansion method totreat flow past a sphere and circular cylinder at low Reynolds number.Goldstein (1956, 1960) and Imai (I 957a) gave the first correct extensionof Blasius' boundary-layer solution for the semi-infinite flat plate. SeeTing (1959) solved the riddle of the course of the viscous shear layer Notebetween two streams of different speeds. 16

Following this developmental period, the method of matchedasymptotic expansions was applied to a variety of problems in fluid

77

78 V. The Method of Matched Asymptotic Expansions 5.2. Nonuniformity of Straightforward Expansion 79

.-\s indicated in Fig. 5.1, this is a good approximation except withinthe "boundary layer" \vhere .\' = 0(1:-)' Introduction of a magnifiedinner coordinate X appropriate to that region by setting

Ho\\-evcr, setting 1-: - °reduces the dift'erential equation to first ordcr, sothat both boundary conditions cannot be satisfied unless it happens that11-- I. The exact solution shows that the condition at .\' -~ 0 must bedropped. Then the approximate solution for small 1-- is

1 e J'U:

f(x; e) = (1 -- a)-- - ax1 -- e- 1:,

derivative in boundary-layer theory is given by Friedrichs (1942) as

(5.2)

(5.1 )

(5.3)

(5.4)

(5.5)

f(l) ~ I

F(..!..jE:

.\' = ~e

f(O) = 0,

F(O) = 0,

f(x; e) -- (1 - II) - ax

dl.- --=-- = adx '

f(x; e) = F(.\'; e),

d~F dFdX~ -- d.';;: = ae,

The exact solution is

transforms the original problem (5.1) to

mechanics. I\Iost of the earlier applications were to viscous flows. Forexample, Germain and Guiraud (1960, 1962) and Chow and Ting (1961)calculated the effect of curvature upon the structure of a shock wave.Murray (1961) and Ting (1960) found the effect of external vorticityupon the boundary layer near and far from the leading edge of a flatplate. Chang (1961) clarified the behavior of viscous flow far from afinite body. Flows at low Reynolds numbers have been analyzed forellipsoids by Breach (1961), for a spinning sphere by Rubinow andKeller (1961), for a circle in shear flow by Bretherton (1962), and so on.

The method is equally successful for inviscid flows. The precedingchapter gives examples in incompressible flow. Cole and :Ylessiter (1957)have studied transonic flow past slender bodies. Although the methodappears to be less popular in the Soviet Union, Bulakh (1961) has usedit to correct linearized supersonic conical flow and its higher approxima­tions in the vicinity of the bow shock wave. Similarly, Fraenkel andWatson (1964) have attacked the "pseudotransonic" flow past a triangularwing that occurs when the bow \vave lies close to the leading edge. Yakura(1962) has analyzed the entropy layer produced by slightly blunting thetip of a body in hypersonic flow.

Since 1960, applications of the method han proliferated in manyfields of fluid mechanics, as \\'ell as in other branches of applied mathe­matics. Some recent examples are discussed in later chapters of thisbook.

Fig. 5. I. ";olutio!l of model problem.

5.2. Nonuniformity of Straightforward Expansion

Before we discuss the details of the method of matched asymptoticexpansions, it is useful to inquire how singular perturbation problemsarise. What is the source of the nonuniformity ? Can we predict whethera given physical problem will lead to regular or singular perturbations?

The classical warning of singular behavior is familiar from Prandtl'sboundary-layer theory. A small parameter multiplies one of the highestderivatives in the differential equations. Then in a straightforwardperturbation scheme that derivative is lost in the first approximation sothat the order of the equations is reduced. One or more boundaryconditions must be abandoned, and the approximation breaks down nearwhere they were to be imposed. This happens except in the unlikelycircumstance that the original boundary conditions are consistent withthe reduced equations.

It is often helpful to cxamine linear ordinary differential equations asmathematical models that illustrate the essential features of morecomplicatcd problems. A simple model that illustrates loss of the highest

Nx;cJ

1-0

o

Outer...-:

// Exact/

,/

o 0.5x

, - - , Inner

80 V. The Method of Matched Asymptotic Expansions 5.3. A Physical Criterion for Uniformity 81

A perturbation solution is uniformly valid in the space andtime coordinates unless the perturbation quantity e is the (5.7)ratio of two lengths or two times.

5.3. A Physical Criterion for Uniformity

In physical problems a more general warning of singular behaviorcan be based upon dimensional reasoning. We ha\'C seen that inherentnonuniformity will be suppressed by exceptional boundary conditions,so that one can give only necessary and not sufficient conditions fornonuniformity. W~ therefore state the following rule instead as a positivetest for uniformity:

The warning pro\'ided by loss of a highest derinti\'C is more oftenthan not absent from a singular perturbation problem. In the thin-airfoiltheory of Chapter IY, the nonuniformity arises not from the differentialequation but from the boundary conditions. Likewise, for viscous flowat low Reynolds numbers the highest derinti\'Cs are all preserved in theapproximate equations of Stokes; the nonuniformity is associated ratherwith the infinite extent of the fluid. Evidently it would be useful to havea more reliable indication of nonuniformity.

v ~o~ JU /

ward outer variables. Nonuniform behavior is possible only if the para­meters in the problem provide another secondary reference length whoseratio to the first tends to zero or infinity as E vanishes. This second length,if properly chosen, is the basis for the inner variables.

.\ familiar problem involving two disparate lengths is potential flowpast a thin round-nosed airfoil. The geometry is characterized by thechord length except close to the leading edge, where it is dominated bythe nose radius. Since the ratio of these two lengths vanishes with thesquare of the thickness ratio, our criterion (5.7) suggests that the thin­airfoil solution could be singular. In Chapter I V this possibility wasseen to be realized, outer variables being based on the chord and innervariables on the nose radius.

Our criterion shows that a coordinate perturbation is never safe fromnonuniformity in the remaining coordinates. The latter may be madedimensionless using either the primary reference length or the pertur­bation coordinate; and because the ratio of these two lengths tends to zeroor infinity, they provide the scales for an inner and an outer expansion.

The following are examples of parameter perturbations that imoheonly one characteristic length scale, and are therefore necessarily regularaccording to our criterion. Slightly compressible flo\\' past a circle(Section 2.4) contains the radius as the only characteristic length, theperturbation parameter being formed from a ratio of speeds or energiesrather than lengths. The slightly distorted circle in potential flo\\'(Section 2.3) involves two lengths, but they arc of the same order ofmagnitude, their ratio approaching unity rather than zero or infinityin the limit I:: -~ O.

The follO\\'ing parameter perturbations involve t\\·o disparate lengths,and do as a consequence exhibit singular behavior (Fig. 5.2). A liftingwing is characterized by its chord and its span, and their ratio \,wishes

as e -~ 0 with.\' 0 fixed (5.6a)

as e _ 0 with X =:: fixed (5.6b)e

\ (I - a) - ax

f(x; e) -- {(I _ a)(l _ e-X)

If we now set I:: = 0, the solution of the differential equation thatsatisfies the inner boundary condition is any multiple of (1 - e-X

).

Imposing the outer boundary condition would give the multiplicativefactor as unity, but the exact solution shows that this is incorrect. Theouter boundary condition must be abandoned for the inner solutionjust as the inne'r condition was dropped from the outer solution. Instead,the inner solution must be matched to the outer solution using thematching principle (4.36). Thus one finds the uniform first approximationfor small E:

This criterion may be understood by considering first a parameterperturbation. The geometry of the problem will be characterized by atypical major dimension that we may call the primary reference length.Examples are the radius of the circular cylinder in Chapter II, and thechord length of the thin airfoil in Chapter IV. This length is the naturalbasis for forming dimensionless coordinates-a characteristic speed alsobeing required in unsteady flows-and these constitute the straightfc"'- Fig. 5.2. Singuiar perturbation problems inn)l\ing t\\"O disparate lengths.

82 V. The Method of Matched Asymptotic Expansions 5.4. The Role of Composite and Inner Expansions 83

9.9). Thus in conical geometry an angle plays the role of a length.Consequently, nonuniformity is possible in conical flo\\"s if the pertur­bation parameter is the ratio of two angles. Examples are a flat ellipticcone, where the straightforward perturbation solution e\"idently fails atthe leading edges just as in thin-airfoil theory, and a circular cone atinfinitesimal angle of attack, which exhibits near its surface the \"orticallayer of Ferri (1950).

Here the 8,,(E) are an appropriate asymptotic sequence, and x, y, z arethe coordinates made dimensionless using the primary reference length.This expansion is valid \\"herever the functions i" are regular. They willbecome singular at any point within the flow field where phenomenaare dominated by the secondary rather than the primary referencelength. This point lies at infinity if the secondary length is the larger..-\ modified expansion, in order to be uniformly valid, must depend alsoupon the coordinates made dimensionless by the secondary referencelength. Because the ratio of primary and secondary lengths is a function

5.4. The Role of Composite and Inner Expansions

\Ye ha\'e seen that a singular perturbation flow problem typicallyimoh'es t\\O disparate lengths, ,\s a result, the straightforward perturba­tion solution \\"ith coordinates referred to the primary reference lengthfails in regions where the secondary reference length is the rele\"antdimension. The secondary reference length is not always the ob\"iousone. It is clearly the chord for a wing of high aspect ratio, the thicknessfor a flat-nosed airfoil, and the viscous length l' C for flo\\' at loy\" Reynoldsnumber. Howe\"er, at high Reynolds number it is the square rootof the product of the \"iscous and geometric lengths. For a round-nosedairfoil it is not the thickness but the nose radius, \\"hich is propor­tional to the square of the thickness divided by the chord. For a thinairfoil in supersonic flow (Section 6.4) it is the mean radius of curvatureof the profile, which is proportional to the square of the chord divided bythe thickness. The sharp-nosed airfoil in subsonic flo\\" is a delicateborderline case between uniformity and nonuniformity in which theregion of non-uniformity, being exponentially small, is not directlyrelated to any physical dimension ..-\ similar obsen'ation applies to the\"orticallayer on an inclined cone in supersonic flow,

The straightforward perturbation solution yields an asymptoticexpansion of the form

(5.8)as E -> 0 \vith x, J',z fixed

in Prandtl's lifting-line theory (cf. Section 9.2) and becomes infinite inthe slender-\\"ing approximation. ,-\ body in \'iscous flow is characterizedby not only a geometric dimension, but also the \"iscous length l' [';

and their ratio is the Reynolds number, which \'anishes in Stokes flow(Chapter YIIl) and becomes infinite in boundary-layer theory (ChapterVII). This example illustrates that the secondary reference length is notalways a geometric dimension.

The follo\\"ing are some coordinate perturbations that are singular(Fig. 5.2). Viscous supersonic flm\" o\"er a cone or a wedge (including theflat plate) can be sohed approximately for distances from the vertexlarge compared \\ith the \iscous length l' ['. Flow of a gas undergoingvibrational or chemical reactions can be treated similarly using a charac­teristic relaxation length, as can the dleet of slight blunting. The impulsi\estart of a body through \'iscous fluid can be expanded in powers of time,referred to a characteristic length and speed. Viscous flow far from abody can be expanded in in\'erse powers of the radius, referred to atypical dimension.

~-\Ithough the follO\\'ing examples in\"ol\"e t\\"o disparate lengths, theyare ne\"ertheless regular perturbation problems. Potential flow past awa\"y \\'all, a cusp-nosed airfoil, or any thin shape free of stagnation pointsis a regular perturbation in the thickness parameter. C niform shearflow past a circle (Section 2.2) sh()\\'s the superficial symptoms of non­uniformity at infinity, in that the ratio of the perturbation to the basicsolution in (2.11) gro\\'s like fr; but the result is exact and thereforeuniformly \'alid. Howe\'er, any other shear distribution mlliid lead to asingular perturbation problem (Exercises 2.4 and 5.7). In an imiscidfluid the expansion for flo\\' far from a body is regular, as is that for animpulsi\"e start. E\'en the archetypical nonuniformity of Prandtl's bound­ary-layer theory disappears if in place of a fixed solid surface oneprescribes a distensible skin mo\'ing at just the speed of the potentialflow.

.\ problem may in\'oh'e more than two disparate lengths, associatedwith a multiple limit process; then multiple nonuniformities are possible.In the case of three layers one may speak of the ollter, middle, and innere.\pansions. An example is \'iscous flow at high Reynolds number andhigh or low Prandtl number, where the thermal boundary layer is muchthicker or thinner than the \'iscous layer. An inviscid exa~pl~ is analyzedin Section 9.9.

.\ coordinate perturbation may sometimes be replaced by a parameterperturbation. For example, the expansion for distances far from the tipof a blunted wedge (Fig. 5.2) may be regarded instead as the solutionfor a wedge of finite length whose nose thickness tends to zero (Section

84 V. The Method of Matched Asymptotic Expansions 5.5. Choice of Inner Variables 85

of E, this amounts to depending also upon f. Thus a uniformlv validexpansion must ha\e the more complicated form

. Be~ause the perturbation parameter f now appears implicitly in thefunctIOn gil as \\'ell as explicitly in the asymptotic sequence 8//, this isnot an asymptotic expansion in the usual sense, \\'e call it a compositeexpallsioll. Such expansions ha\"e been discussed in connection \vithsingular perturbation problems by ErdeIyi (1961), \\'ho calls them"generalized asymptotic expansions." There are t\\"C) objections toworking \\"ith composite npansions, First, they are difficult to manip­ulate; evidently such familiar operations as equating like pO\vers of E

must be reconsidered and, as \\"ill be seen later, a composite series is notuniquely determined, Second, they unnecessarily combine the compli­cations of both the straightfonvard e\:pansion and the region of non­uniformity (sce, ho\\"e\'er, Sections 10.3 and lOA).

It is simpler to isolate the difficulties associated \\"ith the nonuniformitvby constructing a supplementary inlier e.\pallsion \"alid in its vicinity:.This is accomplished by introducing ne\\' illller coordillates that are ~forder unity in the region of nonuniformity. Then the inner expansionhas the form

\\'e akays denote inner \"ariables by capital letters. Here the asymptotic~equel.lce J,,(E) must 1)(: allo\ved to diHer from the asymptotic sequenceil//(f) for the outer expansion; often they are identical, but Section 6.3gives an example in \\hich they arc different. If the region of non­uniformity is the neighborhood of a point in the finite pla~e, the innercoordinates X, 1', Z arc ordinarih' the coordinates made dimensionlessusing the secondary reference I~ngth. If the nonuniformitv occursalong a line, as in boundarv-Iaver theorv onlv the normal coo~dinate ischanged. If it occurs at i~finitv, the ~~ordi~lates must sometimes bestretched by diHerent function~ of E in different directions, Like theouter expansion, the inner expansion is a conventional asymptotic series,so that the usual operations are valid.

It is often a useful preliminary to introduce dependent as well asindependent variables that are of order unitv in the inner and outerregions, so that the leading terms in the as\:m!1totic seq"uences 8 and

• /I

J II are unity. The degree of stretching is in general different for theindependent and dependent variables. FollO\ving Kaplun (1954) and

as E '-~ 0 \vith X,} " X fixed (5.10)

5.5. Choice of Inner Variables

A crucial step in the method of matched asymptotic expansions is thechoice of inner Yariables. One faces the questions:

Lagerstrom and Cole (1955), we may formalize the procedure bydefining:

Outer 'Z'ariables: Dimensionless independent and dependent nriablesbased upon the primary reference quantities in the problem.

Outer limit: The limit as the perturbation parameter E tends to zero\\"ith the outer variables fixed.

Outer e.\pansion: The asymptotic expansion for E ----+ 0 with outer\"ariablcs fixed. Obtained in principle from the exact result bysuccessive application of the outer limit process in conjunction withan appropriate outer asymptotic sequence.

Inlier -('ariables: Dimensionless indcpendent and dependent variablesstretched by appropriate functions of f so as to be of order unit\'in the region of nonuniformity of the outer expansion.

inlier limit: The limit as E -~ 0 with inner nriables fixed.

illller e.\pallsion: The asymptotic expansion for f ----+ 0 \vith innervariables fixed. Obtained in principle from the exact result bysuccessive application of the inner limit process in conjunction\\'ith an appropriate inner asymptotic sequence.

Composite e.\pansion: Any series that reduces to the outer expansion\\"hen e\:panded asymptotically for E -~ 0 in outer variables, and tothe inner expansion in inner variables.

The technique of matching two complementary asymptotic expansionsreduces a singular perturbation problem to its simplest possible elements.If the first inner problem is ne\'ertheless found to be "impossible," thenone may suppose that the problem itself is intractable. For example,it is clear that extending the thin-airfoil theory of Chapter I \" to subsoniccompressible flo\\' leads, in the case of a round-nosed airfoil, to the innerproblem of subsonic flo\\' past a parabolic cylinder, for which no completesolution is known. Again, \'iscous flow past a cusp-nosed airfoil at highReynolds number leads to the inner problem of yiscous flow past asemi-infinite flat plate, for which only partial solutions exist. An ad­yantage of the technique is that even in these "impossible" situationsone can make use of numerical solutions or even of experimentalmeasurements for the inner solution. Thus in his lifting-line theoryPrandtl adYocates the use of experimental airfoil-section data for whatwill be seen in Section 9.2 to be the inner solution.

(5.9)uniformly as E-~ 0f(x,y. z; E) '" 1 i),,(E)g,,(X,y, z; E)

j(x, J', z; E) --1 J,,(E)F,,(.Y, }, X)

86 V. The Method of Matched Asymptotic Expansions 5.5. Choice of Inner Variables 87

and this is confirmed by examining cithcr the resulting inner equation

vorticallayer on an inclined cone (Munson, 1964). This is illustrated bythe following model problem:

(5.15)

(5.13)

(5.16)

(5.14a)

(5.14b)1//=

f(l) = Ix llt ddf

.- ef = 0,x

\ '/11 df .- f =. °• dX

ef(x; e) ,..... I .;- --- (x l - IIt

._ I) + O(ef,I _. m

This is singular at x ~ 0 for m ;;:: I and at x ~ x for m 0(; 1. It seemslikely that for m T 1 an appropriatc inner coordinate is

Iterating or substituting a series in powers of E yields the straightforward(outer) perturbation solution:

(a) Which independent variables should be stretched?(b) How should they be stretched?(c) How should the dependent variables be stretched?

Answering the first question depends upon recognizing the singularnature of the problem, including the location of the nonuniformity andits "shape"-that is, whether it is the neighborhood of a point, line, orsurface.

The degree of stretching required is usually evident when it is possibleto calculate several terms of the outer expansion. For example, theformal thin-airfoil solution for an ellipse was seen in Section 4.4 to beinvalid within a distance of order f:.2 from the leading edge, and theinner coordinates were magnified accordingly. Physical insight maysuggest or confirm the proper stretching as the scale of the secondaryreference length in the problem.

Otherwise, the stretching can be sought by trial. The guiding principlesare that the inner problem shall have the least possible degeneracy, thatit must include in the first approximation any essential elements omittedin the first outer solution, and that thc inner and outer solutions shallmatch..\s an example, consider the model problem (5.1). Trying anarbitrary stretching of the independent variable only, we set

or the exact solutionl(x; e) = F(.X; e),

The problem becomes

(5.11 ) e' exl - m .f(x; e) = exp( - -I--) exp(.-I--)'-- m - 111

=xE,

III 'I I

111 = I

(5.17a)

(5.17b)

Because the highest derivative \\'as lost in the outer limit, d2F dX2 mustnow be preserved in the inner limit. This means that the factor aCE) Emultiplying dF dX must not become infinite as f: --+ O. If it vanishes, thesolution satisfying the inner boundary condition (\vhich must also bepreserwd) is simply a multiple of X; but this cannot be matched \viththe outer solution (5.3). The remaining possibility is that aCe) E

approaches a constant; this also yields the least degeneratc differentialequation. Taking the constant as unity without loss of generality givesthe prcvious results (5.4) and (5.5). It was unnecessary to stretch thedependent \'ariable in this example because the first il~ner problem ishomogeneous; but in general one must admit separate stretching ofeach dependent as ,veil as each independent variable.

The inner variables are almost always, as in the preceding examples,formed by linear stretching. An exception arises in the problem of the (5.20)

(5.19)

(5.18)x == xel!e

xdt_c!=odX

However, this leaves the transformed differential equation

In the special case In -cc 1, (5.l4b) indicates that the region of non­uniformity near the origin is exponcntially small: x "-= G(e'li e

). Onemight suppose that, just as for the sharp-nosed airfoil of Section 4.7,an appropriate inner variable would therefore be given by the correspond­ing linear stretching:

unchanged, so that simple stretching is ineffective. Instead, the exactsolution (5.l7b) shows that the proper inner coordinate is given by thenonlinear distortion

(5.12)F(O) , 0,d~F aCe) dFd'y 2 - ~- dX

88 V. The Method of Matched Asymptotic Expansions 5.7. Matching Principles 89

SeeNote

5

5.7. Matching Principles

The existence of an overlap domain implies that the inner expansionof the outer expansion should, to appropriate orders, agree with theouter expansion of the inner expansion (Lagerstrom, 1957). Thisgeneral matching principle can be given various specific formulations.The literature shows that the choice of matching principle is somewhata matter of the investigator's taste.

Although the principle of minimum singularity often reduces thenumber of possibilities, it cannot always single out a unique flow pattern.For example, it rules out source eigensolutions in the linearized solutionfor a round-nosed airfoil, but not in the second approximation. This is asit should be, because a source eigensolution must actually occur in thesecond-order outer solution for a smooth profile that differs from anellipse only in the vicinity of the leading and trailing edges (Fig. 5.4).

:\latching is the crucial feature of themethod. The possibility of matching restson the existence of an o7.:erlap domainwhere both the inner and outer expansionsare nlid. By virtue of the overlap, onecan obtain exact relations between finite Fig. 5.4. .-\irfoil hm-ing samepartial sums. This remarkable achieve- linearized solution as ellipse.

ment is possible only for a parameterperturbation that is nonuniform in the coordinates, or for a coordinateperturbation that is nonuniform in the other coordinates. One cannotmatch two dift"erent parameter perturbations, such as expansions forlarge and small Yalues of Reynolds number or of :\lach number.:\either can one match two different coordinate expansions, such asfor small and large time or distance. Such series may o\'erlap in thesense that they have a common region of convergence, but the processof analytical continuation yields only approximate relations from anyfinite number of terms (cf. Section 10.9). .

:\latching may also be contrasted with what \ve shall call numericalpatching. This consists in joining two series by forcing their values andperhaps several derivatives to agree at an arbitrary intermediate boundary..-\lthough the result may be of practical utility----or e\-en numericallyindistinguishable from that of matching-·patching is estheticallydispleasing, and ordinarily no simpler. Also, matching is more systematicthan patching in higher approximations. Our view is that patching shouldbe avoided whenever it can be replaced by matching, which providesan imperceptibly smooth blending of the two solutions.

(5.21 )x df f= 0dX

which transforms the differential equation to

Fig. 5.3. Alternatiye symmetric flows past parabola.

Thus it appears that a fractional-power transformation is required whennonuniformity in an exponentially small region arises not from theboundary conditions but from a homogeneous operator of the formx a/ax in the differential equation.

5.6. The Role of Matching

The method of matched asymptotic expansions involves loss ofboundary conditions..-\n outer expansion cannot be expected to satisfyconditions that are imposed in the inner region; converselv, the inne'rexpansion \\-ill not in general satisfy distant conditions. Thus it is anexce~tional circumstance that the iJ~ner solution for the elliptic airfoil(SectlOn 4.10) happens to satisfy the upstream boundary condition­~he in~er solution for a sharp edge does not (Section 4.11). Henc~Insufficient boundary conditions are generallv aYailable for either theouter or inner expansion. The missing co~ditions are supplied bvmatching the t\VO expansions. .

F~r p.artial differenti.al ~quations a useful preliminary to matching isapplIc~tlOn of the pnnClple of minimum singularity (Section 4.5).Expenence has shO\vn that of the admissible solutions onlv the one thatis least singular in its region of nonuniformitv can be m~tched to thecomplementary expansion. for example, the i~ner solution for a round­nosed thin airfoil \\-as seen in Section 4.9 to be a symmetric flow past theosculating parabola. Figure 5.3 shows the first two of an unlimited

number of possibilities. All but the first give unbounded speeds atinfinity, and consequently cannot be matched with the thin-airfoilexpansion (Exercise 4.2).

90 V. The Method of Matched Asymptotic Expansions 5.8. Intermediate Matching 91

In matching his boundary-layer approximation to the outer inviscidflow, Prandtl tacitly applied what we may call the limit matching principle:

Here m and n are any two integers; in practice m is usually chosen aseither n or n -;- I.

This matching principle appears to suffice for any problem to whichthe method of matched asymptotic expansions can successfully beapplied. It will be used throughout this book. In the following section,however, we describe an alternative principle that provides deeperinsight into the nature of the overlap domain.

Whether this primitive rule is correct, or adequate, depends not onlyon the problem, but also on the choice of independent variables beingmatched. It is evidently valid for the tangential velocity in the boundarylayer, which must for large values of its argument approach the inviscidsurface speed. However, it is invalid for the normal velocity or streamfunction, where the first repeated limit in (5.22) is zero but the secondis infinite. The same difficulty arises in plane flow at low Reynoldsnumber (Chapter VIII).

We can improve this simple rule by describing more precisely thelimiting behavior of the quantity being matched (cf. Sections 3.2 and3.3). Instead of mere limits we use asymptotic representations. Thisgives the matching principle

SeeNote

5

(5.26)

(5.25)o <, (X <': (Xi

5.8. Intermediate Matching

In the outer limit process of thin-airfoil theory (Chapter I V) a pointremains a fixed distance from the leading edge as the thickness ratio f'

tends to zero, whereas in the inner limit pro~ess for the elliptic airfoilthe distance decreases like <:~. It is by

no means obvious that the two limit C ~processes can be interchanged,because there is a gap between theinner and outer regions. That a gapexists is clarified by consideri ng apoint \\-hose distance from the nose C ~

decreases only like t' (Fig. 5.5). This ~

point ultimately emerges from any - -vicinitv of the leading edge, and is atthe sa'me time excl~ded- from theregion of validity of the outer solution. c:::::::: ~

To bridge this gap, Kaplun (1957)has introduced the concept of a Fig. 5.5. Intermediate limit process

continuum of intermediate limits, for elliptic airfoil.

lying between the inner and outerlimits..-\lthough he considers a \'Cry general class of limits, it will sufficefor purposes of illustration to consider only those associated with powersof the small parameter. If s is the outer Yariable associated with a non­uniformity at s '-'--' 0, we introduce an intermediate 'C'ariable

Thus the intermediate problem is that of symmetric flow past a parabolaof nose radius <:"-x.

Here :x --"-' 0 gi\'es the outer and.x = 'Xi the inner limit; for example,':Xi = 2 for the elliptic airfoil. The limit as c -- 0 \vith.~ fixed is called theintermediate limit; and its repeated application in conjunction with anappropriate asymptotic sequence yields the intermediate e.\pansion.

Carrying out the intermediate limit in the differential equations andboundary conditions yields the intermediate problem. Although we haveintroduc~d very man~ limit processes, they lead to only a few differentproblems. All interm~diate limits yield essentially a single intermediateproblem, which is often the same as the inner problem. For example,setting .~ = s fX and .-v - y fX in Eq. (4.33a) for the elliptic airfoil andletting f ------>-- 0 gives

(5.22)

(5.23)

(5.24)

The inner limit of (the outer limit)= the outer limit of (the inner limit).

The nt-term inner expansion of (the n-term outer expansion)= the n-term outer expansion of (the 1Il-term inner expansion).

Inner representation of (outer representation)

= outer representation of (inner representation).

Here the outer (or inner) representation means the first nonzero term inthe asymptotic expansion in outer (or inner) variables. This rule providesmatching in cases ,,-here the limit principle (5.22) gives only a trivialresult. For example, we shall see in Section 8.7 that it suffices for planeflow at low Reynolds number.

The principle is extended to higher approximations by retainingfurther terms in the asymptotic expansions. \Ve must permit the numberof terms to be different in the inner and outer expansions, because thenormal matching order (Section 5.9) requires a difference of one in theeven-numbered steps. Thus we obtain the asymptotic matching principleintroduced in Chapter IV:

SeeNote

4

92 V. The Method of Matched Asymptotic Expansions 5.9. Matching Order 93

The intermediate solution is the solution of the intermediate problem.Its difference from the full solution must yanish uniformly in theintermediate limit. Thus for the elliptic airfoil the intermediatd solutionfor surface speed is, from (4.28),

(5.27)

admit the source eigensolution of (4.32'1). The difference between thetwo expansions is, in interrr.ediate yariables

(5.30)

and expanding gives

5.9. Matching Order

This yanishes to order E-that is, to second order in powers of E -ifCi co, C( I -~- f), C\ ~~ 0, and O_x I. The first t\VO of theseresults were found by asymptotic matching in Chapter IY. The thirdmeans that the outer (werlap domain has shrunk to half its preyiouswidth.

(5.31)

Innerexpansion

Outerexpansion

Numberof

term

1-:<,

D --- U(I -, e) -- U - c U(!-- -~- !-) -1- O(e1c >: e2->:)I , 'I . 2s ' 4. '

.-\11 our preyious discussion suggests complete symmetry between theinner and outer limits, so that the two terms could be interchangedthroughout. Howeyer, we haye heretofore used "outer" ah\-ays to den-otethe straightforward or basic approxi- .mation, and \\-e insist on adhering tothis conYention. :\Iorc precisely, weassign the terms so that the outersolution is, to first order, independentof the inner. The test is to considera first-order change in each, and seewhether the other is affected. Forexample, in thin-airfoil theory thefree stream is disturbed only slightlyby doubling the nose radius, whereasthe flow near the nose is drasticallyaltered by doubling the free-streamspeed.

In general, matching must proceed

step by step as indicated by the Fig. 5.6. :\Iatching order for inner

solid arrows in Fig. 5.6. The basic and outer expansions_

solution dominates the inner solution,which in turn exerts a secondary influence on the outer expansion, andso on. This order is inyiolable in the direct problem of boundary-layertheory, for example.

In some overlap domain the intermediate expansion of thedifference bet\\een the outer (or inner) expansion and the (5.29)intermediate expansion must vanish to the appropriate order.

The denominator cannot be expanded, because the result would not beuniform near the stagnation point. This example illustrates that theintermediate solution is not necessarily the intermediate limit of thefull solution-\\hich is here simply (, ---but may haye a more complexstructure.

.\lthough the gap bet\\een inner and outer limits has been bridgedby the intermediatc solution, it is not yet apparent that there exists anoyeriap domain. This is assured by Kaplun's extension theorem, whichasserts that the range of yalidity of the inner or outer limit extends atleast slightly into the intermediate range. \\'e forego the proof of thistheorem, \\-hose truth \\ill be eyident in specific examples. Thus we canmatch the intermediate expansion \\-ith the outer expansion at one endof the range and \\ith the inner expansion at the other end. Often theintermediate expansion is identical \\ith the inner expansion-as in ourexample of the elliptic airfoil or is contained in it as a special case.Then \\e can simply match the inner and outer expansions in the outero\'erlap domain.

:\Iatching requires that in the O\Trlap domain the difference betweenthe outer (or inner) and intermediate solutions Yanish in the intermediatelimit. Thus for the elliptic airfoil \\-l' match the intermediate solution(5.27) to the outer uniform stream by considering

This yanishes if C i ~ C, and .x < 2. Hence the outer o\'erlap domainis 0 :S; ex :S; 2.

\\'e may call the extension of this rule to higher approximations theintermediate matching principle:

For example, consider t\VO terms each of the intermediate and outerexpansions for the speed on an elliptic airfoil, \vhere in the latter \\'e

94 V. The Method of Matched Asymptotic Expansions 5.10. Construction of Composite Expansions 95

Outer

Fig. 5.7. First-order inner and outer solutions for speed on thin ellipse.

--------- - ---

Inner

,. - Composite (5.36)

\ li/ii) +f~n) [f~")]j"1l

f( nI, fi) -- (5.32). r I I~nl + filii,) -- [h(m)]~n)

notation, where .(,\11I1 means the m-term inner expansion, and so on, therule for additive composition is

o

q

U

One can verify this rule by taking the m-term inner and n-term outerexpansions of both sides. C sing the asymptotic matching principle(5.24) shows that the inner and outer expansions are reproduced in theirrespective regions.

\Vorking \vith differences, though conceptually somewhat different,yields the same rule. The outer expansion is made uniformly valid byadding to it the solution of the inner problem for the difference between

found by inspection. Otherwise, it may be calculated as the inner ex­pansion of the outer expansion, or vice versa. Thus, in an obvious

5.10. Construction of Composite Expansions

Representing the solution of a singular perturbation problem by aninner and an outer expansion may raise awkward practical questions ofwhere to shift from one to the other..-\ crude device would be to changewhere the t\\·o cunes cross, but the result would have spurious corners.:\Ioreover, for the elliptic airfoil, for example, the first-order inner andouter solutions for surface speed do not meet (Fig. 5.7).

Fortunately, since the two expansions have a common region ofvalidity, it is easy to construct from them a single uniformly validexpansion. The result is necessarily more complex than either of itsconstituents, and is in fact a composite expansion in the sense of Section5.4. The construction can in principle be carried out in a variety of ways.The results may be different, because a composite expansion is notunique; but they will all be equivalent to the order of accuracy retained.

Two essentially different methods have been used in practice. Thefirst may be called additi'i'e composition. The sum of the inner and outerexpansions is corrected by subtracting the part they have in common,so that it is not counted twice. The common part can sometimes be

One can sometimes short-circuit the standard matching order. Anobvious case is an initial-value problem, where all the boundaryconditions are imposed in the outer region. Then one can calculate anunlimited number of terms of the outer expansion, as indicated bydotted arrows in Fig. 5.6, and subsequently match with the innerexpansion to complete the solution. This situation can arise in fluidmechanics from inverse formulation of a problem, an example beinggiven in Section 9.9.

The same bypassing of the inner expansion occurs in a more subtlecase, when the nonuniformity is so weak that it does not affect the outerflow. An example is the biconvex airfoil of Sections 4.7 and 4.11. For around-nosed airfoil, on the other hand, the example of Fig. 5.4 showsthat only two terms can be calculated before one must resort to matchingwith the inner expansion.

When the standard order is followed, matching will indicate eachnew term in the asymptotic sequence, which therefore need not beguessed in advance. For example, rewriting any number of terms of thethin-airfoil expansion (4.14b) for the ellipse in inner variables andexpanding for small c sho\\'s at each stage that the next term in the innerexpansion is of the order of the next higher power of c. An examplewhere the asymptotic sequences arc different for the inner and outerexpansions is discussed in Section 6.3.

97Exercises

Here s = I --;- x or I - x according as we correct the nonuniformityat the leading or trailing edge; a truly uniform solution is obtained bytreating each edge in turn (Exercise 5.3). Other kinds of multiplenonuniformity (cf. Section 9.13) can likewise be handled by repeatedapplication of the rules for composition.

A composite expansion has at least the accuracy of each of its constit­uents. Thus (5.35) and (5.36) are in error by no more than 0(.::2) awayfrom the edge and O(e) near the edge. In fact the additive result (5.35) isevidently in error by precisely E at the stagnation point. The multipli­cative result (5.36) has the advantage of being exact there. Extending thecomposite result by using two terms of the inner as well as the outerexpansion leads again to (5.36) for either addition or multiplication.This means that by coincidence the error in (5.36) is actually no greaterthan 0(e2 ) everywhere. Figure 5.7 shows the improvement resultingfrom use of the composite expansion.

(5.33)«1f1.11), c

V. The Method of Matched Asymptotic Expansions

The asymptotic matching principle shows that these are equivalent tothe additive rule (5.32).

The second method may be called multiplicatiu composition. Theouter expansion is multiplied by a correction factor consisting of theratio of the inner expansion to its outer expansion, or the inner expansionis treated similarly. This gives

the exact solution and its outer expansion, or the inner expansion IS

corrected analogously. This may be written symbolically

! fill) -I- [f - fln)]IIII)\ 0 I 0 ~

I (",,) - [f -, fllll)]ln), • z't 0

96

f~lIf:lf1)

[f~")]:II1) = [f~III)]~71)(5.34) EXERCISES

SeeNote

6

The first form is recognized as pro\iding the multiplicative correctionfactor that \vas applied to round-nosed airfoils in Section 4.8. The lastform exhibits the inherent svmmetrv between the inner and outer limit. .processes.

The additive and multiplicati\'e rules of composition are related bythe fact that the ratio of two quantities near unity can be expanded intoa sum using the binomial theorem. The additive rule is usually simplerto apply; the multiplicative one sometimes gins simpler results. Eithercan be used even \vhen the inner problem cannot be soh'ed analytically,the solution being known only from numerical computation or experi­ment.

\Ve illustrate these t\VO methods for the surface speed on a thin ellipticairfoil. From two terms of the outer expansion (4.13) and one term ofthe inner expansion (4.46) we obtain as the uniform first-order perturba­tion solution, by additive composition (5.32)

5.1. Cmform approximatio/l for Friedrichs' model. Form in t\\'o different \\'a)'sa composite expansion from the solution (5.6). Discuss the difference, andcompare \\'ith the exact solution. Consider higher approximations.

5.2. Cmform approximatio/l for bico/l'l-'ex ai/jail. Construct an approximationfor the surface speed on a thin biconvex airfoil in incompressible flow that isuniformly \'alid to order E except at the trailing edge.

5.3. Composite rules for t'iCO /lOlllllllformities. De\'ise rules, analogous to (5.32)and (5.34), for constructing composite expansions in the case of t\\'O separatednon uniformities-as for a thin airfoil \\'ith stagnation leading and trailing edges.Apply your results to the surface speed on an elliptic airfoil, and compare \viththe exact solution.

5.4. Outer, middle, and inner e.\pansions. Show that a perturbation solution ofthe problem

(5.35)y(l) = 1 - .::

and by multiplicative composition (5.34)

q 2s., ,~(I : e) --U '\ 2.\ " e~

(5.36 )

for 0 :(: x :(: I requires thrcc matched expansions. Calculate in succession t\\'oterms of the straightfomard (outer) expansion, two terms of the middleexpansion, and onc term of the inner expansion. Choosc nc,,' magnified variablesas suggested by the preceding expansion, and match at each step.

98 V. The Method of Matched Asymptotic Expansions 15.5. iWadel with 1l01lvallishillg highest deri'vative. Calculate three terms of anapproximate solution in powers of E for

Deduce the location of any nonuniformity for 0 ::( x ::( 1, and the size of theregion involved. Calculate the leading term in the inner expansion, finding theconstant by matching. Calculate the second term, and form a uniformly validsecond approximation.

5.6. Impulsi've motioll of light mass 011 spring. A small mass hangs from aweightless spring with internal damping proportional to velocity. A verticalimpulse I is imparted to the mass by striking it with a hammer. Show thatappropriate choice of variables reduces the problem to

(1 + e).\:~ :i.: = e[(l - e).\y2 -- (1 -~ e)x + y3 + 2ey2], y(l) == 1

Chapter VI

THE METHOD

OF STRAINED COORDINATES

6.1. Historical Introduction

Show how a straightforward approximation for small e breaks down. Calculatean inner expansion, and a composite approximation. Discuss the applicabilityof the method when damping is absent, the canonical equation being eji - y = O.

5.7. Circle in parabolic shear. Complete the solution of Exercise 2.4 to order E

by introducing a supplementary expansion valid far from the circle. Constructa uniformly valid composite approximation.

SeeNote

3

eji --L Y --:.... y = 0, y(O) = 0, ey(O) = 1In the late 1940's Whitham and Lighthill were concerned at :\lan­

chester l'ni\ersity with problems involving the locations of bow shockwaves in supersonic flow. As an outgrowth of that work, Lighthill (1949a)described a general technique for removing nonuniformities fromperturbation solutions of nonlinear problems. The method has sub­sequently been applied to a variety of problems in fluid mechanics. Thusit constitutes an important alternative to the method of matchedasymptotic expansions.

The basic idea of Lighthill's technique is that the linearized solutionmay have the right form, but not quite at the right place. The remedy isto slightly strain the coordinates, by expanding one of them as well asthe dependent variables in asymptotic series. The first approximation­which becomes the same function of the strained variables as it was of theunstrained ones--is thereby rendered uniformly valid. One can proceedsimilarly to higher approximations. The straining of coordinates isinitially unknown, and must be determined term-by-term as the solutionprogresses.

The straining is determined by the principle already introduced inSection 4.12:

Higher approximations shall beno more singular than the first.

(6.1)

This halts the disastrous compounding of singularities that invalidates astraightforward perturbation expansion in a region of nonuniformity.Lighthill notes that sometimes a weak increase in singularity can actuallybe tolerated (Exercise 6.1), but it is simplest to adopt the above principle.

This principle does not by any means determine the straining uniquely;but the nonuniqueness can often be used to advantage. Because both

99

100 VI. The Method of Strained Coordinates 1 6.2. A Model Ordinary Differential Equation 101

.-\ straightforward perturbation expansion in powers of the parameterE yields the formal solution

and Waso\\" (1955) has proved convergence under mild restrictions on thefunctions q and r (with corrections by Lighthill, 1961). We make a simplechoice for these functions, and consider the model problem

This is a familiar situation arising from a singular perturbation problem.The series converges, but the radius of convergence vanishes as xapproaches zero; the expansion is not uniformly valid near x ~ O. Theexact solution is

(6.5)

(6.3)f(I) = 2df

(x -"- ef) -d -;- f = I,x

I'~~-I _L x-- x

f= (-) +2~-+4--'\ E e e

and this is finite at oX = O. The full equation is singular along the linex = - Ef (Fig" 6.1), but linearization transfers the singularity to x = 0;and in higher approximations it is spuriously intensified rather than beingcorrected.

The method of strained coordinates permits the singularity in thelinearized solution to shift toward its true position by straining the

6.2. A Model Ordinary Differential Equation

Because of our devotion to fluid mechanics, we have emphasizedapplication of the method of strained coordinates to partial differentialequations. However, it is useful also for many ordinary differentialequations. Indeed, Lighthill introduces his method by consideringequations of the form

(x - ef)dd'f -;- q(x)f = rex) (6.2)x

avoids the possibility of encountering an "impossible" nonlinear problemfor the first term of the inner expansion. On the other hand, the principleused for determining the straining of coordinates is so crude in compar­ison with the detailed description of the region of nonuniformity providedby an inner expansion that it is scarcely surprising that Lighthill'stechnique is not universally sucessful.

independent and dcpendent \"ariables are expanded, the solution isfound in implicit form, \vith the strained coordinate appearing as aparameter. This implicitness, far from being undesirable, is essential inmost problems. Often one requires only a uniform first approximation.Then the second-order equations need not be solved, but merely in­spected to determine the straining.

.-\n analogous straining of the independent variable was used byPoincare (1892) to obtain periodic solutions of nonlinear ordinarydifl'erential equations. For this reason Tsien (1956), in a survey article,has dubbed it the "PL~ method," the ~ standing for an application toviscous flows undertaken by Kuo (1953, 1956). We prefer to speak ofLighthill's tcchniquc, or of the method of strained coordinates, whichdescribes its cssential fcature.

The technique has proved altogether successful for treating hyperbolicdifferential equations-particularly in problems with waves travelingprimarily in one direction, \\"hich is the sort of problem for which it wasinnnted. Lighthill himself (1949b) applied it immediately to conicalshock waves in steady supersonic flow. Whitham applied it to the patternof shock \\"a\"cs on an axisymmetric projectile in steady supersonic flight(1952) and to the propagation of spherical shocks in stars (1953). Legras(1951, 1953) treated supersonic airfoils, and Rao (1956) sonic booms.

.-\n important generalization of the method has been advanced byLin (1954) for hyperbolic equations in two nriables. The strainedcoordinate of Lighthill may then be interpreted as one of the characteris­tics. Lin adopts characteristic parameters as the basis for a perturbationtheory, \vhich amounts to straining both families of characteristics. Thispermits treatment of \\"a\es traveling in both directions.

Because of thc flexibility of Lighthill's method, it represents aphilosophy of approach rather than a definite set of rules. For this reasonit is difficult c\"er to say that the method has failed. :'\evertheless, itappears that the method is inappropriate for elliptic equations, despiteLighthill's (1951) treatment of round-nosed airfoils in incompressibleflow which we outlined in Section 4.12. Likewise, despite Kuo's attempts,the method appears to be unsuitable for parabolic equations, where ithas even led to erroneous results (\Vu, 1956; Levey, 1959). Thus in alater survey bearing the same title as his original paper, Lighthill (1961)advises that his method be used only for hyperbolic partial differentialequations.

It is illuminating to compare the method of strained coordinates withthat of matched asymptotic expansions. It is a strength as well as a\veakness of the former that only one asymptotic expansion is used forthe dependent nriables. The analysis is simpler as a result, and one

SeeNote

7

coordinate x. We expand both x and 1 in powers of 8 with coefficientsthat are functions of a new auxiliary coordinate, say s. Since the strainingis slight, the leading term in the expansion for x may be taken as s itself.

102 VI. The Method of Strained Coordinates Ii

6.2. A Model Ordinary Differential Equation

Then the equation for 12 becomes

(ll), c= _ [.!...-~ 25 ~. x 2(1-] ,.~ 2 252' S

103

(6.9)

(6.11 )

(6.10)

and the solution is

although the constant could be replaced by any regular function of s.It is characteristic of the method that a further choice of straining is

open. The most obvious choice, X2 = -- (1 --'- 2s) 2s, gives as a uniformfirst approximation

1 -+- 5I(x; e) ""'-' -- + ... (6.12a)

5

1 -+- 25X ""'-'5 - e-2-~- +-... (6.12b)

However, this last solution is not required if we seek only a uniformlyvalid first approximation. The straining X 2 can be found simply byexamining the equation for 12 without solving it.

The straining is to be chosen in accord with the principle (6.1) that12 shall be no more singular than 11' Evidently the simplest way ofaccomplishing this is to annihilate the nonhomogeneous right-hand sideof the equation (6.9) for 12' setting

1 i 25 X2(5)--- T -- = canst

252 sExact

,\\\\ Linearized\\\

\\

""

I

IX=-U I" ,\.I\ .

\ .\ .\ .\', I,.\'"\.

--+'1 - --- - -- - --- ---. x

r.

x=-1Cf '.".

Fig. 6.1. Integral curves for (6.3).I n this simple problem it happens that the parameter s can be eliminated

Thus we set to give the e:-.:plicit first approximation

(6.6a)

where(6. 12c)

e

Substituting into Eq. (6.3) and equating like powers of 8 yields

Alternatively, it may be convenient to make the straining vanish atx = 1, where the boundary condition is imposed. Then we chooseX 2 = (3s2 - 1 - 2s) 2s, and the uniform first approximation is

and so on. The solution for 11 that satisfies the boundary condition is

(sI1)' = 1

(sI2)' = 11'(sx2 ' - X 2 ~ 11) = [x2(1 - 11) - iN]'(6.7a)

(6.7b)

1 --L SI(x; e) ""'-'--

S

352 - 1 - 2sx-s---!......e------

2s

(6.13a)

(6.13b)

(6.8) In this case eliminating s shows that the first approximation is the exactsolution (6.5), so that the series (6.13) terminate.

104 VI. The Method of Strained Coordinates 6.3. Comparison with Method of Matched Expansions 105

At X = 0, where the formal expansion (6.4) diverges, the exactsolution (6.5) or (6.13) gives 1(0; e) = (2/10 +- 4)1/2. The result (6.12)from the first choice of straining gives instead 1(0; c) "'-' (2e +- 1)1/2,which agrees with the exact solution to first order in e.

I t is clear that the boundary condition is an outer one, which shouldbe disregarded and replaced by matching with the outer expansion.We apply the asymptotic matching principle (5.24) with m = n = 1.Following the format introduced in Chapter IV, we have

6.3. Comparison with Method of Matched Expansions

It is natural to inquire as to the relative applicability and merits of themethod of strained coordinates and the method of matched asymptoticexpansions. \Ve therefore seek in this example an inner expansion tosupplement the straightforward outer expansion (6.4) in its region ofnonuniformity.

The outer expansion fails when x2 is as small as c, because then suc­cessive terms do not decrease in magnitude as was assumed. Thisindicates that the region of nonuniformity is where x = 0(e1 / 2). Therethe outer expansion suggests that 1 is of order C 1 / 2 . We thereforeintroduce inner variables X and F, which are of order unity in the regionof nonuniformity, by setting

I-term outer expansion:

rcwritten in inner variables:

expanded for small e:

I-term inncr expansion:

I-term inner e.\pansion:

re\\Titten in outer variables:

f"""'.!..± ,x; (6.19a)x

I= I --, e1 / 2X (6.19b)

I~/iX --, 1 (6.19c)

= el/2X =0 x (6.19d)

I ----f ....... el/2 [V.\'"2 --, 2CI - .\"] (6.20a)

= :::[./ 1 - 2CI~ - I] (6.20b)e 'V x-

x X--If'}'e .-

Rewritten in inner \'ariables, Eq. (6.3) becomes

(6.14)expanded for small e:

I-term outer expansion:

CI .=---.-0"

C1

(6.20c)

(6.20d)

dF(X - F) - +' F = e1 / 2

. dX (6.15)By equating (6.19d) and (6.20d), we obtain C1 = I. Hence the first-ordersolution by the method of matched asymptotic expansions is

As suggested by the term ,<:1,2, this is a case where the asymptoticsequences are essentially different for the inner and outer expansions.\Vhereas the outer expansion (6.4) proceeds in integral powers of e,the inner expansion proceeds by half powers, having the form

(~.--:::, x

f(x; e) ....... ~ ..... 1 • • -• ! X 2 2 X

\/ (-;-)- -;; - -;;

as e ---+ 0 with

as e ---+ 0 with

x fixed

x~ fixede ,-

(6.2Ia)

(6.21 b)

and so on. The general solution for F] is

(6.2Ic)

(6.21d)multiplicative

additive

Further terms in the inner expansion can be found by continuing theprocess.

A uniformly nlid composite approximation can be constructed byeither of the methods described in Section 5.10, which give

(6.16)

(6.18)

(6.17a)

(6.17b)

By substituting into the differential equation (6.15) we obtain

106 VI. The Method of Strained Coordinates 6.4. Nonuniformity in Supersonic Airfoil Theory 107

(6.23)

It appears that in this example the method of strained coordinates isby far the simpler of the two. We defer more detailed comparison toSection 6.8.

6.4. Nonuniformity tn Supersonic Airfoil Theory

A simple Aow problem in which the method of strained coordinatesproves efficacious is that of supersonic Aow past a thin airfoil. The classicallinearized solution of Ackeret is based upon the approximate equation

to first order. To see this it is necessary to actually calculate the secondapproximation. This means solving the nonlinear potential equation(2.19) by successive approximations, starting with .\ckeret's linearsolution. Setting ep o---c x -- ep in (2.19) and grouping linear terms on theleft-hand side puts it into the appropriate form:

B" _-}12 [Y - I (2 2' ")( )'Pyy'- cpJ'J: - " -2-- 'PJ' -- cp", -..- epy 'PJ:X + 'Pyy

--;- (2'{J;,: --,- 'P})'P",;, I- 2(1 --I- cpJ,)'Py'{J;,:y -I- 'P/'PYY]

Fig. 6.2. Flow pattern above supersonic airfoil. (a) Linearized. (b) i\onlinear.

Here'P is the perturbation velocity potential; we take it to be normalizedsuch that the velocity vector is q = C' grad(x + 'P)' Although Ackeret'ssolution is a proper first approximation at and near the surface, it failsat great distances from the airfoil. It predicts disturbances propagatingundiminished along the free-stream :\Iach lines to infinity (Fig. 6.2),

(6.25a)

(6.25b)

(6.25c)

'P,,(x, 0) = eT'(x)

cp(x, y) = ° upstream

Let the upper surface of the airfoil be described as before (Fig. 4.1)by Y --" f- T(x). Then the formal second approximation for the streamwise\'elocitv component above the airfoil is found, by iterating upon (6.23),to be (Van Dyke, 1952)

!!...... = 1 - e !]J1_ e2[J.-(1 _ Y +J. M4.)T'2(C)C B B2 . 4 B" ~

_ Y ; 1 ~~4 yT'(g)T"(g) _ T(g)T"W] --'- O(e3 ) (6.24)

where g x - By. At the surface this reproduces the well-known,second-order solution of Busemann. Far from the airfoil however, thesecond term is not small compared with the first-as ""'as assumed inthe iteration process-because it grows linearly with y along any free­stream :\Iach line :\: - By = constant. Evidently the expansion is invalidin the distant region where y = O(c1

).

The troublesome term in y is proportional to (y - I). This is anindication of the fact that the nonuniformity arises only from the"pseudo-transonic" term (y + 1Vl-14ep.rep ,r,t' that is inherent in the non­linear right-hand side of the differential equation (6.23). That term canbe exhibited explicitly by transforming to the oblique coordinatesintroduced later (sec also Hayes, 1954). The contribution of all othernonlinear terms is uniformlv of second order. Although the pseudo­transonic term is small, it has a first-order cumulative effect. It musttherefore be retained in addition to the linear terms in seeking a uniformlyvalid first approximation. To that order the tangency condition may beimposed on the axis, as discussed in Sections 3.8 and 4.2. Thus theproblem for the first approximation is

(6.22)

(b)

B2 == M2 - I'Pyy - B"'Pxx = 0,

(0 )

whereas in reality the :\Iach lines are not straight or parallel, and shockwaves are formed and decay.

This nonuniformity at large distances is somewhat more delicatethan that arising, for example, in Stokes' approximation for viscousAow at low Reynolds numbers (Chapter VIII). There one can use thefirst approximation to estimate the magnitude of the terms neglected inthe equation of motion, showing that they ultimately dominate thoseretained. However, in the preseat problem the nonlinear terms areactually small everywhere compared with the linear ones; it is only theircontribution that becomes dominant. They have what Hayes (1954) callsa cumulative effect, meaning that over a long stretch their inAuence grows

108 VI. The Method of Strained Coordinates 1 6.5. First Approximation by Strained Coordinates 109

We are now in the fortunate position of having a \yell-posed problemfor the dimensionless streamwise velocity perturbation u', given by

(6.29)

(6.28c)

(6.28a)

(6.28b)

upstream'P=O

The linearized solution is rp = - eT(~)/B, so that rp., = ({'"" '"'" O.Although this is not true in the nonlinear solution, because the wavesdepart slightly from the free-stream :Vlach lines, Hayes points out thatthe terms f(-'I and (P",/ appearing here represent truly second-order effects,like others already neglected. Hence they may be discarded, and theproblem for a uniform first approximation becomes simply

y -1 M4'PO'I -I- 2- 132 'P;'P;; = °

T'(t)'PM, 0) = -e B -

, Lluu == U =r:px = 'P;

\Ve prefer to work with this, because it has more physical significancethan the velocity potential (p. Thus the problem to be soh'ed becomesfinally, with the prime deleted henceforth from II',

It is comenient to make a preliminary transformation of coordinates.There are t\H) distinguished directions in the problem: that of the freestream, and that of the outgoing free-stream Mach lines either above orbelow the airfoil (Fig. 6.3). The airfoil lies nearly along the first, and the

Fig. 6.3. \\'a\l~ patterns in supersonic flow.

wave pattern nearly along the second. The other family of :\lach linesplays only a secondary role. This is true if we restrict consideration to,say, the upper surface of a single airfoil; for multiple bodies both familiesof l\1ach waves are of primary importance, and intersect, forming aScotch-plaid pattern. For this reason the second-order solution (6.24)can be expressed more naturally in terms of the normalized distances

u=O upstream

(6.30a)

(6.30b)

(6.30c)

(6.31a)

(6.31b)

u(t, Y); e) r--- w1(s, t) -'- e2uz(s, t) ., ...

t ~ s -+- etis, t) --:- ... , 'I) = t

The transformation of derivatives is found from

6.5. First Approximation by Strained Coordinates

We apply to this problem the method of strained coordinates. It isclear physically (cf. Fig. 6.2) as well as mathematically that the coordinate~ describing the :Uach lines is to be strained. \Ve therefore introduce aslightly strained variable s in place of ~, and set

(6.26a)(6.26b)

(6.27a)

at 'I) = 0 (6.27b)

(6.27c)

t = x - By'I) = By

perpendicular to the two distinguished directions. \Ve therefore adopthenceforth, following Hayes (1954), these "semicharacteristic" coordi­nates. If both families of waves were present, one would use instead thecharacteristic coordinates x - By and x -I- By, and in the next sectionLighthill's technique would have to be replaced by the generalizationof Lin (1954) mentioned previously.

Rewritten in these oblique coordinates, the problem (6.25) becomes

y -- 1 iVIJ ]

'P';" --:- 2- B2 'P,;'P,;,; =2'P'i~

T'(t)'P; = - e B- ... 'P,}

'P == 0 upstream

llO VI. The Method of Strained Coordinates 6.5. First Approximation by Strained Coordinates III

By substituting the expansions (6.31) into the problem (6.30) for u andequating like powers of f, we find

order velocities. For it can be shown (Van Dyke, 1952) that the slopeof the outgoing family of characteristics is

U lt = 0,T(s)

ul(s,O) = - B' upstream (6.33a) dy .= ~Il _ y -I-~ M"":" Lll~ + ... )dx B\ 2 B2 U (6.38)

so that

(6.39)

1. 'I

ReVised 1characteristic, 1

I, Free -stream1 characteristic

1

y -I- 1 2114 LluX - By = -2-- -B- U Y +- const

and inverting and integrating gives

Fig. 6.4. RCI'ision of :\Iuch lines.

which corresponds to (6.37h), the constant being s.Thus the solution obtained by straining coordinates gives velocity

components that are constant along the revised :\Iach lines, and havethe values gi\'en by linearized theory on the airfoil. This is in contrastto linearized theory, according to which the velocity at any point equalsthat at the foot of the free-stream :\Iach line passing through it (Fig.6.4). Formal second-order theory (6.24) attempts to improve the estimate

by relating values at the foot of the revised :\laeh line to those at the footof the free-stream :\lach line by Taylor series expansion; of course thisbreaks down as y -)- x. The uniformly nlid first approximation obtainedby straining coordinates is thus seen to consist of a simple \\ave (Prandtl­:\Ieyer expansion fan), and may be compared with the solution obtainedon that basis by Friedrichs (1948).

\Vhitham (1952) has taken the abon physical interpretation as thehasis for a more heuristic derivation of the uniform first approximation.He introduces "the fundamental hypothesis that linearized theory givesa valid first approximation to the £10\\' eTerywhere provided that in it theapproximate characteristics are replaced by the exact ones, or at leastby a sufficiently good approximation to the exact ones." The revision(6.39) gives in fact a sufficiently good approximation to the characteristics,

(6.36)

(6.35)

(6.34)

(6.37a)

(6.33b)

(6.37b)

, V T'( )v~U,""e s-l-'"

T'(s)ul(s, t) = - B

y -I- 1 NFl T'( )x-By,-vs-----ey s -'". 2 B2 .

, _ Llu T'(s) ,u =CU~-eB--:-""

Clearly "l is the same function of the strained variable s that it wasin linearized theory of the unstrained variable g:

and

The straining function g2 is now to be chosen according to the principle(6.1) that 11 2 be no more singular than 111 as By = 7) = t becomes infinite.The simplest way of assuring this is again to annihilate the right-handside of the equation (6.33b) for U2 by setting

y -;- I .l{J _ y -i-J. M4 T'(s)g2f = 2- B2 Ul = 2 B3

where the parametric variable s is given implicitly by

This solution has a simple physical interpretation. The lines s =constant are actually the revised Ylach lines, calculated using the first-

For the present it suffices to make the straining vanish at the axisy = 0 hy taking the arbitrary function f(s) = O. It is easily seen fromcontinuitv that to first order the vertical velocity is also the same functionof s that it was of t in linearized theory. Hence, with the original Cartesiancoordinates restored, a uniformly valid first approximation for the twovelocity components is given by

112 VI. The Method of Strained Coordinates 6.6. Modifications for Corners and Shock Waves 113

Fig. 6.5. Simple smooth com'ex wall.

y

tI I II ry. - ,c,

TITITITITITITITITITITITIT"'ITlrl"':I:"'I-=I~.X

(6.42)fY T 1 J14 ,

. (s) c= -27j2' eT(s)T (s) - BT(s)

Fig. 6.6. Region of invalidity behind corner.

treated in their turn to obtain a truly uniform solution. This is the casein the present problem if the airfoil involves corners or shock waves.Although these details arc not essential to our main exposition, wedigress for the sake of completeness to indicate the additional modifica­tions that are required.

The preceding result is uniformly valid for a convex wall whoseslope is continuous and slowly changing (1''' of order unity). However,it is clear physically (Fig. 6.6) that the result is invalid even on the surface

for a distance of order e downstream of a corner, unless the corner lieson the axis y = 0, because the solution singles out the surface slopeahead of rather than behind the corner. The same is true of any regionwhere the cunature is so great (1''' of order c 1) that the slope changesby an appreciable fraction of its total variation over a distance of theorder of the thickness of the airfoil.

This difficulty arises from having imposed the tangency condition onthe axis rather than precisely on the surface, and correspondingly havingmade the straining vanish on the axis. It can be remedied si mply bychoosing the function of integration f(s) in (6.36) such that the strainingvanishes precisely at the surface, so that s = x on the airfoil. This gives

(6.41 )

(6.40b)

x - BylOS = e -------~-

y T 1 !VI41 --,- 10 -2-- lizY

vc

x - Bys = ---------

y -1 M4I ..- 10 ---- --- V, 2 B2-

B Ll~ =U

Then (6.37a) gives the velocity components at any point

which can be inverted to give

and then Whitham's hypothesis leads at once to the solution (6.37) thatwe have derived more formally by the method of strained coordinates.Whitham is more concerned with applying his method to bodies ofrevolution; and his result is the basis for calculating sonic booms.

The implicit nature of the straining is evidently essential to the uniformvalidity of the solution. In sufficiently simple cases, however, thedependence can be inverted to give an explicit solution. For example,consider the smooth convex wall shown in Fig. 6.5, with T(x) ~X2

for x > 0. Equation (6.37b) becomes

y + I M4X - By = s(1 + 10 -2- -B2 Y) (6.40a)

This is equivalent to the result of Egs. (92) and (93) of Whitham (I952),though somc\vhat simpler.

(6.43)x - By '.= s - eBT(s) - Li-~ '~21 [y - eT(s)]eT'(s)

Thus a solution that is uniformly valid on the surface even with rapidchanges in cunature, or corners, is ginn by (6.37a) \vith (6.37b) replacedby

In this example one sees clearly the source of the nonuniformity in theformal thin-airfoil expansion. For any given y the solution can be ex­panded in powers of E, but the result is not uniformly valid near y == w.

6.6. Modifications for Corners and Shock Waves

It may happen that removal of a glaring nonuniformity from a pertur­bation problem brings to light additional difficulties, which must be

Fig. 6.7. Corner as limit of smooth wall.

115

(6.46)

(6.45)y -1 }1.1

.\' -, B)' = Xl' - By -- -...:.- -'- e(v -- y )T'c 2 B2. c

6.6. Modifications for Corners and Shock Waves

Then substituting into (6.37a) gives the velocity components in the fanas

so that

This difficulty is easily remedied by temporarily smoothing the cornerslightly, so that the revised characteristic through e\'ery point againstrikes the surface at a point with a definite slope, as indicated in Fig.6.8. Clearly 5 is approximately x,. e\erywhere in the fan, if the cornerlies at (x,., y,.). lIence 1'(5) is constant in the fan with value T(x,), whereas1" \'aries from 1"(x,.-) to ]"(x,. :'). The value of 1" associated with anypoint (x, y) in the fan is found from (6.43), which gives

1I

VI. The Method of Strained Coordinates114

The solution is now uniformly valid for any smooth convex wall,though shock waves have yet to be inserted for a concave wall. This istrue no matter how rapidly the curvature changes. However, in the limitas the curvature becomes discontinuous (Fig. 6.7), tracing characteristics

However, s is undefined for 0 ../ X - By <: [(y ..L 1) 2J(M4 B2) ty,

which is just the region of the Prandtl-:vIeyer fan.

shows that the solution is undefined in the Prandtl-:vIeyer fan from thecorner. For example, taking T(x) ~~ - xH(x), where H is the Heavisideunit step function, gives the com-ex corner shown in Fig. 6.7. Substitutinginto (6.37b) or (6.43) and soh'ing for s gives, to first order in e

This holds only in the fan, where the \'alue of (6.46) lies bet\\'eenT(.\,.-) and 1"(x,.-).

Finally, if the wall is concan the revised characteristics converge, andthey overlap at sufficiently great distances---immediatcly for a concavecorner-so that the solution is not unique. For example, changing thesign of the deflection in Fig. 6.7 leads to two values of s at each point inthe fan. This multivaluedness must be eliminated by introducing ashock \nne to cut off the :\lach lines before they can o\'erlap (Fig. 6.9).The shock wave is inserted according to the well-kno\\'n principle that to

s=

\ x - By

y -!... 1 JJ~

I. .\' - By - 2- B2 ey

for x - By < 0

, y -;- 1 lVI~for x - By /'> 2-7ey

(6.44)

B~~ =co~. (x - By) - (.\'" - BYe)

.- - ,~ e --,---------[' y _.., 1 ,11,1

--,- -- e(\' - \. )2 B2 . . c

(6.47)

( x,y)

y

tP

I

rfI II

.,i ....~L : ------x

Xc

Shockwave

Fig. 6.8. Smoothing of convex corner. Fig. 6.9. Treatment of overlapping characteristics,

116 VI. The Method of Strained Coordinates 6.7. First Approximation by Matched Expansions 117

first order a shock \yaye bisects the :\Iach directions ahead and behind.Whitham (1952) gi\'es an ingenious graphical construction of the shockpattern for a:\isymmetric Amy based on this principle, which can beadapted to plane flow.

leads without difficulty to the straightforward expansion

(6.49)

(6.50)

(6.51 )

This is an approximate form of the full second-order result (6.24). Inaccord with our convention, discussed in Section 5.9, we call this theouter expansion even though its region of nonuniformity is the vicinityof the point at infinity.

The outer expansion is invalid where r) = O(c I), and in that region11 is of order E. This suggests introducing inner variables such that theinner expansion is

This is a nonlinear partial differential equation. However, it happensthat it can be linearized by interchanging the roles of the independentand dependent variables, and seeking a solution for g( t'l , H). Theequation becomes

Substituting into the differential equation (6.30a) gives for the firstapproximation

Fig. 6.10. Shock pattern for double.wedge airfoil. (6.52)

(6.53a)

(6.53b)

and its general solution is

1:( T' H) (T' ) .-C- Y -,~ ~!~ Co' HSUI' = g 1./ 1 2 B2 1

The function of integration g is found by matching with the outer solution(6.49). It is clear, despite the implicit form of the inner solution, thatthis gives

For simple shapes the resulting form of the shock wans can be reasonedout. For e:\ample, for a double-\yedge profile the bow and stern shockw~ves must, beyond their straight sections, be portions of a parabolaWith the corner as focus and a free-stream :\Iach line as axis, in view ofthe focussing property of a parabolic mirror (Fig. 6.10)..:\Jore generally,~t such remote distances that the airfoil appears simply as a point, this~s tr~e for. any profile. Hence the shock pattern grows asymptotically11l WIdth hke yl/2.

6.7. First Approximation by Matched Expansions

It is of interest again to solve the problem (6.30) by the alternativemethod of matched asymptotic expansions. Substituting the assumedseries

(6.48)

where T'-l is the function inverse to T'. For example, in the case shownin Fig. 6.5 \ve have T'(x) = -x, so that T'-1(-BU1 ) = BU1 , and thensolving (6.53a) for [/1 yields the result (6.41) obtained previously by themethod of strained coordinates.

This inner solution is uniformly valid, because the inner equation(6.51) is in fact the full equation (6.30a). In this simple example nosimplification is introduced by the inner expansion, which accordinglyterminates at one term.

118 VI. The Method of Strained Coordinates Exercises 119

f(l) = I

EXERCISES

showing that a uniformly nlid first approximation has the form

f'--" aeb / s, x ~ s - cc (1 + ds + gs2)ebl8

with appropriate values of the constants a, b, c, d, g. Calculate the value of f(O) ,

6.3. Delayed straining. Show that in application of thc method of strainedcoordinates to the problem

SeeNote

7

SeeNote

3f(l) = e(x 2 + ef) ddf - f = 0,x

arising from an elliptic equation is the logarithmic singularity of incom­pressible thin-airfoil theory at a sharp edge. This was seen in Section 4.11to correspond in reality to a small fractional power. Hence straining oft'oordinates fails, whereas the method of matched asymptotic expansions\\'as successful.

The round-nosed airfoil in incompressible flo\\' is an exception,because the square-root singularity of thin-airfoil theory exists, beingmerel\' shifted from the leading edge to approximately the focus of theoscul~ting parabola (Section 4.12). For this reason Lighthill (1951) wasable to use his technique to find a uniform second approximation forround-nosed airfoils. That this is an exceptional case, hO\yeyer, is indi­CIted by the fact that all attempts haw failed to extend his solution tohigher ~rder, to subsonic compressible flow, or to other nose shapes..\11 these problems have been successfully treated by the method ofmatched asymptotic expansions (Yan Dyke, 1954).

This situation has forced Lighthill (1961) to recommend that histechnique be restricted to hyperbolic partial differential equations. Thisis an unfortunate limitation on a powerful method. It is to be hoped thatfurther stud\· will not only clarify the conditions under \\"hich the existingmethod of ~trained coordinates' applies, but also disclose a refinementthat can be applied with confidence to other problems, particularlythose in\"oh'ing elliptic and parabolic partial differential equations.

6.1. Weaker straining principle. Sho\\' that a uniformly valid solution of theproblem (6.3) on page 101 is obtained by the method of strained coordinatesif the straining principle (6.1) is relaxed to require only that the secondapproximation shall be no more singular than the square of the first.

6.2. A problem of Carrier (1954). Apply the method of strained coordinatesto the problem

6.8. Utility of the Method of Strained Coordinates

'fhe particular merits of the method of strained coordinates areemphasized by contrasting it \yith the alternative method of matchedasymptotic expansions. It appears that, as in the foregoing two examples,the latter method is applicable \yhene\'er the former is. The converse isunfortunately not true. Thus the method of matched expansions is themore reliable and perhaps the more fundamental of the two.

I [()\\t'\"lT, in cases \\here the method of strained coordinates applies,

it is often strikingly simple. This is well illustrated by the precedingexarnples, \yhich demonstrate t\\·o advantages of Lighthill's technique:

(i) It requires soh'ing only the straightforward (outer) perturbationequations, \\'hich are linear.

(ii) It giyes directly a single uniformly nlid expansion.

The second of these is less important, because we haw seen in Chapter Ythat an inner and an outer expansion can easily be cOlnbined in seyeralways to form a single uniformly yalid composit~ expansion. On the otherhand, the first is often a substantial ad\"<lntage, because in nonlinearproblems the equation for the leading term of the inner expansion maybe difficult or e\'en impossible to soh·e. Thus in both our examples thefirst inner problem is nonlinear, \\'hereas the method of strained coordi­nates inyohes only linear equations. It is the glory of Lighthill's tech­nique that it attains its goal \\'hile a\'oiding any detailed examination ofthe region of non uniformity.

For the same reason, hO\yeyer, it is not always successful. \Vhat isworse, it may appear to succeed "'hile gi\'ing an incorrect result. Thecrucial question is, therefore, \\'hen is it safe to apply the method ofstrained coordinates? .-\t present, no general rules can be set forth,but only some indications.

One definite rule is that it is useless to strain the coordinates in aperturbation problem that is singular because the highest deri\'ati\'e ismultiplied by a small parameter. Thus Lighthill's technique cannot beused to treat Prandtl's boundary layer. .·\ttempting to do so usually leadsto a null result, as can be seen by considering the model probiem ofSection 5,2, Cnder special circumstances, howe\'er, it leads to a definitebut erroneous result. This situation has been analyzed by Leyey (1959),who proyides a mathematical model illustrating the essential difficulty.

The method of strained coordinates succeeds when the singularitypredicted by the straightforward first approximation actually exists, b~t

at a slightly different location. It fails when the actual singularity is of adifferent type, or nonexistent. The former situation appears to arise withhyperbolic equations, the latter usually with elliptic ones. An example

SeeNote

7

120 VI. The Method of Strained Coordinates

SeeNote

3

the first straining is determined only by the third-order problem. Comparethe solution \\·ith that obtained by the method of matched asymptotic expansions.

6.4. Pritulo's method. Pritulo (1962) points out that if one has calculated astraightforward perturbation solution and found it to be nonuniform, thestraining of coordinates required to render it uniformly valid can be founddirectly therefrom by solving only algebraic rather than differential equations.Illustrate this by substituting (6.6b) into (6.4) and determining the strainingto order £2.

6.5. Cylindrical zCa'/.Je propagation. Consider the nonlinear model problem

Chapter VII

VISCOUS FLOW

AT HIGH REYNOLDS NUMBER

Find the region of nonuniformity of the straightforward expansion for small£ and y :): I. Show that the method of strained coordinates produces the exactsolution.

(''l' . ~', i:.-'Z;.,,--- -:- - -, t·.,,--- =, 0,()' y ex

v(x, I) == ef(x) 7.1. Introduction

We turn now to the prototype of singular perturbation problems, thatof viscous flow past a body at high Reynolds number. Prandtl's boundary­layer theory was invented to treat this problem. It was realized for sometime thereafter that boundary-layer theory provides the leading term inan asymptotic expansion for high Reynolds number. However, attemptsto calculate further terms in the expansion led at first to considerableconfusion, contronrsy, and error. The problem became straightforwardwith systematic application of the method of matched asymptoticexpansions. Here, using that method, we show how Prandtl's theorycan be embedded as the first step in a scheme of successi\'e approxima­tions. The practical utility of such refinement is suggested by an estimateof Lagerstrom and Cole (1955) that the second approximation maypredict the skin friction accurately down to Reynolds numbers of 10 oreven 5.

For simplicity we consider only plane steady laminar flow of anincompressible fluid past a solid body in a uniform parallel stream. Tocommence the analysis we need the basic solution for infinite Reynoldsnumber. Of course the actual flow becomes unsteady and turbulentabove a certain Reynolds number, but that is irrelevant because ourobjective is an approximation for moderate values. Cnfortunately, theproper limiting solution is unknown for finite bodies, where flow separa­tion presumably occurs. For example, Fig. 7.1 shows three conjecturesas to the relevant solution of the inviscid (Euler) equations for a circle,and there are many other possibilities.Cntil this important question issettled, it is impossible to treat flows with separation, because the wakewould exert a first-order influence even in the flow upstream. Thismeans that the body must be semi-infinite, with the exception of thefinite flat plate and perhaps some very thin airfoils. For further discussionof these matters see Goldstein (1960)_.

121

SeeNote

8

122 VII. Viscous Flow at High Reynolds Number7.2. Alternative Interpretations of Flat-Plate Solution 123

Fig. 7.1. Im'iscid flow past circle, (a) Continuous potential flow. (b) Infinite dead­water rC'gion, (c) FlI1ite cusped wake (natchelor 1936).

Serious difficulties arise from corners and other discontinuities, sothat it \vould be preferable to restrict attention to analvtic bodies. Onthe other hand, Prandtl's equations are readilv solYed ~nlv for certainself-similar flO\vs, most of \\,hich correspond t~ sharp-nos;d bodies. Inthe face of this dilemma, \\e choose to accept the difficulties arising fromsharp edges for the sake of reducing the partial differential equations toordinary ones. \Ye shall in fact make specific application to the finite orsemi-infinite flat plate, \vhile indicating the modifications that arise forother shapes. -

The reader is assumed to be familiar \vith the elements of classicalboundary-layer theory-in particular, with the Prandtl-Blasius solutionfor the flat plate as given, for example, by Prandtl (1935), Goldstein(1938), or Rosenhead (1963).

7.2. Alternative Interpretations of Flat-Plate Solution

The I31asius solution for the flat plate plays a \'ariety of roles inboundary-layer theon·. We describe here three essentially different

i I I I I I I I Iii I i II I Itt!! ! I I I I I It!,

Fig. 7.2. Bodies with Blasius solution valid far downstream.

interpretations. First, it applies to a semi-infinite flat plate. It thenrepresents a coordinate perturbation for large x. :\lore precisely, sincethe problem contains only the viscous length v f', it represents theasymptotic solution for large values of the running Reynolds numberl '.\' v. In this sense it applies also to a round-nosed thick plate, or evento such a growing shape as the parabola (Fig. 7.2).

Second, the I31asius solution applies to a cusp-nosed plate (Fig. 7.3).

x

Fig. 7.3. Body with Blasius solution \'alid near leading edge.

In this case, oddly enough, it represents a coordinate perturbation forsmall x. :\Iore precisely, it is the asymptotic solution for small xL,\\,here L is a characteristic length such as the initial radius of curvatureof the profile. The approximation could be improYed by calculatingadditional terms in a series in powers of x L. To be sure, the result isnot \alid in the immediate vicinity of the leading edge, \vherc ,\' =....0 O(v C).Thus \VC encounter a situation that arises occasionally in other branchesof fluid mechanics--for example, for power-law bodies in hypersonicsmall-disturbance theory (Hayes and Probstein, 1959, Section 2.6)­where a coordinate perturbation is valid for small (or large) distances,but not too small (or large). One might suppose that the nonuniformityat the leading edge could be remoYed by constructing a third expansionfor that region. Howevcr, the limiting problem is evidently just that ofthe semi-infinite plate itself. Of course the flow very near the edge isgiven by the approximation of Carrier and Lin described in Section 3.9.However, that is not an inner solution in the sense of Chapter V;it has no overlap with the I31asius solution and so cannot be matchedwith it (Section 10,9),

Third, Blasius' solution applies over a finite flat plate (Fig. 7.4). Itthen represents a parameter perturbation, being the asymptotic solutionfor large Reynolds number CL v based on plate length. This situationarises only because the basic inviscid solution is the same for a finite anda semi-infinite plate, and because the equations are parabolic, so that

124 VII. Viscous Flow at High Reynolds Number 1 7.3. Outer Expansion for Flat Plate; Basic Inviscid Flow 125

the boundary layer on the plate is not affected by the wake downstreamof the trailing edge. Of course the result is not uniformly valid, butbreaks down at both edges. The nonuniformity at the leading edge hasjust been discussed; that at the trailing edge is probably even morecomplicated.

We seek an asymptotic solution of this problem as the Reynoldsnumber R becomes infinite. Because the perturbation parameter appearsin the problem only as 1/R, one might be tempted to assume that theappropriate asymptotic sequence consists of powers of I R. However,Prandtl's theory shows that this is incorrect. We therefore adopt thesafer course of leaving the sequence unspecified, and assume a straight­forward (outer) expansion in the form

,p(x, y) ,..... y upstream

(7.2e)

(7.2d)

finite platesemi-infinite

for 0 <: x <:::: : 1,. 'lJ,

,py(x, O) c= 0

L= I

v

•0---. u

y

tI

IU=I

c:::::::>I

1':-R

Fig. 7 .4. ~otation for finite flat plate. as R --+ 'lJ with x, y > 0 fixed (7.3)

By substituting into the full problem and taking the limit as R -- ce,we obtain for the upstream condition (7.2d)

The distinction between these three interpretations of the Blasiussolution appears in higher approximations. We consider concurrentlythe first and third viewpoints, for the finite and semi-infinite flat plate.For the second, see "an Dyke (1962b). upstream (7.4)

The equation for the first approximation then becomes that forinviscid flow:

The limit appearing here may be zero, infinite, or finite. If it is zero,the problem is homogeneous and the solution (if unique) is lfl = O. Ifthe limit is infinite, the problem is meaningless. Hence a significantresult is obtained only if the limit is finite. Without loss of generality wetake

7.3. Outer Expansion for Flat Plate; Basic Inviscid Flow

Let L be the length of the finite plate, or an arbitrary reference lengthfor the semi-infinite plate (say the distance from the leading edge to ared line painted on the plate!). The solution must actually be independentof L in the latter case, and will be so to first order in the former.

It is cOl1\'enient to introduce dimensionless variables by referring alllengths to L and velocities to ['. This is equivalent to choosing unitssuch that L = L' =-0 1 (Fig. 7.4). Then in Cartesian coordinates the:\avier-Stokes equations are equivalent to Eq. (2.24) for the dimension­less stream function If, with v replaced by the inverse of the Revnoldsnumber R based on the length L:' .

(7.5)

(7.6a)

Adding the boundary conditions of zero velocity on the plate and auniform oncoming stream gives as the full problem

ULR=-

v(7.1) according to which vorticity is simply convected, its viscous diffusion

being negligible to this approximation. A first integral was given inSection 2.1 as

(7.6b)

'a c 1(,py ~ -,px~ - - v2 )P,p .~~ 0. ex ey R .

,p(x,O) = 0

(7.2a)

(7.2b)

That is, the vorticity - v2lfl is some function WI of the stream functiononly, and therefore constant along streamlines. The form of this functionis determined by the flow far upstream. It vanishes in the present

,126 VII. Viscous Flow at High Reynolds Number 7.4. Inner Expansion; Boundary-Layer Equations; Matching 127

problem, as it does whenevcr the oncoming stream is irrotational. Hencethe problem for the first term of the outer expansion becomes

Generalizing to higher approximations, we assume an inner expansion,\-alid within the boundary layer, of the form

(7.11 )

(7.10)

(7.9b)

(7.12c)

(7.12a)

(7.12b)

so that integrating yields a third-order equation:

lJ'1YYI' '~. P1J,lJ'lYY -- lJ'HP 1J:Y = lex)

The function of integration f(x) is proportional to the inviscid surfacepressure gradient. This will be shown by matching with the basicinviscid flow, which determines f(x) and also provides an outer boundarycondition.

Thus \\-e recover the familiar result that the boundary-layer thicknessis proportional to R-l;2.

Equation (7.10) for the first term of the inner expansion now becomes

This is a perfect differential, and can be written as

f(x, y; R) -.,., LlM~)lJ'l(X, 1') - Ll 2(R)Pz(.v,Y)

T Ll3(R)Pa(x, 1') ...!...... as R ---+ 00 with x, Y fixed (7.9a)

:\gain the limit appearing here may be zero, infinite, or finite. Thefirst t\VO possibilities lead to degenerate solutions that cannot satisfy theinner boundary conditions and match the outer flow. Hence we choosethe third possibility. That is, we apply again as in the pre\-ious sectionthe principle of least degeneracy (d. Section 5.5). Without loss of gener­ality we take the constant limit to be unity, giving

where

The .1" are an asymptotic sequence such that the 0/" are all of orderunity in the boundary layer, where Y = O( I). We determine .1] bysubstituting this expansion into the );"avier-Stokes equation (7.2a),multiplying by Ll I , and letting R tend to infinity. This gives

IPH ,E . . ' P p . "(};T)Pl!'Y'= lill}- [-R-A~Z(-R-) ]p 1 Y1'YI'(.\ (, R "r; "-' 1

(7.8)

(7.7a)

(7.7b)

(7.7c)upstream

l/Jl\',y) =Y

Pf1 =,0

f1(x,0) = 0

f1(x, y) -.., Y

which represents simply the uniform parallel stream. At infinite Revnoldsnumber a flat plate causes no disturbance. The corresponding sr;lutionfor any other plane body free of separation is the inviscid potential flow.

Here the no-slip condition f1!1(x, 0) == 0 has been dropped because it isu~enforceable. The or~er of the differential equation is reduced by oneWith neglect of the \ISCPUS term, and one boundary condition mustconsequently be given up. The question of which boundary conditionto abandon .can, in simple problems, be settled mathematicaliy ; here onemust be gUided by experience and physical insight.

The solution of this problem is

That is, if1 must be magnified by the same factor as y.

7.4. Inner Expansion; Boundary-Layer Equations; Matching

Loss of the highest derivati\c is the classical hallmark of a singularperturbation problem (Section 5.2); and we know that the basic im'-iscidsolution is not \'alid near the surface, \vhere the no-slip condition hadto be. abandoned. Thus the region of nonuniformity is the neighborhoodof,a l~ne, r:lther than of a point as in the thin-airfoil problems of Chapter1\.. CoordInates of order unity in that region will be obtained by magni-fYll1g the normal coordinate y, leaving x unaltered. -

Pran~tl was led to the proper magnification by physical intuition andcompanson with simple exact solutions. To some extent, however, onecan proceed formally. Let the vvidth of the region of nonuniformitv(the boundary layer) be of order .1 1(R), \vhere .1 is a function th;tv.anishes as its argument becomes infinite. Then ~n a;propriate magnified(Inner) normal ~oor~inate is given by Y = Y .1](R), where the stretching~a~tor 1 .1](R) ]S st1l1 to be determined. As for the dependent variable,]t ]s clear at least physically that 11 = .p,! must be of order unity insideas well as outside the boundary laver ~o that .

.; .I ,

I

128 VII. Viscous Flow at High Reynolds Number 7.5. Boundary-Layer Solution for Flat Plate 129

For the finite plate the boundary layers on the top and bottom surfacesmerge at the trailing edge and leave the plate as a wake without separating(Fig: 7.5). Evidently the boundary-layer approximation continues to be

(7.17a)

(7.17b)

(7.17c)

(7.l7d)

(7.16b)

for 0\ I, finite plate

<x< "fi'/00, semi-In mte

'Pl..;,O) = 0

'PlY(x,O) = 0

'PlY(x, 00) = 1

7.5. Boundary-J ,ayer Solution for Flat Plate

For the flat plate the inviscid solution (7.8) gives no pressure gradient,and the boundary-layer problem becomes

lJ'lYYY + lJ'l"lJ'lYY - lJ'lYlJ'v:y = 0

inviscid flow according to Bernoulli's equation. This equation actuallyapplies to any plane body, where x is the distance along the surfaceand y that normal to it, because curvature of the surface affects only thesecond approximation of boundary-layer theory. The relevant boundaryconditions are those of zero velocity at a solid surface, which give, if tPis normalized to vanish on the body,

together with the matching condition (7.15). The latter is retaineddespite its already having been used to evaluate the function of integra­tion; it does double duty because Y = 00 is a singular point.

The boundary-layer equation (7.16a) is parabolic-with x as time-likevariable-although the original ~avier-Stokes equation (7.2a) is elliptic.This change of type (ef. Section 3.11) means that upstream influence issuppressed. For this reason the first-order boundary layer on a flatplate is quite unaffected by the trailing edge and the subsequent wake.

(7.15a)

I-term outer e,\pansioll: ifiy ,...,.., ifill..;, y) (7.I3a)

Yrewritten in inner variables: = ifi1Y(x, V l~) (7.13b)

expanded for large R:y

=, ifil"(x, 0) + V RifiJi/ix,0) (7.13c)

I-term inner expansion: = ifi1Y(x, 0) (7.13d)

I-term inner e.\pansion: ifiv -- 'PJ}'(x, Y) (7.14a)

rewritten in outer variables: = 'PIy(x, vRy) (7.14b)

expanded for large R: = 'PJ}'(.\', 00) , ". (7.14c)

I-term outer expansion: = 'PIY(x, 00). (7.14d)

We apply the asymptotic matching principle (5.24) with m = n = I.It is sufficient, and more convenient at this stage, to match VJu ' whichhas the physical interpretation of tangential velocity, rather than VJ itself.Using the previous results (7.3), (7.5), (7.8), (7.9), and (7.11) we obtainfor any plane body:

Here we have used certain properties of the functions tPl and lJ'1 : wetake advantage in (7.13c) of the fact that tPl('\', y) is analytic in y aty = 0, and in (7.14c) we assume that lJ'1 has a limit as Y approachesinfinity. Equating the two final results gives the required matchingcondition:

This is recognized as the familiar requirement that the tangentialvelocity approach at the outer edge of the boundary layer (Y = Xi) theinviscid speed VJIII(X, 0). If we had matched tP itself, the result wouldhave been

This is obtainable from (7.15a) by integration; and it is in this sense,as the first term of an asymptotic expansion for large Y, that the matchingcondition is properly understood.

The matching condition can be used to find the function of integrationf(x), by evaluating the boundary-layer equation (7.l2c) at Y = 00. Theresult is the conventional boundary-layer equation of Prandtl:

as Y ---+ 00 (7.15b)

Fig. 7.5. Boundary layer and wake on finite plate.

(7.16a)

The right-hand side is the (dimensionless) surface pressure gradient in the

valid in the wake, and the pressure gradient still vanishes. The onlychange required is to replace the no-slip condition (7.l7c) by a require­ment of symmetry

130 VII. Viscous Flow at High Reynolds Number 7.6. Uniqueness of the Blasius Solution I3l

.\s in dimensional analysis ,,\vhere im'ariance under changes of scale isa physical necessity rather than, as here, a mathematical circumstance-­this means that the solution (if unique) cannot im'oln the variables0/1 ' ,\', and Y separately, but only in combinations that are invariantunder the transformation. Some alternatives are (7.22)

earlier references. I t must be integrated numerically. Blasius originallypatched a series solution for small ~) to an asymptotic e"pansion for large~). IIo\vewr, Weyl (1942) points out that this is a dangerous business,because the first series has only a modest radius of convergence (and thesecond probably none). It is safer, and now simpler \\'ith electroniccomputers, to integrate numerically from 'I = 0 to some reasonablylarge \'alue, say Yj - 5 or 10.

This is a t\\'()-point boundary-value problem, which \\'(JUld ordinarilyrequire repeated guessing of I~' (0) to satisfy the condition (7.21c) forlarge ~J. In the present problem, however, this complication can bea\'Oided by e"ploiting another group property of the solution. Thedifferential equation (7.2Ia) and initial conditions (7.21b) are im'ariantunder the transformation

One can therefore integrate the problem (7.21) \vith (7.2Ic) replaced byl;'(O) = I, and subsequently pick the constant c to satisfy the conditionat infinity (Goldstein, 1938, p. 135; Rosenhead, 1963, p. 223).

For our purposes the essential results of the numerical solution arecontained in the expansions for small and large Yj:

(7.18)

(7.19)

(7.17c')

1'-c1'

IX

1'.2l'

,/x·

for I < x

PI IyPI = function of

,Ix

We shall, ho\\'ever, follo\v the boundary layer in detail only back to thetrailing edge, because thereafter the flow is not self-similar.

A self-similar solution e"ists on the plate because the problem has acertain group property that permits its reduction to an ordinary differentialequation. This property is e"pressed by the fact that the problem (7.17)is invariant'" under the transformation

\Ve choose the latter forms, inserting factors of 2 III accord with thestandard Falkner-Skan notation (Schlichting, 1960, p. 143; Rosenhead,1963, p. 222): -- 'I ,- ~I - cxp,

"'I ~, 0.469600

~I = 1.21678

(7.23a)

(7.23b)

Reintroducing the original dimensional nriables at this point showsthat the length L disappears from II andY), in accord \\'ith the factsthat it is arbitrary for the semi-infinite plate and irrelennt for theboundary layer on a finite plate. This is another way of motivatingthe group transformation.

Substituting (7.20) into the boundary-layer problem (7.17) gives

l''I = V2~\:

(7.20) Here "exp" stands for terms that are exponentially small for large 'I.This means that the vorticity generated by shear at the \vall decaysthrough the boundary layer faster than any po\\'('r of 'I. That this mustbe true for any boundary layer is suggested by the analogy with diffusionof heat (Rosenhead, 1963, p. 216) and by certain mathematical arguments(Stewartson, 1957; Chang, 1961). We shall see later that the condition ofexponential decay of vorticity must be enforced in cases where it is notautomatically satisfied.

This is the Prandtl-Blasius problem, free of factors of 2 that appear in

* In the case of the finite flat plate, the length changes from I to c' under this trans­formation. Hence it is essential to recognize that the boundary layer on the plate isunaffected by the wake in order to appreciate the invariance,

K' +IJ;' =0

11(0) = 11'(0) = 0

11'(00) = I

(7.2Ia)

(7.2Ib)

(7.2Ic)

7.6. Uniqueness of the Blasius Solution

The above solution of the problem (7.17) is not unique from a mathe­matical viewpoint. To it may be added anyone of an infinite discrete setof eigensolutions, each of which satisfIes a linear perturbation equation de­rived from (7.17a), together with the zero initial conditions (7.17b,c), andexhibits exponential decay at infinity (Stewartson, 1957; Libby and Fox,1963). These have the form

(7.24a)

,

132 VII. Viscous Flow at High Reynolds Number 7.7. Flow due to Displacement Thickness 133

theory. \Ve therefore proceed to the second term in the outer expansion(7.3).

The nature of the factor 02(R), together with the matching conditions,is found by matching with the boundary-layer solution. ,\t this stage,matching ifiy docs not suffice; one must match if; itself,' or ifi.e as wellas ifiy . Csing all previous results, we find(7.25)

(7.24b)

The first of these eigensolutions is the x-derivative of the Blasiussolution (7.20):

where the em are arbitrary constants, and (Libby, 1965)

'\", =co I, 1.887, 2.814, 3.758, 4.704, ...

This has a familiar physical interpretation in terms of the displace­ment thickness of the boundary layer. Substituting into (7.27) showsthat the outer expansion for the stream function appears to vanish aty = f31R-l/2(2.v')l/2. Thus the boundary layer displaces the outer inviscidflow like a solid parabola of nose radius f31 2,/R. The matching condition(7.29a) is the linearized thin-airfoil approximation, transferred to the

(7.28)

(7.26c)

(7.26<1)

(7.26c)

(7.27c)

(7.26b)

(7.26d)

(7.27a)

(7.27b)

(7.29a)t/;k>::, 0) = - f31 v2x

rewritten in outer variables:

2-term outer expansion:

rewritten in inner variables

expanded for large R:

re\Hitten in inner variables:

expanded for large R:

I-tel'm inner expansion:

2-tenn outer expansion:

I -t/; - V R v241(Y))

I - \/ Rv,-,--, - v2.vfJ/-.--' )

vi R \/2x

I -(\/ Ryco., - v2x -- - f31 --:- exp)

V R\/2x

I /­= y .- V R (31 \ 2x

Y I ,-= --:;- - ":I f31 V 2xyR yR .

t/; - y + o2(R)t/;ix, y)}? }'

= -/- - 02(R)t/;2( x, -)\R - \/R

y= VR --i- o2(R)[t/;lV',0) -- ... ]

Here vve have in (7.26c) used the asymptotic expansion (7.23b) for theBlasius function!l' and in (7.27c) the fact that ifi2' like 1/;1' is analyticin y at y = O.

We now apply the asymptotic matching principle (5.24) with m = Iand n = 2. Comparing (7.26e) and (7.27c) shows that o2(R) must besome multiple of R-l/2; we choose

It then follows that

7.7. Flow due to Displacement Thickness

It was observed in Section 5.9 that the normal matching order ofFig. 5.6 must be followed in the direct problem of boundary-layer

It represents physically a slight uncertainty in the effective location of theorigin of abscissae. Similarly, the higher eigensolutions represent furtheruncertainty about the details of the flow near the leading edge. However,they have no such simple interpretation as the first. Stewartson (1957)has discussed how indeterminacy arises in any downstream expansionfor a boundary layer, because the initial conditions arc not imposed(d. Section 3.10).

We can properly dismiss all eigensolutions at this stage by invokingthe principle of minimum singularity (Section 4.5). It is more illumi­nating, however, to understand why the principle applies. The referencelength and the associated parameter R are artificial; hence the dimension­less uriables must occur only as R1/;, Rx, and Ry in order that L willdisappear from the result. The Blasius solution has this property. Thefirst eigensolution (7.25) has it only if the constant C1 is of order R-1.But this means it belongs to the third approximation (Section 7.11)rather than the first. I n the same way the higher eigensolutions arerelegated to higher approximations. This is a typical argument involvingan artificial parameter; for further discussion see Lagerstrom and Cole(1955), Chang (1961), and Exercise 4.8.

This argument would not apply to a blunt-nosed flat plate (Fig. 7.2),which has a real geometric length I" I ndeed, all the eigensolutions mustbe retained there in the asymptotic expansion of the first-order boundary­layer solution. They are not excluded by the principle of minimumsingularity, because more singular terms appear. For example, for theparabola the expansion contains a term in X-I /2 log x between the Blasiussolution and the first eigensolution. The resulting expression for skinfriction was given as equation (3.27) in Chapter III; and we show inSection 10.9 how the constant C1 can be found approximately by joiningwith another expansion from the nose or patching with a numericalsolution.

134 VII. Viscous Flow at High Reynolds Number 7.8. Second-Order Boundary Layer for Semi-Infinite Plate 135

axis (Section 4.2), to the tangency condition for the displacement

parabola.An alternatiYe form, \\'ith a slightly different physical interpretation,

results from differentiating (7.29a) \\'ith respect to x:

-.tP2...(X,0) = . ~1V 2x

(7.29b)

The condition (7.31 b) of symmetric flow ser\'ts to rille out circulatorymotion. This is the linearized thin-airfoil problem for potential flowpast the displacement parabola y -- fil(2.\' R)I 2. 'Ye require only thesurface speed, \\'hich was found in Section 4.8. Ho\\eyer, the full solutionis olwious from the vie\\'point of complex-Yariable theory:

(7.32)

'Ye exhibit the second term because its coefficient \\'ill be ditl'erent fromzero in any other problem, including that of the finite plate. COIl\'ersely,\\'e have

\\here He indicates the real part.The form of J 2(R) in (7.9a) and a matching condition for 0/2 are

found as before by applying the matching principle \\'ith III '0 11 ~ 2.E:-..tending the procedlll'e of (7.13) giH's as the 2-terl11 inner expansionof the 2-tcnn outer e:-;pansion of di!1 :

(7.33a)

(7.33b)

o\/ f(

2-inncr (2-outer) tPII

(7.30a)

This is the second-order component of normal Yclocity in the outerflow, eyaluated at the surface, \yhere it is required to equal the slope ofthe displacement parabola. From this point of Yie\\', the displacementeffect of the boundan' laver acts like a surface distribution of sources.This is the thin-airf<;il c~ndition of Section 4.3..\ detailed discussionof nrious interpretations of the displacement effect is ginn by Lighthill

(1958).Substituting the outer expansion (7.3) into the full equation (7.2a)

yields a linear equation for ViZ:

(7.30b)

The nonlinear convectiYe terms haye split into t\\'o, as usual in a pertur­bation scheme. The t\\'o terms represent, respeetiYcly, eonYection ofsecond-order yorticity along first-order streamlines and of first-orderyorticity along second-order corrections to the streamlines. :rhe latterterm yanishes in our problem, and \\'heneyer the oncoming stream isirrotational. Yiscositv has no eHect at this stage, the flo\\' outside theboundarv laver bein'g imiscid at least to second order, and to everyorder \\'hen 'the oncoming stream is irrotational. Then (7.30a) has thefirst integral

Ho\\'ever the second-order outer \'()fticitv w,) also yanishes when theonconlln~ stream is irrotational, so that ~2 then satisfies the Laplaceequation.

lIere \\'e ha\e assumed that the limit 0/2 A.\', x) exists; for a case \\'hereit does not, sec Exercise 7.2.

:\Iatching suggests that \\'e Inay in general choose

(7.34)

so that both the inner and outer e,xpansions procecd by imerse halfpo\\'ers of thc Heynolds number. This is true for an analytic body("an Dyke, 1962a), but only to second order for one \\ith corners, as \\'eshall see in Section 7.11. Completing the matching proYides, for ageneral body, a condition on 'P21 (.\,x), which has the physical inter­pretation of the increment in tangential velocity at the edge of theboundary layer. For the plate \\'e haye

7.8. Second-Order Boundary Layer for Semi-Infinite Plate

For the semi-infinite flat plate, the problem for the flow due todisplacement thickness is

because the displacement speed yanishes at y O. An infinite paraboliccylinder has the unique property that in thin-airfoil theory its surfacespeed is just the speed of the free stream.

Substituting the inner expansion (7.9) into the full equation (7.2a)yields again a perfect ditferential as the equation for 0/2 :

(7.35)1fI2 }(x, x) = 0

(7.3la)(7.3lb)

(7.31c)(7.3ld)

o

136 VII. Viscous Flow at High Reynolds Number 7.10. Local and Integrated Skin Friction 137

(7.40)

(7.38)

(7.39)

~~\~' ~ue t~ displ~cement thickness the surface speed is increased slightly.I ~IS IS e\:ldent If one regards the difference between the displacement

thickness In the wake and that for the infinite plate-which has no effectat the surface -as due to a distribution of sinks.

, This thin-airfoil problem can be solved by the methods of Chapter IV.1 hus the second-order matching condition (7.35) is found to be replacedby

'P" 1(.\,J~)' ---.P.L "~ log 1 + VX "C -:12f31 (1 I- ~ _ x" _ ... )V27T VX I v.\' 7T' 3' 5

,[,h.is fllnctiol~ is logarithmically infinite at the trailing edge, and thesenes expansion shown does not cOlwerge there. :\evcrthcIcss, Kuo~oh'cs the second-order boundary-layer equation (7.36) Iw correspond-II1gly expanding 0/2 as .

'Pix, Y) = 2;1 [X1i~fl(1)) - }x~i2f2(1)) ---'--- ...]

The problcm~ for the first nine.t" han been integrated numericallv,and the remall1der approximated. Thus Kuo estimates the sum of theslo\\ly com'ergent series for the integrated skin friction. For one side ofthe plate the result is, T being the sh~ar,

_ J~T dx 1.328 4.12cF ~. --- -- --.-. - -- - ...

'~pU2L VR R

This result is in excellent accord with experiments down to R = 10. How­ever, the coefficient 4.12 will be increased by an additional effect, overlookedby Kuo, that is discussed in the next section.

7.10. Local and Integrated Skin Friction

Consider again the semi-infinite plate. From the two-term innerexpansion (7.37b) we calculate the local coefficient of skin friction:

(7.37a)

(7.37b)

7.9. Second-Order Boundary Layer for Finite Plate

Prandtl himself discussed (Prandtl, 1935, p. 90) how the Blasiussolution could be improyed by successive approximations:

Instead of the simple parallel flow, the flow around a paraboliccYlinder ... should be introduced, which would slightly alterthe pressure distribution. The ... calculation would have to berepeated for this new pressure distribution and if necessary theprocess repeated on the basis of the new measure of displace­ment so obtained.

However, we ha\'e seen that because of a coincidence peculiar to theparabola, this procedure gives no second-order contribution for thesemi-infinite plate.

For the finite plate, on the other hand, Prandtl's procedure gives asignificant result. The details ha\'e been carried out by Kuo (1953)whose analysis is, however, subject to criticism on several points. Becausethe boundary-layer equations are parabolic, Blasius' solution is validahead of the trailing edge. Behind the trailing edge, the displacementthickness of the wake is very difficult to calculate (Rosenhead, 1963,p. 280). Kuo makes the perhaps reasonable assumption that it can betaken as constant at its value at the trailing edge (Fig. 7.6). Then in the

This yanishes only for the flat plate; for any other shape the right-handside will contain terms proportional to the local curvature of the surface

(Van Dyke, 1962a).The surface boundary conditions are 0/2(.\', 0) -~ o/2Y(x, 0) =-' 0 for

any body. Hence for the' semi-infinite flat plate 0/2 satisfies a completelyhomogeneous problem. The solution must yanish if it is unique. Althoughit could be nonunique by any of the eigensolutions (7.24), the argumentof Section 7.6 sho\vs that none can appear until the next approximation.Thus 0/

2(.\', Y) = 0, and the complete second approximation is

SeeNote

8

Here R.r ~.c..:c ex v is the local Reynolds number, based on distance fromthe leading edge. The first term is integrable, and yields the classicalFig. 7.6. Assumed form of displacement thickness for finite plate,

T 0.664 0Cf ==== --- .---.,. -- --l- _ -L •••'~pU2 VR

x' R

xI

(7.41 )

138 VII. Viscous Flow at High Reynolds Number 7.11. Third Approximation for Semi-Infinite Plate 139

Fig. 7.7. :\Iomcntum contour for semi-infinite plate.

SeeNote

8

(7.44)

(-F1--o-(P3Ill - Pu·Pall" !- PuyPay - PnPaxl· - P1nPn )

R -3'"= (PllP1n\( - P1J1Pl(r y - 2P1yr yy) ki!J2 [,13(R)]

layer theory ..-\ctually, it must be the result of an increase in local skinfriction abO\-c the Blasius nl1ue near the leading edge, where the localReynolds number is of order unity. :\othing else is knO\\"ll about theH(m in that region, except that it presumably has the form (3.24).

Because it is a local eHect, the concentrated force will appear also atthe leading edge of the finite flat plate. It \\·as, ho\\·ever, overlooked byl\:uo. \Yith this addition, and the re\·ision suggested pre\·iously, thecoefficient 4.11 in (7.40) is replaced by something like 5.3. The result isstill in reasonable accord with experiment, while leaving some scopc forthird-order dfeets.

One l11a\· ask whether a concentrated force appears also at the trailingedge. Rd"kction wggests that it does, but that \\·hereas the leading edgeis ("posed to the free stream, the trailing edge is sheltered by the reb­ti\ely thick boundary layer. Consequently, the trailing-edge force \YOuldcontribute only a third-order term, proportional to R-a ". Clear indica­tions of such a difference between the leading and trailing edges arc seenin the numerical solutions for a finite flat plate carried out by Janssen( 1958).

7.11. Third Approximation for Semi-Infinite Plate

Third-order bou ndary-Iayer theory has been studied only for the semi­infinite plate..-\t this stage the difficulties arising from the nonanalyticleading edge become severe. For that and other reasons, the first attemptsof .-\lden (1948) and others were erroneous. The correct solution wasgiven by Goldstein (1956) and Imai (1957a), and is described in detail inthe book of Goldstein (1960).

.-\11 those analyses use parabolic coordinates, but we can perseverewith Cartesian coordinates. The third term in the outer expansion isseen to vanish, because it represents the displacement effect of the secondterm in the boundary layer, which is zero. The equation for the thirdapproximation in the boundary layer is found to be (since lJf2 ~ 0).

In order to balance the nonhomogeneous terms on the right, there mustbe a term in the inner expansion (7.9) with L1 a(R) ~ I R32. This yieldsthe problem sohed by Alden. However, the resulting vorticity is foundto decay only algebraically for large 'I, which is unacceptable (Section7.5). This difficulty arises because of the sharp leading edge. On an

(7.43)

(7.42)

1.32g 2.326CF -- -,-. - -- ~- .. ,

V R,. R y

to Jones' treatment of leading-edge drag in thin-airfoil theory (Section4.6). Csing the Blasius solution in the boundary l~yer, and the flow dueto displacement thickness in the outer flow, he finds

It is remarkable that 50 years were required to discover that one term ofthe boundary-layer solution proyides two terms of the friction drag. .

"llthough the -details arc complicated, this resu~t .is almost 0 b\lOUS 111

the light of Section 4.6. The first term in (7.43), anSll1g from ;~e mtegra­tion across the boundary layer, is the claSSical result (7.42). I he secondterm arises from integration around the rest of the contour, where theflow is inviscid. It is therefore just the leading-edge drag (4.22) for thedisplacement parabola of nose radius II ~ (j/ R, the constant 2.326

b · 1 r:I.'elng ~)7TIJ1-' .

The~second term in (7.43) thus represents a force at the le.admg edge.Of course it is concentrated there only on the gross scale of boundary-

The second term in (7.41) \\·ould not be integrable, ex~ept for ~he

circumstance that its coefficient is zero, because the asymptotic expansIOnis not valid near the leading edge. The second term. in t~e integratedski n friction has nevertheless been calculated in an ll1gel1lous \~·ay .byImai (1957a). He avoids the difficulty at the leading edge by consldenngthe balance of momentum in a large contour (Fig. 7.7). This corresponds

value for the coefficient of integrated skin friction on one side of the

plate:

140 VII. Viscous Flow at High Reynolds Number 7.12. The Effect of Changing Boundary-Layer Coordinates 141

Thus the inner expansion is found to have the form

,1,( R) I "J"-! ( ) , 0 :' __.I.'_[lOg /'\' fa~(.Y)t _ f:n(.r.I)] _,' ...'f' x,)'; --- ,IR'-'\'1 Y) ~ -R' \.

. RU . ,/2x ,/2x (7.45)

as R ---+:xJ with x, Y > 0 fixed

(7.48a)

as R ---+:xJ with x, V Ry fixed

(7.48b)

1

l'MX'Y) --r VR.J;k>:."Y)';-'"

.pr--1 -- IJI (\' . / R},) ~ ...vR 1" V ,

optill/al (oordinates, in \vhich the independent variables are altered tomake the boundary-layer solution uniformly valid.

\Ve must first discuss hmv the boundarv-Iaver solution is affected bv, . ,

a change of coordinates, This question was implicit in the precedingsection \vhen we observed that recent investigations of the semi-infiniteAat plate have been carried out in parabolic rather than the traditionalCartesian coordinates.

Suppose that the entire Prandtl boundary-layer analysis is repeatedin a different coordinate system. That is, the :\avier-Stokes equationsare written in the new system, the stream function and coordinatenormal to the surface are magnified by a factor HI 2, the limit processR ----+ 'X' is carried out, and the resulting boundary-layer equation issoh'ed subject to zero nlocity at the surface and matching \\'ith thebasic im'iscid flow. One finds that although the outer expansion is inva­riant, the boundary-layer solution is not, so that it represents an alteredAO\v field. Different coordinate svstems vield boundan'-hn'er solutions.. . .that are identical at the surface and differ only negligibly \\'ithin theboundary layer-so that the skin friction is invariant-but may differwildly outside the boundary layer,

For this reason, it has been customary to disregard the boundary­layer solution outside the boundary layer, where it is replaced by thematched inviscid flow. However, Kaplun's vie\\'point is quite different.He takes advantage of the changes by seeking a special system of coor­dinates in which the boundary-layer solution approaches the outer flowas closely as possible.

It is not necessary to repeat the boundary-layer solution when thecoordinates are changed. If the solution has been calculated in anyconvenient coordinate system, its counterpart in any other system isgiven by a simple rule. Suppose that the boundary-layer solution forsteady plane flow past any body has been computed using coordinates(x, y), which are not necessarily orthogonal, but arc for conveniencechosen such that the body is described by y = O. Then just as for theflat plate in Section 7.8, the classical first three steps of the boundary­layer procedure yield a solution correct to order R -1/2 in the form

Now suppose that a different coordinate system (t, "I) is introducedsuch that again, for convenience, the body is described by "I = O. [The

(7.47)

(7.46)

1.328 2.326c --- -- - --- -

F ,/R,. RJ

•

.~Ithough the problem for fa2 is homogeneous, the solution is not zerobut is the first eigensolution (7.25). Exponential decay of vorticity in f31can be achieved by proper choice of the constant C1 '

.~ final difficultv remains, ho\vever. The solution for 1:11 is nonuniqueby the first eigen;olution, and no \\'ay is known of determining its con­stant, Thus the local coefficient of skin friction is given by

and the integrated skin friction (I mai 1957a) by

analytic bodY, a well-defined boundary layer exists beginning at thestag~ation p;lint, and exponential decay of vorticity is automaticallyass~lred. :\Tear the sharp edge of the plate, on the other hand, the bound­ary-layer approximation is il1\alid. Although it becomes nlid fardownstream, exponential decay is not assured, but must be enforced.

This is achie\'ed by adding to lJIa a term involving log x. Then forthe solution to be independent of L there must be a correspondingterm with

where the constant C\ is unknown. As discussed by Goldstein (1960),similar undetermined constants arise in higher approximations. Thesedepend on certain details of the flow near the leading edge. Whether itis possible, even in principle, to determine them-short of solving theproblem exactly-is at present a mystery. Imai (1957a) has estimatedC1 by patching (7.47) at RJ' ~ 1 with the result of the approximation(3.24) for the leading edge.

7.12. The Effect of Changing Boundary-Layer Coordinates

The boundary-layer solution can be combined with the associatedouter expansion: usi'ng the methods of Section 5.10, to form a compositeresult. \Ve consider instead a significant alternatiYC way of forming asingle uniformly valid expansion. This is Kaplun's (1954) method of

SeeNote

8

142 VII. Viscous Flow at High Reynolds Number 7.13. Alternative Coordinates for Flat Plate 143

present I) must not be confused with the Blasius variable, of (7.20).]. Ifthe transformation is regular at the surface, we have by faylor scnesexpansion

x = x(g, 1)) =-= x(g,O) T 0(1))

Y = y(g, 1)) = 1)y,}(g, 0) T 0(1)2)

(7.49a)

(7.49b)

The result has been simplified in appearance. Furthermore. \\'hereas inCartesian coordinates the boundary-layer vorticity is infinite along theentire \t'rtical line.\' c.. ,, 0, in parabolic coordinates it is singular ol~ly atthe point at the origin. Thus parabolic coordinates are seen tc; be supe'riorto Cartesian coordinates in this problem.

Because the outer expansion is invariant under change of coordinates,its expression in the new coordinates is found simply by introducingthe transformation (7.49) into (7.48a). However, the same is not true ofthe boundary-laver solution. Its new form is found by introducing(7.49) into (7.48b), expanding in Taylor series, and discarding termsthat are of order R- I in the inner variables. Thus the solution in thenew coordinate system is found to be

y

SeeNote

9

~11[.\(g, 1)), y(g, 1))] - \) R IMx(g, 1)), y(g, 1))] --'- ...

1 --\/R PI [.\{g, 0), \/ R1))"i (g, 0)] + ...

The latter of these is I\:.aplun's correlation theorem.

(7.50a)

(7.50b)Shrinking

rectangular

7.13. Alternative Coordinates for Flat Plate

I n Cartesian coordinates (x, y) the boundary-layer solution for thesemi-infinite flat plate is given by (7.37). Let us introduce paraboliccoordinates (t, I)) by setting

SeeNote

9

(7.53)

(7.54b)

4-0 - 1)--' ... (7.54a)

g= x,

-0 =)' ,- y",

Fig. 7.8. Coordinate systems for semi-intinire plate.

It is instructive to consider a third system, which "ill be found to beinfcrior to Cartesian coordinates. \\'hat may be described as "shrinkingrectangular" coordinates (f, 7)), indicated in the l(mer half of Fig. 7.8,arc gi,'cn b,'

Applving the correlation theorem gives for the boundarv-Iayer solutionin tll;::; system: - .

\Ve now examine the behavior in the outer region of the boundary-layersolution in each of these coordinate systems. \Ve form the two-term

(7.51 )

(7.52a)

(7.52b)

outer

Inner

\x=~(e -- 1)2)Iy =c= g1)

These are more natural coordinates for the problem (cf. Section 10.6)because-in contrast to Cartesian coordinates-the whole body andnothing else is described by l) ~ O. They arc therefore preferable forthis as well as other problems in mathematical physics inYl)l\-ing a halfplane. Applying the correlation theorem (7.50) gives the solution inparabolic coordinates as

144 VII. Viscous Flow at High Reynolds Number 7.15. Extension of the Idea of Optimal Coordinates 145

outer expansion of the one-term inner expansion. Using the asymptoticform (7.23b) for the Blasius function gives

Optimal coordinates arc not unique, but the most general system IS

given by

Replacing the abscissa s measured from the leading edge by a newcoordinate a, which \'anishcs \vhen s does, transforms the inner solution(7.58b) to

where F 1 and F2 are arbitrary functions.As an example, consider again the semi-infinite plate, and the bound­

ary-layer solution (7.37) in Cartesian coordinates. From (7.56'1) we find,using the generality of (7 .56b) only to drop irrelevant constant factors,

These are not orthogonal; an orthogonal system is obtained from (7.56b)by taking F2(!J;2) - I VJ2' which yields the parabolic coordinates of(7.51 ).

Other examples are gi\en by Kaplun (1954)..-\n interesting case isf10\\' normal to an infinite plane wall. Cartesian coordinates are 'optimal,and the boundary-layer solution satisfies the full :\a\'ier-Stokes equations.

SeeNote

9

(7.57)

(7.56b)

(7.58a)

(7.58b)

as E: ---+ 0 \\ith s > 0 fixed

as E: ---+ 0 with S c,:::: ~ fixedE:

gOJl! = Rev2(.;::-=:-zy) =" g

110 Jlt ,- • Y = gYJ

gOP1(X, y) =, F\[rPlx, y)]

YJoPt(x, y) =, rPtCX, y)F2[rP2(X, y)]

7.15. Extension of the Idea of Optimal Coordinates

Optimal coordinates probably exist also for second- and higher-orderboundary-layer theory, although the rules for finding them ~re not vetknO\\'ll. They could presumably be found also for three-dimensional ;ndcompressible bound~lry layers .

.-\ more significant generalization would be to other kinds of singularperturbation problems. \Ve 1nay ask, for example, whether optimalcoordinates can be found for the thin-airfoil problems of Chapter IV..-\s a simple test, consider the surface speed on an ellipse of thicknessratio E (Fig. 4.2). Corresponding to (7.48), the 2-term outer and I-terminner expansions (4.13) and (4.46) are

V2~ -(7.55a)Cartesian: y Vi/ Vx -r ...

parabolic: gYJ - L,g-r-'" (7 .55b),/R

shrinking rectangular: 1] ~\ V2' . (7.55c)- vi<.' g -r-'"

7.14. Determination of Optimal Coordinates

Kaplun's search for optimal coordinates was inspired by the discoverythat if the);'avier-Stokes equations are approximated by the linearizedOseen equations (Chapter VIII), the boundary-layer solution is the exactsolution if parabolic coordinates are used (Exercise 8.1). This, togetherwith other considerations already mentioned, suggested that paraboliccoordinates may be preferable for the flat plate also when the full ?'oJ avier­Stokes equations are used; and this has been seen to be true.

The question arises: how can optimal coordinates bc found for othershapes? A ,very simple answer has been given by Kaplun. Carry out thesolution to order R-Ij2 in any convenient coordinate system (x, y); itwill have the form (7.48). Then an optimal coordinate system is given by

Comparing with the corresponding outer expansions (7.37a), (7.52a),and (7.54a) shows that parabolic coordinates have reproduced twoterms, Cartesian coordinates only one, and shrinking rectangularcoordinates none at all. .-\ccordingly, we say that the boundary-layersolution in parabolic coordinates contains not only the basic inviscidflow but also that due to displacement thickness. The boundary-layersolution is uniformly valid to order R 12, so that the outer expansion issuperfluous. Coordinates having this property are said to be optimal.

Cartesian coordinates are not optimal according to this definition.HO\vever, they lead to a boundary-layer solution that contains the basicinviscid flow, and is therefore uniformly nlid to first order. \Ve maytherefore sav that Cartesian coordinates are semioptimal in this problem.Bv contrast, the boundarv-la\'er solution in shrinking rectangularcc;ordinates is utterly use!e;s i~ the outer flow; and no more than thiscan be anticipated in general.

gopt(X, y) ~= rPb, y), YJopt(X, y) = rPt(x, y) (7.56a) (7.59)

146 VII. Viscous Flow at High Reynolds Number Exercises 147

This is the counterpart of Kaplun's correlation theorem (7 .50b). The2-term outer expansion of this is simply unity. Therefore in this exampleany coordinate is semioptimal, but none is optimal.

This matter deserycs further study, together with the question of theconnection between Kaplun's optimal coordinates and Lighthill'sstrained coordinates (Chapter \'J). They have in common the idea ofachie\'ing uniformity by modifying the independent variables..-\ superfi­cial difference is that Kaplun changes inner coordinates whereas Lighthillstrains the outer ones. Ho\ye\'er, this is irrcleyant insofar as there is aduality bet\yeen inner and outer expansions (Section 5.9). A moresignificant difference appears in the \yay in which the modification ofcoordinates is determined, \\'hereas l,-aplun requires that the innersolution be yalid insofar as possible in the outer region, Lighthill imposesthe cruder condition that singularities not be compounded. It is conceiv­able that after appropriate generalization and refinement of both methods,they may be found to represent t\\'o faces of the same coin. See alsoSection 1004.

EXERCISES

7.1. Boul/dary layer 01/ l/ 'iCelZlie, Consider symmetric viscous flow past asemi-infinite \,'edge of semivertex angle frr 2. Calculate the potential flow, anddescrihe hO\I' the upstream condition (7.2d) for the \;a\'ier-Stokes equationsmust be modified. Discuss the applicability of the boundary-layer solutions asa coordinate perturbation and as a parameter perturbation. Sholl', by exploitingits group property, that Prandtl's boundary-layer equation (7.16a) can bereduced to the Falkner-Skan equation

/''' + if" - 13(1 - /'2) = 0

Assuming required numerical properties of i, calculate the flow due to dis­placement thickness. Find orthogonal optimal coordinates. Calculate thesecond-order boundary-layer solution. Is there a concentrated force at theleading edge?

7.2. Boundary layer on fiat plate in shear fiO'ic. As a model of the effect ofexternal vorticity on a boundary layer, consider a semi-infinite flat plate atzcro incidence in a parallel stream \vith constant vorticity, its speed beingU - wy. Derive the problem for the second approximation in the boundarylayer, shOlving that the matching condition has the form

Sho\\' th~lt interaction het\\'een the displacement effect and the external vorticityinduces a second,order pressure gradient on the houndary layer Reduce theprohlem to a third-order ordinary differential equation \vith proper boundaryconditions. [The definiti\'C treatment of this eontro\'ersial problem, together\Iith earlier references, is gi\'en by :\Iurray (1961).]

7,3. lm'erse boul/dary,layer problem. Suppose that \\'e soh'e the inverse ratherthan the direct prohlem in houndary,layer theory, in the follo\I'ing sense:lie seek the hody that produces a ,t;i\'en im'iscid flo\\' outside the houndarylayer. Explain \Ihy the asymptotic sequences for the inner and outer expansionsare not the successive pc)\\ers of R 12 of the direct prohlem. Carry the solutionas far ~lS pr~lctieahle for the upper surface of a hody that has a uniform parallelstream outside the houndary layer.

7.4 ......;((,oIIII-order correlatioll theorem. Sho\I' that the second-order counterpartuf (7 ,SOh) is

I -- 1~J ..... V R P1[x(g, 0), ,/ R7))"i(f, 0)] ~' R~PA]

- VRr;x,/f, O)P1' [] -;- 1,Rr!2y ,;,;(g, O)Pll·[ l:

\\"here all square brackets contain the same arguments as the first.

7.5 ..\/ip slIb/ayer. If slight slip is admittecl at the surface, the seeond of thehoundarv conditions (7 .16h) is replaced hy

('alcut.ne the houndary-Iayer solution for the semi-infinite t1,n plate to order f:

hy perturhing the solution for f: O. \;ote that the perturhation is the }'­deril'ati\'e of the hasic solution (Lin and Schaaf, 1951). Sholl' that this approxima­tion is not yalid in a thin suhlayer near the surface. Construct the first term of asupplementary expansion valid in that region.

SeeNote

9

as Y ----+ Cf)

Chapter VIII

VISCOUS FLOW

AT LOW REYNOLDS NUMBER

8.1. Introduction

\\'c considcr no\\" incompressible flo\\" past a body at lo\\" \{cynoldsnumber, as exemplified by the sphere and circular cylinder (Fig. 8.1).

u= I

~

Fig. 8.1. :\"otation for sphere and circle.

Eyery high-school student learns that :\Iillikan calculated thc drag of anoil drop using the approximation developed by Stokes in 1851:

Vaas R == - ->- 0

v(8.la)

A second approximation \\"as found by Oseen in 1910:

67T 3CD'"'-'!i(l+sR) (8.1 b)

However, only in 1957 was it shown how further terms [d. (1.4)] canbe calculated using the method of matched asymptotic expansions.

The classical warning of singular behavior is absent; the highestderivatives are retained in the ~ avier-Stokes equations in the limitR ---+ O. Howcver, the problem contains two characteristic lengths: theradius a and the yiscous length v. U. Their ratio is the Reynolds number,so that in the limit R ---+ 0 thc viscous length becomes vastly greater than

149

'II

150 VIII. Viscous Flow at Low Reynolds Number 8.2. Stokes' Solution for Sphere and Circle 151

8.2. Stokes' Solution for Sphere and Circle

Stokes reasoned that at low speeds the inertia forces, represented bythe convective terms in the ~avier-Stokes equations, are ineffectivebecause they are quadratic in the velocity. Hence at low Reynolds num­ber the pressure forces must be nearly balanced by viscous forces alone..\s a first approximation, Stokes neglected the convective terms. In planeflow the result is the biharmonic equation for the stream function:

the radius of the body. Hence singular behavior can be anticipatedaccording to the physical criterion advanced in Section 5.3.

Viscous flows at low Reynolds number are easily observed experi­mentally, in contrast with those at high Reynolds number. Fig. 8.2shows the sequence of flow patterns for a sphere or circular cylinder as

, 82 I 8 I 82 , 2

y4t/; ='"' 1----;:;-;;- + - ~ -+- -2 ~iJO) t/; = 0\ or" r ur r OU"

(8.2a)

Ii

Fig. 8.2. Flo\\" patterns for sphere or circle at lo\\" Reynolds number. (a) :\10 eddies.(b) Standing eddies. (c) l'nsteady flo\\".

the Reynolds number increases..\t ,'ery 10\\' speeds the streamlinepattern is almost symmetrical fore and aft..\ closed recirculating wakeor standing eddy makes its appearance at about R =-c 10 for a sphere andR "--' 2.5 for a circle (our Reynolds number being based upon radiusrather than diameter). One may imagine that the eddies always existinside the body, and at these Reynolds numbers penetrate through itssurface (cf. Fig. 8.3). The flo\\' becomes unsteady, \\'ith oscillations of

This follows formally from (7.2a) by letting R -+ O. In axisymmetricflow, the corresponding result is

(8.3a)

(8.2b)

t/;(l,8) = t/;I'(I, 8) ~= 0

and the condition of uniform flow upstream is

Let lengths be made dimensionless by reference to the radius a, andn:locities bv reference to the free-stream speed C (Fig. 8.2). Then theboundary c~nditions of zero velocity at the surface are

(c)(b)(0)

Fig. 8.3. Shape of standing eddy behind sphere.

(8.4)

(8.3b)as r ---+ J)planeaxis\'Inmetric

For the circle a symmetry condition must be added to rule out circulation.Consider first' the sp·here. The upstream condition (8.3b) suggests

separating variables, seeking a solution of the form!/J = sin2 0 fer). Thisleads to

The upstream condition shows that no term in 1'4 can be tolerated, andthat the coefficient of the term in 1'2 is $. Then the surface conditions(8.3a) fix the coefficients of l' and I 1', gi~'ing Stokes' approximation:

36.6

oo~From photograph by

Taneda (/956) , R=36.6

2- term Stokesexpansion for R =co

The first term is the uniform stream, and the third a dipole at thecenter of the sphere, both representing irrotational flows. The second

the downstream part of the wake, at about R = 65 for a sphere andR = 15 for a circle. The flow becomes irregular, with separation ofvortices from the rear of the body, above about R = 100 for a sphereand R = 20 for a circle.

./. 1("" .J 3 I). 'J 8'I' ~ 4 L.r- r -;- --;: SlW (8.5)

152 VIII. Viscous Flow at Low Reynolds Number 8.4. The Oseen Approximation 153

because, as usual, flow disturbances are weaker in three dimensions thant\Hl. (For semi-infinite shapes see Exercise 8.2.) Thus Whitehead (1889)failed in his attempt to improve upon Stokes' approximation for thesphere by iteration. The full :\'avier-Stokes equations give (Goldstein,1938, p. 115):

term, which contains all the vorticity, has been dubbed a "stokeslet"by Hancock (1953), \\'ho used a linear distribution of these three elcmentsto simulate a s\\'imming worm. For im'iscid flow the stokeslet is absent,and the coefficient of the dipole is } instead of ~l. Calculating theskin friction gi\'es two-thirds of the drag (8.1a), the remaining thirdbeing pressure drag (Tomotika and .-\.oi, 1950). The flow pattern issymmetric fore and aft, and is therefore free of eddies as in Fig. 8.2a.

Consider now the circular cylinder. The upstream condition (8.3b)suggests seeking a solution of the form !f; = sin () 1(1'), which leads to where

~ D'l/J = --;-J,--_ (l/Je .r; - l/J.~ ~. 2 cot 0 l/Jr - 2 :b_)' D2l/JR 1'2 S1I1 0 Or ) cO l'

(8.9a)

.1. _' . Q\.3 1 1 /'I' _. S1l1 0/1 ,1' og 1',1', r\ (8.6)

, (;2 sin 0 8 (' 1 (;)[)- "= ----r:o + -.,- "li -;-li 7Q"

or" 1'" Co sIn 0 Cu(8.9b)

We can imoke the principle of minimum singularity (Section 4.5),choosing k = 0 so that the stream function and velocity grow as slo\Y!yas possible with r. This leaves

.\ particular integral that satisfies the surface conditions (8.3a) is easilyfound to be

Substituting the first approximation (8.5) into the convective terms onthe right-hand side of (8.9a) that were neglected by Stokes yields theiteration equation

(8.10)[r2 si 11 0 C . 1 (;] 2 9 2 3, 1 . .,

-0-.,- - -.-, ;0-, (-.- -;c-) l/J = .- - R (----:,- _. ---;-.- ~- -, j S1I1- 0 cos 0cr" 1'" cO sm 0 cO 4, 1'" 1'3 1'"

(8.7)2k) 1 1 1 --;- k 1 k 3] . Q--'- l' og l' - -1' - --- - - -1' S1I1 0. 2 2 l' 2

Imposing the conditions (8.3a) at the surface reduces this to

.1.--- ell' log l' - ~r + ~~) sin 0'I' \ - 2 2 l'

(8.8)-- 2. R (2r 2 - 31' - 1 - ~ -~) sin2 0cosO

32 ' l' 1'2(8.11 )

SeeNote

16

The second term is the uniform parallel stream, the third a dipole at theorigin, and the first a t\\'o-dimensional stokes let containing the vorticity.The solution cannot be completed, however, because no choice of theconstant C satisfies the upstream condition (8.3b). The difficulty is that,in contrast \\-ith the solution for the sphere, the stokeslet is now moresingular at infinity than a uniform stream, and so predicts velocities thatare unbounded far from the body.

8.3. The Paradoxes of Stokes and Whitehead

The nonexistence of a solution of Stokes' equation for unboundedplane flO\\' past any body is known as Stohes' paradox. Stokes himself(1851) regarded it as an indication that no steady flow exists; a bodystarted from rest would entrain a continually increasing quantity offluid. However, this explanation is now believed to be incorrect, forreasons discussed in the next section.

Indeed, analogous difficulties arise with three-dimensional bodies,though they are deferred to the second approximation for finite shapes

Howe\-er, the velocity does not behan properly at infinity, and nocomplementary function can be added to correct it. In the ncxt approxi­mation the velocity would become infinite at infinity, as in the firstapproximation (8.8) for the circular cylinder.

The nonexistence of a second approximation to Stokes' solution forunbounded uniform flow past a three-dimensional body is kn()\nl asIf'lzitelzead's paradox. \\'hitehead himself regarded it as an indicationthat discontinuities must arise in the flow field associated with theformation of a dead-water wake. However, this explanation too is nowknown to be incorrect.

8.4. The Oseen Approximation

Just as d'Alcmbert's paradox was resolved by Prandtl's discovery thatflow at high Reynolds number is a singular perturbation problem, sothe paradoxes of Stokes and \Vhitebead were shown by Oseen to arisefrom the singular nature of flO\\· at low Reynolds number. \Vhereas theregion of nonuniformity is a thin layer ncar the surface of the body athigh Reynolds number, it is the neighborhood of the point at infinity for

SeeNote

16

154 VIII. Viscous Flow at Low Reynolds Number 8.4. The Oseen Approximation 155

SeeXote

10(8.17)

(8.16)

cos B)[ I

(8.18)

2 ? I,-- [io" Rr c_ e~Rrl'o,;IJl": (lRr)]Rev'''' 02 \

3 ~(l2 R

1 \sin 28-------_.--

log(4 R) y -'C-}I 2r~.p.,.

them hy I II,. In plane A0\\' the dimensionless equation (7.2a) for thestream function then becomes

The solution for the circular cyli nder \\as gi yen by Lamb (1911) in termsof Cartesian \'eloeity components. For example, the component normalto the free stream is

This constitutes an ad hoc uniformization, of a sort to be discussedfurther in Section 10.2. The general principle is to identify those terms\\hose neglect in the straightforward perturbation solution leads to non­uniformity, and to retain them after simplifying them insofar as possiblein the region of nonuniformity. If the resulting equations can be soh-cd,the result is a uniformly \'aEd composite approximation, of the sortdisCllssed in Section 5.4.

TIlliS <been's equations pro"ide a uniformly \'alid first appro:-.:imationfor either plane or three-dimensional flow at low Reynolds number. Inprinciple, one could retlne the solution b:-' suecessi\e appro:-.:imations,and th<.: result \\'fluld presumably presern its uniformity at e\'ery stage.In practice, hO\\'e\'er, although the Oseen equations are linear, theirsolution is sutticiently complex that no second approximations arekI1O\\'n. It is simpler to decompose the composite expansion into itsconstituent inner and outer expansions, \\'hich may subsequently berecombined. This process \\·ill be carried out in the follo\\ing sections.

The Oseen equations possess a second, essentially ditlerent, inter­pretation ..\t an arbitrary Reynolds number they describe \'iscous f10\\'

at slleh great distances from a j-jnite body that the \docity has nearlyreturned to its free-stream \alue. From this small-disturbance point of\ie\\', the Oseen approximation has been llsed to study the \\ake farbehind a hody (E:-.:ercises 8.1 and 8.3). In sllch applications, JJ in (8.16)ordin.\rily rq)lTsents a pertmhation rather than the full stream function,the distinction atlecting only the fortn of the boundary conditions. Thissecond interpretation of the Oseen approximation of course remains\'alid at 10\\' Reynolds numbers, and \\'ill be used in \yhat fo11O\\'s.

The solution of thc Oseen equations \\'as gi\(~n for the sphere by Oseenhimself (1910) as

(8.14)

(8.15)

(tI.13)

(tI.12)

r -)ox;as

,IS r _.)0 x;

, ~-) sin~ 8= o( ~-·Ir" rJ

-= O(CRr log r)\'ISCOU5

co\)\'Cctiyc

Stokes neglects the first three terms altogether, but Oseen approximates

Thus the ratio of terms neglected to those retained is

CO~YcC~~\~ = O(Rr)"18COU8

These nonuniformities are the source of the singular beha\'ior of theStokes appro:-.:imation. I n three-dimensional tio\\' the difficulty tends tobe concealed because the first appro:-.:imation is sufficiently \yell beha\ed;in the region of nonuniformity \\'here Rr c--= O( I) the Yelocity has alreadyeffectiwly attained its free-stream \'alue, so that it is possible to imposethe upstream boundary condition. This is an exceptional circumstance,which arose preYiously in the inner solution for a round-nosed airfoil(d. Section 5.6).

This explanation of the difficulties encountered by Stokes and\Vhitehead was given by Oseen (1910), who prescribed a cure at thesame time. Rather than neglect the conYective terms altogether, heapproximates them by their linearized forms valid far from the body,where the difficulty arises. For example, in the x-momentum equationin Cartesian coordinates

.-\lthough this r,ltio is small near the body \"hen R is small, it becomesarbitrarily large at sufficiently great distances, no matter how small Rmay be. Thus the Stokes appro:-.:imation becomes im'alid \\'here Rr is oforder unity. This occurs at distances of the order of)1 C, meaning thatthe \'iscous length is then the significant reference dimension. The sameobjection applies II fortiori to plane fio\\' , \yhere the incomplete Stokesappro:-.:imation (8.8) for the circle suggests the estimate

100y Reynolds number. The source of the difficulty can be undertood byexamining the re1ati\c magnitude of the terms neglected in the Stokesapproximation.

Far from the body the nonlinear convcctiYe terms are seen from theright-hand side of (8.10) to be of order R 1'2 • •-\ typical yiscous term-thecross-product in the left-hand side of (8.10)-is, from (8.5),

156 VIII. Viscous Flow at Low Reynolds Number 8.5. Second Approximation Far from Sphere 157

\\here the first term is the uniform stream. In order to match this, theOseen e\:pansion must have the form

Substituting into the full equation (8.9) yields for lf12 the classical line­arized Oseen equation (8.16) in the form

their asymptotic sequences left unspccified. However, \ve prefer to showIv)\\' m'atching automatically determines the form of each term in,;uccession when, as in this problem, the matching proceeds accordingto the standard order (Section 5.9).

Writing the Stokes solution (8.5) in terms of the Oseen \'ariable (8.19)and e\:panding for small R gives as its 2-term Oseen e\:pansion:

(8.21 )

(8.20)

(8.22'1)

as R - 0 with p fixed

., (sin e E )c'"(.f/- - cos e -;- -- -- -;-e .Y-lf12= 0up P u.

k ) ./ 1]., .., e 3 1 '., e2-0seen (I-Sto es ~)= 2: R2 p- sm- - 4 R p sm-

Here y 0.5772 ... is Euler's constant, and K o is the Bessel function.I n both these solutions the surface conditions were satisfied only

approximately in a manner appropriate to the underlying assumptionthat the Reynolds number is small. We shall reconstruct these resultslater. Solutions for arbitrary Reynolds number were carried out byGoldstein (1929) and 'l'omotika and ,\oi (1950). These more complicatedresults arc of limited \'alue because, contrary to Oseen's own views, theapproximation is qualitatiqJy as well as quantitatively invalid at highReynolds number. For example, the Oseen approximation gi\'es boundaryla\'ers \\·hose thickness is of order R I rather than R 12 as in Prandtl'scorrect theory. This discrepancy may be understood physically as arisingfrom thc fact that in Oseen's approximation the \'()rticity generated at thesurface by shear is conq~cted through rather than along the surface. Thedetailed flo\\' patterns calculated by Tomotika and .-\oi would be ofsome interest had not Yamada (1954) pointed out that numerical inac­curacy invalidates their qualit,lti\'e nature even at low Reynolds number.For example, 'l'omotika and .\oi predict the standing eddies of Fig. 8.2bat arbitrarily 10\\ Reynolds number, \vhereas Yamada sho\vs that theyfirst appear behind the circular cylinder at R ~--, 1.51 in the Oseenapproxi mation.

8.5. Second Approximation Far from Sphere

We now improve Stokes' solution for the sphere by applying themethod of matched asymptotic expansion. Our analysis follows thespirit of Kaplun and Lagerstrom (1957), but more nearly the notationof Proudman and Pearson (1957).

Let Stokes's approximation (8.5) be the leading term in an asymptoticexpansion for small Reynolds number, which we call the .Stokes e.\pan­sian. We have seen that this series is innlid far from the body where ris of order R -1. \Ve therefore introduce an appropriate contracted radialcoordinate p by setting

SeeNote

10

p = Rr (8.19)

\\'here

Setting

reduces Eq. (8.22) to

Seeking as before a solution of the form ¢2 = sin2() f(p) gives

(8.22b)

(8.23)

(8.24)

and any other solution of (8.24) having the proper symmetry is more

and envision a second asymptotic expansion nlid in that distant region.\Ve call it the Oseen expansion, because the flow far from the body is asmall perturbation of the uniform stream. According to the conventionadopted in Section 5.9, the Oseen expansion is the outer, and the Stokesexpansion the inner expansion. \Ve choose our notation accordinglyexcept for the radial variable where, because R is not available, we use p

for the outer and r for the inner variable.We could, as in the last chapter, write down the two expansions with

r - (~- + ~)f = 0

The solution that vanishes at infinity is

f .= c,(·] -;- ~)e-;p- p

(8.25)

(8.26)

158 VIII. Viscous Flow at Low Reynolds Number 8.6. Second Approximation near Sphere 159

singular at the origin, and can therefore be rejected as unmatchable bythe principle of minimum singularity (Section 5.6).

Thus the original equation for (8.22) for !f2 is reduced to

8.6. Second Approximation near Sphere

We proceed to the second term in the Stokes expansion. The 2-termStokes expansion of the Oseen expansion (8.21) is found to be

(8.27) 2-Stokes (2-0seen) .p = ~(2r2 - 3r) sin2 (J -+- ,& Rr"(1 - cos (J) sin 2 (J (8.31)

A particular integral is In order to match this, the Stokes expansion must have the form

(8.28) (8.32)

This matches (8.20) if (2 = 1-\\'e have thus found two terms of the Oseen expansion (8.21). When

rewritten in Stokes \ariables this becomes

where the first term is a potential source at the origin, which has beenadded to cancel the sink in the second term, and so assure zero fluxthrough any surface enclosing the body..\ny other complementaryfunction that possesses this property, gives no velocity at infinity, andhas the proper symmetry will be more singular at the origin and henceunmatchabic.

This is a fundamental solution of the Oseen equation, which describesthe disturbance field produced at great distances by any finite three­dimensional nonlifting body. The constant (2 depends upon certaindetails of the flo\v near the body. We find it by applying the asymptoticmatching principle (5.24). Writing the Oseen expansion (8.21) in Stokesvariables and expanding for small R yields

(8.34)

\Ve can construct a uniformly valid composite expansion by combiningthis with the Stokes approximation (8.5) using the rule (5.32) for additivecomposition. The result gives a uniform approximation to the perturba­tion field. It is found to be just the solution (8.17) of the Oseen equationsgiven by Oseen himself. This confirms the statement in Section 8.4 thathis linearized equations yield a uniform first approximation. )J ear thebody the last term in (8.17) reduces to the stokeslet of the Stokes approxi­mation; it may by analogy be called an "oseenlet."

I-Stokes (2-0seen) .p = ~r" sin2 (J _. c2r sin2 (J

as R __ 0 with Rr fixed

(8.29)

(8.30)

The equation for lfl2 is evidently Whitehead's (8.10) without the factor R.His particular integral (8.11) remains valid, and the original Stokesapproximation (8.5) provides the only complementary function with theproper symmetry that is no more singular at infinity. Thus we set

lTl C (2· j 3 1 ) . ., (J 3 (2 ') 3 I I 1 ') . .) (JT" = 2 r- - r " -, sm- - -- r- - r - - - - ---;- sm- cos (J. r 32 I' 1'2(8.33)

The constant C2 is found by matching. Carrying out the Oseen expansionof (8.33) yields

2-0secn (2-Stokes) .p = ~ ~2- p" sin2 (J

- k(2C2p2 - 136 P2 cos(J - ~pl sin2 (J

and this matches (8.31) if C2 --= 3 32.Thus we have found two tcrms of the Stokes expansion for the stream

function in the vicinity of the sphere:

This vanishes nOL only on the sphere and along the axis of symmetry,but also along the curve

(8.36)

This is the approximate description of the boundary of the standingeddy. It is plotted in Fig. 8.3. The eddy appears only at Reynoldsnumbers so large that one would not have expected the Stokes expansionto have any validity . .:\evertheless, the lower half of Fig. 8.3 shows

160 VIII. Viscous Flow at Low Reynolds Number 8.7. Higher Approximations for Circle 161

Fig. 8.4. Length of eddy behind sphere.

striking agreement with the experimental observations of Taneda (1956)at R 0-= 36.6. The downstream end of the eddy lies at

SeeNote

10

(8.39)as R ---+ 0 with r fixed0/--.. Ll1(R)(r log r - ~r- ll.) sin e..' 2 2 r

The multiplier LIt must be allowed to depend upon Reynolds numberbecause our asymptotic sequence is unspecified. Although this approxi­mation cannot satisfy the condition (8.3b) of a uniform stream at infinity,it can be matched to the uniform stream, regarded as the first term of anOseen expansion (Lagerstrom and Cole, 1955, Section 6.3)..-\gain theOseen variable is taken as p = Rr so that lengths arc referred to the

8.7. Higher Approximations for Circle

Stokes' paradox for plane flow is more striking than \rhitehead's forthree dimensions. Its resolution by the method of matched <lsymptotice.\pansions is correspondingly more dramatic, despite the practicalshortcoming that the solution cannot be carried to nearly as greataccuracy. We treat the typical example of the circular cylinder, usi nga synthesis of the work of Kaplun (1957) and of Proudman and Pearson( (957).

The analysis largely parallels that for the sphere, but interestingdifferences appear. In particular, the matching is marginal, and theasymptotic sequence correspondingly slow. It \\'as for this problem thatKaplun and r.agerstrom (1957) devised their sophisticated apparatus ofintermediate limits and e.\pansions, and the intermediate matchingprinciple (Section 5.8). However, we shall see that the asymptoticmatching principle (5.24) is entirely adequate, although the simple limitmatching princi pIe (5.22) does not hold.

\Ve reconsider the solution (8.8) of the biharmonic equation as thefirst term in a Stokes expansion:

t' nfortunately, this has no real root if three terms are retained.The logarithmic term limits the Stokes expansion to values of R smallcompared with I. Until some method is found of enlarging the range ofapplicability of such a series (cf. Section 10.7), we cannot say whetherthe striking predictions of the second Stokes approximation are morethan coincidence.

Because of symmetry, the second term in square brackets in (8.35)contributes nothing to the drag, which according to the first term is then(I -, ~R) times the Stokes value. But the second term is the particularintegral of Whitehead for the nonlinear terms. For that reason, theOseen approximation, which neglects the nonlinear terms near the body,nevertheless gives the drag correct to second order at least for symmetricshapes (Chester, 1962).

(8.37)

(8.38)

ExptJrimtJnf,TOIItJda (956)\

NUn'ltJrical,JtJnson (1959)

~

rc =,hVl-t3R - 1)

20

60

1 - ~R \- :OR210g R ,- O(R'!.) = 0

40R= Va

#'

Therefore the eddy first appears in the flow field at R 0,"" 8. Despite itsmagnitude, this agrees well with the value of 12 measured by Tanedaand the value of 8.5 calculated numerically by Jenson (1959) using thefull ~a\'ier-Stokesequations. Indeed, Fig. 8.4 shows that good agreement

OL.....:.:.....---------­234

.1a

persists out to R "'" 60, \vhich is about the limit for obser\"<ltion ofsteady flow. These remarkable results call for corroboration throughexamination of the effect of further terms in the Stokes expansion.

Higher approximations can be found by continuing the precedinganalysis. Proudman and Pearson (1957) have carried it far enough toshow that the next Stokes approximation contains a term in R2 log Ras well as R2, and that logarithms are thereby introduced also into theOseen expansion beginning with R:l log R. They have calculated onlythe term in R2 log R in the Stokes expansion. This provides the dragformula (1.4) of Chapter I.

According to these results, the Reynolds number at which the eddyfirst appears is a solution of the transcendental equation

SeeNote

10

162 VIII. Viscous Flow at Low Reynolds Number 8.7. Higher Approximations for Circle 163

Writing the Stokes appro:\imation (8.39) in Oseen variables and expand­ing giyes

(8.45a)

(S,45h)

c,p., ("I 4 -, £l \ ("" (I ),------,-=- - -- - C2 og--- y -.- cos- u, - ) p og P((ps1l18) "" p -"

;-Cj~- --- -- C2 si n 8 cos 8 -- O(p log p)((p cos 8) -

The term in log p is again a potential source at the origin that cancels thesink in the term il1\"oh'ing the Hessel function K o . For small p these areappro:\imately

(8.40)

(8.41)

as R --,. 0 with p :::. 0 fixed

I-Oseen ( I-Stokes) <J;= ~ ';:<.) log~. p sin !J

viscous dimension v '-' rather than the radius a. Then the Oseen expan­sion begins \yith the free stream in the form

<J; -- J_ p sin 8-e- ".R

l'sing this, \Y(' find that the Oseen expansion (8.43) behayes near the bodylike

I-Stokes (2-0seen) f = *-[r sin If - c2JJ(R)(1og~ - y. I \p sin 8] (8.4T)

Then matching \\-ith (8.42) according to the asymptotic matching prin­ciple giyes ("2 ~ 1.

The second term in the Stokes expansion--and indeed the term of anyfinite order-is e\idently again a solution of the biharmonic equation,because the nonlinear terrns of order R are transcendentally small onthe scale of p(mers of Ll 1(R). :\latching, or applying the principle ofminimum singularity, sho\\'s that each is simply a multiple of the tirstapprO:\imation (8.8). It is conyenient, following Kaplun (1957), to makethe second term yanish by choosing the constant k so that (8.42) and(8.47) match perfectly: k -" log 4 - y -§. Then the Stokes expansionassumes the form

Thismatcl~es(8.40)ifJl(R),~(log 1 R)--l,ormoregenerallyifJ 1(R)-c(log I R .,- k) ',where kis any constant; and we shall later exploit thisfreedom.

The oyerlap is so slight in this case that in order to match \ye han hadto accept a n:lati\e error of order J 1 , which is enormous compared withthe error of order R in the Stokes approximation for the sphere. \Ve arethereby committed to a slo\\- e:\pansion in po\yers of J 1 , of which aninfinite number of terms correspond to only the first term for the sphere.

E:\panding the Stokes appro:\imation (8.39) further in Oseen variablesvields

2-0seen (I-Stokes),p ~" *[1 - .:::lJ(R) (log p - k -- ~)]p sin!J (8.42)

This requires, in order to match, that the Oseen expansIOn (8.40)continue as

(8,43)

Substituting this into the full equation (7.2a) shows, of course, that f2satisfies the linearized Oseen equation (8.16). The appropriate solutioncan be found by proceeding as for the sphere (Proudman and Pearson,1957). However, the stream function is disadvantageous here because itean be written only as an infinite series, whereas the velocity componentsare closed expressions.

We evidently seek the plane counterpart of (8.28), the oseenlet repre­senting the disturbances produced by an infinitesimal drag at the origin.This fundamental solution, due to Oseen (Rosenhead, 1963, p. 183),gives as Cartesian velocity components

\\here Euler's constant y ~ 0.5772 ... , and integrating gives

4,p2 -- - - c2(log P - 1 -- Y)p sin 8 - O(p2 log p)

I " I} }" .f -- (Ll -" aLl")( r log r - -. r -,- - -"J sm (J"J ti3"J - 2 '2r

where

4 1-1 3703-1

Ll J = (log R - y -- 2) ~= (log ---'~J

(8,46 )

(8A8a)

(8.48b)

_ C,p2 \ C Iu 2c [logp-;-eJpcosa KoCtp)] -- eJPcoSUKo(tp)\ (8.44a)2 = c(p sin fJ) = 2/E-(p cos fJ)

_ N2 8 ,'- 1 (8 44b)V 2 = - ~( fJ) = 2c2~(. fJ-) [log p T e,pco,a K O(2P)] .

d p cos '-' p sm

Forming a uniformly valid t\\'o-term composite expansion by additivecomposition of (8.43) and (8.48) reproduces Lamb's solution (8.18) of thelinearized Oseen equation.

Kaplun (1957) has carried the process through one more cycle to find

164 VIII. Viscous Flow at Low Reynolds Number Exercises 165

SeeNote

11

SeeNote

15

the coefficient of the third term in the Stokes expansion (8.48) as(l3 "'" -0.87. lIenee he finds for the drag coefficient

(8.49)

The first term is the classical result of Lamb (1911, 1932). Comparisonwith the measurements of Tritton (1959) in Fig. 8.5 shows the limitedutility of this result..-\Iso shO\\"I1 for comparison is Tomotika and Aoi's(1950) full nu merical solution of the linearized Oseen equation (8.16).

from a formal mathematical point of view, we should exhaust all the

powers of LJ 1 in (8.48) before considering nonlinear corrections to theStokes equation, which are relatively of the transcendentally small orderR. In practice, however, such terms are significant. The first such term(Exercise 8.5) is of order R, which is greater than LJ.4 for R > 0.00008.Proudman and Pearson (1957) discuss briefly how these terms could becalculated. They would be needed to show any asymmetry in the flowpattern, such as the emergence of standing eddies.

EXERCISES

SeeNote

11

SeeNote

3

8.1. Oseen solution for flat plate and plane wake. In parabolic coordinates(7.51) the :\avier-Stokes equations give for the stream function in plane flow:

\vhere

8.2. f'iscous flow past slender paraboloid. In paraboloidal coordinates (cf.Exercise 8.1), the Kavier-Stokes equations give for axisymmetric flow

Derive the corresponding linearized equation (8.16) of Oseen. Find by separa­tion of variables the Oseen solution for a semi-infinite flat plate in a uniformstream, and for the flow far from a finite nonlifting body. Show that in thefirst case the boundary-layer approximation in parabolic coordinates is the fullOseen solution. Compare the skin friction with the known value for the :\avier­Stokes equations far downstream. In the second case express the constantmultipliers for both the term representing the wake and that for potentialflow in terms of the drag of the body. Relate the solution to (8.44). [Thefirst case \\'as originally treated in a more complicated way by Lewis andCarrier (1949); for the second, see Imai (1951) and Chang (1961).]

o Experiment,Tritton (/959)

2 termsof (8.49)

\,\\,

\\\\

J\"~\

\ \

\~,, " 1term of

o '" ............. (8.49)

" ..... -o ~ - - __~ 0 .....

o CO..................... Filii Oseen--o 0 -__.....o 0 .....

o

4

2

6

8

/6

18

14

12o

Co: ,'~'pv a

10

Drag of circular cylinder at low Reynolds numher.

oo

Fig. 8.5.

.2 .4 .6 Uo.8R:-

11

10 12 14

Find the Stokes solution for the paraboloid of revolution. Show that it can bematched to the uniform stream in the same marginal way as for the circularcylinder. Calculate the second term in the Oseen expansion. Describe how theprocess would continue. \Vhat light does it shed on the accuracy of the knownOseen approximation for the elliptic paraboloid (Wilkinson, 1955)? How does

166 VIII. Viscous Flow at Low Reynolds Number

SeeNote

3

SeeNote

3

the ratio of skin friction to pressure drag in the Oseen approximation comparewith I : I for the circle and 2 : I for the sphere (Tomotika and Aoi, 1950)?

8.3. Wake ofaxi,')'IIlJlletric body, Calculate the leading term in the expansionof the stream function far from a finite body of revolution in viscous flaIl',identifying the constants \\'ith the drag of the body as in Exercise 8,1.

8.4. /'isco/ls flow past thin parabola. Show that the Stokes approximation fora parabolic cylinder is unmatchable in any way with a uniform stream. Sho\1that it appears to match the Oseen approximation for a flat plate to first order,or that for a parabola to second order; but that continuing the solution bv themethod of matched asymptotic expansions indicates that neither of th~se isthe correct leading term. "'hat is the true Oseen limit? (See Lagerstrom andCole, 1955, p. 877.) ,

8.5. Transcendentally slIlall terllls for circle. Supposing that (8.48a) is kno\\'nto any desired order, find the form of the "next" term, aside from undeterminedconstants, sho\\ing that it is of order R. \Yith which term of the Oseenexpansion \vould it match?

8.6. ,Year s/lccess of strained coordinates. Show that the t\vo-term Stokesexpansion (8.35) could be found using the method of strained coordinates,and straining only the radius, if one knew that the straining should \'anishalong the dO\l'I1stream axis of symmetry.

Chapter IX

SOME INVISCID SINGULAR

PERTURBATION PROBLEMS

9.1. Introduction

.-\.s discussed in Chapters Y and YI, the method of matched asymptoticexpansions was invented to treat singular perturbation problems inviscous flow theory, whereas Lighthill's technique of strained coordinates\\'as developed for problems in wan propagation. It is still unclear whetherLighthill's method can be generalized to handle elliptic and parabolic as\\'ell as hyperbolic equations. On the other hand, the method of matchedasymptotic expansions has been successfully applied to a variety of invis­cid flo\\' problems. rndeed, we introduced it in Chapter 1\' for inviscidthin-airfoil theory, and applied it in Section 6.7 to the hyperbolic equa­tions of supersonic flow.

The present chapter is de\'()ted to three more examples of im'iscidflo\\'s invohing nonuniformities. These cover the gamut of speeds fromsubsonic through transonic to hypersonic. The first is the classicallifting-line theory of Prandtl, which is seen in a fresh light \\'hen itssingular nature is re\'Caled. The second is slightly supersonic flow past aslender cone, where the linearized solution must be corrected in order tofind the position of the shock wave. The third is the entropy layer pro­duced by slightly blunting a wedge in hypersonic flow. vVe treat the firstand third by the method of matched asymptotic expansions, and thesecond by strained coordinates; but it will be useful to ask whether thealternative method could have been chosen in each case,

9.2. Lifting Wing of High Aspect Ratio

Prediction of the flow field produced by a finite lifting wing at subsonicspeeds is one of the most intractable problems in aerodynamic theory.Viscosity must be neglected, except insofar as it provides the Kutta­Joukowski condition at the trailing edge; and it is almost essential tolinearize, so that the effects of thickness, camber, and angle of attack can

167

168 IX. Some Inviscid Singular Perturbation Problems9.2. Lifting Wing of High Aspect Ratio 169

----. x

y

tI

I

(e)(b)

y

tI

....L_--. X

rf:2;I

y

(0)

A is the aspect ratio, and the half-chord h(z) is an analvtic function oforder unity. At first we assume that h vanishes as smoothlv as necessarvat z = ·;.1, in order to avoid additional nonuniformities at' the tips. Th~effects of relaxing this restriction will be faced later.

(9 .1 a)

be treated separately. Then the subsonic problem is equivalent to theincompressible one according to the Gathert rule (Jones and Cohen,1960, p. 49). However, even the remaining problem of inviscid incom­pressible flow past a wing of zero thickness and infinitesimal inclinationto the stream is complicated. Its strict treatment by lifting-surfacetheory requires solution of a singular integral equation involving double

integrals.For a wing of high aspect ratio, Prandtl's lifting-line theory reduces

the problem to the solution of a singular integral equation involvingonly one integration. It is not clear, however, how one could refinePrandtl's analysis to obtain better approximations. For example, thewell-known expression for the lift-curve slope of a flat elliptic wing of

aspect ratio A

(9.3a)

(9.2'1)

(9.2b)

(9.2c)

(9.2d)

(9.3b)

(9.3c)

.-1' I I1(z)x

x = .-1- l h(z)

,p c= /l-lctJ(X, Y, z)

Y =.4y

,p" = ° at y .= 0,

,p ........ x cos ex .... y sin ex

,p.,. < Xc at y = 0,

equation:

tangency:

upstream:

Kutta:

The full problem for the dimensionless velocity potential </> is

Fig. 9.1. Inner and outer limits for flat lifting wing. (a) Full problem. (b) Innerlimit. (c) Outer limit.

where

The version of the Kutta condition given here is a simple way of assuringthe physical requirement that the velocity be finite at the trailing edge.

It is evident physically that as the aspect ratio A becomes infinite theflow at any spanwise station approaches plane flow past a flat platehaving the local chord (Fig. 9.1 b). This can be shown formally by intro­duc~ng magnified inner variables that are of order unity nea; th'e wing,settlllg

These are coordinates referred not to the semispan but to a typicalchord, which is the relevant scale for lengths in sections normal to thespan.

(9.lb)

can, to the accuracy of the approximation, be written as

However, no extension of Prandtl's method will yield the next term,which we shall see is not of order A-2.

Friedrichs (1953) has pointed out that this is a singular perturbationproblem. There are t\\"(l characteristic dimensions, the span being theprimary reference length and the chord the secondary one. Their ratiotends to infinity as the aspect ratio increases. Hence according to ourphysical criterion (Section 5.3) it is possible for this parameter-perturba­tion problem to be singular. This possibility is realized for the lifting wing(and also for the nonlifting one, see Exercise 9.1). The nonuniformitycan be treated by applying the method of matched asymptotic expansions.The second approximation will be found to be equivalent to PrandtI'slifting-line theory. However, application of the matching principle isseen to eliminate the occurence of integral equations, which are reducedto quadratures, so that the analysis is substantially simplified. Further­more, continuing the process makes possible the calculation of higherapproximations.

Consider for simplicity a flat wing of zero thickness, whose planformis symmetric in the streamwise as well as the spanwise direction (Fig.9.la). Take the free-stream speed and the semispan as units of velocityand length. Transfer of the tangency condition is avoided by letting thewing lie in tne plane y = 0, to which the free stream is inclined atangle lX. Then the planform may be described by x = =A-1h(z), where

170 IX. Some Inviscid Singular Perturbation Problems 9.3. Lifting-Line Theory by Matched Asymptotic Expansions 171

lPl

'oX cos c"l: - Im:,lrY-- iY)2 ='71'<"(;)

- h(;:;) log[.\ -:- 1'1' ,,-v(X-=-z'f'")""2='-iI2(Z)]} sin c, (9.5)

\\'hich can be put into real form using elliptic coordinates..-\s it clearlyshould be, this is the solution for plane flo\\' past a flat plate of chord2h(;:;) at angle .\ (Fig. 9.1 b). Thus to a first approximation the spal1\\isecoordinate;:; plays only the role of a parameter. For present purposes\\'e require from this solution only the circulation r, \\'hich is 2"i timesthe coefficient of log(x2 - y2)1 2 in r:f;:

The rt:lati\e error in this first approximation near the \\'lng \\'Otdd,from (9.4a), appear to be of order .-1 2. Ho\\,e\,'r, one cannot simplyiterate to find the ;:;econd approximation. Calculating ({):, from (9.5) gi\esan cxprc;:;;:;ion that heh,\\('s like tan 1 )' X far from the airfoil section,eorre;:;ponding to the circulaton- Ho\\' . .-\ particular integral of the itera­tion equation therefore bchan:;:; like (.\2 __- 1-2), which \\"fJUld o\-cn\ht:lmthe uniform stream at distances of the order of ...1 in the scale of the inner\'ariables. This di\'ergence, analogous to that encountered by \\'hiteheadin attempting to impro\'e Stokes flow past a sphere (Section 8.3), isindicatiYe of a singular perturbation problem. \\' e treat it in the follO\\'ingsection by constructing a complementary outer expansion. :'Ylatehing\\ill be seen to force a term of rclatiYe order Al into the inner expansion,the situation being analogous to that in boundary-layer theory(Section 7.3).

(9.7)

(9.9)

x ]dyO! - (z - ~f '

(9.8)

L.-:1. 1 ( __1:~A0 --[ 1417" • _1.1'2 - (z - O~

as .-l ,-+ x \\'ith x, .1',;:; fixed

The bound "ortices in the wing must be continued as free yorticesfollo\\'ing the streamlines downstream to infinit\'. Henceforth it is aconsiderable simplification to assume that the angie of attack is small, sothat only linear terms in \ need be retained. Then the trailing \'()rticesmay he taken parallel with the x-axis (Fig. 9.1 c). The potential of suchan assemhlage of \"()rtices is readily calculated using the 13iot-Sa\'art la\\".Thus the outer expansion is found to ha\'e the form

has the proper symmetry in )'. Higher multipoles can be disregarded atthis stage, hecause they will pro\'e to be un matchable according to theprinciple of minimum singularity. The yortex strength r \\ill e\'idently\'anish as .--1 -->- x, and may therefore be tentati"ely expanded as

.\t this point \\"e make a significant departure from the classical lifting­line theory of Prandtl, and so achieye an essential simplification. Prandtlfinds the distribution of circulation yJz) by sohing an integral equation.I {O\\"e\er, to the order of accuracy of his theon' the circulation may befound directly by matching \yith· the inner sol~ltion. This can be doneformally, but the result is obyious from the fact that all cunes enclosingthe same \ortex lines ha\'e the same circulation (Fig. 9.1). In the innerlimit the circulation is giyen by (9.6). Because this is independent of thesize of the circuit, matching \\"ith (9.7) gi\'Cs

That is, to first order the bound yorticity is the circulation about a flatplate of the local chord length in plane flo\\" at the actual angle of attack.In Prandtl's theory the circulation corresponds to a reduced "effective"angle of attack, but the difference is of higher order. Thus the two-termouter expansion (9.8) is found as

(9.6)

(9.4a)

(9.4b)

(9.4c)

(9.4d)

upstream

X =.0 h(z)

T --- T j - 217":1.-:1.- 111(;:;)

lP -- X cos C\ T Y sin C\

lPx < 'YJ at Y = 0,

lPxx -c, Wrr ...L A-2lP zz c= 0

lPy=O at }'=O, I,X!<::h(z)

This transforms the full problem (9.2) to

The solution of this problem \\'hen A = x is

9.3. Lifting-Line Theory by Matched Asymptotic Expansions

In the outer limit (Fig. 9.lc) the \\'ing shrinks to a line of singularities,\\hich must be solutions of (9.2a). \-ortices arc the first possibility that

x ..] d~v/x~ T y2 ...L (z - ~r

(9.lOa)

\Ve now return to the inner problem and seek a second approximation.

172 IX. Some Inviscid Singular Perturbation Problems 9.4. Summary of Third Approximation 173

(9.11 )

In carrying out the inner expansion of (9.IOa), meaningless divergentintegrals are avoided by first integrating by parts to give

,p ,-,.. x j- (~y- ~,x,-l-I r h'm[tan- 1 :;~~r, -1 """ S

(9.lOb)

Then introducing inner variables (9.3) and expanding for large A givesas the two-term inner expansion of the two-term outer expansion for ep:

2-inncr (2-outer),p .= .-l-I[X _.!- <x}' -"h(z) tan 1-1-]_ 1:\ -l-2y JI !!..(o.31I

2-- LIZ - ~

The integral must be interpreted as the Cauchy principal nlue. It is,remarkably enough, the integral that was encountered in Chapter IV(4, 10) in treating thin symmetrical airfoils, and which arises also in otherbranches of fluid mechanics, including supersonic slender-body theory.

The first term in (9.11) has already been matched with the innersolution (9.5). The second term demands a correction of relative order"..].-1, so that the inner solution (9.3a) must have the expansion

Fn~m the. inner solution we have used only the relation (9.6) betweenthe circulatIOn and angle of attack, which is equi\'alent to the two­~ime~sional lift-curve slope. Prandtl has suggested replacing the theore­tical hft-cune slope by the experimental value, which \HJuld account forthe effects of thickness and viscosity. This is a practical example of thefact, m.entioned in Sec:ion 5.4, that the method of matched asymptoticexpansIOns can be apphed even when the inner problem is "impossible,"and soluble only numerically or experimentally.

9.4. Summary of Third Approximation

The preceding analysis has been continued through one more cvcle toobtain the third approximation (Van Dyke, I964b). Here we outline thefeatures of interest.

C~rrying out the 3-term outer expansion of the 2-term inner expansionprondes, by the previous physical argument, the correction Y3 to thevortex stre~gth (9 .. 7). In addition, it shows the appearance at this stage ofthe next higher s1l1gularity in the lifting line. \Ve call this a dh.·oriex,althou.gh it is a dipole with \'Crtical axis, to emphasize its physical inter­pretatIOn as the x-derivative of a vortex, representing the first momentof the distributed vorticity on the wing. Thus the 3-term outer expansionIS found to be (9.10), with h modified by the factor (9.13), plus

,p ...... .--1- I <1\(X, }', z) -- .--1- 2<1>AX, }', z) _!­

\\ith X, } " z fixed

as .-l ----->- ::JJ

(9.12)(9.14)

Substituting into the full equation (9.2a) shows that r:[)2 as well as r:[)l

satisfies the t\\"()-dimensional Laplace equation in X and Y. Hence (9.11)shows that (1)2 is simply the result of reducing the angle of attack in thelocal flat-plate solution from its geometric value :\ to the effective value

(9.13)

This is a familiar result from lifting-line theory. The trailing-vortexsvstem induces downwash velocities in the vicinity of the wing that arec~nstant across the chord at each spanwise station,'and so act to decreasethe apparent angle of attack of that section. However, we have herereduced the calculation to quadratures by recognizing that for large Athe dmvnwash angle is small compared with the geometric angle",whereas classical lifting-line theory leads to an integral equation becausethat fact is not exploited. Our result can be extracted as the second stepin solving Prandtl's integral equation by iteration.

The 3-tenn inner expansion of this result can be calculated afterfurther integration by parts. One issurprised to find that the anticipated Y

next term of order A-3 in the inner texpansion (9.12) is preceded by oneof order A-3 1og A. As usual (ci. Sec- "------I~tion 10.5), the logarithmic term is "'---I ./much the easier of the two to calcu- -----~late. It requires a further change in ~---.X

the effective angle of attack (9.13), and "'--- ./also straightening by the plate of -----------streamline curvature induced near the Fig. 9.2. Streamline eurYature inducedwing by the trailing vortex system ncar wing by trailing vortex system,

(Fig. 9.2).The nonlogarithmic term involves these and other more complicated

local flows, all of which can be found by using complex variables. Then

! ;

174 IX. Some Inviscid Singular Perturbation Problems 9.5. Application to Elliptic Wing 175

the coefficient of all terms in log (x2 :. y2)1/2 provides the distributionof circulation or spanwise lift, which is found to be

SeeNote

13

(9.18)-+ ...

I 2 4 log A 3 .- 2,Z2 4 I [5 '.- A - 1T2 ---;;12 - T _ Z2 + rr2 A2 :2 -I- rr- -- log(l .- Z2)

log 2 3 -_. 2z2 1T]';' -- - log. / __ "

1 - Z2 1 - Z2 V 1 - Z2

rr,

The circulation is found from (9.15) as

I - I, -}-1 / !i.(O.!!I --'- I-J.-2 Iog .1(2h'2 --'- 3hh")~.. J -1 Z _, I 4· I

+ }cl- 2) (2 log ~ .- ~)h'2 7 (3 log ~ + ~)hhl'

rr~;

- 2:!... f [,).,,(0/).-=-.J]Ah(~) d~dz • -1 Z - ~

I da .1 ,IT :2 dz3 J-1 h(O[h(z) '-;- hm] sgn (z - ~) log !, Z - ~ II d~\ T ."

(9.15)

I ntegrating this across the span according to (9.16) yields the lift-curveslope quoted as (1.6) in Chapter I. The first two terms constitute ourearlier modification (9.1 b) of Prandtl's result (9.1 a).

Fortunately, this is one of the rare cases where the lifting-surfacesolution has been calculated. Figure 9.4 shows that Prandtl's solution

Here TJ; is the two-dimensional value (9.6), and ().e is given by (9.13).Then according to the Kutta- Joukowski law the lift-curve slope is A= CD ----t.~

6

Fig. 9.4. Lift-curve slope of elliptic wing.

864A

ProMtls2"; opprox

(9,10)

o Krlenes (/940)lifting - surface

theory

.­,,,,/

//

/ 2"; opprox,

" (9.1b)

2

3'; opprox.(/6)

Modified3';opprox.(/O.21) ,," /0

I /,-

\I /

I ,-

'Yo/ ..

I .. Slender - wing/ .' theory/0

/ .'

4

2

OL...------I.---_---L --L ---.J

o

has thc adYantagc of vanishing at A = 0, though with twice the correctslope according to slender-wing theory. Howcver, our cxpanded form(9.1 b) is actually the more accurate above A = 4. Our third approxima­tion is seen to diverge below A ~ 3. Postponement of that catastrophe,shown by the dash-dot curve in Fig. 9.4, is discussed later in Section10.7.

(9.16)

(9.17)

de .1 r(z)__L = 21T I h(z) --- dz

dcx • 0 r",(z)

Fig. 9.3. Flat elliptic wing.

y

z

a»-

9.5. Application to Elliptic Wing

.\n elliptic \ving of aspect ratio "-1 (Fig. 9.3) has the half-chord

4 --h(z) = - viI - Z2

1T

176 IX. Some Inviscid Singular Perturbation Problems 9.6. Slightly Supersonic Flow past a Slender Circular Cone 177

Fig. 9.6. Slightly supersonic Bow past slender cone.

The oblique shock w.ave relations provide t\\'o conditions to be imposedat the unknown locatIOn of the bow wave:

(9.19a)

(9.19b)

-x

lim rlf} = e2x}-;O

r

M>I

c::>u

~f the bow shock wave. That can be found only by retaining some non­Imea: terms: and recognizing the singular nature of the perturbationsolutIOn. ThiS problem has been treated by both the method of strainedcoordi~ates (Lighthill, 1949b) and the method of matched asymptoticexp~nslOns (Bulakh, 1961). We apply the former technique to a problemso simple t~at we can carry the solution one step beyond previous results.. ~Ve con~lder a slender circular cone of semivertex angle c at zerolIlcldence m a stream sufficiently supersonic that the flow is conical(Fig. 9.6). If the flow is only slightly supersonic, the essential phenomena

are described by the transonic small-disturbance equation (Oswatitschand Berndt, 1950):

Here If' is the perturbation velocity potential, such that the velocityvector is.U g~ad(x --L If). In this approximation the tangency conditio~can be IIneanzed and, because the body is smooth transferred tothe axis (Section 3.8) as '

We have violated the restriction to cusped tips. Consequently, justas in the application of thin-airfoil theory to round noses (Section 4.4)or of boundary-layer theory to sharp leading edges (Chapter VII), theresult is not locally valid. We see that (9.18) breaks down when I J:: z is

A2rl-z)

/'/

Fig. 9.5. Limiting problem for vicinity of round tip.

of order A-2. .-\t any fixed number of radii from the tip of the planformthe flow fails to become plane no matter how great the aspect ratio.Clearly yet another asymptotic expansion is required for that region.The first term would represent flow past a flat semi-infinite plate ofparabolic planform (Fig. 9.5), and would be applicable to the tip of anyflat wing \vhose outline is an analytic curve. In this tip solution theoriginal coordinates would all be magnified by a factor A2.

Still other nonuniformities exist in this complicated problem. It is wellknown that a vortex sheet tends to roll up, so that our outer solution isnot valid far downstream where x is of order A :1:. Again, airfoil sectionsother than the flat plate would ordinarily be treated using thin-airfoiltheory for the inner problem, which would lead to the nonuniformities atleading and trailing edges discussed in Chapter IV, and their morecomplicated counterparts at tips. Nonuniformities would also arise at theroot juncture of a swept wing or other discontinuity in planform, at adeflected aileron, and so on. The case of a smoothly swept wing of cres­cent planform has been analyzed by Thurber (1961) using the method ofmatched asymptotic expansions. He finds that a logarithmic term thenappears in the second approximation.

SeeNote

13

9.6. Slightly Supersonic Flow past a Slender Circular Cone

For a slender fusiform body in supersonic flow, the familiar linearizedtheory (e.g., Ward, 1955) provides no information about the strength

'P = 0 1-'P = _~_ J12 sin 2

a - Ix y+ I ;112--

atr- = tan ax

(9.19c)

(9.19d)

T,

178 IX. Some Inviscid Singular Perturbation Problems9.7. Second Approximation and Shock Position 179

On the surface of thc cone the pressure coefficient is given by

(9.20)

where~ y --i- 1G=--­

JV[2 -- I(9.25b)

9.7. Second Approximation and Shock Position

(9.26)

(9.27)

_ ~ s2(sech-1S)2]

2 (1 - S2)3/2

1 ._- 3s2r3s3Vl ',,2

This is the transonic small-disturbance form of the second-order solution(\'an Dyke, 1952), expressed in terms of a strained variable that is aboutto be detcrmined.

Terms of order 1::" in the diHerential equation now gi\'c for the thirdapproximation

(I - S2)/" _f3' = G2[ seeh-----=-.:.... _ sech ,IS __ ~ ~-c-I~=3 .I' 1 --- S2V 1- S2 2 \/ I - S2

I t is at this stage that insurmountable difficulties arise in the absence ofstraining. The first term on the right is singular like G2(l -- S2)-1/2 ats = I, which would lead to a similar singularity in the velocity. Thisdefect is removed by proper choice of the straining function r3(s)~

The simplest choice, constant straining:

Integrating and imposing the tangency condition and the first shock-wavecond ition yields

(9.22)

(9.2Ia)

(9.21b)

:\0 term of order 1: 2 appears in (9.21 b), because we shall find that thestraining vanishes to that order. For the same reason, the description of

the shock \\'ave is taken in the form

This problem has bccn solved numerically by Oswatitsch and Sjiidin(1954). Wc instead proceed analytically, seeking an asymptotic expansionfor small E. In tacitly assuming that the :\Iach number is fixed, we dis­regard the transonic nature of the problem; and our solution will conse­quently be valid only in the upper portion of the transonic regime.

.\ straightfonv<ud pcrturbation expansion leads to difficulties at thefree-stream :\Iach cone. The perturbation potential has a square-rootzero there in the first approximation, which through differentiationintroduces singularities into higher approximations. We accordinglyintroduce a strained conical \'ariable s by setting

II

I' i [

I(9.29)

(9.28a)

(9.28b)

was successfully employed by Lighthill (1949b) to find the position ofthe shock wave, which is as far as he carried the solution. However, thiseliminates the nonuniformitv near the shock wave only to introduceanother on the axis, \vhich' would make it impossible 'to impose thetangency condition there in the next approximation. This difficulty isavoided by choosing instead the linear straining

The second shock-wave condition can now be used to find the depar­ture of the shock wave from the free-stream :Vlach cone. At the shockwave, according to (9.21b), (9.22), and (9.28b)

(9.23)

(9.25a)

\\'hen the npansions (9.21) are substituted into the diHercntialequation (9.1 9a), terms in f2 yield the con\'Cntional linearized equation

for /1 :

The solution satlstYlIlg the tangency condition (9.19b) and the first

shock-\vavc condition (9.19c) is

fl(s) = --(sech-1 .I' - \/1--=-.1'2) (9.24)

The second shock-wave condition (9.19d) is automatically satisfied,which is the reason that no term of order 1

2 is required in (9.22).Terms of order £1 in the diHerential equation no\\' give for the second

approximation

II,I

9.8. Third Approximation for Pressure on Cone

which is the transonic small-disturbance form of the result found bvLighthill (1949b) and Bulakh (1961). This is the extent to which theycarried their solution, \\hich applies to a slender cone of any crosssection. The following section coyers new ground.

181

(9.38)

(9.39)

(9.36)

9.8. Third Approximation for Pressure on Cone

Expressing the results in terms of the original parameters, we findthat the shock wave lies at

,/.1/2 =-1 ~ __ 1 _ ~[(y ~_!)e2]" _ ~[(y --;-.J.~~];l _ ...x 8 .112 - 1 12 .lJ2 - I

~ingularities in the right-hand side and hence also in the fourth-order\c1ocity are ayoided by choosing the next term in the straining as

The factor s again seryes to prennt the introduction of a nonuniformityon the axis..\t the shock wave these results give

s = 1 - - ~-G2el -i- (k:, - ~-OJ)e6 -I ... (9.37)

Then substituting into the second shock-wave condition (9.19d) yields

k =-~OJ:, 12

TI

(9.31)

(9.30)

I I

2 vi - .I"

180 IX. Some Inviscid Singular Perturbation Problems

and substituting into (9.19d) yields

------ 4v'2(~G" -- k3) -i-- G =, G k3

The solution is

With the aboye straining, Eq. (9.27) for the third approximationsimplifies to

t- , ,.~. G-" [ seell -1 .I _. seeh.--1 .I(I - s")f" _·2 .3 .I I -- .12 VI - .12

Integrating and imposing the tangency condition and first shock-wayecondition gi \es

13(.1) = }G"[ (~ - log 2) viI=- .,-2 - (log 2 - I) seeh-1 .I

,---- I -- I"--- \' I -- .I" seeh- I .I - 4-(.-Vl -- .I" T .. .,)(seeh-1s)-V I --- S'

1st approx.

1.2

'0

~ _8~

~C\j

+\.)"I~

.4

\

\\ 0 Numerical rOswalitsch 8 SjOdin 1954)

\\

t" 0Detachment "-

"-2 nd appro~

1.6

2.0

(9.33)

(9.32)I ;'"(seeh-1 s)2]

- 2(l - .1")3 /2

The integral appearing here has the properties

.1 -v1= t2 ,IT- t" 1T"I ~--- log --~~ dt --- 4log" .I - (-2 --L t log2 2 - log 2)• s t t d

- {S2 logs --'--- 0(.12) (9.34a)

--- i(l- .1")3/2 log (1 - .12) + 0[(1 - .12)312]

(9.34b)

Terms of order E8 giye the equation for the fourth approximationwhich, with the right-hand approximated near s ~ I, becomes

(1 -s2)r +1/ ~ - G3[(~ -;- 2.!~-.·) 1 " --'---llog(l -.I") , ... ] (9.35)4 .I 4 G -V 1- .I'

oo

Fig. 9.7. Pressure on slender cone in transonic flow.

8

182 IX. Some Inviscid Singular Perturbation Problems 9.9. Hypersonic Flow past Thin Blunted Wedge 183

(9.41)

(9.42)dif; == pu dy - pv dx

y

t i'

I,I

I III IM=oo

c:::> - __x

Fig. 9.9. Notation for thin hyperbolic shock wave.

The conservation equations are

In an inverse problem the body is most conveniently found from thevanishing of the stream function if;. We introduce it in the usual way forcompressible flow by setting

the wedge is thin. Although this simplifies the analysis, it has the remark­able effect of splitting the entropy layer into two layers, so that we needan outer, middle, and inner expansion.

Consider the flow behind a hyperbolic shock wave (Fig. 9.9). Let itsasymptotic slope 08 be small. We take its nose radius to be 08 3 , in orderthat the first approximation be that for the sharp wedge. Thus we carryout a parameter perturbation for small nose radius, whereas Yakuracarried out a coordinate perturbation for large x. The shock wave isdescribed by

where the shock wave is prescribed and the body sought. This approachwas carried out for flow of a perfect gas at infinite Mach number byYakura (1962) using the method of matched asymptotic expansions. Wespecialize his solution for a blunted wedge to the particular case whenC --- ("I + 1)e2

-;.f + 2 log viM2 - Ie'" (2 log 2 - I) + M"2_. 1

+ (~; - ~)[~;~)~J +... (9.40)

Fig. 9.8. Entropy layer on blunted wedge.

9.9. Hypersonic Flow past Thin Blunted Wedge

It was pointed out in Section 5.3 that slight blunting leads to non­uniformity downstream on a body in inviscid supersonic flow. Away fromthe nose, the flow is nearly that for the sharp body almost everywhere. Atthe surface, however, the entropy-because it is constant along stream­lines-has the value for a normal rather than an oblique shock wave.

As anticipated, these results are asymptotic expansions for large valuesof the transonic similarity parameter ("I + l)s2j(M2 - 1). The firstthree approximations for surface pressure are compared in Fig. 9.7 withthe numerical solution of Oswatitsch and Sjodin.

The surface pressure coefficient (9.20) can be written in accord with thetransonic similarity rule of Oswatitsch and Berndt (1950) as

Certain other flow quantities are necessarily also different there. Thechanges take place across the entropy layer (Fig. 9.8), whose thickness is,for plane flow, of the order of the nose radius.

The effect of slight bluntness was first treated as a singular perturba­tion problem by Guiraud (1958). He considered the direct problem,where the body is given. The analysis is simpler for the inverse problem,

continuity:

x-momentum:

y-momentum:

entropy:

(pu)x + (pv)y == 0

p(uux + vu y) + Px == 0

p(uvx +vvy) + P'I == 0

u(pjp')x + v(pjpY)y == 0

(9.43a)

(9.43b)

(9.43c)

(9.43d)

I I

184 IX. Some Inviscid Singular Perturbation Problems 9.10. Small-Disturbance Solution for Blunted Wedge 185

(9.49)

(9.50)

(9.48)

(9.47a)

(9,47b)

(9.47c)

(9.47d)

(9.47e)

2x + (y - 1)0/1Yl = Y + 1

_tPtPl =-e

y + 1PI = --,

Y - 1

u "" 1 + e2u1(x, tPl) + e4u2(x, tPl) + ..

v "" ev1(x, tPl) + e3v2(x, tPl) + .P "" e2P1(x, tPl) + e4pz(x, tPl) + ..P "" Pl(X, tPl) + e2pz(x, tPl) + .y "" eYl(x, tPl) + e3Y2(x, tPl) + .

Here

Thus there is a singularity at If;I = O. One finds that the pressure isregular there, so that the density is singular at the surface of the bodylike I NI2• Likewise 'I) is regular, but y is singular like 1/If;I and 1I likeINI2 • These nonuniformities are compounded in the third and higherapproximations.

The linear equations for the second approximation can also be solvedin closed form. Our present purpose is served, however, by noting thatthe entropy integral (9.44e') gives for the second approximation

is the stream function referred to a typical thickness of the shock layerfor x = 0(1).

Substituting these expansions into the full problem (9.44) and (9.46)and equating like powers of 8 gives for the first approximation a nonlinearsystem whose solution is simply that for the sharp wedge:

9.10. Small-Disturbance Solution for Blunted Wedge

As the perturbation parameter 8 vanishes, the above problem reducesto the hypersonic small-disturbance problem for a sharp wedge (Hayesand Probstein, 1959, p. 47). The initial conditions (9.46) suggest im­proving upon that basic solution by expanding in powers of 8 Z• Wetherefore make the conventional expansion of hypersonic small-disturb­ance theory (Hayes and Probstein, 1959, Chapter II), setting

(9.46)

(9.45)

(9.44e')

dy exdx vx2 - e2

oy v(9.44a)streamline slope: ox u

continuity:oy

(9.44b)otP pu

momentum normal to streamline: 8v + op = 0 (9.44c)ox otP

Bernoulli integral: u2 + v2 + ~_1!... = I (9.44d)y - I P

entropy integral: ~ =f(tP) (9.44e)

2 2 x2

P = y + Te (I + e2 )x2 - e2

y + 1p=-­y -1

u = 1 _~_e2 x2

y + 1 (1 + e2)x2 - e2

2 xvx2-e2v---e

- y + 1 (1 + e2)x2 - e2

y = evx2 - e2

Hence the entropy integral (9.44e) is evaluated at the shock wave as

the Rankine-Hugoniot relations provide the initial conditions

Let the flow variables be made dimensionless by referring velocitiesand density to their free-stream values U and p:r.; , and pressure top:r.; UZ. This leaves the preceding equations unchanged. Then becausethe slope of the shock wave is

The last of these expresses the fact that entropy is constant along stream­lines between shock waves, which is equivalent to conservation ofenergy. It is useful to exploit this property by taking the stream functionas an independent variable, reversing the roles of y and If; by applying thevon Mises transformation of boundary-layer theory. Then the differen­tial equations (9.43) become

186 IX. Some Inviscid Singular Perturbation Problems9.12. Inner Expansion for Entropy Layer 187

(9.56)

(9.55)

(9.54e)2

u1(x, 00) = - y +I

Yl(X, (J) = y -- 11 {J as {J ->- 00 (9.541')y+

_ 2Yo = y + 1 x,

_ 2 _ y + 1 {J2 1/'1

PI = Y + 1 ' PI = y -1(1 + (J2)

_ 2 2y (1 + {J2)I/Y

U1 = - (y + 1)2 - (y + 1)2 ---;r;-

y - 1 - foo [.1 + t2)1/'1 ] I

Yl = y-t--dtP - ii' (t2 - 1 dt\

CY-l

8{J Pl

Again it can be shown that P2 is regular; hence, in view of (9.55), P2 issingular like (J-(2-2/Y).

This expansion too is invalid at the surface of the body, where thevelocity component u becomes infinite and the density is zero in thefirst approximation. For the second approximation the entropy integral(9.44e') gives

These can be solved in succession, giving

(9.51 )

(9.52)

(9.53a)

(9.53b)

(9.53c)

(9.53d)

(9.53e)

u ,...., 1 + e2u1(x, (J) + e4u2(x, (J) + .v ,...., ei\(x, (J) + e3v2(x, (J) + .P ,...., e2Pl(x, (J) + e4p2(x, (J) + .p - pl"<, (J) + e2p2(x, (J) + .Y ,...., eyo(x, (J) + e2Yl(x, (J) + e3Y2(x, (J) + ...

We call this the middle variable, because still another magnification willbe required to reach the surface of the body.

Comparison with the outer expansion suggests a middle expansion ofthe form

This suggests introducing the new independent variable

9.11. Middle Expansion for Entropy Layer

The straightforward small-disturbance expansion breaks down nearthe surface because the solution for the sharp wedge is not a valid firstapproximation in the entropy layer. We treat the nonuniformity by themethod of matched asymptotic expansions.

Because the nonuniformity occurs along a line, as in boundary-layertheory, only the normal coordinate tPl is to be magnified. According to(9.47d), (9.49), and (9.50) the outer expansion of the density behaves forsmall tPl like

Substituting into the full equations (9.44), and matching with the leadingterms of the outer expansion (9.47) according to the asymptotic matchingprinciple (5.24) gives, for the first approximation,

This suggests introducing the new inner variable

(9.57)

(9.58)

9.12. Inner Expansion for Entropy Layer

The nonuniformity of the middle expansion shows that a thirdasymptotic expansion is required to complete the solution. According to(9.53d), (9.55), and (9.56), the middle expansion of the density behavesfor small {J like

(9.54c)

(9.54a)

(9.54d)

(9.54b)

2Yo(x, 00) = y +1 x

i!lx, 00) = y ~ 1

Pl(X, 00) = ~-Iy+y+I

Pl(X, 00) = --1y-

°Yo = 00{J ,

_ °YoVI = ax'

188 IX. Some Inviscid Singular Perturbation Problems 9.12. Inner Expansion for Entropy Layer 189

(b)

...~t----L -----I..~

( b)

(a)

(a)

"--L_

Disparate reference lengths for blunted wedges. (a) Thick wedge. (b) ThinFig. 9.11.wedge.

Fig. 9.10. Displacement effect of blunting. (a) Direct problem. (b) Inverse problem.

even though it moves closer near the nose; in the inverse problem(Fig. 9. lOb) the body must move instead. Another similarity betweenentropy layers and viscous boundary layers is evident from (9.54c) and(9.60b): To a first approximation the pressure is constant across thelayer.

In analyzing the nonslender wedge, Yakura (1962) has calculated thenext approximation, which adds a term in X-I to (9.62). This suffices todescribe the body accurately up to within a few radii of its nose. Hisanalysis requires only the outer and inner expansions. Our middle ex­pansion is introduced by the double limit process (cf. Section 5.3) inwhich the slope and nose radius vanish simultaneously. Fig. 9.11 indi­cates how this produces the three disparate lengths that characterize thethree expansions, where we have tacitly assumed L to be of order unity.

where r is the gamma function. The first term is the result for thesharp wedge, indicated by a dotted line in Fig. 9.9. The secondterm is the asymptotic value of the displacement thickness of the entropylayer. As in a viscous boundary layer, the mass flux is reduced becausethe entropy, and hence the temperature, is increased by blunting, andthe density correspondingly reduced. Hence in the direct problem(Fig. 9.lOa) the shock wave is forced away from the body downstream

(9.62)

(9.61)

(9.60c)

(9.60a)

(9.60e)

(9.59a)

(9.59b)

(9.59c)

(9.59d)

(9.5ge)

(9.60b)

(9.60d)

R1 = Y +J. (1 + lfI2)I/Yy-l

2y 1VI = - (y +1)2 (f + lfI2)1/"/

u"" 1 + e2- 2/Y L\(x, lfI) + .v"" eV/-c, lfI) + .P "" e2Pl(X, lfI) + .p .-.., e2/YR1(x, lfI) + .y "" eYo(x, lfI) + e3-2/YYl(x, lfI) + ...

[2 y - 1 2 21 fa:; dt ]

Y ,..., e y + 1 x - y + 1 e - Y 0 (T+ t2)IIY + .

= e[_2_ x _ y - 1 vi; T(y-l - t) e2- 2/y + ]y + 1 y + 1 2 T(y-l)

2Yo = y +1 x,

2PI = y+l'

Y _ _ Y - 1 f"" dt1 - Y+ 1 'P (1 + t2)I/Y

avo = 0alfl '

aPI = 0elfl '

!l - -~_IY - J.IY _~_R1y - y -;- 11y + 1, 1 + lfI2'

U - _ _ L_ Pl1 - Y _ 1 R

1'

aYI81fi R 1 '

which is the stream function normalized with respect to the nose radiusof the shock wave.

Comparison with the middle expansion suggests an inner expansionof the form

Substituting into the full equations and matching with the middleexpansion yields

These equations can be integrated in succession to find

This last expansion is valid at the surface of the body. In particular, itgives the asymptotic shape of the body that supports the hyperbolicshock wave as

190 IX. Some Inviscid Singular Perturbation Problems 9.13. Composite Expansions for Blunted Wedge 191

Although this is equivalent, the previous form would be preferred for itssimplicity. It also corresponds to the result of Yakura for the thickwedge.

For the ordinate y, our three approximations may be written as

4

o\.... Yakura (1962)

p2

o '- Surface Inner _

~ ------,,-,/

I

~

'ShOCk wave III I101I I

hoi/, I

/ IComposite"""" / 0 I

Y I,..Sharp wedge ;",/ 0 Outer

............../

/"/"

./Middle ./

//

//

/I//III

o

y

Fig. 9.12. Density distribution at 24.5 shock radii downstream on 20° wedge,

M = 00, y = 7/5.

Figure 9.12 shows the distribution of density between the shock waveand body given by (9.63) and (9.66) at 24.5 radii downstream from thenose of a hyperbolic shock wave with 8 = 0.437, corresponding to a wedge

(9.63)

(9.64)

(9.65c)

(9.65a)

(9.65b)2 y - 1\ feD [(t 2 + e4)1/Y ] l'y ",--ex + --\1/1 - -- - 1 dt

Y + 1 y + 1{. 'P t2

2 y - 1 feD ( 1'4 )l/YY ",--ex- -- --- dty+l y+l 'P t2 +e6

2 y-lY '" --- ex + --.jJ

y+l y+l

mner:

middle:

outer:

which is valid in both subdivisions of the entropy layer. It is clear howthis expression assumes different forms in the middle and inner limits.Repeating the process to combine this with the outer solution leaves theresult unchanged. It is therefore the desired uniform approximation,valid in all three regions.

If instead we apply the additive rule (5.32), the uniform approximationis found as

'" y + 1[( .jJ2 )l/Y + (.jJ2 + e6 )l/Y _ (£...)l/Y]P Y - 1 .jJ2 + e4 e4 e4

9.13. Composite Expansions for Blunted Wedge

In order to display the variation of flow quantities throughout thedisturbed region between the shock wave and body, we want to combinethe separate expansions into a single composite expansion. This providesa useful illustration of the technique of forming a composite expansionfrom more than two components (d. Exercise 5.4).

Consider first the density. Applying the multiplicative rule (5.34) tothe middle and inner solutions yields

The result of either additive or multiplicative composition is unneces­sarily complicated. A simpler form is found by inspection as

and this corresponds to the result that Yakura found by additive com­position of his two expansions.

SeeNote

6 2 y - 1:' feD [(t 2 + e4)1/Y ] Iy ",--ex + --.jJ - --- - 1 dtl,

Y + 1 y+ 1. 'P t2 + 1'6, 1(9.66)

Iof 20-degree semivertex angle. Also shown are the outer, middle, andinner approximations, and the result of Yakura in which the wedge isnot assumed to be thin. The discrepancies in surface location indicatethe inaccuracy arising from such a large value of e.

Yakura (1962) has applied the method of matched asymptotic expan­sions also to the axisymmetric problems of the blunted cone that supportsa hyperboloidal shock wave, and the body that produces a paraboloidal

192 IX. Some Inviscid Singular Perturbation Problems Exercises 193

shock wave at M = 00. The latter problem has been treated with equalsuccess by Sychev (1962) using the method of strained coordinates.However, it appears that the solution for the blunted wedge cannot befound by straining the coordinates.

EXERCISES

9.1. Elongated nonltfting body. As a model of thickness effects for a wing ofhigh aspect ratio, consider incompressible potential flow normal to the axisof a slender smooth body of revolution (Fig. 9.13). Show that to a first approxi-

,

$'z

Fig. 9.13. Flow normal to slender body of revolution.

mation the flow is plane at each section, but that straightforward iterationleads to a difficulty analogous to Whitehead's (Chapter VIII). Find the nextapproximation using the method of matched asymptotic expansions. Applyyour solution to the ellipsoid of revolution, checking with the rule of Munk(1929) that the surface speed on any ellipsoid is the projection of the maximumvelocity onto the tangent plane. Apply your solution also to the pointed parabolicspindle. Explain why no correction is necessary at the tips of the ellipsoid. Deviseand apply a correction for the tips of the spindle.

9.2. Lifting wing with sharp tips. Calculate from (9.15) and (9.16) the second­order spanwise loading and lift-curve slope for the wing of lenticular planformbounded by two parabolic arcs. Devise a correction that renders the solutionuniformly valid near the tips. Discuss the corresponding correction for themiddle of a diamond planform.

9.3. Near-equilibrium flow over a wavy wall. Consider steady plane flow ofan inviscid gas in vibrational or chemical nonequilibrium. At an equilibrium

i

Mach number of V~, the linearized equation for the perturbation velocitypotential is (Vincenti, 1959)

where 0 < A < I, and e is proportional to the relaxation time. Iterate uponthe familiar result for equilibrium flow (e = 0) over the wavy wall y = T sin xto find the correction of order eT. It is helpful to use the oblique coordinatesg = x - y and T) = y (cf. Section 6.4). Correct the result in its region ofnonuniformity far above the wall. Form a uniformly valid first approximation.Compare with Vincenti's solution.

SeeNote

3

i.. II I

Chapter X

OTHER ASPECTS

OF PERTURBATION THEORY

10.1. Introduction

This last chapter is devoted to various special topics in perturbationtheory . We first consider several methods, other than the two standardones employed in previous chapters, for dealing with singular perturba­tion problems. Next, we discuss the curious role of logarithms in per­turbation expansions. We then examine in some detail the fascinatingquestion of extracting accurate results from a divergent or slowlyconvergent series. Finally, we describe ways of joining two different per­turbation expansions in cases where they cannot be matched in the senseof the method of matched asymptotic expansions.

10.2. The Method of Composite Equations

We saw in Chapter VIII how Oseen corrected the nonuniformity inStokes' approximation for flow at low Reynolds numbers by partiallyincluding the neglected convective terms. The same idea can be appliedto any singular perturbation problem in which the nonuniformity arisesfrom the differential equations. One would proceed as follows:

(a) Identify the terms in the differential equations whose neglect inthe straightforward approximation is responsible for the non­uniformity.

(b) Approximate those terms insofar as possible while retaining theiressential character in the region of nonuniformity.

(c) Try to solve the resulting composite equations.

This ad hoc procedure has been successfully applied also to the positionof shock waves by Lighthill (1948), and to the vorticallayer on an inclinedcone by Cheng (1962).

However, the method of composite equations seems always to be

195

I

, II

I

I

'II

196 X. Other Aspects of Perturbation Theory 10.3. The Method of Composite Expansions 197

This is invalid in the boundary layer, where applying Prandtl's procedureto the full equation yields instead

(10.5)

(10.6)

v[e,pn 'I 'I'I - 27) :7) (,p;; + ,p'ill)] + [g2(,p~ :7) - ,pn :g)

+ 2(Nn- 7),p~*,p;" + ,p'ln) = 0

But surely no amount of insight would have suggested this choice.

multiple of the potential equation. Thus, knowing the solution (lOA),we c~n constru.ct an unlimited number of alternative composite equationsthat It does satIsfy, one of the simplest being

10.3. The Method of Composite Expansions

A more promising method of directly obtaining a uniformly validsolution is to substitute an assumed composite expansion (cf. SectionSA). Latta (1951) has shown that one can often deduce the requiredform of the expansion from an examination of the inner ("boundarv­lay~r") solution. This provides a finite number of special functio~s,which are taken as the basis for the expansion.. We illus~rate this ~ethod for .Friedrichs' model problem (5.1). Themner solutIOn (5.6b) IS found to Involve the function ex/e. Because thisreproduces itself upon differentiation, no other special functions arerequired. Latta therefore assumes a composite expansion in the form

Substituting into (5.1) and collecting like powers of e and factors ofe X

/' yields

II' = a, h' =0, 11(0) + h1(0) = 0, Ul) = 1I(1O.7a)

1m' == - f:n~l =0, h",' = h;~_l = 0, InJO) + hm(O) = 0, 1",(1) = 0,

m?;;2 (1O.7b)

(10.1 )

(10.3)

inferior to the two more systematic methods discussed earlier. Itsdisadvantages are apparent in flow at low Reynolds numbers (ChapterVIII), where the inner and outer expansions are relatively simple com­pared with the composite solution of the Oseen equations. Thus it hasbeen superseded in the shock-wave problem by the method of strainedcoordinates (Section 9.6; Lighthill, 1949b), and in the vortical-layerproblem by the method of matched asymptotic expansions (Munson,1964).

The shortcomings of the method of composite equations may beillustrated by reconsidering the boundary layer on a semi-infinite flatplate (Chapter VII). We profit from experience (Section 7.14) and useparabolic coordinates. The Navier-Stokes equations give for the streamfunction the equation of Exercise 8.1. The straightforward approxima­tion consists in setting v =--= 0, which yields

We now seek a composite equation that includes both of these approxi­mations. Only the first term in (10.2) is absentfrom (10.1). Adding it to(10.1) yields as a composite equation

Because our coordinates are optimal for this problem (Section 7.13), wemight suppose that the appropriate solution of this equation is Blasius'solution expressed in parabolic coordinates:

(lOA) Solving these gives!l = (1 ~ a) + ax, hI = -(I ~ a), and!m = hill = O.Hence the solution is

This result is supposed to be uniformly valid in the interval of interesto ~ x ~ 1, with an error smaller than any power of e. This is seen tobe true by comparison with the exact solution (5.2).

However, trial shows that this is is not a solution. In fact, no simplesolution exists. In tackling simultaneously the difficulties of the innerand outer regions, we have set ourselves an intractable problem.

Of course, the composite equation is not unique. To our form (10.3)may be added any of the neglected terms in the full equation, and any

f(x; e) ~ (1 - a)(1 - e-XIe ) + ax (10.8)

198 X. Other Aspects of Perturbation Theory lOA. The Method of Multiple Scales 199

Xis as well as x itself. We therefore assume an asymptotic expansion ofthe form

Although these are nominally partial differential equations, the first isformally identical with the first inner equation [(5.5) with I:: = 1], andthis is always the case. Its solution is

(10.12)

(10.13)

(10.14)

(10. lOa)

(10. lOb)

(1O.lla)

(1O.1Ib)

m;?3 (1O.llc)

x == xjs

Each approximation shall be no more singular than itspredecessor-or vanish no more slowly-as s -+ 0 forarbitrary values of the independent variables. The sameshall be true of all derivatives.

flXX + flX = 0

f2XX + j~x = a - 2flXX - fIX

fm/IX + fmx = - 2f(m-llxx - f(m-I)x - f'm-2Ixx,

where

This expansion is intended to hold uniformly not only for 0 ~ x ~ Ibut also for 0 ~ X ~ 00. Certain restrictions will have to be imposed toassure uniformity for X -+ 00, corresponding to s -+ 0 with x > 0; andthis is in fact the crux of the method.

Of course we have replaced ordinary by partial differential equations,but it will be seen that no real complication has been introduced. Substi­t~ting into (5.1) and equating like powers of s yields the system of equa­tIOns

The analogy with Lighthill's principle (6.1) is evident.Just as in the method of strained coordinates, it is possible to achieve a

uniform first approximation by examining the second-order equation,without solving it. In our example, the second-order equation (10.11 b)becomes

where the functions of integration cl(x) and dl(x) are still arbitrary.At this point the argument assumes a striking resemblance to that of

Lighthill's method of strained coordinates (Chapter VI). If our com­posite expansion (10.10) is to be uniformly valid, the ratios f2Jl'f312' and so on, must remain of order unity as E ---+ O. We thereforerequire that

, I-~

The change of argument permitted by the function g is essential in thiscase, whereas it was unnecessary in the previous example. In fact, onefinds thatg(x,y) must be! [(x2 + y2)1/2 - x], !the square of the paraboliccoordinate y), which leads naturally to the optimal coordinates of Kaplun(Section 7.13). The reader can complete the solution, or see Latta (1951)for details.

Further complications may arise, particularly in nonlinear problems.In the model problem of Section 6.2, the rapidly varying functionssuggested by the inner solution (6.18) and its derivatives are infinite innumber. Also, examination of the inner solution does not always indicatethe correct form of the special functions.

lOA. The Method of Multiple Scales

The difficulties just mentioned can be avoided by assuming a moregeneral form for the composite expansion. As discussed in Section 5.3,a perturbation solution is singular because it involves two disparatelength scales for one of the coordinates. Accordingly, Cochran (1962)and Mahony (1962) suppose that the solution depends upon that coor­dinate separately in each of its two scales. That is, the sensitive coordinateis replaced by a pair of coordinates, thus increasing the number ofindependent variables. One can then assume a conventional asymptoticexpansion. A similar idea has been advanced by Cole and Kevorkian(1963).

We again use Friedrichs' model (5.1) for illustration. It is clear fromthe inner solution or other considerations that the solution depends upon

tjJ(x, y; R) ,....., [fl\:, y) + R-wUx, y) + ...]+ erfc[g(x, y);R-l]1/2[hl (x, y) + R-1/2h2(x, y) + ...]+ exp[g(x, y)jR-l][kl(x, y) + R- 1/2k 2(x, y) + ...] (10.9)

In more complicated problems the inner solution, together with itsderivatives, may suggest more than one rapidly varying special function.For example, if we consider the linearized Oseen equation (8.16) inCartesian coordinates for a semi-infinite flat plate, the boundary-layersolution contains the complementary error function erfc(yj2R-1/2xl/2);and we must also include its derivative exp( -y2j4R-l x), higher deriv­atives giving nothing new. Also, it may be necessary to generalize theargument of the special functions, allowing it to be a smooth functionof all the coordinates that is to be determined in the course of theanalysis. As a result of these two modifications, the composite expansionfor the stream function in Oseen flow past the flat plate is taken in the form

SeeNote

14

200 x. Other Aspects of Perturbation Theory 10.5. The Prevalence of Logarithms 201

10.5. The Prevalence of Logarithms

The devotee of perturbation methods is continually being surprisedby the appearance of logarithmic terms where none could reasonablyhave been anticipated. The recent history of fluid mechanics records anumber of well-known investigators who fell victim to the plausible

assumption that their expansion proceded by powers of the smallparameter. As sugge.sted in Chapter III, one must be ready to suspectthe presence of loganthms at the first hint of difficulty. Their presence inboth parameter and coordinate perturbations has been amply illustratedby examples in previous chapters. See, for example, (1.2), (1.4), (1.6),(3.24), (3.27), (7.47), (8.38), (8.48), (9.18), and (9.40). AlthoughStewartson (1957, 1961) has encountered loglog's in various viscousflow problems, they are blessedly rare.

Logarithmic terms arise from a variety of sources. In some problemsthey appear naturally as a result of cylindrical symmetry. This is true,for e~ample, of s~en~er-body theory (Section 9.8), where logarithmsdescnbe the essential s1l1gular nature near the axis of such functions as theBessel. function K o and the inverse hyperbolic function sech- I thatoccur 111 more complete theories.

Anot?er common sourc.e of logarithms is a small exponent. This isexe~p~lfied. by (4.50), which shows how the expansion is nonuniform.T~ls .sltuatlOn alwa~s arises at a slight corner where the equations areelliptiC, as for the biconvex airfoil of Section 4.7. \Vhen one is confidentthat this is the correct diagnosis, it may be possible to render the solutionuniformly valid simply by replacing the logarithms with near-integralpowers (Exercise 9.1; ~1unson, 1964).

In inverse coo.rdinate expansions for viscous flow, logarithms seemoften to be reqUIred to ensure exponential decay of vorticity (Section7.11). In other problems their source is even more obscure. :\Ianv ofthese .ar~ singular p~rturbationproblems. One can only philosophize 'that?escnptl~n by fractIOnal powers fails to exhaust the myriad phenomena111 the Ul1lverse, and logarithms are the next simplest function.

Expansions. in this latter group typically begin with simple powers ofthe pertur.batlOn quantity, its logarithm entering linearly only in thesec.ond, thud, or even fifth term. Thereafter logarithms appear regularlyto 1I1tegral powers that increase successively, though often only at alter"­n.ate st~ps. In a si~gular pertur.bation problem the appearanc~ of loga­nthms 111, say, the 1I1ner expansIOn forces them into the outer expansionat a l~ter stage .through the shift in order of terms that takes place inchang1l1g from 1I1ner to outer variables.

\Vh~n. a logarithn: thus makes its first appearance in a higher-orderterm, It IS accompal1led by. an algebraic term containing the same powerof the perturbatIOn quantity. The logarithmic term is much easier toc~lculate.than its ~Igebraic companion, because it satisfies a homogeneousdIfferentIal eq~atl.on. Nevertheless, the two terms must be regarded ast?gether constltutmg a sll1gle step in the process of successive approxima­tion. That the two terms are intimately related is evident from the fact

Ii

(10.15)g(O) = 0

where g is a smooth function that is positive when its argument is posi­tive, and is determined in the course of the analysis. The reader canverify that in the present example the principle (10.13) requires thatg'(x) = I. In general, of course, g must vanish at the location of thenonuniformity, € may be replaced by some other power or function ofthe perturbation parameter, and the asymptotic sequence may involveother functions of c than integral powers. Partial differential equations areincreased in degree by only one as a result of introducing the new variableX, but the function g must depend upon all the original independentvariables. Some illuminating examples are given by Cochran (1962).

This method and the one discussed in the previous section are seento convert the inner solution of the method of matched asymptoticexpansions into a uniformly valid approximation. They therefore ap­parently provide an answer to the question of how to generalize theconcept of optimal coordinates (Section 7.15). It is the role of the func­tion g to provide the freedom necessary to make the coordinates optimal.It is possible that, with further development, one of these methods willreplace those used previously as the most versatile and reliable techniquefor treating singular perturbation problems.

Any particular integral will include a term (a - c1')X. This would makethe second approximation more singular than the first as X -+ 00, andwill therefore be removed by choosing c1'(x) = a. Similarly, the re­maining particular integral will include a term - d1' Xe- x, which wouldmake the derivative f' vanish more slowly in the second approximationthan the first as X -+ CIJ. We therefore annihilate the right-hand side bychoosing also d1'(x) = O. The values of the constants C1 and d1 can nowbe found by replacing X with X;C in (10.12) and imposing the boundaryconditions in (5.1). The result is just (10.8), which was found by Latta'smethod.

At least one generalization is required in more complicated problems.As in the preceding method, it may be necessary to generalize the magni­fied coordinate (10.1 Ob) by setting

X = K(·:L- ,e

202 X. Other Aspects of Perturbation Theory 10.6. Improvement of Series; Natural Coordinates 203

(10.17)

(10.16)

(10.21 )

11 S e1(Sn) e12(Sn) e13(Sn)nI 4.00000002 2.6666667 3.16666673 3.4666667 3.1333333 3.14210534 2.8952381 3.1452381 3.1414502 3.1415993 (10.18)5 3.3396825 3.13968+5 3.14164336 2.9760462 3.14271297 3.2837385

:L = ! _ ~(1 _ ) _ ~(1 _ )2 + 155(1 _ )3 _ 15,235(1 _ )4e 2 3 yJ 18 yJ 54 7J 648 7J

+ 35,416(1 _ )5 _ 5,656,6~(1 _ )6 + ...243 7J 5832 yJ

the first seven terms actually contain 1T to more than six significantfigures, as is shown by forming the following array:

Here the third column has been formed from the second by applyingthe nonlinear transformation

e1(Sn) = Sn+1Sn-1 - Sn2

(10.19)Sn+l + Sn-1 - 2Sn

Repeating the process yields the subsequent columns, the last of which iscorrect to six figures. Applying the transformation once more along thelower diagonal improves the accuracy further. These and related trans­formation are discussed in detail by Shanks.

An important preliminary consideration in a perturbation problem isthe proper choice of the expansion quantity (cf. Section 3.1). Forcoordinate expansions, this means using a system of natural coordinates.This is by no means as precise a concept as that of optimal coordinates.However, some problems are clearly adapted to a particular coordinatesystem, which is generally found to yield the most satisfactory results. Agood example is the superiority of parabolic to Cartesian or othercoordinates in problems involving parabolic boundaries. For the limitingcase of a semi-infinite flat plate, the role of parabolic coordinates inboundary-layer theory was discussed in Chapter VII.

A second example is the inverse blunt-body problem (Fig. 3.5) for aparaboloidal shock wave. We introduce parabolic coordinates g, 7Jaccording to

x + (y = lb[(~ + iYJ)2 + 1] (10.20)

so that a shock wave of nose radius b is described by 7J = 1. ThenCabannes' series (3.25) for the stream function near the axis, with itserratic changes of sign, is recast (Van Dyke, 1958b) into. one that alter­nates smoothly after the first two terms:

j I

deL = 21T(1 - 0.314 - 0.074 + 0.088 + ...)dlX

whereas Krienes (1940) calculates the exact value as 4.55. Two termsgive 6 per cellt error, the logarithmic term increases that to 15.per cent,but its algebraic companion reduces it again to 3 per cent. Thus 1t app~arsthat the final terms in such series as (1.2) and (1.4), though of theoreticalinterest, have no practical value until the next term i~ calculated. Similarremarks apply to the Nth power of the logarithm, wh1ch must be groupedas a single term with its N companions containing lower powers of the

logarithm. . .We saw in Section 8.6 that the occurrence of loganthm1c terms can

severely limit the range of applicability of a perturbation series. Wereturn to this question later, in Section 10.7.

which converges, but with painful slowness. The seventh partial sum,shown below in the second column, is correct to only one figure, and400,000 terms would be required for six-figure accuracy. Nevertheless,

10.6. Improvement of Series; Natural Coordinates

One can calculate only a few terms of a perturbation expansion,usually no more than two or three, and almost never mor.e than seven.The resulting series is often slowly convergent, or even d1vergent. Yetthose few terms contain a remarkable amount of information, which theinvestigator should do his best to extract. . .

This viewpoint has been persuasively set forth 10 a dehghtful. paper.byShanks (1955), who displays a number of a~azing examples, l~clud1Ogseveral from fluid mechanics. A simple one 1S the numencal senes

that modifying the perturbation quantity (Se~tion 3.1) ~ransfers a con­stant from the logarithmic term to its algebraic compamon.

For practical values of the perturbation quantity, its logarithm does notdiffer greatly from unity, so that the algebraic term-thou?h of smallermathematical order-may actually have the greater magmtude. More­over, experience suggests that the two terms are in~ariably of oppositesign, and that their sum is often much smaller than e1ther of them alone,so that retaining only the logarithmic term may worsen the accuracy. Anexample is the expansion (1.6) for the lift of an elliptic wing. ForA = 6.37 it gives

SeeNote

15

204 X. Other Aspects of Perturbation Theory 10.7. Rational Fractions 205

(bl

Fig. 10.1. (b) Parabolic coordinates.

423

5

7

.50 """-----------------,

.25

-.25

-.50 L-. --'

'"e 0 r------..........----~~----____l

.50

(Sh~'+ waveI

---.x.25

2 te,ms

'",1 0

Figure 10.1 shows that among the advantages of this form is a clearindication that the expansion diverges at the nose of the body, thoughless wildly than its Cartesian counterpart. Further improvement of thisseries is discussed in the next section.

10.7. Rational FractionsFig. 10.1. Series expansion for stream function on axis behind paraboloidal shock

wave at 1VI = 2, y ~ 7/5. (a) Cartesian coordinates.

A third example is the Blasius series (1. 5) for the skin friction on aparabola. Recasting it into parabolic coordinates [with b in (10.20)replaced by a] yields (Van Dyke, 1964a)

Numerical tests (Exercise 10.3) tend to confirm a conjecture, based ontheoretical arguments, that the radius of convergence has been increasedby almost 50 per cent.

Applying Shanks' nonlinear transformation (10.19) to the first threeterms of a power series I + aE + bE2 + ... yields a simple rationalfraction:

(10.23)a + (a2 - b)e

a - be

This is often a more accurate approximation to the sum of the seriesthan the original three terms. For example, it yields the exact sum if theoriginal is a geometric series, whether convergent or divergent. Thisfact tends to explain the success of Shanks' transformation with a seriessuch as (l0.17), which is evidently "nearly geometric."

When more than three terms of a power series are known, Shanks(1955) suggests forming rational fractions of higher order. Thus from

i-

ii(10.22)

tvR Cf = 1.23259g - 1.72643e + 2.11376g5

- 2.44192e + 2.73149g9- 2.99343gn + ...

SeeNote

12

206 x. Other Aspects of Perturbation Theory 10.8. The Euler Transformation 207

the first five terms of Goldstein's series (1.3) for the drag of a spherein the Oseen approximation, he obtains

Here R is the Reynolds number based on radius. This agrees well withmore exact calculations out to R = 10, whereas the original series isuseless for computation above R = 1. The denominator of the rationalfraction (10.24) vanishes at R = -1.97 and - 54.4. Although the secondof these is too large to be significant, the first suggests that the conver­gence of the original series (1.3) is limited by a singularity at R = -2.This is plausible because, as is evident from (8.17) and (8.18), the naturalparameter for Oseen flow is !R, in terms of which the singularity wouldlie at -1 in the complex plane (d. Section 3.5).

In the same way, Van Tuyl (1960) has formed rational fractions fromCabannes' series (3.25) for the flow behind a paraboloidal shock waveat M = 2, and its counterpart (10.21) in parabolic coordinates. Thelatter gives

(10.27)

SeeNote

15

SeeNote

2

..B.- c =-- 73,920 + 66,600R + 10,880R261T D·"-=' 73,920 + 38,880R + 689R2

2ljJ I - 0.73878(1 - 1)) - 37.827(1 - 1))2 + 72.098(1 - 1))312 ~ 1 + 4.5946(1 - 1)) - 13.212(1 - 1))2 - 3.5958(1 - 1))3

(10.24)

(10.25)

furthe.r terms in the ~xpansior:. As suggested in Section 3.9, the agree­TfoIent IS poorer for aXlsymmetnc flow, where the rational fraction (10.26)gives 0.214 compared ~ith a numerical value of 0.128 for the sphere.<?ther t~sts ~f thiS techmque of extrapolating a direct coordinate expan­sIOn to mfimty are proposed in Exercises 10.7 and 10.8.

A technique analogous to that of rational fractions is needed toimprov: ~he utility of series containing logarithmic terms, such as (1.4).No .stnkmg results hav.e yet been achieved. We give an example ofpartial su~cess. The ,senes (1.6) for the lift of an elliptic wing can, byanalogy With Prandtl s result (9.1a), be rewritten as

deL 21T

d-;- ~ -~ + ~ + 16(\og 1TA _ 2) _IA 1T2\ 8 A2

This is, in !act, of the f~rm tha~ arose naturally (with different, incorrect,~onstants) III the analySIS of Knenes (1940). The accuracy is considerablyImproved, the error being reduced from 11 to 1 per cent at A = 2.55,~or e~amp~e. However, this modification has the defect of still becomingmfilllte. (FI~. 9.4), though at a smaller value of A than the original series.There IS eVidently room here for further improvement.

SeeNote

13

This approaches 0.429 for plane flow (n = 1), which is far closer thanthe original result to the accurate numerical calculation of 0.377 for acircular cylinder. This success makes it seem worthwhile to compute

Whereas the original series diverges at the body (Fig. 10.1), this agreeswell with numerical solutions, apparently giving the stand-off distance Llcorrect to four significant figures. The superiority of parabolic coordinatesis strikingly confirmed by the discovery that in Cartesian coordinatesthe corresponding rational fraction becomes infinite in the flow fieldbetween the shock wave and the body.

\Ve can make a bolder application of the same technique to Cabannes'other attack on the blunt-body problem. From his two-term expansion(3.26) in powers of time for the stand-off distance following an impulsivestart, we can form a rational fraction that remains bounded at infinitetime, and so hope to estimate the steady-state value. This gives forM = 00 and y = 7/5:

LI-<=::::!a 7 ( Vt )1 + (n - I)--

15 a

(10.26)

I:

10.8. The Euler Transformation

~he ~ransforn:~tions used in the previous section suffer-despitethel~ eVlde?t ~tlltty-fr~m the objection that they must usually be~pplted arbltranly and blmdly, with no understanding of the mechanismmvolved. This is inevitable when one knows neither the nature nor theloc~tion of the singularity that limits convergence. It also goes withoutsaYlllg that one does not always achieve such remarkable improvementas in the preceding examples.

With further insight into the source of divergence, one can proceedmore rationally and hence more successfully. Thus one may know thelocation of the singularity, though not its character. As discussed inSectio~ 3.5, it .often happens in applied mechanics that a power-seriesexpansIOn havmg physical significance only for positive real values ofthe perturbation quantity is restricted by a singularity elsewhere in thecomplex plane, usually on the negative axis, and at -1 if the variableshave been chosen in the most natural way. That this situation existsmay be known from fundamental considerations, or be suggested by arational fraction as in (10.24), or merely be suspected.

It was suggested in Section 3.5 that under these circumstances it is

,208 X. Other Aspects of Perturbation Theory 10.8. The Euler Transformation 209

usually possible to improve the series by recasting it in powers of thenew parameter

The coefficients now decrease more rapidly than 1;n2, suggesting thateven far downstream the series converges faster than ~ l!n2 = 1T

2J6. Ifso, it should at g = 00 reproduce Blasius' value of 0.664 for the flatplate (Section 7.10). The successive partial sums are

(10.31)

(10.32)

(10.33)

(10.34)LIb = 0.0950, 0.0973, 0.0978,

1 _ 3 [e ,13( e)3 177 ( e)4 ]- YJo - "8 e 1 + ~ , 80 T+~ + 2240 1 + e +".

1 - Tlo = ~e2(1 - e -'- 93 e2 _ 63!e3,' .,,)./ 8 '80 448 '

rate of convergence is discouragingly slow. At y = 7/5, where theperturbation parameter (y - 1)!(y -+- 1) is only -1, the se~ies gives

LI 1b = 6(1 - 0.667 + 0.433 - 0.306 + .,,)

There are various indications that this choice of s is the most natural oneincluding the fact that the series now oscillates almost like a geometri~one with a singularity at s = -1. :Making the appropriate Euler trans­formation (10.28) then yields

which is useless for computation.

Following our recommendation of natural coordinates (Section 10.6),we first recast this in parabolic coordinates (l0.20). The value of 1] at thenose of the body is then given by

This result is an unqualified success. The coefficients of the last two termsdec.rease faster than the inverse square of their exponents, suggestingrapid convergence even at s = 00. However, even y = 00 wouldcorrespond only to s = 1.633; and for actual gases y cannot exceed 5 3,so that s is less than 0.817, and s(l + s) less than 0.450. .

For y = 75, the successive partial sums of (l0.33) give the stand-offdistance as

(10.28)_ ee ::= ---

I + e

The result is a special case of a rational fractions. The purpose of thisEuler transformation is to transfer the singularity from s = -1 to thepoint at infinity. If there are no other singularities in the complexplane, the radius of convergence is thereby made infinite. Variousapplications in applied mechanics have been discussed by Bellman(1955). We give three examples drawn from Chapter 1.

First, consider once more the Blasius series (1.5) for skin friction on aparabola, which was improved somewhat by transforming to the morenatural parabolic coordinates (10.22). The latter form clearly has unitradius of convergence, no doubt as a consequence of the singularity atg = i in the conformal mapping for the parabola. The appropriateEuler transformation is therefore an expansion in powers of g2/(1 + e),after extraction of a factor g because the skin friction is an odd functionof g. This yields (Van Dyke, 1964a)

SeeNote

12

and closer examination leaves little doubt that these are indeed convergingto the Blasius value. Thus the radius of convergence of the Blasius serieshas been extended to infinity.

As a second example, we undertake to improve Chester's Newtonianseries (1.7) for the stand-off distance of the blunt body that produces aparaboloidal shock wave at M = 00. There is reason to believe (VanDyke, 1958b) that this promising method yields an expansion thatconverges for y < 11/5, a value not attainable in reality. However, the

1.743, 1.045, 0.894, 0.827, 0.789, 0.765, (10.30) Applying Shanks' transformation (10.19) to these yields 0.0979. Variousnumerical calculations have given values between 0.0980 and 0.0990. Itis interesting to observe that this approximation, though based on theassumption that y is close to 1, can now be used at y = 00, where itpredicts a stand-off distance about three times as great as at y = 7J5.Pr~vious less s.ucce.ssful attempts to improve Chester's series, usi'ngratIOnal approximatIOns, are given by Van Dyke (l958b) and Van Tuyl( 1959).

As a third example, we reconsider Goldstein's series (1.3) for theOseen drag of a sphere. As discussed in Section 10.7, both theoretical

210 X. Other Aspects of Perturbation Theory 10.9. Joining of Coordinate Expansions 211

and numerical considerations suggest that there is a singularity atR = - 2. The appropriate Euler transformation then yields

R -I 1 - R 19( R)2 1 ( R )3CD ,...., 371(2 + R) [I - 4(2 + R) - 80 2 + R - 64 2 + R

2459 ( R)4 9469 ( R)5 ]- 134,400 2 + R - 537,600 2 + R -'" (10.35)

This has the appearance of converging for all R, though less spectacularlythan the previous two examples. The result is less accurate than therational fraction (10.24) at R = 20, but has the advantage of remaininguseful at R = 00, where (10.24) gives CD = 0 but the next rationalfraction, with a cubic numerator, predicts negative drag. At R = ro thesuccessive partial sums of (10.35) give We now evaluate this same quantity from the direct expansion (10.29),

using the fact that the series

(10.37)Ux 1/2 log (1 + g2) - C

1_ ) -

(-) C .- 0 399 -----,...., 0 664(1 -+- -- + ...)v f • I ..L g2 . _ I 1 + g2

:\-lore often one makes a direct and an inverse expansion for small andlarge values of a coordinate. The latter is usually not convergent, butonly asymptotic, and contains logarithmic terms and undeterminedconstants (Section 3.10). Nevertheless, the two series can be joined-andthe constants thereby found--if the direct series has an infinite radiusof convergence. We demonstrate this for the boundary layer on a para­bola.

Recasting the inverse expansion (3.27) into parabolic coordinates(10.20) and transferring the logarithmic term to the left side gives

(10.36)CD:::::: 9.42, 7.07, 4.83, 4.68, 4.51, 4.34,

SeeNote

2

Although these decrease somewhat irregularly, it is not unlikely that theyare approaching from above the value 3.33 calculated by Stewartson(1956).

(10.38)

converges for g2 ~ ro. Together with the fact that e = 2x on theparabola, this gives

To be sure, the result is found only approximately as the partial sum ofan infinite series; but it is obtained by asymptotic joining rather thannumerical patching at some large finite value of g. Patching with anumerical integration from the nose at g = 6.8 gives C1 = 1.83.

(10.39)

(10.40)

e ,6

-- 001l(---) -'". 1 + gz

- Ux 1:2 log (I + g2) (e e 2

(-v-) Cf - 0.399 -'1 + g2- = 1.34 1 + e) - 0.499(1 T g2)

g2 3 gz 4

.- 0.084(1+ g2) -.0.034(1+ g2)

f .- 0.Ql8(1 ; g2)"

for all e ~ CD. Then expanding formally for small 1/(1 T e) andcomparing with (10.37) yields the undetermined constant C1 as (VanDyke, I964a)

C1 = I ~ (-1.34 + ).00 -[- 0.25 ..L 0.13 + 0.09

T 0.07 T ''')/0.664:::::: 1. 90

10.9. Joining of Coordinate Expansions

Matching, as it is used in the method of matched asymptotic expan­sions, connects two parameter expansions for the same limit process thatare valid in adjacent regions of space-time. Occasionally it is possible tojoin other kinds of expansions in analogous fashion. We discuss in thisand the last section two kinds of problems where this is possible.

Different coordinate expansions for the same problem can sometimesbe joined. If they are both convergent power series, the process is that ofanalytic continuation, which is simple in principle. The practical com­plications are illustrated by Cabannes' expansion for the stream functionbehind a paraboloidal shock wave, recast in parabolic coordinates (10.21).That series converges only about three-quarters of the way to the body(Fig. 10.1), because of a limit line in the physically nonexistent upstreamcontinuation of the flow field (Van Dyke, 1958b). The series can be usedto calculate, say, halfway to the body, and a new series from there willreach the body. However, the new series cannot be obtained directlyfrom the original truncated series, because it would be equivalent to itand so have the same limit of convergence. A remedy (Lewis, 1959)is to use the original series only to obtain new initial data at theintermediate station, and to calculate further coefficients using thedifferential equation.

. I

212 X. Other Aspects of Perturbation Theory Exercises 213

To convert this double expansion for small e and ]\![2 into a singleexpansion for small e, we must sum the series in lVI2. This unlikely taskis accomplished simply by appealing to the second-order similarity rulefor subsonic small-disturbance theory (Van Dyke, 1958a), according towhich

This sort of joining is not possible when the direct series has only afinite radius of convergence, or is merely asymptotic. In the lattercategory, for example, is the approximation (3.24) of Carrier and Lin(1948) for viscous flow near the leading edge of a semi-infinite flat plate,which cannot logically be joined with the higher-order boundary-layerexpansion (Section 7.11). Imai (1957a) has patched the skin friction atUX/V = 1, but the validity of this step is questionable.

10.10. Joining of Different Parameter Expansions

Expansions for different limits of a single parameter (e.g., for smalland large Reynolds number) could be joined as in the preceding sectionif one of them were convergent over the whole range. No examples areknown, although (10.35) comes close to providing one.

Expansions for two different parameters can be joined in finite termsonly if one obeys an appropriate similarity rule. Thus the thin-airfoil orslender-body expansion for subsonic compressible flow can be extractedfrom the Janzen-Rayleigh expansion in powers of M2, but the converseis not possible.

Consider the simplest example, plane flow past the parabolaY = -+- e(2x)lf2, having nose radius e2. The expansion of the velocitypotential in powers of M2 has been carried out by Imai (1952) includingterms in (y + I )M4. Expanding that complicated result formally forsmall e yields

, ., . 1] - 2e 9 ' g2 + 1]2</> ,......, x - e1] + !1\;J-e1]2 ---- - iNJ-e2Iog----

, g2 + 1]2 4e2

(10.41a)y + 1 [ 1]4 . e + 1]2 ]-16- 1lJ4e2 4(g2+ 1]2)2 T log-4;2- + O[M4, (y+ 1)M6, e

3]

where [d. (10.20)]

EXERCISES

SeeNote

3

SeeNote

3

(1O.43a)

(1O.43b)

f(O) = 1'(0) = 1

t2 = vx2 + f32y2 + X

ij2 = Vx2 + fj2y2 - X

ef" + f = 1,

where

whereas the method of matched asymptotic expansions breaks down.

10.3. Estimate of radius of convergence. Estimate the radii of convergence ofthe series (1.5) and (10.22) using Cauchy's ratio test, and compare with thevalues (s/a) = 17/4 and g2 = 1 suggested by the conformal mapping for thepotential flow. Do the same for (1.7) and (10.21), and compare with the sin­gularities of successive rational approximations.

10.1. Lighthill's equation by multiple scales. Solve the problem (6.3) using themethod of multiple scales, showing that the result is not (6. I 2c) but an equallyacceptable alternative. Try solving also by Latta's method of composite expan­sIOns.

where f3 --- (l - M2)1/2 and each function has the same arguments asthe first. The previous double expansion (10.41) is compatible withthis rule only if it can be replaced by

I M2 -2 f2 + -2</> ,......, x + !- ij - - -- e2(4 -::;3__ + log __--l )

f3 4 (32 g2 + ij2 4(32e2

y + 1M4 r ij4 f2 + ij2]- ---e2 4 -_--- + log --- + O(e3 )

16 (34 (g2 + ij2)2 4f32e2

10.4. Lift of elliptic wing. Calculate the first three approximations from (1.6)and (10.27) for A = 8/17 (axis ratio of 2 : I). Estimate the complete sum, andcompare with Krienes' value of 2.99.

10.5. Rational approximations for friction on parabola. Form rational approxi­mations to (1.5) and (10.22), and thereby try to extract the limiting value fardownstream.

10.2. Unmatchable problem. Show that the method of multiple scales issuccessful for the problem

and this is the second-order thin-airfoil solution. The correspondingjoining for the paraboloid of revolution is given by Van Dyke (l958c).

(10.42)

(1O.41b)g2 = Vx2 + y2 + X + O(e2 )

1]2 = Vx 2 + y2 - x + O(e2 )

</>(x,y; 1\;1, y, e) ,......, X + e2!I(x, (Jy; (Je) + e4[f21( )

+ 1H%2( ) + (y + 1) j~24 f23( )] + O(e6)

214 X. Other Aspects of Perturbation Theory

10.6. Critical JJach number for circle. The critical Mach number (whereSee local sonic speed is first attained) is approximated by successive partial sums

Note of Simasaki's series (l.l) as

3

SeeNote

3

SeeNote

3

0.4659, 0.4206, 0.4090, 0.4043, 0.4020, 0.4008,

Estimate the true value, and compare with Simasaki's own estimate of 0.40.Estimate the radius of convergence of (1.1), compare with Simasaki's 0.50,and contrast with the critical ;\Iach number. Consider the possibility of applyingan Euler transformation to (1.1).

10.7. Friction at nose of impulsh'ely started cylinder. Try to extrapolate (3.23)to infinite time, and compare with Hiemenz' value for steady flow (Schlichting,1960, p. 78).

10.8. Separation on impulsi'1Jely started circle. On a circular cylinder of radiusa started moving impulsively through viscous fluid with speed V, the skinfriction varies initially with time in proportion to (Goldstein and Rosenhead,1936)

Vt . ( Vt)2I + 2.8488 cos 6(-a-) - [0.8795 cos2 6 - 0.2390 sin2 6] -a-

where 6 is the angle from the forward stagnation point. Find the location ofthe separation point as a function of time. Extrapolate to infinite time, andcompare with the known value of 6 = 109 degrees (Schlichting, 1960, p. 154).

NOTES

Note 1. Introduction

These Notes have been added to bring the text of 1964 up to date.They briefly describe significant advances in the subject, and addrelevant references to the literature through mid-1975. Further lists ofreferences, for applications to fluid mechanics and other fields, can befound in the book of Nayfeh (1973).

Each note is keyed to one or more sections of the main text, wherean indicator of the form shown here at the right appears in the marginof the page. Whenever possible, related comments have been com­bined into a single Note, with the aim of making this section lessfragmentary.

Note 2. Computer Extension of Regular Perturbations

Since the late 1950s it has been feasible to delegate to an electronicdigital computer the rapidly mounting arithmetic labor involved inextending a regular perturbation series to high order. This procedurehas been applied to a variety of problems in fluid mechanics, includ­ing several discussed in this book. The history of the idea, practicaldetails, and a number of applications have been surveyed by VanDyke (1975, 1976).

The Janzen-Rayleigh expansion (Section 1.3) has been extended toorder Ml 2 by Hoffman (1974a) and to order M l 6 by W. C. Reynolds(unpublished). The result for maximum surface speed, with y = !, is

qrnax = 2.00000 + 1.16667 M2 + 2.57833 M4 + 7.51465 M6U

+ 25.59041 M8 + 96.26329 MlO + 387.92345 Ml2

SeeNote

I

I I

!

+ 1646 Ml4 + 7450 Ml6 + ... (N.l)

Here the coefficients of M4 through MlO differ slightly from those ofSimasaki in Equation (1.1) because the value ofy is different. The last

215

More than thirty years later Stokes added two more terms, and con­jectured that the expansion converges for the highest wave, which hassharp peaks of 120-degree angle. Subsequently Wilton carried thecalculation for deep water to tenth order, but made errors at eighthorder. Using the computer, Schwartz (1974) has extended the seriesto 117th order. Analyzing the coefficients, he showed that Stokes'sexpansion fails to converge to the highest wave because the coef­ficient a is not a monotonic function of the wave height. He eliminatedthis defect by reverting the series to give an expansion in powers ofwave height, and was then able to calculate the maximum waveheight to five figures.

Some other computer extensions at low Reynolds number aredescribed in Note 10.

two coefficients are relatively inaccurate because they were computedonly in single-precision arithmetic. Calculating the successive ap­proximations to the critical Mach number (Exercise 10.6) and thenrepeatedly applying the nonlinear transformation of Shanks (Eq.10.19), Hoffman estimates the critical Mach number as 0.3983± 0.0002. This agrees with the value 0.39852 ± 0.0002 calculatednumerically by Melnik and rves (1971).

Goldstein's series (1.3) for the Oseen drag of a sphere in powers ofReynolds number has been extended by Van Dyke (1970). Termsthrough order R23 were calculated in one minute of computer time.The graphical ratio test of Domb and Sykes (Note 15) shows that thenearest singularity, which was conjectured on the basis of rationalfractions (p. 206) to lie at R = - 2, is actually located at R =

- 2.09086. Applying the Euler transformation (Section 10.8) thatmaps that singularity away to infinity yields a new series that con­verges out to infinite R and, with some further manipulation, repro­duces Stewartson's value (p. 210) of CD = 3.33 at R = 00.

A noteworthy application of this technique is to plane periodicwater waves. Early in his career Stokes expanded the solution inpowers of the coefficient a of the linearized approximation, calculat­ing the second approximation for constant depth and the third ap­proximation for deep water. Thus he found the surface of the wave indeep water described, for an observer moving with the wave, by

It was mentioned in the original Preface that the Exercises at theend of each chapter contain much additional information in conciseform. We point out here that some of them have been subsequentlysolved in the literature or otherwise extended, that several can bebetter posed, and that it is interesting to try alternative techniques ona few.

Exercise 2.2. Slightly porous circle. It is better to ask for the velocitypotential, because for it the effect of slight compressibility can bededuced from Equation (2.2), whereas it is tedious to extract from itthe corresponding result for the stream function.

Exercise 2.3. Corrugated quasi cylinder. Davey (1961) has used thissolution to illustrate how a stagnation point can be a saddle point ofattachment in inviscid flow.

Exercise 2.4. Circle in parabolic shear. As preparation for the modi­fication of Exercise 5.7 described below, it would be useful to consideralso the parallel stream with profile u = U COSh(£1I2 yla).

Exercise 4.5. Exact solution for biconvex airfoil. The discrepancypointed out in the last two sentences is discussed in Note 4 below andin the reference given there.

Exercise 5.7. Circle in parabolic shear. The problem as posed is un­necessarily complicated by nonlinearity. The essentials are exhibitedmore simply if, in the parallel flow far upstream, the parabolic profileis replaced by the hyperbolic-cosine profile u = U cosh(£112 y I a),which is almost the same near the cylinder.

Exercise 6.2. A problem of Carrier. The factor multiplying theexponential in the expression given for x can be simplified to a linearform if we weaken Lighthill's principle as described in Exercise 6.1.Another way of treating the problem is to interchange the roles ofthe independent and dependent variables, which transforms it into aregular perturbation problem for xCi). The solution using matchedasymptotic expansions, discussed by Carrier, is more complicated.

Exercise 6.4. Pritulo's method. Pritulo's idea of introducing thestraining of coordinates a posteriori has several times been discoveredanew (Martin 1967, Usher 1968). Crocco (1972) illustrates its effec­tiveness for several problems in gas dynamics.

217;\"otc 3. Comments on the Exercises

Note 3. Comments on the Exercises

(N.2)

Notes

1 3Y = a cos x - "2a2 cos 2x + Sa3 cos 3x +

216

218 l'\otes "'ote 3. Comments on the Exercises 219

Collins and Dennis (1973b) also computed the time-dependent flownumerically, using a method of series truncation, and found that theycould not continue the integration beyond (Uti a) = 1.25. Telionis

Exercise 10.8. Separation on impulsively started circle. The extensionof Collins and Dennis (1973a) gives, in the first-order boundary-layerapproximation, the friction proportional to

- (0.0105 + 0.0285 cos 2B + 0.0414 cos 4B + 0.0052 cos 6B)(Utla)6

+ (0.0117 cos B + 0.0207 cos 3B + 0.0560 cos 5B

(N.3)

(N.4)

- 0.0127(U1t)3 + 0.0254(U1t)4 - 0.00566(U1t)5

- 0.00134(U1t)6 + 0.00095(U1t)7 + ... ]

be ~. Krienes's value has been improved to 2.944 ± 0.004 by Medan(unpublished calculations using the method of Medan, 1974).

Exercise 10.6. Critical Mach number for circle. As discussed in Note2, Simasaki's series (1.1) has been extended by computer, and thecritical Mach number (for y = ~) estimated accurately from that aswell as other methods.

Exercise 10.7. Friction at nose of impulsively started cylinder. The expan­sion in powers of time has been extended by Collins and Dennis(1973a). The boundary-layer approximation (3.23) for the frictionnear the front stagnation point is

+ 0.0335 cos 7B)(Ut/~f +

1 + 2.8488 cos B(Utla) - (0.3202 + 0.5592 cos 2B)(Utla)2

- (0.0767 cos B + 0.0246 cos 3B)( Uti a).3

+ (0.0661 + 0.1982 cos 2B + 0.1427 cos 4B)(Utla)4

~ (0.0332 cos B + 0.0722 cos 3B + 0.0755 cos 5B)(Utla)5

Pritulo (1969) points out that a parameter can be treated similarly.The idea of slightly straining a parameter so as to suppress non­uniformity was invented by Lindstedt (1882) to avoid secular terms incelestial mechanics and, as mentioned on page 100, was generalizedby Poincare. Nayfeh (1973) has called this the method ofstrained param­eters. Pritulo shows how parameter straining can be introduced aposteriori in the example of the fourth-order thin-airfoil expansion ofDonov (1939) for plane supersonic flow. Divergence of the approxima­tion as the free-stream Mach number M becomes large can be sup­pressed by straining the compressibility factor (M2 - 1)1/2 so thateach term in the expansion for the pressure coefficient remainsbounded for large M.

Exercise 8.2. Viscous flow past slender paraboloid. The first step ofthe solution was given by Veldman (1973). As discussed in Note 4, thesecond step involves a failure of the asymptotic matching principlebecause of a "forbidden region," which is avoided by proceeding tothe third step, matching two terms of the Oseen expansion to twoterms of the Stokes expansion.

Exercise 8.3. Wake of axisymmetric bo4J. This problem is the firststep in the solution of Viviand and Berger (1964). In the second ap­proximation, the failure of the vorticity to decay exponentially (p.131) was shown by Berger (1968) to require, much as for the boundarylayer on a semi-infinite plate (Section 7.11), introduction of a loga­rithmic term; and then its algebraic companion term is undetermined.Berger also deduces the form of the third and higher terms. This solu­tion, and much other work on laminar wakes, is discussed in the bookby Berger (1971).

Exercise 8.5. Transcendentally small terms for circle. As described inNotes 10 and 11, the solution of this exercise is included in the work ofSkinner (1975).

Exercise 9.3. Near-equilibrium flow over a wazy wall. It is instructivealso to show that the method of strained coordinates leads very simplyto a definite result, but it is erroneous because the solution does notdecay at infinity. See Note 7; also Note 6.

Exercise 10.3. Estimate of radius of convergence. The graphical ratiotest of Domb and Sykes, described in Note 15, is the best techniquefor making these estimates.

Exercise lOA. Lift of elliptic wing. As pointed out in Note 13, thefraction ~ in Equation (1.6) should be ~, and the ~ in (10.27) should

220 :"otes Note 4. Asymptotic Matching Principle 221

and Tsahalis (1973; see also Sears and Telionis, 1975) have also inte­grated the boundary-layer equations numerically, using a finite­difference scheme, and at about (Uti a) = 0.65 discern a singularityin the interior of the boundary layer that has the properties of thesquare-root singularity at the surface in steady separation (Goldstein,1948, Stewartson, 1958).

11 Note 4. The Asymptotic Matching Principle

Occasionally the counting involved in the asymptotic matchingprinciple (4.36, 5.24) may seem ambiguous; then an alternative ver­sion of the rule is useful. One may feel uncertain how to count terms,for example, in the following four situations: (i) if the asymptoticsequence has gaps (e.g., the series proceeds by integral powers of eexcept that the coefficient of e3 vanishes identically); (ii) if the asymp­totic sequences are essentially different for the inner and outer ex­pansions (e.g., one expansion proceeds by powers of e, the other byhalf-powers); (iii) if logarithmic terms arise (e.g., en log e appears a<;well as en); or (iv) if eigensolutions exist.

In all such cases, uncertainty is avoided by adopting the followingunambiguous version of the rule:

The inner expansion to order ~ of (the outer expansionto order 8) = the outer expansion to order 8 of (N.5)(the inner expansion to order ~)./

~

Here ~(e) and 8(e) are any two gauge functions (not necessarily thesame), which mayor may not actually appear in the asymptoticsequences for the outer and inner expansions. Of course this alterna­tive is equivalent to the original version (5.24) whenever it is clearhow to count terms. Similar alternative rules based on orders ratherthan counting have been introduced by Fraenkel (1969b, c, pp. 246,265) and Crighton and Leppington (1973, p. 317).

In proposing the asymptotic matching principle the author in­tended merely to provide a specific procedure embodying Kaplun'sideas, and supposed that it was equivalent to the intermediate match­ing principle (5.29). However, in an important study of the matchingprocedure Fraenkel (1969a, b, c) has shown that the two rules aresignificantly different. He points out that the asymptotic rule is easierto use than the intermediate one, which requires a search for therange of the intermediate variable and the order of overlap in thatrange. Moreover, the asymptotic rule is more economical, because incertain problems where the overlapping is weak a constant in the

inner expansion can be found by matching with fewer terms of theouter expansion than are required with the intermediate rule. ThusFraenkel has demonstrated with several examples (Fraenkel, 1969b, c,pp. 251, 269) that, contrary to what was widely believed, the asymp­totic matching principle can be applied to inner and outer expan­sions that contain too few terms to overlap to the order of the termsbeing matched. However, Lagerstrom (1975) points out that thisshortcoming can be circumvented by working with the correctionboundary layer-the inner expansion of the difference between thefull solution and its outer expansion.

Unfortunately, the asymptotic rule can fail when the series containlogarithms. This has been analyzed by Fraenkel, who has discoveredtwo restrictions, one or the other of which is to be imposed on theasymptotic matching principle, depending on how the logarithmsappear. These are no mere mathematical quibbles, for in each case wenow know that ignoring the restriction has led to error in a realisticproblem in fluid mechanics. A majority of the many solutions ofphysical problems carried out in the last decade using the method ofmatched asymptotic expansions are based on the asymptotic rule. Itis therefore of practical importance to recognize these two restrictions:

I. Don't cut between logarithms. Fraenkel insists on normalizing e asthe ratio of the inner and outer scales. (We have not always adheredto this convention; for example, we took the ratio as e2 in Eq. (4.34)for the round-nosed airfoil, and even as e-1/f in Eq. (4.48) for thesharp-edged airfoil.) Then condition (iii) of his Theorem I (Fraenkel,1969a, p. 223) serves as a warning that the asymptotic matchingprinciple may fail if applied to an expansion truncated by separatingterms that differ by less than any power of e. In particular, when theseries proceeds in powers of e and of its logarithm, all the powers oflog E multiplying a given power of e should be treated as a single term.

This warning was violated by Proudman and Pearson (1957) who,as described on page 160, calculated the term of order R2 log R in theStokes expansion for the sphere while disregarding its companionterm of order R2. Apparently no harm was done in this case, however,because Chester and Breach (1969) have confirmed that term by ex­tending the series to one higher power of R (and in so doing havethemselves violated Fraenkel's warning at the next stage).

On the other hand, Crighton and Leppington (1973) have foundthat violating the restriction leads to an erroneous result for the dif­fraction of long waves by a semi-infinite plate of finite thickness. Theouter expansion has the form

222 Notes l\;ote 4. Asymptotic Matching Principle 223

.]p('ij'2 log 1j2 _1j2 + 1) (N.10)

[R(l + P)/~](TIpU2) = ~1 - (log 2 - y)Ll/

- [2 log 2 + 77 2/12 - (log 2 - y)2]~13 + ... (N.ll)

w~ere ~ = ~/\VI ,!)~/2 and fJ = 1j/(vl U)1I2. Matching these deter­mines the multlpher In the Stokes approximation as ~1 = (log 1IR )-1,where the Reynolds number is R = Ualv. The solution has been car­ried to this stage by Veldman (1973). (This result, with an incorrectfactor ~1 independent of Reynolds number, was first obtained bySampson (1891) by taking the limit of the Stokes solution for anellipsoid of revolution.)

. ~he next term can then be calculated in the Oseen expansion,gIVing

where E.l is the exponential integral. (Veldman also gives this result,but he Imposes the surface boundary condition to find the Oseenapproximation, rather than matching.) This can almost be matchedto. (N. 7) with! =1 1 ~nd B = 0, but the sec.ondary terms give themlsma.tch lo?, 1j. = 2": SIn~e log 1] appears only hnearly, the asymptotic~atchIng~nnClple IS vahd for 1m - nl ~ 1. Thus we are at this stageIn the forbIdden region, which is again just the line m = n.

As in the case of the circle (p. 163), the higher terms in the Stokesexpansion are multiples of the first, so that the two-term expansionhas the form

We have now moved out of the forbidden region, and this matchesperfect.ly with (N.9), giving A = 1, B = 0, and c = y - log 2.

.Davls and Werle (1972) have calculated an approximation to thethIrd term of the Stokes expansion that becomes correct far down­stream, as ~ .... 00. They suggest that it is nearly correct for all ~ andconfirn: this by comparing their numerical solutions with the res~ltingexpanSlOn for the local skin friction:(N.7)

(N.6)

where the Stokes variables are ~ = ~/a1l2 and 1j = 1]la1

/2

. The firstterm of the Oseen expansion is just the uniform stream, with

where e is the ratio of plate thickness to wave length, and hence of theinner to outer scales. Matching to order e log e as well as to order egives the correct result; but matching to order e2 log2 e yields a <P3 thatis recognized as being incorrect only because this linear problem pos­sesses a reciprocal theorem that is violated. This trouble is avoided bymatching the three terms in e2 as a block. Crighton and Leppingtondiscuss the mechanics of this restricted matching principle.

2. Forbidden regions in the purely logarithmic case. The restriction justdiscussed cannot be met when, as for the circle at low Reynolds num­ber (Sec. 8.7), the outer and inner expansions proceed in inversepowers of log e (or, equivalently, of log e + k). Fraenkel (1969a, p.226) finds that in this case the asymptotic matching principle (5.24)is valid unless m and n fall within a certain "forbidden region" whoseshape in the mn-plane can be calculated in the course of solution.

It is convenient to uniformize the counting by agreeing that forboth the inner and the outer expansions the kth term is of order(log e)-k (so that typically the "zeroth-order" term is absent fromthe inner expansion). Then in the common case that the inner expan­sion of the outer expansion is a polynomial in log x of degree N, thematching holds for 1m - nl ~ N. Fraenkel (1969c, p. 276) illustratesthis restriction with a one-dimensional model of the circle at lowReynolds number. The forbidden region is just the line m = n, so thatmatching can be carried out successfully by choosing, for example,n = m + 1 (or n = m - 1).

A more realistic example is the problem in Exercise 8.2 ofaxisym­metric viscous flow past a slender paraboloid. For the paraboloid ofnose radius a, separation of variables yields the Stokes approximation

(N.8) We would expect that, just as for the circular cylinder (Note 15), theaccuracy of the Stokes expansion could be improved by telescoping

224 Notes Note 5. The Theory of Matching 225

(N.13)

the first two terms into one. This means changing the definition ofthe gauge function to ill = (log 2/R - y)-l, so that c = 0 in \N.10).Then Davis and Werle's approximation (N.11) assumes the sImpler

form

[R(l + ~2)/~KT/pV2) = ill - (2 log 2 + 7721l2)il l3 + ... (N.12)

which was given by Mark (1954). This modification does impro~e theagreement with numerical results somewhat, but not so dramatIcallyas for the circle (Fig. N.4). .

In the problem of the circular cylinder the log pm Eq. (8.47) me~nsthat the I-term Stokes expansion and the 2-ter~ ?seen exp~nslOn(the inner and outer expansions to order ill) he m the forbIddenregion, which is again the line m = n. Nevertheless, we were able tomatch them correctly on page 163, much as Pro.ud~an and. Pearson(1957) did, using the asymptotic matching p~mclple. ThIs ~eansthat, just as in the case of powers of.e an~ log e dIscussed above, vlOlat­ing Fraenkel's warnings will not mvanably lead to error: (On ~heother hand, the Stokes and Oseen expansions to order umty, whIchare 0 and 1 respectively, obviously do not match.)

Are Poincare expansions necessary? Fraenkel's proof of ~he restrictedasymptotic matching principl~ is based on.the ~ssumptlOn t~at boththe outer and the inner expanslOn are of Pomcare form-that IS, of theform (3.l0a), where the coefficients cn ordinarily depend on space ortime variables but are independent of e, for example,

N

J(x; e) "" L cn(x) 8n(e)n=l

Sheer (1971) points out that for a sharp-edged airfoil the inner solu­tion (4.49) is not of this form. He attributes to this fact the apparentfailure of the matching pointed out in Exercise 4.5 when the finalcomparison is made in inner rather than outer :ariables. His rer.nedyis to work with the logarithm of the velocity, whIch converts the mnersolution to Poincare form without significantly changing the outerexpansion. He goes on to show that putti~g all of the u~boundedpartinto the first approximation leads to a umform expanslOn, so that themethod of matched asymptotic expansions, having served its purpose,is unnecessary. .

Sheer does not discuss the remarkable fact that matchmg thevelocity itself gives the correct second-order result (4.53) if.the finalcomparison is made in outer variables, as on page 70. Is thIS related

to the fact that the outer expansion does have Poincare form? Weexamine the next approximation, matching three terms each of theouter (4.24) and inner (4.49) expansions. (However in (4.49) we must,for the semi-vertex angle, replace the thin-airfoil approximationtan- l 2e by its exact value 2 tan- l e for the circular-arc airfoil becausethe difference, though only of order e3 , makes a change of order e2

through the factor e- lh .) Making the final comparison again in outervariables yields

(4) 12 4 77] }+ - - 6 log 2 + 6 - - + - + - + ... (N.l4)77 77 77 2 3

That this is correct is confirmed by expanding the exact solutionquoted in Exercise 4.5. This might tempt us to conjecture that theasymptotic matching principle remains valid when only one or theother of the inner and outer expansions is of Poincare form, providedthe final matching is carried out in the corresponding variables.

It seems, however, that the resolution of the difficulty is muchsimpler. We did not in fact properly apply our matching rule on page70. The matching principle requires the inner expansion to be ex­panded for small e and truncated while it is written in outer vari­ables. We did not carry out that expansion and truncation completely,because the form of Vi was not prescribed at the stage of Eqs. (4.52c)and (4.52d). Consequently, (4.52d) contains some third-order termsin e2 that should have been eliminated.

The remedy is to work with a specific expansion for Vi' which(4.51d) shows to be Vi = A + Be + .. '. Then the rule can be ap­plied properly, and the discrepancy mentioned in Exercise 4.5 disap­pears. Similarly, in the next approximation using Vi = A + Be+ Ce2 + . . . leads to the correct result (N.14) whether the finalmatching is made in outer or inner variables. Thus it appears that, atleast in this example, the expansions need not both be of Poincareform. However, Note 5 deals with examples where the asymptotic rulefails unless the expansions are cast into Poincare form.

Note 5. The Theory of Matching

For practical application of the method of matched asymptoticexpansions to specific problems, it is ordinarily most efficient to work

226 :"rotes :\'ote 6. Alternative Composite Expansions 22i

Here n corresponds to the number of space dimensions (2 for thecircle, 3 for the sphere), and the factor 8 is zero for the model of in­compressible flow and unity for compressible flow. This problem hasbeen analyzed by Lagerstrom (1961, 1970), Cole (1968, p. 61), Bush(1971), and Lagerstrom and Casten (1972).

The incompressible model, like the actual incompressible problem(Chapter 8), can be treated using limit-process expansions. The Oseenexpansion is a regular perturbation (cf. Note 7). In the compressiblemodel, on the other hand, the Stokes equation is nonlinear, and isnot contained in the Oseen equation. This is the situation in theactual flow problem as well (Lagerstrom, 1964).

Nonlinearity of the Stokes equation has the further consequencethat the solution cannot be found using limit-process expansionsbased on the outer variable r and inner variable rle, for they do notmatch at all. Instead, one must use an inner expansion characterized

with limit-process expansions, and to match using the asymptoticmatching principle (Note 4). That is the method followed in thisbook, and in most of the examples in the literature. The choice oftechnique is certainly a matter of personal taste, however, and Cole(1968), while using limit-process expansions, always matches them bytaking intermediate limits (Section 5.8). However, it is clear that theseapproaches gloss over many subtleties, some of which have practicalimplications.

For a deeper understanding of the ideas underlying matching, ex­pansions characterized by their domains of validity are more funda­mental than limit-process expansions. They may also be essential forconstructing the expansions in some cases. That is, the idea of apply­ing limits to the equation and then finding solutions of the resultingapproximate equations is more basic than looking for limits of thesolution. This view, originating in the work of Kaplun and Lagerstrom(Kaplun and Lagerstrom, 1957, Kaplun, 1957, Kaplun, 1967), hasbeen set forth fully by Lagerstrom and Casten (1972). The readerwill find there a detailed discussion of the heuristic ideas underlyingthe method of matched asymptotic expansions.

In particular, Lagerstrom (1961, pp. 87, 109) has proposed a modelproblem (different from that of Fraenkel mentioned in Note 4) thatillustrates the mathematical structure of viscous flow past the circleand sphere at low Reynolds number:

d2u n - 1 du du t-(dU)2- + --- + u- + 0 - = 0,dr2 r dr dr dr

u(e) 1, u(oo) = 0

(N.15)

by its domain of validity, with the inner variable ri'1J(e), where 11 « 1.Alternatively, Bush (1971) shows that one can use an inner-limit ex­pansion with the inner variable changed to (log l/r)/(log lie).

The same situation arises in a turbulent boundary layer. Millikan(1939) deduced from physical arguments that there are two differentregions, a thin layer near the wall where viscous stresses are important,and a thicker outer layer where only Reynolds stresses are significant.For incompressible flow, this idea has been formalized as an asymp­totic theory for Reynolds number tending to infinity by Yajnik (1970)and Mellor (1972). Bush and Fendell (1972) have extended the anal­ysis to higher approximations using eddy-viscosity models. For com­pressible flow, however, Melnik and Grossman (1974) and Adamsonand Feo (1975) find that limit-function expansions do not match.Again the remedy is either to consider domains of validity or to makea change of variable that leads to matching.

Note 6. Alternative Rules for Composite Expansions

J. Ellinwood (unpublished) has observed that the additive rule(5.32) for composition is the basic one. It can be used to generate avery general family of alternative rules. If Fif) is any sufficientlysmooth functional (generally without poles) that has an inverse F-l(j),then a composite is given by

(N.16)

The additive rule corresponds to F(j) = f and the multiplicative rule(5.34) to F(j) = log]

The rules corresponding to F(j) = r,J, log}; II}; and 1/r yield amonotonic sequence, approximately evenly spaced, at any fixed argu­ment; and this variation might be exploited to improve the agree­ment with exact or experimental results. The composite solution(9.66) that was formed by inspection for hypersonic flow past ablunted wedge corresponds to F(j) = log(ajlal/J), where only theconstant of integration need be fixed by inspection.

Schneider (1973) has pointed out that the multiplicative rule in­volves the risk of dividing by zero, in which case the result-far frombeing uniformly valid-becomes infinite within the region of interest.In fact, he shows that this happens in just the example treated onpage 96, the thin elliptic airfoil. The multiplicative composite (5.36)is valid on the surface, but ahead of (or behind) the airfoil it breaksdown even on the dividing streamline.

223 l\'otes

--Note 7. Utility of Strained Coordinates 229

In the notation of Section 4.9 (with s the distance downstream fromthe leading edge of an ellipse of chord 2 and thickness ratio e), theexact speed on the x-axis is found by conformal mapping as

Note 7. Utility of the Method of Strained Coordinates

Although the method of strained coordinates (Chapter 6) has nowbeen applied to a considerable variety of problems in fluid mechanics(and a few in other fields), the question of when it can be relied uponis not settled. Mathematicians have studied the method only forordinary differential equations (e.g., Comstock, 1972), which providesscant guidance for partial differential equations. From time to time asolution previously calculated by straining is found to be erroneous bya worker using a more pedestrian but reliable method (e.g., Jischke,1970).

It seems amply confirmed that no difficulties arise if the governingequations are hyperbolic, as they are in the majority of applications.The generalization of Lin (p. 100) that consists in successively refin­ing two families of characteristic lines has been further developed byOswatitsch (1962a, b) as the analytic method if characteristics, and ap-

The outer and inner expansions can be extracted from this by expand­ing for small e in terms of sand S = e2s, respectively. Thus it is seenthat the denominator of the multiplicative rule vanishes at about halfa nose radius ahead of the edge. For example, the 2-term outer expan­sion of the I-term inner expansion is 1 - e( - 2s )-1/2, which vanishesat s = -e2 /2.

Similarly, W. Reddall (unpublished) found in working Exercise 9.3that the multiplicative composite formed from two terms of the outerand one term of the inner expansion has a denominator that vanishesat (1 - A )ey = 2. To avoid this danger, the additive rule (5.32) shouldordinarily be preferred, as it largely has been in the literature, despitesome attractive features of the multiplicative rule.

At the stagnation point of the ellipse, with two terms of the outerexpansion and one of the inner, Ellinwood's alternative compositesbased onr, 1, and log] give velocities of order e1/2 , e, and 0, respec­tively. This illustrates that different composite expansions may havedifferent accuracies, and that the error may exceed that of either ofthe constituent expansions.

U 1 ( I-S)u = T-=-; 1 - \;_ 2s + s2 + e2 's<o (N.17)

plied by him and his colleagues to a number of problems III gasdynamics (cr. Stuff, 1972).

The straining must, however, be carried out for the physical("primitive") variables-velocity, pressure, temperature, etc.-ratherthan for such constructs as the velocity potential or stream function.For example, treating the problem of supersonic flow past a thin air­foil using the perturbation velocity potential (6.28) rather than thestreamwise velocity increment (6.30) yields only half the correctstraining function in Eq. (6.35).

Strained coordinates cannot generally be applied to parabolic equa­tions, as in boundary-layer theory. However, they prove effective in awake. Crane (1959) showed that straining provides an attractivealternative to the logarithmic terms that Stewartson (1957) had tointroduce into Goldstein's (1933) expansion for the laminar wake farbehind a flat plate in order to maintain exponential decay of vorticity.(This requirement, discussed on page 131, would appear to be dif­ferent from Lighthill's principle of Eq. (6.1), but non-exponentialdecay leads to singularities at the next step.) Berger (1968) showsthat straining succeeds in the same way for the axisymmetric far wake.

The situation is similarly unclear for elliptic equations. The criti­cism (p. 119) that Lighthill's success with a round-nosed airfoil inincompressible flow cannot be extended to higher order has beenrefuted by Bollheimer and Weissinger (1968) and Hoogstraten (1967),who show that it is only necessary to strain both coordinates slightly,or in fact the single complex variable x + ry. However, Hoogstratenfinds that difficulties persist for other nose shapes.

Although the equations of supersonic flow are hyperbolic, theybecome elliptic inside the Mach cone in conical flow. Lighthill(1949b) found the conical shock wave using strained coordinates; butMelnik (1965b) emphasizes that the method fails near the triple pointwhere the plane shock wave from a supersonic edge meets the nearlycircular conical wave from the vertex. There he shows that themethod of matched expansions can be used to join five differentboundary layers (which were studied earlier by Bulakh, 1961).

Melnik also shows how the vortical layer on a cone in supersonicflow can be matched directly to the outer flow. (He gives a usefulcomparison with other less accurate solutions, including that ofMunson, 1964.) However, in the hypersonic thin-shock-Iayer approxi­mation he finds (Melnik 1965a) that there is no overlap until themethod of strained coordinates is applied to minimize the nonuni­formity of the outer solution near the surface (as an alternative tointroducing an intermediate third region). This idea of combining

--230 Notes ","ote 8. Flat Plate at High R; Triple Decks 231

Fig. N. I. Schematic picture of matching regions for flow past finite flat plate at highReynolds number; primary regions shown by solid lines and secondary regions by dashed lines.

way, as shown in Fig. N.1 (which is a composite of the drawings ofMessiter and Stewartson). There is also a small circular core in which,

Upper deck,R' I/ 2 poten~ial f~~w

Potential flow

R' 5/8 Ma.in··deck, inv'i'sci,d

Blasius boundary layer L~"···.i...!:n.d;~~:-Ia-y.-J~\--In-n-~-rG-O-Id-~-te-in-w-.a-k-e

just as in the corresponding vicinity of the leading edge, the full equa­tions apply. (These, and another circular region of size R-1I2 aboutthe trailing edge, are shown dashed because they affect only higherapproximations. )

This structure contributes a correction to the Blasius drag that isof order R-7I8, and hence slightly more important than the displace­ment effects of order R-l calculated by Kuo (p. 136) and lmai (p.138). Thus Equation (7.40) is corrected to

Note 8. Flat Plate at High Reynolds Number; Triple Decks

Much effort has been devoted in the past decade to improving thesolutions for viscous flow past the semi-infinite flat plate and the finiteplate at large Reynolds number.

The eigensolutions discussed in Section 7.6 have been studied fur­ther by Murray (1965, 1967), Brown (1968), and Ting (1968). A solu­tion for the semi-infinite plate can be evaluated by how well itpredicts the coefficient Cl of the first of these eigensolutions in theexpansions (7.46), (7.47) for skin friction far downstream, as well asthe undetermined constants in the Stokes expansion (3.24) near theleading edge, which (with those constants redefined) gives the localskin friction as

the method of strained coordinates with the method of matchedasymptotic expansions has been applied also to two problems in gasdynamics by Crocco (1972) and to a hypersonic boundary layer byMatveeva and Sychev (1965).

c = A R -112 + B R 1/2 - ~A 2 R +r J: x 32 x(N.18)

1.328 2.661 (1 1 1)cF = Rl/2 + R7I8 + 0 "R' R7I6' R5/4 (N.19)

The most reliable values are given by Botta and Dijkstra (1970) who,refining the procedure of van de Vooren and Dijkstra (1970), carriedout a careful finite-difference solution of the Navier-Stokes equationsin parabolic coordinates, using three different mesh sizes, and beingguided by all that is known of the analytical structure of the solution.They were only able to estimate 2.2 < Cl < 2.5, but found A= 0.75475 ± 0.00005 and B = 0.041. [A less detailed finite-differencesolution (Yoshizawa, 1970) gave Cl = 1.6, A = 0.748, and B = 0.044.]This value of A means that the skin friction near the leading edge is14 per cent higher than the Blasius value.

I,Finite plate. Our understanding of viscous flow past a finite flat plate

(Section 7.9) has been revolutionized by the simultaneous discoveryof Stewartson (1969) and Messiter (1970) that at high Reynolds num­ber the flow near the trailing edge has a compound structure thatStewartson calls the.triple deck.:... It comprises a region around the trail­ing edge whose extent is of order R-3/8 times the length of the plate,and which matches upstream with the classical Blasius solution andits associated outer flow and downstream with the two-layered wakeanalyzed by Goldstein (1930). It consists of three layers, in each ofwhich the Navier-Stokes equations can be approximated in a different

The coefficient 2.661 was evaluated numerically by Melnik and Chow(1975); Jobe and Burggraf give instead 2.694, and Veldman and vande Vooren (1974) find 2.651 ± 0.003. Of the subsequent terms, that oforder R-l arises from the trailing-edge region of size R-3/8 as well asfrom the displacement effects, that of order R-7I6 from the region ofsize R-1I2, and that of order R-5/4 from the region of size R-3/4.

That last small region was treated approximately by Stewartson(1968), and later Dijkstra (1974) solved the full Navier-Stokes equa­tions numerically. They both found that the Stokes solution (3.24)of Carrier and Lin (1948) applies locally at the trailing as well as theleading edge. Dijkstra evaluated the coefficient of that square-rootsingularity in local skin friction, but was unable to find the coefficientof the term in the drag (N.19) of order R-5/4.

Jobe and Burggraf find that the two-term approximation (N.19)agrees unexpectedly well with both experimental measurementsUanour, 1947) and numerical solutions of the Navier-Stokes equa­tions (Dennis and Dunwoody, 1966, Dennis and Chang, 1969). Thismeans that the subsequent terms must either have much smallercoefficients than the first two, or tend to cancel one another.

232 Notes Note 9. Extension of Optimal Coordinates 233

(N.21)

Then optimal coordinates are given by the generalization of (7.56b)

~oPt(x, y) = F1[l/;z(x, y) + R-I/Z l/;3(x, y) + R-l l/;4(x, y) + ... ]

l1opt(x, y) = l/;l(x, y) Fz[l/;z(x, y) + R-I/Z l/;.3(x, y) + ... ]

If logarithms of R appear in the outer expansion, they are simply tobe included here in the obvious way.

Legner has verified this rule to second or higher order in a numberof specific examples. Davis (1974) points out that, in contrast toKaplun's result, this rule is not completely general. However, headopts it in showing how optimal coordinates can prove advantageousin numerical solution of the Navier-Stokes equations. On the otherhand, L. E. Fraenkel (unpublished) has questioned the demonstration

(N.20)

l/; ,...,. l/;l (x, y) + R-lIz l/;z(x, y) + R-l l/;3(x, y)

+ R-3IZ l/;4(x, y) + ...

stream. He promised to remove those restrictions III a later paper,but did not live to write it.

Legner (1971) has shown that Kaplun's rule (7.56b) for constructingoptimal coordinates applies unchanged to axisymmetric flows, tv on··coming streams containing vorticity, to boundary layers with noadjacent walls, such as jets and wakes, to free-convection boundarylayers, to compressible fluids, and to uncoupled and coupled thermalboundary layers. It is implicit in his reasoning that Kaplun's correla­tion theorem (7.50b) also applies to all those flows. In addition,Legner has extended the rule for constructing optimal coordinates tounsteady boundary layers, and to three-dimensional ones in terms ofClebsch's pair of stream functions. He gives examples for many ofthese situations.

Kaplun also limited consideration to the first-order boundarylayer plus the resulting outer flow due to displacement. We have giventhe second-order correlation theorem in Exercise 7,4, and it clearlyapplies also to the first group of problems just mentioned, and couldno doubt be extended to the second group.

Legner has also extended Kaplun's rule for optimal coordinates toarbitrarily high order. Let the solution be carried out in any con­venient coordinate system x, y. The outer expansion will be a con­tinuation of the form (7,4Ba):

Note 9. Extension of the Idea of Optimal Coordinates

Kaplun (1954) limited his study of the role of coordinate systemsin boundary-layer theory to incompressible, steady, plane flow past asolid body without separation, and with an irrotational oncoming

The triple-deck structure has been extended to a plate at an angleof attack of order R-1I16 by Brown and Stewartson (1970), to an air­foil with a finite trailing-edge angle of order R-1/4 by Riley andStewartson (1969), and to a three-dimensional wing with a pair ofeddies at the trailing edge by Guiraud (1974).

Separatedfiow. The multistructured boundary layer that Stewartsoncalls the triple deck was discovered at the same time for the flat plateand for the region of free interaction near a separation point in super­sonic flow. The latter problem was' analyzed independently byStewartson and Williams (1969) and by Neiland (1969), who showthat the primary regions have the same dimensions as in Fig. N.1.Neiland (1971) later matched that local solution to a four-layeredregion of separated flow farther downstream, and found agreementwith experimental measurements of the "plateau" pressure.

The same structure persists near separation at hypersonic speeds ifthe hypersonic interaction parameter X = M/ RI/Z is small; but other­wise Neiland (1970) has shown that the region of free interaction is ofthe order of the length of the body. It depends on an eigensolutionthat grows from the leading edge as a high power of distance. Kozlovaand Mikhailov (1970) show that a similar non-uniqueness arises foran infinite yawed wing, and in the case of an infinite triangular wingpermits the flows on the two halves to meet smoothly on the center­line.

Sychev (1972) has explained how the structure of Fig. N.1 appliesalso to separation from a smooth body in incompressible flow. Anadverse pressure gradient of magnitude Rl/8 acts over the interactionregion of length R-3/8, and in the limit the local pattern is the inviscidfree-streamline flow that has bounded curvature. Sychev shows thatthis structure is compatible with flows upstream and downstream,which have also been studied by Messiter and Enlow (1973). Messiter(1975) discusses how this model, though formally self-consistent, isnot yet complete because of our ignorance of the global structure ofthe recirculating wake, which could, for example, induce an infinitesequence of vortices below the separation streamline.

_J\lljhese problems, and others involving multistructured boundarylayers, have been comprehensively surveyed by Stewartson (1974).

234 Noles "iole 10. Sphere and Circle at Low R 233

6'1T[ 3 9 9 (5 323)CD = If 1 + SR + 40R2 log R + 40 y + 3' log 2 - 360 R2

Note 10. The Sphere and Circle at Low Reynolds Number

Chester and Breach (1969) have, in a difficult computation, con­tinued the analysis of Proudman and Pearson (1957) to include onehigher power of Reynolds number. Thus they extend the series (1.4)for the drag coefficient of a sphere by two terms to find

that (N.21) yields optimal coordinates. Thus the extension to secondand higher order remains to be resolved.

The "shrinking rectangular" coordinates of Fig. 7.8 are an unneces­sarily artificial choice to illustrate that the boundary layer may notcontain even the basic outer flow. Polar coordinates, which are anattractive system for the semi-infinite plate, also display this total lackof optimality.

+ 27 R3 log R + O(R3)]80

(N.22)

ability of the drag for expansion in terms of Reynolds number, andattempts to recast it into more appropriate form. Clearly some suchreinterpretation of the terms containing log R is necessary in orderthat the result be useful for R greater than unity. (Cf. Note 15.)

We may ask whether Chester and Breach's last term is correct,since they have violated the warning of Fraenkel (1969a), discussed inNote 4, against separating like powers of R that differ only by powersof log R. To be sure, their solution itself shows that Proudman andPearson happened to escape unscathed in calculating the term in R2log R while neglecting that in R2. However, it is just at the next stagethat ignoring Fraenkel's warning is known (Note 4) to lead to errorin the simpler problem of diffraction by a thin semi-infinite plate.

It was mentioned on page 155 that whereas the Stokes approxima­tion is a singular perturbation, the Oseen expansion appears to be aregular one. [Kaplun (1957) observed that this is a coincidence thatdisappears in compressible flow, where the Stokes approximation isnonlinear (Lagerstrom, 1964).] The second approximation for a spherewas calculated by C. R. Illingworth in 1947 (unpublished), who foundfor the drag coefficient

Thus iterating on the solution of the Oseen equations extends theagreement with the Navier-Stokes drag from relative order R to R2log R, and subsequent iterations would no doubt produce furtheragreement.

Separated flOw. Dorrepaal, Ranger, and O'Neill (1975) have dis­covered that the Stokes approximation is rich enough to predictseparation on a finite body, provided it is concave. Furthermore,Ranger (1972) shows that in the diverging Jeffery-Hamel flow be­tween plane walls the Reynolds number for separation is predictedwithin three per cent by two terms of the Stokes expansion, and forthe channel of 45-degree half angle to within one and one-half percent by four terms. In that problem the Stokes expansion is a regularperturbation, proceeding in powers of Reynolds number.

This suggests that it is no coincidence that in the singular perturba­tion problem of the sphere the standing eddy is described well by thetwo-term Stokes expansion, as shown in Figs. 8.3 and 8.4. [The

Here y = 0.5722 ... is Euler's constant.This result is disappointing, because comparison with experiment

suggests that the range of applicability has scarcely been increased.Chester and Breach conclude that "the expansion is of practical valueonly in the limited range 0 ,;;;;; R ,;;;;; 0.5 and that in this range thereis little point in continuing the expansion further." Below R = 0.3,however, Dennis and Walker (1971) find that it gives a better ap­proximation than any other asymptotic solution to the drag that theycalculate numerically.

This limited utility contrasts with that of regular perturbation ex­pansions, which have recently been found actually to converge forReynolds numbers considerably larger than unity. For example,Kuwahara and Imai (1969) have calculated by computer (cf. Note 2)eight terms of a power series in R for the steady plane flow producedinside a circle by tangential motion of the boundary, and estimateconvergence up to about R = 32 in three different cases. Again,Hoffman (1974b) has computed 17 terms for the self-similar flow be­tween infinite concentric rotating disks, and finds convergence up toR = 15 for the case of one disk fixed and R = 42 for equal contra­rotation.

In an appendix to the paper of Chester and Breach, Proudmansuggests that the limited utility of their series is due to the unsuit-

+ 38210R3 log R - 0.0034 R3 + ... ) (N.23)

236 Notes Note 11. Transcendentally Small Terms 237

(N.25)

Reynolds number of 8 (based on radius) that it gives for the onset ofseparation may also be compared with the recent values of 10 fromfinite-difference solution of the Navier-Stokes equations (Pruppacher,Le Clair, and Hamielec, 1970), 10.25 from series truncation (Dennisand Walker, 1971), and 8.5 from flow visualization (Payard andCoutanceau, 1971 ).] However, Eq. (8.38) for the Reynolds number atseparation can be extended, using the solution of Chester and Breach(1969), to give

1 9 (5 83)1 - -R + _R2 log R + - log 2 + Y + - + O(R3 log R) = 08 40 3 840

(N.24)

and Ranger (1972) points out that this again has no real root, nor doesthe result of including the term in R3 log R. Once more it seems thatthe logarithm needs reinterpretation.

The solution of Skinner (1975) provides a corresponding equationfor the circular cylinder:

(~1 - 0.87 ~13 + ... ) - 4R( 1 - 4~1 + ... )

- ~R2(1 - ~~1 + ... ) + O(R3) = 096 14

where ~1 = (log 4/R - y + !)-1. Two terms give separation atR = 1.12, and three terms at R = 1.0, which may be compared withthe value 2.88 computed by Underwood (1969) using series trunca­tion (and Yamada's value, quoted on page 156, of 1.51 for the Oseenapproximation). This rough agreement seems satisfactory, since thequantity ~1 that is assumed small is then equal to 0.8.

Note 11. Transcendentally Small Terms

A perturbation solution that yields a formal series in powers ofe will never include such exponentially small terms as e- llt . An ex­ample showing how they may be numerically important is the asymp­totic expansion (Olver, 1964)

j'lT cos nt dt '" '!!.e-n + (_ )n-1r 2'TT 1a t2 + 1 2 l ('TT2 + 1? n2

Even at n = 10 the exponentially small term contributes 15 per centof the total. Dingle (1973) has persuasively argued the case for seekinga "complete asymptotic expansion," including any sets of exponen­tially small terms that may be present.

Regular perturbations seem usually to be free of such terms. Forexample, each of the series discussed in Notes 2 and 10 that has beenextended to high order by computer appears to represent an analyticfunction of the perturbation quantity, having a finite radius of con­vergence in the complex plane. On the other hand, Terrill (1973)has shown that in the singular perturbation problem of laminar flowthrough a porous pipe at high Reynolds number, the first set of ex­ponentially small terms is needed to explain the nature of the solu­tion. A numerical computation showed that there are dual solutionsfor R > 9.1 and none for 2.3 < R < 9.1. Matching a boundary layerto the inviscid core in conventional fashion gave only a single solu­tion in powers of R-1. However, including terms in e-R yields dualsolutions for R > 9.1 and none for 3.3 < R < 9.1, in remarkableagreement with the numerical results. Likewise, Adamson and Richey(1973) found exponentially small terms essential for treating transonicflow through a nozzle with shock waves.

Bulakh (1964) has included exponentially small terms in the outerexpansion of the boundary-layer solution for converging flow betweenplane walls, and checked with the exact Jeffery-Hamel solution. Heshows that such terms will arise also in the higher-order boundary­layer solution at a plane or axisymmetric stagnation point, and inother problems.

In the "purely logarithmic case," where straightforward matchingyields outer and inner expansions in powers of (log lIe)-1 or (logK/e)-l, terms of order e and higher powers often appear. As men­tioned on page 165 (see also Exercise 8.5), such transcendentally smallterms were discussed tentatively by Proudman and Pearson (1957)for flow past a circle at low Reynolds number. Skinner (1975) has sys­tematically examined the first two sets of transcendentally small termsin that problem. He finds that terms in R affect the symmetry of theflow pattern significantly even at R = 0.025, but do not contribute tothe drag. Terms in R2 extend Kaplun's result (8.49) for the drag coef­ficient to

as n-,,>(X) (N.26) (N.27)

238 NotesNote 13. Lifting Wings of High Aspect Ratio 239

(N.28)

(N.30)

. 'OJ(N.29)

dCL "-' 2'lT

da "-' 2 16( 9) 11 + - + - log 'ITA - - -A 'lT 2 8 A2

3 - 2z2 'IT]log _ z2 +1 - z2 VI

_ 10g(1 _ z2) _ log 2(1 - z2)1 - z2

Werle and Davis (1972) have calculated numerically the boundarylayer on a parabola in asymmetric flow. They inevitably encounterthe square-root singularity of Goldstein (1948) whenever the skinfriction vanishes.

The expansion at low Reynolds number for axisymmetric flow pasta paraboloid of revolution is discussed in Note 4. Davis and Werle(1972) and Veldman (1973) have solved the Navier-Stokes equationsnumerically over the whole range of Reynolds number.

dCL [ 2 16 log A 4 (9 ) 1- = 2'lT 1 - - - - -- + - - + 'lT 2 _ 4 log 'IT _ +da A 'lT 2 A2 'lT 2 2 A2

Note 13. Lifting Wing of High Aspect Ratio

Kerney .(1972) has discovered that the integrals in Eq. (9.15) wereevaluat.ed mcorrectly by Van Dyke (1964b) for the elliptic wing. TheexpresslOn (9.18) for the circulation is to be replaced by

Integrating this changes the result (1.6) for the lift-curve slope to

Correspondingly, when this is recast by analogy with Prandtl's result(9.1a) it gives, in place of (10.27),

These corrected results are compared with lifting-surface theory inFig. N.3, which replaces Fig. 9.4. Krienes's results from lifting-surface

:t

I

5toxes limit, R=O (numerical)

9'/"

//

Numerical Isolutions /

I2 nd-order boundary­

layer theory

18f-order boundary­layer theory

.4.1

.6

1.0

1.2

(Ra)I/2-2- .82x pU2

However it is clear from Fig. 8.5 that at least for R > 0.4 this correc­tion is in'the wrong direction, no doubt because it is smaller than theunknown term of order ~14 in Kaplun's expansion.

Fig. N.2. Skin friction at leading edge of parabolic cylindcr.

Note 12. Viscous Flow Past Paraboloids

Paraboloids enjoy many simple and remarkable properties i? fluidmechanics, in both potential and viscous flows. We have seen m Sec­tions 4.6, 4.8, 4.9, 10.6, 10.8, and 10.9 how they serve as models formore complicated flows, or as elements in their solution.

Laminar flow along the axis of a parabolic cylinder has be~~ cal~u­

lated numerically, as a generalization of the flow past a seml-lI~fimte

plate, by several of the groups of workers .~ited in Note 8. Denms andWalsh (1971), Davis (1972), and Botta, DIJkstra,. and .veldman (.1972)independently solved the Navier-Stokes equatlOns m parabohc co­ordinates by finite-difference methods over the range .of ~eynolds

number from zero, corresponding to the flat plate, to mfimty, cor­responding to Prandtl's bound~ry-l~y~r appro~i~ation. T~e threesolutions give the same local skm fnctlOn to withm a fractlOn of aper cent. A striking conclusion is that second-order bound~ry-Iayer

theory, as represented by the friction near th~ leading edge (FIg. N.2),is of little practical utility, for it departs rapIdly from the exact solu­tion as the Reynolds number decreases.

theory have been replaced by the following refined values calculatedby R. T. Medan using the method of Medan (1974):

together with the value 1.7900230 found by Jordan (1971) for thecircular wing (A = 4/'17"). Correcting the error has slightly improvedthe agreement at high aspect ratio, while worsening it at low values.

241

(N.32)

Note 14. The Method of Multiple Scales

r log A 1- = 1 + --fl(z) + -f2(z) + ...roo A A

Fig. N.3 shows that this version provides a very close approximationover the whole range of A. For small values it gives 1. 72 A, comparedwith the exact limit of ('17"/2)A = 1.57 A from slender-wing theory.

Thurber (1965) has treated the more general case of a wing ofsmooth symmetrical swept-back planform, and finds that the loga­rithm of the aspect ratio then appears in the second rather than thethird approximation. Thus the circulation has, instead of (9.15), theform

Notes

1/'17" 2/'17" 8/'17" 16/'17"0.496 ± 0.002 0.969 ± 0.006 2.944 ± 0.004 4.151 ± 0.004

240

(N.34)

(N.33)

X 2 = EX

X 2 = EX,

x = x(1 + C E2 + C E3 + ... )1 2 3 ,

2. The two-variable expansion method. The slower scale is simple, andthe faster scale slightly stretched (linearly), for example,

~nd Thurber gives expressions for evaluating the functionsfl andf2In terms of the geometry of the planform.

Kerney (1971) and Tokuda (1971) independently analyzed thestraight wing with a jet flap. However, their results differ, and thediscrepancies have not yet all been resolved.

Note 14. The Method of Multiple Scales

This book is relatively strong on the method of matched asymptoticexpansions, but weak on the method of multiple scales (Sec. 10.4).The reason is that multiple scales were just being developed when itwas written. That technique has flourished in the last decade, so as tobe second in importance only to matched expansions for fluid me­chanics. It is not possible to redress the balance in these Notes; wemust refer the reader elsewhere.

Cole (1968) devotes a chapter to what he calls "two-variable ex­pansion procedures," but gives no applications to partial differentialequations, and the same is true of the useful article by Kevorkian(1966). The most complete account is Chapter 6 of Nayfeh (1973).He distinguishes three variants of the method:

1. The many-variable or derivative-expansion method. One works with(the first few of) an unlimited number of successively slowersimple scales, for example,

N.31)

Liffing-surface theoryo Medonc Jordan (1971)

Prondt/'s2 nd opprox. _

(9.10) __ - -:.....=-- ~ .........------ ~~ --~--Slender-wing

fh.eory

deL ~ 2'17"

da 2 16 11 + - + -- 10g(1 + '17"e-9/8A)

A '17"2 A2

-­. ...-Modified . /' 9" .- .-

3" opprox. .. /' ~ " "(N.30) /./~:;./"" ,. ,.

, /: " 0 ,."\ /.// ,.I/' /(. . /" 2 nd opprox.

,//,/ ..,ermom I (9.lb)/)1 (N.31)

Ij/0

/0'to 1l....- ----lL------I-------l-------:

e0 24 6A

2

4

6 3 rd opprox.(N.29)

Fig. N.3. Lift-curve slope of elliptic wing (correction of Fig. 9.4).

Germain (1967) has suggested an alternative recasting of (1.6) thatis better for small aspect ratio A. When corrected for the error in (1.6)it becomes

242 :'-lotes Note 15. Analysis and Improvement of Series 243

where the solution by multiple scales holds only for Eed/2 « 1.

This semi-numerical program represents, at least for certain simpleproblems, a more effective use of the computer than finite-differenceor other purely numerical schemes. Greater accuracy can be achievedin shorter computing times, and the solution is found as a functionof the perturbation quantity, rather than for just a few fixed values.Complete success consists in starting with an expansion predicatedon vanishingly small values of a perturbation quantity, and recastingthe series to yield accurate results over the entire range of physicalinterest. Two such triumphs are discussed in Note 2: Schwartz's (1974)recasting of Stokes's series to render it convergent for the highest waterwave, and Van Dyke's (1970) transformation of Goldstein's series forthe Oseen drag of a sphere at low Reynolds number, giving three­figure accuracy at infinite Reynolds number.

Singular perturbation series, especially those involving logarithmsof the perturbation quantity, often seem to cry out for improvement,but no general approach is known. In the "purely logarithmic" case,where the expansion has the form

I. Extend the series to high order by delegating the routinearithmetic labor to the digital computer.

11. Examine the coefficients to reveal the analytic structure of thesolution in the complex plane of the perturbation quantity.

111. Transform the series as suggested by that structure.

Note 15. Analysis and Improvement of Series

Since this book was written the author has learned several ways, inaddition to those discussed in Sections 10.6-10.8, for improving theutility of a perturbation series. These apply mostly to regular pertur­bations, where the solution is found (aside perhaps from a multiplica­tive function) as a formal power series in the perturbation quantity.Then the power of complex analysis can be brought to bear by con­sidering the perturbation quantity in the complex plane, even thoughit usually has physical significance only for positive real values.

For power series we thus have a battery of devices that can be used(Van Dyke, 1974) to unveil in part the analytic structure of the solu­tion, and then exploit that knowledge to increase the accuracy of theseries or extend its range of validity. Together with the growing useof the computer to calculate many terms (Note 2), this process ofanalysis and improvement provides a three-step program for treatingregular perturbations (Van Dyke, 1976):

(N.38)

(N.37)

(N.36)

X 2 = x (N.35)

x + e-d x = 0

No term in clc is ordinarily needed in Xl because it is accounted

for by X 2 •

3. The generalized method. The slower scale is simple, and the fasterscale strained nonlinearly, for example,

and Peyret (1970) slightly stretches both:

Xl = x(1 + clc + (2c2 + .. '),

Here again examples suggest that the term g2(x) could beomitted.

As a matter of fact, most of the problems in fluid mechanics that havebeen solved using the method of multiple scales require onl~ theprimitive version with just two simple scales, say x and cx, eItherstrictly or because the expansion has been carried only to 10';' order.Some typical recent examples are Hinch and Leal (1973), Gnmshaw(1974), and Nayfeh and Tsai (1974).. .

Still other variants are encountered In the lIterature. Levey andMahony (1968) slightly stretch the slower rather than the faster scale:

Any problem that can be solved by match~d expansions can alsobe solved by multiple scales, though less effiCiently. The method ofmultiple scales is therefore reserved for problems where matched ex­pansions do not apply, typically problems involving slowly modulatedoscillations or waves.

Some writers mistakenly suggest that the method of multiple scalesyields results valid over an indefinitely long range. In general, thesolution is valid only to x = D(cl ) if the short and long scales are xand cx. An example is the "aging spring" (Cheng and Wu, 1970),governed by the equation

244 NotesNote 15. Analysis and Improvement of Series 245

40 ~.,-----;-------:-------------,

(NA1)

(NA2a)

(NA2b)eo = limlcn - 11n-+oo en

- ... ]

- 0.0560(xll)3 - 0.0372(xll)4 - 0.0272(xll)5

- 0.0212(xll)6 - 0.0174(xll)7 - 0.0147(x/l)8

More often the signs alternate, as in Eqs. (1.3), (1.5), (1.7), (10.21),and (10.22), and this means that the singularity lies on the negativereal axis. Other more complicated patterns are occasionally encoun­tered, and can be interpreted similarly.

The distance to the nearest singularity-that is, the radius of con­vergence-can be estimated by graphical application of thed'Alembert ratio test, according to which the power series

has radius of convergence

In the case of products of powers of e and its logarithm, we knowof only two attempts at improvement: our recasting (N.30) of theseries (N.29) for the lift-curve slope of an elliptic wing by analogywith Prandtl's first-order result (9.1a), which was further recast byGermain into the form (N.31); and Proudman's proposed reinterpre­tation (Note 10) of the series (N.22) for the drag of a sphere at lowReynolds number.

The nearest singularity; Domb-Sykes plots. The most useful way ofanalyzing the structure of a power series through its coefficients is toinvestigate the nature of the nearest singularity in the complex planeof the perturbation quantity. First, the direction of the nearestsingularity is revealed by the pattern of signs. If the terms of the seriessooner or later settle down to a fixed sign-all positive or all negative­as appears to be the case in Eqs. (1.1) and (10.29), the singularity lieson the positive real axis. A particularly well behaved example dis­playing this pattern is the series of Howarth (1938) for the skin frictionin a laminar boundary layer with decreasing external speed given byU = Uo(1 - xI81):

cf = ~R.x-112[1.328242 - 1.02054(xll) - 0.06926(xll)2

(NAO)

(N.39)

1.2.8R= Uo

]I

.4

fee) = A + c2 +log Kle (log Kle)3

OL-__---.:._-L.. ---I ...L-_--'

o

Effect of telescoping terms for drag of circular cylinder at low Reynolds number.

ji A B + C +(e) = log(l/e) + (log 1/e)2 (log 1/e)3

J

30 J b(3J

/Co /

/3 terms, not

/ telescoped20 i\ / o Experiment,

~ / Tritton

~

"" 0-.. .t term of (8.49) [Lamb}-'--.-10 0 '-.-.-2 'terms, o 00 CD

not', telescoped0 0 o 0 0

Fig. NA.

. : I term, not telescoped(Proudman a Pearson)

greatly improves the accuracy at any stage. In certain linear problems,and for special geometries, an infinite number of terms can in thisway be telescoped into one, as Batchelor (1970) has pointed out forStokes flow past a slender particle.

with K = e-B/ A . As mentioned on page 163, Kaplun chose this form(Eq. 8049) for the drag of a circular cylinder at low Reynolds number,whereas Proudman and Pearson (1957) used the first form. Figure NA(which should be compared with Fig. 8.5) shows that this modification

the first two terms can be telescoped into one, giving

246 Notes Note 16. The Resolution of Paradoxes 247

Fig. N.5. Graphical ratio test for Howarth's series (N.41). (a) Straightforward plot. (b)

Domb-Sykes plot.

I!

Note 16. The Resolution of Paradoxes

The method of matched asymptotic expansions is most strikingwhen it resolves a long-standing paradox. D'Alembert's paradox waseffectively resolved by Prandtl's boundary-layer theory. At the otherextreme of low Reynolds number, the paradoxes of Stokes and White­head, clarified by Oseen, were resolved by Kaplun (1957) usingmatched expansions (Chapter 8). Without straining the definition,we may assert that each of the three problems sketched below stoodas a paradox, "exhibiting an apparently contradictory nature," untilresolved by the same method. Like the paradoxes of Stokes andWhitehead, each involves logarithmic divergence at infinity in a planeproblem.

Filon's paradox. Filon (1928) studied steady viscous flow far from acylinder using the Oseen approximation. In the second approximationhe found the moment on an asymmetric shape as an angular-momen­tum integral that tended logarithmically to infinity as the contourincreased. This difficulty was resolved by Imai (1951), who showedthat the singularity in the second-order wake solution is cancelledby the third-order inviscid solution outside the wake. (The reason­ing is analogous to that later used by Imai for the drag of a semi-

the asymptote. Thus Fig. N.5b clearly shows a slope corresponding toa = ~, and this agrees with the local analysis of Goldstein (1948) andStewartson (1958), which shows a multiple singularity at the separa­tion point dominated by a square root.

The Domb-Sykes plot is seldom so straight as in this example.Other weaker or more distant singularities can make it curved,kinked, oscillatory, zig-zag, sluggish, or irregular. Examples of thesepossibilities, drawn from problems in viscous flow, are shown in VanDyke (1974), together with examples of zero and infinite radius ofconvergence. Some useful modifications and refinements are describedby Gaunt and Guttmann (1974) in the context of the thermodynamicsof lattice models of crystals (the subject from which we have borrowedthe Domb-Sykes plot).

A quite different alternative way of investigating the nature of thenearest singularity, and others in the complex plane, is to form ra­tional fractions (Section 10.7), more commonly called Pade approxi­manis, and examine the zeros of the denominator (p. 206). The useof Pade approximants for improving series has become very popularamong physicists. Useful references are Baker (1965, 1975), Hunterand Baker (1973), and Gaunt and Guttmann (1974).

(N.43b)

(N.43a)

(b)-1.0(0)

{

(co ± cY\ a =1= 0, 1, 2,f(c) = const X

(co ± c)a log(eo ± e), a = 0, 1, 2,

-1.0

1.0 - (0.9585r l, ," ,

1.0 0.9585-cn , ,

cn_. - , ,cn_1 ,

cn ,01 • I 0

0 04 8

n

then

so that the Domb-Sykes plot is straight. It seems that in many physicalproblems the nearest singularity starts with this algebraico-Ioga­rithmic form, so that the plot is asymptotically straight. Then theradius of convergence is readily estimated as the reciprocal of theintercept at lin = 0. In Fig. N.5b that value clearly corresponds tothe singularity at the separation point, which is known from nu­merical calculations (Leigh, 1955) to lie at xll = 0.9585. At the sametime the nature of the nearest singularity is revealed by the slope of

(1957) recommend inverting both scales, plotting Icn1cn-ll versus lin.This has the obvious advantage of requiring extrapolation to theorigin rather than infinity. More important, the extrapolation is oftennearly linear, as in Fig. N.5b. The reason for this is that if

A straightforward plot of Icn_/cnl versus n is ineffective, as illustratedfor Howarth's series (N.41) in Fig. N.5a. Instead, Domb and Sykes

infinite plate, discussed on page 138.) Chang (1961) has simplifiedthis resolution of Filon's paradox by systematically applying themethod of matched asymptotic expansions.

Planing at high Froude number. Green (1936) calculated the planeinviscid free-streamline flow for a flat plate of chord L planing on thesurface of deep water at high Froude number F = (Ug/ L)1/2 = 1h2•

His approximation suffers the defect that, because gravity wasneglected, the free surface drops off logarithmically to infinity withdistance. Rispin (1966) has resolved this defect by embedding Green'sapproximation into a systematic perturbation scheme as the first termof a local expansion, and matching to a distant expansion based onlinear wave theory. The expansion near the plate is found to involvethe logarithm of the Froude number from the second step, havingjust the form of the expansion (N.6) for diffraction near a semi­infinite plate. Ting and Keller (1974) have improved the solution byincluding the impact on the free surface of the jet thrown up by theplate.

Vortex rings. Whereas rectilinear vortices are conveniently idealizedas having concentrated cores, a curved vortex induces upon itself avelocity that increases logarithmically with the inverse of its coreradius. Some cut-off device therefore has to be introduced to simulatea vortex ring (Batchelor, 1967, p. 253). This difficulty was resolved byTung and Ting (1967), who match the outer solution for a con­centrated ring vortex to the inner solution for a diffusing rectilinearviscous vortex. As a result they find that as its core diffuses with thesquare root of time, the speed of the vortex ring decreases logarithmi­cally with time. Saffman (1970) has corrected some algebraic errors.

Laminar mixing. In contrast to these successes, we note that "theriddle of the course of the viscous shear layer between two streams ofdifferent speeds" has apparently not, as stated on page 77, been solvedby the matching of Ting (1959). Klemp and Acrivos (1972) showthat when all higher-order effects are considered the lateral positionof the mixing zone remains indeterminate by an amount of the orderof its own width.

248 Notes

r

REFERENCES AND AUTHOR INDEX

The numbers in square brackets show the pages on which the references appear, so thatthis list serves as an author index.

ACRIVOS, A. See Klemp and Acrivos.ABBon, I. H., and VON DOENHOFF, A. E. (1959). Theory qf Wing Sections. Dover, New York. [47]ADAMSON, T. C., JR., and FEO, A. (1975). Interaction between a shock wave and a turbulent

boundary layer in transonic flow. SIAM J. Appl. Math. 29, 121-145. [227]ADAMSON, T. C., and RICHEY, G. K. (1973). Unsteady transonic flows with shock waves in two­

dimensional channels. J. Fluid Mech. 60, 363-382. [237]ALDEN, H. L. (1948). Second approximation to the laminar boundary layer flow over a flat

plate. J. Math. and Phy.... 27, 91-104. [139]Am, T. See Tomotika and Aoi.

BAKER, G. A., JR. (1965). The theory and application of the Pade approximant method. Advances

in Theoretical Physics, (K A. Brueckner, ed.), pp. 1-58. Academic Press. New York andLondon. [2471

BAKER, G. A., JR. (1975). Essentials if Pad! Approximant.... Academic Press, New York. [247JBAKER, G. A., JR. See Hunter and Baker.BATCHELOR, G. K (1956). A proposal concerning laminar wakes behind bluff bodies at large

Reynolds number.J. Fluid Meeh. 1,388-398. [122]BATCHELOR, G. K, (1967). An Introduction to Fluid Dynamics. Cambridge Univ. Press, London

and New York. [248]BATCHELOR, G. K, (1970). Slender-body theory for particles of arbitrary cross-section in Stokes

f1ow . .J. Fluid Meeh. 44,419-440. [244]BELLMAN, R. (1955). Perturbation methods applied to nonlinear dynamics. .J. Appl. Mech.

22, 500-502. [2081BEI.I.MAN, R. (1964). Perturbation Techniques in Mathematics, Physic." and Engineering. Holt.

Rinehart & Winston, New York; Dover, New York, 1972. [xi]BERGER, S. A. (1968). The incompressible laminar axisymmetric far wake.). Math. and Ph)'S. 47,

292-309. [218,229)BERGER, S. A. (1971). Laminar Wakes. American Elsevier, New York. [218]BERGER, S. A. See Viviand and Berger.BERNDT, S. B. See Oswatitsch and Berndt.BOLLHEIMER, L., and WEISSINGER, J. (1968). An application of the "PLK method" to conformal

mapping and thin airfoil theory. Zeit. Flugwiss. 16,6-12. [229]BOTTA, E. F. F., and DIJKSTRA, D. (1970). An improved numerical solution of the Navier-Stokes

equations for laminar flow past a semi-infinite flat plate. Rept. Math. Ins/. Univ. GroningenNo. TW-80. [230J

BOTTA, E. F. F., DIJKSTRA, D., and VELDMAN, A. E. P. (1972). The numerical solution of theNavier-Stokes equations for laminar, incompressible flow past a parabolic cylinder.). Eng.Math, 6, 63-81. (238)

BREACH, D. R. (1961). Slow flow past ellipsoids of revolution.). Fluid Mech. 10,306-314. [78]BREACH, D. R. See Chester and Breach.BRETHERTON, F. P. (1962). Slow viscous motion round a cylinder in a simple shear.). Fluid

Mech. 12, 591-613. [77, 78]

249

I

I

BRODERICK, J. B. (1949). Super.;onic flow round pointed bodies of revolution. Quart. J. Meeh.

App/. .'vIath. 2,98-120. [29]BROWN, S. N. (1968). An asymptotic expansion for the eigenvalues arising in perturbations

about the Blasius solution. Appl. Sci. Res. 19, 111-119. [230]BROWN, S. N., and STEWARTSON, K. (1970). Trailing-edge stall. J. Fluid Meeh. 42,561-584. [232)Bm.AKH, B. M. (1961). On some properties of supersonic conical gas flow. j. API'I. Math. Meeh.

25,708-717. [78,177, 180, 229JBm.AKH, B. M. (1964). On higher approximations in the boundary-layer theory. j. Appl. Math.

Meeh. 28, 675-681. [237]BUROERS,.J. M. See von Karman and Burger.;.BURGGRAF, O. R. See .lobe and Burggraf.BURNS, J. C. (1951). Airscrews at supersonic forward speeds. Aero. Quart. 3, 23-50. [36]BU'SH, W. B. (1971). On the Lagerstrom mathematical model for viscous flow at low Reynolds

number. SIAM J. AfiP/. Math. 20, 279-287. [226, 227]BUSH, W. B., and FENnELL, F. E. (1972). Asymptotic analysis of turbulent channel and

boundary-layer flow. J. Fluid Mech. 56, 657-681. [227]CABANNES. H. (1951). Etude de I'onde de choc attachee dans les ecoulements de revolution.

Reeh. Aero. No. 24, pp. 17-23. 19, 39]CABANNES, H. (1953). Etudes du depart d'un obstacle dans un flu ide au repos. Reeh. Aero.

No. 36, pp. 7-12. [40]CABANNES. H. (1956). Tables pour la determination des ondes de choe detachees. Rech. Aero.

No. 49, pp. 11-15. [39]CARRIER, G. F. (1954). Boundary layer problems in applied mathematics. Comm. Pure App/.

Marh. 7,11-17. [119]CARRIER, G. F. See Lewis and Carrier.CARRIER. G. F., and LI:';, C. C. (1948). On the nature of the boundary layer near the leading

edge of a flat plate. Quart. Appl. Math. 6,63-68. [39,212,231]CASTEN, R. G. See Lagerstrom and Casten.CHAI>G, G.-Z. See Dennis and Chang.CHANG, I. D. (1961). Navier-Stokes solutions at large distances from a finite body. j. Math.

Meeh. 10,811-876.[38.41.41,7:),78,131,132,165,248]CHENG, H., and Wu, T. T. (1970). An aging spr;ng. Studies Appl. Math. 49, 183-185. [242)CHENG, H. K. (1962). Hypersonic flows past a yawed circular cone and other pointed bodies.

J. FlUid Merll. 12,169-191. [195]CHESTER, W. (1956a). Supersonic flow past a bluff body with a detached shock. Part I. Two­

dimension.!1 body. J. Fluid Meeh. I, 353-365. [29]CIIESTER, W. (1956b). Supersonic flow past a bluff body with a detached shock. Part II. Axi­

symmetrical body. J. Fluid Meeh. 1,490-496. [61CHESTER, W. (1960). The propagation of shock waves along ducts of varying cross section.

Advances in Appl. Mech. 6, 119-152. [3]CHESTER, W. (1962). On Oseen's approximation.J. Fluid Mech. 13,557-569. [161]CHESTER, VV., and BREACII, D. R. (1969). On the flow past a sphere at low Reynolds number.

J. Fluid Alech. 37,751-760. [221,234,236]CHOW, R. R., and TI>lG, L. (1961). Higher order theory of curved shock. j. Aerospace Sci. 28,

428-430. [78]CHOW, R. See Melnik and Chow.COCHRAN,.J. (1962). A New Approach to Singular Perturbation Problems. Ph.D. Dissertation. Stanford

Univ. [198, 200]COilE", D. See Jon!;s and Cohen.

COLE, J. D. (1957). Newtonian flow theory for slender bodies. J. Aeronaut. &i. 24,448-455.[22, 22]

COLE, J. D. (1968). Perturbation Methods in Applied Mathematics. Blaisedell, Waltham, Mass.[xi, 226, 226, 24 I]

COLE, J. D. See Lagerstrom and Cole.

COLE, J. D., and KEVORKIAN, J. (1963). Uniformly valid asymptotic approximations for certainnon-linear differential equations. Nonlinear Differential Equations and Nonlinear Mechanics, (J. P.LaSalle and S. Lefschetz, eds.), pp. 113-120. Academic Press, New York. [l98J

COLE, J. D. and MESSITER, A. F. (1957). Expansion procedures and similarity laws for transonicflow. I. Slender bodies at zero incidence. Z Angew. Math. Phys. 8, 1-25. [78]

COLES, D. (1957). The laminar boundary layer near a sonic throat. Preprints. Heat Transfer Fluid

Mech. Inst., 1957, pp. 119-137. Stanford Univ. Press, Stanford, California. [77)COLLINS, W. M., and DENNIS, S. C. R. (1973a). The initial flow past an impulsively started

cylinder. Quart. J. Mech. Appl. Math. 26, 53-75. [219, 219]COLLINS, W. M., and DENNIS, S. C. R. (1973b). Flow past an impulsively started circular cylin­

der. J. Fluid Mech. 60, 105-127. [219]COMSTOCK, C. (1972). The Poincare-Lighthill perturbation technique and its generalizations.

SIAM Rev. 14, 4:33-446. [228JCOUTANCEAU, M. See Payard and Coutanceau.CRANE, L. J. (1959). A note on Stewartson's paper "On asymptotic expansions in the theory of

boundary layers." J. Math. and Phys. 38, 172-174. [229JCRICHTON, D. G., and LEPPINGTON, F. G. (1973). Singular perturbation methods in acoustics:

diffraction by a plate of finite thickness. Proc. Roy. Soc. Ser. A 335, 313-339. [220, 221]CROCCO, L. (1972). Coordinate perturbation and multiple scale in ~asdynamics. Phil. Trans.

Roy. Soc. A 272,275-301. [217, 230]CURLE, N. (1956). Unsteady two-dimensional flows with free bounC:aries. I. General theory.

II. The incompressible inviscid jet. Proc. Roy. Soc. Ser. A 235,375-395. [41]DAVEY, A. (1961). Boundary-layer flow at a saddle point of attachment. J. Fluid Mech. 10,

593-610. [217)DAVIS, R. T. (1972). Numerical solution of the Navier-Stokes equations for symmetric laminar

incompressible flow past a parabola. J. Fluid Mech. 51, 417-433. [238]DAVIS, R. T. (1974). A study of the use of optimal coordinates in the solution of the Navier­

Stokes equations. Rept. Dept. Aerospace Eng. Univ. Cincinnati No. AFL 74-12-14. [233JDAVIS, R. T. See Werle and Davis.DAVIS, R. T., and WERLE, M. J. (1972). Numerical solutions for laminar incompressible flow

past a paraboloid of revolution. AIAA J. 10, 1224-1230. [223, 239]DENNIS, S. C. R. See Collins and Dennis.DENNIS, S. C. R., and CHANG, G.-Z. (1969). Numerical integration of the Navier-Stokes equa­

tions for steady two-dimensional flow. Phys. Fluids, Suppl. II, 12, 88-93. [231]DENNIS, S. C. R., and DUNWOODY, J. (1966). The steady flow of a viscous fluid past a flat plate.

J. Fluid Mech. 24,577-595. [231JDENNIS, S. C. R., and WALKER, J. D. A. (1971). Calculation of the steady flow past a sphere at

low and moderate Reynolds numbers. J. Fluid Meeh. 48, 771-789. [234, 236]DENNIS, S. C. R., and WALSH, J. D. (1971). Numerical solutions for steady symmetric viscous

flow past a parabolic cylinder in a uniform stream. J. Fluid Mech. 50,801-814. [238]DIJKSTRA, D. (1974). The solution of the Navier-Stokes equations near the trailing edge of a

flat plate. Doctoral Dissertation. Univ. Groningen. Nova Press, Groningen. [231]DI/KSTRA, D. See Botta and Dijkstra; Botta, Dijkstra, and Veldman; van de Vooren and Dijkstra.DINOI.E, R. B. (1973). Asymptotic Expansions: Their Derivation and Interpretation. Academic Press,

London and New York. [237J

251References and Author IndexReferences and Author Index250

252 References and Author Index References and Author Index 253

DOMB, C., and SYKES, M. F. (1957), On the susceptibility of a ferromagnetic above the Curie

point. Pmc. Roy. Soc. Ser. A 240,214-228. 1246]DONOV, A. E. (1939). A flat wing with sharp edges in a supersonic stream. NA.G.A. Tech. Memo.

No. 1394 (1956); transl. from Iz". Akad. Nauk SSSR, 603-626. [218JDORREPAAL, M., RANGER, K. B., and O'NEILL, M. E. (1975). Stokes flow past a body with a

re-entry region. To be published. [235]DUNWOODY, j. See Dennis and Dunwoody.ECKHAUS, W. (1973). Matched A~ymptotic Expansions and Singular Perturbations. North Holland,

Amsterdam and London; American Elsevier, New York. [xi]ENLOW, R. L. See Messiter and Enlow.ERDELYI, A. (1956). As;'mptotic t:xpansions. Dover, New York. [26,33]ERDF.I.YI, A. (1961). An expansion procedure for singular perturbations. Aui Accad. Sci. Torino

CI. Sci. Fis. Mat. Nat. 95, 651-672. [84]FDIOEI.l., F. E. See Bush and Fendell.FEO, A. See Adamson and Feo.FERRI, A. (1950). Supersonic flow around circular cones at angles of attack. Tech. Note NA.G.A.

No. 2236. [83)FERRI, A., NESS, N., and KAI'L1TA, T. T. (1953). Supersonic flow over conical bodies without

axial symmetry. J. Aeronaut. Sci. 20,563-571. [9]Fn.oN, L. G. N. (1928). On the second approximation to the "Oseen" solution for the motion

of a VISCOUS fluid. Phil Trani R~I' Soc A 227,93-135. [247]

{

RAENKEL'L. E (1969a). On the method of matched asymptotic expansions. Part I: A matchingprInciple. Proc Camb. Phil Soc 65,209-231 [220, 221, 222, 235]

FRAF.NKEL, L. E (1969b). On the method of matched asymptotic expansions. Part II: Someapplications of the composlle series Proc. Camb. Phil. Soc. 65,233-261. [220, 220, 221]

FRAENKEL, L. E. (1969c) On the method of matched asymptotic expansions. Part III: Twoboundary-value problems. Proc Camb Phil Soc. 65, 263-284. [220, 220, 221, 222]

FRAENKEL, L. E., and WATSOI', R. (1964). The formulation of a uniform approximation for thinconical wings with sonic leading edges. Proc. Symp. Transsonicum (K. Oswatitsch, ed.), pp. 249­

263. Springer-Verlag, Berlin. [78JFRIEDRICHS, K. O. (1942). Theory of viscous fluids. Fluid Dynamics, Chapter 4. Brown Univ. [79]FRIEDRICHS, K. O. (1948). Formation and decay uf shock waves. Comm. Pure Appl. Math. I,

211-245. [IIi]FRIEDRICHS, K. O. (1953). Special Topics in Fluid Dynamics, pp. 120-130. New York Univ. [77, 168]FRIEDRICHS, K. O. (1954). Special Topics in Analysis. New York Univ. [77]FRIEDRICHS, K. O. (1955). Asymptotic phenomena in mathematical physics. Bull. A mer. Math.

Soc. 61, 485-504. [451GARABEDIAN, P. R. (1956). Calculation of axially symmetric cavities and jets. Pacific J. Math. 6,

611-684. [3, 23)GAUNT, D. S., and GUTTMANN, A. j. (1974). Series expansions: analysis of coefficients. Phase

Transitions and Critical Phenomma, (C. Domb and M. S. Green, eds.) 3. Academic Press, New

York. [247, 247)GERMAI", P. (1967). Recent evolution in problems and methods in aerodynamics. J. Roy.

Aeronaut. Soc. 71, 673-691. [2401GERMAIN, P., and GUiRAUD,j.-P. (1960). Conditions de choc dans un fluide doue de coefficients

de viscosite et de conductibilite thermique faibles mais non nuls. G. R. Acad. Sci. Paris 250,

1965-1967. [78]GERMAIN, P., and GUIRAUD, j.-P. (1962). Conditions de choc et structure des ondes de choc

dans un ecoulement stationnaire de fluide dissipatif. G.NE.R.A. Pub!. No. 105. [78]

GoLDSTEIN, S. (1929). The steady flow of viscous fluid past a fixed spherical obstacle at smallReynolds numbers. Proc. Roy. Soc. Ser. A 123, 225-235. [5, 156)

GOLDSTEIN, S. (1930). Concerning some solutions of the boundary layer equations in hydro­dynamics. Proc. Camu. Phil. Soc. 26, I-30. [230]

GOLDSTEIN, S. (1933). On the two-dimensional steady flow of a viscous fluid behind a solidbody.-I. Proc. Roy. Soc. Ser. A 142, 545-562. [229)

GOLDSTEIN, S., ed. (1938). Modern Developments in Fluid Dynamics. Oxford Univ. Press, Londonand New York. [122, 131, 153]

GOLDSTEIN, S. (1948). On laminar boundary-layer flow near a position of separation. Quart. J.Mech. Appl. Math. 1, 43-69. [220, 239, 247)

GOLDSTEIN, S. (1956). Flow of an incompressible viscous fluid along a semi-infinite flat plate.Tech. Rep. Eng. Res. Inst. Univ. Calif. No. HE-150-144. [29, 77, 139)

1'. GOLDSTEIN, S. (1960). Lectures on Fluid Mechanics. Wiley (Interscience), New York. 177, 121,139,140)

GOl.DSTEIN, S., and ROSENHEAD, L. (1936). Boundary layer growth. Pmc. Cambrid,ge Philos. Soc.32, 392-401. [38, 214J

GREEN, A. E. (1936). The gliding of a plate on a stream of finite depth. Part II. Pmc. Camb. Phil.

Soc. 32, 67-85. [248]GRIMSHAW, R. (1974). Internal gravity waves in a slowly varying, dissipative medium. Geophys.

Fluid Dyn. 6, 131-148. [242]GROSSMAN, B. See Melnik and Grossman.GUlRAUD, j.-P. (1958). Ecoulement hypersonique d'un fluide parfait sur une plaque plane

comportant un bord d'attaque d'epaisseur finie. G. R. Acad. Sci. Paris 246, 2842-2845. [182]GUIRAUD, j.-P. (1974). Ecoulement decolle au voisinage du bord de fuite d'une aile mince tri-

dimensionnelle. J. Mecanique 13, 409-432. [232]GUIRAUD, J.-P. See Germain and Guiraud.GUTTMANN, A. J. See Gaunt and Guttmann.HALL, I. M. (1956). The displacement effect of a sphere in a two-dimensional shear flow. J.

Fluid Mech. 1, 142-162. [13]HAMIELEC, A. E. See Pruppacher, Le Clair, and Hamielec.HANCOCK, G. j. (1953). The self-propulsion of microscopic organisms through liquids. Proc.

Roy. Soc. Ser. A 217,96-121. [152]HANTZSCHE, W. (1943). Die Prandtl-Glauertsche Naherung als Grundlage fUr ein Iterations­

verfahren zur Berechnung Kompressibler Unterschallstromungen. Z Angew. Math. Mech. 23,

185-199. [5,29)HANTZSCHE, W., and WENDT, H. (1941). Die laminare Grenzschicht an einem mit Uberschall­

geschwindigkeit angestromten nicht angestellten Kreiskegel. Jahrbuch der deutschen Lriftfahrtfor­schung 1, 76-77. The laminar boundary layer on a circular cone at zero incidence in asupersonic stream. Rep. and Trans., British M.A.P. ( English Transl.) No. 276 [9)

HAYES, W. D. (1954). Pseudotransonic similitude and first-order wave structure. j. Aeronaut.

Sci. 21,721-730. [106, 107, 108]HAYES, W. D. (1955). Second-order pressure law for two-dimensional compressible flow. j.

Aeronaut. Sci. 22, 284-286. [76]HAYES, W. D., and PROBSTEIN, R. F. (1959). Hypersonic Flow Theory. Academic Press, New York.

[3, 22, 22, 123, 185, 185]HINCH, E. j., and LEAL, L. G. (1973). Time-dependent shear flows of a suspension of particles

with weak Brownian rotations. J. Fluid Mech. 57, 753-767. [242)HOFFMAN, G. H. (1974a). Extension of perturbation series by computer: symmetric subsonic

potential flow past a circle. J. Mecamque 13,433-447. [2151

254 References and Author Index References and Author Index 255

HOFFMAN, G. H. (1974b). Extension of perturbation series by computer: viscous flow betweentwo infinite rotating disks.). Compo Ph)'S. 16, 240-258. [234 1

HOOGSTRATEN, H. W. (1967). Uniformlv valid approximations in two-dimensional subsonic

thin airfoil theory.). Eng. Malh. 1,51-65.12291HOWARTII, L. (1938). On the solution of the laminar boundary layer equations. Proc. Roy. Soc.

Sa. A 164, 547-:>79. 12451HUNTER, D. 1.., and BAKER, G. A., JR. (1973). Methods of series analysis. I. Comparison of

current methods used in the theory of critical phenomena. P~y.... Rev. B 7, 3346-3376. [247]IMAI, I. (1951). On the asymptotic behaviour of viscous fluid flow at a great distance from a

cylindrical body, with special reference to Filon's paradox. Proc. R~y. Soc. Ser. A 208,487-516.

1165, 247JIMAI, I. (1952). Application of the M 2-expansion method to the subsonic flow of a compressible

fluid past a parabolic cylinder. Proc. lsI Japan Nat. Congo Appl. Mech. pp. 349-352. [212]IMAI, I. (1953). On the asymptotic behaviour of compressible fluid flow at a great distance from

a cylinder in the absence of circulation. .J. Phys. Soc. Japan 8, 537-544. [41,41]IMAI, I. (1957a). Second approximation to the laminar boundary-layer flow over a flat plate.

J. Aeronaul. Sci. 24, 155-156. [29, 54, 77,138,139,140, 140,2121IMAJ, I. (1957b). Theory of bluff bodies. Tech. Note Inst. Fluid IJ.ynamics and Appl. Math., Un i/!.

Maryland. No. BN-I04. [23)IMAI, I. See Kuwahara and Imai.IVES, D. C. See Melnik and Ives.JANOUR, Z. (1947). Resistance of a plate in parallel flow at low Reynolds numbers (trans!. from

Czech.). N.A.G.A. Tech. Memo. No. 1316, 1951. [231]JANSSEN, E. (1958). Flow past a flat plate at low Reynolds numbers.J. Fluid Mech. 3,329-343.

[139]JEFFREYS, H. (1926). On the relation to physics of the notion of convergence of series. Philos.

Ma.fi' [7] 2, 241-244. 130]JENSO!'/, V. G. (1959). Viscous flow round a sphere at low Reynolds numbers «40). Proc. R~.

Soc. Ser. A 249, 346-366. [160]JISCIIKE, M. C. (1970). Asymptotic description of radiating flow near stagnation point. AIAA

J. 8, 96-101. [2281JOBE, C. E., and BURGGRAF, O. R. (1974). The numerical solution of the asymptotic equations

of trailing edge flow. Proc. Roy. Soc. Ser. A 340,91-111. [231]JONES, R. T. (1950). Leading-edge singularities in thin-airfoil theory.]. Aeronaut. Sci. 17, 307-

310.155]JONES, R. T., and COHEN, D. (1960). H~fih Speed Wing Theory. Princeton Univ. Press, Princeton,

New Jersey. [46, 48, 168]JORDAN, P. F. (1971). The parabolic wing tip in subsonic flow. AIAA paper No. 71-10; Air Force

Office Sci. Res. Rep/. No. AFOSR-TR-71-0075. [240JKAPUTA, T. T. See Ferri, Ness, and Kaplita.KAPLUN, S. (1954). The role of coordinate systems in boundary-layer theory. Z. Angew. Math.

Phys. 5, 111-135. [64, 77, 84, 140, 145, 232]KAP~U~, S. (1957). Low Reynolds number flow past a circular cylinder.]. Math. Meeh. 6,595­

603. [23, 29, 77, 161, 163, 163, 226, 235, 2471KAPI.UN, S. (1967). Fluid Mechanics and Singular Perturbations, (P. A. Lagerstrom, L. N. Howard,

and C. S. Liu, cds.) Academic Press, New York and London. [226JKAPLUN, S., and LAGERSTROM, P. A. (1957). Asymptotic expansions of Navier-Stokes solutions

for small Reynolds numbers.]. Malh. Mech. 6,585-593. [77, 156, 161,226]KAWAMURA, R., aDd '[SIEN, F. H. (1964). Aerodynamic stability derivatives of axisymmetric

body moving at hypersonic speeds. Proc. 3rd Intern. Congo Aero. Sci. Spartan Books, Baltimore. [9]

KEU.ER, J. B. See Rubinow and Keller; Ting and Keller.KELLX, R. E. (1962). The final approach to steady, viscous flow near a stagnation point following

a change in free stream velocity.]. Fluid Mech. 13,449-464. [41[KERNEY, K. P. (1971). A theory of the high-aspect-ratio jet flap. AIAA]. 9,431-435. [241]KER~EY, K. P. (1972). A correction to "Lifting-line theory as a singular perturbation problem."

AIAA]. 10, 1683-1684. [239]KEVORKIA~, J. (\966). The two variable expansion procedure for the approximate solution of

certain nonlinear differential equations. Lectures in Appl. Math. 7,206-275. Amer. Math. Soc.,Providence, Rhode Island. [2411

KEVORKIAN, .J. See Cole and Kevorkian.KLB1P, J. B., and ACRIVOS, A. (\972). A note on the laminar mixing of two uniform parallel

semi-infinite streams. J. Fluid Mech. 55,25-30. 1248]KOZLOVA, I. G., and MIKIIAILOV, V. V. (1970). Strong viscous interaction on triangular and

slipping [yawed] wings. Mekh. Zhidk. Gaza, No.6, 94-99; English trans!': Fluid Dyn. 5,982­986. 1232]

KRIENES, K. (1940). Die elliptische Tragfliiche auf potentialtheoretischer Gn·ndlage. Z Angew.

Math. Mech. 20,65-88. The elliptic wing based on the potential theory. Tech. Memo. N.A.G.A.

(English Trans/.) No. 971. [202, 207]Kuo, Y. H. (1953). On the flow of an incompressible viscous past a flat plate at moderate

Reynolds numbers.]. Math. and Phys. 32,83-101. [100, 136)Kuo, Y. H. (1956). Viscous flow along a flat plate moving at high supersonic speeds.]. Aeronaut.

Sci. 23, 125-136. [l00JKUWAHARA, K., and 1MAI, I. (1969). Steady, viscous flow within a circular boundary. p~)'s.

Fluids, Suppl. II, 12,94-101. [234]LAGERSTROM, P. A. (1957). Note on the preceding two papers..J. Math. Mech. 6,605-606. [77,891LAGERSTROM, P. A. (1961). Methodes asymptotiques pour l'etude des equations de Navier­

Stokes. Lecture notes, Institut Henri Poincare, Paris. [226, 226JLAGERSTROM, P. A. (1964). Laminar flow theory. High Speed Aerod)'namics and Jet Propulsion (F. K.

Moore, cd.) 4, 20-285. Princeton Univ. Press, Princeton, N.J. [226, 235[LAGERSTROM, P. A. (1970). Some recent developments in the theory of singular perturbations.

Problems in Analpis: A Symposium in Honor q! Salomon Bochner, 261-271. Princeton Univ. Press,Princeton, N.J. [226J

LAGERSTROM, P. A. (1975). Forms of singular asymptotic expansions. Presented at Conf. on NewDirections in Singular Perturbations, Flagstaff, Ariz. To be published in Rocky Mtn. J. Math.

(221)LAGERSTROM, P. A. See Kaplun and Lagerstrom.LAGERSTROM, P. A., and CASTEN, R. G. (1972). Basic concepts underlying singular perturbation

techniques. SIAM Rev. 14,63-120. [226, 226]LAGERSTROM, P. A., and COLE, J. D. (1955). Examples illustrating expansion procedures for the

Navier-Stokes equations. J. Rat. Mech. Anal. 4, 817-882. [64, 77, 85, 121, 132, 161, 166]LAMB, H. (1911). On the uniform motion of a sphere through a viscous fluid. Philos. Mag. [6J

21, 112-121. 1155, 164JLAMB, H. (1932). HydrodynamICs, 6th ed. Cambridge Univ. Press, London and New York. [13, 164]LATTA, G. E. (1951). Singular perturbation problems. Ph.D. Dissertation. Calif. Inst. Tech.

[197, 198JLEAL, L. G. See Hinch and LeaLLE CLAIR, B. P. See Pruppacher, Le Clair, and Hamielec.LEGNER, H. H. (1971). On optimal coordinates and boundary-layer theory. Ph.D. Dissertation.

Stanford U niv. [233 J

256 References and Author Index References and Author Index 257

LEGRAS, J. (1951). Application de la methode de Lighthill a un ecoulement plan supersonique.C. R. Acad. Sci. Paris 233, 1005-1008. [lOOJ

LEGRAS, J. (1953). Nouvelles applications de la methode de Lighthill a I'etude des ondes dechoc. O.NE.R.A. Publ. No. 66. [100]

LEIGH, D. C. F. (1955). The laminar boundary-layer equation: a method of solution by meansof an autnmatic computer. Proc. Camb. Phil. Soc. 51, 320-332. [2461

LEPPINGTON, F. G. See CrighlOn and Leppington.LEVEY, H. C. (1959). The Ihickness of cylindrical shocks and the PLK method. Quart. App!.

Math. 17, 77-93. [100, 118]LEVEY, H. C., and MAHONY, J. J. (1968). Resonance in almost linear systems. I Inst. Math.

Applic. 4, 282-294. (242)LEVINSON, N. (1946). On the asymptotic shape of the cavity behind an axially symmetric nose

moving through an ideal fluid. I. Ann. of Math. (2) 47, 704-730. [42)LEWIS, G. E. (1959). Two methods using power series for solving analytic initial value problems.

Ph.D. Thesis. New York Univ. [21OJLEWIS, J. A., and CARRIER, G. F. (1949). Some remarks on the flat plate boundary layer. Quart.

Appl. Math. 7,228-234. [165JLIBBY, P. A. (1965). Eigenvalues and norms arising in perturbations about the Blasius solution.

AIAA I 3,2164-2165. [l32JLIBBY, P. A., and Fox, H. (1963). Some perturbation solutions in laminar boundary-layer theory.

Part I. The momentum equation.J. Fluid Mech. 17,433-449. [131]LIEPMAI<N, H. W., and PUCKETT, A. E. (1947). Introduction to Aerodynamics rf a Compressible Fluid.

Wiley, New York. [3JLIGHTHILL, M. J. (1948). The position of the shock-wave in certain aerodynamic problems.

Quart. J. Mech. Appl. Math. 1,309-318. [I, 1951LIGIITHILL, M. J. (1949a). A technique for rendering approximate solutions to physical problems

uniformly valid. Philos. Ma,!!,. [7J 40,1179-1201. [71, 99]LIGIITHII.L, M. J. (1949b). The shock strength in supersonic "conical fields." Philos. Mag. [7140,

1202-1223. [100,177,179,180, 196, 229J .LIGHTHILl., M. J. (1951). A new approach to thin aerofoil theory. Aero. Quart. 3, 193-210. [46,61,

73, 100, 119]LIGHTHILL, M. J. (1958). On displacement thickne,s. J. Fluid Mech. 4, 383-392. [1341LIGHTHILL, M. J. (1961). A technique for rendering approximate solutions to physical problems

uniformly valid. Z Flugwiss. 9,267-275. [100, 101, 119]LIN, C. C. (1954). On a perturbation theory based on the method of characteristics. I Math.

and Phys. 33, 117-134. [100, 108JLII<, C. C. See Carrier and Lin.LIN,T. C., and SCHAAF, S. A. (1951). Effect of slip on flow near a stagnation point and in a

boundary layer. Tech. Note NA.C.A. No. 2568. [36]LINIJSTEDT, A. (1882). Beitrag zur Integration der Differentialgleichungen der Storungstheorie.

AM. K. Akad. Wiss., St. Petersburg, 31, No.4. [218J

MAHONY, J. J. (1962). An expansion method for singular perturbation problems. J. Austral.

Math. Soc. 2, 440-463. [l98JMAHONY, J. J. See Levey and Mahony.MARK, R. M. (1954). Laminar boundary layers on slender bodics of revolution in axial flow.

Tech. Memo. Guggenheim Aeronaut. Lab. Calif Inst. Tech. No.2!. [2241MARTIN, E. D. (1967). Simplified application of Lighthill's uniformization technique using

Lagrange expansion formulas. I Inst. Math. Applic. 3, 16-20. [217]MATVEEVA, N. S., and SYCHEV, V. V. (1965). On the Iheory of strong interaction of the bound­

ary layer with an inviscid hypersonic flow. J. Appl. Math. Mech. 29, 770-784. [230]

MEDAN, R. 1'. (1974). Improvements to the kernel function method of steady, subsonic, liftingsurface theory. NASA TM No. X-62,327. [219, 240]

MELLOR, G. L. (1972). The large Reynolds number, asymptotic theory of turbulent boundarylayers. Int. I Eng. S,i. 10, 851-873. [227J

MELNIK, R. E. (1965a). A conical thin-shock-Iayer theory uniformly valid in the entropy layer.Rep. Air Force Flight Dyn. Lab. No. FDL-TDR-64-82. [229J

MELNIK, R. E. (1965b). A systematic study of some singular conical flow problems. Ph.D. Dis­sertation, Polytech. Inst. Brooklyn. [229]

MELNIK, R. E., and CHOW, R. (1975). Asymptotic theory of two-dimensional trailing edgeflows. Aerodynamic Analyses Requiring Advanced Computers. NASA SP No. 347. [231]

MELNIK, R. E., and GROSSMAN, B. (1974). Analysis of the interaction of a weak normal shockwave with a turbulent boundary layer. AIAA Paper No. 74-598. [227J

MELNIK, R. E., and IVES, D. C. (1971). Subcritical flows over two-dimensional airfoils by amultistrip method of integral relations. Proc. 2nd Internat. Con! Numer. Methods Fluid Dyn.,

Berkeley, Calif, 1970, Lecture Notes in Physics 8, 243-251. Springer-Verlag, Berlin and NewYork. (216]

MESSlTER, A. F. (1970). Boundary-layer flow near the trailing edge of a flat plate. SIAMI App!.Math. 18,241-257. [230J

MESSlTER, A. F. (1975). Laminar separation-a local asymptotic flow description for constantpressure downstream. AGARD Symposium on Separated Flows. To be published. [232]

MESSlTER, A. F. See Cole and Messiter.MESSlTER, A. F., and ENLOW, R. L. (1973). A model for laminar boundary-layer flow near a

separation point. SIAM J. Appl. Math. 25, 655-670. [232JMIKHAILOV, V. V. See Kozlova and Mikhailov.MILLIKAN, C. B. (1939). A critical discussion of turbulent flows in channels and circular tubes.

Proc. 5th Internat. Congo Appl. Mech., 386-392. Wiley, New York. [227JMILNE-THOMPSON, L. M. (1960). Theoretical Hydrodynamics, 4th ed. Macmillan, New York. [73, 74JMIYAGI, 1'. See Tamada and Miyagi.MaTT-SMITH, H. M. (1951). The solution of the Boltzmann equation for a shock wave. Phys.

Rev. 82, 885-892. [3]MUNK, M. M. (1922). General theory of thin wing sections. Rep. NA.C.A. No. 142. [71]MUNK, M. M. (1929). Fluid Dynamics for Aircraft Designers. Ronald Press, New York. [192]MUNSON, A. G. (1964). The vortical layer on an inclined cone. J. Fluid Mech. 20,625-643. [87,

196,201, 229]MURRAY, J. D. (1961). The boundary layer on a flat plate in a stream with uniform shear.

J. Fluid Meeh. 11, 309-316. [78, 147]MURRAY, J. D. (1965). Incompressible viscous flow past a semi-infinite flat plate.J. Fluid Meeh.

21, 337-344. [230JMURRAY, J. D. (1967). A simple method for determining asymptotic forms of Navier-Stokes

solutions for a class of large Reynolds number flows. J. Math. and Phys. 46, 1-20. [2301NAYFEH, A. H. (1973). Perturbation Methods. Wiley, New York. [xi, 215, 218, 241]NAYFEH, A. H., and TSAI, M.-S. (1974). Nonlinear acoustic propagation in two-dimensional

ducts. I Aeoust. Soc. Amer. 55, 1166-1172. [242]NEILAND, V. YA. (1969). Theory of laminar boundary layer separation in supersonic flow. Mekh.

Zhidk. Gaza, No.4, 53-57; English trans!.: Fluid Dyn. 4, 33-35. [232JNEILAND, V. YA. (1970). Propagation of perturbation upstream with interaction between a

hypersonic flow and a boundary layer. Mekh. Zhidk. Gaza, No.4, 40-49; English transt.: FluidDyn. 5, 559-566. [232]

NEILAND, V. YA. (1971). Flow behind the boundary layer separation point in a supersonicstream. Mekh. Zhldk. Gaza, No.3, 19-25; English trans!.: Fluid Dyn. 6,378-384. [232J

NESS, N. See Ferri, Ness, and Kaplita.

258 References and Author Index References and Author Index 259

OLVER, F. W. J. (1964). Error bounds for asymptotic expansions, with an application to cylinderfunctions of large argument. Asymptotic Solutions (if Differential Equations and Their Applications,

(C. A. Wilcox, ed.), 163-183. Wiley, New York. 1236]O'MALLEY, R. E., JR. (1968). Topics in Singular Perturbations. Advances in Math. 2, fascicule 4,

365-470. Academic Press, New York and London. [xi]O'MALLEY, R. E., JR. (1974). Introduction to Singular Perturbations. Academic Press, New York. [xi]O'NEIl.l., M. E. See Dorrepaal, Ranger, and O'Neill.OSEEI'<, C. W. (1910). Ueber die Stokes'sche Formel, und iiber eine verwandte Aufgabe in der

Hydrodynamik. Ark. Math. Astronom. Fys. 6, No. 29. [154, 155]OSWATITSCH, K. (1956). Gas Dynamics. Academic Press, New York. (15)OSWATITSCH, K. (l962a). Die Wellenausbreitung in der Ebene bei kleinen Storungen. Arch.

Mech. Stos. 14,621-637. [2281OSWATITSCH, K. (1962b). Das Ausbreiten von Wellen endlicher Amplitude. Zeit. Flugwiss. 10,

130-138. [228]OSWATITSCH, K., and BERNDT, S. B. (1950). Aerodynamic similarity at axisymmetric transonic

flow around slender bodies. KTII Aero TN 15. Roy. Inst. Tech., Stockholm. [177, 182]OSWATITSCH, K., and SJODIN, L. (1954). Kegelige Uberschallstromung in Schallnahe.

Osterreich. Ing. - Arch. 8, 284-292. [178]PAYARD, M., and COUTANCEAU, M. (1974). Sur I'etude experimentale de la naissance et de

l'evolution du tourbillon attache a I'arriere d'une sphere qui se deplace, a vitesse uniforme,dans un fluide visqueux. G. R. Acad. Sci. Paris 278, serie B, 369-372. [236]

PEARSON, J. R. A. See Proudman and Pearson.PEYRET, R. (1970). Etude de I'ecoulement d'un fluide conducteur dans un canal par la methode

des echclles multiples. J. Mecanique 9, 61-97. [242)POINCARE, H. (1892). Les methodes nouvelles de la mecanique celeste, Vol. I, Chapter 3. Dover, New

York. [100]PRANDTL, L. (1905). Uber Fliissigkeiten bei sehr kleiner Reibung. Verh. IIIInternat. Math. Kongr.,

Heide/bog, pp. 484-491. Teubner, Leipzig. Reprinted: Vier Abhandlungen zur Hydrodynamikund Aerodynamik (L. Prandtl and A. Betz, cds.). Gottingen, 1927. Motion of fluids with verylittle viscosity. Tech. Memo. N.A.G.A. (English Transl.) No. 452, 1928. [77J

PRANDTl., L. (1935). The mechanics of viscous fluids. Division G. Aeroqynamic Theory (W. F.Durand, ed.), Vol. 3, pp. 34-208. Springer, Berlin. [122, 136]

PRANDTI., L., and TIETJENS, O. G. (1934). Applied Hydro- and Aeromechanics. McGraw-Hili, NewYork. [55)

PRITULO, M. F. (1962). On the determination of uniformly accurate solutions of differentialequations by the method of perturbation of coordinates. J. Appl. Math. Mech. 26, 661-667.[73, 120]

PRITULO, M. F. (1969). The flow pa<t a yawed delta wing. Fluid Dynamics Transactions, (W.Fiszdon, P. Kucharcyzk, and W. J. Prosnak, eds.) 4, 261-275. PWN-Polish Sci. Pubs.,Warsaw. [218]

PROBSTEIN, R. F. See Hayes and Probstein.PROUDMAN, I., and PEARSON, J. R. A. (1957). Expansions at small Reynolds numbers for the

flow past a sphere and a circular cylinder.J. Fluid Mech. 2,237-262. [5,29,29,77, 156, 160,161, 162, 165, 221, 224, 234, 237, 244]

PRUPPACIIER, H. R., LE CLAIR, B. P., and HAMIELEC, A. E. (1970). Some relations betweendrag and flow pattern of viscous flow pa<t a sphere and a cylinder at low and intermediateReynolds numbers. J. Fluid Mech. 44, 781-790. [236]

PUCKETT, A. E. See Liepmann and Puckett.RANGER, K. B. (1972). The applicability of Stokes expansions to reversed flow. SIAM J. Appl.

Math. 23, 325-333. [235, 236]RANGER, K. B. See Dorrepaal, Ranger, and O'Neill.

RICHEY, G. K. See Adamson and Richey.RILEY, N., and STEWARTSON, K. (1969). Trailing edge flows. J. Fluid Mech. 39, 193-207. [232]RISPIN, P. P. (1967). A singular perturbation method for non-linear water waves pa" an obstacle.

Ph.D. Dissertation. Calif. Inst. Tech. [248]RAo, P. S. (1956). Supersonic bangs. Aero. Quart. 7, 21-44, 135-155. [1001RAYLEIGH, Lord (1916). On the flow of a compressible fluid past an obstacle. Philos. Mag. [6J

32, 1-6. [16]ROSENHEAD, L., ed. (1963). Laminar Boundary La.yers. Oxford Univ. Press, London and New

York. [122, 130, 131, 131, 136, 162]ROSENHEAD, L. See Goldstein and Rosenhead.RUBINOW, S. 1., and KELLER, J. B. (1961). The transverse force on a spinning sphere moving in

a viscous fluid. J. Fluid Meeh. 11, 447-459. [78JSAFFMAN, P. G. (1970). The velocity of viscous vortex rings. Studies Appl. Math. 49,371-380. [248]SAMPSON, R. A. (1891). On Stokes's current function. Phil. Trans. R~y. Soc. Sfr. A 182,449-518

(esp. 504). [223JSCHAAF, S. A. See Lin and Schaaf.SCHLIGHTING, H. (1960). Bounda~y Layer Theory, 2nd English ed. McGraw-Hili, New York. II, 18,

19, 130,214, 214]SCHNEIDER, W. (1973). A note on a breakdown of the multiplicative composition of inner and

outer expansions. J. Fluid Meeh. 59, 785-789. [227]SCHWARTZ, L. W. (1974). Computer extension and analytic continuation of Stokes' expansion

for gravity waves.] Fluid Mech. 62, 553-578. [216, 243]SEARS, W. R., and TELIONIS, D. P. (1975). Boundary-layer separation in unsteady flow. SIAM

J. App!. Math. 28,215-235. [219]SERBIN, H. (1958). The high speed flow of gas around blunt bodies. Aero. Quart. 9,313-330. [23JSHANKS, D. (1955). Non-linear transformations of divergent and slowly convergent sequences.

J. Math. and Phys. 34, 1-42. [5, 202, 205]SHEER, A. F. (1971). A uniformly asymptotic solution for incompressible flow past thin sharp­

edged aerofoils at zero incidence. Proe. Camb. Phil. Soc. 70, 135-155. [224]SIMASAKI, T. (1956). On the flow of a compressible fluid past a circular cylinder. II. Bull. Univ.

Osaka Prefecture Ser. A. 4, 27-35. [4, 214]SJODIN, L. See Oswatitsch and Sjodin.SKINNER, L. A. (1975). Generalized expansions for slow flow past a cylinder. Quart. .J. Meeh.

Appl. Math. 28, part 3. [218, 236, 237]SPREITER, J. R. (1953). On the application of transonic similarity rules to wings of finite span.

Rep. N.A.C A. No. 1153. [221SPREITER, J. R. (1959). Aerodynamics of wings and bodies at transonic speeds.] Aerospace Sci.

26, 465-487. [3]STEWARTSON, K. (1956). On the steady flow past a sphere at high Reynolds number using

Oseen's approximation. Phdos. Mag. [8] I, 345-354. [210]STEWARTSON, K. (1957). On asymptotic expansions in the theory of boundary layers.] Math.

and Phys. 36, 173-191. [29, 131, 131, 132,201,229]STEWARTSON, K. (1958). On Goldstein's theory of laminar separation. Quart.] Mech. Appl. Math.

11,399-410. [220, 247]STEWARTSON, K. (1961). Viscous flow past a quarter-infinite plate.] Aerospace Sci. 28, 1-10. [201 JSTEWARTSON, K. (1968). On the flow near the trailing edge of a flat plate. Proc. R~. Soc. Ser_ A

306,275-290. [2311STEWARTSON, K. (1969). On the flow near the trailing edge of a flat plate II. l\1athematika 16,

106-121. [230]I\STEWARTSON, K. (1974). Multistructured boundary layers on flat plates and rclated bodies.I Adv. App!. Mech. 14, 145-239. [232] /

STEWARTSON, K. See Brown and Stewartson; Riley and Stewartson.STEWARTSON, K, and WILLIAMS, P. G. (1969). Self-induced separation. Proc. Roy. Soc. Ser. A

312, 181-206. [232]STOKER, J. J. (1957). Water Waves. Wiley (Interscience), New York. [42]STOKES, G. G. (1851). On the effect of the internal friction of fluids on the motion of pendulums.

Trans. Camb. Phil. Soc. 9, Part II, 8-106. Reprinted: Math. and Phys. Papers 3,1-141. Cambridge

Univ. Press. [152]STONE, A. H. (1948). On supersonic flow past a slightly yawing cone.]. Math. and Phys. 27,

67-81. [9]STUFF, R. (1972). Closed form solution for the sonic boom in a polytropic atmosphere.]. Aircraft

9, 556-562. 1229]SYCHEV, V. V. (1962). On the method of small disturbances in the problem of the hypersonic

gas flow over thin blunted bodies (Russian). Zh. Prikl. Mekh. i Tekh. Fiz. No.6, 50-59.

Grumman Aircraft Eng. Corp., Res. Dept., English Trans!. TR-25. [192]SYCHEV, V. V. (1972). [On ] laminar separation. Mekh. Zhidk. Gaza, No.3, 47-59; English trans!.:

Fluid Dyn. 7, 407-417. [232]SYCHEV, V. V. See Matveeva and Sychev.SYKES, M. F. See Domb and Sykes.TAMADA, K and MIYAGI, T. (1962). Laminar viscous flow past a flat plate set normal to the

stream, with special reference to high Reynolds numbers.]. Phys. Soc. Japan 17, 373-390. [29]TANEDA, S. (1956). Studies on wake vortices (III). Experimental investigation of the wake

behind a sphere at low Reynolds numbers. Rep. Res. Inst. Appl. Mech. Kyushu Univ. 4,99-105.

[160]TEI.Io!':ls, D. P. See Sears and Telionis.TELlO!': IS, D. P., and TSAIIAI.IS, D. TH. (1973). Unsteady laminar separation over cylinders

started impulsively from rest. Presented at 24th Internat. Astronaut. Cong., Baku. To be

published in Acta Astronautica. [219]TERRILL, R. M. (1973). On some exponentially small terms arising in flow through a porous

pipe. Quart.). Mech. Appl. Math. 26, 347-354. [237]THURBER, J. K (1961). An a~ymptotic method for determining the lift distribution of a swept­

back wing of finite span. Ph.D. Dissertation. New York Univ.THURBER, J. K (1965). An a~ymptotic method for determining the lift distribution of a swept­

back wing of finite span. Comm. Pure Appl. Math. 18, 733-756. [241]THWAITES, B.. ed. (1960). Incompressible Aero,!}'namics. Oxford Univ. Press, London and New York.

[46]Tn:TJENs, O. G. See Prandtl and Tietjens.TIFFORD, A. N. (1954). Heat transfer and frictional effects in laminar boundary layers. Part 4.

Universal series solutions. Tech. Rep., Wright Air Dev. Center No. 53-288. [6, 18 1

TINC, L. (1959). On the mixing of two parallel streams.]. Math. and Phys. 38, 153-165. [77,248]TING, L. (1960). Boundary layer over a flat plate in presence of shear flow. Ph}'s. Fluids 3,78-81.

[78JTI!':G, L. (1968). Asymptotic solutions of wakes and boundary layers.]' Eng. Math. 2,23-38. [230]

TING, L. See Chow and Ting; Tung and Ting.TING, L. and KELLER, J. B. (1974). Planing of a flat plate at high Froude number. Phys. Fluids

17, 1080-1086. [248]TOKUDA, N. (1971). An asymptotic theory of the jet flap in three dimensions.]. Fluid Meeh. 46,

70.1-726. [2411TOMOTIKA, S., and Am, T. (1950). The steady flow of viscous fluid past a sphere and circular

cvlinder at small Reynolds numbers. Quart.]. Meeh. Appl. Math. 3, 140-161. [152, 156, 164, 165]TR~UCOTT, S. C. (1962). Evaluation of indeterminate second-order effects for the Rayleigh

problem. Phys. Fluids 5,1125-1126. [54]

TRITTON, D. J. (1959). Experiments on the flow past a circular cylinder at low Reynolds num-bers.). Fluid Mech. 6,547-567. [1641

TSAIIALlS, D. T. See Telionis and Tsahalis.TSAI, M.-S. See Nayfel, and Tsai.TSIEN, F. H. See Kawamura and Tsien.TSIEN, H. S. (1956). The Poincare-Lighthill-Kuo method. Advances in Appl. Mech. 4,281-349. [1001TUNG, C., and TING, L. (1967). Motion and decay of a vortex ring. Phys. Fluids 10, 901-910. [248/UNDERWOOD, R. L. (1969). Calculation of incompressible flow past a circular cylinder at moder-

ate Reynolds numbers.]. Fluid Mech. 37, 95-114. [236]USHER, P. D. (1968). Coordinate stretching and interface location. II. A new PL expansion.

). Compo Pkys. 3, 29-39. [217]VAN DE VOOREN, A. I. See Veldman and van de Vooren.VAN DE VOOREN, A. I., and DIJKSTRA, D. (1970). The Navier-Stokes solution for laminar flow

past a semi-infinite flat plate.]. Eng. Math. 4, 9 ·27. [230]VAN DYKE, M. D. (1951). The combined supersonic-hypersonic similarity rule.]. Aeronaut. Sci.

18, 499-500. 122JVAN DYKE, M. D. (1952). A study of second-order supersonic flow theory. Rep. lV.A.C.A.

No. 1081. [36, 75,107,111,179]VAN DYKE, M. D. (1954). Subsonic edges in thin-wing and slender-body theory. Tech. Note

NA.C.A. No. 3343. [119JVAN DYKE, M. D. (1956). Second-order subsonic airfoil theory including edge effects. Rep.

NA.C.A. No. 1274. [49]VAN DYKE, M. D. (l958a). The similarity rules for second-order subsonic and supersonic flow.

Rep. NA.C.A. No. 1374. [212JVAN DYKE, M. D. (1958b). A model of supersonic flow past blunt axisymme'ric bodies with

application to Chester's solution. .J. Fluid Mech. 3,515-522. [203, Z08, 209, 210]VAN DYKE, M. D. (1958c). The paraboloid of revolution in subsonic flow.]. Math. and Ph}'s. 37,

38-51. [75, 213JVAN DYKE, M. D. (1959). Second-order slender-body theory-axisymmetric flow. Tech. Rep.

NA.S.A. No. R-47. [74]VAN DYKE, M. D. (1962a). Higher approximations in boundary-layer theory. Part. 1. General

analysis.). Fluid Mech. 14,161-177. [29,135,136]VAN DYKE, M. D. (1962b). Higher approximations in boundary-layer theory. Part 2. Appli­

cation to leading edges.]. Fluid Mech. 14,481-495. [124]VAN DYKE, M. D. (l964a). Higher approximations in boundary-layer theory. Part 3. Parabola

in uniform stream.]. Fluid Mech. 19,145-159. [6,39,204,208,211]VAN DYKE, M. D. (1964b). Lifting-line theory as a singular perturbation problem. Arch. Mech.

Stos. 16, No.3. [173 JVAN DYKE, M. (1970). Extension of Goldstein's series for the Oseen drag of a sphere.). Fluid

Mech. 44,365-372. [216, 243JVAN DYKE, M. (1974). Analysis and improvement of perturbation series. Q!Jart.). Mech. Appl.

Math. 27, 423-450. [243, 247]VAN DYKE, M. (1975). Computer extension of perturbation series in fluid mechanics. SIAM

]. Appl. Math. 28, 720-734. [215]VAN DYKE, M. (1976). Extension, analysis, and improvement of perturbation series. Proc. 10th

Naval Hydrodynamics Symposium, /974. To be published. [215, 243]VAN TUYL, A. (1959). The use of rational approximations in the calculation of flows with

detached shocks. Rep. U. S. Naval Ord. Lab. No. 6679. [209JVAN TUYL, A. (1960). The use of rational approximations in the calculation of flows with

detached shocks.]. Aerospace Sci. 27, 559-560. [206]

260 References and Author Index References and Author Index 261

I

IIII

262 References and Author Index

SUBJECT INDEX

VF.LDMA~, A. E. P. (1973). Thc numerical solution of the Navier-Stokes equations for laminarincompressible flow past a paraboloid of revolution. Computers and Fluids 1, 251-271. [218,223, 239]

VELDMA~, A. E. P. Sa Botta, Dijkstra, and Veldman.VF.J.D\1A~,A. E. P.. and VAt' DE VOORE~, A. I. (1974). Drag of a finite flat plate. Proe. 4th Internal.

Con! Numfr. Methnd, Fluid D)'II., Souldfr, Colo., Lecture Notes in Physics 35, 423-430. Springer­Verlag, Berlin and New York. [231]

VINCE~TI, W. G. (1959). Non-equilibrium flow over a wavy wall.} Fluid Meeh. 6,481-496. [19:)]VIVIA~O, H .. and BER(;ER, S. A. (1964). Incompressible laminar axisymmetric near wake behind

a very slender cylinder in axial flow. Rep. Ilis/. Eng. Res. Univ. Calif No. As 64-5; AIAA }3 (1965) 1806-1812. [2181

VON DOE~HOFF. A. E. See Abbott and von Doenhoff.VO~ K.\R\1..\~, T (1947). The similaritv law of transonic flow.} Math. and Phys. 26, 182-190. [21]VO:>J K.\RMA:>J, T, and BURGERS, J. M. (1935). General aerodynamic theory-perfect fluids.

Division E. Aerod)'namie Theor)' (W. F. Durand, ed.), Vol. 2. Springer, Berlin. [561WALKER, J. D. A. See Dennis and Walker.WALSH, J. D. Sa Dennis and Walsh.WARD, G. N. (1955). Linearized Theory' oiS/ead;' High-Speed Flow. Cambridge Univ. Press, London

and New York. [42, 176]W ASOW, W. (1955). On the convergence of an approximation method of M. J. Lighthill.} Ra/.

Meeh. Anal. 4,751-767. Jl01]WEISSI:>JGER, J. See Bollheimer and Weissinger.WEt"DT, H. (1948). Die Jansen-Rayleighsche Naherung zur Berechnupg von Unterschallstr6-

mungen. Sitzsber. Heidelberg. Akad. Wiss., Ma/h.-Naturwiss. KI., 19487,147-170. [4]WEt"DT, H. See Hantzsche and Wendt.WERLE, M. J. See Davis and Werle.WERLE, M. J., and DAVIS, R. T (1972). Incompressible laminar boundary layers on a parabola

at angle of attack: a study of thc separation point.} Appl. Meeh. 39, 7-12. [239]WEY!., H. (1942). On the differential equations of the simplest boundary-layer problems. Ann.

If Ma/h. 43, 381-407. Jl3l]WHITEHEAD, A. N. (1889). Second approximations to viscous fluid motion. Quart. } Math.

23, 143-152. [153]WIllTIIAM, G. B. (1952). The flow pattern of a supersonic projectile. Comm. Pure Appl. Math. 5,

301-348. [100, Ill. 113, 116JWIIlTIIAM, G. B. (1953). The propagation of weak spherical shocks in stars. Comm. Pure Appl.

Math. 6,397-414. [100]WILKINSON, J. (1955). A note on the Oseen approximation for a paraboloid in a uniform stream

parallel to its axis. Quar/.} Meeh. Appl. Math. 8,415-421. [1651Wn.LIAMs, P. G. See Stewartson and Williams.Wu, T. T. See Cheng and Wu.Wn, Y. T (1956). Two dimensional sink flow of a viscous, heat-conducting, compressible

fluid; cylindrical shock waves. Quar/. App!. Math. 13, 393-418. [100]YAJNIK, K S. (1970). Asymptotic theory of turbulent shear flows.} Fluid Meeh. 42,411-427. [227]YAKURA, J. K. (1962). Theory of entropy layers and nose bluntness in hypersonic flow. Hyper­

sonic How Research (F. R. Riddell, ed.), pp. 421-470. Academic Press, New York, 19, 78, 183,189, 191]

YA\1ADA, H. (1954). On the slow motion of viscous liquid past a circular cylinder. Rep. Res. Inst.App!. Mech. K)'ushu Uni, .. 3, 11-23. [156)

YOSHIZAWA, A. (1970). Laminar viscous flow past a semi-infinite flat plate.} Ph)'s. Soc. Japan 28,776-779. [230]

Accuracy of asymptotic serics, 30-:)2composite expansion, 97, 228effect of logarithmic terms, 202improvement of, 202-210multiple scales, 242

Additive composition, 94, 190, 227, 228Aging spring, 242Airfoil integral, 49, 172Airfoil theory,

singular perturbation problems, 45-76supersonic, 106-117

Analysis of pert urbation series, 243-247Analytic continuation, 31-32, 210Analytic method of characteristics, 228Angle of attack, effective, 172, 173An~les, ratio of, 83Approximations,

irrational, 2rational, 2

role of in fluid mechanics, 1-8Artificial parameter, 75, 132Asymptotic expansion, see Asymptotic seriesAsymptotic form, 26, 90Asymptotic matchin~ principle, 64, 90, 128,

161, 186,220-225restricted, 221-224

Asymptotic representation, 26, 90Asymptotic sequence,

alternative, 28, 244choice of, 29-30, 35, 125definition, 28difference between inner and outer,

84,104logarithms in, 5, 29, 30, 33, 35, 47, 57, 59,

69, 200-202, 207, 218, 220-224, 229,235, 236, 237, 243-245

for thin-airfoil theory, 47, 73Asymptotic series,

accuracy of, 30-32, 97, 202-210, 228, 242complete, 237convergence of, 27, 30-32definition, 26error in, 31power series, 27properties of, 32-34uniqueness of, 33-34

263

Asymptotic solution, 3Axisymmetric, .Iee Body of revolution,

Blunt-body problem, Paraboloid ofrevolution

Backward influence, 38, 39, 42Basic solution, 4, 93Biconvex airfoil, 56-59, 68, 74, 97, 217,

224-225Blasius series for boundary layer,

on cirele, 18-19inverse, for parabola, 41, 211on parabola, 6, 39, 204, 208

Blasius solution for flat plate, 129-131uniqueness, 131-132

Block matching, 74, 222Blunt-body problem, 6, 39, 40, 203, 206,

207, 208-209Blunted wedge in hypersonic flow.

182-192, 227Body of revolution,

nose-correction rules for incompressibleflow, 74

paraboloid, 75, 165, 213, 239slender-body theory, 74spindle, 74

Boundary conditions,discontinuities in, 42loss of, 80, 88, 126perturbation of, 9, II, 13transfer of, 36-37, 43, 47

Boundary layer

on circle, 17-19,38,214,219,219change of coordinates, 140-145, 232-243coordinates, 17first-order equation, 128on flat plate, 129-140, 142-145, 196-197,

230-232higher approximations, 121, 134, 136, 139,

147, 233, 237, 238-239multistructured, 230-232optimal coordinates, 141, 144-146, 200,

232-234in Oseen approximation, 156

-on parabola, 6, 204, 208, 211-212,238, 239

264 Subject Index Subject Index 265

separation, 19,214,219,220, 232,239, 247

theory, 17, 121-147,230-232thickness, 127triple deck, 230, 232turbulent, 227unsteady i;rowth, 38-39, 214, 219

Change of characteristics, 42, 43-44, 228-229Change of type, 42, 129Characteristics, change of, 42, 43-44, 228-229Circular cylinder, 9-20

boundary layer on, 17-19,38,214,219,219

compressible flow, 4,15-16,214,215-216, 219

corrugated, 19,217distorted, 13-14drag at low Reynolds number, 164,

237-238, 244inviscid irrotational Row, 10-11Janzen-Rayleigh approximation, 4, 15-16at low Reynolds number, 161-164,

222-224, 226, 236, 237-238in parabolic shear, 20, 98, 217, 217porous, 19,217pulsatini;, 19separation, 19, 121-122, 150, 214, 219, 236skin friction, 18, 219Stokes approximation, 152-153in uniform shear Row, 11-13viscous Row, 16-19, 152-153, 161-164,

222-224, 226, 237-238Complementary solution, 12Complete asymptotic expansion, 237Composite e}.pansion, 83-85, 98, 163,

197-198, 199accuracy of, 97, 228alternative rules, 227-228for blunted wedge, 190-192, 227construction of, 94-97, 227-228failure of multiplicative, 227-228

Composite expansions, method of,197-198, 213

Composition,additive, 94-190, 227, 228multiplicative, 96, 190, 227-228

Compressible Row, see Hypersonic Row,Subsonic flow, Supersonic flow,Transonic flow

Computer extension, 215-216, 234Cone, circular,

perturbation of solution for supersonicflow, 9

in slightly supersonic Row, 176-182Conical shock wave, 100, 229Conservation principle, global, 53Converi;ence

of asymptotic series, 27, 30-32of coordinate perturbation, 31of expansions in powers of Reynolds

number, 234improvement of, 202-210, 243-247of Janzen-Rayleigh expansion, 4, 214, 219radius of, 32, 206, 207-208, 213, 218, 234,

237, 245-246of Stokes's series for water waves, 216value of, 30

Coordinate perturbation, 3boundary layer on flat plate, 123canvergence of, 31direct expansion, 37-40for elliptic equations, 39-40inverse expansion, 41-42joining of, 210-212nonuniformity of, 81replacement by parameter perturbation,

82-83singular, 82

Coordinates,alternative, for flat plate, 142-144, 234change of, in boundary layer, 140-145natural, 203-204, 232-234optimal, 141, 144-146, 200, 232-234parabolic, 142, 203, 206, 209, 211, 230, 238semicharacteristic, 108semioptimal, 144, 146shrinking rectani;ular, 143-144, 234

Corner on supersonic airfoil, 112-115Correlation theorem, 142, 147, 233, 233Corrugated quasi cylinder, 19, 217Critical Mach number, 4, 214, 216, 219Cumulative effect, 106, 107

D'Alembert's paradox, 153, 247Derivative-expansion method, 241Direct coordinate expansion, 37-40Disparate lengths, 81, 82, 168, 189, 198Displacement thickness

of entropy layer, 189

flow due to, 132-135, 137, 138, 139for wake of finite Rat plate, 136-137

Distorted circle, 13-14Divortex, 173Domain of validity, 226Domb and Sykes, graphical ratio test, 216,

218, 245-247Double limit process, 21, 189Drai;

of circle at low Reynolds number, 164,237-238, 244

leading-edge, 54-56, 138of sphere at low Reynolds number, 5, 149,

161, 206, 209-210, 216, 234-235, 243Dual solutions, 237

Eddy, behind circle and sphere, 150, 156,159-160, 232, 235-236

behind sharp trailing edge, 232Effective angle of attack, 172, 173Eigensolutions

in flat-plate boundary layer, 131-132,230,232

in inverse coordinate expansions, 41, 42in thin-airfoil theory, 52-54, 58, 62-67,

73,89Ellipsoid of revolution, 74, 192Elliptic airfoil,

block matching, 74composite solution, 96-97, 227-228incompressible thin-airfoil solution,

50-52, 72inner and outer solutions, 94inviscid complex velocity, 73inviscid surface 3peed, 52local solution near edge, 62shifting correction, 72-73subsonic thin-airfoil expansion, 5

Elliptic equations,direct coordinate expansion, 39initial-value problem, 39inverse coordinate expansion, 41method of strained coordinates, 100,

118-119, 167, 229Elliptic wini;, 6, 168, 174-176, 207, 213,

239-241Entropy layer, 182, 186-189

displacement effect, 189Euler transformation, 32, 33-34, 207-210, 216Exercises, comments on, 217-220

Exponentially small terms, see also

Transcendentally small termsin asymptotic sequence, 29, 236-2:~8

Exponential decay of vorticity, 1:1I, 139-140,201, 218, 229

Extension theorem, 92

Filon's paradox, 247Finite flat plate,

first-order boundary layer, 123-124,129, 130

inclined, in potential flow, 169-170inclined, in viscous Row, 232

second-order boundary layer, 136-137skin friction, 137, 1:~9, 231trailing edge, 136, 230-231triple deck, 230-232

First approximation, 4First-order solution, 4Flat plate in viscous Row,

alternative coordinates, 142-144, 234­alternative interpretations of Blasius

solution, 122-124boundary-layer solution, 129-140,

142-145, 196-197eigensolutions, 131-132,230,232finite, 123-124, 129, 130, 136-137,230-232finite-difference solution, 230, 231flow due to displacement thickness,

132-134leading edge, 39, 138-139, 230method of composite equations, 196-197optimal coordinates, 1+4, 145Oseen approximation, 165, 198second-order boundary layer, 134-136skin friction, 137-139, 140, 230, 231third-order boundary layer, 139-140

Flow due to displacement thickness,132-135, 137, 138, 139

Forbidden regions, 222-224Free-streamline flow, 42, 232

Gauge functions, 23-28choice of, 24-25

Generalized asymptotic expansion, see

Composite expansionGeneralized method (multiple scales), 242General particular integral, 36, 75Global conservation principle, 53

266 Subject Index Subject Index 267

Graphical ratio test of Domb and Sykes,216, 218, 245-247

Group property, 129, 131, 146

Hif:her-order boundary layer. 121, 134, 136,139. 145, 147, 233-234, 237, 238

Hyperbolic equations,change of characteristics, 42, 43-44, 229direct coordinate expansion, 38inverse coordinate expansion, 41method of strained coordinates, 100, 118,

167, 228-229Hyperbolic shock wave, 183Hypersonic flow,

blunt body, 6, 40, 206-207, 208-209blunted wedge, 182-192,227boundary-layer separation, 232similarity rule, 22, 43

Hypersonic small-disturbance theory, 22,35, 185

"Impossible" inner problem, 85,101,173Improvement of series, 202-210. 243-247

containing logarithms, 243-245Impulsive motion,

blunt body at hypersonic speed, 40,206-207

circle in viscous now, 214, 219, 219cylinder in viscous flow, 38, 214, 219

Indeterminacy, see also Eigensolutionsin boundary-layer solution, 132, 232in inverse coordinate expansions, 41

Initial-value problem for ellipticequations, 39

Inner and outer expansions, see Matchedasymptotic expansions, method of

Inner expansion, 64, 84for boundary layer, 127for boundary layer on semi-infinite flat

plate, 140definition, 85distinction from outer expansion, 93for entropy layer, 187-189for lifting wing of high aspect ratio, 172

Inner limit, 85Inner variables, 64, 84

choice of, 85-88, 221definition, 85for entropy layer. 187for lifting wing of high aspect ratio, 169

ratio to outer variables, 221Intermediate limit, 91, 161,226Intermediate matching, 91-93

principle, 92-93, 161, 220Intermediate problem, 91Intermediate solution, 92Intermediate variable, 91Inverse Blasius series, 41, 211Inverse coordinate expansion, 41-42Irrational approximation, 2-4Iteration, 15

advantages and disadvantages, 34-35, 235on Stokes approximation for sphere, 155

Janzen-Rayleigh approximation,for circle, 4, 15-16, 35, 36, 214,

215-216, 219for parabola, 212

Jeffery-Hamel flow, 235, 237Jet flap, 241Joining

of coordinate expansions, 210-212of parameter expansions, 212-213

Joukowski airfoil, 54-56

KUlla-Joukowski condition, 42, 169

Laminar mixing, 77, 248Leading edge,

round-nosed airfoil, 52, 53, 54-56, 59-68,71-73, 229

sharp-nosed airfoil, 56-57, 68-70, 74,224-225

square-nosed airfoil, 57-59, 73viscous flow over flat plate, 39,

138-139, 230Leading-edge drag, 54-56, 138Leading·edge thrust, 56Least degeneracy, principle of, 86, 127Lifting·line theory, 6, 167-176, 192, 239-241Lighthill's principle, 71, 99, 229Lighthill's rule, 61Lighthill's technique, see Strained

coordinates, method ofLimit matching principle, 90, 161Limit process, 21-23

double, 21, 189multiple, 21, 82

Limit-process expansions, 21, 226-227Logarithms, 5, 200-202

as gauge functions, 25in asymptotic expansions, 5, 29, 30, 33,

35, 47, 57, 59, 68, 69, 73, 140,160-161,162,200-202,207,218,221-224,229, 235, 236, 243-245

in inverse coordinate expansions, 41, 42in lifting-line theory, 173, 176, 241log log, 25, 29, 201purely logarithmic case, 222, 237, 243source of, 68, 201relation to algebraic companion,

201-202,221telescoping, 223, 244in thin-airfoil theory, 57, 59, 68, 69, 73in viscous-now solutions, 140, 160-161,

162. 221-224, 233, 235, 236, 237Log log, 25, 29, 201

M2-expansion method, see Janzen-Rayleighapproximation

Mach lines, revised, 110-111Mach number, critical, 4, 214, 216, 219Many-variable expansion methorl, 241Mass on spring, 98Matched asymptot:. expansions, method

of, 7, 77-98blunted wedge in hypersonic flow,

183-192,227circle at low Reynolds number, 161-164,

222, 226, 236, 237failure, 213lifting wing of high aspect ratio, 167-176,

239-241sphere at low Reynolds number, 156-161,

221, 226, 234-236Matching, 88-89, 89-90, 210

and principle of minimum singularity,73,88

of circulation in lifting-line theory, 171theory o~ 225-227in thin-airfoil theory, 64-70order, 93-94to determine eigensolutions, 53

Matching condition for boundary layer,128, 129

Matching principle, 89-90asymptotic, 64, 90, 128, 161, 186,

220-225, 226intermediate, 92-93, 161, 220limit, 90, 161

Method of composite expansions, \'{f

Composite expansions, method ofMethod of matched asymptotic expansions.

see Matched asymptotic expansions,method of

Method of multiple scales, see Multiplescales, met hod of

Method of series truncation, .lee Seriestruncation, method of

Method of strained coordinates, .leeStrained coordinates, method of

Method of strained parameters, 218Middle expansion, 82, 97, 183, 186-191,229Middle variable, 186Minimum singularity, principle of, 53, 64,

65, 73, 88, 132, 158, 163Model problem for viscous flow, 78-80,

222, 226Multiple limit process. 21, 82, 189Multiple scales, method of, 198-200,213,

241-242accuracy, 242primitive version. 242relation to matched expansions, 242variants, 241-242

Multiplicative composition, 96, 190. 227failure of, 227

Multiplicative correction,round edges, 59-61, 74sharp edges, 74

Natural coordinates, 203-204Navier-Stokes equations, 124, 153, 165. 230,

231, 233, 238Nearest singularity

from graphical ratio test, 245nature of, 246from Pade approximants, 247

Newton-Busemann theory, 6, 22, 23Newtonian approximation, 43. 208Nonequilibrium flow over wavy wall,

192-193Nonuniformity, 33, see also Singular

perturbation problemfor blunted wedge in hypersonic now,

185, 187in conical flow, 83. 100, 229of coordinate perturbations, 81in direct coordinate expansions, 39

268 Subject Index Subject Index 269

for elliptic airfoil, 50in high-a~pect-ratio wing theory, 170, 176model equation, 78-80, 222, 226multiple, 82physical criterion, 80-8:3, 168, 189

prediction of, 78region of, see Region of nonuniformityat sharp stagnation edge, 57source of. 78-80at square edge, 58in supersonic airfoil theory, 106-109

Nonuniqueness, .\W al.w Eigensolutionsof Blasius solution for flat plate,

131-132,230of boundary layer on triangular wing, 232of hypersonic boundary layer, 232of thin-airfoil solution, 52-54

Nose-correction rules,bodies of revolution, 74round edge, 59-68, 71-73sharp edge, 68-70, 74two edges, 97

Optimal coordinates, 141, 144-146, 200extension of. 145, 232-234

Order symbols 0 and 0, 23-29operations with, 2"1-26relation to physical order of magnitude, 25

Osculating parabola, 56, 59-60, 63-64, 66,

67, 73Oseen approximation, 5, 153-156

for boundary layer, 156for flat plate, 165, 198for paraboloid, 165, 222-223for plane wake, 165for sphere, 5, 155, 206, 209-210, 216, 243

Oseen expansion, 156, 157-158, 159, 165,222-223

Oseenlet, 158,,162Outer expansion, 64

for boundary layer on flat plate, 124-126

definition, 85distinction from inner expansion, 93for entropy layer, 185for lifting-line theory, 171, 173

Outer limit, 85Outer variables, 64, 85

ratio to inner variables, 221Overlap domain, 89, 90, 92, 220-221

Pade approximants (rational fractions),205-207, 247

Parabola,boundary layer, 6, 204, 208, 211-212,

238, 239drag, 55-56exact complex velocity, 71Janzen-Rayleigh expansion, 212osculating, 56, 59-60, 63-64, 66, 67, 73Oseen and Stokes approximations, 166potential flow, 55-56, 88skin friction, 6, 204, 208, 213surface speed, 60thin-airfoil approximation, 60, 71viscous flow, 238, 239

Parabolic coordinates, 142, 203, 206, 209,211,230,238

Parabolic equations,direct coordinate expansion, 38inverse coordinate expansion, 41method of strained coordinates, 100,

118-119, 167, 229Parabolic shear, circle in, 20, 98, 217, 217Paraboloidal shock wave, 6, 40, 203-204,

206, 208-209Paraboloid of revolution,

potential flow, 75,213viscous flow, 165, 239

Paradox,d'Alembert's, 153, 247Filon's, 247laminar mixing, 77, 248planing at high Froude number, 248resolution of, 247-248Stokes's, 77, 152, 247Vortex rings, 248Whitehead's, 152-153, 247

Parameter, artificial, 75, 132Parameter perturbation, 3

convergence of, 31joining, 212-213regular, 81singular, 81-82

Parameters, method of strained, 218Particular integral, 12, 36

finding, 36general, 36, 75

Patching, 54, 89, 140, 211, 212Pattern of signs, 245

Perturbation expansion,examples, 4-6regular, 6-8, 9-20, 81, 82, 215, 234, 235,

237, 243singular, 6-8, 33, 45-76, 78, 81-82, 153,

167-193Perturbation quantity, 2, 22-23

choice of, 22-23Physical criterion for uniformity, 80-83, 168Pi, series for, 202-203Planing flat plate, 248Plate, see Finite flat plate, Flat plate in

viscous flowPLK method, see Strained coordinates,

method ofPoincare expansions, 224-225Porous circle, 19, 217Pressure coefficient,

cone in slightly supersonic flow, 178, 182subsonic thin-airfoil theory, 75

Primary reference length, 80, 168Principle of lea~t degeneracy, 86, 127Principle of minimum singularity, 53, 64,

65, 73, 88, 132, 158, 163Propeller, linearized supersonic theory, 36Pseudo-transonic flow, 78, 107Pulsating circle, 19Purely logarithmic case, see Logarithms

Radius of convergence, 32, 206, 207-208,213, 218, 234, 237, 245-247

Rational approximation, 2-4Rational fractions (Pade approximants),

205-207, 247Ratio test, graphical, 216, 218, 245-247Rectangular airfoil, 56-59, 73Reference length,

primary, 80, 168secondary, 81, 82, 83, 168

Region of nonuniformity, 104in boundary layer, 126exponentially small, 57, 87-88at low Reynolds number, 153-154in thin-airfoil theory, 51, 57, 58

Regular perturbation problem, 6-8, 9-20,81, 8~ 215, 23~ 235, 237, 243

Reynolds number, 16high, 121-147,230-232infinite, limiting flow for, 121-122, 232

low, 149-166, 234-236Round edge,

local solution, 62-68multiplicative correction, 59-61, 74nonuniformity in thin-airfoil theory, 52,

53, 54-56, 227-228, 229shifting correction, 71-73

Secondary reference length, 81, 82, 83, 168Second-order boundary layer, 121, 134, 136,

147, 233, 238Self-similar solution, 1-2, 4, 129Semicharacteristic coordinates, 108Separation of flow,

incompressible, 19, 121-122, 150, 159-160,214, 219, 232, 232, 235-236

in Stokes approximation, 235singularity at, 220, 239, 247supersonic and hypersonic, 232

Series expansion, 34-35extension by computer, 215-216, 234improvement of convergence, 202-210,

243-247successive approximations by, 34

Series truncation, method of, 219, 236, 236Shanks transformation, 203Sharp edge, 56-57,68-70, 74, 224-225Shear flow

past circle, 11-13, 20, 98, 217, 217past flat plate, 146-147

Shifting correction for round edges, 71-73Shock wave on supersonic airfoil, 115-116Shrinking rectangular coordinates,

143-144, 234Signs, pattern of, 245Similarity parameter, 21-22

for hypersonic flow, 22for transonic flow, 21, 182

Similarity rule,hypersonic, 22, 43second-order subsonic, 76, 212transonic, 21, 182

Singularity at separation,steady, 239, 247unsteady, 220

Singular perturbation problem, 6-8, 33, 45in airfoil theory, 45-76in coordinate perturbation, 82at high and low Reynolds number, 153

270 Subject Index Subject Index 271

inviscid, 167-193in parameter perturbation, 81-82prediction of, 7Bslight bluntness, 182

Skin friction,on circle, 18, 219on finite flat plate, Ll7, 139, 231on parabola, 6, 204, 208, 211-212, 213on scmi-infinite flat plate, 137-139,

140,230Slender-bodv theory, 5, 37, 74, 75Slightly distorted circle, 13-14Slight shear flow past circle, 11-13, 19-20,

98, 217, 217Slip, 147Source

distribution for airfoil, 49, 59, 71plane. 49

Spheredrag, 5, 149, 161. 206. 209-210. 216,

234-235, 243higher approximations at low Reynolds

number, 159-161,234-235at low Reynolds number, 5. 149, 161,226,

234-235Oseen solution, 5, 155, 206, 209-210pulsating, 43second approximation at low Reynolds

number, 159-161separation, 150, 159-160,219,235-236third approximation at low Reynolds

numbcr, 221, 234-235Spindle, body of revolution, 74Square edge, 57-59, 73Standoff distance for blunt body, 6, 40,

208-209Stokes approximation, 5

for circle, 151-152for compressible flow, 226, 235for flat plate, 39, 212, 230for parabola, 166for paraboloid, 165, 222separated flow, 235for sphere, 151-152

Stokes expansion, 39, 156, 159-161, 161-164,221, 223

Stokeslet, 152, 158Stokes's paradox, 77, 152, 247Strained coordinates, method of, 7, 99-120,

228-230

a posteriori application, 73, 120, 217for blunted bodies in hypersonic flow, 192combined with method of matched

expansions, 229-230comparison with method of matched

asymptotic expansions, 100-101,104-106, 118-119

for conical shock wave, 229for elliptic and parabolic equations, 100,

118-119,167,229for hyperbolic equations, 100, 118, 167,

228-229inapplicability of, 118, 192, 229relation to optimal coordinates, 146resemblance to method of multiple

scales, 199for slightly supersonic flow past cone,

177-182for supersonic airfoil theory, 109-112utility of, 118-119, 228-230

Strained parameters, method of, 218Stream function,

for blunt body in supersonic flow, 39-40,203-204, 206

for circle in plane potential flow, 10for circle in shear flow, 12for plane compressible flow, 183for plane incompressible flow, 10for slightly distorted circle, 14straining inapplicable to, 229for thin-airfoil theory, 73for viscous flow near leading edge of flat

plate, 39Subsonic flow

past body of revolution, particularintegral, 36

past circle, 4, 15-16, 214, 215-216, 219past parabola, 212-213past paraboloid, 75past thin airfoil, 75-76potential equation for, 15second-order similarity rulc, 76, 212

Successive approximations, 34-36Supersonic flow,

airfoil theory, 106-117past blunt body, 6, 39-40, 203, 20r;, 207,

208-209past slender cone, 176-182past slender body of revolution,

particular integral, 36

Sweptback wing, 176, 241

Tangency condition, see Boundary conditionsThin-airfoil theory, 5, 29, 33, 35, 45-76, 213,

218, 224-225Trailing edge

of finite flat plate, 230-232sharp, 232

Transcendentally small term, 34, 163, 165,166,218, 236-238

Transfer of boundary conditions, 36-37,43,47

Transonic flow, 35pseudo-transonic, 78, 107similarity rule, 21, 182slender cone, 176-182small-disturbance theory, 177

Triple deck, 230-232Turbulent boundary layer, 227Two-variable expansion method, 241Type, change of, 42, 129, 229

Uniformity, physical criterion for, 80-83, 168Uniformly valid expansion, see Composite

expansionUniqueness

of Blasius solution for flat plate, 131-132of thin-airfoil solution, 52-54

Upstream influence, 129

Velocity potentialfor circle in incompressible flow, IIfor circle in subsonic flow, 15-16for plane source, 49in subsonic flow, equation for, 15

straining inapplicable to, :229Viscous flow

past circle and sphere, 150, 234-236at high Reynolds number, 121-147,

see also Boundary layernear leading edge of flat plate, 39,

138-139,230at low Reynolds number, 149-166, see also

Stokes approximation, Oseenapproximation

past paraboloid, 165, 218, 239Vortex rings, 248Vortical layer, 83, 87, 196, 229Vorticity,

exponential decay in boundary layer, 131,139-140,201,218,229

external, effect of, 146

Wake,axisymmetric, 166, 218, 229of bluff body, see Eddyof finite flat plate, 136-137, 230plane, 165

Water waves, 216Wedge,

blunted, in hypersonic flow, 182-192boundary layer on, 146potential flow past, 69

Whitehead's paradox, 152-153, 247Wing,

elliptic, 6, 168, 174-176,207, 213, 239-241of high aspect ratio, 6,167-176,239-241with jet flat, 241sweptback, 241thickness effects, 192, 232

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