McGRAW-I4ILL SERIES IN MECHANICAL ENGINEERING

JACK r. IIOLMAN, Southern. Methodist University Co1tsu1lin.g Editor

RARRON . Cryogenic Sys tem ISC:KERT . hzlroduclion lo Heat and M a r Tran.fer ECKERT AND DRAKE . Ana1y.ri.r of Jim1 and Mos,r 7i-nnsfr.r ECKEK.~ AND DRAKE - Ifen1 attd A4ass 7ian. fer

HAM, CRANE, AND RODERS . Mechanics of Machinery HARTENRERO AND DENAVIT . Kinen~nlic Synlhesis of Ihkages

rrmm . Turbulence jmonsm AND AYRE . EtlGqineering Vihralions

~ ~ v 1 N A l . i . Ettgitteering Consideraliot~.~ n/.Ylrc.~r, .ylroi~, ntzd Slretzgth

KAYS . Co~tvecliue Heal and Mass Trcrtzsfir

LICIIIY . f h ~ b t ~ s / i o n Engine' P r o c e ~ ~ e s

M A R ~ N . Kil~~tnalics and Dytian~ks a/ machine.^

IVIELAN . I)!/~lan~ics qf Machinery PIIELAN . ~l l~ ld f l t l l e t l l~ l~ ,~ of n/fecharlim/ h.rigtl

RAVEN . Aulotnnlic Corrlrol En.gineerirtg

SOHP,N(:K . 7'hroric.r ?f Engitteering Expwir~lenlnlio~l

noundary -layer Theory

Dr. HERMANN SCHLICHTING Profresor J3rncrit.11~ nt, tlrc Ihgincrrirrg U ~ ~ i v r r ~ i t . ~ of ~ ~ ~ I I I I R ( . ~ I w c ~ ~ , O r r ~ ~ l n r l ~

Forrner 13ircctor of thc Arrodynnrninclre Vcr~rrclrsnnslnlt (;iittirrgcn

Dr. J . KESTIN I'rofe~sor at ljrown Univrmity in Providcncr, Rliodc Ialand

McGRAW-HILL BOOK COMPANY New York - St. Louis . S a n Francisco . Auckland . BogotL .

Diisselilorf . Johannesburg . London . Madrid . Mexico . Montrenl . New Uelhi - Pa~iarno . Pnri~l . Siio I'nulo . Singtrporo Sydnoy Tokyo . Toronto

Con tents

Library of Congress G t n l o g i ~ ~ g in 1'11blirntio11 Data

I A. I'IBII~:IIII~III~~ lrtws of n ~ o l i a n for a visrnr~r fluid

x i i i X V

nv i i x i x sx i

I

Virsl p ~ t l ~ l i s l ~ r < l i n 1111. ( : rrnl:~n longungc: 11mIrr 1.111: fril,lc "(:llI':N;SSCIIl(~ll'~-~~II1501

vi Contents Contcntn vii

CIIAPTEII V. Exnct ~olut iona of the Nnvier-Stokes eqnationa

a. Parallel flow 1. Pnrnllel flow through n straight channel and Couetto flow 2. The Hagen-Poiseuille theory of flow through a pipe 3. The flow between two concentric rotnting cylinders 4. The n ~ ~ d d e n l y accelerated plone wall; Stokes's first problcrn 5. I%w forn~at,ion in Couet,tc motion 6. Flow in n pipe, start,ing f ro~n rest 7. 'The flow near nn oscillating flat plate; Stolccs's second problem 8. A genernl class of non-steady solutions

b. Other exact solr~t.ions 9. S t n g r d o n in plane flow (FIie~nenz flo~v)

9a. Two-tiimensiond IIOII-steady stagnntion flow 10. Stagnntion in three-dimensional flow 11. Flow near a rotating dink 12. k'low in convergent nnd divergent cl~nnnels 1:). Conc l~~d ing r e~nnrk

R e f r r ~ n c e s

CIIAYL'ER VI . Very slow motion

n. The d ~ f i r c n t i n l eqrmtions for t he rase of very slow motion b. I'nrallel flow pnst n sphere c. The I~ydrodynnrnic theory of Iubricnt,ion d. The l le lc-Sl~aw flow Rcfcrrnrrs

Fort B. Lnnninnr boundary layers

CHAPTER V l l . l3011ntlary-lnycr equntion for tun-dirnrnaionnl inrompreusible flow; houndnry lnyer on n plntc

n. Derivation of bortntlnry-lnyer equations for two-dimcnsional flow b. The scp~ri i t ion of a I)o~mdary layer c. A ren~nrlc on t , l~e integration of t l ~ c bortntlary-layer eqr~ntions d. Skin friction e. 'The 1)oundnry lnycr d o n g a flat. platc I. I b ~ ~ n c l n r y Inyer of I ~ i g i ~ e r order

11 r l r rcnrrs

a. l)ejwn(lrncc of t , l~e cl~nrc~ctcrist.ics of n 11onnd:~ry lnycr on tltc I l e y ~ ~ o l d s ~ ~ n ~ n b c r b. "Sin~ilnr" solnt~ioos of the ho~n~dnry - l aye r ~qnnLions r. 'l'rnnnforn~nt~ion of t h bo~~nt lary- laycr cqontions into t,ho hcnt-conduction

oqna l , i o~~ (1. 'I'l~e ~non~cn t ,um nnd cnorgy-int.cgrnl equations for t,lw l ~ o ~ ~ n d n r y laycr I~.clorcnrcn

' 1 1 1 1 1 I 15unrt s o l ~ ~ t i o n s of the s teady-s tnk bountlnry-lnyrr rquntions in two- tlirnensinnnl n ~ o t i o ~ ~

n. I%\r pnst a wrdgr t

b. Flon in n convergent cl~annel c . Flow pnuta nylindcr; nymmet.rical cnso ( B l n s i ~ ~ s series) ( I . J h ~ n t l n r y lnyer for the potential flow given by U ( x ) = Uo - axn e. Flow in the mn.lte of flat, p l a b a t zero inridcnce 1. 'rho t .mo- t l i~n r~~s io~~n l lnn~innr jet

g. I'arnllrl s l .~cnn~n in Inminnr llow

11 Flou in tlw inlrt I r ~ ~ g t h of n straight cl~nnnol i. Tlw rnctl~od or finite dillcrcncrs j 13oundory lnycr of second order Rcfrr rnrc~s

\ l l X. , \pproxi~nntc ~nc t l~o t l s for t l ~ r solnt,ion of t , l~e ' two- t l i~~~r~~n ion : i l , strncly l ) o ~ ~ ~ ~ ~ I : ~ r y . l ~ i y c r ~ r p n t i n n s

n. ,\p!)lii.nt,ion of tlw I I I ~ I I I ~ ~ I I ~ I I I I I e q ~ ~ n l i n t ~ t.o the flow pnnL n fI11t pllbt,e nt mro incdcnce

b. The npproxi~natc method due to 'I%. vou I

A I l l I I I . I . I I I I I~II : I~ Iio1111(1nry l i~y r r s in cwnprrssil)lr flow

1 : I . I Z o ~ ~ ~ ~ c l : ~ r y - l : ~ y r r rontrol ill Inluit~nr flow

n. hlrtlwcls of l ) o ~ i ~ ~ ~ l : ~ r y - l : ~ y r r c o ~ ~ t r o l I . hI(it,im~ of tllr solid \ d l 2. ~ \ w v I r r : ~ l in11 o r t IN* Iio1111(1ary h y v r (l)Io!vi~~g) 3. SIIVI inn 4. It~jcvl irm of n clilliwwt, gns 6 . I 'rrvrt~l in11 t ~ f trn~lsit ion hy I I I C provisint~ o f s ~ ~ i ( n l ~ l r s l~npr s . I , n ~ ~ ~ i t ~ n r 11cr0fni1s I;. ( ' o n l i ~ ~ g of I I I ~ wtll

I). l t o ~ ~ ~ ~ ( l ~ ~ r y - I ~ t j c r s ~ w t i n n I . Tl~rorvl iw~l rc:s~~lln

I. I . I ~ ' I I I I I ~ : I I ~ ~ I I ~ I ~ ~ C O I I ~ ~ ~ O I I R I .2. 15xnct S O I I I I ~ O I I S I.:!. ,\1qiroxi111:1lr . m l ~ ~ t i o ~ ~ s

2. l C x ~ i c ~ r i ~ ~ ~ v n I : ~ l vrsttlts oil s ~ ~ c t i o u 2.1. I w r r a s r in lift. 2.2. I)vrrrnsc? in clr:tg

c,. 111jrr.tiri11 of ;I dill'rrcv~l g;ln (I

Y Contents

A " I XI X 'I'lirorrt.irnl nsnnn~ptiona for Ilio cnlcul:~t.ion of turbr~lcnt flows

('Il~\Pl'lCI1 XX. l'nrlwlrnt flow t l~ror~gh pipru

a. Exprrin~cntnl restllts for 811100th piprs h. J

'I'i~blr 17.1 : Ikl~cwclcnrc of rrilirnl Rcyriolcln I I I I I I I ~ I ~ of vnlocil.y yrofilrn will1 nuotioll on din~c~~~~ionIcs$ n~~ction vol11111e f:wIor E, :~.fter Ulr icI~ [24:1J

'T:ll,le 20.1: Iht.io of Inearl to 1nnxi111u111 vc1ot.it.y i l l pipe lir,w in t,ertns of tho expone~~t n of t,hc vclocity cliatribution, according to eqri. (20.fi)

'I':hlr 21.1: T

xvi I'orcworcI

Thc result was t . 1 1 ~ l)onk or 483 pages and 206 figuros publisl~ctl in 1061 in the Gcrrnnn Inng~tagn. \Vhcn t,llis book bcca.mc Icnown t,o rcscarcll workers and educntors in t,llc Unit,oti St.ntcs, t.l~cro was nn inunctlint.c request from srvrral quarters for an l h g - MI t.mnslal.ion, sinro no rnrnpar:~blc book was nvnilnhlc in the 1Snglish Inngungr.

'I'hc tcc:l~t~ical contcnt. of t h present. I':t~glisl~ etlit.io~t is dcscril~ctl in t,hc nr~thor's prcf:~c.e. 'l'hc c~npl~:mis is 0 1 1 t,l~o ftlntl;rmrl~l;rI pllysiral itlras rntd~cr ~ , I I :L I I on mntl~c- tn:lt.ic.:tl rrfinrlnrllt,. RIt:t.l~otls of t,llcort:t,icnl n n : ~ l ~ s i s qrc sot forth dong with s11c:l1 rxlwrirnont~d tln.t,n as arc pcrt.in(-nt t,n (Icfinc the regions of applic:tbilit,y of 1I1o I~llcwrc:l~icnl rcsult.s or t.n givr: 1i11ysic:rl it~sifillt, i ~ ~ t , o t , I ~ t : p l ~ c ~ ~ o m c n n .

\Vasl~ingtot~ I). C., 1)corml)cr 1064 I l u g l ~ I,. I ) ry t l rn

Author's Preface to the Seve~itl~ (English) Edition

Whcn J decided in 1075 to writ,c n new rclit,ion of t.11i.q boolc I cnmc t,o t,llc con- clusion t.l~nt, t.l~o prccrt l i~~g sc:qncnrc of n (lcvlnnn rdit,ioll followc:tl I)y nn 15nglisl1 c:clit.iotl was no longrr prscticnltlc. 'I'hc rcnsotl for it wn.8 lllc 11cnvily incronsrtl cost, of p.int,ing. Conscqncntly, I suggrst.rcl 1.0 the bwo publishing cornlmnics, G . llrnun in I

xviii Ar~t,l~or's Prefncr t.o tho Seventh (Englisl~) Er1it.ion

Along with this ncw material, I fee1 t,hat I ought, t o niention the topics which I spcoifioally omit,t,ctl l,o include. I do not, discuss t,he effect of chemieal reactions on flow processes in boundary laycrs as they occur in the presence of hypersonic flow. The sarnc applios t.o I)onndnry Inycrs in rna.gncto-fl~~itl-clytin~~tics, low-dcnsitty flows and Rows of non-Nowt,onian fluids. I still t.11onght that T ought to refrain from giving a.n rxposit,ion of t,lir st,at,ist,ical t,heory of t,ttrl)~~lenrc in this etlit,ion, as in t,hc prcviolls OIICR, hrrnusc n o ~ ~ ~ d n y s t.l~crc arc avnilnblc otlrcr, good prcscnt,nt.ions in I,oolr form.

Oncc again, t,hc lists of refcrenccs have bcen expanclcd considcrahly in many rhnpt,crs. The nurnl~cr of illust,rations increasctl by about G5, hut 20 old ones havc been omit,t.cil; the number of pages increased hy about 70. In spite of t,his, I hope that t,he original character of t,liis book has becn retained, and that it, still can provide tlie reader wit,l~ a bird'.?-e?y view of this important branch of the physics of fluids.

As I worlrrd on the new manuscript I once more enjoyed t,hc vigorous assistance that I rcccivetl from scvrral of my professional collcagues. Professor K. Gersten con- t.tihutctl sect,ions on boundary layers of second orrlcr t,o the part on laminar boundary lnycrs (Seas. VIIf ant1 I X j ) . This is a special field which he successfully worked out in rccent ycnrs. l'rofcssor T. K. Fnnneloep contributed the completely reformulated sc-ct.ion on the nurncrical inkgration of t,hc boundary-layer equations included in Scc. I X i . In t.hc part on turbulent boundary layers, Professor E. Truckenbrodt provitlcrl me witall a new version of the largest portion of Chapter X X I I on two- dimensional and rot~ationally symmetric boundary layers Dr. 1,. M. Mack of the California Institute of Technology was good enough to contribute a new section on the stability of boundary layers in supersonic flow, Sec. XVIle. Dr. J. C. Rotta tliorougl~ly reviewed Part I) on turbulent boundary layers and made many additions to it*. For the Russian litcrxtnre I rccrivstl nlurh help from Professor Milrhailov. The translation was once again cnt,rustctl to Professor J. Kestin's competent pen I ex- press my sincerc thnnlrs l o all tliose gcntlcmen for thcir valuable cooperation.

I should also like to rcpcat my aclrnowlcdgemcnt of thc hclp I rcceived from scvernl professional friends whcn I worked on the fifth (German) edition Nat.urally, their contributions havc now bcen rctaincd for tlie seventh edition. This is the ex- tcnsivc contribution on comprcssiblc hminar bountlary layers inChapter XIIT written by Dr. F .W. Rirgcls, Profcssor I

From Author's Preface to the First (German) Edition

Since :t,I~o~tt, the Ocgit~ning of 1 . 1 1 ~ twrrcnt ccnt,ttr.y niot1t:rn rcsr:~rclt in t , I t ( * l i t* l t l of fluitl clyn:rntics has :~clticvctl grcat sut:ccsscs nntl Itas h : n able to provitlc :I Cllc:. oretiral clarific:ttion of obscrvc:tl pltcnonwna which tJlc scicncc of rlnssirnl Ilytlro- tlyn:imics of t,ltc ~)rocctling c:cntnry failctl t,o (lo. 1kcnLi:~lly t.llrt:c br:tnc:ltcs of l l r ~ i t l tlyr~:~rnic,s 11:~vc bccomc p:~rticnlarly well clcvelopctl during t,hc last fift,y years; tllcy inclutlc hountlary-layer tl~cory, gas dynamics, and acrofoil Lllcory. 7'1m prcscrit t~ook is conccrncd with the branch knnwn as 1)ountl:~ry-layer thcory. This is the oltl(:st branch of modern fluitl dynamics; it w:is fou~~tlctl by 1,. I'mntltl in 1904 wllcn Ilc: succcedctl in showing how flows involving fluids of very s~nnll viscosity, in particular wntm ant1 air, the most imporl;:~nt, oncs from the point of vicw of applications, c:ln 11c m:dc :~tncn:~blc! lo rnnthomr~t,icnI r~nnly.qix. 'l'llis wris rdliovotl by ttiking I.II(: cl1i:c:l.s of friction into account only in regions whcrc they arc rsscnt,i:d, namely in tho thin boundary layer which exists in t h irntnctliatc ncigt~bourl~oocl of a solid body. This concopt matlc it possible to clarify many pllcnomona wliich occur in flows and which Imtl prcvionsly bccn incotl~pmllcrtsit)le. Most important of d l , it, Itas bcconto possiblr t o subject problems connected with thc occurrenor: of drag to a tllcorctical an:tlysis. r , l l te scicnco of aeronautical engineering was making rapitl progress ant1 was soon &ble to utilize these t,l~coretical results in pract.ical applications. I t tfitl, furthertnorc, pose many problcms which could be solvctl with the aid of the ncw bonntla.ry-layc:r theory. Arronautical engineers have long sinco matlc: the conccpt of a tmuntlary layer one of everyday use and i t is now unt.hir:Icable tm d o without it,. In other fieltls of lnaclline design in wltich problems of flow occur, in particular in the design of t,url~ornacl~incry, the theory of boundary layers made much slower progress, I,trt, in motlcrn tin:es t,hc:sc rlcw conccpt,~ Itavo come to t,hc fore in suc:I~ applic.ztions as well.

r 7 I he prcwnt 1)ooIt Ims hcrn writhxi principally for cnginecrs. I t is thc olzt.comc: of a course of Icct,urcs which the Author tlclivcrctl in t,llc Winter Scmcstcr of 1941/42 for the scinnt,ific worltcrs of tho Aoronauf.ical Itcscarch Institut,c in I3r:~11nscl1wcig. Tho stll)jtwt. mnttcr has bcclrt utili;r.ctl aftcr tho war in nlany spc(:i:d 1cct11rt:s 11cld a t t l ~ c ISngitleering Univcrsit,y in 13munschwcig for s t t d c n b of rnccl~anical engineering ant1 physirs. Dr. IT. IIahnclnann prepared a set of locturc notes :iftcr the first sorics of lectures \rat1 been givcn. 'L'lrcsc were rcad mxd amplifier1 by t.hc Autlior. They wwc subscq~lcr~tly published in mimeograpltctl form by the Office for Scicrltific I)ocu- rncnt,at.ion (Zontmlc fiir wisscnschaft~lichcs 13cricltt~swc.scn) nntl tlist.rit~nt.ntl 1.0 :t lirnit,crl circle of intcrcstctl scicntifir: workors.

Several years after the war tho autdtor tlecitlcd con~plkt~cly to re-edit, this older c:ompilat.ion and to publish it in the form of a book. 'l'hc t h o sccrnctl ~~art icular ly propitious becausc i t appeared r i p for thc publication of a comprel~cnsivc I)ook, and because t.hc results of tltc research work carried oul, tlt~ring fhc last trn t.o twcnt,y yrxm rounrlctl off trltc wltolc ficld.

'She book is tlivitlctl i n k four main ptrts. 'L'hc first, part contains two intro- tluct,ory ch:tpters in which t,hc funtlamcnti~ls of 1)onntlary-layer theory arc cxpoundetl without, the use of mathematics and then proccccls to prepare tho matl~ernatical and physical jllstification for the t l~cory of laminar bourulary laycrs, and includes the theory of thormal boundary Iaycrs. Tho t,llird part is concerned with the plleno- menon of transition from laminar to t,urbulent flow (origin of turbulence), arid the fourth pert is devoted t o tnr l~ulcnt flows. It is now possible to take the vicw tha t the theory of laminar boundary laycrs is complete in its main outline. Tho physical relations have been complctcly clarifictl; the meifhods of calculation have been largely worked out and have, in many cascs, bccn simplified to such a n extent, tha t they should present no difficulties to engineers. Jn discussing turbulent flows use has been made essentially only of t,hc scmi-empirical thcorics which derive from Prant l t l '~ mixing length. T t is true that according to present views these theories possess a number of shortcomings but nothing superior has so far been devised to take their plate, nothing, tha t is, which is useful to the engincer. No accour~t of the slstistical theories of tr~rbulcnce has been inclutlcd because they have not yet attained any pract.ical significance for engineers.

As int,imat,cd in the t.itle, the emphasis has bccn laid on thc thcorcticnl trcatmcnt of problems. An attcmpl, has bccn made t.o hring thcse consiclcrations into a form which can he rasily graspctl by engineers. Only a small numl~cr of results has hccn quoted from among Ifhe very voluminous oxperimcntal material. They have bccn chosen for their suitability to give a clear, physical insight. int,o the phenomena and to proviclc direct rcrific:rtion of thc t.l~cory prcsentcd. Some examples have been chosen, namely those a~sociat~ctl with tur1)nlcnt flow, because they constitute the fonntlation of the semi-empirical theory. An attempt was made to tlcmonstrat,e that essential progress is not, mndc through an accum~~lat ion of extensivc exprrirnental rcsriltn but rather through a small number of fundamental cxperiment,~ hacked by theoretical consitlerat,ions.

Brarmschweig, October 1050 - I I e r m a n n S c h l i c h t i n g

Introduction

Towards the end of the 19th ccntury the science of fluid mechanics began $0 dcvclop in two tlircctions which had pmct,ically no points in common. On t,hc onc side therc was the science of theoretical hydrody~tamics which was evolvctl from Euler's equations of motion for a frictionless, non-viscous fluid and which achieved a high degree of completeness. Since, however, the results of this so-called classical science of hydrodynamics stood in glaring contradiction t o experimental results - in particular as regards the very important problem of pressure losses in pipcs and channels, as well as with regard to the drag of a body which moves t,hrongh a mass of fluid - i t had litt,lc practical importance. For this rcason, practical cngincers, prompted by thc need to solve the i~npor t~an t prok~lcms arising from the rapid progress in t,echnology, developed their own highly empirical science of hydraulic^. The scicnce of hydranlics was based on a large number of cxperinlentd tlal,a :mt l difl'ercd greatly in its mct,l~ods antl in its objccts from the scicncc of t.hcorct,icnl hydrodynamics.

At the beginning of the present cent.ury L. Prandtl clisti~lguished himself by showing how t o unify thcse two divergent I~ranchcs of fluitl dynamics. H e achieved a high degree of correlation between theory and experiment and paved the way t o the remarkably successful development of fluid mechanics which has taken place over tlhe past sevent,y years. It had bcen realized even bcfore l'randtl tha t the discre- pancies between t,he results of classical hydrodynamics and experiment. were, in very many cases, due t o the fact tha t the theory neglected fluid friction. Moreover, the complete equations of motion for flows with friction (the Navier-Stolres equa- tions) ha.d been known for a long time. However, owing to the great mathematical difficulties connected wit,ll the solution of t,llcse equations (with the exception of :L small uuniber of particular cascs), tho way t o a thcorcticnl treatment of viscous fluid motion was barred. Furthermore, in the case of the two most important fluids, n:~mcly water antl air, the viscosity is very small and, conseqnently, tho forccs due t o viscous friction are, generally speaking, very small compared with the remaining forces (gravity and pressure forces). For this reason i t was very difficult to comprehend that t,he frictional forces omitted from thc classical theory influenced thc motion of a fluitl to so large an extent.

In a pzpcr on "Fluid Motion with Very Small Friction", read bcfore the Mathc- maticd Congress in IIeidelberg in 1004, I,. Prandt,lt showed how i't was possible tJo analyze viscous flows precisely in cascs which had great pmctica.1 importance. Wit,h

Abl~anrllnngc~~ rur

II. Schlicl~ling and II. U6rtlcr. r o l I1 p p ' 15 -584 .

the aid of thorct ical considcmtiotis anti scvcrnl simplo oxperimenk, ho provcd t.hat the flow about n solid body can be dividod into two regions: a very thin lnycr in t , l~e neighbourhood of t.he body (ho~~mla.r?j lmycr) whcrc friction plays an essential part, and thc remaining region o n t d c this laycr, where frict,ion may be ncglcctcd. On tho basis of Lhis Ilypot,l~esis I'mntltl succccdctl in giving a physically pct~rt~rating nxplnt~at~ion of tlrt: iml)ort,n.ncc of viscous flows, achicvit~g a t tho samc timc n maximum tlegrcc of simplification of the attcntlant rnatltemntical rlifficnlties. The t,heorct,ical considerations werc even tJ~cri snpport,cd by simplc cxpcrimcntn pcrformcxl in a small water tonncl which Prn.ridL1 built, wil,h his own hands. I l c thus took the first step towards a re~tnification of tl~cory and pmcticc. This boundary-layer theory proved cxtrcmely fruit,ful in that, i t provided an cKcctive tool for the tlevelopmcnt of fluid tlynamivs. Since the 1)cginning of the cnrrcnt century the new theory has been tlcvc- loprd a t n vcry fast r:lta untlcr t,hc atlditionnl st~imulns obtained from the recently fountlctl science of aerodynamics. In a vcry short time i t hecame one of thc fo~~ndat,ion stonrs of modern Ilnid clynamics t,ogcthcr with thct other very inlportant tlevclop- m e n k -- t h acrofoil theory and thc sciencc of gas dynamics.

In more recent t,imcs a good deal of at,t,ent,ion has been devotctl to s t d i e s of the mntlirmatirnl just.ification of boundary-layer theory. According to tllcse, boundary- layer theory provitlcs us wit,h a first approximation in the framework of a more general t,hcory designed t,o ca1culat.e ~ s y m p t o t ~ i c expansions of t,he solutions to the complet,e equat,ions of motion. The l ~ r o l ~ l c ~ n is retlucetl to :L so-called singular pertur- bation which is then solved by t.hc mct.liod of mat,chcd asymptotic expnnsions. I3ountlary-layer t.hrory t,hus providcs us with n cIassic example of the npplication of thc met,liotl or singular pcrt,nrbnt,ion. A general presentation of pert>urbation rnct,horls in flnid mechanics was prepared by M. Van Dykt:t. The basis of these rnat,hotls can be Itraced to 1,. J'raritlt.I's early co~lt~ribut~ions.

'I'lic 11onntlar.y-layer tlicory finds its application in the nnlcnlxtion of t,he skin- friction dmg whic:h ac1.s on a body as i t is moved t,hrongh a fluid : for example the rlr:lg cxpcricncctl by a flat p1n.t~ at,xcro incitlcnce, tilo t1m.g of a ship, of an aeroplane wing, aircraft, t~acrl l r , or t,rrrl)ine I)latlc. 13o11ndnry-layer flow 11:~s t ,I~c peculiar property t.ll:~t, untlor ccrt.airl conditions lhe flow in the imnictliat,c ncighbonrhood of a solid wall 1)ccomcs rcvcrscd causing the I~ountlary laycr to separate from it. This is accom- pnnirtl 11y a morc or lrss pronouncctl fonnat,ion of eddies in the wake of t,hc body. 'J'hus t.hc prcssnrc distribution is rltangcd and differs marlrctlly from tha t in a frict.ionl(\ss strcnm. ?'hc tleviation in prcssurc tlist,ribution from the ideal is the canse of form drag, antl its cniculat.ion is t h l ~ s made possible with the aid of bouriclary- laycr t.llnory. 13ountlary-hycr t,heory gives an answer t,o the vcry i r n p ~ r t ~ a n t question of' w11n.t shape mnst, a hotly t ~ o given in orclrr to avoid t.llis dct.rimarital scpn.ration. Scpnr:rI.ion m n also oc.c:ltr in l.llc int.crn:tl flow t.hrorrg11 R (:11nnnc1 ant1 is not, confined to rst,rrnnl Ilows past solitl I~otlicx. I'rol~lrms conrrcct,crl with l .11~ How of fluids throilgl~ t,hc c l~m~ncls f t m ~ ~ c t l by t.hc blntlcs of t,urhomachines (rotary compressors ant1 t,url)inos) ran also he 1,rrntrtl wit,ll tho n.itl of 11ountl:~ry-hycr t,Jlcory. I'r~rt.llcrmore, ~ ~ l w ~ l o n ~ c n n wllic:l~ occur at, t,llc point of rnn.xirnnm hft, of nn acrofoil and which arc assocht,t:tl with s t . : ~ l l i t l ~ (:;I.II 1)c 11ntlcrsI.oot1 only on th r 11n.sis of I~onntlary-layer

theory. I h d l y , problrms of l ~ c a t transfer I)ctwcwl n solitl hody ant1 n fluitl ( p s ) flowing past i L also bclong to thc class of problems in wltic41 l)o~~t~tl:~ry-l:~yc.r 1)11vno- mrnn play n dccisivc pnrL.

At, first the l~ountlary-layer theory was devclopotl rnn.inly for t l ~ c two of 1:~nlin:lr flow in an incon~prcssil)lc fluitl, RR in 1.llis c:uw t h ~ ) l ) ~ t ~ o ~ n t : t ~ o I o ~ i t : : l . l I~j,~)oI.I~t-sis for shr:~ring st.rrsscs a1rr:ttly cxistctl in thc form of Sto1tt.s'~ I:\w. ' l ' l ~ i s t , t , l , i t : W:IS sul)scqucntly tlcvclopctl in a 1:Lrgc ~lurnhcr of rcsonrclt p:Lpcrs :LII(I rt::1(:11vtl s1tt~11 a stagc of pcrfoct.ion I h t a t prcscnt tltc problem of Intninar llow c:1.11 III: consitlt~rctl to h:lvc hccn solved in its main oullinc. 1,:llcr the Ll~cory w:ls cxl.ot~clt:tl 1.0 int:lurlc turl~ulcnt, incornprcssil)lc bountlary layers which are morc irnportzmt from (ht: poitlt, of vicw of practical applications. I t is true that in tltc cnsc of t ~ t r l ) ~ ~ l c n t . flows 0. Iloy- nolds introduced the fundamentnlly important conccpt of nppnrcnt, or virt,~tnl tltrl)u- lent stresses as far back as 1880. IIowevcr, this conccpt was in it,sc.lf itisuffioirnt tso mn.ke tltc theoretical analysis of turbulent flows possible. Great progress was acllicvecl with the intmtlnction of I.'randtl's mixinglcrtgt.l~ thcory (1025) which, t,ogol,hrr wit,li systematic cxperimcnt.s, paved the way for the thcorctical ttrcntmcnt of turl1111c1tt flows wit,l~ the aid of boundary-ln.yrr t.hcory. llowevcr, a rational theory of fully developed turbnlcnt flows is st,ill noncxist.cnt,, antl in vicw of the cxl,rtmc com- plexit,y of sucll flows i t will remain so for a consitlcmhlc time. Onc cannot even be ccrtain tha t science will cvcr be successfnl in this t,aslr. Tn modern times tho phcno- mena which occur in the boundary laycr of R cornprcssiblc flow have becomc the subject of intensive investigntions, the impulse having Iwcn provided by thc rapid incrcasc in tllc spcctl of flight of motlcrn aircmft,. In atltlition to a velocity 1)oitntlary laycr suc:h flows dcvclop a tllcrrnal bonntlnry h y c r ant1 its cxist~cncc phys :I.U irn- portant part in the process of heat txansfcr bctwceri the Iluitl and the solitl body past which i t flows. At vcry high Mach numbers, the surface of Lhc solid wall bccornrs heatetl to a high t,cmperature owing t o the protlnct.ion of frictional heat ("tllcrrnnl barrirr"). This phenomenon prcscnts a tliffic:nlt analytic problem whose ~ o l ~ t t i o n is import.ant in n.ircmft tlcsign antl in the ~~ritlcrsl~anding of the motion or satellites.

r 7 1 he phenomenon of tmnsit,ion from liltninar to turbulcnt flow which is ~ ~ I ~ ~ : L I I I B I I - t.aI for t,he science of fluid tlynamirs was first investigated a t thc end of t l ~ c I0t.11 cen- t,nry, naniely by 0. 12eynoltls. I n 3914 1,. 1'm.ndtl cnrrictl out, his fnmous expcrimrnts with sphcrrs antl ~uccccdc~I in showing that the llow in Ihc 1)ountlnry layer car1 also I)c either laminar or turbulcnt and, furthermore, that, tltc problem of separnt,ion, ant1 hence the problcm of the calculat~ion of dmg, is govcrnctl by this t ran~it~ion. Y'hcoreti- rat invest,igations into t,he process of transilion from laminar to tnrbulcnt flow are basctl on t.110 acceptlance of Iteynoltls's 11ypot11o~is l,liat tohe la t tm occurs ns a con- scclucncc of an instability dcvolopcd by Ihc 1nminn.r 1)onntlary layer. 1'rnntlt.l ittif.int.ctl his thcorc1.icn.l investigntion of trnnsition in tllc ycar 1921 ; after marly v:rin cflort.~, succcss came in the ycar 1920 wlicn W. Tollmicn compntrd theorct,icnlly t,hc crit,ic:aI Reynolds numbor for transition on a flat plate at zero incidence. Ilowcvrr, nlorc t.lran ten years werc to pass 1)efore l'ollmicn's theory coialtl ho vtdficd throngl~ tho vcry carcful experin~enLs performed by 11. 1,. 1)rytlcn antl his coworltcrs. Tho stn1)ilit.y tltcory is capsblc of taking into account the cKcct of a nurnhcr of parnmctcrs (pmssurc gradient,, suction, Mach numlter, transfcr of heat,) on tmnsition. This theory has found m ~ n y important applications, among them in t l ~ c dosign of scrofoils of' very low drag (1aminn.r ncrofoils).

Modc:rn invcstigalions in id~c ficld of fluitl dynamics in general, a s well as in t(11c ficld of bountlary-hycr rcscarch, are characterized by a vcry close relation bc!twcen theory ant1 cxpcrimcnt,. The most important steps forwards have, in most cases, barn t,nltcn as a result of a s m d i numl~cr of f i~ndamcntd cxpcrimcnt,~ bacltetl by t,hcorot,icnl considcrat,ions. A rcvicw of t J ~ c tlcvclopmcnt of boundary-layer t.Ileory wllich st~rcsscs tllc rnuf,nal cross-fertilization bctwccn theory and cxpcrirncnt, is containctl in an n.rliclc writtrn 11y A. lk tz? . Vor about, twenty years aft,er its inccption I)y T,. I'randtl in 1904 thc bonndnry-la.ycr tllcory was being developed nln~ost exclwivcly in his own institute in Goettingen. One of the reasons for this st.nt,c of nffnirs may well havc been root,cd in the circum~t~ancc that, J'randtl's first pnblionthn on boundary-layer theory which appeared in 1904 was very dimcult to understantl. This period can be said to have ended with I'randtl's Wilbur Wright Meniorial I,ect,ureo which was dclivcrcd in 1927 a t a meeting of the Royal Aeronautical Society in 1,ontlon. In later years, roughly since 1930, other research worlters, par- ticularly t,hosc in Grent nrit.ain and in tllo U.S.A., also took an active pn,rt in its tlevrlopmcnt. Toclay, the study of boundary-layer theory has spread all over thc world; together with o t h r branches, it constitutes one of t,he most import,ant pillars of fluid mechanics.

Tho first survey of this I~mnch of science was given by 1%'. Tollmien in 1931 in two short articles in the "llan~lbnch dcr ISxpcrirncnt~alpl~ysiIr" :. S11orl~I.v aftcr- wartls (1936), Prnrdtl p~~l)lishcd a cotnprnl~cnsivc presentation it1 "Aerodynamic 'J'hcory" ctlitctl I,y W . I?. Durands. lluring t.he intcrvcning four dccndcs tllc volume of rescarch into this subject has grown cnorrnonsly$. According to a review published by 11. I,. Drydcn in 195.5, t,hc rate of publication of papers on boundary-layer theory reached one hundred per a.nn7r.m a t that time. Now, some twenty years later, this rate has more than tripled. Like several other fields of research, the t,heory of ho~rntlary layers has reachetl a volume which is so enormous that an individual scientist., even one working in this field, cannot be expected to master all of its specializctl subtlivisions. I t is, tl~rrcforc, right that, the task of describing it in a nlotlcrr~ Ilanclboolt has been cnt.rustcd t,o several authorst. The hist,orical develop- ment, of bountlary-layer theory has recently been traccd by I. Tani*.

. . " 1,. J'mllrlI,l, Tho goncmlion of vortiron ill fluirls ofatn.zII viscosit,y (15td1 Wilbilr Wright Memorial

Jfir(llr% 1!J27). J . Jtoy. Aoro. Soc. 31, 721-741 (1!)27). : (!/. tho bildiogr.zl~hy on 11. 780. : I,. l'r:~n(ll,l, T11c 111ecl1a.11irs ol' v i~coun fluids. Arrodynamii~ tl1oory (W. I?. Ihrm~d, rd.), \'ol. 3,

34 208, I%crlin, 1935. 6 11. Schlirh~ing. So~ne tlrvcloprncn(.s of I~oundnry-layer rcsearch in the past thirty years (The

'I%ird L~t lc l~r~ lc r Metnorin.l J,rcture, I!W)). J . h y . Aero. Soc. 64, 03- 80 ( l%U) . Srr nl?lo: 11. Srlilicl~l ing, Rccrrtt progress in houndn.ry-lnycr research (The 37th Wright. Brothers I ~ ~ i t r i : 1 t t 1 1 r 1 , ! 7 ) \ I . J t t i r i : ~ l 1 427 - 440 (1!)74).

* I . 'I':\t~i. Ilislory of I~o~~nrlnry-lilyor rmrnrcl~. A n ~ ~ r i n l Itrv. rrf Izluid Mwhnnirs 9, 87- 11 t (1977).

Part A. Fundamental laws of motion for a viscous fluid

C H A P T E R I

Outline of fluid motion with friction

Most t.Ileoret.ica1 invcst,igat,ions in the ficld of fluid dynamics arc based on the concrpt of a perfect,, i. c. frictlionlcss antl incompressible, fluid. I n the motion of s u c l ~ a perfect flnid, two cont,act.ing layers cxpcricnrc no tnngcntinl forccs (sl~caring st,rcssrs) b ~ ~ l , act on tach other wit.11 normal forccs (j)rcssums) only. This is cqr~ivalcnt, t.o s t a l . i ~ ~ g tl~nf, a pcrfvct, fluitl olrcrs no inl.crria1 rc~isI.antx to a c11angc in S I I : I ~ O . The t l~cory describing !,hc motion of a pcrft:cl. lluitl is ~ n a t l ~ c ~ ~ ~ ~ n t . i c : ~ l l y vcry far tlnvclopctl ant1 supplies in many cases a satisfactory dcscril;t,ion of real motions, such as e. g. tlle motion of surface waves or the formation of liquid jets in air. On the ot.her hand the theory of perfect fluids fails completely to account for the drag of a body. In this connrxion i t leads t o thc statement tha t a I~otly wllich moves uniformly t,llrongh a fluid which cxt.ends t,o infinity experienccv no drag (tl'Alcmbcrt.'s pamtlox).

'Pliis unacceptable result of thc thcory of a pcrfect Iluid can be traccd t o the fact that. t.11e inner layers of a real fluitl tmnsmit t,angent,ial as well as normal stresses, this lxing also the case ncar n solitl wall wetted by a fluid. Thesc tangential or frict,ion forccs 111 a rrxl Ilnitl arc conncctctl with a propertry which is callctl the viscosil?/ of thc Ilnid.

IZccai~sc of tho almnce of t,angcnt,ial forccs, on the 1)oundary bctwccn a perfect llnitl : ~ t ~ t l a. solitl wnII Lhcrc cxist,s, in gcnt~rnl, :I. tlilrrrcncc in rc~l:~l.ivc t,:~ngrnl.i:il vrloc.it.ics, i. c. t.11crc is slip. On t,hc other hi~ntl, in r(::11 l l ~ ~ i ( l s the cxi~t.cn(:t~ of int.cr- molecular att,ractions causcs thc flnitl to adl~crc to a solitl wall antl t,his gives risc l,o slrraring stmsscs. . , 1 hc exist,cncc of tangcnlial (sl~caring) s,,rcssc:s n r ~ l lhc condiliols 01 ,to d i p n(::~.r solitl walls const.itut1e the essential tliffcrcnccs bctwccn a perfect and a real fluid. Clert,ain fluids wl~ich arc of great, practicd imporl,ance, such as water and air, havc vcry smnll coefficients of viscosity. In many instances. t l ~ c motion of such llwids ol sn~nl l viscosity a.grccs vcry well wit.11 tha t of a perfect Iltritl, bccausc in most cases the - - shearing stressc?~ arc vcry small. For this reason the cxist,cncc of viscosit,y is corrlplctcly nrglcct.cd in the t,heory of perfect fluids, ma.inly bcca.11se this introdnccs a far-reacl~ing simplificatiott of the equations of mot,ion, as a result of

' ext.cnsivc niathe- matical theory I~ecomcs possilh. 11 is, I~owcvcr, islpm&, ss the fact that,

even in fluitls wit,lt vcry srnall viscosit,ics, unliltc in pcrfwt. fluiels, t.he rontlit.ion of no slip near n, solill I~oundary prevails. 'l'l~is c:ot~dil~ion of no slip int,rotlures in many (::~sos very hrgc tliscrcpar~cics in t,hc laws of moLiorl of perfect an(\ ronl fluids. In pnr- t.icular, t h vcry largc tliscrcpel~cy 1)ctwccn Llle v d r ~ o of' drag in a rral ant1 a pnrkct, Iltti(1 I1:w its pl~ysical origin in the contlil,ion o f no slip n w r :L wall.

'I'l~is 11oolc t1r:rls wil,l~ 1 . 1 1 ~ rnot,ior~ of l l r l i t l s of'sm:~II visrosil,y, I)(-r:~llsr of t . l~c grc:~L I~ :~c t , i ca l itnporl.ance of' the problcln. Ihrirtg 1,llc course of lhc s t ~ d y i t will l~cconlc clear how this p:trtJy consistent ant1 p:l,rl.ly tlivcrgcnt I)cl~aviour of pcrfrct and real fluids can be cxpl:tinotl.

h. Viscosity

'I%(: I I : L ~ , I I ~ C of' v i~rosi t~y can 11cst I)c v i ~ r d i z c d with the :lid of t,ltc following cx- ~wrimnnt,: Consitlcr the ~not~ion of a fluid l)cl,\vccrt two very long pn.rallnl ~)latcs, one of wl~inh is a t rrst, the other moving wit,l~ n, constant velocity pnrallcl t,o i t d f , as sl~owu in Fig. 1 . l . 1,ct tJ1o clist.anco h c t w c c ~ ~ thc plates bc h,, the prrssnre Iwing const,nnt

t . l ~ r o l ~ g l ~ o t ~ t tllc fluid. Exprrintcnt t.c~:rcltcs t.l~:rt t.11~ fluitl atll~rrcs l.o l)ot.l~ ~valls, so I,II:II, it,s vclovity :rI, the lownr p1:~t.c is zero, : I , I I ( ~ t,11:1t 3.t Lltc ltplwr ph1.c: is rt111al to t . 1 1 ~ vcloeit,y of the plate, I J . Ir'rtrt~l~ermor~, I.llc vclocit.y tIist,ril)r~t,ion ill t,llc fluid I)ct,wccn the pIat,cs is linear, so that, the fluid vclocit,y is proport,ion:ll tto t.ltc tlist,ancr ?/ from t 11c. lowvr platr, :~ntl we h:tvr

In ortlnr 1.0 s~ lppor t t , l~e motmion it is necessary to apply a I~n~~gc.nt,ial forcn t,o thn tlpprr l)lnto, tho force 1)cing in cc~t~ilibriurn with t l ~ c f'rict~ional forces in t , l~c fluid. I t is Icnown from expcrimont,~ t , l~at tJtis forcc (ta.l~cn per unit awn of t , l~c plal,c) is proprt.ion:~.I to t,hc velocity 1J of the 11l11)er plat.c, ant1 invcrsrly proport,ion:~l to lhc tlist,:r.nrc~ h. 'l'llc 1'ricI.ion:ll force por nit, area, tlcnotctl by t (Srict.ional shearing sl,rcw) is, t,licreCore, proport.ionn1 1.0 lJ/h, for which in general we may als? ssulist.itr~t,c tlii/tl?/. 'l'ltc: 1)ro1)01.t~io11:rIil.y far:l.or I)ct,wcnn t ant1 d71 tly, wl~iclr we s l~al l d c ~ ~ o t , c I)y ,u, I t lc11~1(1s or1 tho r~al~llrc of 1.110 l l ~ ~ i ( l . 11, is ~rna.ll for. "lhiri" fluids, s11c11 nk wal.cr or :~l(:ol~ol, I ~ u t I : q n in the case of vcry viscous liquids, srtclt as oil or glyccrinc. 'I'hl~s wc 11;tve ol)t,:~inctl t,llc ftl~~tl:rrncnl,al rclnt,ion for fluid frict.ion in t,lte form

> du

= f L ~ I Y . (1.2)

T l~ r quantity p is n propertry of thc fluid and depcntls t o n great cxl.cnt on it,s tern- pcrnt,rlrc. It is n rneasuro of tho i)i.~co,qit~y OF the fl~iid. '1 '11~ I:LW of' friction givrtl by cqn. ( I .2) is 1znow11 :LS Nrwtotc's 1rr.v~ of friction. ICqn. (1.2) cnn bc rrg:~relvrl :I.R t,llc c1rlinil.ion of visc:osit.jy. It. is, Ilowevcr, nccxssary to st.ross that the cxnrnplc cot~siclcrc:d in IGg. 1.1 (:onstit~~t.rs :L p:~rt,ic~~larly simple case of fluit1 motion. A gnncr:~liz:~l,it,r~ of this sitn111v e:rsc is cont,:~.inc(l in Stolccs's I:IW of fridion (cf. (!II:L~. I I I ) . ' 1 ' 1 1 ~ ~ l i m c ~ ~ ~ s i ~ ) n ~ of visrosi1,y c:all I IC tlotl~rc:c:tl wit,hol~t, diFlicull.y from cqn. (1.2)-I-. '1'110 sl~c:nritlg s1,rcw is ~ncnsurcd in N/m2 =I J'n nrld tltc vcloc:it,y grntlicnt du/tl?y in ~ o c I . ~ I V I I ( Y *

wllcre tho square 1~r;~(:Iccts arc I I S C ( ~ to ( I c r ~ o t ~ 11ni1.s. ' 1 ' 1 ~ :L~)OVC is not. 1hc o ~ ~ l y , or even the most, witlcly, employctl unit of viscosit,y. l'riblc? 1 . I lists t,he various t~ni ts togct.lrcr with t h i r conversion factors.

.15qn. (1.2) is rc1:rtctl t.o IIooltc's law for all c~l:ist,ic: solicl I)otly in w11ic:h rasc: t l ~ c shearing sCrcss is proport,ional t o the strain

I l r rc (: denotes lhe n~oclnlus of shear, y the change in anglc bct.wc.cn tfwo linrs wlliclt were originally nt right anglcs, nntl 6 tlcnotcs t.110 clisplr~ccmcnt, in t11c tlircc:t.ion of a1)scissae. Wllcrcas in thc cnsc of an elastic solid th : sl~caring strcss is proporl.ional t,o the nw~gniturle of the strain,, y , expcricnrc tcacl~cs tll:~t in t l ~ c case of fluitls it is proport,ionnl t.o the vale of chnnrlc. of strain tly/tll. If' we put

we s1r;~ll obtain, as bcforr, a11

t ' f l ?I!/

bccausc 5 = XI. Jlowcvcr, this analogy is not, complctc, I~cca~lsc t.llc: st,rc:ssas in :r flt~itl tlepcntl on one const,atlt., t , l~c viscosit.y ti, wllcw-:is tllose i r l :tn iso1,ropic vI:~sLic: solicl tlnpcntl on two.

8 I. Ont,line of fluid lnotion with friction

Table 1 . l . Visco~ity conversion factors Numerical values: In t,lrc case of liquids the vi~cosit~y, / t , is nearly indcpcndent, of pressurc and tlccreascs a t a high raLc with increasing tcmpcrat,urc. 111 thc case of gascs, to n first npproximat,ion, thc vi~cosit~y cnn be talrcrr to be intlcpcntlcnt of prcmitrc bi~t, it irrrrcnscs wil,lr l,cmllcrnl,rtrc. 'I'Iio Itinc?~nal,ic v i ~ c o s i l , ~ , 11, for litl~~itl.q has t,hc s m w type of t,cmpcrrat.i~ro tloj~otttlottc:o as p, I)ct.n.~tso 0l1r tltwsit,y, 0, ( - I I J L I I ~ ~ S only ~liglrtly with tcnrpomI,urc, Ilowcvcr, in t h caw of gn.scs, for whiclt C, tlcc:ro:t..qc~ consitlcrsbly with incrc:~sitig tc1npcrn1,11rc, 11 incrcascs rnpitlly willit (,cmpcmt.urc. Table 1.2 contains some numerical values of Q, p and v for water and air.

Table 1.3 contains some additional lisefitl tlat,a.

n. A l d l ~ t e viscosity 11

kp soc/m2 I

kp hr/m2 I'n see kg/m lir J Ibf sec/ft2 Ihf hr/ft2 Il,/ft scc

l'ahle 1.3. I

10 I . Or~llinr of lluicl rnot,ion with fricl.ir~n

111 ortlvr 1,o i~nswrr 1,Irc ~ I I I C S ~ ~ ~ O I I of wI~t:i.l~rr it is ncccssiwy 1.0 l.aI

1. Orrl.li~rr: of firtit1 motion with rrirtiotr

ILln. (1.1 I ) statcs t h i ~ t thc volumc r;.tc of flow is propnrtiounl to tllc first Ix)wcr the 1)rC"Urc (1'0p ppr unit lrngth (pl-p2)/l irnd to 1.h.: fourt,ll powor of tllc ra(jills of

thc pipe. Lf the mran velocity over tho cross-scctioa li = 112 is intrr)~llllsrl, eqn. (1.1 1) can bc rrwrittcn as

v 3 % n ( 1 - 1 1) can be ~ t , i l i ~ ( : d 6'. the cxperimcnt.al dctern~jna ' io~ i l f ~ J I C viSCmit,y, ,;.

I h c nICtllo(1 corlsisl* in thc mcanartw~cnt of tlrc rate of flow ilnd of (,llc pressurn (jmp across a fixall portion of ~1 cn(dl:lry tube of know11 m;llur. Thus cIlollg~l dnt,a rite provided to dctcrtninc 11 from ecrn. 1 I .11). . \ - - , -

The type of flow to'which cqns. (1.10) and (1.1 I ) apply exists in reality only for rclativcly small radii arid flow v~locit~ics. For larger vclorities and radii the character of tho motion changes complctcly: thc prcssurc drop ceases to bc proportional to the first powcr of thc rncan volocit,y as indicated by eqlt. (1.12), but becomes approxi- mately proportionnl to the second power of u . The velocity distrib~lt~ion across a secbiori hccomcs much more ur~iforln and thc well-ordered laminar ]notion is replaced hy a flow in which irregular and fluctuat,ing radial and axial velocity com- porlcnts arc supcritnposcd on thc main motion, so that , consequently, irlt,crlsivc mixing in a radial djrcction takes placc. In such cases Newton's law of frict.iorl, eqn. (1.2), ceases to be applicable. This is the case of lurbule?tt flow, to l)c tliscnsscd in great tl(:t,ail latcr in Chap. XX.

e. 15nciple of ~irnilarit~; the Roynolda and Much nurnbern 13

In the present section we shall endeavour to answcr a very fundamcntol qllcstiorr, ~lamcly tha t conrcrnrd wibh the conditions under which flows of diffcrcnt fluids about two gcomct,ricellg sinrilnr bodics, and with identical initial How dircctions ~lisl,lity gcomnt,rically similar strc!ntnlincs. Such mol.iorrs which havc gconrot~rirnlly

strrcarnlincs arc cnllctl tb?l,atrm.icctbl?y sirnilrr.r, or .qimilnr /10111~9. J k r two Ilowa nl)ont, grornotrici~lly aimili~r Iwtlicv (!:. y . irbout two spltorca) wiI.11 ~lill'(:r(:ttL ( I u i ~ la, tlillilrcllt vclocitics r~rltl tiillkrcni; iincw tlirncrtsior~s, to bo ~irnilar, it,, is cvidcnLly . . .... - . ~~ccessary i h a t the f o l I o ~ v ~ ~ ~ g ~ q ! ~ ~ ~ t , i ~ n .. shoulcl be satislic(l ;.st ? l l ~ g ~ , ~ ~ m e _ t r , ~ ~ ~ ! y sirni,l.r Point$ thC f6FCCS acting on a fluid particlc must !car a fixccl ~ t i o - l t cvcry instant . . . - . . . . . df t.iiiG,

\Vc shall now cdnsicter the irn~~ort,nrrt casc whcn only f'rict,ional and inert,ia forces are prcscnt,. IClaslic forces which may bc duc to clrangcs in volrnnc will hc cxcllltlcd, i. c. i t will bc assumed that tho flnid is incompressible. Gmvit:r.t.ior~:rl forces will also be cxcludccl so th:~t, conscqucntly, frcc surfaces are not adtnittctl, anti in the interior of thc fluid the forcc of gravity is assumed to be bal:~r~cccl 1)y buoyancy. Undcr thcsc assumptions the condit,ion of similarity is satisficcl only if a t all points the ratio of incrtia ant1 friction forccs is thc satnc. In a mot,ion pnrallel t o the x-axis thc inertia force pcr unit volume has the magnit,urlc of g l)lr/l)l, whcrc u ~Icnotcs tlrc componctlt of vclocity in tlrc x-dircctiorr and I)/1)1 clcnot,cs the sribstantivc dcrivativc. I n tho casc of stcady flow wc can replace it by e aslax - dx/dt = e v a@x, where all/ax dcnotcs thc r:hangc in vc1ocit.y with position. 'I111us the incrtia forcc per unit, volumc is cqui~l to C, u aulax. For thc fric- tion force i t is easy t o deduce a n cxprcssion from Newton's law of friction, cqn. (1.2). Considering a fluid pnrt,iclc for which tho x-direction coincides with thc dircct.ion of motion, Fig. 1.3, i t is found tha t the rcsnltant of shcaring forccs is equal to

a~ = - d x d y d ~ . a~

Hence the friction force per unit volumc is equal to atlay, or by eqn. (1.2), top a2u/ay2.

Consequently, the condition of similarity, i. e. the condition t h a t a t all corres- ponding points the ratio of the inertia t o the friction force must be constant, can be written as:

Inertin fxcc - - 2 !L =,on, t . Friction force p a2u/aya

1.. I'rinciplc of similarity; the Reynolds a d Mach numbers

I Thc typr of fluid n~ot~ion cliseussnid in tho preenling Scct,ion wr. very simple bcnasc evcry fluid part,icle ninvcd utr(lcr the infl~lcnec of friotior~al and pressure h rcas orrl.y, incrtia b r c r s laing cvcrywhcre cqurl in zero. 111 a divergent or convergent ch~~nn(:l fl11i11 p:wticlrn arc n.rtcd uport by inert.ia forces in atl(lif.io11 t o pressure and frirtiorr forrrs.

Fig, 1.3. Frictional forces acting on,a fluid particlc

I t is now necessary t o investigate how these forces are changed when the magnitudes which determine the flow arc varied. The latter includc the density e , the viscosit,y p, a representative velocity, e. g. the frcc stream velocity V , and a characteristic linear dimension of the body, c. g. the diamctm d of the sphcrc.

The vclwil y I L a t some point i11 tlrc velocit,y field is proportional to tlte free strrnm velocity IT, l,he vcloci0y gratlicnt au/ar is proportional to Vld, antl similarly a2 t r /~y2 is proporlional to V/d2. Ilcnce the ratio

Thereforc, tllc condition of ~irnilarit~y is sat,isfictl if the q l~an t i l~y p V d / p f ~ a s the same value in bol,l~ flows. The ( p n t i t y p V d / p , which, with 11.1~ = v , can also I N wriLt,cn ns V d/v, is a tlimcnsiotlloss nnrnl)cr \>cen.tlsc i t is the mt.io of t , l~c t,wo forces. I t is known as t.110 Ilayitnk1.c ~slr.?ttl)ar, R. Thus t,wo flovs arc similar when the lt:lin three crqna(.ions :

F : )I -4- 0 :: 0 ,

the solution of wlticl~ is

Din~ctlsint~lcss quantities: 'I'hn reasoning followctl in tho precetling drrivi~f ion of the Rcynoltls numl~er can be e ~ t ~ e n t l e d to inclndc the casc of d i f fc re~~t Itrynolrls numbers in the consitlerat,ion of the velocity ficltl ant1 forccs (normn.l :mtl tangont.i:rl) for flows wiLh geornetrica.lly sitnilar boundaries. Let t h r position of :L point in (.he space around the gcomctrically similar bodies bc intlica1,cd by thc coortlin:tl.t~s 1, !/, z; t~llen tho rat,ios z/d, y/d, z/tl arc its tlinicnsiotlless coortlirt:~l,cs. T l ~ c vc~loc:il.y c:otlt- poncnt,s arc lnatlc dimensionloss by relirrring tllern to the free-stream vch:iI,y V , thus 711 V, 111 V, w / V, and lhc normal and st~caring strosscs, p :~ritl t , can bo mn.clcr tlirnct~- sionlcss by reforring thorn t,o Lllc tloubfc of t,llc tlyrtatnic lieatl, i. e. t o p V 2 t.hus: p/p 1'" and t /p V2. The previously cn~~ncia tcd principle of dynnmical sinlilarit,y can Im c~x1)rt~s- sod in :Ln alternative form by asserting t h t for the two gcornctricnlly similar sys1,cnls with equal Reynolds numbers the dirncnsionless quantitics 141 Y , . . ., p/p V 2 i ~ n d t/e V 2 depend only on the dimensionless coortlinatcs x/d, y/d, z/d. If, Ilowcvcr, the two systems are geometrically, but not dynamically, similar, i. c. if t.lleir Rcynoltls numbers are different, t,llen the tlimensionless quantit,ies under consideratlion innst, also depend on the chamctcristic quantities V, d , Q, 14 of the two ~ystcrns. Applying the principle t,llat physical laws must be independent of the sys tcn~ of nnit.s, it. fi~llows tha t t l ~ e tlimensionless quarifities u/ V, . . ., p/e V2, T/Q VZ can only depend on a dimcrlsionless combinatlion of V , d , Q, and 11. which is unique, being the Itcynolds number R = V d e/p. Thus we are led t o the conclusion tha t for t01c two gcon~cbrically similar systmns which have different Rcynolds numbers antl which arc bring compared, the dimensionless quantities of the field of flow can only be funcI.ions of tlic tthree din~ensionless space coordinates z/d, y/d, z/d and of t.lw Rcynolcls number R.

The precc(ling dirr~cnsinnal annlysin can bc ~~li l izct l to tu:~ltc a n irnport,:r.ttt, assertion about the t.otal force excrtcd l)y a fluid strealn on an imrncrsotl hotly. 7'11c force acting on tho bocly is the surface intcgral of all normal and ~l lcar ing stmsst:s acting on it . If P denotes the component of the resultant force in any given direction, i t is possil~le to write a tlirncnsionless forco coefficient of the form P/d2 Q V2, 1 ~ 1 1 , in- stead of the a,re:b d2 i t is cnstomary to clloose a diKcrcnt charactcrist.ic aro:l, A , of t,he immersed body, e. g. the frontal a.rea exposed by the botly to tile flow tlircct

c. Principle of si~nileril.~; 1110 Ilcynolds nnti Mach numlwxs 17

of I,ho resultant forcc parallcl to the unciisturbctl initial vrlority is referred to as t11e drag I), and the component perpencliculnr to that tlircct.ion is callctl lift, 5. Hencc the dimensionless cocfficicnts for lift and drag become

C - L I ) , - A nnd C, = - - - - - - , 1 8 V S A (I . I d )

if the tlynnn~ic: 11cad 4 Q V 2 is S C I C C ~ C ~ for rcfcrcrlce instcatl of t,hc tlunnt,ity e V2. Thus tho argumcnt leads to the conclusion tha t the tlimcnsionless lift a,nd drag coefficients for geometrically similar systems, i. c. for geometrically sirnilar bodies which have t h same orientatmion with respect to the free-8trea.m direction, are functions of orie variable only, nmnoly the Reynolds numhcr:

c,,=/,(R); C D = / ~ ( R ) . (1.15) I t is ncccssary to strcss once more that this importmt conclusion from Reyr~olds's

principle of similarity is valid only if the assumptions undcrlying i t are satisfied, i. c. if the forces acting in the flow arc due t o friction and inertia only. I n the casc of compressible fluids, whcn elastic forccs arc important, and for motions with free surfaces, whcn gravitational forccs must be taken into consideration, eqrrs. (1.15) d o not apply. In such cases i t is ncccssary t o deducc diKerent similarity principles in which the tlimensionless Froudc n u m l w F = v/G~ (for gravity and inertia) and the c1imensionless Mach number M == V / c (for compressible flows) are included.

The importance of the similarit,y principle given in eqns. (1.14) and (1.15) is very great ns far as the scicnccs of th~orct~icsl and cxpcrimcntal fluid mechanics are concerned. First, the dimcnsior~lcss cocfficicnts, C,,, C,, and R are irlclependent of the system of unilm. Secondly, their use leads to a considerable sirnplificntion in the cxtcnt of expcrimcntal worlc. In most cases i t is impossible to tlcterminc the func(.ions f,(R) and /,(R) throrctic:ally, antl exporimcnt,:~i ~ncthotls must be 11sot1.

S ~ ~ p p o s i n g tl~ali i t is tlcsirccl to tlrtcrrnino thc t l r : ~ ~ cocfficicr~t ITI, for a spot,ilic:tl s11:q)c of h l y , c. g. a sjhcrc, tllcn w i t l ~ o t ~ t the application of Lhc principle of sirni1:~rit.y i t wo111tl hc? ncccssary to carry out drag mcasuremcnt.~ for four indepcntler~t variables, V, d , Q , and p, antl this would const,itute a trcmondous programme of work. I t follows, however, Lhat t2he drag cocfficicnt for sphcros of diKcrcnt tlinmctors with different stream vclocitics antl tliffcrcnt fluicls clcpcntls solcly on onc v:~ri:~l)lc, 1 . h ~ Reynolds r1urn1)cr. Fig. 1.4 rcprcscnts thc dmg cocfficicnt of circular cplintlcrs as a fi~nct~ion of the Itoynolds number antl shows the exccllcnt agrccment hetwceri expcrimcnt antl Reynolds's principle of similarity. The cxperimentnl point,s for the drag cocfficicnt, of circular cylinders of widely differing diameters fall on a single curve. 'The same applies to points ohtnined for the drag cocfficicnt of spheres plotted against t,ho Iteynoltle number in Fig. 1.5. The sutltlcn decrease in the value of thc drag coefficient which occurs near R = 5 x lo5 in the case of circular cylinders and near R = 3 x 10"n the casc of spheres will be discussed, in n ~ o r e detail, later. Fig. 1.6 reproduces photographs of the stream$nes about circular cylinders in oil taken by P. JIomann [7]. They give a good idea of the changes in the ficld of flow associated with various Reynolds numbers. For small Reynolds numbers the wake is laminar, but a t increming Rcynolds numbers a t first very regular vortex patterns, known as Khrmhn's vortcx &recta, are formed. At sLill higher Reynolds numbers, not shown here, tho vortex patterns become irregular and turbulent in character.

2 = V'J

Fig. 1.4. Drag coefficient for circular cylinrlcrn n, a function of tlie Jleynoltls n~~nibcr

4 00

700

C~ roo 80 60 10

7 0

10 8 G L

7

I 08 0 6 0 4

0 2

0 1 08 0 a

Fig. 1.5. Drag coefficient for spheres aa n fiulction of tho Reynolds nulnbcr Curve (1): Stokcs's theory, eqn. (6.10); curve (2): Oseen'a thcory, eqn. (0.13)

e . Principle of 8irnilnrit.y; t,he Itcynolds ntld Mac11 nlt~r~bcrs 19

Fig. 1.6. Firld of flow of oil n h o ~ ~ t n c irr~~ lnr rylintlor at wrying IZrynolcln n~c~nbnrs nltcr Homnnn 171: Irnnnition from lnrninnr flow t,o n vortrx ntrrt-t, i l l I :~n~innr f h v . T l ~ r freqrwnry rnngr for

R = 65 t.o R - 281 I)c tnltrn from Fig. 2.9 \V. Jonm, J . J. Cillotta and 12. \V. \Val. krr [a]

20 I. Outlinc of fluid rnolim with frict.iot~ f. Comparison hrtween thc theory of pcrfcct 11r;itls anti cxpcrin~cnt 21

f. Compnrison between t l ~ c theory of pcrfcct fluids n t d rxperi~nrt~t

In the cases of t,hc motion of water ant1 air, wllich arc the most ilnport.ant ones in engineering applications, the Itcynoltls nurnl)crs arc vcry Inrgc l ) rml~sc of thc very low viscositics of thcsc fluids. 1.t wor~ld, thorcforc, apl)c:tr rcasonal)lc t,o c-xpccL very good :tgrecrncnt 1)clwccn cxperin~cnt, and a 1,hcory in which t l ~ c itlllllcncc of viscosity is ncglcctcd alt,ogcthcr, i. c. with the thcory of pcrfcct fluitls. In any case it secms uscful to bcgin thc comparison with experiment by rcfcrcnce to t h o r y of perfcct fluids, if only on ncconnt of tho large num1)er of cxist,ing explicit mnthe- matical solnt,ions.

I n fact, for certain clnsscs of problems, st~clr as wave formation and tidal motion, exccllent results werc obtained wit01 t,hc aid of this theoryt. Most problems to bc rliscusscd in this book consist in I,hc study of the motion of solid 9odics through fluids a t rcst, or of lluitls flowing through pipes a.nd channels. In such cascs t,hc use of the theory of pcrfcct fluids is limited because it,s solutions do not satisfy thc con- I tliLion of no slip a t the solid surfacc which is always the case! with rcal fluids even a t very small viscositics. I n a pcrfcct tluitl thcro is slip a t a yal l , and tJlis circam- st,ance inLroduccs, cvcn for s lndl viscositics, such funtl:~.mc:~t.al tliKcrcnccs that i t

I is rather surprising to find in somc cascs (e. g. in the case of vcry slender, stream-linc bodies) that thc two solutions display a good measure of agreement. The greatest tliscrepancy betwccn the theory of a perfcct fluid and experiment exists in the consitlcration of drag. The perfcct-fluid theory leads t o the conclusion that when an n.rhit,mry solid body movcs through a n infinitely extended fluid a t rcst i t ex- pericnccs no forcc acting in the clircction of motion, i. e. tha t its drag is zero (dlAlem- bcrt's paradox). This rcsult is in glaring cont.radiction t o observed fact, as drag is mcnsurod on all bodics, evcn if i t can bccome vcry smaU in the case of a stream- line body in stcady flow parallcl to its axis.

By way of i l l~~strat ion we now propose t o make some remarks concernirlg tlhe flow about a circular cylinder. The arrangcmmt of streamlines for a perfcct fluid is given in Fig. 1.9. It follows a t once from considerations of symmetry t h a t the resultant forcc in the direction of motiorl (drag) is equal t o zero. The pressure clistributiou according to the theory of frictionless motion is given in Fig. 1.10, togcther with the results of measurements a t three values of the Reynolds numbcr. At the leading edge, all measured pressure distributions agree, to a certain extent, with that for a perfcct fluid. At, the trailing end, the discrepancy between theory and measurement becomcs large because of the large drag of a circular cylinder. The pressure distri- bution at, the lowest, sobcritical Reynolds numbcr R = 1.0 x 105 diffcrs most from that given by potential theory. The measurements corresponding to the two largest Rcynolds numbers, R = 6.7 x 105 and R = 8.4 x 106, are closer to the potential curve t,han those performed a t t,hc lowest Reynolds number. The large variation of pressure distril)ution wit,l~ Rcynolcls numbcr will be discussed in detail in the next cl~apt~er. A corresponding pressure-distrih~t~ion curbe around a meridian section of o, spl~cre is rcproduccd in Fig. 1.11. Here, t,oo, measurements show large differences for the two Reynolds numbers, and, again, the smaller Reynolds numbcr lies in the range

Fig. 1.0. Frictionlcss flow about a circular cylinder Pig. 1.10

Pig. 1.10. Pre~strre distribution on a circular cylinder in the suhcrit.icnl and er~pcrcriLic~ll range of 1 Reynolds nnnihers after t,he ~neasurements of 0. Flncl~sbnrt [4] and A. Roahko [13]. qm - - e 1''

is the stagnntion pressure of the oncoming flows - - - - frlctionlerur flow Flacl,s,,nrl R = 1.9 x 10' . . . . . R - 8.4 x 10' Itonlikn (1001)

1 i:G

Fig. 1.1 1. Pressure distribution around n sphere in the suhcri- tical and supercriticnl range of . Reynolds numbers, aa mea- sured by 0. Flachsbart [3]

of largc clrng cocfficicnl,~, whrrcas tho Irwgcr valuc lics in lhc rmgo of srnnll c l r q coefficients, Pig. 1.5. I n this case the n,czsnrcd prcssltre-cli~t~ributior~ curve for tho largo Reynolds number approximat,es the theorct~cai di:rvo of frictionless flow very well over the greatest part of the circumfcrcnce. ,

Considerably better agrcemcnt between the theo~etical and measured pressure distribution is obtained for a streamline body in a flow parallel bo its axis [5], Fig. 1.12. Good agreement exists here over almost the whole length of the body, with the exception of a small region near its trailing end. As will be shown later this circumstance is a consequence of the gradual pressurc increash in the down- stream direction.

Although, generally speaking, the theory of perfect fluids does not lead t o useful results as far as drag calculations are concerned, the lift can be calculated from i t v ~ r y successfully. Fig. 1.13 represents the relation between the lift cocfficicnt and angle of inritlcncc, as nteasurctl hy A. Bctx [2] in thc caso of a Zhukovsltii :iwofoil

22 I . Outliric of fluid motion with friction

Fig. I .12. Prmsr~re distrihnt,ion n l m ~ t n ~trenw-line body of rcvolntion: cornpnrison bct- neon tllcory arid mrnsuremcnt. nftcr Fuhr~nann [ 5 ]

Fig. 1.13. Lift nnd drag roeffi- cicnt of n Zliukovnkii profile in plnnm flow, ns ~nenaurod by l k t z 121

a. The hollndary-laycr concept 25

plate, with t>hc tlimensiorls across i t considerably cxaggcratctl. I n front of the leading edge of the plate t,he vrlocit,y elistribrttion is rtnifornl. With increasing distattrc from thc leading edge in the downstrmm direrlion the thiclrness, cf, of t,lle retardetl layor incrrasrs continrlor~sly, nn ilicrrnsing qunnlitira of h i t 1 I)oc*onlo t1TTcy-lrtl. 15vitlcr1tly tho lhiclrnrss or the 1)ountl:~ry Inycr t1wrcvw.s wit11 Oc~crrasir~~ viwosity.

Outline of boundary-layer theory

a. Thc boundary-layer concept

tn tho casc of fluitl motions for which the measured pressure distribution nearly agrcrs with the perfect-fluid thcory, such as the flow past the streamline body in Fig. 1.12, or the aerofoil in Fig. 1.14, the influence of viscosity a t high Reynolds numbers is confined to a very thin layer in the immediate neighbourhoocl of the solid wall. I f tho condition of no slip were not to be sat,isfit:d in the casc of a real fluitl there wollltl 1)c no appreciable tliKcrcncc between the field of flow of thc real fluitl as comparcd with tha t of a pcrfcct fluitl. The fact thaL a t t,hc wnll thc fluid adlicres to i t means, howcvcr, tha t frictional forces rctarcl the motion of the fluid in a thin laycr near the wall. In that, thin layer the velocity of the fluid increases from zero a t thc wall (no slip) t o its full value which corresponds t o external frictionless flow. The layer under consideration is called the boundary layer, and the concept is duo to L. Prantltl 1263.

Figurc 2.1 reproduces a picturc of the motihn of water along a thin flat plate in which the s!,rcamlincs wcrc made visible bjr the sprinkling of particles on the surfn.cc of thc water. The traces lcft by the particles arc proportional to the velocity of flow. Tt is scen that there is a very thin laycr near the wall in which the velocity is' considorably smallcr t,han a t a 1n.rgcr distance from it.. The thickness of this holtntlary laycr incrc,ascs along thc plate in a downstream direction. Fig. 2.2 repre- ~ n n b tliagrammatically the vclocity distribution in such a boundary layer a t the

Fig. 2.2. Sketch of borlntlnry - - - layer on a flat plate in pnr- allel flow at zero inciclcnce -

On the other hand, even with very small viscosities (large Reynolds numbcrs) t.hc frictional shearing strcsses T = /c au/a!j in the 1)oundary laycr arc consitlcrnblc bccnusc of the Inrgc vclocily gr~diont , across lllo Ilow, wllcrct~s o~tl~sitlo tho I~ou~~t l t t ry layer t11cy arc very small. This physical p i c t ~ ~ r c suggcst~n tha t the field of flow in t . 1 ~ casc of lluids of small viscosil.y can I)c tlivitlctl, for tho purpose or matliornnt,icnl annlysis, into two regions: thc t.llin boundary laycr near the wnll, in whic:h rriction must be taken into account, antl the region outside t h r boundary layer, whcrc the forces due to friction are small antl may be ncglcct~cd, and where, thcrcforc, the perfect-fluid theory offers a very good approximation. Such a division of the field of flow, as we shall see in more detail It~tcr, brings about a considerable simplification of the ~nat,l~ematical theory of the motion of fluids of low viscosity. In fact, t,he t,heoretical study of such motions was only made possible by Prandt.1 whorl he introclucctl this concept.

We now propose t o explain the basic concepts of boundary-layer thcory wit11 the aid of purcly physical ideas antl without the nsc of ~nat~hcmatics. The rnathcrn:~t.i- cal bor~ntlary-layer tllcory which forms the main topic of this book will bc tlisc~~sscel in the following chaptcrs.

The dccrlcratctl fluid pnrticles in thc boundary laycr (lo not, in all cnscs, rrmnin in the thin lnycr which atlhcrcs to th r I~ody along thc whole wcttcd l c ~ ~ g l h of ~ I I P wall. I n some cases the boundary layer increases its thickness considerably in the downstrcarn tlirection and the flow in tho boundary laycr beconics revcrscd. 'l'his causes the decclcratcd fluid particles to be forced outwards, which rnmns illat thc boundary h y c r is scpnrated from t11c wall. Wc thcn spcalr of boundniy-ltryer sepalation. This phenomenon is always associatrd with the formation of vortircs and with largc energy losses in the wake of the body. It o_ccur_sprjmarjly nrar blunt bodies, such %s circular cylinders ~ n c l ~ s p h _ c - ~ ~ . Behind such a body thcrc exists a region of strongly dccrleratrtl flow (so-calletl wake), in whicl~ the pressure distribution deviates considerably from that in a frictionless fluid, as seen from Figs.l.10 arlcl 1 11 in the ~ r s p r c t i w cnscs of a rylindcr and a sphere. The large drag of such bodics can be explained by the existence of this large deviation in pressure distribution, which is, in turn, a consequence of boundary-layer separation.

2 (i T I . O~~tlittr of Imun~lnry-lsyw throry

E ~ t i n ~ n I i n ~ t of houndnry-lnyer thickllr~s: 'rhc t,l~ickness o fa boundary layor whir11 llas riot sepnrnlrtl can I)(! casily rst,irnnLrtl in thc following way. Whcrcas friction forccs can be ncglcctctl with rcspoct t.o incrt,ia forccs out,side tho bourltlary Ixynr, owing to low viscosit,y, thry arc of a comparable order of magnitrldc inside it. 'rhc inert,ia forcc prr nit v o l u n ~ is, as cxplninctl in Scct,ion l e, equal to Q 71 & L / ~ x . For a pIat,o of longlh 1 tho gr:ttlinnt arr/a:r is proportional to ll/l, where I J tlrnotes t h r velocil,y onLsitlv the! I)ountl:wy Inyrr. I l c t~rc Ihc irlnrl,in forcc is of tho ortlcr I, 1J2/1. On the othcr l~ant l the friction forcc per n r ~ i t volurnc is equal to at/@/, wllirll, on tho assurnpt~ion of lnrninnr flow, is cqunl t,o 11, a21t/i)?/2. The velocity gratliont al~/ay in a tlirrcLion prrl~rnrliculnr t,o t . l~c wall is of t,lm ordcr Ill6 so that thc friction forcc ])or ~ t i ) i l ~ ~ o l t ~ t n v is i)~/&y - lI/d2. Proni the cotdit.iorl of equality of the friction :md inertia forcrs tho following rc.l:ll ion is obhined:

U e UZ 1 t4 82 -

or, solving for I Itr Imuntl ;~r~-layrr tlriclcr~rss Ot:

The I~nlnr,ric:nl f:~rt,or wl t id~ is, so f:w, st.ill untlct,crn~ined will be drduc:ctl Iatcr (C!l~:lp. VII) from tho exact solut,ion givcn by [ I . 13lasius 141, and i t will turn out t.llnt i t is cqrlal 1.0 5 , al)proxinlatcly. l lrncc for lnmiarrr flow in the bountlary layer wn hnvo

(2.1 a)

'rho tlinlrt~sionlc~ss 1,our~lnr~-lnyer thirknrss, rcfcrrctf to the length of the plate, 1. twronles .

wllorr R, clrnotcs tho ltcynoltls nunlber rclatod to the Icngth of the plat.c, 1 . T t is won from cqn. (2.1) tallat thc boundary-layer thickness is proportional in 4; and t,o I . If I is ropla.cetl hy the variable t l is t~nce z from the leading edge of the plate, i t is seen that d increases proporti~nxt~ely to ii. On tho other hand tho relative boul~(~ary-Iaycr t,I~ickncss O/i d e c r e m s with increasing Reynolds number as I I ~ R so t h a t in tho limiting case of frictionless flow, with R -+ oo, tllc boundary-layer t.lrickness vanishes.

We are now in a position to estimate the shearing stress zo on the wall, and consrq~~ontly, t.hr t,ot,ni drag. According to Newrton's law of friction (1.2) we have

- - --

t A ~~lore rigororts tlrfiniliott of Im~lrtclnry-Iayrr thicknrsn in given s t the end of lhia section.

wherc s l ~ b s c r i p ~ 0 tlenotes the value a t the wall, i. e. for y = 0. Witll thc estimate ( a u / a ~ ) ~ - U / d we obtain 7 , - ,u U / d and, inserting the value of d from cqn. (2.11, we have

We cart now f o r ~ n a dirncnsionlrss sl,rcss with rcTrrnrlrc l o I, l l z , ns c~xpl:~ittc~cl in Cltnp. I, ant1 obtain

1 c,, - = - - I'q . The numrric:ll fartor follows from 11 Blasius's cxart solution, atttl is I 328, so t l l :~~, the drag of a ~ ~ l n t r in parallrl 1nmin:~r flow 1)rromc.s

Tltc following nt~mrrical rxamplc will serve t,o il11tst~~rt.c: t.hr l)rec:rcling c:st,i~rt:~ t.iolt : Laminar flow, stipulntctl here, is obt:~it~rtl , as is known r'ronl exprritnctlt,, for Itcy- nolds numbers CJllv not cxceccling :d)outt 6 x 10Ql.o 10% lpor 1nrgc.r I

28 TI. Ot~(,linc! of bor~ndnry-layer thoory b. Srparation antl vortex fortnn(.ion 20

Dalinition of Imnndnry-layer thickness: Thc clefinition of lhc bountlary-laycr t.lrickncss is to a ccrtain extent arbitrary l)ccausc transitsion from the velocity in t , l~c borlndary t,o that o~~t.sitlc i t t,:~.ltcs plncc asympt,olically. Tlris is, I I O W ~ V C ~ , of no pract,icn.l import,ancc, I~ccnusc t,hc vclocil~y in thc bor~ntlnry laycr at.t,:iins :I. vnl~lc whic:h is vrry c:losc t,o fho cxl,crt~n.l vcloriLy drcatly at, a small tlistancc frotn the wnll. 11, is Ijossil~ln to tlcfino Lhc I)o~lnd;~~.y-l:~yc:r thioltncss :IS l . l~nl rlis1,:~noo from l l l c : wnll wllorc: t,hc vclonity tlilTcrs 11y I pcr ct:111 from the oxt,crnn,l vrlociLy. \ V i l . l ~ titis dnfinition the rtrtmcric:d f:~.ct.or in cqn. (2.2) has thc value 5. [nst,ead of t,hc bonntlary- laycr t.lricknc~s, anotlrcr qunnt.it$y, thc dinplr~cement thickness a,, is somct.imcs used, Fig. 2.3. I t , is dcfinetl by thc cqnntion

(2.6)

' I l c displnccment tl~icltncss indicates l.llc tlistancc by which the external strcam- lines arc shift,cd owing to tire fonnat,ion of t , l~c boundary Iaycr. I n the case of a plate in parallel flow nntl a t zcro incidcncc tlrc tlisplaccmrnt thickness is about & of the bountlary-layer IJ~icltncss 0 givcn in cqn. (2.1 a).

b. Srpamlion and vortcx forrnntion

.. l l te bo~~ntln.ry laycr ncnr a f h t plate in par:~llcl flow and al, zcro incitlencc is part,icrllarly sirnplc, Ijccausc the static prcssurc remains conshnt in the whole field of Ilow. Sincc orlt,sitlc the 1m11ntI:~ry lnyrr tho vclocily rcnmins constant t,hc samc qjplics to the p r c s s ~ ~ r e l~ecausc in the frictiorrlcss flow Bcrl~orrlli's cquation remains vnlitl. Furthcnnorc, tlrc prcssnrc rcmnins scnsibly constnnt over thc width of t,hc \ )o~~~rr la ry layer a t a givcn rlist.ancc x. 1Icncc tlrc prossurc over thc widt.11 of tlrc 1)ountlary Iaycr has tlrc snmc mngnittrtlc ns out.sitle t.hc boundary laycr a t the samc tlist.ancc, ant1 the same applies lo cnscs of arbit,mry body pl~n.pcs whcn tho prcssnrc o~rt.sitlc 1 . h ~ I)o~ln(l:~ry I:~yt:r vnrics along t,lrc wall wit11 t , l~c 1cngl.h of arc. 'l'his fnct is cxprcsscd by saying 1,h:~L t,lrc cstcrnnl prcssnrr is " i~n~rcssct l" on thc boundary Inycr. Ilcncc in the cnsc of the motion p s t a plate l,hc prcssnrc rcmains constant. througIrouL t,llr: bountlnry Inycr.

'j'lrr phrnonrrnon of 1murrtl:~ry InycrsrpnraLiot \ ~nrt~tiot~c~tlprc~viously - - . - - isi!rtinral~ly c~onnrclctl wrtl~ tlrr prcssurc t1istril)ution in ti16 orintlary layrr In the boundary lnycr on a plate rro srpnmlion takrs p h r r as no back-fldw occurs

In ortlcr to r\plnitr t I I V very import nrrt pl~rnornrr~on of bountlary-lnycr s~para t ion let us rorrritlrr 1 hr Ilow :~ljouI n Ijlrrnt hotly, r g abont, a rirrnlar rylintlrr, as shown i t 1 IClg 2 4 111 ft ic.1 inl~lcw flow, t l ~ c flu~tl par1 irlrs nrr :~rc.rlrmlrtl on tlw npstmam

half frorn D to E, and decelerated on the downstream half from E to F. Ifcnce the pressure decreases frorn D to E antl increases from ii' to F. Wltcrl the flow is stmtcd u p the motion in the first inst,arlt is very nearly frict,ionlcss, ant1 rcmains so as I m g as t h bounthry lnycr remains thin. Outsitlo lhc I~onntl:~ry lrtycr lllcro is n tprr~l~s~ornlctl.io~t of pressure into 1tincl.ic energy idong 11 R, 1.110 rcverso hlting pl:~c:o r~lottg IC I(', so IJtaL IL parlidc n r r ivo~ I L L 11' with Llto H I L I I I ~ > vclocil,y 11s it, I I I L ~ nl, J) . A l I r c i ( l ~~:~rl.iclt: wltich lrroves in IJIC i~nmctlinlo vioi~til~y of tho wtdl in I,llc bo~lntl:r.ry I:~.yor rc:~n:iit~s under the influence of the same pressure field as that existing outside, I)crause the external pressure is imprcssctl on the boundary layer. Owing tlo tlrc large friction forces in the thin boundary layer such a psrtic:lc consumcs so much of its kinbtic

Fig. 2.4. Doundary-layer scpara- tion ~ind vortex forrnntion on a circular cylinder (dingran~n~atir) S - point nf s c l ~ n r n l l o ~ ~

energy on its pat.h from D to E tha t thc remaintlcr is too slnall to srlrmount t.hc "pressure hill" from E to F. Such a parLicle cannot move far into t,hc region of' increasing pressure between lC antl P antl its molion is, evcntunlly, arrcst,ed. The external pressure causcs it t,lrcrl t,o move in tho opposite clircction. Tlrc p l ~ o t o g r a ~ l ~ s reproduced in Fig. 2.5 il1nstrat.e the sequence of cvent.s near the downstrcarn side of a round body when ,z fluid flow is started. The prcssurc increases along t,Ile I,otly contour from left t,o right, the flow Ilnving been ma.tlc visil)lc by sprinltlitrg nlrtminirlm drrst on tho surface of thc water. Tlrc boundary layer can be casily rccognizetl by rcfcrcncc t o tlte short traces. In Fig. 2.5s, Lakcn shortly aftcr the s tar t o f lhc rnot,iorl; the rcvcrsc motmion has just begun. In Fig. 2 .5b the rcvcrsc nrotion lrns pci~-t,r:.tctl a consitlcrablc distancc forward : ~ n d l , l~c boundary Iayor lrns tllicltcnctl n.pprcci:~l)ly. Fig. 2 . 5 ~ shows how this rcvcrsc mot,ion givcs risc to a vortex, whoso sizc is incrc,iscd still f u r t h x in Fig. 2.6tI. 'l'hc vorLcx bccorncs scp:~mlctl shortly a f L c r ~ : ~ r ~ I s n . t d rnovc!s tlow~~strearn in tho fluid. This circnn~stancc changcs complctcly blrc fiolcl of flow in tho waltc, and Lhc prcssnrc clisLrib~lI,ion suKcrs a rntlical change, as cornparctl with frictio~rlcss Ilow. 'L'llc f ind statc of nrotion can I)(> inrcrrctl from Wig. 2.6. In t,he eddying region bclrind tlic cylinder there is consitlcrable suction, as sccri fro111 the pressure distribution curve in Fig. 1.10. This suction causes a large prcssurc drag on t.he body.

1 At a larger distance from the body i t is possible to discern a rcgul:~r patt,ern

of vorticcs which move alternately clockwise and courrt~crclocltwise, and wllich is known as a IGirmiin vortex strect [20], Fig. 2.7 (scc also Fig. 1.6). In Fig. 2.6 a vortex moving in a clockwise direction can be seen t o be about to detach it,sclf from the body before joining the pattern. I n a further pzpcr, von Kilrmhn [21] proved that such vorticcs are gcncrally nrrstablc with rcspcct to small t l i~t~urbancrs pnrallcl

Fig. 2.5b Fie. 2.511

Fig. 2 . 5 ~ Fig. 2.5d

to thr1ns14vt:s. 'I'lrc only nrmngnncnt which shows ncnt.ral cqoilil,rium is t,hat with - . 0.281 ([Cia. 2.8). vort,ex sl.rcet moves with n vcloc:it,y I L , which is slnallc\r

I,II:I.II t.Ilc flow vrIorii,y I I in front of t,ho body. I t cnn l)c r c p d e t l as a highly idealized p ic t ,~~r r of t.hc mot,ion in the wake of (,hc body. The kinetic energy cont,ainetl in the vrlocit,y ficltl of the vortcx strect must be continually created, as the body moves t.llrongh tile fnitl. On the basis of this rcpresentrn.,tion it is possible t,o deduce a n exprrssion for t.hc drng from the perfect-fluid theory. I t s ~nngnit,utle per nnit lengt,h of tllr eYlindric:~l hotly is given hy

Fig. 2.7. KhrmQn vortex strcct, from A. Tirn~nc [38]

Fig. 2.8. Strrnmlinm in nvortrx strrrt (hll = 0 28). Thr fluid i8 nt rc~t, nt infinity, and t h ~ vortrx street move8

C i r c d u r cylittder. 'l'hc frequency wit,lr which vor1,irc~s a r r shrtl in a I

32 I I. 011t.linc of boundary-leycr theory

Fig. 2.9. The Stroul~nl nurnher, 5 , for thc I

1). Scp~ralion and vortex iormntion 35

S - point orscpnrnt.ion

T'ig. 2.12. I)ingmnitnnt,ic represell- t.nf,ion of flow i l l t,lw 11o1lt)tlnry layer near n point, of wpnrnt.ion

Fig. 2.14. Flow with 1)ortnrlnry- In.yor srlc(.iott on upper w d l of Irighly tlivcrgetlt clln~~nrl

Fig. 2.1.5. Flow wit,lt honndnry- layer ~uction on 110th wall8 of highly divergent channel

src1.s t,hc wall a t a tlcfinitc angle, ant1 t , l~c point of s~p:iri~t,ion it,sclf is cl~:tern~inctl by tltr ro~trlitinn that t.hc velocil,y grarlicnt. normal to the wall vanisltcts t.htrc:

Scparal.ion, as clrsc:ril)ctl for l l ~ c r : ~ : of a c:irc~~liir ctyli~~tlcr, ciin :LISO occur in a highly divergent rhxnncl, Fig. 2.13. In fror~t of the t.ltroat t . 1 ~ prcssnre tlccrcasrs in thc dirrctiol~ of flow, atltl thc flow atlhcrcs complclcly t.o thc walls, as in a fricf,ionIcs?i fl11id. Jlowcvcr, bcl~intl t,ho throat t.hc tlivcrgcncc of the cl~anncl is so Inrgc? t . I~ : i t . t . 1 1 ~ bountlary layer becomes scparatetl from both walls, rind vorticcs arc l'nrmcd. YYIC stream fills now only a srnall portion of the cross-scct.iona1 area of t.11~ cl~anncl. l low- ever, separation is prevented if boundary-layer suction is npplictl n.t t,ltc wall (Ipig~. 2.14 ant1 2.16).

?'lm photograpl~s in Figs. 2.16 nnd 2.17t j)rovc t.hat the atlvrrsr 1)1vss1irt: gr:dicnt t,ogct,llcr wilh fricl.ion near t.lra wall tlctcrn~inc the proccas of sc~):~r:iLion which is intlcpcntlcnt of such other circumstance as c. g. tltc curvnture of thc wall. 'Jlhc first pictme shows the mot,ion of a fluid against a wall a t right angles to i t (planc stagnnt.ion flow). Along thc streamline in t . h ~ - ~ d a n e of symmetry which lm,tls ho t,hc st,agnat,ion point tllcrc is a cot~sitlcrablc prcssllre incrcnsc i n t,hc clircclion of flow. No separation, howcver, occurs, because no wall friction is prescnt. 'I'herc is no sepnmt,ion near the wall, either, because here t,he flow in thc boundary laycr takes place i n the direction of decreasing pressure on both sides of the plnnc of symmetry. If now a t l ~ i n wall i~ placed along thc planc of syrnmctry a t right anglcs to thc first, wdl, Fig. 2.17, the ncw boundary laycr will show a pressure increase in t,hc direction of flow. Conscqurnt.ly, scparnt,ion now occurs nm,r 1,Ite planc wall. 'L'hc incitlcnce of scpnmt.ion is often rattler scnsitivc to srnall chnngc?~ in the s h p c of t.he solid botly, parl.ic:~~lnrI~. witen th r prcssrm tlistribut,ion is strongly affcct.ct1 by this char~gc in shape. A very instructive exnrnplc is given in Lhc pit:t,urcs of Fig. 2.18 whicl~ show photogrnpl~s of the flow fioltl altout a n~otlrl of :I mot.or vehicle (t,hc Volkswa~gcrl clclivcry van), 123, 351. Whcn t,ho nosc was Il:kt, giving i t an angular shape (a), the flow past thc: fairly slmrp corners in front causcd largo su&ion followed by :L large pressure incrcnsc along the sidc walls. This led to ronlplcte scpnration and to the formati011 of a wide wake behind the body. Thc drag coefficient of the velricle with this angular shape had a valnc: of C , .= 0.76. Thc liwgc: suction nrar the front cnd i d l h scp:~ri~t.ion along t l ~ c side walls were clinlinat,c:tl when the shape wa9 chnngctl by a.rltling t h : round nose shown a t (I)). Simultm~corts l~, tho drag cocfticienl became rna.rltrtlly smaller and had a value of CD = 0.42. Further rcscarch on such vchiclcs have beell performed by 11'. H. IIucho [In] for the rase of a non-~yrnrnct~ric strcam.

t Fig. 2.16. and 2.17. have I)een tdten from Llte "Strom~~ngrn i n I)antpfkossrln~~lnfcn" by TI. FocLthgcr, Mittcilltngcn tlcr Vercinip~oc! Ilr*.'IUUU:ICrsqelbenit,7.e.r, No. 73, p. Ihl (1!)39).

IGg. 2.16. Frrc stagnation flow witl~o~~tarpn. Fig. 2.17. 1)rcrlrrated 8Lag11:~tiorl flow with

ration, au pliotogmphrtl by Fotttingrr scprntion, ns pllotogrnphed by Focttingcr

I fa1 Anaubr nose 1 I

I (b) Round nose I I

- ( z z 0. %? - 0 - no separation

I

IFig. 2.18. I'low n.l,orrl, n ~ n ~ ( l c l of a motor vrl~inlc (Volltsw:i.gc:n tlclivrry vrm). nftrr 15. Morller 1231. n) Angulrrr noso wi1I1 mpnmtcd flow nlong tho whole of the aidc wall nnd lnrge drag cod- ficicr~t (C,, = 0.70); h) l t o r d iionc with no ~cpnrntion nntl small clrng cocflhic~lt (CD = 0.42)

b. Separation and vortcx formation 3 7

Separation is also important for the lifting properties of nn aerofoil. At small incidence anglcs (up to about lo0) the flow does not separate on either side antl closely approximates frictionless contlitions. The prcssurc distrihntion for slleh a cnsr ( " S ~ I I I I ~ " flow, Vig. 2.11)n) WILR givo11 in Vig. 1.14. Will1 inoron~ing i~tcitlo~~cn t,lrc\rc* is tlangcr of srparnt,ioti on t h sucI,ion side of tho nerofoil, I )ce r~~~so t,l~e l ) rcss~~re i l l . crcnw bccomcs sleepcr. Por n given angle of incidenc~, which is nljout l!jO, ~cparat ion Litinlly occurs. The scpwation point is located fairly closely behind the lcading cdge. Thc wr-kc, Fig. 2.19b, shows a large "(lead-water" nrca. The friclionless, lift-creating flow patter-n has Iwcornc dislurbcd, and the drag has become very largo. The ,he- ginning of scpnrat.ion nwrly coincidcs with the occurrence of maximum lift of the acrofoil.

Structural oerodynomics. Flow around land-bnsed bluff bodies, suc11 as struc- tures antl buildings, is consiclcral~ly more complex than flow around streamlined botlies and aircraft. The principal cause of complication is the presence of the ground ant1 the shear created in the turbulent wind as a consequence. The interaction between the incident shcar flow and the stsruct,ure produces coexisting static and tlynamic loads [8, 9, 101. Tlie fluctuating forces produced by vortex formation and shedding can induce oscillat,ions in thc structures nt. their natural frcql~cncics.

The flow patterns observed on a tlctachcd rectangulnr building is shown sahrmali- cally in Fig. 2.20. I n front of the building there appears a bound vortrx whirh arises from the interaction of the boundary layer in t,he sheared flow (d V/dz > 0) ant1 the ground. There is, furthermore, strong vortex shedding from the sharp corllcrs of the building and a complex wake is created behind it. So far no theoretical mcthotls have been developed t o cope with this ext,remely complicated flow pattern. It is, therefore, necessary to rcsort to wind-tunnel studies with the aid of adequately scalctl models.

3 8 If . 011tli11e of boundary-layer throry

Fig. 2.20. Overall view of flow pat,tmn (schematic) around a rcctnngular st.ruc- ture [MI. a) Side view with foreward hound vortex in the stagnation zonr and a ~cperatod roof lmtntlnry layer; h ) ~tpwitd fme and vortex ~hcdding from the t hn windward rornrr of thr roof

\

Y?\

Fig. 2.21. Acrofoil and cir- cular cylinder drawn in such relation to each other as to produce the same drag in parallel flows (parallel to axis of svrnmetry of awofoil)

circuhr cr/linder: Drag

T o conclude this section, we wish t,o tlisc~iss n particr~ln.rly telling example of enectively it is possible to reduce the drag of a body in n st,rearn w l ~ e t ~ the srl)nrntioll of the boundary layer is completely elirninatrtl antl when, in ntltlit,iol~, the I ~ o t l ~ itsrlf is given a shape which is contlucivc to low rcsist.nncr. Pig 2.21 ill~lstrnt.cs tllr c.i~(:ct, R fnvvrnble sllnpe (strenndine body) on drag: it syintrteLrlc ncrofoil n ~ ~ t l a rirc-lllar c:ylintlcr (thin wire) have brrn drnwr~ hrrc to n relative scdo wllicl~ rtssr1rc:s c:clrlnl tIrng in slwnms of cqnnl velocit,~. The cylinder has a tlrag corfficicnt (:I, % 1 wit,l~ rc?spct, to it,s frontd arcn (scr also Fig. 1.4). 0 1 1 t.hc otllcr hnnrl, l . 1 ~ (Irag cocfficic:t~t.oft I I ( , ;I(.I.o- foil, rcferrctl to iLs cross-seclionnl a rm, has the very low vnl~lc* of f:,, - 0.00(;. 'I'll!: cxt.romrly low tllxg of thc ncrofoil is ncl~icvetl ns n rcsctlt, of n cnrt$r~ll~ cltosc.~~ ,)l.olilc~ which assures llmt the boundnry Inycr rernnins laminar ovcr nlmost t,l~c \vl~olc of its wett.ed Irngth (Inminnr ncrofoil). Tfit,l~is conncxion, Chap. XVf l nt~tl, c!s~)rc.i:tll~, Icig. 17.14, s l ~ o ~ ~ l t l Ije consult.cd.

c. Turhulertt llnw in n pipe and in n bot~ntlnry layer

hlensnren~cnt,s show t11n.t the t.ypc of mol,iorl t l ~ r o ~ ~ g l r n rirwlnr pipr which was calculal.cd in Section l d , and in wl~ich 1 . 1 1 ~ vclocily tlislril)trt.ion w:~s p:wnbolic, exists only a t low and n~odcrnte Reynolds numbers. The fact tha t in thc laminar motion tinder disoussion fluid Inminno slide over each other, and l l ~ i ~ t tllcrc: aro no rndial vclocit.y romponrnt.s, so t.hnt t.he prcsslrre clrop is proportiot~:~l t,o the firs1 power of t.he lncnn flow vrlocit.y, const.itmtrs nn esscnt.in1 c:l~arnrt.rristic: of this t.ypc of flow. This cI~arnrt.rrist,ic of the motion can bc mntlc rlrnrly visil,lo 1,s inlrotl~lcit~g a dye into the st.rmm and by tliscl~nrging i t t l ~ r o u g l ~ a t,llin t ~ ~ l ) c , Fig. 2.22. At, t , l~c motlernt,~ Rrynolds nunlhers associntcd wit,l~ Intnit~nr flow t l ~ e tlyc is visit)lr in l h r form o i a clearly tlefinetl t,l~read ext,cnding ovcr thr. wllolc Irngt l~ of t,hc pip., Fig. 2.22a. 13y increr~sing tlte flow velocity it, is pnssil~lc 1.0 rmch a stngr. .vheii t.hc Ruid pnrtic!les cease to move alor~g st,m.igl~t linrs antl t . 1 ~ rcgrllnrity oC the mot.ior~ brrnks down. l ' l ~ c colourcd Lltrencl bcc:o~nc~ mixed wit,\) the flltitl, its sharp out.li~tc? becomrs blurred ant1 nvcmt.11a.ll.y thc whole cross-srrtioll Iwrotnrs colortrrtl, Pig. 2.221). On t.lw n,xinl n~otion t,hcrc are now s ~ ~ p r ~ . i ~ n j m o t l irrc~gr1l:tr rntlial Il~~ct.rt:~t.iot~s wl~irlt clli.c.t the mixing. Such a flow pnttern is cnllccl l ~ ~ f i u l e ~ r ! . 'l'l~r tljw cxl~crilnrnt was first carried out by 0. Reynolds 1291, who nscertninctl t l~nt , the taansitsic.n honl the laminar to Llle t ~ ~ r h ! c n t t,ypc of motion ttaltcs pl:rcc a t a tlcfinit.~ v:t.lnr of I I I V I

In t,hc t r~r l~u lon t region the pressure tlrop becomes approximately pr~port~ional t,o the square of the mean flow velocity. In this case a consiclerably larger pressure tliffcrencc is requirctl in ordcr to pnss a fixed quantit,y of fluid t.hrol1g11 the pipc, ns corrlparocl with laminar flow. l'his follows from t,ho fact tha t t.ho plrcnomcnoll of t.url)ltlrr~t mixing dissipat,cs a largc q~t:tt~t,it,y of' enorgy which c : ~ ~ ~ s c s the rcsist,:tnc:c? 1.0 Ilow t.o incrcasc considcr:tl)ly. lrurl,llcr~norr, in Ihc casr? of Lurl~ulcrlt, llow t,hc volo-

distritlu(.ion over the cross-scct,ior~al arca is much tnoro cvcn t h r l in hminnr flow. 'rhis circumst,ance is also t,o be explained by turbulent mixing which causes an cxc:hangc of momcntum bctwecn the layers near the axis of the tube and those near t,hc walls. Most pipc flows which are encountererl in engineering appliances occur a t such high Reynolds numbcrs tha t turbrllcrlt motion prevails as a rule. Thc laws of turb~llent motion through pipes will be discrlssed in detail in Chap. XX.

111 a way which is similar to the motlion through a pipe, the flow in a boundary laycr along a wall also becomes turbulent when the extcrnal velocity is sufficient,ly largc. ISxpcrimental investigations into the transition from laminar to turbulent flow in the I,ollntlnry Inyer were first carried out by J. M. Burgers [GI and I3. G . vnll (lcr licgge Zijncrl 1171 as wcll as by M. IIansen [lG]. The t,ransit.iorl from laminar to turbulent flow in the boundary layer becomes most clearly discernible by a sutltlcn a.nd largc increase in the boundary-layer thiclrncss ant1 in the shearing stress near the wall. According to eqn. (2.1), with 1 replaced by the current co- ortlinatc s, the dimensionless boundary-layer thickness 6/1/1'27~; becomes constant for laminar flow, and is, as seen from eqn. (2. la) , approximately equal to 5. Fig. 2.23 contains a plot of this tlimcrlsiorllcss boundary-layer thickness agairlst the IZcynoltls number I J , z / v . At R, > 3-2 x 10" very sharp increase is clearly visil)le, and

Fig. 2.23. Boundnry-layer thickness plob- tedr against the Reynolds number based on'the current lcngth z along a plate in pnrnllel flow at zero incidence, ~s mea- sured by llanscn [I61

as sprn from rqn. (2 1 a). l l r ~ l r c to th r rritiral Rrynoltls r~urn l~r r

there corrcspontls R g crlt = 2800. The bountlary Inyrr or1 :I plate is Inr11in:cr near t . l ~ t : leading edge and bcconles turbulent f~lrt.llcr tlowr~st,rca~n. 'I'llc nbscissn r,,,, of t l ~ t point of lrn~lsit~ion can be clctcrminctl from L11c k t l o w ~ ~ v:~lric of R, .,,,. I n t.llc caso of n plate, as in the prcviot~sly discussed pipc flow, the nun~cricnl vaI11o of R,,,, dcpcntls to a ~narkctl degree on the amount of' tlist.~lrl~ancc in tho nxt,crn:tl flow, :111tj the value R, = 3.2 x 10%hot1lcl be regartlet1 ns a lower limit,. With oxccpt.iorl:~Ily (list-rrrbnncc-frcc cxt.crnal flow, valrlcs of R, ,,,, - 10%rlrltl higllrr 11:~vc been :~tt.ail~rtl.

A 1):~rticul:trly rernarltable phcnorncnon connccld with the transit.ioll from laminar to trlrbrllt:nt flow occurs in tJle casc of blunt llotlics, s11cl1 as c i r c ~ ~ l a r cylintlers or spheres. I t will be seen from Figs. 1.4 ard 1.5 t,llaL the tlmg coef'ficierlt o f a circrtlar cylintlcr or a sphcro suffcrs a sutltlcn :d consitlcral~le dccrcasc Ilr:lr Itcynoltls n ~ i m l ~ c r s 1.' I)/v of bout 5 X lo5 or 3 x lo5 rcspccLive1~. This fact was first, obscrvrtl on sphcrcs by G . 1I:iffrl 1141. It. is a conscquerlcc of t,ransition which causes t.he point of separation to movc clownstacam, l)cca~rsc, in the case of a turbulcr~t 1)ountlary laycr, the accelerating influence of the cxt.crn:d flow extmds f u r l h r due t,o t.t~rbulrr~t. mixing. ~Tcncc the point of separation whicll lies