UNIVERSITY OF MINNESOTA

,~' 5"/31/6 "2-

Permanent File Copy St. Anthony Falls Hydraulic Laboratory

ST. ANTHONY FALLS HYDRAULIC LABORATORY LORENZ G. STRAUB, Director

Technical Paper No. 34, Series B

Unsteady, Symmetrical, Supercavitating Flows Past a Thin Wedge in a Jet

by

C. S. SONG

Prepared for OFFICE OF NAVAL RESEARCH

DEilpartrnent of the Navy Wq:shington, D. C.

Contract Nonr 710(24), Task NR 062-052.

IgIltl,(::~ly H!62 Minneapolis, MinnEilsQtq

UNIVERSITY OF MINNESOTA

ST. ANTHONY FALLS HYDRAULIC LABORATORY LORENZ G. STRAUB, Director

Technical Paper No. 34, Series B

Unsteady, Symmetrical, Supercavitating Flows Past a Thin Wedge in a Jet

by

C. S. SONG

Prepared for OFFICE OF NAVAL RESEARCH

Department of the Navy Washington, D. C.

Contract Nonr 710(24), Task NR 062-052

January 1962 Minneapolis, Minnesota

Reproduction in whole or in part is permitted

for any purpose of the United states Government

A B ST R ACT

Problems of symmetrica~ two-dimensional supercavitating flow about

a thin wedge in a finite fluid with two free surfaces are solved Qy means of

a linearized method utilizing the complex acceleration potential. Taking ad­

vantage of the symmetry, the otherwise doubly conl'1ected region is divided in­

to two identical simply connected regions. Conformal mapping technique is

then applied to the upper half of the flow region. An oscillatory-type motion

as well as general types of unsteady motions are considered.

The soluti.on contains no singularity and, as a result, pressure is

ever,rwhere finite. The mathematical condition required for the existence of

a singularity-free solution leads to an equation which gives the relationship

between the cavity length and the cavitation number. The theoretical results

are in good agreement with experImental data for the steady-flow case, but

data for the unsteady case are not yet available.

iii

-----_._----

Abstract • • • • • • • List of Illustrations. Lis~of Symbols. • • •

• • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • · . . . . . . . . . . . . . . . . • • • • • • • • • • • • • • • • •

Page

iii v

vi

I.

II.

III.

IV.

V.

INTRODUCTION • • • • • • • • • • • • • • • • • • • • • • • •

THE ACCELERATION POTENTIAL AND BOUNDARY CONDITIONS • • • • • • •

THE CONFORMAL TRANSFO~~TIONS. • • • • • • • • • • • • • • • • •

METHOD OF SOLUTION . • • • • • • • • • • • • • • • • • • • • • • Ao Infinite-Cavity Case. • · • • · • • • • • • • • • • • • B. Infinite-Fluid Case · • • • • • • • • • • • • • • • • • C. General Case. • • • • • • • • • • • • • • • • • • • • •

CONCLUSIONS. • • • • • • • • • • • • • • • • • • • • • • • • • •

1

2

7

8 11 15 20

23

List of References • • • • • • • • • • • • • • • • • • • • • • • • •• 25 Figures 1 through 12 • • • • • • • • • • • • • • • • • • • • • • • •• 29 Appendix A - Evaluation of the Function ~(t). • • • • • • • • • • •• 39

A. Oscillatory Motion. • • • • • • • • • • • • • • • • •• 39 B. General Motion with Infinite Cavity • • • • • • • • •• L~o

Appendix B - Evaluation of Integrals • • • • • • • • • • • • • • • •• 45 A. Evaluation of 10 • • • • • • • • • • • • • • • • • •• L5 B. Evaluation of LO and Ll " • " " •• " • • • •• 46

iv

;

L T

L

L

Figure

1

2

3

5

6

7

8

9

LIS T 0 F ILL U S T RAT ION S

Flow Configuration and Linearized Boundary Conditions. . . • •

The Conformal Transformations ••••• • • • • • • • • • • • •

Reduced Cavity Length as a Function of Actual Cavity Length

Page

29

30

and Jet Width. • • • • • • • • • • • • • • • • • • • • • • •• 31

The Effect of Jet Width on the Drag Coefficient for the Infinite-Cavity Case •••••••••••••••••• • • •

The Effect of Acceleration on the Cavity Length (0-= 0.1) • •

The Wash-Off Condition •• • • • • • • • • • • • • • • • • • •

The Effect of Cavity Length on the Drag Coefficient for Infinite-Fluid Case. • • • • • • • • • • • • • • • • •• • • •

The Effect of Jet Width and Cavity Length on the Steady Part of the Drag Coefficient ••••••••••••••••••••

The Effect of Jet Width and Cavity Length on the Added Mass' Part of the Drag Coefficient • • • • • • • • • • • • • • • • •

31

32

32

33

34

34

10 The Effect of Jet \l,Tidth and Cavity Length on the Pressure

11

12

Function, L2• • • • • • • • • • • • • • • • • • • • • • • •• 35

Steady Cavity Length as a Function of Cavitation Number. • • •

Steady Drag Coefficient as a Function of Cavitation Number • •

v

------- -~-~~-----

35

36

LIST OF SYMBOLS

A - Displacement of a body.

A -.. A

A - Amplitude of oscillation. o -t a Acceleration.

b - Jet width.

CD - Total drag coefficient.

Coo - Drag coefficient due to velocity.

CDl - Drag coefficient due to acceleration.

CD2 Drag coefficient due to finite cavity.

CDS Steaqy part of drag coefficient.

E(~, k) - Incomplete elliptic integral of second kind.

F(~, k) Incomplete elliptic integral of first kind.

F(z) , Fer) Complex acceleration potential. -t

F . Force.

f l , f2' Fl , F2, G - Functions.

f - Reduced frequency.

Hcr) Homogeneous solution.

h - Vertical coordinate of wedge surface or cavity wall.

I O' I l , LO' L l , etc. Integrals.

i, j Unit of imaginary numbers.

K(kl ) Complete elliptic integral of first kind.

k, kl - Modulus.

t - Cavity length •

.e - Reduced cavity length.

vi

M .. Mass.

o Order of magnitude.

P - Ambient pressure.

p Pressure.

Pc Cavity pressure. -q - Velocity

qc Speed on cavity wall.

Re Real part of.

S Cavity area.

t Time.

U - Ambient speed.

tl - Horizontal component of perturbation velocity or incomplete elliptic integral of first kind.

v - Vertical component of perturbation velocity.

W - Complex velocity.

x, y - Space variables.

x o Reduced jet width 2n 1

[(exp - - 1)- ].

>- r' z, ~, Complex variables. b

~ Half vertex angle of a wedge.

V Gradient operator.

Ao(O, kl ) Heuman's lambda function.

~ - Real part of complex acceleration.

v Negative of the imaginary part of complex acceleration.

~, 11 Real and imaginary part of r. 2

n( a. , k) - Complete elliptic integral of third kind.

2 n(~, a. , k) Incomplete elliptic integral of third kind.

p Mass denSity.

(T - Cavitation number.

vii

T - A variable of integration.

p - Acceleration potential. ,

'lI( t), 'lI (t) - Function of time.

~ - Conjugate harmonic function of acceleration potential.

~l ~ on wedge surface.

w - Angular speed.

viii

U N S TEA D Y, S Y M MET RIC A L -------- -----------~Q~!~2!!!~!~!~Q !~2~~ ~!~~

A THIN WEDGE IN A JET

J. INTRODUCTJ ON

Recentdevelopments in high-speed sea crafthave stimulated research

workon unsteady super cavitating flows. Unfortunately, unsteady supercavitat­

ing problems are considerably more difficult than their steady counterparts,

and the development of an exact theory on general unsteady flows seems very

remote. The only exact theory known to the writer is that due to Von Karman

on a specialtype of unsteady supercavitating flow [lJ* where the time varia­

ble in the velocity potential may be separated from the space variables and

the rear stagnation point occurs on the free surface. More recently, this

special theory was refined and generalized by Yih [2]. Additional difficul­

ties arise when a finite fluid instead of infinite fluid is considered. For

this reason, attention has been directed mainly to solving problems in infin­

ite fluid by means of linearized m~thods. So much progress has been made by

many investigators in tLis field that it is felt timely to take up a problem

of unsteady super cavitating flow in a finite fluid.

Solutions of st.eady supercavitatingflows with zero cavitation num­

ber under the influence of one or two free surfaces can be fOlmd elsewhere.

For example, the problem of an inclined flat plate in a jet of finite width

was solved by Silberman (J] and the problem of an inclined hydrofoil of finite

aspect ratio undera free surface was treated by Johnson [4]. These problems

were solved by the usual conformal mapping technique. When the cavitation

number is not equal to zero, it is more convenient to use the method of dis­

tributing sources and vortices. An excellent example was shown by Cohen and

Tu [5J, who considered a steady supercavitating flow past a thin wedge in a

jet.

Since the known quantity on a free surface is the pressure and not ,

the velocity, unsteady supercavitating problems can be solved more advanta-

geously by considering the acceleration potential rather than the velocity

potential. It has been shown that if the problem can be linearized by using

* Numbers in brackets refer to the List of References on p. 25.

II, ill

2

perturbation velocities, the acceleration potential satisfies the Laplace equa­

tionand, hence, conformal mapping technique can be applied. This is the meth­

od most frequently used and is also the method to be used in this paper. A

supercavitating thin wedge at zero cavitation number symmetrically placed in

a jet and performing symmetrical motions is considered first. Then, the sim­

ilar problem wi tb finite cavity but in infini te fluid is treated. Finally,

the general case of fin:i.te cavity in finite fluid is considered.

This research has been supportedby the Office of Naval Research of

the U. S. Departmentof the Navy under Contract Nonr 710 (24), Task NR 062-052.

The author is indebted to Professor Edward Silberman, who has critically re­

v'iewed the present work. Mr •. Frank Tsai is credited with carrying out some

61'the integrations and Mr. Paul Edstrom has carried out most of the numeri­

cal computations. The manuscript was preparedfor printing by Marveen Minish

and Marjorie Olson under the general supervision of Mr. Loyal Johnson.

II. THE ACCELERATION POTENTIAL AND BOUNDARY CONDITIONS

Newton's equation of motion upon which fluid mechanics stands is

--t --t

F = M a (1)

...... --t

where F, M, and a are force, mass, and acceleration, respectively. In

particular, when the mass distribution is uniform and the system is conserva­

tive, the force per unit mass is derivable from a scalar potential function,

and

CIa)

Itis thus clear that an acceleration potential function exists when

flow is incompressible and frictionless. For a gravitation-free field, the

Euler equation of.motion is

...... aq 1 f! = -- + rq . V )q = - - Vp (lb)

at p

where q is the flow velocity, p is the pressure, p is the density of

the liquid, and t is the time.

3

For a steady, irrotational motion, Eq. (lb) may be integrated to

yield the Bernoulli equation

1 2 p +-p q

2

1 ::: p + _p U2

2

where P and U are the pressure and the speed at infinity, respectively_

Furthermore, for flow containing a cavity, it is known that the pressure in­

side a natural cavity is constant, for example p, except within a small c

region nearthe end of the cavity. The speedon the cavitybOlmdary, qc' is

q =U~ c (2 )

where (f is the cavitation number defined as

1 (f = (p - p )/(-p if)

c 2

For steady flow problems, it is customary to use rather than U as the

reference speed since the flow near the body is of interest. For general un­

steady flows, however, the speed on the cavity boundary is not a constant,

but is a function of time and space. Nevertheless, it is still possible to

use the steady flow value, qc' of Eqs. (2) and (3) as the reference speed.

With this understanding, the velocity of flow q(x, y, t) may be written as

q(x, y, t) = (1 + u, v)

and the Euler equation of motion may be written as

-aq

at

where

P - P

2 P qc

(4)

(5)

(6)

" ,I , ji,l ,'I, I

I,

',1

I" \

II I

I , ,

:i,1 "I , I II, ,', 1:[ 1,1

Iii

ii'

'The addition of P in (6) was possible because P is a constant. This is done merely for convenience.

It is further assumed that the body plus cavity is thin and the mo­tion is such that both u and v are much smaller than unit yin most of the flow region. Then Eq. (5) may be linearized to

¢ -1 = u + u x qc t x' " -1 jU =q vt+v y c x (7)

Hence, it follows from the continuity equation V'. q = 0 that

(8)

Itis thus shown that the linearized acceleration potentialis a harmonic func­tion. This implies the existence of a conjugate harmonic function if; (x, y, t) such that

wi th aj'q 2 = (J.L, v). Therefore, the complex' acceleration potential and the c complex conjugate acceleration defined as

F(z, t) = ¢(x, y, t) + i if;(x, y, t)

aF(z, t) q -2 a(z, t) = c J.L(x, y, t) - iv(x, y, t) = ----

az

are analytic functions of z = x + i y.

Also, ~. (7) may be written as

aF 1 a a = (-- -- + --) W(z, t)

az qc at az

with If denoting the complex conjugate velocity

W(z, t) = u(x, y, t) - iv(x, y, t)

(10)

(11)

It is readily seen that for the steady flow case, F(z) is identical with W(z) up to a constant of integra.tion which may be set equal to zero.

The perturbation problem of general, lIDsteady, supercavitating flow

is now reduced to the problem of finding the analytic flIDction F(z, t) reg­

ular everywhere in the flow field satisfying given boundary conditions. Of

eourse,.it is assumed that there exists a basic steady flow of characteristic

. speed q and that the perturbation speeds defined by Eq. (4) are small com-e

pared to unity in most of the' flow region. Here, the lIDsteady motion is as-

sumed to be due to a perturbation velocity, dA/dt, of the body along the

direction of its basic motion. The boundary conditions are listed below.

(Also see Fig. 1.)

(1) On the free surface, p = P, hence from Eq. (6)

(2 )

~ = 0

On the cavity boundary, p = p c'

(3), and (6)

hen ce from Eqs. (2) ,

(f

~=----2 (1 + (f )

(3) ·When a wedge performs a general motion along the x"':dire c­

tion, the equation of the solid boundary wri.tten for a

coordinate system fixed in the basic steady motion is

hex, t) = ± -r [A(t) + x]

where h is the vertical distance, -r is the half-wedge

angle, and ACt) is the distance between the nose and the

origin of the coordinates at any instant.

The y-componentof the nondimensional velocityat the sol­

id boundary is, up to the first order approximation,

v = q -1 ht + h = ± -r[l + A/q ] c x c

. Note that the velocity A = dA(t)/dt should be small com-

pared with the reference speed, qc' in order that the

linearization be valid, whereas the acceleration A need

not be small. Now, by Eqs. (7), (9), and (15), it fol­

lows that

(12)

(14)

(15)

II11

11\

I 1

i 1

II III II !

11'11 .dl illl il II I Ii II II :1 ~ : II . I

I

I

I II I!

o

on the wedge. By integration,

I where ~ (t) is an arbitrary function of time to be de-termined later. Equation (16) is the boundary condition on the wedge surface.

(4) Since the flow at x = :I: (X) should not be disturbed by the unsteady motion of the wedge, the complex accelera­tion potential must be a constant at x = ± 00. On line oc (y=O, x<-A) andline ef (y=O, x>.e-A), \

V :: 0, by symmetry. Hence,

'" ;::: 0

Here, the constantis chosen to be equal to zero to agree with the steady flow condition. The only exceptional case J_S the case when (1;::: O. In this case, .e -+- + (X) and '" at + 00 is equal to 'V at + (X) which is not equal to zero.

(16)

(17)

The aforementioned boundary conditions are just enough to determine . , the complex acceleration potential uniquely, provided that the function ~ (t) is known.

1 The arbitrary function ~ (t) should be chosen so that the kine-matic conditionon the wedge surface is also satisfied by the velocity field. The integral of the se cond equation of (7), satisfying the conditions tha t v = 0 at x;::: - (X) is, Ref. [1),

x

(18)

Equation (18) is the additional condition requiredto solve the problem (kine­matic and dynamic) uniquely.

7

It should be noted that the so-called closure condition [6, 7, 8]

or the second additional condition [9] is not necessary_ For an unsteady

flow, the cavity boundary is not a material line and the closure condition

would be written

f ah

--dx=O ax

(19)

where the integration is meant to be carried over the closed boundary of the

boQy plus cavity. Upon substitution, Eq. (19) leads to

f ah . J dS -- dx = q vdx = --at c dt

(19a)

where S is the cross-sectional area of the cavity. No additional inforn~­

tion can be derived from Eq. (19a). Wu [10] assumed constant-cavity volume

(dS/dt = 0) but this was unnecessary.

III. THE CONFOR}ffiL T&hl~SFORY~TIONS

The linearized physical plane is shown in Fig. lb. In this case,

the flow is symmetrical with respectto the x-axis and only the region y ~ 0

need be considered.

The Schwarz-Christoffel transfornmtion

1 Z = -- log r - A

b (20)

2n

transforms the upper half strip of the linearized z-planeinto the upper half t

of the r -plane as shown in It'ig. 2a. It is more convenient to work on the t

r-plane which is relatedto the r -plane by the following linear transforma-

tion.

t 2n r = U - 1) / (exp - - 1)

b (21)

The ·resulting transformation does not alter the body size but transforms the

points at - (Xl to (-x, 0) , the nose of the wedge to (0, 0) , and the o

8

I I tail of the cavity to lowing formula s.

(.e, 0). Here, x and.e are given by the fol­o

2n -1 x =: (exp - - 1)

o b' (22)

I .e .. x (exp -- - 1) o

2il.e (23)

b

I where .e may be called the reduced cavity length and lndlcates the extent of free surface effect on cavity length, .e. Equation (23) is plotted in Fig. 3.

I It is interesting to riote that when b-+ 00, then x -+ ro and .e -+.e, o the net result of the transformation is only to shift the origin to the nose of the wedge. Thismeans,for b-+oo, the transformation Eqs. (20) and (21) transform the flow into itself. Hence, it is obvious that the transformation indicates the effect of a finite stream. It should be further noted that the transformation does not specify the boundary condition and, hence, it may be applied both for the flow in a jet and for the flow in a solid channel. How­ever, the boundary conditions are written on the figure for the jet.

IV. METHOD OF SOLUTION

It is required that an analytiC function F( n be found which is regular everywhere on the upper half of the {-plane and satisfying the follow­ing condition on the real axis.

(1)· ~ < - x, ~ = 0 o

(2) -x<~<O ..p=0 o '

(3)

(4) I CT

l<~<.e, ~=----2(1 + 0")

(5) I

.e < ~, ..p = 0

(6) ~ .. :!: 00, ¢ = ..p = 0 except when .e = 00

(24)

On the r -pla.ne, the fl.lnction "'I is

where

f ..

b A

"'I = -'Y 2 2nq c

A log (1 + ~/xo) + 2 ---- + 1

qc

.. '1'( t) = J< t) - A t

qc

•

9

(16a)

(25)

The unknown f1.lnction wet) is shown to be equal to -A/q in Appendix A for c the special cases when the motion dA/dt is of an oscillatory type along the

x-axis or when the cavity is infinitely long.

To solve the mixed boundary value problem of the type of Eq. (24),

the generalized method of solution due to Cheng and Rott [11] may be applied.

The first step is to find the homogeneous solution H(r) satisfy­

ing the following homogeneous boundary conditions:

(1) ~ < - x 0' ¢ = 0

(2) - x < ~ < 1, '" = 0 0

, (24a) (3) 1 < ~ < e , ¢ = 0

(4) ,

e < ~,

Since the pressure must be finite everywhere, the only possible homogeneous

solution is

(26)

Suppose the complete solution of the given problem with boundary conditions

of Eq.'(24) is F(n - ¢(~) + i¥t(~). Then the imaginary part of the function

Gcn defined by

F(n G(t) ,. a p(~, 1]) + i q(s, 7])

H(O (27)

10

is completely specified on the real axis. That is,

-i on the part where ~(s) is known q(s) =

on the part where ~(s) is known

Thus, the problem is reduced to a regular boundary value problem for the func­tion G(r). The solution is

+00

G(r) = -=- f q( T)

dT n T _ r (28)

-00

or by Eq. (27)

+00

F(n = -=- H(n f q( T) dT

n T ... r (29)

-00

provided that F({) satisfies the boundar,r condition at infinity [11]. For the present problem,

(24b)

2(1 + 0")

J ______ 1;.:.-. ___ ,,--__ , 1 < T < t t

~ ( T + X ) (T - 1) (t - T) o

q( T) .. 0 for all other values of T

By substitution, Eq. (29) reads

11

1 ~ . I f 1/11 (T) dT [

1

;or - (r + xo) ( r - 1) (r - f, ) . _

n . 0 ~ (T + xo) ( T - 1) (T - f, ') (T - r)

I

(f $. ] i . . dr ..

1 ~ ( T + xoH T - 1)($ I - T)( T - n (29a) +----

Equation (29a) is an elliptic integral. The resulting drag coef­

ficient will be discussed in the latter part of the paper. For the special

cases of infinite cavity or infinite fluid, the equation is reduced to elemen­

tary integrals and their values can readily be computed.

A. Infinite-Cavity Case ,

Since .e -+ 00 and (f = 0, the complex acceleration potential is

.. ~ ( r + x ) ( r - 1) o

Here, .. q is equivalent to U • c and the function i' is determined to be

-A/qc (see Appendix A).

Since p = P, the drag coefficient may be written as c

1 - A

drag J CD = 1/2PU2 = -4Y ~(x. 0) dx

- A

or in the r-plane

12

21'b

n

where Re means ureal part of."

1

1 Re F(!;)

o

d!;

!;+x o

By substitution it follows that

where

81'2 A

°DO = (1 +-) bI U

0 n

81'2 .. A 2

°D1 = --"2 b 11 n U

with 10 and 11 meaning

1 10g(1 + T Ix )d r 0 =

(31)

(32)

(33)

(4)

di; (36)

1 t 1

1) ] ~ Re ~(!; + x o)(!; - 1) J 11

8n2 (T - t[,)~ (T + xo) ( T - t[,+x o 0 0

The double integral Eq. (35) can be integrated and yields (see Appendix B for

the detail of computa tion)

1 1 I = -- (arc sin )2 o 2n ~1 + Xo

(37)

It should be noted that when flow is steady A = A = 'II = 0 and the

drag coefficient given by Eqs. (33) and (37) is identical to the steady-flow

solution given in Ref. [5]. It is also interesting to note that for infinite­

fluid case

lim

b ~oo

bI = 1 o

13

(38)

and Coo is reduced to the steady-flow solution given by 11. P. Tulin [6].

B.1 changing the order of integration, Eq. (36) is reduced to

1 arc sin ----

1

J log(l

o

~l + x o

1 ( dr J log(l + r/x )

o ~ (r + xo)(l - r) o

dr (J6a)

Integrating the second integral by parts, Eq. (36a) is changed to a more con­

venient form for numerical computation

I :: 1

where

When Xo > 1, 12

The results are

1

( 2 arc sin

1 ~l + x 0

8n2

~o (36b)

1 f log(l + ~/x )d~

12 = J ~ (1 + ~/x ) (~ _ ~) o 0

(39)

1 2

~ log (1 + six )d~ I = 1/2 . 0

3 ~ (1 + ~/x ) (1 - ~) o 0

(40)

-1 may be expanded into infinite series in x • Q

14

4 4 23 176 1,126 ,12 (1/3 - + +

3,46.5x 4 . ) =-

2 945x 3 • • x l.5x 10.5x

0 0 0 0 0

(39a)

2 2 43 76 12,139 13 = (1/3 - + +

3l,18.5x 04

. . . ) 2 lO.5x 0

2 Id9x 3 x 5x 0 0 0

(40a)

For the infinite-fluid case, calculation shows that

lim

b ~oo

-The drag coefficient for the infinite cavity in infinite fluid is, therefore,

• •• A A.

GDoo = -- (1 + - + ---y-) n U U

(42)

For other values of x , x ~ 1, Eqs. (39) and (40) were integra-o 0 ted numerically. The effect of jet width on the drag coefficient is shown in Fig. 4 where both bIo and b2Il are plotted as functions of b. It should be noted that at the other limit where b = 0 both bIO and b2Il are equal to zero.

This section can be concluded by remarking that the added mass of a supercavitating wedge at zero cavitation number is

2 -- p b II

which takes the maximum value of plies that the added mass effect is tion

n

4'Y2 n p in infinite fluid. It also im-

important when the dimensionless accelera-

.. AG

7 (43)

is not too small compared to unity. Here C is the wedge chord. As an ex­ample, if a one-foot-chord wedge is moving at 60 ft per sec, the acceleration

15

term of the drag force can be equal to the steady force when the acceleration

is equal to 111 g.

B. Infinite-Fluid Case

It is readily seen that the transformation given by Eqs. (20) and

(21) degenerate to a parallel translation when b approaches infinity. Hence,

the problem could be solved either on the r-plane or on the z-plane. Since I

$ = .$, the acceleration potential on the r-plane is

F(n • - -=-~(r - 1)( r_ $) 1t ~( r- 1)( r- $)( r - n

(44)

•• 1 A f rdr

J ~(r-l)(r-$){r-n o

(f J -r======-==-d_r --1 ( r - 1)($ - r)( .,. - n

1

When the perturbation velocity A is an OSCillatory motion, '11 is found to . be -A/q (see Appendix A). Only this type of motion is considered here­c after. In order that the acceleration potential given by Eq. (44) satisfy

the conditions at infinity, it is further required that

1 .. 1

A 1 d1' A

S rdr

'Y(l + -) +'Y 2 q ~( T - 1)( r - $) qc ~( r - 1)( r - $) c 0 0

.e (f J

dr = 0 (45)

2(1 + (f) .oJ( r - 1)( 7'" - t) 1

Performing the integrations, this leads to the following forrnulc: which re­

lates the cavitation number and the cavity length.

16

A -{t + 1 A [-{i 21'(1 +-) log +1' 2

qc ..Jt-l qc

~+l ]- nO' + (t + 1) log ::: 0 (4Sa)

..J-e-l 2(1 + 0')

For harmonic oSCillations, by writing A = A sin wt, the equation is changed o to

0' 41' f cos wt)

..Jt + 1 ::: (1 + log

1 + 0' n ..J {- 1

21'12 [-V + (t + 1) .ft+ 1 ] sin wt log

nA ..J e - 1 0

where A = dimensionless amplitude normalized with the chord, and o

A w o --= reduced frequency.

(4Sb)

In particular, when the flow is steady, then A::: A 0 and the condition

(LSa) is simplified to

0' 41' ~ + 1 --- ::: -- log r:==:.-

n ~t-l (4Sc)

1 +, 0'

It can be noted that Eq. (LfSC) is almost identical to an equation developed

by Fabula for his open-cavity model (12J.

To show the effect of acceleration on cavity length, Eq. (L.5b) is

plot/ted in Fig • .5 for the case of u ::: 0.1 and 1'::: IS degrees. The upper

graph shows that the amplitude of cavity-length oscillation increases if A . a decreases when f is kept constant. Two limiting cases when f> > i jA ,

~ ~ 0 small added mass effect, and f < < f jA , large added mass effect, respective-a ly are shown in graph b. The 90-degree phase shift between the two curves

should be observed. The following explanation is offered to facilitate better

understanding of the effect of acceleration on cavity lengtho

...,

-It is known that the acceleration of a solid surface into

a fluid creates a positive hydrodynamic pressure on the

surface due to the added mass effect of the fluid. This

pressure impulse is propagated in all directions with the

speed of sound (which is infinite when the fluid is in­

compressible). Negative pressure is created when the ac­

celeration is. away from the fluid. When the body under­

takes an oscillatory motion, as in the case considered

here, a continuously varying pre ssure field exists. Since

the pressure on a free surface remains constant, the var­

iable pressure field has a similar effect as a variable

cavitation number. Therefore, it is expected that the

cavity is longer when the acceleration is toward the cav­

ity than when the acceleration is toward the liquid. This

is clearly shown by the first curve of Fig. 5b.

17

Moreover, Fig. 5a suggests that the cavity may disappear at a cer­

tain instant if the acceleration te);'m is of such magnitude that the minimum

.e is less than or equal to 1. This is the condition for which the cavity

is t~washed off lt the body. Thus, the wash-off condition, by setting .e = 1

in Eq. (4~b), is

[1 + f cos

This equation leads to

re t-­

A o

",2 f

sin wt]. = 0 nun

A = -----

o -Vl-f'2 (46)

Note that the wash-off condition (46) is independent of q and 'Y. Equation

(46) is plotted in Fig. 6. Since the linear theory is not accurate for small

.e, Eq. (46) must be applied with caution. The drag coefficient is

c = D

1 - A

S (p

-A

1 - A

pc)dx = 2'Yq - 4'Y(1 + q) f ~(x)dx -A

18

or on the r -plane

1

CD = 2'Yq - 4'Y(1 + q) J Re F( l;) dl;

o

Substituting Eq. (44) into Eq. (45a) , it follows that

",here

• A

Coo = -- (1 + --) (1 + q) LO

and

1 [1 La - 1/2 Re 11~ (~ -1) (~- $) ! d1"

~(T - l)(T - t)(T - l;)

1 ~1 L1 = 1/2 Re \ ~ ~ (~ - 1) (~- $) ~ Td.,-

~( T - 1)( T - $)( T - l;)

1

(~ - $) [1 L2 = 1 - ~ J ~ (l; - 1) d1"

~( 1'" - 1) (t - T) ( T - l;) 0

(47a)

(48)

(49)

(50)

(51)

1 d~1 (52)

1 d~ (53)

1 ds (54)

The details of the evaluation of the fil:'st two integrals are shown in the Ap­pendix. The results are

{i + 1 1 L =~log .. +-

1 ~t-l' 4 r -($ - 1)

~ direct integration,

L = 0 2

19

(52a)

log -;===- 2 ~ +1 ]'

vt - l (53a)

(54a)

It is readily seen that when (I = 0 and t-+ 00, Eq. (48) is re­

duced to Eq. (42).

Wu has worked out a problem of an accelerat'ing wedge in infinite

fluid [13]. He considered a case wherein a wedge was given a sudden acceler­

ation of magnitude a at t = 0 and maintained the same acceleration there­

after. He assumed that for small t the velocity potential may be expanded

into a power series of t and, hence, the drag coefficient would be

He then concluded that

+

8,2 Coo = - (1 + (I) (---)

n t - 1

2 ..J t - l ] {t+l - '($ - 1) log _r: . iog_f:

~ t - 1 ~t - 1

Q (-{t --)

'Ya

..Jt-l + 1) log w

-fi - 1

(55)

(56)

(57)

where Q is the rate of cavity-volume increase assumed to be a consta:tlt. He

did not compute CD2 and higher-order terms. It is readily seen that the

present solution agrees with Wu I s solution, (56) and (57), up to the order of

,e-l/2 (order of (I) if Q is set equal to zero.

20

To give an idea of the relative importance of the cavity-length effect, the functions LO and ~ are plotted in Fig. 7. It is seen from Fig. 7 that the cavity-length effect is very small for a long cavity and this is the reason why most of the linear theories give reasonably good prediction of the drag coefficient even though the cavity length is not accurately pre­dicted.

C. General Case

Here, the general case of a finite cavity in a finite jet will be considered. The complex acceleration potential in the r -plane is given by Eq. (29a). Moreover, to satisfy the boundary condition at infinity, F(±oo):: 0,

__ the following equation must hold.

1 dr f \]11 ,"'"

o ..J(e'_T) (1 - T) (T + X ) 0

I t

0- f ~ ($'

dT + = 0

2(1 + 0-) _ T) ( T - 1) (T + X ) 0

Equation (58) may be simplified to

. 21'(1 + A/q ) F(fP, k) c

------- = ----------~----------1 + 0-

'YbA + ---2""'---

nqc K(kl )

K(kl )

1 f log(l + T/X )dr

J -;:=:::::=========o======­..J ~ t I _ r) (1 _ T) (T + X ) o 0

where F(fP, k).= incomplete elliptic integral of first kind, K(kl ) = complete elliptic integral of first kind,

If' = arc sin

I X +.e o

I I .e (x + 1 ) o

argument,

(58)

(58a)

k = 1 +- x

o --="', --- = modulus, and f, + x

o

~ t - 1 kl = I = modulus •

.e + x o

For steaqy supercavitating flows, Eq. (58a) is reduced to

u ---.=-----

21

(59)

The drag coefficient can still be written as the sum of three terms,

as indicated by Eqs. (48) through (.?l). How"ever, the meanings of 10 , 11'

and 12 are different from those given by Eqs. (52), (53) , and (54). The new

quantities are given by the following double integrals.

1

H(~) ~ 1

b J dt-

10 =-Re dl; (60) 4n l; + x (r - l;) H( r)

0 o 0

~ " ~!~ J H(s) ~ log (1 + r/x ) d 1 ds \

0 (61) 8n 0 l; + Xo 0 ( r - s) H(r)

1 b J H(l;) dr

1 == 1 -2 [fJ (t' 1 ds (62) 2n2 l; + x

0 0 _ r) ( r - 1) (T + X ) ( r - l;)

0

where the function H is defined by Eq. (26).

Following a method similar to that used to evaluate Eq. (52) and

shown in AppendixB, 10 of Eq. (60) is found to be

10 = ~ F(CP, k) [2E(CP' 2n

, f, - 1

k) - __ ,=----- F( cP , k) - 2 .e + x o

-f,"-:'-( f,_x~o_+-x-o-) } 600)

22

where E(~, k) = incomplete elliptic integral of second kind.

It is readily checked that when b--+ (X) Eq. (60a) is reduced to Eq. , (52a) and when t --+ 00, it is reduced to bIO of Eq. (33).

The exact value of Ll is rather difficult to compute. Howeve~, since t'·~ t > 1, the integral may be expanded into power series of lit • The result is

where 0 means order of magnitude.

,-2 + oCt )

1 ) a rc sin --;:::=:::::;::­

-Vxo + 1

~61a)

The function L2 ting once with respect to integrals (14].

reduces to the following expression after integra­T and by using an identity formula of elliptic

(62a)

(3

where Ao «(), kl ) = Heuman's lambda function,

fJ • arc Sin~ 1 :OXo ' and

() = variable of integration.

The function 10 , which indicates the effect of jet width and cav­ity length on'the steady part of the drag coefficient, is plotted in Fig. 80 It is noteworthy that, for any given jet width and relatively large t, the effect of cavity length on drag coefficient is very small. The added mass function 11 is computed up to the second term and the result is plotted in Fig. 9. Since the second term is positive and is a monot0nically increasing function of t-l , the added mass is greater when the cavity is shorter. The origin of the function L2 may be traced back to the fact that the pressure

- --------~~~ -- ~~--------

2.3

in the cavity is different from the ambient pressure. It varies from one to

zero as the jet width changes from zero to infinity. This statement has the

following physical significance: for a very narrow jet, the pressure on the

wedge surfaces is almost equal to the ambient pressure and the drag is almost

equal to the pressure differen,ce between the ambient and the cavity. The func­

tion L2 will be called the "pressure function. tI The pressure function ~

is plotted in Fig. 10. It can be observed from Fig. 10 that L2 is a weak

function of cavity length.

There is no theoretical result or experimental data available to

check the unsteady part of the pre,sent theory. Only the steady part of the

solution could be verified by experimental data. The steady cavity length

given by Eq. (59) is plotted in Fig. lla for four different jet widths. Also

Shown in the same figure are Cohen and Tu's [5] curve for b = 10 case and

Wu's curve based on infinite-fluid theory. Some new data were taken in the

free-jet turmel of the St. Anthony Falls Hydraulic Laboratory for two wedges

of different chord and wedge angle. These data are quite similar to those

obtained previously in the same tunnel with a less sophisticated dynamometer

[.3] and are compared with the theor,y in Fig. lIb. Considering the fact that

the cavity length has always been difficult to measure and predict, the agree­

ment is fairly good. The steady part of the drag coefficient given by the

present theory is plotted in Fig. l2a for a IS-degree wedge. Cohen-Tu and

WU IS results are very close to the present result and they are not shown in

Fig. l2a. Figure l2b shows the comparison of the data with the theory. Here, ~k

the agreement is very good.

v. CONCLUSIONS

A linearized acceleration-potential method has been applied and a

singularity-free solution has been obtained for a problem of unsteady super­

cavitating flow about a wedge in a finite jet when the flow is symmetrical

with respect to the x-axis. The most important conclusion that can be drawn

from the present study is the fact that a singularity-free solution (finite

* For a steady infinite cavity an exact theory due to Siao and Hubbard is available [.3]. For ~ = 15 degrees and b = 10, CD = 0.140 by this theory andthis result maybe compared with the present theoryinFig.12b. (CD = 0.140 is somewhat larger than the result quoted in Ref. [3], a numerical error hav­ing been found in the computation for that reference.)

i

I

I

24

pressure everywhere) exists even with linearization. The existence of a sin­

gularity-free solution requires a new condition (mathematical but not physical)

which gives the relation between the cavity length and the cavitation number.

By this method, not only the need for the additional physical condition or

the closure condition is eliminated, but also the process is considerably sim­

plified. Since a steady-flow problem is a special case of an unsteady-flow

problem, and in view of the good agreement between the theory and steady-flow

data, the present method can also be applied to solving other steady, super­

cavitating-flow problems with advantage.

[1]

[2]

[3]

[4]

[5]

[6]

[7J

LIST OF REFERENCES

Von Karman, Theodore. "Accelerated Flow of an Incompressible Fluid with Wake Formation, II Annali di Maternatica Pura ed Applicata, Ser­ies IV, Vol. 29, pp. 247-249. 1949. Collected Works, Vol~ IV, pp. 396-398.

Yih, C. S. UFinite Two-Dimensional 9avities," Proceedings of the R~~al Society of London, (A), 25tl, 1292, pp. 90-100. October 19 o.

Silberman, Edward. "Experimental Studies of Supercavitating Flow about Simple Two-Dimensional Bodies in a Jet," Journal of Fluid Mech­anics, Vol. 5, Part 3, Appendix B. 1959.

Cohen,

pages.

Hirsh and Tu, Yih-O. A Comparison of Wall Effects on Supercav­Hating Flows Past Symmetric Bodies in Solid Wall Channels and Jets. Rensseler Polytechnic Institute, RPI Math Report No. 5. October 1956. 13 pages.

Min, M. P. Steady Two-Dimensional CavitrloWS About Slender Bodies. David Taylor Model Basin Report 34. May 1953. 29 pages.

Parkin, B. R. Full -Cavitatin H drofoils in Nonstea fornia Institute of Technology, Report No. 74 pages.

Motion. Cali-5-2. July 1957.

[8] Timman, R. itA General Linearized Theory for Cavitating Hydrofoils in Nonsteady Flow ,It Second Symposium on Naval Hydrodynamics, edi­ted by R. D. Cooper, Office of Naval Research. August 1958.

[9] Geurst, J. A. A Linearized Theory for the Unsteady Motion of a Super­cavitating HydrQfoil. Institut voor Toegepaste Wiskunde, Technische Hogeschool,Delft, Nederland, Report No. 22. April 1960. 83 pages.

[10] Wu, T. Yao-tsu. A Linearized Theory for Nonsteady Cavity Flows. Cal­ifornia Institute of Technology, Report No. 85-6. September 1957 • 28 page s.

[11] Cheng, H. K. and Rott, N. IlGeneralizations of the Inversion Formula of Thin Airfoil Theory, II Journal of Rational Mechanics and Anal­ysis, Vol. 3,No. 3. 1954. Indiana University, Bloomington, Indiana.

[12] Fabula, A. G. A lication of Thin-Airfoil Theo to drofoils with Cut-Off Ventilated Trailin~ Edge. NAVWEPS Report 7 71, NOTS TP 2547, September 1960. 0 pages.

26

[13] Wu, T.

[14] Byrd, P. F .. and Friedman, M. D. Handbook of Elliptic Integrals for Engineers and Physicists. Lange, Maxwell, and Springer Ltd., New York. 1954.

FIGURES -------(1 through 12)

29

y

p=p 0'

b a

b '2

u c 0 h (x,t) '>

p= Pc e f x

b ~I~ i: '2 ht) =1 II

! ~ p=p

(a) True z - Plane

y

0' 4>=0 b a

cr

1J1=0 1J1=1/I, d 4> = 2(1+cr) 1J1=0 o I e f c x

S Atl) ~ ----1 ~(t) -I

4> :0

(b) Linearized z - Plane

Fig. 1 - Flow Configuration and Linearized Boundary Conditions

30

a

a

'1]'

cp=O 0/=0 -I" 0/='" , o· b, e 0

r-- I I~ 211"

ell 21Tl

ell

(a) ~' - Plane

cp = 0 --r+- 1/1=0

x = o

b,c

I-- Xo --·t4<---- I --121ft

2 Xo ----.!.r4--111 - i' = e2: -I eli -I

2w eb-I

(b) ~ - Plane

+ d

~I

Fig. 2 - The Conformal Transformations

cp = 2(~CT) + 0/ =0

e f ,(

~

f

"0< ..

.£: -0' C

~ ~ > 8 "C Q) (,) ::I

"C Q) a::

1-1

N.o

0 1-1 .J:l

40

20

10

6

4

2

1.0

0.9

0;8

0.7

2 4 6 8 10 20 40 100

Cavity Length, l

Fig, 3 - Reduced Cavity Length as a Function of Actual Cavity Length and Jet Width

~ F-IJ ";:7

~

1 ~ ,...;

b2 I I/

/ ViIa I ~ 0.6

0.5

V /

0.4 If II

0.3 I 2 4 10 20 40 100 1000

Jet Width b

Fig. 4 - The Effect of Jet Width on the Drag Coefficient for the Infinite-Cavity Case

31

32

26

24

22

20

.18 ~ -CI

Ii 16 ..J

:t:­.:;; 14 c (.)

12

10

8

6

~

~~-\

.........

~ ~ --- ---~ ~ ...........

i\ -............ -\

"'-r--V

r-. I / \.3 Ao=0.02

/ \ / 1\

/ 2'A"O.!~ :! /l~ --- __ I --- V ~~~ ---

// ~ V Steady Cavity

1 V (a) f =0.1 Case

18'---~--~--~--~--~--~--~--~---T--~--~--~ -.c 16 j----t---''''''-:-;!.-. -CI c: j 14~~r===~~~==;=~~~~===r===f~~==~~~~~ >.

~ 12t---t---tl==f===~~~~jt~~~~---+---+---1--~ (.)

100 30 60 90 120 150 180 210 240 270 300 330 360 Phase, wt

Fig. 5 - The Effect of Acceleration on the Cavity Length (0- = 0.1)

5

2 V-

.,,/ -2 IxlO 0-4 2 I x I .

, ,

/'" I--'

Wash-off Region ,,/V

~- 2- I"

Note: Three Points ,,/

Correspond to Finite Cavity Region Three Conditions

IIII1 I in Fig. 5 (a) - - -5 Ixl0 3 2 5 Ixl022 5 IxlOI 2

Dimesionless Amplitude, Ao

Fig. 6 - The Wash-Off Condition

2.2

2.0

...J

-g c o

1.8

1.6

...J 1.4

1.2

1.0

0.90 0.1 0.2 0.3 0.4 0.5 0.6 0.7 I

T Reciprocal Cavity Length,

Fig. 7 - The Effect of Cavity Length on the Drag Coefficient for Infinite-Fluid Case

0.8 0.9 1,0

'vJ VJ

2.0 I 1.6

b= en

1.8 1.4

1.6 1.2 , , I

I

1.4 1.0

0 C\l,o

...J 1.2 I 0.8 J

1.0 0.6

L

/ II

b=i

/ 0.8 0.4

0.6 0.2 //

/' ~ ~ I~ ------OA 08 o 0.2 0.4 0.6 . 1.0

o o 0.2 0.4 0.6 0.8 r.o Reciprocal Cavity Length,-t

Fig. 8 - The Effect of Jet Width and Cavity Length on the Steady Part of the Drag Coeffi c ient

Reciprocal Cavity Length,t

Fig. 9 - The Effect of Jet Width and Cavity Length on the Added Mass Part of the Drag Coefficient

1.6

1.4

1.2

1.0

C\l ...J 0.8

0.6

0.4

0.2

a

b=O

I

1

b .. 2 --- r- b=5 I

~

I---I

---- i ------- b=IO -- r----. b=20 :---~ -

b= en - -o 0.2 0.4 0.6 0.8 1.0

Reciprocal Cavity Length, t Fig. 10 - The Effect of Jet Width and Cavity length on the Pressure Function, l2

\JJ .r=-

-.-.~".~i!li

0.5

014

b

.: 0.3 CD

.Q

E ::s ;Z

c: 0

+= 0.2 0 .-'> 0 0

0.1

0.5

014

b .: .8 0.3 E ::s

Z

c: o

~ 0.2 o o

0.1

35

co .. ~ L _ E 59 2,b=10 A q. 3,b=5

~ V 4,b=2 A / --- Wu, b=<» '

~ / --- Cohen-Tu,b=IO ~ /

/ /' d ~ V L

if' I ~ V V l / /

/ I ~ ~ / V

I A 1/ I

~ V V I I ./

i / ~ V V V

/ , I

~ V V /,~ ./

~ V (0) Comparison of Theories for r=15°

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Reciprocal Cavity Length, t v

0 r = 15°, b=IO V t--- 0 r =7.5°, b=5

V - Eq.59 /'

V /

/ V

(")

/V 0

l.....--::: f-'

0

00 /

V V ~ n ........-0 y V V-0 1M --v: 0 V ~ d J::j0 3--

It y y (b) Comparison of Theory and Data

·1 I 0.1 0.2 0.3 0.4 0.5

Reciprocal Cavity Length, t 0.6 07

Fig. 11 - Steady Cavity Length as a Function of Cavitation Number

0.8

OJ ~I ----~----+_----+_----~--~----_4

0.6 rl -T-+-+--L- I o I J b=o:> u I b=IO t.5 ~g:~ I

'(3

;::-0.4 1 ~ ~ /1

U

gO.3 1 I#ZI ~ M9 A

o

0.2 I :/9'lf A

0.1

0.2 0.4 0.6 0.8 Cavitation Number, (j

1.0

(a) Drag Coefficient for y=15°

o U .. -c: Q)

'(3

O~ ~I-----.----~-----.-----..-----r---~

0.3 I j"{

y=15°, b=IO :E 0.2 Q) o

U

CI o 10. o

" " 0.1 I 1 ....... ,.eJ t

y=7.5°, b=5

00 0.1 0.2 0.3 0.4 Cavitation Number, (j

0.5

(b) Comparison of Theory and Dota

0.6

Fig. 12 - Steady Drag Coefficient as a Function of Cavitation Number

__ ~" •• ,~ ________ ?_~ ___ ~ ___ c_~~·

w a-

f F I [

APPENDIX A --------EVALUATION OF '!HE FUNCTION iT(t)

39

APPENDIX A

EVALUATION OF THE FUNCTION wet)

The evaluation of wet) for the most general case is rather in­

volved and will not be attempted here. When the unsteady motion is of oscil­

latory type or when the cavity is infinitely long, then time and space vari­

ables may be separated and the problem is particularly simple.

A. Oscillatory Motion

If the unsteady motion is of oscillatory type and is given by

A(t) = AO exp(jwt)

the imaginary part of the complex acceleration potential may be written as

~ = ~O(x, y) exp(jwt)

where j stands for ~ (but ij f -1) and Eq. (18) is reduced to

x wx ( wr

V :: - exp(-j-) J exp(j-) qc qc

-00

Integrating by parts, Eq. (A-I) is reduced to

w WX V ::: - ~(x, y) + j -- exp(-j --)

qc qc

01/1 -dr

{).,.

x , WT ) ~exp(j --)d r

qc -00

(A-I)

(A-la)

Therefore, since ~ = 0 on (y = 0, x < -A), the condition (18) applied at

the nose of the wedge is

A 2A 'Y (1 + -) = +-- + w) (A-lb)

qc

Hence

['

40

•

\}I(t) = - - (A-2)

B. General Motion with Infinite Cavity

It is more convenient to rewrite the complex acceleration potential

given by Eq. (30) as

where

2A ,::::t~ fl (t)·-;; '1'(1 + - + \}I)

U

'YbA f 2(t) :; -~

2nU2

1 F 2 (n := - ~ ({' + x 0) ({' - 1)

n

1

1 ( log(l + r/x ) dr

J ~(r ± X ) (T - ~) (r - () o o

With these notations, Eq. (18) is reduced to

-00

X T

- - +_\ i

X T dF2 - - + -) Im(-)dr

q q dz c c

(A-3)

(A-5)

Again integrating Eq. (A-5) by parts and setting x:= -A and y = 0, the

additional condition is reduced to

• A

1'(1 + -) = u

-A

J afl <3f2 fl(t) - Im (- F + F2)dr aT 1 ar (A-6)

41

Since the imaginary parts of Fl and F 2 are both equal to zero on (y = 0,

- 00 <x <-A), the integral of Eq. (A-6) is zero if fl and f2 are differ­

entiable. Under this circumstance, Eq. (A-6) is identical to Eq. (A-lb) and,

hence, the function tJ;( t) is given by Eq. (A-2) 0

!.!:!:!!iQI! B

EVALUATION OF INTEGRALS

45

APPENDIX B ~------- -

EVALUATION OF INTEGRALS

A. Evaluation of 10

Here, the integral in Eq. (35) is to be evaluated. It is more con­

venient to rewrite the equation as

(B-1)

By changing the order of integration, it may also be written as

(B-2)

Since

1

Equation (B-2) may be separated into two integrals. That is,

4n 10 = Re l!~_l T--+ x

o 0

1 1

J d -Re

.,J ( T + X )(.T - 1) o 0

S dJ;

~ (J; + x )(~ - 1) o 0

Interchange T and J; and it yields

46

He!l rp+-X10 [~o· ---;=dT=. =1) 1 d'; ~ ';;Xo J. (1" - .;) ~(T + Xo) (T -

(B-3)

Finally, Eqs. (B-1) and (B-3) lead to

1 ( ) 2 arc sin -;:===-

~l + Xo

(B-4)

B. Evaluation of LO and Ll

Straight-forward double integrations of LO and Ll are quite in­

volved. Here, similar techniques used for the evaluation of IO can also be

used and reduced the labor of calculations considerably.

By changing the order of integration in Eq. (52), Lom;ay be writ-

ten as

1

1 S 1 LO = --;- He 0 ~ (1 _ T)( t _,.)

1 J ..J (1 - ~)(t - ~)d, d1'

l' - ~ (B-5)

o

Now, by introducing partial fractions, the inte grand of t he first integral

may be separated into two terms in two different ways. They are

~ (1 - ~) (t - ~) =8€ 1 -1" (B-6)

7 - .; 1 - .; 1 - ~ l' - .;

and

..J (1 - ';)($ - ~)

B~ t -i

- $ - ~ + t - ~ (B-7) -r: - ~ " - ~

Using Eq. (B-6) , Eq. (B-5) becomes

1

1 =~S o 2 o

-r====d=T===-f -~ d1; ...j(l-T)(t-T) o~

1 +- dT

2

Using Eq. (B-7) , Eq. (B-5) becomes

1 1 =-o 2

1 +- Re

2

1 1 dT

J 1Ed" ~ (1 - T) (t - T) .e - '{:: ... 0 0

~.re:-r J~~ o

---dT Jf1=T ~~ dl;

o

47

(B-8)

(B-9)

By interchanging the variables l' and l; and changing order of integration,

it can be shown that the second term of Eq. (B-3) is equal to the negative of

the second term of Eq. (B-9). Therefore, by eliminating the second terms be­

tween Eq. (B-8) and (B-9), 10 is found to be

1

~ dT

o ...j (1 - t) (.e - 1')

~+l =-{elog --­

{i - 1 (B-IO)

Since 10 is knovm, Ll can be most conveniently evaluated by

first writing it as

'{ ...... , <,

"'--

48

1:t = La + 2- Re J~ (1 - ~) (t - ~) J.~ d" d~ 2 0 o~ 1'- ~

(B-ll)

By changing the order of integration and by partial fractions, the second term

of Eq. (B-ll) is reduced to

1

J~ 1Jg ~ Re J~ (1 - ~)(t d,..

- ~) d~ = ---. d~

2 0 T - ~ 2 t - 'J: 0 o ."

!~ 1 J~: : 1

~ ) 'Ii (1 -r)(t - r) . . dr + - Re dtd~ (B-12)

t -,.. 2 ~ ,.. - ~

0 0 0

Again, the second term of the right-hand side of Eq. (B-12) is equal to the

negative of the left-hand side and, thus,

+ ~(i f1-7 d'V2

4 ~~ o

V+l 1 _r: {"i+12 =-{i log {i + - [V.e - (t - l)log -{i ]

t - 1 4 t - 1 (B-13)

Copies

SPONSOR'S DISTRIBUTION LIST FOR TECHNICAL PAPER NO. 34-B

of the St. Anthony Falls Hydraulic Laboratory

Organization

51

5 Chief of Naval Research, Department of the Navy, Washington 25, D. C., Attn:

3 - Code 438 1 - Code 463 1 - Code 466

1 Commanding Officer, Office of Naval Research Branch Office, 346 Broadway, New York 13, New York.

1 Commanding Officer, Office of Naval Research Branch Office, The John Crerar Library Building, 86 East Randolph Street, Chicago 1, Illi­nois.

1 Commanding Officer, Office of Naval Research Branch Office, 1000 Geary Street, San Francisco 9, California.

25 Commanding Officer, Office of Naval Research, Navy 100, Fleet Post Office, New York, New York.

6 Director, Naval Research Laboratory, Washington 25, D. C., Attn: Code 2000.

4 Chief, Bureau of Naval Weapons, Department of the Navy, Washington 25, D. C., Attn:

1 - Aero and Hydro Branch 1 - Research Division 2 - Underwater Missile Branch

Chief, Bureau of Ships, Department of the Navy, Washington 25, D. C., Attn:

1 - Code 310 1 - Code 312 1 - Code 420 1 - Code 554 1 - Code 345

5 Commanding Officer and Director, David Taylor Model Basin, Washing-ton 7, D. C., Attn:

1 - Code 142 1 - Code 500 1 - Code 513 1 - Code 526 1 - Code 591

1 Commande~, U. S. Naval Ordnance Test Station, China Lake, Califor­nia, Attn: . Code 753.

1 Officer-in-Charge, Pasadena Annex, U. S. Naval Ordnance Test Sta­tion, 3202 E. Foothill Boulevard, Pasadena, California, Attn: Code P890962.

<ii-V, \ ""--

52

Copies Organization

1

1

1

1

1

3

Commanding Officer and Director, U. S. Naval Engineering Experiment Station, Annapolis, Maryland.

Commander, Naval Proving Ground, Dahlgren, Virginia, Attn: Tech­nical Library Division (AAL).

Commanding Officer, U. S. Naval Underwater Ordnance Station, New­port, Rhode Island, Attn: Research Division.

Commander, U. S. Naval Ordnance Laboratory, White Oak, Maryland, Attn: Library Division (Desk HL).

Mr. W. I. Niedermair, Coordinator of Research, Maritime Administra­tion, 441 G Street, N. W., Washington 25, D. C.

National Bureau of Standards, Washington 25, D. C., Attn: 1 - Fluid Mechanics Section 1 - Dr. G. B. Schubauer 1 - Dr. G. H. Keulegan

1 National Academy of SCiences, National Research Council, 2101 Con­stitution Avenue, N. W., Washington, D. C.

1 Superintendent, U. S. Naval Academy., Annapolis, Maryland, Attn: Librarian.

1 Superintendent, U. S. Naval Postgraduate School, Monterey, Ca:!Lifor­nia, Attn: Librarian.

1 Superintendent, U. S. Merchant Harine Academy, Kings Point, Long Island, New York, Attn: Captain L. S. McCready, Head, Department of Engineering.

1 Air Force Office of Scientific Research, Mechanics Division, Wash­ington 25, D. c.

1 Commanding Officer, Office of Ordnance, Box CM, Duke Station, Dur­ham, North Carolina.

5 Director of Research, National Aeronautics and Space Agency, 1512 H Street, N. W., Washington 25, D. C.

2 Mr. J. B. ~arkinson, Langley Aeronautical Laboratory, National Aero­nautics and Space Administration, Langley Field, Virginia.

1 Director, Engineering Sciences Division, National Science Founda­tion, 1520 H Street, N. W., Washington 25, D. C.

10 Document Service Center, Armed Services Technical Information Agen­cy, Arlington Hall Station, Arlington 12, Virginia.

1 Office of Technical Services, Department of Commerce, Washington 25, D. C.

53

Copies Organization

4 California Institute of Technology ~ Pasadena 4, California, Attn: 1 - Professor M. S. Plesset 1 - Professor T. Y. Wu 1 - Professor A. Acosta 1 - IVdro Lab

3 University of California, Berkeley 4, California, Attn: 1 - Department of Engineering 1 - Professor H. A. Schade 1 - Professor J. V. Wehausen

1 Director, Scripps Institution of Oceanograptv, University of Cal­ifornia, La Jolla, California.

1 Professor M. Albertson, Department of Civil Engineering, Colorado State University, Fort Collins, Colorado.

2 Iowa Institute of HydrauliC Research, State University of Iowa, Iowa City, Iowa, Attn:

1 - Professor H. Rouse, Director 1 - Professor L. Landweber

2 Harvard University, Cambridge 38, Massachusetts, Attn; 1 - Professor G. Birkhoff, Department of Mathematics 1 - Professor G. F. Carrier, Division of Engineering and

Applied Physics

3 Massachusetts Institute of Technology, Cambridge 39, Massachusetts, Attn:

1 - Department of N. A. and M. E. 1 - Professor A. T. Ippen, Hydro Laborator:r 1 - Library

4 UniverSity of Michigan, Ann Arbor, Michigan, Attn: 1 - Professor R. B. Couch, Department of N. A. and M. E. 1 - Professor C. S. Yih, Department of Engineering Mechanics 1 - Professor V. Streeter, Department of Civil Engineering 1 - Librar:r

1 Director, St. Anthony Falls Hydraulic Laborator:r, Univt::rsity of Min­nesota, Minneapolis 14, Minnesota.

1 Director, Alden Hydraulic Lab ora tor:r , Worcester Polytechnic Insti­tute, Worcester, Massachusetts.

1 Director, Ordnance Research Laboratory, Pennsylvania State Univer­sity, University Park, Pennsylvania.

1 Director, Institute of Mathematical SCiences, New York UniverSity, 25 Waverly Place, New York 3, New York.

1 Professor J. J. Foody, Engineering Department, New York State Uni­versity, Maritime College, Fort Schulyer, New York.

~~1.2_

54

Copies Organization

1

1

1

1

1

1

3

Technical Library, Webb Institute of Naval Architecture, Crescent Beach Road, Glen Cove, Long Island, New York.

Professor S. Corrsin, Chairman, Mechanical Engineering Department, The Johns Hopkins University, Baltimore, ~...aryland.

Commanding Officer, Office of Naval Research Branch Office, 1030 East Green Street, Pasadena 1, California.

Director, Woods Hole Oceanographic Institute, Woods Hole, Massachu­setts.

Society of Naval Architects and Marine Engineers, 74 Trinity Place, New York 6, New York.

Engineering Societies Library, 29 West 39th Street, New York 18, New York.

Stevens Institute of Technology, Davidson Laboratories, 711 Hudson Street, Hoboken, New Jersey, Attn:

1 - Dr. J. Breslin 1 - Mr. D. Savitsky 1 - Library

1 Dr. J. Kotik, Technical Research Group Incorporated, 2 Aerial Way, Syosset; New York.

1 Director, Institute for Fluid Mechanics and Applied Mathematics, University of Maryland, College Park, Maryland.

1 Division of Applied Mathematics, Brown University, Providence 12, Rhode Island.

1 Hydrodynamics Laboratory, National Research Council, Ottawa, Canada.

1 Professor L. M. IViilne-Thomson, Mathematical Research Center, 1118 W. Jor..nson Center, Madison 6, "Wisconsin.

1 Dr. J. M. Robertson,Department of Theoretical and Applied Mechanics, College of Engineering, University of IllinOiS, Urbana, Illinois.

2 Stanford University, Stanford, California, Attn: 1 - Professor J. K. Venard, Civil Engineering Department 1 - Applied Mathematics and Statistics Laboratory

1 Professor J. B. Herbich, Civil Engineering Department, Lehigh Uni­versity, Bethlehem, Pennsylvania.

1 Dean J. S. McNown, Department of Engineering Mechanics, University of Kansas, Lawrence, Kansas.

1 Professor A. G. Strandhagen, fupartment of Engineering MechaniCS, Uni versi ty cif Notre Dame, Not re Dame, Indiana.

Copies

2

1

1

1

1

1

1

1

1

1

1

1

2

1

1

1

1

1

55

Organization

Polytechnic Institute of Brooklyn, Department of Aeronautical En­gineering and Applied Mechanics, 333 Jay Street, Brooklyn 1, New York, Attn:

1 - Professor A. Ferri 1 - Professor H. Reissner

Professor H. Cohen, IBM Research Center, P. O. Brne 218, Yorktovm Heights, New York. .

Professor D. Gilbarg, Applied Mathematics and Statistics Laboratory, Stanford University, Stanford, California.

Mr. Leo Geyer, Chief of Preliminary Design, Grurmnan Aircraft Engin­eering Corporation, Bethpage, Long Island, New York.

Mr. W. P. Carl, Jr., ~namic Developments, Inc., Babylon, Long Is­land, New York.

EDO Corporation, College Point, Long Island, New York.

Mr. H. E. Brooke, Hydrodynamics Laboratory, Convair, San Diego 12, California.

Miami Shipbuilding Corporation, 615 S. W. Second Avenue, Miami 36, Florida.

Baker Manufacturing Company, Evansville, Wisconsin.

Gibbs and Cox, Inc., 21 West Street, New York 16, New York.

Dr. H. Reichardt, Max-P1anck-Institut fuer St roemungsfors chung , Goettingem, Boettingerstrass 6/8, West Germany.

Director of Research, National Aeronautics and Space Administration, Lewis Research Center, 21000 Brookpark Road, Cleveland 35, Ohio.

Hydronautics, Inc., 200 Monroe Street, Rockville, Maryland, Attn: 1 - Mr. Phillip Eisenberg 1 - }fJ.!'. M. P. Tulin

Commanding Officer and Director, U. S. Naval Civil Engineering Lab­oratory, Port Hueneme, California, Attn: Code L54.

Micro-Tech Research Company, 629 Massachusetts Avenue, Cambridge, Massachusetts, Attn: Mr. Cohoon.

Professor J. E. Cermak, Department of Civil Engineering, Colorado State University, Fort Collins, Colorado.

Mr. Blaine Parkin, The Rand Corporation, 1700 Main Street, Santa Monica, California.

Cleveland Pneumatic Ind. Inc., Advanced Systems Development Divi­sion, 1301 E. El Segundo Boulevard, El Segundo, California.

56

Copies

1

1

1

1

Organization

Mr. George H. Pedersen, Turbomachinery Division, Curtiss-vJright Corporation Research Division, Quehanna, Penn~lvania.

Mr. Kenneth E. Hodge, Hydrodynamics Research, Lockheed Aircraft Corporation California Division, Burbank, California.

Dr. Byrne Perry, Department of Civil Engineering, Stanford Univer­Sity, Stanford, California.

Professor H. G. Flynn, Department of Electrical Engineering, The Univ6'rsity of Rochester, College of Engineering, River Campus Sta­tion, Rochester 20, New YorkQ