Two dimensional, steady, cavity flow about slender bodies in channels of finite breadth

Lab

v.

Scheepsbouwtcunde

4

Tehn1HeSCh00

_{Paper No.}

'

56A-29

Two-Dinnsional, Steady, Cavity Flow

About Slender Bodies in Channels

of Finite Breadth

ARCHIEF

The steady, cavitating flow past slender symmetrical

bodies placed in a solid-wall channel is studied by means

of the linearizedtheory of Tulin. The free-boundary

con-dition is linearized and boundary concon-ditions are applied on

the line of symmetry of the flow in analogy with

thin-air-foil theory. A singular integral equation formulation of

the boundary-value problem is obtained and can be solved

to yield expressions for cavity length, maximum cavity

width, and drag coefficient as functions of the cavitation

number and the channel breadth. These expressions are

given for an arbitrary body -and evaluated for the case of a

wedge.

IN

A recent report (1 ),3 M. P. Tulin has introduced a linearized theory of steady, cavity flow. Until the introduction of this notion, borrowed from the linearized theory of compressible flow past a thin airfoil, information on incompressible flows with finite cavities was obtained from the models of Riabouchinsky and Wagner. -These models have provided quite satisfactory agreement with experimental data but the computations which they lead to are often extremely involved. T.ilin has succeeded in producing a theory in which the condition at the free boundar' of the finite cavity is linearized in terms of the small thickness of a slender body. The boundary-value problems are then written in terms of integral equations and approximate solutions may be found. The method employed allows the calculation of cavitating flows for arbitrary forebody shapes.

The linearized theory has in common with other finite-cavity solutions the difficulty that it is physically invalid near the trailing

-end of the cavity.

In this way the Brillouin paradox (2) which predicts nonexistence of flow about closed bodies with constant pressure afterparts is avoided. If the condition is imposed that the slope of the cavity must be continuous with the slope of the body; a unique correspondence between cavity length and cavita-tion number, and therefore a unique flow, will be obtained. This last requirement has been referred to by Tuliñ as the juncture condition.

-A discussion of two-dimensional finite-cavity flows past bodies placed in two-dimensional channels has been given by Birkhoff

'Visiting Associate Professor of Mathematics, Rensselaer Polytech-nic Institute, Troy, N. Y.

I Project Mathematician, Carnegie Institute of Technology, Pitts-burgh, Pa.

Numbers inparentheses refer to Bibliography at end of paper. Contributed by the Applied Mechanics Division for presentation at the Annual Meeting, New York, N. Y., November 25-30, 1956, of

Tax AMERICAN SocIETY OF MECHANICAL ENGINEERS.

Discussion of this paper should be addressed to the Secretary, ASME, 29 West 39th Street, New York, N. Y., and will be accepted until one month after final publication of the paper itself in the

Jotra-NAL OF APPLIED MECHANICS.

NOTE: Statements and opinions advanced in papers are to be understood as individual expressions of their authors and not those of the Society. Manuscript received by ASME Applied Mechanics Division, June 13, 1956. Paper No. .56A-29.

B HIRSH COHEN' AND ROBERT GILBERTI

Plesset, and Simmons (3). It was felt that some of the difficulties of computation indicated in their work might be avoided by ap-plying the linearization just mentioned. Accordingly, the flow past slender two-dimensional bodies placed syinmetrically in channels of finite breadth is discussed in this paper. In particu-lar, results are computed for the shape, maximum width and length, and drag of the finite-cavity afterpart associated with a wedge-shaped forebody. Birkhoff, Plesset, and Simmons have computed the case of the finite cavity attached to a flat plate, placed perpendicular to the channel walls and the flow direction. The results obtained in the present work are not strictly compara-ble since the angle of the wedgeshaped body cannot be allowed to approach ir/2. However, for both the flat plate and the wedge, it is found that the width of the finite cavity increases as the chan-nel walls are brought closer together. The cavities become in-finitely long at a value of the cavitation number greater than zero. For each cavitation number there exists a ratio of maximum height of the forebody to channel breadtli which is the maxi-mum attainable. This is the blockage ratio described in (3).

The lengthening of the cavity with decreasing cavitation num-ber; the change in cavity width ith cavitation number, nd the dependence of the drag on the various parameters of the flow are described analytically and by the curves.

THE LINEABIZED BOUNDARY-VALUE PROBLEM

- I -would be well to list here the symbols used in the following

work and to describe the flow by means of Fig. 1. Let = uniform velocity at , parallel to z-axis

perturbation velocity potential

q= perturbadon velocity

U, V = x, y components of perturbation velocity

U, = component in x-direction of velocity on cavity wall = U,0 + u(x, Y,)

P0 = pressure on body PC = cavity pressure

P,0 = static pressure of stream at infinity

= cavitation number =

FIG. 1

1

Yo( Vo(s)

z.-c zO

In order to describe the flow, a harmonic function ço is sought such that grad cc =

, and grad cc .0 as (x, ii) =. According

to Tulin (1), the linearized boundary conditions to be satisfied on the x-axis are

dYa(x) v(x, 0) dx

-

U0

2u(z,0)

O<x<1

[2]

The condition that the body be closed is stated as

ft

(x, 0)dx = 0.

[3]

J_coy

and for the channel flow the wall condition

ip/

h\

is added. A distribution of sources of strength m(x) along the x-axis for c < x < 1 plus the same distribution along the lines y

= ±nh(n = 1. 2, 3. .

..) for the same interval in xprovides a

flow with the proper behavior at infinity, the necessary sym-metry, and horizontal streamlines which can be taken for channel

walls at i = ±h/2

cc(x, y) = -

f m()

{ln

/[(

-

.., c

c<x<O

[1]

-

+ y9

+

in

_{/[(x - E)2 + (y - nh)2]}

+

in '/[(x_)2 +

(Y_nh)2]}

d [4]

Computing the velocity components u and v

1

r

u(z, y) =

- J

m()

+

_.n=

_{(x_)2+(y_nh)2}

±E

(x - )! + (y - nh)2

[5]

v(x,y) = -fmcE) [(x_2

_+y2

+

_{n_±o (x_)2+(y_,nh}

ynh

+

_{(x - E ± (y}

nh)2] d ... [6]

On the center line of the channel

u(z, f.) =

m() coth

ir(x -

d [71

m'(x')

where the Cauchy-prineipal value of the integral is understood. Along the channel walls'

/

h\

and along the center line

nz(x)

v(x, 0) =

No*, along the-center line in the interval --c <x < 0, the source distribution has the form

dYo(x)

rn(x) = 211,

dx

from Equation [1].

For the interval 0 < x < I Condition. [21

yields

Jm()coth

h

I,'

7r(x)

0

_2Uf d0

ccth '[8]

and adding the closure condition, Equation [3]

-fm()d =

_2Uf dYodE

= 2U,Y0(0).... [9,1

the boundary-value problem is set in the form ofa singular ifite-. gral -equation together with the Closure Condition [9] and the previously mentioned ji.fncture condition.

SoLUTION FOR TBR Souacx DIsTRIBJTxoN

The problem is thus reduced to finding the solution of the

singular integral Equation [81. -II the substitutions

+ (x' + 1);

_=--(i±1);

h'=2f',

c'=i+

2c

are introduced, Equation [8] becomes

1 _{( ,-} ,

-,r( x')

oU,,,

Jm(x)coth

h'

dx=

2

_cf'

dYocth1r(!7)d,

_[101

The solution of this siligular integral equation has been given by Dörr (4) and discussed further by Nickel (5). It is therefore-possi-ble to write immediately -that

hU,,r

1

I1 x')

sinh

r(.1 - x')]

k . 7T

*x'

2

i(1')

- suih

cosh

-

cosh

h'.

-

/-.

lr(1+17)

f1-.!

p= constant fluid density

h= channel breadth

c/h, = ratio of forebody chord to channel breadth c/i, = ratio of forebody chord to c.vity length Yo(x) = ordinate of forebody

where f(n)

k is a constant to be determined by the juncture conditiOn, the condition on the contmuity of the slope of the cavity and the solid body. For convenience, let

ir.

_sinh -. sinh and 2

+

B(x') =

1 - 1 _sinh 2B(.x') I

h'

+

_h'

_7r(1+X')

[14] sinh

h'

To determIrfe the cavity shape the relation dY. v, m(x.) dx

U, - 2U,

is employed. Thus f(,7)coth / . / sinh

h'

Itsinh

_h'

_J

I I- . dy i . sinh

h'

The juncteof body and cavity occ rs at x'

1 and the.

juncture.condition.requires.that.thecoefficient.ot

1

h'

[11]

where Y'(x') is the body or cavity shape in the x', ,-co-ordinate system. &fore computing Y,(x) the condition that the cavity be closed, Equation [9] is imposed. With some computation, we ob-tain from this the relation .

-iT

h'sinh-7

2 2,r ci h - D1 cosh

+ A (-1) sinh -p-.

[16] 0 4 h where

:1

- 4U,J

m'(x')dx' [15] [12] D, =

± r)

di D2(x')

=

f

dYO coth

The integrals involved in the computation may be evaluated by using trarf9rmations of the brm

IT

irz'

in

X coth c

= coth

r = coth

The integrals thus derived in terms of X, , r may be found in in-tegral tables. [We have made use of the fact here that

in m'(x') he equated to zero.

Application Of this requirement see reference (1)1 This may be rewritten in terms of the

origi-yields nal variables as

kh' . 1 I h

- 1)

sinh

_h'

cosh sinh

_-

f°/

sinh

- di... [17]

[

cosh h

2B(l)sinh

h f13J _{This is one of -the desired resultsthe unique relation between u}

and 1 with h as parameter. If the cavity is allowed to be infinitely The distribution of sources along the center line of the flow can long, a value of = c is attained greater than zero and dependent

then be written as on the ratio c/h

2A cosh -- ----2B(---r1) sinh , o h

10 dY0

I(

-- -

eOsh-\

m'(x') =

cosh --

-

I I - I di. .. [18] h- o 4

j .

dl i h' cosh- -1 + -j--

\

2 sinh

1

Ash/2-a, u-.O

- --- cosh

ir(1 - ')

The o, lre1aton is used in the computation of

I

Jr.

ir(1.+x')

i---fl

J VL"

-

h'

sinh

to arrive at the expression (again the transformations indicated after Equation [16] are used)

h'r.

(X2+w

Y(x ) - Y(-1)

-

1 +!

2 4r [' 1\X(, + 1)

/

X2_w\1

_Y0(')r.

/ x+

- sfl_I I

I I -I--

I sin' I

\X(c-1)/J

2ir _L

\X(w±1)

/

Xzw\1

1

P1dY0

.

+

'\X(w

- 1))] - r

J_,

Slfl

X(w - T))

1.

irx'\

Yo(-1)

(

sinh

-- \

-

sin I

1+-ir

I.

in-

2

sinh

--The second and fourth terms annul themselves. In ternis of the original variables

ir.

mx

ir(lx)

I1 [Slflh - smh h h Y,(x)

tan-'

o 2ir

and Smee

/

irl\

in

tan-'

sinh

_j-)

-and since experimentally o is always4 < 0.5, Y, mcannot exceed h/8. The parameters a = c/h; $ = c/i may be introduced and the maximum cavity width xpanded for the case a small with respect

to$

c

[-ira

i (ira\'

mft

o'2maL2$

6 \2$)

Although this goes over to the expression given by Tulin for a-ø0, it must not be concluded that Y, ___ decreases for decreasing h. For h finite, the relation between o and I is such that Y,

in-creases as h dein-creases. This is described by the curves in Fig. 2 for the case (dYo)/(dx) = const. The result should be compared with results given by Armstrong and Tadman (6).

4Reference (2), p. 57.

ir(2x - 1)

cosh 2/i / mt .,

ir(lx)

2 o , sinh --- smh

+ -

- tan'

cit [19] in cit

- 1)

h h

The first term provides a symmetrically shaped curve with zero

values at x = 0 and x = 1.

The second term decreases in value from Y0(0) at x = 0 to zeio at x = I and is relatively small come pared with the first. The maximum value of Y, is obtained by setting the cavity source strength equal to zero. As in the case of infinite stream, the maximum occurs at a value where

1 1

___c<x<

_-i

Y)

tan'

(sinh

--)

[20]

!4

0 0.05 0.10 0.15

0.20

0.25

o.

FIG. 2 MAXIMUM CAVITY WIDTH AS A FUNCTION or CAVITATION NUMBHR

If the cavity is allowed to become infinitely long, the following expression is obtained for the cavity shape

ir

f

tan'

_LP

- 1

Isinh

-- - cosh

dYo

(

2sinh h dt

-

f --- tan'

As x -

this yields

2sinh-- cosh _K2sinh--]

o.

h[231

1+-i-The cavity does not becomeinfinitély wide at an infinite distance downstream from the forebody unless h/2 becomes infinitely large. In Equation [22] if his allowed to become infinite and then x -+ , the result of Tulin is obtained; namely

Y(x) 2

('°dY0

If

i\

hm

-

i

- i--iclt

-/(x)

in J c cit

\

£ / Js

The Cavity Drag. Thea frag thay. b calculated withi the linearized theory. The expression

[Y,(x)]

=

dY,

[

exp

'II

(

lr(t_x))]

[22]

Y,(x) - Y,(0) =

D2f(Po_P)dt

yieId the formula

D = 2pU,

Yo(0)

-

u,_e(t, Yo)

where u,,-o is the x-component of that, part of the disturbance velocity on the body which is induced by the cavity source dis-tribution. From Equation [7]

1

(

Ir(xt)

u,_0(x, Y0)

. j

m(x) coth h dl

The calculations are somewhat lengthy but involve only

ele-men tary functions and lead finally to the following expression for the drag D

_-

(i+--)

° dYo art'

J_ dl 41

ar(tl)

cosh--

C _sinh h

j

/

q\2

art

,'j-)

tanhj

4

+--art cosh art smh h _dl

ir

_sinh

irl'f0dYo

_{/ dl} sinh

--j

_. 'dl %j

_sinh rl h

1)

h

/

q\8

art

(0

dY0 h

dl +

rl C

coshT

J_cdl

-/[sinh-,r1121

\,.

art

cosh - I

1) smh dY0

2h\l

/

2h d

_(1.

.ar

ar(tl)

Lshh1hT5mh

h

,///////////////// ///

tz

/ /(/'.

FIG. 3 WEDGE-SHAPED FOREBODY OF SMALL ANGLE 7 [24]

This may be rewritten with

arc' art' ar 21

G' =coth-7; r =coth--; X== coth--;

1'

--1

in the form D

2(1±)

Cd10 Ar

d'

pU2

cosh!

J

.

..j..._

T

8/

tT\2 IT

ar 1dY0f X+r \

12

L8i'hTJa, dr

4 /

q\2

_{2ir rr'dY0/ rX±1 .\}

1

+-r) siflh7

_jj

_dr

/

o\ fdY0f rX+1

+

%1 +-j)

_{J_, di}

r"-dY0f,

\

-J

, d

k/[X2_

)

di

[251

For the cavity of infinite length the expression for drag becomes

D

2(1

+

h

[o/(

irLCOsh!)

art - 2

cosh-2(1+--)

0 dYo

+

ii

dt -/(sinh-)

o

coshj-dt

1 +

2

f

dYo

.I[siñ11'- (ainh

- cosh

srnhTdi

& -/r .

/

arI\1

[261

iLhIT .srnh_-,_coshT)J

For h/2 -

, both of these

relations reduce to the results of Tulin.

The Slender Wedge, As an example of the cavity flow past an object placed in a channel, the flow past a wedge' is considered.

- Let the wedge have chord length c, maxizurn thickness T, and

half angle yas shown in Fig 3. Taking y to be small enough so that (dYo)/(dx) may be set equal to 'y, calculations may be made for the relation between o and 1, and for the drag.

From Equation [18]', it may be shown that for $ =' c/i = 0 q 2

4y - . growth Of Y with decreasing a can be obtainel. Y m versus a a

-

cosh '[e

1 [271 _{is plotted in Fig. 2 for-various values of 'ith y fixed at 15deg.}

1 +

. Similarly setting (dYo)/(dx) = 'y allows computation of the drag for the case = 0, the infinitely long cavity

D

₂₍₁₊

/

a\2h

(i"°

.'

dr

1

72<1

PU2

2 / 72

J

-/[i - rJ(1 - r2)

For $> 0, the relation

a

/coth..tanh1

\

holds. The a versus '1 curves are plotted for various values of a in Fig. 4. The value a = .0, of course,. corresponds to Tulin's result for the infinite stream

-a

7

02

O. I

-Jr.

.

ira(1+$)

L"

ira sinh - _{,ra(-1 + 2fl)} cosh 2

Jr.

.

+tanh'

ira(1 + fl.)

srnh 29 and

2h /

+ - I -1+

_ir\

.2 1.0 0.4 0.2

\i

rdr

I

IJ_G

.v'lii.i(1--=-472h(

\2

_f

TIlT 1r2

±T)

JGJ11_r(1--T2)

r

J-a "El - (1

72

(1

)2 [sinh_J

_(G

-irc where G

coth

-is given by the Relation [27] and we have D

p

cD=1

-pUO2T

CD is plotted against for -various values of a in Fig. 5.

4a7

0

0.1 0.2 0.3 -

- 0.4

7,radiov

FIG. 5 DRAG COEFFICIENT AS A FurcTIoN OF WEDGE ANGLE FOR A CAVITY OF INYINITE LENGTM

0

0.1 0.3

a4

0'

Fia. 4 CAVITY LENOTU AS A FUNCTiON OF CAVITATION NUMBER FOR

- i WEDGE ANGLEOF 15 DEn

For a given value of ,the value -of a for which

d(fi .= 0)is

attainedis the maximum value of a For larger values of a,:there 'can be no further -lengthening of the cavity and this represents the blockage conditon-described in reference (3).

Using the a, 1 relation for y const, an idea of the rate of

0.8

D

0

CONCLUSION

The linearized theory hs been used to obtain information re-garding length, width, and drag of finite steady cavities attached to slender bodies placed in channels. These relations have been obtained for the case of the general slender forebody, but curves are given for the case of the wedge. In all cases the results re-duce to those obtained by Tulin fOr the case a = 0. FOr each value of a, there exists a Ii ting value-of the cavitation number, , at which the cavity length becomes infinite: For increasing a, it is shown that tl?e cavity width is increased, if o is held con-stant.

ACKNOWLEDGMENT

A portion of this work has been supported ,by the Mechanics Branch of the Office of Naval Research:

BIBLIOGRAPHY

'Steady Two DimensiOnal Cavity Flows About Slender Bodies" by M. P. Tulin, David Taylor Model Basin Report 834,

May, 1953. .

--2 "Hydrodynamics," by Garrett Birkhoff, Princeton University Press, Princeton, N. J., 1950.

3 "Wall Effects in Cavity FlowI and II, by Garrett Birkhoff, M. Plesset. and N. Simmons, Quarterly of Applied Mathematics, Part I, vol. 8, July, 1950. pp. 151-168; Part II, vol. 10, January, 1952, pp. 413-421.

4 "Strenge Losung dér Integralgleichung für die Stromung, durch

em senkrechtes Flugelgitter," by J. Dorr, Ingenieur-Archiv, vol. 19,

1951, p. 66.

-5 "Zusatz zu J. Dörr: Strenge Lsung der Intergralgleichung für em Flugelgitter," by J. Nickel, Ingenieur-Archir, voL 20, 1952, p. 6.

6 'Wall Corrections to Axially. Symmetric .Cavities in Circular Tunnels and Jets," by A. Armstrong and K. Tadman,- Armament Research Establishment Report No. 7/52, March, 1953.