Wall effects in cavity flow - Part 2

V o l . I X , N o . 4, J a n u a r y , 1952

WALL EFFECTS IN CAVITY FLOW—n* • ^

-B y G . -B I R K H O F F , M . P L E S S E T A N D N . S I M M O N S (Harvard University; Naval Ordnance Test Station,

Pasadena; Minislrtj of Supply)

1. Introduction. I n Part I of the present study,** the problems of flow about a cavitatmg body symmetrically placed in a channel or in a free jet have been solved in the case where the cavity extends to infinity do^-nstream. The infinitely long cavity occurs, in each configuration, at one particular cavitation number which is a function of blockage ratio in the first case and is zero in the second. At greater values of the

* denotes origin y ' / I + c? D C B W=c>o C' -oo 0 + -5'-- ^ . 7 ï E 2 -5'-- p l a n e 9= ' . W=-oo B Ac» W-pl one ' ^-plane

\

riG.

I

- CASE A.

cavitation number, the cavity is of finite extent and a different analysis is necessary The solutions of the corresponding problems with finite cavities are given in the present

The configurations examined are again two-dimensional, this permitting the employ-ment of conformal transformation technique. The body is taken in the form of a finite

•Received October 18, 1950. * * Q . Appl. M a t h . 8, 151-168 (1950).

N O T E S [Vol. I X , No. 4

lamina perpendicular to the stream, so that the physical features of the flow may not be obscured by mathematical difficulties. Explicitly, the cases treated are

A. The cavitating lamina in an infinite stream; B. The same in a channel of finite width;

C. The same in a free jet of finite width. ,. k ^ r,n+or

2. Case A. The lamina in an infinite stream. This ca^e, where the hquid has no outer boundaries, is taken first to provide a standard for the other two So utions of this case have been given previously by Riabouchinsky [1] and by Fisher m an unpublished British Admiralty report, but the present treatment is much simp er than either.

i k e the density of the hquid as unity, the velocity at mfimty as unity, and the width of the strip forming the body as 2 units, so as to avoid unnecessary symbols Thi strip is disposed between the points {-4 ± i) in the .-plane (Fig. D- The free boundaries, there shown in broken line, start from the edges of the stnp and . ^ f o ™ downstream on a similar, conventional, solid strip, extending between the Points (4 ± j ) S s device, which is due to Riabouchinsky, avoids the closure jets and t f u l e n c e that would othenvise have to be taken into account. I n f - -'ay, the mathematical

ad-antages of a symmetrical problem are obtained merely by modification of the down-stTeam conditioW^to which the flow aromid the cavitatmg body is kno^^vn to be m-ensitive That this is so, has been clearly demonstrated by Gilbarg Rock and Za-rantone lo f who, in an as yet unpublished analysis of the similar problem with down-s ^ r k,down-sure hy a re-entrant jet, find for low and moderate cavitation numbei^ cavity

ies and drag coefficients virtually indistinguishable from those that result below. The flow being symmetrical about the axis, consideration is restncted to the upper half .-plane. The corresponding regions in the W and f-planes are shown m Fig. 1, together with the auxiliaiy plane of u. Symbolism is as m Part I

Proceeding by Kirchhoff's method for discontinuous flows, the transformation rela¬ tions are found:

Si + a a n h g t a n ^ T ^ ' (1) J- = - ( 1 + Q) j ^ ^ _ .^^^^ ^ t a n w j ' dz f l - ^'tanh 0tanu[^' (2") du where (3) ^ = i log (1 + Q)

and SAP) can be evaluated in terms of standard Jacobian elliptic functions of modulus k = sech P as

SAP) = jf^-+Ë^-^W- ^ ^ The integration of (2) between appropriate limits then yields the cavity half-length and half-width o:

1 = SM = E - k"K k" + E' - k'K" (5) s = SAP)

2

X = SAP) cosh' l3E(k, J TT - ö ) - sinh' ^ ^ - e

sin Ö

(tanh' /3 + tan' ö ) ' ' " J ' ^'^^

sec

e

2/ = SAP) sinh' /?, ;

^^Ktanh' /3 + tan' Ö)'

where ƒ ƒ are the standard elliptic integrals of the second and first kinds respectively d u c ü ï : t l V ' ° " a f t e r ' o m l re:

where = + (8)

'S4(/3) 2(g^ - fc'gO

fc" + i?' - fc'if'- (9) Referred to the velocity on the cavity boundary, (1 + Q)'^^, the drag coefEcient

I S

= SAP). (10)

™OML tïf : ' ' ; ? ' ^ " '^'^^^ ^"'''^^^^^^ '^^d there is no difficulty

f n ' h d i 0 is S Ï Ï H ' T numerically given case. The sub-class of cLes

Z^l^:^^^

a - 1 =

7 ^ (1 + <2) + o(a'),

416 N O T E S

[Vol. I X , No. 4

The cavity contour becomes

^ = 2 r^^^^^ 0 e _ g d - ( i . - e ) ] + o(Q'),

TT + 4 L ₍₁₂₎

= ( c o s e c Ö - 1) +O(Q^).

In the limit, as Q - 0, (11) and (12) become the classical results for the lamina in an infinite stream. i/> He Aco

A

cl

_D.

E

211

E'

N;.

Goo

F

E" " D * C B A«.

W - p l o n e 1 Hoo -OO - / / ) oo ' ^-plane , i' , H A

B

\

V . E

riG. 2 - CASE B.

dn M ' UöJ

dz ^ 2;tfc sn A m'u - ik' sn M cn u

du ,r(l + Q)'^' 1 - fc'sn' A^^SST' (14) where

^ - = ^ ^ ' - (15) w ! ; r r e " / . ' ^ " ^ " ^ ^ ""^^^ ^^e end to conform

The integration of (14), between appropriate hmits, then yields after reduction the following expressions for the geometrical characteristics! reduction the

2h = k ^ f^^'^(^) + (S' - K')A] + ^ COS- (cd A) - kK' sn A, (16)

~ ^ K T T Q P f^"" ^ ~ '^)]- (18) The cavity shape is obtained in Cartesian parametric form as

^ h2 + Q

{ F s n A cd A - Z(A)]v + - b g ^ d ^ j t A L ^' ^ 2 ^ e , ( A - 2 ; ) J '

_ ^ Q ^ , ^^^^ ~ TT 1 + Q ^^^'^ ^ - t a n - {k' sc A)],

where the parameter v runs from 0 to 2K.

Again, after considerable reduction, one finds for the drag coefficients

C^ = 2 l + Q - ^ t a n - ( A ; ' s c A ) l , , [ h 0 1 ^^^^

^f'= - - r + Q t ^ " ' (^' sc A) .

4 - & -

mtegration of (14) and the velocity vector at any point is then giver! by (13) ' ' ' ^ ' " ^

418 N O T E S [Vol. IX, No. 4

The numerical solution for any given case involves some complication. Given Q and h the value of h must be found from (16); successive approximation is the indicated method. At the same time am A is found from (15); the remaining results can then be evaluated. ih

.---""^

FIG. 3 - CASE C.

I t is soon found, on trial, that solutions do not exist for all combinations of Q and/i. For each value of Q, there is a limiting value of the blockage ratio l/h that cannot be exceeded: this limiting value is given by

+ »0 - (1 + QY i + Q Q + Q tan' Q 2(1 + QY (21) I t is easily verified that, in the hmiting condition, the length of the cavity is infinite, so that the solution degenerates to that of Part I . Moreover, the liqmd at infinity down-stream is on the point of cavitating. Hence the limitation is an inherent physical one. I t bears some analogy with the choking phenomenon in a transonic wind-tunnel.

The limitation, at low ca-\'itation numbers, is extremely harsh, e.g. at Q = 0.05, the blockage ratio cannot exceed about 1/1500 (cf. Part I , Sect. 2). Alternatively, for a blockage ratio of 0.05, the minimum cavitation number obtainable is 0.6.

When numerical values are considered, i t is found that for admissible solutions, the drag coefficients for any given cavitation number are virtually the same as in Case A: this is due to the very low blockage. The cavity tends to be larger than with infinite fluid, i.e., in effect, the cavitation number is decreased by the fixed boundaries, especially when conditions are nearly critical, but comparison of calculated cavity contours shows that this effect becomes appreciable only at points substantially downstream.

4. Case C. The lamina in a free jet of finite width. Take the same arrangement as in Case A, but with the body symmetrically placed in a free jet whose width at infinity is 2h units. Still restrictmg consideration to the upper half ^-plane, the region in the W-plane is again an infinite strip and the transformation W-planes of f and u are as shown in Fig. .3. In these planes, it is necessary to take into account the points J, M at which the boundary stream-lines inflect.

Taking account of these singularities and proceeding along the same lines as in the two previous cases, one finds the transformation relations

dz 2hk where

t- = -(1 +

ny/^r^iC^

^ = - log (1 -H Q) (24) TT

and /ƒ, is the Jacobian theta-function constructed, like cd u, with modulus k. k in turn must be found from the complicated integral equation

HI + Qf" f [Hiiu + i0) 2hk

ƒ„

In this expression, the complex radical takes its first quadrant value. In terms of k and jS, the cavity dimensions are now found to be

1/2

sn iu du. (25)

_ hk H{u + i f f )

+

^ - . ( 1 + Qf'"- i„ {H{u + imiu - i f f ) 1 " ^2^^ hk H(u

+

-

" " ' = ^ K T T Ö P io i{Hiu + im(u-ip)V-''''' (27) The intrinsic equation of the cavity boundary is

= 2/t 1 + k ^ HI + Q)''' dnv + k cnv' tme= H(v + ip) - H(v - ip)

(28)

i{H{v + i0) + H{v - i f f ) } '

420 N O T E S _[Vol.

_IX,

The drag and lift coefficients reduce to

(29) 2fe/c g(^•;3 + m) - E{i.p - w) sn

^ ^ ~ x ( l + Q)'''io !ff(i/3 + m ) / ? ( t / 3 - i w ) } " ' i

These relations comprise the solution of Case C. I t is readily shown that, as A; —> 0, the solution degenerates to that of Case A. This however corresponds to very great values of h. and is not of practical interest. I n the general case, (24) and (25) must be solved simultaneously for fc and ;8 by a method of successive approximation. The re-maining results can then, with some trouble, be evaluated.

The case Q small, which is of the greatest practical interest, can be approximately solved in explicit terms. For one finds that this case corresponds to fc 1, so that the elliptic and theta-functions approach degenerate fomis. Thus K is logarithmically large and PIK small in comparison with unity, One develops the solution in powers of Q and retains terms of order Q. Then (24) and (25) become

(30) (1 + Q ) ' " n TT where SS) = 1 - cos /3 - f i s i n ^ log /•Oi\

SM = IT sin /3 log sec /3 + 2[/(tan |,3) - / ( - t a n |/J)]

- cos ^[/(sin /3) - / ( - s i n /?)], in which

a tabulated function [2]. Hence, for given values of Q and h, /S is readily determined. The simplified forms of (26), (27), (28), (29) are, respectively,

2h

1 = n ^ n ^ v 2 ^sin ;ö - cos ^ log sin ^ log tan 5 /3 ,

2h

^ (1 - cos ,3) - sin /3 log (1 + sin j3) ,

« - 1 = n IV.^v^ ^ (1 - cos/3) - sm /3 log (1 + sin &) , (34)

T ( I

+

6 = cot"' (cot /3tanh v) - - v TT

= 2h{\ + Qf" 1 - cos /S + 2 sin log sec

^1,

— ^

r (30) ~ (1 ^ ' O y f ^ [_1 ~ cos/3 4-— sm/3 logsec (9 .

The foregoing general solution for Q small is bound by the condition that fi should not be small m companson w t h Q. This merely implies an upper limit to the permissible width of jet and is no handicap in practice. Within the practical range of blockage ratio.s and caA'itation numbers, the solution holds good.

When Q is very small, the follo%ving first approximations may be used;

\ =

sm,

1 = sin fi,

(37) a = Q- (1 - cos fi),

Cr> = C, = 2 / ï ( ] - cos/ï).

When Q 0, those results become those for the infinite cavity discussed in Part I It^is not part of the present object to give detailed numerical results for application to arbitrary configurations: these i t is hoped to present elsewhere.

Acknowledgement is made to the Chief Scientist, British Ministry of Supply for permission to publish Part TI of this paper. The %aews expressed in the paper are those of the authors.

R E F P ; R E . \ C E S

1. D . P . Riatwuchinsky, Proc. London M a t h . Soc. (2) 19, 202-215 (1920)

2. K-Mitchell Tahhs of Ihe function - y-' log | 1 - , dy wilh an acc.au,ü oJ some properties of thU

and reUded funchons. Phil. Mag. (7), 40, 351-368 (1949). propenus oJ this

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