Linearized two-dimensional cavity flows about symmetric bodies

TECNiSc:E HOGESCHOO.L

Inst.ituut voor Toegepaste

7iiskunde

Jaffalaan 1.62

De1ft

Lfl1EARIzED TWO-DTh1EN5IoL

_{CAVIY FLOWS}

ABOtIT SYMMTRIc BODIES,

by

-J..&.

Geurst

Rapport 9,

1956.

.

h&che Hogesco3J

beift

Linearized two-dimensional _{cavity -flows}

about symmetric bodies,.

by J.i. Geurst.

Summary:

In: this paper .sothe resultV by _{LP. Tülin concerning}

cavita-tional flow about syrnetri.c bodies, _{i.e. a formula giving the} relation betjeen the cavitation number and the cavitation length arid anexpression for the _{cavitation drag, are derive in a quite.} different way b _{means of functiontheoretical methods.}

la Introduction, V

V V

V

An incompressible fluid flowing about _{a. symmetric body wIth}

uniform velocity Uc., _{auses behind that body a wake. In the}

physical Idealization' of _{potential flow the wake is assumed th}

be

an area of constant pressure, i.e. the

_{per resije}

_{f the} fluid. An exact theory of this-flow on the base of free stream-. line.s (Helmholtz flow) is Only possible on the assumptIon of an infinitewake. This is the result _{of BriIlouiri's paradox for} finite wakes (see Birkhoff (1) _{). Acording to linearized theory,} however, a finite wake behind _{a. symmetric body can exist,}

Knowing the points where the Symmetrically shaped cavity adheres at the body and requiring that at these pblnts the tangent to the body should pass Over continuously _{into the tangent to.the cavity} (juncture condition) it is- _{possible to obtainan unique relation}

-

pc_fl-p

ship between the cavitation _{number o = and the length of} the cavity (see TulIn (2) ). .

-oj2.

This reCuit together with _{a formula fqr the cavitational drag} will now be derivedVby _{a method adapted by R. Timman and the} author in an ana1ogou case (see

₍₃₎

_).

_{For this method the} evaluation of several .integals is _{riot needed..}

2, viathematica1 forthulation of the _problem,

At a _{yrnmetricai1y shaped two-dimensional slender body with} symmetry axis along the x-axis of a rectangular coordinate system and-extending from x -

c to x = b,

_{adheres at x o a cavity}

extendingunto x= e

_{This cavity is caused by a stteam with}

uniform velocity U _{at infinity, directed along the}

_{positive. x}

_axi (see fig. 1)

-IT

d.f dx df - dx

-c<xcO

-C(X

<0

-_-;' -* - fig. 1.

The dIsturbance velocity field (u,v) _{has to satisfy the followin} two equations

u+ v

= 0 _{continuity equation for incompressible flow.} -

u

- v.. 0 irrotational flow.

On the assumption that the thickness of the _{body and the cavity,} and the slope of the tangent to the body _{and the cavity are small} compared with the length of the

bodr

_{plus cavity, a linearized}

theory is possible.

ernoulli's equation. takes the form

pc_, - P

-u. Therefor the boundary conditions are :

1)

u----O, v_O at infinity:

U.

+)

moreover'v' must be continuous at x o. This ho].ds for both sides of the 'x axis separately.

'ty

-o _B

.C.u.

-

'2

2

c-:.

fig, 2. .."

,

Solution of the problemq calculatjon of the drag

w

U - iv is an analytic function of the complex variable

= x ly.

The mixed' boundary value problem' for w can be

formu-lated immediately as 'a Riemann-Hilbert problem. (see

Muskheiishvili

₍₊₎

). 'Fird

a. sectlonally holombrphic function w(z) with the x axis as line of discontinuity. Between

_{w and w,}

the limiting, values of w when. z approaches the. x axis from the

upper or lower side, mere :must hold

:

w+w__O

w w -

2i

'

- c

<x

<

0

w

₊

_{w 2.}

+

w - w

=O

2<.x

<cs;..

(u and V has been made dimension-less by.means of U ).'

The solution of the homogeneous prbblem with

_{the right ha-nd sides}

of these linear relations equal, to 'zero,- Is

Wh

P(z)

where P(z) is a polynomial in z with real

coefficients..

2' I

ccx

y=o

p__ - p where -.

Taking (z) = z , we have in Infinity and contiious atz 0.

For -- _{the) linear relationsThow become} w +

(L)

-. _{-&, < x c} Wh Wh. +

-(L)

_-

(L)

= _{- 21} -c z x <.0

()-

2 +

-1

W__2?ri

(.L) -

(L)

Wh

Using the Plernelj

T(to)-with (z) 271:1 formula

T

(to) (to) ( q?(t)at t-z L -.

the solution can be writter as

f

21

r

+ j-

+ solution horn. probi.

df \ /

J_\/____ iv_

7 _k/

ZX-z

dx + + x < c-:i

as ?olution, finite at°

/

This Solution is finite at infinity and _{continous at z=O. Taking} into account w 0 at iiifiñity we have to; put

So C

f

_..

_{Remark: the branch of\/' is determined}

_{by the}

The

relation bet'wê

condtton that the

the Thrm (7'- vdx =

the contour of the Or

- Ifli'Wdz= 0

i.e.

Re [residue w at z' =c- ₌ apting thIs condition', one finds

U)

aequivalegto(j6)

of Tulin (1).

U2

_C)

en 6 and

2

is now found by Imposing the

cavity must be closed. This _{conditIon takes}

0 where th _{interatjon is performed along} body plus the cavity in clock1se _d1rection.

fig.

3.

/2,g.

vhich is'

-fIg.

3).

The drag D is calculated _{as follows (see}

D = -

p)

d = u v dx =

_u2

dx body "°2 2dz

,3U-gYuvax

2 body. body+cavity _bod.y+cavity Re {res.w2 at z

°df

(

dx L

I

by Y = 0

(dr

J

dx

-x

+ d.-c dx df dx,

-2

/tJ

(7L..LJ

thagreémeit with (.19) of Tulin, (1).

+

df

_dx

References: .

G. Pjrkheff, Hydrodynamics, _{Princeton 1950.}

..

M.P. Tulin,. Steady .tw-dimensiona1 _cavity

_{ab.ut slender}

,bedies, Report

₈₃₊

_{David W. Tayler model, basin,}

Washington 193. ;

(3)J, Geurs.t and R. Timman, _{Linearized theory of flow with}

C init _{cavities about a wing.}

Report 7, Inst. voor Toegepaste Wiskunde, Deift,196. N.i. MUSkheljshvjlj, Singular _{integral equations,} Groningen i93. .