A vortex model of the potential for the incompressible and inviscid flow past a rigid body

13 JUNI 1979

ARCHIEF

y.

Scheepsbouwkun

Technische Hogeschool

Dem

KUNGL. TEKNISKA HOGSKOLAN

I STOCKHOLM

HYD ROM E KA NIK

A VORTEX MODEL OF THE POTENTIAL FOR THE INCOMPRESSIBLE AND INVISCID FLOW

PAST A RIGID BODY

OVE SUNDSTRÖM

SUMMARY

TRITA-HYD-79-01

A VORTEX MOVEL OF TI-lE POTENTIAL FOR THE 1NCOMPRESSILE

ANV INVISCIV FLOW PAST A R1GIV BOVY

by

o. Sundström

An equation is derived for vorticity distribution over the surface of a rigid body in potential flow. An iterative technique is used for the solution of the equation. Convergence of the sequence is studied by means of the concept of kinetic energy. The theory is applied to the uniform flow past slender bodies.

INTRODUCTION

The application of potential theory to problems involving a rigid body moving in an ideal fluid is well known in hydro-dynamics. The surface of the body is usually represented by a simple distribution (sources and sinks) or double distribution (dipoles). S.J. Andersson has called the author's attention to the fact that it should sometimes be more convenient to use a vorticity distribution.

EQUATION OF THE VORTICITY DISTRIBUTION

Let

{1,2,3}

be an orthonormal basis of the Euclidean

space R . The corresponding Cartesian coordinates are x1 , x2 and x3 . We write the position vector x.é.

The physical body corresponds to a bounded and open domain D'

with a boundary S consisting of a finite number of regular

surface elements, cf. }Kellog (5). Let D denote the

complement-ary set R3-(DUS) , which is assumed to be corìriected.

Let () be the velocity potential of the flow of an ideal

incompressible fluid. We assume that 2() is harmonic in a

domain containing D'US .

Let q() be the perturbation

poten-tial for the body. Then the velocity potenpoten-tial of the disturb-ed flow becomes

+

On the surface S of D' we have the boundary condition

().Vc()

O,

where

/x.

and ñ() is the outward unit normal.

i i

be two

functions harmonic and continuously differentiable in a closed regular region EUE , then

O =

J

[v1().v2())

+ aE -J

dS()

This result follows by the divergence theorem.

Let

and

2

Using the notations

we have

() -V,)

Set i=i , =c and apply (2.1) to the domain D-O() , where

(2.1)

and

/.

V) vQ()

+

(x()x()) dS()

,

z V2() + V() z

Q ,

The perturbation velocity becomes

V)

=

J+()()

_dS()

where

F() =

()x()

is the vorticity distribution on the surface S

-2-(2.2)

O(i) is a small sphere with center and the radius E

Then, if

Vq()j

O uniformly as , we obtain at the

limit c=+O

dS()

_xD()V)

+

J+/n

dS()

where

/n ñ()V()

, and _X denotes the characteristic

function of the set E

Ti, XCE,

xE(x) z

O, E.

S+ _{signifies that the integral is taken over the exterior side}

of the surface S which bounds D

Since AQzO

throughout DUS

, formula (2.1) applied to

DO() yields

(x(ñ()xQ())

dS()

_XD(X)

+

Summing these equalities, and using

D/Bn=O

on S , we deduce

3. AN ITERATION PROCEDURE

We now seek a solution of equation (2.2) by a method of successive approximations. Therefore we define the iterated

functions . by

VQ()

+

ds(, ji,

with o = , .

-Q

when ±

Applying the divergence theorem to the domain DO() wo

obtain

+

Js

()

DQ()/n dS()

and we note that the normal derivative is continuous across

S . The iteration procedure is repeated and it follows that

+

() .()/n dS()

J _S

For the normal derivatives, we obtain

(

()/n) ±

= +

ñ()(,)

.1()/n dS().

Let us introduce

u.() =

-Then Hence

Vu.()

f

()

(Bu. 1)/BnJ dS()

J _S J

By means of these functions we can write

VQ() + V.() = VQ() +

Vu.()

,

V.(x)

1=1 1

VQ() + V.() = Vu.() ,

The sequence of normal derivatives of the functions u.

becomes, using the notation p.()

(Bu.()/Bn),

pi+1 () + _I

dS()

J

Using the notation

Tp(i) -2J

)ñ()) dS()

we can rewrite the normal derivatives

It is known that if S _{is a Liapunov surface, then a continuous}

and bounded function p _{on this surface generates a Hölder}

con-tinuous function Tp (cf. Günter (4) ) . Accordingly, in this

class of surfaces ñ'V4. remains continuous across S

J

4. ENERGY RELATIONS

Let p and be L'-integrable simple distributions on S

which generate the potentials

=

dS(),

=

-J

,p() dS().

We introduce the symmetric bilinear form

1515

Let

(u/n)

and (u/an) denote the normal derivatives of

u at interior and exterior points of S respectively. Then

(u/n)

(1-T)p ,

(u/nJ

-(1+T)p

By the divergence theorem it follows that

fVu.Vv dV

_J

y

(u/n)

dS =

J:vusvv dV

-J

y (3u/n) dS z and hence

JVuVv dV

_(p,)

R3

Because of symmetry we obtain ((1±T)p,v) =

p,(1±T))

, hence

We write

(Tp,v) = To(p,v)

It follows immediately that

(1±T)op,p) O _,

Ki,i)

O

Hence

Further, for every real number o. , +o.v,p+o.) O . Hence

the Schwartz inequality

<

If f(T) is a polynomial in T , we can form the simple distribution f(T)p , and obtain the inequalities

(1±T)o(f(T,f(T))

= (1±T)f(T)2o(,p) O.

We have the identity 1-T2 ((1+T)(1-T)2 + (1-T)(1+T)2) and

it follows immediately that (1-T2)o(i,p) O or

KTp,Tì-')

Applying the energy integral to the recurrence formula

(1-T)p. , Q/n , we obtain

(1-T)2o(p.,1).

The inequalities (Tp,p)j(p,p) and (Tp,Tp)(p,i) give

and

(1_T)op11,i11)

(1-T)o(i,i1), (1+T)o(p.1,p.41) (1+T)o(p.,i.),

(1-T)o(p.,p.),

24(1+T)o(p.1,.1)

(1-T)o(.,p.),

2(1+T)o(p.

_j+1

,p.1)

(1-T)o((p.,p.)

-Hence 2

(1+T)o(p.1,.1)

1=1 (24.1) (24 . 6)

Since

(l-T)1p1()

,

i1

, (4.7)

it is easily seen that

(p.,p.) O , (1±T)o(p.,p.) 1 0

. (4.8)

The energy of the perturbation velocity outside the body becomes

(Vj dv

= (1+T)o( p. , p. _{) =} D _{i=1 i=1} =

(p,pi

-1=1 Since, by (4.7), (1+T) p. _p -i=1 J

As a matter of fact, the kinetic _{energy is} p _{times this}

expression, p _{is the density of the fluid. For the two cases}

¿=22. even and j=22.+1. _{odd, we obtain by (4.7),}

I

(V.)2 dV

_{((l-T)oKp.,p.) +} 1=1 - _{((1-T)o(p.,p.)} + (p1,p1)) , r

_{(V)2 dV}

((1-T)o(p.,p.) + J 1=1 ((1-T)o(p.,p.) + (p.,p.)) + _{(l+T)o(p2.1,p2.1).}

Theorem 1: If D - is a sphere, then To(p,p) 1 0

Proof: Let the radius of the sphere be R . We have

2Tp()

_{-J ñ((-) -3p() dS()}

where ñ()

/R and 2 =

= R2 , if the center is origin

of coordinates, and we find

4RTp()

= JS

dS()

=

_dS()

Hence 2 R(Tp,p) =

dS()

,P()

_dS()]

_o .

I

-6

(4.9)

co tH H (D

Z

(o Ç)

0

ZH

HM

-

(JJ -

o

H i-HO

-- -o t'i H. H

00H

_H

-)te subdomains of D and D such

. <

H _us d _{about any point}

cD1UD lies + C-»

0

-a H + _H k-O

Jjt'1

HZ Q

O

_{H H}

H F

-i.

_{tends to zero uniformly}

_in

_D1(d).

O 'i

u0

H

H _{n lIN.}

t of D1(d) , then as harmonic

ithmetic means throughout _spheres oy are harmonic Cn C')

()/

dV()

t-1 - _£ L'I inequality gives __i'__' i)

ji1IU

JK

(u.()/)2

dV()

(3/4îd3) (1+T)oi.,p.) _{ó (3/rd3)} If _E24rd3/3 , then < e for _ilN(6) .

I

J-By (.8) it follows for

the perturbation velocity

I

(Vj2 dV

J dV _J

(Vj2 dV

when ij

. (5.i) 1 D ' D Set M _11m

f

D

(V)2 dV

sphere,

the inequality (1.5) _yields

(1-T)o(.,p.)

, and it follows that

harmonic functions we infer

imme-ary positive number. The conditions

to each 6>0 there is a N(6) such

Theorem 3: If M<,

then the sequence

V. converges

uniformly

in D1(d) to a harmonic function V

Proof: Given >0 . By (5.1) it follows that to each 5>0

there is a N(5) such that M-<...

J

(Vj2 dV

J

(V.)2 dV

M D 1 D

when jiN(6) .

Let be any point in D1(d) . By the mean

value theorem for harmonic functions, we have

-

.()/x

(3/4d3)J (./

-

./) dV()

where 2=1,2,3 . The Schwarz inequality and (5.1) give

-

.(/x)2

(3I4d)J (v

-

vJ2

dV < (3I4d3)

If -241Td3/3 , then

-

.(/x

< when

j,iN(&)

Hence

a./3x-

uniformly in D1(d) .

By Harnacks

theorem it follows that the limit function is harmonic.

I

6. SOLUTION OF THE PROBLEM FOR THE SPHERE We have shown that

V() + V.()

VS)

+

Vu.()

,

VÇ2() + V.()

Vu.(x) , XED

The conditions (4.9) and (4.10) give

(v)2

dV 2

(u1)

3 ₁₌₁

This inequality holds whenever To(p,p) 0 . In this special case we may also derive that the perturbation velocity is

integrabel over the outside and inside of the surface

S . Here

the sequence of normal derivatives becomes

2R.1()

Rp.() - u.()

and we find

r p dS R p dS _-

((1_T)o(1,)

₊ i+1 is Thu s

il

1J dB ii dS 1+1

By means of the divergence theorem the following equalities

can be shown

1-T)o(p.,p.) ((Vu.)2

- 2(Vu.)

û.Vu.) dS()

(1+T)o(p.,p.)

f

_{( 2(.Vu.)}

_{Vu. -}

(Vu.)2 ñ.)

_dS()

13+ J_

i

Hence, since =

Rñ()

4' 1 _{(Vu )2 dS}

r (Vu )2 dS 4 p aS

Js_

)3+

s

We infer immediately that V

Vu. - VO

is integrabel

forall

j.

Using Schwarz inequality, we find

( dS (

J(vu

dS

and it follows that 1

(V.(fl2 dS()

S

(p())2 dS()

is+

i

Then,

by Schwarz inequality, the perturbatiion velocity is integrabel over the sphere.

Since (i.,pi) - O as i -- , it follows that (1-T)o(p.,p.) O

Theorem 2 can therefore be extended to the domain D1(d) Finally, the condition M<

in theorem

3

is

fulfilled.

7. CONVERGENCE IN THE CASE OF A REGULAR REGION

Let DUS be a regular region (cf. Kellog (5) ). We shall prove the following theorem,

Theorem 4: Let LY be a regular region and D2UD a sphere. If AQ=O in the closed sphere, then the iteration procedure is convergent.

Proof: Set D3

D-(D2UD)

and let _{be the surface of the}

sphere.

According to previous result, the problem for the sphere

D2UD has a solution

+ vq() ,

v

() =

o

+ v() ,

cD2UD' with J (V )2

dV<

R3

Now, we form a new sequence W. , defined by

Vw.()

_- , W.(cx) ₌ Q

Since V O _{in the sphere, we obtain for xD2UD}

Vw.()

v)

+

dS()

For points D' we have

_{VW.() = Vu.()}

Obviously, the normal derivative is continuous across and

the jump of W. is independent of its index j . Therefore,

(VW.)2 dV

_{ñ().VW.() (w7() -w)) aS(r)}

₊ D3 R3

-1

ñVW. W

dS + I

ñVW.

(wT

-w

₎

as

j-1 Hence f (VW.)2 dV = f I)

We now obtain the inequality

2f

(7W.)2 dV = (1_r)o(p.,p.) D

j-1

-

(VW)2 ) dV - 1_o -+

).Vw.() (wT() -W;()) dS()

Thu s

r (i-T)o(p.,.)

J

(VW)2 dV

_- limJ (VW.)2 dV j=2 D

j--D

where

J

(VW1)2 dV

2f

(V)2 dV

+ J (VQ)2 dV < D D Using (4.»4), we obtain by (4.9),

r

+ (i-T)o(p.,p.) 1=1 + (Pl,Pl) +

+ r

+ 'D

(2

<

Hence, lim(p.,p.) = O and limJ

(V.)2 dV

M <

j-D

The theorem now follows from the theorem 3.

8.

APPLICATION TO THIN OR SLENDER BODIES

Let D represent a body in a homogeneous flow wfth the

velocity VQ = Uë1 . If the surface normal ñ is mainly almost

orthogonal to ,

i.e. ñ

o(i) ,

we may consider as a

good approximation of . We obtain for ED

v) v()

U1 +

UJ

(x(ñ()x1)

dS()

= UV(x1

- JD

(ci-Xi)

i-- dv())

and as

Ux1 +

(U/1)V(D)

x1

For a slender body, we set = and obtain an

approxima-tion for the volume integral. Let the area of the

cross-section of the body be A(i) where -LF1L .

Assuming

A(L) = A(-L) O , we have, on integrating by parts

Ux1 - (U/4)j

((1-x1)2

-

x

+ x)

dA(1)

-L

(i-T)o(. ,p1)

I

ID

When there is a solution, we can write

+ VR(,ct)

with V V +

x(x) dS

If the remainder R is minimized by ci. in the following way

J(VR)2

dV

we obtain, since

Vc0

in D

J((1-)V

+ aVu1)2 dV

In that sense we obtain a still better approximation by using

VÇ instead of

V1

ACKNOWLEDGEMENT

Professor B.J.

Andersson

has given many valuable suggestions. In particular, he demonstrated theorem 1 which made it possible to realize this theory.

LITERATURE

ANDERSSON, B. J _{"On the Mathematical Model of a Ship,'t} International seminar on wave resistance. The Society of Naval Architects of Japan.

(1976)

BATCHELOR, G. K. "An Introduction to Fluid Dynamics,"

Cambridge University Press

₍₁₉₆₇₎

COURANT, R. "DirichletTh Principle,"

Interscience

Publishers, New York

₍₁₉₅₀₎

. GUNTER, N. M. "Die Potentialtbeorie," Teubner, Leipzig

₍₁₉₅₇₎

KELLOG, O. D. "Foundation of Potential Theory," Dover Publications, New York

₍₁₉₅₃₎

LAMB, H. "Hydrodynamics," Sixth edition.

Cambridge University Press

₍₁₉₆₃₎