Image system in an ellipse due to an external source

ARCHIEF

IVAGE IN AN ELLIP3E DUE TO AN EXTERNAL SOURCE

by

Natilde Macagno

Iowa Institute of Hydraulic Research State University of Iowa

Iowa City

y.

Tcchiche

_IIc-e:chooI

Defj

Office of Naval Research

IMAGE SYSTEM IN AN ELLIPSE DUE TO AN EXTERNAL SOURCE

by

Matilde Macagno

Iowa Institute of Hydraulic Research Stato University of Iowa

Iowa City

Introduction

One of the most interesting theorems in hydrodynamics Is known as the Sphere Theorem [lii. This theorem gives the flow about a sphere inmiersed in an arbitrary potential flow.

To find methods for solving similar and more practical problems for elongated bodies, we tried at first to determine the images in a spheroid due to an external source. We obtained a solution, expressed In terms of Legendre functions, but it still remains to interpret this as a distribution of singu-larities.

To get some ideas, we decided to begin with a similar problem in two-dimensions, a problem that also has interesting applications in hydrody-namics, that of determining the image system in an ellipse due to an external

source. This has the advantage that it can be solved by conformal mapping. This method, and the known image system in the circle due to an extern1 source, enables us to get the desired results.

Statement of the Problem

Our problem is to find the image system that a source of strength m induces in an ellipse of semi-axes a

and b

Since the problem Is a

two-dimensional one, we may use the method of conformal mapping.

The image system in the circle Is given by another source located at the inverse point with respect to the circle and a sink located at its

center.

By the Joukowsky transformation

-4-that maps an ellipse in the z-plane into a circle in the -plane, we can find the image system in an ellipse (Fig. 1). This transformation (1) is not one-to-one. For each value of z there are two values of ; hence,

the -plane is mapped into a fliemann surface of two sheets over the z-plane. On separating real and imaginary parts, Pe(z) and Irn(z) , we have, with ' re' ,

X = a cos Q a = r + l/r

(2)

y = b sin Q b = r - l/r

An ellipse with semi-axes a and b is mapped into a circle r = const. For every r i the two circles of radii r

and hr ,

are the mappings of the same ellipse of the z-plane, corresponding to the two sheets of the Riemann surface. The circle of radius r = i corresponds to the line segment from x = -2 to x = +2 , in the z-plane, i.e., it is the mapping of the segment joining the corrmon foci of all the ellipses; this segment is the connection between the two sheets.

For our problem, we assume that the correspondence between the

z-and the ¿' -planes is such that, in the first sheet, the exterior of the ellipse of semi-axes a and b maps into the exterior of the circle of radius ro, and the interior of the ellipse into the ring 1< r <r0

Analysis of the Flow

The complex potential in the ' -plane, due to the source of strength m , in the presence of a circle of radius r0 , with center (0,0), consists of the given source m , located at A , another source ni , located at B , the inverse point of A in the circle, and a sink -m , located at the origin. The flow pattern is given in Fig. 2.

Assume now that the source is located in the z-plane on the real

axis. The complex potential in the -plane due to the source at (d,o) is:

where cl indicates the location of the source outside the circle and r02/d

that of the inverse point.

A conseq.uence of the selected correspondence between the and z-planes is that the sink at the origin = O has no counterpart in the

z-plane. If

_1r021/dI>l

the source at that point in the z-plane will map into a source on the real axis, lying within the ellipse between the given

source and the nearer focus; otherwise this source, as well as the sink, will be located in the second sheet of the Riemann surface. There is a critical point on the major axis extended, at a distance r02 +

hr02

from the origin. Only a source closer to the ellipse than this critical point will have an image in the ellipse at a point corresponding to the inverse point in the

circle.

There are other singular points that are shown by the complex veloc-ity

This indicates that there are singularities at = i corresponding to branch

points at z=±2.

The unit circle maps into the segment (or slit) between the foci x = -2 and. x +2 in the z-plane. In the mapping the streamlines that pass through the unit circle and go to its center correspond to streamlines that pass through the slit, to the second sheet of the Riemann surface.

The general pattern of the flow due to a source outside the ellipse shows streamlines that start from that source (one of them containing the

ellipse) and go to infinity. _{Within the ellipse the flow pattern depends upon} irhether the mapping of the inverse point = r02/d lies on the first or sec-ond sheet of the Riemann surface. In the first case streamlines emanate from the source at this point, pass through the slit, and continue _{to the sink at} infinity in the second sheet. In the second case streamlines from the source in the second sheet enter the ellipse on the first sheet through one part of the slit, return to the second sheet through another part of the slit, and

LW

cW

ct

= (4)

-4-.

continue to the sink at infinity. In either case the flow pattern Indicates the presence of a distribution of singularities along the slit, of sinks in the first case, and sources and sinks in the second.

Iniae Slstern

Within an Elliysc due to a Uniform Strewn

To gain an insight into the method of solution, first consider the problem of the source distribution for uniform flow about an ellipse.

Using the Joukowsky transformation (i) we map the circle of radius

r0 Into the ellipse of semi-a;:es a and b a r0 + l/r0

b r0 - l/r0 and semi-focal distance e = \/a2 - b2 = 2.

In the ¿ -plane, the complex potential of the circle and the urform

stream is given by:

W(

U()

(5)

Subtracting the part corresponding to the parallel flow we obtain

z

W, =

-U

U(+)

-U(+*)

k',= U(tl).+

i.e.,

the complex potential of the image within the ellipse. The complex velocity is

¿W,

_a-L?J=

Uk-t)

d

₌

_-U

k0z,

cL3

-

OLZ

We are interested in the conditions of the flow in going throuíh the slit, i.e., in the -plane, in crossing the unit circle. There = e19,

¿w _U/z1)(16)

and

=

z

= -(-i)

_z

c6

Assume u and y are the velocity components at a point P of the slit as it is approached from the upper side, and

ue and the

veloc-ity components at the point when approached from the lower side; i.e., u u(x, + O)

y

= v(x, +

_o)

u u(x, -

o)

v v(x, -

o)

Then defining the differences

L u=u,.. -u

_u

LXv=v -v

_u

we obtain

L\v-=_U(-i) co-te

(6)

Thus, there is a jump in the v-component of the velocity. This discontinuity can be attributed to a distribution of sources on the x-aris, between the foci, and Its strength is civen by X , as is shown in treatises on potential theory [3].

By means of the Joukowsky transformation (i), we obtain

2CcO

j_ftzZ

_{' o)}

ì_ k

By substituting into (6), we get for the strength of the distribution

_À

_the expression

-k--

-

U

i

-

1F

-6-Hence the image within the ellipse due to a uniform flow is a source-sink distribution on the segment between the foci of intensity

2

given by (7).

The analysis of the distribution of singularities in the focal seg-ment can also be formulated as the following boundary-value problem:

Values of the normal component of the velocity y are prescribed on the x-axis between the points x = -2 and x +2, to be v(x, +0) =

-v(x, -o); also v(x,

o)

o for

_Ix

> 2:

The problem consists now of finding a potential function y in the plane (or w, the velocity potential), with the Civen boundary conditions, which vanishes at infinity.

This function y is antisymmetric; i.e., v(x,y) -v(x,-y) .

There-fore, the problem may be stated for the upper half o' the z-plane, with y prescribed along the x-axis.

The velocity potential for this problem, as is well known, is given by a distribution of sources on the focal segment, of strength per unit length given by (7).

Imnge System in an Ellipse due to an External Source A. Source on x-axis

We assume again a source m, located at

(a1, o),

d1> a.

Accord-ing to the transformation (i), there will be a source ru in the -plane, located at (d,o), d1 = d + 1/a.

The velocity potential in the )' -plane, as we have already seen, is given by

W) =

gtrt

()(k0z/)

With =

reG,

we obtain for the velocity potential

or

-e

2 _IL

(8)

The image system consists of another source located at the point

2

corresponding to = r0 /d, (which may lie on either the first or second sheet of the Riemann surface) and of an unknown distribution of singularities extending between the foci. The latter will be obtained, as in the case of

the ellipse and a uniform flow, from the expressions for the velocity compo-nents at the slit. These are

iA=

(_\

(i

òfl /fL\

4I

-7-tt.. -

()/('-)

'1

'

Differentiating (2) and (a), we have

= -2s&nO

(--)

₄

_ìtrl

2À'm

-

Íid.ws9

t -

("/a)cose

(-2d.cC-sO±cV

+

(1

_--)

_{= flL}

t O

I

/ d

e

= ¡ _{I -} a o t

-

a

o'/) OS

_{O +} (9)

-O-Introducing these valuas in (9) we obtain for u and y the following expressions:

I a.

¿

_tIdcß3Q+d.z

Z$.'-'tS t. f-2d.CÁ59+

+

and the jump in the velocity components is

1A = O

-

r

i - (ka/aa)

-

£,n9 t.

i-zOcsG1-d

This shows that in the seOnent between the foci [x = -2, x +2] there is

a distribution of sources of intensity given by

A

-

m

_{I2(07d)c5+(/a.2) j}

i

(ii)

As we know, the condition

r02/aj

i decides the characteristics of the distribution. If Iro2/dI> i , Eq. (11) is negative for 0< (Tt'; hence

the image system will consist of a point source within the ellipse, and a line distribution of sinks of intensity per unit length given by (ii). Otherwise,

if r2/dk 1,

Eq. (li) takes positive as well as negative values and the image system will consist only of a linear distribution of sources and sinks of intensity also given by (11), and the point source will be located in the

second sheet of the Riemann surface.

Notice that this problem can also be solved as a boundary-value

prob-lem. On the real axis, to the left an right of the foci, dw/dz is real and y = O . The conditions prescribed on the real axis are as follows:

iT=O

(io)

ji

f

g-cLcs9

₊

-9-These boundary values are satisfied by a linear distribution of sources and sinks of strength 7 , extending between the foci.

Source on the vertical axis

In this problem, the source is located at (O, h1). The image sys-tem i.rlll consist, once again, of a distribution of singularities, corresponding to the lina of discontinuities between the foci.

If, in the circle plane, the source is located at (o, h) the com-plex potential of the system will be:

(12)

az for we obtain

e. (13)

¡ _z ,t.

To analyze the distribution of singularities on the focal segment, we write again(/r) ana(Ja/9)for r = 1 . Substituting in (9) we finally

obtain for the discontinuities

-G [

g (i+)

(t/)q/)

+

(+- 4

The discontinuity in y shows that there is a distribution of sources and sinks of strength given by

A=ff,

as in the former problem, and the dis-continuity in the u component corresponds to a line distribution of vortices, with positive sign counterclockwise and strength per unit length

27T Source at an arbitrary point

We now treat the general case when the source is located at an arbi-trary point z0 corresponding to in the y-plane.

lo

-The complex potential Is now given by the expression

f

W()

L-pe

)(4eQc)

1'

Notice that this expression is also the Green's function G(x,y) of the second kind for the ellipse, by means of which more complicated Neumann problem with arbitrary values of the normal derivative of the potential, can be solved

From the potential which in this case is given by

T z

-we obtain the velocity components.

As in all the previous cases, the image system will consist of a distribution of singularities between the foci and, if

r02> d

, a point

located at the point of the z-plane corresponding to 'ç,

=

e

From the Jumps in the velocity components given by

LI U

'nii1c

Ç

e _{l(pt,a_2?}

o-cÇ+- 2?c5(&fc

('Ì)

((c +

- 2

(c/p) e5c.)

(t+(k/ft) -2('/f)co5 (9-)J[I

(ko«/) -Z(/p)cs(+«)}

.. Ç

2.S'n9

_{(.I2-2ycc5(e))[H?2_2?}

C5(e+c()3

('-

)

-(h.o/) _2(koZ/f)c& cosc

-2

c,s (9f)')

we observe that, in the general case, the image system consists of a distri-bution of sources and sinks of strength

2t=&r/2rr

and a distribution of vortices of intensity

crU/zrr,

with positive sense counterclockwise, both distributions lying between the foci.

References

-11-[i] Weiss, P., "On Hycirodynarnical Images: Arbitrary Irrotational Flow

Disturbed by a 3plicrc," Proceedings, Cambridge Philosophical Society, Vol. 51, Part 2, 1955.

Kellogg, O. D., Foundations of Potential Theory, Murray Printing Co., New York, 1929.

Cheng, H. K., and Pott, N., "Generalization of the Inversion Formula of Thin Airfoil Theory," Jour. of Rational Mech. and Analysis, Vol. 3,

No. 3.

Acknowledgments

This work has been carried out in the Iowa Institute of Hydraulic Research under the sponsorship of the Office oiT Naval Research, Contract Nonr 1611(01). The author rishes to express her appreciation to Professor

Louis Landweber, under whose Guidance this work was done, for his continuous advice and valuable suggestions.

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