Title: Motion of a flexible hydrofoil near a free surface
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Title: Motion of a flexible hydrofoil near a free surface
Physical Description: xii, 102 leaves : illus. ; 28 cm.
Language: English
Creator: Reece, Joe Wilson, 1935-
Publication Date: 1963
Copyright Date: 1963
 Subjects
Subject: Hydrodynamics   ( lcsh )
Planing hulls   ( lcsh )
Waves   ( lcsh )
Engineering Mechanics thesis Ph. D
Dissertations, Academic -- Engineering Mechanics -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of FLorida, 1963.
Bibliography: Bibliography: leaves 99-101.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
General Note: Vita.
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Bibliographic ID: UF00097960
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000568378
oclc - 13649198
notis - ACZ5109

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MOTION OF A FLEXIBLE HYDROFOIL

NEAR A FREE SURFACE












By

JOE WILSON REECE













A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY











UNIVERSITY OF FLORIDA
December, 1963














ACKNOWLEDGMENTS


The author wishes to thank Dr. J. Siekmann, Chairman of his

Supervisory Committee, for suggesting this problem and for his con-

tinued encouragement and counsel.

He is grateful to the other committee members, Dr. W. A.

Nash, Head of the Advanced Mechanics Research Section, Dr. S. Y.

Lu, Associate Professor in Engineering Mechanics, Dr. I. K. Ebcioglu,

Assistant Professor in Engineering Mechanics, and Dr. C. B. Smith,

Professor of Mathematics, for their encouragement and helpful criti-

cism of the manuscript.

The author is also indebted to Dr. I. C. Statler, Head of the

Applied Mechanics Department, Cornell Aeronautical Laboratory, for

his timely suggestions and advice.

The author thanks Dr. Nash and Mr. B. M. Woodward, Man-

ager of the University of Florida Computing Center, for providing

financial assistance for the use of the IBM 709 digital computer.

To the Ford Foundation he expresses profound gratitude for

supporting his graduate studies for the past twenty-seven months.

Finally, he is grateful to his wife, Nancy, for her support and

understanding during a difficult time. She has been a "rock" in a sea

of "free surface waves."














TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS ........ .......... ii

LIST OF TABLES . . . . . . . . ... .. v

LIST OF FIGURES . . . . . . . . . .. vi

LIST OF SYMBOLS ................... vii

ABSTRACT . . . . . . . . . . . . x

INTRODUCTION . . . . . . . . . . 1

Chapter

I. FORMULATION OF THE PROBLEM . . . . 5

1. 1 The Motion of the Plate
1. 2 The Plate Model
1. 3 The Velocity Potential
1. 4 The Integral Equation

II. SOLUTION OF THE INTEGRAL EQUATION . . . 23

2. 1 Form of the Solution
2. 2 Solution for the Infinite Set of Equations

III. THE EFFECT OF THE FLOW ON THE PLATE . . 30

3. 1 Pressure Distribution
3. 2 Lift and Moment
3. 3 Drag and Thrust

IV. NUMERICAL EXAMPLE ............... 46

4. 1 Quadratically Varying Amplitude Function
4. 2 Solution of the Algebraic Equations
4. 3 The Resulting Thrust














TABLE OF CONTENTS--Continued


Page

V. SUMMARY AND CONCLUSIONS . . . . .... 58

Appendixes

A. THE FREE SURFACE POTENTIAL . . . .. 63

B. COEFFICIENTS FOR THE INFINITE SET OF
EQUATIONS ... .. . . . .. .. . .. .83

C. A p(e,t) FOR THE CASE IN WHICH D--oo . . .. 91

LIST OF REFERENCES .................. 99

BIOGRAPHICAL SKETCH ................. 102














LIST OF TABLES


Table Page

1. Free Surface Flow Conditions . . . . .. 19

2. Real and Imaginary Parts of Bn and Cn . . .. 52

3. Thrust Coefficient . . . . . . . .. .54

4. Thrust Coefficient . . . . . . . . 54

5. Variation of O, j = 1, 2, 3, 4, With kF2 ...... .68

6. G(x,D;) for x oo . ........... ..... 74

7. G(x, D; ) for x - 00 . . . . . . . .. .79

8. Variation of Sj, j = 1, 2, 3, 4, With kF2 . . . .. 81















LIST OF FIGURES

Figure Page

1. The Flexible Plate . . . . . ... 6

2. The Plate Model . . . . .. . ... 6

3. Single Vortex Line . . . . . . . .. .12

4. Vortex Sheet With Special Coordinates . . . .. 12

5. Thrust Coefficient Versus Reduced Frequency . . 55

6. Thrust Coefficient Versus Reduced Frequency
for Small Reduced Frequencies . . . . .. 57
/
7. Contours of Integration for the Ij Integrals .. . 69

/
8. Contour of Integration for II . . . . . .. 70

9. Contours of Integration for Ij Integrals . . . .. 76

10. Contour of Integration for Il............. 77














LIST OF SYMBOLS


Bn


Cn


Cmn


Cmn
,0
Cmn

CT

C(k)


Co, C Cz

D

D

Dj


dn


F

G(x, y; )


Coefficient of the Fourier series giving the downwash on
the plate

Coefficient of the series expansion for the pressure distri-
bution Ap(x, t)

Coefficient of the Fourier series giving the displacement
function of the plate

Coefficient of the Fourier series giving the slope of the
plate

Complex element of the coefficient matrix of the infinite
set of algebraic equations

Real part of Cmn

Imaginary part of Cmn

Thrust coefficient

Theodorsen function

Coefficients of the quadratic plate amplitude function

Depth of the plate compared to a semichord of 1

Drag

Constants related to the free surface radiation conditions
(j = 1, 2, 3,4)

Auxiliary coefficient used in the series for the pressure
distribution

Froude number

Auxiliary function related to the free surface potential











g Acceleration due to gravity
(z)
Hn () Hankel function of the second kind, nth order, with argu-
ment z

h(x, t) Displacement function for the oscillating plate

Jn(s) Bessel function of nth order with argument a

Kn(Z) Modified Bessel function of the second kind, nth order,
with argument a

k Reduced frequency of the plate motion

L Lift per unit span

M Moment per unit span

p Pressure

T Thrust per unit span

T Time average value of the thrust

t Time

U Magnitude of the free stream velocity

u Magnitude of the x-component of the perturbation velocity

v Magnitude of the y-component of the perturbation velocity

Y(x, t) Displacement in the y-direction of the plate

Yn(s) Neumann function of nth order with argument s

C< Wave number of the plate motion

/8 Auxiliary dissipative constant

r Circulation

< Phase angle of the plate motion


viii









E(x, t)

6(x,t)



(x, t)

















(x, y, t)












Re fj
x, yt)
,(x, y, t)





i).




Re~f}

m{f]


Vortex strength distribution for the plate vortex sheet

Vortex strength distribution for the wake vortex sheet

Coordinate of a vortex sheet in the y-direction

Vertical displacement of the free surface relative to its
mean position

Independent variable related to x by x = cos e

Wave length of the plate motion

Dissipative constant

Coordinate of a vortex sheet in the x-direction

Mass density of the fluid

Period of the plate motion

Total velocity potential function

Velocity potential due to the plate vortex sheet and its wake

Velocity potential due to the image system

Spatial part of (x, y, t)

Circular frequency of the plate motion

External force potential

Imaginary unit

Denotes real part of f

Denotes imaginary part of f















Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy

MOTION OF A FLEXIBLE HYDROFOIL
NEAR A FREE SURFACE

By

Joe Wilson Reece

December, 1963


Chairman: Dr. Julius Siekmann
Major Department: Engineering Mechanics


This study consists of a theoretical investigation of the effect

of free surface waves on the swimming of a thin, flexible, two-

dimensional plate immersed in an ideal, incompressible fluid. The

fluid is assumed to be infinitely deep, and the surface waves consid-

ered are those generated by the motion of the plate itself. It is as-

sumed that the plate moves through the fluid at a constant rate, while

executing lateral motion consisting of waves which travel from front

to rear with increasing amplitude. The amplitude of the plate motion

is made to depend, not only on time, but on the distance which the dis-

placement wave has progressed from the leading edge of the plate.

This type motion is used in order to simulate the swimming of a thin










fish near the surface of the water. Of particular interest is the thrust

developed by the swimming action of the plate.

The mathematical theory used is based on velocity potential

methods as employed in unsteady hydrofoil theory. In this theory the

effect of the plate on the flow field is replaced by a continuous vortex

sheet system of oscillating strength. An image system of vortices is

postulated to lie out of the water just above the plate so that the net

effect of the two systems causes no surface disturbance. This allows

the potential of the surface waves to be investigated as a separate func-

tion through the free surface boundary conditions. A kinematic condi-

tion at the plate between the motion of the fluid and the motion of the

plate leads to an integral equation for the vortex strength distribution.

This distribution is governed by the integral equation in a spatial man-

ner only as it has already been assumed oscillatory in the time varia-

ble due to the harmonic nature of the plate motion. The wake vortices

shed at the trailing edge of the plate are accounted for in the analysis.

They are assumed to lie in the plane of the plate and to be convected

downstream with the free stream velocity.

The solution of the integral equation for the vortex strength

distribution is effected by assuming a Glauert type of series result.

This transforms the integral equation into an infinite system of in-

homogeneous algebraic equations. These equations are solved by











truncating the set so as to ascertain as many of the unknowns as are

needed for accuracy in the final solution. The resulting expression for

the vortex strength distribution insures smooth attached flow at the

plate trailing edge, thus satisfying the Kutta condition.

From this vortex strength distribution the hydrodynamic pres-

sure acting on the plate is found through the use of the unsteady Euler

equation. The lift and moment on the plate are calculated by integrat-

ing the pressure difference between the top and bottom sides of the

plate along the chord of the plate. The thrust is obtained by integrat-

ing the product of the pressure difference and the local slope of the

plate along the chord. It is demonstrated that for the case of swim-

ming at infinite depth the results of this study and those obtained by

other researchers are in agreement.

A numerical example is given for the case of plate motion with

displacement waves whose amplitudes vary in a quadratic manner along

the length of the plate. The time average value of the thrust over a

period of the plate motion is computed for two different depths and sev-

eral frequencies of plate motion. It is found that the influence of the

free surface waves acts to decrease the available thrust. The decrease

is more pronounced at high frequencies than at low frequencies.













INTRODUCTION


Of late, an increasing number of researchers have been inquir-

ing into the matter of sea animal locomotion. The manner in which a

fish swims has been viewed both as a biological and an engineering

problem. Attention is called to the papers of Gawn [1, 2], Taylor [3, 4],

Gray [5], Richardson [6], and Dickmann [7].

Two recent papers by Siekmann [8] and Wu [9] have established

identical results for a fish model swimming horizontally with a con-

stant velocity in an infinite, ideal, incompressible fluid. Both authors

have used a flat plate of zero thickness, infinite span, and finite chord

to approximate a thin fish. The plate experiences a traveling wave be-

ginning at its leading edge and progressing toward its trailing edge with

increasing amplitude. Wu attacks the problem using Prandtl's accel-

eration potential theory, while Siekmann approaches the problem through

the airfoil flutter theory developed by Kissner and Schwarz [10], with

particular reference to the paper by Schwarz [11].

Smith and Stone [12] have partially solved the same problem in

an elliptic coordinate system, but they have failed to consider the wake

effects. Their solution has recently been modified and extended to in-

clude the wake effects by Pao and Siekmann [13]. Results identical to

those of Siekmann [8] and Wu [9] have been obtained.











In the present work the effect of a free surface on the swimming

of the thin plate model is sought. As the plate approaches the vicinity

of the surface, a disturbance arises on the surface which has an effect

on the motion of the plate. Siekmann, Wu, and Pao have shown that a

positive thrust is developed in case the frequency of the plate oscilla-

tion is sufficiently large. Some questions immediately arise when free

surface effects are considered. Will the presence of the free surface

provide an increase or a decrease in the thrust? Just how close to the

surface does the plate have to be in order for the thrust to be influenced

to any great extent?

As a starting point in the analysis, the particularly illuminating

hydrofoil report by Crimi and Statler [14] is used. This report, con-

cerning the motion of an oscillating rigid hydrofoil near the free sur-

face of a liquid, is based on two papers by Tan [15, 16]. Tan has ob-

tained the velocity potential of a point source or vortex line of fluctu-

ating strength moving near the free surface of a liquid. In his second

paper Tan investigates the behavior of the free surface and provides

valuable information regarding the nature of the waves produced by a

disturbing point source of fluctuating strength moving at a uniform rate

near the surface. Crimi and Statler use essentially an integrated form

of Tan's results for a moving point vortex of fluctuating strength. This

idea is connected with airfoil flutter theory in that one way of approach-

ing the problem is by replacing the thin plate by a vortex sheet of










fluctuating strength [17]. In such flutter theory and in the sea animal

locomotion paper by Siekmann [8], an integral equation results for the

strength of the vortex sheet. The formulation of the problem in an in-

finite flow results in a fairly simple integral equation of the form


ftd GQ) e -id- ,

-- -
where H( ) is the sought for function related to the vortex distribution

strength, and \ and CT are constants. This equation has been dis-

cussed by SBhngen [18], and the solution has been modified and used by

Schwarz [11]. The presence of the free surface alters the form of the

analogous integral equation and makes it much more complicated.

Crimi and Statler have employed a method of solution giving

results in approximate but extremely accurate form. In their solution

none of the effects of the free surface are lost. A similar equation will

result in the following analysis, and the same method will be used to

solve it.

Other investigators have studied the effects of the free surface

on an oscillating rigid hydrofoil, but several simplifying assumptions

have been made in their work. Chu and Abramson [19] assumed an in-

finite Froude number allowing for no surface waves. Their solution is

in the form of a chord-to-depth ratio expansion and therefore holds

only for relatively high velocities and large submergence depths. Kap-

lan [20] has analyzed the same problem considering the primary effects






4




of a finite Froude number, but his model consists of a single vortex

and a doublet and is not as precise as the model of Crimi and Statler.

It should be mentioned that the problem is formulated here and

solved within the bounds of linear theory. The results obtained are

perhaps as pertinent to the field of hydrofoil hydrodynamics as they

are to sea animal locomotion.













CHAPTER I


FORMULATION OF THE PROBLEM


Consider the two-dimensional flow of an incompressible, invis-

cid fluid around a flexible solid plate of zero thickness. The flow is

influenced by a free surface of the fluid. We allow that the plate spans

from x = -1 to x = +1 and lies at a distance D beneath the mean free

surface as shown in Figure 1.

An axes system is taken fixed in the plate with the x-axis in the

direction of the flow. The flow, with constant velocity U, is allowed to

stream over the undulating plate in the positive x-direction or in a left-

to-right sense.


1. 1 The Motion of the Plate

The deflection of the solid deformable plate is taken as

Y = h(6, ) 4 X + 1 (1. 1)

and it is assumed that these deflections are small with respect to the

chord of the plate, i. e.,


<./ < I < << (1.2)


If we assume that the motion of the plate consists of a traveling

wave of small amplitude, we may write



















-1


Y = h(x,t)


Mean Free Surface


Figure 1.


The Flexible Plate


Image


Plate Vortex Sheet of
Strength Y(x, t)


Figure 2. The Plate Model


U---


U---


7


I


I


W r


m I









Y= h(x,t) = -(x) cos (x- it + 9S -1 x 1 (1.3)

in which f(x) is the amplitude function of the wave, oc is the wave num-

ber, W is the circular frequency, and 4 is the phase angle. Upon

insertion of the wave length X

2 = T (1.4)

the period '

S--- (1.5)

and the propagation velocity V

*V ^= W (1.6)
cx.
the plate displacement can be written as


y = t)= (x) cos t + (1. 7)

Such motion can also be described generally by

y H (x,) e. t (1.8)

if the real part is used for a physical interpretation. It should be noted

that H(x) may be complex.


1. 2 The Plate Model

The deformable solid plate is replaced by a vortex sheet of os-

cillating strength. As the circulation is time varying, vorticity must

be shed into the wake in order to satisfy Kelvin's circulation theorem

[21]. These shed vortices are carried downstream with the free

stream velocity U. Hence, the model of the plate and its total effect

on the flow is represented by the plate vortex sheet and its trailing










vortex wake. It is also assumed that the shed vortex trail lies in the

plane y = 0. This is not precisely the case, but the assumption greatly

simplifies the analysis.

In order to satisfy the free surface conditions, an image sys-

tem is located 2D units above the plane y = 0. This image system con-

sists of a vortex sheet along with its trailing shed vortices identical to

the plate system located in the plane y = 0 (see Fig. 2). Thus, the

combined effect of the plate system of vortices and its image is to

make the free surface appear as undisturbed. As the surface will ob-

viously be disturbed, a means must be found which will allow the de-

termination of the effect of the disturbed surface on the flow.


1. 3 The Velocity Potential

An obvious method of attack is seen in the use of potential the-

ory in the case of nonaccelerated flow. The potential due to the vortex

sheet of oscillating strength over the plate as well as that of its image

may be written down. The same holds true for the vortices in the wake

and their images. Then, a potential function for the effect of the free

surface may be postulated and a partial differential equation found for

its determination. Thus, a total perturbation velocity potential ((x, y, t)

is postulated which takes into account the effect of the plate system of

vortices and its wake, the image system of vortices and its wake in

addition to the effect of the disturbed free surface.










We assume that the velocity vector q of the flow is given by

S= tV +ut, V} I, (1.9)

where u and v are the x and y perturbation components of the velocity


LL = (1. 10)

This means that we have assumed irrotational flow. Upon substituting

(1. 10) into the continuity equation

d;v t O (1. 11)

for incompressible flow, we see that the velocity potential, < must

satisfy the Laplace equation

2 = 0. (1. 12)

Hence, is a harmonic function of the space variables in the field of

the flow.

The potential, ( must also satisfy certain boundary condi-

tions. If the free surface disturbance given by


j = (X, t)

is to be a stream line, then

) ) -' + t .-
d t ^t Y. d1

If the total velocity vector is taken in the form give

(1. 13) becomes


w n U +on) oy +

which, in the bounds of linear theory, is altered to


, D.


(1. 13)


n by (1.9), equation


0 )


(1. 14)









4 V j- 0. (1.15)

The unsteady Bernoulli equation


['. i 4. s wct) y+ 0, (.1t6)

must also be satisfied at the free surface. In equation (1. 16), p is the

pressure at the free surface, P is the density of the fluid, w(t) is an

arbitrary function of time which can be specified by the flow conditions

at infinity, and A. is the external force potential given by

An. .

If the velocity vector q and the force potential .1 are inserted into

equation (1. 16), and if the pressure is required to equal that at infinity,

i. e., a constant, the equation becomes

uU .U1^ ) 1 = ,y D (1. 17)

where g is the acceleration due to gravity. Again, due to the linear

theory, this boundary condition may be simplified to read

+ = -- 0 (1. 18)

For the third boundary condition the face that the plate is a solid im-

penetrable body is used. The corresponding kinematic condition

'()Y V + (1. 19)

must be met. This relation is also linearized to

V" P O + U3 0 (1. 20)









where Vp is the fluid velocity in the vertical direction of a particle ad-

jacent to the plate.

It should also be noted that the Kutta condition should be satis-

fied at the trailing edge of the plate.

Having set forth the fundamental equations governing the be-

havior of the velocity potential, we are now in a position to proceed to

the development of the various potential functions.

Consider the single vortex line of fluctuating strength located

at the origin as in Figure 3. The velocity potential of this vortex line

as given in polar coordinates is


Sr,et) (1. 21)

or in rectangular coordinates

-XP(.) Y- LT 3 (1.22)

where P(t) is the total time varying circulation.

For a vortex sheet of strength Y(, t) located at a 4 < b, y =

as in Figure 4, the contribution to the infinitesimal element of the ve-

locity potential di at the point (x, y) due to the infinitesimal element of

the sheet dE is given by

(d t +t' dde (1.23)

and the potential due to the whole sheet is


(I= C(,t) ta ' (1.24)
V J Q
























I 1


(r, e)
lx, Y)


eo


-I


rct)


Figure 3.


Single Vortex Line













(x, )

^-i^


f=Q. .


4


Figure 4. Vortex Sheet With Special Coordinates


j1


.I


I


X









The circulation around the vortex sheet is given by


(t) / x,t) dX (1.25)


and is taken positive in a clockwise direction.

Since the circulation is time varying, it is obvious that vorticity

must be shed into the wake to satisfy Kelvin's theorem. Denoting the

strength of these vortices shed at the trailing edge of the plate by E(x, t)

as in Figure 2, and observing that they are carried away by the free

stream velocity U, we write

E(,t) = (x to t- to) o= A- (1.26)
or

E(XX,t)= E 1 1.) ; > 1 (1.27)
UV

The change in circulation around the plate in time dt is

dr dt (1. 28)
dt
and is of equal magnitude and opposite in sign to the vortex

(1,tt)dx d = Udt (1.29)

shed at the trailing edge. Thus,


-(rt)d =-U (1,t) dt


d( t) d dt (1.30)

-f
Using (1. 27), the vortex strength in the wake becomes

r1

/-1










Now it seems reasonable to assume that

x,t) = Y(X) e (1. 32)

since the motion of the plate is harmonic. Hence, it follows from

(1. 31) that


E (X, t) = v(Ce) d




U
-f



i-1

If we recall (1. 25) defining the total circulation around the plate and

use the standard form for the reduced frequency

K = (1.34)
ET
the wake vortex strength becomes

E(Xt) = -/ e ( e (1. 35)

where
1
r -. Y ,) dx (1.36)

so that

re r. (1. 37)

Upon applying equation (1. 24) for the plate vortex sheet and again for

its wake and adding, there results the potential due to the entire vor-

tex sheet which is denoted by














+ K eK ta~ t1 -

For the image system we write correspondingly
For the image system we write correspondingly


, 2"I
& 2W


-1 X


+iKr I K
JI


-tat" D Ii


The two potential functions and az are clearly harmonic.

It is not unreasonable to expect the third potential function 3, due to

the disturbed surface, to be harmonic also. Hence, we write

c, ,1,) Re P + a v- e93 (1.40)
where

^( 't) ) e L= U 1 ,,
and
S(x, y, t). To investigate the nature of Ip we combine the boundary

conditions (1. 15) and (1. 18) by eliminating 'y (x, t) to get

+ 8 -a ( a =D. (1.41)


If equations (1. 38) and (1. 39) are rewritten using


L e
~, = ~r~ e- ,


(1. 38)


(1. 39)


Lwt
gE= (PX, )e





16



and substituted into equation (1. 41), it is not difficult to obtain a par-

tial differential equation for cq3 in the form


93 + z' P 3L _9_ 9_3





S7Tf F


LFK J c ) + o

1

where the Froude number F is given by
-j
F = (1.43)

Tan [15] has solved a partial differential equation similar to

(1.42) in the case of a single vortex of oscillating strength. Hence,

according to Crimi and Statler [14], the solution to (1.42) can be ob-

tained by taking an integral form of Tan's results as



3(?,) = __ f (x) X +
-I
00

+ K r G(; x, ) d (1. 44a)
tT- Fz
e
+I


where








S S(?- D) ee L S(- P)-- r s_

0

4- D- e + Dz e +




T e.s[j-ZD- LX- + D) eS4.- zo -J(x- (1.44b)

and

(s-s )(s- ZK)S K'

(Cs-53)(- S) = S- + Zk)S + (I.44c)

We refer to Appendix A for a more detailed explanation.

The terms of (1. 44) involving the double integration give a par-

ticular solution to (1.43), and the remaining terms containing the D1,

D2, D3, and D4 are the complementary solution. Obviously, the solu-

tion is incomplete in this form since we do not yet know the Dj. The

constants Dj will have to be determined by the free surface flow condi-

tions at infinity.

A detailed analysis of the free surface flow conditions is pro-

vided in Tan's paper [16] for the case of a single source of fluctuating

strength, and a similar analysis is outlined in the paper of Crimi and

Statler [14] for a vortex sheet of fluctuating strength. The method em-

ployed was first used by Rayleigh [22] and constitutes the introduction

of a small fictitious dissipative force -/ grad I >/O. The results

are then examined as 7 tends to zero in the limit.










Crimi and Statler have shown that the constants Dj may be as-

signed to provide the correct free surface flow conditions depending

upon the following combination of the Froude number and the reduced

frequency: kF2

The values of the Dj as so determined are reproduced in tabu-

lar form as a convenience in Table 1. Appendix A offers an analysis

in some detail leading to these constants.

In examining the table we note that for kF2 > 0, S3 and S4 will

be real and positive so that at least two harmonic wave trains always

exist on the surface. Their wave lengths are 27T/S3 and 27T/S4. They

propagate downstream from the plate and cannot exist physically up-

stream of the plate.

For kFZ > 1/4, the roots S1 and S2 are complex conjugates and

therefore produce damped waves which do not influence the flow at

great distances from the plate. In case kF2 < 1/4, S1 and S2 are real

and positive. The group velocity of the wave arising from S1 is less

than U, requiring that this wave trail the plate. The S2 wave has a

group velocity greater than U and will lead the plate. Thus, there are

two harmonic waves propagating upstream for 0 < kF2 < 1/4, one

leading and one trailing the plate.

In the special case kF2 = 0 (steady motion), the only roots

present are S1 and S3. The solutions combine to give a single standing

wave of wave length 27TU2/g downstream of the plate. Let us refer to













" o -i '*



9')












a a NI O









( I
m 0 m
z*-4


H 0

X04


0)
>


0 N



*-4
a0
-4 W

W >o
ua
^! <
i-<


"-4
rl
0




0
H


a) a)






> O






u.. O $- -
LL0 0 0



0 01


0U-
_ _ _ p4 _
-Y Y N
N~ f.v lt


bO


0

0


0
d






0
a



bD


















4.


0
0
CO























4.
o
04
0


nl








4-
0,


,jI-"

C


n


= Ito
wi










the paper by Isay [23] for a solution to the problem of a flat plate in

steady motion moving near the free surface.

When kF2 = 1/4, the integral associated with G( ; x, y) does

not exist. The roots Sl and S2 do exist, however, but the group ve-

locities of the associated waves are the same, creating a resonance

condition. Thus, for a given flow velocity U and, hence, for a given

Froude number F, there exists a critical frequency, critical, for

which (kcritical)F2 = 1/4, and for which no solution exists within the

framework of the present formulation.


1. 4 The Integral Equation

The last step in the actual formulation of the problem requires

that the boundary condition on the plate be used. From the last section

we see that with the use of equations (1. 38), (1. 39), and (1. 44) the total

potential given by

I (x',) = Rei. e LWt

is known except for the vortex strength distribution Y( ). If the

boundary condition (1. 20) relating the velocity of a fluid particle near

the plate to the motion of the plate is used, there results an integral

equation for the determination of the vortex strength distribution. If

we keep in mind the type of motion which is to be imposed on the plate,

namely that given by equation (1. 8), then the boundary condition (1. 20)

becomes









= U H'x + L K H() (1.45)

where

H() dH
dX
Upon substituting the determined value of in (1.45), there results

the integral equation

1 1 f

so r) x-f ii r -a o


2z J1 x-l (.x-)+ ,+o F poJ +



+ vl 'x + LKH(~ -o= -IX $ (1.46)

where the Cauchy principal value is to be taken for the integral on the

left-hand side, and the quantity is found to be from (1.44b):

00 Z s -scx -



,l-zo D(x- ) 3 s,-z0 -,(x- S)L( ]

4- SD, e 1 D, eQ


In addition to the requirement that Y(F) satisfy the integral equation

(1. 46), X ( ) must produce a velocity field which allows a finite ve-

locity at the trailing edge of the plate.

If D is allowed to approach infinity in the integral equation,

another integral equation results, governing the value of (r ) for the










case of infinite flow about the oscillating plate. In this case the equa-

tion becomes



ifT ) Z xKP e d 4- G/ x)


which is seen to be equivalent to the equation investigated by Sbhngen

[18] and applied by Siekmann [8] to the study of sea animal locomotion.

With the derivation of the integral equation (1. 46), the formu-

lation of the problem is complete.













CHAPTER II


SOLUTION OF THE INTEGRAL EQUATION


As mentioned in the Introduction and demonstrated at the end of

Chapter I, the formulation of the problem of an oscillating hydrofoil or

thin plate in infinite flow (D-oo) leads to a well-known integral equa-

tion. The presence of a free surface obviously has complicated the

integral equation considerably, as evidenced by equation (1.46). An

approximate method of attack seems to be the most expedient way of

solving this equation. It is possible to make the final results as accu-

rate as desired through the use of high-speed computing techniques.


2. 1 Form of the Solution

Crimi and Statler [14] have been successful in solving an inte-

gral equation much like (1. 46). Equation (1. 46) differs from the one

solved by Crimi and Statler in that an infinite number of terms enter

due to the plate boundary condition.

We first assume that the vortex strength distribution X(e) is

given by the Glauert trigonometric series


Y (e) = 2SU aI co+. CLsix eX ,9 (2.1i)
'V's f









where x = coso -1 \ x v+1, O ,8 27T, and the c2 may be

complex.

It is easy to find the total circulation around the plate with this

form of the vortex strength distribution. From (1. 36) and (2. 1), f is

calculated as



-1 0
Sy (,X,)dX, = o ,e) sie de



S2 U [ o co- e 4 Qc stV e siw e de



z=7rvc U) (2.2)


Substituting (2. 1) into the left side of the integral equation

(1.46), there results after the integration


X C CO Inc) (2.3)

-1
Now, if the right-hand side of (1. 46) can be expressed as a Fourier

cosine series, like coefficients of cos ne can be equated to obtain an

infinite set of inhomogeneous algebraic equations involving an infinite

number of unknown Q, The expansion of the right-hand side of (1.46)

involves much manipulative work. Fortunately, the results of Crimi

and Statler can be used, as the only differing terms are incorporated

in the inhomogeneous terms of the infinite set of equations:










^ C= Z' C n + a o0, 1,2... (2.4)

Crimi and Statler have considered the motion of a hydrofoil with two

degrees of freedom, pitching and plunging, whereas, the inhomogene-

ous terms of (2.4) reduce to only two, r. and r,. The oscillating plate

dealt with in this work has an infinite number of degrees of freedom.

Thus, as might be expected, an infinite number of inhomogeneous

terms are involved.

The technique used in expanding the integral expressions in the

right-hand side of (1. 46) deserves mention. Consider the integral
00
I1 eJ (ix-4) r (2. 5)
2 (Tr f + 4 D"

where x = cos 9. Now in transform theory, the Laplace transform

of a function F(s) is


M) = 3W(s)} f e~ FCs) ds ,

and in particular

a IVs Qs (2.6)
CLI + CLL
In the fraction of the integrand in (2. 5)

(cos e +
(zD), + (cose +
let a = (cos e ) + and q = 2D when applying (2. 6). Thus,









00
Cos f --ZDS -Ls(J+Cose) +js(t+cos e)
cos( e = e e (2.7)
(Zo) +t (f+cose)z 2 1 7
0


Letting


_cose+ . b cos ie
(2 D)' + (+ +cos e) 2
n-=1

and employing (2. 7), the bn Fourier coefficient is given by

o S 7 -S r -Scos

b e e Cos ->e de
TrT
f0 0


7T
-Ls, r s cos e


0


00 -DS +Ls"
= e T (S) e



In the last step the integral formula


J(S) -L e cos 7
7r


7TT
iS cos 8
7 f
77"
0


CoS 'r


os v e de I cIS


- (-i dS S


9e de



ae de


for the Bessel function Jn of the first kind and order n has been used.

Inserting the total circulation T from (2. 2), the integral I1 can now

be written as


(2. 8)


(2. 9)





27



00

I LKU ( +. e b cos n d0 (2. 10)
n=1

Finally, if bn from (2. 9) is placed in (2. 10), and the ) is integrated

out, Il can be written as


z,


4- b h cos vIt 9
1A I


(2. 11)


where


00
f -2DS

0


',(s, [


k-S


S

8+S


ds -


Other of the integrals are less difficult to expand.


Consider


the integral


2 2= T
Z, Tr


S00 ik 1)
e

IT


where x = cos 9 Assume that


1 -
-2 ~ 2


Then,


C = _ U
L7
7r


( e COS n e




ff Cos q 8
o + ( e CoSne
A"0" cose
of


(2. 13)





dyde .


Interchanging the order of integration and integrating with respect to ,

we obtain


00

C Z-1aU (a ) i)

I


e ( -1)


(2. 14)


(2. 12)


I, = U(a









When expansions for all the integrals on the right-hand side of (1. 46)

are found, the like coefficients of cos n9 are equated, giving expres-

sions for the Cmn terms of (2. 4). These are listed by Crimi and

Statler and are reproduced in Appendix B for convenience.

Return now to the inhomogeneous terms of the infinite set of

equations. If the "downwash"

- [H'(x) +. i\ H (A ] eP
is assumed equal to the Fourier expansion


-cU e 4- +Zcosn e (2. 15)


then it is easy to see that the inhomogeneous terms of (2. 4) are

Ar


r = n I, 3, ... (2.16)

Note that the An are assumed known, as are the Cmn. The integrals

involved in the Cmn cannot generally be expressed in closed form, but

they may be calculated for various flow conditions by utilizing a high-

speed computer.

We are now in a position to consider the infinite set of equations

oo

s.a = C aN + r- f (2.17)

seeking a solution for the 0. 's.










2. 2 Solution for the Infinite Set of Equations

Some of the fundamental theorems dealing with a system of

equations like (2. 4) are presented in the book by Kantorovich and

Krylov [24]. In certain cases it is possible to show that a solution

exists and is unique. Such is not the case here, but we expect that a

solution exists due to the behavior of the Cmn and the r for increas-

ing m and n. The Cmn decrease rapidly when either m or n increases

or when both m and n increase. In the particular example to be con-

sidered in Chapter IV, it will be shown that the t, decrease also with

increasing m.

Assuming that a solution does exist, the procedure leading to

an approximate solution is as follows [24]. We first find the solution

to a reduced finite system of equations


m C + r 0 2,..,N (2. 18)

7n=o

and then repeat the process for N + 1. This, of course, introduces an

additional unknown, ON but at the same time limiting values for Oo ,

., CL . ,MN become apparent. It develops that only a finite num-

ber of the am will be needed to define the lift and moment on the plate,

but the thrust involves an infinite sum of the m. which converge very

rapidly. Also, it becomes apparent that approximately N + 4 complex

equations of the set (2. 4) need be solved to ascertain N of the am to a

reasonable degree of accuracy.














CHAPTER III


THE EFFECT OF THE FLOW ON THE PLATE


In this chapter the forces and moments acting on the oscillating

plate are discussed. Due to the unsteadiness of the motion, a pres-

sure difference results between the top and bottom of the plate. This

pressure difference is not constant, but varies with time and the x-

coordinate along the plate in the chordwise direction. The first task

is to derive an expression for this pressure difference distribution

along the plate.


3. 1 Pressure Distribution

If the pressures at the top and bottom sides of the plate are de-

noted by p+ and p- respectively, the pressure difference is given by

a p = p P ,p (3.1)

where it is assumed that the Ap will be positive in a sense implied by

equation (3. 1). This is in conformity with the negative direction for

the downwash assumed in deriving the integral equation (1.46) govern-

ing the vortex distribution along the plate.

Considering the velocity vector q to be given as in (1. 9), the

unsteady Euler equation









___ u\ iL vI p
t- + V -- (3. 2)


may be linearized to read

S+ U (3.3)
^t T < {\

Writing the last equation for the top side of the plate,

U + -:1 P'

and for the bottom side,

M _j = P_
Yt >KC X
and subtracting one from the other, we obtain

+('-) 4- j C-- .-) 1 (3.4)
^t ^X C X

Now u+ u~ is exactly the strength, X(x, t), of an element of the vor-

tex sheet which we have allowed to replace the plate. Hence, we can

rewrite (3. 4) as

a + L- 1 A (3.5)
S^x e x

If the total vortex sheet (x, t) is divided into bound vortices

(x,t) and free vortices (x,t), as in the paper by Schwarz [11], so

that

u t) i *(x,t) -+ E*Cx (-1 (3. 6)

we may substitute this expression in the second term of (3. 5) to get

SX + ( \ _i aP (3.7)
T-6 X










Now, according to the Kutta-Joukowski law


(3. 8)


Using this in (3. 7), the equation becomes


-E"
= O (3. 9)

giving a relation between the total vortex sheet and the free vortices.


Integrating (3. 9) for


E (x,t),


x

n (dt) i n - r l 3t) w

and inserting the result in (3. 6), we obtain
and inserting the result in (3. 6), we obtain


(x,t) + 1
v-


; -1 ;x < I J


aLDt) f


In Chapter II we assumed the vortex distribution Y(x, t) to be

given by the Glauert trigonometric series


C, cot e
2


L t


QKs;v\ v\e


o7
4- E
^%z I


where x = cos e Placing this expression in (3. 10), there results


~e(t)= Ke,t) + -c) E

ero


L+ t eif + I
&
+21w I


(3. 10)


Y(x, t) =


1Ye,t) = zu


= (e, t)


(3. 11)


-e

L / Cot^ Sc v (p dq>



aK Si1 n s;I( d










Carrying out the indicated integration for I, and 12, we obtain
*x-
I,= ^o( i- ne


Z LQ s~iv\(j-1 a sN1(v\j j-1)e 1
SYl ,
hr


giving for t (e ,t)

Y*(9t) =


S(et) + 2 iL e


+ (3. 12)


Finally, after a little manipulation, the pressure distribution Ap(e, t)

is obtained from (3. 8) as

ZCV P Liot f- co
ap(et) = zl e ao cot
-+


ct,
00

- n \ wzn
Cn2-'


- a_ + 2(2az+0tc)(-1
h-I


It is pertinent to note that this expression reduces to that de-

rived by Siekmann [8] for the case D-0 (See Appendix C.)


3. 2 Lift and Moment

With the expression for Ap(e t) given by (3. 13) at hand, it is

easy to compute the lift and moment acting on the plate. In computing

the lift, the pressure should be allowed to act in a normal direction to

the surface of the plate (which is constantly changing), but as small


ALt (I + YG


) J Y1 10\ (3. 13)










deformations have been assumed, it is within keeping with the linear

theory to assume that the pressure acts normal to the x-axis. Hence,

the lift is given by


L p(X, t) dc. (3. 14)

J-1
and the moment by


M = A p(x,, ) x (3. 15)
J.I

where the moment is considered positive in a clockwise sense and is

calculated with respect to the origin, x = 0, y = 0. Inserting Ap from

(3. 13) and changing the variable according to x = cos E we may

write
7T

L = o v Let) s;y e de (3.16)
o

and


M = f (0,tg) s;viecose S (3.17)
0
x-
It remains now only to use the expression derived for X (G t) from

Section 3. 1 and carry out the indicated integration with respect to e.

The resulting expressions for the lift and moment are respectively


L 2r e ( + and (s (3. 18)

and










M, TUe )I (3. 19)


In calculating the moment, the pressure difference Ap has again been

assumed to act normal to the x-axis, and the leading edge suction

force [8] has been neglected.


3. 3 Drag and Thrust

The drag or thrust experienced by the undulating plate will re-

sult due to the x-component of the hydrodynamic pressure difference

A p. It can readily be seen that the horizontal hydrodynamic force on

a plate element of projection dx on the x-axis is given by


Re (ap] Re dx. (3. 20)

The real parts, denoted by ResU, are taken because nonlinear terms

enter the product which involve mixed expressions in a. Thus, over

the whole plate the hydrodynamic force in the positive x-direction is

given by
1
D = Op+ Ds = e [Ap }Re }dX (3. 21)
-1
where Ds is the leading edge suction force.

The plate displacement Y(x,t) given by equation (1. 8) is an even

function of e where x = cos due to the fact that the plate is very

thin, thus allowing a point on the bottom side of the plate to be repre-

sented by the same coordinates as the adjacent point on the top side of










the plate. Hence, Y(x,t) can be expanded in a Fourier cosine series

of 9 as


y(it) =


Lot


where


= H () Cos 'e de


+flu cosrhe) ] e L ,



TT
77-
= H G) cosne de .
"


The slope, .Y can also be expressed as a Fourier cosine series

since the plate is infinitesimally thin. First, note that from


y(e,t)


= (e) e


there results


(e,9t)
X


and as x = cos e we have


where


de
de


Now we can write


00


77"

f H (e)
o4


Cos e*A
S~v\Q0


CLt


(3. 26)


(3. 22)


(3. 23)


(3. 24)


- I _
s;nG ~G


H e


0


(3. 25)


0(e,t'


where


C.h


de


(3. 27)


D
O
2


SO- [ +










There is obviously a relation between the C's and the B's. This

follows from (3. 22) with


= -2 (3. 28)


Upon comparing (3. 28) and (3. 26), we get

C C = -2 (3. Z9)

or

00
C -2 .(Z-ni) + J -01, 2... (3.30)
2yt Z ZwA+ I
L1Z.
00
CZ -2 1(2n 4--Z) z+ f, Z ,... (3.31)


In Section 2. 1 it was assumed that the portion of the downwash given by

[ H C')O +- LK HCx eWut (3. 32)

was expressible as a Fourier series like


L0 + A os "-n e (3. 33)


This assumption, together with equations (3. 22) and (3. 26), allows the

conclusion that the following relation must exist between the A's, B's,

and C's:

h = C + L k (3.34)

As only the real parts of Ap and are used in computing the

horizontal force D, the notation is best altered to denote this fact.

Following Siekmann [8], we define






38


*t i ~L t -
An e (AV + A,) e = A, L A (3. 35a)

Liot ,, t -, _< 1


-- II
8 e. = ( B,+ S A e. = 6 + L y, (3. 35b)

aL.3', iot --


where

A = AL Cos Lt s A (3.36a)

II II
A = A" Cos wt + A sv (3. 36b)

and so forth. Then, for the real part of ./ we have

-i0
Re c + Cos -n (3. 37)
--?7 V I
nsi
The notation will be simplified somewhat if the pressure dis-

tribution Ap( ,t) is written from (3. 13) as


ap(,t) = 2 z a.ocot + d1 Sie + d, s. -ne (3.38)
L--i

where

d, = I K La30 a, + _3 (3.39)


da= + Q. + (zC Q.-1)lvi .(3. 40)

Analogous to equations (3. 35) and (3. 36), we define


dae O ( 4 dt) e L = d 4- ( (3. 41a)


d, e =- (d' + c ^)e ^ d, +td, (3.41b)


where









Oa Cos Wt,


+- S;i Lot .


Then the real part of Ap(e ,t) is given by


Re p7l =


Z'r -'
2I o, cot a + d. s;ye
Z


The portion of the horizontal force D denoted by Dp follows

from integrating


77

2 e

0


0o cot +
Z


d, sI; +
d 5 8


+ d:
}-'z


sin V, e *


C0 cs
'',z^0 00


siVeI de. (3.44)


Multiplying sin e with the first braced quantity and employing the

trigonometric identity


we may write (3.44) as
we may write (3. 44) as


D= 2e


07
o


1 [cos (n-) cos(+))oe ,


C co siv+e + 4 S;n e +


+ dL [cos (.-v-ie -cos('it-1)r6
31: Z


O -
+ C coM C me de

-m 1-


SI
0.1h


CII II
c, = aCz CoS (t


(3.42a)


(3. 42b)


00
+


sI19 ne


(3.43)


{ I+
2


(3.45)










Using the idea of the Cauchy product in the integrand of (3. 45) and

interchanging the order of integration and summation, Dp may be

written as







C Co d cos (.-1)e cos (-nIcI)e d e +

I 0
+D(p.0 Cvn Co4 i2i on o ',,e o "- J s.n
-0


To. T0











So I ( cc3.46)
t d C e V Yos a +



-t I Z [, COI ( Ni 1) 9 - COS C -ni 1 e2 c - O Csa

Ir = (3.46)


The definite integrals in equation (3. 46) are evaluated as

T"





o




cos ( &-i)O de a= o -
0


(3.47a)




(3.47b)




(3.47c)

















.icot-I- s5ie cos tye de .=


7-
cot Si e CoS y e d =
0





of Si 9 Cos rriede


77-

cos (-ni1) cos ede



C5S(I+1) e cos 'vO e de


7rT


-)i = h-7




n? = 'i+ 7


Using these definite integral values in equation (3. 46), we write


Dp as


P =r \U i I5
Dp = 70~(U Co(ao


00

+ tEj i,


dl
2


-C
yt*1jC,


(3.48)


Recalling equation (3. 29), the last term in (3. 48) may be

written as


oo


- -n .
)1Z


(3. 47d)




(3.47e)


"m = "1

.W.= ?
v>n=2
>v2


o
Il-


0
O


(3.47f)


(3. 47g)




(3. 47h)


(3.49)










From equations (3. 39) and (3. 40), keeping in mind the notation indi-

cated by equations (3. 41) and (3. 42), we may write
ii
d = ,- K( o + 0 T ) (3. 50)

and

+ n 1 2 J (3.51)

Using (3. 50) and (3. 51) in equation (3. 48), the horizontal force Dp is

written as












+ +
~ 1 -- -, \
















The suction force due to the singularity at the leading edge of

the plate is discussed by Siekmann [8]. The contribution of this singu-

larity to the horizontal force acts in the negative x-direction, hence
-1 Y-na B + X k( a C^ +- C// > +





0 "










the terminology "suction force. Its magnitude is

Os = 2T ( 1 C .o (3. 53)


Therefore, the total horizontal force acting in the positive x-direction,

or the drag, is given by










0 = D- D = Dp 2r ( .


(3.54)


The thrust is clearly the negative of this quantity and is written from

equations (3. 52) and (3. 53) as

T o a.Co + )+


2 O


-vI
-Y, 5_' G


, ( -a, 1 -I1 -") I/





y--i2.


I- (za.+ a, ) Z -:*' (3. 55)
?2._

Instead of this expression for the thrust, it is more meaningful

to calculate its time average value over a period Z. The time aver-

age value of a function -f(t) is defined as


S- (3.56)
0


Because of the








-
*

a,


relations (3. 36), need for the following will arise:


o Cos Ot dt = I
2






I siV Ot cos t d t = O (3.57)
o


00


h>iZ










Keeping equations (3. 36) and (3. 57) in mind, the


various products in the time average value of the thrust,


T = T(t) dt

are
are


= (' Ca




- i
2 K;


II
- Co ')


S'


:3 )


Bra .


0f


S'(aIt )- B'


2.
nzz
2 M2


B, dt


00
- .i (-1) (o~- a 8 3 ).(3.59e)
-2 .


,I __
C'


(3.58)


Z"

0
Jo



r

0


czlu


(3. 59a)





(3. 59b)


(3.59c)


B ld
1 )1


0 200
o k=-2-


S-i0

0
9, Q 0-7)
o >=1


(3. 59d)


where = 2 7T/ .


- .,) d t =










Inserting these representative values in (3. 55), the time aver-

age value of the thrust is given by


S-7 a. C + C,





l tru c CT is efne a








00



o -C




--2. s-Z





aC ( (3. 61)
77 9 a

Substituting (3. 60) in (3. 61) gives the thrust coefficient in terms of the

Q's (the solution of the infinite set of algebraic equations), the B's (the

Fourier coefficients of the deflection function's expansion), the C's

(the Fourier coefficients of the slope function's expansion), and the

flow parameters U, and k.














CHAPTER IV


NUMERICAL EXAMPLE


In accordance with the idea of representing the motion of a thin

flat fish with a two-dimensional thin waving plate, we consider an ap-

propriate displacement function and calculate numerical results for the

thrust coefficient in this chapter. It is desired to compare these re-

sults with those of Siekmann [8] for the case of swimming at infinite

depth so as to effectively isolate the influence of the free surface on

the thrust.


4. 1 Quadratically Varying Amplitude Function

Photographic analyses of swimming fish indicate that the mo-

tion of the fish is perhaps best represented by a quadratically varying

amplitude function like


Y = h (X,1t) = (C e C + CXz) Cos (ox-cWt), (4.1)

where the c's are constants. The notation here is in conformity with

equation (1. 3). The phase angle S has been taken as zero.

In the last chapter the thrust coefficient was shown to be a func-

tion of the Fourier coefficients, Bn and Cn, and the 0. which are the

solution to the truncated set of equations represented by equation (2. 18).











Also, the inhomogeneous terms of this set of equations are related to

the Bn and Cn coefficients through equations (2. 16) and (3. 34). Thus,

we first calculate the Bn and Cn coefficients associated with an ampli-

tude function like that given by equation (4. 1). Corresponding to the

notation

Y(x,t) = H () e


H(x) is given by

H( x = (co + c1x + xC ) e

With x = cos e we have

H e) = C- c cose + ccos e

Therefore, from equation (3. 23) we have


(4. 2)


(4.3)


_IT
Lox cos .
S = 2 (C C cos + c, cos e) a cos vY de. (4.4)
-Tr
0


Using the integral formula [25]

.-ey 6 cosa5
o<) e cos e de ,
7r
-o

there results





i c, J J (.^1

where the identity


cose cos Me =- -- [cos (i+ e 1 cos (n-r) e


(4. 5)


(4.6)









has been used. A similar calculation starting with equation (3. 27)

yields the following for the Cn:

C L r[ [ (O)7 -, (T -


^.Z7r[ c, S] 3o( + zc,[r1o -
Iz VI [ U,- +1

-3IT(.] CW [ T^ T.( ] ) (4.7)

To solve the truncated set of algebraic equations for the Q,,

we will separate the set of N complex equations into two sets of N real

equations each. This operation is necessary because the coefficients

and the inhomogeneous terms are themselves complex. We have al-

ready indicated the complex nature of the Bn and Cn coefficients

through the primed notation of the last chapter. Adhering to this no-

tation, the real part of the Bn is

= -1 Zc(CoT (-) + c,[T.(- T C-) + T.-a }] (4. 8)
0 O, 2, o....

S.E-f, c,[()- o (a) T (4.9)

The imaginary part of the Bn is

a K C J I (4. 10)
1-n+

c, T c[_)]} + (4.11)
1 J 'i ., 5Z...

For purposes of calculation it is desirable to eliminate the possibility

of a negative order for any of the Bessel functions. Using the following










recursion formula [26]

(o) + (-X = JT(oc)

as many times as necessary, there results

S- ) + 2 (c c.) T + (4. 12)
So 9~ 0 ,...


8 = (-i) 2c C- ) -T. ) (4. 13)
"- 1 3, ,5, ...
for the real part of Bn, and for the imaginary part


Ii J = o, z,v,..


B; (-1C 2 ( ,,, n(T-1 C1

Similarly, the real part of Cn is

u+z
S = (-I) a c ('C +1C ( (-n -+ +) -)T

S-n ~l 0 Z
c-if; af-rwc c,, + C K,)] J^ Oc)


and the imaginary part reduces to


C= (-1)z 2 iTr C+ C- ) T) +


C = C- 2 c +1) )


Next we discuss the calculation of the An coefficients.

equation (3. 34) there results

A' C x <
V% /


S(4. 15)
S1, 3, 5, ..-


(4. 16)



(4. 17)




(4. 18)
o, ZIi, ...

(4. 19)

From


(4. 20)










and


AnI/
n, C


+ k B


(4. 21)


4. 2 Solution of the Algebraic Equations

The set of N complex algebraic equations given by
N
a J cM M n + r V, 0, -2 N

M=o

is separated into two sets, using the following notation:


Cr = Cr + iCC

m +


(4. 22)


(4. 23)


We obtain

01
N






= C in + C ) + ; .1,v,..,n/ (4. 25)

I ii

To solve these equations for the (, and Qa, we use an itera-

tive method, whereby we assume some arbitrary values for the un-

knowns in the first equation of the first set and compute an initial value

for the first unknown to the left of the equation. This value will be

used in the second equation and so on until initial values of the first

2N unknowns are found. Then the equations are solved again for a

second set of values for the 2N unknowns, making use of the first set










for starting values. This procedure is continued until the difference

between values for the unknowns obtained from two successive itera-

tions is less than some preassigned value. The computer program

used for this operation is designed to provide this check for accuracy

in solving the equations. This iteration method was found to be effec-

tive for computing the 0a and aM for all values of the reduced fre-

quency greater than the critical frequency. For frequencies in the

range 0 k < critical, the method fails. This failure is noted for

depths that are small compared to the half-chord only. The method

is satisfactory for all frequencies at large depths. Note that critical

is the value of k for which no solution exists.

For the range 0 < k < critical and for D < 4, the Crout method

[27] was used to obtain a solution for the equations. The Crout method

will give satisfactory results for all frequencies, but is somewhat

slower in regard to computer time than the iterative procedure de-

scribed above.

All calculations were done by utilizing a high-speed digital

computer. Values for the real and imaginary parts of the Bn and Cn

coefficients are given in Table 2. These coefficients are not functions

of the reduced frequency k as are the An coefficients. The An values

can be obtained through the simple relations (4. 20) and (4. 21) and are

not tabulated here.











TABLE 2

REAL AND IMAGINARY PARTS OF Bn AND Cn


/ # I //
n Bn Bn Cn Cn


0 -0.04084412 -0.02390768 -0.10066456 0.08960791
1 0.03316639 0.02193889 0.12262104 -0.08028761
2 -0.04143261 0.00205140 -0.03433178 0.13348571
3 0.01402839 -0.02095180 -0.04310939 -0.07208199
4 0.00361431 0.01181532 0.04983858 0.00888488
5 -0.00574890 -0.00155564 -0.01419494 0.02244056
6 0.00142237 -0.00204627 -0.00765042 -0.00778154
7 0.00058169 0.00061490 0.00287355 -0.00211474
8 -0.00019193 0.00013840 0.00049327 0.00082705
9 -0.00002839 -0.00004819 -0.00019737 0.00009970
10 0.00001023 -0.00000513 -0.00001779 -0.00004045
11 0.00000083 0.00000189 0.00000728 -0.00000284


Note: a = 7, co = 0.023, c, = 0.042,


c = 0. 034.


4.3 The Resulting Thrust

The thrust coefficient at infinite depth is independent of the flow

velocity U as shown in Appendix C. For finite depths (D < 4) the flow

velocity enters the calculations through the Froude number. As the

time required for the execution of the computer program is fairly

great, only one flow velocity was considered. All the results reported

here are for a flow velocity of U = 3 units per second.

With the Q., Bn, and Cn in hand, the thrust coefficient may be

computed from equations (3. 60) and (3. 61). The first attempt to calcu-

late the thrust coefficient was made using a depth of D = 50. 0. It was










correctly presumed that this depth was large enough to simulate an

infinite running depth. The results obtained are shown in Table 3 and

Figure 5. Also listed in Table 3 are the results obtained by Siekmann

[8]. The infinite sums involved in the expression for the thrust were

found to converge fairly rapidly so that six of the a. gave results

comparable to those obtained by Siekmann at an infinite depth. The

first six a. were computed to within 1 per cent accuracy. Looking

at equation (3. 60) for the time average value of the thrust, it is seen

that the last two sums involve only the first two a., but an infinite

number of the Bn. As the Bn are fairly easy to calculate and require

little computer time, these two sums are evaluated for n = 11. For

higher n, the Bn are so small that their contribution is negligible.

Several attempts were made, using successively smaller depths

beginning with D = 2. 0. Some differences were noted in the thrust co-

efficient for depths just below 1. 0, but the depth must be smaller than

this for a clearly discernible deviation from the thrust at infinite depth

to be observed. Indeed, to register the effect of the free surface waves

on the thrust for the full range of reduced frequencies of interest, it

was necessary to decrease the depth to as little as D = 1/4. The re-

sults for this depth and for several values of the reduced frequency are

shown in Table 4 and Figure 5.

The asymptotic behavior of the thrust in the neighborhood of the

critical frequency requires the observation that in the interest of




















000000000000
I I I I I I I I


H

u


rA~
0"

cI~

H-


000
I I I


n Noo- rM\
"o so o Moo n o o t I
000000
0000000


00000000000000
O 4 O0000000000
O0OO000-NLL4O


0000000c000000
00o r 0 o00000000000

6 6 6 6 6 6 6 6 (\ C4 tn so 06


E- 5
2c

Uc


O o




C1



H




















c. = 0. 023
c, = 0. 042
cz = 0. 034


critical = 0. 8935


0.06-




0.04-



D = 50. 0

0. 02


D = 1/4


0. 00-





-0.02 I l
0 2 4 6 8
k


Figure 5. Thrust Coefficient Versus Reduced Frequency


10










forward motion with as little effort as possible, the "fish" had best not

swim at this frequency. This singular phenomenon also indicates that

a hydrofoil experiencing vibration with an infinite number of degrees of

freedom at just the right frequency might expect to encounter a high

drag coefficient for the proper flow velocity. Again, note that the

critical frequency is a function of the flow velocity alone. It is not a

function of the depth.

The interaction of complicated wave trains on the surface pro-

vides an interesting oscillatory behavior for the thrust coefficient at

frequencies in the range of 0N k < critical. This effect is shown

separately in Figure 6 where the scale has been lengthened. Note also

the wave drag [28] for zero frequency. At infinite or large depth the

thrust is zero at zero frequency, as required by D'Alembert's paradox.

















CT
co = 0. 023
c, = 0. 042 critical = 0. 8935
c, = 0. 034
S= T U=3
0. 02-





0. 01 -





0.00- D = 50. O0





-0.01- /4


D = 1/4

-0. 02
0 0.5 1.0
k



Figure 6. Thrust Coefficient Versus Reduced Frequency
for Small Reduced Frequencies














CHAPTER V


SUMMARY AND CONCLUSIONS


In this study the influence of free surface waves on the swim-

ming of a thin, flexible, two-dimensional, oscillating plate was inves-

tigated. The motion of the plate was assumed to approximate the

swimming action of a thin flat fish near the free surface of a body of

water. The fluid was assumed to be perfect and infinitely deep.

The propulsive effect is due to the motion of the plate, consist-

ing of traveling waves emanating from the "fish" nose which move to-

ward the tail with increasing amplitude. The waves are harmonic with

respect to time.

The "fish" is replaced with a mathematical model consisting of

a vortex sheet of harmonically varying strength. Such a model re-

quires that vortices be shed from the tail of the plate and carried

downstream with the flow. These wake vortices are adequately taken

into account by the theory presented. The assumption that they always

lie in the plane of the oscillating plate simplifies the theory, but it is

still an open question as to whether this assumption is the correct one.

Actually, photographs of an oscillating model indicate that the wake

vortices deviate in their downstream path from the plane of the plate











by perhaps as much as a quarter of the chord at a distance of only a

chord length behind the plate. Taking into account the fact that the

vortices in the wake are discharged at the tail of the plate with a ve-

locity vector which is not in alignment with the free stream velocity

means that the vortices will continue to diverge from the plane of the

plate, or y = 0.

In addition to the plate system of vortices and its wake, an

image system identical to the plate system was postulated to lie just

above the surface so as to make the effect of the whole model on the

surface null. This sort of model was postulated to allow the use of

an exact potential theory. The velocity potential of the plate system

and its image was written down in terms of their unknown vortex

strength distribution. Then a third potential function was introduced

which took into account the free surface waves. This function was in-

vestigated through a partial differential equation resulting from the

application of the free surface boundary conditions. Finally, the

boundary condition concerning the impenetrability of the oscillating

plate was used to obtain an integral equation for the vortex strength

distribution. The integral equation was solved by the assumption that

the vortex strength distribution could be represented by a Glauert

trigonometric series. This introduced a system of infinite algebraic

equations. The pressure distribution and the thrust coefficient are










functions of the solution to this system of equations. The equations

were solved by the approximate scheme of truncating the set and solv-

ing repeatedly while increasing the number of equations solved until

values for the first N unknowns become apparent. The thrust coeffi-

cient was then evaluated for a flow velocity of U = 3 and several dif-

ferent depths.

The results of the calculations for a quadratically varying am-

plitude function with coefficients which are thought to approximate the

swimming of a fish indicate the following conclusions.

1. The theory produces accurate results for the thrust coeffi-

cient versus reduced frequency variation for a depth (D = 50. 0) large

enough to simulate an infinite depth when compared to figures obtained

by a different method.

2. The influence of the free surface is not a marked factor for

relatively shallow depths. One must decrease the running depth to

about one-fourth the half-chord in order to note a finite difference in

the thrust from that observed at an infinite depth.

3. The influence of the free surface waves causes a varying

difference in the thrust coefficient. At high reduced frequencies the

difference is greater than at low frequencies, provided k is greater

than the critical frequency. At frequencies between zero and the criti-

cal frequency the thrust (or in this case, drag) coefficient varies in an





61



oscillatory manner. The point of zero thrust does not occur at a re-

duced frequency equal to the wave number of the plate motion, but at

a slightly higher value for small depths.

4. As a natural consequence of the theory used for this study,

the wave drag for zero reduced frequency is obtained without difficulty.

This might be of interest to designers working in the field of hydrofoil

craft equipped with slightly bent wings.




































APPENDIXES












APPENDIX A


THE FREE SURFACE POTENTIAL


To solve the partial differential equation


Sx


I,
4- 1
P1z


Io


[


iF-
,rFL


(A1)


e-- ---) d+l ,
CK- 0)1 + D12


we proceed as follows, using the operator notation


M *~


A particular integral of (Al) is written in symbolic form as


1
c3 -Q .S'}


S()1-- o)
"(X- f)? + 'Dz


So -K-
lK' e (x-l d()

1


Z 2 il &- k2.
Fz


zP3
^&x


(A2)


where


(A3)


f2 ( t j ) =


(A4)


k~3






64



Now it can be established that


)4- -(-D


I x- F + L(t-ZD)2


dS, y-zo

Then it follows that


= Re


0


00

0^^-D


S;ih S(X-Fl


oo

S-e s(A 6
0


and that the particular

(see [29, p. 103])


P
cp3


1
g( ,.s')


KP
ZTr -


P
integral P3 may be written symbolically as











Se e(A7)

o0


We also observe that for an equation like


a.x+ b4


the particular solution consists of terms of the type


P 1 a.x+ b,


00

= L eS

o


dS}


I


(x- W)Z+(-20)









P
so that the particular integral q3 may be written as

IP

^~~2T n 6Cs > ^


K I
z TrF j


where


pG, ( x,; ) = & -
(s-s,K(s-s,5
00L


LS(,-) -)
e ds
(s-s) (s-s,)


and


&z -QI-
-S-


For the complementary solution we write (see [29, p. 102])
1 -,
L= 1Jz4 1 G)x, C Jd) e (A9)
ZTTF~ Z rrF'
-1 1


where


Gc(x, ; ) =


D,

+ D. 3


SOz e +
T.b--zo (.X-.)] S
+ D, e
Lb 20 (Y-)3S


tij-2O-.(- 3 *K~)15


and the Sj are defined as in (A8). Upon adding (A8) and (A9), the gen-

eral solution of the partial differential equation (Al) appears as it is

given by equation (1. 44) in Chapter I.


(A8)


zK)

ZK)


+- KZ 7

+

G,(x,~;~~ dF ,


(s-s,)(s-s, =

(s-S3)Cs- S =









To fix the values of the constants Dj, we proceed as follows:

If a small dissipative force

-/1 gra.d (A10)
is introduced into the analysis, the external force potential .1 in equa-

tion (1. 16) becomes

a. -= 9 +0 /4 (All)

This leaves the Laplace equation

aZ = 0 (A12)
unaltered, but makes the boundary condition given by (1.41) become


"Vu*e K^t' ^K ^'^K-1 4 y= Z (A 13)
Substituting from equations (1.38), (1.39), and (1.40) into (A13),

we get

,3 + aK(p+) ?3 + +KZL3 -1)+P_
b xz Fz b


tg) cf- f K / e (X- d (A14)
Zf CK-Pz+ D2 7,r 0<-z + D2

which governs the behavior of the spatial portion %3(x, y) of the free

surface potential 3(x,y,t). In equation (A14) the Froude number F,

the reduced frequency k, and a new parameter related to the dissi-

pative coefficient are as follows:

F ,K =
Ir Uv zaO









A particular integral of (A14) is obtained just as for the partial

differential equation (Al) already treated. However, the function

G(x, y; ) of equation (A8) evaluated on the free surface y = D becomes

(x, M s S-o --J edS, (A 15)
J L(5-o;)(s-oG) cs-oa (s-cr,)

where



(s-o~(Us-o,) = s5-[ + ZK(1-ip s -+ K 1(- /) .


As both the integrals in (A15) are singular at 0C, Cauchy prin-

cipal values must be considered. The O' are given by

ad Z= Fl 1- zKFZ(-3)_ (l- K^9F/F-) (A16)

and

0-3 7 2F r-^) 1+ KF p -KF (A 17)

When / = 0, the roots ^,z and (,/, become

SoZ = -- 7-ZFZ (K-- F F) (A18)

and

S3 1 [ Y-ZKFZ + 4 KF)i (A19)

Now it is the behavior of the roots 0' when 3-*0 which is criti-

cal to this analysis. Upon examining equations (A16) and (A17) as P-*-0,

the results in Table 5 are indicated.






68



TABLE 5

VARIATION OF CO j = 1,2,3,4, WITH kF2
o


kF 2 Real Imaginary

a; a U o3 al a, 0 o 09

0 1/F2 0 1/F2 0 0+ 0 0- 0

0< kF2< 1/4 (+) (+) (+) (+) 0+ 0- 0- 0-

1/4
l/2

Notes:

These results may be found in Tan's paper [16] except for an
error which has been corrected here. Tan indicated that lim m m a=
2 'e-o
0+ for 0 < kF2< 1/4. This led him to the conjecture that no waves
could exist upstream of the disturbance, or in this case the plate.


The notation "0+" or "0-" indicates that as -r 0, the imaginary

part of the root indicated approaches the real axis from the positive or

negative side respectively.

Return now to the integral in (A15). By writing


(s -aC,)S -a~)


0- 1 [


7
S-a;


(A20)


- S-,


and doing the same for the term involving CO and O-, we may write


the function G(x, D; ) as





69



G(,O;, ) 3 I --- r;I > (A21)


where
00

i 1 dS = Z (A22)



CI L-o-S(x-)) S
S = e ds q3 (A23)
S O


These integrals may be evaluated by contour integration in the

complex z = S + m plane and use of Cauchy's integral theorem. The
-os
proper contours must be chosen with care. Due to the term e in

the integrals, no part of the negative S-axis may be used. The choice

of the remaining first or fourth quadrants depends on the sign of (x ).

This will be evident later. The contours to be used are those shown in

Figure 7.


Lyn
(a) For x > 0


,, Efor Il',I2
'for 13, 12



(b) For x- < 0




Sfor 13,14


Figure 7. Contours of Integration for the IIntegrals, j 2, 3, 4
Figure 7. Contours of Integration for the Ii Integrals, j = 1, 2, 3, 4











When a pole corresponding to C0 lies inside the contour of in-

tegration, the result obtained using Cauchy's integral theorem includes

the residue at this pole. As we will see later, by allowing x to ap-

proach ; these residue terms are responsible for harmonic wave

trains which propagate to infinity.




Lnv

LR


SRC

\^-^"R


Figure 8.


Contour of Integration for I
Contour of Integration for I1


As an example of the integration procedure, consider I1 for the

range x J > 0. According to the scheme in Figure 7, the contour

El is to be used in this case. Integrating around this contour, shown

in Figure 8, and using Cauchy's integral formula, we have

R
e -o+x-ys L o t(x-t-)l-
e dS + e d> -
S- -- a -
SCR.


R



rr~1-
0


[d (- 27- 1r 1
ef 7-r L e i


(A24)











where it has been assumed that OI does lie within the contour E1

In order for this to be so, the real part and the imaginary part of C0

must be positive. Whether or not this is true may be ascertained from

Table 5. Now, allow R to approach oo in equation (A24). The second

integral can be shown to vanish: We observe that


S= e C= S < 1 i-or >0
D




and if R is sufficiently large,



I Z -0- I


so that

[-o+ L(x-- ]z.

E-O
R
where = x Now


77


-a.R sy e
S ,Iz


(A25)


oC< 7T
z


so that


r/z
-DoZ- i(.X-) ;Ir


R e
and as 6-0 for R -o the integral

tain from (A24)


S TTL e


za. R.
Tr = 9.___- CL1-
a. L &I

clearly vanishes. Thus, we ob-



00
L- D +i 10


0











Proceeding in the same manner, taking care to observe the proper
/ / /
contours of integration, we obtain similar results for IZ, 13, and 14

for the case x > 0:

00
I- o+. C(- )3a r c-D4- <-^
I, = 2T e e advn ; a=7,2 (A26)
'- + CT.
0


-D-LCx-^ ^ /00 L-D- LCx-piC- 0

I^+ = -2(A27 e ) e
3- 3


For x 4 < 0, or upstream from the plate, we obtain

ro+i O(-pIO, f +-DLCYx-F)l(-i v"
oo

_^ =1 -2nT le o e d-_rn ;j1/ (A28)





00 L
ZTr e C4- fa3C*.. (AZ9)



Since what is sought are the flow conditions at infinity, the next

step is to note that the definite integrals in equations (A26), (A27),

(A28), and (A29) decrease rapidly for large x Thus, they con-

tribute only local disturbances around the plate. Further, note that an

infinite propagation of waves can occur only when



P-o

and even then a wave results only when the OC. is contained in the con-

tour of integration.










It is helpful to write a compilation of the results for G(x, D; ~)

as x--,o. This is shown in Table 6. A reminder of the contour of

integration used for each integral making a contribution to G is also

included in the table.

It is a simple matter now to compile the values of G+ and G_

for the ranges of the flow parameter kF2, i. e., from Tables 5 and 6

we have

(a) For kF2 = 0

i1-0'(x-F)30o E [-o+;(x-F)]0
)- e -v ec



G = o. (A30)

(b) For 0 < kF2 < 1/4

SC -0, -tLx-:lo z rr. -p
G+ -r_ r--


* ~ L


a,- e

1/4
= -- o--
3-^


1/2


6_ =


(A31)


t.- -. Cx -).Jo;C;f


S.

< kF < o


O .. rT-0-Cx-P]0' L-0-L( <-op. -"
e- e J


J


(A32)


(A33)


for G(x, D; ) we are to write

S4--


(A34)


6-

(c) For

G,


G =


(d) For

.


- (.






74













i *-







I 0



0 1
fQ)








b
















Q
*



4 o

x *-
sO
--























"^ ^- 3
r-- ^ 1 ,









It would appear that for the range 1/4 < kF2 < 1/2 another wave

train should be included since both 0- and 0. lie within the accepted

contours of integration. However, lirm m 0 0 0, and the resulting

wave trains are damped out as x-,-;o. For example, suppose OC is

written as

C7 = O<, t- i ^ oc, I, + 0.


The corresponding G contribution would be


3_ y, _2___-

+ 'Ll Y,-DYOt,(KL-.Y

(Oi + L Po

and as x -- +oo the contribution damps out. The same argument holds

for Cj in this range of kF2. Hence, the two ranges, 1/4 < kF2< 1/2

and 1/2 < kF2< oO could just as well be considered as one range, i. e.,

1/4
Now consider the function G(x, D; ) as it is written from equa-

tion (1.44b). Again using partial fractions, this function may be writ-

ten as


L1G (X, D-) 4--


Die e DZ e +

-DD-3x-lSa t-O-L(.<-)3 S


+ D, e


(A35)










where

SC-D + OL--)IS

e- ds ; i.= 12 (A36)


00 5-

0 e S-dS S =3 (A37)
o S s

In evaluating the above Ij integrals, the same quadrant scheme

for contours is adhered to as was used in evaluating the Ij integrals of

G(x, D; ). One exception is made in the form of the contour. It has

already been shown that no wave train exists at infinity unless the

roots O- or Sj are real and positive. Thus, we use contours like

those shown in Figure 9 when working with the real positive Sj singu-

larities. A semicircular indention of radius E is taken about the pole

Sj. Again, use of one contour or the other is determined by whether

x is greater than or less than zero.











o S
o S) R


Figure 9. Contours of Integration for Ij Integrals, j = 1, 2, 3, 4









-Yn

LR


Figure 10. Contour of Integration for 11


As an example of this type of calculation, consider Ii for

x > 0. Using the contour shown in Figure 10 and Cauchy's inte-

gral formula, we write


j.~ C-D+ L^- s r [-So + L x-
e s + e d +
S- S, f a- 5,
o I


I L-D +(X-L)s(


s e s,




o L n S1
0


+ e + ((.<-



o-.
= 0


The fourth integral vanishes as R-*o The first and third integrals

combine to give I1 as R--o The second integral is calculated for

E 0 as follows. On the contour


S= e-- + S1,










so that


f --o + (x- r3 (S,+ Se )
7 -i e

'T-


0


/-


L-D4-, (X--) Fe
de.


Now we expand the integrand in a Taylor's series like

te
,.E e a- a .E
e e + o. e ...


where 0. = -D


+ i(x ). Then


e a de
-Tr


and


6 dtS7
*- S,


= -77r e


-0Thu

Thus,


I =S ri e d+ .S


As x -t + o the definite integral above diminishes rapidly, and we

are left with a result similar to the one obtained for I1. The other Ij

are then calculated for x > 0 and for x < 0. The resulting

values are combined according to (A35) and are listed in Table 7.

Again a reminder of the integration contour is provided.


STr




























nI
oo
rI-




0








!



I-J




I
.3










4-








I:
0



(0
-i>
~tS




.J


+
0




I




uJ.
+





t


~
v,
rr
--r
~4r
I
~tS
.3
I
Q
I
LI


8





HH
+I





a O






x"
0


t0
I '-

















6


Q) -













I
ri-







I*A


t o
xP

















-I


.3
0





rI-

0
l-7







t


w
*^I
^0




r5l


I-i










r.
1

i,




IC
4-




I

U)


4m









rU,


4X

4-
Q
LJ

'I-
Q4



IWO
N+








Table 8 indicates the behavior of the Sj roots as the flow pa-

rameter kF2 varies.


TABLE 8

VARIATION OF Sj, j = 1, 2, 3,4, WITH kF2



Real Imaginary

S1 S2 S3 S4 Sl S2 S3 S4

0 1/F2 0 1/F2 0 0 0 0 0

0< kF2< 1/4 (+) (+) (+) (+) 0 0 0 0

1/4 < kF2 < 1/2 (+) (+) (+) (+) (+) (-) 0 0

1/2 < kF< oo (-) (-) (+) (+) () (-) 0 0


The following compilation indicates the behavior of the G(x, D; )

function in an analogous manner to equations (A30), (A31), (A32), (A33),

and (A34).

(a) For kF2 = 0


G_ = 0.

(b) For 0 < kF2 1/4


6rL =-e
+- Sr e


G_ i e.
S,-SZ


+- e
S 3


(A38)


+ s[-e l e 'j

(A39)









(c) For 1/4
G+ o. e
S -S

G = O.

(d) For 1/2



G = 0,

where G(x, D; ) has to be taken as

a [-2+(X-O]Si


(A40)


(A41)


41 [-D-LOnSi


G= G, + G_ + L D. D e (A4Z

Upon comparing equations (A30) through (A34) with equations

(A38) through (A42), the appropriate values of the constants Dj can be

assigned. They are as listed in Table 1 of Chapter I.


)


-L( 4^






>-L(4(-)2S









APPENDIX B


COEFFICIENTS FOR THE INFINITE SET OF EQUATIONS1


R e [0 --e z D SfZ{[ + 1 K S


S f t I _ _
S (K+S)(s-s)(s-s) (- s-s)S-j))s-S 3 s),, s

_ _-__ _c/ 1 i C p 20^-^ ^JfS kcos SZ1




S Se sJ ( s) { s-s, ) (s-]s- )] (s)+Kcos




dS 2 (-s)c s-sx ( +)(s-, 5)) f -
F(S-s,5z) k +L S k +









S-ZDS -ZOSS

1Terms containing e and e are omitted when 5S and SL
are complex.








Re1 -T le-s(0siFS T 1(a-^ (^ SS-SAS k
ZL -SJ,) -Ss- (S K-s)(ss,(-S (-~)(5 -S3) 5-,,) -s'-
0

J i[ -1

,-r -e-2DS,_T_(S _-KS Co + -ZS. .J,( kct cos Stl


7T Z -zos AS3coSiS3] -z 04[q W KScos1
7 L- (JT)/ (5f)- -0 -(" -('
F'(S-s9)3e Ls) (K- z 3(S+) L Z(K + y)J







1 /f -
z co s[ ,F k-S(-s )(-S- ) -k S)(S-S-)(S-fS jds

A r"cos rk(-7)3
2 fJ

k Tr S, e-" osg( s;)in Sf 7 e-zs 2 S-)s inSt
eF'(sf,-Ss) i- St K -S

ki 7 e-Dss 1C3)s-3 S, ezot(sy) sS,
2FZ (S,3-Sy) L-S











+ e(-1) [-(-" e s ) e- S) (S T 4











S71 (-.1- J-(- es ) (s)~1 cs- ss



F CSr T'z 2) [e C SJ %)e- seD3Ls,) eT ,)J s]



(SZ- 5 t[e) S3 Jt Oe Z y)


for v, Z, 3, ...








Re f -(-i) [ Lt- e-jI s) + {[ (-( s (ss)] (s


Kcrs[s + (s)L17 s-s) ( -Fdss- ] A


1


4- [^


^ I- JS + Zic(-I)
F2 (I s)(s-sO(s -s( ) (s)(s-s()Cs v, -iJ) ((i77- T'


S(1[) [S, eS's + sz -zos )(.S) c -


(s, -30 k- S,3 e- (,T.,s,) kca S -Syet"tesz k, C~s S~\


-_ K- r {,[s, e- T (+s,1) S t7) e (o4
,~,i~i-Y 2f! S)[ ~ s ~ ) Xsy )


for vn = Y ,


+(IS( r


e~"{T.~VS I F -s,(s- s -,- s- )


kLS;I]
42s./


( Js,} e--T-- Z -/J(s < s (D c s ,) L its S l











{ (-k e- .(s){ Ls,(~I +)(] s -J(s)



IF' S(s- s3)(S-SO,) ( R-S) (K-SLK (-J )
+ -t) 1--r7] e-"^sM ^[)(s- ,)s-) s-,s- s j ts +r,,


cos [--)15- )
V7--7


ft-1) I e o 3 s-,(s,- X +.s) a, k4 SS


-- r ) ( cos ,) -s e-2 DSyJ (s)(J.(s,


+C(-,s ++.(-I) V1 l15,e" ( ) ')+ s e ((J(s( )
I ~k4kS1


-
(S -SO)


s e. ZS3J(s,5. ( (sj+ 9Sa) O-SieK(s


for -yn 1, 2,3, ..


(. f tt ___ I),
F+ ((-s)(s-S3)(s-s,) (<+s(s-s 1(. -s.)1ds + 2k


V-




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