• TABLE OF CONTENTS
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 Title Page
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 List of Symbols
 Introduction to the problem
 General theory of unsteady motion...
 Application of the theory to a...
 Summary and conclusions
 Appendix
 Reference
 Biographical sketch
 Copyright














Title: theoretical study of the swimming of a deformable waving plate of arbitrary finite thickness.
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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
        Page iv
    List of Tables
        Page v
    List of Figures
        Page vi
    List of Symbols
        Page vii
        Page viii
        Page ix
    Introduction to the problem
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
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    General theory of unsteady motion of flexible bodies of arbitrary finite thickness
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    Application of the theory to a symmetric joukowski base profile with a linearized thickness parameter
        Page 53
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    Summary and conclusions
        Page 104
        Page 105
        Page 106
    Appendix
        Page 107
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        Page 120
    Reference
        Page 121
        Page 122
        Page 123
    Biographical sketch
        Page 124
        Page 125
        Page 126
    Copyright
        Copyright
Full Text











A THEORETICAL STUDY OF THE SWIMMING

OF A DEFORMABLE WAVING PLATE OF

ARBITRARY FINITE THICKNESS












By
JOHN PAUL ULDRICK


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
August, 1963















ACKNOWLEDGMENTS


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

.Supervisory Committee, for suggesting this problem and for his

encouragement and counsel throughout the course of this research.

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

Head of the Advanced Mechanics Research Section, Professor W. L.

Sawyer, Head of the Department of Engineering Mechanics, Dr. I. K.

Ebcioglu, Assistant Research Professor in Engineering Mechanics, and

Dr. R. G. Blake, Associate Professor of Mathematics, for their encourage-

ment and criticism of the manuscript. Also,. the author wishes to thank

Dr. Nash for providing financial assistance for the use of the IBM 709

electronic computer.

To the National Science Foundation, he expresses profound

gratitude for supporting his graduate studies for the past twenty-seven

months.

Finally, to his wife Johnnye, he is grateful not only for her

encouragement and understanding during a trying time but also for her

assistance in editing and typing the rough drafts and the final

manuscript of this dissertation.















TABLE OF CONTENTS


Page

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

LIST OF TABLES . . .. . . . v

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

,IST OF SYMBOLS . . . . . . . . . . vii


Chapter

I. INTRODUCTION TO THE PROBLEM . . . . . .

1.0 Statement of the Problem
1.1 Method of Approach
1.2 Review of Related Literature

II. GENERAL THEORY OF UNSTEADY MOTION OF FLEXIBLE BODIES
OF ARBITRARY FINITE THICKNESS . ...... 9

2.1 Mathematical Formulation
2.2 Pressure Distribution on the Profile Surface
2.3 Potential of the Uniform Base Flow
2.4 Velocity Potential Satisfying the Unsteady
Perturbations of the Profile Surface
2.5 Pressure Distribution of Source Potential
2.6 Circulatory Potential Function
2.7 Circulatory Pressure Distribution
2.8 Lift and Moment
2.9 Thrust Formulation

III. APPLICATION OF THE THEORY TO A SYMMETRIC JOUKOWSKI
BASE PROFILE WITH A LINEARIZED THICKNESS
PARAMETER .... . ......... .... 53

3.1 Calculation of the "Downwash" Velocity for
Any Given Flapping Function
3.2 Calculation of the Pressure Distribution on
the Mean Base Profile
3.3 Calculation of the Lift and Moment
3.4 Calculation of the Thrust and Drag
3.5 Time Average Value of Thrust
3.6 Numerical Example
iii









Chapter Page

IV. SUMMARY AND CONCLUSIONS . . . . . 104


APPENDICES . . . . . . . . . 107

LIST OF REFERENCES ......... .. . . .. . . 121

BIOGRAPHICAL SKETCH . ... .. . ..... 124


iv


















LIST OF TABLES


Table

1.

2.

3.

4.

5.

6.

7.

8.

,9.

10.

11.

12.

13.


Bessel Functions .. . . .

Real Part of the Theodorsen Function .

Imaginary Part of the Theodorsen Function

Thrust Coefficient CT . . . .

Coordinates of the Base Profile . .

Real Part of Bn Coefficients o

Imaginary Part of Bn Coefficients ...

Real Part of On Coefficients . .

Imaginary Part of on Coefficients . .

Real Part of AO Coefficients . . .

Imaginary Part of AO Coefficients . .

Real Part of A1 Coefficients . . .

Imaginary Part of A1 Coefficients . .


Page

98

100

101

103

112

113

114

115

116

117

118

119

120


. .

. .

. .

* .

. .

* .

* .

. .

* .

. .

* .

. .

. .















LIST OF FIGURES


Figure Page

1. Stretched-Straight Configuration of the Fish . . 3

2. Displaced Configuration of the Fish . . . . 3

3. General Profile Configuration . . . . . 13

4. Circle Plane for General Profile Configuration . 13

5. Profile Plane (z-plane) . .. . . . 15

.6. Circle Plane (r -plane) . . . . . . 15

7. Circle Plane with Point Source . . .. . 31

8. Circle Plane with Vortex Pair . . . . . 38

',9. Stretched-Straight Configuration with Pressure
Distribution . . . . . . . . 50

10. Stretched-Straight Configuration for Several
Thickness Parameters . . . . ... . 55

11. Thrust Coefficient Versus Reduced Frequency . . 102















LIST OF SYMBOLS


Symbol

h(x,t)



p
t

P

F= +ivy

z ax + iy




FFo

Fl

F2






U

v

w = + iv
A
w

f ()


Description

Displacement function

Velocity vector of fluid particle

Mass density

Time

Pressure


Complex

Complex


velocity potential

coordinate in the profile plane


Complex coordinate in the circle plane


Complex velocity potential of uniform base flow

Complex velocity potential of source distribution

Complex velocity potential of vortex distribution

Velocity potential

Stream function

Magnitude of the x-component of velocity

Magnitude of the y-component of velocity

Complex velocity in the profile plane

Complex velocity in the circle plane

Mapping function of circle into a profile

Thickness parameter









Symbol

(i



Imi


TI (e,t)





Fn


bn

cn

H*(()


D(z,t)

G(e)


Description

'Real part of' operator for the space imaginary unit

'Real part of' operator for the time imaginary unit

'Imaginary part of' operator for the space imaginary
unit

Space imaginary uhit

Time imaginary unit

Pressure distribution on surface of profile

Circulation

Vortex distribution

Coefficient of the Fourier series expressing the
unsteady velocity of a fluid particle normal to
the circle

Coefficient of the Fourier series expressing the
"downwash" velocity on the profile

Coefficient of the Fourier series expressing the
space variation of the displacement of the
profile surface

Coefficient of the Fourier series related to the
displacement of the profile surface

Coefficient of the trigonometric series of the
pressure distribution on:the surface of the
profile

Auxiliary coefficient related to the Bn's

Auxiliary coefficient related to the Cn's

Trigonometric series for space variation of the
displacement function

Displacement of the profile surface

Trigonometric series expressing the space variation
of normal velocity on the circle

Normal velocity at surface of the circle


viii









Symbol

Aq

qn

qt

U

LA)

k =- /U

T(e,jk; e)


Si =. f+j gg
L

M

T

Q

P(jk; )

Kn(z)


H(2)(z)


J,(x)



T

CT


AAAA I


Description

Tangential velocity of fluid at surface of the circle

Velocity normal to the profile surface

Velocity tangent to the profile surface

Free stream velocity in the profile plane

Circular frequency of harmonic oscillations

Reduced frequency

Function associated with the effect of the wake on
the pressure distribution

Theodorsen function

Lift

Moment

Thrust

Auxiliary function associated with the wake

Auxiliary function associated with the wake

nth order modified Bessel function of the second
kind with argument z

nth order Hankel function of the second kind with
argument z

nth order Bessel function with argument x

Wave number

Time average value of thrust

Thrust coefficient

Auxiliary thickness parameters


Coefficients of quadratic amplitude function


dl, d2s d3















CHAPTER I


INTRODUCTION TO THE PROBLEM


Have you ever watched a fish swim? This is a fascinating

sight. A fish can glide through water at flashing speed or it can

idle in an almost completely immobile state. Throughout history man

has observed the flight of birds and the swimming of fish and dreamed

of flying and swimming himself. Some of these observations have been

the stimulus for man's development of flying vehicles which, in many

ways, surpass the flight of birds. Likewise, studies of sea animal

locomotion will, no doubt, bring about much-improved designs of sea-

faring vehicles.


1.0 Statement of the Problem

Consider a flexible plate of constant depth (chord), of infinite

length (span), and of arbitrary finite thickness (profile) immersed in

an inviscid, incompressible fluid. The flow field is assumed to be

infinite in all directions away from the plate. The assumption of

infinite span and finite chord implies that the flow field around the

plate can be treated as two dimensional. Further, the plate is assumed

to move approximately along a straight line with constant forward speed

and at the same time to execute a perturbation motion of a small ampli-

tude in the transverse direction. The configuration of the plate when

there is no fluctuation is assumed to be symmetric with respect to the









chord, as shown in Figure 1. Henceforth, this shape will be identified

as the stretched-straight configuration or the base profile. The flow

field around the stretched-straight configuration will be referred to

as the base flow field. Shown in Figure 2 is one configuration during

flapping. A rounded nose and sharp tail are used for the model of the

fish.

With the x-y rectangular coordinates fixed in the plate the

mean camber line is defined by



Y, h(X(Xt) X,1 t) + YLX,ti XL- X XT (1.1)


where YU and YL are the ordinates of the upper and lower surfaces of

the plate respectively, and XL and XT are the leading and trailing

edge projections on the x-axis, respectively. The function h(x,t) will

be referred to as the flapping function.

Obviously, as a result of the distortion of the plate, the

velocity of a particle of fluid on the upper and lower surface at the

same x-coordinate will have different magnitudes. This velocity

difference gives rise to a corresponding pressure difference and, as

a consequence, there results a net unsteady hydrodynamic force which,

depends upon the distortion and rate of distortion (flapping) of the

plate.

The component of this force along the x-axis will result in

either a drag or a thrust for the fish.

The thrust is assumed to be generated by a train of displace-

ment waves, which are not standing waves, passing from the leading















U -


Figure 1


Stretched-Straight Configuration
of the Fish


U -


Figure 2

Displaced Configuration
of the Fish


r


I~------


1 .









edge to the trailing edge of the plate. The magnitude of this thrust

depends upon the propagation velocity of these waves.

The purpose of this investigation is to calculate the forces

acting on the plate for any given flapping function h(x,t).


1.1 Method of Approach

In Chapter II of this study the general theory for a flexible .

thick body undergoing preassigned undulations is developed on the

basis of the complex velocity potential method. In this method, the

body profile of the stretched-straight configuration in the physical

plane is mapped by a suitable transformation into a circle and the

unsteady boundary conditions are satisfied by a source distribution on

the circle. The problem is linearized by assuming a small unsteady

perturbation theory. Due to the presence of a sharp trailing edge,

the velocity induced at the tail by the source distribution has a

mathematical singularity in the physical plane. This singularity is

removed by introducing a fluctuating vortex distribution along the wake

stream line of the steady base flow such that the induced velocities of

the source and vortex distributions combined vanish at the tail. This

is the so-called Kutta condition of smooth attached flow at a sharp

trailing edge. From the base flow potential, the source potential, and

the vortex potential the pressure distribution on the base profile is

computed by employing the unsteady Bernoulli equation / .7 With

the pressure distribution known, the hydrodynamic forces acting on

the plate are computed.



*Numbers in brackets denote entries in the List of References.









The thickness enters the problem through the mapping function

in the form of a small thickness parameter. Finally, in computing the

forces acting on the plate, it was convenient to linearize all func-

tions in the thickness parameter. This is carried out in Chapter III.


1.2 Roview of Related Literature

In a search of literature related to this subject, it was

found that most of the investigations in this field have been made in

the last decade. A number of publications have appeared concerning sea

animal locomotion both from a biological and an engineering standpoint.

This study was limited to an investigation of engineering interest

since only these were deemed of significant value to the investigation

herein pursued.

Taylor /2 7 paved the way for new problems in hydrodynamics

with a study concerning the action of waving cylindrical tails in

propelling microscopic organisms in a viscous fluid. In this study he

assumed the tail of the organism to be a flexible cylinder which is

distorted by waves of lateral displacement propagated along its length.

Taylor assumed that the viscous forces played the leading role in

propelling the organism. In a subsequent study, Taylor [3J_ investi-

gated the swimming of long animals such as snakes, eels, and marine

worms by considering the equilibrium of a flexible cylinder immersed in

water when waves of bending of constant amplitude travel down it at a

constant speed.

As another approach to the problem of propulsion of sea animals,

Siekmann /4_y discussed the hydrodynamics and propulsive properties

when a jet of fluid is ejected from the opening of a tube. He provides









calculations for the thrust and the basic equations for the horizontal

rectilinear motion of a rigid torpedo-like body. Siekmann's result

may be applied to investigate in an elementary way the locomotion of

certain aquatic animals belonging to the class of cephalopods, particu-

larly squids, octopuses, and cuttlefish.

Lighthill /.5]7 discussed briefly the swimming of slender fish

at the Forty-eighth Wilbur Wright Memorial Lecture. In a later publi-

cation /-637 he considered the swimming of slender fish in which he

employed as a model a slender cylindrical snake-like configuration

immersed in a uniform flow field along the stretched-straight configura-

tion of the model. Lighthill assumed that the propulsion was generated

by very small fluctuating lateral displacements of the flexible cylinder.

These displacements were of the form of waves which travel down the fish

from the nose to the tail with amplitude increasing from zero over the

front portion to a maximum at the tail. Lighthill found that the most

efficient oscillatory movements were for the fish to pass a wave down

its body at a speed of approximately 5/4 of the desired forward speed.

His theory goes back to Munk's work 7_] on flow about airships.

All of the above studies dealt with a three dimensional axis-

symmetric type flow problem.

Of particular interest to this study is a paper by Siekmann [-87

in which he discusses the propulsive forces generated by an undulating

flexible plate of infinitesimal thickness and infinite aspect ratio in

two-dimensional flow. The general theory used is essentially that due

to Schwarz /9_7 and to Kissner and Schwarz /-10_ in which the thin

plate and its wake is replaced by a vortex distribution of fluctuating

strength. Siekmann's formulation of the problem led to an integral









equation for determining the vortex distribution. Of special

significance in this study was the calculation of the thrust

produced.

Wu /i113, in an independent study, considered essentially

the same problem as Siekmann -8_7. Wu employed Prantl's accelera-

tion potential to determine the forces acting on the plate. In a

subsequent paper /-12_7 he investigated the two-dimensional potential

flow around a flexible, waving, infinitely thin plate which executes a

rectilinear swimming motion, the forward velocity of the plate being

assumed an arbitrary function of time. The general formulae for the

thrust given by Siekmann / 8_7 and Wu /11_7 are in agreement.

At about the same time, Smith and Stone / 13_/ discussed the

swimming of an infinitely thin plate in two-dimensional flow where

the plate was represented in elliptic cylindrical coordinates. They

Satisfied the unsteady boundary conditions by solving the Laplace

equation for the velocity potential and satisfied the Kutta hypothesis

of smooth attached flow at the tail by adding a circulation around the

plate of fluctuating strength such that the net induced unsteady

velocity at the tail vanished for all time. Smith and Stone, however,

failed to consider the effect of the wake and, as such, their theory

is incomplete and not in agreement with Siekmann /8j and Wu /Ll 1].

Recently, Pao and Siekmann -/14_7 considered the Smith-Stone

problem and included the effect of the wake. Their results are in

agreement with those of Siekmann /8]/ and Wu /11J7.

Bonthron and Fejer / 15_ studied the two-dimensional problem

of fish locomotion by employing as a model three infinitely thin

rigid plates hinged together where both rotational and translational






8

oscillations were imposed upon the plates. They employed Theodorsen's

theory [16J for a system of finite degrees of freedom and solved the

dynamic equilibrium equations.

Kelly [l17J measured experimentally the propulsive force

produced in an undulating, thin, two-dimensional plate and found that

the theory given by Siekmann [8j and Wu ll_7 was in agreement with

experimental evidence when allowance was made for skin friction.















CHAPTER II


GENERAL THEORY OF UNSTEADY MOTION OF FLEXIBLE

BODIES OF ARBITRARY FINITE THICKNESS


The solution of the problem of two-dimensional incompressible

steady potential flow around bodies of arbitrary shape has been

treated by many authors. Most of the exact solutions of the flow

field have been obtained by the application of complex variable

theory. The usefulness of the theory depends on a theorem in conformal

representation stated by Riemann almost a century ago. Basically, the

theorem is equivalent to the statement that it is possible to trans-

form the region bounded by a simple curve into the region bounded by a

circle in such a way that all streamlines and equipotential lines of

the first region transform respectively into those of the circle. This

theory for the case of steady potential flow has been developed most

elegantly by Theodorsen and Garrick /18_7.

Recently Kissner and Gorup / 19_/ employed the complex variable

theory for the case of unsteady motion of a rigid profile of arbitrary

finite thickness immersed in a uniform stream. Here the motion of the

profile was limited to two degrees of freedom, i.e., a rotary and

translator movement. The problem was linearized by assuming small

unsteady perturbations about the mean position of the base profile.

Theodorsen -16_7 treated the problem of infinitesimal unsteady

oscillatory perturbations of an infinitely thin airfoil immersed in a









uniform stream. He was interested in the aerodynamic instability of

the airfoil and the mechanism of flutter. He divided his solution into

two parts. First, the boundary conditions on the surface of the plate

were satisfied by an appropriate distribution of sources and sinks just

above and below the line representing the airfoil. Second, a pattern

of vortices was put on this line, with counter-vortices along the wake

to infinity, in such a way that Kutta's hypothesis is fulfilled without

disturbing the boundary conditions at the airfoil. These vortices were

distributed in such a way that the circulation of the whole flow field

was preserved. Each of these parts were obtained by a conformal

transformation of the infinitely thin rigid profile into a circle.

The theory for a flexible arbitrary finite thick body immersed

in a uniform stream undergoing small preassigned undulations normal to

the surface of the profile is treated in this study. The undulating

displacements are assumed to vary along the chord with a small ampli-

tude at the nose and with an increasing amplitude toward the tail.

These displacements are harmonic in time and are of the form of waves

which pass down the chord from the nose to the tail. The theory

presented is an extension of Theodorsen's work / 16_/ to include the

effect of thickness and the effect of an arbitrary flexible motion.

Also, it is an extension of Kissner and Gorup's theory / 197 to

include the effect of flexible displacements and a calculation of

net thrust or drag caused by the undulatory motion.








2.1 Mathematical Formulation

The equations governing the two-dimensional motion of an

incompressible, inviscid fluid are the following:

Continuity equations


div V 0 (2.1, 1)

where V= ULLV is the velocity vector of a fluid particle.

Motion equation:


dV
p T grcd p (2.1, 2)


where p is the mass density and p the hydrodynamic pressure. In the

Euler equation (2.1, 2) the body forces have been neglected.

In the regions of the flow field where the flow is irrotational


curl V = 0 (2.1, 3)

Then, in these regions there exists a scalar point function # (X,yt)
defined by the equation


V = Ojroid (2.1, 4)

where (x,y,t) is the velocity potential.

Substituting equation (2.1, 4) into equation (2.1, 1) leads to

the Laplace equation


(2.1, 5)


VZ j = 0








Also, from equation (2.1, 1) a stream function 'J(x,y,t) can be

defined as


8y _y
(2.1, 6)

v =-- ---
Sx 6y

where UL-. and V = are defined by equation (2.1, 4).

Equations (2.1, 6) are the familiar Cauchy-Riemann differen-

tial equations defining an analytic function F(Z,t)= (xy,t) + i 1(x,y,t)

of the complex variable. = x+ iy From equation (2.1, 6) the

complex velocity, w= u+iv can be found from the complex velocity

potential F(Z,t) for any time t, as



W= a- (2.1, 7)



where the bar denotes the complex conjugate.

Therefore, F(2,t) completely determines the flow field. It

is a well known fact in the theory of complex variables that an

analytic function preserves its analyticity under a conformal trans-

formation of coordinates.

Lot Z=f(r) be a conformal transformation of the exterior of

a uni circle in the -= + i1 plane to the exterior of the profile

in the e x+ iy plane, as shown in Figures 3 and 4. The only

limitation on this function is that at large distances from the origin

the flow in the two planes differs at most by a constant, i.e.,


limn w(z,)= lim ( ',t) (2.1, 8)
.oo 10.- Coo
















U


Figure 3


General Profile Configuration


a--9


Figure 4

Circle Plane for General
Profile Configuration


9









Quantities in the s -plane are denoted by a circumflex. The

coefficient 1/2 in equation (2.1, 8) is selected such that the chord

of the base profile will be approximately two units.

Such a function can be developed in a Laurent series as


0n
=f(t)= ) -" (2.1, 9)
n=-1


where the an's are constant.

As a model for the fish, it is reasonable to require that the

profile have a rounded leading edge and a sharp trailing edge and that

it be symmetric when in the stretched-straight configuration. For this

basic configuration a symmetric Joukowski profile is used. This

profile can be developed in a thickness parameter as



+f 21 +E (1 QE 2nr (2.1, 10)



where C is a small positive quantity. The thickness d of the fish at

its mid-chord is approximately 26. Since the length A of the fish is

approximately two, the thickness ratio at the mid-chord is of the

order F .

Clearly, this function satisfies the requirements of a sharp

trailing edge since



df[ (1-e)


The two planes are shown in Figures 5 and 6 for the stretched-

straight configuration.















--L


Figure 5
Profile Plane ( I-plane)


Figure 6


Circle Plane ( -plane)


__


~


U 4









Since the potential function F(L i) is a point function, it

is invariant uncer the transformation (2.1, 9), i.e.,


S= (2.1, 12)


The velocities of a particle in the two planes are related by



S= w (2.1 13)



Let the location of the downstream stagnation point P0 in the

r -plane be given by


= e (2.1, 14)


where T is the argument of the line OP.

Any point P on the profile in the z-plane is mapped into the
A
point P in the -plane and is given by


z: (eie (2.1, 15)


Since the unsteady perturbation displacements of the profile

boundary are assumed to be small compared to the chord length, it is

assumed that only the linear effects of the unsteady motion are of

importance in determining the flow pattern and the forces acting due

to the undulation of the plate. With this small perturbation theory

the velocity potential can be decomposed as follows:


F(z,t = Fo (7) + FI(i ) + Fz (z,t) (2.1, 16)


where FO is the potential of the uniform flow around the stretched-

straight plate, Fl is the contribution due to the unsteady motion of








the profile, and F2 is the potential required to satisfy the Kutta

condition at the sharp trailing edge. Therefore, the problem is to

determine each of these potentials and to superimpose the results.


2.2 Pressure Distribution on the Profile Surface

In order to determine the thrust, lift, and moment acting on

the plate it is necessary to calculate the pressure distribution on

the surface of the base profile. This can be accomplished by employ-

ing the unsteady Bernoulli equation /-l_, which reads



T + t +() (2.2, 1)


The arbitrary function q(t) can be determined at upstream

infinity since it is assumed that the disturbance due to the unsteady

perturbations vanishes there. Thus,


( i)= 2 uz (2.2, 2)


Therefore, equation (2.2, 1) becomes


+ (2.2, 3)
P +Z t 2 P


where



L( F (2.2, 4)








Obviously, the velocity potential (x,y,t) on the base

profile surface is



2 i[F()L t] (e) (2.2, 5)


where &i denotes the 'real part of' operator for the space imaginary

unit i.

Let


-- O= u + u ." (v0 + V' + V"
(2.2, 6)
00 :w iv- -- = v"
u+ U.o+Vo V- IV' I -


where (ul~aV. ',v') and (.",V") are the x and y components of

the velocity of a fluid particle associated with the base flow poten-

tial, the source potential, and the vortex potential, respectively.

Then,




S= LU+ +vi + Z( UoU' v+ v'l + 2 ( oU" + VoV") +

+ Z(uL'u"+v'v") +-u.2 +v'2 *+iA" + V (2.2, 7)


or, in terms of the complex potentials,



d F dFo, d Fo d F "*dFo A
Tz d-e I ae j a J


6[L 1 F, F + (F F)
I a J 6 f- z e 3+ (2.2, 8)








Substituting equations (2.2, 4) thru (2.2, 8) into equation
(2.2, 3) gives


-P-' Z -
P P Z1 3tL

1 cF, dE, fcot0 L F
2 dE. de a

6Fa 1F 1iF~_
F J Z IF~


+ Fz(Cz.,i)11 +


+ +



6M 3z J


Under the small unsteady perturbation theory the last two
terms of the above equation may be neglected since they are of second
order in the perturbation velocities. The first and third terms are
time independent and, as such, do not contribute to the forces acting

on the plate. Therefore, the remaining terms give the desired
unsteady pressure at a point P on the base profile as


'P 1}z%4(+Fz



or


oPr in tdFe irm iof th a n


or, in terms of the argument 8,


TT (e, t)= p [z(el, t]


(2.2, 10)






(2.2, 11)


(2.2, 12)


(2.2, 9)









2.3 Potential of the Uniform Base Flow

Consider the configuration of the profile shown in Figure 3

with the approaching stream along the positive x-axis with a constant

velocity U. Since the mapping function for the profile is assumed to

be known, it is only necessary to calculate the potential for a flow

around the circle and to transform this flow field to the profile.

Generally, a circulation ro will be present around the profile and

hence around the circle. Therefore, from any text in fluid mechanics

l_/ employing complex variable theory, the potential can be written
as



0FocO)= l ) + log (2.3, 1)


The complex velocity in the C -plane is



S (2.3, 2)



and on the circle the velocity becomes

ee2lee
w(e) = (1-e21 -e (2.3, 3)



Since the velocity vanishes at <=ei1 the circulation o

can be found to be


= -Z U Sin t: (2.3, 4)


For a sharp trailing edge the Kutta condition requires that the flow

leave the profile smoothly. Therefore, to satisfy this condition the









downstream stagnation point must be located at the sharp tail. For

the symmetric base profile, as shown in Figure 5, the sharp tail

coincides with the x-axis and the downstream stagnation point PF in

the r -plane is located on the F axis. Thus, for this case the

circulation Q[ vanishes.

From the above results, the velocity potential of the base

flow becomes



UFo() + (2.3, 5)


The base flow velocity is tangent to the circle and is



d^ -e -iU san8 e'e (2.3, 6)


A
The velocity of a particle on the streamline emanating from PO

is given by


LX) (--) (2.3, 7)



The velocity of the same particle in the profile plane is


Sx f (2.3, 8)
i()-I









2.4 Velocity Potential Satisfying the Unsteady
Perturbations of the Profile Surface

Any very small displacement of a point P on the profile in a

direction normal to the boundary at this point will cause a small

displacement of the corresponding point P on the circle in a radial

direction.

The relationship between corresponding displacements in the

two planes is given by


d = f'() d (2.4, 1)


Let the unsteady displacement of a point P on the profile be

D(z,t). Then the position of point P at any time t is


2p = Zop + D(E,t) (2.4, 2)


where o,, is the coordinate of the point on the stretched-straight

configuration.

The boundary condition is given by the fact that the surface of

the profile is a material impenetrable body, i.e., the velocity of a

fluid particle in a direction normal to the boundary must be equal to

the velocity of the corresponding point on the boundary in this direc-

tion. Thus the complex velocity vector of point P is given by the

material derivative of zp with respect to time, i.e.,


dup bD bD d2
dt t ( + dt (2.4, 3)
dt 3t z dt


where is the complex velocity vector of a fluid particle at this
dt
point.









In order to linearize this boundary condition, the assumption

is made that the velocity of a fluid particle near the surface differs

very little from the base flow velocity. This assumption seems

reasonable at all points of the boundary except in a small region around

the stagnation points.

Using the above assumption, the linearized boundary condition

becomes


d OD dF 8D
d -- (2.4, 4)
dit 6-Lt di bz*


Next, consider the velocity of point P in a direction normal

to the surface. The point P in the z-plane maps conformally into the

point P in the -plane. The tangent to the circle at point P maps

into the tangent to the profile at point P, and the normal to the

circle maps into the corresponding normal on the profile. Point P is

located at -=e and the unit tangent vector is


S= e (2.4, 5)



Since P on the circle is located by specifying the argument 6, point P

on the profile can be expressed in terms of the single parameter e.

Therefore, the tangent vector on the profile is given by the deriva-

tive of z with respect to b The complex unit tangent vector is

given by


d d 1 d
C = "?) d (2.4, 6)








The complex unit normal vector 1i in an outward direction is
900 clockwise to the above complex unit tangent vector, i.e.,


-'


(2.4, 7)


The velocity of point P on the profile in the above normal
direction is simply the scalar product of -_ and Therefore
d


(2.4, 8)


In the r -plane the corresponding velocity is normal to the
circle and is given by


(2.4, 9)


Substituting equations (2.4, 7) and (2.4, 8) into equation
(2.4, 9), the velocity in the radial direction on the circle is found

to be


A d j


e (6,t)-Im d -d


(2.4, 10)


8jr(Q{ cei~, [ ~


(2.4, 11)


(2.4, 12)


A (6, -L)= 9 -, V~'( 'e


,.,,tl; d t









In a similar manner, the tangential velocity on the circle due

to this perturbation is found to be



A [ dt] (2.4, 13)


The undulatory movement of the profile is described by the flap-

ping function h(x,t). Here the x-coordinate refers to the base configu-

ration. For a physical representation of fish swimming, this function

is taken to be imaginary in the space variables, i.e., the displacement

is perpendicular to the real axis. Therefore, the displacement function

can be written as


D(.,t) =i h(x,t) x,L x x, (2.4, 14)


Further, it is assumed that the propulsion is generated by a

train of waves progressing astern with an amplitude depending on the

spatial chord variable x. From photographs made of swimming fishes

20_q it seems reasonable to assume that this amplitude has its small-
est value at the head and its maximum value at the tail. Furthermore,

the time variable enters the problem as a harmonic function. For this

type motion, the displacement function may be written in the form



D(S,t) = i H(X) cos(cx -w-t +A, (2.4, 15)



where oc is the wave number, ) is the circular frequency (which is

taken to be positive throughout this work), A, is an arbitrary phase

angle, and H(x) is the arbitrary amplitude of the wave motion. It








is convenient to write this motion in the general form


(2.4, 16)


where = is the imaginary unit for the time variable t. It is

not to be confused with the spatial imaginary unit i. Eventually, the

real part in the time imaginary unit must be taken for physical inter-

pretation.

From equation (2.1, 10) the x and y coordinates of point P on

the base profile are given by


and


x Lo B+m n-1
X 2 [C0S0+0 (1-E)?-" cos nBl







co
1 ~siO 4C irl Z)n ,,n ne]
Y = [i 2


(2.4, 17)








(2.4, 18)


Since x is an even function of any function of x is even in Q .

Hence


(2.4, 19)


(2.4, 20)


D (Z~t)= 1 H"'(x)e i '


h (xi ) h [ x (), t] h [x(- e), ] = h, ( 8,+L


H*00 = 5 iu~x(e)] = HCIY[(-e)]:t








Thus, let


(2.4, 21)


Substituting equations (2.4, 16) and (2.4, 20) into equation
(2.4, 4) yields


(2.4, 22)


Therefore, from equations (2.4, 12) and (2.4, 22) the radial
velocity on the circle in the r -plane resulting from the undulatory
motion is



,tio- .-I, o el, w d dz iZj zc(eio ) (2.4, 23)



Equation (2.4, 23) can be written in a compact form as


0 (OU) U C(O)&(o


(2.4, 24)


where


Ij ( d1A;)


(2.4, 25)


H~(O =B" n C t, 0 5 n 1


cis iiU.() + TFo H" 0)I~ceb eiwt
d -L (e;0






28

But, from equation (2.1, 10), evaluated for Z = f (ei0)


d F. d E. i
dz d'e d 0


dHI(O) dI-,(Q d
d_ d e d 0


-~ ie'0 f'(e-'8)
6 E


(2.4, 26)


d i e! fI(e O)
do


(2.4, 27)


Also, from equation (2.3, 5), evaluated for = -(eie)


86 dV do


[-u,,ei]Ii-


(2.4, 28)


Substituting these last expressions into equation (2.4, 25)


yields


G () M-IM [ee e') jk H"(8) 4 d0ko)]


(2.4, 29)


where k =- is called the reduced frequency referred to the radius

of the circle or approximately the half-chord of the base profile.
Equation (2.4, 29) can be written in a more convenient form as


() -Ii j e k.f,(e-.J H( f'(e0) de 4 ,


(2.4, 30)









For a symmetrical base profile, as is considered here, it can

be shown that the function G(8 ) is an odd function. Therefore, it is

convenient to write



Qt( 2 Y Pn sin n (2.4, 31)
n- 1


where the Fourier coefficients are

if

P, = i (O) sin r\6 dO. (2.4, 31a)
o

Note that, according to equations (2.4, 4) and (2.4, 30) the

coefficients Pn, like Bn, depend upon the mapping function and, as

such, contain the thickness parameter e The relationships between

the nP s and B 's are found from the solution of equation (2.4, 30).

Due to the complex nature of this equation, the recurrence relations

between the Pn's and B 'S will not be attempted until a specific

example is presented in Chapter III of this study. It is to be noted

also that the coefficients Pn are complex in the time imaginary unit j.

From equation (2.4, 13) .:d equation (2.4, 22), the tangential

velocity resulting from the undulatory motion can be found in a

similar manner as the above radial velocity to be


O O6.t) O U 2 (6)e (2.4, 32)


where



S,(f '(e-) )- HAG1h 1 (2.4, 33)








For profiles of small thickness ratios, it is obvious that the

above tangential velocity is much smaller than the corresponding normal

velocity and, as such, will be neglected in what follows.

Following Theodorsen l/ 16_, the boundary condition for the

normal velocity given by equation (2.4, 24) can be satisfied by a

distribution of sources along the circle in the -plane or a corres-

ponding distribution of sources along the surface of the profile. Let

the strength of this source distribution per unit arc length in the

S-plane be denoted by Q(6,-) This source distribution can be

related, in an obvious way, to the normal velocity & (60{ by


iQ(,t)= 2 c,,-t) (2.4, 34)


From the definition of a point source, it is readily seen that the

potential at a point due to a point source of strength Q located

at i is


F(1 ,t )= lo i (2.4, 35)


Therefore, from Figure 7 the contribution to the total potential

function F,(,t) due to a point source at s=e e is given by


dF,(( ,{) '- if "'s,') log(f-ei')dl- (2.4, 536)



and the total potential at < due to all the sources becomes


F,(,-)= Z.()ocj(-e' ) (2.4, 37)
0









By employing equation (2.4, 24) the source potential becomes


2ir

r, (r,1 ~I i~ w
0r


2.


Figure 7

Circle Plane with Point Source


(9) log(r-e' 2d .


(2.4, 38)


= I-,'


r, =e i


c
U ---P
--
2









ri(li) evaluated on the circle is given by


F, (e et) =


2rr

ZCL 109,t k .oe ei d9.

0


e-9
r8-y- eXPj8- z JT


(2.4, 40)


Hence,


2Cr

F,(eie ^\= 2CL O-t]Lo sin e -zP z d^(2.4, 41)
0


Equation (2.4, 41) can be written as


2 VOP2 ,
F, (e ie) CL-19i to eZ:Sin z r] d 0

0


(2.4, 42)


Thus, the velocity potential associated with the source distribution

evaluated on the circle is


2zr

(ei = n 9,th loe ism e d1 t
o


The tangential velocity on the circle is found to be


a L 0a

0


(2.4, 43)


(2.4, 44)


But


(2.4, 39)


cP,~9it) cot z d~l









Employing equation (2.4, 24), this tangential velocity becomes


2qr

0


(2.4, 45)


The integral in equation (2.4, 45) is a function of the .ngl

6 the so-called conjugate function. According to Robinson and

Laurmann /-21_7 this function is defined as


0~i'


~ (~=~1Li-'e-0d


(2.4, 46)


The integral on the right-hand side is singular at 0=0 and it is

understood that the Cauchy principal value must be taken, i.e., the

integral f is to be interpreted as
og


0


G-4

(IM?
A~ -- 0 f


c 2}r

+ f i-


(2.4, 47)


From equation (2.4, 31)

it is readily seen that


and the definition of the conjugate function


(2.4, 48)


r4(e) P, cos ne
M=J


The circulation


around the circle due to the source potential is

2ir

S= I (oat) d 9 (2.4, 49)
0









Substituting equations (2.4, 45), (2.4, 46), and (2.4, 48)
A
into equation (2.4, 49) it is observed that F vanishes. Therefore,

in the remaining discussions of this study the effects of the source

distribution will be referred to as non-circulatory effects.


2.5 Pressure Distribution of Source Potential

The unsteady pressure distribution resulting from the source

potential can be found by substituting the source potential function

into the linearized Bernoulli equation. Thus, from equations (2.4, 38)

and (2.2, 11) this pressure becomes
2'i

1- +- F(e) 2 Ue [- d i (2.5, 1)
1 .t Wf(e'O Z J qrrj
0

Performing the indicated differentiation leads to
22


1o
f F'(ei'0) Uei-3 ^ d1

2
t ,i) r? i" a -o~e; I(2.5, 2)
whr (e O: 2) .
Substituting equation (2.3, 6) into equation (2.5, 2) and observing

that the 61i operator and the integral operator are commutative in the

above equation, the source pressure distribution becomes









sin
IfT ej i9 4 e' F (2.5, 3)

%where T .
U









But, from equation (2.4, 40)


e -e = Zsm- exp 1 2
L J


(2.5, 4)


Hence,


Ri 105 1 e ''ll = toog IZ z


Siee 1
ei z e i*o-


S t n W 1
1 cosU- 6)


Cot
z


(2.5, 5)


(2.5, 6)


Substituting equations (2.5, 4) thru (2.5, 6) into equation (2.5, 3)

gives


21r

f& {0iog[2sini
0


+ -s 'e C ot


(2.5, 7)


Also,


TT, (e, t)=
2 it


l~e e 1% 1$1 zim 9 V.








Equation (2.5, 7) can be written in the convenient form



T .zt) = P^ ^ ( -c e-)) + se co- d

2'r(

+ ( --1 og 2(l- cos( -0)) C- cot i dO (2.5, 8)

where the following substitution has been made.

[Z sin z [-cos(e-19] .


Changing the variable of integration from i to -1 in the
second integral of equation (2.5, 8) yields

Aci(1) [lo rZ.(1- cos(98-)) + ,eiz cot d 9 =


= 9 (-[r log Z(i-cos(+i9)) +1 cot d? .Ot
-11
But, since -C(-9)= (I) and cot-1-f: =cot -0 this
2 Z
last expression becomes
2

(W) [


=-J (I)[alo2(a cos(O+*)) -I_ I co- d9 .
-2 f









The integrand here is of period 2n. Thus, the limits can be

changed from -2n to 0 and -i to n. Using this fact, equation (2.5, 8)

becomes



P uZeitP e) 1 -cos(e-19)
0

2s ,no Sin
\t'(e 2 co e-cosiJ d (2.5, 9)



2.6 Circulatory Potential Function

For arbitrary undulations of the profile the tangential

velocity induced by the sources will not, in general, vanish at the

sharp trailing edge and, as a consequence, the Kutta hypothesis is

violated. As in Theodorsen's theory /-16_7, a continuous vortex sheet

is introduced along the wake streamline of the base uniform flow with

a sheet of counter-vortices distributed along the chord of the base

profile. These vortices and counter-vortices are introduced in such a

way that Kelvin's theorem / 22_ of total circulation is satisfied and

that the net induced velocity on the boundary of the base circle due to

these vortices is tangent to the circle. In addition, Helmholtz's

law / 22_/ of persistence of vortex strength following a fluid particle

is applied to a vortex element moving with a velocity equal to the

local steady base flow velocity of a particle of fluid along the

streamline emanating from the downstream stagnation point. Finally,

Kutta's condition is satisfied by requiring that the tangential velocity

at the downstream stagnation point induced by this vortex distribution








be just sufficient to cancel the velocity at this point on the circle

due to the source distribution as given in equation (2.4, 45).

According to Theodorsen / 16_/, the first two requirements are

fulfilled by a distribution of point vortex pairs of equal strength,
A
one of strength Lr= Yd~0 located at s= o on the trailing

edge streamline, and the other of strength -Ar --YWdto located

at the image point r Notice that the angular rotations at

these points are assumed to be positive in a clockwise sense.

Let the total potential resulting from this vortex distribu-

tion be denoted by F2(r ,t). Then the contribution to the total

potential due to the vortex pair located as shown in Figure 8 is


AF,= r- toc (2.6, 1)

where AP is the circulation strength in the T-plane.











&rr
S/













Figure 8

Circle Plane with Vortex Pair









The total velocity potential due to the vortex distribution

therefore becomes



F,(. t) f U(,t) 1og o (2.6, 2)
1

where it is assumed that the unsteady motion has been going on for an

infinite time.

According to Wagner [-23_7, the circulation distribution in

the z-plane is related to the circulation distribution in the C-plane

by


u(0,,t)d(, = U(x0,t)+dx, (2.6, 2a)


and from the mapping function, equation (2.1, 10), it follows that

dxo = '( d .

Employing these results the circulatory velocity potential function

can be written in an alternate form as
0-

F2( ),t) ( l f '( o d o (2.6, 2b)


where S(Q ,t) is the vortex distribution in the physical plane.

In accordance with Helmholtz's law 22_/ of constancy of vortex

strength following a fluid particle, the equation governing the vortex

distribution in the physical plane is


Y(xo,) = const. (2.6, 3)


following a fluid particle.








Therefore, the substantial derivative is


dt Y (2.6, 4)
dt at ax. dz


However, since the unsteady perturbations are harmonic in time the

vortex distribution can be written as

Y(x,)e= '(x) ei (2.6, 5)


Combining equations (2.6, 4) and (2.6, 5) yields


a X(.) + dx( ((x 0 (2.6, 6)


where the substitution


dF. dF. d
F-- (2.6, 7)


has been made.

Separating the variables in equation (2.6, 6) and changing

the variable of integration to o it follows that



fd,- (- db (2.6, 8)
f^ ~ t--


Integration of this expression gives



S) Yoexp[-i L d(2.6, 9)
1








where Yo is the strength of the vortex element located at X.(to)
the physical plane. Obviously, Z is the vortex strength when this
particular vortex element was shed from the tail. From equations
(2.1, 10) and (2.3, 5),


"( V((t. 1


where k= 4
U


is the reduced frequency as defined above and


f [o
d to


(2.6, 11)


Combining equations (2.6, 10) and (2.6, 5) with equation
(2.6, 2b), the circulatory potential function becomes


00
itp e.jktvj'(t'j 105 dt.


(2.6, 12)


Finally, in order to satisfy the Kutta hypothesis the
tangential velocity on the circle must vanish at the downstream
stagnation point which is located at r=1 This restriction implies
that


(2.6, 13)


But, from equation (2.4, 38)


- ^i-J < r


(2.6, 14)


(2.6, 10)


'r [ f, T, ) *fi, T l] =


FFI I
ar =









A
Substituting the trigonometric series for G(19 given by

equation (2.4, 31) into equation (2.6, 14) and separating the real

and imaginary parts in the space imaginary unit i, this equation

becomes


U ____ne i d V
ar 2 5 nzI I o


(2.6, 15)


Employing the integrals

2T

Z /sn nsi di = O n =1,2,3S,**-)
1


and


21
sinn3 =

0


the velocity at the rear stagnation point in the r-plane due to the

non-circulatory potential is


(2.6, 16)


Ca
- i ei"' P,
fl1


Now the velocity induced at the downstream stagnation point by

the circulatory potential is investigated. From equation (2.6, 12), it

follows that


CO
TF(o' d'40

~`c .


(2.6, 17)









Combining equations (2.6, 16) and (2.6, 17) with equation

(2.6, 13) yields


Vrr J - -
je4"tife 'W" 0
IQa -ij
2n z


(2.6, 18)


where


Co
Q =-UeLjtE P,
nTt



Therefore, from these results the constant 1o is found to be


a
P(ijk;e)


(2.6, 19)


co

IP (i k ) 2 %-


(2.6, 20)


Finally, substituting the results given in equations (2.6, 19)

and (2.6, 20) into equation (2.6, 12), it follows that the circulatory

potential function becomes


Fz i k f'Q 1 10, d t.
P(l k;0) & e
I.


(2.6, 21)


where









Recall that Q can be determined from the boundary conditions,

i.e., it depends upon the flapping function, base flow velocity, and

the mapping function; and that F(jk; ) depends upon the mapping

function and the reduced frequency. These quantities are assumed to

be known. Thus, the total complex velocity potential function


F(T,t) = F(V) +F,(jt) +F,(T,tl (2.6, 22)

is known for any prescribed flapping function and thickness parameter.


2.7 Circulatory Pressure Distribution

To determine the pressure distribution on the mean base

profile surface due to the vortices in the wake, it is convenient to

compute the pressure ATTr due to a single vortex pair of equal strength

located at and 1 in the '-plane or the corresponding points in

the z-plane. From equation (2.6, 1) the complex velocity potential of

this pair is



AF' ,t)= og 0I-j (2.7, 1)



According to Helmholtz's law /22_ of persistence of

vorticity following a fluid particle, the strength A' is constant

referred to a coordinate system moving with the fluid particle. Here

it is assumed that the velocity of a fluid particle in the wake is

equal to the steady local velocity of the base flow. This streamline

coincides with the positive x-axis in the profile plane. Therefore,


-AF = A o d- (2.7, 2)
dtI a at,








where x0 is the coordinate of the vortex element in the profile

plane.

From equation (2.2, 11) the unsteady pressure due to this
vortex pair is


A1T2l, \ &C pf dtoL6 F (eW O) 1
Ld. d-- .' -(en I I (2.7, 3)


where

_o ,(F '(2.7, 4)


is the local velocity of the vortex element AV along the wake

streamline in the profile plane.

Substituting equation (2.7, 1) into equation (2.7, 3), the

unsteady pressure distribution resulting from this vortex pair is


A (pr { f 10



+ a i log = (2.7, 5)


Recall, from equation (2.1, 10) {'(e;))\z and If'( ,~)

are real in the space imaginary variables. Also, from equation

(2.3, 5)



which is real, and


F'(e"') = -Usst'i iO.'0








Employing these last results and carrying out the indicated
differentiation, equation (2.7, 5) can be written as


6T -r(, (o) 2sne u s, "( (2.7, 6)
+t Z f(\ ^l-2cos 6 Cf'(e )1(t.+I-2,cosb)')


To determine the effect of the entire wake vortex sheet, the
vortex element A' is replaced by


Ox' -- ,d^4, (2.7, 7)

where Y, denotes the circulation distribution per unit length in the
S-plane. Substituting equation (2.7, 7) into equation (2.7, 6) and
integrating over the entire wake, the time dependent circulatory
pressure distribution becomes



jt) f + 21n(2.7, 8)
SV\\ 1z-2Qcos '(eo)\2l(+1 -z.cose o (2.7, 8)
1

Combining equations (2.6, 2a) and (2.6, 9) with equation

(2.7, 8), it follows that


PTZ OM Joe iJ Fj([ Zsin e
1T2 (^'( 2n + l-2 -,co0 G
1 (2.7, 9)

+ Ussine( -1) 1 i klV d)
fV (eiGlk( +I -z2 cose)









Eliminating the constant Yo by way of equation (2.6, 19), the

unsteady circulatory pressure distribution becomes

T tz(e,t~ -- -p T(e, ik;i) (2.7, 10)


where
F-
fFo. ^sn9 UsZsei-1 E) k-ik
jki l_, 43 -2t.cose \f(e' o)\ (,+1-2 zcosei d
T(rik; = 0 O (2.7, 1i)


f- I
1
With the previous results the total unsteady pressure distribu-

tion on the surface of the base profile is obtained by adding the

effects of the non-circulatory flow to that of the circulatory flow.

Hence,

TT(6,t) TT(el) +.(0,~+) (2.7, 12)


Combining equations (2.5, 9), (2.7, 10), (2.7, 11), and

(2.7, 12) it follows that


Ir
t U2ewj [ 1[-cos (8-t )




\(e coseino 1S p QT(,iK;) (2.7, 13)
-y (eio)\ cose -cosO








Substituting the trigonometric series for (1A9) given by

equation (2.4, 31) and employing the so-called Glauert integral [-21_7



1 stn19 Slhn d9 = cos ne
coI Co Co.s5


and



j j i- cos(i-e~) + n
0


into equation (2.7, 13), there results




Tp U~-p e [ z g,. P s- 2 o, Si c, P os +


-pQT(6 k;0 (2.7, 14)

Finally, inserting the value of Q from equation (2.6, 18) into

the above equation the time dependent pressure distribution becomes




TT(,t)= 2z pU et P rP in e ,z cos he +

r T()..k) 1


+ (2.7, 15)









For the special case of a flat plate of infinitesimal

thickness, the parameter C vanishes and equation (2.7, 15) reduces

to (see Appendix A)
ir
__- l-cos(0+1 san_
TT(e,t) 10- lo-C. o) + +
ot C-cos(e- ) on@(coS -cose)



+ C.ot [cot8 -os (] ( )d (2.7, 16)


With some manipulations this can be expressed in identical form

with that given by Kissner and Schwarz [l107, Schwarz 9_7, and

Siekmann [8J. The function .(K is the so-called Theodorsen

function [16J defined by
S3


rd(k) = =

1

J~(k) + .(k) (2.7, 17)


where (k) and HZ)() are Hankel functions of the second kind of

order zero and one, respectively.


2.8 Lift and Moment

Equation (2.7, 15) expresses the unsteady pressure distribution

at points along the mean or stretched-straight configuration of the

profile. Let the complex coordinate of a point on the mean configura-

tion be denoted by


Zo fI( e) .2


(2.8, 1)














(C)


-T d


Figure 9

Stretched-Straight Configuration
with Pressure Distribution


Referring to Figure 9 it is seen that the forces acting on a

small arc element ds are


dFx = -TTdo d dF = TdXo (2.8, 2)


These are the forces exerted by the fluid on the.profile.

The forces F, and F, in equation (2.8, 2) are positive along

the positive x and y coordinate axes, respectively. Equations (2.8, 2)

can be combined to give

d Fx -iFy) = -i TTdio (2.8, 3)


The forces exerted on the profile can be obtained by integra-

ting equation (2.8, 3) along the contour (C) of the base profile.

Hence, the lift becomes


L Fy TTdi (2.8, 4)
(C)









But


d= 'de= de (2.8, 5)

and
zvr
L ti- d ld (2.8, 6)
o

Referring again to Figure 9 it is seen that the moment about

the origin of the z-plane due to the elemental forces acting on the

element d5 is


dM = TT( xdx Vjd) (2.8, 7)


taken in a counterclockwise sense (nose down).

Equation (2.8, 7) can be written as


dM = i TT aodi~ (2.8, 8)



The total moment is found to be

Zqf
M= iTT f(e) 6-d& (2.8, 9)
0

The above equations for the lift and moment are essentially

those developed in the Blasius theorem / l/. It is to be noted

that these equations neglect the change in shape of the profile since

it is assumed that the pressure acts in a direction normal to the

surface of the base profile.









2.9 Thrust Formulation

In order to calculate the x-component of the resultant hydro-

dynamic force acting on the plate, the change in shape of the profile

must be taken into consideration. An analogous consideration was

made by Siekmann / 8_/, Wu / 11_7, Smith and Stone / 132_, and

recently by Pao and Siekmann / 14_/ for the infinitesimally thin plate.

The important fact here is that nonlinear terms are involved and, as

such, there are mixed terms involving the time imaginary unit j.

Therefore, the real part of the time imaginary unit must be taken for

physical interpretation.

The hydrodynamic forces can be computed here from equation

(2.8, 3) if the differential element dEo in that equation is replaced

by the exact differential element d- where


di = d ie d H (e) (2.9, 1)


as given by equations (2.4, 16) and (2.4, 20).

It is to be noted that the coefficients Bn in equation

(2.4, 21) are generally complex in the time imaginary unit j. Thus,

from equations (2.8, 3) and (2.9, 1) the x-component of the hydro-

dynamic force becomes

2VT
F,= e;- TTt [ i, e do de) (2.9, 2)
Sdo de



where j is the 'real part of' operator for the time imaginary

unit j. A positive Fx will indicate a net drag.















CHAPTER III


APPLICATION OF THE THEORY TO A SYMMETRIC JOUKOWSKI BASE PROFILE

WITH A LINEARIZED THICIKESS PARAMETER


In Chapter II the general theory for the unsteady motion of a

flexible body of finite thickness immersed in an incompressible, ideal

fluid was developed. The problem was linearized by assuming very

small displacements and displacement rates of the surface of the base

profile. The steady-state boundary condition on the surface of the

base profile was satisfied by developing the base flow complex velocity

potential F0. The unsteady boundary condition was satisfied by a

source distribution along the surface of the base profile and associa-

ted complex velocity potential F1 was derived. The Kutta hypothesis

for smooth attached flow at the sharp trailing edge was satisfied by

a distribution of vortices in the wake and the complex velocity

potential F2 of this vortex distribution was developed.

The pressure distribution on the stretched-straight configura-

tion was found by linearizing the unsteady Bernoulli equation ~1_7

by assuming a small unsteady perturbation theory. As can be seen

from equation (2.7, 15), the unsteady pressure distribution is

harmonic in the time variable and it depends upon the function G( )

which, according to equation (2.4, 30) is related to the amplitude

function H1(G ) of the flapping. Also, it was observed that the









unsteady pressure distribution contains certain functions related to

the mapping function z = f( ).

In order to estimate the effect of thickness on the thrust,

lift, and moment, it is convenient to linearize the mapping function in

the thickness parameter 6. From equation (2.1, 10) the e-linearized

mapping function becomes


.- -l .? (3.0, 1)


Figure 10 shows the configuration of the base profile for

several thickness parameters. The numerical values for the coordi-

nates are determined from equations (2.4, 17) and (2.4, 18). These

values are tabulated in Appendix B.


3.1 Calculation of the "Downwash" Velocity for
Any Given Flapping Function

The "downwash" velocity on the boundary of the circle is given

by equation (2.4, 24). The corresponding "downwash" velocity on the

base profile as given by equation (2.4, 9) is


^ (e,3t)
=n '(eo( (3.1, 1)



Recall, from equations (2.4, 24) and (2.4, 30)


o[(e, =U (e)(e (3.1, 2)


where


G(e) = -Im











JrJ ____


S= 0.05


6= 0.10







Y


C= 0.15


6= 0.20


Figure 10

Stretched-Straight Configuration for
Several Thickness Parameters


-r


~--~-


~1







Taking the derivative of equation (3.0, 1) gives


'(e I) = [l- e-21 + (e-2iO-e-'sie]
f'z I I -


(3.1, 4)


Separating the real and imaginary parts and changing i to -i,
equation (3.1, 4) becomes


te [c, o+


(3.1, 5)


where


e = 1- cos20 +26(c.os28 cos30)
J. = -s2O +2 6(sin20 stn36)


Also, from equation (2.1, 10)


[ f1(ee')]-


1 J1-~2 -12
- t- L e'0-e I3


(3.1, 6)


This can be written as


[' (e'T)]I


I e4 -eZL e eA61 + Z


(3.1, 7)


Neglecting the 62 term and simplifying, equation (3.1, 7) becomes


'(e eiOlZee .ic- (3.1, 8)
eiO le. e-iG +?.c- e-:19








After some algebraic and trigonometric manipulations,

equation (3.1, 8) can be written in the form


[f (el]e-1


= ei, ~ i
Z.(1 -ae) sii2


(3.1, 9)


where


r' = 1 -2 Ecose -(l-Zl cos26

4 = 26 sin e -(0-z&) sinze


Substituting equations (3.1, 5) and (3.1, 9) into equation

(3.1, 3) yields


sin e(e, +iAj d148j}
2 [1 2 sinnZ 8


(3.1, 10,


Taking the 'imaginary part of' in the space imaginary unit i

of equation (3.1, 10) leads to




^)- (1cos, -tstne dH (()]
() (a cos - 2ese)s H(ne d1 ,*-(3.1, 11)
2 2(1-ejsine d








Combining the values of ., C '1, and I given in

equations (3.1, 5) and (3.1, 9) with equation (3.1, 11) gives


()= -f (-sinze +Zesm.e -Zesin38)cose +


cos20 +Ze2os 2z --2ecos3e) sin ] Hr(O) (3.1, 12)

S(Ze sn -(-Ze)n )cos -sin8(1 -Zecose -UI-ZaeosZO)sine dH(
21 (-Ze)sirn 7


This last expression can be simplified to read


(e) =cr (1-e)s)nO+EsmnZe] H:(8)+[ +26- cose] d-_e


Recall that from equations (2.4, 21) and (2.4, 31)


00
VA7(1 = BO + ?. B ir tcie
n=1


(3.1, 13)


(3.1, 14)


and


a(e) =CPs~n
n-1.


(3.1, 15)


where


I1
Pn = I ) rnned
If f&)SInnedG


(3.1, 16)








Substituting equations (3.1, 14) and (3.1, 13) into equation

(3.1, 16) and performing the indicated integration leads to the recur-

rence relation between the P 's and B 's as
S n n


P, eqB- ^ ^ ( ^ -n


+ '-"' hB, +1n+1]5 ,,j)
1 -26 18 1-265


(3.1, 17)


In computing the forces acting on the profile due to the

unsteady pressure distribution it is convenient to use, instead of

G( ), a new function defined by


AlE sin no--
ee e e e e A, te A ccosng


where the Fourier coefficients are related according to


A n- An+. P=Z


(3.1, 18)


(3.1, 19)


Combining equations (3.1, 13) and (3.1, 18), it follows that


(-Zc I -2f I sine de


(3.1, 20)


It is expedient to write the first term in the last expression

as a trigonometric series in the form


(1-e tZecose)H1(O) = bo+Z bcosne (3.1, 21)
nzl


(nz1.ZA...-) .









where the Fourier coefficients are found to be


b, =(l-085o +2B 3


(3.1, 22)


b, z (1-05 +E(B,,+, + Br%-


n i.


Also, the second term in equation (3.1, 20) can be written in

the form


I d i*(O)
sinO dO


'07 in nesrne
r-L


C0 + ?EZ C rt.c'S


(3.1, 23)


where the recurrence relation for the Fourier coefficients is


Cn-i -Crn+i = ?-tASr


n i .


(3.1, 24)


Finally, the third term in equation (3.1, 20) written in a

Fourier cosine series becomes


+- cos E C. + aZ Cccos tie = AO + ?-T4 cosn e
1-2n o-


(3.1, 25)


where the coefficients are



1+e 1- 26
wo 'a n-z- -- 0 + l fe .1


Z. -- 1+6


(3.1, 26)


n l 7 .









Combining equations (3.1, 18), (3.1, 21), (3.1, 23), and

(3.1, 25) with equation (3.1, 20), it is seen that

An -b + ,c, (3.1, 27)


These results give

A -o-[Ci- ~e 18. Z Co + t

(3.1, 28)
An =~ (1-e)B, +e(B 6n--) n (C+., +C- n>l



It should be noted that the trigonometric series in which the

coefficients are the An's is related to the downwash velocity on the

surface of the base profile in the physical plane, whereas the series

containing the Bn's and the Cn's are associated with the flapping and

the distortion of the plate, respectively. Hence, equation (3.1, 28)

gives the downwash in terms of the displacement and displacement rate

of the fish.


3.2 Calculation of the Pressure Distribution
on the Mean Base Profile

The unsteady pressure distribution on the surface of the

stretched-straight configuration of the fish is given by equation

(2.7, 15) as

IT(8,t) = pUze n e (e 'iz cos ne +
nti


ST(jk) (3.2, 1)
U "f








In order to completely determine this pressure distribution

it is first necessary to approximate the difficult wake effect as

contained in the function T(0jk;e) Consider the velocity of a

fluid particle along the streamline emanating from the sharp trailing

edge as computed from the base flow potential FO. This velocity is

given as


u 1 -
'(.0 2= (3.2, 2)
1


At.the trailing edge


1-
ul m l U I.I2M U(-e) (3.2, 3)




Equation (3.2, 3) reveals that the velocity of a fluid particle

shed from the tail is reduced by a factor of (1 ) from the free

stream velocity. Of course, the velocity of a particle in the wake

approaches the free stream velocity at a large distance downstream

from the trailing edge. To take this slowing-up effect into account

in the wake function T-(,jki;) it is assumed that the pressure at

a point on the mean base profile resulting from a vortex element in

the wake is approximately that induced at a point located an infini-

tesimal distance above and below the x-axis. According to this

assumption the fish can be represented by its mean chord line immersed

in a uniform flow field with a velocity 6U where 6 represents the

slowing-up effect due to the thickness. Obviously, the parameter 6









,depends upon the thickness parameter E For the special case of

6 .= 0, 6.= 1. The relationship between & and 6 is determined

by requiring that the time-dependent pressure vanish at the tail.

From equation (3.0, 1), it follows that the actual chord

length of the stretched-straight fish is

x, -x Z(l-6) (3.2, 4)


This chord line can be mapped onto the same unit circle by the

transformation


E f()= (+ r +3) (3.2, 5)


The values of A and j can be determined as follows:


XT= 1- =A(2)
(3.2, 6)


or


= - B -2 (3.2, 7)


Employing these results, the base flow potential in this flow

field becomes


Fo M= ( r (3.2, 8)


and


F. (1- ) (3.2, 9)








Employing the transformation (3.2, 5) it follows that


(3.2, 10)


'(eiO)z = 4Azs inZe


(3.2, 11)


Substituting equations (3.2, 10) and (3.2, 9) into equation
(2.6, 11) yields


az
1 Z' [6Lgo.0+
I 2 Lz


(3.2, 12)


It is expedient at this point to introduce a new variable

defined by


o = e (3.2, 13)

Hence, expressing equations (3.2, 9), (3.2, 10), and (3.2, 12)

in terms of the new variableX it follows that


FO U4=


* (X) nh


(3.2, .14)


(3.2, 15)


4?1


(3.2, 16)


and


-1_)










Combining these results with the expression for T(CjKi;) as

given by equation (2.7, 11), and changing the variable of integration

gives


(3.2, 17)


Uhh' If+ -2e~cs e
0


st rnh 7-
sine


ez- +I -z&eZcose&


e-i.k cosh'X dX


where A =


, This expression, after some algebraic manipulations,


can be simplified to read


00



UP
A -sinO f*


0


Smn2b9 + s Inhz X e-ik Cos hX d'
cosh)L cose


-%+1 sirnh) e-jIKA COSO dj
ex -1


Employing the identities


srnhX + sinze = cosh2W -cos2O


and


+ +I
e lnh costhX~ + 1


e" +1


0
OO



o


elAkcosh5d%


TW8,jk;6 -


(3.2, 18)









equation (3.2, 18) reduces to


00

U f

T k) 0o
0


Cos hX+ Cose G) ,e-t1A Cos dx


lcoshX +l1 eihAk cashX d4V


It is well known in the theory of Bessel functions /-24_7 that


(3.2, 20)


CO
f eiz coh'x cos h nW dY-
0


where K,(1l is the nth order Bessel function of the second kind which

is valid if IQrgel0 Let


Z = njAk -- ,jl) 9(3.2, 21)


Thus, if &(r 70 it

(3.2, 19) and (3.2, 20)


U
T(e,ik) = _
AsinO


follows that <70 Combining equations

gives the result


K1(jAk1 cos e Kj,(iA1
Kj(jAk) + Ko(jAk)


(3.2, 22)


(3.2, 23)


T(O' i k; Q = U (I C jnSO tj(jAk + COSE
I((Ak) +KjAk)


(3.2, 19)








The ratio containing the circular functions in equation

(3.2, 23) is the Theodorsen function /s16_7 J(Ak) Employing the

relation Z-24/7



K, = |(j Arr "-' ,1(Ak1 (3.2, 24)


where H(AkM are Hankel functions of the second kind and order n,

it follows that


(Ak) 4 '. (A Q F (kA Ak i- (A k (3.2, 25)


It is true that in the development of equation (3.2, 25) it was

necessary to require that A > 0 However, according to Luke and

Dengler [25 equation (3.2, 25) has no need for such a restriction

and therefore, by the method of analytic continuation one can argue

that equation (3.2, 25) is valid for all /A In the present case

/= 0 Substituting equation (3.2, 25) into equation (3.2, 23)

gives


T(BiOk; = _K (AV) + ( X(ACkt) cos 0 (3.2, 26)
U A sine L *


This determines the function T(0, iK() except for the

parameter Al which, as mentioned above, must be found by requiring

that the unsteady pressure vanish at the tail.









Replacing in equation (3.2, 1) the wake function as given in

equation (3.2, 26), the time-dependent pressure distribution becomes


TT(8,t U = EpU e Pn Irsnne sne




+ uaAk) *(I -lA os o


From equation (2.1, 10) it follows that


cosn8 -I


(3.2, 27)


(3.2, 28)


With some algebraic and trigonometric manipulations this expression can

be written in the form


1 1+46+-* -4e4 (I *rz)cosE) +Zc-lcosZo
Ifl-) l Zlj- e) + 4 0 s ne


(3.2, 29)


Neglecting terms involving orders of .. of two and higher, equation

(3.2, 29) reduces to


1I 4e cos E
~U(eieZl Sjn2=


(3.2, 30)


Combining equations (3.2, 27) and (3.2, 30), there results


TT(,t)= 2p:U'ei'V P < snne (I-4fcose) c osnne
t~t n (1- Es) sne

+ [K (Ak) +(I (Akl)cos] (3.2, 31)









It can be seen from this expression that the unsteady pressure

possesses a singularity at the tail where e = 0 and at the nose where

9 = 1T. The singularity at the tail is removed by satisfying the

condition


IT-( -e) smne 0 I = 0 (3.2, 32)


This can be satisfied for all time only if


i---- (3.2, 33)
i-Ze A

which defines A and 6 in terms of the thickness parameter e6 The

singularity at 0 = r produces a concentrated force at the nose, the

so-called suction force arrived at in airfoil theory /"21_7, which

must be added to the hydrodynamic force computed by integrating the

pressure distribution around the profile.

For the subsequent calculations it is convenient to express

the unsteady pressure distribution in a Glauert trigonometric series

as

do
TT(O,t) =pUe- el [tanIk +aE oa,.sinne] (3.2, 34)

where the coefficients are given by (see Appendix A)


o = -- (Ao*-Al(1-46)(Ak) -A, +4e Ao

(3.2, 35)

S- +A An- An^ n- A.) n 1
I-Zo n -e n









Dividing out terms in the denominator containing 6 and

linearizing the results in 6 these coefficients become


o = r, 2(r. + A -ZAo
(3.2, 35a)

on An.-,-A Z -2( AnAn-Aj.(-An..* niI


where

.o = tA, + A.) (Ak) -A,



3.3 Calculation of the Lift and Moment

With the pressure distribution known, the lift can be computed

from equation (2.8, 6) as

2'r
L -= .i f TT (et)d (3.3, 1)
0

The differential complex vector df must be found from the mapping

function, Therefore, from equation (3.1, 1)


d4 [-(L.-)sln 6 -esnZe -it(cosO-cosZ)] d8 (3.3, 2)


Since TT(Ot) is real in the space imaginary unit i, the operator e4;

and the integral operator in equation (3.3, 1) are commutative. Thus,

combining equations (3.3, 2) and (3.2, 34) with equation (3.3, 1), it

follows that

2-T
L=-p aei"' t Z C, slnslnnO[(l-e)sln e ir snZ8 d. (3.3, 3)
0








Integration of this equation leads to the result that


L Z-ep U'jz e [ (-ze) ., +(1-)C1 e Cz, (3.3, 4)


Combining equations (3.2, 35) and (3.3, 4) gives the lift in terms of

the downwash velocity coefficients as


L= -2pU2 (A 4 A00 U-44c(LAk)A Aj +4e Ao +

LA Ao A, + AzA -
Z 1-6 )

6 A AJ + A Z A ] (3.3, 5)
i-Ze 1-2 .

For the case of a flat plate of zero thickness ( 6 = 0), the

lift .becomes

L= -ZtPpU ej I *(A (Ak) + c Ao--] (3.3, 6)


where A = 1. This is in agreement with the results given by

Siekmann 8/J. In making the comparison between this work and that

of Siekmann it must be observed that


A,= (-I.) A, (3.3, 7)

where the 7. correspond to Siekmann's coefficients for the downwash
velocity. The difference in signs arises from the fact that Siekmann

employed a pressure differential across the plate of Ap= p--p+ to

compute the lift, whereas in this work the analogous pressure differen-

tial is Ap=p '-p .






72

The moment of the forces acting on the profile with respect

to the origin is given by equation (2.8, 9) as
21w
M = f6l. TT (,t) f(teied (3.3, 8)
o
0

where positive moment is counterclockwise (nose down).

Combining equations (3.0, 1) and (3.3, 2) and linearizing the

results in 6, it is readily seen that

i I{(e'e)] =[_ 1-2e
fl [f(e )d Z s%"Z6tecosesinZa -|-Smiecos0Z]d0.(3.3, 9)


Substituting equations (3.2, 34) and (3.3, 9) into equation

(3.3, 8), the moment becomes
21T

T 0
M, pUrei. (o,t+an| n ^ ,Han ie)( %-r S00 +


Gcos 8 sin2z -s nrcos2.e)de


Integration of this expression leads to the result that

M -n'pUZei* [ (1-6)a. (i-Ze) %, +e (a. +s) .


In terms of the downwash velocity, the moment becomes

M = pU e *- (1- )( -Zer+2 A, -.Aol +


-(1-2e)[ AZ +c- A14 +2 (AZ- A-As) +


(3.3, 10)




(3.3, 11)







(3.3, 12)


e[A,+A, +0(AA- A A2 A4. +2*2(Aj-l* A.-zN-Lt -A41]i









For the special case of a flat plate, the moment about the center of

the plate reduces to



M P op UiW* [(Ao + A) (lCk -A-A Az - A,] (3.3 13)



This result is in agreement with that obtained by Siekmann /8_7.

3.4 Calculation of the Thrust and Drag

As mentioned in Chapter II of this study the most interesting

part of this problem is the thrust experienced by the fish due to the

flapping. The net thrust or drag is given by the total hydrodynamic

force acting on the plate in the x-direction. The x-component of the

hydrodynamic force imposed on the plate by the pressure distribution

is given by equation (2.9, 2) as

2if

Fx = JdO
0

Due to the singularity in the pressure distribution at the

nose, the force as given in the above equation must be supplemented

by the so-called suction force which is concentrated at the nose.

It is convenient for calculation purposes to decompose the

thrust into three parts as follows:


F F +F F3 (3.4, 2)


where



R = Ri-1 aiT'(e) d9o (3.4, 3)
OdO











TrdH~eY

0


(3.4, 4)


and 3 is the suction force. The force x above represents the

streamwise force computed by integrating the pressure distribution

.along a path defined by the stretched-straight configuration of the

fish, whereas the force F 2 represents the streamwise force computed

by considering the distortion and distortion rate of the fish.

Before proceeding with these calculations, it is necessary to

define the following quantities:


Y, + i e Irn


A n A n +j r

- I -i ~

13 6 4~ itI


- b- jb,


= Aneiw~t =(Aln 4j A~ eiwt






r,, e 4Wt =(C r. C) e4""
A,eiwi = (9, + kr1Ll) ejwtf
B e~= (5', ~e;'Ot


~e jrae


(3.4, 5a)


(3.4, 5b)


(3.4, 5c)


(3.4, 5d)


(3.4, 5e)


(3.4, 5f)


(3.4, 5g)


CL" 4 j a-' ) eiw









Employing equations (3.4, 5a) and (3.2, 34), it follows that

o0
4i Ce(O,t)-- pUz[d'aanl + 2. ', sfnnl] (3.4, 6)


Also, combining equations (3.4, 5d) and (2.4, 21) it can be

shown that



Ri de) = -2E Bn an ne (3.4, 7)


This last expression can be written in a more convenient form as

O --
O F3' nsinne 00
s.e sie = (C, + e C,,cosnef)sin (3.4, 8)

-t g
where the relationships between the and C, are given in

equation (3.1, 24).

Now, substituting equations (3.3, 3) and (3.4, 6) into

equation (3.4, 3) gives

2i-

r
+x1 as-P (nS)(cos0-cosae) (3.4, 9)



From the integral

f sinn0 cosmne =

the second term in equation (3.4, 9) vanishes. To evaluate the contri-

bution to F due to the term


-epU- '.tanl- (cos6-cosZe )de
o









the Cauchy principle value of the integral must be taken, i.e.,


f1 = m -.p Ua:[ f+cnt (cose -cos20) +Cfan(cose-cos2e) dO (3.4, 10)



Employing the identity

+qn (cosO-cosZe)= -Ztc rv +3simO-sin2&


this expression becomes

ws-a 2fr

4-* 0 z. O fe 0


Integration of this equation leads to the result that


FX z tEj.-. Coit lo 1
4--wo I COS(O) C05Q


(3.4, 10b)


Therefore, the hydrodynamic force Fx vanishes.

Next, combining equations (3.4, 4), (3.4, 6), (3.4, 7), and

(3.4, 8) the most interesting part of the hydrodynamic force is


2zi

0
F -rIPU (aitan +2E C-' stnmBK-ZE hB'snne)d .


Employing the integrals


jf qn| sinn6d = (-1)"' zn h
o


zrr

I stnmesirnr6dO
0


(3.4, 11)


i' r m n


m n









the force F2 becomes


F -ZYpu1 U (-1)nel(-Zn6n) Z ian B (3.4, 12)
ft-_i hl

But, according to equation (3,1, 24)


-2 -l rnB, = C - (3.4, 13)


Inserting this expression into equation (3.4, 12) yields


F2 U na (3.4, 14)


To the above force must be added the concentrated force at the

nose. Since the leading edge suction force arises from the singular

pressure at the leading edge, it is necessary for its determination

to take into account the non-linear terms in the expression for the

pressure distribution in the neighborhood of the leading edge. This

can be accomplished most readily by considering the behavior of the

velocity as the leading edge is approached and then employing the

Blasius formula /lj/ to a small circle of radius 6o surrounding

the nose. The velocity at the leading edge can be computed from the

complex velocity potential by equation (2.1, 13) as


m,, t) =. \. 4 d (3.4, 15)

where


is the complex velocity in the r-plane. Since ^ is bounded as









-the leading edge is approached, equation (3.4, 15) can be written

as


hrm w(rf,*) lZ(- dr
r -3 -1 de -


(3.4, 15a;


But, from equation (2.1, 10) it follows that


r-t~~(E) -;- (j~~~?)L7


(3.4, 16)


to z yields
to z yields


. Differentiating this expression with


e
dr z-z


(3.4, 17)


11M -L10
I'-.- d= __ __ _


(3.4, 17a)


From these results it is seen that the velocity in the physical

plane asymptotically approaches an infinite value as


lm ,(i:t) r'(-1,{) w^m rm r i
Z'-V 0 ZI-00 C


(3.4, 18)


where


(3.4, 18a)


' --P +1 -E


and as

respect


and









According to Milne-Thomson /1iJ the extension to Blasius's

formula for the case of unsteady flow can be written as


(3.4, 19)


F3-iF. 3-i wt2dt -di
F I.1q I


Here the contour Ae is taken to be a small circle around the

leading edge with a radius 6, Since the velocity potential j and

Sis bounded at the leading edge, the last contour integral in

equation (3.4, 19) vanishes for the contour under consideration. Thus,

combining equations (3.4, 18) and (3.4, 19) gives


F% -i i p w ij (-t). d- I (3.4, 20)


A small circle around the nose can be written in complex

notation as


zl. S esw


(3.4, 21)


and, from equation (3.4, 18a)


(3.4, 21a)


Substituting these last results into equation. (3.4, 20) and

performing the indicated integration yields
21T
I (-1+etf F oe d.

C


s- -PI-C-Ge) W(-1,t)1 (


d;E = da' = S. ; e'v d-Y


(3.4, 22)









Thus, F vanishes and F: clearly represents a thrust,

i.e., a force directed along the negative x-axis.

To complete the calculation of the suction force, the complex

velocity w(-I,tl in the. -plane must be found from the complex

velocity potential F=o + F1, F This can be done by consider-

ing separately the contribution to this velocity by each potential

function and adding the results. Thus, from equation (2.3, 6)


Wo (-lt) = (-ot) =0 (3.4, 23)



From equation (2.4, 38) it follows that

2W

SUeJ 0) a d 9 (3.4, 24)



Substituting the trigonometric series for C(i) given .by equation

(2.4, 31) into equation (3.4, 24) and separating the real and imaginary

parts in the space imaginary unit i yields



-- P,'snl-ne I" ij d (3.4,.25)
0



Employing the integrals
2ir
Ssinn? dv9 = 0 n l,
and
alt
i Si sn dn39 s= i-1n nd1
2 srr 1+cost
o








equation (3.4, 24) becomes


(3.4, 25a)


WI(Alt UeIw+ ZE (-1) n* ,


Combining the recurrence relation given by equation (3.1, 19) with

equation (3.4, 25a), there results


(-1,t) -iU e' (Ao-A,)


(3.4, 26)


The velocity_ due to F- can be found from equation (2.6, 21)


to be


40

W2 r(At)d
yir P(ik;Ol L+ I


(3.4, 27)


Inserting the expression for P(ik'&) from equation (2.6, 20) and

using the results given in equations (3.2, 5) thru (3.2, 15), it

follows from equation (3.4, 27) that


Q
Wz'l.i) Q


0


e-iACco:Sh*X (coshKY 1) dc


.-jAk coshX (cos h'e 1) d-


(3.4, 28)


Recalling the definition of the Bessel function of the second kind,

equation (3.4, 28) becomes


z (-~,1' = [-KIAkt -K.(iAk)
K,(iAk) + Ko1iAk)J (3.4, 29)









Combining the expression for Q given in equation (2.6, 18) and the

recurrence relation (3.1, 19) with equation (3.4, 29) and simplifying,

it follows that


(3.4, 30)


Adding the results of equations (3.4, 26) and (3.4, 30) the

velocity at the leading'edge.in the T -plane is


W'(-1,11 -i ZU e "W [ (A.+ A S (Ali) -A1]


(3.4, 31)


It then follows from equations (3.4, 22) and (3.4, 31) that the

suction force becomes


FU3 =L


(3.4, 32)


where


( (0+gA1 W(Ak) (A^ .t ;l %1(AIk1 A


Finally, by adding the results given in equations (3.4, 14)

and (3.4, 32) the net thrust becomes


T = ZirrpU1 c4'o(Co ,C: )Zl ni +(i-1(?d8
n r-


(3.4, 33)


Equation (3.4, 33) can be expressed in still another convenient

form for computational purposes by eliminating the i~ 's. Substituting


wz, MAI 1 -U e4-+ (A,* ,) I. Z Y (A 1, -1]









equations(3.1, 24), (3.1, 27), and (3.2, 35a) into equation (3.4, 5),

the following relationships can be derived.


I -c- S"
n- n n5-n~


(3.4, 34a)


(3.4, 34b)


where


S ( --o tokf I
;.--- A, + A, ) RAI AOA + Kjn) t( )-A,


-Cn 3(: C


(3.4, 34c)


(3.4, 34d)


Combining equations (3.4, 34a) and (3.4, 34d) with the second

term in equation (3.4, 33) it follows that


ZZ n
Malt


+4kZ n
XI.


+e ; -,


- e ak n n +


-ek~,, +
T. zn In 7n-1 r% n2)C-n-
nnj.

+r-


(3.4, 35)


C -1-C' zt% (


P. -- ?. 6 [ F. + ?- 0


L [ 61 -JI-Bt)
Zn t "B,,


k + + Z'I'n-zl' Eznn
-k[ 2, B -8~h i ~ n








Substituting equations (3.2, 35) and (3.4, 35) into equation
(3.4, 33) yields

T p- CP +' -i) + z 6. + 4kE n6 8, -





-4[ k'-'( : B,' -6b',~' +kC n( -, 6 ,,) *'* +

u I Cl- (n-l_ (3.4, 36)


For the case of an infinitely thin plate ( 6 = 0), it follows
from equation (3.4, 36) that the thrust reduces to


T =2r P U [(f;,+ I(E -CO ) +kag K,+4kE nBBj. (3.4, 37)


According to equation (3.4, 34b) for the case in which e
vanishes, s = This result is in agreement with Siekmann [-8]
if it is observed that

S= (-) and C, (-I)"~. ~
B, = -1)" 16, and nZ(1 Zn


where the 's and 's are those used by Siekmann.









3.5 Time Average Value of Thrust

Of particular interest in an investigation of the propulsion

of fish is the average thrust experienced over a period of time,

2nf
TO 2 The time average value of an arbitrary function of time

I (t) is defined as


T.

0


(3.5, 1)


Thus


AisTa = f
0




Ti
2 f -p j s r r wt 1 =
0


'p
n3-: sr\wtco w dt 0
-


(3.5, 2a)


(3.5, 2b)


(3.5, 2c)


Recall from equation (3.4, 5) that the real and imaginary parts

in the time imaginary unit j of a coefficient, say Bn, can be written as


SBcosut snt


- nI


and similar formulae for the other coefficients.


(3.5, 3)









Thus, from equations (3.5, 1) thru (3.5, 3) it follows that


0
To,


SC1di
f


T,


6. BB'di


To




j,,
0








0



rf 4;,dt
0


TO

'ra


; C1di


'If -B'I 1' d
0


oJ
0


=


1 z





z


tC2

~+
(C~Y r' + c c(t




+~c~c coc




cv CO, +,L; C." )






(a a7.~Q Cj )



CL., Co' t c;' Co"


Z~(a, C:I + CL" C;
z a 0 1~C


(3.5, 4a)


(3.5, 4b)


(3.5, 4c)


(3.5, 4d)


(3.5, 4e)


(3.5, 4f)


(3.5, 4g)


(3.5, 4h)


(3.5, 4i)


(3.5, 4j)


(3.5, 4k)


(3.5, 4Q)


(3.5, 4n)


S. (BlZ


di b,5( 5B,' -t B












GO
n~1'=.


TC (ri n,,sB, 6,d+ n [( Dn,l+ 6+. r Bn- IV,, U

O nGO

jr c ( -r Ci~d Z [c;,c
0
-'(C C
ZMr In-e n n n i C+~t; C~~4n irn-21 M


-C" ~,(C*-C' -C C"


. (3.5, 4q)


Substituting these results into equation (3.4, 36), it follows

that the time average value of the thrust becomes


T rrpu (r., *C(r(r+'-Cr,


+k2 (BB,': + B," B: +


-t[Zt
-1(r,' 2 A: 2A;)(C~ -Ci) +2 (r,"+aign -2 A: (C;; -C 4-I1


3 C, C., _t C C,, C., I C'D. Z + r.
3( C, +ro r +



-E~kZ (BO B +B 52 -B- Biz
B,-+k B+


and


0


(3.5, 4o)


(3.5, 4p)


(3.5, 5)









where


(3.5, 5a)


n=i


C
4n In2 (Cn-1. -C'n41l C'11%-21 (C h'-1


- n2 (C G C,,4 O c) 2 (C: -C6 1 ]


Y' '4A)i(A k) (A" + A') tL(A k) -A


: (A4A';) F(Ak) + (AO' + A') NaL(A k) -A'


Following Siekmann L 8_/ the thrust coefficient is defined as



C- T p, (3.5, 6)


Thus, the thrust coefficient can be determined by equation (3.5, 5)

by employing the definition given in equation (3.5, 6).


and


(3.5, 5b)


(3.5, 5c)









3.6 Numerical Example

In order to compare the theory developed with available

experimental data /87 for a very thin plate, a displacement

function with a quadratically varying amplitude is used. Thus,

consider the displacement function

D(a,t) = I h(x, (3.6, 1)


where

h(x,t) = (do -dx +d, X2 ) e-i' ei~ (3.6, la)


Here the phase angle Lo is set equal to zero.

From equations (3.2, 5) and (3.2, 7) it follows that

x = (i-e)cose +* (3.6, 2)


Combining equations (3.6, la) and (3.6, 2.) with equations (2.4, 16)

thru (2.4, 21) gives


H (8)-= do+d(l- )cose6 +dz[(l-6)cose*+ e-l (-cose+ (3.6, 3)



By employing the relations





and


co058 c I U + Cos 2-0
2






90

equation (3.6, 3) can be written in the form


H'(9) = (d,+d + kdz)(cos isn ) +


+ (1-r)(d+edza)(cos -jsin ) cose +


(1-GfAz(cos -ji )cos0 e-d()1-)S (3.6, 4)

From this equation the Fourier coefficients Bn can be computed by the
relation

6 = Hi(0) cosne de. (3.6, 5)
0

Employing the integral -237
iT
Jn(.X) = j-n / eix.o0e cosned
0
where Jn(K) denotes the Bessel function of order n, and combining
equations (3.6, 4) and (3.6, 5) the Bn coefficients become



tI +


Sir(- -(dl+-)(d, ) dz)(cos -isn ) J ) +





+i R) (-" (1-0(dl+edz)(cosy -isn )] + (




( 'n- (1_),-2[(1--'- ()o,-t sin.,-,)] Jn-?(,,) (3.6, 6)
z z.








where


and


Cos m eC~os Ile = [cos(m+n) + cos m-nlO


have been used.

Using the following recurrence relations Z247 between the

Bessel functions,

Jn() + J"I.(W) = --Jn( (H)


n




)CZ Z 2 in (X ) in*l
I J(x W


and


Jn+z(XI +Jn.Z(X)


equation (3.6, 6) simplifies to



B (-1) {(+ d.L +(1- )2)(cos- -i sin) J'n(i +


-( (i-ed4.+e dz)(C OS n. -


+(- (Ccos- s 7)' ) I .X
z z xz" i


J(-)i) = 1) n,()


(3.6, 7)




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