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 twentyseven
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. StretchedStraight 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 (zplane) . .. . . . 15
.6. Circle Plane (r plane) . . . . . . 15
7. Circle Plane with Point Source . . .. . 31
8. Circle Plane with Vortex Pair . . . . . 38
',9. StretchedStraight Configuration with Pressure
Distribution . . . . . . . . 50
10. StretchedStraight 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 xcomponent of velocity
Magnitude of the ycomponent 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 muchimproved 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 stretchedstraight configuration or the base profile. The flow
field around the stretchedstraight 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 xy 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 xaxis, 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 xcoordinate 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 xaxis 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
StretchedStraight 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 stretchedstraight 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 socalled 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 torpedolike 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 Fortyeighth 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 snakelike configuration
immersed in a uniform flow field along the stretchedstraight 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
twodimensional 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 twodimensional 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 twodimensional 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 SmithStone
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 twodimensional 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, twodimensional 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 twodimensional 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 countervortices 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 twodimensional 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 CauchyRiemann 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
a9
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 stretchedstraight 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 midchord is approximately 26. Since the length A of the fish is
approximately two, the thickness ratio at the midchord is of the
order F .
Clearly, this function satisfies the requirements of a sharp
trailing edge since
df[ (1e)
The two planes are shown in Figures 5 and 6 for the stretched
straight configuration.
L
Figure 5
Profile Plane ( Iplane)
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 zplane 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 de 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 xaxis 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) = (1e21 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 xaxis 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 stretchedstraight
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 6Lt di bz*
Next, consider the velocity of point P in a direction normal
to the surface. The point P in the zplane 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 xcoordinate 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 wt +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 n1
X 2 [C0S0+0 (1E)?" 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 () MIM [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 halfchord 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
Splane 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(fei')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(re' 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
e9
r8y eXPj8 z JT
(2.4, 40)
Hence,
2Cr
F,(eie ^\= 2CL Ot]Lo sin e zP z d^(2.4, 41)
0
Equation (2.4, 41) can be written as
2 VOP2 ,
F, (e ie) CL19i 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 socalled conjugate function. According to Robinson and
Laurmann /21_7 this function is defined as
0~i'
~ (~=~1Li'e0d
(2.4, 46)
The integral on the righthand 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
G4
(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 noncirculatory 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) Uei3 ^ 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(e19] .
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(icos(+i9)) +1 cot d? .Ot
11
But, since C(9)= (I) and cot1f: =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(e19)
0
2s ,no Sin
\t'(e 2 co ecosiJ 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 countervortices distributed along the chord of the base
profile. These vortices and countervortices 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 Tplane.
&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 zplane is related to the circulation distribution in the Cplane
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)
 ^iJ < 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 rplane due to the
noncirculatory 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 zplane. From equation (2.6, 1) the complex velocity potential of
this pair is
AF' ,t)= og 0Ij (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 xaxis 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(\ ^l2cos 6 Cf'(e )1(t.+I2,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
Splane. 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\\ 1z2Qcos '(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 + l2 ,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 UsZsei1 E) kik
jki l_, 43 2t.cose \f(e' o)\ (,+12 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 noncirculatory 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 (8t )
\(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 socalled Glauert integral [21_7
1 stn19 Slhn d9 = cos ne
coI Co Co.s5
and
j j i cos(ie~) + 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
__ lcos(0+1 san_
TT(e,t) 10 loC. o) + +
ot Ccos(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 socalled 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 stretchedstraight 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
StretchedStraight 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 zplane 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) 6d& (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 xcomponent 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 xcomponent 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 steadystate 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 stretchedstraight 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 elinearized
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
StretchedStraight Configuration for
Several Thickness Parameters
r
~~
~1
Taking the derivative of equation (3.0, 1) gives
'(e I) = [l e21 + (e2iOe'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'0e 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. eiG +?.c e:19
After some algebraic and trigonometric manipulations,
equation (3.1, 8) can be written in the form
[f (el]e1
= ei, ~ i
Z.(1 ae) sii2
(3.1, 9)
where
r' = 1 2 Ecose (lZl cos26
4 = 26 sin e (0z&) 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(1ejsine 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 UIZaeosZO)sine dH(
21 (Ze)sirn 7
This last expression can be simplified to read
(e) =cr (1e)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
n1.
(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 1265
(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
(1e tZecose)H1(O) = bo+Z bcosne (3.1, 21)
nzl
(nz1.ZA...) .
where the Fourier coefficients are found to be
b, =(l085o +2B 3
(3.1, 22)
b, z (105 +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
rL
C0 + ?EZ C rt.c'S
(3.1, 23)
where the recurrence relation for the Fourier coefficients is
Cni 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
12n o
(3.1, 25)
where the coefficients are
1+e 1 26
wo 'a nz  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 =~ (1e)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
stretchedstraight 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 slowingup 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 xaxis. 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
slowingup 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 timedependent pressure vanish at the tail.
From equation (3.0, 1), it follows that the actual chord
length of the stretchedstraight fish is
x, x Z(l6) (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&
ei.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 eik Cos hX d'
cosh)L cose
%+1 sirnh) ejIKA 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) ,et1A 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 Z24/7
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 timedependent 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) +ZclcosZo
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 (I4fcose) 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)
iZe 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
socalled 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(146)(Ak) A, +4e Ao
(3.2, 35)
S +A An An^ n A.) n 1
IZo 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( AnAnAj.(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(cosOcosZ)] 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
2T
L=p aei"' t Z C, slnslnnO[(le)sln e ir snZ8 d. (3.3, 3)
0
Integration of this equation leads to the result that
L Zep 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 U44c(LAk)A Aj +4e Ao +
LA Ao A, + AzA 
Z 16 )
6 A AJ + A Z A ] (3.3, 5)
iZe 12 .
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= pp+ 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)] =[_ 12e
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* [ (16)a. (iZe) %, +e (a. +s) .
In terms of the downwash velocity, the moment becomes
M = pU e * (1 )( Zer+2 A, .Aol +
(12e)[ AZ +c A14 +2 (AZ AAs) +
(3.3, 10)
(3.3, 11)
(3.3, 12)
e[A,+A, +0(AA A A2 A4. +2*2(Ajl* A.zNLt 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 AA 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 xdirection. The xcomponent 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 socalled 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 = Ri1 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 stretchedstraight 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 asP (nS)(cos0cosae) (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 (cos6cosZe )de
o
the Cauchy principle value of the integral must be taken, i.e.,
f1 = m .p Ua:[ f+cnt (cose cos20) +Cfan(cosecos2e) dO (3.4, 10)
Employing the identity
+qn (cosOcosZe)= Ztc rv +3simOsin2&
this expression becomes
wsa 2fr
4* 0 z. O fe 0
Integration of this equation leads to the result that
FX z tEj.. Coit lo 1
4wo 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' stnmBKZE 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 nonlinear 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 rplane. 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
rt~~(E) ; (j~~~?)L7
(3.4, 16)
to z yields
to z yields
. Differentiating this expression with
e
dr zz
(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 ZI00 C
(3.4, 18)
where
(3.4, 18a)
' P +1 E
and as
respect
and
According to MilneThomson /1iJ the extension to Blasius's
formula for the case of unsteady flow can be written as
(3.4, 19)
F3iF. 3i 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 PICGe) W(1,t)1 (
d;E = da' = S. ; e'v dY
(3.4, 22)
Thus, F vanishes and F: clearly represents a thrust,
i.e., a force directed along the negative xaxis.
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,'snlne I" ij d (3.4,.25)
0
Employing the integrals
2ir
Ssinn? dv9 = 0 n l,
and
alt
i Si sn dn39 s= i1n 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' (AoA,)
(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
eiACco: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 +(i1(?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 n5n~
(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 7n1 r% n2)Cn
nnj.
+r
(3.4, 35)
C 1C' zt% (
P.  ?. 6 [ F. + ? 0
L [ 61 JIBt)
Zn t "B,,
k + + Z'I'nzl' 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 (nl_ (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 Ine n n n i C+~t; C~~4n irn21 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 4I1
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 (Cn1. 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 ) ei' 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 = (ie)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[(l6)cose*+ el (cose+ (3.6, 3)
By employing the relations
and
co058 c I U + Cos 20
2
90
equation (3.6, 3) can be written in the form
H'(9) = (d,+d + kdz)(cos isn ) +
+ (1r)(d+edza)(cos jsin ) cose +
(1GfAz(cos ji )cos0 ed()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) = jn / 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) (" (10(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 mnlO
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 +
( (ied4.+e dz)(C OS n. 
+( (Ccos s 7)' ) I .X
z z xz" i
J()i) = 1) n,()
(3.6, 7)
