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## Material Information- Title:
- A theoretical study of the swimming of a deformable waving plate of arbitrary finite thickness.
- Series Title:
- theoretical study of the swimming of a deformable waving plate of arbitrary finite thickness.
- Creator:
- Uldrick, John Paul
- Place of Publication:
- Gainesville FL
- Publisher:
- University of Florida
- Publication Date:
- 1963
- Language:
- English
## Subjects- Subjects / Keywords:
- Base flow ( jstor )
Circles ( jstor ) Fish ( jstor ) Flow distribution ( jstor ) Hydrodynamics ( jstor ) Mathematical independent variables ( jstor ) Pressure distribution ( jstor ) Sine function ( jstor ) Swimming ( jstor ) Velocity ( jstor )
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright John Paul Uldrick. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 021889124 ( alephbibnum )
13409805 ( oclc )
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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 encouragement and criticism of the manuscript. Also,. the author wishes to thank Dr. Nash for providing financial assistance for the use of the IBM 709 ol1. troni c 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 . e . . o . . . . . . . Ii LIST OF TABLES . o o o o .- o' o o . o . o - o . o . o o v LI1ST OF FIGURES o 0 . . . . . 0 . 0 0 a . 0 0 . . . * . . 0 vi ,IST OF SYMBOLS . o o o . . o . . . . . . . . . . . 0 . . . vii Chapter I. INTRODUCTION TO THE PROBLEM o.o.e.o.o.o 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 * o * o o o o o o o . 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 3o4 Calculation of the Thrust and Drag 3o5 Time Average Value of Thrust 3.6 Numerical Example Chapter Page IV. SUMMARY AND CONCLUSIONS . . 104 APPENDICES o o o . . . . * * . . . . . . . . 107 LIST OF REFERENCES . . . . . . . . . . . . . . . . . . 121 BIOGRAPHIOALS KETCH ,. o . . -. . . . . . . . . . . . . . 124 iv LIST OF TABLES Table 1. 2. 3. 4. 5. 6. 7. 8. -9. 10. 11. 12. 13.'0 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 . . Imaginary Part of Bn Coefficients . Real Part of On Coefficients . Imaginary Part of on Coefficients . . Real Part of A Coefficients . . . Imaginary Part of AO Coefficients .o. Real Part of A1 Coefficients . . . . Imaginary Part of A1 Coefficients . . . . Page 98 100 101 103 112 113 114 115 116 117 118 119 120 0 . 0 0 0 0 o . 0oo0 . . o o * . o o o. o * o o * . . o . * o o *o . o o o B o g i e. * . o e . . o o . o o. 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 (Y -plane) . . . . . . . . . . * ï¿½ 15 7. Circle Plane with Point Source . . . . . . . . . o 31 8. Circle Plane with Vortex Pair . . . o o . . . . . o 38 -9. Stretched-Straight Configuration with Pressure Distribution . . . . o ï¿½ . . . . . . . * . . . 50 10. Stretched-Straight Configuration for Several Thickness Parameters . . . . . . & . *. . . . 55 11. Thrust Coefficient Versus Reduced Frequency . . . 102 LIST OF SYMBOLS Symbol h(x, t) t P F zï¿½ z x + iy 4 t +~ F 0 Fl F2 U w U + iv A w f M~* Description Displacement function Velocity vector of fluid particle Mass density Time Pressure Complex comp lex velo city 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 a -, Im~ TrI (e,t) bn cn H*(() 1 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 Fo urier 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 En'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 qG qn qt U k = W/U r(O , jk; e L M T Q PF(jk; e) H(2) (z) Jn(x) T CT ,,.,6,A, x 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, d2, 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 seafaring vehicles. 1.0 Statement of the ProblemConsider 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 amplitude 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 def ined by Y"' (X,-t) Y11 M t) + YI M XL X XT where Y U and Y L are 'the ordinates of the upper and lower surfaces of the plate respectively, and X L and X T are the leading and trailing edge projections on the x-axis, respectively. The function h(xt) 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 displacement waves, which are not standing waves, passing from the leading Figure 1 Stretched-Straight Configuration of the Fish Figure 2 Displaced Configuration of the Fish edge to the trailing edge of the plate. The magnitude of this thrust depends upon the propagation veloc ity of these waves. The purpose of this investigation is to calculate the forces acting on the plate for any given flapping function h(xt). 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, a nd the vortex potential the pressure distribution on the base profile is computed by employing the unsteady Bernoulli equation Z-1.,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 funotions in the thickness parameter. This is carried out in Chapter III. 1.2 fleview of Relatod Literatur2 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 [3]7 investigated 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, Biekmann [4]7 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, particularly squids, octopuses, and cuttlefish. Lighthill [s1Z5 discussed brief ly. the swimming of slender fish at the Forty-eighth Wilbur Wright Memorial Lecture. In a later publication [-6.J 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 configuration 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 axissymmetric type flow problem. Of particular interest to this study is a paper by Siekmann Z[8in 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 [79 and to Kmissner and Schwarz [0_7 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 [-112, in an independent study, considered essentially the same problem as Siekmann [-8]. Wu employed Frantl's acceleration potential to determine the forces acting on the plate. In a subsequent paper [-12], 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 L-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 [s and Wu -117. 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 -16_ for a system of finite degrees of freedom and solved the dynamic equilibrium equations. Kelly [17J 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 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 ofb 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 transform 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 [l8-7., Recently Kiassner 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 translatory movement. The problem was linearized by assuming small unsteady perturbations about the mean position of the base profile. Theodorsen J_16_j treated the pro blem 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 amplitude 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 ICUssner and Gorup's theory Z-19-7 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 equation: cliv V 2 0 (2 1) where V= LvI is the velocity vector of a fluid particle. Motion equation: d p : c-d 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 curt V = 0 (2.1, 3) Then, in these regions there exists a scalar point function (xy t) defined by the equation where (x,yt) 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 ï¿½ Also, from equation (2.1, 1) a stream function 11(x,yjt) can be defined as oy (2.1, 6) 6x y where, LL and V are defined by equation (2.1, 4). Equations (2.1, 6) are the familiar Cauchy-Riemann differential equations defining an analytic function F (Zt)= (xy,t) + i (X, YJ) of the complex variable - x+ i . From equation (2.1, 6) the complex velocity, w= utAi v , can be found from the complex velocity potential (Z~t) for any time t, as W= a" (2.1, 7) where the bar denotes the complex conjugate. Therefore, F(Z,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 transformation of coordinates. Lot ?= f() be a conformal transformation of the exterior of a uni- circle in the = + i v 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., lir w(z,M)= . ira(','I . (2.1,8S) ~l -.oo 1.b00 Figure 4 Circle Plane for General Profile Configuration Figure 3 General Profile Configuration Quantities in the " '-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 a~s where the Oan~ 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 -E.as + in1 E~ (1 Q 2n] (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 _ 0 (211) The two planes are shown in Figures 5 and 6 for the stretchedstraight configuration. L Figure 5 Profile Plane ( I-plane) Figure 6 Circle Plane ( -plane) Since the potential function F(~i is a point function, it is invariant uneer the transformation (2.1, 9), i.e., (2.1, 12) The velocities of a particle in the two planes are related by W L (2.1, 13) Let the location of the downstream stagnation point P 0 in the S-plane be given by ~ (2.1, 14) where eL is the argument of the line OF%. Any point P on the profile in the z-plane is mapped into the A point F in the -plane and is given by z (eia(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: where F0 is the potential of the uniform flow around the stretchedstraight plate, F1 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 employing the unsteady Bernoulli equation flj, which reads +T CL + M (2.2, 1) The arbitrary function .9(t) can be determined at upstream infinity since it is assumed that the disturbance due to the unsteady perturbations vanishes there. Thus, 9f= luz i-'-1 (2.2, 2) Therefore, equation (2.2, 1) becomes - - (2.2,3) P T (2.2 3) where a __ F_22,4 L Z TZ 2.,4 Obviously, the velocity potential .(xy,t) on the base profile surface is e2= FL ()I ) (eiO (2.2, 5) where &i denotes the !real part of' operator for the space imaginary unit i. Let "Uo + LL' + u" I (v. +V' +V (2.2, 6) uï¿½ + I:O L! i V' I- ' v where (u,,V)O , [',V) and (u."V") are the x and y components of the velocity of a fluid particle associated with the base flow potential, the source potential, and the vortex potential, respectively. Then, 0 + Lo Z (U LAOL +vv'l + Lt." + V0V") -i+ Z(UL'U.+VV'" iU !2 +V,2 4_.!"? + V,, (2.2, 7) or, in terms of the complex potentials, T dF 7 -I 6L __].TF,+O,6F Substituting equations (2.2, 4) thru (2.2, 8) into equation (2.2, 3) gives P P Z 1 cF, d, f F Fa 1 1 IF ~ Jz z v + Fz(Cz.,i)11 + + + 6M 3z J Under the small unsteady perturbation theory the last two tc:ms 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 F on the base profile as 'P 1} F UJ(+e% (z or +_ d F, (Z) F__ , or, in terms of the argument 8, IT(Oe* = 1P [z W ei, 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 t1% will be present around the profile and hence around the circle. Therefore, from any text in fluid mechanics [1] employing complex variable theory, the potential can be written as 10((=+ o (2.3, 1) The complex velocity in the -plane is W, d - = U(2.3,9 2) and on the circle the velocity becomes e~lee Since the velocity vanishes at can be found to be r. -zly U sin (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 dov;stream 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 F in the r -plane is located on the - axis. Thus, for this case the circulation Q vanishes. From the above results, the velocity potential of the base flow becomes The base flow velocity is tangent to the circle and is SU sir ei . (2.3, 6) A The velocity of a particle on the streamline emanating from P0 is given by = LX(-g~ (2.3, 7) The velocity of the same particle in the profile plane is U _ //%- (2.3, 8) f - 1 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 relatio nship between corresponding displacements in the two planes is given by d i' V d(2.4, 1) Let the unsteady displacement of a point F on the profile be D(z,t). Then the position of point P at any time t is 2 P Z.P + (2.4, 2) where E,, 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 direction. Thus the complex velocity vector of point P is given by the material derivative of z Pwith respect to time, i.e., M D D dz (2.4, 3) dt t zdt where d:is t.he complex velocity vector of a fluid particle at this 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-p = D- + 6 D (2.4, 4) 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 and the unit tangent vector is dr * ï¿½(2.4, 5) Since P on the circle is located by specifying the argument 0, 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 derivative of z with respect to b ï¿½ The complex unit tangent vector is given by dz 1 d(. C -c = ' (6. ) The complex unit normal vector i in an outward direction is 900 clockwise to the above complex unit tangent vector, i.e., -' t (2.4, 7) The velocity of point F on the profile in the above normal direction is simply the scalar product of __ and W . Therefore d -, (2.4, 8) In the rj -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 AdF dtj C,(6, t) -IM d -- (2.4, 10) 8jr(Q{) (2.4, 11) (2.4, 12) A (6, -L) = 9-, V 9-r o,.,.,(.e,~~~ tl=& ' "d .In a similar manner, the tangential velocity on the circle due to this perturbation is found to be CL [dO t]1 (2.4,9 13) The undulatory movement of the profile is described by the flapping function h(x,t). Here the x-coordinate refers to the base configuration. 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 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 swi=mming fishes L20J it seems reasonable to assume that this amplitude has its smallest 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) = 1(x) cos(ox -wï¿½ +A.) (2.4,1 15) where ot, is the wave number, w is the circular frequency (which is taken to be positive throughout this work), &, is an arbitrary phase angle, and Ii(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 interpretation. From equation (2.1, 10) the x and y coordinates of point P on the base profile are given by and rO CO Y = ! sin 0 n=iI (2.4, 17) (2.4, 18) Since x is an even function of 0, any function of x is even in ï¿½ Hence (2.4, 19) (2.4, 20) D (Z ')- = H"(x) e"] t h (xi ) = h [ x (6), t] = h [x (- e), ] = h, ( 8,L) H*00 = Hiu ï¿½(e] H'; X(-e = ,.] 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 .-I "ddo ,(lio ., dz jz ,: (eJ . (2.4, 23) Equation (2.4, 23) can be written in a compact form as 0L- (OJ- U ^ (o)e 'e (2.4, 24) wher e G(j~ d1A 4(e (2.4, 25) B" n C, 0 5 n 19 U.[ H) + - 'F 0 eiwt 28 But, from equation (2.1, 10), evaluated for f = (e'0) dz de d =H(O dI-,(E)) d d -_ d e d0 a _- _ie'0 f'(e-'8) 6dE (2.4, 26) do = i e!0f do (2.4, 27) Also, from equation (2.3, 5), evaluated for Z = J( e) jF jFdo _ = 86 6V' do u~ (2.4, 28) Substituting these last expressions into equation (2.4, 25) yields G(&) M -Im e'ie(e'0iYk'-4(6) - k H(8) (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 (O:-lmj { e ,Tkf,(e'0eJ H,( - f'(e0) (2.4, 30 (2.4, 30) For a symmetrical base profile, as is considered here, it can be showm that the function G(e ) is an odd function. Therefore, it is convenient to write Q(W m2 sin in (2.4, 31) where the Fourier coefficients are 0 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 -c, The relationships between the Fn's and B n's are found from the solution of equation (2.4, 30). Due to the complex nature of this equation, the recurrence relations betw een the Fn'S and B nI 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) 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-O U () G'(2.4, 32) where e{ L '(e jl - HAG) ,. (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 Z-16_7, 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 corresponding distribution of sources along the surface of the profile. Let the strength of this source distribution per unit arc length in the -plane be denoted by Q(,) ï¿½ This source distribution can be related, in an obvious way, to the normal velocity (6(O4{ by Z (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 located at j is , 0 log''- ) (2.4, 35) Therefore, from Figure 7 the contribution to the total potential function due to a point source at {=e is given by d F, ( ',{) '- if Q*,'') (r-ei la- (2.4, 36) and the total potential at < due to all the sources becomes F,( Z c.0(0o Cj('-e' d (2.4, 37) 0 By employing equation (2.4, 24) the source potential becomes 0 2. Figure 7 Circle Plane with Point Source - 2 (2.4, 38) = + I-,', r, -= e i -r(VLt) evaluated on the circle is given by F, (e ie t) = 2r oo d ZCL 109 k .oe _ ei d9. 0 e-9 (2.4, 40) Hence, zCr 2i(4 0 Equation (2.4, 41) can be written as 2 V F, (e ie) C i tog en z] d 0 0 (2.4, 42) Thus, the velocity potential associated with the source distribution evaluated on the circle is The tangential velocity on the circle is found to be 0 0 (2.4, 43) (2.4, 44) But (2.4, 39) -S L cPAi)cot9z dLl ï¿½Lploying equation (2.4, 24), this tangential velocity becomes 2 oqiw 0 9)cta,& (2.4, 45) The integral in equation (2.4, 45) is a function of the angle a , the so-called conjugate function. According to Robinson and Laurmann [21_] this function is defined as 0 (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 0l F 0 (IM c 2ir (2.4, 47) From equation (2.4, 31) it is readily seen that and the definition of the conjugate function r (e) = P, cos ne f=l The circulation around the circle due to the source potential is = f Vo(,ï¿½) d 9 - (2.4, 49) 0 (2.4, 48) Substituting equations (2.4, 45), (2.4, 46), and (2.4, 48) into equation (2.4, 49) it is observed that r 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 2U 1TO~ 61~~ i EL +, F (elo) 2 i Ue'~f()19re1i (2.5, 1) 43 Lt Wf(e'OAZ 2' _qr 0 Performing the indicated differentiation leads to G ee jl(2.s, 2) Substituting equation (2.3, 6) into equation (2.5, 2) and observing that the 6Lj operator and the integral operator are commutative in the 2 2 Ibov eqtion th sjorc e pessure ditiuin beco-es oz 0 + ,id12 (2.5, 3) here q : t-e . U. But, from equation (2.4, 40) e'8-e s Zsx- e I 1 (2.5, 4> Hence, '40 ' 0-19 iee 1 - Cos - Cot -e ? (2.5, 5) (2.5, 6) Substituting equations (2.5, 4) thru (2.5, 6) into equation (2.5, 3) gives fGP) 0iog [ sini 0 sin "c t + - -J C a (2.5, 7) Also, TT~ (e t)= le e 1% 1$1 zim 9 V. Equation (2.5, 7) can be written in the convenient form T (Ot) =- + -' +r Zo-o .l- cos(e- ))-o 2 Tr where the following substitution has been made. Changing the variable of integration from - to -0 in the second integral of equation (2.5, 8) yields ,%1 CO) [-(loo?(I-cos(O--)l) + , cot d -1' But, since (-j) (X(9) and cot-1-- o 1Co! -ï¿½ this Z last expression becomes 2%r -QW. -10 -O(-0 sn t d The integrand here is of period 2ni. Thus, the limits can be changed from -2o to 0 and -ii to n. Using this fact, equation (2.5, 8) becomes WPM - P U frAl F -COS(e-19) + 0 + t('01 n doecsi) (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 Z22j 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 L22_/ 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'sdurce 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, oine of strength ~r 'd0 located at on the trailing edge streamline, and the other of strength - Afl=-Y"d~ located at the image point rz- 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 distribution be denoted by F 2(C ,t). Then the contribution to the total potential due to the vortex pair located as shown in Figure 8 is AF2 r--r (2.6, 1) where APis the circulation strength in the r-plane. Figure 8 Circle Plane with Vortex Fair The total velocity potential due to the vortex distribution therefore becomes IF,(. 't) ilf A(L0t) 19(2.6, 2) where it is assumed that the unsteady motion has been going on for an infinite time. According to Wagner [-23_j, the circulation distribution in the z-plane is related to the circulation distribution in the C-piane by ~(~0+)d~= ~'x0,+dx0(2.6, 2a) and from the mapping function, equation (2.1, 10), it follows that dxo Employing these results the circulatory velocity potential function can be written in an alternate form as F2 Y 01(2.6 2b where 5wt3)is the vortex distribution in the physical plane. In accordance with Helmholtzs law L22_/ of constancy of vortex strength following a fluid particle, the equation governing the vortex distribution in the physical plane is Y"XM =c.0f5t. (2.6, 3) following a fluid particle. Therefore, the substantial derivative is Id Y W * a :'a (2.6, 4) However, since the unsteady perturbations are harmonic in time the vortex distribution can be written as ~fxt~ ' x~eit . (2.6, 5) Combining equations (2.6, 4) and (2.6, 5) yields a X -K.)+ dx 9(x,, 0 (2.6, 6) where the substitution F dr(2.6, 7) has been made. Separating the variables in equation (2.6, 6) and changing the variable of integration to , it follows that fdJ - F d .(2.6, 8) Integration of this expression gives iL do](2.6, 9) 1 where Yo is the strength of the vortex element located at x.( ï¿½) the physical plane. Obviously, 4 is the vortex strength when this particular vortex element was shed from the tail. From equations (2.1, 10) and (2.3, 5), _/2 V((2U,1 where k= 4 U is the reduced frequency as defined above and f o j~~ toV (2.6, 11) Combining equations (2.6, 10) and (2.6, 5) with equation (2.6, 2b), the circulatory potential function becomes 00 top_ j kt'"(t'o 05----_ t. (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=i ï¿½ This restriction implies that (2.6, 13) But, from equation (2.4, 38) 0 (2.6, 14) (2.6, 10) 'r f,( T t)* f, (, t] " = FFIar r= I A Substituting the trigonometric series for Q(O) 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 _ Uei- __ - i n di (2.6, 15) Employing the integrals Tr tsin n~di = 0 n,,,-- and 21? sin n3 s i n 0 the velocity at the rear stagnation point in the r-plane due to the non-circulatory potential is (2.6, 16) - -iUeZ~aP.ï¿½ fl-1 Now the velocity induced at the downstream stagnation point by the circulatory potential is investigated. From equation (2.6, 12), it follows that 1. (2.6, 17) Combining equations (2.6, 16) and (2.6, 17) with equation (2.6, 13) yields Vrr J - IQ - j (2.6, 18) where Therefore, from these results the constant Y. is found to be a P(,jk;e) (2.6, 19) P~j =e1A o f * ,, o (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 IZ~)JJ P(l k;0& f 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; C ) depends upon the mapping function and the reduced frequency. These quantities are assumed to be known. Thus, the total complex velocity potential function F Tt F0C(V + Fjr~'t) + F2(T~t) (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 tATTZ due to a single vortex pair of equal strength located at . and - in the '-plane or the corresponding points in the z-plane. From equation (2.6, 1) the complex velocity potential of this pair is AFZ'j,t) o~g~ 0oj (2.7, 1) According to Helmholtz's law [22j of persistence of vorticity following a fluid particle, the strength Al' 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 coincidos with the positive x-axis in the profile plane. Therefore, __ - g A~ 64Fl (2.7, 2) Tt dt 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 Af4 L% &C dtoL F~e6 , 1O 1s1~(,ï¿½ -- I; I 1-- (2.7, 3) where ~LL~ol(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 - i , e o i f lo'=9e (2 .7 , 5) Recall, from equation (2.1, 10) I (e;)I2 and f(o are real in the space imaginary variables. Also, from equation (2.3, 5) which is real, and F'(e"e) =-UsstiO iOe Employing these last results and carrying out the indicated differentiation, equation (2.7, 5) can be written as AT(ey- _ [ (o) 2s6,t , (2.7, 6) To determine the effect of the entire wake vortex sheet, the vortex element A is replaced by (2.7, 7) where Y,, denotes the circulation distribution per unit length in the r -plane. Substituting equation (2.7, 7) into equation (2.7, 6) and integrating over the entire wake, the time dependent circulatory pressure distribution becomes j~j) ~+ 1 *(2.7, 8) Z*-~cs V{(eO)IzQ. +1-. -~COS8 1 Combining equations (2.6, 2a) and (2.6, 9) with equation (2.7, 8), it follows that IT M=P(O eiuJojZ aSin e V(2.7, 9) + U , ' sine 1) 1Z(. +i - o ) _r.,IV W -(k2 +I. -zt0cose)] Eliminating the constant Yo by way of equation (2.6, 19), the unsteady circulatory pressure distribution becomes - T(E),ik ) (2.7, 10) where F.__ Zsin E) k{e0\(0+-~cs Jk lk ', *-Et, ose W '(e!O)\IZ .,+1 - z ,oos e-l ' T(04ik;4 0 CO (2.7, 11) Iï¿½- I With the previous results the total unsteady pressure distribution 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 (Q t) - T(T ï¿½O,) +T.(Q ) . (2.7, 12) Combining equations (2.5, 9), (2.7, 10), (2.7, 11), and (2.7, 12) it follows that qr to U"e~wli-cos (e-, ) ei)=- PUvewji () Io -Cos i9*) + inE) 1 0 - p QT(eiK;C ï¿½ (2.7, 13) -y (eio)l cose-COOI Substituting the trigonometric series for (CA9) given by equation (2.4, 31) and employing the so-called Glauert integral Z-21-7 1. 39 Sn lh5 ds9 = cos ne % Coe- c.o5 a and -;of j - -cos(E+-& I nO 0 into equation (2.7, 13), there results Uz IC p S Sin,,. - ,, oso ] + n - W(eio)12 -pQT(O' k; ï¿½ (2.7, 14) Finally, inserting the value of Q from equation (2.6, 18) into the above equation the time dependent pressure distribution becomes CO T =zPZUe[r a ) Sne Cos ha + +. (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) r IT (e,) UeT10(0+1-Co + It,+ - i - "_ [ C l I - o K ( k ( @- ( 2 . 7 , 1 6 ) With some manipulations this can be expressed in identical form with that given by Kssner and Schwarz Z-IO_, Schwarz [-93, and Siekmann Z8_2. The function X(ik is the so-called Theodorsen function l16_7 defined by FX- -*I -k Oa + j 1 ( r ,-(k) +j %(k) (2.7, 17) where k2kk) and H z (k) 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 configuration be denoted by (2.8, i) (C) iT d o 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 =-Tfdjo d Fj =-JXdx. (2.8, 2) These are the forces exerted by the fluid on the.profile. The forces FX and I 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(PF -iFy) = -i TTd% . (2.8, 3) The forces exerted on the profile can be obtained by integrating equation (2.8, 3) along the contour (C) of the base profile. Hence, the lift becomes L=Fy= T , =o (2.8, 4) (C.) But de ae,= - de D d (2.8, 5) and Z'"r 0 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 = IT xdx *jd) (2.8, 7) taken in a counterclockwise sense (nose down). Equation (2.8, 7) can be written as d M = 6L " z.dif (2.8, 8) The total moment is found to be I- TT (e.') do (2.8, 9) 0 The above equations for the lift and moment are essentially those developed in the Blasius theorem I_/. 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 hydrodynamic force acting on the plate, the change in shape of the profile must be taken into consideration. An analogous consideration was made by Sielanann L8_, Wu Lll1_, Smith and Stone [132, and recently by Pao and Siekmann L14_J 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. IThe hydr odynanic forces can be computed here from equation (2.8, 3) if the differential element d~in that equation is replaced by the exact differential element d~,where di ~ ~ K' ~i1~N(6) (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 hydrodynamic force becomes !L-- c@.3e4 d E)o (2.9, 2) [ do -d where 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 SYMMIETRIC JOUKOWSXI BASE PROFILE WITH A LINEARIZED THICIMESS 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 di-placements 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 associated complex velocity potential F, 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 configuration was found by linearizing the unsteady Bernoulli equation 1 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( e') which, according to equation (2.4, 30) is related to the amplitude function H*(G ) of the flapping. Also, it was observed that the unsteady pressure distribution contains certain functions related to the mapping function z = f( N. 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, i0) the e-linearized mapping function becomes -. r(3.0, 1) Figure 10 shows the configuration of the base profile for several thickness parameters. The numerical values for the coordinates 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 "dovnwash" 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 Cn. (e ) . (3.1, ) Recall, from equations (2.4, 24) and (2.4, 30) , U (oe)ue (3.1, 2) where mi(L i V (a-,)., (3.1, 3) JrJ ____ = 0.05 6. 0.10 C= 0.15 6= 0.20 Figure 10 Stretched-Straight Configuration for Several Thickness Parameters i Taking the derivative of equation (3.0, I) gives V'(e [I= - eZie +Z(e2iOese] (3.1, 4) Separating the real and imaginary parts and changing i to -i, equation (3.1, 4) becomes " e -[ + ] (3.1, 5) where S=1- cosze +Z (rosZU - cos 36) -ir\2O t26 (sin2 -Stn3) Also, from equation (2.1, 10) [ f1(e'e]- - e'- 13- (3.1, 6) This can be written as [{' (e&D T - eZ-e -1+] Z (3.1, 7) Neglecting the 62 term and simplifying, equation (3.1, 7) becomes ? 4 ei OZeeiG+ï¿½.e-o] (3.1, 8) After some algebraic and trigonometric manipulations, equation (3.1, 8) can be written in the form [f' (e;le-1 ~t+i,~ti Z.(1-aE)siri2e (3.1, 9) where Er =-gCo0s --Zec-Cos Z8 Substituting equations (3.1, 5) and (3.1, 9) into equation (3.1, 3) yields sin e-enI e de J} (3.1, 10, Taking the 'imaginary part of' in the space imaginary unit i of equation (3.1, 10) leads to AL-- ose- *sre, w1 0 4 ï¿½COS - 'i'sn dHI(eJl ##- (1-asne de ] . (3.1, 11) L:--jsie d Combining the values of 4, C , 1., and CI, given in equations (3.1, 5) and (3.1, 9) with equation (3.1, 1i) gives (0y- - [e-srnze, +zsine -ZesinSO)cose + - C(-cos .0 +ZeCosze -zecos39) S ine] H*(o) (3.1, 12) (zestne -(I-Ze)SInze)coso -(I-Zecos -Ue-ZecosZ0)sInG dv"( +-Ze)sind This last expression can be simplified to read ?(e} =c" (1-6)sinO+sinZei W.(G) + - +26.CO 50 al r L1-ao . -4, de Recall that from equations (2.4, 21) and (2.4, 31) HC)=BO + . S co, tie (3.1, 13) (3.1, 14) and a(P) t i =nB n-1, (3.1, 15) where Pn = If8)snnede f (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 recurrence relation between the Pn's and B 's as n n + '-' riB, - ' [-1B_ +1.n 1]5,n.1) ï¿½ 1-26 1-26 (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(O ), a new function defined by = A* Z. AcosnO where the Fourier coefficients are related according to A n--An+. = ZPn (3.1, 18) (3.1, 19) Combining equations (3.1, 13) and (3.1, 18), it follows that rIAZe I 2 I i deW(e) cj(0}: a-(1-e* cose)H-'(6) -i j- cOSO sine de (3.1, 20) It is expedient to write the first term in the last expression as a trigonometric series in the form (I- e Ze cose)H.(O) - bo +ZZ, b~cosne (3.1, 21) n= $ (n- 1,ZA. -) . where the Fourier coefficients are found to be 6=(I -0o+?e 1 (3.1, 22) brz 1- 5M+EB,+, +Br* I n>, I.- Also, the second term in equation (3.1, 20) can be written in the form I d e1(6) 5inO) dEO 00rne Co + ?-.E C t.coS n a (3.1, 23) where the recurrence relation for the Fourier coefficients is CCn+i = -Zn S n~~i . (3.1, 24) Finally, the third term in equation (3.1, 20) written in a Fourier cosine series becomes +- CE + c .C, cosno ,=O+Z, . cos n -o C (3.1, 25) where the coefficients are +-e Co C2 ,co = - -z1--Z 2. 4n Izc-c% *1-ze(C- +' A+I (3.1, 26) n 7;l Combining equations (3.1, i8), (3.1, 21), (3.1, 23), and (3.1, 25) with equation (3.1, 20), it is seen that A -, = (r b % . (3.1, 27) These results give A. o-[C -1.6 +Z ]+ 2(3.1, 28) Art- i (1-e)+.e(B. . -jnC% CA+ C1 . . +C n>, 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 Cnis 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 ITt(8t)=PU [ Sne si(e' cos ND + 1T( e)] (3.2, 1) U efl In order to completely determine this pressure distribution it is first necessary to approximate the difficult wake effect as contained in the function T(0jjk;e) . Consider the velocity of a fluid particle along the streamline emanating from the sharp trailing edge as computed from the base flow potential F0. This velocity is given as -t o)t (3.2, 2) At.the trailing edge -4 km= I Ii.~ MUe U (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(1,ik;E) , 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 infinitesimal 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 0 0, 6.= 1. The relationship between 6 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 IX-V- XLrz)(3.2, 4). This chord line can be mapped onto the same unit circle by the transformation (3.2, 5) The values of A and j can be determined as follows; XT= 1- =A() (3.2, 6) or A-- (3.2, 7) Employing these results, the base flow potential in this flow field becomes and F. (u T(3.2, 9) Employing the transformation (3.2, 5) it follows that (3.2, 10) '(eiO)jz i4 Azs inZ 0 (3.2, 11) Substituting equations (3.2, 10) and (3.2, 9) into equation (2.6, 11) yields ï¿½ z 1Z Z' [gLo.+ z (3.2, 12) It is expedient at this point to introduce a new variable defined by ey ï¿½(3.2, 13) Hence, expressing equations (3.2, 9), (3.2, 10), and (3.2, 12) in terms of the new variable.X , it follows that o I4 h s(nhx. (3.2, .14) (3.2, 15) 2Csy--1 (3.2, 16) and Combining these results with the expression for T(OjKie) as given by equation (2.7, II), and changing the variable of integration gives (3.2, 17) o 00 .i smO e 2%- e eT l- I- z &-co;e& e-iAk cosh' dX where A = P This expression, after some algebraic manipulations, can be simplified to read 00 Up A -snO f* 0 S nb 9 sInhzX e"iAkCos X d cosh' - cose e-+ sinh)c e-jIkA CoS1 ex -1 Employing the identities strnhX + sinzi = coshX -cos2O and e-C0hX ++I e'X +1 e",-[ 1 , 0 e"Ak Cos % T8,jk;6- (3.2, 18) equation (3.2, 18) reduces to CO U f A 0 sine 0 Cos hx+ Cos G) e-j1Acosl" dx (coshX + 1) ei-Ak coahx dV- It is well known in the theory of Bessel functions Z-24_7 thai (3.2, 20) f ezCO~h'xcos hnWX dY0 where K,(t is the nth order Bessel function of the second kind which is valid if Icrgel-4 . This requires that *();0 Let - k -- j) 9 (3.2, 21) Thus, if &(W7 0 it (3.2, 19) and (3.2, 20) U T(e, k;e0 = U AstnO follows that 1470 . Combining equations gives the result K(jAk1 * cos e Kc,,iAk) Kj(jAk) + Ko(jAk) (3.2, 22) (3.2, 23) T(e.,ik;) U [ (1 - coSO) K(jAk) COSEj A.sina Kj{jAk) +K=(jA'k) I ï¿½os (3.2, 19) The ratio containing the circular functions in equation (3.2, 23) is the Theodorsen function [-16_] t(Ak) ï¿½ Employing the relation Z-24] = (A rj r ' N ,1(Ak1 (3.2, 24) where (AM are Hankel functions of the second kind and order n, it follows that K.(Ak) 1H(A Q (Ak( - Ak . (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 / . In the present case /4- 0 . Substituting equation (3.2, 25) into equation (3.2, 23) gives U A sine This determines the function T(0jit(e) 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, i) the wake function as given in equation (3.2, 26), the time-dependent pressure distribution becomes Mn C, 2(eic) a Asto L. 0) *(i-%t a From equation (2.1, 10) it follows that cos no . (3.2, 27) r (3.2, 28) With some algebraic and trigonometric manipulations this expression can be written in the form I I+4e + e - 4e(1 0) cosE Zc-cos Z -iO - Zrz) + 4 0- sinLe (3.2, 29) Neglecting terms involving orders of E of two and higher, equation (3.2, 29) reduces to ee)l(i-Zl Sn2 (3.2, 30) Combining equations (3.2, 27) and (3.2, 30), there results 1T(O t)~ 2 (& L 4n+ ~ (Ak) + (I - KA1cos ell (3.21 31) It can be seen from this expression that the unsteady pressure possesses a singularity at the tail where 0 = 0 and at the nose where = =. The singularity at the tail is removed by satisfying the condition T(o L - 0 (3.2, 32) This can be satisfied for all time only if I-4e -- (3.2, 33) i-Ze A which defines A and 6 in terms of the thickness parameter e ï¿½ The singularity at 0 = v produces a concentrated force at the nose, the so-called suction force arrived at in airfoil theory [21_, 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 where the coefficients are given by (see Appendix A) = "- [.(A.- A. 1-46)=&(Ak) -A, +4 Aol (3.2, 35) Qn An+ An-% - A,%*j 26 ,A. i A ) r I-ze I n _ -ze Dividing out terms in the denominator containing 6 and linearizing the results in E , these coefficients become ct =r -Ze(r.+ZA,-ZA.1 (3.2, 35a) o, A,.,-,-Ar ? - 2(An-A. -An.) n; iI where 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'rr 0 The differential complex vector df must be found from the mapping function, Therefore, from equation (3.1, 1) S -(.-sn e .O nZe -e(cosO-cCosZ)] d8 (3.3, 2) Since 1T(eO) is real in the space imaginary unit i, the operator ; 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 2Lre 0 Integration of this equation leads to the result that L = -Z1Vp u Z e ( ?[i- 6) 0.0 + (1-"6aLieAz1 (3.3, 4) Combining equations (3.2, 35) and (3.3, 4) gives the lift in terms of the downwash velocity coefficients as .- -AAA - . ? A, +Az --4 Az + _A - A3 ZC A N (3.3, 5) e -e i-2e . For the case of a flat plate of zero thickness ( 6 = 0), the lift .becomes L = -ZvPUZ ejw I(IA k1)Y_(Ak) + cr07P~ (3.3, 6) where = 1. This is in agreement with the results given by Siekmann Z78J. In making the comparison between this work and that of Siekmann it must be observed that A.= (-.)' A, (3.3, 7) where the 1K, 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 differential is AP-pV+-P " 72 The moment of the forces acting on the profile with respect to the origin is given by equation (2.8, 9) as 2w M 6; f TT (et) f eIe d4 (3.3, 8) 0 where positive moment is counterclockwise (nose down). Combining equations (3.0, 1) and (3.3, 2) and linearizing the results in , it is readily seen that [-- sinze -tecoses'nSZ - --siE) caZ].(3.3, 9) Substituting equations (3.2, 34) and (3.3, 9) into equation (3.3, 8), the moment becomes 21T M= P Ue 4i f (Cota n!2 -Z Q +)(- z o 0 ecos sinZO - s-slrOcosZO) dO Integration of this expression leads to the result that M= rpoUZeiw*[.i-)ca. -(i-Z&)c, ,ec.1 ocs ]. In terms of the downwash velocity, the moment becomes M = Ip a -A)(-z A + -(:k-Z.)~ A [A - A-5 - 1-0 Z +0* -4 ,Z6(AZ-Aj-A3)1 + '(3.30 10) (3.3, 11) (3.3, 12) [A, +A, (A- 2Az. + A2 ~A4. +2&.(Al*A.Oz-N.A4)1 For the special case of a flat plate, the moment about the center of the plate reduces to M -U ZeiW*t (Ao , + .,,) L -Az A% A-A,_ (3.3, 13) This result is in agreement with that obtained by Siekmann [-8J7. 3.4 Calculation of the Thrust and Dray 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 x ~ ~ ~ R e ,) - - (3.4, 1) JdO 00 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: 3 (3.4, 2) where 2 l Ra do (3.4, 3) 0d 20Tr fde Je. 0 (3.4, 4) and 3 is the suction force. The force f above represents the streemwise force computed by integrating the pressure distribution .along a path defined by the stretched-straight configuration of the fish, whereas the force F2 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: + i e I, n An An +jAh - I 13 6 4 i M n - - I - 0 r, e4 -t= (C r. C jr ) e "" 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) ,L 4 .- eiUwt Employing equations (3.4, 5a) and (3.2, 34), it follows that R iT(0,10 = p U2'[ 1 a tn~a + Zj ai sinflnJ0] (3.4, 6) Also, combining equations (3.4, 5d) and (2.4, 21) it can be shown that e , = - rn, (3.4, 7) dO n- j This last expression can be written in a more convenient form as nsO ( +'. Z CncosnO) sinre (3.4, 8) sine$ 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 00 -pU If (a.'n C a. ansinvn)(coO -cosZ1) (3.4, 9) 0 From the integral f Sinn& cosMGe =0 the second term in equation (3.4, 9) vanishes. To evaluate the contribution to due to the term -ep Uao r,'o e -(cos O-cosZO)dO the Cauchy principle value of the integral must be taken, i.e., =X1tm -,Cpua:[fci- C (cosa -cos2)* + ank(cos-cos20) dO (3.4, 10) Employing the identity +Qne (cosO-cosZe)= -Ztc rv +3siO- 5,nz2 this expression becomes F 11 -PU U44Z +~ 3sinO-5irnZ8de f(zianvi +351no - SI in*z)db] (3.4, 10a) ng 0 on tohez r Integration of this equation leads to th",e result that FX z ti!Ej.A. Coitlog 1 4--w I CS(O - C5Q-I~a1 (3.4, lob) 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 2 I -e~~u f=44t+Z ~s, mn)(-zS. 'lB'5,no)d& 0 i 0 =-U at,-9EC stnM A Z ', siO Employing the integrals f 0qne stnndi) (_,,n 1 zir hoI 0 I stmmOsrnOdO= 0 (3.4, 11) V1 rn z n mrn the force becomes -tZrrPU1- rv , (-1) ~(-zn6#,) - Z. A "e n ] o(.4, 12) But, according to equation (3,1, 24) -a -i~ n~ C' (3.4, 13) Inserting this expression into equation (3.4, 12) yields --2 (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 [lIj to a small circle of radius 6. surrounding the nose. The velocity at the leading edge can be computed from the complex velocity potential by equation (2.1, 13) as ,. "(3.4, 15) where is the complex velocity in the r-plane. Since W^ is bounded as -the leading edge is approached, equation (3.4, 15) can be written as hrm w(r,*) = Z(-1,t hm d _4W -1 de i (3.4, 15a) But, from equation (2.1, 10) it follows that (fE)2 - (j~)L 3'- E 2. (3.4, 16) to z yields . Differentiating this expression with -r - I t z --r (3.4, 17) lam -146 "'-.-i d ___- _________- e (3.4, 1 7a) From these results it is seen that the velocity in the physical plane asymptotically approaches an infinite value as im w(ilt) : '(-1,{) nm w W^ . ii Z'-o 0z-,,o0" (3.4, 18) where (3.4, 18a) + - and as respect and 4. According to Milne-Thomson [iJ the extension to Blasius's formula for the case of unsteady flow can be written as (3.4, 19) F3 - i F 1 p wl~dz - -dJ O -qeI Here the contour A. is taken to be a small circle around the leading edge with a radius 6, ï¿½ Since the velocity potential j and is 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% - 'if W i I.Z- (3.4, 20) A small circle around the nose can be written in complex notation as z'= Soe;y (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 211 F F - ~PLI-G-) tW(-1,t)1(342) d;E = da' = S. ; e;W 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(-Itl in the. r-plane must be found from the complex velocity potential Fo F1 +F . This can be done by considering separately the contribution to this velocity by each potential function and adding the results. Thus, from equation (2.3, 6) w.(1%)-F(-,)- 0 (3.4, 23) From equation (2.4, 38) it follows that Ue' 0() al9 (3.4, 24) Substituting the trigonometric series for 4(i9) 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 0 - Z- Z.Pn~nO I is'~coOOad (3.4, 2 5) Employing the integrals 21r i---osnnd d 0 nsl and alt i. Si nt nn3sinD 1 s0Tr. 1 +cost equation (3.4, 24) becomes (3.4, 25a) WIAl,) ueI Z. (-1) *1 Pn Combining the recurrence relation given by equation (3.1, 19) with equation (3.4, 25a), there results (-1,t) =- U ei'' (Ao-A,) (3.4, 26) / The velocity_ due to -F, can be found from equation (2.6, 21) to be W2 (At) r y 2i P~i;Of ' VQ. (3.4, 27) Inserting the expression for P(iki) 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 , Qf 0 e-iAkkC5 h*X (coshKY - 1) dc -7jAk Cos hX (cosh v1)d- (3.4, 28) Recalling the definition of the Bessel function of the second kind, equation (3.4, 28) becomes 07 (- 1'il = KItiAk-K(iAk)1 K,(Ak) + Ko1Ak 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"'(-1.1-i) -Z U e~ WA 0 A+1.j :(Ali) -A 1 (3.4, 31) It then follows from equations (3.4, 22) and (3.4, 31) that the suction force becomes S-Z pU (3.4, 32) where (~'0+g1W-C1k) - ~.~,(Akc1 A' Finally, by adding the results given in equations (3.4, 14) and (3.4, 32) the net thrust becomes T~~ZirpU1 G4oo ,~ Z~ln +(i1?8 n r. (3.4, 33) Equation (3.4, 33) can be expressed in still another convenient form for computational purposes by eliminating the F 's. Substituting 1% wz, MAI 1 U e4-+ (A,* A,) I. Z Y. (A 1,0 -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 ',=fN -" (3.4, 34a) (3.4, 34b) where . -( At, + A, )LFRAI AO+K- (A0 iP'(k) - A - P - - (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~6 Ml t4k n nXI. +e(~B; -~:, ~~]+ - e k n . , + dokA '-,-<,,(<-- ,, - ek +l- ( < - L+e.l (3.4, 35) C f -1 - C,, A ' - z t'% P. ?. 6[F. - -, 01 L [ 61 -JI _.!: Zn k + + Z'I'n-zl - En4z Zn Substituting equations (3.2, 35) and (3.4, 35) into equation (3.4, 33) yields ,T=Z.z60 44k nE'% * - k~pu( ( + do'C-B~ '~ +kZ 6B 4 B-- + 00 MI~: For the case of an infinitely thin plate ( 6 0), it follows from equation (3.4, 36) that the thrust reduces to T r P U[ + )+ KB' + 4 .3 According to equation (3.4, 34b) for the case in which e vanishes, This result is in agreement with Siekmann [-8] if it is observed that n*i * B, = -1)" 16, and nZ(1 Zn where the I,'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, To 20 The time average value of an arbitrary function of time (I(t) is defined as 0 (3.5, 1) Thus Ai~~fCos?,Wia j 0 iT f S~U~ rOtat= (3. 5, 2a.) (3.5, 2b) (3. 5, 2 c) 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 ~> cos wt sin wt - nI and similar formulae for the other coefficients. (3.5, 3) Thus, from equations (3.5, 1) thru (3.5, 3) it follows that TO 0 F. I,' dt If d -i Ir 'C'd-t r o 0 o 0 0 C-.' d i 0 0 0 o Ef 4Cdt 0 TO Ar 'ro a' Cl d I z 4 -+ (r"C - i ' + c') ( C O, +c%; 0 C; -- ' + 0 0 z + 1 di B' 52BB (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, 4i) (3.5, 4n) E-k . nzj'=. " n %+ 0 ~( Cl-.1C Cz Cn'l (C' CIM,.i (c,,. cn,%)t + cr'%4,(Cn,,. - ,) -f ,Z (C. - .1 . (3.5, 4q) Substituting these results into equation (3.4, 36), it follows that the time average value of the thrust becomes 7 & -C( T r-rpU= (r'; -)., . * -"'C~r +k2 ( BB:' + , B -e1(rj - 2 A:-2A;)(.c -Ci) +2C+aig -2A)(C '-C', 4 CC,_' , C" _ C . " Z ,2.+ -3 ( BO B + B.'Br - o-Biz -' + r(fl B50 --- ( + . and 0 (3.5, 4o) (3.5, 4p) (3.5, 5) -where (3.5, 5a) -s B . B1 t, n= 4n I- (C'n. -C'n41l C'1%-21 (CI- - * ( G, C-C1) - c2 (C: " - Ch 1 Y ('o' A'1),(A k) - (A" + A 3.) ,t (Ak) -Al ~ (AoA .F(Ak) + (AO +A') NtAk) -A' Following Siekmann L 8_/ the thrust coefficient is defined as C T , (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 [8 for a very thin plate, a displacement function with a quadratically varying amplitude is used. Thus, consider the displacement function D( = I h(x'-E (3.6, 1) where (d -+d3.x +d, 2 )e-iLX e j' ï¿½ (3.6, la) Here the phase angle o 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*(O)= {do+d4I(I-4-)cose 6 *dZ[(-6)co5O* i] l e)o (3.6, 3) By employing the relations and co2 c (i + cosZ6) Z 90 equation (3.6, 3) can be written in the form ++ + (1- )(d .,eda(cos - sLn ) cosO + C14I-Gf (co Lc-jsr~ ~ ~ ec (.3.6, 4), From this equation the Fourier coefficients Bn can be computed by the relation = f H .(0) cosne de. (3.6, 5) 0 Employing the integral [237 iT jn(X ) = - e"xï¿½O colesn dO 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 il(I~n(d.kd +(--- d)(cs?_.- + z + ï¿½ e + (Ie(d e0)5 -~sinJ z + (jjn-Z (,),-2[(1-e)2dl .(cos2 -' sin -)] J n-?01) (3.6, 6) -1 z where and Cosne cosle = - [cos(m+n)e + cos(m-nI ] have been used. Using the following recurrence relations [-24J between the Bessel functions, J (H) + J"I(x) = n- jn(,H) 2n = J() 2 in J (X) i XW and Jn{Z(X +Jn.t(X) equation (3.6, 6) simplifies to B+C1-E) T}(cosT - sin ) n.(id * - )(i-&K4+e d)(C-OS .c -x n (I _g - (3 J,7) (-)i) = (- 1) n (3.6, 7) |

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PAGE 1 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 COUNOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA August, 1963 PAGE 2 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 encourageâ€¢ 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 ed~ting and typing the rough drafts and the final manuscript of this dissertation. ii PAGE 3 ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES .,IST OF SYMBOLS TABLE OF CX>NTENTS . . . . . . '. . . . . . . . . . . . . . . . Chapter I. INTRODUCTION TO THE PROBLm.1 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 .Page ii V vi vii 1 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 11 Downwash 11 Velocity for .A:ny 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 PAGE 4 Chapter IV. SUMMARY AND CONCLUSIONS APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . LIST OF REFERENCES BIOGRAPHIOA~ SKETCH t t t e I t 1â€¢ e t e e e t t e e t e t e e iv Page -104 107 121 124 PAGE 5 LIST OF TABLES Table Bessel Functions Real Part of the Theodorsen Function. Page 98 100 3. Imaginary Part of the Theodcrsen Function 101 4. Thrust Coefficient CT 103 5. Coordinates of the Base Profile 112 6. Real Part of Bn Coefficients 113 7. Imaginary Part of. Bn Coef'f'icients 114 10. Real Part of On Coefficients Imaginary Part of On Coefficients Real Part of Ao Coefficients 115 116 117 11. Imaginary Part of' Ao Coefficients 118 12. Real Part of A 1 Coefficients 119 13~ Imaginary Part of A 1 Coefficients 120 V PAGE 6 LIST OF FIGURES Figure Page l. 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. 5. Profile Plane (z-plane) .6. Circle Plane ( 'f -plane) 7. Circle Plane with Point Source a. Circle Plane with Vortex Pair ,,.-. Stretched-Straight Configuration with Pressure 13 15 15 31 38 Distribution 50 10. 11. Stretched-Straight Configuration for Several Thickness Parameters Thrust Coefficient Versus Reduced Frequency 55 102 PAGE 7 Symbol h(x,t) V I' t p Z X + iy u V w .. u + iv LIST OF sn{BOLS Description Displacement function Velocity vector of fluid particle Mass density Time Pressure Complex velocity potential Complex 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 they-component of velocity Complex velocity in the profile plane Complex velocity in the circle plane Mapping function of circle into a profile . Thickness parameter vii PAGE 8 Symbol Gti i -v-r j .. r-i 1T (8,t) r 'll" B . , n D(z,t) ~(8) 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 unit 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 11 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 PAGE 9 Symbol u k ,..w/U T( 8, jk; f: ) M T Q P( jk; '= ) ~(z) a T Description Tangential velocity of fluid at surface of the circle Velocity normal to the profile surface Velocity tangent to the profile surface Free stream veloc~ty 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 ix PAGE 10 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 1 PAGE 11 2 chord, as sho'wn 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 \ (Ll) where Yu and YL are 'the ordinates of the upper and lower surfaces of the plate respectively, and X 1 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 PAGE 12 " u __.., u 3 y Figure 1 Stretched-Straight Configuration of the Fish y â€¢'fu(X 1 t) 'lrn=h{X,t) XL YLtlC~ XT Figure 2 Displaced Configuration of the Fish X X PAGE 13 4 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). l.l Method of AEEroach 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 tho vortex potential the pressure distribution on the base profile is computed by employing the unsteady Bernoulli equation ["1J*. With the pressure distribution known, the hydrodynamic forces acting on the plate are computed. *Numbers in brackets denote entries in the List of References. PAGE 14 5 The thickness enters the problem through the mapping function in the form of a small thiclmess 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. l.2 R~view of Related Liter~ture 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 1ast 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 J 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 sJ investi gated the swimming of long animals sue~ 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 J discussed the hydrodynamics and propulsive properties when a jet of fluid is ejected from the ope~ing of a tube. He provides PAGE 15 6 calculations for the thrust and the basic equations for the horizontal rectilinear motion of a rigid torpedo-like body. Siekmann 1 s result may ~e app~ied to investigate in an elementary way the locomotion of certain aquatic animals belonging to the class of cephalopods, particu larly squids, octopuset, and cuttlefish. Lighthill "sJ discussed briefly . the swimming of slender fish . at the Forty-eighth Wilbur Wright Memorial Lecture. In a later publication "6J 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 L-7J 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 f"aJ 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 "9J and to Kussner and Schwarz J in which the thin plate and its wake is replaced by a vortex distribution of fluctuating strength. Siekmann 1 s formulation of the problem led to an integral PAGE 16 7 equation for determining the vortex distribution. Of special significance in this study was the calculation of the thrust produced. Wu J, in an independent study, considered essentially the same problem as Siekmann [:aJ. Wu employed Prantl's accelera tion potential to deten:iine the forces acting on the plate. In a subsequent paper ["12J he investigated the two-dimensional potenti~l 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 [:aJ and Wu L-llJ are in agreement. At about the same time, Smith and Stone L-13J 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 [:aJ and Wu .1J. Recently, Pao and Siekmann J considered the Smith-Stone problem and included the effect of the wake. Their results are in agreement with those of Siekmann :aJ and Wu [:nJ. Bonthron and Fejer L-15J 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 PAGE 17 8 oscillations were imposed upon the plates. They employed Theodorsen 1 s theory [:1sJ for a system of finite degrees of freedom and solved the dynamic equilibrium equations. Kelly J measured experimentally the propulsive force produc e d in an undulating, thin, two-dimensional plate and found that the theory given by Siekmann [:aJ and Wu J was in agreement with . experimental evidence when allowance was made for skin friction. PAGE 18 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 L-laJ. Recently Kussner and Gorup L-19J 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 translatory movement. The problem was linearized by assuming small unsteady perturbations about the mean position of the base profile. Theodorsen ["1sJ treated the problem of infinitesimal unsteady oscillatory perturbations of an infinitely thin airfoil immersed in a 9 PAGE 19 10 uniform stream. He was interested in the aerodynamic instability of the airfoil and the mechanism of flutter. He divided his solution into two ~arts. 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 ntream 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 1 s work L-16J to include the effect of thickness and the effect of an arbitrary flexible motion. Also, it is an extension of Kussner and Gorup 1 s theory L-19J to include the effect of flexible displacements and a calculation of net thrust or drag caused by the undulatory motion. PAGE 20 11 2.1 Mathematical Formulation The equations governing the two-dimensional motion of an. incompressible, inviscid fluid are the following: Continuity equation: where V=-[u.)v] is the velocity vector of a fluid particle. Motion equation: (2,l, l) where pis 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 -= O (2.1, 3) Then, in these regions there exists a scalar point function ~\X,y>t} defined by the equation (2.1, 4) where ~(x.y 1 t) is the velocity potential. Substituting equation (2.1, 4) into equation (2.1, l) leads to the Laplace equation (2.1, 5) PAGE 21 12 Also, from equation (2.1, l) a stream function '' (x,yJt} defined as V --~ti_ ox oy where u.-:. 2.i and V = 0 d! a.re defined by equation (2.1, 4). 0 :, J can be (2.1, 6) Equations (2.1, 6) a.re the familiar Cauchy-Riemann differen tial equations defining an analytic function F (i, t): ~(x,y,t} + i 'l' (x, y, t} of the complex variable . .c. : x + i 'J From equation (2.1, 6) the complex velocity; w =lA+I V , can be found from the complex velocity potential F(z,t) for any time t, as (2.1, 7) where the bar denotes the complex conjugate. Therefore, F(i,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 transâ€¢ formation of coordinates. Lot ::-f ( t) be a conformal transformation of the exterior of' a uni circle in the t -:: s + i l'\ plane to the exterior of' the profile in the c "' x + i y 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., lim W(l,t): \im w({,-t) (2.1,:a) i:-co r-00 PAGE 22 iii> u __ ,. .bL 2. 13 y Figure 3 General Profile Configuration Figure 4 Circle Plane for General Profile Configuration X PAGE 23 14 Quan ti ties in the -plane are denoted by a circumflex. The coefficient 1/2 in equation (2.1, 8) is selected such th~t the chord of the base profile will be approximately two units. ' Such a function can be developed in a Laurent series as 00 l=Ht)-=-[ ans-" (2.1, 9) n-=-1 where the o.n' 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 ..:E as (2.1, 10) where .: is a small positive quantity. The thickness d of the fish at its mid-chord is approximately 2 : Since the length JI. of the fish is approximately two, the thickness ratio order .~ at the mid-chord is of the Clearly, this function satisfies the requirements of a sharp trailing edge since (2.1, 11) The two planes are shown in Figures 5 and 6 for the stretched straight configuration. PAGE 24 15 y ' --f" u ---!> X -9" Figure 5 Profile Plane ( i!. -plane) ... !!. Po 2 Figure 6 Circle Plane ( s -plane) PAGE 25 16 Since the potential function ~{z>t) is a point function, it is invariant uncor the transformation (2.1, 9), i.e., The velocities of a particle in the two planes are related by .. (2.1, 13) Let the location of the downstream stagnation point P 0 in the { -plane be given by (2.1, 14) where 't' is the argument of the line OP 0 Any point Pon the profile in the z-plane is mapped into the .... point Pin the { -plane and is given by (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: (2.1, 16) where F 0 is the potential of the uniform flow around the stretched straight plate, F 1 is the contribution due to the unsteady motion of PAGE 26 17 the profile, and F 2 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 determino the thrust, lift, o.nd 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 U."lsteady Bernoulli equation f1J, which reads (2.2, 1) The arbitrary function g{t} can be determined at upstream infinity since it is assumed that the disturbance due to the unsteady perturbations vanishes there. Thus, :. i uz + 2 p r (2.2, 2) Therefore, equation (2.2, 1) becomes (2.2, 3) where (2.2, 4) PAGE 27 18 Obviously, the velocity potential ~(x,y,i:.) on the base profile surface is (2.2, 5) where &'tj .denotes the !real part of 1 operator for the space imaginary unit i. Let oF _ , ,. . , , "\ 0 x Uo + ll + U. -+ I \ V 0 + V + V 1 0 t=i I , ;;._.t =u.+ IV c)? (2.2, 6) and (u.", 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, + 2 ( u'u." + v'v''J + u! 2 + 2 + u''z. + v" z. (2.2, .7) or, in termo of the complex potentials, (2.2, 8) PAGE 28 19 Substituting equations (2.2, 4) thru (2.2, 8) into equation (2.2, 3) gives = !M + -' &, {!t [ F, <â€¢. t) + F, <â€¢. t)]} + _ !_ _ IR,, {d Fo [~ + d \\ ... ]} 2 o'l di die. oi or. + (2.2, 9) -m.i{cr1 0F2} _ [aF, ~fz.] at... dz Z O'l o:: ch oz Under the small unoteady perturbation theory the la.st two tc:::ns 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 Pon the bace profile as (2.2, 10) or (2.2, 11) or, in terms of the argument 9, (2.2, 12) PAGE 29 20 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 !mown, it is only necessary to ce.lcule.te the potential for a flow around the circle and to transform this flow field to the profile. Generally, a circulation r 0 will be present around the profile and hence around the circle. Therefore, from any text in fluid mechanics J employing complex variable theory, the potential can be written as (2.3, 1) The complex velocity in the~ -plane is ro -1--::::,L'f( '{ (2.3, 2) and on the circle the velocity becomes (2.3, 3) Since the velocity vanishes at ~=ei't , the circulation \o can be found to be ro: -2.1'1' U Sin 'C ( 2.3, 4) For a sharp trailing edge the Kutta condition requires that the flow leave the profile smoot.~ly. Therefore, to satisfy this condition the PAGE 30 21 downstream stagnation point must be located at the sharp tail. For the symmetric base profile, as shown in Figure 5, the sharp tail A coincides with the x-axis and the do~mstream stagnation point P 0 in the s -plane is located on the axis. Thus, for this case the . circulation ro vanishes. From the above results, the velocity potential of the base flow becomes (2.3, 5) The base flow velocity is tangent to the circle and is dF;,I -A ie 0 80 -:: I LJ S In 0 e . dt {-:e' 0 L (2.3, 6) .... The veiocity of a particle on the streamline emanating from P 0 is given by (2.3, 7) The velocity of the same particle in the profile plane is (2.3, 8) PAGE 31 22 2.4 Veloci t~r Potential Sn.tic.fying the Unsteady Perturbations of the Profile Surface Any very small displacement of a point Pon the profile in a direction normal to the boundary at this point will cause a small displacement of the corresponding point Pon the circle in a radial direction. The relationship between corresponding displacer:ients in the two planes is given by (2.4, 1) Let the unsteady displacement of a point Pon the profile be D(z,t). Then t.~e position of point Pct any time tis (2.4, 2) is the coordinate of the point on the stretched-straight con:Cigura tion. 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 t.e 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 Pis given by the m!l.tcrinl derivative of zp with respect to time, i.e., (2.4, 3) d::: where _::.. is the complex velocity vector of a fluid particle at this dt point. PAGE 32 23 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 (2.4, 4) Next, consider the velocity of point Pin a direction normal to the surface. The point Pin the z-plane maps conformally into the A point Pin the X -plane. The tangent to the circle at point P maps into the tangent to the profile at point P, and the normal to the A circle maps into the corresponding normal on the profile. Point Pis located at 5= e ie and the unit tangent vector is A ds = , eie d& (2.4, 5) Since Pon the circle is located by specifying the argument e, 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 derivative of z with respect to 6. Tne complex unit tangent vector t. is given by (2.4, 6) PAGE 33 24 The complex unit normal vector n in an outward direction is 90 clockwise to the above complex unit tangent vector, i.e., n -=it (2.4, 1) The velocity of point Pon the profile in the above normal direction is simply the oca.lar product of d l?p and Therefore dt '' (2.4, 8) In the J-pla.ne the corresponding velocity is normal to the circle and is given by (2.4, 9) Substituting equations (2.4, 7) and (2o4, 8) into equation (2.4, 9), the velocity in the radial direction on the circle is found to be (2.4, 10) or " (Si:)_ If\.[ dr ~J Lr J \f'L.1 I de d-1: (2.4, 11) or (2.4, 12) PAGE 34 25 In a similar manner, the tangential velocity on the circle due to this perturbation is found to be (2.4, 13) The u.~dulatory 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 (2.4, 14) Further, it is assumed that the propulsion is generated by a train of waves progressing astern with an ar.ipli tude depending on the spatial chord variable x. From photograph::; made of swllll'lling fishes L-20J it seeos 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 (2.4, 15) where Ol is the wave number, w is the circular frequency (which is taken to be positive throughout this work), /j. 0 is an arbitrary phase angle, and H(x) is the arbitrary amplitude of the wave motion. It PAGE 35 26 is convenient to write this motion in the general form (2.4., 16) where d-=-Ft is the imaginary unit f'or the time variable t. It is not to be confused with the spatial i~ginary unit i. Eventually, the real part in the time imag:i.nary unit nust be taken for physical inter pretation. From equation (2.1, 10) .:,he x and y coordinates of point P on the base profile are given by X-= [c.os fJ + I: cn-l (1-E.)2n cos n6] h'l (2.4, 17) and (2.4., 18) Since X is a."'!. even function of 0 ., any function of X is even in C1 Hence (2.4., 19) and (2.4., 20) PAGE 36 27 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 S -plane resulting fron the undulatory motion is Equation (2.4, 23) can be written in a compact form as (2.4, 24) where (2.4, 25) PAGE 37 28 But, from equation (2.1, 10), evaluated for = f (ei 8 ) (2.4, 26) and d i i e f '( i e; --=1e e de (2.4, 27) Also, from equation (2.3, 5), evaluated for (2.4, 28) Substituting these last expressions into equation (2.4, 25) yields G "(e)--1. [-i6r'{-iB1,kH*(6)isinedH:(e}] m 1 e Te 1j f'(e) de (2.4, 29) 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 ~ore convenient form as (2.4, 30) PAGE 38 29 For a Sym!!l.etrical base profile, as is considered here, it can be sh01m that the function G( e ) is an odd function. Therefore, it io convenient to write ' ,. co Gi(S)-=2. L Pt\ sin ne (2.4, 31) n:::1 where the Fourier coefficients are " G,(6} s1nn6d8 (2.4, 31a) Note that, according to equations (2.4, 4) a.~d (2.4, 30) the coefficients Pn, like B, depend upon the mapping function and, as n suc.1,., contain the thickness pars.meter ~~.. Tl:le relationships betvreen the P 1 s and B 1 s are found fro~ the solution of equation (2.4, 30). n n Due to the complex nature of this equation, the recurrence relations bcti:een the Pn I s and Bn I c will not be attempted until a '..lpecific example is presented in Chapter III of this study. It is to be noted also th~t the coefficients P are comnlex in the time imaginary unit 3. . n Fror:i. equation (2.4, 13) ::. .. :.d. equr..tion {2.4, 22), the tangential velocity resulting from the undulatory motion can be found in a oimilar ma.~ner as the above radial velocity to be (2.4, 32) where (2.4., 33) PAGE 39 30 Fo~ profiles of sms.11 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-16J., the boundary condition for the normal velocity given by equation (2.4, 24) co.n be satisfied by a distribution of sources along the circle in the S -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(e,t) This source distribution can be rela. ted, in "'n obvious way. to the normal velocity o (a J.\ by .__. Lr vi .. , Q PAGE 40 31 By employing equation (2.4, 24) the source potential becomes (2.4, 38) 0 -,u Y. 5 . Figure 7 Circle Plane with Point Source PAGE 41 32 F1 (r, t) evaluated on the circle is given by 2'il" But Hence, l', PAGE 42 33 Em?loying equation (2.4, 24), this tangential velocity becomes t e-'19 d co 19-. 2 (2.4, 45) Tho integral in equation (2.4, 45) io a function of tho ~nglo _6 , the so-called conjugate function. According to Robinson and Laurmann "21J this function is defined as 2'i1" p. 1J .... e-19 ~(8)= l-n' C".:f('\9-)c.ot-y-d~ (2.4, 46) 0 The integral on the right-hand side is singular at i$2:. e and it is understood that the Cauchy principal value mu~t be taken, i.e., the J ;z'lf. integral is to be interpreted as 0 i T( J (2.4, 47) 0 From equation (2.4, 31) and the definition of the conjugate function it is readily seen that a, & (6) :. L Pn C.05 ne (2.4, 48) n =-1 The circulation around the circle due to the source potential is 211' r = J c\o (19-,;t) dl9(2.4, 49) 0 PAGE 43 34 Substituting equations (2.4, 45), (2.4, 46), and (2.4, 48) ,. into equation (2.4, 49) it is observed that r 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 Performing the indicated differentiation leads to 2'1f 1f 1 (e,t) : &; P { u f": ;"'tf G (1'-l Io~ [ ei "] 2 d ,a. + fo' (ei 0 ) u e-it.>tr,.. 0 [ 2 ] } + \fle'oJl2. G(lSl-) eif>_eii3l d~ (2.5, 2) Substituting equation (2.-3, 6) into equation (2.5, 2) and observing that the 6ti operator and the integral operator are commutative in the above equation, the source pressure distribution becomes (2.5, 3) where PAGE 44 and Also, 35 _ But, from equation (2.4, 40) [l e-~ ]2 S\t"\2 Hence, s,n('!J-e) 1 cos ('19--e) t i,_ e Co -2 (2.5, 4)-, (2. 5, 5) (2.5, 6) Substituting equations (2.5, 4) thru (2.5, 6) into equation (2.5, 3) gives 2'1( 11,(e,tl = P ~â€¢;;"''! G(ii) {.,-10~( 2 s, n 8 /]2 + 0 (2.5, 7) PAGE 45 36 Equation (2.5, 7) can be written in the convenient form + J; (,9) [ crlo'i 2 ( 1cos (8-1')) + ;'.;~JI' cot 19 z 0 ] di?} 'IT' (2.5, 8) where the following substitution has been made. [ 8 vi ] 2 ,1 2 s1nT = 2.[1 -cos(6-t1)J Changing the variable of integration from 1' to rJ in the second integral of equation (2.5, 8) yields 2'i1 f & (-Jl [ cr lo PAGE 46 37 The integrand here is of period 2n. Thus, the limits can be changed from -2n to O and -ti ton. Using this fact, equation (2.5, 8) becomes 7f 1T l6 t) 't;, _ P U 2 e 3 "'+. f f~(t>} [ a\o~ 1-cos (0 1 ' i,r l . -1-c:.o~(B+-0) 0 + (2.5, 9) 2.6 Circulatorv 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 1 s theory sJ, 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 J 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 L-22J 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 stream.line emanating from the downstream stagnation point. Finally, Kutta 1 s condition is satisfied by requiring that the tangential velocity at the downstream stagnation point induced by this vortex distribution PAGE 47 38 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 L-l6J, the first two requirements are . fulfilled by a distribution of point vortex pairs of equal strength, one of strength on the trailing edge streamline, and the other of strength -Ar'-=-Ywd~ 0 , located 1 at the image point .r= 'fo . 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 F 2 ( ( , t). Then the contribution to the total potential due to the vortex pair located as shown in Figure 8 is LlF 2 = l Ar \oar-~ 2-n J r-so (2.6, 1) ... where 6.r is the circulation strength in the <-plane. u 2 ,. f._j Ar Figure 8 Circle Plane with Vortex Pair PAGE 48 39 'i'ne total velocity potential due to the vortex distribution therefore becomes (2.6, 2) where it is assumed that the unsteady motion has been going on for an infinite time. According to Wagner L23J, the circulation distribution in the z-plane is related to the circulation distribution in the \-plane by (2.6, 2a) and from the mapping function, equation (2.1, 10), it follows that Employing these results the circulatory velocity potential function can be written in an alternate form as (2.6, 2b) where <>w (\ 0 J t) is the vortex distribution in the physical plane. In accordance with Helmholtz's law L-22J of constancy of vortex strength following a fluid particle, the equation governing the vortex distribution in the physical plane is (2.6, 3) following a fluid particle. PAGE 49 40 Therefore, the substantial derivative is (2.6, 4) Howevor, since the unsteady perturbations are harmonic in time the vortex distribution can be written as (2.6, 5) Comb~ning equations (2.6, 4) and (2.6, 5) yields (2.6, 6) where the substitution (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 (2.6, 8) Integration of this expression gives (2.6, 9) PAGE 50 41 where. 'to is the strength of the vortex element located at X 0 {t 0 ) in the physical plane. Obviously, ~o is the vortex strength when this part~cular vortex element was shed from the tail. From equations (2.1, 10) and (2.3, 5), I (2.6, 10) where is the reduced frequency as defined above and (2.6., 11) Combining equations (2.6, 10) and (2.6, 5) with equation (2.6, 2b), the circulatory potential function becomes (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 s= 1 . This restriction implies that (2.6, 13) But, from equation (2.4, 38) (2.6, 14) 0 PAGE 51 42 A Substituting the trigonometric series for G(~) 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 , (2.6, 1~) Employing the integrals -=0 and San rJ l cos \?d \9 = 1 (n:123-) , ' ' . the velocity at the rear stagnation point in the r -plane due to the non-circulatory potential is (2.6, 16) Now the velocity induced at the downstream stagnation point by the circulatory potential is investigated. From equation (2.6, 12), it follows that (2.6, 17) PAGE 52 43 Combining equations (~.6, 16) and (2.6, 17) with equation (2.6, 13) yields iQ . ' (2.6, 18) where c,:, Q -= U e1wt'[ 2. Pn l\'S\ Therefore, from these results the constant '( 0 is found to be (2.6, 19) where (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 (2.6, 21) PAGE 53 44 Recall that~ can be determined from the boundary conditions, i.e., it depends upon the flapping function, base flow velocity, and the mapping function; and that P(jk; ~) depends upon the mapping function and the reduced frequency. These quantities are assumed to be known. Thus, the total complex velocity potential function (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 ~TT 2 due to a single vortex pair of equal strength located at \: ~o and ! in. the t -plane or the corresponding points in !)o, the z-plane. From equation (2.6, 1) the complex velocity potential of this pair is (2.7, 1) According to Helmholtz's law J of persistence of . 4 vorticity following a fluid particle, the strength Ar 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, (2.7, 2) PAGE 54 45 where x 0 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 where is the local velocity of the vortex element ~r .::treamline in the profile plane. (2.7, 4) along the wake Substituting equation (2.7, 1) into equation (2.7, 3), the unsteady pressure distribution resulting from this vortex pair is are real in the space imaginary variables. Also, from equation (2.3, 5) which is real, and (2.7, 5) PAGE 55 46 Er:lploying these last results and carrying out the indicated differentiation, equation (2.7, 5) can be written as To determine the effect of the entire wake vortex sheet, the ,. vortex element Ar is replaced by (2.7, 7) where ~ denotes the circulation distribution per unit length in the r -plane. Substituting equation (2.7, 7) into equation (2.7, 6) and integrating over the entire wake, the time dependent circulatory pressure distribution becomes (2.7, 8) Combining equations (2.6, 2a) and (2.6, 9) with equation (2.7, 8), it follows that 2s1n e + (2.7, 9) PAGE 56 47 Eliminating the constant 'to by way of equation (2.6, 19), the unsteady circulatory pressure distribution becomes where c:o (2. 7, 10) fr[ F~ttl 2 Sin 6 j l \i'( ~o\l' ~! 1-l-f~ocose U s1na(t!-1) _7 , Jo -ik~"n. + \f'(ei0}\2.(~:+1-2tocos0~f(-.") ClSo Tm 1 \N) = 1 --------------------, I o:, (2.7, 11) f l With the previous results the total unsteady pressure distribu tio~ 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, Combining equations (2.5, 9), (2.7, 10), (2.7, 11), and (2.7, 12) it follows .that Tr Tr < 0/cl ::. U t -iw1 [ P e ~(1?) alo'i 1-cos (0-~) 2.'IT 1-cosl8-\-'9-) 0 + (2.7, 12) (2.7, 13) PAGE 57 48 Substituting the trigonometric series for ~(19) given by equation (2.4, 31) and employing the so-~alled Glauert i~tegral ["21J and 'Tl' ~1 0 -r( s,n\9 s,nn19Co$ 6 c.os 1.J \ O O 1 COS ( e -~) "' J 1. c..o s t e + '19l 0 into equation (2.7, 13), there results C.o'S ne 2. su, ne n (2.7, 14) Finally, inserting the value of Q from equation (2.6, 18) into the above equation the time dependent pressure distribution becomes QO 1T{e,t) = 2fUZe-i"'-tI: Pn[ os~ne h-::. l. (2.7, 15) PAGE 58 49 For the special case of a. flat plate of infinitesimal thickness, the parameter vanishes and equation (2.7, _ 15) reduces to (see Appendix A) -t-C.oi:~ [co-1:e + i-c.ose J::.(l<>J} ~(19)d'\9 2 ~,ne (2. 7, 16) W ith some manipulations this can be expressed in identical form with that given by Kussner and Schwarz f:10J, Schwarz L-9J, and Siekmrum L-SJ. The function J::(t<) is the so-called Theodor sen function ["1sJ defined by .f"(k) + j ~( k) (2.7, 17) where H~ 2 >(k) and H~Z)(k) a.re Hankel functions of the second kind of order zero and one, respectively. 2.8 Lift and M oment 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 denot e d by (2.8, 1) PAGE 59 50 d::. r-Tid~. iTdx 0 Figure 9 Stretched-Straight Configuration with Pressure Distribution )C Referring to Figure 9 it is seen that the forces acting on a small arc element ds are (2.8, 2) These are the forces exerted by the fluid on the.profile. The forces Fx 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 (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 : IR.t J TT d ~o (C.) (2.e, 4) PAGE 60 51 But (2.8, 5) and 2'Ii" L &, J rr ~de (2.8, 6) 0 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 as is dM :1T(xdx-+~d~) (2.8, 7) taken in a counterclockwise sense (nose down). Equation (2.8, 7) can be written as (2.8, 8) The total moment is found to be 2'1f M = tt,J TI -f cei 0 } ~! de (2.8, 9) 0 The above equations for the lift and moment are essentially those developed in the Blasius theorem L-lJ. 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 or the base profile. PAGE 61 52 2.9 Thrust Formulation In order to calculate ~~ex-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 ma.de by Siekmann L-8J, Hu ["11J, Smith and Stone L-l3J, and recently by Pao and Siekma.nn J for the infinitesimally thin plate. The important fact here is ~~at nonlinear terms are involved and, as such, ~ere 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 di. 0 in that equation is replaced by th.e exact differential element di , where (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 -11:"e .0 -:iw-1: df-l~(OJJ de} 3 de (2.9, 2) where G.j is the 1 real part of 1 operator for the time imaginary unit j. A positive Fx will indicate a net drag. PAGE 62 CHAPTER III APPLICATION OF THE THEORY TO A SYMMETRIC JOUKOWSKI BASE PROFILE WITH A LINEARIZED THICKNESS PARA.\fETER 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 proble~ was linearized by assuming very small dis]lacements and displacement rates of the surface of the base profile. The steady-state bou.~dary condition on the surface of the base profile was satisfied by developing the base flow complex velocity potential F 0 The unsteady boundary condition was satisfied by a source distribution along the surface of the base profile and associa ted complex velocity potential F 1 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 F 2 of this vortex distribution was developed. The pressure distribution on the stretched-straight configura tion was found by linearizing the unsteady Bernoulli equation LlJ 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( e) which, according to equation (2.4, 30) is related to the amplitude function H~( e ) of the flapping. Also, it was observed that the 53 PAGE 63 54 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~From equation (2.1, 10) the ~-linearized mapping function becomes (3.0, 1) Figure 10 shows the com~iguration of the base profile for several thickness parameters. The numerical valuesfor 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 unownwash 11 Velocity for Any Given Flapping Function The II downwash" velocity on the boundary of the circle is given by equation (2.4, 24). The corresponding 11 downwash 11 velocity on the base profile as given by equation (2.4, 9) is ,._ '\."" (e;t) f' (e'O) (3.1, 1) Recall, from equations (2.4, 24) and (2.4, 30) (3.1, 2) where (3.1, 3) PAGE 64 55 = 0.05 E = 0.10 y = 0.20 Figure 10 Stretched-Straight Configuration for Several Thickness Parameters PAGE 65 56 Taking the derivative of equation (3.0, 1) gives (3.1, 4) Separating the real and imaginary parts and changing i to -i, equation (3.1, 4) becomes where PAGE 66 57 After some algebraic and trigonometric manipulations, equation (3.1, 8) can be written in the form where : 't1 4i :!.1 2.<1-2.E.)s,n 2 9 (3.1, 9) Substituting equations (3.1, 5) and (3.1, 9) into equation (3.1, 3) yields (3.1, 10, Taking the 1 imaginary part of' in the space imaginary unit i of equation (3.1, 10) leads to (3.1, 11) PAGE 67 58 Combining the values of .J.., e I ,J. l' and el' given in equations (3.1, 5) and (3.1, 9) with equation (3.1, 11) gives + (2l:s1n8 -U-2e->~â€¢n28)cos8 -(1-2ec.o~d)-t1-2~)c.os28)51rie dH~(e) 2(1-2E:)S1n8 cfa. This last expression can be simplified to read (3 .1, 13) Recall that from equations (2.4, 21) and (2.4, 31) 00 H:(e) '!: Bo T 2.L Bn c.os ne (3.1, 14) n:.l and (3.1, 15) where (3.1, 16) PAGE 68 59 Substituting equations (3.1, 14) and (3.1, 13) into equation (3.1, 16) and performing the indicated integration leads to the recur renc~ relation between the Pn's and Bn's as In computing the forces acting on the profile due to the unsteady pressure distribution it is convenient to use, instead of G( e ) , a new function defined by (3.1, 18) where the Fourier coefficients are related according to (3.1, 19) Combining equations (3.1, 13) and (3.1, 18), it follows that (3.1, 20) It is expedient to write the first term in the last expression as a trigonometric series in the form co ( 1-t:: 12: c.os e} H~l8) : bo + 2[ bnC.OS ne (3.1, 21) n-:.1 PAGE 69 60 where the Fourier coefficients are found to be (3.1, 22) Also, the second term in equation (3.1, 20) can be written in the form (3.1, 23) where the recurrence relation for the Fourier coefficients is (3.1, 24) Finally, the third term in equation (3.1, 20) written in a Fourier cosine series becomes (3.1, 25) where the coefficients are (3.1, 26) PAGE 70 61 Combining equations (3.1, 18), (3.1, 21), (3.1, 23), and (3.1, 25) with equation (3.1, 20), it is seen that (3.1, 27) These results give (3~1, 28) It should be noted that the trigonometric series in which the coefficie~ts are the .\i 1 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 1 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 co Tr(6,t) =ZplJze-iw*L Pn[crs~ .... ne h:1 PAGE 71 62 In order to completely determine this pressure distribution it is first necessary to approximate the difficult wake effect as contained in the function T(6>jk;f} Consider the velocity of a ' fluid particl~ along the streamline emanating from the sharp trailing edge as computed from the base flow potential F O This velocity is given as At.the trailing edge uU)= 1,m U ~o-1 = (3.2, 2) =U(1-e:) 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(e,1kj~) , 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 o PAGE 72 63 ,depends upon the thickness parameter E For the special case of' E: -"" O, o. = l. The relationship between o and c 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 This chord line can be mapped onto the same unit circle by the transformation (3.2, 5) The values of )\ and p can be determined as follows: . XT: 12 :,\(2-tt3) X1.. -:.-1+'= = )d-2+,9) (3.2, 6) or (3.2, 7) Employing these results, the base flow potential in this flow field becomes (3.2, 8) and PAGE 73 64 Employing the transformation (3.2, 5) it follows that (3.2, 10) and (3,2, ll) Substituting equations (3.2, 10) and (3.2, 9) into equation (2.6, 11) yields (3.2, 12) It is expedient at this point to introduce a new variable defined by 1' So e (3.2, 13) Hence, expressing equations (3.2, 9), (3.2, 10), and (3.2, 12) in terms of the new variable_')l , it follows that (3. 2, -14) (3. 2, 15) and 4-;..,2 r = T (coshX-1) (3.2, 16) PAGE 74 65 Combining these results with the expression for Tl6,1k;E-) as given by equation (2.7, 11), and changing the variable o~ integration gives ' (3.2, 17) J 0 This expression, after some algebraic manipulations, can be simplified to read s,n 2 8 + s,nh 2 'X e--iAkcosh'X. d'X co~h')C. co~ 8 Employing the identities and (3.2, 18) PAGE 75 66 equation (3.2, 18) reduces to co I (coshX+c.os8} e-iA\(.Cosh'kd'X U o;.....,..------------A s1n8JQ) . (coshX +l)~-1AkC.oshX d~ (3.2., 19) 0 It is well known in the theory of Bessel functions /:24J that 00 Knl'l)-::. J e-tCo'$h.'X.c.o-shn'X d')C 0 (3.2, 2p) where K"(i> is the nth order Bessel function of the second kind which is valid if IQrCj~l PAGE 76 67 The ratio containing the circular functions in equation (3.2, 23) is the Theodorsen function f:1sJ J::.CAkl ~ploying the relation J where l-\~\(A k) are Hankel functions of the second kind and order n, it follows that J:.(Ak) (3.2, 25) It is true that in the development of equation (3.2, 25) it was necessary to require that fi > 0 However, according to Luke and Dengler J, 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 .PIn the present case Substituting equation (3.2 1 25) into equation (3.2, 23) gives (3.2, 26) This determines the function T(S,1\<.jE:) except for the parameter A ~ which, as mentioned above, must be found by requiring that the unsteady pressure vanish at the tail. PAGE 77 68 Replacing in equation (3.2, 1) the wake function as given in equation (3.2, 26), the time-dependent pressure distribution becomes co 11(8,t) 2.pU'ei"':~, Pn [ n,~ne -+ J\;,ne [ ~(Akl -,. (t U.t\k)lco, e]] From equation (2.1, 10) it follows that .1. 1 (3.2, 28) With some algebraic and trigonometric manipulations this expression can be written in the form 1 1 ... 4E. 2 ... ~4, 4E: ( 1 +E:Z) c.os e + 2 1 c.os 2& \f'(e' 8 )\2. = ~â€¢n 1 B {12e:) + 4 '=i. s "" 8 (3.2, 29) Neglecting terms involving orders of . E: of two and higher, equation (3.2, 29) reduces to 1 -46 c.os 6 (1-2~) S\n 2 6 (3.2, 30) Combining equations (3.2, 27) and (3.2, 30), there results {1-4:c.ose) c.osne (1-2.E:) Sm 8 PAGE 78 69 It can be seen from this expression that the unsteady pressure possesses a singularity at the tail where e c O and at the nose where 0 =,n. The singularity at the tail is removed by satisfying the condition This can be satisfied for all time only if 1.-4e12.f: 1 A (3.2, 32) (3.2, 33) which defines A _ and o in terms of the thickness parameter _ E: The singularity ate c n produces a concentrated force at the nose, the so-called suction force arrived at in airfoil theory "21J, 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 00 1T(8Jt) ': pU 2 e 1 U)-I:[ Clo+an~ +2.~1. 01..,s,nhe] (3.2, 34) where the coefficients are given by (see Appendix A) (3.2, 35) PAGE 79 70 Dividing out terms in the denominator containing and linearizing the results in E, these coefficients become (3.2, 35a) where 3.3 Calculation of the Lift o.nd Moment With the pressure distribution known, the lift can be computed from equation (2.8, 6) as 211' . L-=~if TICB,-t:)df (3.3, 1) 0 The differential complex vector df .must be found from the mapping function, Therefore, from equation (3.l, 1) is real in the space imaginary unit i, the operator ~i and the integra.1 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 211" L-=-p\J 2 e 1 t,Jf[ Qo+Qn~ +2t,_Q"Sll"\ne][c1-~)s,ne -i-&s,n2e] de. 0 (3.3, 3) PAGE 80 71 Integration of this equation leads to the result that Combining equations (3.2, 35) and (3.3, 4) gives the lift in terms of the downwash velocity coefficients as (-( A2. i -2.E: For the case of a flat plate of zero thickness ( = o), the lift.~becomes (3.3, 6) where A = 1. 'This is in agreement with the results given by Siekmann {"aJ. In making the comparison between this work and that of Sielanann it must.be observed that where the Ar\ correspond to Sielanann I s coefficients for the downwash velocity. The difference in signs arises from the fact that Sielanann employed a pressure differential across the plate of Ap = pp+ to compute the lift, whereas in this work the analogous pressure differen PAGE 81 72 The moment of the forces acting on the profile with respect to the origin is given by equation (2.8, 9) as 2-rr M = R~J iT(e,t)f(ei 8 )df 0 whero pooi tive mo?t , e~t is counterclo.ckwis o (nor;e down) . . Combining equations (3.0, l) and (3.3, 2) and linearizing the results in ~, it is readily seen that Substituting equations (3.2, 34) and (3.3, 9) into equation (3.3, 8), the moment becomes 2 wf 2 1f e 1-2~ M= p\) e 1 (o. 0 tan 2 -t2L., Qn'$\n\)8)(-y s,n28 + l'\=1 0 Integration of this expression leads to the result that In terms of the downwash velocity, the moment becomes M = 11" p uâ€¢ e;w~[ (1&)(r. -2â€¢(r.+2 A, -2 A.ll + (3.3, 12) PAGE 82 73 For the special ease of a flat plate, the moment about the center of the plate reduces to This result is in agreement with that obtained by Siekmann ["aJ. 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-direetion. The x-component of the hydrodynamic force imposed on the plate by the pressure distribution is given by equation (2.9, 2) as 211' F, = 6t{J t!t;1H8;t.l[ (3.4, 1) 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: (3.4, 2) where 21'1" r; = IR.j{-{ IR. 1 1rce,ti j! da} , 0 (3.4, 3) PAGE 83 74 2'11' F! : Gl+ J IR.;"11(6, tl [i GI.; ei"'t d :~el) de} , 0 (3.4, 4) and F; is the suction force. l The force F x above represents the stream.wise force computed by integrating the pressure distribution .along a path defined by the stretched-straight configuration of the fish, whereas the force r! 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, (3.4, 5a) Yn -, . r., e 1 "'1; = ( r~ -t,j r,'.;)e1w-t (3.4, 5b) : rn + 1 rn ::. An=_, _,, A"eiwi l f\ . A" l ,iw (3.4, 5c) An i' 'l An = n-+1 ne B _, .. 1 e: B e;w-t le' B" ) .;u>t (3.4, 5d) Bn :: " n ni1 I'\ e -â€¢ . CneiiA>t tC: . iwt (3.4, .5e) Cn -:. ch -t 1 C.n = """1"e -b -bâ€¢ .... _b,. -b n e 1 u.>-l -( b 0 n ... " "-''n) e;wt h lo\ ,. ,1 I'\ -. ,1 U (3.4, 5f) (3.4, 5g) PAGE 84 75 Employing equations (3.4, 5a) and (3.2, 34), it follows that 00 . ~ 1 1Tt6J-t:): puz.(o.~+an~ -t2}: &~ s,nne]. t'\':.\ (3.4, 6) Also, _ combining equations (3.4, 5d) and (2.4, 21) it can be shown that (3.4, 7) This last expression can be written in a more convenient form as (3.4, 8) _, where the relationships between the Bn are given in / equation (3.1, 24). Now, substituting equations (3.3, 3) and (3.4, 6) into equation (3.4, 3) gives 211' F~ :: -E.puif (a~tan 0 00 + 2L o.~ Slt\ \'\8)(co~ e c.os28) n~1 From the integral 2.1i' I s111ne c.osme -::. 0 0 (3.4, 9) the second term in equation (3.4, 9) vanishes. To evaluate the contri1 bution to Fx due to the term 2-rr -"-pUi o.~t PAGE 85 76 the Cauchy principle value of the integral must be taken, i.e., ~-4 1'1'1' -~ p uâ€¢ o:; [ J tan t (cos 8 -cos28l {+an!: Ceo, e-co,2eJde 0 ff+6 J Employing the identity this expression becomes 211' + J<-ztc.ni +3s,n8-sin28)d&], 1T+6 Integration of this equation leads to the result that F ~ Therefore, the hydrodynamic force vanishes. (3.4, 10) (3.4, loa) (3.4, lOb) 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 Employing the integrals 2.-rt J tctni s,nn8d& = (-1)"+' 2-rr n~ 1 0 and m=n (3.4,. 11) PAGE 86 77 .the f'orce f:; becomes (3.4, 12) But, accord~g to equation (3~1, 24) (3.4, 13) Inserting this expression into equation (3.4, 12) yields (3.4, 14) To the above f'orce 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 J to a small circle of radius 6 0 surrounding the nose. The velocity at the leading edge can be computed from the complex velocity potential by equation (2.1, 13) as (3.4, 15) where is the complex velocity in the r -plane. Since w is bounded as PAGE 87 78 ~the leading edge is approached, equation (3.4, 15) can be written as (3.4, 15a) But, _from equation (2.1, 10) it follows that (3. 4, 16) and as j"--1 , il--1 +: E Differentiating this expression with respect to z yields (3.4, 17) and = 00 (3.4, 17a) From these results it is seen that the velocity in the physical plane asym.ptotically approaches an infinite value as (3.4, 18) where .. (3.4, 18a) PAGE 88 79 According to Milne-Thomson LlJ the extension to Blasius's formula for the case of unst~ady flow can be written as (3.4, 19) Here the contour ,k 0 is taken to be a small circle around the leading edge with a radius 0 0 Since the velocity potential . :l~ and of c}t is 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 (3.4, 20) A small circle around the nose can be written in complex / notation e.s (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 Zff F 1 . F3 >t I ) '1'. i,c, =l. i \ we-, tl\'L c-, +E:, '~ d'Jt, 2 p ' ,-E:2.Soeâ€¢'Jll 0 (3.4, 22) PAGE 89 80 Th F : us, 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(-1,-t) in the. s-plane must be found from the complex 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) (3.4, 23) From equation (2.4, 38) it follo:ws that (3.4, 24) " Substituting the trigonometric series for G(1') 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 Employing the integrals 2fl' i-rrl s,n n~ d..9 ::. 0 n~ 1 . 0 and (3.4,. 25) PAGE 90 81 equation (3.4, 24) becomes wl(-1, t) :. i u e ,iw-1: 2.f: (-1} P\+l Pn (3.4, 25a) n ... 1 Combining the recurrence relation given by equation (3.1, 19) with equation (3.4, 25a), th.ere results (3.4, 26) , The velocity _w.,.: due to t=-zcan be found from equation (2.6, 21) to be w (-1 t) = ~z l-t,tl a I ~'f (3.4, 27) Inserting the expression for P(-ikjt:) from equation (2.s, 20) and using the results given in equations (3.2, 5) thru (3.2, 15), it follows from equation (3.4, 27) that OI:> J eiAkcosh'X. {cosh~ -1)d1(. w 2 (-l,i.) =iQ -:-----------! ei.l\k cash~ (c.osh'x-+l)d"X. 0 (3.4, 28) Recalling the definition of the Bessel function of the second kind, equation (3.4, 28) becomes (3.4, 29) PAGE 91 82 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 ' Adding the results of equations (3.4, 26) and (3.4, 30) the . velocity at the leadingâ€¢edge .. in_the '!-plane is It then follows from equations (3.4, 22) and (3.4, 31) that the suction force becomes where (3.4, 31) (3.4, 32) Finally, by adding the results given in equations (3.4, 14) and (3.4, 32) the net thrust becomes ct) T : 2.-rr p u2. o.'o ( c~ c:) ZL n a'n en + (l-(.l(r:) 2 n:.1. Equation (3.4, 33) can be expressed in still apother convenient 1 -, I form for computational purposes by e iminating the 4n s. Substituting PAGE 92 83 equations(3.l, 24), (3.1, 27), and (3.2, 35a) into equation (3.4~ 5), the following relationships can be derived. (3.4, 34a) (3.4, 34b) where (3.4, 34c) and _, _, _, _, Cn 3 E: ( C,.. Cn-, C n+1) (3.4, 34d) Combining equations (3.4, 34a) and (3.4, 34d) with the second term in equation.(3.4, 33) it follows that -ZE na'"B~ = \<. 1 [e~B1 +~(B~B~ -B:B'. +s'o6~)] + nc1 00 -t-4kL nB~ 6~ 2 kL n ( B~1 "'"B~+ 1 ) B~ + l'\=1 l'\=-1 co -E: kL in ( E~'n-1, c:~+7. )( c',.._1 C~+1> + "''"'\ (3.4, 35) / ' PAGE 93 84 Substituting equations (3.2, 35) and (3.4, 35) into equation (3.4, 33) yields :r 2 'ltp uâ€¢ {: c:~ \l?~ -l~) + 1< 2 e: ii: cO +4kL n8~B~ -t 1"1=1 ClO -E:[ ki.'( a: s: 5'1 5'1 6~ 6'~ \ + k L n ( 6~-'L e'~+1 l a'., + n::.1 For the case of an infinitely thin plate ( c 0), it follows from equation (3.4, 36) that the thrust reduces to T = 2 -rrp Uz. [ PAGE 94 85 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, T 2ri 0 "' w. The time average value of an arbitrary function of time (\ ( t) is defined as Thus 'l'o Ji. = ~.J 1Hlldt 0 " 1J z. 1 .!1 1 :: 'fo c.os w-\: dt =. 2 0 rt; n, : J SI n oot d t 0 . To n. J s,n wt c.osw-1:dt = 0 0 (3. 5, l) (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 (3. 5, 3) and similar formulae for the other coefficients. PAGE 95 86 Thus, from equations . (3.5, 1) thru (3.5, 3) it follows that 'r. { J r,' l~ di = C to' l; + v-o"C; > 0 "' tf c~ c:dt = i (c.~ c~ + c~c; > 0 ,._ 1j c-â€¢ r 'd.1 -!.z ( r' z + Co"2 } o '-o vo 0 'l'o tf c~ c.~dt : 1 cc.; 2 Tc/) 0 'l'o if a; c:dt 0 'l', 1f -, f' dt = Oi,. 0 1 !To ' -, To ci 0 C 0 dt 0 1 ( â€¢C' â€¢C.") 2 Q., 01-a.. 0 1 JTo -, 1 ( , c' ,, (," l To a; C.1 d t -= 2 a. t + a.o t 0 "l'o . t f e' 1 B~ di: = C 6~1. + B'~ 2 ) 0 'l'o 1 J esd.J. 1 l 0.' 0.â€¢ U" B") 't'o o 2. \. = Z Uo Vz. -t Uo 0 (3.5 1 4a) (3. 5, 4b) (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, 4m) (3.5, 4n) PAGE 96 87 and (3.5, 4o) (3. 5, 4p) (3.5, 4q) Substituting these results into equation (3.4, 36), it follows that the time average value of the thrust becomes (3. 5, 5) PAGE 97 88 -where DO -tt6 -= n [ (S " B " \ a. ' ' 0 ' B ' ' a. "] L., Z: 1'1-1 + l'\-ti ul'\ \ Ui,-1 -t n+l Ju., (3.5, 5a) n=1 0:, C,: L 4 1 n[ c.'in-2.\ (C.'n-1 -c'n .. 1) c',n-tl (C~-1 -c.;+1> + h-:.1 (3.5, 5b) and (3. 5, 5c) Following Siekmann L-aJ the thrust coefficient is defined as (3.5, 6) Thus, the thrust coefficient can be ~etermined by equation (3. 5, 5) by employing the definition given in equation (3.5, 6). PAGE 98 89 3.6 Numerical Example In order to compare the theory developed with available . expe~imental data aJ for a very thin plate, a displacement 7 function with a quadratically varying amplitude is used. Thus, consider the displacement function where Here the phase angle 6, 0 is set equal to zero. From equations (3.2, 5) and (3.2, 7) it follows that X-::. (1-E)COS 0 + (3. 6, l) (3.6, la) (3.6, 2) Combining equations (3.6, la) and (3.6, 2.) with equations (2.4, 16) thru (2.4, 21) gives By employing the relations and PAGE 99 90 equation (3.6, 3) can be written in the form From this equation the Fourier coefficients Bn can be computed by the relation 'i1 Bt'\ = ;J H1 ca> c.os ne de 0 Employing the integral ["23J 1T Jn(J<) = t" ~J e-i> PAGE 100 91 where and c.o~me c.osn8 = i [ cos(m+n)8 + co~(m-n)8] have .been used. Using the following recurrence relations J between the Bessel functions, Jn-1<><) +Jn ... 1<><) = 2 x,n JnlHl, and equation (3.6, 6) simplifies to (3.6, 7) PAGE 101 92 Separating the real and imaginary parts of Bn in the time imaginary unit j leads to the relationships which follow. The real part of Bo is ' (3.6., 8a) The imaginary part'of B 0 is B; = Cdo +: d1 + (1-e)2.iz.} Sin~ J 0 (~) + (3.6., 8b) The real part of the odd Bn's are . Cl l2dz. c. CHâ€¢ [c1 2('Z.n+l) 2(2n-+1)')J 2 J ( )l\ + -G z .alnz xz. )(2. 2n+1 + X Zn 'k JJ (3.6., Sc) PAGE 102 . 93 The imaginary part of the odd Bn's are The real part of the even Bn's are (3.6, 8e) The imaginary part of the even Bn 1 s ~re PAGE 103 94 Next, consider the Fourier series defined by ....L d~(e) -=s,ne de oO Co + z.E Cn c.os ne n=1 where the coefficients are given by 11' 1f 1 d\.\~Ce> cl'\= ;jr"" sin'a de co5 ne de 0 From equation (3.6, 4) it follows that (3.6, 9) (3.6, 9a) (3.6, 10) Substituting equation (3.6, 10) into equation (3.6, 9a) and comparing the results with equations (3.6, 5) thru (3.6, 7), it is seen that (3.6, 11) PAGE 104 95 After some straightforward manipulations, the real and imaginary parts in the time imaginary unit j of the On coefficients may ~e written as follows. The real part of c 0 is c;: [-<1-E:Hd1"':dilc.os;"' +x{do~!d1+C1.-E:l~z.1~,n~""]Jo(){) 4 (3.6, 12a) The imaginary part of c 0 is (3.6, 12b) The real part of the odd Cn 1 s are PAGE 105 96 The imaginary part of the odd en's are The real part of the even en's are The imaginary part of the even en's are PAGE 106 97 With the coefficients Bn and On known, the An coefficients can be computed from equation (3.1, 28) as follows: Notice that the Bn's and en's depend upon the thickness parameter~ , the coefficients d 0 , d 1 , and d 2 , and the various orders of Bessel functions. / (3.6, 13) For the example under consideration, the following numerical values are used in accordance with Sielanann L~BJ: a "' n. Shown in Table 1 are the values of the Bessel functions f:26J for five values of the thickness parameter. PAGE 107 f = 0.00 = 0.05 n Jn(x) Jn(x) 0 -0.30424400 -0.25475200 1 0.28461200 0.34481999 2 o.48543400 o.48582499 3 0.33345900 0.30631199 4 0.15143599 0.12998900 5 0.05214199 0.04210000 6 0.01454600 0.01108300 7 0.00342000 0.00246300 8 0.00069600 0.00047500 9 0.00012500 0.00008000 10 0.00002000 0.00001200 11 0.00000300 0.00000200 12 0.00000000 0.00000000 13 0.00000000 0.00000000 14 0.00000000 0.00000000 TABIE 1 BESSEL FUNCTIONS E:= 0.10 Jn(x) -0.19613700 o.40054200 o.47946499 0.27777100 0.10995100 0. 033li.4200 0.00828999 0.00173800 0.00031599 0.00004999 0.00000700 0.00000100 0.00000000 0.00000000 0.00000000 = 0.15 J (x) n -0.12922200 o.45055600 o.46667299 0.24848900 0.09165499 0.02609700 0.00607800 0.00119700 0.00020500 0.00003099 0.00000399 0.00000000 0.00000000 0.00000000 0.00000000 (:= 0.20 Jn(~) -0.05496600 . o.49377800 o.44790500 O> 0.21907800 0.07509900 0.01996800 o. ooli. 35100 0.00080400 0.00012900 0.00001799 0.00000200 0.00000000 0.00000000 0.00000000 0.00000000 PAGE 108 99 With the values of the Bessel functions Jn as shown in Table 1, the numerical values of the coefficients Bn, C, and A can be n -n comp~ted for the given displacement fu.~ction and wave number. These values are tabulated in Appendix C. The remaining coefficients r 0 and r 0 depend upon the An's and the real and imaginary parts of Theodorsen 1 s function sJ. Notice that the argument of the Theodorsen function contains a thiclmess parameter and the reduced frequency. The real part of Theodorsen 1 s function is tabulated {"24J in Table 2 and the imaginary part of this function is tabulated in Table 3 for five selected thickness parameters e and for ten reduced frequencies. With these values, equations (3.6, 8), (3.6, 12), (3.6, 13), and (3.5, 5) were programmed and runs were made on an IBM 709 electronic computer. The values of the thrust coefficient for ten values of the reduced frequency and the five selected thickness parruneters is tabulated in Table 4. A plot of the thrust coefficient versus reduced frequency is shown in Figure 11 for the five selected thickness parameters. PAGE 109 TABLE 2 REAL PART OF TEE TBEODORSEN FUNCTION E = 0.00 f = 0.05 = 0.10 = 0.15 = 0.20 k cl: (/\k) .t: (J\k) (. (Ak) ct: (Ak) cl: (Ak) 0 l.00000000 1.00000000 l.00000000 l.00000000 1.00000000 l 0.53943500 0.53349999 0.52545600 0.51630100 0.50628000 2 0.51295500 0.51084900 0.50792300 0.50471900 0.50168499 I-' 0 3 0.50628000 0.50510900 0.50367100 0.50219600 'O, 50076099 0 4 0.50367100 0.50293399 0.50212900 0.50124700 0.50052500 5 0.50239699 0.50176200 0;50131200 0.50080600 0.50038700 6 0.50168499 0.50129700 0.50096000 0.50059500 0.50024799 7 0.50124700 0.50099199 0.50070900 0.50051399 0.50014700 8 0.50096000 0.50076099 0.50058699 0.50043300 0.50012100 9 0.50076099 0.50061200 0.50052500 0.50035200 0.50009499 10 0.50061800 0.50055999 0.50046400 0.50027100 0.50006900 PAGE 110 TABLE 3 IMAGINARY PART OF THE THEODORSEN FUNCTION f = 0.00 '= = 0 .05 E = 0.10 = 0.15 E = 0.20 k ~(Ak) <&(J\k) ~(Ak) ~(J\k) ft (Ak) 0 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 1 -0.10027300 -0.09213199 -0.08090200 -0.06474999 -0.04000399 ..... 2 -0.05769099 -0.05249400 -0.04486600 -0.03462200 -0.02059600 0 ..... 3 -0.04000399 -0.03596800 -0.03049599 -0.02350999 -b.01381600 4 -0.03049599 -0.02723600 -0.02314699 -0.01770499 -0.01120599 5 -0.02459899 -0.02105100 -0.01815099 -0.01421800 -0.00934500 6 -0.02059600 -0.01803900 -0. 01552200 -0.01213700 -0.00748400 7 -0.01770499 -0.01577700 -0.01333600 -0.01105100 -0.00635300 8 -0.01552200 -0.01381600 -0.01203400 -0.00996500 -0.00541200 9 -0.01381600 -0.01236900 -0.01120599 -0.00888000 -0.00478800 10 -0.01244700 -0.01167100 -0.01037899 -0.00779399 -0.00416499 PAGE 111 0.06 0.04 CT .. 0.02 !! q 0.00 I ,_ 0.02 0 2 102 do = .023 dl = .042 d2 = .034 a = fl Theory Experiment 0 4 k Figure 11 e = o E: = 0.05 E:::0.10E. == 0.15 E: = o. 20 6 0 8 Thrust Coefficient Versus Reduced Frequency 0 10 PAGE 112 TABLE 4 THRUST COEFFICIENT CT k E = 0.00 E = 0.05 <= = 0.10 <:= 0 .15 E:= 0.20 1 -0.00436836 -0.00511078 -0.00554143 -0.00575810 -0.00588071 2 -0.00304028 -0.00383011 -0. 004 305 91~ -0.00457571 -0.00477146 ..... 0 3 -0.00067421 -0. 0015871+8 -0.00219911 -0.00262206 -'O. 00299391 4 0.00264988 0.00155845 O .00071~348 0.00008702 -0.00055165 5 0.00692289 0.00560035 0.00451762 0.00354955 0.00255496 6 0.01214319 0.01053566 0.00912058 0.00776429 0.00632766 7 0.01831052 0.01636446 0.01455352 0.01272949 0.01076497 8 0.02542489 0.02308680 0.02081422 0.01844777 0.01586781 9 0.03348634 0.03070231 0.02790244 O. 02492011+ O .02163!+99 10 0.04249495 0.03921030 0.03582023 0.03214766 0.02806850 PAGE 113 C'rlAPTER IV SUMMARY AND CONCLUSIONS In this study a mathematical theory was developed for predicting the forces generated by the swir:ming motions of a thick two-dimensional fish. The general treatment was based on unsteady airfoil theory. It was assumed that the for, mrd motion of the fish ia uniform and that the propulsion is generated by lateral displace ments of the fish I s body. These displacements were ta.ken in the form of displacement waves of small a::npli tude which pass down the body of the fish from head to tail. The maximum amplitude of these displace ments were assumed to have its :smallest value at the h c ~ d and to very along the length of the fish to its largest value at the tail. Furthermore, these displacements were assumed to be oscillatory, which vary harmonically with time. Of particular interest in this investi gation was the effect of the thickness of the fish on the forces generated by these ' displacement uaves. For the mathematical formulation of the proble:::i, complex potential theory was employed. In this theory the flow field around an infinite cylinder was described by developing the complex velocity potential, and the flow field around the fish was found by trans forming the flow field around the cylinder into the flow field around the fish. The mapping function used for this transformation contained 104 PAGE 114 105 a thickness parameter. By asm.l.Illing a small unsteady perturbation theory it was possible to decompose the complex potential function into three parts and to superimpose the results. First, the time independent potential function describing the flow field,around the stretched-straight configuration was developed. Second, a time dependent potential function was derived to satisfy the unsteady boundary condition on the surface of the body. Finally, a time dependent potential function was derived associated with the trailing vortices behind the fish. In the treatment of the wake it was assumed that vortex elements move downstream with a velocity somewhat less tha.~ the free stream velocity depending upon the thickness of the fish. From these potential functions the pressure distribution acting on the surface of the fish was found by employing the unsteady Bernoulli equation. The lift and moment uere calculated by integra ting the pressure distribution along a path described by the stretched straight configuration. The propulsive force or thrui::t was computed by taking into consideration the distortion and distortion rate of the fish and integrating the pressure distribution along a path which varies in space and time. Finally, all the functions involving the thickness were linearized in the small thickness para.meter. In order to estimate in a quantitative manner the effect of thickness on the thrust, a quadratic amplitude function was selected and the time average value of the thrust was computed on an IBM 709 electronic computer for several values of reduced frequency and for several values of thickness parameter. These numerical results are plotted in Figure 11. PAGE 115 106 Based upon the theory developed and the numerical results obtained,the following observations can be made. (1) The thiclmess of the two-dimensional fish tends to reduce the available thrust generated by the swimming motion. Thia result can most readily be seen in Figure 11 by comparing different values of the thiclmess parameter. This phenomenon indicates that a slender fish has more available propulsive force in which to maneuver than does a comparable size stocky fish. Of course, in order-to obtain the net thrust available the viscous effects must be considered. (2) The thickness effect is more pronounced at higher reduced frequencies than at smaller ones. (3) The thrust coefficient depends upon all of the Fourier coefficients Bn and "n for a finite thick fish insteady of only two Bn's and two Cn's as in the case of a fish of zero thickness. (4) The argument of the Theodorsen function is increased by an amount depending upon the thickness of the fish. This results from the slowing-up effect of fluid particles in the wake. (5) This theory gives identical result, for the lift, moment, and thrust with existing thin p_late theories when the thickness parameter vanishes. / . PAGE 116 APPENDICES PAGE 117 APPENDIX A CALCULATION OF THE PRESSURE DISTRIBUTION OOEFFIOIENTS an From equation (3.2, 31) the pressure distribution can be written in the form (A-1) where (A-2a) and 0) e = \ z P. [ c.os8cosn8 _ ctU\kl +C1-X:lAkllcos8] t L "' 1-2f: !,ne . s,n & n-:.t (A-2b) It is necessary to make the following expansions 00 'lP Co".in8: (A -A.) co,6 +(A -A)~ +CA -A )Cos39 + L " s1n8 1 Slt'\8 1 J su,e . 2 4 s,'l'\e n~1 where equation (3.1, 19) has been used. 108 PAGE 118 109 Employing the identity c.os(n -\-1)8 -cosCn-1)9 = -2 S\l"\08 s,ne _. equation (A-3) can be expressed in the form , Thus, Next, from equation (3.1, 19) it follows that Combining equations (A-4) and (A-6) with equation (A-2a) there results Combining equations (A-2b) and (3.1, 19) gives (A-5) PAGE 119 110 This expression can be simplified to read 8 = c.os8 [Aocote + A, cos28 _ A, cos6 -cos38 + t. 1-,t: su,8 . s,n 0 -~ A c.os(n-1)8 .. eos(n+l)e _ (A -:.,A\ .,(Akh(1-t PAGE 120 111 Thus, e, becomes 1 ~:.{[ A,, -(Aoâ€¢AL\.tlAk)]tQn~ A 0 s,"8 + f A., [ s1n(n+1le + -s,"'tn-ue]} n~1 (A-10) Substituting the resu~ts of equations (A-7) and (A-10) into equation (A-1), the unsteady pressure distribution becomes (A-11) Therefore, the coe:f'ficients an of the pressure distribution series are (Aâ€¢l2) / . PAGE 121 APPENDIX B TABLE 5 COORDINA'IES OF THE BASE PROFILE t = o'.05 E= 0.10 t = 0.15 E= 0.20 X r r X y X y X y 0.9750 *0.0000 0.9500 *0.0000 0.9250 .0000 0.9000 *0.0000 0.92~5 i-0.0008 o.9o60 i-0.0015 o.S834 i-0.0023 0.8608 i-0.0030 N 0.79?,6 .0056 0.7781 i-0.0112 0.7627 i-0.0168 0.7472 i-0.0225 0.5834 o.0167 0.5790 i-0.0333 0.5746 i-0.0500 0.5702 o.o667 0.3186 .0329 0.3281 i-0.o657 0.3377 i-0.0986 0.3472 i-0.1314 0.0'250 .0500 0.0500 i-0.1000 0.0750 i-0.1500 0.1000 i-0.2000 -0.2686 .0622 -0.2281 i-0.1245 -0:1877 .1867 -0.1472 .2490 -0.5334 i-O.o642 -o.4790 o.1285 -0.4246 .1927 -0.3702 *0.2570 -0.7436 .0532 -0.6781 i-0.1063 -0.6127 i-0.1595 -0.5472 -1-0.2127 -0 . .'8785 i-0.0301 -0.8059 i-0.o603 -0.7334 i-0.0904 -0.6608 *0.12o6 -0.9250 -0.0000 -0.8500 i-0.0000 -0.7750 .0000 -0.7000 .0000 PAGE 122 APPENDIX C TABLE ,6 REAL PART OF B 0 COEFFICIENTS n E = O.OO '= 0.05 E = 0.10 = 0.15 E = 0.20 0 , -0. 02042213 -0. 016 31201 -0.01149067 -0.00619175 -o.ooo66838 1 -0.01658323 -0.01641405 -0.01620035 -0.01593077 -0.01560780 2 -0.02071626 -0.02009586 -0.01938574 -0.01860847 -0.01777410 3 -0.00701416 -O~oo664987 -0.00623394 -0.00575343 . -0.00520659 4 o.0018o700 0.00173444 0.00168785 0.00164140 0.00158089 .... .... 5 0.00287487 0.00251651 0.00214837 0.00178961 0.00144807 (,I 6 0.00071118 0.00049779 o.00033o63 0.00020567 0.00011612 7 -0.00029087 -0.00023737 -0.00018641 -0.00014114 -0.00010253 8 -0.00009594 -O.OOOo6241 -0.00003887 -0.00002317 -0.00001305 9 0.00001419 0-.00001089 0.00000786 0.00000535 . 0.00000350 10 0.00000508 0.00000293 0.00000161 0.00000085 0.00000036 11 -0.00000039 -0.00000024 -0.00000017 -0.00000014 -0.00000007 12 0.00000003 0.00000002 0.00000001 0.00000000 0.00000000 PAGE 123 TABLE 7 ~GINARY PART OF Bn COEFFICIENTS n E: = 0.00 E = 0.05 = 0.10 E = 0.15 E= 0.20 0 .. _0.01195370 -0.01289889 -0.01393920 -0.01504216 -0.01618090 1 -0.01096925 -0.01262985 -0.01405314 -0.01525960 -0.01627067 2 0.00102586 . 0. 00077970 0.00053075 0.00030765 0.00013722 3 0.01047589 0.00964978 0.00889388 0.00819226 0.00752476 .... 4 0.00590707 0.00536432 0.00478891 0.00418416 0.00357294 .... 5 0.00077774 0.00051427 0.00029233 0.00011583 -'o.00001227 6 -0.00102317 -o.ooo86450 -0.00070807 -0.00056158 -0.00043101 1 -0.00030740 -0.00020841 -0.00013556 -o.oooo8366 -0.00004877 8 0. 00006920 . 0.00005446 0.00004107 0.00002956 0.00002038 9 0.00002410 0.00001473 0.00000867 0.00000504 0.00000262 10 -0.00000257 -0.00000189 -0.00000130 -0.00000086 -0.00000052 11 -0.00000098 -0.00000066 -0.00000031 0.00000007 0.00000004 12 0.00000012 0.00000007 0.00000004 0.00000000 0.00000000 PAGE 124 TABLE 8 REAL PART OF en COEFFICIENTS n = 0.00 E = 0,05 E = 0.10 c = 0.15 = 0.20 0 0.05033188 0.05070063 0.05077887 0:05036583 0.04934839 l o.o6130994 o.o6147164 0.06067148 0.05919310 0.05725645 2 0.01716541 0.01787253 0.01837803 0.01850426 0.01813260 3 -0.02155470 -0.01891211 -0.01687211 -0.01524085 -0.01383965 .... 4 -0.02491790 -0.02202579 -0.01903071 -0.01601655 -0.01310699 .... 5 -0.00709790 -0.00503541 -0.00337709 -0.00210962 -0.00119238 en 6 0.00382531 0.00313424 0.00247315 0.00187890 0.00137363 7 0.00143669 0.00093861 0.00058908 0.00035268 0.00020028 8 -0.00024663 -0.00018791 -0.00013614 -0.00009360 -0.00006133 9 -0.00009868 -0.00005800 -0.00003272 -0.00001798 -0.00000906 10 0.00000893 o.ooooo632 0.00000417 0.00000264 0.00000153 11 0.00000371 0.00000225 0.00000104 0.00000000 -0.00000001 12 -0.00000039 -0.00000025 -0.00000011 0.00000000 0.00000000 PAGE 125 TA13IE 9 IMAGINARY PART OF en COEFFICIENTS n = 0.00 E= 0.05 !: = 0.10 E = 0.15 = 0.20 0 -o.o448o435 -0.03512095 -0.02643027 -0.01870675 -0.01184833 1 -o.o4014401 -0.03649979 -0.03255134 -0.02841835 -0.02427445 2 -0.06674290 -o.o6038o65 -0.05453684 -0.04922600 -0.04438993 3 -0.03604089 . ~0.03338065 -0.03042782 -0.02718762 -0.02372656 ..... 4 -0.00388694 -0.00248173 -0.00117400 -0.00007238 0.00075867 ..... en 5 O.Ol.122163 0.00953924 0.00786183 0.00628559 'o. 0048565 3 6 o.0038908o 0.00266211 0.00174414 0.00108673 0.00063589 7 -0.00105745 -o.ooo83591 -0.00063157 -0.00045721 -0.00031593 8 -0.00041346 -0.00025751 -0.00015379 -o.oooo8782 -0.00004754 9 0.00004983 0.00003695 0.00002559 0.00001664 0.00001034 10 0.00002015 0.00001113. 0.00000589 0.00000300 0.00000129 11 .-0. 00000138 -0.00000083 , -0.00000057 -0.00000043 -0.00000019 12 -0.00000010 -0.00000004 .,:: --0.00000001 0.00000000 0.00000000 PAGE 126 TABI.E 10 . REAL PART OF Ao COEFFICIENTS E = 0.00 E = 0.05 E:= 0.10 l: = 0.15 E: = 0.20 k J.6( k) ___ Ao ( k) A~( k) A~( k) A~( k) 0 -0.05033188 -0.05232055 -0.05465307 -0.05737539 -0.06052581 l -0.03837818 -0.03880361 -0.03929716 -0.04001167 -0'.04107282 -:a ;;;0.02642448 -0.02528668 -0.02394124 -0.02264795 -0.02161982 3 ;0.01447077 -0.01176974 -0.00858533 -0.00528422 -0.00216682 4 -0.00251707 0.00174718 0.00677059 0.01207949 0.01728617 5 0.00943663 0.01526412 0.02212650 0.02944322 O 03673916 6 0.02139033 0.02878106 0.03748242 0.04680694 0.05619216 1 0.03334404 0.04229799 0.05283833 0.06417066 0.07564516 8 0.04529774 0.05581492 0.06819426 0.08153439 0.09509815 9 0.05725145 0.06933186 . 0.08355017 0.09889811 0.11455114 10 0.06920515 0.08284879 0.09890608 0.11626183 0.13400414 PAGE 127 TABLE 11 IMAGINARY PART OF Ao COEFFICIENTS E = 0.00 = 0.05 ~= 0.10 ' = 0.15 E= Q.20 k A~( k) A" ( k) 0 A 11 ( k) 0 A" ( k) 0 A~( k) 0 -o.o6130994 -o.o6790729 -0.07477867 -0.08248793 -0.09201924 1 -0.05034068 -0.05530297 -o.o6078999 .;.o.o6730708 -0.07579395 .... 2 -0.03937143 -0.04269865 -0.04680131 -0.05212624 -0,05956867 .... (X) 3 -0.02840218 -0.03009433 -0.03281263 -0.03694540 -0.04334339 4 -0.01743292 -0.01749001 -0.0.882395 -0.02176456 -0.02711811 5 -O.Oo646367 -0.00488569 -0.00483527 -O.Oo658371 -0.01089282 6 0.00450558 .00771862 0.00915340 0.00859712 0.00533246 7 0.01547483 0.02032294 0.02314208 0.02377796 0~02155773 8 0.02644409 0.03292726 0.03713076 0.03895880 0.03778302 9 0.03741334 0.04553159 0.05111943 0.05413965 0.05400830 10 0.04838260 0.05813590 o.o6510811 o.o6932049 0.07023358 PAGE 128 TABLE 12 ~PART OF A 1 COEFFICIENTS E = 0.00 ~= 0.05 f:= 0.10 E = 0.15 ~= 0.20 k Ai( k) A~( k) A~( k) Ai_ ( k) ~( k) 0 0.04480435 0.03691892 0.02820379 0.01855323 0.00751369 1 0.02438222 0.01978109 0.01462211 0.00851101 0.00073587 0.00396009 0.00264326 0.00104044 -0.00153120 -0.00604195 2 tO 3 -0.01646204 -0.01449455 -0.01254123 -0.01157342 -0. 01281978 4 -0.03688417 -0.03163237 -0.02612291 -0.02161564 -0.01959760 5 -o.0573o630 -0.04877019 -0.03970458 -0.03165786 -0. 026 37544 . '6 -0.07772844 -0.06590801 -0.05328625 -0.04170008 -0 .. 03315326 7 -0.09815057 -0.08304583 -0.06686793 -0.05174229 -0. 03993109 8 -0.11857271 -0.10018366 -0.08044960 -0.06178451 -0.04670892 9 -0.13899484 -0.11732148 -0.09403127 -0.07182673 -0.05348674 10 -0.15941697 -0.13445930 -0.10761295 -0.08186895 -0.06026457 PAGE 129 TABLE 13 IMAGINARY.PART OF A 1 COEFFICIENTS E::: o.oo = 0.05 &::: 0.10 E = 0.15 (:= 0.20 k Ai_( k) Ai( k) Ki( k) Ai( k) A'i( k) 0 0.04014401 0.03727745 0.03463721 0.03213027 0.02980281 1 0.02356078 0.01986369 0.01696925 0.01486909 0.01362807 h) 2 0.00697754 0.00244995 -0.00069870 -0.00239210 -0,.00254667 0 -0.00960568 -0.01496378 -0. 018 36666 . .::0.01965329 -0.01872141 4 -0.02618891 -0.03237753 -0.03603462 -0.03691448 -0.03489915 5 -0.04277214 -0.04979127 -0.05370258 -0.05417567 -0.05107089 6 -0.05935538 -0.06720501 -0.07137053 -0.07143686 -0.06724563 7 -0.07593861 -0.68461875 -0.08903849 -0.08869805 -0.08342037 8 -0.09252185 -0.10203250 -0.10670645 -0.10595924 -0.09959511 9 -0.10910507 -0.11944624 -0.12437440 -0.12322043 -0.11576986 10 -0.12568831 -0.13685999 -0.14204236 -0.14048162 -0.13194460 PAGE 130 LIST OF REFERENCES 1. L. M.Milneâ€¢Thomson, Theoretical-Hydrodynamics, 2nd edition, The Macmillian Company (1960). 2. Sir Geoffrey I. Taylor, "The action of' waving cylindrical tails in propelling microscopic organisms," Proc. Roy. Soc. Lond., A 211 (1951), PPâ€¢ !25-239. 3. Sir Geoffrey I. Taylor, "Analysis of' the swimming of long and narrow animals," Proc~ Roy. Soc. Lond., A 214 (1952), pp. 158183. 4. J Siekmann, 11 0n a pulsating jet from the end of a tube, with application to the propulsion of' certain aquatic animals," Jour. Fluid Mech., vol. 15 (1963), PPâ€¢ 399-418. ,<"' / 5. 6. M. J. Soc., M. J. Jour. Lighthill, "Mathematics and aeronautics," Jour. Roy. Aero. vol.' 64 (1960), PPâ€¢ 373-394. Lighthill, "Note on the swimming of slender fish," Fluid Mech., vol. 9 (1960), PPâ€¢ 305-317. M. M. Munk, "The aerodrnamic forces on airship hulls," Tech. Report, 184 ( 1924). N.A.C.A. 8. J. Siekmann, "Theoretical studies of sea animal locomotion, Part l," Ing. Arch., Bd. 31, H~ 3, Berlin (1962), PPâ€¢ 214-228; Part 2, 'Ing.-Arch., Bd. 32, H. l (1963), PPâ€¢ 40-50. 9. L. Schwarz, "Berechnung der Druckverteilung einer harmonich sich verformenden Tragflache in ebener Stromung," Luftfahrforschung, 17 (1940), PPâ€¢ 379-386. 10. H. G. Kiissner and L~ Schwarz, "The oscillating wing with aerodynami cally balanced elevator," Luftfahrforschung, 17 (1940), PPâ€¢ 337â€¢ 354. (English Translations N.A.C.A. T.M. 991 (1941)). 121 PAGE 131 11. 12. 13. 14. 15. . 16. 122 T. Y. Wu, "Swimming of a waving plate," Jour. Fluid Mech., Vol. 10 (1961), pp~ 321. T. Y. Wu, "Accelerated swimming of a waving plate," Fourth S}:poâ€¢ sium on Naval Hydrodynamics, Washington, D. C. (August, 1962 , PPâ€¢ 430-446. E. H. Smith and D. E. Stone, 11 Perfect fluid forces in fish propulsions the solution of the problem in an elliptic cylinder co-ordinate system," Proc. Roy. Soc. Lond., A 261 (1961), .â€¢ pp. 316-328. S. K. Pao and J. Sielanann, "Note on the Smith-Stone theory of.fish propulsion, 11 submitted to the Royal Society. R. J. Bonthron and A. Fejer, "A hydrodynamic study of fish locomo tion," Illinois Institute of Technology, Chicago. Th. Theodor sen, 11 A general theory of' aerodynamic instability and the mechanism of' flutter," NACA Rept. 496 (1935). 17.., ... H. R. Kelly, "Fish propulsion hydrodynamics," Proc. 7th Midwestern Conference on Fluid Mechanics, Michigan State University, Sept. 1961, Plenum Press, Inc. (1962). . 18. Th. Theodorsen and I.E. Garrick, "General potential theory of' arbitrary wing sections," NACA Rapt. 452 (1933). 19. H. G. Kilssner and G. V. Gorup, "Instationare linearisierte Theorie der Flugelprof'ile endlicher Diche in inkompressibler Stromungt" Mitt. Max-Planck-Institut Strom. Forsch., 26, Gottingen (1960J. 20. Sir James Gray, 11 How fishes swim," Sci. Am. (August 1957), pp. 48-54. 21. A. Robinson and J. A. Laurmann, Wing Theory, Cambridge University Press (1956). 22. Sir Horace Lamb, Hydrodynamics, 6th ed., Dover Publications, New York (1932). PAGE 132 123 23. H. Wagner, "Ueber die Entstehung der dynamischen Auftriebes von Tragflugeln,â€¢ A. Angew. Math. Mech., 5, (1925), pp. 17-35. 24. G. N. Watson, A Treatise on the Theo 2nd ed., Cambridge University Press of Bessel Functions, 1944 25. Y. Luke and M. A. Dengler, "Tables of the Theodorsen circulation function for generalized motion," Jour. Aero. Sci., vol. 18 (1951), PPâ€¢ 478-483. 26. Staff of the Computation Laboratory, Cambridge, Mass., Tables of the Bessel Functions of the First Kind, Harvard University Press (1947). PAGE 133 BIOGRAPHICAL SKETCH John Paul Uldrick was born April 11, 1929, in Donalds, South _Carolina. He was graduated from Donalds High School in May, 1946. In . .â€¢ August, 1950, he received the degree of Bachelor of Science in Civil Engineering from Clemson College. Then, for a period of eight months Mr. tJldrick was employed by the South Carolina Highway Department as a . Bridge Construction Engineer. In April, 1951, he entered the United States Air Force and served as an Instructor in the Officers' Communi cations School at Scott Air Force Base in Illinois until his discharge in April, 1953. Mr. Uldrick then worked as a Field Engineer for J.E. Sirrine Company, Engineers, until September, 1954, when he re-entered Clemson College. He worked as a graduate assistant in the Department of Engineering Graphics until June, 1956, at which time he received a Bachelor of Science degree in Mechanical Engineering. He then accepted the position of Instructor in Engineering at Clemson College. Mr. Uldrick entered the Clemson Graduate School in June, 1956, and was awarded the degree of Master of Science in Mechanical Engineering in 1958. From 1958 to 1960 he served as an Assistant Professor of Engineering Mechanics at Clemson College. During the summer months, he worked for the J. E. Sirrine Company, Engineers, and in the summer of 1957 was a Faculty Associate for Lockheed Aircraft Company in Atlanta, Georgia. In September, 1960, he accepted a position as Interim Instructor in Engineering Mechanics at the University of Florida and also entered the University's Graduate 124 PAGE 134 125 School at this time. From June, 1961, until the present time he has, as a National Science Foundation Fellow, pursued his work toward the degree Doctor of Philosophy. John Paul Uldrick is married to the former Johnnye Steven Murdock and is the father of one child. He is a Registered Engineer in the State ot South Carolina and is a member of the .American Society for Engineering Education ~d th e American Society of Mechanical Engineers. PAGE 135 This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of that committee. It was submitted to .;the Dean of the College of Engineering and to the Graduate Council, .. and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 10, 1963 d!::~ck~~. Dean, College of Engineering:::> Dean, Graduate School Supervisory Committees Chairman vJ~~~ ~4&t>c< |