Seminar Report

A REVIEW ON HYDRODYNAMICS OF FISH-LIKELOCOMOTION AND A STUDY ON EFFECT OF

CHORDWISE FLEXIBILITY ON HYDRODYNAMICPROPULSION OF A PITCHING HYDROFOIL

First Annual Progress Seminar

by

NAMSHAD T

144100005

Under the Guidance of:

Prof. Atul SharmaProf. Amit Agrawal

Department of Mechanical EngineeringIndian Institute of Technology Bombay

DEPARTMENT OF MECHANICAL ENGINEERING

INDIAN INSTITUTE OF TECHNOLOGY, BOMBAY

August 2015

Abstract

Various aspects of the physics of swimming has been reported in the literature, howeverthe effect of chordwise flexibility on hydrodynamic propulsion of pitching hydrofoil hasnot been studied so far. This is studied here, using level-set immersed boundary methodbased on an in-house 2D code. While the Navier-Stokes equation governing the fluidflow is solved using fully implicit finite volume method, level-set equation governing themovement of the hydrofoil is solved using finite difference method. It is found that thethrust generation is based on a lift based mechanism for pure pitching hydrofoil and acombination of press ure suction mechanism and lift based mechanism for pitching aswell as undulating motion of hydrofoil. The present study is undertaken for static-head-linear-motionkinematics (of fish-like locomotion) for the combined pitching and undulat-ing motion of hydrofoil. The wavelength of a wave travelling over the hydrofoil is foundas a parameter which leads to transition from pure pitching to combined pitching andundulating motion.

Keywords: Fish-like locomotion, Flexible hydrofoil, Rigid hydrofoil

Contents

List of Figures iii

List of Tables v

Nomenclature vi

1 Introduction 11.1 Fish-like locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objective and Scope of this Study . . . . . . . . . . . . . . . . . . . . . . . 31.3 Governing Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Literature Survey 122.1 Fish-like Locomotion: Undulating Motion . . . . . . . . . . . . . . . . . . 12

2.1.1 Inviscid Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.2 Anguilliform Swimming . . . . . . . . . . . . . . . . . . . . . . . . 132.1.3 Carangiform Swimming . . . . . . . . . . . . . . . . . . . . . . . . 202.1.4 Thunniform Swimming . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Fish-like Locomotion: Pitching and/or Heaving . . . . . . . . . . . . . . . 252.3 Flow Control in Hydrofoils . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4 Flow Control Using Fins . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.5 Simulation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5.1 Tethered Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5.2 Self-propelled Simulation . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Effect of Chordwise Flexibility on Hydrodynamic Propulsion of a Pitch-ing Hydrofoil 343.1 Physical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1.1 Pitching motion of hydrofoil . . . . . . . . . . . . . . . . . . . . . . 343.1.2 Undulatory as well as Pitching motion of hydrofoil . . . . . . . . . 353.1.3 Performance parameters . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Numerical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3.1 Instantaneous flow patterns for pure pitching and combined motion

of hydrofoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.2 Effect of chordwise flexibility on performance parameters . . . . . . 40

4 Conclusions and Scope for Future Work 48

ii

List of Figures

1.1 Swimming modes (A) Anguilliform (B)Subcarangiform (C)Carangiform(D) Thunniform (Lindsey 1978). . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Amplitude variation in anguilliform kinematics (Borazjani and Sotiropoulos2009) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Wake structure in anguilliform swimmer obtained by Carling et al. (1998) . 142.3 Proto-vortex formation in anguilliform swimmer (Müller et al. 2001) . . . . 152.4 Different types of vortex shedding observed by Müller et al. (2001). A)

Body and tail vortices of same sign merged B) Body and tail vortices ofopposite sign merged C) Body and tail vortices shed at different instances. 16

2.5 Vortex shedding behind eel observed by Tytell & Lauder (2004) . . . . . . 172.6 Double and single row of vortices A) Hypothesis by Müller et al. (2001) B)

Numerical results by Borazjani & Sotiropoulos (2009). . . . . . . . . . . . 182.7 Thrust generation mechanism in anguilliform swimmer (Borazjani & Sotiropou-

los 2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.8 Amplitude variation in carangiform kinematics (Borazjani & Sotiropoulos

2008). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.9 Suction and pressure regions in carangiform swimming (Müller et al. (1997)). 212.10 Development body vortex and tail vortex observed by Wolfgang et al. (1999)). 222.11 Vortex within a vortex structure (Borazjani & Sotiropoulos 2010). . . . . . 232.12 Thrust generation mechanism in anguilliform swimmer (Borazjani & Sotiropou-

los (2009)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.13 Thunniform swimming kinematics (Lindsey 1978). . . . . . . . . . . . . . . 252.14 Reversed von Karman vortex street. . . . . . . . . . . . . . . . . . . . . . . 272.15 Modes of vortex formation in oscillating hydrofoils behind a D-section

cylinder obtained by Gopalkrishnan et al. (1994) (1,2,3: foil vortices;A,B,C,D,E,F: cylinder vortices). . . . . . . . . . . . . . . . . . . . . . . . . 28

2.16 Effect of vortices shed by upstream D-cylinder on propulsion performanceproposed by Liao et al. (2003). . . . . . . . . . . . . . . . . . . . . . . . . 29

2.17 Effect of pectoral fin vortices for St = 0.1 (Yu et al. 2011). . . . . . . . . . 30

2.18 Higher local pressure at St = 0.8 (Yu et al. (2011)). . . . . . . . . . . . . . 302.19 Different types of caudal fin models used by Chang et al. (2012). A) Cres-

cent shaped B) Semicircular shaped C) Fan shaped. . . . . . . . . . . . . 312.20 Anguilliform creature showing its segement of the body with polar coordi-

nates (r (i) , φ (i)) with reference to the centre of mass (ò) (Carling et al.1998) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1 Computational domain for hydrodynamics of translating, pitching and un-dulating NACA0012 hydrofoil. . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Pitching NACA 0012 hydrofoil. . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Streamwise varying transverse displacement of undulating NACA0012 hy-

drofoil to obtain its intial wavy/fish-like shape; at the onset of locomotion. 363.4 For pure pitching motion, temporal variation of instantaneous (a1-d1) ve-

locity contours and (a2-d2) pressure contours within one time period atAmax = 0.1 (θmax = 5.74o), St=0.7 and Re = 400. Sub-figure (e) repre-sent the temporal variation of thrust and lift coefficients with the symbolscorresponding to the time instant for the instantaneous plots. . . . . . . . 39

3.5 For combined motion, temporal variation of instantaneous (a1-d1) velocitycontours and (a2-d2) vorticity contours within one time period at λ=1.0,Amax = 0.1, Re = 400 and St=0.7;. Sub-figure (e) represent the temporalvariation of thrust and lift coefficients with the symbols corresponding tothe time instant for the instantaneous plots. . . . . . . . . . . . . . . . . . 41

3.6 For combined motion, variation of mean thrust coefficients with increasingwavelength, for Amax = 0.1, Re = 400 and St = 0.7. . . . . . . . . . . . . . 42

3.7 Temporal variation of thrust and lift coefficients within one time period ofundulation at Amax = 0.1, Re = 400 and St = 0.7 for (a) λ = 2.0 and (b)λ = 7.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.8 Variation of viscous thrust coefficient with St for different wavelengths atAmax = 0.1 and Re = 400. . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.9 (a) (i) Vorticity contour and (ii) Variation of local viscous thrust coefficientalong the right surface for Re = 400, Amax = 0.1, St = 0.7 and λ = 1.0.

(b) Variation of tangential velocity profile along the normal at the point ofmaximum viscous thrust on the upper surface. . . . . . . . . . . . . . . . . 45

3.10 Variation of pressure thrust coefficient with St for different wavelengths atAmax = 0.1 and Re = 400. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.11 Variation of pressure thrust coefficient with St for different wavelengths atAmax = 0.1 and Re = 400. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.12 Variation of propulsive efficiency with St for different wavelengths atAmax =

0.1 and Re = 400. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

iv

List of Tables

1.1 Summary of the literature survey . . . . . . . . . . . . . . . . . . . . . . . 4

Nomenclature

amax Maximum amplitude of undulationHa Heaving amplitudek Frequency of oscillationL Length of fishPa Pitching amplitudeRe Reynolds numberSt Strouhal numberu∞ Swimming speed

Greek symbols

θ Feathering parameterΩ1 Strength of oncoming vortices

Chapter 1

Introduction

Solving of complex human problems by drawing inspiration from natural systems and ele-ments has been started years ago; leading to an evolution of a subject called as biomimetic.One of them is the proposition of fish-like locomotion as an efficient propulsion system;for aquatic vehicles and autonomous underwater vehicle (AUV). The major motivationbehind this proposition is the surprisingly higher propulsive efficiency of fish-like loco-motion compared to conventional aquatic propulsion system. The body of fish propel bypushing the fluid backward with minimum energy loss.

Eventhough the basic mechanism of thrust generation is almost same in all types offishes, the optimum swimming condition depends on the shape and kinematics. The shapeand kinematics depends on the purpose and requirement of fish swimming. For example,high speed swimming fishes will have different form and kinematics compared to low speedfishes. The mechanism of fishes can be best understood by a detailed flow analysis. Thiswill help in designing aquatic vehicles or AUVs for a wide range of applications.

1.1 Fish-like locomotion

Most of the aquatic animals use body/caudal fin for swimming. Based on the swim-ming kinematics used, aquatic animals are classified into four categories; anguilliform,sub-carangiform, carangiform, and thunniform. Figure 1.1 shows the four different cate-gories. In anguilliform swimming, the whole body is undulating whereas in thunnuiformswimming only tail fin is undulating. Amplitude of undulation in the anterior part of thefish is maximum for anguilliform type and minimum for thunniform type.

The shape of fishes also differs for different categories. The body of anguilliformswimmers are characterised as thin long and laterally compressed towards posterior part(Lindsey 1978). Carangiform swimmers are having a streamlined anteriorpart and acaudal fin with narrow peduncle. Thunniform swimmers are usually heavy towards theanterior side. The cross section of the tail is usually hydrofoil shaped and it follows a

1

Figure 1.1: Swimming modes (A) Anguilliform (B)Subcarangiform (C)Carangiform (D)Thunniform (Lindsey 1978).

2

pitching and heaving motion.The fish-like locomotion can be considered as a proper combination of undulatory and

heaving motion. The type of motion used depends on the requirements of the locomo-tion. Any system desgning baed on fish-like locomotion should use any of the fish-likelocomotion which gives better performance.

1.2 Objective and Scope of this Study

Lighthill (1969) was probably the first to review the fluid flow analysis of fish-like locomo-tion. In his review, different types of aquatic swimmers, different swimming kinematics,and some inviscid flow analyses were discussed. Triantafyllou et al. (2000) presented adetailed review on fish-like locomotion. This review includes the two-dimensional compu-tational fluid dynamics (CFD) analysis of oscillating foils and few experimental studieson hydrofoils. Sfakiotakis et al. (1999) reviewed the kinematics and mechanisms of thrustgeneration in different types of aquatic animals. A recent review on hydrodynamic anal-ysis on fish-like swimming including 3D analysis is not found in the published literature.

This study includes a literature review which covers different types of fish-like locomo-tion. The summary of the literature review conducted is shown in table 1.1. This studyalso includes a comparison of rigid (pitching) and undulating hydrofoils.

1.3 Governing Parameters

There are many parameters influencing the propulsive performance of fish-like locomotion.Like any other fluid flow problem, the first parameter is the Reynolds number

(Re = u∞L

ν

)which is usually calculated based on the swimming velocity at constant speed swimmingcondition (u∞) and the maximum length of the fish (L). The second important parameteris the Strouhal number

(St = 2amaxk

u∞

)which is the non-dimensional representation of the

frequency of undulation of the fish. The maximum amplitude of undulation (amax) istaken as the length scale for calculating the Strouhal number. In the case of pitching andheaving hydrofoils, the maximum amplitude of pitching (Pa) and maximum amplitude ofheaving (Ha) also plays a significant role in the propulsive performance. In this study,effect another paramter called the wavelength of the travelling wave of undulatory motion(λ) is also considered.

3

Table 1.1: Summary of the literature survey

Sl No. Year, Authors, Title Parameters, Remarks

11933, Gray. J, Directionalcontrol of fish movement

• Kinematics of fish-likelocomotion.

• Mechanism of thrust generation.

21960, Lighthill, Note on theswimming of slender fish

• Inviscid solution for oscillatingslender fish.

• Conditions for high propulsiveefficiency.

31969, Lighthill, Hydromechanicsof aquatic animal propulsion

• Reviewed modes of aquaticanimal propulsion.

• Mechanism of thrust generation isdiscussed.

41970, Lighthill, Aquatic animalpropulsion of highhydromechanical efficiency

• θ=0, 0.2, 0.4, 0.6, 0.8

• Inviscid solution for differentswimming modes.

Sl. No. Year, Authors, Title Parameters & Remarks

5 1972, Weihs, Semi-infinite vortextrails, and their relation tooscillating airfoils

• St=0.001, 0.335, 0.6

• Vorticity control in oscillatingairfoil

6 1978, Lindsey, Form, functionand locomotory habits in fish • Form and kinematics and

swimming mechanism in variousaquatic animals

7 1983, Weihs & Webb,Optimization of locomotion • Concept of fish schooling

introduced

8 1984, Videler & Hess, Fastcontinuous swimming of twopelagic predators, saithe(Pollachius virens) and mackerel(Scomber scombrus): akinematic analysis

• Kinematics analysis of Mackereland Saithe.

9 1984, Tokoumaru & Dimotakis,Rotary oscillation control of acylinder wake

• Re=15000; St=0-3.5; Ω1=0-16

• Vorticity control in rotatingcylinder.

5


101992, Bilckhan et al., Generationof a vortex chain in the wake ofa Suhundulatory swimmer

• 3D vortex structure of rainbowtrout swimming usinglaser-Doppler anemometry

11

1993, Triantafyllou et al.,Optimal thrust development inoscillating foils with applicationto fish propulsion

• St=0-0.4

• Thrust generation in pitching andheavind hydrofoil

• Optimal efficiency obtained formaximum spatial amplification(0.25 < St <0.35

121994, Gopalkrishnan et al.,Active vorticity control in ashear flow using a flapping foil

• Re=0-700; Ha=0.5-0.833, Pa=15o-45o

• Votrticity control in oscillatinghydrofoil behind a D-sectioncylinder.

• Interaction of cylinder and foilvortices

• Maximum thrust when two samesign vortices merges.

131996, Liu et al., A computationalfluid dynamics study of tadpoleswimming

• Re=500 -100000; St=0.72

• Hydrodynamics of tadpole andfish swimming.

6


141996, Streitlien et al., On thrustestimates for flapping foils

• Ha=0.4-0.833; Pa=0.2-0.785

• Interaction of oscillating foilvortices with incoming vortices.

• Maximum efficiency when foilmoves in proximity withoncoming vortices.

15

1997, Müller et al., Fish footprints: morphology andenergetics of the wake behind acontinuously swimming mullet(Chelon labrosus Risso)

• Re=27500; St=0.34

• Hydrodynamics of mulletswimming using PIV.

• Pressure and suction flowmechanism.

161998, Anderson et al., Oscillatingfoils of high propulsive efficiency

• St=0-0.6; Re=1100

• Hydrodynamics of oscillating foilsusing DPIV.

• Optimum thrust production for0.25<St<0.4, large heaveamplitude to chord ratio.

17

1998, Carling et al.,Self-propelled anguilliformswimming: simultaneous solutionof the two-dimensionalNavier-Stokes equations andNewton’s laws of motion

• Re=83

• Transient flow characteristics of2D anguilliform swimming.

• Thrust generation is only due tobody motion and tail motioninduces drag.

7


181999, Wolfgang et al., Near-bodyflow dynamics in swimming fish

• St=0.45

• Wake structure in carangiformswimmer using experimental andnumerical studies.

• Body bound and tail generatedvorticity.

192000, Triantafyllou et al.,Hydrodynamics of fishlikeswimming

• Review of vorticity control,oscillating foils, and fishswimming.

20

2001, Müller et al., How thebody contributes to the wake inundulatory fish swimming flowfields of a swimming eel(Anguilla anguilla)

• Re=11900-17300

• Proto-vortex, body vortex andtail vortex formation.

• Interaction of body and tailvortices.

21

2003, Liao et al., The Kármángait: novel body kinematics ofrainbow trout swimming in avortex street

• Swimming behind a D-sectioncylinder.

• Effect of cylinder vortices onthrust generation.

222004, Lau et al., Flowvisualisation of a pitching andheaving hydrofoil

• Re=870-3480; St=0.06-1.37

• Hydrodynamics of pitching andheaving hydrofoil.

8


232004, Tytell & Lauder, Thehydrodynamics of eel swimmingI. Wake structure

• Re=60000; St=0.314

• Formation of secondary vortex.

• Formation of lateral jet betweenprimary and secondary vortices.

24

2005, Blondeaux et al.,Numerical experiments onflapping foils mimicking fish-likelocomotion

• Re=164; St=0.175- 0.35

• Hydrodynamics of 3D foilmimicking tail motion ofcarangiform swimmer.

25

2005, Gilmanov & Sotiropoulos,A hybrid Cartesian/immersedboundary method for simulatingflows with 3D, geometricallycomplex, moving bodies

• Immersed boundary method formoving boundary problems.

• Moving boundary with prescribedkinematics.

262005, Yu, A DLM/FD methodfor uid/flexible-body interactions

• Fluid-structure interactionproblem.

272006, Kern & Koumoutsakos,Simulations of optimizedanguilliform swimming

• Re=3857; St=0.59-0.67

• Comparison of burst swimmingand efficient swimming.

• Burst swimming uses tail forthrust generation whereas efficientswimming uses anterior part also.

28

2006, Sarkar & Venkatraman,Numerical simulation ofincompressible viscous ow past aheaving airfoil

• Re=10000; k=4-8

• Thrust generation in asymmetricheaving airfoil.

9


29

2008, Borazjani & Sotiropoulos,Numerical investigation of thehydrodynamics of carangiformswimming in the transitional andinertial flow regimes

• Re=300, 4000,∞; St=0.1-0.7

• 3D analysis of hydrodynamics ofcarangiform swimming.

• Single and double row vorticesformation.

30

2009, Borazjani & Sotiropoulos,Numerical investigation of thehydrodynamics of anguilliformswimming in the transitional andinertial flow regimes

• Re=300, 4000,∞; St=0.1-0.7

• 3D analysis of hydrodynamics ofanguilliform swimming.

• Thrust generation mechanism incarangiform and anguilliformswimmers.

31

2010, Borazjani & Sotiropoulos,On the role of form andkinematics on thehydrodynamics of self-propelledbody/caudal fin swimming

• Re=300, 4000,∞; St=0.3-1.1

• Effect of body shape andkinematics on propulsiveperformance.

• Comparison of anguilliform andcarangiform swimmers.

32

2010, Yu et al.,Three-dimensional numericalsimulation of hydrodynamicinteractions between pectoral-finvortices and body undulation ina swimming fish.

• Re=33000; St=0.2-0.8

• Effect of pectoral fin onpropulsion performance.

• Pectoral fin vortices reduces thepower requirement of swimming.

10


33

2012, Chang et al., Numericalstudy of the thunniform mode offish swimming with differentReynolds number and caudal finshape

• Re=7100-710000; St=0.1-0.5

• Effect of turbulence on thrustgeneration.

• Turbulence reduces form drag bymaking the flow separation weak.

34

2013, Shrivastava et al., A NovelLevel Set-BasedImmersed-Boundary Method forCFD Simulation ofMoving-Boundary Problems

• New immersed boundary methodfor fluid-structure interactionproblems.

11

Chapter 2

Literature Survey

Fluid dynamic analysis of fish-like locomotion has been a subject of several studies in thepast few decades. Studies on fish-like locomotion probably started with Gray (1933). HeObserved the swimming kinematics of different aquatic animals and tried to explain thethrust generation mechanism. Later, inviscid analysis was used to propose theories onfish propulsion, considering hydrodynamics of pitching and/or heaving and undulatingbody (Lighthill 1960, OHAsHi & ISHIKAWA 1972, and Triantafyllou et al. 1993). Recentstudies on fish-like locomotion are carried out with the help of advanced experimentaltechniques and numerical simulations (2D and 3D).

2.1 Fish-like Locomotion: Undulating Motion

A review on flow analyses performed on undulating bodies and real fishes are presentedhere.

2.1.1 Inviscid Analysis

Lighthill (1960) was probably the first one to study analytically the fish-like locomotion.He predicted the thrust generation using inviscid theory and slender body approximation.An analytical expression was obtained for the propulsive efficiency using momentum bal-ance equation. The kinematics which gives maximum efficiency was found to satisfy thefollowing conditions:

• The lateral motion should be restricted to the posterior part.

• The wave velocity should be near the free stream velocity.

• The wave form should have a positive and negative phases on regions of substantialamplitude so that recoil motion can be minimized.

12

Lighthill (1970)studied the effect of modifications in undulatory mode of swimming onthrust development and propulsive efficiency using inviscid approximations. Differentmodes of swimming were considered in this study. He found that in anguilliform swimmingmode, the propulsive efficiency decreases if the momentum at the trailing edge is badlycorrelated with the edge’s lateral momentum. He also found that, the restriction oflateral motion to the posterior part and reduced depth of posterior section in carangiformswimmers is to reduce the recoil movement. The wasted energy in the wake was foundminimum when the yawing axis of the caudal fin is at three-quarter-chord point.

Cheng et al. (1991) used linear three-dimensional waving plate theory to analyse theundulatory motion of rectangular and triangular plates. From the three-dimensional anal-ysis, it was concluded that better performance is obtained for undulating plate with con-stant span and slightly varying amplitude. Fishes with anguilliform kinematics also holdsthe same characteristics. it was also found that triangular plate gives better performancefor carangiform kinematics. The explains the advantage of necking in carangiform swim-mers.

2.1.2 Anguilliform Swimming

The force required for the movement of a fish is produced by undulatory movement of thefish which passes a transverse wave along the body in the opposite direction of swimming(Müller et al. 2001). The amplitude and frequency of the transverse wave depends on themode of swimming. Aquatic animals with high maneuverability ranging from nematodesto eels uses a swimming mechanism called anguilliform swimming (Müller et al. 2001).Anguilliform is purely undulatory mode of swimming in which most or all of the lengthof the body participates in thrust generation (Lindsey (1978)). Figure 1.1 (A) shows thesame. The body of anguilliform swimmers are characterized as thin long and laterallycompressed towards the posterior (Lindsey 1978). Gray (1933) conducted a detailedstudy of kinematics of fish swimming and concluded that anguilliform type of swimmersmoves with a backward traveling wave whose amplitude linearly varies from head to tailof the fish. Borazjani & Sotiropoulos (2009) used the following equation for swimmingkinematics of anguilliform body.

h(z, t) = a(z)sin(kz − ωt) (2.1)

where z is the axial direction , a(z), k, and ω are the amplitude, wave number, andfrequency of the wave respectively. The amplitude envelope is shown in fig. 2.1.

Wake Structure

Carling et al. (1998) studied the hydrodynamics of anguilliform swimming using two-

13

Figure 2.1: Amplitude variation in anguilliform kinematics (Borazjani and Sotiropoulos2009)

dimensional computational fluid dynamic analysis performed on a creature. They founda large vortex ring formed around the body as shown in fig. 2.2. They concluded thatthe thrust generation in anguilliform simmer is entirely by the body movement and thetail tip produces only drag.

A contrary result was obtained by Müller et al. (2001) using an experimental studyconducted on eel swimming. Two types of vortices were identified in the wake of eel; bodyvortices, and tail vortices. Eel moves in the fluid by passing a backward moving waveswith amplitudes increasing from head to tail. high and low pressure regions forms along

Figure 2.2: Wake structure in anguilliform swimmer obtained by Carling et al. (1998)

14

Figure 2.3: Proto-vortex formation in anguilliform swimmer (Müller et al. 2001)

the crest and troughs of the fish. The pressure difference makes flow from crest to troughas shown in fig. 2.3. The flow from crest to posterior trough is stronger than that of theflow from crest to anterior trough. The nature of rotation of the semicircular flow formedwill be same on right and left sides which resembles like the potential part of vortexwith centre inside the body of the eel. Figure 2.3 shows the vortex formation due to thissemicircular flow called as proto vortex. When body flow coincides with peak velocityof the tail, it shed off as body vortices. In addition to the body vortices, the oscillatorymotion of the tail produces tail vortices. The nature of thrust generation depends on thephase angle between body vortices and tail. Müller et al. (2001) concluded that there arethree types of vortex shedding in the wake as shown in fig. 2.4.

1. Body and tail vortices of same rotational sense merges and form a single reversevon-Karman vortex street with maximum thrust generation (Figure 2.4.A).

2. Body and tail vortices of opposite sense of rotation merges and form a weak reversevon-Karman vortex street with maximum propulsive efficiency (Figure 2.4.B).

3. Body and tail vortices shed at discrete instances (Figure 2.4.C).

Due to some experimental limitations and uncertainty in the movement of eel, Mülleret al. (2001) could not perform the experiment for various speed conditions and theresults obtained was not accurate enough to give adequate informations about the flowcharacteristics.

A more accurate study was conducted by Tytell & Lauder (2004) using high resolutionPIV techniques. They conducted a series of experiments and they could obtain the flowcharacteristics for constant swimming speed conditions. The mechanism of formation ofvortices are explained in a detailed way. Two types of vortices are obtained in this study;primary vortices and secondary vortices. Figure 2.5 shows shows the same. Primary vor-tices are the stop/start vortices formed by the combination of stopping vortex of previoushalf cycle and starting vortex of the subsequent half cycle (Lau et al. 2004). These vorticesare shed when the tail changes the direction. When the tail is at the extreme position,

15

Figure 2.4: Different types of vortex shedding observed by Müller et al. (2001). A) Bodyand tail vortices of same sign merged B) Body and tail vortices of opposite sign mergedC) Body and tail vortices shed at different instances.

16

Figure 2.5: Vortex shedding behind eel observed by Tytell & Lauder (2004)

there will be a maximum velocity region at certain distance anterior to the tail tip. Thismaximum velocity pulls the fluid and suction region is formed. This bolus of fluid willshed off and stretches the primary vortex and forms a shear layer. Due to the instabilityof the shear layer, it will rolls up and sheds as vortices which are called secondary vor-tices. This primary vortex from one half tail beat and secondary from the next form thelateral jet. Each full tail beat produces two lateral jets, one to each side and two vorticesseparating them. The component of lateral jets formed in the swimming direction wasjust balancing the upstream momentum excess formed by the head movement, which isthe evidence of constant speed swimming condition. In contrast to Müller et al. (2001),Tytell & Lauder (2004) found that the component of thrust production due to body un-dulation is very less. Even though Tytell & Lauder (2004) could find the formation ofproto vortices, the phase lag between proto vortices and primary vortices was found smalland shed as a single vortex.

A complete three-dimensional study on anguilliform swimming was presented by Kern& Koumoutsakos (2006) to find the hydrodynamics of optimum swimming condition. Thecomputational study was conducted for three different swimming kinematics; maximumswimming efficiency, maximum swimming speed, and kinematics used by Carling et al.(1998). From the wake characteristics obtained for two-dimensional and three-dimensionalstudies for thye reference condition corresponding to Carling et al. (1998), it was foundthat the vorticity break up into secondary vortices are present only in three-dimensionalstudy. From the three-dimensional study, it was found that primary and secondary vor-tices formed per half tail beat cycle resulted in two lateral jets similar to the experimentalresults obtained by Tytell & Lauder (2004). The formation of secondary vortices werefound absent in two-dimensional study. Kern & Koumoutsakos (2006) compared themaximum speed and efficiency conditions also. The major differences between maximumspeed and maximum efficiency conditions are as follows:

1. Undulatory motion of whole the body is responsible for thrust production in maxi-mum efficiency condition whereas only the tail oscillation is used for thrust genera-

17

Figure 2.6: Double and single row of vortices A) Hypothesis by Müller et al. (2001) B)Numerical results by Borazjani & Sotiropoulos (2009).

tion in maximum speed condition.

2. The lateral and longitudinal components of vortices formed were higher for maxi-mum speed condition.

3. The vortex rings formed were found stronger and elongated for maximum speedcondition.

Double row of pairs of vortices forming two lateral jets per tail beat cycle obtainedby Müller et al. (2001) was verified by Borazjani & Sotiropoulos (2009) using a three-dimensional computational study conducted on lampray swimming. Formation of doublerow vortices were found directly related to the Strouhal number which is a measure of theratio of lateral velocity to longitudinal velocity. Single row vortices of drag producing typewas obtained for low Strouhal number. Anguilliform swimmers always swims at higherStrouhal number and double row vortices will be always present in the wake. Borazjani& Sotiropoulos (2009) confirmed the formation of vortex rings for single row and doublerow of vortices hypothesized by Müller et al. (2001). Figure 2.6 shows the vortex ringfor both the cases. Borazjani & Sotiropoulos (2009) also studied the effect of Reynoldsnumber on the wake structure. Efficiency of anguilliform type of swimmers was found

18

Figure 2.7: Thrust generation mechanism in anguilliform swimmer (Borazjani &Sotiropoulos 2009).

maximum with in the transitional Reynolds number.

Thrust Generation Mechanism

Hydrodynamics of fish-like locomotion was first studied by Lighthill (1970). Accordingto Lighthill’s slender body theory, the thrust generation is entirely due to tail movement.The inviscid theory excluded the effect of body undulation of anterior part of the bodyon thrust generation. Later, Müller et al. (2001) presented an experimental investigationon swimming characteristics of eel. They explained the mechanism of thrust generationmechanism for anguilliform swimmers based on the experimental results. As per Mülleret al. (2001), thrust is generated along the whole body by undulatory pumping in whichthe speed of the fluid adjacent to the body is increases from head to tail as a result ofthe backward wave traveling along the body. A detailed description of the mechanismproposed by Müller et al. (2001) is given by Borazjani & Sotiropoulos (2009).

Consider a fluid particle near the body of the anguilliform fish as shown in fig. 2.7.The relative velocity of the fluid particle will be given by U-V, where U is the swimmingvelocity and V is the velocity of the backward traveling wave along the body. Thisrelative velocity results in forces acting in normal and tangential directions (Fnand Ft).Components of these forces in the swimming direction results in the thrust force. Theundulatory pumping mechanism will be present only if the fluid is viscous, because therelative velocity of the fluid near the body will be zero in the case of inviscid fluid. So,for very high Reynolds number, thrust generation will be entirely due to tail motion.

19

Figure 2.8: Amplitude variation in carangiform kinematics (Borazjani & Sotiropoulos2008).

2.1.3 Carangiform Swimming

Carangiform swimming is the mode of propulsion used by most of the fishes in whichonly the posterior portion of the body is capable of wide flexure (Lindsey 1978). Fishessuch as Mackerel uses one of the fastest mode of swimming, the carangiform swimming(Borazjani & Sotiropoulos 2008). The approximate kinematics of this type of swimmingis shown in fig. 1.1 (B). the movement of caudal fin is responsible for majority of thrustgeneration.

Gray (1933) and Videler & Hess (1984) conducted a detailed study on the kinematicsof different types of fishes. According to them, the undulatory motion in carangiform typeof swimmer is in the form of a backward traveling wave with maximum amplitude at thetail. The head is having very small amplitude and it decreases to zero up to the pivotpoint and increases up to the tail. Borazjani & Sotiropoulos (2008) used the kinematicsobtained by Videler & Hess (1984) by considering only the first term in the Fourier series.The equation describing the undulatory movement of the carangiform swimmer used inBorazjani & Sotiropoulos (2008) is same as Eqn. (2.1). The amplitude variation a(z) isshown i fig. 2.8.

Wake Structure

Blickhan et al. (1992) hypothesized that the undulatory movement creates suction andpressure flows which forms a circulatory flow around the inflection point of the body

20

Figure 2.9: Suction and pressure regions in carangiform swimming (Müller et al. (1997)).

and is shed when the inflection point reaches the tail. Experimental study conductedby Müller et al. (1997) confirmed the hypothesis. The undulatory movement of the fishwas found to produce suction and pressure regions on the convex and concave surfacesas shown in fig. 2.9. The maximum velocity at the points of maximum lateral excursionis alternated by a minimum velocity region at the inflection points and the points wheremid-lines crosses, which forms vortices. The vortices with centre as inflection points ormidline crossing points moves towards the tail and combines with tail vortices and shedin the wake. The velocities in the suction and pressure regions was found to increaseas the centre of the vortices moves down the body. Two vortices are shed per tail beatcycle, clockwise vortices are shed to the left side and anticlockwise vortices are shed to theright side. A jet flow is formed between these two jets which is a signature of the thrustproduction. 50 percentage of the thrust generated was contributed by the undulatorypumping mechanism and the remaining 50 percentage by the oscillation of the tail.

Wolfgang et al. (1999) could give an explanation to the two-dimensional wake struc-ture formed in the carangiform swimmers using experimental and computational studyconducted to obtain the propulsive characteristics of giant danio swimming. Formationof the body vortices and tail vortices were observed in the experiments. Figure 2.10 showsthe wake structure obtained in their experiments. When the tail tip is at the left mostposition, a clock-wise vortex with centre at tail tip is formed as a result of the vorticitygenerated due to the undulatory motion. During the rightward motion, tail tip vortices ofthe same sign will be formed and combines with the body shed vortex. A counterclockwisevortex will be formed at a position approximately mid-body of the fish by the undulatorymotion which will be shed in the next cycle.

Three-dimensional wake structure of a carangiform swimmer for various Reynoldsnumber and Strouhal number was studied by Borazjani & Sotiropoulos (2008). Theyobtained two different types of wakes with single row and double row vortices. They alsodifferentiated drag producing and thrust producing wakes. Double row of vortices were

21

Figure 2.10: Development body vortex and tail vortex observed byWolfgang et al. (1999)).

22

Figure 2.11: Vortex within a vortex structure (Borazjani & Sotiropoulos 2010).

obtained for high Strouhal number where the ratio of the lateral velocity of tail undulationto the swimming velocity is less. Double row of vortices were considered as a signature ofthree-dimensionality. The vortex structure was found to become more complex and andpresence of small scale vortices were obtained for higher Reynolds number. Borazjani &Sotiropoulos (2008) also studied the effect of Reynolds number and Strouhal number onthrust generation. For low Reynolds number, thrust producing vortices were obtained athigher Strouhal number. The propulsive efficiency was found maximum in the inertialflow regime.

Borazjani & Sotiropoulos (2010) observed a different types of vortex formation atthe tail tip of a carangiform swimmer. A vortex with in as vortex vortex structure wasobtained as shown in fig. 2.11. One vortex was formed at the leading edge of the tail andanother at the trailing edge.

Barrett et al. (1999) compared the drag in robo tuna under self-propelled and towingconditions. It was found that drag is lesser for fish-like moving body compared to rigidbody towing at the same swimming speed. The drag reduction was found maximumfor Strouhal number corresponding to maximum efficiency and ratio of wavelength toswimming speed greater than 1. It was concluded that laminarization of the boundarylayer and controlling of the vorticity by the tail which manipulate body-shed vorticity tocreate reverse karman street are responsible for the drag reduction.

Thrust Generation Mechanism

Hydrodynamics of fish-like locomotion was first studied by Lighthill (1970). Accordingto Lighthill’s slender body theory, the thrust generation is entirely due to tail movement.The inviscid theory excluded the effect of body undulation of anterior part of the body

23

Figure 2.12: Thrust generation mechanism in anguilliform swimmer (Borazjani &Sotiropoulos (2009)).

on thrust generation. Later, Müller et al. (2001) presented an experimental investigationon swimming characteristics of eel. They explained the mechanism of thrust generationmechanism for anguilliform swimmers based on the experimental results. As per Mülleret al. (2001), thrust is generated along the whole body by undulatory pumping in whichthe speed of the fluid adjacent to the body is increases from head to tail as a result ofthe backward wave traveling along the body. A detailed description of the mechanismproposed by Müller et al. (2001) is given by Borazjani & Sotiropoulos (2009).

Consider a fluid particle near the body of the anguilliform fish as shown in fig. 2.12.The relative velocity of the fluid particle will be given by U-V, where U is the swimmingvelocity and V is the velocity of the backward traveling wave along the body. Thisrelative velocity results in forces acting in normal and tangential directions (Fnand Ft).Components of these forces in the swimming direction results in the thrust force. Theundulatory pumping mechanism will be present only if the fluid is viscous, because therelative velocity of the fluid near the body will be zero in the case of inviscid fluid. So,for very high Reynolds number, thrust generation will be entirely due to tail motion.

2.1.4 Thunniform Swimming

Aquatic animals like butterfish uses a different type of swimming in which the undulatorymotion is restricted to the peduncle and tail fins (Lindsey 1978). In this type of swim-ming called thunniform swimming, the thrust is generated exclusively by a stiff caudal finmounted on an extremely narrow peduncle. Figure 2.13 shows the schematic represen-tation of the thunniform swimmer. Body of the thunniform swimmers are usually heavytowards the anterior side and is beautifully streamlined. The cross section of the tail isusually hydrofoil shaped and it follows a pitching and heaving motion. The mechanism ofthrust generation is more or less similar to that of pitching and heaving hydrofoils. Most

24

Figure 2.13: Thunniform swimming kinematics (Lindsey 1978).

of the thunniform swimmers are meant to swim at higher velocities.Chang et al. (2012) carried out a numerical simulation of the hydrodynamics of tuna-

like fish swimming at different Reynolds numbers (Re = 7100, 71000, 710000). Theystudied the propulsion characteristics at cruising condition using laminar and turbulentmodels. Effect of Reynolds number and turbulence on the drag coefficient was studied.Turbulence effect was found predominant for higher Reynolds number. At Re = 71000,turbulence was found reducing the form darg by making the flow separation weak. Forvery high Reynolds number (Re = 710000), even though the form drag is comparativelylesser, skin friction drag was found increased due to increase in turbulence. Chang et al.(2012) concluded that it is appropriate for thunniform swimmers to swim at relativelyhigher speed.

2.2 Fish-like Locomotion: Pitching and/or Heaving

Majority of fishes swimming at higher speeds uses either carangiform or thunniform kine-matics in which the oscillatory motion of caudal fin is the main source of thrust generation.Studies conducted on pitching and/or heaving foils could explain the thrust generationmechanism using caudal fin.

There has been many studies on hydrodynamic analysis of pitching and/or heavinghydrofoils using inviscid analysis. Weihs (1972) presented an analytical method to predictthrust generation in oscillating hydrofoil using semi-infinite reverse von Karman vortexstreet behind the oscillating body. Triantafyllou et al. (1993) investigated the pitchingand heaving motion of hydrofoil using experimental techniques and inviscid theory. Theyfound that the vortices formed at the trailing edge are convectively unstable. For certainrange of parameters, the shear layer rolls up into vortices and a reverse von Karman vortexstreet is formed.The maximum efficiency condition was obtained for Strouhal numberbetween 0.25 to 0.35 at which the spatial amplification is maximum. Chopra & Kambe

25

(1977) studied the lunate tail motion using inviscid theory. Different shapes of lunate tailswere considered. They found that the contribution of thrust due to leading edge suctionis lesser in tail with curved leading edge which is advantageous because very high leadingedge suction causes thrust reduction due to boundary layer seperation.

OHAsHi & ISHIKAWA (1972) carried out an experimental study on hydrodynamicsof heaving hydrofoil. They analysed the effects of frequency of oscillation and heavingamplitude on propulsive performance for Re ranging from 104to 4 × 104. At very lowfrequency, as the as the wave length associated with the heaving motion is small com-pared to the chord length, the wake deforms very less and a wavy profile is formed. Athigher frequency, the vortices are formed and detach from the trailing edge to form re-verse von Karman vortex street (jet). The frequency at which jet flow forms will reducewith increase in heaving amplitude. the convective velocity of vortices formed plays agreat role in the formation of jet flow. Jet flow is formed when the ration of convectivevelocity of vortices to the free stream velocity is greater than 1. Koochesfahani (1989)experimentally investigated the effects of amplitude, frequency, and shape of wave formon propulsive performance of pitching airfoil. He found that axial flow in the cores of wakeincreases with increase in amplitude of pitching. Anderson et al. (1998) studied experi-mentally the wake characteristics and performance characteristics of pitching and heavinghydrofoil for various St, heave amplitude to chord ratio, and maximum angle of attack.He found that the maximum efficiency is obtained when the phase angle between pitchingand heaving is such that leading edge vortices amalgamate with trailing edge vortices. Theheaving motion was introduced to increase the maximum angle of attack which increasesthe thrust generation till the leading edge vortex formation starts. Hover et al. (2004)experimentally studied the effect of diferent angle of attack profiles on propulsive perfor-mance of pitching and heaving hydrofoils. They concluded that the cosine profile givesbetter performance at higher St. Guglielmini & Blondeaux (2004) numerically studiedthe hydrodynamics of pitching and heaving elliptical foil. They concluded that heavingmotion is preferred when the fluid motion is present while pitching motion can be usedeven in still fluid. Schouveiler et al. (2005) conducted an experimental study on pitchingand heaving hydrofoil for different angle of attack. They found that the contribution ofthrust generation due to leading edge suction increases with increase in maximum angle ofattack. Sarkar & Venkatraman (2006) numerically studied the effect of asymmetricity insinusoidal heaving motion of hydrofoil on propulsive performance. They concluded thatasymmetric sinusoidal motion gives better thrust and propulsive efficiency compared tosinusoidal heave motion.

26

Figure 2.14: Reversed von Karman vortex street.

2.3 Flow Control in Hydrofoils

The greater part of the thrust generated in carangiform and thunniform swimmers areby using the mechanism of pitching and heaving hydrofoils. Tail part of these type ofswimmers are following a pitching and heaving motion. Vorticity control in the pitchingand heaving hydrofoils to produce thrust by making a jet behind the foil was studied byTriantafyllou et al. (1993). They explained how the constrained motion of the hydrofoilgenerate thrust. The hydrofoil was found to force the fluid downwards to produce jet.For certain range of frequency of oscillation, the harmonic forcing of the foil was foundto produce unstable waves which rolls up into vortices and a reverse von Karman vortexstreet was obtained. Figure 2.14 details the same. This reverse vortex street is a signatureof thrust production and jet flow. At low frequency of oscillation, the interaction of solidmotion on body was found to produce drag producing von Karman vortex street. At veryhigh frequency, leading edge flow separation was found to destroy the jet profile. Thethrust producing type of flow control was obtained for Strouhal number between 0.25 to0.35. Anderson et al. (1998) experimentally verified the vorticity control methods in oscil-lating foils. The wake structure was analyzed for different heave amplitude to chord ratio,pitch amplitude, phase angle between heave and pitch and Starouhal number. Strongreversed von Karman vortex street was obtained for certain combination of parametersby the combination of leading and trailing edge vortices.

From the mechanism of thrust generation explained in the previous chapter, it wasconcluded that thrust generation in fish-like locomotion can be explained using the for-mation of body vortices and tail vortices. Gopalkrishnan et al. (1994) demonstratedthe concept of vorticity control in fish-like locomotion using experiments conducted onpitching and heaving hydrofoils behind a D-section cylinder. The formation of vorticesfrom the cylinder and foil were compared with the body and tail vortices in fish-like lo-comotion. The active control of vortices using the controlled motion of the cylinder andfoil were explained. Characteristics of the vortices generated from the cylinder and foildecides the propulsion performance. From the experiments conducted for Re = 550, threedifferent types of flow patterns were obtained for various pitching and heaving amplitudes,

27

Figure 2.15: Modes of vortex formation in oscillating hydrofoils behind a D-section cylin-der obtained by Gopalkrishnan et al. (1994) (1,2,3: foil vortices; A,B,C,D,E,F: cylindervortices).

Strouhal number, and angle of attack. Three different modes are listed below.

1. Pairs of counter rotating vortices; one from the cylinder and other from the airfoil(Figure 2.15.A).

2. Cylinder vortices repositioned by foil and interacts with vortices from foil of oppositesign to produce resultant weak vortices forming reversed von Karman vortex street(Figure 2.15.B).

3. Cylinder vortices repositioned by foil and interacts with vortices from foil of samesign to produce strong vortices forming reversed von Karman vortex street (Figure2.15.C).

Studies conducted by Muller et al. (2001) on anguilliform swimming could also getthe same results (Figure 2.3).

The concept of improving propulsion performance using vorticity control was inspiredfrom nature such as schooling of fishes (Weihs and Webb 1983). The main idea amongall these methods is to effectively use the energy of the body and flow to generate thrust.If the motion of the body could be controlled in such a way to extract energy withminimum loss, the efficiency can be improved. Streitlien et al. (1996) investigated thevorticity control method in pitching and heaving hydrofoils in an inflow consists of uniformflow and staggered array of vortices. From the study conducted using inviscid theory forvarious parameters, it was concluded that maximum efficiency condition is obtained whenthe phase angle is in such a way that the foil motion brought in close contact with theindividual vortices in the incoming flow.

28

Figure 2.16: Effect of vortices shed by upstream D-cylinder on propulsion performanceproposed by Liao et al. (2003).

2.4 Flow Control Using Fins

The shape of a real fish is not as simple as an airfoil. Different types of fins such aspectoral fin, pelvic fin, dorsal fin etc. are present to get efficient and controllable motion.Each fins uses different ways of flow control. out of all these fins, pectoral fin is presentin almost all aquatic animals using body/caudal fin for propulsion. the main objectiveof pectoral fin is to minimize the energy required for propulsion by making use of thevortices formed from the fin (Yu et al. 2011). The mechanism of vortex control usingpectoral fins is similar to swimming behind a cylinder. Liao et al. (2003) experimentallystudied the effects of swimming behind a D-cylinder and a hypothesis was proposed forthe hydrodynamics and thrust generation enhancement. Figure 2.16 describes the same.The vortices formed from the cylinder was proposed to produce a lift (light green arrow)normal to the flow direction (grey arrow). The thrust force is proposed as the componentof lift force in swimming direction (purple arrow).

Yu et al. (2011) investigated the effects of pectoral fins in a carangiform swimmerusing three-dimensional numerical study conducted for a Reynolds number of 3.3 × 104.The vortices formed from the pectoral fin was found to reduce the power consumptionby enhancing the lateral movement. The mechanism is explained in fig 2.17 using thepressure contours obtained for St = 0.1. When the tail is moving towards the right,concave surface will be formed on right side and convex surface will be formed on leftside. The vortices formed from the pectoral fins will shed and low pressure region willbe formed on right side. This low pressure reduces the effort required for the undulatorymovement. At the same time, abducted pectoral fin cause extra form drag to produce. So,the abducted pectoral fin will be of advantage only if the effect of formation of low pressuredue to pectoral fin vortex shedding is significantly larger. Yu et al. (2011) concluded thatthe effect is significant only for lower range of St. For higher St, the undulation produces

29

Figure 2.17: Effect of pectoral fin vortices for St = 0.1 (Yu et al. 2011).

Figure 2.18: Higher local pressure at St = 0.8 (Yu et al. (2011)).

very high localised pressure as shown in fig. 2.18. The localised pressure prevents vortexshedding and hence the advantage of pectoral fin vortex is not present for higher St.

Caudal fin plays a major role in the thrust generation mechanism for carangiform andthunniform type of swimmers. So, the shape and size of caudal fin will be a critical pa-rameter for the propulsion characteristics. Chang et al. (2012) studied the effect of caudalfin shape on tuna-like swimming by considering three different fin shapes; crescent, semi-circular, and fan shaped. Figure 2.19 shows the different models. The thrust generationobtained was higher for fan shaped fin and lower for crescent shaped fin. Higher thrustgeneration in fan shaped fin is attributed to the increased area of the fin which results in astronger and wider jet flow. Propulsive efficiency was found higher for crescent shaped finwhich is present in most of the fishes. This higher efficiency is obtained by the optimizedflow control with minimum energy loss.

30

Figure 2.19: Different types of caudal fin models used by Chang et al. (2012). A) Crescentshaped B) Semicircular shaped C) Fan shaped.

2.5 Simulation Methods

Like any fluid flow problem, the fish-like swimming can be analyzed by solving the math-ematical equations governing the fluid flow and forces. An analytical solution for slenderfish body swimming was obtained by Lighthill (1960)using inviscid approximation. Due tothe limitations of inviscid theory to predict the accurate propulsion efficiency and thrustforce, it is necessary to take care of the effects of viscosity (Carling et al. 1998). CFD anal-ysis of fish-like locomotion including viscous effects started with two-dimensional studiesof pitching and heaving airfoils (Triantafyllou et al. 1993, Blondeaux et al. 2005, Sarkar& Venkatraman 2006) and two-dimensional bodies mimicking fish swimming kinematics(Liu et al. 1996, Carling et al. 1998). Recent developments in CFD methods on analy-sis of moving boundary problems (Gilmanov & Sotiropoulos 2005, Yu 2005, Borazjani &Sotiropoulos 2008, Shrivastava et al. 2013) made the complete three-dimensional analysispossible (Borazjani & Sotiropoulos 2008, Yu et al. 2011).

Selection of any simulation method is a compromise between accuracy of results andcomputational cost. Likewise, analysis of fish-like locomotion can be carried out in twodifferent ways; self-propelled simulation and tethered simulation (Carling et al. 1998).Even though the self-propelled simulation is more realistic and accurate, tethered sim-ulation was found sufficient to predict the average forces and propulsive efficiency withminimum error (Borazjani & Sotiropoulos 2008).

2.5.1 Tethered Simulation

Due to the complexity of the problem, it is in appropriate to simulate the real fish swim-ming condition without any simplification. Straight-line swimming is the best assumptionmade in most of the studies. In addition to that, many studies uses constant speed swim-

31

ming condition because, the transient condition occurs only for a short duration (Carlinget al. 1998). With these two assumptions, fish-like locomotion can be simulated by con-sidering fish as a bluff body with undulatory motion in the lateral direction moving alonga straight line with constant speed. This type of simulation method is called tetheredsimulation in which the forces acting by the fluid on the body is not directly coupled withthe motion (Borazjani & Sotiropoulos 2008).

The assumption of tethered simulation resulted in the simplification of code develop-ment and reduction in computational time. Also, the constant speed assumption will bevalid in quasi-steady state swimming condition, because the variation of swimming veloc-ity is very small compared to the average swimming speed (Carling et al. 1998). In orderto predict the realistic swimming condition using tethered simulation, the frequency ofundulatory motion and swimming velocity should be adjusted in such a way that the netforce acting in the swimming direction is zero. The non-zero force in swimming directionindicates the acceleration or deceleration of the body which is not realistic in the constantspeed swimming condition (Chang et al. 2012).

2.5.2 Self-propelled Simulation

The most realistic method to capture the fluid flow characteristics of aquatic swimmingis to solve the Navier-Stokes equation and Newton’s second law of motion together. Fishbody starts undulating in a stagnant fluid which results the fluid to act some forces onthe body by which the fish starts moving. After certain time, the mean of net force actingon the body will become zero and the mean velocity will become constant (Carling et al.1998). Simulation of this realistic condition can be called as self-propelled simulation.

The major difference between tethered and self-propelled simulation method is in theway of coupling between solid and fluid motions. In case of tethered simulation, the motionof the body was independent of the fluid forces whereas the fluid forces determines theswimming velocity in self-propelled simulation. Even without considering the coupling offluid and solid motions, constant speed swimming condition can be well explained. So,the additional burden of coupling the equations governing the fluid flow and movementof body are appreciated only if the analysis can explain the problem in more detailedand accurate manner. Most of the studies regarding the hydrodynamic characteristics offish-like locomotion were conducted using tethered simulation by adjusting the frequencyof undulation to give zero net force (Borazjani & Sotiropoulos 2008, Yu et al. 2011).

The complete coupling of fluid flow and fish movement needs the force balancing inswimming and transverse directions and momentum balancing. Carling et al. (1998)explained the coupling of Newton’s second law of motion with Navier-Stokes equation fora two-dimensional creature swimming with anguilliform kinematics. They explained theproblem of self-propelled simulation using a two-dimensional body as shown in fig. 2.20.

32

Figure 2.20: Anguilliform creature showing its segement of the body with polar coordi-nates (r (i) , φ (i)) with reference to the centre of mass (ò) (Carling et al. 1998)

Fluid forces and moment will be acting on the body due to undulatory motion. The forceswill be acting in the swimming direction (x) and transverse direction (y). Using Newton’ssecond law of motion, the forces acting in x- and y-directions can be obtained as:

Fx = − d

dt

n∑i=1

4mixi (2.2)

Fy =d

dt

n∑i=1

4miyi (2.3)

where 4mi is the mass of ith segment of the body and xi and yi are the velocities inx- and y-directions. The moment acting on the body with respect to the centre of masscan be obtained as:

M =d

dt

n∑i=1

4mir2i αi (2.4)

The net force and moment acting on the body results in change in linear and angularvelocities. Change in linear velocity can be obtained as:

vx =

−n∑i=1

4mixi

m(2.5)

vy =

n∑i=1

4miyi

m(2.6)

Rotational velocity of the body about the centre of mass can be obtained as:

vr = ˙θm +

n∑i=1

4mir2i αi

I(2.7)

Where I =n∑i=1

4mir2i is the moment of inertia of the body and ˙θm denotes the angular

rotation of the axis generating the body slope.

33

Chapter 3

Effect of Chordwise Flexibility onHydrodynamic Propulsion of a PitchingHydrofoil

The literature survey conducted reveals that there are numerous studies on pure pitchingand superposition of an undulatory over the pitching motion. The superposition is doneby considering a chordwise flexibility - represented in terms of wavelength λ of a wavetravelling across the foil. Thus, there are no study for the effect of the flexibility (λ) onthe hydrodynamics and propulsive performance for a pitching foil; and is the objective ofthe present work. The wavelength is considered as infinite for only pitching and non-zerofor the combined motion. Furthermore, the objective is to study the mechanism of thrustgeneration for the pitching as compared to the combined motion (at various non-zerovalues of λ).

3.1 Physical Description

Computational domain for the problem simulated here is shown in Fig. 3.1. The figureshows a pitching as well as undulating NACA 0012 hydrofoil (with chord length c). Notethat the pitching (undulatory) motion is used to model tail (body) motion, for a fish-likepropulsion. The translatory motion of the hydrofoil is modelled by making the hydrofoilstationary in x-direction and fluid flowing over the foil with uniform velocity u∞.

3.1.1 Pitching motion of hydrofoil

Figure 3.2 shows the schematic for modelling of pitching hydrofoil; mimicking tail motionof fish. In this study, the foil is allowed to pitch about the head. The angle of pitching isgiven as:

34

Figure 3.1: Computational domain for hydrodynamics of translating, pitching and undu-lating NACA0012 hydrofoil.

Figure 3.2: Pitching NACA 0012 hydrofoil.

θ = θmaxsin

(2π

Stτ

2Amax

)(3.1)

Furthermore, θmax is the maximum pitching angle, Amax is the maximum non-dimensionaltransverse displacement and τ is the non-dimensional time (≡ tu∞/c). Furthermore,St (≡ 2amaxf/u∞) is the Strouhal number which is the non-dimensional form of frequencyof undulation f and amax is the dimensional maximum amplitude.

3.1.2 Undulatory as well as Pitching motion of hydrofoil

An initial shape of the hydrofoil - mimicking the wavy shape of the body of fish, atthe onset of locomotion - is modelled as shown in fig. 3.3. The hydrofoil is given anon-dimensional displacement ∆Y in the transverse direction (y), which varies along thelength of the body. The general expression for ∆Y is given as:

35

Figure 3.3: Streamwise varying transverse displacement of undulating NACA0012 hydro-foil to obtain its intial wavy/fish-like shape; at the onset of locomotion.

4Y = A (X) sin

(2πX

λ

)(3.2)

For modelling the undulatory as well as pitching motion, Eq. (3.2) is given a time wisevariation. The undulatory motion is modelled by a wave travelling over the fish body inthe direction opposite to the swimming direction. Thus, for the combinded motion, Eq.(3.2) gets modified as:

4Y = A (X) sin

[2π

(X

λ− Stτ

2Amax

)](3.3)

where, St = 2amaxfu∞

.

3.1.3 Performance parameters

The main objective of the study is to compare the thrust generation mechanism forthe pitching and the combined motion. The thrust generated is quantified using thrustcoefficient, given as

CT =FT

1/2ρu2∞c(3.4)

where FT is the thrust force and ρ is the density of the fluid. The viscous and pressurecomponent of the thrust coefficient is given as

CTv =FTv

1/2ρu2∞c, CTp =

FTp1/2ρu2∞c

(3.5)

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where FTv and FTp are viscous and pressure components of thrust, respectively.

3.2 Numerical Description

The moving boundary problem of pitching and the combined motion are solved usinglevel set based immersed boundary method; proposed by Shrivastava et al. (2013). Thehydrodynamics of the flow field is governed by Navier-Stokes equation while the solid-fluidinterface is tracked using level set equations.

Navier-Stokes Equations:

Continuity : ∇.−→U = 0

Momentum : ∂−→U +∇.

(−→U−→U)

= −∇P +1

Re∇2−→U (3.6)

Level Set equations:

Advection : ∂φ∂τ

+−−→Uadv.∇φ = 0

Reinitialization :∂φ

∂τs+ Sε (φ0) (|∇φ| − 1) = 0 (3.7)

where−→U is the non-dimensional velocity of the fluid,P (≡ p/(1/2ρu2∞)) is the non-

dimensional pressure and−−→Uadv is the advecting velocity of the solid-fluid interface. The

level set function φ is defined as the normal distance function which is positive in the fluidpart and negative in the solid part. In the reinitialization equation, τs is the pseudo timeand Sε (φ0) is the smoothened sign function.

Navier-Stokes equations and level set equations are solved simultaneously. A fully im-plicit pressure correction method using collocated Cartesian grid system is used for Navier-Stokes equations while level set equations are discretized using 3rd order Runge-Kuttamethod for temporal term and 5th order WENO (weighted essentially non-oscillatoryscheme) for spatial terms. Results presented in this work are based on an in-house codedeveloped by Shrivastava et al. (2013). A grid size of 716×328, obtained from a grid inde-pendent study, is employed in the present work. The grid size used is coarser (∆ = 0.25)

away from the body; and finer(δ = 0.005) near the body.

3.3 Results and Discussion

Other than the Reynolds number, the non-dimensional governing parameters are thenon-dimensional frequency and maximum amplitude of pitching; for the pitching foil.

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The superposition of the undulatory motion leads to an additional governing parameteras the non-dimensional wavelength λ.

Effect of the wavelength is studied in the present work, varying as from λ = 1 to 7 insteps of 1.

Although the combined motion – with a larger value of λ – almost corresponds to thepitching foil, the simulation is also done for pure pitching foil. The study is done for St0.2 to 0.7 and Re = 400, and Amax = 0.1. From Amax = 0.1 and c = 1, the maximumpitching angle is obtained as θmax = 5.74o; for the pitching foil.

3.3.1 Instantaneous flow patterns for pure pitching and combined

motion of hydrofoil

Figure 3.4 shows the velocity and pressure contours for the pitching hydrofoil. The liftbased mechanism (Borazjani & Sotiropoulos 2009) can be well explained with the help ofthis figure. It is interesting to note from fig. 3.4 (i) that the thrust coefficient is maximumat the maximum value of upward/downward lift coefficient and minimum at an interme-diate value of upward/downward lift coefficient. When the foil moves upwards from themean position, Fig. 3.4(a1-c1) shows that the fluid on the bottom side is accelerated,resulting in formation of a jet (Pedro et al. 2003). Similar jet is formed on the upper sidewhen the foil moves downwards. This will result in a clockwise and anti-clockwise vortexon the top and bottom sides, respectively; which will result in a reverse von Karmanvortex street. The vortices are produced when the trailing edge is at maximum lateralexcursion, as a combination of stopping vortex from the previous cycle and starting vortexfrom the next cycle.

The pressure contours in fig. 3.4a2 show that when the foil moves towards the bottomextreme, the foil decelerates which causes positive pressure on the top side and negativepressure on the bottom side. This will produce maximum value of downward lift as wellas thrust coefficient.

Figure 3.5 shows the temporal variation of velocity and vorticity contours for the flowfield of pitching as well as undulating foil. Thrust generation in this case is based on thepressure-suction mechanism (Müller et al. 1997). When the wave (passing across the bodyof the foil) is travelling backwards, the fluid is pushed on one/rear side (pressure side)and sucked on the other/front side (suction side). This pressure and suction flow togetherwith increasing amplitude of undulation (from head to tail) increases the velocity of fluidand a jet is ejected behind the hydrofoil. When the fluid is pushed, a semi-circular flow isformed. Due to suction flow on other side, flow of same sense of rotation is formed. Thesetwo flows resemble the potential part of a vortex which is called as proto vortex (Mülleret al. 2001). The proto vortex formation can be clearly seen in the vorticity contours. As

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Figure 3.4: For pure pitching motion, temporal variation of instantaneous (a1-d1) velocitycontours and (a2-d2) pressure contours within one time period at Amax = 0.1 (θmax =5.74o), St=0.7 and Re = 400. Sub-figure (e) represent the temporal variation of thrust andlift coefficients with the symbols corresponding to the time instant for the instantaneousplots.

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the wave is travelling backwards, the proto vortex moves towards the tail and shed off.The transverse motion of the hydrofoil results in a transfer of momentum (of the solid

body motion) to the fluid, which increases the velocity of the fluid and forms a jet. Themomentum of the fluid increases in x- as well as y-directions. Maximum thrust generationwill occur when there is an increase in the x-momentum. This is possible when the foilstake a "C" shape, shown in figs. 3.5(b) & 3.5(d); the area to push forward is maximum.At these time-instant, b & d in Fig. 3.5(e) shows that the lift force is at an intermediatevalue and thrust force is maximum. When the hydrofoil takes an "S" shape, shown in figs.3.5(a) & 3.5(c), most of the momentum transfer is contributed to y-momentum; thus, atthese time instant, a and c in Fig. 3.5(e) shows that the upward/downward lift coefficientis almost maximum and the thrust coefficient is minimum.

3.3.2 Effect of chordwise flexibility on performance parameters

In the previous section, thrust generation mechanisms in pitching and the combined mo-tion of hydrofoils are studied by considering pure pitching. In this section the effect ofchordwise flexibility is studied by varying the wavelength λ for various St.

Figure 3.6 shows the viscous, pressure, and total components of thrust for differentwavelength at St = 0.7 and Re = 400. It is interesting to note that with increase in λ,thrust coefficient increases and asymptotes to the value for pitching foil. Thrust generationis based on pressure-suction mechanism for small wavelength and lift based mechanismfor higher wavelength. For intermediate wavelength, thrust generation will be based on acombined mechanism.

Figure 3.7a shows the temporal variation of thrust and lift coefficients for an interme-diate wavelength λ = 2.0, at St = 0.7 and Re = 400. As expected, the maximum thrustcoefficient is found in between maximum and zero lift coefficients. As we increase thewavelength again, the lift based mechanism will become more dominating and maximumthrust coefficient and upward/downward lift coefficient will be almost in phase. Figure3.7b shows the temporal variation of CT and CL for a higher wavelength λ = 7. It isevident from the figure that the trend is almost similar to that of pure pitching hydrofoil.

Figure 3.8 Shows the variation of viscous thrust coefficient with St for λ = 1.0, 1.5,

and pitching condition while the values of Amax and Re kept fixed at 0.1 and 400. It canbe noticed from this figure that viscous drag is higher for pitching hydrofoil compared toundulating hydrofoil. This can be attributed to the participation of viscosity in thrustgeneration for the undulating case which can be explained with the help of fig. 3.9. Figure3.9a shows the protovortex formation and the variation of local viscous drag along theright surface of the foil. It can be seen from the figure that viscous thrust is producedin region of proto vortex formation. Figure 3.9b shows the tangential velocity profile ona point on the right surface where the viscous thrust is maximum. The suction of fluid

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Figure 3.5: For combined motion, temporal variation of instantaneous (a1-d1) velocitycontours and (a2-d2) vorticity contours within one time period at λ=1.0, Amax = 0.1,Re = 400 and St=0.7;. Sub-figure (e) represent the temporal variation of thrust andlift coefficients with the symbols corresponding to the time instant for the instantaneousplots.

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Figure 3.6: For combined motion, variation of mean thrust coefficients with increasingwavelength, for Amax = 0.1, Re = 400 and St = 0.7.

causes negative velocity gradient which results in thrust force. It is also important tonote from fig. 3.8 that the rate of increase of viscous drag with St is lower for undulatinghydrofoil. This is due to the enhancement of proto vorex formation with St and hencemore thrust is contributed by the viscous part. The rate of increase in viscous drag withSt was found increasing with wavelength.This is due to the disappearance of proto vortexwith increase in λ as the motion asymptotes to pitching.

Figure 3.10shows the variation of pressure component of thrust coefficient with St forλ = 1.0, 1.5 and pitching conditions for constant values of Re = 400 and Amax = 0.1.It can be seen that pitching motion produces highest thrust coefficient. It is also clearfrom the figure that the rate of increase of CTp with St is lower for undulating foil andincreases with increase in λ. This indicates that the effect of increasing thrust generationis higher for lift based mechanism compared to pressure-suction mechanism. Figure 3.11shows the total thrust coefficient variation with St. The trend is almost similar to that ofpressure thrust coefficient. It is also important to note from the figure that St at whichthrust generation occurs is lower for pitching hydrofoil and it decreases with increase inwavelength. This is matching with the findings of Borazjani & Sotiropoulos (2010) whofound that the St corresponding to the self propulsion velocity of carangiform swimmerswhich produces major part of the thrust using pitching and heaving motion of caudal fin islesser compared to anguilliform swimmers which produces major thrust using undulatorymotion. Figure 3.12shows the propulsive efficiency variation for λ = 1.0, 1.5 and pitching

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Figure 3.7: Temporal variation of thrust and lift coefficients within one time period ofundulation at Amax = 0.1, Re = 400 and St = 0.7 for (a) λ = 2.0 and (b) λ = 7.0.

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Figure 3.8: Variation of viscous thrust coefficient with St for different wavelengths atAmax = 0.1 and Re = 400.

conditions. It is worth noticing from the figure that for higher St, efficiency of hydrofoil ismore for low wavelength and for lower St, efficiency is more for higher wavelength. Also,the maximum efficiency is higher always for low wavelength.

It can be concluded from the previous figures that pitching hydrofoil produce morethrust with lesser efficiency and undulating foil produce lesser thrust with higher efficiency.In lower St, pitching is the best while for higher St, undulation gives better performance.So, any system which demands higher thrust can be designed based on pitching andheaving motion. Undulatory motion can be preferred in applications which needs higherpropulsive efficiency. The proper combination both types of motion can be used to obtaindesired performance.

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Figure 3.9: (a) (i) Vorticity contour and (ii) Variation of local viscous thrust coefficientalong the right surface for Re = 400, Amax = 0.1, St = 0.7 and λ = 1.0.(b) Variation of tangential velocity profile along the normal at the point of maximumviscous thrust on the upper surface.

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Figure 3.10: Variation of pressure thrust coefficient with St for different wavelengths atAmax = 0.1 and Re = 400.

Figure 3.11: Variation of pressure thrust coefficient with St for different wavelengths atAmax = 0.1 and Re = 400..

46

Figure 3.12: Variation of propulsive efficiency with St for different wavelengths at Amax =0.1 and Re = 400.

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Chapter 4

Conclusions and Scope for Future Work

A detailed literature survey was conducted on hydrodynamic analysis of fish-like locomo-tion. This study started with the observation of kinematics of fish swimming and someanalytical solutions using inviscid and slender body theories. Studies were also conductedon pitching and heaving hydrofoils mimicking tail motion of the fish. Later on, after ad-vancements in experimental techniques, real fish simulations started. These studied couldexplain the basic mechanism of thrust generation in different types of fishes. Experimen-tal techniques were limited to very small range of parameters and getting constant speedswimming condition was difficult. Developments in computational fluid dynamics tech-niques for moving boundary problem made the complete analysis of fish-like locomotionpossible for a wide range of parameters. Recent studies in fish-like locomotion is mainlyfocused on efficiency improvement methods.

Most of the studies on fish-like locomotion were based on propulsive characteristicsof constant speed swimming condition. Tethered simulation method was used to studythis condition of swimming. Very few attempts have been made to analyze the transientcharacteristics of fish propulsion. A detailed hydrodynamic flow analysis can be conductedfor finding the self propulsion characteristics of fish-like locomotion.

The literature explained the mechanism of thrust generation in different types ofaquatic animals. Formation of proto vortices and body vortices due to the body un-dulation and tail vortices due to the oscillation of the tail were elucidated. Body andtail vortices were not distinguished in those studies. The formation of proto-vortices andinteraction of body and tail vortices can be studied in detail and thrust generation dueto different mechanisms can be explained. Effect of various parameters such as frequencyof undulation, wave length, and Reynolds number can be also studied.

The propulsive characteristics of anguilliform and carangiform types of fishes werestudied in the literature. The propulsive efficiency variation was found different for differ-ent swimmers. Maximum efficiency for anguilliform swimmer was obtained in transitionregime whereas carangiform swimmer was found efficient in inertial regime. A study can

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be conducted to find the role of hydrodynamics in this different behaviour.Most of the recent studies related to fish-like locomotion are concentrated on efficiency

improvement techniques. The literature covered certain such studies related to pitchingand heaving hydrofoils with oncoming vortices and effects of different fins on propulsioncharacteristics of fish-like locomotion. Reduction in power requirement by extracting en-ergy from the incoming flow in an optimum way is common in all these studies. Swimmingbehind a cylinder is the best example to study the effect of incoming vortices on propulsioncharacteristics.

A comparitive study on pitching and undulating hydrofoils was also conducted. It wasconcluded that the mechanism for thrust generation is lift (pressure suction) based forthe pitching (combined) motion of the hydrofoil; with the maximum (minimum) value ofthrust coefficient occurring at time instants corresponding to almost maximum value ofupward/downward lift coefficient. For the first time in the published literature, Wave-length corresponding to the chordwise flexibility is proposed as a parameter for a transitionfrom the combined to pure pitching motion of the hydrofoil. With increasing wavelength,the thrust coefficient for the combined motion increases and asymptotes to the value forpitching motion.

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Bibliography

Anderson, J., Streitlien, K., Barrett, D. & Triantafyllou, M. (1998), ‘Oscillating foils ofhigh propulsive efficiency’, Journal of Fluid Mechanics 360, 41–72.

Barrett, D., Triantafyllou, M., Yue, D., Grosenbaugh, M. & Wolfgang, M. (1999), ‘Dragreduction in fish-like locomotion’, Journal of Fluid Mechanics 392, 183–212.

Blickhan, R., Krick, C., Zehren, D., Nachtigall, W. & Breithaupt, T. (1992), ‘Genera-tion of a vortex chain in the wake of a suhundulatory swimmer’, Naturwissenschaften79(5), 220–221.

Blondeaux, P., Fornarelli, F., Guglielmini, L., Triantafyllou, M. S. & Verzicco, R. (2005),‘Numerical experiments on flapping foils mimicking fish-like locomotion’, Physics ofFluids (1994-present) 17(11), 113601.

Borazjani, I. & Sotiropoulos, F. (2008), ‘Numerical investigation of the hydrodynamics ofcarangiform swimming in the transitional and inertial flow regimes’, Journal of Exper-imental Biology 211(10), 1541–1558.

Borazjani, I. & Sotiropoulos, F. (2009), ‘Numerical investigation of the hydrodynamics ofanguilliform swimming in the transitional and inertial flow regimes’, Journal of Exper-imental Biology 212(4), 576–592.

Borazjani, I. & Sotiropoulos, F. (2010), ‘On the role of form and kinematics on the hy-drodynamics of self-propelled body/caudal fin swimming’, The Journal of experimentalbiology 213(1), 89–107.

Carling, J., Williams, T. L. & Bowtell, G. (1998), ‘Self-propelled anguilliform swimming:simultaneous solution of the two-dimensional navier-stokes equations and newton’s lawsof motion’, The Journal of experimental biology 201(23), 3143–3166.

Chang, X., Zhang, L. & He, X. (2012), ‘Numerical study of the thunniform mode of fishswimming with different reynolds number and caudal fin shape’, Computers & Fluids68, 54–70.

50

Cheng, J.-Y., Zhuang, L.-X. & Tong, B.-G. (1991), ‘Analysis of swimming three-dimensional waving plates’, Journal of Fluid Mechanics 232, 341–355.

Chopra, M. & Kambe, T. (1977), ‘Hydromechanics of lunate-tail swimming propulsion.part 2’, Journal of Fluid Mechanics 79(01), 49–69.

Gilmanov, A. & Sotiropoulos, F. (2005), ‘A hybrid cartesian/immersed boundary methodfor simulating flows with 3d, geometrically complex, moving bodies’, Journal of Com-putational Physics 207(2), 457–492.

Gray, J. (1933), ‘Directional control of fish movement’, Proceedings of the Royal Societyof London. Series B, Containing Papers of a Biological Character pp. 115–125.

Guglielmini, L. & Blondeaux, P. (2004), ‘Propulsive efficiency of oscillating foils’, Euro-pean Journal of Mechanics-B/Fluids 23(2), 255–278.

Hover, F., Haugsdal, Ø. & Triantafyllou, M. (2004), ‘Effect of angle of attack profiles inflapping foil propulsion’, Journal of Fluids and Structures 19(1), 37–47.

Kern, S. & Koumoutsakos, P. (2006), ‘Simulations of optimized anguilliform swimming’,Journal of Experimental Biology 209(24), 4841–4857.

Koochesfahani, M. M. (1989), ‘Vortical patterns in the wake of an oscillating airfoil’,AIAA journal 27(9), 1200–1205.

Lau, T. C., Kelso, R. M. & Hassan, E. R. (2004), Flow visualisation of a pitching andheaving hydrofoil, in ‘Proceedings of the Fifteenth Australasian Fluid Mechanics Con-ference, editors M. Behnia, W. Lin and GD McBain, University of Sydney’.

Liao, J. C., Beal, D. N., Lauder, G. V. & Triantafyllou, M. S. (2003), ‘The kármángait: novel body kinematics of rainbow trout swimming in a vortex street’, Journal ofexperimental biology 206(6), 1059–1073.

Lighthill, M. (1960), ‘Note on the swimming of slender fish’, Journal of fluid Mechanics9(02), 305–317.

Lighthill, M. (1969), ‘Hydromechanics of aquatic animal propulsion’, Annual review offluid mechanics 1(1), 413–446.

Lighthill, M. (1970), On waves generated in dispersive systems to travelling forcing effects,with applications to the dynamics of rotating fluids, in ‘Hyperbolic Equations andWaves’, Springer, pp. 124–152.

Lindsey, C. (1978), Form, function, and locmotory habits in fish, inW. Hoar & D. Randall,eds, ‘Fish Physiology’, Academic Press, New York.

51

Liu, H., Wassersug, R. & Kawachi, K. (1996), ‘A computational fluid dynamics study oftadpole swimming’, The Journal of experimental biology 199(6), 1245–1260.

Müller, U. K., Smit, J., Stamhuis, E. J. & Videler, J. J. (2001), ‘How the body con-tributes to the wake in undulatory fish swimming flow fields of a swimming eel (anguillaanguilla)’, The Journal of experimental biology 204(16), 2751–2762.

Müller, U., Van Den Heuvel, B., Stamhuis, E. & Videler, J. (1997), ‘Fish foot prints:morphology and energetics of the wake behind a continuously swimming mullet (chelonlabrosus risso).’, The Journal of experimental biology 200(22), 2893–2906.

OHAsHi, H. & ISHIKAWA, N. (1972), ‘Visualization study of flow near the trailing edgeof an oscillating airfoil’, Bulletin of JSME 15(85), 840–847.

Pedro, G., Suleman, A. & Djilali, N. (2003), ‘A numerical study of the propulsive efficiencyof a flapping hydrofoil’, International journal for numerical methods in fluids 42(5), 493–526.

Sarkar, S. & Venkatraman, K. (2006), ‘Numerical simulation of incompressible viscous flowpast a heaving airfoil’, International journal for numerical methods in fluids 51(1), 1–29.

Schouveiler, L., Hover, F. & Triantafyllou, M. (2005), ‘Performance of flapping foil propul-sion’, Journal of Fluids and Structures 20(7), 949–959.

Sfakiotakis, M., Lane, D. M. & Davies, J. B. C. (1999), ‘Review of fish swimming modesfor aquatic locomotion’, Oceanic Engineering, IEEE Journal of 24(2), 237–252.

Shrivastava, M., Agrawal, A. & Sharma, A. (2013), ‘A novel level set-based immersed-boundary method for cfd simulation of moving-boundary problems’, Numerical HeatTransfer, Part B: Fundamentals 63(4), 304–326.

Triantafyllou, G., Triantafyllou, M. & Grosenbaugh, M. (1993), ‘Optimal thrust devel-opment in oscillating foils with application to fish propulsion’, Journal of Fluids andStructures 7(2), 205–224.

Triantafyllou, M., Triantafyllou, G. & Yue, D. (2000), ‘Hydrodynamics of fishlike swim-ming’, Annual review of fluid mechanics 32(1), 33–53.

Tytell, E. D. & Lauder, G. V. (2004), ‘The hydrodynamics of eel swimming i. wakestructure’, Journal of Experimental Biology 207(11), 1825–1841.

Videler, J. & Hess, F. (1984), ‘Fast continuous swimming of two pelagic predators, saithe(pollachius virens) and mackerel (scomber scombrus): a kinematic analysis’, Journal ofExperimental Biology 109(1), 209–228.

52

Weihs, D. (1972), ‘Semi-infinite vortex trails, and their relation to oscillating airfoils’,Journal of Fluid Mechanics 54(04), 679–690.

Wolfgang, M., Anderson, J., Grosenbaugh, M., Yue, D. & Triantafyllou, M. (1999),‘Near-body flow dynamics in swimming fish’, The Journal of Experimental Biology202(17), 2303–2327.

Yu, C.-L., Ting, S.-C., Yeh, M.-K. & Yang, J.-T. (2011), ‘Three-dimensional numericalsimulation of hydrodynamic interactions between pectoral-fin vortices and body undu-lation in a swimming fish’, Physics of Fluids (1994-present) 23(9), 091901.

Yu, Z. (2005), ‘A dlm/fd method for fluid/flexible-body interactions’, Journal of compu-tational physics 207(1), 1–27.

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