Fish swimming optimization and exploiting multi-body
ARCHIVES
hydrodynamic interactions for underwater navigation
by
MA SSACHUSETTS INSTITUTE
OF TECHNOLOLGY
Audrey Maertens
APR 15 2015
Dipl6me de l'Ecole Polytechnique (2009)
S.M., Massachusetts Institute of Technology (2011)
LIBRARIES
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2015
@
Massachusetts Institute of Technology 2015. All rights reserved.
Signature redacted
............. ....... .... .................
A uthor ...............
Department of Mechanical Engineering
December 5, 2014
Signature redacted
Certified by........
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Michael S Triantafyllou
Professor of Mechanical and Ocean Engineering
Thesis Supervisor
Signature redacted
A ccepted by .........................
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David Hardt, Professor of Mechanical Engineering
Chairman, Department Committee on Graduate Theses
Fish swimming optimization and exploiting multi-body
hydrodynamic interactions for underwater navigation
by
Audrey Maertens
Submitted to the Department of Mechanical Engineering
on December 5, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Mechanical Engineering
Abstract
When walking, driving or riding a bicycle, we mostly rely on vision to avoid obstacles
and evaluate optimal paths. Underwater, vision is usually limited, but flow structures
resulting from the hydrodynamic interactions between inert and/or living bodies contain
rich information, which fish can read through a dedicated sensory system, the lateral line.
Fish can even extract energy from these flow features.
Immersed Boundary Methods (IBMs) are particularly well suited to simulate flows resulting from several moving bodies. In this thesis, the difficulty of most existing IBMs to
accurately handle Reynolds numbers higher than 103 is discussed, and a second order boundary treatment that significantly improves the accuracy at intermediate Reynolds number
(103 < Re < 105) is presented.
Using this new numerical method, object identification using a lateral line is first investigated. It is shown that the boundary layer of a gliding fish can amplify the hydrodynamic
disturbance due to a nearby obstacle and thus help object detection and identification. With
their lateral line, fish can also identify coherent structures in turbulent flow and measure
flow features generated by their own swimming motion. In particular, fish have been shown
to use their lateral line as a feedback sensor to optimize their motion in both turbulent and
quiescent flow.
Two mechanisms by which fish can minimize the energy expanded when swimming are
presented: gait optimization and schooling. The Strouhal number, pitch angle and angle of
attack at the tail are identified as the key parameters determining swimming efficiency in
quiescent flow. By optimizing the undulatory gait, a quasi-propulsive efficiency of 57% is
attained for a foil undulating in open-water (34% for a fish) at Reynolds number Re = 5000.
Fish often travel in schools, and it is shown that significant energy savings are possible
by exploiting energy from coherent turbulent flow structures present in fish schools. By
properly timing its motion, a foil undulating in the wake of an other foil can reach an
efficiency of 80%.
Thesis Supervisor: Michael S Triantafyllou
Title: Professor of Mechanical and Ocean Engineering
3
Acknowledgments
I would like to thank my advisor Prof. Michael Triantafyllou for giving me the opportunity
to follow my own scientific curiosity and sharing with me his enthusiasm and knowledge
about fish swimming hydrodynamics. I would also like to acknowledge my committee members, Prof. Dick Yue and Prof. Pierre Lermusiaux, for their interest in my project and
constructive feedback. I am particularly indebted to Gabriel Weymouth for introducing
me to Computational Fluid Dynamics and sharing his code with me: it has been a great
privilege to collaborate with him.
Of course, these long years of graduate school would not have been as pleasant without
the happy presence of my labmates. Heather Beem, Gabriel Bousquet, Audren Cloitre,
Jahson Dahl, Jeff Dusek, Dixia Fan, Vicente Fernandez, Amy Gao, Jacob Izraelevitz, Leah
Mendelson, David Rival, James Schulmeister, Stephanie Steele, Dilip Thekkodan, Fangfang
Xie: thank you for the fun times and scientific discussions around lunch, as well as during
crazy South-East Asian adventures and ski trips.
Finally, I would like to thank my parents for always encouraging me to pursue the not
very feminine but exciting engineering route. Of course, I won't forget my amazing fiance
(Dr.) Fabien, without whom I would probably never have done a PhD, who always believes
in me and encourages me to relentlessly aim higher in all aspects of life. And, as bonus,
with him came my Providence friends who certainly contributed to making my New England
years memorable.
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Contents
Introduction
1.1 Research motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Chapter preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Accurate Cartesian-grid simulations of near-body flows at intermediate
Reynolds numbers
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The Boundary Data Immersion Method revisited . . . . . . . . . . . . . .
2.2.1 Smooth multi-domain coupling . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
2.2.2 Evaluation of the convolution
2.2.3 Application to a one-dimensional channel flow . . . . . . . . . . . .
2.2.4 Flow solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
2.3 Application to fluid-solid systems
2.3.1 Two-dimensional flow past a stationary cylinder at low Reynolds num.....................................
ber........
2.3.2 Flow around a stationary SD7003 airfoil . . . . . . . . . . . . . . .
2.3.3 Flow around a heaving and pitching NACA0012 airfoil . . . . . . .
2.3.4 Multi-body example inspired by fish sensing . . . . . . . . . . . .
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Exploiting information from the flow: object identification using a lateral
line
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Symbols and dimensionless numbers . . . . . . . . . . . . . . . . .
3.2.2 Towing tank experiments . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Viscous numerical simulations . . . . . . . . . . . . . . . . . . . .
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3.2.4 Potential flow model
3.2.5 Linear stability analysis of the boundary layer . . . . . . . . . . .
3.3 R esults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Viscous and inviscid pressure traces . . . . . . . . . . . . . . . . .
3.3.2 Flow field around a foil passing a cylinder: viscous effects . . . . .
3.3.3 Convective instability in the foil boundary layer . . . . . . . . . .
3.3.4 Enhancing potential flow predictions with instability results . . . .
3.4 D iscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 The boundary layer: filter or amplifier? . . . . . . . . . . . . . . .
3.4.2 Lateral line stimulus and effect of swimming speed . . . . . . . . .
3.4.3 Can the boundary layer facilitate object identification? . . . . . .
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Exploiting energy from the flow: how efficiently can fish swim?
4.1 Introduction .......
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4.2 Fish swimming: modeling considerations . . . . . . . . . . . . .
4.2.1 Physical model and kinematic parameters . . . . . . . . .
4.2.2 Governing equations and dimensionless quantities . . . . .
4.2.3 On the importance of recoil
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4.2.4 Imposed deformation, mid-line displacement and curvature . . .
4.2.5 Trailing edge pitch and angle of attack . . . . . . . . . . . . . .
4.3 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Fluid/body coupling: numerical implementation . . . . . . . . .
4.3.2 Force and power calculation . . . . . . . . . . . . . . . . . . . .
4.3.3 Feedback controller . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Numerical method validation . . . . . . . . . . . . . . . . . . . .
4.4 Definition of efficiency for self-propelled bodies . . . . . . . . . . . . . .
4.4.1 Net propulsive efficiency
. . . . . . . . . . . . . . . . . . . . . .
4.4.2 Propulsor efficiency . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Quasi-propulsive efficiency . . . . . . . . . . . . . . . . . . . . .
4.4.4 Example: anguilliform vs carangiform gaits . . . . . . . . . . . .
4.5 Gait optimization for a self-propelled undulating foil in open-water
. .
4.5.1 Reynolds number, Strouhal number and slip ratio . . . . . . . .
4.5.2 Optimization of Gaussian envelopes with A
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1
4.5.3 Optimization of quadratic envelopes with A
1 . . . . . . . . .
4.5.4 Optimization of Gaussian envelopes with A
0.65 . . . . . . . .
4.5.5 Optimization of an escape gait with A = 1 . . . . . . . . . . . .
4.6 Energy saving by swimming in pair . . . . . . . . . . . . . . . . . . . .
4.6.1 Kirmdn gaiting and Weihs' schooling theory . . . . . . . . . . .
4.6.2 Flow in the wake of a self-propelled undulating foil . . . . . . . .
4.6.3 Effect of phase and distance for two undulating foils in a line . .
4.6.4 Effect of undulation frequency for two undulating foils in a line
4.6.5 Foil undulating in the reduced velocity region of the wake . . .
4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7.1 Efficiency and the notion of drag/thrust on a self-propelled body
4.7.2 Measure of performance for optimizing velocity and body shape
4.7.3 Proposed schooling theory and comparison with Weihs' theory
4.7.4 Application to three-dimensional fish shapes . . . . . . . . . . .
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Conclusions
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5.1 Accurate Cartesian-grid simulations of bear-body flows at intermediate Reynolds
numbers ....
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131
5.2 The boundary layer instability of a gliding fish helps rather than prevents
object identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.3 Swimming efficiency and drag increase for an undulating fish . . . . . . . . 133
5.4 Swimming optimization for a fish in open-water . . . . . . . . . . . . . . . . 134
5.5 Energy saving by swimming in pair . . . . . . . . . . . . . . . . . . . . . . . 135
5.6 Summary and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
A Convolution evaluation at sharp corners
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B Derivations for the one-dimensional channel flow
141
B.1 Exact solution ..........
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B.2 Direct forcing solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
B.3 Lim iting case v = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
C Varying-coefficient model
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D Validation: flow-induced vibration of a circular cylinder
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10
List of Figures
2-1
2-2
2-3
2-4
2-5
2-6
2-7
2-8
2-9
2-10
2-11
2-12
2-13
2-14
2-15
2-16
2-17
2-18
3-1
3-2
3-3
3-4
3-5
Velocity profile and its derivative at the wall in a one-dimensional channel
flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Smoothing across the immersed boundary. . . . . . . . . . . . . . . . . . . .
25
Integration kernel and its zeroth and first order moments. . . . . . . . . . .
27
L,, (a) and L 2 (b) norms of the velocity error in the channel as a function
of the grid spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
Exact, direct forcing, 1st order and 2nd order BDIM velocity profiles at the
wall (located at y = 0) in an unsteady one-dimensional channel flow. ....
30
Flow past a stationary cylinder at Re = 100. . . . . . . . . . . . . . . . . .
33
L,, and L 2 norms of the velocity and pressure error versus grid size and
kernel radius fro flow past a cylinder. . . . . . . . . . . . . . . . . . . . . . .
34
Flow past a stationary SD7003 airfoil at 40 angle of attack and Re = 10000.
36
Convergence of 1st order (*) and 2nd order (o) BDIM for flow past a stationary SD7003 airfoil at Re = 10000. . . . . . . . . . . . . . . . . . . . . .
36
Flow around a SD7003 airfoil at 4' angle of attack and Re = 10000 with
I = 1. Time-averaged velocity magnitude and streamlines. . . . . . . . . . .
37
Average pressure (C.) and skin friction (Cf) coefficients around a SD7003
airfoil at 4' angle of attack and Re = 10000 with h = 1. dy = dx/4 for (b).
37
Three-dimensional flow around a SD7003 airfoil at 4' angle of attack and
R e = 22000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
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Definition of heaving and pitching motion. . . . . . . . . . . . . . . . . . . .
Lift and drag coefficients on the heaving and pitching NACA0012 at Re = 105. 40
Instantaneous vorticity fields during heaving and pitching motion of a NACA0012
at Re = 10 5 for the 1st and 2nd order formulations (Ao = c, ao = 100, # =
r/2, k = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
Three dimensional flow geometry. . . . . . . . . . . . . . . . . . . . . . . . .
43
Pressure around the axisymmetric fish in open water at Re = 6000. . . . . 43
Instantaneous pressure perturbation field around the axisymmetric fish passing the cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
Experim ental set-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Boundary layer fit for three Reynolds numbers. . . . . . . . . . . . . . . . .
Pressure traces at the three sensor locations shown in figure 3-1. . . . . . .
Snapshots at t = 0.3 (a, c, e) and t = 0.9 (b, d, f) as a NACA0012 foil passes
near the cylinder C 1 (r = 0.1, d = 0.1) at Re = 6 250. . . . . . . . . . . . . .
Experimental flow visualization as a NACA0018 foil passes near a cylinder
at R e = 75000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3-6
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Pressure coefficient changes along a NACA0012 foil passing near the cylinder
C 1 at Re = 6 250, as a function of space and time . . . . . . . . . . . . . . .
Properties of the mean boundary layer velocity profiles computed from viscous simulations, as a function of the location along the foil and wavenumber.
(a): Pair of counter rotating vortices observed in the boundary layer of a
NACA0012 passing near the cylinder C 1 at Re = 6 250. (b): Principal mode
for x = 0.7 and kr = 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Viscous simulations of pressure coefficient changes as a function of time t and
space x along a NACA0012 foil passing near the cylinder C1 for (a) Re = 2000
and (b) Re = 20 000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pressure coefficient changes along a NACA0012 foil passing near the cylinder
C 1, as a function of wavenumber and time, for three Reynolds numbers. . .
Amplification coefficient 6(k, t) estimated from viscous simulations as a function of wavenumber and time for Re = [2000, 6250, 20000]. . . . . . . . .
Pressure coefficient changes along a NACA0012 foil passing near the cylinder
C 1, as a function of wavenumber and time, for three Reynolds numbers. (a-c):
residual, (d--f): fitted model . . . . . . . . . . . . . . . . . . . . . . . . . . .
) estimated from poMagnitude of the pressure coefficient changes (max I
tential flow (a-c) and viscous simulations (d-f). . . . . . . . . . . . . . . . .
Difference between the pressure coefficient changes due to two cylinders. . .
Snapshots at t = 0.9 showing the velocity field and pressure coefficient disturbances as a NACA0012 foil passes near three different cylinders. . . . . .
Schematic showing the fish model parameters. . . . . . . . . . . . . . . . . .
Carangiform and anguilliform motion for f = 1.8 and ao = 0.1 at Reynolds
number Re = 5000 with recoil. . . . . . . . . . . . . . . . . . . . . . . . . .
4-3 (a) Linear and angular momentum and (b) corresponding velocities for a
neutrally buoyant self-propelled NACA0012 with carangiform motion at frequency f = 1/T = 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-4 Quasi-propulsive efficiency as a function of frequency for the carangiform and
anguilliform motions with and without recoil. . . . . . . . . . . . . . . . . .
4-5 (a) Typical displacement amplitude envelope for a swimming saithe or mackerel. (b) Typical curvature amplitude envelope for a swimming saithe or
mackerel. Adapted from Videler [179]. . . . . . . . . . . . . . . . . . . . . .
4-6 Flow configuration for the undulating NACA0012 simulations. The vorticity
field for the carangiform motion with f = 1.8 and zero mean drag is shown
as an exam ple. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-7 Comparison of the WW, LL3 and PL3 wall laws. PL3 uses WW's outer-layer
with a square-root buffer layer. . . . . . . . . . . . . . . . . . . . . . . . . .
4-8 Time-averaged drag and power coefficients for an undulating NACA0012 as
a function of frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-9 (a) Drag and (b) pressure coefficient on an undulating NACA0012 with
carangiform motion at f = 1/T = 2.1. . . . . . . . . . . . . . . . . . . . . .
4-10 Time-averaged power coefficient as a function of undulating frequency for (a)
the zero drag and (b) the fixed amplitude configurations. . . . . . . . . . .
4-11 Net propulsive efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-12 Propulsor efficiency estimated from Wu's potential flow theory. . . . . . . .
4-1
4-2
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4-13 Quasi-propulsive efficiency. (a): Comparison of towed estimates with selfpropelled values (Re = 5000). (b): Comparison of efficiency for Re = 2500
90
and Re = 5000 (self-propelled). . . . . . . . . . . . . . . . . . . . . . . . . .
4-14 Definition of the parameters for a Gaussian envelope. . . . . . . . . . . . . .
92
4-15 Chart of a typical optimization procedure. . . . . . . . . . . . . . . . . . . .
92
4-16 Strouhal number as a function of (a) frequency f and (b) sr/(1 - sr) for a
self-propelled undulating NACA0012 . . . . . . . . . . . . . . . . . . . . . .
93
4-17 Quasi-propulsive efficiency as a function of (a) the undulation frequency f
and (b) the Strouhal number St for a self-propelled NACA0012 at Re = 2500
94
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and Re= 5000.........
4-18 Relationship between friction drag coefficient CDf, Reynolds number Re and
Strouhal number St for an undulating NACA0012. . . . . . . . . . . . . . .
95
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4-19 Relationship between sr/(1 - sr), amplitude a and Reynolds number. . . .
4-20 Optimized (a) prescribed deformation envelopes and (b) displacement en96
velopes for the Gaussian parameterization. . . . . . . . . . . . . . . . . . . .
4-21 Snapshots of vorticity for optimized gaits at t/T = 0 (mod 1). . . . . . . . .
97
4-22 7IQp as a function of x, and 6 near the optimum for Gaussian envelopes. . .
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4-23 Superimposed body outlines over one undulation period for three frequencies. 99
4-24 Drag and power coefficients as a function of time for the optimized Gaussian
envelopes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4-25 Snapshots of pressure field with arrows showing the body velocity. . . . . . 101
4-26 rQp as a function of A(0) or f and A(1/2) for quadratic envelopes. . . . . . 102
4-27 Optimized (a) prescribed deformation envelopes and (b) displacement envelopes for the quadratic parameterization. . . . . . . . . . . . . . . . . . . 103
4-28 Snapshots of vorticity for gaits with polynomial envelope at t/T = 0 (mod 1). 103
4-29 Drag and power coefficients as a function of time for quadratic envelopes. . 104
4-30 Optimized gait with Gaussian envelope for A = 0.65 (a): prescribed deformation envelope aoA(x) and displacement envelope g(x); (b): drag and power
coefficients as a function of time. . . . . . . . . . . . . . . . . . . . . . . . . 105
4-31 Snapshot of vorticity for the optimized gait with Gaussian envelope and
106
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A = 0.65 at t/T = 0 (mod 1). .....
4-32 Snapshot of vorticity for the optimized escape gait with Gaussian envelope
and A= 1 att/T=0 (mod1). . . . . . . . . . . . . . . . . . . . . . . . . . 106
4-33 Optimized escape gait with Gaussian parameterization for A = 1 (a): prescribed deformation envelope aoA(x) and displacement envelope g(x); (b):
drag and power coefficients as a function of time. . . . . . . . . . . . . . . . 107
4-34 Wake behind a self-propelled undulating foil for the optimized gait with Gaussian envelope and A = 1 at frequency f = 1.5 . . . . . . . . . . . . . . . . . 110
4-35 Vorticity phase in the wake of a self-propelled undulating foil. . . . . . . . .111
4-36 Time-averaged power coefficient Cp and amplitude ao for (a) the upstream
foil as a function of distance d and (b) the downstream foil as a function of
phase A 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4-37 Snapshot of the vorticity field for two foils undulating at f = 1.5 with separation distance d = 1 and optimal phase A0 = 0.83. . . . . . . . . . . . . . 113
4-38 Snapshot of the velocity and pressure field for two foils undulating at f = 1.5
with separation d = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4-39 (a) Amplitude ratio ar and (b) quasi-propulsive efficiency ?Qp as a function
of frequency f and phase A0 for two foils swimming in a line. . . . . . . . . 115
13
4-40 Snapshot of the vorticity, velocity and pressure field for two foils undulating
at f = 1.8 with separation distance d = 1 and optimal phase AO = 0.87. . .
4-41 Snapshot of the vorticity field for two foils undulating at f = 2.1 with separation distance d = 1 and phase A0 = 0. . . . . . . . . . . . . . . . . . . . .
4-42 Snapshot of the (a) vorticity and (b) pressure field for two foils undulating
at f = 1.8 with separation distance d = 1 and phase AO = 0.38. . . . . . . .
4-43 Ratio of undulation amplitude ao and time-averaged power coefficient Cp
as a function of phase for two foils undulating at f = 1.5 with longitudinal
separation distance d = 1. In-line foils and foils with offset Ay = 0.17 are
com pared. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-44 Snapshot of the vorticity field for two foils undulating at f = 1.5 with longitudinal separation distance d = 1, transverse separation Ay = 0.17 and
optimal phase AO = 0.65. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-45 Snapshot of the x-velocity and pressure field for two foils undulating at f
1.5 with longitudinal separation distance d = 1, transverse separation dy
0.17 and optimal phase A = 0.65. . . . . . . . . . . . . . . . . . . . . . . .
4-46 Snapshot of the vorticity field around a two-dimensional foil with a separate
tail. ......
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......
..
4-47 Three-dimensional fish geometry based on a giant danio. Simulations are
run 6 x 3 x 3 with constant velocity ' = U, on the inlet, a zero gradient exit
condition with with global flux correction and periodic boundary conditions
along y and z boundaries. The Cartesian grid is uniform near the fish with
grid size dx = dy = dz = 1/100 and uses a 4% geometric expansion ratio
for the spacing in the far-field. . . . . . . . . . . . . . . . . . . . . . . . . .
4-48 ?Qp as a function of xi and 6 near the optimum for (a) 2D and (b) 3D
geom etries with f = 2.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-49 Prescribed deformation envelope aoA(x) and displacement envelope g(x) for
(a) carangiform gait with f = 3 and (b) optimized gait with f = 2.4. . . . .
4-50 Superimposed body outlines over one undulation period for (a) the carangiform motion and (b) the optimized gait. . . . . . . . . . . . . . . . . . . . .
4-51 Snapshots of the flow around a three-dimensional fish with a carangiform and
optim ized gait. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-52 (a,c,e) Side-view and (b,d,f) top-view of the vortex structures at several timesteps for the carangiform gait. (a,b): t/T = 0.1 (mod 1); (c,d): t/T =
0.4 (mod 1); (e,f): t/T = 0.7 (mod 1). A red line shows the formation of a
vortex ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-53 (a,c,e) Side-view and (b,d,f) top-view of the vortex structures at several
time-steps for the optimized gait. (a,b): t/T = 0.1 (mod 1; (c,d): t/T
0.4 (mod 1; (e,f): t/T = 0.7 (mod 1. A red line shows a vortex shed from the
tail that never fully develops into a ring, while green lines show the vortices
shed from the body. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A-1
D-1
116
116
117
118
118
118
123
124
124
125
125
126
127
128
Schematic showing the variables used in the derivation of the convolution
evaluation at sharp corners. . . . . . . . . . . . . . . . . . . . . . . . . . . .
137
Sketch of the flow-induced vibration problem. . . . . . . . . . . . . . . . . .
145
14
List of Tables
2.1
2.2
Mean drag and lift and shedding frequency on a circular cylinder at Re = 100. 35
41
Mean drag coefficient on the heaving and pitching NACA0012 at Re = 10 5 .
3.1
3.2
Fitted parameters for the velocity profiles at x = 0.8. . . . . . . . . . . . . .
51
Average and standard deviation of the training data set inputs and test error. 62
4.1
Mean and maximum amplitude of power coefficient, amplitude of drag coefficient and undulation amplitude for a NACA0012 with carangiform amplitude
at f = 2.1 and 0 drag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
Parameters and properties of gaits with Gaussian envelopes. . . . . . . . . .
98
Parameters and properties of gaits with polynomial envelopes . . . . . . . . 104
Parameters and properties of the optimized gait for a Gaussian envelope with
A = 0.65.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Parameters and properties of thrust producing gaits. . . . . . . . . . . . . . 107
Parameters of the gaits used in the wake vorticity phase estimate and fitted
phase and wavelength for the vorticity in the wake. . . . . . . . . . . . . . .111
Efficiency and drag amplification for various gaits at Reynolds number Re =
120
. .....................
................
5000..........
Efficiency for a pair of undulating foils in various advantageous configurations. 122
Parameters and properties of 3D undulating gaits. . . . . . . . . . . . . . . 125
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
C.1 Number of time steps used for each cylinder radius r and distance d. . . . .
144
D. 1 Amplitude and frequency of vibration, average in-line force coefficient and
maximum cross-flow force coefficient for a flexibly mounted cylinder. . . . .
146
15
16
Chapter 1
Introduction
1.1
Research motivation
Around Christmas 1968, the astronauts aboard Apollo 8, the first humans to contemplate
Earth as a whole planet, were stricken by the vivid blue color of our planet. If Earth looks
like a blue marble, it is because over 70% of its surface is covered by the ocean.
Despite this ubiquity of water, the depth of the ocean remains mostly unknown, because
exploring it - or even simply navigating in it - is technically extremely challenging. Until
World War I, submarines relied solely on their periscope for navigation [181]. After then, the
sonar system was developed [48, 176], and similar to dolphins, whales and bats, submarines
can now use sound transducers to communicate and map the terrain around them [94]. And
it is not until 1960 that the Challenger Deep, the deepest known point in the Earth's seabed
hydrosphere, with a depth of almost 11 km, was reached. Only 9 years before a human first
set foot on the Moon, almost 400 000 km from Earth.
Yet, and despite what was once thought (cf. Edward Forbes' Abyssus Theory, 1843),
the ocean is not a hostile environment for who is properly equipped. It is in the ocean
that life is believed to have emerged, some 3.5 billion years ago, and in the ocean again
that life evolved into animals, almost 600 million years ago. Even today, out of the 60 000
known species of vertebrates, half of them are fish and many more spend a significant
portion of their time underwater. In 500 million years of evolution and natural selection,
fish have learned to use to their advantage the heavy fluid in which they live. This fluid is,
by definition, continually in motion; a motion that is affected by temperature and salinity
gradients, winds, as well as interactions with inert and living bodies. Underwater animals
have developed very effective sensory systems through which they can measure flow motion.
For example, most marine mammals have a set of whiskers which allows them to blindly
detect prey long after they have passed [40]. Another example is the lateral line, present in
most fish and amphibians [112]. This flow sensor, usually used in conjunction with other
sensory modalities more familiar to us like vision, smell and hearing, has been shown to
play a major role in most fish behaviors. For instance, the lateral line has been shown to be
instrumental to prey and predator detection [142, 38], obstacle identification [75, 180, 187],
fish schooling [120, 96] and efficient swimming [206, 96]. The lateral line is so effective that
some fish like the blind Mexican cave fish (Astyanax fasciatus), living in dark underwater
caves, have evolved to rely almost exclusively on it for obstacle and prey detection [113].
It has been recently suggested that they have developed tricks such that mouth suction in
order to increase the strength of the signal measured by their lateral line when detecting
17
obstacle [79].
Underwater animals interact with their environment trough water. Flow structures in
water can mediate information: through their lateral line, fish can gather information about
nearby obstacles or the motion of other fish (fellow, prey or predator). Flow structures can
also transport energy. Using the lateral line as a feedback sensor, fish can minimize the
energy wasted when swimming or extract energy from unsteady flow structures generated by
solid bodies (Ka'rmnn gaiting [97, 98, 3]) or other fish (schooling [90, 186, 121, 92]). Whereas
there is a growing interest in developing pressure and flow sensors that will mimic the
function of the lateral line for underwater vehicles [209, 208, 154, 107], the hydrodynamics
resulting from the interaction between several bodies is still very much unknown. The
goal of this thesis is to help understand how water mediates information and energy in the
multi-body interactions at play in underwater navigation.
The numerical investigation of multi-body hydrodynamic interactions is very promising. Indeed, Computational Fluid Dynamic (CFD) is mature enough to provide accurate
estimates of entire pressure and velocity fields, which are impossible to measure experimentally. Moreover, the freedom associated with simulations makes it easy to independently
vary parameters and estimate the effect of each. However, the numerical simulation of the
flow around several deforming and/or moving bodies is very challenging. In order to avoid
the difficulties of mesh generation and deformation, the state of the art for such problems
are immersed boundary methods, in which the computational grid does not have to conform
to the problem geometry [111]. While these methods have been very successful at simulating fluids with complex geometries such as a cephalopod-like deformable body [193] and
flexible insect wings [159, 150], their use has been mostly limited to low Reynolds numbers
(Re < 1000). In order to accurately estimate the flow features resulting from the interaction
of a fish such as the blind Mexican cave fish with its environment (2500 < Re < 6500 [157]),
I built upon an existing method [195] and improved the treatment of the boundary. Using
this method, I investigated two examples of interaction between a fish and its environment.
In the first example I describe the flow features generated by a fish passing a cylinder and
discuss how these features can be used to identify the cylinder. In the second example, I
investigate mechanisms by which cruising fish can save energy, whether swimming alone or
in groups.
1.2
Chapter preview
In chapter 2, an accurate Cartesian-grid treatment for intermediate Reynolds number fluidsolid interaction problems is described. We first identify the inability of existing immersed
boundary methods to handle intermediate Reynolds number flows to be the discontinuity
of the velocity gradient at the interface. We address this issue by generalizing the Boundary Data Immersion Method (BDIM [195]), in which the field equations of each domain
are combined analytically, through the addition of a higher order term to the integral formulation. The new method retains the desirable simplicity of direct forcing methods and
smoothes the velocity field at the fluid-solid interface while removing its bias. Based on a
second-order convolution, it achieves second-order convergence in the L 2 norm, regardless
of the Reynolds number. This results in accurate flow predictions and pressure fields without spurious fluctuations, even at high Reynolds number. A treatment for sharp corners is
also derived that significantly improves the flow predictions near the trailing edge of thin
airfoils. The second-order BDIM is applied to unsteady problems relevant to ocean energy
18
extraction as well as animal and vehicle locomotion for Reynolds numbers up to 105
In chapter 3, we investigate an example of flow structures resulting from multi-body
hydrodynamic interaction that mediates information. Inspired by the function of the lateral
line in aquatic animals, we study the shape identification of a stationary cylinder through
pressure measurements made by sensors located on the surface of a steadily moving foil,
modeling a fish gliding in close proximity to an object. Comparing experimental results,
potential flow predictions, and viscous simulations, we first show that the pressure in the
boundary layer of the foil is significantly affected by unsteady viscous effects, especially in
the posterior half of the foil. Therefore, even after the effects of the boundary layer thickness
are accounted for, potential flow predictions are inaccurate. Subsequently, we show that the
spatial features of the unsteady patterns developing when the foil is moving near a cylinder
can be predicted accurately through linear stability analysis of the average boundary layer
velocity profile under open water conditions. Because these unsteady patterns result from
amplification of the potential flow-like disturbance caused in the front part of the foil,
they are specific to the cylinder that generated them and could be used to identify its
shape. We develop and demonstrate a methodology to calculate the unsteady pressure
based on combining potential flow predictions with results from linear stability analysis
of the boundary layer. The findings can be useful for object identification in underwater
vehicles, and support the intriguing possibility that the significant viscous effects caused by
nearby bodies on the fish boundary layer, far from preventing detection, could actually be
used by animals to identify objects.
In chapter 4 we show how the flow structures generated by the interaction between a
swimming body and the fluid can be used by the body itself or by other nearby bodies to
minimize energy expenditure. We discuss two means by which undulating bodies can save
energy without modifying the body itself. After showing that the quasi-propulsive efficiency
is the only rational non-dimensional metric of the propulsive fitness for self-propelled bodies,
we use this measure to optimize the undulating motion of a fish in open-water. Efficient
undulation is characterized by a deformation envelope with a peak around 80% from the
trailing edge, corresponding to the peduncle section. By increasing the sharpness of the peak
with increasing undulation frequency, the optimal Strouhal number, pitch angle and angle
of attack at the trailing edge can be obtained regardless of the frequency. Then, we show
that even more energy can be saved by fish swimming in an organized group when properly
timing their motion to use the periodic forces from other individuals' wakes. Finally, we
apply these results, based on two-dimensional simulations of a NACA0012 foil with free
recoil carried out at Reynolds number Re = 5000, to a three-dimensional fish shape based
on a giant danio.
19
20
Chapter 2
Accurate Cartesian-grid
simulations of near-body flows at
intermediate Reynolds numbers
2.1
Introduction
Immersed Boundary (IB) methods have become popular in the last ten years for simulating
flows with complex geometries and moving boundaries. IBs remove the effort needed to
generate a body-fitted grid and enable the use of efficient numerical methods that can be
easily solved in parallel (see [11] for a review of IB methods). This relative simplicity makes
IBs particularly attractive for engineering applications and the study of animal locomotion.
However, these applications are often characterized by large Reynolds numbers, which we
will show are particularly challenging for IB methods.
Introduced by Peskin in the 1970s [125] to simulate heart valves, IB methods were first
developed to solve the coupled motion of an elastic boundary immersed in a viscous fluid
on a fixed Cartesian-grid. The effect of the IB on the surrounding fluid is simulated by
the addition of a force density (which represents the force of the surface of the object on
the fluid) to the Navier-Stokes equations [126]. These methods have then been extended to
fluids with solid boundaries by defining artificial body forces [67]. Many options have been
explored for defining the forcing (structure attached to an equilibrium with a spring [14],
explicit feedback controller [67], porous medium [5]) but all require user specified parameters
and are subject to severe stability constraints due to their stiffness [54].
To overcome these limitations, Fadlun et al. [54] proposed a formulation in which
the forcing is directly estimated from the discrete problem such as to impose the desired
velocity on the boundary. This method and the many variations that have subsequently
appeared in the literature are direct forcing methods and have been widely used for flows
in which the motion of the boundary is prescribed. A well known issue with this class of
algorithms is their tendency to introduce large non-physical pressure oscillations (see [71]
for example). Muldoon [114] even showed that the pressure could locally increase without
bound as the time step goes to zero. These oscillations are caused by the lack of smoothness
of the velocity across the boundary before the projection step [72]. A related issue is that
these methods account for the boundary in the momentum conservation equations but not
in the mass conservation equation. Uhlmann [170] proposed an alternative direct-forcing
formulation in which the forcing is first computed on Lagrangian markers, then spread onto
21
the neighboring Eulerian nodes. While not directly addressing the mass conservation issue,
this formulation later generalized by [175] and [127] has been shown to significantly reduce
undesirable force oscillations.
In sharp-interface approaches, the communication between the moving boundary and
the flow solver is usually accomplished by explicitly modifying the computational stencil
near the IB. Unlike forcing methods, sharp-interface approaches alter both conservation
equations, usually using a ghost-fluid [64] or ghost-cell method [110, 165]. But Seo et
al. [144] showed that even with such treatment, local mass conservation is violated which
produces pressure fluctuations. Instead, they suggest the use of cut-cell finite volumes that
reshape the cells in the vicinity of IBs [210, 169, 139]. However, cut-cells in three dimensions
can produce seven different polyhedral control volumes and arbitrarily small cells. The small
cells need to be merged to avoid stability problems and "freshly cleared" cells that appear
with moving boundaries need a careful treatment in order to avoid pressure fluctuations
[169, 139]. Due to all these considerations, sharp-interface approaches, and especially cutcell methods have lost the simplicity which was the main appeal of IB methods.
An alternative approach, called the Boundary Data Immersion Method (BDIM), has
been proposed by Weymouth and Yue [195]. Similarly to Uhlmann's direct forcing formulation [170, 175, 127], BDIM relies on convolving the equation governing the motion of the
immersed boundary with the Navier-Stokes equations. The boundary, however, is represented by a distance function rather than Lagrangian points. The fundamental difference
between BDIM and direct forcing methods, though, is an additional modification to the
pressure term analogous to the discrete operator adjustments of sharp-interface methods
such as [169] and [64], avoiding the projection issues discussed in [72]. Unlike sharp-interface
methods, BDIM alters the analytic equations near the embedded boundary (and not the
discrete operators), which makes the method easy to implement in existing flow solvers regardless of the geometry being simulated. This enables BDIM to predict a smooth pressure
field even for flow featuring moving boundaries, while retaining the simplicity that makes
IBs attractive. This method has proved its versatility by successfully simulating a variety
of low Reynolds number and multi-phase flows [194, 196, 191, 193].
BDIM can also be compared with the volume-penalization IB method of Kajishima
[89], wherein the interpolating function represents the volumetric fraction of the fluid in
the computational cell. BDIM could reproduce this property using a linear kernel, but in
practice we use a smoother kernel to help avoid spurious force oscillations as discussed in
[207]. Additionally, because in BDIM the interpolation coefficient is only a function of the
distance to the boundary and is independent of the grid, its calculation is trivial compared
to the volume fraction. Finally, like direct forcing methods, Kajishima's method estimates
the pressure without taking the solid into account whereas, as will be discussed in 2.2.4,
BDIM also modifies the pressure equation.
The next big challenge for IB methods lies in moderate to high Reynolds number flows,
which give rise to fundamental problems for existing approaches [111, 81]. The source of
these problems is illustrated here by considering a one-dimensional unsteady channel flow
in which flow of kinematic viscosity v with uniform x-velocity U suddenly enters a channel
with opening 0 < y < L at time t = 0. Figure 2-la shows the velocity profile u(y, t)/U
near the boundary for two Reynolds numbers Re = L 2 /(ut) = 100 and 1000 computed on
a body-fitted grid and Figure 2-1b shows the corresponding derivatives. The solution is
uniformly zero in the solid domain (y < 0) whereas the solution in the fluid (y > 0) has a
non-zero slope at the interface. Therefore, even though the velocity field is continuous across
the boundary, its first derivative is not. Guy [72] showed that the pressure fluctuations in
22
1
18
Re=100
Re=1000
-
16
0.8
-
Re=100
Re=1000
14
_J
0.6
12
10
8
0.4
6
r
-
--
--
4
0.2
2
-0.02
0
0.02
0.04
y/L
0.06
0.08
U,
0.1
-0.02
0.02
0.04
y/L
0.06
0.08
0.1
(b) Velocity derivative
(a) Velocity profile
Figure 2-1:
0
Velocity profile and its derivative at the wall in a one-dimensional channel flow of
height L for Re = L 2 /(Vt) = 100 and 1000.
direct forcing methods are caused by the incompatibility of smooth IB methods with this
discontinuity in the first derivative of the velocity. The higher the Reynolds number, the
larger the jump in the velocity derivative, exacerbating this problem and requiring special
techniques for accurate simulation.
In all direct forcing methods as well as in BDIM, a weighted average between the fluid
and solid velocities is used to estimate the fluid velocity near a solid boundary. Such a
treatment will be referred to as 1st order in the rest of this chapter. While a 1st order
treatment of the boundary can allow accurate predictions at low Reynolds numbers, they
are not appropriate for flows characterized by a thin boundary layer.
In this work we extend [195] by using the analytic BDIM formulation to establish a higher
order formulation of the near-boundary interaction between the fluid and solid domains. The
addition of the higher order term improves the accuracy of the method at high Reynolds
number while generating a smoother velocity profile that reduces pressure fluctuations.
We show through high resolution simulations that this analytically derived first-moment
correction enables accurate simulations of high speed flows without introducing any new
model parameters. 2.2 develops the new second-order BDIM approach. Specifically, 2.2.2
proposes an analytical equation that generalizes the convolution evaluation of [195] through
the addition of higher order terms. Comparison of the new formulation with that of [195]
and a direct forcing method is detailed in 2.2.3 in the context of a one-dimensional channel
flow. Finally, a finite volume implementation of the proposed method is presented in 2.2.4.
The generalized BDIM formulation is then applied to two and three-dimensional fluidsolid systems in 2.3, which demonstrate its improved accuracy for intermediate Reynolds
number flows. Two-dimensional flow past a cylinder is used in 2.3.1 to assess the numerical
properties of the method as well as present and validate the force calculation method.
The new formulation is then tested on flows relevant to applications of practical interest:
unsteady two and three-dimensional viscous flows past a stationary and a moving airfoil in
2.3.2 and 2.3.3, as well as a three-dimensional multi-body application in 2.3.4. This last
example illustrates a case for which an Immersed Boundary method is more appropriate
than a body-fitted one. An improved treatment of sharp edges, essential for thin airfoils, is
also derived in Appendix A.
23
2.2
The Boundary Data Immersion Method revisited
In this work we consider a two-domain interaction problem in which the domain Qf is
occupied by an incompressible viscous fluid and the domain Qb by a solid or deforming
body with prescribed velocity V (5, t). The governing equation in the solid body is simply
given by
U= V
(2.1)
whereas the fluid is governed by the incompressible Navier-Stokes equation
(2.2)
7+1-v2=
P
p
a+
ot
After integration of Eq. 2.2 over a time step At, the fluid and body equations can be written
in the form
for X Gb
(2.3)
u = f(U),
for X' E Qf
with
(2.4a)
b= V
fG, t
+At)=U(to)+
]
tO+At
-(-V)
+ vV 2 j dt-
tO+st 1
-Vp dt
~tto
p
(2.4b)
where 3 PAt is the pressure impulse over At and RAt (1-) accounts for all the non-pressure
terms.
2.2.1
Smooth multi-domain coupling
lB methods aim at solving Eq. 2.3 using a grid that does not conform to the boundary
between Qf and Qb. The approach proposed in [195] to solve Eq. 2.3 consists in convolving
the continuous equations with a nascent delta kernel in order to combine them in a smooth
meta-equation.
Eq. 2.3 can be written as a single equation
t) = b(t),
((,,
b)
+ f(i, ,t)lQf(5)
for
Gc
Q
(2.5)
where 1A is the indicator function of subspace A. Convolving both sides with a nascent
delta kernel K, with spherical support of radius c yields the following smoothed equation
iE(X,t)
j U(', t)K ( S') d'w = b, (x, t) + f(,
QIS t) for x E (2.6)
where
bS,t)
Lb
b(yb,t)KE(S,Sb) dzb
ucx(IQ
u, t) = Xf, ) t) K, (X,
Xf)
(2.7a)
dXf2.b
Thus the general equation Eq. 2.6 smoothly transitions from the fluid equation to the solid
equation as illustrated in Figure 2-2. In the dark gray area, fe
0, such that U" = be.
24
'C
kernel
support of the kernel
boundary
Figure 2-2: Smoothing across the immersed boundary. The equations valid in each domain are
convolved with a kernel of radius c and added together. The gradient of gray illustrates how the
contribution of b, and f, to the smoothed equation changes in the boundary region. The kernel at
a point (marked by a dot) that belongs to the boundary region is represented.
fE. The black dot is within distance E of the
Similarly, in the white area, bE = 0, so ,i=
point intersects both Qf and Qb. At that
at
that
boundary, therefore the kernel centered
point, both b& and fE contribute to UE.
2.2.2
Evaluation of the convolution
Eq. 2.6 is very general and can be applied to any multi-domain problem by replacing Eq.
2.4 with the appropriate equation for each domain. In order to solve Eq. 2.6 numerically,
we need to estimate the integrals on a grid. We wish a formulation that is grid independent,
so that it can easily be implemented on any grid with little computational overhead, even
for moving three-dimensional objects. In order to minimize the dissipative effects on the
solution, it is necessary to ensure smoothing only occurs where it is needed to alleviate
the discontinuity discussed in the introduction, ie on the boundary region in the normal
direction. Therefore, two requirements will be kept in mind while discretizing Eq. 2.6: (i)
smoothing only occurs near the boundary and (ii) smoothing occurs across the boundary
but not along it.
The naive way of discretizing Eq. 2.6 would be to approximate the continuous convolution by its discrete counterpart
bE (It) =
b(,t)KE(,
db =
(,t)KE(
d-
b(-tK(-X
,
)A4 + O(Ax 2 )
(2.8)
b+OA
where Ax denotes the grid size and d the dimensionality (2 or 3). However, this formulation
violates the first requirement. Since the kernel K, (Y, Y) has a finite support, the integral
can alternatively be evaluated using a Taylor expansion
(2.9a)
t)K(F, 4) dz4
be (Y, t) = jb(-,
b(x, t) + Vb(, t) (z
=(&,t)
L
-
z)) Ke(z,5b) d'b + 0(c2)
KE (F, b) d- + V6(y, t)
25
(F
-
)K(,Fb) d-e + O(E2)
(2.9b)
(2.9c)
p~
,)I -, F, ii)
(,Y,
where O(e2) appears on the right hand side to indicate the order of the error introduced by
this linearization. Note that if the velocity within the support of the kernel is not smooth
enough for the Taylor expansion to provide a valid approximation, local grid refinement (as
in [81]) can be used, which will also greatly improve the accuracy of the discrete differential
operators.
In order to compute Eq. 2.9 in the boundary region, the body velocity is extended into
the fluid domain. In the case of non-uniform body velocity, this is done using simple linear
extrapolation. Note that this means the prescribed velocity may not be divergence free, but
the modified pressure equation maintains divergence free flow in the fluid domain. Since we
now have a smooth velocity field U4, the fluid equation can also be extended into the body
part of the boundary region. Defining ft and f, respectively the normal and tangent to the
closest point on the fluid-solid interface, we express Eq. 2.9c as:
be (X, t) ~ b(Y, t)
+
(Z ,t)
K (-,
db+
Xb) -(7,
an
(5 b -
t)
X) - h
Kc (X, Xb) db
(2.10)
1
( F - ) - f K ( , F) d -b
In order to satisfy the second requirement, a kernel that only depends on the distance
to the boundary is used
(2.11)
Ke(-,W) = Ke (- - h h, Y h h)
If the radius of curvature of the interface is large compared to the grid size, the boundary
can be locally approximated by its tangential plane [89], which significantly simplifies the
calculation. Indeed, assuming the boundary is locally flat (h is constant across the support
of the kernel), the tangential component of the integration can be eliminated and the kernel
K, (7,Y) be replaced by a one-dimensional kernel E(- - h, b - h)
/
K(5- h ,
Ke (d, W)Ld
b~
d
h h)
-
d
-
(2.12)
The convolution then simplifies to:
b
V
where pk'a and pAB are respectively the zeroth and first moments of the one-dimensional
kernel #e over Ob. Similar expressions can be obtained when the boundary is not locally
flat. For example, the derivation in the presence of a sharp corner can be found in A.
The same simplification holds for f6 (ile, z, t):
(2.14)
f
26
I~
0.8
*
E
P
,
0.6-
pEF
0.4
\\
0.2-1.5
-1
-0.5
0
d/E
1
0.5
1.5
Figure 2-3: Integration kernel and its zeroth and first order moments. Outside of the boundary
0. In the fluid for d > c,
0 and
= 1.-Similarly, in the body for
region (!d > c), /4
d < -6, p,F = 0 and p",B = 1. Within the smoothing region (Idl < ) all values are non-zero.
where pig,
and p
are the moments over Qf. The kernel is chosen symmetric such that
0 outside of the boundary region. Consequently, i4 = f= f in the fluid,
,
Pi _
away from the boundary and vice versa in the solid. It is also chosen positive in order to
guarantee the convergence of a broad spectrum of algorithms traditionally used to solve the
Navier-Stokes equations. Note that extending the method to higher than second-order will
require dealing with the non-zero second moment of positive kernels outside the smoothing
region.
Unlike distributed forcing methods [207], the solution proposed here is rather insensitive
to the exact form of the kernel. In particular, as long as the kernel is continuous, changing
it does not affect the possible pressure fluctuations or convergence properties. The following
kernel will be used in the rest of this chapter:
= (1 + cos(jx - y|7r/e))/(2E)
0
#e x, y) =
if |x - yj
fx-y
0
c(2.15)
if Ix - yJ > E
Using this kernel and defining d(xF) the signed distance from F to the fluid-body boundary
(d > 0 in the fluid, d < 0 in the body), we find:
[1 +
P6(d)
1( 0
where pt
0=
[
[f
and
p,F
1
+ -sin (7r)]
0
2 1(dsin ( 7r) +
I = PF
1'
(I + cos
for
Idl < e
for
for
d < d >c
7r)
for
for
(2.16a)
Idl <c
|d| > c
(2.16b)
For the complementary domain Qb, we simply have
1 - d) p(d) and yt (d) = -p(-d) = -pc(d). The one-dimensional
P '(d)=
kernel # an its zeroth and first order moments are shown in Figure 2-3. Since these
have been calculated analytically in the continuous domain, BDIM would belong to the
'continuous forcing' approach according to Mittal's definition [111]. This also means that
the formulation derived here can as easily be used on a Cartesian grid (uniform or not) as
on a tetrahedral or unstructured mesh.
27
Combining the simplified convolution Eqs 2.13 and 2.14 with Eq. 2.6 we finally obtain
the new meta-equation
S(X) = PC f + (1l-p) b
(2.17)
-(f-b)
This meta-equation generalizes the one from [195] by adding the first-order term in the
expansion of the convolution in Eq. 2.9. As noted in [195], pC can be interpreted as
an interpolating function acting on the governing equations. The new P' term increases
the order of interpolation, improving accuracy in the presence of a large discontinuity in
the velocity gradient. As will be illustrated in 2.2.3, the P1 term further smoothes the
transition and results in quadratic convergence in the external flow. Therefore, the present
formulation, based on a second-order convolution, will be referred to as 2nd order BDIM,
whereas the formulation presented in [195] will be referred to as 1st order BDIM.
2.2.3
Application to a one-dimensional channel flow
In order to illustrate the role played by the first moment correction, the simplistic example
of the one-dimensional unsteady channel flow presented in the Introduction is considered
again. This example has been chosen because, except for the mass conservation equation
that is absent, its treatment is very similar to that of the full three-dimensional unsteady
Navier-Stokes equations and an exact solution can be calculated (see Appendix B1). A
forward Euler scheme is used such that the transition from time to to time to + At have a
very simple expression:
b =0
(2.18a)
f (u, y, to + At) = u(y, to) + At V
a2 u(y, to).
(2.18b)
ay 2
The transition equation for 2nd order BDIM is given by
U,(y, to + At)
[Pf+
pi
[1
+ At4
v 2u,(y, to),
(2.19)
which we write in matrix form
u,(nAt)
([p4+ pED] [I + AtvD
]
U,
(2.20)
where D and D(2 ) are tridiagonal matrices resulting from the second-order central differencing of the first- and second-order derivatives respectively. The moments PE and P' are
given by Eq. 2.16 where the distance function is
d(y)
L
2
L
2- _y
(2.21)
for this channel geometry.
We will compare the error in the 2nd order BDIM solution to two formulations from
the literature; the 1st order BDIM from [195] and a direct forcing method adapted from
the approach in [207]. The 1st order BDIM transition matrix can be recovered by setting
28
P6 to zero in Eq. 2.20. The direct forcing formulation, which we derive in Appendix B2, is
[1 + At v D(2)1
u. (nAt) = ([1 - ((T]
)
(2.22)
U,
where ( is a column vector defined by (,(y) = (#E(d(y), 0) + #E(d(y - L), 0)) dy for the
kernel q$ defined by Eq. 2.15. Calling uo the exact solution and u, an approximate solution
calculated on grid g, we define the L,
error with parameter p for grid spacing dy
eo(dy,p) = max
max
luo - Ug1
(2.23)
,
gqEG(dy) [yE[p,L-p]J
where p is the location of the first point away from the wall included in the error metric,
and the L 2 error
e 2 (dy) =
max
2699
.GEG(dy)
(UO - ug) 2 dy,
yjo
(2.24)
where 699 is the 99% boundary layer thickness (see Appendix Bi) and 10 grids of similar
spacing but different offset are used in g(dy). For all cases, we ensure that the simulations
are converged in time by using 10 7 time steps.
Figure 2-4 shows that for Reynolds number Re = L 2 /(vt) between 100 and 10000, e'
and e2 are only functions of the ratio between the grid spacing dy and 699. The L, norm
of all three methods converges at first order when the points in the smoothing region are
included. However, excluding the first point off the boundary (setting p
e in Eq. 2.23),
or using the L 2 norm, the 2nd order BDIM shows quadratic convergence.
(a)
(b)
10
10
.
linear
-
:' .
linear
10
10
10
/
-2
/
10-31 20
102
/
-.--
/
2
/
-j
quadratic
direct forcing
ar0
i1
dy/699
---
quadratic
/
-2
10-3
10
100
1o
1'
100
dy1699
-
1st order BDIM
*
2nd
order BDIM
Figure 2-4: L, (a) and L 2 (b) norms of the velocity error in the channel as a function of the grid
spacing normalized by 699, the 99% boundary layer thickness as defined in Appendix B1. Error bars
show the spread of the error calculated for Reynolds numbers Re = [100, 1000, 10000]. In (a), the
black solid curves show the L, norm across the whole channel (p = 0), whereas the red dashed
curves show the error away from the transition region (p = E). Note that as dy increases, so does
the region excluded from the em (dy, f) error, causing the value to decrease artificially when the grid
is so coarse that most of the boundary layer is excluded.
Analysis of the limiting case v = 0, detailed in Appendix B3, can help understand these
convergence results. 1st order BDIM and direct forcing have the same fixed point solution
29
uc(y) = 0 for Id(y)I < e. The smoothed solution calculated with these methods is as sharp
as the exact solution, with the discontinuity displaced e into the fluid. This phenomenon
can also be observed at Re = 1000 in figure 2-5. At large Reynolds number, interpolation
is not enough to ensure appropriate smooth transition from the solid velocity to the fluid
one, which results in linear convergence even outside of the transition region. Increasing
the width of the kernel would not provide much additional smoothing, as these first order
methods cannot take advantage of the whole transition region, and would increase the kernel
dependent error.
For the proposed 2nd order BDIM, the fixed point solution to the infinite Reynolds
number case is
uE(y) = exp (
E
y - 1)
for Id(y)i <e.
(2.25)
to uE(y) = 1 at y = e. The
normal derivative term allows 2nd order BDIM to fully take advantage of the 2e wide transition region, resulting in a smoother velocity at the interface. This results in a significantly
reduced error and second-order convergence in the external flow. The smoothness also
ensures that discrete differential operators still approximate their continuous counterpart,
which helps prevent artificial pressure fluctuations and flow instabilities as will be illustrated
in 2.3.2.
This solution transitions smoothly from UE(y) = 0 at y = -c
118
0.8
'~ 2
'10
0.6
0.4
0
---------- exact
-g-2-E-direct forcing
1st order
---
0.2
-0.02
-...........
exact
---- direct forcing
1storder
-2nd order
16
0
0.02
0.04
0.06
0.08
6
10
2nd2
0.1
order
-0.02
y/L
0
0.02
0.04
0.06
0.08
0.1
y/L
(b) Velocity derivative
(a) Velocity profile
Figure 2-5: Exact, direct forcing, Ist order and 2nd order BDIM velocity profiles at the wall
(located at y = 0) in an unsteady one-dimensional channel flow of height L for Re = L 2/(Vt) = 1000.
The solid region is colored in gray, the fluid region in white. The smoothing region that extends
from y = -0.01L to y = 0.01L is represented by a gradient of gray. The new 2nd order method
predicts a velocity that very closely matches the exact solution outside of the smoothing region. The
boundary is halfway between two grid points.
Finally, we note that, in this one-dimensional flow, 1st order BDIM can be considered
a type of direct forcing method (though different from that described in [207]) because this
example is missing a mass conservation equation. Indeed, figure 2-5 shows that they result
in very similar velocity profiles, whereas the 2nd order BDIM profile is much closer to the
exact solution, especially as v -+ 0.
30
2.2.4
Flow solver
We will now apply the new 2nd order BDIM formulation to our fluid-solid interaction
problem. Substituting Eq. 2.4 into Eq. 2.17 results in the momentum conservation equation
for a general fluid-solid interaction problem
i4(to
Y +p
+ At) =
+ P a-
+ RAt(u-,) -
-
PaAt)
(2.26)
Since the fluid is incompressible, the velocity field has to be divergence-free in the fluid
region at all time
V- = O for XcGQf
(2.27)
The corresponding equation in the body is trivial
-Z
XEC=Qfor
b
-
(2.28)
If the divergence of the velocity is zero inside the body, then the full meta-equation
automatically enforces a divergence-free i'. However, for general deforming bodies (like
the shrinking cylinder example from [192]), the smoothing equation needs to be applied in
order to resolve the discontinuity in V - U. Applying Eq. 2.17 to V -U- leads to the following
generalized equation
(1 )- 4 - (Vp-) for -EQ (2.29)
b
Vi
Note that the mass conservation equation is applied to V - iU,, whereas direct forcing methods
usually apply it to . Substituting in Eq. 2.26 gives the following mass conservation
equation
-.
(p-t
(
=
(ii2
+
+(Ct(e )) +P i +RAt (UE)
n -
(Y - V) Po -
P(2.30)
p)
where we define a modified pressure P implicitly as
- +P61 0PAt
t) = -pPAt
p38(P
(.31)
We use P instead of P on the left hand side of Eq. 2.30 to avoid inversion of a non-symmetric
third-order pressure equation. As this variable substitution does not affect the right hand
side, it does not introduce any error in the velocity field divergence and the error in the
pressure field will not propagate in time. Eq. 2.31 shows that P only differs from P through
a second-order term. The difference P1 between P and P can be approximately estimated
from P by solving the following equation
v
P
V
an
PA
(2.32)
As will be shown in @ 2.3.1, Pi is indeed quadratically driven to zero with grid refinement.
31
- VPO
Eqs. 2.26 and 2.30 form the smoothed governing equations of the coupled fluid-solid
system. They ensure exact mass and momentum conservation as well as the smoothness
necessary for the discrete operators to approximate the continuous ones. The governing
equations Eqs. 2.26 and 2.30 can be implemented with any computational scheme. We
have implemented them using an Euler explicit integration scheme with Heun's corrector.
The following equations are solved in order to calculate U' = i(to + At) from U-0 = i(to)
and V = V(to + At).
Euler integration with pressure correction
i'
=
p
('do + rAt (A))+ (1 - P0) Z + p4 y (do + i
At V -
i VPO
t(o)
-
- ' - (1 -- /1) V. V
=
(2.33c)
U)
(2.33a)
(2.33b)
t
P
Heun's corrector
# 0' d + F ())+
At V.
(1 - P06) f + P6
(- i
i
G
V 'd' - (1 -
a d)-?
)
.- V
23
(2.34b)
P
SAt
U2 =
-
P
1
(i
2
U =
+
6 Vp1
(2.34c)
U2)
(2.34d)
where for incompressible Navier-Stokes,
-A (U)
At
('d-
[
)
i+
vV2
(2.35)
These equations have been implemented in an Implicit Large Eddy Simulation (ILES) code
(see [2] for a discussion on ILES). They are posed on a staggered mesh and central differences are used for all spacial derivatives except in the convective term in r-At which uses a
flux limited QUICK scheme for stability. When the local flow is well resolved, these equations automatically revert to a non-dissipative (central difference) scheme. The only novel
computations required by the 2nd order formulation are in steps 2.33a and 2.34a to add
the /4 derivative term on the right hand side. This normal derivative term is computed by
calculating the gradient using a second-order central difference at all points. The gradient
is then projected on the outward normal to the closest boundary. Our experience is that
this makes up less than 1% of the simulation cost and, as shown in the next section, enables
accurate predictions of high Reynolds number flows.
2.3
Application to fluid-solid systems
In this section, two and three-dimensional flows relevant to animal and vehicle locomotion
are used to demonstrate the versatility and accuracy of the 2nd order BDIM formulation
32
1st order
2nd order
1 -1
w
0as c aR10
i v ft t
Figure 2-6: Flow past a stationary
cylinder
at
Re
= 100, instantaneous vorticity for the 1st and
2nd order BDIM formulations.
and its suitability for practical intermediate Reynolds number applications.
First a grid convergence study is carried out on the canonical case of two-dimensional
flow past a static cylinder at low Reynolds number in 2.3.1. A method to calculate the
forces on a body is also presented and tested on that flow. The two following examples focus
on airfoils at Reynolds numbers between 10 4 and 10 5 , since many potential applications of
IBs (from industrial applications to the study of animal locomotion) involve airfoils for
producing lift or thrust. The first example in 2.3.2 consists in a SD7003 airfoil at 4'
angle of attack and Reynolds numbers Re = 10000 and 22000. In this very challenging
example we show that a careful treatment of sharp edges dramatically improves the flow
predictions near the trailing edge. In 2.3.3, a heaving and pitching NACA0012 airfoil at
Reynolds number Re = 10 5 is simulated. Finally, in 2.3.4, the method is applied to a
three dimensional example for which a body-fitted simulation would be highly impractical.
2.3.1
Two-dimensional flow past a stationary cylinder at low Reynolds
number
The canonical case of two-dimensional flow past a static cylinder is first considered in order
to assess the numerical properties of the proposed method. The flow is simulated in a 29 x 29
diameter D domain, constant velocity u = U on the inlet, upper and lower boundaries, and
a zero gradient exit condition with global flux correction. The grid is uniform near the
cylinder with spacing dx/D = 1/120 and uses a 1% geometric expansion ratio for the
spacing in the far-field. In this example and in the rest of the thesis, the radius of the
smoothing kernel is chosen as twice the grid size e = 2 dx. The following studies in this
section show that this level of smoothing is ideal.
As shown in figure 2-6, both 1st and 2nd order BDIM methods show the same characteristic vortex shedding pattern on this simple example. In order to quantify the error
evolution with grid refinement, the grid size is parametrized by parameter h such that the
spacing is dx/D = h/120. Since an exact solution for this flow does not exist, we use the
solution computed on a highly resolved grid (h = 1) as a baseline for computing the error.
The same flow is then computed for h = [3, 4, 6, 8, 12], and the velocity and pressure
errors are shown on log-log plots in figure 2-7 for 1st and 2nd order BDIM, as well as
direct forcing. Also included on the figure are dotted lines denoting linear convergence and
dashed lines denoting quadratic convergence. On all plots, the errors for 1st order BDIM
and direct forcing are at least twice as large as the error for 2nd order BDIM. The order of
convergence of the direct forcing method is between 1 and 1.5 for both velocity and pressure
in the L 2 and L,, norms. For 1st order BDIM, the order of convergence of the velocity
in L,, norm is also between 1 and 1.5, whereas in L 2 norm and for the pressure in both
33
(b)
u
100
0
a)
(c )
u
10-1
-
(a)
10
u
2
1
_j
0_2
-JN
-j
10
/ ,/
/
,
2
4
8
102
16
2
8
4
(d)
0.5
16
(e)
2/
2
1
4
E/dx
h
h/
(f)
p
10
p
2
-2
10
10
3/
0
10
207
2
0.5 1
10
/
03
-J
-1
4
70
2
8
4
16
2
4
8
-
-----
16
E/dx
h
h
direct forcing
pressure correction
-E
1st order BDIM
---------. linear
*
--
-
2nd order BDIM
quadratic
Figure 2-7: L, and L 2 norms of the velocity (a-c) and pressure (d-f) error versus grid size (a-b,
d-e) and kernel radius (c, f). The grid spacing is dx/D = h/120.
norms, the convergence is close to second-order. Note also that whereas 1st order BDIM
and direct forcing have almost the same velocity error, the pressure error decreases faster
with 1st order BDIM than direct forcing. Finally, the 2nd order BDIM errors all converge
quadratically but for the pressure error in L 2 norm that converges linearly in the range of
grid spacing used. In figures 2-7d,e, the norm of the pressure correction P1 defined by Eq.
2.32 is also plotted and converges quadratically for both norms. We remark here that in this
practical case the orders of convergence do not exactly match those estimated in 2.2.3 on
the one-dimensional channel example. The addition of the pressure term, the smaller range
of grid spacing and the impact of time convergence are potential reasons for the observed
differences.
We have also tested how the width of the kernel affects the accuracy of the computed
solution. For h = 4, figures 2-7c,f show the L 2 norm of the velocity and pressure error for
e/dx from 0.5 to 4. For e/dx greater than 2, the error decreases quadratically with the
kernel radius. Decreasing e/dx from 2 to 1 hardly reduces pressure error and increases the
velocity error; further decreasing e/dx results in an increased error. These plots show that
the choice of e/dx is a trade-off between limiting the diffusion caused by a large kernel radius
while ensuring enough smoothness for the discrete differential operators to be accurate. A
radius of two grid points is the best compromise and has been used throughout this thesis.
Forces are calculated using a one-sided Derivative Informed Kernel (DIK) derived in
[189]. The expression for the pressure force uses the Neumann pressure boundary condition
34
Table 2.1: Simulated and experimental measurements of the shedding frequency (St) and the mean
drag (CD) and lift (CL) on a circular cylinder at Re = 100 compared to 1st and 2nd order BDIM
with dx/D = 1/120 and e/dx = 2. Experimental St is from [116] with an estimated uncertainty of
0.8%; experimental CD is from [164], with an estimated error of 6%.
Exp.
[119]
0.164
0.165
1.25
1.33 + 0.009
[77]
[30]
[29]
0.164
0.167
1.33
1.34
1.35
BDIM 1
BDIM 2
0.167
0.167
1.31 0.009
1.31 + 0.009
CL
CDp
CDf
-
+0.332
0.99
0.0082
0.34 + 0.0010
-
-
-
0.32
0.011
0.012
+0.315
+0.303
+0.321
+0.313
1.00
-
1.01
1.00
0.0085
0.0081
CLf
+0.042
-
-
-
-
CLp
+0.295
-
CD
-
St
-
Source
0.30
0.30
0.0008
0.0007
+0.292
+0.285
0.035
+0.034
and has the form
Fp=f
p-d
hf dZ
where 6+ is a kernel designed to sample in the fluid near the surface.
(2.36)
For this work we
used 6, (d) = Or(d, e)/(1 + d/R) with #, the kernel defined in Eq. 2.15 and R the radius of
curvature. Similarly, the friction force is estimated using
Ff=
p v dh - (V
V+
iUT)6
6d
(2.37)
The advantage of the DIK method is that it evaluates the unsteady forces on the body
in one step without a surface grid. Note however that the forces are evaluated at a distance
e from the body. While this does not affect the accuracy of the pressure force, the friction
force will be underestimated if the width of the linear region of the boundary layer is less
than e. However, as noted by Pourquie [130], this limitation is common to most IB methods.
The mean drag (CD) and lift (CL) coefficients calculated using the DIK method, as well
as the individual contribution from pressure and friction (indicated by subscripts p and f
respectively) are compared to body-fitted simulations from Park [119] and Henderson [77]
in Table 2.1. In the same table, results are also compared to other Cartesian-grid methods
[30, 29] and experimental measurements [116, 164]. For this simple geometry low Reynolds
number example, both 1st and 2nd order BDIM compare well with documented results,
validating the force calculation approach. The mean forces have also been calculated using
a control volume, and the results match the DIK average values exactly.
2.3.2
Flow around a stationary SD7003 airfoil
Next the 1st and 2nd order formulations of BDIM are applied to a more challenging
high Reynolds number streamlined body case. The two-dimensional flow past a stationary SD7003 airfoil at 4' angle of attack is computed in a 15 x 20 chord lengths domain.
Constant velocity ' = U on the inlet, upper and lower boundaries, and a zero gradient exit
condition with global flux correction are used. The grid spacing is set to 200/h points per
chord length near the airfoil (corresponding to 16/h points across the thickness of the airfoil)
with a 1% geometric expansion ratio for the grid spacing in the far-field. The thin boundary layer, low curvature separation and very sharp trailing edge make this case extremely
35
L
1st order
0.4
0.2 -
Y
~
0.4 -2nd order
-
0.2
0
0
-0.2
-
-0.2
-0.4
-0.4
0
0.5
1
x
1.5
2
0
0.5
1
x
1.5
2
Figure 2-8: Flow past a stationary SD7003 airfoil at 40 angle of attack and Re
10000 with grid
size dx = 1/100 (h = 2). Instantaneous vorticity for the 1st and 2nd order BDIM formulations.
Only the 2nd order method successfully predicts the regular vortex shedding pattern expected for
this test case.
0.15
0.8
3
Q0------------------*
*
0
0.6
0.1
2
*
0.05 -e-
CY0.4
0
- - --------------0.2
0
2
4
h
6
8
00
-
()
0
1
o
*
0
4
h
6
8
00
4
h
6
8
Figure 2-9: Convergence of 1st order (*) and 2nd order (o) BDIM for flow past a stationary
SD7003 airfoil at Re = 10000. Drag (CD) and lift (CL) coefficients, as well as reduced vortex
shedding frequency (k = wc/(27rU)) are compared to values from Uranga [171] (dashed lines).
challenging for Cartesian-grid methods. We first consider a Reynolds number (based on the
chord c) Re = 10000 at which the boundary layer is expected to remain laminar along 94%
of the chord and the wake to display periodic vortex shedding [171, 27]. Since [171] reported
that at this Reynolds number 2D and 3D curves for average pressure and stream-wise skin
friction coefficients are indistinguishable, two-dimensional simulations are used.
Figure 2-8 shows instantaneous vorticity fields computed by both BDIM formulations for
h = 2 and e/dx = 2. Whereas the 2nd order method shows laminar separation and periodic
vortex shedding as expected at this Reynolds number (detailed in [171, 27]), the 1st order
one shows vortices forming on the upper surface of the foil. This example compared to the
previous one illustrates the fact that the local accuracy assumes a much greater importance
at high Reynolds number, especially when low curvature separation is involved. At low
Reynolds number, figure 2-6 shows that the low and higher order methods predict similar
results. However, on this more challenging high Reynolds number example, the lower order
method fails to predict the proper qualitative behavior because a higher order treatment
of the boundary is necessary to address the large discontinuity in the velocity derivative
illustrated in figure 2-1b.
A grid refinement study has been performed in order to establish the convergence properties of the methods on this high Reynolds number streamlined body case. The grid spacing
parameter h is decreased from h = 6 to h = 0.5, corresponding to 33 and 400 points per
chord length respectively (2.5 and 32 points across the thickness of the airfoil respectively),
and e/dx = 2 is used. Figure 2-9 shows the time-averaged drag and lift coefficients (re-
36
0.2
0
0
-0.2L
V
U.0
0.2
0.2
0
0.5
1
0
1.5
0.5
x
lvi:
x
0.19
0.00
1.11
0.93
0.74
0.56
0.37
1 .30
Figure 2-10: Flow around a SD7003 airfoil at 4' angle of attack and Re
Time-averaged velocity magnitude and streamlines.
(a) 1
(b) 0.06
0.5
0.
10000 with h
1.
body-fitted [171]
direct forcing
1st order BDIM
2nd order BDIM
0.04
.---.--..
o 0.02
0
.., ............ ..... od -..te 1--']
-0.5
-0.02
-
-1
-N
0
-- - - direct forcing
--1st order BDIM
2nd order BDIM
0
0.2
0.4
0.6
0.8
1
x/c
0
0.2
0.4
0.6
0.8
1
x/c
Figure 2-11: Average pressure (Ce) and skin friction (C1 ) coefficients around a SD7003 airfoil at
40 angle of attack and Re = 10000 with h = 1. dy = dx/4 for (b).
spectively CD and CL) on the airfoil as well as the reduced vortex shedding frequency k
(based on the chord) compared to body fitted ILES results from Uranga [171]. The 1st
order method does not seem to converge to the expected solution until the finest level.
Indeed, despite the smoothing introduced by po the zeroth oder moment of the kernel, the
solution remains in the wrong regime, even with 200 grid points per body length, due to
the large jump in the velocity derivative at the immersed boundary. On the other hand, the
2nd order method converges steadily towards values that are consistent with Uranga [171]
(force coefficients and separation location reported by other authors [27] are within 5% of
values from [171]).
The grid size h = 1 is chosen to further investigate the challenges associated with this
example and the importance of carefully treating sharp corners with IB methods. The
formulation derived above (Eq. 2.13) assumes that the IB is locally flat but it can easily be
extended to account for a sharp corner (see derivation in Appendix A). Figure 2-10 shows
the time-averaged velocity magnitude field and streamlines for four formulations: a) direct
forcing, b) 1st order BDIM, c) the uncorrected 2nd order derived in the previous sections,
37
and d) the 2nd order with the sharp corner treatment derived in Appendix A. The direct
forcing used here applies the 1st order BDIM equation for the momentum conservation and
the unmodified mass conservation equation (Eq. 2.30 with p'
1 and p' = 0)). As has
been observed earlier, the lower order methods (a and b) are unable to provide appropriate
smoothing at this Reynolds number, causing instabilities to develop over the upper surface
of the wing instead of further down in the wake. As a result the separation bubble is swollen
and the velocity above it remains high until the trailing edge. In contrast, the second-order
formulations show that the velocity rapidly recovers after the leading edge, as observed in
[171]. However, the velocity field of the uncorrected 2nd order method shows that without
a careful sharp corner treatment, the low velocity region extends too far downstream with
unphysical fluctuations and streamlines. The simple analytic extension for sharp corners
derived in Appendix A solves these issues and the 2nd order velocity fields and streamlines
closely match those reported by Uranga [171] and Castonguay [27]. This example makes it
clear that the second-order formulation of BDIM significantly improves flow predictions in
cases where the grid does not fully resolve the geometry nor the boundary layer.
Figure 2-11a shows the average pressure coefficient along the airfoil for h = 1. The
pressure coefficient predicted by the 2nd order BDIM compares very well with Castonguay
[27], whereas for the same grid size, the first order methods (direct forcing and 1st order
BDIM) show large pressure fluctuations on the high pressure side and a plateau around
mid-foil on the low pressure side that is not expected at this Reynolds number [171, 27].
Since calculation of the skin friction is carried out e away from the boundary, it is
accurate only if the viscous sublayer is thicker than e, which is not the case in the simulations
above. In order to accurately calculate the skin friction coefficient and separation point for
this stationary, low angle of attack airfoil, a finer grid in the cross-flow direction is needed.
Here we used the h = 1 grid and increased the density in the y direction by a factor of 4.
Correcting for the fact that u,(0) = pco1u,/n (from substituting exact f and b solutions
in Eq. 2.17), we calculated the skin friction as
2v'~~~i
sez ei)
2
++c)
pi(0)(
U2C
-
Cof
=
(2.38)
where - (X- + en) is linearly interpolated from the grid and pi(0) = 0 for 1st order BDIM
and direct forcing. As shown in figure 2-11b, only 2nd order BDIM compares well with
[171]. We are able to accurately predict the skin friction, the location of separation (around
x/c = 0.38 versus x/c = 0.37 for Uranga [171]) and even the transition of the boundary
layer to turbulent as indicated by the sudden dip in Cf around x/c = 0.94.
This example has been chosen to illustrate the main challenges faced by Immersed
Boundary methods: accurately simulating thin boundary layers separating over a low curvature surface and ending with a very sharp edge. We show here that, unlike 1st order
methods, 2nd order BDIM is able to accurately predict the pressure distribution around
the airfoil, without spurious fluctuations, in a way that steadily converges with grid refinement. Using a finer grid in the cross-flow direction, it also provides good prediction of the
skin friction and separation. If skin friction is of interest, local grid refinement [81] or a
wall-model [138, 28, 25] can also be used instead of a global grid refinement in order to
reduce the computational cost. However, in practice, IB methods are of interest to simulate
moving boundaries, in which case the friction forces are much smaller than the pressure
forces.
Next, a Reynolds number Re = 22000 is considered, at which the separated boundary
38
0
0.04
0.0
0.0
00
5
10
15
20
Time
(a) Time history of pressure drag coefficient for two grid resolutions.
-
(b) Instantaneous span-wise vorticity iso-surfaces
computed with h = 1.
Figure 2-12: Three-dimensional flow around a SD7003 airfoil at 4' angle of attack and Re = 22000.
layer is expected to become turbulent around x/c = 0.7 [171]. In this regime, a threedimensional simulation is required to capture the main flow features and a domain of 0.3c
is used in the span-wise direction. As illustrated in the time history of pressure drag
coefficient for h = 1 and h = 2 in figure 2-12a, 2nd order BDIM naturally predicts the three
dimensional wake as soon as the grid spacing is smaller than the viscous sublayer (dy+
7
for h = 1). Figure 2-12b shows instantaneous span-wise vorticity iso-surfaces computed
using 2nd order BDIM and the grid size h = 1.
Compared to the canonical low Reynolds number flow past a cylinder, the present example combines three additional complexities: (i) a high Reynolds number, (ii) a low curvature
and (iii) a sharp edge. This flow presents most of the difficulties encountered in practical
fixed or rotating wings applications while being well documented, which makes it an excellent benchmark. We have shown that this flow is very sensitive to the treatment of the
IB: the first order IB treatments (either BDIM or direct forcing) are unable to capture the
physics of the flow. An appropriate treatment of the sharp trailing edge also dramatically
improves the flow predictions and allows 2nd order BDIM to accurately capture the integrated forces, pressure distribution, flow separation and skin friction for this challenging
test case.
2.3.3
Flow around a heaving and pitching NACA0012 airfoil
In order to illustrate the application of BDIM to moving IBs, we apply it to a heaving
and pitching NACA0012 airfoil at Reynolds number Re = 105 (based on the chord c and
free-stream velocity U). The combination of heaving and pitching motion of a foil, known
as flapping, is at the core of aerial and underwater animal locomotion. The position of the
foil at time t is defined by A the vertical position of its pitch axis located at mid-chord of
the foil and a its angular displacement (see figure 2-13), which are expressed as
{Ac
cos kt
a' = =
ao Ao
cos(kt
+)
(2.39)
where the reduced frequency k is expressed in radians and the time t nondimenionalized
by U and c. AO represents the amplitude of the heaving motion, ao the pitching amplitude
and # the phase between the two.
39
y
U
x
Figure 2-13: Definition of heaving and pitching motion.
'
4.
'
1.2
'
'
'
1st order
2nd order
0.8
2
0.40 -0
-0.4
-
-2
-0.8
,
20
,
.
'
'
22
'
' '
24
'
'
-1.2 L
20
Time
22
24
Time
Figure 2-14: Lift and drag coefficients on the heaving and pitching NACA0012 at Re
10 5
.
-4
A high thrust producing case investigated by [166] is chosen:
AO = 1,
ao = 10',
# = 7r/2,
k = 1.
(2.40)
The grid spacing is set to 67 points per chord length (8 points across the thickness of the
foil) in the region swept by the flapping foil with a 1% geometric expansion ratio for the
grid spacing in the far-field. The domain extends 6 chord lengths upstream, 8 downstream
and 9 chord lengths on either side in the cross flow direction. As in the previous examples,
constant velocity -= U on the inlet, upper and lower boundaries, and a zero gradient exit
condition with global flux correction are used.
The mean drag coefficient (drag force normalized by pU2 c/2) estimated with the 1st
and 2nd order BDIM are compared to direct forcing and [166] for this case and two other
parameter sets in Table 2.2. The role of the pressure equation is more important in dynamic
cases than in static cases, therefore BDIM performs much better on this example than direct
forcing. However, as noticed by Isogai [83], the detailed treatment of the boundary is much
less important for this high frequency flow than for the steady case studied in the previous
section. Indeed, figure 2-14 shows that the (dimensionless) drag and lift predictions from the
1st and 2nd order formulations are very similar. There are however two main discrepancies
between the two: (i) in the lift around t = 22 and (ii) in the mean drag as shown in Table
2.2. Whereas the 1st order formulation underestimates the drag by 20%, the value from
the 2nd order formulation is within 5% of the drag reported by Tuncer [166].
A more detailed analysis of the flow structures is necessary in order to assess the quality
of the simulations. To do so, figure 2-15 shows instantaneous vorticity fields. During the
40
Table 2.2: Mean drag coefficient on the heaving and pitching NACA0012 at Re = 10 5 for various
combinations of the flapping parameters defined in Eq. 2.39. Based on comparisons with experiments
and other simulations, the error of the of the results reported in [166] is estimated to be within 15%.
Flapping parameters
Mean drag coefficient
k
0
ao
Ao
direct forcing
1st order
2nd order
Tuncer [166]
1
1
1.34
900
750
750
100
70
100
1
0.75
0.75
-0.174
-0.061
-0.214
-0.358
-0.173
-0.338
-0.418
-0.224
-0.394
-0.446
-0.29
-0.446
upstroke (figure 2-15, t = 21) both the 1st and 2nd order formulations show the formation
of a positive starting vortex due to the high angle of attack of the foil with respect to the
flow. However, as the foil reaches the top of its trajectory, the 1st and 2nd order predictions
diverge (figure 2-15, t = 22 and 22.5). Only the 2nd order formulation is able to predict
that the second (positive) vortex remains attached until it reaches the tail of the foil (as
observed in [166] and [83]). This accounts for the slight discrepancy in the lift between the
two methods shown in figure 2-14.
We have demonstrated the ability of the 2nd order formulation of BDIM to capture
the main flow features generated by a flapping airfoil at Reynolds number Re = 10 5 . The
method has proved to provide accurate force predictions free of spurious fluctuations. This
example validates the use of BDIM for the study of flapping foils and more generally highly
unsteady flows.
2.3.4
Multi-body example inspired by fish sensing
The validation cases presented above are typically solved using body-fitted simulations,
and therefore do not feature complexities such as interfering bodies in the fluid domain. In
this section we demonstrate the ability of our method to handle several three-dimensional
bodies with different velocities by simulating a fish-like body passing a circular cylinder as
shown in figure 2-16. Fish can sense pressure changes due to passing objects through an
organ called the lateral line [37] and use the information to detect and identify obstacles.
Similar to [199], our 'fish' is represented by an axisymmetric body of revolution based on a
NACA0013 airfoil and a Reynolds number of 6000 is chosen (Reynolds number based on the
airfoil length L and free stream velocity U). The computational frame is attached to the
fish (whose axis of rotation is {y = 0, z = 0}) such that the fish is represented as stationary
on the grid and the cylinder moves with the free stream. The minimum separation distance
between the cylinder and the vehicle as well as the radius of the cylinder are chosen to be
equal to the thickness of the fish (0.13L). The grid spacing is set to 100 points per chord
length near the fish with a 1% geometric expansion ratio for the grid spacing in the far-field
and the computational domain has 10L x 4L x 4L size. Constant velocity - = U on the
inlet and y boundaries, periodic boundary conditions in z and a zero gradient exit condition
with global flux correction are used. The method easily generalizes to multiple immersed
bodies by applying Eq. 2.17 with b(Z) the velocity of the closest body and d() the distance
to it (used to calculate the pE terms).
Figure 2-17a shows the pressure field around the axisymmetric fish in open water. For
easier comparison with body fitted simulations, the pressure coefficient C along the surface
of the fish are compared to Windsor [199] in figure 2-17b. Our Cartesian-grid method slightly
41
t = 21.0
t=
t = 22.0
1st order
1st order
*
N-V
-1
01
1
-1
2
1
0
0
-1
-1
-
1
0
1
1
2-
2nd order
2nd order
-
A
y 0
0
0
-1
-1
-1
-1
0
1
2
-1
0
1
2
1
2
2nd order
-
yc0
22.5
1st order
0
1
2
1
0
x
Figure 2-15: Instantaneous vorticity fields during heaving and pitching motion of a NACA0012 at
Re = 105 for the 1st and 2nd order formulations (A 0 = c, ao = 10', # = 7r/2, k = 1).
underestimates the stagnation pressure at the front of the fish but the agreement with the
body fitted simulations from Windsor [199] (dots in the figure) after the front 2% of the fish
is very good. Our method also shows very small pressure fluctuations where the boundary
crosses the Cartesian grid but, as shown in figure 2-11 on a two-dimensional example, these
fluctuations are significantly smaller than with 1st order methods.
Figure 2-18 shows the instantaneous pressure perturbation field around the fish passing
the cylinder (compared with the open water pressure) in the slice at z = 0. The figure
shows that as the fish passes the cylinder distinctive pressure changes can be felt by the
fish.
This kind of simulation is extremely challenging for body-fitted algorithms and the cost
of deforming and/or regenerating the grid as the computational domain undergoes large
deformations can exceed the cost of solving the Navier-Stokes equations. BDIM avoids these
issues and the complexity and deformation of the geometry do not affect the efficiency of
the new second-order simulation method.
42
a = 0.13L
o U
1
U
2a
U
z
y
Ly
x
Figure 2-16: Three-dimensional flow geometry: projection onto the z = 0 and x = 0 planes.
0
0.4
-0.2
0.2
-Cp
Y
-0.4
-0.6
-0.2
-0.8
-0.4
-
1
p:
0
-0.5
-0.05
005
0
x
0.1
0.15
0.5
1.5
0
0.2
0.8
0.6
0.4
x/L
0.2
0.25
0.3
(b) Time-averaged pressure coefficient.
2nd order BDIM (solid line) is compared to
Windsor [199] (dots) which has a reported
estimated error of 7%.
0.35
(a) Pressure field in the slice z = 0.
Figure 2-17: Pressure around the axisymmetric fish in open water at Re = 6000.
0.6
0.6
I
0.4
y 0.2
0
-0.2
K-.
-1
0.6
0.4
y 0.2
0
I
-0.5
0
0.5
0.40.20
-0.21
I1
-1
dp: -0.005
.
-0.5
-0.003
-0.001
I
0
0.5
0.001
0.003
1
0.005
-0.2
-1
-0.5
0
x
0.5
Figure 2-18: Instantaneous pressure perturbation field around the axisymmetric fish passing the
cylinder (compared with the open water pressure) in the slice at z = 0. Pressure is normalized by
pU
2
.
-0.5I
..I
Q czzzzz~
I
43
44
Chapter 3
Exploiting information from the
flow: object identification using a
lateral line
3.1
Introduction
The development of smaller, inexpensive autonomous underwater vehicles is rapidly expanding with the emergence of new actuators, sometimes inspired by marine animals [32]. The
goal is to enable these vehicles to navigate the ocean and conduct complex tasks, such as inspecting offshore and submerged structures, and patrolling harbors autonomously. However,
unlike the intricate sensory systems that allow aquatic animals to map their environment,
currently available sensors for engineered vehicles, such as sonars, require large amounts of
power and cannot fit in small spaces. Vision is not a reliable modality either, as underwater
environments are often dark and turbid. To design more efficient and robust sensors, one
can turn to the sensory systems of aquatic animals for inspiration.
In order to detect prey, predators, or mates, and navigate through obstacles in the
dark, fish can rely on several modalities: chemical sensing [34], electric sensing [103], and
flow sensing through their lateral line [37, 35]. Because of its versatility, flow sensing is a
very widely used modality. It can be used passively to detect moving bodies, or actively
to detect stationary objects by exploiting the fish's own motion. Biotic or abiotic bodies
can be identified, as well as flow features, including those generated by the fish itself. A
prime example is the blind Mexican cave fish (Astyanax fasciatus) that is known to rely on
its lateral line for most of its behaviors [113], including object detection and identification
[180, 75]. Implementation of artificial lateral lines in underwater robots has already shown
promising results such as station holding in a laboratory setting [140]. However, a fundamental understanding of the complex hydrodynamics at play is necessary to implement
more complex and robust behaviors in unpredictable environments.
The lateral line consists of hundreds of flow sensing units, called neuromasts, located
either directly on the fish skin, in direct contact with the flow, or embedded in canals
connected to the external fluid through pores on the skin [182]. The surface neuromasts
act as local flow velocity or skin friction sensors [109]. When the fish moves rapidly or
is holding station in external flow, these sensors are believed to saturate and lose their
sensitivity to perturbations [53]. The canal lateral line, identified as measuring pressure
gradient, is considered to be the major subsystem involved in prey detection and obstacle
45
identification [36]. A convenient implementation of the canal subsystem consists of an array
of pressure sensors [140]. As discussed by [198], the pressure remains constant across the
thickness of the boundary layer, therefore previous studies relied on potential flow models
ignoring viscous effects to estimate the pressure sensed by a fish. The inviscid flow theory
can indeed very accurately model prey localization with a vibrating dipole [39, 68].
A gliding fish can use its lateral line to sense changes in the patterns of its self-generated
flow, caused by interaction with stationary obstacles [201]. [74] studied the disturbances
caused by circular cylinders with a potential flow model, but viscous simulations of a fish
gliding toward or parallel to a wall showed that viscous effects are substantial [199, 200]. [56]
showed that the flow disturbances caused by a stationary cylinder actively interact with the
boundary layer of a sensing body, rendering potential flow predictions inaccurate. Despite
some encouraging results obtained using a potential flow model in a Bayesian framework
for cylinder identification [56], accuracy is limited because of the rapid breakdown of the
potential flow model. While the difficulty of solving the Navier-Stokes equations makes their
real-time application computationally infeasible, accounting for viscous effects is essential
for accuracy.
In order to develop procedures to model the development of unsteady disturbances,
we rely on methods of linear stability analysis that have proven effective in predicting the
dominant features of unsteady flows, such as wakes [161, 117] and separating boundary
layers [105]. The methods are widely used to study the transition of airfoil boundary layers
from laminar to turbulent [134], and can also predict the selective intensification that occurs
when certain disturbances interact with the boundary layer [205]. In this paper, we show
through a combination of experiments, potential flow and viscous two-dimensional flow
simulations, that linear stability analysis can accurately model the interactions between
the boundary layer of a foil and the flow disturbances caused by a circular cylinder. More
specifically, we show that the boundary layer of the foil acts as a tuned amplifier whose
properties can be predicted using open-water flow results. The accuracy of the potential
flow approximation can therefore be significantly improved by combining it with a linear
amplifier whose properties depend only on the Reynolds number, and which models the
boundary layer effects. Unlike previous work, such as [80] and [158], we consider how a
signal of interest, as opposed to noise, is amplified by the boundary layer. By amplifying
the disturbance caused by an object, the boundary layer could help fish detect and identify
obstacles using their lateral line.
The model problem, the experimental setup, and the numerical methods used are described in @
3.2. Specifically, the symbols and dimensionless numbers are defined in 3.2.1,
followed by a description of the experiments ( 3.2.2), and the viscous ( 3.2.3) and potential
flow ( 3.2.4) simulations. The methodology of the boundary layer linear stability analysis
is presented in 3.2.5. In 3.3, we first describe the viscous effects on a foil passing an
elliptical or circular cylinder ( 3.3.1 and 3.3.2). We then show that the boundary layer
of the foil is subject to a convective instability that can amplify the pressure disturbances
caused by a passing cylinder ( 3.3.3) and present a method for incorporating this knowledge
to improve potential flow predictions ( 3.3.4 and Appendix C). In 3.4, we consider the
implications of the results presented in 3.3. In particular, we reconcile our results with
the well accepted role of the boundary layer as a mechanical filter affecting the lateral line.
( 3.4.1). Finally, we provide numerical results for the typical frequency of the disturbance
along a blind Mexican cave fish, and discuss how the boundary layer instability could help
fish detect ( 3.4.2) and identify ( 3.4.3) objects.
46
3.2
Materials and methods
We define a model problem with two-dimensional geometry to study the sensing mechanisms
of a fish detecting an object: The gliding fish is modeled as a rigid foil of chord length L,
while the object to be detected is represented by a stationary cylinder of elliptical or circular
cross-section.
3.2.1
Symbols and dimensionless numbers
The Reynolds number is based on the foil length L and the foil gliding speed U, as defined
in figure 3-1, such that for kinematic viscosity v, Re = UL/v. In the simulations, the
reference frame is attached to the foil, and the free-stream U, = UZi* defines the positive
x direction. All lengths are normalized by L, velocities by U, and times by L/U. The
results are presented in dimensionless units where the leading edge of the foil is located at
x = 0, its trailing edge at x = 1 and t = 0 corresponds to the time when the center of the
cylinder is at x = 0. The pressure coefficient Cp is calculated from the pressure p such that
Cp = 2p/(pU,2), where the pressure is 0 at infinity. The average velocity field around the
foil in open water is noted i-0 and the instantaneous velocity field is ' = io + u'. Similarly,
the instantaneous pressure is decomposed as p = po + p'. C denotes a circular cylinder with
radius r that passes at distance d from the foil (projected onto the y axis). C1 is the cylinder
characterized by r = 0.1 and d = 0.1.
3.2.2
Towing tank experiments
Experiments were conducted in the SMART (Singapore-MIT Alliance for Research and
Technology) Center testing tank in Singapore that has dimensions 3.6 x 1.2 x 1.2 meters. A
NACA0018 foil with chord length L = 15 cm and span s = 60 cm was towed past a stationary cylinder using an x - y gantry system supplied by Parker Engineering and controlled
using Parker motor controllers and proprietary motion control software. The foil was cast
with internal 3.18 cm PVC tubing to transmit pressure from taps at the foil mid span to the
top. Honeywell 19C015PG4K pressure sensors were mounted on top of the foil, and measurements were collected at a sampling rate of 500 Hz via a NI USB-6289 data acquisition
card. The experimental set-up and the location of the sensor ports are shown in figure 3-1.
The foil was towed at speed U, = 0.5 ms- 1 (corresponding to Reynolds number Re =
75000) past a stationary cylinder of circular or elliptical cross-section. At its closest point,
the foil was d = 5 mm to 10 mm (0.03L < d < 0.07L) away from the cylinder. A laser sheet
and particle tracking system were used to visualize the flow as shown in Figure 3-1c.
Note that the parameters used in the experiments do not match the values typically
found in nature. Whereas blind cave fish are best modeled by foils of 12 - 13% thickness
[199], a thicker foil was necessary to fit the pressure sensors and tubing. The Reynolds
number was one order of magnitude larger than values typically found in the cave fish to
ensure a large signal to noise ratio. Effects of Reynolds number and foil thickness are at
most moderate in this subcritical regime and do not change qualitatively the results.
3.2.3
Viscous numerical simulations
In order to map the entire velocity and pressure fields, we performed two-dimensional viscous simulations on a Cartesian grid using the boundary data immersion method (BDIM)
described in [195] and [104]. In BDIM, the prescribed body kinematics and Navier-Stokes
47
(a)
Stationary
y
cylinder
0
d
Moving
foil
laser
S
', U4
sheet'
L
L1
z
camera
Figure 3-1: (a) Sketch of the cross-section of the experimental apparatus showing the location of
the pressure ports (Si, S2 and S3). (b) Picture of the experimental set-up. (c) Picture showing the
laser sheet in a particle tracking set-up.
equations are integrated over the fluid and solid domains with a kernel of finite radius E.
The resulting blended equations are valid over the complete domain and enforce the noslip boundary condition at the fluid/solid interfaces. Problems previously studied with this
robust immersed boundary method include ship flows and flexible wavemaker flows [190],
shedding of vorticity from a rapidly displaced foil [197], and a cephalopod-like deformable
jet-propelled body [193]. In [104] we demonstrate the ability of BDIM to handle several
moving bodies and generalize the original method to accurately simulate the flow around
4
streamlined foils at Reynolds numbers on the order of Re = 10 . The numerical details of
the simulation method follow those presented in Chapter 2.
In the present simulations, the sensing vehicle is represented by a NACA0012 foil of
unit length L = 1 attached to the computational frame, whereas a circular cylinder moves
with the free-stream and passes at a distance d from the foil as illustrated in figure 3-1.
The computational domain extends lOL upstream of the foil, 12L downstream and 5.5L on
either side. Constant velocity i= U., on the inlet, upper and lower boundaries and a zero
gradient exit condition with global flux correction were used. The grid spacing was set to
200 points per chord length near the foil (corresponding to 24 points across the thickness
of the foil) with a 1% geometric expansion ratio for the grid spacing in the far-field.
Blind cave fish have typical lengths from L = 5 cm to 10 cm and swim at speeds between
U = 5 cm s 1 and 15cm s-1, corresponding to Reynolds numbers exceeding Re = 2500
[157]. In the present paper, Reynolds numbers ranging from 2000 to 20 000 are considered.
Since the parameters in the simulations were chosen to match the values found in nature
rather than those of the experiments, experimental and viscous simulated results will only
be qualitatively compared ( 3.3.1 and 3.3.2).
3.2.4
Potential flow model
In order to calculate the potential flow approximation to the problem, we implemented in
Matlab a two-dimensional constant source panel method. Using the same notations as in
the previous sections, we consider a free-stream U, and we denote by V the prescribed body
48
velocity, which is a function of the location X' for multiple or deforming bodies. fi is the
unit normal vector to the fluid/solid interface. The velocity potential 4 is decomposed as:
<b-
and satisfies V 2 4
=U -
X+
X(),
(3.1)
0 in the fluid with the boundary conditions:
09n
(
- Us) - h
along the fluid/solid boundary,
(3.2)
at infinity.
0
We solve this boundary-value problem by uniformly discretizing the periphery of each object.
The foil and the cylinder were discretized into 200 and 50 segments, respectively.
If the viscous velocity field V- is known, the potential flow model can be improved by moving the fluid/solid interface * in the direction of its normal vector f. P* is the displacement
thickness, calculated as:
Jo0
(1 - v(V(6)} dy,
(3.3)
'
6
where y is the distance to the boundary and v the component of &tangential to the boundary.
6 is the overall thickness of the boundary layer, which we define as the distance y normal
to the wall where:
dv(y) = 0.01dv (0).
(3.4)
dy
dy
If
ment
used,
since
the boundary layer velocity is approximated by its time average, a constant displacethickness 6* is used. In cases where the instantaneous displacement thickness *(t) is
the rate of change of the displacement thickness directly impacts the source strengths
the normal velocity of the boundary used in Eq. 3.2 is:
d6*
where 7b denotes the solid body velocity. Therefore, viscous effects responsible for dynamic
changes in the displacement thickness directly affect the pressure on the surface of the foil.
3.2.5
Linear stability analysis of the boundary layer
A gliding fish, or a foil in steady motion develops a boundary layer and steady state flow
characterized by the velocity field U-0 and pressure field po. This basic flow field satisfies
the incompressible Navier-Stokes and continuity equations. The presence of a solid object
in the neighborhood of the foil causes a perturbation to that basic flow characterized by
velocity field U' and pressure field p'. Writing the Navier-Stokes equation for the total flow
field leads to the following disturbance equations:
.au
I V2
0#9+2 -#v-
+V9 - V
V U = 0.
p'=0
(3.6a)
(3.6b)
Assuming that the radius of curvature is significantly larger than the boundary layer
thickness, we can omit curvature effects. As discussed in [134], inclusion of curvature and
49
non-parallel effects improves the predictions only marginally. Assuming that the perturbation is small and keeping only the first order terms, we get the linear disturbance equation
for parallel flow. We then express the velocity field u' in terms of the streamline function
that we write as a superposition of normal modes:
40(x, y, t) =
(37)
,(y)ei(kx-wt)
where w is a complex frequency and k a complex wave number. Their real parts W, and
kr represent the physical frequency and wavenumber of the disturbance, respectively, while
their imaginary parts wi and ki represent the time and space growth (or decay) rates,
respectively. The resulting linearized disturbance equation is the Orr-Sommerfeld equation:
(uo -w/k)
(
2
k2
kie
d 2O
4
+ 2k 2 d
+ k
=0
(3.8)
with boundary conditions p(y)
0 and dp/dy = 0 at y = (0, + o).
u is the component
of i- tangent to the boundary and y is the normal distance to the boundary. The pressure
perturbation associated with mode p is then given by:
p(y) = (w/k - uo)
+
+p
dy
dy
(3.9)
2d2o ).
d
k Redy
d
We discretized Eq. 3.8 using Chebyshev polynomials, which are particularly well suited to
solve the Orr-Sommerfeld equation [118], though other discretization are also commonly
used [158]. For a given wavenumber k, the corresponding eigenvalues W and eigenmodes
p can be identified by solving the resulting eigenvalue problem. We used a Matlab code
adapted from [185] to solve Eq. 3.8 with 128 points across a domain of length 1. The
eigenvalue (frequency) with largest imaginary part corresponds to the most unstable mode,
called principal mode. The frequency and wavenumber of principal modes are linked by the
dispersion relation that we write D(w, k, Re) = 0.
The basic boundary layer is computed using the Navier-Stokes solver described in 3.2.3.
In order to easily compute the velocity and its derivatives at any point in the boundary
layer, even at the wall where the immersed boundary method is least accurate, the maximum
velocity Ue and boundary layer thickness 699 are estimated and a profile of the form:
uo (y) = U, tanh a Y + b (
699
2+ C (
699
3
699
(3.10)
is fitted for each x location and Reynolds number. The number of data points used to fit
the profiles and associated errors are shown in figures 3-2a and 3-2b respectively for various
Reynolds numbers and locations along the foil. Only locations x > 0.3 are shown as this
ensures enough data points to properly fit the profile. Moreover, locations x < 0.3 will not
be needed in this study (see figure 3-7 for instance). Examples of profiles at x = 0.8 are
shown in figure 3-2c and the corresponding parameters can be found in table 3.1. These
profiles are used to compute uo and its second order derivative in Eq. 3.8.
50
Re=20000
(a)
(b)
x 1 o-
(C)
0
*
0 Re
=20000
0
20-
0.8
0
Re = 2000
Re= 6250
*
0
Re = 2000
Re = 6250
1
8
0
0
o15
00
E
-L
10
0.2
2
02
04
0.6
0.8
1
-.
0.4
0
0
x
5
/.
00.4
0.6
x
0.8
1
0
0.05
0
x
0.1
y
Figure 3-2: Boundary layer fit for Re
[2000, 6250, 20000]. (a): Number of data points used
to fit each boundary layer profile. (b): Maximum error between the viscous simulation data points
and the fitted profiles (normalized by Ue). (c): Boundary layer velocity profiles (solid lines) fitted
from viscous simulations data points (.) at x = 0.8.
Re
Ue
699
a
b
2000
1.054
0.094
0.741
1.123
0.841
6 250
20 000
1.042
1.038
0.060.
0.040
6 438
0.394
1.214
0.797
0.878
1.130
c
Table 3.1: Fitted parameters for the velocity profiles at x =
3.3
3.3.1
0.8.
Results
Viscous and inviscid pressure traces
We first compare pressure traces recorded in the experiments and those simulated with
BDIM with potential flow estimates. Figures 3-3a-c show traces of the pressure at the three
sensor locations indicated in figure 3-lb as the NACAOO18 passes an elliptical cylinder at
three different orientations. For all orientations, as the foil passes the cylinder (0 < t < 1),
significant differences arise between the pressure recorded by the sensors and the inviscid
theory predictions.
The value of the pressure coefficient measured by the first sensor increases as the foil
approaches the cylinder and slowly returns to its initial value after t =0. A displacement
of the stagnation point toward the cylinder is responsible for the pressure increase and this
pressure trace is similar to what would be expected from an inviscid fluid.
The pressure measured by the second sensor decreases as it approaches the cylinder
and the fluid accelerates as it has to go through the channel formed by the foil and the
0.2), the potential flow model
cylinder. Once the sensor has passed the cylinder (t
predicts that the pressure slowly returns to its initial value. However, the experimentally
measured pressure recovers much faster, before undergoing potentially large oscillations.
This feature, consistently observed across all experiments, cannot be accounted for by the
potential flow theory.
At the third sensor location, the potential flow model predicts that the pressure slightly
decreases as the foil approaches the cylinder, which is experimentally observed. After the
front of the foil has passed the cylinder (t
0), the inviscid model predicts that the pressure
returns to its initial value, before slightly decreasing and increasing again as the sensor passes
51
(a)
0 = n/2, d = 0.03
0.4-
0.2-a-
(b)
--- sensor 1
-- sensor 2
sensor 3
0.2 1
-a-
0
0
0
-
-
0
0 = ir/4, d = 0.07
-0.4-0.4
(C)
0
t
-0.5
0.5
0 = 3n/4, d = 0.07
(d)
0.4
0.12
0.2 F
0.08
-a.
0.04
0
0
0
-0.2
d = 0.1 (simulation)
XI
I
'1%
0
4
I,
0.04
^ A
0
0.08,
-0. 5
0.5
0.5
0
.
0
e
.
-0 .5
/
-0.6
0.5
1
t
Figure 3-3: Pressure traces at the three sensor locations shown in figure 3-1. A vertical black
dashed line at t = 0.2 shows a visual indication of when the potential flow pressure starts diverging
from the viscous pressure. (a-c): Experimental (solid lines) and potential flow (dashed lines) traces
for a NACAOO18 (Re = 75000) passing an elliptical cylinder at various orientations 0 and distances
d indicated on the plots. The ellipse has major radius 0.3 and minor radius 0.2. (d): Traces from
viscous (solid lines) and potential flow (dashed lines) simulations of a NACA0012 (Re = 6250)
passing cylinder C 1 (r = 0.1, d = 0.1). In the potential flow simulation of (d) only, the foil has
been augmented by its displacement thickness 6* (it is much thinner and not exactly known in the
experimental cases).
52
the cylinder. The experimental pressure, however, keeps decreasing until the sensor passes
the cylinder (t ~ 0.5), and only then does it rapidly recover its initial pressure. As has been
observed for the second sensor, the experimentally measured pressure roughly matches the
inviscid pressure until the second sensor passes the cylinder (t ~ 0.2), but afterwards the
two pressures differ significantly.
The experiments have shown consistently similar results for various orientations of the
ellipse. In figure 3-3d we compare the pressure traces calculated by the viscous code with
potential flow estimates for the circular cylinder C 1 at Re = 6 250. Despite differences in
the cylinder size and geometry, and the Reynolds number and foil thickness, the viscous
pressure traces present the same features as observed in the experiments. The mechanism
responsible for the discrepancy between an ideal and a viscous fluid appears to have only
a weak dependence on the geometry and Reynolds number. Therefore, results from the
present case are assumed to be representative of most configurations and will be used in
the remaining part of this study to illustrate the discussion.
3.3.2
Flow field around a foil passing a cylinder: viscous effects
Figures 3-4a,b show the velocity and pressure coefficient fields at two different times for a
NACAOO12 passing near the cylinder C1 (r = 0.1, d = 0.1) at Re = 6250. The presence of
the cylinder deflects the streamlines, and in particular, figure 3-4a shows that at t = 0.3 the
flow between the foil and the cylinder is accelerated and the pressure decreased. In order
to better visualize the changes due to the cylinder, figures 3-4c,d show the instantaneous
velocity and pressure fields from which the steady state has been subtracted. At t = 0.3,
the cylinder pushes the flow near the leading edge toward the upstream direction, resulting
in a stagnation point shifted toward y > 0 and an increase in pressure on the cylinder side.
Just downstream of the cylinder, the flow is accelerated toward the trailing edge, resulting
in a faster flow and therefore a decrease in pressure. As the cylinder moves downstream
(t = 0.9), the cylinder keeps accelerating the flow between itself and the foil, causing a
decrease in pressure. Even though the pressure drop near the cylinder is much weaker at
t = 0.9 than at t = 0.3, the amplitude of the pressure drop on the foil is not significantly
reduced. Upstream of the low pressure region on the foil, there is also a high pressure region
(x ~ 0.55) that does not appear to be directly caused by the cylinder.
While in figure 3-4e magnification does not reveal any additional features, figure 3-4f
shows a pair of counter-rotating vortices in the foil boundary layer, around x = 0.55 and
x = 0.8. These vortices correspond to the high and low pressure areas along the foil and
are responsible for the discrepancies observed earlier between viscous and inviscid pressure
predictions. Similarly, swirling flow along the rear half of the foil passing near a cylinder
has been observed experimentally, as illustrated in figure 3-5.
The presence of vortices implies that the pressure changes along the surface of a foil
passing close to a cylinder, cannot be accounted for solely by inviscid theory. A second
component, resulting from the boundary layer dynamics and containing memory effects,
is needed to complement the potential flow model. We argue that if the changes in the
boundary layer thickness are known, a potential flow model accounting for them can provide
pressure predictions in good agreement with experiments and viscous simulations.
Figure 3-6a shows the pressure changes along the foil passing near the cylinder C1 as
a function of time and space, simulated with the potential flow code, augmenting the foil
thickness by its steady state boundary layer displacement thickness at Re = 6 250. Vertical
sections of this plot at x = [0.03, 0.3, 0.57] would result in the pressure traces of figure
53
t = 0.9
t = 0.3
U
(a)
>,
0.3
0.3
0.2
0.1
0.2
0.1
0
0
-0.1
-0.1
0.2
0
0.4
0.6
0.8
CP
0.2
0
-0.2
1
0
0.2
0.4
x
(d)
aU-o
(W
0.3
i
(b)
x
0.6
0.8
-0.4
1
U-Un
C -CPO
0.3
0.2
-0.1
0.2
-0.1
--
-
0
0.05
0
0
- ---
-0.1
-0.05
-0.11
0
0.2
0.4
(e)
x
0.6
0.8
1
0
U-GO
0.2
0.4
(M)
0.1
0.1
>0. 05
>'0.05
X
x-U
U.b
U.t
I
CP-CPO
G-GIn
0.05
-0.05
00
0.2
0.6
0.4
0.8
1
0
0.2
0.6
0.4
0.8
1
x
x
Figure 3-4: Snapshots at t = 0.3 (a, c, e) and t = 0.9 (b, d, f) as a NACA0012 foil passes near the
cylinder C1 (r = 0.1, d = 0.1) at Re = 6250. (a-b): Velocity field and pressure coefficient. (c-d):
The steady fields i'o and C,, have been subtracted and the displacement thickness &*(t) is shown
by a solid grey line. (e-f): Magnified view of the area enclosed within the dashed line of (c-d).
Figure 3-5: Experimental flow visualization as a NACA0018 foil passes near a cylinder at Re =
75000. (a-b): Particles pathlines from t = 0.88 (blue) to t = 1.08 (red). The locations of the foil at
the start and end times are represented with their respective color (the intersection is purple). (b):
Magnified view of the swirling flow region. (c): A representative pathline from t = 0.1 to t = 1 is
represented in green with arrows showing the direction of motion.
54
(a)
Potential flow +6
(b)
1.5
Cp-Cpo
1.5
0.05
1
-
0.5
0
-0.51
0
(C)
Viscous simulation
1.5
0.05
0.5
0
0
-0.05
0.5
x
Cp-Cpo
1
-
0
Viscous simulation
-0.5
1
de-0.05
0
(d)
d8 /dt
0.03
0.5
x
Cp-Cpo
Potential flow + a (t)
1.5
0.02
1
0.05
1
0.01
0.5
0
-
0.5
0
-0.01
0
0
-0.05
-0.02
-0.5-
U
-0.03
U.0
x
-0.51
0
0.5
1
x
Figure 3-6: (a, b, d): Pressure coefficient changes along a NACA0012 foil passing near the cylinder
C 1 at Re = 6 250, as a function of space and time. (a): Calculated from potential flow using the steady
state displacement thickness; (b): calculated with the Navier-Stokes solver; (d): calculated from
potential flow using the instantaneous displacement thickness. (c): Rate of change of displacement
thickness. Dashed line: cylinder location projected onto the x-axis; dotted line: location of a
hypothetical feature moving along the foil at half the free-stream. Positive areas are enclosed within
a solid red line and negative ones within a blue dashed line.
3-3d. As has been discussed previously, the cylinder causes an increase in pressure near the
leading edge, followed by a decrease in pressure around the thickest part of the foil. Along
the thinner rear half of the foil, the pressure changes are much weaker.
Comparing the corrected potential flow prediction with the viscous simulation of figure
3-6b, we see very good agreement downstream of the cylinder, but very poor agreement
upstream of it. A strong low pressure region moving at about half the free-stream (dotted
line) characterizes the viscous pressure changes along the rear half of the foil (top right
quadrant). The amplitude of this secondary contribution to the pressure changes varies
with Reynolds number and cylinder geometry, but it is noteworthy that it moves at about
half the free-stream velocity for all the cases we have tested. In order to assess how much improvement to the potential flow model can be gained from knowledge of the boundary layer
dynamics, the instantaneous displacement thickness has been estimated from the viscous
simulations.
Figure 3-6c shows the rate of change of displacement thickness as a function of space
and time. Here again, we can distinguish two components: a direct contribution from the
cylinder following its displacement (dashed line), and a delayed contribution moving at half
55
the free-stream (dotted line). Even though the velocities involved are only a few percent of
the free-stream velocity, their contribution to the local source strength can be significant,
as they are normal to the boundary. In other words, although the boundary layer thickness
* (t) does not vary much, its changes result in additional sources and sinks that significantly
impact the pressure field.
Finally, in 3-6d we show the pressure changes estimated from the potential flow using
the instantaneous displacement thickness *(t). Even though the amplitude is slightly
underestimated, the potential flow model is now able to predict the main features observed
on figure 3-6b, including the low pressure region moving at half the free-stream. This
confirms our hypothesis that if the changes in the boundary layer thickness are known, a
potential flow model accounting for them can provide pressure predictions in good agreement
with viscous simulations.
3.3.3
Convective instability in the foil boundary layer
In the previous section, we showed that the discrepancy between the inviscid and viscous
pressure estimates can be accounted for by the dynamics of the boundary layer. In this
section we show that these dynamics can be predicted simply from the average shape of
the foil boundary layer in open water flow. In particular, we explain why the secondary
perturbation always moves at half the free-stream velocity and discuss the effects of the
Reynolds number.
As described in 3.2.5, the Orr-Sommerfeld equation identifies the eigenvalues W and
eigenmodes p for a given boundary layer profile and wavenumber k. A boundary layer profile
is unstable if for a real wavelength kr, the imaginary part of the eigenvalue corresponding to
its principal mode, wi, is positive. The boundary layer acts as an amplifier for the selected
waves that grow exponentially in time while traveling with phase velocity c, = Wr/kr and
group velocity cg = Owr/Ok, determined by the dispersion relation D(w, k, Re) = 0.
Figures 3-7a-c show iso-contour plots of positive wi as a function of space and wavelength. At Re = 2000, only the posterior 10% of the foil boundary layer is unstable for
wavenumbers between 10 and 20. However, as the Reynolds number increases, the unstable
region grows larger and encompasses more wavenumbers. At Reynolds number 20 000, as
much as half of the foil boundary layer span is unstable and the most unstable wavenumbers are between 30 and 40. For all three Reynolds numbers, unstable waves are convected
downstream as they grow in time, which is referred to as convective instability [13].
A wave with wavelength 27r/k and initial amplitude Ao(k) at onset time to and position
iO, has amplitude A(k, t, x) as it evolves in time and propagates downstream. The amplitude
ratio a is given by:
a (k, t, x (t)) = A/Ao = exp [Iowi (k,
T,
x (r)) drl
= exp [jX wi (k, t( ), x) /cg dj
.
(3.11)
So if we denote by x1 (k) the location where w (k) = 0, the maximum amplification is:
amax (k, x) = max(A /Ao) = exp
Figures 3-7d-f show iso-contours of positive
[f
wi(, k)/c 9 d1
.
(3.12)
ainax. Similarly to what has been observed in
56
W1
(a)
50
40
0
0
0
C~I
II
0
301
Wj<0
201
10'
0.4
0.8
0.6
In(amax)
(d)
50 r
50
40
40
' 30
' 30
20-
20
10
1
0.4
0.6
(
(a
II
s'
W<0
40
40
' 30
20
20
0.6
0.8
1
10
II
0
0.4
0.6
0.8
1
10
50
(I)
50
40
40
40
30
2 30
()
0.6
0.350.4
0.6
0.8
1
0.8
1
10
0.45
'-----.---
20
j<0
0.4
C.45
30
--
10
I
20
50
20
1
x
(C
c~J
0.8
' 30
-.,.
x
0
0
0
0
0.6
(h)
50S
30
0.4
0.4
x
5U
0
10'
1
(e
b)
50i
40
10
0.8
0.45
x
x
0
tO
Cr
(g)
20
0.6
0.4
0.8
1
10
0.35
0.4
0.6
0.8
1
x
Figure 3-7: Properties of the mean boundary layer velocity profiles computed from viscous simulations, as a function of the location along the foil and wavenumber. (a-c): Iso-contours of positive wi
for the principal modes (0.5 between successive contours). The dotted line shows the most unstable
wavenumber at each location. (d-f): Iso-contours of the maximum amplification in logarithmic scale
(0.25 between successive contours). The dotted line shows the wavenumber of largest amax at each
location. (g-i): Iso-contours of the group velocity (0.1 between successive contours).
57
figures 3-7a-c, as the Reynolds number increases, the unstable region starts earlier in space
and encompasses a wider frequency range. At Reynolds number 2000, the most amplified
wavenumber at the trailing edge is about 13 but Umax remains small (less than 1.1). At
Re = 6 250, the most amplified wavenumber at the trailing edge is close to 23 and Umax is
now about 2. At Reynolds number 20 000, the most amplified wavenumber is 36 with amax
now greater than 12. When the foil passes near a cylinder, principal modes of its boundary
layer get excited with an amplitude depending on the cylinder size and distance. The
unstable modes are amplified and propagate at the velocity determined by the dispersion
relation, resulting in the secondary perturbation observed in 3.3.2.
Since for the range of Reynolds numbers considered disturbances with certain wavenumbers will grow faster (exponentially, due to the instability of the boundary layer) than with
other wavenumbers, figures 3-7d-f show that the boundary layer acts as a wavenumberselective signal amplifier. The amplification rate and preferred frequency range strongly
depend on the Reynolds number: The larger the Reynolds number, the larger the frequency
and amplification rate. There is, however, one property of the boundary layer that remains
constant across our range of Reynolds numbers: the phase velocity of the amplified waves.
As shown in figure 3-7g-i, the phase velocity of the most unstable modes is always between
0.45 and 0.55, explaining the observation made earlier that the secondary perturbation
moves at half the free-stream velocity.
On figure 3-8a we show a close-up of the pair of counter rotating vortices observed in
figure 3-4f in the boundary layer of the NACA0012 foil. Next to it, on figure 3-8b, is shown
the principal mode from the linear theory at x = 0.7 for kr = 15 (Re = 6 250). Despite
the finite angle of the airfoil boundary with respect to the free-stream and a noticeable
difference between the strength of the two vortices in figure 3-8a, figures 3-8a and 3-8b
are strikingly similar. This similarity is another indication that despite the approximations
employed, the linear stability theory is able to capture the dynamics of the boundary layer
responsible for the secondary pressure perturbation. As observed by [31], linear stability
theory has been shown "by serendipity" to provide accurate predictions of the frequency
and wavenumber of basically nonlinear flows. This proves once more to be the case in this
problem.
Finally, figure 3-9 illustrates how the boundary layer properties discussed above impact
the pressure distribution along a foil passing near a cylinder. Similarly to figure 3-6b, figure
3-9 shows the pressure changes along the foil passing near the cylinder C1, now at Reynolds
number Re = 2000 and 20 000. The primary disturbance, in the bottom half of the figure,
is very similar for both Reynolds numbers, but the secondary perturbation, characterized
by a low pressure in the top right quadrant, changes with Reynolds number. Whereas at
Re = 2000 the secondary perturbation is small, at Re = 20000 the amplification is such
that by the time the instability reaches the trailing edge, its amplitude is larger than the
disturbance that caused it.
3.3.4
Enhancing potential flow predictions with instability results
We have shown that the pressure changes along a foil passing near a cylinder have two main
components: the first part can be approximated accurately by potential flow model, while
the second part can be accounted for by the dynamics of the boundary layer, acting as an
amplifier. The properties of the latter can be predicted from the average boundary layer
shape in open water, but the resulting convective instability amplifies the features of the
unsteady flow initially predicted by inviscid theory.
58
(b)
Viscous disturbance
t=AQ
(a)
Linear theory principal mode
x=(17 k.=1S ia-=62
0.1
0.
0.08
>- 0.
0.06
0.04
0.02
0.5
0.6
0.7
0.8
0.9
0.5
x
0.7
0.6
0.9
0.8
x
Figure 3-8: (a): Pair of counter rotating vortices observed in the boundary layer of a NACA0012
passing near the cylinder C1 at Re = 6250. Arrows show the disturbance to the velocity field and
colors the perturbation to the pressure field (the color scale is the same as in figure 3-4f). (b):
Principal mode for x = 0.7 and kr = 15 shown above a boundary located at y = 0.04. The color
scale has been chosen to roughly match that of (a).
(a)
Re = 2000
(b)
1.5.
Cp-Cpo
1.51
0
0.5
0
-0.5 H
0
Cp-Cpo
0.05
0.05
1
Re = 20000
0
0.5
-0.05
0
-0.05
-0.51
0
0.5
x
Figure 3-9: Viscous simulations of pressure coefficient changes as a function of time t and space
x along a NACA0012 foil passing near the cylinder C1 for (a) Re = 2000 and (b) Re = 20000.
Dashed line: cylinder location projected onto the x-axis; dotted line: location of a hypothetical
feature moving along the foil at half the free-stream. Positive areas are enclosed within a solid red
line and negative ones within a blue dashed line.
59
Let us consider a cylinder C causing a change in pressure coefficient Cv, (X, t, C). We
denote by P(k, t, C) the discrete Fourier transform of Cy from the space to the wavenumber
domain, using for our numerical results 128 points. We decompose the pressure changes
into two components:
P(k, t,C) = kI(k, t, C) + P 2 (k, t, C).
If we denote by
(3.13)
Pi the pressure changes estimated by inviscid theory augmenting the foil by
6*:
P, (k, t, C) ~_Pi(k, t, C).
(3.14)
Following linear stability analysis, the secondary pressure changes can be approximated by:
P 2 (k,
t, C) ~ Pi(k, to, C) exp (jwi(k,r)d-r) lt>t0 ~ Pi(k, to, C) a(k, t)
(3.15)
for an appropriate time to. This expression is valid until the disturbances reach the trailing
edge and are shed into the wake.
Using simulations for cylinders ranging in radius from 0.025 to 0.8 and placed at distances ranging from 0.05 to 0.8 (see Appendix C), an estimated amplification coefficient
&(k, t) has been calculated, where:
IP(k, t, C) - Pi(k, t, C)|
=(k,
t) pi (k, to, C)I + e.
(3.16)
to = 0.15 has been chosen, which roughly corresponds to the time when the amplitude
of the inviscid disturbance reaches its maximum. Equations as (3.16) are referred to as
varying-coefficient models, which arise in many scientific areas, and numerous algorithms
have been developed in the last 20 years to estimate their parameters [76, 55]. The main
advantage of these models over the more general form:
P(k, t, C) = f (k, t,ji(k, t, C), Pi (k, to, C)) + e
(3.17)
is that they can handle large dimensions, especially when their use is physically motivated
as here. Details about varying-coefficient models and the algorithm used to estimate &(k, t)
are found in Appendix C.
Figure 3-10 shows the pressure coefficient changes along a NACA0012 foil passing near
the cylinder C 1 , as a function of the wavenumber k and time t, for three Reynolds numbers
2000, 6250, and 20000. Note that since the length of the foil is L = 1, the resolution
in the wavenumber domain is limited to 2-r. As in the space domain plots, the potential
flow pressure changes shown in figures 3-10a-c are similar for all Reynolds numbers and are
only measurable as the cylinder passes the front half of the foil (0 < t < 0.5). The viscous
simulations shown in figures 3-10d-f predict a distinctive second component to pressure
changes that appears later in time, is stronger in magnitude, and spans a wider range of
wavenumbers for larger Reynolds numbers.
Figure 3-11 shows iso-contours of the estimated coefficient &(k, t) for three Reynolds
numbers. The results are consistent with the observations from 3.3.3: The higher the
Reynolds number, the larger the amplification rate and the value of the most amplified
wavenumber are. The values of the most amplified wavenumbers are also very close to
those predicted by linear theory: around k = 18 at Re = 2000, k = 26 at Re = 6250 and
k = 37 at Re = 20 000. The values found for amplification, however, are much smaller than
the upper bound amax found from linear theory and plotted in figures 3-7d-f. Whereas
60
Ra
(a)
=200o
1.5
* D
0
0.5
1.5
1
1
0.5I
0.5,
0
0
-0.5
20
30
50
0
20
30
40
-0.5
50
1.5.
1
1
30
40
50
0
50
V
-0.5
I1
0.5
0
01
20
40
30
0.5
-0.5
U) -0.5
20
k
1.5.
0
0.5
k
(e)
-5-0.5
1
Ou
-0.5
k
(d)
1.5
E
40
Re =20000
Mc)
1.5
0
m
Re = 6250
(b)
20
40
30
-0.5
50
20
30
0
50
k
k
k
40
Figure 3-10: Pressure coefficient changes along a NACA0012 foil passing near the cylinder C1, as
a function of wavenumber and time, for three Reynolds numbers, 2000, 6250, and 20000. (a-c):
1pi (k, t, C) inviscid simulation; (d-f): IP(k, t, C) viscous simulation. Resolution in the wavenumber
domain is limited to 27r.
according to the linear stability theory 6(k, t) could reach 2 for Re = 6250 and 10 for
Re = 20000, the values found here do not exceed 1 and 3 respectively. This difference
can be explained by the fact that pressure changes near the leading edge of the foil largely
contribute to Pi (k, to, C), whereas they do not contribute much to P2 (k, to, C) due to the
high stability of the boundary layer near the leading edge.
We define the test error as the average value of the residual e as defined by Eq. 3.16,
calculated on a test set randomly chosen from the available data set and not used to estimate
(k, t). The remaining part of the data is referred to as the training set (see Appendix
C for details). Table 3.2 shows the average and standard deviation of the inviscid and
viscous pressure changes for three Reynolds numbers and compares the test error to the
(a)
(b)
Re = 2000
Re = 20000
(C)
Re = 6250
1.5
1.5
1.5
1.
0.5*
0
-0 .5-05220
9 .
r l
0
30
40
40
50
50
0.5
0.5
0
0
-0.5-
20
30
40
k
50
-0.5L
20
30
40
50
k
Figure 3-11: Amplification coefficient h(k, t) estimated from viscous simulations as a function of
wavenumber and time for Re = [2000, 6250, 20000].
61
Re
A
1Al
1p - p /41l
IeI/N
1E/P - Pil
2000
0.123
0.149
0.668
0.272
0.409
(+0.009)
(+0.008)
( 0.051)
(+0.016)
(+0.025)
6250
0.096
0.168
0.704
0.284
0.404
(+0.007)
(+0.008)
(t0.032)
(+0.015)
(+0.017)
20000
0.092
0.251
0.772
0.389
0.504
(+0.006)
(+0.010)
(+0.018)
(t0.020)
(+0.025)
Table 3.2: Average and standard deviation of the training data set inputs and test error.
difference between the viscous and potential flow models. While the magnitude of the
inviscid perturbation decreases with increasing Reynolds number due to a thinner boundary
layer, increased boundary layer instability induces a larger viscous disturbance. As a result,
even after using the steady displacement thickness, the average error in the potential flow
model is between 65% and 80%. By adding the Reynolds number dependent component
&(k, t)Pi(k, to, C) to the model, we are able to reduce the error substantially, bringing it
down to values between 25% and 40%.
Figure 3-12 compares the viscous residual IP(k, t, C) - Pi(k, t, C)I (figures 3-12a-c) to the
fitted function &(k, t)[pi(k, to, C) (figures 3-12d-f). The simple form of Eq. 3.16 is able
to enhance the potential flow model and reproduce the basic Reynolds number dependent
features. In particular, the increased amplification of high frequencies and increased delay of the secondary perturbation at high Reynolds number are captured by the model.
The memory and amplification effects due to the boundary layer are thus added to the
inviscid perturbation without significant complexity. Thanks to its simplicity and the wide
range of cylinder sizes and distances used to fit it, the model is likely to generalize well to
other obstacle shapes. A more complex model could, however, achieve better quantitative
agreement with the viscous simulations.
3.4
3.4.1
Discussion
The boundary layer: filter or amplifier?
We showed that the fish boundary layer acts as an amplifier of the pressure disturbance
caused by a nearby cylinder. In the lateral line literature, the boundary layer is often
viewed as a filter [109], damping mostly low frequencies. The two views, however, are not
contradictory as they refer to different problems.
The superficial neuromasts are located on the skin of the fish, contained within the
boundary layer. Therefore, the measured velocity is attenuated depending on the neuromast
height to boundary layer thickness ratio. When the fish is stationary, the thickness of the
boundary layer that develops due to an external stimulus reduces with increasing stimulus
frequency; hence the boundary layer acts as a high-pass filter for the superficial neuromast
excitation [109]. The neuromasts located inside the canals have been shown to respond to
the gradient of pressure below typically a few hundred Hertz [172]. In the absence of freestream velocity, it is possible to predict the stimulus detected by the canal neuromasts using
a potential flow model [39, 68, 132]. Indeed, the thin boundary layer that develops over the
fish does not affect the pressure, as there is no pressure gradient across the thickness of the
boundary layer.
All these studies have been conducted assuming a stationary fish next to a vibrating
62
R=2000
(a)
R=62'ICA
(b)
1.5
R
(M
1.5
=20000
p-i
0.8
1.5
10.6
0.5
0. 5
0
00
-0.5
20
30
(d)
40
50
k
-0.5
0.4
0.5
02
0.2
20
30
(e)
40
50
k
-0.5
20
30
M
1.5
1.5
1.5
0.5
0.5
0.5
40
50
k
!0.
0.8
0.4
-0.5
20
30
40
50
0.5
20
30
40
50
-0.5
20
30
40
50
Figure 3-12: Pressure coefficient changes along a NACA0012 foil passing near the cylinder C 1 , as
a function of wavenumber and time, for three Reynolds numbers, 2000, 6250, and 20000. (a-c):
viscous residual J(k,t,C) -Pi(k,t,QC), (d-f): fitted model 6(k,t)JPi(k,to,QC). Resolution in the wave
number domain is limited to 27r.
sphere, when the fish boundary layer is solely due to the vibrating dipole. Several studies
have shown that moving fish can also use their lateral line to discriminate stationary objects
or arrangements [180, 75], but little is known about how the viscosity affects the measured
signal. In this paper, the fish is assumed to be gliding, resulting in a thicker boundary layer
and typically much higher velocities than for a stationary fish. In this configuration, the
surface neuromasts may be already saturated and hence unable to detect external stimuli,
but the canal neuromasts are still able to detect external stimuli [53]. [199, 200] recently
showed that inviscid simulations significantly underestimate the pressure changes along a
fish gliding toward or parallel to a wall. Indeed, viscosity causes the fish to displace more
water, which can be modeled by adding a stationary displacement thickness. We have shown
here that the dynamics of the boundary layer can further enhance the signal measured by
the lateral line, as the boundary layer amplifies certain wavelengths and frequencies.
In conclusion, depending on the specific problem studied, the boundary layer can either
act as a filter or an amplifier.
3.4.2
Lateral line stimulus and effect of swimming speed
In order to assess the effect of the fish gliding speed we consider specific examples. We
start with a fish of length L = 10cm gliding at two body lengths per second, with a
Reynolds number Re = 20 000. A large portion of its boundary layer will be convectively
unstable with the most unstable wavelength equal to 17mm (figure 3-7f), corresponding
to a frequency around 5 Hz. If the same fish glides at 0.8 body lengths per second, with a
Reynolds number Re = 6 250, the most unstable wavelength is 27mm (figure 3-7e), with a
frequency of 1.5 Hz.
We estimate next the magnitude of the stimulus caused by a nearby cylinder. Figure
63
Re = 2000
(a)
0.8
0
Re
(b)
0.6
0.4
=
6250
0.8
0.6
0.6
0.4
-0.4
0.2
Re = 20000
(C)
0.8
0.2
0.2
0
C
0
0.4
d
0.6
0.8
0.2
0.4
d
(e)
0.8
0.6
0.8
(f)
0.6
0.6.
0.4
d
0.2
0.4
d
0.8
0.6
0.8
0.6
-0.4
-0.4.
0.2
-
0.2
(d)
0.8
0.4
ci)
0
0.2
0.2
0.2
0.4
d
0.6
0.8
0.2
0.2
0.4
d
0.6
0.8
0.6
0.8
Figure 3-13: Magnitude of the pressure coefficient changes (max 1C,|) estimated from potential
flow (a-c) and viscous simulations (d-f).
3-13 shows the magnitude of the pressure coefficient changes due to a cylinder as a function
of its radius r and distance d. The upper threshold in the detectable distance depends
on the background noise, so the iso-contour lines represent detectability limits for various
noise levels. As expected, a cylinder is more likely to be detected the larger its radius is
and the closer it gets. Figures 3-13a-c show that according to the potential flow model, the
signal from a given cylinder is slightly stronger at lower Reynolds number due to a thicker
boundary layer, but the increase in detectability from Re = 20 000 to Re = 2000 is no more
than 30%.
For Reynolds numbers Re = 2000 and Re = 6 250, the amplitude predicted by the
potential flow model is in good agreement with the viscous simulations, also within 30%.
Indeed, despite qualitative differences between the pressure signal predicted by the two
methods, the amplification factor for these Reynolds numbers is less than 1, as shown in
figure 3-11, so the boundary layer does not increase the amplitude of the pressure signal.
However, at Re = 20000, we have estimated amplification factors greater than 2 for
wavelengths between 0.1 and 0.2. As a result, the amplitude of the viscous pressure signal
can be much larger than what would be predicted within an inviscid fluid. If, for example,
we assume that pressure coefficient changes larger than 0.4 can be detected, without the
effects of the pressure amplification by the boundary layer, a cylinder of radius r = 0.3
could only be detected at a distance less than 0.06. Because of the amplification, however,
it can be detected from twice as far.
It is important to note that since only two-dimensional flows are considered in this
paper, pressure changes are overestimated compared to a three-dimensional case. However,
the same methodology can be applied to three-dimensional flow, and the results qualitatively
transfer to three dimensions. Moreover, comparison between two- and three-dimensional
viscous and inviscid simulations of a fish approaching a wall suggest that viscosity impacts
the magnitude of the pressure changes more importantly than three-dimensionality [108].
64
3.4.3
Can the boundary layer facilitate object identification?
As we have shown, the boundary layer acts as a pressure signal amplifier in the posterior
part of the foil, whereas in an inviscid fluid a cylinder would cause significant pressure
changes only along the anterior half of the foil. Since both the anterior and posterior parts
of the foil are subject to large pressure variations, object detection becomes easier in a
viscous rather than in an inviscid fluid. An important question is whether viscous effects
can also help with shape identification.
Figure 3-7 shows that the boundary layer amplifier has a large bandwidth. The bandwidth, defined as the range of wavenumbers for which the amplification is at least 1/v'2
of the maximum amplification, is indeed around 20 for both Re = 6 250 and Re = 20 000.
Therefore, the frequency content of the original signal is largely preserved in the pressure
signal after amplification, preserving information concerning the size, distance, and shape
of the cylinder.
For example, figure 3-14 shows the difference between the pressure coefficient along a
NACA0012 foil passing near the cylinder C1, with r = 0.1, d = 0.1, and a slightly larger
cylinder, placed further away, cylinder C 2 , with r = 0.125, d = 0.12, both at Reynolds
number Re = 6250. Stimulus differences estimated by potential flow (figure 3-14a) and by
viscous simulations (figure 3-14b) are provided. Both estimates agree that the magnitude
of the stimulus due to C1 is larger, as indicated by the positive difference at (x = 0, t = 0)
followed by a negative difference around (x = 0.15, t = 0.1), corresponding to the positive
and negative peaks of the stimulus as seen in figure 3-6, respectively. The peaks associated
with C 2 are also wider, as evidenced by the negative difference on either side of the positive
difference, and vice versa. Along the rear half of the foil, however, the inviscid difference
is much weaker and has a much lower frequency than the viscous one. In the viscous
simulation, the information that the stimulus is stronger but narrower is amplified by the
boundary layer, and hence a negative difference (blue), surrounded with positive difference
(red), is seen moving along the posterior half of the foil at half the free-stream velocity. This
demonstrates that the boundary layer, in addition to amplifying the pressure signal due to
a cylinder, also amplifies the difference between signals caused by two different cylinders,
therefore facilitating object identification.
This difference in the amplified disturbance due to different cylinders is even stronger
at higher Reynolds number, Re = 20 000, where the amplification is larger. Figures 3-15a-c
show disturbances due to three different cylinders, with radius r = 0.10, 0.25, 0.50; and
placed at distances d = 0.06, 0.10, and 0.12, respectively. All three figures contain a characteristic clockwise (blue) vortex, corresponding to a pressure coefficient drop of about 0.25.
The typical width of this main vortex increases from about 0.07 on figure 3-15a, to 0.1 on
figure 3-15b, to 0.12 on figure 3-15c. Whereas the small wavelength of the clockwise vortex on figure 3-15a leaves room for a counter-clockwise vortex of comparable strength, this
second vortex is much weaker on figure 3-15b, and hardly exists on figure 3-15c. It appears
clearly from these three figures that the wavelength of the amplified disturbance increases
with the distance and radius of the cylinder. The difference in the frequency content of the
disturbance caused by different objects, as convected by the unstable boundary layer, can
clearly be used to distinguish between the objects.
65
(a)
Potential flow + 8n
1.5*
(b)
Viscous simulation
AC,
0.01
1.5m
0.005
0.5
0
-0.5
0
-0.51
-0.51
0
0
x
1
0.5
x
-0.01
3-14: Difference between the pressure coefficient changes due to two cylinders: (1) C 1
0.1,d = 0.1) and (2) C2 , Cp(r = 0.125,d = 0.12) using potential flow predictions (a) and
simulations (b); both at Re = 6250. Areas of positive difference are enclosed within a solid
and areas of negative difference within a blue dashed line.
,
Figure
CP(r =
viscous
red line
-0.005
0
(a) d=0.06,r=0.10
0.22
(b) d=0.10,r=0.25
(c)
d=0.12,r=0.50
0.2
0.2
Cp-Cp0
0.2
0.15
0.15
0.15
0.1
0.1
0.1
0.05
0.05
0.05
0.1
0
-0.1
-0.2
x
x
x
Figure 3-15: Snapshots at t = 0.9 showing the velocity field and pressure coefficient disturbances
as a NACA0012 foil passes near three different cylinders with radius r at a distance d: (a) d = 0.06,
r = 0.10; (b) d = 0.10, r = 0.25; (c) d = 0.12, r = 0.50; all at Re = 20 000.
66
Chapter 4
Exploiting energy from the flow:
how efficiently can fish swim?
4.1
Introduction
The grace and agility of fish and marine mammals have excited the curiosity of scientists
for centuries. From the Northern pike (Enox Lucius) that can reach an acceleration of up
to 25g [73] to the European eel (Anguilla Anguilla) that annually swims across the Atlantic
Ocean (over 5000 km) while fasting [65], fish using body undulation as their primary means
of propulsion greatly surpass all engineered vehicles in terms of accelerating, cruising and
maneuvering capabilities. In the hope of unveiling the secrets of fish extraordinary performance, biologists, hydrodynamicists and engineers have tirelessly observed fish swimming
[69, 179, 167], measured their metabolism [6, 183], proposed hydrodynamic principles and
scaling laws [63, 99, 62, 174], and even built robots mimicking fish [162, 152, 143, 82].
In 1933, Gray [69] provided the first detailed analysis of eel swimming kinematics, as
well as force and energy estimates. A similar analysis applied to dolphins [70] led to the
conclusion that dolphins cannot produce enough power to overcome the drag on their body.
Even though Gray's paradox was largely resolved by better data on speed and muscle
performance [6, 183], it is an influential paradox that still inspires researchers, as illustrated
by Bale's recent paper [9]. If Gray's paradox has been so influential, it is because of its
intrinsic relation to the question that has obsessed scientists and engineers for decades: how
efficient is fish swimming? One of the challenges behind this apparently simple question
lies in the difficulty of measuring the swimming power: global methods such as oxygen
consumption rates do not specifically measure the swimming power, and direct measures in
muscle only give local measurements [51].
With large computational power now available to all, computational fluid dynamics
(CFD) provides an alternative means of studying fish swimming that promises new avenues
[41]. Since the first viscous simulations of a two-dimensional self-propelled anguilliform
swimmer by Carling in 1998 [26], a variety of methods have been developed to simulate
fish swimming. These methods range from arbitrary Eulerian-Lagrangian methods with
deformable mesh [91], to immersed boundary methods [16, 149, 101, 12], to multiparticle
collision dynamics methods [136] and viscous vortex particle methods [50]. Thanks to recent
improvements in Particle Image Velocimetry (PIV), it is now possible to visualize near
body and wake flows in great detail [167, 115, 4] and even estimate the thrust generated
by swimming fish [60], but CFD is a unique compliment that gives access to full three67
dimensional flow structures as well as local forces and power. CFD makes it easy to estimate
the hydrodynamic power of swimming, but a more philosophical question still needs to be
answered before the efficiency can be calculated: what is the useful power of a self-propelled
fish? We will address this question in 4.4.
The application of CFD to the study of fish swimming is still in its infancy, and a number
of modeling decisions also need to be made. For instance, 2D or 3D model [91, 45], towed
with imposed kinematics or self-propelled with free recoil [136], actively deformed fins or
elastic fins [12], etc. In addition to modeling considerations, if recoil is allowed, the coupled
body-fluid motion needs to be carefully handled, in particular to ensure stability of the
numerical scheme [61, 211, 33]. Once these modeling and numerical questions are resolved,
CFD becomes a very powerful tool allowing unmatched freedom. Unlike with real fish, it is
particularly easy to alter the body geometry or the swimming kinematics and measure the
influence of each parameter on the swimming performance. Taking advantage of this new
freedom, there has recently been a number of publications reporting efforts in optimizing
fish shape and/or swimming motion [91, 173, 52, 160].
In addition to optimizing their motion with respect to self-generated flow structures, fish
might be able to use each other to save energy. Whether energy saving is one of the reasons
for-and benefits of-schooling has long been controversial. Weihs [186] argued, in one of
the only papers proposing a hydrodynamic theory of schooling, that fish can save energy
by taking advantage of the reduced velocity area found, on average, between two propulsive
wakes. However, Partridge [122] later showed that saithe, herring and cod do not swim in
the diamond pattern predicted by Weihs, which led Pitcher [128] to write that "no valid
evidence of hydrodynamic advantage has been produced, and existing evidence contradicts
most aspects of the only quantitative testable theory published." Yet, as pointed out by
Abraham [1], such conclusions are premature because they ignore the potential trade-offs
involved in school functions. Indeed, despite the difficulty of assessing the importance of
energy saving in schooling due to the dynamic nature of schools, there has been experimental
evidence that the fish in the back of a school spend less energy than those in the front [92].
A recent paper even suggested that in a fish school, individuals in any position have reduced
costs of swimming, compared to when they swim at the same speed but alone [106]. Finally,
the recent finding that ibises in a flock position themselves and phase their motion such
that they can take advantage of the vortices left by the ibis in front of them suggests that
a similar mechanism might be at play in fish schools [129].
The goals of optimizing fish body shape, swimming kinematics, and school organization
are two-fold. From a scientific stand point, comparing the optimal parameters to those
observed in various species can help shed light on the evolution process that led to each
species. From an engineering stand point, the goal is to discover new design principles,
inspired by efficient living organisms [173], and potentially even exceed the performance of
these organisms [173]. In 4.2, we discuss modeling considerations for the simulation of
fish swimming and define the model of a two-dimensional NACA0012 used in the rest of
the chapter. We then present and validate the numerical details specific to fish swimming
simulations in 4.3. After defining the quasi-propulsive efficiency as the only rational way
to measure the performance of fish swimming in 4.4, we optimize the gait of an undulating
foil in open-water ( 4.5) and the positioning and timing for a pair of undulating foils ( 4.6).
68
4.2
Fish swimming: modeling considerations
Fish in nature present a wide variety of designs and propulsion modes. However, most fish
generate thrust by bending their bodies into a backward-traveling wave that extends to the
caudal fin, a type of swimming often classified as body and/or caudal fin (BCF) locomotion
[145]. In this chapter, we investigate the efficiency of BCF propulsion, with particular
examples drawn from eels that undulate their whole body (anguilliform motion), as well as
saithe and mackerel that only undulate the aft third of their body (carangiform motion)
[21]. Since the swimming motion is two-dimensional, we first use a two-dimensional model.
4.2.1
Physical model and kinematic parameters
We represent a swimming fish by a neutrally buoyant two-dimensional undulating NACA0012
foil of length L = 1, as illustrated in figure 4-1. The foil propels itself at speed U, in viscous
fluid by oscillating its mid-line in the transverse direction y. The leading edge of the foil is
located at x = 0 and its trailing edge at x = 1. The lateral displacement h(x, t) of a point
located at x along the foil is given at time t by:
h(x, t) = ho(x, t) + B(x, t) + y1(x)
= aoA(x) sin (27r(x/A - f t + 0)) + B(x, t) + y(x)
= g(x) sin (27 (ft +
4 (x))) + y1(x)
(4.1)
where A(x), with A(1) = 1, is the envelop of the prescribed backward traveling wave of
wavelength A and frequency f,
B(x, t) = (ar + brx) sin (27r(f t + 0r))
(4.2)
is the recoil term due to the hydrodynamic forces on the foil, and
Yi(X)
=
C(x 2 +
-yx + 3)
(4.3)
can be used for steering (see 4.3.3) by adding camber to the foil, while -y and 3 ensure
that linear an angular momentum are conserved through the deformation. yi is necessary
to ensure stability but, in steady regime, yi < ao and it can be ignored.
The parameter ao determines the amplitude of ho at the trailing edge. It will either be
kept constant (ao = 0.1), or adjusted through a feedback control loop to ensure that the
average net drag on the foil is 0, as described in 4.3.3. ho(x, t) can be used to prescribe the
full kinematics of the swimmer, in which case h(x, t)
ho(x, t), or only the deformation for
y
fA
Us
0
a
X
L
Figure 4-1: Schematic showing the fish model parameters. A foil of length L undulates in a flow
of speed U, with a wave traveling backward at speed fA and amplitude a at the trailing edge.
69
1
01carangiform
-carang
0.8
anguil
--
0.6
-0.1
0.1
01
-
anguilliform
0.4
0.1
0
-0.1
0.21
0
0
0.2
0.4
x
0.6
0.8
0
1
(a) Prescribed amplitude envelopes
Figure 4-2: Carangiform and anguilliform motion for
Re = 5000 with recoil.
f
0.6
0.8
x
(b) Mid-line displacement
0.2
0.4
1
= 1.8 and ao = 0.1 at Reynolds number
a freely moving foil on which the recoil is prescribed by hydrodynamic forces. In the latter
case, the envelope of the actual displacement is given by g(x), with peak to peak amplitude
at the trailing edge given by a = 2g(1).
The prescribed kinematics of a carangiform swimmer, based on the experimental observation of steadily swimming saithe [179, 178], is often modeled as:
ao = 0.1,
2
A(x) = 1 - 0.825(x - 1) + 1.625(x
-
1),
A = 1,
B(x, t) = 0.
(4.4)
This motion is for example used in [16, 44]. Experimental observations of American eels
[168] suggest that anguilliform motion can be represented by:
ao =0.1,
A(x)= 1 + 0.323(x - 1) + 0.310(x 2
-
1),
A =1,
B(x, t)= 0.
(4.5)
This motion matches very closely that used in [17].
Figure 4-2a shows the prescribed envelope A(x) for the carangiform (resp. anguilliform)
swimmer defined in Eq. 4.4 (resp. Eq. 4.5). Figure 4-2b illustrates the resulting mid-line
displacement in the presence of the recoil term.
4.2.2
Governing equations and dimensionless quantities
In a self-propelled swimming model, the body motion is prescribed by the coupling between
the fluid and body dynamics. The physical values are non-dimensionalized by the fish body
length L, its intended average cruising speed U, and the density of water p.
The fluid is governed by the Navier-Stokes equations, which we express in a nondimensional form:
t
\
= -
/
-
Re
U
Vp
(4.6a)
(4.6b)
V - U'= 0
where U- is the velocity in the fluid normalized by U8 , the pressure p is normalized by pU,2I
70
and the Reynolds number is defined as:
Re =Us Lv,
(4.7)
where v is the kinematic viscosity of the fluid. Unless specified otherwise, the Reynolds
number used is Re = 5000. The other important dimensionless parameter for swimming is
the Strouhal number, defined as:
St =fa
Us,
(4.8)
where a = 2g(1) is the peak to peak amplitude of the trailing edge displacement.
In a reference frame moving at velocity U, with respect to the fluid, the motion of
the fish can be decomposed into a rigid body motion and a deformation. The rigid body
motion is defined by the translational and rotational velocities, Ve and Wb respectively. The
deformation is defined by velocity V* in a local frame attached to the solid body with origin
at its center of mass (COM). The foil motion is written as:
x*,
(4.9)
fh ds,
(4.10)
V c+V*+
where * is the position vector in the COM frame.
The net force on the body is defined as:
Fh
where fh are the hydrodynamic forces on the foil and
corresponding moment is:
Jc=
0Qb is the surface of the foil. The
ds,
(4.11)
-fh - Vds.
(4.12)
x*xfh
and the swimming power is:
Pin =
We define the net drag on the body as the x-component of the force and the net thrust as
its opposite:
D n = Fhx,
T = -Fh.
(4.13)
From these values, we define the dimensionless power coefficient Cp and thrust coefficient
CT:
C
=
Pi"
1pU3L
and
CT =p71.
1pU2L'
(4.14)
-CT, as well as the friction (CDf) and
We similarly define the drag coefficient CD
CDf + CDp. The time-averaged values of
pressure (CDp) drag coefficients such that CD
these quantities are represented by an overbar. Finally, we call R the resistance (drag) of
the rigid towed foil at speed U, and define the quasi-propulsive efficiency 7Qp and the net
propulsive efficiency 7Th:
R)=-
(4.15)
,
= (TFin
in
4.4.
details
in
in
which are discussed
In order to simplify the equations of motion, we consider a planar motion in the (x, y)
IQ P
71
plane, such that ic is a two-dimensional vector (vX, vg) and wb = W' We then define the
generalized velocity V, location X, and force F vectors, as well as the generalized mass
matrix M:
V =
Vezd
V
Fx
1b
X
F
M=
F
7
Mt
7n 0
0
0
0
IC
m
0
,
0 (4.16)
where m, is the mass of the body which has density Pb = p and I, its moment of inertia
with respect to the COM. The motion of the body is governed by:
d
~(MV) =F.
{
(4.17)
The prescribed deformation V* should also conserve linear and angular momentum, such
that:
b
PfdQb~0
pf
p fx *
dQb=,
(4.18a)
(4.18b)
where Qb represents the volume occupied by the foil at time t. Eqs. 4.17 and 4.18 determine
the recoil B(x, t) resulting from the prescribed motion ho(x, t) and the hydrodynamic forces
F. Due to the significant added complexity incurred by the recoil term, most of the earlier
simulation studies neglected it [16, 44]. However, the amplitude of this term, and its impact
on the estimated swimming power are substantial [136], as will be shown in the next section.
4.2.3
On the importance of recoil
We consider here the carangiform motion of Eq. 4.4 with frequency f = 2.1. Figure 43a shows the dimensionless linear and angular momentum for the fully self-propelled foil
(recoil determined by hydrodynamic forces and adaptive amplitude ao). The angular and
transverse momentum are larger than the longitudinal momentum, but the three amplitudes
are comparable. However, the moment of inertia of the foil is much smaller than its mass:
m = 0.081,
Ic = 0.0045,
(4.19)
where the mass and moment of inertia are non-dimensionalized by the length L and density
p. Therefore, whereas the linear momentum results in velocities smaller than 3% of the
free-stream Us, the rotation of the foil generates velocities at the trailing edge up to 40%
of the free-stream, as shown in Figure 4-3b. This observation suggests that, whereas the
longitudinal motion of the foil might be negligible, the transverse motion, and specifically
the motion due to the free-rotation, are probably important.
In order to further illustrate this result, figure 4-4 shows the quasi-propulsive efficiency
as a function of frequency for the carangiform and anguilliform motions with and without
recoil. The figure shows that, at all frequencies, the undulation with recoil requires more
power than the undulation without recoil. Therefore, simulations that do not allow for
recoil are likely to underestimate the swimming power, as discussed in [135]. The figure
also shows that the optimal frequency without recoil might differ from the optimal frequency
with recoil. In the cases studied here, the optimal frequency for the carangiform undulation
72
4x 1
0.5
2
E
-.
0
E0
E
0
~vC
-0.5
-v_
-
C
C
C
L
-4
0.2
0.6
0.4
0.8
0.2
0
1
0.4
0.6
0.8
1
t/T
/T
(b) Velocity and rotation rate
(a) Linear and angular momentum
Figure 4-3: (a) Linear and angular momentum and (b) corresponding velocities for a neutrally
buoyant self-propelled NACA0012 with carangiform motion at frequency f = 1/T = 2.1.
0.7
carang, recoil
-+--anguil, recoil
-00.6-
carang, no recoil
anguil, no recoil
--
0.50
-
0.4
-0
0.3
0.2'
1
1.5
2
2.5
3
3.5
f
Figure 4-4: Quasi-propulsive efficiency as a function of frequency for the carangiform and anguilliform motions with and without recoil.
without recoil is around f = 1.6, while with recoil it is around f = 2.1.
We have shown here that the pitch motion of the swimming fish is significant and its
impact on swimming performance is large. In order to estimate meaningful values of fish
swimming efficiency, it is critical to allow for recoil.
4.2.4
Imposed deformation, mid-line displacement and curvature
As explained above, the lateral displacement h(x, t) of a point located at x along the foil is
the sum of ho (x, t), the imposed deformation, and B(x, t), the recoil term resulting from the
hydrodynamic forces on the foil. If B(x, t) was negligible, the envelope of the displacement,
g(x), would be proportional to A(x). However, we just showed that the magnitude of B is
comparable to that of A, such that A(x) is not representative of the displacement envelope.
Since the recoil is a linear function of x, the curvature can be expressed as
h"(x, t) = ao [(A"
A
+)
sin (27r(x/A - ft)) + A'
73
cos (27r(x/A - ft))1.
(4.20)
2
Curvature
(b)
Displacement
(a)
0.1
5
0.08-
4
0.06
3
E 0.04-
E 2
0.02
0
1
0
0.2
0.4
0.6
0.8
0
1
0
0.2
0.4
x
0.6
0.8
1
x
Figure 4-5: (a) Typical displacement amplitude envelope for a swimming saithe or mackerel. (b)
Typical curvature amplitude envelope for a swimming saithe or mackerel. Adapted from Videler
[179].
So if A is smooth,
h"(x, t) oc A(x) sin (27r(x/A - ft)).
(4.21)
Therefore, A(x) is representative of the amplitude of the curvature at point x.
Videler [179] analyzed the kinematics of mackerel and saithe swimming and, for both
species of carangiform swimmers, he measured the lateral displacement and body curvatures. Typical envelopes of the displacement and curvature are shown in Figure 4-5. The
displacement is minimum around x = 0.25, with a slight increase in amplitude around the
leading edge and a steady increase until the trailing edge. The curvature is smallest at
the leading edge, with a sharp increase between x = 0.7 and x = 0.9, corresponding to the
peduncle, followed by a sharp decrease in the tail section. The significant difference between
the two curves is another indication that, in swimming fish, the recoil is indeed substantial.
The displacement curve of figure 4-5a can reasonably be approximated by a second
order polynomial, which is why A(x) is usually chosen as a quadratic function when the
displacement is imposed. However, if recoil is allowed, figure 4-5a is representative of
g(x), whereas A(x) is better represented by figure 4-5b. Therefore, when one imposes the
deformation rather than the displacement, a second order polynomial is not appropriate any
more, and a Gaussian function might be better suited. In 4.5, A(x) will be optimized in
order to maximize the quasi-propulsive efficiency of the undulating foil. The performance
of quadratic functions, inspired by the carangiform displacement curve, and of Gaussian
functions, similar to the carangiform curvature envelope, will be compared. If the gait
of living fish corresponds to some optimal motion, we expect the optimization to yield
results that are similar to those shown in figure 4-5: Gaussian deformation and polynomial
displacement.
4.2.5
Trailing edge pitch and angle of attack
The propulsive performance of rigid flapping foils has been extensively studied [133]. The
parameters that describe the symmetric motion of a flapping foil are the heave amplitude,
Strouhal number, maximum angle of attack and phase angle between heave and pitch. Triantafyllou et al. [163] showed that the propulsive efficiency is maximum when the Strouhal
number is between 0.25 and 0.35. Read et al. [133] recorded maximum thrust coefficient
around angle of attack 35' and maximum efficiency for angle of attack 15'.
74
For an undulating foil, we define the Strouhal number, heave amplitude, pitch angle
and angle of attack at the trailing edge. While the motion cannot be characterized by
these parameters alone, they very likely play an important role in the swimming efficiency.
Changing the amplitude of motion and Strouhal number is easy through parameters like
ao and f, but the pitch amplitude 0 max and maximum angle of attack amax cannot be
directly controlled. Therefore, when optimizing the swimming gait, it is important to
choose a parametrization that allows the pitch and angle of attack amplitudes to be adjusted
independently of the heave amplitude and Strouhal number.
The pitch is a function of the envelope shape A(x) and the undulation wavenumber
k = 27r/A. For example, if we consider a motion without recoil (to keep things simple) and
assume the angles are small, the instantaneous pitch angle is given by:
0(t) ~
(
(A'(1) + ik (1))ei(2rft-k)
1, t) = ao Im
(4.22)
The angle of attack is approximated by:
a(t) ~
I lh
U_
(X = lit) -
Oh
= 1, t),
-(X
(4.23)
such that
Omax ~ G' ~ ao /A'(1) 2 + k 2 A(1) 2 ,
amax
ao /A(1)
2
(k
-
27rf/U) 2 + A(1)' 2 .
(4.24)
Therefore, the pitch and angle of attack are best adjusted by changing the derivative of
the prescribed envelope A(x) at the trailing edge. As will be discussed in 4.5, while
parameterizing A(x) by a second order polynomial makes it easy to change the amplitude
of motion at the leading edge and at mid-chord, a Gaussian parameterization spans a wide
range of values for the derivative at the trailing edge without running into degenerate
envelopes.
75
4.3
4.3.1
Numerical method
Fluid/body coupling: numerical implementation
.
In order to solve the coupled fluid/body problem described above, we adapted the boundary
data immersion method (BDIM) presented in chapter 2. In the present simulations, constant
velocity ' = U, is used on the inlet (x = -6), periodic boundary conditions on the upper and
lower boundaries (y = 2.4) and a zero gradient exit condition with global flux correction
(x = 7). The Cartesian grid is uniform near the foil with grid size dx = dy = 1/160 and
uses a 2% geometric expansion ratio for the spacing in the far-field, as illustrated in figure
4-6
The fluid and body equations (Eqs. 4.6 and 4.17 respectively) are integrated over the
fluid and body domains (respectively Qf and Qb) with a kernel of radius c = 2 dx. The
BDIM equations for the smoothed velocity field i4 are valid over the complete domain
Q = Qf U Qb and enforce the no-slip boundary condition at the interface. We repeat here
these equations which, integrated from time t to time t+ = t + At, are:
~file(t+) = (t+)+ ((d)
V
(t+)+ At
(ii(t) -
+ p(d)
--
aPAt)
(4.25a)
(4.25b)
-d (t+) =0
where V' is the velocity field associated with the closest body, h the unitary normal to the
closest fluid/solid boundary (pointing toward the fluid), and d the signed distance to the
closest boundary (d > 0 within the fluid, d < 0 inside a body). The pressure impulse aPAt
and Rst accounting for all the non-pressure terms are defined in Eq. 2.4.
The coupled dynamic equations are discretized using a sequentially staggered Euler
explicit integration scheme with Heun's corrector. Sequentially staggered schemes are computationally efficient, but for large added mass they become unconditionally unstable [61],
regardless of the particular scheme used. In order to stabilize the numerical scheme, we
introduce the virtual added mass matrix:
mii
m12
Ma =m21
m22
(m
m13
m13
m33)
m32
31
(4.26)
.
The virtual added mass, which will be used in an implicit added mass scheme [33, 212, 124],
2-
homogeneous
fine grid region
-2
I
-6
I
I
-4
I
-2
iii
I
|
I
I
I|
2
0
I
I
4
6
x
Figure 4-6: Flow configuration for the undulating NACA0012 simulations. The vorticity field for
the carangiform motion with f = 1.8 and zero mean drag is shown as an example.
76
can eliminate the instability due to large added mass, but its exact value will not affect
the results, as will be shown in 4.3.4 . In the case of an undulating foil, the coefficients
of the matrix can be estimated from the added mass of the foil at zero angle of attack,
or heuristically tuned to avoid instability. In the present simulations, the following virtual
added mass matrix has been used:
Ma = m
0
0
(0
0
11
0
0
0 .
13)
(4.27)
We also define the total mass as:
(4.28)
MT = M + Ma.
With these new definitions, we integrate Eq. 4.17 over a time-step At in the form:
t+At F +Ma dV dr.
1
V(t + At) = V(t) + MT-
(4.29)
At each time step t7 , the fluid and body velocities, ii4= Ui(t,) and '7 = (t,) respectively,
are calculated from the velocities and forces at the previous time steps.
Explicit Euler integration is first applied to Eq. 4.29 in order to calculate an intermediate
body velocity V':
=
update-body (Va, tn+1 - tn, F"', t
V
-
tn
1
, V 7 - 1 , Va).
(4.30)
The new generalized body location Xn+ 1 is then calculated:
{Xn+ 1 } = move-body (Xn, tn+1 - tn, Vn, V'
(4.31)
.
The moments p' and p' are updated accordingly, and an intermediate body velocity field
V'n+1 is calculated following Eq. 4.9, using Vn 1 and the prescribed deformation 6*+.
Explicit Euler integration is then applied to Eq. 4.25 in order to calculate intermediate
velocity un+1 and pressure p'+ 1 fields:
{Un+1, Pn+1}
n+1, tn+1 - tn)
update-fluid U,, Un,
(4.32)
.
Heun's corrector step is then applied to the body, updating V' n+1 accordingly, and to
the fluid:
V" 1 } = update-body Vn+,i
{
'n+1
-
tt , F'
Pn+1= update-fluid (in,
U
, t
n+1,
1
- tn, V1,
V'n+1, tn+1 -
,
tn,
(4.33)
(4.34)
where Fn l are the hydrodynamic forces associated with u/n+1, v/n+1, and p' 1 (detailed
in 4.3.2). F" 1 , the hydrodynamic forces associated with u"n+1, v'n+1, and p"+1 are then
77
calculated for the next time step. Finally, the new velocities are calculated:
Vn+1 =
nl+1
2
+
1
,
V 1 71 + Vifr7,+1
2
'
Vn+1 =
-~
U7+1
Uf +
+
2
n+1
(.5
(4.35)
The equations used to update the body/fluid system are:
{V}=
update-body (Vo, At, F, Ato, V1 , V2):
V=Vo+AtM 1 - 1 (F+Ma V2AV1
Ato
{X}
move-body (Xo, At, V, Vi):
X = X0 +At
{i7, p}
(4.36)
(4.37)
2
update-fluid (i-o, U1, V, At):
TA
At
9 =U+po+
Ut
V-
(PO
(i
1
- il
PE
6
=Ip
+UV2rU1 (4.38a)
1
U
- 6+
FAt
))(43b
1 - V3
- At p" 0=Vp
4.3.2
(4.38b)
(4.38d)
Force and power calculation
The hydrodynamic forces and torque on the body are at the core of the fluid/body system
dynamics. In the study of fish swimming, the power associated with these forces is also of
primary interest as it relates to the power expanded by a swimming fish.
The forces, moment and power defined in Eqs. 4.10, 4.11 and 4.12 are calculated using
a one-sided Derivative Informed Kernel (DIK) derived in [189]. The advantage of the DIK
method is that it evaluates the unsteady forces on the body in one step without a surface
grid. The expression for the hydrodynamic forces has the form:
Fh =
(4.39)
6+ dQ.
The associated torque is expressed as:
.cA =
(*[
d )x
65 dQ,
(4.40)
d.
(4.41)
and the swimming power:
J
=-
-
The stress 9' results from the sum of the pressure and friction components, respectively a
78
.
-.
-
10
/
5
/
------ LL3
/
-ww
- - -
0
0
10
square-root buffer layer
20
30
40
50
y
Figure 4-7: Comparison of the WW, LL3 and PL3 wall laws. PL3 uses WW's outer-layer with a
square-root buffer layer.
and ory, where the pressure term is expressed as:
=)-.
d
(4.42)
In order to calculate the friction term, we use a wall model, which allows us to retain a good
accuracy when the boundary layer is not fully resolved. Though wall models can be used
to adjust the velocity in the boundary region, here we only use it to calculate the friction
force on the body.
The most common wall-law assumes that the near-wall layer consists of a fully viscous
sublayer with linear velocity profile and a fully turbulent logarithmic superlayer (LL2). To
account for the smooth transition between the linear and the logarithmic region in the
buffer layer, a logarithmic fit can be used in the buffer layer, resulting in the three-layer
(LL3) law [23]. The log-laws are transcendental and require an iterative inversion for the
wall shear stress T_ [156, 22]. A simpler two-layer approximation, proposed by Werner and
Wengle [188], is based on the assumption of a 1/7th power-law outside the viscous sublayer,
interfaced with the linear viscous sublayer (WW). Unlike the log-laws, the power-law can
be transformed into an explicit definition of the wall shear stress. However, the power-law
for the outer layer is not valid in the buffer layer and therefore yields wrong values of Tw.
For this purpose, we fit a power velocity profile to the logarithmic buffer layer from LL3.
The resulting model blends a square-root transition layer with the linear viscous sublayer
and the 1/7th power-law for the turbulent superlayer:
U+(y+)
jTj +=Re{T
Iy.
(4.44)
y+
2.5
if y+ < 6.25
if 6.25 < y+ < (8.3/2.5)14/5
8.3(y+)1/7
if (8.3/2.5)14/5 <Y+
where we are using the scaled variables (u and y being already normalized by U
respectively):
U+
(4.43)
and L
Figure 4-7 compares the velocity profiles for WW, LL3 and our three layer power law (PL3).
The wall shear stress can then be calculated explicitly from the velocity u at distance y
79
fron the boundary:
-r (y, i, Re)
( u(y Re)- 1
(u/2.5)4/ 3 (y Re)- 2 / 3
(u/8.3) 7/ 4 (y Re)- 1 / 4
if u y Re < 6.252
if 6.252 < u y Re < 2.5(8.3/2.5)21/5
if 2.5(8.3/2.5)21/5 < uy Re
Also correcting for the fact that in BDIM u,(d
calculate the friction force using:
d Re (4.46)
=+
(4.45)
0) = p&ue/On (from Eq. 4.25), we
W
4.3.3
Feedback controller
In steady state, the time-averaged velocity of a swimming fish is constant and the mean
forces on the swimmer are 0. In order to ensure that the system converges toward a
steady state in which the swimming velocity is the prescribed velocity U8, we designed
a proportional-integral-derivative (PID) controller that adjusts the thrust by tuning the
amplitude of the swimming gait ao. If the foil is fully self-propelled, the time-averaged
linear momentum in x is used as feedback (referred to as displacement control). However,
we have shown in 4.2.3 that the amplitude of the oscillations in vx is very small, so in
most cases we actually fix the foil in x in order to reduce the PID convergence time. In
this case (referred to as force control), the time-averaged drag is used as feedback. We will
show in 4.3.4 that both approaches result in the same swimming power estimate.
We have also shown in 4.2.3 that it is important to let the fish free to heave and pitch
under the effect of the hydrodynamic forces. In order to ensure stability of the fish in heave
and pitch, the time-averaged linear momentum in y is used as the input to a PID controller
that tunes the camber parameter C of the y1 (x) function defined in Eq. 4.3.
For a self-propelled fish with flapping frequency
f
= 1/T, we define the error as:
x:mv(tk)(tk+1
n-i
f
() = -tk),
(4.47)
k=no
where no is the first index k such that tk > t, - T. If the x motion is fixed and force control
is used, Fhx replaces mvo in the calculation of ex.
The integral of the error is calculated as:
n
e-
(tn)
(4.48)
(tk)(tk+1 - tk),
=
k=0
and its derivative is:
ed(tn) =t-
(no - (tn - T))*(tnO-1) + ((tn
-
tno.0
-
T)
-
tno-_13 (tno)
(4.49)
At the beginning of each time step, the parameters ao from Eq. 4.1 and C from Eq. 4.3
80
are updated as:
{ao(t,)
max [ao(t,) + (t, - tn_ 1)(Kje (t,) + KfeT(tn) + Kde'(tn)), 0],
C(tn) =
(4.50a)
(4.50b)
(KpeY(tn) + Kiye '(tn) + Kdey(tn)),
where e' and ey denote respectively the x and y components of the error vector C.
The gain coefficients used in this study are
Kp = 5,
fx= 5,
Kx = 5,
(4.51)
Kx = 100,
(4.52)
Kdy = 12,
(4.53)
for force control in x,
Kp = 5,
Kix
1,
for displacement control in x, and
KPY = 8,
Kty =10,
for displacement control in y.
4.3.4
Numerical method validation
.
Problems previously studied with BDIM include ship flows and flexible wavemaker flows
[190], shedding of vorticity from a rapidly displaced foil [197], and a cephalopod-like deformable jet-propelled body [193]. In chapter 2 we have demonstrated the ability of BDIM
to handle several moving bodies and generalized the original method to accurately simulate
the flow around streamlined foils at Reynolds numbers on the order of Re = 10 4
We first validated the fluid/body coupling routine presented in 4.3.1 by computing
the motion of a flexibly mounted cylinder and the results are presented in appendix D. In
order to validate the code for simulating undulating foils, the force and power resulting
from a fully imposed kinematics are then compared with results reported in the literature.
Finally, a convergence study and sensitivity analysis on a self-propelled undulating foil are
performed.
Undulating NACA0012 with fully imposed kinematics
Using a fully imposed carangiform undulation:
h(x, t) = (0.1 - 0.0825(x - 1) + 0.1625(X
2
- 1)) sin (27r(x -
f t)),
(4.54)
the undulation frequency is varied from f = 0.5 to f = 2 and the resulting time-averaged
force and power coefficients are compared to the values from [44] in Figure 4-8. Note that
in these simulations the kinematics is fully imposed, not allowing for recoil.
Similarly to [44], we find that the average power coefficient, slightly negative at f = 0.5,
increases to around 0.25 at f = 2, and that the drag is positive for f < 1.6 and negative
for f > 1.6. The good agreement between our method and the results from [44] serves as a
validation of the force and power calculation routines for an undulating foil.
81
0.3
- 0.- - C , Dong (2009)
-. 25D, Dong (2009)
0.2
-- a-
0.15 . .
, BDIM
U- , BDIM
0.1
0.05
0-0.05-0.1
0
0.5
1
1.5
2
2.5
f
Figure 4-8: Time-averaged drag and power coefficients for an undulating NACA0012 as a function
of frequency, compared with values from [44].
Self-propelled undulating NACA0012
We now ensure that the simulation results presented here are independent of the grid
parameters. In this section we consider the carangiform motion with frequency f = 2.1.
Figure 4-9 shows the evolution of power and drag coefficients during an undulation
period T=1/f for various configurations. By comparing the free undulation and the fixed
x case, we first notice that fixing the x location of the foil does not impact the power,
confirming the observations from [8]. The amplitude of the drag oscillations are a bit larger
for the case with fixed x location, as would be expected, but this does not impact any of the
results discussed in this paper. On the other hand, precluding all recoil completely changes
the phase and amplitude of the power and drag coefficients. Figure 4-9 also shows that
the power and drag coefficients estimated on grid 1 (introduced in 4.3.1) are very close to
those estimated on a grid twice as fine (grid 2, dx = dy = 1/320) and a grid twice as large
(grid 3, x C [-12, 14], y C [-4, 4]). Table 4.1 summarizes the mean and maximum power,
maximum drag, and undulation amplitude ao for all these cases.
These results confirm that, while fixing the x location of the foil will not impact our
swimming efficiency estimates, the foil should be let free to heave and pitch. Therefore, a
foil fixed in x, free to heave and pitch under the influence of the hydrodynamic forces will
be used throughout this chapter. Moreover, the estimates on grid 1 being very close to
those on a finer and larger grid, grid 1 (5 points across the boundary layer) will be used for
the optimization procedures with a fish in open-water, whereas grid 2 (10 points across the
boundary layer) will be used for visualization and for a swimming pair.
82
(a)
(b)
(b)
0.1
0.3 r
0.2-
0.05
0.1
0
0
0
0
-
-0.1
-
-0.05
-0.2
-0.1
0
0.2
0.4
t/T
0.6
0
0.8
-
free, grid 1
- fixed x, grid 1
no recoil, grid 1
- fixed x, grid 2
fixed x, grid 3
0.2
0.4
0.6
0.8
1
t/T
Figure 4-9: (a) Drag and (b) pressure coefficient on an undulating NACA0012 with carangiform
motion at f = 1/T = 2.1. Various grids and constraints are compared. Grid 2 is twice as fine as
grid 1, while the computational domain of grid 3 is twice as large as that of grid 1.
Case
free, grid 1
fixed x, grid 1
no recoil, grid 1
fixed x, grid 2
fixed x, grid 3
Cp
0.124
0.125
0.093
0.112
0.125
(Cp
- CP)max
0.153
0.155
0.087
0.165
0.156
(CD)max
a0
0.054
0.065
0.054
0.068
0.065
0.100
0.100
0.065
0.097
0.099
Table 4.1: Mean and maximum amplitude of power coefficient, amplitude of drag coefficient and
undulation amplitude for a NACA0012 with carangiform amplitude at f = 2.1 and 0 drag.
83
4.4
Definition of efficiency for self-propelled bodies
Let us consider the general case of a self-propelled body of mass m moving with acceleration
ac and velocity U, (both averaged over a period) along the x-direction. Efficiency is defined
as the ratio of useful work over expended energy, measured over a specific time interval.
The useful work, for a body moving at constant speed within a viscous medium, is the work
needed to overcome the resisting fluid forces (drag). However, except in very few, limiting
cases, this work cannot be measured because the drag of a self-propelled body depends not
only on its shape and speed, but also on the type of propulsor used, and, in particular,
the body-propulsor hydrodynamic interaction. Keeping in mind that efficiency is also a
normalized measure of performance and that the objective is to minimize the expended
power for a given swimming speed, let us define several measures of efficiency.
4.4.1
Net propulsive efficiency
Considering the system {body + propeller} as a whole, the efficiency (referred to as net
propulsive efficiency 71) in its strictest definition is the ratio of the power output Pu1 to
the power input Pin:
=
Pin
tn.
(4.55)
The power output is given by the rate of change of kinetic energy (averaged over a period)
of the body:
d
1
2
Po5=dt=
macUs = TnUs,
(2mU)2
(4.56)
with T, the net thrust produced by the {body + propeller} system, such that:
TIn =
Tn US
(4.57)
Pin
This definition of efficiency is also used to measure the performance of an isolated propeller.
Going back to the intuitive definition of efficiency, which is the ratio of useful work to
total work, different configurations can be compared. A propeller in isolation is meant to
produce thrust that will balance the drag on the hull of a ship, so TsU. is a reasonable
measure of useful power output. Similarly, for a fish performing a C-start or an escape
maneuver [43, 101], its goal is to accelerate, such that 'qn is still a reasonable measure of
efficiency that quantifies how much work is needed to attain a certain speed in a given
amount of time. However, once the cruising speed is reached and the body moves at
constant speed, the total average hydrodynamic force son the body must be zero, so using
the definition of Eq. 4.57, the net efficiency is 0. As pointed out by Schultz & Webb [141]
among others, "unless a fish is trying to 'stir up the water,' it performs no useful work"
when swimming at constant speed:
?I = 0
for a self-propelled body in steady state.
(4.58)
This measure of efficiency becomes meaningless when the goal of the system is not to
accelerate or produce thrust.
84
4.4.2
Propulsor efficiency
Under special circumstances, one could still define a propulsor efficiency, jp, by separating
the propulsor thrust Tp from the body drag (one balancing the other when Tn = 0):
i =TUS
Pin
(4.59)
For flexible self-propelled bodies, such as undulating fish, where the distinction between
thrust and drag cannot be made, obtaining Tp, is much more challenging than for a propeller
mounted on a rigid body. The distinction can be seen as arbitrary, like when it is obtained by
separating the positive longitudinal forces from the negative ones in time [16] or in space [12].
It is still possible, in some cases, to estimate the thrust produced by a swimming fish. Indeed,
when the Reynolds number is sufficiently high and uncontrolled flow separation effects are
of limited extent, inviscid methods can be used to provide an estimate of the power needed
for propulsion, as well as the developing thrust that must equal the resistance. This can be
quite accurate if separation effects, other than vorticity shed from body edges and from fin
trailing edges, are small, and interaction of the body with shed vorticity is insignificant. For
instance, [100], [204], [46], [123], [202], and [213] employ inviscid methodologies to estimate
the thrust generated and power expended by swimming fish.
However, the main problem with this definition of efficiency is that, even in rigid bodies
such as ships and submarines, one is not interested in the propulsor efficiency, but the
power needed to sustain a certain speed. Indeed, it is possible that a very efficient propulsor
may cause a large increase in the total drag when attached to the vessel, due to adverse
hydrodynamic interference, and hence an increase in the required thrust Tp. Then, although
the propulsor efficiency is high, the system efficiency is low because the "fuel" needed may be
excessive over another propulsor that may be less efficient in isolation but does not increase
the resistance. What should be important in terms of the energetics of a certain fish is to
employ a swimming mode that minimizes the power needed for propulsion; whether this
mode is hydrodynamically "efficient" is secondary.
4.4.3
Quasi-propulsive efficiency
The goal of propulsion optimization is set as follows: For a given shape and size vehicle,
find the propulsor that will require the least amount of power to drive the vehicle at a given
speed Us. We intend to minimize the "fuel" consumption under certain size and velocity
constraints and not the hydrodynamic efficiency of the system. This is exactly what the
cost of transport (COT), traditionally used in life sciences, measures, since it is defined as
the energy spent per unit distance traveled:
COT =
Us,
(4.60)
where Pot is the total metabolic power consumed by swimming at speed Us. However, the
COT is a dimensional quantity, and there is no natural way to normalize it. For instance,
Kern [91] normalized the COT by mUf /2, Liu [102] used mLf 2 , Eloy [52] chose pQ2/ 3 U2,
and Tokid [160] normalized it by mgU. For the first two normalizations, two gaits with
different flapping frequencies f would result in different values of normalized COT even
if they have the same cost of transport, which is undesirable. On the other hand, for a
85
given fish, the last normalization is the only one that ensures that two gaits have the same
normalized COT if and only if they have the same COT. While this is a nice property, this
normalized COT is not an efficiency-like quantity since it does not have a natural unit scale.
Hence, we propose to normalizes the COT by the towed resistance R. Since here we only
consider the hydrodynamic efficiency and not the internal losses, qp, is employed, defined
as:
(4-61)
QP = U
nP
where Pin is the power required by the propulsor to drive the vehicle at speed U, under
steady-state conditions (zero total hydrodynamic force) and R is the towed resistance at
speed Us. In the case of a flexible body, the towed resistance must be measured or estimated
in a straight configuration, i.e. not allowing any bending of the body.
Indeed, at constant speed, the role of the propeller (for a ship) or of the swimming motion
(for a fish) is to compensate for the drag such as to keep the cruising velocity constant. In
an ideal fluid, there would be no drag on the body and no work would be needed to sustain
velocity U,: gliding would be enough. However, since water is a real fluid, if the fish was
not swimming, or the propeller not rotating, the body would lose kinetic energy at a rate
of:
d
d_
(4.62)
Pos = d
m U)2
-RUs < 0,
where, again, R is the towed resistance at speed U8 without a propeller (or a swimming
motion). The goal of the propeller - or of the swimming motion - is to prevent this loss of
kinetic energy due to the drag on the rigid body. Since the goal in this case is to compensate
for the resistance R and prevent the kinetic energy loss P10,,, a reasonable definition of useful
power is:
Puse = Pout - P0oss = (Tn + R)Us,
(4.63)
which we use to generalize the quasi-propulsive efficiency q)p to cases where the net thrust
is not 0:
(4.64)
jQP = PusePin = (Tn + R)Us/Pin.
Eq. 4.64 shows that the quasi-propulsive efficiency is the ratio of the useful energy over the
expanded energy, where the goal of swimming is to overcome the drag and prevent kinetic
energy losses. For the case of a self-propelled body moving at constant speed, Ta = 0, such
that the definition of propulsive efficiency proposed in Eq. 4.64 is the same as Eq. 4.61. The
power Pin is either experimentally measured, or evaluated numerically as the time-average
of the power needed to actuate the body. Finally, since towed experiments or simulations
are often preferred to self-propelled ones for practical reasons, we will show in 4.4.4 that
Eq. 4.64 can provide good estimates of the self-propelled quasi-propulsive efficiency under
towed conditions.
There are fundamental differences between the propulsive efficiency of Eq. 4.59 and the
quasi-propulsive efficiency of Eq. (4.61): First, in the "useful" power of Eq. 4.61, one uses
the towed resistance of the vehicle measured under steady towing conditions at speed U
and without a propulsor attached; hence, this definition does not suffer from any ambiguity
as to what the force should be. Second, in Eq. 4.59 all quantities used refer to the same
(self-propulsion) test; in Eq. 4.61 the numerator refers to a towing experiment, while the
denominator to a self-propulsion experiment, conducted at the same speed.
It is not difficult to see that, if we maximize the efficiency TjQp, we simply minimize the
86
expended power Pi, (since the numerator is independent of the propulsor), in agreement
with the original intent. The advantage of qp is that the towed resistance captures the
essential hydrodynamic features of the specific hull or body, and can be used to compare
the performance of dissimilar vehicle shapes, and for devising scaling laws. An apparent
disadvantage is that the quasi-propulsive efficiency is not strictly an efficiency: it is not
necessarily less than one. If the propulsor causes the resistance of the ship to drop substantially - for example by reducing flow separation - then the self-propelled power will possibly
be less than the power needed to tow the bare hull, resulting in a value of T Qp higher than
100%.
A distinctive advantage of the quasi-propulsive efficiency is its universality. Unlike
propulsor efficiencies relying on inviscid thrust models, the quasi-propulsive efficiency is
as appropriate for low-Reynolds-number swimming motions as for large-Reynolds-number
ones. Becker et al. [11] define and use a system efficiency which is the same as the quasipropulsive efficiency definition herein; they study a three-link micro-propulsor, employing
flexing of the links to achieve locomotion at very low Reynolds numbers. In the words of the
authors, "We define a swimming efficiency as the power necessary to pull the straightened
swimmer along its axis at the average speed of the actual swimmer, relative to the average
mechanical power generated by the actual swimmer to achieve that speed." It is important
to note that the useful power is defined in terms of the towed straightened swimmer. In fact,
for very low Reynolds number, it is impossible to distinguish thrust from drag, since viscous
forces produce both forces, making the use of the quasi-propulsive efficiency essential. Microswimmers have, typically, less than a few percent efficiency.
4.4.4
Example: anguilliform vs carangiform gaits
In order to illustrate the discussion above, we will show through an example why the
quasi-propulsive efficiency is the only meaningful way of measuring propulsive efficiency for
self-propelled fishes or vehicles. In this example, we use
A(x) = 1 + (x - 1)ci + (x 2
-
1)c2
(4.65)
as the envelop of the prescribed traveling wave of wavelength A and frequency f. ao, the
amplitude of ho at the trailing edge, will either be kept constant (ao = 0.1) or adjusted
through a feedback control loop to ensure that the average drag on the foil is 0, as described
in 4.3.3.
We consider the carangiform and anguilliform envelopes characterized by A = 1 and:
{carangiform:
anguilliform:
c = -0.825,
c2 =
cl =
c2
0.323,
1.625,
(4.66a)
= 0.310.
(4.66b)
These envelopes are illustrated in figure 4-2a. The performance of both gaits at various
frequencies f in a towed and self-propelled configuration will be compared.
Figure 4-10a shows that the self-propelled undulating NACA0012 foil travels with the
least energy when using the anguilliform gait with frequency f = 1.6, in which case Cp =
0.10. If the carangiform gait was chosen, the most efficient frequency would be f = 2
with a power coefficient of Cp = 0.13. Though dimensionless, the power coefficient is
not an intuitive measure of efficiency and does not allow easy comparison between various
geometries.
87
Zero mean drag: C= 0
(a)
(b)
Fixed amplitude: cO = 0.1
0.16
3
-0-
0.15
-.--
-e- carang
carang
anguil
2.5
0.14
I
2
Ioa- 0.13
1.5
-
0.12
1
0.11
0.5
0.1
anguil
-
1.5
2
2.5
U1
3
--
- ---- --------- - ---3
2
f
4
5
f
Figure 4-10: Time-averaged power coefficient as a function of undulating frequency for (a) the
zero drag and (b) the fixed amplitude configurations.
From the prescribed undulation ho(x, t) and the recoil B(x, t) calculated by the viscous
BDIM simulation, Wu's potential flow theory [203, 204] can estimate the input power and
propulsor thrust Two ~ T,. Using the input power Pim and the net thrust Tn estimated
from the BDIM simulation, as well as thrust and power estimates from Wu's theory, we will
now compare the efficiency of the various gaits using the three measures defined above.
Net propulsive efficiency
As discussed in 4.4.1, the net efficiency qn = TPU/Pn is zero when the mean drag on
the foil is 0, which is the case for the self-propelled cases in Figure 4-11. In these cases,
it is therefore impossible to compare the performance of the two gaits or of the various
frequencies using q, As soon as the mean drag is non zero in the towed simulations
(ao = 0.1), it becomes clear that the anguilliform undulation is more efficient than the
carangiform but, with values ranging from -0.6 to 0.3, these undulating foils seem to be
very poor propellers.
It is interesting to notice that, at low frequency, the net efficiency is negative due to a
net drag on the undulating foil. What is the meaning of this negative efficiency? If we were
considering a propeller, a net drag on the propeller would be counter productive and the
0.4
>
0.2
0
-0.27
-0.6
.
SI
carang, 0 drag
anguil, 0 drag
carang, ao=0.1
-
ang
m. u , a
Nl =% .1
.0
---
-
-A
e
-*
e
-
-0.47
3
f
2
4
Figure 4-11: Net propulsive efficiency.
88
5
ship might "perform" better without the propeller, so one intuitively expects the efficiency
to be negative. However, in the case of a self-propelled undulating foil, an undulation
is counter productive only if it increases the drag, not merely because it is not able to
completely overcome it. Since in the present case the drag on the towed undulating foil is
less than on the towed rigid foil, one would intuitively expect the efficiency to be positive.
The quasi-propulsive efficiency solves this paradox by offering a measure of efficiency that
is compatible with intuition.
Now, if the goal is to accelerate the foil, a net thrust is needed. According to the net
propulsive efficiency, the optimal undulating frequency is around f = 2.5 (,q = 0.27) for the
anguilliform motion and f = 3.5 (i, = 0.21) for the carangiform motion. These frequencies
minimize the work required to attain a given acceleration. However, once the cruising speed
has been reached and the goal is to minimize the power spent swimming in steady state,
there is no guarantee that these frequencies are optimal. Indeed, these optimal frequencies
are different from those selected from figure 4-10.
Potential flow propulsor efficiency
In order to calculate the hydrodynamic efficiency of the undulating foil in the stationary
regime, the thrust produced by the swimming motion needs to be estimated independently
of the drag on the foil. This thrust can, for example, be estimated by one of the numerous
inviscid methods. Here we use Wu's two-dimensional theory [203] which has an analytical
expression for thrust and power. The dependency of 7 w, = TwuU/Pw, on the undulating
frequency f, shown in Figure 4-12, is qualitatively consistent with figure 4-10.
Similarly to what had been observed from the viscous power estimates, Wu's method
suggests that, in general, the anguilliform motion is more efficient than the carangiform
one. The maximum efficiency for the anguilliform gait is 7w, = 0.69 at f = 1.6 whereas
the carangiform gait is most efficient at f = 2 with ?lw, = 0.64. However, this approach
might overestimate the efficiency by rewarding high thrust, which is also synonym of enhanced drag. Whereas here the most efficient gait and frequency according to Wu's theory
correspond to the gait and frequency with least power, there is no guarantee that this will
be true in general.
0.7
--
0.68 .
+
carang, 0 drag
anguil, 0 drag
0.66
0.640.62-
0.6
1
1.5
2
2.5
3
3.5
f
Figure 4-12: Propulsor efficiency estimated from Wu's potential flow theory.
89
(a) 0.5
(b) 0.5
0.40.4
0.3
0.2
-e-
0.1
-e-
0
1
1.5
2
2.5
carang, 0 drag
anguil, 0 drag
carang,a=O.1anguil, a=O.1
3
3.5
f
0.2
-
0.3
-e-
carang, Re=5000
anguil, Re=5000
-O- - carang, Re=2500
-+- - anguil, Re=2500
-
0.1
0
1.5
2
2.5
f
Figure 4-13: Quasi-propulsive efficiency. (a): Comparison of towed estimates with self-propelled
values (Re = 5000). (b): Comparison of efficiency for Re = 2500 and Re = 5000 (self-propelled).
Quasi-propulsive efficiency
Finally, TQp = (R + Tn)Us/Pjn, with values comprised between 0.2 and 0.5, provides an
intuitive and meaningful measure of the efficiency for the two undulating gaits at the various frequencies. Figure 4-13 shows that the carangiform gait, requiring less power, is an
energetically better choice for a cruising undulating foil, and the best frequency is f = 1.6
with an efficiency of 43%. For the carangiform undulation, the maximum efficiency drops
to 35% for the frequency f = 2.1.
Since self-propelled experiments and simulations are often more challenging than towed
ones, it is of high practical interest to be able to estimate the quasi-propulsive efficiency
from towed experiments. Figure 4-13a also shows that the estimates obtained by keeping the
amplitude ao constant instead of ensuring 0 mean drag are very close to the self-propelled
values (except at the very low frequencies).
Within the same hydrodynamic regime, the values of 77Qp for different Reynolds numbers
are also of comparable amplitude, on a natural unit scale. For instance, figure 4-13b compares the efficiency of the same self-propelled undulating motion for two different Reynolds
numbers: Re = 2500 and Re = 5000. Even though the power coefficient increases by 50%
from Re = 5000 to Re = 2500, the difference in efficiency between the two Reynolds numbers is no more than 7% and their trends are very similar. This result therefore corroborates
what the intuition would expect: within a given hydrodynamic regime, the efficiency only
weakly depends on the Reynolds number. This also illustrates that, even though both Cp
and 1/?Qp are normalized versions of the swimming power, Cp is not very convenient to
use due to its strong dependence on Reynolds number.
Finally, we would like to remark that, as the thrust produced by the undulating foil
increases, TjQp converges to rn. Indeed, if T > R, then ?7Qp, ~TaU/i,.
Since this is
typically the case for a propeller, the drag on the hull being much larger than that of the
propeller, TQp can be seen as a generalization of the traditional propeller efficiency to the
low thrust regime.
90
4.5
Gait optimization for a self-propelled undulating foil in
open-water
As stated in 4.4.3, the goal in this section is to find an undulatory gait that requires the
least amount of power (P,) to drive a NACA0012 at speed Us, such that the Reynolds
number is Re = 5000. In other words, we want to maximize the quasi-propulsive efficiency
r1Qp of the undulating foil and identify the key parameters under the constraints if fixed
body size and Reynolds number. To do so, we first consider the carangiform and anguilliform gaits introduced above and investigate the relationship between Strouhal number,
undulation frequency and Reynolds number, and how these numbers relate to the friction
drag coefficient and quasi-propulsive efficiency in 4.5.1.
Then, for several values of undulation frequency f and wavelength A, we optimize the deformation envelope A(x) in 4.5.2-4.5.5. Unlike Eloy [52] and Tokic [160] who combined an
evolutionary algorithm with Lighthill's potential flow slender-body model to simultaneously
optimize the shape and kinematics, with respectively 22 and 9 parameters, we parametrize
the amplitude A(x) by only two parameters. While the reduced numbers of parameters
allows us to find an optimum with a reduced number evaluations, it also facilitates the
visualization and interpretation of the results. Following our observation that the envelope
of curvature amplitude in saithe and mackerel has a distinctive peak around the peduncle
section ( 4.2.4) and our assumption from 4.2.5 that a good parametrization needs to span
a wide range of tangent values at the trailing edge, we first use a Gaussian function:
A(x) = exp (-
(
)
+ (
I)),
(4.67)
where x1 parametrizes the location of the peak and 6 its width, as shown in figure 4-14. In
order to estimate how important the choice of the parametrization is, we also try in 4.5.3
the traditional polynomial envelope, parametrized by ci and c2:
A(x) = 1+ cI(X - 1) + c2 (x 2 - 1).
(4.68)
With the Gaussian function, it is easy to change the pitch and angle of attack amplitudes
at the tail by adjusting the location and width of the peak. The fact that the Gaussian
envelope is always positive is particularly convenient, as most of the (ci, 6) space can be
used to search for an optimal gait without running into degenerate gaits. The quadratic
parametrization makes it easy to change the amplitude of motion at the leading edge and
at mid-chord, but only a narrow band of the (Cl, c 2) space yields reasonable gaits. For
example, making the very non conservative assumption that 0 < A(0) < A(1)/2 results in
the following constraints on ci and c2: 1/2 < C 1 + C2 < 1. Instead of using this thin band,
the results in 4.5.3 are presented in terms of A(O) and A(1/2).
For each parametrization and frequency, the envelope A(x) is optimized using derivativefree optimization [137]. We apply the BOBYQA algorithm that performs bound-constrained
optimization using an iteratively constructed quadratic approximation for the objective
function [131]. For each set of parameters, the viscous simulation is run for 15 nondimensional time units, and the average power coefficient Cp across the last 10 undulation
periods is calculated. Based on the values of Cp, the implementation of BOBYQA provided
by the NLopt free C library [85] interfaced with Matlab computes the next set of parameters. In order to avoid finding a local minimum due to numerical noise, after the algorithm
91
x,
ao exp((1-x,) 2/6 2
-.0.78 a, exp((1-x,)2
/
<oC
ca
2
)
.
)
0.1
a0
0. 0r
0
0.2
0.4
0.6
0.8
1
x
Figure 4-14: Definition of the parameters for a Gaussian envelope.
Choose
- fish model geometry and swimming speed
- wavelength X=1
- frequency f
Set
Optimization loop
- deformation envelope
parameters x1 , 6
Calculated from d(MV)/dt = F
recoil parameters a,, b,
Adjusted by PID such that CT
converges towards 0
- amplitude ao
t = t+dt
ift<1 5
Output
C,()
Navier-Stokes/
body motion
solver
if t1: 5 and converged to C = 0
Output to minimize
if optimization not converged
P
if optimization converged
Optimal gait
x 1, 6
Figure 4-15: Chart of a typical optimization procedure.
92
(a)
(b)
-carang, Re=5000
-anguil, Re=5000
- e - carang, Re=2500
0.55
V)
Q
0.5 0.45
0.65
0.6
- * - anguil, Re=2500
CO
0.55
0.5
P
-
0.65
0.6
'~0.45-A
0.4
0.4
0.35
0.35
1
1.5
2
2.5
3
3.5
0
1
2
f
Figure 4-16:
Strouhal number as a function of (a) frequency
f
3
4
Sr /(1 -Sr)
5
6
and (b) Sr/(1 - s,) for a self-
propelled undulating NACA0012 at Re = 2500 and Re = 5000, where s, is the slip ratio defined as
Sr = U,/(Af).
has converged, it is run again, using the previously found minimum as a starting point. The
optimization procedure is summarized in figure 4-15.
4.5.1
Reynolds number, Strouhal number and slip ratio
For a given undulatory gait and frequency, the amplitude of motion for a self-propelled fish
or foil is enforced by the chosen velocity: there is at most one amplitude that will allow the
fish or foil to swim at the designated speed. Therefore, the Strouhal number St = fa/U,
which is often considered as one of the key factors of fish swimming efficiency [163], is not
a free parameter but a function of the swimming gait, speed and frequency.
Figure 4-16 shows that the Strouhal number depends on both the Reynolds number
and the gait. Indeed, the carangiform gait requires a larger amplitude than the more
efficient anguilliform motion in order to sustain the chosen velocity with a given undulation
frequency. At lower Reynolds number, a larger amplitude is also needed to overcome the
larger friction drag [49]. At low undulation frequency, the swimming amplitude increases
significantly and the Strouhal number is proportional to sr/(1 - Sr), where sr is the slip
ratio defined as sr = U 8 /(Af). However, once the frequency increases above the optimal
frequency (f = 2.1 for carangiform and f = 1.6 for anguilliform, as shown in figure 4-13),
the Strouhal number remains almost constant and independent of the undulation frequency.
Figure 4-17 shows that, for a given gait, the optimal undulating frequency only weakly
depends on the Reynolds number. However, the corresponding Strouhal number strongly
significantly increases with decreasing Reynolds number. Indeed, as the Reynolds number
decreases, the drag on the foil increases, requiring larger oscillations to overcome it. Figure
4-18 shows that this increase in Strouhal number results in increased skin friction: this is
why the quasi-propulsive efficiency slightly decreases with decreasing Reynolds number.
Indeed, it has been shown in the literature that, in the laminar regime, body undulations
cause the skin friction drag to increase [100, 184, 66, 177, 49]. The examples used here
suggest the following scaling law for the friction drag coefficient, as illustrated by Figure
4-18a:
2 CDfo. (4.69)
+
CDf
93
0.3
-e- anguil, Re=5000
0.1
0
0.2-
-0-- carang, Re=5000
0.2
-
1
G
0.3
0CL
0.1
e - carang, Re=2500
* - anguil, Re=2500
1.5
2
f
2.5
-
(b) 0.5
(a) 0.5
0
3
0.35
0.4
0.45
St
0.5
0.55
Figure 4-17: Quasi-propulsive efficiency as a function of (a) the undulation frequency
0.6
f
and (b)
the Strouhal number St for a self-propelled NACA0012 at Re = 2500 and Re = 5000.
And since CDfo
~ 2/V'-, then:
Re,
CDf ~ 2(St + 1)/
(4.70)
as shown in Figure 4-18b. This scaling law holds for both gaits (carangiform and anguilliform), both Reynolds numbers (Re = 2500 and Re = 5000), as well as towed and
self-propelled configurations.
Finally, Figure 4-19 shows that, for a self-propelled foil, the amplitude of the trailing
edge displacement multiplied by the Reynolds number to the power of 1/4 is proportional
to the square-root of sr/(1 - s,), where s, is the slip ratio. While the slope seems to depend
on the gait, it is absolutely independent of the Reynolds number for the values Re = 2500
and Re = 5000 tested here. For the anguilliform motion:
a/L Re'/ 4
1.5
(4.71)
r
1
-s
After reorganization of the terms, this relationship can be expressed as:
St ~
1.5
1
5 -.
Rel/ 4 Sr
s,
I
-(s,
.
(4.72)
Since we showed with figure 4-17 that the optimal slip ratio is mostly independent of the
Reynolds number, Eq. 4.72 suggests that the Strouhal number scales like Re-1/4, as found
by [62] from animal data.
The results presented here suggest that, for a given gait, the Strouhal number is not
the key parameter defining the efficiency of undulatory swimming: the slip ratio seems to
be a more relevant parameter. Specifically, the plots of quasi-propulsive efficiency versus
undulation frequency estimated for a towed and a self-propelled foil collapse, as shown in
figure 4-13a, whereas plotted as a function of Strouhal number, the plots would not collapse.
Similarly, the optimal swimming frequency is almost independent of Reynolds number, while
the optimal Strouhal number changes, as shown in figure 4-17. As a result, in the linear
regime, the Strouhal number scales like Re/4, similarly to the results presented in [174, 62].
94
2.5
_ _
1__
*
2-
0
0
1.5-
0
*
00
_1
carang, a=0.1
anguil, a =0.1
carang, Re=5000
anguil, Re=5000
carang, Re=2500
anguil, Re=2500
y = 2x
0.5
0
0
10
0.2
0.4
0.6
0.8
St
(a) Using the towed friction drag.
4.5
e
1_1_1
carang, a=0.1
4
0
0
*
anguil, Re=5000
3.5-
0
carang, Re=2500
anguil, Re=2500
anguil, aO=0.1
o carang, Re=5000
*
P4
3y = 2(x+1)
2.52
0
0.2
0.6
0.4
0.8
1
St
(b) Without reference to the towed friction drag.
Figure 4-18: Relationship between friction drag coefficient CDf, Reynolds number Re and Strouhal
number St on an undulating NACA0012. Fixed amplitude ao = 0.1 at Re
5000, as well as selfpropelled foils at Re 2500 and Re = 5000 are considered.
4
.0
=1.5x
y
(D 2-
3
---*
0
0.5
1
carang, Re=5000
- anguil, Re=5000
e - carang, Re=2500
- anguil, Re=2500
2
1.5
(
-s))
Sr/(1
Figure 4-19: Relationship between sr/(1
-
95
2.5
2
sr),
amplitude a and Reynolds number.
(a)
(b)
0.12
0.12
0.1
0.1
-
0.08
-f=1.5
- - f=1.8
- --.-. f=2.1
---- f=2.4
0.08
/
---
/f
/
/
004
f=2.7
0.06
/>
.
C 0.06
I
0.04
0.02
0
0.2
0.4
0.6
0.8
1
0
x
0.2
0.4
0.6
0.8
1
x
Figure 4-20: Optimized (a) prescribed deformation envelopes and (b) displacement envelopes for
the Gaussian parametrization. A = 1 and f = [1.5, 1.8, 2.1, 2.4, 2.7].
4.5.2
Optimization of Gaussian envelopes with A = 1
We now fix the Reynolds number to Re = 5000 and optimize the parameters x1 and 6.
The Gaussian envelopes resulting from the optimization for A = 1 and five frequencies
ranging from f = 1.5 to f = 2.7 are shown in figure 4-20a. For all the frequencies, the
optimized deformation envelope A(x) looks qualitatively similar to the curvature envelope
from Videler [179] shown in figure 4-5b, with a small amplitude at the leading edge, a peak
10 to 30% from the trailing edge, and a sharp decrease in amplitude at the trailing edge. We
also observe that the location of the peak amplitude moves aft as the undulation frequency
increases, from x1 = 0.73 at f = 1.5 to x1 = 0.88 at f = 2.7. At the same time, the width
of the peak decreases from 6 = 0.52 at f = 1.5 to 6 = 0.21 at f = 2.7. For all frequencies,
however, the amplitude of the peak is very close to 0.1.
The corresponding displacement envelopes g(x) are shown in figure 4-20b. The displacement envelopes look qualitatively similar to the carangiform displacement envelope
from Videler [179] shown in figure 4-5a, with a minimum amplitude around x = 0.25 and a
maximum amplitude at the trailing edge. While the amplitude at the leading edge decreases
by a factor of two from f = 1.5 to f = 1.8, it remains almost constant for f from 2.1 to 2.7
with a value g(0) = 0.02 very close to that of figure 4-5a. At the trailing edge, on the other
hand, the amplitude varies roughly proportionally to N/sr/(1 - S.), as already observed in
figure 4-19.
Figure 4-21 shows the deformed foil and vorticity snapshots for the five optimized gaits
at t/T = 0 (mod 1), where T = 1/f is the undulation period. For all gaits, the boundary
layer remains attached to the foil as previously observed for waves traveling faster than
the free stream [153, 147], and a reverse Kirmdn vortex street forms in the wake. The
width and wavelength of the reverse Kairmin vortex street decreases with increasing undulation frequency, and secondary small vortices, present at low frequency, disappear at
higher frequency. As expected from figure 4-20, while at f = 1.5 the entire length of the
foil undergoes noticeable deformation and displacement, at higher frequency the front half
of the foil undergoes virtually no deformation. It is also interesting to notice that, whereas
the undulation wavelength is A = 1 for all frequencies, as the peak of the Gaussian becomes
sharper, the curvature due to the envelope becomes predominant over the curvature due to
the wave in the peduncle section. This phenomenon is particularly noticeable for f = 2.7,
at which frequency the undulations are mostly restricted to what would be the peduncle
96
0.4
(a
[(a)
-
(b)
0
>.0
-0.4[
0.4
(d)
(c)
-0.41
0
0.4- (e)
0
1
1.5
2
2.5
X
e
W:
-0.4 1
0.5
(f)
1 1 1 11
0
0.5
1
1
x
1.5
2
-20 -16 -12
-8
-4
0
4
8
12
16 20
2.5
Figure 4-21: Snapshots of vorticity for optimized gaits at t/T = 0 (mod 1).
f = 1.8, (c): f = 2.1, (d): f = 2.4, (e): f = 2.7, (f): colorbar.
(a):
f = 1.5, (b):
and tail sections for a fish.
Table 4.2a summarizes the parameters and properties of the five optimized gaits. The
quasi-propulsive efficiency qQp of these undulatory gaits is of prime interest. The efficiency
reaches 57% for f = 2.7, whereas the least efficient frequency, f = 1.5 reaches 7Qp = 49%.
An other important parameter is the Strouhal number, which oscillates around St = 0.35.
We showed in 4.5.1 that the Strouhal number varies with Reynolds number and that, for
a given envelope, the undulation frequency seemed a more relevant parameter than the
Strouhal number. The consistency of the Strouhal number for the optimized envelopes
across frequencies suggests that, for a given Reynolds number, there exists an optimal
Strouhal number that can be reached with a large range of frequencies. Like the Strouhal
number, the maximum pitch angle 0 max and maximum angle of attack amax are almost
constant across the five optimized gaits, with a value close to 9 max = 31' and amax = 170.
The phase angle between the heave and pitch of the trailing edge that allows these values
of angle of attack is 0 = 82'. The results from this optimization show that, like for
rigid flapping foils, the efficiency of undulating foils is primarily driven by the Strouhal
number, pitch angle, angle of attack and heave-pitch phase angle. There are however other
parameters affecting the efficiency of an undulating foil, since the efficiency ranges from
TjQp = 0.49 at f = 1.5 to qQp = 0.57 at f = 2.7. This result runs contrary to the
observations from Shen et al. [147] that a slip ratio around Sr = 0.8 (f = 1.2) is optimal.
However, in our case, the undulations at higher Reynolds number are confined to a small
section of the foil, thus reducing the losses due to undulation of the front part of the foil
and to increased recoil.
The location and sharpness of the envelope peak need to be tuned for each frequency,
such that the optimal Strouhal number, pitch angle and angle of attack are attained. Indeed, the optimal envelope A(x) for one frequency can result in a highly inefficient gait
at a different frequency. Figure 4-22 shows the efficiency as a function of xi and 6 in the
neighborhood of the optimal envelope for the five frequencies considered above. The migration of the most efficient region from the top-left corner at low frequency to the bottom
97
f
x1
6
ao
a
Omax(o)
Ginax ( )
1.5
1.8
2.1
2.4
2.7
0.73
0.77
0.81
0.87
0.88
0.52
0.36
0.28
0.23
0.21
0.084
0.066
0.062
0.079
0.073
0.23
0.18
0.16
0.15
0.13
31
28
29
35
34
17
19
20
16
15
0
O()
82
82
81
82
84
St
Cp
?Q P
0.34
0.33
0.35
0.37
0.36
0.093
0.089
0.087
0.083
0.081
0.49
0.52
0.53
0.56
0.57
(a) Optimized envelopes at several frequencies.
f
Xi
6
ao
a
Omax(0)
cf max(o)
()
St
Cp
?IQP
1.8
1.8
1.8
1.8
1.8
1.8
0.65
0.78
0.80
0.85
0.90
0.90
0.50
0.49
0.29
0.31
0.37
0.25
0.050
0.068
0.082
0.101
0.103
0.186
0.17
0.18
0.22
0.22
0.21
0.32
22
26
36
37
35
53
22
21
17
16
19
10
82
78
84
82
78
87
0.30
0.32
0.40
0.40
0.38
0.57
0.097
0.093
0.096
0.095
0.094
0.140
0.45
0.47
0.46
0.46
0.46
0.31
(b) Examples of envelopes around the optimal gait at
f
= 1.8.
Table 4.2: Parameters and properties of gaits with Gaussian envelopes. Motion parameters are
the frequency f, peak location x 1, peak width 6 and amplitude ao. Properties are the peak to
peak displacement amplitude at the trailing edge a, maximum pitch angle at the trailing edge 0max,
maximum angle of attack amax, heave and pitch phase angle 0, Strouhal number St, time-averaged
power coefficient Cp and the quasi-propulsive efficiency 'rQP.
right corner at high frequency appears very clearly in this figure. The figure also shows
that, for all frequencies, the efficiency decreases very rapidly as the width of the peak 6 is
decreased bellow its optimal value, while the efficiency is much less sensitive to increases
in 6. Moreover, as the frequency increases and the peak of the optimal envelope becomes
sharper, the optimal region becomes narrower, especially as far as 6 is concerned. Finally,
while for all frequencies it is possible to find a region in the (xl, 6) space that reaches an
efficiency close to 50%, it appears that the envelope that is most efficient at f = 1.5 is quite
inefficient at f = 2.7, and vice-versa. Increasing the curvature of the foil at the base of the
tail when the frequency increases allows the deformation of the foil to match the curvature
of the trailing edge trajectory and thus avoid the efficiency loss associated with a large angle
of attack. Indeed, figure 4-23 shows that, as the length of the stride decreases with increasing frequency, a larger curvature at the peduncle is necessary for the body deformation to
match the trailing edge trajectory.
In order to better understand the impact of x, and 6 on the gait properties, table 4.2b
summarizes these properties for several values of xi and 6 near the optimum for f = 1.8.
As the location of the peak moves aft and its width decreases, the portion of the foil
undergoing significant deformation reduces, therefore a larger amplitude is necessary to
ensure that enough thrust is produced. As a result, the Strouhal number and maximum
pitch angle increase. This observation also allows us to interpret the optimization results.
We observed in figure 4-16 that, for a fixed envelope A(x), the Strouhal number of a selfpropelled undulating foil increases with decreasing frequency. In order to mitigate this
effect, an envelope with small xi and large 6 that can produce the same thrust with smaller
amplitude makes it possible the reach the optimal Strouhal number even at low frequency.
98
(b)
________(e)
Figure 4-22: qp as a function of x1 and 6 near the optimum for Gaussian envelopes. (a): f = 1.5,
(b): f = 1.8, (c): f = 2.1, (d): f = 2.4, (e): f = 2.7, (f): colorbar. The black dots show the location
of the points that have been used to build the thin-plate smoothing spline (tpaps function in Matlab
with smoothing parameter p = 0.999) represented in color.
(a)
f=1.5
(b)
f=2.1
(C)
f=2.7
Figure 4-23: Superimposed body outlines over one undulation period for three frequencies.
99
(a)
(b)
0.2
0.4
0.2
0.1
f=1.8
-0.2-0.1
------
f=2.1
f=2.7
-0.2
0
0.2
0.4
0.6
0.8
1
0
t/T
0.2
0.4
0.6
0.8
1
t/T
Figure 4-24: Drag and power coefficients as a function of time for the optimized Gaussian envelopes.
Similarly, at high Reynolds number, a large x1 and a small 3 make it possible to produce the
required thrust at the optimal Strouhal number. When the undulation frequency reaches
about 2.5, we noticed in figure 4-16 a plateau in the Strouhal number, the same happens
with the optimal envelope which reaches x1 a 0.9 and 6 - 0.2.
Figure 4-24 shows the drag and power coefficients as a function of time for the five
optimal gaits described above. For all gaits, the period of the drag and power is half the
undulation period T, with times of positive drag and negative power alternating with times
of negative drag and positive power. As the frequency increases from f = 2.1 to f = 2.7,
the time of maximum drag (minimum power) shifts from t/T = 0.12 to t/T = 0 (mod 0.5),
whereas the amplitude of drag and power oscillations reach a minimum at f = 2.1. The
amplitude of drag oscillation is twice as large for f = 1.5 and f = 2.7 as it is for f = 2.1,
while the amplitude of power oscillation is three times as large for f = 2.7 as for f = 2.1.
Moreover, f = 2.1 is the only frequency for which the swimming power is always positive.
In order to understand the reasons for the change in phase and amplitude of the drag
and power coefficients, figure 4-25 shows the pressure field and body velocity for the optimized envelopes with frequency f = [1.5, 2.1, 2.7] at their respective time of minimum
and maximum power. For f = 1.5 (figures 4-25a,b) and f = 2.7 (figures 4-25e,f), there
are three distinct sections along the upper side of the foil: high pressure near the leading
edge, low pressure in the middle and high pressure near the trailing edge (and the opposite on the other side). In figures 4-25b,f, these sections almost exactly match transverse
velocity of respectively positive, negative, and positive sign, resulting in a very large instantaneous swimming power. Conversely, in figures 4-25a,e, the sign of the transverse velocity
is reversed, resulting in a significant negative swimming power. For f = 2.1, the pressure
changes along the foil are smaller, and the pressure is close to zero along a large portion of
the foil. Moreover, unlike for f = 2.7, the sign changes in pressure do not match the sign
changes in transverse velocity. For instance, at t/T = 0, the pressure along the bottom side
of the foil near the trailing edge is positive (not shown here), which would result in a positive
swimming power. Therefore, the minimum power is reached at a later time t/T = 0.12, at
which point the amplitude is largest in areas where the pressure is close to zero, resulting
in a very small power. Similarly, the maximum power reached at t/T = 0.34 is not as large
as for f = 2.7 because the sections of high pressure do not exactly match the sections of
large transverse velocity.
100
p
0.4 -(a
[(a)
0.4
(b)
0.32
0
0.24
1
-0.4
0.16
0.4-
(d)
(C)
0
0.08
X
0
-0.08
-0.4
0.4
-0.16
(e)
(f)
-0.24
0a
-U.4
0
-0.32
0.5
1
1.5
0
2
X
-0.4
0.5
1
1.5
2
X
Figure 4-25: Snapshots of pressure field with arrows showing the body velocity. (a, b): optimized
Gaussian envelope at f = 1.5; (c, d): optimized Gaussian envelope at f = 2.1; (e, f): optimized
Gaussian envelope at f = 2.7. (a, c): t/T = 0.12 (mod 1) (minimum power for f = 1.5 and f = 2.1);
(e): t/T = 0 (mod 1) (minimum power for f = 2.7); (b, d): t/T = 0.34 (mod 1) (maximum power
for f = 1.5 and f = 2.1; (f): t/T = 0.29 (mod 1) (maximum power for f = 2.7).
4.5.3
Optimization of quadratic envelopes with A = 1
We showed in the previous section that, by changing the location and width of the peak in
a Gaussian deformation envelope, a very efficient gait can be designed for a large range of
undulation frequencies. Since carangiform and anguilliform gaits are often represented by
polynomial envelopes, we investigate here the swimming efficiency reachable by optimizing
such envelopes. The two parameters to be optimized are A(0) and A(1/2), the envelope
amplitude at the leading edge and mid-chord respectively (the amplitude at the trailing
edge being constrained to A(1) = 1).
First, we fix the undulation frequency to f = 1.8 and optimize the quadratic envelope
A(x), restricting A(0) to positive values. Figure 4-26a shows the efficiency as a function of
A(0) and A(1/2). The carangiform envelope used in previous sections is indicated by a black
square and the anguilliform gait is shown by a diamond. As long as the envelope is concave,
most gaits are rather efficient, and the performance is not very sensitive to the exact values
of the parameters. Therefore, the very convex carangiform envelope is very inefficient,
whereas the anguilliform envelope, which is close to a straight line, is much more efficient.
Among the concave envelopes, A(0) = 0 seems best, together with 1 < A(1/2) 5 1.7, where
the efficiency reaches 48%. The facts that the most efficient gaits lie on the boundary
A(0) = 0 and that rqp = 0.5 cannot be reached suggest that the quadratic parametrization
is not optimal. Indeed, quadratic functions with a sharper peak, similar to the optimal
Gaussian envelope, could be obtained by further decreasing A(0) to negative values, but
the envelope should, by definition, always be positive.
Since the optimal quadratic gait saturates the constraint A(0) > 0, we then fix the
leading edge amplitude to A(0) = 0 and optimize the undulation frequency f and the
101
(a)
1
(b)
1QP
1.5
2
0.5
0.45
1.5
0.4
0.5
0.35
0.5
0''
0
0.2
0.4
1.2
1.4
1.6
1.8
2
0.3
f
A(0)
Figure 4-26: IQP as a function of A(0) or f and A(1/2) for quadratic envelopes. (a): fixed
frequency f = 1.8; (b): fixed leading edge value A(0) = 0. The black dots show the location of
the points that have been used to build the thin-plate smoothing spline (tpaps function in Matlab
with smoothing parameter p = 0.999) represented in color. In (a), the carangiform and anguilliform
motions investigated in 4.4.4 are respectively represented by a black square and a black diamond,
and a dashed line shows the location of linear envelopes (points bellow this line correspond to convex
envelopes, above it the envelopes are concave).
second envelope parameter A(1/2). Figure 4-26b shows the efficiency as a function of f and
A(1/2). Here again, the efficiency is not very sensitive to the exact value of f and A(1/2)
around f = 1.6 and A(1/2) = 1 where efficiencies of 49% are attained. It is interesting
to notice that, since quadratic envelopes can only result in functions with a wide peak,
they can reach the same efficiency as the wide peak Gaussian envelopes at low frequency
(f = 1.5), but not at high frequency (f = 2) where a sharp peak is advantageous.
Figure 4-27 shows the prescribed amplitude aoA(x) and the corresponding displacement envelope g(x) for the two optimal quadratic envelopes described above, as well as the
carangiform and anguilliform envelopes from 4.4.4 at their respective optimal frequency.
Whereas the anguilliform and carangiform envelopes A(x) are convex, the optimized envelopes are clearly concave, reaching a maximum around x = 0.7, similarly to the optimized
Gaussian envelopes at low frequency. The resulting displacement envelopes, however, are
not concave. For the carangiform and anguilliform envelopes, the displacement is minimum
around x = 0.25 and maximum at the trailing edge with a sharp increase in amplitude in
the last 20% of the foil. The two optimized gaits have a displacement envelope very close to
the prescribed anguilliform envelope aoA(x), with a minimum amplitude g(0) = 0.04 at the
leading edge and a maximum displacement amplitude g(1) ~ 0.1 at the trailing edge. The
increase in displacement amplitude in the last 20% of the foil is also milder than for the
anguilliform and carangiform envelopes, which is consistent with the displacement envelope
observed by Videler [179] and represented in figure 4-5.
The deformed foil and vorticity snapshots at t/T = 0 (mod 1) for the four gaits described
above are shown in figure 4-28. For the carangiform gait at f = 2.1, shown in figure 4-28a,
the deformation of the front half of the foil is very small, similarly to the deformation of
the optimized gait at the same frequency in figure 4-21c. However, whereas the optimal
Gaussian envelope leads to large curvature along the rear half of the foil, this is not the
case here. This less efficient gait also produces a reverse Kirmain vortex street with vortices
of larger magnitude. On the other hand, the optimized gaits shown in figures 4-28c,d, are
quite similar to the low frequency gait with optimized Gaussian envelope of figure 4-21a.
These gaits are characterized by large undulations visible along the whole length of the foil.
102
(a)
0.12
0.1
0.1
-
f=1.8
0.08
0.08
-.
--
carang, f=2.1
anguil, f=1.6
.
---- optim (f=1.58)
.
optim,
(b)
0.12
< 0.06
CD 0.06
0.04
0.04
0.02
0.02
Ce
0
0.2
0.6
0.4
0.2
U0
0.8
--
-..
0.4
0.6
0.8
1
x
Figure 4-27: Optimized (a) prescribed deformation envelopes and (b) displacement envelopes for
the quadratic parameterization with A = 1. Carangiform and anguilliform envelopes from 4.4.4 at
their respective optimal frequency are shown, as well as the optimal envelope for f = 1.8 and the
optimal envelope for A(O) = 0, which has frequency f = 1.58.
0.4 (a)(
0.4
(d)
(C)
00
I-
-0.4
00
0.5
1
1.5
2
2.5
x
0
0.5
1
1.5
2
2
2. 5
x
Figure 4-28: Snapshots of vorticity for gaits with polynomial envelope at t/T = 0 (mod 1). (a):
carangiform, f = 2.1; (b): anguilliform, f = 1.6; (c): optimized with f = 1.8; (d): optimized with
A(O) = 0 (f = 1.58). The color axis is the same as in figure 4-21.
The gait resulting from the anguilliform envelope (4-28b) has a mix of features from the
carangiform and the optimized gaits: noticeable undulation along the whole length of the
body, but rather small curvature.
The properties of gaits with various quadratic envelopes are summarized in table 4.3.
Similarly to what has been observed in 4.5.1, the Strouhal number decreases with increasing undulation frequency and reaches a plateau at high frequency. In the meantime, the
pitch angle also decreases, but since the heave-pitch phase angle decreases too, the angle
of attack does not decrease. The inefficient carangiform envelope is characterized by large
Strouhal number, large pitch angle and large angle of attack. The inefficiency of this envelope is mostly the result of the phase angle being very so from 90': when the tail is zero
(0(t) = 0), the the heaving velocity of the tail is large, displacing a lot of fluid but producing
much thrust. The much more efficient anguilliform envelope reaches the optimal Strouhal
number identified in the previous section, St = 0.35, at f = 1.6, as well as a pitch amplitude
of 310. However, the phase angle / is still quite small, resulting in an angle of attack too
large to be optimal. The optimized gaits, with a phase angle around 80' reach a Strouhal
number, pitch amplitude and angle of attack in the range of optimal values found with the
103
optim, f=1.8
2
.9
)
7
f
A(0)
A(1/2)
ao
a
Oiax (O)
amax( 0 )
V'(O)
St
CF
1.6
1.8
2.1
2.6
0.20
0.20
0.20
0.20
0.19
0.19
0.19
0.19
0.158
0.126
0.100
0.077
0.29
0.23
0.18
0.15
44
39
34
28
34
33
33
34
63
61
59
56
0.45
0.41
0.38
0.36
0.127
0.117
0.114
0.118
0.36
0.40
0.41
0.39
1.3
1.6
2.0
2.4
0.37
0.37
0.37
0.37
0.61
0.61
0.61
0.61
0.146
0.096
0.070
0.057
0.32
0.22
0.17
0.14
39
31
25
22
24
24
27
30
72
68
64
61
0.42
0.35
0.33
0.32
0.110
0.100
0.104
0.113
0.42
0.46
0.44
0.41
1.8
1.6
0.00
0.00
1.40
1.06
0.054
0.079
0.17
0.21
23
29
21
20
81
77
0.31
0.34
0.095
0.094
0.48
0.49
]QP
Table 4.3: Parameters and properties of gaits with polynomial envelopes A(x). Motion parameters
are the frequency f, A(0), A(1/2) and amplitude ao. Properties are the peak to peak displacement
amplitude at the trailing edge a, maximum pitch angle at the trailing edge 9 max, maximum angle
of attack aimax, heave and pitch phase angle, Strouhal number St, time-averaged power coefficient
Cp and the quasi-propulsive efficiency rQP. Lines 1-4: carangiform envelope; lines 5-8: anguilliform
envelope; lines 9-10: optimized envelopes.
Gaussian envelopes.
Finally, figure 4-29 shows the drag and power coefficients as a function of time for the
four envelopes discussed previously. It is interesting to notice that the least efficient gait
(carangiform) is also the gait that has the smallest amplitude of oscillations in drag. While
these oscillations are often perceived as detrimental to efficiency, this figure shows that
reducing their amplitude is not necessarily an indication of improved efficiency.
Optimization of Gaussian envelopes with A = 0.65
4.5.4
In the previous sections, a wavelength of A = 1 has been imposed. Even though Videler
measured a body wavelength for saithe and mackerel consistently roughly equal to the
body length [179], values reported for eel, trout or goldfish typically range from 0.6 to 0.76
(b)
(a) 0.2
0.2
0.1
C)
0.4
0
U
0
-0.2 [
-0.1
-
:(Y
-
-0.4
-0.2'
0
0.2
0.6
0.4
0.8
carang, f=2.1
anguil, f=1.6
---- optim (f=1.58)
0
1
-
0.2
0.4
0.6
0.8
1
t/T
t/T
Figure 4-29: Drag and power coefficients as a function of time for quadratic envelopes.
104
f
x1
6
ao
a
Omax(c)
2.27
0.86
0.38
0.064
0.16
33
amax( 0
)
15
V)(O)
St
CP
r7QP
90
0.36
0.085
0.54
Table 4.4: Parameters and properties of the optimized gait for a Gaussian envelope with A = 0.65.
.
times the fish length [167, 183, 71. Here we choose a wavelength of A = 0.65 and optimize
the Gaussian envelope parameters x1 and 6, as well as the undulation frequency f. The
resulting deformation envelope aoA(x) and displacement envelope g(x) are shown in figure
4-30a, while the parameters and properties of the optimized gait are summarized in table
4.4.
The frequency of the optimized gait is f = 2.27. The associated slip ratio is s,
Us/(Af) = 0.68 for A = 0.65 whereas it would be sr = 0.44 for A = 1. The location of the
envelope peak corresponds to the optimal peak location for the same undulation frequency
with A = 1, but the width of the peak is closer to the optimal value for lower frequencies
at A = 1. Indeed, the smaller wavelength results in a larger curvature, so the envelope does
not need to provide as much extra curvature for the body deformation to follow the tail
path. As a result, the height of the peak is now close to 0.07 and not 0.1 as for A = 1. The
Strouhal number, however, is the same as the optimal Strouhal number identified in the
case A = 1, which confirms the existence of an optimal Strouhal number that is a function
of the Reynolds number only. The pitch angle is in the optimal range identified earlier, but
since the phase angle is exactly 9 0 TC, the angle of attack is smaller than for the A = 1
cases. Finally, as pointed out by Lighthill and others [99, 183], there is less recoil with the
smaller wavelength, resulting in a smaller displacement amplitude at the leading edge
A snapshot of the deformed foil and vorticity field at t/T = 0 (mod 1) for this optimized gait is shown in figure 4-31. The wake and body deformation are very similar to
those observed for the corresponding gaits with A = 1 in figure 4-21c. In particular, the
deformation along the front half of the foil is very small, while it is quite large in the rear
half. Even though the peak of the deformation envelope aoA(x) is smaller than for A = 1,
the curvature is roughly the same because of the smaller wavelength.
Finally, we see from the figure 4-30b that the drag and power coefficients have very
(b)
(a)
0.12
'.2
0.1
0.08.
0.06E
0.04
0
-0.1
a0 A
0.02/
0.2
0.6
0.4
0.8
-
9~
0
-0.2
1
0
0.2
0.6
0.4
0.8
1
t/T
x
Figure 4-30: Optimized gait with Gaussian envelope for A = 0.65 (a): prescribed deformation
envelope aoA(x) and displacement envelope g(x); (b): drag and power coefficients as a function of
time.
105
0.4
-0.4
0
1
2
3
I
4
x
Figure 4-31: Snapshot of vorticity for the optimized gait with Gaussian envelope and A
t/T = 0 (mod 1). The color axis is the same as in figure 4-21.
0.65 at
0.4-
-0.4
0
X
1
2
3
4
x
Figure 4-32: Snapshot of vorticity for the optimized escape gait with Gaussian envelope and A
at t/T = 0 (mod 1). The color axis is the same as in figure 4-21.
1
small amplitude oscillations, just like for the optimized gait with f = 2.1 and A = 1, but
the phasing of these coefficients is similar to that observed with the higher frequency gaits
for A = 1.
4.5.5
Optimization of an escape gait with A = 1
All the gaits presented so far have been designed with the goal to minimize the power
consumed by the self-propelled undulating foil in steady state. These gaits have been
obtained by maximizing the quasi-propulsive efficiency rqp and are a good choice for longdistance travel and migration. However, there are also times when fish need to produce a
lot of thrust in order to accelerate. This is for instance the case of a fish attempting to
escape from a predator or chasing a prey [73]. In these configurations, the goal of the fish
is not to minimize its swimming power, but to maximize the acceleration it can reach with
its available power. In other words, the fish wants to maximize its net propulsive efficiency.
Here, the velocity of the foil is still fixed, but it produces a net thrust (T $ 0) which
would result in acceleration if it was not towed. The frequency f as well as the Gaussian
parameters x1 and 6 are optimized, whereas the maximum amplitude of the envelope is
fixed to maxXE[oJ1(aoA(x)) = 0.1.
The snapshot in figure 4-32 shows that, unlike the gaits optimized for distance, the gait
optimized for thrust has a large deformation along the whole length of the foil. The wake is
also wider and the vortices produced are much stronger, in order to produce a large thrust.
Whereas the wake produced by the undulating foil in a self-propelled configuration (T = 0)
is symmetric with respect to the centerline, figure 4-32 shows an asymmetric, deflected
wake pattern. To the knowledge of the authors, such a pattern has never been reported
for undulating foils, but it is consistent with previous findings for a thrust-producing rigid
flapping foil [86, 95]. Both studies found that for values of irSt > 1, the wake appears
asymmetric or "deflected". Since the value of this parameter for the current simulation is
1.95, the asymmetry in the current flow is not unexpected.
106
(a)
,_,_,_,_(b)
2
0.120.1
1
0.08\
61
0-
0.02-
0_
_
01
0
0.2
0.4
0.6
0.8
-21
1
0
0.2
0.4
x
0.6
0.8
1
t/T
Figure 4-33: Optimized escape gait with Gaussian parameterization for A = 1 (a): prescribed
deformation envelope aoA(x) and displacement envelope g(x); (b): drag and power coefficients as a
function of time.
Figure 4-33a shows that the width of the peak of the prescribed deformation envelope
aoA(x) is much larger than the peak for any of the low energy gaits. The resulting displacement envelope, g(x), is very similar to that of the anguilliform motion (figure 4-27b), with a
minimum displacement around x = 0.25 and a displacement amplitude at the leading edge
about half the displacement at the trailing edge. The properties of the gait optimized for
thrust are summarized in table 4.5 and compared to the carangiform and anguilliform gaits
at their respective frequency of maximum net propulsive efficiency. The gait optimized for
thrust and the anguilliform gait have almost the same frequency, motion and pitch amplitude. The phase angle 4, however, is much smaller for the anguilliform motion, resulting
in a much lower thrust and power coefficients, and an overall lower efficiency q, = 0.28
instead of 7, = 0.32. Here again, the carangiform motion is much less efficient. With its
large frequency and small phase angle resulting in a very large angle of attack and Strouhal
number, the thrust coefficient of the carangiform gait is the same as for the optimized gait
but the power is much larger, resulting in a much lower efficiency 77 = 0.21.
f
4.0
2.4
2.5
x1
6
carangiform
anguilliform
0.75
0.74
ao
a
Omax(0)
0.100
0.100
0.089
0.19
0.24
0.25
35
33
32
armax(
52
39
37
0
)
()
St
CT
Cp
54
64
73
0.75
0.56
0.62
0.22
0.13
0.23
1.07
0.47
0.72
7,
0.21
0.28
0.32
Table 4.5: Parameters and properties of thrust producing gaits. Properties are the peak to peak
displacement amplitude at the trailing edge a, maximum pitch angle at the trailing edge Omax, maximum angle of attack amnax, heave and pitch phase angle 0, Strouhal number St, time-averaged thrust
coefficient CT and power coefficient Cp, and the net propulsive efficiency Tin. The optimized escape
gait is compared to the carangiform and anguilliform gaits at their respective optimal frequency for
in-
Finally, this optimal thrust gait requires almost 10 times as much power as the low
energy gaits discussed above. For practical purposes, the optimization would probably be
constrained by the maximum power muscles (or motors) can produce. Moreover, figure
4-33b shows that the amplitude of oscillation of the drag and power coefficients is also one
order of magnitude larger than for the low energy gaits. A possible consequence is that
107
there would be large oscillations in the acceleration.
108
4.6
Energy saving by swimming in pair
We now consider a pair of undulating foils. We have shown in the previous section that
a foil, undulating in open-water, can attain a quasi-propulsive efficiency of almost 60% by
optimizing its gait. The goal in this section is to determine whether, by working as a pair,
fish can further reduce the power required to travel at constant speed U,.
4.6.1
Kairmin gaiting and Weihs' schooling theory
It has been well documented that fish swimming in the Kirmin vortex street behind a
cylinder tend to synchronize their motion to the vortices, which allows them to significantly reduce the energy spent to hold station [98, 97, 3]. This phenomenon, known as
Kirmain gaiting, has been explained by the faculty of fish to harness the energy of the
vortices. We have shown in the previous sections that undulating foils also produce rows of
coherent vortices, however, it is still unclear whether fish can harness the energy of vortices
produced by other fish. As Liao summarized it in his review of fish swimming in altered
flows, "no hydrodynamic or physiological data exist to evaluate the hypothesis that fish can
increase swimming performance by taking advantage of the wake of other members" [96].
Due to the difficulty to experimentally measure the swimming power of individual fish in a
school, simulations can provide extremely valuable information to help solve this mystery.
There are two major differences between the wake of an undulating foil and that of a
D-section cylinder as used to study Krman gaiting. The first one is that the undulating
foil produces a reversed Kdirmain gait, in which the time-averaged flow along the centerline
is faster than the free-stream, and not slower. The second one is that, while the wavelength
of the Kairmain vortex street behind the D-section used in experiments is larger than the
fish body length, the wavelength of the reverse Kdirmin street produced by an undulating
foil is smaller than the foil length.
Whereas KArmAn vortex streets are unsteady by nature, Weihs' schooling theory only
considers time-averaged flows [186]. According to this theory, a fish swimming directly
behind another fish would experience higher relative velocity and would therefore have to
spend extra energy. On the contrary, a fish swimming between two wakes would experience
a reduced relative speed, allowing it to save energy. According to Weihs, the energy benefit
of schooling results from flow refuging and not vortex capture.
Even if the average flow in the wake of a cylinder is slower than the free stream, the fact
that fish change their kinematics and synchronize with the wake suggests that they actually
use individual vortices to save energy [3]. Moreover, Boschitsch et al. [19] recently showed
that the net propulsive efficiency of a pitching foil located behind a similarly pitching foil
could be anywhere between 0.5 and 1.5 times that of an isolated foil, depending on the
phase. These results suggest that, by properly timing its undulation with respect to the
incoming vortices, a fish swimming in a wake of another swimming fish might be able to
save energy despite being, on average, in a jet. If that is the case, the hydrodynamic theory
of schooling, proposed by Weihs, will need to be revised into a theory that accounts for
individual vortices and not only the average flow.
4.6.2
Flow in the wake of a self-propelled undulating foil
The flow in the wake of a self-propelled undulating foil consists of vortices of alternating sign.
The vorticity snapshot in figure 4-34a shows that the vortices decrease in strength under
109
(a)
W
0.4
U
0
1.4
1.24
1.08
0.92
0.76
0.6
0
4.0
0
-4.0
-12.0
20.0
-0.4
i
(C)
0.4-
(d)
5.0
3.0
1.0
-1.0
-
0
-04
(b)
20.0
12.0
5
1-3..5
2
2.5
3
3.5
-5.0
1.5
2
2.5
3
3.5
ti
1.2
1.12
1.04
0.96
0.88
0.8
x
x
Figure 4-34: Wake behind a self-propelled undulating foil for the optimized gait with Gaussian
envelope and A = 1 at frequency f = 1.5. (a): instantaneous vorticity field; (b): instantaneous
x-velocity field; (c): time-averaged vorticity field; (d): time-averaged x-velocity field.
the effect of diffusion, but this is a slow process and the wake is primarily characterized
by its periodicity. Figure 4-34b shows that the vortices are arranged in such a way that
the flow along the y = 0 axis is faster than the ambient flow, whereas the flow away from
the centerline moves slower than the ambient flow. As a result, the time-averaged vorticity
field in figure 4-34c is characterized by four shear layers of alternating sign, resulting in a
jet along the centerline with strips of slowed down flow on either side, as shown in figure
4-34d.
The reverse Karm.n vortex street behind a self-propelled undulating foil is characterized
by its periodic structure with vortices moving parallel to the y = 0 axis in stable formation.
The vorticity at longitudinal distance d from the trailing edge in the wake of an undulating
foil can be modeled as:
w(d, y, t) = Wy (y, t) Wd(d) sin (27r (#1(d) - ft) ),
# 1(d) = d/Aw + #Ow,
(4.73)
where the frequency f is given by the undulation frequency and the wavelength A, and
phase #w of the wake need to be determined. For the five optimized gaits with Gaussian
envelope and A = 1 presented in 4.5.2, we estimated from the vorticity field the phase #1
at several distances d along the wake. In figure 4-35a we show the phase #1 as a function
of the distance d, as well as the least squares linear fit for each swimming gait. The
coefficients for the linear fit are summarized in table 4.6. For all the gaits, the phase is
essentially proportional to the distance d, with a coefficient of proportionality very close to
the undulation frequency f. Since A. = fce, where c, is the speed at which the vortices
travel in the wake, this result shows that the vortices travel at the same speed as the freestream. Moreover, for the five gaits considered, the phase at d = 0 is around 0.25, which
means that the vortices are shed by the foil when the trailing edge has maximum transverse
velocity. Finally, we confirm these observations by plotting #1 as a function of fd in figure
4-35b. Assuming c. = 1, the least-squares estimate (+ standard deviation) of the phase
0" is:
(4.74)
OW = 0.24 + 0.02.
From now on, A, = 1/f and Ow = 0.25 will be used to estimate the phase 01 encountered
by a downstream foil whose leading edge is located at distance d from the upstream foil.
110
(b) 6
f=1.5
4
4
3
f=1.8
-f.-.f=2.1
5
U
f=1.5
*
f=18
f=2.1
f=2.4
f=2.7
o
.
5 -a
-- 0-- f=2.4
---f=2.7
+
-4
4
./.--
.0e
A'
'
(a) 6
3
2
2
N
y=x+0.25
1
0
0. 5
1
d
1.5
0
2
2
4
6
fd
Figure 4-35: Vorticity phase in the wake of self-propelled undulating foil as a function of (a)
distance and (b) distance times frequency. For each frequency, the optimized gait with Gaussian
envelope is used.
01 linear fit
Gait parameters
c.
= 1
f
x1
6
/A1 ,
$w
$1 -fd
1.5
1.8
2.1
2.4
2.7
0.73
0.77
0.81
0.87
0.88
0.52
0.36
0.28
0.23
0.21
1.52
1.77
2.07
2.37
2.65
0.19
0.26
0.30
0.29
0.29
0.22
0.24
0.26
0.26
0.25
Table 4.6: Parameters of the gaits used in the wake vorticity phase estimate and fitted phase and
wavelength for the vorticity in the wake. An estimate of the phase 0,, assuming a phase speed
cW = Aw/f = 1 is also provided.
111
(a)
(b)
1
1.8
R(CP)
1.6--
)
- - - R(a 0
M*: 0.9(cC)1.41.2
0 Ix0.8
0.7
--.
R(C P)
- - - R(a0 )
0.5
1
1.5
1
0.8
0
2
0.5
1
A$
d
Figure 4-36: Time-averaged power coefficient Cp and amplitude ao for (a) the upstream foil as
a function of ditance d and (b) the downstream foil as a function of phase AO. The solid (resp.
dashed) line marks the average value with respect to the phase (resp. distance) and the shaded area
indicates the range of values reached across the various distances (resp. phases).
4.6.3
Effect of phase and distance for two undulating foils in a line
Here we consider two foils following each other and undulating at frequency f = 1.5 with the
optimized gait for this frequency, as illustrated in figure 4-37. The amplitude of undulation,
ao, is adjusted independently for each foil to ensure that both foils are in a stable position
and produce zero net thrust on average. We vary the distance d between the trailing edge
of the upstream foil and the leading edge of the downstream foil, as well as #, the phase of
the downstream foil motion as defined in Eq. 4.1. An important parameter will be A#, the
phase difference between the motion of the downstream foil leading edge and the vortices
it encounters: AO = # - 01 (d). In order to measure the impact of the pair configuration on
each foil, we define R(Cp) (resp. R(ao)), the ratio of the power coefficient (resp. amplitude)
in the pair configuration over the power coefficient (resp. amplitude) for the same gait in
open water.
Both foils can greatly benefit from swimming in pair, but the patterns are very different. The swimming power and amplitude of the upstream foil is virtually independent of
the phase of the downstream foil, as shown in figure 4-36a. For large separations d, the
downstream foil does not impact the upstream foil and its efficiency is almost the same as in
open-water. However, as the downstream foil gets close (d < 0.5), the high pressure around
the leading edge of the downstream foil 'pushes' the upstream foil, regardless of their phase
difference. As a result, the upstream foil can reduce its swimming amplitude, expending less
power than it would in open-water. At d = 0.25, the undulation amplitude is reduced by
10%, resulting in 28% energy saving, corresponding to a quasi-propulsive efficiency (based
on the towed drag on open-water) of rqQP = 69%.
For the downstream foil, the situation is very different. Even when the upstream foil
is several chord lengths ahead, the downstream foil encounters its wake. The performance
of the downstream foil is determined by its interaction with the vortices of the wake. It
appears from 4-36b that the swimming power of the downstream foil depends primarily on
the phase difference A# between its undulation and the encountered vortices. Regardless
of the distance d, the swimming power of the downstream foil is low if AO is between
0.7 and 1, and it is high if AO is between 0.1 and 0.5. Like for the upstream foil, the
reduced swimming power results from a reduced undulation amplitude ao, but despite a
more significant reduction in amplitude (27% for A# = 0.8), the power reduction does not
112
0.5
d
0
-0.5 F
lop
-2
-1
0
1
2
3
4
X
Figure 4-37: Snapshot of the vorticity field for two foils undulating at f
1.5 with separation
distance d = 1 and optimal phase AO = 0.83 at time t/T 0.25 (mod 1). The color axis is the
same as in figure 4-21.
exceed that of the upstream foil. For the downstream foil, a maximum energy saving of
24% is reached at AO = 0.85, corresponding to an efficiency of qp = 65%.
Figure 4-37 shows the vorticity field around the two foils undulating with frequency
= 1.5 at distance d = 1 for the phase AO = 0.83 that minimizes the swimming power
of the downstream foil. At t/T - 0 = 0.25, the downstream foil approaches the negative
vortex (at x = -0.1 on its left) as it is turning its "head" (leading edge) to the right. This
acceleration of the head causes a low pressure on the left side of the head, as shown in
figure 4-38e. Due to its position, the approaching negative vortex causes an increase in
the longitudinal velocity, as shown in figure 4-38b, which results in an increased stagnation
pressure (figure 4-38f). However, this vortex also generates a large transverse velocity
with negative sign, as shown in figure 4-38d. As a result, the effects of the head motion
are amplified by the incoming vortex, displacing the stagnation point downstream on the
right side and deepening the low pressure on the left side (figure 4-38f). While the energy
required by the foil to rotate its head is increased, the drag is decreased, despite the faster
flow encountered by the foil.
f
At the same time, the positive vortex located at x = 0.2 on the right side of the foil
thickens the boundary layer and significantly accelerates the flow in a region where the foil
undulation already accelerates it (figure 4-38a,b). This interaction between the vortex and
the foil results in a very large pressure drop around x = 0.3 that also contributes to the
reduction in drag while increasing the swimming power. The vortices are convected downstream at a speed which is substantially lower than the phase speed of the foil deformation.
Further downstream, the distance between the vortices and the foil increases, and their
interaction becomes weaker. At the trailing edge, the phase between the vortices and the
foil motion is close to 7r, such that the positive vortex reaches x = 1 as the trailing edge of
the foil is at its leftmost position. This vortex will be shed just upstream of the same sign
vortex shed by the downstream foil. The resulting wake configuration is unstable and it
takes several body lengths for the wake to reorganize into two pairs of opposite sign vortices
per cycle.
The results presented in this section are consistent with the experimental results from
thrust producing rigid pitching foils in an in-line configuration [19]. We found that, for
small separation distance, the propulsive efficiency of the upstream foil is greatly improved.
We also showed that the efficiency of the downstream foil only weakly depends on the
separation distance, the primary parameter being the phase difference between the wake
from the upstream foil and the undulating motion of the downstream foil. If the undulation
amplitude was fixed, the downstream foil would experience an increased drag and decreased
power for 0 < A# < 0.5, whereas it would experience a decreased drag and increase power
113
(a) 0.4
_
(b)
0
-0.4
(d)
(C) 0 4
0
-0.4[
-e)0.4-M
-2
-1.5
-1
0
-0.5
x
0.5
1
1.5
2
x
Figure 4-38: Snapshot of the velocity and pressure field for two foils undulating at f
1.5 with
separation d = 1 and optimal phase AO = 0.83. (a,b) x-velocity; (c,d): y-velocity; (e,f): pressure
and arrows showing the velocity of the foil. (a,c,e): upstream foil at t/T = 0.25 (mod 1); (b,d,f):
downstream foil at t/T - # = 0.25 (mod 1). The same color axis as in figure 4-25 is used for the
pressure, and the same as in figure 4-34b is used for the velocity (centered in 0 for the y-velocity).
for 0.5 < 4 < 1. For a self-propelled foil, the energetic benefits of a reduced amplitude
resulting from a reduced drag overcome the power increase caused by the vortices.
4.6.4
Effect of undulation frequency for two undulating foils in a line
We now fix the distance between the two foils to d = 1 and vary their undulation frequency.
For frequencies f = [1.5, 1.8, 2.1], the optimized Gaussian envelope is used, for which the
parameters are summarized in table 4.6.
Figure 4-39 shows that most of the conclusions drawn in 4.6.3 still hold as the undulation frequency is increased. While the upstream foil is mostly unaffected by the presence
and phase of the other one, the undulation amplitude of the downstream foil is largest for
0.5 and smallest for 0.5 < AO < 1. However, the correlation between amplitude
0 < AO
and efficiency is not as strong any more and the exact value of the optimal phase depends
on the frequency. For instance, at f = 2.1, the amplitude is minimum for AO = 0.85, but
the quasi-propulsive efficiency is minimum for AO = 0. Whereas with f = 1.5 most phases
result in an increased amplitude and power coefficient, with f = 1.8 and f = 2.1, the amplitude and power coefficient of the downstream foil never exceeds that of the upstream foil.
Therefore, at these frequencies, it is always beneficial to swim in the wake of an undulating
foil, which is in contradiction with Weihs' theory.
At frequency f = 1.8, the situation at the optimal phase, shown in figure 4-40, is
very similar to that described in the previous section for f = 1.5, with the vortices from
the upstream foil reinforcing the effect of the body undulation. However, the wake at
this frequency is narrower, therefore the vortices are closer to the foil and they lose more
strength through their interaction with the boundary layer. Moreover, since the distance
between two consecutive vortices is proportional to the undulation frequency, the vortices
are closer to each other. The resulting wake is dominated by the two single vortices from the
114
k)
R(ao)
0.87
.
kV
1.3
.
08
.0
1.6
1.8
f
(p
O.8
1.3
01.2
1.2
060.
1.6
2
1.8
2
f
Figure 4-39: (a) Amplitude ratio a, and (b) quasi-propulsive efficiency rQp as a function of
frequency f and phase AOp for two foils swimming in a line at distance d = 1. The amplitude
ratio a, is defined as the ratio of the undulation amplitude of the upstream and downstream foils:
ar = ao(upstream)/ao(downstream). The black dots show the location of the points that have been
used to build the thin-plate smoothing spline (tpaps function in Matlab with smoothing parameter
p = 0.999) represented in color.
downstream foil, each accompanied by a weaker vortices of opposite sign from the upstream
foil.
As the frequency gets higher, the vortices from the upstream foil lose more energy
through interaction with the boundary layer of the downstream foil, therefore the wake
behind the two foils turns into a pair of single vortices, as shown in figure 4-41. Moreover,
since the efficiency of the downstream foil mostly depends on the phase of the leading edge
with respect to the reverse Kairmain vortex street from the upstream foil, the phase of the
trailing edge with respect to the incoming vortices in the optimal configuration changes
with frequency.
Figure 4-42 illustrates the cases of the downstream foil undulating with the worst phase
with respect to swimming efficiency for f = 1.8. In this configuration, the vortices from
the upstream foil counteract the effects of the undulating motion of the downstream foil.
As the foil turns its head to the right, displacing the stagnation point to the right and
causing a low pressure on the left side of the head, the positive y-velocity caused by the
approaching positive vortex has the opposite effect. The high velocity regions caused by
the vortices along the foil correspond to low velocity regions from the undulating motion.
Finally, when the vortices from the upstream foil reach the trailing edge, they merge with
the same sign vortices from the downstream foil. The resulting wake is very stable and is a
classical reverse Kairmain vortex street much wider than the one behind a single foil.
For all the frequencies considered here, a self-propelled foil can save energy by undulating
behind another self-propelled foil undulating at the same frequency, reaching efficiencies
close to qQP = 70%. By properly phasing its motion with respect to the incoming vortex
street, the vortices can reinforce the effect of the undulation. Whereas for a fixed amplitude
this phase would result in an increased swimming power, the reduction in drag results in
an overall decreased swimming power.
4.6.5
Foil undulating in the reduced velocity region of the wake
We have so far considered the case of the pair swimming in an in-line configuration. Since
our fish model has a robust feedback controller ensuring its stability in y, it is also possible
to impose an asymmetric configuration with an offset in the y direction. Indeed, according
115
(b)
(a) 0.4
-0.4'
(C)
0.4
_K
_K
4
(d)
-o
0
-0.4
(e) 0.4
AWL
0
0
-0.4
(h)
(g) 0.4
0
-0.4
-2
-15
-1
-0.5
0
0.5
1
1.5
2
2.5
x
x
Figure 4-40: Snapshot of the vorticity, velocity and pressure field for two foils undulating at f = 1.8
with separation distance d = 1 and optimal phase AO = 0.87. (a,b) vorticity; (c,d) x-velocity; (e,f):
y-velocity; (g,h): pressure and arrows showing the velocity of the foil. (a,c,e,g): upstream foil at
t/T = 0.25 (mod 1); (b,d,f,h): downstream foil at t/T - # = 0.25 (mod 1). The same color axis
as in figure 4-25 is used for the pressure, and the same as in figures 4-34a and b for vorticity and
velocity (centered in 0 for the y-velocity).
0.5 r0
-0.5 [I
I I I I I IIIIII I I I I
-2
-1
I
0
1
I
2
I
3
x
Figure 4-41: Snapshot of the vorticity field for two foils undulating at f = 2.1 with separation
distance d = 1 and phase AO = 0. The same color axis as in figure 4-34a is used.
116
(a) 0.5
()-0.5 -U
0.5
(b) -0.5
0
0.5
-
-0.5
(C)
0
-0.5
-2
-1
0
1
2
3
Figure 4-42: Snapshot of the (a) vorticity and (b) pressure field for two foils undulating at f = 1.8
with separation distance d = 1 and phase AO = 0.38 at t/T - 0 = 0.25 (mod 1). The same color
axis as in figure 4-34a is used for the vorticity and the same as figure 4-25 for the pressure.
to Weihs' theory [186], the only way for a fish to save energy in a school is to swim in the
region of reduced velocity located between two wakes. Figure 4-34d shows that, even with
a single foil upstream, the flow on either side of the wake is slower than the free stream:
for f = 1.5 the average flow is slowest at y = 10.17. With the downstream foil offset from
the upstream foil by Ay = 0.17, we vary the phase difference A# in order to see if the
downstream foil can also save energy when swimming at this location.
Figure 4-43 shows that, even when the downstream foil is not directly in the vortex
street, its swimming performance greatly varies with the phase. However, it is easier for the
foil to save energy in this region of reduced flow velocity than directly behind the upstream
foil. Directly behind the upstream foil, only 30% of the phases result in energy savings, but
by using the unsteadiness of the wake, the quasi-propulsive efficiency can be brought down
from 50% to 60%. When undulating in the region of reduced flow velocity, it is easier to
save energy since over 70% of the phases result in energy savings. The energy savings can
even be very large since 'rQp = 80% is possible for AO = 0.65.
As illustrated in figure 4-44, with this phase the leading edge of the downstream foil
reaches its leftmost position at the same time as it reaches a positive vortex. Figure 4-45b
shows that the leading edge of the downstream foil exactly passes through the region where
the longitudinal velocity is smallest. As a result, the stagnation pressure is greatly reduced
(figure 4-45d). Moreover, the region of accelerated flow between the negative vortex and the
foil (x = 0.4) reinforces the accelerated region caused by the undulation, which we showed
earlier is beneficial.
117
(b)
(a) 1.8
1.8,
downstream foil
-4-in line
1.6
1.6
offset
----
1.4
1.4
0
1.2
1
1
upstream foil
0.8 -e - in line
- -offset
-~
0.8.
-
06
-e-~
ar,
-
0.6
1.2
%
5
0.2
0
0.4
Ac
0.6
0.8
1
0
0.2
0.4
0.6
1
0.8
*
Figure 4-43: Ratio of undulation amplitude ao and time-averaged power coefficient Cp as a function
of phase for two foils undulating at f = 1.5 with longitudinal separation distance d = 1. In-line foils
and foils with offset Ay = 0.17 are compared.
I
i
*0
0
I
I
I
I
I
I
I
0
-1
-2
*
zZZZ~~rf~
i
i
i
2
i
I
3
x
Figure 4-44: Snapshot of the vorticity field for two foils undulating at f = 1.5 with longitudinal
separation distance d = 1, transverse separation Ay = 0.17 and optimal phase AO = 0.65 at time
t/T - # = 0.1 (mod 1). The color axis is the same as in figure 4-21.
(b)
(a)0.5
>
0
I
-0.5
(c)0.5
11,
(d)
. .
; j
-2
-1.5
x
-1
-0.5
I
I
I
I
0
0.5
1
1.5
I
~
2
2.5
x
Figure 4-45: Snapshot of the (a,b) x-velocity and (c,d) pressure field for two foils undulating at
f = 1.5 with longitudinal separation distance d = 1, transverse separation dy = 0.17 and optimal
phase AO = 0.65. (a,c): upstream foil, t/T = 0.1 (mod 1); (b,d): downstream foil, t/T - # =
0.1 (mod 1).The same color axis as in figure 4-25 is used for the pressure, and the same as in figure
4-34b for the velocity.
118
4.7
4.7.1
Discussion
Efficiency and the notion of drag/thrust on a self-propelled body
Schultz & Webb [141] discussed the difficulty of establishing a system propulsive efficiency
for self-propelled bodies. They applied the concept of propulsor efficiency to define the
system efficiency; since the net force is zero (as it must be in every self-propelled body), the
system efficiency defined in this manner is zero as well; this is not a helpful result, because
any system, however wasteful its propulsor may be, will be deemed equally (in)efficient as
any other.
The difficulty of establishing a propulsive efficiency stems from the impossibility to
separate drag and thrust since they balance on average and pressure (resp. viscosity) is
the primary source of both at large (resp. low) Reynolds number. Inviscid approaches
propose thrust estimates, but these remain controversial due to the blurry definition of
thrust for a self-propelled body. For instance, it is sometimes argued that Lighthill's model
overestimates the thrust [78, 4, 149]. The quasi-propulsive efficiency moves away from the
ill-defined notion of drag on a self-propelled body, using the well defined drag on a towed
body instead. It results in an intuitive measure of efficiency that can be used to minimize
the "fuel" consumption rather than the hydrodynamic efficiency.
Although the notion of thrust is ill-defined, attributing high (resp. low) quasi-propulsive
efficiencies to a drag reduction (resp. enhancement) is a possible way of interpreting the
performance of a propulsion system. Indeed, if one considers a {body+propeller} system,
a low quasi-propulsive efficiency is either the result of an inefficient propulsor, or adverse
hydrodynamic interactions between the propeller and the body (or a combination of both
factors). Adverse hydrodynamic interactions between the body and the propulsor can be
interpreted as an increase in drag due to the propulsor:
7
RUs
=QP
-
Pn
=Pin
R
TP
(4.75)
&
where Tp/R is the drag amplification.
This drag increase due to body undulations, which has often been made in the literature,
is at the core of a century long controversy opposing the drag reduction proponents [63, 59,
57] in the wake of Gray and his famous paradox [70], to the drag enhancement advocates
[100, 184, 66, 177]. While the latter have long conjectured that body undulations must
significantly increase the skin friction along the body due to what is often referred to as the
Bone-Lighthill boundary-layer thinning hypothesis [100], such an increase has never been
confirmed. Instead, experimental visualization of the boundary layer of dead towed and
live self-propelled fishes showed that the skin friction on a fish, undulating or not, was just
higher than the drag on a fiat plate [4]. Similarly, theoretical analysis from Ehrenstein
Eloy [49] suggested an increase in the skin friction drag on the order of 1.2, well bellow the
Bone-Lighthill hypothesis values of 3 to 5 [100].
Our viscous simulations of undulating self-propelled foils in which power, friction and
pressure forces are simultaneously estimated can help shed a new light on this controversy.
Using Wu's potential flow theory to estimate the propulsor efficiency, the drag amplification
due to the undulating motion can be estimated as the ratio between the propulsor efficiency
and the quasi-propulsive efficiency. Table 4.7 shows that, for the range of examples considered in this chapter, the drag amplification is between 40% and 60%. This drag increase
is traditionally attributed to an increase in the friction drag, and the amplification of the
119
gait
f
St
1.8
carangiform
2.1
2.6
anguilliform
r/QP
7iWU
CDf CDfo
rlWu/'rQP
0.41
0.40
0.63
1.45
1.60
0.38
0.36
0.40
0.39
0.64
0.62
1.41
1.39
1.58
1.59
1.3
1.6
2.4
0.42
0.35
0.32
0.42
0.46
0.41
0.66
0.69
0.64
1.49
1.38
1.34
1.58
1.50
1.57
optimized
1.5
2.1
2.7
0.34
0.35
0.36
0.49
0.53
0.57
0.76
0.75
0.79
1.42
1.46
1.44
1.54
1.41
1.39
optim, A = 0.65
2.27
0.36
0.54
0.79
1.64
1.45
Table 4.7: Efficiency and drag amplification for various gaits at Reynolds number Re
this Reynolds number, the friction drag accounts for 65% of the towed drag.
5000. At
friction drag is indeed of the same order. However, while the friction drag increases with
increasing Strouhal number and wavenumber, the total drag amplification does not follow
these trends. In general, increases in friction drag alone cannot account for low swimming
efficiencies. It seems from table 4.7 that the least efficient gaits are the result of an inefficient
propulsor and a drag amplification (mostly pressure drag).
By optimizing the undulatory gait, we were able to bring the propulsor efficiency up to
80% and the drag amplification down to 40%, resulting in a quasi-propulsive efficiency of
close to 60%. It might, be possible for drag reduction mechanisms to further mitigate the
drag amplification and result in highly (quasi-propulsive) efficient swimming gaits. Such
mechanisms used by fish and mammals, either passive or active, are reviewed in [57]. For
instance, experiments on a robotic tuna by Barrett et al. [10] suggested that, especially
at high Reynolds number, it is possible for the undulating motion to interact beneficially
with the drag on the body and obtain quasi-propulsive efficiencies larger than 1. Barrett et al. [10] directly measured the power needed to drive the tuna-like motion of a
robotic mechanism under self-propulsion conditions. Inviscid theory provided values for
the self-propulsion power very close to the experimentally measured values [10, 88, 151].
The quasi-propulsive efficiency, estimated as proposed herein, provided values up to 150%,
well in excess of 100%, which simply means that the resistance of the actively swimming
body was less than the drag under straight-towing conditions. The measurements were at
the transitional Reynolds number of around Re = 800 000 where re-laminarization of the
boundary layer and separation suppression is possible. Indeed, simulations [147] and experiments [155] on an actively flapping two-dimensional sheet demonstrated clear turbulence
reduction, in addition to flow separation suppression, which was noted earlier by Taneda
[153]. This can explain the drop in drag under self-propulsion conditions and hence the
high quasi-propulsive efficiency values; indeed Barrett et al. [10] found the equivalent drag
coefficient of the actively swimming mechanism to be closer to laminar boundary layer values, whereas the drag coefficient of the straight-towed mechanism was close to turbulent
boundary layer values.
120
4.7.2
Measure of performance for optimizing velocity and body shape
We have shown through examples that quasi-propulsive efficiency r7Qp is the only rational
measure of the efficiency for a self-propelled body in steady motion. There is no theoretical
guarantee that T/Qp will be smaller than 1, and it can indeed be greater than 1 for very
efficient propulsion [10]. However, it gives an intuitively meaningful number that allows the
comparison of various geometries and propulsion systems. It can, for instance, be used to
compare the efficiency of man-made systems and biological ones. It can not, however, be
used to compare or optimize the performance of hull or body shapes [87, 173], or swimming
velocities [102].
One should also keep in mind that the mechanical efficiency, considered in this paper,
is only the last link in a series of processes involved in swimming. As [51] explains in
his short review of Webb's contributions, "For fish, just as with engineered vehicles, fuel
consumption is the most obvious measure of power input." Fuel comes in the form of
metabolic energy, and the efficiency of converting this chemical energy to mechanical energy
plays an important role in the final measure of swimming efficiency.
A more general goal than that of Section 4.4.3 can be expressed as: For a given mass
m, find the body shape, propulsor and velocity that will require the least amount of energy
to drive the vehicle from point A to point B in a fluid of kinematic viscosity V and density
P.
In other words, the goal is to minimize the energy per unit length traveled for a mass
m in a given fluid. For this problem, the natural units are:
mn
mass: m,
length :
1/3
-
(M)2/3
time : v
,
P
-
P
.
(4.76)
If the average swimming power is Pin and the average velocity is Us, the average energy E
spent per unit length (using the length unit defined above) is:
nm/3(4.77)
=
Usp1/3
The corresponding dimensionless coefficient, which we will call energy coefficient CE, is:
__
CE =
Pin
"
-
pUsv2~
(4.78)
Unlike the quasi-propulsive efficiency, this energy coefficient is convenient for comparing
various geometries and propulsion strategies. However, CE is decreasing with Reynolds
number, therefore any optimization would conclude that a swimming speed of zero is optimal since it does not require energy. Indeed, the coefficient CE takes into account the
hydrodynamic power spent to travel from A to B, but nothing ensures that the travel will
be accomplished in a finite time, which is why the total metabolic power Pit, needs to be
used instead of Pin. The metabolic power, which includes a cost Pm proportional to time,
can be expressed as [102]:
Pt = P + Pm,
/3
(4.79)
where 3 is the muscle power efficiency, Pm is the standard metabolic rate independent of
swimming speed and P is the hydrodynamic power (similar to the definition of Pin in Eq.
121
f
d
Ay
A05 (mod 1)
qQP/(upstream)
?Qp(downstream)
1.5
1.8
2.1
1.5
1.5
1
1
1
0.25
1
0
0
0
0
0.17
0.83
0.84
0.00
0.83
0.65
0.52
0.55
0.56
0.67
0.51
0.60
0.61
0.62
0.66
0.81
Table 4.8: Efficiency for a pair of undulating foils in various advantageous configurations. The
undulation frequency f, longitudinal separation d, transverse distance Ay and the phase difference
A0 between the leading edge of the downstream foil and the vortices in the wake of the upstream
foil are considered.
4.12).
We now define the performance index:
COT'
C77
(4.80)
that can be used to solve the very general problem of optimizing the body shape, swimming
speed and propulsion system. Co, is very similar, in spirit, to Toki's normalized cost
of transport mg/COT, and for a given fluid and fish mass, optimizing C,, is equivalent to
optimizing Tokis normalized COT. Even though the performance index could also be used
to solve the optimization problem presented in Section 4.4.3, its order of magnitude varies
widely with Reynolds number: C, ~ 106 for the examples considered in this paper. The
quasi-propulsive efficiency, with a natural scale going from 0 to 1, is much more intuitive
and easy to work with.
4.7.3
Proposed schooling theory and comparison with Weihs' theory
To the knowledge of the author, the only existing hydrodynamic theory of schooling has
been proposed by Weihs in 1973 [186]. Although this theory proposes some useful insight,
most arguments are based on time-averaged flows. By ignoring the unsteady nature of
fish wakes, Weihs probably over simplified the problem and overlooked key aspects of fish
schooling hydrodynamics.
Whether a fish swims directly behind another fish or with an offset that allows it to
benefit from a reduced flow velocity, A0 the phase difference between its undulation and the
wake vortices determines whether its drag is reduced or enhanced. By using the transverse
velocity of the individual vortices to accentuate the effects of its undulating motion, the
fish can reduce its drag, even when swimming in a region where the averaged flow is faster
than the free-stream. Conversely, swimming in the region of the wake where the flow is,
on average, slower, does not guarantee a reduced drag. However, we observed that it is
easier to reduce drag and save energy by undulating in the region of reduced velocity than
directly behind an other fish: up to 80% quasi-propulsive efficiency can be reached for a
foil undulating with proper phase in the region of reduced flow velocity, as summarized in
table 4.8.
If reduced drag means reduced undulation amplitude, the correlation with energy saving
is not as straightforward in a school as in open-water. Indeed, the vortices impact both
122
0.4
'
'
-0.4 ''
0
0.5
1.5
1
2
2.5
X
Figure 4-46: Snapshot of the vorticity field around a two-dimensional foil with a separate tail.
the drag and the swimming power. If the undulation amplitude was kept constant, phases
0 < AO < 0.5 would result in an increased drag and decreased power, and the reverse would
apply to phases 0.5 < AO K 1. The energy benefits of a reduced amplitude generally more
than compensate the increased swimming power, such that drag reductions tend to result
in power reductions. However, the phase resulting in the smallest amplitude is usually
not exactly the optimal one. In particular, the drag is mostly governed by the interaction
between the head of the fish and the vortices, whereas the power is mostly governed by
the interaction between these vortices and the tail where the transverse velocities are much
larger. The exact value of the optimal phase therefore depends on the undulation frequency
and the gait.
A fish undulating in a vortex street cannot be considered as a rigid body with a propeller
located in a jet. Regardless of the exact location of the fish in the vortex street, constructive interactions between the undulation and the individual vortices can result in enhanced
thrust, while destructive interactions result in enhanced swimming power. The exact value
of the optimal phase depends on the gait details, but in general the drag reduction configurations are the most advantageous, and it is easier to reduce drag when undulating in a
region of averaged reduced flow velocity.
4.7.4
Application to three-dimensional fish shapes
We have so far modeled a fish by a two-dimensional foil. However, fish have a highly
three-dimensional geometry. In particular, most carangiform and thunniform swimmers
are characterized by a region of reduced depth, around 20% from the trailing edge, called
peduncle. In order to model this region of reduced added mass with a two-dimensional
geometry, it might be more appropriate to model a fish with a separate foil for the tail, as
illustrated in figure 4-46. The fish model shown in this figure undulates with the optimized
gait at frequency f = 2.4 identified earlier, and the performance (rcp = 0.54) is very close
to that obtained with a single foil, indicating that the results are robust to changes in the
geometry.
In the rest of this section, we consider a simplified three-dimensional fish shape, shown
in figure 4-47, which is based on a giant danio (Devario aequipinnatu). For this geometry,
we fix the undulation frequency to f = 2.4 and optimize a Gaussian envelope for quasipropulsive efficiency (for a fixed swimming speed U, we minimize the expanded power P.
In figure 4-48 we compare how qQp changes with the envelope parameters x1 and 6 for a
two-dimensional foil and for the three-dimensional shape. The efficiency is generally lower
with the three-dimensional shape, but the dependency on x1 and 6 is very similar for both
geometries: the most efficient gaits are for 0.8 < x, < 0.9 and 0.2 < 6 < 0.3 with a sharp
decrease in efficiency for 6 < 0.2. This shows that, even though three-dimensional effects
123
(a)
(b)
Figure 4-47: Three-dimensional fish geometry based on a giant danio. Simulations are run 6 x 3 x 3
with constant velocity - = U on the inlet, a zero gradient exit condition with with global flux
correction and periodic boundary conditions along y and z boundaries. The Cartesian grid is uniform
near the fish with grid size dx = dy = dz = 1/100 and uses a 4% geometric expansion ratio for
the spacing in the far-field.
(a) 0.6
05
(b) 0.6
0.4
0.3
0.4
0.4
UO
rto
0.3
0.-3
0.35
0.2
0.1
0.6
0.5
0.45
0.5
0.8
X1
1
0.
06
10.3
02
0.-2
0.8
.
1
0.2
X1
Figure 4-48: 'rQp as a function of x, and 6 near the optimum for (a) 2D and (b) 3D geometries
with f = 2.4. The black dots show the location of the points that have been used to build the thinplate smoothing spline (tpaps function in Matlab with smoothing parameter p = 0.999) represented
in color.
reduce the swimming efficiency, most of the conclusions drawn from the two-dimensional
study extend to three-dimensional shapes for two-dimensional undulations.
The parameters and properties of the optimized gait for f = 2.4 are compared to those
of the carangiform gait in table 4.9. Like in the 2D case, the optimization decreases the
power consumption by 50% compared with the carangiform gait. As in 2D, the optimized
gait manages to bring the phase angle between the heave and pitch motion of the trailing
edge close to 900, which significantly reduces the angle of attack. As a result, the optimized
gait for the three-dimensional fish shape have a pitch angle, phase angle and angle of attack
very close to the optimized gait for the two-dimensional foil. However, since the 3D effects
reduce the thrust produced by the undulating motion, the Strouhal number is higher than
in 2D, especially for the carangiform gait.
Figure 4-49 shows the deformation envelope A(x) and the displacement envelope g(x)
for the carangiform gait at f = 3 and for the optimized gait. Despite a different mass
and added mass repartition along the length of the body, the displacement envelope for
the carangiform motion is very similar to the envelope observed in 2D (figure 4-27). The
superimposed body outlines for the optimized gait shown in figure 4-50b also look very
similar to the body outlines of the optimized motions in 2D: the deformation of the tail
follows the trajectory of the trailing edge, resulting in an efficient low angle of attack. The
body outlines for the carangiform motion, on the other hand, show that the pitch of the
tail is out of phase with its velocity (phase angle different from 90') which results in a very
inefficient gait with a large angle of attack.
124
f
3.0
2.4
6
x1
carangiform
0.84
0.26
ao
a
Omax(0)
amax( 0 )
0( 0 )
St
CP
?IQP
0.099
0.085
0.18
0.18
34
37
41
17
59
87
0.53
0.43
0.035
0.023
0.22
0.34
Table 4.9: Parameters and properties of 3D undulating gaits. Properties are the peak to peak
displacement amplitude at the trailing edge a, maximum pitch angle at the trailing edge Omax,
maximum angle of attack amax, heave and pitch phase angle 0, Strouhal number St, time-averaged
power coefficient Cp, and the quasi-propulsive efficiency rqp. The optimized gait at f
2.4 is
compared to the carangiform gait at f 3.
(a)
(b)
0.12
a A
0
g
0.08
L0.06
0.04
,,,
0.02
.
0
aA
E
E
0.04
0.02
-
0.1 --
g
0.1 --
0.08
O0.06 -
0.12
0.2
0.4
0.6
0.8
0
1
0
0.2
0.6
0.4
0.8
1
x
x
Figure 4-49: Prescribed deformation envelope aoA(x) and displacement envelope g(x) for (a)
carangiform gait with f = 3 and (b) optimized gait with f = 2.4.
Finally, we show the flow structure around the 3D fish model for both gaits in figure 4-51.
The performance difference between the two gaits is accompanied by noticeable differences
in the wake structure of the two swimmers. For both gaits, figures 4-51a and 4-51b show
wakes comprised of two interconnected vortex loops per cycle, together with other smaller
structures. In particular, the structure in the wake of the optimized motion is complex, with
many vortex tubes interlaced with each other. Indeed, as can also be seen in the vorticity
field at z = 0 (figure 4-51d), the deformation at the peduncle is quite large for the optimized
gait, resulting in vortex tubes separating from the main body and then interacting with the
structures shed from the tail.
Borazjani et al. [18] also observed in their 3D simulations that, for Strouhal number
greater than St = 0.3, the wake structure observed in 2D, dominated by a single vortex
pair (or ring in 3D), transitions to vortex loops wrapping around each other. Dong et al.
[45] showed that the same phenomenon happens to elliptical flapping foils of finite aspect
ratio: at low aspect ratio/large Strouhal number, two vortex rings are shed each cycle. As
the aspect ratio increases or the Strouhal number decreases, the tip vortices do not merge
together any more and the wake consists of interconnected loops. As the Strouhal number
(a)
Figure 4-50:
(b)
Superimposed body outlines over one undulation period for (a) the carangiform
motion and (b) the optimized gait.
125
(b)
(a)
(d)
(c) 0.4
--
10
02
1-2
-0.4 -
(e)
10
p
(f)
0.4[
0.15
I-
>0
0.09
0_________________
0.03
-0.03
-0.09
-0.15
-0.4
(h)
(g)M0.4- -h
0
-0.4
0
1
0.5
1.5
L
2''
0
1
0.5
1.5
2
x
x
Figure 4-51: Snapshots of the flow around a three-dimensional fish with (a,c,e,f) a carangiform
and (b,d,g,h) an optimized gait . (a,b): Three-dimensional vortical structures visualized using the
A2 -criterion; (c,d): z component of the vorticity in the z = 0 plane; (e,f): pressure in the z = 0
plane; (g,h): pressure in the z = 0.06 plane.
further decreases or the aspect ratio increases, the three-dimensional effects become even
weaker and the linkage between tip vortices disappears. At this point, the 3D wake looks
similar to the (reverse) Kdrma'n vortex street observed in 2D.
In the carangiform example shown here, the tip vortices merge, while with the optimized
gait, which has a lower Strouhal number and angle of attack, they do not. At higher
Reynolds number, the Strouhal number would be smaller and a wake similar to that observed
in 2D would probably emerge. Figures 4-51c and 4-51d show that near the tail, the vorticity
in the z = 0 plane looks very similar to what can be seen behind a 2D foil. However, under
the influence of the tip vortices, the vortex sheets shed by the tail do not evolve into two
strong vortices as in 2D. As a result, whereas the pressure field around the undulating fish
shape is very similar to the pressure around an undulating airfoil, the pressure signature in
the wake in the plane z = 0 is very weak. The pressure signature in the plane z = 0.06,
just above the peduncle, is much stronger, and could still be used by a downstream fish to
reduce its swimming energy.
Figure 4-52 shows a magnified view of the vortex structures generated by the carangiform
motion. A red line shows the formation of a clear vortex ring at the trailing edge of the
tail between figures 4-52a and 4-52c. In figure 4-52e, the vortex ring is fully formed and
detached from the tail. Since the vortex rings are oblique, they produce a large transverse
velocity, which is inefficient and is waste of energy. We also see a spanwise narrowing of the
126
(a)
(b)
(c)
(d)
z
y
0.5
1
1.5
0.5
1
1.5
x
x
Figure 4-52: (a,c,e) Side-view and (b,d,f) top-view of the vortex structures at several time-steps for
the carangiform gait. (a,b): t/T = 0.1 (mod 1); (c,d): t/T = 0.4 (mod 1); (e,f): t/T = 0.7 (mod 1).
A red line shows the formation of a vortex ring.
vortex rings as they convect downstream, as also observed in the simulations of Blondeaux
et al. [15] and Dong et al. [45] for a respectively rectangular and elliptical pitching and
heaving foil.
Figure 4-53 shows a magnified view of the vortex structures generated by the optimized
gait. The structure of the wake is more intricate than for the carangiform motion. In
particular, instead of one set of interconnected vortex tubes, there are two sets of tubes,
marked in red and green in the figure. The loop marked in red is the same as observed for
the carangiform gait, but at this lower Strouhal number, it never fully closes into a clearly
defined vortex ring. The tubes marked in green are formed upstream of the tail and are shed
from the body as a result of the large curvature at the peduncle. The resulting vortex tubes
are interlaced with the vortex loops from the tail with which they have a phase difference
of close to 1800.
For a three-dimensional fish shape with two-dimensional undulation, as for a twodimensional foil, the Strouhal number, pitch angle, angle of attack and phase angle at
the trailing edge are the key parameters for efficient swimming. The optimization results
in a low Strouhal number and angle of attack, which reduces the three-dimensional effects
observed behind the non-optimized gait, such as an inefficient oblique vortex ring. With
the optimized gait, the production of thrust is also distributed between the body and the
tail, both shedding vortex structures with opposite phase. Distributing thrust production
(or energy capture) is often used to increase the efficiency in turbines, fish might use the
same technique to improve their swimming efficiency. Finally, while we used a simplified
fish geometry with a two-dimensional body and caudal fin undulation, fish can also rely on
three-dimensional motion of their dorsal and pectoral fins to save energy [93, 47].
127
(a))
((d)
zy
0.5
1
1.5
0.5
1
1.5
x
x
Figure 4-53: (a,c,e) Side-view and (b,d,f) top-view of the vortex structures at several time-steps
for the optimized gait. (a,b): t/T = 0.1 (mod 1; (c,d): t/T = 0.4 (mod 1; (e,f): t/T = 0.7 (mod 1.
A red line shows a vortex shed from the tail that never fully develops into a ring, while green lines
show the vortices shed from the body.
128
Chapter 5
Conclusions
For engineered vehicles, navigating in the ocean is very challenging. Indeed, senses and
tools traditionally used above ground to help navigate tend to fail underwater. Even when
the water is clear, natural light does not penetrate more than 100 m. Spotlights can still be
used, but they need to have a very high intensity (drawing a lot of power) and their efficacy
is often limited by the scattering caused by particles in suspension. The Global Positioning
System (GPS) cannot be used either as the electromagnetic waves attenuate very quickly
underwater. Therefore, submarines rely almost entirely on sonar to detect obstacles. While
sonar has almost given eyes to submarines, especially in murky water, active sonars are often
bulky and expansive devices, that draw from limited power resources and are potentially
harmful to marine mammals [42, 84]. Moreover, sonar is far-sighted, with a blind zone all
around the vehicle. The lateral line of fish, on the other hand, is short-sighted. Therefore,
a sensory system combining both approaches would be able to map the entire space.
Another challenge, for underwater vehicles, is turbulence that disrupts their trajectory
and causes noise in sonar readings. Indeed, compared to air, water is a very heavy fluid and
its interaction with bodies results in complex flow motion, often associated with significant
pressure fluctuations. Whereas these fluctuations are often considered as noise by manmade vehicles, a better understanding of them would allow us to use them as fish do. For
instance, fish are able to save energy when swimming in a stream by taking advantage of
the pressure gradients and coherent vortices caused by obstacles or other fish [96]. They can
also detect prey and identify obstacles by measuring flow features through their lateral line
[142, 187]. These flow features, used by fish, result from interactions between several (often
deforming) bodies and water. They are all the more complex that the Reynolds number is
high because a very wide range of space and time scales are involved. Therefore, studying
these flows is challenging and requires the development of new methods.
5.1
Accurate Cartesian-grid simulations of bear-body flows
at intermediate Reynolds numbers
Chapter 2 generalizes the Boundary Data Immersion Method proposed in [195] by establishing a higher order analytic meta-equation. 2nd order BDIM provides a robust and accurate
treatment of IBs in high Reynolds number fluid/solid interaction problems. Our method
addresses the issues encountered by first-order methods (including direct forcing methods)
at high Reynolds number by adding a higher-order term to the traditional averaging used to
estimate velocities near the boundary. The resulting algorithm is both simple to implement
129
in existing Navier-Stokes solvers and computationally efficient.
Applications with Reynolds numbers ranging up to 105 are presented: viscous flow past
a static SD7003 airfoil and past a flapping NACA0012 airfoil, as well as flow around an
axisymmetric fish passing a cylinder. It is shown that the predictions of flow around a
slender body with a sharp trailing edge is very sensitive to the IB treatment and that
2nd order BDIM, with its new sharp edges treatments, can successfully predict it. 2nd
order BDIM has also demonstrated its ability to simulate highly unsteady flows without
encountering grid-locking issues as is the case with direct forcing methods [24]. The final
examples illustrate the ease of our method to handle three dimensional complex geometries
with moving boundaries.
A limitation of the present method lies in the necessity to resolve the boundary layer
using a Cartesian-grid in order to accurately predict the skin friction. Considerations of
computational cost caused us to limit our studies to Re < 10 5. Combining BDIM with
a local grid refinement technique [81] could improve computational efficiency for the thin
shear layers of higher Reynolds number flows. The use of a wall-layer model [138, 28, 25]
or tangential force model [139] to reduce the required near-wall resolution for very high
Reynolds numbers is an active area of research. At higher Reynolds number, the interactions
between wall-layer approximation and subgrid-scale model also need to be investigated in
the context of immersed boundary as has been done by Temmerman [156] for body-fitted
grids.
The ability of 2nd order BDIM to accurately simulate the viscous flow around complex
geometries up to Reynolds numbers of at least Re = 10 5 enables a wide range of exciting
applications from ocean energy extraction to animal and vehicle locomotion. The robust
and smooth simulation of pressures and forces are also especially important in resonant
marine systems such as tank sloshing, vortex-induced-vibration reduction, and investigation
of biological hydrodynamic sensors such as the lateral line and seal vibressa. As such we
believe this simple Cartesian-grid approach based on a strong analytical framework to be a
significant contribution to the accurate study of these and other highly non-linear viscous
flow systems.
5.2
The boundary layer instability of a gliding fish helps
rather than prevents object identification
The inspiration for the study presented in chapter 3 derives from the reported function
of the fish canal neuromasts for detecting pressure gradients. The model problem used in
the study, that of a rigid two-dimensional foil moving at a steady speed near a stationary
cylinder, is intended to represent a gliding fish mapping a stationary object, as observed by
[180].
In an inviscid formulation, potential flow can predict accurately the pressure induced
by the object on the foil and hence continuous pressure measurements at a finite number of
locations can yield the shape of the object [74, 56]. The experiments we conducted, however,
show that potential flow predictions are accurate only over the front half of the body and
deviate substantially over the posterior half, with large pressure oscillations present, as
shown in figure 3-3. Hence, potential flow predictions, although easy to obtain even in real
time, cannot be used. Whereas under certain conditions the pressure along the body of
the fish is not influenced by the viscosity [132], when moving in the proximity of objects,
it is affected by the viscous interaction between the body and the surrounding flow. When
130
moving toward or gliding parallel to a wall, the inviscid assumption predicts the correct
shape of the pressure changes but underestimates them [199, 200]. We show that for objects
of general shape, dynamic interactions between the boundary layer and the object generate
flow and pressure features that do not exist in an inviscid fluid. While there is no pressure
gradient across the thickness of the boundary layer, the velocity profile selectively amplifies
unsteady perturbations in the form of large vortices traveling at half the free-stream velocity,
creating large unsteady pressure variations.
Linear stability analysis of the average boundary layer profile in open water, i.e. in the
absence of any nearby obstacle, shows that the large pressure fluctuations in the boundary
layer consist of the pressure disturbances induced by the object, amplified through a convective instability of the flow; hence they are predictable. A methodology is established whereby
the potential flow predictions are used to drive an amplification function derived through
stability analysis. Without significant additional computations, the resulting model adds
to the potential flow pressure prediction a Reynolds number-dependent component caused
by the passing cylinder, featuring memory and amplification effects. The predictions agree
with viscous simulation results, reducing the error by a factor of 2 compared to the potential
flow model, even when the latter includes the steady displacement thickness. Such a model
can dramatically improve the performance of existing object identification algorithms [56]
and the ability of underwater vehicles to identify objects by measuring pressure fluctuations.
The unsteady pressure fluctuations predicted by linear stability analysis can enhance
detectability of the object only if they are combined with potential flow results, because
one must know the features of the signal that is being amplified. Therefore, the devised
methodology places importance on both the potential flow and the linear instability results.
While the features of the pressure induced by a stationary object within the potential flow
theory are rather simple and intuitive, the features of the selectively amplified disturbances
in the boundary layer are not, and hence would require animal learning in order to be
used for detection. There are, indeed, examples of animals training themselves to perform
complex tasks, such as trouts holding place in the vortical wake of bluff cylinders in steady
flow [97]. Given the simple decomposition of the pressure signal we show in this paper, it
is plausible that live fish could employ such self-training to detect and map nearby objects.
5.3
Swimming efficiency and drag increase for an undulating
fish
In chapter 4, an undulating tow-dimensional foil is used to model a fish swimming at
Reynolds number Re = 5000 and determine how efficiently fish can swim. But what is a
proper measure of efficiency for swimming fish? The net propulsive efficiency 7 1 , defined as
the ratio of the change in kinetic energy over the work done by the undulating fish, is a good
measure of performance for accelerating gaits. However, in steady state, ?In = 0, making
this measure useless to compare the performance of different propulsion modes or swimming
gaits. The hydrodynamic efficiency is not a good measure of optimality either, because it
relies on the ill-defined notion of drag and, far more importantly, its value depends on the
propulsion mode employed. The optimal propulsor for a self-propelled system is the one that
minimizes fuel consumption for a given body size and speed. If we consider that, in steady
state, the goal of the swimming motion is to keep the swimming speed constant (prevent
kinetic energy losses due to the drag on the non-swimming body), a natural measure of
efficiency is the quasi-propulsive efficiency rQp, defined as the ratio of the energy needed
131
to tow the rigid fish straight at a given speed divided by the power to self-propel itself
at the same speed. This is a rational non-dimensional metric of the system propulsive
fitness, which extends the definition of the net propulsive efficiency to low thrust cases.
A distinctive advantage of i]Qp is that it can easily be estimated in towed configurations
since the function rlQp(f) is mostly independent of the net thrust. For a given body and
swimming speed, maximizing qQp is equivalent to minimizing the expanded power P.
The notion of drag on a self-propelled body is ill-defined because pressure contributes to
both drag and thrust and there is no general way to separate one from the other. However,
at the Reynolds numbers considered, the friction drag is well defined. Whether the friction
drag increases or reduces when a the body undulates is a very controversial question. We
found that at Reynolds number Re = 5000, the friction drag increases with Strouhal number.
For a self-propelled foil or fish, the friction drag is about 50% higher than for a rigid towed
foil or fish. We also found that the swimming efficiency is not correlated with friction drag.
5.4
Swimming optimization for a fish in open-water
&
Whereas the displacement envelope of the body observed in swimming saithe and mackerel
can be approximated by a convex quadratic function, this displacement is the combination
of a recoil term and backward traveling wave with a peak deformation at the peduncle. We
show that, for a two-dimensional foil as well as a Danio-shaped body, such a deformation
envelope is necessary for efficient undulatory swimming. By changing the location and
width of the deformation peak, the Strouhal number, pitch angle and angle of attack can be
adjusted independently to ensure a low angle of attack. This result suggests that undulating
fish and underwater mammals separately evolved with a peduncle as a means to allow great
flexibility in this region, which allows them to swim efficiently. Widespread undulations are
good for producing a lot of thrust, which is particularly useful for accelerations or when a
low undulation frequency is used. Undulations localized at the tail are more efficient when
the undulation frequency is large and acceleration is not needed. By tweaking the width and
location of the undulation amplitude peak, the foil can undulate very efficiently regardless
of the frequency: at fixed Reynolds number, for all frequencies, the Strouhal number, pitch
angle and angle of attack of the optimal gait are the same. While the quasi-propulsive
efficiency for the gait traditionally used to model carangiform swimmers does not exceed
40%, the optimized gait reaches qQp = 57%, thanks to an improved propulsor efficiency
and a reduced pressure drag amplification. We observed that the friction drag enhancement
due to body undulations, around 40% at Re = 5000, increases with Strouhal number and
wavenumber, but is not directly correlated with the efficiency of a specific gait.
The swimming efficiency of an undulating (3D) fish is also governed by the Strouhal
number, pitch angle, angle of attack, and the phase angle between the heave and pitch
motion of the tail. Therefore, the qualitative conclusions about efficiency drawn from a
2D foil also apply to actual fish. The resulting wake has a periodic 3D structure with
coherent vortices that another fish can use to save energy by properly timing its motion,
just like ibises have been observed to do [129]. However, the three dimensional flow around
a fish is much more complex than the flow around a two-dimensional flow. Since the
three-dimensional effects mostly result in a loss of efficiency, the optimization reduces these
effects while distributing the production of thrust between the body and the tail (resulting
in 1QP = 34%). But there are more ways for a fish to save energy, as reviewed by Fish
Lauder [57] and Fish [58]. In particular, there is evidence that fish move their pectoral [93]
132
and dorsal [47] fins in a complex manner in order to generate vortex structures that will
interact with their caudal fin. And since all these fins are flexible, fish can further improve
their efficiency by controlling the flexibility of their fins [212, 12]. Therefore, a lot can be
learned by combining hydrodynamic simulations with structural and muscle models as in
Toki6 et al. [160].
5.5
Energy saving by swimming in pair
The swimming power can be further reduced by swimming in a group. To test this hypothesis, we considered two foils undulating in various configurations. Whereas the widely used
schooling theory from Weihs [186] predicts that a fish swimming directly behind another
fish would experience increased drag and have to expend more power than in open-water,
we show that this is only partially true: it all depends on the phasing of the undulating
motion with respect to the vortex street. Regardless of the location of the foil in the wake,
when vortices constructively interact with the foil motion, its thrust increases, which can
be used to increase the efficiency, but a bad timing leads to enhanced drag and swimming
power. However, results partly confirm Weihs' intuition, since we found that energy savings
are maximized when simultaneously extracting energy from individual vortices and taking
advantage of averaged reduced flow velocity. For foils undulating at the non-dimensional frequency f = 1.5, the most efficient gait identified in open-water has an efficiency 1Qp = 0.49.
The swimming efficiency can go up to 7Qp = 0.64 when undulating directly behind a similar
foil, and to qQP = 80% when swimming in the reduced-velocity region of the wake.
5.6
Summary and future work
Using a robust Immersed Boundary Method, I have derived and implemented a second
order boundary treatment that significantly improves the accuracy of intermediate Reynolds
number simulations. The new method, referred to as 2nd order BDIM, allows accurate
simulation of flow past a streamlined body, deforming or not, at Reynolds number up to
Re = 105. It can even predict the transition of the flow from laminar to turbulent along the
immersed boundary, which is a substantial achievement. Being able to accurately simulate
the flow around several three-dimensional moving bodies at Reynolds number up to Re =
10 5 is a significant improvement over existing methods, but further development is need to
achieve even higher Reynolds numbers. Indeed, the Reynolds number of underwater vehicles
and most adult fish and marine mammals is still several orders of magnitude larger than
that. The constant increase in computational power will make it possible to simulate higher
Reynolds numbers, but appropriate numerical methods will also be necessary. Turbulence
models that perform well near walls, in particular, are still missing. Moreover, unless a
reliable way to model turbulent boundary layers with relatively coarse grids is developed,
local grid refinement techniques will be needed.
Using 2nd order BDIM, I have first investigated hydrodynamic aspects of object detection through a lateral-line-like sensor. I have identified two mechanisms through which a
stationary object can affect the signal measured by the lateral line of a fish passing next
to it and proposed a numerically inexpensive method for estimating them, which could
be useful for real-time object identification. The first mechanism can be entirely modeled
by a potential flow. This inviscid disturbance, caused by the deflection of the streamlines
around the object, can then excite unstable modes in the boundary layer of the gliding
133
foil. As a result, some frequencies get amplified, but since it happens over a short distance,
the amplification does not single out one frequency. The resulting signal, measured by the
lateral line, is a function of the boundary layer, that can be predicted by linear stability
analysis of the boundary layer, and the object properties (size, location, shape). In other
words, the boundary layer works as a wide-band signal amplifier, where the signal to be
amplified can be estimated by the potential flow theory, and the amplifier properties result
from the linear stability analysis of the boundary layer. The boundary layer, despite causing
turbulence, could facilitates object detection and identification. We are proposing here a
tractable and accurate hydrodynamic model of the interaction between a foil and a cylinder.
The method could easily be applied to three-dimensional vehicle (or fish) shapes and more
complex geometries, since changing the geometry does not alter the hydrodynamic mechanism. Coupled with surface mounted pressure sensors or an artificial lateral line, this model
opens the possibility of obstacle local detection and identification for underwater vehicles.
However, this is a very arduous task: even blind cave fish, that excel at using their lateral
line, sometimes fail to avoid swimming into walls. An artificial lateral line would be best
used as an additional sensor. Merging input from sonar, vision and local pressure could
give access to a complete knowledge of the environment. Fish themselves are known to rely
on several sensory cues for most their behaviors, hearing, vision, smell and the lateral line
being among the major ones.
I have then used 2nd order BDIM, to study efficient swimming of single and paired
fish. Using the quasi-propulsive efficiency as a performance indicator I have optimized the
undulation gait under the assumption of given body size and swimming speed. I have shown
that recoil effects are significant and that, when recoil is allowed, the deformation is largest
at the peduncle for efficient swimming. The exact shape of the optimal envelope depends on
the wavelength and frequency chosen: for low frequency, large oscillations along the whole
body (widespread oscillations) are necessary in order to generate enough thrust; at higher
frequency, large deformation only near the peduncle region (deformation localized at the
tail) are best. Through modifications of the amplitude envelope peak width and location,
the three parameters driving undulation efficiency can be adjusted independently. These
parameters are the same as those identified for rigid flapping foils, namely heave, pitch and
angle of attack at the tail. Quasi-propulsive efficiency up to 57% can be reached for a foil
in open water (34% for a fish), whereas for the envelope traditionally used to carangiform
swimming, it cannot exceed 40% (22% for a fish). Swimming in pair allows further increase
in efficiency. By simultaneously taking advantage of the regions of reduced velocity and
extracting energy from the vortices, up to 80% efficiency can be reached. This requires that
both fish undulate at the same frequency and that the downstream fish carefully phases its
motion with respect to the wake vortices.
Fish have developed many mechanisms to save energy and exchange information through
hydrodynamic interactions; I have contributed to shed light on two of them. Other interesting questions include how the interactions between the various fins of a fish can be used for
enhanced efficiency and maneuverability, and how tweaking the stiffness of the fins can help
take advantage of the flow/structure interaction. Fish schools are also known to respond
very quickly to the presence of a predator. How information travels in a school is another
intriguing question. Unveiling how fish take advantage of their surrounding medium can
help discover new paradigms for the design of robust and efficient underwater robots, but
there are still many open questions.
134
Appendix A
Convolution evaluation at sharp
corners
In Section 2.2.2, kernel moments are analytically evaluated in order to accurately immerse
the boundary data from the solid mechanical system onto the fluid mechanical equations of
motion. At corners (like sharp trailing edges), the smooth interface assumption introduced
at Eq. 2.13 does not hold. Here, the case of a geometry with sharp corners is locally treated
as the intersection of two component planar geometries.
Let us consider a point 7 near a two-dimensional corner defined by two planes. We call
wi (with normal n', and distance dl from 7) the wall closest to 7 and w 2 (with normal n-2
and distance d2 > d, from 7) the other wall. The angle between the two planes is 0. Figure
A-1 shows a schematic of the geometry and variables.
d
nj
Figure A-1: Schematic showing the variables used in the derivation of the convolution evaluation
at sharp corners.
Let us define a local coordinate system centered in 7 such that . =
this local coordinate system, the equation of w2 is:
1 - a2 x + ay + c = 0
(
ni and i -i2 > 0. In
(A.1)
.
with normal n
1 - a2 , a). Therefore, a = i 1 n2. We also know that the point -d 2i
belongs to w 2 , therefore c = d2
135
We can now write:
(U+,
=
fc
Job
b(-,
Xb) bX
b
(
b(it,Y)
~(,z
b
- ) K(5,
, )d b
(K(, d-
d-b+ (,x)
KE(,)
(A.2a)
-db
-
)d
,)b (A.2b)
(A.2c)
B
eB
Using the previously defined local coordinates and assuming a 2D problem (and kernel), we
can write (use d = di):
puoGF) =
Lb KE (zsb) dXb
oB(z)
jidi
(A.3a)
1
ji22
y-eF fx=-VE2_y2
=@(y)#0,(y)
22
(X,Y)<0
(Y)
2
2VE -
dx dy
(A.3b)
y2
(A.3c)
dy
where
1
d2(X,Y)<O =
lx<-(ay+c)/
-a
2
(A.4)
therefore
/min(Ne2_-2,
Jmin(-
2
-(ay+c)/
e2_y , -(ay+c)/v'
=max 02+min
'2
I
ay -c
()
1-aI) 2
(A.5a)
1=dx
62
_2
(A.5b)
(2'
2Vce2 __
922/1I -2
)]
For most functions 4, the integral above does not have a closed form solution. To simplify
2 __-y 2 by E. This is equivalent to
the equation, the integral is simplified by replacing
assuming a kernel with square support instead of circular. Therefore
(y) ~ max [0, - + min
2
(-,2'
1-
U2
2cv/1 -a2
(A.6)
where
0(y) = 0.5 (1 + cos (yrr/c))
136
(A.7)
Similarly,
-Z1(-) =
r
pc'B(_)
=
(A.8a)
(Yb - -) K(i, -b) d-b
V
y=-e
2
x=-
x i + yj)
d2(Xy)<
2
()
e2 -_y 2
2
1-i(i(y)i + O.Eyj
(A.8b)
dx dy
(A.8c)
0,(y) dy
where
-
2
min
(1
-
(ay + c) 2
a2)(2 y 2
-
(A.9a)
11
)
'
2
0
0,
'
C*
-min
2
(ay + c)2
(1 - a2)C2
1i
(A.9b)
These equations minimize the modeling error near sharp corners, as shown in 2.3.2.
137
138
Appendix B
Derivations for the
one-dimensional channel flow
Here we detail some derivations for the unsteady one-dimensional channel flow example
analyzed in Section 2.2.3.
B.1
Exact solution
The exact solution to the unsteady one-dimensional channel flow at Reynolds number Re =
L 2 /(Vt) is
u(y, t) = U E
e-(2k+1)2 r2 /Re sin
((2k + 1)7ry/L).
(B.1)
k=O
The sum converges very rapidly and summing over the first 50 terms guaranties an error
smaller than 10-10. For Reynolds numbers larger than 100, the velocity in the middle of the
channel is not affected by the boundaries, and the 99% boundary layer thickness is given
by
(B.2)
699 = 3.65/vRe.
B.2
Direct forcing solution
We will now derive a direct forcing formulation (described in [207, 170]) of this example.
Using the same notations as in Section 2.2.3, the velocity in direct forcing is expressed as
nC(y, to + At) = f(u, y, to + At) + g(u, y, to + At)
(B.3)
where g is a volume force ensuring that u, = 0 at the boundary (d(y) = 0). The volume
force is evaluated using a regularized delta function 6,
G(Y)6(y
g(u, y, to + At)=
-
Y) =-
E
YC{O,L}
YE{O,L}
139
F(Y, to + At)6E(y
-
Y)
(B.4)
where capital letters (F and G) refer to values at the boundary. These values are interpolated from the Cartesian grid points using the regularized delta function
F(Y, to + At)
f(y', to + At)& (y' - Y)
=
(B.5)
In summary, the direct forcing formulation is
Ue(y, to + At)
f(u, y, to + At) -
f(y', to + At)& (y' - Y)
6S(y - Y)
YE{O,L}
(B.6)
Y
where the discrete delta function is & (y) = #f (d(y), 0) dy for the kernel 0, defined by Eq.
2.15. Defining the column vector (
6 (y) + 6, (y - L), the matrix form of the direct forcing
formulation is:
(B.7)
U6 (nAt)
([I - (c(f] [I + At vD (2)]) U.
B.3
Limiting case v
0
The fixed point solution of the limiting case v = 0 for 1st order BDIM is given by the
equation
(B.8)
Pouf.
Similarly, the fixed point solution for the direct forcing method is given by
UC =
[I - (C(T] UE.
(B3.9)
In both cases, the fixed point solution verifies u,(y) = 0 for Id(y) < e.
For the proposed 2nd order BDIM, the fixed point solution is given by
UE
=
pE
+ PC
UE
(B. 10)
In this case, the fixed point solution for jd(y)j < e is
that
(B. 11)
we (y) = A exp
y
where the constant A = exp(-1) ensures that u,(c) = 1.
140
,
Appendix C
Varying-coefficient model
Varying-coefficient models form a locally parametric family of structured models that assume the form of the multivariate regression function as:
g(s, z) = zt a(s).
(C.1)
Varying-coefficient models can be seen as a generalization of linear regression in which
the coefficients are adjusted locally. By reducing the dimensionality of the functions that
need to be identified, structured models are a popular way to avoid the difficulties of large
dimensions. If the data to model indeed has the structure of Eq. C.1, then a varying
coefficient model can significantly decrease the variance while not increasing the bias, and
is therefore expected to fit better than an unstructured model such as Eq. 3.17. There are
several approaches to estimate a(s) in Eq. C.1, among which kernel-local linear regression
[55], which has been selected here for its simplicity.
The model from Eq. C.1 is estimated from a data set consisting of n triplets (si, z , yj).
3
The local linear estimator &(s) is calculated by minimizing:
n
[yj - z'a - z'B(sj - s)
L(a, B) =
2
Kh(WIsj - s11),
(C.2)
j=1
-
where Kh(t) = K(t/h)/h with K the unit Gaussian kernel. In the case of Eq. 3.16, sj
(k3 ,tj) with dimensionality ds = 2, z3 = Pi(kj, tj, Cj) with d, = 1, and y3 = IP(k3 ,ti, C)
Pi(k3 , tj, Cj)1. a is a dz-vector while B is a dz x d, matrix. The smoothing parameter h is
chosen by 10-fold cross validation.
Let us define:
n..,z ], y =[yi,. .n ., it
Z = zi, .
S= [si - s,.. . ,s7 - s], w, = diag[Kh(JIsI - s),.
F = [Z,diag[S,(1,:)]Z,. .. ,diag[S,(d,,:)]Z]
..
,Kh(1s
- sI)]
The local linear estimator solution to the optimization problem Eq. C.2 is:
a(s) = [Id2 , 0(d
where Id. is a size dz identity matrix and
8,)]
(tw ,f,)
8
-
,w,, y
(C.3)
0(dz,ds) a size dz x (dz * d,) matrix with each entry
being 0.
141
d
0.8
0.4
0.2
0.1
0.05
0.025
5
9
17
27
39
49
5
9
14
21
27
33
5
7
11
14
17
18
U.4
U.8
4
6
7
9
4
6
-
.2
-
U.1
-
-
M.UD
-
I
Table C.1: Number of time steps used for each cylinder radius r and distance d.
The viscous simulations used to estimate & consist of cylinders ranging in radius r
from 0.025 to 0.8 and in distance d from 0.05 to 0.8. For each cylinder, wavenumbers
k = 27r[2, 3,..., 8] are used, as well as a number of time steps proportional to the distance
between the centre of the cylinder and the foil. Table C.1 shows the size and distance of
the cylinders used to estimate h, as well as the number of time step for each cylinder. This
represents a total of 363 x 7 = 2541 data points, half of which were randomly assigned to
the learning set, while the other half was only used for testing and to compute the test error
reported in table 3.2.
142
Appendix D
Validation: flow-induced vibration
of a circular cylinder
The flow-induced vibration of an elastically constrained two-dimensional cylinder is a canonical fluid-structure interaction problem. In this example, the cylinder of diameter D and
density Pc is constrained to move transversely to a uniform free stream U, as illustrated in
Figure D-1. The structural stiffness and damping ratio are designated by k and b, respectively. In the following, all the physical variables are normalized by the cylinder diameter
D and the oncoming flow velocity U. The non-dimensional cylinder displacement, velocity
and acceleration are denoted by
(, (
and
(,
respectively. The sectional in-line and cross-flow
force coefficients are defined as
C
=
1
(D.1)
CY =
and
U
DU2
- DU2'
where Fx and F. are the in-line and cross-flow dimensional sectional forces exerted on the
body by the flow. The body dynamics is a governed by a forced second-order oscillator
equation, which can be expressed in the following non-dimensional form:
(D.2)
m*+ b* + k*( = Cy,
where
b
,
pUD'
7rPC
m*=__b
2 p'
b
k*=
k
k
pU2
.
y
U, p, V
k
b
/T//1
/T
Figure D-1: Sketch of the flow-induced vibration problem.
143
(D.3)
Study
Shiels et al. [148]
Shen et al. [146]
Bourguet & Lo Jacono [20]
BDIM, ma = 0
BDIM, ma = 10
(')max
f
CX
(Cy)nax
0.58
0.57
0.57
0.57
0.58
0.196
0.190
0.188
0.193
0.194
2.22
2.15
2.08
2.04
2.06
0.77
0.83
0.88
0.76
0.76
Table D.1: Maximum amplitude of vibration, vibration frequency, time-averaged in-line force
coefficient and maximum cross-flow force coefficient for a flexibly mounted cylinder at Re = 100, for
m* = 2.5, b* = 0 and k* = 4.96, from previous work and BDIM with several virtual added mass
values m,.
With the cylinder centered (on average) at (x, y) = (0, 0), a computational domain extending from x = -8D to x = 16D and y = -10D to y = lOD is used. The Cartesian grid
is uniform near the undulating cylinder with grid size dx = dy = 1/60 and uses a 2%
geometric expansion ratio for the spacing in the far-field. Constant velocity u = U is used
on the inlet, periodic boundary conditions on the upper and lower boundaries, and a zero
gradient exit condition with global flux correction. The flow and structural parameters used
in this study are:
Re = UD/v = 100,
m* = 2.5,
b*
0,
k* = 4.96.
(D.4)
The results are compared to a vortex method [148], the spectral/hp element method [20]
and an immersed boundary method [146] in table D.l.
While the theoretical added mass ration for a circular cylinder is 1, this value is small
enough that the fluid/structure interaction scheme is stable without the use of virtual added
mass (ma = 0). Table D.1 shows that the actual value chosen for the virtual added mass
does not significantly impact the results as the values found using m" = 10 are very similar
to those calculated with ma = 0.
144
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