Swimming without a Spine: Hydrodynamics and Swimming
Performance of some Marine Invertebrates
By Zhuoyu Zhou
A dissertation submitted to Johns Hopkins University in conformity with the requirements for the degree of Doctor of Philosophy
Baltimore, Maryland January, 2018
© 2018 Zhuoyu Zhou All Rights Reserved ABSTRACT
The primary focus of the present study is to employ computational modeling to investigate the hydrodynamics of free-swimming marine invertebrates. A high-fidelity computational tool based on a sharp interface immersed boundary method (ViCar3D), is developed, and this solver incorporates a non-inertial reference frame treatment in order to significantly reduce the computational cost of these simulations. The analysis includes the details of the wake characteristics and correlation with thrust generation mechanisms, as well as the swimming performance evaluated by coefficient variety of metrics.
Simulations of free-swimming of three distinct marine invertebrates - the Spanish Dancer,
Aplysia and the marine Flatworm, are performed. These animals are known to be active and effective swimmers and exhibit swimming gaits including body/mantle undulation and/or body bending, which are generally representative of a wide range of soft-bodied swimmers. Simulation show that despite a somewhat ungainly swimming motion, the swimming speed and propulsive efficiency of the Spanish Dancer are quite comparable to other more proficient swimmers. For the
Spanish Dancer, a body planform with a wider caudal region has better swimming performance and this might explain the planform shape typically found for these animals.
This importance of body-bending become apparent when examining free-swimming in marine
Flatworms with two kinematic models- pure lateral flapping (LF) and combined lateral flapping and body pitching (CFP). While the body pitch magnitude is quite small (with maximum deflection angle of 15°), this small addition results in a significantly larger swimming speed (with increments ranging from 16% to 121% depending on the phase differences between lateral flaps and body pitch) and a higher Froude efficiency with increases ranging up to 43%.
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For the Aplysia, it is found that animals swimming with kinematics that match field videos have high propulsive efficiency and a relatively high swimming speed. By examining the Froude efficiency and power coefficient for various body planforms and kinematics, it is found that animals that employ the LF gait fall into one group, whereas other animals that employ the CFP gait fall into another, regardless their body shape. In general, the CFP gait is found to be more effective for swimming than the LF gait.
Wake characteristics of the free-swimming of these animals are also analyzed. A bifurcated train of vortex rings is identified within the wake of the Spanish Dancer, and these vortex rings are found responsible for thrust generation. Other vortices in the wake are found to be drag producing.
In the wake of the Aplysia, three distinct trains of vortex rings are identified in addition to other spanwise vortex structures resembling the Karman vortex street. For some models of Flatworms, the addition of a small body pitch was found to significantly change the wake topology, and a well- organized wake with distinct vortex rings is found to be associated with improved swimming performance.
The current study provides a first-of-its-kind view of swimming in invertebrates and the comparative analysis performed here could provide insights into how these animals have adapted for life under water. The current research could also provide data and insights for the design of bioinspired soft swimming robots.
Advisor: Rajat Mittal
Reader: Jung-Hee Seo
Cynthia Moss
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ACKNOWLEDGEMENTS
Now, as I write down this very last part of the thesis, I realized that my student-life has come to an end. Memories of the pleasures and vexations over the years have come alive. I can’t be more grateful to those loving, kind people that have helped me through the difficult and confusing times.
First and foremost, I would like to express my utmost gratitude to my advisor Prof. Rajat Mittal for his continuous support over the years, this piece of work would not have been possible without his insightful guidance, vast patience, and constant encouragement. He has set an example of excellence as both a scientist and a mentor. As a scientist, an expert in computational fluid dynamics, Prof. Mittal has not only a very detailed know-how of the mathematical formulas and
CFD algorithms but also a profound understanding of the fundamental physics. As a mentor, he always puts the interests and growths of students in the first place. He cares about the future of his students. He taught me the importance of working with a can-do attitude. I simply could not imagine a better advisor.
Second, I would also like to thank Dr. Jung-Hee Seo at the Flow Simulations and Analysis
Group, who also served as my thesis reader, and Dr. Kourosh Shoele, for many invaluable, thought-provoking discussions throughout my doctoral research. I am also thankful to Dr. Chao
Zhang for walking me through the code when I first joined the FSAG. Thanks also go to other fellow graduate students, colleagues in the group and department, as well as our collaborators at
Georgia Tech. In addition, I would like to thank Drs. Jung-Hee Seo and Cynthia Moss for providing valuable comments on the thesis.
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Third, I would like to thank Mr. Kuan Xing who always replied promptly to my requests for the sketches of the animals presented in this work, Mr. Harry Blalock for allowing us to use his video of the Spanish Dancer, Mr. Simone Carletti, Mr. George Skeparnias, and Dr. Matteo Charlie
Ichino for their videos of the Aplysia.
Finally, special thanks are given to my family for their unconditional love through my life.
They have always been supportive of my decisions and encouraged me to pursue a higher education.
This work was supported by NSF Grant PLR-1246317. The flow solver developed here benefited from support from NSF CBET-1511200 and IIS-1344772, and the JHU Provost
Discovery Award.
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TABLE OF CONTENTS
ABSTRACT ...... II
ACKNOWLEDGEMENTS ...... IV
TABLE OF CONTENTS ...... VI
LIST OF TABLES ...... IX
LIST OF FIGURES ...... XI
CHAPTER 1. INTRODUCTION ...... 1
CHAPTER 2. NUMERICAL METHOD ...... 14
2.1. GOVERNING EQUATIONS AND DISCRETIZATION SCHEME...... 14
2.2. IMMERSED BOUNDARY TREATMENT ...... 17
2.3. FLOW-INDUCED MOTION ...... 20
2.4. VALIDATIONS OF THE NUMERICAL METHOD IN NON-INERTIAL REFERENCE FRAME . 22
2.4.1. Inline oscillations of a circular cylinder...... 23
2.4.2. Freely falling sphere under gravity ...... 26
CHAPTER 3. METRICS FOR SWIMMING PERFORMANCE ...... 31
CHAPTER 4. SWIMMING HYDRODYNAMICS OF THE SPANISH DANCER .... 35
4.1. INTRODUCTION ...... 35
4.2. METHODOLOGY ...... 36
4.2.1. Kinematic Model ...... 36
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4.2.2. Pitching Moment Balance Study ...... 44
4.2.3. Simulation Setup ...... 47
4.2.4. Grid Convergence ...... 48
4.3. RESULTS ...... 50
4.3.1. Effect of Stokes Number on Swimming Performance ...... 50
4.3.2. Wake Topology ...... 54
4.3.3. Implications of Planform Shape on Propulsion ...... 59
4.4. SUMMARY ...... 63
CHAPTER 5. THE WAKE TOPOLOGY AND SWIMMING PERFORMANCE OF
THE SEA HARE (APLYSIA)...... 64
5.1. INTRODUCTION ...... 64
5.2. METHODS ...... 67
5.2.1. Body Geometry and Swimming Kinematics ...... 67
5.2.2. Simulation Setup and Grid Refinement Study ...... 71
5.3. RESULTS ...... 73
5.3.1. Wake Topology ...... 73
5.3.2. Surge Force Distribution ...... 80
5.3.3. Effects of Kinematic Parameters on Swimming Performance ...... 83
5.3.4. Ground Effect...... 87
5.4. SUMMARY ...... 93
CHAPTER 6. SWIMMING PERFORMANCE AND HYDRODYNAMICS OF
MARINE PLATY(FLAT)HELMINTHES(WORMS) ...... 95
6.1. INTRODUCTION ...... 95
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6.2. METHODS ...... 98
6.2.1. Body Geometry and Swimming Kinematics ...... 98
6.2.2. Simulation Setup ...... 102
6.3. RESULTS ...... 102
6.3.1. Wake Topology and Swimming Performance of Flatworms ...... 103
6.4. SUMMARY ...... 117
CHAPTER 7. PROPULSIVE EFFICIENCY OF MARINE INVERTEBRATES .... 118
7.1. SUMMARY ...... 124
CHAPTER 8. SUMMARY ...... 125
LIST OF REFERENCES ...... 128
VITA ...... 144
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LIST OF TABLES
Table 1-1 Swimming Classification of Marine Invertebrate Animals ...... 8
Table 4-1 Kinematic parameters employed in the modeling the swimming of the Spanish Dancer.
...... 42
Table 4-2 Mean pitching moments computed with various control parameters ...... 45
Table 4-3 Results of grid refinement study for Re = VL/ν = 645 ...... 50
Table 4-4 Comparison of terminal velocities and Froude efficiency at different Stokes numbers
...... 52
Table 4-5 Comparison of terminal velocities and computed Froude efficiencies for different
planforms ...... 62
Table 5-1 Mean thrust coefficients and RMS values for various grid sizes ...... 73
Table 5-2 Kinematic parameters employed in modeling the swimming of Aplysia and the
corresponding swimming performance...... 83
Table 5-3 Swimming performance of Aplysia at different ground proximities ...... 92
Table 6-1 Kinematic Parameters of Flatworms Swim by Pure Lateral Flapping (LF) ...... 102
Table 6-2 Kinematic Parameters of Flatworms Swim by Combined Flapping and Pitching (CFP)
...... 102
Table 6-3 Comparison of swimming performance for various kinematics ...... 117
Table 7-1 Kinematic Parameters for BP1 model...... 119
Table 7-2 Kinematic Parameters for BP2 model...... 119
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Table 7-3 Details of Kinematic Models and their Corresponding Representative Morphology
Investigated in this Study ...... 120
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LIST OF FIGURES
Fig. 1-1 Biomimetic autonomous underwater vehicles (AUV’s)...... 1
Fig. 1-2 Fish swimming modes classified based on movement characteristics and grouped by
propulsive structures: (a) BCF propulsion and (b) MPF propulsion. Shaded areas are the
propulsors. Figure from Sfakiotakis et al. (1999)...... 2
Fig. 1-3 Cladogram illustrating the relationship between various species of marine gastropods and
Flatworms...... 7
Fig. 1-4 Classification of swimming gaits of marine gastropods: (1) Pleurobranchus membranaceus
(http://www.nudibranch.org/Scottish%20Nudibranchs/html/pleurobranchus-
membranaceus-05.html), nf; (2) Hexabranchus sanguineus (Blalock 2008), lb and nf; (3)
Aplysia depilans (http://www.projectnoah.org/spottings/2138017), nf; (4) Melibe leonine
(https://www.joelsartore.com/keyword/hooded-nudibranch/), lb; (5) Limacina helicina
(https://en.wikipedia.org/wiki/Limacina_helicina#/media/File:Limacina_helicina_7.jpg
...... 9
Fig. 1-5 Examples of bioinspired soft-robots for underwater applications...... 11
Fig. 2-1 Schematic describing the naming convention and location of velocity components employed
in the spatial discretization of the governing equations (From Mittal et al. 2008). . 16
Fig. 2-2 Schematic of two-dimensional spatial discretization and immersed boundary detection
(From Mittal et al. 2008)...... 17
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Fig. 2-3 Schematic of simulation setup, thick edges denote the outer boundary of the computational
domain, brown shade is the immersed body with surface B , and the unit normal vector nˆ
pointing from the fluid into the body...... 21
Fig. 2-4 Time history of the drag coefficient (solid line) from the present simulation compared with
that of Kim & Choi (2006) (dotted line)...... 23
Fig. 2-5 Comparison of the absolute inline velocity component (ua) and vertical velocity component
(va) at three different positions at the same phase angle 2 ft obtained from the
current study (solid line) and the reported data by Kim & Choi (2006) or Dutsch et al. (1998)
(dotted lines): (a) inline velocity at xa = -0.6D; (b) inline velocity at xa = +0.6D; (c) inline
velocity at xa = 0; (d) vertical velocity at xa = -0.6D...... 24
Fig. 2-6 Pressure contours (negative values dashed) in the inertial reference frame at Re = 100 and
KC = 5. Left column: present study; right column: Dutsch et al. (1998). Top: phase angle ϕ
= 0°; bottom: ϕ = 288°...... 25
Fig. 2-7 Vorticity contours (negative values dashed) in the inertial reference frame at Re = 100 and
KC = 5. Left column: present study; right column: Dutsch et al. (1998). Top: phase angle ϕ
= 0°; bottom: ϕ = 288°...... 26
Fig. 2-8 Temporal evolutions of the velocity in the gravitational direction for three cases with various
density ratios and Reynolds numbers: (a) sf/ = 7.85, Ret = 435; (b) = 2.56, Ret =
369; and (c) = 2.56, Ret = 42...... 28
Fig. 2-9 Instantaneous vortex structures at Ret = 435 (a-c) and Ret = 42 (d-e): (a-c) tuc / d = 90.0;
91.6; and 93.2; (d-e) =48.1 and 72.1...... 29
Fig. 2-10 Instantaneous vortex structures at Ret = 435: (a) =335.4; (b) 352.5; and (c) 366.2.
Figure from Kim and Choi (2006)...... 29 xii
Fig. 4-1 External morphology of the Spanish Dancer (Hexabranchus sanguineus)...... 36
Fig. 4-2 Sequence of the Spanish Dancer swimming with the flapping frequency ~0.5 Hz. Pictures
showing every 0.2s, from left to right, top to bottom (Blalock 2008)...... 38
Fig. 4-3 Schematic of the reconstruction of the kinematics model based on a rectangular plate: (a)
surface of the flat plate consisting of a central structure with a head and a trunk performing
dorso-ventral bending, and a large mantle margin responsible for wave-like flapping; (b)
plate shape described by eq. (2) & (3) of the pitching motion at t = 0; (c) plate shape described
by eq. (1) of the undulatory motion at t = 0; (d) Superimposition of the undulatory and the
pitching modes. Red dots denote the fix point of the prescribed motion which is the
intersection of the body centerline and the pitching axis...... 41
Fig. 4-4 Comparison of the recreated motion of the Spanish Dancer (column 2 & 4) with the
screenshots of a freely swimming H. sanguineus (column 1 & 3) from Blalock (2008). The
central axis of the body is nearly in a straight line at the beginning of the stroke and the stroke
begins with the pitching down of the head and the simultaneous pitching down of the trunk.
The mantle has a characteristic “cosine” shape at this initial stage and as the head and trunk
approach each other, the mantle wave propagates backwards...... 43
Fig. 4-5 Time history of the pitching moment coefficients. Solid line: prescribed baseline case in
initially quiescent fluid; dashed line: case 7 with optimized kinematics. The mean pitching
moment of for the baseline case is about -0.16, whereas that of the optimized case is less than
-0.01...... 45
Fig. 4-6 Comparison of the swimming kinematics of two cases at 5 consecutive stages: (a) baseline
case with visually-matched kinematics; (b) case 7 with smaller pitching amplitudes in head
and tail such that the cycle-averaged pitching moment is close to zero...... 47
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Fig. 4-7 Time history of force and moment coefficients CL, CT, and CM for four different grid sizes
of a prescribed Spanish Dancer in uniform incoming flow at Re ≈ 645. Blue: ∆x = 0.04L;
red: ∆x = 0.025L; orange: ∆x = 0.015L; magenta: ∆x = 0.010L...... 48
Fig. 4-8 Comparison of the time history of (a) surge and (d) heave velocity for different Stokes
numbers. Blue: Σ = 2,200; Green: Σ = 3,140; Red: Σ = 4,400...... 52
Fig. 4-9 Comparison of the time history of force coefficients: (a) thrust coefficient; (b) lift coefficient
for various Stokes numbers. Blue: Σ = 2,200; Green: Σ = 3,140; Red: Σ = 4,400. .. 53
Fig. 4-10 (a) Perspective view of vortex topology for Σ = 3,140 at the phase where the tail is at its
lowest position and is about to pitch upward. (b) Contours of mean streamwise velocity at
the steady state for Σ = 3,140 along the spanwise symmetry plane. The thick black arrows
denote the direction of travel...... 55
Fig. 4-11 Vortex topology for AR = 2.55 foil at the phase where the foil is at the lowest point in its
heaving motion and starting to move up (figure from Dong et al. 2006)...... 56
Fig. 4-12 Contours of mean streamwise velocity for the AR = 2.55 foil along the spanwise symmetry
plane, the inclination angle of each branch of the wake from the symmetry plane is about 16o
(figure from Dong et al. 2006)...... 57
Fig. 4-13 Correlation between vortex structures and momentum flux in the wake for Σ = 3,140. (a)
Side view showing the vortex structures and the velocity vectors on the symmetry plane. (b)
Perspective view of isosurfaces of (u|u|), which is a simple measure of the streamwise
momentum flux. Purple line: outline of the Spanish Dancer; blue line: centerline of the
Spanish Dancer; colored contours/isosurfaces: momentum flux in the streamwise direction;
white shade on the right figure: vortex structures...... 58
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Fig. 4-14 Time-averaged thrust coefficient per unit area on the body for Σ = 3,140, with head to the
left and trunk to right...... 60
Fig. 4-15 Trapezoidal planform used to investigate effects of body shape on swimming performance.
(a) Case T1 with increased width of the caudal mantle; (b) Case T2 with increased width of
head mantle. The shaded area indicates the baseline rectangular shape...... 60
Fig. 4-16 Side view of the vortex topology for Σ = 4,400 of the (a) new baseline rectangular planform;
(b) case T1 and (c) case T2...... 61
Fig. 4-17 Time-averaged thrust coefficient per unit area for Σ = 4,400 with a smaller flapping
o amplitude ψ0 = 100 for the (a) new baseline rectangular planform; (b) case T1; and (c) case
T2...... 62
Fig. 5-1 Undulatory locomotion in batoid fish: (a) mobuliform mode (less than a half wave) of manta
ray, figure from Fish et al. (2016); and (b) rajiform mode (more than one wave) of stingray,
figure from Blevins & Lauder (2012)...... 65
Fig. 5-2 (a) external features of free-swimming sea hare (A. fasciata) in a streamlined posture; (b)
the planform used in this study adopted from a dissected A. brasiliana with the dorsal visceral
mass removed (Porten et al. 1982)...... 67
Fig. 5-3 Comparison of the reconstructed motion of the Aplysia with screenshots of a freely
swimming A. fasciata (Carletti 2011). (a) fully closed; (b) beginning of opening phase; (c)
fully open; (d) beginning of closing phase. The blue arrow denoted the direction of
swimming...... 70
Fig. 5-4 Temporal profile of thrust coefficients for four grid sizes of an animal swimming in a
uniform inflow with the same prescribed kinematics of the case C5...... 72
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Fig. 5-5 Vortex topology for the baseline case λ / L = 2.0: (a) perspective view; (b) dorsal view; (c)
side view. The visualized vortex structures corresponding to the isosurface of Q = 2.0 are
colored by vorticity in the spanwise direction. The black arrow denotes the direction of
swimming and the brown arrows depict the direction of rotation of these vortices. 74
Fig. 5-6 Vortex topology for AR=2.55 foil at the phase where the foil is at the lowest point in its
heaving motion and starting to move up (figure from Dong et al. 2006)...... 75
Fig. 5-7 Wake formation behind steadily swimming fishes: (a) fishes without distinct propulsor and
undulate using almost the whole body, such as eel (Auguilla rostrate), producing lateral
vortex rings that are partially linked; (b) fishes with discrete caudal fins, such as mackerel
(Scomber japonicas) and bluegill sunfish (Lepomis macrochius), producing linked vortex
rings with a downstream jet. Figure from Lauder & Tytell (2005)...... 76
Fig. 5-8 Wake topology of a manta ray subject to uniform incoming flow. Figure from Fish et al.
(2016)...... 77
Fig. 5-9 Correlation between vortex structures and momentum flux in the wake for the baseline case
λ / L = 2.0: (a) top-view slice intersecting the paired vortex rings (Rp and Rs); (b) slice
intersecting the top set of vortex rings (Rd), the colored contours are the streamwise
momentum flux uu ; (c) Perspective view of isosurfaces of superposed on vortex
structures; (d) ventral view of (c); (e) spanwise vorticity on the symmetry plane, red dashed
lines indicate the sliced location of (a) and (b)...... 79
Fig. 5-10 Cycle-averaged surge force coefficient per unit area plotted on the flat planform for the
baseline case λ / L = 2.0, higher curvature in the right parapodium leads to higher thrust.
...... 80
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Fig. 5-11 Instantaneous surge force distribution at four representative instants of the baseline case λ
/ L = 2.0: (a) fully closed, t / T = 0.10; (b) during the open phase (or downstroke), t / T =
0.36; (c) fully open, t / T = 0.60; (d) during the closing phase (or upstroke), t / T = 0.80.
...... 81
Fig. 5-12 Dorsal view of parapodial cross-section at four representative instants of the baseline case
λ / L = 2.0: (a) fully closed, t / T = 0.10; (b) during the open phase (or mid-downstroke), t /
T = 0.36; (c) during the closing phase (or mid-upstroke), t / T = 0.80. Red curved lines denote
the intersection of the body structure and a horizontal plane risen from the flat planform by
a distance of 0.2 BL...... 82
Fig. 5-13 Representative kinematic models at the closing phase with wavelength λ / L = 2.0: (a)
Baseline case, ψ0,l = 97.5° and ψ0,r = 165°; (b) C3, ψ0,l = ψ0,r = 90°; (c) C5, ψ0,l = ψ0,r =
75°...... 84
Fig. 5-14 Comparison of the time history of (a) swimming speed and (b) thrust coefficients for the
kinematic parameters investigated in the current study...... 85
Fig. 5-15 Schematic of a Sea Hare swim with a wall proximity D...... 88
Fig. 5-16 Time history of surge velocity at different ground proximity (left); zoomed in at quasi-
steady state (right)...... 89
Fig. 5-17 Time history of surge force coefficient at different proximity (left); zoomed in at quasi-
steady state (right)...... 89
Fig. 5-18 Vortex structures at different ground proximity: (a) D = 3.0 BL; (b) D=0.50 BL; (c) D =
0.25 BL; and (d) D = 0.10 BL...... 90
Fig. 5-19 Vorticity (left column) and pressure (right column) contours at the mid-plane for different
ground proximity...... 91
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Fig. 6-1 Marine Flatworms showing bold colors and stunning patterns: (a) Pseudobicero bedfordi
(Laidlaw n.d.); (b) Psedobiceros gratus (Anon n.d.); (c) Pseudobiceros hancockanus
(Petersen n.d.); (d) Thysanozoon nigropapillosum (Wills n.d.)...... 95
Fig. 6-2 (a) Pseudobiceros hancockanus swimming by lateral flapping (LF). Margin shows more
than two full waves and the central axis of the body does not exhibit any significant
undulation or deflections. (b) Thysanozoon Nigropapillosum swimming with lateral flaps
combined with dorso-ventral pitching of the body (CFP). The mantle in this case shows about
1.5 wavelengths...... 97
Fig. 6-3 Perspective view of kinematic model set (1) with pure lateral flaps (LF). Left:
kL/ 1.5 ; right: k = 2.0. Thick black arrow indicates the swimming direction.100
Fig. 6-4 Perspective view of kinematic model set CFP with combined motion of lateral flaps and
dorso-ventral deflections for four phase differences. From left to right: (a) φ = 0; (b) φ = 0.5π;
(c) φ = π; and (d) φ = 1.5π. First row: lateral flap wavenumber k = 1.5; second row: k = 2.0.
Thick black arrow indicates the swimming direction...... 101
Fig. 6-5 Left Column: Instantaneous vortex structures at quasi-steady state of the pure flapping case
with a wavenumber of k = 1.5: (a) Perspective view; (b) Side view; (c) Dorsal view. Vortices
are colored by the spanwise vorticity. Purple color shows the body structure...... 103
Fig. 6-6 Wake topology of free-swimming manta rays simulated using a potential flow model: (a,b,d)
vortex structures in dorsal, side, and perspective views respectively; (c) time-averaged
streamwise velocity iso-surface. Figure from Fish et al. (2016)...... 104
Fig. 6-7 Vortex structures of cases with a combined body pitching and mantle flapping (CFP): φ = 0
(left column); φ = 0.5π (right column). From top to bottom: perspective, side, and dorsal
views. Black arrows point to the top and bottom sets of vortex rings, Vt and Vb, and the two
xviii
‘contrails’, Vlat, respectively. Red arrows point to the lateral semi-circular vortex rings Vlat.
The wavenumber of lateral flaps is k = 1.5...... 106
Fig. 6-8 Vortex structures of cases with a combined body pitching and mantle flapping: φ = π (left
column); φ = 1.5π (right column). From top to bottom: perspective, side, and dorsal views.
Black arrows point to the top and bottom sets of vortex rings, Vt and Vb, and the two
‘contrails’, Vlat, respectively. Red arrows point to the lateral semi-circular vortex rings Vlat.
The wavenumber of lateral flaps is k = 1.5...... 107
Fig. 6-9 Cycle-averaged momentum flux ( uu ) iso-surfaces superimposed onto instantaneous
vortex structures at a quasi-steady state for cases with kinematics of a combination of body
pitching and mantle flapping: left column, φ = 0; right column, φ = 0.5π. Green indicates
positive momentum flux in the downstream and brown indicates negative flux; white shades
denote the vortex structures. Thick brown edges denote the outlines of the body shape. The
wavenumber of lateral flaps is k = 1.5...... 109
Fig. 6-10 Cycle-averaged momentum flux iso-surfaces ( uu) superimposed onto instantaneous
vortex structures at a quasi-steady state for cases with kinematics of a combination of body
pitching and mantle flapping: left column, φ = π; right column, φ = 1.5π. Green indicates
positive momentum flux in the downstream and brown indicates negative flux; white shades
denote the vortex structures. Thick brown edges denote the outlines of the body shape. The
wavenumber of lateral flaps is k = 1.5...... 110
Fig. 6-11 Instantaneous velocity vectors superimposed on momentum flux contours at the body
symmetry plane for cases with combined kinematics of body pitching and mantle flapping:
(a) φ = 0; (b) φ = 0.5π; (c) φ = π; (d) φ = 1.5π. Arrows represent the flow velocity magnitude
and direction. Momentum flux is shown in color in the background, vortex structures are
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shown as white shades, and the animal is outlined in light purple. The wavenumber of lateral
flaps is k = 1.5...... 112
Fig. 6-12 Digital particle image velocimetry (DPIV) results of 2D wake flows. (a) Representative
flow field behind an eel at 90% of the tail beat cycle, lateral jets are observed perpendicular
to the body axis. Black arrows denote the velocity magnitude and flow direction, the eel’s
tail is shown in blue at the bottom. (Figure from Tytell & Lauder 2004) (b-c) Velocity vectors
fields behind the pectoral fin of a bluegill sunfish on the vertical and horizontal light sheets.
Yellow arrows represent the velocity vectors. A central jet oriented downstream is identified
between two counter-rotating vortices. (Figure from Lauder & Drucker 2002) ... 113
Fig. 6-13 Vortex structures of cases with a combined body pitching and mantle flapping: φ = 0 (left
column); φ = 0.5π (right column). From top to bottom: perspective, side, and dorsal views.
The wavenumber of lateral flaps is k = 1.5...... 114
Fig. 6-14 Vortex structures of cases with a combined body pitching and mantle flapping: φ = π (left
column); φ = 1.5π (right column). From top to bottom: perspective, side, and dorsal views.
The wavenumber of lateral flaps is k = 2.0...... 115
Fig. 6-15 Time history of surge velocity (left) and surge force coefficients (right) for the pure lateral
flap case (red line) and four cases with various phase differences between the body margin
undulation and dorso-ventral deflection. The wavenumber of lateral flaps is k = 1.5 for all
five simulations...... 116
Fig. 7-1 Froude efficiency against swimming speed at the quasi-steady state. Magenta diamonds:
Spanish Dancer with combined swimming modes of body deflection and mantle margin
flapping; empty magenta diamonds: variations in planforms of Spanish Dancers; filled
magenta diamonds: variations in Stokes numbers. Green diamonds: Flatworms with
xx
combined modes of body deflection and body margin flapping; empty green diamonds:
wavenumber k 1.5 ; filled green diamonds: wavenumber k 2.0 . Red circles: Aplysia with
various flapping amplitudes and wavenumbers. Blue circles: Flatworms with pure lateral
flaps of various amplitudes and wavenumbers. Cyan squares: self-propelled flat plates with
various body deflection amplitudes, wavenumber is k 0.16 . Black squares: self-propelled
flat plates with various body deflection amplitudes, wavenumber is k 1.5 ...... 121
Fig. 7-2 Power coefficient against swimming speed at the quasi-steady state. Magenta diamonds:
Spanish Dancer with combined swimming modes of body deflection and mantle margin
flapping; empty magenta diamonds: variations in planforms of Spanish Dancers; filled
magenta diamonds: variations in Stokes numbers. Green diamonds: Flatworms with
combined modes of body deflection and body margin flapping; empty green diamonds:
wavenumber k 1.5 ; filled green diamonds: wavenumber k 2.0 . Red circles: Aplysia with
various flapping amplitudes and wavenumbers. Blue circles: Flatworms with pure lateral
flaps of various amplitudes and wavenumbers. Cyan squares: self-propelled flat plates with
various body deflection amplitudes, wavenumber is k 0.16 . Black squares: self-propelled
flat plates with various body deflection amplitudes, wavenumber is k 1.5 ...... 122
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Numerous approaches have been employed to study the locomotion from a variety of complementary perspectives such as biomechanics, engineering physics, physiology, ecology and behavior. However, it’s not until the 1960’s that quantitative work was done to illuminate the energetics of swimming ---the relationship between force and energy generated by propulsive movements. However, despite all the subsequent work, the physics of swimming in marine animals continue to bewilder and confound.
Fig. 1-2 Fish swimming modes classified based on movement characteristics and grouped by propulsive structures: (a) BCF propulsion and (b) MPF propulsion. Shaded areas are the propulsors. Figure from Sfakiotakis et al. (1999).
In order to investigate the hydrodynamics of swimming, swimming gaits that share common features are grouped together. According to the classification of fish swimming modes (Fig. 1-2) for aquatic locomotion proposed by Sfakiotakis et al. (1999), depending on the propulsive
2 structures aquatic animals utilize, the swimming gaits can be categorized as body and/or caudal fin (BCF) movements or median and/or paired fin (MPF) propulsion; whereas depending on the number of waves present on the propulsive structure, the motion can be divided into two major categories: oscillatory or undulatory.
Analytical models were proposed to investigate the swimming performance of fish undulatory motion as this mode of swimming is observed in a wide variety of fishes and marine mammals.
Elongated-body theory (EBT) (Lighthill 1970; Lighthill & Blake 1990), which is an inviscid flow model, was first applied to theoretically estimate the thrust and energy cost of undulatory fishes with a slender body moving through water. This theory remains one of the most fundamental analytical models for evaluating the hydrodynamic forces acting on a slender body and its propulsive efficiency. Many improvements to the EBT have been proposed over the years to cover other major factors involved, such as body thickness, variations in body depth, and presence of dorsal, ventral and caudal fins. To list a few, Newman & Wu (1973) took into account the effects of fins, Cheng & Blickhan (1994) extended the theory to include the effects of the slope at the tail end, and recently, Candelier et al. (2011) had extended the EBT to three-dimensional body shapes.
Following on this initial work, Cheng et al. (1991) proposed the 3-D waving plate theory with variable amplitudes and shapes which can be modified and applied to anguilliform and carangiform modes. Blade-element theory (Blake 1983a) has also been applied to evaluate hydrodynamic forces of MPF movements. All these approaches make simplifying assumptions that may only be met marginally by animals in nature (Yates 1983).
Aside from these analytical models proposed, much experimental work has been devoted to the maximum performance (e.g., limits of swimming speed and linear acceleration) of fish locomotion (Plaut 2000; Videler & Weihs 1982; Wardle 1975; Webb 1978), as these are a
3 reflection of animals’ ability to deal with critical incidents that are far from the most natural environment. However, the ecophysiological relevance of critical swimming performance is disputable (Plaut 2000; Pough 1989), although it might have relevance to bioinspired engineering designs. Other studies have focused on stability, maneuverability (turning radius) and agility
(turning rates) aspects (Fish 2002; Webb et al. 1996; Webb 1994). Recent studies also reveal that fish swimming in schools leads to considerable average energy savings (Hemelrijk et al. 2015;
Marrs et al. 2014).
Development in flow visualization techniques, especially particle image velocimetry (PIV)
(Lauder 2015; Taylor et al. 2014; Prasad 2000) which enables quantitative measurements of both the direction and magnitude of the flow velocity, has led to numerous experiments devised to examine the wake patterns of swimming animals and to investigate thrust generating mechanisms for various body shapes and kinematics. Two-dimensional PIV measurements on the body symmetry plane, for instance, were employed to examine the wake patterns of undulatory swimming in eel (Müller et al. 2001; Videler et al. 1999), mullet (Müller et al. 1997), trout (Liao et al. 2002), and labriform swimming in surfperch (Embiotoca jacksoni) and bluegill sunfish
(Lepomis macrochirus) (Drucker & Lauder 2000). New techniques such as three-dimensional stereoscopic digital particle image velocimetry was also later developed and employed to visualize the wake of rainbow trout (Nauen & Lauder 2002).
From an engineering perspective, the remarkable ability of swimming animals to maneuver and accurately control body posture even in turbulent flows has compelled researchers to design biomimetic AUV’s incorporating structures and/or mechanisms from nature for better performance. Attempts have been made to design biorobotic AUV using pectoral-like fins inspired by the bluegill sunfish (Lauder et al. 2005; Singh et al. 2004), slender-fish robots based on
4
Lighthill’s model (Zhou et al. 2009), tail-actuated robotic fish (Wang & Tan 2013), and turtle-like swimming robot (Kim et al. 2013).
In addition to the analytical and experimental studies, recent advances in computational fluid dynamic (CFD) codes and computing power have enabled researchers to simulate realistic conditions and predict flow patterns around animals with complex body morphology and swimming kinematics. Simulations were performed to investigate the swimming performance including but not limited to the undulatory movements associated with: (1) BCF propulsion anguilliform mode in eel (Adkins & Yan 2006; Borazjani & Sotiropoulos 2010; Kern &
Koumoutsakos 2006) and lamprey (Borazjani & Sotiropoulos 2010; Tytell et al. 2010), carangiform mode in tuna (Adkins & Yan 2006; Zhu et al. 2002), giant danio (Zhu et al. 2002), and mackerel (Borazjani & Sotiropoulos 2010); (2) MPF propulsion employing rajiform mode in stingray (Bottom et al. 2016) and manta (Fish et al. 2016); as well as oscillatory movements of pectoral fin employing labriform mode in bluegill sunfish (Tangorra et al. 2010; Mittal et al. 2006).
Flow around swimming cetaceans (whales, dolphins and porpoises (Weber et al. 2009)) and humans performing dolphin kick (von Loebbecke et al. 2009) were also simulated to investigate their performance.
All these aforementioned studies, however, have focused on swimming in vertebrate animals either due to practical reasons (e.g., availability and sizes of fishes, easy to hold and record in the lab) or physiological reasons (e.g., presumably high swimming performance) or both. In contrast, studies on the hydrodynamics of marine invertebrates are few, and most focus on jet propulsion of jellyfish (Megill 2002; Dabiri 2005; Dabiri et al. 2010) and squid (Anderson & Grosenbaugh 2005;
O’Dor 1988; Mohseni 2006). Yet a vast number of invertebrates exhibiting a wide variety of morphology and swimming gaits with a worldwide distribution from the near Arctic and Antarctic
5 zones to the tropics, exist, although much of the observations/recording of the swimming in these animals have been by amateurs and enthusiasts. Thus, the swimming performance and hydrodynamics of these animals remains virtually unexplored.
The current research focuses on aquatic gastropods and Flatworms. These are two phylogenetically distant groups of animals are often confused (Newman & Cannon 2003) because they share the following common features
1. they are soft-bodied invertebrates;
2. many species from both categories are notoriously poisonous or/and brightly colored;
and
3. they swim by some form of undulatory movements of their propulsive structures.
Fig. 1-3 shows the cladogram of the organisms within the scope of the present study.
Opisthobranchs, which means “gills behind the heart,” are a large and diverse group of gastropods including both terrestrial and aquatic snails and slugs within the phylum Mollusca. Some species, such as Hexabranchus sanguineus, a large and colorful sea slug in the clade of Nudibranchis
(“naked gills”), have lost their calcified shell in the evolutionary process and hence their gills are exposed to the environment. The species Limacina helicina within the clade of Thecosomata and the species Alpysia depilans within the clade of Aplysiomorpha, in contrast, retain calcified shells and both swim by flapping their two wing-like parapodia. Aquatic Flatworms such as
Pseudobiceros bedfordi and Thysanozoon nigropapillosum are in the clade Turbellaria belonging to the phylum Platyhelminthes (which literally means Flatworm), and they all have a bilaterally flattened, symmetric body and swim by undulating the outer margin of their body.
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the ‘breast stroke’ movement, and
jet propulsion (Table 1-1 and Fig. 1-4).
According to the classification of Farmer, for the species of interest in the current work, the
Spanish Dancer (Hexabranchus sanguineus), for instance, swims with a complex gait that combines dorso-ventral bending (lb) with a bilaterally synchronous, large amplitude progressive wave passes down its mantle (nf). The Aplysia, on the other hand, swims by flapping of its wide, wing-like parapodia (pf). The marine Flatworm, seems capable of swimming in a linear path by undulating its bilaterally flattened body (ub), dorso-ventral deflection of the body (lb) however is also observed from time to time.
Table 1-1 Swimming Classification of Marine Invertebrate Animals
(Adapted from Farmer, 1970)
Abbreviations Swimming Example Type of Swimming in Fig. 1-4 in Fig. 1-4 1. Flapping structures (paired) A. Notal nf 1, 2, 7 B. Parapodial pf 3, 5 C. Metapodial mf 2. Undulation A. Entire body ub 7 B. Foot uf C. Parapodial upro 3. Lateral bending of the body lb 2, 7 4. The ‘breast stroke’ movement bs 5. Jet propulsion jp 6
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The study of swimming in marine invertebrates has merit for a number of reasons. As pointed out earlier, studies of underwater locomotion have been dominated by investigation of the swimming of fish and cetaceans and these animals are constrained in their movement by the presence of a vertebra skeleton. Among fish, much of analysis is of the Teleost (ray finned fish) since they make up 96% of all the extant species of fish, and in these fish, all their propulsive surfaces (fins) are supported (and constrained) by cartilaginous rays. In contrast, the bodies and propulsive surfaces of invertebrates are completely unconstrained and unsupported by hard bony/cartilaginous structures (exception is the shells of some gastropods). Thus, marine invertebrates have “solved” the problem of swimming using an “apparatus” that is completely different from fish and cetaceans. One consequence of this is that marine invertebrates have developed a variety of highly complex and bizarre swimming gaits, and studying the swimming of this animals will provide unique insights into the hydrodynamics of aquatic locomotion and add to our relatively extensive knowledge of swimming in fishes and marine mammals.
Interest in the locomotion of these animals also comes from anthropologically induced changes in the overall “health” of the planet. In particular, it is expected that ocean acidification (OA), resulting in undersaturation of calcium carbonate, will become widespread by 2050. OA poses a particularly serious and immediate threat to organisms such as Limacina helicina (pteropod) that precipitate calcium carbonate from seawater (Ries 2012; Lischka et al. 2011; Fabry et al. 2008;
Comeau et al. 2009; Bednaršek et al. 2012). With ocean acidification, the pteropod shell will thin as the aragonite is highly soluble in acid. As the shell thins, it changes the mass distribution and buoyancy of the animal, which will affect locomotion and through it, all locomotion dependent behavior such as foraging, mating, predator avoidance and migrations. A lower shell weight will also be counterbalanced by a smaller mucus web of these animals, potentially decreasing ingestion
10 rates. A lighter shell will also alter the balance of forces (buoyancy and hydrodynamic) and adjustments will be therefore be needed in the swimming behavior for an animal to maintain its position in the water column. A lower shell weight could also affect maneuverability, thereby making these animals more susceptible to predators. Hence, fluid mechanical analysis together with biological studies of pteropod swimming behavior could help predict the effect of OA on these animals.
Fig. 1-5 Examples of bioinspired soft-robots for underwater applications.
The final motivation for this work comes from biorobotics, and particularly, the newly emerging field of soft-robotics (Huang et al. 2015; Suzumori et al. 2007; Shepherd et al. 2011;
Trivedi et al. 2008; Tolley et al. 2014) . Advances in materials (Jaggers & Bon 2017), 3D printing
(Phamduy et al. 2017), and stretchable electronics are making it possible to design robots that have no (or very few) hard components and while much of the initial work in this arena has been on
11 terrestrial robots, several novel designs of soft swimming robots have been developed (see Fig.
1-5). A soft-bodied sheet-like robot was designed to mimic the flapping motion of marine
Flatworms (Kazama et al. 2013) . Another biohybrid robot with both organic actuation and organic motor-pattern control inspired from Aplysia californica was recently designed by Webster et al.
(2017) and this continues to be a rapidly evolving field (Shen et al. 2017; Zhou et al. 2016; Shintake et al. 2016; Bicchi & Burgard 2018; Park et al. 2016). However, what is missing from these studies so far is a detailed analysis of the swimming hydrodynamics, and in particular, the effect of body shape and kinematics on swimming performance.
In this work, we employ computational modeling to study the hydrodynamics of swimming in various species of marine gastropods and Flatworms as well as simple canonical geometries covering a wide variety of swimming gaits. The effect of variations in body shape, kinematics, and size on swimming performance, including terminal speeds, thrust generation, propulsive efficiency and power coefficient, is examined. This study of locomotion in marine invertebrate animals also serves as a starting point to investigate how swimming behavior is affected by morphological changes due to ocean acidification. A better understanding of the swimming mechanisms and performance of these animals could also provide benchmarks for design evaluations of soft-bodied robots and insights for further improvements.
Chapter 2 describes the numerical methods that is employed for the simulations here including the development and validation of the non-inertial reference frame model developed here to simulate self-propelled swimming in an efficient manner. In Chapter 3, we outline the various metrics employed for assessing swimming performance including propulsive efficiency and terminal velocity.
12
Starting with Chapter 4, we focus on the modeling and analysis of swimming in three distinct animals, with this chapter focusing on H. sangiuneus or the Spanish Dancer, which swims with a combined body pitching and lateral flapping. Chapter 5 turns the focus on the Sea Hare or Aplysia, which swims with a large flapping movement of its mantle, and Chapter 6 described the modeling and analysis of the swimming of marine Flatworms, that also employ flapping, but with kinematics that are quite different from those of Aplysia. In Chapter 7, we conduct an extensive analysis of other synthesized gaits to provide a ‘big picture’ view of the effect of body kinematics on swimming performance. Finally, Chapter 8 summarizes the current work and provides some topics for future work.
13
CHAPTER 2. NUMERICAL METHOD
The approach employed to simulate the coupled flow-body system is described in this chapter.
This includes a discussion of an immersed boundary method (ViCar3D) in a non-inertial reference frame that is fixed to the body and detailed validation examples of both forced and flow-induced motion.
2.1. Governing Equations and Discretization Scheme
Free swimming is simulated by solving the equations for flow and the linear acceleration of the body in a coupled manner. The equations governing the flow are the three-dimensional (3D) unsteady, viscous incompressible Navier-Stokes (NS) equations with constant properties. In this work, in order to facilitate the numerical simulations of self-propelled swimming, the governing equations are expressed in a non-inertial reference frame that translates with the body-fixed point.
To obtain good numerical stability (Kim & Choi 2006) , these equations are written in the conservative form as
u i 0 (2-1) xi
uuui() u j V j 1 p ii (2-2) t x j xi x j x j
where ij, 1,2,3, ui is the fluid velocity in the inertial reference frame, V j is the translational velocity of the reference point on the body, p is the pressure, and and are the fluid density and kinematic viscosity, respectively.
14
The above NS equations are discretized on a Cartesian grid with a cell-centered, collocated
(non-staggered) arrangement of the primitive variables ( ui , p ). A second-order accurate fractional-step method, which consists of three sub-steps, is employed for time-advancement of the NS equations. In the first sub-step, a modified momentum equation is solved to obtain
* intermediate cell-center (cc) velocities ui , in that, a second-order, Adams-Bashforth scheme is employed for the convective terms and an implicit Crank-Nicolson scheme is used for the diffusive terms to eliminate the viscous stability constraint. The modified momentum equation is discretized at the cell-nodes given as
* n n uu 1 1 p 1 iin n 1* n 3NNi i DDi i (2-3) tx22 i
where Ni Uj V j u i / x j and Di // xj u i x j are the convective and diffusive
terms respectively, / x j is a second-order central difference operator, U i is the face-center (fc) velocity used to compute volumetric flux, and only the face velocity component normal to the cell- face is involved as in the fully staggered arrangement (Fig. 2-1). Once Eq. (3.3) is solved using a line-SOR scheme, face-center velocities at the intermediate step U * are obtained using the following averaging procedure
** * U1w u 1 P 1 w u1 W (2-4)
** * U2s u 2 P 1 s u2 S (2-5)
** * U3b u 3 P 1 b u3 B (2-6)
where w , s , and b are the weights corresponding to linear interpolation for the west, south and back face velocity components, respectively. The above procedure is necessary to eliminate the
15 checkboard issue of large pressure variations in space. In the second sub-step, the pressure correction equation
uun1* 1 p' ii (2-7) tx i
n1 is solved with the constraint that the final velocity ui be divergence-free. This gives the following Poisson equation
11p' U * i (2-8) xi x i t xi and a Neumann boundary condition imposed on this pressure correction at all boundaries. This
Poisson equation is solved with a highly efficient geometric multigrid method with a Gauss-Siedel line-SOR smoother.
Fig. 2-1 Schematic describing the naming convention and location of velocity components employed in the spatial discretization of the governing equations (From Mittal et al. 2008).
Once the pressure correction is obtained, in the third sub-step, the pressure and velocity are updated as
pnn1' p p (2-9)
16
' 1 p un 1* u t (2-10) ii xi cc
' n1* 1 p U U t (2-11) ii xi fc where cc and fc denote gradients computed at cell-centers and face-centers respectively. These separately updated face-velocities satisfy discrete mass-conservation to machine accuracy, and the above collocated scheme is relatively simple to implement compared to the conventional staggered mesh scheme while maintaining good discrete kinetic energy conservation properties.
2.2. Immersed Boundary Treatment
Fig. 2-2 Schematic of two-dimensional spatial discretization and immersed boundary detection
(From Mittal et al. 2008).
The current immersed boundary method (IBM) is designed to simulate flows over arbitrary complex 2D and 3D stationary or moving bodies “immersed” into the Cartesian volume grid, the
17 surfaces of these bodies are represented by unstructured mesh with triangular elements. The boundary condition on the immersed boundaries (IB) is enforced by the use of “ghost cells”. The schematic of an IB on a 2D Cartesian grid is shown in Fig. 2-2.
The method proceeds by first identifying fluid cells that are outside the body and solid cells inside the solid body, while ghost cells are defined as cells in the solid that have at least one neighboring cell in the fluid. Subsequently, an interpolation scheme is constructed such that the boundary condition on the IB is implicitly satisfied. As illustrated in Fig. 2-2, to obtain the appropriate values of variables at the ghost cells, a line segment is extended from the node of these
GCs into the fluid to an ‘image-point’ (denoted as IP) such that it intersect normal to the IB and the boundary intercept (denoted as BI) is the midpoint between the GC and the IP. Once the BI and the corresponding IP for each ghost cell have been identified, for a 3D case, the value of a generic variable ϕ at the IP can be computed by a trilinear (bilinear in 2D) interpolation using the eight nodes (four nodes in 2D) surrounding the image-point
(,,)xxx123 Cxxx 1123 Cxx 212 Cxx 323 Cxx 413 Cx 51 Cx 62 Cx 73 C 8 (2-12)
T Since the values of variables { } { 1 , 2 ,..., 8 } at the surrounding points are known, the unknown coefficients are then obtained as
1 {}{}CV (2-13) where V is the Vandermode matrix corresponding to the trilinear interpolation scheme with the form
18
xxx123 xx 12 xx 23 xx 13 x 1 x 2 x 3 1 1 1 1 1 1 1 1 xxx xx xx xx x x x 1 1232 12 2 23 2 13 2 1 2 2 2 3 2 xxx xx xx xx x x x 1 1233 12 3 23 3 13 3 1 3 2 3 3 3 V (2-14) xxx1238 xx 12 8 xx 23 8 xx 13 8 x 1 8 x 2 8 x 3 8 1 where the subscripts denote the indices of the surrounding points. Once the coefficients {}C are computed, the value of ϕ at the image-point can be obtained from the following equation
IP i i (2-15) where βi’s are the interpolation weights determined by the coordinates of the IP as well as the corresponding coefficients {C}.
A linear approximation is then applied along the normal probe to satisfy the prescribed boundary condition at the BI. Thus, for the velocity variables, Dirichlet boundary conditions are imposed and expressed as
GC2 BI IP (2-16) whereas for the pressure Poisson equation, Neumann boundary conditions are imposed and a second-order central –difference scheme is given as
GC l p IP (2-17) n BI
where lp is the length from the ghost cell to the image point. The above two equations can be written in the following implicit form
GC i i 2 BI (2-18)
GC i i l p (2-19) n BI
19 so as to be solved in a fully coupled manner together with discretized governing Eqs. (2-3) and (2-
8) for fluid cells, and the trivial equation ϕ = 0 is given for the internal solid cells. Following this procedure, second-order accuracy is achieved locally and globally for both velocity and pressure field while the pressure gradient filed is locally first-order near the boundary. Since the pressure gradient is multiplied by Δt during the velocity correction procedure, and Δt = O(Δ) due to the
CFL number constraint, the velocity is expected to maintain second-order accuracy.
In this work, in order to treat the immersed moving boundary as a sharp interface, a mixed
Eulerian-Lagrangian approach is employed, wherein the immersed boundaries are explicitly tracked as surfaces in a Lagrangian mode, while the flow computations are performed on a fixed
Eulerian grid. The body motion relative to the non-inertial reference frame is prescribed by a kinematic model that will be discussed in later sections. This relative motion is accomplished by moving the nodes of the surface triangles in a prescribed manner. For a given kinematic model, at
n n time step n, the velocity of a node m located at X m is known asUm , after a time Δt, the node is moved to the location
n1 n n Xm X m tU m (2-20)
To advance the field equations from time level n to n + 1, once the boundary is moved to a new location, repeat the aforementioned procedures to update the ghost-cells, body-intercepts, and image-points, and then the coupled system of discretized governing equations and the corresponding implicit boundary conditions formula can be solved. Further details about ViCar3D can be found in Mittal et al. (2008).
2.3. Flow-Induced Motion
While the body deformation (motion relative to non-inertial reference frame) of the animal is prescribed by some kinematic model, as the body deforms, forces are exerted on the body by its
20 surrounding fluid and thus leading to forward motion of the animal. This flow-induced motion is solved by the body dynamics equation. Once the primitive variables (ui, p) are computed at one time-step, by integrating the pressure and shear stress over the immersed boundaries, the force and moment exerted by surrounding fluid on the body are obtained follows:
F () pnˆ dS (2-21) B
M r () pnˆ dS (2-22) B where B denotes the surface the body, the shear stress acting on the surface, and nˆ the unit normal vector pointing out of the fluid control volume. Fig. 2-3 illustrates the convention of these notations.
Fig. 2-3 Schematic of simulation setup, thick edges denote the outer boundary of the computational domain, brown shade is the immersed body with surface B , and the unit normal vector pointing from the fluid into the body.
21
Given the observation that these animals swim along a nearly linear path, all flow-induced rolling, yawing and pitching movements of the body are neglected, whereas the flow-induced translational velocity of the body is estimated by solving Newton’s second law for the animal
dV SF (2-23) b dt where ρb is the constant mass per unit area of the body, and S is the surface area of the body as in these simulations the animals are modeled as zero-thickness membranes. The forward Euler method is employed to discretize the above equation
nn1 VV n SF (2-24) b t
Thus, the above body dynamics equation is advanced in a coupled but sequential manner with the flow equations, and this explicit coupling is found to be stable and accurate for the given ration of animal to water density. Validation of the numerical formulation for simulating body movement in a non-inertial frame is presented in the next section.
2.4. Validations of the Numerical Method in Non-inertial Reference
Frame
In this section, two unsteady flow problems: (a) inline oscillations of a circular cylinder; and
(b) a settling sphere under gravity are solved with Cartesian coordinate systems in a non-inertial reference frame fixed to the body, to show that the present numerical solver is able to accurately predict the forces acting on a moving body.
22
2.4.1. Inline oscillations of a circular cylinder
To validate the accuracy of the current method in non-inertial reference frame, we herein consider the case of a circular cylinder that is forced to oscillate in an initially quiescent fluid. The translational motion of the cylinder is given by a harmonic function:
xc ( t ) A0 sin 2 ft (2-25) where xc is the location of the cylinder center, A0 is the amplitude of the oscillation, and f is the frequency of the oscillation. The flow induced by the moving cylinder is therefore characterized
by two non-dimensional parameters, the Reynolds number ReUdm / and the Keulegan-
Carpenter number KC Um / fd . The simulation is performed at Re = 100 and KC = 5 so as to compare with results in the literature.
Fig. 2-4 Time history of the drag coefficient (solid line) from the present simulation compared with that of Kim & Choi (2006) (dotted line).
Fig. 2-4 plots the temporal profile of the drag coefficient along with the numerical results of
Kim & Choi (2006), it clearly shows that the present computed force is in very good agreement with the reported results. Fig. 2-5 plots the inline (ua) and vertical (va) velocity profiles at three
23 different locations at the phase angle2 ft , together with the computational results of Kim
& Choi (2006) and both the computational and experimental results of Dutsch et al. (1998). Again, the velocity profiles are in good agreement.
Fig. 2-5 Comparison of the absolute inline velocity component (ua) and vertical velocity component (va) at three different positions at the same phase angle 2 ft obtained from the current study (solid line) and the reported data by Kim & Choi (2006) or Dutsch et al. (1998)
(dotted lines): (a) inline velocity at xa = -0.6D; (b) inline velocity at xa = +0.6D; (c) inline velocity at xa = 0; (d) vertical velocity at xa = -0.6D.
The isolines of the pressure and spanwise vorticity in the inertial reference frame for two different phase positions ϕ = 0° and ϕ = 288° are plotted in Fig. 2-6 and Fig. 2-7. The vortex
24 structures are characterized by two counter-rotating vortices shed behind the cylinder when it oscillates. These vortex structures, as well as the pressure contours, are found to be the same as shown in Dutsch et al. (1998) and Kim & Choi (2006), indicating that the present method can accurately describe the pressure and vorticity field.
Fig. 2-6 Pressure contours (negative values dashed) in the inertial reference frame at Re = 100 and KC = 5. Left column: present study; right column: Dutsch et al. (1998). Top: phase angle ϕ
= 0°; bottom: ϕ = 288°.
25
Fig. 2-7 Vorticity contours (negative values dashed) in the inertial reference frame at Re = 100 and KC = 5. Left column: present study; right column: Dutsch et al. (1998). Top: phase angle ϕ
= 0°; bottom: ϕ = 288°.
2.4.2. Freely falling sphere under gravity
To further demonstrate the capability of this method in predicting forces and body movements in flow induced motion, we consider the case of a sphere falling under gravity. The motion can be
fully determined by the density ratio of the solid body to the fluid ( sf/ ) and the fluid viscosity
(ν). Three simulations are performed: (a) = 7.85 and the Reynolds number based on the terminal velocity (Vt) and diameter (D): Ret = VtD / ν = 435; (b) = 2.56, Ret = 369; and (c)
= 2.56, Ret = 42. The computational domain is of size 20DDD 20 20 , the total number of grid points is 97(x ) 257( y ) 97( z ) , with 67 217 67 grid points uniformly distributed
26 across the sphere, the sphere is falling in the y-direction. More grid points are applied in the y- direction to capture the flow features in the wake.
Fig. 2-8 shows the temporal evolutions of the velocity of the sphere in the falling direction, along with experimental data of Mordant & Pinton (2000) and numerical results of Kim & Choi
(2006) and Uhlmann (2005). Results are normalized by the reference values defined as
vref gD and tref D/ g respectively. For relatively higher Reynolds numbers as in cases (a) and (b), oscillations are observed in the velocity profiles of the present study, these oscillations might be caused by vortices shedding from the sphere (Kim & Choi 2006). On the other hand, this oscillation was not observed in Mordant & Pinton (2000) in case (a), which was averaged based on a set of ten experiments and thus the oscillation might be smeared out. In case (b), while no oscillation was observed in the averaged profile, the velocity fluctuation of the present study shows similar frequency as that of a single falling bead in the experiment. Fig. 2-9 plots the vortex structures of two cases Ret = 435 and Ret = 42 at various instants. It can be readily observed from
Fig. 2-9(a-c) that, for the higher Reynolds number case, the vortices shed behind the sphere loses the planar and axi-symmetry, meanwhile the shedding direction changes irregularly as seen in Fig.
2-10 by Kim & Choi (2006). For much lower Reynolds number case (c), however, the sphere accelerates monotonically and reaches a terminal velocity very shortly. Meanwhile, as observed from Fig. 2-9(d-e), vortex structure of this case are steady and maintain axi-symmetric with time.
These vortex structures are identified using the Q-criterion introduced in CHAPTER 3.
27
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Thus, the current numerical method is validated for both forced as well as flow-induced motion.
All the fluid-structure interaction problems considered in this study produced good agreement with the previous numerical and experimental results, indicating the capability and accuracy of the current method for simulations of flow around moving bodies.
30
CHAPTER 3. METRICS FOR SWIMMING PERFORMANCE
The 3-D wake structures generated by the self-propelled swimmers are identified by the Q-
1 criterion (Cucitore et al. 1999), the quantity Q is defined as Q ΩS22 , where S and Ω 2 are the symmetric and anti-symmetric parts of the velocity gradient tensor u , respectively. A vortex is defined as a region where the rotational motion dominates the rate-of-strain and Q > 0.
Vortices are visualized by plotting isosurfaces of Q values.
Swimming speed is crucial to many basic behaviors aquatic animals and can be classified in three categories depending on the speed: sustained, prolonged and burst swimming. Sustained swimming refers to long periods of movement without fatigue, whereas burst swimming is the highest speed attainable by the fish for only short periods of time. Prolonged swimming lies in between (Blake 2004) these two. The critical swimming speed, which is based on fish fatigue and hence prolonged swimming speeds, is the most common way to measure the swimming performance (Tritico & Cotel 2010; Plaut 2000), although it’s still controversial (Dekerle et al.
2005). Thus one quick metric for assessing swimming performance is via the terminal swimming speed. In this study, the instantaneous swimming speed V is non-dimensionalized as VVTL* / and the mean value V * may be viewed as the number of body-lengths traveled per cycle. The wake
Strouhal number associated with the swimming is calculated as St AmT/ V T , where S is the peak-
to-peak amplitude of the flapping or pitching motion of the animal and VT the mean terminal swimming speed.
31
The forces on the swimmer are computed through direct integration of the surface pressure and shear at the centroid of each surface triangular element, and the total force on the body is obtained by a simple trapezoidal integration scheme using Eqn. (2-21) while the total moment computed using Eqn. (2-22). All forces and moments are presented as non-dimensional coefficients, which are computed as
FM CC, , (3-1) FM11 VS22V SL 22TT where F and M are the force and moment components, respectively, CF and CM are the corresponding coefficients, and S is the planform area,. The force components are defined as thrust
(force in the surging direction), denoted by CT, and lift (force perpendicular to both the surging and the spanwise direction, denoted by CL. The calculation of the propulsive (or Froude) efficiency follows the method described in von Loebbecke et al. (2009) and is calculated as
T P() t dt Wuseful 0 useful T , (3-2) Wtotal P() t dt 0 total
where T is the time period of the stroke, Puseful denotes the work done to push the swimmer forward
and Ptotal denotes the total power consumed. The Froude efficiency is applied for terminal swimming conditions, when the cycle-averaged thrust force is balanced exactly by the drag force and thus cycle-averaged propulsive force is equal to zero. The instantaneous total power consumed
is computed as the summation of the dot product of the total force Ftk ()exerted by each triangular
body element on the surrounding fluid and the corresponding velocity Utk () of that surface triangle. That is
32
N Ptotal ()()() t Fk t U k t , (3-3) k 1 where k is the element index and N is total number of surface triangles. The surface force has
surge sway heave components FFFFk k,, k k , and depending on whether the surge component of the force
surge Ftk () is negative or positive, it could contribute either hydrodynamic drag or thrust to the forward motion. At the terminal state, since the total propulsive force averaged over one cycle is zero, to separate these two contributions, the force component in the direction of swimming and the instantaneous useful power is computed as
N surge surge Fkk()() t F t P() t Usurge () t , (3-4) useful k k 1 2
surge where Uk is the component of absolute velocity (with reference to a stationary reference frame) of each body triangle in the direction aligned with the swimming direction.
It should be noted that the propulsive efficiency of self-propelled swimmers has been estimated by a number of different methods (Lighthill 1970; Tytell & Lauder 2004; Fish et al.
2016). Experiments do not usually have access to forces on the body of the swimmer and employ estimates of power from measurements of wake velocity and pressure. Computational models such as the current one, on the other hand, have access to the local force and velocity distributions on the body and the above definition is formulated to be universally applicable to computational models for any mode of swimming. This same definition has been previously been applied to estimate propulsive efficiencies of human undulatory swimming (von Loebbecke et al. 2009), as well as carangiform swimming in cetaceans (von Loebbecke et al. 2009) and fish (Bergmann et al. 2014).
33
Schultz & Webb (2002) thus pointed out the controversy on defining and estimating the propulsive efficiency of a self-propelling animal and have suggested that the total non-dimensional power expended by the animal to maintain its terminal velocity is itself a measure of propulsive performance, and is equivalent to an effective drag coefficient of the swimming animal, that the body works against to maintain its terminal speed. We therefore also calculate a power coefficient at the terminal state as
P D C total effective , (3-7) P 11 VSVS32 22TT
where CP is the power coefficient (or equivalently, the effective drag coefficient) and Deffective is the effective drag force of the swimming body, and also compare this metric along with the Froude efficiency for the various cases simulated in this study.
34
CHAPTER 4. SWIMMING HYDRODYNAMICS OF THE SPANISH
DANCER
4.1. Introduction
The focus of this chapter is on the swimming hydrodynamics of the Spanish Dancer
(Hexabranchus sanguineus), a nudibranch that swims with a complex gait that combines dorso-
ventral body undulation with a bilaterally synchronous, large amplitude progressive wave that
passes down its mantle. While these animals spend much of their time crawling over coral-reefs
(Gohar & Soliman 1963), they also occasionally swim. Edmunds (1968) notes that they are
“powerful swimmers” and Eliot (1904) discovered some of them swimming in the surface waters
up to half a mile off shore.
Based on the classification of Farmer (1970), the Spanish Dancer swims with a complex gait
that combines notal flapping with a body undulation. While the neurobiology of these animals has
been relatively well-studied by virtue of the simplicity, accessibility, and robustness of the
neuronal circuits responsible for their movement (Kristan 2008, Wentzell et al. 2009), less is
known about their propulsive mechanism and swimming energetics. An extensive search of
literature uncovered a single fifty year old paper (Edmunds, 1968) that addressed the swimming
of this animal. H. sanguineus inhabits marine habitats in many parts of the world and presents a
variety of anatomical and behavioral traits that have direct implications for swimming performance.
The effects of these traits, particularly, the effect of body planform on swimming performance is
also unknown.
35
In the current study, we employ computational modeling to study the hydrodynamics of swimming in the Spanish Dancer. We employ high-fidelity computational fluid dynamics to model the realistic swimming in these animals and use the simulation to gain insights into the flow features and energetics of locomotion. The simulations are also used to explore the effect of simple variations in body planform, on the swimming hydrodynamic.
4.2. Methodology
4.2.1. Kinematic Model
Notal margin
Gills
Head Trunk
Fig. 4-1 External morphology of the Spanish Dancer (Hexabranchus sanguineus)
The body of the Spanish Dancer consists of a central structure with a head and trunk and this is surrounded by a large, relatively thin, membranous mantle. Fig. 4-1 shows the external morphology of the Spanish Dancer with a highly ruffled mantle margin (parapodia) with numerous folds, gills on the dorsal surface of the tail, and a foot partially hidden on the ventral side (Newman
36
1994). In the current model, the entire body of the Spanish dancer is modeled in a simple manner as a zero-thickness deformable membrane. The premise for this choice is that the thin mantle dominates the total surface area of the animal and drives much of the hydrodynamics of the locomotion, and approximation of the body as a plate simplifies the modeling significantly while retaining the feature most important to the hydrodynamics. The planform of these animals can vary among the sub-species and the baseline body shape is chosen to be rectangular with an aspect-ratio of L : W = 2 : 1, where L and W are the body-length and width respectively. This is a fair representation of the observed body shape of these animals and in a later section, the effect of planform is explored in detail.
37
Fig. 4-2 Sequence of the Spanish Dancer swimming with the flapping frequency ~0.5 Hz.
Pictures showing every 0.2s, from left to right, top to bottom (Blalock 2008).
As mentioned earlier, the Spanish Dancer exhibits a complex swimming gait that combines a large-scale dorso-ventral undulation with a wave like flapping of the mantle. Since these animals are difficult to maintain in laboratory conditions, controlled studies of their swimming kinematics are not available. Given this, the kinematics of the body and mantle are based on a field video of
38 an animal recorded swimming off the coast of the island of Saipan (Blalock 2008), which seems to be reasonably representative of the swimming of these animals. In this video (Fig. 4-2), the animal is observed to move in a nearly linear path and this video therefore serves as a suitable basis to model linear swimming in these animals. The movement of the body and mantle is prescribed with respect to a fixed point on the body which is taken to be the intersection of the body midline and the pitching axis shown as the red dots in Fig. 4-3. The available video indicates that the head is roughly 30% of the body length (L) and this is the value used in the current model.
The following sinusoidal functions describe the dorso-ventral angular deflections of the head and trunk about this fixed point
h(t ) h01 h sin(2 t / T ) (4-1)
t(t ) t01 t sin(2 t / T ) (4-2) where T is the period of the locomotory movement, θh0 and θt0 are the equilibrium positions of the head and trunk, and θh1 and θt1 are the respective amplitudes. The time-dependent angular deflection associated with the travelling wave along the mantle with respect to the axis-of- symmetry of the body is prescribed as
1 (x , y , t ) 1 tanhy y / K sin 2 (x / t / T ) (4-3) m 2 00 where x is the axial coordinate extending from the head to the tail ( 0 xL), y is the spanwise coordinate extending from the plane-of-symmetry to the mantle margin ( W/ 2 y W / 2 ), ψ0 is the amplitude of mantle deflection, and ϕ is the phase difference between the body undulation and mantle movement. The hyperbolic tangent term generates a smooth but rapid spanwise variation of the mantle deflection ψm from 0 to its maximum value ψ0 and this variation is
39 controlled by choosing appropriate values of the parameters K and y0, which have dimensions of length.
The body is made of a mesh with triangular elements (Fig. 4-3) where each element is attached to a virtual joint that can rotate about hinges either parallel to the pitching axis or the centerline, locomotion of the swimmer can then be generated by smoothly rotating these joints following the
Denavi-Hartenberg convention (Jazar 2010). With the kinematics of the body prescribed about the fixed body point, simulations are carried out where the acceleration and movement of the body are computed by direct coupling of the body dynamics equations with Navier-Stokes equations, thereby allowing us to simulate the free-swimming of these animals. Fig. 4-3b illustrates the deformed pitching plate at t = 0, and the definition of the positive pitching directions of the head and the tail. Fig. 4-3c illustrates the shape of the deformed undulating plate at time t = 0, and the definition of the mantle angle ψ. Fig. 4-3d illustrates the constructed motion with the combination of these two modes.
40
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The above kinematic model introduces nine non-dimensional parameters (θh0, θh1, θt0, θt1, ψ0,
λ / L, ϕ, K / L, y0 / L) and these parameters have to be chosen so as to provide a reasonable match to the observed swimming. These parameters were initially estimated from selected still frames of the video and then an iterative process was used to modify the parameters until a good visual match was obtained. The final parameter set used in our simulations is shown in the Table below.
Head Trunk Mantle Parameter θh0 θh1 θt0 θt1 ψ0 λ / L ϕ K / L y0 / L Value 60o -90o 30o 30o 120o 1.0 135o 0.1 1/6 Table 4-1 Kinematic parameters employed in the modeling the swimming of the Spanish Dancer.
Fig. 4-4 shows a series of snapshots of the reconstructed swimming kinematics compared to the screenshots of the freely swimming Spanish Dancer and we note that the match is quite reasonable. The choice of the mantle wavelength λ is worth noting since there is some disagreement on this issue in the published literature. Edmunds (1968) noted that while his observations of H. marginatus and those of Gohar and Soliman (1963) for H. sanguineus indicated one full wave (i.e. λ / L = 1) on the mantle, the observations of Morton (1964) for H. marginalis suggest two complete waves (i.e. λ / L = 2) on the mantle. Our own analysis of the kinematics suggests that λ / L = 1 provides the best match to the observed kinematics and is therefore in-line with Edumnds (1968) and Gohar and Soliman (1963). It is interesting to note that the current consensus among zoologists is that H. marginatus and H. marginalis are in-fact not distinct species
(Valdés 2002) but variants of or synonymous with H. sanguineus and therefore, the observations of Edmunds (1968) and Morton (1964) have direct relevance to the current work.
42
Z Z
푡 푇 = 0 Y X 푡 푇 = 0.08 Y X
Z Z
푡 푇 = 0.35 Y X 푡 푇 = 0.44 Y X
Z Z 푡 푇 = 0.56 푡 푇 = 0.52 Y X Y X
Z Z
푡 푇 = 0.79 Y X 푡 푇 = 0.90 Y X
Fig. 4-4 Comparison of the recreated motion of the Spanish Dancer (column 2 & 4) with the screenshots of a freely swimming H. sanguineus (column 1 & 3) from Blalock (2008). The central axis of the body is nearly in a straight line at the beginning of the stroke and the stroke begins with the pitching down of the head and the simultaneous pitching down of the trunk. The mantle has a characteristic “cosine” shape at this initial stage and as the head and trunk approach each other, the mantle wave propagates backwards.
43
4.2.2. Pitching Moment Balance Study
Since measurements of the size and mass for the H. sanguineus are not available, mass properties of a related nudibranch, the Aplysia brasiliana (Donovan et al. 2006), are used to set the value of the area density to 20 kg/m2. As discussed in Chapter 3, the body-flow coupled system is advanced explicitly in time, and the solver is found to be stable and accurate for the given ratio of animal to water density. In our model, surge (forward) and heave (vertical) movements are modeled using Eq. (3.5), however given the bilateral symmetry of the kinematics, sideways drift
(sway) of the body as well as the roll and the yaw are neglected. The only other degree-of-freedom
(DoF) is pitch and handling of this DoF in our model requires some care
Fig. 4-5 plots the temporal profile of the pitching moment coefficient of the baseline case (solid line) with kinematics matching the field video moving in initially quiescent fluid and for this model, the cycle-averaged pitching moment coefficient is found to be about -0.16. A mean pitching moment on the body would result in a net pitch rotation of the body, which would have to be compensated by some feedback control process by the animal in order to enable it to swim along a straight path. Since we do not incorporate any such feedback process in our modeling, model consistency requires that the swimming kinematics selected here do not generate large magnitudes of pitching moment. A series of simulations were performed to find, in an iterative manner, the set of kinematic parameters that results in a cycle-averaged moment that is close to zero. In this analysis, the body centroid is fixed at one location in the computational domain and moves with prescribed kinematics in a initially quiescent fluid. This reduces the computational expense of this analysis and make it more manageable.
44
tT/
Fig. 4-5 Time history of the pitching moment coefficients. Solid line: prescribed baseline case in initially quiescent fluid; dashed line: case 7 with optimized kinematics. The mean pitching moment of for the baseline case is about -0.16, whereas that of the optimized case is less than -
0.01.
Table 4-2 Mean pitching moments computed with various control parameters
Head Pitch (deg) Tail Pitch (deg) Undulation (deg) Case ↓ θh0 θh1 θt0 θt1 ψ0 shift M z Baseline 15 -120 45 45 120 135 -0.16 1 30 -120 45 45 120 135 -0.20 2 0 -120 45 45 120 135 -0.25 3 15 -120 45 45 120 120 -0.22 4 15 -120 45 45 120 165 -0.20 5 15 -120 45 45 120 105 -0.20 6 15 -120 60 15 120 135 -0.11 7 60 -90 30 30 120 135 -7.8e-3
45
Table 4-2 lists the parameters controlling body kinematics explored in this study, as well as their corresponding mean pitching moment coefficients over a cycle. Based on our simulations, in cases 1 and 2, the mean pitching moment increases with change in the equilibrium position of the head; in cases 3, 4, and 5, little difference is observed by varying the phase difference between the pitching motion of the body and the undulatory motion of the mantle margin; in cases 6 and 7, however, the mean pitching moments are significantly reduced by modifying the equilibrium position and the pitching amplitudes of both the head and the tail. In particular, the mean pitching moment coefficient of case 7 has been reduced by a factor of 23 compared to the baseline case, which would result in a very small net rotation. The temporal profile of the pitching moment coefficient of case 7 is plotted in Fig. 4-5 and can be compared to the baseline case. As seen from the figure, the upstroke (t / T = 0.1 ~ 0.6, Fig. 4-6) generates a positive moment while the downstroke (t / T = 0.6 ~ 1.1, Fig. 4-6) generates a balanced negative moment. Case 7 is therefore selected for all the self-propelled simulations performed in the rest of this chapter, and the selected kinematics do indeed correspond well to linear swimming and as evident from Fig. 4-6, the gait of the body for case 7 remains quite similar to the baseline case.
While the selected gait has a significantly reduced pitching moment, for a self-propelled model, even a small magnitude of net pitching in every cycle integrates to non-negligible values over multiple cycles, and would lead to noticeable pitching of the body. In the current study, we choose to eliminate any such flow-induced pitching of the body by eliminating the degree-of-freedom in pitch from the model.
46
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of Σ = 3,140, which is high enough to be representative of these animals, and low enough to allow adequate resolution of the flow features and vortices.
4.2.4. Grid Convergence
Fig. 4-7 Time history of force and moment coefficients CL, CT, and CM for four different grid sizes of a prescribed Spanish Dancer in uniform incoming flow at Re ≈ 645. Blue: ∆x = 0.04L; red: ∆x = 0.025L; orange: ∆x = 0.015L; magenta: ∆x = 0.010L.
The computational domain used in the current study is cubic in shape and of size
20LLL 20 20 . The Spanish Dancer is placed at the center of the domain. A systematic grid
48 refinement study was carried out for a Spanish Dancer swimming in a uniform flow of velocity V
= 1.29 BL s-1, which matches the terminal velocity and the swimming Reynolds number Re = VL/ν
= 645 of the nominal case. The uniform, isotropic grid sizes in the cuboid containing the immersed body for these four meshes are ∆x = 0.04L, 0.025L, 0.015L, and 0.010L. These correspond to total numbers of grid points of 0.8, 1.1, 4.5, and 8.5 million, respectively, and therefore spanned one full order-of-magnitude in grid size.
The simulations are performed with a fixed Δt equal to 0.001T (CFL=0.13, 0.20, 0.35 and 0.53).
The time history of thrust, lift, and moment coefficients are plotted in Fig. 4-7 and we note that the magnitude of the overall magnitude of thrust is about three times smaller than that of the lift.
Thus it is not surprising that the variations in thrust with grid size are more noticeable than that of the lift force.
Table 4-3 summarizes the mean and root-mean-square (RMS) values of several key hydrodynamic quantities for the grid refinement study. As expected, the mean lift and thrust coefficients are nearly zero. The RMS values of lift show less than variation 10% for all resolutions, and less than 3% between the nominal and the fine grids, indicating that the lift is not quite sensitive to the grid resolution. On the other hand, the mean thrust coefficients demonstrate a higher sensitivity to the resolution as seen in Fig. 4-7(b), and the difference of the RMS. values between the nominal and fine grids is about 10%. As explained before, this is due to the overall lower magnitude of thrust. Therefore, in order to obtain accurate results while maintaining relatively lower computational cost, in free-swimming simulations, the grid spacing ∆x = 0.015L and time-step size Δt = 0.002T are used for the lower and intermediate Reynolds numbers of 427 and 645 with the maximum CFL numbers no greater than 0.7, whereas the same grid spacing with smaller Δt = 0.001T are used for the highest Reynolds number of 931 such that the maximum CFL
49 is no greater than 0.5. Far-field (Neumann) boundary conditions are applied for both pressure and velocity on all the outer boundaries.
Table 4-3 Results of grid refinement study for Re = VL/ν = 645
Case CT CT rms CL CL rms CM CM rms ∆x = 0.04L 0.153 0.482 -0.031 1.468 -0.003 0.282
∆x = 0.025L 0.092 0.413 -0.032 1.525 -0.004 0.230
∆x = 0.015L 0.029 0.386 -0.027 1.628 -0.005 0.322 (Nominal) -0.016 0.348 -0.027 1.579 -0.024 0.322 ∆x = 0.010L
4.3. Results
Results from current simulations are presented in this section. Simulations are initiated with the Spanish Dancer at zero velocity. As the flapping motion ensues, the animal accelerates and eventually reaches a terminal swimming state. In all cases, it takes about 10 swimming cycles to reach this terminal state and simulations are continued for about 5 additional cycles to accumulate statistics.
4.3.1. Effect of Stokes Number on Swimming Performance
The first set of simulations are used to explore the effect of Stokes number on the swimming performance. Stokes number represents the size and/or frequency of flapping and the size of
Spanish Dancers varies significantly from about 1.5 cm for juveniles to 50 cm for adults. Similarly, flapping frequency for adults ranges from 0.25 to 0.5 Hz but is generally expected to be higher for
50 smaller specimens. In the current study, keeping in consideration the increased resolution requirements for simulations at high Stokes numbers, we have conducted simulations for three different Stokes numbers: 2200, 3140, and 4400 to explore the effect of this parameter. To provide some context, the lower value of 2200 could represent a 1.8 cm juvenile swimming with a flapping period of one second, whereas the higher value of 4400 could correspond to a small, 5 cm adult swimming with a period of 3.5 seconds.
The direction of swimming for the model is not prescribed a-priori and is the result of the net hydrodynamic force on the body. Once the swimmer reaches a terminal swimming state and is swimming along a nominally linear path, the direction of this linear path is designated as the surge direction and the direction normal to it, the heave direction. The velocity and forces are then decomposed along these two normal directions. Fig. 4-8 shows the time variation of the surge velocity from the initiation of the swimming to the terminal swimming state for the three cases.
This plots show that in self-propelled motion, the animal model is initially at rest. Swimming movements generate hydrodynamic thrust and heave forces, leading to acceleration of the body and an increase in the drag. The velocity continues to increase until the cycle-averaged thrust is balanced by the cycle-average drag, and the average swimming velocity becomes constant and the mean terminal speeds for the various cases are listed in Table 4-4. The non-dimensional terminal
* velocity for these three cases varies from VT =1.22 for the lowest Stokes number case to 1.33 for the highest. Thus, doubling of the Stokes number increases the terminal swimming speed by about
10%. For the field video employed here, the time period of the stroke is about 2 seconds and the velocity in body-lengths per second (BL/s) for the highest Stokes number would therefore be 0.67
BL/s. This in the same range as slow swimming eels (0.5 BL/s; Tytell 2004) but is slower than most fish, that have speeds in the range of 1 to 4 BL/s. The speed of H. sanguineus also compares
51 favorably to another soft-bodied invertebrate, the jellyfish (Aurelia aurita), which have been measured swimming at 0.5 BL/s (McHenry 2003).
(a) (b) Fig. 4-8 Comparison of the time history of (a) surge and (d) heave velocity for different Stokes numbers. Blue: Σ = 2,200; Green: Σ = 3,140; Red: Σ = 4,400.
Table 4-4 Comparison of terminal velocities and Froude efficiency at different Stokes numbers
* Σ V T St η CP 2,200 1.22 0.30 0.43 2.42 3,140 1.29 0.29 0.50 1.94 4,400 1.33 0.28 0.57 1.66
Fig. 4-9 shows the time-variation of the thrust and heave forces over three cycles in the terminal swimming state. The asymmetry of the thrust profile over the pitch-down and pitch-up movements is quite evident. The heave force is more symmetric during the up and down pitch motions and is also of a higher overall magnitude than the thrust force. The lift force is quite insensitive to the
Stokes number but the thrust force shows some variation with Stokes number for both the pitch-
52 up and pitch-down motions. The variation of the thrust coefficient is also not symmetric between the pitch-down and pitch-up motions and analysis shows that downstroke generates a small net positive thrust (0.026, 0.051 and 0.058 for the Σ = 2,200, 3,140, and 4,400 cases respectively and this is balanced by an equal amount of net drag during the upstroke.
Down Up Down Up
(a) (b) Fig. 4-9 Comparison of the time history of force coefficients: (a) thrust coefficient; (b) lift coefficient for various Stokes numbers. Blue: Σ = 2,200; Green: Σ = 3,140; Red: Σ = 4,400.
Table 4-4 lists computed efficiencies and Strouhal numbers for the various cases simulated here. It is noted that for all three cases, the Strouhal numbers are very close to 0.3. Numerous experimental and numerical studies of swimming and flying animals suggest that the propulsive efficiency peaks within the narrow range of St from 0.2 to 0.4 (Taylor et al. 2003). The propulsive efficiencies in the current simulations range from 43 to 57% and the higher value is quite comparable to propulsive efficiency computed in the same way for cetaceans (55.8%; von
Loebbecke et al. 2009). For rainbow trout, efficiencies based on elongated body theory, which typically overestimates the true efficiency (von Loebbecke et al. 2009), are found to be in the range of 50 to 85% (Webb et al. 1984). Efficiency of anguilliform propulsion in eels from
53 experimental measurements (Tytell and Lauder 2004) indicates values ranging from 43 to 54%, but due to the way power is computed from the wake profile, and the different expression used for efficiency in the study of Tytell and Lauder (2004), it is difficult to make a direct comparison.
Even if one chooses to ignore these differences and make a direct comparison, the efficiency of the Spanish Dancer is well within the efficiency range estimated for eels.
The trends for the power coefficient are consistent with the Froude efficiency for these three cases; as efficiency goes up with increasing Stokes number, the power coefficient goes up, indicating, not surprisingly, that the animal is a more effective and efficient swimmer at higher
Stokes numbers. The overall conclusion from the above analysis therefore is that despite a somewhat unorthodox and ungainly gait, the swimming efficiency of the Spanish Dancer, an animal that does not rely primarily on swimming for locomotion, is comparable to other animals that exclusively locomote via swimming.
4.3.2. Wake Topology
This section focuses on the salient features of the wake-topology for the self-propelling
Spanish Dancer. We describe the wake topology for the intermediate Stokes number case (Σ =
3,140) only since the key features do not vary for the range of Stokes numbers studied here. Fig.
4-10 a shows the perspective view of the vortex structures in the terminal swimming state when the tail is at its lowest position and is about to pitch upward. The vortex structures are identified by plotting an isosurface of the imaginary part of the complex eigenvalue of the instantaneous velocity gradient tensor. The magnitude of the isosurface plotted here is 3.0 and it is chosen to clearly illustrate the significant vortex structures. The plot suggests that the wake topology of this very complex swimming motion that combines mantle undulation and body pitching, is very similar to that of a simple low-aspect-ratio flapping foil (Dong et al. 2006), as well as that of self-
54 propelled anguilliform swimmers at intermediate Reynolds number (van Rees et al. 2013).
Comparison of Fig. 4-10 with the corresponding plot of the elliptic flapping foil with aspect-ratio
2.55 (Fig. 4-11) shows some common features including a bifurcated vortex wake. Fig. 4-10, we identify two vortex rings R1 and R3 in the upper set and one ring R2 in the lower set. The rings in each set have the same direction of rotation as illustrated by lines and arrows in the figure. It’s also observed that each vortex ring has two sets of thin vortex ‘contrails’ at both its upstream and downstream ends that extended to its two adjacent counter-rotating rings. Despite these similarities in vortex rings, unlike the vortex structures of the flapping foils, vortices formed at the head of the
Spanish Dancer are quickly dissipated, and therefore, these vortices have no noticeable contribution to the vortex ring formation.
R3 ~15° R1
R2
(a) (b) Fig. 4-10 (a) Perspective view of vortex topology for Σ = 3,140 at the phase where the tail is at its lowest position and is about to pitch upward. (b) Contours of mean streamwise velocity at the steady state for Σ = 3,140 along the spanwise symmetry plane. The thick black arrows denote the direction of travel.
55
Fig. 4-11 Vortex topology for AR = 2.55 foil at the phase where the foil is at the lowest point in its heaving motion and starting to move up (figure from Dong et al. 2006).
56
Fig. 4-12 Contours of mean streamwise velocity for the AR = 2.55 foil along the spanwise symmetry plane, the inclination angle of each branch of the wake from the symmetry plane is about 16o (figure from Dong et al. 2006).
Fig. 4-10b shows the contours of the mean streamwise velocity at the steady state along the streamwise symmetry plane for the case. Again, similar to the topology of the flapping foils in
Dong et al. 2006, a bifurcated jet is observed. The inclination angle of each of the two branches of the wake to the centerline is about 15°, which is very close to the jet angle (16°) of flapping foils with AR = 2.55 (Fig. 4-12). Moreover, the mean wake can be considered to be symmetric with respect to the centerline within statistical uncertainty.
57
Drag producing portion Oblique negative flux “Hull” drag
Thrust producing Oblique positive flux portion (a) (b) Fig. 4-13 Correlation between vortex structures and momentum flux in the wake for Σ = 3,140.
(a) Side view showing the vortex structures and the velocity vectors on the symmetry plane. (b)
Perspective view of isosurfaces of (u|u|), which is a simple measure of the streamwise
momentum flux. Purple line: outline of the Spanish Dancer; blue line: centerline of the Spanish
Dancer; colored contours/isosurfaces: momentum flux in the streamwise direction; white shade
on the right figure: vortex structures.
We further examine the correlation between the vortex structures and the thrust production with the aid of Fig. 4-13a and b. Fig. 4-13a shows the velocity vectors superposed on a side view of the vortex structures as well as contours of (u|u|), which is indicative of the streamwise momentum flux induced by the vortices. It can be seen that while the vortex rings induce strong downstream (positive in the current convention) oriented momentum flux, the other parts of the elongated vortex structures contribute an upstream (negative) oriented streamwise momentum.
Thus, some portions of the vortex contrail are thrust producing, while others are drag inducing.
This notion becomes clearer in Fig. 4-13b, where isosurfaces of (u|u|) are superposed on
58 isosurfaces of the vortex contrail behind the animal. The two oblique regions of positive streamwise momentum flux are found to coincide very precisely with the centers of the vortex rings, confirming that the vortex rings are indeed the thrust generating features of the wake. The negative momentum flux appears in two regions: the first in a set of obliquely oriented regions outside thrust regions and these are associated with the elongated vortices attached upstream of the vortex rings. The second region of low momentum fluid lies in the center of the wake and this is associated with the “hull” drag of the body of animal.
4.3.3. Implications of Planform Shape on Propulsion
As mentioned earlier, there is some variability in the planform shape of the animals and from the perspectives of functional morphology as well as bioinspired design of soft swimming robots, it is useful to examine the effect of planform on the propulsive performance on this animal. Fig.
4-14 shows the time-averaged streamwise force per unit area on the body of the animal. The figure shows that while most of the thrust is generated by the notal margin near the caudal region of the animal, most of the drag is generated by the head of the animal. Interestingly, the few published pictures of the planform shape of individual specimens (Gohar and Soliman 1963, Edumnds 1968) indicate that the mantle margin is wider in the caudal regions compared to the region near the head.
This shape might improve the propulsive performance of the animal and we investigate this by simulating two additional trapezoidal planform shapes shown in Fig. 4-15a and b. In the first case, the head mantle is reduced in width by 25% and the caudal mantle is increased in width by the same amount, and in the second case, this change is reversed. It is noted that the overall area remains the same as the baseline rectangular case. The amplitude of the mantle undulation is
o reduced from ψ0 = 120° to 100 in order to avoid the intersection of the two mantle margins at the
59 ends of each half-stroke. The baseline case with the same ψ0 has also been simulated in order to provide a clearer comparison.
Fig. 4-14 Time-averaged thrust coefficient per unit area on the body for Σ = 3,140, with head to the left and trunk to right.
Head Trunk Head Trunk
(a) (b)
Fig. 4-15 Trapezoidal planform used to investigate effects of body shape on swimming performance. (a) Case T1 with increased width of the caudal mantle; (b) Case T2 with increased width of head mantle. The shaded area indicates the baseline rectangular shape.
60
Simulations for these two additional cases are conducted at the highest Stokes number of 4,440 and Fig. 4-16 shows the vortex topology for these two cases as well as the corresponding baseline case. It is noted that the slight change in planform has a noticeable impact on the wake topology; the T1 case generates the most complex wake whereas the T2 case, creates a simpler wake dominated by vortex rings. This is connected with the fact that the caudal region of the body has the highest overall excursion and velocity and it therefore generates the most energetic vortex structures. Increase in the size of this region (as in T1) increases the vorticity released in the wake, leading to a more complex wake.
(a)
(b) (c)
Fig. 4-16 Side view of the vortex topology for Σ = 4,400 of the (a) new baseline rectangular planform; (b) case T1 and (c) case T2.
61
Table 4-5 Comparison of terminal velocities and computed Froude efficiencies for different
planforms
* Shape V T St η CP Baseline 1.29 0.29 0.57 1.40 T1 1.37 0.27 0.57 1.36 T2 1.12 0.33 0.47 2.14
(a) (b) (c)
Fig. 4-17 Time-averaged thrust coefficient per unit area for Σ = 4,400 with a smaller flapping
o amplitude ψ0 = 100 for the (a) new baseline rectangular planform; (b) case T1; and (c) case T2.
Table 4-5 shows the key propulsive characteristics for these two cases and comparison with the baseline case shows that increasing the width of the caudal mantle does marginally improve the propulsive performance of the animal. While the propulsive efficiency is the same as the baseline case, the terminal velocity for T1 is about 6% higher. On the other hand, reducing the size of the caudal mantle and increasing the size of the head mantle significantly diminishes both the efficiency (-17.5%) and the terminal speed (-13.2%) of the animal. Fig. 4-17 shows the surface distributions of thrust coefficient per unit area for the new baseline rectangular case as well as the two trapezoidal models and the increase in the width of the caudal mantle clearly enhances the thrust producing region of body. Thus, the current simulations generally support the notion that
62 the slightly wider caudal mantle observed in Spanish Dancers is an adaptation for enhancing swimming performance.
4.4. Summary
Simulations have been used to explore the swimming performance and wake characteristics of the nudibranch Spanish Dancer. Simulations show that despite its unorthodox swimming gait, the swimming performance of this animal, as measured by the propulsive efficiency and the terminal swimming speed, compares favorably with other swimming animals. The wake of these animals is dominated by a bifurcated train of vortices that bears significant similarity to simple, low-aspect- ratio flapping foils (Dong et al. 2006). Analysis of the wake indicates that vortex ring-type structures are associated with thrust production and this balances the hull drag and the drag associated with other vortex structures. Analysis of swimming performance with slight modified planforms suggests that the slightly wider caudal mantle observed in these animals enhances swimming performance. The current study provides an understanding of the swimming of an animal for which, very little is known so far. Given that these animals are soft bodied invertebrates that have a simple neuromuscular system, they could serve as inspiration for the design of efficient, soft swimming robots.
63
CHAPTER 5. THE WAKE TOPOLOGY AND SWIMMING
PERFORMANCE OF THE SEA HARE (APLYSIA)
5.1. Introduction
All Aplysia are benthic animals, and among the 37 recognized species, seven (A. brasiliana, A. depilans, A. extraordinaria, A. fasciata, A. morio, A. pulmonica, and A. tanzanensis) are good swimmers (Carefoot 1987) that can cross large distances in one swim episode and also maintain a fairly straight path against natural currents. A. brasiliana and A. fasciata are two related species
(Medina et al. 2001) whose swimming has been investigated in some detail. All species of Aplysia swim by a similar parapodial flapping (Farmer 1970) that is actuated by a wave of muscle contraction from the anterior to the posterior of each parapodium; the parapodia are folded to cover the mantle area during the upstroke (closing phase) and fully extended to expose the mantle area for the downstroke (opening phase). The amplitude of flapping is typically very large with the two parapodia overlapping at the end of the upstroke.
The simple neural system with large neurons and axons has made the Sea Hare, or Aplysia, a valuable laboratory animal for investigations into the nervous systems and brain function and neuronal control of swimming in these animals has been studied by von der Porten et al. (1980),
Porten et al. (1982), McPherson & Blankenship (1991) and others. Efforts have also been made to study the energetics of these animals by evaluating the cost of transport (Donovan et al. 2006).
The propulsive mechanism for the mode of swimming adopted by these animals is however still unclear. Proposed theories include sculling, jet propulsion, and hydrodynamic lift (Carefoot 1987;
Donovan et al. 2006). Porten et al. (1982) suggested that the leading edge of the parapodia presents
64 as an “airfoil” that produces lift on both power and recovery strokes. They suggested that while the anterior third of the parapodia produces thrust, the rear two-thirds provides pitch stability during swimming.
(a) Manta ray (b) Stingray
Fig. 5-1 Undulatory locomotion in batoid fish: (a) mobuliform mode (less than a half wave) of manta ray, figure from Fish et al. (2016); and (b) rajiform mode (more than one wave) of stingray, figure from Blevins & Lauder (2012).
In the classification scheme of swimming gaits in marine gastropods put forth by Farmer
(1970), the Aplysia employs paired flapping of its parapodia for swimming. However, in the larger context of swimming animals, we note that the swimming gait of Aplysia is most similar to the so- called rajiform and mobuliform modes of propulsion adopted by batoid fish such as electric rays, sawfishes, guitarfishes, skates and stingrays (Fig. 5-1) (Rosenberger 2001). In these modes, thrust is generated by passing a synchronized undulatory traveling wave along the pectoral fins. The feature that distinguishes these two modes is the wavenumber (number of waves per body length) of the undulatory wave; in the rajiform mode, the wavenumber is greater than one whereas in the mobuliform mode, the wavenumber is less than half. Stingrays swim using the rajiform mode with
65 a mean wavenumber of about 1.1 whereas fish such as cownose rays (Rhinoptera bonasus) are found to have 0.4 waves on their fins (Rosenberger 2001). As we will show later, the Aplysia has wavelengths that compare to the mobuliform mode, but the amplitude of flapping in Aplysia significantly exceeds anything seen in batoid fishes. It is expected that this exaggerated movement of the parapodia will general hydrodynamical features that are different from those observed for batoid fish, and this movement and associated hydrodynamics will also impact the swimming performance in ways that has not been elucidated in previous studies of swimming in batoid fish.
The large amplitude movement of the parapodia in Aplysia as compared to batoid fish is undoubtedly associated with the lack of a skeleton in these marine gastropods, which allows significant bending in their appendages and parapodia. This versatility in movement is also one factor that makes these animals attractive for consideration as bioinspired or biohybrid soft robots, and recently, attempts have been made to develop soft biohybrid robots with both organic actuation and organic motor-pattern control inspired by the locomotion of the non-swimming species
Aplysia californica (Webster et al. 2016; Webster et al. 2017). Thus, this novel application in the area of biohybrid robotics generates additional interest in the swimming performance of soft- bodied animals such as Aplysia.
In the current study, we employ high-fidelity computational fluid dynamics to model free- swimming in these animals and use the simulation to gain insights into the flow features, propulsive mechanisms, and energetics of locomotion. Simulations are also conducted to explore the effects of variations in kinematics, such as parapodial flapping amplitude and undulatory wavelength. Understanding of the hydrodynamics of these animals may also help in the development of softbodied bioinspired or biohybrid robots that utilize living neuron/tissue to swim
(Webster et al. 2016).
66
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Aplysia is modeled as a zero-thickness deformable membrane. The planform of these animals is adopted from Fig. 7 in Porten et al. (1982) of a dissected Aplysia brasiliana with the dorsal visceral mass removed. Bebbington & Hughes (1973) described the differences in the shape of the pedal sole/parapodia of three swimming species: A. depilans, A. fasciata, and A. punctata, and while the shape of the parapodia varies slightly among species, we consider the flat, distended shape of A. brasiliana (see Fig. 5-2b) as a good representation of these animals.
As reported by Carefoot and Pennings (2003), when swimming at the surface, the sea hare exhibits a complex swimming gait that combines a head-bobbing with each parapodial flapping, similar to the swimming gaits of the Spanish Dancer. The bobbing movement is however, usually not observed when swimming below the surface, and this also agrees with the observation by
Porten et al. (1982). In this study, Since these animals are difficult to maintain in laboratory conditions, the kinematics of the body for the current study are obtained from a field video of an animal recorded swimming close to the seabed (Carletti 2011). In this video, the animal is observed to swim by passing synchronous waves along the two parapodia. The animal moves in a fairly straight path, and this video therefore serves as a suitable basis for modeling linear swimming of these animals. It’s worth pointing out that Bebbington & Hughes (1973) noted a slight phase difference between the movements of the two parapodia in A. fasciata, whereas the measurements of Porten et al. (1982) indicated little phase difference in flapping. The video of Carletti (2011) and out model is inline with the latter study. Therefore, the time-dependent angular deflection of the parapodia with respect to the axis-of-symmetry is described by a synchronized traveling wave as
1 (x , y , t ) 1 tanhy y / K sin 2 (x / t / T ) , (5-1) m 2 00
(0x L , W / 2 y W / 2)
68 where ψ0 is the angular amplitude of the parapodial deflection, x and y are the axial and spanwise coordinates, respectively (L and W are the length and width of the body), T is the cycle period.
Since the shape of the paired parapodia is bilaterally symmetric, but the movements are asymmetrical with one parapodium closing over the another during the closing phase, the flapping amplitudes of the two sides ψ0,l and ψ0,r (where the subscripts l and r denote the left and right sides, respectively) are set to be slightly different to capture this feature. The hyperbolic tangent function prescribes the spanwise variations with two parameters K and y0, both of which have dimensions of length.
Once the shape of parapodia is determined, the membrane representing the body is divided into triangular elements where each element is attached to a virtual joint that can rotate about a hinge parallel to the axis-of-symmetry. The locomotion of the swimmer defined by Eqn. (5-1) can then be generated by smoothly rotating these joints following the Denavi-Hartenberg convention (Jazar
2010). Fig. 5-2b shows the distended planform meshed with triangular elements. With the kinematics of the body prescribed about the axis-of-symmetry, simulations are carried out where the acceleration and movement of the body are computed by direct coupling of the Navier-Stokes equations (see Eqns. 2.2 and 2.3) with the body dynamics equations (see Eqn. 2.5).
69
Fig. 5-3 Comparison of the reconstructed motion of the Aplysia with screenshots of a freely swimming A. fasciata (Carletti 2011). (a) fully closed; (b) beginning of opening phase; (c) fully open; (d) beginning of closing phase. The blue arrow denoted the direction of swimming.
The above kinematic model introduces five non-dimensional parameters: the flapping amplitudes of the two parapodia ψ0,l, ψ0,r, the wavelength λ / L, K / L and y0 / L. These parameters were initially estimated from selected still frames of the video to achieve a good visual match. The parameters controlling the spanwise variations are chosen as K / L = 0.1 and y0 = W / 4 for all
Aplysia models investigated in this study. The rest of the parameters used in our baseline case are
ψ0,l = 97.5°, ψ0,r = 165° and λ / L = 2.0. As pointed out earlier, the amplitude of undulation in
70
Aplysia is significantly higher than that observed for batoid fish. For instance, cownose rays
(Rhinoptera bonasus) which swims by using mobuliform mode exhibits the highest flapping amplitude among the eight species studied by Rosenberger, but this still only corresponds to an angular amplitude of about 35⁰ per the current kinematic description. Thus, the hydrodynamics of swimming in Aplysia is expected to be distinct from batoid fish. Effects of flapping amplitude and wavelength are explored in detail in later sections. Fig. 5-3 shows four representative stages of the reconstructed swimming structure in comparison with the screenshots of a freely swimming A. fasciata from the field video and we note that the reconstruction provides a reasonable match.
5.2.2. Simulation Setup and Grid Refinement Study
As discussed in CHAPTER 2, the hydrodynamics and swimming performance of the animal can be fully determined by the geometric and kinematic parameters, as well as the density ratio,
and the Stokes number 2/fLAm , where f is the flapping frequency, L is the body length, Am is the flapping amplitude. The Stokes number can also be interpreted as a flapping velocity based
Reynolds number. Statistical analysis (Donovan et al. 2006) of 25 fully grown specimens , with live body mass ranging from 34 to 506 g and the corresponding parapodial area (single parapodium) from about 20 to 200 cm2, indicated Stokes numbers from O(104) to O(105). In the current study we choose a nominal value of Σ = 3,450, which is high enough to be representative of these animals, but low enough to allow adequate resolution of the flow features and vortices. For context, this
Stokes number could represent an Aplysia of parapodial area of 25 cm2 (single parapodium only) based on extrapolation of the data of Donovan et al. (2006).
71
Fig. 5-4 Temporal profile of thrust coefficients for four grid sizes of an animal swimming in a uniform inflow with the same prescribed kinematics of the case C5.
The computational domain used in this study is of size 10LLL 8 7 and the head of the Aplysia is placed at (3LLL ,4 ,3.5 ) . The configuration is chosen such that the domain is large enough that the outer boundary can be considered nearly undisturbed, and far-field (Neumann) boundary conditions are applied for both pressure and velocity on all the outer boundaries. A systematic grid refinement study was carried out for an animal swimming in a uniform inflow with prescribed kinematics of C5 described in section 6.3.2. Grids in this refinement study had a high-resolution region of size 2.5LLL 1 0.6 with uniform grid spacing of 0.03L, 0.015L, 0.012L, and 0.010L, with corresponding total grid points of 0.8, 3.0, 7.5, and 9.0 million, respectively. The force coefficients were examined to assess grid dependency and as shown in Fig. 5-4 the temporal profile of the nominal grid 0.012L shows good agreement with the finest grid spacing of 0.010L. Table
5-1 summarizes the key hydrodynamic quantities for this grid refinement study and it’s found the
difference of both the mean thrust coefficients CT and their r.m.s values CT rms between the nominal and fine grids is within 1%.
72
Table 5-1 Mean thrust coefficients and RMS values for various grid sizes
0.012L x 0.030L 0.015L 0.010L (nominal)
CT 0.013 0.002 -0.002 -0.004
CT rms 0.072 0.069 0.069 0.069
5.3. Results
Simulation results of free-swimming are presented in this section. For all simulations, animals are initially released in a stationary fluid and as the flapping ensues, the animals accelerates and eventually reach a quasi-steady terminal swimming state. In all cases, it takes about eight cycles to reach this quasi-steady state and simulations are continued for about five additional cycles to accumulate statistics.
5.3.1. Wake Topology
In this section, we focus on the vortex wake of a self-propelled Aplysia. We describe here the wake topology for the baseline case with λ / L = 2.0 only since the key features do not vary much with the kinematic parameters studied here. Fig. 5-5 shows the vortex structures for the terminal swimming condition. In this plot, we identify three sets of vortex rings (curved arrows) in the wake, with a pair of vortex rings (Rs and Rp) oriented in the spanwise direction and connected by two horizontal vortices (straight solid arrows), and a third set of vortex rings (Rd) orientated along the streamwise direction. The dash dotted arrows denote the orientation of these vortex rings. A close examination of the vortex formation process indicates that while the vortex rings Rs and Rp are generated by the downstroke of the caudal tip of the parapodia, the dorsal vortex ring Rd is generated by the upstroke of the parapodia at the caudal end. With respect to this dorsal vortex 73
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Fig. 5-6 Vortex topology for AR=2.55 foil at the phase where the foil is at the lowest point in its heaving motion and starting to move up (figure from Dong et al. 2006).
A comparison of this wake with those of flapping foils (Fig. 4-11) and other swimmers (Fig.
5-7 and Fig. 5-8) provides a useful context for interpreting the current results. The paired vortex rings on the port and starboard side of the animal (Rp and Rs, respectively) produced by parapodial undulation resemble the oblique vortex rings shedding by a finite-aspect ratio flapping foil (Fig.
4-11), as well as the linked vortex rings of freely swimming eels (anguilliform), mackerel
75
(subcarangiform), and manta ray (rajiform), all of which present two, and only two sets of vortex rings. Despite these similarities, the paired vortex rings of Aplysia, are shed simultaneously in a non-staggered pattern at the parapodia tips while the vortex rings from these other foils/swimmers are shed alternately from the two ends and arrange in a staggered fashion.
(a) (b)
Fig. 5-7 Wake formation behind steadily swimming fishes: (a) fishes without distinct propulsor and undulate using almost the whole body, such as eel (Auguilla rostrate), producing lateral vortex rings that are partially linked; (b) fishes with discrete caudal fins, such as mackerel
(Scomber japonicas) and bluegill sunfish (Lepomis macrochius), producing linked vortex rings with a downstream jet. Figure from Lauder & Tytell (2005).
76
Fig. 5-8 Wake topology of a manta ray subject to uniform incoming flow. Figure from Fish et al. (2016).
A more interesting feature that distinguishes the current observed wake pattern from those previously reported, especially as compared to batoid fish, is the formation of the third distinct train of vortex rings that is arranged along the dorsal end of the wake. There is little similarity between the vortex topology of the current work and that of rajiform or mobuliform swimmers such as rays, skates (Clark & Smits 2006) and mantas (Fish et al. 2016). In Fish et al. (2016), flow simulations of a manta at Reynolds number comparable to the current case were conducted and the simulation (Fig. 5-8) showed the formation of two pairs of vortex loops for one full stroke.
These vortex loops arrange in a wake that expands in the dorso-ventral direction, but not in the
77 lateral direction. The Aplysia on the other hand forms a pair of laterally arranged vortices at the end of the downstroke and a single, dorsally arranged vortex loop due to the coordinated upstroke of the parapodia. This wake is seen to expand both in the lateral direction as well as the dorso- ventral direction. Finally, the wake also contains a vortex street comprised of pairs of counter- rotating vortices identified in Fig. 5-5a as Vk that are aligned along the spanwise direction and connect the laterally arranged vortex rings in the near wake.
The correlation between the vortex structures and the thrust production is elucidated in Fig.
5-9. We pick two slices intersecting the centers of the laterally paired vortex rings and the dorsal vortex rings, respectively. Fig. 5-9a and b show 2-D slices selected from the dorsal view. In these plots, the vortex structures are illustrated as white shades, and velocity vectors are superposed onto the contours of uu , which is indicative of the streamwise momentum flux induced by the vortices. It can be seen that while the dorsal vortex rings induce strong downstream (positive in the current convection) oriented momentum flux which signifies thrust generation, the two horizontal (spanwise) vortices behind the body center that connect the paired vortex rings contribute an upstream (negative) oriented momentum flux. Thus, all three sets of vortex rings are thrust producing whereas the horizontal vortices are drag inducing.
78
(a) (b)
(d) (c) “Hull” drag
(e)
Thrust producing
Karman Vortex Street (Vk)
Fig. 5-9 Correlation between vortex structures and momentum flux in the wake for the baseline case λ / L = 2.0: (a) top-view slice intersecting the paired vortex rings (Rp and Rs); (b) slice intersecting the top set of vortex rings (Rd), the colored contours are the streamwise momentum flux uu ; (c) Perspective view of isosurfaces of superposed on vortex structures; (d) ventral view of (c); (e) spanwise vorticity on the symmetry plane, red dashed lines indicate the sliced location of (a) and (b).
We further examine this correlation by superimposing the isosurfaces of onto the isosurfaces of the vortex structures as presented in Fig. 5-9c (perspective view) and Fig. 5-9d
(ventral view). The three streaks of positive streamwise momentum flux are found to be precisely
79 located along the centers of the three sets of vortex rings, confirming that the vortex rings are indeed the sources of thrust. The negative momentum flux associated with the “hull” drag of the animal, however, is located between the two horizontal vortices as observed from the ventral view in Fig. 5-9d. By plotting the spanwise vorticity on the symmetry plane (Fig. 5-9e), these horizontal
(spanwise) vortices are found to resemble a Karman vortex street that is associated with drag generation. Thus, the self-propelled Aplysia produces a wake that shows a clear spatial separation in thrust and drag.
5.3.2. Surge Force Distribution
Fig. 5-10 makes it clear that these aforementioned vortex rings are generated by parapodial flapping, whereas the drag-based spanwise vortices (identified as Vk) are shed behind the non- deforming portion of the body. This implies that the two parapodia act as the ‘propellers’ while the rest part of the body acts as the ‘hull’. Fig. 5-10 shows the cycle-averaged streamwise force per unit area on the body. It can be readily seen from this figure that the anterior portion, in particular, the first third of the parapodia generates thrust whereas the rest of the body portion generates drag on average.
Fig. 5-10 Cycle-averaged surge force coefficient per unit area plotted on the flat planform for the baseline case λ / L = 2.0, higher curvature in the right parapodium leads to higher thrust.
80
Fig. 5-11 Instantaneous surge force distribution at four representative instants of the baseline case λ / L = 2.0: (a) fully closed, t / T = 0.10; (b) during the open phase (or downstroke), t / T =
0.36; (c) fully open, t / T = 0.60; (d) during the closing phase (or upstroke), t / T = 0.80.
As suggested by Porten et al. (1982), the parapodial flapping of the Aplysia may be analogous to wing movements in birds that the leading edge of parapodia is configured into the shape of an
“airfoil” and thus the lift acting on it has a positive component in the swimming direction. Fig.
5-11 plots the instantaneous distribution of surge force coefficient at four representative time steps which corresponds to two peaks (Fig. 5-11b and d) and two troughs (Fig. 5-11a and c) of the surge
81
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streamwise force distribution on the parapodia, as well as in the isosurfaces of momentum flux and vortex structures, results from the asymmetry in flapping motion.
5.3.3. Effects of Kinematic Parameters on Swimming Performance
As mentioned earlier, the amplitude of the parapodia undulation is found to vary significantly for these animals with energetic swimmers showing overlapped (and therefore high amplitude) flapping and tired animals not able to overlap the two parapodia during the flapping (Donovan et al. 2006). Furthermore, our observation of the kinematics suggests an undulation wavelength of λ
/ L ≤ 1 and the baseline kinematic model of λ / L = 0.5 shows fair agreement with the video. A quantitative analysis of the effect of parapodial wavelength on swimming performance, which might be interesting from the viewpoint of bioinspired robotics, is not available. We have therefore conducted simulations to explore the effects of both the undulatory amplitude and wavelength on the swimming performance and results from this study are presented in this section.
Table 5-2 Kinematic parameters employed in modeling the swimming of Aplysia and the
corresponding swimming performance.
Case ↓ ψ0,l ψ0,r λ / L V T η CP Baseline 97.5⁰ 165⁰ 2.0 0.81 23.8 0.96 C1 97.5⁰ 165⁰ 3.3 0.89 20.6 1.00 C2 97.5⁰ 165⁰ 1.5 0.65 22.7 1.28 C3 90⁰ 90⁰ 2.0 0.75 22.6 1.04 C4 90⁰ 90⁰ 1.0 0.45 19.2 2.14 C5 75⁰ 75⁰ 2.0 0.65 21.3 1.10 C6 75⁰ 75⁰ 1.0 0.42 22.7 1.94
83
Fig. 5-13 Representative kinematic models at the closing phase with wavelength λ / L
= 2.0: (a) Baseline case, ψ0,l = 97.5° and ψ0,r = 165°; (b) C3, ψ0,l = ψ0,r = 90°; (c) C5, ψ0,l
= ψ0,r = 75°.
Table 5-2 summarizes of the kinematic parameters for the various cases investigated here along with the metrics for swimming performance obtained from the simulations. To provide some context, the baseline case with the highest flapping amplitudes ψ0,l = 97.5° and ψ0,r = 165°, as well as cases C1 and C2, as discussed in the kinematic modeling section, correspond to the scenario where the left parapodium closes over the right, and thus, mimics active swimmers; cases C3 and
C4 with the intermediate flapping amplitudes ψ0,l = ψ0,r = 90° correspond to the scenario where the two parapodia barely touch each other at the top of the upstroke; in cases C5 and C6 with the lowest flapping amplitudes ψ0,l = ψ0,r = 75°, the parapodia are far apart and these could model
‘tired’ swimmers (Donovan et al. 2006). Fig. 5-13 shows the three representative postures during the closing phase.
84
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efficiency as well. Experimental measurements (Donovan et al. 2006) show that the velocity of freshly rested animals is 0.096±0.014 m s-1 (initial speed), and it drops to 0.067±0.014 m s-1 (final speed) after four hours of continuous swimming. For an animal of 10 cm length, as the scale studied in this work, these speeds are equivalent to about 1.0 and 0.7 BL s-1, respectively. Thus, our simulation results are quite comparable to the experimental results.
Swimming speeds vary significantly among vertebrate and invertebrate swimming animals. As reported by Tytell and Lauder (2004), the speeds of eels range from 0.5 to 2 BL s-1. Thus, the swimming speeds of Aplysia fall in a range comparable to eels, which employ the so-called anguilliform mode of propulsion. The highest speed of Aplysia does not match that of the manta ray, a rajiform swimmer, which can swim at 1.5 BL s-1 (Fish et al. 2016). For Stingrays, Bottom et al. (2016) reported swimming speeds between 1.5 BL s-1 and 2.5 BL s-1, which are significantly faster than the Aplysia. Aplysia also cannot match the speeds of fish which easily exceed 3.0 BL s-
1 (Cotterell & Wardle 2004; Korsmeyer et al. 2002). Interestingly, the Aplysia is also found to underperform in terms of speed of swimming compared to the closely related species of marine gastropods, the Spanish Dancer (Hexabranchus sanguineus). As studied in Chapter 5, this other nudibranch swims with a complex gait that combines dorso-ventral body undulation with a bilaterally synchronous, large amplitude progressive wave that passes down its mantle margin, and simulations results have provided estimates for terminal speed of 1.33 BL/cycle for these animals.
Comparison of swimming efficiencies are more difficult due to the ambiguity of defining efficiency for an animal swimming at a terminal speed (Schultz & Webb 2002). For instance, the
Froude efficiency of eels calculated based on the wake power estimation (Tytell, 2004, Tytell and
Lauder, 2004), is around 50%. For manta rays, the efficiency evaluated by employing Tytell and
Lauder’s expression is about 48% (Fish et al. 2016). On the other hand, Bottom et al. (2016)
86 estimated the Froude efficiency (as defined in Tytell and Lauder 2004) of stingrays with fast swimming velocity of 2.5 BL s-1 to be 34.1% and slow swimming velocity of 1.5 BL s-1 to be
22.9%. An easier comparison of Aplysia’s swimming efficiency is to animals for which the swimming efficiency has been calculated using the same expression as in the current paper. von
Loebbecke et al. (2009) estimated the swimming efficiency of a cetacean using the same numerical method and same definition of efficiency to be 56%. For the Spanish Dancer, a species closely related to Aplysia, as discussed in Chapter 5, the efficiency for an animal of comparable scale was also estimated to be 57%. Thus, Aplysia seems to lag both in terms of swimming speed and efficiency compared to other swimming animals. However, it would seem that the reduction in swimming performance comes with the benefit of decreased kinematic complexity, especially when compared to the Spanish Dancer, and this might have implications for design of swimming softbodied robots that inspired these marine gastropods. In closing, it should be pointed out that the swimming speed and swimming efficiency of elite, Olympic level swimmers performing the underwater dolphin kick is found to be 0.47 BL s-1 and 29% respectively (von Loebbecke et al.
2009). Thus, the Aplysia, with its simple neuromuscular control system, still manages to outperform the best human swimmers.
5.3.4. Ground Effect
It has long been known that broad propulsors operating in close proximity to an underlying surface, usually 10% of the chord of the main wing or less, can lead to significant hydrodynamic benefits. Wing-in-ground effect vehicles, for instance, have a higher lift-to-drag ratio since a higher pressure zone is formed beneath the lifting surface due to flow deceleration, meanwhile induced drag is reduced due to suppression of 3D modes, such as wing-tip vortices
(Rozhdestvensky 2006). ‘Unsteady ground effects’ of canonical geometries, such as pitching
87 airfoils and flapping flexible plates near ground, have also been studied in recent years. It is found that with rigid propulsors, the increase in lift translates to an increase in thrust as well in these unsteady cases under certain conditions (Quinn et al. 2014a; Quinn et al. 2014b; Tang et al. 2016).
A number of studies describing the advantages of birds and fish gliding near the substrate/ground
(Blake 1983b; Hainsworth 1988; Rayner 1991; Nowroozi et al. 2009; Park & Choi 2010) have also been conducted. Since these effects scale with the ground proximity normalized by the chord, fishes that are more dorsoventrally compressed, such as batoids and flatfishes, are thus expected to benefit more when swimming near the ground. Many species of marine molluscs, Aplysia and
Flatworms in particular, habitually swim close to the seabed or in shallow water, hence resulting in potential hydrodynamic consequences.
D Ground
Fig. 5-15 Schematic of a Sea Hare swim with a wall proximity D.
In this section, we present the results on the self-propulsion of Aplysia with fixed baseline kinematics for various ground proximities. Fig. 5-15 illustrates the schematic of an animal swimming at a distance D from the bottom surface of the animal to the ground, the ‘ground’ is modeled by a Dirichlet boundary condition with zero-velocity in the inertial reference frame.
Simulations are performed at four distances D = 3.0, 0.5, 0.25, and 0.1 body-length (BL) respectively. Fig. 5-16 and Fig. 5-17 plot the temporal profiles of surge velocities and surge force
88
Fig. 5-16 Time history of surge velocity at different ground proximity (left); zoomed in at quasi- steady state (right).
Fig. 5-17 Time history of surge force coefficient at different proximity (left); zoomed in at quasi- steady state (right).
coefficients for those four distances respectively. It can be readily seen from these plots that, while both swimming speeds and surge force coefficients increase monotonically with decrease in
89 ground proximity, unlike the previously reported studies for 2D airfoils and flexible flapping plates, these lines almost collapse together for the three cases D = 3.0, 0.5, and 0.25 BL, and only a slight difference is observed for the case very near the ground D = 0.1 BL.
Fig. 5-18 Vortex structures at different ground proximity: (a) D = 3.0 BL; (b) D=0.50 BL; (c) D
= 0.25 BL; and (d) D = 0.10 BL.
The wake patterns at the same instance, including vortex structures, spanwise vorticity and pressure contours, of self-propelled animals swimming near the ground are plotted in Fig. 5-18 and Fig. 5-19. In Fig. 5-18, again, very little difference is seen between the three cases with
90
Fig. 5-19 Vorticity (left column) and pressure (right column) contours at the mid-plane for different ground proximity.
relatively larger distances, whereas for the smallest distance D = 0.10 BL, the spanwise vortices
Vk connecting the paired vortex rings identified in section 5.3.1 are found diminished as they convect downstream due to the interaction with the boundary layer developed near the ground. No significant difference is observed in the vortex rings. A further look at the vorticity and pressure contours in Fig. 5-19 Vorticity (left column) and pressure (right column) contours at the mid-plane for different ground proximity. reveals that there is a slight increase in pressure for the cases D =
91
0.50 BL and 0.25 BL, whereas for the case D = 0.10 BL, the negative (brown) spanwise vortices are diminished significantly due to interaction with the ground, and a low pressure region is developed between the bottom surface of the animal and the ground. Meanwhile, the pressure at the leading edge is found to be smaller compared to other cases, and therefore, variations in lift and thrust are expected for this case.
Table 5-3 summaries the statistical data of swimming performance at the quasi-steady state, and it is found that the improvement in the terminal swimming speed, Froude efficiency, and power coefficient of the case with the closest wall proximity (D = 0.1 BL) is within 1% of the other case, and hence not considered significant. One reason that the ground effect is not noticeable in these simulations might be that in the present study, the kinematics of Aplysia are fixed, while real benthic animals may adjust their body movements accordingly for better performance near the ground. For instance, hovering swimmers, such as darters, expand their ventral halves of the pectoral fins and contract the dorsal halves when hovering near the substrate (Carlson & Lauder
2010). Mottled sculpin, a negatively buoyant fish, holds station by orienting their bodies upstream and extending their large pectoral fins laterally to generate negative lift (Coombs et al. 2007).
Plaice are also found to increase tailbeat frequency with swimming speed at the bottom, and this small change in swimming kinematics presumably reduces cost of transport comparing to cod maintaining the same kinematics near the ground (Webb 2002).
Table 5-3 Swimming performance of Aplysia at different ground proximities
D (BL) 3.00 0.50 0.25 0.10 V T 0.813 0.813 0.813 0.816 η 0.204 0.204 0.204 0.206
CP 0.93 0.93 0.93 0.94
92
Studies of canonical geometries have also been conducted to investigate the parameter space of near-ground benefits. Experimental studies show a monotonic increase in swimming speed of self-propelled flexible rectangular panels up to 25% as they approached the ground, while the energy cost maintained constant (Quinn et al. 2014b). Numerical studies of self-propelled flapping flexible plate near the ground reveal that the swimming performance depends on the ratio of the heaving frequency to the natural frequency F = f / fn, and a beneficial regime 0.5 < F < 1.0 was identified (Tang et al. 2016). As a result, all those previously reported hydrodynamic benefits involve changes in body kinematics. It’s also worth pointing out that (Blake 1979) reported substantial energy savings (up to 60%) for a negatively buoyant species, the mandarin fish, hovering near the substrate. However, the efficiency was found to be essentially the same.
Furthermore, he suggested that the effect of ground on power consumption during forward progression is unlikely to be large.
5.4. Summary
Simulations have been conducted to explore the wake characteristics and swimming performance of the Aplysia. The unique wake generated by self-propelled Aplysia consist of two sets of vortex-rings that extend obliquely away from a centerline in a horizontal plane and another set of vortex rings that extents away in the dorsal direction in the wake. Analysis of the wake shows that the thrust and drag signature of the wake structures is distinct, with the three sets of vortex ring generating thrust and a Karman vortex wake generated by the flow over the body of the animals that is associated with drag. Analysis of the swimming performance indicates the baseline case, with kinematics chosen to match the field video, is able to obtain high propulsive efficiency and relatively high swimming speed. The swimming performance of animals flapping
93 near the ground with fixed kinematics, however, shows very little improvement. Given that these animals are soft bodied invertebrates that have a simple neuromuscular system, and live successfully in an environment subject to large changes in both temperature and salinity, they could serve as inspiration for the design of robust and versatile, bioinspired/bio-hybrid soft swimming robots (Webster et al. 2017). Finally, for the kinematics employed here, vicinity to the ground does not seem to significantly impact the swimming performance of these animals.
94
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also present pseudo-tentacles that resemble nudibranchs. Some species, such as Thysanozoon nigropapillosum and Pseudobiceros hancockanus, swim by passing ruffling waves along the body margin like a stingray. These polyclads have been called ‘scarf-‘ or ‘skirt-dancers’ (Newman &
Cannon 2003) and their swimming gaits have some resemblance to the Spanish Dancer and Aplysia that are described in earlier chapter. However, Flatworm are phylogenetically quite distant from nudibranchs; while nudibranchs belong to the Phylum Mollusca, Flatworms are in the Phylum
Platyhelminthes. Marine Flatworms range from 10 to 50 mm in length, approximately half as broad, and only about 1 mm thick. Most marine Flatworms are benthic and known to feed on a wide range of invertebrate animals such as gastropods and barnacles. Our interest here is in these swimming
Flatworms, and particularly in comparing their swimming performance to the other animals studied here.
While some efforts have been made to identify and document different species of marine
Flatworms, their phylogeny (Baker et al. 2013; Baker et al. 2015; Gregory et al. 2000; Jie et al.
2016; Newman & Cannon 2003), and their mating and reproductive behaviors (Schärer et al. 2004;
Vreys & Michiels 1997), very little has been done to understand the mechanisms of locomotion in these animals. Nevertheless, marine Flatworms, with no skeleton, exhibit well-organized swimming behaviors by virtue of their multi-layers of circular and longitudinal muscles around the body, as well as the powerful dorso-ventral muscles that allow them to fully control their body deformation. These animals are capable of unidirectional swimming, hovering, and making quick turns as commonly seen in vertebrate animals (Zoneku1 2014).
96
(a) (b)
Fig. 6-2 (a) Pseudobiceros hancockanus swimming by lateral flapping (LF). Margin shows more than two full waves and the central axis of the body does not exhibit any significant undulation or deflections. (b) Thysanozoon Nigropapillosum swimming with lateral flaps combined with dorso-ventral pitching of the body (CFP). The mantle in this case shows about 1.5 wavelengths.
Interest in Flatworms has also come from the field of robotics. Kazama et al. (2013) constructed a soft-bodied sheet-like robot inspired by the polyclad Flatworm that was controlled by three motors, two for the lateral flaps and a third one for the body axis. The locomotion of the robot was found to depend on the phase difference between the lateral flaps and body deflection that, the robot was found to swim unidirectionally for phase differences of 0.5π and π, swim backwards for phase difference of 0, and hover for the phase difference of 1.5π. Kano et al. (2012), inspired by Flatworms’ capability of moving over various terrains by changing their locomotory pattern, such as crawling on the substrate and swimming in water, attempted to develop a two- dimensional, sheet-like, multi-terrestrial robot utilizing an autonomous decentralized control scheme that could move adaptively to its environment. The resulting robot was found to be able to crawl over irregular surfaces and the designers claimed that the robot could also swim.
Thus, Flatworms offer an interesting model of swimming by a soft-bodied invertebrate that can complement our study of nudibranchs. This chapter describes our modeling and analysis of
97 the locomotion of marine Flatworms and culminates with a comparison of swimming performance of all the animals studied in this research.
6.2. Methods
6.2.1. Body Geometry and Swimming Kinematics
Marine Flatworms are bilaterally flattened and symmetric. Large species have an elongated body with a flat ribbon-like or oval shape as seen in Fig. 6-1 and Fig. 6-2. Thus, similar to the treatments for Spanish Dancer, the body of a Flatworm can be modeled by a zero-thickness rectangular membrane. The planform used for the Flatworm is of aspect-ratio L:W=3:1, where L and W are the body-length and width respectively. An extensive literature search gives very meagre information about their swimming kinematics. Newman & Cannon (2003) described that the
Flatworms swim by undulating the edges of their body, but no quantitative descriptions were provided in this or other aforementioned studies. Kazama et al. (2013) suggest that the locomotion of Flatworms is generated by the flaps of both their lateral sides and the body axis in order to achieve various swimming behaviors such as forward and backward swimming, hovering and turning. Based on our observations of available field videos (Basler 2012; Zoneku1 2014), we can make the following observations:
(a) there are usually from one to three waves present on the body depending on
species or body sizes (e.g. two waves on Pseudobiceros hancockanus in Fig.
6-2a, one and a half waves on Thysanozoon nigropapillosum in Fig. 6-2b);
(b) the flapping amplitudes vary from case to case; and
(c) in addition to the lateral flaps of the body margin, considerable deflections of
the body axis are also very often present, while the head of the animal is
98
usually held steady (Fig. 6-1 and Zoneku1, 2014). Given that these animals
are swimming in a dynamic environment, it is unknown whether the body
deflection is employed for stability or propulsion.
In order to investigate the propulsive mechanisms of Flatworms, and how their gait affects their swimming performance, we examine two kinematic models: (1) pure lateral flapping (LF) ;
(2) combined lateral flapping and body (CFP) pitching. The motivation to employ this different models is two-fold: first, there is not sufficient data available to determine the most appropriate gait parameters for these animals, and second, we are interested in exploring these parameters for possible application to bioinspired soft swimming robots.
The following sinusoidal functions are used to describe the swimming gait: for the dorso- ventral deflections
pp ,0 sin(2 tT / ) (6-1)
where θp,0 is maximum deflection angle at the tail, T is the cycle period of the motion. For the pitching motion, the leading edge of the body acts as the nodal point and it therefore does not oscillate. For the pure lateral flap case, θp,0 and φ are set to be zero. For the cases with dorso-ventral body undulation,
pp ,0 sin 2 x / t / T (6-2)
For the flapping of the body margin flaps, the movement is described by
1 1 tanhy y / K sin 2 (x / t / T ) (6-3) ff2 0 ,0 where x is the axial coordinate extending from the head to the tail ( 0 xL), y is the spanwise coordinate extending from the plane-of-symmetry to the body margin ( W/ 2 y W / 2 ), θf,0 is the flapping amplitude, λ is the undulation wave length, φ is the phase difference between the
99 pitching and the flapping. The spanwise variations are prescribed by the hyperbolic tangent term, and the parameters K = L and y0 = W / 8 are used for all free-swimming Flatworm simulations performed in this chapter.
Since field videos suggest that there are anywhere from one to three waves present on the body margin, two intermediate wavenumbers: kL/ 1.5 and 2.0 are considered, and a representative flapping amplitude of θf,0 = 75° is used to investigate the propulsive mechanisms for both the LF and CFP models. In addition to the flapping parameters, the pitching amplitude θp,0 =
15° and four phase differences φ = 0, 0.5π, π, and 1.5 π are chosen for the CFP models. Additional kinematic parameters are considered in the later section 7-3.2 for a broad study of swimming efficiency.
Fig. 6-3 Perspective view of kinematic model set (1) with pure lateral flaps (LF). Left:
kL/ 1.5 ; right: k = 2.0. Thick black arrow indicates the swimming direction.
100
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6.2.2. Simulation Setup
The simulation setup is similar to that of the Spanish Dancer and Aplysia in that the animal’s area density is assumed to be 20 kg/m2 , and the corresponding Stokes number 2/LA T is
about 3,800, where AL tan p,0 is the amplitude of body deflection. The computational domain for this set of simulations is of size 12LLL 8 6 , and the grids employed are about 7.5 million with a minimum grid spacing xL0.012 .
6.3. Results
Table 6-1 Kinematic Parameters of Flatworms Swim by Pure Lateral Flapping (LF)
θf,0 (deg) 75 75 45 30 25 20 17.5 15 10 k 1.5 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0
Table 6-2 Kinematic Parameters of Flatworms Swim by Combined Flapping and Pitching (CFP)
θp,0 (deg) 15 15 15 15 15 15 15 15
θf,0 (deg) 75 75 75 75 75 75 75 75 k 1.5 1.5 1.5 1.5 2.0 2.0 2.0 2.0 φ 0 0.5π π 1.5π 0 0.5 π π 1.5π
Results of free-swimming simulations of Flatworms are presented in this section. Also presented are the metric of swimming performance across different species of marine invertebrates
(nudibranchs and polyclads) that have been investigated in this research. Table 6-1 and Table 6-2 summarizes the kinematic parameters for Flatworms that swim using pure flapping (LF) and a combined flapping and pitching CFP) respectively, which are explored in this chapter. As with our previous simulations, the animals are initially released in a quiescent fluid and as body movements
102 ensues, the animal start surging forward, and simulations are conducted beyond where a terminal swimming state is reached.
6.3.1. Wake Topology and Swimming Performance of Flatworms
Fig. 6-5 Left Column: Instantaneous vortex structures at quasi-steady state of the pure flapping case with a wavenumber of k = 1.5: (a) Perspective view; (b) Side view; (c) Dorsal view.
Vortices are colored by the spanwise vorticity. Purple color shows the body structure.
Right Column: Cycle-averaged momentum flux ( uu ) isosurfaces superimposed onto instantaneous vortex structures for the corresponding views. Colored contours: momentum flux; white shades: vortex structures. Brown line shows the outline of the body.
In this section, we focus on the wake topology and performance of the Flatworms models that are simulated here. Left column of Fig. 6-5 shows the vortex structures of the baseline LF case for which the undulatory wavenumber is kL/ 1.5 . The animal is found to shed a series of semi-
103 circular vortex rings linked up from each side of the mantle margin as the wave propagates down to its tail, leads to a staggered pattern in each lateral set of vortices. The center of the body, however, sheds spanwise vortices that connect these lateral vortices.
Fig. 6-6 Wake topology of free-swimming manta rays simulated using a potential flow model:
(a,b,d) vortex structures in dorsal, side, and perspective views respectively; (c) time-averaged streamwise velocity iso-surface. Figure from Fish et al. (2016).
As seen in Fig. 6-5 the right column, four streaks of positive momentum flux ( uu) are identified, and these aligning with the centers of lateral semi-circular vortices and as a result, produce positive a surge force. This suggests that the mantle margin flaps acts as the ‘propeller’ for this animal. In contrast, most of the negative momentum flux is found to wrap the body of the animal and a weak negative momentum flux is found intersecting the spanwise vortices, indicating that most of the drag is produced on central structure of the body that experience little deformation.
This wake pattern is found to resemble that of manta rays that swim by the flapping of their large pectoral fins as shown in Fig. 6-6. For these animals, two sets of interlocked vortex rings are found
104 originating from the fin-tips while no spanwise vortices are observed. Moreover, two distinct velocity jets are found, which align with the centers of the vortex rings indicating thrust production
(Fish et al. 2016).
We now turn to the CFP cases and focus on the effect of phase difference between pitching and flapping. The other parameters for these four cases are highlighted in red in Table 6-2. Fig.
6-7 and Fig. 6-8 show the vortex topology of these four cases in self-propelled motion. It can be readily observed from these plots that, unlike the pure flapping case seen in , adding a small pitching movement (with a maximum deflection of 15°) result in significant changes in the wake topology. For cases with phase differences of φ = 0 and φ = 1.5π, the vortex topology resembles that of the Spanish Dancer described in section 4.3.2. There is a relatively we-defined bifurcated vortex wake that the two sets of vortex rings (Vt and Vb) shed when the tail changes direction and they are designated as ‘primary’ vortices by Tytell & Lauder (2004). These vortex rings are connected by two sets of thin vortex ‘contrails’ (Vc) shed by the tail as it sweeps up and down, indicating that the wake topology is dominated by the body deflection rather than the lateral flaps.
For the case of phase difference φ = 1.5π, in addition to the staggered array of vortex rings, semi- circular vortex rings (Vlat) shed from the body margin undulation are also observed near the first two to three staggered vortex rings, but these dissipate quickly as they convect downstream (see right column in Fig. 6-8). These lateral semi-circular vortex rings Vlat are more obvious in the cases with φ = 0.5π and φ = π; they form in between the upper and lower sets of vortex rings vertically, but outside the two vortex ‘contrails’ laterally, which leads to a much more complex vortex pattern.
Meanwhile, the size of the lower and upper sets of vortex rings shed due to the body deflection are found to be much smaller in these two cases.
105
Fig. 6-7 Vortex structures of cases with a combined body pitching and mantle flapping (CFP):
φ = 0 (left column); φ = 0.5π (right column). From top to bottom: perspective, side, and dorsal views. Black arrows point to the top and bottom sets of vortex rings, Vt and Vb, and the two
‘contrails’, Vlat, respectively. Red arrows point to the lateral semi-circular vortex rings Vlat. The wavenumber of lateral flaps is k = 1.5.
106
Fig. 6-8 Vortex structures of cases with a combined body pitching and mantle flapping: φ = π
(left column); φ = 1.5π (right column). From top to bottom: perspective, side, and dorsal views.
Black arrows point to the top and bottom sets of vortex rings, Vt and Vb, and the two ‘contrails’,
Vlat, respectively. Red arrows point to the lateral semi-circular vortex rings Vlat. The wavenumber of lateral flaps is k = 1.5.
To further analyze the propulsive mechanism such as thrust generation, we compare the cycle- averaged momentum flux (uu) pattern of this CFP models to the LP case, as well as wake of the
Spanish Dancer and Aplysia. The cycle-averaged momentum flux patterns of these four cases are plotted together with a representative instantaneous vortex pattern at quasi-steady state in Fig. 6-9 and Fig. 6-10. For cases with phase differences of φ = 0 and φ = 1.5π, since a clearly bifurcated vortex topology is generated, the momentum flux pattern is also similar to that of the Spanish
107
Dancer in that two streaks of positive momentum flux are identified aligning with the centers of the staggered vortex rings; whereas negative momentum fluxes are found intersecting those vortex contrails (left column of Fig. 6-9 and right column of Fig. 6-10). On the other hand, for the case with phases difference of φ = 0.5π, similar to the pure lateral flaps case shown in Fig. 6-5, four streaks of positive momentum flux are found aligning with the lateral semi-circular vortex rings
Vlat instead, indicating most thrust is generated by the lateral flaps. For the case with phase difference of φ = π, two additional streaks of positive momentum flux are identified that are aligned with the staggered vortex rings as well, while the rest vortices correspond to negative fluxes.
Another observation worth pointing out is that negative momentum flux regions are found to wrap about 2/3 of the anterior portion of the body, positive momentum fluxes, in contrast, are only observable at the posterior where body deflection is relatively large. This is surprising since although the body pitch is negligible at the anterior, the amplitudes of lateral flaps are constant along the body axis (Fig. 6-3). This small pitch, as a consequence, seems to be of considerable significance in propulsion.
108
Fig. 6-9 Cycle-averaged momentum flux (uu) iso-surfaces superimposed onto instantaneous vortex structures at a quasi-steady state for cases with kinematics of a combination of body pitching and mantle flapping: left column, φ = 0; right column, φ = 0.5π. Green indicates positive momentum flux in the downstream and brown indicates negative flux; white shades denote the vortex structures. Thick brown edges denote the outlines of the body shape. The wavenumber of lateral flaps is k = 1.5.
109
Fig. 6-10 Cycle-averaged momentum flux iso-surfaces ( uu) superimposed onto instantaneous vortex structures at a quasi-steady state for cases with kinematics of a combination of body pitching and mantle flapping: left column, φ = π; right column, φ = 1.5π. Green indicates positive momentum flux in the downstream and brown indicates negative flux; white shades denote the vortex structures. Thick brown edges denote the outlines of the body shape. The wavenumber of lateral flaps is k = 1.5.
The wake flows on the body symmetry plane of these four cases are plotted in Fig. 6-11. For steady swimming when thrust and drag are balanced on cycle-average, as pointed out by Tytell &
Lauder (2004), a momentum balance in the wake is expected. The wake topology of self-propelled 110
Flatworms with phase differences φ = 0 and φ = 1.5π, as well as that of the Spanish Dancer, for instance, is found to have strong downstream momentum jets aligned with the centers of those primary vortex rings or lateral semi-circular vortex rings. These downstream jets are balanced by the upstream momentum jets behind the center structure of the body held nearly motionless as seen in Fig. 6-11(a) and (c). The magnitude of momentum flux of the case φ = π, however, is smaller than that of case φ = 0, as the radii of the top and bottom vortex rings are smaller, and moreover, some positive momentum flux is contributed by other lateral semi-circular vortex rings as discussed before. This wake morphology with spatial separation in thrust and drug, is therefore similar to that of previously described carangiform and pectoral fin swimming (Fig. 6-12b, c)
(Lauder & Drucker 2002; Drucker & Lauder 2000; Müller et al. 1997). For the case φ = 1.5π, the momentum flux separation is less distinguishable. More interestingly and in contrast to the other cases, for the case φ = 0.5π, no significant downstream jets are observed. Rather vertical jets are formed due to the vertically oriented vortex rings as observed in wakes of swimming eels’ (Fig.
6-12a) (Tytell & Lauder 2004).
This spatial separation in momentum flux, and hence in thrust and drug, is due to the separability of the thrust-producing structures (propellers such as caudal/pectoral fins) from the drag-producing body. Fig. 6-11, together with the body envelopes shown in Fig. 6-2 reveal that, the posterior portion of the body shows considerable movements in cases a, c, and d, and therefore acts as the counterpart of caudal fins in carangiform swimmers. For case b for which φ = 0.5π, and there are k = 1.5 waves on the body margin, the body pitching and flapping envelope is 90 degrees out-of-phase and this results in a much smaller sweeping area at the tail. Thus, the deformation in the posterior part is no greater than the anterior part, the body margin at the anterior therefore would contribute no less to the thrust generation compared to the posterior portion.
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(a) (b)
(c) (d)
Fig. 6-11 Instantaneous velocity vectors superimposed on momentum flux contours at the body symmetry plane for cases with combined kinematics of body pitching and mantle flapping: (a)
φ = 0; (b) φ = 0.5π; (c) φ = π; (d) φ = 1.5π. Arrows represent the flow velocity magnitude and direction. Momentum flux is shown in color in the background, vortex structures are shown as white shades, and the animal is outlined in light purple. The wavenumber of lateral flaps is k =
1.5.
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Fig. 6-12 Digital particle image velocimetry (DPIV) results of 2D wake flows. (a)
Representative flow field behind an eel at 90% of the tail beat cycle, lateral jets are observed perpendicular to the body axis. Black arrows denote the velocity magnitude and flow direction, the eel’s tail is shown in blue at the bottom. (Figure from Tytell & Lauder 2004) (b-c) Velocity vectors fields behind the pectoral fin of a bluegill sunfish on the vertical and horizontal light sheets. Yellow arrows represent the velocity vectors. A central jet oriented downstream is identified between two counter-rotating vortices. (Figure from Lauder & Drucker 2002)
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Fig. 6-13 Vortex structures of cases with a combined body pitching and mantle flapping: φ = 0
(left column); φ = 0.5π (right column). From top to bottom: perspective, side, and dorsal views.
The wavenumber of lateral flaps is k = 1.5.
Fig. 6-13 and Fig. 6-14 show the wake topology of four cases with wavenumber k = 2.0, and phase differences between the lateral flaps and body deflection are φ = 0, φ = 0.5π, φ = π, and φ =
1.5π respectively. In comparison with the cases with k = 1.5, as there are a half more waves on the body, as illustrated in Fig. 6-2, the body pitching and flapping envelopes of phase difference φ for wavenumber k = 1.5 is found to be similar to that of phase difference φ + π for wavenumber k =
2.0. We would therefore expect the wake morphology for any phase difference φ for wavenumber
114 k = 1.5 to resemble the wake morphology of phase difference φ + π for wavenumber k = 2.0. Not surprisingly, Fig. 6-13 and Fig. 6-14 confirm this conjecture.
Fig. 6-14 Vortex structures of cases with a combined body pitching and mantle flapping: φ = π
(left column); φ = 1.5π (right column). From top to bottom: perspective, side, and dorsal views.
The wavenumber of lateral flaps is k = 2.0.
Fig. 6-15 show the temporal profiles of surge velocity and surge force coefficients from the initiation of the swimming to the terminal state for the LF case together with other four CFP cases with various phase differences between the body margin undulation and dorso-ventral deflection.
For all five simulations, the wavenumber of lateral flaps is k = 1.5 and other parameters are listed
115 in Table 6-1 and Table 6-2 highlighted with red color. As observed in Fig. 6-15, for the pure flapping case and the case φ = 0.5π, the variations in both surge velocity and force coefficients magnitudes are quite small compared to other cases. This observation, again, can be explained by
Fig. 6-1 and Fig. 6-2 of body kinematics and vortex topology discussed before. For the case of pure lateral flaps, this constant flapping amplitude leads to equal contribution to thrust generation along the body axis. Whereas for the case with a combination of flapping and pitching, due to the phase difference of φ = 0.5π between these two modes, the deformation in the posterior region is not as significant as other cases. Thus, larger body deformation or moving amplitudes would result in higher fluctuations in forces or velocities.
Fig. 6-15 Time history of surge velocity (left) and surge force coefficients (right) for the pure lateral flap case (red line) and four cases with various phase differences between the body margin undulation and dorso-ventral deflection. The wavenumber of lateral flaps is k = 1.5 for all five simulations.
The corresponding metrics of swimming performances for these cases are summarized in Table
6-3. As listed in the table, all four cases with combined flapping and pitching (CFP) modes outperform the pure flapping (LF) case in terms of terminal swimming speed and Froude efficiency.
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The power coefficient of the pure flapping case, also has a relatively small value compared to the two cases φ = 0.5π and φ = π. The final speeds and Froude efficiencies of these two cases are also found to be significantly smaller compared to cases φ = 0 and φ = 1.5π, which have relative well- organized wake morphologies with two distinct sets of vortex rings. In summary, a small pitch in combination with body undulation can greatly improve swimming performance in both swimming speed and efficiency, and a well-organized wake topology is likely to indicate higher swimming performance.
Table 6-3 Comparison of swimming performance for various kinematics
CFP Case LF φ = 0 φ = 0.5π φ = π φ = 1.5π
* Vt 0.55 1.22 0.75 0.64 1.01 0.20 0.43 0.25 0.25 0.39
CP 1.60 0.69 2.21 2.88 1.11
6.4. Summary
Simulations have been conducted to explore the performance and wake topology of marine
Flatworms. Simulations results show that addition of a small magnitude of body pitch results in significant changes in the wake pattern and greatly improves the swimming performance in comparison with the pure flapping case. A well-organized wake with distinct vortex rings is found to be a signature of high swimming performance. The swimming performance is also affected by the phase difference between the lateral flaps and dorso-ventral deflection of the body.
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CHAPTER 7. Propulsive Efficiency of Marine Invertebrates
The efficiency of self-propelled biological or bioinspired swimmers has been of great interest
(Maertens et al. 2015; Bale et al. 2014; Tytell 2010; Borazjani & Sotiropoulos 2010; Tytell &
Lauder 2004). Unlike propeller driven engineered vehicles, biological and bioinspired swimmers employ very disparate and complex modes/gaits for propulsion and comparing the swimming performance across different swimmer is a non-trivial exercise. This is particularly so due to two factors: (1) for a body swimming at its terminal velocity, the net average force on the body is zero and (2) it is usually difficult to separate the ‘propeller’ from the ‘hull’ of the swimmer. As explained earlier, we have employed three metrics to assess swimming performance: terminal velocity in body lengths per second, the Froude efficiency and the power coefficient. Using these three metrics we not compare the swimming performance across the large number of simulations that have been carried out here.
So far, we have modeled combined mantle flapping and body pitching (CFP) inspired from
Spanish dancer and Flatworms, and pure lateral flapping (LF) inspired from Aplysia and
Flatworms. In order to cover as large a range of kinematics as possible, we also add to these models, two other models that only involve body pitching (BP). The first case is a pitching plate (BP1) which is essentially modeled using Eqn. (6-1) without any mantle flapping. The parameters for which simulations have been conducted for this model are listed in the table blow.
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Table 7-1 Kinematic Parameters for BP1 model.
p0 /6 /8 /12 / 24
k p 1.5 1.5 1.5 1.5
The second model (BP2) involves an undulatory pitching wave that propagates from the head to the tail and is given by Eqn. (6-2), and the simulations for this model are for the following parameters.
Table 7-2 Kinematic Parameters for BP2 model.
k ' p 0.16 ------
All in all, a total of 38 simulations are included in this meta-analysis of swimming performance and Table 7-3 summarizes the details of all the cases simulated here.
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Table 7-3 Details of Kinematic Models and their Corresponding Representative Morphology
Investigated in this Study
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Fig. 7-1 Froude efficiency against swimming speed at the quasi-steady state. Magenta diamonds:
Spanish Dancer with combined swimming modes of body deflection and mantle margin flapping; empty magenta diamonds: variations in planforms of Spanish Dancers; filled magenta diamonds: variations in Stokes numbers. Green diamonds: Flatworms with combined modes of body deflection and body margin flapping; empty green diamonds: wavenumber k 1.5 ; filled green diamonds: wavenumber k 2.0 . Red circles: Aplysia with various flapping amplitudes and wavenumbers. Blue circles: Flatworms with pure lateral flaps of various amplitudes and wavenumbers. Cyan squares: self-propelled flat plates with various body deflection amplitudes, wavenumber is k 0.16 . Black squares: self-propelled flat plates with various body deflection amplitudes, wavenumber is k 1.5 .
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Fig. 7-2 Power coefficient against swimming speed at the quasi-steady state. Magenta diamonds:
Spanish Dancer with combined swimming modes of body deflection and mantle margin flapping; empty magenta diamonds: variations in planforms of Spanish Dancers; filled magenta diamonds: variations in Stokes numbers. Green diamonds: Flatworms with combined modes of body deflection and body margin flapping; empty green diamonds: wavenumber k 1.5 ; filled green diamonds: wavenumber k 2.0 . Red circles: Aplysia with various flapping amplitudes and wavenumbers. Blue circles: Flatworms with pure lateral flaps of various amplitudes and wavenumbers. Cyan squares: self-propelled flat plates with various body deflection amplitudes, wavenumber is k 0.16 . Black squares: self-propelled flat plates with various body deflection amplitudes, wavenumber is k 1.5 .
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Fig. 7-1 plots the Froude efficiencies against the corresponding swimming speeds at the terminal state, which includes swimming modes of pure flapping, pure pitching, and a combination of both. For both undulatory (flapping) and deflecting (pitching) motion along the body axis, the amplitudes and wavenumbers are varied in the previously presented models of Aplysia and
Flatworms. In addition, the dorso-ventral body nodal point can also be prescribed; the deflection point for the Spanish Dancer, for example, lies at about 30% of the body length near the anterior; whereas the deflection points of the Flatworms and pitching flat plates are placed at the leading edge of the body axis. A large set of kinematic models are therefore obtained by varying these kinematic parameters. As observed from Fig. 7-1 and Fig. 7-2, the swimming speeds range from
0.02~1.77 BL/s, and thus span about two order-of-magnitude.
According to Fig. 7-1 and Fig. 7-2, the Spanish Dancer is found to outperform all other swimmers in both swimming speeds, Froude efficiencies, and power coefficients. The filled magenta diamonds denote the set of simulations with the same baseline rectangular shape used in
CHAPTER 4, and the Froude efficiency and swimming speeds are both found to increase with
Stoke number. The empty magenta diamonds denote the set of simulations with the same Stokes number (flapping/pitching amplitudes are thus the same) but the planform is varied among these cases. As found before, the planform with a wider caudal region is found to have the highest Froude efficiency. The Flatworm models where we adopt a combined modes of flapping and pitching similar to that of Spanish Dancer, but without head pitching are show in the plot as green diamonds.
The filled green diamonds denote lateral flaps of wavenumber k 1.5 and the empty diamonds denotes k 2.0 . The plot shows that the smaller wavenumbers with same flapping and pitching amplitudes result in larger swimming speeds as the traveling waves on the body have larger wave velocities. Since the Spanish Dancer and Flatworms have similar swimming kinematics, their
123
Froude efficiencies and power coefficients are also observed as fall along the same trend on this plot.
The Aplysia with parapodial flapping and Flatworms performing pure lateral flapping are also found to follow a similar trend. The Froude efficiency and speed are found to approach zero as the flapping amplitude is decreased. The two sets of pure pitching simulations are found to fall in two different categories: the case with oscillatory pitch about the leading edge (BP1) generated some of the highest swimming speeds but with low efficiency. On the other hand, the pitching undulatory wave (BP2) generated some of the lowest swimming speed but with high efficiency. Note that BPI is more akin to the s-called carangiform swimming exhibited by fast swimmers such as tuna, dolphins and others (Dewar & Graham 1994; Fish & Hui 1991; Donley & Dickson 1997), whereas
BP2 is similar to anguilliform swimming typical of slower swimmers like eels. Thus the above results are consistent with what is known about the swimming of these animals.
The plot of power coefficient versus terminal velocity show trends are more difficult to interpret due to the large range of the power coefficient. Nevertheless, it too shows that pure lateral flapping or both pitch undulation are not very effective and that pitching about the leading-edge could be an effective way to swim. Of course, we note that such a swimming is difficult to achieve in reality since it requires the generation of torque about the leading-edge.
7.1. Summary
Propulsive efficiency including Froude efficiency and power coefficient are examined for various swimming invertebrates well as canonical geometries with a wide range of kinematic models. It’s found the pure lateral flaps of different planforms (Flatworms and Aplysia) fall into the same category, while the combined flapping and pitching motion (Spanish Dancer and
Flatworms with pitching) falls into another.
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CHAPTER 8. SUMMARY
This work had used self-propelled Spanish Dancer, Aplysia, and marine Flatworms as model organisms to explore a variety of issues associated with the hydrodynamics of free-swimming in invertebrate animals, including swimming speeds, wake patterns, propulsive mechanisms, and propulsive efficiency. This is important from the perspectives of both evolutionary biology and potential engineering applications in soft-bodied robots.
A high-fidelity immersed boundary method (ViCar3D) was employed as a basic tool to simulate the free-swimming of these animals. The solver was modified so as to solve the flow equations in a conservative form in a non-inertial reference frame that translates with the animal.
The animal-fluid coupled system was solved in a sequential way. Multiple benchmark cases of flows with moving bodies, such as two-dimensional forced inline oscillations of a cylinder and flow induced motion of a settling sphere, were used to validate and assess the capability and accuracy of the current method. Given that the definition of efficiency of self-propelled motion remains ambiguous, especially for situations where the ‘propeller’ is inseparable from the ‘hull’ of the swimmer, two metrics, the Froude efficiency and power coefficient, are used to evaluate swimming performance were proposed.
Analysis of swimming hydrodynamic of three marine invertebrates: the Spanish Dancer,
Aplysia and marine Flatworm was then presented. For all these simulations, first the corresponding free-swimming kinematic were identified from field videos, and then described by appropriate mathematical functions to create the corresponding kinematic model. The Spanish Dancer is found to swim with a complex gait that combines notal flapping and with body bending. Simulation
125 reveal that the propulsive efficiency and the terminal swimming speed of the nudibranch Spanish
Dancer is quite comparable to other swimming animals and it also increases with Stokes number.
The wake of the Spanish Dance was found to be dominated by a bifurcated train of vortex rings that resemble the wake pattern of simple, low-aspect-ratio flapping foils. These vortex ring-type structures were found to align with jets that were directed downstream, and thus are associated with thrust production. There were also other vortices that were associated with the hull drag that was balanced by the thrust over a cycle at the terminal state. Analysis of the effects of variations in planform shape indicated that a planform with a slightly wider caudal region enhances swimming performance, and this might explain the presence of this feature on actual animals.
The Aplysia, on the other hand, was found to swim by flapping of their two large, wing-like parapodia. Analysis of its wake characteristics shows that the wake is quite different compared to most other swimming animals. This unique wake consisted of three distinct trains of vortex rings with two extending obliquely away from the centerline of the horizontal plane and a third one extending away from the dorsal side. These vortex rings, were found to be associated with thrust generation. A Karman vortex wake was also identified behind the central structure of the animal and was found to be associated with the “hull” drag. Effects of variations in kinematics, such as flapping amplitudes and wavelengths, on swimming performance were investigated, and it was found that the baseline case with kinematics chosen to match the field video has relatively high propulsive efficiency and relatively high swimming speed. While the Aplysia is very often observed swimming near the sea substrate for feeding, and it has a relatively dorsal-ventrally flattened body, no significant improvement in swimming performance was found while swimming near the ground.
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For marine Flatworms, two kinematic models, pure lateral flapping (LF) and combined lateral flapping and body pitching (CFP), were studied, since these animals are employing a wide variety of body kinematics. Comparison of the wake characteristics of these two models show that the
CFP model, with the addition of a small magnitude of body pitch results in significant changes in the wake pattern and noticeable improvements in swimming performance compared to the LF model. The phase differences between the body margin undulation and pitch of the CFP model were also found to affect the swimming performance. Furthermore, the simulations suggest that a well-organized wake with distinct vortex ring structures generally signifies a high swimming performance.
Finally, the free-swimming performance, as evaluated by Froude efficiency and power coefficient, was examined for a wide variety of body shapes and kinematics associated with these marine invertebrates. This comparison found that the LF model of different planforms (e.g.,
Flatworms and Aplysia) falls into the same category of generally low swimming performance, whereas the CFP model (e.g. Spanish Dancer and Flatworms with pitch) falls into a category with higher swimming performance.
The current research represents one of the first detailed investigations of swimming hydrodynamics of these marine invertebrates and many future avenues of research in this area exists. The three animals studied here represent a very small cross-section of the swimming gaits that are observed in this group of animals. Many other bizarre but seemingly effective swimming gaits remain to be explored (Farmer 1970). Similarly, soft swimming robots are starting to be developed by various research groups, and computational modeling of the type employed here, could be used to optimize the design of these robots.
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Vita
Zhuoyu Zhou was born in Ningxiang, Hunan, P. R. China in 1990. She received her B.S. degree in Thermal Science and Power Engineering from the University of Science and Technology of China, P. R. China in 2011, and M.S. in Mechanical Engineering from Michigan Technological
University in 2013. She enrolled in the Ph.D. program in the Mechanical Engineering Department at the Johns Hopkins University and joined Professor Mittal’s group in 2013. Her research focuses on employing computational modeling to investigate the hydrodynamics of free-swimming marine invertebrates. Starting in March 2018, Zhuoyu will work for Simerics Inc., where she will conduct
CFD simulations/analysis on “real” pumps and motors.
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