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How to make a tuna burst : the role of angle of attack in the production of thrust Whale, James Callum Andrew 2016

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HOW TO MAKE A TUNA BURST:  THE ROLE OF ANGLE OF ATTACK IN THE PRODUCTION OF THRUST  by  James Callum Andrew Whale  B.Sc., The University of British Columbia, 2012   A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF SCIENCE  in  THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES  (Zoology)   THE UNIVERSITY OF BRITISH COLUMBIA  (Vancouver)  July 2016   © James Callum Andrew Whale, 2016   ii  Abstract Tuna—along with whales and lamnid sharks—utilise thunniform locomotion, a mode of swimming that optimises efficiency at high speed and isolates thrust production to the caudal fin.  Thunniform performance is controlled by adjustments in the way the caudal fin interacts with fluid flow which, in turn, determines thrust and efficiency.  The effect of tail motion on performance provides insight into the link between locomotor muscle biomechanics and hydrodynamics; these insights can be used to mimic and optimise animal motion in a robotic context.   This study focuses on how the maximum angle of attack (    ), contributes to tuna cruising and bursting, and the corresponding effects on fluid flow.  I hypothesise that cruising tuna do not adjust      to modulate thrust but instead vary amplitude via Strouhal number.  I also hypothesise that      affects thrust by changes in vorticity shed by the tail.  To study these phenomena, I constructed a tuna tail model 3-D printed from CT scan data of a tuna tail.  I then oscillated this model in a water tunnel across a range of biologically relevant motions.  I calculated thrust and efficiency from direct measurements of force and torque and then used ink-flow visualisation and particle image velocimetry to reveal the resulting flow structures.  The results indicate that the efficiency optimum of      peaks around 15° with the thrust optimum beyond 30°.  Mechanistically, an increase of      increases the magnitude of the resultant force but angles it to the side, increasing the amount of wasted lateral energy.  Increasing      increases the size and strength of shed vortices eventually causing shedding of an additional leading edge vortex at midstroke.   iii   These results, paired with red muscle work loop data, suggest that during cruise the      undergoes minimal variation, and suggest that in order to take advantage of the additional thrust that high values of      provide, white burst muscles need to advance peak force timing.  In addition to contributing to a better understanding of the hydrodynamics of swimming and the associated musculature, these results also offer insight into the field of biomimetics and the construction of fish-mimicking robots such as AUVs.  iv  Preface This thesis is original, unpublished work by the author, J. Whale. The tuna tail model CT scan data in Section 2.2.1 was from M. Delepine.  The motion system hardware Section 2.2.5 was designed primarily by A. Richards, O. Barannyk, and M. Delepine.  The acroBASIC program in Section 2.2.5 was modified from a version written by A. Richards and M. Delepine.  The LabVIEW program in Section 2.2.6 was modified from a version written by B. Bocking.  All data processing was done with MATLAB programs that were written by myself.  The force data and ink flow-vis data were collected by me.  The PIV data were collected with the help of M. Rahimpour. The data analyses are my original work.  v  Table of contents Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iv Table of contents ............................................................................................................................v List of tables................................................................................................................................ viii List of figures ................................................................................................................................ ix List of symbols and abbreviations .............................................................................................. xi Acknowledgements .................................................................................................................... xiii Dedication .....................................................................................................................................xv Chapter 1: Introduction ................................................................................................................1 Chapter 2: Research chapter ........................................................................................................7 2.1 Theory ............................................................................................................................. 9 2.1.1 Swimming kinematics ............................................................................................... 10 2.1.2 Flow conditions, the choice of swimming speed and tail scaling ............................. 14 2.1.3 Motion regime summary ........................................................................................... 17 2.1.4 Strouhal number and the heave waveform................................................................ 18 2.1.5 Pitch waveform and angle of attack profile .............................................................. 20 2.1.6 Phase angle................................................................................................................ 23 2.1.7 Tail material and stiffness ......................................................................................... 25 2.2 Methods......................................................................................................................... 27 2.2.1 Tail model design ...................................................................................................... 27 2.2.2 Rod effect model ....................................................................................................... 30 2.2.3 Tail apparatus ............................................................................................................ 31 vi  2.2.4 Flow channel ............................................................................................................. 34 2.2.5 Motion system ........................................................................................................... 36 2.2.6 Force measurement ................................................................................................... 39 2.2.7 Post-processing and analysis..................................................................................... 40 2.2.8 Ink-flow visualisation ............................................................................................... 43 2.2.9 Laser PIV visualisation ............................................................................................. 49 2.3 Results ........................................................................................................................... 50 2.3.1 Statics ........................................................................................................................ 50 2.3.2 Dynamics: Direct measurement ................................................................................ 52 2.3.3 Dynamics: Flow visualisation ................................................................................... 62 2.3.4 Effect of apparatus .................................................................................................... 69 2.4 Discussion ..................................................................................................................... 73 2.4.1 Mechanics of angle of attack and Strouhal number .................................................. 73 2.4.2 Muscular implications of angle of attack control ..................................................... 76 2.4.3 Flow characteristics of αmax ....................................................................................... 83 2.4.4 Comparison with similar studies ............................................................................... 90 Chapter 3: Conclusion .................................................................................................................94 3.1 Future improvements and directions ............................................................................. 94 3.2 Concluding remarks ...................................................................................................... 98 Bibliography ...............................................................................................................................100 Appendices ..................................................................................................................................106 Appendix A Equipment details ............................................................................................... 106 A.1 VeroWhitePlus RGD835 material properties ......................................................... 106 vii  A.2 Water tunnel ............................................................................................................ 106 A.3 Load cell datasheet .................................................................................................. 107  viii  List of tables Table 1 Summary of key kinematic parameters ........................................................................... 18   ix  List of figures Figure 2.1 Lift-based thrust in sailboats, illustrating two coordinate decompositions. ................ 11 Figure 2.2 a. Lift-based thrust for a static foil; b. Lift-based thrust for a flapping foil. ............... 13 Figure 2.3 Illustrations of different possible pitch-heave phase angles. ....................................... 24 Figure 2.4 Foil designed to measure the effect of the apparatus rod ............................................ 31 Figure 2.5 Comparison of bubble entrainment between three iterations of the apparatus ........... 32 Figure 2.6 Design of third iteration tail apparatus visualised in SolidWorks ............................... 34 Figure 2.7 Flowchannel and motion system with apparatus (second iteration) and tail. .............. 36 Figure 2.8 Simplified schematic of motion system ...................................................................... 37 Figure 2.9 a. Ink visualisation tail components showing inside and outside faces ....................... 45 Figure 2.10 a. Ink visualisation tail being epoxied; b. Ink flow in operation with all ports open. 46 Figure 2.11 Statics results for tail angle of attack = -30° to 30° at 1.4m s−1, Re = 50,000 ........... 51 Figure 2.12 Performance as a function of Strouhal number and maximum angle of attack ......... 52 Figure 2.13 Data from Figure 2.12 plotted as thrust and efficiency contours .............................. 53 Figure 2.14 Average power and performance ............................................................................... 55 Figure 2.15 Resultant vectors over one tail beat showing displacement and force through time . 57 Figure 2.16 Thrust vector component over one tail-beat .............................................................. 58 Figure 2.17 Instantaneous values over the course of one stroke ................................................... 59 Figure 2.18 Simulated work loops of lateral force and torque for different αmax ......................... 61 Figure 2.19 Ink flow visualisation for St = 0.35 at 1 BL s−1 ......................................................... 63 Figure 2.20 Particle image velocimetry for St = 0.35 at 2 BL s−1................................................. 64 Figure 2.21 Vertical ink flow-visualisation for St = 0.35 of the ventral lobe of the tail ............... 66 Figure 2.22 Comparison between ink and PIV flow visualisation ............................................... 68 x  Figure 2.23 The effect of the vertical strut on the thrust coefficient and the efficiency of a foil . 71 Figure 2.24 Time series of interaction between rod and dorsal lobe ............................................ 71 Figure 2.25 PIV visualisation of wake interaction between rod and dorsal lobe ......................... 72 Figure 2.26 Simplified illustrations of von Kármán wake street types......................................... 84  xi  List of symbols and abbreviations Symbol Unit Property   °, radians Angle of attack      °, radians Maximum angle of attack CL — Coefficient of lift of tail CT — Coefficient of thrust of tail ηp — Propulsive efficiency of tail f Hz Tail-beat frequency FLat N Lateral force Fx N Force measured by load cell parallel to the tail orientation axis Fy N Force measured by load cell perpendicular to the tail orientation axis   m Heave amplitude (at the pitching-axis)  ̇ m s−1 Heave velocity (at the pitching-axis) L m Characteristic length MAC m Mean aerodynamic chord; 0.0358 MZ Nm Torque ψ °, radians Phase shift by which pitch waveform leads heave waveform ϕ °, radians Relative flow angle Re — Reynolds number ρ kg m−3 Density; 998.2 for freshwater at 20℃ s m Span length of tail; 0.201 S m2 Planform area of tail; 0.00632 Swb m2 wetted surface of body St — Strouhal number xii  Symbol Unit Property Strod — Strouhal number of pitching axis Sttip — Strouhal number of tail tip t s Time T N Thrust TBF Hz Tail-beat frequency   °, radians Pitching angle  ̇ ° s−1, radians s−1 Angular velocity U m s−1 Free stream water velocity υ m2 s−1 Kinematic viscosity; 1.004×10-6 for freshwater at 20℃   xiii  Acknowledgements First off, I thank my supervisor, Bob Shadwick, who inspired me daily with his seemingly effortless creativity and ability, while teaching me to balance both working and enjoying my life as a grad student.  Thanks to my committee members:  Doug Altshuler, who has watched me transition from someone who studies things that flap and honk to things that flap and burble; John Gosline, for his warm and knowledgeable nature; and Colin Brauner, for his friendly presence and his quick editing.  I thank my defence committee for their questions, which were simultaneously challenging, astute, and quite a lot of fun.  Thanks too to Margo, whose scientific dedication and ingenuity are an inspiration to me. UBC Zoology is truly a great place to work, and I am honoured and grateful to have spent time here in the company of so many brilliant, affable folk.  A big thank you to Marc Delepine for his original foray into thunniform research.  I am indebted to Peter Oshkai at UVic for his collaboration and trust, and am grateful for the help of Mostafa Rahimpour in all things laser.  Thanks to Gabor Szathmary for his enthusiastic generosity and thought-provoking conversation. I would not have made it through with my sanity intact without my labmates—reminding me that there is a time for steadfast focus, and a time for home-brewed beer at the Pecker and Peduncle, and frisbee in the sunshine.  I‘m particularly grateful to the office crew: Benny for always providing insight, balance and good cheer; Kelsey for her quick laugh and generous wit; Trisha for her dependable friendship and good company; and Hannah for her good advice and sonorous caw.  xiv  Above all, I am grateful for the support of my parents, my sisters and my close friends, who patiently heard out my various fears and frustrations throughout, and helped me to see my way through to the other side.  In the process, my loved (and tolerant) ones learned far more than they ever expected about thunniform locomotion, which I am certain will come in very handy at parties. xv  Dedication It is easy enough to be pleasant,      When life flows by like a song,  But the man worth while is one who will smile,  When everything goes dead wrong.  –Ella Wheeler Wilcox   1  Chapter 1: Introduction For tuna, life is movement: they prowl the vast empty corridors of the open ocean without rest, and upon encountering prey can give chase at over 20 body lengths per second (Fierstine and Walters, 1968).  The full story behind this feat of acceleration is yet unknown, but we know that kinematics play a central role.  My focus is the role of two kinematic parameters of tail motion – maximum angle of attack and Strouhal number – in cruising and the transition to bursting.  To this end I directly measure the forces and torque in play for a replica tuna tail undergoing simulated tuna swimming motion, and look at how different motions affect these values throughout a tailbeat.  With force and torque measurements, I quantify performance and consider how it is affected by kinematics.  Using ink and laser flow visualisation techniques, I investigate how angle of attack impacts the development of vortices.  My story builds on the story told in Delepine (2013), adding new visualisation techniques and an upgraded apparatus. Tuna are not alone in their high speed abilities; they are among three diverse taxa that all swim using thunniform locomotion: the tunas (Thunnus), the lamnid sharks (Lamnidae), and the whales (Cetacea) (Donley et al., 2004).  After hundreds of millions of years evolving along different pathways, these groups betray their evolutionary convergence through their physiology, hydrodynamics, and lifestyle.  Physiologically, each taxon has developed a version of endothermy to keep its cruising muscles warm even during sojourns to the chilly depths of the ocean (Amano and Miyazaki, 1993; Musyl et al., 2003; Sepulveda et al., 2004; Shadwick and Syme, 2008).  All taxa have honed a bauplan which shifts their muscle mass forward so that their tail fin has less of an inertial burden in its endless oscillations; motion is restricted to the posterior third of the streamlined, tear-drop shaped body; the peduncle is equipped with keels to 2  ease its passage through its propulsive arc; and the high aspect ratio tail is shaped like a crescent moon and displays a perfect hydrofoil profile in cross-section (Donley et al., 2004; Pabst, 2000; Syme and Shadwick, 2011).  Hydrodynamically, these taxa are united in their use of ―lift-based‖ thrust, characterised by the hydrofoil tail which establishes bound circulation and creates force through regions of high and low pressure, similar to an airplane or an insect‘s flapping wings (Vogel, 2013).  In terms of lifestyle, thunniform locomotors spend their lives cruising the empty open ocean in search of prey.  So great is their dependence on movement that the tuna and the lamnids have both lost the ability to actively ventilate and must swim in order to breathe by ram ventilation (Bernal et al., 2001).  They represent the apex predators of the open ocean and dwarf their relatives, from the massive tuna to the great white shark, and upward to the behemoth of the ocean, the blue whale.  All three taxa rely on cruising and bursting locomotion: cruising for their long-distance foraging and bursting for the rapid acceleration necessary to chase down and catch prey (Biton Porsmoguer et al., 2014).  A popularly recognised example of burst capability can be seen in the antics of dolphins, who can effortlessly fling themselves up to 15 feet above the water, a feat requiring a fantastic degree of acceleration (Lang, 1975). The first kinematic variable I explore in this study, angle of attack ( ( )), is the angle between the tail itself and the fluid it encounters, and has been shown to have a strong effect on performance (Delepine, 2013; Read et al., 2003; Read et al., 2003; Schouveiler et al., 2005)    ( ) is a simple matter in steady-state aerodynamics: in the case of a level flying airplane,  ( ) is a fixed value of the angle between the wing and the oncoming airflow (McCroskey, 1982).  In flapping foil scenarios,  ( ) is more complex because it varies throughout the oscillation cycle.  Since it is dependent on different kinematics, it is possible to adjust those other kinematics to produce an  ( ) that does not vary for a tailbeat stride; however, in that case it is still necessary 3  for the  ( ) to reverse on the return stroke.  This pattern is called ―square wave profile‖ and has the advantage of steady  ( ) but the rather severe disadvantage of requiring infinite angular acceleration at the end of the tailbeat in order to flip and reverse the angle for the return stroke (Esfahani et al., 2013).  In my experiment I vary the  ( ) sinusoidally, a point explained in further detail in the theory and methods sections.  Being a simple sinusoid in phase with the tail-beat frequency, the entire  ( ) profile can be represented by its maximum value, defined as     .  Previous experiments have shown that this value has a large effect in the production of thrust and the optimisation of efficiency (Delepine, 2013; Hover et al., 2004; Read et al., 2003; Schouveiler et al., 2005).  Changing the      of the foil has two main effects: it changes the magnitude of the resultant force vector that the tail generates and it changes the angle of the vector, producing a different ratio of thrust to lateral force.  Using a similar setup, Delepine (2013) shows that motion with a low     has relatively low efficiency and poor thrust, and at high     the thrust is greatly increased but the efficiency still suffers.  The intermediate range of     = 10° to 15° shows an optimum for efficiency, with thrust in an intermediate range.   The second kinematic variable, Strouhal number (St), is a central parameter in unsteady fluid dynamics, being strongly associated with the behaviour of unsteady flow phenomena (Rott, 1992).  It was originally developed to describe oscillating flow mechanisms, principally in the context of vortex shedding.  When the frequency of oscillation and the forward speed are fixed, as they are in this study, the St is directly proportional to the tailbeat amplitude and therefore maximum lateral velocity.  This partly explains the relationship between St and thrust.  Low St results in low maximum lateral velocity of the tail and, since hydrodynamic force is proportional to the square of velocity, the tail produces less force and lower thrust.  Regarding efficiency, St 4  generally does not reach an optimum like     in the ranges I consider.  As can be seen in Delepine (2013) it has a smaller influence on efficiency than     .   These kinematic variables and their role in thunniform locomotion are also of interest to engineers, thanks to the burgeoning field of biomimetics: technology that is biologically inspired.  Part of the logic of biomimetics is that nature conducts her own kind of research and development, through trial and error over unimaginable timescales (Schouveiler et al., 2005).  Fish present a tantalising subject of biomimetics study: Du et al. (2015) assert that there are over 400 publications concerning fish swimming and some 10 robotic fish used in research.  One potential application is the development of fish-mimicking robots which could be used for ocean surveying, exploration and even military subterfuge (Barrett et al., 1996).  Research in this field has already produced the ―RoboTuna‖, a thunniform-inspired robot at M.I.T. (Barrett et al., 1996).  The material I present in this thesis would have direct applicability to further development of such a project.  Given the multitude of kinematics available for research, it is natural to wonder why I am focusing on the variables of      and St.  The answer is that I am aiming to investigate flapping in a biological context.  This is unlike most engineering flapping foil studies which are typically not constrained by biological realism – if the aim is development for human use, then adding extra biological constraints is not necessary.  While that is true, it does not tell us about the animals that inspired the studies in the first place which is what I wish to investigate.  In order to do so, I constrain the kinematic parameters to values observed in living tuna.  Most fall into very narrow ranges which can be approximated by a single value; an example is the tail-beat frequency, which has a tight linear relationship with swimming speed.  My two variables of study,      and St, are two for which precise values have not been found.  This is because either 5  the value is highly variable in nature or it has so far proved too difficult to measure, or both.  Preliminary data from similar biomimetics studies provide us trends to work from, and results from other flapping foil studies provide further insight.  Several engineering studies have also investigated the roles that       and St play in flapping foil performance, but there are still many knowledge gaps.  Most kinematic variables are interdependent; in order to study one, several others must be held constant.  The number of kinematic variables leads to a vast parameter space which includes Reynold‘s number, Strouhal number, frequency, forward velocity, angle of attack profile, pitch profile and heave profile: there are infinite combinations available.  Furthermore, the mechanical characteristics of the foil itself introduce another slew of variables, with studies investigating the effect of chord length, aspect ratio, flexibility and resonance.  And most engineering studies consider flow in a 2-D context: they use foils with very high aspect ratios or foils that have end-plates to prevent span-wise flow, whereas the lunate tails of thunniform swimmers experience several 3-D phenomena, such as spiral leading edge vortices (Borazjani and Daghooghi, 2013).  All of this suggests that there is still much to be learned in this field, especially in a biological context. In this study I emulate thunniform motion with a model-based system, using a tuna-like tail and tuna-like swimming motions.  I focus on the following hypotheses: (i) at cruising speeds tuna do not use maximum angle of attack (    ) to modulate thrust and instead rely on St, (ii)      increases thrust by changes in the vorticity shed by the tail.  I fix all kinematic variables to biologically relevant values except      and Strouhal number (St), for which I test thirty permutations.  I consider swimming speeds at the lower and upper ends of the range at which tuna cruise.  The performance of each set is evaluated in terms of thrust, lateral force, torque, input power, and output power; but primarily in terms of propulsive efficiency (the ratio of 6  useful hydrodynamic power to input power) and the thrust coefficient (normalised thrust).  In particular I consider areas of the parameter space which represent surplus thrust and thus acceleration and mode transition, noting that my setup is limited to cruising speeds and so I am seeing only the beginning of the acceleration transition.  Finally, I investigate the flow phenomena at different      through multiple flow visualisation techniques.   7  Chapter 2: Research chapter  For the emulation of tuna tail motion, a flow system apparatus capable of flapping my model tail in a water tunnel is used.  A realistic tail model is generated by first scanning an actual tuna tail and then using that scan data to 3-D print a scaled-down model.  To simulate motion of the tuna tail I use a two-axis servomotor motor system mounted on water tunnel; flow in the water tunnel simulates forward motion.  Kinematics are generated from empirical values; unknown kinematics are generated from realistic ranges gleaned from biological and engineering studies.  A three-axis load cell mounted between the motion system and the tail captures data for forces in two axes as well as torque.  Using these measurements along with position data from the motion system I calculate the other performance variables.  An important consideration is my isolation of the tail without representing the tuna body. One of the defining characteristic of thunniform locomotion is the so-called decoupling of ‗motor‘ and ‗propeller‘:  the muscle (motor) is shifted forward away from the caudal fin (propeller), and delivers power along a drive-train (massive tendons), and as a result the caudal fin is responsible for upwards of 90% of thrust production (Fierstine and Walters, 1968; Syme and Shadwick, 2002).  In addition, motion is confined to the posterior ⅓ of the body, meaning that the bulk of the tuna does not interact with flow other than as a bluff (albeit highly streamlined) body (Shadwick et al., 2013).  This characteristic of thunniform locomotors is in stark contrast to other undulatory modes, for which much of the length of the body serves as both the ―motor‖ and ―propeller‖.  This is vividly demonstrated in caudal fin ablation experiments: Gray (1933) commented that whiting with full caudal fin amputation can move with ―relatively undiminished speed‖, and Webb (1973) found that salmon, which swim in the carangiform 8  mode, were still capable of achieving 84% of critical speed after having the majority of their caudal fin ablated.  A similar study in tuna found the exact opposite: finned tuna were almost unable to swim, reinforcing the notion that the caudal fin is absolutely necessary to generate thrust (Fierstine and Walters, 1968).  My study capitalises on this unique feature of thunniform locomotion, and takes the uncoupling of motor and propeller one step further by considering the tail alone.  Although the tuna‘s body is certainly responsible for flow phenomena and influences the performance of the caudal fin, it is my prediction that, given the tuna‘s uncoupled tail and highly streamlined body, this influence is small.    In order to further investigate the biological significance of my findings, I extrapolate the muscle requirements of     and St and consider the performance implications. At this point, tuna white muscle mechanics are virtually unstudied, as is the role of angle of attack in bursting thunniform fish.  I use lateral force and torque as a proxy to study muscle effort, mainly considering the onset of peak force and torque and what that may mean for associated musculature.  I also create simulated muscle work loop diagrams using the lateral force and torque. To further our understanding of the actual flow phenomena responsible for my performance results, I use two flow visualisations techniques: ink flow visualisation (flow-vis) and digital particle image velocimetry (PIV).  Although an older method, ink flow-vis has an advantage in that it shows flow patterns in real time and it is not restricted to a two dimension plane; it can be viewed from any angle, offering a more intuitive understanding of fluid flow.  My ink delivery method is novel in biomimetic thunniform motion studies.  I also use PIV, which time averaged data from many synchronised tail-beats.  This powerful technique is 9  effective for revealing average flow conditions and offers opportunities for further digital processing, such as the quantification of vorticity, shear, and turbulence.  In this thesis, I investigate the relationship between     , St, efficiency and thrust, regarding steady-state cruising as well as high thrust regions typical of transition from cruising to bursting.  I find that for cruising speeds, efficiency is maximised at     = 15° and is stable over the entire range of St, suggesting that St may be the best way to adjust thrust during cruising.  In terms of thrust alone, changes in St and      have a similar (and additive) effect on thrust, suggesting that it may be advantageous to utilise both during bursting; however, I find that although effective at increasing thrust, the higher angles of     require substantially more heaving power than St to produce the same thrust at a loss of efficiency.  Increasing       is only possible by a distinct advance of peak force, corresponding to a distorted work loop, whereas St has almost no effect on peak force timing whatsoever.  Regarding flow phenomena, I find that      has a major influence on vortex size and distribution as well as span-wise flow, with     = 15° forming a traditional reverse von Kármán vortex street and     = 25° initialising the shedding of a second leading edge vortex prior to stroke reversal.  2.1 Theory This section introduces swimming kinematics and the fundamentals of flow, and then goes on to describe the rationale in the selection of kinematic values: Strouhal number, angle of attack, and phase.  The mathematical basis of pitching and heaving motion is discussed and the section concludes with an explanation of the choice of material for the tail model.  10  2.1.1 Swimming kinematics Thunniform locomotion uses a unique method of ―lift-based‖ thrust production that sets it apart from other undulatory swimming modes, such as anguilliform (used by eel-like fish) and carangiform (used by salmon-like fish).  The latter two modes use ―drag-based‖ thrust, which involves essentially slithering through the water and using the length of the body as a surface to generate thrust.  This is analogous to snakes on land, except because water is a fluid there is a high degree of backwards slippage (Vogel, 2013).  Functionally these modes can be compared to a row-boat: oars shove water backwards, with drag providing the traction; in fish, the coils of the slithering body function as the oars, dragging water backwards.  In marked contrast to this, the ―lift-based‖ thrust employed by thunniform locomotors does not use the whole of the body as a surface to produce thrust; instead the task of thrust production is delegated to the caudal fin in an interesting example of uncoupling (Syme and Shadwick, 2011).  The idea of uncoupling is this: in most swimming modes the length of the body is used to produce thrust, and it is also the site of the muscle used to produce force for motion – this is a coupled system.  However, in thunniform locomotion the site of force generation and the site of thrust production are separate.  This can be seen in the numerous adaptations that specialise the anterior region for generating force and the posterior region for producing motion and thus thrust: (i) anterior placement of locomotory muscle, evident in the tear-drop shaped body, (ii) a specialised musculotendinous system so that red muscle contraction transmits force posteriorly and while minimising local body flexure, (iii) dorsoventral compression of the posterior region, seen as a caudal peduncle no thicker than the underlying vertebrae, (iv) caudal keels on the peduncle which slice through the water as the peduncle heaves to and fro, and (5) a stiff, lunate caudal fin that displays the profile of a hydrofoil (Shadwick and Syme, 2008). 11   Figure 2.1 Lift-based thrust in sailboats, illustrating two coordinate decompositions.  While drag-based thrust can be compared to a row-boat‘s oars physically shoving themselves against the water, the lift-based thrust employed by thunniform locomotion is better compared to a sailboat sailing upwind.  The sail acts as an aerofoil projecting vertically, angled to deflect oncoming wind (Dedekam, 2000).  This deflection generates a pressure surface on the side of the sail angled into the wind, and a suction surface on the opposite side; the sum of these two pressures results in an aerodynamic force, as seen in Figure 2.1.  The sail is orientated so that part of this aerodynamic force faces the desired direction of travel.  Importantly, the direction and speed of the wind is not the only factor to consider when orienting the sail: the motion of the boat through the water means that the sail sees another source of wind which results from forward motion.  The vector addition of the actual wind and this relative wind produces the apparent wind.  The angle of attack, which is the angle between the sail and the 12  apparent wind, determines how the sail behaves: at 0° there is no deflection and thus no aerodynamic force, while at 90° the wind is not just deflected but completely obstructed, and the only force the sail experiences is drag.  It is at intermediate angles that the aerodynamic force is produced.  Like any vector the force can be broken into components, such as lift and drag; these are the forces respectively perpendicular and parallel to the oncoming flow and referenced in the term ―lift-based‖.  However, in the sailing context it is more useful to decompose the aerodynamic force into thrust, the component oriented in the direction of travel, and the lateral force, the force perpendicular to travel.  The thrust drives the boat forward while the lateral force causes the boat to tilt sideways or ―heel‖ away from the wind, and can be considered wasted energy.    In the context of the thunniform locomotor, the sailboat analogy most accurately represents the state of the tail at the midpoint of heave motion, when the caudal fin crosses the centreline.  In Figure 2.2 we see a tail frozen in the midpoint of a tailbeat cycle, when it is at maximum heaving velocity.  Like the sail, the caudal fin experiences an oncoming flow that is angled away from the ―true wind‖.  One component of the oncoming fluid velocity is from the forward motion of the fish through the water, and the other component is that seen by the foil as a result of being heaved sideways through the water.  This resultant flow forms an angle with the tail: the angle of attack.  The angle between the tail and the body of the fish is θ: the pitching angle.  The aerodynamic force is now underwater and thus a hydrodynamic force, but it similarly decomposes into components of thrust, aligned with the direction of travel, and lateral force, which is cast off from side to side.  The symmetry of tailbeat motion ensures that the lateral forces cancel out one another so the net motion is forward.  Although the lateral force allows for 13  dramatic manoeuvring (Schouveiler et al., 2005), in forward motion it serves to wag the body side to side and is considered wasted energy.   Figure 2.2 a. Lift-based thrust for a static foil; b. Lift-based thrust for a flapping foil.  In addition to these forces, another important factor is the moment of force, or torque.  The pressures acting on a foil result in the hydrodynamic force, and the point at which this force acts is called the centre of pressure.  When the centre of pressure is aligned with the pitching axis of the foil then there is no tendency of the foil to rotate, while misaligned there is a moment of force that tends to rotate an object about a point, felt as torque.  While the sailboat analogy is relevant for the tail at midpoint and useful for conceptualising forces, the thunniform tail foil is an unsteady system: both the pitch and heave motions are continuously changing (pitching refers to pivoting and heaving refers to sideways motion).  As a result of this, every component of the system is in flux, including the angle of 14  attack.  The angle of attack is based on the angle between the resultant flow and the tail, but a component of the resultant flow relates to heaving speed which is always varying, and in addition the tail is continuously pitching and so theta is also varying.  Although the number of factors at work here can seem overwhelming, the pitching and heaving motions (and as a result the oncoming flow velocity and angle of attack and other kinematic variables such as speed and frequency) are all governed by mathematical relationships.  By understanding these relationships I can use forward speed, tail-beat frequency, and the pitch and heave profile as a way to control the angle of attack profile and other variables such as the Strouhal number.  2.1.2 Flow conditions, the choice of swimming speed and tail scaling Reynolds number (Re) is a critical parameter in fluid mechanics, serving to predict the conditions in which turbulence occurs (Vogel, 2013).  It is a dimensionless measure of the relative influence of inertial forces as opposed to viscous forces.  For example, bacteria experience life at very low Re dominated by viscous forces, so swimming through water them feels like swimming through honey would to us; whereas whales experience life at very high Re dominated by inertial forces, so a single tail flip will send one gliding through the water for a long distance (Kim and Steager, 2012; Vogel, 2013).   Mathematically, Re is expressed as:                                    (1) which simplifies to:             (2) 15  where   is the speed of the fluid,   is the characteristic length, taken to be chord length in hydrodynamics, and   is the kinematic viscosity, defined as the ratio of the dynamic viscosity to the density of the fluid and held constant in these experiments.  An important concept in biomechanics is Re scaling, in which scaling laws are used to mimic flow conditions in different systems; an excellent example is the work by Dickinson et al. (1999) which replicated the flow phenomena of a fly‘s wing by using a large scale model that was operated in a tank of viscous mineral oil.  However, before considering the necessary Re scaling it was necessary to determine the values of U and L.  In choosing a relevant swim speed at which to test, I had to consider the swimming speed of tuna in the wild, the scaling relationship between tuna body length and tail span, and the influence of Reynolds number (Re).  In considering the swimming speed, I used measurements of normalised swimming speed where the forward speed of the animal is divided by its body length to give speed in units of body lengths per second, BLs−1.  Using cruising speed as a starting point, studies by Marcinek et al. (2001) indicated that migrating Pacific bluefin tuna of 80-120cm body length have cruising speeds of 1 to 1.34 BLs−1.  Blank et al. (2007) found that the minimum gross cost of transport for 70-80cm Pacific bluefin tuna in a water tunnel occurs at 1 to 1.34  BLs−1.  In contrast, Tsuda (2009) used speed logging archival tags on 70-80cm tuna in situ and found them to have a mean cruising speed of 2 BLs−1.  From this range of values I chose a parameter space of 1-2 BLs−1 to test the kinematics of cruising, although in this thesis I focus on the results at 2 BLs−1.   The next step was to convert this normalised body-length speed into an actual flow speed.  The water tunnel I used for the experiment was found to have a maximum practical operating speed of 1.4 m s−1 when configured with the motion system.  For this speed to correspond to a 16  cruising speed of 2 BL s−1, the size of the simulated tuna would have to be 70 cm.  Happily, this simulated size matched the size of the individuals from which the normalised swimming speed was derived, and it also matched the previous study by Delepine (2013).  Knowing this, I returned to the concept of Re scaling, and found that in order to scale the flow interactions to that of an adult blue fin tuna of average length 2m, it would require a 2.85-fold increase of flow speed or a 2.85-fold decrease in kinematic viscosity.  The latter was not possible in my setup, and the former would require scaling down the testing swim speed to biologically-unrealistic levels.  In the end I decided not to apply Re scaling.  Unlike many studies that employ scaling, this study had the luxury that even in its un-scaled form the experiment was still relevant to tuna (but juveniles rather than adults).  Since most experiments on tuna involve juveniles because the adults are massive and unwieldy, further data on juvenile tuna was seen to be a valuable contribution to the scientific community.    Basing the tail size and flow conditions on juvenile tuna comes with one caveat, which is that juvenile tuna have slightly different swimming performance to adults, due in part to a change of tail morphology with growth.  Bhoopat (1997) found that among mackerel, a close relative of the tuna, larger individuals generally swim more efficiently than small.  Donley and Dickson (2000) confirmed this trend in tuna.  From this we can predict that the efficiency found in this experiment is probably a slight underestimate to that with the Re scaled to adult tuna.   With the simulated body length known, morphological relationships provided the corresponding tail span of 20 cm (Tičina et al., 2011).  To calculate actual Re it was necessary to determine the characteristic length of the tail.  In most engineering flapping foil studies, characteristic length is simply the chord length of the foil.  Things become more complex when considering a foil that is not straight and rectangular.  In such cases, it is common practise to use 17  the mean aerodynamic chord (MAC), a term that originated in aeronautics for aircraft with complex trapezoidal wings (Abbott and Doenhoff, 1959).  The MAC is a weighted chord length which takes into account the aspect ratio and shape of the wing and is described by the formula:          ∫           (3) in which   is the planform foil area,   is the span of the foil, and   is the chord at coordinate   along the span.  Using this I found the MAC of the tail to be 3.58 cm.  For the two speeds tested, 0.7 m s−1 and 1.4 m s−1, the tail had a minimum Re of 25,000 and 50,000, respectively.  Since these Re were less than 105 the effects were due to change monotonically between the two values tested, so no intermediate values were necessary (Delepine, 2013; Vogel, 2013).    2.1.3 Motion regime summary  In order to test the effects of kinematics, I created permutations of different kinematic values.  Certain kinematics were held constant, such as the angle of attack profile, the heave waveform and the phase shift between pitching and heaving; some kinematics were held at one of two values, such as the tail-beat frequency and the forward flow speed; and some kinematics were varied through an interval of values, such as the maximum angle of attack and the Strouhal number.  Each kinematic was selected based on either empirical observations from live animals or on empirical test results from engineering flapping foil studies:   18  Table 1 Summary of key kinematic parameters Kinematic Symbol Value or interval Units Reference Heave waveform   ( ) sine radians (Fierstine and Walters, 1968; Knower, 1998) Angle of attack profile   ( ) cosine radians (Hover et al., 2004) Phase shift    90 degrees (Fish and Hui, 1991; Hover et al., 2004; Read et al., 2003; Schouveiler et al., 2005) Swimming speed  of 70 cm tuna   0.7, 1.4 m s−1 (Blank et al., 2007; Marcinek et al., 2001; Tsuda, 2009) Tail-beat frequency   1.95, 3.12 Hz (Blank et al., 2007) Maximum angle  of attack       5 : 5 : 30 degrees (Hover et al., 2004; Read et al., 2003; Schouveiler et al., 2005) Strouhal number  St 0.2 : 0.05 : 0.4 — (Borazjani and Daghooghi, 2013; Eloy, 2012; Lentink et al., 2010; Rohr and Fish, 2004; Triantafyllou et al., 1993)   2.1.4 Strouhal number and the heave waveform   Tailbeat amplitude is governed by two kinematics in particular: the Strouhal number (St) and the heave waveform.  The St is responsible for the maximum excursion of the tail-tip while the heave waveform determines the exact shape that the peduncle traces through the water.  The Strouhal number is a dimensionless value that describes oscillatory flow mechanisms and originally pertained to the pattern of vortices shed behind a bluff body (Rott, 1992), described by the formula             (4) 19  where   is frequency of oscillation,   is the width of the vortex street (which can be represented by the tail tip amplitude), and   is the flow velocity.  St is a strong predicator of flow type and is particularly relevant in the establishment of wake vortices, such as those found behind swimming fish; consequently it is highly conserved and is rarely found outside of the range of 0.15 to 0.8 from animals ranging from tadpoles to blue whales, with the majority falling in the range of 0.2 to 0.4 (Eloy, 2012; Nudds et al., 2004).  Studies in flapping foil locomotion suggest that the best performance is found in the range of 0.25 to 0.35 (Hover et al., 2004; Read et al., 2003; Triantafyllou et al., 1993); however, studies of odontocete locomotion by Rohr and Fish (2004) indicate that the majority of swimming occurs in the range of 0.15 to 0.25.  I chose a range of 0.2 to 0.4 to match the majority of these studies and opted to adjust the St after initially testing, if necessary.  The Strouhal sets the maximum tailbeat amplitude, but the heave waveform controls the actual movement of the peduncle.  Fierstine and Walters (1968) and Knower (1998) both report that the tail flaps sinusoidally; studies by Shadwick and Syme (2008) on another thunniform, the mako shark, reports likewise.  Triantafyllou et al. (2004) detail that sinusoidal motion is a fundamental component in the development of a propulsive vortex street wake.  Taking this into account the heave was defined as a simple harmonic motion, where  ( ) is the heave as a function of time,    is the maximum heave, relating to St,   is time, and   is the angular velocity in radians, defined as where   is the tail-beat frequency.   ( )           (  )  (5)           (6) 20  Tail-beat frequency has a tight relationship with normalised swimming speed: Blank et al. (2007) swam juvenile pacific bluefin tuna in a water tunnel and determined the following relationship: where       is normalised swimming speed in body lengths per second.  Using this relationship I determined the tail-beat frequency in this experiment to be 1.95 Hz and 3.12 Hz for the low and high cruise speeds, respectively.  2.1.5 Pitch waveform and angle of attack profile Pitching motion together with heaving motion is responsible for moving the tail in such a way that it efficiently produces thrust.  A critical component to this process is the angle of attack ( ( )), which determines how the tail interacts with oncoming flow in the production of propulsive vortices.  In propulsive flapping motion  ( ) can never be fixed as a constant: one tail-beat is made of two mirrored strokes, so for half of the tail-beat  ( ) must reverse direction; thus the existence of the  ( ) profile, which maps the angle as a function of tail-beat phase.   There are infinite  ( ) profiles available, but Hover et al. (2004) narrows the selection down to four with potential profiles: (i) that which results from simple harmonic motion in pitch and heave, (ii) a square-wave, (iii) a symmetric sawtooth wave, and (iv) a cosine function.  The first case is the simplest and was the method used in early flapping foil experiments: a phase difference is set between the pitch and heave axes and then they are both oscillated as simple sinusoids.  While convenient, this method suffers from a lack of explicit control over angle of attack.  In a setup where the pitch-heave phase shift is 90°, method (i) will generally produce a                        (7) 21  smoothly sinusoidal  ( ) which performs quite well while operating at low St; however, as St increases above 0.15, the  ( ) profile begins to degrade and higher harmonics appear, resembling the addition of a square wave and an out-out-of-phase sinusoid.  Hover et al. (2004) show that these added fluctuations are detrimental to thrust at high St, corrupting the propulsive wake with additional vortices. The second option, the square wave profile, involves adjusting the pitch so that  ( ) does not vary through one full stroke.  A rather unpardonable feature of the square wave profile is that it requests infinite angular acceleration in order to instantly reverse for the return stroke, so at best a square wave profile is only approximated.  Hover et al. (2004) find this case to produce significantly higher efficiency than (i), and Esfahani et al. (2013) find it to actually produce more thrust than the other cases.  However, the presence of a velocity discontinuity suggests poor biological applicability.  Compared to the square wave case the sawtooth wave (triangular wave) produces the highest levels of thrust but lacks efficiency over all.  It seems that both square wave and sawtooth wave profiles suffer from their sharp profiles. The fourth and final profile is the cosine profile, in which pitch and heave are modulated so that  ( ) smoothly approaches a maximum value at midstroke, varying harmonically without sudden interruptions as per a cosine function, Hover et al. (2004) find this profile produces the best combination of efficiency and thrust, showing a clear improvement on all three other profiles.  It also produces the cleanest propulsive wake across the entire St range.  Since it is a simple sinusoid, it can be defined by its maximum value,     , which is a constant within each test motion.   ( )         (  )   (8) 22  At low St the heave and pitch profiles for a cosine  ( ) profile both resemble simple sinusoids, but at higher St more complex waveforms are required of the pitching waveform in order to maintain the same  ( ) profile.  Since the heave waveform was held as a simple sinusoid in (5), it was necessary to adjust the pitch waveform to maintain the angle of attack.  In doing so the first step was to make use of the relative flow angle  ( ).  It describes the angle of the resultant flow on the tail, similarly to  ( ) but relative to the stream-wise axis rather than the tail, where  ̇( ) is the heave velocity, defined as the derivative of heave position: As shown in Figure 2.2b, the pitch waveform  ( ) is defined as:   Substituting in the preceding three equations results in the pitch waveform as a function of time and three constants: maximum heave amplitude, maximum angle of attack, and flow speed,  Further calculations were used to convert the desired St into Strod, the former of which uses the amplitude of the tail-tip, and the later which uses the heave amplitude  ( ) and can thus be incorporated into (12).    ( )       ( ̇( ) )   (9)   ̇( )           (  )   (10)   ( )    ( )   ( )   (11)   ( )         (  )       (        (  ) )   (12) 23  2.1.6 Phase angle  Phase angle refers to the degree of synchrony between the pitch and heave motions, with one complete oscillation (or tail-beat) being defined as 360°.  Figure 2.3 shows the four extremes: the first row demonstrates a 90° phase angle, where pitching motion leads heaving motion by ¼ of a tail-beat cycle; the heaving motion seems to gently draw the front of the foil from side to side, with the rear edge trailing obediently behind.  In the second demonstrates antiphase, where the tailing edge seems sluggish in comparison and traces a line of small amplitude.  In the third row the phase angle leads by 270°, presenting the maximum planform area and presumably maximum drag to oncoming flow.  Finally the forth row illustrates the effect of in-phase, synchronised pitch and heave, where maximum heave excursion coincides with maximum outward rotation and the tail appears to be delivering a backhanded slap.  Although (Anderson et al., 1998) found optimal efficiency at a phase angle of 75°, studies by (Prempraneerach et al., 2003), Read et al. (2003), Hover et al. (2004), and Schouveiler et al. (2005) all suggest that 90° provides the best propulsive performance.  Fish and Hui (1991) report the same phase shift in dolphins, another thunniform locomotor.  In this study I follow their lead and apply a 90° phase angle. 24   Figure 2.3 Illustrations of different possible pitch-heave phase angles.    25  2.1.7 Tail material and stiffness  Foil flexibility is a feature seen in all animals that practise flapping foil propulsion, from fruit flies to whales (Lucas et al., 2014).  I can confirm this in the case of members of the other thunniform taxa, the lamnid sharks and cetaceans, who possess tails which border on floppy (personal observation).  Furthermore, numerous studies demonstrate that adding flexibility to a flapping foil system improves performance: Katz and Weihs (1978), Shinde and Arakeri (2010), and Barannyk et al. (2012) are three of several studies that find that flexibility results in increased efficiency.  However, except for Barannyk these studies also find a trade-off in decreased thrust for the price of efficiency.  Shinde and Arakeri (2014) have suggested that flexibility increases efficiency because it prevents the free stream generated by the tail from meandering quite so much.  All of these facts suggest that flexibility is a factor that should be taken into consideration.  However, in this thesis I chose not to test flexibility for three reasons: resonance tuning, material complexity, and intrinsic stiffness.    Resonance is a phenomenon in which an oscillating body is driven to ever greater amplitudes at a certain frequency.  The simplest analogy is pushing someone on a swing: if you push at the resonant frequency it takes minimal effort to achieve impressive height, taking advantage of the natural frequency of the system (at which the system ―wants‖ to oscillate); however, increase or decrease your pushing frequency and the experience rapidly degrades into one person in an almost stationary swing, being shoved in an unpleasant and seemingly arbitrary manner.  Every system possesses resonant frequencies and flexible foils are no exception; in the case of flexible foils the foil is the swing and the heaving axis is the pusher.  Richards and Oshkai (2015) found that matching flapping frequency to the intrinsic resonant frequency of the 26  foil resulted in improved performance.  Thus, to introduce flexibility into a system one must take into account the resonant frequency of the flapping foil and how that may impact results.  Resonant frequency is a product of several factors, including flexural stiffness and distribution of mass (Richards, 2013).  In biological systems these factors are complex and intertwined.  In contrast to an engineering model which may use an isotropic, homogenous material with a known young‘s modulus (a normalised measure of stiffness), biological tails can be extraordinarily complex systems rife with anisotropic (direction dependent) properties and composite materials with sophisticated variations in flexibility, sometimes even modulated by the living organism (Flammang, 2010; Vogel, 2013). Matching the flexural stiffness of a 3-D printed tail to an actual specimen an arduous task of trial and error; matching resonance frequency would exponentially complicate this process.  Regarding tuna, the necessity of such lengths become questionable.  Relative to other fish, and even to other thunniform taxa, tuna have a particularly stiff caudal fin (Shadwick et al., 2013).  The tail is made up of bony fin rays embedded in collagen and has reduced intrinsic tail musculature relative to other bony-rayed fishes (Delepine, 2013; Fierstine and Walters, 1968).  The natural stiffness of the tail approaches that of rigid nylon: Delepine (2013) undertook the task of replicating the flexural stiffness of tuna caudal fin and found that there was no significant difference in performance between his boil-realistic models and a stiff nylon version.    In this experiment I chose not to focus on flexibility as a performance variable, partly because of the complexity of doing so and partly so that the effects of motion could be investigated in greater depth.   27   Overall, we see that the motion regimes are constructed by holding one set of kinematics fixed and then using different combinations of      and St.  The fixed set is based off of empirical observations of swimming tuna while the      and St are constrained within ranges deemed both relevant and realistic for thunniform locomotors.  Having discussed the theory behind tuna motion and tail selection, I now proceed to the discussion of the methods used to carry out the experiment.    2.2 Methods This section covers the design and execution of the dynamic experiments, beginning with the construction of the tail model and tail apparatus, followed by an introduction to the flow channel and motion system, an explanation of the programmatic control and post processing of force and torque data, and the design of ink and PIV flow visualisation experiments.  2.2.1 Tail model design Obtaining and scanning specimens This study used an Atlantic bluefin tuna (Thunnusthynnus) tail provided by Tony MacDonald, of Tony’s Tuna Fishing in PEI, from an adult measuring 2 meters in fork length, 125kg in weight and with a tail span of 54cm.  The raw tail could not be oscillated in the flapping foil apparatus due to being too large and terminating in a hydrodynamically unrealistic stump, as well as the risk of putrefaction from lack of refrigeration.  As an alternative, I designed a 3-D printed scaled tail model constructed from computed tomography (CT) scans of the original tail.  Computed tomography is a type of x-ray that takes planar x-ray images from many angles and then assembles these images in 3-D.  Multiple tails were scanned, but this thesis 28  presents a subset of data focused on tuna swimming and used the scan data collected by Delepine (2013).  The CT scan was performed by Gabor Szathmary at FP Innovations in Vancouver, and Delepine used ScanIP for conversion to .stl.   Design of tail model For modification of the .stl file, I chose a free, open-source program commonly used by artists for computer animation: Blender version 2.76b.  The strength in Blender over other software is its ability to import and modify complex organic shapes; most other computer-aided design (CAD) software is specialised for creating objects from the ground up using shapes and properties that can be represented by mathematical equations.  When faced with importing a complex organic shape these programs will seize up, whereas an animation program like Blender allows not only import, but also offers sculpting and other tools ideal for cleaning and modifying mesh.   In order to make the nose cone to replace the stump, I first sliced off any mesh forward of the pivot point.  To make the nose cone as unobtrusive as possible, but for it to also match the morphology of the tail, I modelled it from the tail itself.  The ―spin‖ function in Blender allows for the creation of an item as if on a lathe; an arbitrary line is extruded about an axis.  The spin function was applied to a segmented line which had been traced along the exposed right edge of the remaining mesh.  In one of the more tedious but unavoidable steps, I manually stitched the front half of the nose cone to the tail mesh, rebuilding substantial sections on the unmatched side when necessary (the caudal keels were slightly uneven).     Before the final steps of model creation was the decision of how large to make the tail; what scaling factor to use.  Having a large tail presents several advantages, namely the reduction 29  of signal noise thanks to the larger forces involved as well as a Reynolds number more closely matched to an adult tuna, but is limited by the width of the tunnel; simulating a larger body means simulating a wider tailbeat.  A second disadvantage of simulating a larger body is that the normalised-body-length swimming speed will have a lower maximum, as the tunnel cannot exceed 1.6 m s−1 in our configuration.  To study higher speeds a small tail would be vital, but the diminished forces will eventually be masked by signal noise.  In the end I settled on dimensions matching the tail used in Delepine (2013), 20.1cm along the z-axis (the tail-span), which provided quite a clear signal but also represented a reasonably sized juvenile tuna and allowed for testing up to 2 BL s−1.  The rudimentary tail model was now complete in shape, but still rough in texture and without mounting holes for attachment to the apparatus.  The ―smooth modifier‖ was set to increase the number of faces on the model by several-fold until the surface appeared sleek.  At this point I used the sculpting functions to remove more deformities.  To create the space for mounting the apparatus I used Boolean subtraction: I built a simple geometric model that matched the negative space that would be occupied by the apparatus and mounting screw, and then subtracted those regions from the tail model.  The completed model was exported as an .stl file.  The final step was to convert the tail from pixels on a screen into physical material using the game-changing technology of PolyJet3-D printing; for this I employed the services of Corbel3D, a 3-D printing company in Vancouver.  As flexibility was not a variable being tested, I opted on the durable VeroWhitePlus RGD835 nylon material, of elastic modulus 2000-3000MPa (see datasheet in appendix A.1).  The tails were printed with a matte finish.    30  2.2.2 Rod effect model Knowing that the vertical strut of the apparatus would disturb the flow upstream of the tail, I designed a model to quantify the magnitude of this effect.  The model was a half-foil that could be oriented in two ways: facing upwards, the foil operated in the eddies shed by the vertical strut; facing downwards, the foil operated in undisturbed free-flow.  By comparing the two over a subset of motion regimes I planned to quantify the extent and nature of the impact.  Following the example of Abbaspour and Ebrahimi (2014), Bandyopadhyay et al. (2012), and Schouveiler et al. (2005), I chose a NACA0012 aerofoil and scaled it so that the chord length matched the MAC of the tuna tail model.  The height was adjusted so that the total planform area matched that of half the tuna tail, 31.7 cm2 (equivalent to one tail lobe).  The pivot point was chosen so that if the tuna tail and the foil were mounted simultaneously, the foil would intersect the tuna‘s leading edge at the span of the MAC.  The tail apparatus is designed for a tail with sweepback where the front of the tail is significantly forward of the pivot point.  Since the foil had no sweepback angle but was set back to match the leading edge of a swept back foil, it was situated too far back to firmly contact the apparatus.  To give it some footing, I made an additional forward projecting portion designed as a short section of enlarged NACA0012 foil as seen in Figure 2.4.  31   Figure 2.4 Foil designed to measure the effect of the apparatus rod; can be mounted in a facing-up or facing-down orientation.  See associated apparatus, Figure 2.6.  2.2.3 Tail apparatus The apparatus forms the vital connection between the load cell and the tail itself.  The connection between tail and apparatus proves challenging: not only must it be as stiff and stable as possible, but it must have minimal or no interaction with water upstream or downstream of the tail.  Such interactions result in flow structures that will interfere with the flow structures produced by the tail and artificially augment or diminish propulsive performance to some extent.  In the most basic sense the apparatus is a rod or mast, and in fact the first iteration of the apparatus took the form as exactly that: a rod aligned with the pivot axis that was permanently glued into the tail. The main problem with that first ―rod‖ apparatus was its alignment with the pitching axis: the pitching axis is located just before the main flair of the caudal fin lobes, and so having the rod in that location put it in very close proximity with the leading edge of the fin, guaranteeing interaction between the two.  A further problem was posed by the upper region of the apparatus 32  which passed through the air-water interface.  The upper part of the rod was made out of a much thicker rod of metal which shed a considerable wake into the channel water flow.  When operated at low flow speed this caused little problem, as the thicker part was elevated some distance above the top of the tail, so the eddies in the current behind them were convected above the tail without interference.  However, when operated at our maximum speed of 1.4 m s−1, the large negative pressure behind the rod resulted in ―entrainment‖: suction of air bubbles from the surface.  The entrainment revealed a strong downwash from the rod which injected eddies into lower regions of the tunnel with certain implications of flow interference.  Figure 2.5 Comparison of bubble entrainment between three iterations of the apparatus all executing the same motion and phase matched.  Entrainment is particularly severe in b. but apparently non-existent in c.  Motion blur is exacerbated in b. because of flexure of the apparatus. I attempted to solve this through the application of a crank-shaft type model: I designed a simple system in which the lower part of the vertical strut was displaced forward 8 inches and so for most of the tailbeat its wake was shed far afield of the tail.  In order to reduce entrainment, which is directly related to the width of an object obstructing flow as well as its relative velocity, I shortened the thick upper rod and elongated the crankshaft rod so it passed through the air-water interface instead.  This design ran into practical problems of vibrational resonance, but even more problematic was the high speed of the displaced rod during pitching: being 8 inches 33  away from the pitching axis gave it rather terrific angular velocity which resulted in a mess of eddies and considerable surface entrainment as it slashed back and forth.  Despite these flaws, the second iteration did have the advantage of being to fit interchangeably with tails, so not only was the system more easily transported, but multiple tails could be printed and then tested on the same apparatus.   The third and final iteration applied the lessons learned from the first two: (i) it used a crankshaft style vertical rod but shifted it forward only 3 inches, and (ii) to deal with surface entrainment and vibrational resonance simultaneously it maintained the original length of the stocky rod but added a large circular plate whose radius matched the horizontal distance from pivot to tail tip: 5.5 inches.  The plate‘s function was to block entrainment and it was successful in doing so (verified with laser particle image velocimetry).  A comparison of the entrainment of each apparatus under the same flow and motion conditions is presented in Figure 2.5. I used SolidWorks, a computer aided software (CAD), to design the new apparatus, as seen in Figure 2.6.  Construction of the apparatus was carried out by Max Rukosuyev, a researcher and machinist at UVic.  The vertical rods and lower plate were made from stainless steel; the disk and remaining components were milled from aluminium.  Although the new design was considerably heavier than previous designs, the subtraction method used in data processing removed the effects of this additional mass; comparison with trials of the other trials confirmed this. 34   Figure 2.6 Design of third iteration tail apparatus visualised in SolidWorks, with an approximate tail model added for perspective  2.2.4 Flow channel For the experiments conducted in this thesis I used the water tunnel and motion system at the Oshkai fluids mechanics laboratory at the University of Victoria (UVic), as shown in Figure 2.7.  The high flow requirements of the project were fulfilled by the Engineering Laboratory Design Inc. 4,715 L re-circulating water tunnel (model 504), powered by a 25 horsepower motor 35  (see appendix for details).  After flowing through the water channel, water is directed through turning vanes into a large return flow pipe and then sucked through the impeller, after which it arrives in the settling chamber via the expansion cone.   The settling chamber contains honeycomb flow straighteners which remove flow turbulence; smoothed water accelerates through the contraction nozzle and re-enters the working section.  The 45cm × 45cm × 2.5m working section of the tunnel is made from thick transparent acrylic on the sides and bottom, while the top is open.  Although the tunnel can be lidded and pressurized, the motion system frame design requires open access to the tunnel through the free surface.  During operation the free surface develops standing waves which limit the maximum flow speed to about 1.6 m/s.  These waves can interact with the end of the working section, where the un-lidded section ends.  Because this interaction can exacerbate fluctuations in flow speed, I lowered the water height until the standing waves at the rear of the working tunnel were unobstructed (approximately 40 cm depth).   The tunnel flow speed is controlled by a variable frequency drive that communicates with the impeller motor.  The flow speed was initially set by consulting a calibration chart relating the flow speed to the RPM of the impeller motor, but this was found to be susceptible to variation, especially as a function of water height.  To account for this variability the tunnel was recalibrated each runtime using a Höntzsch HFA U363 flow probe.  As a consequence of imperfect flow smoothing, free-surface interactions and edge effects, flow speed will vary at different positions in any water channel; in order to calibrate the part of the channel relevant to the tail, I designed a frame which aligned the flow probe in the middle of the channel, at the same height and position as the front of the test tail.  The flow probe provided measurements of 36  flow speed time averaged over ten seconds.  I adjusted the motor controller until these time-averaged measurements settled around the desired flow speed for at least two minutes.   Figure 2.7 Flowchannel and motion system with apparatus (second iteration) and tail.  Visible in the foreground is the motion system power supply and controller, shown attached with thick yellow cables.   2.2.5 Motion system In dynamic experiments the motion system oscillated the tail through a combination of pitching (rotating) and heaving (translating) motions, using a two-servo motion system developed by Peter Hannifin Corporation.  The heaving motion was generated by an timing belt driven actuator (OSPE..B) and the pitching motion was generated by a brushless servo motor (SM series) geared down with a gear drive (Gen II Stealth Gearhead "In-Line" PX).  A pair of serial digital servo drives (Parker Aries AR-xxAE) powered the motors.  In order to hold the motors in place, I mounted them into a frame made with the 80/20 Inc. T-slotted aluminium 37  framing system.  The frame attached to the top of the water tunnel via bolts, as seen in Figure 2.7.  Forces were measured by a load cell mounted on the output shaft of the pitching servo motor. This type of motor system is not controlled directly from a computer terminal – rather by a motor controller specialised for fast, accurate motion, which in my setup was a Parker ACR9000.  The controller operates in ―servo mode‖, which is a closed loop control system: incremental optical encoders in the motor provide continuous position feedback that the motor controller uses to ensure that the actual position matches the commanded position.  Motor error of this system was assessed by Richards (2013), who found an overall heave uncertainty of ±0.122 mm in heave motion and ±0.333° in pitching motion.  For a more complete treatment of the uncertainty and accuracy in the motion system see work by Andrews (2013) and by Delepine (2013).    Figure 2.8 Simplified schematic of motion system Control of the ACR9000 system is through a programming language called acroBASIC which, in order to maximise controller speed and responsiveness, is limited to a simple syntax and a small memory size.  As an assembly language, acroBASIC is incapable of most high level 38  functions seen in other programming languages, and relies instead on simple commands, numbered bits and flags.  However, acroBASIC has one function which allows the execution of complex motions in multiple axes: the cam table.  In electronic camming, a function produces a profile from the master axis which the slave axis follows.  A mechanical analogy is the valve system in a combustion engine: the camshaft is the master axis, the cam is the function, and the rod attached to the cam follower is the slave axis.  A cam table is a motion profile that has been discretised into points.  The motion controller interpolates these points, in effect ―filling in the gaps‖, and commands the motor to smoothly follow the filled-in motion profile.  One execution of the cam table represents one cycle, so the period of the table was set to match the desired tailbeat frequency.   In my case there was one cam table for each axis (pitch and heave); each cam table was a sinusoidal function discretised into 100 points.  In order to avoid commanding the motor to go from a dead stop to a high velocity (requiring near-infinite acceleration – not possible and very uncomfortable for the motors), the program did two things: (i) Before cam table execution the motors were commanded to move to their extreme negative positions for the motion regime (in sinusoidal motions the velocity goes to zero at either extreme) (ii) Motion began with a ramping function which increased the amplitude of the motors from 10% of their final amplitude up 100% over ten cycles.  Once in operation at full amplitude the motion system executed another ten cycles so that the flow conditions could stabilise, and then at the beginning of the eleventh cycle it activated the trigger output, which was recorded by the data collection computer.  39  2.2.6 Force measurement Forces on the tail were measured with a NovoTech 3-axis load cell (model F233-Z 4346), one end which mounted onto the output shaft of the pitching motor and the other which connected to the flapping apparatus.  The load cell captured the instantaneous force in both horizontal axes and instantaneous torque in the vertical axis at a sampling frequency of 1000Hz.  The x-axis was aligned parallel with the chord of the tail model and the y-axis perpendicular to the chord.  A consequence of the design was that the load cell force axes rotated with the tail, so during processing I transformed the forces from the coordinate system of the tail to the fixed coordinate system of the tunnel.  Once mounted into the frame, the weight of the apparatus passively exerted forces and torque on the load cell.  In order to remove this bias I assembled the complete apparatus and took a no-flow measurement before each set of trials; this measurement was subtracted from the dynamic recordings during processing. The small force and torque measurements were amplified with Mantracourt Electronics SGA/A load cell amplifiers, on a ±10V scale.  An analog-to-digital converter (model NI-PXI4472) converted the output from the amplifiers into a signal that could be processed in a program developed in National Instruments Corporation‘s LabVIEW software.  The program sampled the data at 1000Hz and wrote raw values for each axis into a LabVIEW measurement file (.lvm).  In addition to the force and torque measurements, the program also sampled a trigger input; the trigger signal was produced by the motor controller and facilitated synchronisation the motion and force systems.    40  2.2.7 Post-processing and analysis In order to convert the raw data traces into meaningful values I wrote a processing script in MATLAB R2013b.  It produced values of performance variables through a several step process of filtering, transformation, subtraction and calculation.  The annotated code can be found in the appendix.   The first order was to import the force and position data, after which MATLAB‘s filtering function removed the noise from the force signals. Following Delepine (2013) I applied a 12th order Butterworth filter with a cut-off frequency of 10.5 Hz. A zero-phase filter prevented the data from being shifted in time, which was vital given the necessary synchronization with the motion system.  The motion system position data was also imported but, given the lack of noise, filtering was unnecessary. The force recording system recorded each full trial, including the ramp-up and ramp-down segments, so the next step was to extract the relevant, full amplitude cycles and convert them into a usable form.  The program detected the point of time at which the trigger signal crossed a one volt threshold and then selected the corresponding segment of the force and torque traces.  The number of cycles multiplied by the cycle length determined the end point.  The position data did not require this extraction (the acroBASCI code only recorded the relevant segment), but it did require being interpolated by a factor of two, as it was sampled at 500 Hz,  half of the force sampling rate.  Since the load cell rotated with the tail, its force readings were in a rotating frame of reference relative to the water tunnel; the next step was to transform those forces into a fixed frame of reference.  A trigonometric calculation used cosine and sine to decompose the forces    and   into components that were parallel to the tunnel (stream-wise) and perpendicular 41  (transverse), and then these components were added.  The stream-wise component corresponded to thrust and the perpendicular component corresponded to lateral force.  Due to the dynamic nature of the system this was repeated for every time point, according to the following formulas:   ( )     ( )     ( ( ))     ( )     ( ( ))   (13)      ( )     ( )     ( ( ))    ( )     ( ( ))   (14) where  ( ) is the angle between the load cell and the water tunnel.  The torque was unaffected by rotation so this transformation only applied to   ( ) and   ( ). For each trial I recorded between 10 and 40 tailbeat cycles, but calculations and plotting required a single representative tail-beat cycle so the cycles were phase averaged.  This involved cutting each force recording into segments the exact length of one cycle, using vector addition to add these cycles together, and then reducing the resulting vector by a factor equal to the number of cycles.   The load cell system is unable to measure force on the tail alone without also measuring forces on and due to the apparatus connecting it to the tail; the solution was to record all motion trials twice, once with the tail attached to the apparatus and once with the apparatus alone.  The difference between these trials represents the forces on and due to the tail alone, and so the program used vector subtraction to take the difference between the ―tail and apparatus‖ and the ―apparatus alone‖ trials.    With the thrust, lateral force and torque values phase averaged and isolated from the apparatus, I could calculate the performance variables.  The first was the coefficient of thrust,   , calculated as the ratio of mean thrust to the dynamic pressure.  It can be thought of as the amount of dynamic pressure that is converted into thrust (Gülçat, 2011), 42                (15) where   is the mean thrust over one cycle,  is the planform area of the tail,   is the density of water, and  is the flow speed.   I calculated the instantaneous output power by taking the product of thrust and flow velocity,          ( )   ( )     (16) The instantaneous input power was in two parts: power to pitch the tail and power to heave the tail.  These were found by multiplying the lateral force by the heave velocity and the torque by the angular velocity,        ( )      ( )   ̇( )   (17)        ( )    ( )   ̇( )   (18) where        and       are the instantaneous pitching and heaving power,      is the lateral force,  ̇ is the velocity of the heave axis,  is the torque, and  ̇ is the angular velocity of the pitch axis. The final major performance variable to calculate was the propulsive efficiency,   , which is taken as a ratio of the usable output power to the power required to move the tail,                                           (19) where the mean power output was instantaneous power output integrated with respect to time and averaged by the period,  ,            (∫  ( )        )   (20) 43  and the mean power input was the sum of mean power to pitch and mean power to heave, treated similarly, The force traces were all subject to a degree of asymmetry due to structural asymmetries in the tail; for the purpose of plotting these asymmetries were removed using a function I wrote called ―symmetrise‖ (in appendix).  This function split the forces and torque in a tailbeat into two halves, one per stroke.  The function then time averaged the two halves, although since the lateral force and torque are mirrored in the second half they had to be flipped.  The averaged parts were then reassembled into a symmetrical version of the original data.  2.2.8 Ink-flow visualisation Compared to PIV, ink flow-visualisation offers a more qualitative, intuitive understanding, as well as an opportunity to investigate 3-D flow phenomena.  The system is also simpler, consisting of an ink source, delivery tubes, and ink-ports near or on the model to eject ink.  Inspired by the study by Bandyopadhyay et al (2012), I chose to eject the ink from the tail itself with a different colour per side.  Setting up the experiment required several steps: creating the tail, connecting the ink, making the ink, masking the background, and finally video capturing the system in operation.  In this type of dynamic system, one of the main challenges is the delivery of ink: pipes affixed to the outside of the model itself will distort its profile and interfere with flow.  To account for these issues, I designed a tail with a system of internal ink channels that terminated in ports along either side of the leading edge.  I modelled the ink pipes in Blender by creating           (∫     ( )   ̇( )     ∫   ( )   ̇( )    )  (21) 44  and manipulating vertices, filling in and extruding faces, sculpting, and finally applying a smoothing filter.  The pipes were created as a solid object, resembling an octopus with several extra arms, which was then ―subtracted‖ from the tail model using the Boolean subtraction method.   In placing the exit ports I chose seven paired locations spaced evenly along the leading edge.  The exception was the central port which had to be placed farther back because of the caudal keel.  Smooth and even ink delivery was a concern, so I designed the long pipes that reached to the tail tips to have a smaller radius to promote an even pressure drop, according to the Hagen-Poiseuille equation,                (22) from which we see the relationship between length and radius is     , all else being equal.   45   Figure 2.9 a. Ink visualisation tail components showing inside and outside faces, as well as the jig that functions as an ink tank septum; b. Close-up of exit port and channel; c. Close-up of ink tank and entry port  3-D printing the completed tail was challenging because the type of printing uses an infill material which must be removed from the completed model.  In my case the internal ink channels were too narrow to realistically remove infill, so I designed the tail in two halves.  In the outer edges of the tail that are only 4 mm thick it was already necessary to offset the ink channels since there was not room for two side-by-side.  These areas were not a problem for reassembly of the halves; however, there was a problem in the central tanks where the ink entered before being ferried down individual channels, as these had to be kept separate for the purpose of using two colours.  To solve this problem I made a third piece, seen in Figure 2.9a 46  which formed a septum in the ink tank, splitting it in two.  The septum had an additional advantage: it served as a jig during tail assembly, keeping the two outer pieces aligned.  Figure 2.10 a. Ink visualisation tail being epoxied; b. Ink flow in operation with all ports open.  The three pieces of the model were 3-D printed using the same printer and material as the other tail models.  I chose an epoxy with a set time of 2 hours and, using a very fine paintbrush, applied a thin layer on the mating surfaces of the model pieces, being careful to avoid the channel dugouts.  Several clamps set around the periphery and centre of the model held the sides together as the epoxy cured (see Figure 2.10a).  To make the ink used for visualisation I combined water, glycerol (glycerine), and food grade colouring.  Having tried several different proportions of glycerol to water, I found that too low a concentration of glycerol immediately billowed into diffuse clouds that were ineffective at revealing turbulence, whereas too high a concentration of glycerol had such viscosity that the tubing fittings blew off, spraying ink across the laboratory.  A mixture of 1:3 parts water to 47  glycerol balanced these two effects: it maintained ink cohesion for a sufficient time but diffused in the presence of strong turbulence, and it did not compromise the integrity of the tube fittings.  A 120 ml mixture received approximately 12 drops of food colouring (Americolor brand soft gel paste), either red or blue.  The ink mixture was stored in two 60 ml syringes which, when manually depressed, delivered ink to the model through a 2.3 m of paired tubing.  Since the tail apparatus was subject to variable heaving motion, I chose highly flexible and durable silicon tubing with an inside diameter of 2.4 mm to facilitate the viscous nature of the glycerin-rich ink mixture.  The proximal-ends of the silicon tubes each plugged into a 1-way male luer slip stopcock, while the distal-ends inserted into corresponding openings in the tail model and glued in place with silicon caulking.  The other end of the stopcock was male luer lock and screwed onto the tip of the syringe; this allowed quick detachment of the syringes for refill.  The tubes leading to the model were fixed to the vertical strut, one in front and behind, and wrapped in electrical tape so as to cause minimal disturbance to the flow.  In most trials ink flow was limited to one pair of ports by covering the others with white electrical tape.  Viewed from below, the background of the system was confounded by the large aluminum disc of the tail apparatus and the free surface of the flow tunnel; this provided poor contrast for visualising the dye.  To create a uniform white background and hide the free surface, I made a 100 cm × 45 cm rectangle of white 4mm corrugated plastic board, reinforced the leading edge with a 40 cm steel ruler, and tethered it just behind the tail apparatus using string attached to the to the front section of the flow channel.  Setting the backdrop below the tail apparatus disc provided the best background; however, even at the lowest speed the water flow drove the backdrop downwards with great force, risking damage to the tail model.  Therefore the 48  backdrop was set above the dark tail apparatus disc, and the disc was matched to the backdrop with a double layer of white electrical tape.  I limited experiments to the low speed flow because even with the background set above the apparatus disc, the higher flow speed proved too forceful on the large surface of the background and threatened to damage it.  Perhaps more important than a uniformly white background is adequate lighting, to which end I used four high powered lights.   The primary light was an ALZO 3000 High Intensity LED Light which produced 8000 Lux at 1m at 5300K (daylight white), equivalent to a 1000W incandescent bulb.  An 8 inch par reflector fitted with silk diffusion fabric softened the light output, which was aimed to illuminate the model and the backdrop behind.  The remaining light was provided by three Einstein E640 Flash Units, each of which produced 250W of warm white light.  Frosted white Pyrex domes softened the light produced by the bayonet-style quartz bulbs.   Ink flow phenomena were video captured with a Canon digital single lens reflex (DSLR) camera, model EOS 5D II, fitted with a macro zoom lens.  Although the PIV Photron APX-RS camera is capable of much faster frame capture, I chose the Canon camera for its colour-capable imaging, which allowed for differentiation of red and blue ink streams.  Video was recorded at 30 frames-per-second.  In capturing the highest quality images I had to compromise between image sharpness, image brightness, and depth of field (DOF).  Ideally, sharpness and brightness will be maximised and DOF minimised (to blur the background), but brightness is at odds with sharpness and DOF.  Sharpness is increased by raising the shutter speed and lowering the ISO number; brightness is increased by lowering the shutter speed, widening the lens aperture (lowering the f-stop), increasing the ISO number, and lowering the focal length; and DOF is decreased by narrowing the aperture and increasing the focal length.  After some 49  experimentation I settled on keeping the shutter speed quite high, above 1/800 of a second, raising ISO to about 500, and then narrowing the aperture as much as the brightness would allow.  To extract still frames from the video recordings I used the PotPlayer video player by Kakao Corporation.  I used MATLAB for image editing, which included correcting colour balance, cropping, rotating and tiling the images.    2.2.9 Laser PIV visualisation In addition to ink flow-vis I also used laser particle image velocimetry (PIV).  Laser PIV is a technique which allows for a quantitative assessment of flow conditions at all points in a plane.  The method uses a computer algorithm to trace the path of particles in the water flow.  The algorithm calculates the speed and direction of water flow and uses this to reconstruct the instantaneous flow conditions, revealing relative flow and velocity among many other variables. The particles take the form of microscopic beads that are made from glass or another highly reflective material; a pulsed laser sheet brightly illuminates a plane of water seeded with these particles.  A high speed camera captures images in pairs, with each pair only an instant apart.  In processing the algorithm divides the field of view into a grid of ―interrogation windows‖; it then matches those windows in the subsequent image and calculates the mean displacement.  This is repeated with smaller and smaller windows, as processing power allows.  Finally, the relative trajectory of these matched images from one frame to the next is used to calculate velocity and other relevant measurements.   The PIV trials in this experiment used silver-coated hollow glass spheres as flow tracers.  The beads were an average diameter of 13 µm and density of 1.6 g/cc were used as flow tracers.  50  Illumination was provided by a 25mJ Nd:YLF dual diode-pumped laser (Darwin-Duo series by Quatronix) operated at 10A with a sheet thickness of approximately 1.5 mm.  Image analysis and calculations were done with LaVisionDaVis 8.2 software.  For image capture I used a Photron APX-RS high-speed digital camera with CMOS sensor.  Background image blur was achieved by using a lens that produced a shallow depth of field, in this case a Nikon 50mm f/2.8 Nikkor AF-series lens set to minimum f-stop (maximum aperture diameter). Considering the location for the plane, I rejected the midpoint of the tail because it was likely to function largely as a bluff body (Peter Oshkai, pers. comm.).   Instead I chose the midpoint of the ventral lobe, since that is near the point of the mean aerodynamic chord (MAC) and is unobstructed by the apparatus.  It also matched one of the ink-pore locations for the ink flow-vis model, providing a better means of comparison.  2.3 Results 2.3.1 Statics The results displayed in Figure 2.11 demonstrate that the lift force far exceeds the drag force for the angles I am concerned with, reaching a plateau at an approximate 15°–20° angle of attack.  They show a maximum lift-to-drag ratio at 4.8°, which is not predicted in dynamic experiments.   51   Figure 2.11 Statics results for tail angle of attack = -30° to 30° at 1.4 m s−1, Re = 50,000, showing force coefficients of lift and drag as well as lift-to-drag ratio. Bifurcation among lift and drag data is due to structural asymmetry in the tail.  Error-bars represent 95% confidence intervals.   52  2.3.2 Dynamics: Direct measurement  Figure 2.12 Performance as a function of Strouhal number and maximum angle of attack at 2 BL s−1, Re = 50,000.  Error-bars represent 95% confidence intervals calculated from population of 40 tail-beats during one trial. The results in Figure 2.12 show how St and     affect hydrodynamic performance in terms of the propulsive efficiency and coefficient of thrust.  They demonstrate that      has a narrow band of high efficiency performance, peaking at 15°, whereas St is less volatile with a gradual increase of efficiency over the tested range.  With the exception of      at low St, increase of either kinematic results in an increase of thrust; however, St develops thrust in a smooth and even manner relative to     , which shows a parabolic trend.  The highest thrust levels are achieved at combined maximum of      = 30° and St = 0.40; the highest efficiency levels are achieved at      = 15° and St = 0.25.  To further highlight trends in      and St, I converted the data into contour map form. 53      Figure 2.13 Data from Figure 2.12 plotted as thrust and efficiency contours at 2 BL s−1, Re = 50,000 Figure 2.13 displays the coefficient of thrust and propulsive efficiency contours over the tested range of      and St, fitted with cubic splines.  The efficiency contour suggests two efficiency maxima: one at      = 15° and St = 0.25 and the other at      = 14° and St = 0.35; the latter location corresponds to a higher thrust output, as seen in Figure 2.13a.  It should be noted that contour maps do not display uncertainty.  To investigate the individual values of power that make up efficiency I plotted average power as a function of      and St.  The results in Figure 2.14a demonstrate that     is closely tied to the amount of useful power the caudal fin produces, as well as the power required from pitching and heaving.  Surprisingly, high values of      require less torque power, despite the expectation of higher torque at higher angles of attack.  This decrease in required pitching-power seems more than 54  compensated by a greater heave-power requirement.  Panel b shows that variation in St (seen as the shaded area) has a small effect on efficiency relative to     , for which there is a clear optima at 15°.  In panel c. the relative input and output power is plotted as a function of St, with the effect of      visible as the height of the shaded region.  St shows a positive correlation with all power involved, although in general the power required to heave is several-fold larger than the power required to pitch.  Panel d. further emphasises that efficiency is relatively stable regardless of St, with a slight optima seen at St = 0.25.  Comparison of panel b. and d. shows the similar effect both kinematics have on thrust.  The results so far consider values that are averaged over a tailbeat; to explore the kinematic performance as a function of time I plotted instantaneous values over a tailbeat.   55   Figure 2.14 Average power and performance at 2 BL s−1 as functions of αmax and St, shaded area = min and max values.     56  In Figure 2.15 I plot instantaneous force magnitude and direction with relation to heave position. Comparison of the top and bottom row demonstrates that, relative to     , St has very little influence on the distribution and layout of the force generated by the tail; its influence is applied evenly over the whole tailbeat and manifests as a general increase in the magnitude of the resultant force with a slight increase in forward tilt.  Comparison from left to right shows that       is responsible for much larger changes in the way the tail interacts with flow. The resultant force distribution at low      and high St is unusually sinuous, even pointing backwards during parts of the tailbeat.  An increase of      manifests an increase in the magnitude of the resultant force coupled with a rearward tilt; at high      the vector magnitude is particular high at the midpoint of the tailbeat with increased sideways tilt.  This tilt illustrates the trade-off of efficiency for thrust, as most of the force that the tail produces is ejected from side to side.  Since the angled resultant vectors somewhat obscure the size and placement of the thrust component, I generated another set of vector diagrams with the thrust component alone, shown in Figure 2.16.   57    Figure 2.15 Resultant vectors over one tail beat showing displacement and force through time .  Tail is moving right to left, positive (thrust) force extends to the left.  Heavy black line presents heave through time.  Every ¼ stroke is represented by foil; vectors corresponding to foils are coloured red.  Outer line represents path traced by tip of resultant vector.  Forces are scaled 1:1 so that one length of background grid is 5 Newtons.    58   Figure 2.16 Thrust vector component over one tail-beat , with every ½ stroke represented by a foil, see Figure 2.15 for additional details Isolating just the thrust in Figure 2.16 shows the positive correlation with     and St, with increases seen from the top to bottom rows and left to right columns.  Again, St is seen to have very little effect on the distribution of thrust throughout the tailbeat.  Low values of      display regions of negative thrust at starting just after stroke reversal, likely contributing to the poor efficiency seen in Figure 2.12.  In general the thrust peak occurs between the mid- and end-stroke of a tailbeat, advancing forward at higher      while having almost no relation to St.  This advance of peak force is not predicted by quasi-statics, so in order to investigate it further I plotted the instantaneous force, torque and power traces as a function of tailbeat phase. 59   Figure 2.17 Instantaneous values over the course of one stroke (half a tailbeat), showing a. Heave and pitch waveforms b. Phase of peak force and torque as a function of angle of attack c. Instantaneous power in and out (shaded region denotes positive and negative work)   60  In Figure 2.17 we see the relationship of force, torque and power to heave and pitch over the course of one tail stroke (half of a tailbeat).  The top row demonstrates how pitch and heave vary as the      increases, with the pitch waveform displaying an increase in higher harmonics resulting in a depressed peak, and the heave showing a small increase in amplitude.  The middle row highlights the effect      has on the timing of peak force and torque, which all advance as       increases.  In addition, the disparate peaks converge on a single timing at       = 25°; investigations of the timing at      = 30° reveals they continue to advance, showing that the alignment of forces at       = 25° is likely coincidental and not due to a common maxima being reached.  The large regions of negative thrust in the vector diagram (Figure 2.16) are matched by the negative thrust occurring at low     .  The third row is the product of the middle row and the derivative of the top row (the heaving velocity and angular pitching velocity), as show in equations (16) to (18); it demonstrates the instantaneous output power and input power in the form of pitching and heaving power.  The trend in peak power is similar to the middle row, excepting the pitching power which also advances, but generally occurs later in the stroke.  This row also shows the difference in relative contribution of pitching and heaving; high      is powered mainly by heaving power.  The advances of force and power raised question of how this might apply to apply to locomotory muscles.  To better understand this I plotted the force and torque data as simulated work loops.   61   Figure 2.18 Simulated work loops of lateral force and torque for different αmax . Numbers represent successive time points over one stroke (half a tail-beat).  Red dash represents force and torque acting in the direction of extension and therefore representing positive work done on the foil by surrounding fluid. The results in Figure 2.18 demonstrate ―work loop‖ analogs that I constructed with the lateral force and torque data.  The work loops simulate muscle from one side of the fish only, responsible for heaving the tail in the negative direction; the corresponding muscle on the other side of the fish is assumed to be responsible for heave in the positive direction.  The x-values are plotted as distance rather than strain since the total length of the simulated muscle is unknown; however this does not affect the shape of the loops.  Muscle activation leads to force in the direction of shortening, so I do not consider the negative forces in the direction of extension.  Every loop is counter-clockwise and represents positive work being done.  The work loop at       = 5° displays a segment of force opposing motion, progressing roughly from point 3 to point 4.  This somewhat counterintuitive result indicates that the tail is being acted on by fluid flow and explains the negative thrust seen after stroke reversal in Figure 2.16.  The relation between tailbeat phase and peak force and torque is again demonstrated by the shape of the work loops; at low      the peak is at near maximum heave, while at high      the peak is at 62  midstroke.  The work loops also illustrate the manner in which required heaving power increases with     .  2.3.3 Dynamics: Flow visualisation In Figure 2.19, ink plumes illustrate the influence of      on vortical flow structures produced during the tail-beat. The top row demonstrates that       = 5° produces a relatively laminar wake with the exception of one small leading edge vortex (LEV) which is shed at stroke reversal in the fourth panel.  Detailed inspection of the shape of ink vortices over time reveals the direction of rotation: a train of counter clockwise vortices appear in the wake prior to role reversal, whereas the rotation of the LEVis clockwise.  Vortex structure in the plume following vortex shedding matches its clockwise rotation.  In the middle row of      = 15° the situation is similar, but the magnitude of the LEV shed is greater, as demonstrated by ink plumes that are well mixed and diffused.  The presence of the LEV is clear in the third panel, and is just shed in the fourth panel.  Vortex rotation matches the upper row but is more distinct.  In the third row of     = 25° the phenomena described above are still present, but exaggerated.  In addition, as early as panel one we see the formation of an additional LEV which is shed just after panel two.  In panel three a second, larger LEV forms and is shed in the following panel.  To get a more precise and quantitative view I repeated the visualisation with particle image velocimetry (PIV). 63   Figure 2.19 Ink flow visualisation for St = 0.35 at 1 BL s−1.  Images progress from left to right.  Arrows indicate first LEV.   Tail-beat phase is approximate.   64   Figure 2.20 Particle image velocimetry for St = 0.35 at 2 BL s−1 .  Blue indicates clockwise vorticity; red counter clock-wise. 65  The time averaged results of PIV in Figure 2.20 show the same trends as the prior figure but with additional details.  At      = 5° we see that prior to LEV release, the wake takes the form of a series of small vortices: the clarity of their presence after time averaging indicates that their spacing and size is very consistent between tail-beats.  The LEV seen in panels three and four as a small elongate vortex is even weaker than it appears in the ink flow-vis.  The vortices immediately preceding the LEV take the form of a tightly bound von Kármán drag street.  In the middle row at      = 15° the tidy chains of vortices are replaced by smoothed, stretched out versions, which may indicate more variation in vorticity between tail beats.  The clockwise and counter clockwise vorticity of this trail is represented by an overall greater blue or red characteristic, respectively.  Relative to      = 5° the shed LEV is much more distinct.  In the lower row at      = 25° there is a marked increase in vortex size and distinctiveness.  All traces of the vortex chains at low      are gone, replaced by large, distinct vortices.  The edge of the additional midstroke vortex is visible in panel one; in panel two that vortex has been shed and convected half way through the field of view.  Visible here but not in the ink flow-vis are the trailing edge vortices (TEVs) in blue that are appear to alternate with the LEVs forming a von Kármán drag street in panel two.  The TEVs are seen to fade in strength more quickly than the LEVs, leaving an overall reverse von Kármán street.  Having investigated horizontal, planar phenomena, I next return to the ink flow-vis method to look at vertical and span-wise phenomena.   66   Figure 2.21 Vertical ink flow-visualisation for St = 0.35 of the ventral lobe of the tail.  Columns match Figure 2.19.  In this image series the tail is seen moving away from the observer.  67   The results of ink flow-vis in the vertical plane are displayed in Figure 2.21.  A trend common to all      is the criss-crossing of ink dye streams, although the effect is most evident at       = 25°.  The LEV is revealed by a blue ink plume that gathers on the tail from panels one to three and is subsequently shed in panel four and convected downstream in panel five.  Following the release of the LEV there is a criss-cross event in the opposite orientation to the first, showing movement of flow towards the tail tip on the side facing the camera and towards the midline on the rear side.  As the LEV is convected downstream it takes the form of an angled cloud of ink that matches the profile of the trailing edge.  The ink stream at the trailing edge of the dorsal lobe illustrates the TEV which sucks trailing edge flow forwards over the tail.  This effect is only visible in the second and third rows of      = 15° and 25°, and is only apparent in the third row at      = 25°.  The two techniques used in this section visualise flow using very different means; in order to compare these methods I synchronised flow image and compared them side by side.  68   Figure 2.22 Comparison between ink and PIV flow visualisation.  Both rows at 1 BL/s, St = 0.35,      = 15°; approximate phase matching is shown.69   Figure 2.22 provides a comparison of the ink-flow method and the PIV method taken at      = 15°.  Direct comparison between the two is hampered by the different frame rates each was recorded at and the lack of a triggering signal, so their alignment is approximate.  In order to provide a more objective comparison I use PIV data taken at 1 m s−1, matching the speed of the ink trial; however, the PIV data have a much smaller sample size of 50 images compared to 300 images in Figure 2.19, which is why the flow appears more fragmented and chaotic.  Additional caution must be taken in this comparison, as the ink-flow model uses blue and red ink to distinguish which side of the tail the flow originated on, whereas the PIV uses red and blue colours to indicate the direction of vorticity, with blue meaning clockwise and red counter clockwise.  The comparison confirms that the ink flow-vis correctly identifies the vorticity preceding the LEV.   It also illustrates the problem in ink flow-vis: strong turbulence causes ink to diffuse more rapidly, reducing visibility.  2.3.4 Effect of apparatus Figure 2.23 quantifies the flow disturbance that results from the vertical strut of the apparatus through the comparison of a foil that was behind the strut to a foil that encountered uninterrupted free flow.  The results over several trials have been compared with a two-sample t-test.  The upper panel shows that the strut causes a small but statistically significant detriment to the thrust coefficient which increases with     .  In the lower panel we see that the same trend exists for efficiency but to a lesser extent.  At an       of 5° there is considerable variation and no significant difference, whereas at 25° we see a 0.035 reduction in the thrust coefficient and a 0.027 reduction in propulsive efficiency.  Since the actual tail has one lobe behind the strut and one lobe in free flow, the overall effect should be halved.  It is useful to know that the effect is 70  only detrimental in nature and does not augment performance; thus my results are lightly underestimated.    A moment of direct interaction is observed in Figure 2.24.  The bluff body wake street being shed by the vertical strut crosses the path of the dorsal lobe tail; this is visible as a kink in both ink plumes in panel two which evolves into two counter clockwise rotating vortices.  By panel 4 they have mixed, decreasing visibility.  This same phenomenon was not observed in the ventral lobe.    The same interaction is captured in more detail using PIV in Figure 2.25.  A von Kármán drag street is captured behind the vertical strut and its lack of distinct vortices indicates that it is chaotic.  The wake street shed by the rod flows alongside the wake shed by the tail but does not directly overlap; nonetheless the size of the vortex street indicates interaction.  At stroke reversal in the third panel down the apparatus wake street is seen to cross the tail.  Curiously there is no counter clockwise vortex seen as in Figure 2.24.  However, this could very well be due to the chaotic flow cancelling out such structures in the time averaging step.   71    Figure 2.23 The effect of the vertical strut on the thrust coefficient and the efficiency of a foil for 5 trials, showing either foil up, in which the foil is obstructed by wake from the rod, or foil down, in which the rod is in free flow without interaction with the rod.  Asterisks represent significant difference as found by two-sample t-test.   Figure 2.24 Time series of interaction between rod and dorsal lobe, visible as paired vortex in both ink streams.  Location of rod is shown for the first panel.  Passage of interference from rod tracked by dashed circle. 72   Figure 2.25 PIV visualisation of wake interaction between rod and dorsal lobe, as viewed from below, showing the location of the rod as well as the wake from both rod and tail.  Images are a composite of two sets of data taken to increase the field of view. 73  2.4 Discussion In summary of the results section, I find that the static lift-to-drag ratio is maximised at an angle of attack of 4.8°.  In dynamic experiments the highest propulsive efficiency is found at a maximum angle of attack (    ) of 15° over several Strouhal numbers (St), while the highest thrust is found at the maximum value of both kinematics.  Increasing      requires many times more heaving power than pitching power, and results in an advance of peak thrust timing.  This correlates to an advance of required peak lateral force and torque.  Flow visualisations show that increasing      results in larger leading edge vortices and eventually the release of a second leading edge vortex around midstroke.  2.4.1 Mechanics of angle of attack and Strouhal number The static data in Figure 2.11 and the vector diagrams in Figure 2.16 provide a mechanistic explanation for the effect of Strouhal number (St) and maximum angle of attack (    ) from the quasi-static perspective, which views the tail-beat as a series of steady-state events. The vector diagrams illustrate how the St increases thrust without sacrificing efficiency (Figure 2.16, from upper row to lower row).  With regard to the effect of St on the hydrodynamic resultant force that occurs at the midpoint of heave: an increase in St causes the resultant to lengthen and tilt slightly forward, indicating a greater ratio of thrust to lateral force.  The profile of the hydrodynamic force is unchanged, seen as traced along the vector tips.   Statics, which detail the performance of a foil at various angles of attack, pertain more to      than to St.  They show at what angles the foil produces maximum lift and drag, and the 74  angle at which the lift-drag ratio is maximised (where the resultant force has the greatest forward tilt).  The static results suggest that an angle of 5° produces an optimum lift-to-drag ratio and thus maximum efficiency is expected.  Interestingly, this is not corroborated by our dynamic results: an      of 5° produces the worst performance in thrust and second or third worst in efficiency (depending on St).  Instead I find that the optimum      for efficiency is 15°, as seen in Figure 2.12.  The static results show that at 15°, lift is just sub-maximal but still several times larger than drag.  Considering the average (rather than the maximum) angle of attack over one stride still fails to elucidate this.  At an      of 15°, the average angle is ~10°, exceeding the predicted efficiency optimum by 5°.  This is likely due to the unsteady phenomena of delayed stall and vortex shedding which are not accounted for in quasi-statics (Gülçat, 2011).   Despite this inaccuracy, the statics data offer some mechanistic understanding for the performance of     : as the angle of attack increases, the lift-to-drag ratio becomes smaller and therefore the resultant vector tilts sideways, shedding ever more force laterally.  In addition to resultant tilt, the magnitude also increases, so there is a two-part increase in lateral force.  The vector diagrams illustrate these phenomena, showing the largest lateral component at the greatest value of     .  Unlike a change in St, the shape of the hydrodynamic force traced out along the vector tips is wholly dependent on     .    Understanding the effect of different motion parameters on      and St allows us to relate them to the swimming modes of cruising and bursting.  Cruising swimming emphasises efficiency: being obligate ram-ventilators means tuna can never rest, so their ability to cruise affects every moment of their lives (Shadwick et al., 2013).  Applied over a lifetime, even tiny efficiency savings add up to immense net energy savings, so there is a strong pressure for efficient motion.  Cruising is largely a steady state, so of the two performance variables 75  efficiency is tantamount while thrust must merely meet the relatively low minimum drag requirements.  The results in Figure 2.12 show that a maximum efficiency is met at an      of 15° over a range of Strouhal values.  The contour plot in Figure 2.13b provides closer examination: a clear efficiency peak extends through several Strouhal values, deviating only slightly in      by 1-2° from the lowest to the highest St.  This suggests that if thrust needs to be modulated during cruising, then St is the most logical way to do so, either by increasing tail beat amplitude or frequency.  Since St is defined by amplitude as well as frequency, these trends should still exist if frequency instead of amplitude were manipulated.  Studies by Read et al. (2003) and Schouveiler et al. (2005) corroborate my results, with both studies finding the optima efficiency to occur at the same      value of 15°.  The results of this and other studies provide strong evidence for an invariant      during cruising. Although during cruising the optimal values of      and St are clear, during bursting several factors come into play and the question becomes ambiguous.  Burst swimming is analogous to sprinting: it is a high intensity, short duration activity that maximises speed (Shadwick et al., 2013).  Relative to the endlessness of cruising, bursting takes place over miniscule durations, with thrust rather than efficiency as the dominant factor for success.  In the transition from cruising to bursting, higher thrust enables greater acceleration on which successful prey capture and predator evasion depends.  Figure 2.13a shows that the highest thrust available exists at the maxima of both      and St.  Taken in combination with Figure 2.13b, it suggests that to transition from cruise-level thrust up to burst-level thrust, the most effective path is to first maximise St and then maximise     .  Medium levels of bursting performance can be achieved with St alone, while the highest levels of bursting performance are achieved with the assistance of     . 76   2.4.2 Muscular implications of angle of attack control In a broad sense, muscle effort results in force which produces action; therefore knowing force one can estimate muscle effort (Andrews, 1983).  Like my experimental system, a tuna‘s caudal fin is confined to two axes of motion that are approximated by pitch and heave (Schouveiler et al., 2005).   Even without knowing the precise mechanism responsible for heaving and pitching the tail, one observes that the work and forces applied to the axes (lateral force and torque) must be matched by whatever powers tail motion, as per Newton‘s third law.  To this end I extrapolated my results to predict the relationship between muscle effort and kinematics in different swimming conditions.    In representing muscle effort I considered both instantaneous force and power, paying particular attention to the timing of peak values.  An advantage of instantaneous power over force is that it demonstrates the type of work being done: muscles tend to perform differently depending on whether they are doing positive or negative work (Vogel, 2013).  However, it is still useful to consider force alongside power since, being a product of force and velocity, power vanishes during isometric contraction.   While the relationship between       and muscle effort is complex, St and muscle effort present a more straightforward case.  St is directly proportional to tailbeat frequency and tailbeat amplitude, so an increase in St can be enacted by either (i) increasing muscle shortening and thus heave (peduncle amplitude) and tip amplitude, or (ii) increasing muscle contraction frequency.  In contrast to St, the relationship of     to muscle effort is previously unexplored. Increasing       from 5° to 25° requires large increases in force and power, as seen in Figure 2.17.  Less predictably, an increase of       also requires an advance in the timing of 77  peak force and power.  Comparison of Figure 2.17b and Figure 2.17c show that at higher      there seems to be more isometric force applied for pitching in contrast to low      which has more isometric force applied to heaving.  Furthermore, there is actually a net decrease in pitching power required at high     , in opposition to the 6-fold increase seen in heaving power.    To get a better idea of what is required from the perspective of the musculature I used the force and displacement data to make simulated work loops, as seen in Figure 2.18.  I rely on the fact that in swimming tuna, muscles on either side of the body contract in anti-phase, alternating in action.  Knower (1998) shows that muscle force occurs predominately during shortening and that muscle shortening on one side of the body is associated with lateral movement of the tail towards that side.  I designed my work loops to consider only positive forces because negative forces (those in the opposite direction) are necessarily accounted for by muscles on the opposite side of the body.  Cruising My results suggest that during cruising       is held constant and does not play a part in modulating thrust.  As described in 2.4.1, cruising requires stable, continuous low-level thrust and depends on an optimised efficiency.  In terms of kinematics, there is no demonstrable benefit to modulating      from the value that produces the highest propulsive efficiency, which is 15° as seen in Figure 2.12.  The study by Shadwick and Syme (2008) investigated the duration and onset of muscle stimulus in aerobic red muscle, and found that there is a narrow window of onset stimulus that produces maximum positive work output.  Given that cruising relies heavily on a state of optimal efficiency, the ideal state is likely one of continuously maximised positive work output and thus stable muscle stimulus onset time.  As shown in Figure 2.15, a change in      is 78  associated with a change in the timing of peak force, which would require a change in onset of muscle stimulus and thus is unlikely to occur during cruising.   An additional point for a fixed      is shown by Knower (1998), who observes that in a cruising tuna, muscle force consistently peaks as the tail crosses the midline of the body, indicative of unchanging force timing.  In vitro data by Shadwick et al. (1998) suggest that during cruising, the shape of the work loop of cruising musculature is quite stable.   My simulated muscle work loops in Figure 2.18 show that the simulated work loop undergoes large shape changes in response to     ; further reason to suspect that       is held constant.  These results in combination with muscle studies predict that factors other than       are varied when modulating thrust during cruising.    Bursting In the red muscle dominated cruising mode, the likelihood of      being used to adjust swimming kinematics is low; however, there is more to consider in the white muscle dominated bursting mode.  Burst swimming is dominated by white muscle for two reasons: (i) the high tail-beat frequencies demand correspondingly high muscle contraction frequencies at which red slow-twitch muscle is unable to produce meaningful power (Shadwick and Syme, 2008), (ii) the thrust must overcome a drag force that is proportional to the square of the velocity, thus demanding a much larger power source: white muscle makes up about 90% of skeletal muscle mass and is specialised for high intensity, short duration activities such as bursting (Freund, 1999).  Observations by Fierstine and Walters (1968) suggest that      has a high rate of variation of up to 100°.  These data were taken for tuna swimming around in circular pen and would benefit from a corroboration using modern methods.  Nonetheless, there is a case to be made for      as a control variable for thrust. 79  One of the challenges in fish muscle biomechanics is the complexity inherent in fish muscle layout: long trains of interlocking cones of muscles are wrapped in tendinous sheaths that intertwine with each other and the axial skeleton, forming elaborate pulley systems whose relationship to pitch and heave is still not fully understood (Gemballa and Treiber, 2003).  It is difficult to ascertain whether the pitching motion is under active control; as of yet there is no proven mechanism for how it might be controlled independently of heave.  This is not to say that      is invariant, but its response may be involuntarily coupled to the lateral motion of the peduncle.  That being said, it is simply not known whether pitching and heaving motion is coupled while under white muscle control; one hypothesis could be that anterior musculature is tied more to heaving motion and posterior musculature to pitching.  Shadwick and Syme (2008) show that tuna red muscle is physically uncoupled from local body bending; unlike red muscle, white muscle is coupled to body bending, which may manifest as differences in the motion profile of the tail.  Shadwick and Syme, (2008) show that the phase relationship between muscle strain and activation dictates muscle function, allowing for different swimming modes.  Given that the relationship between white muscle and burst kinematics is unknown and likely distinct from the red muscle used during cruising, there may be a role for      in the control of thrust. When thrust is boosted with      then peak force must be advanced, as seen in Figure 2.12b.  This suggests an advance in muscle activation onset.  Altringham and Johnston (1990) show that advanced stimulus increases peak force but at the expense of overall power.  This shows that there may be some occasion for stimulus to be advanced.  In red-muscle dominated cruising we see little evidence for such a change, so the possibility of it in white muscle depends largely on the similarity of red muscle and white muscle behaviour.  Future knowledge of the 80  actual work loop characteristics of burst muscle will be a great help in understanding the limitations in       control.  Regarding our simulated muscle work loops in Figure 2.18, the shape at low     shows force being exerted on the tail in opposition to motion.  This suggests that hydrodynamic force is not a perfect proxy for muscle force and that the biological system is probably not capable of moving the system exactly as in this experiment; this warrants further exploration.    Cavitation Another factor that may come into play at burst speeds is cavitation: the formation of vapour cavities (bubbles) due to the local pressure dropping below the vapour pressure of the liquid.  When subjected to higher pressures the voids collapse violently and emanate shockwaves.  Continuous exposure to these implosions can cause significant damage, even pitting and eroding hard metal propellers (Carlton, 2012).  In a biological context, calculations by Iosilevskii and Weihs (2008) show a cavitation potential for dolphins near their maximum speeds, suggesting that it may put a speed-limit on thunniform swimmers due to pain and tissue damage.   A high      relates to cavitation because of its effect on the hydrodynamic force: relative to St, an increase of       results in a larger hydrodynamic force, as seen in Figure 2.15.  The high angle of attack generates a larger lateral force component for a given thrust component, resulting in a larger resultant force, and thus a greater range of pressure.  As the tail experiences regions of further decreased pressure, cavitation becomes increasingly likely.   This would seem to count against the idea of using      to increase thrust at high burst speeds; however there are two caveats to consider.  The first is that, given the short duration of a 81  burst, the associated pain of cavitation may be bearable.  Delepine (2013) accounts a marine biologist, Chris Harvey-Clark, who heard clicking among bursting tuna suggestive of cavitation, meanwhile Iosilevskii and Weihs (2008) report that cavitation-related damage has been observed on tuna caudal fins, meaning that the damage and potential pain is borne to some extent.  The second caveat is that cavitation is a depth-dependent phenomenon.  It depends on local pressure approaching the vapour pressure of the liquid, but at depth there is a huge increase in the local hydrostatic pressure which must be overcome for cavitation-inception, greatly decreasing its potential (Carlton, 2012).  Tuna are known to dive to forage—Atlantic Bluefin have been recorded at a depth of 817m (Musyl et al., 2003)—at such depths there is likely no potential for cavitation.  It may be that thrust maximization via       is only used while swimming at depth.  Physiological limits Heave and frequency impose physiological limits on the maximum values of      and St.  Although St can be used to increase thrust without sacrificing efficiency, its relationship with heave and frequency (4) may constrain it during bursting.  Considering first frequency: Blank et al. (2007) show that frequency is proportional to swimming speed, while Shadwick and Syme (2008) show that past the optimum muscle contraction frequency there is a steep drop-off in muscle performance.       may provide an alternate way to increase thrust and maximum burst speed without requiring tail-beat frequency to far exceed its optimum.   Limitation of heave is more complex as it relates to both       and St to differing extents.  An increase in heave may be beneficial, with Shadwick and Syme (2008) showing that muscle performance increases with strain (up to a point), but there is a hard upper limit to how far any system of joints can articulate.  The tail-tip of fish generally moves within a peak-to-peak 82  amplitude of 20% of the body length (Bainbridge, 1958; Shadwick et al., 2013); the heave of the peduncle will be slightly less than this due to the phase difference between the peduncle and the tail tip.  St‘s relationship to heave is implicit in its definition (4); what is less obvious is that      also effects peduncular heave amplitude.  The key is the way the two kinematics affect heave in relation to thrust: I find that for an equivalent increase of thrust, increasing St and holding       fixed requires an increase of heave amplitude three times larger than increasing      and holding St fixed.  This suggests that in situations that require the maximum amount of thrust, increasing       in preference to St may enable the system to reach a higher total thrust within the limits of peduncular heave amplitude and muscle contraction frequency.  If that is the case, then there is good reason to predict that sub-maximal burst speeds will be reached through a combination of increased St and     .  It is known that burst swimming should predominately involve white muscle, but the precise mechanism is yet undetermined; this study has provided evidence that unlike red muscle, white muscle may utilise       in order to access higher levels of thrust.  This depends closely on the capability of the burst musculature system to vary peak force production relative to tailbeat phase, with more thrust requiring earlier onset.  Although cavitation is more of a limiting factor for      than St, it may pose no issue at depth where tunas regularly spend time.  With consideration of the physiological limits associated with each kinematic,      may allow the system to reach higher thrust before hitting the upper limits of muscle performance.  In conclusion, during cruising the control of thrust is probably dominated by St, whereas during bursting there is evidence that       may play an important role, and likelier still that St and       are used in conjunction to maximise performance. 83   2.4.3 Flow characteristics of αmax Development of Vortex Street  Flapping propulsion produces thrust through unsteady hydrodynamics (Gülçat, 2011).  The quasi-static analysis above represents steady forces by painting the system as a series of isolated moments in time, during which the system is considered to have steady-state flow conditions.  However, this highly simplified method can only partly explain the performance results; the remaining forces are due to unsteady hydrodynamics—the continuously evolving fluid flows that stream from the moving tail.  In Section 2.1.1 I explain the mechanism of lift-based propulsion, but that explanation is incomplete without considering unsteady hydrodynamic effects.   Unsteady hydrodynamics produce thrust by the opposite mechanism by which a bluff body experiences vortex-related drag (Vogel, 2013).  In certain flow conditions predicted by the Strouhal number (St), a bluff body will shed a train of eddies.  The eddies alternate from one side to the other, taking turns swirling around either edge of the bluff body and rolling up into vortices.  The end result is a staggered train of vortices that are convected down-stream away from the bluff body, as in Figure 2.26a.  Between each pair of vortices there is a bulk flow of fluid – a jet angled forward towards the bluff body.  Downstream of the bluff body, the time-averaged flow profile is in the form of a jet directed straight towards the bluff body (Read et al., 2003).  The momentum in this jet is equal and opposite to the drag felt by the bluff body (Durbin and Medic, 2007).  In fact, one offshoot of flapping foil research is the development of flapping foil turbines that harvest energy from the vortices responsible for this drag force in order to produce electricity (Kinsey and Dumas, 2008). 84   Figure 2.26 Simplified illustrations of von Kármán wake street types with vortex rotation shown in red and blue and jet direction shown in green.  Vortices are convected in the bulk flow.  Black curves represent the interference fringes seen in soap film visualisation (based on data from Anderson et al. (1998)). In flapping foil propulsion an almost identical scenario plays out, but rather than passively generating vorticity, the tail actively establishes the aptly-named Reverse von Kármán street, made up of a wake of vortices that rotate in the opposite direction to those in a von Kármán vortex street (Jones and Platzer, 2009).  The average velocity now takes the form of a jet directed away from the propulsor, and the momentum imparted to this jet is experienced by the propulsor as a thrust force.  Different motion profiles of the flapping foil produce variations in 85  the pattern of shed vortices, altering jet angle, degree of vorticity, and even the layout and number of vortices (Triantafyllou et al., 1993).  This results in vortex configurations of varying levels of thrust and efficiency, with poorly chosen kinematics producing net drag.   The formation of these staggered vortices cleverly makes use of stall while managing to avoid the catastrophic loss of lift that usually follows.  In static operation an aerofoil smoothly and continuously deflects oncoming air, with flow streamlines closely following the shape of the aerofoil as they are rerouted (McCroskey, 1982).  Critically, flow streamlines remain attached – their course is changed but their passage smooth.  Stall occurs when the angle of attack becomes so high that the flow on the rearward surface completely detaches from the foil, leaving a region of chaotic, drag-producing turbulence.  As the angle continues to increase, the lift drops to zero and drag becomes the dominant force.  This scenario has spelled doom for many an aircraft; however, the flapping foil system neatly evades the problem by taking advantage of a phenomenon called dynamic stall, which uses a rapid onset of high angle of attack in order to exceed the amount of static lift available (Gülçat, 2011). When the angle of attack rapidly increases it produces a leading edge vortex (LEV) which rolls off the leading edge onto the outer surface of the aerofoil and off of the trailing edge.  The presence of this transient vorticity augments the pressure difference on the foil, increasing the magnitude of the lift force (Pitt Ford and Babinsky, 2013).  However, this bounty is short-lived: the force lasts only so long as the vortex is attached to the surface of the foil.  This which is not long: convected by free-stream fluid flow it quickly rolls off the trailing edge.  Lift drops far below predicted levels as the turbulent, unattached stall condition establishes itself.  Up until that point the flapping foil has managed to ‗cheat‘ stall and produce unusually high lift past the angle at which stall normally occurs, if only briefly.  Lift conditions remain bewildered until the 86  angle of attack has decreased below the static stall angle, at which point flow can reattach and the steady-state lift condition re-establish (Gülçat, 2011).  However, the flapping foil does not have patience for this: as the LEV is rolling down its back the foil has already reversed its angular velocity, timed so that it flicks the LEV off its trailing edge at the reversal of the tailbeat.  As the foil begins to move in the opposite direction the whole series of events is repeated.  Left behind in the wake is a trail of staggered vortices: a reverse von Kármán street (see Figure 2.26).  Flow visualisation Ink flow-visualisation (flow-vis) and particle image velocimetry (PIV) are two methods with which to visualise flow events that occur during flapping propulsion, such as the reverse von Kármán street; using them I observed that      influences the size and distribution of propulsive vortices as well as drag flow.  In addition, the ink flow-vis from a different angle illuminated 3-D effects that occur along the span of the tail in response to     .   A phenomenon visible in all trials was the bluff body drag street, although its form varied with     .  Not normally present in basic illustrations of propulsive flow mechanisms, the drag trail manifested in ink as a relatively laminar region with several weak vortices visible in the centre of the trail (Figure 2.19a); phased-averaged in PIV, it resolved into either an elongate region of low vorticity in one direction or a tightly knit von Kármán drag street (Figure 2.20a).  While the tightly staggered formation is characteristic of drag, the elongate region is less obvious.  The rotation direction of this region was seen to depend on stroke direction and imparted some net vorticity to the water.  Given that the lowest      produces thrust in parts of the tailbeat, it may be that the elongate region is a ‗biased‘ drag street that imparts a net propulsive momentum to the water.  A LEV is not clearly defined in the PIV data, but can be 87  seen in the fourth panel of the ink data Figure 2.19a.  The elongate region perhaps contains a manifestation of an LEV, as seen in PIV Figure 2.20. The drag phenomenon is visible at every value of     , but its relative duration or duty cycle differs: at low      the drag trail is present through most of the tailbeat, whereas at high      it is disrupted into approximately ⅓ of the tailbeat, as seen in Figure 2.19.  The presence of a von Kármán vortex street type-region may explain the negative thrust I measure at the beginning of the tailbeat: the release of the LEV accounts for some of the thrust drop-off, and perhaps this drag street is responsible for the thrust dipping below threshold values. In addition to the drag phenomenon, higher values of      are associated with stronger LEVs: in PIV they appear larger and more vibrant, whereas in ink flow-vis their presence is announced by an absence of ink; the high turbulence and pressure in the stronger vortices tear the ink plumes apart, diluting them to varying degrees of transparency.  The small LEVs at low      appear more reluctant to be shed, which may contribute to an explanation of why at low      there is a delay in the peak and subsequent decay of the resultant force and torque per tailbeat, as seen in Figure 2.17.  The large LEVs also support the notion that the      of theoretical maximum thrust has not been exceeded: Read et al. (2003) find that very high values of      cause strong LEV separation, weakening the thrust capabilities of the wake; it appears those have not been reached here.   One of the most fascinating phenomena was the influence of      on the shape and distribution of the wake street.  The reverse von Kármán vortex street described in the previous section actually represents the simplest state, defined as 2S, meaning ―two single vortices‖ as seen in Figure 2.26b.  This type of the vortex street is seen at medium and low values of     .  At high      the vortex street changes form: an additional LEV is shed at each midstroke before 88  the primary LEV.  The rotation of this additional LEV matches the rotation of the subsequent LEV shed at stroke reversal.  This pattern has been classified as 2P, meaning ―two paired vortices‖, and curiously is also seen in the wake behind anguilliform swimmers such as eels (Ohashi and Ishikawa, 1972; Tytell and Lauder, 2004).  These paired vortices may offer further explanation for the increased lateral force at high     .  Triantafyllou et al. (2004) reports that at higher values of St there is an evolution in 3-D vortical wake structure.  The 3-D wake structure normally associated with 2S streets resembles a series of connected hoops or doughnuts; however in 2P wake streets it appears as paired branches that diverge over time and take the form, in cross-section, of four vortices per cycle.  Triantafyllou et al. discuss this only as a consequence of St, not in relation to     .  Here I find that this 2P wake structure is also dependent on     , showing traditional 2S patterns for values of 5-15° and only progressing to the divergent pattern at 25° The final observation was a product of ink flow-visualisation alone and involved the presence of flow along the vertical span-wise axis.  Each      presented the same general pattern: vertical flow that pulsed in phase with heave velocity.  The effect was greatest at mid-stroke and drew the ink streams outwards towards the nearest tail tip on one lateral surface of tail, and inwards towards the tail midline on the other lateral surface, as seen in Figure 2.21.  The ink stream on the trailing surface was always drawn outwards, and on the leading surface inwards.  At low      this manifested as gentle, alternating fluctuation of the ink streams; however, at high      the effect became exaggerated to the point such that streams appeared to criss-cross one another, forming a trail of X-shapes that alternated in orientation of red and blue ink streams.  Since the effect was alternating and the streams continuous, they ought to have reached a point of being nearly parallel between successive criss-crosses; but this point was 89  concurrent with the shedding of the LEV, during which the ink streams billowed into diffuse clouds.  These clouds were yet informative, nonetheless, and illustrated the outwards flow associated with strong LEVs.  Further, the ink spun into plumes along the contour of the leading edge, such that the resulting shape took the form of a croissant.  This effect is corroborated by the 3-D PIV carried out by Flammang et al. (2011), in which they found the wake of swimming sunfish to take the form of a series of interconnected hoops.  In the setup used here the ink flow visualised the forward edge of those hoops.  This sheds light on the form of ink billows seen in Figure 2.21, as what was seen was not purely 2-D flow, which would have appeared circular, but rather a portion of a curving cylinder which appeared elliptical.  Regarding wasted energy, the phenomenon of stream-wise flow may also offer insight into the loss of efficiency evident at higher     , as it may be a manifestation of lateral force or some other form of discarded momentum.    Comparison of methods The methods of this study presented the opportunity to compare the efficacy of two flow visualisation methods: ink-flow visualisation (ink flow-vis) and particle image velocimetry (PIV).  These methods differ in the type of data they present, with PIV showing quantitative data restricted to a plane and generally phase-averaged, and ink-flow showing qualitative data not limited to a plane and occurring instantaneously, as seen in Figure 2.22.  PIV is restricted by the placement of complex and bulky equipment; in my setup imaging was possible only in the transverse or dorso-ventral plane, while ink flow visualisation can be viewed from any angle at which a camera can be placed. Although PIV is restricted to a plane of imagery it can quantify phenomena throughout that plane, except those locations obscured by the shadow of the laser; 90  ink flow-vis has no shadows but is restricted to the path traced out by the ink, which fills only a small region of the field of view (albeit the most important region).  In addition, while PIV can visualise precise vortex details at any speed, ink flow-vis is limited to lower vorticities: very turbulent events rapidly disperse the ink, announcing the presence of turbulence and vorticity while simultaneously obscuring the details.  Both methods have their advantages and disadvantages, and the combination presents more insight than sole use of either.  Overall this study finds that the kinematic variable      has a strong influence on flow phenomena, influencing the magnitude and distribution of shed vortices.  The two methods detailed here permit visualisation of unsteady flow phenomena, presenting additional evidence and explanation of the direct force and torque measurements.  I find that low      is dominated by drag, presenting a minimal reverse von Kármán street and very little span-wise flow; whereas at high     , the LEVs are greater in size and number and there is a strong criss-cross pattern formed by span-wise flow.  Analysis of these data is on-going and will be presented in future publications.  2.4.4 Comparison with similar studies Comparison with engineering studies  Comparison between flapping foil studies is somewhat confounded by the multitude of factors that are known to affect propulsive performance.  Chief among them are the Reynolds number (Re), but also influential are the foil aspect ratio (the presence of endplates simulates an infinite aspect ratio), the angle of attack profile, the combination of heave and pitch motion used to control the angle of attack, the phase angle between pitching and heaving motion, the 91  combination of heave and frequency used to control the Strouhal number, and the foil flexibility (and resonance tuning).  Some of the most comparable studies are: (i) Read et al. (2003), (ii) Hover et al. (2004), and (iii) Schouveiler et al. (2005).  These three studies all use symmetric NACA-type rigid foils with thickness to chord length ratios ranging from 12-15%.  They work within Re of 30,000-40,000, which is within the range studied here of 25,000-50,000.  All use a cosine angle of attack profile, although contrary to this study, to control angle of attack they fix pitch as a simple sinusoid and vary the heave waveform.  Furthermore, to adjust St they fix heave amplitude as a proportion of chord length and then vary frequency.  The phase angle is matched at 90°.  The results of these studies shows strong agreement with the results found here.  The        of peak propulsive efficiency found in (i) and (ii) is 15°, matching my result exactly, while (ii) finds a peak at 20° but concludes that the range of useful efficiencies is between 15° and 20°.  There is some discrepancy between all three studies as to the value of St at which efficiency is maximised, for (i) St = 0.16 at a heave-chord ratio of 0.75, and St = 0.2-0.5 at a heave-chord ratio of 1.0, (ii) St = 0.35, and (iii) St = 0.25; the last which agrees with this experiment.  The value of maximum efficiency is 64-74% in these studies versus 64% here.    All studies agree with my results regarding maximum thrust, which is found by maximizing both       and St, to the detriment of efficiency.  In terms of the value of the maximised coefficient of thrust (CT): at       = 30° and St = 0.40, (i) finds CT = 0.62, (ii) finds CT = ~0.91, and (iii) finds CT = ~0.84.  These encompass my finding of CT = 0.62.  This similarity is notable because of the many differences between studies.  In particular it seems to suggest a diminished role of tail shape, as not only where the foils in those studies rectangular, they also had plates which simulated infinite aspect ratio (by blocking 3-D flow in 92  the span-wise direction).  At this juncture it certainly seems that, relative to kinematics, the effect of tail shape on performance is slight.  Further studies on tail shape would benefit to consider a finer resolution of kinematic parameters than that used here.  Comparison with Delepine (2013)  The experiment conducted by Delepine (2013) provides an excellent comparison: it was conducted with the same apparatus using most of the same kinematic parameters.  The primary difference between our setups is the flapping foil apparatus (rod) and the maximum test speed, and thus Re: he tested up to 1.05 m s−1 whereas I tested up to 1.4 m s−1, with maximum Re of 34,000 and 50,000, respectively.    Our static test data shows the same trends but differ in absolute value: Within the 0°−30° range Delepine shows a maximum lift coefficient of 0.51 whereas my data show 0.76.  More on this discrepancy follows.  In terms of the angle of maximum lift-to-drag ratio, however, our data show near perfect agreement: 6° in Delepine to 5° here.    Considering dynamics data, the raw force and torque traces show agreement in shape and trends, with those in Delepine showing a similar advance in peak forces as       increases, as well as regions of negative thrust following stroke reversal.  There is full agreement in thrust occurring at maximum       and St, although in his study the maximum St tested is 0.35.  While the trend is identical, the actual thrust coefficient diverges, with Delepine reporting 0.35 at           and St = 0.35 in comparison to my 0.56 with the same parameters.  This echoes the difference in static force coefficients.  The       of maximum efficiency is similar between the studies from mid St onwards, with a value of 15°.  For St = 0.25, Delepine reports the      to be 93  10°.  In terms of the actual value of propulsive efficiency, Delepine finds a maximum of 54% compared to my maximum of 63%.  There are a number of possible causes for the discrepancies between these experiments.  The foremost is the effect of the new apparatus, which was found to reduce the negative impact of wake interference from the vertical strut.   In addition to the apparatus, the motion system was outfitted with a new load cell between the experiments conducted by Delepine and those conducted here, and so an intrinsic flaw in the load cell may have contributed.  The planform area calculation certainly affected the results: Our tails are the same size, but Delepine took the planform area to be the projected area of the entire model at 0.0070 m2 whereas I excluded the area attributed to the nose-cone, resulting in a planform area of 0.0063 m2, 90% of his value.  Since the planform area appears in the denominator of the coefficient of thrust as in (15), this can explain 11% of the increase in my values.  A source of error that may exacerbate this effect is the flow speed of the water tunnel.  As discussed in the methods, a calibration curve was initially used but the flow speed was found to be unreliable, which is why I eventually employed a flow probe to recalibrate per trial.  Despite the differences in absolute values, both Delepine and my results demonstrate the same trends.  There is also some comparable flow visualisation in the form of two PIV trails, although the PIV in Delepine (2013) is presented as preliminary data and is thus quite minimal.  Comparison with my PIV results shows the same broad trends, with the LEV developing earlier at higher     .  Delepine‘s PIV results also reveal the same ―biased‖ drag street trail which could be a mechanism of propulsion.    94  Chapter 3: Conclusion 3.1 Future improvements and directions  Apparatus The extensive process of designing, creating, refining and finally testing the apparatus has afforded considerable insights into possible improvements.  One the main criticisms of this kind of apparatus is the presence of the vertical strut because it interferes with flow upstream of the foil.  Multiple iterations of the apparatus in this experiment attempted to address this, and although successful, the effect could be further mitigated.  Rather than having a single vertical strut placed forward of the tail, I recommend an arrangement of two vertical struts set on either side of the pivot axis, spaced sufficiently apart so that there is no overlap between the wakes shed by the rods and that shed by the tail.   Another factor that could be improved is the method of shielding surface effects.  The current iteration makes use of a circular disc set well above the tail model which effectively blocks vertical flow from the surface.  However, being affixed to the motion axes reduces the maximum tailbeat amplitude that can be tested, due to reduced clearance between the apparatus and the tunnel walls.  In addition, the uninhibited free surface limits the maximum speed of the tunnel because of the development of standing waves.  A solution that would kill two tuna with one stone would be to install a modified pressure lid.  The motion system used in this thesis was inherited from previous research and incompatible with a pressure lid; design of a new motion system layout and creation of a modified pressure lid would allow the tunnel to be filled to capacity and would eliminate the free surface, negating the need for the disc in the apparatus. Eliminating the free surface would have a number of other beneficial effects:  It would reduce the fluctuation in flow speed that is exacerbated by surface waves, thus reducing the 95  required sample size of tail-beats required per trial.  The tunnel could be run at its maximum velocity of 2 m s−1 which, using the tails from this experiment, would allow for testing speeds equivalent to 3 BL s−1.  Running the tunnel in open surface mode reduces visibility at higher speed because air bubbles are sucked down through the turning vanes and into the impeller which produces a cloudy mist; a lid would also prevent this.  Another problem in having a free surface is the potential for gravitational waves caused by sub-surface movement; in my setup these were presumed to be present (although difficult to ascertain because of the surface wake shed by the apparatus).  Schaefer and Fuller (2010) show that tuna generally swim at depths below the threshold for gravitational waves, so eliminating such waves from the testing apparatus would provide a more accurate picture of actual tuna swimming conditions. Free surface elimination would also benefit flow visualisation.  Having a uniform white background is important in ink trial experiments; the method used here (a tethered plastiboard backdrop) was limited by flow tunnel speed, but a white-painted lid would suffer no such limitation.  PIV is even more sensitive to background noise than ink flow-vis; the turbulent free surface was a source of much frustration in the PIV trials I conducted.  In order to reduce its effect, I had to prioritise reducing depth of field to blur the background, but this limited me to a lens of focal length 50mm which had such a small field of view that I was only able to capture half of the tailbeat and the near wake effects.  If a smooth background were obtained then one could use a 20mm lens and simultaneously capture the entire foil motion, the far wake, and the leading edge vortex formation at the front of the foil.      96  High speed testing  A great boon to this topic would be the actual testing of burst-like motion kinematics, which this experiment was unable to achieve due to equipment limitations.  The first step would involve the above-mentioned design of a lid allowing for the tunnel to be used at full speed; however, but that still limits testing to 3 BL s−1.  It would be ideal to test speeds equivalent to 10 BL s−1, which could only be achieved through one or both of the following options: (i) using a scaled down tail.  Testing with a tail of half the chord length would double the simulated speed.  Attempts made in this thesis to do so were confounded by signal noise, so to utilise this method it would be necessary to increase load cell sensitivity and reduce noise; (ii) Changing to a higher capacity flow system.  The MIT Marine Hydrodynamics Laboratory is home to a recirculating water tunnel that operates at speeds up to 10 m s−1, which would more than meet bursting requirements.  Another option is a tow-tank such as the one used by Hover et al. (2003) at the MIT Department of Ocean Engineering, which consists of a 30 m long, 2.6 m wide rectangular channel.  Rather than fixing the motion system and pumping water past, this type of system uses still water which the motion system is dragged through on a mobile carriage; thus the maximum speed is limited by how quickly the carriage can be towed.  The maximum speed of the MIT tow-tank may not be much higher than the UVic tunnel, but there are tow-tanks used in ship engineering capable of much higher speeds of 10 m s−1 such as the National Research Council‘s towing tank in St. Johns.  Muscle research  This thesis has explored the relationship between muscle function and swimming kinematics and has proposed several mechanisms to explain the experimental results.  Several of 97  these hypotheses could be tested simply by knowing more about the biomechanics of tuna swimming muscle.  At the time of writing there has been little research conducted on the properties of tuna white burst muscle, so one important future direction is to fill some of those gaps.  Particularly telling would be a work loop study on tuna white muscle to determine the relationship of cycle frequency, muscle strain and muscle activation phase to peak force and power.  Concerning the role of       in thrust production, during bursting it would be particularly pertinent to know how the performance of white muscle varies with peak force onset.    There are still many questions regarding the relative roles of red and white muscle in the different swimming modes.  It has been shown in multiple studies that red muscle is primarily responsible for producing power during cruising, with explanations of force transmission through the tendinous AOT-ITL-POT pulley system, but a mechanism for white muscle power during bursting is not yet available (Gemballa and Treiber, 2003; Nakae et al., 2014. Acronym stands for ―Anterior Oblique Tendon to Inter-Tendon Length to Posterior Oblique Tendon‖).  Such an explanation would aid understanding the level of control that tuna are able to exercise over pitch: whether they can control pitch by modulating some combination of bending of the axial body and tension of the great lateral tendons and myorhabdoid tendons, or whether pitch angle is a passive phenomenon that is inextricably coupled to heave motion.    In situ experimentation In elucidating the questions posed here, perhaps the most useful experiments would be live, in vivo measurements from tuna swimming across a range of speeds.  In vivo measurements have been collected for tuna in previous studies, such as Knower (1998) and Shadwick and Syme 98  (2008), but they are limited to tunnel-bound cruising speed and measurements of strain and EMG.  With the break-neck pace of new technology available to researchers, the feasibility of obtaining many more measurements in a natural setting is increasing all the time.  It should be possible to set up a self-contained data logger worn by a tuna; this could be connected to EMG sensors, sonomicrometry and force buckles, creating a system to gauge exactly how muscle performs in situ.  Not only that, but tiny electronic gyroscopes, like the kind available in all manner of personal electronics, could be used to record motion kinematics.  A setup with one gyroscope tethered to the caudal fin, one to the peduncle, and another attached to the mid-body could provide the data necessary to determine frequency, tail-tip amplitude, heave amplitude, pitch angle, and angle of attack, precisely answering the question of tuna kinematics.  The beauty of such a setup is that it could record data from all manner of cruising and bursting swimming, elucidating the whole swimming speed range, and furthermore it would not be limited to juvenile tuna as most studies are.  Although the logistics of programming and executing such a setup would be daunting, it is fully achievable and even affordable with the technology available today.  3.2 Concluding remarks The hydrodynamics and biomechanics behind thunniform locomotion remain to be fully understood, but this study offers some explanation of the role of angle of attack in the propulsive performance of a tuna-like tail moving according to tuna-like motion.  The lack of predictive power in static tests demonstrates the high degree of unsteady hydrodynamics present in such a system.  Dynamic tests show that peak efficiency is reached at an      of 15° over a range of St values.  Coupled with muscle work loop data, the results indicate that during red muscle 99  dominating cruising, tuna maintain a stable     , relying on St to modulate thrust.  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Biol. 59, 565–582.  106  Appendices Appendix A  Equipment details A.1 VeroWhitePlus RGD835 material properties  ASTM Units Metric Units Imperial Tensile strength D-638-03 MPa 50-65 psi 7250-9450 Elongation at break D-638-05 % 10-25    Modulus of elasticity D-638-04 MPa 2000-3000  psi  290,000-435,000 Flexural Strength D-790-03 MPa 75-110  psi  11000-16000 Flexural Modulus D-790-04 MPa 2200-3200 psi  320,000-465,000 HDT, °C @ 0.45MPa D-648-06 °C 45-50  °F 113-122 HDT, °C @ 1.82MPa D-648-07 °C 45-50  °F  113-122 Izod Notched Impact  D-256-06  J/m  20-30  ft lb/inch  0.375-0.562 Water Absorption D-570-98 24hr  %  1.1-1.5    Tg DMA, E»  °C  52-54  °F  126-129 Shore Hardness (D)  Scale D Scale D  83-86    Rockwell Hardness  Scale M  Scale M  73-76    Polymerized density  ASTM D792  g/cm3  1.17-1.18   Ash content USP281 %  0.23-0.26      A.2 Water tunnel Engineering Laboratory Design, Inc water tunnel specifications: MAX FLOW RATE  2 m/s 107  TEST SECTION  45.0cm × 45.0cm × 250.0cm OVERALL DIMENSIONS 7.360m × 174.5cm × 1206.0cm WEIGHT   6,550 kg full POWER   20HP VOLUME   4,715 L  A.3 Load cell datasheet F233-Z 4346 loadcell Fx and Fy calibrated to 80N compression and tension. Mz calibrated to 4Nm CW and CCW 4 metre flying lead. Sealed to IP68, 1metre H2O (tested for 24 hours). 350R bridge resistance: Fx and Fy. 700R bridge resistance: Mz. Nominal output Fx, Fy 0.35mV/V, Mz 0.65mV/V Aluminium alloy construction. Standard datasheet for guidance performance only. Dimensions as per F233-Z3712 loadcell.    

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