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Functional design and swimming energetics of the freshwater pufferfish, Tetraodon fluviatilis Varley, Robert Mark 1989

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FUNCTIONAL DESIGN AND SWIMMING ENERGETICS OF THE FRESHWATER PUFFERFISH, TETRAODON FLUVIATIUS by ROBERT MARK VARLEY B .Sc, The University of British Columbia, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Zoology) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1989 © Robert Mark Varley In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of ^Z^KPL^O  The University of British Columbia Vancouver, Canada Date DE-6 (2/88) ABSTRACT Measurements of morphometric characteristics pertinent to hydromechanical analysis were recorded, transformed where necessary and regression analysis was performed to relate the morphometric characteristics to standard body length. Terminal velocity measurements were recorded for a series of drop tank experiments. The data was converted into drag coefficients and Reynolds numbers and regression analysis was performed- to establish the specific relationships between those two hydromechanical parameters which were compared to theoretical estimates calculated from hydromechanical theory. High speed cineTilms of pufferfish fin and body motions made during forward swirriming were recorded and subsequently digitized onto a computer with a frame analyzer. The data was converted to distance and time from which the kinematic parameters of fins and body motions were calculated and compared to values found for other aquatic propulsive systems. A modified Actuator-Disc model was employed to estimate propulsive power and efficiency during steady forward swiiiiming based on the morphometric, kinematic and hydromechanical parameters calculated for the pufferfish. Comparisons of the experimental estimates for drag and power were made with theoretical estimates and with estimates found for other aquatic propulsive systems. The efficacy of the modified Actuator-Disc model was ii discussed with respect to negating factors found during this study for the application of the model to multiple fin aquatic propulsive systems. i i i TABLE OF CONTENTS ABSTRACT ii LIST OF TABLES v LIST OF FIGURES vi ACKNOWLEDGEMENT xi GENERAL INTRODUCTION 1 CHAPTER ONE: HYDROMECHANICS 9 CHAPTER TWO: MORPHOMETRY AND DRAG Introduction 24 Materials and Methods 25 Results 29 Discussion 75 CHAPTER THREE: KINEMATICS Introduction 122 Materials and Methods 123 Results 126 Discussion 135 CHAPTER FOUR: POWER AND EFFICIENCY Introduction 144 Methods 148 Results 154 Discussion 156 SUMMARY 167 LITERATURE CITED 169 APPENDIX 176 iv LIST OF TABLES TABLE I. Surface area measurements of spheres 33 TABLE II. ANOVA and Tukey test results for spheres 34 TABLE III. ANCOVA results for fm ray lengths 47 TABLE IV. Permissible roughness calculation results 80 TABLE V . Fineness Ratios of aquatic animals 82 TABLE VI. Propulsive fin kinematic parameters 129 TABLE VII. Power and efficiency values 155 v LIST OF FIGURES FIGURE 1. Streamline flow around a body 21 FIGURE 2. Pressure gradient around an assymetrical body 21 FIGURE 3. Couette flow pattern 21 FIGURE 4. Velocity profile at fluid-solid interface 22 FIGURE 5. Boundary layer relative to distance from leading edge 22 FIGURE 6. Boundary layer flow reversal 22 FIGURE 7. Schematic wake width of a separated boundary layer 23 FIGURE 8. Maximum body depth relative to body length with 95% confidence limits 36 FIGURE 9. Maximum body width relative to body length with 95% confidence limits 37 FIGURE 10. Maximum body depth relative to maximum body width with 95% confidence limits 38 FIGURE 11. Snout to maximum depth. distance relative to snout to maximum width distance with 95% confidence limits 39 FIGURE 12. Snout to maximum depth distance relative to body length with 95% confidence limits 40 FIGURE 13. Snout to maximum width distance relative to body length with 95% confidence limits 41 FIGURE 14. Body depth at dorsal/anal fm region relative to body length with 95% confidence limits 42 FIGURE 15. Body width a dorsal/anal fin region relative to body length with 95% confidence limits 43 FIGURE 16. Snout to dorsal/anal fin region relative to body length with 95% confidence limits 44 vi FIGURE 17. Body depth relative to body width at dorsal/anal fin region with 95% confidence limits 45 FIGURE 18. Body surface area relative to body length with 95% confidence limits 52 FIGURE 19. Wetted surface area relative to body length with 95% . confidence limits 53 FIGURE 20. Total surface area relative to body length with 95% confidence limits 54 FIGURE 21. Sum of fin surface areas relative to body length with 95% confidence limits 55 FIGURE 22. Cross-sectional or projected area relative to body length with 95% confidence limits 56 FIGURE 23. Body volume relative to body length with 95% confidence limits 57 FIGURE 24. Body surface area relative to body length with 95% confidence limits 58 FIGURE 25. Pectoral fin anterior ray length relative to body length with 95% confidence limits 60 FIGURE 26. Dorsal fin anterior ray length relative to body length with 95% confidence limits 61 FIGURE 27. Anal fin anterior ray length relative to body length with 95% confidence limits 62 FIGURE 28. Nare height relative to body length with 95% confidence limits 63 FIGURE 29. Nare distance from snout relative to body length with 95% confidence limits 64 FIGURE 30. Nare height relative to distance from snout with 95% • confidence limits 65 vii FIGURE 31. Drag force relative to Terminal velocity. Series 16 fins on 69 FIGURE 32. Drag force relative to Terminal velocity. Series 16 fins off 70 FIGURE 33. Drag force relative to Terminal velocity. Series 23 fins off 71 FIGURE 34. Drag force relative to Terminal velocity. Series 16 fins on, 16 fins off, 23 fins off 72 FIGURE 35. Drag coefficient relative to Reynolds number. Series 16 fins on, 16 fins off, 23 fins off, minimum laminar, minimum turbulent, total laminar, total turbulent 73 FIGURE 36. Drag coefficient relative to Reynolds number. Experimental results from assorted fish. minimum laminar, minimum turbulent 74 FIGURE 37. Minimum power relative to Reynolds number. Series 16 fins on. measured, minimum laminar, minimum turbulent, total laminar, total turbulent 105 FIGURE 38. Minimum power relative to Reynolds number. Series 16 fins off. measured, minimum laminar, minimum turbulent, total laminar, total turbulent 106 FIGURE 39. Minimum power relative to Reynolds number. Series 23 fins off. measured, minimum laminar, minimum turbulent, total laminar, total turbulent 107 FIGURE 40. Minimum power relative to Reynolds number. Comparison of series 16 fins on ( ), 16 fins off ( ), 23 fins off ( ) 108 FIGURE 41. Drag coefficient relative to Reynolds number. For different Fineness ratios. Based on the total laminar drag coefficient 109 viii FIGURE 42. Drag coefficient relative to Reynolds number. For different Fineness ratios. Based on the total turbulent drag coefficient 110 FIGURE 43. Drag/Body volume ratio relative to Fineness Ratio and flow conditions. For Reynolds number=1500 114 FIGURE 44. Drag/Body volume ratio relative to Fineness Ratio and flow conditions. For Reynolds number=6000 115 FIGURE 45. Drag/Body volume ratio relative to Fineness Ratio and flow conditions. For Reynolds number=10500 116 FIGURE 46. Drag/Body volume ratio relative to Fineness Ratio and flow conditions. For Reynolds number=15000 117 FIGURE 47. Drag/Body volume ratio relative to Fineness Ratio and flow conditions. For Reynolds number=30000 118 FIGURE 48. Propulsive fin frequency (cycles/s) relative to . specific swimming velocity (lengths/s) 131 FIGURE 49. Propulsive fin specific amplitude (mean amplitude/fin base length) relative to specific swimming velocity (lengths/s) 132 FIGURE 50. Propulsive fin specific wavelength (mean wavelength/ fin base length) relative to specific swimming velocity 133 FIGURE 51. Fin frequency (cycles/s) relative to specific swimming velocity (lengths/s) for pufferfish with triggerfish and mandarin fish estimates 134 FIGURE 52. Power output relative to specific swimrning velocity compared with theoretical estimates 163 FIGURE 53. Power output relative to specific swimming velocity compared to other MPF swimmers 164 FIGURE 54. Propuslive efficiency relative to specific swimming velocity 165 ix FIGURE 55. Propulsive efficiency relative to specific swimming compared to other MPF swimmers 166 x ACKNOWLEDGMENT I am grateful for the advice, support and friendship afforded me by my thesis supervisor, Dr. R. W. Blake. Thanks are due' also to my most humble assistant and friend, Mr. M. D. Smith, for his help in the lab and in the printing of this document. Last but certainly the opposite of the least, I thank my family and frierids, especially my wife Lynn, for their support and encouragement. xi GENERAL INTRODUCTION The study of aquatic animal locomotion is as diverse and complex as the range of organisms there is to analyze. From the human sperm to the sperm whale, from the water beetle to the squid, and from the flying fish to the giant rays, the reservoir of subjects has barely been tapped. It seems the only limitations are creativity, a solid biological background blended with sufficient fluid mechanics theory and a decent research grant. Of these three, I suspect the second ingredient has more of an effect in preventing or discouraging more participants in the classical functo- morpho- loco- field of investigation. As Sir James Lighthill (1975) has written, It is therefore when a zoologist and a hydrodynamicist have got to know each other well enough to be able to talk together about the problems, and gradually to learn enough of each other's language so as to be able to communicate effectively, that collaborative progress involving hydrodynamically sound analysis of zoologically significant motions becomes possible. Interest in aquatic animal locomotion has a long reach back in time, to about the sixth or fourth century B.C. in Europe (Webb, 1975; Blake, 1983d) from whence come some of the first recorded references, attributed to Aristotle, to the possible functional basis of tail fin propulsion. The single most significant advancement came when cin6film was first employed to record the propulsive motions of fish (Marey, 1894). Since then, other notable advances arose when 1 attempts were made to estimate the drag and power output of fish swimming against a load to which they where tethered via a pulley or fulcrum (Houssay, 1912; Magnan, 1930). Another approach, the first recorded instances of which are from Magnan (1930) and Magnan & Saint-Lague (1930), is to time the rate of descent of dead or anaesthetized fish down a column of water. This technique has been used by many researchers since then and is employed in this study as well. The development of hydrodynamic models heralded a new era in analysing aquatic propulsion. Gray (1936) developed a conundrum when he calculated, from rigid body hydrodynamic theory, the power required by a swimming dolphin to overcome drag. When he compared these estimates to alternate estimates he made based on muscle power output, he concluded that there was not sufficient power available to overcome drag, much like the bumble-bee that cannot fly. This problem became known as "Gray's Paradox." While further refinements were being made on the power output of mammalian muscle (Hill, 1938, 1939), other studies were made of stability, control and fin kinematics (Harris 1936). Using updated estimates for muscle power output and improving upon the hydrodynamic theory employed by Gray, Bainbridge (1961) found that for most fish and cetaceans, Gray's Paradox, was not valid. Patterns of fish propulsion are so diverse that it is unlikey one model can be applied to all fish propulsory systems. The primary goal of analysis of aquatic locomotion in fish 2 and other organisms is to estimate the cost of locomotion. There are two main avenues of approach to the problem: one is based on estimates of drag, the other on estimates of thrust. Drag-based estimates can come from theoretical equations based on hydromechanical theory and empirical observations of technical bodies of revolution. Or, they can come from direct force-distance-time measurements like the early techniques described previously. Thrust-based estimates can come from hydrodynamic models or from metabolic power calculations based on the rate of oxygen consumption during simming (Marr, 1960; Blazka et al, 1960; Brett, 1963, 1964). Hydrodynamic models combine the kinematic parameters of propulsion with hydromechanical theory. The earliest models, termed quasi-static or resistive, integrate the instantaneous forces for each segment during a propulsive cycle (von Holste & Kuchemann, 1942; Parry, 1949; Gero, 1952; Taylor, 1952; Gray, 1953b). Advancement in hydromechanical approaches came with the development of reactive models which consider the rate of change of momentum of a mass of water affected by the body segment during a propulsive cycle (Gadd, 1952; Lighthill, 1960, 1969, 1970, 1971; Wu, 1961, 197 Id). This so-called elongated body theory has been refined, modified and widely applied over the years. For example Blake (1983b) modified the model to accomodate the undulatory fin swirnming in the knifefish, Xenomystis nigri. Other models based on the momentum priciple include 3 Blade-Element theory and Actuator-Disc theory. Pectoral fin rowing has been analysed with blade element theory, which arbitrarily divides the fin span into a number of segments for which the normal force and thrust force are calculated. The sum of the thrust impulses is equated with the drag impulse acting on the body and a mean stroke power is calculated (Blake, 1979b, 1980a, 1981a, c). The Actuator-Disc model has been applied to aquatic locomotion of the mandarin fish (Synchropus picturatus), seahorse {Hippocampus hudsonius) and the electric fish (Gymnarchus niloticus), (Blake, 1979d, 1980a, 1980b). It is this model which this thesis applies to the median and paired fin propulsion of the pufferfish, Tetraodon fluviatilis (Linnaeus, 1758; Hamilton, 1822). 4 The Fish, T. fluviatilis This species of freshwater pufferfish is one of approximately 330 species included in the order Tetraodontiformes, which currently consists of 8 families: Balistidae (triggerfishes) Diodontidae (porcupine puffers) Triodontidae (threetooth puffers) Tetraodontidae (freshwater puffers) Molidae (sunfishes) Ostraciodontidae (boxfishes) Triacanthodidae (spikefishes) Triacanthidae (triplespines) As will be immediately apparent to those readers familiar with the terms terra- and -odont, members of the family Tetraodontidae share the common feature of four bony dental plates; two plates fused to each other on both the upper and lower jaws. These bony plates, when combined with the powerful masticatory musculature, enable members of this family to crush the shells of the small invertebrates upon which they characteristically feed. Another distinguishing feature of the family is the locomotory apparatus which consists of independent propulsory fins that provide a level of manoeuvring unsurpassed among the aquatic vertebrates and which is roughly analogous to helicopters or VTOL-aircraft. In general, the spatio-temporal environment of the genus 5 Tetraodon, and indeed for many of the members of the family Tetraodontidae, is a relatively complex one requiring a high degree of dexterity. For the marine species, coral reefs present a maze in which the prey may find shelter, so the ability to easily move forwards and backwards, and in and out of the convolutions in the reef is a highly valuable one. For the fresh-water species such as T. fluviatilis, the brackish, estuarine environment, consisting of submerged roots such as those from mangroves and aquatic plants, similarly presents a relatively complex environment in which speed is of little value and in which a high degree of manoeuvrability is required in order to root-out food items. The geographical distribution of the freshwater pufferfishes is such that they are common in the African region and they are considered marine-derived, saltwater dispersants (Moyle & Cech, 1988). According to a review of the genus Tetraodon (Dekkers, 1975), the genus is widely distributed throughout the Ganges river system, and the species T. fluviatilis is known from India, Ceylon, Bangla Desh, Burma and Borneo but not from the intermediate Malaya, Sumatra, Java, Thailand, Cambodia or Vietnam. 6 The thesis Consideration of morphological characteristics in a functional context allows general predictions concerning locomotor strategy and mode of life. Quantification of the size and shape of a body and its propulsive elements, when combined with principles of fluid mechanics, provides the hydromechanical parameters necessary to define and standardize the fluid flow regime about an aquatic animal. The establishment of hydrodynamic similitude is indeed the first requisite towards assessing the performance of a particular locomotor strategy or pattern and allowing comparisons with other aquatic organisms and man-made objects in terms of the common currencies of energetic cost and efficiency. The following flow chart outlines how the different chapters of this thesis are related. morphometries fluid mechanics hydromechanical defini tion and standardization kinematics of locomotion propulsive models power & efficiency estimates success or fitness of locomotor strategy & mode of life 7 This thesis examines the morphology, hydromechanics, kinematics and swimming energetics of T. fluviatilis with respect to steady forward, rectilinear swimming. The first chapter consists of a presentation of hydromechanical theory pertinent to the analysis and characterization of the fluid flow regime surrounding the fish. The second chapter deals with the morphometric and hydro mechanical characteristics relevant to the estimation of the parameters which are required to establish the terms of hydrodynamic similitude by which the pufferfish may be defined and compared with other fish. In the third chapter, the values of kinematic parameters of the propulsive fins are derived, described and compared to those for other fish. In the final chapter, estimates of power output and propulsive efficiency generated by the Actuator-Disc model are compared to theoretical minima and to values found for some other fish. 8 CHAPTER ONE: HYDROMECHANICS Fineness Ratio and Shoulder Position are the two primary shape parameters used in hydromechanical analysis to characterize body form for the purpose of drag estimation. Fineness Ratio (FR) is defined as the length of an object divided by its maximum diameter (1/d) and is a measure of the degree to which a body is streamlined. The degree of streamlining affects the amount of surface area relative to body volume and the magnitude of the pressure gradients in the boundary layer. Shoulder Position (SP) is defined as the ratio of the distance from the leading edge (snout) to the position of maximum diameter divided by the length of an object. The position of maximum diameter indicates the general region where the pressure gradient changes from favourable to adverse (streamwise) and may affect the proportion of the body that experiences laminar flow in the boundary layer. The term "streamlined" is a general descriptive term, borrowed from the concept of streamlines, which is used to describe the resultant paths along which fluid particles travel downstream, usually relative to the surface of an object (Rouse, 1946). A fluid "particle" is an arbitrarily defined element which is small in mass and volume relative to the overall flow field being considered but large relative to the molecular size of the fluid (Vogel, 1981; Blake, 1983d). The magnitude of distortion of a streamline gives a qualitative indication of flow disturbance 9 caused by an object in opposition to a fluid. Streamlines are related to the principle of continuity, a geometrical construct in hydromechanics which allows the assertion that the fluid volume flux (0 in a field of flow is a constant related to the cross-sectional area of the stream-tube (A), and the flow velocity (LO, A, U, = A, U2 = Q The stream tube walls can be material in the form of a pipe or non-material in the form of an imaginary set of streamlines which bound a region of finite cross-section. The principle applies in both situations where it is considered that the fluid is inviscid and incompressible and that there is no exchange in mass between streamlines. Where streamlines constrict, continuity predicts the flow velocity will be increased in order to maintain a constant rate of fluid volume flow. Conversely, where streamlines diverge the increased area results in a drop in velocity. The importance of the velocity changes in the fluid particles is seen in a theorem named for Bernoulli (1738) who described the inverse relationship between momentum flux and pressure for an ideal fluid, which is considered to be inviscid, isothermal and incompressible. The theorem states that for constant flow along a streamline, the total pressure (P^ of a fluid, the sum of the dynamic and static pressures, is constant 1/2 pU2+ p + pgh = PT 10 or 1/2 p{U\-U\) + (pfp2) + pg(\-h2) = 0 (where p=fluid density, £7=fluid velocity, /?=internal pressure, g=gravitational acceleration, /i=height of the fluid above a reference point). Schematically, as the streamlines compress together and curve past an object in symmetrical flow conditions (Fig. 1), the principle of continuity predicts fluid volume flux is conserved by an increase from free-stream velocity (U^) in the frontal region (F), to a maximum velocity (U2) in the shoulder region (T and B) where the streamlines are most constricted. Concomitant with the increase in fluid velocity, as predicted by Bernoulli's theorem, the pressure (p^ decreases over the front portion of the body to a minimum (p2) in the shoulder region. The reverse situation occurs over the posterior portion of the body where the pressure restores to as the velocity of the fluid particles decreases to the free stream velocity (L^). The unfavourable, particle-retarding pressure gradient around the posterior portion of the body is exactly balanced by the favourable, particle-accelerating pressure gradient around the anterior portion of the body. The momentum of the fluid particles is completely conserved so that, as first described by d'Alembert for whom this paradox is named, there is no net resistance as an ideal fluid slips past the surface of an object, even for the case of an object with asymmetric longitudinal section (Fig. 2). Although the adverse pressure gradient of the 11 posterior body portion is spread over a greater area than is the favourable pressure gradient, there is no net difference in pressure, hence the conservation of momentum in an ideal fluid is independent of object shape. To overcome d'Alembert's dilemma, Prandtl (1904) proposed the boundary layer concept to explain the behaviour of real fluids at a solid-fluid interface. A real or Newtonian fluid, unlike an ideal fluid, has a characteristic time-dependent resistance to deformation indicated by dynamic viscosity (u., kg m"1 s"1). For the theoretical case of a fluid bounded by two parallel plane surfaces (Fig. 3), each of negligible mass and a distance (/) apart, dynamic viscosity is defined as the force (F) per unit area (A) (shear stress) required to maintain constant velocity (CT) of the moveable top plane relative to the fixed bottom plane _ FIA _ (shear stress) ^ dU 161 (shear rate) y is defined as the force (F) per unit area (A) (shear stress) required to maintain constant velocity (U) of the moveable top plane relative to the fixed bottom plane _ FIA _ (shear stress) ^ dU/dl (shear rate) that1" the velocity of a fluid particle at the fluid-solid interface is zero relative to the velocity of the surface, that is, the fluid particle is deemed "attached" to the surface. This 12 assumption is termed the "no-slip condition" and is applicable at any fluid-solid interface regardless of the pathic nature of the fluid or surface, excepting rarefied gases (Goldstein 1938, Vogel 1981). The boundary layer concept divides the fluid flow past the surface of a solid (such as a flat plate or rigid streamlined body oriented parallel to the direction of flow) into an "inner" and an "outer" region. The inner and outer regions are continuous and the border between them is a statistical convention examined in the following paragraphs. In the inner region the velocity gradient (dU/dl) is high, as the fluid particle velocity increases rapidly from zero at the surface to approach that of the free-stream velocity (Fig. 4) (modified from Prandtl & Tietjens 1934b). This steep velocity gradient is due to the high shear stresses which occur as a result of the fluid viscosity. The thickness of the inner region in laminar flow continues to grow in parabolic fashion in proportion to the square root of the distance (x) from the leading edge (x4^5 for turbulent flow) as more fluid is affected (Fig. 5). In the outer region the fluid particle velocity is essentially that of the free stream and as such the viscous effects are negligible. There are numerous definitions of boundary layer thickness (Prandtl & Tietjens 1934b) from which a commonly used one called the velocity thickness (8) is defined as the distance from the object surface to the region where the fluid velocity differs from that of the free-stream by 1% (Fig. 4). Blasius (1908) 13 calculated the velocity thickness for laminar and turbulent boundary layers as a function of the Reynolds number (Re) and the distance downstream from the leading edge (x) | - 5 Re, "°-5 ; | « 0.37 Re, ^ respectively, for a smooth flat plate oriented parallel to the flow (Rouse, 1946). The type of flow regime in the boundary layer can be laminar, turbulent or transitional between the two. The flow condition for an object of given size and shape is dependent on the relative magnitude of inertial and viscous forces acting in the boundary layer. The ratio of these two forces was proposed by Reynolds (1883) who first described the phenomenon of transition while investigating the factors which appeared to have an effect on the nature of flow in fluids. By altering the dimensions of an object and the velocity, density and viscosity of a fluid, singly or in concert, Reynolds found that transition from laminar to turbulent flow could be induced or predicted. The inertial force of the fluid can be recognized as that of the dynamic force or rate of change in momentum of the fluid particles as seen earlier in the equation calculated by Bernoulli F(inertial)=pU2A The viscous force of the fluid will be familiar from the earlier definition of viscosity relating force per unit of area with the velocity gradient F(viscous)=\iAU/l 14 The ratio of these two forces provides a non-dimensional index of the conditions of flow around a body called the Reynolds number (Re) Re= P^24,„ = = Ul inert ial \iU(All) \i v viscous which is defined by the kinematic viscosity of the fluid (v=|i/p), the velocity of the fluid (U) and some characteristic length (/) such as the body length parallel to the direction of flow or the distance from the leading edge to the position of interest. Kinematic viscosity relates the dynamic viscosity of a fluid to its density and as such gives an indication of the propensity or ability of a fluid to damp out irregular or non-uniform fluid particle trajectories (Batchelor 1967). For instance, in the case of two fluids of equal density, the fluid with the greater dynamic viscosity will be better suited to damp out disturbances caused by errant fluid particle trajectories by converting the momentum of the fluid particles into heat energy. The importance of the Reynolds number is in establishing a condition called hydrodynamic similitude wherein for objects of like shape and orientation, the flow conditions around them are identical when their Reynolds numbers are equal. Bodies of revolution which are ten times different in size will have identical Reynolds numbers if the products of their velocity and length are equal, given constant body shape and orientation, and constant .temperature, viscosity and density of the fluid in the uniform outer flow field conditions. The whole business of model testing rests squarely on this fundamental principle of 15 hydrodynamic similitude. The condition of a boundary layer for an object of given size and shape in a fluid of given kinematic viscosity, temperature, density and uniform flow are in general indicated by the Reynolds number where a sub-critical Reynolds number (<5xl05) suggests fully laminar flow and a super-critical Reynolds number (>5xl06) suggests fully turbulent flow, with the values in between denoting a mix of transitional laminar and turbulent conditions in the boundary layer (Prandtl & Tietjens 1934b). The Reynolds number at which transition occurs is termed the critical Reynolds number and is related to velocity thickness id) d . = 5(Re .)"°5 cm cnt Flow conditions in the boundary layer are also affected by roughness elements, such as external nares, eyes, opercula, denticles, appendages and the like, which protrude through the thickness of the boundary layer to disturb the outer flow. The maximum permissible height (h) of roughness elements can be related to a Reynolds number as Reh = UhN where transition is expected to occur at values of Reh. s900 and slOO for single and distributed roughness elements respectively. The nature of the flow in the free stream can also influence the boundary layer since a turbulent free stream will transfer energy to the boundary layer which may cause transition to occur at a lower Reynolds number (closer to the leading edge) or it may induce boundary layer separation sooner than in a uniform laminar flow field. 16 The importance of flow conditions in the boundary layer is that the status of the boundary layer has a significant effect on the amount of drag force experienced by a body in opposition to a fluid. The total drag force experienced by a sufficiently submerged body of rotation in a steady fluid flow is the sum of the frictional and pressure drags. A body submerged at least three body diameters below the surface is considered to be unaffected by wave drag (Hertel, 1966). At lower Reynolds numbers where form or pressure drag is minimal and laminar flow is expected, frictional drag due to shear stresses is the major source of drag. Given fluid viscosity and object length and velocity, the amount of frictional drag is directly related to the amount of surface area (Hoerner 1965) and as such surface area minimizing shapes like a sphere should theoretically incur the least amount of frictional drag. As the Reynolds number increases so too does the amount of drag attributable to the inertial effects of the fluid. In an attached boundary layer, relative to the direction of free-stream travel, the rate of change of momentum in the fluid particles around the anterior portion of an object is not equalled by the rate of change of momentum around the posterior portion of the body due to the shear stresses in a real fluid, thus the origin of pressure drag is ultimately due to the viscosity of the fluid. There is a point along the deceleration region on the posterior body surface at which the particle velocity will fall to zero. The adverse pressure gradient imposed upon the boundary layer by 17 the outer flow field (Bernoulli's theorem) as the streamlines diverge past the shoulder will force the particles within the boundary layer to reverse flow in the upstream direction (Fig. 6) (Rouse 1946). A discontinuity in flow results, the flow field in the outer region becomes distorted, and the boundary layer separates from the body surface creating a zone of low pressure (wake) on the rear of the body surface (Fig. 7). This boundary layer separation gives rise to a dramatic increase in pressure drag, since the amount of pressure or form drag is directly related to the width of the separated wake (Shapiro, 1964). Turbulence in the boundary layer results in a more uniform velocity distribution throughout the major portion of the layer and a greater thickness due to the exchange of momentum between the fluid particles in random trajectories. This momentum exchange causes a turbulent boundary layer to be more stable, ie. better able than a laminar boundary layer to absorb and dissipate separation-inducing perturbations such as turbulence in the free stream or roughness elements on the body surface. However, turbulence in the boundary layer also produces a steeper velocity gradient in the region immediately adjacent to the body surface which results in a greater shear rate and larger viscous drag than that for a laminar boundary layer (Shapiro, 1964). The pressure drag in separated laminar flow is usually far greater than that for the same body in turbulent flow since the point of discontinuity or flow separation in the less stable laminar boundary layer will occur closer to the leading edge 18 producing a much wider wake than in a turbulent boundary layer. Pressure drag is directly related to the diameter of the wake so a turbulent boundary layer which is less prone to separation can enhance or compliment the separation-delaying nature of streamlining to minimize pressure drag. A streamlined body can be defined as one which limits the distortion of flow in order to minimize disturbances to the outer fluid field. By gently curving the anterior surface to the shoulder position and tapering the posterior surface to the tail, boundary layer separation is prevented or delayed to the posterior-most possible position. An optimal fineness ratio for a streamlined body of rotation is the result of a compromise between a number of conflicting concerns. In order to delay (ideally to prevent) boundary layer separation and eliminate form drag, the body should be greatly elongated downstream of the shoulder position but since friction drag increases with the amount of surface area, minimization of total drag is achieved when both form and friction drag combined are a minimum (Rouse, 1946). Also at issue are the design criteria by which a body shape is constrained, such as maximization of volume versus maximum speed, for minimum drag. For streamlined bodies of revolution required to maximize volume and rmnimize surface area, a fineness ratio of around 4.5 is considered to be optimum for minimizing total drag (Von Mises 1959). However, a departure from the optimum within a range of fineness ratio of approximately 2.5 to 7 results in a drag penalty of about 10% or less (von Mises, 19 1959), allowing considerable latitude in body design. For bodies of revolution which move through fluids at higher velocities and higher Reynolds numbers the major contribution to total drag comes from form drag. Thus a small increase in friction drag as a result of streamlining may more than pay for itself if the result is a substantial saving in pressure drag. The form drag of a streamlined body of revolution is less than 5% of that of a sphere of equal diameter (Rouse, 1946; Vogel, 1981). Bodies which move through a fluid at lower velocities (and Reynolds Numbers) experience mostly friction drag, thus more rotund shapes with lower surface areas are favoured, especially for bodies which are required to maximize volume. A body which travels at higher velocities likely to promote boundary layer separation may benefit by inducing turbulence in the boundary layer so as to trade off the increase in frictional drag incurred over the anterior portion of the body against the reduction in pressure drag which results from a delayed boundary layer separation and narrower wake at the tail end of the body. A sphere has the minimum surface area per unit volume hence the least frictional drag but potentially the largest pressure drag penalty should the boundary layer separate. These hydrodynamic principles allow objects of different shape, size and orientation, under different flow conditions, to be compared to a standard shape and flow condition in order to determine the total drag upon the object in opposition to the flow. 20 FIGURE 1. Streamline flow around a body. 21 FIGURE 4. Velocity profile at fluid-solid interface. FIGURE 5. Boundary layer relative to distance from leading edge. ' ^ - ~ ^ ( Laminar Transition \ Turbulent boundary boundary zone I layer l a / e r laminar sub-layer FIGURE 6. Boundary layer flow reversal. 22 F I G U R E 7. Schematic wake width of a separated boundary layer. 23 CHAPTER TWO: MORPHOMETRY AND DRAG INTRODUCTION In this chapter the morphometric parameters relevant to the characterization of the pufferfish body are defined (ie. Fineness Ratio, Shoulder Position, surface areas and fin dimensions) and related to some characteristic length, usually standard body length (/s= distance from snout to caudal peduncle). The results are compared to other fish forms. Following that, the results of the drop tank experiments are combined with the results from the morphometric analysis to estimate the relationship between the two parameters which allow the principle of hydrodynamic similitude to be established: the experimentally determined drag coefficient and the Reynolds number. These parameters are compared to theoretical values and estimates for other fish. Estimates of power output are made based on the experimentally determined drag coefficient and are briefly considered with respect to the kinematic parameters of the propulsive fins. 24 M A T E R I A L S A N D M E T H O D S The fish Specimens of the Asian pufferfish, Tetraodon fluviatilis, were obtained from a conimercial aquarium and over the course of approximately one and one half years were grown from an initial length of about 3 cm to a range of lengths with a maximum of about 7 cm. The fish were reared in a 420 litre glass aquarium (pH, salinity and temperature were 7.5, 0.5 % and 25°C respectively) with commercial aquarium gravel as substrate and various artificial aquatic plants and other objects to break up the field. Water quality was maintained by an external foam and charcoal filter, regular partial water changes and periodic tank and substrate cleaning. The fish received a varied diet of beef liver or heart and brine shrimp. Specimens were fixed in 37% formaldehyde and preserved in 40% isopropyl alcohol. Morphometries Lengths were measured with a standard micrometer (Mitutoyo, ± .005 cm). Fin surface areas were measured by coating the fins with toluidine blue and blotting the spread out fins onto paper which was cut out, weighed and compared to the weight of a piece of the same paper of known area. The surface area of the body was measured by sowing a wrap of clear plastic tightly about the body. The excess was trimmed off and the weight of the plastic 25 was compared to that of a piece of plastic of known area. To determine the accuracy and precision of the body surface area measurement method, three different diameter spheres were subjected to the same treatment as described for the fish. Two additional methods were applied; wrapping in aluminum foil and dipping in liquid soap. The procedure employing foil was analogous to that for plastic. For the soap film procedure the weight of the sphere was measured before and after dipping in the liquid, once the excess soap had been allowed to drop. The difference in weight was compared to that for standard glass slides which were subjected to the same treatment. Al l weights were determined with electronic balances (Mettler PK300 ± .00 lg and Mettler M3 ± l|ig). Drag Terminal velocity estimates were^  obtained by dropping dead fish of a given size and weight down a glass column (30 x 32 x 120 cm) filled with the same fluid in which they were preserved (40% isopropyl alcohol, 20.5 °C, kinematic viscosity 3.65 x 10"6 m s ) to avoid any error which might arise from interaction between fluids of different density and viscosity. A speed trap consisting of two horizontal fields of infrared light beams was attached to the outside of the column. Each field was created with opposing rows of photo-electric emitters and sensors which were spaced 1.27 cm apart and recessed 0.4 cm in a strip of black plexiglass 1.9 cm thick and 29 cm long. The emitters and sensors were connected to a digital timer which displayed the time 26 elapsed for an object to travel between the horizontal fields which were spaced a known vertical distance apart. Both fields could be independently adjusted along the column to ensure that the object had reached terminal velocity before triggering the timer. The trigger threshold for the photo-electric sensors was adjustable for both fields and the trigger mechanism had to be reset for each pass. The digital timer was connected to a lap top computer (Zenith model ZFL-181-93) to which each elapsed time measurement was sent by depressing either a button on the face of the timer or a foot treadle. For a fish of given size and weight, a series of elapsed times was obtained and saved as a separate file on the computer. Submerged weights of the fish were progressively increased by inserting small lead rods through the mouth into the pharyngeal region and were measured directly with an electronic balance equipped with an adaptor which suspended the fish or rods in the fluid. A vertical descent through the column was ensured by attaching a dan flight (wetted surface area = 44.4 cm2) to the posterior end of the fish with a shaft made from thin piano wire (diameter = .43 mm, length = 17.1 cm) which was twisted in a double strand and inserted through the caudal peduncle parallel to the spinal column leaving approximately 10 cm extending from the trailing edge of the caudal fin. In addition the pectoral, dorsal and anal fins were amputated flush with the body, except on one series, to eliminate any fin flutter effects and to ensure a vertical descent. A calibration curve of terminal velocities for the dart 27 flight and shaft was obtained from a series of elapsed times for different submerged weights which were progressively increased by sequentially rolling paper thin sheets of lead onto the leading end of the flight shaft. The leading end of the shaft could pass through both fields undetected if it travelled between the infrared beams but the dart flight always triggered the sensors. Four possible combinations of triggering order for the respective start and stop fields were: shaft-flight (over-estimated time), shaft-shaft and flight-flight (correct time), and flight-shaft (under-estimated time). These four combinations meant that three populations of elapsed time were sampled so the data which fell into the first and fourth combination were rejected .before the mean terminal velocities were calculated. This was accomplished by a visual examination of the frequency distribution of elapsed times which clearly revealed the three populations. The terminal velocity curve for the fish alone was calculated by subtraction of the curve for the flight alone from the curve for the fish with flight attached. Results were analysed on a micTO-computer according to established statistical procedures for transformations and regression analysis (Sokal & Rohlf 1981, Zar 1984) (see APPENDIX I. for a regression summary table). 28 RESULTS Linear Morphometry Body profile is conserved throughout the observed range of body lengths: both maximum body depth and maximum body width are linearly related to standard body length (Is, measured from the snout to the caudal peduncle; unless otherwise stated subsequent references to body length refers to standard body length), (Figs. 8 & 9); maximum body depth and maximum body width are not significantly different for fish of a given 1» (Fig. 10); the positions (relative to the snout) of maximum body depth (Xrf) and maximum body width (X )^ are coincident for fish of ,a given Is (Fig. 11) and are both directly related to Is (Figs. 12 & 13). These relationships indicate that the body has a circular cross-section at the point of maximum thickness. This circular profile extends from the snout to the region between the shoulder position and the dorsal/anal fins where the body becomes laterally compressed near the caudal peduncle. In this posterior region; body depth, body width, and distance from snout to dorsal and anal fins are linearly related to body length (Figs. 14 to 16). Also, the posterior depth to posterior width ratio remains constant (mean=1.40, s=.134) with body length (Fig. 17). The mean Fineness Ratio (FR = total body length/maximum thickness) which describes the degree of streanilining in an object is 3.37 (s=.260, n=19). The Shoulder Position (SP = distance from snout to point of maximum thickness/body length) 29 which roughly indicates the region of minimum pressure along an object has a mean value of 0.430 (s= 0.043, n=20). The body profile is well conserved over the range of body lengths measured. The overall body shape is that of a teardrop with moderate lateral compression occurring posteriorly (in the dorsal fin to caudal peduncle region). 30 Surface area and volume Surface arsa estimates employing the three different methods of foil, plastic and soap film are presented in Table I. For all three spheres the soap film method is the least precise (consistently highest standard deviation) while the foil method is the most precise (consistently lowest standard deviation) but the least accurate. The plastic wrap proved to be the most accurate method since in all cases estimates were the closest to the surface area of a sphere based on diameter (S = 7td2 ). Analysis of variance was applied for each sphere to the three treatment methods. For sphere 3, all three methods give significantly different results while for spheres 1 and 2 the plastic and soap film methods are not significantly different according to a Tukey multiple comparison test (Table II). Both the soap film and foil wrap methods are prone to over-estimation. Due to the vagaries of manipulation during application, the aluminum foil accumulates wrinkles which cause a proportional amount of over-estimation. The soap film over-estimation increases with the degree to which the surface of a body departs from being smooth, ie, convolutions, cavities, projections, etc. On the whole the plastic wrap method is considered to be the best and it is this method which is employed to determine the body surface areas for the fish. Body surface area (SJ (Fig. 18) and fin surface area (S,) b f measurements are used to calculate wetted surface area (S ) (Fig. w 19), the sum of body surface area and the area of any fins present. The total surface area (Fig. 20) is comprised of the 31 body surface area and the surface area of all fins (Fig. 21). For all relations of surface area versus body length and for projected area measurements versus body length the regression coefficient is not significantly different from 2 (Appendix I). Projected area is the transverse cross-sectional area of the body at the point of maximum thickness which is sometimes employed in the calculation of drag coefficients in place of wetted surface area (Fig. 22). The volume of a submerged fish is equal to the volume of fluid displaced by the fish which is calculated by dividing the mass of the displaced fluid (the difference between the weight of the fish in air and the submerged weight) by the density of the fluid. Volume is related to body length and surface -area (Figs. 23 & 24), providing slope coefficients similar to those expected 3 2/3 (volume « length , volume « surface area). The mean density of the preserved specimens is 1.125 (s=.036). 32 TABLE I. Surface Area Measurements sphere Diameter (cm) nd2 (cm2) method surface area (cm2) mean s n 1 2.614 21.466 foil 25.707 0.601 5 plastic 21.897 1.012 5 soap 19.678 2.630 5 2 3.715 43.358 foil 51.563 0.945 5 plastic 45.495 1.437 5 soap 46.983 2.478 5 3 4.423 61.459 foil 74.654 1.413 5 plastic 65.525 1.709 5 soap 69.957 2.634 5 33 TABLE II. ANOVA and Tukey test results for three spheres. sphere method F P method pair S.E. q 1 foil 16.799 <.0005 f-s 0.744 8.104 plastic f-p 5.121 soap p-s 2.983 NSD 2 foil 16.494 <.0OO5 f-s 0.779 7.792 plastic f-p 5.881 soap p-s 1.911 NSD 3 foil 26.373 <.0005 f-s 0.889 10.269 plastic f-p 5.285 soap p-s 4.985 34 FIGURE 8. Maximum body depth relative to body length with 959c confidence limits. FIGURE 9. Maximum body width relative to body length with 95% confidence limits. FIGURE 10. Maximum body depth relative to maximum body width with 95% confidence limits. FIGURE 11. Snout to maximum depth distance relative to snout to maximum width distance with 95% confidence limits. FIGURE 12. Snout to maximum depth distance relative to body length with 95% confidence limits. FIGURE 13. Snout to maximum width distance relative to body length with 95% confidence limits. FIGURE 14. Body depth at dorsal/anal fin region relative to body length with 95% confidence limits. FIGURE 15. Body width a dorsal/anal fin region relative to body length with 95% confidence limits. FIGURE 16. Snout to dorsal/anal fin region relative to body length with 95% confidence limits. FIGURE 17. Body depth relative to body width at dorsal/anal fin region with 95% confidence limits. 35 Maximum body depth vs Body length n - 20 PC 2 - J93 3.2 3.0 4 4.4 4A 5J2 5.0 O 0.4 Body length (cm) .424(X)-.208 I — i — i — i — i — i — i — i — i — i — i — i — i — i — i — r (wo) u)dep Apoq u;nu/;xe/v 38 Xd vs Xw n - 20 RT2 - .87 vo o 8-6 8 2.0 Xwt snout to maximum width (cm) • observed .809(X>+.454 Xd vs Body length n-20 PT2-.7Q o Body length (cm) • observed 307(X>+270 Xw vs Body length n - 2 0 FT2-.7Q 3.2 3.0 4 4.4 4.6 5J2 5.0 0 Body length (cm) • observed 396(X)-J095 . Body length (cm) J301(X)-J392 Body width <§> Median fins vs Body length n-20 FT 2 - .94 3.2 3.0 4 4.4 4.6 52 5.0 O Body length (cm) .201(Xr-.267 Snout to Median fins vs Body length n - 20 PC2 - J97 5 -, 3J2 3.0 4 4.4 4A 52 5.0 0 0.4 Body length (cm) .06MXH204 Body depth vs body width at Median fins n- 20 PT2 - .92 Fin morphometry The anterior chord length (fin base to ray tip) for the pectoral, -dorsal and anal fins is linearly related to body length (Figs. 25 to 27), there is no significant difference between the regressions and as such a common regression is calculated for the three fins (Table III). Mean fin chord lengths are calculated for each fin from the average of the anterior, medial and posterior fin ray lengths and are regressed against body length (Appendix I). There is no significant difference between the curves for the dorsal and anal fins; however there is statistically a slight difference between the elevation of the pectoral fin curve from that for the dorsal and anal fins which is likely attributable to the fact that the pectoral fins appear to be slightly less rectangular than the dorsal and anal fins which taper from the leading, anterior ray towards the posterior of the fin. Fin surface areas for all fins are not significantly different and regressions against standard body length provide slopes not different from 2 (Appendix I). The mean angle of incidence (a) with incident flow for the pectoral fin base is calculated to be 48 ± 2 degrees above a horizontal line from the mouth through the middle of the caudal peduncle and is independent of body length. 46 TABLE III. ANCOVA results for anterior fin rays for the pectoral, dorsal and anal fins relative to body length. F DFnum DFden F(.05,l) slopes 0.635 2 51 3.18 NSD elevations 1.942 2 53 3.18 NSD overall 1.275 4 51 2.56 NSD Tukey test for slopes diff SE q DFp q(0.05,3,40) 1 vs 2 -0.034 0.023 -1.491 51 3.442 NSD 1 vs 3 -0.028 0.023 -1.234 51 3.442 'NSD 2 vs 3 0.006 0.023 0.257 51 3.442 NSD Tukey test for elevations diff SE q DFc q(0.05,3,40) 1 vs 2 0.057 0.022 2.650 53 3.442 NSD 1 vs 3 0.012 0.022 0.576 53 3.442 NSD 2 vs 3 0.045 0.022 2.074 53 3.442 NSD X= 0.566 Y= 4.441 common slope= 0.197 a = -0.311 Y' = 0.197(X) -0.311 1- pectoral fin 2- dorsal fin 3- anal fin 47 Surface roughness-Nares Nare height (h) and distance from snout (x) are linearly related to body length (Figs. 28 & 29) and nare height is linearly related with distance from snout (Fig. 30). Regressions are calculated for the three relationships (Appendix I). Permissible height calculations indicate that the combination of nare height and location exceeds that required for an element to have no effect on the boundary layer flow condition (Table IV). The local Reynolds Number based on the nare height, Re^ = ^ , will cause transition in the boundary layer to occur when Re^ exceeds about 900 for a single roughness element and about 120 for distributed elements (Webb, 1975). Likewise, after Vogel (1981), by comparison of the observed nare > height to location ratios with Reynolds Numbers based on distance from the snout, the ratio exceeds the permissible values calculated for both pointed (h/x< 9.5Re ~15) and rounded (h/x< \2.1Re'n$) protruding objects. Thus it is likely that the protrusion of the nares into the outer flow field has some effect on the boundary layer flow conditions, at least over the dorsal region. 48 TABLE IV. Permissible Roughness calculation results Series Re local Re point round h/x 16 on 3.59E+03 350 0.117 0.151 .363 4.27E+03 416 0.103 0.132 5.01E+03 489 0.091 0.117 5.73E+03 559 0.083 0.106 6.84E+03 667 0.072 0.093 7.63E+03 744 0.067 0.086 8.24E+03 804 0.063 0.081 16 off 4.06E+03 396 0.107 0.137 6.51E+03 635 0.075 0.096 7.50E+O3 732 0.068 0.087 8.10E+O3 791 0.064 0.082 8.60E+O3 839 0.061 0.078 9.47E+03 924 0.057 0.073 1.01E+04 986 0.054 0.069 1.08E+04 1055 0.051 0.066 1.15E+04 1123 0.049 0.063 1.28E+04 1245 0.045 0.058 23 off 1.07E+04 834 0.061 0.079 .254 1.16E+04 901 0.058 0.074 1.24E+04 967 0.055 0.070 1.33E+04 1034 0.052 0.067 1.41E+04 1100 0.050 0.064 1.50E+04 1167 0.048 0.061 49 TABLE IV. Continued Series Re local Re point round h/x 1.58E+04 ' 1234 0.046 0.059 1.67E+04 1300 0.044 0.056 1.75E+04 1367 0.042 0.054 1.84E+04 1433 0.041 0.052 50 FIGURE 18. Body surface area relative to body length with 95% confidence limits. FIGURE 19. Wetted surface area relative to body length with 95% confidence limits. FIGURE 20. Total surface area relative to body length with 95% confidence limits. FIGURE 21. Sum of fin surface areas relative to body length with 95% confidence limits. FIGURE 22. Cross-sectional or projected area relative to body length with 95% confidence limits. FIGURE 23. Body volume relative to body length with 95% confidence limits. FIGURE 24. Body surface area relative to body length with 95% confidence limits. 51 Body surface area (cm"2) a 3 f3 5 3 S S S ^  H ^ ^ ^ o 53 (EJUS) BSJB eoejjns 54 [Zjaio) B&JB eoBjjns uu 55 Projected area vs Body length n- 20 FT 2 - .90 OS -\ , , , - i \ 1 1 1 1 1 1 I I r 32 3.0 4 4.4 4.6 52 5.0 O Body length (cm) .OGO(Xr221l Body length (cm) • observed .04(Xr321 Body surface area vs Body volume n - 74 R~2 - .99 30 -. _ ' 3 5 7 9 71 13 Body volume lcm~3) • observed 5.04(Xrj07 FIGURE 25. Pectoral fin anterior ray length relative to body length with 95% confidence limits. FIGURE 26. Dorsal fin anterior ray length relative to body length with 95% confidence limits. FIGURE 27. Anal fm anterior ray length relative to body length . with 95% confidence limits. FIGURE 28. Nare height relative to body length with 95% confidence limits. FIGURE 29. Nare distance from snout relative to body length with 95% confidence limits. FIGURE 30. Nare height relative to distance from snout with 95% confidence limits. 59 LPECT ant.ray length vs Body length n-W RT2 - .67 IJ Body length (cm) • observed J62(Xr-20G DORSAL ant. ray length vs Body length n- 19 RT2 -.77 02 H 1 1 T 1 1 1 1 1 1 1 1 1 i r 32 3.0 4 4.4 4A 52 5.0 0 Body length (cm) • observed 210(X>-358 ANAL ant. ray length vs Body length n-W TC2 - JM 12 Body length (cm) • observed 2KXX)-J374 Nare height vs body length n -12 FT2 - .79 32 3.0 4 4.4 4A 52 5J0 O 0.4 Body length (cm) • observed J5(XX)-J53 • observed Body length (cm) 1.4CXX) -157 Nare height vs distance from snout n - 74 FT2 - AO Distance from snout (mm) o observed 32(X)+20 Drag Estimates The relationship between drag force and terminal velocity for the fish is obtained by subtracting the curve for the flight from that for the fish with flight (see Figs. 31 to 34). This is possible because the drag forces of the component parts are considered to be additive and it is assumed there are no interactive effects between the fish and the flight when they fall together in the tank. In addition, it is considered that when terminal (constant) velocity is reached, the total drag force encountered by an object is exactly equal to the force produced by acceleration due to gravity acting on the submerged mass of the object. The predictive force-velocity curves for the fish alone have slope values which fall between 1 and 2 (Appendix I) as would be expected from Newtonian dimensional analysis. These curves permit the generation of Drag Coefficient (CD ) versus Reynolds number (Re) curves which are compared to theoretical curves for minimum and total C D for laminar and turbulent boundary layers . (Fig. 35). The C D values obtained for the pufferfish are compared to those for other median and paired fin (MPF) swimmers (Fig. 36). From the force-velocity relationships Minimum Power estimates are calculated and related to Reynolds Number for different boundary layer types (Figs. 37 to 40). Theoretical values of C D are compared for different Fineness Ratios and boundary layer types (Figs. 41 and 42) and finally, theoretical Drag/Body Volume ratios are related to Fineness Ratio 66 at different Reynolds Numbers and types of boundary layer flow condition (Figs. ~;3 to 47). 67 FIGURE 31. Drag force relative to Terminal velocity. Series 16 fins on. FIGURE 32. Drag force relative to Terminal velocity. Series 16 fins off. FIGURE 33. Drag force relative to Terminal velocity. Series 23 fins off. FIGURE 34. Drag force relative to Terminal velocity. Series 16 fins on, 16 fins off, 23 fins off. FIGURE 35. Drag coefficient relative to Reynolds number. Series • 16 fins on, D16 fins off, + 23 fins off, 1- minimum laminar, 2-rjtinimum turbulent, 3-total laminar, 4-totaI turbulent. FIGURE 36. Drag coefficient relative to Reynolds number. Experimental results from assorted fish. minimum laminar, minimum turbulent. . 68 Force vs Velocity 16, fins on OJ023 0J022 -0J021 -0.02 -0J01G -0.016 -0.017 -0JD10 -0015 -0.014 -0J013 -0XJ12 -oxjn -0J01 -OJ009 -0XXD6 -0J007 -OJOOO -OJOOB -0J0O4 -OJOOO -0J002 0J6 Terminal velocity (m/s) + night o O c c Q 0J032 0.03 -0.026 -0.020 0.024 0.022 -| 0.02 0.016 -| 0J010 0.014 -0.012 -0.01 -0.006 -OJOOO -0.004 0.002 02 Force vs Velocity 10. fins off — I — 03 total 0.4 — I — 05 0.0 Terminal velocity (m/s) + flight fish total Terminal velocity (m/s) + night o fish Drag coefficient vs Reynolds number Drag Coef f ic ient vs Reynolds N u m b e r 1 . 0 0T c a) o g 0.10 o o 0.01 1000 # Angel fish, open fins O Blue gourami, open fins A Angel fish, no fins A Blue gourami, no fins H 1 1 1 1 (—f 1E4 Reynolds Number • Electric fish • Boxfish H Seahorse V Pufferfish, open fins 1E5 O Pufferfish, no fins • Pufferfish, no fins Cd (tur) Cd (lam) DISCUSSION Surface Area Determination Methods The purpose in examining surface area estimation springs from the question of applicability of common methods employed in predicting surface area based on body length. Numerous authors appear to accept as a general rule relationships such as Sw=0.4L2 for moderately streamlined fish and cetaceans (Gray, 1936b; Parry, 1949; Webb, 1975a) which, according to Webb, is more likely to be a high rather than low estimate. The effect of this approximation is to under-estimate the drag per unit area, which in turn under-estimates the empirically deteiTriined drag coefficient. This leads to an under-estimation of the drag force calculations based on the coefficient and ultimately to an under-estimation of minimum power calculations. The results of the regression analysis of wetted surface area and body length for this study ( Sw=.79L21) support the concern that the above relationship for predicting surface area is not appropriate for less streamlined shapes such as T. fluviatilis. In fact the estimates are about 2.5 times lower than those obtained from the the presents analysis. The result is that the drag coefficient would be over-estimated by about 2.5 times and thus cause the more rotund shape of the puffer to appear hydrodynamically disadvantaged with a lower efficiency rating. The increase in the intercept value may seem somewhat anomalous at first glance in that a more streamlined salmonid 75 shape has a higher per unit volume surface area than a more rotund shape such as a puffer. However, the relation is based on unit length, which for more streamlined shapes is relatively greater than for stubbier shapes when related to the amount of surface area. Wetted surface area, for the purpose of estimating drag coefficients in this study, is defined as the sum of the surface area of the body plus the surface area of any fins. The four propulsive fins were amputated for the majority of drop tank experiments in order to ensure a steady vertical descent with no flutter of body parts. This decreases the surface area under the influence of the fluid and thus the drag force experienced by a fish in a dead-drop experiment. It is not clear to 'what extent the propulsive fins contribute to the overall drag of a fish swimming in the tetraodontiform mode, although an attempt at accounting will be made later in the section discussing drag. Morphometry Diversity in body shape and mode of propulsion are characteristic of higher teleost fishes. The species in this study, Tetraodon fluviatilis, is propelled by means of undulatory median and paired fins. Some other families swim similarly, eg. Diodontidae (marine pufferfish.es), Balistidae (triggerfishes, filefishes), Synathidae (seahorses, pipefishes) and Ostraciidae (boxfishes, cowfishes, trunkfishes). Breder (1926) was the first to classify fish on the basis of locomotory pattern and early on defined three general categories: 76 Anguilliform (eel-like, after Anguilla), Carangiform (trout-like, after Carangid-c) and Ostraciiform (caudal fin sculling, after Ostraciidae). Subsequent work continued to describe locomotory patterns after a particular species. Hence, on the basis of the position of the propulsive fins for median and paired fin swimmers, terms such as Tetraodontiform (short dorsal and anal fins, after Tetraodontidae), Diodontiform (short pectoral fins, after Diodontidae), Balistiform (extended dorsal and anal fins, after Balistidae), Labriform (paddling or flapping pectorals, after Labridae), Ostraciiform, Rajiform (continuous enlarged pectorals, after Raja), Gymnotiform (extended anal fin, after Gymnotidae) and Amiiform (extended dorsal fin, after Amia) were used to describe the locomotory pattern typical of the. functional group. (See Lindsey, 1978 for a detailed review). Blake (1983) has criticized this approach to undulatory median and paired fm swimmers for lack of a functional basis and proposed in its stead a general system of classification based upon the amplitude, frequency and wavelength characteristic of the propulsive fins. This system is, of course, a continuum of relative measures anchored at one extreme by fish displaying waveforms of large amplitude, low frequency and large wavelength (termed Group 1 forms) and at the other extreme by fish presenting • waveforms of small amplitude, high frequency and small wavelength (Group 2 forms). Tetraodontiform fishes (the freshwater pufferfish) should fall somewhere in between these two extremes. The' fish in this group (Tetraodontiform) are relatively slow 77 swimmers and for the most part could be described as stout or less streamlined when compared with faster swimming fusiform fish such as salmonid (trout, salmon) and scombrid (tuna, mackerel) fishes. The freshwater pufferfish is propelled by means of undulatory paired pectoral and median dorsal and anal fins. The caudal fin during routine forward rectilinear progression is collapsed fan-like and held rigidly in the dorsoventral plane. The caudal fin system is active as a rudder during complex turning or reversing manoeuvres, wherein it is expanded fan-like, and as the sole propulsive unit in escape responses during which the usual propulsive fins (pectoral, dorsal and anal) are collapsed and tightly adducted against the body. The importance of defining body form in terms amenable to hydrodynamic analysis has been mentioned in the introduction. The current section is concerned with providing answers for the following questions. 1) What are the hydrodynamically relevant morphometric characteristics of the basic body form of Tetraodon fluviatilis! 2) Are these characteristics maintained over a range of body length as the fish grows? 3) Is the body shape a reasonable analogue of an axes-symmetric body of revolution (eg. a prolate spheroid)? The basic body shape of the specimens observed in this study can be described as a tear-drop with slight lateral compression occurring in the posterior region from the median fins (dorsal and anal) towards the caudal peduncle. 78 Analysis of the general body morphology has shown that Fineness Ratio (FR) and Shoulder Position (SP) are constant over the range of body lengths measured in this study (mean FR=3.37, s=.259; mean SP=.43, s=.044; also Figs. 8 - 13). Compared with some faster swimming pelagic forms, the freshwater pufferfish is relatively rotund and thus considered less streamlined (see Table VI for a list of FR values for some other species). For example, some fish and cetaceans which swim in the carangiform mode have FR values in the order of 3.5 to 5 while other fish said to swim in the subcarangiform mode tend to have higher values ranging from around 5.5 to 7 (Hertel, 1966; Webb, 1971a; Aleyev 1977). Faster swimming fish generally tend to have higher FR values. When compared to man-made objects, the pufferfish body resembles the fuselage of some airships, bombs and. boat hulls, objects presumably designed with maximum volume for mirrimum surface area, having FR values in the order of 3 to 5 (Hoerner, 1965). The value of the Fineness Ratio, as mentioned previously, is a relative measure of the degree of streamlining present in a body, which is inversely proportional to the extent to which a body will disturb the fluid through which it travels, and as such provides some information as to the magnitude of the pressure gradient and the type of boundary layer the body is likely to encounter, given the Reynolds Number at which it operates. Lower FR values suggest that when boundary layer separation does occur, the wake will be wider and therefore cause a greater amount of 79 pressure drag than would be experienced by a more streamlined body, all else being equal. Higher FR values mean that the influence of the favourable pressure gradient is extended along the body; encouraging the maintenance of laminar boundary layer flow over the major portion of the body. However, lower FR values also mean that forms such as the pufferfish present less surface area and greater volume for a given body length than a more streamlined fish. For example, the following data illustrates how, for a prolate spheroid, the ratio of surface area to volume increases with Fineness Ratio. FR 2.8 3.4 6.5 7.3 9.5 S /Volume 3.5 3.7 4.4 4.6 * 5.0 While faster swimming scombrid fishes . (Scombridae, Thunnidae, Katsuwonidae; Webb, 1975a) have body shapes which can maintain a high proportion of laminar flow at higher Reynolds Number (>10°), a more rotund fish such as the pufferfish may enjoy some advantages over a more streamlined form at lower swimming speeds, such as lower surface area drag and lower drag per unit body volume, as will be discussed in a later section concerning drag estimates. The position of maximum thickness (SP) in the freshwater pufferfish remains constant (mean SP=.43) over a range of body lengths (Figs. 11-13) for which the greatest length is roughly double that of the least (*3.2 to 6.5). Also, the SP invariably coincides with that of the opercular opening (and the pectoral 80 TABLE V. Species Fineness Ratios of various aquatic organisms Vd 1/w x/1 authority Puffer (Jetraodon fluviatilis) Tuna (Euthynnus affinis) Rainbow trout (Salmo gairdneri) Whiting (Gadus merlangus) Perch (Perca fluviatilis) Perch (Psettodes erumei) Halibut (Hippoglossus hippoglossus) Greenland halibut (Reinhardtius hippoglossoides) Plaice (Pleuronectes platessa) Plaice (Pleuronectes platessa) Goldfish (Carassius auratus) Trout (Salmo irideus) Dace j (Leuciscus leuciscus) Bream Goldfish (Carassius auratus)' 3.4 3.4 0.43 this study 4.0 4.9 0.5 Magnuson (1970) 5.4 7.3 0.39 Hertel (1966) 6.5 8.9 0.24 Haslett (1962) 5.1 7.5 0.3 Kipling (1957) 5.1 12.3 — Norman (1934) 4.6 12.2 — de Groot (1970) 5.8 12.3 — de Groot (1970) 4.4 15.2 — de Groot (1970) 4.4 13.6 0.24 Arnold & Weihs (1978) 4.5 Bainbridge (1960) 6.3 ibid. 6.7 ibid. 2.9 8.4 0.42 Bainbridge (1963) 3.3 6.0 0.32 ibid. 81 TABLE V. continued Species 1/w x/1 authority Dace (Leuciscus leuciscus) 4.5 8.1 0.41 ibid. Dolphin 4.0 — — Hertel (1966) Swordiish (Xiphias gladius) 4.2 — — ibid. Blue whale 4.8 — — ibid. Greenland shark (Somniosus microcephalus) 3.8 — — ibid. Tuna (Thunnus sp) 3.6 — — ibid. Barracuda (Sphyraena sp) . 6.3 — 0.40 ibid. Shark (Lamnidae) 5.6 — 0.44 ibid. Smooth dogfish (Mustelus canis) 7.1 — 0.45 ibid. Pike (Esox sp) 5.6 — 0.55 ibid Alligator gar (Lepisosteus sp) 9.1 — 0.70 ibid. d is the mean diameter of depth and width 'data from scale drawings 82 fins) which is similar to what Houssay (1912) found for a variety of fish. It is possible that this could assist in venting gill effluent, since the SP indicates the general region of the body which experiences maximum fluid velocity, hence from Bernoulli's theorem, minimum pressure. For example, for a FR of 5.7, Hoerner (1965) states the following correspondence between shoulder position and the location of minimum pressure for a body of revolution. Shoulder Position .3 .4 .5 Minimum pressure location .2 .35 .6 Allen (1961) has shown that for some smaller, fish gill effluent can be a turbulence-causing disturbance which can disrupt the boundary layer flow and cause it to separate. The portion of the body which encounters laminar boundary layer flow is dependent upon numerous factors besides FR and SP, some of which are the characteristics of the incident flow, surface roughness (eg. nares, opercula, scales) and Reynolds Number. So without flow visualization data it is difficult to state one way or another what will be the effect of imposing the opercular opening, hence gill effluent, upon the position of maximum thickness. The Shoulder Position of the pufferfish (.43) contrasts with that for some faster swimming pelagic forms such as scombroid fishes which have the position of maximum girth set back along the body about .6 to .7 of the body length (Walters, 1962; 83 Hertel, 1966) presumably to maximize the portion of the body which encounters a positive pressure gradient and laminar boundary layer flow. In some fishes the body profile may change as the fish grows. For example, the Shoulder Position of Trachurus mediterraneus migrates posteriorly from about .3 to .45 (as FR increases from roughly 3.31 to 4.23) as the fish grows from about 1 cm to 40 cm in length (Burdak, 1969). It is likely that during earlier stages of development, swimming speeds, Reynolds Numbers and drag are relatively low and that development of guts and gonads is of paramount importance. If velocity, Reynolds Number and drag increase as the fish grows in length to reach the adult form in which it spends the major portion of its life, posterior-ward migration of the Shoulder Position and an increase in Fineness Ratio would increase the probability of maintaining an intact laminar boundary layer thereby minimizing drag forces. This same selective pressure would not be imposed upon slower swimming fish and this correlates with the maintenance of the basic body profile during growth over a range of body lengths in the pufferfish. The' relationships between body volume with area and length are not significantly different from those expected on the basis of geometry and scaling in that surface area «= length2, volume <*= length3 and surface area « volume2/3 (Figs. 23 & 24, Appendix I). Comparison of the values of body volume and surface area for the pufferfish with theoretical values calculated for an axes-symmetric body of revolution (prolate spheroid) using 84 one-half standard body length and one-half maximum body depth as the respective major and minor semi-axes a and b, reveals that the pufferfish specimens average 85% and 91% of the respective volume and surface area values for a prolate spheroid of same semi-axes. Values were calculated from the following standard formulae (CRC Handbook of Mathematical Sciences, 5th ed.). eccentricity e=J 1- b W volume A/3nab2 surface area Conclusions to be drawn from this section include 1) The Fineness Ratio (3.37) and Shoulder Position (.43) are independent of body length. These parameters allow comparisons to be made with other fish when combined with Reynolds Number and Drag Coefficient data. 2) The basic body profile of the pufferfish specimens measured in this study remains constant as the fish grow over a range of sizes. Thus, at least for the range of body lengths observed and likely also for lengths outside the range observed (except perhaps for larval stages), the body dimension regressions- calculated (Appendix I) can be used with confidence. 3) The body form of the pufferfish is reasonably analogous to a rigid, axes-symmetric body of revolution on the basis of shape, volume and surface area and also on the basis of the 85 locomotory pattern of the fish wherein during routine forward rectilinear progression the body is held roughly in a "stretched-straight" position while thrust is generated by undulatory paired pectoral and median fins. The implications of these factors on drag estimates are discussed in the following section. 86 Drag Estimates The purpose of the current section is to examine the following questions. 1) What are the drag forces acting upon the fish and are the estimates obtained valid? 2) What are the drag coefficient estimates and how do they compare with theoretical estimates and those of other median and paired fin swimmers? 3) What type of boundary layer flow conditions are expected? 4) What are the estimates of power requirements? 5) How is thrust compensation achieved as the fish grows in size? There are four different techniques which are commonly employed to directly measure the drag forces acting on a body in opposition to fluid flow: terminal velocity, deceleration in glide, towing tank and water or wind tunnel. (For reviews see Bainbridge, 1961; Webb, 1975a; Blake, 1983d). The first two techniques are distinguished from the latter as they are passive; the object is unrestricted and estimates are inferred from distance and time measurements. The second two techniques are active in that the object is tethered and estimates are obtained from direct force measurements. Both types of technique have advantages and disadvantages. Towing techniques (Houssay, 1912; Magnan, 1930; Sundnes, 1963; Kent et al, 1961; Kempf & Neu, 1932; Denil, 1936), water tunnel (Brett, 1963; Webb, 1970) and wind tunnel techniques with 87 either frozen specimens (Blake, 1980b) or models (Harris, 1936) are easier to control experimentally but at the cost of some degree of uncertainty due to the interactive effects of the line or spar with the object to which it is attached. Terminal velocity (Magnan, 1930; Richardson, 1936; Gero, 1952; Blake, 1979a,b, 1981a,c) and deceleration-in-glide (Magnan, 1930; Gray, 1957a, 1968; Lang & Daybell, 1963) techniques are harder to control in a spatiotemporal sense, especially with live animals but the inferred drag estimates should be unbiased since the body is unfettered. Both types of technique are subject to problems such as body and or fm flutter (Hertel, 1966; Brett, 1963; Webb, 1970) and deviations from true rectilinear progression. An • alternative to direct drag measurements is to calculate theoretical values from the hydromechanical equations (chapter one) established by combining empirical observations of technical bodies of revolution with hydromechanical theory. The- significantly lower levels of cost and complexity for experimental apparatus are a major advantage of passive over active techniques and with the addition of electronic timing devices to drop tanks, terminal velocity experiments provide precise and accurate data. The main assumptions in obtaining drag estimates from the drop tank data are that 1) free stream flow is steady 2) the boundary layer is attached and laminar 3) flow around the dead fish is mechanically similar to that for an actively swimming fish 4) that the drag contributions of the body and the 88 stabilizing flight are independent, additive and taken together, equal the total drag. The first assumption is likely met in the drop tank as the fluid is motionless before the specimen is dropped, acceleration to terminal velocity is smooth and the velocities are sub-critical. The second assumption has, for all non-dead fish, some associated difficulties. For fish which swim by means of large amplitude oscillations of their body and or fins (ie. anguilliform, carangiform, sub-carangiform), the estimates of dead drag will grossly under-estimate the drag encountered by an actively swimming fish. However, for fish which hold their bodies rigid and swim by relatively small oscillations of their propulsive fins (ie. balistiform, gymnotiform, tetraodontiform), the rigid body estimates should be fairly close. The fourth is assumption is most likely met as care was taken to position the trailing dart flight at a great enough distance from the trailing edge of the fish to prevent the flow around the body from interfering with that around the flight. Subtraction of the flight drag curve from the total drag curve results in the fish-alone drag curve (Figs. 31-33, see also Fig. 34). The third assumption has been approached in the morphometries section and is discussed further here. The drag estimates inferred from the tenninal velocity experiments are estimates of the total drag force acting on the fish. The total drag force, as mentioned in the hydromechanics 89 section, is comprised of numerous components including friction and pressure drag. Fish in the wild are somewhat more animated than a dead fish, with propulsive fins amputated, in a drop tank experiment and as such the values calculated are considered minimum estimates. The validity of the drag estimates stems from the fact that, for the freshwater pufferfish, the body is rigidly held in a "stretched-straight" position during routine forward rectilinear progression, unlike fish and cetaceans for which thrust is produced by body and caudal fm oscillations. One difficulty arises when considering the drag force encountered by an actively oscillating fin. In the terminal velocity experiments using fish with the propulsive fins (pectoral, dorsal, and anal) intact, the fins act as rigid fixed wings. The drag on the dorsal, anal and caudal fins is mainly due to friction drag since they are parallel to the incident flow and as such pressure drag is negligible. However, the pectoral fins are oriented laterally in a broadside-on fashion and are inclined vertically at about 48 degrees above the axis of incident flow which produces a component of lifting force normal to the axis of progression and a retarding pressure drag in addition to the friction 'drag. The placement of "plates" (fins) at the shoulder of a streamlined body can increase the drag encountered by a streamlined body by up to five times that for the same body without fins (Hoemer, 1965; von Mises, 1959). Blake (1981a) has shown that regardless of fm shape (circular, rectangular, square or triangular) the dead drag on a fish with open pectoral fins 90 perpendicular to incident flow is greater with the fins attached directly to the body than with fins attached distally via spars, due to interference of the fins with the flow over the body (Blake, 1979b). The pressure drag arising from the fins will contribute a major portion to this increase in drag (Blake, 1981a). In this study, amputation of the four propulsive fins decreases the drag by nearly one half of that measured for the same fish with fins intact. With respect to an actively swimming pufferfish, the drag values obtained with the propulsive fins amputated will be somewhat under-estimated while those values for fish with propulsive fins intact will be over-estimated. The true value likely falls in between the two curves; fms intact and fins amputated (Fig.34) and the curves for fish with propulsive 'fins amputated can be considered as minimum estimates of total drag encountered by an actively swimming fish. Although the drop tank model requires the free stream to be stable and laminar, the real life fluid environment for the pufferfish is likely unstable and somewhat turbulent. The pufferfish habitat is a brackish, riverine and estuarine one (Dekkers, 1975) where flow conditions may vary widely, both spatially and temporally. Although the free-stream flow may be partly turbulent, it is likely that the boundary layer will remain intact at such relatively low Re values (^104) since the fluid should damp out the disturbances and prevent separation. With an ' intact boundary layer, pressure drag is negligible and friction drag is the main component of drag resisting forward 91 movement. 92 Drag Coefficients The Drag Coefficient (CD(ex P )) vs Reynolds Number (Re) curves are presented in Fig. 35. The dependence of Crxexp) on Re for series 16 (fins on and off) is apparent and contrasts with the relative independence of CD(ex P ) on Re for series 23. The two curves for series 16, fins intact and fins amputated, follow a trend similar to the theoretical estimates (CD(the)) for a flat plate of equivalent surface area in laminar boundary layer (BL) flow conditions. The curve for series 23 shows a trend similar to the theoretical estimates for an equivalent flat plate in turbulent BL flow conditions. The following equations are used to calculate the theoretical estimates of C D (taken from Hoerner, 1965). Minimum frictional C D : laminar flow BL ; C/(iam)=1.328/?e"'5 turbulent flow BL; C/(tur)=.074/?e""2 Total C D : laminar flow BL; CT(iam)=C/(Um)[l+(d/l)15]+0.11(d/l)2 turbulent flow BL; CT(tur)=C/(mr)[l+1.5(d/l)1'5+7(d/l)3] According to Hoerner (1965) the equation for Crrum) is applied between Re values of 104 to 105. It is assumed the equation is also applicable at lower Re values (103-104) with laminar BL flow conditions. The equation for Cr(tur) is applied at higher Re values (>105) where the BL is likely transitional or 93 fully turbulent. The presence of the propulsive fins (series 16) elevates the CD-Re curve above that for which fins are amputated by an average of about 1.5 times but the same general form of the curve is maintained- This indicates that the BL flow conditions are not greatly affected by the presence of the propulsive fins and that the BL flow conditions are largely laminar since the curves are similar in form to the theoretical laminar curves. For series 16 with fins intact the values of C D ( e x P ) range from about .075 (Re=4x\03) to .038 (/?e*8xl03) and with fins amputated the values of CD(exp) range from about .04 (i?e=4xl03) to .02 (/?e*1.3xl04). At lower Re values ( « 1 0 5 ) the BL flow conditions are generally considered to be laminar and CD(ihe) is dependent upon Re. The flatness of the CD-Re curve (CD-.028, /te*l.lxl04-2.1xl04) for series 23, fins amputated, suggests that the BL flow condition^  in the terminal velocity experiments may be turbulent to some extent even though the Re values do not exceed 2.2xl04. Another possibility is that the BL is laminar but the system is operating in a region of Re where the CD is relatively independent of the the Re. This would be the "saddle" of the laminar portion in the lower Re region of the theoretical CD-Re curves which can be seen in most fluid dynamics texts (ie. Hoemer, 1965). At higher, super-critical Re values (>5xl06) the BL flow conditions are usually turbulent and the CD<the) progressively becomes relatively independent of Re. It is possible to induce turbulence in the BL at lower Re values by protruding fms, 94 nares, eyes, etc., through the BL into the free-stream flow however the fluid viscosity tends to damp out the turbulence before it causes the BL to separate. Permissible Roughness calculations (Table IV) indicate that the two small nares between and anterior to the eyes exceed the height allowable for the local Re. The protrusion of the nares through the BL into the free-stream will cause a disturbance in the free-stream flow which may in turn introduce some turbulence into the boundary layer. In order to comment more specifically on the degree of turbulence that may be introduced into the BL a detailed flow visualization analysis is required (Allen, 1961; McCutcheon, 1977). When compared with theoretical values, the experimentally determined estimates of C D exceed by varying amounts the theoretical minima for a flat plate of equivalent surface area in both laminar and turbulent BL flow conditions (Fig. 35). For series 16, fins intact, Crxexp) estimates exceed CD(the) values for laminar BL flow conditions by an average factor of 2.9 and 1.7* (C/(iam) and C r ( U m ) , respectively) and with fins amputated factors average 1.7 times greater and 0.9 times less than C /bam) and C /(wr). For a larger specimen such as in series 23 (fins amputated), CD(exp) values exceed Crxihe) values by average factors of 2.6 and 1.2 for C / ( i a m ) and C/(wr), respectively. Although the CD(ex P) estimates are noticeably higher than the C/oam) values, it must be borne in mind that the values of 95 CD(ihe) are based on the two-dimensional rriinimum frictional drag on a flat plate of equivalent surface area which exists without the concerns of locomotion, growth and reproduction (respiration, guts and gonads) and which, at such low Re values, encounters negligible pressure drag. Also, the CD(the) values are, by necessity, derived from empirical analyses for axes-symmetric bodies of revolution which, for the Re range involved in this study (#e^l03-104) are nearly impossible to come by (Hoerner, 1965). In comparing the CD(CXP) estimates for the pufferfish to those found for some other MPF swimmers, the slopes of the curves are generally consistent and the elevation of the CD(exP) curve due to the presence of the pectoral fins is similar to that found for the Angelfish (Pterophyllum eimekei) and the Blue gourami (Trichogaster trichopterus), (Fig. 36), (see Blake, 1983d). For electric eels and knifefishes which swim by passing a propulsive wave down continuous fins, Blake found similar slopes and elevations (Blake, 1983b) at comparable Reynolds numbers (ie. CD*.05 at Re*\03 and Cd*.02 at Re*10*). For some sub-carangiform and anguilliform swimmers, drag estimates exceed the theoretical minima by three to five times (Lighthill, 1971; Alexander, 1967). Based on a number of fish species Bainbridge (1960, 1961) states that the total drag ( C T ( w r ) ) is. approximately 1.2 times greater than the theoretical minimum frictional drag ( C / ( i a m ) ) . This differs considerably from the results of the calculations for the Re range and FR values relevant to this study. 96 Using the CD equations for lower Re values (<105) the CD ratios of Crcum) to C/(Um) are approximately 1.6, 1.7, 1.9, 2.1 and 2.2 at their respective Re values of 3, 5, 10, 15, and 20 (xlO3). The reason for the difference from a factor of 1.2 is found in the sensitivity of the equation (drum)) to both FR and Re. (Bainbridge employs the equation Cr(tur) for total drag at Re values >105). For a given FR, the CD ratio ( C T ( i a m ) / C / ( i a m ) ) increases with Re, for example, with a FR=3.37, the ratio roughly doubles from 1.6 (7?e=3000) to 3.5 (Re=\05). At a given Re, the CD ratio decreases ' with an increase in FR, for example, with Re=lOs, the ratio decreases from 3.5 (FR=3.37) to 2 (FR=5). The point to be made here is that contrary to practices sometimes found in the literature, it is not always appropriate to assume the total theoretical drag is simply a 1.2 multiple of the minimum theoretical friction drag of an equivalent flat plate without first considering the effect of the FR of the body in question and the Re range in which it operates. This is especially important for the lower Re range in which many MPF swimmers typically operate (<105). Another issue concerns the applicability of the equations used to calculate theoretical drag coefficients. At /?e=1.5xl04, the CD values produced by C/(Um) and C/(wr) are equal (CD=.0108). Below J?e=1.5xl04, C/(mr) produces values which are actually lower than those produced by the analogous equation for laminar flow, C/(Um). It seems unlikely that at the lower Re 97 values under consideration (assuming an intact BL), the theoretical minimum drag for turbulent BL flow conditions (frictional or total) would be less than that for laminar BL flow conditions. However, that is exactly the result (Figs.35, 36). Because of the higher energy losses associated with a turbulent BL, it is expected that C D values for a turbulent BL would be higher in terms of frictional drag. Since at such low Re values BL separation is unlikely, the total turbulent drag should also be higher than the total laminar drag since the major component' of drag at these lower, viscous-dominated Re values is almost completely comprised of frictional drag and it is unlikely that the BL will separate. One possible reason for the disparity regarding the relative elevations of the experimental and theoretical curves is that the equations • are based on empirical relationships which were obtained at higher Re values than those which are appropriate to this particular study. Unfortunately, in the classical empirical CD-Re studies there is a dearth of information for the Re range 103-104 (Hoerner, 1965). So what predictions can be made regarding the flow conditions surrounding the fish at these Re values? A reasonable guess would seem to be that at the lower end of the Re range (3000-6000) the BL is laminar. As the Re values increase towards the upper end of the range (*104), some disturbance is probably introduced into the BL as a result of roughness elements such as fins, nares, eyes, and opercula at various points along the body. It is most likely that the BL is still attached and has some 98 regions of turbulence associated with the nares, eyes and pectoral fins. With respect to the distribution of Cr>-Re data points for other MPF swimmers, the best estimate of a Cn-Re correlation for the pufferfish should be based on a combination of the curves found for series 16 and 23 (fins amputated) which, as a minimum estimate, gives a useful lower bound for estimating the drag on the pufferfish body. 99 Power Output The minimum power output (/\exp)) necessary to overcome drag is calculated as the product of drag force (ospU SWCD) and velocity for the three cases: series 16, fins intact; series 16, fins amputated; series 23, fins amputated (Figs. 37 to 40) and is compared for each case with theoretical curves (Athe)) based upon the four CD values: C/aam), C/W), Croam) and Crcwr). The form of the curves is in general agreement with expectations based on dimensional analysis wherein power is proportional to the cube of velocity as the slopes of the curves fall between 2 and 3. The presence of the propulsive fins requires roughly double the power output over the Re values measured. Again it must be remembered that the power estimates are based on the drop tank drag measurements in which the propulsive fins act as flat plates or wings as opposed to actively undulating propulsors in a swimming fish. While the drag on the median fins (dorsal and anal) is essentially frictional, the open pectoral fins should incur considerable pressure drag due to their 48 degree angle above the incident flow, in addition to friction drag. The pressure drag on an undulating pectoral fin is likely far less than that for a static fm since the velocity of the fluid ejected by the fin is higher than that of the incident flow (free-stream) however the higher fluid velocity across the fin will increase the friction drag on the fin. With the fins amputated, the power required to overcome the frictional drag of the fins is missing so the estimates in these cases are low. As a 100 result the true estimate of power requirement should fall somewhere between the two curves for fins intact and fins amputated. The minimum power curve for series 23 (fins amputated) picks up where the curve for series 16 (fins amputated) ends and continues on in the same fashion over its range of Re (Fig. 40). This suggests continuity in the hydrodynamic requirements of the basic body shape of the pufferfish since there are, in this study, fish of different body lengths (total body lengths; 6.95, 9.82 cm) which seem to follow the same basic curve, not unlike that expected on the basis of dimensional analysis. Thus, as an approximation, the two curves can be considered as estimates of the same curve, obtained over different Re ranges with different specimens assumed to be hydrodynamically equivalent. 101 FIGURE 37. Minimum power relative to Reynolds number. Series 16 fins on. • measured, • minimum laminar, + minimum turbulent, A total laminar, X total turbulent. FIGURE 38. Minimum power relative to Reynolds number. Series 16 fins off. • measured, £ minimum laminar, -4- minimum turbulent, A total laminar, X total turbulent. FIGURE 39. Minimum power relative to Reynolds number. Series 23 fins off. D measured, £ minimum laminar, -f minimum turbulent, A total laminar, x total turbulent. FIGURE 40. Minimum power relative to Reynolds number. Comparison of series 16 fins on ( • ), 16 fins off ( -f ), 23 fins off (0). FIGURE 41. Drag coefficient relative to Reynolds number. For different Fineness ratios. Based on the , total laminar drag coefficient. FIGURE 42. Drag coefficient relative to Reynolds number. For different Fineness ratios. Based on the total turbulent drag coefficient. 102 0.000 Power vs Reynolds Number 10. Fins on 0.0O5 A 0.004 A 0.003 A 0JO02 A OJOOI A o 35 —r— 45 T — 1 1— T — 55 05 (Thousands) Reynolds Number Power vs Reynolds Number 10. Fins off (Thousands) Reynolds Number 0.04 0.035 0.03 H 0.025 0.02 A 0J015 A 0.01 -] OJOOO 10 Power vs Reynolds Number T 12 1 14 T 10 (Thousands) Reynolds Number Power vs Reynolds Number 0.035 0.03 0.025 A 0.02 4 0D15 A 0.01 A OJOOB • PexpWon T 1 r It 13 (Thousands) Reynolds Number + PexpIO o Pexp23 3 0.055 c q> o o CJ 8> c Q 0.045 0.035 0.025 A O Drag Coefficient vs Reynolds Number T 10 F.R.: • 3 + 3J5 based on CT(lam) x 2 0 (Thousands) Reynolds Number 4 A 4.5 V 5.5 40 o oo C o Q FR.: a Drag Coefficient vs Reynolds Number ~r 10 based on CT(turb) OS 1 20 (Thousands) Reynolds Number ' A 4.5 40 5.5 Drag, Volume and Fineness Ratio For FR values ranging from 3 to 5.5, values of Croum) and CT(tur) are compared over a range of Re values (Figs. 41 & 42). The decrease in C T with increase in FR is apparent for both BL types. For the laminar BL (Fig. 41) at Re=\tf the Croam) for a FR=3 is approximately 1.5 times greater than that for a FR=5 (.0280 vs .0187). When the Re value is tripled to 3xl04, the CD ratio is about 1.7. A similar trend exists for the turbulent BL (Fig. 42) where the Oxtur) for a FR=3 is about 1.3 times greater than that for a FR=5, at both Re=\tf and tfe=3xl04. Other trends apparent are that the decrease in CT with increased FR is non-linear; as FR increases the decrease in CT diminishes, and that CT is inversely proportional to Re. It has generally been assumed that lower drag coefficient values imply higher performance hence, a measure of success. However, for slower swimming fish such as MPF swimmers, it may be that what matters more than a low coefficient of drag is the ratio of drag per unit volume. Throughout the hydrodynamic and fish locomotion literature it has generally been regarded that the optimum FR with respect to maximum volume for minimum drag falls around 4.5 (von Mises, 1959; Hoerner, 1965; Shapiro, 1964; Webb, 1975a; Vogel, 1981). For a series of prolate spheroids, theoretical values of total drag per unit volume are related to FR at different Re values (Figs. 43 to 47). It is apparent that for the lower end of the Re range (1500) an optimum FR is approximately 2.3 and that a departure to a FR=4.5 results in an increase in Drag/Volume of 109 about 2 5 % . As the Re values increase, the optimum shifts towards higher F R value; and the penalty for departing from the optimum diminishes. Near the upper end of the Re range (1.5xl04) the optimum F R is about 3 . 5 and the penalty for a departure to a F R = 2 or 6 is in- the order of 1 5 % . The points to be made here are: that the optimum F R is dependent upon Re at lower Re values and that the optimum F R ranges from approximately 2.5 to 3.5 (7?e*3xl03-1.5xl04 respectively) which happens to be the range of F R and Re in which the pufferfish in this study are placed. It makes a certain amount of sense that one of the most highly evolved or specialized ray-finned fishes (Moyle and Cech, 1 9 8 8 ) would be shown to approach an optimal morphology based on the appropriate hydromechanical drag analysis. It is also a reminder that speed with stamina is but one of a number of successful strategies possible. 110 FIGURE 43. Drag/Body volume ratio relative to Fineness ratio and flow conditions. For Reynolds number= 1500. FIGURE 44. Drag/Body volume ratio relative to Fineness ratio and flow conditions. For Reynolds number= 6000. FIGURE 45. Drag/Body volume ratio relative to Fineness ratio and flow conditions. For Reynolds number= 10500. FIGURE 46. Drag/Body volume ratio relative to Fineness ratio and flow conditions. For Reynolds number= 15000. FIGURE 47. Drag/Body volume ratio relative to Fineness ratio and flow conditions. For Reynolds number= 30000. Ill Drag / Volume vs Fineness Ratio Re-1500, Cd based on wetted surface • • • • • • • • • • + + 1 1 1 1 3 5 Fineness Ratio Total laminar + Total turbulent 00 Drag / Volume vs Fineness Ratio Re-OOOO. Cd based on wetted surface O I < 1 q> O > \ q> o c o 8> c Q 50 A 40 A 30 20 A • • • • • • + + • • • • io A o 3 5 • Total laminar Fineness Ratio + Total turbulent o I < .6 O > \ fl> O C o k s> c Q 150 HO -130 -120 -HO -100 90 oo A 70 00 -| 50 AO -30 -20 -10 -0 Drag / Volume vs Fineness Ratio Re-10500. Cd based on wetted surface • • + + • • • • / 3 "T" 5 • [] • • Total laminar Fineness Ratio + Total turbulent on O I ( 6^ Q> E O > \ 0) o (-o u. 8> Q 260 200 -240 -220 -200 -160 100 -| 140 120 -100 -60 00 -40 -20 -0 Drag / Volume vs Fineness Ratio Re-15000. Cd based on wetted surface • • • • + + / 3 + • • 5 • 0 + • Total laminar Fineness Ratio + Total turbulent Drag / Volume vs Fineness Ratio Re-30000. Cd based on wetted surface • • • • • • • • • + + 1 1 1 1 3 5 Fineness Ratio • Total laminar + Total turbulent Fin Morphometry and Thrust Production As an organism grows in size, its surface area increases in proportion to the square of the body length (as does the friction drag) and volume increases in proportion to the cube of the body length. Thus at the very least, the thrust power requirements increase in proportion to the skin friction, necessitating some kind of thrust compensation in order to at least achieve the same swim speeds. Thrust compensation can be accomplished in one of two ways: morphometrically or kinematically. For morphometric compensation to occur, as a fish grows in size the propulsive apparatus (pectoral, dorsal and anal fins) must increase in at least an isometric fashion since thrust production is directly related to the square of the span of the propulsive fin. Fin morphometry analysis reveals that the relationship between fin ray length and body length is negatively allometric and as such, thrust compensation appears not to be achieved morphometrically. This is not unusual as the mechanics of loading for the fin rays argues against morphometric compensation since the bending moments on the fin rays are also proportional to the square of the span and as a result quickly become rather large, necessitating and exponential increase in the strength of the load bearing components of the fin (ie. fin rays). Since morphometric compensation is not indicated in the present analysis, it is predicted that thrust compensation is 117 accomplished through adjustments in the kinematic parameters, frequency, amplitude and wavelength of the undulatory propulsive fins, either singly or in concert. Conclusions to be drawn from this section include 1) Although pressure drag is always present due to the viscous nature of fluids, it is considered that the major component of the total drag encountered by the body is frictional drag. 2) The true estimate of the total drag curve likely falls between those for fins intact and fins amputated, the latter being a minimum estimate of the total drag encountered by an actively swimming fish. 3) The drag estimates are valid since the drop tank protocol is reasonably matched by an actively swimming fish wherein the body is rigidly held in a "stretched-straight" position during steady forward, rectilinear progression and the hydrodynamic similitude between specimens of all sizes has been demonstrated (in the morphometries section). 4) The majority of the Crxexp) estimates (fins amputated), based on the drag measurements, fall between .02 and .03 (/?<>-5000-2xl04) and are, in the main, centered around the C T ( i a m ) curve. The CD(exP) estimates are comparable to those found for some other MPF swimmers (fins amputated and fins intact) over a similar range of Re values. 5) It is expected that at the lower end of Re range (-3000) the BL is laminar and that as the Re values rise, elements such 118 as nares, eyes, opercula and fins introduce some regions of disturbance into the BL which for the most part are damped out as a result of the fluid viscosity since the Re values at which the system operates are still relatively low (^104). 6) Power requirement curves are similar to those expected on the basis of dimensional analysis and are minimum estimates (fins amputated) of the power requirements for an actively swimming fish. The two different curves for specimens with fms amputated are considered to be estimates of the same curve. 7) Morphometric analysis indicates thrust compensation is accomplished through alteration of one or more of the kinematic parameters; frequency, amplitude and wavelength for one or more of the propulsive fins. 119 CHAPTER THREE: KINEMATICS INTRODUCTION The earliest methods of classification of swimming modes are based on the location rather than the kinematic or functional aspects (Breder, 1926). This approach has resulted in some rather different 'fish, in kinematic terms, being lumped into the same category. For instance, both the seahorse (Hippocampus) and the electric fish (Gymnarchus) have their propulsive fins located dorsally. However, the seahorse oscillation frequency is about twenty times greater than that of the electric fish. The seahorse dorsal fin is relatively small and the dorsal fin of the electric fish most of the length of the body. A more recent approach is to classify fish based on their fin kinematics (Blake 1979a, 1983d). This system results in the electric fish being included in the so-called Group I forms which are characterized by fin oscillations showing low frequency, large amplitude and long wavelength. The seahorse is included with Group II forms which typically show high frequency, small amplitude and short wavelength propulsive waves on their fins. This chapter is concerned with the fin kinematic parameters of the pufferfish: fin beat frequency, amplitude and wavelength. The results of the cineTilm recordings made of actively swimming fish are presented and comparisons with other fish are made. 120 MATERIALS AND METHODS Fish were placed singly in a 180 litre recirculating flow tank equipped with two submerged mirrors (14 x 76 cm) oriented in a metal framework at 45 degrees above and below the fish to provide top, bottom and side views. The tank was constructed with 6 mm thick clear plexiglas and measured 22 cm wide by 185 cm long by 52 cm tall. A horizontal partition separated the opposing fluid flow except at the ends where the water could recirculate between top and bottom. A stream baffle constructed with tightly packed drinking straws (20 cm long) oriented parallel to the flow was inserted upstream of the fish. ' An electric motor with driveshaft and propeller circulated the water. The flow velocity could be varied between 0 and 0.5 m/s and was controlled by a rheostat connected to the propeller motor. The tank was illuminated with four 600 W tungsten filament lamps and images were recorded on Kodak 7276 Plus-X reversal 16 mm film at 200 frames per second with a high-speed cinecamera (Redlake Lo-cam II, model 51-0003). Data for fin and body motion analysis was obtained from the cinefilms by digitizing successive frames with an image analyser (PCD Model ZAE-3C) and stored on a micro-computer disk for subsequent analysis. 121 Data Analysis Methods The angle swept by the propulsive fin rays (8) is calculated by applying solid analytical geometry to the digitized data taken from the cin6film records. Specifically, the relative positions of the fin ray base and tip are measured in two (x, y) of the three dimensions in which they travel. The span of the fin ray (R) is known so the positions of the fin ray base and tip in the third (z) dimension can be calculated by rearranging the distance formula R2=(x2.xi)2-r(y2-yi)2+(z2-zi)2 to (z2-z1)=[R2-(x2-xi)2-(y2-yi)2]1/2 By defining the fin ray base as the origin {x'=y =z =0) it follows that z=(R2-x2-y2)1/2 and the position of the fin ray tip in three dimensions at any point along the subarc (s) traced by the fin ray tip can be calculated. The- straight-line distance (L) between the two endpoints of the subarc traced by the fin ray tip over the course of a half-cycle is also calculated from the distance formula and the angle swept (6) is obtained through the application of the law of cosines L2= a2+b2-2ab(cos9) and since, in this case, a=b=R cos9= 1-(L2/2R2). 122 The amplitude (A) of the waveform present on the fin is the subarc traced by the fin tip during the course of a half-cycle. Fin oscillation frequency is derived from the number of cineTilm frames elapsed during a complete fm-beat cycle divided by the cin6film framing rate. Propulsive wavelength is obtained from measurements of fin waveforms traced onto transparencies from the digitizer screen. 123 RESULTS Of the three kinematic parameters examined, the mean frequency (f = average cycle time"1 for a given velocity) varied proportionally with specific velocity (lit = velocity/body length) (Fig. 48, Append.I). Values for specific amplitude (As = mean amplitude/fin base length) and specific wavelength (Ks = mean wavelength/fin base length) present a large amount of variability and although As does tend to increase with Ut, neither As nor Xs vary with velocity in a statistically significant fashion (Figs. 49 & 50). The frequency, amplitude and wavelength of the dorsal and anal fins are virtually identical and the frequency for the pectoral fins is matched by those of the dorsal and anal fins (Fig. 48, Append.I). Derived values of kinematic parameters are summarized in Table VI. 124 During steady forward, rectilinear swimming the fish is propelled by undulatory, paired pectoral and median fins while the caudal fin is held in the dorsoventral plane which passes through the median longitudinal axis of the body. The bases of the median dorsal and anal fms lie in this dorsoventral plane and the rays of both fins oscillate symmetrically about their bases in such a fashion as to cause a propulsive wave to travel along the length of the fins in the anteroposterior direction thereby generating forward-directed thrust. The pectoral fin bases are located laterally on the left and right sides of the fish immediately posterior to and at the same median, dorsoventral position as the opercular openings (which coincides with the position of maximum thickness, the shoulder position) and are oriented at a mean angle of incidence of 48° above a horizontal plane passing through the median longitudinal axis of the body, along an axis about which the fin rays symmetrically oscillate to produce a propulsive wave which travels posteriorly and ventrally thereby generating a forward-directed thrust component. A vertically-directed thrust component provides the lift force required to overcome any excess weight over buoyancy. The dorsal and anal fins oscillate with similar frequency, amplitude and wavelength in a synchronous manner which virtually eliminates the tendency to rotate the body about its median longitudinal axis since the fin rays for both fins rotate laterally in opposite directions in planes at an angle of 42° to the median longitudinal axis of the body. Were it not so the fish would tend to rotate about the median longitudinal axis in one direction during the first half-cycle and then back in the 125 opposite direction during the second half-cycle. However, the very same synchronicity which provides roll stability about the median longitudinal axis produces moments of force about the center of mass (located approximately at the shoulder position) which tend to cause lateral yawing movements at the head and tail. The depth of section at the caudal fin and shoulder position helps to resist these yawing motions. The right and left pectoral fins are 180° out of phase with each other so that completion of the dorsoposterior (upward) arc of the fight pectoral fin rays coincides with completion of the anteroventral (downward) arc of the left pectoral fin rays and vice versa. The phase shift between the pectoral fins results in forces which tend to cause rotation about the median longitudinal axis of the body. However, the caudal fin section and the section and action of the median fms resists such rotation. The 180° phase shift between the pectoral fins allows the dorsal fin to be simultaneously in phase with both the left and right pectoral fins such that the direction of rotation of the pectoral fin rays with respect to the median longitudinal axis of the body is always matched by that for the dorsal fin rays and is the opposite of the anal fin. The period for all four propulsive fms is similar so that completion of the upward and downward arcs of the respective right and left pectoral fin rays coincides with completion of the half-cycle rotation of the dorsal (and anal) fin rays from the right side to the left side of the body. During the next half-cycle the dorsal (and anal) fin rays rotate from the left side to the right side of the body while the left and right pectoral fin rays swing upwards (dorsoposteriorly) and 126 downwards (anteroventrally) respectively. 127 TABLE VI. Summary of fin kinematic parameters Pectoral fin Us n ? s A s As X s X* Q s Sd 2.31 12 11.5 2.54 0.75 0.087 1.25 1.50 0.174 3.04 1.37 0.200 0.490 3.39 5 11.9 0.36 0.74 0.040 1.59 1.50 0.082 3.05 1.22 0.076 0.509 2.95 8 12.4 1.85 0.78 0.029 1.67 1.61 0.060 2.96 1.30 0.056 0.541 4.47 7 14.0 0.46 0.83 0.046 1.78 1.54 0.086 3.23 1.40 0.093 0.583 4.65 13 15.1 0.93 0.74 0.068 1.60 1.64 0.150 3.14 1.23 0.131 0.515 3.68 21 14.5 0.85 0.79 0.049 1.71 1.68 0.104 2.96 1.33 0.095 0.554 4.10 31 14.2 0.96 0.83 0.089 1.78 1.71 0.185 3.32 1.40 0.186 0.585 128 TABLE VI. Summary of fin kinematic parameters Dorsal fin Us n J s A s As "k s X s 6 s Sd 2.31 18 11.5 2.19 1.05 0.095 1.50 1.42 0.068 3.51 1.72 0.213 0.838 3.39 6 11.8 0.99 0.92 0.061 2.05 1.37 0.524 3.23 1.51 0.128 0.685 2.95 8 12.7 2.25 0.89 0.044 2.00 1.33 0.066 3.45 1.45 0.089 0.662 4.47 7 13.8 1.06 0.98 0.033 2.19 1.45 0.049 3.31 1.64 0.072 0.744 4.65 12 15.4 0.97 1.00 0.047 2.23 1.41 0.067 3.53 1.67 0.105 0.762 3.68 21 14.4 0.97 1.01 0.023 2.27 1.33 0.030 3.60 1.71 0.051 0.779 4.10 31 14.1 1.14 1.06 0.084 2.37 1.49 0.117 3.67 1.84 0.230 0.836 Us: specific velocity (l/s) / : mean frequency (Hz) A: mean amplitude (cm) At: specific amplitude k: mean wavelength (cm) Xs: specific wavelength 0: half-cycle angle swept by fin Su disc area swept by fin, full-cycle 129 FIGURE 48. Propulsive fin frequency (cycles/s) relative to specific swimming velocity (lengths/s). FIGURE 49. Propulsive fin specific amplitude (mean amplitude/fin base length) relative to specific swimming velocity (lengths/s). FIGURE 50. Propulsive fin specific wavelength (mean wavelength/ fin base length) relative to specific swimming velocity. FIGURE 51. Fin frequency (cycles/s) relative to specific swimming velocity (lengths/s) for pufferfish with triggerfish and mandarin fish estimates. 130 Frequency vs Velocity A pufferfish pectoral fin V pufferfish dorsal fin 3 4 Specific Velocity (l /s) zil GO TD CD O — • < o o —• • CO Specific Amplitude cn 1 • —1 ' • < -! • < • < • < • < A pectoral V dorsal fi —1—_ 1-• < A pectoral V dorsal fi —1—_ 1-• < • < > — • CL CD < CO < o o —mm • r-r-Wavelength vs Velocity 5.0 T • — UJ UJ _ C CD > a o *o 0) CL CO 3.0 2.0 V V . • V A V A V A pufferfish pectoral fin V pufferfish dorsal fin 3 4 Specific Velocity (l/s) Fin Frequency vs Velocity + O Triggerfish, dorsal fin # Triggerfish, anal fin A Mandarin fish, pectoral fin A Pufferfish, pectoral fin Q Pufferfish, dorsa l /ana l fin 1 H 2 3 4 Specific Velocity (lengths/s) DISCUSSION Pufferfish Fin Kinematics The results of the fin kinematics analysis place the pufferfish somewhere in the intermediate zone of the broad range encompassed the by the functional classification of fish into Group I and Group II forms (ie. pufferfish /*5-15 Hz, seahorse /=40 Hz, electric fish /-2.5 Hz) in the company of other undulatory MPF swimmers such as triggerfishes, mandarin fish and angelfish (f«13'Hz, 8 Hz, 5 Hz respectively). The significance of the coordination of the cycle time and rotational direction of the dorsal and anal fin ray motion with that of the pectoral fins is open for speculation. It is clear that individual fins are capable of operating independently with respect to frequency, amplitude, wavelength, direction of the propulsive wave and direction of the thrust output relative to the fin base axis. As an example, for a complex manouevre wherein the fish at once descends while swimming in reverse and simultaneously turning to the right, the body and caudal fin are arched to the left, the dorsal fin reverses the direction of the propulsive wave as does the anal fin, although at a lower amplitude and frequency than the dorsal fin. The right pectoral fin also reverses the propulsive wave direction (with a frequency and amplitude greater than that of the dorsal fin). The left pectoral fin, on the inside of the turn, acts to keep the longitudinal and lateral axes of the body in 4 an orientation close to horizontal by reversing the direction of the propulsive wave and changing the effective direction of 135 the thrust from antero-ventral to ventral. During steady forward, rectilinear swimming the dorsal fin is out of the immediate downstream wash generated alternately by the left and right pectoral fins during their dorso-posterior half-cycle' movements. This coordination avoids a turbulent stream of fluid, flowing in a direction antagonistic to that of the dorsal fin motion, which would presumably increase fin friction and interference drag. With respect to stability, while contra-rotating pectoral fins would eliminate any tendency to roll about the median longitudinal axis of the body, such a system would introduce a displacement-causing moment in the vertical plane against which the pufferfish has no defence apart from the maximum girth of the body. The energy required to sinusoidally accelerate the body in the dorso-ventral plane would be much greater than that required to slightly roll the body about its median longitudinal axis. Coordination of multiple appendages during locomotion suggests the existence of a central nervous system motor-control program. The coordination of fin kinematics and correlation with swimming velocity is not unlike a gait pattern as seen in many terrestrial organisms. Although the master control of the overall locomotory process is determined by the brain, there is ample evidence from a wide variety of organisms which indicates that the neuronal organization of the spinal cord is responsible for the basic patterns of locomotory movements (Roberts 1981). 136 Comparisons with other species Of the relatively few studies which have examined the kinematics of propulsive systems for fish which undulate or oscillate median and or paired fins, the fin oscillation frequency varies positively with velocity of swimming. Values for specific amplitude, As, and specific wavelength, X*, tend to show a considerable degree of non-linear variability. Fin frequency is related to specific velocity for the pufferfish. and two other undulatory MPF swimmers; the mandarin fish, Synchropus picturatus, and the triggerfish, Rhinecanthus aculeatus in Figure 51. The mandarin fish swims by passing undulatory waves down its paired pectoral fins, in a mode termed labriform by Breder (1926). Triggerfishes swim by passing undulatory waves down their paired dorsal and anal fins, in the balistiform mode of locomotion (Breder, 1926). For the mandarin fish, Blake (1979c) found that pectoral fm frequency ranges linearly from about 4 to 8 Hz over a U* range of approximately 1.5 to 3.0 1/s. For the triggerfish, / increases linearly from about 3.5 Hz to 13 Hz for the dorsal fin and from about 3 Hz to 8 Hz for the anal fin, over a range of Ut<*\.0-3.0 (Blake, 1978). Half-wavelength, amplitude and wave velocity were found to vary during slow forward progression (U**\) but the total number of propulsive wavelengths (ca.l) present on the fin at any given time at any given velocity remains relatively constant. For the pectoral and dorsal/anal fins of the pufferfish, frequency increases linearly with Ut from about 10 Hz (C/i*2.25) to 15 Hz (£/i*4.75). Although At tends to increase with 137 Us, the regression coefficient is not significantly different from zero. Likewise, the curve for Xs remains flat over the Us range. The range of Us and / values during steady forward, rectilinear swimming for the pufferfish, mandarin fish and triggerfish are comparable, with the values of / for the pufferfish being the highest, followed by the triggerfish dorsal fin, triggerfish anal fin and mandarin fish pectoral fins (Fig. 51). Employing the functional categorization of undulatory fin swimmers suggested by Blake (1979c), all three of these fish fall between the Group One and Group Two forms, with intermediate values for frequency. Blake (1980b) compared the dorsal fin kinematics of the electric fish, Gymnarchus niloticus, and the seahorse, Hippocampus hudsonius, two undulatory fin swimmers with markedly different fin kinematics and mode of life. After Breder (1926), both the electric fish and the seahorse are classified as members of the same swimming mode, the amiiform mode, due to the use of a dorsal fin as the main propulsive unit. Morphologically and functionally, this is clearly absurd and is likely what led Blake (1979c) to propose a more function-orientated classification system. The short based dorsal fin of the seahorse is less than one sixth (=15%) of the total body length while the long, ribbon-like dorsal fin of the electric fish extends over more than half (=55%) of the total body length, over 3.5 times the ratio of -fin to body length of the seahorse. The extremely high fin-beat frequency of the seahorse (41 Hz at £/i=1.21) contrasts sharply with that of the electric fish (2.4 Hz at f/«=.57). The 138 seahorse undoubtedly represents the upper margin of Group Two forms while the electric fish likely represents the lower end (Group One forms) of the continuum which encompasses the more intermediate kinematic forms such as the pufferfish, mandarin fish and triggerfish. Although the / values for the pufferfish are between those of the seahorse and electric fish, the values for As and Xs are far greater for the pufferfish. Values of As for the pufferfish (pectoral As=1.5, dorsal As*2.0) contrast with those of the seahorse (dorsal As*.3) and the electric fish (dorsal As-.08). Xs for the pufferfish (pectoral \**3, dorsal Xs*3.5) are considerably larger than for the seahorse (dorsal Xs*.29) and the electric fish (dorsal ta«.51). The reason for the high values for the pufferfish is suggested by the fin morphology. The propulsive fins of the pufferfish are triangular in shape with the rays decreasing in size from the anterior, leading edge to the posterior, trailing edge. The fin base length is small relative to the length of the anterior rays which is related to the amplitude of the fin oscillation, since for a given amount of rotation, a longer fin ray results in a larger amplitude. In contrast, the dorsal fin base length of the electric fish is long relative to the fin ray length and the ratio for the seahorse is in between the pufferfish and the electric fish. For the purposes of Actuator-Disc theory (discussed in the following chapter), the relevant fin kinematic parameter is the area swept by the fin rays, which is related to the fin ray length and the amplitude of oscillation. The pufferfish disc areas are in the order of 1.0 cm2 and 1.3 cm2 for the pectoral 139 and dorsal fins, respectively. Values for the seahorse and electric fish arc .5 cm2 and 1.24 cm2 respectively. An indication of the rate at which the disc area is swept by the fin rays during a complete fin-beat cycle can be seen in the product of the oscillation frequency and the disc area. On this basis, the seahorse (<*20 cm2/s) is followed by the pufferfish (*15 cm2/s) and the electric fish («3 cm2/s). It is anticipated that the reverse may be apparent with respect to values of Froude efficiency since it is energetically less costly to accelerate a large mass of fluid to a low eventual velocity than it is to accelerate a small mass of fluid to a high eventual velocity. Webb (1973b) has studied the kinematics of lift-based oscillatory fin propulsion in the shiner sea perch, Cymatogaster aggregata, another of the so-called labriform swimmers (Breder, 1926). The propulsory mechanism of the sea perch differs from that of the undulatory fin swimmers mentioned previously in that the pectoral fins act in a fashion analogous to the flapping of wings in bird flight. Rather than passing an undulatory wave down the fin, the fin acts much like a rigid lift-producing span that is oscillated across the incident stream flow. Two patterns of pectoral fin motion were found which differed in the length of the propulsive wave present on the fm. In fin pattern A, the wavelength is shorter, approximately twice the trailing edge length, resulting in a phase difference of about 7t between the movements of the anterior and posterior fin rays. In fin pattern B, the wavelength is much longer and results in a phase difference of about 7t/5 between anterior and posterior fin rays. Fin pattern A persists up to swimming speeds of approximately 2 140 l/s while fin pattern B predominates at t/s>2.8. Webb also found that at Ui>\.5, frequency and amplitude increase with Us and that the product of frequency and amplitude is linearly related to Us. Over a similar range of swimming speeds, pufferfish frequencies are in the order of 5.5 times (£/«*2.5) and 3 times (C/s-4.5) higher than the sea perch frequencies. The mechanics of swimming for three representatives of the ostraciiform swimming mode have been examined. Blake (1977) studied two species of boxfish, Lactoria cornuata and Tetrasoma gibbosus, which propel themselves by oscillating five propulsive fins; dorsal, anal, caudal, and two pectoral fins.Blake found that fin-beat frequency is higher at greater swimming velocities (2.7, 3.8, 4.5, 3.3 Hz for the pectoral, dorsal, anal and caudal fms, respectively, at £/s=l). He also found that as much as three-quarters of a wavelength could be found on all the propulsive fins except the caudal fin which oscillates from side to side in a sculling fashion without producing an undulatory wave. These / values are relatively low compared to the pufferfish and the other fish examined except for the electric fish (2.4 Hz at Us*.6). The other representative of the ostraciiform mode is a species of trunkfish, Ostracion lentiginosum, for which / is proportional to Us over a wide range of / from 5.5 Hz (£/s=l.l) to 16.6 Hz (£/s=4.7) (Blake 1981b). The propulsive fin kinematics for this trunkfish can, with a twist of imagination, be considered the ultimate sub-carangiform (sic) swimmer since the body is rigidly held and the propulsion comes exclusively from the sculling motion of the caudal fin, which is analogous to an 141 oscillating rudder on a boat. The / and Us values for the pufferfish are similar to those for the trunkfish (U%>2). Most fish locomotion studies have examined fish species which produce thrust by oscillating various portions of the body and caudal fin; the anguilliform, carahgiform and sub-carangiform modes. Bainbridge (1958) found for carangiform and sub-carangiform swimmers such as dace (Leuciscus leuciscus), goldfish (Carassius auratus), and trout (Salmo gairdneri), that tail-beat frequency is related to swimming velocity (at fe5 Hz), As increases with / up to <*5 Hz whereafter it remains relatively constant and the product of / and amplitude is linearly related to Us. Webb (1971a) found similar results for rainbow trout. The values of the pufferfish frequencies are in the order of double those found in the above mentioned studies over similar Us ranges. For a cod, Gadus morhua, another carangiform swimmer, Videler (1981) found that frequency (range 2.2-5 Hz) and maximum A% (range .08-. 10) are related to swimming velocity (Us range 1.3-2.8) and that the propulsive wavelength of the body is relatively constant (Videler & Wardle, 1978). It is interesting to note the results found by Fish (1984) for the muskrat, Ondatra zibethicus. In this species thrust is generated during surface swimming by alternate oscillations of the hind feet. It was found that the arc through which the hind feet travel (ie. amplitude) increases with s w i m m i n g velocity while the oscillation frequency remains constant at 2.5 Hz over a swimming velocity range of 0.2 to 0.75 m/s, not unlike the results found for some carangiform and sub-carangiform swimmers 142 over the lower regions of their Us values. Perhaps there is a unifying principle relating fin oscillation / , A, X and Us to optimum contraction velocity for propulsive musculature. The common factor which appears to be consistent throughout all of these studies is that frequency is proportional to swimming velocity during steady forward swirriming. At the lower end of the swimming velocity range, f/«<1.0-1.5, / varies with Us in a non-linear fashion. For the most of the undulatory and oscillatory MPF swimmers (ie. pufferfish, mandarin fish, triggerfish, seahorse, electric fish, trunkfish, boxfish) that have been studied, there seems to be a lack of detailed information concerning As and X* relative to Us and the data available appear to have a fair amount of variability in As and Xs not explained by variation in Us. For the BCF swimmers (ie. trout, cod, goldfish, dace) that have been studied, As appears related to Us over the lower Us regions, whereafter it remains relatively constant and the product of / and As is proportional to Us. 143 CHAPTER FOUR: POWER AND EFFICIENCY INTRODUCTION Different methods for estimation propulsive power output can be divided into two groups: drag-based and thrust-based. The former group can be further sub-divided into two groups: measured drag and theoretical drag. As mentioned in chapter two, drag values are measured from terminal velocity, deceleration-in-glide, towing tank and water and wind tunnel experiments. Theoretical drag is calculated from equations combining empirical observations of technical bodies of revolution with hydromechanical theory. Both measured drag and theoretical drag assume that 1) flow is mechanically similar to that for a technical body of revolution, 2) flow is steady, 3) the attached BL is laminar and 4) the primary source of drag is friction drag. The thrust-based approach to estimating propulsive power output can also be subdivided into two categories: direct and indirect measurements. Direct measurements involve connecting a specimen to a force measuring balance (Houssay, 1912; Gero, 1952; Lang & Daybell; 1963; Gray, 1968) and measuring the rate of displacement and force exerted during swimming. This approach is inaccurate since only the power in excess of the drag power of the fish and the power associated with the force measuring balance operation (ie. 144 inertia, friction and line drag) can be measured. There are two basic methods for indirect estimation of propulsive power output: hydrodynamic models, which result from the combination of the kinematic parameters of locomotion (ie. frequency, amplitude and wavelength) with hydromechanical theory; and metabolic power estimates which result from the application of oxycalorific conversion factors to oxygen consumption data that is measured with respect to swimming velocity. Propulsive power is calculated as the difference between total metabolic power and standard (resting) metabolic power. The variables involved in estimation metabolic power with respect to swimming velocity are manifold and their relations are complex. (For reviews see Webb 1975a, 1978) . Hydrodynamic models can be divided into two general categories: resistive and reactive. Both the resistive and reactive models divide the body into a series of segments and consider the motion of the segments as simple harmonic motion, and that the BL is laminar. Resistive theory considers 1) the fluid in which the segments operate to be viscid, 2) the resistive force acting on any given segment at any given time to be the same as the steady-state force, 3) the steady-state force to be dependent only upon the instantaneous velocity and the angle of of attack of a segment with respect to the fluid, and 4) that the total resistive force acting on the segment is the sum of all the steady-state, resistive forces acting on the segment during the cycle (von Holste & Kuchemann, 1942; Parry, 1949; Gero, 1952; 145 Taylor, 1952; Gray, 1953b). Reactive theory, in contrast to resistive theory, considers the inertial forces acting on a segment to be proportional to the rate of change in resultant velocity of a mass of water (virtual mass) affected by the body (Lighthill, 1960, 1969, 1970, 1971; Wu, 1961, 1971a,bc,d; Newman &Wu, 1973, 1974). Reactive theory has been applied mostly to fish which swim in the anguilliform, carangiform and sub-carangiform modes, however, Blake (1983b) has modified the theory to apply it to undulatory fin swirnming of electric eels and knifefish. A special application of the momentum principle which considers the rate of change in momentum of a mass of fluid passing through an idealized disc is called the Actuator-Disc model and it has been applied to the analysis of MPF swirriming in the seahorse, electric fish (Blake, 1980b) and in the mandarin fish (Blake, 1979c,d). For all the approaches listed in the preceding paragraphs, an estimate of propulsive efficiency can be calculated by dividing the minimum power output necessary to overcome drag, either by measured or calculated, by the total propulsive power output estimate generated by the particular approach chosen. In this chapter, Actuator-Disc theory is applied to the morphometric, hydrodynamic and kinematic parameters from the previous chapters to estimate propulsive power and efficiency during steady forward, rectilinear swimming. Estimates obtained from the model are compared with those obtained from hydromechanical theory. Power and efficiency estimates from a 146 range of other fish locomotion studies are compared to the results for the pufferfish. The strengths and weaknesses of the model are discussed and comparison is made estimates made from other models. 147 METHODS Actuator-Disc Theory The Actuator-Disc model is based upon the Rankine-Froude Momentum Theory which considers the fin to be an idealized device, roughly analogous to a ship propeller or a helicopter rotor, which produces a sudden pressure rise in a stream of fluid as it passes through the propulsive disc area. Application of the model requires the following assumptions to be made. 1) The pressure change and thrust loading are constant across the disc. 2) There are no rotational velocity energy losses in the wake. 3) The velocity profile across the disc is uniform. 4) A definite boundary separates the flow passing through the disc from the flow outside it. 5) The static pressure in and out of the wake is equal to the free-stream static pressure both in front of and behind the disc. • Applied here, the disc is actually a sector area which is prescribed by the fin ray base and the end points of the subarc traced by the fin ray tip. There is probably some departure from a uniform velocity profile across the disc since, owing to the sinusoidal nature of the propulsive fin oscillations, the fin ray 148 must, at the completion of every half-cycle, decelerate, reverse direction, and then accelerate to reach peak velocity, roughly midway through each half-cycle. For the very same reasons it is likely that there is created some degree of rotational velocity in the wake. The magnitude of the energy requirements attributable to a variable velocity profile across the disc and to rotational velocities in the wake are unknown (Blake, 1983a). However, the results of flow visualization (Blake, 1976,c,d) and anemometry (Blake, unpublished) experiments indicate that the flow patterns observed about the fins of S. picturatus and H. hudsonius, two undulatory fin swimmers, in general concur with those predicted or required for the Actuator-Disc model. The model does not account for two other important potential sources of energy loss: the energy requirements associated with overcoming the frictional drag of the fins and with the fin tip effects. Values for the fin friction power have been estimated to be as low as 5% of the induced power required to hover for 5. picturatus (Blake, 1979d). Values for energy losses due to tip effects are unknown for fish although estimates of 15% have been made for. helicopter rotors and for propellers (Bramwell, 1976) to which the Actuator-Disc model applies, but the difference in kinematics between a fin system and such man-made systems suggests caution when applying this estimate to fin systems. The effect of our uncertainty over the various unaccounted-for energy losses will be that the power estimates will be underestimates 149 and as such may prove useful as lower bounds of the actual power requirements (Blake, 1983d). The advantages of a model which does not require detailed fin-fluid kinematics must be balanced against the disadvantages of uncertainty regarding underestimation of total thrust power output. As long as any energy losses associated with the fin system, ie. rotational, non uniform velocity, fin profile and tip effects are not significant relative to the kinetic energy injected into the wake, then the power estimates provided by the model may serve as a useful comparison with other estimates of the power requirements of the fish swirriming by means of undulatory fin systems for which the Actuator-Disc model is appropriate (Ellington, 1978; Blake, 1983d). In applying the model, the rate of change in momentum imparted to the fluid, integrated over the disc area, is related to the thrust required to overcome the drag of the body. Four discrete regions of stream flow are defined for the purpose of this model: far upstream, immediately upstream of the fm, immediately downstream of the fin, and far downstream. Recalling Bernoulli's theorem o.spU2+ pgh + p= constant and assuming the static head pressure term (pgh) is constant, far upstream -the relative velocity of the fluid is U and the pressure is p o + o.spU2. Immediately adjacent to the upstream side of the 150 disc (just before the fin) the velocity is considered to have increased to U+v and the pressure to have decreased to p + o.5p(f/+vi)2. Immediately adjacent to the downstream side of the disc (just after the fin) the velocity is assumed to be unchanged as the pressure is considered to have, by the action of the fin, increased by Ap to p + Ap + ospiU+vJ . Far downstream of the fin the velocity is assumed to have increased to U+v and the pressure to have returned to po. Equating the two upstream conditions p 0 + o.5pf/2= p + osp(U+v)2 and the two downstream conditions p + Ap + o.sp(U+v)2= p 0 + o.5p(C/+v2)2 and subtracting the upstream from the downstream we have Ap= p(U + osv2)v2 The thrust generated by the fin is calculated from the product of the change in pressure (Ap) and the area of the disc (Sd= QR , where 6=angle swept by fin rays and R=fm ray length) as T= SdAp= Sdp(£7 + o.sv2)v2 The rate of mass flow through the area of the disc for one fin is pS (U + 'v ) therefore the rate of change of momentum of the fluid 151 stream passing through S D is p$d(U + v^ v which is equal to the thrust so pSd(L/ + Vl)v2= pSd(L/ + o.sv2)v2 and 2v = v. 1 2 In order to circumvent the formidable difficulties encountered in obtaining accurate estimates of v and v2, Blake (1980b) devised a method to calculate the values instead, providing that the drag of the body under investigation is known. By defining a nondimensional inflow factor, E=vJU, and recalling 2v = v2, thrust becomes Thrust and drag are equal in magnitude during steady forward T= 2pS U2(E + E2) 2 swimming, D- T= o.spU S , so that CD= 2pS df/ 2(E + E2) and - 4£ 2 + 4H - (S C D / S > 0 w a o. spU2S w -4 ± [16 + 16(S CD/S J ] w d 8 0.5 SO = [(l + sWCD/sdrJ-1]/2 By defining the slipstream velocity (v) as v = (U + v> U(\ + 2S)= £7(1 + S CD/S V s 2 w d ,0.5 we find v = (v - LO and v= o.5(v- LO-The useful power output (Po) is given by 152 Po= TU= 2pSdL>3E(l + E) while the power input (Pi) is given by 7 3 - / , . -\2 Pi= T(U + Vj)= 2pSd(7 JE(l + E) so the ideal efficiency is found from The ideal efficiency is an estimate of the propulsive efficiency of the system to which it is applied. 153 R E S U L T S The estimates of propulsive power output (/W)) range from a low of ==5.7xl0~5 Watts at specific velocity (Us) of "2.3 to a high of *2.4xl 0"4 Watts at I/i«4.7 (Fig. 52, Table VD). Estimates of propulsive power input (/'(in)) for the pectoral fm range from *7.2xl0"5 Watts at £A*2.3 to 2.8x10^ Watts at U**4J, and for the dorsal fm from 6.5xl0' 5 Watts to 2.7X10" 4 Watts over the same Us' range. These values are in the order of 1.2, 1.5 and 1.4 (P(OM), pectoral P(in) and dorsal P(in) respectively) times greater than the theoretical minimum for total power (Pcum)) of a rigid body equivalent in laminar flow conditions. Similar comparisons with P(XUT) yield values of 2, 2.5 and 2.3 for /'(out), pectoral P(in) and dorsal /'(in), respectively. In all cases the / ' ( in) of the . pectoral fin is greater than that for the dorsal fin. The reverse is true for the efficiency (T|) values where dorsal T) ranges from .87 to .89 while the pectoral r\ ranges from .79 to .85 (Table VII). Compared with power outputs for some other fish, the puffer P(om) is an order of magnitude lower that the nearest curve at comparable Us (Fig. 53). The efficiency values increase slightly over the Ut range of 2.3-4.7. for both fins (Fig. 54). Efficiency estimates for the puffer rank at the top of the range of values achieved by some of the other undulatory M P F swimmers that have been studied (Fig. 55). 154 TABLE VII. Power and Efficiency estimates l/s Foul Fin Fin Finn Fmr pectoral • fin dorsal fin 2.31 0.57 0.72 0.79 0.67 0.87 0.49 0.29 2.95 0.79 0.95 0.83 0.92 0.85 0.70 0.41 3.39 1.10 1.32 0.83 1.27 0.87 1.01 0.61 3.68 1.35 1.59 0.85 1.52 0.88 1.26 0.77 4.10 1.75 2.03 0.86 1.95 0.89 1.68 1.04 4.47 2.17 2.51 0.87 2.45 0.89 2.13 1.33 4.65 2.39 2.79 0.85 2.67 0.89 2.35 1.48 Uf. specific velocity (l/s) Fout: power output (x 10"4 Watts) Fin: power input (x 10"4 Watts) T): ideal efficiency Fi«m: rriinimum power output (x 10"4 Watts), rigid body equivalent, laminar flow Par. minimum power output (x 10"4 Watts), rigid body equivalent, turbulent flow 155 DISCUSSION Estimates of P ( o m ) , P ( i n ) and rj are calculated by supplying the Actuator-Disc model with the relevant morphometric and kinematic parameters; swimming velocity, fin span, angle swept and drag coefficient. Theoretical drag coefficients are calculated based on the pufferfish mean FR (3.37) and combined with the wetted surface area and forward swimming velocity of the pufferfish in the standard hydromechanical drag equation to calculate theoretical minimum power curves for laminar P ( iam) and turbulent P(wr) flow conditions. The departure of the pufferfish F(out) estimates from the theoretical minima (Fig. 52) is a common occurrence among a variety of fish species and therefore unsurprising. In fact, the maximum departure of <*1.2 times (lam) and «*2 times (tur) the theoretical minima is low when compared with factors of 3 to 5 times (Webb, 1975a) the theoretical minima for some carangiform and sub-carangiform swimmers (/?e*104-106) or even up to 10 times (Yates, 1983) the theoretical minima. Richardson (1936) found ratios of =2.5 for mackerel and herring from terminal velocity measurements (/te=2xl04). The P c e x p y P d h e ) ratios obtained for pufferfish from the Actuator-Disc model are similar to those found in the CD(exP)/CD(ihe) ratios calculated in chapter II. Although the CD(the) values for a flat plate equivalent are not, in and of themselves terrifically meaningful, they do provide a 156 convenient method to compare the relative magnitudes of kinematic parameters or morphological factors which can elevate the CD(exp) and thus, the drag and P(out) estimates above their Peine) minima. Of the eight curves in Figure 55, only the rainbow trout is a BCF swimmer, added to provide contrast with the remaining 7 fish which swim by means of oscillatory or undulatory motions on the propulsive fins.. Swimming in the sub-carangiform mode where the posterior 1/3 to 1/2 of the body (and caudal fin) oscillated laterally across the median longitudinal axis, the trout encounters a substantial increase in swimming drag from the increased friction drag which results from the thinning of the BL, since the lateral velocity of the propulsive segment is greater than the free stream velocity, and from the increased pressure drag which results from the separation of the BL over the posterior portion of the body. Of the seven other fish in Figure 53, four are analyzed with the Actuator-Disc model: the seahorse and electric fish (Blake, 1980b), the mandarin fish (Blake, 1979c) and the pufferfish. Two fish are analyzed with elongated body theory: the trunkfish (Blake, 1981b) and the knifefish (Blake, 1983b). The sea perch (Webb, 1975b) is analyzed by calculating metabolic power output from oxygen consumption and swimming velocity. The eight curves in Figure 53 are bounded (at £/s*3) by rainbow trout with a P(oui) value of «8xl0"2 Watts at the upper end and at the lower end by the pufferfish P(om) value of *6xl0"5 Watts, spanning three orders of magnitude. Webb (1971b) 157 found good agreement between P(om) estimates made by conversion of oxygen consumption values and by applying elongated body theory (Lighthill, 1969) to the propulsive kinematics of the rainbow trout. In the present study, there is no alternative, independent estimate of F(om) with which to compare the Actuator-Disc model estimates, other than the theoretical minimum for a flat plate of equivalent area. Such an estimate is highly desirable, as will be discussed in a following segment. Webb (1975b) estimates the propulsive efficiency for sea perch at between .6-.65, which he considers poor in relation to r) estimates for trout (=.7) and salmon (-.9). T| estimates for p p steady swimming (=.8-. 9) and for burst and glide swimming (=.55-.6) are calculated for Zebra Danio by McCutcheon (1977). Surface swimming in muskrat (Fish 1984) is estimated at rj =.27-.33. Blake (1980a) estimates T J at .15-.3 for angelfish p p pectoral fin rowing. Sea lion fore flipper Ti is estimated at «.8 at £/s=2. There appears to be a wide range of propulsive efficiency estimates for aquatic propulsive system. It also appears that to a large extent, depending upon the model or method employed and the assumptions that are made, a range of r|p estimates can be calculated. Based on the results of the Actuator-Disc model applied to pufferfish swimming, the propulsive efficiency of the pufferfish is fairly high (=.86) and increases slightly over the velocity range measured (t/s=2.3-4.7). During the course of this study it became apparent that the 158 simplicity and ease with which the modified Actuator-Disc model (Blake, 1980b) could be applied to undulatory M P F propulsory systems is mitigated by the lack of an independent estimate of thrust power requirements and as such, the model does not provide for the construction of a force balance, supplied on either side by independent estimates of thrust and drag. The. Actuator-Disc model, prior to this study, has never been applied to multiple fin propulsive systems of aquatic organisms. Only single fin propulsive systems have been analysed previously (Blake, 1979c, 1980b). The Actuator-Disc model is an idealized device based on the principal of momentum which considers and instantaneous pressure rise, integrated over the disc area, to be created by an actuating element, (ie. a fin) in the steam of fluid passing through the disc. The thrust force is proportional to the rate of change of momentum and the minimum power required to create the induced velocity is the product of the thrust force and the free stream velocity. Based on these principles, Blake (1980b) offers a method to calculate the induced velocity of the fluid through the disc by equating thrust with drag during steady forward swimming, provided that the disc area swept by the actuating element and the wetted surface area and drag coefficient of the body are known. Two problems arise however, for a multiple fin system. One stems from the fact that the inflow factor (the ratio of the fluid velocity through the disc to the free steam velocity) is 159 dependent upon the ratio of total wetted surface area to the disc area, and as it turns out, there seems to be no way to partition the thrust or drag between multiple propulsive units. The other problem is that the power estimates are intrinsically dependent upon the whole body surface area and drag coefficient, no independent thrust balance can be obtained. It follows from this that a power balance (drag power = thrust power)cannot be computed from the model. The / '(out) estimates • are with respect to the power required to overcome whole body drag and there is no problem here. However, the model is such .that the P(in) estimates, for example with respect to the dorsal fin, treat the system as if the dorsal fin is the only propulsive unit in operation. Therefore the P(in) estimates are blind to the power inputs from the other propulsive fins. Nevertheless, it is useful to compare the estimates produced by the model to those for other fish using the same model and for estimates obtained through alternate methods of analysis. The points to be taken from this chapter include 1 ) / '(out) estimates are * 1 . 2 - 2 . 5 times greater than Poam) and are 1 -2 orders of magnitude less than those found for other MPF swimmers at similar velocities ( / ' ( o u t ) * 1 0 ' 4 Watts, £/i-2.3-4.7). 2 ) Ideal efficiency values increase slightly over the velocity range to a maximum value of <*.89 which is similar to the values obtained for efficient BCF swimmers and not unlike the efficiency values for well designed screw-type propellers. 3) It is apparent that the ease with which the modified form 1 6 0 of the Actuator-Disc model can be applied is mitigated by the difficulties discovered during the course of this study. Specifically, /'(out) and P(in) estimates appear to be indivisible between multiple propulsive units and calculation of thrust and power balances are not afforded by the model as it now stands. 161 FIGURE 52. Power output relative to specific swimming velocity compared with theoretical estimates FIGURE 53. Power output relative to specific swimming velocity compared to other MPF swimmers FIGURE 54. Propuslive efficiency relative to specific swimming velocity FIGURE 55. Propulsive efficiency relative to specific swiniming velocity compared to other MPF swimmers 162 Power vs Velocity 3 E - 4 T 0-1 • 1 ' 1 1 1 2 3 4 5 Specific Velocity (l/s) Power Output (Watts) m I CD -rj N) CD o. — h — • • o < o o m o o 4 d . c ^ CD n -CD 0) zr Q O 5 o %J o c o o o :> CD < CO < o o £91 Ideal Efficiency p o o p o ro V CD N> H 1 1 1 1 1 1 H-CD n — h < CD_ O o C/5 > + o CO o a. TJ o CD Q O t U O CM c m c m c m am o m m CD* 13 O < GO < o o ON 1.0 Efficiency vs Velocity 0.8 + 0.6 c o LU 0.4 + 0.2 0.0 n • • _ r 3 l • O o o • o i i I I i g CP o o o A o A A A H h O mandarin fish # electric fish A seahorse O trunk fish • knife fish • pufferfish pectoral V pufferfish dorsal H i\ • 1 , 1 . 0 2 3 4 Specific Velocity (l/s) SUMMARY 1) The morphometric parameters relevant to hydromechanical analysis are well defined and regressions relating these parameters to body length are provided in the Appendix. 2) The basic body shape of the pufferfish remains constant as the fish grows in size.FR (3.37) and SP (.43) are independent of body length. 3) the shape of the pufferfish body is a good analogue for an axes-symmetric technical body of revolution. Specimens average 85% and 91% of the respective volume and surface area values for a prolate spheroid of equal semi-axes. 4) It is expected that during routine forward swimming, the BL is laminar and attached. 5) Terminal velocity drag estimates are valid and provide useful minimum estimates for the drag on an actively swimming fish. 6) The majority of experimentally determined CD values fall between .02 and .03 (7?e*5xl03-2xl04) and range from *1.7-2.6 times greater than CD(the> values. 7) Analysis of fin morphometry suggests thrust compensation is accomplished kinematically. 8) The common factor apparent from comparisons of propulsive fm kinematic parameters is that fin oscillation frequency is proportional to swimming velocity (except over lower velocity ranges t/s<l-1.5 for some species). For MPF swimmers, a 167 considerable amount of variability in As and is not explained by variation in swimming velocity. For BCF swimmers, the product of frequency and amplitude is linearly related to swimming velocity. 9) P(ou\) estimates are « 1 . 2 - 2 . 5 times greater than P(ihe)and are =^ 1-2 orders of magnitude less than those found for other MPF swimmers at similar velocities. (PcouoadO"4 Watts, I/.-2.3-4.7). 10) Propulsive efficiency estimates increase slightly with velocity to a maximum of *.89 which is similar to values obtained for efficient BCF swimmers and not unlike values for a well designed screw propeller. 11) The utility of the modified Actuator-Disc model is mitigated by the weaknesses discovered during the course of this study with respect to analysis of pufferfish locomotion. Specifically, The P(pat) and P(in) -estimates are not divisible among multiple propulsive units and calculation of thrust and power balances are not possible with the model as it now stands. 168 LITERATURE CITED Alexander, R. McN. (1977) Functional design in fishes. London: Hutchinson. Aleyev, Yu. G. 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(1983) Hydromechanics of body and caudal fin propulsion. In Fish biomechanics, eds P. W. Webb & D. Weihs. New York: Praeger. Zar, J.H. (1984) Biostatistical analysis. 2nd ed., New Jersey: Prentice-Hall Inc. 175 APPENDIX I. Summary of regressions Relation a Sy.x b Sb n RA2 Ho .05 1 Dep/1 -0.208 0.117 0.424 0.027 20 0.93 B=0 rej 2 Wid/1 ' -0.035 0.107 0.379 0.024 20 0.93 B=0 rej 3 dep/wid -0.109 0.111 1.084 0.064 20 0.94 B=0 NSD 4 Xd/length 0.276 0.212 0.367 0.048 20 0.76 B=0 rej 5 Xw/length -0.095 0.220 0.398 0.050 20 0.78 B=0 rej 6 Xd/Xw 0.454 0.158 0.869 0.080 20 0.87 B= 1 NSI 7 Pdep/1 -0.392 0.074 0.361 0.017 20 0.96 B=0 rej 8 Pwid/1 -0.287 0.071 0.261 0.016 20 0.94 B=0 rej 9 Xp/length 0.264 0.125 0.688 0.028 20 0.97 B=0 rej 10 Pdep/Pwid 0.072 0.108 1.308 0.091 20 0.92 B=l rej 11 FR(dep)/l 3.814 0.248 • -0.080 0.048 20 0.14 B=0 NSD 12 SP(dep)/l 0.480 0.043 -0.011 0.010 20 0.07 B=0 NSD 13 Sw/length *-0.102 0.035 2.117 0.089 18 0.97 B=2 NSD 14 SwT/1 *-0.029 0.045 2.120 0.114 18 0.96 B=2 NSD 15 SwB/1 *-0.119 0.034 2.087 0.082 19 0.97 B=2 NSD 16 SwF/1 *-0.719 0.106 2.166 0.271 18 0.80 B=2 NSD 17 SwPect/1 *-1.493 0.121 2.248 0.308 18 0.77 B=2 NSD 18 SwDors/1 •-1.376 0.139 2.008 0.355 18 0.67 B=2 NSD 19 SwAnal/1 *-1.570 0.133 2.221 0.338 18 0.73 B=2 NSD 20 SwCaud/1 *-1.236 0.085 2.132 0.216 18 0.86 B=2 NSD 21 ProjA/1 *-1.099 0.046 2.211 0.111 20 0.96 B=2 NSD 22 SwB/Vol * 0.751 0.024 0.671 0.023 14 0.99 B=.67 NSD 176 APPENDIX I. continued Relation a Sy.x b Sb n RA2 Ho .05 23 Volume/1 *-1.353 0.034 3.206 0.102 14 0.99 B=3 NSD 24 Wt(aix)/1 M.251 0.036 3.124 0.109 14 0.99 B=3 NSD 25 AFRPect/1 -0.266 0.070 0.182 0.017 19 0.87 B=0 rej 26 AFRDors/1 -0.358 0.117 0.216 0.029 19 0.77 B=0 rej 27 AFRAnal/1 -0.374 0.090 0.210 0.022 19 0.84 B=0 rej 28 alpha/1 0.902 i 0.052 • -0.012 0.010 21 0.08 B=0 NSD 29 PFB/1 -0.029 0.080 0.130 0.016 20 0.79 B=0 rej 30 DFB/J -0.113 0.075 0.147 0.015 20 0.85 B=0 rej 31 AFB/1 -0.120 0.085 0.136 0.017 20 0.79 B=0 rej 32 Nare: h/x 0.196 0.281 0.321 0.038 14 0.86 B=0 rej 33 Nare:h/1 -0.530 1 0.288 0.498 0.082 12 0.79 B=0 rej 34 Nare:x/1 -1.574 l 0.769 1.398 0.218 12 0.80 B=0 rej 35 FV Fit 23 *-1.520 0.011 1.478 0.017 18 0.99 B=0 rej 36 FV Fit 16 *-1.485 0.019 1.476 0.035 10 0.99 B=0 rej 37 FV Tot 23 *-0.979 0.003 1.798 0.011 10 0.99 B=0 rej 38 FVTotl6on *-1.173 0.016 1.292 0.050 7 0.99 B=0 rej 39 FVTotl6of M.282 0.062 1.374 0.144 10 0.92 B=0 rej 40 FV 23 M.121 0.001 1.971 0.002 10 0.99 B=0 rej 41 FV 16on *-1.451 0.001 1.175 0.002 7 0.99 B=0 rej 42 FV 16off *-1.706 0.001 1.238 0.002 10 0.99 B=0 rej 43 CDRe 23 •-1.415 -0.035 44 CDRe 16on * 1.811 -0.825 177 APPENDIX I. continued Relation a Sy.x b Sb n RA2 Ho .05 45 CnRe 16of * 1.369 -0.762 46 MPow 23 *-14.226 2.965 47 MP l-6on M0.761 2.175 48 MP 16off *-l 1.285 2.238 49 CoRe a * 1.295 0.049 -0.542 0.057 16 0.86 B=0 rej 50 CDRe b * 1.142 0.044 -0.516 0.057 17 0.84 B=0 rej 51 CDRe c * 0.712 0.069 -0.499 0.064 26 0.72 B=0 rej 52 CDRe d * 0.706 0.032 -0.463 0.044 13 0.91 B=0 rej 53 CDRe e * 1.298 0.041 -0.451 0.058 8 0.91 B=0 rej 54 CDRe f * 1.679 0.018 -0.331 0.026 6 0.98 B=0 rej 55 CDRe g * 1.972 0.030 -0.391 0.041 12 0.90 B=0 rej 56 /pect/t/s 6.444 0.948 1.849 0.406 8 0.78 B=0 rej 57 /dors/tA 6.198 1.121 1.910 0.479 8 0.73 B=0 rej 58 /«•]/£/• 6.203 0.974 1.994 0.453 8 0.75 B=0 rej 59 A.cpeco/f/, 0.805 0.131 0.012 0.056 8 0.01 B=0 NSD 60 A,(don)/Us 0.824 0.150 0.071 0.064 8 0.17 B=0 NSD 61 As(anaiy[/s 0.837 0.126 0.069 0.053 8 0.20 B=0 NSD 62 X*(peco/f7s 3.461 0.156 0.004 0.067 8 0.00 B=0 NSD 63 Uiorsyt/s 2.850 0.118 0.072 0.051 8 0.25 B=0 NSD 64 k(anaiy£/s 2.791 0.134 0.096 0.072 8 0.21 B=0 NSD •denotes: log a 178 APPENDIX I. continued Key to regressions Number Description 1 maximum body depth 2 maximum body width 3 maximum body depth / maximum body width 4 snout to maximum depth 5 snout to maximum width 6 snout to maximum depth / snout to maximum width 7 posterior maximum depth 8 posterior maximum width 9 snout to posterior maximum depth 10 posterior maximum depth / posterior maximum width 11 fineness ratio 12 shoulder position 13 wetted surface area 14 total surface area 15 body surface area 16 total fin surface area 17 pectoral fin surface area 18 dorsal fin surface area 19 anal fin surface area 20 caudal fin surface area 21 maximum projected area 22 body surface area / body volume 179 APPENDIX I. continued Number Description 23 body volume 24 body weight (in air) 25 anterior fin ray pectoral 26 anterior fin ray dorsal 27 anterior fin ray anal 28 pectoral fm base angle of above longitudinal median axis 29 fin base length - pectoral 30 fin base length - dorsal 31 fin base length - anal 32 nare height / distance from snout 33 nare height 34 nare distance from snout 35 force / velocity - flight 23 36 force / velocity - flight 16 37 force / velocity - flight and fish 23 fins off 38 force / velocity - flight and fish 16 fins on 39 force / velocity - flight and fish 16 fins off 40 force / velocity - fish 23 off, net 41 . force / velocity - fish 16 on, net 42 force / velocity - fish 16 off, net 43 drag coefficient / Reynolds number - fish 23 off 180 APPENDIX I. continued Number Description 44 drag coefficient / Reynolds number - fish 16 on 45 drag coefficient / Reynolds number - fish 16 off 46 minimum drag power / Reynolds number - fish 23 off 47 minimum drag power / Reynolds number - fish 16 on 48 minimum drag power / Reynolds number - fish 16 off 49 drag coefficient / Reynolds number - angelfish fins on 50 drag coefficient / Reynolds number - blue gourami fins on 51 drag coefficient / Reynolds number - angelfish fins off off 52 drag coefficient / Reynolds number - blue gourami fins off 53 drag coefficient / Reynolds number - electric fish 54 drag coefficient / Reynolds number - seahorse 55 drag coefficient / Reynolds number - boxfish 56 mean frequency / specific velocity - pectoral fin 57 mean frequency / specific velocity - dorsal fm 58 mean frequency / specific velocity - anal fin 59 specific amplitude / specific velocity - pectoral fm 60 specific amplitude / specific velocity - dorsal fin 61 specific amplitude / specific velocity - anal fin 62 specific wavelength / specific velocity - pectoral fin 181 APPENDIX I. continued Number Description 63 specific wavelength / specific velocity - dorsal fin 64 specific wavelength / specific velocity - anal fm Note: unless otherwise indicated, all relations have standard body length as the independent variable. 182 APPENDIX II. Plate of top, bottom and side view of pufferfish in filming tank. 1 cm2 grid. 183 184 

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