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 degree at the thesis in University of partial fulfilment of of department this thesis for or by his or requirements British Columbia, I agree that the freely available for reference and study. I further copying the representatives. an advanced Library shall make it agree that permission for extensive scholarly purposes may be her for It is granted by the understood that head of copying my 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 hydromechanical necessary and of morphometric analysis were regression characteristics recorded, analysis was pertinent transformed performed to to where relate the morphometric characteristics to standard body length. Terminal velocity measurements were recorded for a series of drop tank coefficients performed- experiments. and to The Reynolds establish data numbers the was and specific converted regression relationships into drag analysis between was those two hydromechanical parameters which were compared to theoretical estimates calculated from cineTilms of pufferfish hydromechanical theory. High speed 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 propulsive based on Actuator-Disc model power and the efficiency during morphometric, was employed steady kinematic to forward and estimate swiiiiming 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 ii Actuator-Disc model was discussed study for respect with the application to negating of the propulsive systems. iii factors model to found multiple during fin this aquatic 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 Results 25 29 Discussion 75 CHAPTER THREE: KINEMATICS Introduction 122 Materials and Methods Results 123 126 Discussion 135 CHAPTER FOUR: POWER AND EFFICIENCY Introduction Methods Results 144 148 154 Discussion SUMMARY 156 167 LITERATURE CITED APPENDIX 169 176 iv LIST OF TABLES TABLE I. Surface area measurements of spheres 33 TABLE II. ANOVA and Tukey test results for spheres TABLE III. ANCOVA results for fm ray lengths 47 TABLE IV. Permissible roughness calculation results TABLE V . Fineness Ratios of aquatic animals TABLE VI. Propulsive fin kinematic parameters TABLE VII. Power and efficiency values v 155 34 82 129 80 LIST OF FIGURES FIGURE 1. Streamline flow around a body 21 FIGURE 2. Pressure gradient around an assymetrical body FIGURE 3. Couette flow pattern 21 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 confidence limits 52 with 95% 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 length with 95% confidence limits area 56 relative to body 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 fins on 69 relative to Terminal velocity. Series 16 FIGURE 32. Drag force fins off 70 relative to Terminal velocity. Series 16 FIGURE 33. Drag force fins off 71 relative to Terminal velocity. Series 23 FIGURE 34. Drag force relative to Terminal velocity. Series fins on, 16 fins off, 23 fins off 72 16 FIGURE 35. Drag coefficient relative to Reynolds number. 16 fins on, 16 fins off, 23 fins off, minimum laminar, minimum turbulent, total laminar, total turbulent 73 FIGURE Series 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 and frierids, the opposite especially my of the least, I thank my family wife encouragement. xi Lynn, for their support and 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 squid, of and from the flying subjects are fluid these three, effect in classical a solid I suspect preventing functo- morpho- background and a decent the second or It seems the only limitations biological mechanics theory to the to the giant rays, the reservoir has barely been tapped. creativity, sufficient fish beetle research ingredient discouraging more loco- of field blended with grant. Of has more of an participants in the 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 when cin6film functional basis of tail fin propulsion. The single most was first (Marey, employed 1894). Since significant advancement to record then, came the propulsive other 1 notable motions advances of arose fish when 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, first recorded 1912; Magnan, 1930). Another approach, instances of which are from Magnan (1930) the 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 alternate estimates drag. When he compared he made based was not concluded that there overcome drag, much like the on muscle sufficient bumble-bee these estimates to power power that output, he available to 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 fin Using updated stability, control estimates for and muscle kinematics power output (Harris 1936). 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 2 locomotion in fish 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 estimates drag, the other on can come from hydromechanical theory and bodies of revolution. force-distance-time estimates of theoretical empirical Or, they measurements thrust. equations based observations can like Drag-based of come the on technical from early direct techniques described previously. Thrust-based estimates can come from hydrodynamic models or from metabolic consumption power during calculations simming based (Marr, on the 1960; rate Blazka of oxygen et al, 1960; Brett, 1963, 1964). Hydrodynamic propulsion termed with models combine hydromechanical quasi-static or the kinematic theory. The resistive, integrate parameters earliest the of models, instantaneous forces for each segment during a propulsive cycle (von Holste & Kuchemann, 1942; Parry, 1949; Gero, 1952; Taylor, 1952; Gray, 1953b). Advancement development in hydromechanical approaches came with the of reactive models which consider the rate of change of momentum of a mass of water affected during a propulsive 1970, 1971; theory has Wu, been cycle 1961, refined, (Gadd, 1952; by the body segment Lighthill, 197 Id). This so-called modified and widely 1960, 1969, elongated applied body 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 3 momentum priciple include Blade-Element theory and Actuator-Disc theory. Pectoral theory, fin rowing has been analysed which arbitrarily divides the fin span segments for which the normal force impulse acting on the body blade into and calculated. The sum of the thrust impulses drag with a number of thrust force is equated and a mean element are with the stroke power is calculated (Blake, 1979b, 1980a, 1981a, c). The locomotion Actuator-Disc of the mandarin {Hippocampus niloticus), model thesis the pufferfish, fish applies 1980a, to the median Tetraodon been applied (Synchropus hudsonius) and (Blake, 1979d, this has the 1980b). electric 4 seahorse fish (Gymnarchus It is this and paired aquatic picturatus), model which fin propulsion of fluviatilis (Linnaeus, 1822). to 1758; Hamilton, The Fish, T. fluviatilis This species approximately of freshwater 330 species pufferfish included is one in of 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 with will the terms Tetraodontidae plates; be immediately apparent to terra- -odont, members share two plates and the common fused to each lower jaws. These bony plates, masticatory the musculature, shells of the enable small those feature other of on readers of four both familiar the family bony dental the upper and when combined with the powerful members of invertebrates this family upon to crush which they characteristically feed. Another locomotory fins distinguishing apparatus which that provide a level aquatic vertebrates feature consists of of the family independent of manoeuvring unsurpassed and which is roughly analogous to is the propulsory among the helicopters or VTOL-aircraft. In general, the spatio-temporal 5 environment of the genus Tetraodon, and indeed Tetraodontidae, is a for many relatively of the members complex one of the family 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 brackish, such fresh-water estuarine as those species such environment, from as consisting mangroves and value and in which a high of fluviatilis, the submerged aquatic presents a relatively complex environment little T. plants, in which degree roots similarly speed is of 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, 1988). According 1975), river the saltwater dispersants genus Bangla intermediate & to a review of the genus Tetraodon is widely distributed throughout system, and the species T. fluviatilis Ceylon, (Moyle Desh, Malaya, Burma Sumatra, and Borneo Java, Vietnam. 6 the Cech, (Dekkers, Ganges is known from India, but Thailand, not from the Cambodia or The thesis Consideration functional of context locomotor strategy morphological allows and mode characteristics in general predictions life. Quantification of of a concerning the size and shape of a body and its propulsive elements, when combined with principles of parameters fluid necessary to mechanics, define and provides the standardize hydromechanical the fluid flow regime about an aquatic animal. The establishment of hydrodynamic similitude is performance allowing indeed of a the first particular chart organisms pattern and cost and flow aquatic or the objects in terms of the common currencies of energetic following other strategy assessing man-made The with locomotor towards and efficiency. comparisons requisite outlines how the 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 different This thesis kinematics and examines the swimming energetics morphology, of hydromechanics, T. fluviatilis with respect to steady forward, rectilinear swimming. The first chapter hydromechanical consists theory pertinent of a presentation of to the analysis and characterization of the fluid flow regime surrounding the fish. The second chapter mechanical characteristics parameters which hydrodynamic are deals with the relevant required similitude by morphometric to the and estimation to establish the which the pufferfish may hydro of the terms of be defined and compared with other fish. In the the third chapter, the propulsive fins are values of kinematic parameters of derived, described and compared to those for other fish. In propulsive the final efficiency chapter, generated estimates by the of power Actuator-Disc output model and are compared to theoretical minima and to values found for some other fish. 8 CHAPTER ONE: HYDROMECHANICS Fineness Ratio shape parameters and used Shoulder in Position hydromechanical are the two analysis to primary characterize body form for the purpose of drag estimation. Fineness Ratio (FR) is defined divided by its maximum diameter as the length of an object (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 maximum diameter gradient affect the length indicates the of proportion of the object. The position of general region where the pressure changes from favourable the an to adverse (streamwise) body that experiences and may laminar flow in the boundary layer. The borrowed describe term from the downstream, 1946). is the general streamlines, along to "particle" is a of paths usually relative A fluid is concept resultant the an which surface fluid (Vogel, streamline but large 1981; Blake, gives a descriptive which fluid of an arbitrarily defined small in mass and volume relative being considered a "streamlined" relative to to the the indication 9 particles object to travel (Rouse, element which molecular of used overall flow 1983d). The magnitude qualitative is term, size of field the of distortion of flow disturbance caused by an object in opposition to a fluid. Streamlines geometrical are related construct to in the principle hydromechanics of continuity, which allows a the assertion that the fluid volume flux ( 0 in a field of flow is a constant area related to the cross-sectional of the stream-tube (A), and the flow velocity (LO, A, U, = A , U = Q 2 The stream tube walls can be material in the form of a pipe or non-material in the form of an imaginary set of which The principle bound a region of applies in both situations finite cross-section. where it is considered streamlines 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 streamlines rate of diverge fluid the volume increased area flow. Conversely, results in a where drop in velocity. The importance particles is described of the constant changes in the fluid seen in a theorem named for Bernoulli (1738) who the inverse relationship pressure for an ideal fluid, isothermal velocity and flow momentum flux and which is considered to be inviscid, incompressible. along between The a streamline, the theorem total states pressure fluid, the sum of the dynamic and static pressures, is constant 1/2 pU + p + pgh = P 2 10 T that (P^ for of a or (where 1/2 p{U\-U\) + p=fluid g=gravitational f £7=fluid density, acceleration, + pg(\-h ) = 0 (p p ) 2 2 velocity, /i=height of /?=internal the pressure, fluid above a reference point). Schematically, as curve past an object principle by the increase compress predicts from fluid free-stream together and conditions (Fig. 1), in symmetrical flow of continuity an streamlines volume velocity flux is (U^) in the conserved the frontal region (F), to a maximum velocity (U ) in the shoulder region (T 2 and B) where the streamlines are most constricted. Concomitant predicted over by portion region. portion of the increase theorem, The reverse velocity, pressure (p^ as decreases 2 occurs the pressure particles fluid to a minimum (p ) in the situation where fluid in the of the body of the body velocity the Bernoulli's the front shoulder with over restores decreases to the posterior to the as the free stream velocity (L^). The around the the unfavourable, the posterior favourable, anterior particles resistance even section portion of the body particle-accelerating portion of the body. is completely d'Alembert particle-retarding for whom conserved this for the case of an is exactly pressure so that, with (Fig. 2). Although the adverse 11 described by is no net of an object, asymmetric pressure around of the fluid there slips past the surface object balanced by as first is named, gradient gradient The momentum paradox as an ideal fluid pressure longitudinal gradient of the posterior body favourable portion pressure pressure, hence is spread gradient, the over a there conservation of greater is area no net momentum in an than is the difference in ideal is fluid independent of object shape. To the overcome boundary fluids at unlike resistance m" layer a an d'Alembert's concept solid-fluid ideal dilemma, to explain interface. fluid, to deformation the theoretical Prandtl A has or of Newtonian characteristic by proposed behaviour real a indicated the (1904) dynamic real fluid, time-dependent viscosity (u., kg s" ). 1 1 For plane (/) surfaces apart, area (Fig. dynamic (shear (A) case 3), each viscosity stress) of is a of fluid bounded negligible defined required to as mass the by parallel a distance and force maintain two (F) constant per unit velocity (CT) of the moveable top plane relative to the fixed bottom plane ^ y is area defined (A) as (shear the force stress) _ (shear stress) (shear rate) per (F) required unit to maintain constant velocity of the moveable top plane relative to the fixed bottom plane (U) ^ that " 1 the interface is, _ FIA dU 161 the velocity is fluid zero _ FIA dU/dl of relative particle is a to _ (shear stress) (shear rate) fluid the deemed 12 particle at the fluid-solid velocity of the surface, that "attached" to the surface. This assumption any is termed fluid-solid fluid or the interface surface, "no-slip condition" and is applicable at regardless of excepting rarefied the gases pathic nature of the (Goldstein 1938, Vogel 1981). The boundary layer concept divides the surface body of a solid (such oriented parallel to as a the flat fluid plate or flow rigid direction of flow) past the streamlined 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. as In the the fluid surface to (modified in approach is of leading (Fig. the to fluid flow to the 4 essentially velocity of & the 5 In the that of the viscosity. The for to of turbulent the which in the flow) region free stresses grow root outer This thickness the stream of fluid and velocity as a inner region fashion in from the (x) fluid is particle as the (Fig. 4) occur parabolic more high, at steep the distance as zero velocity 1934b). shear is (dU/dl) rapidly from free-stream Tietjens high square gradient increases continues edge (x ^ 5). the that Prandtl due laminar proportion region particle velocity from gradient result inner such affected velocity the is viscous effects are negligible. There are numerous definitions of boundary layer thickness (Prandtl & Tietjens 1934b) from which a commonly used one called the velocity object from thickness surface that of (8) is to the region the free-stream defined where by 13 1% as the the distance fluid (Fig. 4). from the velocity differs Blasius (1908) 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) Re, | - 5 respectively, for a "°- ; 0.37 Re, | « 5 smooth flat plate ^ oriented parallel to the can be flow (Rouse, 1946). The laminar, type of turbulent flow or regime in transitional the boundary between the layer 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 while investigating the factors which appeared to of transition have an effect on the nature of flow in fluids. By altering the dimensions of an object or and in the concert, velocity, density and viscosity of a fluid, singly Reynolds that found 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)=pU A 2 The viscous force earlier definition of of the viscosity fluid relating will force with the velocity gradient F(viscous)=\iAU/l 14 be familiar per unit from the of area 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= which is defined the velocity of P^ 4,„ = 2 \iU(All) Ul inert ial v viscous = \i by the kinematic viscosity of the fluid the fluid (U) and some characteristic (v=|i/p), 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 to its density viscosity relates the of trajectories two fluids by (Batchelor of errant a fluid propensity to damp out irregular or non-uniform equal dynamic viscosity will caused viscosity of and as such gives an indication of the or ability of a fluid particle dynamic 1967). For density, be better fluid the fluid suited particle instance, to in with trajectories by the the damp out fluid case greater disturbances converting the momentum of the fluid particles into heat energy. The importance condition like called shape and identical when revolution identical length which of the Reynolds number hydrodynamic orientation, their the Reynolds conditions numbers different Reynolds numbers if the products equal, ten flow wherein times are are given constant body constant .temperature, viscosity uniform outer flow field testing similitude rests squarely and is in establishing a are for objects of around them are Bodies of equal. in of size their shape and density of the will have velocity and orientation, fluid in and the conditions. The whole business of model on this 15 fundamental principle of hydrodynamic similitude. The condition of a boundary size and shape temperature, in a fluid layer for of an object of given kinematic viscosity, given density and uniform flow are in general indicated by the Reynolds number where a sub-critical Reynolds number (<5xl0 ) 5 suggests fully (>5xl0 ) suggests 6 between laminar flow fully denoting a and a super-critical Reynolds number turbulent mix of flow, with transitional the laminar values and conditions in the boundary layer (Prandtl & Tietjens in turbulent 1934b). The Reynolds number at which transition occurs is termed the critical Reynolds number and is related to velocity thickness id) . = 5(Re d conditions in the roughness elements, such denticles, appendages and thickness of maximum the as boundary permissible boundary the 5 cnt cm Flow .)"° layer are external nares, also affected eyes, like, which protrude layer to disturb the height (h) of roughness by opercula, through the flow. The outer 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 frictional rotation and in a pressure body diameters steady drags. below the fluid A flow body surface is the submerged is considered sum at to of the least be three unaffected by wave drag (Hertel, 1966). At lower Reynolds numbers where form or pressure drag is minimal and laminar flow is shear stresses is the major and object length expected, frictional drag due to source of drag. Given fluid viscosity 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 attached travel, to boundary the around the rate layer, of inertial relative change of effects to the momentum anterior portion of an object rate of change body due the of the fluid. direction in of the fluid In an free-stream particles is not equalled by the of momentum around the posterior portion of the to the thus the origin of pressure drag is ultimately due to the viscosity of the fluid. There is a point body surface at shear stresses in a real fluid, along the which the deceleration particle region on velocity will the fall posterior to zero. The adverse pressure gradient imposed upon the boundary layer by 17 the outer diverge flow past field the (Bernoulli's shoulder will theorem) force as the the streamlines particles within the boundary layer to reverse flow in the upstream direction (Fig. 6) (Rouse the 1946). A discontinuity in flow outer region separates from becomes results, distorted, the body surface and separation drag, since gives the rise amount to of flow the field in boundary layer creating a zone of low pressure (wake) on the rear of the body surface layer the a (Fig. 7). This boundary dramatic pressure or increase form in drag pressure 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, steeper turbulence velocity gradient in the in the boundary region layer immediately the body surface which results in a greater viscous drag than that for a laminar also produces adjacent a to shear rate and larger boundary layer (Shapiro, 1964). The greater point laminar pressure drag in separated laminar flow is than that for the same body in turbulent of discontinuity boundary layer or will flow separation occur 18 closer in to usually far flow the the since the less stable leading edge producing a much wider wake than in a turbulent boundary layer. Pressure wake so drag a separation is directly turbulent can related to boundary layer or compliment enhance the diameter which is less the of prone the to 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 fluid gently field. By shoulder position boundary layer and curving tapering separation the the is anterior posterior prevented to the surface to the to the tail, delayed to the surface or outer posterior-most possible position. An optimal fineness is the result concerns. In of a compromise between order to separation and elongated downstream drag increases ratio for a streamlined body of rotation delay eliminate with of the (ideally to form the a number drag, amount of prevent) the shoulder of body position surface boundary should but area, conflicting layer be greatly since friction 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, speed, for such as maximization of minimum drag. For volume versus maximum 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 19 10% or less (von Mises, 1959), allowing considerable latitude in body design. For bodies of revolution which move through fluids at higher velocities total and higher drag friction comes Reynolds numbers from form drag. the major Thus a contribution to small increase in 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 Reynolds rotund Numbers) shapes experience with lower mostly surface at lower velocities (and friction areas drag, are thus favoured, more especially for bodies which are required to maximize volume. A body which travels at higher velocities likely boundary layer the boundary drag separation may benefit by layer so as to trade off the incurred over the anterior to promote inducing turbulence increase portion of the in body in frictional 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 shape, hydrodynamic size and orientation, be compared to a standard determine the principles total drag under allow objects different flow of different conditions, to shape and flow condition in order to upon the flow. 20 object in opposition to the 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 boundary l a / e r Transition \ zone I ( Turbulent boundary layer l a m i n a r sub-layer FIGURE 6. Boundary layer flow reversal. 22 FIGURE 7. Schematic wake w i d t h o f a separated 23 boundary layer. CHAPTER TWO: MORPHOMETRY AND DRAG INTRODUCTION In this chapter characterization Ratio, of Shoulder related to length (/ = the the body are areas and surface characteristic distance s pufferfish Position, some morphometric parameters from length, snout to relevant defined fin to (ie. the Fineness dimensions) and standard body usually caudal peduncle). The results are compared to other fish forms. Following that, the results of the drop tank experiments combined estimate the with the the results from relationship between principle of hydrodynamic experimentally determined number. parameters These the the two parameters similitude to be coefficient and drag are morphometric compared to analysis which to allow established: the are the Reynolds theoretical values and estimates for other fish. Estimates experimentally considered with of power output determined respect drag to are made coefficient the propulsive fins. 24 kinematic based and are parameters on the briefly of the MATERIALS A N D METHODS The fish Specimens were obtained approximately length 7 (pH, salinity various field. with years commercial Water quality heart in a aquarium and partial The and brine water fish grown other shrimp. glass 0.5 % as substrate to a Specimens and varied were of aquarium and break an external changes of an initial a maximum objects by received from 420 litre gravel fluviatilis, the course with 7.5, was maintained regular cleaning. were were plants and over of lengths reared temperature aquatic filter, or aquarium 3 c m to a range were Tetraodon pufferfish, a conimercial and substrate liver Asian one and one half artificial charcoal the c m . The fish respectively) and from of about about the of 25°C and up foam and periodic tank diet beef of fixed in 37% formaldehyde and preserved in 40% isopropyl alcohol. Morphometries Lengths ± were measured .005 cm). F i n surface with which toluidine blue areas of the same paper o f known body. by sowing T h e excess a a standard were and blotting was cut out, weighed measured with the measured spread and compared was trimmed of clear out fins area of plastic the fins onto paper of a piece the body was tightly off and the weight 25 (Mitutoyo, by coating to the weight area. T h e surface wrap micrometer of about the the plastic was compared to of a piece of plastic of known area. that To determine the accuracy and precision of the body surface area measurement subjected to additional methods dipping the in analogous method, same to treatment were liquid three diameter as described for spheres the were fish. Two applied; wrapping in aluminum foil soap. that different for The procedure plastic. For the employing soap film and foil was procedure the weight of the sphere was measured before and after dipping in the liquid, once difference slides the in excess weight which were soap was subjected were determined with had been compared to allowed to the that same electronic balances to for drop. standard treatment. The glass A l l weights (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 m s ) to between with avoid fluids consisting attached alcohol, 20.5 °C, kinematic viscosity of to any of two the error different density horizontal fields outside opposing rows which of the might and of 6 arise from viscosity. A infrared column. 3.65 x 10" Each of photo-electric emitters interaction light field and speed trap beams was was created 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 26 which displayed the time elapsed for an object which were could be independently the object timer. spaced had The adjustable a both travel between the known vertical distance adjusted reached trigger for to along the terminal threshold fields for and the apart. fields Both fields column to ensure that velocity the horizontal before triggering the sensors was photo-electric trigger mechanism had to be reset for each pass. The digital timer was connected to a lap top computer (Zenith model ZFL-181-93) measurement to which each elapsed time 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 file on of elapsed times was obtained and saved as a separate the progressively computer. increased Submerged by weights inserting small of lead the rods fish were 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 attaching vertical descent a dan flight through (wetted the column surface area was ensured = 44.4 cm ) 2 to by 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 the strand and inserted through caudal peduncle parallel to the spinal column leaving approximately 10 cm extending from the trailing edge of the caudal fin. dorsal and anal fins were amputated In addition flush with the on one series, to eliminate any fin flutter effects the pectoral, body, except and to ensure a vertical descent. A calibration curve of terminal 27 velocities for the dart flight and shaft different was obtained from a series of elapsed times for submerged sequentially weights rolling paper which were progressively thin sheets of lead increased onto the by leading end of the flight shaft. The leading end of the shaft undetected dart if it flight travelled always between of stop were: shaft-flight flight-flight (correct and (under-estimated the triggered combinations fields could pass through both fields triggering time). the order These for infrared beams sensors. Four the respective (over-estimated start and combinations the possible time), time), four but and shaft-shaft flight-shaft meant that three populations of elapsed time were sampled so the data which fell into the mean by first terminal and fourth velocities a visual examination combination were were calculated. of the rejected .before This frequency was the accomplished distribution of elapsed times which clearly revealed the three populations. The calculated terminal by velocity subtraction curve of the for curve for the the fish alone was flight alone from the curve for the fish with flight attached. Results established were analysed statistical on a procedures micTO-computer for according transformations to 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 snout related to the references to standard body caudal peduncle; length (Is, measured from the unless otherwise to body length refers stated subsequent 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 (X ) and rf 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 cross-section at the indicate point that of the body has maximum thickness. a circular This circular profile extends from the snout to the region between the shoulder position and the laterally compressed dorsal/anal near the fins where caudal the peduncle. body In becomes this posterior region; body depth, body width, and distance from snout to dorsal and anal fins are linearly related to body length (Figs. 16). ratio remains Also, the posterior depth to posterior width 14 to constant (mean=1.40, s=.134) with body length (Fig. 17). The mean Fineness Ratio (FR = total body length/maximum thickness) which object 3.37 distance is from describes (s=.260, snout the degree n=19). to point of 29 The of streanilining Shoulder Position maximum thickness/body in an (SP = length) 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 measured. The moderate lateral conserved overall body compression over the shape is range that of of occurring posteriorly fin to caudal peduncle region). 30 body lengths a teardrop with (in the dorsal Surface area and volume Surface of employing the foil, plastic and soap film three spheres (consistently is estimates arsa the the least accurate. surface treatment according of soap to sphere For film a Tukey estimates on sphere all 3, while for are to precise foil method be were the sphere spheres to 1 but most closest to 7td 2 ). the three methods give and significantly multiple comparison test the (S = three not least deviation) diameter each methods 2 the different (Table II). Both and foil wrap methods are prone to over-estimation. the vagaries aluminum foil accumulates of proved based the standard applied for results to amount wrap the while lowest methods in Table I. For all is deviation) plastic a different the soap film Due The methods. and method (consistently variance was significantly plastic film since in all cases area of are presented standard precise method Analysis soap highest most accurate the the three different of manipulation wrinkles over-estimation. The during application, the which cause a proportional 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 measurements f are used to calculate wetted surface area (S ) (Fig. w 19), the present. sum of body The total surface surface area area and the (Fig. 20) 31 is area of any fins comprised of the body surface area and the surface area of all fins (Fig. 21). For all relations of surface projected area coefficient is not Projected area is area measurements versus significantly the versus transverse body body different length length from cross-sectional 2 area the and regression (Appendix of for the I). 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 mass of the displaced fluid which is calculated by dividing (the difference between the the weight of the fish in air and the submerged weight) by the density of the fluid. 23 Volume is related to & 24), providing slope 3 (volume « body length coefficients and similar surface -area (Figs. to those expected 2/3 length , volume « surface area). The mean density of the preserved specimens is 1.125 (s=.036). 32 TABLE I. sphere 1 Surface Area Measurements Diameter nd method 2 (cm) (cm ) 2.614 21.466 foil soap 3.715 43.358 foil plastic soap 3 4.423 61.459 foil plastic soap 33 2 mean 2 plastic 2 surface area (cm ) s n 25.707 0.601 5 21.897 1.012 5 19.678 2.630 5 51.563 0.945 5 45.495 1.437 5 46.983 2.478 5 74.654 1.413 5 65.525 1.709 5 69.957 2.634 5 TABLE II. ANOVA and Tukey test results for three spheres. sphere method F P method S.E. q pair 1 foil 16.799 <.0005 plastic foil 16.494 <.0OO5 2.983 0.779 f-p soap foil f-s plastic f-s f-p soap p-s 34 NSD 7.792 5.881 1.911 p-s 26.373 <.0005 8.104 5.121 p-s plastic 3 0.744 f-p soap 2 f-s 0.889 10.269 5.285 4.985 NSD 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 with 95% confidence limits. to maximum body width 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 n - 20 3.2 3.0 4 4.4 vs Body length PC2 - J93 4A Body length (cm) .424(X)-.208 5J2 5.0 O 0.4 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 o 86 vo 8 2.0 • Xwt snout to maximum width (cm) observed .809(X>+.454 Xd vs Body length n-20 PT2-.7Q o • observed Body length (cm) 307(X>+270 Xw vs Body length n - 2 0 3.2 3.0 4 4.4 FT2-.7Q 4.6 5J2 5.0 Body length (cm) • observed 396(X)-J095 . 0 Body length (cm) J301(X)-J392 Body width <§> Median fins vs Body length n-20 3.2 3.0 4 4.4 FT2 - .94 4.6 Body length (cm) .201(Xr-.267 52 5.0 O Snout to Median n - 20 fins vs Body length PC2 - J97 5 -, 3J2 3.0 4 4.4 4A Body length (cm) .06MXH204 52 5.0 0 0.4 Body depth vs body width at Median fins n- 20 PT2 - .92 Fin morphometry The anterior pectoral, -dorsal (Figs. 25 to regressions chord and 27), base there is no significant the lengths and difference average of the and the anal are fins; between that for the fact rectangular ray tip) to difference Mean fin chord lengths anterior, regressed against I). There is no significant difference dorsal to for body the length between the and as such a common regression is calculated for the fin from fin ray (fin anal fins is linearly related three fins (Table III). each length the however elevation dorsal and of anal fins that the pectoral than the dorsal fins curves for fin likely appear anal the pectoral to posterior length (Appendix statistically which is fins and body is the calculated for medial and between there are a the slight curve from attributable be slightly which taper to less from the leading, anterior ray towards the posterior of the fin. Fin different surface and areas regressions for all against fins are standard not body significantly length provide slopes not different from 2 (Appendix I). The mean angle of incidence (a) pectoral fin base is calculated to be 48 ± horizontal line from the mouth through peduncle and is independent of body length. 46 with incident flow for the 2 degrees above a the middle of the caudal TABLE III. ANCOVA results for anterior fin rays for the pectoral, dorsal and anal fins relative to body length. F slopes 0.635 elevations 1.942 overall DFnum 1.275 2 DFden 51 F(.05,l) 3.18 NSD 2 53 3.18 NSD 4 51 2.56 NSD Tukey test for slopes diff SE q 1 vs 2 -0.034 0.023 -1.491 51 3.442 1 vs 3 -0.028 0.023 -1.234 51 3.442 'NSD 0.257 51 3.442 2 vs 3 0.006 0.023 DFp q(0.05,3,40) NSD NSD Tukey test for elevations diff SE q DFc 1 vs 2 0.057 0.022 2.650 53 3.442 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 q(0.05,3,40) NSD Surface roughness-Nares Nare related linearly to height (h) and body length related with distance (Figs. distance 28 from & from snout 29) snout (x) are and nare (Fig. 30). linearly height is 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 about 120 for Vogel (1981), distributed the snout, for both pointed protruding (h/x< objects. the and 15 it is nares into the outer flow field observed permissible 9.5Re ~ ) Thus the after location ratios with Reynolds Numbers based on distance from exceeds of 1975). Likewise, to ratio comparison (Webb, nare > height the by elements element and likely values rounded that the has some effect layer flow conditions, at least over the dorsal region. 48 calculated (h/x< protrusion \2.1Re' ) n$ of the on the boundary TABLE IV. Series 16 on 16 off 23 off Permissible Re Roughness calculation results local Re point round 3.59E+03 350 0.117 0.151 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 4.06E+03 396 0.107 635 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 1.07E+04 834 0.079 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 .363 0.137 6.51E+03 0.061 h/x .254 TABLE IV. Series Continued Re local Re point round 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 h/x FIGURE 18. Body surface area relative to body length confidence limits. with 95% 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 length with 95% confidence limits. relative to body 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 a 3 f3 5 3 S area SS^H (cm"2) ^ ^ ^ o 53 (EJUS) BSJB eoejjns 54 [Zjaio) B&JB eoBjjns 55 uu Projected area- vs Body length n- 20 OS -\ 32 , , 3.0 , -i 4 \ 1 1 4.4 FT 2 1 .90 1 4.6 Body length (cm) .OGO(Xr221l 1 52 1 I 5.0 I r O • observed Body length (cm) .04(Xr321 Body surface area vs Body volume n - 74 R~2 - .99 _ 30 -. ' 3 • 5 observed 7 9 Body volume lcm~3) 5.04(Xrj07 71 13 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- 02 H 32 1 T 1 3.0 • 1 4 observed 1 1 19 1 4.4 RT2 -.77 1 4A 1 1 1 52 Body length (cm) 210(X>-358 1 5.0 i r 0 ANAL ant. ray length vs Body length n-W 12 • observed TC2 - JM Body length (cm) 2KXX)-J374 Nare height vs body length n -12 32 3.0 • 4 observed 4.4 FT2 - .79 4A 52 Body length (cm) J5(XX)-J53 5J0 O 0.4 • observed Body length (cm) 1.4CXX) -157 Nare height vs distance from snout n - 74 FT2 - AO o observed Distance from snout (mm) 32(X)+20 Drag Estimates The for the from relationship fish is obtained that for the fish possible because the considered to interactive effects fall between be together when terminal force drag by an it velocity the is component assumed and the addition, velocity object terminal curve for the fish In the of and the and flight (see Figs. 31 to 34). This is forces tank. (constant) encountered produced with flight additive the force by subtracting between in drag is is is reached, exactly are are no there flight it parts when they considered that the equal total drag the force to 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 the from Newtonian dimensional analysis. These curves permit generation of Drag Coefficient ( C D ) 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 estimates the are force-velocity calculated and relationships related to Minimum Reynolds Power Number for different boundary layer types (Figs. 37 to 40). Theoretical values of C D are compared for different Ratios and theoretical boundary layer types (Figs. 41 Drag/Body Volume ratios 66 and 42) are related Fineness and finally, to Fineness Ratio at different condition Reynolds (Figs. ~;3 to Numbers and types of boundary 47). 67 layer flow force relative to Terminal velocity. Series 16 force relative to Terminal velocity. Series 16 force relative to Terminal velocity. Series 23 FIGURE 34. Drag force relative to Terminal fins on, 16 fins off, 23 fins off. velocity. Series 16 FIGURE 35. FIGURE 31. Drag fins on. FIGURE 32. Drag fins off. FIGURE 33. Drag fins off. FIGURE 36. Drag coefficient relative to Reynolds number. • 16 fins on, D16 fins off, + 23 fins off, 1- minimum laminar, 2-rjtinimum turbulent, 3-total laminar, 4-totaI turbulent. Series Drag coefficient relative to Reynolds number. Experimental results from assorted fish. minimum laminar, minimum turbulent. . 68 Force vs Velocity 16, fins on OJ023 0J021 - 0J022 0J01G0.02 0.017 0JD10 0015 0.014 0J0130XJ12 oxjn 0J01 0.016 0XXD6- OJ009 0J007 - OJOOOOJOOB - OJOOO0J0O4 0J002 0J6 Terminal velocity (m/s) + night Force vs Velocity 10. fins off 0J032 0.03 0.026 0.020 0.024 0.022 -| 0.02 O o c c Q 0.016 -| 0J010 0.014 0.012 0.01 0.006 OJOOO 0.004 0.002 —I— — I — 02 03 total 0.4 0.0 05 Terminal velocity (m/s) + flight fish Terminal velocity (m/s) total + night o fish Drag coefficient vs Reynolds number Drag C o e f f i c i e n t vs R e y n o l d s N u m b e r 1.00T c a) o 0.10 g o o 0.01 # O A A H 1 1000 Angel fish, open fins Blue gourami, open fins Angel fish, no fins Blue gourami, no fins 1 1 1 (—f 1E4 • • H V Reynolds N u m b e r Electric fish Boxfish Seahorse 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=0.4L 2 for moderately Parry, streamlined 1949; Webb, fish 1975a) and which, cetaceans (Gray, according to 1936b; Webb, is more likely to be a high rather than low estimate. The effect approximation in turn is to under-estimate under-estimates coefficient. This calculations leads based to on the the an the drag per unit area, empirically which drag deteiTriined under-estimation coefficient of this of and the drag ultimately force to an under-estimation of minimum power calculations. area The results and body concern is that not of the length the for above appropriate fluviatilis. In fact for the those obtained from regression this analysis study relationship less estimates ( for about wetted surface Sw=.79L ) support 21 predicting streamlined are of shapes 2.5 surface such times the area as T. lower than the the presents analysis. The result is that the drag coefficient would and thus cause the be over-estimated by about more rotund shape of the puffer 2.5 times to appear hydrodynamically disadvantaged with a lower efficiency rating. The anomalous increase at first in the glance intercept value in a that 75 more may seem somewhat streamlined salmonid 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 coefficients surface area, for the purpose of estimating in this study, is defined as the sum of the surface area of the body plus the surface area of any fins. propulsive drag fins experiments were amputated for the majority The four of drop tank 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. the propulsive swimming in fins the contribute It is not clear to 'what extent to the tetraodontiform overall mode, drag although of an a attempt fish at accounting will be made later in the section discussing drag. Morphometry Diversity characteristic study, in of body shape higher and teleost Tetraodon fluviatilis, mode fishes. The Some other families Diodontidae pufferfish.es), filefishes), Synathidae propulsion species is propelled by means median and paired fins. (marine of (seahorses, this of undulatory swim similarly, eg. Balistidae pipefishes) in are (triggerfishes, and Ostraciidae (boxfishes, cowfishes, trunkfishes). Breder (1926) was the first to classify fish on the basis of locomotory pattern and early on defined 76 three general categories: Anguilliform after (eel-like, Carangid-c) Ostraciidae). and after a position of the Anguilla), Ostraciiform Subsequent patterns swimmers, after work particular terms such as Tetraodontidae), after Diodontidae), after Balistidae), Labriform after Labridae), Ostraciiform, Gymnotidae) used to Balistiform Raja), and describe the or Rajiform locomotory and the paired fin and anal pectoral and fins, anal fins, flapping pectorals, (continuous enlarged (extended anal dorsal fin, after pattern of dorsal dorsal after locomotory basis (short (paddling (extended sculling, the (short (extended Gymnotiform Amiiform on Diodontiform (trout-like, describe median Tetraodontiform after after for fin to Hence, fins fins, pectorals, (caudal continued species. propulsive Carangiform typical of fin, after Amia) were the. functional to undulatory group. (See Lindsey, 1978 for a detailed review). Blake (1983) has criticized this approach median and paired fm swimmers for lack of a functional basis and proposed upon the in the its amplitude, propulsive relative fins. measures waveforms (termed stead general frequency system and This system anchored at is, one of Group 1 wavelength forms) characteristic of a course, extreme at the by 2 forms). should fall of fish displaying and large wavelength other extreme Tetraodontiform somewhere based continuum of of small amplitude, high frequency (Group pufferfish) and classification wavelength of large amplitude, low frequency presenting • waveforms freshwater a by and small fishes in between fish (the these two extremes. The' fish in this group (Tetraodontiform) 77 are relatively slow swimmers and for the most part could be described as stout or less streamlined when compared with faster such as salmonid (trout, salmon) and swimming fusiform fish scombrid (tuna, mackerel) fishes. The freshwater undulatory caudal paired fin collapsed The pectoral during fan-like caudal turning pufferfish fin propelled and median routine and held system is or reversing is dorsal and forward rectilinear rigidly active manoeuvres, by in as the a of anal fins. The progression dorsoventral rudder wherein it means is plane. during complex 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 occurring described in the as a posterior tear-drop with slight region from the and anal) towards the caudal peduncle. 78 lateral median compression fins (dorsal 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 freshwater with some pufferfish is faster swimming relatively rotund pelagic and forms, thus the 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 resembles objects compared the to fuselage presumably man-made of designed some the pufferfish body airships, bombs and. boat hulls, with objects, maximum volume for mirrimum surface area, having FR values in the order of 3 to 5 (Hoerner, 1965). The a value of the Fineness Ratio, as mentioned previously, is relative body, measure which is of the degree of streamlining present inversely proportional to body will disturb the fluid provides some information gradient and the the extent to in a which a through which it travels, and as such as to the type of boundary magnitude layer the of body the pressure 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 79 cause a greater amount of pressure drag body, all else influence the than of would be being the equal. favourable body; encouraging the experienced Higher by FR pressure a more values gradient maintenance streamlined mean is that the extended along boundary layer of laminar flow over the major portion of the body. However, lower FR values also mean that forms area and such as the pufferfish greater volume for streamlined fish. For how, for a prolate present less a given body length example, the following than data spheroid, the ratio of surface surface a more illustrates 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 Number (>10°), enjoy high some proportion a more advantages of laminar rotund over fish flow such a more as at higher the streamlined Reynolds pufferfish form at may 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 pufferfish lengths double position of remains constant (Figs. 11-13) that of the coincides with that maximum of (SP) in SP=.43) over a range of body (mean for least thickness which (*3.2 the the to 6.5). opercular 80 greatest Also, opening the length the (and freshwater is roughly SP invariably the pectoral TABLE V. Fineness Ratios of various aquatic organisms Species Vd 1/w x/1 Puffer (Jetraodon fluviatilis) 3.4 3.4 0.43 this study Tuna (Euthynnus affinis) 4.0 4.9 0.5 Rainbow trout (Salmo gairdneri) 5.4 7.3 0.39 Hertel (1966) Whiting (Gadus merlangus) 6.5 8.9 0.24 Haslett (1962) Perch (Perca fluviatilis) 5.1 7.5 0.3 Perch (Psettodes erumei) 5.1 12.3 — Norman (1934) Halibut (Hippoglossus hippoglossus) 4.6 12.2 — de Groot (1970) Greenland halibut (Reinhardtius hippoglossoides)5.8 12.3 — de Groot (1970) Plaice (Pleuronectes platessa) 4.4 15.2 — de Groot (1970) Plaice (Pleuronectes platessa) 4.4 Goldfish (Carassius auratus) 4.5 Trout (Salmo irideus) 13.6 0.24 Arnold & Weihs (1978) Bainbridge (1960) 6.3 ibid. 6.7 ibid. Dace j (Leuciscus leuciscus) Bream Goldfish (Carassius auratus)' 81 2.9 8.4 3.3 6.0 authority Magnuson (1970) Kipling (1957) 0.42 Bainbridge (1963) 0.32 ibid. TABLE V. continued 1/w x/1 4.5 8.1 0.41 ibid. 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. . 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. Species Dace (Leuciscus leuciscus) Dolphin Barracuda (Sphyraena sp) d is the mean diameter of depth and width 'data from scale drawings 82 authority fins) which is similar to what Houssay of fish. It effluent, is since possible the which experiences that this SP indicates found for a variety (1912) could assist in venting gill the general region of the body maximum fluid velocity, hence from Bernoulli's theorem, minimum pressure. For example, for a FR of 5.7, Hoerner (1965) states the position and the following location of correspondence between minimum pressure shoulder for a body of revolution. Shoulder Position .3 .4 .5 Minimum pressure location .2 .35 .6 Allen effluent (1961) can disrupt the has be shown a that for turbulence-causing boundary layer flow some smaller, fish disturbance and cause it to which gill can 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 roughness without or the (eg. flow another opening, characteristics of the incident flow, surface nares, opercula, scales) and Reynolds Number. So visualization data it what hence will gill be the effluent, is difficult effect upon to state one way of imposing the opercular the position of maximum thickness. The Shoulder Position of the pufferfish that for some faster (.43) contrasts with 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 83 body length (Walters, 1962; Hertel, 1966) presumably to which encounters a maximize positive the pressure portion of gradient the body and laminar boundary layer flow. In some fishes grows. For example, migrates mediterraneus the body profile the may change Shoulder Position posteriorly from about .3 as the fish of Trachurus to (as FR .45 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 and that development drag are relatively low gonads is of paramount importance. If velocity, of guts and 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 an intact Ratio would increase the probability of maintaining 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 «= length , volume <*= 2 length and surface area « volume 3 2/3 (Figs. 23 & 24, Appendix I). Comparison of the values of body volume and surface area for the pufferfish axes-symmetric with body theoretical of values revolution 84 calculated (prolate for spheroid) an using one-half standard body length and one-half maximum body depth as the respective major the pufferfish specimens average volume and semi-axes. and minor semi-axes a and b, reveals that surface Values 85% and 91% of the respective area values for were calculated a prolate from spheroid of same the following standard formulae (CRC Handbook of Mathematical Sciences, 5th ed.). e=J 1- b W eccentricity volume A/3nab 2 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 body of sizes. observed and (except perhaps Thus, likely at least also for for for lengths larval the range outside stages), the of the lengths range observed body dimension regressions- calculated (Appendix I) can be used with confidence. 3) The body form to a shape, rigid, of the pufferfish is reasonably analogous axes-symmetric volume and surface body of area and 85 revolution on also on the the basis basis of of the locomotory pattern rectilinear progression "stretched-straight" of the fish the wherein body position while is during routine forward held thrust roughly is in generated a by undulatory paired pectoral and median fins. The implications of these factors discussed in the following section. 86 on drag estimates are 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 as they are passive; inferred from techniques from the latter the object is unrestricted and estimates are distance are are distinguished active and time measurements. in that the object The second two is tethered estimates are obtained from direct force measurements. and 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 are frozen easier degree specimens to of (Blake, control 1980b) experimentally uncertainty due to the or but models at (Harris, the interactive cost effects 1936) of of some the line or spar with the object to which it is attached. Terminal velocity (Magnan, Blake, 1979a,b, 1981a,c) 1952; 1930; Gray, 1957a, harder to animals control but and Richardson, a spatiotemporal inferred 1936; deceleration-in-glide 1968; Lang & Daybell, in the 1930; 1963) Gero, (Magnan, techniques sense, especially drag estimates should be are with live 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 theoretical one) to values established direct from by the drag measurements hydromechanical is to calculate equations (chapter combining empirical observations of technical bodies of revolution with hydromechanical theory. The- significantly experimental active apparatus techniques devices to lower are and drop levels a with tanks, of major the cost advantage addition terminal and of of velocity complexity passive electronic experiments for over timing provide precise and accurate data. The main drop tank data assumptions are that boundary layer is attached fish is mechanically fish 4) that the in obtaining 1) free stream estimates from flow and laminar 3) flow similar drag drag to that contributions 88 for of an the is steady around 2) the the the dead actively swimming body and the 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 to is motionless terminal before velocity the is specimen smooth is dropped, acceleration and the velocities are sub-critical. The associated second assumption difficulties. amplitude For fish oscillations anguilliform, has, of for which their carangiform, all non-dead fish, swim by means body and sub-carangiform), or the some of large fins (ie. 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 propulsive swim by fins (ie. relatively small balistiform, oscillations gymnotiform, of their tetraodontiform), the rigid body estimates should be fairly close. The fourth is taken to position assumption is most likely met as care was the trailing dart flight at a great enough distance from the trailing edge of the fish to prevent the flow around the body Subtraction of results from the flight in the fish-alone interfering with that drag curve from the drag curve (Figs. around the total flight. drag curve 31-33, see also Fig. 34). The third assumption has been approached in the morphometries section and is discussed further here. The experiments fish. drag estimates are estimates of The total drag force, inferred from the tenninal velocity the total drag force acting on the as mentioned 89 in the hydromechanics section, is comprised of numerous components including friction and pressure drag. Fish in the wild are fish, with propulsive fins and as values such the somewhat more amputated, in a calculated animated drop are than a dead tank experiment considered minimum estimates. The validity of the drag estimates stems from the that, for the a "stretched-straight" rectilinear freshwater pufferfish, the position progression, unlike body is during fish and rigidly held in routine cetaceans fact forward for which thrust is produced by body and caudal fm oscillations. One difficulty encountered by arises an when actively velocity experiments using (pectoral, dorsal, anal) and considering oscillating fish fin. with intact, the the In the fins drag the force terminal propulsive act as fins rigid fixed wings. The drag on the dorsal, anal and caudal fins is mainly due to friction as are such drag since they pressure oriented vertically drag laterally at about is are parallel to the incident flow and negligible. However, the fins in a 48 broadside-on degrees above fashion the pectoral and axis of are inclined incident flow which produces a component of lifting force normal to the axis of progression friction of a and 'drag. a retarding The placement streamlined streamlined body body by can up pressure of "plates" increase to drag the 1965; von Mises, shown of or triangular) regardless the dead fm drag (fins) drag shape on 90 to at shoulder the the same body rectangular, with open the by a 1959). Blake (1981a) (circular, a fish addition encountered five times that for without fins (Hoemer, that in has square pectoral fins perpendicular to incident flow directly to due interference to (Blake, the body 1979b). contribute 1981a). a In than of The the fins fins portion study, greater with pressure major this is to the this the decreases the drag by nearly one half the body the in four attached via spars, over from increase of fins distally flow arising amputation the attached with drag with fins drag will (Blake, propulsive fins of that measured for the same fish with fins intact. With values respect obtained to an with the somewhat under-estimated propulsive fins likely falls intact in actively swimming propulsive while will between those be the the fins pufferfish, the amputated will drag be values for fish with over-estimated. The true value two curves; curves for fms fish intact amputated (Fig.34) and with amputated can be considered as minimum estimates and fins propulsive 'fins of total drag encountered by an actively swimming fish. Although the drop tank model requires the free stream to be stable and laminar, pufferfish is pufferfish habitat (Dekkers, 1975) spatially and the likely real life unstable is a where and brackish, flow riverine and may widely, both vary it remain intact such relatively low Re values (^10 ) should With an ' intact friction drag is boundary out the boundary the main disturbances layer, pressure component 91 of and prevent drag is drag may layer 4 damp The one turbulent, fluid the the estuarine the free-stream flow that for turbulent. partly at likely environment somewhat conditions temporally. Although is fluid be will since the separation. negligible and resisting forward movement. 92 Drag Coefficients The Drag Coefficient vs (CD(ex )) P Reynolds Number (Re) curves are presented in Fig. 35. The dependence of Crxexp) on Re for series 16 (fins on and off) the relative independence of curves for trend series similar plate of 16, to the equivalent is on Re for series 23. The two CD(ex ) P fins apparent and contrasts with intact and fins amputated, theoretical estimates (CD(the)) surface follow for a a flat 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) ]+0.11(d/l) 15 2 turbulent flow BL; CT(tur)=C/(mr)[l+1.5(d/l) ' +7(d/l) ] 1 5 According applied equation to Hoerner between Re values is also laminar BL flow applicable conditions. (1965) of at the 10 4 to lower 3 equation 10 . 5 for It is Re values The equation for Crrum) assumed is the (10 -10 ) with Cr(tur) 3 4 is applied at higher Re values (>10) where the BL is likely transitional or 5 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 greatly affected the BL flow indicates fins the by the presence BL flow conditions are not of the propulsive fins and that conditions are largely laminar since the curves are similar in form with that to the theoretical laminar curves. For series 16 intact the values of CD(ex ) P range from about .075 (Re=4x\0 ) to .038 (/?e*8xl0 ) and with fins amputated the values 3 of CD(exp) 3 range from about .04 (i?e=4xl0 ) to .02 (/?e*1.3xl0 ). 3 At lower Re values («10 ) 4 the 5 generally considered to be laminar and BL flow is dependent upon CD(ihe) Re. The flatness of the CD-Re curve (CD-.028, for series 23, fins condition^ in the amputated, suggests terminal velocity conditions are /te*l.lxl0 -2.1xl0 ) 4 that experiments the BL 4 flow may be turbulent to some extent even though the Re values do not exceed 2.2xl0 . 4 Another possibility is that the BL is laminar but the system is operating in a region of Re where the C D is independent of the the Re. This would be the relatively "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 conditions higher, are super-critical Re usually turbulent becomes relatively independent turbulence BL at in the of lower 94 values and (>5xl0 ) 6 the Re. It is Re values the CD<the) possible by BL flow progressively to induce protruding fms, nares, eyes, however etc., through the fluid the viscosity B L into tends to the damp free-stream flow 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 through Re. The protrusion the BL into the free-stream the free-stream will cause of the nares a disturbance in flow which may in turn introduce some turbulence into the boundary layer. In order turbulence to comment more specifically on the degree of that may be introduced into the B L a detailed visualization analysis is required (Allen, 1961; flow McCutcheon, 1977). When compared determined with estimates theoretical minima theoretical C D exceed of for a flat values, by the experimentally varying plate of equivalent amounts surface the area in both laminar and turbulent BL flow conditions (Fig. 35). For series 16, fins intact, Crxexp) estimates exceed CD(the) values for laminar B L flow conditions by an average factor of 2.9 and 1.7* amputated than 23 factors C/bam) (fins average and (C/(iam) and average C/(wr). amputated), factors of respectively) Cr(Um), 1.7 times greater For a larger CD(exp) 2.6 values and 1.2 and with and 0.9 times fins less specimen such as in series exceed for Crxihe) C/(iam) values and by C/(wr), respectively. Although the C/oam) the CD(ex ) P estimates are noticeably higher than values, it must be borne in mind that the values of 95 are based on the on a flat plate of equivalent the concerns CD(ihe) guts and of locomotion, gonads) pressure necessity, derived bodies revolution study drag. growth and empirical nearly which exists reproduction (respiration, values CD(the) analyses the without low Re values, encounters the which, for are 4 area such Also, from (#e^l0 -10 ) 3 surface and which, at negligible of two-dimensional rriinimum frictional drag for Re range impossible to are, by axes-symmetric involved in come by this (Hoerner, 1965). In comparing the estimates CD(CX ) P for the pufferfish to those found for some other MPF swimmers, the slopes of the curves are generally consistent and the elevation of the curve CD(ex ) P due to the presence of the pectoral fins is similar to that found for the Angelfish (Trichogaster electric eels wave down elevations (Pterophyllum trichopterus), and knifefishes continuous (Blake, (Fig. 36), and (see the at Blake found comparable Blue Blake, which swim by passing fins, 1983b) eimekei) gourami 1983d). a propulsive similar slopes and numbers (ie. swimmers, drag Reynolds CD*.05 at Re*\0 and Cd*.02 at Re*10*). 3 For some estimates exceed (Lighthill, 1971; sub-carangiform and the minima Alexander, species Bainbridge (CT(wr)) is. minimum the results theoretical frictional of the 1967). Based 1961) states (1960, approximately drag anguilliform 1.2 times calculations for relevant to this study. 96 three to a number on that greater This (C/(iam)). by the than differs the five of total times fish drag theoretical considerably from the Re range and FR values For Using the CD equations ratios of Crcum) to for lower Re values (<10 ) the C D 5 are C/(Um) approximately 1.6, and 2.2 at their respective Re values of 3, 5, 1.7, 1.9, 2.1 10, 15, and 20 (xlO ). 3 The reason for the difference from a factor of 1.2 is found in the sensitivity of the (Bainbridge employs the equation (drum)) equation to for Cr(tur) both FR and Re. total drag at Re values >10 ). 5 For a given FR, the ratio CD increases (CT(iam)/C/(iam)) with Re, for example, with a FR=3.37, the ratio roughly doubles from 1.6 (7?e=3000) to 3.5 (Re=\0 ). At a given Re, the 5 ratio CD decreases ' with an increase in FR, for example, with Re=lO , the s ratio decreases from 3.5 (FR=3.37) to 2 (FR=5). The point sometimes to found be in made the here is that literature, it is contrary not always to assume the total theoretical drag is simply a the minimum theoretical without first question friction considering the and the Re drag effect range in to practices appropriate 1.2 multiple of of an equivalent of the FR of which it flat the operates. plate body in This is especially important for the lower Re range in which many MPF swimmers typically operate (<10 ). 5 Another used the to issue calculate values CD (CD=.0108). Below concerns the theoretical applicability drag produced by J?e=1.5xl0 , 4 coefficients. C/(Um) C/(mr) flow, C/(Um). It seems 97 and produces actually lower than those produced by the laminar of unlikely the At equations /?e=1.5xl0 , 4 C/(wr) are values which equal are analogous equation for that at the lower Re values under theoretical (frictional consideration drag minimum (assuming for an turbulent intact BL BL), flow the conditions or total) would be less than that for laminar B L flow conditions. However, that is exactly the result (Figs.35, 36). Because turbulent of the higher energy losses associated that C D values for BL, it is expected with a a turbulent B L would be higher in terms of frictional drag. Since at such low Re values BL separation also be higher is unlikely, than the the total total laminar turbulent drag drag should 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 elevations of equations • are the reason for the experimental based on disparity regarding and theoretical empirical the curves relationships is relative that which the were obtained at higher Re values than those which are appropriate this particular study. Unfortunately, CD-Re studies there is a dearth in the classical to empirical of information for the Re range 10 -10 (Hoerner, 1965). 3 4 So what predictions conditions surrounding the can fish be at made regarding these Re values? the A flow 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 (*10 ), 4 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 B L 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 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 the pufferfish body. 99 for estimating the drag on for Power Output The minimum power output (/\exp)) necessary to overcome drag is calculated velocity as the product of cases: series for the three drag 16, fins fins amputated; series 23, fins amputated compared each for upon the four The case expectations of based intact; series 16, (Figs. 37 to 40) and is theoretical curves (Athe)) based C/aam), C/W), Croam) and Crcwr). C D values: form with (ospU SWCD) and force the on curves is dimensional in general analysis agreement wherein with 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 in a wings as opposed swimming fish. While anal) is essentially incur considerable above the pressure than ejected that the drag pressure flow, a static the drag addition increase amputated, the the friction than drag on (dorsal and fins friction drag. of angle The far less the fluid of the incident flow velocity the should 48 degree velocity that fluid fins fin is likely the higher propulsors pectoral to pectoral since by the fin is higher however open due to their in fm undulating on the median on an undulating for (free-stream) will actively frictional, incident drag to across fin. With the power required to overcome the frictional the fin the fins drag of the fins is missing so the estimates in these cases are low. As a 100 result the somewhere true estimate between the of two power requirement curves for fins should fall and fins intact 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 basic body study, fish shape of of the different hydrodynamic requirements pufferfish body lengths of since there are, in (total body lengths; the this 6.95, 9.82 cm) which seem to follow the same basic curve, not unlike that expected on the approximation, the the basis of dimensional analysis. Thus, as an two curves can be same curve, obtained over different considered Re ranges with specimens assumed to be hydrodynamically equivalent. 101 as estimates of different 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 Power vs Reynolds Number 10. Fins on 0.000 0.0O5 A 0.004 A 0.003 A 0JO02 A OJOOI A o 35 —r— 45 T — 1 1— 55 (Thousands) Reynolds Number T— 05 Power vs Reynolds Number 10. Fins off (Thousands) Reynolds Number Power vs Reynolds 0.04 0.035 0.03 H 0.025 0.02 A 0J015 A 0.01 -] OJOOO T 10 12 1 14 T 10 (Thousands) Reynolds Number Number Power vs Reynolds Number 0.035 0.03 0.025 A 0.02 4 0D15 A 0.01 A OJOOB T • PexpWon 1 r It 13 (Thousands) Reynolds Number + PexpIO o Pexp23 Drag Coefficient vs Reynolds Number based on CT(lam) 0.055 0.045 c q> o 3 0.035 o CJ 8> c Q 0.025 A O F.R.: • 3 + 3J5 T x 10 20 40 (Thousands) Reynolds Number 4 A 4.5 V 5.5 Drag Coefficient vs Reynolds Number based on CT(turb) C o oo o Q ~r 1 10 FR.: a OS 20 40 (Thousands) Reynolds Number ' A 4.5 5.5 Drag, Volume and Fineness Ratio For FR values ranging from 3 to 5.5, values of Croum) and are compared over a range of Re values (Figs. 41 & 42). CT(tur) 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 3xl0 , the C D 4 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=3xl0 . 4 Other trends apparent are that the decrease increased FR is non-linear; as FR increases in C T with the decrease in C T diminishes, and that CT is inversely proportional to Re. It values has generally imply higher been assumed performance that hence, lower a drag measure of coefficient 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; For a series of Shapiro, 1964; Webb, 1975a; Vogel, 1981). 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 optimum F R the is about upper 3.5 end of the Re range (1.5xl0 ) the 4 and the penalty for a departure to a F R = 2 or 6 is in- the order of 1 5 % . The points optimum F R is dependent upon Re at lower Re values and that the optimum FR ranges from to be made here are: approximately 2.5 that the to 3.5 respectively) which happens to be the range of F R (7?e*3xl0 -1.5xl0 3 4 and Re in which the pufferfish in this study are placed. It highly makes a certain evolved or amount of sense that one specialized ray-finned fishes of the (Moyle most and Cech, 1 9 8 8 ) would be shown to approach an optimal morphology based on the appropriate reminder that hydromechanical speed with drag stamina successful strategies possible. 110 is analysis. but one It of is a also a number of 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 3 Total laminar • 1 5 Fineness Ratio + Total turbulent Drag / Volume vs Fineness Ratio Re-OOOO. Cd based on wetted surface 00 • • O I • < 1 • • 50 A 40 A • • + + • • • q> O > \ 30 q> o c o 8> 20 A c Q io A o 3 • Total laminar 5 Fineness Ratio + Total turbulent Drag / Volume vs Fineness Ratio Re-10500. Cd based on wetted 150 surface HO [] 130 - • 120 - o I HO - < .6 • 100 • • • • 90 O > \ fl> O C o k s> c Q oo A 70 00 + + -| 50 AO 30 - 20 - 10 0 / "T" 5 3 Fineness • Total laminar Ratio + Total turbulent • • Drag / Volume vs Fineness Ratio Re-15000. Cd based on wetted surface 260 O I ( ^6 - 240 - 220 - 200 - 0 • O 140 o (o. u 8> Q • • • • 160 100 -| \ 0) • • Q> E > on 200 + + 120 - + 100 60 00 - 40 - 20 - 0 5 3 / Fineness • Total laminar Ratio + Total turbulent + Drag / Volume vs Fineness Ratio Re-30000. Cd based on wetted surface • • • • • • • • + 1 1 1 3 • Total laminar 1 5 Fineness Ratio + 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 increase in at the very proportion least, to the the skin thrust power friction, requirements 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 size the must propulsive increase production in apparatus at least is directly related (pectoral, an to dorsal isometric the and fashion square of the grows in anal fins) since span thrust of the propulsive fin. Fin morphometry between fin ray and such, as analysis length thrust and reveals body that length compensation is appears the relationship negatively allometric not to be achieved morphometrically. This is not unusual as the mechanics of loading for the fin rays argues moments the against morphometric on the fin rays are span necessitating and and as a compensation the also proportional to the result exponential since quickly increase become in bending square of rather large, the strength of the not indicated in the load bearing components of the fin (ie. fin rays). Since present morphometric analysis, it is compensation predicted 117 is that thrust compensation is accomplished frequency, through adjustments amplitude and in the wavelength of kinematic the parameters, undulatory propulsive fins, either singly or in concert. Conclusions to be drawn from this section include 1) Although viscous nature pressure of fluids, drag is always present it is considered due that to the the major component of the total drag encountered by the body is frictional drag. 2) The between being true those estimate of fins intact for a minimum estimate the total and of the drag fins curve amputated, likely falls the latter total drag encountered by an actively swimming fish. 3) The drag estimates are valid since the drop tank protocol is reasonably body is steady matched by an actively rigidly held forward, similitude in rectilinear between specimens a swimming fish "stretched-straight" progression of and all sizes wherein the position the has during hydrodynamic been demonstrated (in the morphometries section). 4) based The on majority the (/?<>-5000-2xl0 ) 4 curve. The drag of the Crxexp) measurements, estimates fall between (fins .02 amputated), and and are, in the main, centered around the CD(ex ) P estimates are comparable to those .03 CT(iam) 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 eyes, opercula nares, 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 (^10 ). 4 6) Power requirement curves are similar to those expected on the basis of dimensional analysis and are minimum estimates (fins amputated) fish. of the power requirements The two different for an actively curves for specimens with fms swimming amputated are considered to be estimates of the same curve. 7) Morphometric accomplished parameters; analysis indicates thrust compensation is through alteration of one or more of the kinematic frequency, amplitude and wavelength for one or more of the propulsive fins. 119 CHAPTER THREE: KINEMATICS INTRODUCTION The based earliest methods of classification of swimming modes are on the location rather than the kinematic or functional aspects (Breder, 1926). This approach has resulted in some rather different 'fish, category. electric kinematic For instance, fish dorsally. in terms, both the (Gymnarchus) However, the being the same seahorse (Hippocampus) have seahorse lumped into their oscillation propulsive and the fins frequency is located 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 fin more recent approach is to classify kinematics (Blake 1979a, 1983d). This fish based on their system results in the electric fish being included in the so-called Group I forms which are characterized large amplitude by fin and long oscillations wavelength. showing The low seahorse frequency, 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 fluid flow tall. A horizontal partition separated the opposing 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 driveshaft upstream of the fish. ' An and propeller circulated the electric water. motor The flow with 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 an from the image analyser cinefilms by digitizing successive frames with (PCD Model ZAE-3C) micro-computer disk for subsequent analysis. 121 and stored on a Data Analysis Methods The angle swept by the propulsive fin rays (8) is calculated by applying solid analytical geometry from the cin6film records. to the digitized data taken 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 R =(x .x ) -r(y -y ) +(z -z ) 2 2 2 2 to 2 i 2 i 2 i (z -z )=[R -(x -x ) -(y -y ) ] 2 2 1 2 2 i 2 2 1/2 i By defining the fin ray base as the origin {x'=y =z =0) it follows that z=(R -x -y ) 2 2 2 and the position 1/2 of the fin in three dimensions at any point along the subarc (s) ray tip traced by the fin ray tip can be calculated. The- straight-line distance the subarc half-cycle traced is also by the (L) between the two endpoints of fin ray tip calculated from the over distance the course of a formula and the angle swept (6) is obtained through the application of the law of cosines L = a+b-2ab(cos9) 2 2 2 and since, in this case, a=b=R cos9= 1-(L /2R ). 122 2 2 The amplitude (A) of the waveform present on the fin is the subarc Fin traced by the oscillation frames cin6film frequency elapsed during a framing measurements fin tip during the rate. is derived from complete Propulsive fm-beat course of a half-cycle. the number of cineTilm cycle wavelength is divided by obtained of fin waveforms traced onto transparencies from digitizer screen. 123 the from the RESULTS the Of three kinematic parameters examined, the mean frequency (f = average cycle time" for a given velocity) varied proportionally (lit 1 with specific velocity = velocity/body length) (Fig. 48, Append.I). Values for specific length) and specific length) present does tend to amplitude (As = mean amplitude/fin base wavelength a large increase (Ks = mean wavelength/fin amount with of base variability and although As Ut, neither As nor Xs vary with velocity in a statistically significant fashion (Figs. 49 & 50). The frequency, amplitude and wavelength anal fins pectoral (Fig. are fins 48, is virtually identical and the of the dorsal and frequency for the matched by those of the dorsal and anal fins Append.I). Derived values summarized in Table VI. 124 of kinematic parameters are During propelled the and by caudal through the steady is in swimming pectoral the and and of anal dorsoventral both fms fins lie in oscillate the median plane longitudinal axis of the dorsal rays rectilinear paired held median median the undulatory, fin the forward, fish fins is while which passes body. The bases of this dorsoventral symmetrically plane about their bases in such a fashion as to cause a propulsive wave to travel along the thereby length generating bases are fish immediately horizontal the on posterior as position and are oriented fins laterally position the the the the of passing left and and at opercular maximum anteroposterior thrust. the to at a mean plane in forward-directed located dorsoventral with of and to produce ventrally the component. propulsive thereby A fin right sides the the of same median, (which coincides shoulder position) angle of incidence of 48° above a through a pectoral openings median body, along an axis about which the oscillate The the thickness, direction wave generating vertically-directed longitudinal axis of fin rays symmetrically which a travels posteriorly forward-directed thrust component thrust provides the lift force required to overcome any excess weight over buoyancy. The dorsal and anal fins oscillate with similar amplitude and wavelength in a synchronous manner eliminates the longitudinal laterally tendency axis since in opposite to the rotate fin frequency, which virtually the body about its rays for both fins directions in planes at an angle median rotate of 42° to the median longitudinal axis of the body. Were it not so the fish would tend direction to rotate about the during the first median half-cycle 125 longitudinal axis in one and then back in the opposite very direction same median center during synchronicity longitudinal of mass the second which axis provides produces (located half-cycle. roll moments approximately at The depth of section at the the stability about the force about the of the shoulder which tend to cause lateral yawing movements tail. However, caudal position) at the head and 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 anteroventral pectoral fin rays coincides (downward) vice versa. The phase arc of shift the with completion of the left pectoral fin rays and 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 with completion of the half-cycle pectoral fin rays coincides 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 126 upwards (dorsoposteriorly) and 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 Specific Amplitude 1 -! ' —1 • < < • • > GO CD O — < • TD — • • < < • o o —• • 1- A pectoral V dorsal f i —1—_ • • • cn CD < < • CO CL < < < CO < o o —mm • r-r- Wavelength vs Velocity 5.0 T • — _C CD > UJ UJ V a V o *o 0) CL V V V 3.0 CO . • A A A pufferfish pectoral fin V pufferfish dorsal fin 2.0 3 4 Specific Velocity (l/s) Fin Frequency vs Velocity O # A A Q H + Triggerfish, dorsal fin Triggerfish, anal fin Mandarin fish, pectoral fin Pufferfish, pectoral fin Pufferfish, d o r s a l / a n a l fin 1 2 3 4 Specific Velocity (lengths/s) DISCUSSION Pufferfish Fin Kinematics The results pufferfish of the fin kinematics place the somewhere in the intermediate zone of the broad range encompassed the by the functional Group I and Group II forms (ie. /=40 analysis Hz, electric fish /-2.5 classification of fish into pufferfish /*5-15 Hz, seahorse Hz) in the company undulatory MPF swimmers such as triggerfishes, of other 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 independently with direction the output of relative complex to manouevre that individual respect to propulsive the fins fin wherein are frequency, wave and capable of amplitude, direction base axis. As the fish at an of operating wavelength, the example, once thrust for a descends while swimming in reverse and simultaneously turning to the right, the body the and caudal fin are reverses the direction of arched to left, the dorsal fin the propulsive wave as does the anal fin, although at a lower amplitude and frequency than the dorsal fin. The right pectoral direction the (with fin a frequency dorsal fin). The left also reverses and amplitude pectoral fin, on the propulsive wave greater the than that of inside of the turn, acts to keep the longitudinal and lateral axes of the body in 4 an orientation close of wave and changing the propulsive to horizontal 135 by reversing the effective the direction direction of 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 half-cycle' movements. of fluid, flowing dorsal fin motion, fins during their This coordination avoids in a direction which a turbulent stream antagonistic would presumably dorso-posterior to that of the increase fin friction and interference drag. With fins respect would to eliminate stability, any while tendency to contra-rotating roll about pectoral 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 suggests the of multiple existence of appendages a central program. The coordination of during locomotion nervous system motor-control fin kinematics and correlation with swimming velocity is not unlike a gait pattern as seen in many terrestrial locomotory organisms. Although the process is determined master by the control of brain, there evidence from a wide variety of organisms the neuronal organization of the spinal is ample which indicates that cord is responsible the basic patterns of locomotory movements (Roberts 1981). 136 the overall for Comparisons Of with other species the kinematics oscillate relatively of few propulsive median and studies systems or which for paired fish amplitude, As, and specific examined which fins, frequency varies positively with velocity specific have the undulate fin the or oscillation of swimming. Values for 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, found 1978). Half-wavelength, to vary during slow amplitude and wave velocity were forward progression (U**\) but the total number of propulsive wavelengths (ca.l) present on the fin at any given time at constant. For the pectoral pufferfish, frequency increases linearly with Ut from about 10 Hz (C/i*2.25) any given velocity and dorsal/anal to 15 Hz (£/i*4.75). Although 137 At remains fins relatively of the tends to increase with Us, the from zero. Likewise, regression coefficient is not significantly the curve for Xs remains different over the Us flat range. The range rectilinear swimming triggerfish are pufferfish being fin, 51). for the Employing fin the swimmers suggested during pufferfish, with highest, anal values the comparable, triggerfish between Us and / of the mandarin values followed by and mandarin functional steady of the fish fish / of and for triggerfish pectoral categorization forward, the dorsal fins (Fig. undulatory fin by Blake (1979c), all three of these fish fall the Group One and Group Two forms, with intermediate values for frequency. Blake electric (1980b) fish, compared the dorsal fin kinematics Gymnarchus niloticus, and the of the seahorse, Hippocampus hudsonius, two undulatory fin swimmers with markedly different fin kinematics both the electric fish and mode of life. After Breder (1926), and the seahorse are classified as members of the same swimming mode, the amiiform mode, due to the use of a dorsal fin functionally, (1979c) to as the this is clearly absurd propose system. The short one sixth (=15%) ribbon-like dorsal than half main a based of fin propulsive unit. Morphologically and and is likely more what function-orientated led Blake classification dorsal fin of the seahorse is less than the of total the (=55%) of the total body electric body length fish while extends length, over the long, over more 3.5 times the ratio of -fin to body length of the seahorse. The extremely high fin-beat frequency of the seahorse sharply with that of the electric (41 Hz at fish 138 £/i=1.21) contrasts (2.4 Hz at f/«=.57). The seahorse represents the undoubtedly forms while the electric fish upper likely margin of Group Two represents the lower end (Group One forms) of the continuum which encompasses the more intermediate fish kinematic and forms triggerfish. Although are between those of for As and Xs are for the those pufferfish of the As-.08). Xs*3.5) are morphology. The shape (dorsal for the the the fins is the related to given the amount amplitude. electric ratio the of the contrast with electric fish \**3, dorsal seahorse (dorsal the long relative seahorse is suggested the pufferfish in size the from for the by the are triangular the to fin the fin anterior, which is since results in for a a larger base length of the ray length and the pufferfish and the (discussed in the parameter is the fin ray fin between rays oscillation, fin ray dorsal in anterior fin a longer contrast, is the the values trailing edge. The fin base length of of rotation, fish of length amplitude In for to for the Values of As (pectoral is decreasing posterior, relative pufferfish pufferfish (dorsal ta«.51). The reason leading edge to the small fish, As*2.0) and mandarin the pufferfish. dorsal than for electric the pufferfish rays pufferfish, values As*.3) larger propulsive with the / As=1.5, electric fish for the greater for (pectoral Xs the values in far as seahorse and considerably Xs*.29) and high the seahorse (dorsal such the electric fish. For the following area length purposes chapter), swept and by the the the fin of Actuator-Disc theory relevant rays, amplitude of fin kinematic which is related oscillation. areas are in the order of 1.0 cm 2 139 and 1.3 to The cm 2 the pufferfish disc for the pectoral and dorsal fins, respectively. electric fish arc .5 cm for and 1.24 cm 2 of Values the seahorse and respectively. An indication 2 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 seahorse and (<*20 cm /s) is 2 the reverse frequency electric may efficiency fish be since large mass of and the disc followed by («3 apparent it fluid is cm /s). the pufferfish It 2 with area. On this is respect energetically to a low less basis, the (*15 anticipated to values costly to eventual velocity cm /s) 2 that of the Froude accelerate a than it is to accelerate a small mass of fluid to a high eventual velocity. Webb (1973b) has studied oscillatory fin propulsion in aggregata, another of the the the shiner so-called kinematics sea of perch, labriform lift-based Cymatogaster 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 length, resulting in a phase difference twice the trailing edge of about 7t between the movements of the anterior and posterior fin rays. In fin pattern B, the difference wavelength of is much longer and results in a phase 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 that at Over are frequency Ui>\.5, the product a the range order t/s>2.8. Webb also found and that Us and amplitude is linearly related to Us. of of at and amplitude increase with of frequency similar in B predominates swimming speeds, 5.5 times (£/«*2.5) pufferfish and 3 frequencies times (C/s-4.5) three representatives of the higher than the sea perch frequencies. The mechanics of swimming for ostraciiform studied swimming two species gibbosus, which mode of themselves fins; dorsal, that fin-beat (2.7, 3.8, 4.5, 3.3 Hz for fms, respectively, propulsive to side wave. of fins in is higher at the a wavelength the sculling and two £/s=l). / Blake He other pectoral greater also found could be fin which caudal are oscillating without and Tetrasoma five propulsive fins.Blake found swimming velocities that as much as all the oscillates from side found on producing relatively fish (1977) pectoral, dorsal, anal and caudal fashion values the examined. by and at except a These pufferfish caudal, frequency three-quarters been boxfish, Lactoria cornuata propel anal, have low examined an undulatory compared except for to the the electric fish (2.4 Hz at Us*.6). The species other of representative trunkfish, of the ostraciiform Ostracion lentiginosum, mode for is which a / proportional to Us over a wide range of / from 5.5 Hz (£/s=l.l) to 16.6 Hz (£/ =4.7) (Blake 1981b). The propulsive fin kinematics for s this trunkfish the ultimate can, a twist of sub-carangiform (sic) swimmer rigidly held and sculling motion of with the the propulsion caudal fin, 141 imagination, comes which since be the exclusively is considered body from analogous to is the an is oscillating rudder on a boat. The / and pufferfish are similar to those for the trunkfish Most fish locomotion studies Us values for the (U%>2). have examined fish species which produce thrust by oscillating various portions of the body and caudal fin; the modes. anguilliform, carahgiform and sub-carangiform Bainbridge sub-carangiform goldfish (1958) swimmers found such as (Carassius auratus), and tail-beat frequency is related to As increases with / constant for carangiform dace trout (Leuciscus and leuciscus), (Salmo gairdneri), that swimming velocity (at fe5 Hz), up to <*5 Hz whereafter it remains relatively and the product of / and amplitude is linearly related to Us. Webb (1971a) found similar results for rainbow trout. The values of the pufferfish those found in the frequencies above are in the order of double 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 1.3-2.8) .08-. 10) are related and that the to swimming propulsive wavelength velocity of (Us range the body is relatively constant (Videler & Wardle, 1978). It is for the interesting muskrat, generated during to note the results Ondatra zibethicus. surface swimming by found by Fish (1984) In this alternate species thrust oscillations of the hind feet. It was found that the arc through which the hind feet travel (ie. amplitude) increases with swimming velocity while the oscillation frequency remains constant at 2.5 Hz over a swimming results velocity range of 0.2 to 0.75 m/s, not unlike the found for some carangiform and sub-carangiform swimmers 142 is over the unifying lower regions principle of relating Us values. their fin oscillation / , Perhaps there is a 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, trunkfish, boxfish) that fish, triggerfish, have been seahorse, studied, there electric 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 product of / and As is proportional to Us. 143 constant and the 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 deceleration-in-glide, experiments. measured towing Theoretical combining empirical from tank and drag is terminal water and calculated observations of velocity, wind from technical tunnel equations 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 output thrust-based can also be approach to subdivided estimating into two propulsive categories: power direct and indirect measurements. Direct measurements involve connecting a specimen to a force measuring balance (Houssay, 1912; Gero, 1952; and measuring the rate of Lang & Daybell; 1963; Gray, 1968) 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 144 (ie. 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 of combination frequency, is kinematic parameters conversion measured power is power and with calculated respect as standard factors the to to from oxygen (ie. the application consumption swimming difference (resting) locomotion with hydromechanical theory; power estimates which result oxycalorific that the amplitude and wavelength) and metabolic of of velocity. Propulsive between metabolic data total power. metabolic 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 categories: models resistive and can be divided reactive. Both into the two general resistive reactive models divide the body into a series of and segments and consider the motion of the segments as simple harmonic motion, and that the BL is laminar. Resistive theory segments operate considers 1) to be viscid, 2) the fluid in the resistive which force the 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 to the fluid, a segment with respect resistive force steady-state, acting resistive on forces the and 4) segment is acting on the the that the total sum of segment all the 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 an all estimate the approaches listed in of propulsive efficiency the preceding paragraphs, can be dividing the minimum power output necessary calculated by to overcome drag, either by measured or calculated, by the total propulsive power output estimate generated by the particular approach chosen. In this chapter, morphometric, previous during from hydrodynamic chapters steady the Actuator-Disc to hydromechanical estimate forward, model and are theory. kinematic propulsive rectilinear compared Power theory and 146 is parameters power swimming. with applied Estimates those efficiency and to the from the efficiency obtained obtained estimates from from a 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 Momentum device, Theory roughly based which considers analogous to a the ship upon the Rankine-Froude fin to be an idealized 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 losses in the across the disc. 2) There are no rotational velocity energy 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 disc. and behind the • 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 sinusoidal velocity nature of profile across the disc the propulsive fin 148 since, oscillations, owing to the the fin ray must, at direction, the completion and then of every accelerate half-cycle, to reach decelerate, reverse velocity, roughly peak 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 profile variable velocity velocities in the wake are unknown (Blake, 1983a). However, the results of flow (Blake, unpublished) across requirements attributable to a visualization the (Blake, experiments disc 1976,c,d) indicate that observed about the fins of S. picturatus undulatory fin swimmers, and to rotational and anemometry the flow patterns and H. hudsonius, two 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 kinematics Actuator-Disc model between a fin applies, system and but such the difference in 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 149 be underestimates 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 of kinematics uncertainty must be regarding output. As long system, ie. effects are as balanced underestimation any energy significant of the total velocity, relative to fin disadvantages thrust losses associated rotational, non uniform not against power with the fin profile and tip 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 undulatory requirements fin systems of the fish for which swirriming the by means of Actuator-Disc model is appropriate (Ellington, 1978; Blake, 1983d). In applying the imparted to the model, the fluid, rate of change integrated over the disc in momentum area, is related to the thrust required to overcome the drag of the body. Four discrete regions of this model: far stream flow upstream, are defined for the purpose of immediately upstream of the fm, immediately downstream of the fin, and far downstream. Recalling Bernoulli's theorem o.spU + pgh + 2 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.spU . 2 Immediately adjacent to the upstream side of the 150 disc (just increased before to o.5p(f/+v) . 2 i the U+v and Immediately disc (just after fin) the the velocity pressure adjacent to to the is considered to have have decreased to p + downstream side of the 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 + o.5pf/ = p + 2 0 osp(U+v) 2 and the two downstream conditions p + Ap + o.sp(U+v) = p + o.5p(C/+v ) 2 2 0 2 and subtracting the upstream from the downstream we have Ap= p(U + osv )v 2 The thrust generated by the fin 2 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= S Ap= S p(£7 + o.sv )v d d 2 2 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$ (U + v^v which is equal to the d thrust so pS (L/ + )v = pS (L/ + o.sv)v d Vl 2 d 2 2 and 2v = v. 1 In 2 order encountered (1980b) to circumvent the formidable in obtaining accurate estimates of devised a method to calculate v difficulties and v , Blake 2 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 = v , thrust becomes 2 T= 2pS U (E + E ) 2 2 Thrust and drag are equal in magnitude during steady forward 2 swimming, D- T= o.spU S , so that 2pS f/ (E + E ) 2 2 d CD= and - o. spU S 2 4£ 2 + 4H - (S C D / S > w a w -4 ± [16 + 16(S C D / S J ] SO 8 w 0.5 d = [(l + s CD/s r -1]/2 J d W By defining the slipstream velocity (v) as v = (U + v> s U(\ + 2 S ) = £7(1 + S C D / S w 2 we find v = (v - LO and v= o.5(v- LO- The useful power output (Po) is given by 152 d ,0.5 V 0 Po= TU= 2pS L> E(l + E) 3 d while the power input (Pi) is given by 7 3J - / , . -\2 Pi= T(U + Vj)= 2pS (7 E(l + E) d so the ideal efficiency is found from The ideal efficiency is an estimate efficiency of the system to which it is applied. 153 of the propulsive RESULTS The a low high of estimates of of ==5.7xl0~ *2.4xl 0" the fm than rigid to These values body (Fig. the Watts 5 in the of P(in) for 4 VD). to from U**4J, and for Watts the over 1.5 and respectively) flow power 2, 1.4 times (Pcum)) conditions. of a Estimates range 1.2, total values from "2.3 fm at order laminar yield Table 2.7X10" range of (Us) pectoral to dorsal in (/W)) Watts minimum equivalent 52, 2.8x10^ and P(XUT) output velocity for are theoretical with specific (/'(in)) 6.5xl0' P(in) power I/i«4.7 from the comparisons at £A*2.3 pectoral greater at input at Us' range. (P(OM), a power Watts dorsal same Watts 4 5 propulsive Watts 5 propulsive *7.2xl0" of of Similar 2.5 and 2.3 is greater for pectoral P(in) and dorsal /'(in), respectively. /'(out), In all that for (T|) values cases the the dorsal where of /'(in) fin. The dorsal T) the . pectoral reverse ranges fin is true for the from .87 to .89 other fish, than efficiency while the pectoral r\ ranges from .79 to .85 (Table VII). Compared P(om) is an with power order of outputs magnitude for some lower that the the nearest puffer curve at range of for the comparable Us (Fig. 53). The 2.3-4.7. puffer the efficiency for rank other values both at the fins top undulatory increase (Fig. of MPF 54). the range swimmers 55). 154 slightly over Efficiency of values that the Ut estimates achieved have been by some studied of (Fig. TABLE VII. l/s Power and Efficiency estimates Foul Fin Finn Fin Fmr dorsal fin pectoral• 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" Watts) 4 T): ideal efficiency Fi«m: rriinimum power output (x 10" Watts), rigid body equivalent, 4 laminar flow Par. minimum power output (x 10" Watts), rigid body equivalent, 4 turbulent flow 155 DISCUSSION Estimates the of Actuator-Disc kinematic and model parameters; drag and rj are calculated P(in) P(om), with the swimming coefficient. relevant velocity, calculated based on the pufferfish morphometric fin Theoretical by supplying span, drag and angle swept coefficients are mean FR (3.37) and combined with the wetted surface area and forward swimming velocity of the pufferfish in calculate the standard theoretical hydromechanical drag equation minimum power curves for laminar to and P(iam) turbulent P ( w r ) flow conditions. The departure theoretical variety of the pufferfish minima (Fig. 52) of fish species maximum departure of is F(out) a common and therefore estimates from occurrence unsurprising. In <*1.2 times (lam) the among a fact, the 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*10-10) or even up to 10 times 4 (Yates, ratios 1983) of =2.5 measurements pufferfish found for and of mackerel 4 from in theoretical (/te=2xl0 ). the Although the in the the CD(the) minima. ratios PcexpyPdhe) Actuator-Disc P Richardson (1936) found and herring from terminal velocity The CD(ex )/CD(ihe) 6 model ratios are plate themselves terrifically meaningful, 156 similar calculated values for a flat obtained in to chapter equivalent for those II. are not, they do provide a convenient method to compare the relative magnitudes of kinematic parameters or morphological factors and thus, the drag and which can elevate the estimates P(out) above their CD(exp) 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 the 1/2 of the median body (and caudal fin) longitudinal axis, the oscillated trout laterally across encounters a substantial increase in swimming drag from the increased friction drag which results of from the the thinning of propulsive segment the is BL, since greater the than lateral velocity 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 (Blake, theory: the trunkfish (Blake, 1981b) and the knifefish 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 rainbow trout with a end and *6xl0" 5 at the P(oui) lower are bounded (at value of «8xl0" end by the 2 £/s*3) by Watts at the upper pufferfish P(om) value of Watts, spanning three orders of magnitude. Webb (1971b) 157 found good agreement between P(om) estimates made by conversion of oxygen theory consumption (Lighthill, rainbow trout. independent values 1969) In to the estimate and by the present of applying elongated propulsive kinematics study, there with which F(om) is to Actuator-Disc model estimates, other than the for a flat no body of the alternative, compare the theoretical minimum 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) p steady swimming and salmon (-.9). T| estimates for p (=.8-. 9) and for burst and glide swimming (=.55-.6) are calculated for Zebra Danio by McCutcheon (1977). Surface swimming rj =.27-.33. p Blake in muskrat (1980a) (Fish estimates 1984) T J at p is .15-.3 pectoral fin rowing. Sea lion fore flipper Ti estimated for at angelfish is estimated at «.8 at £/ =2. s 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, is high fairly the (=.86) propulsive and increases efficiency slightly of over the the pufferfish velocity range measured (t/s=2.3-4.7). During the course of this study it became apparent that the 158 simplicity and ease (Blake, 1980b) systems is thrust for could which be mitigated power the with the applied of an independent estimate such, the model not lack requirements and as a force balance, MPF model undulatory the of Actuator-Disc to by construction modified propulsory does of provide supplied on either side study, has never been by independent estimates of thrust and drag. The. Actuator-Disc applied Only to multiple single fin model, fin prior to propulsive propulsive this systems systems have of been aquatic organisms. analysed previously (Blake, 1979c, 1980b). The Actuator-Disc principal rise, of momentum integrated actuating through of induced the (ie. disc. velocity is the idealized considers area, fin) thrust and an disc a The momentum is which over element, the change model in the force the is of based on the and instantaneous pressure to be by created steam of proportional minimum product device power fluid to the an passing rate of required to create the thrust force and the free (1980b) offers a the stream velocity. Based calculate the equating these that wetted principles, induced thrust provided the on velocity with the drag disc Blake of the fluid during area swept surface area and problems arise however, steady by drag through the the forward actuating coefficient of method disc to by swimming, element the and body are system. One known. Two stems fluid from velocity the fact that the through the disc for a inflow to 159 the multiple factor free fin (the steam ratio of velocity) the is 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 thrust from this that a power balance (drag power = power)cannot estimates • are with be computed respect to the from the power model. The /'(out) 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, propulsive treat the unit in system as if operation. the dorsal fin Therefore the is the only 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 (/'(out)*10' 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 160 of the Actuator-Disc model can be applied is difficulties discovered Specifically, between /'(out) multiple and during P(in) propulsive the estimates units course appear mitigated of to and calculation this be of study. indivisible thrust and power balances are not afforded by the model as it now stands. 161 by the 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 3E-4T 0-1 2 • 1 3 ' 1 4 Specific Velocity (l/s) 1 1 5 Power Output (Watts) m I m CD o N) -rj :> CD o. —h — • o CD • < < CO o o < o o o 4 d. c ^ CD n Q O - 5 o J % CD 0) zr o c o o o £91 Ideal Efficiency p o N> H o ro 1 1 o 1 V 1 p o CO CD 1 1 H- O m CM CD n —h < c m < CD_ O GO o C/5 > CD* 13 O c m + o a. T J o CD Q t U c m O am o m < o o Efficiency vs Velocity 1.0 n• 0.8 + • _r3l O 0.6 •i • o o LU I I o o o oA 0.4 + A A A 0.2 0.0 H 0 i g CP o o c ON i h H i\ • 1 O # A O • • V 2 3 4 Specific Velocity (l/s) mandarin fish electric fish seahorse trunk fish knife fish pufferfish pectoral pufferfish dorsal , 1 . SUMMARY 1) analysis The morphometric parameters are well defined relevant and to hydromechanical 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 times greater than 7) CD(the> (7?e*5xl0 -2xl0 ) 3 4 and range from *1.7-2.6 values. Analysis of fin morphometry suggests thrust compensation is accomplished kinematically. 8) The common factor apparent from comparisons of propulsive fm kinematic proportional ranges to t/s<l-1.5 parameters swimming for is that velocity some fin (except species). 167 oscillation For over MPF frequency lower is velocity swimmers, a 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(ihe)and other estimates P(ou\) are «1.2-2.5 times greater than are =^1-2 orders of magnitude less than those found for MPF swimmers at similar velocities. Watts, (PcouoadO" 4 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 Specifically, among respect The multiple to P(pat) analysis and propulsive pufferfish -estimates P(in) units of and are calculation locomotion. not of thrust and power balances are not possible with the model as it now stands. 168 divisible LITERATURE CITED Alexander, R. McN. (1977) Functional design in fishes. London: Hutchinson. Aleyev, Yu. G. (1977) Nekton. The Hague: Junk. Allen, W. H. (1961) Underwater flow visualization techniques. U. S. Nav. Ord. Test Sta. Tech. Publ. 2759: 28p. Bainbridge, R. (1958) The speed of swimming of fish as related to size and to the frequency and amplitude of the tail beat. J. Exp. 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S. Hoar & D. J. Randall, New York: Academic Press. Wu, T. Y. (1961) Swiniming of a waving plate. / . Fluid Mech. 10:321-344. Wu, T. Y. (1971a) Hydromechanics of swimming propulsion. Part 1. Swimming of a two-dimensional flexible plate at variable forward speeds in an inviscid fluid. J. Fluid Mech. 46: 337-355. Wu, T. Y. (1971b) Hydromechanics of swinirning propulsion. Part 2. Some optimum shape problems. J. Fluid Mech. 46:521544. 174 Wu, T. Y. (1971c) Hydromechanics of swimming propulsion. Part 3. Swimming and optimum movements of slender fish with side fins. / . Fluid Mech. 46:545-568. Wu, T. Y. (1971d) Hydromechanics of swirnming fishes and cetaceans. Adv. Appl. Math. 11:1-63. Yates, G. T. (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. Relation Summary of regressions a Sy.x b Sb n R2 Ho .05 A 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 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 0.454 0.158 0.072 0.108 0.869 0.080 20 0.87 1.308 0.091 20 0.92 B=1 NSI 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 B=2 NSD 13 Sw/length *-0.102 0.035 2.117 0.089 18 0.97 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 B=2 NSD 18 0.80 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 B=2 NSD 20 SwCaud/1 *-1.236 0.085 21 ProjA/1 22 SwB/Vol *-1.099 0.046 * 0.751 0.024 18 0.73 2.132 0.216 18 0.86 2.211 0.111 20 0.96 B=2 NSD B=2 NSD 0.671 0.023 14 0.99 B=.67 NSD 176 APPENDIX I. Relation continued a Sy.x 23 Volume/1 *-1.353 0.034 24 Wt(aix)/1 M.251 0.036 b Sb n R2 A 3.206 0.102 14 0.99 3.124 0.109 14 0.99 Ho .05 B=3 NSD 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 i0.052 •-0.012 0.010 21 0.08 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 14 0.86 0.321 0.038 B=0 NSD B=0 rej 33 Nare:h/1 -0.530 10.288 0.498 0.082 12 0.79 B=0 rej 34 Nare:x/1 -1.574 l0.769 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 0.001 1.971 0.002 10 0.99 B=0 rej B=0 rej 40 FV 23 M.121 41 FV 16on *-1.451 0.001 1.175 0.002 42 FV 16off *-1.706 0.001 1.238 0.002 10 0.99 43 CDRe 23 •-1.415 -0.035 44 CDRe 16on * 1.811 -0.825 177 7 0.99 B=0 rej APPENDIX I. continued Relation a Sy.x 45 CnRe 16of * 1.369 b Sb n R2 Ho .05 A -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 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 /pect/t/s 59 A.cpeco/f/, 0.805 0.131 0.012 0.056 60 A,(don)/Us 0.824 0.150 8 0.01 0.071 0.064 B=0 8 0.17 NSD 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. Number continued 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 cm grid. 2 183 184
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Functional design and swimming energetics of the freshwater pufferfish, Tetraodon fluviatilis Varley, Robert Mark 1989
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Title | Functional design and swimming energetics of the freshwater pufferfish, Tetraodon fluviatilis |
Creator |
Varley, Robert Mark |
Publisher | University of British Columbia |
Date Issued | 1989 |
Description | 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 cinefilms 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 swimming 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 discussed with respect to negating factors found during this study for the application of the model to multiple fin aquatic propulsive systems. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-08-24 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0097569 |
URI | http://hdl.handle.net/2429/27678 |
Degree |
Master of Science - MSc |
Program |
Zoology |
Affiliation |
Science, Faculty of Zoology, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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