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

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FUNCTIONAL DESIGN AND SWIMMING ENERGETICS OF THE FRESHWATER PUFFERFISH, TETRAODON  FLUVIATIUS  by ROBERT MARK VARLEY B.Sc, The University of British Columbia, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Zoology) We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA October 1989 © Robert Mark Varley  In  presenting this  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,  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 + )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/ + 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. Biol. 35: 109-133. Bainbridge, R. (1960) Speed and stamina in three fish. J. Exp. Biol. 37: 129-153. Bainbridge, R. (1961) Problems of fish locomotion. In Vertebrate locomotion, Symp. Zool. Soc. Lond. 5: 1332.  Bainbridge, R. (1961) Problems of fish locomotion. In Vertebrate Locomotion. Symposia of the Zoological Socie of London, (ed.) J.E.Harris, 5:13-32. Batchelor, G. K. (1967) An Introduction to Fluid Dynamics. Cambridge, U.K.:Cambridge University Press. Bernoulli, D. (1738) Hydrodynamica.  (in Hoerner, 1965)  Blake, R. W. (1976) On seahorse locomotion. J. Mar. Biol. Assoc. U. K. 56: 939^949. Blake, R. W. (1977) On ostraciiform locomotion. J. Mar. Biol. Assoc. U. K. 57: 1047-1055. Blake, R. W. (1979b) The mechanics of labriform locomotion. I. Labriform locomotion in the Angelfish (Pterophyllum eimekei): an analysis of the power stroke. / . Exp. Biol. 82: 255-271. Blake, R. W. (1979c) The swirriming of the mandarin fish Synchropus picturatus (Callinyiidae: Teleostei). J. Mar. Biol Assoc. U. K. 59: 421-428. Blake, R. W. (1979d) The energetics of hovering in the Mandarin fish (Synchropus picturatus). J. Exp. Biol. 82: 25-33. Blake, R. W. (1980a) The mechanics of labriform locomotion. II. An analysis of the recovery stroke and the overall fin-beat cycle efficiency in the Angelfish. / . Exp. Biol. 85: 337-342.  169  Blake, R. W. (1980b) Undulatory median fin propulsion of two teleosts with different modes of life. Can. J. Zool. 5t: 2116-2119. Blake, R. W. (1981a) Influence of pectoral fin shape on thrust and drag in labriform locomotion. J. Zool. (London). 194: 53-66. Blake, R. W. (1981b) Mechanics of ostraciiform locomotion. Can. J. Zool. 59: 1067-1071. Blake, R. W. (1981c) Mechanics of drag-based mechanisms of propulsion in aquatic vertebrates. In Vertebrate locomotion, Symp. Zool. Soc. Lond. 48: 29-52. Blake, R. W. (1983a) Median and paired fin propulsion. In Fish Biomechanics, eds P. W. Webb & D. Weihs. New York: Praeger. Blake, R. W. (1983b) Swimming in the electric eels and knifefishes. Can. J. Zool. (61) 6: 1432-1441. Blake, R. W. (1983d) Fish locomotion.  London: Cambridge.  Blasius, H. (1908) Boundary layers in fluids of small viscosity. Zeitschr. Mathematik Physik, vol. 56. Blazka, P., M. Wolf and M. Cepela. (1960) A new type of respirometer for the determination of metabolism in fish in an active state. Physiol. Bohemoslov. 9: 553-558. Bramwell, A. R. S. (1976) Helicopter dynamics. Edward Arnold.  London:  Breder, C. M. (1926) The locomotion of fishes. Zoologica 159-256.  4:  Brett, J. R. & Glass, N. R. (1973) Metabolic rates and critical swimming speeds of sockeye salmon (Oncorhynchus nerka) in relation to size and temperature. J. Fish. Res. Board Can. 30: 379-387. Brett, J. R. (1963) The energy requirement for swirriming by young sockeye salmon with a comparison of the drag force on a dead fish. Trans. R. Soc. Can. 1, Ser. TV: 441457. Brett, J. R. (1964) The respiratory metabolism and swimming performance of young sockeye salmon. / . Fish. Res. Board Can. 21:1183-1226. Burdak, V. D. (1969) The ontogenetic development of the scale cover of the mullet Mugil saliens Risso. J. Zool. 47: 732-738.  170  C.  R. C. Handbook of chemistry and physics, ed. R. C. Weast. CRC Publishing Co.  Dekkers, W. J. (1975) Review of the Asiatic Freshwater Puffers of the genus Tetraodon Linnaeus, 1758 (Pisces, Tetraodontiformes, Tetraodontidae. Zool. Genootshap "Natura Artis Magistrata." Bijdragen Tot de Dierkunde. (45) 1: 87-142. Ellington, C. P. (1978) The aerodynamics of normal hovering flight:three approaches. In Comparative physiology: water, ions and fluid mechanics, ed. L. Bolis, K. Schmidt-Nielsen & S. H. P. Maddrell. Cambridge University Press. Fish, F. E. (1984) Mechanics, power output and efficiency of the swimming muskrat (Ondatra zibethicus). J. Exp. Biol. 110: 183-201. Gadd, G. E. (1952) Some hydrodynamic aspects of swimming in snakes and eels. Philos. Mag. 58: 663-760. Gero, D. R. (1952) The hydrodynamic aspects of fish propulsion. Am. Mus. Novit. No. 1601, pp. 1-32. Goldstein, S. (1938) Modern developments in fluid dynamics, vol.2. Oxford: Clarendon Press. Gray, J. (1936) Studies in animal locomotion VI. The propulsive powers of the dolphin. J. Exp. Biol. 13: 192-199. Gray, J. (1936) Studies in animal locomotion. VI. The propulsive powers of the dolphin. / . Exp. Biol. 13:192-199. Gray, J. (1953b) Undulatory propulsion. / . Microsc: Sci. 94:551-578. Gray, J. (1957) How animals move. London: Penguin Books. Gray, J. (1968) Animal locomotion. London: Weidenfeld & Nicholson. Hamilton, F. (1822) An account of the fishes found in the river Ganges and its branches. Edinburgh: A. Constable & Co. Harris, J. E. (1936) The role of fins in the equilibrium of swimming fish. I. Wind-tunnel tests on a model of Mustelus canis (Mitchell). J. Exp. Biol. 13: 476-493. Hertel, H. (1966) Structure, form and movement. New York: Reinhold.  171  Hill, A. V. (1938) The heat of shortening and the dynamic constants of muscle. Proc. R. Soc. Lond. Ser. B Biol. Sci. 126: 136-195. Hill, A. V . (1939) The transformation of energy and mechanical work of muscles. Proc. Phys. Soc. 51: 1-18 Hoerner, S. F. (1965) Fluid-dynamic drag. New Jersey: published by the author. Houssay, F. (1912) Forme, puissance et stabilite des poissons. Paris: A. Hermann. Lang, T. G. & Daybell, D. A. (1963) Porpoise performance tests in a seawater tank. Nov. Ord. Test Stat. Tech. Rep. 3063: 1-50. Lighthill, M. J. (1960) Note on the swinirning of slender fish. /. Fluid Mech. 9: 305-317. Lighthill, M. J. (1969) Hydromechanics of aquatic animal propulsion: a survey. Ann. Rev. Fluid. Mech. 1: 413446. Lighthill, M. J. (1970) Aquatic animal propulsion of high hydromechanical efficiency. J. Fluid Mech. 44: 265-301. Lighthill, M. J. (1971) Large-amplitude elongated-body theory of fish locomotion. Proc. R. Soc, Ser. B 179: 125138. Lighthill, M. J. (1975) Mathematical Biofluiddynamics. Philadelphia: Society for Industrial and Applied Mathematics. Lindsey, C. C. (1978) Form, function and locomotory habits in fish. In Fish physiology, vol. 2, eds. W. S. Hoar & D. J. Randall. New York: Academic Press. Magnan, A. & Sainte-Lague, A. (1930) Resistance f l'avancement et puissance des poissons. Bull. Tech. Serv. Aerotech., Paris. 71: 1-107. Magnan, A. (1930) Les caracteristiques geometriques et physiques des poissons. Ann. Sci. Nat. Zool. (10) 13: 355-489. Marey, E. J. (1894) Le mouvement.  Paris: Mas son.  Marr, J. (1959) A proposed tunnel design for a fish respirometer. Tech. Memo. No. 58-3, pp. 1-13. Pac. Nav. Lab., Esquimalt.  172  McCutcheon, C. W. (1977) Froude propulsive efficiency of a small fish measured by wake visualization. In Scale effects in animal locomotion, ed. T. J. Pedley, London: Academic Press. Moyle, P. B. And J. J. Cech, Jr. (1988) Fishes: an introduction to ichthyology. New Jersey: Prentice-Hall. Newman, J. N. & Wu, T. Y. (1973) A generalized slender-body theory for fish-like forms. J. Fluid Mech. 57: 673-693. Parry, D. A. (1949) The swimming of whales and a discussion of Gray's paradox. / . Exp. Biol. 26:24-34 Prandtl, L. & Tietjens, O. G. (1934b) Applied hydro- and aeromechanics. Unabridged 1957 ed., New York: Dover Publications, Inc. Reynolds, O. (1883) An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and the law of resistance in parallel channels. Trans. Roy. Soc. Lond. 174:935-982. Richardson, E.G. (1936) The physical aspects of fish locomotion. J. Exp. Biol. 13:63-74 Roberts, B. L. (1981) The organization of the nervous system offishesin relation to locomotion. Symp. Zool. Soc. Lond. 48: 115-136. Rouse, H. (1946) Elementary mechanics of fluids. New York: John Wiley & Sons. Schlichting, H. (1968) Boundary-layer Theory. McGraw-Hill. Shapiro, A.H.  6 ed. New York:  (1964) Shape and Flow. London: Heinemann.  Sokal, R.R. And F.J.Rohlf. (1981) Biometry. Francisco: W.H.Freeman & Co.  2nd ed., San  Sundnes, G. (1963) Energy metabolism and migration of fish. Northwest Atlantic Environmental Symposium, Special Publication, 6: 743-746. Taylor, G.(1952) Analysis of the swimming of long narrow animals. Proc. R. Soc, Ser. A 214: 158-183. Videler, J. J. & Wardle, C. S. (1978) New kinematic data from high speed cine film recordings of swimrning cod (Gadus morhua). Neth. J. Zool. 28: 465-484. Videler, J. J. (1981) Swimming movements, body structure and propulsion in cod Gadus morhua. In Vertebrate locomotion, Symp. Zool. Soc. Lond. 48: 1-27. 173  Vogel, S. (1981) Life in moving fluids: the physical biology of flow. Boston: W. Grant Press. Von Holste, E . & Kuchemann, D. (1942) Biological and aerodynamic problems of animal flight. / . R. Aeronaut. Soc. 46: 44-54. Von Mises, R. (1959) Theory of Flight. New York: Dover Publications. Walters, V. (1962) Body form and swirnming performance in scombroid fishes. Am. Zool. 2: 143-149. Webb, P. W. (1971a) The swimming energetics of trout. I. Thrust and power at cruising speeds. J. Exp. Biol. 55: 489-520. Webb, P. W. (1971b) The swimming energetics of trout. U. Oxygen consumption and swinirning efficiency. J. Exp. Biol. 55: 521-540. Webb, P. W. (1973b) Kinematics of pectoral fin propulsion in Cymatogaster aggregata. / . Exp. Biol. 59: 697-710. Webb, P. W. (1975a) Hydrodynamics and energetics of fish propulsion. Bull. Fish. Res. Board Can. No. 190. Webb, P. W. (1975b) Efficiency of pectoral fin propulsion of Cymatogaster aggregata. In Swimming and flying in nature, vol.2, eds. T. Y. Yu, C. J. Brokaw, & C. Brennan. New York: Plenum. Webb, P. W. (1975c) Acceleration performance of rainbow trout Salmo gairdneri (Richardson) and green sunfish Lepomis cyanellus (Rafinesque). / . Exp. Biol. 63: 451-465. Webb, P. W. (1978) Hydrodynamics: nonscombroid fish. In Fish physiology, vol.7, eds. W. 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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