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Steady swimming in the pufferfish Diodon holocanthus : propulsive momentum enhancement is an adaptation… Chan, Keith 2010

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STEADY SWIMMING IN THE PUFFERFISH DIODON HOLOCANTHUS: PROPULSIVE MOMENTUM ENHANCEMENT IS AN ADAPTATION FOR THRUST PRODUCTION IN UNDULATORY MEDIAN AND/OR PAIRED FIN SWIMMERS by  Keith Chan B.Sc., University of British Columbia, 2007  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Zoology)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) February 2010  © Keith Chan, 2010  ABSTRACT Lighthill and Blake (1990) developed a form of elongated body theory appropriate for the analysis of undulatory fins attached to a rigid body and proposed a possible benefit due to momentum enhancement relative to the fins “on their own.” In this study, the magnitude of this momentum enhancement was determined by particle image velocimetry (PIV). This momentum enhancement depends on the ratio between half body depth for the major body axis s and the distance from the body midline to the fin tip l (s/l) and the half body depth for the minor axis i and s (i/s). The theoretical momentum enhancement factor for Diodon holocanthus were of the order of 2.2 and 2.7 for the median and pectoral fins respectively and compared well with inferred values based on thrust determined from PIV wake measurements with factors ranging from 2.2 2.4 and 2.7 - 2.9. Mean theoretical thrust for the caudal fin with no provision for momentum enhancement was compared to those measured from PIV and no significant difference was found (P > 0.05). There was no significant difference between the average sum of the total fin thrust and fish drag (P > 0.05). Pectoral fin thrust was half that of the median and caudal fins due to high jet angles, low circulation and momentum. Pectoral fin lift was insufficient to overcome the excess weight over buoyancy of the fish, implying weight support from the swim bladder and/or the body. Vortex rings generated by the fins were elliptical and depended on fin chord and amplitude. Hydrodynamic advantages are likely universal among rigid bodied organisms propelled by undulatory fins with the extent of momentum enhancement dependent on fin number, position, s/l and i/s. However, a trade-off between momentum enhancement and thrust generation sets a practical limit to the former. When s/l and i/s are large (small fins), whilst momentum  ii  enhancement is high, low absolute thrust is produced to propel a body of large surface area with consequent high drag. Limits to enhancement are also set by reductions in propulsive force associated with progressive reductions in fin wavelength.  iii  TABLE OF CONTENTS ABSTRACT........................................................................................................................ ii TABLE OF CONTENTS................................................................................................... iv LIST OF TABLES...............................................................................................................v LIST OF FIGURES ........................................................................................................... vi LIST OF NOTATIONS .................................................................................................... vii ACKNOWLEDGEMENTS................................................................................................ x 1 INTRODUCTION ............................................................................................................1 2 MATERIAL AND METHODS........................................................................................2 2.1 Morphometrics...................................................................................................2 2.2 Drag....................................................................................................................3 2.3 Steady swimming kinematics ............................................................................4 2.4 Particle image velocimetry ................................................................................5 2.5 Momentum enhancement...................................................................................9 2.6 Statistics ...........................................................................................................12 3 RESULTS .......................................................................................................................13 4 DISCUSSION .................................................................................................................29 4.1 Kinematics .......................................................................................................29 4.2 Wake momentum .............................................................................................29 4.3 Density .............................................................................................................32 4.4 Momentum enhancement.................................................................................33 5 CONCLUSION...............................................................................................................35 REFERENCES ..................................................................................................................36  iv  LIST OF TABLES Table 1: Frequency and amplitude for the pectoral, dorsal, anal and caudal fins.............15 Table 2: Momentum enhancement for the dorsal and pectoral fins..................................16 Table 3: Vortex ring measurements for the dorsal, pectoral and caudal fins in the frontal plane...................................................................................................................................26 Table 4: Vortex ring measurements for the dorsal, pectoral and caudal fins in the parasagittal plane ...............................................................................................................27  v  LIST OF FIGURES Figure 1: Schematic diagram of experimental set-up .........................................................8 Figure 2: Diodon holocanthus cross-section ....................................................................11 Figure 3: Drag versus swimming speed and drag coefficient versus Reynolds number for dead specimens of Diodon holocanthus ............................................................................17 Figure 4: Displacement versus time for the dorsal, anal, pectoral and caudal fins ..........18 Figure 5: Water velocity vector fields for the dorsal fin...................................................19 Figure 6: Fluid vorticity components for the dorsal fin....................................................20 Figure 7: Water velocity vector fields for the pectoral fin ...............................................21 Figure 8: Fluid vorticity components for the pectoral fin.................................................22 Figure 9: Water velocity vector fields for the caudal fin..................................................23 Figure 10: Fluid vorticity components for the caudal fin .................................................24 Figure 11: Theoretical and experimental momentum enhancement factor for Diodon holocanthus ........................................................................................................................28 Figure 12: Vortex rings in the wake for the dorsal, anal, pectoral and caudal fins ..........31  vi  LIST OF NOTATIONS L  body length  θ  half amplitude (angle between the most extreme lateral position of the tip of the fin to the midline)  a  amplitude (distance between the most extreme lateral positions of the tip of the fin from one side to the other)  l  distance from the midline of the body to the tip of the pectoral, dorsal and anal fins  s  half body depth  af  coordinate from the anterior to the posterior of the fin  κ  local twist of each fin  Af  frontally projected area  Ao  boundary of a small region area  A  area of the projected area of the vortex ring  Wa  weight in air  W  weight in water  ρf  fish density  ρ  water density  M  vortex momentum  Mf  momentum of the fin per unit length  Mo  momentum of the fin on its own  D  drag  Fx  thrust  vii  Fy  lift  Fz  medial component of the force  P  pressure force acting across a plane at the anterior of the fin  U  swimming speed  Ut  terminal velocity  u  velocity component parallel to the flow field x  v  velocity component parallel to the flow field y  w  velocity component parallel to the flow field z  Uv  tangential component of the velocity  V  propulsive wave velocity  ω  fin angular velocity  ɳ  swimming efficiency  tp  fin stroke period  f  fin frequency  CD  drag coefficient  Re  Reynolds number  β’  theoretical momentum enhancement factor  β  momentum enhancement factor  ν  kinematic viscosity of water  ε  vorticity  ε  vorticity component  Γ  circulation  C  curve enclosing the vortex  viii  dl  incremental path element vector  φ  jet angle  ix  ACKNOWLEDGEMENTS I would like to thank my supervisor, Dr. Robert Blake, for his support, encouragement and patience during the time of my project. It has been wonderful to have worked with Bob during these few years and it has been an excellent experience. He has been a great mentor. I would also like to thank the members of the Blake lab for their support and encouragement throughout the process. I had a great time. I would like to thank my committee members: Drs. Dana Grecov, John Gosline and Robert Shadwick for helpful comments and discussions and Dr. Sheldon Green for help with interpretation of the results.  x  1 INTRODUCTION Lighthill and Blake (1990) and Lighthill (1990a,b) employed a form of elongatedbody theory appropriate for the analysis of undulatory fins attached to a rigid body and proposed that undulatory median and paired fins (UMPF) swimmers might benefit from a hydromechanical interaction between their body and fins due to momentum enhancement relative to the fins “on their own.” Over the last 30 years or so, a number of features in functional design and swimming performance have been identified in fish propelled by UMPF. Examples occur at the level of morphology, kinematics and physiology (e.g. Blake, 1983a,b, 2004; Webb, 1984, 1998; Webb and Blake, 1985). A number of examples follow. Fin forms and kinematics are associated with high hydrodynamic efficiency at low speeds with a low small propulsive intrinsic fin muscle mass (Blake, 1983a,b) and cost of transport (Gordon et al., 2000; Korsmeyer et al., 2002). Precise manoeuvrability is agented by multiple fins capable of reversing fin waveform direction and these allow for equally efficient forward and backward motion (Blake, 1983a,b, 2004). Low speed swimming releases many UMPF swimmers from the morphological constraints associated with minimizing resistance (streamlining), allowing for a broad variety of rigid body forms with reasonable drag, an absence of drag enhancement due to bodily oscillation, projecting structures (e.g. large protruding mobile eyes, protective nare flanges and spines) and cryptic adaptations (Blake, 2004). Reduced recoil forces result from the action of vortical flows generated by the body (Bartol et al., 2003, 2005, 2008) and side force cancellation of fin pairs (Blake, 2004). Whilst some of the aforementioned adaptations are wide spread among UMPF swimmers (e.g. high hydrodynamic efficiency at low speeds, precise manoeuvrability),  1  others are particular to given forms (e.g. carapace generated vortical flows in boxfishes, high frequency fin motions in certain seahorses where hydrodynamic efficiency is sacrificed in favour of crypsis; Blake, 1981). It is likely that hydrodynamic advantages (thrust enhancement at no cost to hydrodynamic efficiency, reduction of side forces minimizing energy wasting yawing motions and body drag; Lighthill and Blake, 1990) are universal among rigid bodied organisms propelled by undulatory fins (e.g. many teleosts, cuttlefish, squids). Particle image velocimetry (PIV) (Willert and Gharib, 1991) has been successfully applied to assess near-body flow patterns (Anderson et al., 2001), the wake structure of fish propelled in the undulatory body and caudal fin modes (Müller et al., 1997; Drucker and Lauder, 1999; Nauen and Lauder, 2002a,b; Standen and Lauder, 2007) and the flow fields produced by caudal fin swimmers (Drucker and Lauder, 1999, 2001; Standen and Lauder, 2005, 2007; Tytell et al., 2008). We apply this technique to the UMPF swimming in the pufferfish Diodon holocanthus (diodontiform mode, sensu Breder, 1926) to assess the possibility of fin thrust enhancement (Lighthill and Blake, 1990; Lighthill, 1990a,b,c). We hypothesized that theoretical momentum enhancement calculated from the model would not be significantly different from experimental values (PIV) determined for thrust.  2 MATERIAL AND MATERIALS 2.1 Morphometrics Body length L (tip of the rostrum to the trailing edge of the caudal fin), distance from the body midline to the tip of the pectoral, dorsal and anal fins l, the corresponding  2  half body depth s (major axis; i.e. l=s+fin length) and distance of the half body depth for the minor axis (i.e. thickness) i for the fins were measured (30 cm caliper, Helios +0.05 cm, Mebtechnik, Germany). Frontally projected body area Af (determined from digitized tracings of “head on” photographs; Houston Instruments Hipad digitizer, Houston, Texas, U.S.A.), weight in air (Wa) and water (W) (Mettler PK300 scale, +0.001 g, with manufacturer’s suspension apparatus; Columbus, Ohio, U.S.A.) were determined. Fish density (ρf) was calculated from: ρ f = (Wa − W ) −1Wa ρ , where ρ is water density. 2.2 Drag Drag: D =  1 ρ Af U 2CD (where U and CD are forward velocity and drag 2  coefficient respectively) is a function of Reynolds number (Re = LU/ν, where ν is the kinematic viscosity of water). To determine the drag force, dead pufferfish D. holocanthus (n=6) were dropped to terminal velocity (Ut, ten times per fish, N=60) down a large water filled Plexiglass column (1.3 m x 1.3 m x 2.6 m) against a 1 cm x 1 cm grid. To eliminate drag due to fin flutter and ensure a stable drop path, a dart-shaped stabilizer was used to stabilize the fish which were stiffened by a fine steel wire running through the length of the body and caudal fin with paired and median fins removed at the level of their fin base (Webb, 1975; Blake, 1983a). Fish buoyancy was altered by placing lead pellets into their mouths which were sewn closed giving a Ut range. Fish were filmed (60 Hz; Troubleshooter high-speed camera, Model TS500MS, Fastec Imaging, San Diego, CA, U.S.A.) and video sequences were selected where the fish fell straight and achieved terminal velocity. At terminal velocity, D = W =  1 ρ ACDU t 2 . Drag at Ut was calculated 2  by subtracting the drag of the stabilizer from that of the stabilizer and fish together. Data  3  were analyzed frame by frame using ImageJ (National Institutes of Health; http://rsb.info.nih.gov/ij).  2.3 Steady swimming kinematics Fish swam in a sea water filled Perspex recirculating flow tank (250 cm x 25 cm x 60 cm). Flow was produced by a 0.5 HP electric motor which rotated a propeller (15 cm diameter). A flow rectifying grid (20 cm x 20 cm) made of straws (0.55 cm diameter) was placed in front of the propeller. A current meter (A. OTT, Kempton, Germany, TYP. 12.400, accurate to 0.50 cm s-1) was located 30 cm from the bottom and 75 cm down stream from the flow rectifying grid. Temperature was maintained (25+1o C) by a recirculating heater (Brinkmann instruments, Model IC-2, 1000 W). Two acrylic mirrors (35 cm x 0.5 cm x 200 cm) were glued together at 90o forming a triangular shaped mirror that was placed into the working section of the tank with its apex pointing away from the camera. This allowed the movements of the dorsal, anal, pectoral and caudal fins to be simultaneously recorded with top, bottom and horizontal views of the fish through a 2 cm x 2 cm reference grid drawn outside of the front surface of the tank (Fig. 1). Fish (n=5) were acclimatized for one week to experimental lighting conditions (three Berkey Coloran Halide 650 W bulb) and for half an hour before filming. They were filmed (250 Hz; Troubleshooter high-speed camera, Model TS500MS, Fastec Imaging, San Diego, CA, U.S.A.) at six swimming velocities (0.5 L s-1, 1.0 L s-1, 1.5 L s1  , 2.0 L s-1, 2.5 L s-1 and 3.0 L s-1). Four measurements were made per fish at each  velocity (N=20 for each variable). Video segments for analysis where the fish swam steadily with little movement relative to the grid (< 0.05 L), in the centre of the tank (i.e.  4  uninfluenced by the mirrors) and directly in line with the optical axis of the camera with no distortion of the grid due to parallax through 10 complete tailbeats (constituting one trial) were selected. Body and fin outlines were analyzed with ImageJ (National Institutes of Health; http://rsb.info.nih.gov/ij). Fin oscillation frequency (f, fin excursion from one side to the other and back again divided by duration), half amplitude (θ, angle between the most extreme lateral position of the fin tip to the midline) and amplitude (a, distance between the most extreme lateral positions of the fin tip from one side to the other) were measured. Fin length was divided by the time between peaks of fin lateral displacement (lateral displacement for the time taken for the wave of undulation to pass) to give the propulsive wave velocity V. The angle of inclination α (the average angle between the anterior and posterior axis of the fin) was determined.  2.4 Particle image velocimetry Particle image velocimetry (time resolved PIV, Dantec Dynamics, Denmark) measurements were made on pufferfish (n=4) in the re-circulating flow tank described above at 0.5 L s-1, 1.0 L s-1 and 1.5 L s-1. Six measurements were made at each velocity per fish and fin. Steady swimming motions (with velocity variation within < 0.05 L) in the centre of the flow tank away from the walls (> 10 cm) were analyzed. Silver-coated spherical glass particles (mean diameter 20 µm, density 1.3 g cm-3, PSP Polyamid Seeding Particles, PSP-20, Dantec Dynamics, Denmark) were seeded at a concentration of 14 mg l-1. The particles remained suspended in the water during the course of each experiment. Focused light sheets (3 W continuous-wave Dantec Dynamics argon-ion laser, approximately 1-2 mm thick and 25 cm wide) were “fired” to illuminate the motion  5  of the particles in two orthogonal planes (frontal (xz) and parasagittal (xy)) in separate experiments for each of the dorsal, pectoral and caudal fins (Fig. 1). PIV measurements of the anal fin were not made because of the requirement for the video images to be obtained from directly below the re-circulating tank where the view would be obstructed by the bottom and central dividers of the tank giving blurry images of the laser light sheet and inaccurate velocity field measurements. It was assumed the flow field generated by the anal fin would be equal in magnitude and opposite in direction to that of the dorsal fin. For the frontal plane, the laser sheet was fired directly from the side. A 45o mirror was placed above the tank to align the laser to the parasagittal plane. Experiments were filmed with two synchronized cameras (500 Hz; AOS X-Pri, AOS Technologies AG, Baden Daettwil, Switzerland). One was situated perpendicular to the laser sheet and recorded particle images while the other, parallel to the sheet, determined its location. Pairs of successive video images were digitized, downloaded to hard disk for computational image interrogation (FlowManager, version 3.4; Dantec Dynamics, Denmark). Particle displacement was analyzed by two-frame cross-correlation giving two-dimensional flow fields (Raffel et al., 1998). One frame at the end of the fin-stroke cycle and the other immediately before it were selected to allow for the computation of the circulation and projected area of the vortex rings. Post-processing of PIV data began with the validation of the velocity vectors using a dynamic mean value algorithm to reject each vector with velocity much greater than the average velocity of its eight nearest neighbours (Raffel et al., 1998). Any remaining validated vectors that were clearly inconsistent with the actual flow fields (usually occurring near fin edges where velocity differed from that of the surrounding fluid) were manually deleted. The accuracy of the  6  PIV system was determined by comparing the average free-stream velocity calculated by cross-correlation with that determined by tracking individual particles in the video record. PIV estimation of velocity was within 5 % of the true value. Hydrodynamic variables were calculated using the velocity vectors. Vorticity (ε) and vorticity components ( ε ) were computed as:  dΓ dAo  ε=  (1)  where Γ is circulation (a measure of vortex strength) around the boundary of a small region area Ao and:   ∂w  ∂v ∂u  ∂w ∂v  ∂u   ε = − , − , −   ∂y ∂z ∂z ∂x ∂x ∂y   (2)  where u, v and w are the velocity components parallel to the flow fields of x, y and z respectively (Lighthill, 1986). Circulation is:  Γ = ∫ U v dl  (3)  C  where U v , C and dl are the line integral of the tangential component, curve enclosing the vortex and incremental path element vector respectively (Batchelor, 1967). The average free-stream velocity was subtracted from each vector in both planes.  7  M irr or  C am e ra Fish  M irro r  B  A  D  E  C  F  Figure 1: i. Set-up of kinematic videography experiments with one camera and two mirrors joined at 90o. ii. Schematic diagram of two perpendicular planes, frontal and parasagittal, illuminated by laser light for PIV visualization for the frontal plane of the dorsal (A), pectoral fin (B) and caudal fin (C) and parasagittal plane for dorsal (D), pectoral fin (E) and caudal fin (F) are shown.  8  Fin forces from circulation values were calculated from the “near fin” circulation of vortices in the wake and this method is justified by the equivalence in magnitude of bound and shed circulation (based on Kelvin’s theorem) during the impulsive start of a hydrofoil (Milne-Thomson, 1966; Dickinson, 1996; Dickinson and Gӧtz, 1996). Vortex momentum (M) is: M = ρΓA  (4)  where A is the area is the projected area of the vortex ring and is measured at the end of the stroke. The force (F) is: F = dM / dt  (5)  Combining (3) and (4) gives the time-averaged force ( F ): F = ρΓ A / t p  (6)  where tp is the fin stroke period. The jet angle ( φ , angle between the horizontal axis and the mean orientation of the central water jet of the vortex ring) was measured. The thrust ( Fx ) in the frontal and parasagittal planes ( F cos φ ), the vertical component ( Fy ) in the  parasagittal plane ( F sin φ ) and the medial component of the force ( Fz ) in the frontal plane ( F sin φ ) were calculated.  2.5 Momentum enhancement Lighthill and Blake (1990) employed an extension of large-amplitude elongatedbody theory to determine energy savings associated with fin movements attached to a deep, essentially rigid body. Following Lighthill and Blake (1990), the momentum enhancement factor is expressed as a function of the ratio of the body semi-depth to the sum of the fin depth plus body depth (s/l). Fig. 2 shows the case of two equal fins  9  attached to an elliptical cross-section (major axis to s, minor axis to i) with the fins extending a distance l – s beyond each end of the major axis where l is the distance from the centre of the elliptic cross-section to the tip of each fin. The momentum budget is given by: L  d M f dx f = − Fx + P − U ( M f ) x f = 0 dt ∫0  (7)  where Mf, P and xf are the momentum of the fin per unit length, pressure force acting across a plane at the anterior of the fin and the coordinate from the anterior to the posterior end respectively. The left hand side of equation (7) has a mean of zero because the time derivative is integrated with respect to time over a cycle as is the case for body undulations. Therefore, the mean propulsive force is: Fx = −U ( M f ) x f = 0 + P  (8)  The velocity of a coordinate z bounded by -l < z < l can be expressed as f(z) with the inflexible body of –s < z < s and f(z) = 0 for ( z < s) and f(z) = ω ( z − s ) for (s < z < l) (Fig. 2). The momentum per unit fin length can be found by solving the twodimensional Laplace equation for velocity potential outside the plate giving: l  M f = 2 ρ ∫ (l 2 − z 2 )0.5 f ( z )dz  (9)  −l  Solving the integral gives:  1  M f = 2 ρω  (l 2 − z 2 )0.5 (2l 2 + s 2 ) − sl 2 cos −1 ( s / l )  3   (10)  The momentum of the fin on its own Mo is: 1 M o = πρω (l − s )3 4  (11)  10  z  Figure 2: Fish cross-section with distance from the midline of the body to the tip of the pectoral, dorsal and anal fins l, the corresponding half body length s (i.e. l=s+fin length) and distance of the half body depth for the minor axis (i.e. thickness) i for the pectoral, dorsal and anal fins. The direction for the co-ordinate z is also shown.  where ω is fin angular velocity:  ω = Vκ  (12)  κ is local twist of each fin:  11  κ = 2πθ / λ  (13)  where the fin’s undulation motion takes the form of a sinusoidal variation of with amplitude θ and wavelength λ . The theoretical momentum enhancement factor (β’) can be determined when Mf is divided by Mo and substituting z = l at the tip of the fin:  8(− sl 2 cos −1 ( s / l )) β'= π (l − s)3  (14)  Equation (14) does not take account of the thickness to depth ratio of the body i/s. When this is done (see Lighthill, 1990a, equations (32) to (37)), β ' can be represented as a function s/l and i/s (see Lighthill, 1990a, Fig. 7). Thrust measurements ( Fx ) when obtained from PIV can be used with kinematic measurements and substituted to determine the magnitude of the momentum enhancement factor (β). The propulsive force ( Fx ) is dominated by −U ( M ) a =0 because  P is negligible (Lighthill and Blake, 1990) and is given by: 1 Fx = U β M o (1 − η )sin α av 2  (15)  where η = U / V is swimming efficiency (defined as swimming speed divided by fin propulsive wave velocity) and:  1  β = Fx  UM o (1 − η ) sin α av  2   −1  (16)  2.6 Statistics The effect of swimming speeds and “fins” on ring momentum angle, jet angle, ring radius, ring area, mean vortex circulation, ring momentum, Fx , Fy and Fz were compared by one-way ANOVA with post-hoc Student Newman-Keul’s test (SPSS 13.0  12  for Windows). Comparisons between theoretical and experimental momentum enhancement factors (β’ and β respectively) and between drag values and thrust measurements for all fins and were compared using one-sample t-tests (Zar, 1999). The null hypothesis was rejected at α = 0.05 in all cases.  3 RESULTS Fish density was measured to be 1030+5 kg m3 (S.G.=1.03; n=10). Pufferfish drag increased with the square of water velocity (D = aV2 + bV + c, r2 > 0.5) ranging from ≈ 0.5 - 2.3 N for speeds of 0.2 - 1 m/s. Drag coefficients ranged from 0.15 - 0.75 for Reynolds numbers of 30000 - 150000 (Fig. 3). Dorsal, anal, pectoral and caudal fin frequency and amplitude increased significantly with velocity (P < 0.05; Table 1). Dorsal, anal and caudal fin frequency at any given speed was not significantly different and ranged from ≈ 3 - 5.5 Hz over a speed range of 0.5 - 3.0 L s-1 (P > 0.05). The fish held their pectoral fins against the sides of the body for speeds > 2.0 L s-1 (Table 1). Movements of the dorsal, anal and caudal fins were synchronized at all swimming velocities. Synchronization was in phase for the dorsal and anal fins which were out of phase with the caudal fin (Fig. 4). The degree of phase shift between the anal and dorsal fins and the caudal fin increased with increasing speed (Fig. 4). Hydrodynamic efficiency, angle of inclination and the momentum of the fins on their own were determined from kinematic measurements and they increased with swimming velocity for both the dorsal and pectoral fins (Table 2). Local fin twist and hydrodynamic efficiency between the dorsal and pectoral fins were not significantly different (P > 0.05). The sine  13  of the angle of inclination for the pectoral fin was significantly greater than the dorsal fin while momentum of the fin on its own was less (P < 0.05). Water velocity vector fields for the frontal and parasagittal planes of the dorsal, pectoral and caudal fins have one vortex pair (Figs. 5 to 10). The dorsal fin vortices create forces in the x direction (i.e. backwards relative to the direction of travel) and in the z direction (i.e. medially towards the body) in the frontal plane. In the parasagittal plane, forces are generated backwards in the x direction and vertically downward in the y direction (Figs. 5 and 6). For the pectoral fin in the frontal plane, the vortices generate forces backwards in the x direction and away from the body in the z direction. Forces are generated backwards and downwards in the parasagittal plane (Figs. 7 and 8). Force generation for the caudal fin was backwards and towards the body in the frontal plane and backwards in the parasagittal plane (Figs. 9 and 10). In the frontal plane, the vortex ring momentum ( ρΓA ) and jet angles for the dorsal fin are significantly different from those of the pectoral or the caudal fin (P < 0.05; Table 3), differing by a factor of about 2. Ring radius (≈ 2 x 10-2 m) and area (≈ 12.6 x 10-4 m2) for the three fins were independent of water velocity and similar (P > 0.05; Table 3). Mean vortex circulation and momentum increased with velocity (P < 0.05). Dorsal and caudal fin circulation and momentum were significantly larger than that of the pectoral fin corresponding to larger forces (P < 0.05; Table 3). Pectoral fin thrust (in the x direction; Fx ) was about half of that generated by the dorsal or caudal fin. Half the force generated by the pectoral or the caudal fin produced side forces away and towards the body in the z direction. Thirty percent of the force generated by the dorsal fin produced side force towards the body in the z direction (Table 3).  14  Table 1: Frequency and amplitude for the pectoral, dorsal, anal and caudal fins during steady swimming at different speeds Swimming Speed  Pectoral Fin  Dorsal Fin  Anal Fin  Caudal Fin  (L s-1)  f (Hz)  a (cm)  f (Hz)  a (cm)  f (Hz)  a (cm)  f (Hz)  a (cm)  0.5 1.0 1.5 2.0 2.5 3.0  2.83+0.18 3.33+0.14 4.08+0.16 -  2.18+0.20 3.06+0.17 3.16+0.18 -  2.97+0.22 3.18+0.14 3.40+0.12 4.56+0.22 5.17+0.18 5.68+0.18  4.20+0.06 4.60+0.08 4.72+0.06 5.32+0.08 5.38+0.06 5.42+0.12  3.01+0.22 3.20+0.14 3.43+0.18 4.68+0.18 5.16+0.20 5.68+0.18  4.40+0.08 4.70+0.08 5.30+0.10 5.54+0.12 5.78+0.10 5.78+0.12  3.01+0.22 3.26+0.14 3.35+0.14 4.66+0.20 5.19+0.12 5.68+0.18  1.10+0.16 1.34+0.12 1.54+0.12 1.77+0.10 1.75+0.06 2.05+0.10  Values are mean + 2 s.e.m.; N =20 in all cases  15  Table 2: Momentum enhancement (β) at different swimming speeds for the dorsal (D) and pectoral (P) fins  16  Figure 3: Drag versus swimming speed (A: y=0.0301x2+0.201x, r2=0.59) and drag coefficient versus Reynolds number (B: y=26332x-1.02, r2=0.55) for dead specimens of the porcupine pufferfish (D. holocanthus).  17  Amplitude (cm)  3  A  2 1 0 -1  0  0.1  0.2  0.3  0.4  0.5  0.1  0.2  Time (s)  0.3  0.4  0.5  0.1  0.2 Time (s)  0.3  0.4  0.5  Time (s)  -2 -3  Amplitude (cm)  3  B  2 1 0 -1  0  -2 -3  Amplitude (cm)  3  C  2 1 0 -1  0  -2 -3  Phase Shift (o)  180 150 120 90 60 30 0 0  0.5  1  1.5 2 Speed (L s-1)  2.5  3  3.5  Figure 4: Displacement versus time for the pectoral fin (green triangle), dorsal fin (blue diamond), anal fin (pink square) and caudal fin (black circle) at swimming speeds of 1 L s-1 (A), 2 L s-1 (B) and 3 L s-1 (C) for D. holocanthus.  18  A  B  Figure 5: Water velocity vector fields of the frontal (A) and parasagittal plane (B) for the dorsal fin of D. holocanthus at 0.5 L s-1. Flow patterns are shown for the end of the stroke cycle.  19  A  Vorticity (s-1)  B  Vorticity (s-1)  Figure 6: Fluid vorticity components of the frontal (A) and parasagittal plane (B) for the dorsal fin of D. holocanthus during swimming at 0.5 L s-1. Red or orange coloration indicates positive vorticity in the counterclockwise direction and blue or purple coloration indicates negative vorticity in the clockwise direction. Green regions represent locations where vorticity is absent.  20  A  B  Figure 7: Water velocity vector fields of the frontal (A) and parasagittal plane (B) for the pectoral fin of D. holocanthus at 0.5 L s-1. Flow patterns are shown for the end of the stroke cycle.  21  A  Vorticity (s-1)  B  Vorticity (s-1)  Figure 8: Fluid vorticity components of the frontal (A) and parasagittal plane (B) for the pectoral fin of D. holocanthus during swimming at 0.5 L s-1. Red or orange coloration indicates positive vorticity in the counterclockwise direction and blue or purple coloration indicates negative vorticity in the clockwise direction. Green regions represent locations where vorticity is absent.  22  A  B  Figure 9: Water velocity vector fields of the frontal (A) and parasagittal plane (B) for the caudal fin of D. holocanthus at 0.5 L s-1. Flow patterns are shown for the end of the stroke cycle.  23  A  Vorticity (s-1)  B  Vorticity (s-1)  Fig. 10. Fluid vorticity components of the frontal (A) and parasagittal plane (B) for the caudal fin of D. holocanthus during swimming at 0.5 L s-1. Red or orange coloration indicates positive vorticity in the counterclockwise direction and blue or purple coloration indicates negative vorticity in the clockwise direction. Green regions represent locations where vorticity is absent.  24  The ring momentum of the caudal fin in the parasagittal plane is about twice that of the dorsal or the pectoral fin and its jet angle is about 7.5 times less than that of the dorsal or the pectoral fin (P < 0.05; Table 4). Ring radius (≈ 2.2 x 10-2 m) and area (≈ 17.5 x 10-4 m2) for the dorsal and caudal fin were significantly greater (P < 0.05) than that of the pectoral fin (≈ 1.7 x 10-2 m and 10 x 10-4 m2 respectively). Dorsal fin vortex circulation and momentum was significantly higher than that of the pectoral and caudal fins (P < 0.05) and both increased significantly with swimming speed (P < 0.05; Table 3). Fx for the dorsal or the caudal fin (≈ 4 mN) was about twice that of the pectoral fin. The lift force Fy for the dorsal fin was positive and was negligible for the caudal fin (≈ 0.2 - 0.7 mN; Table 3). There was no significant difference between the average sum of the thrust forces for all fins in the x direction for the frontal and parasagittal planes relative to measured drop tank drag for all three speeds (P > 0.05). Drag measurements at 0.5, 1.0 and 1.5 L s-1 are 12.6, 25.2 and 37.8 mN respectively. The average thrust Fx between the frontal and parasagittal planes for the dorsal and anal fins (assuming their thrust to be comparable given their morphological and kinematic similarities) was: 7.9+1.6, 14.5+2.5 and 23.3+2.4 mN. Values for both pectoral fins were: 3.2+1.2, 6.5+0.6 and 9.3+2.0 mN and 2.6+0.6, 5.0+0.4 and 9.0+1.7 mN for the caudal fin for speeds of 0.5, 1.0 and 1.5 L s-1 respectively. Therefore, the total thrust forces are 13.7+2.4, 25.9+2.9 and 41.6+4.3 mN which are not significantly different from the drag measurements (P > 0.05).  β values ranged from 2.2 - 2.4 and 2.7 - 2.9 for the dorsal and pectoral fins respectively (Table 2) and not significantly different from that predicted by Lighthill and Blake (1990) (P > 0.05; Fig. 11).  25  Table 3: Vortex ring measurements from perpendicular flow-field sections of the wake during swimming at three different speeds (0.5 L s-1, 1.0 L s-1 and 1.5 L s-1) for the dorsal (D), pectoral (P) and caudal (C) fins in the frontal plane Measurement  Fin  Speed 0.5 L s-1  1.0 L s-1  1.5 L s-1  Ring momentum angle (degrees)  D P C  58.25+3.31 a,A 22.50+3.40 a,B 27.63+6.16 a,b,B  61.75+3.23 a,b,A 30.00+2.95 b,B 23.25+6.26 a,B  65.13+2.84b,A 33.13+2.94b,B 35.88+6.41b,B  Jet angle (degrees)  D P C  -31.75+3.31a,A -67.50+3.40a,B -62.38+6.16a,b,B  -28.25+3.23 a,b,A -60.00+2.95b,B -66.75+6.26a,B  -24.88+2.84 b,A -56.88+2.94 b,B -54.13+6.41 b,B  Ring radius (x 10-2 m)  D P C  1.84+0.09 a,A 1.90+0.15 a,A,B 2.05+0.10 a,B  1.85+0.09 a,A 2.06+0.07 a,B 2.09+0.15 a,B  1.87+0.09a,A 2.03+0.07a,A,B 2.17+0.17a,B  Ring area (x 10 -4 m2)  D P C  10.80+0.10 a,A 11.76+1.46a,A,B 13.31+1.29 a,B  10.91+1.09 a,A 13.45+1.01 a,B 14.13+1.84 a,B  11.13+1.04a,A 13.04+0.93 a,A,B 15.25+2.38a,B  Mean vortex circulation (x 10-4 m2 s-1)  D P C  17.50+3.62 a,A 14.47+4.77 a,A 17.52+4.94 a,A  30.00+4.01 b,A 19.88+3.16 a,B 45.97+7.71 b,C  42.14+3.19c,A 20.49+3.43a,B 46.64+11.46 b,A  Ring momentum (x 10 -4 kg m s-1)  D P C  20.38+5.18 a,A 17.87+6.75 a,A 25.38+8.19 a,A  34.05+5.44 b,A 26.43+3.80 b,A 70.05+15.82 b,B  47.90+5.88c,A 27.53+4.76b,B 70.05+16.20 b,C  F x (mN)  D P C  5.03+0.64 a,A 1.93+0.78 a,B 3.00+1.00 a,B  9.36+1.43 b,A 4.39+0.74 b,B 6.22+0.71 b,C  14.67+1.90c,A 5.92+1.07 c,B 11.45+2.82c,A  F z (mN)  D P C  3.25+0.47 a,A 4.61+1.77 a,A,B 6.69+2.33 a,A  5.21+1.17 b,A 7.32+1.07 b,A 21.03+5.24 b,B  6.77+1.05b,A 9.45+1.72b,A 20.17+5.24b,B  Values are mean + 2 s.e.m.; N = 24 for all cases Significant effects of swimming speed and fin effects were compared using ANOVA with post-hoc Student Newman-Keul's test. Significant differences (P <0.05) between speeds and fins are indicated by different lower case and upper case letters respectively.  26  Table 4: Vortex ring measurements from perpendicular flow-field sections of the wake during swimming at three different speeds (0.5 L s-1, 1.0 L s-1 and 1.5 L s-1) for the dorsal (D), pectoral (P) and caudal (C) fins in the parasagittal plane Measurement  Fin  Speed 0.5 L s-1  1.0 L s-1  1.5 L s-1  Ring momentum angle (degrees)  D P C  41.63+3.45a,A 35.00+7.52a,A 82.88+3.06a,B  50.25+7.41a,A 36.25+2.65a,B 84.38+4.80a,C  63.25+7.90b,A 38.38+9.65a,B 84.63+1.53 a,C  Jet angle (degrees)  D P C  -48.38+3.45a,A -55.00+7.52a,A -7.13+3.06a,B  -39.75+7.41 a,A -53.75+2.65 a,B -5.63+4.80 a,C  -26.75+7.90 b,A -51.63+9.65 a,B -5.38+1.53 a,C  Ring radius (x 10-2 m)  D P C  2.09+0.13 a,A 1.66+0.10 a,B 2.53+0.09 a,C  2.27+0.09 a,A 1.73+0.04 a,b,B 2.36+0.08 a,A  2.32+0.20a,A 1.87+0.15b,B 2.43+0.15a,A  Ring area (x 10 -4 m2)  D P C  14.10+1.87a,A 8.81+1.03 a,B 20.29+1.53a,C  16.33+1.22a,A 9.44+0.45 a,b,B 17.68+1.19a,A  17.67+3.15a,A 11.37+1.89b,B 19.02+2.18a,A  Mean vortex circulation (x 10-4 m2 s-1)  D P C  11.41+3.90a,A 7.35+2.92 a,A,B 3.21+0.88 a,B  12.75+4.11a,A 10.48+2.82a,b,A,B 6.47+1.39 b,B  17.54+2.98a,A 12.93+2.69 b,A,B 9.80+1.52 c,B  Ring momentum (x 10 -4 kg m s-1)  D P C  14.88+4.85a,A 6.71+2.91 a,B 7.20+2.41 a,B  20.17+5.77a,A 9.85+2.48 a,b,B 11.62+2.62a,B  30.86+7.72b,A 14.34+3.25b,B 19.43+4.22b,B  F x (mN)  D P C  2.84+0.87 a,A 1.30+0.67 a,B 2.09+0.69 a,A,B  5.15+1.81 a,A 2.06+0.69 a,b,B 3.67+0.84a,A,B  8.63+2.11b,A 3.38+1.06b,B 6.56+1.43b,A  F y (mN)  D P C  3.33+1.18 a,A 1.29+0.52 a,B 0.4+0.2a,B  3.28+0.88 a,A 2.37+0.50 a,A 0.21+0.09 a,b,B  4.98+2.08a,A 4.01+1.34b,B 0.66+0.22b,B  Values are mean + 2 s.e.m.; N = 24 for all cases Significant effects of swimming speed and fin effects were compared using ANOVA with post-hoc Student Newman-Keul's test. Significant differences (P <0.05) between speeds and fins are indicated by different lower case and upper case letters respectively.  27  5  Momentum Enhancement Factor  4  3  2  1  0 0  0.2  0.4  0.6  0.8  1  s/l  Figure 11: Momentum enhancement factor β versus s/l (the ratio between body radius and the sum of the body radius and fin depth) predicted by the mathematical model of Lighthill and Blake (1990) and Lighthill (1990a) with t/s = 0.75 (dark line) and experimentally values determined for the dorsal fin (square) and pectoral fin (triangle) of D. holocanthus at swimming velocities of 0.5 L s-1 (blue), 1.0 L s-1 (pink) and 1.5 L s-1 (red). Error bars represent + 2 s.e.m.  28  4 DISCUSSION 4.1 Kinematics Kinematic measurements show that the dorsal and anal fins are 90o out of phase with the caudal fin at 1.0 L s-1, increasing with swimming speed, approaching 180o at 3 L s-1 (Fig. 4). When the dorsal and anal fins are displaced maximally towards one side, the caudal fin is maximally displaced to the opposite side, implying that the side forces (z direction) generated by the median fins sum and are in the opposite direction to those generated by the caudal fin minimizing yawing motions. Were the median fins to be in phase with the caudal fin, all of the side forces generated would sum in the same z direction producing a large energy wasting turning moment about the centre of mass of the fish. The side forces generated by the pectoral fins are equal in magnitude and opposite in z direction and do not contribute to yawing motion.  4.2 Wake momentum In the frontal plane, ring radius and area for all fins are similar (Table 3). However, vortex circulation and ring momentum are greater for the median and caudal fins relative to the pectoral fins at higher speeds (i.e. speeds > 1.0 L s-1). Greater circulation and momentum are associated with higher force generation. In addition, useful thrust in the x direction is calculated based on the cosine of the jet angle and so a low angle is favourable for thrust generation. Pectoral fin thrust was about half of that of the median and caudal fins due to a higher jet angle coupled with lower circulation and momentum (Table 3). About half of the total force generated by the pectoral and caudal fins is wasted in generating side forces in the z direction producing yawing moments. About 30 percent of the force generated by the median fins was wasted in this way.  29  In the parasagittal plane, ring radius and area for all fins are also similar (Table 4). Vortex circulation and ring momentum for the median fins are higher than those of the pectoral and caudal fins, resulting in higher force production. Thrust and lift generation are calculated based on the cosine and sine of the jet angle respectively and, therefore, a high angle is favourable for lift. The lift for the median fin and pectoral fins are significant, while that of the caudal fin is negligible (i.e. low jet angle; Table 4). Fig. 8 shows a schematic representation of the likely wake structures based on PIV measurements for the frontal and parasagittal planes. A vortex ring was generated by each of the median, pectoral and caudal fins. The vortex rings generated by the dorsal and anal fins are assumed to be similar in magnitude and size (based on their morphological and kinematic similarities) and moving in opposite y direction. The caudal fin vortex ring is almost vertical in the xy plane with no lift generation. Therefore, the pectoral fins are the only fins generating vortex rings with a vertical component. Vortex ring size was similar to the fin chord in all cases. The magnitude of the fin excursion in the z direction determined the distance between the vortices in the xz (frontal) plane. The shape of the vortex rings in all cases was elliptical, independent of velocity and departs from the axisymmetrical shape of an ideal vortex ring according to Helmholtz’s theorem characterized by a central jet flow which extends perpendicularly to the plane of the vortex ring (Milne-Thompson, 1966; Webster and Longmire, 1997; Raffel et al., 1998).  30  Fz Fy Fx  Fz Fz  Fx  Fy Fx  Fx Fy Fz  Figure 12: Lateral view representation of the vortex rings (blue) in the wake of D. holocanthus. During steady motion, the dorsal, anal, pectoral and caudal fins shed vortex rings that are associated with a central jet of high momentum and forces in the x,y and z component (Fx, Fy and Fz with Fz pointing out of the page). It is likely that the three dimensional wake structure of undulatory fin swimmers will be variable, depending, among other things, on fin position, morphology, kinematics, forward velocity and whether or not the caudal fin is operational. Studies to date have shown substantial differences in the proposed wake structures for axial and pectoral fin swimmers (reviewed by Lauder and Tytell, 2006). In pectoral fin swimming (labriform locomotion) of the bluegill sunfish Lepomis macrochirus, a single vortex ring is generated on the downstroke of each pectoral fin at low speeds (Drucker and Lauder, 1999), and this is also the case for the undulatory pectoral fins of D. halocanthus. In the bluegill sunfish, both the downstroke and upstroke generate a pair of linked vortex rings at higher speeds. The pectoral fins are not deployed at higher speeds in D. halocanthus. Surfperch Embiotoca jacksoni pectoral fins shed two linked vortex rings (Drucker and  31  Lauder, 2000). The eel (anguilliform locomotion) is hypothesized to generate a series of partially linked lateral vortex rings that separate in the far wake characterized by momentum jets oriented laterally with a negligible downstream component (Müller et al., 2001; Tytell and Lauder, 2004; Lauder and Tytell, 2006). The caudal fin wake of D.  halocanthus is similar to that of trout, mackerel and sunfish, which show considerable streamwise momentum (Lauder et al., 2002, 2003; Nauen and Lauder, 2002a,b). Not surprisingly, there are differences in the flow fields generated by median fins employed for stability and control (e.g. rainbow trout Oncorhynchus mykiss) and those that play a major role in generating propulsive forces (e.g. D. holocanthus). In the former, the median fins shed a linear array of vortex centres with strong lateral jets to each side (Drucker and Lauder, 2001; Lauder and Drucker, 2004), making only a small contribution to propulsion through affecting the incident flow to the caudal fin (Drucker and Lauder, 2001). In the latter, propulsive median and paired fins produce a thrusting wake.  4.3 Density The density of the pufferfish (≈ 1035 kg m3) corresponds to a submerged weight of about 11.2 mN. Lift forces generated in the parasagittal plane by the dorsal fin and anal fin are likely of equal magnitude and opposite in the y direction (Table 4) and, therefore, do not contribute to the net lift production. A net lift force (2.58 - 8.02 mN for speeds of 0.5 - 1.5 L s-1) is generated by the pectoral fins (Table 4). Presumably, the balance of force needed to support the weight of the fish in the water column is provided by the swim bladder and any hydrodynamic lift generated by the body.  32  4.4 Momentum enhancement Lighthill and Blake’s (1990) model was developed for the simple case of a rigid body propelled by undulatory median and/or paired fins. As in this case, at least at a certain speed range, undulatory fin propulsion may be augmented by the action of caudal fin motions that are not subject to potential momentum enhancement. In these instances, drag is overcome by the joint contribution of the undulatory and caudal fin. The thrust contribution of the latter must be subtracted from the total thrust to allow for an assessment of the magnitude of any momentum enhancement to the undulatory fins. This is possible given that the magnitude of the thrust of all of their fins is known from PIV measurements. When the caudal fin thrust contribution is subtracted from the total, the force required from the undulatory fins to overcome the drag is 10.1, 20.3 and 28.8 mN at speeds of 0.5, 1.0 and 1.5 L s-1 respectively. Assuming no momentum enhancement for the fins (i.e. β = 1), their contribution would be 4.5 mN (0.5 L s-1), 9.9 mN (1.0 L s-1) and 18.2 mN (1.5 L s-1). Therefore, when β = 1, force contributions from the four fins are insufficient to overcome the drag, implying a overall momentum enhancement factor of about 2.2, 2.1 and 1.6 at the three speeds. Lighthill and Blake’s (1990) approach implies momentum enhancement for a fin attached to a body relative to that for a fin on its own is based on elongated-body theory. A similar approach can be applied to the low aspect ratio rigid caudal fin (ostraciiform tail) of D. holocanthus. Based on Blake (1981), the mean thrust force Fx of the caudal fin can be written as:   ∂h Uh   ∂h 1  ∂h Uh   Fx = m  − −  −   c   ∂t 2  ∂t c   x = L f  ∂t  (17)  33  where c is the fin chord at its trailing edge and m is the added mass per unit length which 1 can be estimated as: m = π c 2 ρζ where ζ is a non-dimensional shape parameter ≈ 1 4 (Lighthill, 1970). Equation (17) makes no provision for momentum enhancement (i.e. β = 1). Caudal fin thrust enhancement relative to the posterior of the body would be implied if the PIV force measurements were to be significantly greater than those calculated from equation (16). Calculated thrust values from equation (17) of 3.8, 7.2 and 11.3 mN exceed those determined experimentally from PIV (2.6, 5.0 and 9.6 mN) at 0.5, 1.0 and 1.5 L s-1 respectively, implying no momentum enhancement between the caudal fin and the posterior of the body. Momentum enhancement of the caudal fin would require a greater calculated PIV thrust value than that of equation (17), and this is not the case. The extent of momentum enhancement for the undulatory fins of D. holocanthus is of the order of 2.5. Values for other undulatory median and paired fin swimmers will be different depending on the number of fins, the position of their placement, s/l and i/s. At first sight, it may seem that natural selection should operate to maximize the value of  β, which in turn implies high values of s/l and i/s. In a limit, short fins attached to forms of narrow deep section (i.e. high β values) should be common. However, there is a tradeoff between momentum enhancement and the rate of generation of momentum (thrust force). When s/l and i/s are large, small fins, whilst subject to good momentum enhancement, will produce low absolute thrust to propel a body form of large surface area and therefore high drag. Conversely, when s/l and i/s are small, a deep fin would produce high thrust to propel a small body but with little momentum enhancement. Indeed, as s/l and i/s tend to zero, the limit of “a fin on its own” is reached. In addition, there are likely limits to the extent of momentum enhancement set by short-wavelength  34  limitations arising from reductions in propulsive force associated with progressive reductions in fin wavelength (Lighthill, 1990b). In particular, when the body is of small width and large depth, elongated-body theory predicts that the enhancement factor would be very large. However, the aforementioned short-wavelength limitation places a practical upper limit on β which may be calculated based on a simple limiting case. If the fin depth l − s has an arbitrary magnitude in comparison with k = 2π / λ , the overall depth of fish 2l is large compared with k −1 . The result of this simple calculation for 2lk → ∞ can be approximated as: β = 1.53(k (l − s ))  −  1 2  (Lighthill, 1990b).  5 CONCLUSION The approach taken here to assess the thrust production and its enhancement of D.  holocanthus could be applied to other fish propelled by median and paired fins. Caudal fin actively produces propulsive thrust together with the undulatory median and paired fins in D. holocanthus, and its overall contribution to thrust was assessed independently from that of the median and paired fins to determine overall momentum enhancement. This is not the situation in routine swimming in certain other UMPF swimmers (e.g. ostraciiform mode, where the caudal fin is deployed as a rudder; Blake, 1977) or where the caudal fin is absent or poorly developed (e.g. gymnotiform mode). From a practical standpoint, dealing with these forms will be simpler than the case studied here. 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