Fluid Boundary Condition Abstract .Full

8/2/2019 Fluid Boundary Condition Abstract .Full

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When a body moves relative to a surrounding fluid, aboundary layer exists very close to the body surface as a resultof the no-slip condition and viscosity (Prandtl, 1904).Consider an object held stationary in a uniform oncoming flowwith velocity U. The fluid in direct contact with the bodysurface adheres to the surface and has zero velocity. The fluid

just above the surface is slowed by frictional forces associatedwith the viscosity of the fluid. The closer the fluid is to thesurface, the more it is slowed. The result is a thin layer wherethe tangential velocity, u, of the fluid increases from zero atthe body surface to a velocity close to U. This velocity at theouter edge of the boundary layer, Ue, depends on the shape ofthe body (Schetz, 1993). By definition, the boundary layerextends from the objects surface,y=0, to a positiony=, wherethe tangential velocity relative to the objects surface is 0.99Ue.The curve representing the continuous variation in tangentialvelocity from y=0 to y= is commonly referred to as theboundary layer profile or, more specifically, the u-profile

(Fig. 1). Normal velocity relative to the surface also variesfrom zero at the body surface to some external value, Ve,generating what is known as the v-profile (Fig. 1). A thirdprofile, the w-profile, usually exists in the flow over three-dimensional surfaces, where w is tangential to the wall andperpendicular to u. Note that ifu, v or w is not specified, theterm boundary layer profile generally refers to the u-profile.

The shapes of the boundary layer profiles above a particularposition on a surface depend on the shape of the body, surfaceroughness, the upstream history of the boundary layer, thesurrounding flow field and Reynolds number. Flow in theboundary layer can be laminar or turbulent, resulting inradically different classes of profile shapes. Prandtl (1952),Schlichting (1979) and Batchelor (1967) provide thoroughdescriptions of the boundary layer concept. The behavior of abody moving relative to a real fluid cannot be accuratelydescribed without an understanding of the boundary layer.Since the work of Prandtl (1904), great strides have been made

81The Journal of Experimental Biology 204, 81102 (2001)Printed in Great Britain The Company of Biologists Limited 2001JEB2919

Tangential and normal velocity profiles of the boundarylayer surrounding live swimming fish were determinedby digital particle tracking velocimetry, DPTV. Twospecies were examined: the scup Stenotomus chrysops, acarangiform swimmer, and the smooth dogfish Musteluscanis, an anguilliform swimmer. Measurements were takenat several locations over the surfaces of the fish andthroughout complete undulatory cycles of their propulsivemotions. The Reynolds number based on length,Re, rangedfrom 3103 to 3105. In general, boundary layer profileswere found to match known laminar and turbulent profilesincluding those of Blasius, Falkner and Skan and the lawof the wall. In still water, boundary layer profile shapealways suggested laminar flow. In flowing water, boundarylayer profile shape suggested laminar flow at lowerReynolds numbers and turbulent flow at the highestReynolds numbers. In some cases, oscillation betweenlaminar and turbulent profile shapes with body phase

was observed. Local friction coefficients, boundary layerthickness and fluid velocities at the edge of the boundarylayer were suggestive of local oscillatory and mean

streamwise acceleration of the boundary layer. Thebehavior of these variables differed significantly in theboundary layer over a rigid fish. Total skin friction wasdetermined. Swimming fish were found to experiencegreater friction drag than the same fish stretched straightin the flow. Nevertheless, the power necessary to overcomefriction drag was determined to be within previousexperimentally measured power outputs.

No separation of the boundary layer was observedaround swimming fish, suggesting negligible form drag.Inflected boundary layers, suggestive of incipientseparation, were observed sporadically, but appeared to bestabilized at later phases of the undulatory cycle. Thesephenomena may be evidence of hydrodynamic sensingand response towards the optimization of swimmingperformance.

Key words: undulatory swimming, fish, boundary layer, friction,drag, separation, hydrodynamics, digital particle image velocimetry,

digital particle tracking velocimetry.

Summary

Introduction

THE BOUNDARY LAYER OF SWIMMING FISH

ERIK J. ANDERSON, WADE R. MCGILLIS AND MARK A. GROSENBAUGH*

Department of Applied Ocean Physics and Engineering, Woods Hole Oceanographic Institution, Woods Hole,

MA 02543, USA*Author for correspondence (e-mail: [email protected])

Accepted 17 October; published on WWW 5 December 2000


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in understanding fluid forces acting on bodies. Nevertheless,the hydrodynamics of undulatory swimming remains elusive.

Drag, thrust and power in undulatory swimming have not beendefinitively determined. This is, in part, because no definitivemeasurements of boundary layer flow over a swimming fish orcetacean have been performed.

Few attempts have been made to characterize the boundarylayers of undulatory swimmers, and none has producedboundary layer velocity profiles. Most recently, Rohr et al.(1998b) have suggested that the relative intensity ofbioluminescence around a swimming dolphin may be linkedto the thickness of the boundary layer. In a set of earlierinvestigations, Kent et al. (1961) and Allen (1961) achieved aqualitative description of flow in the near-field and possibly theboundary layers of fish using the Schlieren technique. Thenear-field is the region of flow around the fish affected by thepresence of the fish and its swimming motions. In contrast, theso-called far-field is the region in which the impact of the fishhas decayed essentially to nothing. While the boundary layercan certainly be considered part of the near-field flow, to aidin the discussion, we use the term near-field to refer to theregion dominated by the presence of the fish, but outside theboundary layer.

The understanding of drag mechanisms in undulatoryswimming has been impeded significantly by this lack ofboundary layer data. Both form drag and friction drag on a

body depend on the nature of the boundary layer. Unlike thedrag on a rigid body, such as an airplane wing, the drag on aswimming fish cannot be measured by simply placing a fish-shaped model in a wind or water tunnel. The boundary layerof a swimming fish is complicated by the motion of the bodyand is unquestionably different from that over a rigid model.

Furthermore, since the drag- and thrust-producing mechanismsof a swimming fish are coupled, even the use of an activelyswimming model requires indirect means to determine drag(Barrett et al., 1999). Gray (1936) was clearly skeptical of theextension of the so-called rigid-body analogy to thedetermination of drag on a swimming dolphin but, left withno alternative, he used rigid-body drag as a tentativeapproximation. Webb (1975) catalogues the rigid-body dragcalculations and measurements on fish that ensued, butreiterates the warning concerning the weakness of the analogy.The reservations of Gray (1936) were affirmed when Lighthill(1960, 1970, 1971) published his reactive model of fishpropulsion, which predicted thrust in steady swimming to be

as much as 35 times greater than the theoretical rigid-bodydrag. While the reactive thrust model of Lighthill (1960, 1970,1971) is considered to overestimate thrust, it is widely believedthat the drag on a swimming fish is, indeed, greater than rigid-body drag. Weihs (1974) determined that it was possible forfish to capitalize on this state of affairs by burst-and-coastswimming.

Lighthill (1971), citing discussions with Q. Bone, claimsthat this enhanced friction drag may be the result of boundarylayer effects resulting from the lateral movements of the bodysegments of swimming fish. The rate at which vorticity isproduced as the body surface is thrust into the surroundingfluid is likely to be higher than the outward diffusion of

vorticity that occurs during the retreat of the body surface. Theresult of this mechanism would be a boundary layer that isthinner and of higher shear than would be expected over therigid body. Our fish boundary layer data substantiate thishypothesis and reveal an additional mechanism of friction dragenhancement mean streamwise acceleration of the near-fieldflow.

Lighthills (1971) prediction of enhanced friction dragfurther confused the already troubled field of energetics inundulatory locomotion. Gray (1936) and Gero (1952), amongothers (Webb, 1975), made measurements that suggested thatthe power required to overcome rigid-body drag for dolphinsand certain fish was greater than their muscle mass was capableof producing. This spawned a search for mechanisms thatcould reduce the drag on an undulatory swimmer to levelsbelow the rigid-body drag. If, as Lighthill (1971) suggested,the drag on a swimming fish was actually several times therigid-body drag, the situation became even more problematic.It was clear that Lighthills (1971) model over-predicted thrust,that swimming performances had been exaggerated or that theestimates of available muscle power were too low.

Investigators of undulatory swimming hydrodynamics andmuscle physiology have studied each of these alternatives inan attempt to resolve the discrepancies. Thrust and power

E. J. ANDERSON, W. R. MCGILLIS AND M. A. GROSENBAUGH

y

x

u = u(y)

Ue

u() = 0.99Ue

v = v(y)

y

v

Object surface

Fig. 1. Tangential and normal velocity profiles in the boundary layerover the surface of an object. Tangential velocities are represented byhorizontal vectors and normal velocities by vertical vectors.Tangential velocity, u, above any given position,x, along the surfacevaries from 0 to Ue, the tangential fluid velocity at the edge of theboundary layer, with normal distance,y, from the surface. Boundarylayer thickness, , is defined as the normal distance between thesurface of the object and the point at which u=0.99Ue. The curve

connecting the tips of the tangential velocity vectors is known as theu-profile. The plot of fluid velocity normal to the body surface, v, asa function ofy, displayed to the right of the diagram, demonstratesthe conventional presentation of the normal velocity profile, orv-profile. There would be a set of velocity profiles for every position,

x, along the surface of the object in this two-dimensional example. Itis important to note that all velocities are measured with respect to acoordinate system fixed to the body surface. Therefore, the samebasic profile shapes are obtained whether the object is held stationaryin a flow or whether the object moves through still water.


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83The boundary layer of swimming fish

were estimated from velocity measurements of the wake of aswimming mullet Chelon labrosus (Mller et al., 1997).These investigators used techniques that were developed tocalculate thrust and minimum muscle power output in birdand insect flight, where they were met with varied success(Rayner, 1979a,b; Ellington, 1984; Spedding et al., 1984;

Spedding, 1986, 1987). In their preliminary work, Mller etal. (1997) report thrust estimates even higher than thetheoretical values of Lighthill (1971). At the same time,claims of extraordinary performances of undulatoryswimmers have been toned down somewhat (Lighthill, 1969;Rohr et al., 1998a) and estimates of available muscle powerhave been refined (Bainbridge, 1961; Webb, 1975; Weis-Fogh and Alexander, 1977; Fish, 1993; Rome et al., 1993;Coughlin et al., 1996). In the light of such findings, it appearsless incumbent upon fish and cetaceans to possessextraordinary drag-reducing secrets (Fish and Hui, 1991).Still, the problem has not been unequivocally resolved.Excised fish muscle driven at rates equal to those measured

in vivo has given relatively low power outputs (Rome andSwank, 1992; Coughlin et al., 1996; Swank and Rome, 2000;Rome et al., 2000). These studies suggest that maximumpower output measurements recorded during non-physiological stimulation and strain are not applicable.

Despite the dearth of available boundary layer data andLighthills (1971) prediction of drag enhancement based ontheoretical thrust, theories of drag reduction by boundary layermanipulation abound. The most notable proposed mechanismsfall under the categories of laminar boundary layermaintenance, turbulent drag reduction, utilization of shedvorticity and the delay of separation. Theories of dragreduction in undulatory swimming are reviewed and critised

in Webb (1975), Webb and Weihs (1983) and Fish and Hui(1991). One recent experimental study using a robotic fishclaims to have substantiated drag reduction in undulatoryswimming (Barrett et al., 1999). Earlier works, on the flowover waving plates, have also demonstrated mechanisms thatmay act to reduce drag, especially form drag. Taneda andTomonari (1974) observed that the flow over a waving platewith wave speed c, less than the oncoming flume speed U,resulted in separation of flow and turbulent recirculationregions in the wave troughs. When wave speed was increasedso that c/U1, flow remained attached over the entireplate. In some cases, boundary layer flow was completelylaminarized. In others, it oscillated between turbulent andlaminar.

Here, we present the first description of boundary layer flowin swimming fish based on high-resolution velocity profiles.We report the unsteady spatial distribution of boundary-layer-related variables over the surface of swimming fish and discussmechanisms responsible for the observed behaviors. Thedistribution of wall shear stress, determined from the boundarylayer, is used to determine the total friction drag and thepower necessary to overcome it. Theories of boundary layermanipulation, drag reduction and friction drag enhancementare re-examined.

Materials and methods

Fish

Scup Stenotomus chrysops (N=9) and smooth dogfishMustelus canis (N=1) were caught in traps or by hook and linein Nantucket Sound, off Woods Hole, MA, USA. The animalswere kept in 750 l tanks with a constant flow of fresh sea water

from Nantucket Sound. All fish kept longer than 2 days werefed frozen squid biweekly. Fish were transferred to and fromtheir tanks in 30 l buckets or 60 l coolers. Followingexperiments, fish were killed by cervical transection. The bodylength, L, of scup averaged 19.51.8cm (mean S.D.). Thedogfish measured 44.4cm.

Swimming conditions

Scup were observed swimming both in still water and in aflume. In still water, scup were observed swimming at340cms1 at water temperatures of 11C or 2225 C,depending on the season during which the experiments wererun. In the flume, scup were observed swimming at

1065cms1 at 2223C. The dogfish was observedswimming at 2065cm s1 in the flume at 2223C.

In flume trials, observations from three positions along themidline of each fish were made at one or more speeds. In scup,the measurements were made atx=0.50L, 0.77L and 0.91L. Indogfish, the measurements were made at x=0.44L, 0.53L and0.69L. The majority of flume data for scup were acquired at aswimming speed of 30cm s1 (18 swimming sequences). Atthis speed, scup were observed to use primarily caudal finpropulsion with infrequent strokes by their pectoral fins.Recordings of transverse velocity showed continuousundulatory swimming during all acquired sequences. In stillwater, scup tended to swim more slowly, frequently using their

pectoral fins and gliding. Therefore, in our analysis of thefish boundary layer, we have concentrated on the flumeexperiments and the fastest of the still-water swimmingsequences. The majority of the flume data for the dogfish wereacquired at a swimming speed of 20 cm s1 (22 swimmingsequences). Rigid-body measurements in dogfish were made attwo positions, x=0.44L and 0.69L at 20cms1. The moreforward positions on the dogfish were chosen because it wasdifficult to acquire sufficient data in the posterior region wherethe body wave amplitude increases dramatically with position.At positions posterior tox0.75L, the fish surface was capturedinfrequently in the small field of view of the flow-imagingcamera. The swimming speeds of 30 cm s1 in scup and20cms1 in dogfish were chosen because at these speeds thefish swam steadily for long periods without tiring.

Still-water trials were performed in a large rectangular tank(2.5m1.2m0.5m). Water depth was 20 cm. A channel 20cmwide was constructed along one of the long glass walls of thetank. The midpoint of the channel was used as the test section.The flow-imaging camera was partially submerged in a glassenclosure to prevent free surface optical distortion. Fish swamdeeply and slowly enough so that free surface wave effectswere negligible. Flowing-water trials were performed in alarge, recirculating, open-channel flume capable of speeds up


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to 70cms1. The racing-oval-shaped flume, with straightsections 7.6m long, is paddle-driven by a conveyor beltmechanism. The flume channel is 78 cm wide and 30cm deep.Water depth during fish swimming trials was 16cm. The testsection used was constructed against one of the glass walls ofthe flume, 20cm wide and 80cm long. The free surface was

eliminated using a sheet of acrylic. Honeycomb flow-throughbarriers bounded the test section, confining the fish to the testsection, and damping out large-scale flow disturbances. Thebarriers were 12.7cm in streamwise length with a tubediameter of 1.3cm. Turbulence intensity in the test sectionmeasured by laser Doppler anemometry (LDA) was 46 %over the range of experimental flow speeds. Without thehoneycomb barriers, turbulence intensity measured 78%.Velocity measurements outside the fish boundary layerdemonstrated scatter in agreement with the measured testsection turbulence intensity. Still-water trials showed little tono scatter in velocity outside the boundary layer. In both still-and flowing-water trials, fish swam far enough from the wall

generally about 10cm that wall effects are expected to beminimal.

Fluid flow around the fish was illuminated by a horizontallaser sheet, 0.5mm thick, and imaged from above with ahigh-resolution digital video camera (Kodak ES 1.0,1008pixels1018pixels) (Fig. 2). The flow was seeded withneutrally buoyant fluorescent particles, 2040m in diameter(Johns Hopkins University). Macro photographic lenses(Nikon, Micro-Nikkor, 60mm) were used to obtain high-quality, high-magnification images of particles in the flow overthe fish surface (Fig. 3). Fields of view used with the particle-imaging camera were 12cm on each side. The resultingimages had a scale of 50100pixelsmm1. Our fish boundary

layers measured 0.512mm in thickness. The laser (New WaveResearch, Nd:YAG, dual pulsed) was operated at low powerto prevent irritation to the animal and to minimize glare. Thetime delay, t, between laser pulses, i.e. between exposures ofthe flow, was set at 210 ms depending on swimming speed.The measured displacement of particles between exposures isdivided by this time to obtain particle velocities. The laser andthe particle-imaging camera were synchronized using a digitaldelay triggered by every second vertical drive signal of thecamera. The vertical drive signal is a TTL pulse that signalsthe moment between two exposures. When triggered, thedigital delay triggered laser 1 of the dual laser to fire t/2before, and laser 2 to fire t/2 after, the next vertical drivesignal of the camera, which was ignored by the digital delay.The camera was operated at approximately 30Hz, and 100sequential images were acquired per swimming sequence.Therefore, pairs of exposures, or image pairs, were acquired at15 Hz, and continuous sequences of 50 pairs were acquired.Two standard video cameras were used to obtain simultaneousrecordings of whole-body motion in lateral and dorsal views.This allowed fish boundary layer flow to be compared withrelevant instantaneous whole-body kinematic variables.

Measurements were confined to positions on the fish wherethe body surface was essentially perpendicular to the laser

sheet. As the angle between the laser sheet and the fish surfacedeviates from 90, boundary layer velocity profiles aredistorted, tending to give an incorrectly low wall shear stress.Images in which the fish surface is perpendicular to the lasersheet are easily distinguished from images in which the surfaceis at an angle to the sheet. In the former, the fish surface appears

as a sharp edge. In the latter, depending on the direction of tilt,either the intersection of the beam and the fish surface is notvisible or the features of the fish surface beneath the sheet arevisible, dimly illuminated by reflected laser light. Only imagesof the former type were used in the analysis.

In both still-water and flume trials, all three video cameraswere fixed with respect to the frame of the test section duringimage acquisition. In still water, the fish swam through the testsection. They therefore swam through each cameras field ofview at their swimming speed, U, and flow velocity outside thefish boundary layer was nearly zero. In the flume, fish heldstation in the test section without significant streamwisemotion with respect to the fields of view. The flow outside theboundary layer of the fish therefore moved through the fieldsof view at the approximate flume speed, U. Apart from theambient turbulence of the flume flow, the two situations areequivalent from the standpoint of fluid dynamics. Bothtechniques proved useful to the analysis of the fish boundarylayer. Still-water trials revealed actual boundary layerdevelopment over particular fish in undisturbed flow, whereasflume trials revealed the phase-dependent aspects of theboundary layer at selected positions on the fish. The flume wasalso used to look at boundary layer development by recordingseveral sequences from various streamwise positions.


Fig. 2. Sketch of the experimental arrangement for digital particleimaging velocimetry (DPTV) image pair acquisition. The illustration

depicts a still-water trial. Uand the arrow represent the velocity ofthe fish through the test section. In flume experiments, fish wereobserved to hold station in the flow through the test section. In suchcases, U represents the flume speed, and the flume direction wouldbe opposite to the arrow shown.

Dual pulsedlaser system

High-resolution

digital video

camera

U


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Rigid-body drag

In general, the dogfish swam very close to the bottom of theflume, and it was possible to measure the boundary layer of thedogfish at the same streamwise position and flume speed forboth swimming and resting. Three image sequences of thedogfish boundary layer were acquired while the dogfishconveniently rested motionless on the bottom of the flume. Theflume speed and water temperature were 20cm s1 and 23C.The resting data were used to determine rigid-body frictiondrag for the dogfish.

It was important to confirm that the bottom boundary layerof the test section did not affect the rigid-body measurementssignificantly. LDA showed that the boundary layer of the testsection bottom was thinner than 1.5 cm. Dogfish boundarylayer data were taken between 1.2 and 1.8cm. Flowvisualizations were therefore made outside, or at the outer edgeof, the flume bottom boundary layer, where small changes inthe height would not be expected to have a significant effecton the flow velocities at the outer edge of the fish boundary

layer, Ue. Velocities measured by particle tracking confirmedthis. Ue values in both the swimming and rigid-body cases werefound to be essentially the same at x=0.44L.

Digital particle tracking velocimetry

The acquisition and analysis of image pairs for digital

particle image velocimetry (DPIV) and digital particle trackingvelocimetry (DPTV) are now common practice amongengineers, chemists and a growing number of biologists. Forthis reason, the details of these techniques will be left to thenumerous existing works on the subject; the reader is referredto Adrian (1991), Willert and Gharib (1991) and Stamhuis andVideler (1995). Here, we report the variations on the themesof DPIV and DPTV necessary to capture and resolve the fishboundary layer. Flow velocities around the fish were quantifiedprimarily by semi-automatic DPTV (Stamhuis and Videler,1995). Particle pairs are located manually with a cursor on thecomputer screen. The term particle pair refers to the twoimages of the same particle that occur in an image pair. A

particular image pair typically has tens to hundreds of particlepairs depending on seeding density. Once the particle pairshave been located, a computer program then determines thecentroids of the particles and calculates displacement andvelocity. Conventional DPIV and automatic particle-trackingcode were sometimes used to resolve the outermost regions ofboundary layer flow, but they often failed to resolve the flowvery close to the moving surface of the fish.

The fish surface was located using an edge-detectionalgorithm developed in the study of squid locomotion(Anderson and DeMont, 2000; Anderson et al., 2000). Thealgorithm was further developed during the course of thepresent work to match surface features in sequential images and

thereby calculate the precise motions of the animal surface. Thismotion was conveniently described by a tangential and a normaldisplacement. Deformation and rotation of the fish surface werefound to be negligible for any image pair because of the shorttime separating the images and the small field of view. Trialsduring which the fish rested motionless on the bottom of thetank revealed the accuracy of this wall-tracking algorithm to bebetter than 0.5pixels. At our magnifications, this represents1020m error in displacement and, after smoothing,negligible error in surface slope. For a typical swimming trial,say U=20cms1 and t=5ms, this translates to less than 2%error in the measurement of tangential flow velocity relative tothe fish surface. Average maximum error in normal velocity is210%, depending on the magnitude of the transverse bodyvelocity. Since wall shear stresses were determined from theslope of the boundary layer profile near the body surface, sucherrors in velocity relative to the fish surface do not affect ourcalculated skin friction. Instead, these errors impact less criticalmeasurements, such as outer edge velocity, boundary layerthickness and their fluctuations. In general, these variables werelarge enough that errors were insignificant to negligible.

Tangential and normal velocity calculations

To construct tangential and normal velocity profiles from the

Fig. 3. A double exposure showing examples of particle pairs used todetermine fluid velocities in the boundary layer around a swimmingscup. An example of a particle pair is labeled with white arrows. Theparticles in the image were moving roughly from left to right. Scale

bar, 1 mm. The camera angle was as shown in Fig. 2. The bodysurface of the scup appears as a sharp, bright edge in the lower halfof the image. The position on the scup shown is x=0.55L on themidline of the fish, whereL is body length. The scup was swimmingat 8.3cms1 through still water, roughly to the right in the field ofview (black arrow). The body surface was moving laterally1.7cms1 in the direction away from the region of fluid shown herein the upper portion of the image. Note that the particles closer to thefish move a greater distance than the particles farther away from thefish. This is because the fluid closest to the fish is most influenced bythe motion of the fish through the fluid. However, in the frame ofreference of the fish, the particles closest to the fish are moving moreslowly than the particles farther from the fish, resulting in boundarylayer profiles similar to those shown in Fig. 1. The double exposure

was constructed by simply adding successive video images. Theimage was swept of approximately half of its original particles andthreshold-filtered for clarity of presentation.


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image pairs of flow over the fish surface, the motion ofparticles in the image pairs must be viewed from the referenceframe of the fish. Unless the surface can be described by astraight line, this requires the construction of axes normal andtangential to the fish surface for each particle. Assuming thatthe velocity profiles do not change significantly over the

relatively small field of view, this method results in the desiredboundary layer profiles. The separate profiles are built up fromthe normal and tangential components of velocity determinedfor each particle, with respect to the fish, plotted against normaldistance of the particle from the fish surface.

Normals from particles to the fish surface were determinedthrough a standard minimization of the distances from theparticles to the fish surface. The radius of curvature of the fishsurface was always larger in scale than the field of view. Thisensured convergence of the minimization process. The fishbody surface was found to be fitted well by a cubic polynomial.This was used as a means to smooth surface roughness,reducing needless scatter in the minimization process. The

normal velocity, v, of a particle with respect to the fish wascalculated by:

where t is the time between laser pulses, and y1 and y2 arethe lengths of the normals for the particle in the first and secondimages respectively. This simple equation can be used because,as mentioned above, the deformation and rotation of the fishsurface was negligible over the time between images, t.

The calculation of tangential velocity also began bydetermining normals to the fish surface from points in the fluidusing the same distance minimization. In this case, however,

the normals were determined from the midpoint of a particletrack to the average position of the fish surface in the twoimages. The slope of the average fish surface was determinedat the intersection of the normal and the average fish surface.The slope was used to construct a unit tangent vector, t, of theaverage fish surface, in a streamwise sense, with respect tothe camera pixel coordinates. That is, the vector lies in thehorizontal plane of the laser sheet, is at a tangent to the fishsurface and points roughly in the caudal direction. The velocityof the particle, Vp, and the velocity of the fish surface, Vs, weredetermined in the same coordinate system. The tangentialvelocity, u, of the particle with respect to the fish was thendetermined by the vector operation:

u = (VpVs) t , (2)

i.e. u is the component of the velocity of the particle, relativeto the fish surface, in the direction of the surface unit tangentvector in the plane of the laser sheet. Therefore, u=0 at the fishsurface and u=Ue at the edge of the boundary layer. The normalvelocity of the particle with respect to the fish can bedetermined in a similar manner, but normal velocitiescalculated from equation 1 are more accurate since fish surfaceaveraging is sidestepped. In some instances, conventionalDPIV was used to resolve the outer boundary layer and

near-field, reducing the tedium of semi-automatic DPTVprocessing. The better the seeding, the closer to the fish DPIVcould be used with confidence. DPIV nodes were treated as thepositions of virtual particles in the first image, and the locationsof correlation peaks were treated as virtual particle positionsin the second image. This use of DPIV was made only well

beyond the linear sublayer of the boundary layer and onlywhen particle densities allowed. The linear sublayer is theregion of the boundary layer closest to the body surface inwhich the tangential velocity profile is linear. It will be shownbelow that an accurate determination of velocities in the linearsublayer is critical to the analysis of skin friction. As expectedin instances of proper seeding, cross checks of such DPIV databy DPTV showed negligible differences in velocitiescalculated in the outer regions of the boundary layer.

DPTV errors

Absolute errors in DPTV depend on camera pixel resolution,field of view dimensions, particle shape, size, centroid analysis

and image quality. Relative errors are magnified by decreasedparticle displacements, which depend on tand the field of viewdimensions. We estimate average maximum DPTV errors oftangential velocities in the linear sublayer of the fish boundarylayers to be between 5 and 15%. This range arises fromconservative estimates of sub-pixel accuracy and particledisplacements of the order of 10pixels. These errors tend to beunbiased since they depend on the images of individual particles.Therefore, if enough particle pairs are sampled in a given imagepair, the error in wall shear stress determined for that image pairtends to be unbiased. Wall shear stress is determined from alinear fit of the u-profile in the linear sublayer.

Increased scatter was commonly observed in our v-profile

data compared with the u-profile data. This is probably dueto DPTV errors magnified by generally shorter normaldisplacements. Turbulence, wall tracking errors, variation inthe profile over the streamwise length of the field of view andcross-stream surface curvature may also contribute to scatterin our profiles. In still water, very little scatter was observedin our u-profiles, especially outside the boundary layer, whereparticles are nearly stationary in the field of view. This is strongsupport for setting our DPTV error towards the lower end ofour estimated 515% mentioned above.

Boundary layer profile analysis

Shapes of actual boundary layer profiles have beendetermined over the years both theoretically using theNavierStokes equations and experimentally using techniquessuch as hot-wire anemometry. Prandtls student Blasius (1908)determined the first boundary layer solution from theNavierStokes equations. Blasius (1908) used numericalmethods to determine the velocity profiles for the simplest flowgeometry laminar flow over a flat plate with no streamwisepressure gradient. Blasius solution shows excellent agreementwith experimental data. Since Blasius, several other so-calledexact solutions of the NavierStokes equations have beendetermined for laminar boundary layers, including accelerating

(1)y2y1

tv = ,



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and decelerating flows (Falkner and Skan, 1930), and for three-dimensional flows (Sowerby, 1954). It is important to notethat these are solutions, not theory, and are, therefore, validdescriptions of boundary layer behavior despite their age.

Knowledge of turbulent boundary layer profiles comesmainly from experimental data. Time-averaged measurements

of turbulent flow over flat plates with no pressure gradient haveconveniently revealed what is known as the law of the wall(Schlichting, 1979). When appropriately non-dimensionalized,the tangential velocity data follow a universal profile. Theeffects of streamwise pressure gradients and variousgeometries on this universal profile are well documented(Schetz, 1993). Tangential velocity, u, and distance from thewall,y, are non-dimensionalized for the law of the wall usingthe following definitions:

where is the kinematic viscosity of the fluid, o is the wallshear stress and is the fluid density. The defined intermediate,u*, is known as the friction velocity. Traditionally, the non-dimensionalized tangential velocity, u+, is plotted as a functionof log10(y+). Fig. 4A shows the law of the wall plotted in thismanner. Two distinct curves are evident. Closest to the wall,which can be thought of as running parallel to the u+ axis, theprofile is linear, with u+=y+. Note that, on a semi-logarithmic

plot, it does not look linear. This curve represents the linearsublayer, which is commonly referred to as the viscoussublayer in the analysis of turbulent boundary layers. Fartherfrom the wall, the profile follows a logarithmic curve. Flow isturbulent in the logarithmic region and laminar in the linearsublayer; a region called the transition zone separates the two.Unlike the linear sublayer, the shape and position of thelogarithmic region of the time-averaged profile may verysignificantly as a result of surface roughness and streamwisepressure gradients (Schetz, 1993). For this reason, data in thelogarithmic region cannot be used to determine wall shearstress on an undulating fish. The linear sublayer must be used.Nevertheless, the general shape of the logarithmic region isstill useful for distinguishing between turbulent and laminarprofiles. Boundary layer profiles were fitted to the law of thewall using the linear sublayer. The profile was then classifiedas turbulent or laminar on the basis of the profile shape outsidethe linear sublayer. For example, if the Blasius boundary layeris plotted using the non-dimensionalization of equation 3, themajority of the boundary layer profile follows the linear curveand is poorly fitted by the logarithmic curve (Fig. 4B).

It should be noted here that, for turbulent boundary layers,it is the time-averaged profile at a given streamwise positionthat is described by the law of the wall. This dependence of

the analysis of turbulence on sampling time is due to thefluctuating nature of turbulent flow. If the sampling time is tooshort, the instantaneous boundary layer profile could appear tobe laminar and not necessarily Blasius-like even if the flowwere turbulent. It is only when several instantaneous boundarylayer profiles over a particular point in a turbulent boundarylayer are drawn overlapped that the average curve drawnthrough the combined profiles follows the law of the wall. Ourprofiles, at most, can be considered time averages over aneffective sampling period ofTs=l/U, where l is the streamwisedimension of the field of view and Uis the swimming speed.Ts in our experiments ranged from 0.02 to 1.0s, much shorterthan traditional sampling periods, and led to uncertainty in thedesignation of certain profiles as turbulent. Nevertheless,several of our fish boundary layer profiles at high Reynolds

(3)

u

u*u+

u*

,

o

yu*

y+

100 101 1020

10

20

30

u+

y+

u+ = 5.75log(y+)+5.2

u+ = y+

A

100 101 1020

10

20

30

y+

u+

U= 40 cm s-1

x = 10 cmT= 15C

B

Fig. 4. Tangential boundary layer profiles presented as isconventional for the law of the wall. u+ and y+ are non-dimensionalized tangential velocity and normal distance from thebody surface, respectively. The relationship between these plots andthe tangential profile defined in Fig. 1 is best illustrated in B. If theaxes of B were switched and the y+ axis were changed from alogarithmic scale to a standard scale, the shape of the profilerepresented by the circles would be essentially the same as thetangential profile shown in Fig. 1. In other words, one can imaginethe object surface to be parallel to the u+ axis and the scaled

boundary layer tangential velocity vectors to span from they+

axis tothe plotted curves. (A) The time averaged profile of the law of thewall for turbulent boundary layer flow over a flat-plate with nostreamwise pressure gradient plotted in non-dimensional wall unitson a semi-logarithmic graph. (B) The tangential velocity profile ofthe laminar, zero streamwise pressure gradient, flat-plate Blasiusboundary layer (open circles) scaled as for the law of the wall. Thevalues used for velocity, U, streamwise position,x, and temperature,T, are within the experimental ranges of the present study.


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88

numbers showed excellent agreement with the law of the wall.More importantly, in the neighborhood of a particular surfaceposition, the shapes of u-profiles in the linear sublayer of aturbulent boundary layer are less variable than those in thelogarithmic region. Therefore, it is safe to assume that ourmeasurements of wall shear stress, which are based on the

linear sublayer, are accurate.Throughout the discussion of the measured boundary layers,the quantity,Rex, or the length Reynolds number, is used.Rexis the Reynolds number based on position, x, i.e. RexUx/.Rex is commonly used (Fox and McDonald, 1992) in detailedanalyses of fluid phenomena that depend on streamwiseposition, such as boundary layer thickness, wall shear stressand the transition of boundary layer flow from laminar toturbulent. For example, the position at which laminar flowmakes the transition to turbulent flow over a flat plate does notdepend on the total length, L, of the plate. Instead, transitiontends to occur at Rex=3.5105 to 5105, for any flat plate orrelatively similar surface (Schlichting, 1979), regardless ofL

and, subsequently, the standard Reynolds number based onlength,Re. Note thatRex atx=L is the same asRe.

Wall shear stress and friction drag

The tangential component of wall shear stress, o, in theplane of the laser sheet was determined using:

where is the dynamic viscosity of the fluid, u=u(y) is thetangential component of fluid velocity over the object in the

plane of the laser sheet and y is in the direction of the localoutward normal of the surface. In the linear sublayer of bothlaminar and turbulent boundary layers, the value of the partialderivative the normal gradient of u is constant and can bedetermined by a simple linear fit. This use of experimental datato determine wall shear stress has been termed the near-wallmethod by sterlund and Johansson (1999). Their wall shearstresses calculated from equation 4 using hot-wire velocitymeasurements show excellent agreement with theory andconcurrent measurements of shear stress by the oil-filmtechnique. They also determined and verified fluctuating shearstress measurements, due to the unsteadiness of turbulent flow,with Micro Electro Mechanical Systems (MEMS) hot films.

The wall shear stress distribution, o, over an object can beused to calculate the total friction drag,Df, using:

where S is the three-dimensional function defining the bodysurface of the fish, dA is the incremental area over which aparticular shear stress applies, and is the angle between thebody surface tangent in the laser plane and the streamwisedirection. The coefficient of friction for any object is defined as:

where is the fluid density,A is the total wetted surface areaof the body and Uis the relative velocity of the object throughthe fluid. To obtain accurate values of friction drag and the

coefficient of friction for a swimming fish, a large number ofmeasurements of wall shear stress at different positions and atdifferent phases of the undulatory motion must be taken.

For comparative purposes, a local coefficient of friction, Cfx,was defined as:

By this definition, Cf is the area average ofCfx over the fishsurface. Therefore, Cf for a given fish falls between themaximum and minimum values ofCfx determined over the fishbody. Rough time averages of Cfx and other boundary layervariables were determined by simply averaging thesequantities over sufficient boundary layer realizations to coverthe entire locomotory cycle.

Undulatory phase

Boundary layer data were taken on one side of the fish forany given trial. The fish surface oscillated in the field of viewof the particle-imaging camera because of the transversemotion of the body. We will use the term crest to describethe instance when the section of the fish surface in view hasmoved to its full amplitude in the direction of the outward-pointing surface normal, i.e. the positive y-direction. We usetrough for the instance of full amplitude in the negative y-

direction. Phase is set to 90 at the crest and 270 at the trough.Transverse wall velocity as a function of time determined fromwall tracking was fitted with a sine function. The phase of thebody surface transverse position was determined by integratingwall velocity or simply by subtracting 90 from the phase oftransverse wall velocity.

Detailed phase analysis was only applied to flume data. Still-water trials result in a more complicated mix of phase andposition. The propulsive wave of the fish travels streamwise ata speed slightly greater than the swimming speed, U (Gray,1968). Since, for still water, the field of view is fixed with respectto the bulk fluid in the tank, phase appears to change more slowlythan if observed in a flume. If the wave speed were nearly equal

to the swimming speed, almost no change in phase would beobserved. Therefore, still-water trials give information at variousphases at various positions. In contrast, flume trials giveinformation at one position as a function of phase.

Results

Fish boundary layer profiles

More than 70 swimming sequences of scup and 30sequences of dogfish were acquired, yielding hundreds ofusable image pairs for boundary layer realization. Tangential

(7)o

GU2,Cfx

(6)Df

GAU2,Cf=

o dAcos , (5)Df=

S

(4)o = ,y=0

u

y



9/22


and normal velocity profiles were determined for more than270 image pairs from 36 swimming sequences with high imagequality over the full range of experimental speeds. Only onedogfish has so far been examined, so generalizationsconcerning anguilliform swimmers must be considered

tentative. Nevertheless, the quantity and consistency of thedogfish data suggest that the conclusions regarding thespecimen observed are well founded.

Fish boundary layer profiles tended to resemble the solutionsof either Blasius or the law of the wall (Fig. 5). Profiles thatdeviated from these two types often exhibited good agreementwith the FalknerSkan solution (Fig. 6). The FalknerSkansolution can describe either an accelerating (Fig. 6A,B) or adecelerating (Fig. 6C,D) boundary layer depending on thechoice of a coefficient in the FalknerSkan differentialequations. Boundary layers are classified as accelerating ordecelerating on the basis of their u-profiles. However, in theinstantaneous profiles of a boundary layer, the evidence ofacceleration or deceleration is found in the v-profile. Negativenormal velocity at the outer edge of the boundary layer(Fig. 6B) reveals that there is a net normal flow of fluid, ornormal flux, into the boundary layer characteristic of anaccelerating boundary layer. In contrast, the Blasius solutionalways shows positive normal velocity at the edge of theboundary layer (Figs 5B, 6B) and is, therefore, a deceleratingboundary layer.

The connection between normal flux and accelerationinvolves the incompressibility and continuity of water. Imaginea constant-diameter pipe carrying water with a prescribed

upstream volume input. If the pipe is tapped, so that water canbe pumped in or out, the downstream volume flow of the pipecan be changed. Incompressibility and continuity require thatthe flow speed must also change. If water is pumped in, flowmust accelerate in the pipe in the vicinity of the tap. If we pump

water out, the pipe flow decelerates.Fish boundary layer profiles occasionally resembled

strongly decelerating FalknerSkan profiles characterized byhighly inflected u-profiles with low wall shear stress (Fig. 6C).The v-profiles of these realizations revealed flow out of theboundary layer characteristic of boundary layer deceleration(Fig. 6D). Inflected boundary layers of this type are often asign of incipient separation (Batchelor, 1967). No profilesindicative of separation were observed.

Flow condition in the boundary layer

In still-water trials, boundary layer profile shapes alwayssuggested laminar flow. This is not entirely surprising sinceReynolds numbers, Re, were 3103 to 6104 lower thanthe standard critical range for boundary layer transition,Rex=3.5105 to 5105. Recall that, atx=L,Rex is at a maximumand equal to Re. In flume trials, however, both laminar andturbulent profile shapes were observed even though Reynoldsnumbers did not quite reach the critical value. The criticalvalues for boundary layer transition assume quiet incomingflow over smooth rigid surfaces. The ambient turbulence of theflume, the roughness of the fish surface and the unsteadinessof the flow over the fish might be expected to trip turbulenceat lower than critical Reynolds numbers. The boundary layer

Fig. 5. Two representative boundary layerrealizations illustrating the distinction betweenlaminar-like and turbulent-like boundary layers. Eachdata point represents information calculated from one

particle pair of the image pairs used for the givenrealizations. The first realization shown (AC) isfrom x=0.50L, where L is body length, on a scupswimming in the flume at 42 cms1,Rex4104. Thesecond (D) is from x=0.53L on the dogfishswimming in the flume at 20cms1, Rex4104.(A) The u-profile of the first realization showingagreement with a Blasius fit drawn as a solid curve.(B) The v-profiles of the first realization and theBlasius fit of A. (C) The u-profile of the firstrealization compared with the law of the wall byfitting the linear sublayer. The boundary layerdistinguishes itself as laminar-like, as outlined in Fig.4. (D) The dogfish boundary layer realization

showing good agreement with the law of the wall,distinguishing the profile as turbulent-like. Note theslight shift in the logarithmic region. The fit exhibitssharp contrast to the fit of the profile shown in C.

Rex, the length Reynolds number based on x, thestreamwise position on the body measured from the leading edge; u, tangential velocity;y, normal distance from the body surface; v, normalvelocity; u+,y+, non-dimensionalized tangential velocity and normal distance, respectively.

0 10 20 30 40 500

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2

3

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y(mm)

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90

over scup swimming in the flume at 30 cm s1,Re=6104, wasapparently always laminar over the entire body. The boundarylayer over a dogfish swimming at 63 cm s1, Re=3105,measured at x=0.63L, Rex=1.9105, appeared to be primarilyturbulent. In some cases, at Reynolds numbers between thesetwo values, the boundary layer apparently oscillated betweenlaminar and turbulent. When this was observed, turbulentprofile shapes tended to appear at the crest phase of the bodywave. The boundary layer generally returned to a laminar

shape during the crest-to-trough motion.The rigid-body case of the dogfish revealed an interesting

effect. Flow appeared laminar at x=0.44L and turbulent atx=0.69L. For the swimming dogfish, boundary layer flowappeared to be laminar at x=0.44L and x=0.69L for most ofthe time, with some evidence of oscillation between laminarand turbulent flow at x=0.69L. The observation of laminarboundary layer flow at x=0.69L during swimming suggests astabilization process. The same phenomenon was observed byTaneda and Tomonari (1974) comparing the boundary layerflow for the rigid body and various swimming cases of awaving plate.

Local friction coefficients

Posterior tox=0.8L in scup andx=0.5L in dogfish, the time-averaged local friction coefficients, Cfx, of both speciesincrease above the flat-plate laminar and turbulent values (Figs7, 8). This increase in friction is much more dramatic in theanguilliform swimmer. Local friction coefficients in the rigid-body case of the dogfish do not show this increase and remainbetween the laminar and turbulent flat-plate values, i.e. thefriction drag on the swimming dogfish is higher than that onthe dogfish stretched straight in the flow.

In many cases, the values ofCfx, Ue and versus relative


u (cm s-1)

y(mm)

v (cm s-1)

y(mm)

u (cm s-1)

y(mm)

v (cm s-1)

y(mm)

0 10 20 300

1

2

3

A

-5 0 50

1

2

3

B

0 5 10 150

2

4

6

8

C

-2 -1 0 1 20

2

4

6

8

D

Fig. 6. Two representative boundary layer realizations

that are fitted well by the FalknerSkan solution. Thefirst realization (A,B) is from x=0.50L, where L is bodylength, on a scup swimming in the flume at 30 cm s1,

Rex3104. The second (C,D) comes from very close tothe body trailing edge of a scup swimming in still waterat 14cms1 and decelerating at 10cms2, Rex2104.(A) The u-profile of the first realization with aFalknerSkan fit drawn as a solid curve. The dashedcurve is the Blasius solution with the same wall shearstress. (B) The v-profiles of the first realization, theFalknerSkan solution and the Blasius solution. (C) Theu-profile of the second realization. (D) The v-profile ofthe second realization. The solid curve is theFalknerSkan fit.Rex, the length Reynolds number based

onx, the streamwise position on the body measured fromthe leading edge; u, tangential velocity; y, distance fromthe body surface; v, normal velocity.

2104 4104 6104 810410-3

10-2

10-1

L

T

Rex

Cfx

Fig. 7. Time-averaged local friction coefficients, Cfx, versus localReynolds number, Rex, on scup () and dogfish (), including therigid-body case of the dogfish (). Cfx is plotted versus Rex becauseit is known that geometrically similar objects all have the samedistribution of Cfx with Rex regardless of size, speed or fluidenvironment. Plotting the local friction coefficient versus Rex istherefore the best way to compare the distribution of friction over aset of objects in varying conditions of size, speed and viscosity. Thedata were averaged over several locomotory cycles from severalswimming sequences at the same flume speed for each species at2223C: U=20cms1 for the dogfish, U=30cms1 for scup. Thelines labeled T and L are flat plate friction for turbulent (T) andlaminar (L) boundary layer flow with no streamwise pressuregradient. On average, each data point shown, representing a whole-cycle average, represents eight boundary layer realizations for scup,34 realizations for the swimming dogfish and eight realizations forthe rigid dogfish. Error bars are based on the maximum percentageerrors in the determination of the slope of the linear sublayer, i.e. thewall shear stress, for the boundary layer profiles contributing to eachdata point. U, swimming speed.


11/22


position,x/L, were observed to depend both on species and onthe sign of the transverse velocity of the fish surface (Fig. 8).Cfx increases out of the range of flat-plate friction moreforward on the body of the dogfish than on the scup(Fig. 8A,B). In both species, local friction oscillates in phasewith transverse body velocity (Fig. 8A,B). In the dogfish, thetime average ofUe increases with streamwise position on thebody (Fig. 8D), suggesting a mean acceleration of both theboundary layer and the near-field flow over the fish. In thescup, the time average ofUe is close to Ufor the entire regionthat was measured (Fig. 8C). In both species, Ue oscillates inphase with transverse body velocity (Fig. 8C,D) and localfriction (Fig. 8A,B), suggesting local oscillatory accelerationand deceleration in the near-field and boundary layer. Theboundary layer thickness over the posterior region of thedogfish, where local friction increases above flat-plate friction,oscillates 180 out of phase with transverse body velocity (Fig.8F). Oscillatory effects in Cfx, Ue and are more pronouncedin the anguilliform swimmer than in the carangiform swimmer.Finally, the behavior ofCfx, Ue and in the rigid-body case isopposite to that in the swimming dogfish (Fig. 8B,D,F), whilescup data show some similarity to the rigid dogfish case.

Uncertainties in Cfx, Ue and were determined to be

approximately 31 %, 6% and 21 %, respectively, withsome variation among trials depending on the quality of theflow realizations. For example, the rigid-body case of thedogfish has lower than average uncertainty in Cfx (19%) asa result of the large number of images of the same eventacquired; i.e. many particle pairs were sampled. Uncertaintieswere often greater in one direction than another. For instance,the uncertainties in Cfx for the swimming dogfish were +42%and 21 %. Where appropriate, error bars are used to displaythe unique uncertainties of data points.

Data from scup swimming in the flume at swimming speedsranging from 30 to 60cm s1 at a water temperature of 23.3 Cshow that, in the neighborhood ofx=0.5L, Cfx falls within therange of values expected for flat plates (Fig. 9). The effectsof transverse body surface velocity at this position areconsistently small compared with more caudal positions(Fig. 8A). Therefore, at some positions on the fish,Rex appearsto be sufficient to predict local friction, whereas at otherpositions local friction deviates from flat plate friction andoscillates significantly. Boundary layer data from still-watertrials in scup suggest that the similarity to a flat plate mayextend over the majority of the anterior half of the fish(Fig. 10). Boundary layer acceleration due to the finite

0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

x/L

Cfx

A

LT

0.4 0.6 0.8 10

0.01

0.02

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x/L

Cfx

B

L

T

0.4 0.6 0.8 1

0.8

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x/L

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/U

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/U

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8

10

x/L

(mm)

E

0.4 0.6 0.8 10

2

4

6

8

10

x/L

(mm)

F

Fig. 8. Plots of time-averaged local frictioncoefficients, Cfx, normalized tangential velocity atthe edge of the boundary layer, Ue/U, andboundary layer thickness, , as a function ofrelative streamwise position, x/L, for the samedata presented in Fig. 7. Time averages overperiods during which the fish transverse bodyvelocity was positive or negative are denoted by() and (), respectively. Data from scup are

presented in A, C and E. Dogfish data arepresented in B, D and F. The rigid body case isdenoted by circles connected by dashed lines.Turbulent and laminar flat plate friction, labeledT and L, are included in A and B forcomparison. On average, the data points for theopposite directions of transverse velocity () and() represent half as many realizations as for thewhole-cycle averages of Fig. 7. Error bars arebased on the maximum percentage errors in thedetermination of the variables presented for theboundary layer profiles contributing to each datapoint.


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92

thickness of the scup leading edge, in contrast to a flat plate,is the likely explanation of a spike in local friction observednear the anterior end (Fig. 10).

Oscillatory behavior of the boundary layer

Oscillations in Cfx, Ue and were highly correlated to thetransverse velocity of the body surface (Fig. 8). Local frictionand Ue tend to be highest when the fish surface is moving intothe fluid and lowest when the surface is retreating from thefluid; behaves in the opposite manner. A more highly resolvedpicture of the relationships between Cfx, Ue, and Ve versusbody phase was obtained using polar phase plots for the dogfishswimming at 20cms1 (Figs 11, 12). Cfx and Ue are roughly inphase. Boundary layer thickness is roughly 180 out of phasewith Cfx. Normal flux oscillates roughly 180 out of phase withtransverse body velocity. In addition to these previouslydescribed trends, the phase plots reveal a clockwise processionof maximum Cfx, Ue, and possibly Ve with increasing relativeposition,x/L. This procession suggests that the distributions ofthese variables can be characterized as waves travelling alongthe body of the fish with wavelengths and speeds different fromthose of the body wave. The details of these distribution waveswill be discussed below.

Oscillation of normal velocity

Not only was Ve observed to oscillate with body motion, butsequences of normal velocity profiles in both scup and dogfishswimming in the flume also revealed oscillation throughout theentire profile (Fig. 13). In both species, the sign of the normal

velocity throughout the boundary layer is 180 out of phasewith transverse body surface velocity, vw. As the body surfacemoves into the fluid, normal velocity is negative. Duringretreat, it is positive. At this short distance from the surface ofthe fish, incompressibility and continuity predict that thisbehavior is not simply a relative velocity effect. Furthermore,

if the effect were due strictly to relative motion, the v-profileswould be expected to exhibit velocities equal to the transversewall velocity throughout the boundary layer.

Incipient separation

While no boundary layer separation was observed in the fishstudied, incipient separation was seen in six swimmingsequences. Figs 14 and 15 show examples of incipientseparation in scup in both still and flowing water. The examplefrom still water (Fig. 14) dramatically demonstrates the highlyinflected, low shear boundary layer profile shape of incipientseparation. Our data show that incipient separation occurs afterwall velocity, vw, becomes negative, and that frictionessentially drops to zero where the inflected profiles occur.

In the flume, a time sequence of the boundary layer behaviorwas obtained that included incipient separation (Fig. 15). Asin the still-water example (Fig. 14), incipient separation occursclose to where wall velocity, vw, becomes negative. Localfriction decreases noticeably. The time sequence suggests thatthe inflected boundary layers, which occur at troughs, arestabilized as the body phase cycles towards the subsequentcrests. In the flume, instances of inflected boundary layers wereobserved twice in separate sequences of scup swimming at30cms1 and once in the dogfish swimming at 20cm s1.


Fig. 9. Time-averaged local friction coefficients, Cfx, versus lengthReynolds number, Rex, at x=0.50L, where L is body length, fromseveral scup swimming sequences ranging in swimming speed from30 to 60cms1. No lines are drawn connecting these data points(diamonds) since they do not represent the distribution ofcoefficients of friction along the body of a scup. The data at eachRexrepresent 910 boundary layer realizations. Error bars are based onthe maximum percentage errors in the determination of the slope ofthe linear sublayer, i.e. the wall shear stress, for the boundary layerprofiles contributing to each data point. Rex, the length Reynoldsnumber based onx;x, position along the body.

L

T

2104 4104 6104 810410-3

10-2

10-1

Rex

C

fx

0 0.2 0.4 0.6 0.8 10

0.005

0.01

0.015

0.02

0.025

0.03

Cfx

x/L

T

L

Still water

Flume

Fig. 10. Local friction coefficients, Cfx, representing 18 scupboundary layer realizations compiled from five still-water trials atswimming speeds of 1040cms1 (circles). Each point representsdata from one boundary layer realization. The lines labeled T andL are flat-plate friction for turbulent and laminar boundary layerflow, respectively, with no streamwise pressure gradient. The whole-cycle-averaged local friction for scup swimming at 30cm s1 in theflume (Fig. 8A) is included and denoted by diamonds connected bysolid lines. Sequences displaying incipient separation, i.e. Cfx0,were not included. Error bars are omitted for clarity. The uncertaintyin Cfx is 31 %.x, position along the body;L, body length.


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Total skin friction and friction coefficients

Table 1 presents calculations of total body friction drag andcorresponding friction coefficients for scup (swimming) anddogfish (swimming and rigid). The power required to overcomefriction drag is also presented. In Fig. 16, the coefficients offriction are plotted versus Re together with flat-plate friction forcomparison. The coefficients of friction for swimming scup andthe rigid dogfish fall within the range of flat-plate friction forlaminar and turbulent flow. The coefficient of friction for theswimming dogfish lands above this range.

Discussion

The nature of the fish boundary layer

In the most general sense, the boundary layer of swimmingfish can be characterized by streamwise trends and local

oscillations in Cfx, Ue, , Ve and overall profile shape.Streamwise trends proved to be highly dependent onswimming mode (Fig. 8). Local oscillations of boundary-layer-related variables occurred similarly in both the dogfishand scup, although the amplitudes of oscillation were greaterin the dogfish. The data reveal that all these behaviors canbe understood from the perspective of two superimposedfluid accelerations: mean streamwise acceleration and localoscillatory acceleration that is correlated to the transversemotion.

The streamwise increase ofUe in the dogfish is evidence ofmean streamwise acceleration of the near-field and boundarylayer flow. The time-averaged values of Cfx increase and ofdecrease, as would be expected in a boundary layer under anaccelerating exterior flow. No significant mean streamwiseacceleration was observed in scup; however, the near-field flow

Fig. 11. Phase plots of local friction coefficients, Cfx, and boundary layer thickness, , from 10 swimming sequences of the dogfish at the sameswimming speed, U=20cms1, at three streamwise positions, representing 100 boundary layer realizations. The three positions along the bodyexamined were, x=0.44L, 0.53L and 0.69L, where L is body length. Each phase plot presents the behavior of a particular boundary layervariable versus body phase, , measured at a particular position along the fish. The crest of the body surface corresponds to =90; the troughcorresponds to =270 . Time and phase increase in the counterclockwise direction, and radial distance expresses the magnitude of theboundary layer variable plotted. The radial scaling is printed between the angular positions 60 and 90 . A solid radius is drawn on each phaseplot to mark the phase of the maximum value of the variable displayed. Consider the plot ofCfx atx=0.69L. At =0 , the body is cycling fromtrough to crest, and Cfx is equal to 0.033. The highest positive transverse body velocities occur near this phase. As the phase reaches 90, thebody reverses direction. Cfx decreases, reaching a minimum of 0.005 near =150. At the trough, =270 , friction is increasing and reaches amaximum near =330 as the body is thrust towards the fluid. The cycle then repeats itself. The set of three plots for each variable are drawn tothe same scale so that magnitudes as well as phase relationships can be compared. For example, one can observe the mean streamwise increasein Cfx, noting the progressive increase in area enclosed by the plotted curves. These curves are fourth-degree polynomial fits of the boundarylayer data. They are constrained to be periodic, but not sinusoidal, by equalizing function values and slopes at the beginning and end of thecycle. This method of fitting the data allows for asymmetric phase plots.

Cfx

0.02

0.04

30

210

60

240

90

270

120

300

150

330

180 0

0.02

0.04

0.02

0.04

x = 0.44L

(

mm)

2.5

5

x = 0.53L

2.5

5

x = 0.69L

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94

was not observed to decelerate either. The absence of meanacceleration over the scup follows from the tendency ofcarangiform swimmers to produce the majority of their thrust atthe caudal fin. Mean streamwise acceleration is a sign of thrustproduction anterior to the caudal fin. The difference betweenscup and dogfish in this regard can be understood by consideringthe relatively small wave amplitudes present in carangiformswimmers. Studies of swimming performance after completecaudal fin amputation (Breder, 1926; Gray, 1968; Webb, 1973)

show that carangiform swimmers are able to compensatesurprisingly well for the loss of fin thrust by increasing bodywave amplitude and frequency. The observed differences inamplitude and frequency after complete amputation suggest achange in swimming mode on the part of the fish. Meanstreamwise acceleration of the near-field might be expected tooccur over a larger portion of the body in these fish since theyhave only their bodies to produce thrust in the amputated state.However, it does not follow that carangiform swimmers actuallydo use their body wave to produce a significant amount of thrustforward of the caudal fin. When the caudal fin is amputated, onewould not expect the fish to use the same body motion to swimas it did with the caudal fin intact. Therefore, it would be tenuousto conclude that, since a carangiform swimmer with its caudalfin amputated uses body-based thrust to swim, the same is truewhen the tail has not been removed. Our data suggest low body-based thrust in scup compared with caudal-fin-based thrust sincemean streamwise acceleration of the near-field fluid forwardof the peduncle, which would be the evidence of the bodyproducing thrust forward of the peduncle, was not observed.

In both scup and dogfish, Ue and were observed to oscillate180 out of phase with each other. Cfx behaves as would beexpected according to the first-order approximation oUe/(equation 4). This and the concurrent oscillation of the v-profile

(Fig. 13) reveal a cycle of local tangential acceleration anddeceleration of the boundary layer at any given position alongthe fish. As explained above, positive and negative normalvelocity relative to the body at the edge of the boundary layerare evidence of normal flux out of and into the boundary layer,respectively. In general, tangential flow accelerates as the bodycycles from trough to crest, and decelerates as the body cyclesfrom crest to trough.

One might argue that normal flux exhibited by the v-profile

is simply the observation of relative motion due to the surface-fixed coordinate system, but that would be true only if one werefocusing on the far-field, where there is negligible impact onthe flow due to the fish. Allen (1961) apparently uses this far-field concept to explain his supposed observation of boundarylayer thickness oscillation. In contrast, the normal flux revealedin Fig. 13 occurs at the level of the near-field and boundarylayer. Therefore, it is not merely relative fluid motion. This isindicated by the fact that across the boundary layer, normalvelocity, v, remains well below vw in magnitude. Near thesurface of the fish this is the necessary result of the continuityand incompressibility of water and the no-flux boundarycondition at the surface. The fact that the same is observedat the edge of the boundary layer indicates tangential and/orcross-stream boundary layer acceleration.

Wave-like distributions of boundary layer variables and

pressure

As mentioned above, the oscillatory behavior ofCfx, Ue, and Ve with relative position along the fish suggests that thestreamwise distributions of these variables can be representedas travelling waves moving in the same direction as the fishbody wave. The clockwise procession of maximum values inthe phase plots reveals an ever-increasing downstream shift in


Table 1. Total drag calculations based on measured wall shear stress distributions over scupStenotomus chrysops and dogfishMustelus canis

M. canis S. chrysops

Flume, rigid body Flume Still water Flume

Swimming speed, U(cm s1) 20 20 10 30Temperature, T(C) 22.8 22.8 23.3 23.3Lateral body area,A (m2) 0.0213 0.0213 0.0206 0.0206Body length,L (cm) 44.4 44.4 19.5 19.5Mass,M(kg) 0.218 0.218 0.166 0.166Measured friction drag,Df(N) 0.0033 0.0064 0.0013 0.0067Theoretical rigid-body friction drag,Dft (N) 0.0018 0.0018 0.0007 0.0044Measured friction drag coefficient, Cf 0.0076 0.0146 0.0127 0.0071Theoretical friction drag coefficient, Cft 0.0041 0.0041 0.0068 0.0047

Df/Dft 1.8 3.6 1.9 1.5Df/measured rigid-body friction drag 1.0 1.9Power required to overcomeDf(mW) NA 1.3 0.13 2.0Mass of red muscle per mass of fish 0.0209*Mass of red muscle (kg) 0.0035Power required per mass red muscle (W kg-1) 0.6

*Taken from Zhang et al., 1996.NA, not applicable.


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the streamwise distributions of these variables with respect

to the phase of the body travelling wave (Figs 11, 12). Theregular periodic behavior of these variables at fixed positionson the fish reveals that these distribution waves and the bodytravelling wave have the same frequency, f. Since c=f, theincreasing streamwise phase shift of the variable distributionswith respect to the body wave is therefore due to thedistribution waves having a longer wavelength, , and higherwave speed, c, than the body travelling wave.

Wave speeds and wavelengths of the distribution waves canbe determined from the streamwise rate of procession. Theprocession of local friction is approximately 30 asx changesfrom 0.44L to 0.53L. Betweenx=0.53L and 0.69L, processionis approximately 60. Therefore, the ratio of procession tochange in body position, i.e. the rate of procession, is roughlyconstant (approximately 354 per body length). Thus, thewavelength and wave speed of the streamwise distribution oflocal friction are roughly constant. If the rate of processionwith streamwise position were variable, the wavelength andwave speed would be variable. In the dogfish swimming at20cms1, the measured body wavelength, , was 27cm andbody length,L, was 44.4 cm. Therefore, the rate of processiongiven above translates to approximately 215 per bodywavelength, i.e. when the body wave travels one bodywavelength, the friction distribution travels 1.6 body

wavelengths. In other words, the friction distribution travels

1.6 times faster along the body than the body wave. Boundarylayer thickness exhibits the same rate of procession. Ue appearsto have the same rate of procession despite the larger phaseshift betweenx=0.44L and 0.53L. There is so little variation inUe atx=0.44L that it is possible that there is significant errorin the determined phase of the maximum. Finally, normal flux,Ve, exhibits the same rate of procession between x=0.44L and0.53L, but very little procession occurs between x=0.53L and0.69L.

Taken together, the general procession of all four variables(Figs 11, 12) is evidence of a travelling pressure distributionover the fish. Boundary layer thinning, negative normal fluxand the increase in Ue can be understood as being linked toaccelerations of the boundary layer and near-field flow. Theseaccelerations, in turn, can be thought of as being driven, at leastin part, by pressure gradients. We assume here that maxima inboundary layer acceleration, as shown by Cfx, Ue, and Ve, areindicative of maxima in pressure gradient. Therefore, thepressure distribution around the fish behaves like thedistributions of these variables in wavelength and wave speed.

From this assumption, the data in Figs 11 and 12 suggest thatthe travelling wave of the pressure distribution and the bodywave are approximately 60 out of phase at x=0.44L, withpressure lagging behind the body wave. This means that, when

Fig. 12. Phase plots of tangential and normal velocity at the edge of the boundary layer, Ue and Ve, respectively, for the same 10 swimmingsequences of dogfish swimming (20 cms1) as in Fig. 11. The details of the construction of the phase plots are described in the legend ofFig. 11. For Ve, the solid lines represent positive values, or outflow, and the dashed lines, negative values, or inflow. The increasing areaenclosed by the plots ofUe show mean streamwise acceleration as shown in Fig. 8D.

17.5

35

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a body crest is atx=0.44L, there is a pressure maximum (1/6)upstream of the body crest. As these two waveforms progressdown the fish, the faster wave speed of the pressure distributioncauses the pressure maximum to move past the body wave crest.

By the time the body wave crest has reached the peduncle ofthe fish, the pressure maximum resides on the rearward-facing

surface of the caudal fin. In the same way, pressure minima shiftto positions on the forward-facing surfaces of the posterior bodyand caudal fin. This orientation of pressure maxima and minimawould result in thrust production over the posterior half of thebody. On the anterior half of the body, pressure maxima occuron the forward-facing surfaces and pressure minima occur onthe rearward-facing surfaces, as is normally the case for a non-

thrust-producing body moving through a fluid. Therefore, thepressure distribution suggested by the behavior of the boundarylayer, independent of the assumption of thrust production, is inelegant agreement with the expected hydrodynamics in fish.This suggests that analysis of the boundary layer may be aninvaluable tool in the investigation of the hydrodynamics ofundulatory swimming.

Drag enhancement and drag reduction

Friction drag on swimming fish is higher than rigid-bodyfriction drag, as predicted by Lighthill (1971). The data inTable 1 reveal that the friction drag on a swimming dogfishis 3.6 times the theoretical rigid-body friction drag and 1.9

times the measured rigid-body friction drag. The differencebetween these two ratios can be explained by the turbulent


0

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Fig. 13. Time series of normal velocity profiles, v-profiles, from scup(A) (x=0.77L, U=30cms1) and dogfish (B) (x=0.53L, U=20cms1)swimming sequences in the flume, together with transverse bodysurface velocity, or wall velocity, vw. The dashed vertical linesrepresent the v=0 axis for each profile and are positioned at the timesof the realizations. These times correspond to the times at which vwwas determined. Velocities within the profiles can be determined on

the basis of the velocity scale bar shown and the respective v=0 axis positive to the right, negative to the left (see Fig. 5B). L, bodylength; U, swimming velocity; y, distance normal to the bodysurface.

Fig. 14. Boundary layer development (i.e. u-profiles), transverse body surface velocity, vw, andlocal friction coefficients, Cfx, over a swimmingscup showing incipient separation. The u-profilesshown were observed 5cm above the center lineof the fish and spanned from the leading edge tothe trailing edge of the body for a chord length,

Lc, of 9 cm. The dashed vertical lines represent theu=0 axis, where u is the tangential fluid velocity,for each profile and are positioned at the relativestreamwise position on the fish,x/Lc, of the givenrealization. These positions correspond to thepositions at which vw and Cfx were determined.Velocities within the profiles can be determinedon the basis of the velocity scale bar shown andthe respective u=0 axis positive to the right,negative to the left (see Fig. 5A). The decreasingdistance between successive u=0 axes revealsthat the fish was decelerating. The very quiet, oruniform, flow just outside the boundary layershows that this is a still-water trial. y, distancenormal to the body surface.


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boundary layer flow observed over the posterior region of therigid dogfish measured in the flume. Our theoretical drag iscalculated assuming laminar flow to compare our measureddrag enhancement with that predicted by Lighthill (1971).Drag enhancement for the dogfish falls within his predicted35 times. In scup, drag enhancement was observed to be lesspronounced than that observed in the dogfish, and measuredfriction drag was calculated to be only 1.51.9 times thetheoretical rigid-body drag. Interestingly, the behaviors of

Cfx, Ue and for the swimming scup are not dramaticallydifferent from those for the rigid dogfish (Figs 7, 8). This isreasonable considering that a carangiform swimmer deviatesless from a rigid body than does an anguilliform swimmer(Breder, 1926).

Fig. 8 reveals that enhanced friction drag can be linked toboundary layer thinning in both species, thus supportingthe hypothesis of Bone and Lighthill (in Lighthill, 1971).However, greater values of Ue in the swimming dogfishcompared with the rigid-body case suggest that meanstreamwise acceleration of the near-field is a second,independent mechanism of enhanced friction drag. While thehypothesis of Bone and Lighthill requires transverse motion tooperate, any streamwise acceleration of the flow around a bodycan result in increased friction drag, regardless of transversemotion. Of course, there would be no acceleration of the flowaround a fish if the fish were not waving its body, but thehypothesis of Bone and Lighthill is more closely linked to thetransverse body motion than this second hypothesis. Thestreamwise acceleration hypothesis is linked to the mean flowfield arising from a propulsive system that generates thrust overa significant portion of the body. This would occur whether ornot the body used undulatory propulsion as long as the thrust-producing elements were close to the body surface. A non-

undulatory example in animal swimming might be the squid,for exampleLoligo pealei. This organism propels itself usinga high-velocity jet that exits beneath its arms. The acceleratedflow over the surface of the arms undoubtedly leads toenhanced friction drag.

It might be argued from the dogfish data atx=0.69L that onlythe BoneLighthill hypothesis is acting. At this position,boundary layer thickness over the swimming dogfish isapproximately one-quarter of that over the rigid dogfish(Fig. 8F), while the local friction on the swimming dogfish isfour times that on the rigid dogfish (Fig. 8B). The linearapproximation oUe/ suggests that boundary layer thinningalone is enough to explain the enhanced friction drag. This

0

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2104 4104 6104 810410-3

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Fig. 15. Time series of tangential velocity profiles,u-profiles, showing incipient separation atx=0.77L,

near the peduncle of a scup swimming at 10 cm s1in the flume. Approximately one locomotory cycleis shown, as revealed by the plot of transverse wallvelocity, vw. Incipient separation occurs mostclearly in the two profiles measured between 0.7and 0.8 s. The data at the start of the time series,although they are of poor quality, are attached andstable. The dashed vertical lines represent the u=0axis for each profile and are positioned at the timesof the realizations. Velocities within the profilescan be determined on the basis of the velocity scalebar shown and the respective u=0 axis positive tothe right, negative to the left (e.g. Fig. 5A).L, bodylength; y, distance normal to the body surface;

Cfx, local friction coefficient; u, fluid velictytangential to the body surface.

Fig. 16. Total coefficients of friction, Cf, versus Reynolds number,Re, calculated for scup () and dogfish () (Table 1), including therigid body case of the dogfish (). Turbulent (T) and laminar (L)flat-plate total friction coefficients are included.


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arguments fails, however, because the boundary layer atx=0.69L on the rigid dogfish was clearly turbulent, while theboundary layer on the swimming dogfish at this positionappeared to be laminar for the majority of the time. The shapesof laminar and turbulent profiles are radically different, and theapproximation oUe/ breaks down. The BoneLighthill

hypothesis does not include the effects of such differences inboundary layer condition. Their estimate that boundary layerthinning can lead to swimming friction drag that is 35 timesgreater than rigid-body friction drag is made assuming that theboundary layer flow condition is the same in both theswimming and rigid-body cases. If the boundary layer on therigid dogfish were laminar, rather than turbulent, it would havebeen up to 40 % thinner and would have had a lower localfriction. Therefore, the calculated drag enhancement atx=0.69L would be greater than fourfold, while the boundarylayer thinning would be less than fourfold. Then, by the linearapproximation of o, the degree of boundary layer thinningbetween the rigid and swimming cases would not be enough

to account for the increase in local friction. Furthermore, thefact that Ue atx=0.69L for the swimming dogfish is 1.6 timesthat for the rigid dogfish (Fig. 8D) makes it very difficult toargue that mean streamwise acceleration has no impact on thelocal friction at this position.

In scup, no obvious mean streamwise acceleration of thenear-field flow was observed (Fig. 8C), while frictionincreased by a factor of almost 2 between x=0.77L andx=0.91L. Fig. 8E reveals that boundary layer thicknessdecreases by almost 50% between these two positions, and theBoneLighthill hypothesis can account for the streamwiseincrease in local friction. The lack of mean streamwiseacceleration in sc

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Fluid Boundary Condition Abstract .Full