AbstractFishes are found in a great variety of body formswith tail shapes that vary from forked tuna-like tails tothe square-shaped tails found in some deep-bodied species. Hydrodynamic theory suggests that afish’s body and tail shape affects undulatory swimming performance. For example, a narrow caudalpeduncle is believed to reduce drag, and a tuna-like tail to increase thrust. Despite the prevalence ofthese assertions, there is no experimental verification of the hydrodynamicmechanisms thatmayconfer advantages on specific forms.Here, we use amechanically-actuated flapping foilmodel tostudy how two aspects of shape, caudal peduncle depth and presence or absence of a forked caudalfin,may affect different aspects of swimming performance. Four different foil shapes were eachmade ofplastics of three different flexural stiffnesses, permitting us to study how shapemight interact withstiffness to produce swimming performance. For each foil, wemeasured the self-propelling swimmingspeed. In addition, wemeasured the forces, torques, cost of transport and power coefficient of eachfoil swimming at its self-propelling speed. Therewas no single ‘optimal’ foil exhibiting the highestperformance in allmetrics, and for almost allmeasures of swimming performance, foil shape andflexural stiffness interacted in complicatedways. Particle image velocimetry of several foils suggestedthat stiffnessmight affect the relative phasing of the body trailing edge and the caudalfin leading edge,changing the flow incident to the tail, and affecting hydrodynamics of the entire foil. The results of thisstudy of a simplifiedmodel offish body and tailmorphology suggest that considerable caution shouldbe usedwhen inferring a swimming performance advantage frombody and tail shape alone.
1. Introduction
Fishes are remarkably versatile swimmers, exhibitinghigh performance in many different aspects of aquaticlocomotion. Some species can migrate long distances,crossing oceans with limited fuel reserves, while othersuse rapid acceleration to catch prey. Still other speciesare able to maneuver through spatially complexhabitats such as mangroves or coral reefs. Given thisremarkable diversity of swimming behavior, it ishardly surprising that fish morphology is also highlyvaried. Body and tail shape are thought to be amongthe chief determinants of swimming performance,and particular shapes are thought to be advantageousfor different swimming behaviors. Yet, surprisingly,controlled experimental investigations of the effects ofbody and tail shape on swimming performance arescarce.
Convergence of many distantly-related fishes on asmall set of distinct body shapes raises the question ofwhether those shapes are advantageous for specificaspects of swimming performance—either by havingthe lowest cost of transport (CoT), the highest self-propelling speed, or the best maneuverability. Forinstance, several species of pelagic, highly-active cruis-ing fishes have converged on a body plan with a nar-row caudal peduncle (the region just in front of the tailwhere the body narrows) and a forked or semi-lunatetail. This suggests that such a body shape providesenhanced swimming economy, either by increasingthrust or reducing drag (Webb 1984, 1988, Wolfganget al 1999, Blake et al 2009).
The proximate physical mechanisms by whichbody shape might confer a hydrodynamic advantage,however, remain uncertain. There are many covaryingfeatures of biological propulsors, such as their
differing evolutionary history and physiology, thatmake it difficult to determine the specific effect ofshape (as an isolated single factor) on performanceusing studies of living biological systems. Simplemechanical models, while not a replacement for directstudy of biological phenomena, facilitate the reductionof such sources of unexplained variance, therebyallowing the researcher to drawmore direct inferencesabout the effects of the variable in question on specificparameters of performance.
Several hydrodynamic theories suggest mechan-isms by which narrow caudal peduncles and swept-back tails could enhance swimming performance.Most of these are motivated by slender-body theory(Lighthill 1975), which implies that undulating bodiesexhibit an inherent tradeoff between thrust and drag:bodies and tails with large surface area have a greaterability to generate thrust, but in doing so, the large sur-face area incurs an energetic cost due to increaseddrag. Fusiform bodies reduce this drag cost, but stou-ter fishes typically have more muscle and surface areaavailable for thrust production (Webb 1984). Thesetheories remain limited by the extent of our knowl-edge of the hydrodynamics of bodies of varying stiff-ness, activation, and kinematics. Basic knowledge ofthese body-fluid interactions continues to grow, andcomputational fluid dynamic studies are beginning tofill this gap in knowledge (see, for example, BorazjaniandDaghooghi 2013).While promising, these simula-tions must be tested and replicated in the real world(e.g. Borazjani et al 2012).
Tail structure and kinematics may also mitigate afusiform body’s theoretical low thrust production.Fish with fusiform bodies have long, narrow ped-uncles to separate the tail from the body. This nar-rowing means that, if all else were equal, a fusiformbody could not shed as much energy into the wake asa stout body. However, the separation between thetail and body allows the tail to oscillate withoutresulting in energetically costly inertial recoil thatwould arise from high amplitude side-to-side oscil-lation of the body (Lighthill 1975, Lindsey 1978,Magnuson 1978,Webb 1992,Wolfgang et al 1999). Asemi-lunate or forked tail may allow a fusiform bodyto produce high thrust by via the generation of lead-ing edge suction (Chopra and Kambe 1977, Magnu-son 1978, Karpouzian et al 1990). This mechanismhas not yet been observed in live fishes, but is pre-dicted by computational models of caudal fin kine-matics (Borazjani and Daghooghi 2013) where aleading edge vortex (LEV) on the fish tail has beenproposed to increase thrust.
Differences in fish body and tail shape also appearto be coupled with differences in body stiffness(Webb 1984). Thunniform and carangiform swim-mers with their deep bodies, narrow caudal pedunclesand semilunate tails appear to have stiffer bodies andtails than fishes with more generalized body shapessuch as trout or bluegill sunfish (Webb 1984). The co-
occurrence of particular shapes with particular stiff-nesses will complicate attempts to determine experi-mentally in live fishes how shape and stiffness mayinteract during locomotion.
Recently, controlled studies using simple mechan-ical and robotic ‘flapping foil’ models have providedfor the removal of these confounding factors, allowingthe study of how fundamental physical traits affectswimming performance. The non-linear effects oftraits such as stiffness and length on swimming of flex-ible foils or strip-like panels have been measured, ashave resonant phenomena resulting from the foil-fluid interaction, and the effect of near-wall swimmingand center of mass oscillations (Lauderet al 2011, 2012, Alben et al 2012, Dewey et al 2013,Wen and Lauder 2013, Shelton et al 2014, Quinnet al 2014a, 2014b).
These flapping-foil models may appear somewhatdistant from the biological systems they attempt toemulate, but the ease of their control and manipula-tion makes it possible to address comparative andcomplex biological questions with targeted experi-ments (Lauder et al 2012, Shelton et al 2014). We usethis framework here to focus on the specific questionof how fish-like peduncle and tail shape may affectswimming performance in a simplified system. Whilethe results of studies using this model system may notbe directly applicable to biological fish swimming,they can shed light on causal hydrodynamic phenom-ena that would make particular shapes effective, sug-gesting further avenues for investigation in live fishsystems. Using flexible flapping foils designed to spana range of observed fish peduncle and tail shapes andstiffnesses, and using foil materials that match therange of known fish body stiffnesses, we measuredhow differences in body and caudal fin shape affectswimming speed, hydrodynamics, and CoT. Using amechanical controller, we were able to precisely con-trol the leading edge motion of these flexible foilshapes, and to quantify the self-propelled swimmingspeeds of each shape and stiffness.
Following the hypotheses outlined by Webb andothers (Lighthill 1975, Lindsey 1978, Magnuson 1978,Webb 1984, Borazjani and Sotiropoulos 2010), wehypothesized that foils with narrow caudal peduncleswould be the most economical swimmers, while foilswith deep peduncles might produce more thrust atgreater energetic cost. In addition, we suspected thatthe presence of a forked tail in conjunction with a nar-row peduncle might further increase thrust, withoutadding an offsetting economic cost. Stiffness wasexpected to interact with these effects by modulatingthe timing of interactions between the body trailingedge and the tail leading edge, and we expected that anintermediate stiffness would provide the most effec-tive phasing of the body and tail and hence enhancethrust.
2.1. Foil design and experimental setupTomodel the effects of caudal peduncle depth and tailshape on swimming performance, we created four foilshapes each with either a narrow or a deep caudalpeduncle, and a forked or unforked tail shape(figure 1). Foils were laser cut from three thicknesses/stiffnesses of plastic shim stock, and for ease ofidentification we refer to these by the color of the shimstock used as in our previous paper (Sheltonet al 2014). Foil identification conventions and theflexural stiffness ranges of each foil are given in table 1.The flexural stiffness of the foils used here encom-passes a wide range of the stiffnesses observed in realfishes (Lauder et al 2012, Shelton et al 2014). Long et al(2002) measured the stiffness of hagfish (Myxineglutinosa) bodies at a value of 3 × 10−4 Nm2.McHenryet al (1995) derived flexural stiffness values for sunfishbodies of ∼1 × 10−3 Nm2 near the head to1 × 10−6 Nm2 near the tail. Hereafter, specific foils arenamed by combining the first letter of the color of thematerial with the number of the foil shape as in table 1,e.g. C1 is the relatively inflexible coral-colored foilwith a narrow peduncle and forked tail (figure 1,table 1).
Foil body aspect ratio was calculated as follows:
=AR l A/ , (1)2
where l is foil body length (18.5 cm), and A is thefoil area.
We used a Riemann sum to approximate the sec-ond moment of area for each foil shape, as a shapemetric, to describe the distribution of area along thefoil. Briefly, each foil shapewas divided into 37–0.5 cmthick trapezoidal segments from the anterior edge tothe end of the foil (18.5 cm from the leading edge). Foreach foil segment, j, the average height was calculated.Each segment was then assumed to be a rectangle hav-ing width of 0.5 cm, and height equal to the averagesegment height. Using this approach, the shape-descriptor second moment of area of the foil with
respect to the leading edge axis (S) was approximatedas follows,
∫ ∑= ≈=
S r A A rd , (2)j
j j2
1
372
where rj is the distance from the centroid of foilsegment j to the leading edge.
Foil flexural stiffness was calculated for the deepestand the narrowest point of the ‘body’ portion of thefoil, the leading edge and the peduncle, respectively.Young’s modulus (E) values for each of the three foilmaterials, white, yellow, and coral, were available fromcollaborators’ earlier work (Quinn, personal commu-nication; Quinn et al 2014a, 2014b). The secondmoment of area of the foil about the axis of foil bend-ing (I) was calculated at each of the two locations asfollows:
=I b h/12, (3)3
for a rectangular cross-section with neutral axisvertical through the centroid, where b is the foilthickness and h is the height of the foil at the locationof interest. Note that this is an entirely separatecalculation from that of S, which was used to describethe foil shape over the long axis. Flexural stiffness (EI)was calculated the product ofE and I.
Foils were moved using a computer-controlledmechanical actuator designed to flap flexible foils inoncoming flow. This device is the same one used inour earlier research on aquatic propulsion (see Lauderet al 2011, 2012, Quinn et al 2014a, 2014b, Wen
Figure 1. Foil shapes and descriptive shapemetrics. Units areas follows: area (cm2), aspect ratio (AR, unitless), shape-descriptor secondmoment of area (S, cm4). Identifying shapenumber is in large print on the foil ‘tail’. Also see table 1 fordetails of foil properties, and for color codes that identify thematerial stiffnesses studied.
Table 1. Foil name abbreviations, shapes,flexural stiffnesses, andcolor code used in this paper. Seefigure 1 for images of foil shapes.W indicates thewhite foil color, Y yellow, andC coral color. Colorversion is available online.
et al 2013, 2014). Each foil was clamped by a roundshaft fitted with an ATI-Nano17 six-axis force–torquetransducer (ATI Industrial Automation, Apex, NC,USA) at the leading edge, and attached to a carriageplaced on a recirculating swimming flume. A customLabVIEW program (National Instruments Corp.,Austin, TX, USA) controlled a heavemotor on the car-riage, moving the shaft with ±1 cm sinusoidal heave at2 Hz and 0° pitch. These parameters approximate thekinematics of the posterior body region of swimmingfishes (Lauder andMadden 2006, Shelton et al 2014).
A second custom LabVIEW program monitoredfore-aft forces as the foil was flapping. Flow speed waschanged manually until the observed fore-aft forceswere within 0.005 N of 0 N. The flow speed at whichthis occurred was recorded. This procedure was repe-ated seven times, the highest and lowest recordedspeeds were removed, and the self-propelling speedwas calculated as the mean average of the remainingfive speeds recorded for each foil.
2.2. Force analysisTen flapping trials for each foil were performed at thatfoil’s self-propelling speed. For each trial, heave-position, forces and torques were recorded continu-ously for ten seconds (e.g. figure 2). Fore-aft forceswere filtered in IgorPro (Wavemetrics, Inc., Portland,OR,USA) using a customnarrowband-pass to remove2 Hz noise that resulted from the imposed heavemotion. The force in the fore-aft direction (Fx) wasexpected to have a dominant 4 Hz signal (i.e. twice theheave frequency) based on our previous work. Allforce and torque traces were smoothed for ease ofanalysis.
A custom IgorPro program was written to calcu-late derived measures of performance from the origi-nal three-force axes (Fx, Fy, Fz), self-propulsion speed(Ueq), and foil heave position (Ypos). Foil powercurves were calculated by multiplying the values ofthe instantaneous heave velocity and the forceapplied in the direction of the heave axis (Fy), as fol-lows:
=P tY
tF( )
d
d. (4)y
pos
The net work done on the foil by the motor wascalculated as the integral of the power curve,
∫=W P t t( )d . (5)net
The work per heave cycle (hereafter, ‘work percycle’) was calculated by dividing the net work doneover a 10 s trial by the number of cycles in that timeperiod (20 cycles). The foil power coefficient was cal-culated following Read et al (2003), where ρ is the fluiddensity, c is the mean chord, and s is the mean span ofthe foil.
ρ=C P U cs2 ¯/ . (6)p eq3
CoT was calculated in two ways. First, CoT wascalculated as the net work done over the course of each10 s trial divided by the total distance traveled (Ueq *10 s). Then, mass-specific CoT was calculated bydividing the first measurement by the foil mass. Tor-que oscillation of the foil about the rod (Tz) was calcu-lated as the average (Tmax –Tmin) for 20 heave cycles.This torque can be interpreted as the tendency forbody and tail oscillation to cause a yawing moment atthe anterior end of the foil.
2.3. StatisticsThe Shapiro-Wilk W test for normality and Levene’stest for homogeneity of variance were conducted inJMP Pro 9 (SAS Institute Inc., Cary, NC, USA). Allmetrics were heteroscedastic with non-normal distri-butions, so all data were transformed using an aligned-rank transform in ARTool v. 1.5.1 (Wobbrocket al 2011). Comparisons among foil shapes andstiffnesses were then conducted using two-way ANO-VAs in JMP Pro 9, following the procedure detailed inWobbrock et al (2011) (table 1). Significant differenceswere determined following a false detection ratecorrection, to reduce the chance of type I error frommultiple testing, with a maximum allowable falsedetection rate of 5% (see Benjamini andHochberg 1995).
2.4. Particle image velocimetry (PIV)While flapping at the self-propelling speed, flowaround foils was filmed in ventral view, via a 45°mirror, with a high speed camera (Photron PCI-1024;each framewith 1megapixel resolution) at a frame rateof 1000 Hz. Near neutrally buoyant particles wereilluminated using a Coherent 10Watt laser, andanalyzed using DaVis v. 7.2.2 (LaVision GmbH,Goettingen, GER) PIV software. The start of eachflapping cycle was defined as when the leading edgewas at its right-most lateral excursion. Still frameswere taken when an visible trailing edge vortex wasformed off the trailing edge of the foil ‘body’, andwhenthat vortex had moved down the foil far enough tointeract with flow at the ‘tail’ leading edge. For opaqueflapping foils, the shadow of the foil blocked visualiza-tion of flow on the right side of the foil. For ease ofinterpretation, these unusable shadow areas weremasked using CorelDRAW X5 (Corel Corp.,Ottawa, CAN).
2.5.Midline kinematicsMidline envelopes were digitized from high-speedvideos by tracing the foil midline every 0.125 s fromthe start of one flapping cycle to the start of thesubsequent cycle, for a total of eight traces, using acustom MATLAB program (The MathWorks, Inc.,Natick,MA,USA).
3.1. Force and swimming speedSwimming foils exhibited a sinusoidal thrust profile,with two thrust peaks for every foil oscillation cycle(figure 2).With every thrust peak, the foil power curvedips slightly negative, showing that for a brief period,the undulating foil is actually doing work on the rod,instead of the rod andmotor working on the foil. Foilsare self-propelling, and hence Fx averages to zero overaflapping cycle (figure 2).
There were significant interactions between shapeand stiffness for every swimming performance metricmeasured except for the self-propelled swimmingspeed, Ueq (figure 3, table 2) for which the interactionterm was not significant. Self-propelled swimmingspeed varied significantly with both foil shape andstiffness, with the yellow intermediate-stiffness foil inthe Y4 (deep peduncle) shape exhibiting the fastestswimming speed overall. The stiffest (coral) foil mate-rial with the C3 and C4 shapes showed the slowestswimming speeds. For two of the three materials
Figure 2.Representative force, torque, and calculated data from foils C1 (left) andW4 (right) when each is swimming at their self-propelled speed. Note that the swimming foils are self-propelling, and so the Fx curves average to zero over a cycle.
tested, white and coral, Ueq did not change sub-stantially with shape (figure 3). For the medium-stiff-ness, yellow foils, it appeared that Ueq is highest in thefoils with deep peduncles, and higher still in the deep-peduncle foil with an unforked tail (figure 3).
For all foils, the energy cost per heave cycle depen-ded on foil material more than foil shape. The stiffestfoils (coral) had higher energetic costs per heave cyclethan the yellow foils, and the flexible white foils hadthe lowest costs per heave cycle. The CoT, however,exhibited opposite trends depending on whether ornot mass was incorporated in to the calculation(figure 3; table 2). Coral foils exhibited the lowestmass-specific CoT, and the white foils had the highestmass-specific CoT, while the opposite was true whenmass was not accounted for (figure 3; table 2). Thisdiscrepancy is the result of less-flexible coral foilsbeing much heavier than the white flexible foils, whilethe yellow foils were of an intermediatemass.
The most flexible flapping foils tended to havesimilar power coefficients across all shapes. The powercoefficients of the medium-stiffness yellow foils ten-ded to be lower in shapes with greater surface area. Thestiffest, coral foils, however, tended to have higherpower coefficients with higher surface area.
Torque also variedwith foil stiffnessmore than foilshape (figure 3). The coral foils all had similar, hightorques, and the white foils had similar low torques.An interesting exception to this trend is foil Y4, themedium stiffness foil with a deep peduncle and anunforked tail. Y4 exhibited the highest torques of anyfoil, as well the highestUeq.
Figure 3. Swimming performancemetrics for self-propelling foils. Rawdata from each of thefive trials per foil are shown. Points aretranslucent to show any overlap. Statistical analyses of these data are given in table 2.
Table 2. Summary of two-wayANOVAon aligned ranks for fourvariables.
Variables
Source of
Variation df F-ratio Prob.
Ueq Shape 3 35.60 <0.0001*
Stiffness 2 12.78 0.0007*
Shape x
stiffness
6 0.016 0.8989
Mass-specific Shape 3 4.22 0.01a
COT Stiffness 2 31.07 <0.0001a
Shape x
stiffness
6 4.49 0.0011a
PowerCoefficient Shape 3 5.03 0.0041a
Stiffness 2 43.88 <0.0001a
Shape x
stiffness
6 11.52 <0.0001a
TzOscillation Shape 3 14.23 <0.0001a
Stiffness 2 81.25 <0.0001a
Shape x
stiffness
6 47.87 <0.0001a
a Shows significance following false detection rate correction (after
There was considerable variance observed in thework, power, and CoT of some foils, particularlyamong the yellow and coral foils (figure 3). The forcemeasurements used in the derivation of these valueswere very sensitive to small changes in the initial con-ditions, and the observed scatter may result from evenminor variation in the forces measured from trial totrial.We have reported the raw data for thesemeasuresin order to accurately convey this scatter.
3.2.Midline kinematicsFoil kinematics vary considerably with both foil shapeand foil material (figure 4). Material stiffness (table 1)governed the number of wavelengths on each foil, withthe white foils exhibiting approximately 1.5 wave-lengths, while the other materials only supportedabout 0.5 wavelengths. Shape, too, had an effect onmidline kinematics, by modulating the lateral excur-sion of any particular point along the foil. For instance,foils with narrow peduncles tended to exhibit greaterlateral excursion of the peduncle notch than foils withdeep peduncles (figure 4). Interestingly, while shapedid affect lateral excursion, it did not appear to changethe position of nodes and antinodes along the body.
3.3. Flow visualizationPIV revealed that flow patterns at the peduncle notchof each foil varied considerably, while midline flowpatterns were far more consistent across stiffnessesand shapes (figures 5 and 6). Of particular interestwere flow patterns around the caudal peduncle(through the peduncle notch), where flow off thetrailing edge of the upstream ‘body’ segment of the foilappeared to greatlymodify the flow incident on the tailleading edge. For foils with narrow peduncles, in theplane of the peduncle notch, there was obvious flow
through the gap between the body and the tail. Thisflow was not observed in the plane of the foil midline.Depending on the foil’s shape and stiffness, flowthrough the peduncle notch could either increase thetail’s effective angle of attack, or result in almost noflow immediately anterior to the tail (figures 5 and 6).
Phase differences between the body trailing edgeand the tail leading edge were observed, and likelywere caused by the interaction of tail shape and stiff-ness. These phase differences produced interestingchanges in the flow incident on the tail as well—dictat-ing whether or not flow off the body trailing edgewould interact with the tail, or merely pass the tail by(figures 5 and 6). We noted the presence of a LEV onthe tail leading edge of foil C1, a foil of the stiffestmaterial with both a narrow peduncle and a tail fork,which appears to be a product of such a fluid-structureinteraction (hollow arrow, figure 6). A bound LEVwasnot observed on any other foil.
4.Discussion
4.1. Propulsion of differently-shaped flexible foilsFish tail fin shape and its impact on locomotorfunction has been the subject of research formore thana century (e.g., Ryder 1886, Breder 1926, Affleck 1950,Webb 1978). From observations of the evolutionarytransition from heterocercal to homocercal tails andtheir potential for lift generation by the tail(Ryder 1886, Grove and Newell 1936) to earlymechanical studies of tail function (Grove and New-ell 1936, Affleck 1950), it was clear that tail shape hasthe potential to impact swimming performance. Usingmechanical models to isolate the effects of particularshapes was an important first step, and the paper byAffleck (1950) is a classic in this regard. Although these
Figure 4.Midline envelopes showing the lateral (side-to-side) amplitude of foilmotion along the length for self-propelling foils ofdifferent shapes and stiffnesses. Scale bar represents 5 cm for all foil shapes.
early studies used stiff plates, and were largely qualita-tive, they proved that shape can exert a large influenceon the forces experienced by a fish during undulatoryswimming. A later study performed a more directmanipulation of fish tails—determining how partialamputation of the tail affected swimming perfor-mance (Webb 1973). These manipulations, however,appeared to have little effect on the swimming
performance of trout with altered tails. More recentstudies (e.g. Plaut 2000, Blake et al 2009, Law andBlake1996, Webb and Fairchild 2001) have used naturalvariation in fish body and tail shapes and a compara-tive approach to investigate the effect of changes in fishmorphology on locomotor performance.
Lauder et al (2012) summarized previous resultsobtained using the flapping foil apparatus used in this
Figure 5. Flow visualization around self-propelling flexible, white foils,W1 (top panel) andW4 (bottompanel). The top row in eachpanel shows flow at the foilmidline, the bottom row shows flow at themiddle of the caudal ‘notch’. Time is given as percentcompletion of oneflap cycle, where 0% is at the lowest heave point. Timeswere chosen to show the appearance of a vortex at thetrailing edge of the body (column 1), and the point at which that vortex reaches the fore-aft position of the tail leading edge (column2). Vortices from the body trailing edge are indicatedwithwhite arrows. Scale bars represent 5 cm. Vorticity (blue, clockwise rotation;red, counterclockwise rotation) is also shown on each image. Color version is available online.
study, and showed differences in swimming perfor-mance between flexible foil models with differenttrailing-edge shapes. Their study, which employedseveral differently shaped foils, including a simplemodel of a homocercal (symmetrical tail) and anotherwith a shark-like (asymmetrical) tail, demonstratedthat 3D flow over the entire tail was more complicatedthan the section of flow observed in the plane of the
foil midline. They also demonstrated that even simplechanges in trailing edge shape could effect large chan-ges in the forces produced by the foil (Lauderet al 2012). Subsequent work using this same flappingfoil mechanism attempted to determine how length orstiffness alone affect swimming performance (Sheltonet al 2014). Data from Alben et al (2012) suggest thatspecific combinations of foil length and stiffness can
Figure 6. Flow visualization around the stiff, coral foils, C1 (top panel) andC4 (bottompanel). The top row in each panel showsflowat the foilmidline, the bottom row shows flow at themiddle of the caudal ‘notch’. Time is given as percent completion of one flapcycle, where 0% is at the lowest heave point. Times were chosen to show the appearance of a vortex at the trailing edge of the body(column 1), and the point at which that vortex reaches the fore-aft position of the tail leading edge (column 2). Vortices shed from thebody or upstream trailing edge are indicatedwithwhite arrows, while bound vorticity on the tail leading edge is indicatedwith ahollow arrow. Scale bars represent 5 cm. Color version is available online.
exhibit multiple performance optima. However, thoseexperiments were conducted on rectangular foils, anddid not include shape manipulations. The work ofDewey et al (2013) and Quinn et al (2014b) addressedthe scaling of propulsion using flexible panels or foils,and focused on resonant effects and derivation of scal-ing laws. Resonant effects could certainly be relevantto the propulsion of the different tail shapes studiedhere, especially when comparing the similar shapesthat differ in stiffness. However, given that we onlystudied one swimming speed (the self-propelledspeed), it is difficult to ascribe any particular differencesin swimming shapes among foils (figure 4) to a reso-nant effect. The question of how changing mass dis-tribution along the foil length (figure 1) impactsresonant effects during locmotion is an intriguing onefor future study.
One additional area to consider in flexible foil pro-pulsion is the oscillation that would naturally occur inthe center of mass if the foils were truly unconstrainedto move freely in a fluid environment. Under the cur-rent experimental constraints, foils cannot oscillate inthe upstream–downstream (x) direction as the heavemotor constrains the leading edge to lateral motiononly. In this study as in the other papers cited above,the swimming flexible foils do not exhibit uncon-strained center of mass motion as observed in freely-swimming fishes (Xiong and Lauder 2014), and thuswill show momentary small imbalances in the forcesduring propulsion. Wen and Lauder (2013) addressedthis constraint by allowing controlled x-directionmotion and varying the extent of this movement todetermine the effect on thrust forces of center of massmotion. They found that reductions in the magnitudeof thrust force oscillation could be achieved by allow-ing the swimming foils to oscillate axially during pro-pulsion. This study is a continuation of this overallbody of work on flexible panel propulsion, and we usecontrolled manipulations of a simple experimentalmodel to remove confounding factors and ask a spe-cific question: how might two aspects of shape,namely, the narrow caudal peduncle and the forkedtail so often associatedwith economical cruising, affectswimming performance. The long-held hypothesis,largely based on hydrodynamic principle instead ofexperimental data, was that both a narrow peduncleand a forked tail would reduce swimming CoT. In par-ticular, we expected the combination of narrow ped-uncle and forked tail to maintain high speed at a lowerenergetic cost (Brill 1996).
The results, however, do not agree with this simpleassertion. The effects of both tail shape and tail stiff-ness interacted, such that it was difficult to predict anygeneral performance difference between a deep ped-uncle and a narrow one, and between forked andunforked tails (figure 3, table 2). Even stiffness aloneseemed to generate unexpected changes in foil swim-ming performance. Within any of the three given stiff-nesses/materials, the effects of shape on performance
were irregular. Z-torques increased as material flex-ural stiffness increased—the coral foils had the highesttorques, and the white foils had the lowest (figure 3,table 2).
In sum, our results suggest that both shape andstiffness are important in determining the propulsiveperformance of undulating foils and that complexinteractions between these two parameters occur.
4.2.Hydrodynamics of differently-shaped foilsPIV of the different foils complemented the perfor-mance measures in that flow, too, behaved in acomplex manner depending on shape and stiffness. Inparticular, flow off the midline axis was highlyvariable, and provided new information about howthe foil interacted with the surrounding fluid that isnot captured in the plane of the midline. The off-midline flow was complex, especially in the regionbetween the body and the tail parts of the foil due tothe sharp edges encountered by the flow. This suggeststhat three-dimensional flow surrounding the tail isdependent on shape—especially that of the peduncle—and that studying midline flow alone fails touncover much of the variation in the hydrodynamicsof different shapes. This reinforces the views of Tytellet al (2008), and more recent studies taking advantageof volumetric PIV (Flammang et al 2011) and CFD(Borazjani and Daghooghi 2013): fishes and otherflapping bodies with irregular shapes do not operate inflatland. Their moving, three-dimensional shapesinfluence hydrodynamic flowpatterns significantly.
One way that shape and stiffness may be workingin tandem to modulate performance is by alteringkinematics, specifically, by modulating the phase rela-tionship of the body and the tail (Lighthill 1970;figures 5 and 6). The presence of a body trailing edgeand a tail leading edge, which varies with shape(figure 1), allows flow from the body to interact withor even dictate the flow incident on the tail. Body andtail flow interaction appears to be modulated by foilkinematics. For instance, with foil C1, the kinematicswere such that the vorticity off the trailing edge of thebody interacts with the leading edge of the tail, chan-ging the tail’s effective angle of attack (figure 6). Inother foils, such as foil W4, the leading edge of the tailwas nowhere near the vorticity shed by the body whenthat vorticity passed the tail (figure 5). Shape and stiff-ness thus interact to produce varying kinematics. Thekinematics may be what ultimately drives swimmingperformance in these foils by altering the phasing offlows in the gap between the body and tail regions ofthe foils (also see Drucker and Lauder (2001), Akhtaret al (2007) and Standen and Lauder (2007) for discus-sion of flow interactions among fins in fishes). Indeed,the optimal kinematics (see Eloy 2013) for a givenbody shape vary considerably—even within the lim-itations of elongated-body theory. Taken as a whole,these results suggest that body-tail phase relationships
may be a useful potential determinant of swimmingperformance.
The presence of a LEV on foil C1 is intriguing,given that leading edge vorticity is suspected to play arole in the presumed benefits of forked and semi-lunate tails (Chopra and Kambe 1977, Karpouzianet al 1990, Borazjani and Daghooghi 2013). This vor-tex appears to be the product of the effective angle ofattack created by the interaction of vorticity off thebody and on the tail, forming a weak, but discernablerearward jet (figure 6). A LEV on the foil tail surfacemay enhance propulsion via leading edge suction inthe same manner suggested by Borazjani andDaghooghi (2013) in their computational fluiddynamic study of fish tail function. We note that foilC1 had a higher Ueq than the other foil shapes of thesame material (figure 3). The placement of thebound LEV on foil C1 suggests that it would producesuction to pull the tail forwards and augment thrust.If that is the case, it suggests a narrow, stiff pedunclemight be required for LEV thrust enhancement: thenarrow peduncle to ensure a distinct body trailingedge and tail leading edge, and the stiffness to createthe proper phase relationship between body and tailduring undulatory propulsion. Whether thismechanism plays a role in fish swimming has yet tobe determined for live fish, but remains a tantalizingpossibility.
4.3. Implications forfish tail shape functionThe foils used in this study are a simplified models ofactual fish tails, and yet, even their performanceappears to be dictated by a complex interaction ofshape and stiffness. It was difficult to determine anypredictive relationship between shape and perfor-mance, and there was no single shape or stiffnesswith the best performance for all performancemetrics. The flow pattern produced by a given tailshape was governed by the interaction of the bodyand tail flows, suggesting that tail shapes cannot bestudied in isolation of the body. Because the flowincident on the tail is in large part determined by themovement of the body in front of it, an isolated tailfoil, without a body component, may not accuratelymodel tail hydrodynamics in the freely-swimmingfishes.
The foils used in this study are not intended toexactly replicate fish motion, but rather to investi-gate the complexity of shape as factor affectingundulatory locomotor dynamics, and suggest futureavenues of research in biological systems of undula-tory propulsion. The changes produced by varyingshape of the foils—including the interaction of flowbetween anterior and posterior regions of the foils—suggest the possibility of similar interactions havinga role in fish locomotion. A few notable studies haveobserved interactions among median fins in live fishsimilar to those of the foils in the present study
(Drucker and Lauder 2001, Standen and Lau-der 2007, Tytell et al 2008). Future research in biolo-gical systems may reveal the importance of suchshape-based hydrodynamic interactions in fishswimming.
The complexity of this study’s findings alsodemonstrate that even one shape can behave differ-ently depending on the kinematics with which itsmoved, how the body in front of the tail is shapedand moves, and the body and tail’s material proper-ties. Many of the foils in this study contradicted thesimplistic hypotheses about function. For example,the forked tail with the narrow peduncle region (themost ‘tuna-like’ tail) did not display the highest Ueq
or the lowest energetic cost at all stiffnesses(figure 3). All of this is not to say that existinghypotheses about how fish body and tail shape affectswimming performance are wrong. Rather, it sug-gests that extrapolating any performance advantagefrom morphology alone is a risky venture. Theassumptions behind the claims of adaptivemorphol-ogy may be correct, but until the implied mechan-istic links between morphology and performance areproven, they remain assumptions. Morphology andperformance often have complicated interrelation-ships. Until there are data demonstrating that a mor-phological feature directly affects a specific metric ofswimming performance, equating morphologicaldifferences with performance differences ispremature.
Acknowledgments
This work was supported by an NSF GraduateResearch Fellowship under grant DGE-1144152 to K.F., and National Science Foundation grant EFRI-0938043, Office of Naval Research grant N00014-09-1-0352 (monitored by Dr Thomas McKenna) to G.V.L., and ONR MURI grant N00014-14-1-0533 (mon-itored by Dr Bob Brizzolara). Special thanks to DanQuinn and members of the Lauder Lab for manyhelpful discussions on foil propulsion, and to KelseyLucas in particular for comments on an earlier draft ofthismanuscript.
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