Understanding how the shape and motion of an aquaticanimal affects the performance of swimming requiresknowledge of the fluid forces that generate thrust and drag.Despite recent advances towards understanding thebiomechanics of locomotion (see Dickinson et al., 2000 for areview), these forces are poorly understood in swimminganimals that are a few millimeters in length. The large diversityof larval fish and marine invertebrates at this scale generatehydrodynamic force that is dependent on both the viscosity andthe inertia of the surrounding water. To understand the relativecontribution of inertial and viscous forces to the generation ofthrust and drag, theoretical models have been developed for thehydrodynamics of swimming at this scale (e.g. Jordan, 1992;Vlyman, 1974; Weihs, 1980). However, little experimentalwork has attempted to test or refine these theories (exceptionsinclude Fuiman and Batty, 1997; Jordan, 1992). The goalof the present study was to test hydrodynamic theory bycomparing the predictions of theoretical models with
measurements of the speed of freely swimming animals andthe forces generated by tethered animals.
Swimmers that are millimeters in length generally operatein a hydrodynamic regime characterized by Reynolds numbers(Re) between 100 and 103, which is a range referred to as theintermediate Re in the biological literature (e.g. Daniel et al.,1992). Re(Re=ρuL/µ, where u is mean swimming speed, L isbody length, ρ is density of water, andµ is dynamic viscosityof water) approximates the ratio of inertial to viscous forcesand suggests how much different fluid forces contributeto propulsion. At intermediate Re, a swimming body mayexperience three types of fluid force: skin friction, form forceand the acceleration reaction. Skin friction and form force arequasi-steady and therefore vary with the speed of flow. Inprevious studies on intermediate Re swimming, these forceshave collectively been referred to as the ‘resistive force’ (e.g.Jordan, 1992). However, we will consider these forcesseparately because the present study is concerned with how
Understanding how the shape and motion of an aquaticanimal affects the performance of swimming requiresknowledge of the fluid forces that generate thrust anddrag. These forces are poorly understood for the largediversity of animals that swim at Reynolds numbers (Re)between 100 and 102. We experimentally tested quasi-steady and unsteady blade-element models of thehydrodynamics of undulatory swimming in the larvae ofthe ascidian Botrylloides sp. by comparing the forcespredicted by these models with measured forces generatedby tethered larvae and by comparing the swimmingspeeds predicted with measurements of the speed of freelyswimming larvae. Although both models predicted meanforces that were statistically indistinguishable frommeasurements, the quasi-steady model predicted thetiming of force production and mean swimming speedmore accurately than the unsteady model. This suggeststhat unsteady force (i.e. the acceleration reaction) does
not play a role in the dynamics of steady undulatoryswimming at Re≈102. We explored the relativecontribution of viscous and inertial force to the generationof thrust and drag at 100<Re<102 by running a series ofmathematical simulations with the quasi-steady model.These simulations predicted that thrust and drag aredominated by viscous force (i.e. skin friction) at Re≈100
and that inertial force (i.e. form force) generates a greaterproportion of thrust and drag at higher Re than at lowerRe. However, thrust was predicted to be generatedprimarily by inertial force, while drag was predicted to begenerated more by viscous than inertial force at Re≈102.Unlike swimming at high (>102) and low (<100) Re, thefluid forces that generate thrust cannot be assumed to bethe same as those that generate drag at intermediate Re.
The hydrodynamics of locomotion at intermediate Reynolds numbers:undulatory swimming in ascidian larvae (Botrylloides sp.)
Matthew J. McHenry1,*, Emanuel Azizi2 and James A. Strother1
1Department of Integrative Biology, University of California, Berkeley, CA 94720, USA and 2Organismic andEvolutionary Biology Program, 221 Morrill Science Center, University of Massachusetts, Amherst, MA 01003, USA
*Author for correspondence at present address: The Museum of Comparative Zoology, Harvard University, 26 Oxford St, Cambridge,MA 02138, USA (e-mail: [email protected])
Accepted 10 October 2002
328
they individually contribute to the generation of thrust anddrag.
Skin friction is generated by the resistance of fluid toshearing. This is a viscous force, which means that it increasesin proportion to the speed of flow. Skin friction (also called the‘resistive force’ by Gray and Hancock, 1955) dominates theundulatory swimming of spermatozoa (Re!100; Gray andHancock, 1955) and nematodes (Gray and Lissmann, 1964)and has been hypothesized to contribute to thrust and drag inthe intermediate Re swimming of larval fish (Vlyman, 1974;Weihs, 1980) and chaetognaths (Jordan, 1992).
The form force is generated by differences in pressure on thesurface of the body and it varies with the square of flow speed(Granger, 1995). This inviscid force is equivalent to the resultantof steady-state lift and drag acting on a body at Re>103. Theform force is thought to contribute to the generation of thrustand drag forces at the intermediate Reswimming of larval fish(Vlyman, 1974; Weihs, 1980) and may dominate forcegeneration by the fins of adult fish (Dickinson, 1996).
The acceleration reaction [also referred to as the ‘reactiveforce’ (Lighthill, 1975), the ‘added mass’ (Nauen andShadwick, 1999) and the ‘added mass inertia’ (Sane andDickinson, 2001)] is generated by accelerating a mass of wateraround the body and is therefore an unsteady force (Daniel,1984). This force plays a negligible role in the hydrodynamicsof swimming by paired appendages at Re<101 (Williams,1994) but is considered to be important to undulatoryswimming at intermediate Re (Brackenbury, 2002; Jordan,1992; Vlyman, 1974) and dominant in some forms ofundulatory swimming at Re>103 (Lighthill, 1975; Wu, 1971).Although it is assumed that the acceleration reaction does notplay a role in undulatory swimming at Re<100 (Gray andHancock, 1955), it is not understood how the magnitude of theacceleration reaction varies across intermediate Re.
Weihs (1980) proposed a hydrodynamic model thatpredicted differences in the hydrodynamics of undulatoryswimming in larval fish at different intermediate Re. Heproposed a viscous regime at Re<101, where viscous skinfriction dominates propulsion, and an inertial regime atRe>2×102, where inertial form force and the accelerationreaction are dominant (also see Weihs, 1974). For the range ofRebetween these domains, thrust and drag were hypothesizedto be generated by a combination of skin friction, form forceand the acceleration reaction. Although frequently cited inresearch on ontogenetic changes in the form and function oflarval fish (e.g. Muller and Videler, 1996; Webb and Weihs,1986), it remains unclear whether Weihs’ (1980) theory, whichis founded on measurements of force on rigid physical models,accurately characterizes the forces that act on an undulatingbody (Fuiman and Batty, 1997).
The present study used a combination of empiricalmeasurements and mathematical modeling of the larvae of theascidian Botrylloidessp. to test whether the hydrodynamics ofswimming in these animals is better characterized by a quasi-steady or an unsteady model. By taking into account theacceleration reaction, skin friction and form force generated
during swimming, models were used to formulate predictionsin terms of the speed of freely swimming larvae andforce generation. By comparing these predictions withmeasurements of force and speed, we were able to determinewhether larvae generate thrust and drag by accelerationreaction (the unsteady model) or strictly by form force and skinfriction (the quasi-steady model). Ascidians are an ideal groupfor exploring these hydrodynamics because the larvae ofdifferent species span nearly two orders of magnitude in Re[e.g. ≈5×100 in Ciona intestinalis(Bone, 1992); Re≈102 inDistaplia occidentalis(McHenry, 2001)].
Materials and methodsColonies of Botrylloidessp. were collected in the months of
August and September from floating docks (Spud Point Marina,Bodega Bay, CA, USA) in water that was between 14°C and17°C. Colonies were transported in coolers and placed in arecirculating seawater tank at 16°C within 2 h of collection. Tostimulate release of larvae, colonies were exposed to brightincandescent light after being kept in darkness overnight(Cloney, 1987). Released larvae were used in either forcemeasurement experiments, free-swimming experiments or formorphometric analysis. In all cases, observation tanks wereequipped with a separate outer chamber into which chilledwater flowed from a water bath equipped with a thermostat(1166, VWR Scientific) that kept larvae at 16°C.
Force measurements
Larvae were individually attached to a calibrated glassmicropipette tether in order to measure the forces that theygenerated during swimming. Each larva was held at the tip ofthe tether using light suction (Fig. 1) from a modified mouthpipette. This micropipette was anchored at its base with a rubberstopper that provided a flexible pivot. No bending in themicropipette was visible under a dissecting microscope whenloaded at the tip of the tether. We therefore assumed that themicropipette was rigid and that deflections at the tip were dueentirely to flexion at the pivot. The small deflections by thetether were recorded during calibration and larval swimming bya high-speed video camera (Redlake Imaging PCI Mono/1000SMotionscope, 156 pixels×320 pixels, 1000 frames s–1) mountedon a compound microscope (Olympus, CHA), which wasplaced on its side at a right angle to the micropipette (Fig. 1).Video recordings of tether deflections made at the objective ofthe compound microscope were translated into radialdeflections at the pivot of the micropipette (ϕ) using thefollowing trigonometric relationship:
where δ is the linear deflection (away from its resting position)of the tether measured at the objective, and hobjective is thedistance from the tether pivot to the objective (Fig. 1A). Inorder to avoid changing the mechanical properties of the tether,room temperature was held at 22.2°C throughout experiments.
(1)
δhobjective
ϕ = arctan ,
M. J. McHenry, E. Azizi and J. A. Strother
329Undulatory swimming at intermediate Reynolds numbers
The tether was modeled as a pendulum, with input forcegenerated by the tail of a swimming larva (F) at the end and adamped spring at the pivot (Fig. 2). According to this model,the moments acting at the pivot were described by thefollowing equation of motion (based on the equation for adamped pendulum; Meriam and Kraige, 1997a):
where t is time, kdamp and kspring are the damping coefficient
(with units of Nms rad–1) and spring coefficient (with unitsof Nm rad–1), respectively, I tether is the moment of inertia ofthe tether, mtetherand mbody are the mass of the tether and thebody of the larva, respectively, g is the acceleration due togravity, hcm is the distance from the pivot to the center ofmass of the tether, and htip is the distance from the pivotto the tip of the pipette. I tether was calculated using thestandard equation for a hollow cylinder (Meriam and Kraige,1997a):
where rtether is the inner radius of the micropipette. Wecalculated the force generated by tethered larvae by solvingequation 2 for F, using the measurements of tether deflections.We found that adding second- and third-order terms toequation 2 had a negligible effect (<0.5% difference) on forcemeasurements. This suggests that any variation in stiffnessor damping with strain or strain rate did not influence ourmeasurements.
To calibrate the tether, we measured its stiffness anddamping constants in a dynamic mechanical test. This testconsisted of pulling and releasing the tether and then recordingits passive movement over time (Fig. 2A). The tether oscillatedlike an underdamped pendulum (Meriam and Kraige, 1997a)with a natural frequency (101 Hz) well outside the range of tail-beat frequencies expected for ascidian larvae (McHenry,2001). Using the equation of motion for the tether (equation 2,with F=0), its oscillations were predictable if the mass and thestiffness and damping coefficients were known. Conversely,we solved for the stiffness and damping coefficients fromrecordings of position and a measurement of the mass of thetether (see Appendix for details).
We examined how errors in our measurement of stiffnessand damping coefficients were predicted to affectcalculations of the force generated by larvae (Fig. 2C–H).By simulating the input force generated by a larva as a sinewave with an amplitude of 20µN, we numerically solvedequation 2 (using MATLAB, version 6.0, Mathworks) forthe position of the tether over time at 1000 Hz (the samplingrate of our recordings). From these simulated recordings oftether position, we then solved equation 2 for F, the forcegenerated by the larva. This circular series of calculationsdemonstrated that our sampling rate was sufficient to followrapid changes in input force (Fig. 2C). Furthermore, wefound that a minimum of 92% of the instantaneous momentsresisting the input force were generated by the stiffness of thetether (i.e. the weight and damping of the tether provided amaximal 8% of the resistance to input force). If the valuesof stiffness and damping coefficients used in forcemeasurements differed from those used to simulate tetherdeflections, then measured force did not accurately reflect thetiming or magnitude of simulated force (Fig. 2D). Thissituation is comparable with using inaccurate values ofstiffness and damping coefficients for measurements of forcein an experiment.
Ventral perspective fromhigh-speed video camera #1
Edge of micropipette viewed from high-speed video camera #2
hobjective
δ
A
B
C
Fig. 1. The experimental set-up for tethering experiments. (A) Werecorded the tail motion of a larva and the deflections of the tether towhich a larva was attached. The larva is illustrated with theorientation that allowed for the recording of lateral forces: thelongitudinal axis of the body is perpendicular to the direction ofdeflections. To measure thrust, the longitudinal axis was alignedparallel to the deflections of the tether. (B) The ventral perspective ofa larva was recorded with video camera #1 mounted to a dissectingmicroscope mounted beneath the glass tank. (C) Deflections of theglass tether (δ) were recorded by video camera #2 mounted to acompound microscope.
330
By varying the difference between the stiffness and dampingcoefficients used to simulate changes in tether position overtime (i.e. the actual coefficients) and those used for forcemeasurements (i.e. the measured coefficients), we exploredhow inaccuracy in measured coefficients was predicted to alterthe timing and magnitude of measured force (Fig. 2E–H). Wesimulated changes in force at 18 Hz, to mimic oscillations inforce at the tail-beat frequency (McHenry, 2001), and at180 Hz, to simulate rapid changes in force. Within the level ofprecision (i.e. ±2 S.D.) of our measurements of stiffness anddamping coefficients, measured force was not predicted toprecede or lag behind simulated force by more than 1 ms,which is just 1.8% of an 18 Hz tail-beat period (Fig. 2E,F).Error in the damping coefficient may have causedmeasurements to overestimate rapidly changing force by asmuch as 7.5% (Fig. 2G). Within the precision of measuredstiffness coefficients, measured forces may have differed fromactual values by as much as 2.0% (Fig. 2H). These findingssuggest that our measurements accurately reflect the timing of
force generated by larvae, but the magnitude of force may beinaccurate by as much as 7.5%.
Midline kinematics
The ventral surface of the body was recorded duringtethered swimming (Fig. 1A) with a high-speed videocamera (Redlake Imaging PCI Mono/1000S Motionscope,320 pixels×280 pixels, 500 frames s–1) mounted to a dissectingmicroscope (Wild, M5A) beneath the glass tank containing thetethered larva. The video signal from this camera was recordedby the same computer (Dell Precision 410, with Motionscope2.14 software, Redlake Imaging) as was used to recordmicropipette deflections, which allowed the recordings to besynchronized.
Coordinates describing the shape of the midline of the tailwere acquired from video recordings, and the motion of the tailof larvae of Botrylloides sp. was characterized using themethodology presented by McHenry (2001). A macro program(on an Apple PowerMac G3 with NIH Image, version 1.62)
M. J. McHenry, E. Azizi and J. A. Strother
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Fig. 2. The precision and accuracy of force measurements. (A) An example of the measurements of the passive movement of the tether afterbeing pulled and released. These data were used to measure the coefficients of stiffness and damping (see Materials and methods). (B) A free-body diagram illustrates the forces acting on the tether during an experiment. The input force generated by a swimming larva is resisted by acomponent of the weight of the tether and the stiffness (illustrated by the spring) and damping of the pivot (illustrated by the dashpot).(C,D) Measurements of input force (filled circles) at 1000 Hz from the deflections of a tether (not shown) calculated from the simulated changesin force (red lines). (C) The input force measured from deflection measurements using accurate values for the stiffness and dampingcoefficients. (D) The input force measured using a damping coefficient that is less than the actual value (by 2×10–6Nms rad–1). (E,F) The timelag between simulated and measured input force for varying degrees of error in the damping (E) and stiffness (F) coefficients. (G,H) The ratioof maximum measured to maximum simulated input force for varying degrees of error in the damping (G) and stiffness (H) coefficients.
331Undulatory swimming at intermediate Reynolds numbers
found 20 midline coordinates that were evenly distributedalong its length (see McHenry, 2001 for details). In order touse the measured kinematics in our hydrodynamic models atany body length, we normalized all kinematic parameters tothe body length of larvae (L, the distance from the anterior toposterior margins of the body) and the tail-beat period of theirswimming (P; note that asterisks are used to denote non-dimensionality). According to McHenry (2001), the followingequations describe the temporal variation in the change in theposition of the inflection point along the length of the tail (z*),the curvature of the tail between inflection points (κ*), and thetrunk angle (θ, the angle between the longitudinal axis of thetrunk and the third midline coordinate, located at 0.15 taillengths posterior to the intersection point of the trunk and tail):
z*( t*) = ε* t* , (4)
θ = χsin(2πt*) , (6)
where t* is non-dimensional time, ε* is the wave speed ofinflection point, α* is the amplitude of changes in curvature,γ* is the period of change in curvature, and χ is the amplitudeof change in trunk angle. Propagation initiates at the base ofthe tail after a phase lag of ζ* from the time when the trunkangle passes through a position of zero.
Morphology and mechanics of the body
We measured the shape of the body to provide parametervalues for our calculations of fluid forces and to estimate thebody mass, center of mass and its moment of inertia. Theperipheral shape of the body was measured (with NIH Imageversion 1.62 on an Apple PowerMac G3) using digital stillimages of larvae from dorsal and lateral views that werecaptured on computer (7100/80 PowerPC Macintosh withRasterops 24XLTV frame grabber) using a video camera(Sony, DXC-151A) mounted on a dissecting microscope(Nikon, SMZ-10A). These images had a spatial resolutionof 640 pixels×480 pixels, with each pixel representingapproximately a 6µm square with an 8-bit grayscale intensityvalue. Coordinates along the peripheral shape of the body wereisolated by thresholding the image (i.e. converting fromgrayscale to binary; Russ, 1999). We found coordinates at 50points evenly spaced along the length of the trunk and 50 pointsevenly spaced along the length of the tail (using MATLAB).From images of the lateral view, we used the same method tomeasure the dorso-ventral margins of the trunk, cellular tailand tail element. By the same method, we measured the widthof the trunk from the dorsal view.
By assuming that the trunk was elliptical in cross-sectionand that the cellular region of the tail was circular in cross-section, we calculated the body mass, center of mass andmoment of inertia using a program written in MATLAB fromreconstructions of the body’s volume. These calculationsdivided the volume of the body into small volumetric elements
(each having a volume of ∆wi, where i is the element number)with the position of each element’s center located at xi andyi
coordinates with respect to the body’s coordinate system. Thissystem has its origin at the intersection between the trunk andtail, its x-axis running through the anterior-most point on thetrunk, and its orthogonal y-axis oriented to the left of the body,on the frontal plane (as in McHenry, 2001). The tail fin wasassumed to be rectangular in cross-section, with a thickness of0.002 body lengths (measured from camera lucida drawings oftail cross-sections; Grave, 1934; Grave and Woodbridge,1924). The mass of the body was calculated as the product ofthe tissue density (ρbody) and the sum of volumetric elementsthat comprise the body:
where q is the total number of volumetric elements. Theposition of the center of mass (B) was calculated as (Meriamand Kraige, 1997a):
The moment of inertia for the body about any arbitrary axis ofrotation was described by the inertia tensor (I), calculated withthe following equation (Meriam and Kraige, 1997a):
We calculated the forces generated by accelerating the mass ofthe tail in tethered swimming. This tail inertia force (Finertia)was calculated with the following equation:
where V i is the velocity of the tail element. In order to removefrom the measurements any force not generated by fluid forces,we subtracted the tail inertia force from the measured force inour comparisons with predicted forces.
In order to test the effect of tissue density, we ransimulations (see ‘Modeling free swimming’ below) with themean kinematics and morphometrics at high tissue density(ρbody=1.250 g ml–1, the density of an echinopluteus larva of anechinoid with calcareous spicules; Pennington and Emlet,1986) and low tissue density (ρbody=1.024 g ml–1, the densityof seawater at 20°C; Vogel, 1981). All other simulations wererun with a tissue density typical of marine invertebrate larvaenot possessing a rigid skeleton (ρbody=1.100 g ml–1;Pennington and Emlet, 1986).
Kinematics of freely swimming larvae
Freely swimming larvae were filmed simultaneously with twodigital high-speed video cameras (recording at 500 framess–1)
(10)Finertia= ρbody ∆wi^q
i=1
dV i
dt,
(9)I = ρbody ∆wi .^q
i=1
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− xiyi
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mbody
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i=1
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t*
γ*κ* = ,− cos 2π + 1
α*
2
332
using the methodology described by McHenry and Strother (inpress). These cameras (Redlake PCI Mono/100S Motionscope,320pixels×280pixels per camera, each equipped with a 50mmmacro lens, Sigma) were directed orthogonally and both werefocused on a small volume (1cm3) of water in the center of anaquarium (with inner dimensions of 3cm width × 3cm depth ×6cm height). Larvae were illuminated from the side with twofiberoptic lamps (Cole Parmer 9741-50).
We recorded the swimming speed of larvae by tracking, inthree dimensions, the movement of the intersection betweenthe trunk and tail during swimming sequences. From the meanvalues of swimming speed (u ), we calculated aRe of the bodyfor freely swimming larvae using the following equation:
Hydrodynamic forces and moments generated by the tail
We modeled the hydrodynamics of the tail using a blade-element approach that divided the length of the tail into 50 tailelements and calculated the force generated by each of theseelements. Each element was dorso-ventrally oriented, meaningthat the length of each element ran from the dorsal to theventral margins of the fin. For each instant of time in aswimming sequence, the force acting on each element (Ej,where j is the tail element number) was calculated by assumingthat it generated the same force as a comparably sized flat platemoving with the same kinematics. Our models assume thateach tail element generates force that is independent ofneighboring elements. This neglects any influence that flowgenerated along the length of the body may have on forcegeneration. The total force generated by such a plate is the sumof as many as three forces: the acceleration reaction (Eja), skinfriction (Ejs) and the form force (Ejf; Fig. 3). The contributionof each of these forces to the total force and momentinstantaneously generated by the tail was calculated by takingthe sum of forces and moments generated by all elements (seeAppendix). Dividing the tail into 75 and 100 tail elements didnot generate predictions of forces or moments that werenoticeably different from predictions generated with 50 tailelements, but models with 25 tail elements did generatepredictions different from models with 50 elements. Therefore,we ran all simulations with 50 tail elements.
We modeled the swimming of larvae with both quasi-steadyand unsteady models. In the quasi-steady model, the forcegenerated by the tail (F) was calculated as the sum of skinfriction (Fs) and the form force (Ff; F=Ff+Fs), and the totalmoment (M ) was calculated as the sum of moments generatedby skin friction (Ms) and the form force (M f; M=M f+Ms).According to this model, the force acting on a tail element isequal to the sum of the form force and skin friction acting onthe element (Ej=Ejf+Ejs; Fig. 3B). In the unsteady model, theforce generated by the tail was calculated as the sum of all threeforces (F=Ff+Fs+Fa, where Fa is the acceleration reactiongenerated by the tail), and the total moment was calculatedas the sum of moments generated by all three forces
(11)Re= .ρLu
µ
M. J. McHenry, E. Azizi and J. A. Strother
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Fig. 3. Schematic drawing of the quasi-steady and unsteadyhydrodynamic models. (A) The force generated by a single tailelement (Ej) is drawn on the silhouette of the body of a larva from adorsal perspective. The force generated by this element hascomponents acting towards thrust and laterally. The force generatedby the whole tail was calculated instantaneously as the sum of forcegenerated by all tail elements. The position vector of the element (Rj)with respect to the center of mass describes the lever arm used by thetail element to generate a moment about the center of mass.(B,C) Each of the models is illustrated by the vectors that comprisethe force generated by the tail element. (B) The force acting on tailelements (Ej) in the quasi-steady model was calculated as the sum ofthe form force (Ejf) and skin friction (Ejs). (C) The force acting on tailelements in the unsteady model was the sum of the quasi-steadyforces and the acceleration reaction (Eja). (D) The coefficient of forceacting normal to the surface of a flat plate (cj norm) oriented normal toflow. The form force (in green; see equation 21) is found as thedifference between the total force (in black; see equation 18) and theforce generated by skin friction (in violet; see equation 19). The totalforce is generated primarily by form force at height-specific Reynoldsnumbers (Rejl) of ≈103, skin friction is dominant at Rejl<100, but thenormal force is a combination of the two at intermediate Revalues.
333Undulatory swimming at intermediate Reynolds numbers
(M=M f+Ms+Ma, where Ma is the moment generated by theacceleration reaction). According to the unsteady model, theforce acting on a tail element is equal to the sum of the formforce, skin friction and acceleration reaction (Ej=Ejf+Ejs+Eja;Fig. 3C).
The acceleration reaction
The acceleration reaction generated by a tail element wascalculated as the product of the added mass coefficient (cja),the density of water (ρ) and the component of the rate ofchange in the velocity of the element that acts in the directionnormal to the element’s surface and lies on the frontal plane ofthe body (V j norm; Lighthill, 1975):
The added mass coefficient was estimated as (Lighthill, 1975):
where lj is the distance between dorsal and ventral margins ofthe fin (height of a tail element), and ∆s is the width of the tailelement. Note that this is the added mass coefficient forinviscid flow and is assumed not to vary with Re.
Skin friction
At Re<102, skin friction may generate force that is bothnormal and tangent to a surface. Therefore, the equationfor skin friction on a tail element combines analyticalapproximations for skin friction acting tangent (Schlichting,1979) and normal (Hoerner, 1965) to the surface of a flatplate:
where V j tan is the tangent component of the velocity of theelement, s is the distance along the tail from the tail base to theelement, and Rejs is the position-specific Reynolds number fora tail element. This Reynolds number was calculated as:
where sj is the position of the element down the length of thetail, vj is the time-averaged value for tail element speed overthe tail-beat cycle.
Form force
The form force acts normal to a surface and varies with thesquare of flow speed, as expressed by the following equation(Batchelor, 1967):
where vj norm is the magnitude (or speed) of the normal
component of the velocity of the tail element, and cjf is theforce coefficient for the form force. At Re≥102, the force actingnormal to the surface of a plate is dominated by the form force(Granger, 1995; Sane and Dickinson, 2002), so cjf may beconsidered equivalent to the coefficient of force measurednormal to the surface of the plate, cj norm. This coefficient maybe calculated from measurements of force on a flat plate withthe following equation:
where Fnorm is the force measured on the plate in the normaldirection, cj norm=3.42 is an appropriate approximation for tailelements at high Re(Dickinson et al., 1999).
The contribution of the form force to the total force actingon a flat plate is predicted to change with Re (Fig. 3D). Usingthe form of the curve-fit equation for changes in the forcecoefficient on a sphere at different Regiven by White (1991),the following equation gives the force coefficient generated byboth form force and skin friction (cj s+f norm) over intermediateRe(100<Re<103):
where Rejl is the height-specific Reynolds number of the tailelement (described below). The first and last terms in thisequation describe the force generated at high (Rejl<102) andlow (Rejl<100) Reynolds numbers, respectively, and the secondterm is an intermediary fit to the experimental data reviewedby Hoerner (1965). In the viscous regime (Rejl<100), skinfriction dominates the force acting on a plate. The forcecoefficient in the normal direction for a tail element generatedentirely by skin friction is given by the following equation(Lamb, 1945):
The height-specific Reynolds number of a tail element wascalculated as:
Subtracting the contribution of skin friction (equation 19) fromthe coefficient for the total normal force (equation 18) yieldsthe coefficient for the form force for a tail element:
Hydrodynamic forces and moments generated by the trunk
The force acting on the trunk (T) was assumed to be thesame as that acting on a sphere with the same kinematics anda diameter equal to the length of the trunk. At intermediate Re,this force is equal to the sum of skin friction (Ts) and the form
(21)cjf = 3.42 − .1
1 + Rejl!
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µ
(19)cjs norm= .64
πRejl
(18)cj s+f norm= 3.42 − ,+1 64
πRejl1 + Rejl!
(17)cj norm=2Fnorm
ρljvj normV j norm∆s
(16)Ejf = − ρljcjfvj normV j norm∆s ,1
2
(15)Rejs = ,ρsjvj
µ
(14)Ejs = − µ∆s ,V j norm+ 0.3232
πlj
sj
RejsV j tan!
(13)cja = πlj2∆s ,1
4
(12)Eja = −cjaρ .dV j norm
dt
334
force (Tf). The form force varies with the square of the velocityof the trunk (P; Batchelor, 1967):
whereρ is the density of water, S is the projected area of thetrunk, p is the speed of the trunk and kf is the coefficient of theform force on a sphere, which varies with Re in the followingway (with skin friction subtracted; White, 1991):
where Rea is the Reynolds number of the trunk (calculatedusing equation 11 with the length of the trunk, a, used as thecharacteristic length). The skin friction acting on a sphere ispredicted by Stokes law (Batchelor, 1967):
Ts = 3πaµP . (24)
Given the relatively low value for the added mass coefficientof a sphere (0.5) and the low accelerations expected by thetrunk during steady swimming, we assumed negligible forcegeneration by the acceleration reaction acting on the trunk. Thetrunk generated a moment (O) about the center of mass, whichwas the sum of moments generated by the form force and skinfriction acting on the trunk:
O = D × Tf + D × Ts, (25)
where D is the position vector for the center of volume of thetrunk with respect to the body’s center of mass.
Modeling free swimming
Using the equations that describe the hydrodynamics ofswimming, we modeled the dynamics of free swimming tocalculate predicted movement by the center of mass of aswimming ascidian larva. The acceleration of the body (A) wascalculated as the sum of hydrodynamic forces acting on thebody, divided by body mass:
The angular acceleration about the center of mass wascalculated using the following equation (based on Symon,1960):
where V is the rate of rotation vector about the center ofmass, and IB is the inertia tensor given in the body’s coordinatesystem (with the center of mass as its origin). The velocity andposition of the body’s center of mass were calculated in twodimensions from the respective first and second time integralsof equation 26, and the rate of rotation and orientation ofthe body were calculated from the respective first and secondtime integrals of equation 27. In order to calculate these
integrals, models were programmed in MATLAB using avariable-order Adams–Bashforth–Moulton solver forintegration (Shampine and Gordon, 1975). This is a non-stiffmultistep solver, which means that it uses the solutions at avariable number of preceding time points to compute the currentsolution.
We calculated the percentage of thrust and drag generatedby the form force and skin friction in order to evaluate therelative importance of these forces to propulsion. Thispercentage was calculated individually for the trunk and tailand for both thrust and drag. For example, the followingequation was used to calculate the percentage of thrustgenerated by the form force on the tail (Hf tail):
where Ff′ and Fs′ are the form force and skin friction,respectively, generated by the tail in the direction of thrust (i.e.towards the anterior of the trunk). Similar calculations werealso made for the percentage of skin friction generated by thetail, form force generated by the trunk, and skin frictiongenerated by the trunk.
In order to examine how the relative magnitude of formforce and skin friction changes with the Re of the body, we rana series of simulations using model larvae of different bodylengths. Each simulation used the mean morphometricsand kinematic parameter values. The non-dimensionalmorphometrics and kinematics were scaled to the meanmeasured tail-beat period and the body length used in thesimulation. This means that animations of the body movementsin the model appeared identical for all simulations (i.e. modelswere kinematically and geometrically similar), despite beingdifferent sizes.
Statistical comparisons between measurements andpredictions
We tested our mathematical models by comparing themeasured forces and swimming speeds of larvae with modelpredictions. We measured the mean thrust (force directedtowards the anterior) and lateral force generated by a tetheredlarva and used our model to predict those forces using the samekinematics as measured for the tethered larva and the meanbody dimensions. Such measurements and model predictionswere made for a number of larvae, and a paired Student’s t-test (Sokal and Rohlf, 1995) was used to compare measuredand predicted forces. Such comparisons were made withpredictions from both the quasi-steady model and the unsteadymodel.
Predictions of mean swimming speeds from both modelswere compared with measurements of speed. Modelpredictions of swimming speed were generated usingthe mean body dimensions and the tail kinematics ofindividual larvae measured during tethered swimming. Thisassumes that the midline kinematics of freely swimminglarvae were not dramatically different from that of tethered
(28)Hf tail = × 100% ,Ff′
Fs′ + Ff′ + Fs′ + Ff′
(27)
dIB
dt= I–1 ,(M + O) − V(IB ·V) − ·V
dV
dt
(26)A = .F + T
mbody
(23)kf = + 0.4 ,6
1 + Rea!
(22)Tf = − ρSkfpP ,1
2
M. J. McHenry, E. Azizi and J. A. Strother
335Undulatory swimming at intermediate Reynolds numbers
larvae. Mean swimming speeds were measured on a differentsample of freely swimming larvae, and an unpaired t-test(Sokal and Rohlf, 1995) was used to compare predictions ofswimming speed with measurements. We verified thatsamples did not violate the assumption of a normaldistribution by testing samples with a Kolomogorov–Smirnov test (samples with P>0.05 were considered to benormally distributed).
ResultsHydrodynamics at Re≈102
Tethered larvae
Using measured kinematics (Fig. 4, Table 1), we tested theability of hydrodynamic models to predict both the timing andmean values of forces generated by larvae. The magnitude ofpredictions of form force and the acceleration reaction(Fig. 5A–C) were approximately two orders of magnitude
0 20 40 60 80 100 120Time (ms)
–0.5
0
0.5
Tru
nk a
ngle
θ (r
ad)
0
0.5
1.0
Pos
ition
of i
nfle
ctio
n po
int
z * (
body
leng
ths)
0
0.2 α
0.4
Cur
vatu
re, κ
*(r
ad b
ody
leng
th–1
)
Right-directed Left-directed
26 ms 34 ms 42 ms 50 ms 58 ms 66 ms 74 msB
C
D
E
γ
P ζ
A TrunkBase of tailInflection points
Concave-left bend
Concave-right bend
Trunk angle, θ
Fig. 4. The undulatory motion typical of tethered larvae. (A) The shape of the tail of a larva at any instant was described by the trunk angle (θ),position of inflection points on the tail (z) and the curvature of the tail between the inflection points (κ∗ ). The trunk angle was positive when thetail was bent to the right and negative when bent to the left of the body. (B) Changes in the midline of the tail of a larva over a single tail beat.Notice that as time progresses (to the right), inflection points (filled circles) move down the midline in the posterior direction (away from thebase of the tail). (C–E) Points represent measurements of each kinematic parameter and the curves are found by a least-squares fit to functionsdescribed in the Materials and methods. (C) The curvature of bends between inflection points (with a period γ) in both concave-left andconcave-right bends (having an amplitude α). (D) The propagation of inflection points begins at the tail base with a phase lag of ζ with respectto a zero value of the trunk angle. (E) The trunk angle oscillates with time (with a period P). The vertical gray bands show when the trunk angleis directed towards the left side of the body, and white bands occur when the trunk angle is directed to the right.
336
greater than the predictions for the tail inertia and skin frictionforces (Fig. 5D,E). Due to the low magnitude of skin friction,the tail force predicted in the lateral direction by the quasi-steady model (F=Ff+Fs) was qualitatively indistinguishablefrom the prediction of form force (Fig. 5F). The prediction forthe tail force in the lateral direction by the unsteady model hadthe addition of the acceleration reaction (F=Ff+Fs+Fa), whichgenerated peaks of force when the form force was low inmagnitude (Fig. 5F). These force peaks were not reflected inthe measurements of lateral force (Fig. 5G). This measuredforce oscillated in phase with trunk angle (θ; phase lag mean± 1 S.D.=0.03±0.02 tail-beat periods,P=0.230, N=11; Figs 5H,6), unlike the acceleration reaction, which was predicted to beout of phase with trunk angle. Both quasi-steady and unsteadymodels predicted mean thrust and mean lateral force that wasstatistically indistinguishable from measurements (Table 2).
The force predictions by the quasi-steady model moreclosely matched the timing of measurements than those of theunsteady model (Fig. 7). The force predicted by the quasi-
steady model (F=Ff+Fs) oscillated in phase with measuredlateral forces. However, the unsteady model (F=Ff+Fs+Fa)predicted peaks of force generation by the acceleration reactionacting in the direction opposite to the measured force (Figs 5,7). At instants of high tail speed, the form force was large andwas followed by the acceleration reaction acting in the oppositedirection as the tail decelerated and reversed direction.Although both models accurately predicted mean forces (Table2), the timing of force production suggests that the accelerationreaction does not generate propulsive force in the swimmingof ascidian larvae.
Freely swimming larvae
Simulations of free swimming allowed the body of larvae torotate and translate in response to the hydrodynamic forcesgenerated by the body. As such movement could contributeto the flow encountered by a swimming larva, the forcesgenerated by freely swimming larvae were not assumed to bethe same as those generated by tethered larvae. Therefore,
M. J. McHenry, E. Azizi and J. A. Strother
Table 1.Swimming kinematics of tethered larvae
L P ε∗ α∗ γ∗ χIndividual (mm) (ms) (body lengths per tail-beat period) (rad per body length) (tail-beat periods) (rad)
Mean ± 1 S.D.= 1.93±0.13 42.7±2.9 1.21±0.12 0.97±0.12 1.21±0.12 0.21±0.08
L, body length; P, tail-beat period; ε, wave speed of inflection point; α, amplitude of tail curvature;γ, period of tail curvature; χ, amplitude oftrunk angle.
All data are time-averaged values for the duration of at least three tail beats.
Table 2.Model verification in tethered and freely-swimming larvae
Model predictions
Quasi-steady UnsteadyMeasurements (F=Ff+Fs) P (F=Ff+Fs+Fa) P N
All values are means ± 1 S.D. P values are the results of a Student’s t-test that compared measurements with predictions. These were pairedcomparisions of force and unpaired comparisons of speed.
337Undulatory swimming at intermediate Reynolds numbers
simulations of free swimming were a closer approximation ofthe dynamics of freely swimming larvae and provided a testfor whether the results of tethering experiments apply to freelyswimming larvae.
The results of these simulations support theresult from tethering experiments that theacceleration reaction does not play a role in thehydrodynamics of swimming. The quasi-steadymodel (F=Ff+Fs) predicted a mean swimmingspeed that was statistically indistinguishable frommeasured mean swimming speed. By contrast, theunsteady model (F=Ff+Fs+Fa) predicted a meanswimming speed that was significantly differentfrom measurements (Table 2). We found small(<4%) differences in predicted mean speedbetween models using a high (ρbody=1.250 g ml–1)and low (ρbody=1.024 g ml–1) tissue density,suggesting that any inaccuracy in the tissue density
used for simulations (ρbody=1.100 g ml–1) had a negligibleeffect on predictions.
Reynolds number values varied among different regions ofthe body (Table 3). The mean Reynolds number for the wholebody (Re=7.7×101) was larger than the Reynolds number forthe trunk (Rea=2.8×101) because the whole body is greater inlength than the length of just the trunk. The mean height-specific Reynolds number (Rejl) and the mean position-specificReynolds number (Rejs) were larger towards the posterior(Table 3).
Time (ms)0 20 40 60 80 100 120
–20
0
20
–20
0
20
–20
0
20
–0.25
0
0.25
–0.04
0
0.04
–0.2
0
0.2
–20
0
20
Tai
l for
ce,
FM
easu
red
forc
e (µ
N)
Form
forc
e,F
f
Acc
eler
atio
nre
actio
n, F
a
Tai
l ine
rtia
,F
iner
tia
Ski
n fr
ictio
n,F
s
Tru
nk a
ngle
,θ
(rad
)
F
G
B
C
D
E
A+
+AnteriorPosterior
Left
Right
H
Pred
icte
d ta
il fo
rces
(µN
)
F=Ff+Fs
F=Ff+Fs+Fa
Fig. 5. Measured and predicted forces for a tetheredlarva. (A) The legend for the direction of force data(B–H). Violet traces represent lateral forces that aredirected to the right of the body when negative and to theleft when positive. Green traces show force along theantero-posterior axis of the trunk that is directed towardthe anterior when positive and toward the posterior whennegative. (B–H). As in Fig. 4, the vertical gray bandsshow when the trunk angle (θ) is directed toward the leftside of the body, and white bands occur when the trunkangle is directed to the right. Note that (B) the form forceand (C) the acceleration reaction are on the same scale asthe measured force, but both (D) the tail inertia force and(E) skin friction are plotted on smaller scales. (F) The tailforce predicted for quasi-steady (F=Ff+Fs, heavy line)and unsteady models (F=Ff+Fs+Fa, thin line) illustratethe differences between these models. (G) Forcegenerated by the larva against the tether, in the lateraldirection for unfiltered (points) and filtered (line) data(see text for details). (H) Variation in trunk angle withtime (the line is filtered data and the points areunfiltered).
0 0.2 0.4 0.6 0.8 1.0
Time (tail-beat cycles)
–20
0
20
Lat
eral
forc
e (µ
N)
–0.4
0
0.4
Tru
nk a
ngle
,θ
(rad
)
B
A
Fig. 6. The phase relationship between lateral force generation andtail kinematics. Positive values are directed to the left of the bodyand negative values are directed to the right. Each blue curve showsthe mean values over four tail beats for a single larva, with timenormalized to the tail-beat period. The black solid lines represent themean, and the black dotted lines represent ±1 S.D. for all larvae(N=11). As in Fig. 4, the gray band shows when the trunk angle isdirected towards the left side of the body, and white bands showwhen the trunk angle is directed to the right. Measurements of lateralforce (A) are plotted above trunk angle (B).
338
Hydrodynamics at 100<Re<102
Predictions by the quasi-steady model showed how thrustand drag may be generated differently by form force and skinfriction at different Re. At Re≈100, both thrust and drag werepredicted to be dominated (>95%) by skin friction acting onthe trunk and tail (Fig. 8A,B). At Re≈101, most drag (63%) wasgenerated by skin friction acting on the trunk, and most thrust(69%) was generated by skin friction acting on the tail(Fig. 8C,D). At Re≈102, drag was generated by a combinationof skin friction and form force, but thrust was generatedalmost entirely by form force acting on the tail (Fig. 8E,F). Byrunning simulations throughout the intermediate Re range(100<Re<102), we found that form force gradually dominatesthrust generation (up to 98%) with increasing Re. Although theproportion of drag generated by form force increases with Re,skin friction generates a greater proportion of drag (>62%) thandoes form force, even at Re≈102.
DiscussionThe acceleration reaction
It is surprising that our results suggest that the accelerationreaction does not contribute to thrust and drag in the steadyundulatory swimming of Botrylloides sp. larvae. Vyman’s(1974) model for the energetics of steady swimming in fishlarvae assumes that the acceleration reaction should operate atthe Reynolds number at which these larvae swim (Re≈102).Although the energetic costs of locomotion predicted byVyman (1974) show good agreement with measurements,these predictions from an unsteady model have not beencompared with the predictions of a quasi-steady model.Furthermore, the hydrodynamics assumed by Vyman (1974)have yet to be experimentally tested. By contrast, Jordan(1992) did compare quasi-steady and unsteady predictionswith measurements of the startle response behavior of thechaetognath Sagitta elegans. This study found that the
M. J. McHenry, E. Azizi and J. A. Strother
A
–20
–10
0
10
20
–20
0
20
0
0
0.2 0.4 0.6 0.8 1.0
1.0
Time (tail-beat cycles)
Time (tail-beat cycles)F=Ff+FsL
ater
al fo
rce
(µN
)
Pred
icte
d fo
rce
(µN
)
–20 0 20
Measured force (µN)
B
C
–20
–10
0
10
20
–20
0
20
0 0.2 0.4 0.6 0.8 1.0
Time (tail-beat cycles)
F=Ff+Fs+Fa
Lat
eral
forc
e (µ
N)
Pred
icte
d fo
rce
(µN
)–20 0 20
Measured force (µN)
D
Fig. 7. Comparison of predicted and measured lateral forces. Graphs to the left (A and C) show mean measured lateral forces (dark gray line)±1 S.D. (light gray fill) of measured forces (the same data as in Fig. 5A; N=11) and the mean (solid black line) ±1 S.D. (dotted black line) ofpredicted lateral forces for the same 11 larvae. Graphs on the right (B and D) present the same data, but the measured forces are plotted againstpredicted forces for each instant of time in the tail-beat cycle. Points vary in color from blue to red as the tail-beat cycle progresses. The greenregression line was calculated by a least-squares solution to a linear curve fit of the data (slope=0.32, y-intercept=0, r2=0.50 in B; slope=0.13,y-intercept=0, r2=0.05 in D). The gray line has a slope of 1, which represents a perfect match between measured and predicted data. (A,B) Theforces predicted for the lateral force by the quasi-steady model compared with measurements. (C,D) The forces predicted for the lateral forceby the unsteady model compared with measurements.
339Undulatory swimming at intermediate Reynolds numbers
unsteady model better predicted the trajectory of swimmingthan did the quasi-steady model, which suggests that theacceleration reaction is important to undulatory swimming atintermediate Re.
This discrepancy between our results and Jordan (1992) onthe relative importance of the acceleration reaction may bereconciled if the acceleration reaction coefficient varies withRe. The acceleration reaction is the product of the accelerationreaction coefficient (which depends on the height of the tailelement), the density of water and the acceleration of a tailelement (equation 12). Both Jordan (1992) and the presentstudy used the standard inviscid approximation (equation 13)
for the acceleration reaction coefficient (used in elongatedbody theory; Lighthill, 1975). However, chaetognaths attainRe≈103 and more rapid tail accelerations than ascidian larvae.If the actual acceleration reaction coefficient is lower than theinviscid approximation at the Re of ascidian larvae (Re≈102),then predictions of the acceleration reaction would be smallerin magnitude. The chaetognath may still generate sizeableacceleration reaction in this regime by beating its tail withrelatively high accelerations.
Although swimming at Re>102 has not been reported amongascidian larvae, numerous vertebrate and invertebrate speciesdo swim in this regime. We predict that as Reapproaches 103,
0
20
40
60
80
100
Tail Trunk
–6.0
0
6.0
–6.0
0
6.0
–50
0
50
–50
0
50
–0.30
0
0.30
Tai
l for
ce(µ
N)
–0.30
0
0.30
0 40 80 120 160 200Time (ms)
0 40 80 120 160 200Time (ms)
Skinfriction
Tru
nk fo
rce
(µN
)
Perc
enta
ge o
f tot
al fo
rce
Re≈100
A B
C D
E F
Formforce
Skinfriction
Formforce
Skinfriction
Formforce
Skinfriction
Formforce
Skinfriction
Formforce
Skinfriction
Formforce
Tai
l for
ce(µ
N)
Tru
nk fo
rce
(µN
)
0 40 80 120 160 200Time (ms)
Tai
l for
ce(µ
N)
Tru
nk fo
rce
(µN
)
Drag Thrust
0
20
40
60
80
100
Perc
enta
ge o
f tot
al fo
rce
Drag Thrust
0
20
40
60
80
100Pe
rcen
tage
of t
otal
forc
e
Drag Thrust
Skin friction Form force
Total force
Re≈101
Re≈102
Fig. 8. Thrust and drag predicted bythe quasi-steady model to act on thebody of a freely swimming larva atdifferent Reynolds numbers (Re).The graphs on the left (A,C,E)show a representative time series ofthe skin friction (violet lines) andform force (green lines) acting onthe trunk and tail for approximately4.5 tail beats with the same non-dimensional tail kinematics anddifferent body lengths. Thrust actsin the positive direction and dragacts in the negative direction. Thetotal force is the sum of skinfriction and the form force. Thegraphs on the right (B,D,F)illustrate the percentage of the totalthrust and drag that is generated byskin friction and form force thatacts on the trunk and tail. Error barsdenote ±1 S.D., which is variationgenerated by running simulationswith different kinematic patterns(N=5). Re was varied by changingthe body length of model larvae.
340
the acceleration reaction contributes more to the generation ofthrust in undulatory swimming. Although it remains unclearhow the magnitude of the acceleration reaction changes withRe, the unsteady models proposed here (F=Ff+Fs+Fa) andelsewhere (Jordan, 1992; Vlyman, 1974) should approximatethe hydrodynamics of undulatory swimming at Re≈103.
Skin friction and form force
In support of prior work (e.g. Fuiman and Batty, 1997;Jordan, 1992; Vlyman, 1974; Webb and Weihs, 1986; Weihs,1980), our quasi-steady model (F=Ff+Fs) predicted that therelative magnitude of inertial and viscous forces is different atdifferent Re. At Re≈100, skin friction (acting on both the trunkand tail; Fig. 8) dominated the generation of thrust and drag(Fig. 9). This result is consistent with the viscous regimeproposed by Weihs (1980) for swimming at Re<101. Also inaccordance with Weihs (1980) are the findings that form forcecontributes more to thrust and drag at high Rethan at low Re(Fig. 9) and that thrust (Fig. 8) is dominated by form force atRe≈102. However, it is surprising that drag was generated moreby skin friction than form force at Re≈102 (Figs 8, 9). Contraryto Weihs’ (1980) proposal for an inertial regime at Re>2×102,this result suggests that the fluid forces that contribute to thrustare not necessarily the same forces that generate drag. This isunlike swimming in spermatozoa (at Re!100), where boththrust and drag are dominated by skin friction acting on boththe trunk and flagellum (Gray and Hancock, 1955), or someadult fish (at Re@102), where thrust and drag are bothdominated by the acceleration reaction (Lighthill, 1975; Wu,1971).
Our results suggest that ontogenetic or behavioral changesin Recause gradual changes in the relative contribution of skinfriction and form force to thrust and drag. As pointed out byWeihs (1980), differences in intermediate Rewithin an orderof magnitude generally do not suggest large hydrodynamicdifferences. Although it has been heuristically useful to
consider the differences between viscous and inertialregimes (e.g. Webb and Weihs, 1986), it is valuableto recognize that these domains are at opposite endsof a continuum spanning three orders of magnitudein Re. This distinction makes it unlikely that larvalfish grow through a hydrodynamic ‘threshold’ whereinertial forces come to dominate the hydrodynamicsof swimming in an abrupt transition with changingRe (e.g. Muller and Videler, 1996).
In summary, our results suggest that theacceleration reaction does not play a large role in the
hydrodynamics of steady undulatory swimming at intermediateRe(100<Re<102). Our quasi-steady model predicted that thrustand drag are generated primarily by skin friction at low Re(Re≈100) and that form force generates a greater proportion ofthrust and drag at high Re than at low Re. Although thrust isgenerated primarily by form force at Re≈102, drag is generatedmore by skin friction than form force in this regime. Unlikeswimming at Re>102 and Re<100, the fluid forces that generatethrust cannot be assumed to be the same as those that generatedrag at intermediate Reynolds numbers.
L, body length; a, trunk length; s, distance along the tail from thetail base to the element; l, height of tail element; Rea, Reynoldsnumber of the trunk; Re, Reynolds number of the whole body; Rejl,height-specific Reynolds number of a tail element; Rejs, position-specific Reynolds number of a tail element. N=14 for allmeasurements.
0
20
40
60
80
100
0
20
40
60
80
100
100 101 102
Reynolds number of the whole body (Re)
Perc
enta
ge o
f dra
gPe
rcen
tage
of t
hrus
t Skin friction
Form force
A
B
Fig. 9. The percentage of thrust and drag generated byskin friction and form force predicted by the quasi-steadymodel. Reynolds number of the whole body (Re) wasvaried by running a series of simulations over a range ofbody lengths. Lines show the percentage of (A) thrustand (B) drag generated by skin friction (violet) and formforce (green).
341Undulatory swimming at intermediate Reynolds numbers
We thank M. Koehl for her guidance and advice, S. Sanefor his wisdom on hydrodynamics, and A. Summers, W. Korffand W. Getz for their suggestions on the manuscript. Thiswork was supported with an NSF predoctoral fellowship andgrants-in-aid of research from the American Society ofBiomechanics, Sigma Xi, the Department of IntegrativeBiology (U.C. Berkeley) and the Society for Integrative andComparative Biology to M. McHenry. Additional supportcame from grants from the National Science foundation (#OCE-9907120) and the Office of Naval Research (AASERT #N00014-97-1-0726) to M. Koehl.
AppendixTether calibration
We used a least-squares method (described by Hill, 1996) tofind the stiffness and damping constants of the tether fromrecordings of its position when allowed to oscillate without anylarva attached. This method uses the equation of motion for thetether given any position measurement (ϕe):
Moments generated at the pivot of the tether may be calculatedwith a version of this equation with different parameter valuesfor each instant of time in a series of e position recordings.Such a time series of equations may be represented by thelinear expression:
Aq = r , (30)
where
Best fits for values of c and k were found by solving thefollowing equation:
q = (ATA)–1ATr , (32)
where (ATA)–1 is the inverse of the product of A and thetranspose of A. Solutions to this equation were found usingMATLAB. This method was verified by analyzing fabricatedposition data that were generated by numerical solutions toequation 2 (a fourth-order Runge–Kutta in MATLAB) withknown values of k and c.
Calculating tail force
The total force generated by the tail of a larva was calculatedas the sum of forces acting on all elements of the tail. For
example, the total acceleration reaction generated by the tailwas found as the sum of acceleration reaction forces acting ontail elements:
where n is the total number of tail elements. Similarly, themoment generated by these forces was calculated as the sumof cross products between the vector of the position of the tailelement with respect to the body’s center of mass (Rj) and theacceleration reaction acting on tail elements (Meriam andKraige, 1997b):
The same calculations were used to determine the total forceand moment generated by skin friction and form force for eachinstant of time in a swimming sequence.
List of symbolsa length of the trunkA acceleration of the bodyB position of the center of masscja added mass coefficientcjf coefficient of force on tail element due to form
forcecj norm coefficient of total force on tail element in the
normal directioncjs coefficient of force on tail element due to skin
frictioncj s+f norm coefficient of force on tail element in the normal
direction due to form force and skin frictionD position of the center of volume of the trunkEj total force acting on a tail elementEja acceleration reaction on a tail elementEjf form force on a tail elementEjs skin friction on a tail elementF total force generated by the tailFa tail force generated by acceleration reactionFf tail force generated by form forceFf′ tail force generated by form force in the direction
of thrustFinertia tail inertia forceFnorm force in the normal direction measured on a plate Fs tail force generated by skin frictionFs′ tail force generated by skin friction in the
direction of thrustg acceleration due to gravityhcm distance from the tether pivot to the center of
mass of the tetherHf tail percentage of thrust generated by form force on
the tailhobjective distance from the tether pivot to the objective
(34)Ma = Rj × Eja .^n
j=1
(33)Fa = Eja ,^n
j=1
A = ϕ2 , q = ., r = −mghcmsin(ϕ2) − I
A A A
dϕ2
dt
d2ϕ2
dt2
ϕw −mghcmsin(ϕw) − Idϕw
dt
d2ϕw
dt2
ϕ1 −mghcmsin(ϕ1) − Idϕ1
dt
d2ϕ1
dt2
c
k
(31)
c + kϕe = −Itether − mtetherghcmsin(ϕe) .dϕe
dt
d2ϕe
dt2(29)
342
htip distance from the tether pivot to the tip of the pipette
i volumetric element numberI inertia tensor for the body of a larvaIB inertia tensor for the body in the body’s
coordinate systemItether moment of inertia of the tetherj tail element numberkdamp damping coefficientkf coefficient of the form force on the trunkkspring spring coefficientl height of a tail elementL body lengthmbody mass of the body of a larvaM total momentMa moment generated by the acceleration reactionM f moment generated by form forceMs moment generated by skin frictionmtether mass of the tetherO moment generated by force on the trunkp speed of the trunkP tail-beat periodP velocity of the trunkq total number of volumetric elementsRj position of the tail element with respect to the
center of massRe Reynolds number for whole bodyRea Reynolds number of the trunkRejl height-specific Reynolds number of a tail elementRejs position-specific Reynolds number for a tail
elementrtether inner radius of the micropipettes distance along the tail from the tail base to the
elementsj position of the element down the length of a tailS projected area of the trunkt timeT force acting on the trunkTf form force acting on the trunkTs skin friction acting on the trunku mean swimming speedv mean tail element speedV i velocity of a tail elementvj norm speed of the normal component of the velocity of
a tail elementvj tan speed of the tangent component of the velocity of
a tail elementV j norm normal component of the velocity of a tail elementV j tan tangent component of the velocity of a tail
elementxi x-coordinate of volumetric elementyi y-coordinate of volumetric elementz position of inflection point along the length of the
tailα amplitude of change in curvatureχ amplitude of change in trunk angle
δ linear deflection of the tether∆s width of a tail element∆wi volume of a volumetric elementε wave speed of inflection pointϕ radial deflection of the tether at its pivotϕe measurement of tether deflectionγ period of change in curvatureκ tail curvatureµ dynamic viscosity of waterθ trunk angleρ density of waterρbody density of tissueΩ rate of rotation about the center of massζ phase lag of inflection point relative to trunk
angle* non-dimensional quantity
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