The Journal of Experimental Biology 201, 2171–2181 (1998) 2171 Printed in Great Britain © The Company of Biologists Limited 1998 JEB1472

BURROW VENTILATION IN THE TUBE-DWELLING SHRIMP SUBTERRANEA (: THALASSINIDEA)

III. HYDRODYNAMIC MODELLING AND THE ENERGETICS OF PLEOPOD PUMPING

EIZE J. STAMHUIS* AND JOHN J. VIDELER Department of Marine Biology, University of Groningen, PO Box 14, 9750 AA Haren, the Netherlands *e-mail: [email protected]

Accepted 5 May; published on WWW 25 June 1998

Summary The process of flow generation with metachronally The energetics of the tube and the pump are derived from beating pleopods in a tubiform burrow was studied by the forces, and the mechanical efficiency of the system is designing a hydrodynamic model based on a thrust–drag the ratio of these two. Adjusted to standard Callianassa force balance. The drag of the tube (including the shrimp) subterranea values, the model predicts a mean flow velocity comprises components for accelerating the water into the in the tube of 1.8 mm s−1. The mean thrust force, equalling tube entrance, for adjusting a parabolic velocity profile, for the drag, is 36.8 µN, the work done by the pleopod pump accelerating the flow into a constriction due to the shrimp’s per beat cycle is 0.91 µJ and the energy dissipated by the body and another constriction due to the extended tail-fan, tube system is 0.066 µJ per cycle. The mechanical efficiency for shear due to separation and for the viscous resistance is therefore 7.3 %. Pump characteristics that may be varied of all tube parts. The thrust produced by the beating by the shrimp are the beat frequency, the phase shift, the pleopods comprises components for the drag-based thrust amplitude and the difference in pleopod area between the and for the added-mass-based thrust. The beating pleopods power and recovery strokes. These parameters are varied are approximated by oscillating flat plates with a different in the model to evaluate their effects. Furthermore, the area and camber during the power stroke and the recovery moment of added mass shedding, the distance between stroke and with a phase shift between adjacent pleopod adjacent pleopods, the number of pleopods and the total pairs. The added mass is shed during the second half of the tube drag were also varied to evaluate their effects. power stroke and is minimized during the recovery stroke. A force balance between the pleopod thrust and the tube drag is effected by calculating the mean thrust during one Key words: burrow ventilation, tube-dwelling shrimp, Callianassa beat cycle at a certain flow velocity in the tube and subterranea, ventilation energetics, laminar flow, hydrodynamic comparing it with the drag of the tube at that flow velocity. modelling.

Introduction Most tube-living use the thrust generated by suggest even higher ventilation capacities (e.g. de Vaugelas, beating limbs to ventilate their burrow (Atkinson and Taylor, 1985; Scott et al. 1988). The energy involved in burrow 1988; Allanson et al. 1992). The thalassinid decapods beat ventilation has been estimated from the ventilation flow rates three or four pairs of pleopods metachronally (Farley and Case, in a few shrimp only (Gust and Harrison, 1981; 1968; Dworschak, 1981). Burrow ventilation is assumed to be Allanson et al. 1992; Stamhuis and Videler, 1997b). These energetically expensive and is always a periodic event estimates of the dissipated energy are based on the resistance (Atkinson and Taylor, 1988; Farley and Case, 1968; Torres et of the tube, without taking the mechanism of flow generation al. 1977; Felder, 1979; Dworschak, 1981; Mukai and Koike, into account. They therefore underestimate the total energy 1984; Scott et al. 1988; Forster and Graf, 1995; Stamhuis et invested by the pumping shrimp (Stamhuis and Videler, al. 1996). The ventilation flow rates of the vary 1997b). between 0.6 and 10 ml min−1 (Koike and Mukai, 1983; Mukai The thalassinid shrimp creates and and Koike, 1984; Forster and Graf, 1995; Stamhuis and inhabits an extensive burrow in the seabed (Witbaard and Videler, 1997b), and those of the Upogebiae vary between 33 Duineveld, 1989; Atkinson and Nash, 1990; Rowden and and 50 ml min−1 (Dworschak, 1981; Koike and Mukai, 1983). Jones, 1995; Stamhuis et al. 1997). The shrimp ventilates the Descriptions of sediment resuspension in tropical burrow on average every 14 min for approximately 1.3 min, Callianassidae and respiratory ventilation in Upogebiae which represents 8 % of its time budget (Stamhuis et al. 1996). 2172 E. J. STAMHUIS AND J. J. VIDELER

The ventilation flow is produced by active pumping using three (1>Cc>0). The cross-sectional area in the third part of the tube pairs of pleopods in an ad-locomotory metachronal pattern at equals that in the first part (A3=A1). a frequency of approximately 1 Hz. The pleopods are spread The total drag force consists of the following components during the power stroke and retracted during the recovery (Prandtl and Tietjens, 1957; Schlichting, 1979; Munson et al. stroke. The areas of the pleopods differ by a factor of 1994), where U is velocity, ρ is the density and µ is the approximately two between these strokes (Stamhuis and dynamic viscocity of sea water: (a) the force related to 2 Videler, 1997a). Although the beat pattern is rhythmical, the accelerating water into the tube entrance = gρU1 A1; (b) the flow generated in the burrow is non-pulsatile and completely drag associated with adjusting a parabolic velocity profile = 2 laminar. In the vicinity of the beating pleopods, the main flow 1.2×gρU1 A1; (c) the viscous drag of the tube section upstream is confined to the lower part of the tube, owing to a constriction of the = 8πµL1U1; (d) the force required to accelerate 2 caused by the tail-fan (Stamhuis and Videler, 1997b). The the flow into the first constriction = gρ{U1[(A1/A2)−1]} A2; (e) kinematics of the structures involved in burrow ventilation and the viscous drag of the first constricted section = 8πµL2U2; (f) the resulting flow patterns in the vicinity of the shrimp are the force required to accelerate the flow into the second 2 reasonably well understood, but the mechanisms of thrust constriction = gρ{U2[(A2/Ac)−1]} Ac; (g) the drag involved in production and the energy involved in ventilation need more the flow separation directly behind the second constriction = investigation. The present study therefore concentrates on 2 2 gρUc (1−Ac/A3) A3; and (h) the viscous drag of the tube mapping the thrust and drag forces generated by Callianassa section downstream of the animal = 8πµL3U3. subterranea and on the energy involved in ventilation by The tube is divided into three sections with lengths L1, L2 applying hydrodynamic modelling. The dissipated mechanical and L3 (see Fig. 1). The drag of the upstream section is the sum energy of pumping is estimated by adapting a more generally of the drag components a, b and c. Components d, e and f sum applicable model to the pumping characteristics of the shrimp. to form the drag of the middle section, and the sum of g and h The effects of varying characteristic variables on the flow represents the drag of the downstream section. The drag forces velocity in the tube, the dissipated mechanical energy and the (F1,d,tube–F3,d,tube) of these sections can be written as specific pump efficiency are evaluated. equations by expressing all velocities U and tube diameters D – as multiples or fractions of U and Dtube, where Dtube is the Materials and methods diameter of the tube: The tube-pump model –2 2 2.2ρU πDtube – F1,d,tube = + 8πµL1U , (2) A shrimp ventilating its burrow is essentially a pump driving 8 water through a tube. When the flow velocity is constant, the thrust-producing pump is counteracted by the resistance of the F2,d,tube = tube. The mean thrust will equal the mean drag, and the forces –2 2    2    are in equilibrium: πρU Dtube 1 1 1 2   − 1 Cbody +  −  Cc – – 8  C   C C  F = − F . (1)  body c body  thr,pump d,tube – U Both the thrust and the drag are a function of the flow + 8πµL2 , (3) velocity in the tube. To determine the magnitude of the thrust Cbody and the drag, equation 1 must be solved numerically with the – flow speed in the tube as an iteration variable. πρU2D 2  1  2  tube 2 – F3,d,tube =  − 1 (1 − Cc)  + 8πµL2U . 8  C  Tube drag  c  The drag of the tube, including the shrimp’s body, is (4) calculated by summing the drag contributions of the separate The total tube drag is: parts of the tube, applying the approach of Stamhuis and Videler (1997b). The flow in the tube is assumed to be laminar Fd,tube = F1,d,tube + F2,d,tube + F3,d,tube . (5) (Hagen–Poiseuille flow) with a mean flow velocity. The tube The energy dissipated by the tube and the water flow during containing the animal is abstracted to be a straight cylindrical one beat cycle of the pleopods Wtube,T is the product of the drag tube, of length Ltot and diameter D1, with two constrictions force, the mean velocity and the duration of the beat cycle T: (Fig. 1). The first constriction is a reduction of the tube cross- × – × sectional area A to A2=CbodyA1, where Cbody is the coefficient Wtube,T = Fd,tube U T . (6) of tube area reduction due to the shrimp’s body, over a distance 2 L2, representing the shrimp’s body (1>Cbody>0, A1=sπD1 ). Pleopod-generated thrust The second constriction, immediately behind the first, reduces The thrust generated by an oscillating pleopod pair is the tube area to Ac=CcA1, where Cc is the coefficient of tube calculated by summing the drag forces and the forces required area reduction due to the tail-fan constriction. It represents the to accelerate the added mass of water around the pleopod. A reduction in cross-sectional area by the uropods and the telson pleopod pair is assumed to resemble a slightly cambered flat Hydrodynamic modelling of burrow ventilation flow 2173

β t r ∆ D 1 r ∆ D 3 D 2 F

ω ∆FN t D c

L1 L2 L3

Ltot Fig. 1. Diagram of a model ventilating Callianassa subterranea in a tube, showing hydrodynamic events (dashed lines), cross-sectional ventilatory profiles (horizontal arrows) and variables used in the tube-pump model (see text). D, diameter; L, length; subscripts 1, 2, 3 and c refer to the respective tube sections and the narrow constriction; Ltot, total length of the tube; FN, normal force; r, fraction of pleopod length; βt, angle between tube axis and pleopod; ωt, time-dependent angular velocity of a pleopod pair. plate attached to the tube wall by a hinge. Each pair has its mass of a section of an oscillating flat plate, being the mass of own sinusoidal motion pattern with time-dependent angular the volume of a cylinder with a diameter equal to the width velocity ωt. The relative velocity between a pleopod and the wpp and a height ∆r (equalling the body of revolution around water depends on the phase of the oscillation as well as on the the longitudinal central axis) (Blake, 1979; Morris et al. 1985). distance to the pleopod hinge. The force ∆F exerted on the d(ωr) water in the direction of the flow by a horizontal slice of a 2 ∆Fam = ρ(sπwpp ∆r) × × sinβ , (11) pleopod (height ∆r and width wpp) at distance r from the hinge dt (Fig. 1) can be expressed as the sum of a drag force F and an d where t is time. added mass force F : am The added mass forces are not constant throughout the beat ∆F = ∆Fd + ∆Fam . (7) cycle because they decrease with increasing Reynolds numbers (Vogel, 1994) and as a result of shedding of the water mantle The drag force ∆F is derived from Newton’s Resistance Law d (Daniel, 1984). Added-mass shedding at sinusoidally expressed as a Bernoulli-like dynamic pressure equation oscillating pleopods can be predicted from the limb kinematics incorporating a drag coefficient C (Prandtl and Tietjens, d and the resulting forces. The normal drag forces have been 1957): dealt with separately and are neglected here. ∆Fd = 0.5ρCd(wpp∆r) × |ωr − Uysinβ| × (ωr − Uysinβ)sinβ , The added mass of a pleopod pair Ma,pp is assumed to equal sρπ 2 (8) wpp R, where R is pleopod length, with its centre of mass at R/2 (Fig. 3). At the start of a power stroke, the oscillating pleopod where Uy is the velocity component perpendicular to the tube exerts a force Fam=(dω/dt)×Ma,pp×(R/2) on the added mass, axis and β is the angle between the pleopod and the axis. resulting in an acceleration d(ωR/2)/dt in the tangential direction. The drag coefficient Cd of a pleopod is assumed to equal the Ma,pp is subjected to an additional radial (centrifugal) force drag coefficient of a flat plate perpendicular to the flow Cd,fp, corrected for camber using a multiplication factor fcam: 1.6 d 1.4 Cd = Cd,fp × fcam , (9) 1.2 w where C is a function of the Reynolds number Re based on pp d,fp 1.0 the plate width (after Vogel, 1994):

cam Concave side f 0.8 −1.21 Cd,fp = 1.17 + 17.1Re (10) 0.6 6 2 (N=7, 1

Fd,par gradually. This process was simplified to implement it in our model by incorporating a coefficient Cshed setting the moment of instantaneous added mass shedding. Because the water mantle is smaller during the recovery stroke than during the power stroke, the added-mass forces are assumed to be smaller during the recovery stroke than during the power stroke. Ma,pp + The thrust Fthr,t produced by the whole pleopod pair at time t is found by integrating the thrust forces from the hinge to the + tip of the pleopods:

vshed,t vt=ωt × R/2 ⌠ R Fthr,t =  dF . (13) ⌡0 – The mean thrust Fthr during one motion cycle with period T is found by integrating Fthr,t over a whole period divided by T: Frad ⌠ T⌠ R – 1 F =   dFdt . (14) Fig. 3. Diagram depicting the radial forces on the added mass M thr a,pp T ⌡0 ⌡0 of an oscillating pleopod, modelled as a flat plate. Velocity vectors in the radial and tangential directions are indicated. Vectors are not The rate of doing work per pleopod pair slice is the product of drawn to scale. The light shaded area indicates Ma,pp at the start of the velocity (ωr−Uysinβ) and the force dF. The total work per the power stroke (t=0), the dark shading is for t≈0.15T after t=0, beat cycle per pleopod pair is then found by integrating over where t is time and T is the duration of the beat cycle. The centre of the pleopods from hinge to tip, and over time from 0 to T: mass of Ma,pp is depicted as a circle with a central plus sign. vt, ⌠ T⌠ R velocity of Ma,pp in the tangential direction with respect to the W =   (ωr − Uysinβ)dFdt . (15) pleopod; vshed,t, radial velocity of Ma,pp in the radial direction; Frad, thr,t ⌡ ⌡ centrifugal force on the added mass of the pleopod in the radial 0 0 direction; Fd,par, drag force parallel to the pleopod during added mass The mechanical efficiency η of the pump-and-tube system shedding; ωt, time-dependent angular velocity of the pleopod pair; R, described can be found by expressing the energy required to pleopod length. accelerate the water into the tube and to overcome the tube drag as a fraction of the total energy produced by the pump: 2 2 Frad=Ma,pp×vt /(R/2)=Ma,pp×ωt R/2, where vt is the velocity of Wtube,T Ma,pp in the tangential direction with respect to the pleopod. This η = . (16) W force is the reactive force due to inertia because Ma,pp resists thr,T changes in the direction of movement (Daniel, 1984). Thus, while A pump system consisting of more than one pleopod pair can Ma,pp is accelerated in the tangential direction, it tends to move be treated as an array of single pleopod pairs. However, phase radially and is shed from the pleopod due to Frad. However, as shifts between adjacent pleopod pairs or very small distances soon as Ma,pp starts to move in the radial direction with respect between the pairs may result in direct interactions. These to the pleopod, this motion is resisted by an induced drag force interactions will influence the net thrust forces and the energy 2 parallel to the pleopod: Fd,par=gρCd,parwppRvshed,t . vshed,t is the produced as a result of masking effects. In this model, a pleopod velocity of the added mass in radial direction, with Cd,par being pair is assumed not to contribute to drag or thrust as long as it approximated by the drag coefficient of a flat plate parallel to the is touching a more rostrally situated neighbouring pair. flow (after Vogel, 1994): −0.91 Cd,par = 0.004 + 12.4Re (12) Applying the tube-pump model to Callianassa subterranea (N=7, 1

14 A PP1 12 PP2 10 PP3 8 6 4 2 0 Distance from telson (mm) 0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

100 B Fam Fthr 80 Fd

60 N) µ 40 Force ( 20

0

−20 0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

1.0 C 0.8 J)

µ 0.6 Fig. 4. The positions of the pleopod tips with respect to the telson (A), the generated thrust and 0.4 its components (B) and the cumulative mechanical ( Work 0.2 work (C) from the tube-pump model mimicking a ventilating Callianassa subterranea. To increase 0 the clarity of the figure, one beat cycle is displayed twice. PP1–PP3, pleopods 1–3; Fam, added mass force; Fthr, thrust produced by one pleopod pair; 0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Fd, drag force. Time (s) Hydrodynamic modelling of burrow ventilation flow 2177

0.035 20 100 0.0035 35 5 Flow velocity Flow velocity ) ) 18 90 1 0.030 Mechanical efficiency 1 Mechanical efficiency −

− 0.0030 30 Work 16 80 Work 4 0.025 14 70 0.0025 25 J) J)

12 60 µ 0.020 0.0020 20 3 µ 10 50 0.015 8 40 0.0015 15

Work ( 2 6 30 Work ( 0.010 0.0010 10

Flow velocity (m s 4 20 0.005 Flow velocity (m s 1 2 10 0.0005 5 Mechanical efficiency (%)

0 0 0 Mechanical efficiency (%) 0 0 0 0 1 2 3 4 5 6 7 8 9 10 0 15 30 45 60 75 90 Frequency [Hz] α – (degrees) Fig. 5. The mean flow velocity in the tube U, the mechanical Fig. 7. The model predictions of the mean flow velocity in the tube efficiency η and the work done per beat cycle W as a function of – pp,T U, the mechanical efficiency η and the work done per beat cycle the pleopod beat frequency f, predicted by the tube-pump model. Wpp,T as a function of the maximum pleopod angle swept α.

at small values of α due to decreasing βmax. At higher values The distance between adjacent pleopod pairs cannot be of α, the η curve reflects the curve of the pleopod drag changed by a pumping C. subterranea, because it is completely coefficient Cd as a function of Reynolds number due to the determined by its morphology. In juveniles growing towards higher angular velocities. The work done by the pleopods per adulthood, however, the distances between the pleopod pairs beat cycle Wpp,T increases with the square of amplitude. increase with animal size. This may affect not only the absolute Fig. 8 shows that an increase in pleopod area during the flow velocity in the tube but also the efficiency of the pump recovery stroke App,recovery results in a gradual decrease in flow system and the work per beat cycle. Fig. 10 shows the effects velocity in the tube. The value of η, however, decreases steeply of an increase in the distance between the adjacent pleopod – at relatively larger pleopod areas during the recovery stroke, pairs on U, η and Wpp,T as predicted by the model for a resulting in an increase in the work per beat cycle Wpp,T. ‘standard’ C. subterranea. All three variables increase with In the model, the added mass that is accelerated during the increasing distance between the pleopod pairs, up to power stroke is assumed to be shed when the pleopod pair is approximately 8 mm. Above 8 mm, all pleopod pairs are able half-way through the power stroke. This assumption is based to perform an almost complete power stroke and mask one on flow studies (Stamhuis and Videler, 1997b). Fig. 9 shows another only during part of the recovery stroke. At distances the effect of varying the moment of shedding during the power greater than 11 mm, the pleopods no longer interact. – stroke expressed as a fraction of T in Cshed, on U, η and Wpp,T. The number of pleopod pairs cannot be varied by C. Shedding of the added mass at Cshed=0.25 (half-way through subterranea. Some other thalassinid species, however, have the power stroke) seems to yield the highest flow velocity in four pairs of pleopods, and some tube-living invertebrates the tube and the highest mechanical efficiency. A delayed have many more paddling structures which are used for shedding, at Cshed≈0.35 results in a reduction in the work per ventilation. The implications of more pairs of pleopods has cycle, because shedding takes place at the moment when the net thrust force approaches zero. 0.0020 10 1.3 ) 1

0.0030 20 1.4 − Flow velocity 8 1.2 ) 18 0.0015 1 Mechanical efficiency 1.3 − 0.0025 Work 16 1.2 J)

Flow velocity 6 1.1 µ 0.0020 14 1.1 J) 0.0010 Mechanical efficiency

12 µ 1.0 Work 4 1.0

0.0015 10 Work ( 0.9 8

Work ( 0.0005 0.0010 6 0.8 (m s Flow velocity 2 0.9 Flow velocity (m s (m velocity Flow

0.7 Mechanical efficiency (%) 0.0005 4

2 Mechanical efficiency (%) 0.6 0 0 0.8 0 0 0.5 0 0.2 0.4 0.6 0.8 1.0 0 60 120 180 240 300 360 App,recovery/App,power Phase shift, θ (degrees) Fig. 8. The model predictions of the mean flow velocity in the tube – – Fig. 6. The mean flow velocity in the tube U, the mechanical U, the mechanical efficiency η and the work done per beat cycle efficiency η and the work done per beat cycle Wpp,T as a function of Wpp,T as a function of the pleopod area ratio App,recovery/App,power, the phase difference θ between adjacent pleopods, predicted by the where power and recovery refer to the power and recovery strokes tube-pump model. respectively. 2178 E. J. STAMHUIS AND J. J. VIDELER

0.0025 12 1.2 0.007 50 3.0 Flow velocity Flow velocity

) 360°/n Mechanical efficiency 1 0.006 Mechanical efficiency 2.5 10 1.0 − ) 0.0020 } 40 1 Work Work − 0.005

8 0.8 2.0 J) 30 µ 0.0015 J) 0.004 µ 1.5 6 0.6 0.003 20

° Work ( 0.0010 120 1.0 4 0.4 Work ( 0.002

Flow velocity (m s } 10 0.001 0.5 Flow velocity (m s 0.0005 Mechanical efficiency (%) 2 0.2 Mechanical efficiency (%) 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 Number of pleopod pairs C – shed Fig. 11. The mean flow velocity in the tube U, the mechanical Fig. 9. The model predictions of the mean flow velocity in the tube efficiency η and the work done per beat cycle W as a function of – pp,T U, the mechanical efficiency η and the work done per beat cycle the number of pleopod pairs n at a fixed phase shift θ=120 ° and at Wpp,T as a function of the added mass shedding coefficient Cshed, θ=360 °/n, predicted by the tube-pump model. which is expressed as a fraction of cycle duration. been evaluated by varying the number of pleopod pairs n from efficiency has quadrupled, at only double the cost compared 1 to 10. Since in C. subterranea the phase shift between the with the situation with θ=120 °. pleopod pairs θ=360 °/n, the phase shift was set to vary along Burrow size is determined by the inhabiting C. subterranea. with n. For comparison, n was also varied without varying θ, The effect of decreases as well as increases in burrow size are but θ was kept at the ‘standard’ C. subterranea phase shift of assumed to affect the burrow resistance. Changes in burrow 120 °. resistance have been tested by multiplying the tube drag Fd,tube With θ fixed at 120 °, the possession of more than one by a factor b between 0 and 5 (Fig. 12). Since the flow velocity pleopod pair is clearly advantageous, but the flow velocity does in the tube at standard settings (b=1) is only a fraction of the not increase as much with n when there are more than two pairs mean velocity of the pleopods, a decrease in burrow size (Fig. 11, lower set of curves). At n>2, the pump system does results in an increase in the flow velocity in the tube. The total not change very much, and little additional thrust is added with work done by the pleopods decreases and the mechanical every additional pair of pleopods, as reflected in the gradual efficiency increases dramatically with decreasing burrow size. – rates of increase of U, Wpp,T and θ. When varying θ with n, An increase in burrow size does not result in dramatic the system changes with every additional pair of pleopods. The changes, since all the variables shown are close to saturation. mechanical efficiency η increases more steeply than linearly The flow velocity and the mechanical efficiency decrease with increasing n up to approximately eight pairs of pleopods slightly and the work done increases slightly. Apparently, the (Fig. 11, upper set of curves). Above n=8, η still increases, flow velocity in the burrow and the work done by the pleopods – although its slope as well as the slopes of U and Wpp,T start to do not change very much above a critical burrow size, which decrease. Note that at, for example, n=8 and θ=360 °/n=45 °, is approximately 0.6 times the total tube length (Ltot) in our the flow velocity in the tube has doubled and the mechanical model.

0.008 50 1.2 0.0030 18 1.2 ) 1.0 1

) 16 40 − 1 0.006 − 1.1 0.0025 14 0.8 30 J) µ J) Flow velocity 1.0 µ 0.004 0.6 12 Mechanical efficiency 20

Work Work ( 0.9 0.4

0.0020 10 Work ( 0.002 Flow velocity (m s velocity Flow 10 Flow velocity (m s velocity Flow 8 0.8 0.2

Mechanical efficiency Mechanical efficiency (%) Work Mechanical efficiency (%) 0.0015 6 0.7 0 0 0 0 0.005 0.010 0.015 0 1 2 3 4 5 Distance between pleopods (m) b Fig. 12. The model predictions of the mean flow velocity in the tube – – Fig. 10. The mean flow velocity in the tube U, the mechanical U, the mechanical efficiency η and the work done per beat cycle efficiency η and the work done per beat cycle Wpp,T as a function of Wpp,T as a function of the burrow size, expressed as b, where b is a the distance between adjacent pleopod pairs, predicted by the tube- constant varying from 0 to 5 by which Fd,tube, the drag force of the pump model. tube, is multiplied. Hydrodynamic modelling of burrow ventilation flow 2179

Discussion note that the actual ratio of the pleopod areas of C. subterranea Our tube-pump model predicts a mean flow velocity in the of approximately 0.5 represents a critical value. The tube of 1.8 mm s−1 for a ventilating C. subterranea at mechanical efficiency and the flow velocity start to decrease ‘standard’ model settings. This is very close to the mean steeply around that value, and the work per beat cycle starts to velocity of 2.0±0.1 mm s−1 actually generated by C. increase drastically at this point (Fig. 8). A ratio closer to 1 subterranea in an artificial burrow with characteristics similar would therefore be disadvantageous. to the model settings (Stamhuis and Videler, 1997b). Shedding of added mass has not been incorporated Therefore, the model predictions for the mean thrust and drag quantitatively in previous models of paddling propulsion, forces, for the work done and for the mechanical efficiency are although its importance has been pointed out (Daniel, 1984, expected to be realistic. 1995; Morris et al. 1990). Added mass shedding at the The mean beat frequency during ventilation of C. sinusoidally oscillating pleopods of C. subterranea can be subterranea is approximately 1 Hz. According to the model, predicted from their limb kinematics and the resulting forces the mechanical efficiency increases steeply in the frequency on the added mass. We incorporated shedding of added mass range 0–2 Hz and then levels off. The optimum beat frequency as an instantaneous process in our model because this might therefore seems to be approximately 2 Hz. However, the work mimic the real situation closely: the added mass is accelerated per beat cycle increases with the square of frequency, so that during the first half of the power stroke and subsequently shed doubling the frequency will quadruple the work per beat cycle. with only very slight deceleration during the second half of that Sometimes the shrimp removes suspended sediment from its stroke (Stamhuis and Videler, 1997b). Variation of the moment burrow by heavy ventilation movements lasting approximately of shedding in our model suggests that shedding the added 10 s at an estimated frequency of 3–5 Hz (Stamhuis et al. mass at a Cshed (=tshed/T, where tshed is the time of added mass 1996). The work per beat cycle during such a pumping bout is shedding) of 0.25–0.30, approximately half-way through the up to 25 times the work during normal ventilation. Such an power stroke, is indeed most efficient. extreme effort will probably lead rapidly to energy depletion. In our calculations of the added mass component of the The phase shift between adjacent pleopod pairs in C. pleopod thrust, we assumed that the initial velocity of the subterranea during normal ventilation is approximately one- added mass was zero. The velocity of the pleopod at that third of the cycle. The model predicts the highest flow rates moment is zero (ω=0) and, by definition, the added mass is the and mechanical efficiencies at small phase shift angles, at only water mantle attached to the pleopod (Vogel, 1994). When the a small increase of the work per cycle (25 %). It therefore pleopod accelerates during the first part of the power stroke, seems optimal to apply an almost simultaneous beat pattern for the added mass must be accelerated as well. This process might all pleopod pairs. Such a pattern has been observed during be assisted by the flow in the lower part of the tube due to the propulsion of the copepods Pleuromamma xiphias and high local flow velocities, but this contribution is hard to Acanthocyclops robustus (Morris et al. 1985, 1990). These quantify. We did not take this into account and hence assume species apply a small phase shift between the swimming legs that the added mass forces are independent of the flow in the during the power stroke only and perform a simultaneous tube. recovery stroke. This results in maximum propulsive force for We varied the number of pleopods at different phase shifts all swimming legs during the power stroke. The disadvantage in order to compare the thrust-producing system of C. of such a system, however, is that the thrust is generated in subterranea with that of other tubiculous invertebrates and pulses, resulting in a jumpy swimming pattern. In C. with that of free-swimming paddlers. With more than three subterranea, pulsatile thrust generation would probably lead to pleopod pairs or other paddling structures, a phase shift smaller a pulsating flow in the tube, which is energetically than 120 ° becomes advantageous. With more (pairs of) limbs, disadvantageous (Stamhuis and Videler, 1997b). The the mechanical efficiency of the propelling system increases hydrodynamic resistance of a tube increases dramatically with significantly with the number of limbs at phase shifts of increasing pulsatility of the flow in the tube (Caro et al. 1978). 360 °/n. This is observed, for example, in the brine shrimp A phase shift angle of 120 °, as found in C. subterranea, might Artemia swimming with up to 11 paddling thoracic therefore not be the theoretically optimal phase shift, but most appendages. The number of appendages depends on the probably helps to keep the flow in the tube steady and can be developmental stage. The phase shift between pairs of trunk regarded as energetically advantageous. limbs is approximately one cycle divided by the number of A maximum angle swept of approximately 45 ° seems a active pairs, but minimally approximately one-ninth of a cycle good compromise between maximizing the flow velocity and (Barlow and Sleigh, 1980). Our model, although not originally minimizing the energetic costs. Differences in pleopod area developed for this type of free-swimming paddler, predicts an between the power stroke and recovery stroke are determined increase in the mechanical efficiency of trunk–limb propulsion mainly by the shrimp’s morphology, and it is not surprising with an increase in the number of trunk limbs during that a relatively smaller pleopod area during the recovery development towards adulthood. The model may, with a few stroke should increase the flow velocity and the mechanical modifications, be applicable to many other paddling , efficiency, at decreasing costs. It is, however, interesting to tubiculous or free-swimming. 2180 E. J. STAMHUIS AND J. J. VIDELER

List of symbols α0 maximum angle swept by a pleopod (degrees) A cross-sectional area of tube section (m2) β angle between pleopod and tube axis (degrees) 2 App area of pleopod pair (m ) βmax maximum angle between pleopod and tube b constant with a value of 1–5 axis (degrees) Cbody coefficient of tube area reduction due to the βt angle between pleopod and tube axis at time shrimp’s body t (degrees) Cc coefficient of tube area reduction due to the β0 angle between pleopod and tube axis at rest tail-fan constriction (degrees) Cd drag coeffient η mechanical efficiency −1 −1 Cd,fp drag coefficient of a flat plate perpendicular µ dynamic viscosity (kg m s ) to the flow ρ density (kg m−3) Cd,par drag coefficient with the object parallel to the θ phase shift angle (degrees) flow θn phase shift of the nth pair of pleopods −1 Cshed added mass shedding coefficient ω angular velocity (rad s ) d pleopod camber depth (m) ωt time-dependent angular velocity D diameter Dtube diameter of tube section (m) Subscripts ƒ frequency (Hz) 1, 2, 3 tube sections 1, 2 and 3, respectively C multiplication factor correcting for camber fcam d c narrow constriction of the tube F force (N) recovery recovery stroke F added mass force (N) am power power stroke Fd drag force (N) F drag force parallel to pleopod during added d,par We cordially thank Professor T. J. Pedley (University of mass shedding (N) Cambridge) for guidance on the fluid dynamics of tube flows, F drag force of the tube (N) d,tube especially on entrance effects and oscillatory flow. FN normal force, perpendicular to the pleopod (N) Frad centrifugal force on the added mass of the pleopod, in the radial direction (N) References Fthr thrust force produced by beating pleopods (N) ALLANSON, B. R., SKINNER, D. AND IMBERGER, J. (1992). Flow in Fthr,pump thrust force produced by the pump (N) prawn burrows. Est. Coast. Shelf Sci. 35, 253–266. Fthr,t thrust force produced by a pleopod pair at ATKINSON, R. J. A. AND NASH, R. D. M. (1990). Some preliminary time t (N) observations on the burrows of Callianassa subterranea (Montagu) L length of tube section (m) (Decapoda: Thalassinidea) from the west coast of Scotland. J. nat. Ltot total length of tube (m) Hist. 24, 403–413. Ma,pp added mass of pleopod (kg) ATKINSON, R. J. A. AND TAYLOR, A. C. (1988). Physiological ecology n number of pleopod pairs of burrowing decapods. In Aspects of Decapod Biology P power (J s−1) (ed. A. A. Fincham and P. S. Rainbow). Symp. zool. Soc. Lond. 59, r fraction of pleopod length R (m) 201–226. Oxford: Clarendon Press. R pleopod length (m) BARLOW, D. L. AND SLEIGH, M. A. (1980). The propulsion and use of water currents for swimming and feeding in larval and adult Re Reynolds number Artemia. In The Brine Shrimp, vol. 1 (ed. G. Persoone, P. t time (s) Sorgeloos, O. Roels and E. Jaspers), pp. 61–73. Wetteren, Belgium: tshed time of added mass shedding (s) Universal Press. T duration of the beat cycle, period (s) BLAKE, R. W. (1979). The mechanics of labriform locomotion. I. − U velocity (m s 1) Labriform locomotion in the angelfish (Pterophyllum eimekei), an −1 Uy velocity perpendicular to the tube axis (m s ) analysis of the power stroke. J. exp. Biol. 82, 255–271. vshed,t velocity of Ma,pp in radial direction with CARO, C. G., PEDLEY, T. J., SCHROTER, R. C. AND SEED, W. A. (1978). respect to the pleopod (m s−1) The Mechanics of the Circulation. Oxford: Oxford University vt velocity of Ma,pp in the tangential direction Press. with respect to the pleopod (m s−1) DANIEL, T. L. (1984). Unsteady aspects of aquatic locomotion. Am. Zool. 24, 121–134. wpp pleopod width (m) DANIEL, T. L. (1995). Invertebrate swimming: integrating internal and W work (J) external mechanisms. In Biological Fluid Dynamics (ed. C. P. Wpp,T work done per beat cycle (J) Ellington and T. J. Pedley), pp. 61–89. Cambridge: The Company Wthr,t work done by a pleopod pair at time t (J) of Biologists Ltd. Wtube,T total energy dissipated by the tube and water DE VAUGELAS, J. (1985). Sediment reworking by Callianassidae mud flow during one beat cycle (J) shrimp in tropical lagoons: a review with perspectives. Proc. Fifth α angle swept by a pleopod (degrees) Int. Coral Reef Congr. Tahiti 6, 617–621. Hydrodynamic modelling of burrow ventilation flow 2181

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