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Transcript
A.MER. Zoou, 24:121-134 (1984) Unsteady Aspects of Aquatic Locomotion1 THOMAS L. DANIEL2 Department of Zoology, Duke University, Durham, North Carolina 27706 SYNOPSIS. Virtually all animals swim unsteadily. They oscillate appendages, undulate, and produce periodic propulsive forces so that the velocity of some part of their bodies changes in time. Because of their unsteady motion, animals experience a fluid force in addition to drag—the acceleration reaction. The acceleration reaction dominates the forces resisting rapid accelerations of animals and may be responsible for generating thrust in oscillating appendages and undulating bodies. The ever-present unsteady nature of animal swimming implies diverse applications of the acceleration reaction. INTRODUCTION a force occurs and cannot be summarily neglected. This force, called the "acceleration reaction," depends on the instantaneous acceleration, not velocity, of an object relative to its fluid environment (Batchelor, 1967). While drag is resistance to motion through a fluid, the acceleration reaction is resistance to changes in the velocity of that motion. The acceleration reaction has been critical in understanding the mechanics of aquatic animal locomotion. LighthilPs (1960, 1970) "reactive elongated-body" theories consider the propulsive force of undulating fish to arise from the acceleration reaction. Similarly, analyses of caudal and pectoral fin propulsion (Blake, 1981a, b) and specialized swimming behaviors such as turning (Weihs, 1972) and rapid starts in fish and crayfish (Webb, 1979a, 1982) are all based, in part, on the acceleration reaction. The acceleration reaction is likely to be important in low speed flight and hovering as well as in swimming mechanics. Thus, Vogel (1962) demonstrated that the frequency of the unsteady, periodic motion of Drosophila wings is influenced by the density of the air in which they beat. His results are indicative of a role for the acceleration reaction in insect flight. Ellington (1980, 1981) showed that the lift generated by the flapping wings of insects depends strongly on unsteady flow phenomena. 1 While the acceleration reaction is clearly From the Symposium on Biomechanics presented at the Annual Meeting of the American Society of relevant to animal locomotion we know Zoologists, 27-30 December 1982, at Louisville, Ken- surprisingly little about its practical behavtucky. 2 Present address: Department of Zoology, Univer- ior and effects. In this paper, I develop simple methods for dealing with the accelsity of Washington, Seattle, Washington 98250. Whenever animals move relative to a fluid environment, forces are generated. Drag and lift predominate in analyses of the mechanics of animal locomotion. In practice, drag and lift commonly refer to steady state motion and the instantaneous velocity of an animal relative to the fluid. But animals and their appendages do not often move in a steady manner. Animals that swim with oscillating fins and flippers; animals that accelerate to escape predation or attack their prey; eels, worms, snakes and fish which swim by undulations; and animals such as jellyfish and squid which swim with a pulsatile thrust force—all produce and encounter unsteady flows. In part, the almost complete lack of wheels in animals (Gray, 1968; LaBarbera, 1982) gives rise to this diverse array of undulating and reciprocating thrust producing mechanisms. In biology, steady motion appears to be more the exception than the rule. For accelerating and decelerating animals a quasi-steady analysis might yield estimates of the relevant forces. In such an analysis, steady-state forces (measured or calculated) are integrated to obtain the average lift or drag acting on an animal or its propulsive appendages. Explicitly or tacitly, a quasi-steady approach ignores any forces associated with acceleration per se. But for objects accelerating in a fluid, such 121 122 THOMAS L. DANIEL eration reaction and explore applications of the acceleration reaction in animals that swim unsteadily or move appendages in an unsteady fashion. Thus I will describe (1) the behavior and limitations of the acceleration reaction, (2) how the acceleration reaction can be used to produce thrust, and (3) how the acceleration reaction resists the motion of animals undergoing periodic propulsion. WHERE UNSTEADY FLOWS OCCUR In the strictest sense, unsteady flows arise whenever the density, viscosity or velocity of a fluid at any point in space changes with time (Streeter, 1975). Time-dependent changes in viscosity and density are likely to be rare in an aquatic environment and, therefore, need not be discusssed here. Let us consider only those cases in which the velocity of an animal or an appendage changes in time and thus generates an unsteady motion of the fluid around it. "The world is full of unsteady time-varying flows" (Vogel, 1981). Animals that swim with body undulations, reciprocating appendages, and pulsatile propulsive forces all produce and encounter unsteady flows. Indeed, animals need not locomote to encounter unsteady flows. For example, barnacles, limpets, and snails living in rocky intertidal regions experience the periodic impact of breaking waves with accelerations of more than 100 m/sec 2 (Denny et al., in preparation). Also, sea anemones are subject to subsurface wave surge (Koehl, 1977). And unsteady flows are by no means limited to aquatic environments. Vogel (1981) suggests that vortex shedding (periodic release of vortices) might aid spore release in certain mosses. Vortex shedding gives rise to flows which can induce oscillation in flexible structures, and thereby create even more unsteadiness in the flow. Still other examples of unsteady flows can be found in the ever present periodic flows within circulatory and respiratory systems. While unsteadyflowsare important in flight (Ellington, 1980, 1981) and may be important to organisms in gusty winds and surging waves or even sessile creatures shedding vortices, I will focus on unsteady flows about swimming animals. I use swimming animals to illustrate one basic aspect of unsteady flow—the acceleration reaction. T H E ACCELERATION REACTION AND ITS COEFFICIENT The theory underlying the acceleration reaction is part of a vast body of theory concerning ideal fluids. Ideal fluids have density but no viscosity and are, thus, said to be inviscid. In ideal flows drag does not exist, but the acceleration reaction does occur. In real fluids (those with some viscosity) both drag and the acceleration reaction will exist. If a body moves through a real fluid at constant velocity, drag will resist its motion. If, however, the body moves at constant velocity in an ideal fluid it will experience no net force since, in inviscid flows, the pressure distribution will be symmetric about the plane of the body perpendicular to its direction of motion (Batchelor, 1967). But the body displaces fluid and, as it moves, fluid must be moved from in front of the body to its rear, going around and not through the body. Motion of the body, therefore, induces some motion in the ideal fluid. Even though the body induces motion, it experiences no net force. This situation is known as "d'Alembert's paradox" (Batchelor, 1967): no force is required to move a body at constant velocity in an ideal fluid. But fluids are, for the most part, not ideal. They have some viscosity. Viscosity robs fluid of its momentum as it moves around a body and gives rise to pressure drag (an asymmetric fore/aft pressure distribution) and skin friction (viscous stresses at the surface of the object). Based on the above arguments one would incorrectly assume that to increase the velocity of a body in an ideal fluid would require no force other than that required to accelerate the mass of the body. But in accelerating the body we must also accelerate fluid around it. Work must be done to increase the kinetic energy of the fluid around it as well as to increase the kinetic energy of the body. In short, when an object accelerates in a fluid, a force must be applied to increase the kinetic energy of both object and fluid. The force required to increase the kinetic 123 UNSTEADY SWIMMING energy of the fluid is called the acceleration reaction. In real flows, the acceleration reaction acts in addition to drag to resist motion. The acceleration reaction (G) depends on the size, shape and acceleration of the body. To an external observer, it has the same effect as an increase in the mass of the body; thus it is defined in the following way: G = -apV(du/dt) (1) where a is the added-mass coefficient, p is the density of the fluid, V is the volume of the body and (du/dt) is its acceleration relative to the fluid far away from it. With a dimensionless, G has dimensions of force. Equation (1) may be used to define two additional terms: the added-mass and the virtual-mass of an object. The added-mass is simply the product apV, which represents the additional mass of fluid that must be accelerated in order to accelerate a body surrounded by fluid. The virtual-mass is the sum of the mass of the body and the added-mass. Size dependence of the acceleration reaction is given by the volume factor: larger bodies will experience a greater force. The sign of (du/dt) indicates the direction in which the acceleration reaction will act. When a body accelerates, (du/dt) is positive and G will act in a direction opposite to the body's motion (in the same direction as the drag force). When a body decelerates, (du/dt) will be negative and G will act in the same direction as the body's motion (in a direction opposite to that of the drag force). Hence the acceleration reaction resists both acceleration and deceleration of a body. Shape dependence of the acceleration reaction is given by the size-independent added-mass coefficient, a. For spheres, the added-mass coefficient is 0.5; for cylinders whose long axes are normal to the direction of motion, 1.0. The added-mass coefficient depends as well on the orientation of an object relative to its direction of motion. For example, an oblate spheroid ("M&M" shape) traveling in a direction parallel to its axis of rotation (moving broadside) has a higher added-mass coef- .a 2.0 - 20 30 length/diameter FIG. 1. A plot of the theoretical added-mass coefficients as a function of the length to diameter (fineness) ratio for spheroids (solid line) and elliptical cylinders (broken line). The length to diameter ratio is the ratio of the axis parallel to the direction of motion to the axis perpendicular to the direction of motion. The axis of rotation for ellipsoids is the length axis (from Daniel, 1983). Note, an infinite added-mass coefficient does not imply infinite resistance to acceleration. The reaction in the fluid is proportional to the addedmass (a-7n), m — 0 as a — oo, the limit of the product is finite and size dependent. ficient than a sphere. When traveling in a direction perpendicular to that axis (moving sideways), its added-mass coefficient is less than that for a sphere. Added-mass coefficients for a variety of spheroids and elliptical cylinders are given in Figure 1. Analytical solutions for the added-mass coefficients of spheroids, cylinders, toroids and cubes can be found in Lamb (1932), Birkhoff (1960), Kochin et al. (1964) and Batchelor (1967). For more complicated shapes, analytical solutions to the addedmass coefficients are difficult to obtain. An empirical method for evaluating the coefficient is given by Daniel (in preparation). We see, then, that the acceleration reaction differs from drag in three important aspects. (1) The acceleration reaction is directly proportional to the volume of an object while drag is more nearly proportional to its surface or cross-sectional area. (2) The acceleration reaction depends on changes in the velocity of an object while 124 THOMAS L. DANIEL drag depends on the instantaneous value of the object's velocity. The acceleration reaction resists both acceleration and deceleration while drag resists acceleration and augments deceleration. (3) The coefficient associated with drag (drag coefficient, Crf) differs from that associated with the acceleration reaction (added-mass coefficient, a). The drag coefficient is a function of the size, shape and velocity of an object and is usually presented as a function of Reynolds number (Re, a ratio of inertial to viscous forces). The added-mass coefficient depends only on shape. Also, the way in which these coefficients depend on shape is quite different. For example, the added-mass coefficient for a thin flat plate moving broadside is infinite while the drag coefficient relative to the surface area of the plate is about 1.0 (in a Re range of 10 4 -10 5 ). For a flat plate moving edge-on, the added-mass coefficient is zero (Batchelor, 1967) and the drag coefficient is 0.05 over a similar Re range (Hoerner, 1965). Note, however, that an infinite added-mass coefficient does not imply infinite resistance to acceleration. Recall that the reaction is proportional to the product of a and V. For a thin circular disc moving broadside, V is zero. The limit of the product aV is 8a 3 /3 where a is the radius of the disc (Batchelor, 1967). There are two important limitations on applying equation (1) to movement ofanimals in fluids. (1) Swimming animals rarely maintain a rigid, non-flexing shape. Swimming movements such as undulations or reciprocation of appendages result in time-varying shapes and, hence, time-varying added-mass coefficients. Therefore, changes in geometry must be considered in mechanical analyses of swimming. This same limitation also applies to analyses of drag on animals and has been a stumbling block in analyses of fish swimming (Lighthill, 1975; Wu, 1977; Daniel, 1981). (2) The acceleration reaction and its coefficient are derived from ideal flow theory which admits no vorticity (ultimately, formation of vortices in theflow).But unsteady flow near a boundary generates some vorticity which diffuses away from the boundary by the action of viscosity (Batchelor, 1967). As an object accelerates from rest, vortices are formed and subsequently shed, and the pattern of flow will thus change in time. When vortex formation and shedding occurs, one can no longer blindly trust analytical solutions for the added-mass coefficient. Vorticity, though, diffuses into the flow about an object over some finite amount of time and, for low amplitude, short duration accelerations, vorticity will not significantly affect the flow. Birkhoff (1960) suggests a rule of thumb: ideal flow theory is applicable for the first few diameters of travel. For a sphere of diameter D, we can use an added-mass coefficient of 0.5 as the sphere accelerates through a distance of about 3D. Beyond about three diameters of travel, the acceleration reaction may be quite large relative to drag but we can no longer determine its magnitude analytically. At the very least, time averaged empirical estimates of added-mass coefficients are needed to determine the unsteady forces acting on animals or appendages undergoing accelerations in motions of large amplitude. BIOLOGICAL APPLICATIONS Can a body propel itself by virtue of the acceleration reaction alone? Certainly, if an animal accelerates a limb in one direction, it will propel itself in the opposite direction. Unfortunately, the limb cannot be accelerated forever; at some point it must return to its initial position. Thus we must ask: can an animal propel itself with periodic deformations of its body? These questions are surrounded by considerable controversy. With a generalized momentum theorem, Wu (1976) lays the groundwork for analyzing cyclic averages of forces due to periodic deformations. Benjamin and Ellis (1966) and Saffman (1967) have shown that a body can propel itself with periodic deformations so long as there is some fore/aft asymmetry in the deformations. In the following, I argue that cyclic averages of the acceleration reaction are zero only under some fairly weak symmetry conditions. There is an endless list of biological situations in which the acceleration reaction UNSTEADY SWIMMING 125 will be an important force. Let us consider three broad classes of unsteady motion in swimming animals: (1) paddles, fins and flippers, (2) undulators and (3) jet propulsors. Paddles, fins and flippers A casual survey of swimming mechanisms reveals a remarkable variety of animals which use some sort of oar-like device to produce thrust. Fish tails and pectoral fins, polychaete parapodia, and hindlimbs of aquatic beetles, bugs, and frogs are a few examples of appendages used for swimming. All of these appendages reciprocate to produce thrust and their motion is, FIG. 2. A diagram of the components of the acceltherefore, unsteady. Consider two well eration reaction acting on one of the paired hindlimbs a dytiscid beetle. In panel i, the limbs are just beginstudied examples: the caudal fin of ostra- of ning to accelerate posteriorly; decelerating at the end ciiform fishes and hindlimbs of dytiscid of the stroke in panel ii with the limbs ending parallel beetles. Two points emerge: (1) the accel- to the long axis of the body. The motion is reversed eration reaction on an oscillating appen- in panels iii and iv. A positive sign by the curved arrow dage may generate thrust and (2) there indicates acceleration; negative, deceleration. exists a distinct set of kinematic parameters which maximize the thrust produced by oscillation of paired appendages. Aquatic insects. A variety of aquatic insects acceleration reaction is zero also—the beeswim by rowing hindlimbs which are tle cannot swim by either scheme. fringed with long setae (Alexander, 1968). In reality, the hindlimbs of dytiscid beeDytiscid beetles are the most extensively tles oscillate about an axis tilted approxistudied of these insect rowers (Hughes, mately 2ir/3 radians (120 degrees) relative 1958; Nachtigall, 1960, 1980; Nachtigall to the long axis of the animal (Nachtigall, and Bilo, 1975), with excellent measure- 1960). Consider only the acceleration reacments of limb and body kinematics and drag tion acting on paired hindlimbs oscillating coefficients (Nachtigall, 1960). These have about an axis behind the plane perpendicbeen used in mechanical analyses of swim- ular to the animal's motion. Two questions ming in which thrust was assumed to arise arise: (1) can the acceleration reaction genfrom the drag which resists the motion of erate thrust? (2) how do the amplitude and the hindlimbs (Alexander, 1968; Nachti- axis of oscillation affect the magnitude of gall, 1980). But, because the limbs oscillate the acceleration reaction? The first, qualback and forth, their motion is unsteady; itative, question is answered by inspecting and an analysis of thrust production which the components of force acting on a hindignores the acceleration reaction may be limb. inappropriate. Figure 2 is a diagram of the components Consider a beetle with hindlimbs that of the acceleration reaction acting on one oscillate in a horizontal plane back and of a pair of hindlimbs. As these limbs accelforth about a line in that plane (the stroke erate backwards at the beginning of a stroke axis) perpendicular to the animal's motion. (Fig. 2i), the acceleration reaction will act If the shape of the hindlimbs does not approximately normal to their surface with change, a quasi-steady analysis finds that some forward and medially directed comthe net force for a cycle of oscillation is ponents. Since paired limbs move in conzero. Moreover, because these paired limbs cert, medially directed components will oscillate symmetrically about an axis per- cancel and forward components will add. pendicular to the animal's motion, the net As the limbs decelerate at the end of the 126 THOMAS L. DANIEL Stroke angle (ir ) FIG. 3. (a) The average thrust force arising from the acceleration reaction acting on a hindlimb is plotted against the stroke angle (7,) for a variety of midstroke positional angles (72). A beat frequency of 3 Hz and limb dimensions of 1 cm in length and 0.5 cm in width were used in generating this graph. These values correspond to those presented by Nachtigall (1960) for dytiscid beetles. The equation that describes the force 0 " [-cos(. - ( * 72 where a is the width of the limb; L its length, and u the frequency of oscillation. Angles are defined in Figure 3b. / , is a Bessel function of order 1. first half cycle and accelerate at the beginning of the second half cycle (Fig. 2i, iii), the lateral components again cancel and the rearward components will add, reducing the net thrust. But if the limbs are perfectly parallel to the long axis of the body at this stage, this rearward component will be zero. Finally, at the end of the stroke, deceleration of the limbs gives rise to a forward directed component of the acceleration reaction. We see, then, the crucial role of the rearward axis of the stroke. And, as Saffman (1967, p. 389) points out, propulsion may occur only when "the shape of the body when the recoil [acceleration reaction] is sending it forward is different from the shape when recoil is sending it back." Figure 3a summarizes the effects of stroke angle and axis of rotation on the average thrust force resulting from the acceleration reaction (angles are defined in Fig. 3b). Notice that a stroke angle and axis of oscillation 2ir/3 radians maximizes the average thrust force. And, indeed, Nachtigall (1960) found that the stroke angle and axis of rotation for swimming dytiscid beetles were both 2ir/3 radians (no error indicated). His data suggest that dytiscid beetles move their appendages in a way that would maximize propulsive force if the acceleration reaction were the dominant source of thrust. The analysis should apply, as well, to many other aquatic insects. The acceleration reaction provides a mechanism by which an oscillating appendage can produce thrust. But its quantitative application to swimming in dytiscids encounters two important limitations. First, the hindlimbs of dytiscids change shape within each cycle of oscillation. The setae on these limbs expand during the power stroke (first half cycle of oscillation) and fold during the recovery stroke (second half cycle of oscillation). The effect of this asymmetry on the net drag and acceleration reaction is unknown. Second, the large amplitude of the oscillations of the hindlimbs confounds a rigorous analysis of the acceleration reaction. As stated above, large amplitude oscillations give rise to vortex shedding and, therefore, to an unknown added-mass coefficient. Even within these limitations, consideration of the unsteady nature of limb motion reveals a novel mechanism for thrust production which has rarely been considered (for an exception, see Blake, 1981a, b, c) and which is entirely consistent with the observed motions. Fish tails. Ostraciiform swimming is an extreme mode in the continuum of locomotory patterns found among fishes (Breder, 1926). This mode, named after the family Ostraciidae (boxfishes and cowfishes), is characterized by pendulum-like oscillations of a rigid tail on a rigid body (Blake, 1981a). On the assumption of steady or quasi-steady drag forces this form of motion can generate no thrust (in the absence of vorticity) (see Fig. 4a). But the motion of a wagging tail is unsteady, with its lateral velocity varying sinusoidally (Smith and Stone, 1961; Bainbridge, 1963; Blight, 1977; Blake, 1981a). Consider the application of the acceler- UNSTEADY SWIMMING 127 Quasi - steady I Unsteady FIG. 4. A diagram of the instantaneous force vectors on the caudal fin of a fish. The fin is oscillated in a symmetrical fashion about a line along the long axis of the fish, (a) A quasi-steady analysis of the forces acting on the fin which the acceleration reaction is neglected. In the left-hand panel, the tail is moving from an extreme lateral position to the midline; from the midline to the opposite position in panel ii. The motion is reversed in panels iii and iv. Notice that all components of force cancel for this model, (b) An unsteady analysis of the forces acting on the tail in which steady forces are neglected. The motion of the tail is identical to that of the above. Here, the forward components of the forces add. ation reaction to unsteady fin movements (for simplicity, neglecting drag) (Fig. 4b). As the fin accelerates from an extreme lateral position to the midline, the acceleration reaction will resist its motion. The acceleration reaction vector will, like drag, act roughly normal to the surface of the fin with components directed laterally and forward (Fig. 4bi). As the fin decelerates from the midline to the opposite extreme, the acceleration reaction will be negative—that is, it will resist deceleration of the fin, and try to keep it moving. Here, then, lateral components will cancel and forward components will add over the first half cycle of oscillation. The second half of the cycle gives an identical result, forward components again adding. Over an entire fin beat cycle, the net lateral component is zero and the forward-directed component is positive, therefore providing a net thrust. Thus neglect of the consequences of the unsteadiness of fin motion leads to the incorrect conclusion that no thrust can be produced. An unsteady model provides a possible mechanism for thrust generation by a rigid oscillated fin (see the example Smith and Stone, 1961; Blake, 1981a). In reality, some combination of quasi-steady and unsteady phenomena determine the actual thrust produced. The above analysis of caudal fin propulsion is not intended to be a description of the total thrust produced by fin oscillation; it only serves to point out a potential effect of the acceleration reaction. In reality, fish locomotion is dominated not by forces acting parallel to the direction of fluid motion (such as drag and the acceleration reaction) but by forces acting perpendicular to the direction of fluid motion—lift forces (see for example, Wu, 1971a, b, c). Caudal fins move unsteadily and the lift 128 THOMAS L. DANIEL These generalizations apply to many swimming animals including fish using pectoral fins, polychaetes using parapodia, frogs, and turtles. Reduced frequency (a-fl/U'l FIG. 5. A diagram showing the decrease in lift on a tail as a function of the reduced frequency parameter a =f-l/U where/is the frequency of lateral oscillation, / is the length of the tail, and U is the forward speed of the fish (see inset). Lo is the lift on the tail without any lateral oscillations (calculated from Lighthill, 1975, p. 96). produced by their motion is strongly affected by the degree of unsteadiness. A common measure of unsteadiness for hydro- and airfoils is called the reduced frequency parameter, a (Wu, 197la, b, c; Lighthill, 1975): a=f-l/U (2) where/is the frequency of lateral oscillation, I is the length of the tail, and U is the forward speed of the animal. When the reduced frequency parameter is greater than 1.0, the thrust produced by lift is reduced to one half of its quasi-steady value (see Fig. 5). For many fish, the reduced frequency parameter exceeds 1.0 (Yates, 1983). It may be possible for the acceleration reaction to offset any reduction in lift due to unsteady motion of fins. The interaction between lift and the acceleration reaction remains unexplored. Three generalizations may be drawn from the above examples: (1) The acceleration reaction on an oscillated appendage generates thrust in a direction parallel to the axis of oscillation for the appendage. (2) Oscillation of paired appendages generates thrust as long as their axes of rotation are tilted away from the plane perpendicular to the direction of the animal's motion. (3) A stroke angle and axis of rotation of 2TT/3 radians maximizes the thrust produced by paired appendages. Undulators Less obvious examples of unsteady flow to which the acceleration reaction might apply are found in the sinuous swimming movements of snakes, worms, eels and sharks. Early theoretical analyses of the fluid mechanics of their complex swimming patterns assumed that thrust arose from the "resistance" of water to lateral undulations (Taylor, 1952). Resistance depended on the velocity of a segment of an animal relative to the fluid around it, and thrust arose from the drag acting on the undulated portions of the body. In reality, body undulations generate an unsteady flow with segments of an animal accelerating laterally at different phases of a swimming cycle. This unsteady view of fish swimming was formalized by Lighthill (1960) as a "reactive theory" in which it was shown that accelerated segments of an undulating fish experience a reactive force that depends on the lateral acceleration, not the velocity, of each segment. This force, the acceleration reaction, provides thrust for fish which swim by undulation. The magnitude of the thrust generated by any laterally accelerated segment is directly proportional to the added-mass coefficient of that segment. Lighthill's (1971) "elongated-body theory" is an extension of his reactive theory. The elongated-body theory relies exclusively on the acceleration reaction as the source of thrust for swimming fish. It also shows that increases in the amplitude of swimming undulations augment the total thrust generated through the acceleration reaction. In Lighthill's theoretical work, the acceleration reaction is critical to fish propulsion. In addition to generating propulsive forces, the acceleration reaction determines, in part, the kinematics of swimming undulations. For example, the head and tail of a swimming shark undergo a high degree of lateral oscillation while the center of the body remains laterally stable. Weihs (1980, 1981) showed that this pat- UNSTEADY SWIMMING tern of motion is a direct result of the acceleration reaction. Viewing the body of a shark as a series of elliptical cylinders, Weihs argues that the added mass of the head and tail is low and that of the central portion of the body is comparatively high. Since the acceleration reaction is directly proportional to the added mass, reistance to lateral acceleration of the head and tail will be lower than for the central portion of the body. 129 the animal and its acceleration reaction: F = ma + ama (3) Rearranging equation (3) gives the acceleration as: a = F/m{\ + a) (4) The added-mass coefficient of G. vertens is approximately 1.0. Thus its initial acceleration should be only about 0.5 m/sec 2 which is in close agreement with measured values for acceleration of G. vertens (Daniel, Jet propulsors: Escape and swimming in preparation). Neglect of the acceleraThe acceleration reaction is also impor- tion reaction overestimates the perfortant as a force resisting the motion of an mance of an accelerating animal. animal which starts from rest or one which When escape from predation is imporswims with a periodic propulsive force. tant to the survival of an animal, we expect Cephalopod molluscs, cnidarian medusae, to see adaptations which give rise to high salps, siphonophores, and a few aquatic accelerations. Certainly maximizing the insect larvae (dragonfly) swim by a jet reac- propulsive force will maximize acceleration mechanism in which thrust results from tion during escape. Also, reductions in the the forceful expulsion of water from some virtual mass ([m + am]) will increase accelinternal cavity (Hughes, 1958; Johnson et eration during escape. Those medusae utial, 1972; Daniel, 1980, 1983; Bone and lizing escape responses are generally proTrueman, 1982). These animals use jet late with length/diameter ratios greater propulsion both for impulsive escape than 2.0 and, consequently, added-mass maneuvers and for normal swimming. In coefficients less than 0.2 (Donaldson et al., escape, achieving rapid acceleration is 1980; Daniel, in preparation). Squid and clearly critical to the survival of an animal. dragonfly larvae also use escape responses In routine swimming, the periodic nature and are also prolate with length/diameter of the jet gives rise to a discontinuous thrust ratios exceeding 3.0 (Hughes, 1958; Packforce.Their simple swimming movements ard, 1960; Ward and Wainwright, 1972). and simple shape make jet propulsors ideal Virtual mass can be reduced, as well, by cases in which to determine the effect of reducing body mass; Webb (19796) argues the acceleration reaction on locomotion. that reduced skin mass in certain fishes is Birkhoff (1960) considered the role of an adaptation for rapid acceleration. But the acceleration reaction in determining reducing the mass of a jet propelling anithe initial upward acceleration of a hydro- mal does not necessarily result in a higher gen balloon following release. Here, I adapt acceleration during escape. The propulBirkhofFs example to explore the mechan- sive force is proportional to the mass of the ics of escape swimming for jet propelling animal because it is proportional to the volanimals. For example, if an animal of mass, ume of water contained within some interm, develops a propulsive force, F, we would, nal cavity. by Newton's second law, predict its initial Routine swimming of medusae and squid acceleration to be F/m. A medusa such as is accomplished by repeated jetting of Gonionemus vertens undergoing an escape water. The intermittent nature of their response has an average mass of about thrust gives rise to an unsteady motion 0.002 kg and develops a propulsive force in which the animal undergoes repeated of about 2 mN (Daniel, 1980, 1983). Hence, accelerations and decelerations. While the we would predict its initial acceleration to cyclic average velocity of a medusa may be be about 1 m/sec 2 . But we have neglected constant, its instantaneous velocity will the acceleration reaction. The propulsive oscillate about some mean value. The aniforce must overcome both the inertia of mal constantly accelerates and decelerates 130 THOMAS L. DANIEL with the acceleration reaction changing sign. The acceleration reaction averaged over one jetting cycle will be zero for an animal swimming with a constant cyclic average velocity. Hence, the average velocity of medusae and squid is determined not by the acceleration reaction, but by the drag acting on the animal. The instantaneous force resisting the animal's motion is, however, dominated by the acceleration reaction (Daniel, 1983). Therefore, the degree to which the instantaneous velocity varies from the average is determined by the acceleration reaction, as well as drag. Mechanical analyses ofjet propulsion must consider both steady and unsteady fluid flow phenomena. RELATIVE IMPORTANCE OF THE ACCELERATION REACTION The acceleration reaction applies to many aspects of animal swimming. What remains to be shown is how important the acceleration reaction is as a force acting on oscillated appendages or on animals which swim unsteadily. Recall that in the simple case of an oscillating rigid appendage drag cancels. In such cases the acceleration reaction is the only force we need consider. And, as Lighthill (1971, 1975) has shown, the acceleration reaction overwhelms the effects of drag for animals which swim with large amplitude undulations. For animals accelerating from rest, such as medusae and squid undergoing escape maneuvers and fish predators lunging for their prey, neither the acceleration reaction nor drag cancel. For these animals, it is possible to estimate the relative importance of the acceleration reaction using a simple model which is applicable to all forms of unsteady locomotion. The ratio of the average acceleration reaction to the average total resistive force represents the relative importance of the acceleration reaction. The average acceleration reaction for an animal accelerating over some time interval (T) is: (1/T) X apV0(du I dt) dt. (5) The average total resistive force is simply the sum of the average drag and acceleration reaction: {[apV0(du/dt)} + [0.5pSCdu2]} dt. (6) The ratio of equation (5) to equation (6) gives the relative contribution of the acceleration reaction to the total resistive force: -X X apV0(du/dt) dt/ [apV0(du/dt) • 0.5pSCdu2] dt (7) which can be solved numerically using existing data for drag and added-mass coefficients. The ratio can have a value anywhere between 0 and 1.0—at 1.0, the acceleration reaction is the only force acting; at 0, only drag acts. A ratio of 0.5 indicates equal contributions of drag and the acceleration raction to the total resistive force. Values of the ratio (equation (7)) are given in Table 1 for a variety of jet propelled animals accelerating from rest. Notice that in all cases the ratio is about equal to or exceeds 0.5. This result shows that the acceleration reaction is the predominant force experienced by these animals when starting from rest. There are, however, cases in which the ratio will be small. Thus, as a sphere is accelerated (at a constant acceleration) from rest, the acceleration reaction will maintain some constant value. Drag will be initially zero and then increase as the square of the sphere's velocity (for 102 < Re < 105). Since velocity increases linearly for constant acceleration, drag will increase in proportion to the square of elapsed time. Hence, for constant accelerations the ratio will drop with time. This result is summarized in Figure 6 for a sphere starting from rest with a variety of accelerations. Notice that, the ratio depends on the mag- 131 UNSTEADY SWIMMING TABLE 1. animals. Values for the relative importance of the acceleration reaction (R, see equation (7)) for various jet propelled Animal Mass (g) Acceleration (cm/sec') Duration (sec) 3 2,000 0.08 0.48 Packard, 1969 1 700 0.08 0.65 Donaldson et al., 1980 100 0.1 0.8 Daniel, 1983 23 0.1 0.92 Bone and Trueman, 1981 Authority Squid Loligo vulgans Medusae Aglantha digitale Gonwnemus vertens 2 Salps Abylopsis tetragona 1 Dragonfly larvae Anax imperator 1 0.1 data or 0.67 Values not reported explicitly.1 Estimated300* from author's figures. Hughes, 1958 -2 I cms nitude as well as the duration of the acceleration. The ratio is highest for accelerations of short duration. CONCLUSIONS: BIOLOGICAL IMPLICATIONS The acceleration reaction is an important determinant of the mechanics of swimming in fluids. We may, in a general way, extend the arguments presented here to explore some ecological and evolutionary aspects of unsteady swimming. Predator-prey interactions Rapid linear accelerations during attack or escape and rapid angular accelerations during tight turns while an animal is pursued or in pursuit are two situations in which the acceleration reaction, and thus the shape and size of an animal will determine, in part, the success or failure of an animal in a predator-prey interaction. Where linear acceleration is important to the survival of an organism we may expect adaptations for a reduction in virtual mass. Webb (1979b) has claimed that reduced skin mass is one such adaptation among fishes. Streamlining and elongation of an animal also serve to reduce its virtual mass. We Time (s) may, therefore, expect the overall shape of an animal to be determined in part by the FIG. 6. The relative importance of the acceleration importance of rapid acceleration to its surreaction (/?, see equation (7)) is plotted against time vival. for a sphere 1 cm in radius starting from rest. Accelerations of 1, 10, 100, and 1,000 cms" 2 are shown. The ability of an animal to rapidly change 132 THOMAS L. DANIEL direction while being pursued by a predator or while in pursuit of prey should be of some selective advantage. High angular accelerations, a result of a rapid change in direction, can be achieved if the reaction in the fluid to sideslip is high. Thus high added-mass coefficients associated with motion perpendicular to the long axis of the animal would be advantageous. During a turn, an animal must also rotate and, as with linear translation, there is an addedmass coefficient associated with rotation. Where maneuverability is important to the survival of an animal we may expect resistance to impulsive rotation to be minimized and resistance to sideslip to be maximized. Dorsal fins, and lateral compression serve to augment the added-mass coefficient associated with sideslip. Anteroposterior compression serve to reduce the added-mass coefficient associated with rotation. By considering added-mass coefficients for forward swimming, Webb (1983) showed that turning radius of trout and bass scales in direct proportion to some length of a fish (a volume/area ratio). Depending on the relative magnitudes of translation and rotation during a turn, there may be other scaling factors: in particular, the added-mass coefficient for rotation, which is size dependent (see Weihs, 1972, for a detailed analysis of fish turning)Mechanical constraints Many theories for animal swimming assume that the deformation of a body or an appendage is independent of the fluid forces resulting from their motion (see for example, Lighthill, 1975; for an exception, Katz and Weihs, 1977). In reality, fins and other propulsive appendages are, to varying degrees, flexible. The amount of deformation they undergo is proportional to the load they experience. The load, in turn, depends on their motion through the fluid. For reciprocating appendages, this load is determined, in part, by the acceleration reaction. Recall, however, that the acceleration reaction depends on the shape of an appendage. Thus we encounter a circular argument: the shape of an appendage depends on the load it experiences; the load, in turn, depends on the shape. Animal swimming may, therefore, be viewed in terms of two constraints. Consider, for example, reciprocating appendages. For given structural properties of an appendage (such as stiffness in bending) there must exist constraints on its motion. Failure or large deformations may occur when the fluid forces become sufficiently large. We can expect this particularly in unsteady flows where the acceleration is large. Alternatively, for given thrust requirements, there exist upper limits on the flexibility of a propulsive appendage: if an appendage is too flexible it will not retain the appropriate shape for producing thrust. Both the structural design of an appendage and the loading history due to its unsteady motion will be important determinants of the mechanics and energetics of aquatic locomotion. ACKNOWLEDGMENTS I take great pleasure in thanking Drs. R. W. Blake, H. F. Nijhout, S. Vogel, P. Webb, T. Y. Wu and G. T. Yates for critical readings of the manuscript. I am indebted to Dr. M. Denny for making the Biomechanics symposium and my participation possible. I thank the support of the Cocos Foundation at Duke University and the Bantrell Foundation at the California Institute of Technology. REFERENCES Alexander, R. McN. 1968. Animal mechanics. Univ. Wash. Press, Seattle. Bainbridge, R. 1963. Caudal fin and body movements in the propulsion of somefish.J. Exp. Biol. 40:23-56. Batchelor, G. K. 1967. An introduction to fluid dynamics. Cambridge Univ. Press, London. Benjamin, T. B. and A. T. Ellis. 1966. Deformation of solids by impact of liquids. Phil. Trans., A 260: 221-240. Birkhoff, G. 1960. Hydrodynamics: A study in logic, fact and similitude. Princeton Univ. Press, Princeton. Blake, R. W. 1981a. Mechanics of ostraciiform propulsion. Can. J. Zool. 59:1067-1071. Blake, R. W. 19816. Influence of pectoral fin shape on thrust and drag in labriform locomotion. J. Zool. London 194:53-66. Blake, R. W. 1981c. Mechanics of drag-based mechanisms of propulsion in aquatic vertebrates. Symp. Zool. Soc. London 48:29-52. UNSTEADY SWIMMING 133 the body of Dytiscus marginalis (Dytiscidae, Blight, A. R. 1977. The muscular control of verteColeoptera). In T. Y. Wu, C. J. Brokaw, and C. brate swimming movements. Biol. Rev. 52:181Brennen (eds.), Swimming andflyingin nature, pp. 218. 585-595. Plenum Press, New York. Bone, Q. and E. R. Trueman. 1982. Jet propulsion of the calycophoran siphonophores Chelophyes and Nachtigall, W. 1980. Mechanics of swimming in water beetles. In H. Y. Elder and E. R. Trueman (eds.), Abylopsis. J. Mar. Biol. Assoc. U.K. 62:263-276. Aspects of animal movement, pp. 107—124. CamBreder, C. M. 1926. The locomotion of fishes. Zoobridge Univ. Press. London. logical 159-297. Packard, A. 1969. Jet propulsion and the giant fibre Daniel, T. L. 1980. Jet propulsion in hydrozoan response of Lohgo. Nature 221:875-877. medusae. In C. Hui (ed.), Advisory workshop on animal swimming, pp. A93-A117. ONR report Saffman, P. G. 1967. The self-propulsion of a deformable body in a perfect fluid. J. Fluid Mech. 062-653. 28:385-389. Daniel, T. L. 1981. Fish muscus: In situ measureSmith, E. H. and D. E. Stone. 1961. Perfect fluid ments of polymer drag reduction. Bio. Bull. 160: forces in fish propulsion. Proc. Roy. Soc. A 216: 376-382. 316-328. Daniel, T. L. 1983. Mechanics and energetics of Streeter, V. L. and E. B. Wylie. 1975. Fluid mechanics. medusan jet propulsion. Can. J. Zool. 61:1406McGraw-Hill, New York. 1420. Donaldson, S., G. O. Mackie, and A. O. Roberts. 1980. Taylor, G. 1952. Analysis of the swimming of long narrow animals. Proc. Roy. Soc. A 214:158-183. Preliminary observations on escape swimming and giant neurons in Aglantha digitate (Hydromedu- Vogel, S. 1962. A possible role of the boundary layer sae: Trachylina). Can. J. Zool. 58:549-552. in insect flight. Nature 193:1201-1202. Ellington, C. P. 1980. Vortices and hovering flight. Vogel, S. 1982. Life in movingfluids.Willard Grant In W. Nachtigall (ed.), Instationdre Effekte an Press, Mass. Schwingenden Tierflugeln, pp. 66-101. Akademie Ward, D. V. and S. A. Wainwright. 1972. Locoder Wiss. und der Literatur. Franz Steiner, Wiesmotory aspects of squid mantle structure. J. Zool. baden. London 167:437-449. Ellington, C. P. 1981. The aerodynamics of hovering Webb, P. W. 1979a. Mechanics of escape response animal flight. Thesis, Cambridge Univ. in crayfish (Oronectes vinhs). J. Exp. Biol. 79:245— 263. Gray, J. 1968. Animal locomotion Weidenfeld and Nicholson, London. Webb, P. W. 1979A. Reduced skin mass: An adapHughes, G. M. 1958. The co-ordination of insect tation for acceleration in some teleost fish. Can. movements. J. Exp. Biol. 35:567-583. J. Zool. 57:1570-1575. Johnson, W., P. D. Soden, and E. R. Trueman. 1972. Webb, P. W. 1982. Fast-start resistance of trout. J. A study in jet propulsion: An analysis of the Exp. Biol. 96:93-106. motion of the squid, Lohgo vulgans. J. Exp. Biol. Webb, P. W. 1983. Speed, acceleration and ma56:155-165. noeuverability of two teleost fishes. J. Exp. Biol. Katz, J. and D. Weihs. 1977. Hydrodynamic pro(In press) pulsion by large amplitude oscillation of an airfoil Weihs, D. 1972. A hydrodynamical analysis of fish with chordwiseflexibility.J. Fluid Mech. 88:485turning manoeuvres. Proc. Roy. Soc. B 182:59— 497. 72. Kochin, N. E., I. A. Kibel, and N. V. Rose. 1964. Weihs, D. 1980. A series of energy-saving mechaTheoretical hydrodynamics. John Wiley, New York. nisms in animal swimming. In C. Hui (ed.), AdviKoehl, M. A. R. 1977. Effects of sea anemones on sory workshop on animal swimming, pp. A241-A270. the flow forces they encounter. J. Exp. Biol. 69: ONR report 062-653. 87-105. Weihs, D. 1981. Body section variation in sharks— LaBarbera, M. 1982. Why the wheels won't go. Am. an adaptation for efficient swimming. Copeia Nat. 121:395-408. 1981:217-219. Lamb, H. 1932. Hydrodynamics. Dover, New York. Wu,T. Y. 1961. Swimming of a waving plate. J. Fluid Lighthill, M.J. 1960. Note on the swimming of slenMech. 10:321-344. der fish. J. Fluid Mech. 9:305-317. Wu, T. Y. 1971a. Hydromechanics of swimming proLighthill, M. J. 1970. Aquatic animal propulsion of pulsion. Part I. Swimming of a two-dimensional high hydrodynamical efficiency. J. Fluid Mech. flexible plate at variable forward speeds in an 44:265-301. inviscid fluid. J. Fluid Mech. 46:337-355. Lighthill, M. J. 1971. Large-amplitude elongatedWu, T. Y. 1971 b. Hydromechanics of swimming probody theory of fish locomotion. Proc. Roy. Soc. pulsion. Part 2. Some optimum shape problems. B 179:125-138. J. Fluid Mech. 46:521-544. Lighthill, M. J. 1975. Mathematical biofluiddynamics. SIAM, Philadelphia. Wu,T. Y. 1971c. Hydromechanics ot swimming propulsion. Part 3. Swimming and optimum moveNachtigall, W. 1960. Uber Kinematic, Dynamik und ments of slender fish with sidefins.J. Fluid Mech. Energetik des Schwimmens einheimischer Dytis46:545-568. ciden. Z. Vergl. Physiol. 43:48-180. Wu, T. Y. 1976. The momentum theorem for a Nachtigall, W. and D. Bilo. 1975. Hydrodynamics of 134 THOMAS L. DANIEL deformable body in a perfect fluid. SchifFstechnik Yates, G.T. 1983. Hydromechanics of body and cau23:226-232. dal fin propulsion. In P. W. Webb and D. Weihs Wu, T. Y. 1977. Introduction to the scaling of aquatic (eds.), Fish bwmechanics. Praeger, New York. (In animal locomotion. In T. J. Pedley (ed.), Scale press) effects in animal locomotion, pp. 203-232. Academic Press, New York.