## Abstract

Physical limits on swimming speed of lunate tail propelled aquatic animals are proposed. A hydrodynamic analysis, applying experimental data wherever possible, is used to show that small swimmers (roughly less than a metre long) are limited by the available power, while larger swimmers at a few metres below the water surface are limited by cavitation. Depending on the caudal fin cross-section, 10–15 m s^{−1} is shown to be the maximum cavitation-free velocity for all swimmers at a shallow depth.

## 1. Introduction

In a curious example of converging evolution, all of the fastest marine swimmers have similar propulsion systems that are based on a narrow crescent caudal fin, better known as the ‘lunate tail’ (Lighthill 1969). Dolphins, tunas and mackerel sharks are obvious members of this group. Since the pioneering work of Gray (1936), the maximal speed attainable by these swimmers has been repeatedly debated (e.g. Wardle 1975; Wardle & Videler 1980), undoubtedly fuelled by reports of yellowfin tuna and wahoo swimming faster than 20 m s^{−1} (Walters & Fierstein 1964) and anecdotal reports of dolphins overtaking fast vessels. Yet undisturbed measurements of dolphin swimming (Fish & Rohr 1999) have never resulted in speeds in excess of 15 m s^{−1}, suggesting that reports of much higher speeds could have been biased by proximity to the observing vessel (Weihs 2004).

In this study, we show that a swimming speed in excess of 15 m s^{−1} is hardly possible at a shallow depth due to cavitation of the caudal fin. To this end, we examine possible limits on the maximal speed attainable by lunate tail propulsion, which result from (i) maximal power that can be produced by the swimmer's muscles, (ii) a combination of the hydrodynamic stall of the fin and the (physiological) limit on the maximal tail beat frequency, and (iii) from an onset of cavitation.

Stall, and the associated loss of hydrodynamic lift and rise of drag, is caused by boundary layer separation from the foil surface (Batchelor 1990). The boundary layer separates either due to an unfavourable pressure gradient over the leeward surface of the fin (henceforth referred as ‘standard’ stall) or due to appearance of vapour bubbles, forming whenever the surface pressure drops below the vapour pressure of the liquid in which the fin moves. The latter phenomenon is known as cavitation (Batchelor 1990). These vapour bubbles move downstream into the area of higher pressure and collapse. If the collapse occurs at the fin surface (and it depends on operating conditions), damage may occur to the fin surface. If the collapse occurs downstream of the trailing edge, it suggests a cavitation-triggered stall. The unfavourable pressure gradient increases with the increasing angle of incidence of the fin. The surface pressure drops both with the increasing angle of incidence and the swimming speed. Since a high angle of incidence is necessary to increase the speed for a given tail beat frequency, both stall and cavitation limit the thrust produced by the fin and hence set limits on the maximal swimming speed.

Prerequisites to our analysis of the speed limits are relations between the angle of incidence of the fin (or, rather, its lift coefficient), its velocity relative to the fluid, the swimming velocity and, of course, the power required to move the fin. In principle, these relations can be extracted from any of the numerous studies published during the last 40 years; the works of Lighthill (1970), Chopra (1974, 1976), Chopra & Kambe (1977) and Cheng & Murillo (1984) are pertinent examples. Nonetheless, we prefer using none of them ‘as is’. Being aimed at calculating the propulsion efficiency and the tail feathering as accurately as possible, they are too detailed for the basic analysis we perform here. Hence, we begin this study by recapitulating the analysis of lunate tail propulsion using as simple an approach as we believe practical.

## 2. Analysis

### 2.1 Balance of forces

Consider a fish of length *l* moving with a constant velocity *u* in a fluid of density *ρ* by caudal fin propulsion. The caudal fin has an area *S*_{c}; it oscillates laterally with an amplitude and a frequency *f*, moving with a lateral velocity *v*. The lift *L* and drag *D* of the caudal fin can be expressed in terms of its respective lift and drag coefficients, *C*_{L} and *C*_{D},(2.1)Likewise, the drag of the body, without the fin, can be expressed in terms of its drag coefficient, *C*_{D,b}, by(2.2)Note that all hydrodynamic coefficients defined thus far, lift and drag of the caudal fin and the drag of the body, use the caudal fin area as the reference area.

In order to simplify the following derivations, it will be assumed that the lateral velocity, *v*, as well as the lift and drag coefficients, *C*_{L} and *C*_{D}, of the caudal fin are constants, with *v* and *C*_{L} changing side each half period (figure 1). Limitations of this assumption will be discussed later on.

Now, let be the ratio of lateral-to-forward velocities. Under the present assumptions, this ratio is closely related with the stride length —the distance in body lengths travelled during one period. In fact, under present assumptions, *v*=4*hf* and, therefore,(2.3)

For constant speed swimming, the forward thrust produced by the caudal fin should balance the drag of the fish and the fin combined. Hence, after rearrangement(2.4)or, eventually,(2.5)

### 2.2 Power

The power spent by the caudal fin during the stroke is given by(2.6)Applying the longitudinal force balance (2.5), equation (2.6) can be recast as(2.7)where(2.8)is the power required to move the fish body (caudal fin excluded) at velocity *u*, and the term in parentheses can be interpreted as the reciprocal of a propulsive efficiency *η*, i.e.(2.9)

### 2.3 Drag coefficients

In order to make use of the above results, the actual values for the drag coefficients are needed. The value of coasting (stretched straight) *C*_{D,b} for scombrids can be estimated from the data presented in tables V, VI, VII, X and XI of Magnuson (1978); it turns out to be approximately 0.2.1 The value of *C*_{D,b} for delphinids and lamnids should be slightly higher owing to the drag of their fixed dorsal and pectoral fins. When strenuously swimming, the average body drag may increase (Fish & Rohr 1999; Weihs 2004). We shall avoid addressing this issue specifically by providing estimates of swimming velocity limits for different drag coefficients.

The drag coefficient of the caudal fin *C*_{D} can be estimated based on the standard parabolic relation (Nicolai 1984),(2.10)where *C*_{D,0} and *k* are constants. *C*_{D,0} is estimated to be approximately 0.01 and is known to be rather insensitive to the particular fin ‘design’ (Jacobs 1931*b*); *k* depends mainly on the fin aspect ratio (Nicolai 1984) and is bounded between 0.05 and 0.1 for all the swimmers mentioned earlier.2

### 2.4 Stall and maximal lift coefficient

Any foil can keep the flow attached to both its surfaces only up to a certain angle of attack. Above that angle, the flow on the leeward (suction) surface of the foil separates due to an unfavourable pressure gradient (with pressure increasing towards the trailing edge), causing a loss of lift and an increase in drag; this phenomenon is called ‘stall’. At high Reynolds numbers in air, the lift coefficient obtained on the verge of stall is the maximal lift coefficient, *C*_{L,max}, which can be generated by the foil. It varies between 1.0 and 1.2 (Jacobs 1931*a*,*b*) for both scombrids, which have a relatively thin caudal fin cross-section with a thickness-to-chord ratio of approximately 0.09 (F. Fish 2006, personal communication), and delphinids and lamnids, which have a much thicker caudal cross-section with thickness-to-chord ratio of up to 0.2 (Lingham-Soliar 2005).

### 2.5 Cavitation

At sufficiently high swimming velocity, the pressure due to acceleration of the flow around the leading edge may locally drop below the vapour pressure, causing vapour-filled cavities (bubbles) to appear. This phenomenon is known as cavitation (Batchelor 1990). The bubbles are carried downstream by the flow into the rear high-pressure region mentioned above, where they collapse. If the collapse occurs on the surface of the foil, it can damage the surface of the foil. If the collapse occurs downstream of the training edge, it suggests a fully separated flow regime, which can be referred to as a cavitation-induced stall. In apparent contrast with the standard pressure-gradient-induced stall, the lift coefficient obtained on the verge of cavitation-induced stall is not necessarily the maximal possible lift coefficient at that velocity. This maximal (cavitating) lift coefficient is probably almost the same as the non-cavitating *C*_{L,max}. The main difference between the two is in the associated drag coefficient, which is an order of magnitude larger in the cavitating flow.

The lift coefficient *C*_{L,c} at which cavitation appears is derived in appendix A. It is given, approximately, by(2.11)where *τ* is a parameter depending on the foil section and *u*_{d} is a parameter depending both on the foil section and on the fluid pressure at the swimming depth. For example, for a Joukowski profile (Milne-Thomson 1973) of thickness *t* and chord *c* moving at depth *d*,(2.12)(2.13)where(2.14)is the shape parameter; *p*_{0} is the atmospheric pressure; and *g* is the acceleration owing to gravity. For different cross-section shapes, the expression for *C*_{L,c} has a similar form, but with a slightly different relation between *σ* and the thickness ratio *t*/*c*. Typical values of *τ* range from 0.28 to 0.93—the former is the characteristic of the 8–9% chord thick sections of scombrids (F. Fish 2006, personal communication), and the latter is the characteristic of the 20% chord thick sections of delphinids and lamnids (Lingham-Soliar 2005). Associated values of *u*_{d}, when swimming near the sea surface, range between 22 and 15 m s^{−1}, respectively.

Equation (2.11) indicates that cavitation will precede the standard stall whenever the fin velocity exceeds , i.e. approximately 6 m s^{−1} for scombrids and 10 m s^{−1} for delphinids and lamnids near the sea surface. Cavitation becomes imminent at any lift coefficient once the fin velocity exceeds *u*_{d}. However, since the fin velocity is always greater than the swimming velocity and swimming requires a finite (non-zero) lift coefficient to produce thrust, the swimming velocity at which cavitation develops will always be lower than *u*_{d}, as, indeed, will be shown below.

## 3. Swimming velocity limits

### 3.1 Power limit

Given any swimming velocity *u*, there are an infinite number of combinations of the lift coefficient and the lateral velocity satisfying equation (2.5). For each combination, the power required to move the fish through water can be computed using equation (2.7). The power is infinite for both vanishingly small and infinitely large lift coefficients— turns infinite for the former by equation (2.5), and *C*_{D} turns infinite for the latter by equation (2.10). Hence, the power has a minimum *P*_{min}=*P*_{b}/*η*_{max} for a certain finite lift coefficient (at which the propulsion efficiency *η* reaches its maximum *η*_{max}). At the same time, there exists a (physiological) maximum *P*_{max} on the available power in the fish muscles. Combination of (hydrodynamic) *P*_{min} and (physiological) *P*_{max} yields a limit(3.1)on the maximal swimming velocity.

Lift coefficient yielding a minimum of *P* (or, equivalently, a maximum of the propulsion efficiency *η*) can be found by differentiating equation (3.1), subject to equation (2.5), with respect to *C*_{L} and equating the result to zero. With details found in appendix B, the result is shown in figure 2. Here, and are the best hydrodynamic efficiency (lift-to-drag ratio) of the caudal fin and the lift coefficient at which this efficiency is achieved. Typical values of the optimal lift coefficient for all swimmers addressed herein vary between 0.25 (low aspect fins) and 0.3 (high aspect fins) with corresponding efficiencies of 0.86–0.89. These values are on the higher side, since they exclude unsteady effects (Lighthill 1970).

The maximal available power can be always expressed as a product(3.2)of the maximal available power per unit mass of the swimmer, , and the swimmer's mass, *ρ*_{b}*V*_{b}; *ρ*_{b} and *V*_{b} being the body density and volume, respectively. The body volume for scombrids can be estimated from the data collected by Magnuson (1978); based on his table VII and a typical body shape found in his fig. 10, *V*_{b} is approximately 0.02*l*^{3} for *Thunnus*, *Euthynnu*s and *Auxis*, and approximately 0.01*l*^{3} for *Scomber*. The body density can be found in his table III, but for the sake of simplicity it can be set roughly equal to that of water.

We could not find a consensus value in the literature for the maximal available power per unit mass, . It is obviously species and conditions dependent, increasing with body temperature. In the following discussion, we have bracketed with values ranging from 10 to 160 W kg^{−1} (Azuma 1992). The resulting values of *u*_{max,P} are shown in figure 6 at the end of this paper. In the interim, we note that since caudal fin area changes, approximately, with the length of the swimmer squared, whereas the maximal available power varies, approximately, with the length of the swimmer to the third power, equation (3.1) shows that the maximal velocity due the available power limit increases with cube root of the swimmer's length. Hence, insofar as this limit is concerned, the fastest swimmers should have large-volume (high *P*_{max}) streamlined bodies (low *S*_{c}*C*_{D,b}) and high aspect ratio tails (high *η*_{max}). They should preferably have high body temperature (high ). Indeed, all are distinctive features of delphinids, scombrids and lamnids.

### 3.2 Cavitation limit

Substituting equations (2.10) and (2.11) into equation (2.5) results in the equation(3.3)for the reduced lateral tail velocity (or its reciprocal, which is proportional to the stride length) enabling swimming at velocity *u* with caudal fin on the verge of cavitation. Conversely, it can be solved to obtain the cavitation incipient swimming velocity at a given stride length(3.4)

The solution of equation (3.3) is shown in figure 3. First, it is apparent that above a certain velocity (henceforth referred as ‘*u*_{c}’), swimming cannot be sustained without cavitation. Below that threshold, equation (3.3) has two solutions, and . Cavitation can be avoided only if is kept between the two—moving the tail slower requires higher lift coefficient and hence invokes cavitation, and moving the tail faster increases the apparent flow velocity and invokes cavitation as well.

The solutions of equation (3.3) for a given *u* have no closed analytical form in the general case. Yet(3.5)(3.6)and(3.7)provide good approximations for the maximal swimming velocity, the corresponding stride length and the associated lift coefficient (figures 3 and 4). Likewise,(3.8)and(3.9)provide good approximations for the minimal lateral velocity (maximal stride length) and the corresponding lift coefficient at no-cavitation boundary (figures 3 and 4).

It was mentioned earlier (see paragraph following equation (2.14)) that cavitation will precede the standard stall only if the fin velocity is larger than . Equation (3.9) allows finding the corresponding swimming velocity *u*_{s}. In fact, setting *C*_{L,1}=*C*_{L,max} therein and solving it for *u*/*u*_{d} yields an estimate,(3.10)for the maximal swimming velocity possible with no cavitation when the fin is on the verge of stall.

The maximal possible swimming velocity with no cavitation, *u*_{c} is shown in figure 4 for representative values of the body drag coefficients; it is somewhere between 10 and 15 m s^{−1}. This velocity is insensitive to the fin planform (indeed, approximation (3.5) for *u*_{c} is independent of *k*); it increases as the fin area increases (lower *C*_{D,b}) and it has a maximum for thickness-to-chord ratio of 0.06–0.1. Perhaps a coincidence, but these values are characteristic for scombrids.

The maximal swimming velocity with no cavitation at *C*_{L,max}, *u*_{s}, is also shown in figure 4. It almost equals *u*_{c} for delphinids and lamnids (having a thickness ratio of 0.2), but it is less than half of *u*_{c} for scombrids. In other words, delphinids can swim on the verge of stall increasing speed by increasing the tail beat frequency up until cavitation appears. There remains very little to gain in the maximal speed by reducing the lift coefficient and increasing the beat frequency. Scombrids can accelerate on the verge of stall only up to a relatively small velocity—almost one-third of their maximal cavitation-free velocity. The latter can be reached only by significantly reducing the lift coefficient. It can be achieved only through excessive (when compared with delphinids) flexibility of the tail joint.

### 3.3 Maximal tail beat frequency limit

Given the lift coefficient of the fin, *C*_{L}, equation (2.5), subject to equation (2.10), can be solved to yield the reduced lateral velocity of the tail, , required to sustain swimming velocity at that lift coefficient. Although this solution cannot be expressed in a closed analytical form in the general case, for all practical values of lift and drag coefficients,(3.11)provides a very good approximation (figure 5*a*). The associated stride length, , is shown in figure 5*b*.

At the same time, there exists a (physiological) maximum *f*_{max} on the possible tail beat frequency, and hence there exists a constraint on the maximum possible lateral tail velocity. Combining the (hydrodynamic) requirement of and (physiological) limit *v*_{max} of *v* yields a limit(3.12)on the maximal swimming velocity at *C*_{L}. Obviously, a prerequisite of reaching the cavitation limit addressed in §3.2 is the ability of the swimmer to move its tail fast enough. Formally, it is required that *v*_{max} should exceed .

Equation (3.11) implies that tends to infinity as *C*_{L} tends to either zero or infinity. Hence, has a minimum ; it is shown in appendix C that, in the non-cavitating flow regime, this minimum is obtained at the highest possible value of the lift coefficient, i.e. at *C*_{L,max} or *C*_{L,1}, whichever is smaller.

We could not find a consensus value in the literature for the maximal beat frequency; hence, we shall avoid substituting any numbers. The trends however are of interest. Assuming for a moment that a fish consists of an elastic material having an effective Young modulus *E*_{b} (e.g. Collinsworth *et al*. 2002) and density *ρ*_{b}, let be the longitudinal wave propagation velocity (e.g. Graff 1975). Response time of an elastic body to an impulse should be proportional to the ratio of the characteristic length to the propagation velocity, i.e. the ratio *l*/*a*.

The contraction of a muscle is triggered by Ca^{++} ion concentration (Johnston 1983; Vander *et al*. 1985). Hence, there is a time delay between the arrival of the nerve signal (action potential) and the beginning of the muscle motion. Combining the (physiological) time delay with the (elastic) response time suggests that the maximal beat frequency can be approximated by the ratio(3.13)where *l*_{0} is a certain constant associated with the time delay.

A ‘real’ muscle is composed of cells that are approximately of the same size for small and large fishes alike. Hence, the propagation velocity (directly dependent on the cells-averaged value of the effective Young modulus) should be insensitive to the body length. At the same time, both Ca^{++} ion concentration and the propagation velocity are governed by a series of enzymatic reactions (Vander *et al*. 1985). The reaction rates increase with temperature, increasing *a* and decreasing *l*_{0}, and hence increasing *f*_{max}.

Combining equation (3.13) with equation (3.12), it appears that, insofar as the maximal beat frequency limit on the swimming velocity is concerned, the fastest swimmers should be large and warm bodied. Yet, for very large swimmers (for which propagation time is large when compared with the time delay) the maximal beat frequency limit on the swimming velocity (whether associated with the onset of cavitation or with the onset of stall) turns out to be independent of the body length.

## 4. Discussion

Power and cavitation limits on the swimming velocity have been combined in figure 6. Maximum tail beat frequency limit has not been shown owing to the uncertainty in the particular value of that frequency.

Since the maximal power-limited velocity increases, roughly, with the cube root of the body length, and since both cavitation and tail beat frequency limits (for large fishes) are independent of the body length, all fishes are power limited when small and either cavitation or tail beat frequency limited when large. The particular body length at which the available power limit is no longer the most severe constraint is conditions (body temperature and depth) dependent. Large swimmers at depth may have their top speed limited by the combination of the standard stall of the caudal fin and the maximal tail beat frequency; large swimmers near the water surface may have their top speed limited by cavitation.

In fact, cavitation poses a real limit on warm-bodied large swimmers at shallow depth, with 10–15 m s^{−1} being the maximal cavitation-free velocity. Above that speed cavitation is imminent. Lacking pain receptors on their caudal fins, scombrids may temporarily cross the cavitation limit, and cavitation-induced damage has been observed (Kishinouye 1923); on the other hand, delphinids probably cannot cross it without pain (Lang 1966).

We have tacitly avoided unsteady hydrodynamic effects and assumed that the caudal fin alignment—and possibly flex—is adjusted so as to provide constant lift coefficient during the beat cycle. For a given lift coefficient, chord-wise flexibility (Katz & Weihs 1978) will increase the leading edge suction causing cavitation at lower swimming speeds. Unsteady effects will increase the drag coefficient of the tail, but since its drag is normally small when compared with that of the fish body, it will only have a small effect on the velocity limits as discussed above.

## Footnotes

↵Based on table V, the body drag coefficient is estimated to be approximately 0.005 when referred to the body surface area. Assuming similarity in body shape among the scombrids (table VII), their surface area can be approximated as 0.45

*l*^{2}(indeed, a 44 cm long skipjack tuna has a surface area of 840 cm^{2}(table VI), whereas 40 cm long Kawakawa has the surface area of 720 cm^{2}(table XI). At the same time, caudal fin area, for most scombrids, is approximately 0.011*l*^{2}(table X). Hence, the representative value of*C*_{D,b}is approximately 0.2—0.005 times the body area divided by the caudal fin area.↵For symmetrical cross-sections at the pertinent range of Reynolds numbers,

*C*_{D,0}is typically found between 0.008 and 0.012 (Jacobs 1931*b*). The value of*k*can be estimated using semi-empirical formula*k*=1/(*πAe*)+*δ*, relating it with the aspect ratio*A*of the fin, span-wise loading correction constant*e*, typically approximately 0.9, and profile parasite drag rise constant*δ*, typically approximately 0.01 (Jacobs 1931*b*; Nicolai 1984). Since 4<*A*<8 for most scombrids (Magnuson 1978), therefore 0.05<*k*<0.1.- Received April 8, 2007.
- Accepted May 24, 2007.

- © 2007 The Royal Society