• swimming
• flight
• biolocomotion
• drag

Swimming and flying animals across many species and scales cruise in a relatively narrow range of Strouhal numbers 0.2<St<0.4 (1, 2). The Strouhal number St=2fA/U is a dimensionless parameter that relates stroke frequency f, stroke amplitude A, and forward speed U. It has been hypothesized that for animals that range widely or migrate over long distances, natural selection should favor swimming and flying motions of high propulsive efficiency, and so the kinematics, described by the Strouhal number, should be tuned for high propulsive efficiency. Indeed, the cruising range of Strouhal numbers observed in nature overlaps the range of Strouhal numbers experimentally shown to result in high propulsive efficiency for simple propulsors (1, 3, 4).

A typical efficiency curve for a simple propulsor is shown in Fig. 1. We see that at low Strouhal numbers, the efficiency rapidly rises with increasing Strouhal number, reaches a maximum, and then falls off relatively slowly with further increases in Strouhal number. Here, the propulsive efficiency η is defined as η=TU/P, where T is the mean net thrust that propels the animal forward, U is the mean forward cruising speed, and P is the mean mechanical power required to create the thrust.

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Fig. 1.

A typical efficiency curve showing efficiency η as a function of St. Data are for a heaving and pitching NACA0012 foil (5) (A/L=0.19, and heaving leads pitching by 90°).

What dictates the Strouhal number that leads to maximum efficiency? Three prevailing theories have been proposed. The first (1, 6) argues that peak efficiency occurs when the kinematics result in the maximum amplification of the shed vortices in the wake, yielding maximum thrust per unit of input energy; this phenomenon has been termed “wake resonance” (7). The second theory (8) argues that the preferred Strouhal number is connected with maximizing the angle of attack allowed, while avoiding the shedding of leading edge vortices. The third (9) holds that, for aquatic animals, the ratio of the tail beat amplitude to the body length essentially dictates the Strouhal number for cruise, since it requires a balance between thrust and drag.

Here, we offer a simple alternative explanation for the observed peak in efficiency, and we also explain the rapid rise in efficiency at low St and the more gradual decrease at high St. Our explanation highlights the important role that fluid drag plays in determining the efficiency behavior.

Consider a cruising animal, one that is moving at constant velocity. We make the assumption that the thrust is produced primarily by its propulsor (for example, caudal fin for a fish, fluke for a mammal, wing for a bird) and that the drag is composed of two parts: the drag due to its body (Db, proportional to the body surface area), and an “offset” drag due to its propulsor (Do, proportional to the propulsor frontal area projected over its range of motion). More details are given below.

This decomposition is illustrated in Fig. 2, where the thrust-producing propulsor is separated from the drag-producing body and represented by an oscillating airfoil (10). To be clear, fliers are distinct from swimmers in that fliers’ propulsors need to produce lift to combat gravity, in addition to thrust to propel themselves forward. As far as steady forward cruising is concerned, however, the physics of forward propulsion is not affected by the additional requirement of lift (10).

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Fig. 2.

Swimmers and fliers can be decomposed into thrust-producing (orange) and drag-producing (blue) parts, with the propulsor aptly represented by an oscillating airfoil.

We also simplify the motion of the propulsor to model it as a combination of heaving (amplitude H) and pitching (amplitude Θ). Biologically relevant motions are ones where the heaving and pitching motions are in phase or where the heaving motion leads the pitching motion by 90° (4). In cruise, our model requires that the thrust produced by the propulsor balances the total fluid drag experienced by the body and the propulsor.

We now consider the performance (thrust, power, and efficiency) of an isolated propulsor. For the net thrust T, we use the scalingT∼ρSpV2−Do,[1]where ρ is the density of the fluid, Sp is the area of the propulsor, and V (∼fA) is the characteristic speed of the transverse motion of the propulsor. The V2 scaling is derived in SI Appendix, where it is also shown to be representative of biologically relevant flapping motions. In addition, the scaling is supported by theory (11, 12), empirical curve fits on fish performance (13, 14), and the performance of a large group of swimming animals (15). As indicated above, we will assume that for a cruising animal the net thrust of the propulsor balances the drag of the body Db, where Db∼ρSbU2, and Sb is the surface area of the body. Hence, for a negligible offset drag,St2∼Sb/Sp.[2]Previous work has proposed that this thrust–drag balance alone yields a constant Strouhal number (15). However, Eq. 2 shows that this conclusion implicitly assumes that Do=0 and that the area ratio Sb/Sp remains constant, which will not hold across the many different species that cruise in the preferred range 0.2<St<0.4. To arrive at a more general result, we need to understand the energetics determining swimming and flying. The net thrust of the propulsor at peak efficiency then sets the cruising speed.

For the power expended, we adopt the scalingP∼ρSpfLV2−VhVθ,[3]where L is a characteristic length scale of the propulsor, and Vh and Vθ are the transverse velocity scales characteristic of the heaving and pitching motions, respectively. This scaling is derived in SI Appendix, where further details are given. It is based on established theory and analysis (11, 16, 17), and it is corroborated by a large set of experiments (4). It derives from the nonlinear interaction of the power produced by the propulsor velocity and its acceleration, an interaction that is critical to our understanding of the large-amplitude motions observed in nature.

We now consider the offset drag—that is, the drag of the propulsor in the limit of vanishing f—which scales asDo∼ρU2Spg(Θ).[4]Here, Θ is the amplitude of the pitching motion, and the function g(Θ) is positive when Θ=0 and increases with Θ (3, 4). The offset drag can be viewed as scaling with the projected frontal area of the propulsor, as in bluff body flows (18).

Hence, we arrive atη=TUP∼V2U−b1U3gfL(V2−VhVθ),[5]where the constant b1 sets the relative importance of the drag term compared with the thrust term (in general, we expect b1 to be a function of Reynolds number Re=ρLU/ν, where μ is the fluid viscosity). The efficiency can be recast in terms of the Strouhal number St=2fA/U and a dimensionless amplitude A*=A/L, so thatη∼A*St2−b1gSt31−H*Θ*.[6]The other nondimensional terms, H*=H/A and Θ*=LΘ/A, represent, respectively, the amplitudes of the heaving and pitching motions relative to the total amplitude of motion.

We see immediately that to achieve high efficiency, the dimensionless amplitude A* should be large. This observation is consistent with the argument put forth by R. M. Alexander, where he proposed that large-amplitude motions are more efficient than small-amplitude motions (19). However, there are two potential limiting factors. First, as A* becomes larger, the instantaneous angle of attack increases, dynamic stall effects may become important, and the drag model given here for Do will be invalidated. Second, animal morphology naturally sets a limit as to how large they can make A*. For efficient cruising, therefore, A* should be as large as an animal’s morphology allows, while avoiding dynamic stall at all times. Our argument is consistent with the experimental observations made by Saadat et al. (9) in what we called the third theory. The author of ref. 8 (the second theory) similarly argues for large-amplitude motions, although she argues that large-amplitude motions are connected to the optimal Strouhal number, whereas we argue that, all else fixed, the amplitude sets the total efficiency, but it does not dictate the optimal Strouhal number.

What about the optimal Strouhal number? When there is no offset drag (b1=0), the efficiency increases monotonically as St decreases, and the optimal efficiency is achieved in the limit St→0. However, in the presence of offset drag (b1≠0), the efficiency will become negative as St→0 because the drag dominates the thrust produced by the propulsor. In general, Eq. 6 gives negative efficiencies at low St, a rapid increase with St to achieve a positive peak value at St=3b1g, and a subsequent slow decrease with further increase in St as the influence of drag becomes weaker. The comparison between the form given by Eq. 6 and the data originally shown in Fig. 1 makes this clear, as displayed in Fig. 3. The offset drag is crucial in determining the low St behavior and in setting the particular St at which the peak efficiency occurs. Note that the maximum value of the efficiency is directly related to the value of the drag constant b1, which further emphasizes the critical role of the drag term in determining the efficiency behavior. The amplification of shed vortices described in the wake resonance theory (the first theory) may simply arise as a signature of the efficient production of net thrust, but this is purely speculative.

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Fig. 3.

Efficiency η as a function of St. Data are as given in Fig. 1 for a heaving and pitching NACA0012 foil (5). Solid lines are given by Eq. 6 with a fixed proportionality constant of 0.155. The drag constant, b1, is set to 0.5, 0.35, 0.23, 0.15, 0.1, and 0.05 as the colors vary from dark to light, and we have set g(Θ)=Θ. The proportionality constant and the value of b1 corresponding to the experimental data were calculated by a total least squares fit to the data.

Finally, we consider the composition of the motion—that is, the relative amounts of heaving and pitching. As shown in SI Appendix, for biologically relevant flapping motions, the denominator of Eq. 6 is minimized (and hence efficiency is maximized) when H=LΘ. In other words, optimally efficient propulsors should have heaving and pitching motions that contribute equally to the total motion. When we also take the numerator of Eq. 6 into account, we actually expect the heaving contribution to be a little larger because the offset drag is dominated by pitch. We are not aware of biological measurements that would allow us to test the optimal heaving and pitching balance, so at this point it remains a hypothesis.

We leave the reader with a final thought. We expect that the relative importance of the drag, captured by b1, will depend on the Reynolds number. Our drag model is similar to that for a bluff body, such as a sphere or cylinder, so we expect b1 will be large at small Reynolds numbers and decrease as the Reynolds number increases until it reaches about 1,000, above which the drag will be almost constant (at least for Re<2×105, although biological measurements imply that the drag may remain constant up to Re=108) (15, 20). Therefore, at low Reynolds numbers, the location of the peak efficiency will change with Reynolds number: As the Reynolds number increases, the optimal St will decrease, until b1 reaches its asymptotic value at a sufficiently high Reynolds number. Our conclusion is consistent with biological measurements (at least for swimmers), where the preferred Strouhal number appears to decrease as the Reynolds number increases, until it reaches an asymptotic value (15). This further substantiates our claim that the presence of fluid drag on the propulsor is the crucial factor in creating an efficiency peak, which dictates the cruising conditions of swimming and flying animals. In other words, energetic considerations set the kinematics of the propulsor to the most efficient one, and the net thrust of the propulsor at peak efficiency balances the drag of the body to set the cruising speed.