# Efficient cruising for swimming and flying animals is dictated by fluid drag

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Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved May 29, 2018 (received for review April 05, 2018)

## Significance

Almost 30 y ago, researchers discovered that a great variety of efficient swimmers cruise in a narrow range of Strouhal numbers, a dimensionless number describing the kinematics of swimming. Almost 15 y later, separate researchers discovered that fliers (bats, birds, and insects) also cruise in the same narrow range of Strouhal numbers. Attendant experiments on flapping airfoils have shown that this narrow range of Strouhal numbers gives rise to the most efficient kinematics. Here, we explain why this range of Strouhal numbers is the most efficient.

## Abstract

Many swimming and flying animals are observed to cruise in a narrow range of Strouhal numbers, where the Strouhal number

Swimming and flying animals across many species and scales cruise in a relatively narrow range of Strouhal numbers

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

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

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 (

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).

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

We now consider the performance (thrust, power, and efficiency) of an isolated propulsor. For the net thrust T, we use the scaling*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 **2** shows that this conclusion implicitly assumes that

For the power expended, we adopt the scaling*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 as

Hence, we arrive at

We see immediately that to achieve high efficiency, the dimensionless amplitude

What about the optimal Strouhal number? When there is no offset drag (**6** gives negative efficiencies at low **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

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 **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

## Materials and Methods

The experimental setup is the same as described by Van Buren et al. (4). Experiments on a heaving and pitching airfoil were conducted in a water tunnel with a

Heaving motions were generated by a linear actuator (Linmot PS01-23 × 80F-HP-R), pitching motions about the leading edge were generated by a servo motor (Hitec HS-8370TH), and both were measured by encoders. The heaving and pitching motions were sinusoidal, as described in *SI Appendix*, Eqs. **S1** and **S2**, with frequencies

The forces and moments imparted by the water on the airfoil were measured by a six-component sensor (ATI Mini40) at a sampling rate of 100 Hz. The force and torque resolutions were

## Acknowledgments

This work was supported by Office of Naval Research Grant N00014-14-1-0533 (program manager, Robert Brizzolara).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: dfloryan{at}princeton.edu.

Author contributions: D.F. and T.V.B. designed research; D.F. and T.V.B. performed research; D.F. and T.V.B. analyzed data; and D.F. and A.J.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

See Commentary on page 8063.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1805941115/-/DCSupplemental.

Published under the PNAS license.

## References

- ↵
- ↵
- ↵
- Floryan D,
- Van Buren T,
- Rowley CW,
- Smits AJ

- ↵
- ↵
- ↵
- ↵
- ↵
- Wang ZJ

- ↵
- Saadat M,
- Fish FE,
- Domel AG,
- Di Santo V,
- Lauder GV,
- Haj-Hariri H

- ↵
- ↵
- Garrick IE

- ↵
- Lighthill MJ

*Proc R Soc B*179 125–138. - ↵
- Bainbridge R

- ↵
- ↵
- ↵
- Theodorsen T

- ↵
- Sedov LI

- ↵
- White FM

- ↵
- Alexander RM

- ↵
- Schlichting H

## References

- ↵
- ↵
- ↵
- Floryan D,
- Van Buren T,
- Rowley CW,
- Smits AJ

- ↵
- ↵
- ↵
- ↵
- ↵
- Wang ZJ

- ↵
- Saadat M,
- Fish FE,
- Domel AG,
- Di Santo V,
- Lauder GV,
- Haj-Hariri H

- ↵
- ↵
- Garrick IE

- ↵
- Lighthill MJ

*Proc R Soc B*179 125–138. - ↵
- Bainbridge R

- ↵
- ↵
- ↵
- Theodorsen T

- ↵
- Sedov LI

- ↵
- White FM

- ↵
- Alexander RM

- ↵
- Schlichting H

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