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# Energy efficiency and allometry of movement of swimming and flying animals

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved April 8, 2014 (received for review June 3, 2013)

## Significance

Is a whale or a tuna more efficient? The answer depends on the definition of efficiency. If one uses cost of transport, defined as the energy required to move unit distance, then the tuna wins. If energy of transporting unit mass over unit distance is used then the whale wins. We address this ambiguity by rationally deriving a new efficiency measure called the energy-consumption coefficient (*C*_{E}), which is a nondimensional measure of fuel consumption. *C*_{E} is a fundamental metric to quantify efficiency of self-propelled bodies analogous to the drag coefficient (*C*_{d}) to quantify aerodynamic shapes of vehicles. The analysis also leads to allometric scalings of frequency and velocity of swimming and flying organisms over more than 20 orders of magnitude of mass.

## Abstract

Which animals use their energy better during movement? One metric to answer this question is the energy cost per unit distance per unit weight. Prior data show that this metric decreases with mass, which is considered to imply that massive animals are more efficient. Although useful, this metric also implies that two dynamically equivalent animals of different sizes will not be considered equally efficient. We resolve this longstanding issue by first determining the scaling of energy cost per unit distance traveled. The scale is found to be *M*^{2/3} or *M*^{1/2}, where *M* is the animal mass. Second, we introduce an energy-consumption coefficient (*C*_{E}) defined as energy per unit distance traveled divided by this scale. *C*_{E} is a measure of efficiency of swimming and flying, analogous to how drag coefficient quantifies aerodynamic drag on vehicles. Derivation of the energy-cost scale reveals that the assumption that undulatory swimmers spend energy to overcome drag in the direction of swimming is inappropriate. We derive allometric scalings that capture trends in data of swimming and flying animals over 10–20 orders of magnitude by mass. The energy-consumption coefficient reveals that swimmers beyond a critical mass, and most fliers are almost equally efficient as if they are dynamically equivalent; increasingly massive animals are not more efficient according to the proposed metric. Distinct allometric scalings are discovered for large and small swimmers. Flying animals are found to require relatively more energy compared with swimmers.

Cost of transport (COT), defined as the energy spent per unit distance traveled, is often used as a measure of the energy efficiency of movement. However, a dimensionless efficiency measure enables comparison across animal sizes for which the scale of COT is required. The weight of the animal could be the scale to nondimensionalize COT (1, 2), but there is no basis in mechanics to do so. Here, we derive the scaling of COT. To that end, we note that it is frequently assumed that a swimming animal spends energy to overcome drag. The assertion that an animal spends power to overcome drag lacks direct evidence in undulatory propulsion. It is well known that animals swimming at low Reynolds number spend power in producing undulatory kinematics (3⇓–5). Even at finite Reynolds numbers it is believed that swimming animals spend power to produce swimming kinematics (6). In this work we formally show that an undulatory swimmer spends power to produce the undulatory kinematics rather than to overcome drag. This result will be used to obtain the scaling of COT. This scale will be used to nondimensionalize COT, which is the new energy-consumption coefficient proposed here. Our analysis also leads to allometric scalings of different variables.

## Power Spent During Swimming

Swimming animals achieve locomotion through the undulatory kinematics of their body. The undulations of the body are predominantly in a direction lateral to the direction of swimming (7⇓⇓–10). In consequence, we hypothesize that, during steady undulatory swimming, most of the power is spent by a swimming animal to generate movement in the lateral direction *u* is the axial swimming speed. This implies*SI Appendix*, section S2 for derivations, and the sign convention is summarized at the end of this article). Because there is no net external force on a swimming body, **1**, then the hypothesis that all of the net power is spent to undulate laterally would be true. To test this we analyzed data from simulations of steady free-swimming larval zebrafish and black ghost knifefish. The simulations were based on experimentally obtained kinematics and the numerical solutions were validated with experimental data (*SI Appendix*, section S1). We computed each power term in Eq. **1** based on simulations. For the larval zebrafish, we found that

Here,

## A Comment on Froude Efficiency

The efficiency of swimming is often quantified by the Froude efficiency (11). It is defined as

## Energy-Consumption Coefficient

Metabolic and mechanical COT are two commonly used definitions of COT; the former is based on the metabolic power and the latter on the mechanical power spent by an animal. Here we focus on the mechanical COT, which is given by

We can choose the wave speed to nondimensionalize the swimming speed: *f* and λ are the frequency and wavelength of undulations, respectively. Note that using the body size or length *L* as the length scale would not fundamentally affect our results or the rationale for scaling. We choose λ because it is associated with the wave speed of undulations.

Because we have shown that the power is spent to produce lateral oscillations, the power of lateral undulations will be used to obtain the scaling of power. Additionally, it has been shown (7) that, for swimmers such as eels, trout, and bluegill sunfish, the axial oscillations during swimming are very small compared to lateral oscillations. This provides a kinematic basis to use the power of lateral undulations as the scale for power. The power scaling *a* is the amplitude of undulations, ρ is the density of the fluid, and

The scaling of COT follows from the ratio of the scaling of power spent to the scaling of speed:**2** is very similar to the equation of drag coefficient, *U* is the speed of the body relative to the fluid, and *A* is the area normal to the fluid velocity. Thus, *SI Appendix*, section S4 and Fig. S8.

Our next goal is to estimate *M*, for swimming and flying animals. First we need to derive an allometric scaling for COT in terms of the mass of the animal **2** are required. The scalings of all of the variables except frequency are known [

## Allometry of Movement

In this section we interrogate the allometric scaling of frequency, speed, and COT along with some additional variables.

The metabolic power, *f* is given by

Viscous forces dominate the inertia forces at low *SI Appendix*, section S3)*L* is a linear measure of the swimmer’s size.

Consider the metabolic power

The power produced through the swimmer’s metabolism must be at least same order as the power required to move the swimmer’s body. Equating *M* is derived as (*SI Appendix*, section S5)*a* and *b*. We give a range of exponents because the value of α ranges from 2/3 to 3/4 in the literature. We can also arrive at the frequency scaling in the inertia regime using Hill’s estimate of the muscle power **3**), which gives

Swimming speed *u* scales as

In Fig. 2*A* we compare theoretical predictions of the frequency scaling to data of swimming animals ranging from a bacterium to a sperm whale. It is seen that small animals with mass between *B*. Many microanimals use cilia for locomotion. For such animals the speed scaling we derived for undulatory propulsion may not be applicable (Fig. 2*B*).

Flying animals flap their wings normal to the direction of flight. Hence, fliers too spend their power laterally like undulatory swimmers. The frequency and velocity scalings of flying animals are derived similar to that for swimming (*SI Appendix*, section S6) to give

Next, we interrogate whether the speed of animals observed in nature is energetically optimal. For this we consider the speed of locomotion (*u*) that minimizes gross cost of transport (GCOT) (24), where GCOT is based on gross power. The gross power is the sum of the basal metabolic power and the power for locomotion (*SI Appendix*, section S7). Note that COT, defined earlier, is based only upon the power for locomotion. The optimal speed that minimizes GCOT is derived to be (*SI Appendix*, section S7):*B* and 3*B*), which suggests that animals move with speed scales that also minimize GCOT.

We also note that the Strouhal number, *SI Appendix*, section S8). In both cases where α = 2/3 and α = 3/4, as expected

The allometric scaling of COT can now be derived by substituting the mass scaling of the variables in Eq. **2** (*SI Appendix*, section S9):

The allometric scalings of COT, frequency, speed, optimal speed, and Strouhal number are summarized in Table 1. We see that the regression fit scaling is generally aligned with theoretical predictions. Specifically, the COT scaling is in excellent agreement with predictions. Additional effects may lead to deviations from predicted scalings; however, our analysis captures the primary mechanisms that cause the basic trends in a physically consistent framework.

*C*_{E} Applied to Swimming and Flying Animals

It follows from Eq. **2** that the energy-consumption coefficient *SI Appendix*, section S9). By using experimentally available values of COT we get empirical measurements of *SI Appendix*, section S9) (22) therefore,

Fig. 5 shows that swimming animals above 1 kg and almost all fliers show no meaningful trend in *M* (*SI Appendix*, section S9 for a discussion on how to covert this to a trend with respect to **7**). This implies that small-mass swimmers are also dynamically equivalent and equally efficient, but according to the viscous scaling. ^{−2} kg, this transition ends around mass of 1 kg. Flying animals are found to require relatively more energy compared with swimmers. This is presumably because flying animals spend energy to not only move forward but also to overcome gravity.

The *SI Appendix*, Fig. S9), respectively. Note that the apparent transition in *Re* for low *Re* trend to transition toward a constant value between ^{−6} and 10^{−4} kg. The dip in ^{−4} to 10^{−6} kg, is an indicator of the transition (Fig. 5). Similar to *SI Appendix*, section S9.2). We also note that undulatory or flapping swimming ends and ciliary or flagellar swimming takes over for mass less then 10^{−6} kg (Fig. 2); the *Re* corresponding to this mass is between 10 and 20, which has been shown to be the critical *Re* for transition between ciliary and flapping swimming in *Clione antarctica* (28).

In summary, *SI Appendix*, section S9). We anticipate that the concept of energy-consumption coefficient will be applicable to terrestrial locomotion, automotive vehicles, and self-propelled vehicles in general.

## Sign Convention

Swimming velocity is positive for swimmers swimming in the positive directions of the axes. Power spent by the swimmer on the fluid is positive. Forces presented in this work are the forces on the fluid by the swimmer. Thus, for a swimmer swimming along positive *x* direction drag force along *x* axis is positive.

## Acknowledgments

This work was supported by National Science Foundation (NSF) Grants CBET-0828749, CMMI-0941674, and CBET-1066575 (to N.A.P). Computational resources were provided by NSF’s TeraGrid Project Grants CTS-070056T and CTS-090006, and by Northwestern University High Performance Computing System – Quest.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: n-patankar{at}northwestern.edu.

Author contributions: R.B. and N.A.P. designed research; R.B., M.H., A.P.S.B., and N.A.P. performed research; N.A.P. led the computation method development; A.P.S.B. developed the computational tool; R.B., M.H., A.P.S.B., and N.A.P. analyzed data; and R.B. and N.A.P. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1310544111/-/DCSupplemental.

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