Unifying constructal theory for scale effects in running, swimming and flying

Running, flying and swimming occur in very different physical environments. Not surprisingly then, the mechanics of moving a body on legs that contact solid ground are vastly different from what is required to achieve weight support in air, or to move a neutrally buoyant body through liquid(Alexander, 2003). Despite these differences there are strong convergences in certain functional characteristics of runners, swimmers and fliers. The stride frequency of running vertebrates (Heglund et al.,1974; Alexander and Maloiy,1984; Heglund and Taylor,1988; Gatesy and Biewener,1991) scales with approximately the same mass exponent(M^-0.17) as swimming fish(Drucker and Jensen, 1996). Velocity of running animals (Pennycuik, 1975; Iriarte-Diaz, 2002) scales with approximately the same mass exponent (M^0.17) as the observed and theoretically predicted speed of flying birds(Pennycuick, 1968; Tucker, 1973; Lighthill, 1974; Greenewalt, 1975; Bejan, 2000). Force output of the musculoskeletal motors of runners, swimmers and fliers conforms with surprisingly little variation to a universal mass specific value of about 60 N kg^-1 (Marden and Allen,2002). What explains these consistent features of animal design?

In the absence of a theory that unifies design features across different forms of locomotion, biologists have concentrated on potentially common constraints. For example, Drucker and Jensen(1996) hypothesized that the scaling of muscle shortening velocity for maximal power output during oscillatory contraction (M^-0.17; Anderson and Johnston, 1992)might explain the common scaling of stride frequency. Numerous authors have hypothesized that scale effects in locomotion are caused by constraints related to biomechanical safety factors and the need to avoid premature structural failure (McMahon, 1973, 1975; Biewener and Taylor, 1986; Biewener, 2005; Marden, 2005), or to maintain dynamic similarity (Alexander and Jayes,1983; Alexander,2003).

Here we take the different approach of starting not with constraints but with general and presumably universal design goals that can be used to deduce principles for optimized locomotion systems. Our approach is approximate(order of magnitude accuracy) and is not intended to account for all forms of biological variation. Rather it predicts central tendencies. Furthermore, such a theory is not mutually exclusive of other hypotheses such as common constraints, because constraints have evolved within a design framework, i.e. perhaps theory can provide explanations for the nature of constraints.

The theory presented here follows from the more general constructal theory of the generation of flow structure in nature (Bejan, 1997, 2000, 2005). According to the constructal law, in order for a flow system to persist (to survive) it must morph over time (evolve) in such a way that it accomplishes the most based on the amount of power or fuel consumed. The latest reviews show that the constructal law accounts for spatial and temporal flow self-optimization and self-organization in animate and inanimate natural flow systems (Bejan, 1997, 2000; Poirier, 2003; Bejan and Lorente, 2004). Examples include river basins, lung design, turbulent structure,vascularization, snowflakes and mud cracks.

How can constructal theory be applied to the streams of mass flow called running, flying and swimming? In the same way that it has been applied to design features of inanimate flow systems such as the morphing of river basins and atmospheric circulation (Bejan, 1997, 2000), by examining how locomotion systems can minimize thermodynamic imperfections (friction, flow resistances) together, such that at the global level the animal moves the greatest distance while destroying minimum useful energy (or food, or `exergy'in contemporary thermodynamics; Bejan,1997). We show that this theoretical approach delivers in surprisingly simple and direct fashion the body-mass scaling relations for running, flying and swimming–the complete relations, the slopes and the intercepts, not just the exponents of the body mass.

There is a long and productive history of optimality models in analyses of animal locomotion (e.g. Tucker,1973; Alexander, 1996, 2003; Ruina et al., 2005), so much so that the term maximum range speed is part of the common vocabulary of the field and instantly brings to mind a U-shaped curve of cost vs speed on which there is one speed that maximizes the ratio of distance travelled to energy expended. Our approach is similar in that it predicts maximum range speeds, but differs from previous efforts by simultaneously predicting stride/stroke frequencies and net force output, while being general across different forms of locomotion. Like other optimality models, our theory does not maintain that animals must act or be designed in the predicted fashion,only that over large size ranges and diverse taxa predictable central tendencies should emerge. Ecological factors will often favour species that move in ways other than that which optimizes distance per cost, for example where energy is abundant and the risk of being captured by active predators is high. Evolutionary history and the chance nature of mutation can also restrict the range of trait variation that has been available for selection. These and other factors should act primarily to increase the variation around predicted central tendencies.

The horizontal loss W₂/L depends on what friction effect dominates the horizontal drag. Here we consider three different drag models, and show that because they are of similar dimension,the choice of friction model does not affect the predicted optimal speed significantly.

The horizontal loss term W₂/L∼μM_bgcould have been evaluated more directly by recognizing M_bg as the vertical force exerted by the animal body on the ground, μM_bg as the horizontal friction force, andμ M_bgL as the work destroyed by ground friction along the travel distance L.

Consider finally the model of a highly deformable ground surface such as sand, mud or snow of density ρ (Fig. 1B). A `high enough' speed means that the accelerated terrain material does not have time to interact by friction with and entrain its neighbouring terrain material. This model is analogous to the behaviour of a pool of fluid that is hit by a blunt body at a sufficiently high speed. In summary, we assume that the sand behaves as an inviscid liquid when it is suddenly impacted by a blunt body (the foot), Fig. 1B.

The foot contact surface is A. The foot hits the ground with the vertical Galilean velocity V_y∼(gL_b)^1/2. By analogy with drag in high-Reynolds flow, the vertical force felt by the foot during impact scales as F_y∼ρV_y²AC_D,where C_D is a constant of order 1. The work done by the foot to deform the sand vertically is F_yδ, whereδ is the depth of the sand indentation. This work is the same as W₁, hence F_yδ∼M_bgL_b.

The simplest reading of this result makes use of the rough approximation that animal bodies are geometrically similar (especially when compared over large size ranges). In this case(ρA^3/2/M_b)^1/3 is a factor of order 1 that does not depend on M_b.

In sum, the effect of the horizontal friction model is felt through a factor that is a dimensionless constant in the range 1–10, namely(ρ_b/ρ_a)^1/3 for air drag,μ ^-1 for hard ground, and(ρA^3/2/M_b)^1/3 for deformable ground. The optimal running speed is a remarkably robust result,always on the order of

The analysis is summarized by the observation that we have taken into account all the forces that the ground places on the leg and which dissipate through friction all the work done by the animal. We had to do this fully,without bias, without postulating that the forces are aligned with the leg or some other direction. The ground forces have one resultant, with two components, horizontal and vertical. The work dissipated by the horizontal component (W₂) was estimated in three ways in the preceding analysis. The work dissipated by the vertical `friction' forces(W₁) is known exactly: it is the kinetic energy stored in the body at the peak of its cycloid-shaped trajectory. We did not have to model the friction process on the vertical because we know its total effect: W₁. This feature alone cuts through a lot of would be modelling, which is not relevant to the minimization of what counts, namely the total dissipation per cycle(W₁+W₂).

In conclusion, the force produced by the leg while running at optimal speed is a multiple (of dimension 1) of the body weight. Below we show that the same theoretical force characterizes flying and swimming. Fig. 2C shows that these predictions are supported by the large volume of data on the maximal force produced by animal motors over sizes ranging from small insects to large mammals (Marden and Allen,2002).

Swimming exhibits the same body-mass scaling as running and flying, not because of a coincidence, but because swimming is thermodynamically analogous to running and flying. Swimming is another example of optimal distribution of imperfections in time, or the optimization of intermittency (cf. chapter 10 in Bejan, 2000). The analogy with flying is shown in Fig. 3.

The new aspect of the present analysis of swimming is the vertical loss, W₁∼M_bgL_b. This work is spent by the fish in order to lift above itself the body of water(M_w, the same as the fish mass because the fish and the water have nearly equal density) that it displaces during one cycle. The theory is that the product M_wL_brepresents W₁/g, not that during each cycle the fish lifts the mass M_w to the height L_b. The duration of the cycle, t∼(Lg^-1)^1/2, is the time in which the lifted water mass falls, to occupy the space just vacated by the fish. During this time, the fish (M_b) and its water-body partner(M_w) can be thought of as a `big eddy' that will dissipate W₁ in time and space, in the wake. The fish mass M_b is as much a part of the eddy as the water mass M_w.

The net force output for travel at the speed that minimizes work per distance traveled is 2gM_b. The force 2gM_b plotted in Fig. 2C is the order of magnitude of the average force exerted by the fish, which is remarkably close to the maximum force indicated by the empirical data in Fig. 2C. The average force scale 2gM_b also holds for flying (eqn 9.49 in Bejan, 2000, p. 239), and is comparable with the maximum force estimated in this article for running(H/y₁)gM_b. This is why in Fig. 2C the line F=2gM_b is compared with the force data for all forms of animal locomotion. [Note that this differs from the∼6gM_b figure for motor force output(Marden and Allen, 2002; Marden, 2005) because in the present case M_b refers to total body mass rather than motor mass, and animal motors average about 20–75% of body mass].

To put swimming in the same theory with flying and running(Fig. 2) may seem counterintuitive, because fish are neutrally buoyant and birds are not. This`intuition' has delayed the emergence of a theory that unifies swimming with the rest of locomotion. In reality, there are gravitational effects in swimming just as in flying and running. Water in front of a moving body can only be displaced upward, because water is incompressible and the lake bottom and sides are rigid. Said another way, the only conservative mechanical system(the only spring) in which the fish can store (temporarily) its stroke work W₁ is the gravitational spring of the water surface that requires a work input of size W₁∼M_bgL_b.

Elevation of the water surface has been demonstrated and used in the field of naval warfare, where certain radar systems are able to detect a moving submarine by the change in the surface water height (termed the Bernoulli hump) as it passes (unpublished US Naval Academy lecture; www.fas.org/man/dod-101/navy/docs/es310/asw_sys/asw_sys.htm)and is also evident in the data from recent studies that have examined water movement patterns around swimming fish. A two-dimensional study of water movement around the body of swimming mullet(Müller et al., 1997)shows positive pressure in front of and around the head of the fish(Fig. 4C), and suction on alternating sides of vortices that form along the fish's posterior and in its wake. Regardless of depth, this pressure around the head must raise the water surface (at a very low angle except when the body is near the surface) over a large area centered near the anterior end of the fish, and some of this raised water subsequently falls into the vortices of the wake. A three-dimensional study of the wake of fish with a homocercal (symmetrically lobed) tail(Nauen and Lauder, 2002) found that there is a measurable downward force in these wake vortices, amounting to about 10% of the thrust force, and that there is a downward force on the head,which we interpret as the reaction force to the elevated water surface(Fig. 4B).

Previous analyses of wave effects on swimming(Hertel, 1966; Webb, 1975; Webb et al., 1991; Videler, 1993; Hughes, 2004) have focused on swimming in shallow water, where there are additional drag costs from the formation of surface waves. Surface waves(Fig. 4D) are horizontal motion of water away from the high pressure and elevated water surface height above a swimming fish (the Bernoulli hump; our W₁). As depth increases, fish must still displace water upward, but the effect of that submerged wave on the water surface becomes less and less perceptible because the area over which the free surface rises in order to accommodate W₁ is very large, and net horizontal motion of the water surface and associated frictional costs become negligible. Thus, even though the cost of surface waves decreases with increasing depth, the scales of water movement in the vicinity of the fish are dictated by the fish size (mass,length), not by the depth under the free surface. The vertical work W₁ is non-negligible near the surface(Fig. 4D), where it is the cause of horizontal surface waves that impose additional swimming costs. What has not been appreciated previously is that this vertical work is non-negligible and is fundamental to the physics of swimming at all depths.

The swimming cycle sketched in Fig. 3B is identical to that of a shallow-water gravitational wave of depth L_b, wavelength 2L_b and horizontal speed V∼(gL_b)^1/2, which is also the speed of a hydraulic jump. This is not a coincidence. The fact that this wave speed is the same as the optimized swimming speed(⁠

To summarize, in order to advance horizontally by one body length, the fish lifts the equivalent of a body of water of the same size as its body, to a height equivalent to the body length. What the fish does (tail flapping) is felt by the hard bottom of the lake. The hard crust of the earth supports all the flappers and hoppers, regardless of the medium in which the particular animal moves.

So far we have only visually compared the empirical data against predictions from the theory (Fig. 2). In order to examine the fit between data and theory more rigorously, we show in Table 1the mass scaling exponents estimated from regression slopes of log₁₀ transformed mass vs velocity, frequency, and net force output of runners, fliers and swimmers. This table shows also the mean log₁₀ difference between empirical data and predicted values(Fig. 2). Together, these comparisons address how well the data fit the theory in terms of both scaling exponents and magnitude. All of our predicted mass scaling exponents are based on the assumption of geometric similarity(L=[M/ρ_b]^1/3), but small and statistically significant deviations from this assumption tend to be the rule rather than the exception throughout the literature on animal scaling. Wingspan of birds shows a particularly large divergence from geometrical similarity, scaling as

A new theory predicts, explains and organizes a body of knowledge that was growing empirically. This we have done by bringing the cruising speeds,frequencies, and force outputs of running, flying and swimming under one theory. This theory falls within the growing field of constructal theory,which has been used previously to account for form and design of inanimate flow structures (Bejan, 2000). Animal locomotion is no different than other flows, animate and inanimate:they all develop (morph, evolve) architecture in space and time(self-organization, self-optimization), so that they optimize the flow of material. In the past it made sense to describe the flapping frequencies of swimmers and flyers in terms of the Strouhal number (St). It made sense because such animals generate eddies, and because St is part of the language of turbulent fluid mechanics. After this unifying theory of locomotion, one can also talk about the Strouhal number of runners,St=t^-1L_b/V_opt,which in view of the first part of the analysis turns out to be a constant in the 0.1–10 range, just as for swimmers and flyers. The St constant is an optimization result of the theory, and it belongs to all flow systems with optimized intermittency, animate and inanimate.

All animals, regardless of their habitat (land, sea, air) mix air and water much more efficiently than in the absence of flow structure. Constructal theory has already predicted the emergence of turbulence, by showing that an eddy of length scale L_b, peripheral speed V and kinematic viscosity ν transports momentum across its body faster than laminar shear flow when the Reynolds number L_bV/ν exceeds approximately 30 (cf. chapter 7 in Bejan, 2000). This agrees very well with the zoology literature, which shows that undulating swimming and flapping flight (i.e. locomotion with eddies of size L_b) is possible only if L_bV/ν is greater than approximately 30(Childress and Dudley,2004).

And so we conclude with a promising link that this simple physics theory reveals: the generation of optimal distribution of imperfection (optimal intermittency) in running, swimming and flying is governed by the same principle as the generation of turbulent flow structure. The eddy and the animal that produces it are the optimized `construct' that travels through the medium the easiest, i.e. with least expenditure of useful energy per distance traveled.

The scale analysis presented in this paper is based on the simplest geometrical model for a body that runs, flies or swims: the body geometry is represented by one length scale, L_b∼(M_b/ρ_b)^1/3. We covered a large territory with this simple first step, and we can do more if we adopt a slightly more complex model. One reason for trying this next step is that some of the scatter (discrepancies) between the present formulae and speed and frequency data can be attributed to changes in body shape as body mass increases. Another reason is to show how the present theoretical approach can be used in future studies of more complicated living systems and processes.

In conclusion, the accuracy of the theoretical approach presented in this paper can be improved by basing it on more realistic (multi-scale) body geometries.

List of symbols and abbreviations:

 
• a

  the exponent describing how λ scales with body mass

 
• A^1/2

  frontal length scale of a foot

 
• C_D

  drag coefficient

 
• F

  force

 
• F_D

  drag force in air

 
• F_X

  drag force on foot sliding on the ground

 
• F_y

  force normal to a horizontal surface

 
• g

  gravitational acceleration

 
• H

  vertical height deviation during a cycle of locomotion

 
• L

  horizontal distance traveled during a cycle of locomotion

 
• L_b

  a characteristic length of an animal

 
• m

  a mass scaling exponent that equals 1+a

 
• M

  mass

 
• M_b

  body mass

 
• M_w

  water mass

 
• s

  foot sliding distance

 
• St

  Strouhal number

 
• t

  time

 
• t_c

  duration of foot contact with the ground during one cycle

 
• \(t_{\mathrm{opt}}^{-1}\)

  optimal cycle frequency

 
• V

  velocity

 
• V_opt

  optimal velocity; maximizes distance per total work expended

 
• V_y

  vertical velocity

 
• W₁

  work expended in the vertical plane

 
• W₂

  work expended in the horizontal plane

 
• y

  vertical distance above the ground

 
• Y_b

  characteristic body length along the dorso-ventral axis

 
• μ

  coefficient of friction

 
• δ

  depth of ground indentation during foot impact

 
• ξ

  horizontal sliding distance by a foot impacting the ground

 
• λ

  ratio of characteristic lengths that describes body shape

 
• ν

  kinematic viscosity

 
• ρa

  air density

 
• ρb

  body density

We thank the following colleagues for sharing data: J. Iriarte-Diaz, F. Fish, J. Rohr, G. Taylor. This work was supported by National Science Foundation grant CTS-0001269 to A.B. and IBN-0091040 and EF-0412651 to J.H.M. We thank the organizers of the 2004 Ascona Conference (published in J. Exp. Biol.208, part 9, 2005): we are very grateful for having been invited, because the ideas developed in this paper were formed in Ascona.