## SUMMARY

Biologists have treated the view that fundamental differences exist between
running, flying and swimming as evident, because the forms of locomotion and
the animals are so different: limbs and wings *vs* body undulations,
neutrally buoyant *vs* weighted bodies, etc. Here we show that all
forms of locomotion can be described by a single physics theory. The theory is
an invocation of the principle that flow systems evolve in such a way that
they destroy minimum useful energy (exergy, food). This optimization approach
delivers in surprisingly direct fashion the observed relations between speed
and body mass (*M*_{b}) raised to 1/6, and between frequency
(stride, flapping) and , and
shows why these relations hold for running, flying and swimming. Animal
locomotion is an optimized two-step intermittency: an optimal balance is
achieved between the vertical loss of useful energy (lifting the body weight,
which later drops), and the horizontal loss caused by friction against the
surrounding medium. The theory predicts additional features of animal design:
the Strouhal number constant, which holds for running as well as flying and
swimming, the proportionality between force output and mass in animal motors,
and the fact that undulating swimming and flapping flight occur only if the
body Reynolds number exceeds approximately 30. This theory, and the general
body of work known as constructal theory, together now show that animal
movement (running, flying, swimming) and fluid eddy movement (turbulent
structure) are both forms of optimized intermittent movement.

- design in nature
- animal locomotion
- optimality theory
- optimal speed
- maximum range speed
- optimal frequency
- stride frequency
- wing beat frequency
- Strouhal number
- force output
- scaling
- allometry
- turbulence
- gravitational wave
- constructal theory

## Introduction

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.

## Running

Consider the cyclical motion of running,
Fig. 1A. In order to maintain a
constant horizontal speed *V*, the animal must perform work to account
for its two mechanisms of work destruction. One is the vertical loss
*W*_{1}: the destruction of the gravitational potential energy
accumulated at the peak of each jump,
*W*_{1}=*M*_{b}*g**H*,
where *M*_{b} is body mass and *H* is vertical height
deviation during the jump, which is destroyed during each landing (for
simplicity, we neglect elastic storage during landing). The other is the
horizontal loss, *W*_{2}, which occurs because of friction
against the air, the ground, or internal friction. Internal friction is
diffuse and not readily described by theory; for that reason we present a
simplified theory in which all work to overcome friction is external. The
constructal law calls for the minimization of the total destruction of work
per distance traveled *L*:
1

Throughout this paper we use the method of scale analysis
(Bejan, 2004), which consists
of solving the appropriate conservation equations as algebraic equations, with
the additional simplification that dimensionless factors of order 1 are
neglected. A first illustration of this method is in estimating the vertical
loss term *W*_{1}/*L*, where *L*=*Vt*, and
*t* is the time scale of frictionless fall from the height of the run
(*H*), namely
*t*∼(*H g*

^{-1})

^{1/2}. Since the types of animal motion that we are considering are cyclical, with motion of body parts along a roughly circular or oblong path and re-establishment of starting positions at the beginning of each cycle, it follows that height deviations, in this case the height of the run

*H*, scales with the body length scale

*L*

_{b}=(

*M*

_{b}/ρ

_{b})

^{1/3}, where ρ

_{b}is the body density. In conclusion,

*H*∼

*L*

_{b}and the vertical loss term becomes

*W*

_{1}/

*L*=

*M*

_{b}

*g**H*/

*Vt*=

*M*

_{b}

*g**H*/

*V*(

*H*

**g**^{-1})

^{1/2}, which yields: 2

The horizontal loss *W*_{2}/*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.

Assume first that the drag is dominated by air friction. The air drag
*F*_{D} is on the order of:
3
where *C*_{D} is a numerical factor of order 1 (neglected, as
we would do for any value of the drag coefficient within an order of magnitude
of 1) and ρ_{a} is the air density. The horizontal loss of useful
energy is *W*_{2}∼*F*_{D}*L*, which
means that *W*_{2}/*L* is replaced by
*F*_{D} in Eqn 1.
The total loss per unit length traveled is:
4
This total loss function can be minimized with respect to *V* by
solving the equation d(*W*/*L*)/d*V*=0, which yields the
scaling equation
.
The result is the optimal speed:
5
where (ρ_{b}/ρ_{a})^{1/3} ≅10, becauseρ
_{b} ≅10^{3} kg m^{-3} andρ
_{a}≅1 kg m^{-3}. A compilation of velocity data
(Fig. 2A) for animals running
over a variety of terrains shows that the speeds and their trend are
anticipated well by Eqn 5. Note
that the same optimal speed as in Eqn
5 is obtained if one sets equal (in an order of magnitude sense)
the two terms appearing on the right side of
Eqn 4. In this way, we see that
to run at optimal speed is to strike a balance between the vertical loss (the
first term) and the horizontal loss (the second term). Optimal running means
optimal distribution of losses (imperfections) during locomotion. In this
regard it is noteworthy that human runners recover approximately 50% of
external kinetic energy and gravitational potential energy stored in elastic
tissues (Ker et al., 1987),
which means that about 50% is lost to friction, both internally in the tissues
and externally to the environment. Thus, our simplified theory that considers
only external friction yields an estimate of friction loss that is close to
what occurs in actual elastic systems.

Consider next the case of friction dominated by contact with the ground. If
the surface is flat and hard, we may use a Coulomb friction model and write
that the horizontal loss is *W*_{2}∼μ*Fs*, where
the coefficient of friction μ is a number of order 1, *F* is the
normal force during the foot contact time *t*_{c}, and
*s* is the foot sliding distance, *s*=*Vt*_{c}.
The contact time scale is dictated by the impact that the body experiences in
the vertical direction, such that when the body makes contact with the ground
it is decelerated from its free fall velocity (*gH*)^{1/2} to
zero. Writing Newton's second law of motion,
*F∼M*_{b}(*gH*)^{1/2}/*t*, and using
*H∼L _{b}*, we find

*W*

_{2}∼μ

*Fs*∼μ[

*M*

_{b}(

*gH*)

^{1/2}/

*t*]

*Vt*∼μ

*VM*

_{b}(

*gL*

_{b})

^{1/2}, such that Eqn 1 becomes: 6

The horizontal loss term
*W*_{2}/*L*∼μ*M*_{b}** g**
could have been evaluated more directly by recognizing

*M*

_{b}

**as the vertical force exerted by the animal body on the ground, μ**

*g**M*

_{b}

**as the horizontal friction force, andμ**

*g**M*

_{b}

*g**L*as the work destroyed by ground friction along the travel distance

*L*.

The monotonic function of *V* obtained for the total loss in
Eqn 6 indicates that there is a
lower bound for the optimal running speed. Minimum work per distance is
achieved when *V* exceeds the scale:
7
which is essentially the same as the scale predicted in
Eqn 5 and tested against animal
data (Fig. 2A).

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}∼ρ*V*_{y}^{2}*AC*_{D},
where *C*_{D} is a constant of order 1. The work done by the
foot to deform the sand vertically is *F*_{y}δ, whereδ
is the depth of the sand indentation. This work is the same as
*W*_{1}, hence
*F*_{y}δ∼*M*_{b}*gL*_{b}.

The foot also moves horizontally to the distance ξ through the sand. The
horizontal drag force is
*F*_{x}∼ρ*V*^{2}*A*^{1/2}δ*C*_{D},
where *C*_{D}∼1 and *A*^{1/2}δ is the
frontal area of the foot as it slides horizontally through the sand. The work
destroyed by horizontal deformation of the sand is
*W*_{2}∼*F*_{x}ξ, whereξ
=*Vt*_{c}, and the time of contact with the ground is
*t*_{c}∼δ/*V*_{y}. Putting these
formulae together, we find
*W*_{2}∼*V*^{3}*M*_{b}^{2}/ρ*A*^{3/2}
(*gL*_{b})^{1/2}, and
Eqn 1 becomes:
8

### Vertical loss Horizontal loss

The optimal running speed for minimal work per unit of length traveled is: 9

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
(ρ*A*^{3/2}/*M*_{b})^{1/3} is a
factor of order 1 that does not depend on *M*_{b}.

Eqn 9 shows that the optimal running speed or deformable ground is: 10 This formula, as does Eqn 5, agrees in both trend and magnitude with the numerous speed data compiled by empirical studies (Fig. 2A; note that Eqn 5 and 10 are plotted on the figure, i.e. the lines on the plot are predicted from theory and are not regression fits).

The corresponding stride frequency scale is
*t*_{opt}∼*V*_{opt}/*L*_{b},
which yields:
11
This agrees well with the observed proportionality between stride frequency
and body mass raised to –0.14 in the most rigorously defined study of
scale effects on stride frequency of runners
(Heglund et al., 1974).
Quantitatively, the predicted frequency is approximately 10 s^{-1}
*M*_{b}^{-1/6}, when *M*_{b} is
expressed in kg (Fig. 2B).

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
(ρ*A*^{3/2}/*M*_{b})^{1/3} for
deformable ground. The optimal running speed is a remarkably robust result,
always on the order of
.
We propose that it is this robustness that accounts for the allometric law
connecting *V* with and
frequency with in animals of
so many different sizes and habitats (Fig.
2A,B).

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 (*W*_{2}) was estimated in three ways in the
preceding analysis. The work dissipated by the vertical `friction' forces
(*W*_{1}) 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:
*W*_{1}. 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
(*W*_{1}+*W*_{2}).

Additional support for this running theory comes from the calculation of
the vertical force *F* that propels *M*_{b} to the
height *H* during each cycle (Fig.
1C). The force *F* acts during a short time
*t*_{1}, when the leg makes contact with the ground, and the
movement of *M*_{b} upward is governed by Newton's second law
of motion:
12
Integrating this from the initial conditions,
13
we find that the body altitude and vertical speed at the time
*t*=*t*_{1} are:
14
15

When *t* exceeds *t*_{1}, the body continues to move
upward, reaching *y*=*H* at *t*=*t*_{2},
where d*y*/d*t*=0. Integrating
Eqn 12 with *F*=0, and
satisfying the continuity conditions,
16
we find that
17
The body falls to the ground during the time
18
Next, we note that *t*_{1}<*t*_{2}, so that
in an order of magnitude sense
*t*_{2}≡*t*_{3}. Eliminating
*t*_{1} and *t*_{2} from the results derived
above, we obtain:
19
Because *y*_{1} scales with *H*, and
*H*>*y*_{1}, the final result is:
20

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).

## Flying

The proportionality between speed and
has been examined previously
using constructal theory, to predict speeds of animal and machine flight
(Bejan, 2000). That the same
proportionality rules optimal running is not a coincidence; rather it is an
illustration of the fact that a universal principle is involved. Running
requires least food when during each cycle a certain amount of work is
destroyed by vertical impact, and a certain amount by horizontal friction. The
same balancing act is responsible for optimal flight: *W*_{1}
is the work (*M*_{b}*g**H*) required to
lift the body that had fallen to the vertical distance *H* during the
cyclic time interval
*t*∼(*H*/*g*^{-1})^{1/2}.
During the same period, the work spent on overcoming drag is
*W*_{2}∼*F*_{D}*L*, where
and C_{D}∼1. Cycles in which the vertical and horizontal losses
(*W*_{1}, *W*_{2}) alternate in order to
maintain cruising at constant altitude are sketched in
Fig. 3A. The total work spent
per distance traveled is:
21
The altitude increment (*H*) achieved during each stroke of the wing is
dictated by the wing length scale, which is the length scale of the flying
body, *H∼L*_{b}. From Eqn
21 we learn that the spent work is minimal when
(Fig. 2A):
22
The wing flapping frequency that corresponds to this optimal flying speed is
*t*^{-1}∼*V*_{opt}/*L*_{b},
or:
23
which is the formula shown in Fig.
2B. The correspondence between observed wing beat frequencies and
this prediction (Fig. 2B) is
not as satisfying as that for velocities or stride frequencies of runners or
swimmers, but nonetheless the agreement is generally within one order of
magnitude. The dimensionless frequency is the Strouhal number:
24
which for optimal flight becomes a constant:
St∼(ρ_{a}/ρ_{b})^{1/3}∼10^{-1}.
This agrees with the large volume on St data on animal flight
(Taylor et al., 2003).

## Swimming

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,
*W*_{1}∼*M*_{b}*g**L*_{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*_{w}*L*_{b}
represents *W*_{1}/** g**, not that during each
cycle the fish lifts the mass

*M*

_{w}to the height

*L*

_{b}. The duration of the cycle,

*t*∼(

*L*

*g*^{-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

*W*

_{1}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}.

During the same time interval, the fish also overcomes drag by performing
the work *W*_{2}∼*F*_{D}*L*, where
*L*∼*V*_{t}∼ρ_{b}*V*_{2}*L*_{2}*C*_{D}
and *C*_{D}∼1. The fish body density ρ_{b} is
the same as the water density. In sum, the total work spent per unit travel
is:
25
The optimal swimming speed is:
26
which agrees well with empirical data for animal swimming speeds
(Fig. 2A). The optimal
undulating frequency of the body is:
27
(Fig. 2B), with the
corresponding Strouhal number, which agrees in an order of magnitude sense
with all known observations.

The net force output for travel at the speed that minimizes work per
distance traveled is 2*g**M*_{b}. The force
2*g**M*_{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 2*g**M*_{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*/*y*_{1})*g**M*_{b}.
This is why in Fig. 2C the line
*F*=2*g**M*_{b} is compared with the force
data for all forms of animal locomotion. [Note that this differs from the∼
6*g**M*_{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
*W*_{1} is the gravitational spring of the water surface that
requires a work input of size
*W*_{1}∼*M*_{b}*g**L*_{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 *W*_{1}). 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
*W*_{1} 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
*W*_{1} 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 2*L*_{b} and
horizontal speed
*V*∼(*g**L*_{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
()
and the observed swimming speeds of fish
(Drucker and Jensen, 1996) and
marine mammals (Rohr and Fish,
2004) (Fig. 2A)
provides additional support for the theory that, fundamentally, swimming is an
optimized intermittent movement in the gravitational field, like flying and
running.

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.

## Comparison of model predictions against empirical data

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 1
the mass scaling exponents estimated from regression slopes of
log_{10} transformed mass *vs* velocity, frequency, and net
force output of runners, fliers and swimmers. This table shows also the mean
log_{10} 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
(Rayner 1987). Because larger
fliers have relatively longer wings, they should have a more negative mass
scaling of wingbeat frequency than predicted by our model. The observed
scaling exponents in Table 1
fall near the predicted values of 0.167 and –0.167 for velocities and
frequencies, with the exception of wing beat frequency of fliers
(–0.26), which differs from our model in the correct direction given the
unusual scaling of avian wing dimensions (see Appendix 1). In general,
empirical scaling slopes vary because of the identity of the species sampled,
taxonomic differences in the scaling of body dimensions, variation between
studies in animal behaviour and methodology, and statistical assumptions
(Martin et al., 2005). Thus,
although we do not find statistical support across the board for the exact
scaling slopes predicted from the theory, in all cases the scaling exponents
derived from theory fit the data about as well as could be expected. Regarding
magnitude, it is not possible to statistically examine the fit between
dimensional predictions and data, other than the expectation that the data
should fall within a range that spans about 0.1 to 10-fold of the predictions.
As shown in Fig. 2 and
Table 1, that is what we find
for speeds, frequencies and force outputs. Many biologists will dismiss as
meaningless a theory that has plus or minus one order of magnitude accuracy,
but keep in mind that this theory uses only density, gravity and mass, without
any fitting constants, to make these predictions.

## Concluding remarks

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*^{-1}*L*_{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*_{b}*V*/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*_{b}*V*/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.

## Appendix

### Bodies with two length scales

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.

Consider the analysis shown for flying in Eqn
21,
22,
23, but instead of the
one-length body description (*L*_{b}), recognize that the
geometry of a large bird with its wings spread out (flapping or gliding) is
better captured by two length scales: the wing span *L*_{b},
which is horizontal, and the body or wing thickness, *Y*_{b},
which is vertical. By making this change, we are saying that the flying bird
looks more like a flying saucer (volume
) than a sphere
(volume ). The body mass scale is:
A1
where λ is the geometric shape ratio of the two-scale body:
A2

Next, we redo the analysis leading to Eqn
21. As before, for the *W*_{1}/*L* term we
use *H∼L*_{b}. To estimate *F*_{D}, we use
L*b*_{Y}*b* _{in} place of
*L*_{b}^{2}, so that the second term on the right side
of Eqn 21 readsρ
_{a}*V*^{2}*L*_{b}*Y*_{b}.
In place of Eqn 22 and
23 we find:
A3
A4

Flying bodies have smaller shape ratios (λ) when they are larger.
This is true of insects, birds and airplanes alike. The fluid mechanics
reasons for why this trend must exist deserve a study of their own. Here, we
accept empirically the notion that λ decreases as *M* increases,
and write that in a narrow enough range of large *M* values, λ
behaves as:
A5
where a>0. This means that the group (*M*/λ), which appears
in Eqns A3 and
A4, behaves as
*M*^{m}, where m>1 because m=1+a. In conclusion, the
*M* effect in Eqn A3 and
A4 is:
A.6
A.7
meaning that the modulus of the exponent of *M* should become larger
than 1/6 when *M* increases. This conclusion agrees with
*V*_{opt} data for birds (e.g. fig. 9.13 in
Bejan, 2000), which shows that
the exponent of *M* in Eqn
A6 becomes greater than 1/6 as *M* increases. The same
conclusion is in agreement with observations that wing beat frequencies are
more closely proportional to
rather than
(Table 1;
Rayner, 1987).

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
- optimal cycle frequency
- V
- velocity
*V*_{opt}- optimal velocity; maximizes distance per total work expended
*V*_{y}- vertical velocity
*W*_{1}- work expended in the vertical plane
*W*_{2}- 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

## ACKNOWLEDGEMENTS

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.

- © The Company of Biologists Limited 2006