Locomotion rhythm makes power and speed

This article addresses two questions, why certain animals (frogs, breaststroke swimmers, hovering fliers, jellyfish) push rapidly against the surrounding fluid and then reach forward slowly, and whether this rhythm of propulsion is a manifestation of the universal phenomenon of design evolution in nature. Emphasis is on the distribution of time periods of locomotion in which, during the driving phase of cyclic movement (the motive stroke, phases 1 and 2, in alternating sequence with the dissipative stroke, phase 3), the work is generated (phase 1) and dissipated (phase 2). The relative lengths of the characteristic times t1 and t2 of the phases 1 and 2, are predicted. The relative duration of the proposed three phases of a cycle is the ‘rhythm’. The analysis is based on a model of how the effective cross-sections of the stroking body parts impact the surrounding medium, water, or air, and the total power required to account for the kinetic energy losses during phases 2 and 3, which are due to drag forces posed by the surrounding medium. The body configuration (limbs' cross-sections) determines the limbs' velocities that maximize mean power, and the times t1 and t2 within the motive stroke. Emphasis is placed on the freedom to change the evolving design. Freedom is represented in two ways: the number of degrees of freedom in changing the dimensions of the model and its deformation in time, and the effect that evolutionary changes have on the access that the body has to its available space. Freedom to change the locomotion design leads to greater power and speed.


Abbreviations
Limb length, m f Frequency, s -1 f Function, Eqs.(10) and (21), (m/s) Ambient density, kg/m 3 Have you noticed that frogs and breaststroke swimmers push the water back quickly, and reach forward slowly?Did you also notice that after the push & reach motion their bodies 'glide' during a time interval comparable with the first two?Why is this happening?What is the connection between this rhythm and the evolution of animal design and its movement on earth?
Locomotion (or transportation, for humans) is movement with purpose, to offer access to moving animals and people, from one location to another.Movement is against a medium-in water, on land, and in the air.Power drives movement, and all the power is dissipated as the moving body advances against the opposition posed by the medium.This is the thermodynamics of locomotion, in a nutshell.
Thermodynamics is broader than fluid dynamics 2 .Thermodynamics means heating (therme, Gr.) and power (dynamis, Gr.).Between the flows of heat and power there are macroscopic flow configurations (designs) that convert the heating into power and the movement driven by that power (called engines, animals, vehicles), or convert the power and movement into heating (called coolers, refrigerators, brakes, dissipators).The 'heating' originates from processes (combustion, metabolism) that consume measurable quantities known as food, fuel, exergy, free energy, caloric value, etc.The rate of consumption (e.g., metabolism) must not be confused with the power generated for causing movement (e.g., locomotion).Between the two there is the flow design that converts partially the first into the second.
Essential in thermodynamics and this article is the flow configuration-the design-its movement in space and time, and its evolution.Evolution means change after change in a direction discernible to the observer.
Regarding locomotion, fluid dynamics teaches the relation between the speed of the mover and the forces that the mover must overcome.Together, thermodynamics and fluid dynamics teach the relation between movement and the power required (and dissipated) to drive the movement.It is here that the human need is most evident.Because the power from fuel is not freely abundant, the motive behind most of the advances in fluid dynamics has been to decrease the power requirement when the needed speed and size of the mover are specified.The fluid dynamics of locomotion and transportation has generated a rich catalog of body configurations (designs), such as the hydrodynamic shape of fish, birds, hulls, keels, submerged wings, turbine blades and vanes, trucks, trains, automobiles, airplane fuselages, wings, and engine and wheel covers [1][2][3][4][5] .
The science of animal design benefits from physics, and transportation design finds inspiration in the animal realm [6][7][8][9] .These advances continue to attract questioning and revision.For example, the work of Hirt et al. 18 was addressed in Günter et al. 20 and Bejan et al. 6 .
Figure 1.The two-step cycle of the theory in Refs. 1,4.

Background
The universal scaling laws of locomotion in all media came from the idea 1,4,5 that for a body to move forward it must perform work and dissipate work (W y and W x , Fig. 1).The body does work (W y ) to lift its weight to a height that matches its linear dimension (length), and while falling forward it dissipates work (W x ) to displace the medium in front of it.In swimming, the lifted weight is that of the parcel of water displaced by the body as it moves forward.
Unlike in the background reviewed below, the theory presented in this article is about the rhythm of motion during W y alone, not during W y + W x .It is about what happens during the motive stroke (t y ) alone, which in Section "Rhythm" will be divided into two time intervals, doing work, t 1 , followed by dissipation, t 2 , in other words t y = t 1 + t 2 .
In the original theory 1,4,5 the moving body has one length, mass M, or weight Mg.The time interval of 'motive' time, t y , must match the time interval of work dissipation during falling forward, t x .The single time interval (t y , or t x ) is dictated by the body weight (Mg), namely (M/ρ) 1/6 g -1/2 , or (L B /g) 1/2 , where ρ is the body density and L B is the single body length, L B ~ (M/ρ) 1/3 .Conversely, the frequency of the lift and fall rhythm must be lower in bigger bodies.This is in accord with animal locomotion and competitive athletics 4,42 .
The freedom in the design is in the speed of locomotion, V, which is free to vary while decreasing the total effort (W y + W x ) made to cover the distance L x .The resulting speed is proportional to M 1/6 (ρ/ρ m ) 1/3 , where M is the body mass and ρ and ρ m are the body and medium densities.The predicted (theoretical) lines are drawn in Fig. 2 for fliers and swimmers.
In sum, bigger movers should be faster (Fig. 2).Fliers should be approximately ten times faster than swimmers of the same size, because the factor (ρ/ρ m ) 1/3 in water is 1, and in air is 10.The frequency of the lift & fall cycle (f ~ V/L x ) must be proportional to M -1/6 , therefore the bigger movers should move their limbs less frequently.The work per unit travel must scale as (ρ m /ρ) 1/3 Mg.Consequently, flying is approximately ten times more economical than swimming, because (ρ m /ρ) 1/3 in air is 1/10, and in water is 1.
The factor (ρ m /ρ) 1/3 is the physics of why bio designs spread from water to land and later in air.The 'food chain' is a consequence of the same physics.The bigger animal is faster and catches the smaller animal.The flier is faster and preys on the slower movers of the same size in water.The present article is not about legged locomotion, although the data covering terrestrial locomotion are included comparatively in the original theory 4,5 .For legged locomotion, the reader is directed to Günther et al. 20 .
These predictions unite the voluminous and highly diverse body of observations.Figure 2 shows the latest compilation over the body size range 10 -6 -10 8 kg.The two straight lines are the theoretical scaling laws for fliers and swimmers.The spacing between the lines (fliers vs. swimmers) is the theoretical factor (ρ m /ρ) 1/3 = 10, which unites locomotion in water with locomotion in air.5][6] ).www.nature.com/scientificreports/because the air density for airplanes is lower than for animal fliers.The density difference was taken into account in the upper graph, which is the most recent 7 .Figure 2 is instructive for an additional reason: it puts on display the physical meaning of the concept of 'scale' .The segments on the abscissa and the ordinate cover many different orders of magnitude of their respective measurements.Coming from the Latin scalae, the word scale means flight of stairs, staircase, ladder, or a sequence of steps.The steps are marked as segments on rulers and tape measures used historically in design and manufacturing, from tailor shops to construction sites.An old slide rule is segmented in centimeters, therefore, it is a 'centimeter scale' to be used for measuring centimeter-scale bodies.The tape measure used in landscaping and construction is marked in meters, therefore, it is a meter-scale.The two scales are physically as different as the objects measured with them.That is why Fig. 2 is a multi-scale collection of measurements of animals with locomotion.
The observed animal body is an object with many features, one large, some small, and many much smaller.If distinguishable, these features are different length scales.The large scale (say, L) is related to the body mass of the animal (M), therefore, if M is measured then the large length scale is known, L ~ (M/ρ) 1/3 , where ρ is the body density.For a person L is of order 1 m; smaller features (palms and feet) are of order 10 cm; many much smaller features (fingers, toes, feathers, hairs) are of order 1 cm and 1 mm.These three examples represent different length scales, cf., Figs. 3 and 4.
The body is not the object of analysis in science: it cannot be because it is too complicated to describe, manipulate, and argue about.The object of analysis is a simple facsimile of the observed object, which is called model.Models are diverse, few simple and many complicated, or few single-scale and many multi-scale.The simplest model of the animal body (L B , or M) is the 'one-length' model used in Fig. 1 and the construction of Fig. 2. Adding more features to the one-scale model makes the model more realistic and multi-scale.The first reviewer of the original manuscript of this article pointed out that the added features are all dictated by the body size (M), because on a small animal (e.g., frog) all the features are small.Well, they are all 'small' from the viewpoint of the observer, who is certain that a frog is smaller than a man.Yet, as explained in the preceding paragraphs, the smaller features are different length scales.
To avoid confusion, we used the advice provided by the reviewer and called the smaller features 'lengths' , not length scales.In the model with multiple features, the smaller scales are accounted for in terms of aspect ratios.For example, the characteristic palm length divided by the body length is one aspect ratio.If in the ensuring analysis the aspect ratio is free to vary then that ratio is one degree of freedom in the design and performance based on that model.Degrees of freedom are also the ratios of time intervals in the rhythmic movement of the animal.
Advances continue to be made by endowing the design of the moving body with more degrees of freedom.If the model of the bird or airplane is endowed with several lengths (Fig. 3) instead of the single length associated with the body size M [namely L B ~ (M/ρ) 1/3 ], then, as we predicted analytically in Ref. 19 based on the constructal law, it is possible to determine the four shapes that account for the flying architecture: fuselage cross section A, fuselage profile L/D, wing cross section t/L w , and wing span divided by fuselage length, S/L. Figure 3 shows that the S/L proportionality predicted from theory is in accord with the evolution of commercial jet aircraft 19 .
The predicted architecture holds equally for animal flight, which is why airplanes have evolved to look like birds 43 .They evolved with freedom to acquire greater access as they move through the medium.They did not emerge from 'mimicking' bird design.The evolution of civil and military helicopters was predicted the same way as for airplanes, by evolving the multi-length architecture with freedom and consistently toward greater www.nature.com/scientificreports/access, in accord with the constructal law.For example, the rotor diameter should match the body length scale of the helicopter 44 .Boats with sails, big or small, should look the same: the height of the sails should match the length of the hull and the mast 37 .
Movers on earth come in all sizes because they must scan the surface of the earth completely: they achieve this with hierarchies of body sizes and body numbers.Airplanes, birds, and ships should be organized as hierarchical traffic routes populated by many small and few large movers 22,45 , as is suggested qualitatively in Fig. 4. The biggest moves on earth-the river basins-share the same hierarchy, and for the same reason.Water and air currents evolve into hierarchies to have access on the globe, and their organization is predictable 22 .Extensive literature reviews are in Refs. 4,5,15.

Rhythm
The following theory is to regard locomotion from a 'more liberated' viewpoint.The idea is to endow the motive stroke with an additional degree of freedom: intermittency.The work motion is 'push and reach' .The added freedom is the rhythm in progress forward.Rhythm is the duration of 'push' relative to 'reach' during the motive stroke (t y , Fig. 1).
The analysis is presented in terms familiar to the reader: frog-like swimming (Fig. 5).The rhythm can be adjusted freely so that the time-averaged swim is faster per unit of power spent, or more economical (with reduced power) when the speed is specified.
The motion of the swimming frog is symmetric in space and periodic in time.The legs and the front members kick back symmetrically.The legs complete a cycle, the front members do the same, and together they complete the motive stroke of pushing forward (t y , Fig. 1).
The new feature in the present article is that the motive stroke is composed of two unequal time steps, t 1 and t 2 .The back kick (t 1 ) is shorter than the forward reach (t 2 ).Why is this evolutionary design present in amphibians and in humans (breaststroke, Fig. 5)?
Speed is one objective, because a greater speed is essential for life.Amphibians swim away from danger, and they are not the only ones.Soldiers and freedom lovers swim across the river to save themselves.Athletes swim against the clock to win medals, fame, and opportunities for a better life after their competitive sports careers.The other objective is economy of effort, which is documented in many studies in zoology and biophysics [8][9][10][11][12][13][14][15][16][17] .
Size makes speed (Fig. 2) and gives the impression that the speed question has been answered.True, bigger bodies swim faster in the animal realm, where evolution (the freedom to change the design) had a very long time to run its course.It is not true in the swimming pool, where speed records do not come from the big bodies of obese swimmers.Speed records come from two distinct features of design, the body (size, shape, svelteness) and the rhythm in the morphing of the body shape.Consider the simplest model that captures the main features of the moving design (Fig. 5).The body of the swimmer is represented by its mass M (kg), the single scale D (m) and the time-averaged forward speed V B (m/s).That V B is related to M is not the issue here: the challenge is to maximize V B by varying freely the rhythm in a more realistic description of the body architecture.
As noted already, the two steps of the motive stroke (frog, breaststroke) are represented by two time intervals, t 1 (back kick) and t 2 (forward reach).In the frame of reference attached to the moving body, the back kick is executed with the speed V 1 to a distance (the reach) of size L, which is comparable with the body dimension D, therefore t 1 ~ L/V 1 .The body pushes against the water with a blunt member (leg, arm) with frontal cross section D 2 , which is fast so that the Reynolds number (based on D and V 1 ) exceeds 10 2 , and consequently the drag coefficient C D is a constant of order 1.In the following analysis we neglect factors of order 1 in accord with the rules of scale analysis 46 .
The same model captures the main features of the second step.In the frame of reference attached to the body, the forward reach is to a distance L of order D with the speed V 2 during the time interval t 2 ~ L/V 2 .The cross section of the blunt member (leg, arm) is D 2 , and the drag coefficient is constant and of order 1.
During t 1 the legs and arms push back against the water and propel the body forward.The force felt by the body is where ρ is the water density and C D ~ 1, which in this analysis is neglected along with the 1/2 factor.The speed V 1 refers to the movement of the arms and lengths in the frame of reference of the swimmer.If the length of the swing (arm, leg) is L, the work done by the moving members is During t 2 the legs and arms moved forward to the distance L. Their speed in the frame attached to the swimmer is The relative speed between the moving members and the water is V 2 + V B , therefore the drag force on the members is The work done by the swimmer during t 2 has the effect of a 'brake' , and acts against propulsion: The periodic motion has several degrees of freedom that will become clear soon.Adjusting the variable features against each other is the key.In the present analysis the objective is a greater speed V B , which means greater net power ( Ẇ) spent on moving forward, (1) www.nature.com/scientificreports/or where F B is the averaged drag force on the body moving at speed V B : From Eqs. ( 7)-( 9) we conclude that Ẇ is proportional to V 3 B , which means that in the remainder of this analysis we focus on maximizing the function f defined below: First, we consider V 1 and V B fixed, and vary V 2 by noting that the function f is equal to V 3 1 f , where f is the dimensionless function where f (x) is maximum at x = 1 2 − β , which means that V 2 must be smaller than half of V 1 : The maximum power forward that corresponds to Eq. ( 13) assumes the form which, in view of 8) and ( 9), leads to one equation for V B /V 1 : Although the exact solution is V B /V 1 = 0.3194, for simplicity we will use Next, from Eqs. ( 13) and ( 16) we obtain In conclusion, we can predict that for greater speed in swimming the reaching motion should be slower than the kicking motion.This agrees with observations of swimmers, as we will show in Sections "Frog swimming" and "Breaststroke".
The total power Ẇtot spent by the swimmer is greater than Ẇmax because the morphing body performs work in both directions, pushing back (W 1 ) and reaching forward (W 2 ).The scale of Ẇtot is derived by replacing Eq. ( 7) with Ẇtot ∼ (W 1 + W 2 )/(t 1 + t 2 ), and then using the results of the Ẇmax analysis above, namely Eqs. ( 16) and (17).The result is Dividing Eqs. ( 18) and (18'), we find that the net power spent on moving forward is roughly 1/4 of the power spent by the swimmer.
Finally, the scale Ẇ ∼ = D 2 ρV 3 B represents both efforts, Eqs. ( 18) and (18').Inserting in this Ẇ scale the V B scale predicted by minimizing effort per unit travel 3,4 , namely V B ~ (gD) 1/2 , we find that the work spent per unit travel is of order Mg (or ρD 3 g), which is in accord with the earlier theory of locomotion 1,4,5 (cf., Section "Background").

Two lengths
The one-length model (D) of the body with rhythmic movement in water (Fig. 5) can be extended to more accurate descriptions.The simplest step in this direction is in Fig. 6.There are at least two lengths that capture the body construction, the frontal cross section of limbs that push against the water, D 2 2 , and the frontal cross section of the torso (shoulders, head, etc.), D 2 1 .The length of the movement of the limbs (during kicking and reaching forward) continues to be represented by the body scale D 1 .
In the two-length model the new degree of freedom is the relative size of the moving parts, namely the ratio ε = D 2 /D 1 , where ε < 1. Interesting and potentially useful in the conceptual design of artificial locomotion is to discover what aspect ratio D 2 /D 1 characterizes a better design, the faster swimmer, and the more economical aquatic robot.
The analysis follows in the same steps as in Eqs.
(1)-( 18) while keeping in mind the difference between the two lengths pictured in Fig. 6, namely D 1 and D 2 (or εD 1 ).Equations ( 1) and ( 5) are replaced by Equations ( 10), ( 14) and ( 15) are replaced by Equation ( 23) is plotted in Fig. 7.When ε approaches 1 the solution reproduces Eqs. ( 16)-(18') of the singlelength model.More relevant is the opposite limit, ε << 1, because in swimmers the ratio D 2 /D 1 is of order 1/10.According to Eq. ( 23), in the small-ε limit V B /V 1 becomes negligible relative to 1/2 (inside the parentheses), and Eq. ( 23) reduces to, in accord with Fig. 7 , In place of Eqs. ( 17) -(18') we obtain where " + …" indicates higher order terms in ε, which are neglected.In the limit D 2 < < D 1 the ratio Ẇmax / Ẇtot approaches 3/5, indicating a higher fraction of power allocated to forward propulsion than in the result based on the one-length model [read under Eq.(18').( 19) The main results of this section are compared in Table 1 against the corresponding results based on the onelength model (Section "Rhythm").According to both models, Ẇmax and Ẇtot have the same scale when based on the larger dimension (D, or D 1 ).The results for Ẇmax are close, indicating that the simpler model captures sufficiently well the value of the rhythm-maximized power for forward motion.
The results for Ẇtot show that the total power responsible for the same Ẇmax is sizably smaller (by 46 percent in the limit ε << 1) in the two-length model, relative to the one-length model.This is an important finding because the change from the one-length model to the two-length model represents the direction toward more degrees of freedom in morphing the flow architecture of the swimming body.In the direction toward more freedom to change, the performance of the multi-length design is the recipient of rewards from freedom.Such rewards are a common occurrence in the evolution and performance of flow architectures with freedom to morph 22,47,48 .We return to this conclusion in Section "Discussion and conclusions".

Frog swimming
The frog derives most of its swimming power from the hind legs.The frog 'motor' is at the back.The swimming cycle is composed of three time intervals, in order: Reach The knees bend and move into the body of the frog.This time interval is t 2 in the terminology of Section "Rhythm".Kick The legs push back and become straight.This interval is t 1 in the terminology of Section "Rhythm".Glide The whole body shapes itself hydrodynamically (elongated, without abrupt changes in its cross section perpendicular to the swimming direction), and moves as if rigid while drag dissipates the kinetic energy acquired during t 1 .The time to glide is called t 3 .
The three time intervals are available for measuring in several videos of swimming frogs.The measurements reported in Fig. 8 are from watching a slow-motion version of the video available at 49 .The measured rhythm is such that if the total cycle (t 2 + t 1 + t 3 ) has a duration of 7 time units, then t 2 ≃ 3 units, t 1 ≃ 2 units, and t 3 ≃ 2 units.Important are the ratios that these intervals reveal: • t 1 and t 2 have the same scale, and t 1 /t 2 ≃ 2/3 < 1 , or V 2 /V 1 < 1 .This ratio is in accord with the one- length and two-length analyses of Sections "Rhythm" and "Two lengths".Table 1.The effect of the number of model lengths on the maximized external power for swimming forward, relative to the total internal power spent by the morphing body.
One length, D Two lengths, D 1 >> D 2 • The motor time interval is t 2 + t 1 .In the earlier scaling of locomotion 3,4 the gliding time t 3 is the time of 'falling forward' against drag (t x in Fig. 1).The earlier theory predicted that the motor time (t y ) and the gliding time should have the same time scale.This is confirmed by the measurements plotted in Fig. 8, where t 3 /(t 2 + t 1 ) is of order 1.

Breaststroke
The breaststroke swimmer benefits from the power furnished by two 'motors' , one in front (the arms), and the other at the back (the legs).Each motor goes through the reach & push cycle described in Fig. 8 for the single motor of the frog.The two cycles of the arms and the legs are followed by an interval of gliding to dissipate the kinetic energy acquired from the motors.The time intervals of reach, push and glide are available on many videos of swimming lessons for amateurs and athletes.The measurements summarized in Fig. 9 came from watching the video available at 50 .The units of time on the horizontal scale are not important, as they depend on speed selected for the slow-motion examination of the video.Important are the relative sizes of the cycles and their internal reach & push intervals.
One complete cycle consists of one 'arms cycle' , followed smoothly by one 'legs cycle' and then by one glide interval.The complete cycle (motor + sliding) repeats itself nearly identically as one watches the video.For example, a typical complete cycle began at time 0 and exhibited these changes: 0: arms begin to push back; 7: arms stop pushing back and begin to reach forward; legs begin to bend, heels approaching buttocks; 13: legs start to kick back; 16: legs are completely extended; sliding begins 17: arms stop reaching forward; 26: sliding ends, arms begin to push back, and so on.
Note that the arms cycle and the legs cycle overlap, and that together they constitute the motive interval that started at time 0 and ended at 16.This sequence of changes is illustrated in Fig. 9, where we plotted the averages of observations of four nearly identical cycles.The dissipative interval started at time 16 and ended at 26.Three conclusions follow.First, the time ratio for push/reach during the arms cycle, t 1 /t 2 , is less than 1, in accord with the theory and with the measurements of rhythm in frog swimming.Second, the same conclusion holds for the push/reach time ratio of the cycle executed by the legs.Third, the total motor interval is of the same order as the sliding interval, in accord with the earlier theory of locomotion (Fig. 1, Section "Background").

Hovering
Flapping flight (Fig. 10) is an opportunity to extend the theory to a science and technology that is even wider than its aquatic counterpart.In hovering, the body maintains its altitude in two steps, spreading the wings and pushing the air fast downward, and closing the wings and raising them above the torso.The net work performed during one cycle is represented by W y in Fig. 1.In hovering the flapping is happening in place, which means that L x = 0 and W x = 0.
The work done during the first step (t 1 , downward) must be greater than the work done while raising the wings (t 2 ).The wing (its face area) and downward speed must be greater than during the return motion (upward).Unknown, and potentially useful for designs of hovering flight are the rhythm (t 1 /t 2 ) and the effect of changing the frontal area when the wing changes direction.
The simplest model is sketched in Fig. 11, where the frame of reference is attached to the body (D B ).This model differs from that of Fig. 6 in three respects: it has three lengths not two, the motion is oriented vertically,   The significance of the expression (1 − ε 2 x 2 ), and how to calculate it, is explained next.The maximum work available for maintaining the altitude of the body during one cycle is found from Eq. ( 29) by substituting in it the solution obtained for x(ε) by solving Eq. ( 31) , W max equals the work required to lift the body back to a height equal to the body length scale: The body density ρ is significantly greater than the air density ρ a .Setting W max ~ W lift , we obtain where η = D 1 /D B < ∼ 1, and Ṽ1 is the downward wing speed nondimensionalized relative to the Galileian speed of free fall to the level D B , ( 27) swimmers 58 and fliers 59 exhibit St values in the same range 1-10.The predicted frequencies that stand behind the theoretical lines in Fig. 2 agree with that universal Strouhal number range 1-10.
The Strouhal number is a dimensionless way of expressing the frequency of vortex shedding (or meandering) of turbulent (i.e., relatively inviscid) streams: jets, wakes, plumes.Its empirical discovery and relative universal value in the range 1-10 (independent of Re) was labeled St by its discoverer Vincenc Strouhal in 1878.It is a feature of large scale turbulence structure.The universal St range was predicted 60 from pure theory in 1981, and reviewed in chapter 4 of Bejan's 1982 thermodynamics book 52 .
All bodies moving in a medium (air, water) shed behind them vortices migrating with a particular frequency downstream as meandering wakes.The flows generated behind moving animals are recent examples in a body of St literature that validates through frequency measurements the universality of the St prediction made for turbulent flow in general 46,52,60 .
Although this confirmation should be enough, the theory 4 revealed an additional part of the physics that continues to be overlooked in biology.The studies of wake frequencies behind swimmers 58 are based on the classical view that gravity does not matter to the fish.The study of wakes behind fliers 59 found that the Strouhal number for fliers is comparable with the Strouhal number for swimmers.Such a coincidence should have been questioned before, because obviously gravity does matter to the bird.Gravity is one of the two major forces in flight (the other is drag).The theory 4,5 predicted this coincidence even before the question arose.We showed 1,4,5 that the reason for the universality of the Strouhal number (or frequency) correlation is that gravity is as important in swimming as in flying.
Relevant to the reference made by the reviewer to Tucker 56 is the tested prediction 4,5 that the work/distance requirement for the swimmer is greater than for the runner and the flier (Section "Background").This prediction is in accord with the history of the biosphere: the effort required for locomotion decreased chronologically in the sequence water-land-air.See Fig. 12 for the evolution and spreading of locomotion in the biosphere 61 : Big history contradicts Tucker 56, who claimed that the energy cost of locomotion decreases from running to swimming and, finally, to flying.Clearly, Tucker made a mistake, which was taken as truth by many including his Duke colleague Vogel 10 .
Our objective is not to criticize the work and legacy of our Duke colleagues.For them we have the greatest respect because of their contributions to the science of animal design [8][9][10] and to the fame of our university.The disagreement stems from how the 'cost' of locomotion was evaluated.There are two ways to do this.
Our way, here and in the original theory 1,4,5 , was to focus on the thermodynamics (physics) of locomotion, the body and its actual movement against the ambient.The required work is due to a finely tuned cycle of two efforts, the work needed to lift the body, and the work needed to move forward by getting the environment out of the way.These two efforts are easy to rationalize, based on the simplest dynamics analysis.The derivation of W y and W x lies naked on the table 1,4 .
The other approach is older, and it is preferred in biology.It begins with the notion that life is due to the 'free' energy embodied in food, and that the power generated by metabolism drives locomotion and other functions inside the body.This approach is opaque because between the food intake and the movement of the body on earth there is an 'engine' that propels the body intermittently, not in sync with the food intake, which is also intermittent.An engine is a flow system with configuration (design).This, and the design and performance of the engine are not revealed.Coming from thermodynamics 2,52,53 we know what is missing, starting with the concept of free energy (exergy, availability) that dates from Maxwell and Gibbs, not Schrödinger one hundred years later.
Tucker 56 followed the second approach.He plotted on the ordinate the group P i /(WV), where W is the body weight (Mg in our article), V is the speed, and P i is "the rate of energy release called metabolic rate", which he called "the power input P i , a term more consistent with engineering usage".No, it is not consistent with engineering, because both animal and vehicle do not move constantly while eating constantly.Furthermore, in a vehicle AbbreviationsC D Drag coefficient D Fuselage diameter, m D Length of one scale model, m D B Body length scale, m D 1 Torso length, m D 2 Limb length, m f Frequency, s -1 f Function, Eqs.(10) and (21), (m/s) 3 f Dimensionless function F B Body drag force, N F 1 , F 2 Forces on two-length model, N g Gravitational acceleration, m/s 2 L Fuselage length, m L Reach, m L B Body length, m L w Swept length of wing, m L x Travel, m, Fig. 1 M Body mass, kg S Wing span, m t 1 Push time, s t 2 Reach time, s t 3 Glide time, s t x , t y Time intervals, s, Fig. 1 t Wing thickness, m V Velocity, m/s https://doi.org/10.1038/s41598-023-41023-6

Figure 3 .
Figure3.The proportionality between wingspan and body length is predictable19 .

Figure 4 .
Figure 4. Animals and vehicles sweep the surface of the earth organized in hierarchies of body size and body numbers 22 .

3 BFigure 6 .
Figure 6.Body with two lengths scales: the body scale D 1 , and the moving organs D 2 .

Figure 7 .
Figure 7.The relation between V B /V 1 and D 2 /D 1 .

Figure 8 .
Figure 8. Three time intervals in the frog swimming cycle: reach, kick, glide.

Figure 9 .
Figure 9. Three intervals in the breaststroke cycle: arms cycle, legs cycle, and sliding.

Figure 10 .
Figure 10.Two time intervals in flapping flight: closing and raising the wings above the torso, and spreading the wings and pushing the air fast downward.

Figure 12 .
Figure 12.Timeline of the evolution and growth (spreading) of locomotion on earth.