THE PHYSICAL ASPECTS OF FISH LOCOMOTION

BY E. G. RICHARDSON, B.A., PH.D., D.SC.

(Armstrong College, Newcastle on Tyne.)

(Received January i, 1935.)

(With Eight Text-figures.)

I. INTRODUCTION.

THE mechanism by which a fish is able to dart through a heavy medium like waterwith little apparent effort has long been a matter of interest to zoologists. Beforethe arrival of the cinematograph in its modern form, Marey (1884) obtained a seriesof " chronophotographic" records of fishes in movement to demonstrate thesinusoidal motion, while more recently Magnan (1930) and Gray (1933) have takencinematograph pictures at the rate of 10-20 per sec, in which the fish is seen swim-ming behind a reticule, thus making detailed study of the mode and rate of pro-gression possible. Houssay (1912) and Breder (1926) have also made contributionsto this problem. Their work will be noted later. With the general adoption ofstreamlined form for all bodies which have to move rapidly through air or water,the attention of workers in hydrodynamics has been focused on the same problem.Indeed, the extensive researches of Magnan were subventioned by the French AirMinistry. It is from this aspect that the author is interested in the subject.

There appear to be two possible explanations to account for the rapidity ofmovement of a fish through the water, viz. (1) that in virtue of the mucus on itssurface it may possess a resistance lower than that of the same shape whenunlubricated, (2) that in virtue of its flexibility it cleaves the water with less re-sistance than if rigid. If either of these facts can be substantiated it is obviously ofgreat importance to aircraft and ship design.

II. RESISTANCE MEASUREMENTS.

Although Houssay made measurements of the resistance of wooden models offish towed through the lake of the ficole Militaire at Paris, the only measurements onfishes were those of Magnan deduced from cinema photographs of the weightedbodies of fish freely falling in a water tank. The photographs are scarcely clearenough for the exact deduction of the rate of fall. Indeed Magnan decides that thespace : time curves are exact parabolas, which as he rightly deduces, implies thatthe resistance is independent of the velocity. He does not, however, complete theargument which is that if the parabolic law is followed the velocity increases withoutlimit, a condition only fulfilled on free fall in a vacuum. A body falling in a

64 E. G. RICHARDSON

resisting medium must sooner or later reach a constant velocity, determined by thebalance between gravity and resistance, which can then be determined as a functionof the terminal velocity. The author decided to determine the rate of fall of fishesover various distances by a direct chronographic method.

A tank 8 ft. high and 1 ft. 6 in. square was filled with water and equipped withstarting and stopping devices for the object whose fall was to be investigated. Theformer device was arranged so that the fish could be started from a little below thesurface, the latter was a tray which could be lowered by cords passing over pulleysto counterpoise weights. The fish was held in an electromagnetic grip by its tail.Breaking the electric circuit released the fish and simultaneously released the brakeof a Wood's chronograph. When the nose of the fish hit the lower stage, which wasa pivoted gauze network, it broke a contact, which stopped the chronograph again.As will be seen from the diagram of the apparatus (Fig. 1) this contact had to bearranged in a vessel above water, otherwise it did not break smartly, the water beinga conductor of electricity. With this apparatus the time could be read to 0-002 sec,while the distances were measured by telescope to 1 mm. The freshly killed fish,deprived of all its fins except the caudal fin, was loaded by lead shot in the mouth.Trouble was experienced at distances of fall exceeding 2 ft. owing to the flexibilityof the fish which often led it to describe a curved trajectory. Later a wire stiffenerwas pushed into the body to keep the fall vertical. At any given distance, times wererepeated until a minimum had been established. After each fall the lower stage wasraised to retrieve the fish. When a series of falls at various distances and weights hadbeen made, the fish was carefully measured and a template made. Subsequently awooden model was made to the template, loaded in the scooped-out interior by leadshot and provided with an aluminium tail. The time of fall of the model in the tankwas then determined under three conditions of surface, i.e. (a) roughened, (b)smoothed and polished, (c) greased with oil. Writing the equation of motion of thefalling body in the form ,

m dt=mg~R'

where the first term represents the downward force on the body at the instant whenits velocity is v, the second term is the weight of the body in the water, or buoyancy,and the third is the resistance. We will write the latter as a function of v in the formkvn, so that ,

m =m g + kvn.at

n may have, in practice, values between o and 2; (a) if n = o, v = ( ) t, the\m ml

acceleration is constant and we have the case postulated by Magnan; (b) if n = 1(Stokes' law), fc

the terminal velocity is m'glmk; (c) if n = 2 (Newton's law), v = c tanhgjc, where cis the terminal velocity =Vm'glmk. In case (c) the terminal velocity is of course less,

The Physical Aspects of Fish Locomotion 65

ceteris paribus, and reached in a shorter time. On Fig. 2 are plotted the displace-ment : time curves for a mackerel and a herring loaded by different weights to showthat a terminal velocity, indicated by the graph becoming a straight line, is in

To starting switchof chronograph

[*o stopping switchof chronograph

Fig. 1. Tank for time of fall of fishes and models.

practically every case attained within the depth of the tank. Probably Magnan'sfailure to detect a terminal velocity is due to his not having carried the distances farenough. (Although he had at his disposal a tank 3 metres deep, the displacement:time curves on which he bases his n = o deduction seem not to extend beyond half ametre.) Table I gives the resulting data in the form of weight in water (W) andcorresponding terminal velocity (V). These are subsequently plotted in the form

JEB-XIIli 5

66 E. G. RICHARDSON

log W: log V on Fig. 3. The slope of these curves gives the appropriate value of nat each value of V.

3 4 5

Time (sees.)

Fig. 2. Rate of fall of fishes

Table I. Resistance measurements.

Mackerel.

Weight in air 1OC3 gm., in water 267 gm., volume 123 c.c.W gm. 267 40-5 480 650Fcm./sec 6-65 n-o 125 13-5

Rough wooden model. Weight in air 151 gm., in water 267 gm., volume 130 c.c.W 267 41-5 48-2 65-0V 83 100 i2-o 13-3

Wooden model smoothed and varnished.W 267 41 -5 45-0V 8-3 12-0 12-5

Herring.

Weight in air 864 gm., in water 13-8 gm., volume 72-6 c.c.W 13-8 230 31-2V S'O 100 12-0

Rough wooden model. Weight in air 84-2 gm., in water 13-9 gm., volume 70-3 c.c.W 13-9 23-0 31-0V 5-0 9-0 no

Wooden model smoothed and varnished.W 13-9 188 23-0 31-3V 5"25 8'0 IO-O I2'2

The general conclusion to be drawn from this table and the graphs is that the fishhas a slightly less resistance than the rough wooden model, but that a carefullysmoothed and polished wooden model can improve on this, particularly at the lowspeeds. The exponent n given by the slope of the curves of Fig. 3 varies from i-oto i-8 for the fishes used, just below the turbulence regions to which Newton's lawapplies. It is probable that the normal velocity range covered by a fish lies belowthe turbulent regime except when it is spurred to violent movement.

The Physical Aspects of Fish Locomotion 67

The experiments with the smoothed model covered with a light oil previous toimmersion to imitate the mucus on the body of the fish did not indicate any marked

1-0

1-2

1-0,0-6

+4

+2

I-2

• fisho rough modelX varnished model

0-8 1-0 1-2

log terminal V

Fig. 3. Resistance of fishes.

1-4

Paraffin

Interface between / / / liquid

0 I 2

Velocity (cm./sec.)

Fig. 4. Velocity gradients in two liquids in relative motion.

improvement on the resistance of the model at speeds below the critical. Boththeory and practice, however, concur in indicating that the presence of a viscouslayer, i.e. more viscous than the main body of the fluid in which it is immersed,

5-2

68 E. G. RICHARDSON

delays the onset of turbulence. It is in fact the shearing force in the "boundarylayer " of fluid close to the solid surface of an immersed obstacle that gives rise toinstability which in turn leads to turbulence in the wake. The viscosity of the mucuswas determined in a cup-and-ball viscometer and found to be about five times thatof water at ordinary temperatures. Below critical speeds the added viscous layerwall simply follow the Stokes' type of viscous flow and cannot lead to any improve-ment in resistance if the velocity relative to the surface falls from its mid-streamvalue to zero at the surface without a break. It has been suggested, however, thatthere may be sliding of two liquids one over the other at their common interface,although it is established that none occurs at a solid-liquid boundary. Some experi-ments which the author performed some time ago give the lie to this suggestion(Richardson, 1933). One liquid (paraffin) was made to pass over another (water) ina tank, and by means of a hot-wire anemometer the velocity gradient on each side ofand across the interface was measured. Typical results are shown in Fig. 4 for dif-ferent surface speeds of the paraffin. No jump in the velocity occurs at the interface,merely a change of slope consequent upon the difference of viscosity. Except at highspeeds then we may dismiss the idea that a fish has a small resistance because of itsoiled surface.

III. EXPERIMENTS ON LOCOMOTION.

Having failed to detect any morphological peculiarity in the fish which shouldgive it a lowered resistance, we turn to the more difficult question of whether thereis something inherent in the sinusoidal motion which should produce this effect.Difficult because it is not possible to consider the resistance apart from the pro-pulsion. Just as with a screw propeller, measurements of the resistance at zero rateof advance, i.e. when the screw is acting as a mill, are not sufficient data for derivingthe efficiency of the screw, so the resistance of the flexing fish should not bemeasured only when it is being towed or drifting in the stream. As it is not easy tomake measurements on a live fish, one has recourse to a model, held stationarywhile the medium flows past it.

Before describing such experiments, one should say at the outset that there isknown to aeronautics a means of reducing the resistance of models which is pertinentto the fish problem. I refer to the Katzmayr (1922) effect, viz. the reduction ofresistance sometimes experienced by an aerofoil when, instead of meeting the windat a fixed angle of incidence it is given a periodic see-saw motion through a smallangle. Alternatively, the object may be held still, while the incident fluid is givena sinusoidal motion by means of oscillating guides, this being the form that experi-ments on the Katzmayr effect usually take. Theory shows that the reduction in thedrag component will be most apparent when the object is such that the lift com-ponent (or cross-force) varies rapidly with angle of incidence, and may in some casesattain a negative value, so that the aerofoil actually propels itself against the stream,some of the energy used to oscillate it thus becoming available for its propulsion.Owing to the difficulty of seeing how to apply the Katzmayr effect to full-scaleaeronautics it has remained of merely academic interest, and an exact theory of the

The Physical Aspects of Fish Locomotion 69

effect is not available. It seems, however, to be an effect which cannot be disre-garded in reference to fish locomotion, although an actual fish differs from theaerofoil model in that it is not rigid, and cannot oscillate rigidly about an axis as inthe aeronautical experiments just described. The conditions of the experiments arehowever approximately fulfilled in the locomotion of those fishes whose bodies aretoo stiff to show a true sinusoidal motion, but merely flap to and fro, such as themullet, cited by Magnan. Further, such fishes have bodies of the requisite form toexhibit a large change of cross-force as the body flaps to and fro, or, what amountsto the same thing, a small change of the angle of incidence of the stream on a longslim body will probably provoke a large change in the direction of the total (re-sisting) force.

To study the resistance of a completely flexible body it was again necessary toresort to a model, this time a flexible rubber flag or "lamina". The experiments in

fr-%s o\—L- \»

Fig. 5. Wind channel and balance for resistance of flexible laminae.

the water channel having shown that the free-fall method was scarcely suited to apliant model, owing to the tendency towards a curved trajectory, it was decided tomeasure the resistance in a wind-tunnel. The axis of the latter had to be vertical inorder that the lamina, supported on a horizontal arm could hang vertical. The re-sistance of the flag was then measured by the position of a counterpoise weight onthe balance arm (W, Fig. 5). The oscillation of the flag stalk (S) (corresponding tothe head movements of the fish) was effected by making the stalk a thin steel rod ofstreamline section and fixing near it a little electromagnet (M), clamped rigidly tothe balance beam (B). A dashpot (D) prevented vibration being imparted to thebalance beam. The electromagnet was fed by the interrupted current from a Muir-head vibrator, whose frequency could be fixed at any value between 20 and 50vibrations per sec, thus vibrating the flag head at a known and constant frequencyand amplitude. The latter depended on the strength of the current supplied. Therubber was also on occasion weighted at the lower end by an aluminium strip toalter its natural frequency of vibration. In order to measure the velocity of thewaves along the strip, it was illuminated intermittently by a ray of light shining

70 E. G. RICHARDSON

through the stroboscopic disc and window in the channel as shown on the left ofFig. 5. By suitable adjustment of the speed of rotation of the stroboscopic disc thewaves on the flexible strip could be "held" stationary and the wave-length ob-served through the travelling telescope X seen on the right. This wave-length variedslightly with the channel wind-speed, as did also the amplitudes at the head (theaverage value of the latter was 3 mm.). Knowing the frequency and the wave-length, the velocity of the waves along the lamina could be calculated. These corre-spond roughly with the channel wind velocities at zero resistance.

Before discussing the results let us consider the theory of this type of fish pro-pulsion considered as a motor.

Let v = velocity of fish relative to water, and w = velocity of waves along fish,relative to the fish. Let the mass of the water set in motion per second—this willdepend mainly on the amplitude of the wave passing down the body and pushingthe water back—be mw. Let F= propulsive force = change of momentum,m (w — v) per sec. Then the energy usefully employed = Fv = mwv (w — v) persec, while the kinetic energy rejected in the discharge = \mw (w — v)*, so that thetotal energy given to the fluid = Jmai {(w — v)* + zv (w — v)}. The theoretical efficiencyis then zv {w — v)j(w'i — vi) = zvj{w + v). The actual efficiency will be less than thisby an amount depending on the energy wasted in causing rotation of the water. Itwill be noted that though increase of w raises the force of propulsion it lowers theefficiency. The fish which approaches nearest to mechanical perfection is one whichproduces a slow wave of large amplitude along its body thus making m a maximum.Eddy production is likely to be increased if the fish tries to impart a high velocityto the water through a fast-moving wave as in lashing its tail.

The results obtained with the flexible lamina are shown in Figs. 6 and 7 in theform of resistance/speed curves. It will be noticed that at two wind velocities thelamina shows negative resistance when subjected to forced vibrations. At higherwind speeds the resistance becomes positive, as might be expected since the velocityof the waves passing over the surface of the lamina approximates to or exceeds thevelocity of the wind stream. So long as the conditions under which the fish ismoving approximate to those of the lamina in these experiments it is obvious thatthe resistance overcome by the fish cannot be assumed to be the same as that over-come by a rigid fish moving at the same speed.

The ordinary flapping of a flag in the wind is, of course, the correspondingdynamo effect to the fish-motor. The latter advances through the fluid by impartingenergy to it, whereas the former is stationary and receives energy from the wind, sothat any slight instability of the surface of the flag is passed along it as a wave ofincreasing amplitude. We find another analogy in the edge-tones formed at themouth of a flute organ pipe where the air debouching from the slit hits the lip ofthe pipe and in doing so oscillates from side to side in flag fashion. Indeed, cine-matograph films made by the author (Richardson, 1930, 1931 a, b) of the motion ofsuch a jet, rendered suitably opaque, bear a striking similarity of movement tothat in Gray's films of fish swimming.

The late Lord Rayleigh treated the instability of flags and jets mathematically,

The Physical Aspects of Fish Locomotion 71

but his results are scarcely useful in the present discussion. In the case of the edge-tone, the jet (owing to its extreme sensibility to disturbance) is set into strongrotational motion. Much of this energy goes into forming vortices, which are the

i +4

- 4V

20 60Air Velocity v

Fig. 6

Model 33 MI.

^Unloaded (mean

^~">L«ded (mean a

tz——•

1 1

100cm./sec.)

long

w — 5 cm |iec,

• 15 cm /we.)

Modtl 17 on. long

mean tr — 40 cm j

(mean w — 20 cm /iec )

20 60 100

Air Velocity v (cm./sec.)Fig. 7.

Figs. 6 and 7. Resistance of flexible laminae; continuous lines = free vibration,broken lines = forced vibration at 25 cycles/sec.

cause of the sound, and these vortices roll between the jet and the stationary fluidon each side with half the velocity of the jet (as when a coin is rolled between thetable and a card above). Whenever, in fact, relative motion of two fluids or of aflexible solid and a fluid occurs, there will be wave motion along the boundary, andvice versa. Breder constructed a mechanical wave machine which fish-like propelleditself through water in accordance with this principle.

IV. THE JET PROPULSION OF THE OCTOPUS.

Of the available aids to translation, the majority of marine creatures employsinusoidal motion to a greater or less extent, unless great acceleration is required,when a single flick may be given to the tail, or for reversing, when the pectoral finsmay be used as oars.

E. G. RICHARDSON

Table II.

v ft./sec.

12-52223

u> ft./sec.

o9

21

Propulsion lb. wt.

Theoretical

0-0510-1660016

Actual

o-°330-044o-oio

Efficiency

Theoretical

o-oo0-58o-93

Actual

000030o-54

The members of the octopus family occasionally employ jet propulsion. Theyuse the reaction derived from expelling a jet of fluid to propel themselves violentlyin the opposite direction. The theory of the jet propeller is similar to that already

I 0

-1

-2

40 50 ucm./sec.

Fig. 8. Resistance of jet propellers,

-o 15 cm. model; x x x 22 cm. model; 10=30 cm./sec. approx.

given for the wave propeller, save that w now stands for the velocity of the jetrelative to the model, and m is the mass ejected per second. Some experimentsmade by the writer (1931 b) some years ago on air-jet propellers with a view to theirpossible adoption on aircraft are germane to this type of fish propulsion. A modelwith a streamlined head of wood, neatly fitted to a brass tube, was mounted in awind channel with the tube pointing downstream. Air was led into the interior of

The Physical Aspects of Fish Locomotion 73

the tube through the hollow strut which connected the model to the usual forcebalance. After passing along the barrel, the air was ejected from the stern. Theweight of the model was taken up by a counterpoise on the balance beam, the de-flection of which acted as a measure of the propulsive force, which could be com-pared with that indicated by the propeller theory given above. Using a hot-wireanemometer and a Pitot manometer, the integrated velocity and the pressure dropbehind the model were calculated, and hence the work done in overcoming frictionin the apparatus was computed. This in turn allowed the actual efficiency to becompared with the theoretical value. Results for various jet and wind speeds areshown in Table II and on Fig. 8.

The propulsive forces realised were smaller than the theoretical values, but thetrend of the results was in agreement with theory. When v = w, the theoreticalefficiency is 1, but the propulsive force is o. This case is nearly realised in the thirdexperiment in Table II. Larger jet speeds could not be attained with the compressoravailable. Losses in friction inside the model were considerable in the arrangementchosen. Fig. 8 shows how the actual propulsive force varies (for a constant jet pres-sure) with the channel wind speed, (a) for a model having a 6-in. tube, and (b) for amodel with a 9-in. tube.

The size of the orifice has no effect on the efficiency, for although a greater areaof discharge will produce a greater reaction, yet more energy must be supplied tomaintain the velocity of discharge.

The experiments with the model jet propellers verify in a general way the pro-peller theory as it applies to creatures or machines which use reaction for loco-motion.

V. SUMMARY AND ACKNOWLEDGMENTS.

1. The resistance of an inert fish moving in water is approximately the sameas that of a wooden model similar in form and moving at the same velocity.

2. When a lamina is made to vibrate in a current of air, it exerts a thrust in thedirection in which the vibrations are travelling over the lamina; as long as thevelocity of movement of the vibrations exceeds that of the external current of air,the resistance of the lamina is negative.

3. Attention is drawn to the significance of the Katzmayr effect in respect tothe locomotion of fish.

I wish to express my best thanks to Dr H. O. Bull of the Dove Marine Laboratoryfor his help during the course of this work, and for his criticism of the manuscript.I am also indebted to Dr J. Gray for helpful criticism and to Dr R. Curry and MrW. Ridley for their help in making the observations.

74 E. G. RICHARDSON

REFERENCES.

BREDER, C. M. (1026). Zoologica, N.Y., 4, 159.GRAY, J. (1933). J. exp. Biol. 10, 88, 391; Proc. roy. Soc. 113, 115.HOUSSAY, F. (1912). Forme, Puistance et Stability des Poissotu.KATZMAYR, Z. (1922). Flugtech. Bibl. 13, 65, 80.MAGNAN, A. (1930). Arm. Set. nat. 13, 355.MAREY, F. (1884). Le Moiwement.RICHARDSON, E. G. (1930). Proc. roy. Soc. Arts, Jan.

(1931 a). Proc. phys. Soc. Lond. 43, 394.(1931 b). J.R. aero. Soc. Jan.(i933)- Kolloidzschr. 65, 32.

Other references to Fish Locomotion in Russell, E. S. and Bull, H. O. (1933), J. Cons. int. Explor.Mer. 7, 255.