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                     Swimming locomotion, how does it work ?

Hydrodynamics of swimming locomotion in aquatic vertebrates

Up to now, ideas concerning the fast locomotion of aquatic vertebrates were based on a misunderstanding about the generation of propulsion and the function of the tail fin. This fact has led to erroneous statements that cannot be harmonized with physical laws. This new investigation of the problem based on hydromechanical considerations comes to a satisfactory explanation.

The regularities of fluid mechanics relevant for the swimming locomotion, although not completely new, are explained once more, since there were considerable misinterpretations. Any swimming locomotion is based on the principle of a rearward acceleration of water which leads to a forward movement of the swimmer. Fluid dynamic laws are utilized by aquatic animals in a variable manner and to a variable extent, depending on the requirements of size, speed range, and main food source. Propulsion for a fast forward locomotion can be generated only by alternately bending the trunk and by the resulting rearward acceleration of water, but never by the tail fin itself. The obvious function of the tail fin as a stabilizer and rudder is clarified.

Any swimming animal has strictly to stick to the limits given by fluid dynamic laws. The overwhelming majority of small and medium-sized swimmers is constrained by laminar flow requirements. Few genera only such as lamnids, tunnies, and cetaceans with a semilunate tail fin have entered the turbulent flow region. Exclusively forms with a well developed tail fin are capable of living in the open seas due to a favourable exploitation of energy. Presence and shape of the tail fin are reliable indicators of the swimming performance available.

Introduction

The locomotion of swimming vertebrates can easily be studied in Recent forms. Nevertheless, there are still remarkable obscurities about the exact mechanisms of locomotion which have caused many erroneous ideas. In particular, so far the physical background concerning the generation of propulsion in fast swimmers with a pronounced tail fin has not satisfactorily been cleared up.

Ideas about the generation of propulsion can be traced back far to the nineteenth century. These ideas could only satisfy certain aspects such as the locomotion of eel-shaped forms. Various workers have dealt again and again with the problem of swimming locomotion, and it has recently found a renewed and increased interest, e.g. by Hertel (1963). McGowan (1992) has summarized further aspects, mainly based on work by Webb (1988). Lighthill (1975) has attempted a theoretical approach from a mathematical viewpoint. Although he considered mathematical, biological, and certain aerodynamic aspects, nevertheless up to now important factors of hydrodynamics were poorly understood or were widely disregarded for other reasons. A consistent solution for all kinds of fast swimming locomotion which actually must be presumed had not been found so far.

As in all other vertebrates biomechanical requirements are decisive factors also in swimmers as to the formation of remarkable features and the adaptation to various lifestyles and environments. For the elucidation of different functional aspects in swimming locomotion regularities of aerodynamics and hydrodynamics can advantageously be applied, since water and air have similar properties, although the densities are very different. Hertel (1901 - 1982), an expert in fluid dynamics as I am, has already dealt with this subject, considering technical aspects for the evaluation of properties and capabilities of swimming vertebrates. However, he did not succeed in finding a comprehensive solution for the mechanism of propulsion in fast swimmers.

Here I present a solution of the problem which shows a continuous transition from anguilliform to thunniform locomotion. Several misunderstandings and misinterpretations will be corrected. These had been the results of traditional views and uncritically accepted erroneous conceptions. Different physical requirements upon an animal in a given environment find their expression in its shape in a regular way (Nursall 1958), thus also in swimmers. On the other hand, the fact of a strong adaptation of aquatic vertebrates to requirements of hydrodynamic laws allows an unmistakable judgement of fossil forms as to their preferred locomotory habits and their attainable maximum or continuous speeds, in comparison to modern forms.

1. Physical pre-conditions for the swimming locomotion

In order to be able to move optimally in water, that is with sufficient speed at the lowest expenditure of energy, an animal must take account of the given conditions and physical requirements of the agent as perfectly as possible. Principally, locomotion of all aquatic animals in water is effected by a backward acceleration of water to as high as possible speeds which as a reaction leads to a movement in the forward direction. It is essentially the same principle which is applied in rocket propulsion. The difference is that the rocket propellant contained in the tanks is burned and expelled backwards at high speed. Thus the total mass decreases, whereas a swimmer accelerates the water immediately surrounding its body, with its mass remaining unchanged. This requires high density of the agent in connection with low viscosity. This requirement is fulfilled by water in an almost ideal manner. Propulsion cannot easily be generated in this way in an agent with low density such as air. Therefore, flyers in the air need a low weight and in addition large wings in order to be able to generate propulsion (and lift) by the acceleration of a sufficiently big mass of air at a relatively high beat frequency.

1.1 Physical properties of a streaming agent

There are various possibilities for the indispensable acceleration of the water immediately surrounding the swimmer. Following the history of biogenetics, the first monovesicular beings have already developed the ciliary motion and thereby a mechanism for a change of location. For such tiny forms water represents a viscous agent, like honey for example, which is however not sticky. This property has turned out very advantageous for the development of a swimming capability, since the ciliary movement led to a recognizable change of position and thus actually opened a path for locomotion. In the course of evolution the ciliary motion was diversely modified, improved, and adapted to the respective requirements. By shifting of cilia to certain marginal regions their activities could be bundled and coordinated to form undulating strings. These underwent a further development to undulating fins within different aquatic genera.

formuldAn increase in size of swimming animals is accompanied by a change of hydrodynamic properties, that is, water becomes less and less viscous. This fact requires modified mechanisms of locomotion. Conditions and properties of a flow are adequately described by the Reynolds-number Re. This number denotes the ratio of inertial to frictional forces in a flow. It is defined as  A growing Reynolds-number is indicative of increasing inertial forces, in particular growing centrifugal forces along a curved surface. In general it does not matter if a swimmer is moving in standing water or if it is at rest in streaming water. Growing speed as well as increasing length make the Reynolds-number increase. Details about the range of Re-numbers usually occurring in swimming or flying animals are cited by Nachtigall (1977).

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Fig. 1. Influence of the Re-number on the hydrodynamic properties of a matter with an unusually acute leading edge at high angle of attack, exaggerated for purposes of illustration. 

a: laminar flow at a low Re-number (<102). The flow can follow any contour without separation.

b: laminar flow at distinctly higher Re-number (>104). Because of the inertial forces present there is a flow separation and formation of vortices. The central stream line terminates where the flow comes to rest, and there the dynamic pressure has its maximum

 

Inertial forces do not play an important role as long as the speed is low and the matter in a flow is small, and accordingly Re is low. Frictional forces predominate, although these are low, too. The flow does not lose considerable energy by friction at the surface and it can follow any contour like honey, without a hazard of separation. This flow behaviour is demonstrated in fig.1 for very low Re-numbers and for comparison for somewhat higher ones. Low Re-numbers are present in very small young fishes or in flying insects, in which the smooth plate is completely sufficient as an airfoil section. With increasing inertial forces the flow can no more follow any abrupt change of contour. Therefore, the growing hazard of flow separation has to be considered. Wing sections have to be thickened and stream-lined, as changing thickness and curvature of bird wings with increasing specific size demonstrate. For the same reason the fins of big swimmers such as whales are thickened, and the leading edge is rounded.

At Re-numbers less than approximately 5*105 the flow is laminar, that is, it moves in parallel layers one upon another, without influencing each other. When exceeding the critical Re-number the flow changes its character from laminar to turbulent. The turbulent flow is characterized on the one hand by the presence of (small) vortices, the so-called turbulence. On the other hand, the turbulence allows a considerable exchange of energy between flow layers and thus makes it remain rich in energy for a longer time than the laminar flow. Therefore, the flow separation occurs later. However, the small vortices within the turbulent flow must not be equated or mixed up with the vortices occurring in a separated flow. The turbulence in a flow is not disadvantageous in any case, there are even undoubted advantages at high Re-numbers. Although the frictional drag in a turbulent flow is higher than in a laminar one and should be avoided for this reason, unfortunately a laminar flow cannot be maintained at high Re-numbers. For technical applications an early transition to turbulence is even often artificially induced, because in any case a laminar flow with separation is much worse than the turbulent boundary layer. The separated laminar flow must be strictly avoided, since it results in high drag and, even worse, in a large area in the wake with disturbed flow which makes a downstream control by fins or rudders impossible (fig.1b).

According to its definition the Re-number grows at a given speed V with the running length L along a surface, counting from the tip. In any case the flow is laminar in the beginning, except if turbulence is artificially induced. The laminar flow leads to the lowest frictional drag coefficient, which in addition diminishes with growing speed, but also with increasing running length, since the boundary layer becomes thicker and thicker near the surface due to retardation. Thus, disturbances of a smooth surface protrude less and less from the boundary layer, thereby decreasing the local drag coefficient. However, because of the inevitable energy loss by friction the boundary layer becomes increasingly unstable. The disadvantage consists in relatively high energy losses, which because of the layer structure of the flow cannot be compensated for by an energy supply from the surrounding sound flow, and the result is a tendency of flow separation with increasing running length.

As long as the cross-section of the swimmer increases, and consequently there is a pressure drop along the surface, this property remains unproblematical, since the pressure drop effects an acceleration of flow. Probably this fact is the reason that in many fast moving small to medium-sized fishes moving in the laminar flow region the maximum cross-section can be found in the rear part of the trunk, often accompanied by a backward shift of ventral and dorsal fins.

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Fig. 2. Characteristics of a flow under different conditions, the pressure development along the contour, and the pressure               loss along a spindle-like matter which determines the frictional drag (schematic presentation).     

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Fig. 3. The basically possible different conditions and properties of the medium flowing along a body. Note that the local distribution of conditions may be quite different, that is the change of laminar to turbulent flow may happen earlier or later, depending on conditions.

The situation is completely changed when the maximum cross-section is reached. From now on the flow is decelerated, since the local pressure and consequently the local flow velocity depend on the thickness distribution along the matter. Now the flow must run against a pressure rise, because in an ideal case, that is, without any drag occurring, the flow must rise to the ambient pressure again at the end of the matter. Unfortunately, the laminar flow is not well suited to withstand a pressure rise, and now there is a strong tendency to separation and extensive formation of vortices which are indicative of considerable drag. Therefore, it is preferable that in the region of the critical Re-number the change to turbulence is attained before the maximum cross-section is reached. In that case the flow can experience an increase of energy from outside the boundary layer.

In fig.2 the various possible flow conditions along a spindle-like body are schematically demonstrated. Fig.2a shows the pressure distribution along the contour for an ideal flow without any frictional losses, which leads to a complete pressure recovery at the end. In fig.2b the conditions are shown for a real laminar flow with small losses by friction. In fig.2c and d a comparison of the flow behaviour is depicted for the laminar flow separated from the contour respectively for undisturbed turbulent flow. The respective outlined pressure losses at the end of the matter illustrate the drag amounts. It should be noted that (beyond the critical Re-number) the change from laminar to turbulent flow can never completely be avoided, it can only be delayed. For entirely even and smooth surfaces the critical Re-number is about 5*105, but never more than 106. At this Re-number the change of flow characteristics happens (fig.3). The exact point of flow change apart from speed  is dependent on further parameters such as surface roughness. Thus, under real conditions the flow changes its character at lower Re-numbers. Because of the unstable conditions of a laminar flow it does not happen always exactly at the same Re-number and location. Insofar there is a transitional area. However, contrary to statements in the literature, e.g. Reif (1981a), the flow itself cannot be transitional, it is either laminar or turbulent. The change occurs abruptly. The change can be delayed within certain bounds by a refinement of contour and surface as well as by a backward shift of the maximum cross-section. But ultimately it must happen. However, a refinement of roughness can be advantageous only to a certain degree, since the thickness of the boundary layer increases with the running length. For Re-numbers close to the critical one the flow change should happen before the maximum cross-section is reached in order to make an energy supply from outside possible and overcome the following pressure rise. The maximum velocity allowable for maintaining the laminar flow condition in swimmers can be estimated using the total length. By modifying the Re-number definition we obtain

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This equation implies that swimmers must reduce their maximum velocity with growing length to remain in the laminar flow region. This is a very unpleasant limitation which, however, is surely strictly observed by small fishes. For example the flow change in a fish swimming at a speed of 1 m/s occurs after a laminar running length of 58 cm. At a speed of 10 m/s the change would already occur after 5,8 cm. Thus, the conservation of laminar conditions in fishes such as trouts for example makes sense only as long as the total length does not exceed 30-50 cm, in order to avoid flow separation at the maximum cross-section. On the other hand, the transition to the turbulent flow region does not represent an insurmountable barrier. Hunters that have entered this region gain considerable advantages since their prey is constrained as to its maximum speed.

Shape and position of the maximum cross-section are excellent indicators of the flow characteristics valid in the locomotion of a fish. The maximum cross-section is often found in the rear part of swimmers in the laminar region, for example in pikes which might come close to the critical Re-number during rapid acceleration, and a strong decrease of cross-section behind it is avoided. On the other hand, in big and fast forms such as whales the conservation of laminar flow conditions is possible only in a very small area near the head and, therefore, can no more play an important role. In big tunnies (Thunnus) and sharks the maximum cross-section is located in the anterior part of the trunk, since the transition to turbulence happens early, and a rearward shift of the maximum cross-section would not be connected with an advantage (fig. 4). A quoted thickness position of 70 % by Riess (1986) is not generally valid, only for small forms which can actually stay in the laminar flow region, such as the small tunny of 30-40 cm length depicted by Hertel (1963: fig. 166). Big forms do not have a noteworthy backward shift of the maximum cross-section. They use to swim in the turbulent flow region.       

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Fig. 4. Typical shapes of fishes as an indication of adaptation to different speed regions.   

a: contour of a fast form swimming in the laminar flow region (Lepisosteus, Recent), redrawn from Abel (1912),                                    

b: shape of a fast big shark, swimming in the turbulent  flow region. Arrows indicate directions of locomotion which can be generated by trunk and fins.   

 

A comparison of fish shapes with a laminar wing section as shown by Hertel (1963) is only qualitatively correct and useful to demonstrate general tendencies in the formation of streamlined swimmers. The term laminar spindle used by Hertel (1963) is incorrect and misleading.The flow along a wing section or a spindle-like body is not automatically laminar corresponding to the shape. Laminar flow cannot be equated with low drag in any case, just as little as turbulent flow with high drag. The qualities of laminar (air-)foil sections which differ from other sections mainly by a backward shift of the maximum cross-section have been evaluated experimentally by wind tunnel tests using straight wings and are valid in a strict sense only for a two-dimensional flow. However, the technical application of such sections is restricted to few purposes such as the wings of sailplanes, because of strong cleaness requirements. An extrapolation of qualities to the threedimensional conditions of a spindle-like trunk of a swimmer is therefore allowable only with certain restrictions, as admitted by Hertel (1963). The trunk shape of fishes is not primarily determined by a thrive for laminar flow. The evolutionary aim is in any case a minimization of hydrodynamic drag. This aim is even more decisive for high speeds than for moderate ones. Consequently, a laminar flow at speeds with supercritical Re-numbers has not been proved in dolphins (Reif 1981a). Probably, this statement applies also to ichthyosaurs (Klima 1992). Nevertheless, an optimal adaptation of the skin of all big swimmers to the existing requirements of locomotion is most likely.

1.2 Hydrodynamic drag

For reasons of energy conservation for any location along a matter this equation is valid :

     Pges = Pstat + Pdyn   , with Pges = total pressure [kp/m2], ambient pressure, Pstat = hydrostatic pressure, Pdyn = hydrodynamic                                                                                                                                                                                                  pressure

In a flow without any losses the total of static and dynamic pressure remains constant. The lowest static pressure is reached at the thickest cross-section, that is, where the highest local speed occurs. After having passed it an increasing deceleration happens, accompanied by a corresponding pressure rise. Somewhere behind the matter the overspeed is completely reduced in a real flow. At the end of the matter the ambient pressure can only be recovered in case of an ideal flow (fig.2a). Unfortunately, a dragless flow does not exist in reality. Any matter moving in a flow consumes a certain portion of energy, which results in a downstream force, the hydrodynamic drag. The vortices occurring in the separated flow along a car passing by can well be felt as a turbulent wind. In front of the matter the ambient pressure of the flow is somewhat higher than behind it (fig.2b, c, d). The pressure loss effects the drag force. Therefore

formulc1

The reference area comprises that part of the flow where the pressure loss happens. Since the evaluation of this area as well as that of the pressure loss is hardly feasible it is usual to take measurements of the drag force directly or by using models. It cannot be calculated theoretically, only be estimated with a certain probability. Apart from size and properties of the matter drag is dependent on speed and density of the agent. In order to be able to extrapolate empirical results of measurements to different conditions the drag coefficient cw has been introduced which refers the drag force to a characteristic reference area and makes it independent of matter size and of speed within a certain well defined range. For extrapolations beyond this range the measured drag must be divided into its components which must individually be estimated. For differing Re-numbers for example corrections must be applied. Only the frictional drag component is dependent on the Re-number. It must be related to the surface area, but the shape drag to the maximum cross-section area.

formulb

Since drag increases by the square of speed, its minimization is fundamentally important to all swimming animals. With the exception of the density of water the other parameters of the above equation can be influenced by modifications to a different extent. The simplest possibility for a swimming animal consists in a restriction to low speeds. This applies to many swimmers such as reef dwellers or big plankton eaters such as Rhiniodon or Mola, since a fast locomotion is commonly not required and does not yield additional advantages. For predators this eventuality is generally not practicable as little as for migrating whales. A minimization of the cross-section is possible only within certain limits. However it is rather doubtful whether this aspect plays a role in slender swimmers such as eels (Anguilla).

The primary modification consists in the minimization of the drag coefficient. It is dependent on various parameters such as shape and surface roughness. Under most favourable conditions drag consists of frictional drag only. This drag is unavoidable, because the flow must be slowed down to zero immediately at the surface. This implies a requirement for minimizing the ratio of surface to volume. However, the minimum would be a ball, which is an impossible optimum for a swimmer. Obviously, apart from frictional drag further requirements must be fulfilled which may result from different aspects. Only the simultaneous fulfilment of all requirements leads to a true optimum. Nevertheless, the frictional drag represents the main drag portion in well adapted swimmers. All swimmers attach great importance to its minimization. This can be attained by a suitable development of the surface and by pressing not continuously needed parts such as dorsal and ventral fins tightly to the trunk during fast locomotion. The aim must always be the avoidance of vortices and the suppression of disturbancies as perfectly as possible. Probably, the viscous skin of fishes serves primarily this purpose (Reif 1981a). Unfortunately, boundary layer flow can hardly be described by analytical methods. Despite various efforts a complete solution is not in sight. Partial successes cannot be generalized, although some algorithms for applications in limited areas have been formulated. However, in any case vortices represent considerable drag, contrary to statements by Riess (1986), who presumed that vortices occurring during the sinuous locomotion of certain swimmers were useful for propulsion, if present at all to an extent worth mentioning. The wind produced by a passing car does never take part in propulsion, but is an indication of considerable drag. Furthermore, vortices are not stationary as flow sketches might suggest, they always move downstream. The so-called Karman vortex street occurs in matters with great drag such as a flagstaff in the wind, and it is per se an indication of separated flow. Flyers and swimmers cannot be interested in the generation of vortices happening in partially or completely separated flow. However, they are unavoidable in any case. In a flow without vortices and thereby without drag swimmers would be able to attain higher speeds.

    reibung4

 

Fig. 5 Relationship between the coefficient cw   of the frictional drag and the Re-number. With increasing length or speed the drag coefficient decreases These values were experimentally gained using hydraulically smooth plain surfaces. The transition from laminar to turbulent flow in reality happens in the brownish area, differently from the ideal conditions of the diagram. As can easily be recognized, the transition from laminar flow (blue line) to turbulent flow (green line) leads to a considerable increase in drag coefficient.

While in airplanes the portion of frictional drag of the total drag is only about 5%, in fishes the drag consists almost exclusively of frictional drag. This means that the transition to turbulent flow erects a high fence that can only be crossed by relatively few large forms.

 

Another smaller drag portion is caused by imperfectly streamlined parts of the trunk such as mouth, eyes, etc. Yet, this portion is presumably rather small in swimmers in the laminar flow range. Certainly, these forms are optimally adapted to all relevant requirements. High speeds are not always enormously important and in rather few cases a primary evolutionary aim. Only in big fast swimmers such as lamnid sharks, tunnies, dolphins, whales, and ichthyosaurs with a semilunate tail fin is the drag minimization of special importance and finds its expression in convergent shapes.

Finally there is a drag portion resulting from the variable shape during locomotion. Yet, it may just as well be regarded as a propulsive loss. This is a matter of taste, since it cannot exactly be quantified.

2. Propulsion

As mentioned above the propulsion of swimmers is produced by a backward acceleration of as large as possible quantities of water to as high as possible speed. The change of the propulsive impulse per time unit is

formula

This relation makes clear that propulsion is dependent on the mass moved per second and on the speed increase which an animal can give this mass. In an analogous manner air is accelerated backwards in flying animals, however because of the required aerodynamic lift a considerable additional downward component is included (Ebel 1996). In swimming animals propulsion results from a reaction to the water acceleration. It has its maximum at the beginning, since the forward speed V0 is still zero. On the other hand, the maximum speed is attained when V2 equals V0, thus no further acceleration of water can be achieved. At constant speed there is a balance between propulsion and hydrodynamic drag. The agent must receive just the energy which it loses by drag. This is the only possibility to preserve an attained speed. In addition, in forms heavier than water such as sharks a lift force must be produced, but in general in relation to the propulsion to a low extent only. The most important prerequisite for an effective and fast locomotion consists in keeping the drag as low as possible. The second need is the production of propulsion at as low as possible energy consumption. Furthermore, good manoeuvrability and position control are required. The adaptation to various lifestyles and habitats requires taking account of these conditions, setting quite different priorities.

Braun & Reif (1985) have defined a fundamental difference between forms with continuous and those with discontinuous propulsion. Here I only regard swimmers with continuous propulsion. Discontinuous propulsion is mainly restricted to animals not permanently living in the water or which did not yet complete the adaptation or do not need it. Such forms produce propulsion analogous to a swimming man moving in the water by rowing or paddling with their extremities. This means moving the broadside of the limbs through the water and thereby utilizing drag for propulsion. Paddling and rowing are based on the same principle as parachuting in the air. A locomotion based on the drag of the propulsive organs is inefficient. Thus, it is generally not used by permanently swimming vertbrates. Much more efficient is the hydrodynamically generated propulsion by a forced stream along a profiled section, with different speeds on upper and lower side and thus different pressures along the contour due to a certain angle of attack. Contrary to rowing or paddling in this case the slender side of an hydrofoil is moved through the water. Thereby a force is generated which acts perpendicularly on the foil (fig. 6). Obviously, in the past this fact has caused most errors and problems of understanding. The phenomenon of generating a force by a flow along a hydrofoil or an airfoil is adequately known and understood by the production of lift in flyers in the air, while apparently in swimmers investigations so far have been insufficient. It should be noted that the direction of the generated force is of considerable importance to the argumentation following later on.

Propulsion of swimming animals is always effected by the generation of pressure differences along the contour which lead to an acceleration of the adjacent water mass in the direction of the pressure decrease. This can be accomplished essentially only in two ways, either by unpaired fins or by alternately bending the trunk. The first kind of locomotion is restricted to slow locomotion, but fast swimming is possible only using the second method. Propulsion by trunk bending is based on the same principle as undulating fins. It was considerably modified and improved in the course of evolution. This kind of locomotion developed step by step from the original anguilliform type to the perfect thunniform one. The differences are not fundamental, only gradual. They apply to different frequency, amplitude, and wave length of the undulation, and they are dependent on body size, skeletal features and degree of slenderness of a swimmer. Although similar performances can be attained in different ways nevertheless the applied basical principle is always the same, in this case a backward acceleration of water.

2.1 Propulsion by undulation

Propulsion by undulating fins is a very efficient means of slow locomotion or position control in small to medium-sized fishes, rays or in cephalopds such as Sepia. By varying the direction of undulation, even in partial areas of fins, any direction of movement can be achieved. Such fins are well suitable for manoeuvring. Water is moved in the direction of undulation, and a locomotion in the opposite direction is obtained. Since such fins have to be very mobile, they are restricted as to their stiffness and thus allow small forces only as well as relatively low masses and speeds. Such fins are the ideal mechanism for the slow locomotion of small forms, since the water receives a forced guidance, even if flow separation occurs in partial regions. However, with increasing running length the thickness of the laminar boundary layer grows and, therefore, there is a decrease in the efficiency of undulation towards the end. Thus the effective length of undulating fins is restricted.

The undulation sequence must strictly be followed. It must begin at the posterior part of the fin to start an acceleration of water, since in the beginning only a small change of the original condition must occur; it must start with small quantities of water being moved. Then the undulation is extended to further parts of the fin, and finally the movement is completely in progress and a controlled movement is going on. Now the wave runs from the front to the rear through the fin. Any flexible fin works this way. Locomotion by this kind of propulsion generation is unequivocally defined by the direction of the undulation of the fins and an additional steering device is practically not necessary, since a reversion of undulation is possible.

Closely related to the undulation by fins is the undulating locomotion by means of the trunk. In this case also water is moved rearwards by undulation. This kind of locomotion is mainly used by very slender fishes such as eels or water-dwelling snakes, but also in a modified way by sharks with a weakly developed tail fin living near the bottom, for example Scyliorhinus. The attainable speed in these forms is always moderate, compared to fast swimmers such as big tunnies. The expenditure of energy in European and American eels on the way to their spawning places in the Atlantic ocean is surely very high. The propulsive undulation is more effiicient in roundish trunks if a water exchange from one flank to the other can be prevented by a high fin fimbris as in eels. But it does not become seriously inefficient even in large vertebrates, since several forms such as crocodiles for example utilize a locomotion by an undulating tail. A long and flexible tail represents a precondition for this mode of propulsion. High speeds are not feasible.

From a hydromechanical viewpoint the peculiar feature of the undulating locomotion consists in the fact that the wave-length is considerably shorter than the length of the trunk or the undulating fin (Lighthill 1975). This leads to the favourable effect that all side forces normal to the path compensate each other. Pettigrew (1875) had already recognized that in eels there are always pairs of curvature (~~ ). In forms using this kind of locomotion obviously a caudal fin is not necessary. Although the occurrence of side forces is not advantageous they are, however, unavoidable because the acceleration of the adjacent water can be initiated and maintained in an alternating way only by the respective convexly curved flank. An exact direction of movement can be achieved only if a definite compensation of side forces is possible. Since the water cannot but flow along the body surface its direction is not just backwards but includes considerable transverse components (fig.6). Only an undulation with a small waveength allows the complete compensation of lateral movements. For this reason additional devices for directional control are not required.

2.2 Propulsion by thoracic fins

Another possibility to generate propulsion in small as well as in larger vertebrates consists in the utilization of the thoracic fins respectively modified arms.They are mostly used for propulsion at low speed by fishes living on reefs. However, in larger vertebrates these fins serve predominantly purposes such as manoeuvring and position finding, in sharks additionally for generating some lift. Moreover, ‘thoracic fins’ are used for generating propulsion as well as for manoeuvring by some diving birds and seals which have secondarily adapted to an aquatic lifestyle. This also applies to some fossil reptiles, for example plesiosaurs. This mode of propulsion resembles the undulation of fins as to its effect, in so far as a negative pressure is produced at the surface resulting in a rearward acceleration of water. It demonstrates that the feasibilities of locomotion in the water are principally limited. The wings that have evolved from modified arms are used for the pursuit of fishes under water by birds such as aulks or puffins as well as by penguins with a corresponding course of movement as by birds flying in the air. In order to generate a propulsive force in the forward direction the fins must be moved forward downward. This course of movement can be well studied in turtles because of their low frequency. The generated force of variable size acts always perpendicular to the upside of the fin (fig. 6). The downward movement during pronation results in a propulsive force. Former authors, e.g. Abel (1912), presumed that the swimming locomotion of turtles might be a rowing one. However, the design of the anterior extremities shows clearly that such fins never are paddles, rather hydrodynamically acting wings. This applies likewise to those of mosasaurs, ichthyosaurs, and plesiosaurs. Hydrodynamical lift forces can be much greater than forces of paddles based on hydrodynamic drag.

The utilization of thoracic fins for propulsion is, however, possible only within narrow limits. Since an increase of length (L) leads to an increase of volume and mass with the third power (L3), but only to a change of the corresponding cross-section by the second power (L2), a growing size must inevitably lead to a decrease of speed respectively the expenditure of energy required becomes uneconomically high. This context must be taken into account when regarding fossil forms such as nothosaurs, plesiosaurs, placodonts, and similar forms as to their possible swimming performance. An estimation of the total mass can yield indications whether a speed corresponding to penguins appears obtainable or whether the thoracic fins were preferably used for manoeuvring or stabilization as in most dolphins. During manoeuvring also remarkable forces can occur which can considerably exceed the directional control forces. In that case the thoracic fins do not function as a propeller transmitting energy to the surrounding agent, but on the contrary as a windmill taking up forces from the environment and transmitting them to the skeleton. The special development of the shoulder joint in the Amazonas dolphin Inia geoffrensis (Klima et al. 1980) apparently takes particularly account of an optimum force transmission during manoeuvring, taking up energy. The largest animals using this kind of propulsion nowadays are underwater predators such as penguins and sea lions. For reasons discussed above the attainable speed and endurance by propulsion by the forelimbs must necessarily be considerably lower in a sea lion (Zalophus) than in penguins. But sea lions use additionally propulsion by trunk bending as well as even do otters (Enhydra).

2.3 Propulsion by the trunk  ( next page)

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