Floating locomotion in vertebrates, how does that work?

The locomotion of swimming animals can be easily observed in recent forms. Nevertheless, even today, there are still considerable uncertainties regarding the exact mechanism of movement. Ideas about the locomotion of vertebrates and especially of fish go back in part to the 19th century. Wrong views were never recognized and have been accepted without criticism until today. For example, it is repeatedly claimed that the caudal fin in fish, especially in large ones such as tuna, lamnid sharks and whales, or in fossil reptiles such as ichthyosaurs, produced the propulsion. It may look like the tail fin drove the animal, but in fact, it is quite different. The caudal fin is not involved in propulsion at all! The video of successful thrust by a simplified artificial fish body further down can serve as an unquestionable proof. In reality, the thrust is caused by the alternating curvature of the trunk and the resulting forced flow around the body. More details on the following page. Various amateurish efforts to research, imitate and technically exploit fish propulsion on a false basis have therefore all failed miserably so far, despite the actors’ full-bodied expectations of success

The locomotion of swimmers was also illuminated in the 20th century by various editors and has recently found increased interest again. J. Lighthill (1975) attempted a theoretical approach to the problem from the mathematical side. Mathematical, biological, and certain aerodynamic aspects were addressed, but elements of hydrodynamics were not adequately understood or were ignored mainly for other reasons. A uniform mechanism for all types of swimming, which is actually to be assumed, had not yet been found. Up to now, ideas of swimming locomotion could only partially do justice to the actual conditions, e.g., the movement of very slim, eel-shaped forms. However, when generating the propulsion of fast swimmers with pronounced caudal fins, the solid background could not yet be satisfactorily explained and was still misrepresented.

To get closer to the different functional aspects of locomotion, laws of aerodynamics and also of flight mechanics can be applied analogously when studying aquatic animals, since water and air are similar in their properties, even though they differ considerably in density. H. Hertel (1901-1982), like me, an expert in aviation, had already dealt with this subject by trying to introduce aspects from technology when determining the properties and capabilities of swimming vertebrates. Ultimately, however, he was unable to achieve a comprehensive clarification of the problem due to a mental error. The traditional idea of propulsion by the caudal fin had probably blocked his better knowledge.

In the following, I will now explain a uniform solution, which shows continuity from the anguilliform to the uniform type of movement. Various misinterpretations and misunderstandings can be corrected. However, I want to go into more detail about the relationship between hydrodynamics and locomotion, also at the risk of repeating some of the things I know. Some statements at different publications were formulated so misunderstandable, contradicting, and bumpy, that clarification by someone familiar with this metier can’t hurt.

1. physical requirements for swimming locomotion

To be able to move optimally in water, i.e., with the least amount of energy and at sufficient speed, an animal must take into account the conditions and physical requirements of this medium as correctly as possible. All aquatic animals have a common task in locomotion, namely to accelerate the most significant reasonable amount of water against the direction of movement to the highest possible speed, so that a change in the intended direction is achieved as a reaction. Humans do the same. We try to push as much water as possible backward with our hands and feet, unfortunately with only moderate success, and even flippers cannot do much about it. This principle is based on the exploitation of hydrodynamic resistance and is similar to paddling, so it is very ineffective. Excellent swimmers, on the other hand, make full use of the properties of the water and accelerate the water immediately surrounding them by body curvature. The outstanding feature of water is, on the one hand, its high density and, on the other hand, its excellent mobility. If the mass of air is low, for example, propulsion could not be generated in this way.

Actively flying animals, therefore, need large wings with the lowest possible body weight to be able to accelerate sufficiently large air masses at a relatively high flapping frequency. The propulsion mechanisms of both groups of animals are in principle related. Only in this way could a transition from swimming locomotion to flying in the air be made possible!

1.1 Physical properties of the flowing medium

There are various possibilities for the acceleration of the water adjacent to the float, which is essential for locomotion. In the history of development, a mechanism for locomotion has already been developed in the first unicellular organisms with the ciliary movement. For such small forms, water is a robust medium like honey, but it is not sticky. This property has had a beneficial effect on the development of buoyancy, insofar as the movement of the flagella led to a recognizable change of position or location, thus opening up an original path to targeted locomotion. The circular movement of the flagella has been modified, improved, and adapted to the particular requirements of multicellular organisms in the course of evolution. By relocating flagellate cells to some peripheral regions, the activities of the cilia could be bundled and coordinated so that that undulating bands could be formed. These could be further developed into undulating fins in very different aquatic animal groups.

Later on, the movement always remained a circular movement.

With an increasing body size of floating organisms, however, water becomes more and more mobile, less viscous, and requires modified mechanisms for locomotion, which take into account the changing properties. The state and features of a current are described sufficiently precisely by the size of the Reynolds number Re. This indicates the ratio of inertial forces to frictional forces in a flow. It is defined as

At low velocities and thus small re-numbers, a flow can follow every contour change without the risk of flow separation. The frictional forces predominate. An increasing Re-number is an expression of increasing inertial forces, i.e., centrifugal forces occurring on a curved contour. It is generally irrelevant whether the medium flows around a body at rest or whether the entity itself moves through the resting medium. Both increasing speed and increasing body length lead to a rising Re-number.

At small bodies and speeds and correspondingly low Re-number, inertial forces do not yet play a role, and frictional forces predominate. However, these forces themselves are also tiny. The flow still loses very little energy due to friction on the surface of the body around which it flows and can flow around any desired contour, similar to honey, without the risk of separation. In fig.1, this behavior is shown for very small or, in contrast, for somewhat higher Re-numbers. Short Re-numbers occur e.g., with tiny young fish or also with flying insects, where the formation of the wing as a flat plate is entirely sufficient for the generation of lift. With the increase of inertia forces, however, the flow can no longer follow every abrupt change of contour. Therefore, the growing danger of detachment must be taken into account, so that profiles must be thickened, curved more gently and streamlined, as can be seen very clearly from the changed thickness and curvature of bird wings as their size increases. For this reason, the fins of large swimmers such as whales are also thickened and rounded at the front as their body size increases.

At Re-numbers below approx. 5*105 the current behaves laminar, i.e., it moves in parallel layer packets, which glide over each other without disturbing or influencing each other. If the critical Re-number is exceeded, the transition to turbulent flow takes place, which on the one hand is associated with a certain amount of vortex formation, i.e., turbulence, and on the other hand permits a greater exchange of energy between the individual layers of flow, and therefore remains energetically costly longer than the laminar flow and thus does not tend to detach as quickly. This vortex formation, however, must not be confused or equated with a large-area vortex area due to detached flow. Turbulence in itself is not necessarily a disadvantage; at high Re-numbers, it has advantages due to a lower drag coefficient. At low Re-numbers, the resistance in a turbulent flow is higher than in a laminar flow, and for this reason, it should be avoided. Still, at high Re-numbers, a purely laminar flow is unfortunately no longer possible. In technology, turbulence is often even artificially induced early on by tripping edges, and possibly even in swimmers such as some sharks, because in any case more disadvantageous than the turbulent boundary layer is the laminar flow with detachment, which must be avoided at all costs, since it leads to high resistance as well as a huge area of disturbed flow in the wake, and is therefore no longer controllable by control surfaces located further downstream (Fig.1b).

By definition, the Re-number increases at a given speed according to the running length L reached a given location, calculated from the tip of the body. The flow always has a laminar run-up distance if turbulence has not been artificially forced. Laminar flow causes the smallest possible resistance coefficient due to friction, which decreases with the increasing length of run or speed, since the boundary layer becomes thicker due to deceleration at the body surface, and unfortunately also increasingly unstable due to energy loss. Disturbances of the smooth surface then protrude less and less from the boundary layer, so that the local resistance decreases. The disadvantage is the relatively significant loss of energy, which cannot be compensated for by the surrounding healthy flow due to the stratification of the flow packages, and the associated tendency to detachment with increasing running length. As long as the cross-section of the body around which the flow passes increases, i.e., there is a pressure gradient in the direction of flow, this behavior of the stream is not yet problematic, as the pressure gradient accelerates it. Probably, for this reason, many fast-swimming small to medium-sized fish that move in the laminar flow range show a shift of the broadest cross-section or its incipient reduction far backward, often in connection with a backward turn of dorsal and ventral fins.

The situation changes when the most extensive cross-section is reached. As the local speed and thus also the local static pressure at a body around which flow occurs depends on its thickness distribution, the current is decelerated from here on and must now start against an increase in demand, as the cross-sectional area decreases and the weight must ideally, i.e., if there were no resistance at all, reach the resting pressure of the environment again at the end of the body. The laminar flow can only cope with an increase in strength with difficulty, and it now tends strongly towards detachment and extensive turbulence, which is a measure of the resistance.

Loss-free and lossy flow

Consideration of hydrodynamic requirements for fish

Which flow condition plays a decisive role in fish can often be seen from the reserve of the most extensive cross-section alone. In the case of floats in laminar flow, the area of the most extensive cross-section extends far to the rear, as in the case of a pike, and a sharp reduction of the cross-section behind it is avoided. With extensive and fast forms such as whales, however, the laminar position of the current can no longer play a role.

A comparison of fish forms with a laminar profile (H. Hertel 1963) is only qualitatively correct and useful to show general tendencies in the flow-favorable design of floats. In contrast, the term used by Hertel for a laminar spindle is misleading. A wing profile or a spindle-shaped body is not always flowed around in a laminar manner according to its design; laminar does not automatically mean low resistance, fierce not only high resistance. The properties of laminar profiles, which differ from other wing patterns primarily by a backward displacement of the most extensive cross-section, were experimentally determined by measurements in the wind tunnel on straight sides with such cross-sections and strictly speaking is only valid for two-dimensional flow. Their technical application, however, is limited to a few possibilities due to the required high surface quality and cleanliness and is particularly well suited for the slender wings of gliders. A transfer to the three-dimensional conditions of the spindle-shaped fuselage of a float is therefore only conditionally permissible, as Hertel himself admitted. The striving does not primarily determine the fuselage shape in fish for laminar flow. An evolutionary goal is the minimization of hydrodynamic resistance, but this is not the only one. This goal applies even more to high speeds than to low ones. A sometimes suspected laminar flow, even at supercritical speeds, could not be proven in dolphins (W. Reif 1981). It is also unlikely to have been present on the skin of ichthyosaurs (M. Klima 1992). Physics cannot be cheated. Nevertheless, an optimal adaptation of the coat of large swimmers to all occurring requirements is very likely. However, considerations about the mysterious lack of resistance of fish surfaces are probably based primarily on the false idea of propulsion by the tail fin.

1.2 Hydrodynamic resistance

For reasons of energy constancy, the following relationship applies to the pressure at any location of a body around which a flow is taking place.

At lossless flow, the sum of static and dynamic pressure is constant. Lowest static pressure is achieved at the thickest part of the body flowed around, i.e., at locally highest speed. After exceeding that speed, the flow becomes increasingly decelerated with a corresponding increase of static pressure. After some distance behind the body, Overspeed is reduced to null again. At the end of the body, the resting pressure is completely present still, if the flow around the body does not have any resistance (Fig.2a). In reality, unfortunately, a flow around the body without resistance does not occur. Everybody around which flow occurs withdraws a specific part of its energy from the stream, which is shown as a force in the direction of flow, as hydrodynamic resistance. The disturbed flow around a passing vehicle due to turbulence can be felt as the airstream. In front of the body, the tide has a higher resting pressure than directly behind it (Fig. 2 b,c,d). The pressure loss causes the resistance to force W in the direction of flow. So the following applies

The area comprises the part of the flow around the pipe where the pressure loss occurs. Since the determination of this area, as well as that of the pressure loss after flowing around a body, is rarely practicable, the resistance force is generally determined directly in the test or on models. In theory, it cannot be calculated precisely, but only estimated. In addition to the dimensions and nature of the body, the resistance depends on the density of the medium and the speed. To be able to use empirical measurement results for other applications, the resistance coefficient CW was introduced, with which the force is related to a specific area and is thus independent of the speed and size of the model within a defined range. For extrapolations, the measured value must first be divided into its components, and the proportions must be estimated.

Unfortunately, the flow in the boundary layer is hardly accessible to analytical methods. Despite various efforts, no comprehensive solution is in sight, since lossy flows can practically only be approached empirically. Partial successes can hardly be generalized, even if algorithms for some applications in limited areas have been derived. However, vortex formation always means resistance, in contrast to J. Riess (1986), who assumed that vortices occurring during meandering locomotion could serve to generate propulsion if they are present at all to any significant extent. This author had come up with an unconventional interpretation of hydrodynamics. One would like to shout to him and many others: “Cobbler, stick to your last. The airflow of a car is never involved in propulsion but is always a sign of considerable resistance. Vortices are also not stationary, as flow-pictures might lead people without expert knowledge to assume, but they still move helically downstream. So-called Karman?s vortex-street occurs at bodies with high resistance when flow around, e.g., at a flagpole within the wind, and by itself it?s always a sign of high shape-resistance due to detached flow. Flyers or swimmers may not be interested in the generation of vortices caused by partial flow separation. Nevertheless, they cannot always be avoided. Without vortex formation and thus without resistance, floats could reach higher speeds.

Further, less resistance comes up by fact, the shape of the body can not be shaped optimum flow-conform in any case, e.g., in area of mouth, gills, or eyes. Nevertheless, for swimmers in the laminar flow area, this proportion of resistance should below. They are certainly optimally adapted about all relevant requirements. High speed is not always of paramount importance and is only sought in relatively few cases. Only with large swimmers with a semilunate caudal fin, such as the lamnid sharks, tunas, dolphins, whales, or even ichthyosaurs, minimizing the form drag is of particular importance and then manifests itself in convergent forms. Finally, there is also a resistance component, which results from the changing contour during propulsion generation. However, it can also be interpreted as a loss of propulsion. This is almost purely a matter of taste, especially since it cannot be measured precisely.

However, one should not overestimate the evolutive minimization of resistance in swimming vertebrates. The surface of swimmers can be very differently formed, and yet they can all swim well. Much more important are elasticity and perfect contour retention during the hull movement. They ensure optimal locomotion. The high surface quality was previously greatly overestimated because the exact propulsion mechanism of fast fish had not been recognized.

2. generation of propulsion

The propulsion of swimmers is achieved by moving water at the highest possible speed and in the most significant reasonable quantity against the direction of the floating body, as already mentioned.

This relationship shows that propulsion depends on the mass moved every second, and the increase in speed that an animal can impart to this mass of water. In the case of actively flying animals, the air is moved backward in a similar way, with a considerable component downwards due to the required aerodynamic lift. The propulsion of swimmers is a reaction to the acceleration of water. It is at its highest when the animal accelerates from rest, as the velocity V0 is then still zero. On the other hand, the maximum speed is reached when the outflow speed is the same as the inflow speed, i.e., acceleration of the surrounding water can no longer be achieved. With stationary locomotion, there is a state of equilibrium between propulsion and hydrodynamic resistance. The medium water must be supplied with exactly the amount of energy that it loses due to resistance. Only in this way can a speed once reached be maintained. Also, in the case of forms which are more substantial than the medium, such as sharks, a buoyancy force must be generated, but usually only to a minimal extent in comparison with the propulsion force.

Although the formula for propulsion seems quite simple, its determination is tough. Both the cross-sectional area of the moving water and the speed of the accelerated water are challenging to determine. This is probably one reason why hardly any reliable measurement data is available. Indirect statements must remain unreliable.

A prerequisite for achieving effective and fast locomotion is to keep the hydrodynamic resistance as low as possible. The second necessity is to generate as much propulsion as possible with as little energy input as possible. Also, there is also a desire for excellent maneuverability and position control. The adaptation to different lifestyles and biotopes requires the consideration and emphasis of these conditions with very different weightings.

The hydrodynamic resistance of fish is meager. For example, flying fish, which are related to pike, can cover a distance of about 200 m out of the water at an initial speed of V0 = 70 km/h until the rate is reduced due to friction to such an extent that they splash back into the water.

The big problem for understanding fish propulsion has always been to recognize and understand how the thrust or acceleration of the water occurs. The propulsion of swimming animals is based solely on the fact that negative pressures are generated at the contour, which leads to an acceleration of the adjacent water masses in the direction of the resulting pressure gradient. The pressure differences between the front and back set the water in motion. This can be achieved by continuous propulsion in essentially only two ways, namely by undulling unpaired fins or by alternating curvature of the hull itself, whereby the first type of movement serves exclusively for slow swimming. In contrast, fast movement is only possible with the second.

The propulsion by the curvature of the hull is based on the same principle as with undulling fins but has been considerably modified and improved in the course of evolution. This type of propulsion developed seamlessly from the original anguilliform type to the completed uniform. The occurring differences are not fundamental. They only concern different frequency, amplitude, and wavelength of the undulation and depend on body size, skeletal development, and the slenderness of the swimmer. The torso muscles can generate higher forces to create the torso curvature than undulating fins and ligaments ever could. In nature, similar performances can be achieved in different ways, but the physical principle applied is always the same.

2.1 Propulsion generation by undulation

In small to moderately large fish such as rays and also in cephalopods such as cuttlefish, movement using undulating fins is a very economical way of slow locomotion and position control. By varying the direction of undulation, often even in parts of the fins, any course of movement can be achieved. They are very well suited for maneuvering. By undulation, water is moved in one direction, and locomotion in the opposite direction is achieved. Since such fins must be very mobile, however, they are limited in their stiffness and thus allow only limited forces and relatively low moving masses and speeds. They are the ideal drive for the slow movement of relatively small and tiny shapes, as the water is forced to move even if the flow should become detached in some areas. However, as the length of the run increases, the thickness of the laminar boundary layer increases, and the effect of undulation decreases increasingly towards the rear. The sufficient length of fin seams is therefore limited.

The course of the undulation must be followed very precisely. It has to start at the end of the fins first so that a speeding up of the water can be initiated. This must, therefore, be done at a point where initially only a slight change of the initial state is caused, i.e., only a small amount of liquid is moved. The undulation is then extended further forward until the movement is finally fully initiated, and a clear direction of movement is achieved. However, the wave then runs from the front to the back through the fin, i.e., against the direction of motion. All flexible fins work this way. With this type of propulsion generation, the direction of movement is determined by the direction of movement of the ridges, and an additional control device is not necessary, as the direction of undulation can also be reversed.

Closely related to the fin undulation is the meandering locomotion using the whole hull. Also, here water is transported backward by fluctuation. This type of movement is used primarily by very slender fish such as eels, moray eels, or aquatic snakes, but in a modified form also by sharks living close to the bottom with a weakly developed tail fin such as the catshark (Scyliorhinus). In these forms, the available speeds are always moderate. The amount of energy required for the migration of European and American eels to their spawning grounds in the Atlantic is undoubtedly very high. Undulation remains most effective when, in the case of rounded bodies, water exchange between the two sides of the body can be prevented by an upper fin seam, as in the case of eels. However, it does not become ineffective even with large vertebrates, because crocodiles, for example, use locomotion by the undulation of the tail. A precondition is the possession of a long and very flexible tail. High speeds are not possible.

2.2 Propulsion generation by pectoral fins

A further possibility of generating propulsion in small as well as larger vertebrates is the use of the pectoral fins or modified arms. Reef fish often use them for propulsion at low speed. For most larger fish, however, they are mainly used for maneuvering and positional stability, sharks also use them for low buoyancy generation. For some diving birds and seals, which have adapted to life in the water as a secondary means of survival, they are used primarily for generating propulsion, but also for maneuvering. This also applies to some fossil reptiles such as plesiosaurs. This type of propulsion is similar to the undulation of fins in that it also generates negative pressures on the moving surface and accelerates water backward. However, it also shows, possibilities of movement within stream are limited in principle. The arms transformed into wings are used by penguins as well as many water birds such as aukes or puffins underwater when hunting for fish with a very similar motion sequence as the wings of birds in the air. The fins must be moved forward and down according to the forward thrust. This motion sequence can be observed very well in marine turtles because of their low frequency. The force of variable magnitude produced in this process always acts perpendicular to the top of the fin (Fig.6). The movement downwards during pronation results in a directly forward-directed propulsion force. In the past, it was assumed that the swimming movement of turtles could be oars,. However, the formation of the front extremities clearly shows that all such fins are hydrodynamically acting wings, thus also those of mosasaurs, ichthyosaurs, and plesiosaurs. They can generate or absorb considerably higher forces than paddles. Rowing and paddling, on the other hand, is based on the exploitation of resistance and is an order of magnitude less favorable in its effect.

The outstanding importance of pectoral fin propulsion in the broader sense of the term lies in the fact that small aquatic vertebrates were able to develop wings that created the conditions that later enabled dinosaurs, birds, and bats to leave the water and fly in the air. This was the only way to develop the ability to fly; underwater flight led over to flying in the air.

However, the use of pectoral fins for propulsion is subject to narrow limits. As the volume or body mass increases with the third power (L3) when the length is changed, but the relevant muscle cross-section only increases with the other power (L2), the achievable speed must inevitably decrease as the size of a swimmer increase, or the energy required becomes uneconomically high. This correlation must also be taken into account when assessing fossil forms such as the nothosaurs, placodontians, plesiosaurs, and marine turtles about their possible swimming performance. Some time ago, when I read in a press report about the discovery of a giant plesiosaur in South America that this unusually large animal indeed plowed through the sea formally, I found this assessment between body size and thus presumed high speed only amusing. The rate of these animals was probably always quite modest and did not play a significant role for them. By estimating the total mass, it can be checked whether a corresponding speed could be achieved in comparison to penguins, for example, or whether the fins were not mainly used for stabilization and maneuvering, as is the case with dolphins. The limits of performance seem to be challenging to understand.

Considerable forces can also occur during maneuvering, which can considerably exceed the steering forces for directional control. The pectoral fins do not then act like a propeller that releases energy to the surrounding medium by power input, but conversely, so to speak, like a wind turbine driven by the movement in the medium, which must absorb forces from the environment and dissipate them as well distributed as possible into the axial skeleton. The unique design of the shoulder joint of the Amazon dolphin Inia geoffrensis (M. Klima et al. 1980) seems to be specially designed for the requirements of optimal force application during maneuvering. Today, the most abundant animals with this type of drive are underwater hunting penguins and sea lions. In the case of the sea lion (Zalophus), attainable speed and endurance during propulsion through the forelimbs are necessarily considerably lower than in penguins for the above reasons. However, it also uses at least additionally propulsion by the curvature of the hull, just like even the sea otter (Enhydra).

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