2.3 Propulsion by the trunk

Aquatic vertebrates with a spindle-like trunk have developed a much more efficient, but also more complicated and in a way more demanding method of producing propulsion by alternately bending the rear part of the trunk, in contrast to undulation of the whole trunk. Although this kind of propulsion is used by all fishes with a well developed tail fin in the same way, it can be most efficient only in large forms moving at high speed in a turbulent flow. As mentioned above, the turbulent flow contains more energy than the laminar flow, it can better overcome a pressure rise, and therefore does not tend to separate early. The contour may have a stronger curvature before interferences of the boundary layer and flow separation occur. Thus larger propulsive forces are possible. This flow property is used by all fast swimmers for the generation of propulsion by trunk bending. Presumably, high speeds in large vertebrates have become possible only by this kind of propulsion. It is the only possibility for really fast and continuous locomotion. However, this kind of propulsion does not differ principally from propulsion by undulation. Propulsion continues to be generated by acceleration of water, but now the strongest muscular contraction is concentrated to the posterior half or even the third part of the trunk (Lighthill 1975). The highest acceleration happens in the region of the strongest curvature and thus at the lowest local pressure. It requires a particularly supple and flexible nature of the contour and a very lithe vertebral column, that is a high number of vertebrae. The well developed tail fin is an excellent indicator of this kind of propulsion. But this fin never serves for propulsion or exclusively for directional control as presumed for instance by Riess (1986) for certain ichthyosaurs with semilunate caudal fin. The propulsion is generated immediately at the contour, moreover in any case in front of the tail fin. For generating propulsion it is not sufficient to bend the trunk only by muscular contraction and to create a convex surface at the opposite flank. In addition, by creating a negative pressure on the surface of the lengthened flank the flow shall attain a backward direction. For this purpose during a half cycle the point of strongest curvature must move forward. For an optimum effect the propulsion in this case also is generated by a very exactly coordinated succession of muscular activities of the trunk, in fishes as well as in ichthyosaurs from the left to the right flank because of an easier lateral motility, in aquatic mammals because of a better dorso-ventral motility of the vertebral column up and down (Klima 1992). It is possible to imitate technically this method of generating propulsion if the relevant preconditions are sufficiently observed.

It is by no means the task of the trunk musculature to move the tail fin, as frequently mentioned in the literature. This fin is normally moved in a more or less passive manner. The fact that the caudal fin cannot be involved in the generation of propulsion can be derived from the position of the relevant musculature. This musculature is in any case situated in front of the tail fin which itself is not very muscular, although for example in whales stout and elastic (Klima 1993). The musculature is located along the trunk, and it is subdivided into several separated systems (Alexander & Goldspink 1977). In cetaceans it can be regarded as a remainder of the phylogenetically ancient structure of the trunk musculature (kind information by Prof. M. Klima). In fish this musculature is subdivided by myosepta into multiple segments called myomeres (Carroll 1993). The probable function of this segmentation as outlined in fig.5 was poorly understood so far (Reif 1981b). It finds, however, a satisfying explanation by the fact that, as mentioned above, the course of curvature must happen by an exactly coordinated sequence to achieve the most efficient propulsion. It is basically the same effect which generates propulsion in eels. The muscular contractions run through the trunk in an alternating manner, in this case from the rear end forward (Carroll 1993).


Fig. 5. Sketch of the musculature generally present in fishes along the trunk with <-sized myomeres, subdivided by myosepta into segments. Redrawn from Carroll (1993). 

Thus, initially a small quantity of water is moved at the end of trunk giving a starting direction of the accelerated water. With the trunk bending proceeding forwards further portions of the adjacent water are included in the motion. Thus a locomotion towards a well defined direction is achieved. An important precondition for such a locomotion consists in the presence of a very flexible vertebral column with a large number of vertebrae which puts up a great resistance to the muscular contraction and reacts to the shortening of the contour by muscular contraction on the one flank with a corresponding lengthening on the other. In analogy to the undulating locomotion during a cycle propulsion is produced only on the convexly curved side of the trunk. It is well known that a continuous propulsion as in a rotating propeller is not feasible in animals. Therefore, a comparison with a propeller is inappropriate.

videomodellfischanimationRecently, I received a video of a test vessel powered by an artificial fish. Statements of this page have been used for the development of this kind of propulsion, and thereby it demonstrates the correctness of my ideas.

   Video  fish propulsion

       Movement of the fish      


2.4. Gray’s Paradoxon

The problem of propulsion in fast fishes has caused the reatest uncertainty and complete misunderstanding in biologists in the past, following from tradional, though incorrect ideas, which have never been seriously doubted. James Gray who in the thirties of the past century dealt with the physics of swimming vertebrates even created his paradoxon. He could not understand how the power could be generated needed for the speed he observed in fish. As any such a paradoxon this also results from poorly understood physical relationships. It was based on the old conception that propulsion had to be produced by the tail fin. Certainly, Gray knew that something was wrong concerning his idea of propulsion, but he was unable to find an explanation. Although Hertel (1963) recognized a participation of the trunk in the motion, he also could not abandon the idea of propulsion by the tail fin respectively he did not seriously question it. He opined that fishes move with minimum expenditure of energy, which finally is true, or they use the principle of “ingenious laziness”.

If in fish the propulsion would be generated by a propeller a power

                       L  =  W x  V                  with  W = drag   and   V0 = forward speed

would be required. However, fishes do not at all generate a propulsive force, only a tranverse one occurs by bending the trunk (fig.7). They only activate alternately the lateral musculature, in a faster or slower mode, and beyond that they do nothing at all. In particular, they do not spend power for propulsion. Curiously, propulsion is completely independent of body engagement, only dependent on the bending frequency. There is no work in the direction of path, the work is

                       A  =  K  x s   = 0               with   A = work,   K = force,  and   s = path

This means that only a small fraction of the power by propeller propulsion is needed. It is a wonderful trick which fishes have discovered for their motion, so concealed that biologists were completely unable to recognize it. The alternating bending of the trunk and the resulting pressure distribution along it makes the water flow past the body and initiates a movement in the opposite direction. As in many such cases finally the problem of this paradoxon now is settled to everyone’s satisfaction.

3. Function of the caudal fin during fast locomotion

Propulsion might also be produced without the tail fin present, but an intended exact direction would not be possible. Experiments with the tail fin of fishes amputated have demonstrated that they continued to generate propulsion, however at an increased frequency (Gray 1933), presumably because of the missing damping effect by the tail fin. Curiously enough, even this result did not initiate a new thorough reasoning about the mechanism of propulsion, which obviously could not depend on the tail fin. On the other hand, this fact demonstrates how difficult it appears to seriously doubt a traditional, but nevertheless erroneous text book opinion. It is even more likely that the results of observations are questioned, as in this case by Hertel (1963). In contrast to such a procedure, a solution that takes account of physical laws respectively of their effects on skeleton and shape of animals is free of contradiction and does no more allow an entirely different alternative. Moreover, it is then in full agreement with all carefully obtained observational results.

So far, generally a distinction was drawn between two propulsive modes by the trunk, namely on the one hand the movement using propulsion by the whole undulating trunk as in eels and propulsion by the tail fin with the trunk relatively little moving on the other hand (Carroll 1993). This distinction is not particularly significant, since there is only a gradual difference between these two modes; they are connected by transitions. Mechanism and function of the tail fin have hitherto thoroughly been misunderstood. According to traditional views, e.g. Abel (1912), propulsion in fast moving fishes is performed by strong deflections of the caudal fin. Up to now this view is widespread. For example it is adopted by Lighthill (1975), Robinson (1976), Braun & Reif (1985), Sander (2000). Although Hertel (1963) has presumed a cooperation of trunk and tail fin regarding the generation of propulsion, his ideas did not meet the real conditions, since he did not correctly recognize the mechanism of thrust generation. While the anguilliform propulsion is long well understood, in forms with a well developed caudal fin curiously an entirely different mechanism was presumed which up to now could never be clarified.



Fig. 6. Locomotion of a fish.  This is the traditional view of generation of propulsion by the tail fin, comparable to the propeller of a ship, from O. Abel (1912). Although this sketch is amost 100 years old, there has not been much progress in the mean time.


The movement of the tail fin alone cannot result in a locomotion of the animal. For example the killerwhale (Orcinus orca) employs its tail fin for stunning individuals in densely crowded herring swarms. This happens without a forward movement. However, of course the fin plays an important role during fast locomotion. At first view the impression of propulsion being generated may arise, since it moves with the same frequency as the trunk. But for this task the tail fin is not at all suitable. On the other hand without it a directed locomotion would not be possible. Although all previous statements were surely based on very careful observations, nevertheless these were erroneously interpreted. Abel (1912) wrote: ”Wenn wir eine Bachforelle beobachten ..., so sehen wir deutlich, dass diese rapide Bewegung durch starke, seitlich ausgeführte Schlaege der vertikal stehenden Schwanzflosse erzielt und von einer schlaengelnden Bewegung des Koerpers in Form einer halben 8 in der Horizontalebene unterstuetzt wird”. (Observing a trout... we can clearly see that this rapid locomotion is obtained by strong lateral deflections of the tail fin which is supported by an undulating movement of the trunk describing half an eight in the horizontal plane.) Obviously, even the most accurate observations are per se not sufficient to guarantee a correct functional explanation. Apparently the function of the tail fin cannot be cleared up without a thorough knowledge of the underlying hydrodynamic laws. Thus a mix-up of cause and effect is unavoidable. As long as misunderstandings persist new proposals are brought forward again and again, for instance recently by Loercher (1999) who presumes a propeller analogy in Jurassic ichthyosaurs. This idea had previously been proposed by Kripp (1954) for ichthyosaurs and by Kuekenthal (1908) for whales.

In contrast to the undulating locomotion in slender swimmers such as eels with a body length exceeding the wave length of undulation considerably, in fishes with a well developed caudal fin the wave length of the trunk motion in any case is larger than the body length (Lighthill 1975). The corresponding fin deflection is relatively small. This movement cannot really be called an undulating one. Only the hindmost part of trunk is involved in the movement. This difference from the undulating locomotion leads to an aggravating difference of physical conditions which requires consequences, because in this case the occurring side forces due to trunk bending are not automatically compensated. The forces occurring during generation of propulsion do not only have an average axial direction, but at times also considerable transverse ones, since the water is accelerated along the tapering contour only on the respective convexly curved side. Therefore, the compensation of forces required for a controlled locomotion has now to be done in another way. This is the primary function of the tail fin. This is also the reason why it is present only in forms in which the wave length of trunk bending is definitely greater than the flexible trunk length. In all other forms a tail fin is not necessary. Unequivocally, it has the task to give the flow an intended direction. Thus, it serves exclusively directional control and steering in the fast trunk-driven locomotion. If the occurring forces are not entirely compensated the animal swims a curve. However, the tail fin is not capable of accelerating water for the generation of propulsion. It only can change the direction of water which has been accelerated.

A consideration as to the feasibilities to generate a force in a flow along a more or less cambered plane under a certain angle of attack makes rapidly clear that only a force acting vertically on this wing area can be produced hydrodynamically or aerodynamically. Although an analogy between the tail fin of swimming vertebrates and the wings of birds or flying insects has previously been considered, e.g. by Lighthill (1975), and which is surely present, nevertheless the misinterpretation of the tail fin as to its function and effects has obviously completely been overlooked so far, since a function other than generation of propulsion appeared impossible. But a check of the acting forces and their directions would have clarified the real facts much earlier. This aspect was previously not at all or incorrectly treated. Unfortunately, Hertel (1963) made a mistake about the direction of the acting forces and did not consider that a force must act vertically on a wing section, since it represents the resultant pressure distribution of upper and lower surface. The total force is then splitted up into its components of drag and lift (fig. 7).



Fig. 7. Flow along a profiled wing area or a fish trunk and direction of the generated force as well as its components of lift and drag. For a better clearness the angle of attack is considerably exaggerated compared to real conditions.          


Different physical properties in flyers and swimmers, based on the same principle

Two different physical effects are utilized for flying in the air and swimming in the water, which nevertheless are both present in the flow along a cambered wing section. These effects are the local acceleration of the agent on the one hand and the generation of a force on the other. Although these effects are causally connected with each other, their importance is, however, extremely different for flyers and swimmers. The airfoil section of a flyer is more or less rigid as shown in fig.7. The differences in speed on upper and lower sides create a force analogous to the side force. Lift as its vertical component with respect to the flow direction compensates for the weight of a flyer during level flight. The aim of a flying wing consists exclusively in the generation of a continuously acting lift force. The axial component of drag is disadvantageous except for slowing down, but it is unavoidable. A propulsive force cannot be generated in this way. In airplanes it must additionally be produced by engines, in birds by pronation and supination of the wing beat. The fact that during lift generation the airflow is accelerated on the upper side of the wing is of very little importance to the process of flying. The occurring forces and moments must be balanced by the tail unit. As in fish the lift in flyers is gratis, because the work in the direction of path is zero.

The trunk of a fish presents a basically analogous behaviour as a wing section, and its periodically changed camber results in an alternating pressure distribution along the contour. But in contrast to flyers in the air the occurring side force is disadvantageous, although unavoidable. Also in swimmers it has nothing to do with the production of propulsion, as forces erroneously attached to the tail fin suggest again and again in the literature. In this case only the acceleration of the adjacent water which is about 800 times denser than air is utilized for locomotion (fig. 8). As a reaction the swimmer moves in the opposite direction. Thus, we have two very different effects of the same physical principle. This relationship is not so obvious and has not been recognized so far. I must admit that it was a surprising realization for me as a former specialist in aerodynamics, too. A comparison of a tail fin with the wing of a bird is only correct in so far as both represent areas where forces occur in a flow. But function and effect of the tail fin can only be compared to the empennage of an airplane or with the tail feathers of a bird, which are used for steering and compensation of moments. The tail fin acts as a rudder, and nothing else is possible



Fig. 8.  Speed V2 generated by the trunk at the location 2 and its direction. A corresponding turn back VF with a transverse component VFvert must be generated by the tail fin in order to obtain a resulting forward speed VR.

Fig. 8 shows schematically the speeds present at trunk and tail fin. Following from the camber attained at this point of time water is accelerated from the swimmer’s speed V0 to V2 at the end of the trunk. This leads to the outlined direction with a transverse component V2 vert, which must be compensated for by the tail fin. For this purpose it must change its angle about the trunk axis and turns the flow back. Thus the transverse component is compensated. The resulting speed VR is remaining, which together with the accelerated water mass produces the instantaneous thrust S along the track.

According to fig. 8 the thrust is


The instantaneous value of thrust is dependent on time.

Apart from the consideration of the occurring speeds and their directions an evaluation of the balance of forces leads the same result concerning the function of the tail fin. As fig. 9 demonstrates, at any time of the propulsive stage corresponding to the variable camber a side force variable as to size, direction, and location is present on the contour. This force does not necessarily act in the centre of gravity and thus strives to turn the animal out of its direction of movement. In order to prevent such a turn the tail must continuously produce a corresponding counter-acting force which balances the moments. In addition, fig. 8 makes clear that the two acting forces can only have retarding axial components. Fig. 9 yields the relation for balanced moments about the centre of gravity

         Sk * a - Fk * b = 0, with Sk = side force, Fk = tail fin force

For the stated reasons the fin force must also be vertical on the ”wing” tail fin. This hydromechanical effect has the consequence that the anterior part of the fish can maintain its alignments relative to the direction of motion. Nevertheless, the side force effects a slight shift of the whole body in the direction of the acting side force. Therefore, the centre of gravity does not move exactly on a straight line, but on a slightly sinuous one. Possibly, this fact is the reason that certain cetaceans during foraging close to the bottom use to swim with a flank down. Presumably, this behaviour facilitates maintaining a constant distance from the ground.



Fig. 9.  Compensation of the side force by the the tail fin.

         Sk = side force caused by trunk bending,

         Fk = counteracting fin force, CG = centre of gravity


Fig. 9 shows that the moment generated by the tail fin (force Fk times lever-arm b) depends on the size of the force as well as on its distance from the centre of gravity. The required force of the tail fin for the compensation of the side force can therefore be diminished to the same degree as the distance from the centre of gravity increases. Since the distance of the tail fin from the centre of gravity is always larger than that of the occurring side force, the force required for compensation is considerably smaller than the side force. Without a caudal fin a fast and controlled locomotion by trunk bending is not possible, because a balance of side forces would not be feasible.

The rear end in certain cetaceans is slim from the dorsal view but relatively high. This fact might suggest the impression mentioned by Hertel (1963) that this trunk part can contribute little only to propulsion. This impression is however incorrect. Generation of propulsion by bending the posterior part of the trunk is a fundamentally valid principle. The validity can therefore not be constrained by divergences from an as ideally assumed shape. It applies to cetaceans as well as to all other forms with a semilunate tail fin. Propulsion does not originate only from the posterior trunk section, but similar to the lift generation in a bird wing is a consequence of the flow along the whole contour and the emerging pressure differences between the differently curved surfaces. On the one hand the great length of the trunk offers the advantage that the fluke can remain relatively small, because of its large distance from the centre of gravity  (fig.9). On the other hand the large forces needed to bend the trunk in dorsoventral direction require an as large as possible lever-arm for the trunk musculature (Ebel 2000: fig. 8), and thus a large distance from the axis of the vertebral column, and a resulting big height of the posterior trunk.

Since the only function of the tail fin consists in a turn of the flowing agent, respectively in the balance of moments, it cannot have another function, not even secondarily. If it is weakly developed as for example in the Recent lung fish Neoceratodus it does not play an important role for locomotion. A function beyond the directional control cannot be present also in fast and big swimmers, while in small forms an additional vertically acting undulation is possible (fig. 4a). The shape of the tail fin is primarily given by the aim of maximum efficiency with the smallest fin area. Moreover the load above and below the strut should be equal to avoid a distortion.

Among others Alexander & Goldspink (1977: 238, 239) also are of the opinion that the propulsion is generated by a strong lateral deflection of the tail fin. These authors show an illustration of a fin deflection which cannot be harmonized with observations of the really occurring fin angle, just as little as can a figure shown by Hertel (1963: Abb. 168). Such a deflection can be present only by chance at the beginning of the locomotion when the flow is not yet established.

Lighthill (1975) regretted to be unable to formulate an algorithm describing the thrust produced by the tail fin. Although he was aware of the problem in general he did not recognize that the physical pre-conditions were incorrect. He mentioned an unexplicable violation of the principle that locomotion by swimming is accomplished by a backward acceleration of water, caused by the lateral deflection of the fin. He could not make a statement about the remaining side force, and he presumed that this force was negligible. Moreover, he presumed that the propulsive force was caused by a negative pressure at the leading edge of the tail fin, which in fact could only be responsible for a a very small portion of the total force required.

Kripp (1954) already had recognized that in fast swimmers transversal forces are present which must be ”stabilized”, that is, must be compensated for by the tail fin. In addition, he was aware that in fact a propulsive force could not be generated by a lateral deflection of the tail fin. Nevertheless, he was thoroughly convinced that the propulsion was effected by this fin, and therefore he considered a propeller analogy. He opined that the caudal fin for this purpose had only to be distorted similar to a propeller. But actually propulsion cannot be produced this way, since no further acceleration of fluid takes place. A propeller analogy is a wrong idea. The distortion of a propeller is necessary for quite another reason: since at a constant number of revolutions the tangential speed increases with a growing radius to maintain a constant angle of attack, therefore, the angle of incidence has to be reduced with a growing radius. The flow through a propeller is at any radius vectorially composed of forward speed and tangential speed. The distortion takes account of the variable angle of the resulting speed and guarantees optimum angle of attack, optimum flow conditions, and thereby the highest efficiency.

Since up to now the caudal fin has erroneously been regarded as the primary propulsive organ, the different shape in various forms could not correctly be understood and has caused various misinterpretations about its function. Several workers such as Abel (1912) and Lighthill (1975) have supposed that the different shape of caudal fin lappets served the purpose of generating a vertically acting force, for example hydrodynamic lift in forms heavier than water. However, since this force, respectively the moment of force times lever-arm, had to transmitted by the fin strut, the presumed effect cannot be obtained this way. The pectoral fins as for example in sharks are much better suited for this purpose and allow an exact control, because they have a larger effective distance (lever-arm) from the centre of gravity. There have been several presumptions with in part complicated approaches about the semlunate, arrow-shaped or other forms of the caudal fin and its particular function in big swimmers, e.g. by Lighthill (1975). Yet, physical principles used in nature are mostly basically rather simple. The asymmetrical shape of various caudal fins surely has other reasons, for example a primarily bottom-related or a near-surface mode of life in such forms.



Fig. 10. Ideal configuration of trunk end and caudal fin for the generation of maximum propulsion, minimum drag, and optimum deflection. 

a: without an effect on the flow, turbulent flow partly separated,          

b: with vortex generators effecting the flow and thus considerably less vortices due to flow separation.


Presumably, in general it is sufficient that the turn of the flow is passively performed by a suitable flexible attachment of the caudal fin to the vertebral column, but not by an active control.This only becomes necessary in very fast swimmers, and then is supported by special joints (Fierstine & Walters 1968). Musculature of the trunk and stiffness of the caudal fin must be designed to safely perform the required turn of the flow any time. With increasing importance of the thrust propulsion by the trunk the tail fin has to be increased and stiffened, the distance from the centre of gravity must as well be increased. Apparently, the span of the tail fin is closely connected with the maximum height of the trunk with its attached fins. Ideally, the trunk would end like a spindle at the rear end and thus reduce the thickness to a minimum. This would result in a minimum drag of the wake. As shown in fig.9, the tail would follow after a short distance and would generate the necessary turn of the flow. Of course, in reality such a configuration is impossible, but there is a tendency towards it. Apparently, during the fast locomotion of big swimmers at maximum power there can be large areas with turbulent vortices in the wake of the trunk, which must completely be turned back by the tail fin. In order to perform an optimum turn in this region the span of the tail fin should actually be considerably larger ( fig.9). In scombrids, fast swimmers which must be able to move at high speed with maximum power, there is a crest on dorsum and venter formed by finlets, which probably serves the same purpose as so-called dorsal fins and vortex generators on the rear fuselage of certain airplanes. These are used to prevent a flow separation by generating small vortices in advance and thus improving the flow conditions.

All swimmers moving in the turbulent flow regime indicate by the presence of the fully developed caudal fin and the convergent shape that they have reached a stage of adaptation which cannot be further improved concerning the fast locomotion.