New Theory of Propulsion by Tail-Fin Strokes

Twisting of a rudder at the stern of a rigid object. Double row of Eddies flowing away. Eddy formation is the downward stroke of the rear edge. Maximum rudder angle is +/-23 degrees; amplitude a, +/- 30mm; frequency f, 0.4 c/s.
 
The new theory of fish propulsion by tail-fin strokes developed below is based on observation and the simplist laws of fluid dynamics. The fact that this simple solution has not been offered previously demonstrates how little attention is paid by engineers to the natural phenomena associated with fishes.

Tail-fin stroke without forward movement(stationary)
The tail stroke is comprised of "twisting" and "bending," the "twisting" taking place around an axis in the plane of the fin and "bending" being understood as a translatory movement directed obliquely to the plane of the fin. These components of the movements are shown individually and coupled as harmonic oscillations of a fin model. In these theoretical cases essentially only forces perpendicular to the plane of the fin act on the model fin.
 
The following cases are to be taken into consideration theoretically:
  1. Twisting back and forth: This movement at best can effect only a small amount of propulsion through secondary forces. We see that no substantial amount of propulsion is generated from the flow photograph in [the figure above]. This picture shows that a rotation by any significant thrust current directed backward.
  2. Bending up and down: This cannot generate any propulsion, for the hydrodynamic forces are resistances perpendicular to the plate and can have no forward-directed components.
  3. In-phase coupling ( = 0): In this coupling, maximum bending MAX and maximum twisting MAX coincide, and thus is equal to zero when is equal to zero. Propulsive components remain, as in case I, very small (secondary forces).
  4. Coupling with +90 degree phase shift: In this case, rotation leads to bending. In the up stroke, when the bending is zero, the rotation is maximum. When the bending passes through the point = 0, the speed is maximum. The flow separates when the angles of incidence are as large as drawn and hydrodynamic force is perpendicular to the plane of the plate, i.e., ahead and down in the upward stroke, and ahead and up in the downward stroke. Thus, it always has a propulsive component.