The dynamics of carangiform swimming motions

Abstract

An investigation based on the elongated-body theory of Lighthill (1960, 1970) and Wu (1971) is made of the carangiform swimming motions of fish, a mode of propulsion in which the amplitude of body undulation becomes significant only in the posterior half, or even third, of the length of the fish and the anterior part is relatively inflexible. For typical slender fish performing undulatory swimming motions, three hydro-dynamic features are taken into account: (a) the resistance to the lateral undulatory motions as well as longitudinal (tangential) frictional resistance, (b) the forces of interaction with the water associated with its inertial (virtual-mass) response to the lateral motions of the fish and (c) the reaction forces due to the vortex sheets shed from sharp trailing edges to the rear of the section of maximum span (depth), where a great variety of fins are generally found.The active tail oscillations give rise to oscillatory side forces, to which the remainder of the body responds passively. These passive yawing motions are studied to find their amplitude, the yawing axis and any associated energy dissipation. The contribution of each of the above three forces is examined and the effects of the oscillation frequency, a slenderness parameter δ of the body, and the shape of the transverse cross-sections are considered.The present theory is further applied to predict the turning movement when the fish changes direction, the way in which the above forces act during this process being investigated. A small perturbation analysis relative to a uniform rectilinear motion not only reveals whether or not the motion is dynamically stable, but also leads to a full description of the motion that follows a given initial perturbation. For finite perturbations a numerical method is adopted to find the time development of both the direction of motion relative to the direction of the initial uniform motion and the ratio of the kinetic energy of the body to the initial kinetic energy for different magnitudes I0 of the initial impulsive perturbation in the transverse direction. The final angle of turn is obtained in terms of I0 for different values of δ. The loss of kinetic energy during the turn divided by the initial kinetic energy is found to be very small for a small angle of turn.Detailed analyses are made for rigid fish models in which the distribution of depth (or span) along the body length is quadratic with a maximum at the centre. Agreement of the present analyses with observations is fairly good at least qualitatively and some quantitative estimates are made of the side forces, both oscillatory and impulsive, exerted on the body by the tail. It is found that the vortex sheets shed from trailing edges increase the energy loss due to the yawing oscillations, in addition to the resistive dissipation, but contribute to the directional stability of rectilinear motion.