The three-dimensional waving plate theory is developed to investigate the swimming performance of fish undulatory motion. In particular, the propulsive effectiveness is discussed. The unsteady potential flow over model rectangular and triangular flexible plates performing a motion which consists of a progressing wave with variable amplitudes is calculated by the vortex ring panel method. It is found that the undulatory motion can reduce three-dimensional effects. It is this important hydrodynamic phenomenon that may be one of the main reasons why such undulation is widely used as the swimming method by a large number of aquatic animals. When the span of the undulating plate is nearly unchanged and the wave amplitude is constant or increases slightly along the chord, and the wavelength is close to the body length, theoretical results show that the swimming performance is best and the flow around the plate has a quasi-two-dimensional property. This swimming method may be observed in many fishes, especially those with an anguilliform mode of propulsion. The modification of the anguilliform mode of propulsion to the carangiform mode is also discussed. It is confirmed that the pronounced necking of the body anterior to the tail, which acts to improve the propulsive performance, is a major morphological adaptation of fishes using the carangiform mode, in which the characteristic nature of flexural movement confined to the rear part of the body is that the amplitude of undulation increases posteriorly and no complete wavelength is at any time apparent.

A low-dimensional Galerkin method, initiated by Noack and Eckelmann ͓Physica D 56, 151 ͑1992͔͒, for the prediction of the flow field around a stationary two-dimensional circular cylinder in a uniform stream at low Reynolds number is generalized to the case of a rotating and translating cylinder. The Hopf bifurcation describing the transition from steady to time-periodic solution is investigated. A curve indicating the transitional boundary is given in the two-dimensional parameter plane of Reynolds number Re and rotating parameter ␣. Our results show that rotation may delay the onset of vortex street and decrease the vortex-shedding frequency. © 1996 American Institute of Physics. ͓S1070-6631͑96͒00107-9͔The problem of the flow around a uniformly rotating and translating circular cylinder has been investigated by several researchers due to its engineering importance and academic interest. Badr et al. 1 numerically simulated the steady and unsteady flow past a rotating circular cylinder at low Reynolds numbers Re with rotating parameter ␣, in which Re is based on the cylinder radius R and the incoming velocity U ϱ and ␣ϭR /U ϱ , where represents the angular velocity of the rotating cylinder. Ingham, 2 Ingham and Tang, 3 D'Alessio and Dennis 4 considered numerical solutions of the steadystate N-S equation at subcritical Re. The investigations of the unsteady flow for supercritical Reynolds numbers are relatively fewer than the case of the steady-state flow. Badr et al. 5 numerically studied the time-dependent flow past an impulsively rotating and translating circular cylinder started from rest for ReϾ200, while Coutanceau and Menard 6 gave corresponding experimental results. Chang and Chen 7 investigated the same problem at some higher Re for 0р␣Ͻ2, and suggested there are three modes of vortex shedding existing in wakes depending on Re and ␣.Meanwhile, the research on the bifurcation structure in an open-flow at low Reynolds numbers is of great interest. Provansal et al., 8 Sreenivasan et al., 9 and Schumm et al. 10 experimentally studied the onset of 2-D vortex shedding in the wakes behind a stationary circular cylinder and showed that the transition from the steady to the periodic flow is characterized by a Hopf bifurcation and can be described by the Stuart-Landau equation. Jackson, 11 Zebib 12 and Noack et al. 13,14 numerically investigated the onset of vortex shedding in flow past a stationary circular cylinder by applying the linear stability analysis to an autonomous dynamical system.Following Noack's work, 13 a low-dimensional Galerkin method ͑LDGM͒ is generalized to the case of a 2-D uniformly rotating and translating circular cylinder. Although the LDGM cannot compete with grid-based computational techniques for high accurate simulations of the velocity fields or the resolution of far-wake properties, it is confirmed to be an ideal tool for investigations on global stability and chaos-theoretical analysis. 13,14 In the present Galerkin method, the streamfunction is approximated by a finite ex- BRIEF CO...

We present (1) the dynamical equations of deforming body and (2) an integrated method for deforming body dynamics and unsteady fluid dynamics, to investigate a modelled freely self-propelled fish. The theoretical model and practical method is applicable for studies on the general mechanics of animal locomotion such as flying in air and swimming in water, particularly of free self-propulsion. The present results behave more credibly than the previous numerical studies and are close to the experimental results, and the aligned vortices pattern is discovered in cruising swimming.

Fish usually bend their bodies into a ''C'' shape and then beat their tails one or more times to escape from predators (in nature) or stimuli (in experiments). The maneuvering behavior, i.e., the C-shape bending and the return flapping, is called C-start. In this paper, the escaping performance of fishlike C-start motions has been numerically investigated for a flow physics study by the use of a two-dimensional deformable foil bending and stretching quickly. The C-start motions, performed in the quiescent water and based on prescribed deforming modes, are predicted by a numerical method coupling the two-dimensional incompressible Navier-Stokes equations and the deforming body dynamic equations. It has been found earlier that a typical C-start motion consists of (1) a main C-shape bending and (2) a rearward travelling curvature wave which was seldom mentioned in previous studies. In order to reveal the flow control mechanism of the traveling curvature wave in a fish's C-start motion, two kinds of C-start flows with different deforming modes, namely the integrated mode (IM, a C-shape bending plus a travelling curvature wave) and the basic mode (BM, a C-shape bending only) are analyzed and compared in detail. According to the numerical results, it shows that if proper values of the travelling curvature wave parameters are chosen, the foil's escaping maneuverability presented in the IM is much better than that in the BM, i.e. the turn angle and the speed of the center of mass at the end of a C-start in the IM is almost twice as large as those in the BM. Further study shows that the travelling curvature wave not only can enhance the thrust and the centripetal force but also increase the propulsive efficiency. These results suggest that an efficient travelling curvature wave is of great significance in the flow control of a C-start motion. Finally, a parametric study finds that the phase difference between the C-shape bending and the travelling curvature wave (i.e., the initial phase angle in the travelling curvature wave of the deforming model) is a key parameter in the flow control. To achieve the desirable turn angle, escaping speed, and propulsive efficiency in the C-start motions, the initial phase angles must be ranged within specific magnitudes. It is found that for optimum values of the initial phase angle, the foil's flexible deforming process is qualitatively consistent with that of a fish body in nature. The results obtained in this study provide a new physical insight into the understanding of swimming mechanisms of fish's C-start maneuvers.