Theodore Y. Wu

Hydromechanics of low-Reynolds-number flow. Part 5. Motion of a slender torus

In order to elucidate the general Stokes flow characteristics present for slender bodies of finite centre-line curvature the singularity method for Stokes flow has been employed to construct solutions to the flow past a slender torus. The symmetry of the geometry and absence of ends has made a highly accurate analysis possible. The no-slip boundary condition on the body surface is satisfied up to an error term of O(E^2 ln E), where E is the slenderness parameter (ratio of cross-sectional radius to centre-line radius). This degree of accuracy makes it possible to determine the force per unit length experienced by the torus up to a term of O(E^2). A comparison is made between the force coefficients of the slender torus to those of a straight slender body to illustrate the large differences that may occur as a result of the finite centre-line curvature.

A generalized slender-body theory for fish-like forms

A consistent slender-body approximation is developed for the flow past a fish-like body with arbitrary combinations of body thickness and low-aspect-ratio fin appendages, but with the fins confined to the plane of symmetry of the body. Attention is focused on the interaction of the fin lifting surfaces with the body thickness, and especially on the dynamics of the vortex sheets shed from the fin trailing edges. Explicit results are given for axisymmetric bodies having fins with abrupt trailing edges, and calculations of the total lift force are presented for bodies with symmetric and asymmetric fin configurations, moving with a constant angle of attack. (Modified author abstract)

Nonlinear water waves in channels of arbitrary shape

The generalized channel Boussinesq (gcB) two-equation model and the forced channel Korteweg-de Vries (cKdV) one-equation model previously derived by the authors are further analysed and discussed in the present study. The gcB model describes the propagation and generation of weakly nonlinear, weakly dispersiveand weakly forced long water waves in channelsof arbitrary shape that may vary both in space and time, and the cKdV model is applicable to unidirectional motions of such waves, which may be sustained under forcing at resonance of the system. These two models are long wave approximations of a hierarchy set of section-mean conservation equations of mass, momentum and energy, which are exact for inviscid fluids. Results of these models are demonstrated with four specific channel shapes, namely variable rectangular, triangular, parabolic and semicircular sections, in which case solutions are obtained in closed form. In particular, for uniform channels of equal mean water depth, different cross-sectional shapes have a leading-order effect only on the variations of a K-factor of the coefficient of the term bearing the dispersive effects in the model equations. For this case, the uniform-channel analogy theorem enunciated here shows that long waves of equal (mean) height in different uniform channels of equal mean depth but distinct K-shape factors will propagate with equal veolcity and with their effective wavelengths appearing K times of that in the rectangular channel, for which K=1. It also shows that the further channel shape departs from the rectangular, the greater the value of K. Based on this observation, the solitary and cnoidal waves in a K-shaped channel are compared with experiments on wave profiles and wave velocities. Finally, some three-dimensional features of these solitary waves are presented for a triangular channel.

A porous prolate-spheroidal model for ciliated micro-organisms

A fluid-mechanical model is developed for representing the mechanism of propulsion of a finite ciliated micro-organism having a prolate-spheroidal shape. The basic concept is the representation of the micro-organism by a prolate-spheroidal control surface upon which certain boundary conditions on the tangential and normal fluid velocities are prescribed. Expressions are obtained for the velocity of propulsion, the rate of energy dissipation in the fluid exterior to the cilia layer, and the stream function of the motion. The effect of the shape of the organism upon its locomotion is explored. Experimental streak photographs of the flow around both freely swimming and inert sedimenting Paramecia are presented and good agreement with the theoretical prediction of the streamlines is found.

Extraction of Flow Energy by Fish and Birds in a Wavy Stream

This paper makes an attempt to explore the possibility for birds and aquatic animals to extract intrinsic flow energy from a wavy (or even turbulent) stream for enhancing their locomotion through the fluid medium. Sea gulls and pelicans have been observed to skim ocean waves over a long distance without making noticeable flapping of their wings as in ordinary flight. In a study of migrating salmon, Osborne (1960) found that an increased flow rate in a swollen river did not slow down the salmon, while en route to spawn, by nearly as much a margin as would be predicted by the law of resistance in proportion to the square of their velocity relative to the flowing water. A few possible reasons have been proposed, including the prospect that the flow energy associated with the waves and eddies in a stream can be extracted and used to generate thrust for locomotion.

On theoretical modeling of aquatic and aerial animal locomotion

Publisher Summary This chapter discusses the theoretical modeling of aquatic and aerial animal locomotion, several objectives that are focused on exploring how, why, and under what premises such high efficiency and low-energy cost can be achieved in these diverse modes of locomotion as a result of a long history of convergent evolution. The chapter takes an integral viewpoint from the foundation built by the pioneering leaders in the field, such as Herbert Wagner, Theodore von Karman, William R. Sears, and Sir James Lighthill, followed by other researchers through developing various theoretical and experimental methods used in studies on the subject. In subdividing the various classes of hydrodynamic theories and the underlying physical conceptions, it is seen that the generalized slender-body theory is readily capable of expounding the complex interaction between the swimming body and the vortex sheets shed from the appended fins, caudal fins, or lunate tails. Some important nonlinear effects are considered, and separate resort is made for mechanophysiological studies on energetics and hydromechanics of fish propulsion involving the biochemical and mechanical conversions of energy.

A unified theory for modeling water waves

Publisher Summary This chapter presents a unified theory for modeling water waves. The different types of waves in water, varying from ripples on a placid pond, breaking of shoaling waves on a beach, billows on a stormy sea and in the ocean interior, to geophysical waves and devastating tsunamis, are truly extensive. Eulers equations are adopted to describe three-dimensional, incompressible, inviscid long waves on a layer of water of variable depth, which may vary with the horizontal position vector. Four sets of theoretical models for describing fully nonlinear fully dispersive (FNFD) unsteady gravity-capillary waves on water of variable depth in terms of four sets of basic variables are obtained. It is found that for determining solutions to the model equations for initial-boundary value problems with external forcing by surface pressure and seabed motion, effective numerical schemes are useful. It is observed that for two-dimensional irrotational water waves in particular, an alternative closure relation is derived by applying Cauchys contour integral formula. The modeling of FNFD waves in water of uniform depth is also elaborated.

Evolution of long water waves in variable channels

This paper applies two theoretical wave models, namely the generalized channel Boussinesq (gcB) and the channel Korteweg–de Vries (cKdV) models (Teng & Wu 1992) to investigate the evolution, transmission and reflection of long water waves propagating in a convergent–divergent channel of arbitrary cross-section. A new simplified version of the gcB model is introduced based on neglecting the higher-order derivatives of channel variations. This simplification preserves the mass conservation property of the original gcB model, yet greatly facilitates applications and clarifies the effect of channel cross-section. A critical comparative study between the gcB and cKdV models is then pursued for predicting the evolution of long waves in variable channels. Regarding the integral properties, the gcB model is shown to conserve mass exactly whereas the cKdV model, being limited to unidirectional waves only, violates the mass conservation law by a significant margin and bears no waves which are reflected due to changes in channel cross-sectional area. Although theoretically both models imply adiabatic invariance for the wave energy, the gcB model exhibits numerically a greater accuracy than the cKdV model in conserving wave energy. In general, the gcB model is found to have excellent conservation properties and can be applied to predict both transmitted and reflected waves simultaneously. It also broadly agrees well with the experiments. A result of basic interest is that in spite of the weakness in conserving total mass and energy, the cKdV model is found to predict the transmitted waves in good agreement with the gcB model and with the experimental data available

Nonlinear waves and solitons in water

Abstract A new theoretical model is introduced for evaluating three-dimensional gravity-capillary waves in water of uniform depth to various degrees of validity for predicting nonlinear dispersive water wave phenomena. It is first based on two basic equations, one being the continuity equation averaged over the water depth, and the other the horizontal projection of the momentum equation at the free surface. These two partial differential equations are both exact (for flows assumed incompressible and inviscid), but involve three unknowns: the horizontal velocity at the free surface (in two horizontal dimensions), u ; the depth-mean horizontal velocity, u ; and the water surface elevation, Ξ. Clsure of the system for modeling fully nonlinear and fully dispersive water waves is accomplished by finding for the velocity field a third exact equation relating these unknowns. Interesting phenomena in various cases are illustrated with review and discussion of literature.

Propagation of solitary waves through significantly curved shallow water channels

Propagation of solitary waves in curved shallow water channels of constant depth and width is investigated by carrying out numerical simulations based on the generalized weakly nonlinear and weakly dispersive Boussinesq model. The objective is to investigate the effects of channel width and bending sharpness on the transmission and reflection of long waves propagating through significantly curved channels. Our numerical results show that, when travelling through narrow channel bends including both smooth and sharp-cornered 90°-bends, a solitary wave is transmitted almost completely with little reflection and scattering. For wide channel bends, we find that, if the bend is rounded and smooth, a solitary wave is still fully transmitted with little backward reflection, but the transmitted wave will no longer preserve the shape of the original solitary wave but will disintegrate into several smaller waves. For solitary waves travelling through wide sharp-cornered 90°-bends, wave reflection is seen to be very significant, and the wider the channel bend, the stronger the reflected wave amplitude. Our numerical results for waves in sharp-cornered 90°-bends revealed a similarity relationship which indicates that the ratios of the transmitted and reflected wave amplitude, excess mass and energy to the original wave amplitude, mass and energy all depend on one single dimensionless parameter, namely the ratio of the channel width b to the effective wavelength [lambda][sub]e. Quantitative results for predicting wave transmission and reflection based on b/[lambda][sub]e are presented.