Daohua Zhang

On solitary waves forced by underwater moving objects

The phenomenon of a succession of upstream-advancing solitary waves generated by underwater disturbances moving steadily with a transcritical velocity in twodimensional shallow water channels is investigated. The two-dimensional Navier{ Stokes (NS) equations with the complete set of viscous boundary conditions are solved numerically by the nite-dierence method to simulate the phenomenon. The overall features of the phenomenon illustrated by the present numerical results are unanimous with observations in nature as well as in laboratories. The relations between amplitude and celerity, and between amplitude and period of generation of solitary waves can be accurately simulated by the present numerical method, and are in good agreement with predictions of theoretical formulae. The dependence of solitary wave radiation on the blockage and on the body shape is investigated. It furnishes collateral evidence of the experimental ndings that the blockage plays a key role in the generation of solitary waves. The amplitude increases while the period of generation decreases as the blockage coecient increases. It is found that in a viscous flow the shape of an underwater object has a signicant eect on the generation of solitary waves owing to the viscous eect in the boundary layer. If a change in body shape results in increasing the region of the viscous boundary layer, it enhances the viscous eect and so does the disturbance forcing; therefore the amplitudes of solitary waves increase. In addition, detailed information of the flow, such as the pressure distribution, velocity and vorticity elds, are given by the present NS solutions.

Generation of solitary waves by transcritical flow over a step

It is well-known that transcritical flow over a localized obstacle generates upstream and downstream nonlinear wavetrains. The flow has been successfully modelled in the framework of the forced Korteweg-de Vries equation, where numerical and asymptotic analytical solutions have shown that the upstream and downstream nonlinear wavetrains have the structure of unsteady undular bores, connected by a locally steady solution over the obstacle, which is elevated on the upstream side and depressed on the downstream side. In this paper we consider the analogous transcritical flow over a step, primarily in the context of water waves. We use numerical and asymptotic analytical solutions of the forced Korteweg-de Vries equation, together with numerical solutions of the full Euler equations, to demonstrate that a positive step generates only an upstream-propagating undular bore, and a negative step generates only a downstream-propagating undular bore.

Numerical study of nonlinear shallow water waves produced by a submerged moving disturbance in viscous flow

Two‐dimensional solitary waves generated by a submerged body moving near the critical speed in a shallow water channel are studied numerically. The incompressible Navier–Stokes equations in a curvilinear free‐surface‐fitted coordinate system are solved by the finite difference method. The present numerical results are compared with the existing experimental data, and with the numerical solutions of two inviscid‐flow models, i.e. the general Boussinesq equation and the forced Korteweg‐de Vries equation. It is found that the viscous effect in the boundary layer around the body and on the bottom of the channel plays an important role in the generation of solitary waves on the free surface. Hence the Navier–Stokes solutions have a better agreement with the experimental data than those obtained from two inviscid‐flow models. The effect of the submergence depth of the body on the waves generated is also investigated. It reveals that waves are insensitive to the submergence depth of the body, except for a small region quite close to the bottom of the water channel.

Steady transcritical flow over a hole: Parametric map of solutions of the forced Korteweg–de Vries equation

Transcritical flow of a stratified fluid over an obstacle, or through a contraction, can be modeled by the forced Korteweg–de Vries equation, which describes a balance among weak nonlinearity, weak dispersion, and small forcing effects. Here we seek steady solutions with constant but different amplitudes upstream and downstream of the forcing region. Our interest is in the case when the forcing has negative polarity, which represents a hole. The effects of the width of the hole and the amplitude of the hole on these steady solutions are investigated.

Steady transcritical flow over an obstacle: Parametric map of solutions of the forced extended Korteweg–de Vries equation

Transcritical flow of a stratified fluid over an obstacle is often modeled by the forced Korteweg–de Vries equation, which describes a balance among weak nonlinearity, weak dispersion, and small forcing effects. However, in some special circumstances, it is necessary to add an additional cubic nonlinear term, so that the relevant model is the forced extended Korteweg–de Vries equation. Here we seek steady solutions with constant, but different amplitudes upstream and downstream of the forcing region. Our main interest is in the case when the forcing has negative polarity, which represents a hole. The effects of the width of the hole and the amplitude of the hole on these steady solutions are then investigated.