Modulation theory for solitary waves generated by viscous flow over a step

Abstract

The effect of dissipation on the flow of a fluid over a step or jump is examined. This flow has been described theoretically and numerically by using the framework of the forced Korteweg–de Vries–Burgers equation. This model includes the influence of the viscosity of the fluid beyond the Korteweg–de Vries approximation. Modulation theory solutions for the undular bores generated upstream and downstream of the forcing are found. Predictions show that the effect of viscosity plays a key role in determining the properties of waves such as the upstream and downstream solitary wave amplitudes and the widths of the bores. The numerical simulations have shown that the unsteady flow consists of upstream and downstream undular bores, which are connected by a locally steady solution over a step. These numerical solutions are compared with modulation theory results with the excellent agreement obtained. Data from the undular bores are all tabulated. Also, the numerical solutions of the Euler equations are compared with the results of the model, which gave good comparisons. Moreover, the comparisons with the experimental measurements of Lee et al. (1989) together with the Navier–Stokes equations show evidence of the significance of this model.