Steady Two-Dimensional Free-Surface Flow Past Disturbances in an Open Channel: Solutions of the Korteweg–De Vries Equation and Analysis of the Weakly Nonlinear Phase Space

Abstract

Two-dimensional free-surface flow past disturbances in an open channel is a classical problem in hydrodynamics—a problem that has received considerable attention over the last two centuries (e.g., see Lamb’s Treatise, 1879). With traces back to Russell’s experimental observations of the Great Wave of Translation in 1834, Korteweg and de Vries (1895), and others, derived an unforced equation to describe the balance between nonlinearity and dispersion required to model the solitary wave. More recently, Akylas (1984) derived a forced KdV equation to model a pressure distribution on the free-surface (e.g., a ship). Since then, the forced KdV equation has been shown to be a useful model approximation for two-dimensional flow past disturbances in an open channel. In this paper, we review the stationary solutions of the forced KdV equation for four types of localised disturbances: (i) a flat plate separating two free surfaces; (ii) a compact bump, or dip in the channel bottom topography; (iii) a compact distribution of pressure on the free surface and (iv) a step-wise change in the otherwise constant horizontal level of the channel bottom topography. Moreover, Dias and Vanden-Broeck (2002) developed a phase plane method to analyse flow over a bump, and this general approach has also been applied to the three other types of forcing (see Binder et al., 2005–2015, and others). In this study, we use eleven basic flow types to classify the steady solutions of the forced KdV equation using the phase plane method. Additionally, considering solutions that are wave-free both far upstream and far downstream, we compare KdV model approximations of the uniform flow conditions in the far-field with exact solutions of the full problem. In particular, we derive a new KdV model approximation for the upstream dimensionless flow-rate which is conveniently given in terms of the known downstream dimensionless flow-rate.