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.

Termination points and homoclinic glueing for a class of inhomogeneous nonlinear ordinary differential equations

“…on the right hand side, and this does not approach a delta function in the limit → 0. Consequently, branch B 0 can be described by naïvely replacing the right hand side of (1.1) with M δ(x) and then following the type of phase plane analysis reviewed by Binder [1], but the remaining branches B n for n ≥ 1 cannot be described in this way.…”

Classical hydrodynamics for analogue space–times: open channel flows and thin films

“…The Saint–Venant shallow water equations do not take into account the effect of dispersion: the reader will look to the steady and unsteady solutions of the forced KdV equation, which is able to tackle some dispersive properties like the undulation despite the lack of a clear comparison between experiments and numerical calculations [66,68]. Thanks to the combination of the wave and flow phase diagrams (figures 4 and 5) and depending on the flow rate and the initial water depth, the experimentalists can design all sorts of flow configurations [63–69] reproducing non-dispersive hydraulic black holes [58], dispersive hydraulic white holes with [30,46] or without [26,29] undulation and dispersive hydraulic wormholes [50] (the region in-between both horizons).…”

Classical Hydrodynamics for Analogue Spacetimes: Open Channel Flows and Thin Films

“…5). It is currently an open problem for free surface flows past disturbance despite its long history and attempts at classification of types of flows and free surface deformation [63][64][65][66][67][68][69]. In this work, we will not enter into a general discussion on this matter but we review the main type of experiments performed so far.…”

“…The Saint-Venant shallow water equations do not take into account the effect of dispersion: the reader will look to the steady and unsteady solutions of the forced KdV equation which is able to tackle some dispersive properties like the undulation despite the lack of a clear comparison between experiments and numerical calculations [66,68]. Thanks to the combination of the wave and flow phase diagrams (see Figures 4 and 5) and depending on the flow rate and the initial water depth, the experimentalists can design all sorts of flow configurations [63][64][65][66][67][68][69] reproducing nondispersive hydraulic black holes [58], dispersive hydraulic white holes with [30,46] or without [26,29] undulation and dispersive hydraulic wormholes [50] (the region in-between both horizons). The next generation of analogue gravity experiments in Classical Hydrodynamics shall combine both phase and flow diagrams by taking into account hydraulic and dispersive effects in order to design engineered analogue space-times.…”