Non-uniqueness of steady free-surface flow at critical Froude number

Free-surface flow past a disturbance at critical Froude number is commonly found to be unsteady with complex wave patterns both upstream and downstream of the disturbance. Such flows can be undesirable as the waves that are generated can have a negative impact in applications including the erosion of waterway banks and energy loss through wave drag on a ship. This motivates us to develop a new approach to obtain steady solutions at critical Froude number that are wave free in the far field. Under the assumption of two-dimensional, irrotational, incompressible fluid flow, we show that both weakly and fully nonlinear solutions to the problem are non-unique. A range of qualitatively different types of numerical solutions and analytical approximations are discovered, for example for flow over a corrugated channel bottom. Copyright c © EPLA, 2014 Introduction. – The study of steady two-dimensional gravity waves in finite depth channels has a rich and long history [1–11], which has continued to attract attention in more recent years [12–21]. A physical motivation for considering these flow problems is the well-known phenomenon where surface waves occur both upstream and downstream of a disturbance (e.g. ship or obstacle) that is moving at critical speed (fig. 1). Both experimental observations and theoretical predictions have shown that critical flows are in general unsteady with solitons [22] being radiated ahead of the steadily moving disturbance [17,19,23–30]. Unsteady critical flows provide interesting and complex wave patterns to investigate, but from a practical viewpoint the waves generated are undesirable for a number of reasons. One reason is that the wash or wake generated by disturbances moving close to or at critical speed can damage both the integrity of waterway banks, leading to erosion and leaking, and the delicate eco-systems that live alongside them [31–34]. Another reason is that the energy lost in these surface waves is an important component of the overall drag on the disturbance [5,6,35–37]. This provides us with the incentive to develop a general approach to obtain steady critical flows [15,18,19,26,38], with no waves in the far field both upstream and downstream of the H