Numerical simulations of two-layer flow past topography. Part I: The leeside hydraulic jump

Laboratory observations of the leeside hydraulic jump indicate it consists of a statistically stationary turbulent motion in an overturning wave. From the point of view of the shallow-water equations (SWE), the hydraulic jump is a discontinuity in fluid-layer depth and velocity at which kinetic energy is dissipated. To provide a deeper understanding of the leeside hydraulic jump, three-dimensional numerical solutions of the Navier-Stokes equations (NSE) are carried out alongside SWE solutions for nearly identical physical initial-value problems. Starting from a constant-height layer flowing over a two-dimensional obstacle at constant speed, it is demonstrated that the SWE solutions form a leeside discontinuity owing to the collision of upstream-moving characteristic curves launched from the obstacle. Consistent with the SWE solution, the NSE solution indicates the leeside hydraulic jump begins as a steepening of the initially horizontal density interface. Subsequently, the NSE solution indicates overturning of the density interface and a transition to turbulence. Analysis of the initial-value problem in these solutions shows that the tendency to form either the leeside height-velocity discontinuity in the SWE or the overturning density interface in the exact NSE is a feature of the inviscid, nonturbulent fluid dynamics. Dissipative turbulent processes associated with the leeside hydraulic jump are a consequence of the inviscid fluid dynamics that initiate and maintain the locally unstable conditions.