Variational RANS modeling of hydraulic jumps

Abstract

The hydraulic jump is a phenomenon which frequently appears in nearshore, estuarine and riverine regions; it plays a major role for example in the development and migration of sand waves in mobile beds, and in controlling the propagation of infra-gravity water waves from the nearshore into the river through bar-built, shallow estuaries. Although the prediction of these natural flows becomes crucial in many circumstances, it is difficult because hydraulic jumps are complex flows, and their modeling is far from simple. The steady-state hydraulic jump may be undular or broken, and their physical properties differ significantly. Undular jump profiles are dominated by the deviation of the fluid pressure from hydrostatic conditions, with turbulence playing a secondary role. In contrast, broken hydraulic jumps are determined by the differential advection of momentum due to breaking, and turbulence stresses. Between both types, there is a complex transition involving the interaction of non-hydrostaticity and turbulence, which is difficult to mimic with depth-averaged models. Simulations of these flows are typically conducted by resorting to vertically-averaged models either using the Saint-Venant or the Serre-Green-Naghdi equations, but none of them are fully satisfactory. In this work, an alternative approach for hydraulic jumps is proposed by developing a variational Reynolds-Averaged Navier-Stokes (RANS) model which uses Kantorovich-Krylov expansions of the flow variables, and a novel depth-averaged, eddy viscosity approach suitable for non-hydrostatic flows. The model is validated through results of simulations of both undular and broken jumps compared against detailed experiments. New experiments were conducted in a simplified bar-built shallow river inlet; the variational RANS model was found to reproduce well the flow profile over the bar, the hydraulic jump position and dynamic pressures over the bar surface.