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The Shallow Water model (SWM) provides a simplification of the Navier–Stokes model (NSM) for stratified flows over a topography when the depth of the fluid layer is small compared to the horizontal scale of the flow. Nevertheless, the application of SWM is limited to the case of slowly variable bottoms and fails in describing the fluid flow over steep obstacles. In this work, we propose to extend the applicability of SWM when the topography is no longer slowly variable with space, by replacing the topography with an “apparent bottom”. This methodology is tested for the laminar flow of a two-layer fluid over a semi-circular cylinder. Sixteen different steady configurations are investigated in order to assess the influence of the Froude number and the blocking factor corresponding to the ratio between the obstacle height and the fluid layer normal height. Here, the apparent bottom required for SWM is obtained by enforcing the liquid height profile to be the one obtained from full resolution (NSM).

Modeling Open Channel Flows of a Viscous Fluid: Critical Transition and Apparent Bottom