Abstract
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).
Highlights
Introduction a Viscous FluidCritical TransitionThe Shallow Water model (SWM) is largely used to describe the dynamics of stratified flows such as air and water in open channel flows, lakes and oceans, or atmospheric flows, where the air is decomposed into several layers [1,2]
The SWM is derived from the Navier–Stokes model (NSM) in the limit of small fluid layer depth compared to the horizontal scale of the flow
We have compared the steady solutions of a two phase flow on a titled bottom with a semi-circular obstacle for both the 2D Navier–Stokes (NS) equations and the Shallow Water (SW) equations
Summary
The Shallow Water model (SWM) is largely used to describe the dynamics of stratified flows such as air and water in open channel flows, lakes and oceans, or atmospheric flows, where the air is decomposed into several layers [1,2]. The two-layer turbulent flow over a semi-cylindrical obstacle has been studied numerically using the Volume of Fluid Method (VOF) and the standard k − e turbulent model in [27–29], the CLEAR-VOF method and a large eddy simulation (LES) in [30], and an interface-tracking algorithm for computation of the free-surface with a moving grid in [31]. In many problems that are currently of interest, the position and the fluid velocity on the free-surface are given, but the shape and location of the solid bottom are unknown a priori. We introduce a fictitious surface: the “apparent-bottom” (AB), which is the bottom that corresponds to a given position of the free-surface in the SWM This methodology shows how a steep topography should be transformed, in order to obtain reliable predictions of a stratified flow with the SWM.
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