Increasing Accuracy in Shallow Water Flows: Maintaining Vorticity in Presence of Bathymetry

Abstract

The Kelvin-Helmholtz instability promotes the appearance of complex turbulent flow patterns which typically arise in presence of shear layers. Such condition evolves linear perturbations into complex turbulent structures that determine the mass exchange across the different jets/layers in the flow. In order to accurately predict the mass exchange across a shear layer in shallow flows over complex bathymetries, very high order numerical schemes are required. In this work, a WENO-ADER Augmented scheme, called ARL-ADER method, is used to provide accurate predictions of mass exchange processes under a variety of 2D scenarios involving turbulent shear layers. The proposed scheme is assessed in terms of numerical diffusion by means of the analysis of the numerical energy cascade. Numerical results show that the diffusivity of the scheme determines the turbulent structures produced along the shear layer. It is shown that traditional 1-st order schemes are very diffusive and would require the use of very fine meshes, which is computationally inefficient. On the other hand, the ARL- ADER scheme is able to capture the small-scale vortices and reproduce the theoretical energy cascade even in coarse meshes, while preserving the well-balanced property.