Investigation of temporal mixing layer in shallow water with BGK algorithm

Abstract

Shallow water flows are common in natural or man-made hydraulic systems, for example, in the wake of islands and in the downstream of headlands and other protrusions. Experiments, numerical simulations and theoretical analysis are applied to investigate the phenomenon. The present work is devoted to the comparison of nonlinear models and linear stability analysis results. In previous work, nonlinear models focus on the spatial growth of perturbations and linear stability analysis focuses on the temporal growth of perturbations. Comparisons are then performed with Gaster’s transformation, which maps the temporal growth of perturbations to the spatial growth. However, Gaster’s transformation only works well near the neutral curve, at which the flows are marginally unstable. The paper circumvents the use of Gaster’s transformation by comparing linear temporal analysis with nonlinear temporal simulations, both focusing on the temporal growth of perturbations. It is found that within a short time the linear stability analysis underestimates the peak turbulent kinetic energy and the wavenumber at which the flow attains the peak kinetic energy. Over time, the peak energy and the corresponding wavenumber obtained from the linear stability analysis are consistent with the nonlinear simulation results. Also, an initial energy drop is observed in the nonlinear simulations but not in the linear stability analysis results. Such drop may be due to 1.) the initial transient growth which is not captured with the normal-mode based linear stability analysis, or 2.) some energy in the cross-stream velocity perturbations is converted to the potential energy, resulting in a change in the water depth, which is not captured with the rigid-lid assumption in the linear stability analysis.