Modelling the effects of gravity current propagation in a nonlinearly stratified ambient with shallow-water-type equations

Abstract: Variations in solute concentration in a fluid are common in natural scenarios, including salinity differences in the ocean and smoke propagation in bushfires. Flow in such scenarios can be induced by the placement of a heavy fluid into a lighter ambient fluid, where the ambient density may itself vary with height. In this study, a system of shallow water equations (SWEs) has been derived for a gravity current propagating in a power-law distributed ambient density. Through the analysis of the ambient potential energy, it is found that for sufficiently small increases from the unstratified state, the front velocity exhibits self-similar behavior for a given ambient height. However, for larger increases in ambient potential energy, the front velocity asymptotes towards a value matching that of a uniformly stratified ambient with density equal to the maximal density of the power-law distributed ambient. Comparisons have also been made between shallow water predictions and two- and three-dimensional direct numerical simulations (DNSs), showing excellent agreement between both data sets during the slumping phase. Keywords: Gravity currents, Shallow water equations, Navier-Stokes equations