Gravity currents propagating into two-layer stratified fluids: vorticity-based models

Abstract

The vorticity-based modelling approach originally introduced by Borden & Meiburg (J. Fluid Mech., vol. 726, 2013b, R1) is extended to gravity currents propagating into two-layer stratified ambients. Vorticity models are developed for three different flow configurations: no upstream-propagating wave, an upstream-propagating expansion wave only and an upstream-propagating expansion wave and a bore. For a given gravity current height and stratification strength, along with ambient inflow layer thicknesses and velocities, the models yield the gravity current velocity, the bore and expansion wave properties and the ambient outflow layer thicknesses and velocities. We furthermore establish which of the three configurations will occur in a given parameter regime. Since energy-related closure assumptions are not required for any of the configurations, we can determine the dissipation as a function of the gravity current height, for a given set of flow parameters. To investigate which gravity current height is selected in real flows, we carry out two-dimensional Navier–Stokes simulations for comparison. These yield gravity current heights close to the vorticity model solutions for energy-conserving flows. Hence we adopt these energy-conserving solutions as the vorticity model predictions. We subsequently discuss these predictions in the context of earlier models by other authors, and of two-layer stratified flows over obstacles.