Front conditions of high-Re gravity currents produced by constant and time-dependent influx: An analytical and numerical study

The propagation of a Boussinesq gravity current which is created by an inflow at the bottom of a horizontal rectangular container is considered. In particular attention is focused on the front condition of the current which is characterized by the Froude number FrN=uN/(g′hN)1/2FrN=uN/(g′hN)1/2 (where uN,hNuN,hN are the speed and the height of the front, and g′g′ is the reduced gravity). Following a procedure similar to the one given in Benjamin (1968) [1], we have shown that the front condition is significantly influenced by the position of the outflow boundary (i.e., by the direction of flow of the ambient fluid displaced by the current). An expression for the front Froude number (FrNFrN) as a function of the depth ratio a=hN/Ha=hN/H, which takes into account the direction of the ambient flow, is presented. Two limiting cases are considered: no-return ambient flow (when the outflow and inflow are at the opposite end walls) and full-return ambient flow (inflow and outflow on the same end wall). Theoretical considerations suggest that the propagation speed of the gravity current in the no-return configuration will be higher than in the classical full-return configuration of Benjamin. Navier–Stokes simulations of constant inflow currents in no-return configuration were carried out. Overall, the front Froude number (FrNFrN) evaluated from these simulations conforms to the analytical insights and its predictions. Unsteady gravity currents are also simulated by implementing time dependent inflow boundary conditions. Two sets of simulations were carried out that represent accelerating and decelerating currents by implementing waxing (increasing in time) and waning (decreasing in time) inflow rates. It is observed that the front Froude number (FrNFrN) evaluated from the simulations of unsteady fronts is in good agreement with the front Froude number of a steady front for comparable depth ratio.