On constant influx gravity current propagating over a porous bottom

Abstract

Consider the propagation of a gravity current (GC) sustained by a source of a fluid of density ρc and constant volume rate Q into an ambient fluid of height H and density ρa over a permeable bed. The porous layer of a given length Δxp is located at the bottom of the container. Assume Boussinesq and large Reynolds-number flow. We present a new model for the prediction of the thickness h and depth-averaged velocity u as functions of distance x and time t. We show that during the propagation, the GC can develop four main stages: two first stages occur before the dense fluid enters the permeable domain and two follow after that. The second and the fourth stages are steady-states with constant height and velocity; while during the third phase the current is sinking into a substrate: its height is decreasing and its velocity is increasing. We show that the significant parameter of the problem is the length of the gap Δxp. We derive a compact implicit expression which connects the heights of the current during the second and the fourth steady-state phases. Comparisons with published experiments show good qualitative agreement, however, quantitative comparison was not possible.