The relaxation time for viscous and porous gravity currents following a change in flux

Abstract

The equilibration time in response to a change in flux from Q to $\Lambda$Q after an injection period applied to either a low-Reynolds-number gravity current or one propagating through a porous medium, in both axisymmetric and one-dimensional geometries, is shown to be of the form $\tau$ = Tf($\Lambda$) , independent of all the remaining physical parameters. Numerical solutions are used to investigate f($\Lambda$) for each of these situations and compare very well with experimental results in the case of an axisymmetric current propagating over a rigid horizontal boundary. Analysis of the relaxation towards self-similarity provides an illuminating connection between the excess (deficit) volume from early times and an asymptotically equivalent shift in time origin, and hence a good quantitative estimate of $\tau$ . The case of $\Lambda$ = 0 equilibration after ceasing injection at time T is a singular limit. Extensions to high-Reynolds-number currents and to the case of a constant-volume release followed by constant-flux injection are discussed briefly.