Time to Approach Similarity

In a recent article, Ball and Huppert (J. Fluid Mech., 874, 2019) introduced a novel method for ascertaining the characteristic timescale over which the similarity solution to a given time-dependent nonlinear differential equation converges to the actual solution, obtained by numerical integration, starting from given initial conditions. In this article, we apply this method to a range of different partial differential equations describing propagating gravity currents of fixed volume as well as modifying the techniques to apply to situations for which convergence to the numerical solution is oscillatory, as appropriate for gravity currents propagating at large Reynolds numbers. We investigate properties of convergence in all of these cases, including how different initial geometries affect the rate at which the two solutions agree. It is noted that geometries where the flow is no longer unidirectional take longer to converge. A method of time-shifting the similarity solution is introduced to improve the accuracy of the approximation given by the similarity solution, and also provide an upper bound on the percentage disagreement over all time.