Wide flow model for converging gravity currents and the effects of the flow resistance model on the propagation

Abstract

We are investigating flows in the viscous-buoyancy balance regime in a converging channel with a cross section described by a power function y∼xkzr, where x and y are the streamwise and spanwise horizontal coordinates, respectively, and z is the vertical coordinate. We are interested in the different results depending on whether we use a simplified model of the flow resistance law, which varies depending on whether the height of the current is much greater/smaller than the channel width or a somewhat more general model described by the Darcy–Weisbach equation in which the flow resistance law depends on the shape of the cross section through the Fanning friction factor. The simplified models, one of which developed here is original and new, allow a self-similar solution of the second kind, unlike the general model. The general model, to the best of our knowledge applied for the first time to a generic cross section described by a power function, requires numerical integration. However, a comparison of the front propagation of the gravity current according to the different models, performed by numerical integration of the differential problem, shows that the current assumes a self-similar arrangement as a good approximation for the general model. For some channel geometries, the three models give a very similar result, which results in a difficult attribution to a specific model based on experiments. The effects of anisotropy in the vertical direction of the channel cross section are also highlighted by both the numerical and self-similar solutions.