A consistent reduction of the two-layer shallow-water equations to an accurate one-layer spreading model

Abstract

The gravity-driven spreading of one fluid in contact with another fluid is of key importance to a range of topics. These phenomena are commonly described by the two-layer shallow-water equations (SWE). When one layer is significantly deeper than the other, it is common to approximate the system with the much simpler one-layer SWE. It has been assumed that this approximation is invalid near shocks, and one has applied additional front conditions to correct the shock speed. In this paper, we prove mathematically that an effective one-layer model can be derived from the two-layer equations that correctly captures the behaviour of shocks and contact discontinuities without additional closure relations. The result shows that simplification to an effective one-layer model is justified mathematically and can be made without additional knowledge of the shock behaviour. The shock speed in the proposed model is consistent with empirical models and identical to front conditions that have been found theoretically by \eg~von Kármán and by Benjamin. This suggests that the breakdown of the SWE in the vicinity of shocks is less severe than previously thought. We further investigate the applicability of the SW framework to shocks by studying one-dimensional lock-exchange/-release. We derive expressions for the Froude number that are in good agreement with the widely employed expression by Benjamin. The equations are solved numerically to illustrate how quickly the proposed model converges to solutions of the full two-layer SWE. We also compare numerical results from the model with results from experiments, and find good agreement.