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The gravity-driven spreading of one fluid in contact with another fluid is ofkey importance to a range of topics. To describe these phenomena, the two-layershallow-water equations is commonly employed. When one layer is significantlydeeper than the other, it is common to approximate the system with the muchsimpler one-layer shallow water equations. So far, it has been assumed thatthis approximation is invalid near shocks, and one has applied additional frontconditions for the shock speed. In this paper, we prove mathematically that aneffective one-layer model can be derived from the two-layer equations thatcorrectly captures the behaviour of shocks and contact discontinuities withoutany additional closure relations. The proof yields a novel formulation of aneffective one-layer shallow water model. The result shows that simplificationto an effective one-layer model is well justified mathematically and can bemade without additional knowledge of the shock behaviour. The shock speed inthe proposed model is consistent with empirical models and identical to thefront conditions that have been found theoretically by e.g. von K\'arm\'an andby Benjamin. This suggests that the breakdown of the shallow-water equations inthe vicinity of shocks is less severe than previously thought. We furtherinvestigate the applicability of the shallow water framework to shocks bystudying shocks in one-dimensional lock-exchange/lock-release. We deriveexpressions for the Froude number that are in good agreement with the widelyemployed expression by Benjamin. We then solve the equations numerically toillustrate how quickly the proposed model converges to solutions of the fulltwo-layer shallow-water equations. We also compare numerical results using ourmodel with results from dam break experiments. Predictions from the one-layermodel are found to be in good agreement with experiments.

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