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

## TL;DR

This paper mathematically derives an effective one-layer shallow-water model from the two-layer equations that accurately captures shock behavior without extra closure relations, simplifying analysis of fluid spreading phenomena.

## Contribution

It provides a rigorous derivation of a one-layer model that correctly describes shocks in two-layer flows, challenging previous assumptions about the model's limitations near shocks.

## Key findings

- The derived model accurately predicts shock speeds consistent with empirical and theoretical models.
- Numerical simulations show rapid convergence of the one-layer model to two-layer solutions.
- Predictions from the model agree well with dam break experiments.

## Abstract

The gravity-driven spreading of one fluid in contact with another fluid is of key importance to a range of topics. To describe these phenomena, the two-layer shallow-water equations is commonly employed. When one layer is significantly deeper than the other, it is common to approximate the system with the much simpler one-layer shallow water equations. So far, it has been assumed that this approximation is invalid near shocks, and one has applied additional front conditions for 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 any additional closure relations. The proof yields a novel formulation of an effective one-layer shallow water model. The result shows that simplification to an effective one-layer model is well 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 the front conditions that have been found theoretically by e.g. von K\'arm\'an and by Benjamin. This suggests that the breakdown of the shallow-water equations in the vicinity of shocks is less severe than previously thought. We further investigate the applicability of the shallow water framework to shocks by studying shocks in one-dimensional lock-exchange/lock-release. We derive expressions for the Froude number that are in good agreement with the widely employed expression by Benjamin. We then solve the equations numerically to illustrate how quickly the proposed model converges to solutions of the full two-layer shallow-water equations. We also compare numerical results using our model with results from dam break experiments. Predictions from the one-layer model are found to be in good agreement with experiments.