Momentum balance in the shallow water equations on bottom discontinuities

TL;DR

This paper analyzes how to correctly balance the momentum in shallow water equations over bottom discontinuities, emphasizing the importance of energy conservation and proper depth choice to avoid unphysical solutions.

Contribution

It demonstrates that energy-conserving approaches and proper depth selection are essential for consistent momentum balance over bottom steps in shallow water models.

Findings

Energy conservation is crucial for correct momentum closure.

Proper depth choice between upstream and downstream depths avoids unphysical solutions.

Current methods may lead to energy-increasing, unphysical configurations.

Abstract

This work investigates the topical problem of balancing the shallow water equations over the bottom steps of different heights. The current approaches in the literature are essentially based on mathematical analysis of the hyperbolic system of balance equations and take into account the relevant progress in treating the non-conservative form of the governing system in the framework of path-conservative schemes. An important problem under debate is the correct position of the momentum balance closure when the bottom elevation is discontinuous. Cases of technical interest are systematically analyzed, consisting of backward-facing steps and forward-facing steps, tackled supercritical and subcritical flows; critical (sonic) conditions are also analyzed and discussed. The fundamental concept governing the problem and supported by the present computations is that the energy-conserving approach is the only approach that is consistent with the classical shallow water equations formulated with geometrical source terms and that the momentum balance is properly closed if a proper choice of a conventional depth on the bottom step is performed. The depth on the step is shown to be included between the depths just upstream and just downstream of the step. It is also shown that current choices (as given in the literature) of the depth on (or in front of) the step can lead to unphysical configurations, similar to some energy-increasing solutions.