Theory of accelerated flows in a long wave approximation: High-order Shallow Water Equations (HSWE)

TL;DR

This paper develops a high-order long-wave model for accelerated turbulent coastal flows, enhancing shallow water equations to better capture turbulence effects in engineering applications.

Contribution

It introduces a novel high-order long-wave model incorporating turbulence transport, improving the accuracy and stability over classical shallow water equations.

Findings

Model is stable and efficient.

Accurately captures turbulence effects.

Shows promising results in sediment transport simulations.

Abstract

The majority of coastal flows are characterized by turbulence, rendering the application of shallow water equations an inadequate approach for their accurate description. This paper presents a theory for characterizing accelerated coastal flows, offering an enhanced representation of turbulent flows in the long-wave approximation. The fundamental premise of the theory is as follows: Fluctuating velocities present in the water flow are found to correlate and generate additional motion (or fluctuating motion), which is associated with a velocity of the same order and regularity as the mean flow velocity. The resulting high-order long-wave model is both stable and efficient, comprising classical shallow water equations and additional equations that describe the transport of kinetic energy due to turbulence. The model's detailed eigenstructure and Rankine-Hugoniot relations, steady state solutions are presented in this paper. The proposed theory shows promise for addressing a variety of engineering problems, including roll waves, hydraulic jumps, dam breaks, and flooding. The derived model can accurately address the problem of the mixing layer interaction with a free surface and its transition into a turbulent surface jet. Recent application to sediment transport gives interesting results and good numerical convergences.