Moment-enhanced shallow water equations for non-slip boundary conditions

TL;DR

This paper introduces modified shallow water moment expansion models that accurately handle non-slip boundary conditions, improving vertical velocity profile approximation and maintaining hyperbolicity, validated through comprehensive numerical comparisons.

Contribution

The paper proposes modified shallow water moment expansion models that effectively address non-slip boundary conditions while preserving model hyperbolicity and generalizability.

Findings

Modified models perform well under non-slip conditions

Numerical comparisons show improved accuracy over standard models

Models maintain hyperbolicity and are compatible with previous analyses

Abstract

The shallow water equations often assume a constant velocity profile along the vertical axis. However, this assumption does not hold in many practical applications. To better approximate the vertical velocity distribution, models such as the shallow water moment expansion models have been proposed. Nevertheless, under non-slip bottom boundary conditions, both the standard shallow water equation and its moment-enhanced models struggle to accurately capture the vertical velocity profile due to the stiff source terms. In this work, we propose modified shallow water equations and corresponding moment-enhanced models that perform well under both non-slip and slip boundary conditions. The primary difference between the modified and original models lies in the treatment of the source term, which allows our modified moment expansion models to be readily generalized, while maintaining compatibility with our previous analysis on the hyperbolicity of the model. To assess the performance of both the standard and modified moment expansion models, we conduct a comprehensive numerical comparison with the incompressible Navier--Stokes equations -- a comparison that is absent from existing literature.