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.

翻译：浅水方程通常假设沿垂直方向的速度剖面为常数。然而，该假设在许多实际应用中并不成立。为更好地近似垂直速度分布，已有研究提出了如浅水矩展开模型等方法。然而，在非滑移底部边界条件下，由于源项具有刚性，标准浅水方程及其矩增强模型均难以准确捕捉垂直速度剖面。本文提出了改进的浅水方程及相应的矩增强模型，这些模型在非滑移与滑移边界条件下均表现良好。改进模型与原模型的主要差异在于对源项的处理方式，这使得我们改进的矩展开模型易于推广，同时保持了与先前关于模型双曲性分析的兼容性。为评估标准矩展开模型与改进模型的性能，我们与不可压缩Navier-Stokes方程进行了全面的数值比较——此类比较在现有文献中尚属空白。