A new Energy Equation Derivation for the Shallow Water Linearized Moment Equations

TL;DR

This paper presents a systematic derivation of the energy equation for the Shallow Water Linearized Moment Equations, enhancing understanding and potential numerical implementation of these models for free-surface flows.

Contribution

It introduces a new systematic derivation method for the energy equation of SWLME, facilitating extensions and numerical solutions of SWME variants.

Findings

Derived the energy equation for SWLME using a systematic approach.

The derivation includes a skew-symmetric formulation for improved numerical stability.

Provides a foundation for extending SWME models and their computational methods.

Abstract

Shallow Water Moment Equations (SWME) are extensions to the well-known Shallow Water Equations (SWE) for the efficient modeling and numerical simulation of free-surface flows. While the SWE typically assume a depth-averaged vertical velocity profile, the SWME allow for vertical variations of the velocity profile. The SWME therefore assume a polynomial profile and then derive additional evolution equations for the polynomial coefficients via higher order depth integration. In this work, we perform a new systematic derivation of the energy equation for a specific variant of the SWME, called the Shallow Water Linearized Moment Equations (SWLME). The derivation is based on the standard SWE energy equation derivation and includes the skew-symmetric formulation of the model. The new systematic derivation is beneficial for the extension to other SWME variants and their numerical solution.