Conservative and skew-symmetric forms of the incompressible Navier-Stokes equations in sigma-coordinates

TL;DR

This paper develops conservative and skew-symmetric formulations of incompressible Navier-Stokes equations in sigma-coordinates that preserve structural properties and ensure energy conservation or boundedness.

Contribution

It introduces novel formulations that maintain the intrinsic structure of the equations in terrain-following coordinates, improving upon conventional sigma-transform methods.

Findings

Conservative form aligns with general conservation laws.

Skew-symmetric form conserves energy for Euler equations.

Boundary conditions ensure energy boundedness.

Abstract

This study derives conservative and skew-symmetric formulations of the incompressible flow equations in a terrain-following sigma-coordinate system that preserve key structural properties of the Cartesian formulation. Unlike conventional formulations based on the direct application of the sigma-transformation to Cartesian equations, in which metric-induced terms disrupt the intrinsic structure of the governing equations, the proposed formulations are designed to avoid these structural inconsistencies. A conservative form is derived in a manner consistent with general conservation laws, and its modified eigenstructure is analyzed relative to the Cartesian counterpart. A skew-symmetric formulation is then derived by introducing a new set of variables, yielding a form that is energy-conserving for the Euler equations and energy-bounded for the Navier-Stokes equations. Finally, we discuss characteristic-based boundary conditions to ensure energy boundedness of the system.