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arXiv · 2604.22060

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

Jaeyoung Jung·Marco Giometto
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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.

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Jaeyoung Jung, Marco Giometto. 2026-04-23. Conservative and skew-symmetric forms of the incompressible Navier-Stokes equations in sigma-coordinates. https://arxiv.org/abs/2604.22060

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