A Geometric Approach Towards Momentum Conservation

TL;DR

This paper introduces a geometric discretization of the Navier-Stokes equations that treats momentum as a covector-valued volume-form, ensuring exact conservation of mass and momentum with higher order accuracy.

Contribution

It presents a novel geometric approach to discretize Navier-Stokes equations, emphasizing tensor-based momentum conservation and higher order approximation.

Findings

Scheme satisfies mass and momentum conservation exactly

Applicable to Kovasznay flow and lid-driven cavity flow

Resembles a staggered-mesh finite-volume method

Abstract

In this work, a geometric discretization of the Navier-Stokes equations is sought by treating momentum as a covector-valued volume-form. The novelty of this approach is that we treat conservation of momentum as a tensor equation and describe a higher order approximation to this tensor equation. The resulting scheme satisfies mass and momentum conservation laws exactly, and resembles a staggered-mesh finite-volume method. Numerical test-cases to which the discretization scheme is applied are the Kovasznay flow, and lid-driven cavity flow.