Discrete exterior calculus discretization of incompressible Navier-Stokes equations over surface simplicial meshes

TL;DR

This paper introduces a conservative discretization method for incompressible Navier-Stokes equations on surface meshes using discrete exterior calculus, achieving high accuracy and conservation properties.

Contribution

It develops a novel DEC-based discretization of the Navier-Stokes equations that preserves mass and vorticity with high accuracy on surface meshes.

Findings

Second order accuracy on structured meshes

First order accuracy on unstructured meshes

Conservation of mass and vorticity up to machine precision

Abstract

A conservative discretization of incompressible Navier-Stokes equations is developed based on discrete exterior calculus (DEC). A distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product. The governing equations are first rewritten using the exterior calculus notation, replacing vector calculus differential operators by the exterior derivative, Hodge star and wedge product operators. The discretization is then carried out by substituting with the corresponding discrete operators based on the DEC framework. Numerical experiments for flows over surfaces reveal a second order accuracy for the developed scheme when using structured-triangular meshes, and first order accuracy for otherwise unstructured meshes. By construction, the method is conservative in that both mass and vorticity are conserved up to machine precision. The relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step.