A Primitive Variable Discrete Exterior Calculus Discretization of Incompressible Navier-Stokes Equations over Surface Simplicial Meshes

TL;DR

This paper develops a conservative primitive variable discretization of the Navier-Stokes equations using discrete exterior calculus on surface meshes, extending previous methods to include energy-preserving time integration and Coriolis effects for planetary flows.

Contribution

It introduces an extended DEC-based discretization that preserves invariants and energy over time, suitable for simulating rotating planetary flows.

Findings

Second order accuracy on structured meshes

First order accuracy on unstructured meshes

Conservation of invariants over long simulations

Abstract

A conservative primitive variable discrete exterior calculus (DEC) discretization of the Navier-Stokes equations is performed. An existing DEC method (Mohamed, M. S., Hirani, A. N., Samtaney, R. (2016). Discrete exterior calculus discretization of incompressible Navier-Stokes equations over surface simplicial meshes. Journal of Computational Physics, 312, 175-191) is modified to this end, and is extended to include the energy-preserving time integration and the Coriolis force to enhance its applicability to investigate the late time behavior of flows on rotating surfaces, i.e., that of the planetary flows. The simulation experiments show second order accuracy of the scheme for the structured-triangular meshes, and first order accuracy for the otherwise unstructured meshes. The method exhibits second order kinetic energy relative error convergence rate with mesh size for inviscid flows. The test case of flow on a rotating sphere demonstrates that the method preserves the stationary state, and conserves the inviscid invariants over an extended period of time.