Local conservation laws of continuous Galerkin method for the incompressible Navier--Stokes equations in EMAC form

TL;DR

This paper develops new weak form formulations for local momentum and angular momentum balances in incompressible Navier-Stokes equations, demonstrating that continuous Galerkin methods using EMAC preserve these conservation laws discretely.

Contribution

It introduces equivalent weak form conservation laws and proves their preservation in Galerkin discretizations with EMAC, supported by numerical validation.

Findings

Discrete conservation of momentum and angular momentum is maintained by Galerkin methods with EMAC.

New weak form formulations are equivalent to classical conservation laws.

Numerical tests confirm the theoretical preservation of conservation laws.

Abstract

We consider local balances of momentum and angular momentum for the incompressible Navier-Stokes equations. First, we formulate new weak forms of the physical balances (conservation laws) of these quantities, and prove they are equivalent to the usual conservation law formulations. We then show that continuous Galerkin discretizations of the Navier-Stokes equations using the EMAC form of the nonlinearity preserve discrete analogues of the weak form conservation laws, both in the Eulerian formulation and the Lagrangian formulation (which are not equivalent after discretizations). Numerical tests illustrate the new theory.