Longer time simulation of the unsteady Navier-Stokes equations based on a modified convective formulation

TL;DR

This paper introduces a modified convective formulation for the Navier-Stokes equations that conserves energy and other physical quantities, enabling longer and more accurate simulations of unsteady flows.

Contribution

It applies skew-symmetrization with divergence-free velocity approximation to preserve energy and other invariants in finite element discretizations of NSEs.

Findings

Conserves energy, linear momentum, helicity, enstrophy, and vorticity.

Performs comparably to EMAC formulation in long-term simulations.

Outperforms the standard skew-symmetric formulation in accuracy and stability.

Abstract

For the discretization of the convective term in the Navier-Stokes equations (NSEs), the commonly used convective formulation (CONV) does not preserve the energy if the divergence constraint is only weakly enforced. In this paper, we apply the skew-symmetrization technique in [B. Cockburn, G. Kanschat and D. Sch\"{o}tzau, Math. Comp., 74 (2005), pp. 1067-1095] to conforming finite element methods, which restores energy conservation for CONV. The crucial idea is to replace the discrete advective velocity with its a $H(\operatorname{div})$-conforming divergence-free approximation in CONV. We prove that the modified convective formulation also conserves linear momentum, helicity, 2D enstrophy and total vorticity under some appropriate senses. Its a Picard-type linearization form also conserves them. Under the assumption $\boldsymbol{u}\in L^{2}(0,T;\boldsymbol{W}^{1,\infty}(\Omega)),$ it can be shown that the Gronwall constant does not explicitly depend on the Reynolds number in the error estimates. The long time numerical simulations show that the linearized and modified convective formulation has a similar performance with the EMAC formulation and outperforms the usual skew-symmetric formulation (SKEW).