Variational multiscale modeling with discretely divergence-free subscales

TL;DR

This paper presents a stabilized variational multiscale method for incompressible Navier-Stokes equations that ensures discrete mass conservation, achieves optimal convergence, and demonstrates robustness and stability, suitable for turbulence modeling.

Contribution

It introduces a novel residual-based VMS formulation with a discrete pressure field that maintains mass conservation at both coarse and fine scales, extending to nonlinear turbulence modeling.

Findings

Ensures discrete mass conservation for divergence-conforming methods.

Achieves optimal convergence and robustness in different flow regimes.

Numerical results confirm stability and effectiveness in turbulence modeling.

Abstract

We introduce a residual-based stabilized formulation for incompressible Navier-Stokes flow that maintains discrete (and, for divergence-conforming methods, strong) mass conservation for inf-sup stable spaces with $H^1$-conforming pressure approximation, while providing optimal convergence in the diffusive regime, robustness in the advective regime, and energetic stability. The method is formally derived using the variational multiscale (VMS) concept, but with a discrete fine-scale pressure field which is solved for alongside the coarse-scale unknowns such that the coarse and fine scale velocities separately satisfy discrete mass conservation. We show energetic stability for the full Navier-Stokes problem, and we prove convergence and robustness for a linearized model (Oseen flow), under the assumption of a divergence-conforming discretization. Numerical results indicate that all properties extend to the fully nonlinear case and that the proposed formulation can serve to model unresolved turbulence.