Optimal solutions employing an algebraic Variational Multiscale approach Part II: Application to Navier-Stokes

TL;DR

This paper extends the Variational Multiscale method to nonlinear Navier-Stokes equations, providing a high-order, conservative, and computationally efficient framework that closely approximates optimal solutions.

Contribution

It introduces a nonlinear VMS formulation for Navier-Stokes that maintains high-order accuracy and conservation properties, advancing previous linear approaches.

Findings

Method achieves high-order accuracy and conservation of mass, energy, and vorticity.

Numerical results demonstrate robustness and computational efficiency.

Approximates the optimal projection of the continuous solution.

Abstract

This work presents a non-linear extension of the high-order discretisation framework based on the Variational Multiscale (VMS) method previously introduced for steady linear problems. We build on the concept of an optimal projector defined via the symmetric part of the governing operator. Using this idea, we generalise the formulation to the two-dimensional incompressible Navier-Stokes equations. The approach maintains a clear separation between resolved and unresolved scales, with the fine-scale contribution approximated through the approximate Fine-Scale Greens' operator of the associated symmetric operator. This enables a consistent variational treatment of non-linearity while preserving high-order accuracy. We show that the method yields numerical solutions that closely approximate the optimal projection of the continuous/highly-resolved solution and inherits desirable conservation properties. Particularly, the formulation guarantees discrete conservation of mass, energy, and vorticity, where enstrophy conservation is also achieved when exact or over-integration is employed. Numerical results confirm the methodology's robustness and accuracy, while also demonstrating its computational cost advantage compared to the baseline Galerkin approach for the same accuracy.