An Improved Galerkin Framework for Solving Unsteady High-Reynolds Navier–Stokes Equations

The numerical simulation of unsteady, high-Reynolds-number incompressible flows governed by the Navier–Stokes (NS) equations presents significant challenges in computational fluid dynamics, primarily concerning numerical stability and computational efficiency. Standard Galerkin finite element methods often suffer from non-physical oscillations in convection-dominated regimes, while the multiscale nature of these flows demands prohibitively high computational resources for uniformly refined meshes. This paper proposes an improved Galerkin framework that synergistically integrates a Variational Multiscale Stabilization (VMS) method with an adaptive mesh refinement (AMR) strategy to overcome these dual challenges. Based on the Ritz–Galerkin formulation with the stable Taylor–Hood (P2−P1) element, a VMS term is introduced, derived from a generalized θ-scheme. This explicitly constructs a subgrid-scale model to effectively suppress numerical oscillations without introducing excessive artificial diffusion. To enhance computational efficiency, a novel a posteriori error estimator is developed based on dual residuals. This estimator provides the robust and accurate localization of numerical errors by dynamically weighting the momentum and continuity residuals within each element, as well as the flux jumps across element boundaries. This error indicator guides an AMR algorithm that combines longest-edge bisection with local Delaunay re-triangulation, ensuring optimal mesh adaptation to complex flow features such as boundary layers and vortices. Furthermore, the stability of the Taylor–Hood element, essential for stable velocity–pressure coupling, is preserved within this integrated framework. Numerical experiments are presented to verify the effectiveness of the proposed method, demonstrating its ability to achieve stable, high-fidelity solutions on adaptively refined grids with a substantial reduction in computational cost.