# An Improved Galerkin Framework for Solving Unsteady High-Reynolds Navier-Stokes Equations
## 5 Conclusions  
In this paper, an improved Galerkin framework is presented, designed to effectively address the persistent challenges of numerical stability and computational efficiency inherent in the simulation of unsteady, high-Reynolds-number Navier-Stokes equations. The work successfully demonstrates that a deep, synergistic coupling of Variational Multiscale Stabilization (VMS) with an adaptive mesh refinement (AMR) strategy provides a robust and powerful solution. The key contributions of this study are threefold. First, a VMS method is introduced, derived from a generalized \(\theta\)-scheme, which explicitly constructs a subgrid-scale model to quell the non-physical oscillations that plague standard Galerkin methods in convection-dominated regimes. Unlike methods such as Streamline Upwind Petrov-Galerkin (SUPG) that primarily add artificial diffusion, VMS provides a more physically consistent stabilization by directly accounting for subgrid-scale effects, thereby ensuring the physical fidelity of the solution without excessive smearing. This ensures the physical fidelity of the solution. Second, to tackle the challenge of computational expense, a novel and efficient posteriori error estimator is developed based on dual residuals, which is specifically tailored to the VMS formulation. This estimator assesses the residuals of the large-scale stabilized equations, ensuring that the AMR strategy is directly guided by the accuracy of the VMS solution itself. This creates a feedback loop where the stabilization and mesh refinement work in concert, a key aspect of this synergistic framework. Third, and most crucially, the framework is built upon the Taylor-Hood element, whose stability and satisfaction of the essential inf-sup condition are well-established. This preserves the mathematical rigor of the velocity-pressure coupling, as the AMR strategy incorporates local re-triangulation to maintain the mesh quality necessary for the inf-sup condition to hold on the non-uniform, adaptively generated grids. The performance and superiority of the proposed method were rigorously validated. After verifying accuracy with an analytical solution at \(\text{Re}=1\), the framework demonstrated  
Figure 14: Numerical results for classical cylinder flow at \(\text{Re}=5000\) (\(t=1.5000\)). (**a**) Velocity field. (**b**) Pressure field. (**c**) Vorticity field. its robustness on the benchmark lid-driven cavity flow at Re = 1000. Crucially, its full potential was unleashed on a highly complex case of a lid-driven cavity with an offset cylinder at Re = 5000. Furthermore, the framework's capability to accurately and stably simulate the classical cylinder flow at Re = 5000, a benchmark known for its complex von Karman vortex street, further underscores its versatility and robustness. Even in this chaotic regime, this integrated approach maintained stability, captured the intricate multiscale vortex dynamics, and achieved stable convergence on grids that were, on average, significantly coarser than what would be required by traditional methods. This led to a substantial reduction in computational cost, without compromising the physical fidelity of the final solution. Looking forward, the developed framework serves as a powerful and extensible foundation for future research. Immediate next steps include extending the methodology to three-dimensional (3D) simulations. While the extension to 3D introduces significant challenges, particularly concerning the substantial increase in computational cost and the complexity of mesh generation and management (e.g., efficient 3D Delaunay re-triangulation and load balancing for anisotropic meshes), the fundamental principles of VMS and dual-residual error estimation remain applicable. Addressing these challenges will necessitate advanced parallel computing strategies and optimized data structures to fully leverage the benefits of adaptive refinement in a 3D context.