# An Improved Galerkin Framework for Solving Unsteady High-Reynolds Navier-Stokes Equations
## 4 Numerical Experiments
### Classical Cylinder Flow at Re = 5000  
To further demonstrate the robustness and versatility of the integrated VMS-AMR framework, results for the classical two-dimensional flow past a circular cylinder at a high  
Figure 11: Zoomed view of the adaptively refined mesh near the stationary cylinder at Re = 5000. Figure 12: The numerical results of lid-driven cavity with a stationary cylinder (\(T=1.500\)). (**a**) Velocity field. (**b**) Pressure field. (**c**) Vorticity field. Reynolds number of Re = 5000 are presented. The computational domain is a rectangular channel \([-0.5,2.0]\times[-0.5,0.5]\), with the cylinder of radius \(r=0.1\) centered at \((0.0,0.0)\). A uniform inflow velocity of \(u_{i}n=1.0\) is imposed at the left boundary \((x=-0.5)\), while a zero-pressure condition is applied at the right outflow boundary \((x=2.0)\). No-slip boundary conditions are enforced on the top and bottom walls \((y=-0.5,y=0.5)\) and on the cylinder surface. The adaptive mesh refinement strategy ensures high-quality mesh generation, particularly around the curved cylinder boundary and in the wake region as illustrated in Figure 13, to accurately capture the complex flow features in this region. The kinematic viscosity is set to \(\mu=0.00004\) to achieve \(Re=5000\) based on the cylinder diameter \((D=0.2)\). This benchmark is notoriously challenging due to the formation of the von Karman vortex street, which involves complex, non-steady vortex shedding and strong shear layers. Figure 14 displays the computed velocity, pressure, and vorticity fields at \(t=1.5000\). The velocity field clearly shows the development of the von Karman vortex street in the wake of the cylinder, characterized by the alternating shedding of vortices. The velocity magnitude accelerates around the cylinder shoulders and decelerates in the wake, consistent with the inflow velocity. The pressure field exhibits a high-pressure region at the cylinder's stagnation point and low-pressure regions at the shoulders and within the vortex cores in the wake. These pressure values are within physically reasonable bounds, indicating a stable and accurate pressure solution. The vorticity field provides the most direct evidence of the vortex shedding, showing distinct alternating positive (red) and negative (blue) vorticity concentrations forming the vortex street. The sharp gradients in vorticity near the cylinder surface and within the shear layers are well-resolved. This successful simulation further validates the capability of the VMS-AMR framework to accurately and stably resolve highly complex, non-steady, high-Reynolds-number flows. The numerical experiments presented in this section collectively validate the proposed framework from several critical perspectives. The simulation at Re = 1, for which an exact solution is available, provides a quantitative verification of the method's fundamental accuracy (Section 4.1). The steady-state and unsteady adaptive simulations of the lid-driven cavity flow at Re = 1000 (Sections 4.2 and 4.3) further demonstrate the robustness of VMS and the efficiency of the AMR strategy in capturing well-defined vortex structures on a standard benchmark. The framework's full capability is then primarily demonstrated through two highly challenging test cases: the unsteady flow in a cavity with a stationary cylinder at a high Reynolds number of Re = 5000 (Section 4.4), and the classical cylinder flow at Re = 5000 (Section 4.5). In this chaotic regime, the VMS-AMR synergy proved essential. It successfully suppressed numerical oscillations while dynamically concentrating  
Figure 13: Zoomed view of the adaptively refined mesh near the circular cylinder for classical cylinder flow at Re = 5000.  
computational effort on the complex interactions between shear layers, boundary layers, and shed vortices. This led to a stable, high-fidelity simulation of a physically intricate problem with significant computational savings, a task that would be exceedingly difficult for conventional methods.