# An Improved Galerkin Framework for Solving Unsteady High-Reynolds Navier-Stokes Equations
## 4 Numerical Experiments
### Unsteady Adaptive Simulation at Re = 1000  
Having validated the accuracy and steady-state behavior of the solver, the full capability and efficiency of the integrated VMS-AMR framework are now demonstrated. Consider a square cavity size of \(\Omega=[0,1]\times[0,1]\), with the velocity at the top boundary of the region as \(\mathbf{u}=(1,0)\), and the remaining parts as \(\mathbf{u}=\mathbf{0}\). Take \(\
u=0.001\) (i.e., Re = 1000), \(\lambda=0.5\), \(T=5.0\).  
This problem is particularly challenging due to the strong velocity gradients near the moving lid and the formation of a large primary vortex accompanied by smaller secondary vortices in the corners. An effective numerical method must capture these multiscale features accurately without spurious oscillations.  
Figures 6 and 7 powerfully illustrate the adaptive strategy in action. Starting from an initial uniform mesh, the AMR algorithm dynamically refines the grid, correctly concentrating elements near the walls where boundary layers form and in regions of high vorticity. This targeted refinement is guided by the posteriori error estimator. Figures 8-10 show the final computed flow fields on different initial grids. The velocity streamlines clearly depict the large primary vortex and the developing secondary vortices, consistent with the established results in the literature [33]. The pressure contours are smooth and oscillation-free, with the characteristic low-pressure zone at the vortex core, demonstrating the effectiveness of VMS.  
Figure 6: The adaptively refined mesh effect diagram (\(20\times 20\) effect diagram). (**a**) Initial mesh. (**b**) \(t=2.700\). (**c**) \(t=5.700\). (**d**) \(t=15.000\).  
The following Figure 10 shows the operation effect at \(T=10.0\).  
It can be seen that in theory, a uniform mesh of approximately \(129\times 129\) grids [33] would be needed to resolve all scales and converge stably. In contrast, this adaptive method achieves stable convergence using an initial \(30\times 30\) grid, which dynamically refines to a final mesh with a number of nodes comparable to a much finer uniform grid only in localized regions. The total degrees of freedom are thus reduced by approximately \(90\%\) compared to the \(129\times 129\) uniform grid, highlighting the immense computational savings.  
Figure 8: Results at \(\text{Re}=1000\), \(T=5.0\) (\(50\times 50\) grids, initial 60 time steps). (**a**) Velocity field. (**b**) Pressure field.  
Figure 7: The adaptively refined mesh effect diagram (\(30\times 30\) effect diagram). (**a**) Initial mesh. (**b**) \(t=2.500\). (**c**) \(t=6.700\). (**d**) \(t=10.000\).  
This significant reduction in degrees of freedom directly translates to immense computational savings, as the computational effort scales nonlinearly with the mesh size. While the adaptive process introduces overheads such as error estimation and remeshing, these costs are demonstrably outweighed by the benefits of concentrating computational power only where it is most needed, leading to a far more efficient simulation compared to uniformly refined meshes that would be computationally intractable for such complex, multiscale flows.