# An Improved Galerkin Framework for Solving Unsteady High-Reynolds Navier-Stokes Equations
## 4 Numerical Experiments
### Validation Case: Lid-Driven Cavity Flow at Re = 1000  
After confirming the basic accuracy, the solver's ability is validated to handle flows at a moderately high Reynolds number against a widely recognized benchmark. The lid-driven cavity flow at Re = 1000 is simulated, and the steady-state results are compared with the classic reference data provided by [PERSON] et al. [33]. This test is crucial for verifying that the VMS formulation can accurately capture the complex flow physics, such as the location and intensity of the primary and secondary vortices, before assessing the efficiency of the adaptive strategy. The simulation is run on a initial \(40\times 40\) grid until the flow reaches a steady state (\(T=20.0\)).  
Figure 3 displays the computed steady-state velocity and pressure fields. Table 3, Figure 4 and Table 4, and Figure 5 compare the velocity profiles along the horizontal and vertical centerlines of the cavity with the benchmark data from [PERSON] et al. The excellent agreement demonstrates that this method correctly reproduces the established physical features of this challenging flow, providing a solid foundation for the subsequent simulations.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Error Type** & \(\|\mathbf{u}-\mathbf{u}_{h}\|_{\mathbf{e}}\) & \(\|\mathbf{u}-\mathbf{u}_{h}\|_{\mathbf{1}}\) & \(\|\mathbf{p}-p_{h}\|_{\mathbf{0}}\) & \(\|\mathbf{u}-\mathbf{u}_{h}\|_{1}+\|p-p_{h}\|_{\mathbf{0}}\) \\ \hline \(\|\mathbf{u}\|_{0}\) & \(\|\mathbf{u}\|_{0}\) & \(\|\mathbf{u}\|_{1}\) & \(\|\mathbf{u}\|_{0}\) & \(\|\mathbf{u}\|_{1}+\|p\|_{0}\) \\ \hline Error magnitude & 0.0832\% & 0.9785\% & 1.7646\% & 0.9785\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: The result error in the case of Re = 1.  
Figure 2: Results at Re = 1, \(T=0.03\) (\(20\times 20\) grids, initial 10 time steps). (**a**) Velocity field (exact). (**b**) Pressure field (exact). (**c**) Velocity field (FEM). (**d**) Pressure field (FEM).  
Figure 4: The velocity on the horizontal central axis compared with the Ghia benchmark solution.  
Figure 5: The velocity on the vertical central axis compared with the Ghia benchmark solution.  
Figure 3: Steady-state results at Re = 1000 (initial \(40\times 40\) grids). (**a**) Velocity field. (**b**) Pressure field.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \(y\) & 0.0000 & 0.0625 & 0.0730 & 0.1000 & 0.3500 & 0.2260 & 0.2205 & 0.5000 & 0.8000 & 0.6255 & 0.9000 & 0.9502 & 0.9570 & 1.0000 \\ velocity & 0.00000 & 0.02470 & 0.233092 & 0.959906 & 0.394536 & 0.32753 & 0.319689 & 0.025269 & -0.250756 & -0.436970 & -0.5755203 & -0.462410 & -0.346960 & -0.217880 & 0.00000 \\ \hline \(x\) ([PERSON] et al.) [33] & 0.00000 & 0.06250 & 0.070313 & 0.007950 & 0.196250 & 0.226663 & 0.234475 & 0.500000 & 0.804690 & 0.899375 & 0.906250 & 0.948313 & 0.953125 & 0.968750 & 1.00000 \\ velocity ([PERSON] et al.) [33] & 0.00000 & 0.27485 & 0.20012 & 0.32627 & 0.37095 & 0.33075 & 0.32235 & 0.02526 & -0.31966 & -0.42665 & -0.51550 & -0.39188 & -0.33714 & \(-0.21388\) & 0.0000 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The velocity on the vertical central axis compared with the Chia benchmark solution.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \(x\) & 0.0000 & 0.0625 & 0.0730 & 0.1000 & 0.3500 & 0.2260 & 0.2205 & 0.5000 & 0.8000 & 0.6255 & 0.9000 & 0.9500 & 0.9502 & 0.9570 & 1.0000 \\ velocity & 0.00000 & 0.024707 & 0.233092 & 0.959906