# A Novel Cell-Based Adaptive Cartesian Grid Approach for Complex Flow Simulations
## 4 Results and Discussion
### Subsonic NHLP Airfoil  
The NHLP three-element airfoil was composed of a leading-edge slat, main wing, and trailing-edge flap. The deflection angle of the leading-edge slat for the NHLP airfoil was set at 25\({}^{\circ}\), and the trailing-edge flap was deflected at 20\({}^{\circ}\). Extensive testing of this airfoil was conducted in the British Aircraft Corporation (BAC) wind tunnel [40]. The selected test conditions were as follows: the incoming flow Mach number was 0.197, the angle of attack 4.01\({}^{\circ}\), and the Reynolds number 3.52 \(\times\) 10\({}^{6}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Grid Type** & **Dimensionless Time** & **Number of Grids (\(\times\)10\({}^{3}\))** \\ \hline Uni-level = 3 & 2.1 & 64.5 \\ Uni-level = 5 & 32.3 & 1032.2 \\ AMR-level = 3 & 1.0 & 31.6 \\ AMR-level = 5 & 6.3 & 163.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Forward facing step case grid setup. Figure 12: Density distribution at step height \(y=0.5\). The intricate geometries, such as narrow gaps between different wing sections and complex surface features like cutouts, posed significant challenges for the topology design, mesh point distribution, and mesh quality control of body-fitted structured grids; however, the immersed boundary adaptive Cartesian grid method enabled the rapid generation of high-quality computational meshes. The computational domain was set to \([-16,\,25]\times[-20,\,20]\) with the initial Cartesian grid size of \(\Delta y=\Delta x=1.28\). The maximum refinement level was set to 12 layers with 7 buffer layers. The final geometry-adaptive grid, as shown in Figure 13, consisted of approximately \(1.51\times 10^{5}\) cells and was generated in just \(8.6\) s on a desktop computer under single-core conditions. Near-wall treatment was applied using the wall function approach, with the SA turbulence model implemented. Reference points at the same distance were considered near the wall, with an overall average \(y^{+}\) of around 300. The simulation employed the LU-SGS scheme for progression and initiated AMR after the flow field reached preliminary stability. The grid layout before and after AMR is illustrated in Figure 14. The figure reveals that the adaptive criterion refined the wake region to better capture the flow characteristics in that area. Figure 15 presents the curve showing the change in the number of grids during the AMR process, as well as the convergence curve for the density residuals. After eight AMR iterations, the number of grids increased from \(1.51\times 10^{5}\) to \(2.03\times 10^{5}\). The change in grid numbers was relatively small in the last AMR iteration, and the residuals also further  
Figure 14: NHLP three\(-\)segment wing grid distribution before and after adaptation. (**a**) Initial grid; (**b**) after AMR. Figure 13: NHLP three\(-\)segment wing computational grids. (**a**) Global grids layout; (**b**) near-wall grids (the red box indicates the magnification of a local grid). converged, indicating that both the flow field and the grid had been sufficiently developed. Figure 15b displays the surface pressure coefficient curves before and after AMR. Due to the use of the wall function model in this study, errors were inevitably present in areas of flow separation, such as at the leading edges of the slat and the main wing. Moreover, with the mesh involving 8 layers of grids for AMR and 12 layers for the mesh near the wing surface, activating adaptation did not alter the mesh distribution near the wall, thus having a minimal impact on surface quantities. When we selected the maximum \(Cp\) values for the leading edges of the slat, main wing, and tail wing, which were less influenced by flow separation, the discrepancies between the findings of this study and the experimental results were \(-1.9\%\), \(0.1\%\), and \(4.1\%\), respectively. Additionally, the magnitude of error was very similar, regardless of whether AMR was activated or not. However, the primary effect of AMR was to significantly enhance the resolution in the wake region.