# A Novel Cell-Based Adaptive Cartesian Grid Approach for Complex Flow Simulations
## 4 Results and Discussion
### Supersonic Flow around a Sphere  
The sphere had a diameter of \(D\) = 2, with an incoming flow \(Ma\) of 1.9 and a Reynolds number based on the diameter (\(Re_{D}\)) of 1 \(\times\) 10\({}^{5}\). The computational domain was set to 30 \(\times\) 40 \(\times\) 40, as shown in Figure 18a. The initial Cartesian grid size was \(\Delta y=\Delta x=2.0\), with an initial grid count of 6000. A maximum of 8 levels of refinement was set, resulting in a final adaptive mesh counting 5.3 million cells for the flow field. Figure 18b illustrates the space-filling Cartesian grid around the sphere and the triangular mesh on the surface of the sphere, with the sphere surface discretized into 10,278 triangular elements.  
Figure 19 presents the grid before flow AMR and the corresponding Mach number contours, demonstrating that after 1000 steps of evolution, the detached shock in front of the sphere started to emerge. However, due to insufficient grid resolution in regions far from the surface, the flow field features could not be accurately described. With the continuous process of adaptation, the characteristics of the flow field became more pronounced, and shock capturing became finer. By the time of the sixth solution adaptation, the criteria for solution adaptation had effectively captured the bow shock in front of the sphere as well as the expansion shock generated at the rear, indicating that the flow field had been sufficiently developed.  
Figure 17: NHLP three-segment wing flow field distribution of turbulence viscosity coefficient before and after adaptation. (**a**) Initial grid; (**b**) after AMR.  
Figure 18: Sphere case computational domain and grid distribution diagram. (**a**) Global grid layout (the blue line represents the schematic of the computational domain size); (**b**) near-wall grid.  
Figure 20 illustrates the impact of AMR on the pressure coefficient distribution over the surface of the sphere. The figure reveals that, with the given number of surface triangular mesh elements, a smooth representation of surface quantities could be extracted from the Cartesian grid of the flow field for subsequent force integration processing. Additionally, the AMR had a minor effect on the pressure distribution in the stagnation region ahead of the flow but had a more significant impact on the separated region at the rear. When calculated on a coarser grid, the negative pressure region's amplitude and extent at the back were overestimated, affecting the overall drag coefficient. In Table 2, upon comparing the simulation results of this study with the experimental results of [PERSON] and [PERSON] [41], it is evident that thanks to the finer resolution in the wake achieved by the AMR (increasing the overall grid count from 1.38 million to 4.17 million), the overall drag coefficient was brought closer to the experimental values.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Grid Type** & **Number of Grids (\(\times\)10\({}^{4}\))** & \(\mathbf{C_{d}}\) \\ \hline [PERSON] and [PERSON] [41] & / & 0.97 \\ Initial grid & 138 & 1.062 \\ After AMR & 417 & 0.988 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of drag coefficients for supersonic sphere with experimental results.  
Figure 19: Flow field and grid distribution around the sphere in adaptive process. (**a**) Initial grid; (**b**) after two AMR processes; (**c**) after six AMR processes.  
Figure 20: Pressure coefficient distribution diagram of sphere surface. (**a**) The front of the sphere; (**b**) the back of the sphere.