# A Novel Cell-Based Adaptive Cartesian Grid Approach for Complex Flow Simulations
## 4 Results and Discussion  
### A Mach 3 Wind Tunnel with a Forward Facing Step  
To verify the capability of the developed cell-based AMR method in capturing flow features such as shock discontinuities and shear layers, the classic Woodward-Colella test [39], namely the forward facing step flow problem, was selected for simulation studies. The computational domain for the step was a rectangular area of \(L\times H\) = 3.0 \(\times\) 1.0, with the step height being 0.2, positioned at a distance \(x\) = 0.6 from the wind tunnel entrance. The freestream Mach number \(Ma\) = 3.0, and supersonic inflow and outflow boundary conditions, were set at the inlet and outlet, respectively, with slip adiabatic solid wall boundary conditions applied to both the upper and lower boundaries. The initial grid generation is illustrated in Figure 9, where the initial cell size \(\Delta x\) = \(\Delta y\) = 0.05, with a maximum of three layers of refinement, and the initial number of grid cells being 11,397. Figure 8: A schematic of interpolation for the flow filed around an immersed boundary. (**a**) Ghost cell method; (**b**) wall function model. The brown curve: wall surface; \(n\): outward normal; Information at \(r\) is interpolated from cells 1 to 4. Based on three adaptive criteria (velocity vorticity-based, velocity divergence-based, and a combined criterion of velocity vorticity divergence), the grid distribution at the time instant (\(t^{*}=tMa_{\infty}/H=4\)) is shown in Figure 10, with the grid numbers being 22,668, 22,092, and 31,617, respectively. At this moment, the flow details within the region include: (1) three times shock reflection and their paths; (2) expansion waves at the step corner and the weak oblique shock waves induced by its upper surface; (3) minor oscillations in entropy behind the Mach stem directly above the step, followed by the vortex roll-up (Kelvin-Helmholtz instability). It was observed that the velocity vorticity criterion could capture the details of shear layers and vortices in the flow field relatively well, but was weak in capturing shock discontinuities. On the other hand, the velocity divergence criterion had a strong capability in capturing shocks but failed to accurately capture shear layers. The combined criterion of velocity vorticity-divergence was able to more comprehensively capture the flow field features under complex scenarios. Figure 10: _Cont._  
Figure 9: Initial mesh for forward facing step. To compare the effect of different numbers of adaptive refinement layers on the solution efficiency, an additional two layers of adaptation were introduced under the prior background mesh size, thus considering a flow field solution adaptation with five layers. Figure 11 displays the flow field distribution under the five-layer solution adaptation. It is evident, compared to the previous section, that the shock width was narrower, and more refined vortex roll-ups were captured behind the Mach stem. To compare the impact of AMR on the spatial distribution of the flow field, Figure 12 presents the density distribution along the midline at \(y=0.5\) of the step height at the aforementioned time instant. The label \"Uni\" denotes the uniform grid discretization solution using the finest mesh size corresponding to the areas of interest. As can be  
Figure 11: Flow field distribution under different adaptive levels. (**a**) AMR-level = 3; (**b**) AMR-level = 5. Figure 10: Grid distribution under different adaptive criteria for solutions, \(t^{*}=4.0\). (**a**) Based on velocity vorticity; (**b**) based on velocity divergence; (**c**) based on velocity vorticity-divergence. discerned from the figure, activating AMR had a minimal effect on the flow field results, achieving an outcome comparable to the simulation with a uniformly fine mesh.