# A Novel Cell-Based Adaptive Cartesian Grid Approach for Complex Flow Simulations
## 5 Conclusions  
This research developed a technique based on cell-based AMR Cartesian grids. It utilized a DWSL method for the discretization of the finite difference method and adopted IBMs near physical boundaries. This approach facilitated the handling of both transient and steady flow problems under various velocity conditions flexibly and conveniently. Based on the analysis conducted, the following conclusions are drawn:  
1. In the context of supersonic flow over a forward facing step, employing a comprehensive criterion based on velocity, vorticity, and divergence enhanced the capture of flow field characteristics. AMR achieved comparable flow field results to uniform grids, while judicious grid coarsening and refinement reduced computational costs without compromising simulation accuracy. With solution adaptation across three layers, grid count and overall computational cost decreased to 48.9% and 47.6%, respectively, compared to uniform fine grids. Increasing adaptation layers to five further reduced these ratios to 15.8% and 19.5%, respectively.
2. In the simulation of a high Reynolds number turbulent flow over an NHLP three-element airfoil, the immersed Cartesian grid method autonomously generated the required computational grid within seconds. After eight iterations of AMR, the number of grids increased from 1.51 \(\times\) 10\({}^{5}\) to 2.03 \(\times\) 10\({}^{5}\). While activating AMR had a minor effect on surface pressure coefficients, it significantly enhanced the resolution of the simulation in the wake region.
3. For the three-dimensional supersonic flow simulation around a sphere, the initial grid resolution was inadequate for accurately portray the features of the flow field. Through six iterations of solution adaptation, the accuracy in capturing shock waves was significantly improved, particularly at the front and rear of the sphere. Although this process had a minimal impact on the pressure distribution in the stagnation region, it notably improved the prediction of the separation zone, avoiding the overestimation of negative pressure regions, and thus, making the predictions of drag coefficients closer to experimental results.  
In the future, our research endeavors will focus on pioneering higher-order numerical algorithms and efficient parallel technologies. These advancements aim to enhance simulation accuracy, reduce computational costs, and ultimately address the escalating demands posed by increasingly complex and larger-scale engineering problems.  
Conceptualization, X.Y. and Z.T.; methodology, C.L. and L.B.; formal analysis, C.L., D.Z. and W.W.; investigation, S.M. and W.W.; writing--original draft preparation, C.L. and S.M.; writing--review and editing, C.L., D.Z. and L.B.; supervision, X.Y. and Z.T.; funding acquisition, L.B. and C.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Key Research and Development Program of China (Grant No. 2019 YFA0405204) and the Postgraduate Scientific Research Innovation Project of Hunan Province (Grant No. CX20230237).  
Not applicable.  
The data presented in this study are available in the article.  
The authors declare no conflict of interest.