## 2 Materials and Methods
### Numerical Approach
#### 2.2.4 Mesh Adaptation Criteria  
When using the AMR technique, the mesh refinement level refers to the maximum number of times a mesh cell can be refined during the computation process. A refinement level of one for a volumetric mesh cell represents the division of a parent grid cell into eight sub-grid cells, as presented in Figure 2. Figure 2a illustrates an initial grid plane schematic within the yellow box, whereas Figure 2b-d correspond to refinement levels 1, 2, and 3, respectively, because the mesh was refined in that region. Herein, a refinement level 2 was applied to the continuum-based model.  
Commonly, refining the grid and establishing adaptive grid criteria is a challenging task. When dealing with a high angle of attack flow around a submarine, complex phenomena, such as boundary layer separation and vortex shedding, come into play. Therefore, choosing the novel omega vortex identification method as the criterion for adaptive grid refinement could lead to better vortex capture, thereby enhancing the accuracy of the solution.  
The novel omega vortex identification approach was first proposed by [PERSON] et al. (2016) in 2016. In contrast to conventional vortex identification methods, the new omega method has physical significance and normalization characteristics. It is able to simultaneously capture both strong and weak vortices, without the need for significant threshold adjustments. The new omega-based approach is mainly constructed based on the decomposition of the eddy field into its rotating component, denoted by \(R\), and its non-rotating component, denoted by \(S\), as in the following form:  
\[\omega=R+S \tag{9}\]  
Typically, the directions of \(R\) and \(\omega\) are dissimilar. By introducing the parameter \(\Omega\), which represents the ratio of the magnitude of the rotating vortex to the magnitude of the total vortex, the corresponding calculation formula is given by the following:  
\[\Omega=\frac{\left\|B\right\|^{2}_{F}}{\left\|A\right\|^{2}_{F}+\left\|B \right\|^{2}_{F}}, \tag{10}\]  
where \(A\) and \(B\) represent the symmetric and antisymmetric tensors of the velocity gradient, which are expressed in Equations (9) and (10) as follows:  
\[A=\frac{1}{2}(\
abla V+\
abla V^{T})=\begin{bmatrix}u_{x}&\frac{1}{2}(u_{y}+v _{x})&\frac{1}{2}(u_{z}+w_{x})\\ \frac{1}{2}(u_{y}+v_{x})&v_{y}&\frac{1}{2}(v_{z}+w_{y})\\ \frac{1}{2}(u_{z}+w_{x})&\frac{1}{2}(v_{z}+w_{y})&w_{z}\end{bmatrix} \tag{11}\]  
\[B=\frac{1}{2}(\
abla V-\
abla V^{T})=\begin{bmatrix}0&\frac{1}{2}(u_{y}-v_{x} )&\frac{1}{2}(u_{z}-w_{x})\\ \frac{1}{2}(v_{x}-u_{y})&0&\frac{1}{2}(v_{z}-w_{y})\\ \frac{1}{2}(w_{x}-u_{z})&\frac{1}{2}(w_{y}-v_{z})&0\end{bmatrix} \tag{12}\]  
It can be seen that the value range of \(\Omega\) is between 0 and 1, which can be understood as the vorticity concentration. The expression \(\Omega=1\) represents the case of fluid rotating with the cylinder, whereas \(\Omega>0.5\) indicates that \(B\) is larger than \(A\), and therefore, \(\Omega\) slightly larger than 0.5 can be used as a vortex identification criterion. In practical applications, to avoid the problem of very large errors when the denominator of Equation (12) is a very small number, a small positive constant (\(\varepsilon\)) is added to the denominator. As a result, the expression \(\Omega\) is modified to the following:  
\[\Omega=\frac{\left\|B\right\|^{2}_{F}}{\left\|A\right\|^{2}_{F}+\left\|B \right\|^{2}_{F}+\varepsilon} \tag{13}\]  
In this study, the value of \(\varepsilon\) was set to 0.001 based on the approximate expression suggested by [PERSON] et al. (2018), and \(\Omega=0.52\) was selected as the threshold value for vortex identification. To control the number of grid points, certain constraints were imposed on the extent of the adaptive mesh in the hull computational domain. Specifically, only an area five times the hull width around the hull was refined.  
Figure 2: Adaptive mesh refinement levels. (**a**) initial mesh; (**b**) level 1; (**c**) level 2; (**d**) level 3.