## 2 Numerical Methods
### Eulerian Approach Based on Continuum Properties
#### 2.1.2 Vortex Core Identification and Vortex Model  
Although partial differential equations are non-dispersive, the difference equations obtained by discretizing the partial differential equations are dispersive and dissipative. This fact implies that the numerical scheme of low order requires more numbers of grids to resolve the complex flow structure more accurately [15]. It is challenging to apply high-order numerical schemes or lots of grids in practical engineering fields because of enormous numerical cost and time, though these methods have been used to investigate relatively simple problems for academic purposes. Due to this difficulty, a lot of previous studies using the CFD techniques such as the RANS solvers have used the grids with local high resolution in the vicinity of the target object or wall to accurately simulate the boundary layer. Figure 3: Computational grids for flow over the entire body and propellers: (**a**) entire grids and rotating part of grids, (**b**) surface grids on Model-A and (**c**) surface grids on Model-B. However, the TVC of underwater propellers typically forms along a long spiral vortex trajectory in the downstream direction away from the propeller surface. This implies that the enormous numerical cost and time are required to simulate the TVC accurately using the RANS solver only. In this study, to deal with this issue, the tip vortex is reconstructed by applying the vortex model to the flow field obtained from the RANS solver. The unknown parameters of the vortex model, the minimum vortex core radius and maximum tangential velocity (or azimuthal velocity), are determined from the pressure and velocity fields of the RANS results. Because the vortex structure of the vortex model is assumed to be axisymmetric with an origin on the vortex core center, the vortex core identification needs to be made before the vortex reconstruction. Several criteria are available to identify the vortex core, such as minimum pressure, maximum vorticity, maximum Q-criterion, and minimum \(\lambda_{2}\)-criterion. In particular, the \(\lambda_{2}\)-criterion has been successfully used for identifying the coherent vortex structure for various types of problems [31]. \(\lambda_{2}\) is defined as the second largest eigenvalue of \(S^{2}+\Omega^{2}\), where \(S\) and \(\Omega\) are the symmetric and antisymmetric parts, respectively, of the flow velocity gradient, \(J\equiv\
abla u\). The symmetric part presents a strain rate tensor of velocity and describes the deformation rate, while the anti-symmetric part is directly related to the rate of rotation of a fluid, but not its deformation at all. Because \(S^{2}+\Omega^{2}\) has a symmetric matrix, it only has real eigenvalues. The symmetric and antisymmetric parts are defined as follows:  
\[J\equiv\
abla\overset{\rightarrow}{u}=\left[ \begin{array}{ccc}\partial_{x} u&\partial_{y}u&\partial_{z}u\\ \partial_{x}v&\partial_{y}v&\partial_{z}v\\ \partial_{x}w&\partial_{y}w&\partial_{z}w\end{array} \right]=S+\Omega \tag{4}\]  
\[S=\frac{J+J^{T}}{2}=\left[ \begin{array}{ccc}\partial_{x}u&\frac{1}{2}\Big{(} \partial_{y}u+\partial_{x}v\Big{)}&\frac{1}{2}(\partial_{z}u+\partial_{x}w)\\ \frac{1}{2}\Big{(}\partial_{y}u+\partial_{x}v\Big{)}&\partial_{y}v&\frac{1}{2} \Big{(}\partial_{z}v+\partial_{y}w\Big{)}\\ \frac{1}{2}(\partial_{z}u+\partial_{x}w)&\frac{1}{2}\Big{(}\partial_{z}v+ \partial_{y}w\Big{)}&\partial_{z}w\end{array} \right] \tag{5}\]  
\[\Omega=\frac{J-J^{T}}{2}=\left[ \begin{array}{ccc}0&\frac{1}{2}\Big{(} \partial_{y}u-\partial_{z}v\Big{)}&\frac{1}{2}(\partial_{z}u-\partial_{x}w)\\ \frac{1}{2}\Big{(}\partial_{x}v-\partial_{y}u\Big{)}&0&\frac{1}{2}\Big{(} \partial_{z}v-\partial_{y}w\Big{)}\\ \frac{1}{2}(\partial_{x}w-\partial_{z}u)&\frac{1}{2}\Big{(}\partial_{y}w- \partial_{z}v\Big{)}&0\end{array} \right]. \tag{6}\]  
Here, \(u\), \(v\), and \(w\) denote the velocity components in the \(x\)-, \(y\)- and \(z\)-directions, respectively.