## 2 Numerical Methodology  
### Turbulence Simulation Methods  
#### 2.1.1 ZDES Based on a Two-Equation k-\(\omega\) SST Model  
In this paper, the k-\(\omega\) SST model proposed by [PERSON] [18] was utilized as the base model to construct the ZDES hybrid method. The conservative forms of the \(k\) and \(\omega\) equations are as follows:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho u_{j}k)}{\partial x_{ j}}\ =\ P-\beta^{*}\rho\omega k+\frac{\partial}{\partial x_{j}}\Bigg{(}(\mu+\sigma_{k} \mu_{t})\frac{\partial k}{\partial x_{j}}\Bigg{)} \tag{1}\]  
\[\begin{array}{l}\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial(\rho u _{j}\omega)}{\partial x_{j}}\ =\ \frac{\gamma_{j}}{\alpha_{j}}P-\beta\rho\omega^{2}+\frac{ \partial}{\partial x_{j}}\Big{(}(\mu+\sigma_{\omega}\mu_{t})\frac{\partial \omega}{\partial x_{j}}\Big{)}\\ +2(1-F_{1})\frac{\partial\omega}{\omega}\frac{\partial k}{\partial x_{j}} \frac{\partial k}{\partial x_{j}}\end{array} \tag{2}\]  
where \(\rho\) is the density, \(k\) is the turbulent kinetic energy, \(\sigma_{k}\) = 0.5, \(\
u_{t}\ =\ \frac{\mu_{t}}{\rho}\) is the turbulent eddy viscosity, \(\mu\) is the molecular dynamic viscosity, and d is the distance from the grid point to the wall. The production term \(P\) is given by  
\[P\ =\ \tau_{ij}\frac{\partial u_{i}}{\partial x_{j}} \tag{3}\]  
where the Reynolds stress is given by  
\[\tau_{ij}\ =\ \mu_{t}\bigg{(}2S_{ij}-\frac{2}{3}\frac{\partial u_{k}}{\partial x _{k}}\delta_{ij}\bigg{)}-\frac{2}{3}\rho k\delta_{ij} \tag{4}\]  
\[S_{ij}\ =\ \frac{1}{2}\Bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}\Bigg{)} \tag{5}\]  
The turbulent eddy viscosity \(\mu_{t}\) and the vortex magnitude \(\Omega\) are given by  
\[\mu_{t}\ =\ \frac{\rho a_{1}k}{\max(a_{1}\omega,\Omega F_{2})} \tag{6}\]\[\Omega\ =\ \sqrt{2W_{ij}W_{ij}},\ \ \ W_{ij}\ =\ \frac{1}{2}\left(\frac{\partial u _{i}}{\partial x_{j}}-\frac{\partial u_{j}}{\partial x_{i}}\right) \tag{7}\]  
ZDES selects the RANS and DES calculation regions based on the requirements: RANS is employed in the attached flow region, and DES is employed in the flow separation region.  
In this study, ZDES is based on the k-\(\omega\) SST model. Modification of the destruction term of the k-equation is necessary in ZDES. After introducing the length scale \(\overset{\sim}{l}_{\text{ZDES}}\), the turbulent kinetic energy transport equation is rewritten in the following form:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho u_{j}k)}{\partial x_{j }}\ =\ \tau_{ij}\frac{\partial u_{j}}{\partial x_{j}}-\frac{\rho k^{3/2}}{\overset{ \sim}{l}_{\text{ZDES}}}+\frac{\partial}{\partial x_{j}}\left((\mu+\sigma_{k}u _{t})\frac{\partial k}{\partial x_{j}}\right) \tag{8}\]  
For the original SST turbulence model, the length scale \(\overset{\sim}{l}_{\text{ZDES}}\) is given by  
\[\overset{\sim}{l}_{\text{ZDES}}\ =\ l_{\text{SST}}\ =\ \frac{\sqrt{k}}{\beta^{*}\omega} \tag{9}\]  
For ZDES, different hybrid length scales need to be defined for different separation types in the separation flow region. Deck [19] classified flow separation into three types: (I) Separation caused by external geometry, where the separation position is fixed; (II) Separation caused by a pressure gradient on the surface, where the separation position is uncertain; (III) Separation caused by the influence of the incoming boundary layer. This study currently focuses on the first two types of flow.