# Dynamic Stall Mechanisms of Pitching Airfoil: IDDES Study Across Different Mach Numbers
## 2 Numerical Methods  
### CFD Methods  
The Reynolds-averaged Navier-Stokes (RANS) equations are employed as the governing equations of airfoil flow, which can be expressed in the following integral form:  
\[\frac{\partial}{\partial t}\int_{\Omega}\textbf{W}d\Omega+\oint_{\partial \Omega}(\textbf{{F}}_{c}-\textbf{{F}}_{v})\cdot\textbf{{n}}dS=0 \tag{1}\]where \(\partial\Omega\) is the boundary of a control volume \(\Omega\); \(\mathbf{n}\) is the unit normal vector of the surface element \(dS\); \(\mathbf{W}\), \(\mathbf{F}_{c}\) and \(F_{v}\) are the vector of conserved variables, convective fluxes, and viscous fluxes, respectively, which are written as  
\[\mathbf{W}= \begin{bmatrix}\rho\\ \rho u\\ \rho v\\ \rho e\end{bmatrix} ,\mathbf{F}_{c}= \begin{bmatrix}\rho(\mathbf{V}-\mathbf{V}_{\mathbf{\omega} })\\ \rho u(\mathbf{V}-\mathbf{V}_{\mathbf{\omega}})+\mathbf{I}_{\mathbf{x}}p\\ \rho v(\mathbf{V}-\mathbf{V}_{\mathbf{\omega}})+\mathbf{I}_{\mathbf{y}}p\\ \rho w(\mathbf{V}-\mathbf{V}_{\mathbf{\omega}})+\mathbf{I}_{\mathbf{z}}p\\ \rho H(\mathbf{V}-\mathbf{V}_{\mathbf{\omega}})+\mathbf{V}_{\omega}p\end{bmatrix} ,\mathbf{F}_{v}= \begin{bmatrix}0\\ \mathbf{I}_{\mathbf{x}}\tau_{xx}+\mathbf{I}_{\mathbf{y}}\tau_{xy}+\mathbf{I}_{\mathbf{z}}\tau_{xz}\\ \mathbf{I}_{\mathbf{x}}\tau_{yx}+\mathbf{I}_{\mathbf{y}}\tau_{yy}+\mathbf{I}_{\mathbf{z}}\tau_{yz}\\ \mathbf{I}_{\mathbf{x}}\tau_{zx}+\mathbf{I}_{\mathbf{y}}\tau_{zy}+\mathbf{I}_{\mathbf{z}}\tau_{zz}\\ \mathbf{I}_{\mathbf{x}}\Theta_{x}+\mathbf{I}_{\mathbf{y}}\Theta_{y}+\mathbf{I}_{\mathbf{z}}\Theta_{z} \end{bmatrix}  \tag{2}\]  
where \(\rho\) denotes the air density, \(p\) the static pressure, \(E\) and \(H\) the total energy and enthalpy per unit mass, respectively. \(u\), \(v\), and \(w\) are the velocity components of the absolute velocity vector \(\mathbf{V}\), and \(\mathbf{V}_{\mathbf{\omega}}\) is the grid velocity vector; \(\mathbf{I}_{\mathbf{x}}\), \(\mathbf{I}_{\mathbf{y}}\) and \(\mathbf{I}_{\mathbf{z}}\) are unit vectors along the \(x\), \(y\), and \(z\) axis. \(\tau_{ij}\) represents the viscous stress, and \(\Theta_{i}\) is the terms describing the work of the viscous stresses and the heat conductive in the fluid. The \(k\)-\(\omega\) shear stress transport (SST) turbulence model is adopted to account for the effects of turbulence. The SST model comprises two transportation equations relating to the turbulent kinetic energy (\(k\)) and its specific dissipation rate (\(\omega\)):  
\[\frac{\partial(\rho k)}{\partial t}+\
abla\cdot(\rho Vk)=P_{k}-\rho\sqrt{k^{3} }/l_{RANS}+\
abla\cdot[(\mu_{l}+\sigma_{k}\mu_{t})\
abla k] \tag{3}\]  
\[\frac{\partial(\rho\omega)}{\partial t}+\
abla\cdot(\rho V\omega)=\frac{\alpha \rho}{\mu_{t}}P_{k}-\beta\rho\omega^{2}+\
abla\cdot[(\mu_{l}+\sigma_{\omega} \mu_{t})\
abla\omega]+2(1-F_{1})\rho\sigma_{\omega 2}\frac{\
abla k\cdot\
abla \omega}{\omega} \tag{4}\]  
where \(F_{1}\) is the blending function. The terms on the right-hand side represent eddy-viscosity production, dissipation, and conservative diffusion, respectively. A hybrid RANS-LES method can be built using the IDDES model based on the SST turbulence model [34], where the RANS length scale in Equation (2) is replaced by the IDDES length scale in the SST IDDES formulations. The IDDES length scale reads as follows:  
\[l_{IDDES}=\widetilde{f}_{d}\cdot l_{RANS}+\left(1-\widetilde{f}_{d}\right) \cdot l_{LES} \tag{5}\]  
Here \(\widetilde{f}_{d}\) denotes the empirical blending function, \(l_{LES}\) denotes the LES length scale. For detailed formulas, please refer to reference [34]. The governing equations are solved using our in-house code CLORNS [35] based on a structured sliding mesh framework. Spatial discretization is achieved via the finite volume method (FVM): inviscid terms are computed using a fifth-order TENO scheme with [PERSON]'s flux difference splitting, while viscous terms employ second-order central differencing.