# Dynamic Stall Characteristics of the Bionic Airfoil with Different Waviness Ratios
## 2 Materials and Methods
### Control Equation and Turbulence Model  
In the numerical simulation case, the fluid flow was considered to be a viscous incompressible turbulent flow. The density of the fluid was constant. The flow control equations are the continuity Equation (4)  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_{i})=0 \tag{4}\]  
Figure 3: Computational domain and grids. (**a**) calculation domain; (**b**) local schematic of airfoil grid; (**c**) overall schematic of airfoil grid.  
and the momentum Equation (5)  
\[\frac{\partial}{\partial t}(\rho u_{i})+\frac{\partial}{\partial x_{j}}(\rho u_{i} u_{j})=-\frac{\partial p}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\Bigg{[}\mu \bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_ {i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_{l}}{\partial x_{l}}\bigg{)} \Bigg{]}+\frac{\partial}{\partial x_{j}}\Big{(}-\rho\overline{u^{\prime}{}_{ i}u^{\prime}{}_{j}}\Big{)} \tag{5}\]  
where \(t\) and \(x_{i}\) are time and space in Cartesian coordinates; \(p\), \(\rho\), and \(\
u\) are the time-averaged pressure, density, and kinematic viscosity; and \(-\rho\overline{u^{\prime}{}_{i}u^{\prime}{}_{j}}\) is the Reynolds stress.  
The URANS equations are solved based on the finite volume method. For the turbulent flow calculations, the SST \(k\)-\(\omega\) turbulence model was chosen due to its ability to capture dynamic flow structures in a wide range of Reynolds numbers [28].  
The SST \(k\)-\(\omega\) turbulence model is a two-equation eddy viscosity model. The turbulent kinetic energy transport \(k\) is governed by Equation (6).  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{j}}\Bigg{[} \rho u_{j}k-(\mu+\sigma_{k}u_{l})\frac{\partial k}{\partial x_{j}}\Bigg{]}= \tau_{tij}S_{ij}-\beta^{*}\rho\omega k \tag{6}\]  
The equation for the specific consumption rate \(\omega\) is Equation (7).  
\[\frac{\partial}{\partial t}(\rho\omega)+\frac{\partial}{\partial x_{j}} \Bigg{[}\rho u_{j}k-(\mu+\sigma_{\omega}\mu_{t})\frac{\partial\omega}{\partial x _{j}}\Bigg{]}=P_{\omega}-\beta\rho\omega^{2}+2(1-F_{1})\frac{\rho\sigma_{ \omega 2}}{\omega}\frac{\partial k\,\partial\omega}{\partial x_{j}\partial x_{j}} \tag{7}\]  
The eddy viscosity model for the Reynolds stress is Equation (8).  
\[\tau_{tij}=2\mu_{t}(S_{ij}-\frac{1}{3}S_{nm}\delta_{ij})-\frac{2}{3}\rho k \delta_{ij} \tag{8}\]  
The function \(F_{1}\) is given by Equation (9).  
\[\left\{\begin{array}{c}F_{1}=\tanh\Phi_{1}^{4}\\ \\ \Phi_{1}=\min\Bigl{[}\max\Bigl{(}\frac{\sqrt{k}}{0.00\omega^{\prime}},\frac{ 500\mu}{\rho\omega^{2}}\Bigr{)},\frac{4\rho k}{\sigma_{\omega 2}D_{\omega}^{+}y^{2}}\Bigr{]}\\ \\ D_{\omega}^{+}=\max\Bigl{(}2\rho\frac{1}{\sigma_{\omega 2}}\frac{1}{\omega}\frac{ \partial k}{\partial x_{j}}\frac{\partial\omega}{\partial x_{j}},10^{-10} \Bigr{)}\end{array}\right. \tag{9}\]  
In the equation system(9), \(y\) is the distance to the next surface and \(D_{\omega}{}^{+}\) is the positive part of the cross-diffusion term. \(\mu_{t}=\rho k\) / \(\omega\) is the eddy viscosity; \(S_{ij}\) is the average velocity strain-rate tensor; \(\beta=0.09\), \(\sigma_{k}=0.2\), and \(\sigma_{\omega}=0.2\) are model constants; and \(\delta_{ij}\) is the Kronecker delta.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Values** & **Values** \\ \hline Airfoil & NACA0015 & NACA0012 \\ Reynolds number (Re) & \(3.0\times 10^{5}\) & \(1.35\times 10^{5}\) \\ Amplitude of oscillation (\(\alpha_{\rm amp}\)) & \(10^{\circ}\) & \(15^{\circ}\) \\ Oscillation frequency (f) & \(1.7507\) & \(1.162\) \\ Mean angle of attack (\(\alpha_{\rm mean}\)) & \(10^{\circ}\) & \(10^{\circ}\) \\ Free-flow velocity (U\({}_{\infty}\)) & \(22\) m/s & \(19.72\) m/s \\ Chord length of airfoil (C) & \(0.2\) m & \(0.1\) m \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters for the computational case.