# Computational Fluid Dynamics-Aided Simulation of Twisted Wind Flows in Boundary Layer Wind Tunnel
## 4 Development Stage  
### Establishment of Numerical Wind Tunnel  
#### 4.1.1 CFD Setup for Numerical Wind Tunnel  
In the CFD simulation, the complex fluid dynamics issues can be simplified for numerical resolution through the Navier-Stokes (N-S) equations. By neglecting the temperature effect, a closed set of governing equations is formed by the continuity equation and the momentum conservation equation. In this study, both the Reynolds-Averaged Navier-Stokes (RANS) model and the Large-Eddy Simulation (LES) model are utilized for the CFD simulations. Figure 7: Twist angle profiles measured within model test region. Vortex filtration is involved in the LES model by considering the feature grid size. Large-scale vortices are solved directly by the N-S equation, and small-scale ones are solved by the sub-grid scale model. The governing equations in the instantaneous state after filtration can be expressed as:  
\[\frac{\partial\widetilde{u}_{i}}{\partial x_{i}}=0 \tag{6}\]  
\[\frac{\partial\widetilde{u}_{i}}{\partial t}+\widetilde{u}_{j}\frac{\partial \widetilde{u}_{i}}{\partial x_{j}}=-\frac{\partial\widetilde{p}}{\partial x_{ i}}+\frac{1}{\mathrm{Re}}\frac{\partial^{2}\widetilde{u}_{i}}{\partial x_{j}^{2}}- \frac{\partial\tau_{ij}}{\partial x_{j}} \tag{7}\]  
where \(\tau_{ij}\) is the sub-grid scale stress, given by \(\tau_{ij}\)= \(\rho u_{i}^{\sim}u_{j}-\rho\widetilde{u}_{i}\widetilde{u}_{j}\) to make the equations closed. The standard Smagorinsky sub-grid model [24] is adopted in this study as the LES turbulence model, expressed as:  
\[\tau_{ij}-\frac{1}{3}\tau_{\mathrm{k}\delta}_{ij}=-2\mu_{t}\widetilde{S}_{ij} \tag{8}\]  
\[\mu_{t}=\left(C_{\mathrm{s}}\Delta\right)^{2}\lvert\overrightarrow{S}\rvert \tag{9}\]  
\[\widetilde{S}_{ij}=1/2\left(\partial\widetilde{u}_{i}/\partial x_{j}+\partial \widetilde{u}_{j}/\partial x_{t}\right) \tag{10}\]  
\[\lvert\overrightarrow{S}\rvert=\sqrt{2\widetilde{S}_{ij}\widetilde{S}_{ij}} \tag{11}\]  
\[\Delta=\left(\Delta_{x}\Delta_{y}\Delta_{z}\right)^{\frac{1}{3}} \tag{12}\]  
where \(\mu_{t}\) is the dynamic viscosity in the sub-grid scale; \(\lvert\overrightarrow{S}\rvert\) is the velocity scale; \(\Delta_{x}\), \(\Delta_{y}\), and \(\Delta_{z}\) are the grid size in the x, y, and z directions, respectively; and \(C_{s}\) is the Smagorinsky constant taken as 0.1 in the present study. On the other hand, statistics are used in the RANS approach to represent turbulence flow as time-averaged and fluctuating parts. When the N-S equations are averaged, the instantaneous velocity is regarded as the sum of the average velocity and the fluctuation velocity. The equations are expressed below:  
\[\frac{\partial\overline{u}_{i}}{\partial x_{i}}=0 \tag{13}\]  
\[\frac{\partial\overline{u}_{i}}{\partial t}+\overline{u}_{j}\frac{\partial \overline{u}_{i}}{\partial x_{j}}=-\frac{1}{\rho}\frac{\partial\overline{ \varphi}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left[v\left(\frac{ \partial\overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline{u}_{j}}{ \partial x_{i}}\right)-\overline{u^{\prime}_{i}u^{\prime}_{j}}\right] \tag{14}\]  
where \(\rho\) is the air density; \(I\) and \(u_{j}\) are the wind velocity components; \(v\) is the coefficient of kinetic viscosity; and \(I_{i}\) and \(x_{j}\) are the coordinates vector of corresponding velocity components. An additional term, namely the Reynolds stress \(\rho\overline{u^{\prime}_{i}u^{\prime}_{j}}\), is contained in the incompressible N-S equations for the RANS simulation. To make the equations closed, several turbulence models were proposed based on different assumptions about this term, including the Standard \(k\)-\(\epsilon\) model [25], the RNG \(k\)-\(\epsilon\) model [26], the Realizable \(k\)-\(\epsilon\) model [27], the Standard \(k\)-\(\omega\) model [28], and the SST-\(k\)-\(\omega\) model [29]. Using CFD techniques, a numerical wind tunnel was established as a replica of the wind tunnel at Chongqing University introduced in Section 3.