# Accurate RANS Simulation of Wind Turbine Stall by Turbulence Coefficient Calibration
## 4 Simulations with Turbulence Coefficient Calibration
### Airfoil Simulation with Different Values of \(\beta_{\infty}^{*}\)  
The airfoil simulation of the S809 airfoil has been made using various values of \(\beta_{\infty}^{*}\). The optimal value can easily be obtained by comparison with the airfoil experimental data. The simulation results show that \(\beta_{\infty}^{*}\) = 0.11 is the optimal value for the S809 airfoil at Re = \(1\times 10^{6}\). The predicted lift coefficient is shown in Figure 12 for \(\beta_{\infty}^{*}\) = 0.09 (the default value) and \(\beta_{\infty}^{*}\) = 0.11. The results for other values of \(\beta_{\infty}^{*}\) are not displayed here for simplicity. In general, a higher \(\beta_{\infty}^{*}\) leads to a lower lift coefficient in the range of light stall.  
Table 6 lists the relative simulation error defined as  
\[\delta=\frac{C_{L,sim}-C_{L,\text{exp}}}{C_{L,\text{exp}}}\times 100\% \tag{1}\]  
in which \(C_{L,sim}\) and \(C_{L,\text{exp}}\) are the lift coefficients obtained from the simulation and the experiment, respectively. An averaged relative error is defined as following to measure the overall error level in a range of angles of attack,  
\[\overline{\delta}=\frac{1}{n}\sum_{n=1}^{i}\left|\delta_{i}\right| \tag{2}\]  
in which \(i\) is the index and \(n\) is the count.  
It can be seen from the figure and the table that \(\beta_{\infty}^{*}\) = 0.11 leads to significantly better results in the range of \(8^{\circ}\) < AoA < \(20^{\circ}\) where light stall occurs. The averaged relative error is reduced to 4.5% from 30.0%. On the other hand, the calibration results in little change of lift coefficient in the range of AoA > \(20^{\circ}\) where deep stall occurs.  
The pressure distributions over the airfoil at four angles of attack of \(8^{\circ}\), \(12^{\circ}\), \(16^{\circ}\), and \(20^{\circ}\) are shown in Figure 13. According to the NREL Phase VI experiment, light stall is about to occur at \(8^{\circ}\), is in progress at \(12^{\circ}\) and \(16^{\circ}\), and converts to deep stall at \(20^{\circ}\). The pressure distributions for \(\beta_{\infty}^{*}\) = 0.11 are consistent with the experimental data at all the four angles of attack. In contrast, \(\beta_{\infty}^{*}\) = 0.09 only performs well at \(8^{\circ}\) where no stall occurs. Figure 14 shows the simulated streamlines around the airfoil at angles of attack of \(16^{\circ}\) and \(20^{\circ}\). At \(16^{\circ}\), the flow patterns for \(\beta_{\infty}^{*}\) = 0.09 and \(\beta_{\infty}^{*}\) = 0.11 are both trailing edge flow separation (light stall). The difference is that \(\beta_{\infty}^{*}\) = 0.11 leads to a larger separation vortex. At \(20^{\circ}\), \(\beta_{\infty}^{*}\) = 0.11 and \(\beta_{\infty}^{*}\) = 0.09 gives complete flow separation (deep stall) and trailing edge flow separation (light stall), respectively.  
Figure 12: The simulation results of lift coefficient for different values of \(\beta_{\infty}^{*}\) compared to experimental data. (Re = 1 \(\times\)\(10^{6}\)).  
The airfoil simulation has also been carried out at the Reynolds numbers of 0.75 \(\times\) 10\({}^{6}\) and \(1.5\times 10^{6}\), which are the lowest and the highest Reynolds number of the OSU experiment [37],  
Figure 14: The simulated streamlines around the airfoil for different values of \(\beta_{\infty}^{*}\) (Re = 1 \(\times\) 10\({}^{6}\)).  
Figure 13: The predicted pressure distributions for different values of \(\beta_{\infty}^{*}\), with a comparison to experimental data (Re = 1 \(\times\) 10\({}^{6}\)).  
respectively. The results are shown in Figure 15. \(\beta_{\infty}^{*}=0.11\) makes the simulation accuracy significantly improved at the both Reynolds numbers, suggesting that the optimal \(\beta_{\infty}^{*}\) calibrated at a certain Reynolds number stays in a relatively wide range of Reynolds numbers.