# Aerodynamic Sensitivity Analysis for a Wind Turbine Airfoil in an Air-Particle Two-Phase Flow
## 4 Aerodynamic Sensitivity Analysis to Particles for S809 Airfoil
### Particle Effects at Different Re Numbers  
Four \(Re\) numbers of \(0.75\times 10^{6}\), \(1\times 10^{6}\), \(1.25\times 10^{6}\) and \(1.5\times 10^{6}\) are computed at a fixed particle concentration \(m=1\%\). Figure 6 shows the variations of lift and drag coefficient increments with the angle of attack \(\alpha\), where the relative increments are calculated from  
\[\begin{split}\Delta C_{L}&=\frac{C_{L,p}-C_{L,g}}{ C_{L,g}}\times 100\%\\ \Delta C_{D}&=\frac{C_{D,p}-C_{D,g}}{C_{D,g}}\times 100\% \end{split} \tag{8}\]  
with the subscripts \(P\) and \(g\) denoting the flows with and without particles, respectively.  
As can be observed from Figure 6, the particle phase leads to a lift decrease and a drag increase; with increasing angle of attack, the lift decrement and drag increment present similar variation trends, i.e., from increasing to decreasing and to increasing again, and there is a most sensitive angle of attack at which the increments of aerodynamic coefficients reach their peak values; with \(Re\) increasing from \(0.75\times 10^{6}\) to \(1.5\times 10^{6}\), the most sensitive angle of attack increases from \(8.6^{\circ}\) to \(10.1^{\circ}\) roughly; in all cases, the maximum increments of lift and drag coefficients are \(-5.74\%\) and \(21.19\%\), respectively. Considering the actual \(Re\) range shown in Figure 2, the more sensitive region of angles of attack is conservatively determined as from \(7.1^{\circ}\) to \(11.1^{\circ}\), which is marked by a square in Figure 6.  
In Figure 6, the aerodynamic coefficient increments change slowly in the range of small angles of attack without flow separation. In an attached flow, the airfoil aerodynamic performance is affected by the boundary layer momentum loss due to particles. Figure 7 illustrates the velocity profiles at three positions on the S809 airfoil surface with relatively large curvatures, where the legends of upper and lower refer to the upper and lower surfaces, respectively. The particle phase makes the velocity in the boundary layer decrease, and the influence near the leading edge is greater.  
Figure 6: \(Re\) effects on lift and drag coefficient increments of S809 airfoil at \(m=1\%\): (**a**) Lift coefficient increment; (**b**) drag coefficient increment.  
With a further increase in the angle of attack in Figure 6, flow separation occurs and the magnitudes of lift and drag increments increase sharply till the most sensitive angle of attack. Figure 8 plots the location of separation point and its dimensionless moving distance defined as  
\[\frac{\Delta x}{c}=\frac{x_{P}-x_{g}}{c}\times 100\% \tag{9}\]  
with \(x_{P}\) and \(x_{g}\) being the \(x\) coordinates of separation point in the air-particle and air flows, respectively.  
As can be found from Figure 8, the particle phase causes the separation point to move upstream, that is to say, flow separation becomes more extensive. In Figures 6 and 8b, the separation point moving distance and the aerodynamic coefficient increments have similar variation trends with the angle of attack, and at each \(Re\) number, there is a about \(1^{\circ}\) angle difference between the maximum upstream-moving distance of separation point and the maximum lift decrement. Consequently, in the case of flow separation, it can be concluded that the particle influences on flow separation dominate the variations of aerodynamic coefficient increments and result in the severer aerodynamic performance degradation in Figure 6.