# A Study of Wind Shear Influences on the Aerodynamic Performances of a UAV Airfoil
## 2 Numerical Simulation Model and Shear Wind Model
### Shear Wind Model  
Numerous velocity distributions usually appear in a natural shear wind, which has a great influence on UAV performances. In this paper, three different wind profiles, gradient-free wind, linear distributed shear wind and logarithmic distribution shear wind are selected to study these influences. The three wind profiles are defined as follows:  
Gradient-free wind: The wind field without gradient is defined in the gradient-free wind model, which is defined in Equation (1).  
\[V_{\rm W}=C \tag{1}\]  
In Equation (1), \(V_{\rm W}\) is the wind speed at height \(h\), which is selected as Mach number 0.3, Mach number \(Ma=V_{\rm W}/a\), and \(a\) is sound speed. The Mach number corresponding to \(C\) in this paper is 0.3. As shown in Figure 2, the velocity has no gradient along the height direction and wind speed is constant.  
Linear distributed shear wind: The velocity magnitude varies linearly along the height direction while the wind direction remains the same. The velocity profile against height is defined in Equation (2), i.e., \(dV_{\rm W}/dh\) is constant. Schematic diagrams of the shear wind field with positive and negative gradients are shown in Figure 3.  
\[V_{\rm W}=V_{Whmin}+\left(\frac{V_{Whmax}-V_{Whmin}}{h_{max}-h_{min}}\right)(h- h_{min}) \tag{2}\]  
Figure 1: NACA0012 airfoil grid.  
Figure 2: Schematic diagram of gradient-free wind.  
where \(h_{min}\) and \(h_{max}\) are heights at the lowest and highest positions in the wind field, \(V_{Whmin}\) and \(V_{Whmax}\) are the corresponding velocities. It is shown that velocity gradient with \(dV_{W}/dh=\pm 3.05\), 2.29 and 1.52 are used for the four wind profiles, while the Mach number M = 0.3 is set for wind speed, as listed in Table 1. A positive and negative gradient velocity gradient is characterized using the \(dV_{W}/dh\), and \(dV_{W}/dh=-3.05\) is used for the negative one.  
Logarithmic distribution shear wind: A logarithmically distributed shear wind profile is established using the experimental wind data proposed by [PERSON] (2018). The observed wind data is fitted and estimated using a particle filtering approach to establish the mathematical model, as defined in Equation (3).  
\[\begin{array}{l}V_{W}=V_{Whmin}+\frac{\Delta V_{Wh}}{2}[1+\text{erf}(4\frac {h-h_{max}}{h_{min}-h_{min}})]\\ \Delta V_{Wh}=V_{Whmax}-V_{Whmin}\\ h_{avg}=(h_{max}+h_{min})/2\end{array} \tag{3}\]  
where erf is the error function; \(\Delta V_{Wh}\) is the velocity difference within the wind field, \(\Delta V_{Wh}=V_{Whmax}-V_{Whmin}\); \(h_{avg}\) represents the average height of the wind field, \(h_{avg}=(h_{max}+h_{min})/2\). The Mach numbers corresponding to \(V_{Whmin}\) and \(V_{Whmax}\) are 0.2 and 0.4, respectively. A typical velocity profile is shown in Figure 4.  
\begin{table}
\begin{tabular}{c c c c} \hline \(Ma\_V_{Whmin}\) & \(Ma\_V_{Whmax}\) & \(dV_{W}/dh\) & \(Ma\_V_{Wharg}\) \\ \hline
0.2 & 0.4 & 3.05 & 0.3 \\
0.4 & 0.2 & \(-3.05\) & 0.3 \\
0.225 & 0.375 & 2.29 & 0.3 \\
0.25 & 0.35 & 1.52 & 0.3 \\ \hline \end{tabular}
\end{table}
Table 1: Parameters of four wind models.  
Figure 3: Schematic diagram of Linear distributed shear wind: (**a**) Positive gradient; (**b**) Negative gradient.