# Numerical Study on Tree Belt Impact on Wind Shear on Agricultural Land
## 2 Methods
### Numerical Method
#### 2.2.3 Boundary Conditions  
The inflow to the domain was a turbulent atmospheric boundary layer. Its velocity profile can be approximated by a power law or a log-law function. Physical measurements of the wind direction (Figure 3a) revealed that the prevailing wind direction was north-northwest NNW (330\({}^{\circ}\)), with a secondary main direction at an azimuth of 200\({}^{\circ}\) south-southwest (SSW). The study presented results from the NNW wind direction, to which the tree belt was nearly perpendicular. For this wind direction, the power law and log-law profiles were used to fit the data from the measurement mast. The power law profile of velocity has the general appearance of [87]:  
\[\frac{u}{u_{ref}}=\left(\frac{z}{z_{ref}}\right)^{\alpha}, \tag{1}\]  
where \(\alpha\) is the profile exponent and the '_ref_' index refers to reference quantities for velocity and elevation. At a reference height of 40 m, the measured velocity was 6.2 m/s, which yielded a profile exponent of 0.196. According to [81], this profile exponent corresponded to a moderately rough (grass and farmland) to rough (park and vegetation) terrain. The migration from a farmland to a rough terrain can probably be explained by the presence of tree belts nearby. The log-law profile has the general appearance:  
\[\frac{u}{u_{*}}=\frac{1}{\kappa}ln\left(\frac{z+z_{0}}{z_{0}}\right), \tag{2}\]  
where:  
\(u*\) = friction velocity (m/s),  
\(z_{0}\) = aerodynamic terrain roughness (m). Curve fitting with the three mast measurement points allowed to determine the friction velocity as 0.5 m/s and the aerodynamic terrain roughness as \(z_{0}=0.298\) m. This value also corresponds to a rough terrain. The log-law and power law curve-fitted profiles together with measured data are plotted in Figure 3b. The two analytical expressions yielded almost full agreement at elevations above the tree belt, which are of main interest for the operation of wind turbines. Therefore, any of the profiles could be used as a boundary condition for the simulation. However, the power law would predict an infinite velocity gradient at zero elevation; therefore, the log-wall expression was used as a boundary condition for velocity. Figure 3: (**a**) Prevailing wind direction at 40 m height based on on-site data, (**b**) Log-law and power law profiles fitted to experimental data. To determine the turbulent kinetic energy profile, the turbulent kinetic energy was first calculated at each measurement point from the formula:  
\[k=\frac{3}{2}(U.I)^{2}, \tag{3}\]  
with \(U\) and \(I\) being the mean measured air velocity and turbulence intensity, respectively. Then, the log-law was applied to the turbulent kinetic energy in the form:  
\[k=\frac{u_{*}^{2}}{\sqrt{C_{\mu}}}\sqrt{C_{1}ln\left(\frac{z+z_{0}}{z_{0}} \right)+C_{2}}. \tag{4}\]  
The curve-fitting coefficients \(C_{1}\) and \(C_{2}\) defaulted to 0 and 1, respectively, which would result in a uniform value of \(k\). However, calculations from measurements via Formula (5) indicated a slight variation in the value of \(k\), and thus the coefficient \(C_{1}\) was chosen as 0.22, while \(C_{2}\) was retained as 1. This change proved to bring a negligible difference in the results. The same coefficient values were also used to assign the profile of the turbulent dissipation rate, given by:  
\[\varepsilon=\frac{u_{*}^{3}}{\kappa(z+z_{0})}\sqrt{C_{1}ln\left(\frac{z+z_{0}} {z_{0}}\right)+C_{2}}. \tag{5}\]  
The high-Re turbulence model requires wall functions to set the boundary conditions at the wall, in this case the ground. Wall constraint was applied to the turbulent viscosity, the turbulent kinetic energy \(k\), and its dissipation rate \(\varepsilon\). As a first step, a standard wall function group was applied, in which the model expression for the turbulent viscosity is:  
\[\
u_{t}=f_{blend}\left(\
u_{t(vis)},\
u_{t(log)}\right), \tag{6}\]  
with  
\[ \begin{array}{c}\
u_{t(vis)}=0\\ \
u_{t(log)}=\
u_{w}\left(\frac{y^{+}x}{ln\left(Ey^{+}\right)}-1\right)\\ y^{+}=C_{\mu}^{1/4}y\frac{\sqrt{k}}{v_{w}},\end{array}