# Numerical Study on Tree Belt Impact on Wind Shear on Agricultural Land
## 2 Methods
### Numerical Method
#### 2.2.3 Boundary Conditions
 \tag{7}\]  
where:  
\(\
u_{t}=\text{{Turbulent viscosity}}\left[\text{m}^{2}/\text{s}\right]\)  
\(\
u_{t(vis)}=\
u_{t}\) computed by viscous sublayer assumptions \(\left[\text{m}^{2}/\text{s}\right]\)  
\(\
u_{t(log)}=\
u_{t}\) computed by inertial sublayer assumptions \(\left[\text{m}^{2}/\text{s}\right]\)  
\(\
u_{w}=\text{{Kinematic viscosity}}\) of fluid near wall \(\left[\text{m}^{2}/\text{s}\right]\)  
\(y^{+}=\text{{Estimated wall}}-\text{{normal height}}\) of the cell centre in wall units \(\left[-\right]\)  
\(\kappa=\text{{von Karman constant}}\)\(\left[-\right]\)  
\(E=\text{{wall roughness parameter}}\)\(\left[-\right]\)  
\(C_{\mu}=\text{{Emerical model constant}}\)\(\left[-\right]\)  
\(y=\text{{wall}}-\text{{normal height}}\)\(\left[\text{m}\right]\)  
\(k=\text{{turbulent kinetic energy}}\)\(\left[\text{m}^{2}/\text{s}^{2}\right]\)  
\(f_{blend}=\text{{Wall}}-\text{{function blending operator}\)  
The wall constraint on the turbulent kinetic energy was essentially a zero-gradient condition, which took care of the peculiarities of the \(k\) equation. The epsilon wall-boundary condition provided a wall constraint on the dissipation rate and on the turbulent kinetic energy production, \(G\). Unlike the \(k\) wall function, in the epsilon wall function the value of the cell center, rather than the value on the cell face, is calculated, and the process involvesstepwise blending between the viscous and inertial epsilon expressions depending on the y\({}^{+}\) distance:  
\[\varepsilon_{vis}=2\omega k\frac{\
u_{w}}{y^{2}}, \tag{8}\]  
\[\varepsilon_{log}=\omega C_{\mu}\frac{k^{3/2}}{\
u_{tw}y}, \tag{9}\]  
\[G=\omega(\
u_{tw}+\
u_{w})\Big{|}n\cdot(\
abla u)_{f}\Big{|}C_{h}^{\frac{1}{2 }}\frac{k^{\frac{1}{2}}}{xy}, \tag{10}\]  
with \(\
u_{tw}\) being the turbulent viscosity near the wall and \(\omega\) the cell-corner weights. In the calculations, the stepwise blending was used, in which \(\varepsilon=\varepsilon_{vis}\) if \(y^{+}<y_{lam}^{+}\) and \(\varepsilon=\varepsilon_{log}\) when \(y^{+}>y_{lam}^{+}\), where \(y_{lam}^{+}\) is the intersection of the viscous and inertial sublayers in wall units. With this set of wall functions, the finest mesh of this study produced y\({}^{+}\) values in the range 49.5-94.5, with an average of 70. The first cell was thus very well positioned in the log layer. After running the model in the absence of tree belt, it became clear, however, that these wall functions significantly overestimated the velocity profile in the lower heights (\(<\)h) compared to the atmospheric logarithmic profile. The reason behind this is, among other things, that these functions inherently lack the aerodynamic roughness height \(z_{0}\) as a parameter, which is only relevant for atmospheric flows. For this reason, a specialized set of wall functions had to be deployed that handle atmospheric boundary layer conditions. The boundary condition for turbulent viscosity was thus modified based on the work of [86]:  
\[\
u_{t(w)}=\
u_{w}\bigg{(}\frac{y^{+}\kappa}{ln(max(E^{\prime},1+10^{-4}))}-1 \bigg{)}, \tag{11}\]  
with  
\[u^{*}=C_{\mu}^{1/4}\sqrt{k}, \tag{12}\]  
\[y^{+}=\frac{u^{*}y}{\
u_{w}}, \tag{13}\]  
\[E^{\prime}=\frac{y+z_{0}}{z_{0}}. \tag{14}\]  
In a similar way to the wall functions before, a wall constraint on the turbulent kinetic energy dissipation rate and on the turbulent kinetic energy production contribution was applied:  
\[\varepsilon=\frac{\omega C_{\mu}^{3/4}k^{3/2}}{\kappa(y+z_{0})}. \tag{15}\]  
with the new atmospheric boundary layer wall functions, the velocity profiles improved significantly, but the dimensionless wall distance y\({}^{+}\) deteriorated to a value of 177 on average.