## 3 Numerical Simulation
### Boundary Conditions and RANS  
The boundary conditions mainly include several parts such as the inlet boundary, the exit boundary, the top boundary, the ground boundary, and the side boundary. The motion of the air was assumed to be an incompressible steady flow in 3D, regardless of the effect of the Coriolis effect force. For simplicity, the effect of temperature on air properties (i.e., air density) was ignored in this CFD simulation study. The dynamic viscosity coefficient of air was taken as 1.7894 \(\times\) 10\({}^{-5}\) N\(\cdot\)S\(\cdot\)m\({}^{-2}\). According to the AIJ code [7] in Japan, if the computational domain is large enough, the side and top boundary conditions have little effect on the results around the target building. The non-viscous wall state (usually the velocity component and the tangential velocity gradient are zero) and the large computational domain would make the simulation process more stable. The outlet condition was adopted as the pressure outlet. The inlet boundary was considered as the velocity boundary condition and was set as the inlet wind velocity profile. The average wind velocity profile was described by a power law. The wind profile was determined by the formula:  
\[U=U_{r}(\frac{z-d}{z_{r}})^{a} \tag{1}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Grid Scheme** & \multicolumn{2}{c}{**Maximum Horizontal Grid Size/m**} & \multicolumn{2}{c}{**Maximum Vertical Grid Size/m**} & **Grid Number** \\ \hline Scheme 1 & 150 & 200 & \(6.3\times 10^{7}\) \\ Scheme 2 & 100 & 100 & \(8.1\times 10^{7}\) \\ Scheme 3 & 50 & 50 & \(1.25\times 10^{8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Grid sensitivity analysis schemes. Figure 6: Computational grids of the CFD model. where \(U_{t}\) is the wind speed at the reference altitude \(z_{r}\); \(a\) denotes the power law index; and \(d\) denotes the zero plane displacement. Parameters of the mean wind speed profile and boundary conditions are summarized in Tables 5 and 6. RANS (Reynolds averaged Navier-Stokes) is widely used in CFD simulation to deal with turbulence because of its high accuracy and low computational cost. The time-average momentum and continuity equations of RANS are shown as:  
\[\frac{\partial\overline{u_{i}}}{\partial t}+\overline{u_{j}}\frac{\partial \overline{u_{i}}}{\partial x_{j}}=-\frac{1}{\rho}\frac{\partial\overline{p}}{ \partial x_{i}}+v\frac{\partial^{2}\overline{u_{i}}}{\partial x_{j}\partial x _{j}}-\frac{\partial\overline{u_{i}^{\prime}u_{j}^{\prime}}}{\partial x_{j}} \tag{2}\]  
\[\frac{\partial\overline{u_{i}}}{\partial x_{i}}=0 \tag{3}\]  
where \(\overline{u_{i}}\) denotes the mean velocity component in Cartesian coordinates (\(x_{j}\), i.e., \(x_{1}\), \(x_{2}\), \(x_{3}\)); \(u_{i}^{\prime}\) denotes the zero-mean turbulent velocity component; \(\overline{p}\) is the mean wind pressure; \(\rho\) is the air density. In the turbulence models of RANS, the \(k\)-\(\varepsilon\) turbulence model is most commonly used in the field of wind engineering. The \(k\)-\(\varepsilon\) turbulence model including the standard \(k\)-\(\varepsilon\) model, the RNG (renormalization group) \(k\)-\(\varepsilon\) model, and the realizable \(k\)-\(\varepsilon\) model. The standard \(k\)-\(\varepsilon\) model [29] was proposed by [PERSON], and it has particular performance in industrial engineering applications. However, the standard \(k\)-\(\varepsilon\) model produces some distortion when simulating flows with large pressure gradients, strong separation flows, strong cyclonic flows, and large curvature flows. The RNG \(k\)-\(\varepsilon\) model [30] and the realizable \(k\)-\(\varepsilon\) model [31] are the modified versions. The RNG \(k\)-\(\varepsilon\) model considers the turbulent flow in the vortex effect, adds an analytical formula for the turbulent Prandtl number, and a differential formula for the effective viscosity, which can better handle flow with a large strain rate and flow curvature. It has good performance in the field of wind engineering for hilly terrain [32].