## 3 Numerical Simulation  
### Methodology  
Fluid is considered to be incompressible viscous flow in anti-wind design of structures. Spatial averaging of transient N-S equation can yield the governing equation in large eddy simulation (LES):  
\[\frac{\partial\overline{\mu_{i}}}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial\overline{\mu_{i}}}{\partial t}+\frac{\partial(\overline{\mu_{i} \mu_{j}})}{\partial x_{j}}=-\frac{1}{\rho}\frac{\partial\overline{p}}{ \partial x_{i}}+v\frac{\partial^{2}\overline{\mu_{i}}}{\partial x_{j}\partial x _{j}}-\frac{\partial\tau_{ij}}{\partial x_{j}} \tag{2}\]  
where \(\rho\) is air density, \(t\) is time, \(v\) is kinematic viscosity coefficient of air, \(\overline{\mu_{i}}\) and \(\overline{\mu_{j}}\) are velocities in three directions after filtering, and \(\tau_{ij}\) is non-closed term in the N-S equation after spatial averaging, i.e., subgrid scale stress.  
\[\tau_{ij}=\overline{\mu_{i}\mu_{i}}-\overline{\mu_{i}\mu_{j}} \tag{3}\]  
Boussinesq approximation was introduced according to Smagorinsky subgrid-scale model based on eddy viscosity assumption. Thus, the subgrid-scale stress is written as  
\[\tau_{ij}-\frac{1}{3}\tau_{ij}\delta_{ij}=-2\mu_{t}\overline{S_{ij}}=-\mu_{t}( \frac{\partial\overline{\mu_{i}}}{\partial x_{j}}+\frac{\partial\overline{ \mu_{j}}}{\partial x_{i}}) \tag{4}\]  
\(e\) tensor of solvable scale, \(\tau_{kk}\) is the isotropic component of subgrid-scale stress, which is contained in the pressure item after filtering, \(\delta_{ij}\) is Kronecker delta function, \(\mu_{t}\) is subgrid-scale turbulence eddy viscosity coefficient, generally using the Smagorinsky assumption:  
\[\mu_{t}=(C_{s}\Delta)^{2}\big{|}\overline{S}\big{|} \tag{5}\]  
where \(C_{s}\) is Smagorinsky constant, generally taken as 0.1-0.23 and being 0.1 in this study. Strain rate tensor \(\big{|}\overline{S}\big{|}=\sqrt{2S_{ij}S_{ij}}\). \(\Delta\) is grid scale, \(\Delta=(\Delta_{x}\Delta_{y}\Delta_{z})^{1/3}\), where \(\Delta_{x}\), \(\Delta_{y}\), and \(\Delta_{z}\) are the grid size in \(x\), \(y\), and \(z\) directions, respectively. This is the standard Smagorinsky subgrid-scale model. Some researchers propose dynamic determination of \(C_{s}\) value to better characterize collision, separation, free shear layer, and vortex shedding of the flow field around the blunt body. It is known as the dynamic Smagorinsky model.  
However, both the Smagorinsky model and dynamic Smagorinsky model are algebraic models, which assume a local equilibrium between the generation and dissipation of subgrid-scale turbulent kinetic energy. These models are not fit for simulating structures with a high Reynolds number, such as the cooling towers. To solve this problem, we proposed a new form of subgrid-scale model based on literature [15]:  
\[\frac{\partial k_{ggs}}{\partial t}+\frac{\partial\mu_{j}k_{ggs}}{\partial x _{j}}=-\tau_{ij}S_{ij}-C_{\varepsilon}\frac{{k_{ggs}}^{3/2}}{\Delta}+\frac{ \partial}{\partial x_{j}}\bigg{[}(C_{d}\Delta v\sqrt{k_{ggs}}+v)\frac{\partial k _{ggs}}{\partial x_{j}}\bigg{]}-\varepsilon_{w} \tag{6}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline No. of working condition & Working condition 5 & Working condition 6 & Working condition 7 & Working condition 8 \\ Template number & 90 & 105 & 120 & 139 \\ Height/m & 146.2 & 165.7 & 185.2 & 210.0 \\ Air inlet diameter/m & 111.21 & 110.61 & 112.61 & 115.81 \\ Minimum shell & 0.375 & 0.370 & 0.370 & 0.370 \\ thickness/m & & & & \\ \hline \end{tabular}
\end{table}
Table 2: _Cont._where, \(K_{gs}\) is the kinetic energy of transportation equation SGS, \(C_{s}\) is Smagorinsky constant, and \(v\) is positive. Here, \(C_{s}\) will reduce the amplitude of \(K_{gsg}\), and a Gaussian filter is needed for finite difference.  
This model is better applied to engineering applications. There is no need for experimental filtering, and the computational load is small. So, it can be used to simulate structures with a high Reynolds number, such as cooling towers [10].