# Effects of Turbulence Modeling on the Simulation of Wind Flow over Typical Complex Terrains
## 2 Numerical Methodologies
On the other hand, in the separation shear boundary layer, the generation of turbulence is greater than the dissipation, and there is a strong nonequilibrium far away from the wall [28]. So, when simulating the separation region on the leeward slope of complex terrain, it is necessary to increase the turbulence generation and decrease the dissipation, and adjusting \(C_{\mu}\) can meet this need. \(C_{\mu}\) is defined as follows:  
\[C_{\mu}=(\frac{u_{*}^{2}}{k})^{2} \tag{3}\]  
where \(u_{*}\) is the friction velocity and \(k\) is the turbulent kinetic energy. (1) Standard SST \(k\)-\(\omega\) model  
The SST \(k\)-\(\omega\) model [29] is described as a shear-stress transport \(k\)-\(\omega\) model, including a \(k\)-\(\omega\) model for the inner layer and a \(k\)-\(\epsilon\) model for the outer layer. Thus, the model combines the advantages of both the \(k\)-\(\omega\) model for the simulations near the boundary layer and the \(k\)-\(\epsilon\) model for the simulations in free flow. Compared with the standard \(k\)-\(\omega\) model, this model modifies the calculation of turbulent viscosity. The model performs better than the other two-equation models in the case of flow separation and can be used for high-precision boundary layer simulations [30]. It is also more accurate than the standard \(k\)-\(\epsilon\) model in simulations of wind flow over complex terrain [9]. In the standard SST \(k\)-\(\omega\) model (Std SST \(k\)-\(\omega\)), the default closure constants are as follows: \(\beta^{*}=0.09\);  
\[\beta_{1}=0.075,\,\sigma_{k1}=1.176,\,\sigma_{\omega 1}=2,\,\gamma_{1}=0.556 \text{ (inner layer)};\]  
\[\beta_{2}=0.0828,\,\sigma_{k2}=1.0,\,\sigma_{\omega 2}=1.168,\,\gamma_{2}=0.44 \text{ (outer layer)}.\]  
(2) Modified SST \(k\)-\(\omega\) model  
The model has also been modified by many scholars in the simulation of the atmospheric boundary layer [31,32]. In this paper, the closure constants of the modified SST \(k\)-\(\omega\) model (Mod SST \(k\)-\(\omega\)) are as follows: \(\beta^{*}=C_{\mu}=0.05\), since \(\beta^{*}/\beta\) is usually 1.2 [20], \(\beta_{1}=\beta^{*}/1.2=0.0412\), and  
\[\gamma_{1}=\beta_{1}/\beta^{*}-\sigma_{\omega 1}\kappa^{2}/\sqrt{\beta^{*}}=0.4757 \tag{4}\]  
in the outer layer  
\[\beta_{2}=(C_{2\epsilon}-1)\cdot C_{\mu}=0.046,\,\,\gamma_{2}=\beta_{2}/\beta ^{*}-\sigma_{\omega 2}\kappa^{2}/\sqrt{\beta^{*}}=0.2764\]  
Finally, the closure constants of the Mod SST \(k\)-\(\omega\) model are summarized as follows:  
\[\beta_{1}=0.0412,\,\sigma_{k1}=1.176,\,\sigma_{\omega 1}=2,\,\gamma_{1}=0.4757 \text{ (inner layer)};\]  
\[\beta_{2}=0.046,\,\sigma_{k2}=1.0,\,\sigma_{\omega 2}=1.168,\,\gamma_{2}=0.2764 \text{ (outer layer)}.\]  
(3) Standard linear pressure-strain RSM model  
Compared to the \(k\)-\(\epsilon\) and SST \(k\)-\(\omega\) models, the RSM model is more complex [33,34], as it requires solving for the Reynolds stresses in each component and avoids the Boussinesq assumption of isotropic turbulence. It is more suitable for complex cases, such as stream-lined curvature, swirl, rotation, and high-strain-rate flows, and has the potential to predict anisotropic turbulence. On the other hand, the RSM model takes an average of 50-60% more time per iteration and requires 15-20% more memory compared to the \(k\)-\(\epsilon\) and \(k\)-\(\omega\) models [34]. In the standard linear pressure-strain RSM model (Std RSM), the default closure constants are \(C_{\mu}=0.09,\,C_{1\epsilon}=1.44,\,C_{2\epsilon}=1.92,\,\sigma_{k}=1.0,\, \sigma_{\epsilon}=1.3,\,C_{1}=1.8,\,C_{2}=0.6,\,C_{1^{\prime}}=0.5,\) and \(C_{2^{\prime}}=0.3\). The first five parameters are the same as in the standard \(k\)-\(\epsilon\) model, the middle two parameters are used to model the pressure-strain term, and the last two parameters are responsible for the wall reflection term in the Reynolds stress transport equation to ensure the redistribution of the normal Reynolds stress next to the wall [34; 35].