# Effects of Turbulence Modeling on the Simulation of Wind Flow over Typical Complex Terrains
## 2 Numerical Methodologies  
The RANS equations express the conservation of mass and the momentum of a Newtonian fluid, assuming an incompressible, steady-state turbulent flow with a constant viscosity (\(\mu\)), and can be expressed as follows [16]:  
\[\frac{\partial U_{i}}{\partial x_{i}}=0 \tag{1}\]  
\[\rho U_{j}\frac{\partial U_{i}}{\partial x_{i}}=-\frac{\partial p}{\partial x _{i}}+\mu\frac{\partial}{\partial x_{j}}(\frac{\partial U_{i}}{\partial x_{j} }+\frac{\partial U_{j}}{\partial x_{j}})+\frac{\partial}{\partial x_{j}}(- \rho\overline{u^{\prime}_{i}u^{\prime}}_{j})+S_{u,i} \tag{2}\]  
where \(p\) is the mean pressure, \(\mu\) is the viscosity, \(S_{u,i}\) is the momentum source (or sink) in the \(i\) direction, which can represent the volume force if there is a wind turbine [17], and \(-\rho\overline{u^{\prime}_{i}u^{\prime}}_{j}\) is known as the Reynolds stress tensor \(\tau_{ij}\)[18], which is expressed as the covariance of the velocity fluctuations, accounting for the transfer of momentum between different eddies or scales within the turbulent flow. The Reynolds stress tensor includes both the normal stresses, which are the variances in the velocity components, and the shear stresses, which are the covariances between different velocity components. The problem of turbulent closure is due to the need to model the time-averaged fluctuating velocity component of the momentum Equation (2), i.e., the Reynolds stress. To solve for the Reynolds stress, the most common method is based on the Boussinesq approximation, which express the Reynolds stress as an expression for the eddy viscosity [19], and this approach assumes that \(\mu_{t}\) is an isotropic scalar. Various turbulence models apply some extra transport equations to account for the turbulence using empirical constants and information about the mean flow, and the most common models are the one-equation and two-equation turbulence models. In this paper, the standard SST \(k\)-\(\omega\) model, the standard linear Reynolds stress model (RSM), and innovative variants of these two models (i.e., the modified SST \(k\)-\(\omega\) model and the modified RSM model) with modified closure constants are used to simulate the wind flow over the complex terrain. The RSM models are not based on the Boussinesq approximation, where the Reynolds stress in each direction is solved by an equation, along with an \(\varepsilon\) equation, for a total of seven additional equations required for this model. The following is a brief description of several RANS turbulence models and the modified methods used in the paper. In the RANS turbulence models, \(C_{\mu}\) is usually derived from Equation (3) [20], which assumes an equilibrium of turbulence generation and dissipation, such as in a zero-gradient pressure turbulent boundary layer. In the inner layer of the turbulence boundary layer, the value of \(k/u_{*}^{2}\) is 5.48, resulting in \(C_{\mu}\) = 0.9. In the standard turbulence model, the default value of the model closure constant \(C_{\mu}\) is 0.09 [21]. However, in the real atmosphere boundary layer, the value of \(k/u_{*}^{2}\) varies from 3.35 to 8.75 in different experiments [22; 23; 24; 25]. On the one hand, in order to reduce the generation of the turbulent kinetic energy (TKE) at the lower layer of the ABL, and to be consistent with the TKE at the inlet conditions, \(C_{\mu}\) is modified from 0.09 to 0.05 in this paper, which is different from other scholars' studies, where \(C_{\mu}\) is usually modified to 0.033 [20; 26]. According to reference [26] and numerical simulation investigations, it was found that the production of turbulence kinetic energy is higher than that given in the inlet when \(C_{\mu}\) is modified to 0.033 (especially near the ground), which leads to inconsistency with the inlet condition; thus, it is necessary to adjust this value to being a little larger. In this paper, \(k/u_{*}^{2}\) is selected to be 4.47, resulting in \(C_{\mu}\) = 0.05, which is also in agreement with the findings in the literature [27].