# Development of a CFD-Based Wind Turbine Rotor Optimization Tool in Considering Wake Effects
## 2 Numerical Methods
### Turbulence Model  
Herein the turbulence model is discussed with the aim to model the nonlinear Reynolds stresses terms in Equations (2) and (3). Accurate predictions of turbulent kinetic energy and dissipation are the key factors in turbulent wake modelling. In the present work, the modified \(k\)-\(\omega\) turbulence model of [PERSON] [41] is used. [PERSON]'s baseline model combines the \(k\)-\(\omega\) model by [PERSON] [42] in the inner region of the boundary layer with the standard \(k\)-\(\epsilon\) model in the outer region and the free steam outside the boundary layer. The modified transport equations are written as follows:  
\[\frac{\partial k}{\partial t}+\frac{\partial(u_{j}k)}{\partial x_{j}}=\frac{1 }{\rho}p_{k}+\frac{1}{\rho}\frac{\partial}{\partial x_{j}}[(\mu+\sigma_{k}\mu _{t})\frac{\partial k}{\partial x_{j}}]-\beta^{*}k\omega \tag{4}\]  
\[\frac{\partial\omega}{\partial t}+\frac{\partial(u_{j}\omega)}{\partial x_{j}} =\frac{\gamma}{\rho v_{t}}p_{k}+\frac{1}{\rho}\frac{\partial}{ \partial x_{j}}[(\mu+\sigma_{\omega}\mu_{t})\frac{\partial\omega}{\partial x_ {j}}]+2(1-F_{1})\frac{\rho\sigma_{\omega 2}}{\rho\omega}\frac{\partial k}{ \partial x_{j}}\frac{\partial\omega}{\partial x_{j}}-\beta\omega^{2} \tag{5}\]  
where the original model constants for the inner region are:  
\[\sigma_{k1}=0.5,\ \sigma_{\omega 1}=0.5,\ \beta_{1}=0.075,\ \beta^{*}=0.09,\ \kappa=0.41,\ \gamma_{1}=0.55\]  
and for the outer region  
\[\sigma_{k2}=1.0,\ \sigma_{\omega 2}=0.856,\ \beta_{1}=0.0828,\ \beta^{*}=0.09,\ \kappa=0.41,\ \gamma_{2}=0.44 \tag{6}\]  
The re-formulation of the original turbulence model contains two parts: (1) change of model constants; (2) added the source terms to maintain the turbulence level. (1) The modified model constants are based on the work of [PERSON] et al. [43] in which most of the constants remain the same except for the following ones  
\[\beta_{1}=0.033,\ \beta^{*}=0.025,\ \gamma_{1}=0.37\]The new constants were suggested according to measurements. With the decrease of dissipation effect, the kinetic energy towards the rotor is effectively increased, which improves the numerical accuracy for wind turbine wake flow modeling [43]. (2) For RANS solvers, the turbulence level is usually specified from the inlet of a computational domain. For the \(k\)-\(\omega\) models, it is the initial \(k\) and \(\omega\) values that are responsible for the turbulence level  
\[k=\frac{3}{2}(u_{0}I_{0})^{2} \tag{7}\]  
\[\omega=\frac{k}{\mu}\left(\frac{\mu_{t}}{\mu}\right)^{-1} \tag{8}\]  
where \(I_{0}\) represents the ambient turbulence level and \(\mu_{t}/\mu\) is the eddy viscosity ratio. A similar approach is implemented in some commercial CFD programs such as the ANSYS Fluent software. In another approach proposed by [PERSON] and [PERSON] [44], the boundary values of \(u_{0},k\) and \(\omega\) are basically a function of an aerodynamic roughness height corresponding to a specific site. This approach does not apply to the current work where an axisymmetric flow is assumed, and no wall boundary is specified. No matter what boundary values are given at inlet, the free decay of turbulence intensity cannot be avoided with the current set of RANS equations. In the free-stream flow, the mean velocity gradient is zero such that the turbulence quantities decay. Therefore, the production terms and the diffusion terms in Equations (4) and (5) can be neglected.