Wilcox k-ω Model

Wilcox proposed a turbulence model similar to the standard k-ε turbulence model but replaced the dissipation rate (ε) equation with the eddy frequency (ω) equation (Wilcox, 2006; Wilcox, 2008). The eddy frequency (ω) is often referred to the specific dissipation rate and is defined as $\omega =\epsilon /k$ . The Wilcox k-ω turbulence model has an advantage over the k-ε turbulence model as the k-ω model does not require any wall functions for the calculation of the velocity distribution near walls. As a result, the k-ω turbulence model has better performance for flows with adverse pressure gradient when compared to the k-ε turbulence models. However, the k-ω model exhibits a strong sensitivity to the freestream boundary condition (Wilcox, 2006) for external flow applications.

Transport Equations

Production Modeling

where $\gamma =\frac{{\beta }_0}{{\beta }^{*}}-\frac{{\sigma }_{\omega }{\kappa }^2}{\sqrt{{\beta }^{*}}}$ , $\beta ={\beta }_0{f}_{\beta }$ , $f_{\beta }=\frac{1+85{\chi }_{\omega }}{1+100{\chi }_{\omega }}$ , ${\chi }_{\omega }=\left|\frac{{\text{Ω}}_{ij}{\text{Ω}}_{jk}{\widehat{S}}_{ki}}{{\left({\beta }^{*}\omega \right)}^3}\right|$ , ${\widehat{S}}_{ki}=S_{ki}-\frac{1}{2}\frac{\partial \overline{u_m}}{\partial x_m}{\delta }_{ki}$ , ${\text{S}}_{ij}=\frac{1}{2}\left(\frac{\partial \overline{u_i}}{\partial x_j}+\frac{\partial \overline{u_j}}{\partial x_i}\right)$ , ${\text{Ω}}_{ij}=\frac{1}{2}\left(\frac{\partial \overline{u_i}}{\partial x_j}-\frac{\partial \overline{u_j}}{\partial x_i}\right)$

Dissipation Modeling

Modeling of Turbulent Viscosity ${\mu }_t$

where $\acute{\omega }=max\left[\omega ,C_{lim}\sqrt{\frac{2{\overline{S}}_{ij}{\overline{S}}_{ij}}{{\beta }^{*}}}\right]$ , ${\overline{S}}_{ij}=S_{ij}-\frac{1}{3}\frac{\partial \overline{u_k}}{\partial x_k}{\delta }_{ij}$ , $C_{lim}=\frac{7}{8}$ ,

Model Coefficients

${\sigma }_k$ = 0.6, ${\sigma }_{\omega }$ = 0.5, ${\beta }^{*}$ = 0.09, ${\beta }_0$ = 0.0708, $\kappa$ = 0.4, ${\sigma }_d=\{\begin{array}{c}0.0 for \frac{\partial k}{\partial x_j}\frac{\partial \omega }{\partial x_j}\le 0\\ \frac{1}{8} for \frac{\partial k}{\partial x_j}\frac{\partial \omega }{\partial x_j}>0\end{array}$ .