# Wilcox k-ω Model

Since all three k-ε turbulence models cannot be integrated all the way to walls, wall damping wall functions must be employed to provide correct near wall behavior. It is also known that the standard k-ε turbulence model fails to predict the flow separation under adverse pressure gradients.

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 $\gamma =\frac{{\beta}_{0}}{{\beta}^{*}}-\frac{{\sigma}_{\omega}{\kappa}^{2}}{\sqrt{{\beta}^{*}}}$ , $\beta ={\beta}_{0}{f}_{\beta}$ , ${f}_{\beta}=\frac{1+85{\chi}_{\omega}}{1+100{\chi}_{\omega}}$ , ${\chi}_{\omega}=\left|\frac{{\text{\Omega}}_{ij}{\text{\Omega}}_{jk}{\widehat{S}}_{ki}}{{\left({\beta}^{*}\omega \right)}^{3}}\right|$ , ${\widehat{S}}_{ki}={S}_{ki}-\frac{1}{2}\frac{\partial \overline{{u}_{m}}}{\partial {x}_{m}}{\delta}_{ki}$ , ${\text{S}}_{ij}=\frac{1}{2}\left(\frac{\partial \overline{{u}_{i}}}{\partial {x}_{j}}+\frac{\partial \overline{{u}_{j}}}{\partial {x}_{i}}\right)$ , ${\text{\Omega}}_{ij}=\frac{1}{2}\left(\frac{\partial \overline{{u}_{i}}}{\partial {x}_{j}}-\frac{\partial \overline{{u}_{j}}}{\partial {x}_{i}}\right)$

## Dissipation Modeling

## Modeling of Turbulent Viscosity ${\mu}_{t}$

where $\stackrel{\xb4}{\omega}=max\left[\omega ,{C}_{lim}\sqrt{\frac{2{\overline{S}}_{ij}{\overline{S}}_{ij}}{{\beta}^{*}}}\right]$ , ${\overline{S}}_{ij}={S}_{ij}-\frac{1}{3}\frac{\partial \overline{{u}_{k}}}{\partial {x}_{k}}{\delta}_{ij}$ , ${C}_{lim}=\frac{7}{8}$ ,

## Model Coefficients

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