# 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

Turbulent Kinetic Energy k (1)
Eddy Frequency (Specific Dissipation Rate) ω (2)

## Production Modeling

Turbulent Kinetic Energy k (3)
${P}_{k}={\mu }_{t}{S}^{2}$
Eddy Frequency ω (4)
${P}_{\omega }=\frac{\gamma \omega }{k}{\mu }_{t}{S}^{2}$

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

## Dissipation Modeling

Turbulent Kinetic Energy (k) (5)
${D}_{k}=-\rho {\beta }^{*}k\omega$
Eddy Frequency (ω) (6)
${D}_{\omega }=-\rho \beta {\omega }^{2}$

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

(7)
${\mu }_{t}=\frac{k}{\stackrel{´}{\omega }}$

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

## Model Coefficients

${\sigma }_{k}$ = 0.6, ${\sigma }_{\omega }$ = 0.5, ${\beta }^{*}$ = 0.09, ${\beta }_{0}$ = 0.0708, $\kappa$ = 0.4, .