# Wilcox's k-omega model

## Kinematic Eddy Viscosity

$\nu _T = {k \over \omega }$

## Turbulence Kinetic Energy

${{\partial k} \over {\partial t}} + U_j {{\partial k} \over {\partial x_j }} = \tau _{ij} {{\partial U_i } \over {\partial x_j }} - \beta ^* k\omega + {\partial \over {\partial x_j }}\left[ {\left( {\nu + \sigma ^* \nu _T } \right){{\partial k} \over {\partial x_j }}} \right]$

## Specific Dissipation Rate

${{\partial \omega } \over {\partial t}} + U_j {{\partial \omega } \over {\partial x_j }} = \alpha {\omega \over k}\tau _{ij} {{\partial U_i } \over {\partial x_j }} - \beta \omega ^2 + {\partial \over {\partial x_j }}\left[ {\left( {\nu + \sigma \nu _T } \right){{\partial \omega } \over {\partial x_j }}} \right]$

## Closure Coefficients and Auxilary Relations

Failed to parse (unknown function\hspace): \alpha ^* = 1 ; \hspace{2cm} = {{0.025 + 10 R_t / 27} \over {1 + 10 R_t / 27}}

$\alpha = {{5} \over {9}}$
$\beta = {{3} \over {40}}$
$\beta^* = {9 \over {100}}$
$\sigma = {1 \over 2}$
$\sigma ^* = {1 \over 2}$
$\varepsilon = \beta ^* \omega k$

## References

1. Wilcox, D.C. (1988), Re-assessment of the scale-determining equation for advanced turbulence models, AIAA Journal, vol. 31, pp. 1414-1421.