Wilcox's k-omega model

==Kinematic Eddy Viscosity ==
:<math>
\nu _T  = {k \over \omega }
</math>

== Turbulence Kinetic Energy ==
:<math>
{{\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]
</math>

== Specific Dissipation Rate==
:<math>
{{\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]
</math>

==Closure Coefficients and Auxilary Relations==
:<math>
\alpha  = {{13} \over {25}}  
</math>
:<math>
 \beta  = \beta _0 f_\beta    
</math>
:<math>
\beta ^*  = \beta _0^* f_{\beta ^* }   
</math>
:<math>
\sigma  = {1 \over 2}  
</math>
:<math>
\sigma ^*  = {1 \over 2}  
</math>
:<math>
\beta _0  = {9 \over {125}} 
</math>

:<math>
f_\beta   = {{1 + 70\chi _\omega  } \over {1 + 80\chi _\omega  }}
</math>

:<math>
f_{\beta ^* }  = \left\{ 

\begin{matrix}
   {1,} & {\chi _k  \le 0}  \\ 
   {{{1 + 680\chi _k^2 } \over {1 + 80\chi _k^2 }},} & {\chi _k  > 0}  \\ 
\end{matrix}


  \right.
 </math>