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Wilcox's k-omega model

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Contents

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


\alpha  = {{13} \over {25}}

 \beta  = \beta _0 f_\beta

\beta ^*  = \beta _0^* f_{\beta ^* }

\sigma  = {1 \over 2}

\sigma ^*  = {1 \over 2}

\beta _0  = {9 \over {125}}

f_\beta   = {{1 + 70\chi _\omega  } \over {1 + 80\chi _\omega  }}

\chi _\omega   = \left| {{{\Omega _{ij} \Omega _{jk} S_{ki} } \over {\left( {\beta _0^* \omega } \right)^3 }}} \right|

\beta _0^*  = {9 \over {100}}

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.

\chi _k  \equiv {1 \over {\omega ^3 }}{{\partial k} \over {\partial x_j }}{{\partial \omega } \over {\partial x_j }}

\varepsilon  = \beta ^* \omega k

l = {{k^{{1 \over 2}} } \over \omega }
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