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Wilcox's k-omega model

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:<math>
:<math>
f_\beta  = {{1 + 70\chi _\omega  } \over {1 + 80\chi _\omega  }}
f_\beta  = {{1 + 70\chi _\omega  } \over {1 + 80\chi _\omega  }}
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</math>
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:<math>
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\chi _\omega  = \left| {{{\Omega _{ij} \Omega _{jk} S_{ki} } \over {\left( {\beta _0^* \omega } \right)^3 }}} \right|
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</math>
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:<math>
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\beta _0^*  = {9 \over {100}}
</math>
</math>
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   \right.
   \right.
  </math>
  </math>
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:<math>
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\chi _k  \equiv {1 \over {\omega ^3 }}{{\partial k} \over {\partial x_j }}{{\partial \omega } \over {\partial x_j }}
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</math>
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:<math>
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\varepsilon  = \beta ^* \omega k
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</math>
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:<math>
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l = {{k^{{1 \over 2}} } \over \omega }
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</math>

Revision as of 10:33, 26 September 2005

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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