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Wilcox's k-omega model

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(→Closure Coefficients and Auxilary Relations)
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==Closure Coefficients and Auxilary Relations==
==Closure Coefficients and Auxilary Relations==
:<math>
:<math>
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\alpha  = {{13} \over {25}}   
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\alpha  = {{5} \over {9}}   
</math>
</math>
:<math>
:<math>
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  \beta  = \beta _0 f_\beta   
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  \beta  = {{3} \over {40}}  
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</math>
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:<math>
+
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\beta ^*  = \beta _0^* f_{\beta ^* }  
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</math>
</math>
:<math>
:<math>
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:<math>
:<math>
\sigma ^*  = {1 \over 2}   
\sigma ^*  = {1 \over 2}   
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</math>
 
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:<math>
 
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\beta _0  = {9 \over {125}}
 
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</math>
 
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:<math>
 
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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|
 
</math>
</math>
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</math>
</math>
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:<math>
 
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f_{\beta ^* }  = \left\{
 
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\begin{matrix}
 
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  {1,} & {\chi _k  \le 0}  \\
 
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  {{{1 + 680\chi _k^2 } \over {1 + 80\chi _k^2 }},} & {\chi _k  > 0}  \\
 
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\end{matrix}
 
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  \right.
 
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</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>
 
:<math>
:<math>
\varepsilon  = \beta ^* \omega k
\varepsilon  = \beta ^* \omega k
</math>
</math>
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:<math>
 
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l = {{k^{{1 \over 2}} } \over \omega }
 
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</math>
 
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== References ==
== References ==
#{{reference-book|author=Wilcox, D.C. |year=2004|title=Turbulence Modeling for CFD|rest=ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc.}}
#{{reference-book|author=Wilcox, D.C. |year=2004|title=Turbulence Modeling for CFD|rest=ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc.}}

Revision as of 09:53, 6 October 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 = {{5} \over {9}}
\beta = {{3} \over {40}}
\sigma = {1 \over 2}
\sigma ^* = {1 \over 2}
\beta _0^* = {9 \over {100}}


\varepsilon = \beta ^* \omega k

References

  1. Wilcox, D.C. (2004), Turbulence Modeling for CFD, ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc..
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