# Wilcox's k-omega model

(Difference between revisions)
 Revision as of 10:34, 26 September 2005 (view source)Zxaar (Talk | contribs)← Older edit Revision as of 09:53, 6 October 2005 (view source) (→Closure Coefficients and Auxilary Relations)Newer edit → Line 16: Line 16: ==Closure Coefficients and Auxilary Relations== ==Closure Coefficients and Auxilary Relations== :$:[itex] - \alpha = {{13} \over {25}} + \alpha = {{5} \over {9}}$ [/itex] :$:[itex] - \beta = \beta _0 f_\beta + \beta = {{3} \over {40}} -$ + - :$+ - \beta ^* = \beta _0^* f_{\beta ^* } +$ [/itex] :$:[itex] Line 29: Line 26: :[itex] :[itex] \sigma ^* = {1 \over 2} \sigma ^* = {1 \over 2} -$ - :$- \beta _0 = {9 \over {125}} -$ - - :$- f_\beta = {{1 + 70\chi _\omega } \over {1 + 80\chi _\omega }} -$ - - :$- \chi _\omega = \left| {{{\Omega _{ij} \Omega _{jk} S_{ki} } \over {\left( {\beta _0^* \omega } \right)^3 }}} \right|$ [/itex] Line 46: Line 32: [/itex] [/itex] - :$- f_{\beta ^* } = \left\{ - - \begin{matrix} - {1,} & {\chi _k \le 0} \\ - {{{1 + 680\chi _k^2 } \over {1 + 80\chi _k^2 }},} & {\chi _k > 0} \\ - \end{matrix} - - - \right. -$ - - :$- \chi _k \equiv {1 \over {\omega ^3 }}{{\partial k} \over {\partial x_j }}{{\partial \omega } \over {\partial x_j }} -$ :$:[itex] \varepsilon = \beta ^* \omega k \varepsilon = \beta ^* \omega k$ [/itex] - - :$- l = {{k^{{1 \over 2}} } \over \omega } -$ - == 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.}}

## 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..