# Wilcox's k-omega model

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 Revision as of 10:29, 26 September 2005 (view source)Zxaar (Talk | contribs)← Older edit Revision as of 10:33, 26 September 2005 (view source)Zxaar (Talk | contribs) Newer edit → Line 36: Line 36: :$:[itex] f_\beta = {{1 + 70\chi _\omega } \over {1 + 80\chi _\omega }} f_\beta = {{1 + 70\chi _\omega } \over {1 + 80\chi _\omega }} +$ + + :$+ \chi _\omega = \left| {{{\Omega _{ij} \Omega _{jk} S_{ki} } \over {\left( {\beta _0^* \omega } \right)^3 }}} \right| +$ + + :$+ \beta _0^* = {9 \over {100}}$ [/itex] Line 49: Line 57: \right. \right. [/itex] [/itex] + + :$+ \chi _k \equiv {1 \over {\omega ^3 }}{{\partial k} \over {\partial x_j }}{{\partial \omega } \over {\partial x_j }} +$ + + :$+ \varepsilon = \beta ^* \omega k +$ + + :$+ l = {{k^{{1 \over 2}} } \over \omega } +$

## 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

$\alpha = {{13} \over {25}}$
$\beta = \beta _0 f_\beta$
$\beta ^* = \beta _0^* f_{\beta ^* }$
$\sigma = {1 \over 2}$
$\sigma ^* = {1 \over 2}$
$\beta _0 = {9 \over {125}}$
$f_\beta = {{1 + 70\chi _\omega } \over {1 + 80\chi _\omega }}$
$\chi _\omega = \left| {{{\Omega _{ij} \Omega _{jk} S_{ki} } \over {\left( {\beta _0^* \omega } \right)^3 }}} \right|$
$\beta _0^* = {9 \over {100}}$
$f_{\beta ^* } = \left\{ \begin{matrix} {1,} & {\chi _k \le 0} \\ {{{1 + 680\chi _k^2 } \over {1 + 80\chi _k^2 }},} & {\chi _k > 0} \\ \end{matrix} \right.$
$\chi _k \equiv {1 \over {\omega ^3 }}{{\partial k} \over {\partial x_j }}{{\partial \omega } \over {\partial x_j }}$
$\varepsilon = \beta ^* \omega k$
$l = {{k^{{1 \over 2}} } \over \omega }$