# Wilcox's k-omega model

(Difference between revisions)
 Revision as of 10:29, 26 September 2005 (view source)Zxaar (Talk | contribs)← Older edit Latest revision as of 16:52, 8 March 2011 (view source)m (→References) (18 intermediate revisions not shown) Line 1: Line 1: + {{Turbulence modeling}} ==Kinematic Eddy Viscosity == ==Kinematic Eddy Viscosity == :$:[itex] Line 15: Line 16: ==Closure Coefficients and Auxilary Relations== ==Closure Coefficients and Auxilary Relations== + :[itex] :[itex] - \alpha = {{13} \over {25}} + \alpha = {{5} \over {9}}$ [/itex] + :$:[itex] - \beta = \beta _0 f_\beta + \beta = {{3} \over {40}}$ [/itex] + :$:[itex] - \beta ^* = \beta _0^* f_{\beta ^* } + \beta^* = {9 \over {100}}$ [/itex] + :$:[itex] \sigma = {1 \over 2} \sigma = {1 \over 2}$ [/itex] + :$:[itex] \sigma ^* = {1 \over 2} \sigma ^* = {1 \over 2} -$ - :$- \beta _0 = {9 \over {125}}$ [/itex] :$:[itex] - f_\beta = {{1 + 70\chi _\omega } \over {1 + 80\chi _\omega }} + \varepsilon = \beta ^* \omega k$ [/itex] - :$+ == References == - f_{\beta ^* } = \left\{ + - + - \begin{matrix} + - {1,} & {\chi _k \le 0} \\ + - {{{1 + 680\chi _k^2 } \over {1 + 80\chi _k^2 }},} & {\chi _k > 0} \\ + - \end{matrix} + + #{{reference-paper|author=Wilcox, D.C. |year=1988|title=Re-assessment of the scale-determining equation for advanced turbulence models|rest=AIAA Journal, vol. 26, no. 11, pp. 1299-1310}} - \right. + [[Category:Turbulence models]] -$ +

## 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

$\alpha = {{5} \over {9}}$
$\beta = {{3} \over {40}}$
$\beta^* = {9 \over {100}}$
$\sigma = {1 \over 2}$
$\sigma ^* = {1 \over 2}$
$\varepsilon = \beta ^* \omega k$

## References

1. Wilcox, D.C. (1988), "Re-assessment of the scale-determining equation for advanced turbulence models", AIAA Journal, vol. 26, no. 11, pp. 1299-1310.