# SST k-omega model

(Difference between revisions)
Jump to: navigation, search
 Revision as of 07:30, 11 October 2005 (view source) (→Closure Coefficients and Auxilary Relations)← Older edit Revision as of 07:32, 11 October 2005 (view source) (→Closure Coefficients and Auxilary Relations)Newer edit → Line 22: Line 22: :$:[itex] CD_{k\omega}=\mbox{max} \left( 2\sigma_{\omega 2} {1 \over \omega} {{\partial k} \over {\partial x_i}} {{\partial \omega} \over {\partial x_i}}, 10 ^{-10} \right ) CD_{k\omega}=\mbox{max} \left( 2\sigma_{\omega 2} {1 \over \omega} {{\partial k} \over {\partial x_i}} {{\partial \omega} \over {\partial x_i}}, 10 ^{-10} \right ) +$ + + :$+ F_2=\mbox{tanh} \right[ \right[ \mbox{max} \left( { 2 \sqrt{k} \over \beta ^* \omega y} , { 500 \nu \over y^2 \omega } \right ) \right ] ^2 \right ]$ [/itex]

## Kinematic Eddy Viscosity

$\nu _T = {a_1 k \over \mbox{max}(a_1 \omega, S F_2) }$

## Turbulence Kinetic Energy

${{\partial k} \over {\partial t}} + U_j {{\partial k} \over {\partial x_j }} = \tau _{ij} {{\partial U_i } \over {\partial x_j }} - \beta ^* k\omega + {\partial \over {\partial x_j }}\left[ {\left( {\nu + \sigma_{k1} \nu _T } \right){{\partial k} \over {\partial x_j }}} \right]$

## Specific Dissipation Rate

${{\partial \omega } \over {\partial t}} + U_j {{\partial \omega } \over {\partial x_j }} = \alpha {\omega \over k}\tau _{ij} {{\partial U_i } \over {\partial x_j }} - \beta \omega ^2 + {\partial \over {\partial x_j }}\left[ {\left( {\nu + \sigma_{\omega 1} \nu _T } \right){{\partial \omega } \over {\partial x_j }}} \right] + 2( 1 - F_1 ) \sigma_{\omega 2} {1 \over \omega} {{\partial k } \over {\partial x_i}} {{\partial \omega } \over {\partial x_i}}$

## Closure Coefficients and Auxilary Relations

$F_1=\mbox{tanh} \left\{ \left\{ \mbox{min} \left[ \mbox{max} \left( {\sqrt{k} \over \beta ^* \omega y}, {500 \nu \over y^2 \omega} \right) , {4 \sigma_{\omega 2} k \over CD_{k\omega} y^2} \right] \right\} ^4 \right\}$
$CD_{k\omega}=\mbox{max} \left( 2\sigma_{\omega 2} {1 \over \omega} {{\partial k} \over {\partial x_i}} {{\partial \omega} \over {\partial x_i}}, 10 ^{-10} \right )$
Failed to parse (syntax error): F_2=\mbox{tanh} \right[ \right[ \mbox{max} \left( { 2 \sqrt{k} \over \beta ^* \omega y} , { 500 \nu \over y^2 \omega } \right ) \right ] ^2 \right ]

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

## References

1. Wilcox, D.C. (1988), "Re-assessment of the scale-determining equation for advanced turbulence models", AIAA Journal, vol. 31, pp. 1414-1421.