# SST k-omega model

(Difference between revisions)
 Revision as of 07:48, 11 October 2005 (view source) (→Closure Coefficients and Auxilary Relations)← Older edit Revision as of 08:11, 11 October 2005 (view source) (→Closure Coefficients and Auxilary Relations)Newer edit → Line 17: Line 17: :$:[itex] - F_1=\mbox{tanh} \left\{ \left\{ \mbox{min} \left[ \mbox{max} \left( {\sqrt{k} \over \beta ^* \omega y}, {500 \nu \over y^2 \omega} \right) , {4 \sigma_{\omega 2} k \over CD_{k\omega} y^2} \right] \right\} ^4 \right\} + F_2=\mbox{tanh} \left[ \left[ \mbox{max} \left( { 2 \sqrt{k} \over \beta^* \omega y } , { 500 \nu \over y^2 \omega } \right) \right]^2 \right]$ [/itex] :$:[itex] - CD_{k\omega}=\mbox{max} \left( 2\rho\sigma_{\omega 2} {1 \over \omega} {{\partial k} \over {\partial x_i}} {{\partial \omega} \over {\partial x_i}}, 10 ^{-10} \right ) + P_k=\mbox{min} \left(\tau _{ij} {{\partial U_i } \over {\partial x_j }} , 20\beta^* k \omega \right)$ [/itex] :$:[itex] - F_2=\mbox{tanh} \left[ \left[ \mbox{max} \left( { 2 \sqrt{k} \over \beta^* \omega y } , { 500 \nu \over y^2 \omega } \right) \right]^2 \right] + F_1=\mbox{tanh} \left\{ \left\{ \mbox{min} \left[ \mbox{max} \left( {\sqrt{k} \over \beta ^* \omega y}, {500 \nu \over y^2 \omega} \right) , {4 \sigma_{\omega 2} k \over CD_{k\omega} y^2} \right] \right\} ^4 \right\}$ [/itex] - :$:[itex] - P_k=\mbox{min} \left(\tau _{ij} {{\partial U_i } \over {\partial x_j }} , 20\beta^* k \omega \right) + CD_{k\omega}=\mbox{max} \left( 2\rho\sigma_{\omega 2} {1 \over \omega} {{\partial k} \over {\partial x_i}} {{\partial \omega} \over {\partial x_i}}, 10 ^{-10} \right )$ [/itex] Line 47: Line 46: :$:[itex] - \sigma = {1 \over 2} + \sigma_{k1} = 0.85 +$ + + :$+ \sigma_{k2} = 1$ [/itex] :$:[itex] - \sigma ^* = {1 \over 2} + \sigma_{\omega 1} = 0.5$ [/itex] :$:[itex] - \varepsilon = \beta ^* \omega k + \sigma_{\omega 2} = 0.856$ [/itex]

## Kinematic Eddy Viscosity

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

## Turbulence Kinetic Energy

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

## Specific Dissipation Rate

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

## Closure Coefficients and Auxilary Relations

$F_2=\mbox{tanh} \left[ \left[ \mbox{max} \left( { 2 \sqrt{k} \over \beta^* \omega y } , { 500 \nu \over y^2 \omega } \right) \right]^2 \right]$
$P_k=\mbox{min} \left(\tau _{ij} {{\partial U_i } \over {\partial x_j }} , 20\beta^* k \omega \right)$
$F_1=\mbox{tanh} \left\{ \left\{ \mbox{min} \left[ \mbox{max} \left( {\sqrt{k} \over \beta ^* \omega y}, {500 \nu \over y^2 \omega} \right) , {4 \sigma_{\omega 2} k \over CD_{k\omega} y^2} \right] \right\} ^4 \right\}$
$CD_{k\omega}=\mbox{max} \left( 2\rho\sigma_{\omega 2} {1 \over \omega} {{\partial k} \over {\partial x_i}} {{\partial \omega} \over {\partial x_i}}, 10 ^{-10} \right )$

$\alpha = {{5} \over {9}}$
$\beta = {{3} \over {40}}$
$\beta^* = {9 \over {100}}$
$\sigma_{k1} = 0.85$
$\sigma_{k2} = 1$
$\sigma_{\omega 1} = 0.5$
$\sigma_{\omega 2} = 0.856$

## References

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