SST k-omega model

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(Difference between revisions)

Revision as of 08:11, 11 October 2005

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

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

SST k-omega model

From CFD-Wiki

Revision as of 08:11, 11 October 2005

Contents

Kinematic Eddy Viscosity

Turbulence Kinetic Energy

Specific Dissipation Rate

Closure Coefficients and Auxilary Relations

References

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@@ Line 17: / Line 17: @@
 :<math>
-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]
 </math>
 :<math>
-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)
 </math>
 :<math>
-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\}
 </math>
 :<math>
-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 )
 </math>
@@ Line 47: / Line 46: @@
 :<math>
-\sigma  = {1 \over 2}
+\sigma_{k1}  = 0.85
+</math>
+:<math>
+\sigma_{k2}  = 1
 </math>
 :<math>
-\sigma ^*  = {1 \over 2}
+\sigma_{\omega 1}  = 0.5
 </math>
 :<math>
-\varepsilon  = \beta ^* \omega k
+\sigma_{\omega 2}  = 0.856
 </math>