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SST k-omega model

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Contents

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 )

\phi = \phi_1 F_1 + \phi_2 (1 - F_1)

\alpha_1  = {{5} \over {9}},   \alpha_2  = 0.44

 \beta_1  = {{3} \over {40}},  \beta_2  = 0.0828

\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.
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