# SST k-omega model

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+ | {{Turbulence modeling}} | ||

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+ | The SST k-ω turbulence model [Menter 1993] is a [[Two equation turbulence models|two-equation]] [[Eddy viscosity|eddy-viscosity]] model which has become very popular. The shear stress transport (SST) formulation combines the best of two worlds. The use of a k-ω formulation in the inner parts of the boundary layer makes the model directly usable all the way down to the wall through the viscous sub-layer, hence the SST k-ω model can be used as a [[Low-Re turbulence model]] without any extra damping functions. The SST formulation also switches to a k-ε behaviour in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the [[Turbulence free-stream boundary conditions|inlet free-stream turbulence properties]]. Authors who use the SST k-ω model often merit it for its good behaviour in adverse pressure gradients and separating flow. The SST k-ω model does produce a bit too large turbulence levels in regions with large normal strain, like stagnation regions and regions with strong acceleration. This tendency is much less pronounced than with a normal k-ε model though. | ||

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==Kinematic Eddy Viscosity == | ==Kinematic Eddy Viscosity == | ||

:<math> | :<math> | ||

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

</math> | </math> | ||

== Turbulence Kinetic Energy == | == Turbulence Kinetic Energy == | ||

:<math> | :<math> | ||

- | {{\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 + \ | + | {{\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_k \nu _T } \right){{\partial k} \over {\partial x_j }}} \right] |

</math> | </math> | ||

== Specific Dissipation Rate== | == Specific Dissipation Rate== | ||

:<math> | :<math> | ||

- | {{\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 | + | {{\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} \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}} |

</math> | </math> | ||

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:<math> | :<math> | ||

- | P_k=\mbox{min} \left(\tau _{ij} {{\partial U_i } \over {\partial x_j }} , | + | P_k=\mbox{min} \left(\tau _{ij} {{\partial U_i } \over {\partial x_j }} , 10\beta^* k \omega \right) |

</math> | </math> | ||

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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 ) | 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> | </math> | ||

- | |||

:<math> | :<math> | ||

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

</math> | </math> | ||

:<math> | :<math> | ||

- | + | \alpha_1 = {{5} \over {9}}, \alpha_2 = 0.44 | |

</math> | </math> | ||

:<math> | :<math> | ||

- | \ | + | \beta_1 = {{3} \over {40}}, \beta_2 = 0.0828 |

</math> | </math> | ||

:<math> | :<math> | ||

- | \ | + | \beta^* = {9 \over {100}} |

</math> | </math> | ||

:<math> | :<math> | ||

- | \sigma_{k2} = 1 | + | \sigma_{k1} = 0.85, \sigma_{k2} = 1 |

</math> | </math> | ||

:<math> | :<math> | ||

- | \sigma_{\omega 1} = 0.5 | + | \sigma_{\omega 1} = 0.5, \sigma_{\omega 2} = 0.856 |

</math> | </math> | ||

- | + | == References == | |

- | + | ||

- | + | ||

- | == | + | #{{reference-paper|author=Menter, F. R.|year=1993|title=Zonal Two Equation k-ω Turbulence Models for Aerodynamic Flows|rest=AIAA Paper 93-2906}} |

+ | #{{reference-paper|author=Menter, F. R. |year=1994|title=Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications|rest=AIAA Journal, vol. 32, no 8. pp. 1598-1605}} | ||

- | + | [[Category:Turbulence models]] |

## Latest revision as of 21:36, 28 February 2011

The SST k-ω turbulence model [Menter 1993] is a two-equation eddy-viscosity model which has become very popular. The shear stress transport (SST) formulation combines the best of two worlds. The use of a k-ω formulation in the inner parts of the boundary layer makes the model directly usable all the way down to the wall through the viscous sub-layer, hence the SST k-ω model can be used as a Low-Re turbulence model without any extra damping functions. The SST formulation also switches to a k-ε behaviour in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the inlet free-stream turbulence properties. Authors who use the SST k-ω model often merit it for its good behaviour in adverse pressure gradients and separating flow. The SST k-ω model does produce a bit too large turbulence levels in regions with large normal strain, like stagnation regions and regions with strong acceleration. This tendency is much less pronounced than with a normal k-ε model though.

## Contents |

## Kinematic Eddy Viscosity

## Turbulence Kinetic Energy

## Specific Dissipation Rate

## Closure Coefficients and Auxilary Relations

## References

**Menter, F. R. (1993)**, "Zonal Two Equation k-ω Turbulence Models for Aerodynamic Flows", AIAA Paper 93-2906.**Menter, F. R. (1994)**, "Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications", AIAA Journal, vol. 32, no 8. pp. 1598-1605.