# SST k-omega model

### From CFD-Wiki

(→Closure Coefficients and Auxilary Relations) |
Malkavian GT (Talk | contribs) m |
||

(27 intermediate revisions not shown) | |||

Line 1: | Line 1: | ||

+ | {{Turbulence modeling}} | ||

+ | |||

+ | 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. | ||

+ | |||

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

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

Line 6: | Line 10: | ||

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

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

- | {{\partial k} \over {\partial t}} + U_j {{\partial k} \over {\partial x_j }} = | + | {{\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 | + | {{\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> | ||

Line 17: | Line 21: | ||

:<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] | |

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

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

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

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

:<math> | :<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\} | |

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

:<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 ) |

</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_{k1} = 0.85, \sigma_{k2} = 1 |

+ | </math> | ||

+ | |||

+ | :<math> | ||

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

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

== References == | == References == | ||

- | #{{reference-paper|author= | + | #{{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.