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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>
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== Turbulence Kinetic Energy ==
== Turbulence Kinetic Energy ==
:<math>
:<math>
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{{\partial k} \over {\partial t}} + U_j {{\partial k} \over {\partial x_j }} = \tau _{ij} {{\partial U_i } \over {\partial x_j }} - \beta ^* k\omega  + {\partial  \over {\partial x_j }}\left[ {\left( {\nu  + \sigma_{k1} \nu _T } \right){{\partial k} \over {\partial x_j }}} \right]
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{{\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>
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{{\partial \omega } \over {\partial t}} + U_j {{\partial \omega } \over {\partial x_j }} = \alpha {\omega  \over k}\tau _{ij} {{\partial U_i } \over {\partial x_j }} - \beta \omega ^2  + {\partial  \over {\partial x_j }}\left[ {\left( {\nu  + \sigma \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}}  
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{{\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>
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\alpha  = {{5} \over {9}}
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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>
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\beta  = {{3} \over {40}}  
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P_k=\mbox{min} \left(\tau _{ij} {{\partial U_i } \over {\partial x_j }} , 10\beta^* k \omega \right)
</math>
</math>
:<math>
:<math>
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\beta^* = {9 \over {100}}
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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>
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\sigma  = {1 \over 2}
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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 )
</math>
</math>
:<math>
:<math>
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\sigma ^*  = {1 \over 2} 
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\phi = \phi_1 F_1 + \phi_2 (1 - F_1)
</math>
</math>
:<math>
:<math>
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\varepsilon = \beta ^* \omega k
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\alpha_1 = {{5} \over {9}},  \alpha_2  = 0.44
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</math>
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:<math>
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\beta_1  = {{3} \over {40}},  \beta_2  = 0.0828
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</math>
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:<math>
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\beta^* = {9 \over {100}}
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</math>
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:<math>
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\sigma_{k1}  = 0.85,  \sigma_{k2}  = 1
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</math>
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\sigma_{\omega 1}  = 0.5,  \sigma_{\omega 2}  = 0.856
</math>
</math>
== References ==
== References ==
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#{{reference-paper|author=Wilcox, D.C. |year=1988|title=Re-assessment of the scale-determining equation for advanced turbulence models|rest=AIAA Journal, vol. 31, pp. 1414-1421}}
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#{{reference-paper|author=Menter, F. R.|year=1993|title=Zonal Two Equation k-ω Turbulence Models for Aerodynamic Flows|rest=AIAA Paper 93-2906}}
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#{{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}}
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[[Category:Turbulence models]]

Latest revision as of 21:36, 28 February 2011

Turbulence modeling
Turbulence
RANS-based turbulence models
  1. Linear eddy viscosity models
    1. Algebraic models
      1. Cebeci-Smith model
      2. Baldwin-Lomax model
      3. Johnson-King model
      4. A roughness-dependent model
    2. One equation models
      1. Prandtl's one-equation model
      2. Baldwin-Barth model
      3. Spalart-Allmaras model
    3. Two equation models
      1. k-epsilon models
        1. Standard k-epsilon model
        2. Realisable k-epsilon model
        3. RNG k-epsilon model
        4. Near-wall treatment
      2. k-omega models
        1. Wilcox's k-omega model
        2. Wilcox's modified k-omega model
        3. SST k-omega model
        4. Near-wall treatment
      3. Realisability issues
        1. Kato-Launder modification
        2. Durbin's realizability constraint
        3. Yap correction
        4. Realisability and Schwarz' inequality
  2. Nonlinear eddy viscosity models
    1. Explicit nonlinear constitutive relation
      1. Cubic k-epsilon
      2. EARSM
    2. v2-f models
      1. \overline{\upsilon^2}-f model
      2. \zeta-f model
  3. Reynolds stress model (RSM)
Large eddy simulation (LES)
  1. Smagorinsky-Lilly model
  2. Dynamic subgrid-scale model
  3. RNG-LES model
  4. Wall-adapting local eddy-viscosity (WALE) model
  5. Kinetic energy subgrid-scale model
  6. Near-wall treatment for LES models
Detached eddy simulation (DES)
Direct numerical simulation (DNS)
Turbulence near-wall modeling
Turbulence free-stream boundary conditions
  1. Turbulence intensity
  2. Turbulence length scale

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

\nu _T = {a_1 k \over \mbox{max}(a_1 \omega, S 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_k \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} \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 }} , 10\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. Menter, F. R. (1993), "Zonal Two Equation k-ω Turbulence Models for Aerodynamic Flows", AIAA Paper 93-2906.
  2. Menter, F. R. (1994), "Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications", AIAA Journal, vol. 32, no 8. pp. 1598-1605.
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