SST k-omega model

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

From CFD-Wiki

(Difference between revisions)

Jump to: navigation, search

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.

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

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.

@@ 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 ==
 :<math>
@@ Line 6: / Line 10: @@
 == Turbulence Kinetic Energy ==
 :<math>
-{{\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]
+{{\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>
 == Specific Dissipation Rate==
 :<math>
-{{\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_{\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}}
+{{\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>
@@ Line 17: / Line 21: @@
 :<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\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 }} , 10\beta^* k \omega \right)
 </math>
 :<math>
-F_2=\mbox{tanh} \mbox{max} { 2 \sqrt{k} \over \beta^* \omega y} , { 500 \nu \over y^2 \omega }
+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>
-F_2=\mbox{tanh} \right[ \right[ \mbox{max} \left( { 2 \sqrt{k} \over \beta^* \omega y} , { 500 \nu \over y^2 \omega } \right ) \right ] ^2 \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>
-\alpha  = {{5} \over {9}}
+\phi = \phi_1 F_1 + \phi_2 (1 - F_1)
 </math>
 :<math>
- \beta  = {{3} \over {40}}
+\alpha_1  = {{5} \over {9}},   \alpha_2  = 0.44
 </math>
 :<math>
-\beta^*  = {9 \over {100}}
+ \beta_1  = {{3} \over {40}},  \beta_2  = 0.0828
 </math>
 :<math>
-\sigma  = {1 \over 2}
+\beta^*  = {9 \over {100}}
 </math>
 :<math>
-\sigma ^*  = {1 \over 2}
+\sigma_{k1}  = 0.85,  \sigma_{k2}  = 1
 </math>
 :<math>
-\varepsilon  = \beta ^* \omega k
+\sigma_{\omega 1}  = 0.5,  \sigma_{\omega 2}  = 0.856
 </math>
 == References ==
-#{{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}}
+#{{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]]

SST k-omega model

From CFD-Wiki

Latest revision as of 21:36, 28 February 2011

Contents

Kinematic Eddy Viscosity

Turbulence Kinetic Energy

Specific Dissipation Rate

Closure Coefficients and Auxilary Relations

References

Views

My wiki

wiki navigation

wiki search

wiki toolbox