Menter's Shear Stress Transport

Menter's Shear Stress Transport turbulence model, or SST, is a widely used and robust two-equation eddy-viscosity turbulence model used in Computational Fluid Dynamics. The model combines the k-omega turbulence model and K-epsilon turbulence model such that the k-omega is used in the inner region of the boundary layer and switches to the k-epsilon in the free shear flow.

History

The SST two equation turbulence model was introduced in 1994 by F.R. Menter to deal with the strong freestream sensitivity of the k-omega turbulence model and improve the predictions of adverse pressure gradients. The formulation of the SST model is based on physical experiments and attempts to predict solutions to typical engineering problems. Over the last two decades the model has been altered to more accurately reflect certain flow conditions. The Reynold's Averaged Eddy-viscosity is a pseudo-force and not physically present in the system. The two variables calculated are usually interpreted so k is the turbulence kinetic energy and omega is the rate of dissipation of the eddies.

SST (Menter’s Shear Stress Transport) turbulence model ^[1]

${\displaystyle {\frac {\partial (\rho k)}{\partial t}}+{\frac {\partial (\rho u_{j}k)}{\partial x_{j}}}=P-\beta ^{*}\rho \omega k+{\frac {\partial }{\partial x_{j}}}\left[\left(\mu +\sigma _{k}\mu _{t}\right){\frac {\partial k}{\partial x_{j}}}\right]}$ 

${\displaystyle {\frac {\partial (\rho \omega )}{\partial t}}+{\frac {\partial (\rho u_{j}\omega )}{\partial x_{j}}}={\frac {\gamma }{\nu _{t}}}P-\beta \rho \omega ^{2}+{\frac {\partial }{\partial x_{j}}}\left[\left(\mu +\sigma _{\omega }\mu _{t}\right){\frac {\partial \omega }{\partial x_{j}}}\right]+2(1-F_{1}){\frac {\rho \sigma _{\omega 2}}{\omega }}{\frac {\partial k}{\partial x_{j}}}{\frac {\partial \omega }{\partial x_{j}}}}$ 

Variable Definition

${\displaystyle P=\tau _{ij}{\frac {\partial u_{i}}{\partial x_{j}}}}$ 

${\displaystyle \tau _{ij}=\mu _{t}\left(2S_{ij}-{\frac {2}{3}}{\frac {\partial u_{k}}{\partial x_{k}}}\delta _{ij}\right)-{\frac {2}{3}}\rho k\delta _{ij}}$ 

${\displaystyle S_{ij}={\frac {1}{2}}\left({\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}\right)}$ 

${\displaystyle \mu _{t}={\frac {\rho a_{1}k}{{\rm {max}}(a_{1}\omega ,\Omega F_{2})}}}$ 

${\displaystyle F_{1}={\rm {tanh}}\left({\rm {arg}}_{1}^{4}\right)}$ 

${\displaystyle {\rm {arg}}_{1}={\rm {min}}\left[{\rm {max}}\left({\frac {\sqrt {k}}{\beta ^{*}\omega d}},{\frac {500\nu }{d^{2}\omega }}\right),{\frac {4\rho \sigma _{\omega 2}k}{{\rm {CD}}_{k\omega }d^{2}}}\right]}$ 

${\displaystyle {\rm {CD}}_{k\omega }={\rm {max}}\left(2\rho \sigma _{\omega 2}{\frac {1}{\omega }}{\frac {\partial k}{\partial x_{j}}}{\frac {\partial \omega }{\partial x_{j}}},10^{-20}\right)}$ 

${\displaystyle F_{2}={\rm {tanh}}\left({\rm {arg}}_{2}^{2}\right)}$ 

${\displaystyle {\rm {arg}}_{2}={\rm {max}}\left(2{\frac {\sqrt {k}}{\beta ^{*}\omega d}},{\frac {500\nu }{d^{2}\omega }}\right)}$ 

The constants β, σ_k, σ_ω are computed by a blend from the corresponding constants via the following formula

${\displaystyle \phi =F_{1}\phi _{1}+(1-F_{1})\phi _{2}}$ 

Constants

K-W Closure

${\displaystyle \sigma _{k1}=0.85}$  , ${\displaystyle \sigma _{w1}=0.65}$  , ${\displaystyle \beta _{1}=0.075}$ 

K-e Closure

${\displaystyle \sigma _{k2}=1.00}$  , ${\displaystyle \sigma _{w2}=0.856}$  , ${\displaystyle \beta _{2}=0.0828}$ 

SST Closure Constants

${\displaystyle \beta ^{*}=0.09}$  , ${\displaystyle a_{1}=0.31}$ 

Boundary and Far Field Conditions

Far Field

${\displaystyle {\frac {U_{\infty }}{L}}<w_{\rm {farfield}}<10{\frac {U_{\infty }}{L}}}$ 

${\displaystyle {\frac {10^{-5}U_{\infty }^{2}}{Re_{L}}}<k_{\rm {farfield}}<{\frac {0.1U_{\infty }^{2}}{Re_{L}}}}$ 

Boundary/Wall Conditions

${\displaystyle \omega _{wall}=10{\frac {6\nu }{\beta _{1}(\Delta d_{1})^{2}}}}$ 

${\displaystyle k_{wall}=0}$ 

Most software implementations like OpenFOAM and ANSYS Fluent do not include the factor of 10 for omega at the wall, following a Wilcox formulation. However in ^[2] F.R. Menter states: "present author found it much easier and as accurate to implement the following boundary condition"

Validation with experimental results

A good agreement between mass-transfer simulations with experimental data were attained for turbulent flow using the SST two equation turbulence model developed by F.R. Menter for rectangular and tubular shapes,^[3] a modified hydrocyclone^[4] and for curved rotating systems^[5] taking into account a curvature correction term.

References

1. Menter, F. R. (August 1994). "Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications". AIAA Journal. 32 (8): 1598–1605. Bibcode:1994AIAAJ..32.1598M. doi:10.2514/3.12149. S2CID 120712103.
2. Menter, F. R. (July 1993). "Zonal Two Equation k/omega, Turbulence Models for Aerodynamic Flows". AIAA Journal: 1993–2906. doi:10.2514/6.1993-2906. S2CID 130535195.
3. Colli, A. N.; Bisang, J. M. (January 2018). "A CFD Study with Analytical and Experimental Validation of Laminar and Turbulent Mass-Transfer in Electrochemical Reactors". Journal of the Electrochemical Society. 165 (2): E81–E88. doi:10.1149/2.0971802jes.
4. Colli, A. N.; Bisang, J. M. (January 2020). "Coupling k Convection-Diffusion and Laplace Equations in an Open-Source CFD Model for Tertiary Current Distribution Calculations". Journal of the Electrochemical Society. 167: 013513. doi:10.1149/2.0132001JES. hdl:11336/150891. S2CID 208732876.
5. Colli, A. N.; Bisang, J. M. (July 2019). "Time-dependent mass-transfer behaviour under laminar and turbulent flow conditions in rotating electrodes: A CFD study with analytical and experimental validation". International Journal of Heat and Mass Transfer. 137: 835–846. doi:10.1016/j.ijheatmasstransfer.2019.03.152. S2CID 132955462.

Notes

• 'CFD Online Wilcox k-omega turbulence model description'. Accessed May 12, 2014. http://www.cfd-online.com/Wiki/Wilcox%27s_k-omega_model
• 'An Introduction to Computational Fluid Dynamics: The Finite Volume Method (2nd Edition)', H. Versteeg, W. Malalasekera; Pearson Education Limited; 2007; ISBN 0131274988
• 'Turbulence Modeling for CFD' 2nd Ed., Wilcox C. D. ; DCW Industries ; 1998 ; ISBN 0963605100
• 'An introduction to turbulence and its measurement', Bradshaw, P. ; Pergamon Press ; 1971 ; ISBN 0080166210