SST (Menter’s Shear Stress Transport)

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

$\frac{\partial (\rho k)}{\partial t} + \frac{\partial (\rho u_j k)}{\partial x_j} = P - \beta^* \rho \omega k + \frac{\partial}{\partial x_j} \left[\left(\mu + \sigma_k \mu_t \right)\frac{\partial k}{\partial x_j}\right]$

$\frac{\partial (\rho \omega)}{\partial t} + \frac{\partial (\rho u_j \omega)}{\partial x_j} = \frac{\gamma}{\nu_t} P - \beta \rho \omega^2 + \frac{\partial}{\partial x_j} \left[ \left( \mu + \sigma_{\omega} \mu_t \right) \frac{\partial \omega}{\partial x_j} \right] + 2(1-F_1) \frac{\rho \sigma_{\omega 2}}{\omega} \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}$

Variable Definition

$P = \tau_{ij} \frac{\partial u_i}{\partial x_j}$

$\tau_{ij} = \mu_t \left(2S_{ij} - \frac{2}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij} \right) - \frac{2}{3} \rho k \delta_{ij}$

$S_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)$

$\mu_t = \frac{\rho a_1 k}{{\rm max} (a_1 \omega, \Omega F_2)}$

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

$F_1 = {\rm tanh} \left({\rm arg}_1^4 \right)$

${\rm arg}_1 = {\rm min} \left[ {\rm max} \left( \frac{\sqrt{k}}{\beta^*\omega d}, \frac{500 \nu}{d^2 \omega} \right) , \frac{4 \rho \sigma_{\omega 2} k}{{\rm CD}_{k \omega} d^2} \right]$

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

$F_2 = {\rm tanh} \left({\rm arg}_2^2 \right)$

${\rm arg}_2 = {\rm max} \left( 2 \frac{\sqrt{k}}{\beta^* \omega d}, \frac{500 \nu}{d^2 \omega} \right)$

Constants

K-W Closure

$\sigma_{k1} = 0.85$ , $\sigma_{w1} = 0.65$ , $\beta_{1} = 0.075$

K-e Closure

$\sigma_{k2} = 1.00$ , $\sigma_{w2} = 0.856$ , $\beta_{2} = 0.0828$

SST Closure Constants

$\beta^* = 0.09$ $a_1 = 0.31$

Boundary and Far Field Conditions

Far Field

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

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

Boundary/Wall Conditions

$\omega_{wall} = 10 \frac{6 \nu}{\beta_1 (\Delta d_1)^2}$

$k_{wall} = 0$

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

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