Menter's Shear Stress Transport

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.

Source:

$\backslash frac\{\backslash partial\; (\backslash rho\; k)\}\{\backslash partial\; t\}\; +\; \backslash frac\{\backslash partial\; (\backslash rho\; u\_j\; k)\}\{\backslash partial\; x\_j\}\; =\; P\; -\; \backslash beta^*\; \backslash rho\; \backslash omega\; k\; +\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\_j\}\; \backslash left[\backslash left(\backslash mu\; +\; \backslash sigma\_k\; \backslash mu\_t\; \backslash right)\backslash frac\{\backslash partial\; k\}\{\backslash partial\; x\_j\}\backslash right]$

\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}

$P\; =\; \backslash tau\_\{ij\}\; \backslash frac\{\backslash partial\; u\_i\}\{\backslash partial\; x\_j\}$

$\backslash tau\_\{ij\}\; =\; \backslash mu\_t\; \backslash left(2S\_\{ij\}\; -\; \backslash frac\{2\}\{3\}\; \backslash frac\{\backslash partial\; u\_k\}\{\backslash partial\; x\_k\}\; \backslash delta\_\{ij\}\; \backslash right)\; -\; \backslash frac\{2\}\{3\}\; \backslash rho\; k\; \backslash delta\_\{ij\}$

$S\_\{ij\}\; =\; \backslash frac\{1\}\{2\}\; \backslash left(\; \backslash frac\{\backslash partial\; u\_i\}\{\backslash partial\; x\_j\}\; +\; \backslash frac\{\backslash partial\; u\_j\}\{\backslash partial\; x\_i\}\; \backslash right)$

$\backslash mu\_t\; =\; \backslash frac\{\backslash rho\; a\_1\; k\}\{\{\backslash rm\; max\}\; (a\_1\; \backslash omega,\; \backslash Omega\; F\_2)\}$

$F\_1\; =\; \{\backslash rm\; tanh\}\; \backslash left(\{\backslash rm\; arg\}\_1^4\; \backslash right)$

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

$\{\backslash rm\; CD\}\_\{k\; \backslash omega\}\; =\; \{\backslash rm\; max\}\; \backslash left(2\; \backslash rho\; \backslash sigma\_\{\backslash omega\; 2\}\; \backslash frac\{1\}\{\backslash omega\}\; \backslash frac\{\backslash partial\; k\}\{\backslash partial\; x\_j\}\; \backslash frac\{\backslash partial\; \backslash omega\}\{\backslash partial\; x\_j\},\; 10^\{-20\}\; \backslash right)$

$F\_2\; =\; \{\backslash rm\; tanh\}\; \backslash left(\{\backslash rm\; arg\}\_2^2\; \backslash right)$

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

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

$\backslash phi\; =\; F\_1\; \backslash phi\_1\; +\; (1-F\_1)\; \backslash phi\_2$

$\backslash sigma\_\{k1\}\; =\; 0.85$ ,

$\backslash sigma\_\{w1\}\; =\; 0.65$ ,

$\backslash beta\_\{1\}\; =\; 0.075$

$\backslash sigma\_\{k2\}\; =\; 1.00$ ,

$\backslash sigma\_\{w2\}\; =\; 0.856$ ,

$\backslash beta\_\{2\}\; =\; 0.0828$

$\backslash beta^*\; =\; 0.09$ ,

$a\_1\; =\; 0.31$

$\backslash omega\_\{wall\}\; =\; 10\; \backslash frac\{6\; \backslash nu\}\{\backslash beta\_1\; (\backslash Delta\; d\_1)^2\}$

$k\_\{wall\}\; =\; 0$

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 F.R. Menter states: "present author found it much easier and as accurate to implement the following boundary condition"

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, a modified hydrocyclone and for curved rotating systems taking into account a curvature correction term.

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• 'CFD Online Wilcox k-omega turbulence model description'. Accessed May 12, 2014. http://www.cfd-online.com/Wiki/Wilcox%27s_k-omega_model
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• '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}}

{{DEFAULTSORT:SST (Menter's Shear Stress Transport)}}

Category:Turbulence models