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The Menter Shear Stress Transport Turbulence Model

This web page gives detailed information on the equations for various forms of the Menter shear stress transport (SST) turbulence model. If any particular variant has been overlooked, please report it to the page curator.

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"Standard" Menter SST Two-Equation Model (SST)

The reference for the standard implementation of the Menter SST model is:

The two-equation model (written in conservation form) is given by the following:

\frac{\partial (\rho k)}{\partial t} + \frac{\partial (\rho u_j k)}{\partial x_j} = \cal 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} \cal 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}

Note that in the reference, the Lagrangian derivative was used, which is not identical with the proper form of these equations as written by the author and others elsewhere. The equations have been written above to be in proper conservation form, consistent with, e.g., Wilcox (in Turbulence Modeling for CFD, DCW Industries, Inc., La Canada, CA, 2006), Menter et al (in Turbulence, Heat and Mass Transfer 4, 2003, pp. 625-632), and Menter (in NASA TM 103975, 1992).

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)

and the turbulent eddy viscosity is computed from:

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

Each of the constants is a blend of an inner (1) and outer (2) constant, blended via:

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

where \phi_1 represents constant 1 and \phi_2 represents constant 2. Additional functions are given by:

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)

and \rho is the density, \nu_t = \mu_t/\rho is the turbulent kinematic viscosity, \mu is the molecular dynamic viscosity, d is the distance from the field point to the nearest wall, and \Omega is the vorticity magnitude.

Note that it is generally recommended to use a production limiter (see Menter, F. R., "Zonal Two Equation k-omega Turbulence Models for Aerodynamic Flows," AIAA Paper 93-2906, July 1993). In this reference, the term P in the k-equation is replaced by:

{\rm min}(\cal P, 20 \beta^* \rho \omega k)

The boundary conditions recommended in the original reference are:

\frac{U_{\infty}{L} < \omega_{\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}
\omega_{wall} = 10 \frac{6 \nu}{\beta_1 (\Delta d_1)^2}
k_{wall} = 0

where "L is the approximate length of the computational domain," and the combination of the two farfield values should yield a freestream turbulent viscosity between 10-5 and 10-2 times freestream laminar viscosity. Thus, the farfield turbulence boundary conditions are somewhat open to interpretation. Note that the turbulence variables decay (sometimes dramatically) from their set values in the farfield for external aerodynamic problems. See the version (SST_sust) below for an alternative formulation that eliminates this decay, and provides more precise definitions for the boundary conditions.

The constants are:

\gamma_1 = \frac{\beta_1}{\beta^*} - \frac{\sigma_{\omega 1} \kappa^2}{\sqrt{\beta^*}}           \gamma_2 = \frac{\beta_2}{\beta^*} - \frac{\sigma_{\omega 2} \kappa^2}{\sqrt{\beta^*}}
\sigma_{k 1} = 0.85           \sigma_{\omega 1} = 0.5           \beta_1 = 0.075
\sigma_{k 2} = 1.0           \sigma_{\omega 2} = 0.856           \beta_2 = 0.0828
\beta^*=0.09           \kappa=0.41           a_1 = 0.31

 

Menter SST Two-Equation Model with Vorticity Source Term (SST-V)

This form of the SST model is sometimes used because vorticity magnitude \Omega is usually readily available in most Navier-Stokes codes. Furthermore, the vorticity source term is often nearly identical to the exact source term in simple boundary layer flows, and the use of the vorticity term can avoid some numerical difficulties sometimes associated with the use of the exact source term. The reference for this usage is:

The equations are the same as for the "standard" version (SST), with the exception that the term P (in both equations) is approximated with the following:

P = \mu_t \Omega^2 - \frac{2}{3}\rho k \delta_{ij} \frac{\partial u_i}{\partial x_j}

For low-speed flows, the second term on the right hand side of this equation is generally small compared to the first term. (For incompressible flows the second term is identically zero.) A production limiter is still employed for the P term in the k-equation, as described for (SST).
 

Menter SST Two-Equation Model from 2003 (SST-2003)

This form of the SST model has several relatively minor variations from the original SST version (SST). The reference for its usage is:

The main change is in the definition of eddy viscosity, which uses the strain invariant rather than magnitude of vorticity in its definition:

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

where

S = \sqrt{2 S_{ij}S_{ij}}

Another minor difference from (SST) is that the production limiter is used for both k and omega equations, and the constant is changed from 20 to 10. In other words, P in both the k and omega equations gets replaced by:

{\rm min}(\cal P, 10 \beta^* \rho \omega k)

The definition of {\rm CD}_{k \omega} is slightly different in that it uses 10-10 rather than 10-20 for its second term:

{\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^{-10} \right)

Finally, the definitions of two of the constants are slightly different:

\gamma_1 = 5/9           \gamma_2 = 0.44

The first is higher than the original constant definition by approximately 0.43%, and the second is lower by less than 0.08%.
 

Menter SST Two-Equation Model with Controlled Decay (SST-sust)

This form of the SST model eliminates the non-physical decay of turbulence variables in the freestream for external aerodynamic problems, through the addition of sustaining terms to the equations. The reference is:

The equations are:

\frac{\partial (\rho k)}{\partial t} + \frac{\partial (\rho u_j k)}{\partial x_j} = \cal 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] +\beta^* \rho \omega_{\rm amb} k_{\rm amb}
\frac{\partial (\rho \omega)}{\partial t} + \frac{\partial (\rho u_j \omega)}{\partial x_j} = \frac{\gamma}{\nu_t} \cal 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} + \beta \rho \omega_{\rm amb}^2

where everything except the last term in each equation is identical to the standard model (SST). The recommended farfield boundary conditions are somewhat different:

\omega_{\rm farfield} = \frac{5 U_{\infty}}{L}
k_{\rm farfield} = 10^{-6} U_{\infty}^2

Here, L is no longer the "approximate length of the computational domain," like it was for (SST), but rather the defining length scale for the particular problem (usually associated with some feature or scale of the aerodynamic body of interest). In the equations, \omega_{\rm amb} and k_{\rm amb} are taken to be these farfield boundary values. The extra terms have the effect of exactly cancelling the destruction terms in the freestream when the turbulence levels are equal to the set ambient levels. Inside the boundary layer, they are generally orders of magnitude smaller than the destruction terms for reasonable freestream turbulence levels (say, Tu = 1% or less), and therefore have little effect. The farfield boundary condition k_{\rm farfield} = 10^{-6} U_{\infty}^2 corresponds to a freestream Tu level of 0.08165%.
 

Menter SST Two-Equation Model with Controlled Decay and Vorticity Source Term (SST-Vsust)

This form of the SST model combines (SST-V) and (SST-sust). The model is identical to (SST-sust), with the exception that the term P (in both equations) is approximated with the following:

P = \mu_t \Omega^2 - \frac{2}{3}\rho k \delta_{ij} \frac{\partial u_i}{\partial x_j}

For low-speed flows, the second term on the right hand side of this equation is generally small compared to the first term. (For incompressible flows the second term is identically zero.) A production limiter is still employed for the P term in the k-equation, as described for (SST).
 

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