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Turbulence Modeling Resource

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. All forms of the model given on this page are linear eddy viscosity models. Linear models use the Boussinesq assumption for the constitutive relation:

$\tau_{ij} = 2 \mu_t \left(S_{ij} - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij} \right) - \frac{2}{3} \rho k \delta_{ij}$

Unless otherwise stated, for compressible flow with heat transfer this model is implemented as described on the page Implementing Turbulence Models into the Compressible RANS Equations, with perfect gas assumed and Pr = 0.72, Pr_t = 0.90, and Sutherland's law for dynamic viscosity.

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The first version listed (SST) is considered "standard". It is the original published version.

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

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

Menter, F. R., "Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications," AIAA Journal, Vol. 32, No. 8, August 1994, pp. 1598-1605, https://doi.org/10.2514/3.12149.

This model is almost identical to the Menter baseline model. Only one constant ( $\sigma_{k 1}$ ) and the expression for turbulent eddy viscosity are different. 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, https://ntrs.nasa.gov/citations/19930013620).

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

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

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

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 $\Omega = \sqrt{2 W_{ij} W_{ij} }$ is the vorticity magnitude, with

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

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, https://doi.org/10.2514/6.1993-2906). 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:

$\gamma_1 = \frac{\beta_1}{\beta^*} - \frac{\sigma_{\omega 1} \kappa^2}{\sqrt{\beta^*}}$ $\gamma_2 = \frac{\beta_2}{\beta^*} - \frac{\sigma_{\omega 2} \kappa^2}{\sqrt{\beta^*}}$

$\sigma_{k 1} = 0.85$ $\sigma_{\omega 1} = 0.5$ $\beta_1 = 0.075$

$\sigma_{k 2} = 1.0$ $\sigma_{\omega 2} = 0.856$ $\beta_2 = 0.0828$

$\beta^*=0.09$ $\kappa=0.41$ $a_1 = 0.31$

Note: the $(2/3) \overline \rho k \delta_{ij}$ term in the Boussinesq approximation for tau_ij is sometimes ignored in the momentum and energy equations. Similarly, the production term in two-equation turbulence models is often approximated by P = mu_t S². This expression is exact for incompressible flows and is typically considered a very good approximation, except perhaps for very high Mach number flows (see items 4 and 7 on the page Notes on Running the Cases with CFD, and the Implementing Turbulence Models into the Compressible RANS Equations page). There are various ways that these approximations can be implemented:

(SSTm) (most common) - when the $(2/3) \overline \rho k \delta_{ij}$ term is IGNORED in tau_ij in the momentum and energy equations, and the production term is APPROXIMATED by P = mu_t S².
(SSTs) - when the $(2/3) \overline \rho k \delta_{ij}$ term is INCLUDED in tau_ij in the momentum and energy equations, and the production term is APPROXIMATED by P = mu_t S².
(SSTe) - when the $(2/3) \overline \rho k \delta_{ij}$ term is IGNORED in tau_ij in the momentum and energy equations, and the production term is THE EXACT expression.

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

This form of the SST model is sometimes used because vorticity magnitude $\Omega = \sqrt{2 W_{ij} W_{ij} }$ 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:

Menter, F. R., "Improved Two-Equation k-omega Turbulence Models for Aerodynamic Flows," NASA TM 103975, October 1992, https://ntrs.nasa.gov/citations/19930013620.

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:

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

A production limiter is still employed for the P term in the k-equation, as described for (SST).

Note 1: The use of the "V" version (vorticity instead of strain in the production term) is sometimes favored for hypersonic flow applications, particularly when strong bow shock waves are present (see, for example, AIAA-2011-3143, https://doi.org/10.2514/6.2011-3143).

Note 2: When the $(2/3) \overline \rho k \delta_{ij}$ term is ignored in tau_ij in the momentum and energy equations and the production term is approximated by P = mu_t*(Omega²), a modified naming convention should be employed: (SST-Vm).

Menter SST Two-Equation Model with Kato-Launder Source Term (SST-KL) and (SST-KLm)

Note that the above Vorticity Source Term approximation is similar in spirit to the Kato-Launder correction (Kato, M. and Launder, B. E., "The Modelling of Turbulent Flow Around Stationary and Vibrating Square Cylinders," 9th Symposium on Turbulent Shear Flows, Kyoto, Japan, August 1993, paper 10-4), which uses $\mu_t S \Omega$ instead of $\mu_t \Omega^2$ . Implementation of the Kato-Launder correction would be called (SST-KL). If the $(2/3) \overline \rho k \delta_{ij}$ term is ignored in tau_ij in the momentum and energy equations and if the $- (2/3) \rho k \delta_{ij} \partial u_i / \partial x_j$ term in P is ignored, then the naming convention is (SST-KLm).

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

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

Menter, F. R., Kuntz, M., and Langtry, R., "Ten Years of Industrial Experience with the SST Turbulence Model," Turbulence, Heat and Mass Transfer 4, ed: K. Hanjalic, Y. Nagano, and M. Tummers, Begell House, Inc., 2003, pp. 625 - 632.

Note, however, a typographical error existed in this paper that was subsequently corrected by the authors. In the omega equation (2nd part of eqn (1) in the paper), the production term was incorrectly given as $\alpha \rho S^2$ (using the paper's notation). Instead, it should have read $\alpha \tilde P_k / \nu_t$ (again using the paper's notation). In this expression, the P_k term has a tilde over it, which refers to the limited value of the k production term ${\rm min}(\cal P, 10 \beta^* \rho \omega k)$ . See below.

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%.

Note that if the vorticity form of the production term is used with this model, it should be referred to as (SST-V2003).

When the $(2/3) \overline \rho k \delta_{ij}$ term is ignored in tau_ij in the momentum and energy equations and the production term is approximated by P = mu_t S², a modified naming convention should be used: (SST-2003m). Other variants (s and e) follow the naming conventions described above at the end of the SST section.

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

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:

Spalart, P. R. and Rumsey, C. L., "Effective Inflow Conditions for Turbulence Models in Aerodynamic Calculations," AIAA Journal, Vol. 45, No. 10, 2007, pp. 2544 - 2553, https://doi.org/10.2514/1.29373.

The equations are:

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

$\omega_{\rm farfield} = \omega_{\rm amb} = \frac{5 U_{\infty}}{L}$

$k_{\rm farfield} = k_{\rm amb} = 10^{-6} U_{\infty}^2$

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%.

Note that applying the sustaining terms with the inflow values of k and omega that have been typical in external aerodynamics (e.g., as used for the standard model (SST)) will not be effective. The reason is that these values have been adjusted to allow a drastic decay before the fluid approaches the body.

When the $(2/3) \overline \rho k \delta_{ij}$ term is ignored in tau_ij in the momentum and energy equations and the production term is approximated by P = mu_t S², a modified naming convention should be used: (SST-sust-m). Other variants (s and e) follow the naming conventions described above at the end of the SST section.

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

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).

When the $(2/3) \overline \rho k \delta_{ij}$ term is ignored in tau_ij in the momentum and energy equations and the production term is approximated by P = mu_t*(Omega²), a modified naming convention should be employed: (SST-Vsust-m).

Menter SST Two-Equation Model with Rotation/Curvature Correction (SST-RC) and (SST-RCm)

This rotation/curvature form of the SST model is the same as the "standard" version (SST), except that the production term P in both equations gets multiplied by a function $f_{r1}$ , described below.

The reference for SST-RC is:

Smirnov, P. E., Menter, F. R., "Sensitization of the SST Turbulence Model to Rotation and Curvature by Applying the Spalart-Shur Correction Term," ASME Journal of Turbomachinery, Vol. 131, October 2009, 041010, https://doi.org/10.1115/1.3070573.

Although the above reference uses the naming convention "CC", here we use "RC" (rotation and curvature) for consistency with the SA-RC naming convention already established.

The empirical function that multiplies the production term P in both the $k$ and $\omega$ equations is:

$f_{r1} = {\rm max} \left[ {\rm min} \left( f_{rotation}, 1.25 \right), 0.0 \right]$

where

$f_{rotation} = (1 + c_{r1}) \frac{2r^*}{1+r^*}\left[ 1 - c_{r3} {\rm tan}^{-1}(c_{r2} \hat r)\right] - c_{r1}$

All the variables and their derivatives are defined with respect to the reference frame of the calculation, which may be rotating with rotation rate $\Omega^{rot}$ . The remaining functions are defined as:

$r^* = S/W$

$\hat r = \frac{2 W_{ik} S_{jk}}{W D^3} \left( \frac{DS_{ij}}{Dt} + (\varepsilon_{imn}S_{jn} +\varepsilon_{jmn}S_{in}) \Omega^{rot}_m \right)$

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

$W_{ij} = \frac{1}{2} \left[ \left(\frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) + 2 \varepsilon_{mji} \Omega^{rot}_m \right]$

$S^2 = 2 S_{ij}S_{ij}$

$W^2 = 2 W_{ij}W_{ij}$

$D^2 = {\rm max} \left( S^2, 0.09 \omega^2 \right)$

$c_{r1} = 1.0$ $c_{r2} = 2.0$ $c_{r3} = 1.0$

The term $DS_{ij}/Dt$ represents the components of the Lagrangian derivative of the strain rate tensor. The rotation rate $\Omega^{rot}$ is nonzero only if the reference frame itself is rotating.

Note that if the RC correction is added to a different base SST model, the naming of the model should reflect it. For example, if adding RC to (SST-2003), the corrected model should be referred to as (SST-2003RC). If adding RC to (SST-V), the corrected model should be referred to as (SST-VRC).

When the $(2/3) \overline \rho k \delta_{ij}$ term is ignored in tau_ij in the momentum and energy equations and the production term is approximated by P = mu_t S², a modified naming convention should be used: (SST-RCm). Other variants (s and e) follow the naming conventions described above at the end of the SST section.

Menter SST Two-Equation Model with Hellsten's Simplified Rotation/Curvature Correction (SST-RC-Hellsten) and (SST-RC-Hellsten-m)

This simplified rotation/curvature form of the SST model is the same as the "standard" version (SST), except that the destruction term in the $\omega$ equation gets multiplied by the function F₄, described below.

The references for SST-RC-Hellsten are:

Hellsten, A., "Some Improvements in Menter's k-omega SST Turbulence Model," AIAA Paper 98-2554, June 1998, https://doi.org/10.2514/6.1998-2554.
Mani, M., Ladd, J. A., and Bower, W. W., "Rotation and Curvature Correction Assessment for One- and Two-Equation Turbulence Models," Journal of Aircraft, Vol. 41, No. 2, 2004, pp. 268-273, https://doi.org/10.2514/1.9321.

The latter reference discovered that a different value of the constant $C_{RC}$ worked better than the value reported in the first reference, so this newer value is used here.

In the original model, replace the $\omega$ equation destruction term:

$\beta \rho \omega^2$

with

$F_4 \beta \rho \omega^2$

where

$F_4 = \frac{1}{1+C_{RC} R_i}$

$R_i = \frac{W}{S} \left( \frac{W}{S} - 1 \right)$

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

$W = \sqrt{2 W_{ij} W_{ij}}$ $W_{ij} = \frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right)$

$C_{RC}=1.4$

Note that if the RC-Hellsten correction is added to a different base SST model, the naming of the model should reflect it. For example, if adding RC-Hellsten to (SST-2003), the corrected model should be referred to as (SST-2003RC-Hellsten). If adding RC-Hellsten to (SST-V), the corrected model should be referred to as (SST-VRC-Hellsten).

When the $(2/3) \overline \rho k \delta_{ij}$ term is ignored in tau_ij in the momentum and energy equations and the production term is approximated by P = mu_t S², a modified naming convention should be used: (SST-RC-Hellsten-m). Other variants (s and e) follow the naming conventions described above at the end of the SST section.

Special notes for users of OpenFOAM.

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Recent significant updates:
10/04/2024 - added "s" and "e" variant definitions
03/24/2021 - clarifications on use of "m" designation when P=mu_t S² and k term ignored in momentum and energy equations
12/19/2020 - added an additional note regarding inflow values for SST-sust
8/28/2020 - delineate the SST-Vm modification to SST-V
6/30/2015 - mention Pr, Pr_t, and Sutherland's law
8/29/2013 - mention of Kato-Launder correction
7/08/2013 - added SST-RC-Hellsten

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Last Updated: 10/04/2024