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Turbulence Modeling Resource

The Menter Baseline Turbulence Model

This web page gives detailed information on the equations for various forms of the Menter baseline (BSL) 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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"Standard" Menter Baseline Two-Equation Model (BSL) and (BSLm)

The reference for the standard implementation of the Menter BSL 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 SST 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:

$\mu_t = \frac{\rho k}{\omega}$

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:

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

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

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

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

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.

The constants are:

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

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

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

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

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:

(BSLm) (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.
(BSLs) - 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.
(BSLe) - 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 Baseline Two-Equation Model with Vorticity Source Term (BSL-V) and (BSL-Vm)

This form of the BSL 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:

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 (BSL), with the exception that the term P (in both equations) is approximated with the following:

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

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: (BSL-Vm).

Note that this 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 should be called (BSL-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 (BSL-KLm).

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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
6/30/2015 - mention Pr, Pr_t, and Sutherland's law
8/29/2013 - mention of Kato-Launder correction

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Last Updated: 10/04/2024