Langley Research Center

Turbulence Modeling Resource

The Spalart-Allmaras 1-equation BCM Transitional Model

This web page gives detailed information on the equations for the SA-BCM (and earlier version SA-BC) transitional turbulence model. This model is a linear eddy viscosity model. 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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Spalart-Allmaras 1-equation BCM Transitional Model (SA-BCM)

This transition model is a modification update to SA-BC, which had a few issues (see red text below). The SA-BCM version fixes these issues with a different definition of $Term_2$ and $\chi_2$ . This version of the SA one-equation model is based on SA without the ft2 term (SA-noft2). The production term of the SA-noft2 is multiplied with a $\gamma_{BC}$ intermittency function in order to damp turbulence production until some transition criteria is achieved. From that point on, the damping effect of the transition model is disabled and the rest of the flowfield is allowed to be fully turbulent. The proposed $\gamma_{BC}$ intermittency function is calculated using local flow variables and transition onset is determined based on experimental correlations. The references are:

Cakmakcioglu, S. C., Bas, O., Mura, R., and Kaynak, U., "A Revised One-Equation Transitional Model for External Aerodynamics," AIAA Paper 2020-2706, June 2020, https://doi.org/10.2514/6.2020-2706.
Mura, R. and Cakmakcioglu, S. C., "A Revised One-Equation Transitional Model for External Aerodynamics - Part I: Theory, Validation and Base Cases," AIAA Paper 2020-2714, June 2020, https://doi.org/10.2514/6.2020-2714.

Note that "BC" stands for two of the authors' last names.

The SA-BCM transition model is coupled with the SA-noft2 model as follows:

$\frac{\partial \tilde{\nu}}{\partial t}+ u_j \frac{\partial \tilde{\nu}}{\partial x_j} = \boldmath{\gamma_{BC}} c_{b1} \tilde{S} \tilde{\nu} - c_{w1} f_w \left(\frac{\tilde{\nu}}{d}\right)^2 + \frac{1}{\sigma} \left[ \frac{\partial}{\partial x_j} \left( (\nu + \tilde{\nu}) \frac{\partial \tilde{\nu}}{\partial x_j} \right) + c_{b2} \frac{\partial \tilde{\nu}}{\partial x_j} \frac{\partial \tilde{\nu}}{\partial x_j} \right]$

The $\gamma_{BC}$ intermittency function is defined as:

$\gamma_{BC} = 1-exp(-\sqrt{Term_1}-\sqrt{Term_2})$

$Term_1$ is given by:

$Term_1 = \frac{max(Re_{\theta} - Re_{\theta c}, 0.0)}{\chi_1 Re_{\theta c}}$

where

$Re_{\theta} = \frac{Re_v}{2.193}, \phantom{XX} Re_v = \frac{\rho {d}^2}{\mu} \Omega$

and

$Re_{\theta c} = 803.73 {(Tu_\infty + 0.6067)}^{-1.027}$

In the above equations, $\rho$ is the local density, $\mu$ is the local molecular viscosity, $\Omega$ is the vorticity, $d$ is the distance to the nearest wall, $Re_v$ is the so-called vorticity Reynolds number, $Re_\theta$ is the momentum thickness Reynolds number, $Re_{\theta c}$ is the experimental transition onset critical momentum thickness Reynolds number, $Tu_\infty$ is the freestream turbulence intensity in percent, and $\chi_1$ is a calibration constant. Please note that the $Tu_\infty$ is a constant for the entire flowfield, as the SA-noft2 model ignores the turbulent kinetic energy (k).

The physical interpretation of $Term_1$ is that it checks for the onset location of transition by comparing the locally calculated $Re_\theta$ to the experimental correlation $Re_{\theta c}$ . As soon as the so-called vorticity Reynolds number $Re_v$ exceeds a critical value, $Term_1$ becomes larger than 0.0, and the intermittency function $\gamma_{BC}$ is triggered. However, the $Re_v$ is a function of distance to the nearest wall $d$ , thus, the $Re_v$ takes a very low value inside the boundary layer (close to the wall). Because of this, there cannot be intermittency generation inside the boundary layer by using $Term_1$ alone. To remedy this, $Term_2$ is introduced. $Term_2$ is (newly) defined as:

$Term_2 = max\left(\frac{\mu_t}{\chi_2 \mu}, 0.0}\right)$

In these equations, the $\mu_t$ is the turbulent eddy viscosity and $\chi_2$ is a calibration parameter. Finally, $\chi_1$ and $\chi_2$ are given below (with $\chi_2$ modified from its original definition in SA-BC):

$\chi_1 = 0.002, \phantom{XX} \chi_2 = 0.02$

The freestream $\tilde{\nu}$ is set to ${\tilde{\nu}}_{farfield} = 0.015 \nu_\infty : to : 0.025 \nu_\infty$ (or 0.005 times the original SA boundary condition). (The influence of $Tu_\infty$ comes in through $R_{e_{\theta c}}$ , and not through the freestream $\tilde{\nu}$ .)

Spalart-Allmaras 1-equation BC Transitional Model (SA-BC)

This older version of SA-BCM is no longer recommended. The reference is:

Cakmakcioglu, S. C., Bas, O., and Kaynak, U., "A Correlation-Based Algebraic Transition Model," Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 232, No. 21, 2018, pp. 3915-3929, https://doi.org/10.1177/0954406217743537.

It was subsequently uncovered that $\chi_2$ was not coded as originally written in the above paper. Below, $\chi_2$ has been corrected to reflect how it was actually implemented. This model is the same as SA-BCM, except that $Term_2$ is defined as:

$Term_2 = \frac{max(\nu_{BC} - \chi_2), 0.0}{\chi_2}$

where

$\nu_{BC} = \frac{\nu_t}{U d}$

In these equations, the $\nu_{BC}$ term is a proposed turbulent viscosity-like non-dimensional term, where $\nu_t$ is the turbulent viscosity, $U$ is the local velocity magnitude, $d$ is the distance to the nearest wall and $\chi_2$ is a calibration parameter. The $\chi_1$ and $\chi_2$ are given as (calibrated against Schubauer and Klebanoff flat plate test case):

$\chi_1 = 0.002, \phantom{XX} \chi_2 = 5.0/Re$

Warning: the use of velocity in the $\nu_{BC}$ term in this model makes it, strictly speaking, not Galilean invariant. Therefore results will be dependent on the frame of reference. Such a dependence has been avoided in conventional turbulence modeling, and certainly in the original SA model. (Note the use of velocity in the trip term f_t1 of the model SA-Ia, which has similarities to a transition model. In that model, in order to maintain Galilean invariance, $\Delta U$ was used, as the difference between the local velocity and a relevant wall velocity.)

In the original reference, $\chi_2$ was given as 5. However, as coded by the authors, it actually includes the freestream Reynolds number $Re = \rho_{\infty} U_{\infty} L_{ref} / \mu_{\infty}$ as shown above. This inclusion of Re may be problematic, since it brings a possibly non-unique reference length into play. Turbulence models normally avoid such a dependence. The reference length L_ref can be arbitrary for a given problem (for example, it may be wing chord, wing span, body diameter, or any number of things).

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Samet Cakmakcioglu is acknowledged for helping with this webpage.

Recent significant updates:
07/20/2020 - added modified SA-BCM model version
02/19/2019 - mentioned issue of lack of Galilean invariance
02/13/2019 - corrected chi_2 to reflect actual implementation (not in original reference)
01/29/2019 - emphasized warning message about inability to duplicate the results in the original reference

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Last Updated: 11/11/2021