Langley Research Center

Turbulence Modeling Resource

The AFT 3-equation Transitional Model

This web page gives detailed information on the equations for various forms of the Amplification Factor Transport (AFT) transition modeling framework. 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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AFT2017b 3-equation Transition Model Framework with SA (SA-AFT2017b)

This model framework is sometimes shortened to the initialism "AFT," with the version number trailing the initialism. The AFT model is most often coupled with the Spalart-Allmaras model and is formally referred to as SA-AFT2017b. The primary reference for the implementation of the AFT 3-equation Transition Modeling Framework is:

Coder, J.G., Pulliam, T. H., and Jensen, J.C., "Contributions to HiLiftPW-3 Using Structured, Overset Grid Methods", AIAA Paper 2018-1039, January 2018, https://doi.org/10.2514/6.2018-1039.

The default 3-equation model is given by:

$\frac{D\tilde{\nu}}{Dt} = c_{b1} \tilde{S}\tilde{\nu}(1-f_{t2}) - \Big(c_{w1}f_{w} - \frac{c_{b1}}{\kappa ^2} f_{t2} \Big) \Big(\frac{\tilde{\nu}}{d}\Big)^2 + \frac{1}{\sigma}\bigg[\frac{\partial}{\partial x_j}\Big((\nu + \tilde{\nu})\frac{\partial\tilde{\nu}}{\partial x_j}\Big) + c_{b2}\frac{\partial\tilde{\nu}}{\partial x_j}\frac{\partial\tilde{\nu}}{\partial x_j}\bigg]$

$\frac{\partial \rho\tilde{n}}{\partial t} + \frac{\partial \rho u_j \tilde{n}}{\partial x_j} = \rho \Omega F_{crit} F_{growth}\frac{d\tilde{n}}{dRe_{\theta}} + \frac{\partial}{\partial x_j}\Big[\sigma_n(\mu + \mu_t)\frac{\partial \tilde{n}}{\partial x_j}\Big]$

$\frac{\partial \rho \tilde{\gamma}}{\partial t} + \frac{\partial \rho u_j \tilde{\gamma}}{\partial x_j} = c_1 \rho S F_{onset} \Big[1-{{exp}} (\tilde{\gamma})\Big] - c_2 \rho \Omega F_{turb} \Big[c_3 \, {{exp}} (\tilde{\gamma}) - 1 \Big] + \frac{\partial}{\partial x_j} \bigg[\Big(\mu + \frac{\mu_t}{\sigma_y}\Big) \frac{\partial \tilde{\gamma}}{\partial x_j}\bigg]$ <- corrected 10/04/2022

(Note: prior to 10/04/2022 the third equation above had the incorrect sign in front of the c₂ term. This term should have a minus sign in front of it, as shown above.) The source terms of the $\tilde{n}$ equation are based on an estimate of the integral boundary layer shape factor, $H_{12}$ . The production terms are given as

$H_L = \frac{d^2}{\mu} \bigg[ \frac{\partial}{\partial x_i} \Big( \rho u_j \frac{\partial d}{\partial x_j} \Big) \frac{\partial d}{\partial x_i} \bigg]$

The shape factor $H_{12}$ is computed as

$H_{12} = 0.376960 + \sqrt{\frac{H_L + 2.453432}{0.653181}}$

The functions of the production term are defined as

$F_{crit} = 0, Re_{V} < Re_{V,0}$ $F_{crit} = 1, Re_{V} \ge Re_{V,0}$

$Re_{V,0} = k_V Re_{\theta,0}$

$k_V = 0.246175{H_{12}}^2 - 0.141831 H_{12} + 0.008886$

$Re_{V} = \frac{\rho S d^2}{\mu + \mu_t}$

${log}_{10} \Big(Re_{\theta ,0} \Big) = 0.7 \, {tanh} \bigg(\frac{14}{H_{12} - 1} - 9.24 \bigg) + \frac{2.492}{(H_{12} - 1)^{0.43}} + 0.62$

$F_{growth} = D (H_{12}) \frac{1 + m(H_{12})}{2} \, l({H_{12}})$

$D(H_{12}) = \frac{H_{12}}{0.5482 H_{12} - 0.5185}$

$l(H_{12}) = \frac{6.54 H_{12} - 14.07}{{H_{12}}^2}$

$m(H_{12}) = \frac{1}{l(H_{12})} \bigg[0.058 \frac{(H_{12} - 4)^2}{H_{12} - 1} - 0.068 \bigg]$

$\frac{d \tilde{n}}{d Re_{\theta}} = 0.028 \Big( H_{12} - 1 \Big) - 0.0345 \, {exp} \bigg[- \bigg(\frac{3.87}{H_{12}-1} - 2.52 \bigg)^2 \, \bigg]$

In the above, $\rho$ is density, $\mu$ is the molecular dynamic viscosity, $\mu_t$ is the eddy viscosity, $d$ is the wall distance, $S$ is the strain rate magnitude, and $\Omega$ is the vorticity magnitude:

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

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

$\Omega = \sqrt{2 \Omega_{ij} \Omega_{ij}}$

$\Omega_{ij} = \frac{1}{2}\bigg(\frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \bigg)$

The third equation is that of $\tilde{\gamma}$ , which is the natural logarithm of the actual intermittency. This change of variable ensures that the transported $\tilde{\gamma}$ will not be negative and will be physically interpretable across all attainable values, enabling the compatibility of the AFT model with all types of flow solvers. This equation is based on the one suggested by Menter et al. (Menter, F.R., Smirnov, P. E., Liu, T., and Avancha, R., "A One-Equation Local Correlation-Based Transition Model," Flow Turbulence Combustion 95, pp. 583-619 (2015), https://doi.org/10.1007/s10494-015-9622-4). The production terms are

$F_{onset} = {max} \Big[ F_{onset,2} - F_{onset,3}, 0 \Big]$

$F_{onset,1} = \frac{\tilde{n}}{N_{crit}}$

$F_{onset,2} = {min}\Big(F_{onset,1},2 \Big)$

$F_{onset,3} = {max} \Big[1 - \Big(\frac{R_T}{3.5}\Big)^3, 0 \Big]$

Note that the reference paper AIAA 2018-1039 combines $F_{onset,1}$ and $F_{onset,2}$ into its $F_{onset,1}$ . These have been separated here for clarity and consistency with other variants of the AFT model. The destruction term is given as

$F_{turb} = {exp} \bigg[ - \Big( \frac{R_T}{2} \Big)^4 \bigg]$

In the above, $N_{crit}$ is the critical amplification factor from Mack (Mack, L. M., "Transition and Laminar Instability," NASA CR-153203, 1977, https://ntrs.nasa.gov/citations/19770017114) and $R_T$ is the turbulence Reynolds number.

$N_{crit} = -8.43 - 2.4 {ln} \Big( {\frac{\tau}{100}} \Big)$

$\tau =2.5 {tanh}\bigg(\frac{Tu(\%)}{2.5}\bigg)$

$R_T = \frac{\mu_t}{\mu}$

where $Tu(\%)$ is the freestream turbulence intensity and $\tau$ is a limiter suggested by Drela (Drela M. and Youngren H., "User's Guide to MISES 2.63," February 2008, https://web.mit.edu/drela/Public/web/mises/mises.pdf). The AFT model couples with the SA model via a modification to the f_t2 term

$f_{t2} = c_{t3}[1 - {exp}(\tilde{\gamma})]$

The calibration constants for the AFT model are $\sigma_n = 1.0$ , $c_1 = 100$ , $c_2 = 0.06$ , $c_3 = 50$ , $\sigma_y = 1.0$ , and $c_{t3} = 1.2$ (note that c₂ was incorrectly listed as 0.6 prior to 10/04/2022), where $c_{t3}$ comes from the SA model, and $c_1$ , $c_2$ , $c_3$ , and $\sigma_y$ come from Menter et al. (Menter, F.R., Smirnov, P. E., Liu, T., and Avancha, R., "A One-Equation Local Correlation-Based Transition Model," Flow Turbulence Combustion 95, pp. 583-619 (2015), https://doi.org/10.1007/s10494-015-9622-4). The boundary conditions for $\tilde{n}$ and $\tilde{\gamma}$ are

$\tilde{n}_{\infty} = 0$

$\tilde{\gamma}_{\infty} = 0$

$\frac{\partial \tilde{n}}{\partial y} \bigg| _{wall} = 0$

$\frac{\partial \tilde{\gamma}}{\partial y} \bigg| _{wall} = 0$

It is suggested that the modified eddy viscosity ratio be set to 0.1 (actual eddy viscosity ratio of $2.79 {x} 10^{-7}$ ). To determine the transition location, Spalart's turbulence index should be used (Spalart, P. R. and Allmaras, S. R., "A One-Equation Turbulence Model for Aerodynamic Flows," Recherche Aerospatiale, No. 1, 1994, pp. 5-21 (see Spalart-Allmaras Turbulence Model page)

$i_t = \frac{1}{\kappa \sqrt{\nu \Omega}} \frac{\partial \tilde{\nu}}{\partial n} \bigg | _{wall}$

which is designed to be 0 in laminar boundary layers where $\tilde{\nu} = 0$ and 1 in turbulent boundary layers where $\tilde{\nu}/\nu = \kappa y^+$ by design of the SA model.

Note that some implementations limit $\tilde{n}$ and $\tilde{\gamma}$ to prevent runaway behaviors.

AFT2019b 3-equation Transition Model Framework with SA (SA-AFT2019b)

This variant of the AFT model preserves the same form of the AFT2017b variant and features improved correlations for the integral boundary layer shape factor $H_{12}$ , the local shape factor $H_L$ , and $F_{growth}$ . The primary reference for the implementation of AFT2019b is:

Coder, J.G., "Further Development of the Amplification Factor Transport Transition Model for Aerodynamic Flows", AIAA Paper 2019-0039, January 2019, https://doi.org/10.2514/6.2019-0039.

(Note this reference has a typo in its $\tilde{\gamma}$ equation, with incorrect sign on the c₂ term. See the $\tilde{\gamma}$ equation in the SA-AFT2017b model, above, for the correct definition.) The modifications to the source terms are as follows:

$H_L = \frac{d^2}{\nu} \bigg[ \frac{\partial}{\partial x_i} \Big(u_j \frac{\partial d}{\partial x_j} \Big) \frac{\partial d}{\partial x_i} \bigg]$

$H_{12} = 0.26 H_L + 2.4$

$D(H_{12}) = \frac{2.4 H_{12}}{H_{12} - 1}$

$k_V = \frac{1}{0.4036 {H_{12}}^2 - 2.5394 H_{12} + 4.3273}$

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Jim Coder and Jared Carnes of The University of Tennessee, Knoxville are acknowledged for helping put together this webpage.

Recent significant updates:
10/04/2022 - fixed typos in gamma equation and in c₂ coefficient

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Last Updated: 10/04/2022