Spalart-Allmaras Model

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The Spalart-Allmaras Turbulence Model

This web page gives detailed information on the equations for various forms of the Spalart-Allmaras turbulence model. If any particular variant has been overlooked, please report it to the page curator.

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"Standard" Spalart-Allmaras One-Equation Model (SA)

The following equations represents the most commonly-used implementation of the Spalart-Allmaras model (written in non-conservation form). The primary reference is:

Spalart, P. R. and Allmaras, S. R., "A One-Equation Turbulence Model for Aerodynamic Flows," Recherche Aerospatiale, No. 1, 1994, pp. 5-21.

Note that this journal reference had a small typo (appendix only) in its definition of the constant c_w1 (missing square). The typo is corrected below. The original reference made use of a trip term that most people do not include, because the model is most often employed for fully turbulent applications. Therefore, in this "standard" representation the trip term is being left out (see version (SA-Ia) below for the version including the trip term). As a consequence, the farfield boundary condition must be changed from that given in the above reference. The new farfield boundary condition is taken from the following references:

Spalart, P. R., "Trends in Turbulence Treatments," AIAA 2000-2306, June 2000.
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.

In all of the following, a "hat" is used over the turbulence field variable, rather than a "tilde" as given in the references, for the sole practical reason that the "tilde" showed up very poorly on the screen.

The one-equation model is given by the following equation:

$\frac{\partial \tilde \nu}{\partial t} + u_j \frac{\partial \tilde \nu}{\partial x_j} = c_{b1}(1-f_{t2})\tilde S \tilde \nu - \left[c_{w1}f_w - \frac{c_{b1}}{\kappa^2}f_{t2}\right] \left(\frac{\tilde \nu}{d} \right)^2 + \frac{1}{\sigma} \left[ \frac{\partial}{\partial x_j} \left( \left( \nu + \tilde \nu \right) \frac{\partial \tilde \nu}{\partial x_j} \right) + c_{b2}\frac{\partial \tilde \nu}{\partial x_i} \frac{\partial \tilde \nu}{\partial x_i} \right]$

and the turbulent eddy viscosity is computed from:

$\mu_t = \rho \tilde \nu f_{v1}$

where

$f_{v1} = \frac{\chi^3}{\chi^3+c_{v1}^3}$

$\chi = \frac{\tilde \nu}{\nu}$

and $\rho$ is the density, $\nu = \mu/\rho$ is the molecular kinematic viscosity, and $\mu$ is the molecular dynamic viscosity. Additional definitions are given by the following equations:

$\tilde S = \Omega + \frac{\tilde \nu}{\kappa^2 d^2} f_{v2}$

where $\Omega$ is the magnitude of the vorticity, d is the distance from the field point to the nearest wall, and

$f_{v2} = 1 - \frac{\chi}{1+\chi f_{v1}}$ $f_w = g \left[ \frac{1+c_{w3}^6}{g^6 + c_{w3}^6} \right]^{1/6}$

$g = r + c_{w2}(r^6 - r)$

$r = {\rm min} \left[ \frac{\tilde \nu}{\tilde S \kappa^2 d^2}, 10 \right]$

$f_{t2} = c_{t3} {\rm exp} \left( -c_{t4} \chi^2 \right)$

The boundary conditions are:

$\tilde \nu_{wall} = 0$ $\tilde \nu_{farfield} = 3 \nu_{\infty} - 5 \nu_{\infty}$

The constants are:

$c_{b1} = 0.1355$ $\sigma = 2/3$ $c_{b2} = 0.622$ $\kappa = 0.41$

$c_{w2} = 0.3$ $c_{w3} = 2$ $c_{v1} = 7.1$ $c_{t3} = 1.2$ $c_{t4} = 0.5$

$c_{w1} = \frac{c_{b1}}{\kappa^2} + \frac{1+c_{b2}}{\sigma}$

Note: To avoid possible numerical problems, the term $\hat S$ must never be allowed to reach zero or go negative.

Spalart-Allmaras One-Equation Model with Trip Term (SA-Ia)

The form of the Spalart-Allmaras model with the trip term included is given in the following reference:

Spalart, P. R. and Allmaras, S. R., "A One-Equation Turbulence Model for Aerodynamic Flows," Recherche Aerospatiale, No. 1, 1994, pp. 5-21.

The equations are the same as for the "standard" version (SA), except there is an additional trip term on the right hand side of the equation:

where:

$f_{t1} = c_{t1} g_t {\rm exp} \left[ -c_{t2} \frac{\omega_t^2}{\Delta U^2}(d^2 + g_t^2d_t^2) \right]$

$g_t = {\rm min} \left[ 0.1, \frac{\Delta U}{\omega_t \Delta x_t} \right]$

and $\Delta U$ is the difference between the velocity at the field point and that at the trip, $\Delta x_t$ is the grid spacing along the wall at the trip, $\omega_t$ is the wall vorticity at the trip, and $d_t$ is the distance from the field point to the trip.

The farfield boundary condition is:

$0 \leq \tilde \nu_{farfield} < \frac{1}{10}\nu_{\infty}$

Spalart-Allmaras One-Equation Model without f_t2 Term (SA-noft2)

Some implementations of Spalart-Allmaras ignore the $f_{t2}$ term, which was a numerical fix in the original model in order to make zero a stable solution to the equation with a small basin of attraction (thus slightly delaying transition so that the trip term could be activated appropriately). It is argued that if the trip trip is not used, then $f_{t2}$ is not necessary. The equations are the same as for the "standard" version (SA), except that the term $f_{t2}$ does not appear at all. An example reference that uses this form is:

Eca, L., Hoekstra, M., Hay, A., and Pelletier, D., "A Manufactured Solution for a Two-Dimensional Steady Wall-Bounded Incompressible Turbulent Flow," International Journal of Computational Fluid Dynamics, Vol. 21, Nos. 3-4, 2007, pp. 175-188.

Based on studies (see, e.g., Rumsey, C. L., "Apparent Transition Behavior of Widely-Used Turbulence Models," International Journal of Heat and Fluid Flow, Vol. 28, 2007, pp. 1460-1471), use of this form as opposed to the "standard" version (SA) probably makes very little difference, at least at reasonably high Reynolds numbers, provided that the "standard" version use the appropriate boundary condition of $\tilde \nu_{farfield} = 3 \nu_{\infty}$ (or greater).

Spalart-Allmaras One-Equation Model with Rotation/Curvature Correction (SA-RC)

This form of the Spalart-Allmaras model attempts to account for rotation and curvature effects. The reference is:

Shur, M. L., Strelets, M. K., Travin, A. K., Spalart, P. R., "Turbulence Modeling in Rotating and Curved Channels: Assessing the Spalart-Shur Correction," AIAA Journal Vol. 38, No. 5, 2000, pp. 784-792.

The model is the same as for the "standard" version (SA), except that the $c_{b1} \tilde S \tilde \nu$ term gets multiplied by the rotation function $f_{r1}$ :

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

where

$r^* = S/\omega$

$\tilde r = \frac{2 \omega_{ik} S_{jk}}{D^4} \left( \frac{DS_{ij}}{Dt} + (\varepsilon_{imn}S_{jn} +\varepsilon_{jmn}S_{in}) \Omega'_m \right)$

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

$S^2 = 2 S_{ij}S_{ij}$ $\omega^2 = 2 \omega_{ij}\omega_{ij}$

$D^2 = \frac{1}{2} \left(S^2 + \omega^2 \right)$

$c_{r1} = 1.0$ $c_{r2} = 12$ $c_{r3} = 1.0$

The term $DS_{ij}/Dt$ represents the components of the Lagrangian derivative of the strain rate tensor. The rotation rate $\Omega'$ is used only if the reference frame itself is rotating (note that all derivatives should be defined with respect to the reference frame).

Compressible Form of Spalart-Allmaras One-Equation Model (SA-Catris)

This compressible form was developed by Catris and Aupoix, and is given in the following reference:

Catris, S. and Aupoix, B., "Density Corrections for Turbulence Models," Aerospace Science and Technology, Vol. 4, 2000, pp. 1--11.

In this version, the equation is written in conservation form and the diffused quantity is taken to be $\sqrt{\rho} \tilde \nu$ (the transported quantity remains as $\tilde \nu$ ).

$\frac{\partial \rho \tilde \nu}{\partial t} + u_j \frac{\partial \rho \tilde \nu}{\partial x_j} = c_{b1} \Omega \rho \tilde \nu - c_{w1}f_w\rho \left(\frac{\tilde \nu}{d} \right)^2 + \frac{1}{\sigma} \frac{\partial}{\partial x_j} \left(\mu \frac{\partial \tilde \nu}{\partial x_j} \right) + \frac{1}{\sigma} \frac{\partial}{\partial x_j} \left( \sqrt{\rho} \tilde \nu \frac{\partial \sqrt{\rho}\tilde \nu}{\partial x_j} \right) + \frac{c_{b2}}{\sigma} \frac{\partial \sqrt{\rho}\tilde \nu}{\partial x_i} \frac{\partial \sqrt{\rho}\tilde \nu}{\partial x_i}$

There is no trip term. Note that Catris and Aupoix ignore the terms $f_{t2}$ and $f_{v2}$ from the original model; all other functions and constants are the same as for the "standard" version (SA).

Spalart-Allmaras One-Equation Model with Edwards Modification (SA-Edwards)

This form was developed primarily to improve the near-wall numerical behavior of the model (i.e., the goal was to improve the convergence behavior). The reference is:

Edwards, J. R. and Chandra, S. "Comparison of Eddy Viscosity-Transport Turbulence Models for Three-Dimensional, Shock-Separated Flowfields," AIAA Journal, Vol. 34, No. 4, 1996, pp. 756-763.

This version is the same as for the "standard" version (SA), except that $f_{t2}$ is ignored, and the following two variables are redefined:

$\tilde S = S^{1/2} \left[ \frac{1}{\chi} + f_{v1} \right]$

$r = \frac{{\rm tanh} \left[ \tilde \nu/(\tilde S \kappa^2 d^2)\right]}{{\rm tanh}(1.0)}$

Note that this method makes use of $S^{1/2}$ (rather than vorticity $\Omega$ ), where:

$S= \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \frac{\partial u_i}{\partial x_j} - \frac{2}{3} \left( \frac{\partial u_k}{\partial x_k} \right)^2$

Spalart-Allmaras One-Equation Model with f_v3 Term (SA-fv3)

This form of the Spalart-Allmaras model came about as a result of e-mail exchanges between the model developer and early implementers. It was devised to prevent negative values of the source term, and is not recommended because of unusual transition behavior at low Reynolds numbers (see Spalart, P. R., AIAA 2000-2306, 2000). Unfortunately, coding of this version still persists. Because this method came about through private communications, there is no official reference for it. However, see the following for a brief description:

Rumsey, C. L., Allison, D. O., Biedron, R. T., Buning, P.G., Gainer, T. G., Morrison, J. H., Rivers, S. M., Mysko, S. J., and Witkowski, D. P., "CFD Sensitivity Analysis of a Modern Civil Transport Near Buffet-Onset Conditions," NASA/TM-2001-211263, December 2001.

The equations are the same as for the "standard" version (SA), with the following exceptions:

$\tilde S = f_{v3} \Omega + \frac{\tilde \nu}{\kappa^2 d^2} f_{v2}$

$f_{v2} = \frac{1}{(1+\chi/c_{v2})^3}$

$f_{v3} = \frac{(1+\chi f_{v1})(1-f_{v2})}{\chi}$

$c_{v2} = 5$

Strain Adaptive Formulation of Spalart-Allmaras One-Equation Model (SA-salsa)

This form was developed primarily to extend the predictive capabiliy of the model for nonequilibrium conditions. It also makes use of some of the aspects of the (SA-Edwards) version. The reference is:

Rung, T., Bunge, U., Schatz, M., and Thiele, F., "Restatement of the Spalart-Allmaras Eddy-Viscosity Model in Strain-Adaptive Formulation," AIAA Journal, Vol. 41, No. 7, 2003, pp. 1396-1399.

This version is the same as for the "standard" version (SA), except for the following changes. First, $f_{t2}$ is ignored. Second, the term

$\frac{1}{\sigma}\left[\frac{\partial}{\partial x_j}\left(\left(\nu + \hat \nu \right) \frac{\partial \hat \nu}{\partial x_j}\right)\right]$

is written slightly differently as:

$\frac{\partial}{\partial x_j}\left[\left(\nu + \frac{\hat \nu}{\sigma} \right) \frac{\partial \hat \nu}{\partial x_j}\right]$

Third, the following two variables are redefined:

$\hat S = S^* \left[ \frac{1}{\chi} + f_{v1} \right]$

$r = 1.6 {\rm tanh} \left[ 0.7 \sqrt{\frac{\rho_0}{\rho}} \left( \frac{\hat \nu}{\hat S \kappa^2 d^2} \right) \right]$

where $\rho_0$ is the freestream stagnation density, and

$S^* = \sqrt{2 S_{ij}' S_{ij}'}$

$S_{ij}' = \frac{1}{2}\left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij}$

Fourth, the sensitization to nonequilibrium effects comes in through a change in $c_{b1}$ , which is no longer a constant. The source term changes from $c_{b1} \hat S \hat \nu$ to $c_{b1}' \hat S \hat \nu$ and $c_{w1}$ changes from

$c_{w1} = \frac{c_{b1}}{\kappa^2} + \frac{1+c_{b2}}{\sigma}$

$c_{w1} = \frac{c_{b1}'}{\kappa^2} + \frac{1+c_{b2}}{\sigma}$

where the new variable is

$c_{b1}' = 0.1355 \sqrt{\Gamma}$

and

$\Gamma = {\rm min}[1.25, {\rm max} (\gamma, 0.75)]$ $\gamma = {\rm max}(\alpha_1, \alpha_2)$

$\alpha_1 = \left[1.01 \left( \frac{\hat \nu}{S^* \kappa^2 d^2} \right) \right]^{0.65}$ $\alpha_2 = {\rm max} \left[ 0,1-{\rm tanh} \left( \frac{\chi}{68} \right) \right]^{0.65}$

Mixing Layer Compressibility Correction in Spalart-Allmaras One-Equation Model (SA-comp)

This correction improves SA behavior in compressible mixing layers. The reference is:

Spalart, P. R., "Trends in Turbulence Treatments," AIAA 2000-2306, June 2000.

This version is the same as for the "standard" version (SA), except that the following additional term is included on the right hand side of the equation.

$-C_5 \frac{\hat \nu^2}{a^2} \frac{\partial u_i}{\partial x_j} \frac{\partial u_i}{\partial x_j}$

where a is the speed of sound and $C_5=3.5$ .

If used in conjunction with SA-noft2 instead (see for example Forsythe, J. R., Hoffmann, K. A., Squires, K. D., AIAA 2002-0586, 2002.), the model name would become SA-noft2-comp.

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