Langley Research Center

Turbulence Modeling Resource

The SA-noft2-Gamma-Retheta 3-equation Transitional Model

This web page gives detailed information on the equations for various forms of the SA-noft2-Gamma-Retheta 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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SA-noft2-Gamma-Retheta 3-equation Transition Model (SA-noft2-Gamma-Retheta)

This model is result of the coupling of the Spalart-Allmaras (SA-noft2) model with the Langtry-Menter local correlation based transition approach. The SA-noft2 model is augmented by two additional equations, originally developed by Menter et al. (Flow Turb Combust 77:277-303, 2006) in the the k-omega equations context, for transitional flows. The primary references for the implementation of this model are:

D'Alessandro, V., Montelpare, S., Ricci, R., and Zoppi, A., "Numerical Modeling of the Flow Over Wind Turbine Airfoils by Means of Spalart-Allmaras Local Correlation Based Transition Model," Energy, Vol. 130, 2017, pp. 402-419, https://doi.org/10.1016/j.energy.2017.04.134.
D'Alessandro, V., Montelpare, S., and Ricci, R., "Assessment of a Spalart–Allmaras Model Coupled with Local Correlation Based Transition Approaches for Wind Turbine Airfoils," Applied Sciences, Vol. 11, No. 4, 2021, 1872, https://doi.org/10.3390/app11041872.

Like the SST-2003-LM2009 model on which it is partially based, this model is not Galilean invariant, due to its explicit use of the velocity vector.

The transport equations to take into account transition are given by following:

$\frac{\partial (\rho\gamma)}{\partial t} + \frac{\partial (\rho u_j \gamma)}{\partial x_j} = \rho P_\gamma - \rho D_\gamma + \frac{\partial}{\partial x_j}\left[ \left( \mu + \frac{\mu_t}{\sigma_f}\right) \frac{\partial \gamma}{\partial x_j} \right]$

$\frac{\partial (\rho \widehat{Re}_{\theta,t})}{\partial t} + \frac{\partial (\rho u_j \widehat{Re}_{\theta,t}) }{\partial x_j} = \rho P_{\theta,t} + \frac{\partial}{\partial x_j} \left[ \sigma_{\theta,t} \left( \mu + \mu_t \right) \frac{\partial \widehat{Re}_{\theta,t}}{\partial x_j} \right]$

(In the original references, these equations were written in incompressible form, as follows:

$\frac{\partial \gamma}{\partial t} + u_j \frac{\partial \gamma}{\partial x_j} = P_\gamma - D_\gamma + \frac{\partial}{\partial x_j}\left[ \left( \nu + \frac{\nu_t}{\sigma_f}\right) \frac{\partial \gamma}{\partial x_j} \right]$

$\frac{\partial \widehat{Re}_{\theta,t}}{\partial t} + u_j \frac{\partial \widehat{Re}_{\theta,t} }{\partial x_j} = P_{\theta,t} + \frac{\partial}{\partial x_j} \left[ \sigma_{\theta,t} \left( \nu + \nu_t \right) \frac{\partial \widehat{Re}_{\theta,t}}{\partial x_j} \right]$

)

The source terms in the $\gamma$ equation are defined as:

$P_\gamma = c_{a1} S \left[ \gamma F_{onset} \right]^{0.5} \left(1 - c_{e1} \gamma \right) F_{length}$

$D_\gamma = c_{a2} \Omega \gamma F_{turb} \left(c_{e2} \gamma -1\right)$

In $P_\gamma$ , the term $F_{onset}$ is computed as:

$F_{onset} = max \left( F_{onset,2} - F_{onset,3}, 0 \right)$

with

$F_{onset,2} = min \left( max \left( F_{onset,1} , F_{onset,1}^4 \right) , 4\right)$

$F_{onset,3} = max \left[\left( 2 - \left( \frac{R_T}{2.5} \right)^3 \right), 0 \right]$

$F_{onset,1} = \frac{Re_\nu}{2.193 Re_{\theta,c}}$

The terms $Re_\nu$ and $R_T$ are obtained as follows: $Re_\nu = \frac{S d^2}{\nu}$ and $R_T = \frac{\nu_t}{\nu}$ . On the other hand, for the destruction term, $D_\gamma$ , the coefficient $F_{turb}$ is defined:

$F_{turb} = \exp\left(-\frac{R_T}{4}\right)^4$

As regards the source terms in the transport equation for $\widehat{Re}_{\theta,t}$ , the following equation is adopted:

$P_{\theta,t} = \frac{c_{\theta,t}}{T} \left( Re_{\theta,t} - \widehat{Re_{\theta,t}} \right) \left(1- F_{\theta,t} \right)$

The last term is defined as:

$F_{\theta,t} = min \left( max \left( \exp\left(-\frac{d}{\delta}\right)^4 , 1 - \left( \frac{\gamma -1/c_{e2}}{1-1/c_{e2}} \right)^2 \right) , 1.0 \right)$

where

$\delta = \frac{375\Omega \nu \widehat{Re}_{\theta,t} d}{\left| U \right|^2}$

$U = \sqrt{u_i u_i}$

Additionally, the term T appearing in the source term of the $\widehat{Re}_{\theta,t}$ equation is defined as follows:

$T = \frac{500\nu}{\left| U \right|^2}$

The following closure constants are adopted in order to close the transport equations for transition: $c_{a1}=2.0$ , $c_{a2}=0.06$ , $c_{e1}=1.0$ , $c_{e2}=50$ , $c_{\theta,t}=0.03$ , $\sigma_{f}=1.0$ , $\sigma_{\theta,t}=2.0$ .

The main transport equation from SA-noft2 is modified in the following way, in order to take into account the transitional effects:

$\frac{\partial \widehat \nu}{\partial t} + u_j \frac{\partial \widehat \nu}{\partial x_j} = P_{\widehat \nu} - D_{\widehat \nu} + \frac{1}{\sigma} \left[ \frac{\partial}{\partial x_j} \left( \left( \nu + \widehat \nu \right) \frac{\partial \widehat \nu}{\partial x_j} \right) + {c_{b2}}\frac{\partial \widehat \nu}{\partial x_i} \frac{\partial \widehat \nu}{\partial x_i} \right]$

where

$P_{\widehat \nu} = \gamma_{eff} \left( c_{b1}\widehat S \widehat \nu \right)$

$D_{\widehat \nu} = c_{w1} f_{w} \left( \frac{\widehat \nu}{{d}} \right)^2$

(Note: earlier versions had a $max \left( min\left(\gamma, 0.5\right) ,1.0 \right)$ multiplying this term, but this was a holdover from previous code modifications, and is simply unity all the time.)

where $\gamma_{eff} = max \left( \gamma, \gamma_{sep} \right)$

with

$\gamma_{sep} = min \left( 2.0 \cdot max \left[ 0 , \left( \frac{Re_\nu}{3.235 Re_{\theta,c}} \right) -1\right] F_{reattach}, 2.0 \right)F_{\theta,t}$

and

$F_{reattach} = \exp\left(-\frac{\mathrm{R_T}}{20}\right)^4$

The following closure functions complete the definition of the SA-noft2 equation:

$f_{v1} = \frac{\chi^3}{\left( \chi^3 + c_{v1}^3\right)}$

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

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

$f_{w} = g\left[ \frac{1+c_{w3}^6}{g^6 + c_{w3}^6} \right]^{\frac{1}{6}}$

$\widehat S = \left[\Omega + min\left( 0 , S-\Omega \right)\right] + \frac{\widehat{\nu}}{k^2d^2}f_{v2}$

$r = \left\{ \begin{array}{l} r_{max} ...{for}... \frac{\widehat{\nu}}{\widehat{S} k^2d^2}< 0\\ \min \left( \frac{\widehat{\nu}}{\widehat{S} k^2d^2}, r_{max} \right) ...{for}... \frac{\widehat{\nu}}{\widehat{S} k^2d^2} \ge 0 \end{array} \right$

where $\chi=\widehat{\nu} / \nu$ is the dimensionless turbulent variable, $\Omega = \sqrt{2 W_{ij} W_{ij} }$ is the vorticity tensor module, and $S = \sqrt{2 S_{ij} S_{ij} }$ .

The following closure constants are also adopted for the SA-noft2 equation: $c_{b1}=0.1355$ , $c_{b2}=0.622$ , $c_{v1}=7.1$ , $\sigma=2/3$ , $c_{w1}=\frac{c_{b1}}{k^2} + \frac{(1+c_{b2})}{\sigma}$ , $c_{w2}=0.3$ , $c_{w3}=2$ , and $k=0.41$ .

Model Empirical Correlations

The present model contains, as other $\gamma-\widehat{Re_{\theta,t}}$ approaches available in literature, three empirical correlations needed to compute $Re_{\theta,t}$ , $Re_{\theta,c}$ , and $F_{length}$ . In the seminal contribution of Menter et al. (Flow Turb Combust 77:277-303, 2006), these three variables are related to: $Re_{\theta,t} = f\left(Tu, \lambda_\theta\right)$ , $F_{length} = f\left( \widehat{Re_{\theta,t}} \right)$ , and $Re_{\theta,c}= f\left( \widehat{Re_{\theta,t}} \right)$ ,

where Tu is the turbulence intensity and $\lambda_\theta$ is the Thwaites' pressure gradient coefficient:

$\lambda_\theta = \frac{\theta^2}{\nu} \frac{d\left| U \right|}{d s}$

These correlations were not released immmediately since they were proprietary. For this reason a specific literature segment containing several approaches for these correlations was started. In D'Alessandro et al. (Energy 130:402-419, 2017) and Zoppi (PhD Thesis), various approaches, developed within SST k-omega, were tested and compared for the current model. The best performance in terms of $Re_{\theta,t}$ were found using an expression developed by Menter et al.:

$Re_{\theta,t} = \left\{ \begin{array}{l} \left( 1173.51 - 589.428 \cdot Tu + 0.2196/Tu^2 \right) F\left(\lambda_\theta \right) ...{for}... Tu \leq 1.3\\ 331.5\left(Tu - 0.5668\right)^{-0.671}F\left(\lambda_\theta \right) ...{for}... Tu>1.3 \end{array}$

where

$F\left(\lambda_\theta \right) = \left\{ \begin{array}{l} 1 + \left[12.986 \lambda_\theta + 123.66 \lambda_\theta^2 + 405.689 \lambda_\theta^3 \right] \exp\left( - \left( \frac{Tu}{1.5} \right)^{1.5}\right) ...{for}... \lambda_\theta \leq 0\\ 1+ 0.275\left[1-\exp\left( -35 \lambda_\theta \right)\right] \exp\left( -\frac{Tu}{0.5} \right) ...{for}... \lambda_\theta >0 \end{array}$

It is important to remark that in the above correlations $Tu = Tu_\infty$ is used for all the points of the flowfield. Moreover, $Re_{\theta,t}$ is computed by iterating on the value of $\theta_t$ , since $Re_{\theta,t}$ is a function of $\theta_t$ itself through the presence of $\lambda_{\theta}$ .

In the literature, a strong sensitivity of gamma-Retheta models to the $F_{length}$ and $Re_{\theta,c}$ correlations (as well as transition criteria used to compute $Re_{\theta,t}$ ) has been documented. There are many different correlations for $F_{length}$ and $Re_{\theta,c}$ to choose from (see Tables 1 and 2 in D'Alessandro et al., Energy, 130:402-419, 2017). However, it was noted in that reference that the Malan et al. correlation (Malan et al. AIAA-2009-1142, 2009, https://doi.org/10.2514/6.2009-1142) provides the best performance for the flowfields developing around low-Reynolds number operating airfoils. These correlations are:

$F_{length} = min \left( \exp\left( 7.168-0.01173\widehat{Re}_{\theta,t} \right) + 0.5 , 300\right)$

$Re_{\theta,c} = min \left( 0.615 \widehat{Re}_{\theta,t} + 61.5 , \widehat{Re}_{\theta,t} \right)$

Boundary Conditions

Standard boundary conditions are adopted for $\widehat \nu$ , the Spalart-Allmaras variable: $\widehat \nu_\infty=3\nu$ in the freestream and $\widehat \nu_{wall}=0$ . The boundary condition for $\gamma$ at the wall is zero normal flux. At an inlet or freestream, the value of $\gamma$ is equal to 1. The boundary condition for $\widehat{Re}_{\theta,t}$ at the wall is zero flux, while at an inlet or freestream $\widehat{Re}_{\theta,t}$ is calculated from the specific empirical correlation based on the inlet turbulence intensity.

It also is very important to remark that in order to capture the laminar and transitional boundary layers correctly, the grid must have a first cell height, viscous sublayer scaled, y+ of no more than approximately 1.

SA-noft2-LRe-Gamma-Retheta 3-equation Transition Model (SA-noft2-LRe-Gamma-Retheta)

This model is the same as (SA-noft2-Gamma-Retheta) except that the low Reynolds number correction is included by modifying c_w2 (see SA-LRe).

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Andrea Zoppi and Valerio D'Alessandro of Università Politecnica delle Marche (Italy) are acknowledged for helping put together this webpage.

Recent significant updates:
02/06/2023 - clarified the form of the destruction term in the SA equation

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Last Updated: 02/06/2023