Langley Research Center

Turbulence Modeling Resource

SSG/LRR Full Reynolds Stress Model

This web page gives detailed information on the equations for various versions of the blended SSG/LRR full second-moment Reynolds stress models, which use an omega equation for the length scale equation. Full second-moment Reynolds stress models are very different from simpler one and two-equation linear/nonlinear models, in that the latter use a constitutive relation giving the Reynolds stresses $\tau_{ij}$ in terms of other tensors via some assumed relation (such as Boussinesq's hypothesis). On the other hand, full second-moment Reynolds stress models compute each of the 6 Reynolds stresses directly (the Reynolds stress tensor is symmetric so there are 6 independent terms). Each Reynolds stress has its own transport equation. There is also a seventh transport equation for the lengthscale-determining variable.

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.

Return to: Turbulence Modeling Resource Home Page

SSG/LRR-omega Full Reynolds Stress Model (SSGLRR-RSM-w2012)

This model was developed within the framework of the European Project FLOMANIA for application to aeronautical flow problems. It uses a blend of two different pressure-strain models (the LRR part is based on an earlier version of the WilcoxRSM-w2006 model). Its main references are:

Eisfeld, B., Rumsey, C., and Togiti, V., "Verification and Validation of a Second-Moment-Closure Model," AIAA Journal, Vol. 54, No. 5, 2016, pp. 1524-1541, https://doi.org/10.2514/1.J054718.
Eisfeld, B., Rumsey, C., and Togiti, V., "Erratum: Verification and Validation of a Second-Moment-Closure Model," AIAA Journal, Vol. 54, No. 9, 2016, p. 2926, https://doi.org/10.2514/1.J055336.
Cecora, R.-D., Radespiel, R., Eisfeld, B., and Probst, A., "Differential Reynolds-Stress Modeling for Aeronautics," AIAA Journal, Vol. 53, No. 3, 2015, pp. 739-755, https://doi.org/10.2514/1.J053250.
Cecora, R.-D., Eisfeld, B., Probst, A., Crippa, S., and Radespiel, R., "Differential Reynolds Stress Modeling for Aeronautics," AIAA Paper 2012-0465, January 2012, https://doi.org/10.2514/6.2012-465.
Eisfeld, B., "Implementation of Reynolds Stress Models into the DLR-FLOWer Code," Institutsbericht, DLR-IB 124-2004/31, Report of the Institute of Aerodynamics and Flow Technology, Braunschweig, ISSN 1614-7790, 2004.
Eisfeld, B. and Brodersen, O., "Advanced Turbulence Modelling and Stress Analysis for the DLR-F6 Configuration", AIAA Paper 2005-4727, June 2005, https://doi.org/10.2514/6.2005-4727.

Note that the most recent papers (listed first) have different coefficients for $C_2^{(LRR)}$ and $D^{(\omega)}$ than the earlier papers. The newer values are recommended. Some references also have a minor typo on the sign of the production term in the omega equation (corrected here).

The full Reynolds stress model solves directly for the Reynolds stresses $\overline \rho \hat R_{ij}$ (which correspond to $-\tau_{ij}$ as described on the page Implementing Turbulence Models into the Compressible RANS Equations). In other words, using the notation of this website:

$\overline \rho \hat R_{ij} = -\tau_{ij} = \overline {\rho u_i'' u_j''}$

Note that this $\overline \rho \hat R_{ij}$ definition is the negative of the $\overline \rho R'_{ij}$ used in the WilcoxRSM-w2006 model. The six Reynolds stress equations and one length scale equation are given by:

$\frac{\partial \overline \rho \hat R_{ij}}{\partial t} + \frac{\partial (\overline \rho \hat u_k \hat R_{ij})}{\partial x_k} = \overline \rho P_{ij} + \overline \rho \Pi_{ij} - \overline \rho \varepsilon_{ij} + \overline \rho D_{ij} + \overline \rho M_{ij}$

$\frac{\partial (\overline \rho \omega)}{\partial t} + \frac{\partial (\overline \rho \hat u_k \omega)}{\partial x_k} = \frac{\alpha_{\omega} \omega}{\hat k} \frac{\overline \rho P_{kk}}{2} - \beta_{\omega} \overline \rho \omega^2 + \frac{\partial}{\partial x_k} \left[ \left( \overline \mu + \sigma_{\omega} \frac{\overline \rho \hat k}{\omega} \right) \frac{\partial \omega}{\partial x_k} \right] + \sigma_d \frac{\overline \rho}{\omega} {\rm max} \left( \frac{\partial \hat k}{\partial x_j} \frac{\partial \omega}{\partial x_j}, 0 \right)$

The production term is exact:

$\overline \rho P_{ij} = - \overline \rho \hat R_{ik} \frac{\partial \hat u_j}{\partial x_k} - \overline \rho \hat R_{jk} \frac{\partial \hat u_i}{\partial x_k}$

The dissipation is modeled via:

$\overline \rho \varepsilon_{ij} = \frac{2}{3} \overline \rho \varepsilon \delta_{ij}$

where $\varepsilon = C_{\mu} \hat k \omega$ , and $\hat k = \hat R_{ii}/2$ .

The pressure-strain correlation is modeled via:

$\overline \rho \Pi_{ij} = - \left( C_1 \overline \rho \varepsilon + \frac{1}{2} C_1^* \overline \rho P_{kk} \right) \hat a_{ij} + C_2 \overline \rho \varepsilon \left( \hat a_{ik} \hat a_{kj} - \frac{1}{3} \hat a_{kl} \hat a_{kl} \delta_{ij} \right) + \left( C_3 - C_3^* \sqrt{\hat a_{kl} \hat a_{kl}} \right) \overline \rho \hat k \hat S_{ij}^* + C_4 \overline \rho \hat k \left( \hat a_{ik} \hat S_{jk} + \hat a_{jk} \hat S_{ik} - \frac{2}{3} \hat a_{kl} \hat S_{kl} \delta_{ij} \right) + C_5 \overline \rho \hat k \left( \hat a_{ik} \hat W_{jk} + \hat a_{jk} \hat W_{ik} \right)$

where pressure dilatation is neglected, and the anisotropy tensor is:

$\hat a_{ij} = \frac{\hat R_{ij}}{\hat k} - \frac{2}{3} \delta_{ij}$

The pressure-strain coefficients are blended (as described below) between Launder-Reece-Rodi (LRR) (J. Fluid Mech. vol. 68, no. 3, 1975, pp. 537-566, https://doi.org/10.1017/S0022112075001814) near walls (without wall-correction terms) and Speziale-Sarkar-Gatski (SSG) (J. Fluid Mech. vol. 227, 1991, pp. 245-272, https://doi.org/10.1017/S0022112091000101) away from walls. Also:

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

$\hat S_{ij}^* = \hat S_{ij} - \frac{1}{3} \hat S_{kk} \delta_{ij}$

$\hat W_{ij} = \frac{1}{2} \left( \frac{\partial \hat u_i}{\partial x_j} - \frac{\partial \hat u_j}{\partial x_i} \right)$

Neglecting the pressure diffusion component, the diffusion term is modeled via a generalized gradient diffusion model (Daly and Harlow, Phys Fluids 13:2634-2649, 1970, http://doi.org/10.1063/1.1692845):

$\overline \rho D_{ij} = \frac{\partial}{\partial x_k} \left[ \left( \overline \mu \delta_{kl} + D \frac{\overline \rho \hat k \hat R_{kl}}{\varepsilon} \right) \frac{\partial \hat R_{ij}}{\partial x_l} \right] = \frac{\partial}{\partial x_k} \left[ \left( \overline \mu \delta_{kl} + D \frac{\overline \rho \hat R_{kl}}{C_{\mu} \omega} \right) \frac{\partial \hat R_{ij}}{\partial x_l} \right]$

The fluctuating mass flux $\overline \rho M_{ij}$ is neglected.

All of the coefficients are blended (similar to Menter's SST model), via:

$\phi = F_1 \phi^{(\omega)} + (1-F_1) \phi^{(\varepsilon)}$

$F_1 = tanh(\zeta^4)$

$\zeta = {\rm min} \left[ {\rm max} \left( \frac{\sqrt{\hat k}}{C_{\mu} \omega d} , \frac{500 \hat \mu}{\overline \rho \omega d^2} \right) , \frac{4 \sigma_{\omega}^{(\varepsilon)}\overline \rho \hat k} {(CD) d^2} \right]$

$(CD) = \sigma_d^{(\varepsilon)} \frac{\overline \rho}{\omega} {\rm max} \left( \frac{\partial \hat k}{\partial x_k} \frac{\partial \omega}{\partial x_k}, 0 \right)$

Here, d is the distance to the nearest wall. The coefficients are as follows:

$C_{\mu} = 0.09$

The inner (superscript "( $\omega$ )") (near-wall) coefficients are:

$\alpha_{\omega}^{(\omega)} = 0.5556$ $\beta_{\omega}^{(\omega)} = 0.075$

$\sigma_{\omega}^{(\omega)} = 0.5$ $\sigma_d^{(\omega)} = 0$

$C_1^{(\omega)} = 1.8$ $C_1^{*(\omega)} = 0$

$C_2^{(\omega)} = 0$ $C_3^{(\omega)} = 0.8$

$C_3^{*(\omega)} = 0$ $C_4^{(\omega)} = 0.5 (18 C_2^{(LRR)} + 12) / 11$

$C_5^{(\omega)} = 0.5 (-14 C_2^{(LRR)} + 20)/11$ $D^{(\omega)} = 0.75 C_{\mu}$

$C_2^{(LRR)} = 0.52$

(Note that the $C_2^{(LRR)}$ value has been modified to achieve better agreement with the log layer of a zero pressure gradient boundary layer.)

The outer (superscript "( $\varepsilon$ )") coefficients are:

$\alpha_{\omega}^{(\varepsilon)} = 0.44$ $\beta_{\omega}^{(\varepsilon)} = 0.0828$

$\sigma_{\omega}^{(\varepsilon)} = 0.856$ $\sigma_d^{(\varepsilon)} = 1.712$

$C_1^{(\varepsilon)} = 1.7$ $C_1^{*(\varepsilon)} = 0.9$

$C_2^{(\varepsilon)} = 1.05$ $C_3^{(\varepsilon)} = 0.8$

$C_3^{*(\varepsilon)} = 0.65$ $C_4^{(\varepsilon)} = 0.625$

$C_5^{(\varepsilon)} = 0.2$ $D^{(\varepsilon)} = 0.22$

Boundary conditions were not given in the original reference. However, in the farfield they are (ref: private communication with the author):

$\hat R_{ij, farfield} = \frac{2}{3} \hat k_{farfield} \delta_{ij}$ $\omega_{farfield} = \frac{\overline \rho \hat k_{farfield}}{\mu_{t, farfield}}$

with the $\hat k_{farfield}$ and $\mu_{t, farfield}$ set by the user. The former value is a function of farfield turbulence intensity, Tu ( $\hat k_{farfield} = (3/2) (Tu)^2 U_{farfield}^2$ ). Typical values are: Tu=0.001 (= 0.1%), $\mu_{t, farfield}/\overline \mu_{farfield} = 0.1$ . At solid walls:

$\hat R_{ij, wall} = 0$ $\omega_{wall} = 10 \frac{6 \hat \nu}{\beta_{\omega}^{(\omega)} (\Delta d_1)^2}$

where $\Delta d_1$ is the distance from the wall to the nearest field solution point. This latter boundary condition is the same as that recommended in Menter, F. R., AIAA Journal, Vol. 32, No. 8, August 1994, pp. 1598-1605, https://doi.org/10.2514/3.12149.

Boundary conditions for RSM are important to handle correctly at symmetry boundaries:

On an x-symmetry plane, the 12 (xy) and 13 (xz) Reynolds stress components should have Dirichlet conditions of zero.
On a y-symmetry plane, the 12 (xy) and 23 (yz) Reynolds stress components should have Dirichlet conditions of zero.
On a z-symmetry plane, the 13 (xz) and 23 (yz) Reynolds stress components should have Dirichlet conditions of zero.

Other Reynolds stress components should receive the usual symmetric treatment (i.e., zero gradient).

Note that the Reynolds stresses should adhere to the following realizability conditions (see, e.g., J. Fluid Mech. (1994), vol. 278, pp. 351-362, https://doi.org/10.1017/S0022112094003745):

For the diagonal elements:

$\overline {\rho u_i'' u_i''} \geq 0$ $i \in {1,2,3}$ (no summation on i)
For the off-diagonal elements (i not equal to j):

$\left|\overline {\rho u_i'' u_j''}\right| \leq \sqrt{\left(\overline {\rho u_i'' u_i''}\right) \left(\overline {\rho u_j'' u_j''}\right)}$ $i,j \in {1,2,3}$ (no summation on i or j)

Regarding additional modeled terms appearing in the Favre-averaged equations (see Implementing Turbulence Models into the Compressible RANS Equations), the turbulent heat flux is modeled via:

$c_p \overline {\rho u_j'' T''} \approx -\frac{c_p \mu_t}{Pr_t} \frac{\partial \hat T}{\partial x_j}$

where $\mu_t$ is obtained via an "equivalent eddy viscosity": $\mu_t=\overline \rho \hat k / \omega$ (ref: private communication with the author).

The terms associated with molecular diffusion and turbulent transport in the energy equation are modeled as half the trace of the $\overline \rho D_{ij}$ term. For the generalized gradient diffusion model, this is:

$\frac{\partial}{\partial x_k} \left( \overline{\sigma_{ij} u_i''} - \frac{1}{2} \overline{\rho u_i'' u_i'' u_j''} \right) \approx \frac{1}{2} \left( \overline \rho T_{ii} + \overline \rho D_{ii}^{(\nu)} \right)$

where:

$\overline \rho T_{ij} = \frac{\partial}{\partial x_k} \left( D \frac{\overline \rho \hat k \hat R_{kl}}{\varepsilon} \frac{\partial \hat R_{ij}}{\partial x_l} \right)$

$\overline \rho D_{ij}^{(\nu)} = \frac{\partial}{\partial x_k} \left( \overline \mu \frac{\partial \hat R_{ij}}{\partial x_k} \right)$

SSG/LRR-omega Full Reynolds Stress Model with Simple Diffusion (SSGLRR-RSM-w2012-SD)

This model is the same as (SSGLRR-RSM-w2012), except it uses a "simple diffusion" (SD) model rather than the generalized gradient diffusion model. Its main reference is:

Eisfeld, B., "Implementation of Reynolds Stress Models into the DLR-FLOWer Code," Institutsbericht, DLR-IB 124-2004/31, Report of the Institute of Aerodynamics and Flow Technology, Braunschweig, ISSN 1614-7790, 2004.

The diffusion term is modeled via:

$\overline \rho D_{ij} = \frac{\partial}{\partial x_k} \left[ \left( \overline \mu + D \frac{\overline \rho \hat k^2}{\varepsilon} \right) \frac{\partial \hat R_{ij}}{\partial x_k} \right] = \frac{\partial}{\partial x_k} \left[ \left( \overline \mu + D \frac{\overline \rho \hat k}{C_{\mu} \omega} \right) \frac{\partial \hat R_{ij}}{\partial x_k} \right] = \frac{\partial}{\partial x_k} \left[ \left( \overline \mu + \frac{D}{C_{\mu}} \mu_t \right) \frac{\partial \hat R_{ij}}{\partial x_k} \right]$

with:

$D = 0.5 C_{\mu} F_1 + \frac{2}{3} 0.22 (1-F_1)$

SSG/LRR-omega Full Reynolds Stress Model V2019 (SSGLRR-RSM-w2019)

This model is identical to the original model SSGLRR-RSM-w2012, except that an additional term involving a length scale correction (LSC) is included in the omega equation, for the purpose of eliminating a non-physical "backbending" seen in the original model near reattachment of separated flows. Its reference is:

Eisfeld, B. and Rumsey, C. L., "Length-Scale Correction for Reynolds-Stress Modeling," AIAA Journal, Vol. 58, No. 4, 2020, pp. 1518-1528; https://doi.org/10.2514/1.J058858.

In the original model, the following term in the omega equation:

$\beta_{\omega} \overline\rho \omega^{2}$

gets replaced with:

$\left[ 1 - F^{(LSC)}(\chi) \right] \beta_{\omega} \overline\rho \omega^{2}$

where

$F^{(LSC)}(\chi) = \frac{1}{2} \left\{ 1 + \tanh \left[ A \left( \chi - \chi_{T} \right) \right] \right\}$

and

$\chi = \max \left[ \left( \frac{{\cal L}_{t}}{\ell_{log}} - 1 \right) \left( \frac{{\cal L}_{t}}{\ell_{log}} \right)^{2} , 0 \right]$

and

$\frac{{\cal L}_{t}}{\ell_{log}} = \frac{k^{1/2}}{C_{\mu}^{1/4} \kappa \omega d}$

with d=distance to the nearest wall, $A=31$ , $\chi_{T} = 1$ , and $\kappa = 0.41$ . The $F^{(LSC)}(\chi)$ term should be active only near stagnation/reattachment points. When $F^{(LSC)}(\chi)$ is zero, the original model is recovered.

SSG/LRR-omega Full Reynolds Stress Model V2019 with Simple Diffusion (SSGLRR-RSM-w2019-SD)

This model is the same as (SSGLRR-RSM-w2019), except it uses the "simple diffusion" (SD) model rather than the generalized gradient diffusion model (see description under (SSGLRR-RSM-w2012-SD)).

Return to: Turbulence Modeling Resource Home Page

Recent significant updates:
01/25/2022 - removed "/" from naming convention because it sometimes causes problems/confusion
12/11/2019 - added description of new SSG/LRR-RSM-w2019 and SSG/LRR-RSM-w2019-SD
03/24/2016 - mention realizability of Reynolds stresses

Privacy Act Statement

Accessibility Statement

Responsible NASA Official: Ethan Vogel
Page Curator: Clark Pederson
Last Updated: 01/25/2022