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Turbulence Modeling Resource

Transitional SSG/LRR Reynolds Stress Model

This web page gives detailed information on the equations for various forms of the transitional SSG/LRR Reynolds stress model, which is a blend between Langtry-Menter γ-Reθt equations and full second-moment Reynolds stress equations. 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.

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Transitional SSG/LRR Reynolds Stress Model (SSG/LRR-trans)

This model is developed by coupling of the SSG/LRR-w2012-SD full turbulence model with the Langtry-Menter γ-Reθt model. The primary reference for the implementation of the 9-equation transitional RSM is:

Shengyang Nie, Normann Krimmelbein, Andreas Krumbein, et al. "Coupling of a Reynolds Stress Model with the γ−Reθt Transition Model," AIAA Journal, Vol. 56, No. 1, September 2017, pp. 146-157, https://doi.org/10.2514/1.J056167.

Note that the SSG/LRR-ω model almost shares the same length-scale determining equation as the Menter’s k-ω SST model, except the production term. The production of ω in Langtry-Menter’s 4-equation model is linear to the square of the mean strain rate. In contrast to the ω-based RSM coupling with γ-Reθt, the ω production does not vanish in the fully laminar region. Along with this, an inverse-ω scale variable( $\lambda=1/\sqrt[8]{\omega}$ ) is recommended by:

Shunshun Wang, Shengye Wang, Xiang Fu, et al. "A Wall-Boundary-Natural Transitional Reynolds-Stress Model for High-Order Wing-Body Simulations," Physics of Fluids, Vol. 36, No. 8, July 2024, pp. 084-115, https://doi.org/10.1063/5.0219939.

The λ equals 0 strictly on the viscous wall and so avoids the invalidity of generation term upstream the transition onset. Based on this, the model is given by the following:

$\begin{gathered} \frac{\partial\left(\bar{\boldsymbol{\rho}}\hat{R}_{ij}\right)}{\partial t}+\frac{\partial\left(\bar{\boldsymbol{\rho}}\hat{u}_k\hat{R}_{ij}\right)}{\partial x_k}=\bar{\boldsymbol{\rho}}P_{ij}+\bar{\boldsymbol{\rho}}\Pi_{ij}-\bar{\boldsymbol{\rho}}\varepsilon_{ij}+\bar{\boldsymbol{\rho}}D_{ij} \\ \frac{\partial(\bar{\rho}\lambda)}{\partial t}+\frac{\partial(\bar{\rho}\hat{u}_k\lambda)}{\partial x_k}=\bar{\rho}P_\lambda-\bar{\rho}\varepsilon_\lambda+\bar{\rho}D_\lambda+\bar{\rho}C_{D\lambda}+\bar{\rho}G_\lambda \\ \frac{\partial(\bar{\boldsymbol{\rho}}\boldsymbol{\gamma})}{\partial t}+\frac{\partial(\bar{\boldsymbol{\rho}}\hat{\boldsymbol{u}}_j\boldsymbol{\gamma})}{\partial x_j}=P_{\boldsymbol{\gamma}}-E_{\boldsymbol{\gamma}}+\frac{\partial}{\partial x_j}\left[\left(\boldsymbol{\mu}+\frac{\boldsymbol{\mu}_t}{\boldsymbol{\sigma}_f}\right)\frac{\partial\boldsymbol{\gamma}}{\partial x_j}\right] \\ \frac{\partial\left(\bar{\boldsymbol{\rho}}\tilde{R}e_{\boldsymbol{\theta}t}\right)}{\partial t}+\frac{\partial\left(\bar{\boldsymbol{\rho}}\hat{u}_{j}\tilde{R}e_{\boldsymbol{\theta}t}\right)}{\partial x_{j}}=P_{\boldsymbol{\theta}t}+\frac{\partial}{\partial x_{j}}\left[\boldsymbol{\sigma}_{\boldsymbol{\theta}t}\left(\boldsymbol{\mu}+\boldsymbol{\mu}_{t}\right)\frac{\partial\tilde{R}e_{\boldsymbol{\theta}t}}{\partial x_{j}}\right] \end{gathered}$

The source terms in Reynolds stress equations are defined as:

$\bar{\rho}P_{ij}=-\bar{\rho}\hat{R}_{ik}\frac{\partial\hat{u}_j}{\partial x_k}-\bar{\rho}\hat{R}_{jk}\frac{\partial\hat{u}_i}{\partial x_k}$

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

$\bar{\rho}D_{ij}=\frac{\partial}{\partial x_k}\left[\left(\mu+\frac{D}{C_\mu}\mu_t\right)\frac{\partial\hat{R}_{ij}}{\partial x_k}\right]$

$\begin{aligned} \bar{\rho}\Pi_{ij}=& \begin{aligned}&-\left(C_1\bar{\rho}\varepsilon+\frac{1}{2}C_1^*\bar{\rho}P_{kk}\right)\hat{a}_{ij}+C_2\bar{\rho}\varepsilon\left(\hat{a}_{ik}\hat{a}_{kj}-\frac{1}{3}\hat{a}_{kl}\hat{a}_{kl}\delta_{ij}\right)\end{aligned}+\left(C_3-C_3^*\sqrt{\hat{a}_{kl}\hat{a}_{kl}}\right)\bar{\rho}\hat{k}\hat{S}_{ij}^*+C_4\bar{\rho}\hat{k}\left(\hat{a}_{ik}\hat{S}_{jk}+\hat{a}_{jk}\hat{S}_{ik}-\frac23\hat{a}_{kl}\hat{S}_{kl}\delta_{ij}\right)+C_5\bar{\rho}\hat{k}\left(\hat{a}_{ik}\hat{W}_{jk}+\hat{a}_{jk}\hat{W}_{ik}\right) \end{aligned}$

The dissipation rate as well as turbulence viscosity are as follows:

$\varepsilon=C_\mu\hat{k}/\lambda^8,\quad\mu_t=\bar{\rho}\hat{k}\lambda^8$

The boundary conditions of $R_{ij}$ at the solid wall and in the far-field are given as:

$\hat{R}_{ij,\mathrm{wall}}=0,\hat{R}_{ij,\infty}=\frac23\hat{k}_\infty\delta_{ij}$

The details of the λ-scale transport equation are as follows:

$\frac{\partial \left( \bar{\rho }\lambda \right)}{\partial t}+\frac{\partial \left( \bar{\rho }{{{\hat{u}}}_{k}}\lambda \right)}{\partial {{x}_{k}}}=-\text{8}\frac{{{\alpha }_{\omega }}\lambda }{\hat{k}}\frac{\bar{\rho }{{\text{P}}_{\text{k}k}}}{2}+\text{8}{{\beta }_{\omega }}\bar{\rho }{{\lambda }^{\frac{7}{8}}}+\frac{\partial }{\partial {{x}_{k}}}[\left( \mu +{{\sigma }_{\omega }}{{\mu }_{t}} \right)\frac{\partial \lambda }{\partial {{x}_{k}}}]+{{\sigma }_{d}}\bar{\rho }{{\lambda }^{\frac{1}{8}}}\text{min}(\frac{\partial \hat{k}}{\partial {{x}_{k}}}\frac{\partial \lambda }{\partial {{x}_{k}}},0)-\frac{9}{8\lambda }(\mu +{{\sigma }_{\omega }}{{\mu }_{t}})\frac{\partial \lambda }{\partial {{x}_{k}}}\frac{\partial \lambda }{\partial {{x}_{k}}}]$

The boundary conditions of λ at the solid wall and in the far-field are given as:

$\lambda_{\text {wall }}=0, \quad \lambda_{\text {far field }}=\sqrt[8]{\frac{\mu_{t, \text { far field }}}{\bar{\rho} \hat{k}_{\text {far field }}}}$

In the above, k is the turbulence kinetic turbulence energy, $\hat{k} =\hat{R}_{ii}/2$ , the hat represents the Favre average, $\hat{\phi}=\overline{\rho\phi}/\bar{\rho}$ , $\bar{\rho }$ is the non-weighted averaging density, $\mu$ is the molecular dynamic viscosity, $d$ is the distance from the field point to the nearest wall, $S=\sqrt{2 \hat{S}_{i j} \hat{S}_{i j}}$ is the strain rate magnitude, and $\Omega=\sqrt{2 \hat{W}_{i j}\hat{W}_{i j} }$ is the vorticity magnitude, with

$\hat{S}_{i j}=\frac{1}{2}\left(\frac{\partial \hat{u}_i}{\partial x_j}+\frac{\partial \hat{u}_j}{\partial x_i}\right)$

$\hat{W}_{i j}=\frac{1}{2}\left(\frac{\partial \hat{u}_i}{\partial x_j}-\frac{\partial \hat{u}_j}{\partial x_i}\right)$

$\hat{a}_{i j}=\frac{\hat{R}_{i j}}{\hat{k}}-\frac{2}{3} \delta_{i j}$

$\hat{S}_{ij}^*=\hat{S}_{ij}-\frac13\hat{S}_{kk}\delta_{ij}$

Moreover, all the coefficients of the turbulence model are the same as the SSG/LRR-w2012-SD model.

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

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

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

where

$F_{onset1} = \frac{Re_V}{2.193 Re_{\theta c}}$

$Re_V = \frac{\bar{\rho } S d^2}{\mu}$

$F_{onset2} = min \left( max \left( F_{onset1},F_{onset1}^4 \right), 2.0 \right)$

$R_T=\frac{\bar{\rho }\hat{k} }{\mu \omega}$

$F_{onset3} = max \left[ 1 - \left( \frac{R_T}{2.5} \right)^3, 0 \right]$

$F_{onset} = max \left(F_{onset2} - F_{onset3}, 0 \right)$

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

$F_{length} = F_{length,1} \left( 1 - F_{sublayer} \right) + 40.0 F_{sublayer}$

$F_{\text {length }, 1}= \begin{cases}39.8189+\left(-119.270 \cdot 10^{-4}\right) \hat{R} e_{\theta t}+\left(-132.567 \cdot 10^{-6}\right) \hat{R} e_{\theta t}^2, & \hat{R} e_{\theta t}<400 ; \\ 263.404+\left(-123.939 \cdot 10^{-2}\right) \hat{R} e_{\theta t}+\left(194.548 \cdot 10^{-5}\right) \hat{R} e_{\theta t}^2+\left(-101.695 \cdot 10^{-8}\right) \hat{R} e_{\theta t}^3, & 400 \leq \hat{R} e_{\theta t}<596 ; \\ 0.5-\left(3.0 \cdot 10^{-4}\right)\left(\hat{R} e_{\theta t}-596.0\right), & 596 \leq \hat{R} e_{\theta t}<1200 ; \\ 0.3188 & 1200 \leq \hat{R} e_{\theta t}\end{cases}$

${{F}_{sublayer}}=exp\left[ -{{\left( \frac{R{{e}_{\lambda }}}{200} \right)}^{2}} \right]$

$R{{e}_{\lambda }}=\frac{\bar{\rho } {{\lambda }^{8}}{{d}^{2}}}{\mu }$

The momentum thickness Reynolds number Re_θc is recalibrated for the RSM:

$Re_{\theta c}=f_{\theta c}\tilde{R}e_{\theta t}, f_{\theta c}=0.99-0.37\Big\{1-\exp\Big[-\max\Big(0,\frac{\tilde{R}e_{\theta t}+40}{320}\Big)\Big]\Big\}^2$

The source term of the $\tilde Re_{\theta t}$ equation is defined as:

$P_{\theta t} = c_{\theta t} \frac{\bar{\rho}}{T} \left( Re_{\theta t} - \tilde Re_{\theta t} \right) \left( 1.0 - F_{\theta t} \right)$

for which

$T=\frac{500 \mu}{\bar{\rho }U^2}$

$U = \sqrt{ \hat{u}_k \hat{u}_k}$

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

$\delta=\frac{375\Omega\mu\tilde{R}e_{\theta t}d}{\bar{\rho }U^2}$

$F_{wake} = exp \left[ - \left( \frac{Re_{\omega}}{1\cdot 10^{5}} \right) ^2 \right]$

$\lambda_\theta=\frac{\bar\rho \theta_t^2}{\mu} \frac{d U}{d s}$

$Tu=100\frac{\sqrt{2\hat{k}/3}}{U}$

$\frac{dU}{ds} = \frac{\hat{u}_m \hat{u}_n}{U^2} \frac{\partial \hat{u}_m}{\partial x_n}$

${{\operatorname{Re}}_{\theta t}}=\left\{ \begin{array}{*{35}{l}} \left[ 1173.51-589.428Tu-8T{{u}^{3}}+\frac{0.2196}{T{{u}^{2}}} \right]F\left( {{\lambda }_{\theta }} \right), & Tu\le %1.5 \\ \left[ 240{{(Tu+0.01)}^{0.64+0.7Tu}}+150 \right]F\left( {{\lambda }_{\theta }} \right), & Tu>%1.5 \\ \end{array} \right.$

$F \left( \lambda_{\theta} \right) = \left\{ \begin{array}{ll} 1 + \left[ 12.986 \lambda_{\theta} + 123.66 \lambda_{\theta} ^2 + 405.689 \lambda_{\theta} ^3 \right] exp \left( -\left( \frac{Tu}{1.5} \right)^{1.5} \right), & \lambda_{\theta} \leq 0; \\ 1 + 0.275 \left[1 - exp \left( -35.0 \lambda_{\theta} \right) \right] exp \left( - \frac{Tu}{0.5} \right) & \lambda_{\theta} > 0 \end{array} \right.$

IMPORTANT: The expression for $Re_{\theta t}$ is an implicit function of $\theta_t$ through the presence of $\lambda_{\theta}$ since

$Re_{\theta t}=\frac{\bar{\rho}U\theta_t}{\mu}$

(the equations for $Re_{\theta t}$ are typically solved by iterating on the value of $\theta_t$ ). Note that in the nomenclature of AIAA Journal 47(12):2894-2906, 2009, the expression for $Re_{\theta t}$ uses a "local freestream velocity" for U, which is actually intended to be the velocity at the edge of the boundary layer. But in the functionality of the model, this velocity needs to be the local velocity. Although $Re_{\theta t}$ is small for small velocities (i.e. near the wall in a boundary layer), this effect is accounted for in the model with the F_theta_t term. Outside of the boundary layer, the transported $\tilde Re_{\theta t}$ is "attracted" to the equilibrium value ( $Re_{\theta t}$ ) and is physically correct at the edge of the boundary layer. Inside the boundary layer, the attraction is suppressed and the value at the edge is diffused into the boundary layer.

The calibration constants for the Langtry-Menter model are:

$\begin{array}{l} c_{a1} = 2.0 \\ c_{a2} = 0.06 \\ c_{e1} = 1.0 \\ c_{e2} = 50 \\ c_{\theta t} = 0.03 \\ s_1 = 2 \\ \sigma_{f} = 1.0 \\ \sigma_{\theta t} = 2.0 \end{array}$

The boundary conditions for $\gamma$ and $\tilde Re_{\theta t}$ are:

$\frac{\partial \gamma}{\partial n} \vert _{wall} = 0$

$\gamma_{farfield} = 1$

$\frac{\partial \tilde Re_{\theta t}}{\partial n} \vert _{wall} = 0$

$\tilde{R} e_{\theta t, \text { far field }}= \begin{cases}1173.51-589.428 T u-8 T u^3+\frac{0.2196}{T u^2}, & T u \leq 1.5 \% \\ 240(T u+0.01)^{0.64+0.7 T u}+150, & T u>1.5 \%\end{cases}$

The effects of laminar-turbulent transition are introduced to the underlying SSG/LRR model by modifying the source terms as:

$\begin{aligned} &\tilde{P}_{ij}=\gamma_{\mathrm{eff}}P_{ij} \\ &\tilde\varepsilon _{ij}=\operatorname*{min}\left(1,\operatorname*{max}\left(\gamma_{\mathrm{eff}},0.1\right)\right)\varepsilon _{ij} \\ &\tilde{\Pi}_{ij}=\gamma_{\mathrm{eff}}\Pi_{ij} \end{aligned}$

For numerical robustness, the following three limits are enforced:

$-0.1 \leq \lambda_{\theta} \leq 0.1$

$Tu \geq 0.027$

$Re_{\theta t} \geq 20$

This model is not Galilean invariant, due to its explicit use of the velocity vector.

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Shengye Wang is acknowledged for helping with this webpage.

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Last Updated: 04/08/2025