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Turbulence Modeling Resource

GLVY stress-epsilon Full Reynolds-stress Model

This web page gives detailed information on the equations for the GLVY full second-moment Reynolds-stress model that uses an $\varepsilon^*$ equation for the scale-determining variable. Full second-moment Reynolds-stress models are very different from simpler 1- and 2-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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GLVY stress-epsilon Full Reynolds-stress Model (GLVY-RSM-2012)

This model's reference is:

Gerolymos G.A., Lo C., Vallet I., Younis B.A.: "Term-by-term analysis of near-wall second-moment closures" AIAA J. 50(12) (2012) 2848-2864 https://doi.org/10.2514/1.J051654.

This model is the latest evolution of the GV-RSM-2001 [Gerolymos G.A., Vallet I.: AIAA J. 39(10) (2001) 1833-1842 https://doi.org/10.2514/2.1179], and is fully wall-normal-free (the wall-normal distance or direction are not used in the model). This modeling choice renders inhomogeneous terms active also away from solid walls, and indeed independently of the presence of walls (inhomogeneous terms are also active in free flows and should never be dropped when using this model).

The variables of the model are the Favre-avergaged second-moments of velocity $R_{ij}:=\widetilde{u_i''u_j''}$ and the modified dissipation rate $\varepsilon^*$ . Using the notation of this website, in relation to $\tau_{ij}$ which is used in the mean-flow equations, as described on the page Implementing Turbulence Models into the Compressible RANS Equations), we have:

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

Notations are different from the original reference, to conform with this website's practices, sometimes driven by html rendering quality. The symbol $R_{ij}$ is used instead of $\hat R_{ij}$ used in the description of the SSG/LRR-RSM-w2012, or of $-R'_{ij}$ used in the description of the WilcoxRSM-w2006, to avoid any ambiguity with the use of "hat" (Favre-averaging nonlinear operator) or of "prime" (Reynolds-fluctuation linear operator).

The model equations for $R_{ij}$ and $\varepsilon^*$ are (terms in blue are exact, terms in black are modeled; Eq. numbers in red are closures of various terms; in the $\varepsilon^*$ -equation RHS-terms are already modeled)

$({\rm GLVY}.1)\qquad {\color{blue}{\underbrace{\frac{\partial}{\partial t}\left(\overline{\rho}R_{ij}\right) +\frac{\partial}{\partial x_\ell}\left(\overline{\rho}R_{ij}\hat{u}_\ell\right)}_{\displaystyle C_{ij}}}}= {\color{blue}{\underbrace{-\overline{\rho}R_{i\ell}\frac{\partial \hat{u}_j}{\partial x_\ell} -\overline{\rho}R_{j\ell}\frac{\partial \hat{u}_i}{\partial x_\ell}}_{\displaystyle P_{ij}}}}+ {\color{blue}{\underbrace{\frac{\partial}{\partial x_\ell}\left(\breve{\mu}\frac{\partial R_{ij}}{\partial x_\ell}\right)}_{\displaystyle d_{ij}^{(\mu)}}}} +d_{ij}^{(u)}+\Pi_{ij}-\overline{\rho}\varepsilon_{ij}+K_{ij}$

$({\rm GLVY}.2)\qquad {\color{blue}{ \frac{\partial\bar\rho\varepsilon^*} {\partial t }+\frac{\partial\left(\hat u_\ell\bar\rho\varepsilon^*\right)} {\partial x_\ell }}}= \frac{\partial } {\partial x_\ell}\left[C_\varepsilon\frac{\mathrm{k} } {\varepsilon^*}\bar\rho R_{m\ell}\frac{\partial\varepsilon^*} {\partial x_m } +{\color{blue}{\breve\mu\frac{\partial\varepsilon^*} {\partial x_\ell }}} \right]+C_{\varepsilon 1} P_\mathrm{k}\frac{\varepsilon^*} {\mathrm{k} } -C_{\varepsilon 2}\bar\rho\frac{{\varepsilon^*}^2}{\mathrm{k}} +2 \breve\mu C_{\mu} \frac{{\rm k}^2}{\varepsilon^*} \frac{\partial^2 \hat{u}_i}{\partial x_\ell \partial x_\ell} \frac{\partial^2 \hat{u}_i}{\partial x_m \partial x_m}$

$({\rm GLVY}.3)\qquad P_{\rm k} = \frac{1}{2} P_{\ell\ell}\;\; ;\;\; C_\varepsilon=0.18\;\; ;\;\; C_{\varepsilon1}=1.44 \;\; ;\;\; C_{\varepsilon2}=1.92(1-0.3\mathrm{e}^{-{Re^*_{\mathrm{\tiny T}}}^2})\;\; ; \;\; C_\mu=0.09\mathrm{e}^{-\frac{3.4 } {(1+0.02 Re^*_{\mathrm{\tiny T}})^2}}$

$({\rm GLVY}.4)\qquad \mathrm{k}=\frac{1}{2}R_{\ell\ell}\;\; ; \;\; Re_{\mathrm{\tiny T}}^* =\frac{\mathrm{k}^2 } {\breve\nu\varepsilon^*}\;\; ; \;\; \breve\mu:=\mu_{\rm Sutherland}(\hat T)\;\; ; \;\; \breve\nu:=\frac{\breve\mu}{\bar\rho}$

where $C_{ij}$ is the convection tensor, $P_{ij}$ is the production tensor, $d^{(\mu)}_{ij}$ is the molecular-diffusion tensor, $d^{(u)}_{ij}$ is the turbulent-diffusion tensor (by the fluctuating velocities), $\Pi_{ij}$ are the terms containing the fluctuating pressure (velocity/pressure-gradient correlation), $\varepsilon_{ij}$ is the rate-of-dissipation tensor, and $K_{ij}$ are direct compressibility effects (terms proportional to the fluctuating density transport $\overline{u_i''}=\overline{u}_i-\hat{u}_i=-\bar{\rho}^{-1}\overline{\rho'u_i'}$ , hence to the difference between Reynolds and Favre averaged velocities).

The modeled terms in (GLVY.1) ( $K_{ij}$ , $d^{(u)}_{ij}$ , $\varepsilon_{ij}$ , and $\Pi_{ij}$ ) are given by

${\color{red}{({\rm GLVY}.5)}}\qquad K_{ij}=0$

${\color{red}{({\rm GLVY}.6)}}\qquad d_{ij}^{(u)}=\frac{\partial}{\partial x_\ell}\left(-\overline{\rho u_i'' u_j'' u_\ell''}\right)\;\; ; \;\; -\overline{\rho u_i'' u_j'' u_\ell''}=C^{(\mathrm{\tiny S}u)}\frac{\mathrm{k} } {\varepsilon}\left(\bar\rho R_{im} \frac{\partial R_{j\ell} }{\partial x_m} +\bar\rho R_{jm} \frac{\partial R_{\ell i}}{\partial x_m} +\bar\rho R_{\ell m}\frac{\partial R_{ij} }{\partial x_m}\right)\;\; ; \;\; C^{(\mathrm{\tiny S}u)}=0.11$

${\color{red}{({\rm GLVY}.7)}}\qquad \bar\rho\varepsilon_{ij}=\frac{2}{3}\bar\rho\varepsilon\left(1-f_\varepsilon\right)\delta_{ij}+f_\varepsilon\frac{\varepsilon}{\mathrm{k}}\bar\rho R_{ij}\;\; ; \;\; \varepsilon=\varepsilon^*+2\breve\nu\frac{\partial \sqrt{\mathrm{k}}}{\partial x_\ell} \frac{\partial \sqrt{\mathrm{k}}}{\partial x_\ell}\;\; ; \;\; f_\varepsilon=1-A^{[1+A^2]}\left[1-\mathrm{e}^{-\frac{Re_{\mathrm{\tiny T}}}{10}}\right]\;\; ; \;\; Re_{\mathrm{\tiny T}} =\frac{\mathrm{k}^2 } {\breve\nu\varepsilon}$

$({\rm GLVY}.8)\qquad a_{ij}:=\frac{R_{ij}}{\mathrm{k}}-\frac{2}{3}\delta_{ij}\;\; ; \;\; A_2:=a_{ik}a_{ki}\;\; ; \;\; A_3:=a_{ik}a_{kj}a_{ji}\;\; ; \;\; A:=1-\frac{9}{8}(A_2-A_3)$

${\color{red}{({\rm GLVY}.9)}}\qquad \Pi_{ij}=\underbrace{\overbrace{\phi_{ij}^{(\mathrm{\tiny RH})}+\phi_{ij}^{(\mathrm{\tiny RI})}}^{\displaystyle\phi_{ij}^{(\mathrm{\tiny R})}}+ \overbrace{\phi_{ij}^{(\mathrm{\tiny SH})}+\phi_{ij}^{(\mathrm{\tiny SI})}}^{\displaystyle\phi_{ij}^{(\mathrm{\tiny S})}}}_{\displaystyle\phi_{ij}}+\frac{2}{3}\phi_p\delta_{ij}+d_{ij}^{(p)}$

${\color{red}{({\rm GLVY}.10)}}\qquad \phi_p=0$

${\color{red}{({\rm GLVY}.11)}}\qquad d_{ij}^{(p)}=C^{(\mathrm{\tiny Sp1})}\bar\rho\frac{\mathrm{k}^3 } {\varepsilon^3}\frac{\partial\varepsilon^*} {\partial x_i }\frac{\partial\varepsilon^*} {\partial x_j } +\frac{\partial } {\partial x_\ell}\left[C^{(\mathrm{\tiny Sp2})}(\overline{\rho u_m''u_m''u_j''}\delta_{i\ell} +\overline{\rho u_m''u_m''u_i''}\delta_{j\ell})\right] +C^{(\mathrm{\tiny Rp})}\bar\rho\frac{\mathrm{k}^2 } {\varepsilon^2}\breve{S}_{k\ell}a_{\ell k}\frac{\partial\mathrm{k}} {\partial x_i }\frac{\partial\mathrm{k}} {\partial x_j }$

$({\rm GLVY}.12)\qquad C^{(\mathrm{\tiny Sp1})}=-0.005\;\; ; \;\; C^{(\mathrm{\tiny Sp2})}=+0.022\;\; ; \;\; C^{(\mathrm{\tiny Rp })}=-0.005\;\; ; \;\; \breve S_{ij}:=\frac{1}{2}\left(\frac{\partial\hat u_i}{\partial x_j} +\frac{\partial\hat u_j}{\partial x_i}\right)$

${\color{red}{({\rm GLVY}.13)}}\qquad \phi_{ij}^{(\mathrm{\tiny R})}=\underbrace{-C_\phi^{(\mathrm{\tiny RH})}\left(P_{ij}-\frac{1}{3}\delta_{ij} P_{mm}\right)}_{\displaystyle\phi^{(\mathrm{\tiny RH})}_{ij}} \underbrace{+C_\phi^{(\mathrm{\tiny RI})}\left[ \phi^{(\mathrm{\tiny RH})}_{nm}e_{\mathrm{\tiny I}_n}e_{\mathrm{\tiny I}_m}\delta_{ij} -\frac{3}{2}\phi^{(\mathrm{\tiny RH})}_{in}e_{\mathrm{\tiny I}_n}e_{\mathrm{\tiny I}_j} -\frac{3}{2}\phi^{(\mathrm{\tiny RH})}_{jn}e_{\mathrm{\tiny I}_n}e_{\mathrm{\tiny I}_i}\right]}_{\displaystyle\phi^{(\mathrm{\tiny RI})}_{ij}}$

${\color{red}{({\rm GLVY}.14)}}\qquad \phi^{(\mathrm{\tiny S})}_{ij}=\underbrace{-C_\phi^{(\mathrm{\tiny SH1})}\bar\rho\varepsilon^* a_{ij}}_{\displaystyle\phi^{(\mathrm{\tiny SH1})}_{ij}} \underbrace{+C_\phi^{(\mathrm{\tiny SI1})} \frac{\varepsilon^*}{\mathrm{k}}\left[\bar\rho R_{nm}e_{\tsc{i}_n}e_{\tsc{i}_m}\delta_{ij} -\frac{3}{2}\bar\rho R_{ni}e_{\tsc{i}_n}e_{\tsc{i}_j} -\frac{3}{2}\bar\rho R_{nj}e_{\tsc{i}_n}e_{\tsc{i}_i}\right]}_{\displaystyle\phi^{(\mathrm{\tiny SI1})}_{ij}}$

$|\qquad\qquad\qquad\qquad\qquad \underbrace{-C_\phi^{(\mathrm{\tiny SI2})}\bar\rho\frac{\mathrm{k}}{\varepsilon}\frac{\partial\mathrm{k}}{\partial x_\ell} \left[ a_{ik}\frac{\partial R_{kj}}{\partial x_\ell} +a_{jk}\frac{\partial R_{ki}}{\partial x_\ell} -\frac{2}{3}\delta_{ij}a_{mk}\frac{\partial R_{km}}{\partial x_\ell}\right]}_{\displaystyle\phi^{(\mathrm{\tiny SI2})}_{ij}} \underbrace{+C_\phi^{(\mathrm{\tiny SI3})}\left[ \phi^{(\mathrm{\tiny SI2})}_{nm}e_{\tsc{i}_n}e_{\tsc{i}_m}\delta_{ij} -\frac{3}{2}\phi^{(\mathrm{\tiny SI2})}_{in}e_{\tsc{i}_n}e_{\tsc{i}_j} -\frac{3}{2}\phi^{(\mathrm{\tiny SI2})}_{jn}e_{\tsc{i}_n}e_{\tsc{i}_i}\right]}_{\displaystyle\phi^{(\mathrm{\tiny SI3})}_{ij}}$

$({\rm GLVY}.15)\qquad e_{\mathrm{\tiny I}_i}:=\frac{\displaystyle\frac{\partial}{\partial x_i}\Biggl(\frac{\ell_\mathrm{\tiny T}[1-\mathrm{e}^{-{\frac{Re^*_\mathrm{\tiny T}} { 30}}}]} {1+2\sqrt{A_2}+2A^{16}} \Biggr) } {\sqrt{\displaystyle\frac{\partial}{\partial x_\ell}\Biggl(\frac{\ell_\mathrm{\tiny T}[1-\mathrm{e}^{-{\frac{Re^*_\mathrm{\tiny T}} { 30}}}]} {1+2\sqrt{A_2}+2A^{16}} \Biggr) \displaystyle\frac{\partial}{\partial x_\ell}\Biggl(\frac{\ell_\mathrm{\tiny T}[1-\mathrm{e}^{-{\frac{Re^*_\mathrm{\tiny T}} { 30}}}]} {1+2\sqrt{A_2}+2A^{16}} \Biggr) } }\;\; ; \;\; \ell_{\mathrm{\tiny T}}:=\frac{k^\frac{3}{2}}{\varepsilon}$

$({\rm GLVY}.16)\qquad C_\phi^{(\mathrm{\tiny RH})}=\min{\left[1,0.75+1.3\max{[0,A-0.55]}\right]}\;A^{[\max(0.25,0.5-1.3\max{[0,A-0.55]})]}[1-\max(0,1-{\frac{Re_\mathrm{\tiny T}} {50 }})]$

$({\rm GLVY}.17)\qquad C_\phi^{(\mathrm{\tiny RI})}=\max{\left[{\frac{2}{3}-\frac{1}{6C_\phi^{(\mathrm{\tiny RH})}},0}\right]} \sqrt{\frac{\partial}{\partial x_\ell}\Biggl(\frac{\ell_\mathrm{\tiny T}[1-\mathrm{e}^{-\frac{Re^*_\mathrm{\tiny T}} {30 }}]} {1+1.6A_2^{\max(0.6,A)}} \Biggr) \frac{\partial}{\partial x_\ell}\Biggl(\frac{\ell_\mathrm{\tiny T}[1-\mathrm{e}^{-\frac{Re^*_\mathrm{\tiny T}} {30 }}]} {1+1.6A_2^{\max(0.6,A)}} \Biggr) }$

$({\rm GLVY}.18)\qquad C_\phi^{(\mathrm{\tiny SH1})}=3.7 A A_2^{\frac{1}{4}}\left[1- \mathrm{e}^{-\left(\frac{Re_{\mathrm{\tiny T}}} {130 } \right)^2} \right]\;\; ; \;\; C_\phi^{(\mathrm{\tiny SI1})}=\left[-\frac{4}{9}\left(C_\phi^{(\mathrm{\tiny SH1})}-\frac{9}{4}\right)\right]\; \sqrt{\frac{\partial}{\partial x_\ell}\Biggl(\frac{\ell_\mathrm{\tiny T}[1-\mathrm{e}^{-\frac{Re^*_\mathrm{\tiny T}} {30 }}]} {1+2.9\sqrt{A_2}} \Biggr) \frac{\partial}{\partial x_\ell}\Biggl(\frac{\ell_\mathrm{\tiny T}[1-\mathrm{e}^{-\frac{Re^*_\mathrm{\tiny T}} {30 }}]} {1+2.9\sqrt{A_2}} \Biggr) }$

$({\rm GLVY}.19)\qquad C_\phi^{(\mathrm{\tiny SI2})}=0.002\;\; ; \;\; C_\phi^{(\mathrm{\tiny SI3})}= 0.14\;\sqrt{\frac{\partial\ell^*_\mathrm{\tiny T}}{\partial x_\ell} \frac{\partial\ell^*_\mathrm{\tiny T}}{\partial x_\ell}}\;\; ; \;\; \ell_{\mathrm{\tiny T}}^*:=\frac{k^\frac{3}{2}}{\varepsilon^*}$

Concerning the pressure terms, the superscript $(\cdot)^{(\mathrm{\tiny S})}$ denotes slow pressure terms (the original terms that are modeled and the model itself do not depend on the gradients of mean-flow velocity) and the superscript $(\cdot)^{(\mathrm{\tiny R})}$ denotes rapid pressure terms (the original terms that are modeled and the model itself depend linearly on mean-flow velocity-gradients). These apply to pressure-diffusion $d_{ij}^{(p)}$ (the terms with coefficients $C^{(\mathrm{\tiny Sp1})}$ and $C^{(\mathrm{\tiny Sp2})}$ in (GLVY.11) are slow terms, while the term with coefficient $C^{(\mathrm{\tiny Rp })}$ is a rapid one) and to redistribution ( $\phi^{(\mathrm{\tiny S})}_{ij}$ and $\phi^{(\mathrm{\tiny R})}_{ij}$ in (GLVY.9)). Concerning redistribution, the superscript $(\cdot)^{(\mathrm{\tiny H})}$ denotes quasi-homogeneous terms (which do not depend on gradients of $R_{ij}$ or $\varepsilon^*$ ), while the superscript $(\cdot)^{(\mathrm{\tiny I})}$ denotes inhomogeneous terms (which contain gradients of $R_{ij}$ or $\varepsilon^*$ ). All practical flows are inhomogeneous to some extent.

Notes:

The model should be used as a whole, the inhomogeneous terms being active in free-shear flows (there are no correct fixed-points with the homogeneous terms only).
Length scales and turbulence-Reynolds-number are sometimes defined using the system-variable modified dissipation-rate $\varepsilon^*$ (GLVY.2) or the dissipation-rate $\varepsilon$ (GLVY.7). This deliberate choice was made because the behavior of $\varepsilon^*$ or $\varepsilon$ very near the wall is different, so that the corresponding choices impact the asymptotic near-wall behavior (the rate at which each component of $R_{ij}$ approaches 0).
Boundary-conditions at a solid wall are $R_{ij}=0$ and $\varepsilon^*=0$ .
There is no particular limitation for the farfield boundary-conditions, the model behaving very well with both high or low turbulence intensities [Gerolymos G.A., Sauret E., Vallet I.: AIAA J. 42(6) (2004) 1101-1106 https://doi.org/10.2514/1.2257]. It is recommended to choose a physically appropriate turbulence intensity (isotropic or anisotropic), and a length scale $\ell_{\mathrm{\tiny T}_\infty}$ , which, given $\mathrm{k}_\infty$ , fixes $\varepsilon_\infty$ . Just recall that the choice of the length scale fixes the streamwise rate of decay of turbulence, which can be fast, especially at high turbulence intensites, or may be nonegligible if the farfield boundary is a long distance away. There are analytical formulas avialable to compute this decay [Gerolymos G.A., Sauret E., Vallet I.: AIAA J. 42(6) (2004) 1101-1106 https://doi.org/10.2514/1.2257], and the model returns results very close to the analytical formulas. Incidentally, contrary to 1- or 2-equation models, farfield eddy-viscosity is not a relevant parameter.
The pressure terms $\Pi_{ij}$ are the most important part of the model (indeed of second-moment closures in general). This is the major difference with 2-equation models, where only the trace of $\Pi_{ij}$ , appears, and has in general little influence.
Although there are no compressibility corrections other than Morkovin's hypothesis, the model has been validated for shock-wave/boundary-layer interactions up to Mach-number 5.
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).

Implementation into c-RANS: The model-terms required in the page Implementing Turbulence Models into the Compressible RANS Equations are:

The Reynolds stresses are readily obtained from the variables $R_{ij}$ of the model by $\tau_{ij}:=-\overline{\rho u_i''u_j''}=-\overline{\rho}R_{ij}$ .
In the tests run so far, the turbulent heat-flux was obtained by

$c_p\overline{\rho u_j''T''} = -\frac{C_\mu}{Pr_\mathrm{\tiny T}}\overline{\rho}\frac{\mathrm{k}}{\varepsilon^*}c_p\frac{\partial\hat T}{\partial x_j}\;\; ; \;\; Pr_\mathrm{\tiny T}=0.9$

where $C_\mu$ was defined in (GLVY.3).
Finally, the extra turbulence-kinetic-energy diffusion terms in the mean-flow energy equation (Implementing Turbulence Models into the Compressible RANS Equations) should be modelled, approximately, as

$\frac{\partial}{\partial x_j}\Biggl(\overline{\sigma_{ij}u_i''}-\frac{1}{2}\overline{\rho u_i''u_i''u_j''}\Biggr)\approx\frac{1}{2}d_{\ell\ell}^{(\mu)}+\frac{1}{2}d_{\ell\ell}^{(u)}$

where $d^{(\mu)}_{ij}$ is defined in (GLVY.1) and $d^{(u)}_{ij}$ is modeled by (GLVY.6). The approximation stems from the fact that the model, like the majority of Reynolds stress-models, uses the exact term $d^{(\mu)}_{ij}$ with an appropriate definition of $\varepsilon_{ij}$ [(4), p. 1369, Gerolymos G.A., Joly S., Mallet M., Vallet I.: J. Aircraft 47(4) (2010) 1368-1381 https://doi.org/10.2514/1.47538] while what appears in the mean-flow Favre-averaged total energy equation is $d^{(\tau)}_{ij}$ [(3), p. 1369, Gerolymos G.A., Joly S., Mallet M., Vallet I.: J. Aircraft 47(4) (2010) 1368-1381 https://doi.org/10.2514/1.47538].

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Recent significant updates:
6/30/2015 - mention Pr, Pr_t, and Sutherland's law
11/20/2014 - added statement about BC treatment at symmetry planes

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