Langley Research Center

Turbulence Modeling Resource

Stress-omega Full Reynolds Stress Model

This web page gives detailed information on the equations for Wilcox's full second-moment Reynolds stress model, 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.

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Wilcox Stress-omega Full Reynolds Stress Model (WilcoxRSM-w2006)

This model's reference is:

Wilcox, D. C., Turbulence Modeling for CFD, 3rd edition, DCW Industries, Inc., La Canada CA, 2006.

Written in conservation form, the six Reynolds stress equations are solved in terms of the Reynolds stresses $\overline \rho R'_{ij}$ (notation different from the original reference), where $\overline \rho R'_{ij}$ corresponds 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:

$\overline \rho R'_{ij} = \tau_{ij} = -\overline {\rho u_i'' u_j''}$

Note that this $\overline \rho R'_{ij}$ definition is the negative of the $\overline \rho \hat R_{ij}$ used in the SSG/LRR-RSM-w2012 model. The six Reynolds stress equations and one length scale equation are given by:

$\frac{\partial \overline \rho R'_{ij}}{\partial t} + \frac{\partial (\overline \rho \hat u_k R'_{ij})}{\partial x_k} = -\overline \rho P_{ij} - \overline \rho \Pi_{ij} + \frac{2}{3} \beta^* \overline \rho \omega k \delta_{ij} + \frac{\partial}{\partial x_k} \left[ \left( \overline \mu + \sigma^* \mu_t \right) \frac{\partial R'_{ij}}{\partial x_k} \right]$

$\frac{\partial (\overline \rho \omega)}{\partial t} + \frac{\partial (\overline \rho \hat u_k \omega)}{\partial x_k} = \frac{\alpha \overline \rho \omega}{k} R'_{ij} \frac{\partial \hat u_i}{\partial x_j} - \beta \overline \rho \omega^2 + \frac{\partial}{\partial x_k} \left[ \left( \overline \mu + \sigma \mu_t \right) \frac{\partial \omega}{\partial x_k} \right] + \sigma_d \frac{\overline \rho}{\omega} \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}$

The production term is:

$P_{ij} = R'_{ik} \frac{\partial \hat u_j}{\partial x_k} + R'_{jk} \frac{\partial \hat u_i}{\partial x_k}$

with $k=-R'_{ii}/2$ and $\mu_t = \overline \rho k / \omega$ .

The pressure-strain correlation is modeled (without wall-correction terms) via:

$\Pi_{ij} = \beta^* C_1 \omega \left( R'_{ij} + \frac{2}{3} k \delta_{ij} \right) - \hat \alpha \left( P_{ij} - \frac{2}{3} P \delta_{ij} \right) - \hat \beta \left( D_{ij} - \frac{2}{3} P \delta_{ij} \right) - \hat \gamma k \left( S_{ij} - \frac{1}{3} S_{kk} \delta_{ij} \right)$

or equivalently:

$\Pi_{ij} = - C_1 \varepsilon a_{ij} + (\hat \alpha + \hat \beta) k \left( a_{ik} S_{jk} + a_{jk} S_{ik} - \frac{2}{3} a_{kl} S_{kl} \delta_{ij} \right) + (\hat \alpha - \hat \beta) k \left(a_{ik} W_{jk} + a_{jk} W_{ik} \right) + \left[ \frac{4}{3} \left( \hat \alpha + \hat \beta \right) - \hat \gamma \right] k \left( S_{ij} - \frac{1}{3}S_{kk} \delta_{ij} \right)$

with

$D_{ij} = R'_{ik} \frac{\partial \hat u_k}{\partial x_j} + R'_{jk} \frac{\partial \hat u_k}{\partial x_i}$

$a_{ij} = -\frac{R'_{ij}}{k} - \frac{2}{3}\delta_{ij}$

and $P = P_{kk}/2$ and $\varepsilon = \beta^* k \omega$ . The coefficients (listed below) are from the LRR-QI model (J. Fluid Mech (1975), vol 68, part 3, pp. 537-566).

The closure coefficients are:

$\hat \alpha = (8+C_2)/11$ $\hat \beta = (8 C_2-2)/11$

$\hat \gamma = (60 C_2 - 4)/55$ $C_1 = 9/5$

$C_2 = 10/19$ $\alpha = 13/25$

$\beta = \beta_0 f_{\beta}$ $\beta^* = 9/100$

$\sigma=0.5$ $\sigma^*=0.6$

$\beta_0 = 0.0708$

$\sigma_d = 0, \quad for \quad \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j} \leq 0$ $\sigma_d = \frac{1}{8}, \quad for \quad \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j} > 0$

and

$f_{\beta} = \frac{1 + 85 \chi_{\omega}}{1 + 100 \chi_{\omega}}$

$\chi_{\omega} = \left| \frac{W_{ij} W_{jk} \hat S_{ki}} {(\beta^* \omega)^3} \right|$

$\hat S_{ki} = S_{ki} - \frac{1}{2} \frac{\partial \hat u_m}{\partial x_m} \delta_{ki}$

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

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

There are no specific farfield boundary conditions recommended for this model, and there are various wall boundary conditions for $\omega$ . See the Wilcox k-omega page for more details. Also, the approximate wall boundary condition for $\omega$ from Menter (AIAA Journal, Vol. 32, No. 8, August 1994, pp. 1598-1605, https://doi.org/10.2514/3.12149) is sometimes used (see SSG/LRR-RSM-w2012 model). For the Reynolds stresses at solid walls:

$R'_{ij, wall} = 0$

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, nothing specific is recommended beyond what is described on the page: Implementing Turbulence Models into the Compressible RANS Equations.

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Recent significant updates:
03/24/2016 - mention realizability of Reynolds stresses
06/30/2015 - mention Pr, Pr_t, and Sutherland's law
11/20/2014 - added statement about BC treatment at symmetry planes
09/02/2014 - corrected typo in chi_omega equation
04/07/2014 - split SSG/LRR model to separate page
08/29/2013 - included equivalent pressure-strain form in WilcoxRSM-w2006
08/23/2013 - added "RSM" in naming convention for SSG/LRR version
08/22/2013 - added definitions of strain and vorticity tensors for Wilcox
07/22/2013 - corrected sign of production term in omega eqn of SSG/LRR-RSM-w2012

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Last Updated: 11/08/2021