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Turbulence Modeling Resource

The Wilcox k-omega Turbulence Model

This web page gives detailed information on the equations for various forms of the Wilcox k-omega turbulence model. 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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The first version listed (Wilcox2006) is the latest "standard" version of this model.

Wilcox (2006) k-omega Two-Equation Model (Wilcox2006) and (Wilcox2006m)

The references for this model are:

Wilcox, D. C., "Formulation of the k-omega Turbulence Model Revisited," AIAA Journal, Vol. 46, No. 11, 2008, pp. 2823-2838, https://doi.org/10.2514/1.36541.
Wilcox, D. C., Turbulence Modeling for CFD, 3rd edition, DCW Industries, Inc., La Canada CA, 2006.

The two-equation model (written in conservation form) is given by the following:

$\frac{\partial (\rho k)}{\partial t} + \frac{\partial (\rho u_j k)}{\partial x_j} = \cal P - \beta^* \rho \omega k + \frac{\partial}{\partial x_j} \left[\left(\mu + \sigma_k \frac{\rho k}{\omega} \right)\frac{\partial k}{\partial x_j}\right]$

$\frac{\partial (\rho \omega)}{\partial t} + \frac{\partial (\rho u_j \omega)}{\partial x_j} = \frac{\gamma \omega}{k} \cal P - \beta \rho \omega^2 + \frac{\partial}{\partial x_j} \left[ \left( \mu + \sigma_{\omega} \frac{\rho k}{\omega} \right) \frac{\partial \omega}{\partial x_j} \right] + \frac{\rho \sigma_d}{\omega} \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}$

where

$P = \tau_{ij} \frac{\partial u_i}{\partial x_j}$

$\tau_{ij} = \mu_t \left(2S_{ij} - \frac{2}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij} \right) - \frac{2}{3} \rho k \delta_{ij}$

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

and the turbulent eddy viscosity is computed from:

$\mu_t = \frac{\rho k}{\hat \omega}$

where:

$\hat \omega = {\rm max} \left[ \omega, C_{lim} \sqrt{\frac{2 \overline S_{ij} \overline S_{ij}}{\beta^*}} \right]$

$\overline S_{ij} = S_{ij} - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij}$

and $\rho$ is the density and $\mu$ is the molecular dynamic viscosity.

There are no specific farfield boundary conditions recommended for this model. See Menter k-omega SST for farfield values recommended there. Note that the Wilcox k-omega two-equation models exhibit some sensitivity to freestream boundary conditions on $\omega$ (see Wilcox, D. C., Turbulence Modeling for CFD, 3rd edition, DCW Industries, Inc., La Canada CA, 2006 for details). An example of the sensitivity can be found on the 2D Mixing Layer Validation - Wilcox2006 Model Results page.

At solid walls:

$k_{wall} = 0$

There are various wall boundary conditions mentioned for $\omega$ in the references above, including both smooth and rough walls. For smooth walls, the asymptotic behavior is

$\omega_{wall} \rightarrow \frac{6 \nu_{wall}}{\beta_0 d^2}$

as $y \rightarrow 0$ , where d is the distance to the nearest wall. However, according to Menter (AIAA J 32(8):1598-1605, 1994), it is not appropriate to use this asymptotic value for the BC at a wall. Instead, many CFD codes employ the approximate $\omega$ wall boundary condition from Menter for this model (see Menter k-omega SST).

Alternatively, the Wilcox references also specify a so-called "slightly-rough-surface" boundary condition for $\omega$ :

$\omega_{wall} = \frac{40000 \nu_{wall}}{k_s^2}$

where it is important for smooth walls to "select a small enough value" of $k_s$ to insure that $u_{\tau} k_s / \nu < 5$ .

The constants and auxiliary functions are:

$\sigma_k = 0.6$ $\sigma_{\omega} = 0.5$

$\beta^* = 0.09$ $\gamma = \frac{13}{25}$ $C_{lim} = \frac{7}{8}$

$\beta = \beta_0 f_{\beta}$ $\beta_0 = 0.0708$

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

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

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

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

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

Note that the $\gamma$ coefficient (termed $\alpha$ in Wilcox's book) was chosen in order to yield an appropriate value for the Karman constant ( $\kappa = 0.40$ ), via the expression:

$\gamma = \frac{\beta_0}{\beta^*} - \frac{\sigma_{\omega} \kappa^2}{\sqrt{\beta^*}}$

Also note that in the so-called "Pope correction" for this model, the term $\chi_{\omega}$ is zero for 2-D. If the Pope correction is specifically turned off for a 3-D computation, then the model should be referred to as (Wilcox2006-noPope).

Note: the $(2/3) \overline \rho k \delta_{ij}$ term in the Boussinesq approximation for tau_ij is sometimes ignored in the momentum and energy equations. Similarly, the production term in two-equation turbulence models is often approximated by P = mu_t S². This expression is exact for incompressible flows and is typically considered a very good approximation, except perhaps for very high Mach number flows (see items 4 and 7 on the page Notes on Running the Cases with CFD, and the Implementing Turbulence Models into the Compressible RANS Equations page). There are various ways that these approximations can be implemented:

(Wilcox2006m) (most common) - when the $(2/3) \overline \rho k \delta_{ij}$ term is IGNORED in tau_ij in the momentum and energy equations, and the production term is APPROXIMATED by P = mu_t S².
(Wilcox2006s) - when the $(2/3) \overline \rho k \delta_{ij}$ term is INCLUDED in tau_ij in the momentum and energy equations, and the production term is APPROXIMATED by P = mu_t S².
(Wilcox2006e) - when the $(2/3) \overline \rho k \delta_{ij}$ term is IGNORED in tau_ij in the momentum and energy equations, and the production term is THE EXACT expression.

Wilcox (2006) k-omega Two-Equation Model with k-equation Production Limiter (Wilcox2006-klim) and (Wilcox2006-klim-m)

Although the official model described above does not include it, some applications use a production limiter in the k-equation (patterned after the SST model). Everything is identical to the (Wilcox2006) model above except that the term P in the k-equation is replaced by:

${\rm min}(\cal P, 20 \beta^* \rho \omega k)$

There has been only limited experience to date, but the use of this form as opposed to the "standard" version (Wilcox2006) appears to makes very little difference.

When the $(2/3) \overline \rho k \delta_{ij}$ term is ignored in tau_ij in the momentum and energy equations and the production term is approximated by P = mu_t S², a modified naming convention should be used: (Wilcox2006-klim-m). Other variants (s and e) follow the naming conventions described above at the end of the Wilcox2006 section.

Wilcox (1998) k-omega Two-Equation Model (Wilcox1998) and (Wilcox1998m)

The reference for this model is:

Wilcox, D. C., Turbulence Modeling for CFD, 2nd edition, DCW Industries, Inc., La Canada CA, 1998.

The two-equation model (written in conservation form) is given by the following:

and the turbulent eddy viscosity is computed from:

$\mu_t = \frac{\rho k}{\omega}$

Meanings of variables and definitions of boundary conditions are the same as for (Wilcox2006).

The constants and auxiliary functions are:

$\sigma_k = 0.5$ $\sigma_{\omega} = 0.5$

$\beta_0^* = 0.09$ $\gamma = \frac{13}{25}$

$\beta = \beta_0 f_{\beta}$ $\beta_0 = \frac{9}{125}$

$f_{\beta} = \frac{1 + 70 \chi_{\omega}}{1 + 80 \chi_{\omega}}$

$\chi_{\omega} = \left| \frac{\Omega_{ij} \Omega_{jk} S_{ki}} {(\beta_0^* \omega)^3} \right|$

$\beta^* = \beta_0^* f_{\beta^*}$

$f_{\beta^*} = 1, \quad for \quad \chi_k \leq 0$ $f_{\beta^*} = \frac{1+680 \chi_k^2}{1+400 \chi_k^2}, \quad for \quad \chi_k > 0$

$\chi_k = \frac{1}{\omega^3} \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}$

Note that the $\gamma$ coefficient (termed $\alpha$ in Wilcox's book) was chosen in order to yield an appropriate value for the Karman constant ( $\kappa \approx 0.41$ ), via the expression:

$\gamma = \frac{\beta_0}{\beta_0^*} - \frac{\sigma_{\omega} \kappa^2}{\sqrt{\beta_0^*}}$

Also note that in the so-called "Pope correction" for this model, the term $\chi_{\omega}$ is zero for 2-D.

When the $(2/3) \overline \rho k \delta_{ij}$ term is ignored in tau_ij in the momentum and energy equations and the production term is approximated by P = mu_t S², a modified naming convention should be used: (Wilcox1998m). Other variants (s and e) follow the naming conventions described above at the end of the Wilcox2006 section.

Wilcox (1988) k-omega Two-Equation Model (Wilcox1988) and (Wilcox1988m)

The references for this model are:

Wilcox, D. C., "Reassessment of the Scale-Determining Equation for Advanced Turbulence Models," AIAA Journal, Vol. 26, No. 11, 1988, pp. 1299-1310, https://doi.org/10.2514/3.10041.
Wilcox, D. C., Turbulence Modeling for CFD, 1st edition, DCW Industries, Inc., La Canada CA, 1993.

The basic equations for this two-equation model are the same as for (Wilcox1998):

and the turbulent eddy viscosity is computed from:

$\mu_t = \frac{\rho k}{\omega}$

The only difference is in the values taken by some of the variables:

$\sigma_k = 0.5$ $\sigma_{\omega} = 0.5$

$\beta^* = 0.09$ $\beta = \frac{3}{40}$ $\gamma = \frac{5}{9}$

Note that the $\gamma$ coefficient (termed $\alpha$ in Wilcox's book) was chosen in order to yield an appropriate value for the Karman constant ( $\kappa \approx 0.408$ ), via the expression:

$\gamma = \frac{\beta}{\beta^*} - \frac{\sigma_{\omega} \kappa^2}{\sqrt{\beta^*}}$

When the $(2/3) \overline \rho k \delta_{ij}$ term is ignored in tau_ij in the momentum and energy equations and the production term is approximated by P = mu_t S², a modified naming convention should be used: (Wilcox1988m). Other variants (s and e) follow the naming conventions described above at the end of the Wilcox2006 section.

Wilcox k-omega Two-Equation Models with Vorticity Source Term (Wilcox2006-V, Wilcox1998-V, Wilcox1988-V)

This form of two-equation models is sometimes used because vorticity magnitude $\Omega$ is usually readily available in most Navier-Stokes codes. Furthermore, the vorticity source term is often nearly identical to the exact source term in simple boundary layer flows, and the use of the vorticity term can avoid some numerical difficulties sometimes associated with the use of the exact source term. The reference for this usage is:

Menter, F. R., "Improved Two-Equation k-omega Turbulence Models for Aerodynamic Flows," NASA TM 103975, October 1992, https://ntrs.nasa.gov/citations/19930013620.

The equations are the same as for the "standard" versions of the Wilcox models, with the exception that the term P (in both equations) is approximated with the following:

$P = \mu_t \Omega^2 - \frac{2}{3}\rho k \delta_{ij} \frac{\partial u_i}{\partial x_j}$

Note: When the $(2/3) \overline \rho k \delta_{ij}$ term is ignored in tau_ij in the momentum and energy equations and the production term is approximated by P = mu_t*(Omega²), a modified naming convention should be employed: (Wilcox2006-Vm) or (Wilcox1998-Vm) or (Wilcox1988-Vm).

Note that this approximation is similar in spirit to the Kato-Launder correction (Kato, M. and Launder, B. E., "The Modelling of Turbulent Flow Around Stationary and Vibrating Square Cylinders," 9th Symposium on Turbulent Shear Flows, Kyoto, Japan, August 1993, paper 10-4), which uses $\mu_t S \Omega$ instead of $\mu_t \Omega^2$ . Implementation of the Kato-Launder correction would be called (Wilcox2006-KL), (Wilcox1998-KL), or (Wilcox1988-KL). If the $(2/3) \overline \rho k \delta_{ij}$ term is ignored in tau_ij in the momentum and energy equations and if the $- (2/3) \rho k \delta_{ij} \partial u_i / \partial x_j$ term in P is ignored, then the naming convention is (Wilcox2006-KLm), (Wilcox1998-KLm), or (Wilcox1988-KLm).

Low Reynolds Number Version of Wilcox (2006) k-omega Two-Equation Model (Wilcox2006-LRN)

The reference for this model is:

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

This model is the same as the (Wilcox2006) model, with the following changes:

Instead of $\beta^* = 0.09$ use:

$\beta^* = 0.09 \left( \frac{100 \beta_0 / 27 + (Re_T/R_{\beta})^4} {1 + (Re_T/R_{\beta})^4} \right)$

Instead of $\gamma = 13/25$ use:

$\gamma = 13/25 \left( \frac{\alpha_0 + Re_T/R_{\omega}} {1 + Re_T/R_{\omega}} \right) (\alpha^*)^{-1}$

Instead of $\mu_t = \rho k / \hat\omega$ use:

$\mu_t = \alpha^* \frac{\rho k}{\hat\omega}$

Instead of $\hat\omega = {\rm max} (\omega, C_{lim} \sqrt{2 \overline S_{ij} \overline S_{ij} / \beta^*} )$ use:

$\hat\omega = {\rm max} \left(\omega, C_{lim} \sqrt{ \frac{2 \overline S_{ij} \overline S_{ij}}{\beta_0^*/\alpha^*}} \right)$

Instead of $(\mu + \sigma_k (\rho k / \omega))$ in the $k$ diffusion term use:

$\left(\mu + \sigma_k \alpha^* \frac{\rho k}{\omega} \right)$

Instead of $(\mu + \sigma_{\omega} (\rho k / \omega))$ in the $\omega$ diffusion term use:

$\left(\mu + \sigma_{\omega} \alpha^* \frac{\rho k}{\omega} \right)$

with:

$Re_T = \frac{\rho k}{\mu \omega}$ $\alpha^* = \frac{\alpha_0^* + Re_T/R_k}{1 + Re_T/R_k}$

$R_{\beta} = 8$ $R_k = 6$ $R_{\omega} = 2.61$

$\alpha_0 = \frac{1}{9}$ $\alpha_0^* = \frac{\beta_0}{3}$ $\beta_0 = 0.0708$ $\beta_0^* = 0.09$

Special notes for users of OpenFOAM.

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Recent significant updates:
10/04/2024 - added "s" and "e" variant definitions
03/24/2021 - clarifications on use of "m" designation when P=mu_t S² and k term ignored in momentum and energy equations
08/08/2016 - added Wilcox2006-klim naming convention
06/30/2015 - mention Pr, Pr_t, and Sutherland's law
07/08/2014 - mention of equation relating gamma and Karman's constant kappa
04/02/2014 - added Wilcox2006-noPope designation
12/16/2013 - added clarification that chi_w term is zero in 2-D (Pope correction)
8/29/2013 - mention of Kato-Launder correction

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Last Updated: 10/04/2024