Wilcox k-omega Model

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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. If any particular variant has been overlooked, please report it to the page curator.

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

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.
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. 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. The 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 $u_{\tau} k_s / \nu < 5$ . Some CFD codes employ the approximate $\omega$ wall boundary condition from Menter for this model (see Menter k-omega SST).

The constants and auxiliary functions are:

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

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

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

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

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

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

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

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

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

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:

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

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

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

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

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

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

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

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

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

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.
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:

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

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

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.

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:

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

For low-speed flows, the second term on the right hand side of this equation is generally small compared to the first term. (For incompressible flows the second term is identically zero.)

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