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12.5.1 Standard $k$- $\omega$ Model


The standard $k$- $\omega$ model in FLUENT is based on the Wilcox $k$- $\omega$ model [ 403], which incorporates modifications for low-Reynolds-number effects, compressibility, and shear flow spreading. The Wilcox model predicts free shear flow spreading rates that are in close agreement with measurements for far wakes, mixing layers, and plane, round, and radial jets, and is thus applicable to wall-bounded flows and free shear flows. A variation of the standard $k$- $\omega$ model called the SST $k$- $\omega$ model is also available in FLUENT, and is described in Section  12.5.2.

The standard $k$- $\omega$ model is an empirical model based on model transport equations for the turbulence kinetic energy ( $k$) and the specific dissipation rate ( $\omega$), which can also be thought of as the ratio of $\epsilon$ to $k$ [ 403].

As the $k$- $\omega$ model has been modified over the years, production terms have been added to both the $k$ and $\omega$ equations, which have improved the accuracy of the model for predicting free shear flows.

Transport Equations for the Standard $k$- $\omega$ Model

The turbulence kinetic energy, $k$, and the specific dissipation rate, $\omega$, are obtained from the following transport equations:

 \frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\par... ...k \frac{\partial k}{\partial x_j}\right) + G_k - Y_k + S_k (12.5-1)

 \frac{\partial}{\partial t} (\rho \omega) + \frac{\partial}... ...a}{\partial x_j}\right) + G_\omega - Y_\omega + S_{\omega} (12.5-2)

In these equations, $G_k$ represents the generation of turbulence kinetic energy due to mean velocity gradients. $G_\omega$ represents the generation of $\omega$. ${\Gamma}_k$ and ${\Gamma}_\omega$ represent the effective diffusivity of $k$ and $\omega$, respectively. $Y_k$ and $Y_\omega$ represent the dissipation of $k$ and $\omega$ due to turbulence. All of the above terms are calculated as described below. $S_k$ and $S_{\omega}$ are user-defined source terms.

Modeling the Effective Diffusivity

The effective diffusivities for the $k$- $\omega$ model are given by

$\displaystyle {\Gamma}_k$ $\textstyle =$ $\displaystyle \mu + \frac{\mu_t}{\sigma_k}$ (12.5-3)
$\displaystyle {\Gamma}_\omega$ $\textstyle =$ $\displaystyle \mu + \frac{\mu_t}{\sigma_\omega}$ (12.5-4)

where $\sigma_k$ and $\sigma_\omega$ are the turbulent Prandtl numbers for $k$ and $\omega$, respectively. The turbulent viscosity, $\mu_t$, is computed by combining $k$ and $\omega$ as follows:
 \mu_t = \alpha^* \frac{\rho k}{\omega} (12.5-5)

Low-Reynolds-Number Correction

The coefficient $\alpha^*$ damps the turbulent viscosity causing a low-Reynolds-number correction. It is given by

 \alpha^* = \alpha^*_\infty \left( \frac{\alpha^*_0 + {\rm Re}_t/R_k}{1 + {\rm Re}_t/R_k} \right) (12.5-6)

$\displaystyle {\rm Re}_t$ $\textstyle =$ $\displaystyle \frac{\rho k}{\mu \omega}$ (12.5-7)
$\displaystyle R_k$ $\textstyle =$ $\displaystyle 6$ (12.5-8)
$\displaystyle \alpha^*_0$ $\textstyle =$ $\displaystyle \frac{\beta_i}{3}$ (12.5-9)
$\displaystyle \beta_i$ $\textstyle =$ $\displaystyle 0.072$ (12.5-10)

Note that, in the high-Reynolds-number form of the $k$- $\omega$ model, $\alpha^* = \alpha^*_\infty = 1$.

Modeling the Turbulence Production

Production of $k$

The term $G_k$ represents the production of turbulence kinetic energy. From the exact equation for the transport of $k$, this term may be defined as

 G_k = - \rho \overline{u'_{i} u'_{j}} \frac{\partial u_j}{\partial x_i} (12.5-11)

To evaluate $G_k$ in a manner consistent with the Boussinesq hypothesis,
 G_k = \mu_t \,S^2 (12.5-12)

where $S$ is the modulus of the mean rate-of-strain tensor, defined in the same way as for the $k$- $\epsilon$ model (see Equation  12.4-22).

Production of $\omega$

The production of $\omega$ is given by

 G_\omega = \alpha \frac{\omega}{k} G_k (12.5-13)

where $G_k$ is given by Equation  12.5-11.

The coefficient $\alpha$ is given by
 \alpha = \frac{\alpha_\infty}{\alpha^*} \left( \frac{\alpha_0 + {\rm Re}_t/R_\omega}{1 + {\rm Re}_t/R_\omega} \right) (12.5-14)

where $R_{\omega}$ = 2.95. $\alpha^*$ and Re $_t$ are given by Equations  12.5-6 and 12.5-7, respectively.

Note that, in the high-Reynolds-number form of the $k$- $\omega$ model, $\alpha = \alpha_\infty=1$.

Modeling the Turbulence Dissipation

Dissipation of $k$

The dissipation of $k$ is given by

 Y_k = \rho \, \beta^* f_{\beta^*} \, k \, \omega \\ (12.5-15)

 f_{\beta^*} = \left\{ \begin{array}{ll} 1 & \mbox{$\chi_k ... ...+ 400 \chi_k^2} & \mbox{$\chi_k > 0$} \end{array} \right. (12.5-16)

 \chi_k \equiv \frac{1}{\omega^3}\frac{\partial k}{\partial x_j}\frac{\partial \omega}{\partial x_j} (12.5-17)

$\displaystyle \beta^*$ $\textstyle =$ $\displaystyle \beta^*_i \left[1 + \zeta^* F({\rm M}_t)\right]$ (12.5-18)
$\displaystyle \beta_i^*$ $\textstyle =$ $\displaystyle \beta^*_\infty \left( \frac{4/15 + ({\rm Re}_t/R_\beta)^4}{1 + ({\rm Re}_t/R_\beta)^4} \right)$ (12.5-19)
$\displaystyle \zeta^*$ $\textstyle =$ $\displaystyle 1.5$ (12.5-20)
$\displaystyle R_{\beta}$ $\textstyle =$ $\displaystyle 8$ (12.5-21)
$\displaystyle \beta^*_\infty$ $\textstyle =$ $\displaystyle 0.09$ (12.5-22)

where Re $_t$ is given by Equation  12.5-7.

Dissipation of $\omega$

The dissipation of $\omega$ is given by

 Y_\omega = \rho \, \beta \, f_\beta \, \omega^2 (12.5-23)

$\displaystyle f_\beta$ $\textstyle =$ $\displaystyle \frac{1 + 70 \chi_\omega}{1 + 80 \chi_\omega}$ (12.5-24)
$\displaystyle \chi_\omega$ $\textstyle =$ $\displaystyle \left\vert\frac{\Omega_{ij} \Omega_{jk} S_{ki}}{(\beta^*_\infty \omega)^3} \right\vert$ (12.5-25)
$\displaystyle \Omega_{ij}$ $\textstyle =$ $\displaystyle \frac{1}{2} \left ({\partial u_i\over\partial x_j} - {\partial u_j\over\partial x_i}\right)$ (12.5-26)

The strain rate tensor, $S_{ij}$ is defined in Equation  12.3-11. Also,
 \beta = \beta_i \left[1 - \frac{\beta^*_i}{\beta_i} \zeta^* F({\rm M}_t)\right] (12.5-27)

$\beta^*_i$ and $F({\rm M}_t)$ are defined by Equations  12.5-19 and 12.5-28, respectively.

Compressibility Correction

The compressibility function, $F({\rm M}_t)$, is given by

 F({\rm M}_t) = \left\{ \begin{array}{ll} 0 & \mbox{${\rm M... ...2 & \mbox{${\rm M}_t > {\rm M}_{t0}$} \end{array} \right. (12.5-28)

$\displaystyle {\rm M}_t^2$ $\textstyle \equiv$ $\displaystyle \frac{2 k}{a^2}$ (12.5-29)
$\displaystyle {\rm M}_{t0}$ $\textstyle =$ $\displaystyle 0.25$ (12.5-30)
$\displaystyle a$ $\textstyle =$ $\displaystyle \sqrt{\gamma R T}$ (12.5-31)

Note that, in the high-Reynolds-number form of the $k$- $\omega$ model, $\beta_i^* = \beta^*_\infty$. In the incompressible form, $\beta^* = \beta_i^*$.

Model Constants

\alpha^*_\infty = 1,\;\; \alpha_\infty = 0.52,\;\; \alpha_0... ...beta^*_\infty = 0.09,\;\; \beta_i = 0.072, \;\; R_\beta = 8

R_k = 6,\;\; R_\omega = 2.95,\;\; \zeta^* = 1.5,\;\; {\rm M}_{t0} = 0.25,\;\; \sigma_k = 2.0, \;\; \sigma_\omega = 2.0

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© Fluent Inc. 2006-09-20