K–omega turbulence model

In computational fluid dynamics, the k–omega (k–ω) turbulence model is a common two-equation turbulence model, that is used as an approximation for the Reynolds-averaged Navier–Stokes equations (RANS equations). The model attempts to predict turbulence by two partial differential equations for two variables, k and ω, with the first variable being the turbulence kinetic energy (k) while the second (ω) is the specific rate of dissipation (of the turbulence kinetic energy k into internal thermal energy).

## Contents

* 1 Standard (Wilcox) k–ω turbulence model [1]
* 2 Notes
* 3 References
* 4 External links

## Standard (Wilcox) k–ω turbulence model [1]

The eddy viscosity νT, as needed in the RANS equations, is given by: νT = k/ω, while the evolution of k and ω is modelled as:

[math]\displaystyle{ \begin{align} & \frac{\partial (\rho k)}{\partial t} + \frac{\partial (\rho u_j k)}{\partial x_j} = \rho 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], \qquad \text{with } P = \tau_{ij} \frac{\partial u_i}{\partial x_j}, \\\ & \displaystyle \frac{\partial (\rho \omega)}{\partial t} + \frac{\partial (\rho u_j \omega)}{\partial x_j} = \frac{\alpha \omega}{k}\rho 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}. \end{align} }[/math]

For recommendations for the values of the different parameters, see (Wilcox 2008).

## Notes

1. ↑ (Wilcox 2008)

## References

* Wilcox, D. C. (2008), "Formulation of the k–ω Turbulence Model Revisited", AIAA Journal 46 (11): 2823–2838, doi:10.2514/1.36541, Bibcode: 2008AIAAJ..46.2823W
* Wilcox, D. C. (1998), Turbulence Modeling for CFD (2nd ed.), DCW Industries, ISBN 0963605100
* Bradshaw, P. (1971), An introduction to turbulence and its measurement, Pergamon Press, ISBN 0080166210
* Versteeg, H.; Malalasekera, W. (2007), An Introduction to Computational Fluid Dynamics: The Finite Volume Method (2nd ed.), Pearson Education Limited, ISBN 978-0131274983

## External links

* CFD Online Wilcox k–omega turbulence model description, http://www.cfd-online.com/Wiki/Wilcox's_k-omega_model, retrieved May 12, 2014

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