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 a closure 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).

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

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

${\begin{aligned}&{\frac {\partial (\rho k)}{\partial t}}+{\frac {\partial (\rho u_{j}k)}{\partial x_{j}}}=\rho P-\beta ^{*}\rho \omega k+{\frac {\partial }{\partial x_{j}}}\left[\left(\mu +\sigma _{k}{\frac {\rho k}{\omega }}\right){\frac {\partial k}{\partial x_{j}}}\right],\qquad {\text{with }}P=\tau _{ij}{\frac {\partial u_{i}}{\partial x_{j}}},\\&\displaystyle {\frac {\partial (\rho \omega )}{\partial t}}+{\frac {\partial (\rho u_{j}\omega )}{\partial x_{j}}}={\frac {\alpha \omega }{k}}P-\beta \rho \omega ^{2}+{\frac {\partial }{\partial x_{j}}}\left[\left(\mu +\sigma _{\omega }{\frac {\rho k}{\omega }}\right){\frac {\partial \omega }{\partial x_{j}}}\right]+{\frac {\rho \sigma _{d}}{\omega }}{\frac {\partial k}{\partial x_{j}}}{\frac {\partial \omega }{\partial x_{j}}}.\end{aligned}}$

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

Notes

Wilcox (2008)

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References

Wilcox, D. C. (2008), Formulation of the k–ω Turbulence Model Revisited, 46, AIAA Journal, pp. 2823–2838, Bibcode:2008AIAAJ..46.2823W, doi:10.2514/1.36541
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 0131274988

External links

CFD Online Wilcox k–omega turbulence model description, retrieved May 12, 2014

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[1] Wilcox (2008)