K–omega turbulence model

Tool in computational fluid dynamics

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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).

## Standard (Wilcox) k–ω turbulence model

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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:[1]

∂ ( ρ k ) ∂ t + ∂ ( ρ u j k ) ∂ x j = ρ P − β ∗ ρ ω k + ∂ ∂ x j [ ( μ + σ k ρ k ω ) ∂ k ∂ x j ] , with  P = τ i j ∂ u i ∂ x j , ∂ ( ρ ω ) ∂ t + ∂ ( ρ u j ω ) ∂ x j = α ω k ρ P − β ρ ω 2 + ∂ ∂ x j [ ( μ + σ ω ρ k ω ) ∂ ω ∂ x j ] + ρ σ d ω ∂ k ∂ x j ∂ ω ∂ x j . {\displaystyle {\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}}\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{aligned}}}

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

## Notes

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1. ^ Wilcox (2008)

## References

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* Wilcox, D. C. (2008), "Formulation of the k–ω Turbulence Model Revisited", AIAA Journal, 46 (11): 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 978-0131274983

## External links

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