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).

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:

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

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

• 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

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