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).

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:^[1]

${\displaystyle \begin{array}{rl}& \frac{\partial (\rho k)}{\partial t}+\frac{\partial (\rho u_{j}k)}{\partial x_j}=\rho P-{\beta }^{*}\rho \omega k+\frac{\partial }{\partial x_j}\left[(\mu +{\sigma }_k\frac{\rho k}{\omega })\frac{\partial k}{\partial x_j}\right],\text{with }P={\tau }_{ij}\frac{\partial u_i}{\partial x_j},\\ & {\displaystyle \frac{\partial (\rho \omega )}{\partial t}+\frac{\partial (\rho u_j\omega )}{\partial x_j}=\frac{\alpha \omega }{k}\rho P-\beta \rho {\omega }^2+\frac{\partial }{\partial x_j}\left[(\mu +{\sigma }_{\omega }\frac{\rho k}{\omega })\frac{\partial \omega }{\partial x_j}\right]+\frac{\rho {\sigma }_d}{\omega }\frac{\partial k}{\partial x_j}\frac{\partial \omega }{\partial x_j}.}\end{array}}$

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, 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 

• CFD Online Wilcox k–omega turbulence model description, http://www.cfd-online.com/Wiki/Wilcox%27s_k-omega_model, retrieved May 12, 2014 

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