K–omega turbulence model
{{Short description|Tool in computational fluid dynamics}}
{{refimprove|date=June 2014}}
{{Lowercase title}}
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
The eddy viscosity νT, as needed in the RANS equations, is given by: {{nowrap|νT {{=}} k/ω}}, while the evolution of k and ω is modelled as:{{harvtxt|Wilcox|2008}}
\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}
For recommendations for the values of the different parameters, see {{harvtxt|Wilcox|2008}}.
Notes
{{Reflist}}
References
- {{citation |title=Formulation of the k–ω Turbulence Model Revisited |last=Wilcox |first=D. C. |journal=AIAA Journal |volume=46 |issue=11 |year=2008 |pages=2823–2838 |doi=10.2514/1.36541 |bibcode = 2008AIAAJ..46.2823W }}
- {{citation| title=Turbulence Modeling for CFD |edition=2nd | last1=Wilcox | first1=D. C. |publisher=DCW Industries |year=1998 |isbn=0963605100 }}
- {{citation |title=An introduction to turbulence and its measurement |last=Bradshaw |first=P. |author-link=Peter Bradshaw (aeronautical engineer) | publisher=Pergamon Press |year=1971 |isbn=0080166210 }}
- {{citation |title=An Introduction to Computational Fluid Dynamics: The Finite Volume Method |edition=2nd |first1=H. |last1=Versteeg |first2=W. |last2=Malalasekera |publisher=Pearson Education Limited |year=2007 |isbn=978-0131274983 }}
External links
- {{citation |url=http://www.cfd-online.com/Wiki/Wilcox%27s_k-omega_model |title=CFD Online Wilcox k–omega turbulence model description |accessdate=May 12, 2014 }}
{{DEFAULTSORT:K-omega turbulence model}}