K–omega turbulence model explained

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 ν_T, as needed in the RANS equations, is given by:, while the evolution of k and ω is modelled as:

\begin{align} & (\partial (\rho k))/(\partial t) + (\partial (\rho u _{j} k))/(\partial x _{j}) = \rho P - \beta ^{*} \rho \omega k + (\partial)/(\partial x _{j}) \left[\left(\mu + \sigma _{k} (\rho k)/(\right) \omega) (\partial k)/(\partial x _{j}) \right], with P = \tau _{ij} (\partial u _{i})/(\partial x _{j}) , \\ & \displaystyle (\partial (\rho \omega))/(\partial t) + (\partial (\rho u _{j} \omega))/(\partial x _{j}) = (\alpha \omega)/(k) \rho P - \beta \rho \omega ^{2} + (\partial)/(\partial x _{j}) \left[ \left( \mu + \sigma _{\omega} (\rho k)/(\omega) \right) (\partial \omega)/(\partial x _{j}) \right] + (\rho \sigma _{d})/(\omega) (\partial k)/(\partial x _{j}) (\partial \omega)/(\partial x _{j}) . \end{align}

For recommendations for the values of the different parameters, see .