4.4.4 Modeling Turbulent Production in the $k$- $\epsilon$ Models

The term $G_k$, representing the production of turbulence kinetic energy, is modeled identically for the standard, RNG, and realizable $k$- $\epsilon$ models. From the exact equation for the transport of $k$, this term may be defined as

To evaluate $G_k$ in a manner consistent with the Boussinesq hypothesis,

where $S$ is the modulus of the mean rate-of-strain tensor, defined as

When using the high-Reynolds number $k$- $\epsilon$ versions, $\mu_{\rm eff}$ is used in lieu of $\mu_t$ in Equation  4.4-21.