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12.4.4 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

 G_k = - \rho \overline{u'_i u'_j} \frac{\partial u_j}{\partial x_i} (12.4-20)

To evaluate $G_k$ in a manner consistent with the Boussinesq hypothesis,
 G_k = \mu_t S^2 (12.4-21)

where $S$ is the modulus of the mean rate-of-strain tensor, defined as
 S \equiv \sqrt{2S_{ij} S_{ij}} (12.4-22)


When using the high-Reynolds number $k$- $\epsilon$ versions, $\mu_{\rm eff}$ is used in lieu of $\mu_t$ in Equation  12.4-21.

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