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12.7.6 Modeling the Turbulence Kinetic Energy

In general, when the turbulence kinetic energy is needed for modeling a specific term, it is obtained by taking the trace of the Reynolds stress tensor:

 k = \frac{1}{2}\overline{u'_i u'_i} (12.7-28)

As described in Section  12.7.9, an option is available in FLUENT to solve a transport equation for the turbulence kinetic energy in order to obtain boundary conditions for the Reynolds stresses. In this case, the following model equation is used:
 \frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\par... ... + G_{ii}\right) - \rho \epsilon (1 + 2 {\rm M}_t^2) + S_k (12.7-29)

where $\sigma_k = 0.82$ and $S_k$ is a user-defined source term. Equation  12.7-29 is obtainable by contracting the modeled equation for the Reynolds stresses (Equation  12.7-1). As one might expect, it is essentially identical to Equation  12.4-1 used in the standard $k$- $\epsilon$ model.

Although Equation  12.7-29 is solved globally throughout the flow domain, the values of $k$ obtained are used only for boundary conditions. In every other case, $k$ is obtained from Equation  12.7-28. This is a minor point, however, since the values of $k$ obtained with either method should be very similar.

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