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12.7.7 Modeling the Dissipation Rate

The dissipation tensor, $\epsilon_{ij}$, is modeled as

 \epsilon_{ij} = \frac{2}{3} \delta_{ij} ( \rho \epsilon + Y_M ) (12.7-30)

where $Y_M= 2 \rho \epsilon {\rm M}_t^2$ is an additional "dilatation dissipation'' term according to the model by Sarkar [ 315]. The turbulent Mach number in this term is defined as
 {\rm M}_t = \sqrt{\frac{k}{a^2}} (12.7-31)

where $a$ ( $\equiv \sqrt{\gamma R T}$) is the speed of sound. This compressibility modification always takes effect when the compressible form of the ideal gas law is used.

The scalar dissipation rate, $\epsilon$, is computed with a model transport equation similar to that used in the standard $k$- $\epsilon$ model:
 \frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partia... ...} - C_{\epsilon 2} \rho \frac{\epsilon^2}{k} + S_{\epsilon} (12.7-32)

where $\sigma_{\epsilon} = 1.0$, $C_{\epsilon 1} = 1.44$, $C_{\epsilon 2} = 1.92$, $C_{\epsilon 3}$ is evaluated as a function of the local flow direction relative to the gravitational vector, as described in Section  12.4.5, and $S_{\epsilon}$ is a user-defined source term.


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