4.9.7 Modeling the Dissipation Rate

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

where $Y_M= 2 \rho \epsilon {\rm M}_t^2$ is an additional "dilatation dissipation'' term according to the model by Sarkar [ 300]. The turbulent Mach number in this term is defined as

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:

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  4.4.5, and $S_{\epsilon}$ is a user-defined source term.

In the case when the Reynolds Stress model is coupled with the omega equation, the dissipation tensor $\epsilon_{ij}$ is modeled as

Where $\beta^*_{RSM}$ is defined in Section  4.9.4 and the specific dissipation rate $\omega$ is computed in the same way as for the standard $k-\omega$ model, using Equation  4.5-2.