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12.4.6 Effects of Compressibility on Turbulence in the $k$- $\epsilon$ Models

For high-Mach-number flows, compressibility affects turbulence through so-called "dilatation dissipation'', which is normally neglected in the modeling of incompressible flows [ 403]. Neglecting the dilatation dissipation fails to predict the observed decrease in spreading rate with increasing Mach number for compressible mixing and other free shear layers. To account for these effects in the $k$- $\epsilon$ models in FLUENT, the dilatation dissipation term, $Y_M$, is included in the $k$ equation. This term is modeled according to a proposal by Sarkar [ 315]:

 Y_M = 2 \rho \epsilon {\rm M}_t^2 (12.4-27)

where M $_t$ is the turbulent Mach number, defined as
 {\rm M}_t = \sqrt{\frac{k}{a^2}} (12.4-28)

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.

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