Compressible Turbulence
Abstract
We present a model to treat fully compressible, nonlocal, timedependent turbulent convection in the presence of largescale flows and arbitrary density stratification. The problem is of interest, for example, in stellar pulsation problems, especially since accurate helioseismological data are now available, as well as in accretion disks. Owing to the difficulties in formulating an analytical model, it is not surprising that most of the work has gone into numerical simulations. At present, there are three analytical models: one by the author, which leads to a rather complicated set of equations; one by Yoshizawa; and one by Xiong. The latter two use a Reynolds stress model together with phenomenological relations with adjustable parameters whose determination on the basis of terrestrial flows does not guarantee that they may be extrapolated to astrophysical flows. Moreover, all thirdorder moments representing nonlocality are taken to be of the down gradient form (which in the case of the planetary boundary layer yields incorrect results). In addition, correlations among pressure, temperature, and velocities are often neglected or treated as in the incompressible case.
To avoid phenomenological relations, we derive the full set of dynamic, timedependent, nonlocal equations to describe all mean variables, second and thirdorder moments. Closures are carried out at the fourth order following standard procedures in turbulence modeling. The equations are collected in an Appendix.
Some of the novelties of the treatment are (1) new flux conservation law that includes the largescale flow, (2) increase of the rate of dissipation of turbulent kinetic energy owing to compressibility and thus (3) a smaller overshooting, and (4) a new source of mean temperature due to compressibility; moreover, contrary to some phenomenological suggestions, the adiabatic temperature gradient depends only on the thermal pressure, while in the equation for the largescale flow, the physical pressure is the sum of thermal plus turbulent pressure.
 Publication:

The Astrophysical Journal
 Pub Date:
 June 1997
 DOI:
 10.1086/304175
 Bibcode:
 1997ApJ...482..827C
 Keywords:

 Hydrodynamics;
 Turbulence