DENSITY PROBABILITY-DISTRIBUTION IN ONE-DIMENSIONAL POLYTROPIC GAS-DYNAMICS

We discuss the generation and statistics of the density fluctuations i n highly compressible polytropic turbulence, based on a simple model a nd one-dimensional numerical simulations. Observing that density struc tures tend to form in a hierarchical manner, we assume that density fl uctuations follow a random multiplicative process. When the polytropic exponent gamma is equal to unity, the local Mach number is independen t of the density, and our assumption leads us to expect that the proba bility density function (PDF) of the density field is a log-normal. Th is isothermal case is found to be special, with a dispersion aa scalin g as the square turbulent Mach number (M) over tilde(2), where s=In rh o and rho is the fluid density. Density fluctuations are stronger than expected on the sole basis of shock jumps. Extrapolating the model to the case gamma not equal 1, we find that as the Mach number becomes l arge, the density PDF is expected to asymptotically approach a power-l aw regime at high densities when gamma<1, and at low densities when ga mma>1. This effect can be traced back to the fact that the pressure te rm in the momentum equation varies exponentially with s, thus opposing the growth of fluctuations on one side of the PDF, while being neglig ible on the other side. This also causes the dispersion sigma(s)(2) to grow more slowly than (M) over tilde(2) when gamma not equal 1. In vi ew of these results, we suggest that Burgers flow is a singular case n ot approached by the high-(M) over tilde limit, with a PDF that develo ps power laws on both sides. [S1063-651X(98)16909-X].