Density probability distribution in one-dimensional polytropic gas dynamics

We discuss the generation and statistics of the density fluctuations in highly compressible polytropic turbulence, based on a simple model and one-dimensional numerical simulations. Observing that density structures tend to form in a hierarchical manner, we assume that density fluctuations follow a random multiplicative process. When the polytropic exponent gamma is equal to unity, the local Mach number is independent of the density, and our assumption leads us to expect that the probability density function (PDF) of the density field is a log-normal. This isothermal case is found to be special, with a dispersion aa scaling as the square turbulent Mach number (M) over tilde(2), where s=In rho 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 large, the density PDF is expected to asymptotically approach a power-law regime at high densities when gamma<1, and at low densities when gamma>1. This effect can be traced back to the fact that the pressure term in the momentum equation varies exponentially with s, thus opposing the growth of fluctuations on one side of the PDF, while being negligible 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 view of these results, we suggest that Burgers flow is a singular case not approached by the high-(M) over tilde limit, with a PDF that develops power laws on both sides. [S1063-651X(98)16909-X].