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 γ 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 σₛ² scaling as the square turbulent Mach number M², where s≡lnρ and ρ is the fluid density. Density fluctuations are stronger than expected on the sole basis of shock jumps. Extrapolating the model to the case γ≠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 γ<1, and at low densities when γ>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 σₛ² to grow more slowly than M² when γ≠1. In view of these results, we suggest that Burgers flow is a singular case not approached by the high-̃ \~M limit, with a PDF that develops power laws on both sides.