Density probability distribution in one-dimensional polytropic gas dynamics

Abstract

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 lognormal. This isothermal case is found to be singular, with a dispersion $σ_s^2$ which scales like the square turbulent Mach number $\tilde M^2$, where $s\equiv \ln ρ$ and $ρ$ is the fluid density. This leads to much higher fluctuations than those due to shock jump relations. Extrapolating the model to the case $γ\not =1$, 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 $σ_s^2$ to grow more slowly than $\tilde M^2$ when $γ\not=1$. In view of these results, we suggest that Burgers flow is a singular case not approached by the high-$\tilde M$ limit, with a PDF that develops power laws on both sides.