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 $\ensuremath{\gamma}$ 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 ${\ensuremath{\sigma}}_{s}^{2}$ scaling as the square turbulent Mach number ${M}^{2},$ where $s\ensuremath{\equiv}\mathrm{ln}\ensuremath{\rho}$ and $\ensuremath{\rho}$ is the fluid density. Density fluctuations are stronger than expected on the sole basis of shock jumps. Extrapolating the model to the case $\ensuremath{\gamma}\ensuremath{\ne}1,$ 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 $\ensuremath{\gamma}<1,$ and at low densities when $\ensuremath{\gamma}>1.$ 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 ${\ensuremath{\sigma}}_{s}^{2}$ to grow more slowly than ${M}^{2}$ when $\ensuremath{\gamma}\ensuremath{\ne}1.$ In view of these results, we suggest that Burgers flow is a singular case not approached by the high-$\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{M}$ limit, with a PDF that develops power laws on both sides.