Density probability distribution in one-dimensional polytropic gas dynamics

Thierry Passot; Enrique Vazquez-Semadeni

doi:10.1103/PhysRevE.58.4501

paper

Density probability distribution in one-dimensional polytropic gas dynamics

arXiv:physics/9802019 · doi:10.1103/PhysRevE.58.4501

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 $M ~^{2}$ , 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 $γ \neq = 1$ , 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 $M ~^{2}$ when $γ \neq = 1$ . 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.

9 pages + 12 postscript figures. Submitted to Phys. Rev. E

Density probability distribution in one-dimensional polytropic gas dynamics

Abstract

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