363 research outputs found

    Density probability distribution in one-dimensional polytropic gas dynamics

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    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\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 lognormal. This isothermal case is found to be singular, with a dispersion σs2\sigma_s^2 which scales like the square turbulent Mach number M~2\tilde M^2, where slnρs\equiv \ln \rho and ρ\rho is the fluid density. This leads to much higher fluctuations than those due to shock jump relations. Extrapolating the model to the case γ1\gamma \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\gamma<1, and at low densities when γ>1\gamma>1. This effect can be traced back to the fact that the pressure term in the momentum equation varies exponentially with ss, 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\sigma_s^2 to grow more slowly than M~2\tilde M^2 when γ1\gamma\not=1. In view of these results, we suggest that Burgers flow is a singular case not approached by the high-M~\tilde M limit, with a PDF that develops power laws on both sides.Comment: 9 pages + 12 postscript figures. Submitted to Phys. Rev.

    Bioaccessibility of metals in soils surrounding two dismissed mining sites in Northern Italy

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