dorsal/arxiv
View SchemaDensity probability distribution in one-dimensional polytropic gas dynamics
| Authors | Thierry Passot, Enrique Vazquez-Semadeni |
|---|---|
| Categories | |
| ArXiv ID | physics/9802019 |
| URL | https://arxiv.org/abs/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 $\gamma$ 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 $\sigma_s^2$ which scales like the square turbulent Mach number $\tilde M^2$, where $s\equiv \ln \rho$ and $\rho$ is the fluid density. This leads to much higher fluctuations than those due to shock jump relations. Extrapolating the model to the case $\gamma \not =1$, 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 $\gamma<1$, and at low densities when $\gamma>1$. 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 $\sigma_s^2$ to grow more slowly than $\tilde M^2$ when $\gamma\not=1$. 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.
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"abstract": "We discuss the generation and statistics of the density fluctuations in\nhighly compressible polytropic turbulence, based on a simple model and\none-dimensional numerical simulations. Observing that density structures tend\nto form in a hierarchical manner, we assume that density fluctuations follow a\nrandom multiplicative process. When the polytropic exponent $\\gamma$ is equal\nto unity, the local Mach number is independent of the density, and our\nassumption leads us to expect that the probability density function (PDF) of\nthe density field is a lognormal. This isothermal case is found to be singular,\nwith a dispersion $\\sigma_s^2$ which scales like the square turbulent Mach\nnumber $\\tilde M^2$, where $s\\equiv \\ln \\rho$ and $\\rho$ is the fluid density.\nThis leads to much higher fluctuations than those due to shock jump relations.\n Extrapolating the model to the case $\\gamma \\not =1$, we find that, as the\nMach number becomes large, the density PDF is expected to asymptotically\napproach a power-law regime, at high densities when $\\gamma\u003c1$, and at low\ndensities when $\\gamma\u003e1$. This effect can be traced back to the fact that the\npressure term in the momentum equation varies exponentially with $s$, thus\nopposing the growth of fluctuations on one side of the PDF, while being\nnegligible on the other side. This also causes the dispersion $\\sigma_s^2$ to\ngrow more slowly than $\\tilde M^2$ when $\\gamma\\not=1$. In view of these\nresults, we suggest that Burgers flow is a singular case not approached by the\nhigh-$\\tilde M$ limit, with a PDF that develops power laws on both sides.",
"arxiv_id": "physics/9802019",
"authors": [
"Thierry Passot",
"Enrique Vazquez-Semadeni"
],
"categories": [
"physics.flu-dyn",
"astro-ph",
"chao-dyn",
"nlin.CD"
],
"doi": "10.1103/PhysRevE.58.4501",
"title": "Density probability distribution in one-dimensional polytropic gas dynamics",
"url": "https://arxiv.org/abs/physics/9802019"
},
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