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 log-normal. This isothermal case is found to be special, with a dispersion σ2s scaling as the square turbulent Mach number M~2, where s≡ ln ρ and ρ is the fluid density. Density fluctuations are stronger than expected on the sole basis of shock jumps. Extrapolating the model to the case γ≠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 σ2s to grow more slowly than M~2 when γ≠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.
- Publication:
-
Physical Review E
- Pub Date:
- October 1998
- DOI:
- 10.1103/PhysRevE.58.4501
- arXiv:
- arXiv:physics/9802019
- Bibcode:
- 1998PhRvE..58.4501P
- Keywords:
-
- 47.27.Ak;
- 47.40.Ki;
- 95.30.Lz;
- Fundamentals;
- Supersonic and hypersonic flows;
- Hydrodynamics;
- Physics - Fluid Dynamics;
- Astrophysics;
- Nonlinear Sciences - Chaotic Dynamics
- E-Print:
- 9 pages + 12 postscript figures. Submitted to Phys. Rev. E