[Paper Review] Density PDFs of Super-Sonic Turbulence

Aake Nordlund, Paolo Padoan|arXiv (Cornell University)|Oct 6, 1998

Gas Dynamics and Kinetic Theory3 citations

TL;DR

This paper investigates the probability density function (PDF) of gas density in supersonic turbulence using analytical and numerical methods. It finds that isothermal supersonic turbulence produces a log-normal PDF whose width scales linearly with Mach number, while polytropic turbulence with γ < 1 yields a skewed PDF resembling a power law at high densities.

ABSTRACT

The question of the shape of the density PDF for supersonic turbulence is addressed, using both analytical and numerical methods. For isothermal supersonic turbulence, the PDF is Log-Normal, with a width that scales approximately linearly with the Mach number. For a polytropic equation of state, with an effective gamma smaller than one, the PDF becomes skewed and reminiscent of (but not equal to) a power law on the high density side.

Motivation & Objective

• To determine the functional form of the density probability density function (PDF) in supersonic turbulent flows.
• To understand how the equation of state (specifically γ < 1) affects the shape of the density PDF.
• To establish the scaling of PDF width with Mach number in isothermal supersonic turbulence.
• To compare analytical predictions with numerical simulations of turbulent gas dynamics.
• To assess whether the high-density tail of the PDF follows a power-law behavior under polytropic conditions.

Proposed method

• Analytical modeling of density statistics in supersonic turbulence using the log-normal approximation for isothermal gas.
• Numerical simulations of compressible, supersonic turbulence with varying Mach numbers and equations of state.
• Use of a polytropic equation of state with effective γ < 1 to model non-isothermal conditions.
• Statistical analysis of simulated density fields to extract the PDF shape and scaling behavior.
• Comparison of simulated PDFs with theoretical predictions, including log-normal and power-law forms.
• Quantification of PDF width as a function of Mach number for isothermal and polytropic cases.

Experimental results

Research questions

• RQ1 What is the functional form of the density PDF in isothermal supersonic turbulence?
• RQ2 How does the PDF width scale with the Mach number in isothermal supersonic flows?
• RQ3 Does the high-density tail of the PDF in polytropic turbulence exhibit power-law behavior?
• RQ4 How does a polytropic equation of state with γ < 1 affect the skewness and shape of the density PDF?
• RQ5 To what extent do numerical simulations confirm analytical predictions for density PDFs in supersonic turbulence?

Key findings

• For isothermal supersonic turbulence, the density PDF is well described by a log-normal distribution.
• The width of the log-normal PDF scales approximately linearly with the Mach number.
• In polytropic turbulence with an effective γ < 1, the PDF becomes skewed and develops a high-density tail resembling a power law.
• The power-law-like behavior in the high-density regime is not exact but qualitatively similar to a power law.
• The transition from log-normal to skewed PDF occurs when the equation of state deviates from isothermality.
• Numerical simulations confirm the analytical predictions for both isothermal and polytropic cases.