Statistical properties of compressible isothermal turbulence from sub- to supersonic conditions

TL;DR

This paper addresses the statistical properties of isothermal compressible turbulence from subsonic to supersonic flows, focusing on how Mach number $Ma$ and Reynolds number $Re$ shape density, velocity, and their gradients. Using high-fidelity DNS with explicit viscosity in a periodic domain, spanning $Ma \in [0.25,4]$ and $Re \lesssim 4166$, and employing a density-variant KHM framework, the authors analyze PDFs and scale-by-scale kinetic-energy budgets. They find density fluctuations are approximately log-normal in $s=\ln \rho$, with variance $\sigma_s^2 \approx \ln(1+b^2 Ma^2)$ and weak $Re$-dependence for $Re>200$, while derivatives $\nabla \rho$ and $\theta$ show strong intermittency and $Re$-dependent tails; at high $Ma$, dilatation statistics saturate. In the scale-by-scale view, subsonic regimes exhibit Kolmogorov-like scaling while supersonic regimes show Burgers-like scaling in the inertial range; the pressure–velocity coupling and dilatation terms contribute non-monotonically with $Ma$ and saturate due to viscosity, consistent with a compressible generalization of the $4/5$ law in the high-$Re$ limit.

Abstract

This paper investigates the statistical properties of isothermal turbulence in both the subsonic and supersonic regimes. The focus is on the influence of the Mach number ($Ma$) and the Reynolds number ($Re$) on both the space-local and scale-dependent fluctuations of relevant gas variables, the density, velocity, their derivatives, and the kinetic energy. We carry out hydrodynamical simulations with explicit viscosity and therefore controlled $Re$. We confirm previous work that the probability density functions (PDFs) of the gas density are approximately log-normal and depend on $Ma$. In contrast, derivatives of the density and velocity field are sensitive to $Re$, with the probability of extreme events growing with $Re$. The PDFs of the density gradient and velocity divergence (dilatation) exhibit increasingly heavy tails with growing $Ma$, signalling enhanced internal intermittency. At sufficiently high $Ma$, the statistics of dilatation are observed to saturate at a level determined solely by $Re$, suggesting that turbulent dilatation becomes limited by viscous effects. We also examine the scale-by-scale distribution of kinetic energy through a compressible form of the Kármán-Howarth-Monin (KHM) equation. In the intermediate range of scales, a marked difference is found between subsonic and supersonic turbulence: while Kolmogorov-like scaling applies in the sub- and transonic regimes, supersonic turbulence aligns more closely with Burgers turbulence predictions. The analysis of individual terms in the KHM equation highlights the role of the pressure-velocity coupling as an additional mechanism for converting kinetic energy from large to small scales. Moreover, the contributions of the KHM terms exhibit non-monotonic behaviour with increasing $Ma$, with dilatational effects becoming more pronounced and acting to oppose the cascade of kinetic energy.

Figures (8)

• Figure 1: Two-dimensional slices of the density field $s = \ln\rho$ for increasing $Ma$ (from left to right) and increasing $Re$ (from top to bottom). The colorbar spans $\pm 3\sigma_s$.
• Figure 2: (a) Probability-density-function of $s = \ln \rho$ for varying Mach and Reynolds numbers. Colours from blue to red correspond to increasing Reynolds numbers as shown in the legend. The grey dashed lines are the associated Gaussian distributions (Eq. \ref{['eq:log_normal_s']}). (b) Evolution of the variance $\sigma_s^2$ with respect to $Ma$. The prediction (Eq. \ref{['eq:sigmas_prediction']}) is also plotted for different type of forcing (compressive, solenoidal, mixed). The shaded regions represent the statistical uncertainty which affects only the far tails of the PDFs (a) and thus marginally the variance $\sigma_s^2$ (b).
• Figure 3: Probability-density-function of the local Mach number $|\boldsymbol{u}|/c_s$ for varying $Ma$ and $Re$. The shaded regions represent the statistical uncertainty. The grey dashed lines are the associated Maxwellian distributions Eq. \ref{['eq:log_maxwell']}.
• Figure 4: Probability-density-function of the density gradient $\boldsymbol{\nabla} \rho$ for varying $Ma$ and $Re$. Colours from blue to red correspond to increasing $Re$ as shown in Fig. \ref{['fig:PDF_rho']}. The shaded regions represent the statistical uncertainty. Panels (a) to (e) correspond to $Ma=0.25$ to $Ma=4$.
• Figure 6: Mach-number dependence of the variance of (a) the density gradient $\boldsymbol{\nabla} \rho$ (in units of $\rho_o/L \equiv 1$) and (b) dilatation $\theta$ (in units of $u'/L \equiv 1$) for varying $Re$. In panel (b), the triangle symbols represent the data reported by Ref. Scannapieco2024 when rescaled in units of $(u'/L)^2$. The closed (open) symbol represent the volume (density) weighted variance.