Fluctuations of thermodynamic variables in stationary compressible turbulence
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AbstractA large database of new direct numerical simulations of forced compressible turbulence on up to$204{8}^{3} $grids, and a range of Reynolds (${R}_{lambda } $) and turbulent Mach (${M}_{t} $) numbers, is analysed to study the scaling of pressure, density and temperature fluctuations. Small-perturbation analysis is used to study the scaling of variances, and different cross-correlations as well as spectra. Qualitative differences are observed between low and high${M}_{t} $. The probability density functions (p.d.f.s) of pressure and density are negatively skewed at low${M}_{t} $(consistent with incompressible results) but become positively skewed at high${M}_{t} $. The positive tails are found to follow a log-normal distribution. A new variable is introduced to quantify departures from isentropic fluctuations (an assumption commonly used in the literature) and is found to increase as${ M}_{t}^{2} $. However, positive fluctuations of pressure and density tend to be more isentropic than negative fluctuations. In general, Reynolds number effects on single-point statistics are observed to be weak. The spectral behaviour of pressure, density and temperature is also investigated. While at low${M}_{t} $, pressure appears to scale as${k}^{- 7/ 3} $($k$is the wavenumber) in the inertial range as in incompressible flows, a${k}^{- 5/ 3} $scaling also appears to be consistent with the data at a range of Mach numbers. Density and temperature spectra are found to scale as${k}^{- 5/ 3} $for a range of Mach numbers.
