Mach number study of supersonic turbulence: the properties of the density field

Abstract

We analyse the scaling properties of turbulent flows using a suite of three-dimensional numerical simulations. We model driven, compressible, isothermal, turbulence with Mach numbers ranging from the subsonic (⁠|$\mathcal {M} \approx 0.5$|⁠) to the highly supersonic regime (⁠|$\mathcal {M}\approx 16$|⁠). The forcing scheme consists of both solenoidal (transverse) and compressive (longitudinal) modes in equal parts. We confirm the relation |$\sigma _{s}^2 = \ln {(1+b^2\mathcal {M}^2)}$| between the Mach number and the standard deviation of the logarithmic density with b = 0.33. We find increasing deviations with higher Mach number from the predicted lognormal shape in the high-density wing of the density probability density function. The density spectra follow |$\mathcal {D}(k,\,\mathcal {M}) \propto k^{\zeta (\mathcal {M})}$| with scaling exponents depending on the Mach number. We find |$\zeta (\mathcal {M}) = \alpha \mathcal {M}^{\beta }$| with coefficients α = −2.1 and β = −0.33. The dependence of the scaling exponent on the Mach number implies a fractal dimension |$D=2+1.05 \mathcal {M}^{-0.33}$|⁠.