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A numerical investigation on active and passive scalars in isotropic compressible turbulence

In this paper, we investigated the statistical differences between active and passive scalars in isotropic compressible turbulence. In the inertial range, the kinetic energy and scalar have the Kolmogorov spectra, and the Kolmogorov and Obukhov-Corrsin constants are 2.06 and 0.87, respectively. The local scaling exponents computed from the second-order structure functions exist plateaus for velocity and active scalar, while that for passive scalar takes first a minimum of 0.61 then a maximum of 0.73. The mixed third-order structure function of velocity and passive scalar satisfies the 4/3-law. For the scaling exponent, the one of velocity and passive scalar mixing is between those of velocity and passive scalar, while the one of velocity and active scalar mixing is below those of velocity and active scalar. At large amplitudes, the p.d.f. of active scalar fluctuations is super-Gaussian, whereas that of passive scalar fluctuations is sub-Gaussian. Moreover, the p.d.f.s of the two scalar increments are concave and convex shapes, respectively, which exhibit strong intermittency at small scales, and approach Gaussian as scale increases. The active scalar has "ramp-cliff" structures, while the passive scalar seems to be dominated by rarefaction and compression. By employing a "coarse-graining" approach, we study scalar cascades. Unlike passive scalar, the cascade of active scalar is mainly determined by the viscous dissipation at small scales and the pressure-dilatation at large scales, where the latter is substantial in the vicinity of small-scale shocklets but is negligible after space averages, because of the cancelations between rarefaction and compression regions. Finally, in the inertial range, the p.d.f.s for the subgrid-scale fluxes of scalars can collapse to the same distribution, revealing the scale-invariant feature for the statistics of active and passive scalars.