[Paper Review] A numerical investigation on active and passive scalars in isotropic compressible turbulence

Qionglin Ni, Yipeng Shi+1 more|arXiv (Cornell University)|May 11, 2015

Fluid Dynamics and Turbulent Flows40 references3 citations

TL;DR

This study investigates active and passive scalar transport in isotropic compressible turbulence using direct numerical simulations, revealing that active scalars exhibit stronger intermittency and distinct cascade mechanisms dominated by pressure-dilatation and viscous dissipation, while passive scalars follow a solenoidal-dominated, energy-like cascade. Key findings include Kolmogorov $k^{-5/3}$ scaling with $C_{OC} = 0.87$, super-Gaussian active scalar statistics, and scale-invariant subgrid flux statistics across both scalar types.

ABSTRACT

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.

Motivation & Objective

• To understand the statistical and dynamical differences between active and passive scalars in isotropic compressible turbulence.
• To quantify the influence of compressibility on scalar transport and cascade mechanisms.
• To examine scaling laws, intermittency, and structure functions for both scalar types.
• To analyze subgrid-scale flux statistics and their scale-invariant behavior.

Proposed method

• Direct numerical simulation (DNS) of isotropic compressible turbulence with solenoidal forcing at large scales.
• Use of coarse-graining to compute subgrid-scale (SGS) fluxes and variance budgets across scales.
• Computation of structure functions, correlation coefficients, and probability density functions (p.d.f.s) for scalar increments and dissipation rates.
• Application of multifractal analysis to predict scaling exponents and assess intermittency via auto-correlation of dissipation.
• Decomposition of pressure-dilatation into compressive and solenoidal components to isolate their roles in scalar cascades.
• Rescaling of SGS flux p.d.f.s to test for collapse and scale-invariance in scalar transport statistics.

Experimental results

Research questions

• RQ1 How do the scaling exponents of velocity and scalar structure functions differ between active and passive scalars in compressible turbulence?
• RQ2 What is the role of pressure-dilatation and viscous dissipation in the cascade of active scalars compared to passive scalars?
• RQ3 How does the intermittency of active and passive scalars compare, as measured by p.d.f. shapes and dissipation auto-correlation?
• RQ4 To what extent do the subgrid-scale fluxes of active and passive scalars exhibit scale-invariant statistics in the inertial range?
• RQ5 How do the compressive and solenoidal components of velocity influence the transport and cascade of active and passive scalars?

Key findings

• The kinetic energy spectrum and scalar spectra follow the $k^{-5/3}$ power law in the inertial range, with Kolmogorov and Obukhov-Corrsin constants of 2.06 and 0.87, respectively.
• The local scaling exponent of the second-order structure function for passive scalar shows a minimum of 0.61 and a maximum of 0.73, indicating non-Gaussian, intermittent behavior.
• The mixed third-order structure function of velocity and passive scalar satisfies Yaglom’s $4/3$-law, confirming the existence of a forward cascade in scalar fluctuations.
• The probability distribution function (p.d.f.) of active scalar fluctuations is super-Gaussian, while that of passive scalar is sub-Gaussian, reflecting stronger intermittency in active scalars.
• The p.d.f.s of scalar increments are concave (active) and convex (passive), indicating strong small-scale intermittency that diminishes with increasing scale.
• In the inertial range, rescaled subgrid-scale flux p.d.f.s for both active and passive scalars collapse to a single distribution, demonstrating scale-invariant statistics.