Power and nonpower laws of passive scalar moments convected by isotropic turbulence

Transition of fluctuations from Gaussian state to turbulent state

Variation of the statistical properties of an incompressible velocity, passive vector and passive scalar in isotropic turbulence was studied using direct numerical simulation. The structure functions of the gradients, and the moments of the dissipation rates, began to increase at about [Formula: see text] from the Gaussian state and grew rapidly at [Formula: see text] in the turbulent state. A contour map of the probability density functions (PDFs) indicated that PDF expansion of the gradients of the passive vector and passive scalar begins at around [Formula: see text], whereas that of the longitudinal velocity gradient PDF is more gradual. The left tails of the dissipation rate PDF were found to follow a power law with an exponent of 3/2 for the incompressible velocity and passive vector dissipation rates, and 1/2 for the scalar dissipation rate and the enstrophy; they remained constant for all Reynolds numbers, indicating the universality of the left tail. The analytical PDFs of the dissipation rates and enstrophy of the Gaussian state were obtained and found to be the Gamma distribution. It was shown that the number of terms contributing to the dissipation rates and the enstrophy determines the decay rates of the two PDFs for low to moderate amplitudes. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

Extraction of invariant manifolds and application to turbulence with a passive scalar

The reduction of dimensionality of physical systems, especially in fluid dynamics, leads in many situations to nonlinear ordinary differential equations which have global invariant manifolds with algebraic expressions containing relevant physical information on the original system. We present a method to identify such manifolds, and we apply it to a reduced model for the Lagrangian evolution of field gradients in homogeneous and isotropic turbulence with a passive scalar.

Turbulent mixing: A perspective

Mixing of initially distinct substances plays an important role in our daily lives as well as in ecological and technological worlds. From the continuum point of view, which we adopt here, mixing is complete when the substances come together across smallest flow scales determined in part by molecular mechanisms, but important stages of the process occur via the advection of substances by an underlying flow. We know how smooth flows enable mixing but less well the manner in which a turbulent flow influences it; but the latter is the more common occurrence on Earth and in the universe. We focus here on turbulent mixing, with more attention paid to the postmixing state than to the transient process of initiation. In particular, we examine turbulent mixing when the substance is a scalar (i.e., characterized only by the scalar property of its concentration), and the mixing process does not influence the flow itself (i.e., the scalar is "passive"). This is the simplest paradigm of turbulent mixing. Within this paradigm, we discuss how a turbulently mixed state depends on the flow Reynolds number and the Schmidt number of the scalar (the ratio of fluid viscosity to the scalar diffusivity), point out some fundamental aspects of turbulent mixing that render it difficult to be addressed quantitatively, and summarize a set of ideas that help us appreciate its physics in diverse circumstances. We consider the so-called universal and anomalous features and summarize a few model studies that help us understand them both.

Steep Cliffs and Saturated Exponents in Three-Dimensional Scalar Turbulence

The intermittency of a passive scalar advected by three-dimensional Navier-Stokes turbulence at a Taylor-scale Reynolds number of 650 is studied using direct numerical simulations on a 4096^{3} grid; the Schmidt number is unity. By measuring scalar increment moments of high orders, while ensuring statistical convergence, we provide unambiguous evidence that the scaling exponents saturate to 1.2 for moment orders beyond about 12, indicating that scalar intermittency is dominated by the most singular shocklike cliffs in the scalar field. We show that the fractal dimension of the spatial support of steep cliffs is about 1.8, whose sum with the saturation exponent value of 1.2 adds up to the space dimension of 3, thus demonstrating a deep connection between the geometry and statistics in turbulent scalar mixing. The anomaly for the fourth and sixth order moments is comparable to that in the Kraichnan model for the roughness exponent of 4/3.

Scale-similar clustering of heavy particles in the inertial range of turbulence

Heavy particle clustering in turbulence is discussed from both phenomenological and analytical points of view, where the -4/3 power law of the pair-correlation function is obtained in the inertial range. A closure theory explains the power law in terms of the balance between turbulence mixing and preferential-concentration mechanism. The obtained -4/3 power law is supported by a direct numerical simulation of particle-laden turbulence.