Turbulent mixing of a passive scalar

TLDR

The study investigates the statistically stationary state of a passive scalar advected by turbulence with a linear mean gradient in two dimensions using numerical experiments. The authors model turbulence with a synthetic Gaussian velocity field obeying Kolmogorov scaling and a band‑limited Eulerian flow, then examine scalar spectra (5/3 or 17/3) and PDF shapes under hyperdiffusive and physical dissipation, and provide a physical explanation for ramp‑and‑cliff structures. For large Péclet numbers, the scalar gradients exhibit stretched‑exponential PDFs (exponent ~0.6) with O(1) skewness, a ramp‑and‑cliff structure, and exponential tails for physical dissipation, while the dissipation PDF is stretched‑exponential, not lognormal, and the gradient PDFs converge to a limiting form in the white‑noise limit.

Abstract

The statistically stationary state of a turbulently advected passive scalar is studied, with an imposed linear mean gradient in two dimensions, via a number of numerical experiments. For a synthetic Gaussian velocity field, which is generated by a linear stochastic process, and whose spectra and Eulerian correlation time follow Kolmogorov scaling on all scales, the exponents of the scalar spectra are consistent with 5/3 or 17/3 depending on the diffusivity. For large Péclet numbers (Pe), the probability density function (PDF) of the scalar gradients perpendicular to the mean is well fit, from about 0.1–10 times the root-mean-square value, by a stretched exponential with exponent ∼0.6. The PDF for gradients parallel to the mean has similar tails and a 𝒪(1) skewness for all Pe studied. The scalar has a ramp-and-cliff structure similar to that first seen in shear-flow experiments with scalars. A physical picture of the mechanism by which the ramp-and-cliff features form is given. A second model with the velocity evolving under the Euler equations restricted to a band of wave numbers produces the k−1 Batchelor spectrum when the scalar is dissipated with a hyperdiffusivity (∝k4). For physical dissipation (∝k2), the PDF of the scalar has exponential tails, and for gradients less than the cutoff set by the maximum strain, the PDF of the gradients is similar to that obtained with the stochastic velocity model. The PDF of the dissipation is approximately stretched exponential like the gradient PDFs and not lognormal. The skewness of the gradients parallel to the mean decreases with decreasing autocorrelation time of the velocity, and the gradient PDFs assume a limiting form in the white-noise limit.