Small-scale averaging coarse-grains passive scalar turbulence

Open Access

Small-scale averaging coarse-grains passive scalar turbulence

Tobias Bätge and Michael Wilczek

Phys. Rev. Fluids 6, 064503 – Published 28 June 2021

Abstract

Scalar fields which are subject to turbulent mixing typically feature a broad range of scales. When focusing on the large-scale dynamics, it remains a question how to effectively parametrize the small scales. Here, we address this question within the framework of a stochastic, one-dimensional passive scalar model. We show that small-scale averaging, i.e., an ensemble average over small-scale velocity fluctuations, results in an effective diffusivity reminiscent of phenomenological eddy viscosity models, while reducing the effective Reynolds number of the advecting velocity field. Based on that, we establish a filtering procedure that exactly maps second-order statistics of the fully resolved passive scalar field to the one obtained by small-scale averaging. Using fully resolved simulations, we show that small-scale averaging also captures higher-order large-scale statistics of passive scalar fields.

Received 23 October 2020
Accepted 14 May 2021

DOI:https://doi.org/10.1103/PhysRevFluids.6.064503

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Nonequilibrium statistical mechanics Stochastic processes Turbulence Turbulent mixing

Stochastic dynamical systems

Stochastic differential equations

Fluid DynamicsNonlinear DynamicsStatistical Physics & Thermodynamics

Authors & Affiliations

Tobias Bätge and Michael Wilczek ^*

Max Planck Institute for Dynamics and Self-Organization, (MPI DS), Am Faßberg 17, 37077 Göttingen, Germany and Faculty of Physics, University of Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany

^*michael.wilczek@ds.mpg.de

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Vol. 6, Iss. 6 — June 2021

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Images

Figure 1
Fully resolved, filtered, and small-scale-averaged scalar fields (the latter two are defined below) and corresponding gradient fields of an example realization resolved with $N = 2^{15}$ grid points. The extreme events in the form of cliffs in the scalar field are represented as strong peaks in the gradient field. On the large scales, all three scalar fields share a similar structure. The zoom into one cliff structure highlights the differences between the three field types on the scale of the filter length $λ_{c}$ .
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Figure 2
Spectral characteristics of the fully resolved passive scalar system, its filtered version, and the small-scale-averaged version. (a) The top panel shows the spectrum of the advecting velocity field $u$ and illustrates the splitting into small-scale and large-scale components, $u^{'}$ and $U$ , respectively. Here, $k_{c} = 128 Δ k$ denotes the wave number separating small and large scales, which in the following also defines the filter length scale. In the bottom panel, the fully resolved scalar spectrum $C_{k}$ , filtered scalar spectrum ${\tilde{C}}_{k}$ , and small-scale-averaged scalar spectrum ${\hat{C}}_{k}$ obtained from a simulation are compared to the respective analytical solutions (dashed gray lines) [see Eqs. (18, 19, 20)]. For wave numbers larger than the filter wave number, the small-scale-averaged and filtered spectrum drop off relative to the fully resolved one with minor deviations among each other at the end of the spectral cutoff. (b) Individual contributions of the balance between the small-scale-averaged and filtered spectral energy budget, Eq. (26), normalized by the forcing amplitude $F (0)$ . Overall, the sum of the small-scale-averaged spectral energy flux ${\hat{Π}}_{k}$ and the small-scale-averaged spectral energy dissipation ${\hat{ε}}_{k}$ add up to the spectral energy flux of the filtered passive scalar, showing that SSA can capture the spectral characteristics of the filtered spectral energy budget. The minor deviation in the balance of small-scale-averaged flux and dissipation to the filtered flux contribution, Eq. (26), is highlighted with an inset; it can be traced back to the fact that our filter is not a perfect low-pass filter. It commutes with the large-scale forcing only approximately.
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Figure 3
Statistics of the fully resolved, filtered, and small-scale-averaged scalar fields. (a) All three one-point scalar PDFs are close to Gaussian, here marked by a black dashed line. Note that the PDFs of the full and the filtered fields are almost indistinguishable. (b) The gradient PDFs show the characteristic heavy tails and a good agreement between filtered and small-scale-averaged PDFs up to the outer tails, which are significantly less heavy than those of the fully resolved, unfiltered fields. The transparent areas correspond to confidence intervals of 95% obtained by a basic bootstrap [37]. (c) For increments larger than the filter length, all three scalar fields show a similar flatness, approaching the Gaussian value for large increments (gray line). Here, the transparent areas also correspond to the confidence intervals of 95%.
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