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Settling of inertial particles in turbulent Rayleigh-Bénard convection

Vojtěch Patočka1,*, Enrico Calzavarini2, and Nicola Tosi1

  • 1Institute of Planetary Research, German Aerospace Center (DLR), Berlin, Germany
  • 2Université de Lille, ULR 7512-Unité de Mécanique de Lille-Joseph Boussinesq (UML), F-59000 Lille, France

  • *patocka.vojtech@gmail.com
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Phys. Rev. Fluids 5, 114304 – Published 11 November, 2020

DOI: https://doi.org/10.1103/PhysRevFluids.5.114304

Abstract

The settling behavior of small inertial particles in turbulent convection is a fundamental problem across several disciplines, from geophysics to metallurgy. In a geophysical context, the settling of dense crystals controls the mode of solidification of magma chambers and planetary scale magma oceans, while rising of light bubbles of volatiles drives volcanic outgassing and the formation of primordial atmospheres. Motivated by these geophysical systems, we perform a systematic numerical study on the settling rate of particles in a rectangular two-dimensional Rayleigh-Bénard system with Rayleigh number up to 1012 and Prandtl number from 10 to 50. Under the idealized condition of spherically shaped particles with small Reynolds number, two limiting behaviors exist for the settling velocity. On the one hand, Stokes law applies to particles with small but finite response time, leading to a constant settling rate. On the other hand, particles with a vanishing response time are expected to settle at an exponential rate. Based on our simulations, we present a physical model that bridges the gap between the above limiting behaviors by describing the sedimentation of inertial particles as a random process with two key components: (i) the transport of particles from vigorously convecting regions into sluggish, low-velocity “piles” that naturally develop at the horizontal boundaries of the system, and (ii) the probability that particles escape such low-velocity regions without settling at their base. In addition, we identify four distinct settling regimes and analyze the horizontal distribution of sedimented particles. For two of these regimes settling is particularly slow and the distribution is strongly nonuniform, with dense particles being deposited preferentially below major clusters of upwellings. Finally, we apply our results to the crystallization of a magma ocean. Our prediction of the characteristic residence times is consistent with fractional crystallization, i.e., with the efficient separation of dense crystals from the residual lighter fluid. In absence of an efficient mechanism to reentrain settled particles, equilibrium crystallization appears possible only for particles with extremely small density contrasts.

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Phys. Rev. Fluids 5, 114304– Published 11 November, 2020
  • Received 14 April 2020
  • Accepted 8 October 2020
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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.

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