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We investigate the effect of particle inertia on the merger of co-rotating dusty vortex pairs at semi-dilute concentrations. In a particle-free flow, the merger is triggered once the ratio of vortex core size to vortex separation reaches a critical value. The vortex pair separation then decreases monotonically until the two cores merge together. Using Eulerian–Lagrangian simulations of co-rotating particle-laden vortices, we show substantial departure from the vortex dynamics previously established in particle-free flows. Most strikingly, we find that disperse particles with moderate inertia cause the vortex pair to push apart to a separation nearly twice as large as the initial separation. During this stage, the drag force exerted by particles ejected out of the vortex cores on the fluid results in a net repulsive force that pushes the two cores apart. Eventually, the two dusty vortices merge into a single vortex with most particles accumulating outside the core, similar to the dusty Lamb–Oseen vortex described in Shuai & Kasbaoui (J. Fluid Mech., vol 936, 2022, p. A8). For weakly inertial particles, we find that the merger dynamics follows the same mechanics as that of a single-phase flow, albeit with a density that must be adjusted to match the mixture density. For highly inertial particles, the feedback force exerted by the particles on the fluid may stretch the two cores during the merger to a point where each core splits into two, resulting in inner and outer vortex pairs. In this case, the merger occurs in two stages where the inner vortices merge first, followed by the outer ones.
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Instability of a dusty vortex
We investigate the effect of inertial particles dispersed in a circular patch of finite radius on the stability of a two-dimensional Rankine vortex in semi-dilute dusty flows. Unlike the particle-free case where no unstable modes exist, we show that the feedback force from the particles triggers a novel instability. The mechanisms driving the instability are characterized using linear stability analysis for weakly inertial particles and further validated against Eulerian–Lagrangian simulations. We show that the particle-laden vortex is always unstable if the mass loading $M>0$ . Surprisingly, even non-inertial particles destabilize the vortex by a mechanism analogous to the centrifugal Rayleigh–Taylor instability in radially stratified vortex with density jump. We identify a critical mass loading above which an eigenmode $$m$$ becomes unstable. This critical mass loading drops to zero as $$m$$ increases. When particles are inertial, modes that fall below the critical mass loading become unstable, whereas modes above it remain unstable but with lower growth rates compared with the non-inertial case. Comparison with Eulerian–Lagrangian simulations shows that growth rates computed from simulations match well the theoretical predictions. Past the linear stage, we observe the emergence of high-wavenumber modes that turn into spiralling arms of concentrated particles emanating out of the core, while regions of particle-free flow are sucked inward. The vorticity field displays a similar pattern which leads to the breakdown of the initial Rankine structure. This novel instability for a dusty vortex highlights how the feedback force from the disperse phase can induce the breakdown of an otherwise resilient vortical structure.
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- Award ID(s):
- 2148710
- PAR ID:
- 10435744
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 948
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2022.687
