Caustics and clustering in the vicinity of a vortex

Ravichandran, S.; Govindarajan, Rama

doi:10.1063/1.4916583

Cited by 39 publications

(42 citation statements)

References 30 publications

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“…Close to a point vortex, upon neglecting the Basset history, the dynamics of particles has been shown to obey a boundary-layer structure (Ravichandran & Govindarajan 2015;Deepu et al 2017), where particles initially located within a critical radius r c are able to form caustics and evacuate the vicinity of the vortex rapidly, whereas particles initially outside this radius do not form caustics, and so their dynamics may be represented by a velocity field. We define caustics as space-time points where two particles can exist simultaneously with different velocities.…”

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confidence: 99%

Accurate solution method for the Maxey–Riley equation, and the effects of Basset history

2019

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The Maxey-Riley equation has been extensively used by the fluid dynamics community to study the dynamics of small inertial particles in fluid flow. However, most often, the Basset history force in this equation is neglected. Analytical solutions have almost never been attempted because of the difficulty in handling an integro-differential equation of this type. Including the Basset force in numerical solutions of particulate flows involves storage requirements which rapidly increase in time. Thus the significance of the Basset history force in the dynamics has not been understood. In this paper, we show that the Maxey-Riley equation in its entirety can be exactly mapped as a forced, time-dependent Robin boundary condition of the one-dimensional diffusion equation, and solved using the Unified Transform Method.We obtain the exact solution for a general homogeneous time-dependent flow field, and apply it to a range of physically relevant situations. In a particle coming to a halt in a quiescent environment, the Basset history force speeds up the decay as stretchedexponential at short time while slowing it down to a power-law relaxation, ∼ t −3/2 , at long time. A particle settling under gravity is shown to relax even more slowly to its terminal velocity (∼ t −1/2 ), whereas this relaxation would be expected to take place exponentially fast if the history term were to be neglected. An important mechanism for the growth of rain drops is by the gravitational settling of larger drops through an environment of smaller droplets, and repeatedly colliding and coalescing with them. Using our solution we estimate that the rate of growth rate of a rain drop can be gross overestimated when history effects are not accounted. We solve exactly for particle motion in a plane Couette flow and show that the location (and final velocity) to which a particle relaxes is different from that due to Stokes drag alone.For a general flow, our approach makes possible a numerical scheme for arbitrary but smooth flows without increasing memory demands and with spectral accuracy. We use our numerical scheme to solve an example spatially varying flow of inertial particles in the vicinity of a point vortex. We show that the critical radius for caustics formation shrinks slightly due to history effects.Our scheme opens up a method for future studies to include the Basset history term in their calculations to spectral accuracy, without astronomical storage costs. Moreover our results indicate that the Basset history can affect dynamics significantly.

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Accurate solution method for the Maxey–Riley equation, and the effects of Basset history

2019

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“…) We see clearly that starting from an initial profile in which particles are uniformly distributed in space, the particles evacuate rapidly from regions close to the core of the vortex. (The case of σ = 0 is validated against Ravichandran and Govindarajan [25].) Therefore, at some later time (t = 0.6τ η ), Φ is essentially 0 at small values of r before sharply peaking at r .…”

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“…A scale-free version (again following the scaling used in Ref. [25]) where the dependence on the Taylor Reynolds number is apparent is shown in the Appendix. The decrease in the critical Stokes number with increasing stretch- FIG.…”

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Understanding droplet collisions through a model flow: Insights from a Burgers vortex

et al. 2019

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We investigate the role of intense vortical structures, similar to those in a turbulent flow, in enhancing collisions (and coalescences) which lead to the formation of large aggregates in particle-laden flows. By using a Burgers vortex model, we show, in particular, that vortex stretching significantly enhances sharp inhomogeneities in spatial particle densities, related to the rapid ejection of particles from intense vortices. Furthermore our work shows how such spatial clustering leads to an enhancement of collision rates and extreme statistics of collisional velocities. We also study the role of poly-disperse suspensions in this enhancement. Our work uncovers an important principle which, if valid for realistic turbulent flows, may be a factor in how small nuclei water droplets in warm clouds can aggregate to sizes large enough to trigger rain. arXiv:1811.04486v2 [physics.flu-dyn]

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“…They can also attain relative velocities much larger than that of the underlying flow. Dubbed the sling effect [41], these events correspond to the formation of singularities or caustics in the particle velocity field [42,43]. Although clustering and caustics have been tied to the centrifugal ejection of heavy particles out of vortices, they also occur in smooth random flows that are devoid of structure [44][45][46][47][48].…”

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“…One possible explanation is provided by preferential concentration: Heavy particles are centrifuged out of rotational regions, and thus tend to accumulate in straining zones just outside vortices [43]. This causes the number density to increase in straining regions, at the expense of vortical zones, as shown in Fig.…”

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Flow structures govern particle collisions in turbulence

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The role of the spatial structure of a turbulent flow in enhancing particle collision rates in suspensions is an open question. We show and quantify, as a function of particle inertia, the correlation between the multiscale structures of turbulence and particle collisions: Straining zones contribute predominantly to rapid head-on collisions compared to vortical regions. We also discover the importance of vortex-strain worm-rolls, which goes beyond ideas of preferential concentration and may explain the rapid growth of aggregates in natural processes, such as the initiation of rain in warm clouds. * jrpicardo@icts.res.in;

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Caustics and clustering in the vicinity of a vortex

Cited by 39 publications

References 30 publications

Accurate solution method for the Maxey–Riley equation, and the effects of Basset history

Accurate solution method for the Maxey–Riley equation, and the effects of Basset history

Understanding droplet collisions through a model flow: Insights from a Burgers vortex

Flow structures govern particle collisions in turbulence

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