Caustics-induced coalescence of small droplets near a vortex

Deepu, P.; Ravichandran, S.; Govindarajan, Rama

doi:10.1103/physrevfluids.2.024305

Cited by 15 publications

(21 citation statements)

References 23 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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“…Previous efforts to study dynamics of heavy, inertial particles in vortices were made by simulating droplet trajectories in a 15 prescribed velocity field for several simple single vortex models. Such research for the simplest model of a line vortex with stretching was conducted by Markowicz et al (2000) with limitation to horizontally oriented vortices.…”

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“…Images covering the approximately 2 m long section of the sheet at a distance approximately 10 m from the source were taken with a Nikon D3S DSLR 12 Mpix camera. 15 Laser sheet photography was accompanied by high-resolution measurements of small-scale turbulence and cloud microphysics, as described in Siebert et al (2015). Flow turbulence was measured by 3D ultrasonic anemometers operated at 10 Hz, from which the velocity structure functions were calculated using Taylor's frozen-flow hypothesis, and the energy dissipation rates were determined using inertial range scaling.…”

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“…Nonparallel alignment of the gravity vector and vortex axis (θ = 0) destroys the axial symmetry of the system and introduces 15 the presence of other attractors, such as limit cycles and equilibrium points outside the axis.…”

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“…(10) 15 The condition for unstable points near the axis splits in parts. The first part concerns only vortex parameters and the second part particle sizes.…”

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Turbulence Induced Cloud Voids: Observation and Interpretation

Karpińska¹,

Bodenschatz²,

Malinowski³

et al. 2018

Preprint

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<p><strong>Abstract.</strong> The phenomenon of <q>cloud voids</q>, i.e., elongated volumes inside a cloud that are devoid of droplets, was observed with laser sheet photography in clouds at a mountain-top station. Two experimental cases, similar in turbulence conditions yet with diverse droplet size distributions and cloud void prevalence, are reported. A theoretical explanation is proposed based on the study of heavy inertial sedimenting particles inside a Burgers vortex. A general conclusion regarding void appearance is drawn from theoretical analysis. Numerical simulations of polydisperse droplet motion with realistic vortex parameters and Mie scattering visual effects accounted for can explain the presence of voids with sizes similar to that of the observed ones. Preferential concentration and sorting effects in a vortex tube are discussed for reasonable cloud conditions.</p>

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