Attracting fixed points for heavy particles in the vicinity of a vortex pair

Abstract

We study the behaviour of heavy inertial particles in the flow field of two like-signed vortices.In a frame co-rotating with the two vortices, we find that stable fixed points exist for these heavy inertial particles; these stable frame-fixed points exist only for particle Stokes number St < St cr .We estimate St cr and compare this with direct numerical simulations, and find that the addition of viscosity increases the St cr slightly. We also find that the fixed points become more stable with increasing St until they abruptly disappear at St = St cr . These frame-fixed points are between fixed points and limit cycles in character.

Clustering of heavy particles in vortical flows: a selective review

“…The simple picture given here no longer holds in rotating frames (as shown in [29]), leading to the following interesting consequences.…”

“…In particular, these fixed points are within closed streamlines in the rotating frame of reference. In two-dimensional turbulence, vortices often group themselves into twos and threes and these interactions can result in 'non-traditional' clustering of particles [29], as shown below.…”

On the transport, segregation, and dispersion of heavy and light particles interacting with rising thermal plumes

“…When as time progresses, the particledense lines undergo the convective transport towards the top due to the large-scale buoyancy flow, such loops are stretched and deformed. Superimposed on this are the typical effects experienced by non-neutrally buoyant, finite-sized particles: although in the region affected by plume vortices the dominant effect in producing particle motion is represented by convective transport, we must still keep in mind that particle are not passive tracers and as such they are subjected to the centrifugal force, which tends to display them outwards and the gravity force, which tends to pull them down (Eaton and Fessler 31 ; Raju and Meiburg 32 ; Ravichandran et al, 40 ). In general, by having an influence on the "efficiency" by which the convective flow is able to transport them (or, in other words, on the ability with which particles are able to "follow" this unsteady flow), both the mass and size of particles (namely, the density ratio  and the Stokes number in the nondimensional space of parameters) will definitely have an important role also in the "folding process" of the particle-dense lines (discussed before), which finally determines the extension and intricate shape of the two closed circuits discussed above.…”

“…Though, owing to page limits we do not strive to review all aspects relating to the sedimentation of a dispersed particle in a quiescent or non-quiescent medium, we wish to mention here also the excellent analyses by Raju and Meiburg 32,36 (from which we took most of the inspiration at the root of the present work), Dávila and Hunt 37 , Eames and Gilbertson 38 , Chen et al, 39 , Ravichandran et al, 40 and Bergougnoux et al, 41 . Wang and Maxey 13 and Dávila and Hunt 37 demonstrated that the transient nature of turbulence can give rise to increased settling speeds for sufficiently small particles.…”

The patterning behaviour and accumulation of spherical particles in a vibrated non-isothermal liquid

“…Along these lines, several authors confirmed that owing to the finite mass and size of particles their motion tends to deviate appreciably from that of ideal tracers (no mass, vanishing size), which would simply follow the local velocity of the background flow; among such attempts, in particular, it is worth mentioning Balkovsky et al 1 , who clearly indicated that because of finite particles' inertia, the velocity field of a set of spatially distributed particles will violate the incompressibility constraint even if the fluid flow is incompressible. Today this simple principle is regarded as the necessary theoretical prerequisite allowing solid particles (in association with regions of high shear present in the fluid and a more or less pronounced degree of turbulence of the considered velocity field) to demix and form ordered accumulation structures in specific regions of the physical space [1][2][3][4][5][6][7][8][9][10][11][12][13] .…”