Effect of gravity on clustering patterns and inertial particle attractors in kinematic simulations

In this paper, we study the clustering of inertial particles using a periodic kinematic simulation. The systematic Lagrangian tracking of particles makes it possible to identify the particles' clustering patterns for different values of particle inertia and drift velocity. The different cases are characterized by different pairs of Stokes number (St) and Froude number (Fr). For the present study, $0\le \mathrm{St}\le 1$ and $0.4\le \mathrm{Fr}\le 1.4$. The main focus is to identify and then quantify the clustering attractor—when it exists—that is the set of points in the physical space where the particles settle when time goes to infinity. Depending on the gravity effect and inertia values, the Lagrangian attractor can have different dimensions, varying from the initial three-dimensional space to two-dimensional layers and one-dimensional attractors that can be shifted from a horizontal to a vertical position.

Figure 1

Particles' (a) initial distribution and (b) reinjection.

Figure 2

Time evolution of inertial particles for $\mathrm{St}=0.207$ and $\mathrm{Fr}=0.55$ with $t=1$ (top left), $t=10$ (top right), $t=600$ (bottom left), and $t=1200$ (bottom right).

Figure 3

Time evolution of inertial particles for $\mathrm{St}=0.413$ and $\mathrm{Fr}=0.85$ with $t=1$ (top left), $t=5$ (top right), $t=100$ (bottom left), and $t=300$ (bottom right).

Figure 4

Evolution of the particles cloud for $0.548\le \mathrm{Fr}\le 1.34$ and $0.165\le \mathrm{St}\le 0.663$, at $t=300$.

Figure 5

Evolution of particles with increasing St for a given $\gamma$ at $t=300$. Top: $\gamma =0.138$; left to right: $\mathrm{St}=0.100$, 0.165, 0.249, and 1. Middle: $\gamma =0.689$; left to right: $\mathrm{St}=0.124$, 0.165, 0.249, and 0.827. Bottom: $\gamma =2$; left to right: $\mathrm{St}=0.207$, 0.600, 0.827, and 1.

Figure 6

Benchmark for BCM.

Figure 7

Contour plot of the attractor fractal dimension $D$ as a function of (St, Fr).

Figure 8

Box-counting slope for very similar cases for different values of Fr at $\mathrm{St}=0.207$.

Figure 9

Isocontours of (a) $\mathrm{\Delta }$(St,Fr) and (b) $\mathrm{\Delta }(\mathrm{St},\gamma )$ showing different types of clusterings.

Figure 10

(a) Iso-contours of $\mathrm{\Delta }$ as a function of $(\mathrm{St},\mathrm{Fr})$, (b) ${\mathrm{\Delta }}_V/\mathrm{\Delta }$ when $\mathrm{\Delta }\le 0.006$, c) ${\mathrm{\Delta }}_H/\mathrm{\Delta }$ when $\mathrm{\Delta }\le 0.006$.

Figure 11

Flow chart describing the different attractors in relation to the two critical values of $\mathrm{St}$.