Effect of the energy-spectrum law on clustering patterns for inertial particles subjected to gravity in kinematic simulation

We study the clustering of inertial particles using a periodic kinematic simulation. Particle clustering is observed for different pairs of Stokes number and Froude number and different spectral power laws $\left(1.4\le p\le 2.1\right)$. The main focus is to identify and then quantify the effect of $p$ on the clustering attractor—by attractor we mean the set of points in the physical space where the particles settle when time tends to infinity. It is observed that spectral power laws can have a dramatic effect on the attractor shape. In particular, we observed an attractor type which was not present in previous studies for Kolmogorov spectra ($p=5/3$).

Figure 1

Spectral energy distribution for different power laws.

Figure 2

Different characteristic attractor shapes.

Figure 3

$\mathrm{\Delta }$ as a function of $p$ for $\text{St}=0.124$, 0.207, 0.413, and 1 at $t=100$ and without gravity $\text{Fr}=\infty$.

Figure 4

$\mathrm{\Delta }$ as a function of $p$ for $\text{St}=0.124$, 0.207, 0.413, and 1 at $t=100$ and $\text{Fr}=0.49$.

Figure 5

$\mathrm{\Delta }$ as a function of $p$ for $\text{Fr}=0.57$, 0.63, 0.85, 1.2, and $\infty$ at $t=100;\text{St}=0.207$.

Figure 6

$\text{Fr}=0.89$: (a) Isocontours of $\mathrm{\Delta }$ as functions of $(\text{St},p)$. Particle clusters with different power laws of energy spectrum for increasing values of $\text{St}$. (b) $p=2.5$, (c) $p=2.1$, (d) $p=5/3$, and (e) $p=1.5$ and from left to right $\text{St}=0.09$, 0.298, 0.413, and 0.91.

Figure 7

$\text{Fr}=0.59$: (a) Isocontours of $\mathrm{\Delta }$ as functions of $(\text{St},p)$. Particle clusters with different power laws of energy spectrum for increasing values of $\text{St}$. (b) $p=2.4$, (c) $p=2.1$, (d) $p=5/3$, and (e) $p=1.5$ and from left to right $\text{St}=0.1$, 0.207, 0.413, and 0.91.

Figure 8

$\text{Fr}=0.49$: (a) $\mathrm{\Delta }$ as a function of $(\text{St},p)$. Particle clusters with different power laws of energy spectrum for increasing values of $\text{St}$. (b) $p=2.5$, (c) $p=2.1$, (d) $p=5/3$, and (e) $p=1.5$ and from left to right $\text{St}=0.16$, 0.413, 0.70, and 0.91.

Figure 9

$\text{St}=0.207$: (a) Isocontours of $\mathrm{\Delta }$ as functions of $(\text{Fr},p)$. Typical particle clusters for different spectral power laws and decreasing values of $\text{Fr}$: (b) $p=2.5$, (c) $p=2.1$, (d) $p=5/3$, and (e) $p=1.5$ and from left to right $\text{Fr}=1.1$, 0.85, 0.72, and 0.6.

Figure 10

$\text{St}=0.413$: (a) Isocontours of $\mathrm{\Delta }$ as functions of $(\text{Fr},p)$. Typical particle clusters for different spectral power laws and decreasing values of $\text{Fr}$: (b) $p=2.5$, (c) $p=2.1$, (d) $p=5/3$, and (e) $p=1.5$ and from left to right $\text{Fr}=1.1$, 0.85, 0.72, and 0.6.

Figure 11

$\text{St}=1$: (a) Isocontours of $\mathrm{\Delta }$ as functions of $(\text{Fr},p)$. Typical particle clusters for different spectral power laws and decreasing values of $\text{Fr}$: (b) $\text{Fr}=0.67,p=1.67$, (c) $\text{Fr}=0.95,p=1.67$, (d) $\text{Fr}=0.95,p=2$, and (e) $\text{Fr}=0.67,p=2$.