Turbulence Unsteadiness Drives Extreme Clustering

We show that the unsteadiness of turbulence has a drastic effect on turbulence parameters and in particle cluster formation. To this end we use direct numerical simulations of particle laden flows with a steady forcing that generates an unsteady large-scale flow. Particle clustering correlates with the instantaneous Taylor-based flow Reynolds number, and anticorrelates with its instantaneous turbulent energy dissipation constant. A dimensional argument for these correlations is presented. In natural flows, unsteadiness can result in extreme particle clustering, which is stronger than the clustering expected from averaged inertial turbulence effects.

Figure 1

Instantaneous value of $C_{ϵ}$ as a function of $R_{\lambda }$ for the CRC flow at three different spatial resolutions. Each resolution corresponds to a different viscosity and to a different averaged Reynolds number. Inset: $C_{ϵ}$ compensated by $\sqrt{{\mathrm{Re}}_0}$ as a function of $R_{\lambda }$.

Figure 2

Probability density functions (PDFs) of the normalized Vonoroı volumes $\mathcal{V}=V/⟨V⟩$ of inertial particles for CRC and HIT simulations, for (a) $N^3=768^3$, and (b) $N^3=1024^3$. For CRC forcing, the PDFs are time-averaged over long times. A random Poisson process (RPP) is indicated by the dashed line.

Figure 3

Time series of (a) $C_{ϵ}$, (b) $ϵ$, (c) $R_{\lambda }$, and (d) ${\sigma }_{\mathcal{V}}$ for CRC runs with $\mathrm{St}=3$ and 8. Time is normalized by the mean turnover time $⟨T⟩$, and ${\sigma }_{\mathrm{RPP}}$ is indicated by the dotted gray line in panel (d). Inset: cross-correlation of ${\sigma }_{\mathcal{V}}$ and $R_{\lambda }$ as a function of the time lag $\tau$.

Figure 4

Standard deviation of the Voronoı volumes, ${\sigma }_{\mathcal{V}}$, as a function of $R_{\lambda }$ for CRC runs, differentiating branches in which $R_{\lambda }$ increases (solid lines) and decreases (dashed lines), for (a) $\mathrm{St}=3$, and (b) 8. Red arrows indicate the direction of time evolution.

Figure 5

PDFs of ${\sigma }_{\mathcal{V}}$ for HIT and CRC runs with $N^3=768^3$ and for particles with $\mathrm{St}=8$. Note the larger values of ${\sigma }_{\mathcal{V}}$ (i.e., enhanced clustering) in the CRC run, as well as the larger dispersion.