Inertial spheroids in homogeneous, isotropic turbulence

We study the rotational dynamics of inertial disks and rods in three-dimensional, homogeneous, isotropic turbulence. In particular, we show how the alignment and the decorrelation timescales of such spheroids depend, critically, on both the level of inertia and the aspect ratio of these particles. These results illustrate the effect of inertia—which leads to a preferential sampling of the local flow geometry—on the statistics of both disks and rods in a turbulent flow. Our results are important for a variety of natural and industrial settings where the turbulent transport of asymmetric, spheroidal inertial particles is ubiquitous.

Figure 1

Representative plots of the probability density function of the alignment of the orientation vector $\mathbf{p}$ with ${\mathbf{e}}_1$ (black circle $\circ$), ${\mathbf{e}}_2$ (pink cross $\times$), ${\mathbf{e}}_3$ (blue inverted triangle $▿$), and ${\mathbf{e}}_w$ (red asterisk $*$) for (a) $\alpha =0.1$ (oblate), (b) $\alpha =0.5$, (c) $\alpha =1.5$, and (d) $\alpha =2.0$ (rod). These measurements are made for particles with a bare Stokes numbers ${\text{St}}_{\mathrm{s}}=1.0$. (In Table 1, the actual Stokes numbers corresponding to the different particles are listed.)

Figure 2

Representative plots of the probability density function of the alignment of the orientation vector $\mathbf{p}$ of a rod ($\alpha =2.0$) with ${\mathbf{e}}_1$ (black circle $\circ$), ${\mathbf{e}}_2$ (pink cross $\times$), ${\mathbf{e}}_3$ (blue inverted triangle $▿$), and ${\mathbf{e}}_w$ (red asterisk $*$) for (a) $\text{St}=0.0$, (b) $\text{St}=0.152$, (c) $\text{St}=0.76$, and (d) $\text{St}=4.56$.

Figure 3

Plots of (a) $\langle |\mathbf{p}·{\mathbf{e}}_i|\rangle$ and (b) $\langle |\mathbf{p}·{\mathbf{e}}_i{|}^2\rangle$ vs $\alpha$ for ${\mathrm{St}}_{\mathrm{s}}=0$ (red open circles), ${\mathrm{St}}_{\mathrm{s}}=0.1$ (blue upward-pointing triangles), ${\mathrm{St}}_{\mathrm{s}}=0.5$ (magenta squares), ${\mathrm{St}}_{\mathrm{s}}=1.0$ (black asterisks), ${\mathrm{St}}_{\mathrm{s}}=1.5$ (gray diamonds), ${\mathrm{St}}_{\mathrm{s}}=2.0$ (green downward-pointing triangles). The solid and dashed lines are for ${\mathbf{e}}_1$ and ${\mathbf{e}}_{\omega }$, respectively. The insets show the representative plots of the same quantities, at ${\mathrm{St}}_{\mathrm{s}}=1$, for ${\mathbf{e}}_1$ (red open circles), ${\mathbf{e}}_2$ (blue triangles), ${\mathbf{e}}_3$ (black squares), and ${\mathbf{e}}_{\omega }$ (magenta asterisks).

Figure 4

Plot of characteristic decay times, normalized by ${\tau }_{\eta }$, of the correlation functions (a) $\langle \mathbf{p}\left(t\right)·\mathbf{p}\left(0\right)\rangle$ and (b) $\langle \left|\mathbf{p}\right(t)·\mathbf{p}(0\left)\right|\rangle$ vs $\alpha$ for different ${\mathrm{St}}_{\mathrm{s}}$. The color codes are the same as in Fig. 3. The top-left insets show the correlation function decay for a fixed ${\mathrm{St}}_{\mathrm{s}}=1.0$ and changing $\alpha$ from $\alpha =0.1$ (magenta dashed line), $\alpha =0.5$ (red triangles), $\alpha =1.0$ (cyan dashed-dotted line), $\alpha =1.5$ (black open circles), and $\alpha =2.0$ (blue solid line). The bottom-right insets show the correlation function decay for a fixed $\alpha =2.0$ and different ${\mathrm{St}}_{\mathrm{s}}$ (same color code as the main plot). In the $y$ axis of the insets of the left panel, $C=\langle \mathbf{p}\left(t\right)·\mathbf{p}\left(0\right)\rangle$, and the right panel, $C_{\mathrm{abs}}=\langle |\mathbf{p}\left(t\right)·\mathbf{p}\left(0\right)|\rangle -0.5$.