Enhanced and suppressed multiscale dispersion of bidisperse inertial particles due to gravity

Using direct numerical simulation (DNS), we investigate how gravity modifies the multiscale dispersion of bidisperse inertial particles in isotropic turbulence. The DNS has a Taylor Reynolds number $R_{\lambda }=398$, and we simulate Stokes numbers (based on the Kolmogorov timescale) in the range $\text{St}\le 3$ and consider Froude numbers $\text{Fr}=0.052$ and $\infty$, corresponding to strong gravity and no gravity, respectively. The degree of bidispersity is quantified by the difference in the Stokes number of the particles, $\left|\mathrm{\Delta }\text{St}\right|$. We first consider the mean-square separation of bidisperse particle pairs and find that, without gravity (i.e., $\text{Fr}=\infty$), bidispersity leads to an enhancement of the the mean-square separation over a significant range of scales. When $\left|\mathrm{\Delta }\text{St}\right|\ge O\left(1\right)$, the relative dispersion is further enhanced by gravity due to the large difference in the settling velocities of the two particles. However, when $\left|\mathrm{\Delta }\text{St}\right|\ll 1$, gravity suppresses the relative dispersion as the settling velocity contribution is small, and gravity suppresses the nonlocal contribution to the particle dynamics. In order to gain further insights, we consider separately the relative dispersion in the vertical (parallel to gravity) and horizontal directions. As expected, the vertical relative dispersion can be strongly enhanced by gravity due to differences in the settling velocities of the two particles. However, a key finding of our study is that gravity can also significantly enhance the horizontal relative dispersion. This nontrivial effect occurs because fast settling particles experience rapid fluctuations in the fluid velocity field along their trajectory, leading to enhanced particle accelerations and relative velocities. For sufficiently large initial particle separations, however, gravity can lead to a suppression of the horizontal relative dispersion. We also compute the probability density function (PDF) of the particle-pair dispersion. Our results for these PDFs show that even when $\text{Fr}\ll 1$ and $\left|\mathrm{\Delta }\text{St}\right|\ge O\left(1\right)$, the vertical relative dispersion of the particles can be strongly affected by turbulence. This occurs because although the settling velocity contribution to the relative motion is much larger than the “typical” velocities of the turbulence when $\text{Fr}\ll 1$ and $\left|\mathrm{\Delta }\text{St}\right|\ge O\left(1\right)$, due to intermittency, there are significant regions of the flow where the turbulent velocities are of the same order as the settling velocity. These findings imply that in many applications where $R_{\lambda }⋙1$ the effect of turbulence on the vertical relative dispersion of settling bidisperse particles may never be ignored, even if the particles are settling rapidly.

Figure 1

$\langle \parallel {\mathit{w}}^p\left(0\right){{\parallel }^2\rangle }_{\xi }/u_{\eta }^2$ as a function of $\xi /\eta$ for ${\text{St}}_1=1$ and various ${\text{St}}_2$. Solid and dashed lines correspond to the results with $\text{Fr}=0.052\mathrm{and}\infty$, respectively.

Figure 2

FIT and BIT mean-square separation results (with ${\xi }^2$ subtracted) for ${\text{St}}_1=1$ and various ${\text{St}}_2$: (a) and (b) $\xi \in (0\eta ,1\eta ]$, (c) and (d) $\xi \in [3\eta ,4\eta ]$. Solid and dashed lines correspond to the results with $\text{Fr}=0.052\mathrm{and}\infty$, respectively.

Figure 3

FIT and BIT mean-square separation results (with ${\xi }^2$ subtracted) for ${\text{St}}_1=1$ and various ${\text{St}}_2$: (a) and (b) $\xi \in [9\eta ,10\eta ]$, (c) and (d) $\xi \in [19\eta ,20\eta ]$. Solid and dashed lines correspond to the results with $\text{Fr}=0.052\mathrm{and}\infty$, respectively.

Figure 4

FIT and BIT results for $\langle R^2\left(t\right)\rangle \equiv \langle \parallel {\mathit{r}}^p\left(t\right){{\parallel }^2\rangle }_{\xi }-{\xi }^2$, divided by the ballistic prediction $t^2\langle \parallel {\mathit{w}}^p\left(0\right){{\parallel }^2\rangle }_{\xi }$ for ${\text{St}}_1=1$ and various ${\text{St}}_2$ combinations: (a) and (b) $\xi \in (0\eta ,1\eta ]$, (c) and (d) $\xi \in [3\eta ,4\eta ]$. Solid and dashed lines correspond to the results with $\text{Fr}=0.052\mathrm{and}\infty$, respectively.

Figure 5

FIT mean-square separation results in the vertical and horizontal directions from DNS for ${\text{St}}_1=1$, different ${\text{St}}_2$, and with $\xi \in [0\eta ,1\eta ]$. The red line corresponds to the vertical separations and the blue line corresponds to the horizontal separations. The dashed black line corresponds to the results without gravity. The solid black line corresponds to (12) for $\text{Fr}=0.052$.

Figure 6

FIT mean-square separation results in the vertical and horizontal directions from DNS for ${\text{St}}_1=1$, different ${\text{St}}_2$, and with $\xi \in [3\eta ,4\eta ]$. The red line corresponds to the vertical separations and the blue line corresponds to the horizontal separations. The dashed black line corresponds to the results without gravity. The solid black line corresponds to (12) for $\text{Fr}=0.052$.

Figure 7

BIT mean-square separation results in the vertical and horizontal directions from DNS for ${\text{St}}_1=1$, different ${\text{St}}_2$, and with $\xi \in [0\eta ,1\eta ]$. The red line corresponds to the vertical separations and the blue line corresponds to the horizontal separations. The dashed black line corresponds to the results without gravity. The solid black line corresponds to (12) for $\text{Fr}=0.052$.

Figure 8

BIT mean-square separation results in the vertical and horizontal directions from DNS for ${\text{St}}_1=1$, different ${\text{St}}_2$, and with $\xi \in [3\eta ,4\eta ]$. The red line corresponds to the vertical separations and the blue line corresponds to the horizontal separations. The dashed black line corresponds to the results without gravity. The solid black line corresponds to (12) for $\text{Fr}=0.052$.

Figure 9

DNS results for ${\mathcal{P}}_{1,3}^{\mathcal{F}}(r,t|\xi )$ [plots (a), (c), (e)], ${\mathcal{P}}_{1,3}^{\mathcal{B}}(r,-t|\xi )$ [plots (b), (d), (f)] for ${\text{St}}_1=1,{\text{St}}_2=0.5$ with $\xi \in [3\eta ,4\eta ]$ and (a), (b) $t=0.5{\tau }_{\eta }$, (c), (d) $t=2.5{\tau }_{\eta }$, (e), (f) $t=20{\tau }_{\eta }$. The red line corresponds to the vertical separations, the blue line corresponds to the horizontal separations, and the black line corresponds to the results without gravity.

Figure 10

DNS results for ${\mathcal{P}}_{1,3}^{\mathcal{F}}(r,t|\xi )$ [plots (a), (c), (e)], ${\mathcal{P}}_{1,3}^{\mathcal{B}}(r,-t|\xi )$ [plots (b), (d), (f)] for ${\text{St}}_1=1,{\text{St}}_2=0.9$ with $\xi \in [3\eta ,4\eta ]$ and (a), (b) $t=0.5{\tau }_{\eta }$, (c,d) $t=2.5{\tau }_{\eta }$, (e), (f) $t=20{\tau }_{\eta }$. The red line corresponds to the vertical separations, the blue line corresponds to the horizontal separations, and the black line corresponds to the results without gravity.

Figure 11

DNS results for ${\mathcal{P}}_{1,3}^{\mathcal{F}}(r,t|\xi )$ [plots (a), (c), (e)], ${\mathcal{P}}_{1,3}^{\mathcal{B}}(r,-t|\xi )$ [plots (b), (d), (f)] for ${\text{St}}_1=1,{\text{St}}_2=3$ with $\xi \in [3\eta ,4\eta ]$ and (a), (b) $t=0.5{\tau }_{\eta }$, (c), (d) $t=2.5{\tau }_{\eta }$, (e), (f) $t=10{\tau }_{\eta }$. The red line corresponds to the vertical separations, the blue line corresponds to the horizontal separations, and the black line corresponds to the results without gravity.