Preferential Concentration of Free-Falling Heavy Particles in Turbulence

We present a sweep-stick mechanism for heavy particles transported by a turbulent flow under the action of gravity. Direct numerical simulations show that these particles preferentially explore regions of the flow with close to zero Lagrangian acceleration. However, the actual Lagrangian acceleration of the fluid elements where particles accumulate is not zero, and has a dependence on the Stokes number, the gravity acceleration, and the settling velocity of the particles.

Figure 1

(a) Particle distribution in an arbitrary $xz$ plane in simulations with $\mathrm{Fr}=0.23$ and for different Stokes numbers $\mathrm{St}=3$, 8, and 20. Columns form in the direction parallel to gravity. (b) Column width as a function of St, for $\mathrm{Fr}=0.23$.

Figure 2

(a) Particles’ mean vertical velocity as a function of Fr for $\mathrm{St}=6$. Error bars for 95% confidence intervals are shown; standard deviations of fluctuations of $⟨v_z⟩$ in time are less than 2%. The inset shows the anomaly $⟨v_z⟩/v^{*}$. (b) PDFs of particles’ vertical velocity increments. As St increases, the PDFs are closer to Gaussian (see the kurtosis Ku in the inset, $\text{Ku}=3$ is marked as a reference).

Figure 3

(a) PDFs of Voronoï volumes for particles with $\mathrm{St}=6$ and different Fr. (b) Standard deviation of the PDFs of $\nu /⟨\nu ⟩$ as a function of the theoretical terminal velocity $v^{*}$. (c) Mean Voronoï volumes of clustered particles as a function of St and for different Fr.

Figure 4

(a) Joint PDFs of the volumes of cells belonging to clusters ${\nu }_c$, and of the vertical Lagrangian acceleration at the particles’ positions, for $\mathrm{St}=6$, and for two values of Fr (from left to right, $\infty$ and 0.23). (b) Mean value of the vertical Lagrangian acceleration at the clustered particles’ positions, as a function of $\log ({\nu }_c)$.

Figure 5

(a) Mean vertical velocity of clustered particles as a function of the mean vertical velocity of fluid elements at particle’s locations. Velocities are normalized by $v^{*}$; as a result points with $⟨v_z⟩/v^{*}>1$ (most of them) indicate particles falling faster than the Stokes terminal velocity. The dashed line corresponds to Eq. (2). (b) Same for the Lagrangian acceleration of fluid elements normalized by $\mathrm{St}g$ as a function of $⟨v_z⟩/v^{*}$. The dashed line corresponds to Eq. (3).