Settling strongly modifies particle concentrations in wall-bounded turbulent flows even when the settling parameter is asymptotically small

We explore the role of gravitational settling on inertial particle concentrations in a wall-bounded turbulent flow. While it may be thought that settling can be ignored when the settling parameter $\text{Sv}\equiv v_s/u_{\tau }$ is small ($v_s$ is Stokes settling velocity, $u_{\tau }$ is fluid friction velocity), we show that even in this regime the settling may make a leading order contribution to the concentration profiles. This is because the importance of settling is determined, not by the size of $v_s$ compared with $u_{\tau }$ or any other fluid velocity scale, but by the size of $v_s$ relative to the other mechanisms that control the vertical particle velocity and concentration profile. We explain this in the context of the particle mean-momentum equation, and show that in general, there always exists a region in the boundary layer where settling cannot be neglected, no matter how small Sv is (provided it is finite). Direct numerical simulations confirm the arguments and show that the near-wall concentration is highly dependent on Sv even when $\text{Sv}\ll 1$, and it can reduce by an order of magnitude when Sv is increased from $O\left({10}^{-4}\right)$ to $O\left({10}^{-2}\right)$. The results also show that the preferential sampling of ejection events in the boundary layer by inertial particles when $\text{Sv}=0$ is profoundly altered as Sv is increased, and it is replaced by a preferential sampling of sweep events due to the onset of the preferential sweeping mechanism.

Figure 1

Illustration of the system under consideration. The mean flow is in the horizontal direction, and gravity acts in the downward vertical direction. A source of particles at the free surface maintains a steady-state regime in which the statistics of the system depend only on the vertical position $z$.

Figure 2

Plot of $\varrho$ as a function of $z$ for different Sv, and (a) $\text{St}=0.93$, (b) $\text{St}=2.80$, (c) $\text{St}=9.30$, (d) $\text{St}=46.5$.

Figure 3

Plots of the average fluid velocity sampled by the particles ${\langle u^p\left(t\right)\rangle }_z$ as a function of $z$ for different Sv, and (a) $\text{St}=0.93$, (b) $\text{St}=2.80$, (c) $\text{St}=9.30$, (d) $\text{St}=46.5$. The insets correspond to the same plots but in a log-log scale to highlight the behavior for small $z$ (and in these plots the symbols disappear for small $z$ once the quantity becomes negative). Legend is the same as that in Fig. 2.

Figure 4

Plots of the average vertical particle velocity ${\langle w^p\left(t\right)\rangle }_z$ as a function of $z$ for different Sv, and (a) $\text{St}=0.93$, (b) $\text{St}=2.80$, (c) $\text{St}=9.30$, (d) $\text{St}=46.5$. Legend is the same as that in Fig. 2. Note that inset plots showing the results in a log-log scale are not shown for this quantity as the data is very noisy for small $z$ due to the spatial differentiation of the data for ${\langle w^p\left(t\right)\rangle }_z$ involved in computing $D_t{\langle w^p\left(t\right)\rangle }_z$.

Figure 5

Plots of the turbophoretic velocity $-\text{St}{\mathbf{\nabla }}_z\mathcal{W}$ as a function of $z$ for different Sv, and (a) $\text{St}=0.93$, (b) $\text{St}=2.80$, (c) $\text{St}=9.30$, (d) $\text{St}=46.5$. The insets correspond to the same plots (except that the vertical axis is now $\text{St}{\mathbf{\nabla }}_z\mathcal{W}$) but in a log-log scale to highlight the behavior for small $z$. Legend is the same as that in Fig. 2.

Figure 6

Plots of the diffusive velocity $-\text{St}\mathcal{W}{\varrho }^{-1}{\mathbf{\nabla }}_z\varrho$ as a function of $z$ for different Sv, and (a) $\text{St}=0.93$, (b) $\text{St}=2.80$, (c) $\text{St}=9.30$, (d) $\text{St}=46.5$. The insets correspond to the same plots but in a log-log scale to highlight the behavior for small $z$. Legend is the same as that in Fig. 2.