Role of boundary conditions and wall orientation on the transport of settling inertial particles in wall-bounded flows

TL;DR

The paper tackles how gravity and inertia govern the transport of settling inertial particles in wall-bounded turbulence, focusing on the roles of wall orientation and particle-wall boundary conditions. It combines a phase-space theoretical framework with direct numerical simulations to separate explicit ($Sv$) and implicit gravity effects and to quantify their impact on near-wall concentrations. The main findings show that in horizontal channels, particle concentrations are highly sensitive to $Sv$ due to explicit settling, with boundary conditions (elastic vs absorbing) qualitatively changing the outcome; in vertical channels, the explicit gravity effect vanishes and the implicit effect is negligible for $Sv\ll1$, becoming significant only when $Sv\geq O(1)$. Mechanisms near walls reveal that low-speed streaks can suppress settling in vertical channels, while far from walls, preferential sweeping can enhance settling, shaping overall transport and deposition patterns. These insights have implications for environmental and engineering models of particle transport in inclined or stratified flows where gravity and boundary interactions are pivotal.

Abstract

Gravitational settling affects particle transport in turbulent flows in two ways; explicitly, by introducing a finite settling velocity, and implicitly, by modifying how the particles interact with the flow field. For wall-bounded flows, when the wall is horizontal (gravity perpendicular to the wall) both the explicit and implicit effects of settling impact the particle transport towards the wall, whereas when the wall is vertical (gravity parallel to the wall) only the implicit effect plays a role. Surprisingly, it was recently demonstrated that even when the settling parameter $Sv$ is very small, settling can play a very significant role in controlling the near-wall transport in a horizontal channel. In this paper, we use direct numerical simulations to explore how this finding is affected by the particle boundary conditions and whether it also occurs in vertical channels where only the implicit effect of settling plays a role. We show that the sensitivity of the particle transport to $Sv$ depends upon the particle boundary conditions, with elastic-collisions (that generate a zero mean particle flux) and absorbing-wall conditions (that generate a negative mean particle flux) exhibiting qualitatively and quantitatively different sensitivities to $Sv$. It is found that for vertical channels the impact of settling on the particle transport is in fact negligible if $Sv$ is small, and only becomes significant when $Sv\geq O(1)$. Finally, we examine the physical mechanisms governing the settling velocity of particles in vertical channel flows, and find that low-speed streaks can play a significant role in suppressing the settling velocity of the particles near the walls, in contrast to the core region where the preferential sweeping mechanism leads to an enhancement of their settling velocity.

Figures (10)

• Figure 1: The three configurations (a) a horizontal channel with $\Phi=0$ (HZF case), (b) a vertical channel with $\Phi=0$ (VZF case), and (c) a horizontal channel with $\Phi<0$ (HNF case). In these schematics, the shaded planes denote the surfaces where the particles rebound elastically, while the non-shaded planes denote the surfaces where either periodic or absorbing wall boundary conditions apply.
• Figure 2: Fluid and particle statistics during the simulation (red curve corresponds to the friction velocity $u_{\tau}$ as a function of time, shown for reference). (a) $\varrho_{max}=\max[\varrho]$ at $St = 4.65$, $Sv = 10^{-5}$ in HZF case, (b) $\varrho_{max}$ at $St = 46.5$, $Sv = 10^{-5}$ in HZF case, (c) $\varrho_{max}$ at $St = 4.65$, $Sv = 10^{-4}$ in HNF case, (d) $\varrho_{max}$ at $St = 46.5$, $Sv = 10^{-4}$ in HNF case.
• Figure 3: Particle concentration $\varrho$ at different $Sv$ for (a) $St=0.93$, (b) $St=4.65$, (c) $St=9.3$, (d) $St=46.5$ in the HNF case.
• Figure 4: Particle concentration $\varrho$ at different $Sv$ for (a) $St=0.93$, (b) $St=4.65$, (c) $St=9.3$, (d) $St=46.5$ in the HZF case.
• Figure 5: Maximum particle concentration $\varrho_{max}$ as a function of time $t$ for $St = 0.93$ in a HZF case (a) $Sv = 10^{-5}$, (b) $Sv = 10^{-4}$.