Conditional stability of particle alignment in finite-Reynolds-number channel flow

Finite-size neutrally buoyant particles in a channel flow are known to accumulate at specific equilibrium positions or spots in the channel cross-section, if the flow inertia is finite at the particle scale. Experiments in different conduit geometries have shown that while reaching equilibrium locations, particles tend also to align regularly in the streamwise direction. In this paper, the force coupling method was used to numerically investigate the inertia-induced particle alignment, using square-channel geometry. The method was first shown to be suitable to capture the quasisteady lift force that leads to particle cross-streamline migration in channel flow. Then the particle alignment in the flow direction was investigated by calculating the particle relative trajectories as a function of flow inertia and of the ratio between the particle size and channel hydraulic diameter. The flow streamlines were examined around the freely rotating particles at equilibrium, revealing stable small-scale vortices between aligned particles. The streamwise interparticle spacing between aligned particles at equilibrium was calculated and compared to available experimental data in square-channel flow [Gao et al., Microfluid. Nanofluid. 21, 154 (2017)]. The new result highlighted by our numerical simulations is that the interparticle spacing is unconditionally stable only for a limited number of aligned particles in a single train, the threshold number being dependent on the confinement (particle-to-channel size ratio) and on the Reynolds number. For instance, when the particle Reynolds number is $\approx 1$ and the particle-to-channel height size ratio is $\approx 0.1$, the maximum number of stable aligned particles per train is equal to 3. This agrees with statistics realized on the experiments of [Gao et al., Microfluid. Nanofluid. 21, 154 (2017)]. It is argued that when several particles are hydrodynamically connected moving as a unique structure (the train) with a steady streamwise velocity, large-scale hydrodynamic perturbations induced at the train scale prohibit small-scale vortex connection between the leading and second particles, forcing the leading particle to leave the train.

Figure 1

Scheme showing the possible focusing positions of particles on four spots in the cross-section of a square channel (light blue spheres) and their alignment in the streamwise direction (darker blue). $Z$ indicates the flow direction. $X$ and $Y$ are the directions parallel to the walls. In the simulations particles are placed in the symmetry plane with respect to the $X$ direction.

Figure 2

Slip velocity (left panel) and migration velocity (right panel) vs. particle position $Y/H$ for $\text{Re}=13$ (triangles) and $\text{Re}=39$ (squares) with particle size $d_p/H=0.06$. The solid black curve shows the law proposed by Ref. [31]. The vertical dotted line indicates a distance from the wall equal to the particle radius. The horizontal dashed line in left panel represents the Faxen correction.

Figure 3

Lift force scaled by $\rho U_m^2{a}^4/H^2$, acting on a particle in a square-channel flow versus the position of the particle in the $y$ direction (at $x=H/2)$ for different $\text{Re}$. The red and blue symbols are the lift force from our simulation for particle diameter $d_p/H=0.06$ and 0.11, respectively. The symbols are for $\text{Re}=120$ (circles), 38 (squares), and 13 (triangles). The corresponding ${\text{Re}}_p$ is between 0.05 and 1.5. The red dashed and blue solid lines are obtained from the solution of Ref. [32] for particle diameters $d_p/H=0.06$ and 0.11, respectively.

Figure 4

Zoom on the flow streamlines in the $\left(YZ\right)$ frame of a particle at equilibrium in a square-channel flow. The black region illustrates the particle (stretched for convenience). ${\text{Re}}_p=0.5$ (left), 1.5 (center), and 3.0 (right). The two stagnation points (indicated with crosses) get closer to the particle surface as ${\text{Re}}_p$ increases. The fore-aft asymmetry increases with ${\text{Re}}_p$. Red arrows show the flow direction.

Figure 5

Panels (a), (b), and (c) contain the pair-dynamics in a two-particle train for ${\text{Re}}_p=0.5$, 1.5, and 3.0, respectively. The trajectories of the leading particle with respect to the lagging one are overlaid on the streamlines around a single particle. The initial position of the center of the leading particle is shown with asterisks. Red arrows show the flow direction. (d) Flow streamlines (in the frame of leading particle) around a stable pair of particles obtained for ${\text{Re}}_p=3$, showing a stable small recirculation connecting both particles.

Figure 6

(a) Steady configuration of three-particle trains at different Reynolds numbers in the $\left(YZ\right)$ plane. Particles in the same train have the same color and are linked with lines for eye guidance. ${\text{Re}}_p=0.5$ (red), 1.5 (blue), and 3 (black). The $z$ coordinate of the lagging particle in a train is set arbitrarily to 0. The velocity profile is represented on the right of the figure. (b) Plots representing the center positions of the three-particle trains at different Reynolds numbers [same colors as in (a)].

Figure 7

Properties of stable particle trains. (a) shows the distance between the train barycentre and the closest wall $y_T$. (b) shows the average interparticle distance $l$. Red Squares, upward-pointing triangles, and black plus symbols are for 2-, 3-, and 4-particle assembly in square channel with $\frac{d_p}{H}=0.11$. Blue downward-pointing triangles and black circles are for 3- and 4-particle assembly in square channels with $\frac{d_p}{H}=0.14$ and 0.08, respectively. Black cross is for 4-particle in rectangular channel $\left(\frac{d_p}{W}=0.09\right)$. Black stars are from the experiments of [17] realized in square channel flow with $\frac{d_p}{H}=0.11$. The magenta square and cyan triangle are obtained for 2- and 3-particle trains, using a twice finer numerical resolution in a square channel with $d_P/H=0.11$.

Figure 8

(a) Trajectories of the fourth particle placed in front of a stable three-particle train at ${\text{Re}}_p=1.5$ and $d_p/H=0.11$ for different initial positions. These trajectories are relative to the (new) leading particle of the remaining stable three-particle train (at ${\text{Re}}_p=1.5)$. The trajectories are overlaid on the streamlines, in $\left(YZ\right)$ plane. Panel (b) contains the evolution of the relative particle spacing in the train $\left(\mathrm{\Delta }Z\right)$. Black, red, and blue lines correspond to the distance between front-second, second-third, and third-fourth. The $Z$ coordinate corresponds to the streamwise position of the front, second, and third particles for each curve, respectively.

Figure 9

A diagram showing the maximum number of particles in a stable train for different particle Reynolds number ${\text{Re}}_p$ and confinement $d_p/H$ in a square-channel flow, as predicted from the FCM simulations.

Figure 10

Probability of finding trains with three aligned particles, obtained from post-processing the experiments of Ref. [17], versus the suspension concentration. The probability is the highest at low concentration. Different symbols are for different ${\text{Re}}_p$: triangles, stars, and circles are for ${\text{Re}}_p=0.12$, 0.7, and 1.8, respectively. These results are for $d_p/H=0.11$.

Figure 11

Plots of velocity streamlines relative to the front particle in the microchannel in the $XZ$ plane for (a) a single particle, (b) two-particle train, and (c) three-particle train at ${\text{Re}}_p=1.5$. Colored contours show the streamwise velocity perturbation (subtracting the unperturbed channel flow from the instantaneous velocity field) scaled by the average flow velocity. Flow direction is from left to right.

Figure 12

The ratio between the drag force on particle train and forward pressure force applied by the fluid on the front particle. The symbol legend is idem to Fig. 7.

Figure 13

Left Panel: Numerical calculation of the wall-normal force (blue open circles) applied on a particle near a wall in linear flow, and comparison with different theoretical predictions. The force is scaled by $\mu a{V}_{\text{slip}}$. The different theories are from Saffman [39] (blue asterisks), Cherukat and Mclaughlin [30] (red upward-pointing triangle), Lovalenti in the Appendix of Ref. [30] (green square) and Magnaudet [40] (magenta right-pointing triangle). Rigth Panel: streamwise particle slip velocity versus ${\text{Re}}_p$. The red circles are from FCM simulations. The lines are the prediction of the slip velocity from Goldman et al. [29] at $y_p=1.54$ and $2.35a$, respectively.