Discontinuity in the sedimentation system with two particles having different densities in a vertical channel

The two-dimensional lattice Boltzmann method was used to numerically study a sedimentation system with two particles having different densities in a vertical channel for Galileo numbers in the range of $5\le \mathrm{Ga}\le 15$ (resulting in a Reynolds number, based on the settling velocity, approximately ranging between 0.6 and 7). Two types of periodic motion, differing from each other in terms of the size of the limit cycle, the magnitude of the time period, and their changes upon increasing the density difference between particles, are identified depending on whether there is a wake effect. The most prominent features of this system are discontinuous changes in the settling velocity $(6.7\le \mathrm{Ga}<9.7)$ and time period of oscillation $(10.5\le \mathrm{Ga}\le 15)$ at a critical value of the density difference between particles. The first discontinuity results in an abrupt increase in the Reynolds number, associated with a Hopf bifurcation without the presence of vortex shedding. The second discontinuity is accompanied by the disappearance of “abnormal rotation” (referring to the situation in which a particle appears to roll up a wall when settling) of the heavy particle, which directly results from a sharp increase in the amplitude of oscillation induced by the enhanced wake effect at another critical density difference between particles. The wall effects on these discontinuous changes were also examined.

Figure 1

Illustration of the bounce-back boundary conditions in the LBM proposed by Lallemand and Luo [17]. The solid squares denote the solid nodes while the solid circles denote the fluid nodes.

Figure 2

Problem description and notations used in the present study.

Figure 3

Comparison of the drag coefficient for a circular particle settling in an infinite channel ($L=4d$) with the previous numerical results [4, 21, 22].

Figure 4

Limit cycles constructed by $X_1^{\prime }$ and $X_2^{\prime }$ for the periodic motion of particles at (a) $\mathrm{Ga}=5$, (b) $\mathrm{Ga}=6$, and (c) $\mathrm{Ga}=6.7$.

Figure 5

Limit cycles for $\alpha =0.006$ (PMI) and $\alpha =0.09$ (PMII) in the transition regime $(6.7\le \mathrm{Ga}\le 7.15)$.

Figure 6

Time history of the horizontal position of the heavy particle $\left(X_2^{\prime }\right)$ for $\alpha =0.06$ at $\mathrm{Ga}=6$: examination of the effect of particle-particle collision strategy.

Figure 7

Instantaneous flow field during one period $(T^{\prime }\approx 103.9)$ for $\alpha =0.01$ at $\mathrm{Ga}=6.7$ (periodic motion without wake, referred to as PMI): (a) $0.25T^{\prime }$, (b) $0.5T^{\prime }$, (c) $0.75T^{\prime }$, and (d) $T^{\prime }$. The white arrow on each particle is used to visually track the particle's rotation, with the orientation initially being vertical.

Figure 8

Instantaneous flow field during one period $(T^{\prime }\approx 55.9)$ for $\alpha =0.1$ at $\mathrm{Ga}=6.7$ (periodic motion with the wake, referred to as PMII): (a) $0.25T^{\prime }$, (b) $0.5T^{\prime }$, (c) $0.75T^{\prime }$, and (d) $T^{\prime }$. The arrows are the same as those in Fig. 7.

Figure 9

Normalized time period $\left(T^{\prime }\right)$ of the motion of particles as a function of $\alpha$ for $5\le \mathrm{Ga}\le 7.15$.

Figure 10

Limit cycles for the periodic motion of particles at (a) $\mathrm{Ga}=7.8$, (b) $\mathrm{Ga}=9$, (c) $\mathrm{Ga}=10.5$, and (d) $\mathrm{Ga}=15$.

Figure 11

Normalized time period $\left(T^{\prime }\right)$ of the motion of particles as a function of $\alpha$ for $7.15\le \mathrm{Ga}\le 9.5$.

Figure 12

Settling velocity $\left(U_T^{\prime }\right)$ of the two particles as a function of Ga $(6\le \mathrm{Ga}\le 10.5)$ and $\alpha$ ($0.006\le \alpha <0.11$).

Figure 13

Instantaneous vorticity contours $\left({\omega }^{\prime }\right)$ for different α at $\mathrm{Ga}=6.7$: (a) $\alpha =0.01$ (PMI), (b) $\alpha =0.04$ (PMI), (c) $\alpha =0.085$ (steady state), (d) $\alpha ={\alpha }_{C1}=0.086$ (PMII), (e) $\alpha =0.09$ (PMII), and (f) $\alpha =0.1$ (PMII). Results are selected at the time when the distance between particles reaches a minimum for the periodic state of particles. The maximum vorticity $\left(\right|{\omega }^{\prime }{|}_{\mathrm{m}})$ is also presented in each subpanel.

Figure 14

Drag coefficient $\left(C_d\right)$ of the light particle as a function of the Reynolds number (${\mathrm{Re}}_{\mathrm{T}}$) over the whole range of Ga studied $(5\le \mathrm{Ga}\le 15)$.

Figure 15

Comparison of the horizontal displacement of the light particle $\left(X_1^{\prime }\right)$ resulting from different initial arrangements at $\mathrm{Ga}=7$ for (a) $\alpha ={\alpha }_{C1}=0.063$ (PMII) and (b) $\alpha =0.062$ (steady), respectively.

Figure 16

Normalized time period $\left(T^{\prime }\right)$ of the motion of particles as a function of $\alpha$ for $10.5\le \mathrm{Ga}\le 15$. Abnormal rotation or normal rotation here refers respectively to the situation where the heavy particle rotates counterclockwise or clockwise when it settles in the channel.

Figure 17

Time history of the particle orientation with respect to the horizontal direction $\left({\theta }^{\prime }\right)$ for different values of $\alpha$ at $\mathrm{Ga}=12$ for (a) the light particle and (b) the heavy particle, respectively. Both particles are released with the orientation initially being vertical, i.e., ${\theta }_1^{\prime }\left(0\right)={\theta }_2^{\prime }\left(0\right)=0.5$.

Figure 18

Phase diagram at $\mathrm{Ga}=12$ for the two cases: (a) particles are not allowed to rotate and (b) particles rotate freely. The limit cycles are constructed by $X_1^{\prime }$ and $X_2^{\prime }$ (upper), and $\mathrm{\Delta }{X}^{\prime }(=X_1^{\prime }-X_2^{\prime })$ and $\mathrm{\Delta }{Y}^{\prime }\left(={Y^{\prime }}_1-{Y^{\prime }}_2\right)$ (bottom), respectively.

Figure 19

Instantaneous vorticity contours $\left({\omega }^{\prime }\right)$ at the same values of $\alpha$ as those in Fig. 18 for $\mathrm{Ga}=12$ for the two cases. The results are chosen at the time when the distance between particles reaches its minimum. Also shown in each subfigure are the distance between particles $\left(S^{\prime }\right)$ and the maximum vorticity $\left(\right|{\omega }^{\prime }{|}_{\mathrm{m}})$.

Figure 20

Normalized maximum $\left(U_{Tmax}^{\prime }\right)$ and minimum $\left(U_{Tmin}^{\prime }\right)$ values of the settling velocity for each particle for different $\alpha$ at $\mathrm{Ga}=12$. Also shown is the settling velocity (solid line) of a single particle under the same conditions.

Figure 21

Discontinuity in the settling velocity $\left(U_T^{\prime }\right)$ for different values of channel width $(3.5d\le L\le 7d)$. Panels (a)–(e) refer to the steady state of particles (also shown in Fig. 22) under different conditions which will quickly bifurcate into a periodic state if the value of $\alpha$ is increased a little.

Figure 22

Instantaneous vortex contours $\left({\omega }^{\prime }\right)$ at steady state under different conditions [corresponding to the cases of (a)–(e) in Fig. 21]: (a) $L=3.5d$ and $\mathrm{Ga}=14$, (b) $L=4d$ and $\mathrm{Ga}=12$, (c) $L=5d$ and $\mathrm{Ga}=8$, (d) $L=6d$ and $\mathrm{Ga}=6$, and (e) $L=7d$ and $\mathrm{Ga}=5$.

Figure 23

Discontinuity in the time period $\left(T^{\prime }\right)$ at $\mathrm{Ga}=11$ for different values of channel width $(4.25d\le L\le 5.125d)$.