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Density ratio effect on particle sedimentation in a vertical channel
Accepted Manuscript
Density ratio effect on particle sedimentation in a vertical channel Yong Rao , Chaofeng Liu , Huatao Wang , Yushan Ni , Chao Lv , Shuang Liu , Yamei Lan , Shiming Wang PII: DOI: Reference:
S0577-9073(17)30807-9 10.1016/j.cjph.2018.06.007 CJPH 553
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
30 June 2017 6 June 2018 7 June 2018
Please cite this article as: Yong Rao , Chaofeng Liu , Huatao Wang , Yushan Ni , Chao Lv , Shuang Liu , Yamei Lan , Shiming Wang , Density ratio effect on particle sedimentation in a vertical channel, Chinese Journal of Physics (2018), doi: 10.1016/j.cjph.2018.06.007
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ACCEPTED MANUSCRIPT Highlights A particle settling in a vertical channel is studied at different density ratios. The effect of density ratio on the particle settling modes is discussed. The distinct flow regimes are presented at different density ratios. The relation between particle settling modes and flow regimes is analyzed.
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Density ratio effect on particle sedimentation in a vertical channel *
Yong Rao a , Chaofeng Liu b, Huatao Wang c, Yushan Ni d, Chao Lv a, Shuang Liu a, Yamei Lan a and Shiming Wang a a
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College of Engineering Science and Technology, Shanghai Ocean University, Shanghai 201306, China b Shanghai Electro-Mechanical Engineering Institute, Shanghai 200233, China c Research Management Office for Science and Technology , Fudan University, Shanghai 200433, China d Department of Aeronautics and Astronautics, Fudan University, Shanghai 200433, China
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A circular particle settling in a confined vertical channel with static fluid is investigated by lattice Boltzmann method, the effect of density ratio on the particle motion and flow pattern is discussed. It shows that the particle starting from an initial position off-center displays various moving modes accompanied by different flow patterns for 1.003≤γ≤5.0: when γ<1.1, the particle is finally in a stable equilibrium near the centerline and settles uniformly with a symmetrical flow, but there are different modes before reaching the centerline at different density ratios, a monotonic approach appears at γ≤1.005 and an transient overshoot appears at 1.005<γ<1.1; When γ=1.1, a weak oscillation appears in the particle moving but the flow is almost symmetrical, it is in a critical state of the particle moving from stable equilibrium with a transient overshoot to oscillation; When 1.1<γ≤3.0, the particle oscillates regularly around the centerline with the uniform amplitude and frequency, and it turns into irregularity for γ>3.0. Meanwhile the rotation of the particle depends on the distance from a wall; it rolls up the closer wall and stops at centerline.
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Keywords: particulate flow, sedimentation, density ratio effect, Lattice Boltzmann method
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1. Introduction
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The sedimentation of particles in fluid is ubiquitous in nature and engineering field, e.g. the deposition of silt in water conservancy project, falling of hail in meteorology, purification of water in environmental engineering, erythrocyte sedimentation in biological engineering, etc. The particle motion affects the flow pattern, and vice versa. These two kinds of motions are coupled into a complex two-phase flow, which is important both in theoretic research and engineering application, and a lot of investigation has been carried out, both experimentally and numerically. Many previous works have reported the particle sedimentation in fluid under various conditions, including off-center distance, channel width, fluid properties, particle shape, inter-particle interaction etc. Howard H.Hu et al. [1] revealed the effect of vortex shedding on particle motion and reproduced the drafting, kissing, and tumbling scenario of two falling particles in the falling process. J. Feng et al. [2] investigated the initial value problems for the sedimentation of a circular and elliptical particle in a vertical channel, the particle motion at different Reynolds numbers and the non-linear effects of particle-fluid, particle-wall and inter-particle interactions were discussed. Anthony Wachs[3] investigated the sedimentation of polygonal isometric particles with collisions in a Newtonian fluid, the influence of shape on particles sedimentation and the collective behavior of 300 particles settling in a closed box were discussed. Changyoung Choi[4] et al. investigated the flow and motion characteristics of a freely falling square particle in a channel and discussed the effects of the off-center distance and *Corresponding author E-mail: [email protected] 2
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Reynolds number. Sharma and Patankar[5] numerically simulated the free falling motions of disk and plate shaped particles. These studies have well revealed the mechanism of particle deposition and help us to understand the phenomenon, but they are more interested in the motion regimes than in the process and characteristics of particle settling as that of overshoot, which are more helpful for us to reveal the mechanism and understand the phenomenon, that is the focus of this article. In this paper, we will simulate a circular particle free falling in a narrow vertical channel with a static fluid, the trajectory along with translational and angular velocity of the particle and the corresponding streamlines of flow will be shown to reveal the moving modes, process and characteristics of the particle and flow at different density ratios. This simulation will be carried out by the lattice Boltzmann method (LBM). The LBM is a mesoscopic method based on the theory of molecular motion and mainly used in fluid mechanics at first, over the past two decades, it has developed into an effective numerical tool to be applied in many fields, including micro-flow, multiphase-flow, turbulence, porous media flow, magneto-hydrodynamics, thermal conduction, combustion, chemical reaction, diffusion etc. Particle settling in fluid is a two-phase flow and its physical mechanism can be well revealed by the LBM, this has been proved by many published works. Ladd[6,7] successfully applied the LBM to particle–fluid suspensions. Since then many researchers make a lot of efforts to improve the efficiency and accuracy of the LBM in the application of particulate flows and have a good effect[8-15]. The LBM overcame the limitations of the conventional computational methods by using a fixed, non-adaptive (Eulerian) grid system to represent the flow field. It is easier to code because of no requiring re-meshing and has proven to be robust in simulating particulate flows[16-20]. Before the simulations, the LBM is introduced.
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2. Lattice Boltzmann method
The Boltzmann equation with the single relaxation time approximation is [21] f 1 f f f eq , t
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(1)
Where is the micro-particle velocity, f eq is the equilibrium distribution function, and
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is the relaxation time. The velocity space can be discretized into a finite set of points , and the Boltzmann equation (1) becomes [22,23]
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f 1 f f feq . t For the two-dimensional nine-velocity (D2Q9) square lattice shown in Fig.1,
e6
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e7 e8 e4 Fig 1. The two-dimensional nine-velocity square lattice
the discretized velocity is set as e , 3
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1, 2,3, 4 ,
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and the equilibrium distribution functions have the form [22-24] feq [1
3 9 3 (e u) 4 (e u) 2 2 u u] , 2 c 2c 2c
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where c x t , x and t are the lattice constant and time step respectively, is
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0 4 9 1 9 1, 2,3, 4 . 1 36 5, 6, 7,8
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By discretizing equation (2) in space and time, the lattice Boltzmann equation is [22-24] 1 (6) f x e t , t t f x, t f x, t f eq x, t , where t . The macroscopic quantities (such as the fluid density and velocity u ) are the hydrodynamic moments of the distribution function f (7) f u e f .
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The relaxation time is related to the viscosity by 2 1 2 c t (8) 6 2 The speed of sound of this model is cs c 3 , and the pressure is p cs . In the simulation, we set x t 1 3. Description of the Problem
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The schematic diagram of computational domain is shown in Fig.2. A circular particle is placed in a vertical channel filled with a static fluid. Because heavier than the fluid, the particle descends, rotates and translates under the combined action of gravity, buoyancy and hydrodynamics force, and finally reaches an equilibrium state.
L Fig 2. Schematic diagram of a circular particle sedimentation in a vertical channel. 4
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In this paper, the particle diameter is d =0.1 cm and the channel width is 4d. The particle is released at 0.076 cm away from the left wall and 15d from the inlet. As the density ratio increases, the fully developed flow and periodic vortex shedding will appear, and a large domain size is needed, the channel height is set as 32d for γ<1.3, 40d for 1.3≤γ≤1.5 and 55d for γ>1.5. The fluid density and its kinematic viscosity are 1g/cm3 and 0.01cm2/s, respectively. In the lattice Boltzmann simulations, the particle diameter is 26 lattice units and the relaxation time is 0.6. Zero velocities are applied uniformly at the inlet and the normal derivative of the velocity is set to zero at the outlet. The particle surface and channel wall condition are treated by a linear interpolation proposed by Filipova et al.[25] and improved by Mei et al.[23]. The hydrodynamic force and torque exerting on the particle are calculated by stress integration proposed by Inamuro et al.[26]. The translation of the centroid and the rotation of the particle are updated at each Newtonian dynamics time step by using a so-called half-step “leap-frog” scheme [27]. In the following section, the sedimentation of a circular particle in a vertical channel will be simulated at different particle-fluid density ratios for 1.003≤γ≤5.0, the particle motion and resulting flow regimes will be discussed. At first, an example is carried out to validate the code. 4. Validation of the Code
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A circular particle settling in a vertical channel has been investigated extensively. In order to test the code’s ability to handle this kind of problem, an example of density ratio being 1.03 is carried out, the terminal Reynolds number of the particle is 8.33, which is defined as Re dup , where u p is the final velocity of the particle. The settling trajectory, together with translational velocity and angle velocity are shown in Fig. 3, they are in agreement with the published results by finite element method (FEM)[11,13]. It indicates that the present code works with acceptable accuracy and is qualified for the following problems.
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Fig 3. Comparison of particle trajectory and velocity obtained by LBM and FEM. (a) particle trajectory, (b) the horizontal velocity, (c) the vertical velocity, (d) the angular velocity. 5
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The particle exhibits several regimes of motion in a vertical channel at different density ratios. Whenγ≤1.005, it drifts monotonically to the centerline of the channel and then maintains a steady equilibrium, this can be seen from the trajectories (Fig 4(a)). During the settling, the particle first accelerates to a maximum in the horizontal and vertical directions (Fig 4 (b, c)), then the horizontal velocity decreases until zero, which indicates the particle reaches the centerline. The similar variation also can be observed in the angular velocity, but a second accelerated time, which decreases with the increase of density ratio, appears before the angular velocity reaching a maximum(Fig4(d)). The graphs of angular velocity also reveal the particle rolls up the closer wall during settling until the rotation stops. While the vertical velocity decreases to a stable nonzero value after accelerating to a maximum (Fig4(c)), which indicates the particle is in a uniform linear motion in the vertical direction. It is obvious that a bigger density ratio results in a faster equilibrium, a bigger maximum horizontal velocity, a bigger maximum angular velocity with a shorter second accelerated time, and a bigger final vertical velocity.
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Fig 4. Settling trajectories and velocities of the particle at different density ratios for 1.003≤γ≤1.005 (a) particle trajectory, (b) the horizontal velocity, (c) the vertical velocity, (d) the angular velocity.
The above simulation shows that the particle finally is in a steady equilibrium with a monotonic approach and maintains a constant settling velocity at the middle of the channel where the horizontal movement and rotation stop. Meanwhile, a pair of steady symmetric vortex appears on both side of the particle, as show in Fig.5, the flow is steady and guarantees the particle in a uniform linear motion finally. The final flow pattern keeps stable but the vortices enlarge (Fig.5) with the increase of density ratio for γ≤1.03.
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Fig. 5. Streamlines of flow field at different density ratios
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With the further increase of density ratio, the final flow pattern keeps stable and the vortices further enlarge but still symmetric when γ<1.1, and the particle is also in a steady equilibrium finally, however no longer with a monotonic approach but with an overshoot, as shown in Fig.6(a, b). A crest in the trajectory and a trough in the horizontal velocity graph appear before they are horizontal, which indicates the particle moves across the centerline and reaches a position near the right wall, then it returns with a negative horizontal velocity and reaches the middle of the channel. Fig.7 shows the streamline with an overshoot at different density ratios, where the particle is close to the right wall and presses the right vortex to be slender. When γ≥1.01, a vortex shedding appears causing a transient oscillatory in trajectory and velocity. Obviously, the bigger density ratio makes the particle have a greater overshoot and be closer to the right wall. 0.6 (a) overshoot
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Fig. 6. Settling trajectories and velocities of the particle at different density ratios for 1.006≤γ≤1.1. (a) particle trajectory, (b) the horizontal velocity, (c) the vertical velocity, (d) the angular velocity 7
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The angular position of the particle also has an overshoot. As show in Fig.6(d), the angular velocity increases from zero to a maximum and then decreases to zero, it indicates the particle rotates anticlockwise at first and accelerates to the maximum, then the rotation slows down until it stops and turns clockwise. In the following stage, the particle rotation undergoes a similar course with the anticlockwise stage but with a small maximum, which is presented by a trough in the angular velocity graph, it denotes that the particle rolls up the right wall. The angular velocity graph also shows that the rotation finally vanishes when the particle is in an equilibrium for γ<1.1. When γ=1.1, the horizontal and angular velocity along with the trajectory are weak oscillatory (Fig.6(a, b, d), it suggests that the equilibrium position becomes unstable and the motion of the particle is a weak oscillation although the streamlines appear to be stable and symmetrical (Fig.8). This is a critical state of the particle moving from a steady equilibrium with a transient overshoot to an oscillation.
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Fig.7. Streamlines with an overshoot for different density ratios
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Fig.8. Streamlines at different time for γ=1.1 8
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Fig.9. Streamlines at different times for γ=1.3
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Fig.9 presents the streamlines for γ=1.3 at different times, it shows that two slender wakes, not symmetrical but periodical, appear on both sides of the particle, which cause a periodical flow and an oscillatory moving of the particle in horizontal direction, therefore an oscillation with a small amplitude appears in the trajectory, horizontal and angular velocity (Fig 10 (a, b, d)). For the weak asymmetrical effect, no vortex shedding appears in this case.
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Fig 11. Particle trajectory(a), and horizontal velocity(b), vertical velocity(c), angular velocity(d) for different density ratios at 2.3≤γ≤3.0.
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Whenγ>1.3, the asymmetrical effect of wakes increases to form a periodic vortex shedding and cause the particle oscillating, with the increase of density ratio, the oscillation is stronger and stronger with an increased amplitude in trajectory and velocity. As shown in Fig 10 and 11, the particle has a shorter monotonic path and an earlier start to oscillation for a bigger density ratio, the frequency of oscillation is also increased because of the vortex shedding getting faster and faster. Fig 12 shows the streamlines forγ=1.5 at different time, it is obvious that two wakes behind the particle coupled to be a vortex street varying periodically, which causes the particle in
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Fig 12. Streamlines at different time forγ=1.5 10
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an unstable equilibrium and to be oscillatory. Fig 13 is the streamlines forγ=2.5 at different time, it shows that the wake vary periodically and the vortex are more intensive at the same time than that forγ=1.5, which indicates the vortex shedding is faster.
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Fig 13. Streamlines at different time forγ=2.5
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In the interval for 1.3≤γ≤3, the particle is in an oscillation which is stronger and stronger as the increase of density ratio, but it is periodic, the trajectory and velocity oscillate with a stable frequency and amplitude respectively, that is, the particle is in a regular oscillation. When γ>3, the oscillatory periodicity of the particle becomes irregular gradually. As shown in Fig 14, the trajectories of the particle are oscillatory not with a stable frequency and amplitude but a varied one at the same density ratio and more and more irregular as the density ratio increase, in this case, the flow and the particle moving is in an irregular oscillation. 0.6
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Fig 14. Particle trajectory at different density ratios 11
ACCEPTED MANUSCRIPT 6. Conclusion
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A particle settling in a confined vertical channel with a static fluid is a complicated problem, the sedimentation of the particle and the flow pattern of fluid present a complex style because of their interaction, these problems are investigated by Lattice Boltzmann method in this paper. By analyzing the trajectories and velocities of the particle and the streamlines of the flow, it can be seen that the particle sedimentation displays different styles at different density ratios as following: (1) The particle starting from an initial position close to the left wall drifts monotonically to the centerline of the channel and then maintains a steady equilibrium when γ≤1.005. Before reaching the equilibrium position, the particle movement undergoes a stage of acceleration and deceleration, and finally it stops the horizontal movement and maintains the uniform linear motion in the vertical direction. The rotation of the particle also undergoes a stage of acceleration, including a second acceleration before reaching the maximum, and deceleration and finally stops at the equilibrium position. The angular velocity shows that the particle rolls up the closer wall before it stops rotating. Obviously, the bigger density ratio results in a shorter second acceleration stage for rotation and a faster equilibrium for migration. (2) When 1.005<γ<1.1, the particle moving is similar to that for γ≤1.005 but with a transient overshoot. Before reaching the equilibrium position, the particle gets across the centerline to a position close to the right wall where the rotation reverses to be clockwise and the flow is asymmetrical, moreover, a vortex shedding appears at γ≥1.01 and a transient weak oscillation appears in the trajectory along with the horizontal and angular velocity. Then the particle returns to the centerline where the horizontal movement and rotation vanishes, and the particle is in a steady equilibrium. When γ=1.1, a weak oscillation appears in the trajectory along with the horizontal and angular velocity, it indicates the particle moving is weak oscillatory, but the streamlines are almost stable and symmetrical. This case is a critical state of the particle moving from a stable equilibrium with a transient overshoot to an oscillation. (3) The obvious oscillation of the particle moving occurs at γ=1.3, the trajectory along with the horizontal and angular velocities are oscillatory because two wakes behind the particle are asymmetrical and vary periodically, which causes the particle oscillating near the centerline but with a small amplitude and frequency because of no vertex shedding. As the increase of density ratio, the oscillation becomes stronger and stronger with an increased amplitude and frequency, which is stable at the same density ratio for γ≤3, that is, the oscillation is stable and regular. When γ>3, the trajectories of the particle are irregular and the amplitude and frequency are variable at the same density ratio, which indicates the particle moving is an irregular oscillation. Acknowledgments
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This research was financially supported by the Doctoral Scientific Research Foundation of Shanghai Ocean University(Grant No. A-0209-13-0105328), the Cultivation and Sustentation Foundation for Young Teachers of Universities in Shanghai(Grant No. ZZHY13038), the National Natural Science Foundation of China(Grant No. 11572090),the Special Fund for Renewable Energy of the State Oceanic Administration, People’s Republic China (Grant No. SHME2013JS01), the Open Research Fund of State Key Laboratory of ocean engineering, Shanghai Jiao Tong University(Grant No.1303) and the Special Fund for Science and Technology development, Shanghai Ocean University(Grant No. 1416).
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[1] H. H. Hu, D. D. Joseph, M. J. Crochet, Direct simulation of fluid particle motions, Theoretica and Computational Fluid Dynamics 3 (1992) 285-306 [2] J. Feng, H. H. Hu, D. D. Joseph, Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid: Part 1. Sedimentation, Journal of Fluid Mechanics 261 (1994) 95–134. [3] A. Wachs, A DEM-DLM/FD method for direct numerical simulation of particulate flows: Sedimentation of polygonal isometric particles in a newtonian fluid with collisions, Computers & Fluids 38 (2009) 1608–1628 [4] C. Y. Choi, H. S. Yoon, M. Y. Ha, Flow and motion characteristics of a freely falling square particle in a channel, Computers & Fluids 79 (2013) 1–12 [5] N. Sharma, N. A. Patankar, A fast computation technique for the direct numerical simulation of rigid particulate flows, Journal of Computational Physics 205 (2005) 439–57. [6] A.J.C. Ladd, Numerical simulations of particulate suspensions via a discretized Boltzmann equation Part I. Theoretical foundation, Journal of Fluid Mechanics 271 (1994) 285–310. [7] A.J.C. Ladd, Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part II. Numerical results, Journal of Fluid Mechanics 271 (1994) 311–339. [8] Z. G. Feng, E. E. Michaelides, The immersed boundary-lattice Boltzmann method for solving fluid–particles interaction problems, Journal of Computational Physics 195 (2004) 602–628 [9] Z. G. Feng, E. E. Michaelides, Proteus: a direct forcing method in the simulations of particulate flows, Journal of Computational Physics 202 (2005) 20–51 [10] H. B Li, H. P. Fang, Z. F. Lin, S. X. Xu, S. Y. Chen, Lattice Boltzmann simulation on particle suspensions in a two-dimensional symmetric stenotic artery, Physical Review E 69 (2004) 031919 [11] H. B Li, X. Y. Lu, H. P. Fang, Y. H. Qian, Force evaluations in lattice Boltzmann simulations with moving boundaries in two dimensions, Physical Review E 70 (2004) 026701 [12] R. Z. Wan, H. P. Fang, Z. F. Lin, S. Y. Chen, Lattice Boltzmann simulation of a single charged particle in a Newtonian fluid, Physical Review E 68 (2003) 011401 [13] B. H. Wen, H. B. Li, C. Y. Zhang, H. P. Fang, Lattice-type-dependent momentum-exchange method for moving boundaries, Physical Review E 85 (2012) 016704 [14] Z. H. Xia, K. W. Connington, S. Rapaka, P. T. Yue, J. J. Feng, S. Y. Chen, Flow patterns in the sedimentation of an elliptical particle, Journal of Fluid Mechanics 625 (2009) 249–272. [15] H. H. Yi, L. J. Fan, Y. Y. Chen, Lattice Boltzmann simulation of the motion of spherical particles in steady poiseuille flow, International Journal of Modern Physics C 20(2009): 831-846. [16] Z. G. Feng, E. E. Michaelides, Hydrodynamic force on spheres in cylindrical and prismatic enclosures, International Journal of Multiphase Flow 28 (2002) 479–496. [17] Z. G. Feng, E. E. Michaelides, Interparticle forces and lift on a particle attached to a solid boundary in suspension flow, Physics of Fluids 24 (2002) 49–60. [18] A.J.C. Ladd, R. Verberg, Lattice-Boltzmann simulation of particle–fluid suspensions, Journal of Statistical Physics 104 (2001) 1191–1251. [19] H. H. Yi, Y. L. Yao, Lattice Boltzmann Simulation of the Interaction between Sedimenting Spheres in a Newtonian Fluid, Chinese Journal of Physics 49(5) (2011) 1046-1059 [20] H. H. Yi, The Lift Force on a Spherical Particle in Rectangular Pipe Flow, Chinese Journal of Physics 52(1-I) (2014) 174-184 [21] P. L.Bhatnagar, E. P. Gross, M. Krook, A model for collision processes in gas. I. Small amplitude processes in charged and neutral one-component system, Physical Review 94 (1954) 511-525 13
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[22] X. He, L. Luo, Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation, Physical Review E. 56(6) (1997) 6811-6817. [23] R. Mei, L. S. Luo, W. Shyy, An accurate Curved Boundary Treatment in the Lattice Boltzmann Method, Journal of Computational Physic 155 (1999) 307－330. [24] Y.H. Qian, D. d’Humieres, P. Lallemand, Lattice BGK models for Navier–Stokes equation, Europhysics Letters 17 (6) (1992) 479–484. [25] O. Filippova, D. Hanel, Lattice-Boltzmann simulation of gas-particle flow in filters, Computers & Fluids 26, 697 (1997). [26] T. Inamuro, K. Maeba, F. Ogino, Flow between parallel walls containing the lines of neutrally buoyant circular cylinders, International Journal of Multiphase Flow 26(12) (2000) 1981-2004. [27] M. P. Allen, D. J. Tildesley, Computer Simulation of Liquid (Clarendon, Oxford, 1987).
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Density ratio effect on particle sedimentation in a vertical channel Yong Rao , Chaofeng Liu , Huatao Wang , Yushan Ni , Chao Lv , Shuang Liu , Yamei Lan , Shiming Wang PII: DOI: Reference:
S0577-9073(17)30807-9 10.1016/j.cjph.2018.06.007 CJPH 553
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
30 June 2017 6 June 2018 7 June 2018
Please cite this article as: Yong Rao , Chaofeng Liu , Huatao Wang , Yushan Ni , Chao Lv , Shuang Liu , Yamei Lan , Shiming Wang , Density ratio effect on particle sedimentation in a vertical channel, Chinese Journal of Physics (2018), doi: 10.1016/j.cjph.2018.06.007
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Highlights A particle settling in a vertical channel is studied at different density ratios. The effect of density ratio on the particle settling modes is discussed. The distinct flow regimes are presented at different density ratios. The relation between particle settling modes and flow regimes is analyzed.
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Density ratio effect on particle sedimentation in a vertical channel *
Yong Rao a , Chaofeng Liu b, Huatao Wang c, Yushan Ni d, Chao Lv a, Shuang Liu a, Yamei Lan a and Shiming Wang a a
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College of Engineering Science and Technology, Shanghai Ocean University, Shanghai 201306, China b Shanghai Electro-Mechanical Engineering Institute, Shanghai 200233, China c Research Management Office for Science and Technology , Fudan University, Shanghai 200433, China d Department of Aeronautics and Astronautics, Fudan University, Shanghai 200433, China
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A circular particle settling in a confined vertical channel with static fluid is investigated by lattice Boltzmann method, the effect of density ratio on the particle motion and flow pattern is discussed. It shows that the particle starting from an initial position off-center displays various moving modes accompanied by different flow patterns for 1.003≤γ≤5.0: when γ<1.1, the particle is finally in a stable equilibrium near the centerline and settles uniformly with a symmetrical flow, but there are different modes before reaching the centerline at different density ratios, a monotonic approach appears at γ≤1.005 and an transient overshoot appears at 1.005<γ<1.1; When γ=1.1, a weak oscillation appears in the particle moving but the flow is almost symmetrical, it is in a critical state of the particle moving from stable equilibrium with a transient overshoot to oscillation; When 1.1<γ≤3.0, the particle oscillates regularly around the centerline with the uniform amplitude and frequency, and it turns into irregularity for γ>3.0. Meanwhile the rotation of the particle depends on the distance from a wall; it rolls up the closer wall and stops at centerline.
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Keywords: particulate flow, sedimentation, density ratio effect, Lattice Boltzmann method
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1. Introduction
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The sedimentation of particles in fluid is ubiquitous in nature and engineering field, e.g. the deposition of silt in water conservancy project, falling of hail in meteorology, purification of water in environmental engineering, erythrocyte sedimentation in biological engineering, etc. The particle motion affects the flow pattern, and vice versa. These two kinds of motions are coupled into a complex two-phase flow, which is important both in theoretic research and engineering application, and a lot of investigation has been carried out, both experimentally and numerically. Many previous works have reported the particle sedimentation in fluid under various conditions, including off-center distance, channel width, fluid properties, particle shape, inter-particle interaction etc. Howard H.Hu et al. [1] revealed the effect of vortex shedding on particle motion and reproduced the drafting, kissing, and tumbling scenario of two falling particles in the falling process. J. Feng et al. [2] investigated the initial value problems for the sedimentation of a circular and elliptical particle in a vertical channel, the particle motion at different Reynolds numbers and the non-linear effects of particle-fluid, particle-wall and inter-particle interactions were discussed. Anthony Wachs[3] investigated the sedimentation of polygonal isometric particles with collisions in a Newtonian fluid, the influence of shape on particles sedimentation and the collective behavior of 300 particles settling in a closed box were discussed. Changyoung Choi[4] et al. investigated the flow and motion characteristics of a freely falling square particle in a channel and discussed the effects of the off-center distance and *Corresponding author E-mail: [email protected] 2
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Reynolds number. Sharma and Patankar[5] numerically simulated the free falling motions of disk and plate shaped particles. These studies have well revealed the mechanism of particle deposition and help us to understand the phenomenon, but they are more interested in the motion regimes than in the process and characteristics of particle settling as that of overshoot, which are more helpful for us to reveal the mechanism and understand the phenomenon, that is the focus of this article. In this paper, we will simulate a circular particle free falling in a narrow vertical channel with a static fluid, the trajectory along with translational and angular velocity of the particle and the corresponding streamlines of flow will be shown to reveal the moving modes, process and characteristics of the particle and flow at different density ratios. This simulation will be carried out by the lattice Boltzmann method (LBM). The LBM is a mesoscopic method based on the theory of molecular motion and mainly used in fluid mechanics at first, over the past two decades, it has developed into an effective numerical tool to be applied in many fields, including micro-flow, multiphase-flow, turbulence, porous media flow, magneto-hydrodynamics, thermal conduction, combustion, chemical reaction, diffusion etc. Particle settling in fluid is a two-phase flow and its physical mechanism can be well revealed by the LBM, this has been proved by many published works. Ladd[6,7] successfully applied the LBM to particle–fluid suspensions. Since then many researchers make a lot of efforts to improve the efficiency and accuracy of the LBM in the application of particulate flows and have a good effect[8-15]. The LBM overcame the limitations of the conventional computational methods by using a fixed, non-adaptive (Eulerian) grid system to represent the flow field. It is easier to code because of no requiring re-meshing and has proven to be robust in simulating particulate flows[16-20]. Before the simulations, the LBM is introduced.
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2. Lattice Boltzmann method
The Boltzmann equation with the single relaxation time approximation is [21] f 1 f f f eq , t
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Where is the micro-particle velocity, f eq is the equilibrium distribution function, and
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is the relaxation time. The velocity space can be discretized into a finite set of points , and the Boltzmann equation (1) becomes [22,23]
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f 1 f f feq . t For the two-dimensional nine-velocity (D2Q9) square lattice shown in Fig.1,
e6
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e7 e8 e4 Fig 1. The two-dimensional nine-velocity square lattice
the discretized velocity is set as e , 3
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By discretizing equation (2) in space and time, the lattice Boltzmann equation is [22-24] 1 (6) f x e t , t t f x, t f x, t f eq x, t , where t . The macroscopic quantities (such as the fluid density and velocity u ) are the hydrodynamic moments of the distribution function f (7) f u e f .
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The relaxation time is related to the viscosity by 2 1 2 c t (8) 6 2 The speed of sound of this model is cs c 3 , and the pressure is p cs . In the simulation, we set x t 1 3. Description of the Problem
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The schematic diagram of computational domain is shown in Fig.2. A circular particle is placed in a vertical channel filled with a static fluid. Because heavier than the fluid, the particle descends, rotates and translates under the combined action of gravity, buoyancy and hydrodynamics force, and finally reaches an equilibrium state.
L Fig 2. Schematic diagram of a circular particle sedimentation in a vertical channel. 4
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In this paper, the particle diameter is d =0.1 cm and the channel width is 4d. The particle is released at 0.076 cm away from the left wall and 15d from the inlet. As the density ratio increases, the fully developed flow and periodic vortex shedding will appear, and a large domain size is needed, the channel height is set as 32d for γ<1.3, 40d for 1.3≤γ≤1.5 and 55d for γ>1.5. The fluid density and its kinematic viscosity are 1g/cm3 and 0.01cm2/s, respectively. In the lattice Boltzmann simulations, the particle diameter is 26 lattice units and the relaxation time is 0.6. Zero velocities are applied uniformly at the inlet and the normal derivative of the velocity is set to zero at the outlet. The particle surface and channel wall condition are treated by a linear interpolation proposed by Filipova et al.[25] and improved by Mei et al.[23]. The hydrodynamic force and torque exerting on the particle are calculated by stress integration proposed by Inamuro et al.[26]. The translation of the centroid and the rotation of the particle are updated at each Newtonian dynamics time step by using a so-called half-step “leap-frog” scheme [27]. In the following section, the sedimentation of a circular particle in a vertical channel will be simulated at different particle-fluid density ratios for 1.003≤γ≤5.0, the particle motion and resulting flow regimes will be discussed. At first, an example is carried out to validate the code. 4. Validation of the Code
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A circular particle settling in a vertical channel has been investigated extensively. In order to test the code’s ability to handle this kind of problem, an example of density ratio being 1.03 is carried out, the terminal Reynolds number of the particle is 8.33, which is defined as Re dup , where u p is the final velocity of the particle. The settling trajectory, together with translational velocity and angle velocity are shown in Fig. 3, they are in agreement with the published results by finite element method (FEM)[11,13]. It indicates that the present code works with acceptable accuracy and is qualified for the following problems.
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The particle exhibits several regimes of motion in a vertical channel at different density ratios. Whenγ≤1.005, it drifts monotonically to the centerline of the channel and then maintains a steady equilibrium, this can be seen from the trajectories (Fig 4(a)). During the settling, the particle first accelerates to a maximum in the horizontal and vertical directions (Fig 4 (b, c)), then the horizontal velocity decreases until zero, which indicates the particle reaches the centerline. The similar variation also can be observed in the angular velocity, but a second accelerated time, which decreases with the increase of density ratio, appears before the angular velocity reaching a maximum(Fig4(d)). The graphs of angular velocity also reveal the particle rolls up the closer wall during settling until the rotation stops. While the vertical velocity decreases to a stable nonzero value after accelerating to a maximum (Fig4(c)), which indicates the particle is in a uniform linear motion in the vertical direction. It is obvious that a bigger density ratio results in a faster equilibrium, a bigger maximum horizontal velocity, a bigger maximum angular velocity with a shorter second accelerated time, and a bigger final vertical velocity.
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Fig 4. Settling trajectories and velocities of the particle at different density ratios for 1.003≤γ≤1.005 (a) particle trajectory, (b) the horizontal velocity, (c) the vertical velocity, (d) the angular velocity.
The above simulation shows that the particle finally is in a steady equilibrium with a monotonic approach and maintains a constant settling velocity at the middle of the channel where the horizontal movement and rotation stop. Meanwhile, a pair of steady symmetric vortex appears on both side of the particle, as show in Fig.5, the flow is steady and guarantees the particle in a uniform linear motion finally. The final flow pattern keeps stable but the vortices enlarge (Fig.5) with the increase of density ratio for γ≤1.03.
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Fig. 5. Streamlines of flow field at different density ratios
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With the further increase of density ratio, the final flow pattern keeps stable and the vortices further enlarge but still symmetric when γ<1.1, and the particle is also in a steady equilibrium finally, however no longer with a monotonic approach but with an overshoot, as shown in Fig.6(a, b). A crest in the trajectory and a trough in the horizontal velocity graph appear before they are horizontal, which indicates the particle moves across the centerline and reaches a position near the right wall, then it returns with a negative horizontal velocity and reaches the middle of the channel. Fig.7 shows the streamline with an overshoot at different density ratios, where the particle is close to the right wall and presses the right vortex to be slender. When γ≥1.01, a vortex shedding appears causing a transient oscillatory in trajectory and velocity. Obviously, the bigger density ratio makes the particle have a greater overshoot and be closer to the right wall. 0.6 (a) overshoot
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The angular position of the particle also has an overshoot. As show in Fig.6(d), the angular velocity increases from zero to a maximum and then decreases to zero, it indicates the particle rotates anticlockwise at first and accelerates to the maximum, then the rotation slows down until it stops and turns clockwise. In the following stage, the particle rotation undergoes a similar course with the anticlockwise stage but with a small maximum, which is presented by a trough in the angular velocity graph, it denotes that the particle rolls up the right wall. The angular velocity graph also shows that the rotation finally vanishes when the particle is in an equilibrium for γ<1.1. When γ=1.1, the horizontal and angular velocity along with the trajectory are weak oscillatory (Fig.6(a, b, d), it suggests that the equilibrium position becomes unstable and the motion of the particle is a weak oscillation although the streamlines appear to be stable and symmetrical (Fig.8). This is a critical state of the particle moving from a steady equilibrium with a transient overshoot to an oscillation.
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Fig.7. Streamlines with an overshoot for different density ratios
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Fig.8. Streamlines at different time for γ=1.1 8
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Fig.9 presents the streamlines for γ=1.3 at different times, it shows that two slender wakes, not symmetrical but periodical, appear on both sides of the particle, which cause a periodical flow and an oscillatory moving of the particle in horizontal direction, therefore an oscillation with a small amplitude appears in the trajectory, horizontal and angular velocity (Fig 10 (a, b, d)). For the weak asymmetrical effect, no vortex shedding appears in this case.
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Whenγ>1.3, the asymmetrical effect of wakes increases to form a periodic vortex shedding and cause the particle oscillating, with the increase of density ratio, the oscillation is stronger and stronger with an increased amplitude in trajectory and velocity. As shown in Fig 10 and 11, the particle has a shorter monotonic path and an earlier start to oscillation for a bigger density ratio, the frequency of oscillation is also increased because of the vortex shedding getting faster and faster. Fig 12 shows the streamlines forγ=1.5 at different time, it is obvious that two wakes behind the particle coupled to be a vortex street varying periodically, which causes the particle in
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an unstable equilibrium and to be oscillatory. Fig 13 is the streamlines forγ=2.5 at different time, it shows that the wake vary periodically and the vortex are more intensive at the same time than that forγ=1.5, which indicates the vortex shedding is faster.
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Fig 13. Streamlines at different time forγ=2.5
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In the interval for 1.3≤γ≤3, the particle is in an oscillation which is stronger and stronger as the increase of density ratio, but it is periodic, the trajectory and velocity oscillate with a stable frequency and amplitude respectively, that is, the particle is in a regular oscillation. When γ>3, the oscillatory periodicity of the particle becomes irregular gradually. As shown in Fig 14, the trajectories of the particle are oscillatory not with a stable frequency and amplitude but a varied one at the same density ratio and more and more irregular as the density ratio increase, in this case, the flow and the particle moving is in an irregular oscillation. 0.6
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A particle settling in a confined vertical channel with a static fluid is a complicated problem, the sedimentation of the particle and the flow pattern of fluid present a complex style because of their interaction, these problems are investigated by Lattice Boltzmann method in this paper. By analyzing the trajectories and velocities of the particle and the streamlines of the flow, it can be seen that the particle sedimentation displays different styles at different density ratios as following: (1) The particle starting from an initial position close to the left wall drifts monotonically to the centerline of the channel and then maintains a steady equilibrium when γ≤1.005. Before reaching the equilibrium position, the particle movement undergoes a stage of acceleration and deceleration, and finally it stops the horizontal movement and maintains the uniform linear motion in the vertical direction. The rotation of the particle also undergoes a stage of acceleration, including a second acceleration before reaching the maximum, and deceleration and finally stops at the equilibrium position. The angular velocity shows that the particle rolls up the closer wall before it stops rotating. Obviously, the bigger density ratio results in a shorter second acceleration stage for rotation and a faster equilibrium for migration. (2) When 1.005<γ<1.1, the particle moving is similar to that for γ≤1.005 but with a transient overshoot. Before reaching the equilibrium position, the particle gets across the centerline to a position close to the right wall where the rotation reverses to be clockwise and the flow is asymmetrical, moreover, a vortex shedding appears at γ≥1.01 and a transient weak oscillation appears in the trajectory along with the horizontal and angular velocity. Then the particle returns to the centerline where the horizontal movement and rotation vanishes, and the particle is in a steady equilibrium. When γ=1.1, a weak oscillation appears in the trajectory along with the horizontal and angular velocity, it indicates the particle moving is weak oscillatory, but the streamlines are almost stable and symmetrical. This case is a critical state of the particle moving from a stable equilibrium with a transient overshoot to an oscillation. (3) The obvious oscillation of the particle moving occurs at γ=1.3, the trajectory along with the horizontal and angular velocities are oscillatory because two wakes behind the particle are asymmetrical and vary periodically, which causes the particle oscillating near the centerline but with a small amplitude and frequency because of no vertex shedding. As the increase of density ratio, the oscillation becomes stronger and stronger with an increased amplitude and frequency, which is stable at the same density ratio for γ≤3, that is, the oscillation is stable and regular. When γ>3, the trajectories of the particle are irregular and the amplitude and frequency are variable at the same density ratio, which indicates the particle moving is an irregular oscillation. Acknowledgments
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This research was financially supported by the Doctoral Scientific Research Foundation of Shanghai Ocean University(Grant No. A-0209-13-0105328), the Cultivation and Sustentation Foundation for Young Teachers of Universities in Shanghai(Grant No. ZZHY13038), the National Natural Science Foundation of China(Grant No. 11572090),the Special Fund for Renewable Energy of the State Oceanic Administration, People’s Republic China (Grant No. SHME2013JS01), the Open Research Fund of State Key Laboratory of ocean engineering, Shanghai Jiao Tong University(Grant No.1303) and the Special Fund for Science and Technology development, Shanghai Ocean University(Grant No. 1416).
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