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Particle Dispersion Simulation in Turbulent Flow Due to Particle-Particle and Particle-Wall Collisions

Particle Dispersion Simulation in Turbulent Flow Due to Particle-Particle and Particle-Wall... Downloaded from https://academic.oup.com/jom/article/32/2/237/5948549 by DeepDyve user on 09 October 2022 PARTICLE DISPERSION SIMULATION IN TURBULENT FLOW DUE TO PARTICLE-PARTICLE AND PARTICLE-WALL COLLISIONS J.-H. Lin K.-C. Chang Department of Aeronautics and Astronautics National Cheng Kung University Tainan, Taiwan ABSTRACT Simulation of the 3-D, fully developed turbulent channel flows laden with various mass loading ratios of particles is made using an Eulerian-Lagrangian approach in which the carrier-fluid flow field is solved with a low-Reynolds-number k- turbulence model while the deterministic Lagrangian method together with binary-collision hard-sphere model is applied for the solution of particle motion. Effects of inter- particle collisions and particle-wall collisions under different extents of wall roughness on particle disper- sion are addressed in the study. A cost-effective searching algorithm of collision pair among particles is developed. It is found that the effects of inter-particle collisions on particle dispersion cannot be negli- gible when the ratio of the mean free time of particle to the mean particle relaxation time of particle is less or equal to O(10). In addition, the wall roughness extent plays an important role in the simulation of particle-wall collisions particularly for cases with small mass loading ratios. Keywords: Particle-laden flow, Particle-particle collisions, Particle-wall collisions, Channel flow. 1. INTRODUCTION tion are referred to our previous work [1] and briefly described here. Modeling of particle-laden flows has been developed along two distinct (Eulerian-Eulerian and-Lagrangian) 2.1 Governing Equations for Carrier Fluid frameworks. In the first approach, the particle ele- The Reynolds-averaged Navier-Stokes equations ments are also described as a continuous fluid and, coupled with low-Reynolds number k- turbulence therefore, named as the two-fluid model. In contrast, model developed by Abe et al. [2] are given below. the second approach treats the particles as individual entities of which their motions are simulated using the Continuity: Lagrangian tracking method. It is well agreed [1] that u influential extents of collisions between particle-  0 (1) particle and-wall (in a wall-bounded configuration) x become more significant as the particle-loading ratio is Momentum: increased. This study is addressed to how to set up an accurate yet economical numerical procedure in simu- ()  u gg lating the particle dispersion motion in the wall-  ()  uu gg g ji  tx bounded turbulent flows on the basis of Lagrangian tracking method.   x px 2  ()    ku S  tij gg pu ii   xx x x 3 ij j i   2. PHYSICAL MODELING (2) Traditional approach based on the Reynolds- averaged Navier-Stokes (RANS) equation associated Turbulence kinetic energy: with a low Reynolds number version of k- turbulence ()  k model [2] is applied for the solution of the carrier-fluid  ()  uk gg  tx flow field, while the stochastic separated flow model [3] j for the solution of the dispersed-phase (particles) flow    pk   field. The effects of turbulence, inter-particle colli- (3)     PS     kg p   xx  x sions, and particle-wall collisions on particle dispersion ij k j   are taken into account in the study. Detailed modeling Dissipation rate of turbulence kinetic energy: on these physical phenomena of particle dispersion mo- Corresponding author ([email protected]) Journal of Mechanics, Vol. 32, No. 2, April 2016 237 DOI : 10.1017/jmech.2015.63 Copyright © 2015 The Society of Theoretical and Applied Mechanics, R.O.C. Downloaded from https://academic.oup.com/jom/article/32/2/237/5948549 by DeepDyve user on 09 October 2022 ()  where  ()  u gg  tx j 1   uu   pg    p    1.5 (12)        0.135k 0.22k    xx  x ij k j     gg 2 (4)  d  CP C f PS pp  12kk p   kk p (13) 18 where the eddy viscosity,  , and the turbulence produc- tion term, P , are given, respectively, by 2.3 Equations of Particle Motion Equations of particle motion consist of translation  Cf (5) tg  and rotation parts, which are expressed, respectively, as follows.  u  uu g j g ii P  (6)  kt  Translation:  x xx j ij  dU Here f and f , which are appeared in Eqs. (4) and (5),   m FFF F p DS M G (14) dt respectively, are two damping functions accounting for the eddy viscous effects in the near-wall region and are given by Rotation:   dΩ   y Re   p (7) I Τ f  1exp  10 .3exp    p ν    (15) dt 3.1 6.5        where the quasi-steady drag force (F ), Saffman lift     y 5.0 Re force (F ), Magnus lift force (F ), and gravitational   S M f  1exp  1 exp  (8)      0.75 force (F ), while the viscous torque (T ) are considered G v 14 Re 200   t      in modeling the translational and rotational motion of particle, respectively. Calculations of these forces are where referred to our previous work [1] and not repeated here. The momentum inertia I for solid sphere shown in Eq. 2 p yu  k   w (15) is defined by yu,,Re  t   I  0.1 md ppp (16) 2.2 Particle Source Terms in the Carrier Fluid 2.4 Turbulent Dispersion of Particles The particle source term shown in Eq. (2) represents the momentum exchange rate between the carrier fluid Solution of the equations of instantaneous particle and particles and is calculated using the PSI-cell meth- motion, i.e. Equations (14) and (15) require input the od [4] in a specified grid cell as instantaneous velocity of the carrier fluid. Neverthe- less, only the mean quantities of the carrier-fluid veloc- ll ity components are provided in the solutions of the  Su () ()u p ptt p t ui  i (9) i  V RANS governing equations. The fluctuating velocity l1 of carrier fluid along the particle trajectory is generated where N is the total number of particles passing through using the Langevin method [6] as follows. the specified cell within a given time step t, and V is the volume of the specified grid cell. To account for ('uu )  ( ' )RG ('u ) 1R (17) gtt g t i g rms ii i the effects of turbulence modulation, two particle-fluid interaction terms S and S are introduced into Eqs. where G is the Gaussian distribution function with zero pk p th (10) and (11) as the addition source terms, respectively, mean and unit variance for i component of fluid ve- and are modeled as the ones suggested by Lightstone et locity, and R is the autocorrelation function expressed al. [5] as follows. by l Rt  exp( /T ) (18) N *  m  2k  S 1  p  k * (10)  V    where the Lagrangian integral time scale T is estimated l1 p p L  by N * m  2 2 k S 1  p   * (11) T  (19)  V    l1 p p  C  238 Journal of Mechanics, Vol. 32, No. 2, April 2016 Downloaded from https://academic.oup.com/jom/article/32/2/237/5948549 by DeepDyve user on 09 October 2022 In this study, the model constant of C is set equal to 7.0 where f and e are the normal restitution coefficient and suggested by Sawford [7]. frictional coefficient of the particle, respectively, and are given with the values of 0.995 and 0.3, respectively, in the present study; (U ) is the relative velocity vector ij ct 2.5 Inter-Particle Collisions of the particle i to the particle j at the contact point during collision and can be expressed by Searching for inter particle collisions is basically made with the binary-collision hard-sphere model [8]. There are two steps in searching for the particle colli- i ()UUU  Ω  e Ω  e (26) ij ct p p p n p n ij i j sion pair. First, all particles are advanced to the next 22 time step (t t) through solving the equations of motion Here e and e represent the unit vectors along normal n t without taking into account inter particle collisions as and tangential directions, respectively, at the contact shown in Fig. 1. Second, particle collision pair is point of two particles i and j during collision, and are searched by the numerical procedure which will be given by elaborated in the next section. In the hard-sphere collision model, the interaction er  /|r | (27) nt t t t between the collided particles is assumed to be impul- sive, and hence the particles only exchange momentum eU () /|()U | (28) t ij ct ij ct for each other. When a collision is identified, the post-collision ve- The post-collision positions of the collision pair are locities for the collision pair are updated in accordance updated by each corresponding values of [r  (1 t t with the inelastic hard-sphere model as  )tU ]. It is assumed that there happens at most (20) UUJ / m one collision for the particle i in a given t. No fur- pp p ii ther search for the collision pair is, thus, performed in the remaining time step, i.e. (1  )t for the particle i. UUJ / m (21) pp p jj 2.6 Particle-Wall Collisions * i ΩΩ eJ (22) PP n ii 2I It is known [10] that the particle-wall collisions can dominate the particle motion in a channel flow under j certain conditions such as wall roughness and size of ΩΩ eJ (23) PP n jj loading particle. In consideration of the changing wall 2I roughness with time due to constant occurrence of particle-wall collisions, a stochastic model for virtual Here the impulse force vector J is composed of the rough wall (illustrated in Fig. 2) developed by Som- tangential (J ) and normal (J ) components which are t n merfeld and Huber [11] is adopted in this study. The expressed [9], respectively, by probability density function of contact surface angle , P , at a given incident angles of particle  and a col  mm pi pj standard deviation of surface angle, , is expressed as  U  JJ   ct tn ij J  7mm  for (24) t  pi pj JJ  tn  fJ  n P (,,) col   mm  11sin( ) pi pj (29) J(  1 e)(Ue) (25)  max 0 , exp n ij n   mm  2s  in() 2 Δ pi pj      Fig. 2 Schematic of particle collisions on a rough- Fig. 1 Schematic for searching collision pair in bina- wall surface. ry collision process. Journal of Mechanics, Vol. 32, No. 2, April 2016 239 Downloaded from https://academic.oup.com/jom/article/32/2/237/5948549 by DeepDyve user on 09 October 2022 3. SEARCHING ALGORITHM OF COLLISION PAIR AMONG PARTICLES Searching for inter-particle collisions using a deter- ministic, binary-collision hard-sphere model follows the basic approach developed by Bird [8], and is modified in the flowing steps ,(the flowchart is shown in Fig. 3.) 1. All particles are advanced to the next time step (t t) through solving the equations of motion without taking into account inter-particle collisions as shown in Fig. 1. 2. Find the minimal collision time  t, i.e. the first min collision pair in all particles. 3. Update the post-collision velocities for the collision pair at the minimum time using the hard-sphere model as suggested by Tanaka and Tsuji [9]. 4. Repeat the above steps but with the time step of (1  )t until no more collision pair can be found min in the remaining part of t. Note that the time step (t) mentioned above should be much less than the relaxation time of particle ( ) and others time scales such as the turbulence time scale, residence time scale (x / u ), etc. to assure the solu- i pi tion accuracy of equations of motion. Searching for the potential collision pair for a speci- fied particle should be, in principle, checked with all the others particles, which leads to the CPU time for collision-pair searching process is proportional to N(N1)  N . This is cost-ineffective particularly for Fig. 3 Flowchart for particle tracking algorithm which the cases loaded with large N number of particles. In takes account into the inter-particle collisions practice, searching for collision pair for a specified par- with binary collision hard-sphere model. ticle can be confined in a nearby sub-region around the particle. As Fig. 4 shows, all the particles, colored red, which are found within the cube centered at a particle colored blue, are stored. When searching for a colli- sion pair for the particle colored blue, only the particles in the neighbor list (with the N number) need to be nblist scanned. However, the requirement of computational resource (i.e. memory bandwidth and floating-point operations) for constructing the neighbor list for each particle is still proportional to N , since it needs to scan over all other particles location and , then, calculates the distance between the specified particle and others. In consideration of saving more computational ex- penditure, an effective yet economical searching algo- rithm is proposed in the following. The Lagrangian tracking domain for identifying the particles’ locations is divided into a number of blocks which is similar to Fig. 4 Illustration of neighbor list of particles, the red the “cell” concept in the Eulerian finite-volume mesh, particles are stored in the neighbor list of the Next, recording the index of the adjacent cell for each blue particle. cell in a way shown in Fig. 5. Here the adj_cell [:] is an array which records the cell indices of all the adjacent-cell information for each cell. For example, th the i adjacent cell for cell j is termed as adj_cell [adj_cell_index [j]  i]. In this way, we can allocate the specific adjacent cell for a specified cell directly without doing the conventionally searching process. Definition of the cell adjacency can be made with shar- ing either the interface (shown in Fig. 6(a)) or the same node (shown in Fig. 6(b)). To cover more potential Fig. 5 Illustration of data structure for adjacency cell neighboring sub-region, the definition of the cell adja- index. cency shown in Fig. 6(b) is adopted in this study. 240 Journal of Mechanics, Vol. 32, No. 2, April 2016 Downloaded from https://academic.oup.com/jom/article/32/2/237/5948549 by DeepDyve user on 09 October 2022 Fig. 7 Illustration of data structure for particles list for each cell. Fig. 6 Illustration of the adjacent cells, the green cells are the adjacent cells of the yellow one. The definition of adjacent can be made with sharing cpu time ~N either (a) the same interface; or (b) the same 10 2 ~N node. The data structure for the list of particles’ indices is similar to the adjacent cell index. The diagram shown 5 6 in Fig. 7 is the particles list for each cell in which the 10 10 number of particles array of partilce_tag [:] is used to store the indices of th particles for each cell. For instance, the index of i Fig. 8 Averaged CPU time per t versus number of particle in the cell j is recorded in the array particle_tag particles in completing a Lagrangian particle [partilce_cell [j]  i]. The particles data in the specific tracking process. region of Lagrangian cell can be directly allocated without any additional searching process. 4. RESULTS AND DISCUSSION The list of particles’ indices has to be updated before finishing an iterate as follows. The distances between The 3-D fully developed, turbulent downward chan- each particle in the specified cell, colored yellow in Fig. nel flows laden with various mass loading ratios of 6, and the cell centers of each adjacent cell, colored copper particles with 70m in diameter, which were green in Fig. 6, are re-calculated, respectively. If the experimentally investigated by Kulick et al. [12], are new distance between the particle in the specified cell, chosen as the test problem. Simulations of this test colored yellow, and the center of an adjacent cell is problem have been reported in the literature [13-15]. shorter than the original distance between the particle However, none of the existed studies have ever consid- and cell center of the specified cell, colored yellow, the ered both the effects of inter-particle collisions and par- index of the particle will be transferred to the nearest ticle-wall collisions by using the economic yet realistic adjacent cell. models. For instance, Eaton [13] did not take into By means of this list of particles’ indices, the colli- account the inter-particle collisions in his simulation. sion pairs can be directly accessed between the speci- Although the simulations made by Yamamoto et al. [14] fied cell and adjacent cells without scanning all parti- and Zhao et al. [15] considered the mechanism of inter- cles in the flow domain. Thus, the computational time particle collisions in their modelings, both of these two required for this algorithm of Lagrangian particle studies employed a simple but unrealistic smooth-wall tracking processes, including solving the equations of collision model in the simulations. Thus, the simula- motion, searching the collision pair, and updating the tion with the more economic and realistic modelings of list of neighbor particles’ indices is theoretically pro- inter-particle collisions and particle-wall collisions is portional to O(N) only, where N is the total number of conducted in the following. particles. A series of the fully developed channel The computational domain in form of Cartesian co- flows laden with various numbers of copper particles of ordinate is set as 1.0m(L)  0.04m(H)  0.01m(W) as 5m in diameter, ranging from 5,000 to 2,000,000, is schematically shown in Fig. 9. The mean stream-wise tested on a single core of Intel Xeon E5-1620 processor velocity of carrier fluid at the inlet, u , is equal to to estimate the CPU time required for completing a 10.5m/s, which corresponds to Re, based on the hy- Lagrangian particle tracking process with the proposed draulic diameter, of 13,800. Detail of the numerical algorithm in a given time step (t). The “Lagrangian aspects and boundary conditions are referred to our cells” are set the same as the Eulerian grid layout which is composed of 216,000 grid meshes. It is also found previous work [1] and not repeated here for brevity. that the optimal particle number in a searching list had 4.1 Effects of Inter-Particle Collisions on Particle better be in the order of O(10) to yield high computa- Dispersion tional efficiency in the Lagrangian tracking processes together with the search of collision pairs. Figure 8 Three different mass loading ratios of 2, 20 and presents the averaged CPU time per t versus the num- 40 are tested for investigating the effects of inter- ber of particles laden in the flow. Obviously, the re- particle collisions on particle dispersion. Their pre- sults corroborate that the CPU time is approximately dicted mean stream-wise particle velocity (u ) and root- proportional to N. mean-squared fluctuating stream-wise ((u ) ) and p rms Journal of Mechanics, Vol. 32, No. 2, April 2016 241 cpu time per step (sec) Downloaded from https://academic.oup.com/jom/article/32/2/237/5948549 by DeepDyve user on 09 October 2022 wall-normal ((v ) ) particle velocities are presented in between the cases with and without considering inter- p rms Fig. 10. One more simulation for the case of 2 mass particle collisions become more remarkably as the mass loading ratio but without considering the inter-particle loading ratio is increased. Even for the most dilute collisions is purposely preformed, and its predicted re- case (2 mass loading ratio) investigated in the study, sults of u , (u ) and (v ) are also presented in Fig. p p rms p rms there appear observable differences between the two 10. It is noted that the results obtained without con- results predicted between with and without considering sidering inter-particle collisions differ very slightly inter-particle collisions. from each other in the cases with different mass loading ratios in the test problem. Thus, the results obtained The mean collision time,  , of particle for the cases from the case of 2 mass loadings ratio but without with mass loading ratios of 2, 20 and 40 are 2.237, considering inter-particle collisions can serve as a 0.4156 and 0.1892 sec, respectively in contrast to the comparison baseline for the tested cases with three var- particle relaxation time,  , is about 0.1 sec. The rati- ious mass loading ratios. The comparison results os of  over  for the cases with 20 and 40 made from Fig. 10 reveal clearly that the differences loading ratios are O(10 ) while for the case with 2 mass loading is O(10 ). This concludes that the in- ter-particle collisions have to be considered in the mod- eling when  / is less or equal to O(10 ). cp 4.2 Effects of Particle-Wall Collisions on Particle Dispersion Two different wall roughness of  = 0 (smooth wall) and  = 0.02 are tested for each cases of mass loading ratios of 2, 20 and 40. Their predicted u , (u ) and (v ) are presented in Fig. 11. It is p rms p rms clearly seen that the wall roughness extent is an im- portant parameter to affect the particle-wall collisions. The comparison made between the cases with  = 0 and 0.02 for each tested mean loading ratio reveals that the effects of wall roughness extent on particle disper- sion motion become more remarkable as the mass loading ratio is deceased. This can be attributed to a Fig. 9 Schematic of test problem. fact that the particle dispersion is more dominated by the mechanism of inter-particle collisions as the mass loading ratio is increased. (a) 4.3 Comparison of the Model Predictions with Experimental Data The predicted fully-developed distributions of u and k for the single-phase (i.e. 0 mass loading ratio) case, (b) are presented in Fig. 12 and compared with the meas- ured data of Kulick et al. [12]. Here the dimensionless wall distance (y ) is defined by yu /  where u =   1/2 ( / ) and  is the wall shear stress. Good agree- w w ment between the predicted and measured results im- plies that the employed turbulence model performs well. (c) The fully-developed distributions of u , (u ) and p p rms (v ) , predicted with the 20 mass loading ratio and p rms  = 0.02 are presented in Fig. 13 and compared with the measured data of Kulick et al. [12] Reasonable agreement between the predicted and measured results Fig. 10 Comparison of the fully developed distribu- implies that the modeling of particle dispersion has to tions of (a) u ; (b) (u ) ; and (c) (v ) in take into account of the inter-particle collisions and p p rms p rms particle-wall collisions in the flow laden with signifi- the case with wall roughness of  = 0 for cant mass loading ratios. three different mass loading ratios. 242 Journal of Mechanics, Vol. 32, No. 2, April 2016 Downloaded from https://academic.oup.com/jom/article/32/2/237/5948549 by DeepDyve user on 09 October 2022 5. CONCLUSIONS (a) A modeling study in the Eulerian-Lagrangian framework, which is particularly addressed to the parti- cle dispersion in turbulent flow due to inter-particle collisions and particle-wall collisions, is performed. A cost-effective searching algorithm of collision pair (b) among particles is developed by constructing a list of particles indices. It is demonstrated that the averaged CPU time required for completing a Lagrangian particle tracking process, including solving the equation of mo- tion, searching the collision pair, and updating the list of neighbor particles’ indices, is approximately propor- (c) tional to the number of particles (N). It is shown from the simulation results of the fully developed channel flow laden with various mass loading ratios of 70m- in-diameter copper particles that the effects of in- ter-particle collisions on particle dispersion cannot be Fig. 11 Comparison of the fully developed distribu- negligible when . In addition, the tions of (a) u ; (b) (u ) ; and (c) (v ) ; ob- /(  O10) p p rms p rms cp tained with two different wall roughness ex- wall roughness extent plays an important role in the tents in the cases of 2, 20 and 40 mass simulation of particle-wall collisions particularly for the loading ratios. cases with the small mass loading ratios. (a) ACKNOWLEDGEMENTS This works was done with financial support by Na- tional Science Council, R.O.C. under Grant No. NSC 98-2221-E006-132. REFERENCES (b) 1. Lin, J. H. and Chang, K. C., “A Modeling Study on Particle Dispersion in Wall-Bounded Turbulent Flow,” The Advances in Applied Mathematics and Mechanics, 6, pp. 764782 (2014). 2. Abe, K., Kondoh, T. and Nagano, Y., “A New Tur- bulence Model for Predicting Fluid Flow and Heat,” International Journal of Heat and Mass Transfer, 37, Fig. 12 Comparison of predicted (a) u and (b) k with pp. 139151 (1994). the experimental data for the single phase 3. Crown, C. T., Troutt, T. R. and Chung J. N., “Nu- cases. merical Models for Two-Phase Turbulent Flows,” The Annual Review of Fluid Mechanic, 28, pp. (a) 1143 (1996) 4. Crown, C. T., Sharma, M. P. and Stock, D. E., “The Particle-Source-In-Cell (PSI-Cell) Model for Gas- Droplet Flows,” Journal of Fluids Engineering, 99, pp. 325332 (1997). (b) 5. Lightstone, M. F. and Hodgson, S. M., “Turbulence Modulation in Gas-Particle Flows: Comparison of Selected Models,” Canadian Journal of Chemical Engineering, 82, pp. 209219 (2004). (c) 6. Mito, Y. and Hanratty, T. J., “Use of a Modified Langevin Equation to Describe Turbulent Dispersion of Fluid Particles in a Channel Flow,” Flow, Turbu- lence and Combustion, 68, pp. 126 (2002). Fig. 13 Comparison between the predicted and meas- 7. Du, S., Sawford, B. L. and Wilson, J. D., “Estima- ured (a) u ; (b) (u ) ; and (c) (v ) ; in the p p rms p rms tion of the Kolmogorov Constant (C0) for the La- case of 20% loading ratio with  = 0.02. grangian Structure Function, Using a Second-Order Journal of Mechanics, Vol. 32, No. 2, April 2016 243 Downloaded from https://academic.oup.com/jom/article/32/2/237/5948549 by DeepDyve user on 09 October 2022 Lagrangian Model of Grid Turbulence,” Physics of 12. Kulick, J. D., Fessler, J. R. and Eaton, J. K., “Parti- Fluids, 7, pp. 30833090 (1995). cle Response and Turbulence Modification in Fully Developed Channel Flow,” Journal of Fluid Me- 8. Crowe, C., Sommerfeld, M. and Tsuji, Y., Multi- phase Flow with Droplets and Particles, CRC Press, chanics, 277, pp. 109134 (1994). Boca Raton (1998). 13. Eaton, J. K., “Two-Way Coupled Turbulence Simu- 9. Tanaka, T. and Tsuji, M., “Numerical Simulation of lations of Gas-Particle Flows Using Point-Particle Gas-Solid Two-Phase Flow in a Vertical Pipe: On Tracking,” International Journal of Multiphase Flow, the Effect of Inter-Particle Collision,” ASME/FED 35, pp. 792800 (2009). Gas-Solid Flows, 121, pp. 123128 (1991). 14. Yamamoto, Y., Potthoff, M., Tanaka, T., Kajishima, 10. Sommerfeld, M., “Modelling of Particle-Wall Colli- T. and Tsuji, Y., “Large-Eddy Simulation of Turbu- sions in Confined Gas-Particle Flows,” International lent Gas-Particle Flow in a Vertical Channel: Effects Journal of Multiphase Flow, 18, pp. 905926 of Considering Inter-Particle Collision,” Journal of (1992). Fluid Mechanics, 442, pp. 303343 (2001). 11. Sommerfeld, M. and Huber, N., “Experimental 15. Zhao, L. H., Andersson, H. I. and Gillissen, J. J. J., Analysis and Modelling of Particle-Wall Collisions,” “Turbulence Modulation and Drag Reduction by International Journal of Multiphase Flow, 25, pp. Spherical Particles,” Physics of Fluid, 22, pp. 14571489 (1999). 081702-1081702-4 (2010). (Manuscript received January 15, 2015, accepted for publication May 25, 2015.) 244 Journal of Mechanics, Vol. 32, No. 2, April 2016 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Mechanics Oxford University Press

Particle Dispersion Simulation in Turbulent Flow Due to Particle-Particle and Particle-Wall Collisions

Lin, J.-H.; Chang, K.-C.
Journal of Mechanics , Volume 32 (2) – Apr 1, 2016
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Copyright © 2022 Society of Theoretical and Applied Mechanics of the Republic of China, Taiwan
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Downloaded from https://academic.oup.com/jom/article/32/2/237/5948549 by DeepDyve user on 09 October 2022 PARTICLE DISPERSION SIMULATION IN TURBULENT FLOW DUE TO PARTICLE-PARTICLE AND PARTICLE-WALL COLLISIONS J.-H. Lin K.-C. Chang Department of Aeronautics and Astronautics National Cheng Kung University Tainan, Taiwan ABSTRACT Simulation of the 3-D, fully developed turbulent channel flows laden with various mass loading ratios of particles is made using an Eulerian-Lagrangian approach in which the carrier-fluid flow field is solved with a low-Reynolds-number k- turbulence model while the deterministic Lagrangian method together with binary-collision hard-sphere model is applied for the solution of particle motion. Effects of inter- particle collisions and particle-wall collisions under different extents of wall roughness on particle disper- sion are addressed in the study. A cost-effective searching algorithm of collision pair among particles is developed. It is found that the effects of inter-particle collisions on particle dispersion cannot be negli- gible when the ratio of the mean free time of particle to the mean particle relaxation time of particle is less or equal to O(10). In addition, the wall roughness extent plays an important role in the simulation of particle-wall collisions particularly for cases with small mass loading ratios. Keywords: Particle-laden flow, Particle-particle collisions, Particle-wall collisions, Channel flow. 1. INTRODUCTION tion are referred to our previous work [1] and briefly described here. Modeling of particle-laden flows has been developed along two distinct (Eulerian-Eulerian and-Lagrangian) 2.1 Governing Equations for Carrier Fluid frameworks. In the first approach, the particle ele- The Reynolds-averaged Navier-Stokes equations ments are also described as a continuous fluid and, coupled with low-Reynolds number k- turbulence therefore, named as the two-fluid model. In contrast, model developed by Abe et al. [2] are given below. the second approach treats the particles as individual entities of which their motions are simulated using the Continuity: Lagrangian tracking method. It is well agreed [1] that u influential extents of collisions between particle-  0 (1) particle and-wall (in a wall-bounded configuration) x become more significant as the particle-loading ratio is Momentum: increased. This study is addressed to how to set up an accurate yet economical numerical procedure in simu- ()  u gg lating the particle dispersion motion in the wall-  ()  uu gg g ji  tx bounded turbulent flows on the basis of Lagrangian tracking method.   x px 2  ()    ku S  tij gg pu ii   xx x x 3 ij j i   2. PHYSICAL MODELING (2) Traditional approach based on the Reynolds- averaged Navier-Stokes (RANS) equation associated Turbulence kinetic energy: with a low Reynolds number version of k- turbulence ()  k model [2] is applied for the solution of the carrier-fluid  ()  uk gg  tx flow field, while the stochastic separated flow model [3] j for the solution of the dispersed-phase (particles) flow    pk   field. The effects of turbulence, inter-particle colli- (3)     PS     kg p   xx  x sions, and particle-wall collisions on particle dispersion ij k j   are taken into account in the study. Detailed modeling Dissipation rate of turbulence kinetic energy: on these physical phenomena of particle dispersion mo- Corresponding author ([email protected]) Journal of Mechanics, Vol. 32, No. 2, April 2016 237 DOI : 10.1017/jmech.2015.63 Copyright © 2015 The Society of Theoretical and Applied Mechanics, R.O.C. Downloaded from https://academic.oup.com/jom/article/32/2/237/5948549 by DeepDyve user on 09 October 2022 ()  where  ()  u gg  tx j 1   uu   pg    p    1.5 (12)        0.135k 0.22k    xx  x ij k j     gg 2 (4)  d  CP C f PS pp  12kk p   kk p (13) 18 where the eddy viscosity,  , and the turbulence produc- tion term, P , are given, respectively, by 2.3 Equations of Particle Motion Equations of particle motion consist of translation  Cf (5) tg  and rotation parts, which are expressed, respectively, as follows.  u  uu g j g ii P  (6)  kt  Translation:  x xx j ij  dU Here f and f , which are appeared in Eqs. (4) and (5),   m FFF F p DS M G (14) dt respectively, are two damping functions accounting for the eddy viscous effects in the near-wall region and are given by Rotation:   dΩ   y Re   p (7) I Τ f  1exp  10 .3exp    p ν    (15) dt 3.1 6.5        where the quasi-steady drag force (F ), Saffman lift     y 5.0 Re force (F ), Magnus lift force (F ), and gravitational   S M f  1exp  1 exp  (8)      0.75 force (F ), while the viscous torque (T ) are considered G v 14 Re 200   t      in modeling the translational and rotational motion of particle, respectively. Calculations of these forces are where referred to our previous work [1] and not repeated here. The momentum inertia I for solid sphere shown in Eq. 2 p yu  k   w (15) is defined by yu,,Re  t   I  0.1 md ppp (16) 2.2 Particle Source Terms in the Carrier Fluid 2.4 Turbulent Dispersion of Particles The particle source term shown in Eq. (2) represents the momentum exchange rate between the carrier fluid Solution of the equations of instantaneous particle and particles and is calculated using the PSI-cell meth- motion, i.e. Equations (14) and (15) require input the od [4] in a specified grid cell as instantaneous velocity of the carrier fluid. Neverthe- less, only the mean quantities of the carrier-fluid veloc- ll ity components are provided in the solutions of the  Su () ()u p ptt p t ui  i (9) i  V RANS governing equations. The fluctuating velocity l1 of carrier fluid along the particle trajectory is generated where N is the total number of particles passing through using the Langevin method [6] as follows. the specified cell within a given time step t, and V is the volume of the specified grid cell. To account for ('uu )  ( ' )RG ('u ) 1R (17) gtt g t i g rms ii i the effects of turbulence modulation, two particle-fluid interaction terms S and S are introduced into Eqs. where G is the Gaussian distribution function with zero pk p th (10) and (11) as the addition source terms, respectively, mean and unit variance for i component of fluid ve- and are modeled as the ones suggested by Lightstone et locity, and R is the autocorrelation function expressed al. [5] as follows. by l Rt  exp( /T ) (18) N *  m  2k  S 1  p  k * (10)  V    where the Lagrangian integral time scale T is estimated l1 p p L  by N * m  2 2 k S 1  p   * (11) T  (19)  V    l1 p p  C  238 Journal of Mechanics, Vol. 32, No. 2, April 2016 Downloaded from https://academic.oup.com/jom/article/32/2/237/5948549 by DeepDyve user on 09 October 2022 In this study, the model constant of C is set equal to 7.0 where f and e are the normal restitution coefficient and suggested by Sawford [7]. frictional coefficient of the particle, respectively, and are given with the values of 0.995 and 0.3, respectively, in the present study; (U ) is the relative velocity vector ij ct 2.5 Inter-Particle Collisions of the particle i to the particle j at the contact point during collision and can be expressed by Searching for inter particle collisions is basically made with the binary-collision hard-sphere model [8]. There are two steps in searching for the particle colli- i ()UUU  Ω  e Ω  e (26) ij ct p p p n p n ij i j sion pair. First, all particles are advanced to the next 22 time step (t t) through solving the equations of motion Here e and e represent the unit vectors along normal n t without taking into account inter particle collisions as and tangential directions, respectively, at the contact shown in Fig. 1. Second, particle collision pair is point of two particles i and j during collision, and are searched by the numerical procedure which will be given by elaborated in the next section. In the hard-sphere collision model, the interaction er  /|r | (27) nt t t t between the collided particles is assumed to be impul- sive, and hence the particles only exchange momentum eU () /|()U | (28) t ij ct ij ct for each other. When a collision is identified, the post-collision ve- The post-collision positions of the collision pair are locities for the collision pair are updated in accordance updated by each corresponding values of [r  (1 t t with the inelastic hard-sphere model as  )tU ]. It is assumed that there happens at most (20) UUJ / m one collision for the particle i in a given t. No fur- pp p ii ther search for the collision pair is, thus, performed in the remaining time step, i.e. (1  )t for the particle i. UUJ / m (21) pp p jj 2.6 Particle-Wall Collisions * i ΩΩ eJ (22) PP n ii 2I It is known [10] that the particle-wall collisions can dominate the particle motion in a channel flow under j certain conditions such as wall roughness and size of ΩΩ eJ (23) PP n jj loading particle. In consideration of the changing wall 2I roughness with time due to constant occurrence of particle-wall collisions, a stochastic model for virtual Here the impulse force vector J is composed of the rough wall (illustrated in Fig. 2) developed by Som- tangential (J ) and normal (J ) components which are t n merfeld and Huber [11] is adopted in this study. The expressed [9], respectively, by probability density function of contact surface angle , P , at a given incident angles of particle  and a col  mm pi pj standard deviation of surface angle, , is expressed as  U  JJ   ct tn ij J  7mm  for (24) t  pi pj JJ  tn  fJ  n P (,,) col   mm  11sin( ) pi pj (29) J(  1 e)(Ue) (25)  max 0 , exp n ij n   mm  2s  in() 2 Δ pi pj      Fig. 2 Schematic of particle collisions on a rough- Fig. 1 Schematic for searching collision pair in bina- wall surface. ry collision process. Journal of Mechanics, Vol. 32, No. 2, April 2016 239 Downloaded from https://academic.oup.com/jom/article/32/2/237/5948549 by DeepDyve user on 09 October 2022 3. SEARCHING ALGORITHM OF COLLISION PAIR AMONG PARTICLES Searching for inter-particle collisions using a deter- ministic, binary-collision hard-sphere model follows the basic approach developed by Bird [8], and is modified in the flowing steps ,(the flowchart is shown in Fig. 3.) 1. All particles are advanced to the next time step (t t) through solving the equations of motion without taking into account inter-particle collisions as shown in Fig. 1. 2. Find the minimal collision time  t, i.e. the first min collision pair in all particles. 3. Update the post-collision velocities for the collision pair at the minimum time using the hard-sphere model as suggested by Tanaka and Tsuji [9]. 4. Repeat the above steps but with the time step of (1  )t until no more collision pair can be found min in the remaining part of t. Note that the time step (t) mentioned above should be much less than the relaxation time of particle ( ) and others time scales such as the turbulence time scale, residence time scale (x / u ), etc. to assure the solu- i pi tion accuracy of equations of motion. Searching for the potential collision pair for a speci- fied particle should be, in principle, checked with all the others particles, which leads to the CPU time for collision-pair searching process is proportional to N(N1)  N . This is cost-ineffective particularly for Fig. 3 Flowchart for particle tracking algorithm which the cases loaded with large N number of particles. In takes account into the inter-particle collisions practice, searching for collision pair for a specified par- with binary collision hard-sphere model. ticle can be confined in a nearby sub-region around the particle. As Fig. 4 shows, all the particles, colored red, which are found within the cube centered at a particle colored blue, are stored. When searching for a colli- sion pair for the particle colored blue, only the particles in the neighbor list (with the N number) need to be nblist scanned. However, the requirement of computational resource (i.e. memory bandwidth and floating-point operations) for constructing the neighbor list for each particle is still proportional to N , since it needs to scan over all other particles location and , then, calculates the distance between the specified particle and others. In consideration of saving more computational ex- penditure, an effective yet economical searching algo- rithm is proposed in the following. The Lagrangian tracking domain for identifying the particles’ locations is divided into a number of blocks which is similar to Fig. 4 Illustration of neighbor list of particles, the red the “cell” concept in the Eulerian finite-volume mesh, particles are stored in the neighbor list of the Next, recording the index of the adjacent cell for each blue particle. cell in a way shown in Fig. 5. Here the adj_cell [:] is an array which records the cell indices of all the adjacent-cell information for each cell. For example, th the i adjacent cell for cell j is termed as adj_cell [adj_cell_index [j]  i]. In this way, we can allocate the specific adjacent cell for a specified cell directly without doing the conventionally searching process. Definition of the cell adjacency can be made with shar- ing either the interface (shown in Fig. 6(a)) or the same node (shown in Fig. 6(b)). To cover more potential Fig. 5 Illustration of data structure for adjacency cell neighboring sub-region, the definition of the cell adja- index. cency shown in Fig. 6(b) is adopted in this study. 240 Journal of Mechanics, Vol. 32, No. 2, April 2016 Downloaded from https://academic.oup.com/jom/article/32/2/237/5948549 by DeepDyve user on 09 October 2022 Fig. 7 Illustration of data structure for particles list for each cell. Fig. 6 Illustration of the adjacent cells, the green cells are the adjacent cells of the yellow one. The definition of adjacent can be made with sharing cpu time ~N either (a) the same interface; or (b) the same 10 2 ~N node. The data structure for the list of particles’ indices is similar to the adjacent cell index. The diagram shown 5 6 in Fig. 7 is the particles list for each cell in which the 10 10 number of particles array of partilce_tag [:] is used to store the indices of th particles for each cell. For instance, the index of i Fig. 8 Averaged CPU time per t versus number of particle in the cell j is recorded in the array particle_tag particles in completing a Lagrangian particle [partilce_cell [j]  i]. The particles data in the specific tracking process. region of Lagrangian cell can be directly allocated without any additional searching process. 4. RESULTS AND DISCUSSION The list of particles’ indices has to be updated before finishing an iterate as follows. The distances between The 3-D fully developed, turbulent downward chan- each particle in the specified cell, colored yellow in Fig. nel flows laden with various mass loading ratios of 6, and the cell centers of each adjacent cell, colored copper particles with 70m in diameter, which were green in Fig. 6, are re-calculated, respectively. If the experimentally investigated by Kulick et al. [12], are new distance between the particle in the specified cell, chosen as the test problem. Simulations of this test colored yellow, and the center of an adjacent cell is problem have been reported in the literature [13-15]. shorter than the original distance between the particle However, none of the existed studies have ever consid- and cell center of the specified cell, colored yellow, the ered both the effects of inter-particle collisions and par- index of the particle will be transferred to the nearest ticle-wall collisions by using the economic yet realistic adjacent cell. models. For instance, Eaton [13] did not take into By means of this list of particles’ indices, the colli- account the inter-particle collisions in his simulation. sion pairs can be directly accessed between the speci- Although the simulations made by Yamamoto et al. [14] fied cell and adjacent cells without scanning all parti- and Zhao et al. [15] considered the mechanism of inter- cles in the flow domain. Thus, the computational time particle collisions in their modelings, both of these two required for this algorithm of Lagrangian particle studies employed a simple but unrealistic smooth-wall tracking processes, including solving the equations of collision model in the simulations. Thus, the simula- motion, searching the collision pair, and updating the tion with the more economic and realistic modelings of list of neighbor particles’ indices is theoretically pro- inter-particle collisions and particle-wall collisions is portional to O(N) only, where N is the total number of conducted in the following. particles. A series of the fully developed channel The computational domain in form of Cartesian co- flows laden with various numbers of copper particles of ordinate is set as 1.0m(L)  0.04m(H)  0.01m(W) as 5m in diameter, ranging from 5,000 to 2,000,000, is schematically shown in Fig. 9. The mean stream-wise tested on a single core of Intel Xeon E5-1620 processor velocity of carrier fluid at the inlet, u , is equal to to estimate the CPU time required for completing a 10.5m/s, which corresponds to Re, based on the hy- Lagrangian particle tracking process with the proposed draulic diameter, of 13,800. Detail of the numerical algorithm in a given time step (t). The “Lagrangian aspects and boundary conditions are referred to our cells” are set the same as the Eulerian grid layout which is composed of 216,000 grid meshes. It is also found previous work [1] and not repeated here for brevity. that the optimal particle number in a searching list had 4.1 Effects of Inter-Particle Collisions on Particle better be in the order of O(10) to yield high computa- Dispersion tional efficiency in the Lagrangian tracking processes together with the search of collision pairs. Figure 8 Three different mass loading ratios of 2, 20 and presents the averaged CPU time per t versus the num- 40 are tested for investigating the effects of inter- ber of particles laden in the flow. Obviously, the re- particle collisions on particle dispersion. Their pre- sults corroborate that the CPU time is approximately dicted mean stream-wise particle velocity (u ) and root- proportional to N. mean-squared fluctuating stream-wise ((u ) ) and p rms Journal of Mechanics, Vol. 32, No. 2, April 2016 241 cpu time per step (sec) Downloaded from https://academic.oup.com/jom/article/32/2/237/5948549 by DeepDyve user on 09 October 2022 wall-normal ((v ) ) particle velocities are presented in between the cases with and without considering inter- p rms Fig. 10. One more simulation for the case of 2 mass particle collisions become more remarkably as the mass loading ratio but without considering the inter-particle loading ratio is increased. Even for the most dilute collisions is purposely preformed, and its predicted re- case (2 mass loading ratio) investigated in the study, sults of u , (u ) and (v ) are also presented in Fig. p p rms p rms there appear observable differences between the two 10. It is noted that the results obtained without con- results predicted between with and without considering sidering inter-particle collisions differ very slightly inter-particle collisions. from each other in the cases with different mass loading ratios in the test problem. Thus, the results obtained The mean collision time,  , of particle for the cases from the case of 2 mass loadings ratio but without with mass loading ratios of 2, 20 and 40 are 2.237, considering inter-particle collisions can serve as a 0.4156 and 0.1892 sec, respectively in contrast to the comparison baseline for the tested cases with three var- particle relaxation time,  , is about 0.1 sec. The rati- ious mass loading ratios. The comparison results os of  over  for the cases with 20 and 40 made from Fig. 10 reveal clearly that the differences loading ratios are O(10 ) while for the case with 2 mass loading is O(10 ). This concludes that the in- ter-particle collisions have to be considered in the mod- eling when  / is less or equal to O(10 ). cp 4.2 Effects of Particle-Wall Collisions on Particle Dispersion Two different wall roughness of  = 0 (smooth wall) and  = 0.02 are tested for each cases of mass loading ratios of 2, 20 and 40. Their predicted u , (u ) and (v ) are presented in Fig. 11. It is p rms p rms clearly seen that the wall roughness extent is an im- portant parameter to affect the particle-wall collisions. The comparison made between the cases with  = 0 and 0.02 for each tested mean loading ratio reveals that the effects of wall roughness extent on particle disper- sion motion become more remarkable as the mass loading ratio is deceased. This can be attributed to a Fig. 9 Schematic of test problem. fact that the particle dispersion is more dominated by the mechanism of inter-particle collisions as the mass loading ratio is increased. (a) 4.3 Comparison of the Model Predictions with Experimental Data The predicted fully-developed distributions of u and k for the single-phase (i.e. 0 mass loading ratio) case, (b) are presented in Fig. 12 and compared with the meas- ured data of Kulick et al. [12]. Here the dimensionless wall distance (y ) is defined by yu /  where u =   1/2 ( / ) and  is the wall shear stress. Good agree- w w ment between the predicted and measured results im- plies that the employed turbulence model performs well. (c) The fully-developed distributions of u , (u ) and p p rms (v ) , predicted with the 20 mass loading ratio and p rms  = 0.02 are presented in Fig. 13 and compared with the measured data of Kulick et al. [12] Reasonable agreement between the predicted and measured results Fig. 10 Comparison of the fully developed distribu- implies that the modeling of particle dispersion has to tions of (a) u ; (b) (u ) ; and (c) (v ) in take into account of the inter-particle collisions and p p rms p rms particle-wall collisions in the flow laden with signifi- the case with wall roughness of  = 0 for cant mass loading ratios. three different mass loading ratios. 242 Journal of Mechanics, Vol. 32, No. 2, April 2016 Downloaded from https://academic.oup.com/jom/article/32/2/237/5948549 by DeepDyve user on 09 October 2022 5. CONCLUSIONS (a) A modeling study in the Eulerian-Lagrangian framework, which is particularly addressed to the parti- cle dispersion in turbulent flow due to inter-particle collisions and particle-wall collisions, is performed. A cost-effective searching algorithm of collision pair (b) among particles is developed by constructing a list of particles indices. It is demonstrated that the averaged CPU time required for completing a Lagrangian particle tracking process, including solving the equation of mo- tion, searching the collision pair, and updating the list of neighbor particles’ indices, is approximately propor- (c) tional to the number of particles (N). It is shown from the simulation results of the fully developed channel flow laden with various mass loading ratios of 70m- in-diameter copper particles that the effects of in- ter-particle collisions on particle dispersion cannot be Fig. 11 Comparison of the fully developed distribu- negligible when . In addition, the tions of (a) u ; (b) (u ) ; and (c) (v ) ; ob- /(  O10) p p rms p rms cp tained with two different wall roughness ex- wall roughness extent plays an important role in the tents in the cases of 2, 20 and 40 mass simulation of particle-wall collisions particularly for the loading ratios. cases with the small mass loading ratios. (a) ACKNOWLEDGEMENTS This works was done with financial support by Na- tional Science Council, R.O.C. under Grant No. NSC 98-2221-E006-132. REFERENCES (b) 1. Lin, J. H. and Chang, K. C., “A Modeling Study on Particle Dispersion in Wall-Bounded Turbulent Flow,” The Advances in Applied Mathematics and Mechanics, 6, pp. 764782 (2014). 2. Abe, K., Kondoh, T. and Nagano, Y., “A New Tur- bulence Model for Predicting Fluid Flow and Heat,” International Journal of Heat and Mass Transfer, 37, Fig. 12 Comparison of predicted (a) u and (b) k with pp. 139151 (1994). the experimental data for the single phase 3. Crown, C. T., Troutt, T. R. and Chung J. N., “Nu- cases. merical Models for Two-Phase Turbulent Flows,” The Annual Review of Fluid Mechanic, 28, pp. (a) 1143 (1996) 4. Crown, C. T., Sharma, M. P. and Stock, D. E., “The Particle-Source-In-Cell (PSI-Cell) Model for Gas- Droplet Flows,” Journal of Fluids Engineering, 99, pp. 325332 (1997). (b) 5. Lightstone, M. F. and Hodgson, S. 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Sommerfeld, M., “Modelling of Particle-Wall Colli- T. and Tsuji, Y., “Large-Eddy Simulation of Turbu- sions in Confined Gas-Particle Flows,” International lent Gas-Particle Flow in a Vertical Channel: Effects Journal of Multiphase Flow, 18, pp. 905926 of Considering Inter-Particle Collision,” Journal of (1992). Fluid Mechanics, 442, pp. 303343 (2001). 11. Sommerfeld, M. and Huber, N., “Experimental 15. Zhao, L. H., Andersson, H. I. and Gillissen, J. J. J., Analysis and Modelling of Particle-Wall Collisions,” “Turbulence Modulation and Drag Reduction by International Journal of Multiphase Flow, 25, pp. Spherical Particles,” Physics of Fluid, 22, pp. 14571489 (1999). 081702-1081702-4 (2010). (Manuscript received January 15, 2015, accepted for publication May 25, 2015.) 244 Journal of Mechanics, Vol. 32, No. 2, April 2016

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