Modified DLM method for finite-volume simulation of particle flow

Modified DLM method for finite-volume simulation of particle flow A. M. Ardekani, S. Dabiri, and R. H. Rangel Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA 92697-3975, USA A Distributed-Lagrange-Multiplier(DLM)-based method is implemented for the simulation of particulate flow. Initially, we show that our fluid-particle solver produces results which are in good agreement with numerical studies of benchmark viscous flow problems. Subsequently, the bouncing motion of a solid sphere onto a solid plate in an ambient fluid is considered. Comparing results for the coefficient of restitution for different Stokes numbers shows very good agreement with the experimental results. Finally, the effect of the coefficient of restitution for a dry collision on the vortex dynamics associated with a colliding sphere onto a wall is discussed. I. Introduction The interaction between fluid and particles in a dispersed flow occurs in several natural and industrial applications such as sedimentation, combustion systems, crystal growth, and microfluidic devices. For example, the design and utilization of microfluidic devices for biomedical purposes requires a better understanding of particulate flow in low and intermediate Reynolds number flows. Particulate flow simulations under the Stokes-flow assumption have been conducted using Stokesiandynamics techniques. 1 For dilute suspensions, the unsteady motion of two solid spheres has also been analyzed. 2 However, for intermediate Reynolds numbers the use of numerical simulation is generally unavoidable. The flow around a single object can be calculated using a fixed body-fitted grid in a non-inertial reference frame. For multiple objects moving relative to each other, remeshing is required along the moving interfaces. The flow field is usually solved using finite-element 3 or boundary-element 4 methods. An alternative is to use a fixed cartesian grid which has the advantage of permitting the use of fast solvers. Calhoun 5 and separately Russell et al. 6 utilized a cartesian grid using a stream function-vorticity formulation in order to simulate irregular shapes and multiple objects. The non boundary-fitted methods are usually simpler and more efficient and have been used by several researchers. 7 10 Another promising method developed in the last decade is a Distributed-Lagrange-Multiplier (DLM) technique using a fictitious domain as presented by Glowinski et al., 9 who used a finite-element method with a fixed structured grid thus eliminating the need of remeshing required for unstructured boundary-fitted grids. In this method, the entire domain is treated as fluid but the fluid inside the particle domain satisfies a rigidity constraint by using Lagrange multipliers. Patankar et al. 11 formulated their DLM solution by forcing the deformation tensor in the particle domain to be zero thus eliminating U and ω as the variables from the coupled system of equations, where U and ω are the translational and angular velocity of the particle, respectively. This formulation introduces a stress field in the particle domain similar to the pressure in an incompressible fluid. Sharma et al. 10 presented the formulation of DLM for steady Stokes flow using a control-volume method. Rebound of colliding particles has also been studied in recent years. Davis et al. 12 employed an elastohydrodynamic approach and showed that the pertinent parameter for collision in the fluid is not the Reynolds number Re but the Stokes number St = 1 ρ p 9 ρ f Re where ρ p and ρ f are the particle and fluid densities, respectively. No rebound occurs for St lower than a critical value due to the fact that elastic energy stored by the particle deformation is dissipated in the fluid. The experiments by Joseph et al. 13 show that the Graduate Student Researcher, AIAA student member Professor, AIAA senior member 1 of 13

rebound velocity depends strongly on the impact Stokes number and weakly on the elastic properties of the material, where the Stokes number is defined using the approaching velocity of the particles. It has been shown that below a St of 10, no rebound occurs. 13 15 For impact Stokes numbers larger than 500 the coefficient of restitution asymptotes to that for a dry collision. Whereas several experimental studies have been conducted on the influence of the fluid on the collision process, few numerical studies address this issue. In this work, we numerically study the effects of Stokes number on the rebound velocity and obtain results which are in agreement with the experimental results by Gondret et al. 15 We first implement DLM for fixed cylindrical particles in low to intermediate Reynolds number flow. The constraint of zero velocity satisfies the condition of rigid body motion inside the particle. Thus, initially it is enough to set the velocity field inside the particle equal to zero and this can be done by means of a rigidity force field in the particle domain. Even though the advantage of using DLM resides in the use of a fixed grid for moving boundaries, utilizing DLM for the flow around stationary objects provides the opportunity of experimenting with irregular geometries simply by using a rectilinear grid. Furthermore, modification of the flow solver due to the presence of the particle is simple. Subsequently, DLM is implemented for the case of moving particles. The bouncing motion of a solid sphere onto a wall is numerically simulated and the vortex dynamics associated with this problem is discussed. Following the derivation procedure of Patankar et al., 10, 16 we derive the governing equations in Section II. The numerical implication will be described in Section III and the method is verified by applying it to different fluid-particle interaction problems in Sections IV and V. II. Theoretical Development In this section, the DLM method is described for the motion of one particle but the method can be easily extended for additional particles. Let Γ represent the fluid boundary which is not shared with the particle. The solid domain and its boundary are denoted by P and P, respectively. The computational domain is Ω, including both the fluid and the particle. The governing equations in the fluid domain are ρ f Du Dt = σ + ρ f g in Ω \ P (1) u = 0 in Ω \ P (2) u = U i on P (3) σ n = t on P (4) u t=0 = u 0 (x) in Ω \ P (5) in addition to the outer boundary conditions on Γ. In these equations, u is the fluid velocity, g is the acceleration of gravity, ρ f is the fluid density, n is the normal unit vector, U i is the particle velocity, and t is the traction vector on the particle surface. The initial velocity u 0 satisfies the continuity equation, σ = pi + τ is the stress tensor, p is the pressure field, I is the identity tensor, and τ is the viscous stress: The governing equations in the particle domain are τ = 2µD[u] = µ [ u + ( u) T ]. (6) ρ P Du Dt = σ + ρ Pg in P (7) u = 0 in P (8) D[u] = 0 in P (9) u = U i on P (10) σ n = t on P (11) u t=0 = u 0 (x) in P. (12) where ρ P is the particle density. Equation (9) satisfies the continuity equation but in order to facilitate numerical implementation, equation (8) is retained. As pointed out by Sharma et al., 10 the rigidity constraint gives rise to a stress field inside the particle which is a function of three scalar Lagrange multipliers for three dimensional problems. Thus for a two dimensional problem, the stress field can be represented in terms of 2 of 13

two scalars: σ = pi + τ + D[λ] (13) where λ represents the Lagrange multipliers and τ is zero inside the particle due to the rigidity constraint. The governing equations in the entire domain can be combined as: ρ Du = σ + ρg + f in Ω (14) Dt u = 0 in Ω (15) u t=0 = u 0 (x) in Ω (16) where ρ = ρ f ρ P in Ω \ P in P (17) in addition to the outer boundary conditions on Γ. In the above equations, f = D[λ] is zero everywhere except in the particle domain and leads to the rigid body motion inside the particle. III. Numerical Implementation A finite-volume method using a staggered grid for incompressible flow is implemented. The SIMPLE algorithm 17 is used to solve the fluid equations with modifications to account for the presence of particles. The Crank-Nicolson scheme is used for time discretization. The discretised momentum equations in the x and y directions are a i,j u i,j = a nb u nb p I,J p I 1,J δx u V u + b i,j + F x (18) a I,j v I,j = a nb v nb p I,J p I,J 1 δx v V v + b I,j + F y (19) Where V u and V v are the volumes of the u-cell and v-cell, respectively; u and v represent the horizontal and vertical components of the velocity field; I and J are the nodes at the center of the main control volumes; while i and j represent the nodes at the center of the control volumes for u and v, respectively as shown in Fig. 1. Subscript nb refers to corresponding neighboring staggered control volumes; b is the momentum source term which includes the gravity term; and F represents the rigidity force which makes the velocity field inside the particle domain satisfy the rigidity constraint. v I-1,j+2 v I,j+2 u i-1,j+1 u i,j+1 u i+1,j+1 v I-1,j+1 v I,j+1 u i-1,j u i,j u i+1,j P I,J v I-1,j v I,j Figure 1: Staggered grid, u control volume and corresponding volume fraction 3 of 13

1.5 3 1.5 R=0.1 6 Figure 2: Relative position of cylinder and boundaries. Following the SIMPLE algorithm and taking the particles into account, the force F is added as an unknown variable and F = F + F where F is the force predicted at each iteration (the force calculated in the previous iteration) and F is the correction force. At each iteration, the rigidity force is defined as and similarly in the y direction F xi,j = F xi,j + Cρ P A i,j (u i,j u Ri,J ) (20) F yi,j = F yi,j + Cρ P A I,j (v I,j v RI,j ) (21) where C is a dimensional constant which includes an under-relaxation factor; A i,j and A I,j are the area of u-cell and v-cell, respectively; while u R is the velocity vector rigidified inside the particle, equal to u outside the particle domain, and defined as follows u R = (1 φ)u + φu P (22) where φ is the volume fraction occupied by the particle in each control volume, defined separately for u and v-cells. For example, as shown in Fig. 1, φ u = V hatched V u. In the present study, φ is exactly calculated while u P, defined only in the particle domain, can be calculated as follows: u P = U P + ω r (23) where U P and ω are the particle translational and angular velocities. By using conservation of linear and angular momentum for the solid particle, one can calculate the particle translational and angular velocities 10, 16 as follows: M P U P = ρudx and I P ω = r ρudx (24) P Defining a rigidity force as in equations (20) and (21) guarantees that, upon convergence at each time step, u = u R everywhere in the domain. Finally, the modifications due to the presence of the particle can be included by adding F as a source term in the momentum equations. P IV. Results and Discussion A. Flow past a stationary circular cylinder As in the first case, the flow past a stationary circular cylinder is modeled. This case is designed to test the solution process without the added complication of the particle motion. Figure 2 shows the geometry used for this case. The boundary conditions are 4 of 13

v u y = 0, y = 0 on the top and bottom boundaries Du Dt u = 0, x = 0 on the right (exit flow) boundary u = U in, v = 0 on the left (inlet flow) boundary For fixed objects we have u p = 0 and u R = u(1 φ) (25) The drag force can be determined by calculating the summation of F in the particle domain. Table 1: Summary of results for 1 < Re < 40 C D (L) Re 1 4 10 20 30 40 Takami& Keller 18 10.28 4.44 2.75 (0.25) 2.00 (0.93) 1.72 (1.61) 1.54 (2.32) Dennis& Chang 19 2.85 (0.26) 2.05 (0.94) 1.52 (2.35) Fornberg 20 2.00 (0.91) 1.50 (2.24) Calhoun 5 2.19 (0.91) 1.62 (2.18) present work 11.11 4.65 2.88 (0.24) 2.08 (0.90) 1.75 (1.57) 1.55 (2.23) Table 1 shows a summary of results for different Reynolds numbers where C D is the drag coefficient and L represents the ratio of circulation length to the particle diameter. The results compare very favorably with both experimental and previous computational results. For the data presented in Table 1, the mesh size is 1 11 of the particle radius. The geometry used for Re = 1 and Re = 4 is different from the one shown in figure 2. Since the upstream flow is affected by the presence of the cylinder as much as downstream flow, we double the width of the flow domain and the center of the cylinder is moved towards the center of the computational domain. 1.5 1 C D C L C T C D,C L,C T 0.5 0-0.5 0 50 100 150 200 Time Figure 3: Time dependent lift, drag, and torque coefficient at Re = 100 1. Unsteady vortex shedding: Re = 100 For higher Reynolds number, instability occurs and the classic oscillatory wake behind the cylinder can be easily captured. Because our algorithm has a slight built-in asymmetry in its details, there is no need to artificially perturb the flow field to initiate the unsteady behavior. In figure 3, we plot lift, drag, and torque coefficients, C L, C D, C T versus nondimensional time tu in D, respectively, at Re = 100 where D is the diameter of particle. The forces which impose zero velocity field in the particle domain are equal to the reaction forces applied on the cylinder to fix it. Therefore, the lift and drag coefficients can be calculated easily just by 5 of 13

calculating the summation of F y and F x in the particle domain, respectively. The summary of results is shown in table 2. Generally, our results are well within the range of results reported by other researchers. Table 2: Summary of results for Re = 100 C D C L St Braza et al. 21 1.36 ± 0.015 ±0.25 Liu et al. 22 1.35 ± 0.012 ±0.339 0.164 Calhoun 5 1.33 ± 0.014 ±0.298 0.175 Russell et al. 6 1.34 ± 0.007 ±0.276 0.165 present work 1.36 ± 0.0085 ±0.310 0.163 (a) Streamlines (b) Vorticity contours Figure 4: Streamlines and vorticity contours for flow past a biperiodic array. Re= 5 and volume fraction is 0.2. Streamline contour levels are 0:0.1:1 and 0.45:0.01:0.55. Vorticity contour levels are -31:2:31. B. Flow past a biperiodic array Flow past an infinite periodic array of cylinders is considered in this section. The purpose of this example is to show that our method works well for low-reynolds-number flows. Flow past a cylinder array in a unit square with periodic boundary conditions in both the horizontal and the vertical directions is considered. The following boundary conditions are utilized: u y = 0, v = 0 on the top and bottom boundaries Periodic boundary conditions on the downstream and upstream velocities A 280 280 uniform grid is used for Re = 5 and a volume fraction of 0.2. Reynolds number is based on particle diameter and fluid average upstream velocity (U 0 ). Streamlines and vorticity contours are shown in figure 4. The solution was computed on one cylinder and plotted periodically. Comparing our drag coefficient F x µu 0 with the results by Koch and Ladd 23 shows only a 1.2% difference. 6 of 13

V. Numerical Implementation For Moving Objects A. Sedimentation of a spherical particle towards a wall In this section, we examine the motion of a spherical particle moving towards a wall with a non-zero dry coefficient of restitution. All dimensional quantities are in the CGS units unless otherwise stated. Specifically, the collision of a steel sphere with density of 7.8 colliding with a glass wall is investigated. The dry coefficient of restitution for this system is equal to 0.97 ± 0.02 according to the data reported by Gondret et al. 15 The sphere is moving in oil with density of 0.935 and viscosity of 0.1. The sphere diameter is 0.3 and the gravitational acceleration is g = 981. A fixed nonuniform structured grid is employed. The smallest mesh size is 0.0003 0.001 near the contact region where higher resolution is needed. The numerical simulation is performed utilizing axisymmetry. The distance h from the bottom of the sphere to the wall and the particle velocity are shown in figure 5. The Reynolds number based on the approach velocity of the sphere towards (a) Distance between particle surface and the wall (b) Vertical velocity Figure 5: Sedimentation of a spherical particle and its collision with the wall. Re = 63, St = 58, h min = 1.2µm, and e dc = 0.97. the wall is 62.7 and the impact Stokes number is 58. The effective roughness height is h min = 0.932 µm. As it can be seen, rebound trajectories are non-parabolic since the velocity decreases nonlinearly with time. A jump in velocity occurs when the surface of the particle and the wall come into contact. Subsequently, the particle moves away from the wall and a marked decrease in particle velocity is observed. As the sphere moves away from the wall and the gap widens, the velocity varies more slowly. Figure 6 compares the results using our numerical method with the experimental results by Gondret et al. For larger Stokes number, higher rebound is observed, as expected. It can be seen that the coefficient of restitution we obtain numerically is in very good agreement with the experimental measurements. More detailed results can be found in Ardekani and Rangel. 24 B. Vortex dynamics of a colliding sphere onto a wall When a sphere moves towards a wall and stops upon making contact with it, a secondary vortex ring is generated from the wake vorticity. The evolution of vorticity for this problem is considered numerically by Thompson et al. 25 and experimentally by Eames and Dalziel. 26 In both these studies, the sphere sticks to the wall after the collision. In the present work, we discuss the vortex dynamics associated with the collision of a sphere and a wall for non zero coefficient of restitution. First, we compare our numerical results with experimental results by Eames and Dalziel. 26 In this case, a sphere is set into motion from a distance of L = 7.5D above the wall and moves with constant velocity and a Reynolds number of 850. A nonuniform structured grid and a domain size of 10D 20D with 300 350 nodes are employed. Figure 7 shows the vorticity contours when the sphere sticks to the wall after collision. 7 of 13

1.1 1 0.9 0.8 0.7 e/e dc 0.6 0.5 0.4 0.3 0.2 0.1 0 10 0 10 1 10 2 10 3 10 4 10 5 10 6 St Figure 6: Coefficient of restitution normalized by that for dry collision as a function of St. Present results ( ) where h min = 0.7µm. Experimental measurement for different materials by Gondret et al. 15 tungsten carbide ( ), steel ( ), glass ( ), Teflon ( ), Derlin ( ), polyurethane ( ), and Nylon ( ). Roughness in these experimental cases is less than 1µm. Lubrication theory of Davis et al. 12 ( ). In this case, the sphere is moving with constant velocity before the collision. The left and right sides of each frame in the figure show the present numerical results and the experimental results by Eames and Dalziel, 26 respectively. This comparison shows very good agreement. In frames (a,b) of figure 7 the wake vorticity of the sphere approaching the wall can be observed. As the sphere stops, the wake vorticity moves towards the wall due to its inertia. The wake vortex threads over the sphere and generates a secondary vortex ring as seen in frame (c). This coherent structure, composed of the wake and secondary vortices, leaves the sphere and strikes the wall as seen in frame (d). The secondary vortex is stretched by the wake vortex and becomes a sheet-like structure (e-f). Finally, the secondary vortex extends radially and breaks into two parts. One part is advected around the wake vortex while the second component remains trapped close to the point of contact between the sphere and the wall in frame (g). The motion of a sphere falling under gravity is considered next. The sphere is released from a height of L = 5D with a Reynolds number of 510 based on the initial velocity. The Reynolds number increases to 865 when the particle collides with the wall. The vorticity contours are shown in figure 8. The coefficient of restitution for a dry collision is zero in the left half and 0.5 in the right half of each frame of figure 8. In frame (a), both sides correspond to a time before collision and are the same. As time elapses (frame (b)), a secondary vortex ring is generated in both cases while the wake vortex is moving towards the wall due to its inertia. In the right half of frame (b), a secondary vortex is growing and becoming the wake vortex for the up-going sphere. A separated region on the sphere surface with vorticity of the same sign as the primary vortex is observed in both cases. In addition, a region with the opposite sign of the primary vorticiy is generated in both cases at the wall. In left half of the frame (c), a coherent structure leaves the sphere and extends radially whereas in the right side of this frame, since the particle is moving downward, counterclockwise vorticity spreads out below the sphere. The secondary vortex is divided into two portions: one above the sphere on the surface while the larger portion of this secondary vortex remains between sphere and the wall. In frame (d), since the particle is moving downwards, the clockwise vorticity above the sphere is diffused while the counterclockwise vorticity below the sphere grows. The coherent structure which has extended radially and the primary vortex far above the sphere are similar to the case without rebound. In frames (e-f) of figure 8, the vortices have diffused and clockwise or counterclockwise vorticity is observed around the sphere for upward and downward motion of the sphere, respectively. The separation distance versus nondimensional time is shown in figure 9. Figure 10 shows a comparison between zero coefficient of restitution (left half of each frame) and dry coefficient of restitution (right half of each frame). Frames (a-c) are similar to the previous case, except that in this case, a secondary vortex, which is the wake vortex during the upward motion of the sphere, 8 of 13

(a) -τ 0 (b) 0 (c) τ 0 (d) 2τ 0 (e) 3τ 0 (f) 4τ 0 (g) 5τ 0 Figure 7: Collision of a sphere onto a wall. The left and right frames correspond to the present numerical results and the experimental results by Eames and Dalziel,26 respectively. Re = 850, hmin = 9.2 µm, and the time difference between adjacent figures is τ 0 = D U. The vorticity contours are shown and the black line is the streakline. 9 of 13

(a) before collision (b) 0.76τ (c) 1.56τ (d) 2.36τ (e) 3.16τ (f) 3.39τ Figure 8: Collision of a sphere onto a wall. The coefficient of restitution for a dry collision is zero in the left half and 0.5 in the right half of each frame. Vorticity contours are shown. Re = 865, h min = 9.8 µm, and τ D = g. 10 of 13

4 3 h/d 2 1-2 -1 0 1 2 3 Time/τ Figure 9: Nondimensional separation distance. Re = 865, h min = 9.8 µm, and e dc = 0.5 is larger than the one observed at the same time in figure 8. In frame (d-e), the clockwise vorticity below the sphere moves above it due to the change in direction of the motion of the sphere. In frame (f) this clockwise vorticity which remained from the secondary vortex is advected from the sphere and deviates the wake vortex of the downward moving sphere. The coherent structure near the wall is similar to that for the case without rebound. In general, vorticity is stronger for higher dry coefficient of restitution due to the fact that less energy is dissipated during the collision process. The separation distance versus nondimensional time is shown in figure 9. VI. Conclusions We describe a finite-volume algorithm using a Distributed-lagrange-multiplier for solving particulate flow. We show that our fluid-particle solver produces results which are in good agreement with experimental and numerical studies of benchmark viscous flow problems. One advantage of this approach is that the modifications needed to account for the presence of particles can be easily implemented. Comparison of our numerical results for the bouncing motion of a solid sphere onto a wall shows very good agreement with experimental results. The evolution of vorticity due to collision and the effect of the coefficient of restitution are discussed. References 1 Brady, J. F. and Bossis, G., Stokesian dynamics, Annu. Rev. Fluid Mech., Vol. 20, 1988, pp. 111 157. 2 Ardekani, A. M. and Rangel, R. H., Unsteady motion of two solid spheres in Stokes flow, Phys. of Fluids, Vol. 18, 2006. 3 Hu, H. H., Patankar, N., and Zhu, M. Y., Direct numerical simulations of fluid-solid systems using the arbitrary Lagrangian-Eulerian technique, J. Comput. Phys., Vol. 169, 2001, pp. 427 462. 4 Pozrikidis, C., Orbiting motion of a freely suspended spheroid near a plane wall, JFM, Vol. 541, 2005, pp. 105 114. 5 Calhoun, D., A cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular regions, J. Comput. Phys., Vol. 176, 2002, pp. 231 275. 6 Russell, D. and Wang, Z. J., A cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow, J. Comput. Phys., Vol. 191, 2003, pp. 177 205. 7 Sethian, J. A. and Smereka, P., Level set methods for fluid interfaces, Ann. Rev. Fluid Mech., Vol. 35, 2003, pp. 341 372. 8 Esmaeeli, A. and Tryggvason, G., A front tracking method for computations of boiling in complex geometries, Int. J. Multiphase Flow, Vol. 30, 2004, pp. 1037 1050. 9 Glowinski, R., Pan, T. W., Hesla, T. I., and Joseph, D. D., A distributed Lagrange multiplier fictitious domain method for particulate flows, Int. J. Multiphase Flow, Vol. 25, 1999, pp. 755 794. 10 Sharma, N., Chen, Y., and Patankar, N. A., A distributed Lagrange multiplier based computational method for the simulation of particulate-stokes flow, Comput. Methods Appl. Mech. Engrg, Vol. 194, 2005, pp. 4716 4730. 11 patankar, N. A., Singh, P., Joseph, D. D., Glowinski, R., and Pan, T. W., A new formulation of the distributed Lagrange multiplier/ fictitious domain method for particulate flows, Int. J. Multiphase Flow, Vol. 26, 2000, pp. 1509 1524. 12 Davis, R. H., Serayssol, J. M., and Hinch, E. J., The elastohydrodynamic collision of two spheres, J. Fluid Mech., Vol. 163, 1986, pp. 479. 13 Joseph, G., Zenit, R., Hunt, M., and Rosenwinkel, A., Particle-wall collisions in a viscous fluid, J. Fluid Mech., Vol. 433, 2001, pp. 329 346. 11 of 13

(a) before collision (b) 0.76τ (c) 1.56τ (d) 2.36τ (e) 3.16τ (f) 4.76τ (g) 5.56τ Figure 10: Collision of a sphere onto a wall. The coefficient of restitution for a dry collision is zero in the left half and 1.0 in the right half of each frame. Vorticity contours are shown. Re = 865, h min = 9.8 µm, and τ D = g. 12 of 13

Figure 11: Nondimensional separation distance. Re = 865, h min = 9.8 µm, and e dc = 1.0 14 Gondret, P., Hallouin, E., Lance, M., and Petit, L., Experiments on the motion of a solid sphere toward a wall: From viscous dissipation to elastohydrodynamic bouncing, Phys. Fluids, Vol. 11, 1999, pp. 2803 2805. 15 Gondret, P., Lance, M., and Petit, L., Bouncing motion of spherical particles in fluids, Phys. Fluids, Vol. 14, 2002, pp. 643 652. 16 Sharma, N. and Patankar, N. A., A fast computation technique for direct numerical simulation of rigid particulate flows, J. Comput. Phys., Vol. 205, 2005, pp. 439 457. 17 Patankar, S. V., Numerical heat transfer and fluid flow, McGraw-Hill, 1980. 18 Takami, H. and Keller, H. B., Steady two-dimensional viscous flow of an incompressible fluid past a circular cylinder, Phys. Fluids Suppl., Vol. 12, 1969, pp. II51 II56. 19 Dennis, S. C. R. and Chang, G. Z., Numerical solutions for steady flow past a circular cylinder at Reynolds number up to 100, J. Fluid mech., Vol. 42, 1970, pp. 471. 20 Fornberg, B., A numerical study of steady viscous flow past a circular cylinder, J. Fluid mech., Vol. 98, 1980, pp. 819 855. 21 Braza, M., Chassaing, P., and Minh, H. H., Numerical study and physical analysis of pressure and velocity fields in the near wake of a circular cylinder, J. Fluid mech., Vol. 165, 1986, pp. 79 130. 22 Liu, C., Zheng, X., and Sung, C. H., Preconditioned multigrid methods for unsteady incompressible flows, J. Comput. Phys., Vol. 139, 1998, pp. 35 57. 23 Koch, D. L. and Ladd, A. J. C., Moderate Reynolds number flows through periodic and random arrays of aligned cylinders, J. Fluids Mech., Vol. 349, 1997, pp. 31 66. 24 Ardekani, A. M. and Rangel, R. H., A computational method for particulate flow with collisions, in preparation. 25 Thompson, M. C., Hourigan, K., Cheung, A., and Leweke, T., Hydrodynamics of a particle impact on a wall, Applied Mathematical Modelling, Vol. 30, 2006, pp. 1356. 26 Eames, I. and Dalziel, S. B., Dust resuspension by the flow around an impacting sphere, J. Fluid Mech., Vol. 403, 2000, pp. 305. 13 of 13