PARTICLE SEDIMENTATION IN A CONSTRICTED PASSAGE USING A FLEXIBLE FORCING IB-LBM SCHEME

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1 International Journal of Computational Methods Vol. 11, No. 5 (2014) (27 pages) c World Scientific Publishing Company DOI: /S PARTICLE SEDIMENTATION IN A CONSTRICTED PASSAGE USING A FLEXIBLE FORCING IB-LBM SCHEME S. M. DASH,,T.S.LEE and H. HUANG Department of Mechanical Engineering National University of Singapore, Singapore Department of Modern Mechanics University of Science and Technology of China P. R. China sunil.dash@nus.edu.sg Received 2 January 2013 Accepted 9 October 2013 Published 24 February 2014 A novel flexible forcing hybrid immersed boundary-lattice Boltzmann model (IB-LBM) is introduced in the present paper for solving moving boundary problems. In conventional IB-LBM schemes, explicit formulations of force density term may not ensure no-slip boundary condition accurately, which leads to inaccurate force and torque calculations for moving object. Following an implicit force density calculation, a single Lagrangian velocity correction term is advised in this study. The formula for this correction term is much simpler and with the help of flexible number of sub-iteration/forcing, the computational time is significantly saved. The no-slip boundary condition is achieved accurately within a convergence limit. The proposed algorithm shows advantages for unsteady and moving boundary problem, where boundary convergence is satisfied consistently at every time step. In particular, a 2D particulate flow case is simulated in a constricted channel. Interesting observations and results are discussed in this article. Keywords: Immersed boundary method; flexible forcing; lattice Boltzmann method; particle sedimentation; constricted channel. 1. Introduction Fluid-particle interactions occur in many applications, such as chemical, environmental, geological, aerospace, nuclear, oil and gas engineering, and biological science. Transportation of sedimentary particles in aquatic environment, fluidized bed reactor, colloidal suspension, cell transport in arteries are few examples of this Corresponding author

2 S.M.Dash,T.S.Lee&H.Huang branch of physics. In particular, we focus our interest on fluid-particle interaction where surrounding wall effects are significantly important, for example, a constricted section of a flow channel. In real life one may encounter many such practical problems to name a few, sedimentation of sand particles or rising of bubbles in a hourglass, transport of blood cells in constricted arteries, granular flow in a hopper, multiphase mixed flow in a separator etc. Hence, the fluid particles interaction problems have received considerable attentions and there are many relevant experimental and numerical studies reported in the literature. The inclusion of particle suspension in a fluid domain is difficult to analyze in either theoretical or experimental techniques, because the interaction between the fluid flow and the moving particles are complex. Due to the development of computational fluid dynamics, there are many direct numerical studies [Brady and Bossis (1988); Hu (1996); Hu et al. (2001)] on fluid-particle interactions using either finite volume method (FVM) or finite element method (FEM). In the numerical simulations, both hydrodynamics of the fluid and the Newtonian dynamics of the moving particle are considered. In the numerical methods, the fluid phase is described by Navier Stokes (NS) equations, while the description of the particle phase such as position and velocity are traced in Lagrangian frame. Fluid-particle interactions can be coupled through ensuring the no-slip boundary condition on the particle. However, in FEM [Hu et al. (2001)], usually the computational mesh is required to generate at each time step. Generating geometrically adapted mesh and projection of the fluid variables from the old to new mesh make these methods complex and computationally expensive. Another numerical method PHYSALIS was developed by Prosperetti et al. [Zhang and Prosperetti (2003); Prosperetti and Oguz (2001)]. In this method, the analytical solution is used for the near region of the particle surface, with the parameters determined from the outer flow conditions. However, finding the analytical solutions for a complex geometry is difficult and tedious. Hence, the method is only limited to low Reynolds number (Re) flow and circular particles. To avoid the grid regeneration and reduce the computational effort, fixed grid methods like distributed Lagrange multiplier/fictitious domain algorithm (DLM/FD) was proposed by Glowinski et al. [Glowinski et al. (1999); Glowinski et al. (1999); Glowinski et al. (2000)] to simulate the particle sedimentation. The basic idea of this method is that a fictitious domain is used to simulate the particle region combined with the Eulerian fluid domain. Using distributed Lagrange multiplier, constraints of the rigid-body motion is imposed on the fictitious fluid inside the particle. But the form of Lagrange multiplier makes the scheme mathematically complex. On the other hand, a simple immersed boundary method (IBM) proposed by Peskin [1977] is a suitable alternative. A body force density term is added to the flow governing equations to impose the no-slip boundary condition. This force density term is evaluated using different methods, such as penalty method

3 Particle Sedimentation in a Constricted Passage [Peskin (1977)], direct forcing method [Feng and Michaelides (2004)] or momentum exchange method [Niu et al. (2006)]. In the conventional IBM, the no-slip condition is approximately satisfied at the converged solution. That may leads to unphysical streamline penetration into the immersed solid [Kang and Hassan (2011)] and improper force and torque calculation for particulate flows. Shu et al. [2007] found that the no-slip condition is not satisfied because of the pre-calculated force density term. They have proved that adding a force density term in the governing equations is identical as making a velocity correction. Therefore, an implicit approach is proposed in which the velocity correction is unknown [Wu and Shu (2010)]. The unknown one is determined after the corrected velocity field satisfies the no-slip boundary condition [Wu and Shu (2010)]. However, in the implicit approach, [Wu and Shu (2010)] a complex matrix is required to be solved. To avoid the complicated matrix calculation one may follow the multi-direct-forcing scheme as proposed by Wang et al. [Wang et al. (2008); Luo et al. (2007)]. In this scheme, an iterative procedure is used to find out the correct force density term. However, use of fixed number of iteration steps [Wang et al. (2008); Luo et al. (2007)] may not satisfy the no-slip condition accurately, in particular for unsteady and moving boundary flow cases. In addition, the sub-iteration scheme increases computational time and cost. To avoid the matrix calculation and reduce the iteration/computational cost, we try to develop an efficient algorithm which satisfies the no-slip boundary condition and evaluate the force and torque acting on the solid boundary accurately. This is an extension study of our previously proposed flexible forcing scheme for athermal/thermal IB-LBM model [Dash et al. (2013, 2014)] to simulate moving boundary flow problems. In the present algorithm, the flow field is solved using LBM. As an alternative to NS solver, LBM has gained much research attention due to its simplicity, easy implementation and intrinsic parallel nature. Using LBM, the particulate flow has been studied by Ladd [1994a, 1994b], Behrend [1995], Aidun et al. [1998], and Qi [1999] where the interactions between fluid and particle are implemented through momentum exchange (bounce back rule). Also, recently a second order variant of the hydrodynamic boundary condition is advised by Sahu and Vanka [2011] for the no-slip boundary in multiphase flow. However, using the momentum exchange scheme, the force curve as a function of time is not smooth and may oscillate dramatically. Alternatively, to overcome the drawbacks, IBM has been incorporated into LBM by many researchers [Feng and Michaelides (2004); Wu and Shu (2010); Uhlmann (2005); Feng and Michaelides (2005)]. Using IB-LBM schemes, they have studied some 2D and 3D moving boundary flow problems. However, the wall effects are negligible or not important in their studies. Here, we have applied the developed flexible forcing IB-LBM scheme to study the sedimentation of the single and two particles in constricted passages. The effects of

4 S.M.Dash,T.S.Lee&H.Huang density of the solid particles, viscosity of the fluid, shape and size of the constriction are addressed in detail. To the best of our knowledge, this flow situation is not studied in the literature. 2. Numerical Methodology The flow governing equations for an unsteady, incompressible viscous flow involving immersed boundaries are [Peskin (1977)], ρ + ρu =0, t (1) t (ρu)+ (ρuu) = p + ν [ρ u +( u)t ]+f, (2) f(x,t)= F B (s, t)δ(x X B (s, t))ds, (3) X B (s, t) t Γ = u(x B (s, t),t)= Ω u(x,t)δ(x X B (s, t))dx, (4) where the variables: ρ, u,p,t,ν represent density, flow velocity, pressure, time, kinematic viscosity of the fluid respectively. x and X B are Eulerian and Lagrangian co-ordinates, f and F B are the force density acting on the fluid and immersed boundary respectively. δ(x X B (s, t)) is a Dirac delta function. The above Eqs. (1) and (2) represent traditional NS equation with the force density f. Equations(3) and (4) relates the immersed solid boundary Γ and fluid domain Ω by distributing the boundary force to nearby fluid points and computing the boundary velocity from the fluid velocity. In the present study, the developed hybrid flexible forcing IB-LBM algorithm [Dash et al. (2013, 2014)] is applied to solve the above flow governing equations. In the following, the numerical scheme [Dash et al. (2013, 2014)] is extended to moving boundary problems. The lattice Boltzmann equation that recovers Eqs. (1) and (2) is [Wu and Shu (2010)], f α (x + e α δt, t + δt) =f α (x,t) 1 τ (f α(x,t) f eq α (x,t)) + F αδt, (5) where f α (x,t)andfα eq (x,t) are the density distribution functions and its equilibrium part along the discrete lattice direction α e α is the discrete lattice velocity. The discrete velocities of D2Q9 model are shown in Fig. 1. τ is a non-dimensional relaxation parameter in the BGK approximation. fα eq (x,t) is obtained from Taylor series expansion of Maxwell Boltzmann distribution function. f eq α (x,t)=ρw α [ 1+ e α u c 2 + (e α u) 2 (c s u ) 2 s 2c 4 s ], (6)

5 Particle Sedimentation in a Constricted Passage Fig. 1. D2Q9 lattice model with lattice velocity directions. (0, 0), α =0, e α = c (±1, 0), (0, ±1), α =1 4, (±1, ±1), α =5 8. 4/9, α =0, W α = 1/9, α =1 4, 1/36, α =5 8. W α is the weighting coefficient. The sound speed is defined as c s = c/ 3, where c is the lattice speed and is given by δx/δt. δx and δt are the mesh and time step size respectively. To recover NS equation with the body force density an additional discrete forcing term F α is included in the conventional LB equation. Various formulations of F α was investigated by Huang and Krafczyk [2011] and Mohamad and Kuzmin [2010]. They had showed that in case of single phase fluid problems the deviation of results were negligible. In our present simulation, formula suggested by Guo et al. [2002] for F α is adopted. This definition retrieves unsteady second order NS equation accurately without any extra velocity-force divergence tensor terms. ( F α = 1 1 ) ( eα u W α 2τ c 2 + e ) α u s c 4 e α f, (9) s where f in Eq. (9) is the force density at Eulerian fluid node, which is distributed from the force density at Lagrangian boundary points. With the above definition of F α the macroscopic velocity field is shifted [Mohamad and Kuzmin (2010)] and is defined as, (7) (8) ρu = α e α f α + 1 fδt. (10) 2 By defining an intermediate Eulerian velocity u = α e αf α /ρ, and an Eulerian velocity correction term δu =(1/2ρ)fδt, Eq. (10) can be written as, u(x,t)=u (x,t)+δu(x,t). (11)

6 S.M.Dash,T.S.Lee&H.Huang In IBM, the Eulerian velocity correction δu is distributed from the boundary node velocity correction δu B as follows, δu(x,t)= δu B (X B,t)δ(x X B )ds, (12) Γ where the immerse boundary is represented using a set of Lagrangian co-ordinates X B (s k,t) s k (k = 1, 2, 3,...,n) is the position in the Lagrangian co-ordinates. δ(x X B ) is smoothly approximated using 2 point Dirac delta function. { 1 r, r 1, δ(r) = (13) 0, r > 1. D(xij X k B)= 1 h 2 δ ( xij X k B h ) δ( yij Y k B h ), (14) where xij is Cartesian co-ordinates of Eulerian nodes. Using Eqs. (13) and (14), Eq. (12) can be simplified to an algebraic Eq. (15). δu(xij, t) = δu k B (Xk B,t)D(xij Xk B ) s k, (15) k where U k B is the boundary velocit, s k is the arc length of the boundary element. Using the velocity correction at Eulerian nodes, the corrected Eulerian velocity is obtained as, u(xij, t) =u (xij, t)+δu(xij, t). (16) In the view of mathematics, the no-slip boundary condition implies that the fluid velocity at the boundary point must be equal to the boundary velocity at the same position. Using the concept of interpolation, the no-slip condition can be written as, U k B (Xk B,t)= u(xij, t)d(xij X k B ) x y, (17) ij where x and y are the mesh sizes in the horizontal and vertical direction of the computational domain. Substituting Eqs. (15) and (16) into Eq. (17) we get, U k B(X k B,t)= u (xij, t)d(xij X k B) x y ij + δu k B(X k B,t)D(xij X k B) s k D(xij X k B) x y. (18) ij k In the above equation δu k B is kept unknown for an implicit forcing formulation [Wu and Shu (2010)] and is obtained by solving Eq. (18). As we can see the evaluation of δu k B requires a matrix inversion that increases the computational effort. Again it is not easy for the moving boundary problems. Along with high computational storage demand for 3D case, the co-efficient calculation of δu k B needs a sequential coding pattern. Therefore, to avoid these defects a flexible forcing IB-LBM scheme [Dash et al. (2013, 2013b)] is applied

7 Particle Sedimentation in a Constricted Passage The boundary velocity correction is actually a term which corrects the interpolated Eulerian velocity at Lagrangian point to achieve the desired boundary velocity U kd B (e.g., zero velocity for stationary immerse particle). In the proposed algorithm, a single Lagrangian correction term is added in Eq. (17) to satisfy the desired boundary condition. This formulation keeps the basic idea of implicit velocity correction and mathematically simpler than Eq. (18). The computational effort for the new code development is also significantly reduced. So the amount of correction at the boundary node (X k B,t) can be obtained as that shown in Eq. (20). U kd B (Xk B,t)= ij u(xij, t)d(xij X k B ) x y + δuk B (Xk B,t), (19) δu k B(X k B,t)=U kd B (X k B,t) u(xij, t)d(xij X k B) x y. (20) ij Because there are more than one Lagrangian node points involve, satisfying the no-slip condition at all the boundary points in a single turn is impossible using the above procedure. Hence, an additional sub-iteration scheme is imposed to satisfy the no-slip condition within a convergence limit. This further ensures that when the Eulerian velocities, that are interpolated back to the Lagrangian boundary nodes will satisfy the no-slip condition with an accuracy of the order δu k B.The convergence criterion (CC) is set as in Eq. (21), where m is the sub-iteration number until the criterion is satisfied. δu k B (Xk B,t)m (21) It is worth to mention that the CC is a case dependent term which can be adjusted to higher/lower order term based on complexity of the problem, required accuracy of the final result and available computational resources. This approach has certain similarity to multiple forcing schemes [Wang et al. (2008); Luo et al. (2007)]. The difference is that in our scheme, the sub-iteration is dependent on CC and unlike to the fixed number of sub-iterations [Wang et al. (2008); Luo et al. (2007)]. That means the number of sub-iterations can be varied till the exact no-slip boundary condition is satisfied. This makes the scheme flexible and computationally efficient by avoiding the unnecessary sub-iteration. Again in the case of unsteady and moving boundary problems, use of the fixed sub-iteration [Wang et al. (2008); Luo et al. (2007)] may not yield the no-slip condition at each time step. Then the force and torque calculations are questionable. To overcome these defects, the proposed flexible sub-iteration scheme is a suitable alternative. This approach also reduces computational effort and storage demand by avoiding the matrix calculation and ensures the order of accuracy to O(δU k B ). δu k B (Xk B,t) m+1 =(1 ε)δu k B (Xk B,t) m 1 + εδu k B (Xk B,t) m. (22) We tried to further reduce the number of sub-iterations by using a successive relaxation parameter (SRP), ε as shown in Eq. (22). ε is varied in the range of 0 to

8 S.M.Dash,T.S.Lee&H.Huang It is found that ε between , the number of sub-iterations required is reduced significantly. The effects of SRP and CC selection will be discussed later. The force density at the boundary and Eulerian grid points is calculated using Eqs. (23) and (24), respectively. F k B(X k B,t)= m 2ρδU k B(X k B,t) (m) /δt, (23) f(xij, t) = m 2ρδu(xij, t) (m) /δt. (24) 2.1. Kinematics of moving particle Using Newton s laws of motion the net force on the particle is calculated from the combined effects of the gravity, buoyancy, hydrodynamic, and collision forces. ( F net i = 1 ρ ) f M i g F k B ρ (Xk B ) s k + F coll i, (25) p k where M i is the mass of the particle, ρ f and ρ p are densities of the fluid and particle respectively. Similarly from the hydrodynamic force, the torque acting on the particle can be computed as, T net i = k (X k B X R) F k B (Xk B,t) s k, (26) where X R is the center of mass of the particle. To calculate the collision force between particle-wall or particle-particle, a lubrication forcing mechanism is followed [Glowinski et al. (2000)]. This force is repulsive in nature and acts only when the distance between the particle and the nearest wall or another particle is less than λ. In our present study, we set λ =2δx. 0 d i,j R i + R j + λ, F coll i = 1 (27) (X i X j )(R i + R j + λ d i,j ) 2 d i,j <R i + R j + λ, ε p where R i and R j are the radius of the colliding particles. In case of collision with a wall, this force calculation is done by assuming an imaginary particle of same size on the other side of wall. d i,j is the distance between the centers of ith and jth particles with their corresponding center location at X i and X j respectively. ε p is a small positive stiffness parameter followed from Glowinski et al. [2000]. In IBM simulation, the fluid phase is assumed to be present at outside as well as inside of the solid boundary. This internal fluid does not affect the flow field and pressure outside the boundary once the no-slip condition is accurately satisfied. However, part of the body force term is utilized to move the internal fluids, therefore the net force and torque calculation on the particle is compromised. To include this internal loss effect we have followed Feng and Michaelides [2005] formulation for rigid body approximation. The corresponding translational velocity U R and angular

9 Particle Sedimentation in a Constricted Passage velocity Ω calculation is done with added mass effect term. du R n+1 ( ) M i dt = F net ρf du R n i + M i ρ p dt, (28) n+1 ( ) n dω I i dt = T net ρf dω i + I i ρ p dt, (29) where I i is the moment of inertia tensor of the particle. n isthetimestep. Other microscopic variables: Density, pressure, and kinematic viscosity are calculated using the following formula. N ( ρ = f α, p = ρc 2 s, υ = τ 1 ) c 2 sδt. (30) 2 α=0 In summary the solution procedures for the proposed flexible forcing IB-LBM scheme are outlined below. (1) Set the initial flow field and calculate f α and fα eq. (2) Perform streaming using Eq. (5) with initial setting F α,δu = 0 and calculate intermediate velocity u = α e αf α /ρ. (3) From the desired boundary velocity calculate the velocity correction δu k B (Xk B,t) m=0. (4) Update the Eulerian velocity u as in Eq. (16) with updated δu and calculate new δu k B (Xk B,t) m using both the Eqs. (20) and (22). Repeat this step with a sub-iteration loop until the CC in Eq. (21) is satisfied. (5) Calculate f and fα eq using Eqs. (24) and (6). Update the net force and torque on the particle and calculate its new position and velocity using Eqs. (25) (29). (6) Repeat the above steps (2) (5) for time evolution. 3. Numerical Validations In order to validate the proposed numerical scheme, a few benchmark moving boundary problems: Single and two particles sedimentation in a viscous fluid medium are simulated Single particle sedimentation This problem about single particle sedimentation has been extensively studied [Wu and Shu (2010); Wan and Turek (2006)]. To validate our scheme, an identical case as that in Wu and Shu [2010] is simulated. A fluid filled two dimensional rectangular box of 2 cm wide and 6 cm high is selected. The fluid properties: density ρ f and viscosity µ f are set to 1.0 g/cm 3 and 0.1 g/cm s, respectively. A circular particle of diameter d p = 0.25 cm and density ρ p = 1.25 g/cm 3 is initially kept in the rectangular box with its center at (1 cm, 4 cm). Initially both the fluid and particle are at rest. The computational domain is discretized with uniform Eulerian mesh

10 S.M.Dash,T.S.Lee&H.Huang t=0.2s t=0.5s t=0.9s X Fig. 2. (Color online) Instantaneous vorticity contours at different time steps. of size To represent the surface of the circular particle, 50 Lagrangian mesh points are used. The particle starts falling under the effects of gravity once it is released from its initial position. The particle keeps accelerating until it reaches a steady/terminal velocity. Thereafter, it settles with terminal velocity. The temporal evolution of the vorticity contours are shown in Fig. 2 at different time steps. The flow regime is in good agreement with that in Wu and Shu [2010] and Wan and Turek [2006]. In Fig. 3, the longitudinal center co-ordinate y p, longitudinal velocity v p, Reynolds number Re p, and translational kinetic energy E t of the particle as functions of time are shown in (a), (b), (c), and (d), respectively. They are also compared quantitatively with that in Wu and Shu [2010]. The definition for Re p and E t are, ρ p d p u 2 p + v2 p Re p =, (31) µ f E t =0.5M(u 2 p + vp), 2 (32) where M is the mass of the particle. The obtained results agree well with those in Wu and Shu [2010]. The effects of CC and SRP in the simulations are also investigated. In Table 1, the variation of averaged forcing/sub-iteration numbers with change in CC and SRP are shown. As expected, with the decrease in CC, the number of forcing/subiteration increases. We also observed that by suitably tuning SRP, overall computational time can be significantly reduced. If SRP is chosen in the range , the average forcing/sub-iteration number is almost reduced by half in comparison

11 Particle Sedimentation in a Constricted Passage (a) (c) Fig. 3. Temporal evolution of (a) Y center co-ordinate, (b) vertical velocity, (c) Reynolds number, and (d) translational kinetic energy of the particle. Table 1. Variation of sub-iteration\forcing number with respect to CC and SRP. CC SRP NF Re NF Re NF Re (b) (d)

12 S.M.Dash,T.S.Lee&H.Huang to other SRP values. Hence, SPR plays an important role to improve the computational efficiency. It is also found that different CC (from 10 4 to 10 6 ) has little effect on the maximum Reynolds number. The reason of this change is associated with the error allowed in the no-slip boundary convergence. When CC is set as 10 5, the maximum Reynolds number is comparable to that provided by Wu and Shu [2010] and by Wan and Turek [2006] Therefore, CC is followed as 10 5 for rest of our simulations Two particle sedimentation To further validate the proposed algorithm, two particles sedimentation in a rectangular box is simulated. The parameters are identical as those in Wang et al. [2008]. Here, the box dimension is selected as 2 cm width and 6 cm height. A uniform mesh of size is utilized to discretize the computational domain. Two circular particles of same diameter d p =0.25 cm and density ρ p =1.5g/cm 3, is kept in the rectangular box with their initial center location at ( cm, 4.5 cm) and ( cm, 5.0 cm). Surface of the particles are represented by 50 Lagrangian points. The horizontal offset distance between the particles centers acts as a perturbation to breakdown the later equilibrium states as mentioned by Wang et al. [2008]. It is expected that the particles will reproduce the Drafting Kissing Tumbling (DKT) phenomena [Fortes et al. (1987)]. The fluid parameters: density ρ f and viscosity µ f are set to 1.0 g/cm 3 and 0.01 g/cm s, respectively. In Fig. 4 the instantaneous vorticity contours of the settling particles are shown. Two particles are initially at rest and start to settle under the effects of gravity. The leading particle (particle-2) creates a wake of low pressure, and the trailing particle (particle-1) is caught in its wake. Due to the reduction in the drag force on the trailing one, it settles faster than the leading one. With the increased speed of Y 6 t=0.1 s t=0.2 s t=0.28 s t=0.35 s Drafting Kissing Tumbling 0 X 2 Fig. 4. (Color online) Instantaneous vorticity contours at different time steps with DKT phenomena

13 Particle Sedimentation in a Constricted Passage (a) Fig. 5. Temporal evolution of (a) Y center co-ordinate, (b) vertical velocity of the particles. the trailing particle, it drafts towards the leading one. Later the two particles make a contact, also known as kissing. The kissing of particles is an unstable equilibrium state and with the initial perturbation supplied these particles get separated (also known as tumbling). As mentioned by Fortes et al. [1987], the tumbling phenomena is breakdown of the unstable equilibrium state and with different numerical perturbation schemes the results may differ after kissing. There are some discrepancies also observed in our comparisons. Quantitative comparisons of the trajectory of the vertical center co-ordinate y p and vertical velocity v p of the particles as functions of time are made with those in Wang et al. [2008]. Our results as plotted in Fig. 5, are found highly consistent with those in Wang et al. [2008]. These validation studies demonstrate the capabilities of the proposed scheme for accurate numerical simulation of the fluid-particle interactions. In the following, we extend our studies to simulate the particle sedimentation in a constricted passage. 4. Problem Definition To simulate the particle sedimentation in a constricted passage/channel a rectangular box of width W and height H is selected. Two symmetrical semi-circular walls of radius R are placed at the mid plane of the domain to produce constricted passage. The schematic of the problem is shown in Fig. 6. The radius of the semi-circles can be adjusted to alter the constriction gap size. In the present study, the settling circular particle has diameter D and density ρ p.inthecase of two particles sedimentation, the diameters and densities of the particles are kept identical. It is found that the numerical results are affected by the boundary walls of the rectangular box/channel. Therefore, we have performed test case simulations by varying the aspect ratio (W/H) of the channel until the wall effects are minimized. (b)

14 S.M.Dash,T.S.Lee&H.Huang Fig. 6. Schematic of particle sedimentation in a constricted channel for (a) single particle, (b) two particles, (c) division of the channel region, and (d) surrounding spatial domain near the particle. Fig. 7. Variation of wall effects with respect to aspect ratio of the channel. We choose maximum retardation velocity (V R ) of the particle in the constriction zone as a parameter to check the wall effects. From the plot in Fig. 7, the wall effects found reducing with the increase in aspect ratio. For our present study, we chose the aspect ratio of the channel as 1:9. Other parameters used in the simulations are, Width of the channel W =8D. Height of the channel H = 72D. Constriction gap size S = 1.25D 2.0D. Diameter of the particle D =0.25cm

15 Particle Sedimentation in a Constricted Passage Table 2. Grid independence test for particle sedimentation in the constricted passage. V Newgrid R V Oldgrid R Mesh size V R % Error = 100 V Oldgrid R Density of the fluid ρ f = 1.0 g/cm 3. Viscosity of the fluid µ f = 0.1 g/cm s. Density of the particle ρ p = g/cm 3. Grid independence test is also performed to confirm that the results are invariable with grid resolutions. Table 2 shows the change in V R with different grid size for a particular particle density and constriction gap size. From the grid independence results we have selected uniform grid of size to discretize the computational domain. 50 Lagrangian node points are used to represent the surface of the particle. In the present simulation, IBM is also employed for the no-slip boundary condition on the stationary semi-circular constriction wall. The number of Lagrangian forcing points on the surface of semi-circle is uniformly distributed with spacing s k =1.4 x. 5. Results and Discussions 5.1. Single particle sedimentation For the simulation of single particle sedimentation, the circular particle is kept at initial height of 22D from the center of the semi-circular constriction and along the vertical centerline of the box. The density of the particle is varied between g/cm 3 to achieve different Reynolds number of the flow. Also to find the effects of different constriction gap size the radius of semi-circular constrictions is adjusted. We restrict our focus to the constriction gap size in the range of 1.25D 2.0D Flow regime To better illustrate the obtained results, we have divided the whole computational domain into three zones as shown in Fig. 6(c). Zone-1. This zone corresponds to the region of the box above the constrictions. Initially the particle is at rest. With advancement in time the particle starts accelerating due to gravity. From Fig. 8(b), it can be observed that the velocity magnitude is increasing until the particle attains a steady state condition with a constant terminal velocity. At this condition, the net upward force (drag and buoyancy forces) on the particle gets balanced with the net downward force (weight of the particle)

16 S.M.Dash,T.S.Lee&H.Huang (a) (c) Fig. 8. Temporal evolution of (a) Y center co-ordinate, (b) vertical velocity, (c) Reynolds number, and (d) translational kinetic energy of the particle with density 1.25 g/cm 3 and in the constriction gap size of 1.25D. (b) (d) Corresponding to the velocity plot, similar behavior is observed for Reynolds number Re and translational kinetic energy E t of the particle. The definition for Re and E t are followed from Eqs. (31) and (32). Once the particle attains the constant terminal velocity it continues to move with that magnitude till it encounters the constrictions or zone-2. In Fig. 9(a), pressure contours around the particle are shown. The reference pressure is ρ 0 c 2 s. In Fig. 9(a), solid contour lines indicate the positive pressure zone

17 Particle Sedimentation in a Constricted Passage t=0.2 s t=0.4 s t=1.5 s t=2.0 s Fig. 9. (Color online) Instantaneous pressure and vorticity contours at different time steps in zone-1 [(a), (b)] and zone-3 [(c), (d)] for particle density 1.5 g/cm 3. and dotted contour lines are for the negative pressure zone. From the figure, we can see that the pressure distributions below and above the particle are different. The positive pressure zone is near the front face of the particle and the negative pressure zone is near the rear end of the particle. This is because the fluid encounters the particle at the front face first and hence leads to formation of the stagnation point or the maximum positive pressure zone. While at the rear end of the particle, there is development of a wake/flow separation region which produces low pressure zone. The rear end pressure contours spread over time which implies increase in the wake length. This growth of the wake region is also verified in the vorticity contour plot, as shown in Fig. 9(b) at different time steps. The vorticity growth is associated with the continuous transfer of kinetic energy from the particle to the fluid [ten Cate et al. (2002)]. Zone-2. This region of the channel contains the constriction walls. The particle attains the terminal velocity in the zone-1 before entering to the zone-2. The velocity plot in Fig. 8(b), indicates that the particle starts to decelerate with reduction in its velocity magnitude as well as Re and E t values. In the zone-2, presence of the constriction walls significantly reduces the available space for the particle movement. From the literature [ten Cate et al. (2002); Wang et al. (2013)], the drag force exerted on the moving particle is a complex inverse function of channel wall gap. In our present study, the channel width in zone-2 is less than other zones. Hence, the drag force on the particle due to the small gap will be highly significant in zone-2. In Figs. 10(a) and 10(b), pressure and vorticity contours on the particle are plotted, respectively. In this zone, lateral pressure contours are present due to the constriction walls. While the particle tries to move further down in the constriction, it squeezes out the surrounding fluid along the constriction walls. The constriction shape creates an adverse pressure gradient for this squeezed flow field, which leads to a boundary layer separation as shown in Fig. 10(b). The separated flow field forms two recirulating vortices on the constriction walls. At t =0.7s, there is a low

18 S.M.Dash,T.S.Lee&H.Huang t=0.6 s t=0.7 s t=0.8 s t=1.0 s Fig. 10. (Color online) Instantaneous pressure and vorticity contours at different time steps in zone-2 for particle density 1.5 g/cm 3. pressure band behind the particle. This band contains the two separated vortices on the constriction walls and two attached vortices on the particle. In the view of flow physics, the low pressure band will attract the surrounding fluid and spreads its area of influence. Hence, the circulation of the vortices increases as shown in Fig. 10(b). When the attached vortices on the particle are sufficiently large in size, they will shed from the particle. When the particle further settles, the drag force on the particle keeps increasing as the gap converges. The minimum velocity magnitude of the particle will be reached at the narrowest gap, which is referred to as the V R. Beyond the narrowest gap, on the diverging profile of the constriction, the particle again starts accelerating as shown in Fig. 8(b). It is worthy to mention that the retarding and accelerating part of the velocity plot in the constriction zone are not symmetric, which may be due to the presence of lateral pressure contours in the convergence section and change of the available spatial gap in the divergence section as shown in Fig. 6(d). Zone-3. After leaving zone-2, the particle enters zone-3 which is the bottom portion of the channel. The particle keeps accelerating similar to zone-1 till the steady state/terminal velocity condition is attained. The particle continues to move with the constant terminal velocity until it settles on the bottom of the channel. As the particle approaches the channel bottom, the kinetic energy of the particle decays and it starts decelerating by squeezing out the fluid between the particle and the bottom wall. This also generates outward flow as shown in the vorticity plot of Fig. 9(d). In Figs. 9(c) and 9(d), pressure and vorticity contours are plotted at different time steps. The plots are similar to that of zone-1. To realize the collision between the particle and the bottom wall, the lubrication force as described in Eq. (27) is used Effects of constriction gap size In order to find out the effects of the constriction gap size, the radius of the semicircular constriction walls are varied. In the present study, we focused the gap size

19 Particle Sedimentation in a Constricted Passage (a) (c) Fig. 11. Temporal evolution of (a) Y center co-ordinate, (b) vertical velocity, (c) Reynolds number, and (d) translational kinetic energy for different constriction gap size and particle density 1.25 g/cm 3. in the range of 1.25D 2.0D, between the two constriction walls along the horizontal centerline. In Fig. 11, vertical velocity, trajectory, Re and E t of the particle are plotted for various constriction gaps. It is found that with the decrease in the constriction gap size, the V R increases and corresponding changes in Re and E t are also produced. This behavior is associated with the drag force in the narrow gap. In Fig. 12, pressure contours and the instantaneous co-efficient of pressure (C P ) are shown for different constriction gap sizes and when the particle reaches the centerline of the constriction. In the lower array of Fig. 11, the distribution of C P as a function of θ is shown. For the cases of gap size 1.25D and 1.5D, there are two stagnation points. But for the larger constriction gap there is only one stagnation point. With increased stagnation points the pressure recovery for the gap size 1.25D (b) (d)

20 S.M.Dash,T.S.Lee&H.Huang Fig. 12. (Color online) Instantaneous pressure contours on the particle at centerline of the constriction with gap size (a) 1.25D, (b) 1.5D, (c) 1.75D, and (d) 2.0D and corresponding C P distribution for particle density 1.5 g/cm 3. (a) Fig. 13. Comparisons of (a) maximum retardation velocity (V R ) and (b) sedimentation time for different constriction gap size and density of the particle. (b) is less than that of 2.0D. Hence, the drag force on the particle will be higher in case of gap size 1.25D Effects of density of the particle Density of the particle is varied to generate different Reynolds number. For higher density, particle s terminal velocity increases. As the drag force on the particle is an

21 Particle Sedimentation in a Constricted Passage inverse function of the flow velocity, therefore the drag force on the denser particle will be less compared to the lighter particle for a particular constriction gap size. Hence, the magnitude of V R increases with the decrease of particle density. In the Fig. 13(a), V R as a function of density of the particle is plotted for various channel gaps. It is also found that for a particular particle density, magnitude of V R decreases with increase in constriction gap. As we know, with increase in particle density the velocity of the particle increases and thus the sedimentation time gets reduced to reach the channel bottom. The variation in sedimentation time with different constriction gap sizes and densities of the particle are plotted in Fig. 13(b). It is also observed that with the increase in particle density, the constriction wall effects become insignificant and variation of the sedimentation time is negligible. Another interesting finding in the constricted passage sedimentation problem is that there may be formation of a virtual stationary state. The particle attains this state with near zero kinetic energy and slope of the particle trajectory as shown in Figs. 8(a) 8(d). This occurs for certain combination of the particle density and the constriction gap size Two particles sedimentation Two circular particles of same density and diameter are used to study the present sedimentation problem. We named the particles as per their initial order of arrangement: particle-1 and particle-2 for the leading and trailing particles respectively. The initial gap between two particles is 4D. The initial positions of the particles are 18D and 14D for particle-2 and particle-1 respectively, which is measured above the center of the semi-circular constriction and along the vertical centerline of the channel. In the following, the effects of different constriction gap size (1.25D 2.0D) and different particle density ( g/cm 3 ) on the flow regime will be analyzed Flow regime Zone-1. This is the region above the constriction walls of the channel. At time t = 0, both the particles are at rest and start falling under gravity. In Fig. 16, the velocity of the particles is plotted. Due to the accelerating stage, the velocity is increasing with time and particle-2 follows the trend of particle-1 during the initial time steps. But deviation is noticed at the later time steps, which may be due to the interaction between the trailing particle and the wake of the leading particle. At the initial time step, the two particles sedimentation resembles closely as an impulsively started flow over tandem circular cylinders. As shown by Sumner et al. [1999] the temporal development of the flow regime is dependent on the cylinders center distance. For the center distance less than 2D, separated shear layers from the upstream cylinder reattach on the surface of the downstream cylinder. But for the higher center distance this reattachment is avoided. In the present simulation,

22 S.M.Dash,T.S.Lee&H.Huang t=1.6 s t=0.1 s t=0.2 s t=0.3 s Fig. 14. (Color online) Instantaneous pressure and vorticity contours at different time steps in zone-1 [(a), (b)] and zone-3 [(c), (d)] for particle density 1.5 g/cm 3. t=0.4 s t=1.8 s t=0.6 s t=2.1s t=0.7s Fig. 15. (Color online) Instantaneous pressure and vorticity contours at different time steps in zone-2 for particle density 1.5 g/cm 3. we set the initial gap as 4D and as the initial Re of the flow is low, the streamline reattachment is avoided. This can be verified from the vorticity plot in Fig. 14, where the vortices of the two particles do not interact. Again from the pressure contours as shown in Fig. 14, the force exerted on two particles will be different and hence the particle velocities as functions of time will be different. In Fig. 16, trajectories of the particles are also plotted. t=0.8 s t=1.0 s t=1.4 s Zone-2. This is the region of the channel containing the constriction walls. For the two particle sedimentation, it is found that while particle-2 is traveling in zone-1, particle-1 has already entered zone-2. The behavior of the particle-1 is similar to that discussed in single particle sedimentation section. Here, only the behavior of particle-2 will be discussed

23 Particle Sedimentation in a Constricted Passage (a) Fig. 16. Temporal evolution of (a) vertical velocity, (b) Y center co-ordinate of particles with density 1.5 g/cm 3 and in the constriction gap size of 1.75D. As previously noted a low pressure band is formed above the constrictions. When particle-2 enters this low pressure zone with less drag force, it accelerates faster and attains higher velocity magnitude than particle-1 as shown in Fig. 16. With further settling of the particle-2, it encounters narrowing portion of the constriction zone and particle-2 starts decelerating due to the increase of drag force. From Fig. 16, it is found that the velocity of particle-2 may reduce to zero and reverse its movement direction. This means particle-2 may move upward in zone-2 for certain time before it again starts falling. The interesting upward movement may be due to the wake region of particle-1. When particle-1 crosses the centerline of the constriction it starts accelerating, while from Fig. 16, we can find that particle-2 is still decelerating. With forward movements of particle-1 in the diverging part of the constriction, squeezes out the surrounding fluid through its converging part. This squeezed fluid moves in the opposite direction relative to the motion of particle-2 thus helping in its velocity deceleration. With the combined effect of the upward drag force and the oppositely moving squeezed fluid may be sufficient to balance the weight of the particle and makes the particle move upward. Using the vector plots (Fig. 17) of the fluid velocity this motion can be verified. To further verify this flow regime, pressure contours are plotted. It is noticed that there is a critical point in the wake of particle-1 where streamlines are discontinuous. From the definition of Perry and Chong [1987], this discontinuity is associated with formation of a node. Again the critical point is linked with a pressure gradient [Perry and Chong (1987)] and hence leads to a force generation. We noticed that the pressure gradient reduces with further downward movement of particle-1. With the reduction in drag force and under the effects of gravity, particle-2 again starts falling downward and reaches zone-3. The vorticity plots are shown in Fig. 15. (b)

24 S.M.Dash,T.S.Lee&H.Huang t=0.8s t=0.9s t=1.3s Fig. 17. (Color online) Velocity vector plot at different time step for the particle of density 1.5 g/cm 3 at zone-2 with constriction gap size of 1.75D. Zone-3. This is the region of the channel below the constriction walls. The flow regimes behind the particles are much similar to zone-3 of the single particle sedimentation case. The particles attain the terminal velocity before settling on the channel base. Here, one thing should be noticed that the terminal velocity of particle-2 is less than the terminal velocity of particle-1. This is because with movement of particle-1, the fluid in its wake gain kinetic energy and as mentioned by Cate et al. [2013], the momentum diffusive time scale of the fluid is larger than the particle advection time scale. Therefore, when particle-2 moves in the wakes of particle-1, the surrounding flow field still retains certain relative velocity. Thus the terminal velocity is smaller for particle-2. The pressure and vorticity plots are shown in Figs. 14(c) and 14(d). (a) (b) Fig. 18. (Color online) Comparisons of maximum retardation velocity (V R ) for different constriction gap size and density of particles; (a) particle-2 (b) particle

25 Particle Sedimentation in a Constricted Passage Table 3. Sedimentation time lag (in sec) between particle-2 and particle-1 for different constriction gap sizes and density of the particle. Density (g/cm 3 ) Gap size = 1.25D Gap size = 1.5D Gap size = 1.75D Gap size = 2.0D Effects of constriction gap Constriction gap effects are studied by comparing the V R of the particles. With the decrease in constriction gap, the retardation velocity increases for a particular particle density. Again by increasing the density of the particle, the retardation velocity is found decreasing. A comparison plot is shown in Fig. 18 for particle-1 and 2. We have also compared the sedimentation time lag between two particles as shown in Table 3. It is found that with reduction in constriction gap size, the time lag increases significantly. Also with the increase in particle density, the time lag decreases. 6. Conclusions We have discussed a novel flexible forcing hybrid IB-LBM scheme to simulate the particulate flow problems. Following the implicit type forcing term calculation, an unknown single Lagrangian velocity correction is introduced for suitable satisfaction of the no-slip boundary condition within a convergence limit. Use of flexible forcing principle not only avoids the complex mathematics involved in matrix inversion but also satisfy the boundary condition consistently at every time step with same order of accuracy. This is advantageous for unsteady and moving boundary flow problems. Also the algorithm suggested here is simple for new computational code development. The proposed algorithm is validated against the benchmark flow problems, particle sedimentation in a closed channel. The results are found to be in excellent agreement with previous published article. Then we extend the work to study the sedimentation process involved in a constricted channel. In this study, only the semi-circular constrictions are used and other constriction shapes are under future investigation. By varying the constriction gap size and density of the particles, the sedimentation processes are greatly influenced. Detailed investigation on the single and two particle sedimentation processes are provided with comparison plots. In future, this work will be extended to three dimensional studies. Acknowledgments This work is supported by NUS, Singapore research scholar funding. We are thankful to Profesor T. T. Lim for valuable discussions. The authors appreciate the detailed comments from the anonymous reviewers for improvising the paper

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