Inertial migration of spherical particles in circular ... - Semantic Scholar

PHYSICS OF FLUIDS 20, 103307 共2008兲

Inertial migration of spherical particles in circular Poiseuille flow at moderately high Reynolds numbers Xueming Shao, Zhaosheng Yu,a兲 and Bo Sun Department of Mechanics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, 310027 Hangzhou, China

共Received 24 May 2008; accepted 4 September 2008; published online 29 October 2008兲 The inertial migration of spherical particles in a circular Poiseuille flow is numerically investigated for the tube Reynolds number up to 2200. The periodic boundary condition is imposed in the streamwise direction. The equilibrium positions, the migration velocity, and the angular velocity of a single particle in a tube cell are examined at different Reynolds numbers, particle-tube size ratios, and tube lengths. Inner equilibrium positions are observed as the Reynolds number exceeds a critical value, in qualitatively agreement with the previous experimental observations 关J.-P. Matas, J. F. Morris, and E. Guazzelli, J. Fluid Mech. 515, 171 共2004兲兴. Our results indicate that the hydrodynamic interactions between the particles in different periodic cells have significant effects on the migration of the particles at the tube length being even as large as 6.7 particle diameters and they tend to stabilize the particles at the outer Segré–Silberberg equilibrium positions and to suppress the emergence of the inner equilibrium positions. A mirror-symmetric traveling-wave-like structure is observed when the particle Reynolds number is large enough. A pair of counter-rotating streamwise vortices exists at both upstream and downstream of the particle but with different rotating directions. The fluids in the half of the pipe without the particle flow more slowly and most fluids in the other half with the particle move faster with respect to the parabolic profile. The intensity of the structure is influenced by the local particle Reynolds number, the particle motion, and the tube length. In addition, the migration of multiple particles in a periodic tube cell is examined. We attribute the disparity in the critical particle Reynolds number for the occurrence of the inner particle annulus for the experiments and our simulations to the effect of the tube length or the periodic boundary condition in our numerical model. © 2008 American Institute of Physics. 关DOI: 10.1063/1.3005427兴 I. INTRODUCTION

Particle migration in a Poiseuille flow is an important subject that has attracted much interest since Segré and Silberberg1,2 observed that a neutrally buoyant sphere migrated to a stable equilibrium position about 0.6 tube radius from the axis. The Segré–Silberberg effect was subsequently confirmed and experimentally further investigated by others 共e.g., Oliver,3 Jeffrey and Pearson,4 and Karnis et al.5兲. The perturbation method has been widely employed to examine the inertia-induced lift force responsible for particle migration in a shear flow 共e.g., Rubinow and Keller,6 Saffman,7 Cox and Brenner,8 Ho and Leal,9 Vasseur and Cox,10 Schonberg and Hinch,11 Asmolov,12 and Matas et al.13兲. Brenner,14 Cox and Mason,15 and Leal16 presented comprehensive reviews of experimental and theoretical works on particle migration in early years. Recently, Matas et al.17,18 extended the Segré–Silberberg experiments to moderately high Reynolds numbers up to 2400 共based on the tube diameter and average velocity兲. They observed that the equilibrium radial position, which they termed the Segré–Silberberg annulus, moved toward the wall as Re is increased, and that an accumulation of particles on an inner annulus, i.e., at smaller radial position than the a兲

Corresponding author.

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Segré–Silberberg annulus, emerged at Re⬎ 600. More and more particles left the outer annulus and joined the inner annulus, as Re increases. For the particles of radius being 0.068 75 tube radius, the Segré–Silberberg annulus disappeared at Re= 1500,18 which was explained by the strong velocity fluctuations of the particles at this Re for which the system is close to the laminar-turbulent transition.19 In the numerical aspect, Feng et al.20 investigated the migration of a single circular particle in a two-dimensional Poiseuille flow. Joseph and Ocando21 discussed the correlation between the lift force and the angular slip velocity. The lift-off of one or many particles in a two-dimensional Poiseuille flow has been examined by Joseph and his co-workers.22–24 Yu et al.25 reported the numerical results on the radial, angular, and axial velocities for both neutrally buoyant and non-neutrally buoyant spherical particles in a vertical Poiseuille flow at Re up to 300. Pan and Glowinski26 simulated the motion of neutrally buoyant balls in a circular Poiseuille flow. Yang et al.27 studied the correlations for the lift force on a neutrally buoyant particle that rotated and moved forward freely but without radial velocity at Re up to 250. Recently, Chun and Ladd28 investigated the migration of a single and multiple particles in a square duct at Re up to 1000. For the duct-particle size ratio H / d = 9.1 and the volume fraction of 0.1, an initially uniform distribution evolved into three different steady-state distributions depending on

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Re. At Re= 100, the particles migrated to a position near the boundary walls and were strongly aligned in the direction of the flow, making linear chains of more or less uniformly spaced particles, as observed in experiments.29 At Re= 500, the particle trains became unstable and the spacing between the particles was no longer uniform. At Re= 1000, a few particles appeared in the center of the duct and had a substantial diffusive motion in the velocity-gradient plane. Based on the observation that there is a sudden onset of additional new equilibrium positions near the duct center at Re⬎ 700 for a dumbbell of two particles with a stiff spring, Chun and Ladd suggested an explanation for the appearance of the aforementioned inner annulus observed experimentally by Matas et al.17 at a high Reynolds number 共say ⬎700兲: The particle train with the uniform spacing is broken and closely spaced pairs and triplets are formed, and these clusters have additional equilibrium positions near the center of the duct. The aim of the present study is to numerically examine the inertial migration of neutrally buoyant spherical particles in a tube at moderately high Reynolds numbers. The rest of this paper is organized as follows. In Sec. II, we describe briefly the fictitious domain method adopted here and the numerical model for the problem studied. In Sec. III, we present and discuss the results on the radial and angular velocities of a particle at different Reynolds numbers and two different values of the tube length. In addition, some results on the migration of multiple particles are also reported and discussed. The concluding remarks are given in Sec. IV. II. NUMERICAL MODEL

ⵜ 2u ⳵u + u · ⵜu = − ⵜp + ␭ ⳵t Re u = U + ␻p ⫻ r ⵜ·u=0 共␳r − 1兲Vⴱp

共␳r − 1兲

共2兲

P共t兲,

共3兲

冊冕冕

dU g − Fr = − dt g

d共Jⴱ · ␻ p兲 =− dt

␭dx,

共4兲

P

r ⫻ ␭dx.

共5兲

P

In the above equations, u represents the fluid velocity, p the fluid pressure, ␭ the pseudobody force that is defined in the solid domain P共t兲, r the position vector with respect to the mass center of the particle, ␳r the particle-fluid density ratio defined by ␳r = ␳s / ␳ f , Re the Reynolds number defined by Re= ␳ f UcLc / ␮, Fr the Froude number defined here by Fr = gLc / U2c , Vⴱp the dimensionless particle volume define by Vⴱp = V p / L3c , and Jⴱ the dimensionless moment of inertia tensor defined by Jⴱ = J / ␳sL5c . 2. Numerical schemes

A fractional-step time scheme is used to decouple systems 共1兲–共5兲 into the following two subproblems. Fluid subproblem for uⴱ, p: u ⴱ − u n 1 ⵜ 2u ⴱ 1 = − 䉮 p + 关3共u · ⵜu兲n − 共u · ⵜu兲n−1兴 − 2 Re 2 䉭t

A. Fictitious domain method

+

1. Fictitious domain formulation

The recently proposed direct-forcing fictitious domain30 共DF/FD兲 method is employed here to simulate the motion of particles in a tube. The DF/FD method is an improved version of our earlier distributed-Lagrange-multiplier/FD 共DLM/FD兲 code.31–33 The DLM/FD method was originally developed by Glowinski et al.34,35 The spirit of the method is that the interior of the particles is filled with the fluids and the inner fictitious fluids are enforced to satisfy the rigidbody motion constraint via a pseudobody force, which is introduced as a distributed-Lagrange multiplier in the DLM/FD formulation.34 We only briefly describe the algorithm of the DF/FD method in the following, and the reader is referred to Yu and Shao30 for the details of the method. For simplicity of description, we will consider only one particle in the following exposition. Suppose that the particle density, volume and moment of inertia tensor, translational velocity, and angular velocity is ␳s, V p, J, U, and ␻ p, respectively. The fluid viscosity and density is ␮ and ␳ f , respectively. Let P共t兲 represent the solid domain and ⍀ the entire domain including interior and exterior of the solid body. By introducing the following scales for the nondimensionalization: Lc for length, Uc for velocity, Lc / Uc for time, ␳ f U2c for the pressure, and ␳ f U2c / Lc for the pseudobody force, the dimensionless FD formulation for the incompressible fluid can be written as follows:

共1兲

⍀,

in

冉

in

⍀,

in

1 ⵜ 2u n + ␭n , 2 Re

共6兲

ⵜ · uⴱ = 0.

共7兲

A finite-difference-based projection method on a homogeneous half-staggered grid is used for the solution of the above fluid Eqs. 共6兲 and 共7兲. The diffusion problem 共6兲 can be decomposed into tridiagonal systems with the alternative direction implicit 共ADI兲 technique and the Poisson equation for the pressure arising from the projection scheme is solved with the fast Fourier transform 共FFT兲-based fast solver. All spatial derivatives are discretized with the second-order central difference scheme. Particle subproblem for Un+1, ␻sn+1, ␭n+1, un+1:

␳rVⴱp

␳r

冉

冊冕冉

Un+1 Un g = 共␳r − 1兲Vⴱp + Fr + 䉭t 䉭t g

冋

P

Jⴱ · ␻sn+1 Jⴱ · ␻sn = 共␳r − 1兲 − ␻sn ⫻ 共Jⴱ · ␻sn兲䉭t 䉭t +

冕冉 r⫻

P

冊

uⴱ − ␭n dx. 䉭t

冊

uⴱ − ␭n dx, 䉭t

册

共8兲

共9兲

Note that the above equations have been reformulated so that all the right-hand side terms are known quantities and con-


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Inertial migration of spherical particles

In the above manipulations, the trilinear function is used to interpolate the fluid velocity from the Eulerian nodes to the Lagrangian nodes, and to distribute the pseudobody force from the Lagrangian nodes to the Eulerian nodes. For spherical particles, the Lagrangian nodes are distributed on a sequence of spherical surfaces 共see Ref. 30 for the details兲.

the difference in results from two different conditions should be small since the flow flux under the constant pressure gradient condition 共and reverse兲 changes very little during the simulation. The constant pressure gradient condition is adopted in the present study. The computational domain is a cuboid just containing the tube. The no-slip boundary condition on the tube wall is fulfilled by introducing another pseudobody force with the DF/FD technique.26,30 Our DF/FD code has been fully validated in Ref. 30. Particularly, it has been shown that our results on the evolution of the radial position with time for 共Re, ␭a , L / R兲 = 共50, 0.15, 4兲 obtained with the mesh size h = a / 4.8 and time step ⌬t = 0.005R / Um are in good agreement with those of Pan and Glowinski,26 and the equilibrium position 共being around 0.6R兲 also agrees very well with the arbitrary Lagrangian Eulerian 共ALE兲 result of Yang et al.27 In the present study, we choose a finer mesh h = R / 64 corresponding to h = a / 6.4 for ␭a = 0.1 and h = a / 9.6 for ␭a = 0.15, since much higher Reynolds numbers are considered here. For the cases of L = 2R and 4R studied, the number of the mesh points is 128⫻ 128⫻ 128 and 128⫻ 128⫻ 256, respectively. Further refinement in the mesh size and increase in the tube length are limited by our computer resource. We choose time step ⌬t = 0.005R / Um, which has been tested to be a suitable value. The initial pseudobody force ␭ in Eqs. 共6兲–共9兲 is set to be zero. As implied in Eqs. 共8兲 and 共9兲, the initial particle translational and angular velocities are irrelevant to the results for the neutrally buoyant case 共i.e., ␳r = 1兲, and the particle velocities become equal to those of the undisturbed local fluid after Eqs. 共8兲 and 共9兲 are performed for ␭ = 0.

B. Numerical model of the problem

III. RESULTS AND DISCUSSION

Figure 1 delineates the schematic of the flow problem involving one spherical particle. The diameter of the tube and particle is D and d 共the radius being R and a兲, respectively. We denote the particle-tube radius 共or diameter兲 ratio by ␭a = a / R. The periodic boundary condition is introduced in the streamwise direction and the length 共period兲 of the tube is denoted by L. Initially, the following parabolic velocity profile is imposed in the tube:

A. Single particle case

FIG. 1. The schematic for a neutrally buoyant spherical particle in a circular Poiseuille flow.

sequently the particle velocities Un+1 and ␻n+1 are obtained without iteration, unlike the original DLM/FD method. Then, ␭n+1 defined at the Lagrangian nodes are determined from ␭n+1 =

Un+1 + ␻sn+1 ⫻ r − uⴱ + ␭n , 䉭t

共10兲

and finally, the fluid velocities un+1 at the Eulerian nodes are obtained from un+1 = uⴱ + 䉭 t共␭n+1 − ␭n兲.

冉冊

u = Um 1 −

r2 , R2

共11兲

共12兲

where Um is the maximum velocity at the centerline of the pipe. We take the radius of the tube R and the maximum velocity Um as the characteristic length and velocity, respectively. Thus the time is scaled as R / Um, which can be regarded as the inverse of the average shear rate ␥˙ a = Um / R. The Reynolds number in the fluid momentum Eq. 共6兲 is then defined by Re= ␳ f UmR / ␮, called the tube Reynolds number. The particle or shear Reynolds number may be defined by Rep = ␳ f a2␥˙ a / ␮ = ␭2aRe. For the Poiseuille flow with the periodic boundary condition in the streamwise direction, either constant pressure gradient or constant flow flux condition can be used to sustain the flow. For the case of one neutrally buoyant particle,

The primary aim of the present study is to investigate the inertial migration of a single particle in a tube. Only two particle-tube radius ratios ␭a = 0.1 and 0.15 are considered. Since the periodic boundary condition is imposed in the streamwise direction, we actually examine the motion of a regular train of particles. Two values of the tube length or the distance between two consecutive particles L = 2R and 4R are considered. In the following, we will discuss the results on the motion of the particle for four cases: 共␭a , L / R兲 = 共0.15, 2兲, 共0.15,4兲, 共0.1,2兲, and 共0.1,4兲. 1. Equilibrium positions

Case 1: 共␭a , L / R兲 = 共0.15, 2兲. Figure 2 shows the radial equilibrium positions at different Reynolds numbers for all four cases. For 共␭a , L / R兲 = 共0.15, 2兲, the results are reported for Re= 100– 2000. In our simulations, the flows are observed to be of laminar type. The velocity profile is found to be also smooth even for Re= 2400, a Reynolds number at which the flow is expected to become turbulent in the absence of particles. The transition to turbulence in the presence of particles could take place at Re down to around 1500 in the experiments.19 The reason for laminar flows at high Re in our simulations is probably due to the lack of the unstable disturbance to the flow. Thus, we are concerned with the


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FIG. 2. Radial equilibrium position vs Re for ␭a = 0.1 and 0.15, and L / R = 2 and 4.

FIG. 4. Evolution of the radial position of a particle released at different positions in a tube for 共␭a , L / R兲 = 共0.15, 4兲. Solid lines indicate the results for Re= 1000, and dash-dot lines for Re= 760.

motion of particles in a laminar Poiseuille flow in the present study. Nevertheless, the departure of the velocity profile from the parabolic one and the weak secondary flow can be observed in our simulations, as shown later. From Fig. 2, for the case of 共␭a , L / R兲 = 共0.15, 2兲, two branches of equilibrium positions are observed for Re = 100– 2200. One represents the Segré–Silberberg equilibrium positions, occurring for Reⱕ 1300. The other represents the inner equilibrium positions, being closer to the tube axis than to the tube wall and occurring for Reⱖ 1000; note that the critical Re is between 760 and 1000 共see Fig. 8兲 and we have not attempted to determine the accurate value. For 1000ⱕ Reⱕ 1300, there exist two equilibrium positions. As shown in Fig. 3, the particle released at r0 = 0.05 and 0.6 for Re= 1000 migrates to the inner equilibrium position, whereas the particle released at r0 = 0.7 and 0.8 migrates to the Segré– Silberberg equilibrium position. From Fig. 2, as the Reynolds number increases, the Segré–Silberberg equilibrium positions shift closer to the tube wall, whereas the inner equilibrium positions become closer to the tube axis.

Case 2: 共␭a , L / R兲 = 共0.15, 4兲. As the tube length L 共i.e., the distance between two consecutive particles兲 is increased to 4 tube radius while keeping ␭a = 0.15, it is observed in Fig. 2 that the particle migrates to the Segré–Silberberg equilibrium positions for Reⱕ 500, and to the inner equilibrium positions for Reⱖ 640. The critical Re for the occurrence of the inner equilibrium positions is obviously smaller, compared to the case of L = 2R. In addition, as L is increased, the Segré–Silberberg equilibrium positions are closer to the tube axis 共Fig. 2兲. Figure 4 shows that the particle released at different positions migrates to the same inner equilibrium position for Re= 760 and 1000, respectively. Case 3: 共␭a , L / R兲 = 共0.1, 2兲. From Fig. 2, as the particlepipe radius ratio ␭a is decreased to 0.1 while keeping the tube length L = 2R, the particle migrates to the Segré– Silberberg equilibrium positions for Reⱕ 1300, and to the inner equilibrium positions for Reⱖ 1500. Figure 5 shows that the particle released at r0 = 0.2 and 0.8 can achieve the same equilibrium position after several overshoots at Re up to 2200.

FIG. 3. Evolution of the radial position of a particle released at different positions in a tube for 共Re, ␭a , L / R = 共1000, 0.15, 2兲.

FIG. 5. Evolution of the radial position of a particle released at different positions in a tube for 共Re, ␭a , L / R兲 = 共2200, 0.1, 2兲.


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Case 4: 共␭a , L / R兲 = 共0.1, 4兲. From Fig. 2, the particle migrates to the Segré–Silberberg equilibrium positions for Re ⱕ 1000 in case of 共␭a , L / R兲 = 共0.1, 4兲. At Re= 1200, we see in Fig. 6 that there is a stable Segré–Silberberg equilibrium position being around 0.832 while the particle released at r0 = 0.6 experiences an oscillatory motion largely within r = 0.5– 0.7 during the simulation time up to 1400. As Re is increased to 1300, although the particle released near the tube wall can be attracted to the Segré–Silberberg equilibrium position and stay there for a relatively long time, it will eventually loses stability, leaving the Segré–Silberberg equilibrium position and undergoing an oscillatory motion largely within r = 0.4– 0.7, as shown in Fig. 7. No stable inner equilibrium position can be observed as we further increase the Reynolds number in this case. From Fig. 2, the difference between the Segré–Silberberg equilibrium positions for L = 2R and 4R at ␭a = 0.1 is less pronounced than at ␭a = 0.15. From above observations for all four cases, both the particle size and the distance between the particles influence the particle migration at high Reynolds numbers significantly.


The critical Reynolds number for the onset of the inner equilibrium position is larger for a smaller ␭a, which is not surprising since the occurrence of the inner equilibrium position must be related to the inertial effect and the wall repulsive effect and both effects are stronger for a larger ␭a at the same Re. The particle or shear Reynolds number Rep defined earlier is a better Reynolds number to measure the inertial effect. The critical Rep for the occurrence of the inner equilibrium position is around 22.5 共i.e., 1000⫻ 0.152兲 for 共␭a , L / R兲 = 共0.15, 2兲 and around 15 共i.e., 1500⫻ 0.12兲 for 共␭a , L / R兲 = 共0.1, 2兲, and thus is smaller for a lower ␭a. As the distance between the particles L is decreased, the hydrodynamic interactions between the particles are intensified and it is interesting that this renders the particles in the periodic arrangement in favor of the Segré–Silberberg equilibrium position and, on the other hand, against the inner equilibrium position, in the sense that for a smaller interparticle distance the particles can keep staying stably at the Segré–Silberberg equilibrium position until a higher Reynolds number and the occurrence of the inner equilibrium position is also retarded to a higher Reynolds number. The numerical results on the Segré–Silberberg equilibrium positions of Yang et al.27 for ␭a = 0.15 and the experimental results of Matas et al.17 for ␭a = 0.125 are plotted in Fig. 2 for comparison. Our results agree well with both of them. We note that the value of the tube length L used by Yang et al. was not reported, and the error bar for the experimental data of Matas et al. is relatively large, which, as we suspect, is probably caused by the hydrodynamic interactions between particles located nonuniformly in the pipe even at the very low particle concentration; we have seen that the equilibrium position becomes closer to the wall as the particle interdistance is reduced from 13.3 particle diameters to 6.7 particle diameters. Our results are also consistent with the computations of Chun and Ladd28 on the particle migration in a duct. They observed the sole Segré–Silberberg equilibrium positions at Re up to 1000 for a spherical particle with the duct-particle size ratio being 9.1, and, on the other hand, a sudden onset of additional new equilibrium positions near the duct center at Re around 750 for a dumbbell of two particles with a stiff spring, which is effectively a larger particle. 2. Radial migration velocity and angular velocity


We now examine the behavior of the particle migration velocity and angular velocity. At low particle Reynolds numbers, the lift force is balanced with the radial drag force that is proportional to the radial migration velocity, and thus the migration velocity can represent the lift force except for the initial stage when the velocity increases from zero very rapidly 共see Fig. 8兲. For relatively high particle Reynolds numbers, we suppose that the migration velocity can still well reflect the lift force except for the time when the velocity changes very rapidly. Figure 8 shows the radial migration velocity versus radial position for a particle of ␭a = 0.15 released at r0 = 0.05 in the case of Re= 100, 500, 760, and 1000, and L = 2R and 4R, respectively. The initial rapid increase and overshot in the


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FIG. 8. Radial migration velocity vs radial position for a particle of ␭a = 0.15 released at r0 = 0.05 and Re= 100, 500, 760, and 1000, respectively. Solid lines: L = 4R, and dotted lines: L = 2R. The arrow on the left indicates the increase of Re.

velocity after release is related to the initial condition and was shown to be a transient effect in Yu et al.25 since the velocity profiles for the particles released at different positions converged to the unique one after initial overshoot for 共Re, ␭a兲 = 共100, 0.25兲. Thus, in Fig. 8, we could get a more complete and physical velocity profile by removing the initial overshoot and extending the curve to the origin of the coordinates. The experiments14 and computations25,27 revealed that the migration velocity or lift force increases with increasing Re at relatively low Re; however, at Re being sufficiently large, the dimensionless migration velocity or lift force decreases with increasing Re, as shown by the computations of Asmolov12 and Yu et al.25 Figure 8 also shows that the 共dimensionless兲 migration velocity is reduced as Re is increased from 500 to 1000 for r ⬎ 0.2. As shown in Fig. 8, the effect of the particle interdistance L on the migration velocity is very small before the maximum velocity is reached and then becomes pronounced

FIG. 9. Particle angular velocity vs radial position for a particle of ␭a = 0.15 released at r0 = 0.05 and Re= 100, 500, and 760, respectively. Solid lines: L = 4R, and dotted lines: L = 2R.


during the subsequent velocity-decreasing period. The pronounced difference between the migration velocities for L = 2R and 4R takes place at r ⬎ 0.5 for Re= 100, and r ⬎ 0.3 for Re= 500– 1000. It is reasonable that the effect of L is more significant as the particle moves closer to the wall, since the hydrodynamic interactions between the particles are stronger and more likely to be sensitive to the interparticle distance as the particle moves closer to the wall and the particle Reynolds number based on the local shear rate is enhanced. It is interesting to observe in Fig. 8 that for Re = 500 and 760 the migration velocities in the case of L = 4R decrease monotonously from the maximum to zero, whereas in the case of L = 2R they decrease to a minimum above zero and then increase again before finally reaching zero, which is a reminiscence of the characteristics of the lift force on a particle in a plane Poiseuille flow at a high Reynolds number obtained by Asmolov12 using the perturbation theory. Figure 9 delineates the particle angular velocity versus radial position for a particle of ␭a = 0.15 released at r0 = 0.05 and Re= 100, 500, and 760, respectively. The previous computations of Yu et al.25 for relatively low Reynolds numbers indicated that the particle angular velocity was proportional to the local radial position during the migration process, which means that the ratio of the particle and local fluid angular velocities is a constant since the dimensionless local fluid angular velocity is ␻ = r. From Figs. 9 and 10, this is also true for Re up to 1000 during the outward migration process. Not surprisingly, as Re is increased and consequently the inertial effect is enhanced, the particle-fluid angular velocity ratio decreases. Unlike the migration velocity, the angular velocity is affected by L insignificantly during the outward migration process 共Fig. 9兲. We have examined the behavior of the velocities of a particle released near the tube axis and migrating toward the tube wall, and now we inspect the case of a particle released near the tube wall and migrating inward. At relatively low Reynolds numbers, the migration velocity approaches zero from both sides of the equilibrium position and the particle angular velocity is also proportional to the local radial position during the inward migration process.25 In other words, there is no overshoot for the particle motion at the equilibrium position at relatively low Reynolds numbers. However, the pronounced overshoots of the particle radial position, migration velocity and angular velocity can be observed in Figs. 4, 10共a兲, and 11共a兲, respectively, for a particle migrating inward to the inner equilibrium position in case of L = 4R at Re= 1000. For L = 2R, there is no pronounced overshoot since the particles released at r = 0.7 and 0.8 migrated to the outer Segré–Silberberg equilibrium position, as shown in Figs. 3, 10共b兲, and 11共b兲. Note that at initial stage the particles released at r = 0.7 and 0.8 for the case of L = 4R also migrate toward the Segré–Silberberg equilibrium position, with the migration and angular velocities being almost the same as the case of L = 2R 共see Figs. 10 and 11兲; the angular velocities decrease rapidly from the local fluid rotational velocities 共␻ = r兲 to the specific values corresponding to Re = 1000 共i.e., ␻ = 0.72r兲, as indicated in Fig. 10共a兲. The Segré– Silberberg equilibrium position is, however, not stable for L = 4R and Re= 1000. In Fig. 10共a兲, the angular velocities


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FIG. 10. Particle angular velocity vs radial position for a particle of ␭a = 0.15 released at different radial positions 共r0 / R = 0.05, 0.6, 0.7, and 0.8兲 and Re = 1000 in the cases of 共a兲 L = 4R and 共b兲 L = 2R.

overshoot the line of ␻ = 0.72r, accompanied by the rapid increase in the inward migrating velocities. For Re= 1000, the stable particle angular velocity must be equal to 0.72 times the local fluid rotational velocity, thus the particle angular velocities in Fig. 10共a兲 then start to return to the line of ␻ = 0.72r after the maximum departure from the line. A decrease in the magnitude of the migration velocities can be observed Fig. 11共a兲 as the angular velocities approach the line of ␻ = 0.72r. The second overshoot in the angular velocities subsequently takes place due to the substantial particle inertial effect, and the inward migrating velocities start to increase again as the particle angular velocities depart from the line of ␻ = 0.72r to some extent. The particle inertial effect is so strong that the particles can rotate even faster than the local fluid as the particles migrate inward rapidly. Then, the migration velocities begin to decrease. Substantial rates of the decrease in both migration and angular velocities can be seen at r ⬇ 0.2 in Figs. 10共a兲 and 11共a兲. The angular velocities finally return to ␻ = 0.72r without significant overshoot, whereas the migration velocities become comparable to those for a particle directly released near the tube axis after a pronounced overshoot. From above observations, the particle inertial effect plays an important role in the surprisingly large inward migrating velocities for L = 4R.

3. Flow structures

A recent important progress in understanding the transition to turbulence in a pipe is the discovery of the travelingwave solutions which are saddle points in phase space.36–39 A typical traveling-wave structure is characterized by the larger flow velocities at some regions close to the wall and smaller velocities at some regions close to the tube axis with respect to the parabolic profile, and the presence of weak secondary flows 共or streamwise vortices兲. A mirrorsymmetric traveling wave was observed by Pringle and Kerswell39 even at Re down to 773. We now examine the flow fields in the presence of a particle. The contours of the streamwise velocity relative to the parabolic profile and the in-plane velocity vectors in the planes of z = Z p, Z p + L / 4,Z p + L / 2, and Z p − L / 4 共i.e., Z p + 3 / 4L兲 for 共Re, ␭a , L / R兲 = 共1000, 0.15, 2兲 at time when the particle has achieved the equilibrium position are depicted in Fig. 12. Here, Z p represents the streamwise position of the particle and is actually set to be the position of the velocity grid closest to the particle for convenience. In Fig. 12, we observe a mirror-symmetric traveling-wave-like structure that has following features. First, regarding the secondary flow, the fluids flow away from the particle along the azi-

FIG. 11. Radial migration velocity vs radial position for a particle of ␭a = 0.15 released at different radial positions 共r0 / R = 0.05, 0.6, 0.7, and 0.8兲 and Re = 1000 in the cases of 共a兲 L = 4R and 共b兲 L = 2R.


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FIG. 12. Velocity fields in the planes of 共a兲 z = Z p, 共b兲 z = Z p + L / 4, 共c兲 z = Z p + L / 2, and 共d兲 z = Z p − L / 4 for 共Re, ␭a , L / R兲 = 共1000, 0.15, 2兲兲. The contours 共gray scale兲 indicate the streamwise velocity relative to the parabolic profile with the maximum 共brightest兲 and minimum 共darkest兲 levels and the increment being 0.17, −0.11, and 0.02, respectively. The maximum in-plane 共secondary flow兲 velocity as shown in 共a兲 is 0.048.

muthal direction in the particle plane 关Fig. 12共a兲兴; a pair of counter-rotating streamwise vortices exist at both upstream and downstream of the particle, and they have different rotating directions with the flow pointing from the wall toward the axis between the pair for the upstream one and the reverse for the downstream one 关Figs. 12共b兲 and 12共d兲兴. Second, regarding the streamwise velocity, the fluids in the half of the pipe without the particle flow more slowly and most fluids in the other half with the particle move faster with respect to the parabolic profile. This is interesting and anomalous, since one could imagine that the fluids in the half of the pipe without the particle should move faster due to the fact that a particle in fluids lags the fluids 共e.g., Ref. 25兲. Third, since the structures move in the streamwise direction with the particle, they can be regarded as the traveling wave. Nevertheless, the traveling-wave structures in our simulations appear different from those in the absence of the particle. This is not surprising since our structures are induced by the particle. The traveling waves in the absence of the particle are not observed in our simulations and this is also not surprising since they were found numerically only with the special initial conditions.37–39 Figure 13 plots the velocity fields in the particle planes for Re= 100– 1000 and ␭a = 0.15, and Fig. 14 compares the velocity fields in the particle planes for L = 4R and L = 2R at 共Re, ␭a兲 = 共1300, 0.1兲. The particle is released in the x-z plane at x = 0.05 for Figs. 13共a兲–13共c兲, x = 0.8 for Fig. 13共d兲, and x = 0.6 for Fig. 14. We see that the particle largely keeps moving in the x-z plane except for Fig. 13共c兲 in which the departure of the particle away from the x-z plane is obvious.

FIG. 13. Velocity fields in the particle planes for 共a兲 Re= 100, 共b兲 Re= 500, 共c兲 Re= 1000, and 共d兲 Re= 1000. ␭a = 0.15. L = 4R for 共a兲–共c兲 and L = 2R for 共d兲. The velocity representation is the same as in Fig. 12.

One reason for the departure is that the particle is released very close to the tube axis. All flow structures are mirrorsymmetric about the line passing the particle center and the tube axis. From Fig. 13, the traveling wave does not form at Re = 100 and can be observed at Re= 500. It becomes stronger as Re increases 共Fig. 13兲, the particle moves closer to the wall 关Figs. 14共b兲, 14共d兲, and 14共f兲兴, or the particle size is increased 关Figs. 13共d兲 and 14共f兲兴. Thus, the intensity of the traveling wave mainly depends on the local particle 共or shear兲 Reynolds number based on the particle radius and the local shear rate. The traveling wave is also affected by the tube length and the motion of the particle. The comparison between Figs. 14共a兲 and 14共b兲 indicates that the traveling wave develops faster in a longer tube, which might be responsible for the subsequent disparity in the particle migration direction for two tubes. From Figs. 14共c兲 and 14共e兲, the traveling wave is stronger when the particle moves toward the tube axis, compared to the case of the particle moving toward the wall. Clearly, the particle-induced flow structures 共traveling wave兲 can enhance the hydrodynamic interactions between the particles, and the interaction between the flow structures and the particle motion may result in the complex particle migration behavior. B. Multiparticle case

We have examined the effects of hydrodynamic interactions on the migration of particles constrained in the periodic arrangement by changing the interparticle distance and observed that the enhanced hydrodynamic interactions help to stabilize the particles at outer equilibrium positions. Now we


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FIG. 15. Snapshots of 共x , y兲-positions of particles in circular Poiseuille flow. 共Re, ␭a , L / R , N p兲 = 共760, 0.15, 4 , 10兲. ␾ = 1.1%. Two circles with the dashed lines indicate the outer and inner equilibrium positions for a single particle and the same Re and ␭a as shown in Fig. 2.

FIG. 14. Velocity fields in the particle planes for 共a兲 L = 4R , t = 50, 共c兲 L = 4R , t = 150, 共e兲 L = 4R , t = 250, 共b兲 L = 2R , t = 50, 共d兲 L = 2R , t = 150, and 共f兲 L = 2R , t = 450. 共Re, ␭a兲 = 共1300, 0.1兲. The velocity representation is as in Fig. 12.

investigate the effects of the interactions between multiple unconstrained particles on the migration, focusing on the following two points. First, our results have indicated that at Re= 760 and 1000 a single particle of ␭a = 0.15 in a pipe of L = 4R migrates to the inner equilibrium position, whereas it can migrate to the outer equilibrium position as well for L = 2R, and then we wonder whether all particles would migrate to inner equilibrium positions or the particle interactions would stabilize some particles at outer equilibrium positions in the case of multiple particles in the pipe of L = 4R. Ten particles are randomly placed in an axial plane at initial time, as shown in Fig. 15. The volume fraction ␾ is around 1.1%. From Figs. 15 and 16, there is a tendency for particles to aggregate and form trainlike structures due to the hydrodynamic interactions.29,28 A train is more likely to form at the outer equilibrium position in the vicinity of the wall, since the particles have nearly the same axial velocities and the collision between them will not happen during the lateral migration. The chance is slim for a particle to align itself with

other particles in the z-axis direction via entering along the radial direction since the particles have different axial velocities and very small migration velocities and consequently the collision is easily resulted in. One example of such collision is shown in Figs. 17共c兲 and 17共d兲, and the other is shown in Figs. 16共c兲 and 16共d兲, where the collision leads to the breakdown of the formed train and subsequently the particles migrate inward rapidly, as observed early for the case of a single particle at Re= 1000. It is clear that the hydrodynamic interactions can result in the formation of the particle train that can stay some time in the vicinity of the wall, and the particle train at Re= 760 and 1000 is not stable28 and may break down, particularly due to the direct collision with incoming particles, as observed here. From Figs. 15 and 16, fewer particles can be observed near the wall at Re= 1000 than Re= 760, in qualitatively agreement with the experimental results. Second, the inner annulus of particles was observed in the experiments at 共Re, ␭a兲 = 共760, 0.068 75兲,18 and 共Re, ␭a兲 = 共1000, 1 / 17兲,17 respectively. However our computations for a single particle at L up to 4R predict no inner equilibrium

FIG. 16. Snapshots of 共x , y兲-positions of particles in circular Poiseuille flow. 共Re, ␭a , L / R , N p兲 = 共1000, 0.15, 4 , 10兲. ␾ = 1.1%. Two circles with the dashed lines indicate the outer and inner equilibrium positions for a single particle and L = 2R as shown in Fig. 2.


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and the overlap was either caused by more than one contact point of one particle with other particles 共or the wall兲 or the subtle bugs in our code involving the collision model. Nevertheless, the essential change in the particle distribution 共appearance of inner annulus兲 is not expected in a longer simulation. Therefore, we attribute the disparity between the experiments and our simulations to the effect of the tube length or the periodic boundary condition in our numerical model. As observed earlier, the hydrodynamic interactions between one particle and those in other periodic cells can suppress the emergence of the inner equilibrium position. A sufficiently large pipe length L in the numerical model is probably required to reproduce the experimental observations, and we postpone the study to a future time when significantly improved computational resources are available to us. IV. CONCLUSION

FIG. 17. Snapshots of 共x , y兲-positions of particles in circular Poiseuille flow. 共Re, ␭a , L / R , N p兲 = 共1000, 0.1, 4 , 10兲. ␾ = 0.33%. The dashed circle indicates the equilibrium position for a single particle and L = 2R as shown in Fig. 2.

position for such low Re or ␭a 共i.e., low particle Reynolds number兲. We wonder whether the interactions between particles would produce the inner annulus of particles. Two simulation cases are considered: 10 particles are placed randomly in an axial plane, and 30 particles in the entire pipe at initial time at 共Re, ␭a , L / R兲 = 共1000, 0.1, 4兲. The volume fraction ␾ is 0.33% and 1%, respectively. From Figs. 17 and 18, the interactions between particles cannot cause the occurrence of the inner annulus of particles for both volume fractions, although, for the case of ␾ = 1%, one particle is trapped near the tube axis for a long time and the collision between particles causes the breakdown of the particle train 关see Figs. 18共e兲 and 18共f兲兴. The simulations were terminated unexpectedly because the overlap between the particles 共or with the tube wall兲 was detected; the hard-sphere collision model with the restitution coefficient of 0.9 was used in our code,

FIG. 18. Snapshots of 共x , y兲-positions of particles in circular Poiseuille flow. 共Re, ␭a , L / R , N p兲 = 共1000, 0.1, 4 , 30兲. ␾ = 1%.

The inertial migration of spherical particles in a circular Poiseuille flow has been numerically investigated for the tube Reynolds number up to 2200. The equilibrium positions, the migration velocity, and the angular velocity of a single particle in a tube cell are examined at different Reynolds numbers for four cases: 共␭a , L / R兲 = 共0.15, 2兲, 共0.15,4兲, 共0.1,2兲, and 共0.1,4兲. Inner equilibrium positions are observed as the Reynolds number exceeds a critical value, in qualitatively agreement with the previous experimental observations of Matas et al.17,18 The hydrodynamic interactions between the particles in different periodic cells have significant effects on the migration of the particles at the tube length being even as large as 6.7 particle diameters in case of 共␭a , L / R兲 = 共0.15, 2兲, and they tend to stabilize the particles at the outer Segré–Silberberg equilibrium positions and to suppress the emergence of the inner equilibrium positions. An enormous inward migrating velocity is observed for a particle released in the vicinity of the tube wall and migrating toward to the inner equilibrium position for L = 4R, and is explained by the strong effect of the particle inertia and thereby the strong overshoot of the particle angular velocity. A mirror-symmetric traveling-wave-like structure is observed when the particle Reynolds number is large enough. For the secondary flow, the fluids flow away from the particle along the azimuthal direction in the particle plane and a pair of counter-rotating streamwise vortices exist at both upstream and downstream of the particle but with different rotating directions. For the streamwise velocity, the fluids in the half of the pipe without the particle flow more slowly and most fluids in the other half with the particle move faster with respect to the parabolic profile. The intensity of the structure is influenced by the local particle Reynolds number 共the tube Reynolds number, the particle size, and the particle radial position兲, the particle motion, and the tube length. The migration of multiple particles in a periodic tube cell is also examined. We attribute the disparity in the critical particle Reynolds number for the occurrence of the inner particle annulus for the experiments and our simulations to the effect of the tube length or the periodic boundary condition in our numerical model.


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ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China 共No. 10602051兲 and the Major Program of the National Natural Science Foundation of China 共No. 10632070兲, and helpful suggestions from the referees. 1



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