Flow and Drag Phenomena of Tandem Spheroid Particles at Finite

Article pubs.acs.org/IECR

Flow and Drag Phenomena of Tandem Spheroid Particles at Finite Reynolds Numbers Nanda Kishore* Department of Chemical Engineering, Indian Institute of Technology, Guwahati-781039, Assam, India ABSTRACT: Two dimensional steady Newtonian flow over tandem spheroid particles has been numerically investigated by solving the continuity and momentum equations along with appropriate boundary conditions. The main focus is to elucidate effects of the Reynolds number (Re), the particle aspect ratio (e) and the interparticle distance (S) on the flow and drag phenomena of tandem spheroid particles. To avoid numerical artifacts, extensive domain and grid independence studies have been carried out. For the case of tandem spherical particles, that is, for e = 1, the present results are in excellent agreement with existing experimental and numerical results. Further extensive new results have been obtained in the range of conditions 1 ≤ Re ≤ 100, 0.25 ≤ e ≤ 2.5, and 2 ≤ S ≤ 6. For different combination of Re, e, and S, complicated flow structures have been observed around both leading and trailing particles especially for large values of Re and e; and for small values of S. The influence of the leading particle on the drag force of the trailing particle is significantly larger than that of the trailing particle on the leading particle. This observation is consistent with previous experimental observations for the case of two tandem spheres, that is, for e = 1. Furthermore, drag coefficients of leading particles are close to those of unbounded isolated particles. For all values of the Reynolds number and aspect ratio, as the value of the interparticle distance increases, the individual and total drag coefficients increase.

1. INTRODUCTION The relative motion between solid particles and surrounding fluids is an idealization of many natural processes and industrially important settings such as air pollution, wastewater treatment, aerosols, fluidized beds, combustion, floatation, etc.1−6 In such envisaged applications, one often encounters many regular and irregular shaped particles interacting with the surrounding fluid, with each other, and/or with the equipment wall. Therefore, aside from the basic information on the settling behavior of isolated particles, additional information related to the interactions of particles with neighbor particles and/or with the confining walls is germane to the development of suitable frameworks for the analysis of multiparticle systems. The confining wall imposes an extra retardation force on the settling particle to enhance the drag force as compared to the case of the unbounded particle under identical conditions.7,8 On the other hand, hydrodynamic interactions with neighboring particles largely depends on the concentration of particles, relative orientation of particles, shape and size of particles, thermophysical properties of fluid, and operating conditions. However, understanding the flow characteristics in such complicated multiparticle systems involving variety of particles is very difficult because of inherent difficulties. Therefore, many theoretical simplifications have been evolved in the literature such as in-line or tandem or streamwise arrangements, side-byside or parallel arrangements, and inclined particles against the mean flow, etc. In such arrangements of particles, apart from the particle Reynolds number (Re) and the aspect ratio (e), the other important parameters are the separation distance between the particle (S) and the position of particles. These parameters can significantly affect the overall drag force acting on particles and the recirculation flow structures. © 2012 American Chemical Society

For the case of isolated and tandem spherical particles (e = 1) settling in Newtonian and non-Newtonian fluids, adequate theoretical/numerical literature has been accrued.1−16,22−33 However many experimental studies revealed that the spheroid shape is a very good approximation for nonspherical particles.1 The spheroid shape encompasses a wide range of shapes for different values of the aspect ratio (defined as e = b/a, where a and b are equatorial and polar radii of a spheroid, respectively) such as cylinder-like (e > 1) to disk-like (e < 1) particles via spheres (e = 1). Despite this fact, though, the steady axisymmetric flow past unconfined and confined spheroid particles has been investigated in the moderate range of Reynolds number;17−20 the analogous flow behavior of unconfined two tandem spheroid particles has not been investigated beyond the creeping flow regime. Thus, the aim of this work is to fill this gap in the literature. In this work, the effects of the particle aspect ratio (e), Reynolds number (Re), and the interparticle distance (S) on the flow and drag phenomena of tandem spheroid particles have been elucidated in the following range of conditions: 1 ≤ Re ≤ 100, 0.25 ≤ e ≤ 2.5, and 2 ≤ S ≤ 6.

2. PREVIOUS WORK Based on experimental and theoretical studies, reliable information is now available for tandem spherical particles (a special case of spheroid particles with unit aspect ratio) on their flow and drag behavior in the wide range of Reynolds numbers as recapitulated here. Rowe and Henwood21 experimentally Received: Revised: Accepted: Published: 3186

December 2, 2011 January 18, 2012 January 20, 2012 January 20, 2012 dx.doi.org/10.1021/ie202814w | Ind. Eng. Chem. Res. 2012, 51, 3186−3196

Industrial & Engineering Chemistry Research

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creeping flow regime. Thus this work aims to fill this gap in the literature.

measured drag forces on two spheres in tandem and parallel arrangement at Reynolds numbers ranging from 32 to 96. Tsuji et al.22 carried out experiments in the range of Reynolds number of 100−1000. Their results indicate when two spheres are in tandem arrangement, only the vortex of leading sphere changes in structure, and that of the rear sphere is same as the vortex of single sphere without interaction. Their drag results are in reasonable agreement with those of Rowe and Henwood.21 Using a microforce measuring system, Zhu et al.23 carried out experiments in the range of Re = 20−130 and observed that the influence of the leading particle on the drag force of the trailing particle is significantly larger than that of the trailing particle on the leading particle. Liang et al.24 conducted both experimental and numerical studies on flow interactions between spherical particles placed in different configurations in the range of Reynolds number from 30−106. However, no quantitative drag values and explanations of wake flow structure were presented. Chen and Lu25 carried out experiments in the range of Re < 200 and observed that for fixed value of the Reynolds number, as the separation distance increases, the total drag increases in the case of tandem (streamwise) arrangement and reverse trend observed for sideby-side spheres. Folkersma et al.26 numerically obtained flow patterns, drag, and interaction coefficients of two parallel spheres using a finite element method based solver for Re up to 50. Tsuji et al.27 numerically investigated unsteady 3-D interactions at Re = 30, 100, 200, and 250 and reported that the drag force of particles is attenuated when aligned streamwise and augmented when held side by side against the mean flow. Zou et al.28 numerically investigated the flow past two fixed spheres in tandem arrangement using a local mesh refinement virtual boundary method at Re = 250 and found that for separation distance (gap between two spheres divided by the diameter of the sphere) of 1.5, the flow remains axisymmetric even at Re = 250, which is not the case for a noninteracting single sphere. Yoon and Yang29,30 numerically investigated the flow-induced forces on two identical arbitrarily positioned spheres at Re = 300 by employing an immersed boundary method. Prahl et al.31 simulated the flow and drag phenomena of two arbitrarily fixed spherical particles at Re = 50, 100, and 200. Their results indicate that the drag reduction for the secondary or rear sphere is as high as 60−80% for Re = 50−200. Later on Prahl et al.32 extended this study for Re = 300 to investigate the vortex shedding characteristics of two tandem spheres. Jadoon et al.33 numerically studied effects of inflow conditions on dynamic interactions of fixed dual spheres for several configurations at Re = 300 and 600. On the other hand, for the case of multiple spheroid particles, only approximate solutions for Stokes viscous flow are available. Wakiya34 obtained approximate Stokes solution for the case of viscous flow past two mutually interacting spheroids (both tandem and parallel arrangements). Later, Kim35 extended this approach to obtain Stokes solution for arbitrarily oriented unequal spheroids. Gluckman et al.36 developed an approximate method called multipole technique to obtain the flow and drag coefficients of multiparticle systems such as monosize spheres and spheroids. Using this technique, Liao and Krueger37 obtained the Stokes solution of two spheroids of different shapes and volume. Finally, on the basis of the aforementioned discussion, it is safe to conclude that there are no results available on recirculation wake interactions and drag characteristics of two tandem spheroid particles beyond the

3. PROBLEM STATEMENT AND DESCRIPTION A Newtonian fluid of density, ρ, and viscosity, μ, is flowing inside a cylindrical tube (of length L and diameter Dt) in which two tandem spheroid particles are fixed in series at upstream and downstream distances of Lu and Ld, respectively. The two tandem spheroid particles are of identical polar and equatorial radii and are located along the axis of the cylindrical tube (Figure 1). The interparticle distance between two particles is S

Figure 1. Schematic representation of the flow past two tandem spheroid particles.

and is defined as the ratio between the distance connecting the centers of particles and the equatorial diameter of particles. To impose the unbounded particles condition, it is assumed that the cylindrical tube is of large diameter and sufficiently long enough to circumvent the entry and exit effects. This is further ascertained by adopting a moving wall boundary condition along the tube wall. Furthermore, the flow is assumed to be steady and axisymmetric; that is, Vz = 0 and no flow variables depends on the z-coordinate. This flow problem is governed by the continuity and Navier−Stokes equations:

Continuity equation (1)

∇·V = 0

Navier−Stokes equation ρ

DV = −∇P + μ∇2 V Dt

(2)

The appropriate boundary conditions for this flow problem can be prescribed as follows: • uniform axial velocity is prescribed at the inlet Vx = V ;

Vy = 0

(3)

• standard no-slip condition is applied on the surfaces of particles Vx = 0; 3187

Vy = 0

(4)

dx.doi.org/10.1021/ie202814w | Ind. Eng. Chem. Res. 2012, 51, 3186−3196


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Table 1. Summary of Domain Independence Study of Two Tandem Spheroid Particles at Re = 1 S=2 Domain

S=4

Cd2

Cd1

Cd

Domain-1 Domain-2

10.978 10.948

9.516 9.504

20.494 20.452

Domain-1 Domain-2

43.155 43.482

36.063 36.361

79.218 79.843

Cd1

Cd2

Aspect Ratio (e) = 0.25 11.252 10.351 11.262 10.353 Aspect Ratio (e) = 2.5 49.500 42.071 49.981 42.521

S=6 Cd

Cd1

Cd2

Cd

21.603 21.615

11.306 11.313

10.657 10.665

21.963 21.978

91.571 92.502

53.552 54.110

46.119 46.663

99.671 100.773

Table 2. Summary of Grid Independence Study of Two Tandem Spheroid Particles at Re = 200 S=2 grid

S=4

Cd2

Cd1

Cd

G1 G2 G3

0.251 0.238 0.238

0.155 0.155 0.155

0.406 0.393 0.393

G1 G2 G3

2.510 2.512 2.510

−0.414 −0.416 −0.414

2.096 2.096 2.096

Cd1

Cd2

Aspect Ratio (e) = 0.25 0.242 0.179 0.240 0.178 0.238 0.179 Aspect Ratio (e) = 2.5 2.193 0.029 2.192 0.031 2.192 0.032

• pressure outlet boundary condition has been used at the outlet so that the conditions upstream of the exit shall not be influenced ∂Vx = 0; ∂x

∂p = 0; ∂x

∂Vy ∂x

=0

Vy = 0

(5)

∂Vx =0 ∂y

(6)

(7)

The governing eqs 1 and 2 along with aforementioned boundary conditions (3−7) have been solved to obtain the steady velocity and pressure fields around two tandem spheroid particles. Finally, the present results are reported as functions of pertinent dimensionless parameters such as the Reynolds number (Re), the total drag coefficient (Cd), the interparticle distance (S), and the aspect ratio (e); and are defined as follows:

Reynolds number Re =

(2b)V ρ μ

(8)

Total drag coefficient Cd =

2Fd ρV 2A p

= Cdp + Cdf (9)

Aspect ratio e = b/a

Cd1

Cd2

Cd

0.421 0.418 0.417

0.238 0.239 0.239

0.193 0.195 0.194

0.431 0.434 0.433

2.222 2.223 2.224

1.974 1.974 1.974

0.499 0.499 0.500

2.473 2.473 2.474

4. NUMERICAL METHODOLOGY The numerical solution of this problem is obtained by solving the continuity and Navier−Stokes equations using commercial software, ANSYS 12, in conjunction with a mesh generating software, Gambit. Since the detailed numerical procedure has been presented elsewhere,18−20 only salient features are presented here. To obtain the steady state velocity and pressure fields for specified values of Re, e, and S, the semiimplicit method for pressure-linked equations (SIMPLE) solution method along with the quadratic upstream interpolation for convective kinematics (QUICK) scheme for convective terms has been adopted. The tolerance residual value for the continuity, x-velocity, and y-velocity momentum equations is set to 10−9. Once the fully converged steady velocity and pressure fields are obtained, these values are further used to evaluate the near object kinematic and dynamic characteristics such as the individual and total drag coefficients of two spheroid particles, streamlines and vorticity contours, and distributions of the pressure coefficient and the vorticity magnitude along the surfaces of tandem spheroid particles. 4.1. Domain Independence Study. The size of the computational domain has significant effect at low Reynolds numbers. Thus, a detailed domain independence study was carried out at Reynolds number, Re = 1, for the case of two tandem spheroid particles of two extreme values of the aspect ratio, that is, for e = 0.25 and e = 2.5 for all values of interparticle distance, S = 2, 4, 6. Table 1 summarizes the total drag coefficient of two particles for different values of the interparticle distance obtained by using two different domains, namely, Domain-1 and Domain-2. Here the sizes of two domains are scaled using the equatorial diameter of the particle

• symmetry boundary condition has been used along the axis of the tube Vy = 0;

Cd

where a and b are equatorial and polar radii of the spheroid, Fd is the drag force acting on the particle, Ap is the projected area of the particle; Cdp and Cdf are the pressure and frictional components of the total drag coefficient. Further, the dimensionless pressure coefficient is defined as 2(Po − P∞)/ (ρV2), where Po is the pressure on the surface of the particle and P∞ is the pressure at the free stream conditions.

• Moving wall boundary condition has been imposed along the tube wall. The wall is imposed to move with a velocity equal to the inlet velocity V Vx = V ;

S=6

(10) 3188



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Table 3. Comparison of Present Values of Cd2/Cd0 with the Experimental Results of Zhu et al.23 at Re = 50 and for Different Values of S

(2a); thus, the size of Domain-1 is Lu = 40, Ld = 80, Dt/2 = 60 and that of Domain-2 is Lu = 30, Ld = 60, Dt/2 = 40. From Table 1, for all values of e and S, the difference between drag coefficients of two spheroid particles obtained by using Domain-1 and Domain-2 is less than ±0.5% indicating that two domains produce almost identical results. However, for the safer side, the larger domain that is, Domain-1 is chosen for all other calculations. 4.2. Grid Independence Study. Since there are no sharp features (such as dramatic changes in pressure drop, sudden change in drag, etc.) in the present study of viscous flows past tandem spheroid particles, the grid resolution study is sufficient to establish the reliability of numerical methodology.38 Further the effect of grid is more significant at large Reynolds numbers because of sharp velocity gradients and very thin boundary layer on the surface of the particle; thus, a detailed grid independence study was carried out at Re = 200 for two tandem spheroid particles. Three different grids G1, G2, G3 with 120, 150, 200 nodes, respectively, on the surfaces of each particle and 25, 50, 75 nodes between the interconnecting axis for S = 2, S = 4, S = 6, respectively, were used. Table 2 shows a summary of the total drag coefficient of two tandem spheroid particles of extreme values of the aspect ratio for e = 0.25 and e = 2.5 at Re = 200 with interparticle distance S = 2, 4, 6. The difference between drag coefficients obtained by different grids is within ±0.5% to each other. Therefore, a moderately fine grid, G2, with 150 nodes on the surfaces of each particle; 80 and 160 nodes along Lu and Ld, respectively, and 40 nodes along inlet and outlet boundaries is used for all other computations of two tandem spheroid particles.

a

S

present

Zhu et al.23a

2 4 6

0.486 0.629 0.702

0.48 0.62 0.72

Obtained from their empirical correlation.

Table 4. Comparison of Present Values of Cd2/Cd0 with the Numerical Results of Prahl et al.31 for S = 6 and for Different Values of Re Re

present

Prahl et al.31

50 100

0.702 0.655

0.70 0.65

present comparisons, it is safe to conclude here that the present results are accurate and reliable within ±2−4%. 5.2. Streamline Patterns. Figure 2 depicts the streamline contours of two tandem spheroid particles of aspect ratio, e = 2.5 (Figure 2a−c), e = 1.5 (Figure 2d−f), and e = 0.5 (Figure 2g−i) at Re = 100 (upper half, solid lines) and Re = 20 (lower half, dashed lines) with interparticle distance S = 2, 4, 6. From Figure 2a−c, for prolate spheroid particles of aspect ratio e = 2.5, the flow separation behind the leading and trailing particles occurs for both values of the Reynolds number and for all values of the interparticle distance because of large frontal area of prolate particles which is acting as resistance to flow. However, the length of the recirculation wake for Re = 100 is much larger than that at Re = 20. Further, for all values of the interparticle distance, at Re = 100, the recirculation wake of the leading particle is interacting with the trailing prolate, and thus, the particles interaction is significant even S = 6. On the other hand, at Re = 20, the recirculation wake of the leading particle does not reach the trailing particle for S = 4, and S = 6 owing to the reduction in recirculation length (because of reduction in the shearing force at smaller values of Reynolds number); however, for S = 2, the recirculation wake of the leading prolate interacts with the trailing prolate. For the case of Re = 20 and S = 4 (Figure 2b, lower half), for the trailing prolate, there is a frontal recirculation wake at the front stagnation point. This can be attributed to the high pressure region in the front of the trailing particle and low pressure region behind the leading particle.32 Furthermore, regardless of values of the Reynolds number and interparticle distance, the length of the recirculation wake behind the trailing particle is much smaller as compared to that in the case of the single unbounded particle19 due to less shear experienced by the trailing particle because of the presence of the leading particle ahead of the trailing particle in the flow direction. For two tandem prolate spheroid particles of aspect ratio e = 1.5 (Figure 2d−f), at Re = 100 (upper half) and Re = 20 (lower half) the interaction between particles is qualitatively similar to the case of prolate particles of aspect ratio e = 2.5 (Figure 2a− c). However, the lengths of the recirculation wake behind two particles are smaller than the case of e = 2.5 because of substantial reduction in the frontal area. In the case of e = 1.5, for all values of the Reynolds number, the recirculation wake of leading particle do not reach up to the front stagnation of the trailing particle for S = 4 (Figure 2(e)) and S = 6 (Figure 2(d)). However, at S = 2 (Figure 2(f)), the wake of leading particle

5. RESULTS AND DISCUSSION To delineate effects of pertinent dimensionless parameters such as the Reynolds number, aspect ratio, and interparticle distance on the flow and drag phenomena of unconfined two tandem spheroid particles, the following range of conditions have been considered: Re = 1, 10, 20, 50, 100, e = 0.25, 0.5, 1, 1.5, 2, 2.5, and S = 2, 4, 6. The main emphasize of this work is on streamline patterns, surface pressure coefficient, and vorticity magnitude distributions, and individual and total drag coefficients for two tandem oblate and prolate spheroid particles of different aspect ratio as functions of the interparticle distance and the Reynolds number. 5.1. Validation. The solution methodology is extensively validated for various cases such as the drag coefficient of unconfined and confined single spherical and spheroid particles at low to moderate Reynolds numbers and presented elsewhere.18 Hence, in this work, additional validations concerning the case of two tandem spherical particles are shown with existing experimental and numerical results in the literature. The total drag coefficient of leading and rear spheroid particles are denoted by Cd1 and Cd2, respectively, and that of single isolated spheroid particle settling in an unbounded fluid is denoted by Cd0. Table 3 shows a comparison of present values of Cd2/Cd0 with those obtained by using the empirical correlation based on experimental results of Zhu et al.23 at Re = 50 and for different values of the interparticle distance, S = 2, 4, 6. The agreement between two results is within ±2.82%. Table 4 shows a comparison of Cd2/ Cd0 values with the numerical results of Prahl et al.31 for S = 6 and at different Reynolds numbers. Here too, the agreement between two studies is excellent and the difference is within ±0.77%. Therefore, on the basis of previous experience18 and 3189



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Figure 2. Streamline contours of two tandem spheroid particles at Re = 100 (upper half, solid lines) and Re = 20 (lower half, dashed lines): (a−c) e = 2.5, (d−f) e = 1.5, and (g−i) e = 0.5.

Figure 3. Effects of Re and e on the distribution of surface pressure coefficient of two tandem spheroid particles with interparticle distance (a−c) S = 2, (d−f) S = 4, and (g−i) S = 6.

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Figure 4. Effects of Re and e on the distribution of surface vorticity magnitude of two tandem spheroid particles with interparticle distance (a−c) S = 2, (d−f) S = 4, and (g−i) S = 6.

trailing particles increases; however, the recirculation wake of the trailing particle is smaller than those of the leading particle and unbounded single particle. For fixed values of the aspect ratio and Reynolds number, as the value of the interparticle distance increases, the effect of the leading particle on the trailing particle decreases. For fixed values of the Reynolds number and interparticle distance, as the value of the aspect ratio increases, lengths of recirculation wakes behind two particles increases. 5.3. Distributions of Pressure Coefficient. Figure 3 shows the distribution of the pressure coefficient on surfaces of and along the connecting central axes between two tandem spheroid particles of aspect ratio e = 0.5, 1.5, 2, and 2.5 with interparticle distance S = 2 (Figure 3a−c), S = 4 (Figure 3d−f) and S = 6 (Figure 3g−i) at different Reynolds numbers. The positions of leading and trailing particles are, for S = 2, from −1.5 to −0.5 (center at −1) and 0.5 to 1.5 (center at 1), respectively. Similarly for S = 4, the positions of two particles are −2.5 to −1.5 (center at −2) and 1.5 to 2.5 (center at 2), respectively, and for S = 6, the positions are −3.5 to −2.5 (center at −3) and 2.5 to 3.5 (center at 3), respectively. At Re = 1 and S = 2 (Figure 3a), for both leading and trailing oblate spheroids, the pressure coefficient is maximum at front

interact with trailing particle for both values of Reynolds number. Further, for e = 1.5, at Re = 20 and for all values of S, there is no formation of recirculation wake behind the trailing particle, whereas recirculation wake is present behind the leading particle. Here too, regardless of values of the Reynolds number and the interparticle distance, the recirculation wake behind the trailing particle is much smaller as compared to unbounded single prolate spheroid of aspect ratio 1.5.19 On the other hand, for two oblate tandem spheroid particles of aspect ratio e = 0.5 (Figure 2(g−i)), altogether there is no flow separation for both the leading and trailing particles at Re = 20. This trend is consistent with the case of isolated unbounded single oblate spheroid particles.19 However, at Re = 100 and for all values of S, there is a very small recirculation wake behind the leading oblate spheroid. Thus, the interaction of neighboring particles is considerably poor, that is, the effect of the interparticle distance has no influence on the recirculation wake behavior of tandem oblate spheroid particles, but it has significant effect on the drag coefficients of two oblate particles. In summary, for fixed values of the aspect ratio (e > 0.5) and the interparticle distance, as the value of the Reynolds number increases, the recirculation wake behind both leading and 3191



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Figure 5. Effects of Re and S on pressure, friction, and total drag coefficients of two tandem spheroid particles of different aspect ratio: (a−c) e = 0.5, (d−f) e = 1.5, (g−i) e = 2.5.

distance, as the value of Reynolds number increases from Re = 1 to Re = 20 and to Re = 100, the pressure recovery decreases along the connecting central axis. Further, regardless the value of the Reynolds number, for oblate spheroids of aspect ratio 0.5, as the value of interparticle distance increases from S = 2 to S = 4 and S = 6, two oblate particles behave as two independent isolated oblates; thus there is nothing like pressure recovery or loss along their connecting central axis. Furthermore, for tandem prolate particles, at all values of Re and S, there is a rise in the pressure coefficient in the front half (circumferential distance from the front stagnation point to the top of the particle) of the trailing particle because of the interaction of the recirculation wake of the leading particle with the trailing particle; and this effect decreases as the value of the aspect ratio increases. For Re = 20, as the value of the interparticle distance increases from S = 2 (Figure 3b) to S = 4 (Figure 3e) and S = 6 (Figure 3h), qualitatively, the distribution of the pressure coefficient of the leading and trailing particles are similar to that is the case of S = 2, but along the central axis, the pressure coefficient increases because of decreased influence of the recirculation wake of the leading particle on the trailing particle by an increase in the value of interparticle distance. At Re = 100 and S = 2 (Figure 3c), S = 4 (Figure 3f), and S = 6 (Figure 3i),

stagnation, it decreases approximately to zero at the top of the oblate and then decreases to a finite minimum negative value all the way up to the rear stagnation point, that is, there is no recovery of pressure at all. Thus along the connecting central axis, for oblates, at Re = 1, the pressure increases from the finite minimum pressure coefficient (corresponding to the pressure coefficient of leading particle at its rear stagnation) to some finite maximum value of pressure coefficient (corresponding to the pressure coefficient at front stagnation of trailing particle). However, for prolate particles (e > 1), even at Re = 1, there is some amount of pressure recovery in the rear half of the leading particle, and the pressure coefficient at the front stagnation of the trailing particle is not maximum (as in the case of isolated prolate particle) but it is at some intermediate level which increases in the front region of trailing particle and then progressively decreases almost up to the rear stagnation point of the trailing particle before relaxing by small pressure recovery. Therefore, at Re = 1 and for S = 2, for tandem prolate particles, as the value of the aspect ratio increases, along the central axis, the pressure recovery is marginal. This can be attributed to the increasing interference between two spheroids with the increase of aspect ratio. On the other hand, for tandem oblate spheroids, irrespective of value of the interparticle 3192



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Table 5. Comparison of Average Total Drag of Two Tandem Spheroids to the Total Drag of an Unconfined Single Spheroid at Finite Reynolds Numbers e = 0.25 Re

Cd,avg

e = 0.5

Cd,avg/Cd0

Cd,avg

e = 1.5

Cd,avg/Cd0

1 10 20 50 100

10.248 1.544 0.922 0.484 0.305

0.901 0.884 0.873 0.855 0.841

14.155 2.160 1.308 0.705 0.456

0.847 0.831 0.817 0.795 0.780

1 10 20 50 100

10.800 1.627 0.974 0.512 0.324

0.950 0.931 0.921 0.905 0.892

15.470 2.348 1.426 0.771 0.499

0.925 0.904 0.891 0.869 0.854

1 10 20 50 100

10.985 1.661 0.997 0.526 0.334

0.966 0.951 0.943 0.930 0.920

15.885 2.418 1.472 0.798 0.518

0.950 0.931 0.919 0.900 0.885

Cd,avg S=2 27.32 4.259 2.642 1.485 0.991 S=4 31.67 4.889 3.036 1.717 1.156 S=6 33.81 5.203 3.244 1.851 1.264

the distribution of the pressure coefficient along the leading particle through the connecting central axis and the trailing particle are qualitatively similar to those at Re = 20. Finally, for all values of the Reynolds number and the aspect ratio, the distribution of the pressure coefficient along the leading spheroid particle is almost similar to that in the case of a single unbounded spheroid particle; however, the distribution of pressure coefficient along the trailing prolate spheroid particle is altogether different from that of the single isolated prolate in an unbounded fluid. 5.4. Distributions of Vorticity Magnitude. Figure 4 shows the combined effects of the aspect ratio, Reynolds number, and interparticle distance on the distribution of vorticity magnitude along the surfaces of two tandem spheroid particles and their connecting central axis. At Re = 1 and 20, for two tandem oblate spheroid particles of aspect ratio 0.5, regardless the values of the interparticle distance, along both leading and trailing particles, the vorticity shoots up right at the front stagnation point, then gradually decreases almost up to the rear stagnation point, and then suddenly drops to minimum value at the rear stagnation point. As the value of the Reynolds number increases to 100, for both the leading and trailing oblate particles, after a sudden rise to a maximum value at front stagnation point, the vorticity magnitude decreases with a small slope almost up to the rear stagnation point before suddenly falling down to a minimum value. For the case of two prolate spheroids of aspect ratio e = 1.5 and S = 2, at Re = 1 (Figure 4b), the vorticity increases from the front stagnation point to a maximum value up to almost the top of the particle and then decreases all the way up to the rear stagnation point. As the value of Re increases to 20, for two prolates of e = 1.5 (Figure 4b), there is a small peak behind the leading particle (because of the recirculation wake) but not behind the trailing particle; however, there is a very small peak at the front stagnation of the trailing particle presumably because of the frontal recirculation wake (as seen in Figure 2). The reason for this trend of vorticity magnitude can be ascribed to that the shear rate gradually increases from the front stagnation point to a certain maximum value corresponding to the position of the peak of

e=2

e = 2.5

Cd,avg/Cd0

Cd,avg

Cd,avg/Cd0

Cd,avg

Cd,avg/Cd0

0.714 0.702 0.680 0.639 0.599

33.48 5.241 3.274 1.862 1.254

0.679 0.666 0.644 0.600 0.557

39.61 6.215 3.904 2.242 1.524

0.653 0.641 0.619 0.574 0.532

0.828 0.806 0.781 0.739 0.700

38.95 6.026 3.755 2.134 1.427

0.789 0.766 0.738 0.687 0.635

45.78 7.107 4.439 2.525 1.685

0.755 0.733 0.703 0.646 0.588

0.883 0.857 0.835 0.797 0.765

42.02 6.483 4.056 2.329 1.590

0.852 0.824 0.797 0.750 0.707

49.84 7.699 4.823 2.771 1.874

0.822 0.794 0.764 0.709 0.654

the primary hump, and then gradually decreases up to the point of separation. The formation of secondary humps in the rear of particles is because of the formation of the recirculation wake as seen in Figure 2. By further increasing Re to 100, in addition to the recirculation wake behind both leading and trailing prolate particles of e = 1.5 (Figure 4b), there is a small peak at the front stagnation of the trailing particle because frontal recirculation occurred at such small interparticle distance and large Reynolds numbers. As the particle aspect ratio increases to 2.5, for S = 2 (Figure 4c), for Re = 20 and Re = 100, qualitatively similar vorticity distribution has been found as in the case of e = 1.5. However, for e = 2.5 and S = 2, at Re = 1 (Figure 4c), there is a very small peak formation behind the leading particle and at the front stagnation of the trailing particle. As the interparticle distance increases to S = 4 (Figure 4d−f) and S = 6 (Figure 4g−i), qualitatively similar trends are observed as in the case of S = 2 for all values of the Reynolds number and aspect ratio. However, for two prolate particles of aspect ratio e = 2.5, at Re = 1, as the interparticle distance increases to S = 4 (Figure 4f) and S = 6 (Figure 4i), there is no sense of any peak neither behind the leading particle nor at the front stagnation of trailing particle unlike in the case of S = 2 (Figure 4c). Furthermore, as the value of the aspect ratio increases, the peaks of primary and secondary humps decrease. On the other hand, irrespective of values of the Reynolds number, for the aspect ratio and interparticle distance, along the connecting central axis, more or less nothing happens; and the heights of primary and secondary humps of leading particles are always larger than those of the trailing particles because of the recirculation wake phenomena seen in Figure 2. In summary, effects of the aspect ratio and interparticle distance have a complicated influence on the vorticity distribution of two tandem spheroid particles particularly at small values of interparticle distance. 5.5. Drag Phenomena. Figure 5 shows the combined effects of the Reynolds number and the interparticle distance on the pressure drag, friction drag, and total drag coefficients of two tandem spheroid particles of aspect ratio e = 0.5 (Figure 5a−c), e = 1.5 (Figure 5d−f) and e = 2.5 (Figure 5g−i). 3193



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Figure 6. Effect of Re and S on the ratio of pressure, friction, and total drag coefficients between two tandem spheroid particles: (a−c) e = 0.5, (d−f) e = 1, (g−i) e = 2.

(Figure 5g−i), the individual and total drag coefficients of the trailing particles are substantially smaller as compared to the leading particle even for large interparticle distance, that is, S = 6. Furthermore, the average drag coefficient (Cd,avg = 0.5[Cd1 + Cd2]) of two tandem spheroid particles is compared to the drag coefficient of an unbounded single spheroid particle (Cd0) at different values of pertinent dimensionless parameters (Table 5). For a fixed value of the interparticle distance, as values of the Reynolds number and aspect ratio increases, the ratio between Cd,avg and Cd0 decreases indicating the average drag on two tandem spheroids is increasing compared to that of a single isolated spheroid particle of the same aspect ratio. For instance, for two oblate spheroids of e = 0.25 separated by a distance of S = 2, at Re = 1, the average drag on two spheroids of ∼10% is larger than Cd0, and it increases to ∼16% as the Reynolds number gradually increases to 100. Similarly, for two tandem prolate spheroids of e = 2.5 separated by a distance of S = 2, the average drag is approximately 35% more than that of the isolated prolate particle of e = 2.5 at Re = 1, and it increases to ∼47% as Re increases to 100. However, for fixed values of the Reynolds number and aspect ratio, as the interparticle distance increases, the ratio between the average drag on two tandem

Regardless of values of the aspect ratio and the interparticle distance, the characteristic Cd versus Re trend was observed, that is, as the value of the Reynolds number increases, the individual and total drag coefficients of both the leading and trailing particles decrease. Irrespective of values of the Reynolds number and interparticle distance, as the value of the aspect ratio increases, the individual and total drag coefficients of the two tandem spheroid particles increase. For fixed values of the Reynolds number, aspect ratio, and interparticle distance, the drag coefficients of leading spheroid particles are very close to those of single unbounded spheroid particles.19 Further, the individual and total drag coefficients of the leading particles are always greater than those of the trailing particles because reduced stresses act on the trailing particles as compared to those on the leading particle. This trend is consistent with existing experimental and numerical results23,31 for the case of two tandem spherical particles, that is, for e = 1. Furthermore, for all values of the Reynolds number and the aspect ratio, as the value of the interparticle distance increases, the individual and total drag coefficients of two particles increase. Qualitatively, this trend is also consistent with existing experimental studies for the case of two tandem spherical particles.25 For two prolate particles of aspect ratio e = 2.5 3194



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interparticle distance increases, the total and individual drag coefficients of two particles increase. The total and individual drag coefficients of leading spheroid particles are very close to those of single unbounded spheroid particles. For all values of the Reynolds number, aspect ratio, and interparticle distance, the individual and total drag coefficients of the leading particles are greater than those of the trailing particles.

spheroids and that of a single isolated particle increases, indicating that the difference between the average drag on two tandem spheroids and the drag on an isolated unbounded spheroid decreases. Figure 6 shows the combined effects of the interparticle distance and Reynolds number on the ratio between individual drag coefficients and the ratio between total drag coefficients of the leading and trailing particles of different aspect ratios. For two oblate tandem particles of aspect ratio e = 0.5 (Figure 6a) with S > 2, the ratio between pressure drag coefficients of the leading and trailing particles increases with increasing Reynolds number. However, for two oblates of e = 0.5 with S = 2, the value of Cdp1/Cdp2 increases as the Reynolds number increases up to Re = 50 and then decreases as Re further increases to 100. This can be attributed to the increasing influence of the leading particle on the trailing particle with increasing Reynolds number and/or decreasing interparticle distance. Further, for all values of the aspect ratio, the ratio between the friction drag coefficients and ratio between the total drag coefficients of two particles increases with increasing Reynolds number and/or decreasing interparticle distance. For the case of two tandem spherical particles, that is, for e = 1 (Figure 6d−f) and for two prolate particles of aspect ratio e = 2 (Figure 6g−i), qualitatively similar behavior can be seen for all values of interparticle distance and Reynolds numbers. However, for e = 2 and S = 2 (Figure 6g−i), these ratios drastically increase beyond Re = 50 because of a strong influence of the wake structure of the leading particle with the trailing particle. Finally, for fixed values of the interparticle distance, the total and individual drag coefficients of both leading and trailing particle decrease as the value of the Reynolds number increases and/or the aspect ratio decreases. Regardless of the values of the Reynolds number and aspect ratio, as the value of the interparticle distance increases, the total and individual drag coefficients of two particles increase. The drag coefficients of leading spheroid particles are very close to those of single unbounded spheroid particles. Further, the individual and total drag coefficients of leading particles are always greater than those of trailing particles.

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AUTHOR INFORMATION

Corresponding Author

*Email: [email protected]. Tel.: +91 361 2582276. Fax: +91 361 2582291. Notes

The authors declare no competing financial interest.

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NOMENCLATURE a = equatorial radius of the spheroid particle, m Ap = area of the particle, m2 b = polar radius of the spheroid particle, m Cd = total drag coefficient, dimensionless Cd0 = total drag coefficient of single isolated particle, dimensionless Cd,avg = average total drag coefficient of two spheroids, dimensionless Cdf = friction drag coefficient, dimensionless Cdp = pressure drag coefficient, dimensionless Dt = diameter of the tube, m e = aspect ratio, dimensionless Fd = drag force, N L = length of the tube, dimensionless Ld = downstream distance, dimensionless Lu = upstream distance, dimensionless P = pressure, Pa Re = Reynolds number, dimensionless S = interparticle distance, dimensionless V = velocity vector, m/s

Greek Symbols

μ = viscosity of the fluid, kg/m.s ρ = density of the fluid, kg/m3

6. CONCLUSIONS The flow and drag phenomena of two tandem spheroid particles are found to be strong functions of pertinent dimensionless parameters such as the Reynolds number, aspect ratio, and interparticle distance. For two tandem prolate particles of large aspect ratio (say e > 2), at large Reynolds numbers, the recirculation wake of the leading particle is interacting with the trailing particle, and thus, the particles interaction is significant even for S = 6; however, the interaction of particles is stronger for small values of the interparticle distance. On the other hand, for two oblate tandem spheroid particles, the influence of neighboring particles is considerably poor as compared to that of tandem spheres and prolate spheroids. The effects of neighboring particles on the distribution of the pressure coefficient and the vorticity magnitude along the surfaces of particles and their connecting central axis are strongly influenced for small values of the interparticle distance and large values of the Reynolds number. For all values of the interparticle distance, the total and individual drag coefficients of both the leading and the trailing particle decrease as the value of the Reynolds number increases and/or the aspect ratio decreases. Irrespective of values of the Reynolds number and aspect ratio, as the value of the

Subscripts

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x = x-component y = y-component 1 = spheroid-1 2 = spheroid-2

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Flow and Drag Phenomena of Tandem Spheroid Particles at Finite

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