An ellipsoidal particle in tube Poiseuille flow

Transcription

1 J. Fluid Mech. (217), vol. 822, pp c Cambridge Universit Press 217 doi:1.117/jfm An ellipsoidal particle in tube Poiseuille flow 664 Haibo Huang 1, and Xi-Yun Lu 1 1 Department of Modern Mechanics, Universit of Science and Technolog of China, Hefei, Anhui 2326, China (Received 15 September 216; revised 24 April 217; accepted 4 Ma 217) A suspended ellipsoidal particle inside a Poiseuille flow with Renolds number up to 36 is studied numericall. The effects of tube diameter (D), inertia of the particle and the flow, and the particle geometr (both prolate and oblate ellipsoids) are considered. When a prolate particle with a/b = 2 is inside a wider tube (e.g. D/A > 1.9), where A = 2a is the length of the major ais of the particle, the terminal stable state is tumbling. When the prolate particle is inside a narrower tube (1. < D/A < 1.9), log-rolling or kaaking modes ma appear. Which mode occurs depends on the competition between fluid and particle inertia. When the fluid inertia is dominant, the log-rolling mode appears, otherwise, the kaaking mode appears. Inclined and spiral modes ma appear when D/A < 1 and D/A = 1, respectivel. For a prolate ellipsoid with a/b = 4, if 1 < D/A < 1.9, there is onl the kaaking mode and the log-rolling mode is not observed. When an oblate particle is inside a wider tube (e.g. D/A > 3.5), it ma adopt the log-rolling mode. Inclined and intermediate modes are firstl identified in narrower tubes. The phase diagram of the modes is also provided. The modes in the phase diagrams were not found to be affected b the initial state of the particle based on limited observation. Ke words: particle/fluid flow, sediment transport, suspensions 1. Introduction The motion of particles in tubes is ubiquitous in nature and man applications in industries, such as chemical, biological, and mechanical engineering. Man studies on the motion of particles in simple flows have been carried out, such as particles rotational behaviours in Couette flow (Jeffer 1922; Aidun, Lu & DING 1998; Ding & Aidun 2; Qi & Luo 23; Yu, Phan-Thien & Tanner 27; Huang et al. 212b) and sedimentation of particles inside tubes (Xia et al. 29; Huang, Yang & Lu 214). Rosén, Lundell & Aidun (214) investigated the rotational mode of a prolate ellipsoid suspended in a shear flow. Log-rolling, tumbling (Jeffer 1922), inclined rolling, kaaking, inclined kaaking, and stead modes (Qi & Luo 23; Yu et al. 27) are found. In the log-rolling mode, the particle rotates with the evolution ais aligned with the vorticit. In the tumbling mode, the particle rotates with the evolution ais in the flow-gradient plane. For the kaaking mode, the particle performs both precession and nutation around the vorticit ais. address for correspondence: huanghb@ustc.edu.cn

2 An ellipsoidal particle in tube Poiseuille flow 665 The rotation of a neutrall buoant oblate spheroid in a shear flow at small shear Renolds number was studied using the lattice Boltmann method (LBM) (Rosén et al. 215). At small shear Renolds number Re a O(1), the LBM result predicts a bifurcation of the tumbling orbit at aspect ratio λ c.1275, below which tumbling is stable (as well as log-rolling). The value is in qualitative agreement with the analtical results λ c.137, which is derived from an unbounded sstem at infinitesimal Re a (Einarsson et al. 215). Usuall the LBM has a second-order accurac in both space and time (Mei, Luo & Sh 1999). Rosén et al. (215) mentioned that to pinpoint the critical parameter values λ c, more accurate methods are necessar and a high-accurac computational fluid dnamics solver, i.e. the commercial software package STAR-CCM+ is recommended. The inertial effects of fluids and particles on a prolate spheroidal particle in simple shear flow have been investigated b Rosén et al. (216). The showed that the dnamics of the rotational motion can be quantitativel analsed through the eigenvalues of the log-rolling particle. It is also found that the effect on the orientational dnamics from fluid inertia can be modelled with a Duffing Van der Pol oscillator (Rosén et al. 216). Particle dnamics in viscoelastic liquids, such as single particle, two particles, and multiple particles in shear flow, Couette flow, and Poiseuille flow, also has been investigated etensivel (see a recent review article b D Avino & Maffettone (215)). Rheolog of a dilute viscoelastic suspension of spheroids in unconfined shear flow was studied b D Avino, Greco & Maffettone (215). Taking into account the effects of the initial orientations of the particle and confined flow geometries, the dnamics of a neo-hookean elastic prolate spheroid suspended in Newtonian fluid under shear flow was also studied (Villone et al. 215). The above flows involve either solid particles suspension in non-newtonian fluid or a elastic prolate spheroid suspended in a Newtonian fluid. However, here we limited our stud on onl the Newtonian fluid instead of the viscoelastic fluid. The particle is limited to be solid instead of elastic. Segre and Silberberg first studied the migration of neutrall buoant spherical particles in Poiseuille flows eperimentall and found that the particles migrate towards an equilibrium position and equilibrate at a distance of.6 times the radius of the tube from the tube s centre (Segre & Silberberg 1961). On one hand, the particle eperiences the Magnus effect due to rotation of the particle. The rotation of the particle is induced b the shear stress in the Poiseuille flow. When the particle migrates radiall to the wall, the fluid between the wall and the particle is squeeed, and conversel the particle will eperience high pressure on the side facing the wall to prevent it reaching the wall. Hence, the particle will seek an equilibrium position between the ais and the wall where the total radial force is ero. Some numerical studies on an ellipsoid in a two-dimensional (2D) Poiseuille flow (Feng, Hu & Joseph 1994; Qi et al. 22) have been carried out. According to the stud of Feng et al. (1994), a neutrall buoant particle ehibits the Segre Silberberg effect in a Poiseuille flow. The driving forces of the migration have been identified as a wall repulsion due to lubrication, an inertial lift related to shear slip, a lift due to particle rotation and the velocit profile curvature. However, in realit the behaviour of a three-dimensional (3D) ellipsoid in a tube flow ma be ver different from that in the 2D cases. There are numerous studies on neutrall buoant spherical particles in tube flows (Zhu 2; Yu, Phan-Thien & Tanner 24; Yang et al. 25). Yang et al. (25) used a method of constrained simulation to obtain correlation formulas for the lift force, slip velocit, and equilibrium position. Yu et al. (24) studied particle

3 666 H. Huang and X.-Y. Lu migration in a Poiseuille flow using a finite-difference-based distributed Lagrange multiplier/fictitious domain method (DLM/FD) method. Both non-neutrall and almost neutrall buoant cases were investigated. The found that the suppression of the sphere rotation produces significant large additional lift forces pointing towards the tube ais on the spheres in the neutrall buoant cases. A general technique based on the LBM for simulating solid fluid suspensions was proposed b Ladd (1994a). Besides the spherical particles, the behaviours of a non-spherical particle in tubes flows also attract much attention. Karnis, Goldsmith & Mason (1966) studied the migration of non-spherical particles in tubes through eperiments. The observed that for a rod-like particle, the major ais of the particle rotates on the plane passing though the centre of the particle and the tube ais (tumbling state), while for a disk-like particle, it rotates with its minor ais perpendicular to the same plane (log-rolling state). Beon, Seo & Lee (215) also have shown that a prolate ellipsoid ma adopt the tumbling state in the Poiseuille flow in their eperiment. Sugihara-Seki (1996) numericall studied the motions of an inertialess elliptical particle in tube Poiseuille flow using a finite element (FE) method. A prolate spheroid is found to either tumble or oscillate in rotation, depending on the particle tube sie ratio, the ais ratio of the particle, and the initial conditions. A large oblate spheroid ma approach asmptoticall a stead, stable slightl inclined configuration, at which it is located close to the tube centreline. However, in the paper the consider onl the motion where two of the three principal aes of the ellipsoid lie in a plane containing the tube ais and the fluid motion is assumed to be smmetric with respect to this plane. On the other hand, the inertia of the particle, which is ver important in this problem, is neglected. Hence, it is onl a starting point for the analsis of the general motion of an ellipsoid in tube flows. Pan, Chang & Glowinski (28) simulated the motion of a neutrall buoant ellipsoid in a tube Poiseuille flow and investigated its rotation and migration behaviour inside circular tubes. The found its rotation ehibits distinctive states depending on the Renolds number ranges and the shape of particle. However, the stud onl considered circular tubes with fied R/a 2.5, where R is the tube s radius and a is the semi-major ais of the ellipsoid. For the prolate spheroid, a/b = 3, where b is the semi-minor ais. The tube length is short (the length is onl four times the radius R of the clinder) and a uniform pressure gradient along the tube is applied. For cases with Re = 5.4, the prolate ellipsoid s major ais rotates on the plane passing through the clinder ais and its centre of mass. This behaviour is similar to the eperimental results of the rod-like particle moving and rotating in the Poiseuille flow reported in Karnis et al. (1966). This is called the tumbling mode. For cases with Re = and 5.9, besides the tumbling state, the prolate ellipsoid ma ehibit the second different rotational behaviour (log-rolling mode), which depends on the initial orientation and positions. In the log-rolling mode, after reaching its equilibrium distance to the central ais of the tube, the prolate ellipsoid is rotating with respect to its major ais (the evolution ais), which is perpendicular to the plane passing through the central ais of the tube and its centre of mass. The log-rolling state was not reported in Karnis et al. (1966). The bifurcation phenomena ma be attributed to the boundar condition applied in their simulations (Pan et al. 28), which is mentioned above. For the oblate ellipsoid and R/a 3.3, the oblate ellipsoid rotates with its minor ais perpendicular to the plane passing through the central ais of the tube and the centre of mass of the disk. That is a log-rolling mode for Re < 81. The behaviour is similar to the eperimental results for the disk-like particle moving and rotating in the Poiseuille flow reported in Karnis et al. (1966).

4 An ellipsoidal particle in tube Poiseuille flow 667 In this paper, the migration and rotation behaviours of an ellipsoid inside different narrow circular tubes have been investigated. Cases with Renolds number up to 36 were simulated. Onl the cases of suspended particles are investigated, i.e. the gravit effect is neglected. Hence, the main emphasis in this work is to stud the effects of the wall boundar and inertia on the ellipsoid behaviours in circular tube flows. The paper is organied as follows. In 2, the multiple-relaation-time (MRT) LBM and basic equations for the motion of the solid particle are briefl introduced. The flow problem is described in 3. The identified motion modes for a prolate spheroid are discussed in 4. The inertial effect is shown in 5. The motion modes for an oblate spheroid are discussed in 6. Finall, some concluding remarks are given in Numerical method 2.1. Multiple-relaation-time (MRT) lattice Boltmann method The MRT-LBM (Lallemand & Luo 23) is used to solve the fluid flow governed b the incompressible Navier Stokes equations. The lattice Boltmann equations (LBE) (d Humiéres et al. 22) can be written as f ( + e i δt, t + δt) f (, t) = M 1 Ŝ[ m(, t) m eq (, t) ], (2.1) where the Dirac notation of ket vectors smbolie the column vectors. f (, t) represents the particle distribution function, which has 19 components f i with i =, 1,..., 18 because of the D3Q19 model used in our 3D simulations. The collision matri Ŝ = M S M 1 is diagonal with Ŝ (, s 1, s 2,, s 4,, s 4,, s 4, s 9, s 1, s 9, s 1, s 13, s 13, s 13, s 16, s 16, s 16 ), (2.2) where the parameters of Ŝ are chosen as (d Humiéres et al. 22): s 1 = 1.19, s 2 = s 1 = 1.4, s 4 = 1.2, s 9 = 1/τ, s 13 = s 9, s 16 = m eq is the equilibrium value of the moment m, where the moment m = M f, i.e. f = M 1 m. M is a linear transformation matri which is used to map the column vectors f in discrete velocit space to the column vectors m in moment space. The matri M and m eq are the same as those used b d Humiéres et al. (22) and Huang et al. (212b). In (2.1), e i are the discrete velocities. For the D3Q19 velocit model, e i = c , (2.3) where c is the lattice speed, defined as c = / t. In our stud = 1lu and t = 1ts, where lu and ts represent the lattice unit and time step, respectivel. mu is used to denote the mass unit. The macro-variables of fluid flow can be obtained from ρ = i f i, ρu ζ = i f i e iζ, p = c 2 sρ, (2.4a c) where subscript ζ denotes three coordinates. The parameter τ is related to the kinematic viscosit of the fluid: ν = c 2 s (τ.5) t, where c s = c/ 3 is the sound speed. The numerical method used in our stud is based on the MRT-LBM (d Humiéres et al. 22) and the dnamic multi-block strateg (Huang et al. 214).

5 668 H. Huang and X.-Y. Lu (a) (b) FIGURE 1. (Colour online) Schematic diagram of the combination of coordinate transformation from (,, ) to (,, ) with three Euler angles (ϕ, θ, ψ). Line ON represents the pitch line of the (, ) and (, ) coordinate planes. Two coordinate sstems are overlapping initiall. First the particle rotates around the ais with a recession angle ϕ and then the particle rotates around the new ais (i.e. line ON ) with a nutation angle θ. Finall the particle rotates around the new ais with an angle of rotation ψ Solid particle dnamics and fluid solid boundar interaction In our simulation, the ellipsoidal particle is described b 2 a b + 2 = 1, (2.5) 2 c2 where a, b and c are the lengths of the three semi-principal aes of the particle in the, and ais of a bod-fied coordinate sstem, respectivel (see figure 1). The aspect ratio of the ellipsoidal particle is defined as a/b. The bod-fied coordinate sstem can be obtained b a combination of coordinate transformation around the ais with Euler angles (ϕ, θ, ψ) from the space-fied coordinate sstem (,, ) which initiall overlaps the bod-fied coordinate sstem. The combination of coordinate transformation is illustrated in figure 1. The evolution ais alwas overlaps the direction. The migration and rotation of the particle are determined b the Newton equation and Euler equation, respectivel, I dω(t) dt m du(t) dt = F(t), (2.6) + Ω(t) [I Ω(t)] = T(t), (2.7) where I is the inertial tensor, and Ω(t) and T(t) represent the angular velocit and the torque eerted on the particle in the bod-fied coordinate sstem, respectivel. In the frame, I is diagonal and the principal moments of inertia can be written as I = + c 2 mb2, I 5 = + a 2 mc2, I 5 = + b 2 ma2, (2.8a c) 5 where m = 4/3ρ p πabc is the mass of the particle and ρ p is the densit of the particle. It is not appropriate to solve (2.7) directl due to an inherent singularit (Qi 1999). Thus four quaternion parameters are used as generalied coordinates to solve the corresponding sstem of equations (Qi & Luo 23). A coordinate transformation matri with four quaternion parameters (Qi & Luo 23) is applied

6 An ellipsoidal particle in tube Poiseuille flow 669 to transform corresponding items from the space-fied coordinate sstem to the bod-fied coordinate sstem. With four quaternion parameters, equation (2.7) can be solved using a fourth-order-accurate Runge Kutta integration procedure (Huang et al. 212b). In the simulations, the fluid solid boundar interaction is based on the schemes of Aidun et al. (1998) and Lallemand & Luo (23). The accurate moving-boundar treatment proposed b Lallemand & Luo (23) is applied to solve the problem caused b the moving curved-wall boundar condition of the ellipsoid. The general schemes for calculation of interactive force between fluids and particles in the LBM include stress integration, momentum echange and volume fraction models, which has been summaried and analsed b Chen et al. (213). The stress integration scheme ma be good but it is not so efficient (Chen et al. 213). Here the momentum echange scheme is used to calculate the force eerted on the solid boundar, which is accurate and efficient (Chen et al. 213). The forces due to the fluid nodes covered b the solid nodes and the solid nodes covered b the fluid nodes (Aidun et al. 1998) are also considered in the stud. To prevent overlap of the particle and the wall, usuall the repulsive force between the wall and particle should be applied (Huang et al. 214). Here the lubrication force model is identical to that we used in Huang et al. (214) and the validation of the force model has been tested etensivel b Huang et al. (214). Our previous stud on the tumbling mode of an ellipsoidal particle suspended in shear flow partiall validated our three-dimensional LBM code (Huang, Wu & Lu 212a) because the rotational period or orbit is ver consistent with Jeffer s analtical solution (Jeffer 1922). The LBM code has also been validated b Yang, Huang & Lu (215) for the case of migration of a neutrall buoant sphere in a tube Poiseuille flow. The lift force, angular and migration velocities agree well with those in Yang et al. (25). Here three more cases are simulated to validate our LBM code. The first one concerns migrations of a neutrall buoant sphere in tube Poiseuille flows (Karnis et al. 1966). The second deals with an ellipsoid particle sedimentation in a vertical circular tube under gravit. The third one describes the rotation of a prolate spheroid in shear flow. The result is shown in appendi A. The accurac of the LBM is also investigated in Flow problem The motion of a neutrall buoant ellipsoid inside tube flow is illustrated in figure 2, where D = 2R denotes the diameter of the circular tube. In the problem, two kinds of ellipsoids prolate and oblate particles are considered. Particle sies a = 2b = 2c for the prolate particle and a = b/2 = c/2 for the oblate particle are considered. The ais is the evolution ais and it overlaps the major ais of the prolate particle and the minor ais of the oblate particle. To describe the orientation of the particle, α, β, and γ are used to denote the angles between the -ais and the space-fied coordinates -, -, and -aes, respectivel (cos 2 α + cos 2 β + cos 2 γ = 1). The Renolds number (Re) is defined as Re = AU m ν, (3.1) where U m is the central velocit of the flow without the particle and A is the length of the major ais. For the prolate and oblate ellipsoid A = 2a and A = 2b, respectivel. The confinement ratio is D/A. It is noted that there is no gravit in this flow problem.

7 67 H. Huang and X.-Y. Lu FIGURE 2. (Colour online) Schematic diagram of a prolate ellipsoid in Poiseuille flow. The Poiseuille flow is in the -direction. Case A (lu) τ t p = mu/(lu ts 2 ) A A A B B B C TABLE 1. Grid independence and time-step independence studies for a prolate ellipsoid case with Re = 162, D/A = 2, ρ p /ρ f = 1, and tube length L = 8D. The initial orientation is (45, 9, 45 ), and the initial position (, ) = (,.5). For all cases in our stud, the fluid densit is ρ f = 1mu/lu 3 and the length of the tube is L = 8D for Re < 2, For higher-re cases, e.g. Re > 2, a longer computational domain, e.g. L = 12D, is adopted. In this wa, the effects of the inlet/outlet boundar conditions are minimied. In most simulations, the major ais of the ellipsoidal particle is represented b 6lu, i.e. A = 6lu and τ f =.6. For eample, in the case of D/A = 1.2, the total mesh is approimatel 76lu 76lu 68lu. The grid independence stud and time-step independence stud have been performed in 3.1 and it is shown that the mesh sie and the time step are sufficient to obtain accurate results Accurac of the LBM The accurac of the LBM is investigated here. Grid independence and time-step independence studies are performed. As an eample, cases of a prolate ellipsoid with Re = 162 and D/A = 2 were simulated. The ke parameters for the cases are shown in table 1, where t is a normalied time and p is the pressure difference between the two ends of the tube. The corresponding results for grid independence and time-step independence are shown in figure 3. In figure 3(a), the grid sie is labelled, e.g. the legend A = 4lu means 4 grids are used to discretie the particle s major ais. From figure 3(a), it is seen that the result of A = 6lu is ver close to that of A = 72lu. However, for the coarse mesh, i.e. A = 4lu, the period of the

8 An ellipsoidal particle in tube Poiseuille flow 671 (a) lu (b) (c) Spatial accurac (d) Temporal accurac FIGURE 3. (Colour online) Effect of grid sie (a) and time step (b) on the orientation of the prolate ellipsoid (cos γ ). The parameters for the cases are shown in table 1. Log log plots of the error described in (3.2) as a function of grid sie (c) and time step (d), respectivel. rotation has a significant discrepanc with respect to that of A = 72lu. The grid sie with A = 6lu seems small enough to obtain an accurate result. Hence, for cases with Re O(1), the grid sie with A = 6lu is used. For the time-step independence stud, three cases with τ =.575,.6,.64 were simulated (see table 1). The corresponding t = tν/a 2 = , , , respectivel. It is seen from figure 3(b) that the curve for τ =.6 is ver close to that for τ =.575. Hence, the time step with τ =.6 is small enough to get accurate results. To test the accurac of the present numerical method, the case with the finest mesh A = 9lu and the smallest time step t = (Case C) is also simulated. The result is referred to as the accurate result. The cos γ of Case A3 is ver close to that of Case C. Since the error can var with time, a time average over one period is required. After the tumbling state of the ellipsoid reaches a stable periodic state at t 1 with a period of T, the relative error is defined as E 2 = where cos γ (t) is the accurate result. t1 +T [cos γ (t) cos γ (t)] 2 dt t 1 t1 +T [cos γ (t)] 2 dt t 1, (3.2)

9 672 H. Huang and X.-Y. Lu 2 15 Kaaking Log-rolling Spiral Tumbling Inclined Re FIGURE 4. (Colour online) Phase diagram for a prolate ellipsoid (a/b = 2) inside Poiseuille flow (ρ p /ρ f = 1). Group I Group II Group III Group IV (ϕ, θ, ψ ) (ϕ, θ, ψ ) (ϕ, θ, ψ ) (ϕ, θ, ψ ) (, 9, ) (3, 9, ) (6, 9, ) (9, 9, ) (, 9, 3 ) (3, 9, 3 ) (6, 9, 3 ) (9, 9, 3 ) (, 9, 6 ) (3, 9, 6 ) (6, 9, 6 ) (9, 9, 6 ) TABLE 2. Tpical initial orientations. The least-squares data fit in figures 3(c) and 3(d) shows that the fitted slopes are 2.6 and 1.93, respectivel. Hence, the present LBM solver has an approimatel second-order accurac in both space and time. That is consistent with the conclusion on accurac of the LBM in Mei et al. (1999). 4. Motion mode of a prolate spheroid Figure 4 shows the motion mode distribution in the D/A Re plane for ρ p /ρ f = 1. For each point in the figure ecept the cases with D/A 1, at least eight tpical cases with different initial positions and orientations were simulated. In four of the eight tpical cases, we picked one initial orientation from each group listed in table 2 and the initial position is (, ) = (, ). In the other four cases, the initial orientation is chosen similarl, but (, ) = (,.1), i.e. the particle is placed slightl awa from the tube ais. It is found that the terminal rotational mode does not depend on the initial conditions. From figure 4, it is seen that the confinement of the tube plas a critical role for motion mode distribution. When R < a, the prolate ellipsoid can onl adopt the inclined mode to pass through the tube. When R = a, the spiral mode as shown in figure 5(a) is identified. In the inclined mode, the -ais is inclined inside an ai-smmetric plane. When R 1.9a, the confinement effect is completel diminished,

10 An ellipsoidal particle in tube Poiseuille flow 673 (a) (b) (c) (d) FIGURE 5. (Colour online) Motion modes of a prolate ellipsoid in Poiseuille flow. (a) Spiral mode, (b) kaaking mode, (c) log-rolling mode, (d) tumbling mode. The upper image is the side view and the lower image is the corresponding top view. It is noted in the Poiseuille flow the particle is not onl rotating but also moving along with the flow, which is in the -direction. the particle alwas adopts the tumbling mode in figure 5(d), which is independent of the Renolds number (Re < 2). For 1. < D/A < 1.9, when Re is small (e.g. Re = 3), the prolate ellipsoid adopts the kaaking mode (see figure 5b). Suppose the -ais passing through the particle centre and perpendicular to the aismmetric plane, which passes through the tube ais and the centre of the particle. The kaaking mode adopts an intermediate orbit, where the particle performs both precession and nutation around the -ais, resembling the motion of a kaak paddle (Rosén et al. 214). In this confinement regime (1. < D/A < 1.9), the fluid inertial effect is also important. When Re is larger, the particle adopts the log-rolling mode (see figure 5c) instead of the kaaking mode. 5. Inertial effect 5.1. Particle s rotational energ and inertial effect of the fluid In this section we mainl discuss both inertial effects of the fluid and the particle. First, the effect of fluid inertia on the particle behaviour is discussed. Figure 6 shows cos γ ma as a function of Re. γ is the angle between the -ais and the -direction. cos(γ ) ma represents the maimum cos γ in one period in the kaaking mode (see figure 5b). It quantifies the etent of deviation from the log-rolling state. When the particle is in the log-rolling state, cos γ ma should be equal to ero because the major ais ( -ais) is perpendicular to the -direction. It is seen from figure 6 that for cases D/A = 1.7, ρ p /ρ f = 1, cos γ ma is not continuous as a result of the mode transition. When Re is less than 8, cos γ ma.9, the ellipsoid is in the kaaking state. As Re increases above 9, the ellipsoid adopts the log-rolling mode. Hence, as the inertia of the fluid increases, the kaaking

11 674 H. Huang and X.-Y. Lu Kaaking Log-rolling Re 18 FIGURE 6. (Colour online) Transition from the kaaking mode (the shaded area) to the log-rolling mode. cos γ ma and normalied angular velocit ω as functions of Re for D/A = 1.7, ρ p /ρ f = 1. mode ma transit to the log-rolling mode. This character can be understood as follows. When Re increases, both the shear stress acting on the particle and the inertia of the fluid increase. The larger shear stress on the particle increases the angular velocit of the particle (ω ) around the -ais (see figure 6). The energ of the rotational bod is (I ω2 )/2. Hence the rotational energ increases. On the other hand, the larger fluid inertia ma prevent the nutation or the flapping-like movement. Both the increasing rotational energ and fluid inertia are believed to assist the particle reach the log-rolling state without nutation. Although ω increases with Re, the equilibrium radial position r = r/r is approimatel.46, and increases onl slightl with Re (not shown). The ellipsoidal particle with aspect ratio a/b = 4 is also considered in our stud. The phase diagram for a prolate ellipsoid with a/b = 4 inside Poiseuille flow is shown in figure 7. For D/A > 1.9, the tumbling mode still occurs. For 1 < D/A < 1.9, there is onl the kaaking mode and the log-rolling mode is not observed. The significant difference between figures 4 and 7 lies in the log-rolling mode. The disappearance of the log-rolling mode in figure 7 ma be attributed to the significantl smaller rotational energ in cases with a/b = 4. A tpical eample is presented below. Suppose two cases D1 and D2 have identical ke parameters D/A = 1.5, Re = 78.4, ρ p /ρ f = 1, a = 3lu, and τ =.6, but in Case D1, b = c = a/2 and in Case D2, b = c = a/4. That means in Cases D1 and D2, the particle aspect ratios are 2 and 4, respectivel. It is seen from figures 4 and 7 that the particles in Cases D1 and D2 adopt the log-rolling and kaaking modes, respectivel. The behaviour ma be understood as follows. In Case D2 the particle is more rod-like and the principal moment of inertia I in Case D2 is onl 1/16 of that in Case D1 because I = m((b2 + c 2 )/5) = 4/3ρ p πabc((b 2 + c 2 )/5). In the equilibrium state, the normalied average angular velocities ω = (ω A2 )/ν for Cases D1 and D2 are 36.5 and 22.3, respectivel. It is seen that the energ of the rotational bod (I ω2 )/2 in Case D2 would be much smaller than that in Case D1. A smaller energ of the rotational bod ma induce nutation motion when it rotates around the -ais and

12 An ellipsoidal particle in tube Poiseuille flow Kaaking Spiral Tumbling Inclined 15 Re 1 L FIGURE 7. (Colour online) Phase diagram for a prolate ellipsoid (a/b = 4) inside Poiseuille flow with ρ p /ρ f = 1. interacts with fluid. Hence, the particle in Case D2 more easil adopts the kaaking state instead of the log-rolling state. Again, we see that the increased rotational energ indeed assists the particle reaching the log-rolling state without nutation Inertial effect of the particle To investigate the inertial effect of the particle, twelve cases listed in table 3 were simulated. The are classified into four groups. Here ρ p /ρ f is not limited to be unit, i.e. the cases with different ρ p /ρ f were performed. Group I represents a wide tube case with D/A = 3. Groups III and IV denote narrow tube cases (D/A = 1.5). Group II denotes cases with D/A = 1.9. The results of these cases with different densit ratios are presented in table 3 and figure 8(b,c). In Group I, onl the tumbling state is observed. The normalied tumbling rotational angular velocit increases with the densit ratio ρ p /ρ f. The particle s lateral migration is slightl closer to the wall as ρ p /ρ f increases. Specificall, for the case with ρ p /ρ f = 3., the lateral migration is periodicall oscillating and r (.566,.57), i.e. the centre of mass of the ellipsoid takes a wav trajector in the radial direction inside an ai-smmetric plane. Group II in table 3 shows that for the tube D/A = 1.9, again onl the tumbling mode is reproduced. In the tumbling mode, the radial position of the particle s centre of mass is continuousl oscillating periodicall. The magnitude of the oscillation increases with the densit ratio ρ p /ρ f. It is seen that when ρ p /ρ f =.3, the average r =.442 with r (.412,.47), the oscillation magnitude is δr = =.58. When ρ p /ρ f = 1 and ρ p /ρ f = 3., it oscillates with δr =.84 and δr =.163, respectivel, in the radial direction. The oscillation in the radial direction is attributed to the particle s inertia and the strong interaction between the particle and the tube wall. For a specific narrow tube, the magnitude of the radial oscillation increases with the inertia of the particle.

13 676 H. Huang and X.-Y. Lu Group D A Re ρ p Mode Average ω = ω A2 ρ f ν r Average r cos γ ma tumbling N/A N/A I tumbling N/A N/A tumbling (.566,.57).569 N/A tumbling (.412,.47).442 N/A II tumbling (.384,.468).421 N/A tumbling (.344,.57).49 N/A log-rolling N/A N/A log-rolling N/A N/A III kaaking N/A kaaking 2.74 (.42,.449) kaaking 2.64 (.44,.45) log-rolling N/A N/A log-rolling N/A N/A IV log-rolling N/A N/A kaaking (.424,.441) kaaking (.414,.44) TABLE 3. The effect of the inertia of the particle, r = r/r is the equilibrium radial position. cos γ ma is onl applicable for the kaaking mode. (a) Log-rolling (b) Re Tumbling Kaaking Log-rolling Tumbling Re (c) 25 2 Kaaking Re FIGURE 8. (Colour online) (a) Phase diagram in a three-dimensional parameter space. (b,c) Phase diagram in the planes ρ p /ρ f = 3. and ρ p /ρ f =.3, respectivel. Hence, for D/A 1.9, the motion mode is not affected b the particle s inertia (ρ p /ρ f 3). Figure 8 shows the phase diagram in a three-dimensional parameter space. As we do not intend to provide accurate borders between the modes, it is onl a schematic diagram. From figure 8, it is seen that the tumbling mode is still in the

14 An ellipsoidal particle in tube Poiseuille flow Log-rolling mode Inclined mode Intermediate Re FIGURE 9. (Colour online) Phase diagram for an oblate ellipsoid inside Poiseuille flow (ρ p /ρ f = 1). right part of the space (D/A 1.9), which is independent of both Re and ρ p /ρ f under our considered parameter space. Groups III and IV in table 3 are the cases in narrower tubes but with lower and higher Re, respectivel. For the lower Re cases (Group III), in the narrower tubes (D/A = 1.5) the particle adopts the log-rolling mode when ρ p /ρ f =.3 and.6, while ρ p /ρ f = 1., 2. and 3., the state is kaaking. For higher Re cases (Group IV), the particle adopts the log-rolling mode at ρ p /ρ f =.3,.6 and 1., but when ρ p /ρ f = 2. and 3., it still takes the kaaking mode. Hence, for a constant Re, the kaaking mode is preferred when the particle s inertia increases. Hence, for D/A < 1.9, the mode distribution would be affected b the particle s inertia (ρ p /ρ f ). The border separating the kaaking and log-rolling modes ma move upwards or downwards depending on the particle s inertia. Figure 8 shows the approimate result. If the planes with constant ρ p /ρ f are viewed from front to back, the border moves upwards. It means that on a plane with higher ρ p /ρ f, the area of the kaaking mode increases. In other words, due to the inertial effect, the kaaking mode becomes more common. 6. Motion modes of an oblate ellipsoid In this section, the motion modes for an oblate ellipsoid are discussed. Compared to the motion modes of the prolate ellipsoid, the phase diagram for the motion mode of the ellipsoid, i.e. figure 9, is simpler. In figure 9, onl cases with D > A are considered because the oblate ellipsoid is unable to pass through the circular tube when D A. For each point in the figure, similar to that in 4, at least eight tpical cases with different initial positions and orientations were simulated. It is not found that the terminal rotational mode depends on the initial conditions. Figure 9 shows that the oblate ellipsoid alwas adopts the log-rolling mode in wide tubes (e.g. D/A > 3.2). The log-rolling mode is shown in figure 1(a). In this mode,

15 678 H. Huang and X.-Y. Lu (a) (b) FIGURE 1. (Colour online) Motion modes of an oblate ellipsoid in Poiseuille flow. (a) Log-rolling mode, (b) inclined mode. The upper images are two side views for each mode and the lower images are the corresponding top views. It is noted in the Poiseuille flow the particle is not onl rotating but also moving along the flow. The Poiseuille flow is in the -direction. the evolution ais of the oblate ellipsoid (the -ais) is almost perpendicular to the plane passing through the particle s centre of mass and the tube ais. Due to the shear stress applied to the particle, it will rotate and finall reach this equilibrium state. It is also noted that when the tube is narrow, e.g. D/A < 2, the wall effect is also strong and Re is higher, the particle will not onl adopt the log-rolling rotational mode but also perform the circumferential movement simultaneousl, i.e. swirling around the tube ais due to the aimuthal instabilit. From figure 9, it is seen that inside a narrower tube, three modes ma appear, which depend on Re. The three modes are the log-rolling mode, the intermediate mode, and the inclined mode. The inclined mode is shown in figure 1(b). In this mode, the evolution ais of the oblate ellipsoid is inside instead of perpendicular to the plane passing through the particle s centre of mass and the tube ais. Moreover, the evolution ais is no longer perpendicular to the -ais (see figure 1b). Hence it is called the inclined mode. In this mode, the ellipsoid moves solel with the Poiseuille flow in the -direction without rotating. We will discuss wh the inclined mode eists in detail in 6.2. The intermediate mode is a state between the log-rolling mode and the inclined mode. Figure 11 shows the projection of the particle in the (, )-plane. σ denotes the angle between the -ais and the plane passing through the tube ais and the centre of the particle. The angle θ 1 can be computed from the instantaneous (, ) position of the particle, and it is noted that σ = θ 1 θ 2. In the intermediate mode, the angle σ is neither close to (the inclined mode) nor close to 9 (the log-rolling mode). The mode combines the characteristics of both the log-rolling and inclined modes. In implementation, onl the states with σ (2, 7 ) are classified to the intermediate mode, i.e. the threshold is 2. When σ (7, 9 ), the mode is ver close to the log-rolling mode and is classified to the log-rolling mode. When σ (, 2 ), the mode is classified to the inclined mode. In the intermediate

16 An ellipsoidal particle in tube Poiseuille flow 679 o FIGURE 11. (Colour online) Angle between the -ais and the ai-smmetric plane passing through the tube ais and the centre of the particle. Mode γ σ Log-rolling Almost 9 (7, 9 ) Intermediate Slightl deviate from 9 (2, 7 ) Inclined Significantl deviate from 9 (, 2 ) TABLE 4. The characteristics of the log-rolling, intermediate, and inclined modes for the oblate ellipsoid. mode, the oblate particle is not onl inclined, i.e. γ is not close to 9, but also rotates around its -ais. Meanwhile, the particle is swirling around the tube ais. The characteristics of the modes are summaried in table 4. We would like to discuss the intermediate mode in detail. Here we take the cases of D/A = 2 as an eample. In the case for Re = 1, τ =.65, A = 8lu, the orientation of the -ais and trajector position as functions of time are shown in figure 12. Initiall, the particle is tumbling along a major ais due to the initial orientation (ϕ, θ, ψ) = (9, 9, 3 ) and the initial position (, ) = (,.19). After t > 1.25, the oblate ellipsoid enters the intermediate mode. From figure 12, it is seen that in the intermediate mode, cos(γ ) becomes constant with cos(γ ).8 (γ 85 ). Also cos(α) and cos(β) change sinusoidall, but the angle σ is a constant σ For the swirling movement, figure 12(b) shows r = is a constant, which means that the projection of the trajector of the particle s centre in the (, )-plane is a circle. In the intermediate mode, it is found that the angle σ depends on Re. The angle σ as a function of Re is shown in figure 13(a). It is noted that onl the red circles in figure 13 are classified as the intermediate mode. When Re is lower, e.g. Re 4, the oblate ellipsoid adopts the log-rolling mode (σ = 9 ). The angle σ decreases with Re in the intermediate section. Hence, as Re increases, the particle graduall changes from the log-rolling mode to the inclined mode. Figure 13(b) shows the normalied angular velocit ω as a function of Re. When Re (4, 16), ω increases with Re due to the increasing shear stress. However, when Re > 16, the angular velocit decreases with Re because σ decreases and the ( )-plane deviates significantl from the tube s ai-smmetric plane containing oo (see figure 11). The shear stress acting on the particle decreases with σ decreasing, which leads to smaller ω.

17 68 H. Huang and X.-Y. Lu (a) 1. Orientation of the (b) Normalied position Normalied time r FIGURE 12. (Colour online) In the intermediate mode, the orientation of the -ais and trajector position as functions of time (the case of D/A = 2, ρ p /ρ f = 1, Re = 1), where time t is normalied b t = tν/a 2 and position is normalied b D/2. (a) (b) Re Re FIGURE 13. (Colour online) The angle (a) and normalied angular velocit ω = ω A2 /ν (b) as functions of Re. The red circles represent the cases at the intermediate mode (D/A = 2, ρ p /ρ f = 1) The particle inertial effect The approimate phase diagram in a three-dimensional parameter space for an oblate ellipsoid is shown in figure 14. The parabolic clinder consists of a cluster of parabolic curves. Inside the parabolic clinder there is the intermediate mode. It is seen that the mode distribution would be affected b the particle s inertia (ρ p ). For wider tubes, the log-rolling mode is dominant in the phase diagram. For narrower

18 An ellipsoidal particle in tube Poiseuille flow 681 (a) Inclined A E D C (b) Re Re F B Intermediate Log-rolling (c) 35 3 Intermediate Inclined 25 Log-rolling Re FIGURE 14. (Colour online) Phase diagram for an oblate ellipsoid. (a) Phase diagram in a three-dimensional parameter space. The vertical plane ABCD separates the inclined and log-rolling modes. The rotational mode appears inside the parabolic clinder is the intermediate mode. The region of the inclined mode is bounded b the plane ABCD, the curved plane BCEF, and the wall of the bo. (b,c) Phase diagram in the planes ρ p /ρ f = 3. and ρ p /ρ f =.3, respectivel. tubes, the border separating the intermediate and the other two modes ma move upwards or downwards depending on the particle s inertia. If the planes with constant ρ p /ρ f are viewed from front to back, the border moves upwards. It means that on a plane with higher ρ p /ρ f, the area of the log-rolling mode increases. For ρ p /ρ f = 3, the inclined mode almost disappears under our considered parameter space and the log-rolling mode occupies a greater area in the D/A Re plane. In other words, at a slightl higher Re, the inertia of the oblate particle helps it maintain the log-rolling state The inclined mode In the following, the reasons for the eistence of the inclined mode are further discussed. The streamlines around the particle and pressure contours on an oblate ellipsoid are illustrated in figure 15. The pressure is normalied b p = (p c 2 s ρ f )/ρ f U 2 m. In the case for D/A = 2 and Re = 354, the dimensional parameters are τ =.565, tube length L = 192lu and diameter D = 16lu. The oblate ellipsoid has dimensions a = 2lu, b = c = 4lu. Under the pressure difference p = mu/(lu ts 2 ) between the two ends of the tube, the maimum velocit in the Poiseuille flow is U m =.958lu/ts and terminall the particle reaches the inclined mode with a constant velocit U p =.559lu/ts in the -direction without rotation. Figure 15(a) shows the streamlines inside the ai-smmetric plane (plane ABCD, i.e. ( = )-plane) where the -ais stas. The angle between the -ais and -ais is approimatel 67 because cos(γ ).38. The streamlines are drawn in an inertial frame moving with the same velocit as that of the particle. Figure 15(b) shows the

19 682 H. Huang and X.-Y. Lu (a) D A C B Pressure (b) FIGURE 15. (Colour online) Streamlines and pressure contours on the surface of an oblate ellipsoid in Poiseuille flow. The case is D/A = 2, ρ p /ρ f = 1, and Re = 354. The position of the centre of the particle is at = and =.468. The streamlines are drawn in the inertial frame moving with U p, which is the particle s velocit in the -direction in the space-fied coordinate sstem. pressure contours and surrounding streamlines viewed from the -direction. It is seen that the contours and streamlines are almost smmetric about the ( = )-plane. We can qualitativel understand wh the particle is able to maintain the stead state. In figure 15(a), on the lower and upper surface of the particle, the particle eperienced lower and higher pressure, respectivel. That ma generate a positive torque (counterclockwise). On the other hand, from the directions of the streamlines, it is seen that on the whole the shear stress acting on the surface would generate a negative torque (clockwise). Hence the torque due to the pressure difference will be balanced b the negative torque due to the viscous force acting on the surface. In this wa, the inclined mode is able to be an equilibrium state. Because both the log-rolling mode and the inclined mode are stable modes, an intermediate state combining the characteristics of both these stable modes should also possibl eist. 7. Conclusion In this stud, the behaviours of suspended particles in tube Poiseuille flow are numericall investigated. The effects of tube diameter, the inertia of both the particle and the flow, and the particle geometr (both prolate and oblate ellipsoids) are analsed. For prolate ellipsoids, aspect ratios a/b = 2 and 4 are considered, while for the oblate ellipsoid, the aspect ratio is fied to 1/2. When a prolate particle with a/b = 2 is inside a wider tube (e.g. D/A > 1.9), the terminal stable state is tumbling. When 1. < D/A < 1.9, i.e. a prolate particle is inside a narrower tube, the log-rolling or kaaking modes can appear. Which mode the particle adopts depends on the competition between the inertia of the fluid and the particle. When the inertia of the fluid is dominant, the log-rolling appears; otherwise, the kaaking mode appears. It is also found that the inclined and spiral modes ma appear when D/A < 1 and D/A = 1, respectivel. For a prolate ellipsoid with a/b = 4, if 1 < D/A < 1.9, onl the kaaking mode is seen, and the log-rolling mode is not observed. A possible reason is that the rotational energ in cases with a/b = 4 is much smaller than that in cases with a/b = 2. When an oblate particle is inside a wider tube (e.g. D/A > 3.5), it ma adopt the log-rolling mode. It is the first time in the literature that the inclined and intermediate

20 An ellipsoidal particle in tube Poiseuille flow 683 modes have been identified for oblate ellipsoids in narrower tubes (1 < D/A < 3.5). As the inertia of the fluid increases, the oblate ellipsoid ma change from the logrolling mode to the intermediate mode, and then the inclined mode without rotation. The phase diagram of the modes is also provided. Basicall, for wider tubes, our observation is consistent with those in the literature, i.e. the prolate and oblate ellipsoids adopt tumbling and log-rolling modes, respectivel (Karnis et al. 1966; Sugihara-Seki 1996; Pan et al. 28; Beon et al. 215). On the other hand, our finding is slightl different from that of Pan et al. (28). The claimed that the final state ma depend on the initial position and orientation. Here, based on limited observation, it is found that the final state does not depend on the initial states. The difference is attributed to the ver short computational domain in the flow direction and the periodic inlet/outlet boundar condition in their simulations (Pan et al. 28) instead of sufficient long tubes with pressure boundar conditions. Acknowledgements X.Y.L. is supported b National Natural Science Foundation of China (NSFC) grant no H.H. is supported b NSFC: and the Fundamental Research Funds for the Central Universities. Appendi A. Validation A.1. Migration of a sphere in Poiseuille flows To validate our LBM code, the migrations of a neutrall buoant sphere in tube Poiseuille flows were studied. As we know, man etensive eperiments concerning the migration of spheres in Poiseuille flows have been carried out (Karnis, Goldsmith & Mason 1963; Karnis et al. 1966). Here, our LBM results are compared with those obtained in eperiments (Karnis et al. 1966). In the simulations, -ais denotes the tube ais. The radii of the tube and the sphere are R =.2 cm and r =.61 cm, respectivel. The sphere is initiall placed in the (, )-plane. = and /R = 1 represent the tube ais and the wall, respectivel. Two cases with initial positions /R =.21 and.68 were simulated. The densit of the fluid is 1.5 g cm 3 and µ = ρν = 1.2 g cm 1 s 1. The flow rate is Q = cm 3 s 1. Because Q = (πr 2 /2)U m, the corresponding Re = (U m R)/ν =.198, where U m is the maimum velocit on the ais of the tube. To make the simulations more efficient, the multi-block strateg is also used (Huang et al. 214). The fine mesh sie is , the coarse mesh sie is and the tube length is L = 24lu. In the fine and coarse meshes, the relaation times are set as τ f =.9, τ c =.7, respectivel. The radius of the tube is R = 29.5lu. To match their Re, in our simulation U m = lu/ts and the corresponding pressure drop between the inlet and outlet is p = mu/lu/ts 2. In the simulations 1lu =.678 cm, 1ts = s, 1mu = g. The trajectories of spheres released from the /R =.21 (Case 1) and /R =.68 (Case 2) and those measured in Karnis et al. (1966) are illustrated in figure 16. It is shown that the LBM results are in ecellent agreement with the eperimental ones. The approach to an equilibrium position roughl midwa between the centre and the wall is the well-known Segre Silberberg effect. A.2. Sedimentation of an ellipsoid In this section, ellipsoid sedimentation in a circular tube is simulated and compared with the cases in Swaminathan, Mukundakrishnan & Hu (26). In the simulations,

21 684 H. Huang and X.-Y. Lu Case 1 (Karnis et al. 1966) Case 2 (Karnis et al. 1966) Case 1 (LBM) Case 2 (LBM) t (s) FIGURE 16. (Colour online) Migration trajectories of a neutrall buoant sphere in Poiseuille flows. Two cases with different initial positions were simulated and compared with eperimental data (Karnis et al. 1966). The spheres are put in the (, )-plane and released from /R =.21 (Case 1) and /R =.68 (Case 2), respectivel. the densit of the fluid is 1. g cm 3. The gravitational acceleration is g = 98 cm s 2, the viscosit of the fluid ν =.1 cm s 1, tube diameter D =.4 cm. In our LBM simulation, (ρ p ρ f )/ρ f = ρ/ρ f =.1, and the length of the major ais of the ellipsoid A = 2a =.1 cm is represented b 52lu, which means 1lu =.1923 cm. The densit of the fluid is set to be ρ f = 1mu/lu 3 and 1mu = g. In this particular case τ f = 1.2. Hence, 1ts represents s. The initial orientation of the particle is (φ, θ, ψ ) = (9, 9, 45 ), which means the evolution ais is in the (, )-plane and the angle between the ais and the ais is 45. The mesh sie is 216lu 216lu 18lu and the length of the tube is L 9D. The particle is kept in the centre of the domain using the dnamic multi-block strateg. The results for comparison are shown in figure 17. and are the normalied positions on -ais and the -direction (normalied b D/2). It is noted that the Galileo number Ga = ( ρ/ρ f )(ga 3 /ν 2 ) instead of Re is a true control parameter of the flow. However, Ga is not given in Swaminathan et al. (26). For comparison purposes, we have to tr different values of ρ/ρ f to make the simulated Re close to the Re in their cases (Swaminathan et al. 26). Here the cases of Ga = 3.43 and 9.78 are simulated, which have Re =.36 and 1.3, respectivel. For the case of Re.31, our result (Re =.36) agrees well with that in Swaminathan et al. (26). In this case, the particle moves and rotates inside the (, )-plane. It is seen that at 15, it moves across the ais. At 25, the ellipsoid collides with the wall and then moves towards the ais of the tube. After it passes through the -ais again, the spheroid enters the inclined mode (settling off-ais with a constant inclination to the horiontal). The trajector of the oscillator movement in our simulation is highl consistent with the prediction in Swaminathan et al. (26).

22 An ellipsoidal particle in tube Poiseuille flow FIGURE 17. (Colour online) Trajector of the centre of ellipsoids when the sediment in a circular tube at various Renolds number. The centres of the ellipsoids are initiall put in the ais of the tube, with the -ais inside the (, )-plane. The initial orientations are γ = 45., are the normalied positions in the -ais and the -direction (normalied b D/2). In the case of Re 1., both trajectories ( LBM Re = 1.3 and Re =.92 Swaminathan et al. (26) ) oscillate for a while and finall reach an almost identical equilibrium -position. Our LBM simulations are consistent with the data in Swaminathan et al. (26). A.3. Rotation of a prolate spheroid in shear flow To further validate our simulation, the rotation of a prolate spheroid in shear flow was compared with the Jeffer s solution (Jeffer 1922) and those presented in figure 2 of Qi & Luo (23). In our simulations, the streamwise direction of the shear flow is along the -direction. The velocit gradient and the vorticit are oriented in the - and -directions, respectivel. Two walls located at = and = N move in opposite directions with speed U. Periodic boundar conditions are applied in both the - and -directions. The particle Renolds number is defined as Re = 4Ga2, (A 1) ν where the shear rate is defined as G = 2U/N and a is the length of the semi-major ais. The ke parameters in our simulations are listed in table 5. For the case Re = 32, 128, and 2, the parameters are identical to those in Qi & Luo (23). Please refer

23 686 H. Huang and X.-Y. Lu (a) Gt (b) Normalied angular velocit Jeffre Gt FIGURE 18. (Colour online) Figure 2 in Qi & Luo (23) (a) is compared with our result (b). The normalied angular velocit of a prolate spheroid as a function of the dimensionless time Gt at Re =.1, 32, 128, 2 and the analtic result of Jeffer s theor at Re = are shown. The dimensions of both figures are the same. The simulation parameters for each case are listed in table 5. Re (N, N, N ) (a, b, c) τ ν U.1 96, 96, 96 12, 6, , 64, 64 16, 8, , 64, 64 16, 8, , 96, 96 24, 12, TABLE 5. Simulation bo and particle sie and ke parameters for different cases. to table 1 in their paper. Here b and c represent the lengths of minor aes with b = c = a/2. τ and ν are the relaation times in the LBM and the kinematic viscosit, respectivel. In the simulations, the initial orientation is set as (ϕ, θ, ψ ) = (9,, ), which makes the particle quickl enter a tumbling mode. The tumbling rotates inside the (, )-plane. Our LBM results are presented in figure 18(b). To perform the comparison, figure 2 of Qi & Luo (23) is presented in figure 18(a). We note for clarit that in all the LBM results, the time and the angular velocit are normalied b 1/G and G, respectivel. For Re =.1, it is seen that both our result for Re =.1 and that of Qi & Luo (23) agree ver well with the Jeffer s analtical solution. For the cases of Re = 32, 128, and 2, the rotation periods of our results are generall consistent with the corresponding periods of Qi & Luo (23). The peaks of the angular velocities for the cases of Re = 32, 128, and 2 also agree well with those of Qi & Luo (23), but valles have small discrepancies; e.g. for Re = 2, the valles are.17 and.13 at figure 18(a) and 18(b), respectivel. The discrepancies ma be attributed to the moving-boundar condition treatment. Our boundar condition scheme is based on Lallemand & Luo (23), which is more accurate than that in Qi & Luo (23).