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Chemical Engineering Science 65 (2010) 2159–2171

Contents lists available at ScienceDirect

Chemical Engineering Science

0009-25

doi:10.1

� Corr

E-m

(K. Nan

journal homepage: www.elsevier.com/locate/ces

Direct numerical simulations of a freely falling sphere using fictitious domainmethod: Breaking of axisymmetric wake

Rupesh K. Reddy a, Jyeshtharaj B. Joshi a, K. Nandakumar b,�, Peter D. Minev c

a Institute of Chemical Technology, Department of Chemical Engineering, Matunga, Mumbai-400 019, Indiab Cain Department of Chemical Engineering, Louisiana State University, Baton Rouge, USAc Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

a r t i c l e i n f o

Article history:

Received 10 April 2009

Received in revised form

2 December 2009

Accepted 8 December 2009Available online 21 December 2009

Keywords:

Falling sphere

Instability of wake

Particle rotation

Lateral migration

09/$ - see front matter & 2009 Elsevier Ltd. A

016/j.ces.2009.12.009

esponding author. Tel.: +1 225 578 2361; fax

ail addresses: nandakumar@lsu.edu, kumar.na

dakumar).

a b s t r a c t

In the present paper, numerical simulations of the wake generated by a freely falling sphere, under the

action of gravity, are performed. Simulations have been carried out in the range of Reynolds numbers

from 1 to 210 for understanding the formation, growth and breakup of the axisymmetric wake. The

in-house code used is based on a non-Lagrange multiplier fictitious-domain method, which has been

developed and validated by Veeramani et al. (2007). The onset of instability in the wake and its growth

along with the dynamic behavior of a settling sphere is examined at Reynolds number (Re) of 210. It is

found that at the onset of instability the sphere starts to rotate and gives rise to a lift force due to the

break of the axisymmetry in the wake which in turns triggers a lateral migration of the sphere. The lift

coefficient of a freely falling sphere is 1.8 times that of a fixed sphere at a given sphere density of

4000 kg m�3 and sphere to fluid density ratio of 4. This is attributed to the Robin’s force which arises

due to the rotation of the sphere. At this Reynolds number (Re=210) a double threaded wake is

observed, which resembles the experimental observations of Magarvey and MacLatchy (1965).

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Wakes generated by the bubbles, drops and particles in a fluidare important fundamental features in multiphase flows likesedimentation and fluidization (adsorption, leaching, particleclassification and backwashing of down flow granular filters,slurry transport (water lubricated transport of heavy crude andcoal slurries) and hydraulic fracturing (oil and natural gasproduction). The direct computation of such possibly unsteadyflows became possible in the recent years due to the increase inthe computational power and the development of advancedparallel numerical techniques. In the present work, the onset ofthe instability in the wake behind a freely falling sphere, itsgrowth and saturation which leads to the breakup of theaxisymmetric toroidal vortex, is simulated by using an in-housecode. The code which is based on a non-Lagrange multiplierfictitious-domain method, that has been developed and validatedby Veeramani et al. (2007). In this method, the fluid flow isgoverned by the continuity and momentum equations and theparticle motion is governed by the equations of motion of a rigidbody. The flow field around the particle is resolved and the

ll rights reserved.

: +1 225 578 1476.

ndakumar@ualberta.ca

resultant hydrodynamic force between the particle and the fluid iscomputed from the solution itself rather than being modeled byany drag law.

The behavior of the wake behind a sphere has been studied bya number of researchers over a wide range of Reynolds number.The earlier flow visualization experiments have been carried outby Taneda (1956) using a string mounted sphere moving at aconstant velocity in a water tank. He measured the size,separation angle and the center of the steady axisymmetric wakebehind the sphere. He has reported that the size of the vortex ringis proportional to the logarithm of the Reynolds number. He foundthat the Reynolds number at which the axisymmetric toroidalvortex ring begins to form in the rear end of a sphere is Re=24 anda faint periodic motion at the rear end of the vortex ring wasfound to begin at Re=130. The wake generated by liquid drops(carbon tetrachloride and chlorobenzene) in water has beenstudied by Magarvey and Bishop (1961) who also classified thewake, based upon the nature of the tail of the vortex and Reynoldsnumber. Up to Re=210 the wake is steady and axisymmetric andis referred to as a single thread wake. For 210oReo270 thevortex becomes non-axisymmetric and was classified as a doublethreaded wake. In the range of 270oReo290 the doublethreaded wake becomes unstable which leads to a wavy vortextail. Above Re=290, vortex loops begin to shed into the freestream. The formation and structure of the vortices due toaccelerating liquid drops at different intervals of time, at Re=340,

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has been shown experimentally by Magarvey and MacLatchy(1965). They observed that the liquid drops follow a spiral pathwhile settling. As pointed out by Winnikow and Chao (1966) andNatarajan and Acrivos (1993) these experiments with fallingliquid drops in immiscible liquids could be compared with thestandard solid rigid sphere wakes due to the presence of thesurfactants at the liquid–liquid interface which hold the drops in aspherical shape.

Masliyah (1972) has shown the recirculating wakes behind asphere and three oblate spheroids by using flow visualizationtechnique over a Re range from 15 to 100. He analyzed thevariation of the wake length and angle of separation of the stablewake with respect to the Reynolds number. Achenbach (1974) hasstudied the fixed sphere wakes for the range 400oReo5�106.With a help of a sketch, he has explained the periodic formationand release of vortex loops in the free stream, and determined theshedding frequency at Re=3000. The characteristics of the steadywake behind liquid filled spheres have been studied experimen-tally by Nakamura (1976), using dyed water for flow visualizationexperiments. From these experiments he observed that a stableand steady accumulation of dyed water at the rear end of thesphere begins at Re=7.3 and that the shape of the wake changesfrom concave to convex as the Reynolds number increases. Hepointed out that the tracer, aluminum dust, used by Taneda(1956) is not fine enough to seamlessly drift along with the slowfluid stream when Reo30. He also found that the maximumReynolds number at which the toroidal vortex is steady is about190, which is in contrast with the Magarvey and Bishop’s (1961)observation. This early instability in the wake at a Reynoldsnumber of 190 can be attributed to the use of liquid filled sphereswhere the fluid inside is free to move around, thus potentiallyaffecting the sphere’s motion and the wake development.Sakamoto and Haniu (1990) have measured the vortex sheddingfrequencies of a fixed sphere for Reynolds numbers from 300 to40,000 using hotwire anemometry and flow visualization experi-ments. The onset of a hairpin vortex shedding takes place atRe=300. The wake behind a fixed sphere from Re=30 to 4000 hasbeen visualized by using tracers illuminated by laser sheet by Wuand Faeth (1993). They also measured the streamwise velocitiesusing laser velocimetry. Ormieres and Provansal (1999) qualita-tively showed that the double threaded wake of a sphere held by athin metallic pipe was formed at Re=220 and a periodic vortexshedding occurred at Re=300. They also made quantitativemeasurements of the free-stream velocity in a wind tunnel byusing laser Doppler velocimetry (LDV) and hotwire anemometry.Visualizations of the vortex structures and measurements of thestreamwise velocity of a fixed sphere in the range of Reynoldsnumbers from 270 to 500 in a uniform flow channel have beenperformed by Schouveiler and Provansal (2002). Flow visualiza-tion experiments capturing the wake structure behind the risingand falling solid spheres in water have been also performed byVeldhuis et al. (2005) using the Schlieren technique. From theseexperiments they concluded that the wake generated by a movingsphere is different from the wake generated by a fixed sphere.They have demonstrated the formation of a pair of oppositesigned vortex threads and kinks on these threads which cause theformation of the hairpin vortices.

One of the first theoretical stability studies is performed byKim and Pearlstein (1990) who have carried out a two dimen-sional linear stability analyses by using a pseudo spectral methodfor discretization of the Navier–Stokes equation. They predictedthat the primary bifurcation occurs at Re=175 which is signifi-cantly lower than the experimentally observed by Magarvey andBishop (1961) value of 210. Using a modified linear stabilityanalysis, Natarajan and Acrivos (1993) carried out two-dimen-sional simulations of a sphere and a circular disk. They predicted

that the primary bifurcation in case of a sphere occurs at Re=210which is in a perfect agreement with the experimental observa-tion and they reported that the secondary (Hopf type) bifurcationand shedding of vortices occurs at Re=278. Tomboulides (1993)has performed three dimensional numerical simulations of a fixedsphere from Re=20 to Re=1000. He used a numerical method inwhich spectral element decomposition in the axial and radialdirection is combined with a spectral expansion in the tangentialdirection. He observed initial flow separation at Re=20 and asteady non-axisymmetric flow at Re=212. The vorticity of thenon-axisymmetric flow field resembled the double-thread wakeas observed by Magarvey and Bishop (1965) at Re=210. The wakestructure behind a fixed sphere in the range of Reynolds numberfrom 20 to 300 has been studied both, numerically andexperimentally, by Johnson and Patel (1999). They used a fourthorder Runge–Kutta method for the integration of the momentumequations along with a pressure Poisson equation to satisfy thecontinuity equation. They reported that the flow separation andformation of a vortex behind the sphere begins at Re=20 andcomputed the separation angle, length and the position of thevortex. They observed that the axisymmetry is broken at Re=210and explained this phenomenon by presenting three-dimensionalinteractions of the stream lines. They also calculated the values ofthe lift force, which arises after the breakage of symmetry. Inaddition, these authors reported that a periodic vortex sheddingbegins at Re=270 and explained it with the help of vorticitydiagrams. The variations in the drag and lift forces due to vortexshedding were also presented.

Numerical simulations of the flow past fixed oblate spheroidalbubbles at Re ranging from 100 to 1000 has been simulated byMagnaudet and Mougin (2007). Their numerical method is basedon a finite volume discretization on a staggered grid and a thirdorder Runge–Kutta algorithm for solving the velocity fieldcombined with a semi implicit algorithm for the viscous terms.At Re=180 they observed a non-axisymmetric wake, using 3Dparticle paths. They also presented the vorticity isosurfaces atRe=180, 300 and 700. The wake instability of a fixed sphere hasbeen studied by Ghidersa and Dusek (2000) who used a spectralelement method and reported that the primary bifurcation occursat Re=215. A vortex method for numerical simulation of flowspast spheres have been developed by Ploumhans et al. (2002).They presented the isosurfaces of the streamwise vorticity atRe=300, 500 and 1000. The hydrodynamic forces acting on a rigidfixed sphere at the range of Reynolds numbers from 10 to 320have been computed by Bouchet et al. (2006) using spectralelement method. They presented the vortex shedding frequenciesand oscillation amplitude of the drag and lift forces from Re=280to 320.

All these numerical studies throw light on the flow separation,wake instability and vortex shedding of a fixed sphere. However,the lateral migration and rotation of the sphere after axisymmetrybreaking have not been explained by these fixed sphere simula-tions. Understanding of such fundamental dynamic instabilities ispossible only by simulating the settling or fluidizing of a freesphere. Further, there are relatively few theoretical studies of suchphenomena available in the literature. A finite element baseddistributed Lagrange multiplier (DLM) fictitious domain methodfor solid–liquid flows has been introduced by Glowinski et al.(1999 and 2001). This numerical technique is based on theextension of the Navier–Stokes equations into the ‘‘fictitious’’domain occupied by the particles. The no-slip boundary condi-tions on the particle boundaries are enforced by means ofdistributed Lagrange multipliers. These multipliers represent theadditional body force per unit volume needed to maintain therigid-body motion inside the particle boundary. By using thismethod they simulated the sedimentation of 504 and 6400

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R.K. Reddy et al. / Chemical Engineering Science 65 (2010) 2159–2171 2161

circular (2D) particles. Pan et al. (2002) have performed bothsimulation and experiment of 1204 spherical particles at Reynoldsnumber of 1000 in a three dimensional rectangular box having thewidth of one particle diameter. The main aim of this study is toapply the distributed Lagrange-multiplier-based fictitious domainmethod for fluidization of 1204 spheres in a 2D fluidized bed.They have used very course grid (only 11 node points along thediameter of the sphere). They did not report the mechanism of thebraking of axisymmetric wake behind a single freely fallingsphere. Jenny et al. (2003) and Jenny and Dusek (2004) performedboth experiments and simulations of the freely falling sphere inpipe and studied the influence of density ratio on primary andsecondary Hopf bifurcation. They conclude that the density ratiowill not effect the primary bifurcation and its only effectssecondary Hopf bifurcation. Yu et al. (2004) have simulated thesedimentation of a sphere in a vertical tube using a finitedifference based distributed Lagrange multiplier (DLM) method.They presented the flow features, settling velocities, trajectoriesand the angular velocities of the spheres at Reynolds numberranging from 20 to 400. They pointed out that the rotation and thelateral migration of a sphere from the center of the tube to thewall is highly correlated with the Robin’s effect. They have usedonly 16 nodes along the diameter of the sphere and with a timestep of 0.01. They did not report the mechanism of the braking ofaxisymmetric wake. For analyzing this phenomenon very highmesh resolution around the sphere is required. An improvedLagrange multiplier / fictitious domain based technique has beenpresented by Diaz-Goano et al. (2003). This method defines aglobal Lagrange multiplier, l, whose physical meaning is of anadditional velocity field that imposes the rigid body motion. Thisallows complete elimination of the Lagrangian grids used byGlowinski et al. (2001) for the discretization of the distributedLagrange multipliers in their case. This allows for the use ofbasically only one solver on relatively regular grids whichfacilitates the parallel implementation of the method. Further-more, Diaz-Goano et al. (2003) have employed a second orderscheme in space, incremental projection scheme for the resolutionof the generalized Stokes problem and hence achieved better,performance better than the first order scheme. Recently,Veeramani et al. (2007) have developed a non-Lagrange multiplierversion of this technique and simulated the sedimentation of asphere at four different Reynolds numbers (1.5, 4, 12 and 32). Themain new feature of the modified method is that it avoids the useof Lagrange multipliers for implementation of the rigid bodymotion and explicitly resolves the hydrodynamic force betweenthe solid and liquid phases. Sedimentation of 64 sphericalparticles in a closed rectangular box at Reynolds number of 10and the migration of a neutrally buoyant sphere in a 3D Poiseuilleflow is also presented in this paper.

Although, these studies presented adequate numerical techni-ques for multiparticle systems which allow to trace the motion ofand compute the corresponding velocity field around free rigidparticles, they did not focus on fundamental issues like theformation of a steady wake and its growth behind a single fallingsphere, and the onset of instability and breakup of theaxisymmetry. Therefore, the present paper is aimed at addressingthese issues at various Reynolds numbers using the technique ofVeeramani et al. (2007).

3. Fictitious domain formulation

Consider an incompressible fluid having density rL andviscosity mL occupying a bounded region OL and a rigid particleoccupying a domain OS. Let OðOL [OSÞ be the entire computa-tional domain and S be the interface between the fluid and solid

domains. For simplicity, let O be a rectangular domain having anexternal boundary G. Under these assumptions the fluid motion isgoverned by the Navier–Stokes equations

r � uL ¼ 0 in OL

DuL

Dt¼�rpLþ

1

Rer

2uL in OL ð1Þ

Here the gravity term has been incorporated into pressuregradient term. Re is the Reynolds number of the flow based onthe particle size. For length and velocity scale, the spherediameter (dp) and terminal settling velocity ðVS1Þ has been usedfor the non-dimensionalization respectively. For time scaleVS1=dp and for pressure rLV2

S1 has been used.The equations of motion of the rigid solid particle are given by

rS

dU

dt¼ ðrS�rLÞ

1

Fregþ

rL

VF

Idxdtþx� Ix¼ T ð2Þ

where F is the total hydrodynamic force acting on the particle andT is the hydrodynamic torque about its center of mass, I is thetensor of inertia of the particle, and eg is the unit vector in thedirection of gravity. F and T are defined as

F¼Z@OS

r � n ds ð3Þ

T¼

Z@OS

r� rn ds ð4Þ

where r is the stress tensor of the fluid given by

rL ¼�pLdþ1

ReðruLþðruLÞ

TÞ ð5Þ

The main advantage of the fictitious domain methods over theArbitrary Lagrangian–Eulerian (ALE) methods is that it extends thefluid ðOLÞ equations into the domain occupied by the particles ðOSÞ

and imposes the boundary conditions on the particle surface as aside constraint, thus avoiding the need of remeshing OL wheneverthe particles move. This allows for the use of a relatively simplegrid which greatly facilitates the parallelization of the code.Further the Lagrange multiplier method imposes the rigid bodymotion using Lagrange multipliers (see Glowinski et al., 1999;Diaz-Goano et al., 2003). In the present study, we use the set offictitious domain equations derived by Veeramani et al. (2007):

r � uL ¼ 0 in ODuL

Dt¼�rpLþ

1

Rer2uLþ

rS�rL

rL

ðG�FÞ in O

dU

dt¼

1

V

ZOS

F dO ð6Þ

where

F¼

1

Fregþ

rL

rS�rL

F; in OS

0; in OL

8><>: ð7Þ

G¼

1

Freg ; in OS

0; in OL

8<: ð8Þ

In addition, the angular velocity of the particle can becomputed from the no-slip boundary condition on the surface ofthe particle which results in the following equation:

x¼1

2V

ZOS

r � uL dO ð9Þ

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The interaction force F (which plays the role of a Lagrangemultiplier) is to be determined from the condition that enforcesthe rigid body motion in OS i.e.

Uþx� ðx�XÞ ¼ uL in OS ð10Þ

where X denotes the position of the particle’s centroid.The equations above are discretized by means of second order

P2�P1 tetrahedral finite elements. The time discretization isperformed using an operator splitting method which allows tosplit the treatment of the various operators and to resolve the

24

16

16

z

y x

y

z x

Fig. 1. (A) Computational domain used for the single particle simulations.

(B) Mesh used for the bifurcation study (for clarity only one fourth density of

mesh is shown).

interaction force F explicitly. Briefly, the set of time-discreteequations is given by (for details see Veeramani et al., 2007).

Advection–diffusion substep: The center of the mass of theparticle is predicted explicitly by

Xp;nþ1¼Xnþ1

þ2dtUnð11Þ

where dt is the time step. Then we solve for u�L form

t0u�L�1

Rer2u�L ¼�t1 ~u

nL�t2 ~u

n�1L �rpn

LþrS�rL

rL

G in O ð12Þ

u�L ¼ 0 on @O

where the advected velocities in the right-hand side are thevelocities at time levels n and n�1 interpolated at the feet of the

0.1

1

10

100

1 10 100 1000

DR

AG

CO

EFF

ICIE

NT

, CD

, (-)

3

12

0.1

1

10

100

1 10 100 1000

REYNOLDS NUMBER, Re , (-)

PRE

SSU

RE

AN

D S

HE

AR

DR

AG

CO

EFF

ICIE

NT

S

1

2

REYNOLDS NUMBER, Re, (-)

Fig. 2. Drag coefficient (CD) vs. Reynolds number, (A) Variation of CD with grid

spacing: 1, grid spacing of 0.2; 2, grid spacing of 0.1; 3, grid spacing of 0.05; n,

Roos and Willmarth (1971); ’, Pettyjohn and Christiansen (1948). (B) Pressure

and viscous contributions in the total drag: 1, viscous drag; 2, pressure drag.

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corresponding characteristics

xðtn�iÞ ¼ xðtnþ1Þ�ðiþ1Þdtunþ1e ; i¼ 0;1 ð13Þ

The particle velocity is predicted by

t0U�þt1Unþt2Un�1

¼1

V

ZOSðtnþ 1Þ

ðt0u�Lþt1 ~unL�t2 ~u

n�1L ÞdO ð14Þ

Projection substep: On the next substep we impose theincompressibility constraint by solving:

t0ðu��L �u�LÞ ¼�rðp

nþ1L �pn

L Þ in O

r � u��L ¼ 0 in O

u��L � n¼ 0 on @O ð15Þ

Rigid body constraint: The rigid body constraint is enforced bysolving

unþ1L ¼ u��L þ ðU

��u��L Þþ

1

2V

�ZOSðtnþ 1Þ

r � unþ1L dO

��

�ðx�Xp;nþ1Þþ

1

V

rL

rS

ZOSðtnþ 1Þ

ðu��L �U�ÞdO�

lOSin O

Unþ1¼

1

V

rL

rS

ZOSðtnþ 1Þ

u��L dOþ 1�rL

rS

� �U�� ð16Þ

The linear solver used is a preconditioned conjugate gradientsolver.

Table 1Details of the numerical simulations.

S.no. Details Re=1 Re=25

1 Diameter of the particle (mm) 10 10

2 Density of solid (kg m�3) 4000 4000

3 Density of fluid (kg m�3) 1200 1200

4 Viscosity of fluid (kg m�1 s�1) 1.3 0.17

5 Terminal settling velocity (m/s) 0.104 0.361

6 Froude number 0.11 1.328

7 Dimension of the computational domine 16�24�16 16�24�16

8 No of nodes (million) 0.6 0.6

Fig. 3. Stream line patterns around the sphere in the XY plane: (

4. Results and discussion

The present code has been comprehensively validated by Diaz-Goano et al. (2003). The grid independence and sensitivity of thesettling velocity of a sphere on the time step has been discussed inan earlier study (see Veeramani et al., 2007), which also presents acomparison with the experimental results of ten Cate et al. (2002).Furthermore, in the present work, a grid independence study wascarried out for an accurate prediction of the drag coefficientdefined as

CD ¼FD

12rLV2

S1p4 d2

P

ð17Þ

where FD is the drag force in the streamwise direction (the Y axisin all the simulations) which is a combination of pressure (form)and viscous (skin) drag. While computing the drag coefficient thepressure and the stress at all the nodes, on the surface of thesphere have been considered. In the present study the number oftriangular surface elements used to compute the drag and liftcoefficients is about 1680.

For all the simulations the computational domain of adimensionless size 16�24�16 has been used (see Fig. 1A) andthe mesh is shown in Fig. 1B. A moving reference frame was usedat the center of mass of the particle along the flow direction toprevent the particle from getting outside the finely resolvedregion of the domain. At the end of the each time step thecomputed axial velocity (Y axis) of the sphere is subtracted fromthe fluid velocity at each node. Therefore, the sphere movesrelative to the grid in lateral directions (X and Z axes) only.Simulations were performed with four different grid sizes of 0.2,

Re=50 Re=100 Re=150 Re=200 Re=210

10 10 10 10 10

4000 4000 4000 4000 4000

1200 1100 1000 1000 1000

0.105 0.062 0.0435 0.0352 0.034

0.44 0.56 0.66 0.703 0.71

1.98 3.19 4.44 5.037 5.138

16�24�16 16�24�16 16�24�16 16�24�16 16�24�16

0.9 1.5 1.5 2.1 2.1

dp

L

(Xr,Yr)

θ

A), Re=1; (B), Re=25; (C), Re=50; (D), Re=100; (E), Re=200.

ARTICLE IN PRESS

REYNOLDS NUMBER, Re, (-)

DIM

EN

SIO

NL

ESS

WA

KE

LE

NG

TH

(-)

REYNOLDS NUMBER, Re, (-)

0

10

20

30

40

50

60

70

0

SEPA

RA

TIO

N A

NG

LE

, θ, (

-)

Nakamura (1976)

Taneda (1956)

Masliyah (1972)

DNS

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0

Taneda (1956)Magnaudet et al. (1995)DNS

50 100 150 200 250

50 100 150 200 250

Fig. 4. Comparison of the simulated separation angle and wake length values with

experimental results: (A) Separation angle: n, Nakamura (1976); , DNS; m,

Taneda (1956); &, Masliyah (1972); (B) Wake length: , DNS; m, Taneda

(1956); J, Magnaudet et al. (1995).

R.K. Reddy et al. / Chemical Engineering Science 65 (2010) 2159–21712164

0.1, 0.05 and 0.025 (this is the size of a typical element in thevicinity of the particle, normalized with its diameter). The effectof the grid size on the drag coefficient at different Reynoldsnumbers is shown in Fig. 2A. For the grid size of 0.2 and at Re=5, itcan be observed that the predicted drag coefficient (CD) was 12%lower than the experimental results of Roos and Willmarth (1971)and this deviation was more pronounced (39%) at Reynoldsnumber of 200. At Re=5 the predicted CD is in agreement with theexperimental results when the grid resolution increases from 0.2to 0.1, but a deviation of 15% was observed at Re=200 whichsuggests the need for further refinement. At 0.05 grid spacing thedrag coefficient shows good agreement at Re=5 and 200, andremains unchanged (to a high accuracy) if the grid resolution isincreased from 0.05 to 0.025. Hence, for the rest of thesimulations a grid size of 0.05 was selected. At this resolution,after the dynamic meshing of the sphere proposed by Veeramaniet al. (2007) for the purpose of computing the various integralsappearing in the fictitious domain discretization, the number oftriangular surface elements used to compute the drag coefficientis about 1680. The variation of the pressure or form drag (CDP) andviscous or friction drag (CDF) with respect to the Reynolds numberis shown in Fig. 2B. At Re=1 it can be observed that drag due tothe pressure was 33% of the total drag and the rest was a viscousdrag. As flow separation begins (at Re=25) the pressurecontribution in the total drag increases and equals to theviscous drag (CDP) around Re of 180. It can be observed thatafter Re4200 the pressure drag dominates over the viscous drag.

Numerical simulations of a freely falling sphere (dp=10 mmand rS=4000 kg m�3) were carried out in the range of Reynoldsnumbers from 1 to 210. The details of the simulations are given inTable 1. At Reynolds numbers of 1, 25, 50, 100 and 200 the streamline patterns around the sphere in the XY plane are shown inFig. 3. All the streamlines presented in Fig. 3 are shown at thesteady state, after the sphere reaches its terminal settling velocity.It can be observed that at Re=1, the flow around the sphere iscreeping and there is no flow separation (Fig. 3A). At Re=25 asmall and steady wake at the rear end of the sphere is observed(Fig. 3B). As the Reynolds number increases from 25 to 200 thesteady wake grows and stretches in the streamwise direction. Inthe range of Reynolds number from 25 to 200 it is observed thatthe topological flow structure of the wake is the same andchanges only in the location of the flow separation and length ofthe wake (Fig. 3B–E). As described in Fig. 3D that the flow isseparated at an angle ‘‘y’’ from the rear stagnation point andrejoins at a point (Xr Yr) on the axis of the flow to form a closedseparation wake. The computed values of the separation angle (y)and the normalized wake length (L/dP) are shown in Fig. 4. It canbe observed that the predictions are in an excellent agreementwith the experimental values in the literature. However, adeviation is observed when the simulated values of therecirculation length at Re4130 are compared with theexperimental results of Taneda (1956). This can be attributed tothe early unsteadiness in the wake as reported by Taneda (1956)that occurs in his experiment at Re=130. This early unsteadinessis not observed in the experimental results of Magarvey andBishop (1961) and numerical results of Tomboulides (1993),Johnson and Patel (1999), and Ghidersa and Dusek (2000). Thesestudies show that the steady axisymmetric wake exists up toReynolds number of 210. This has also been confirmed by thepresent study. This lower and upper limit of the steady andaxisymmetric regime is consistent with the experimental resultsof Magarvey and Bishop (1961) (freely falling drops) and thenumerical simulations of Tomboulides (1993), Johnson and Patel(1999) (fixed sphere) and linear stability analysis of Natarajan andAcrivos (1993) (fixed sphere). It is also observed that the wakelength and separation angle follow the logarithmic relationship

with the Reynolds number. At all the Reynolds numbers studied(20–200) neither a rotation nor a lateral migration of the sphere isobserved.

At Re=210 it is observed that the 3D wake behind the sphere isno longer axisymmetric. However, the flow still possesses adiscrete (2D) symmetry in a plane containing the axis of thesphere parallel to the walls of the container. This plane israndomly selected by the flow. The 2D projection of the flow ina plane perpendicular to the first one (but still containing thesame axes of the sphere) is non-symmetric. In the presentsimulations, the plane of symmetry is at an angle of 421 withthe positive X-axis. As mentioned by Johnson and Patel (1999) andGhidersa and Dusek (2000), in the case of a fixed sphere, the plane

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of symmetry is selected randomly and its orientation is deter-mined by the initial conditions in perfectly rotationally symmetricconfigurations or sensitive to the slightest initial perturbations ofthe rotational invariance. In the present simulation, the asym-metry grows naturally, with the symmetry plane being chosendue to numerical perturbations in the flow field. This is confirmedby a simulation in which we slightly perturb the vertical positionof the sphere changing the initial position of the centroid from (0,4, 0) to (0, 4.05, 0). This change can introduce only a smallperturbation due to the grid and should not affect the results forthe terminal velocity or drag coefficient. Indeed, the lateral andangular velocities of the sphere are the same in both cases but theplane of symmetry in the perturbed case is at an angle of 2221

Fig. 6. Comparison of the simulated wake with the experimentally observed wake: Sy

(C), Johnson and Patel (1999); Non-symmetric view(D), DNS (non-symmetric view); (E

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

1

Z-A

XIS

2

X-AXIS

-1.5 -1 -0.5 0 0.5 1 1.5 2

Fig. 5. Projected path of lateral migration of sphere on XZ plane (back view): 1,

original sphere and initial position is at (0.0, 4.0, 0.0); 2, shifted sphere and initial

position is at (0.0, 4.05, 0.0).

(1801+421) to the X-axis. The entire flow pattern in these twocases is identical but shifted at an angle of 1801. The projectedpath of the centroid of the sphere on the XZ plane is presented inFig. 5 for both cases where the 1801 shift in the projectedtrajectory can be clearly seen.

In Fig. 6 we visualize the wake behind the sphere in the twomutually orthogonal planes of symmetry (Fig. 6A) and asymmetry(Fig. 6D). The simulated wake structure closely resembles theexperimental results of Magarvey and MacLatchy (1965), Johnsonand Patel (1999), and Schouveiler and Provansal (2002) as shownin Fig. 6B–F. In these flow visualization experiments the dye istrapped in the vortex cores and leads to a two-threaded structureas it can be seen in Fig. 7 where selected streamlines fromdifferent view points are plotted. From this figure, the twocounter rotating vortices which were formed due to the break ofthe steady axisymmetric vortex can be observed.

The contour plots of the pressure drag coefficient

CP ¼P�P1rLV2

S1

ð18Þ

at Re=200 and 210 are shown in Fig. 8A and B, respectively. Formore quantitative information we also show the pressuredistribution along the line MN (Y=4) in Fig. 9. It clearlydemonstrates that the pressure distribution looses its symmetryfor some Reynolds number between 200 and 210. The reason forthe symmetry loss can be attributed to a centrifugal instability inthe wake as the sphere falls freely under gravity. As theinstantaneous Reynolds number increases, the centrifugal forcein the wake increases and causes instability in the pressuregradient. The viscous forces can control it until a certain thresholdrotational velocity of the vortex in the wake (at about Re=210)when the centrifugal perturbation becomes dominant and cannotbe balanced by the viscous forces, thus causing the axisymmetrybreakup. As mentioned above, it is clear that the pressure force onthe sphere dominates over the viscous force only after Re=200(Fig. 2).

In the case of a fixed sphere this primary bifurcation has beenextensively studied by Tomboulides (1993), Natarajan andAcrivos (1993), Johnson and Patel (1999), Ghidersa and Dusek(2000), and Schouveiler and Provansal (2002). However, in the

mmetric view (A), DNS (symmetric view); (B), Magarvey and MacLatchy (1965);

), Johnson and Patel (1999); (F), Schouveiler and Provansal (2002).

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Fig. 7. Different views of stream line patterns of the two counter rotating vortices.

Fig. 8. Contours of pressure coefficient (CP): (A) Re=200; (B) Re=210.

R.K. Reddy et al. / Chemical Engineering Science 65 (2010) 2159–21712166

case of a freely falling sphere it has been observed that the particlerotates and moves laterally after breaking of axisymmetry, whichcannot be explained by fixed sphere simulations. The graphs of thethree components of velocity of the freely falling sphere at Re of 150and 210 with respect to time are shown in Fig. 10. At Re=150 thelateral velocities (X and Z velocities) are very small and only 0.2% ofthe axial velocity of the sphere. While at Re=210, it can be observedthat at t=24.3 (point A in Fig. 10B) the lateral (X and Z) componentsof the velocity start to deviate from zero. This happens exactly whenthe sphere reaches its terminal settling velocity (point B in Fig. 10B).The variation of the angular velocities around the X and Z-axes ofthe sphere with respect to time is shown in Fig. 11. The angularvelocity about the Y-axis is negligible. It is interesting to find thatthe sphere slowly starts rotating at t=4.53 (point A in Fig. 11) whichis much earlier than the time when the lateral migration starts(t=24.3). In order to find the reasons for the non-axisymmetricrotation and the lateral migration of the sphere, we carried outa comprehensive analysis by studying the dynamic behavior of the

wake of a sphere at different time intervals starting from rest(t=0).

The onset of instability, its progressive growth and dynamicbehavior of the wake of the freely falling sphere was examined byusing the instantaneous streamlines passing through four pointsin the core of the wake. These four points were located on the XZ

plane which is at a distance of one particle diameter from thecenter of the sphere (points 1–4 in Fig. 12A). Fig. 12 shows theback views (XZ views) of the streamlines of the freely fallingsphere at various time intervals. At t=3 (Fig. 12B) it can beobserved that there is no azimuthal interaction in the core of thewake. Hence, up to this point of time there is no instability inthe wake of the sphere. As the sphere accelerates more, thecentrifugal acceleration in the core of the toroidal vortex alsoincreases due to the high rotational velocity of the vortex, andcauses an increase in the azimuthal velocities of the fluid particles

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-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

-2

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

-2

NY = 4 M

PR

ES

SU

RE

CO

EFF

ICIE

NT,

CP,

(-)

PR

ES

SU

RE

CO

EFF

ICIE

NT,

CP,

(-)

DIMENSIONLESS DISTANCE, (-)

-1.5 -1 -0.5 0 0.5 1 1.5 2

-1.5 -1 -0.5 0 0.5 1 1.5 2

Fig. 9. Variation of pressure coefficient (CP) along the line Y=4 (shown as KL in

Fig. 8): (A) Re=200; (B) Re=210.

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0

SP

HE

RE

VE

LOC

ITY

(-) 120

TIME (-)

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0

31A

SP

HE

RE

VE

LOC

ITY

(-)

120TIME (-)

2

B

20 40 60 80 100

20 40 60 80 100

Fig. 10. Linear velocities of the freely falling sphere with respect to time. (A) At

Re=150; (B) Re=210: 1, X direction linear velocity of sphere; 2, Y direction linear

velocity of sphere; 3, Z direction linear velocity of sphere.

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0

A

TIME (-)

AN

GU

LAR

VE

LOC

ITY,

ω, (

-)

2

1

3 6 9 12 15 18 21

Fig. 11. Initial angular velocities of the sphere with respect to time at Re=210: 1,

angular velocity around X-axis; 2, angular velocity around Z-axis.

R.K. Reddy et al. / Chemical Engineering Science 65 (2010) 2159–2171 2167

in the core of the wake. At t=4.53 a very small azimuthalinteraction of the streamlines is observed (Fig. 12C) which can beconsidered as a starting point of the wake instability. It should bepointed out that this very small instability is just a starting pointof the breakup of axisymmetry which affects only a very smallcore of the wake and overall the wake has still a closed steady andalmost axisymmetric toroidal shape. But strictly speaking, at thispoint in time (t=4.53) the axisymmetry is broken and the flowaround the sphere has already selected the plane of symmetry

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1

2

3

4

z

y x

z

y x

Fig. 12. XZ views (back views) of four stream lines (1, 2, 3 and 4) of the accelerating sphere at various time intervals: (A) t=0; (B) t=3.0; (C) t=4.53; (D) t=7.8; (E) t=11.7;

(F) t=13.1; (G) t=16.6; (H) t=17.2; (I) t=19.1.

R.K. Reddy et al. / Chemical Engineering Science 65 (2010) 2159–21712168

about which the toroidal vortex is going to break (the solid line inFig. 12C) and the wake is asymmetric in the plane of the dashedline in Fig. 12C. Furthermore, at the same time (t=4.53), therotation of sphere starts (point A in Fig. 11). From this observationit can be concluded that the onset of azimuthal instability in thecore of the wake (at t=4.53) starts the rotation of the sphere. Asthe sphere settling velocity increases, it can be observed that theinstability progressively grows and changes its azimuthaldirection with time of sedimentation of the sphere (Fig. 12D–I).

It can also be seen that the sphere starts to oscillate due to thechange in the direction of the azimuthal interaction. Themagnitude of the oscillations and the change of the sign ofthe angular velocities are shown in Fig. 11. It is observed that, asthe azimuthal instability increases in the core of the wake, thesphere angular velocity also increases and reaches a saturationpoint (t=20.4). At this point, the toroidal vortex is unable tosustain itself, and thus breaks into two counter rotating vortexthreads. At the saturation point (t=20.4) the wake structure,

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+ve Z- axis

Axis of rotation

+ve X-axis

-veZ- axis

Direction ofSphere migration Side view

for Fig. 5A

42°

-ve X-axis

Non-symmetricplane

Symmetric plane

Side view for Fig. 5B

Lift force direction

z

y x

Fig. 13. Wake structure at saturation point (XZ view) at t=20.4.

-0.15

-0.12

-0.09

-0.06

-0.03

0

0.03

0.06

0.09

0.12

0

AN

GU

LAR

VE

LOC

ITY

OF

SP

HE

RE

(-)

3

2TIME (-)

1

4 8 12 16 20 24 28 32

Fig. 14. Variations of angular velocities of the sphere with respective to time. 1,

angular velocity around X-axis; 2, angular velocity around Y-axis; 3, angular

velocity around Z-axis.

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0

2

LIFT

CO

EFF

ICIE

NT,

CL,

(-)

1

80TIME (-)

20 40 60

Fig. 15. Variation of lift coefficient (CL) with respective to time: 1, freely falling

sphere; 2, fixed sphere.

R.K. Reddy et al. / Chemical Engineering Science 65 (2010) 2159–2171 2169

direction of lateral migration, axis of rotation, symmetry planeand non-symmetric plane are shown in Fig. 13. The counterrotating vortices are shown (from different viewpoints) in Fig. 7.Due to the action of the counter rotating vortices which wereformed due to the breakup of the axisymmetric wake (at t=20.4),the particle starts to rotate with much higher angular velocitythan before (see Fig. 14). Note that the angular velocities betweent=0 and 20 in Fig. 11 have a much smaller magnitude incomparison to the angular velocity of the sphere after thebreakup of the axisymmetric vortex. The pair of counterrotating vortices, formed after the breakup, generates a lateralforce on the sphere (a lift force). It is shown (point A in Fig. 10)

that the sphere starts to migrate only after t=24.3. Therefore, thelift force which arises due to break of axisymmetry is the maindriving force for the lateral migration of the particle. This lift forcecan be quantified by the lift coefficient defined as

CL ¼FL

12rLV2

S1p4 d2

P

ð19Þ

FL being the force that acts normally to the streamwise direction.The variation of the lift coefficient with respect to time is

shown in Fig. 15. The time averaged lift coefficient is found to be�0.045 (the sign of the lift coefficient will depend upon theorientation of the plane of axisymmetry which governs the

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-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

10 15 20 25

4

3

1

2

TIME (-)

PRE

SSU

RE

CO

EFF

ICIE

NT

, CP,

(-)

Fig. 16. Variation of pressure at two points with respective to time: 1, (0.4, 4.0,

0.4) freely falling sphere; 2, (�0.4, 4.0, �0.4) freely falling sphere; 3, (0.4, 4.0, 0.4)

fixed sphere; 4, (�0.4, 4.0, �0.4) fixed sphere.

R.K. Reddy et al. / Chemical Engineering Science 65 (2010) 2159–21712170

direction of migration of the sphere. As explained above thisasymmetry grows naturally, with the symmetry plane beingchosen due to numerical perturbations in the flow field). For thecase of a fixed sphere Tomboulides (1993) and Johnson and Patel(1999) have reported that the lift coefficient value is �0.024which is about 1.8 times less at a given sphere density of4000 kg m�3 and sphere to fluid density ratio of 4 (the change indensity of the sphere/ density ratio will result in different velocitygradients and pressure distribution around the sphere. Therefore,the lift coefficient which has been evaluated by integratingpressure and stress on the surface of the sphere will alsochange). The excess lift force can be attributed to the rotation ofthe particle which occurs immediately after breaking ofaxisymmetry (t=20.4). This excessive lift force that arises due toparticle rotation is known as the Robin’s force and has beenqualitatively explained by Yu et al. (2004) in the case of spherefalling in the vertical tube. In order to quantify the effect ofrotation (Robin’s force) on the lift force, we conducted a fixedsphere simulation at Re=210. The variation of the lift force for thefixed sphere is also shown in Fig. 15 which clearly demonstratesthe difference of the lift force between the fixed and the freespheres. It can be also observed that after breaking of theaxisymmetric vortex the lift force in case of a fixed sphere isconstant whereas the freely falling sphere exhibits a largeamplitude fluctuation in the lift coefficient. This unsteady lift atRe=210 can be attributed to the lateral migration of sphere.Moreover, the strength and sign of the counter rotating vorticesvaries periodically which causes the fluctuations in the lift.Furthermore, the variation of pressure at two points for bothcases of a fixed and freely falling spheres are shown in Fig. 16. Thetwo points are situated at (0.4, 4.0, 0.4) and (�0.4, 4.0, �0.4)(shown in Fig. 8 as K and L). It can be noticed that in the case of afreely falling sphere the pressure difference between the twopoints is higher than in the case of a fixed sphere. Again, thisexcess pressure difference around the particle must be due to therotation of the particle (Robin’s effect), that is not present in thefixed sphere case. Bouchet et al. (2006) and Magnaudet andMougin (2007) have also noticed that the lift force of a freelyfalling sphere is different from the fixed sphere but they did not

compute this difference explicitly. The present computationsyield that at Re=210 (which we accept as the point of the firstbifurcation in the flow) the lift coefficient of a freely falling sphereis about 1.8 times higher than the one of a fixed sphere.

5. Conclusions

Numerical simulations of a freely falling sphere have beenperformed in the range of Reynolds numbers from 1 to 210. It isobserved that the computed values of the separation angle (y) and thenormalized wake length (L/dP) are in an excellent agreement with theexperimental values in the literature. From the simulations of a freelyfalling sphere in the range of Reynolds numbers between 1 and 200, itcan be concluded that the wake generated by the sphere is identicalwith the wake generated by a fixed sphere. At Re=210, it is observedthat the axisymmetric toroidal vortex behind the sphere breaks intotwo counter rotating vortices. This breakup causes a lateral migrationof the sphere and it begins to follow a quite chaotic spiral path. Thelift force which arises due to break of axisymmetry is the main drivingforce for the lateral migration of the spherical particle. The presentcomputations reveal that the lift coefficient of a free-falling sphere(�0.045) is 1.8 times more than the lift coefficient of a fixed sphere(�0.024) at a given sphere density of 4000 kg m�3 and sphere to fluiddensity ratio of 4. The excess lift force can be attributed to the rotationof the particle (Robin’s effect) which occurs immediately after thebreakup of axisymmetry.

Notation

eg

unit vector in the direction of gravity F dimensionless total hydrodynamic force acting

on the particle, dimensionless

FD drag force on single particle, kg m s�2

Fr

Froude number ðV2S1=gdpÞ, dimensionless

H

grid size, m

h

distance between particle and wall or particleand particle, m

I

tensor of inertia n unit normal pointing out of the particle N number of particles in the entire domain,

dimensionless

pL extension of the pL to the entire domain O,

dimensionless

pL

dimensionless pressure, dimensionless

r

dimensionless radius of particle, dimensionless Re Reynolds number (dp VsN rL/mL), dimensionless T dimensionless hydrodynamic torque of a

particle, dimensionless

U dimensionless velocity of the particle,

dimensionless

uL extension of the uL to the entire domain O uL dimensionless fluid velocity, dimensionless

V

volume of the particle VS1 terminal settling velocity, m s�1

VS

settling velocity of the particle in the presenceof other particles, m s�1

Greek letters

G1, G2, G3

and G4

boundaries of rectangular domain

mL

molecular viscosity of fluid, kg m�1 s�1

rL

density of fluid, kg m�3

rS

density of solid particle, kg m�3

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R.K. Reddy et al. / Chemical Engineering Science 65 (2010) 2159–2171 2171

r

dimensionless shear stress tensor,dimensionless

S

interface between fluid and solid phases o angular velocity of particle, dimensionless O entire computational domain OL fluid domain OS solid domain

Subscript and superscript

L

liquid phase S solid phase

Abbreviations

ALE

arbitrary Lagrangian–Eulerian DLM distributed Lagrange multiplier DNS direct numerical simulation LDV laser Doppler velocimetry

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