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Lubrication corrections for lattice-Boltzmann simulations of particle suspensions

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Lubrication corrections for lattice-Boltzmann simulations of particle suspensions
N.-Q. Nguyen and A. J. C. Ladd
*
Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611-6005
Received 5 July 2002; published 30 October 2002
The lattice-Boltzmann method has been reﬁned to take account of near-contact interactions between spheri-cal particles. First, we describe a comprehensive solution to the technical problems that arise when twodiscretized surfaces come into contact. Second, we describe how to incorporate lubrication forces and torquesinto lattice-Boltzmann simulations, and test our method by calculating the forces and torques between aspherical particle and a plane wall. Third, we describe an efﬁcient update of the particle velocities, taking intoaccount the possibility that some of the differential equations are stiff.DOI: 10.1103/PhysRevE.66.046708 PACS number
s
: 83.85.Pt, 47.11.
j, 47.15.Pn
I. INTRODUCTION
Lattice-Boltzmann simulations
1,2
are being increas-ingly used to calculate hydrodynamic interactions
3–9
, bydirect numerical simulation of the ﬂuid motion generated bythe moving interfaces. However, as two particles approacheach other the calculation of the hydrodynamic force breaksdown in the gap between the particles, typically when theminimum distance between the two surfaces is of the orderof the grid spacing. For example, it is impractical to usesufﬁcient mesh resolution to resolve the ﬂow in the smallgaps that can result from an imposed shear ﬂow. At highshear rates the rheology of a suspension of spheres is quali-tatively affected by lubrication forces between particles sepa-rated by gaps less than 0.01
a
, where
a
is the particle radius
10,11
. A simulation at this resolution (
10
7
grid points perparticle
is unfeasible for more than a few particles. Thenumber of grid points can be reduced by using body ﬁttedcoordinates
12
or adaptive meshes
13,14
, but the smallzone size in the gap reduces the time step that can be used tointegrate the ﬂow ﬁeld
15
. The problem is exacerbated bythe uniform grid used in lattice-Boltzmann simulations, but itshould be noted that similar techniques, using particles em-bedded in regular grids, are becoming increasingly popularin computational ﬂuid dynamics
16,17
. Some aspects of this work may therefore be applicable to such methods aswell.Simulations of hydrodynamically interacting particlestypically use multipole expansions of the Stokes equations
18,19
. Again the calculations break down when the par-ticles are close to contact, unless an impractical number of multipole moments are used
20
. A solution to this problemis to calculate the lubrication forces between pairs of par-ticles for a range of small interparticle gaps, and then patchthese solutions on to the friction coefﬁcients calculated fromthe multipole expansion. The method exploits the fact thatlubrication forces can be added pair-by pair, and has beenshown to work well in practice
18
. A simpliﬁed version of this approach has already been adopted to include normallubrication forces in lattice-Boltzmann simulations
21
. Inthis paper we extend our previous work to include all com-ponents of the lubrication force and torque, and test the nu-merical scheme for the interactions between a spherical par-ticle and a planar wall. We ﬁnd that the hydrodynamicinteractions can be well represented over the entire range of separations by patching only the most singular componentsof the lubrication force onto the force calculated from thelattice-Boltzmann model. This is simpler than the Stokesiandynamics approach, where the patch is calculated at everyseparation.We begin this paper with a summary of the lattice-Boltzmann algorithm for particle suspensions
1,2
. We dis-cuss recent innovations by other groups
6,22
, and describesome additional improvements to these methods. In particu-lar, we propose a comprehensive solution to the technicaldifﬁculties that arise when particles are close to contact, inparticular the loss of mass conservation. The bulk of ournumerical results are a series of detailed tests of the hydro-dynamic interactions between two surfaces in near contact.We demonstrate that after including corrections for the lubri-cation forces we obtain accurate results over a wide range of ﬂuid viscosities. Finally, we describe an efﬁcient implicit al-gorithm for updating the particle velocities even in the pres-ence of stiff lubrication forces. An explicit solution of thesedifferential equations requires either that the particles aremuch denser than the surrounding ﬂuid
2
, or that verysmall time steps are used to update the particle velocities. Onthe other hand, if the particle velocities are updated implic-itly, the resulting system of linear equations severely limitsthe number of particles that can be simulated. In this paperwe describe what we call a ‘‘cluster-implicit’’ method,whereby groups of closely interacting particles are groupedin individual clusters and interactions between particles inthe same cluster are updated implicitly, while all other lubri-cation forces are updated explicitly. In ﬂuidized suspensionsclusters typically contain less than ten particles, even at highconcentrations, so that the implicit updates are a small over-head. Our simulations efﬁciently cope with clusters of sev-eral hundred particles by using conjugate-gradient methods,and only slow down if the number of particles in the clusterexceeds 10
3
.
II. THE LATTICE-BOLTZMANN METHOD
The lattice-Boltzmann equation describes the time evolu-tion of a discretized velocity distribution function
n
i
(
r
,
t
)
*
Email address: ladd@che.uﬂ.eduPHYSICAL REVIEW E
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/$20.00 ©2002 The American Physical Society
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n
i
r
c
i
t
,
t
t
n
i
r
,
t
i
n
r
,
t
,
1
where
i
is the change in
n
i
due to molecular collisions, and
t
is the time step. This one-particle distribution functiondescribes the mass density of particles with velocity
c
i
, at alattice node
r
, at a time
t
;
r
,
t
, and
c
i
are discrete, whereas
n
i
is continuous. The hydrodynamic ﬁelds, mass density
, mo-mentum density
j
u
, and momentum ﬂux
, are momentsof this velocity distribution,
i
n
i
,
j
u
i
n
i
c
i
,
i
n
i
c
i
c
i
.
2
In three dimensions, isotropy requires a multispeed model;for example, the 18-velocity model described in Ref.
23
,which uses the
100
and
110
directions of a simple cubiclattice. In this work the 18-velocity model is augmented withstationary particles, which enables it to account for smalldeviations from the incompressible limit, although in simu-lations of stationary ﬂows we have found the numerical dif-ferences to be small
24
.A computationally useful form for the collision operator
i
(
n
) can be constructed by linearizing about the local equi-librium
n
eq
23
,
i
n
i
n
eq
j
L
ij
n
jneq
,
3
where
L
ij
are the matrix elements of the linearized collisionoperator,
n
jneq
n
j
n
jeq
, and
i
(
n
eq
)
0. The form of theequilibrium distribution is constrained by the moment condi-tions required to reproduce the inviscid
Euler
equations onlarge length scales and time scales. In particular, the secondmoment of the equilibrium distribution should be equal tothe inviscid momentum ﬂux,
eq
i
n
ieq
c
i
c
i
p
1
uu
. Asuitable form for the equilibrium distribution of the 19-velocity model is
25
n
ieq
a
c
i
j
•
c
i
c
s
2
uu
:
c
i
c
i
c
s
2
1
2
c
s
4
,
4
where
c
s
c
2
/3,
c
x
/
t
,
p
c
s
2
, and the coefﬁcientsof the three speeds are
a
0
13 ,
a
1
118,
a
2
136.
5
In our suspension simulations we use a 3-parameter col-lision operator, allowing for separate relaxation of the ﬁveshear modes, one bulk mode, and nine kinetic modes. Thepostcollision distribution
n
i
*
n
i
i
is written as a series of moments
Eq.
2
, as for the equilibrium distribution
Eq.
4
24
,
n
i
*
a
c
i
j
•
c
i
c
s
2
uu
neq
,
*
:
c
i
c
i
c
s
2
1
2
c
s
4
.
6
The zeroth (
) and ﬁrst (
j
u
) moments
Eq.
2
are un-changed, but the nonequilibrium second moment,
neq
, ismodiﬁed by the collision process,
neq
,
*
1
neq
13
1
v
neq
:
1
1
,
7
where
neq
eq
; the eigenvalues
and
v
are relatedto the shear and bulk viscosities and lie in the range
2
0.The macrodynamical behavior arising from the lattice-Boltzmann equation can be found from a multitime-scaleanalysis
24,26
. A complete error analysis is rather lengthy
24
, but it can be shown that for sufﬁciently low velocitiesthe convergence is quadratic in the lattice spacing. TheNavier-Stokes equations
t
“•
u
0,
t
u
“•
uu
“
c
s
2
“
2
u
v
“
“•
u
8
are recovered in the low velocity limit, with viscosities
c
s
2
t
1
12
and
v
c
s
2
t
23
v
13
.
9
III. SOLID-FLUID BOUNDARY CONDITIONS
Boundary conditions in the lattice-Boltzmann model arestraightforward to implement, even for nonplanar surfaces
1
. Solid particles are deﬁned by a surface, which cuts someof the links between lattice nodes. Fluid particles movingalong these links interact with the solid surface at boundarynodes placed halfway along the links. Thus we obtain a dis-crete representation of the particle surface, which becomesmore and more precise as the particle gets larger
Fig. 1
. Inthe past we have treated the lattice nodes on either side of theboundary surface in an identical fashion, so that ﬂuid ﬁlls thewhole volume of space, both inside and outside the solidparticles. Because of the relatively small volume inside eachparticle, the interior ﬂuid quickly relaxes to rigid-body mo-tion, characterized by the particle velocity and angular veloc-ity, and closely approximates a rigid body of the same mass
24
. However, at short times the inertial lag of the ﬂuid isnoticeable, and the contribution of the interior ﬂuid to theparticle force and torque reduces the stability of the particlevelocity update. For these reasons we have followed the sug-gestion in Ref.
6
, and removed the ﬂuid from the interior of the particles. A somewhat different implementation of thesame idea is described in Ref.
22
.The moving boundary condition
1
without interior ﬂuid
6
is then implemented as follows. We take the set of ﬂuidnodes
r
just outside the particle surface, and for each node allthe velocities
c
b
such that
r
c
b
t
lies inside the particlesurface. An example of a set of boundary node velocities isshown by the arrows in Fig. 1
a
. Each of the correspondingpopulation densities is then updated according to a simple
N.-Q. NGUYEN AND A. J. C. LADD PHYSICAL REVIEW E
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rule which takes into account the motion of the particle sur-face
1
;
n
b
r
,
t
t
n
b
*
r
,
t
2
a
c
b
0
u
b
•
c
b
c
s
2
,
10
where
n
b
*
(
r
,
t
) is the postcollision distribution at (
r
,
t
) in thedirection
c
b
, and
c
b
c
b
. The local velocity of the par-ticle surface,
u
b
U
r
b
R
,
11
is determined by the particle velocity
U
, angular velocity
,and center of mass
R
;
r
b
r
12
c
b
t
is the location of theboundary node. We have used the mean density
0
in Eq.
10
instead of the local density
(
r
,
t
) since it simpliﬁes theupdate procedure. The differences between the two methodsare small, of the same order (
u
3
) as the error terms in thelattice-Boltzmann model. Test calculations show that evenlarge variations in ﬂuid density
up to 10%
have a negligibleeffect on the force
less than 1 part in 10
4
).As a result of the boundary node updates, momentum isexchanged locally between the ﬂuid and the solid particle,but the combined momentum of solid and ﬂuid is conserved.The forces exerted at the boundary nodes can be calculatedfrom the momentum transferred in Eq.
10
,
f
r
b
,
t
12
t
x
3
t
2
n
b
*
r
,
t
2
a
c
b
0
u
b
•
c
b
c
s
2
c
b
.
12
The particle forces and torques are then obtained by sum-ming
f
(
r
b
) and
r
b
f
(
r
b
) over all the boundary nodes asso-ciated with a particular particle. It can be shown analyticallythat the force on a planar wall in a linear shear ﬂow is exact
1
, and several numerical examples of lattice-Boltzmannsimulations of hydrodynamic interactions are given in Ref.
2
.To understand the physics of the moving boundary condi-tion, one can imagine an ensemble of particles, moving atconstant speed
c
b
, impinging on a massive wall orientedperpendicular to the particle motion. The wall itself is mov-ing with velocity
u
b
c
b
. The velocity of the particles aftercollision with the wall is
c
b
2
u
b
, and the force exertedon the wall is proportional to
c
b
u
b
. Since the velocities inthe lattice-Boltzmann model are discrete, the desired bound-ary condition cannot be implemented directly, but we caninstead modify the density of returning particles so that themomentum transferred to the wall is the same as in the con-tinuous velocity case. It can be seen that this implementationof the no-slip boundary condition leads to a small masstransfer across a moving solid-ﬂuid interface. This is physi-cally correct and arises from the discrete motion of the solidsurface. Thus during a time step
t
the ﬂuid is ﬂowing con-tinuously, while the solid particle is ﬁxed in space. If theﬂuid cannot ﬂow across the surface there will be large arti-ﬁcial pressure gradients, arising from the compression andexpansion of ﬂuid near the surface. For a uniformly movingparticle, it is straightforward to show that the mass transferacross the surface in a time step
t
Eq.
10
is exactlyrecovered when the particle moves to its new position. Forexample, each ﬂuid node adjacent to a planar wall has ﬁvelinks intersecting the wall. If the wall is advancing into theﬂuid with a velocity
U
, then the mass ﬂux across the inter-face
from Eq.
10
is
0
U
. Apart from small compressibil-ity effects, this is exactly the rate at which ﬂuid mass isabsorbed by the moving wall. For sliding motion, Eq.
10
correctly predicts no net mass transfer across the interface.Although the link-bounce-back rule is second-order accu-rate for planar walls oriented along lattice symmetry direc-tions, it is only ﬁrst-order accurate for channels oriented at
FIG. 1. Location of boundary nodes for a curved surface
a
andtwo surfaces in near contact
b
. The velocities along links cuttingthe boundary surface are indicated by arrows. The locations of theboundary nodes are shown by open squares, and the ﬂuid nodes bysolid circles.LUBRICATION CORRECTIONS FOR LATTICE- . . . PHYSICAL REVIEW E
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046708-3
arbitrary angles
27,28
. Thus for large channels, the hydro-dynamic boundary is displaced by an amount
from thephysical boundary, where
is independent of channel widthbut depends on the orientation of the channel with respect tothe underlying lattice. Convex bodies sample a variety of boundary orientations, so that it is not possible to make ananalytical determination of the displacement of the hydrody-namic boundary from the solid particle surface. Neverthe-less, the displacement of the boundary can be determinednumerically from simulations of ﬂow around isolated par-ticles. By considering the size of the particles to be the hy-drodynamic radius,
a
hy
a
, rather than the physical ra-dius
a
, approximate second-order convergence can beobtained, even for dense suspensions
2
.The hydrodynamic radius can be determined from thedrag on a ﬁxed sphere in a pressure-driven ﬂow
2
. Galileaninvariance can be demonstrated by showing that the sameforce is obtained for a moving particle in a stationary ﬂuid
2
. Since the particle samples different boundary node mapsas it moves on the grid, it is important to sample differentparticle positions when determining the hydrodynamic ra-dius, especially when the particle radius is small (
5
x
).Rather than averaging over many ﬁxed conﬁgurations, wechose to have the particle move slowly across the grid, atconstant velocity, sampling different boundary node maps asit goes. The changing boundary node maps lead to ﬂuctua-tions in the drag force, as shown in Fig. 2. The force hasbeen averaged over a Stokes time
t
s
so that the relative ﬂuc-tuations in force are comparable to the relative ﬂuctuations invelocity of a neutrally buoyant particle in a constant forceﬁeld. The force ﬂuctuations,
F
(
F
2
F
2
)/
F
, areof the order of 1% for particles of radius 2
3
x
, and areconsiderably smaller for larger particles
Table I
. More so-phisticated boundary conditions have been developed usingﬁnite-volume methods
29,30
and interpolation
13,14,31
.Both methods reduce the force ﬂuctuations by at least anorder of magnitude from those observed here, but even withthe simple bounce-back scheme, the ﬂuctuations in force canbe reduced by an appropriate choice of particle radius. Wehave noticed that ﬂuctuations in particle force are stronglycorrelated with ﬂuctuations in particle volume. Thus wechoose the radius of the boundary node map so as to mini-mize ﬂuctuations in particle volume for random locations onthe grid. It can be seen from Table I that a two fold reductionin the force ﬂuctuations is possible by this procedure, al-though for sufﬁciently large particles the difference is mini-mal. A set of optimal particle radii is given in Table II.The bounce-back rule leads to a velocity ﬁeld in the re-gion of the boundary that is ﬁrst-order accurate in the gridspacing
x
. The hydrodynamic boundary
the surface wherethe ﬂuid velocity ﬁeld matches the velocity of the particle
isdisplaced from the particle surface by a constant,
Fig. 3
,that depends on the viscosity of the ﬂuid
2
. For the range of kinematic viscosities used in this work, 1/6
*
1/1200,
varies from 0 to 0.5
x
Table II
; the dimensionless kine-matic viscosity
*
t
/
x
2
. For small particles (
a
5
x
),
also depends weakly on the particle radius
TableII
. Although the difference between the hydrodynamicboundary and the physical boundary is small, it is importantin obtaining accurate results with computationally useful par-ticle sizes. Calibration of the hydrodynamic radius leads toapproximately second-order convergence from an inherentlyﬁrst-order boundary condition; it will not be necessary whenmore accurate boundary conditions are implemented.The hydrodynamic radii (
a
hy
) in Table II were deter-mined from the drag force on a single sphere in a periodiccubic cell
2
. The Reynolds number in these simulations(
0.2) was sufﬁciently small to have a negligible effect onthe drag force. The time averaged force was used to deter-mine the effective hydrodynamic radius for three different
FIG. 2. The drag force
F
as a function of time, normalized bythe drag force on an isolated sphere,
F
0
6
aU
. Time is mea-sured in units of the Stokes time,
t
s
a
/
U
. The particle is movingalong a randomly chosen direction in a periodic unit cell with Re
1.TABLE I. Variance in the computed drag force
F
F
2
F
2
/
F
for a particle of radius
a
moving along a ran-dom orientation with respect to the grid. The results are averagedfor 50
200
t
s
, after the system reaches a steady state. The dimen-sionless viscosity
*
1/6. The numbers in brackets indicate pow-ers of 10.
a
/
x
2.7 2.5 8.2 8.5
F
5.738
3
1.208
2
4.332
4
5.674
4
TABLE II. Hydrodynamic radius
a
hy
in units of
x
) for vari-ous ﬂuid viscosities;
a
is the input particle radius.
a
/
x
1.24 2.7 4.8 6.2 8.2
*
1/6 1.10 2.66 4.80 6.23 8.23
*
1/100 1.42 2.92 5.04 6.45 8.46
*
1/1200 1.64 3.18 5.31 6.72 8.75N.-Q. NGUYEN AND A. J. C. LADD PHYSICAL REVIEW E
66
, 046708
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046708-4
kinematic viscosities:
*
1/6,
*
1/100, and
*
1/1200. In each case the cell dimensions were ten timesthe particle radius and the corrections for periodic boundaryconditions
about 40%
were made as described in Ref.
2
.The difference between the hydrodynamic radius and theinput radius,
a
hy
a
, is independent of particle size forradii
a
5
x
, and its magnitude increases with decreasingkinematic viscosity
24
. The kinematic viscosity
*
1/6gives a hydrodynamic radius that is the same as the inputradius
for sufﬁciently large particles
, and is the naturalchoice for simulations at low Reynolds number. At higherReynolds numbers a lower viscosity is necessary to maintainincompressibility
2,24
, and for accurate results it is thenessential to use the calibrated hydrodynamic radius.In order to implement the hydrodynamic radius in a mul-tiparticle suspension, all distance calculations are based onthe hydrodynamic radius
as shown in Fig. 3
; the input ra-dius
a
is only used to determine the location of the boundarynodes. It should be noted that not all combinations of particleradius and viscosity can be used. Table II indicates that par-ticle radii less than 3
x
cannot be used with a kinematicviscosity
*
1/6, since the hydrodynamic radius is thenless than the input radius.
IV. PARTICLE MOTION
An explicit update of the particle velocity
U
t
t
U
t
t m
F
t
,
13
t
t
t
t I
T
t
,
14
has been found to be unstable
2
unless the particle radius islarge or the particle mass density is much higher than thesurrounding ﬂuid. In previous work
2
the instability wasreduced, but not eliminated, by averaging the forces andtorques over two successive time steps. Subsequently, an im-plicit update of the particle velocity was proposed
32
as ameans of ensuring stability. Here we present a generalizedversion of that idea, which can be adapted to situationswhere two particles are in near contact.The particle force and torque can be separated into a com-ponent that depends on the incoming velocity distributionand a component that depends, via
u
b
, on the particle veloc-ity and angular velocity
Eqs.
11
and
12
,
F
F
0
FU
•
U
F
•
,
15
T
T
0
TU
•
U
T
•
.
16
The velocity independent forces and torques are determinedat the half-time step
F
0
t
12
t
x
3
t
b
2
n
b
*
r
,
t
c
b
,
17
T
0
t
12
t
x
3
t
b
2
n
b
*
r
,
t
r
b
c
b
,
18
where the sum is over all the boundary nodes
b
describingthe particle surface, with
c
b
representing the velocity associ-ated with the boundary node
b
and pointing towards the par-ticle center. The matrices
FU
2
0
x
3
c
s
2
t
b
a
c
b
c
b
c
b
,
19
F
2
0
x
3
c
s
2
t
b
a
c
b
c
b
r
b
c
b
,
20
TU
2
0
x
3
c
s
2
t
b
a
c
b
r
b
c
b
c
b
,
21
T
2
0
x
3
c
s
2
t
b
a
c
b
r
b
c
b
r
b
c
b
22
are high-frequency friction coefﬁcients, and describe the in-stantaneous force on a particle in response to a suddenchange in velocity.The magnitude of the friction coefﬁcients can be readilyestimated, thereby establishing bounds on the stability of anexplicit update. Apart from irregularities in the discretizedsurface,
FU
and
T
are diagonal matrices, while
F
TU
0. For a node adjacent to a planar wall
i
a
c
i
c
i
2
518
c
2
, where the sum is over the ﬁve directions that crossthe wall. The number of such nodes is approximately4
a
2
/
x
2
, so that
FU
20
9
0
xa
2
t
.
23
Similarly,
T
8
9
0
xa
4
t
.
24
FIG. 3. Actual
solid lines
and hydrodynamic
dashed lines
surfaces for a particle settling onto a wall.LUBRICATION CORRECTIONS FOR LATTICE- . . . PHYSICAL REVIEW E
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, 046708
2002
046708-5

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