Advances in Computational Modelling of MultiPhysics in ParticleFluid Systems
ABSTRACT The current work presents the recent advances in computational modelling strategies for effective simulations of multi physics
involving fluid, thermal and magnetic interactions in particle systems. The numerical procedures presented comprise the Discrete
Element Method for simulating particle dynamics; the Lattice Boltzmann Method for modelling the mass and velocity field of
the fluid flow; the Discrete Thermal Element Method and the Thermal Lattice Boltzmann Method for solving the temperature field.
The coupling of the fields is realised through hydrodynamic and magnetic interaction force terms. Selected numerical examples
are provided to illustrate the applicability of the proposed approach.
 Citations (38)
 Cited In (0)

Article: Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation
[Show abstract] [Hide abstract]
ABSTRACT: An efficient and robust computational method, based on the latticeBoltzmann method, is presented for analysis of impermeable solid particle(s) suspended in fluid with inertia. In contrast to previous latticeBoltzmann approaches, the present method can be used for any solidtofluid density ratio. The details of the numerical technique and implementation of the boundary conditions are presented. The accuracy and robustness of the method is demonstrated by simulating the flow over a circular cylinder in a twodimensional channel, a circular cylinder in simple shear flow, sedimentation of a circular cylinder in a twodimensional channel, and sedimentation of a sphere in a threedimensional channel. With a solidtofluid density ratio close to one, new results from twodimensional and threedimensional computational analysis of dynamics of an ellipse and an ellipsoid in a simple shear flow, as well as twodimensional and threedimensional results for sedimenting ellipses and prolate spheroids, are presented.Journal of Fluid Mechanics 10/1998; 373:287  311. · 2.29 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We present a general method based on a multipoleexpansion theory that allows us to calculate efficiently and accurately the electrostatic forces and the dielectric constant of an assembly of spheres. This method is applied to the study of two aspects which play an important role in the behavior of electrorheological (ER) fluids. The first one concerns the calculation of the principal values ε and ε⊥ of the dielectric tensor of the bodycenteredtetragonal (bct) lattice, and the calculation of the induced dipole on the particles in this lattice. These are rigorous calculations on physical properties of interest in the study of ER fluids. These results support the idea that the columnlike aggregates which have been found in ER fluids should have a bct structure. Although calculations based on the dipolar approach were previously presented, no results are available that confirm this idea rigorously. The second point concerns an exact analytical derivation of an expression describing the manybody electrostatic forces among spherical polarizable particles in terms of the multipole moments. We have compared this force expression, in the case of twoparticle interactions, to some results from the literature. It agrees very well with some analytical twoparticle expressions for perfectly conducting spheres and also with some recent results concerning the interactions between two polarizable spheres. Furthermore, we present results for threeparticle contributions to the electrostatic force and show that these contributions are unexpectedly large. In particular, the rate of divergence of the force between two conducting spheres can be considerably changed by the presence of a third one.Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics 11/1993; 48(4):27212738. · 2.33 Impact Factor  Géotechnique 01/1979; 29:4765. · 1.67 Impact Factor
Page 1
Advances in Computational Modelling of
MultiPhysics in ParticleFluid Systems
Y.T. Feng, K. Han and D.R.J. Owen
Abstract The current work presents the recent advances in computational model
ling strategies for effective simulations of multi physics involving fluid, thermal
and magnetic interactions in particle systems. The numerical procedures presented
comprise the Discrete Element Method for simulating particle dynamics; the Lat
tice Boltzmann Method for modelling the mass and velocity field of the fluid flow;
the Discrete Thermal Element Method and the Thermal Lattice Boltzmann Method
for solving the temperature field. The coupling of the fields is realised through hy
drodynamic and magnetic interaction force terms. Selected numerical examples are
provided to illustrate the applicability of the proposed approach.
1 Introduction
In recent years the modellingof coupledfield problems,in which two or more phys
ical fields contribute to the system response, has become a focus of major research
activity. Among them, the quantitative study of fluid, thermal and magnetic inter
actions in particulate systems encountered in many engineering applications is of
fundamental importance. For instance, the mineral recovery operation in the min
ing industry employs a suction process to extract rock fragments from the ocean or
river bed. The computational modelling of this particle transport problem requires
a fluidparticle interaction simulation. The motion of the particles is driven collect
ively by the gravity and the hydrodynamicforces exerted by the fluid, and may also
be altered by the interaction between the particles. On the other hand, the fluid flow
pattern can be greatly affected by the presence of the particles, and is often of a
turbulent nature. In the nuclear industry, the process of a pebble bed nuclear reactor
essentially involves the forced flow of gas through uranium enriched spheres that
Y.T. Feng · K. Han · D.R.J. Owen
Civil and Computational Engineering Centre, School of Engineering, Swansea University,
Swansea SA2 8PP, UK; email: y.feng@swansea.ac.uk
51
© Springer Science+Business Media B.V. 2011
in Applied Sciences 25, DOI 10.1007/9789400707351_2,
E. Oñate and R. Owen (eds.), ParticleBased Methods, Computational Methods
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52 Y.T. Feng et al.
are cyclically fed through a concentric column in order to extract thermal energy.
In this situation, the introduction of additional field, thermal (heat transfer between
the moving particles in the form of conduction,convection,and radiation, as well as
transfer of heat to the gas stream), poses even more computational challenges. An
other example is the modelling of magnetorheological fluids (MR fluids). An MR
fluid is a type of smart fluid, which consists of micronsized magnetizable particles
dispersed in a nonmagnetic carrier fluid. In the absence of a magnetic field, the
rheologicalbehaviourof an MR fluid is basically that of the carrier fluid, exceptthat
the suspended magnetizable particles makes the fluid ‘thicker’. When subjected to
an external magnetic field, the particles become magnetized and acquire a dipole
moment. Due to magnetic dipolar interactions, the particles line up and form chain
like structures in the direction of the applied field. This change in the suspension
microstructure significantly alters the rheological properties of the fluid. To model
the particle chain formation and the rheological properties of the MR fluid under an
applied magnetic field, the magnetic, hydrodynamicand contact interactions should
be fully resolved.
Thefundamentalphysicalphenomenainvolvedinthesesystemsaregenerallynot
well understood and often described in an empirical fashion, mainly due to the in
tricate complexity of the hydrodynamic, thermodynamic and magnetic interactions
exhibited and the nonexistence of highfidelity modelling capability.
The Discrete Element Method [5], among other discontinuous methodologies,
has become a promising numerical tool capable of simulating problems of a dis
crete or discontinuous nature. In the framework of the Discrete Element Method, a
discretesystem is consideredas anassemblyofindividualdiscreteobjectswhichare
treated as rigid and represented by discrete elements as simple geometric entities.
The dynamic response of discrete elements depends on the interaction forces which
can be shortranged, such as mechanical contact, and/or mediumranged, such as
attraction forces in liquid bridges, and obey Newton’s second law of motion. By
tracking the motion of individual discrete elements and handling their interactions,
the dynamic behaviour of a discrete system can be simulated.
Conventionalcomputational fluid dynamic methods have limited success in sim
ulating particulate flows with a high number of particles due to the need to gen
erate new, geometrically adapted grids, which is a very timeconsuming task es
pecially in threedimensional situations [10]. In contrast, the Lattice Boltzmann
Method [2,36] overcomes the limitations of the conventional numerical methods
by using a fixed, nonadaptive (Eulerian) grid system to represent the flow field. In
particular, it can efficiently model fluid flows in complex geometries, as is the case
of particulate flow under consideration. A rich publication in recent years (see, for
instance, [1,4,10,13,15,16,21,28,29,33]and the references therein) has provedthe
effectiveness of the method.
If an additional field, thermal, is introduced to a particulate system, the Thermal
Lattice Boltzmann Method [25] may be employed to model heat transfer between
particles and between particles and the surroundingfluid. Our numerical tests show,
however, that the Thermal Lattice Boltzmann Method is not efficient for simulat
ing heat conduction in particles. For this reason, a novel numerical scheme, termed
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Advances in Computational Modelling of MultiPhysics in ParticleFluid Systems53
the Discrete Thermal Element Method [14] is put forward. In this approach, each
particle is treated as an individual element with the number of (temperature) un
knowns equal to the number of particles that it is in contact with. The element
thermal conductivity matrix can be very effectively evaluated and is entirely de
pendent on the contact characteristics. This new element shares the same form and
properties with its conventionalthermal finite element counterpart.In particular,the
entire solution procedure can follow exactly the same steps as those involved in
the finite element analysis. Unlike finite elements or other numerical techniques,
no discretisation errors are involved in the Discrete Thermal Element Method. The
numerical validation against the finite element solution indicates that the solution
accuracy of this scheme is reasonable and highly efficient in particular.
The interaction problems considered is often of a dynamic and transient nature.
Althoughthe Discrete Thermal Element Methodis capableof modellingthe steady
state heat conductionin large particulate systems efficiently, it is not trivial to be ex
tendedto transientsituations.Meanwhile,its formulationis notcompatiblewiththat
of the Discrete Element Method which accounts for particleparticle interactions.
Therefore the Discrete Thermal Element Method needs to be modified to realise
thermalparticle coupling. The pipenetwork model is such a modification [15], in
which each particle is replaced by a thermal pipenetwork connecting the particle’s
centre with each contact zone associated with the particle.
For numerical modelling of magnetorheological fluids, in addition to the above
numericaltechniques,themagneticforcesformedbetweenmagnetizedparticlesun
deranexternallyappliedmagneticfieldneedtobeproperlyaccountedfor.Thisturns
out not to be an easy issue since the mutual
The objective of this work is to present our recent developments [13–16,23,24]
on all essential computational procedures for the effective modelling of the above
mentionedmultiphysicsproblemsinvolvingfluid, thermaland/ormagneticinterac
tions in particulate systems. In what follows, the basic formulations of the Discrete
Element Method, the Lattice Boltzmann Method, the Discrete Thermal Element
Method, the magnetic interactions and the coupling techniques, will be outlined.
Selected numerical examples are provided to illustrate the applicability of the pro
posed approach.
2 ParticleParticle Interactions – Discrete Element Approach
Interactions between the moving particles are modelled by the Discrete Element
Method [5], in which each discrete object is treated as a geometrically simplified
entity that interacts with other discrete objects through boundary contact. At each
time step, objects in contact are identified with a contact detection algorithm; and
the contact forces are evaluated based on appropriate interaction laws. The motion
of each discrete object is governed by Newton’s second law of motion. A set of
governing equations is built up and integrated with respect to time, to update each
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54 Y.T. Feng et al.
object’s position,velocityandacceleration.Themain buildingblocksof the discrete
element procedure are described as follows.
2.1 Representation of Discrete Objects
In the Discrete Element Method,discrete objects are treatedas rigid andrepresented
either by regular geometric shapes, such as disks, spheres and superquadrics, or by
irregular geometric shapes, such as polygons, polyhedrons, clustering or clumping
of regular shapes to form compound shapes.
Circularandsphericalelementsare themost used discrereteelements dueto their
geometric simplicity, smooth and continuous boundary. Contact resolution for this
type of element is therefore trivial and computationally efficient. However, ideal
ising materials such as grains and concrete aggregates as perfect disks (or spheres)
is not always realistic and may not produce correct dynamic behaviour. One of the
reasons is that circular and spherical elements cannot provide resistance to rolling
motion. This has led to the introduction of more sophisticated elements to represent
the discrete system more realistically.
Contrary to the circular and spherical elements where only the radius can be
modified, polygonal elements (polygons or polyhedrons) offer increased flexibility
interms ofshapevariation.Since theboundaryofthis typeofelementis notsmooth,
some complex situations such as corner/corner contact, often arise in the contact
resolution.
Higher order discrete elements can be used, such as superquadrics and hyper
quadrics as proposed in [37], which may represent many simple geometric entities
(for instance, disk, sphere, ellipse and ellipsoid) within the framework. However,
this mathematical elegancy may be offset by the expensive computation involved in
the contact resolution.
Preparation of an initial packing configuration of particles is a very important
issue both practically and numerically. There is only limited work reported. See
[8,9] for a very effective packing of disks/ploygons, and for [19] for spheres.
2.2 Contact Detection
In the discrete element simulation of problems involving a large number of discrete
objects, as much as 60–70% of the computational time could be spent in detecting
and tracking the contact between discrete objects. Due to a large diversity of object
shapes, many efficient contact processing algorithms often adopt a twophase solu
tion strategy. The first phase, termed contact detection or global search, identifies
the discrete objects which are considered as potential contactors of a given object.
The second phase, termed contact resolution or local search, resolves the details of
the contact pairs based on their actual geometric shapes.
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Advances in Computational Modelling of MultiPhysics in ParticleFluid Systems55
Some search algorithms used in general computing technology and computer
graphics have been adopted for this purpose. Algorithms such as bucket sorting,
heapsorting,quicksorting,binarytree andquadranttreedatastructureall originated
from general computing algorithms. However, applications of these algorithms in
discrete element codes need modifications to meet the needs of particular discrete
element body representations and the kinematic resolution.
For the detection of potential contact between a large number of discrete ele
ments, a spatial search algorithm based on spacecell subdivision and incorporating
a tree data storage structure possesses significant computational advantages. For in
stance, the augmented spatial digital tree [6] is a spatial binary tree based contact
detection algorithm. It uses the lower corner vertex to represent a rectangle in a bin
ary spatial tree, with the upper corner vertex serving as the augmented information.
The algorithm is insensitive to the size distributions of the discrete objects. Nu
merical experiments in [6] indicate that this search algorithm can reduce the CPU
requirement of a contact detection from an originally demanding level down to a
more acceptable proportion of the computing time.
Anothertypeofthe contactdetectionalgorithmsis the socalledcell basedsearch
[31,32]. The main procedures in these algorithms involve: (1) dividing the domain
that the discrete objects occupy into regular grid cells; (2) mapping each discrete
object to one of the grid cells; and (3) for each discrete object in a cell, checking for
possible contacts with other objects in the same cell and in the neighbouring cells.
Provided the number of cell columns and rows is significantly less than the number
of discrete objects, it can be proved that the memory requirement for the dynamic
cell search algorithm is O(N). Also for a fixed cell size the computational time Top
may be expressed as
Top= O(N + ?)
where ? represents the costs associated with the maintenance of various lists used in
the algorithm. Numerical tests conducted in [22] show that the dynamic cell search
algorithmis even more efficient than the tree based search algorithms for large scale
problems.
2.3 Contact Resolution
The identified pairs with potential contact are then kinematically resolved based
on their actual shapes. The contact forces are evaluated according to certain con
stitutive relationship or appropriate physically based interaction laws. In general,
the interaction laws describe the relationship between the overlap and the corres
pondingrepulsive force of a contact pair. For rigid discrete elements, the interaction
laws may be developed on the basis of the physical phenomenainvolved. The Hertz
normal contact model that governs elastic contact of two spheres (assumed rigid in
discrete element modelling) in the normal direction is such an example, in which
the normal contact force, Fn, and the contact overlap, δ, has the following relation
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56 Y.T. Feng et al.
Fn=4E∗√(R∗)
3
δ3/2
(1)
where
1
E∗=1 − ν2
1
R∗=
1
E1
+1 − ν2
1
2
E2
1
R1
+
R2
with R1andR2beingthe radii;E1,E2, andν1,ν2are theelastic properties(Young’s
modulus and Poisson’s ratio) of the two spheres.
For irregularly shaped particles, such as polygons and polyhedra, the contact
interaction models can be an serious issue where the normal direction may not be
uniquely defined. Energy based contact models, proposed in [10] for polygons and
[11] for polyhedra, provide an elegant solution to the problem. An application to
superquadrics is proposed in [20].
For ‘wet’ particles the interaction laws may include the effects of a liquid bridge.
In other cases, adhesion may be considered.
Energy dissipation due to plastic deformation, heat loss and material damping
etc during contact is taken into account by adding a viscous damping term in the
governing equation.
Friction is one of the fundamental physical phenomena involved in particulate
systems. Althoughthe searchfor a quantitativeunderstandingof the features of fric
tion has been in progress for several centuries, a universally accepted friction model
has not yet been achieved. One difficulty is associated with the nature of the friction
force near zero relative velocity, where a strong nonlinear behaviour is exhibited.
The classic Coulomb friction law is usually employed in engineering applications
for its simplicity. The discontinuous nature of the friction force in this model, how
ever, imposes some numerical difficulties when the relative sliding velocity reverses
its direction and/or during the transition from sliding (sticking) to sticking (slid
ing). The difficulties are usually circumvented by artificially introducing a ‘trans
ition zoneŠ which smears the discontinuity in the numerical computation. Never
theless, the suitability of any friction model should be carefully examined and the
associated numerical issues fully investigatedin order to correctly capture the phys
ical phenomena involved. Proper considering rolling friction is another challenging
issue and many numerical issues remain outstanding [7].
A comprehensive study of the contact interaction laws can be found in [17,18].
2.4 Governing Equations and Time Stepping
The motion of the discrete objects is governed by Newton’s second law of motion
as
?M¨ u + Cd˙ u = Fc
J¨θ = Tc
(2)
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Advances in Computational Modelling of MultiPhysics in ParticleFluid Systems57
where M andCdare respectivelythe mass and dampingmatrices of the system, u, ˙ u
and ¨ u are respectively the displacement, velocity and acceleration vectors, J is the
moment of inertia,¨θ the angular acceleration, Fcand Tcdenote the contact force
and torque, respectively.
The configuration of the entire discrete system is evolved by employing an ex
plicit time integration scheme. With this scheme, no global stiffness matrix needs
to be formed and inverted, which makes the operations at each time step far less
computationally intensive. However, any explicit time integration scheme is only
conditionally stable. For a linear system the critical time step can be evaluated as
?tcr=
2
ωmax
(3)
where ωmaxis the maximum eigenvalue of the system. However, the above result
may not be valid since a contact system is generally nonlinear, as is demonstrated
in [12]. To ensure a stable and reasonably accurate solution, the critical time step
chosen should be much smaller than the value given in Eq. (3).
3 FluidParticle Interactions
The interaction between fluid and particles is solved by a coupled technique: using
the Lattice Boltzmann Method to simulate the fluid field, and the Discrete Element
Method to model particle dynamics. The hydrodynamic interactions between fluid
and particles are realised through an immersed boundary condition. The solution
procedures are outlined as follows.
3.1 The Lattice Boltzmann Method
In the Lattice Boltzmann Method, the problem domain is divided into regular lat
tice nodes. The fluid is modelled as a group of fluid particles that are allowed to
move between lattice nodes or stay at rest. During each discrete time step of the
simulation, fluid particles move to the nearest lattice node along their directions of
motion, where they ‘collide’ with other fluid particles that arrive at the same node.
By tracking the evolution of fluid particle distributions, the macroscopic variables,
such as velocity and pressure, of the fluid field can be conveniently calculated from
its first two moments.
The lattice Boltzmann equation with a single relaxation time for the collision
operator is expressed as
fi(x + ei?t,t + ?t) − fi(x,t) = −1
τ
?fi(x,t) − feq
i(x,t)?
(4)
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58Y.T. Feng et al.
where fiis the density distribution function of the fluid particles with discrete velo
city eialong the ith direction; feq
i
is the equilibrium distribution function; and τ is
the relaxationtime whichcontrols the rate ofapproachto equilibrium.The lefthand
side of Eq. (4) denotes a streaming process for fluid particles while the righthand
side models collisions through relaxation.
In the widely used D2Q9 model [33], the fluid particles at each node move to
their eight immediate neighbouringnodes with discrete velocities ei(i = 1,...,8).
The equilibrium distribution functions feq
i
velocity, v, which are defined in D2Q9 model as
⎧
⎪⎪⎩
in which c = ?x/?t is the lattice speed with ?x and ?t being the lattice spacing
and time step, respectively; wiis the weighting factor with w0= 4/9, w1−4= 1/9,
w5−8= 1/36.
The macroscopicfluid variables, density ρ and velocity v, can be recoveredfrom
the distribution functions as
depend only on the fluid density, ρ, and
⎪⎪⎨
feq
0
= ρ
?
1 −
?
3
2c2v · v
1 +3
?
feq
i
= wiρ
c2ei· v +
9
2c4(ei· v)2−
3
2c2v · v
?
(i = 1,...,8)
(5)
ρ =
8
?
i=0
fi,ρv =
8
?
i=1
fiei
(6)
The fluid pressure field p is determined by the following equation of state:
p = c2
sρ
(7)
where csis termed the fluid speed of sound and is related to the lattice speed c by
cs= c/√3
The kinematic viscosity, ν, of the fluid is implicitly determined by the model
parameters, ?x,?t and τ as
(8)
ν =1
3
?
τ −1
2
??x2
?t
=1
3
?
τ −1
2
?
c?x
(9)
which indicates that the selection of these three parameters should be correlated to
achieve a correct fluid viscosity.
It can be proved that the lattice Boltzmann equation (4) recoversthe incompress
ible Navier–Stokes equations to the second order in both space and time [2], which
is the theoretical foundation for the success of the Lattice Boltzmann Method for
modelling general fluid flow problems. However, since it is obtained by the linear
ised expansion of the original kinetic theory based Boltzmann equation, Eq. (4) is
only valid for small velocities, or small ‘computational’ Mach number defined by
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Advances in Computational Modelling of MultiPhysics in ParticleFluid Systems59
Ma=vmax
c
(10)
where vmaxis the maximum simulated velocity in the flow.
Generally smaller Mach number implies more accurate solution. It is therefore
required that
Ma? 1
i.e., the lattice speed c should be sufficiently larger than the maximumfluid velocity
to ensure a reasonably accurate solution.
(11)
3.2 Incorporating Turbulence Model in the Lattice Boltzmann
Equation
As many fluidparticle interaction problems are turbulent in nature, a turbulence
model should be incorporated into the lattice Boltzmann equation (4).
The Large EddySimulation, amongst other turbulencemodels, solves large scale
turbulent eddies directly but the smaller scale eddies using a subgrid model. The
separation of these scales is achieved through the filtering of the Navier–Stokes
equations, from which the solutions to the resolved scales are directly obtained.
Unresolved scales can be modelled by, for instance, the Smagorinsky subgrid
model [34] that assumes that the Reynolds stress tensor is dependent only on the
local strain rate.
Yu et al. [38] proposed to incorporate the Large Eddy Simulation in the lattice
Boltzmann equation by including the eddy viscosity as
˜ fi(x + ei?t,t + ?t) = ˜ fi(x,t) −1
τ∗
?
˜ fi(x,t) −˜ feq
i(x,t)
?
(12)
where ˜ fiand ˜ feq
functionattheresolvedscale,respectively.Theeffectoftheunresolvedscale motion
is modelled through an effective collision relaxation time scale τt. Thus in Eq. (12)
the total relaxation time equals
i
denote the distribution function and the equilibrium distribution
τ∗= τ + τt
where τ and τtare respectively the relaxation times corresponding to the true fluid
viscosity ν and the turbulence viscosity ν∗defined by a subgrid turbulence model.
Accordingly, ν∗is given by
ν∗= ν + νt=1
32
?
τ∗−1
?
c2?t =1
3
?
τ + τt−1
2
?
c2?t
νt=1
3τtc2?t
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60Y.T. Feng et al.
With the Smagorinsky model, the turbulence viscosity νt is explicitly calculated
from the filtered strain rate tensor˜Sij = (∂j˜ ui+ ∂i˜ uj)/2 and a filter length scale
(which is equal to the lattice spacing ?x) as
νt= (Sc?x)2ˆS
(13)
where Scis the Smagorinsky constant; andˆS the characteristic value of the filtered
strain rate tensor˜S
ˆS =
i,j
??
˜Sij˜Sij
An attractivefeatureof the model is that˜S can be obtaineddirectly fromthe second
order moments,˜ Q, of the nonequilibriumdistribution function
˜S =
˜ Q
2ρScτ∗
(14)
in which ˜ Q can be simply computed by the filtered density functions at the lattice
nodes
8
?
where ekiis the kth component of the lattice velocity ei. Consequently
˜ Qij=
k=1
ekiekj(˜ fk−˜ feq
k)
(15)
ˆS =
ˆ Q
2ρScτ∗
(16)
withˆ Q the filtered mean momentum flux computed from˜ Q
ˆ Q =
?
2
?
i,j
˜ Qij˜ Qij
(17)
3.3 Hydrodynamic Forces for FluidParticle Interactions
The modelling of the interaction between fluid and particles requires a physically
correct ‘noslip’ velocity condition imposed on their interface. In other words, the
fluid adjacent to the particle surface should have identical velocity as that of the
particle surface.
Ladd [28] proposes a modification to the bouncebackrule so that the movement
of a solid particle can be accommodated. This approach provides a relationship of
the exchange of momentum between the fluid and the solid boundary nodes. It also
assumes that the fluid fills the entire volume of the solid particle, or in other words,
the particle is modelled as a ‘shell’ filled with fluid. As a result, both solid and fluid
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Advances in Computational Modelling of MultiPhysics in ParticleFluid Systems61
nodesoneither side ofthe boundarysurfaceare treatedin anidentical fashion.It has
been observed, however, that the computed hydrodynamic forces may suffer from
severe fluctuations when the particle moves across the grid with a large velocity.
This is mainly caused by the stepwise representation of the solid particle boundary
and the constant changing boundary configurations.
To circumvent the fluctuation of the computed hydrodynamic forces with the
modified bouncebackrule, Noble and Torczynski[30] proposedan immersed mov
ing boundary method. In this approach, a control volume is introduced for each lat
tice node that is a ?x × ?x square around the node, as illustrated by the shadow
area in Figure 1a. Meanwhile, a local fluid to solid ratio γ is defined, which is the
volume fraction of the nodal cell covered by the particle as shown in Figure 1b.
Thelattice Boltzmannequationforthose lattice nodes(fully orpartially) covered
by a particle is modified to enforce the ‘noslip’ velocity condition as
fi(x + ei?t,t + ?t) = fi(x,t) −1
τ(1 − β)?fi(x,t) − feq
i
?+ βfm
i
(18)
where β is a weighting function depending on the local fluid/solid ratio γ; and fm
is an additional term that accounts for the bounce back of the nonequilibrium part
of the distribution function, computed by the following expressions:
⎧
⎩
where −i denotes the opposite direction of i.
The total hydrodynamic forces and torque exerted on a particle over n particle
covered nodes are summed up as
??
??
where xcis the coordinate of the particle center.
With this approach, the computed hydrodynamic forces are sufficiently smooth,
which is also confirmed in our previous numerical tests [13,21].
i
⎨
β =
fm
i
γ(τ−0.5)
(1−γ)+(τ−0.5)
= f−i(x,t) − fi(x,t) + feq
i(ρ,vb) − feq
−i(ρ,v)
(19)
Ff = c?x
n
?
βn
?
?
i
fm
iei
??
(20)
Tf = c?x
n
(x − xc) ×
βn
?
i
fm
iei
??
(21)
3.4 Fluid and Particle Coupling
Fluid and particle coupling at each time step is realised by first computing the fluid
solution,andthenupdatingtheparticlepositionsthroughtheintegrationoftheequa
tions of motion given by
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62 Y.T. Feng et al.
(a) Control area of a node(b) Nodal solid area fraction
Fig. 1 Immersed boundary scheme of Noble and Torczynski
?ma + cdv = Fc+ Ff+ mg
J¨θ = Tc+ Tf
(22)
where m and J are respectively the mass and the moment of inertia of the particle;
¨θ the angular acceleration; g the gravitational acceleration if considered;Ffand Tf
are respectively the hydrodynamic force and torque; Fcand Tcdenote the contact
force and torque from other particles and/or boundary walls; cdis a damping coef
ficient and the term cdv represents a viscous force that accounts for the effect of all
possible dissipation forces in the system. The static buoyancy force of the fluid is
taken into account by reducing the gravitational acceleration to (1−ρ/ρs)g, where
ρsis the density of a particle.
This dynamic equation governing the evolution of the system can be solved by
the central difference scheme. Some important computational issues regarding the
solution are briefly discussed as follows:
1. Subcycling time integration. There are two time steps used in the coupled pro
cedure, ?t for the fluid flow and ?tDfor the particles. Since ?tDis generally
smaller than ?t, it has to be reduced to ?tsso that the ratio between ?t and ?ts
is an integer ns:
?ts=?t
ns
where ?·? denotes an integer roundoffoperator. This basically gives rise to a so
called subcyclingtime integrationfor the discrete element part; in one step of the
fluid computation, nssubsteps of integration are performed for Eq. (22) using
thetime step ?ts; whilst thehydrodynamicforcesFfandTfare keptunchanged
during the subcycling.
2. The dynamic equation in the lattice coordinate system. Since the lattice
Boltzmannequationis implementedin the lattice coordinatesystem in this work,
the dynamic equation (22) should be implemented in the same way. It can be de
(ns= ??t/?tD? + 1)
(23)
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Advances in Computational Modelling of MultiPhysics in ParticleFluid Systems63
rived that in the lattice coordinate system Eq. (22) takes the form of
¯ m¯ a + ¯ cd¯ v =¯Fc+¯Ff+ ¯ m¯ g
(24)
where
⎧
⎩
⎨
¯ m = m/ρs?x2¯ v = v/c
¯ a = a?t/c;
¯ cd= c?xcd;
¯ g = g?t/c
¯Ft= Ft/(ρ0c2?x)
4 ThermalParticle Interactions
4.1 Convective Heat Transfer
If an additional field, thermal, exists, the Thermal Lattice Boltzmann Method is
adopted to account for heat exchange between particles and between particles and
the surrounding fluid. In the doublepopulationmodel [25] , in addition to the evol
ution equation for fluid flow (Eq. (4)), an internal energy distribution function is
also introduced to solve thermodynamics, as described by the following evolution
equation:
¯ gi(x+ei?t,t +?t)− ¯ gi(x,t) = −
1
τg+ 0.5
?¯ gi(x,t) − geq
i(x,t)?−
τg
τg+ 0.5fiZi
(25)
where
¯fi= fi+0.5
τf
(fi− feq
i)
(26)
¯ gi= gi+0.5
τg
(gi− geq
i) +?t
2fiZi
(27)
inwhichgiis theinternalenergydistributionfunctionwithdiscretevelocityeialong
the ith direction; geq
i
is the corresponding equilibrium distribution function; τgis
the internal energyrelaxation time which controls the rate of change to equilibrium.
The term Zi = (ei− v) · [∂v/∂t + (ei· ∇)v] represents the effect of viscous
heating and can be expressed as
Zi=(ei− v) · [v(x + ei?t,t + ?t) − v(x,t)]
?t
(28)
For gas flow, the lattice speed c can be defined as
c =
?
3RTm
where R is the gas constant and Tmthe average temperature.
Page 14
64Y.T. Feng et al.
The internal energy equilibrium distribution functions geq
D2Q9 model as
⎧
⎪⎪⎪⎪⎪⎩
in which wiare the weightingfactors with the same valuesas definedin Section 3.1;
and ρ? denotes the internal energy.
The internal energy per unit mass ? and heat flux q can be calculated from the
zeroth and first order moments of the distribution functions as
??
To evaluate the convective heat exchange between a solid particle and the sur
rounding fluid, the following approach is proposed in this work.
Assume that a solid particle is mapped onto the lattice by a set of lattice nodes.
The nodes on the boundary of the solid region are termed boundary nodes. If i is a
link (or direction) between a boundary node and a fluid node, the convective heat
exchange between the solid particle and the surrounding fluid can be evaluated as
?
where gi(x,t+) denotes the post collision distribution at the boundary node x, and
−i is the opposite direction of i.
Our numericaltests show that the ThermalLattice Boltzmann Methodcan model
natural or forced convection in particulate systems well, but is not efficient to sim
ulate heat conductionbetween particles, particularly for systems comprising a large
number of particles. For this reason, a novel numerical approach, termed the Dis
crete ThermalElementMethod[14],is proposed,whichis outlinedin the following.
i
are defined in the
⎪⎪⎪⎪⎪⎨
geq
0= w0ρ?
geq
i
= wiρ?
geq
i
= wiρ?
?
?3
?
−3(v · v)
2c2
2+3(ei· v)
3 +6(ei· v)
?
2c2
+9(ei· v)2
2c4
+9(ei· v)2
2c4
−3(v · v)
2c2
−3(v · v)
2c2
?
(i = 1,2,3,4)
(i = 5,6,7,8)
c2
?
(29)
ρ? =
?
¯ gi −?t
2
?
fiZi;
q =
ei¯ gi− ρ?v −?t
2
?
eifiZi
?
τg
τg+ 0.5
(30)
q =
i
[g−i(x,t) − gi(x,t+)]
(31)
4.2 Conductive Heat Transfer in Particles
Consider a circular particle of radius R in a particle assembly that is in contact
with n neighboring particles, as shown in Figure 2a, in which heat is conducted
only through the n contact zones on the boundary of the particle, and the rest of the
particle boundary is fully insulated. A polar coordinate system (r,θ) is established
with the origin set at the centre of the particle. Each contact zone (assumed to be an
arc) can be described by the position angle θ and the contact angle α in Figure 2b.
In general situations the position angles are well spaced along the boundaryand the
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Advances in Computational Modelling of MultiPhysics in ParticleFluid Systems65
q1
q1
qi
qn
qj
(a) A circular particle in an assembly
θi
θi
( )
iq θ
αi αi
θj
αj
αj
( )
jq θ
Qj
Qn
Qm
Q ( )
θ
ii
ii
i
Rqd
θ α+
∫
θ α−
θ=
(resultant flux)
T
n
0
∂
∂
=
T
n
0
∂
∂
=
(b) Thermal particle representation
Fig. 2 Heat conduction in a simple particle system
contact angles αiare small. The position and contact angles of the n contact zones
constitute the local element (contact) configuration of the particle. Furthermore, if
the heatfluxalongtheithcontactzoneis describedbya (local)continuousfunction
qi(θ), then the heat flux on the boundary of the particle can be represented as
?qi(θ − θi)
The heat flux equilibrium in the particle requires
q(θ) =
θi− αi≤ θ ≤ θi+ αi
otherwise
(i = 1,...,n)
0
(32)
?2π
0
q(θ)dθ = 0 (33)
The temperature distribution T(r,θ) within the particle domain ? = {(r,θ) :
0 ≤ r ≤ R;0 ≤ θ ≤ 2π} is governed by the Laplace equation as:
?κ?T = 0
∂n= q(θ)
in ?
κ∂T
on ∂?
(34)
where κ is the thermal conductivity; ∂? denotes the boundary (circumference) of
the particle; and ∂T/∂n is the temperature gradient along the normal direction to
the boundary. Then the temperature at any point (r,θ) ∈ ? can be expressed as
?2π
where Tois the temperature at the centre, i.e. To= T(0,0).
T(r,θ) = −
R
2πκ
0
q(φ)ln
?
1−2r
Rcos(θ−φ)+
?r
R
?2?
dφ+To, (r,θ) ∈ ?
(35)