Lattice boltzmann method ammar

APPLICATION OF LATTICEAPPLICATION OF LATTICE
BOLTZMANN METHODBOLTZMANN METHOD
Done by: Ammar Al-khalidiDone by: Ammar Al-khalidi
M.Sc. StudentM.Sc. Student
University of JordanUniversity of Jordan
8/Feb/20068/Feb/2006
Jordanian-Germany winter
academy 2006

02/02/15 Jordanian-Germany winter academy 2006 2
Table of contentTable of content
• Introduction to Lattice Boltzmann methodIntroduction to Lattice Boltzmann method
– What is the Lattice Boltzmann Method?What is the Lattice Boltzmann Method?
– What is the basic idea of Lattice Boltzmann method?What is the basic idea of Lattice Boltzmann method?
– Why is the Lattice Boltzmann Method Important?Why is the Lattice Boltzmann Method Important?
– Comparison between Lattice Boltzmann and conventional numerical schemesComparison between Lattice Boltzmann and conventional numerical schemes
– Lattice BoltzmannLattice Boltzmann important featuresimportant features
• Lattice Boltzmann methodLattice Boltzmann method
– Lattice Boltzmann equations.Lattice Boltzmann equations.
– Boundary Conditions in the LBMBoundary Conditions in the LBM
– Some boundary treatments to improve the numerical accuracy of the LBMSome boundary treatments to improve the numerical accuracy of the LBM
• Application of Lattice BoltzmannApplication of Lattice Boltzmann
– Lattice Boltzmann simulation of fluid flowsLattice Boltzmann simulation of fluid flows
– Driven cavity flows resultsDriven cavity flows results
– Flow over a backward-facing stepFlow over a backward-facing step
– Flow around a circular cylinderFlow around a circular cylinder
– Flows in Complex GeometriesFlows in Complex Geometries
– Simulation of Fluid TurbulenceSimulation of Fluid Turbulence
– Direct numerical simulationDirect numerical simulation
– LBM models for turbulent flowsLBM models for turbulent flows
– LBM simulations of multiphase and multicomponent flowsLBM simulations of multiphase and multicomponent flows

– Why Lattice Boltzmann Method Important?Why Lattice Boltzmann Method Important?
– some boundary treatments to improve the numerical accuracy of the LBMsome boundary treatments to improve the numerical accuracy of the LBM
– driven cavity flows resultsdriven cavity flows results
– flow over a backward-facing stepflow over a backward-facing step
– flow around a circular cylinderflow around a circular cylinder
– Lbm models for turbulent flowsLbm models for turbulent flows
– lbm simulations of multiphase and multicomponent flowslbm simulations of multiphase and multicomponent flows

What is the Lattice BoltzmannWhat is the Lattice Boltzmann
Method?Method?
• The lattice Boltzmann method is aThe lattice Boltzmann method is a
powerful technique for the computationalpowerful technique for the computational
modeling of a wide variety of complex fluidmodeling of a wide variety of complex fluid
flow problems including single andflow problems including single and
multiphase flow in complex geometries. Itmultiphase flow in complex geometries. It
is a discrete computational method basedis a discrete computational method based
upon the Boltzmann equationupon the Boltzmann equation

What is the basic idea ofWhat is the basic idea of
Lattice Boltzmann method?Lattice Boltzmann method?
• Lattice Boltzmann method considers a typicalLattice Boltzmann method considers a typical
volume element of fluid to be composed of avolume element of fluid to be composed of a
collection of particles that are represented by acollection of particles that are represented by a
particle velocity distribution function for eachparticle velocity distribution function for each
fluid component at each grid point. The time isfluid component at each grid point. The time is
counted in discrete time steps and the fluidcounted in discrete time steps and the fluid
particles can collide with each other as theyparticles can collide with each other as they
move, possibly under applied forces. The rulesmove, possibly under applied forces. The rules
governing the collisions are designed such thatgoverning the collisions are designed such that
the time-average motion of the particles isthe time-average motion of the particles is
consistent with the Navier-Stokes equation.consistent with the Navier-Stokes equation.

Why Lattice BoltzmannWhy Lattice Boltzmann
Method Important?Method Important?
• This method naturally accommodates a varietyThis method naturally accommodates a variety
of boundary conditions such as the pressureof boundary conditions such as the pressure
drop across the interface between two fluids anddrop across the interface between two fluids and
wetting effects at a fluid-solid interface. It is anwetting effects at a fluid-solid interface. It is an
approach that bridges microscopic phenomenaapproach that bridges microscopic phenomena
with the continuum macroscopic equations.with the continuum macroscopic equations.
• Further, it can model the time evolution ofFurther, it can model the time evolution of
systems.systems.

Comparison between lattice BoltzmannComparison between lattice Boltzmann
and conventional numerical schemesand conventional numerical schemes
• The lattice Boltzmann method is based onThe lattice Boltzmann method is based on
microscopic models and macroscopic kineticmicroscopic models and macroscopic kinetic
equations. The fundamental idea of the LBM isequations. The fundamental idea of the LBM is
to construct simplified kinetic models thatto construct simplified kinetic models that
incorporate the essential physics of processesincorporate the essential physics of processes
so that the macroscopic averaged propertiesso that the macroscopic averaged properties
obey the desired macroscopic equations.obey the desired macroscopic equations.
• Unlike conventional numerical schemes basedUnlike conventional numerical schemes based
on discriminations of macroscopic continuumon discriminations of macroscopic continuum
equations.equations.

lattice Boltzmannlattice Boltzmann importantimportant
featuresfeatures
• The kinetic nature of the LBM introduces threeThe kinetic nature of the LBM introduces three
important features that distinguish it from otherimportant features that distinguish it from other
numerical methods.numerical methods.
1.1. First, the convection operator (or streamingFirst, the convection operator (or streaming
process) of the LBM in phase space (or velocityprocess) of the LBM in phase space (or velocity
space) is linear.space) is linear.
2.2. Second, the incompressible Navier-Stokes (NS)Second, the incompressible Navier-Stokes (NS)
equations can be obtained in the nearlyequations can be obtained in the nearly
incompressible limit of the LBM.incompressible limit of the LBM.
3.3. Third, the LBM utilizes a minimal set of velocitiesThird, the LBM utilizes a minimal set of velocities
in phase space.in phase space.

The LBM originated from lattice gas (LG)The LBM originated from lattice gas (LG)
automata, a discrete particle kinetics utilizingautomata, a discrete particle kinetics utilizing
a discrete lattice and discrete time. The LBMa discrete lattice and discrete time. The LBM
can also be viewed as a special finitecan also be viewed as a special finite
difference scheme for the kinetic equation ofdifference scheme for the kinetic equation of
the discrete-velocity distribution function.the discrete-velocity distribution function.
The idea of using the simplified kinetic equationThe idea of using the simplified kinetic equation
with a single-particle speed to simulate fluidwith a single-particle speed to simulate fluid
flows was employed by Broadwell (Broadwellflows was employed by Broadwell (Broadwell
1964) for studying shock structures.1964) for studying shock structures.

– lbm simulations of multiphase and multicomponent flowslbm simulations of multiphase and multicomponent flows

LATTICE BOLTZMANNLATTICE BOLTZMANN
EQUATIONSEQUATIONS
• There are several ways to obtain the latticeThere are several ways to obtain the lattice
Boltzmann equation (LBE) from either discreteBoltzmann equation (LBE) from either discrete
velocity models or the Boltzmann kinetic equation.velocity models or the Boltzmann kinetic equation.
• There are also several ways to derive theThere are also several ways to derive the
macroscopic Navier-Stokes equations from themacroscopic Navier-Stokes equations from the
LBE. Because the LBM is a derivative of the LGLBE. Because the LBM is a derivative of the LG
method.method.
– LBE: An Extension of LG AutomataLBE: An Extension of LG Automata

EQUATIONSEQUATIONS
),...,1,0()),,((),(),( Mitxftxfttxexf iiii =Ω+=∆+∆+
)),(( txfii Ω=Ω
where fi is the particle velocity distribution
function along the ith direction;
Where Ω is the collision operator which
represents the rate of change of fi resulting from
collision. Δt and Δx are time and space
increments, respectively.

Boundary Conditions in the LBM
• Wall boundary conditions in the LBM were originallyWall boundary conditions in the LBM were originally
taken from the LG method. For example, a particletaken from the LG method. For example, a particle
distribution function bounce-back scheme (Wolframdistribution function bounce-back scheme (Wolfram
1986, Lavall´ee et al 1991) was used at walls to obtain1986, Lavall´ee et al 1991) was used at walls to obtain
no-slip velocity conditions.no-slip velocity conditions.
• By the so-called bounce-back scheme, we mean thatBy the so-called bounce-back scheme, we mean that
when a particle distribution streams to a wall node, thewhen a particle distribution streams to a wall node, the
particle distribution scatters back to the node it cameparticle distribution scatters back to the node it came
from. The easy implementation of this no-slip velocityfrom. The easy implementation of this no-slip velocity
condition by the bounce-back boundary schemecondition by the bounce-back boundary scheme
supports the idea that the LBM is ideal for simulatingsupports the idea that the LBM is ideal for simulating
fluid flows in complicated geometries, such as flowfluid flows in complicated geometries, such as flow
through porous media.through porous media.

Boundary Conditions in the LBM
• For a node near a boundary, some of itsFor a node near a boundary, some of its
neighboring nodes lie outside the flow domain.neighboring nodes lie outside the flow domain.
Therefore the distribution functions at these no-Therefore the distribution functions at these no-
slip nodes are not uniquely defined. The bounce-slip nodes are not uniquely defined. The bounce-
back scheme is a simple way to fix theseback scheme is a simple way to fix these
unknown distributions on the wall node. On theunknown distributions on the wall node. On the
other hand, it was found that the bounce-backother hand, it was found that the bounce-back
condition is only first-order in numerical accuracycondition is only first-order in numerical accuracy
at the boundaries (Cornubert et al 1991, Zieglerat the boundaries (Cornubert et al 1991, Ziegler
1993, Ginzbourg & Adler 1994).1993, Ginzbourg & Adler 1994).

some boundary treatments tosome boundary treatments to
improve the numerical accuracy ofimprove the numerical accuracy of
the LBMthe LBM
• To improve the numerical accuracy of the LBM, otherTo improve the numerical accuracy of the LBM, other
boundary treatments have been proposed. Skordosboundary treatments have been proposed. Skordos
(1993) suggested including velocity gradients in the(1993) suggested including velocity gradients in the
equilibrium distribution function at the wall nodes.equilibrium distribution function at the wall nodes.
• Noble et al (1995) proposed using hydrodynamicNoble et al (1995) proposed using hydrodynamic
boundary conditions on no-slip walls by enforcing aboundary conditions on no-slip walls by enforcing a
pressure constraint.pressure constraint.
• Inamuro et al (1995) recognized that a slip velocity nearInamuro et al (1995) recognized that a slip velocity near
wall nodes could be induced by the bounce-backwall nodes could be induced by the bounce-back
scheme and proposed to use a counter slip velocity toscheme and proposed to use a counter slip velocity to
cancel that effect.cancel that effect.
• Other boundary treatmentsOther boundary treatments

– LBM simulations of multiphase and multicomponent flowsLBM simulations of multiphase and multicomponent flows

SIMULATION OF FLUID FLOWSSIMULATION OF FLUID FLOWS
• DRIVEN CAVITY FLOWS:DRIVEN CAVITY FLOWS:
The fundamental characteristics of the 2-D cavity flow are theThe fundamental characteristics of the 2-D cavity flow are the
emergence of a large primary vortex in the center and twoemergence of a large primary vortex in the center and two
secondary vortices in the lower corners.secondary vortices in the lower corners.
The lattice Boltzmann simulation of the 2-D driven cavity byThe lattice Boltzmann simulation of the 2-D driven cavity by
Hou et al (1995) covered a wide range of ReynoldsHou et al (1995) covered a wide range of Reynolds
numbers from 10 to 10,000. They carefully comparednumbers from 10 to 10,000. They carefully compared
simulation results of the stream function and the locationssimulation results of the stream function and the locations
of the vortex centers with previous numerical simulationsof the vortex centers with previous numerical simulations
and demonstrated that the differences of the values of theand demonstrated that the differences of the values of the
stream function and the locations of the vortices betweenstream function and the locations of the vortices between
the LBM and other methods were less than 1%. Thisthe LBM and other methods were less than 1%. This
difference is within the numerical uncertainty of thedifference is within the numerical uncertainty of the
solutions using other numerical methods.solutions using other numerical methods.

DRIVEN CAVITY FLOWS
RESULTS

FLOWOVER A BACKWARD-
FACING STEP
• The two-dimensional symmetric sudden expansion
channel flow was studied by Luo (1997) using the LBM.
The main interest in Luo’s research was to study the
symmetry-breaking bifurcation of the flow when
Reynolds number increases.
• In this simulation, an asymmetric initial perturbation was
introduced and two different expansion boundaries,
square and sinusoidal, were used. This simulation
reproduced the symmetric-breaking bifurcation for the
flow observed previously, and obtained the critical
Reynolds number of 46.19. This critical Reynolds
number was compared with earlier simulation and
experimental results of 47.3.

FLOW AROUND A CIRCULARFLOW AROUND A CIRCULAR
CYLINDERCYLINDER
• The flow around a two-dimensional circular cylinder wasThe flow around a two-dimensional circular cylinder was
simulated using the LBM by several groups of people.simulated using the LBM by several groups of people.
• The flow around an octagonal cylinder was also studiedThe flow around an octagonal cylinder was also studied
(Noble et al 1996). Higuera&Succi (1989) studied flow(Noble et al 1996). Higuera&Succi (1989) studied flow
patterns for Reynolds number up to 80.patterns for Reynolds number up to 80.
– At Re =52.8, they found that the flow became periodicAt Re =52.8, they found that the flow became periodic
after a long initial transient.after a long initial transient.
– For Re = 77.8, a periodic shedding flow emerged.For Re = 77.8, a periodic shedding flow emerged.
– They compared Strouhal number, flow-separationThey compared Strouhal number, flow-separation
angle, and lift and drag coefficients with previousangle, and lift and drag coefficients with previous
experimental and simulation results, showingexperimental and simulation results, showing
reasonable agreement.reasonable agreement.

Flows in Complex Geometries
• An attractive feature of the LBM is that the no-An attractive feature of the LBM is that the no-
slip bounce-back LBM boundary condition costsslip bounce-back LBM boundary condition costs
little in computational time. This makes the LBMlittle in computational time. This makes the LBM
very useful for simulating flows in complicatedvery useful for simulating flows in complicated
geometries, such as flow through porous media,geometries, such as flow through porous media,
where wall boundaries are extremelywhere wall boundaries are extremely
complicated and an efficient scheme for handingcomplicated and an efficient scheme for handing
wall-fluid interaction is essential.wall-fluid interaction is essential.

Simulation of Fluid Turbulence
• A major difference between the LBM andA major difference between the LBM and
the LG method is that the LBM can bethe LG method is that the LBM can be
used for smaller viscosities.used for smaller viscosities.
• Consequently the LBM can be used forConsequently the LBM can be used for
direct numerical simulation (DNS) of highdirect numerical simulation (DNS) of high
Reynolds number fluid flows.Reynolds number fluid flows.

DIRECT NUMERICALDIRECT NUMERICAL
SIMULATIONSIMULATION
• To validate the LBM for simulating turbulent flows, MartinezTo validate the LBM for simulating turbulent flows, Martinez
et al (1994) studied decaying turbulence of a shear layeret al (1994) studied decaying turbulence of a shear layer
using both the pseudospectral method and the LBM.using both the pseudospectral method and the LBM.
• The initial shear layer consisted of uniform velocityThe initial shear layer consisted of uniform velocity
reversing sign in a very narrow region. The initial Reynoldsreversing sign in a very narrow region. The initial Reynolds
umber was 10,000. they carefully compared the spatialumber was 10,000. they carefully compared the spatial
distribution, time evolution of the stream functions, and thedistribution, time evolution of the stream functions, and the
vorticity fields. Energy spectra as a function of time, smallvorticity fields. Energy spectra as a function of time, small
scale quantities, were also studied. The correlationscale quantities, were also studied. The correlation
between vorticity and stream function was calculated andbetween vorticity and stream function was calculated and
compared with theoretical predictions. They concluded thatcompared with theoretical predictions. They concluded that
the LBE method provided a solution that was accurate .the LBE method provided a solution that was accurate .

DIRECT NUMERICALDIRECT NUMERICAL
SIMULATION ResultsSIMULATION Results

LBM MODELS FOR TURBULENT
FLOWS
• As in other numerical methods for solving the Navier-StokesAs in other numerical methods for solving the Navier-Stokes
equations, a subgrid-scale (SGS) model is required in the LBM toequations, a subgrid-scale (SGS) model is required in the LBM to
simulate flows at very high Reynolds numbers. Direct numericalsimulate flows at very high Reynolds numbers. Direct numerical
simulation is impractical due to the time and memory constraintssimulation is impractical due to the time and memory constraints
required to resolve the smallest scales (Orszag & Yakhot 1986).required to resolve the smallest scales (Orszag & Yakhot 1986).
• Hou et al (1996) directly applied the subgrid idea in theHou et al (1996) directly applied the subgrid idea in the
Smagorinsky model to the LBM by filtering the particle distributionSmagorinsky model to the LBM by filtering the particle distribution
function and its equation infunction and its equation in particle velocity distribution EquationEquation
using a standard box filter.using a standard box filter.
• This simulation demonstrated the potential of the LBM SGS modelThis simulation demonstrated the potential of the LBM SGS model
as a useful tool for investigating turbulent flows in industrialas a useful tool for investigating turbulent flows in industrial
applications of practical importance.applications of practical importance.

LBM SIMULATIONS OF
MULTIPHASE AND
MULTICOMPONENT FLOWS
• The numerical simulation of multiphase and
multicomponent fluid flows is an interesting and
challenging problem because of difficulties in modeling
interface dynamics and the importance of related
engineering applications, including flow through porous
media, boiling dynamics, and dendrite formation.
Traditional numerical schemes have been successfully
used for simple interfacial boundaries .
• The LBM provides an alternative for simulating
complicated multiphase and multicomponent fluid flows,
in particular for three-dimensional flows.

• Method of Gunstensen et al:Method of Gunstensen et al:
Gunstensen et al (1991) were the first to develop theGunstensen et al (1991) were the first to develop the
multicomponent LBM method.multicomponent LBM method.
• It was based on the two-component LG model proposed byIt was based on the two-component LG model proposed by
Rothman & Keller (1988). Method of Shan & ChenRothman & Keller (1988). Method of Shan & Chen
Later, Grunau et al (1993) extended this model to allowLater, Grunau et al (1993) extended this model to allow
variations of density and viscosity.variations of density and viscosity.
Shan & Chen (1993) and Shan & Doolen (1995) usedShan & Chen (1993) and Shan & Doolen (1995) used
microscopic interactions to modify the surface-microscopic interactions to modify the surface-
tension–related collision operator for which thetension–related collision operator for which the
surface interface can be maintained automatically.surface interface can be maintained automatically.

• Free Energy Approach
– The above multiphase and multicomponent lattice
Boltzmann models are based on phenomenological
models of interface dynamics and are probably most
suitable for isothermal multicomponent flows.
– One important improvement in models using the free-
energy approach (Swift et al 1995, 1996) is that the
equilibrium distribution can be defined consistently
based on thermodynamics.
– Consequently, the conservation of the total energy,
including the surface energy, kinetic energy, and
internal energy can be properly satisfied (Nadiga &
Zaleski 1996).

Numerical Verification andNumerical Verification and
ApplicationsApplications
• Two fundamental numerical tests associated with interfacial
phenomena have been carried out using the multiphase and
multicomponent lattice Boltzmann models.
• The first test, the lattice Boltzmann models were used to verify
Laplace’s formula by measuring the pressure difference and surface
tention between the inside and the outside of a droplet . The
simulated value of surface tension has been compared with
theoretical predictions, and good agreement was reported
(Gunstensen et al 1991, Shan and Chen 1993, Swift et al 1995).
• In the second test of LBM interfacial models, the oscillation of a
capillary wave was simulated (Gunstensen et al 1991, Shan & Chen
1994, Swift et al 1995). A sine wave displacement of a given wave
vector was imposed on an interface that had reached equilibrium.
The resulting dispersion relation was measured and compared with
the theoretical prediction (Laudau & Lifshitz 1959) Good agreement
was observed, validating the LBM surface tension models.

SIMULATION OF HEATSIMULATION OF HEAT
TRANSFER AND REACTION-TRANSFER AND REACTION-
DIFFUSIONDIFFUSION
• The lattice Bhatnagar-Gross-Krook (LBGK)The lattice Bhatnagar-Gross-Krook (LBGK)
models for thermal fluids have been developedmodels for thermal fluids have been developed
by several groups. To include a thermal variable,by several groups. To include a thermal variable,
such as temperature, Alexander et al (1993)such as temperature, Alexander et al (1993)
used a two-dimensional 13-velocity model on theused a two-dimensional 13-velocity model on the
hexagonal lattice. In this work, the internalhexagonal lattice. In this work, the internal
energy per unit mass was defined through theenergy per unit mass was defined through the
second-order moment of the distributionsecond-order moment of the distribution
function.function.

• Two limitations in multispeed LBM thermal models severely restrictTwo limitations in multispeed LBM thermal models severely restrict
their application. First, because only a small set of velocities is used,their application. First, because only a small set of velocities is used,
the variation of temperature is small. Second, all existing LBMthe variation of temperature is small. Second, all existing LBM
models suffer from numerical instability (McNamara et al 1995),models suffer from numerical instability (McNamara et al 1995),
• On the other hand, it is difficult for the active scalar approach toOn the other hand, it is difficult for the active scalar approach to
incorporate the correct and full dissipation function. Twoincorporate the correct and full dissipation function. Two
dimensional Rayleigh-Bénard (RB) convection was simulated usingdimensional Rayleigh-Bénard (RB) convection was simulated using
this active scalar scheme for studying scaling laws (Bartoloni et althis active scalar scheme for studying scaling laws (Bartoloni et al
1993) and probability density functions (Massaioli et al 1993) at high1993) and probability density functions (Massaioli et al 1993) at high
Prandtl numbers.Prandtl numbers.
• Two dimensional free-convective cavity flow was also simulatedTwo dimensional free-convective cavity flow was also simulated
(Eggels & Somers 1995), and the results compared well with(Eggels & Somers 1995), and the results compared well with
benchmark data. Two-D and 3-D Rayleigh-Bénard convectionsbenchmark data. Two-D and 3-D Rayleigh-Bénard convections
were carefully studied by Shan (1997) using a passive scalarwere carefully studied by Shan (1997) using a passive scalar
temperature equation and a Boussinesq approximation. This scalartemperature equation and a Boussinesq approximation. This scalar
equation was derived based on the two-component model of Shanequation was derived based on the two-component model of Shan
& Chen (1993). The calculated critical Rayleigh number for the RB& Chen (1993). The calculated critical Rayleigh number for the RB
convection agreed well with theoretical predictions. The Nusseltconvection agreed well with theoretical predictions. The Nusselt
number as a function of Rayleigh number for the 2-D simulation wasnumber as a function of Rayleigh number for the 2-D simulation was
in good agreement with previous numerical simulation using otherin good agreement with previous numerical simulation using other
methodsmethods

Applications in commercial
programs
• ex direct building energy
simulation based on large eddy
techniques and lattice boltzmann
methods

Two orthogonal slice planes of the
averaged velocity field

Turbulent flow, orthogonal slice
planes of the averaged velocity
field

Thank You

Lattice boltzmann method ammar

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