LBM Modelling of Fluid Flow

Mechanical Engineering Department
Jadavpur University
Applications
of Lattice
Boltzmann
Method in
Dynamic
Modelling of
Fluid Flows
Angshuman Pal
B. Mechanical E – IV
001411201048
Guide: Prof. Himadri
Chattopadhyay
Heat Power Specialization

Applications of Lattice Boltzmann Method in Dynamic Modelling
of Fluid Flows
2
Contents
1 Abstract..................................................................................................................................... 3
2 Introduction .............................................................................................................................. 4
2.1 Computational Fluid Dynamics........................................................................................... 4
2.2 The Navier-Stokes equation ............................................................................................... 4
2.3 Discretization..................................................................................................................... 5
2.4 Lattice Boltzmann Laws...................................................................................................... 5
3 The Lattice Gas Automata Theory .............................................................................................. 6
3.1 Cellular Automata .............................................................................................................. 6
3.2 The FHP Model................................................................................................................... 6
3.3 Streaming and Collision Process in LGM ............................................................................. 7
3.3.1 Inter-Particle Collision ................................................................................................ 7
3.3.2 Particle-Solid Collision ................................................................................................ 8
4 The Lattice Boltzmann Method .................................................................................................. 9
4.1 Boltzmann distribution function......................................................................................... 9
4.2 The Boltzmann Transport Equation .................................................................................... 9
4.3 Framework of LBM........................................................................................................... 10
4.3.1 Lattice Arrangements ............................................................................................... 10
4.3.2 Directional Densities and Velocities.......................................................................... 11
4.3.3 Streaming................................................................................................................. 11
4.3.4 Collision.................................................................................................................... 11
4.3.5 Boundary Conditions ................................................................................................ 12
4.4 Reynolds Number............................................................................................................. 13
5 Algorithm ................................................................................................................................ 14
6 Validation - Flow through Circular Pipe.................................................................................... 15
6.1 Poiseuille’s parabolic profile............................................................................................. 15
6.2 Flow Parameters.............................................................................................................. 15
6.3 Velocity distribution......................................................................................................... 16
7 Conclusion............................................................................................................................... 18
8 Future Scope for Work............................................................................................................. 19
9 References............................................................................................................................... 20
10 Acknowledgement............................................................................................................... 20

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3
1 Abstract
Computational Fluid Dynamics techniques are an important development in the study of fluid
behaviour where complicated differential equations involving complex geometries and irregular
flows can be solved using iterative numerical techniques. Such techniques are being used extensively
in the field of fluid mechanics and heat transfer. Solving the Navier Stokes equation for viscous fluids
is of particular importance in this field.
The Lattice Boltzmann Method is an alternative to the commonly used discretization principle for
solving the Navier Stokes equation. It deals with the properties of fluid particles on a micro scale and
subsequently uses them to generate the model of the entire flow domain on a macro scale. The flow
domain is broken up into lattices inhabited by finitely many fluid particles. Governed by rules of
streaming and collision, all the particles together generate the model of the entire flow.
A detailed study is performed on the LBM model and its underlying principles. A program for its
execution is written in MATLAB environment. Simulations are run on a simple geometry involving
steady flow through a circular pipe. The results obtained are analysed and verified against the
expected results given by mathematical models and actual experimentation.
Keywords: Lattice Boltzmann Method, Computational Fluid Dynamics, Navier Stokes equation, BGKW
approximation.

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2 Introduction
2.1 Computational Fluid Dynamics
The analysis of the mechanics of a fluid system (liquid or gaseous) requires the solution of a number
of governing equations involving intensive properties of the fluid which describe the dynamic motion
of the fluid. These governing laws are expressed in the form of differential equations which may be
solved upon imposition of correct boundary conditions to the equation.
For simplified flows the governing differential equations are accordingly simple and can be solved
using conventional techniques. But engineers and mathematicians observed that practical
applications of the equations hardly ever involved equations so simple so the need was felt for
devising an alternate technique which would be capable of solving equations of complex nature.
With the advent of computation and its introduction in engineering problems, an alternative
approach was envisioned. Numerical methods were developed for solving differential equations
using non-analytical approaches. All such methods applied in the field of Fluid Mechanics came to be
established as the subject of Computational Fluid Dynamics.
2.2 The Navier-Stokes equation
Newton’s second law, postulating the conservation of momentum, when applied to fluid flows,
results in the establishment of the Navier-Stokes equation. Named after Claude-Louis Navier and
Goerge Gabriel Stokes, the equation uses the Eulerian approach in fluid mechanics to balance the
forces acting upon a control volume in a flow field including the shear stresses generated between
layers of the fluid. The resulting equation is the governing equation for a viscous flow.
The Navier-Stokes equation for a Newtonian incompressible liquid can be expressed as a differential
equation as follows:
𝜌
𝐷𝑉⃗
𝐷𝑡
= 𝜌𝑔 − ∇⃗⃗ 𝑝 + 𝜇∇2
𝑉
ρ = density of the fluid
𝑉⃗ = velocity field of the fluid
p = pressure
𝜇 = coefficient of viscosity
The Navier-Stokes (NS) equation is the primary step to solving many fluid flow based problems. In
case of flow through a complex geometrical domain or involving high irregularities, computational
methods are employed to obtain solutions to the NS equation.

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2.3 Discretization
The fundamental step in applying computational methods to solving the NS equation is the process
of discretization. In mathematics, discretization is the process of transferring continuous functions,
models, variables, and equations into discrete counterparts. The entire flow domain, which is a
continuous region in reality, is divided into finitely small regions of specified size. Flow properties
like velocity and acceleration are attributed to these finite discrete regions, and the differential
equation is solved on the basis of this assumption.
2.4 Lattice Boltzmann Laws
The Lattice Boltzmann gas laws are based on a probabilistic approach towards the behaviour of
fluids. The entire body of fluid is assumed to consist of finitely small discrete particles and each
particle has a particular value of the parameter being measured associated to it (e.g. temperature,
velocity, etc). On a microscale, these parameters are specified for each particle and a frequency
distribution is created for the specific property. Once the frequency distribution is obtained, the
macroscale properties are calculated by finding the mean of the frequency distribution.
All properties like density, velocity, temperature etc can be calculated in this manner. Density at a
point can be calculated by the simple process of finding the total number of discrete particles
present at a node. Temperature being a scalar quantity, the mean of the frequency distribution of all
particles at a particular coordinate gives the temperature at that location. Since velocity is a vector
quantity, the individual velocity components have to be considered while computing the mean from
the frequency distribution.
Once every lattice point in the domain has been allocated values defining its state, the governing
equations are applied to predict the changes that will occur at that particular lattice after one time
step. The fluid properties at the next time step are accordingly calculated. In this way, the algorithm
progresses over time and temporal variation of fluid properties are recorded.

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3 The Lattice Gas Automata Theory
The Lattice Gas Automata Theory is the precursor to the Lattice Boltzmann Method (LBM). The
Lattice Gas Model (LGM) uses an elegant mathematical technique involving cellular automata theory
to predict and represent the behaviour of discrete particles in a fluid medium.
3.1 Cellular Automata
A cellular automaton (CA) is an algorithmic entity that occupies a position on a grid or lattice point in
the fluid medium and interacts with its identical neighbouring lattices. A cellular automaton
examines its own state and the states of some number of its neighbours at any particular time step
and then resets its own state for the next time step according to simple rules. Hence, the rules and
the initial boundary conditions imposed on the group of cellular automata uniquely determine their
evolution in time.
3.2 The FHP Model
Frish, Hasslacher, and Pomeau in 1986 defined a technique of presenting lattice interactions in two-
dimensional flows that could stimulate the NS equation, which came to be known as the FHP model.
Lattice points were located at a distance of one lattice unit (lu) from each other. Every lattice point
was allotted a maximum of six particles, each with a velocity of magnitude 1 lu/ts (lattice unit per
time step), and making an angle of 60° with each other.
Any lattice point can thus be defined as an eight-bit binary string. The first six digits from left
indicated absence (0) or presence (1) of a particle in each of the six directions at that lattice point.
The seventh bit is used for indicating the presence of a solid at that point. The eighth bit is used to
introduce randomness into the system when two alternatives are equally likely during streaming
process.
Figure 1: FHP Unit Velocity Vectors
Source: Sukop, M. C. "DT Thorne, Jr.
Lattice Boltzmann Modeling Lattice
Boltzmann Modeling." (2006).

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3.3 Streaming and Collision Process in LGM
Every lattice point in the flow domain is assigned an eight-bit string representing the state of
velocity. An eight-bit string can be represented by a decimal number between 0 and 255, thus every
point is assigned a decimal number.
These configurations and their respective alterations over a time step are recorded in matrix form. A
MATLAB code is written for this purpose which will give the user the entire mapping between
elements of the old configuration and the new configuration.
3.3.1 Inter-Particle Collision
For proceeding to the next time step, the collision rules governing the lattices need to be defined.
The primary requirement behind framing the collision rules is conserving the momentum of a lattice
point conserved before and after a time step. Collision between particles can involve two or three
particles. The possible outcomes of the collision for two and three particle events are as follows:
For two-particle collisions, the 8th
bit in the string serves the purpose of selecting one out of the two
possible post-collision outcomes. If the 8th
bit is 0, one of the configurations is selected for the next
time step; if the 8th
bit is 1, the other is selected.
In these combinations, we see a need for change of configuration post collision. In other cases, it can
be observed that if momentum is to be conserved then no change at all is possible for those
orientations after the time step. An example is shown below.
Figure 2: Two and Three particle collisions
Source: Sukop, M. C. "DT Thorne, Jr. Lattice Boltzmann Modeling Lattice

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Therefore for inter-particle collision, these are the collision rules to be followed.
3.3.2 Particle-Solid Collision
The necessity for the 7th
bit in the 8-bit string arises when a solid is present. In such a situation the
7th
bit takes the value 1.
In case a solid is present at a lattice point, all the fluid particles as if bounce back from their original
direction to the exactly opposite direction to it. In the modified combination after proceeding one
time step, the 1st
bit takes the value of the previous 4th
bit, the 2nd
takes the value of the 5th
, the 3rd
takes the 6th
, and so on in a cyclic order.
Figure 3: No change in configuration after collision
Source: Sukop, M. C. "DT Thorne, Jr. Lattice Boltzmann Modeling Lattice

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9
4 The Lattice Boltzmann Method
4.1 Boltzmann distribution function
The LBM simulates the flow patterns of fluids not only in a steady state but also of thermodynamic
systems not in equilibrium. The Boltzmann transport equation describes the statistical behaviour of
such a system in thermodynamic inequilibrium. The equation does not consider the particles
individually, but instead considers all of them together in a probabilistic approach.
A manifestation of this approach can be seen when expressing the particles in terms of the velocities
they carry, and its fractional presence among all the particles present.
When 𝑐 𝑥 𝑐 𝑦 and 𝑐 𝑧 are velocities in the X, Y and Z directions, and 𝑓(𝑐) is the corresponding fraction
of particles having velocity between 𝑐 𝑥 + 𝑑𝑐 𝑥, the total domain may be spanned as
∭ 𝑓( 𝑐 𝑥) 𝑓(𝑐 𝑦)𝑓(𝑐 𝑧) 𝑑𝑐 𝑥 𝑑𝑐 𝑦 𝑑𝑐 𝑧 = 1
The appropriate distribution f can be defined as 𝑓( 𝑐 𝑥) = 𝐴𝑒−𝐵𝑐 𝑥
2
where A and B are constants. Over
the three directions, the products are taken and we obtain 𝑓( 𝑐) = 𝐴3
𝑒−𝐵𝑐 𝑥
2
.
4.2 The Boltzmann Transport Equation
This equation is presented in a modified form when we analyze the effect of external force acting on
a flow domain, with the assumption (for the moment) that no inter-particle collisions are taking
place. The forces are conservative in nature and hence the momentum is conserved.
A system is acted upon by a force F. If 𝑓( 𝑟, 𝑐, 𝑡) be the number of particles having velocity c at
coordinates r at the point of time t when external force F comes to work upon it, assuming there is
no inter-particle collision, the pre- and post-collision states may be related as follows:
𝑓( 𝑟 + 𝑐𝑑𝑡, 𝑐 + 𝐹𝑑𝑡, 𝑡 + 𝑑𝑡) − 𝑓( 𝑟, 𝑐, 𝑡) = 0
However in practical applications there are bound to be collisions between particles, and hence the
collision operator is introduced. The modified form of the equation is
𝑓( 𝑟 + 𝑐𝑑𝑡, 𝑐 + 𝐹𝑑𝑡, 𝑡 + 𝑑𝑡) − 𝑓( 𝑟, 𝑐, 𝑡) = Ω𝑓(𝑑𝑟, 𝑑𝑐, 𝑑𝑡)
This equation can be rewritten in terms of momentum as follows:
𝑓( 𝑥 + 𝑑𝑥, 𝑝 + 𝑑𝑝, 𝑡 + 𝑑𝑡) = 𝑓( 𝑥, 𝑝, 𝑡) + [Γ(+)
− Γ(−)
]𝑑𝑥𝑑𝑝𝑑𝑡
The collision operator Ω can be defined over time dt as the difference of the particles which were
not supposed to arrive at ( 𝑥, 𝑝) but end up there due to collision of particles (Γ(+)
), and the
particles which were supposed to arrive at ( 𝑥, 𝑝) but fail to do so because of collision of particles
(Γ(−)
).

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4.3 Framework of LBM
4.3.1 Lattice Arrangements
While dividing a flow domain into discrete lattices for solution through LBM, a standard model of
nomenclature DmQn is followed, where m refers to the number of dimensions in the problem and n
refers to the number of speeds existing at a lattice point. The most common lattice arrangements
are D1Q3 and D1Q5 for 1-D problems, D2Q5 and D2Q9 for 2-D problems, and D3Q15 and D3Q19 for
3-D problems. The approach used in this project involves two-dimensional problems with D2Q9
arrays.
A random lattice selected from the D2Q9 arrangement looks as follows:
The nine directions are represented as C, E, N, W, S, NE, NW, SW and SE. They are represented by
the numbers 0 through 8. Any particle which is present at a lattice point in the flow is capable of
having a fixed velocity along only one of these nine directions.
 For the direction 0, the fixed velocity of a particle is zero; the particle remains static.
 For the directions 1 through 4, the fixed velocity of a particle is 1 lu/ts, and this velocity acts
along the direction specified by the number.
 For the directions 5 through 8, the fixed velocity of a particle is √2 lu/ts, and this velocity acts
along the direction specified by the number.
Figure 4: D2Q9 Lattice and Velocities
Source: Sukop, M. C. "DT Thorne, Jr. Lattice Boltzmann Modeling
Lattice Boltzmann Modeling." (2006).

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4.3.2 Directional Densities and Velocities
The macroscopic density and velocity at a lattice point is calculated with the help of a parameter
called the directional density of particles. It is defined as the number of fluid particles at a particular
lattice point with velocities along a particular direction among the nine possible directions.
For example, 𝑓1( 𝑥, 𝑦) = 100 signifies that there are 100 particles streaming along positive X
direction, each with a velocity of 1 lu/ts, at the location (x,y).
The directional densities at a point are used to calculate all flow parameters at the location.
 Macroscopic density is calculated as follows:
𝜌 = ∑ 𝑓𝑎
𝑎=8
𝑎=0
 Macroscopic velocity along a particular direction is calculated as follows:
𝑢⃗ =
1
𝜌
∑ 𝑓𝑎 𝑒 𝑎⃗⃗⃗⃗
𝑎=8
𝑎=0
where 𝑒 𝑎⃗⃗⃗⃗ is the component of the directional component of velocity along direction a.
At each time step, the directional densities of the lattices are obtained and the density and velocities
are calculated accordingly.
4.3.3 Streaming
Similar to the LGM theory, it is initially assumed that all interactions between particles are absent,
and the progress of the flow occurs only through the process of streaming, also alternatively called
hopping or propagation. The mechanics can be extremely simple and encapsulated by just the
notions of streaming in space and billiard-like collision interactions.
Mathematically, the streaming process may be defined as follows:
𝑓𝑎( 𝑥 + 𝑒 𝑎Δ𝑡, 𝑡 + Δ𝑡) = 𝑓𝑎( 𝑥, 𝑡)
The equation can be explained in words by saying that directional densities at a particular lattice are
carried over to the next lattice located along the direction in question.
4.3.4 Collision
In reality the flow process is hardly devoid of collisions, and there are a lot of inter-particle
interactions going on which affect the macroscopic motion of the fluid. As a result the streaming
equation needs to be modified considering the collision of particles.
If collisions take place between the molecules there will be a net difference between the numbers of
molecules in an interval. The rate of change between final and initial status of the distribution

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12
function is called collision operator (Ω). The equation for evolution of the number of the molecules
can be written as
𝑑𝑓
𝑑𝑡
= Ω(𝑓)
The directional derivative being a function of coordinates, speed and time, the first order differential
equation can be written as
𝑑𝑓 =
𝜕𝑓
𝜕𝑟
𝑑𝑟 +
𝜕𝑓
𝜕𝑐
𝑑𝑐 +
𝜕𝑓
𝜕𝑡
𝑑𝑡
∴
𝑑𝑓
𝑑𝑡
=
𝜕𝑓
𝜕𝑟
𝑐 +
𝜕𝑓
𝜕𝑐
𝜕𝑐
𝜕𝑡
+
𝜕𝑓
𝜕𝑡
In absence of any external force,
𝜕𝑐
𝜕𝑡
= 0, and the expression can be further simplified to
𝜕𝑓
𝜕𝑡
+ 𝑐∇f = Ω
4.3.4.1 The BGKW Approximation
The Bhatnagar-Gross-Krook-Welander (BGKW) approximation is used to define the behaviour of the
colliding particles in a simplified way. A collision operator replaced as Ω =
𝑓 𝑒𝑞−𝑓
𝜏
, where 𝜏 is the
relaxation factor.
Collision of the fluid particles is considered as a relaxation towards a local equilibrium (𝑓 𝑒𝑞
). For a
D2Q9 arrangement, the local equilibrium can be derived as follows:
𝑓𝑎
𝑒𝑞
( 𝑥) = 𝑤 𝑎 𝜌( 𝑥)[1 + 3
𝑒 𝑎⃗⃗⃗⃗ 𝑢⃗
𝑐2
+
9
2
( 𝑒 𝑎⃗⃗⃗⃗ 𝑢⃗ )2
𝑐4
−
3
2
𝑢⃗ 𝑢⃗
𝑐2
]
where 𝑤0 =
4
9
, 𝑤1,2,3,4 =
1
9
, 𝑤5,6,7,8 =
1
36
4.3.5 Boundary Conditions
The boundary conditions pose a considerable problem for LBM models. In a lot of cases, no-slip
boundary conditions are present at walls where the fluid velocity is equal to the wall velocity. In this
case, two problems arise. Firstly, due to absence of lattice points beyond the boundaries it is not
possible to obtain certain directional densities through streaming process. Secondly, the lattice
points are defined in terms of directional derivatives and not the velocities itself. So it becomes
difficult to implement things such as the no-slip conditions.
Periodical boundary conditions are sometimes applicable for flows such as flow past solid cylindrical
objects placed periodically causing wake formation. If the flow conditions are identical between two
or more sets of points in the flow range, the first set of simulations can be repeated over the regions
which follow.

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4.3.5.1 Inlet
The flow inlet boundary conditions are comparatively simpler to impose. Since the components of
the lattice velocities are known along each direction, the directional densities at the inlet section are
calculated and entered so that density and velocity come out as required at the inlet.
4.3.5.2 Solid walls and Bounceback
At any solid wall, five out of the eight non-zero directional densities are obtained from streaming,
but three remain unknown. The density is also unknown as a result. Additionally, since we are
dealing with the boundary we know the exact value of the velocity that will be observed at the
position from the no-slip condition. This gives four unknowns in total.
On the other hand, we have one equation from the density calculation, two equations from
balancing X-direction and Y-direction momenta, and one additional equation from the equilibrium
function. This gives four equations in total.
Four equations and four unknowns are solved, and all the directional densities are obtained.
4.3.5.3 Outlet
The outlet of the flow domain is an open region. Since the properties are unknown beyond the flow
region, streaming is not completely possible. Since no-slip conditions are not valid as well, the
remaining directional densities cannot be derived normally. Due to these limitations a second order
polynomial interpolation is followed to obtain the densities which cannot be obtained otherwise.
𝑓𝑛 = 2𝑓𝑛−1 − 𝑓𝑛−2
4.4 Reynolds Number
The Reynolds number of the flow that is being simulated is a matter of concern while applying the
LBM technique. It is important to maintain identical Reynolds numbers in the model and the actual
prototype.
𝑅𝑒 =
𝜌𝑢𝐿
𝜇
=
𝑢𝐿
𝜈
In most cases the prototype dimension is too large to be modelled in software; hence the
dimensions and velocity are required to be scaled down. In such cases the kinematic viscosity of the
model is altered.
In the flow development region of the fluid the viscosity value plays an important role. It determines
the length and time duration through which the development will take place.
In case of the collision theory used in the LBM technique, the relaxation factor is related to the
kinematic viscosity according to the BGKW approximation. It is given by 𝜈 =
1
3
(𝜏 −
1
2
)

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5 Algorithm
The basic framework of the algorithm has been executed in many programming languages and all
give suitable results. This particular code has been written in MATLAB environment. Some important
features of the code are given below:
 All the lattice points in the domain are assumed to be forming an array of elements. The
array is of three dimensions.
 In order to specify a particular directional density, the value is specified as 𝑓(𝑥, 𝑦, 𝑎), where
x and y denote the coordinates of the point and a denotes the direction considered.
 The inputs to be specified at the beginning of the iterations are the flow region dimensions,
location of solid boundaries, inlet velocity, viscosity of the fluid medium, and number of
iterations.
The following steps followed in formulating the algorithm for the Lattice Boltzmann Method.
 Dimensions, time step and initialization
The necessary scaling is performed to create the model. Time step is specified in accordance
to the expected velocity of the fluid in the domain. If velocity is considerably high then the
time step needs to be small and vice versa.
Initialization is done by making all directional densities zero at all lattice points. The standard
velocities along the nine directions of a D2Q9 array are specified, along with their
components along X and Y directions.
 Creation of time step loop
The initial velocity is specified through the input boundary conditions (directional velocities)
in the flow domain. The number of iterations sought by which time the flow is expected to
have become steady are entered.
 Calculating densities and velocities
At the beginning of each loop, the current density and velocities in X and Y direction are
calculated using the formulae.
 Streaming
In the streaming step, a new matrix is created for the next time step. The directional
densities at all lattice points for the new matrix are imported from the previous matrix.
Boundary conditions are also incorporated.
 Collision
After all the directional densities and velocities have been calculated for the new time step,
the equilibrium function is calculated and the collision matrix terms are added to the original
streaming matrix.
 Iteration
After the new matrix is completed, the old matrix is replaced by the new matrix and the
code proceeds to the next iteration.

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6 Validation - Flow through Circular Pipe
The first simulation is performed upon the simple case of a steady uniform flow entering a circular
pipe with constant flux. The pipe is long and axisymmetric. The LBM algorithm has been applied to
the problem and the different profiles of velocity along the positive X axis (direction of flow) have
been represented.
The pipe is initially empty and begins to be filled at 𝑡 = 0. The directional densities required to
achieve a particular inlet velocity are precalculated and entered as input. The net Y-component of
the inlet velocity (in radial direction) is kept as zero.
6.1 Poiseuille’s parabolic profile
The following assumptions are considered while solving the problem:
 The fluid is incompressible.
 The flow is laminar.
 The fluid is viscous and follows Newton’s law of viscosity.
 The flow entering the channel has a uniform velocity which does not change over time.
 Effects of gravity are ignored.
If the following conditions are met, then the fluid is expected to create a parabolic profile some
region after its entry into the pipe. This intermediate region is called the region of hydrodynamic
development and the steady profile obtained is called Poiseuille’s parabolic profile. It is given by
𝑣 =
1
4𝜇
𝜕𝑝
𝜕𝑥
( 𝑅2
− 𝑟2)
6.2 Flow Parameters
A Reynolds number lying in the laminar region has to be selected for the development of a parabolic
profile from LBM equations. A test value of 330 is selected.
The following values of flow conditions are selected:
u = 35 lu/ts D = 30 lu ν = 3.2 lu2
/ts
The Reynolds number calculated in these low conditions is
𝑅𝑒 =
𝑢𝐷
𝜈
=
30𝑙𝑢 × 35𝑙𝑢/𝑡𝑠
3.2𝑙𝑢2/𝑡𝑠
≈ 330
The chosen value of Reynolds number falls within the range of laminar flow.

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6.3 Velocity distribution
The simulation of the LBM technique is performed on the model and velocity profiles are obtained at
various locations along the axis. A velocity profile is the component of velocity along the axial
direction plotted against the corresponding diametric span
When the fluid enters the pipe it shows a square shape representing uniform velocity of value as
mentioned previously. The profile is shown below.
After the flow attains a steady nature, as expected the velocity distribution assumes a parabolic
shape. The profile is shown below.
Figure 5: Velocity profile at inlet
Figure 6: Velocity profile after development

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17
By equating the volume flux at different sections of the flow, it can be shown that the peak velocity
of the parabola is 1.5 times the average velocity across the section, which is equal to the uniform
velocity at inlet.
𝑢 𝑚𝑎𝑥 =
3
2
𝑢 𝑎𝑣𝑔
In this case the peak velocity in the parabolic profile comes to be 45.03 lu/ts, while the theoretical
value should be 52.5 lu/ts.

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7 Conclusion
The Lattice Boltzmann Method is one of the up and coming techniques for numerical solution of
problems of heat and mass transfer with concentrated applications in the area of fluid mechanics. It
deals with complicated governing differential equations like Navier Stokes equation, heat conduction
equation, diffusion equation, and provides an alternative method of solution. The bottom-up
approach of modelling the behaviour of the particles in a molecular particle stage and then gradually
incorporating the entire macroscale domain is a novel method in the solution of flow problems.
In this project, a thorough overview of the physics and mathematics behind the Lattice Boltzmann
Method is conducted. The logical sequence to be followed is documented and the various
approximations used in the method are justified.
A program is written in MATLAB environment for execution of the technique. A comparatively
simple system consisting of a steady flow through a circular pipe is simulated and LBM techniques
are used to predict the velocity profiles formed. The results given match the well-known outcomes.
A parabolic profile is established and continuity of flow is maintained.

of Fluid Flows
19
8 Future Scope for Work
The work carried out in this project is by no means exhaustive and there is scope for a lot of future
work on this which could not be completed due to paucity of time. The following are outlined as
possible areas for investigation
 Extensive validation of the LBM technique
In this project a simple condition of steady pipe flow of an incompressible Newtonian fluid
has been carried out. There are several more commonly used models which may be
validated using the LBM technique. Some of them are the flow over a flat plate, boundary
layer flow, lid driven cavities, etc.
 Hydrodynamic development
The region of hydrodynamic development, i.e. the distance the fluid has to travel before it
attains a steady nature, is important while considering applications in Fluid Mechanics
problems. By varying the Reynolds number in flow through a pipe or over a flat plate, the
algorithm can be modified and corrected to make a statement regarding the variation in
length of the development region.
 Heat and Mass Transfer problems
The applications of the Lattice Boltzmann method are not limited to problems of fluid
dynamics. Other flow related problems including forced and free convection problems,
steady or unsteady heat conduction problems, and the variety of problems associated with
heat and mass transfer can be solved using the LBM approach.
 Effect of Reynolds Number
The kinematic viscosity is the determining factor behind the relaxation factor during the
collision of particles. As a result, when combined with the scale of the problem the Reynolds
number plays a very important role in LBM methodology. The effect of variation of Re value
can be a matter for investigation.
 Effect of Knudsen number
The Knudsen number is an important parameter while considering microscale problems.
Knudsen number is defined as
𝐾𝑛 =
𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑚𝑒𝑎𝑛 𝑓𝑟𝑒𝑒 𝑝𝑎𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑
𝑅𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑎𝑡𝑖𝑣𝑒 𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝑠𝑐𝑎𝑙𝑒
In case the value of Kn is near or greater than unity, the conventional laws of continuum
mechanics do not hold and statistical methods need to be used.
The conventional techniques of CFD following the discretization principle follow the
macroscale equations of fluid mechanics irrespective of the length scale. On the other hand
LBM is based on a more fundamental statistical interpretation of the behaviour of fluid
particles. In case of large Knudsen number flows, a comparative statement can be made
regarding the performance of discretization and LBM.
 Complex flow geometries
Since the LBM technique does not involve physical modelling of geometries and domains, it
is much easier to analyse complicated geometries. In case of problems like cross flow heat
exchangers with repeating and symmetrical geometries, LBM method is efficient.

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20
9 References
 Mohamad, Abdulmajeed A. Lattice Boltzmann method: fundamentals and engineering
applications with computer codes. Springer Science & Business Media, 2011.Sukop, M. C.
"DT Thorne, Jr. Lattice Boltzmann Modeling Lattice Boltzmann Modeling." (2006).
 Sukop, M. C. "DT Thorne, Jr. Lattice Boltzmann Modeling Lattice Boltzmann Modeling."
(2006).
 Shan, Xiaowen, and Hudong Chen. "Simulation of nonideal gases and liquid-gas phase
transitions by the lattice Boltzmann equation." Physical Review E 49.4 (1994): 2941.
 Frisch, Uriel, Brosl Hasslacher, and Yves Pomeau. "Lattice-gas automata for the Navier-
Stokes equation." Physical review letters 56.14 (1986): 1505.
10 Acknowledgement
I would like to acknowledge and express my gratitude towards the following individuals and
institutions for giving me the scope of undertaking this Advanced Design Project:
 Prof. Himadri Chattopadhyay, my guide for this Project, for providing me with the primary
idea for the project and guiding me during its execution.
 Ms. Runa Samanta, PhD Scholar, Department of Mechanical Engineering, for her guidance
and help.
 The Board of Studies, Mechanical Engineering Department, Jadavpur University, for the
giving me the chance to undertake this Project.
 The Mechanical Engineering Department, Jadavpur University, for the assistance during the
Project.

LBM Modelling of Fluid Flow

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