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2013 International Conference on Aerospace Science & Engineering (ICASE)
978-1-4799-0993-3/13/$31.00 © 2013 IEEE
Numerical Simulation of Thermal Incompressible
Fluid Flow Using Lattice Boltzmann’s Method
Abdullah Mufti1, Romana Basit1, M. Abdul Basit2
1Department of Chemical Engineering, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad 45650, Pakistan
2Faculty of Mechanical Engineering, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad 45650, Pakistan
Email: pe1102@pieas.edu.pk, basit@pieas.edu.pk, fac055@pieas.edu.pk
Abstract—Objective of this paper is to numerically simulate
thermal fluid flow using Lattice Boltzmann’s method. D2Q9
model is used for simulation of 2D thermal fluid flow. Different
thermal cases such as thermal Poisseuille flow, thermal couette
flow are studied, along with conduction in metals with or
without heat generation source within metal. Results produced
from Lattice Boltzmann’s method are compared with
analytical solutions or with some other CFD technique (Fluent)
results. On comparison one can say that Lattice Boltzmann’s
method is a remarkable tool for simulating and solving
computational fluid dynamics problems.
Keywords—Lattice Boltzman’s method; therma; poisseuille;
couette.
I. INTRODUCTION
Lattice Boltzmann’s method is an explicit technique to solve
fluid flow problem. It is easy to implement and can simulate
a number of cases. Boltzmann’s equation is discretized on
some assumptions which results in lattice Boltzmann’s
equations. Theses equations are capable of solving transport
phenomena problems for example momentum energy and
mass transfer problems. FORTRAN is used as a
programming language.
Figure 1 D2Q9 Model
A. D2Q9 Model
In this paper D2Q9 model is used to study heat transfer
problems. This model is a 2 dimensional model having 9
velocities. Center particles are at rest. 8 directions are for 8
velocity vectors. Lattice width and height is 1 lattice unit.
User can set lattice unit according to the demand of the
problem in hand, and other unit is time step (ts) it is the time
in which one colission and one propogation occurs.
Therefore lattice speed “c” is defined as lattice units per
time step (lu/ts). Vector (1 to 4) have velocity magnitude of
1 lu/ts. Vectors (5to 8) have velovity magnitude of √2 lu/ts.
Directions of the velocity vectors are indicated in the Fig. 1.
B. Diffusion Problems.
Lattice Boltzmann Method is capable of solving heat
diffusion problems in rectangular slabs. As the particles are
not moving so it is not a flow problem and momentum
equation is neglected. Only heat equation is solved all over
the domain. If source term is present within the slab source
can be added. LBM is capable of simulating both transient
and steady state cases
C. Convection Problems
Poisseuille flow and Couette flow both have been
successfully simulated using Lattice Boltzmann’s Method.
Heating effect are then added to simulate thermal heat flow
cases, in both Poisseuille and Couette flow. Temperature
profiles are developed in the domains. Both of the cases
produced satisfactory results for transient and steady state
cases.
D. Abbreviations and Acronyms
LBM Lattice Boltzmann’s method
TLBM Thermal Lattice Boltzamnn’s method
BE Boltzmann’s Equation
LBE Lattice Boltzmann’s Equation
CFD Computational Fluid Dynamics
c Lattice speed
NS Navier Stokes
FPS Foot pound second
MKS Meter Kilogram second
CGS Centimeter gram seconds
E. Units
LBM is capable of solving problems in consistent unit
system, one can use FPS, MKS,CGS or any other system,
Input data must be consistent.
F. Equations
1) Boltzmann’s Equation (BE)
(1)
This is the Boltzmann’s equation, the first term represents
time variation of distribution function, and second term
defines spatial variation of distribution, third term talks
about effect of forces acting on the particles, and Q(f,f)
defines the collision term for particles. Following
assumptions convert BE to LBE.
i. Single relaxation time
ii. Velocity discretization i.e. using finite set of
velocities.
iii. Particles can move only along 8 directions.
iv. Modeling of fluid by many cells of same type.
v. Update of cells at each time step.
2) LBE Density Distribution Function
(2)
This is the discretized form of Boltzmann’s equation for
density distribution. Left side of the equation represents
streaming, while the right side of the equation represents
collision.fiis the density of the particles, ei is the lattice
velocity of the particles, ω is relaxation time.
3) LBE for Temperature Distribution
(3)
This equation is temperature distribution function. Similar
to the above density equation left side of the equation is
streaming and right side is collision.
4) Equilibrium density distribution function
(4)
This equation is Taylor series expansion of equilibrium
density distribution function. wiis the weighting function
(wi=4/9 for i=0, wi=1/9 for i=1,2,3,4 and wi=1/36for
i=5,6,7,8) ei is the lattice speed ith direction, u is the
macroscopic velocity and ρ is the macroscopic density of
the node.
5) Equilibrium temperature distribution function
(5)
This is temperature equilibrium function. T in the term is
temperature of the node. eiis the lattice speed for
ithdirection.wi is the weighting function (wi=4/9 for i=0,
wi=1/9 for i=1,2,3,4 and wi=1/36for i=5,6,7,8) , ei is the
lattice speed ith direction, and u is the macroscopic velocity
6) Macroscopic temperatures
(6)
Macroscopic temperature at any point is equal to the sum of
the temperature distribution function at that point.
7) Macroscopic Density (7)
Macroscopic density at any point is the sum of density
distribution function on that node
8) Macroscopic Velocity
(8)
Velocity is calculated by multiplying density distribution
function with lattice speed and then dividing by total density
at that node.
9) Density Relaxation Parameter
(9)
Density relaxation parameter is calculated by using the
macroscopic viscosity of the fluid.(momentum diffusivity)
10) Temperature Relaxation Parameter
(10)
Temperature distribution function is calculated by using
macroscopic thermal diffusivity.
II. RESULTS AND DISCUSSION
A. Heat conduction Simulation
Consider the heat conduction problem in a slab. Left side of
the plate is at a temperature of 1000 oC(1273K). Top and
right sides are at 500 oC(773 K) and adiabatic boundary
condition at the bottom. When these conditions were set in
LBM conduction code and simulation was run following
temperature contours were obtained (Fig.2).
Figure 2 Temperature Contours for conduction
Same problem was solved on Fluent. Almost same
contours were obtained, but for close comparison data
from LBM contours and from fluent contours were
compared. In Fig. 3 data from LBM solution and
FLUENT solution are compared along one of the lines
on the slab. LBM results match with the fluent’s results
at each point. Percentage error in LBM results is less
than 0.1%
Figure 3 Comparison of Temperature ( Fluent and LBM)
So this means that LBM conduction code is quiet accurate
and one can use it as an alternative technique to solve
conduction problems.
B. Thermal Poisseuille Flow between Parallel Plates
Thermal Poisseuille flow code was developed in two stages.
In the first stage a code was developed that could simulate
simple Poisseuille flow. Velocity profile developed from
this code was then tested against the analytical solution.
Results were quiet accurate. The comparison of LBM and
analytical results are presented in Fig. 4.
Figure 4 Poisseuille Flow Analytical Vs LBM result
In the second stage code was rewritten so that temperature
effects could be added to it. Then it was tested with fluent.
Again the results were a 100% match. Below are the
temperature contours. These contours represent temperature
of water as it passes through heated channel. Boundary
conditions for this problem are periodic boundary
conditions for flow on open ends, and no slip on walls, for
temperature is defined on each wall, and inlet, at outlet
adiabatic condition prevails.
Figure 5 Temperature Contours for Poisseuille Flow
C. Couette Flow
Couette flow is flow between parallel plates with upper
plate moving. The flow profile is developed due to the upper
plate’s motion. Fig. 6 shows the comparisons of Analytical
and LBM results.
Figure 6 Couette Flow Comparison
Themral Couette flow code is developed on same ground as
thermal Poisseuille flow code. Only boundary conditions
were different for the top plate.Temperature Contours were
obtained were then compared with contours obtained from
Fluent. Results were remarkably similar.
Figure 7 Temperature Contours for Couette Flow
Results for these different flow show that LBM is a
remarkable tool for simulating fluid flow problems
with/without heat transfer.
REFERENCES
[1] Gladrow and D.A. Wolf, Lattice Gas Cellular Automata and Lattice
Boltzmann Method An Introduction,Springer, 2000.
[2] M. Jr, C. Sukop & T. Daniel, Lattice Boltzmann Modeling,Florida :
Kripsbv, Meppel, 2006.
[3] A.A. Mohammad, Lattice Boltzmann Method,Springer-Verlag, London,
2011.
[4] S. Succi,The Lattice Boltmann Equation for Fluid Dynamics and
Beyond,Clarendon Press Oxford, 2001.
