SIMULATION OF PARTICLE MOTION IN INCOMPRESSIBLE FLUID BY
LATTICE BOLTZMANN MRT MODEL
MOHAMMAD POURTOUSI
A dissertation submitted in partial fulfilment of the
requirements for the award of the degree of
master of mechanical engineering
Faculty of Mechanical Engineering
Universiti Teknologi Malaysia
JANUARY 2012
v
To my beloved mother and father
vi
ACKNOWLEDGMENT
I would like to thank Dr. Nor Azwadi Che Sidik for his direction, assistance,
and guidance. Special thanks should be given to my friends in UTM university who
helped me in this project.
Finally as far as there is a limit time and paper , it is impossible to mention
all of my friends that have had good impact on my master project . only I would like
to warmly appreciate my parents supports during doing this project and other friends
helps to toward my work.
vii
ABSTRACT
As far as these days developing simulation fluid flow in different geometries
are one the main concern of thermo fluid researchers , This study are going to
employ Lattice Boltzmann Method as a computational method to solve some
different geometries. This approach tends to demonstrate different accuracy and
stability in two Relaxation time for Lattice Boltzmann Method(LBM) in lid driven
cavity .Velocity field for different Reynolds number and aspect ratio in channel fluid
flow are systematically presented to interpret developed vortex in different time. In
this geometries multi particles are simulated for different Reynolds number and it is
found that the percentage of removal particles in different time after stability is
decreased by growing aspect ratio . In final section Multi-relaxation-time based on
Lattice Boltzmann method is applied for simulation of backward-facing step .The
obtained results shows position and length of the vortex. The numerical results
obtained in this paper are in good agreement with the published experimental and
numerical results.
viii
ABSTRAK
Setakat yang hari ini, pembangunkan simulasi aliran bendalir dalam geometri
yang berlainan adalah satu kebimbangan utama penyelidik bendalir termo. Kajian ini
akan menggunakan Kaedah Kekisi Boltzmann sebagai satu kaedah pengiraan untuk
menyelesaikan beberapa geometri yang berbeza. Pendekatan ini cenderung untuk
menunjukkan ketepatan dan kestabilan yang berbeza dalam dua masa Kelonggaran
untuk Kaedah Kekisi Boltzmann dalam yang aliran didorong oleh rongga tudung.
Medan halaju untuk nombor Reynolds yang berbeza dan nisbah aspek dalam aliran
bendalir
saluran secara
sistematik dibentangkan untuk
mentafsir
vorteks
dibangunkan dalam masa yang berbeza. Dalam geometri ini, simulasi pelbagai zarah
dilakukan untuk nombor Reynolds yang berbeza dan mendapati bahawa peratusan
penyingkiran zarah dalam masa yang berbeza selepas kestabilan berkurang dengan
penambahan nisbah aspek. Dalam seksyen akhir kelonggaran masa pelbagai yang
berdasarkan kaedah Boltzmann kekisi digunakan untuk simulasi langkah menghadap
ke belakang. Keputusan yang diperolehi menunjukkan kedudukan dan panjang
vorteks. Keputusan berangka yang diperolehi dalam kertas ini adalah dalam
persetujuan yang baik dengan keputusan uji kaji dan berangka penerbitan.
ix
TABLE OF CONTENTS
CHAPTER
1
2
3
4
TITLE
PAGES
ACKNOWLEDGMENT
vi
ABSTRACT
vii
ABSTRAK
viii
LIST OF FIGURES
i
NOMENCLATURE
i
INTRODUCTION
1
1.Introduction
1
LITERATURE REVIEW
4
2.1 Literature Review
4
2.2.Advantages of Lattice Boltzmann Method
10
2-3 Objective
11
2.4 Problem Statement
11
METHODOLOGY
13
3. Methodology
13
3.1.1
13
Lattice Boltzmann Method for fluid flow with SRT and MRT
3.2.1 Boundary Condition
19
3.2.2 Periodic Boundaru Condition
19
3.2.3 No-slip Boundary Condition
20
3.2.4 Boundary Condition governing equations
20
3.3.1. Multi relaxation Method(MRT)
24
3.4.1 Particle Trajectory Analysis
26
3.4.2
Particles movement through Fluids
27
3.4.3
Particle motion simulation by Lattice Boltzmann Method
30
RESULTS AND DISCUSSION
32
x
4.1 Difference between Single Relaxation Time (SRT) and Multi
Relaxation Time(MRT) in lid driven cavity in term of stability and accuracy
4.2 Channel fluid flow by MRT-LBM
32
41
4.2.1 prediction of vortex by solid particle trajectory in Channel fluid flow
based on MRT-LBM
45
4.3 Multi Particle Channel fluid flow
48
4.3.1 contaminated removal in different Reynolds number
50
4.3.2 contaminated removal in different Reynolds number for AR=4
53
4.4 Multi-relaxation-time Lattice Boltzmann method simulation of
backward-facing step for incompressible flow
4.4.1 Backward-facing step for different Reynolds Number
5
CONCLUSION
61
61
68
5.1.Conclusion
68
5.2
69
Future works
REFERENCES
71
APPENDIXES
74
LIST OF FIGURES
Figures
Title
pages
Figure 3.1. Illustrate the 9 velocity direction in square lattice for D2Q9 LB model 14
Figure3.2 show the streaming process on the D2Q9 LB model
15
Figure 3.3 Shows the collision process on the D2Q9 LB model
16
Figure3.4. Illustration of bounce-back algorithm for the D2Q9 model[24]
20
Figure3.5. f4, f7 and f8 are unknown distribution functions.[24]
21
Figure 4.1. This figure provides information about comparison between SRT and
MRT for Re100,in X direction velocity
33
Figure 4.2. This figure provides information about comparison between SRT and
MRT for Re100,in Y direction velocity
34
Figure 4.3. This figure provides information about comparison between SRT and
MRT for Re400,in X direction velocity
35
Figure 4.4. This figure provides information about comparison between SRT and
MRT for Re400,in Y direction velocity
36
Figure 4.5. This figure provides information about comparison between SRT and
MRT for Re1000,in X direction velocity
37
Fig4.6This figure provides information about comparison between SRT and MRT
for Re1000,in Y direction velocity
38
Figure 4.7. This figure provides information about comparison between SRT and
MRT for Re3200,in X direction velocity
39
Figure 4.8. This figure provides information about comparison between SRT and
MRT for Re3200,in Y direction velocity
40
Figure 4.9. Shows the configuration of streamline in different Reynolds number from
Fang et al. research results
41
Figure 4.10. velocity field with MRT –LB in Reynolds Number =50 and Aspect
ration=1
42
Figure 4.11. velocity field with MRT –LB in Reynolds Number =50 and Aspect
ration=2
42
Figure 4.12. velocity field with MRT –LB in Reynolds Number =50 and Aspect
ration=3
42
Figure 4.13. velocity field with MRT –LB in Reynolds Number =50 and Aspect
ration=4
42
ii
Figure 4.14. velocity field with MRT –LB in Reynolds Number =100 and Aspect
ration=1
43
Figure 4.15. velocity field with MRT –LB in Reynolds Number =100 and Aspect
ration=2
43
Figure 4.16. velocity field with MRT –LB in Reynolds Number =100 and Aspect
ration=3
43
Figure 4.17. velocity field with MRT –LB in Reynolds Number =100 and Aspect
ration=4
43
Figure 4.18. velocity field with MRT –LB in Reynolds Number =400 and Aspect
ration=1
44
Figure 4.19. velocity field with MRT –LB in Reynolds Number =400 and Aspect
ration=2
44
Figure 4.20. velocity field with MRT –LB in Reynolds Number =400 and Aspect
ration=3
44
Figure 4.21. velocity field with MRT –LB in Reynolds Number =400 and Aspect
ration=4
44
Figure 4.22.Particle trajectory in reynolds number 50 and AR=4,(a) LBM and (b)
Fang et al.result.
45
Figure 4.23. Particle trajectory in Reynolds number 50 and AR=4 with Multi
Relaxation Time
45
Figure 4.24.prediction vertex by particle trajectory in Reynolds Number =400 ,
Aspect ration=4 and Time=0.4
46
Figure 4.25. prediction vertex by particle trajectory in Reynolds Number =400 ,
Aspect ration=4 and Time=0.8
46
Figure 4.26.prediction vertex by particle trajectory in Reynolds Number =400 ,
Aspect ration=4 and Time=1.6
47
Figure 4.27.prediction vertex by particle trajectory in Reynolds Number =50 ,
Aspect ration=4 and Time=8
47
Figure 4.28.prediction vertex by particle trajectory in Reynolds Number =50 ,
Aspect ration=4 and Time=8
48
Figure 4.29.Experimental (left) removal process of contaminated cavity fluid in Re=
50 and AR=4 Fang et al
48
Figure 4.30.Fouling removal from cavity in Reynolds number 50 and AR=4
49
Figure 4.31.Fouling removal from cavity in Reynolds number 50 and AR=3
51
Figure 4.32.Fouling removal from cavity in Reynolds number 50 and AR=3
52
Figure 4.33.Fouling removal from cavity in Reynolds number 50 and AR=3
53
Figure 4.34.Fouling removal from cavity in Reynolds number 100 and AR=4
55
Figure 4.35.Fouling removal from cavity in Reynolds number 70 and AR=4
56
Figure 4.36.Fouling removal from cavity in Reynolds number 50 and AR=4
57
Figure 4.37.Particle removal percentage Reynolds number 50 and AR=4
58
Figure 4.38.Particle removal percentage Reynolds number 70 and AR=4
58
Figure 4.39.Particle removal percentage Reynolds number 100 and AR=4
59
Figure 4.40.Particle removal percentage Reynolds number 50 and AR=3
59
Figure 4.41.Particle removal percentage Reynolds number 70 and AR=3
60
Figure 4.42.Particle removal percentage Reynolds number 100 and AR=3
60
iii
Figure 4.43. Velocity field for backward-facing step in Reynolds Number=20
Figure 4.44.Velocity field for backward-facing step in Reynolds Number=30
Figure 4.45.Velocity field for backward-facing step in Reynolds Number=40
Figure 4.46.Velocity field for backward-facing step in Reynolds Number=50
Figure 4.47.Velocity field for backward-facing step in Reynolds Number=60
Figure 4.48.Velocity field for backward-facing step in Reynolds Number=100
Figure 4.49.Velocity field for backward-facing step in Reynolds Number=130
Figure 4.50.velocity field for Reynolds Number 50 for different time
Figure 4.51.velocity field for Reynolds Number 70 for different time
Figure 4.52.velocity field for Reynolds Number 100 for different time
Figure 4.53.different reattachment points for backward-facing step
Figure 4.54.different reattachment points for different Reynolds number
62
62
62
62
62
62
62
64
65
66
67
67
NOMENCLATURE
distribution function in discrete Boltzmann equation
and in the LB fluid flow model
Kinematic viscosity in the lattice Boltzmann method
Dimensionless fluid density in the lattice Boltzmann
method
Single Relaxation time of Boltzmann equation
Macroscopic flow velocity in the lattice Boltzmann
method
Discrete particle velocity in each discrete direction
Speed of sound
Time
Physical molecular mass
Ma
Mach number
Drag Force
Fe
External Force
Fb
Buoyancy Force
Cd
Drag Force Coefficient
CFL
Courant- Friedrichs - Lewy number
AR
Aspect ratio of the channel cavity
Particle diameter
Re
Channel Reynolds number
ii
Rep
Particle Reynolds number
BGK
Bhatnagar-Gross-Krook
CFD
Computational Fluid Dynamics
D2Q9
Two Dimensions nine velocities lattice Boltzmann method
FEM
Finite Element Method
FDM
Finite Difference Method
FVM
Finite Volume Method
LB
Lattice Boltzmann
LBE
Lattice Boltzmann Equation
LBM
Lattice Boltzmann Method
LGA
Lattice Gas Approach
PDE
Partial Differential Equations
MRT
Multi Relaxaion Tim
CHAPTER 1
INTRODUCTION
1.Introduction
Many methods have been recently introduced in order to analyze a laminar
flow and its modeling of hydrodynamic or aerodynamic removal of particles from the
internal surfaces . They have tended to solve physical problems for different geometries in
industries and research laboratories . In this case Lattice Boltzmann method (LBM)
is one of the newest method that has been vastly studied by a huge number of
papers. As a matter of fact, Lattice Boltzmann scheme is one of the numerical
techniques that is normally used to solve the equation of turbulent and laminar flow
which is represented time –dependent fluid flow [1].
Also it should be noted that, LBM is one of the most effective numerical
ways for simulating and modeling complicated physical chemical system with
complex geometry. LBM has introduced as a microscopic numerical method and has
a certainly effect on simulating fluid flow . In particular, the easy implementation of
boundary conditions makes LBM very interesting for the simulation of multiphase
flows and specially flow in complex geometries[2].
To solve Lattice Boltzmann equation partial differential must be considered.
In this regard partial differential equation presents fluid flow through the space and
2
time .As a matter of fact ,certain solutions only exist for a few specific cases with
simple geometries and suitable boundary conditions. It is certainly true that to obtain
simplified equation , the complex phenomena must be ignored. However, nowadays
digital computers have rapidly developed and many researchers prefer to use high
performance computers in their field of study.
Many papers have been presented Lattice Boltzmann in different groups by
researchers and indeed , three groups of them have been broadly developed in their
field of studies . First of all different type of fluid flow respect to the fluid regime
consist of laminar, turbulent and incompressible flow and therefore, different
Reynolds number and changing characteristic of fluid are used by seintic .The
second group wants to indicate different geometries and different aspect ratio in 2D
and 3D modeling patterns.
Finally last group of papers are clearly represented by engineers which
discuss a bout different theoretical ,numerical and experimental methods of solving
the equation and simulation fluid flow in different shapes. Moreover, their results are
compared by exits ones to show the validation.
Many years ago, the modeling of incompressible Laminar fluid flow inside
the different kind of geometries was investigated and there are number of articles
published by researchers in entire the world . The current study tends to present the
incompressible fluid flow in case of laminar by MRT-LB method for different
physical problems such as cavity and channel flow. Furthermore it shows the
discrepancy between this numerical modeling with SRT method.
The present work
is going
to consider the difference between Multi
relaxation time and single relaxation time in terms of accuracy and stability in cavity.
Moreover , the instability of fluid flow is performed by different meshes and
Reynolds numbers.
3
Since a plotting vortex and streamlines for fluid flow are one of the important
concern for scientists ,this study investigates a prediction of vortex structure and
different position of vortex with particle trajectory in channel to show clearly this
phenomenon .
Also a reattachment area for vortex inside the backward facing step flow is
carried out and verified with available benchmark in different time and Reynolds
numbers. To extend this work Multi particles with Lattice Boltzmann based on
Multi Relaxation Time inside the channel are simulated and then agree well with a
existing numerical results.
4
CHAPTER 2
LITERATURE REVIEW
2.1 Literature Review
In many papers, it is clear that the writers tend to give a good definition for
a new method for simulating fluid flow because of poor efficiency of the
conventional numerical methods. Many numerical methods have emerged to model
physical problem in a short time and using simple modeling ways . This literature
review is going to introduce CFD modeling related to Lattice Boltzmann method.
Some methods such as Stokesian dynamic (SD) that was introduced by Brady and
Bossis [3] , focused on different definition of boundary element (BEM).
There are too many articles published regarding to accuracy an stability of
MRT and they carried out this simulation for various geometries . Indeed, Multi
Relaxation Time for Lattice Boltzmann in case of incompressible flow was studied
by Rui and Boachang . Moreover , Lid driven Cavity, steady Poiseuille flow and
double shear flow have been done with MRT modeling during their work . It was
found that Multi Relaxation Time has a good stability in comparison by single
Relaxation Time and they showed it more accurate in different geometries [4].Multi
Relaxation Time for simulation of deep lid driven cavity flows was presented by
Chen and Lin for different Reynolds numbers. They have achieved that MRT is more
5
suitable for parallel computations in comparison by SRT [5]. They also simulated lid
driven cavity from Re= 100 to 7500 and their result was verified completely with
Ghia data. Some engineers have addressed thermal physical problems with lattice
Boltzmann . Therefore , they have broadly proposed to develop Multi Relaxation
time in different terms of thermal problems . Mezrhab, et al [6] used Double MRT to
simulate convective flows . D2Q9 and D2Q5 were applied to model temperature
fields. They heated cavity numerically and then it was observed that Rayleigh
numbers were validated and all fit numerical results .
MRT-LBM has been developed for different shape such as channel flows .
Researchers have tested several methods to solve temperature and velocity field for
this type of geometries and they also develops Lattice Boltzmann method for this
kind of shapes . Yen and Yang [8,7] brightly proposed the simulation of Y-shaped
channel and field synergy principle . As a matter of fact , they focused on the thermal
mixing efficiency after simulation field synergy . Furthermore , in particularly it has
been brightly illustrated that thermal mixing efficiency was improved by obstacles
and increasing the intersection angle between velocity and temperature gradient.
They presented cylinder in the backward-facing step with field on synergy principle
They
represented incompressible steady for low Reynolds numbers for this
modeling . In addition, three different boundary conditions were used for channel
flow which were called Boundary condition of temperature field ,boundary condition
of velocity field and boundary condition of treatment of cylinder respectively .They
worked on convective heat transfer for similar geometries
and they have
successfully used LBM to study about square blockage in the channel flow .In
particular has been demonstrated that field synergy principle applied to present an
interruption within fluid results in decreased intersection angle between the
temperature and velocity gradient[8].
Some research groups worked on channel flow over a cavity with miscible
behavior and different Reynolds numbers were considered by them . They showed
cleverly contaminated removal particles with different aspect ratios . It has to be
noted that they simulated density difference between the channel fluid and cavity .
6
Furthermore , Fang in 2002[9] also did the cavity with effect of mixed convection on
transient hydrodynamic removal of a pollutant .
Since researchers have investigated about accuracy and stability of LBM ,
some kinds of Relaxation times have been rapidly developed by scientists for
different physical problems. Lattice Boltzmann Equation (LBM) is obtained form
Lattice Gas Automata (LGA)
and this statement has been brightly presented by
many alternative investigation . As a matter of fact this method is emerged to solve
several fluid flow problems[6]. Single Relaxation Time that is called Lattice
Boltzmann Bhatnagar_Gross_Krook(LBGK) has been carried out by engineers for
different hydrodynamic problems . Several Geometries and complicated boundary
conditions were simulated wonderfully by this method [10] .
There are several experience numerical instability whenever the Reynolds
number flow is increased and Relaxation time is approached to 0.5 . One good way
to solve this problem is to use Multi Relaxation Time (MRT) , it should be better to
note that for different Reynolds number , MRT is more stable because it has more
degrees of freedom and more accurate than SRT [11] .
One of the most important item that makes instable Fluid simulation to stable
modeling
in Lattice Boltzmann Method is Defining boundary conditions , Since
modeling cavity and channel flow need to use different boundary conditions, Some
researchers
have addressed about velocity and boundary condition of LBM by
Zou and He [12]. They considered number of researches about 2D and 3D Lattice
Boltzmann BGK models as well. Moreover they demonstrated that these conditions
and the simulation results approve the analytical solution of the poiseuille flow
driven by density diversity. Many researches represented modeling the fluid inside
the cavity numerically and experimentally such as Faure, et al work (2006)[13] .
Navier stokes equations for laminar fluid flow is done by Mehta and Lavan
[14] , They solved this equations for unsteady incompressible flow and their work is
done for Reynolds number from 1 to 1500 and different aspect ratio . Furthermore
7
Yao and Cooper[15] presented broadly 3D unsteady incompressible flow by finite
difference scheme in a rectangular cavity.
Shen Chen[15,16] solved Vorticity and stream function for lid-driven cavity
.He represented solving Navier stoks equation by using large eddy simulation based
on LBM .He clearly conduct a comprehensive numerical study of decaying HIT
with LBE-DNS and LBE-LES to establish the suitability of LBM for turbulence
applications. In addition , he worked on effectiveness of the lattice Boltzmann
equation (LBE) as a computational tool to develop direct numerical simulations
(DNS) and large-eddy simulations (LES) of turbulent flows.
Other papers have been proposed for the stability of the lattice Boltzmann
method which is evaluated simulation of subcritical turbulent flows around a sphere.
Some measures was used to decrease the computational cost. and of course Large
eddy simulation was applied to increase the efficient simulation of resolved flow
structures on non-uniform computational meshes [17].
In this study, the modeling of fluid regeneration in a cavity is simulated by a
numerical method of the Navier Stokes equations and and the energy equation was
combined with it to represent transient flows . The researcher broadly showed the
effect of mixed convection on the clearance of particles which is suspended in a
cavity flow.
This study tended to present intently different range of Grashof
numbers between 1-4000 . Moreover it has to be noted that several aspect ration are
successfully demonstrated in this approach . was also tested dnt e
This work results performed that, the rate of the contaminated cavity fluid
that removed was quite high during the unsteady start up of the duct flow and
approaches zero after the flow reaches a steady state.
Furthermore, whenever
Grashof number was changed it had curiously
impact on flow patterns and cleaning efficiency . Also the cleaning process is
interestingly increased by growing
Grashof number. Actually
the interaction
8
between the external duct flow and buoyancy induced flow has direct effect on
Grashof number .
After this investigation , Simulation of solid particles behavior in a
driven cavity flow was broadly
carried out
by Kosinski and Hoffmann [18].
Eulerian–Lagrangian (E–L) was used to simulate particle and this approach
represented that particles are assumed markers moving in the computational domain.
They assumed that hard sphere pattern for modeling solid particle interactions.
In
this research, several types of Reynolds number of the flow and particle momentum
response time were investigated.
The derived data show the action of the particle cloud in the system and
simultaneously indicate the controlling of the fluid flow particles .The sample also
demonstrates that particles tends to have movement toward the walls and this event
is more beneficial to show a higher value of the Stokes number.In addition , this
approach was presented by showing snapshots of particle position ,also the mean
square displacement of the particles.It should be noted that this investigation
indicated that by increasing stokes number and decreasing distance the fluid velocity
is brightly change.
On the other hand Wan and Turek [19] investigated on modeling a large
number of particles with Multigrid fictitious boundary.
Fundamentally MFBM is used Multigrid FEM background mesh
and
commences with a large mesh which can include many of the geometrical fine-scale
details and Furthemore uses rough boundary parameterization that adequately
present all large-scale structures to show clearly the boundary conditions.
whole fine-scale type is clearly demonstrated behavior of internal in a way
that the corresponding elements in all matrices and vectors are categorized as
degrees of freedom that are obviously constructed all iterative steps of solution . An
impotant progress of the MFBM is that it can treat the interaction between the fluid
9
and the moving rigid particles, specially a fixed Multigrid mesh is permitted to be
represented and it is not essential to make new mesh.Here , Two vivid samples of
numerical modeling in a cavity were utilized to clarify that this proses is useful
enogh to model reality of particulate flows
which was accompanied by se particles
movement.
It is interesting to say that Engineers carried out to use ;BM method to
simulate particle-fluid interaction and there are too many obtained result present
these Scientific works.
Heemels and Hagen [20] , performed
a modification of the Lattice
Boltzmann internal fluid modeling. Its behavior is intently modified without any
disturb the dynamics of the particle. And in addition The equations of motion for the
solid particles were simulated curiously that the microscopic conservation laws for
mass and momentum were completely satisfied . They vastly determined and
compared both the time-dependent rotational and translational motion of an isolated
spherical particle and the viscosity of a concentrated suspension of hard spheres
against known results for solid particles as well.
Another study focused on the immersed boundaries of LBM to explain the
solid particle interaction in physical problems. In this method that was intently
investigated by Zhi-Gang and Efstathios [21], The lattice Boltzmann methodn and
the immersed boundary methods were applied to solve problem. A regular Eulerian
grid was simulated to represent the flow field and a Lagrangian grid
by this
method.As a matter of fact particle deformation and fluid structure deformation have
been simulated simulated by their method.
Inamuro et al.[22] performed lattice Boltzmann method to simulate twophase immiscible fluids with large density variances. To illustrated the credit of the
method, they simulated the method to the modeling of capillary waves, binary
droplet collisions, and bubble flows.
10
Since in this work prediction of vortex is presented ,The literature review has
demonstrated separating flow over a backward-facing step by QHD methods. Chao
and
Tzu
showed
separating
flows
over
a
backward-Facing
step
by
quasihydrodynamic(QHD) system of equations and they performed reattachment
area of the vortex for different times and Reynolds number .Furthermore it should be
noted that Separating flow was indicated from laminar flow to turbulent flow in their
study [23].
2.2.Advantages of Lattice Boltzmann Method
The LBM has proved that it does not need to apply the pressure on interfaces of
refined grid actually the implicitly is included in the computational scheme.
In contrast to the convectional techniques which are formulated in terms of
discretized macroscopic continuum equation, Navier Stoke, the LBM is considered
to be based on microscopic model. It may be interpreted as a finite difference
method for the numerical simulation of the discrete velocity Boltzmann equation.
Hence, the LBM does not need to consider explicitly the distribution of pressure on
interfaces of refined grids since the implicitly is included in the computational
scheme.
LBM is clearly shown that simulation of LBM for complex geometries and
complicated boundary condition has done easily than another method and this
method is being to simulate
immiscible and miscible multiphase fluids
[24].Furthermore it is usual to use for thermal effects .
It has been represented that both laminar and turbulent flows have carried out
by researchers [25] . during recent years complex geometries has been investigated
vividly by LBM .
11
LBM scheme has an effect parallelization of the simulations even on parallel systems
with more or less slow interconnection due to the regular lattice. Furthermore, it has
an easy limited dynamics that causes LBM generally just needs the nearest neighbor
information, so in very large systems running the simulation is practical due to the
lack of memory resources and long processing times.
This study has proved that Multi Relaxation Time(MRT) is indicated that This
Relaxation time is more accurate and stable than Single Relaxation Time (SRT).
Current thesis shows modeling obstacle with LBM modeling is more convenient
than other CFD methods.
2-3 Objective
Investigation about accuracy between single relaxation time (SRT) and Multi relaxation
time (MRT) in LBM .
Investigation about stability between single relaxation time (SRT) and Multi relaxation
time (MRT) in LBM .
Prediction of vortex structure by particle trajectory method with Multi relaxation
time(MRT)
Determining vortex position in backward-facing step for incompressible flow in
deferent time
Simulation Multi particles with MRT-LBM in channel flow
2.4 Problem Statement
Single relaxation time has a limitation in term of stability .
12
Prediction vortex by particle is a new idea that have not investigated yet.
Lack of investigation about difference accuracy and stability between SRT and
MRT.
Advantages of Lattice Boltzmann in modeling particle in the channel.
Simulation Multi particles with MRT-LBM in channel flow have not investigated.
SCOPE OF PROJECT
y Using numerical method to solve problem.
y Programming laminar flow fluid with Lattice Boltzmann by Matlab in lid-driven
cavity in difference relaxation time.
y Simulation Multi particles with MRT-LBM in channel flow.
y Solving solid particle in laminar flow with LBM by Matlab software in channel.
13
CHAPTER 3
METHODOLOGY
3. Methodology
3.1.1 Lattice Boltzmann Method for fluid flow with SRT and MRT
LBM is a particle-based method , in which collective behaviour of particles is
presented by a single particle probability distribution function .LBM is a derivative
of the lattice gas automata (LGCA) models in which, evolution of particles on a
fixed lattice simulate the overall macroscopic behaviour.
There are some model for simulation velocity field and Temperature
gradient by Lattice Boltzmann method as a assumption for possible velocity .For this
project
D2Q9 model has been used for simulation . Actually D2Q9 provides
information about position and dimension of the particle respectively.
For the aim of ease without losing generality, reducing the number of
probable particle spatial positions and microscopic momentum from a continuum to
just a handful and similarly discretising time into distinctive steps are to be useful.
There is a assumption that the case of a simple nine velocity on a square lattice, the
nine possible velocities, which are known as the D2Q9 model.
14
The first term in Lattice Boltzmann equation is going to present probability
of being a particular molecule for time and position of molecule which is called
(, ).
Particle will be moved from to + Δ by the time step of + Δ if
collision is not happen. In this equation has been presented by streaming term and
is going to represent velocity magnitude in special direction . It should be noted that
the most important thing about this model is discretization of velocity space. As a
matter of fact this statement is shown velocities are restricted to a limited set of
orientation .At each time step , particle is going to move from one point of mesh
point to the next one. Figure 3.1 has indicated that 9 velocity direction is determined
for D2Q9 .
c7
c3
c6
c4
c1
c2
c8
c5
c9
Figure 3.1 Illustrate the 9 velocity direction in square lattice for D2Q9 LB model
Actually It must be said that the number of particles f(x, t)dxdc
is
completely equal to the transitioned particles which have been demonstrates by
( + , + ) clearly then after combining these two terms with
eachother equation (3.1) is difined.
( + , + ) − (, ) = 0
(3.1)
It has to be noted that magnitude of the distribution function remains unchanged, but
15
due to their movement to a neighboring node according to their direction.
Figure3.2 show the streaming process on the D2Q9 LB model
On the other hand, there is
collision term in term of lattice Boltzmann
equation and it causes to take place between molecules and changed the result in
some phase points starting at (, ) not arriving at ( + Δ, + Δ) and some not
starting at (, ) momentum level and arriving there. In other words It causes a net
difference between the numbers of molecules which redistribute such that the
conservation laws of mass and momentum are satisfied. So we have:
16
( + , + ) − (, ) = Ω()
(3.2)
Figure 3.3 Shows the collision process on the D2Q9 LB model
Ω() is known as the collision operator in equation (3.2) which
demonestrates the rate of change of resulting from collision. Interestingly enough
to say that whether collision is used before streaming or vice versa. Furthermore
they are acquiring alternatively. In particular has been shown that Using Taylor
expansion in time and space, the follow continuum structure of the kinetic equation
accurate to first order has been derived :
+
= Ω()
(3.3)
= Ω( )
(3.4)
Or
+
Where, the equation (3.4) is named as a Boltzmann equation.
Boltzmann equation with a single relaxation time and Multi Relaxation time
method illustrates an approximation and this approximation is used
to show the
essential kinetic of a system regarding to a set of distribution function. Based on
BGK collision model or SRT for the Boltzmann equation(3.5) is indicated:
17
(, , ) − (, , )
+ =
(3.5)
Where, is an equilibrium distribution function and is the relaxation
time. This equation is known as BGK Boltzmann equation which is simulating the
effect of particle collisions with in the viscous fluid in microscopic level.
Finally, the macroscopic properties of fluid are obtained as follow:
Density:
= ∑! (, )
(3.6)
Momentum:
= ∑ (, )
(3.7)
Two researchers Chen and Qian at the same time suggested that the collision
operator be evaluated using a single time relaxation process in which relaxation to
some properly chosen equilibrium distribution occurs at some constant rate. In fact
the collision term Ω is substituted by the single time relaxation approximation, hence
the normal form of the equilibrium distribution function is written as:
= [ " + # . + ( . )$ + $ ]
(3.8)
Where a, b, c and d are the lattice constant. This method can be used only for
small velocities or small Mach numbers u/C% , where Cs is the sound speed. With the
advantage
of the equations (3.6) and (3.7) the coefficient can be obtained
analytically as below:
9
3
= & ' 1 + 3 . + ( . )$ − $ *
2
2
(3.9)
18
Where the weights are obtained by He and Lou (1997) and Abe
simultaneously for nine velocity square lattice as
!
&6,7,8,5 = /6 . Furthermore, =
:;
:
!
!
-
5
&! = , &$,/,-,4 =
and
where c is the sound speed ( = √3>?).
In the D2Q9 model as shown in figure 1, the 9 bit discretized velocities are given by:
= (0,0)
for i=1
= (1,0), (0,1), (−1,0), (0, −1)
for i=2,3,4,5
= (1,1), (1, −1), (−1,1), (−1, −1)
for i=6,7,8,9
In this way, ci for static particle equal to zero, for i=2,3,4,5 equal to 1 lu/ts
and for i=6,7,8,9 equal to √2 lu/ts are determined.
The equilibrium distribution can be chosen in the following form for particles
of each type in simplest implementation:
!
4
3
= '1 − $ *
9
2
$,/,-,4 =
6,7,8,5 =
(3.10a)
1
9
3
'1 + 3 + ( . )$ − $ *
9
2
2
(3.10b)
1
9
3
& ' 1 + 3 . + ( . )$ − $ *
36
2
2
(3.10c)
The relaxation time is related to the viscosity by
=
2 − 1
6
Where, ν is the kinematic viscosity.
(3.11)
19
3.2.1 Boundary Condition
Dynamics of flow, if it is in single phase or multiphase, rely on the nearby
situation. This dependence is mathematically arranged by applying the suitable
boundary conditions (BCs) to the governing equations. This section derives various
fundamental boundary conditions for 2D lattice Boltzmann Method. Boundary
conditions
and Initial
conditions are
essential
for any computational
fluid
dynamic methods. For traditional CFD methods, for every boundary and
initial
conditions Navier-Stokes equations have a unique solution.
Boundary
condition (BC) is a quite complex problem in LBM. The difficulties arise from the
fact that there are no physical perception on the velocities behaviour distribution
function on boundaries. Usually we only have macroscopic information. therefore
we have to translate this macroscopic data on the microscopic distribution functions.
There is no unique way of completing this translation and many authors propose
their own solution. It is important to say that the BC chosen is of primary importance
since it affects the numerical accuracy of the simulation seriously but also its
stability.
3.2.2 Periodic Boundaru Condition
The simplest kind of boundary condition is the periodic one. In this case, the
system is suggested to be isolated in a closed region hence the number of particles
remains unchangeable due to conservation law. The periodic conditions are applied
as a natural part of the streaming operation, so that outgoing particles at one end of
the lattice become incoming particles at the other end. Mass is neither gained nor lost
through the periodic boundaries. At wall nodes, bounce-back describes the
temporary non-equilibrium populations of non-tangent incoming links that would
otherwise be given by streaming.
20
3.2.3 No-slip Boundary Condition
The no slip boundary condition is physically suitable whenever the solid wall
has sufficient roughness to prevent fluid motion at its surfaces. For a rigid wall with
no-slip Conditions, the interactions of the fluid particles with the wall are most easily
described in LBM using a bounce-back scheme. In this plan the leaving direction of
the distribution function are simply specified as the reverse of their received
direction at the boundary sides.
Figure3.4 Illustration of bounce-back algorithm for the D2Q9 model[24]
3.2.4 Boundary Condition governing equations
Boundary conditions used for LB are attributed to two main classes. In the
wet node approach, boundary nodes are wet; they are part of the fluid. So , the
particle populations of such a node act in accordance with the results of the
Chapman-Enskog expansion. They can be split into equilibrium and no equilibrium
part and associated with the macroscopic variables of the flow. In the bounce-back
21
approach, the boundary nodes are located outside the fluid. They implement a
bounce-back dynamics, or a variation thereof, the value of the known particle
populations is copied to their unknown neighbour pointing in the opposite direction.
As these nodes are not part of the fluid, they follow different rules. It is usually not
possible to compute macroscopic variables as moment of particle populations, or to
apply other results of the Chapman-Enskog expansion. It may be taken as a
Lagrangian finite difference method for simulation of the discrete velocity
Boltzmann equation. So, in the case of velocity boundary condition the Zou and He
method can be applied as follow:
Figure3.5 f4, f7 and f8 are unknown distribution functions[24]
If it considered that unknown distribution function relied on north boundary
node after the streaming process as it shown in figure 4, and the vertical and
0
horizontal velocity define as BC = [ ] . According to the figure(3.5) f0, f1,f3 and f2 ,
DC
f5 and f6 are already known they are arrive from other nodes which are locate inside
the wet domain. So, four unknowns appear and four equations need for finding the
unidentified nodes.
If we consider the typical distribution function using the velocities in x and y
direction, the macroscopic velocity could be written as:
= Σ (, ), BC =
!
F
Σ . (3.12)
22
According to velocity matrices values:
0 = ! − / + 4 − 6 − 7 + 8
(3.13)
And
DC = $ − - + 4 + 6 − 7 + 8
(3.14)
Till now three equations come out. For the forth ones we can use the bounceback condition normal to the boundary
$ − $
= - − -
(3.15)
As suggested by Zou and He (1997). These equations lead us to a system of
four equations and four unknown which could be solved. Equations (3.6) and (3.14)
have the related unknown’s f4, f7 and f8, so they can be written by substituting these
variables on the left side:
- + 7 + 8 = − C − ! − $ − / − 4 − 6
(3.16)
And
- + 7 + 8 = $ + 4 + 6 − DC
(3.17)
Considering the right side of the both equations obviously:
− C − ! − $ − / − 4 − 6 = $ + 4 + 6 − DC
And solve for :
(3.18)
23
(3.19)
+ DC = C + ! + $ + / + 4 + 6 + $ + 4 + 6
=
G HI HJ H$(KHL HM)
!HNG
(3.20)
Due to equation (3.15) , we solve for f4:
- = $ − $ + -
$
= $ − DC
(3.21)
/
According to definition of collision formula :
- − $
O
!
5
!
!
!
5
/
$
= O + (−1. DC ) +
!
+ (1. DC ) +
/
!
DC$
$
−
!
6
DC$ −
!
6
(C$ + DC$ )P −
(3.22)
$
(C$ + DC$ )P = − DC
/
By the solve of (3.22) and (3.20) equations the density and f4 could be find.
In addition, by substituting equations (3.14) and (3.21) to solve for f7 and
equation (3.22) for solve f8 :
DC = $ − - + 4 + 6 − 7 − 8 →
2
DC = $ − '$ − DC * + 4 + 6 − 7 − (−
RSSSSSSSTSSSSSSSU
! + / − 4 + 6 + 7 ) →
RSSSTS
3 SSU
V
DC =
2
DC + 24 − 27 + ! − / →
3
W
24
1
1
7 = 4 + (! − / ) − DC
6
2
(3.23)
And for answer of f8
2
(! + / + 4 − 6 + 8 ) − 8 →
DC = $ − '$ − DC * + 4 + 6 − RSSSSSSSTSSSSSSSU
RSSSTS
3 SSU
X
V
DC =
2
DC + 26 − 28 − (! − / ) →
3
!
!
$
6
8 = 6 − (! − / ) − DC
(3.24)
These result calculated due to definition of velocity matrices. So by different
definition in various boundaries these calculation should be repeated by this
procesoure.
3.3.1. Multi relaxation Method(MRT)
In Multi-Relaxation method the collision operator is classified as
fi ( x ct , t t ) fi ( x, t ) [ fi ( x, t ) fi eq ( x, t )]
(3-25)
In this equation is the collision step and this term is changed to momentum
space and it is illustrated as
fi ( x ct , t t ) fi ( x, t ) M 1S[m( x, t ) meq ( x, t )]
(3.26)
S is going to illustrated diagonal Matrix and m( x, t ) and meq ( x, t ) provides
information about vectors of momentum.
D2Q9 model M is defined as following
25
1 1
4 1
4 2
0 1
M 0 2
0 0
0 0
0 1
0 0
1
1
1
1
1
1
1 1 1 2
2 2 2 1
2
1
2
1
0
1
0
1 1 1
0
2
0
1 1 1
1
2
0
0
1 1
2 1
1
1
1
1
1
1
1 0
0
0
0
0
0
1 1
1
1
2
1
1
1
1
1
0
1
The momentum vector here is m ( , e, , jx , qx , j y , qy , pxx , pxy )T
(3-27)
(3-28)
And equilibrium of the moment is
m0eq m1eq 2 ( jx 2 j y 2 )
m2eq 3( jx 2 j y 2 )
m3eq jx
m4eq j x
m5eq j y
m6eq j y
m7eq jx 2 j y2
m8eq jx j y
(3-29)
And jx j y also is defined by following equations:
jx u x fi eq cix
i
j y u y fi eq ciy
i
(3-30)
S is a diagonal Matrix
s diag (1,1.4,1.4, s3 ,1.2, s5 ,1.2, s7 , s8 )
(3-31)
26
3.4.1 Particle Trajectory Analysis
Nowadays, particle trajectory simulation have many important applications in
a diverse range of engineering and scientific fields and have therefore been the
subject of intensive, experimental, theoretical and numerical analysis during these
years . Powder technology, food and chemical industries and biological research
needs grow up in clearance procedure inside the unsmooth channels.
However, the cavity flow simulation in various Reynolds number extract the
attention of many researchers. In this particular problem, there are a range of
methods and approaches for finding the solid-liquid interaction. But most of the
studies carried on the subject of the particles trajectory in driven cavity flow . This
type of flow is crucial for analyzing fundamental features of recirculation fluid In the
case of fluid flow simulation; lattice Boltzmann established itself as a powerful
numerical scheme for solving flow problems. There are a few studies which are
implied this method for solid-liquid interaction investigation.
In present study, the particle trajectory is studied in lid driven cavity flow to
validate the main approach of research because there are some published result from
different authors that could be utilize for the validation of methods by comparison.
For example P. Kosinski et al. carried out a numerical study of particle motion
tracing in Eulerian-Lagrangian approach which has been published in powder
technology journal. In mentioned paper solid particles were studied in a 2D square
cavity flow. The result of the project due to particle trajectory was compared by the
experimental study of Tsorng et al. in case of Reynolds number 470. Because of
that, in present work the case of method validation set accordingly.
27
3.4.2 Particles movement through Fluids
Three forces acting on a particle moving through a fluid. The most common
force is the external force of gravitational or centrifugal act. In addition, the buoyant
force, that acts corresponding with the external force but in the opposed direction
and the force in the direction of flow exerted by the fluid on the solid which is
nominated as the drag force. Drag force appears each time there is relative
movement between the particle and the fluid.
If it is considered that, a particle of mass ( m) moving through a fluid under
the action of an external force Fe ,the velocity of the particle relative to the fluid is u,
and the buoyant force is acting on the particle is Fb and the drag force of fluid flow
define as Fd , then:
Y
= Z − Z\ − Z
(3.32)
The external force can be related on the mass of the particle and the
acceleration of the movement.
Z = . "
(3.33)
And the buoyancy force can be mentioned according to Archimedes'
principle:
Z\ = . D\ . ^
(3.34)
28
Where the is the density of particles and D\ is the volume of particle
which is submerge in fluid and g is the standard gravity on Earth.
According to this definition, if the mass of the object placed into the fluid is
less than the mass of the displaced fluid then the object will be buoyant and will
immerse itself in the water to a point where the mass of the displaced fluid is equal
to the mass of the buoyant object. For reaching this aim the density of the particle
should be equal or particularly equal to the density of fluid.
In present study, the density of particle supposed to be same as the fluid so
the particles assume buoyant and the buoyancy force is neglected.
The main and noticeable force is counted in this research is the drag force
which is acting on particles. Drag force can be state as:
Z = . _. . ` $
(3.35)
Where Cd is the drag coefficient, and V is the relative velocity between the
fluid and particle and A is the area of the particle.
The drag coefficient directly depends on particle Reynolds number .For flow
around the sphere; there are two main definitions for drag coefficient:
Stokes region, Re < 1
=
24
>a
(3.36)
29
And the transient region 1< Re <1000
=
$b
(1 +
!
6
K
>a J )
(3.37)
According to our assumption about the fluid Reynolds number in all cases we
use the equation (3.46).
On the other hand, the Reynolds number of particle could be defined as:
>a = . .
eYgNhe
i
(3.38)
Where the is the fluid density and μ is the dynamic viscosity. , which is
called from velocity profile calculation, is the fluid velocity and Dj is the particle
motion velocity.
So as it mentioned before in the case of the force vector of each particle in
this research, we only considered the drag force acting on a particle and other
mechanisms like buoyancy and lift forces, which can be easily implemented into the
code, are neglected so in jth particle, force vector can be stated as:
ll⃗
k = lll⃗
=
no pq FeY
l⃗gN
llll⃗esY
llll⃗u
r l⃗gN
r
$
(3.39)
30
3.4.3 Particle motion simulation by Lattice Boltzmann Method
The main elements of this solid fluid interaction studies, focus on
formulating the hydrodynamic force on particles under various condition, such as
different Reynolds number, particle situation inside the particular geometry and the
flow field and attempt to solve the equation of motion for particles by applying the
flow field characteristic properties, finding the properties of particle system such as
drag coefficient, particle velocity and so on.
The approach of this research in the first part is to prove the correctness of
the methods by comparing the fluid profile forming with the carried studies and in
the second step evaluating the behave of single particle motion in various flow
movement to study a system of multiple particles under a variety of flow conditions.
According to what mentioned before, the diameter of particles and density of
them should be chosen in the way that it cause the buoyancy and other forces except
drag force could be neglected. Accordingly, it is clear that the stoke number of
particles suppose to be very smaller than one. This fact causes the particles motion
just become the result of the influence of the fluid phase and they move in the
direction of velocity vectors of the current profile.
So, by applying the lattice Boltzmann method for finding the distribution
function in each node of the meshed geometry, from the connection between the
application of LBM and the behaviour of solid particle, it is considered that the LBM
is the best choice to couple with the second Newton’s law for prediction of fluidsolid interaction. Therefore, the purposes of this study are coupling the procedure of
the LBM formulation and solid particle dynamics (Lagrangian-Lagrangian), and to
compare the result of this research by the exits result in other methods of fluid-solid
interaction studies.
31
Furthermore, in this study it is assumed that presence of solid particle gives
no effect to the fluid flow and the particles are far enough so the particle-particle
interaction can be neglected.
32
Chapter 4
Results and Discussion
4.1 Difference between Single Relaxation Time (SRT) and Multi Relaxation
Time(MRT) in lid driven cavity in term of stability and accuracy
In this section lid driven Cavity is used to study about accuracy and stability
between MRT and SRT. So square cavity has been used for this case study .The
movement wall is considered as a movement boundary condition but bounce-back
boundary condition shows
stationary wall. This simulation is developed for
different Reynolds number
from 100 to 3200 .In addition, these Results are
validated excellent agreement with exist numerical results .
33
Figure 4.1 This figure provides information about comparison between SRT and
MRT for Re100,in X direction velocity
This study has proved accuracy of MRT in comparison by SRT is
represented in Figure 4.1 and 4.2 . Indeed, velocity in X and Y direction is validated
by Ghia in both method. Moreover, it has to be noted that Figure 4.1 and 4.2 show
that MRT has a good agreement with Ghia result in preview literatures.
34
Figure 4.2 This figure provides information about comparison between SRT and
MRT for Re100,in Y direction velocity
35
Figure 4.3 This figure provides information about comparison between SRT and
MRT for Re400,in X direction velocity
By increasing the Reynolds number MRT has emerged as a accurate
method apart from SRT ,figure(1.3)and (1.4) is represented accuracy of MRT for
RE 400 .
36
Figure 4.4 This figure provides information about comparison between SRT and
MRT for Re400,in Y direction velocity
37
Figure 4.5 This figure provides information about comparison between SRT and
MRT for Re1000,in X direction velocity
It is interesting to say that by growing Reynolds number SRT will be
approached to instability for example in Re = 1000 SRT is not stable with low mesh
This investigation is presented lid driven cavity by 100*100 mesh for Reynolds
1000.
38
Fig4.6This figure provides information about comparison between SRT and MRT
for Re1000,in Y direction velocity
39
Figure 4.7 This figure provides information about comparison between SRT and
MRT for Re3200,in X direction velocity
40
Figure 4.8 This figure provides information about comparison between SRT and
MRT for Re3200,in Y direction velocity
Multi relaxation time in both accuracy and stability has been shown in this
work. Figure (4.8) is shown that 100*100 meshes are used for lid driven cavity and
furthermore it should be noted that these meshes have a good stability till Re=1000
for MRT . On the other hand SRT will be stable for 400 and after this number the
code will be crashed.
41
4.2 Channel fluid flow by MRT-LBM
This section provides information about velocity along the channel and
position of the vortex in channel in different times. These results illustrats channel
flow in different time and Reynolds number , moreover different Reynolds number is
compared for several aspect ratio.
Figure 4.9 Shows the configuration of streamline in different Reynolds number from
Fang et al. research results
According to obtained results for velocity field in the channel flow ,the
simulations results strongly agree with existing result . The developed vortex from
left hand side of channel has been illustrated in Figure(4.13) for low Reynolds
number. Furthermore it has to be noted that it is not generated by increasing time. On
the other hand in Reynolds number 400 ,the vortex is developed and after a littel
time is moved form left hand side of channel to right hand side of channel. Different
aspect ratio and different Reynolds number are clearly illustrated in Figure (4.104.21) and furthermore , location of vortex have been indicated.
42
Figure 4.10 velocity field with MRT –LB in Reynolds Number =50 and Aspect
ration=1
Figure 4.11 velocity field with MRT –LB in Reynolds Number =50 and Aspect
ration=2
Figure 4.12 velocity field with MRT –LB in Reynolds Number =50 and Aspect
ration=3
Figure 4.13 velocity field with MRT –LB in Reynolds Number =50 and Aspect
ration=4
43
Figure 4.14 velocity field with MRT –LB in Reynolds Number =100 and Aspect
ration=1
Figure 4.15 velocity field with MRT –LB in Reynolds Number =100 and Aspect
ration=2
Figure 4.16 velocity field with MRT –LB in Reynolds Number =100 and Aspect
ration=3
Figure 4.17 velocity field with MRT –LB in Reynolds Number =100 and Aspect
ration=4
44
Figure 4.18 velocity field with MRT –LB in Reynolds Number =400 and Aspect
ration=1
Figure 4.19 velocity field with MRT –LB in Reynolds Number =400 and Aspect
ration=2
Figure 4.20 velocity field with MRT –LB in Reynolds Number =400 and Aspect
ration=3
Figure 4.21 velocity field with MRT –LB in Reynolds Number =400 and Aspect
ration=4
45
4.2.1 prediction of vortex by solid particle trajectory in Channel fluid flow
based on MRT-LBM
As far as prediction of vortex in different geometries is important for
researchers This section is going to present new idea to determine vortex and
streamline in different time by solid particle trajectory . The developed code has
been vividly validated with previous results found in literature and it has been
conclude that particle trajectory is useful to discover position and structure of vortex
flow.
Figure 4.22 Particle trajectory in reynolds number 50 and AR=4,(a) LBM and (b)
Fang et al.result.
Figure 4.23 Particle trajectory in Reynolds number 50 and AR=4 with Multi
Relaxation Time
Figure 4.23 is demonstrated that two particles in the channel flow in different
position for Re 50 and this result has powerful agreement with Fang et al.result
fig(4.22).It is interesting enough to say that at the left hand side of channel solid
46
particle goes around the vortex and another particle is shown streamline at the right
hand side of cavity .
Since the current study has proved different position and structure of vortex
in channel, the obtained results is simulated channel in different time and particle has
been considered in different position due to find position of vortex and structure of
it. Figure (4.24) is performed channel flow at time 0.4 for Re 400 and it is necessary
to say that at this time there is no vortex in the cavity , so the particle follows a flow
fluid streamline .
Figure 4.24 prediction vertex by particle trajectory in Reynolds Number =400 ,
Aspect ration=4 and Time=0.4
Figure 4.25 prediction vertex by particle trajectory in Reynolds Number =400 ,
Aspect ration=4 and Time=0.8
47
Figure 4.26 prediction vertex by particle trajectory in Reynolds Number =400 ,
Aspect ration=4 and Time=1.6
By increasing time the vortex is slowly generated form left hand side of
channel .figure (4.25) shows that particle goes around the vortex meanwhile it should
be to noted that only one particle can goes around the vortex however the others are
rotated and then goes out of the channel
Figure 4.27 prediction vertex by particle trajectory in Reynolds Number =50 ,
48
Aspect ration=4 and Time=8
Figure 4.28 prediction vertex by particle trajectory in Reynolds Number =50 ,
Aspect ration=4 and Time=8
4.3 Multi Particle Channel fluid flow
As far as removing Multi particle in flow fluid is one of the important
problem in industry this investigation is discovered simulation of particles by Multi
Relaxation Time based on LBM in channel fluid flow.
Multi particles have been assumed after fluid flow stability in the cavity for
different Reynolds number and Aspect ratio .In addition , the numerical modelling is
strongly validated by Fang et al figure(4.29).
Figure 4.29 Experimental (left) removal process of contaminated cavity fluid in Re=
50 and AR=4 Fang et al
49
As it is clearly illustrated in Figure (4.30 ) the fouling removal from cavity is
happened in Reynolds 50 and aspect ratio 4 . As a matter of fact in this simulation
multi particles are assumed form unsteady flow to stable flow and these simulation
results agree with Fang et al result.
Figure 4.30 Fouling removal from cavity in Reynolds number 50 and AR=4
50
4.3.1 contaminated removal in different Reynolds number
Figures(4.32-31-33) are going to represent 400 particles inside the channel
with different Reynolds numbers .It is noted that percentage of removal particle is
vividly demonstrated in different time .
These figures show some particles at the left hand side of the cavity that
they go around the vortex however, some others go out . Moreover, they have
indicated that by increasing Reynolds number more particles follow the vortex.
Since by increasing time the vortex is developed along the channel flow , the
huge percentage of particles inside the cavity can not leave cavity . It is necessary to
say that in Reynolds number 100 less than 52 percentage of particles
interestingly removed .
(a)
(b)
are
51
(c)
(d)
Figure 4.31 Fouling removal from cavity in Reynolds number 50 and AR=3
(a)
(b)
52
(c)
(d)
(e)
Figure 4.32 Fouling removal from cavity in Reynolds number 50 and AR=3
(a)
53
(b)
(c)
(d)
(e)
Figure 4.33 Fouling removal from cavity in Reynolds number 50 and AR=3
4.3.2 contaminated removal in different Reynolds number for AR=4
According to figures(4.34,35,36) by increasing aspect ratio high percentage
of particles can not leave the cavity .
54
(a)
(b)
(c)
55
(d)
(e)
Figure 4.34 Fouling removal from cavity in Reynolds number 100 and AR=4
(a)
(b)
56
(c)
(d)
(e)
Figure 4.35 Fouling removal from cavity in Reynolds number 70 and AR=4
(a)
57
(b)
(c)
(d)
(e)
Figure 4.36 Fouling removal from cavity in Reynolds number 50 and AR=4
As far as investigation of removal particle in different time is one of the
main concern of engineers .This behaviour is performed in this
current work
.Removal percentage of particles are presented in figure (4.37-42) for different
Reynolds number and aspect ratio . As a matter of fact ,percentage of removal
particles are grown by increasing time .
58
Figure 4.37 Particle removal percentage Reynolds number 50 and AR=4
Figure 4.38 Particle removal percentage Reynolds number 70 and AR=4
59
Figure 4.39 Particle removal percentage Reynolds number 100 and AR=4
Figure 4.40 Particle removal percentage Reynolds number 50 and AR=3
60
Figure 4.41 Particle removal percentage Reynolds number 70 and AR=3
Figure 4.42 Particle removal percentage Reynolds number 100 and AR=3
61
4.4 Multi-relaxation-time Lattice Boltzmann method simulation of backwardfacing step for incompressible flow
4.4.1 Backward-facing step for different Reynolds Number
This section provides information about generation vortex in backwardspacing step flow by MRT-LBM and length of vortex is illustrated along the
channel. This study investigates flow in a backward space and these results agree
well with suitable numerical results.
In this investigation rectangular mesh(200*40) is used for backspace flow
geometry . Re number is defined as Re=(U.H)/ν
where
is maximum velocity .
Actually for simulating channel flow parabolic velocity is assumed as a entry .
Mostly for solving channel flow it should be better to consider velocity around 0.07
, because Relaxation time can be suitable .The maximum inlet velocity U employed
as characteristic velocity and duct height H is the length scale .This result is going to
presented different reattachment for different Reynolds Number and discrepancy
Time and this result achievement has successful verified with benchmark [7,8].There
are different length of vortex by different Time .Vortex is going to generated form
the first obstacle at the left hand side the channel and actually will be grown time by
time.
As they are clearly illustrated in figures(4.43-49) velocity field for backwardfacing step in different Reynolds number are shown along the channel .As far as
determining stracture of vortex in this section is important . figures(4.43-49) are
going to represent developed vortex after stability in channel fluid flow.
62
Figure 4.43 Velocity field for backward-facing step in Reynolds Number=20
Figure 4.44 Velocity field for backward-facing step in Reynolds Number=30
Figure 4.45 Velocity field for backward-facing step in Reynolds Number=40
Figure 4.46 Velocity field for backward-facing step in Reynolds Number=50
Figure 4.47 Velocity field for backward-facing step in Reynolds Number=60
Figure 4.48 Velocity field for backward-facing step in Reynolds Number=100
Figure 4.49 Velocity field for backward-facing step in Reynolds Number=130
63
Different time and Reynolds number for prediction of vortex are indicated in
Figure (4.50-52). Increasing time and Reynolds number have a big impact on
increasing reattachment area in backward space. Furthermore it is noted that vortex
has been developed rapidly in comparison by high Reynolds number.
64
Reynolds=50
t1
t2
t3
t4
Figure 4.50 velocity field for Reynolds Number 50 for different time
65
Reynolds=70
t1
t2
t3
t4
Figure 4.51 velocity field for Reynolds Number 70 for different time
66
Reynolds=100
t1
` t2
t2
t3
t3
t4
t4
Figure 4.52 velocity field for Reynolds Number 100 for different time
67
Figure 4.53 different reattachment points for backward-facing step
Figure 4.54 different reattachment points for different Reynolds number
Figure (4-53,54) illustrate different reattachment area and length of vortex
along the channel . Moreover it has to be noted that this data is presented for
different time and Reynolds number and it is shown vividly in figure(4.54)
68
CHAPTER 5
CONCLUSION
5.1.Conclusion
Since flow fluid problems are serious in industries , CFD modeling has been
endeavored to solve it. Many researchers have investigated about different ways to
make CFD modeling and up to now there are too many methods which have been
discovered by them . Furthermore ,they have studied about accurate and stable way
to make suitable models in simulation of fluid flow.
As far as suitable model is caused to decrease computing time ,some
scientists have proposed Lattice Boltzmann Method (LBM). Many papers have
proved this method to model flow fluid and they have presented Lattice Boltzmann
with different Relaxation time. Multi Relaxation time has been emerged and
moreover obtained results has powerful agreement with experimental results.
In this work numerical stability and accuracy is investigated and the result
present that MRT model is much stable and accurate than that of the corresponding
BGK.
69
Since One of the applicable fluid flow problem is multi particles in channel
in industries ,current paper discusses about different Reynolds number for Multi
Relaxation time in the channel due to find out removal percentage of particles inside
the channel. The obtained results for multi particles have mentioned that
by
increasing Reynolds number the vortex is developed . So it causes to reduction of
percentage of removal particles.
In final section reattachment area is proved for different Reynolds number
and time .It is conclude that in low Reynolds number length of vortex is so small but
by increasing Reynolds this area will be developed and in addition it is noted that
this area is strongly dependent to several time .
5.2
Future works
As environmental viewpoint Heavy metals pollution have dangerous impacts
on environment which the most important of them is the toxic influence specially in
industrial cities.These days pollution problems and ecosystem damage have been
arised
by industrial activities . These damages have been resulted from some
accumulation of pollutants for instance , Toxic metal metal (chromium, copper, lead,
cadmium, zinc, and etc).
As a matter of fact, heavy metals have played the main role to dangerous
pollutant and they are mostly found in wastewater of different industries processes
such as electroplating ,metal finishing, metallurgical work , and chemical
manufacturing . There are too many methods that have been scientifically proposed
by engineers to eliminate heavy metals from wastewater such as following:
adsorption on miscellaneous adsorbents supercritical fluid extraction Ionexchange,
etc. It should be noted that the most effective way to remove heavy metal is
adsorption method. However , this way is extremely expensive to apply in industries
. In other word , this method is impossible to use everywhere .On the other hand, one
of the cost effective and efficient method that is emerged for industrial is
70
Biosorption. Since Modeling fluid flow before and after biosorption process and
during this physiochemical process needs to use computational fluid dynamic , many
CFD methods have been used to model fluid flow . One of the alternative method
for fluid process is Lattice Boltzmann method . Indeed, this method has been used
for laminar flow and turbulent flow and it has to be noted that LBM has been
proposed for physical chemical system with different geometries and complicated
boundary condition . For future work ,it is suggested that
modeling physical
chemical problems which is caused to find out optimization of sustainable treatment
of wastewater from industries will be considered.
.
71
REFERENCES
1.
Chen S.Y., Wang Z., Shan X.W., Doolen G.D., Lattice-Boltzmann
computational fluid dynamics in three dimensions . Journal of Statistical
Physics, 1992, 68: 379-400.
2.
Doolean C., lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech,
1998, 30:64-329.
3.
Brady J.F., and Bossis G ., Stokesian dynamics. Annual Review
of Fluid
Mechanics, 1988.20: 111-157.
4.
Du R., Shi B and Chen X., Multi-relaxation-time lattice Boltzmann model for
incompressible flow. Physics Letters A, 2006. 359: 564-572.
5.
Lin L .S., Chen Y.C and Lin C.A., Multi relaxation time lattice Boltzmann
simulations of deep lid driven cavity flows at different aspect ratios. Computers
&amp; Fluids, 2011. 45: 233-240.
6.
Mezrhab A, Double MRT thermal lattice Boltzmann method for simulating
convective flows. Physics Letters A, 2010. 374: 3499-3507.
7.
Chen C.K., Yen T.S., and Yang Y.T., Lattice Boltzmann method simulation of
a cylinder in the backward-facing step flow with the field synergy principle.
International Journal of Thermal Sciences, 2006.45: 982-989.
72
8.
Chen C.K., Yen T.S., and Yang Y.T., Lattice Boltzmann method simulation of
backward-facing step on convective heat transfer with field synergy principle.
International Journal of Heat and Mass Transfer, 2006. 49: 1195-1204.
9.
Fang, Effect of mixed convection on transient hydrodynamic removal of a
contaminant from a cavity. International Journal of Heat and Mass Transfer,
2002. 46: 2039-2049.
10. Succi S., The Lattice Boltzmann equation for fluid dynamics and beyond.
clarendon press oxford, 2001.
11. Doolean C, lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech,
1998, 30:64-329.
12. Zou Q.S., He X.Y., On pressure and velocity flow boundary conditions and
bounceback for the lattice Boltzmann BGK model.physics of fluids,1996. 9:
1591-1598.
13. Faure T.M., Visualizations of the flow inside an open cavity at
medium
range Reynolds numbers. Experiments in Fluids, 2007. 42: 169-184.
14. Mehta U.B and Lavan , Flow in a Two-Dimensional Channel with a
Rectangular Cavity. NASA CR , 1969.
15. Yao H, Cooper R, and
Raghunathan S, Numerical simulation of
incompressible laminar flow over three-dimensional rectangular cavities.
Journal of Fluids Engineering-Transactions of the Asme, 2004 .126: 919-927.
16. Chen S and Doolen G.D ., Lattice Boltzmann method for fluid flows. Annual
Review of Fluid Mechanics,1998.30:329-364.
17. Broglia R., Pascarelli R. A., and Piomelli U., Large-eddy simulations of ducts
with a free surface. Journal of Fluid Mechanics, 2003.484: 223-253.
18. Kosinski A.K., Hoffmann A.C., Simulation of solid particles behaviour in a
driven cavity fl ow. Powder Technology,2008.191:327-339.
73
19. Wan D.C and Turek S, An efficient multigrid-FEM method for the simulation
of solid-liquid two phase flows. Journal of Computational and Applied
Mathematics, 2007.203: 561-580.
20. Heemels M.W., Hagen M.H.J and Lowe C.P., Simulating solid colloidal
particles using the lattice-Boltzmann method. Journal of Computational
Physics, 2000.164:48-61.
21. Feng, Z.G. and Michaelides E.E., The immersed boundary-lattice Boltzmann
method
for
solving fluid-particles
interaction
problems.
Journal of
Computational Physics, 2004.195: 602-628.
22. Inamuro T A., lattice Boltzmann method for incompressible two-phase flows
with
large
density
differences.
Journal
of
Computational
Physics,2004.198:628-644.
23. Elizarova T.G., Simulation of separation flows over a backward-facing
step.Computational mathematics and modeling,2004
24. Sukop T, Lattice Boltzmann Modelling. Springer. 2004.
25. Chen S , and Doolen G.D., lattice Boltzmann method for fluid flows. Annual
Review of Fluid Mechanics, 1998. 30: 329-364.
74
APPENDIXES
The program code of simulation
Multi Relaxation Time for multi particle in channel flow
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%
% % % % % % % %%%MULTI RELAXATION TIME FOR MULTI PARTICLE IN CHANNEL
FLOW
%
____________________________________________________________________
_____
% THIS CODE IS WRITEN BY MOHAMMAD POURTOUSI
%%% MASTER OF MECHANICAL ENGINEERING AT UTM UNIVERSITY
%
EMAIL:mo_poortoosi@yahoo.com
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%
clear;clc;
%%%% PHYCICAL definitions FOR CHANNEL FLOW
lx=200;
ly=40;
hight1 =20;
hight2 =20;
lenght1=60;
lenght2=80;
lyInlet
lxInlet
lxOutlet
lyOutlet
lyLeftCavity
lxBottomCavity
lyRightCavity
=
=
=
=
=
=
=
hight1+1:ly;
1:lenght1;
lenght1+lenght2 : lx ;
hight2 + 1 : ly ;
1:hight1;
lenght1:lenght1+lenght2;
1:hight2;
rho(1:lx+1,1:ly)=1;
Uo=0.08;
umax=0.08;
Re=50;
%%%%%%%
75
Nu = Uo*(ly-hight1)/Re;
tau = 3*Nu+0.5;
ITERATIONFLUID=100;
ITERATIONPARTICLE=100;
ITERATIONMODEFRAME=50;
ASPECRATIO=lenght2/hight1;
u=zeros(lx,ly);
v=zeros(lx,ly);
mEq=ones(9,lx,ly);
f=ones(9,lx,ly);
w=[4/9,1/9,1/9,1/9,1/9,1/36,1/36,1/36,1/36];
ex=[0,1,0,-1,0,1,-1,-1,1];
ey=[0,0,1,0,-1,1,1,-1,-1];
S=[1,1.4,1.4,1,1.2,1,1.2,1/tau,1/tau];
M=[1, 1, 1, 1, 1, 1, 1, 1, 1;-4,-1,-1,-1,-1,
4,-2,-2,-2,-2, 1, 1, 1, 1; 0, 1, 0,-1, 0,
0,-2, 0, 2, 0, 1,-1,-1, 1; 0, 0, 1, 0,-1,
0, 0,-2, 0, 2, 1, 1,-1,-1; 0, 1,-1, 1,-1,
0, 0, 0, 0, 0, 1,-1, 1,-1];
InvMS=M\diag(S);
2, 2, 2, 2;
1,-1,-1, 1;
1, 1,-1,-1;
0, 0, 0, 0;
%%%%%%%%%%%%%%%%%%%%%% MULTI PARTICLE PART POSITION
%%%%%%%%%%%%%%%%%%%%%%%%%%%
Pnumber=400;
Dp=zeros(1,Pnumber);
% --------------------------------------------------------------------sum1=60;sum2=1;n=1;dx=1;dy=1
for i=1:4:80
for j=0:1:19
x=i*dx+sum1;
y=dy*j+sum2;
particles(1,n)=x;
particles(2,n)=y;
n=n+1
end
end
% ---------------------------------------------------------------------Particleplot=zeros(3,Pnumber);
Velocityparticles=zeros(3,Pnumber);
rhop=zeros(1,Pnumber);nParticle2=zeros(Pnumber,1);
Coln=zeros(1,Pnumber);
NuL=ly*Uo/Re;
LRatio = 4/lx;
NuRatio = NuL/Nu;
dt = (LRatio)^2*(NuRatio);
mu=37.2e-6; %%% viscosity of the fluid
rhop(:)=1.5; %%% particle density
rhof=1; %%% fluid density
Dp(1:round(Pnumber/2)) = 0.0015;
Dp(round(Pnumber/2):Pnumber)=0.0015;
% % % % % tic
% MULTI RELAXATION TIME MODELING
for t=1:ITERATIONFLUID
76
%%
%%%%%%%
collision
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i=1:lx
for j=1:ly
jx=rho(i,j) * u(i,j);jy=rho(i,j) * v(i,j);
mEq(1,i,j)= rho(i,j);
mEq(2,i,j)= -2*rho(i,j) + 3 * (jx^2 + jy^2);
mEq(3,i,j)= rho(i,j)
- 3 * (jx^2 + jy^2);
mEq(4,i,j)= jx;
mEq(5,i,j)= -jx;
mEq(6,i,j)= jy;
mEq(7,i,j)= -jy;
mEq(8,i,j)= jx^2 - jy^2;
mEq(9,i,j)= jx * jy;
end
end
m = reshape(M * reshape(f,9,lx*ly),9,lx,ly);
f = f - reshape(InvMS * reshape(m-mEq,9,lx*ly),9,lx,ly);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%%%%%%%
streaming
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i=1:9
f(i,:,:) = circshift(f(i,:,:), [0,ex(i),ey(i)]);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%%%%%%
Boundary Condition
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Left Wall (bounce back)
for uu=lyInlet
rhoInput=f(1,1,lyInlet)+f(3,1,lyInlet)+f(5,1,lyInlet)+2*(f(4,1,lyInl
et)+...
f(7,1,lyInlet)+f(8,1,lyInlet))./(1-u(1,uu));
f(2,1,lyInlet)=f(4,1,lyInlet) + 2/3 * rhoInput * Uo;
f(6,1,lyInlet)=f(8,1,lyInlet) + rhoInput * u(1,uu) / 6 ;
f(9,1,lyInlet)=f(7,1,lyInlet) + rhoInput * u(1,uu) / 6 ;
end% %%%% Right Wall (bounce back)
f(2,lx,lyOutlet)= f(2,1,lyInlet) ;
f(6,lx,lyOutlet)= f(6,1,lyInlet) ;
f(9,lx,lyOutlet)= f(9,1,lyInlet);
% -----------------------------------------------------------------------%%%% Bottom Wall (bounce back)
f(3,:,1)=f(5,:,1);
f(6,:,1)=f(8,:,1);
f(7,:,1)=f(9,:,1);
%%%% Top Wall
f(5,:,ly)=f(3,:,ly);
f(9,:,ly)=f(7,:,ly);
f(8,:,ly)=f(6,:,ly);
%%%% Bottom First Obstacle wall
f(3,lxInlet,hight1)=f(5,lxInlet,hight1);
77
f(6,lxInlet,hight1)=f(8,lxInlet,hight1);
f(7,lxInlet,hight1)=f(9,lxInlet,hight1);
%%%% Bottom Second Obstacle wall
f(3,lxOutlet,hight2)=f(5,lxOutlet,hight2);
f(6,lxOutlet,hight2)=f(8,lxOutlet,hight2);
f(7,lxOutlet,hight2)=f(9,lxOutlet,hight2);
%%%% Left Cavity wall
f(2,lenght1,lyLeftCavity)=f(4,lenght1,lyLeftCavity);
f(6,lenght1,lyLeftCavity)=f(8,lenght1,lyLeftCavity);
f(9,lenght1,lyLeftCavity)=f(7,lenght1,lyLeftCavity);
%%%% Right Cavity Wall
f(4,lenght1+lenght2,lyRightCavity)=f(2,lenght1+lenght2,lyRightCavity
);
f(8,lenght1+lenght2,lyRightCavity)=f(6,lenght1+lenght2,lyRightCavity
);
f(7,lenght1+lenght2,lyRightCavity)=f(9,lenght1+lenght2,lyRightCavity
);
%%%% Bottom Cavity Wall
f(3,lxBottomCavity,1)=f(5,lxBottomCavity,1);
f(6,lxBottomCavity,1)=f(8,lxBottomCavity,1);
f(7,lxBottomCavity,1)=f(9,lxBottomCavity,1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%%%%%%%%
rho and Velocity
%%%%%%%%%%%%%%%%%%%%%%%%%%%
rho
= reshape(sum(f),lx,ly);
u(:,:) = reshape(ex * reshape(f,9,lx*ly),lx,ly)./rho;
v(:,:) = reshape(ey * reshape(f,9,lx*ly),lx,ly)./rho;
u(lxInlet,lyLeftCavity)= 0;u(lxOutlet,lyRightCavity)= 0;
v(1,lyInlet) = 0 ;v(lxInlet,lyLeftCavity)=
0;v(lxOutlet,lyRightCavity)= 0;
L = ly-(hight1+1); yinlet = lyInlet-hight1-0.5;
u(1,lyInlet) = 4 * umax / (L*L) * (([0.5:1:19.5].*L)([0.5:1:19.5].*[0.5:1:19.5]));
int2str(t)
figure(2);
Uaverage=sqrt(u(:,:).^2+v(:,:).^2);
imagesc(Uaverage(:,ly:-1:1)'./Uo);
drawnow
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
% %%%%%%%
Particles insert %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % % MULTI PARTICLE LOOP
Removalpresentage=0;
for oo=1:ITERATIONPARTICLE
remind=0;
for n=1:Pnumber
Ureal(:,:)=u(:,:)/(LRatio*NuRatio);
Vreal(:,:)=v(:,:)/(LRatio*NuRatio);
78
xp=particles(1,n); upx=Velocityparticles(1,n);
yp=particles(2,n); upy=Velocityparticles(2,n);
xpp=round(xp);
ypp=round(yp);
if xpp<=lx&&xpp>=1&&ypp<=ly&&ypp>=1
velocityX=Ureal(xpp,ypp); velocityY=Vreal(xpp,ypp);
[xp yp upx upy] =
FparticleRK4(xp,yp,upx,upy,velocityX,velocityY,...
rhof,rhop(n),mu,dt,Dp(n));
particles(1,n)=xp; Velocityparticles(1,n)=upx;
particles(2,n)=yp; Velocityparticles(2,n)=upy;
else
remind=remind+1;
Removalpresentage=abs((((remind)/Pnumber)*100));
end
end
int2str(oo)
% end
% % % % % % % % % % % toc
if mod(oo,ITERATIONMODEFRAME)==0
figure(11)
TIMe=((oo*dt)+(t*dt));
hold on
plot(TIMe,Removalpresentage,'--ro','LineWidth',2,...
'MarkerEdgeColor','k',...
'MarkerFaceColor','g',...
'MarkerSize',5)
title(['REMOVAL
PRECENTAGE''AR=',num2str(lenght2/hight1),'&Re=',num2str(Re)])
xlabel('TIME');
ylabel('Removalpresentage');
hold off
figure(1);
fplot(num2str(hight1),[0 lenght1]);
xplot=lyLeftCavity;xplot(:)=lenght1;yplot=lyLeftCavity;
plot(xplot,yplot);
fplot(num2str(hight2),[(lenght1+lenght2) lx]);
xplot=lyRightCavity;xplot(:)=lenght1+lenght2;yplot=lyRightCavity;
plot(xplot,yplot);
figure(oo)
hold on
for n=1:Pnumber
plot(particles(1,n),particles(2,n),'r.')
end
% hold off
79
title(['AR=',num2str(lenght2/hight1),'&Re=',num2str(Re),'Iteration =
',int2str(oo),...
' && Time = ',num2str((oo*dt)+(t*dt)),'Removal particles
precentage=',...
num2str(Removalpresentage)])
xlabel('X direction');
ylabel('Y direction');
[xm,ym]=meshgrid(1:lx,1:ly);
u1=u';
v1=v';
fill([0,0,60,60,140,140,200,200],[0,20,20,0,0,20,20,0],'g')
fill([0,200],[40,40],'k')
drawnow
axis equal
axis([0 lx 0 ly]);
%
F=getframe(figure(oo));
imwrite(F.cdata,['C:\MULTI PARTICLE RESULT\2' num2str(oo) '.jpg']
,'jpg')
end
end