THE SIMULATION OF DROPLET MOTION BY USING LATTICE
BOLTZMANN METHOD
MOHD RODY BIN MOHAMAD ZIN
A thesis submitted in fulfillment of the
requirements for the award of the degree of
Master of Engineering (Mechanical)
Faculty of Mechanical Engineering
Universiti Teknologi Malaysia
APRIL 2009
DEDICATION
To my father and my late mother and most importantly my beloved wife
and daughter
ACKNOWLEDGEMENTS
All thanks belong to ALLAH, the Most Gracious, the Most Merciful and the
source of this success to complete this thesis.
In preparing this thesis, I have been in contact with various people,
researchers, academicians and practitioners. They have contributed to my knowledge
and understandings of this project. At first I wish to express my deepest gratitude to
my supervisor, Dr. Nor Azwadi Che Sidik for all his encouragement and guidance.
Without his suggestions, helps and criticisms, this thesis would not been as it is
presented now.
Here, I would like to thank to Universiti Teknikal Malaysia Melaka (UTeM)
for giving me the opportunity to do this research by funding my Master Engineering
studies. I am also indebted to Universiti Teknologi Malaysia and lectures especially
in Mechanical Engineering Faculty for providing me with all the knowledge during
my study here.
I would also like to express my thankfulness to my course mates for all their
help and encouragement.
Here I would like to thank to all my family members for always been
supportive during all my endeavors.
Lastly and most importantly, I am also deeply appreciated to my beloved
wife, Nor An binti Ibrahim and my daughter Dyan Nur Qistina for their continuous
moral support, patience and love.
ABSTRACT
SCMP (Single Component Multiphase) - LBM (Lattice Boltzmann Model) scheme
was developed in order to simulate the phenomenon of droplet motion under different
conditions. This study more concern on phenomenon of droplet falling from a flat ceiling
and the movement of droplet on inclined surface. Various type of parameter such as contact
angle, gravitational force and angle of inclined surface are used to interpret the results
obtained in order to explain the phenomenon of droplet dynamics. The basic idea of SCMP
LBM is incorporating the free energy method in lattice Boltzmann governing equation. The
Van Der Waals real gas equation of state is derived to determine different of phases in the
system. The new equilibrium distribution ݂௘௤ is calculated into the SCMP LBM equation.
The capillary and gravitational effects are incorporated into SCMP LBM equation via
pressure tensor and the new velocity in calculation of equilibrium distribution function,
݂௘௤ .Both capillary and gravity-driven flow contributes in different regimes of droplet
shapes. Good agreement was obtained between the present approach and those previous
studies using Navier-Stokes solver and original LBM.
ABSTRAK
Simulasi untuk fenomena pergerakan titisan bendalir kecil dalam pelbagai keadaan
telah dibangunkan menggunakan kaedah SCMP (Single Component Multiphase) - LBM
(Lattice Boltzmann Model). Fenomena titisan bendalir jatuh daripada siling yang rata dan
pergerakan titisan di atas permukaan yang condong dititit beratkan. Pelbagai jenis
pembolehubah seperti sudut lekapan, daya graviti dan sudut permukaan condong digunakan
untuk mentafsirkan fenomena titisan bendalir ini dengan lebih jelas. Idea asas di dalam
SCMP LBM, adalah dengan menggunakan kaedah tenaga terbebas di dalam persamaan
lattice Boltzmann. Persamaan gas nyata daripada Van Der Waals diterbitkan untuk
menentukan setiap fasa yang berbeza di dalam sistem. Dengan menggunakan pendekatan
daripada Brient’s, nilai baru fungsi taburan keseimbangan dikira untuk dimasukkan ke
dalam persamaan SCMP LBM. Kesan kapilari dan graviti telah dimasukkan ke dalam
persamaan SCMP LBM melalui persamaan tekanan lekapan dan nilai baru halaju di dalam
fungsi taburan keseimbangan. Kedua-dua kesan ini memberikan bentuk titisan yang
berbeza. Perbandingan keputusan yang diperolehi daripada pendekatan yang dilaksanakan
dengan kajian lepas yang menggunakan Navier-Stokes dan original LBM mendapati ianya
mencapai persamaan yang ketara jelasnya.
i
TABLE OF CONTENTS
CHAPTER
TITLE
PAGE
TABLE OF CONTENTS
i
LIST OF TABLES
iv
LIST OF FIGURES
v
LIST OF SYMBOLS
vii
INTRODUCTION
1
1.1
Background
1
1.2
Computational Fluid Dynamics
2
1.3
Lattice Boltzmann Model
3
1.4
Classical CFD versus Lattice Boltzmann Methods
4
1.5
Objective
5
1.6
Scope
5
CHAPTER 1
CHAPTER 2
LATTICE BOLTZMANN MODEL
6
2.1
Classical Boltzmann equation
6
2.2
Bhatnagar-Gross-Krook (BGK) Collision model
8
2.3
Boundary Conditions
9
ii
2.3.1
Periodic Boundary Condition
10
2.3.2
Free Slip Boundary Condition
10
2.3.3
Bounceback Boundary Condition
11
2.4
Relaxation Time
12
2.5
The Lattice Boltzmann Equation
13
2.6
Isothermal Lattice Boltzmann Models
14
CHAPTER 3
INTRODUCTION TO MULTIPHASE FLOW
16
3.1
Introduction
16
3.2
Van Der Waals Fluid
22
3.3
Phase (Liquid-Vapor) Separation and Interface
25
Minimization
3.4
Free energy lattice Boltzmann
26
3.4.1
Briant’s Approach
26
3.4.2
Yonetsu’s Approach
28
3.5
Thermodynamics of the fluid
29
3.6
Static wetting
31
3.6.1
Cahn theory
32
3.6.2
Partial wetting in lattice Boltzmann
34
CHAPTER 4
METHODOLOGY
35
4.1
Methodology
35
4.2
Flow Chart
37
CHAPTER 5
RESULT AND DISCUSSION
41
5.1
41
Original LBM Code Validation Analysis
5.1.1
Flow pattern of two rectangular cylinders
41
5.1.2
Bubble Rise
42
iii
5.2
Presence studies
47
5.2.1
Phase Separation
47
5.2.2
Equilibrium system of droplets with varying
49
contact angles
5.2.3
Deformation of droplet on horizontal plate in
54
a gravitational flow
5.2.4
Droplet falling
57
5.2.5
Droplet sliding
59
CHAPTER 6
CONCLUSION AND RECOMMENDATIONS
61
6.1
Conclusion
61
6.2
Recommendations
63
REFERENCES
64
APPENDIX A
69-81
LIST OF TABLES
TABLE NO
TITLE
PAGE
5.1
Parameters used for the simulation of phase separation
47
5.2
Selected fluid properties
49
5.3
Typical experimental and calculated data for various
50
droplets on substrate
5.4
The physical value of state and analysis condition
50
5.5
Simulation conditions for droplet falling from flat ceiling
58
5.6
Simulation conditions for droplet sliding on inclined
59
surface
v
LIST OF FIGURES
FIGURE NO
TITLE
PAGE
1.1
Classical CFD versus LBM
4
2.1
Periodic boundary condition
10
2.2
Free slip boundary condition
11
2.3
Bounceback boundary condition
11
2.4
Time relaxation concept
12
2.5
2-D Lattice structure for density distribution function
15
3.1
Conceptual framework for LBM
17
models[Sukop et a. 2001]
3.2
Isotherm plot of
24
3.3
Time series of liquid-vapor phase separation dynamics
25
3.4
Density gradient at the interface for various values of
31
4.1
Algorithm flowchart
37
5.1
A schematic of the coordinate system and
42
computational domain.
5.2
Streamline plot for Re = 10 and s/d=1.5
43
5.3
Streamline plot for Re=30 and s/d=3
43
5.4
Streamline plots for Re=10 and s/d=5(left)
44
and Re=50 and s/d=5(right)
5.5
Streamline plot for Re=70 and s/d=4
44
5.6
Shape regimes for bubbles in unhindered gravitational
46
motion through liquid [He et al,1999]. and Ra=104
vi
(contour values ranges from 0.01 to 1.0 with 20 levels).
5.7
Time evolution of buble shape at Eo=20 (density
46
distribution;maximum (red) = 4.895, minimum
(blue) = 2.211).
5.8
Snapshots of phase separation from t=500 to t=20000
48
5.9
Density profile at equiblirium condition
48
5.10
Liquid deformation on solid surface. The condition
49
θw<
and θw>
indicates that the solid is wet by the liquid,
indicates non-wetting, with the limits θw=
and θw=180o defining complete wetting and complete
non-wetting, respectively
5.11
Computational model for a droplet in contact with
50
a wetting wall
5.12
Equilibrium system of droplets with various contact angles 51
5.13
The ratio of droplet wet length and droplet height at
52
various contact angles
5.14
A droplet in contact with a wetting wall
52
5.15
Deformation of droplet on horizontal plate,
54
Bo=20,(density distribution; maximum (red) = 4.895,
minimum (blue) = 2.211).
5.16
Shape of droplets on a horizontal plate at an
55
equilibrated state
5.17
The ratio of droplet wet length and droplet height at
56
various Bond number
5.18
Shape of droplet falling
57
5.19
Shape of droplet sliding on inclined surface
59
vii
LIST OF SYMBOLS
SYMBOLS
a
Acceleration
ao
Droplet height
Bo
Bond number
bo
Wet length
c
Micro velocity vector
cs
Speed of sound
D
Dimension
E
Energy
fq
Heat viscous dissipation
f(x,c,t)
Density distribution function
fi
Discretized density distribution function
fi
eq
Discretized equilibrium density distribution function
Ff,g
External force
g
Gravitional force
gi
Discretized internal energy distribution function
gieq
Discretized equilibrium internal energy distribution function
q
Internal heat source
ν
Kinematic viscosity
α
Thermal diffusivity
μ
Viscosity
viii
Constant for surface tension
k
Thermal conductivity
cp
Specific heat
ρ
Density
x
Characteristic length
P
Pressure
Q
Collision operator
Ts
Surface temperature
Tc
Cold temperature
TH
Hot temperature
T∞
Quiescent temperature
β
Thermal expansion coefficient
k
Thermal conductivity of the fluid
R
Gas constant
t
Time
u
Horizontal velocity
u
Velocity vector
U
Horizontal velocity of top plate
Channel inlet velocity
v
Vertical velocity
υ
Dynamic shear
Bulk viscisity
x
Space vector
w
Weight coefficient
τ
Time relaxation
Stress in fluid
χ
Thermal diffusivity
Ω
Collision operator
ix
Abbreviations
BGK
Bhatnagar-Gross-Krook
CFD
Computational Fluid Dynamics
D2Q9
Two Dimensions Nine Velocities
FD
Finite Difference
FDLBE
Finite Difference Scheme Lattice Boltzmann Equation
FE
Free Energy
FEM
Finite Element Method
LB
Lattice Boltzmann
LBE
Lattice Boltzmann Equation
LBM
Lattice Boltzmann Method
LGA
Lattice Gas Approach
MD
Molecular Dynamics
PDEs
Partial Differential Equations
SCMP
Single Component Multiphase
SC
Shan Chen
VOF
Volume of Fluid
2-D
Two Dimensions
3-D
Three Dimensions
Non-dimensional parameter
Grx
Grashof number
Nu
Nusselt number
Pr
Prandtl number
Rax
Rayleigh number
Re
Reynolds number
1
CHAPTER 1
INTRODUCTION
1.1
Background
At this recent day, simulation is a very important as a tool to predict the answer of
the problem in fluid dynamic. The application of computational method promising a good
approximating results to the physical world. A lot of works has been done and still in
discovering for a better computational method to solve the problem and improving the
method that already exist.
There are numerous computational exist in literature. One of them is used to solve
the fluid flow problem. Computational fluid dynamics (CFD) is one of the branches of fluid
mechanics that uses numerical methods and algorithms to solve and analyze problems that
involve fluid flows.
The fundamental law of any CFD problem is the Navier-Stokes equations, which
define any single-phase fluid flow. The classical Navier-Stokes equations have been used
from 150 years ago which describe viscous fluid flow. These equations can be simplified
by removing terms describing viscosity to yield the Euler equations. Further simplification,
2
by removing terms describing vortices yields the full potential equations. They are nonlinear partial differential equations which express mass and momentum conservation for
fluids and can only be easily solved for only simple cases.
These equations can be
linearized to yield the linearized potential equations. Solving this equation is a very
challenging task. A lot of numerical method was introduced by mathematicians and
engineers in CFD, such as Finite Difference Method, Finite Element Method and Finite
Volume Method to solve the Navier-Stokes equation numerically.
1.2
Computational fluid dynamics
In conventional computational fluid dynamics (CFD), scientists and engineers
describe a fluid flow by introducing a representative control-volume element, on which
macroscopic mass and momentum are conserved. The conservation laws of mass and
momentum lead to a "macroscopic" mathematical model, governed by the Navier-Stokes
equation, which is traditionally discretized and applied to a physical domain of interest.
Physical variables such as velocity and pressure at each grid point around the element can
be numerically computed.
Computational fluid dynamics (CFD) is the numerical simulation of fluid flows.
CFD become essential tool in solving the Navier-Stokes Equation, the continuity equation,
the energy equation and equation derived from them. Incompressible Navier-Stokes
equation is the heart of the CFD, which represent a local conservation law for the
momentum in the system. This equation only partially addresses the complexity of most
fluids of interest in engineering applications; it is successfully applied in different areas for
predictions of fluid flows.
The classical approach in CFD, treat of such fluids and describe the new physical
properties in terms of transport phenomena related to a new observable, macroscopic
property. A PDE is written down for the dynamics of this property then is solved by an
appropriate numerical technique. In a fluid with important temperature variations for
3
example, a new observable property, the temperature, is introduced and its dynamics is
described by a heat transport equation.
1.3
Lattice Boltzmann Model
In recent years, the lattice Boltzmann method (LBM) has attracted much interest in
the physics and engineering communities. As a different approach from the conventional
computational fluid dynamics (CFD), the LBM has been demonstrated to be successful in
simulations of fluid flow and other types of complex physical system. In particular, this
method is promising for simulations of multiphase and multicomponent fluid flow
involving complex interfacial dynamics. It is a discrete computational method based upon
the Boltzmann equation. It considers a typical volume element of fluid to be composed of a
collection of particles that are represented by a particle velocity distribution function for
each fluid component at each grid point. It obtains macroscopic flow information based on
integration of probability density function.
Unlike other conventional CFD that directly simulates evolution of the macroscopic
kinetic equation for the single particle distribution function, the time is counted in discrete
time steps and the fluid particles can collide with each other as they move, possibly under
applied forces. The rules governing the collisions are designed such that the time-average
motion of the particles is consistent with the Navier-Stokes equation.
A major advantage of lattice Boltzmann method is the ease and accuracy with which
it enables complicated boundary geometries to be processed, hence, investigating suitable
boundary conditions for lattice Boltzmann simulations has become a highly researched area
in many engineering and scientific applications. Another advantage of using LBM is the
simplicity of programming, the parallelism of the algorithm, and the capability of
incorporating complex microscopic interactions. It is an approach that bridges microscopic
phenomena with the continuum macroscopic equations. Further, it can model the time
evolution of the systems.
4
1.4
Classical CFD versus Lattice Boltzmann Methods
The conventional simulation of fluid flow and other physical processes generally
starts from non linear partial differential equation (PDEs). These PDEs are discretized by
either finite differences, finite element, finite volume or spectral methods. The resulting
algebraic equations of ordinary differential equation are solved by standard numerical
methods.
In LBM, the starting point is a discrete microscopic model governed by Boltzmann
equation. The derivation of the corresponding macroscopic equation requires multi-scale
analysis [Wolf Gladrow, 2000].
Partial Differential
equations (NavierStokes Equation)
Discretized
Discrete Model (Lattice
Boltzmann Models)
Multi-Scale
Analysis
Ordinary Differential
Equation (Solved using
standard numerical
method)
Partial Differential
equations (Navier-Stokes
Equation)
Figure 1.1: Classical CFD versus LBM
5
1.5
Objectives
The objective of this project is to develop Single Component Multiphase Lattice
Boltzmann scheme for simulating the phenomenon of droplet motion.
1.6
Scope
The scopes are:
i.
Numerical simulation of two dimensional lattice Boltzmann
scheme
ii.
Study and apply lattice Boltzmann method in solving single
component multiphase flow problem
iii.
Develop programming code using Fortran 90 base on free
energy approach-multiphase lattice Boltzmann
6
CHAPTER 2
LATTICE BOLTZMANN MODEL
2.1
Classical Boltzmann Equation
The LBM was developed from the improvement at lattice gas automata (LGA)
model, [G. McNamara and G. Zanetti, 1988]. LGA were introduced in 1950 and can be
described as a collection of identical “cells” regularly distributed in space where each cell
has an associated state, which changes during each time step depending on the states of its
neighbors.
In the LGA model, the dynamics of particles consists of two steps:
-
particles at the same site collide according to a set of hard-sphere
particle collision rules that conserve mass, momentum, and energy
(for multispeed models) at each lattice site;
-
after collision, particles advance to the next lattice site in the
direction of their velocities. The small number of discrete velocities
allowed in the LGA models is tightly coupled with the spatial lattice
structure.
7
The problem with LGA is its binary nature where it attempts to simulate individual
particles and consequently produces grainy “black and white” distributions. They can be
smoothed by averaging over larger areas of the LGA grid, but this is makes for inefficient
use of high grid resolution where performance is wasted.
Lattice Boltzmann equation is directly obtained from the lattice gas automata by
taking ensemble average with the assumption of random phase and leads to the following
equation [Wolf Gladrow, 2000];
∆ ,
∆ ,
∆ -
, ,
(2.1)
=
, , , is the single particle distribution function with discrete velocity c. Ω
where
is
the lattice Boltzmann collision operator.
Distribution function
with velocity
, ,
describes the number of particles at position , move
at time . There are two conditions related to the distribution function;
without collisions and with collisions.
At a short time ∆ , each particle moves from
velocity changes from
particle at
to
∆ , where
∆ and each particle
to
is the acceleration due to external forces on a
with a velocity . Hence, the number of molecules
∆ ,
the number of molecules
∆ ,
∆
, ,
is equal to
, for the distribution without
collisions [N. A. C. Sidik, 2007]. Therefore;
∆ ,
∆ ,
∆
-
, ,
=0
(2.2)
There will be a net difference between the number of molecules
∆ ,
and the number of molecules
∆ ,
∆
, ,
if collision occurs between
the molecules. This can be expressed by;
∆ ,
∆ ,
∆
-
, ,
=Ω
(2.3)
8
where Ω
is the collision operator. On dividing by
zero ( ~0 give the Boltzmann equation for
, and letting
;
2.4
Ω
2.2
tends to
Bhatnagar-Gross-Krook Collision Model
The Bhatnagar-Gross-Krook (BGK) collision model is used to model collisions as a
statistical redistribution of momentum which locally drives the system toward equilibrium
while conserving mass and momentum. However, one of the major problems when
associated with the Boltzmann equation is the complicated nature of the collision integral.
The collision function represents the collision of fluid molecules at each node and has the
following form [Bhatnagar et al. 1954]:
1
Ω(f ) = − ⎛⎜ f ( x, c, t ) − f
τ⎝
where f
eq
eq
(x, c, t )⎞⎟
(2.5)
⎠
is the equilibrium distribution function and τ is the relaxation time which is related to
the viscosity of the fluid ( υ =
(2τ - 1) ), where υ
6
is the kinematics viscosity.
From BGK Lattice Boltzmann Equation:
f (x + c Δ t , t + Δ t ) − f (x , t ) = −
1
τ
(f
− f
eq
)
(2.6)
We know that:
f
= f
neq
+ f
eq
(2.7)
9
where;
f
eq
f
neq
2.3
= The equilibrium distribution function
and
= The nonequilbrium distribution function
Boundary Conditions
Since the early 1990s, many papers have proposed and investigated the behavior of
various boundary conditions (Ziegler 1993; Skordos 1993; Inamuro et al. 1995; Noble et al.
1995; Ginzbourg and d'Humieres 1996; Maier et al. 1996; Zou and He 1997; Fang et al.
1998; Verberg and Ladd 2000; Zhang et al. 2002; Ansumali and Karlin 2002; Chopard and
Dupuis 2003). This work continues, though workable boundary conditions for many types
of simulations are now available.
The distribution function at boundary nodes is unknown after each streaming
process. The boundary conditions are responsible for determining these unknown
distributions. In general, there are two ways to define boundary conditions; placing the
boundary on grid nodes or placing the boundary on links (Xiaoyi He et al. 1995). To define
the boundary condition, appropriate selection must be conducted; it depends on the type of
boundary conditions to be applied. LBM have several types of boundary conditions.
10
2.3.1 Periodic Boundary Condition
The simplest boundary conditions is ‘periodic’ in that the system becomes closed by
the edges being treated as if they are attached to opposite edges. Most early papers used this
type of boundary condition along with bounceback boundary condition. In simulating flow
in a slit for example, bounceback boundary condition is applied at the slit walls and
periodic boundary condition is applied to the ‘open’ ends of the slit. Fully periodic
boundary condition is also useful in some cases (for example, simulation of an infinite
domain of multiphase fluids).
Figure 2.1: Periodic boundary condition [Rodzimin et al. 2008]
2.3.2 Free Slip Boundary Condition
If the boundary is smooth with the negligible of friction exerted upon the flowing
gas or liquid, free slip boundary condition is the most suitable choice [S. Succi 2001]. In
this case, the tangential motion of fluid flow on the wall is free and no momentum to be
exchanged with the wall along the tangential component. These boundary conditions reflect
the distribution functions at the boundaries to neighboring position in lattice. This can
reviewed with reference to Figure 2.2. The usual approach is the direct reflection of
particles whenever a particle collide with wall in direction 3 will reflect into direction 5,
and similarly for direction-2 particles reflect into direction 6.
11
Figure 2.2: Free slip boundary condition
2.3.3 Bounceback Boundary Condition
Bounceback boundaries are particularly simple and have played a major role in
making LBM popular among modelers interested in simulating fluids in domains
characterized by complex geometries such as those found in porous media. Their beauty
lies in that one simply needs to designate a particular node as a solid obstacle and no
special programming treatment is required. Thus it is trivial to incorporate images of porous
media for example and immediately compute the flow in them. Bounceback boundaries
come in several variants (Succi 2001) and do not work perfectly (e.g., Gallivan et al. 1997;
Inamuro et al. 1995).
Figure 2.3: Bounceback boundary condition
12
The usual approach is direct reflection of particles that collide with the wall.
Whenever a particle in direction 3 from node B collides with the wall, a direction-6 particle
is sent back to node B in the following time step, and similarly for direction-2 particles
from node C. Consequently, the time average of the population at node A has an equal
number of direction-3 and direction-6 particles and an equal number of direction-2 and -5
particles, and so the average velocity at node A is zero. This result is the basis of the logic
of using direct reflection at the walls.
2.4
Relaxation Time
Time relaxation, τ in BGK collision model is the time taken to reach steady state
solution for transient fluid flow problem. The concept of time relaxation as discussed;
.
.
.
.
Non-equilibrium
.
Equilibrium
Non-equilibrium
Figure 2.4: Time relaxation concept
Figure 2.4 show, value of
0.5 is the limit for the relaxation time. At
0.5, all
particles initially at non-equilibrium state is totally exchange to non-equilibrium state. This
creates the instability in numerical and the iteration will not converge.
Numerical stability and iteration to converge need the particles at equilibrium state.
This can be obtained by manipulating the value of the time relaxation. The more closer time
relaxation to 1, the more number of particles exchange to equilibrium state.
13
2.5
The Lattice Boltzmann Equation
The Boltzmann equation with BGK collision model can be expressed as;
2.8
The Maxwell-Boltzmann equilibrium distribution function is defined as;
1
2.9
2
2
The BGK lattice Boltzmann equation can be derived by further discretise using an Euler
time step in conjunction with an upwind spatial discretization and setting the grid spacing
divided by the time step equal to the velocity;
,
∆
∆
,
,
∆
∆
,
∆ ,
∆ ,
∆
∆
∆
∆
,
∆
,
∆
2.10
2.11
As a result;
∆ ,
∆
,
∆
2.12
14
2.6
Isothermal Lattice Boltzmann Models
Most lattice Boltzmann simulations are used to only simulate the continuity and
Navier-Stokes equations. The temperature is assumed to be constant and the equilibrium
distribution will no longer conserve energy; instead it serves as a thermoset. So, the
evolution of the lattice Boltzmann BGK equation needs to discretised in velocity space in
order to apply the lattice Boltzmann scheme into the digital computer. To find out this
results, expand the Boltzmann-Maxwell equilibrium distribution function up to u².
f
eq
⎛ 1 ⎞
= ρ⎜
⎟
⎝ 2πRT ⎠
D/2
⎧ c 2 ⎫ ⎡ c ⋅ u (c ⋅ u ) 2
u2 ⎤
exp⎨−
+
−
⎬⎢1 +
⎥
RT 2( RT ) 2 2 RT ⎦
⎩ 2 RT ⎭⎣
(2.13)
The macroscopic properties, density ( ρ ) and flow velocity (u), of the nodes is calculated
using the following relations.
∑f =∑f
i
eq
i
∑ cf = ∑ cf
i
=ρ
eq
i
or
= ρu
(2.14)
(2.15)
In this study, the improved incompressible D2Q9i (two-dimensional, nine particles,
incompressible) model is used , which has three types of particles on each node; a rest
particle, four particles moving along x and y principal directions with speeds c i = ±1 , and
four particles moving along diagonal directions with speeds c i = 2 .
Then the expression for the discretized density equilibrium distribution function is
obtained as follow:
fi
eq
2
⎡
9(ci .u )
3u 2 ⎤
= ρω i ⎢1 + 3ci .u +
+
⎥
2
2 ⎥⎦
⎣⎢
(2.16)
15
where;
⎧(0.0)
⎪
ci = ⎨(cosθ i , sin θ i )
⎪
⎩ 2 (cosθ i , sin θ i )
i=0
(2.17)
i = 1,2,3,4
i = 5,6,7,8
and θ i =(i -1)п/2 for i = 1,2,3,4 and (i – 5)п/2 + п/4 for i = 5,6,7,8.
Here, ω i is a weighting factor, ci and u are the microscopic and macroscopic
velocities at each nodes, respectively. Values for the weighting factor and the microscopic
velocities depend on the used lattice Boltzmann model, (D2Q9 model) where the
microscopic velocity, c = 3RT and the weighting factor, ω i = 4/9 for i =1, ω i =1/9 for i =
1,2,3,4, and ω i =1/36 for i = 5,6,7,8. Figure 2.5 shown equivalents to the well-known
D2Q9 model Lattice structure,
c7
c3
c6
c1
c4
c8
c2
c9
c5
Figure 2.5: 2-D Lattice structure for density distribution function.
16
CHAPTER 3
INTRODUCTION TO MULTIPHASE FLOW
3.1
Introduction
Conventional methods for simulating two phase flow consist of numerical
integration of the Navier-Stokes equations and molecular dynamics simulations. These
techniques are extremely computationally intensive and particularly difficult to implement
in random geometry. In recent years, the LBM has become an established numerical
scheme for simulating multi-phase fluid flows. The key idea behind the LBM is to recover
correct macroscopic motion of fluid by incorporating the complicated physics of problem
into a simplified microscopic models or mesoscopic kinetic equations. This intrinsic feature
enables the LBM to model phase segregation and interfacial dynamics of multi-phase flow,
which are difficult to be handled by applying conventional CFD methods or employing the
molecular dynamics (MD) method to incorporate intermolecular interactions at mesoscopic
level. The LBM has demonstrated a significant potential and broad applicability with many
computational advantages including the parallel of algorithm and the simplicity of
programming [Chen and Doolen, 1998].
17
The true strengths of LBMs however lie in their ability to simulate multiphase
fluids. Both single and multi-component multiphase fluids can be simulated. ‘Component’
refers to a chemical constituent such that a ‘single component’ (say H2O) multiphase
system would involve the liquid and vapor phases of water. These are particularly rich
systems to consider as surface tension, evaporation, condensation, and cavitation are
possible. Liquid-vapor behavior in partially saturated porous media can be simulated. In
contrast, a multi-component system can consist of separate chemical components such as
oil and water; such systems have been studied more extensively because of their economic
importance. For example, Darcy’s law-based relative permeability concepts for
multicomponent oil/water-like systems have been investigated using LBM (Buckles et al.
1994; Soll et al. 1994; Martys and Chen, 1996; Langaas and Papatzacos, 2001).
Figure 3.1: Conceptual framework for LBM models [Sukop et al. 2007]
18
At the top left in Figure 3.1, is a single chemical component whose molecules are
not subjected to any ‘long-range’ interaction forces. Adding a long-range attractive force
makes phase separation into a liquid and its own vapor possible. If we add a second
chemical component, we have the possibility of simulating completely miscible fluids
(basically chemical solutions) in the absence of long range interactions (lower left), and
completely immiscible fluids (oil and water for example) when there are long range
repulsive interactions (lower right). This chapter focuses on Single Component Multiphase
(SCMP) models. Early examples of lattice gas SCMP models can be found in Rothman
(1988) and Appert and Zaleski (1990). The lattice Boltzmann implementation of these
models began with Shan and Chen (1993, 1994). There are also so-called ‘‘free energy’’
approaches proposed by Swift et al. (1996), and ‘‘finite density’’ models that use the
Enskog equation for dense gases (Luo 2000; He and Doolen 2002). Zhang and Chen (2003)
have also proposed an approach based on tracking an energy (temperature) component.
Such finite density or energy models seem to hold the key to the ultimate development of
the LBM for practical applications due to the more realistic and consistent treatment of the
equation of state that preserves the essential (molecular) physics of the process.
Several authors have set up lattice Boltzmann schemes for two-phase systems.
Among them, [Gunstensen et al,1991] developed the multiphase LBM in 1991. It was
based on the two-phase lattice gas model proposed by Rothman and Keller4 in 1988. Later,
[Gunstensen et al,1991] extended this model to allow variations of density and viscosity.
However, as pointed by Chen et al., the multiphase LBM by [Gunstensen et al, 1991] has
two drawbacks: first, the procedure of redistribution of the colored density at each site
requires time-consuming calculations of local maxima; second, the perturbation step with
the redistribution of colored distribution functions causes an anisotropy surface tension that
induces unphysical vortices near interfaces
In order to simulate multi-phase fluid flows, [Gunstensen et al, 1991] developed a
multi-component LBM on the basis of two-component lattice gas model. Shan and Chen
presented a LBM model with mean-field interactions for multi-phase and multi-component
fluid flows. Later, [Swift et al,1995 and 1996] proposed a LBM model for multi-phase and
19
multi-component flows using the idea of free energy; [He et al,1999] developed a model
using the index function to track the interface of multi-phase flow.
Although the LBM is a promising method for multi-component/phase flows, one of
disadvantages is that all the schemes listed above are limited to small density ratio (less
than 20) due to numerical instability. Obviously, this is not realistic for most two-phase
systems e.g. the density ratio of liquid–gas systems is usually larger than 100, and even the
density ratio of water to air is about 1000. To overcome this difficulty, [Inamuro et al,2004]
proposed a LBM for incompressible two-phase flows with large density differences by
using the projection method. In this method, two particle velocity distribution functions are
used. One is used for calculating the order parameter to track the interface between two
different fluids; the other is for calculating the predicted velocity field without pressure
gradient. The corrected velocity satisfying the continuity equation can be obtained by
solving a Poisson equation.
In the recent decade there has been significant development in numerical methods
for modelling and simulation of interface dynamics There are several traditional CFD
methods, most of which can be fitted in two categories : the front tracking method and front
capturing method.
In particular, the front-tracking technique has emerged as a superior method.
Systematic comparisons show that front-tracking (FT) method is superior to particle
methods, PLIC-VOF, level sets and capturing, in that order. The front-tracking method uses
a set of discrete marker points connected to each other to form a piecewise linear (in 2D) or
a triangular (in 3D) description of the interface. The marker points (or markers) are
completely independent of the grid system upon which flow fields are computed, and are
evolved in a Lagrangian manner.
Front-capturing methods track the movement of fluid and capture the interface
afterwards. In this method, the two fluids are modelled as a single continuum with
discontinuous properties at the interface. There are three types of front capturing method:
20
the Marker-and-cell (MAC) method, Volume-of-Fluid (VOF) method, and level set
method, based on how the interface propagation is obtain thereafter.
The most important distinction between front-tracking and diffusive interface
capturing is, of course, the treatment of the interface. In the front-tracking method,
interfacial locations are tracked by a set of Lagrangian marker points from which the
interfacial forces are evaluated, while in the diffusive interface capturing method,
interfacial locations are somewhat arbitrarily defined by the contours of some continuous
function which has to be computed throughout the entire domain. This has some important
ramifications.
The existing multiphase lattice BGK diffusive interface capturing method in several
aspects the multiphase LBE method it is due to a non-ideal gas equation of state, which
usually does not consider the surface tension explicitly. In the multiphase LBE method, the
surface tension is a numerical artifact which is difficult to control independently [Alexander
et al, 1993].
There are three types of multiphase lattice Boltzmann model that frequently used.
The first type is so-called colored model for immiscible two-phase flow proposed by
Gunstensen et al. and based on the original lattice gas model by Rothman and Keller.
Gunstenten et al. used colored particles to distinguish between phases. However, the
colored model has some limitation that the model is not rigously based upon
thermodynamics, so it is difficult to incorporate macroscopic physics into the model.
The second type of LB approach used to model multi-component fluids was derived
by Shan and Chen (SC model) and later extended by others. In the SC model, a non-local
interaction force between particles at neighbouring lattice sites is introduced. The net
momentum, modified by interparticle forces, is not conserved by the collision operator at
each local lattice node, yet the system’s global momentum conservation exactly satisfied
when boundary effects are excluded. The main drawback of the SC model is that it is not
21
well-established thermodynamically. One cannot introduce temperature since the existence
of any energy-like quantity is not known.
The third type of LB model for multiphase flow is based on the free-energy (FE)
approach, developed by Swift et al., who imposed an additional constraint on the
equilibrium distribution functions. This free-energy approach provides more realistic
contact angles and fluid density profiles near the vicinity of an impenetrable wall which
cannot be easily obtained by other LBM schemes. The FE model conserves mass and
momentum locally and globally, and it is formulated to account for equilibrium
thermodynamics of nonideal fluids, allowing for the introduction of well defined
temperature and thermodynamics. The major drawback of FE approach is the unphysical
non-Galilean invariance for the viscous terms in the macroscopic Navier-Stokes equation.
Efforts have been made to restore the Galilean invariance to second-order accuracy by
incorporating the density gradient terms into the pressure tensor.
The next chapter discuss the theory of Van Der Waals fluid. The value of density
for both liquid and gas phases at certain pressure and temperature is determined by plotting
the isotherm P-V graph. The free energy approach two-phase lattice Boltzmann model is
also discussed in the next chapter.
22
3.2
Van Der Waals Fluids
The Van Der Waals real gas equation of state is
3.1
n = number of mole
a,b = constant characteristic of particular gas
R = Gas constant
P,V and T = Pressure, Volume and Temperature
We have,
n = 1 is set for convenience. The Van Der Waals equation can also be written in the form
3.2
by equating Eq.(3.3) and Eq.(3.4) to zero
0
3.3
0
3.4
The first and second derivatives for Eq. (3.2) will be
2
3.5
23
2
6
3.6
Solve both Eq. (3.5) and Eq. (3.6) for RTC
2
3.7
3
3.8
Equating the right hand side of both above equation
2
3
3.9
Finally
3
Substitute
3.10
back into Eq. 3.7 will give
8
27
3.11
By substituting these two results into Eq. (3.4) will give
8
27
3.12
24
Define the following reduced quantities:
;
;
3.13
Thus the van Der Waals equation becomes
3
3
1
8
3.14
Figure 3.2 shows the plot of isotherm on a
diagram for various . For T
greater than TC, (T>TC), the graph looks very much like the ideal gas isotherms. The system
separates into two phases, a gas of volume VG and a liquid volume VL when T less than TC .
The gas and liquid phases have same pressure,PLG. The value VG and VL can be calculated
by recalling that equilibrium condition, and the chemical potentials of the two phases must
be equal. By using Maxwell equal area construction the VG and VL can be determined.
Figure 3.2: Isotherm plot of
25
The above graph show the phase area at T=1.2, 1.0 and 0.9. if T = 0.55, the value of VG,
and VL are 0.4523 (or density ρG = 2.221),0.2043(or density ρL=4.895) respectively.
3.3
Phase (Liquid-Vapor) Separation and Interface Minimization
In this case, the phase separation ultimately leads to a single droplet or vapor.
Whether liquid drops or vapor bubbles are formed depends on the total mass in the domain
and consequently on the initial density selected.
Figure 3.3: Time series of liquid-vapor phase separation dynamics [Sukop et al, 2007]
When phase separation occurs, there is a strong tendency for the interfaces formed
to minimize their total area (or length in 2-D). This is a straightforward consequence of free
energy minimization and occurs in part by geometric rearrangement into the minimum
surface area volume (a sphere or in 2-D, a circle). Depending on the initial conditions, this
rearrangement may also involve a significant amount of coalescence of ‘blobs’ of each
phase. In liquid-vapor systems, there can also be condensation and evaporation; bubbles
can simply fill in or grow at the expense of mass elsewhere in the domain.
26
3.4
Free energy lattice Boltzmann
There are two methods have been proposed in FE model. The first one was
proposed by Briant et al. which described the original model to restore Galilean invariance.
The second one was proposed by Yonetsu. He has make a further improved the Briant’s
model where the isotropy of the free- energy two- phase model is considered.
3.4.1
Briant’s Approach
A power series in local velocity is assumed
,
,
,
,
,
3.15
where the summation over repeated Cartesion indices is understood.
The coefficient A, B, C, D and
moments of
are determined by placing constraint on the
. The collision term conserves mass and momentum the first moment of
are constrained by
3.16
3.17
,
The continuum macroscopic equations approximated by evolution equation
correctly describe the hydrodynamics of a one-component, non-ideal fluid by choosing the
next moment of
,
,
. This gives
3.18
27
where
1
∆
2
3
3.19
is the kinematic shear viscosity, τ is time relaxation and
is the pressure tensor. The first
formulation of the model omitted the third term in Eq. (3.18) and was not Galilean
invariant. Holdych et al. showed that the addition of this term led to any non Galilean
invariant terms being of the same order as finite lattice corrections to Navier-Stokes
equations. In order to fully constraint the coefficient A, B, C, D and
, a fourth
condition is needed, which is
,
,
,
3.20
3
The values of the coefficients can be determined by a well established procedure. For the
constraints (3.16)-(3.19) one possible choice of coefficients is:
8
3.21
4
2
,
12
,
12
4
,
8
16
3.22
3.23
4
,
3.24
2
16
12
8
8
8
3.25
3.26
3.27
3.28
4
for all
and
3.29
28
The analysis of Holdych et al. shows that the evolution scheme, Eq. (2.1)
approximates the continuity equations
0
(3.30)
And the following Navier-Stokes level equation:
(3.31)
3
(3.32)
3
3
3
3
The top line is the compressible Navier-Stokes equation while the subsequent lines
are error terms.
We have, then, described a framework for a one component free energy lattice
Boltzmann. The properties of the fluid are determined by the choice of pressure tensor
is discussed for two coexisting phases.
3.4.2
Yonetsu’s Approach
The derivation of the coefficients A, B, C, D and
based on the isotropic tensor approach.
proposed by Yonetsu was
29
Yonetsu et al. claimed that their model could predict well the phenomenon of
bubble shear and give very good agreement with analytical result for the Laplace’ law
pressure of the droplet-gas system.
3.5
Thermodynamics of the fluid
The thermodynamics of the fluid enters the lattice Boltzmann simulation via the
pressure tensor
. The equilibrium properties of a system with no surface (i.e periodic
boundaries) can be described by a Landau free energy functional
,
2
3.33
Subject to the constraint
3.34
where
,
is the free energy density of bulk phase,
is a constant related to the
surface tension, M is the total mass of fluid and the integrations are over all space. The
second term in Eq. (3.42) gives the free energy contribution from density gradients in an
inhomogeneous system. For Van Der Waals fluid, free energy density bulk phase can be
written in the form
,
1
3.35
Introducing a constant Lagrange multiplier, µ, we can minimise Eq.(3.42), giving
a condition for equilibrium as
ρ
0
3.36
30
By multiplying Eq.(3.45) by
/
and integrating once with respect to x, we obtain the
first integral
constant
2
3.37
At equilibrium condition, the chemical potential and pressure of both phases are given by
1
1
2
3.38
3.39
1
respectively. We now define
,
φ
µρ
p, meaning that Eq. (3.36) and Eq.
(3.37) can be rewritten as
3.40
and
3.41
2
By solving Eq. (3.41), we are able to determine the density profile at the interface for
different values of
as shown in Figure 3.4 Fourth order Rungge-Kutta is used to solve
Eq. (3.41) and temperature is set at T=0.55. As can be seen from the graph, the value of
is related to the density gradient at the interface and also affect the width of interface.
31
Figure 3.4: Density gradient at the interface for various value of
The free energy enters the lattice Boltzmann via the pressure tensor
. Since the
free energy function and total mass constraint are independent of position, it follows from
Noether’s theorem that conservation of momentum takes the form
0
By choosing
3.42
this gives
3.43
With
3.44
2
where
is the equation of state of the fluid
32
3.6
Static wetting
The equilibrium shape of the liquid defines a static contact angle with the surface,
θw, which is determined by a balance between the interactions between liquid, gas, and
solid. The wet surface at (θw = 0), partially wet (0< θw < ) or dry (θw = ). According to
Young’s law [52 from Brient’s theses] the contact angle is related to the three surface
tensions
3.45
cos
where
and
are the surface tensions of the solid-gas and solid-liquid interfaces
respectively. As usual
is the liquid-gas surface tension.
The aim of this chapter is to describe how equilibrium wetting can be incorporated
into lattice Boltzmann numerical scheme. In order to synchronize the equilibrium wetting
with lattice Boltzmann numerical scheme, the Cahn theory were used for the case of partial
wetting.
3.6.1 Cahn theory
From Cahn theory, he assumes that the fluid-solid are short-ranged such that they
contribute surface integral to the total free energy of the system. The total free energy
becomes
Ψ
Here, Φ
the surface,
φ
,
2
Φ
3.46
is a surface free energy density function which depends only on the density at
, and S is the surface bonding V.
33
Following Cahn and Gennes the Φ is expanded as power series in a linear term
Φ
1
3.47
From Eq.3.56 the general boundary condition become
1
3.48
The surface tension for solid gas and solid liquid in the case of complete wetting is given as
2
2
2
2
1
Ω 1
Ω
3.49
1
Ω 1
Ω
3.50
where Ω is the wetting potential
Ω
2
2
cos
3
1
cos
3
3.51
and α = arcos(sin2
therefore, by choosing a desired contact angle
calculated.
, the required wetting potential can be
34
3.6.2
Partial wetting in lattice Boltzmann
In this section, the method of incorporating wetting into the lattice Boltzmann
scheme is proposed by including the equilibrium distribution at the wall. The proposed
scheme is illustrated as below by considering a two-dimensional system with a wall at x=0.
i.
Think of an angle between 0 and π, call it
ii.
The general form of boundary condition is
iii.
Putting the dimensional quantities back into Eq.(3.51) to calculate 1
2
iv.
2
cos
2
3
/
1
cos
1
3.52
3
Equation (3.57) is imposed at the wall through the equilibrium distribution
function,
/
. For a wall site the equation for
and
become
1
3.53
1/2 ∆
7
,
1/ ∆
,
1
8
2
1,
,
2,
1
3
∆
1
3.54
3.55
35
CHAPTER 4
METHODOLOGY
4.1
Methodology
This research is carried out to simulate the phenomenon of the droplet under motion
conditions. There are four conditions of droplet motions where:
•
Droplets with varying contact angles at equilibrium state
•
Deformation of droplet on horizontal plate under gravitational effect
•
Phenomenon of droplet falling
•
Phenomenon of droplet sliding
For equilibrium system of droplets with varying contact angle, there is no
gravitational force applied where it is simulated naturally according to the Van Der Waals
theory. The droplet will form its own contact angle at reached the equilibrium state. To
validate this phenomenon, the graph of the dependency of the ratio of wet length between
droplet and a wetting wall (bo/ao) to a droplet height on the contact angle (θω) of a droplet
and compared with F.A.Dulilean et al. as the benchmark.
36
The same procedures were repeated; however in the next problem the gravitational
force is implemented to see the deformation of droplet on horizontal plate. By varying the
value of gravitational force, the droplet deforms until equilibrium state where the Bond
number can be calculated. To validate this result, the graph of the dependency ratio of a wet
length between a droplet and a wetting wall to a droplet on Bond number were plotted and
compared with [Murakami et al.2001].
For the droplet falling, the droplet is stick to the ceiling and leave to fall under
gravitational effect. The pattern of the droplet will investigated and observed. The patterns
of the droplet are compared with [Ozawa et al.2005]
In order to see the effect of capillary force and viscosity of the droplet in more real
situation, the solid plane is set to be inclined at certain degree of angle.
37
4.2
Flow Chart
Start
Set initial value of ρ,u and v
Calculate;
,
,
,
,
,
Assume;
fn =feq
Streaming;
,
,
∆
,
,
∆ ,
Collision;
Δt
f i,n j+ k = f in, j+ k −
( f in, j+ k − f eq )
τ
Calculate;
n+k
∑ f i, j = ρ
n+k
∑ c y f i, j = ρv
n+k
∑ c x f i, j = ρu
NO
Iteration end
YES
Print fn+k, ρ,u
and v
Figure 4.1: Algorithm flowchart.
Stop
38
Figure 4.1 shows the simulation algorithm for multiphase LBM using Fortran 90.
The algorithm begin with the setting at initial conditions, streaming, collision and boundary
conditions. In order to simulate SCMP problem, there are three parameters is set as initial
condition. Where u and v is equal to zero and ρ is set for liquid and gas. From the initial
conditions, the equilibrium distribution function is calculated;
,
,
,
,
,
4.1
By assuming fn = feq , we then calculate the streaming, where G, is the additional
term that incorporating the thermodynamic into lattice Botlzmann streaming equation.
,
,
∆
,
,
∆ ,
4.2
After the streaming process, the particles will collide with each other in its own
behavior by following this collision equation;
Δt
f n+k = f n+k −
( f n + k − f eq )
i, j
i, j
i, j
τ
(4.3)
The profile of distribution function between two neighboring nodes is constructed.
The value of distribution function at new time step is obtained by applying the streaming
and collision process.
There are three types of boundary condition implemented in this simulation. To
define the boundary condition, appropriate selection must be conducted; it depends on the
type of boundary conditions to be applied. LBM have different types of boundary
conditions. The simplest boundary condition is non-slip boundary condition; it defines zero
velocity at the wall by averaging the velocity at the wall before and after collision. These
boundary conditions bring back the distribution functions at the boundaries to original
position in lattice [N. A. C. Sidik 2007]. This is appropriate when the solid wall has a
sufficient rugosity to prevent any net fluid motion at the wall.
39
If the boundary is smooth with the negligible of friction exerted upon the flowing
gas or liquid, free slip boundary condition has been implemented [S. Succi 2001]. In this
case, the tangential motion of fluid flow on the wall is free and no momentum to be
exchanged with the wall along the tangential component. These boundary conditions reflect
the distribution functions at the boundaries to neighboring position in lattice.
Periodic boundary conditions typically intended to isolate bulk phenomena from the
actual boundaries of the real physical system and consequently they are adequate for
physical phenomena where surface effect play a negligible role. Periodic boundary
conditions are applied directly to the particle populations, and not to macroscopic flow
variables. They are generally useful for capturing flow invariance in a given direction. If a
uniform body force is used instead of an imposed pressure gradient, periodic conditions can
be used in place of macroscopic inflow/outflow conditions in the stream wise direction
[Robert S. Maier et al.1996]. This boundary condition can be implemented by bringing the
same distribution function that leaving the outlet to the inlet.
After all the above process completed, the output will be plotted in a contour view
where the density distribution is counted. The equations are as follow;
n+k
∑ f i, j = ρ
(4.4)
n+k
∑ c y f i, j = ρv
(4.5)
n+k
∑ cx f i, j = ρu
(4.6)
One of the important aspects in the numerical simulation is the convergence
criterion. Convergence criterion is used to check whether the solution achieved the steady
state solution or not. It takes thousands of iteration to reach steady state depending on the
value of the density and the set up boundary conditions. For the multiphase flow, the
convergence criterion is set by using equation below:
max |
|
10
where the calculation is carried out over the entire system.
4.7
40
Simulation of the original LBM and the SCMP-LBM was started by coding in the
Fortran 90. Desktop with Intel Pentium(R) 4, 3.40GHz processor and 2.00 GB RAM was
used to compile the code. The results obtained were transfer to workstation Silicon
Graphics 320 for the graphical representation. Software used for the graphical is
AVS/Express Visualization Edition, Version 4.2 R991.
41
CHAPTER 5
RESULT AND DISCUSSION
5.1
Original LBM Code Validation Analysis
In order to verify the LB numerical scheme, two samples of simulation have been
carried out. For a simple flow, the flow pattern of two rectangular cylinders was study in
order to demonstrate the capability of LBM in solving fluid flow. For a multiphase flow,
the phenomenon of bubble rise under buoyancy force is simulated.
5.1.1
Flow pattern of two rectangular cylinders
The flow pattern with a Reynolds number range 10 ≤ Re ≤ 100 has been
investigated numerically with mesh size of 251 x 201 and 351 x 289 lattice. For the
Reynolds numbers considered in this paper, it is known from experiments and other
numerical studies that vortex shedding can be observed and a 2D time-dependent type of
flow. The flow past two obstacles with certain transversal distance between the obstacles, S
where S = 1.5, 2, 3, 4 and 5 times of the obstacle diameter, D has been studied in this part.
The blockage ratio for Reynolds number = 10, 30 and 50 is 0.15 and for the Reynolds
42
number = 70, 80 and 100 is 0.12. The numerical simulation was carried out for a 500000 to
1000000 non-dimensional time iteration to reach a final steady state, requiring 5000 to
10000 time steps with ∆t = 100 for all the analysis.
S
U
D
Y
X
Figure 5.1: A schematic of the coordinate system and computational domain
The boundary conditions in this investigation are as follows. At the inlet, a
parabolic velocity inflow profile is applied. The outflow boundary condition for velocity is
∂u
∂x
= ∂v
∂x
= 0 . No-slip boundary conditions are prescribed at the body surfaces. At the
top and bottom surfaces of the channel, symmetry conditions simulating a frictionless wall
are used (u = v = 0 ) . The normal derivative for the pressure is set to zero at all boundaries.
The normal derivative in the diagonal direction for the pressure is also set to zero at all
corners of the flow field.
In this study, the flow pattern for square cylinders in tandem arrangement is studied
by using Lattice Boltzmann numerical scheme. The numerical simulations of the vortex
shedding for two square cylinders were carried out for Reynolds number of 10<Re<100.
The Reynolds number,Re=10 was first simulated. At s/d=1.5, the symmetrical small wake
appeared in center of x axis of square cylinders. It slowly disappeared with the increment of
Re.
43
5 Streamlline plot forr Re = 10 annd s/d=1.5
Figure 5.2:
Foor s/d=3 andd Re=10 to 100, a sim
milar pattern
n was observved. The tw
win vortex was
w
formed beetween the two squaree cylinders and a smalll wake apppeared downnstream of the
second cylinders.
Figuree 5.3: Stream
mline plot foor Re=30 annd s/d=3
Foor the case of Re=10 and
a s/d=5, small wakee formed affter the upstream cylinnder
and creeping steady flow passes the downnstream squuare cylindeer with no separation
s
t
take
ment of Re, result in the formatiion of largge twin vorrtex
place. Hoowever furtther increm
between thhe cylinderss and follow
wed by a sm
mall wake juust downstreeam the cyliinder as shoown
in Figure 5.4(right).
44
Fiigure 5.4: Streamline
S
p
plots
for Re=10 and s/d
d=5(left) and Re=50 an
nd s/d=5(rig
ght)
Sinnce the sim
mulated flow
w pattern arre similar for
f all the R
Reynolds nu
umber, we can
conclude that the forrmation of twin recircculation zon
ne can be oobserved beetween the two
t
5
which is good
g
agreem
ment with K
Kelkar and Patankar .T
The
square cyllinders at 50<Re<100
recirculatiion zone ap
ppears clearlly and stablle when 3<s/d<5. The downstream
m wake slow
wly
disappear when Re an
nd s/d increeases. All of the flow structure
s
forr the two sq
quare cylind
ders
in this LB
BM simulaation compaared well w
with the prrevious expperimental and numerrical
investigatiion.
Figure 5.5: Streamline plot for Re=70
R
and s//d=4
Thhis study fo
ound that the
t recircullation zonee greatly ddependencess on Reyno
olds
number an
nd gap spacing, s/d of the
t two squaare cylinderrs. The flow
w pattern is similar for low
Reynolds number witth 251 x 20
01 meshing grid. The recirculation
r
n zone clearrly observed
d at
w
the ranges
r
limitt. The mottivation of this study is due to the
Re>50 annd s/d>3 within
scattering of results for
f the flow
w characterisstic especiaally the depeendence of the transversal
45
distance between the length of center of circular cylinders to the depth of square cylinder,
S/D and the value of Re. In conclusion, in order to obtain a much more accurate result, 3D
model analysis and un-uniform grid mesh is recommended for future research interest.
5.1.2
Bubble Rise
In this section, the –two dimensional single bubble rising under buoyancy is
simulated. The density of each phase is taken as 1
4.895 and
2.221. The periodic
boundary condition is employed at all boundaries. Initially, buble is located at the lower
region(one sixth of the height) of computational domain of 161x481. The dimensionless
parameters (Eotvos, Morton number and Reynolds) are defined as;
∆
5.1
∆
5.11
5.12
where g is the gravitational force, ∆ is the density difference for two phase system,
is
the fluid density, U is the velocity of the bubble at equilibrium state, d is the radius of
bubble and
is the surface tension coefficient.
There are six types of bubble shape that can be classified. Which are spherical,
ellipsoidal, wobbling, dimpled, skirted and spherical-cap, depending on the Eo number and
Re number.
46
Figure 5.6: Shape regimes for bubbles in unhindered gravitational motion through liquid
[He et al,1999].
Simulation have been done for Eo=20. Due to the buoyancy force, the bubble
moves upward. In a meantime, the middle part of the bubble encounters a large deformation
due to hit from surrounding water. Eq. (5.1) indicates that the increase of Eo is equivalent
to the decrease of the surface coefficient . For the case Eo=20, the shape of the bubble
changes from the original. This is due to the decrement of surface tension coefficient.
Figure 5.7: Time evolution of buble shape at Eo=20 (density distribution; maximum (red)
= 4.895, minimum (blue) = 2.211)
47
5.2
Presence studies
This subsection starts with simulation of phase separation where the system are in
random phase at initial condition and transient to liquid and vapor phase with minimum
surface tension allowed. Then the droplet spreading and wetting is simulated with various
contact angle. A gravitational effect is implemented to the droplet until it is achieve the
equilibrium state. The shape of the droplet falling from a flat ceiling is simulated and
finally the motion of droplet sliding will observed.
5.2.1
Phase Separation
In this section, the phase separation which is based from the thermodynamic
instability of the Van Der Waals fluid is simulated. As discussed in Sec. 3.2, if the initial
state is set to an isothermal unstable region, according to the equation of state, the system
will automatically separates to the liquid phase and the vapor phase and then achieve the
equilibrium state.
The transient behavior of phase separation was done in order to examine the
validity of Brient’s model. The D2Q9 model with 101× 101 lattice is used and the
simulation was done at T = 0.55 . Other parameters are presented in the Table 5.1
Table 5.1 Parameters used for the simulation of phase separation
Δx
Δy
Δt
τ
κ
1.0
1.0
1.0
1.00
0.0001
48
(a) t=2
(b) t=200
(c) t=500
(d) t=20000
Figure 5.8: Snapshots of phase separation from t=500 to t=20000
Density values for random phase
separation
6
5
density
4
3
2
1
0
0
50
100
150
Figure 5.9: Density profile at equilibrium condition
In Figure 5.8, the density distribution clearly describes the phase separation of
liquid and vapor. High density (4.895) is the liquid phase and low density (2.211) is vapor
phase. At 20000 time iterations the bubble nuclei form a 2D circle minimum surface
tension area.
49
5.2.2
Equilibrium system of droplets with varying contact angles
In this section, the data are taken from [Derrick et .al 2004] to simulate the droplet
under equilibrium of equation of state. The investigated liquids are Silicone oil,
Hexadecane and Glycerine while the solid substrates include Glass, PMMA (poly methyl
methacrylate) and Polystyrene. The droplet is left to spread until it reached equilibrium
contact angle. The decided contact angle are 0o to 180o as shown in Table 5.3.
Table 5.2: Selected fluid properties
Figure 5.10: Liquid deformation on solid surface. The condition θw<90 indicates that the
solid is wet by the liquid, and θw>90 indicates non-wetting, with the limits θw=0 and
θw=180o defining complete wetting and complete non-wetting, respectively
50
Table 5.3: Typical experimental and calculated data for various droplets on substrate
Final Area of
Liquid
Substrates
Spreading,
2
cm
Contact
Contact
Angle
Angle (Cal)
Error
(Exp)
Glycerine
Glass
0.077
28.5
24.78
3.62
Glycerine
PMMA
0.042
70.64
64.76
6.97
Glycerine
Polystyrene
0.0324
104
83.88
20.12
Hexadecane
Glass
0.08656
23.93
23.30
0.63
Hexadecane
PMMA
0.14068
11.55
12.21
-0.66
Hexadecane
Polystyrene
0.20545
6.544
8.08
-1.536
Table 5.4: The physical value of state and analysis condition
4.106
4/7
2.894
0.125
T
c
r
0.4
1.0
20
Δx
1.0
3.5
τ
β
0.0025
1.0
0.1
Δy
Δt
L
H
1.0
1.0
151
51
Figure 5.11: Computational model for a droplet in contact with a wetting wall
51
(a)
(d)
= 8O
(b)
= 24O
(c)
= 70O
= 104O
Figure 5.12: Equilibrium system of droplets with various contact angles
52
50
45
40
bo/ao
35
30
F.A.L.Dullien et al
25
Present
20
15
10
5
0
0
50
100
150
200
θw
Figure 5.13: The ratio of droplet wet length and droplet height at various contact angles
Figure 5.14: A droplet in contact with a wetting wall
Initially, the droplet is set at 180o contact angle or in non-wetting condition. Then
the droplet is left to spread on the wetting wall without applying the gravity to form the
contact angle in Table 5.3. The droplet will form the set contact angle when it reaches the
equilibrium state. The equilibrium state means that the adhesive and cohesive force are
balanced when it reach the contact angle. Figures 5.12 show the equilibrium contact angles
are good agreement with [Derrick et .al. 2004]. In order to verify this result, the ratio of
droplet wet length and droplet height at various contact angles is plotted.
53
The equation for wet length bo over droplet height ao, from F.A.L Dullien are:
2√1
1
1
√1
2√1
1
1
√1
,
,
2
2
5.2
5.21
From the graph in Figure 5.13, it is clearly shown that the droplet contact angle is
in good agreement with theoretical value when the contact angle is larger than 70o or in a
partial wetting condition to non-wetting. However, when the contact angle is lower than
70O the show little agreement with the theoretical value. In conclusion, it can be justify that
the Brient’s approach had lead the equilibrium contact angles satisfy the theoretical value
from F.A.L. Dullien. In the next section, the deformation of droplet under gravitational
force on horizontal plate will be discussed.
54
5.2.3
Deformation of droplet on horizontal plate in a gravitational flow
In order to simulate the droplet under gravitational flow, the gravitational force term
should included in the equation. The gravitational force enters the system from the new
value of velocity at y- direction after the streaming process is done. The value of velocity in
equilibrium distribution function is called back in velocity subroutine. Where the equation
yields;
5.3
(a) 5000 step
(b) 8000 step
(c) 12000 step
(d) 15000 step (equilibrated state)
Figure 5.15: Deformation of droplet on horizontal plate, Bo=20,(density distribution;
maximum (red) = 4.895, minimum (blue) = 2.211).
55
Present (Bo=1.321)
Present (Bo=5.143)
Present (Bo=9.087)
Present (Bo=15.75)
Present (Bo=25.08)
Murakami et al. (Bo=1.313)
Murakami et al. (Bo=5.250)
Murakami et al. (Bo=9.103)
Murakami et al. (Bo=15.46)
Murakami et al. (Bo=25.16)
Figure 5.16: Shape of droplets on a horizontal plate at an equilibrated state
56
12
10
bo/ao
8
6
Murakami et.al
Present
4
2
0
0,000
5,000
10,000
15,000
20,000
25,000
30,000
Bo
Figure 5.17: The ratio of droplet wet length and droplet height at various Bond number
The effect of gravitational flow give a vital roles in determine the shape of droplet
with several of Bond number. The use of the dimensionless Bond number Bo – which
relates capillary and gravitational forces. The dimensionless Bond number reflects the
balance between gravitational and capillary forces and is
5.4
By varying the value of gravity it will affect the Bond number where it is
proportional to ratio of wet length over droplet height, bo/ao. The strength of adhesive
wetting droplet is depending on surface tension between solid and liquid. From Figure
5.16, it clearly shown that the Brient’s approach is in good agreement with the droplet
shape pattern from Murakami et.al. The Bond number also gives an understanding that the
viscosity of the droplet is increase when the Bond number is larger. This argument is
supported by the time of iteration for droplet deformation into equilibrated state in Figure
5.16.
57
5.2.4
Droplet Falling
t=1
t=15000
t=25000
t=30000
(a) Present
Figure 5.17: Shape of droplet falling
(b) Ozawa
58
Table 5.5: Simulation conditions for droplet falling from flat ceiling
0.0025
Δ
1.0
gy
-0.00001
θw
/2
In this section, the droplet is simulated under falling condition where it is stick on
the flat ceiling. The droplet contact angle at the ceiling plane is set to 90o which is the same
initial condition with [Ozawa et al. 2005]. The present result shows that the droplet is start
to fall from ceiling at 15000 iteration times. The droplet tail can be clearly observed at this
time. At 25000 times iteration the droplet is started to form a sphere shape until it fully
developed as a sphere shape when reaches 30000 time iteration. From [Ozawa et al. 2005],
the wetting potential will affect the time period for the droplet to reach steady state. Where,
by decreasing the value of wetting potential and
. From the presented results, current
simulation results can be said to give very good agreement with the benchmark results.
59
5.2.5
Drroplet slidin
ng
θ=45o
θ=4
45o
Figu
ure 5.18: Sh
hape of dro
oplet sliding
g on inclinedd surface
ny droplet problems
p
asssociated with
w gravitattional effectt. One of th
hem
Thhere are man
is the dro
oplet motio
on phenomeenon on innclined surfface. In thiis section, the droplet is
simulated with two conditions
c
o contact aangle on 45o of inclined angle. Ta
of
able 5.6 sho
ows
mulating the droplet slid
ding.
the conditions parameeter for sim
Table 5.6: Simulation
n conditions for droplet sliding on iinclined surrface
κ
0.0025
Δt
1.0
gx
0.0000
01
θw
70o, 10
04o
θ
45o
60
The time evolution for this simulation is taken from 0 to 100000 time iterations. The
parameters that distinguish between these two cases were their contact angle. In Figure
5.18(left) θw was set to70o and in Figure 5.18(right) θw was set to 104o. In Figure
5.18(left) is in negligible wetting condition at the initial state. It gradually changes its shape
until 100000 iterations time. However there are just small changes in it shapes compared to
Figure 5.18(right). The droplet is remaining in negligible wetting condition after
completing the iteration.
As shown in Figure 5.18(right), it is clearly observed that the droplet changes it
partial wetting initial state to almost complete wetting condition at the end time iteration.
Obviously, the changes of droplet shapes are influenced by the contact angle at initial state.
It is understood that the surface tension between droplet and solid surface from different
contact angle gives the strength of droplet viscosity to stick and slide on the surface.
61
CHAPTER 6
CONCLUSION AND RECOMMENDATIONS
6.1
Conclusion
In the first chapter the background of CFD and introduction of LBM is presented. In
chapter two, the lattice Boltzmann theory and the stability of the model related to the time
relaxation was clearly mentioned. Several types of boundary conditions used in the LBM
simulation and also the isothermal model and thermal model are discussed to expose the
reader about LBM. Chapter three discussed the theory of LBM in single component
multiphase flow. In chapter four, the methodology and the algorithm that have been used
for the simulation was explained. In chapter five, results of the numerical simulations for
the flow pattern and bubble rise were demonstrated for the purpose of validation code. The
results gave a good agreement with theoretical benchmark.
The present approach was successfully incorporated the free energy method in
lattice Boltzmann governing equation that derived based on Brient’s approach. The VanDer Waals real gas equation of state was used to determine different phases in the system.
The new equilibrium distribution ݂௘௤ has been applied into the SCMP LBM equation.
62
Present study has proved the capability of lattice Boltzmann model simulating
single component multiphase flow at the microscopic scale. Results show that this method
can indeed be very useful in such studies. The advantages of multiphase lattice Boltzmann
approach are not only capable of incorporating interface deformation and interaction but
also in the interparticle interactions, which are difficult to implement in traditional methods.
In order to verify the proposed approach, the phenomena of phase separation,
deformation of droplet spreading and wetting with gravitational effect, droplet falling and
droplet sliding were studied.
The phase separation phenomenon has been correctly predicted where the value of
density or volume for both phases at equilibrium state are in good agreement with the
isothermal ܲ෨ െ ܸ෨ graph. For droplet spreading and wetting, there were two parameter
influences the droplet phenomenon; the gravitational effect and the contact angle. If the
gravitational flow accounted into the system, the droplet spread on the horizontal flat
surface at equilibrium state. Then, the Bond number has been calculated to demonstrate
different gravitational flow was results in different behavior of droplet. For a droplet falling
phenomenon, the present studies gave a same shape pattern compared to benchmark
solution. From droplet sliding simulation result, it showed that the shape for droplet with
104o contact angle remain its negligible wetting condition where else for 70o contact angle
the droplet almost form a complete wetting condition at the same inclined angle and
iteration times. It can be concluded that, all the tested cases were successfully simulated by
using the free energy single component multiphase lattice Boltzmann method and gave a
good agreement with the benchmark solution.
63
6.2
Recommendations
All the cases were successfully solved by using SCMP LBM. However, there are a
lot of thing can be improved. For future works, there a lot of problem can be solved by
using SCMP LBM where it can also improvised the previous study. Among them are:
i)
Droplet falling and sliding on a curvature plane
ii)
Coupling SCMP LBM with FDM
iii)
Droplet motion on rough surface
iv)
3D droplet spreading and wetting
64
REFERENCES
A. Dupuis and J.M. Yeomans, Lattice Boltzmann modeling of droplets on chemically
heterogeneous surfaces, Future Gener. Comput. Syst. 20 (2004), pp. 993–1001.
A. Dupuis and J.M. Yeomans, Modeling droplets on superhydrophobic surfaces:
equilibrium states and transitions, Langmuir 21 (2005), pp. 2624–2629.
A.J. Briant, P. Papatzacos and J.M. Yeomans, Lattice Boltzmann simulations of contact
line motion in a liquid–gas system, Philos. Trans. Roy. Soc. London A 360 (2002), pp.
485–495.
A.J. Briant, A.J. Wagner and J.M. Yeomans, Lattice Boltzmann simulations of contact
line motion: I. Liquid–gas systems, Phys. Rev. E 69 (2004), p. 031602.
A.J. Briant and J.M. Yeomans, Lattice Boltzmann simulations of contact line motion:
II. Binary fluids, Phys. Rev. E 69 (2004), p. 031603.
A.K. Gunstensen, D.H. Rothman, S. Zaleski and G. Zanetti, Lattice Boltzmann model
of immiscible fluids, Phys. Rev. A 43 (1991), pp. 4320–4327.
Derivation of the Lattice Boltzmann Method by Means of the Discrete Ordinate Method
for the Boltzmann Equation Journal of Computational Physics 131, 241–246 (1997)
Article No.. CP965595
65
Dieter A. Wolf-Gladrow (2000). Lattice Gas Cellular Automata and Lattice Boltzmann
Models, An Introduction. Springer
Dieter Wolf-Gladrow 1 (1995). A Lattice Boltzmann Equation for Diffusion. Journal of
Statistical Physics, Vol. 79, Nos. 5/6, 1995
Donald P. Ziegler (1993). Boundary Conditions for Lattice Boltzmann Simulations.
Journal of Statistical Physics, Vol. 71, Nos. 5/6
F.A.L.Dullien, Characterization of porous media – pore level, Transport in porous
media, 6, (1991), p.581.
F. Kuznik, J. Vareilles, G. Rusaouen, G. Krauss (2007). A double-population lattice
Boltzmann method with non-uniform mesh for the simulation of natural convection in a
square cavity. International Journal of Heat and Fluid Flow 28 862–870
Francois Coron (1989). Derivation of Slip Boundary Conditions for the Navier-Stokes
System from the Boltzmann Equation. Journal of Statistical Physics, Vol. 54, Nos. ¾
G. McNamara and G. Zanetti, Use of the Boltzmann equation to simulate Lattice-Gas
Automata, Phys. Rev. Lett. 61 (1988), pp. 2332–2335.
H. Kusumaatmaja, A. Dupuis and J.M. Yeomans, Lattice Boltzmann simulations of
drop dynamics, Math. Comput. Simul. 72 (2006), pp. 160–164.
H. Takewaki, A. Nishigushi and T. Yabe. Cubic Interpolated Pseudo-Particle Method
(CIP) for Solving Hyperbolic Type equation. National Institute for Fusion Science. NIIElectronic Library Serices.
J.S. Rowlinson and B. Widom, Molecular Theory of Capillarity, Clarendon, Oxford
(1989).
66
J.W. Cahn, Critical-point wetting, J. Chem. Phys. 66 (1977), pp. 3667–3672.
Jonas Tölke, Manfred Krafczyk, Manuel Schulz, Ernst Rank (2000). Discretization of
the Boltzmann equation in velocity space using a Galerkin approach. Computer Physics
Communications 129, pp 91–99
K.Murakami, S.Nomura, M.Shinozuka, Numerical Analysis of Shape of Liquid
Droplet on Solid Surface with SOLA-VOF Method, Japanese J. Multiphase Flow, 12,
(1998),pp.58-56
M. Rohde, D. Kandhai, J. J. Derksen, and H. E. A. Van den Akker (2003). Improved
bounce-back methods for no-slip walls in lattice-Boltzmann schemes: Theory and
simulations. Physical Review E 67
M.R. Swift, W.R. Osborn and J.M. Yeomans, Lattice Boltzmann simulation of nonideal
fluids, Phys. Rev. Lett. 75 (1995), pp. 830–833.
M.R. Swift, E. Orlandini, W.R. Osborn and J.M. Yeomans, Lattice Boltzmann
simulations of liquid–gas and binary fluid systems, Phys. Rev. E 54 (1996), pp. 5041–
5052.
M.C. Sukop, D.T. Thorne, Jr. ,Lattice Boltzmann Modeling, ISBN-10 3-540-27981-4
Springer Berlin Heidelberg New York, Springer-Verlag Berlin Heidelberg 2006, 2007
N. G. van Kampen (1987). Chapman-Enskog as an Application of the Method for
Eliminating Fast Variables. Journal of Statistical Physics, Vol. 46, Nos. ¾
P. A. Skordos (1993). Initial and Boundary conditions for the Lattice Boltzmann
method. Physical review E, Volume 48, Number 6
67
Q.M. Chang and J.I.D. Alexander, Analysis of single droplet dynamics on striped
surface domains using a lattice Boltzmann method, Microfluid. Nanofluid. 2 (2006), pp.
309–326.
Qisu Zou, Shuling Hou and Gary D. Doolen (1995). Analytical Solutions of the Lattice
Boltzmann BGK Model. Journal of Statistical Physics, Vol. 81, Nos. ½
S. Chen and G.D. Doolen, Lattice Boltzmann method for fluid flows, Ann. Rev. Fluid
Mech. 30 (1998), pp. 329–364.
Sauro Succi (2001). The Lattice Boltzmann Equation for fluid dynamics and beyond.
Oxford Science Publications
Takaji Inamuro, Masato Yoshino, and Fumimaru Ogino (1995). A non-slip boundary
condition for lattice Boltzmann simulations. Phys. Fluids 7 (12), American Institute of
Physics
Takashi Yabe, Feng Xiao and Takayuki Utsumi (2001). The constrained interpolation
profile method for multiphase analysis. Journal of Computational Physics 169, pp556–
593
Taku Ozawa, Takahiko Tanahashi, “CIVA(Cubic Interpolation with Volume/Area
coordinates) and AMR (Adaptive Mesh Refinement) method for discrete Boltzmann
Equation”, JSME International Journal B, 48, pp229-234, (2005)
T. Inamuro, T. Ogata, S. Tajima and N. Konishi, A lattice Boltzmann method for
incompressible two-phase flows with large density differences, J. Comput. Phys. 198
(2004), pp. 628–644.
T. Lee and C.L. Lin, A stable discretization of the lattice Boltzmann equation for
simulation of incompressible two-phase flows at high density ratio, J. Comput. Phys.
206 (2005), pp. 16–47.
68
T. Young, An essay on the cohesion of fluids, Philos. Trans. Roy. Soc. London 95
(1805), pp. 65–87.
U. Frish, B. Hasslacher, D. Humieres, P. Lallemand, J. P. Rivet and Y. Pomeau Lattice gas
hydrodynamics in two and three dimensions. Complex Sys. 1, (1987). pp 649-707
URL: http://research.nianet.org/~luo
Xiaoyi He, Qisu Zou, Li-Shi Luo, and Micah Dembo (1997). Analytic Solutions of
Simple Flows and Analysis of Nonslip Boundary Conditions for the Lattice Boltzmann
BGK Model. Journal of Statistical Physics, Vol. 87, Nos. ½
X.W. Shan and H.D. Chen, Lattice Boltzmann model for simulating flows with multiple
phases and components, Phys. Rev. E 47 (1993), pp. 1815–1819.
69
APPENDIX A
Source code
PARAMETER (LX = 101, LY = 51, CD = 8)
REAL*8 RHO(1:LX,1:LY), U(1:LX,1:LY), V(1:LX,1:LY)
REAL*8F(1:LX,1:LY,0:CD), FEQ(1:LX,1:LY,0:CD),FNEW(1:LX,1:LY,0:CD)
REAL*8 CX(0:CD),CY(0:CD)
REAL*8 KAPPA, A, B, T, R, NL, NG, TAU, NU
KAPPA = 0.00075D0
A = 9.0D0/49.0D0
B = 2.0D0/21.0D0
T = 0.55D0
R = 1.0D0
NL = 4.106D0
NG = 2.894D0
TAU = 1.0D0
NU = (TAU-0.5)/3.0D0
CX(0) = 0.0D0
CY(0) = 0.0D0
CX(1) = 1.0D0
CY(1) = 0.0D0
CX(2) = 0.0D0
CY(2) = 1.0D0
CX(3) = -1.0D0
CY(3) = 0.0D0
CX(4) = 0.0D0
CY(4) = -1.0D0
CX(5) = 1.0D0
CY(5) = 1.0D0
CX(6) = -1.0D0
CY(6) = 1.0D0
CX(7) = -1.0D0
CY(7) = -1.0D0
70
CX(8) = 1.0D0
CY(8) = -1.0D0
DO J = 1,LY
DO I = 1,LX
IF((I - LX/2)**2 + (J - 20)**2 .LE. 400) THEN
RHO(I,J) = NL
ELSE
RHO(I,J) = NG
END IF
u(i,j) = 0.0d0
v(i,j) = 0.0d0
END DO
END DO
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
CALL EQUILIBRIUM (LX,LY,CD,U,V,CX,CY,RHO,KAPPA,NU,A,B,T,FEQ)
DO I = 1,LX
DO J = 1,LY
DO K = 0,CD
F(I,J,K)=FEQ(I,J,K)
END DO
END DO
END DO
DO ITER = 1, 1000000
CALL COLLIDE(lx,ly,cd,f,feq,fnew,TAU)
CALL MOVE(lx,ly,cd,f,fnew)
CALL BOUNDARY(lx,ly,cd,f)
CALL MACRO(lx,ly,cd,rho,f,u,v,NG,NL)
IF(RHO(I,J) .GT. NG)THEN
CALL vel(LX,LY,NG,RHO,u,v,NL,tau)
END IF
CALL EQUILIBRIUM (LX,LY,CD,U,V,CX,CY,RHO,KAPPA,NU,A,B,T,FEQ)
IF(MOD(ITER,10) .EQ.0)THEN
WRITE(*,*)ITER
CALL fileoutput(lx,ly,u,v,rho,ITER)
END IF
END DO
71
CALL fileoutput(lx,ly,u,v,rho,ITER)
END PROGRAM
subroutine fileoutput(lx,ly,u,v,rho,ITER)
real*8 x(10301),y(10301)
integer nbool(4,20000)
real*8 U(1:LX,1:LY),V(1:LX,1:LY),RHO(1:LX,1:LY)
nnode = lx*ly
ne = (lx-1)*(ly-1)
do j=1,ly
do i = 1,lx
ID=lx*(j-1)+i
x(ID)=DFLOAT(I-1)+1
Y(ID)=DFLOAT(J-1)+1
IF(I.EQ.1) X(ID)=1
IF(I.EQ.lx) X(ID)=lx
IF(J.EQ.1) Y(ID)=1
IF(J.EQ.ly) Y(ID)=ly
end do
end do
IE=0
do J=1,ly-1
do I=1,lx-1
IE=IE+1
NBOOL(1,IE)=(J-1)*lx+i
NBOOL(2,IE)=NBOOL(1,IE)+1
NBOOL(4,IE)=NBOOL(1,IE)+lx
NBOOL(3,IE)=NBOOL(4,IE)+1
END DO
END DO
open(unit=ITER+100,file='OUTPUT1.inp',status = 'REPLACE',action = 'write',iostat =
ierror)
open(unit=17,file='DATA.DAT',status = 'REPLACE',action = 'write',iostat = ierror)
WRITE(ITER+100,*) NNODE,NE,3,0,0
DO I=1,NNODE
WRITE(ITER+100,900) I,X(I),Y(I),0.0D0
END DO
DO IE=1,NE
WRITE(ITER+100,901) IE,'1 quad',(NBOOL(NA,IE), NA=1,4)
72
END DO
WRITE(ITER+100,*) 3,1,1,1
WRITE(ITER+100,*) 'uvel , _'
WRITE(ITER+100,*) 'vvel , _'
WRITE(ITER+100,*) 'density, _'
do j=1,ly
do i = 1,lx
ID=lx*(j-1)+i
WRITE(ITER+100,902) ID,u(I,j),v(I,j),rho(I,j)
END DO
end do
do j=1,ly
do i = 1,lx
ID=lx*(j-1)+i
WRITE(17,902) ID,u(I,j),v(I,j),rho(I,j)
END DO
end do
900 FORMAT(I6,3E17.8)
901 FORMAT(I6,A10,4I6)
902 FORMAT(I6,3E17.8)
903 FORMAT(I6,E17.8)
CLOSE(16)
CLOSE(17)
return
end
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
subroutine
EQUILIBRIUM(LX,LY,CD,U,V,CX,CY,RHO,KAPPA,NU,A,B,T,FEQ)
REAL*8 FEQ(1:LX,1:LY,0:CD)
REAL*8 RHO(1:LX,1:LY)
REAL*8 U(1:LX,1:LY),V(1:LX,1:LY),CX(0:CD),CY(0:CD)
REAL*8 KAPPA,NU,A,B,T
REAL*8D_RHO_DX,D_RHO_DY,
LAPLACIAN,A2,A0,A1,B2,B1,C2,C1,D1,D2,B0,C0,D0
REAL*8 G2XX,G2YY,G2XY,G2YX,G1XX,G1YY,G1XY,G1YX
REAL*8UX,UY,TMP,UU,PHI,PI,BETA,GAMMA,PC,NC,ANGLE,VIS,ALPHA
PI= 3.141592653589793D0
TAU = 1.0D0
BETA = 0.1D0
GAMMA = 0.3D0
73
PC = 0.125D0
NC = 3.5D0
ANGLE = PI/2.0D0
PHI=2.0D0*BETA*GAMMA*SQRT(2.0D0*KAPPA*PC)*SQRT(COS(ACOS(SIN(ANG
LE)*SIN(ANGLE))/3.0D0)*(1.0D0-COS(ACOS(SIN(ANGLE)*SIN(ANGLE))/3.0D0)))
IF(ANGLE .GT. PI/2) PHI = - PHI
do i = 1,lx
idown = i-1
iup = i + 1
!TOP AND BOTTOM WALL
do j = 1,ly
jup = j + 1
jdown = j-1
UX = U(I,J)
UY = V(I,J)
IF(J .EQ. 1 .AND. I .NE. 1 .AND. I .NE. LX) THEN ! BOTTOM WALL
D_RHO_DX = 0.5D0*(RHO(IUP,J) - RHO(IDOWN,J)) !CENTRAL DIFFERENCE
D_RHO_DY = - PHI/KAPPA
LAPLACIAN =
RHO(IUP,J)-2.0D0*RHO(I,J)+ RHO(IDOWN,J) +0.5D0*(7.0D0*RHO(I,J) +8.0D0*RHO(i,2)- RHO(I,3))+ 3.0D0*PHI/KAPPA
ELSEIF(J .EQ. LY .AND. I .NE. 1 .AND. I .NE. LX)THEN !TOP WALL
D_RHO_DX = 0.5D0*(RHO(IUP,J) - RHO(IDOWN,J)) !CENTRAL DIFFERENCE
D_RHO_DY = 0.5D0*(3.0D0*RHO(I,J)-4.0D0*RHO(I,J-1)+RHO(I,J-2))
LAPLACIAN
=
RHO(IUP,J)-2.0D0*RHO(I,J)+
RHO(IDOWN,J)+RHO(I,J)2.0D0*RHO(I,J-1)+RHO(I,J-2)
ELSEIF(I .EQ. 1 .AND. J .NE. 1 .AND. J .NE. LY)THEN !LEFT WALL
D_RHO_DX = 0.5D0*(-3.0D0*RHO(I,J)+4.0D0*RHO(I+1,J) - RHO(I+2,J)) !CENTRAL
DIFFERENCE
D_RHO_DY = 0.5D0*(RHO(I,JUP)-RHO(I,JDOWN))
LAPLACIAN
=
RHO(I,J)-2.0D0*RHO(I+1,J)+
RHO(I+2,J)+RHO(I,JUP)2.0D0*RHO(I,J)+RHO(I,JDOWN)
ELSEIF(I .EQ. LX .AND. J .NE. 1 .AND. J .NE. LY)THEN !RIGHT WALL
D_RHO_DX = 0.5D0*(3.0D0*RHO(I,J)-4.0D0*RHO(I-1,J) + RHO(I-2,J)) !CENTRAL
DIFFERENCE
74
D_RHO_DY = 0.5D0*(RHO(I,JUP)-RHO(I,JDOWN))
LAPLACIAN=RHO(I,J)-2.0D0*RHO(I-1,J)+
2.0D0*RHO(I,J)+RHO(I,JDOWN)
RHO(I-2,J)+RHO(I,JUP)-
ELSEIF(I .EQ. 1 .AND. J .EQ. 1) THEN
! BUCU BAWAH KIRI
D_RHO_DX = 0.5D0*(-3.0D0*RHO(I,J)+4.0D0*RHO(I+1,J) - RHO(I+2,J))
D_RHO_DY = 0.5D0*(-3.0D0*RHO(I,J)+4.0D0*RHO(I,J+1)-RHO(I,J+2))
LAPLACIAN=RHO(I,J)-2.0D0*RHO(I+1,J)+
RHO(I+2,J)+RHO(I,J)2.0D0*RHO(I,J+1)+RHO(I,J+2)
ELSEIF(I .EQ. LX .AND. J .EQ. 1) THEN ! BUCU BAWAH KANAN
D_RHO_DX = 0.5D0*(3.0D0*RHO(I,J)-4.0D0*RHO(I-1,J) + RHO(I-2,J))
D_RHO_DY = 0.5D0*(-3.0D0*RHO(I,J)+4.0D0*RHO(I,J+1)-RHO(I,J+2))
LAPLACIAN
=
RHO(I,J)-2.0D0*RHO(I-1,J)+
RHO(I-2,J)+RHO(I,J)2.0D0*RHO(I,J+1)+RHO(I,J+2)
ELSEIF (I .EQ. 1 .AND. J .EQ. LY)THEN ! BUCU ATAS KIRI
D_RHO_DX = 0.5D0*(-3.0D0*RHO(I,J)+4.0D0*RHO(I+1,J) - RHO(I+2,J))
D_RHO_DY = 0.5D0*(3.0D0*RHO(I,J)-4.0D0*RHO(I,J-1)+RHO(I,J-2))
LAPLACIAN = RHO(I,J)-2.0D0*RHO(I+1,J)+ RHO(I+2,J)+RHO(I,J)-2.0D0*RHO(I,J1)+RHO(I,J-2)
ELSEIF(I .EQ. LX .AND. J .EQ. LY)THEN ! BUCU ATAS KANAN
D_RHO_DX = 0.5D0*(3.0D0*RHO(I,J)-4.0D0*RHO(I-1,J) + RHO(I-2,J))
D_RHO_DY = 0.5D0*(3.0D0*RHO(I,J)-4.0D0*RHO(I,J-1)+RHO(I,J-2))
LAPLACIAN = RHO(I,J)-2.0D0*RHO(I-1,J)+ RHO(I-2,J)+RHO(I,J)-2.0D0*RHO(I,J1)+RHO(I,J-2)
ELSE ! FLUID NODE
D_RHO_DX = 0.5D0*(RHO(IUP,J) - RHO(IDOWN,J)) !CENTRAL DIFFERENCE
D_RHO_DY = 0.5D0*(RHO(I,JUP) - RHO(I,JDOWN)) !CENTRAL DIFFERENCE
LAPLACIAN=RHO(IUP,J)-2.0D0*RHO(I,J)+
RHO(IDOWN,J)+RHO(I,JUP)2.0D0*RHO(I,J)+ RHO(I,JDOWN)
END IF
G2XX = KAPPA/16.0D0*((D_RHO_DX*D_RHO_DX)-(D_RHO_DY*D_RHO_DY)) +
(NU/8.0D0)* (UX*D_RHO_DX-UY*D_RHO_DY)
G2YY = KAPPA/16.0D0*((D_RHO_DY*D_RHO_DY)- (D_RHO_DX*D_RHO_DX)) +
(NU/8.0D0)* (UY*D_RHO_DY-UX*D_RHO_DX)
G2XY = KAPPA/8.0D0*(D_RHO_DX*D_RHO_DY) + NU/8.0D0*(UX*D_RHO_DY +
UY*D_RHO_DX) !MODIFIED TANAKA
G2YX = G2XY
G1XX = 4.0D0*G2XX
G1YY = 4.0D0*G2YY
G1XY = 4.0D0*G2XY
G1YX = 4.0D0*G2YX
75
VIS = ((RHO(I,J)-NC)/NC)
!PO = (RHO(I,J)*T/(1.0D0-B*RHO(I,J))) - A*RHO(I,J)*RHO(I,J)
PO
=
PC*(((RHO(I,J)-NC)/NC)+1)*(((RHO(I,J)-NC)/NC)+1)*(3.0D0*((RHO(I,J)NC)/NC)*((RHO(I,J)-NC)/NC)-2.0D0*((RHO(I,J)-NC)/NC)+1.0D02.0D0*BETA*GAMMA)
A2=(POKAPPA*RHO(I,J)*LAPLACIAN)/8.0D0+(NU/4.0D0)*(UX*D_RHO_DX+UY*
D_RHO_DY)
A1 = 2.0D0*A2
A0 = RHO(I,J) - 12.0D0*A2
B2 = RHO(I,J)/12.0D0
B1 = 4.0D0*B2
B0 = 0.0D0
C2 = RHO(I,J)/8.0D0
C1 = 4.0D0*C2
C0 = 0.0D0
D2 = - RHO(I,J)/16.0D0
D1 = 2.0D0*D2
D0 =12.0D0*D2
UU = UX*UX+UY*UY
feq(i,j,0) = A0 + D0*UU
FEQ(I,J,1)=A1+B1*(CX(1)*UX+CY(1)*UY)+C1*(CX(1)*UX+CY(1)*UY)*(CX(1)*UX
+CY(1)*UY)+D1*UU
+G1XX*CX(1)*CX(1)+G1XY*CX(1)*CY(1)+G1YX*CY(1)*CX(1)++G1YY*CY(1)*CY
(1)
FEQ(I,J,2)=A1+B1*(CX(2)*UX+CY(2)*UY)+C1*(CX(2)*UX+CY(2)*UY)*(CX(2)*UX
+CY(2)*UY)+D1*UU +
G1XX*CX(2)*CX(2)+G1XY*CX(2)*CY(2)+G1YX*CY(2)*CX(2)++G1YY*CY(2)*CY(
2)
FEQ(I,J,3)=A1+B1*(CX(3)*UX+CY(3)*UY)+C1*(CX(3)*UX+CY(3)*UY)*(CX(3)*UX
+CY(3)*UY)+D1*UU +
G1XX*CX(3)*CX(3)+G1XY*CX(3)*CY(3)+G1YX*CY(3)*CX(3)++G1YY*CY(3)*CY(
3)
FEQ(I,J,4)=A1+B1*(CX(4)*UX+CY(4)*UY)+C1*(CX(4)*UX+CY(4)*UY)*(CX(4)*UX
+CY(4)*UY)+D1*UU+G1XX*CX(4)*CX(4)+G1XY*CX(4)*CY(4)+G1YX*CY(4)*CX(4
)++G1YY*CY(4)*CY(4)
FEQ(I,J,5)=A2+B2*(CX(5)*UX+CY(5)*UY)+C2*(CX(5)*UX+CY(5)*UY)*(CX(5)*UX
+CY(5)*UY)+D2*UU+g2xx*CX(5)*CX(5)+G2XY*CX(5)*CY(5)+G2YX*CY(5)*CX(5)
++G2YY*CY(5)*CY(5)
FEQ(I,J,6)=A2+B2*(CX(6)*UX+CY(6)*UY)+C2*(CX(6)*UX+CY(6)*UY)*(CX(6)*UX
+CY(6)*UY)+D2*UU+g2xx*CX(6)*CX(6)+G2XY*CX(6)*CY(6)+G2YX*CY(6)*CX(6)
++G2YY*CY(6)*CY(6)
FEQ(I,J,7)=A2+B2*(CX(7)*UX+CY(7)*UY)+C2*(CX(7)*UX+CY(7)*UY)*(CX(7)*UX
+CY(7)*UY)+D2*UU+g2xx*CX(7)*CX(7)+G2XY*CX(7)*CY(7)+G2YX*CY(7)*CX(7)
++G2YY*CY(7)*CY(7)
76
FEQ(I,J,8)=A2+B2*(CX(8)*UX+CY(8)*UY)+C2*(CX(8)*UX+CY(8)*UY)*(CX(8)*UX
+CY(8)*UY)+D2*UU+g2xx*CX(8)*CX(8)+G2XY*CX(8)*CY(8)+G2YX*CY(8)*CX(8)
++G2YY*CY(8)*CY(8)
END DO
END DO
RETURN
END
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
subroutine MACRO(lx,ly,cd,rho,f,u,v,NG,NL)
real*8 F(1:LX,1:LY,0:CD)
real*8 NG,NL
real*8 RHO(1:LX,1:LY),U(1:LX,1:LY),V(1:LX,1:LY)
do i = 1,lx
do j = 1,ly
u(i,j) = 0.0d0
v(i,j) = 0.0d0
end do
end do
do i = 1,lx
do j = 1,ly
rho(i,j) =
f(i,j,1)+f(i,j,2)+f(i,j,3)+f(i,j,4)+f(i,j,5)+f(i,j,6)+f(i,j,7)+f(i,j,8)+f(i,j,0)
u(i,j) = f(i,j,1)+f(i,j,5)+f(i,j,8)-f(i,j,3)-f(i,j,7)-f(i,j,6)
v(i,j) = f(i,j,5)+f(i,j,6)+f(i,j,2)-f(i,j,7)-f(i,j,8)-f(i,j,4)
u(i,j) = u(i,j)/rho(i,j)
v(i,j) = v(i,j)/rho(i,j)
IF(RHO(I,J) .LE. NG) RHO(I,J) = NG
IF(RHO(I,J) .GE. NL) RHO(I,J) = NL
end do
end do
DO I=1,LX
u(i,1) = 0.0d0
v(i,1) = 0.0d0
u(i,LY) = 0.0d0
v(i,LY) = 0.0d0
77
RHO(I,LY) = NG
END DO
DO J=1,LY
u(1,J) = 0.0d0
v(1,J) = 0.0d0
u(LX,J) = 0.0d0
v(LX,J) = 0.0d0
RHO(1,J) = NG
RHO(LX,J) = NG
END DO
return
end
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
subroutine colliDE(lx,ly,cd,f,feq,fnew,TAU)
real*8 F(1:LX,1:LY,0:CD),FEQ(1:LX,1:LY,0:CD),FNEW(1:LX,1:LY,0:CD)
real*8 TAU
real*8 tfrac
do i = 1,lx
do j = 1,ly
tfrac = 1.0d0/TAU
fnew(i,j,1) = tfrac*feq(i,j,1)+(1.0d0-tfrac)*f(i,j,1)
fnew(i,j,2) = tfrac*feq(i,j,2)+(1.0d0-tfrac)*f(i,j,2)
fnew(i,j,3) = tfrac*feq(i,j,3)+(1.0d0-tfrac)*f(i,j,3)
fnew(i,j,4) = tfrac*feq(i,j,4)+(1.0d0-tfrac)*f(i,j,4)
fnew(i,j,5) = tfrac*feq(i,j,5)+(1.0d0-tfrac)*f(i,j,5)
fnew(i,j,6) = tfrac*feq(i,j,6)+(1.0d0-tfrac)*f(i,j,6)
fnew(i,j,7) = tfrac*feq(i,j,7)+(1.0d0-tfrac)*f(i,j,7)
fnew(i,j,8) = tfrac*feq(i,j,8)+(1.0d0-tfrac)*f(i,j,8)
fnew(i,j,0) = tfrac*feq(i,j,0)+(1.0d0-tfrac)*f(i,j,0)
end do
end do
return
end
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
subroutine move(lx,ly,cd,f,fnew)
real*8 F(1:LX,1:LY,0:CD),FNEW(1:LX,1:LY,0:CD)
78
!
O DIRECTION
DO J = 1,LY
DO I = 1,LX
II = I ; JJ = J
F(II,JJ,0) = FNEW(I,J,0)
END DO
END DO
!
1 DIRECTION
DO J = 1,LY
DO I = 1,LX-1
II = I+1; JJ = J
F(II,JJ,1) = FNEW(I,J,1)
END DO
END DO
!
2 DIRECTION
DO J = 1,LY-1
DO I = 1,LX
II = I ; JJ = J+1
F(II,JJ,2) = FNEW(I,J,2)
END DO
END DO
!
3 DIRECTION
DO J = 1,LY
DO I = 2,LX
II = I-1 ; JJ = J
F(II,JJ,3) = FNEW(I,J,3)
END DO
END DO
!
4 DIRECTION
DO J = 2,LY
DO I = 1,LX
II = I ; JJ = J-1
F(II,JJ,4) = FNEW(I,J,4)
END DO
END DO
!
5 DIRECTION
DO J = 1,LY-1
DO I = 1,LX-1
II = I+1; JJ = J+1
79
F(II,JJ,5) = FNEW(I,J,5)
END DO
END DO
!
6 DIRECTION
DO J = 1,LY-1
DO I = 2,LX
II = I-1; JJ = J+1
F(II,JJ,6) = FNEW(I,J,6)
END DO
END DO
!
7 DIRECTION
DO J = 2,LY-1
DO I = 2,LX-1
II = I-1 ; JJ = J-1
F(II,JJ,7) = FNEW(I,J,7)
END DO
END DO
!
8 DIRECTION
DO J = 2,LY
DO I = 1,LX-1
II = I +1; JJ = J-1
F(II,JJ,8) = FNEW(I,J,8)
END DO
END DO
RETURN
END
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
subroutine BOUNDARY(lx,ly,cd,f)
real*8 F(1:LX,1:LY,0:CD)
DO I = 1,LX !SLIP BOUNDARY AT TOP WALL
F(I,LY,4) = F(I,LY,2)
F(I,LY,7) = F(I,LY,5)
F(I,LY,8) = F(I,LY,6)
!SLIP BOUNDARY AT BOTTOM WALL
F(I,1,2) = F(I,1,8)
F(I,1,5) = F(I,1,7)
F(I,1,6) = F(I,1,4)
END DO
80
DO J = 1,LY !SLIP BOUYNDARY AT LEFT WALL
F(1,J,1) = f(1,J,3)
F(1,J,5) = f(1,J,7)
F(1,J,8) = f(1,J,6)
!SLIP BOUYNDARY AT RIGHT WALL
F(LX,J,3) = F(LX,J,1)
F(LX,J,6) = F(LX,J,8)
F(LX,J,7) = F(LX,J,5)
END DO
!BUCU ATAS KIRI
F(1,LY,8) = F(1,LY,6)
F(1,LY,4) = F(1,LY,2)
F(1,LY,1) = F(1,LY,3)
!BUCU ATAS KANAN
F(LX,LY,3) = F(LX,LY,1)
F(LX,LY,7) = F(LX,LY,5)
F(LX,LY,4) = F(LX,LY,2)
!BUCU BAWAH KIRI
F(1,1,1) = F(1,1,3)
F(1,1,5) = F(1,1,7)
F(1,1,2) = F(1,1,4)
!BUCU BAWAH KANAN
F(LX,1,2) = F(LX,1,4)
F(LX,1,6) = F(LX,1,8)
F(LX,1,3) = F(LX,1,1)
RETURN
END
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
subroutine vel(LX,LY,NG,RHO,u,v,NL,tau)
REAL*8 RHO(1:LX,1:LY),V(1:LX,1:LY),U(1:LX,1:LY)
REAL*8 G,NG,NL,tau
G= -0.000012D0
do i = 1,lx
do j = 1,ly
81
IF(RHO(I,J) .GT. NG) THEN
v(i,j) = v(i,j) + tau*g/rho(i,j)
end if
end do
end do
return
end