Applications of Lattice Boltzmann Method in Dynamic Modelling of Fluid FlowsAngshuman Pal
Computational Fluid Dynamics techniques are an important development in the study of fluid behaviour where complicated differential equations involving complex geometries and irregular flows can be solved using iterative numerical techniques. Such techniques are being used extensively in the field of fluid mechanics and heat transfer. Solving the Navier Stokes equation for viscous fluids is of particular importance in this field.
The Lattice Boltzmann Method is an alternative to the commonly used discretization principle for solving the Navier Stokes equation. It deals with the properties of fluid particles on a micro scale and subsequently uses them to generate the model of the entire flow domain on a macro scale. The flow domain is broken up into lattices inhabited by finitely many fluid particles. Governed by rules of streaming and collision, all the particles together generate the model of the entire flow.
A detailed study is performed on the LBM model and its underlying principles. A program for its execution is written in MATLAB environment. Simulations are run on a simple geometry involving steady flow through a circular pipe. The results obtained are analysed and verified against the expected results given by mathematical models and actual experimentation.
finite difference Method, For Numerical analysis. working matlab code. numeric analysis finite difference method. MATLAB provides tools to solve math. Using linear programing techniques we can easily solve system of equations. This file provides a running code of Finite difference matlab code
Applications of Lattice Boltzmann Method in Dynamic Modelling of Fluid FlowsAngshuman Pal
Computational Fluid Dynamics techniques are an important development in the study of fluid behaviour where complicated differential equations involving complex geometries and irregular flows can be solved using iterative numerical techniques. Such techniques are being used extensively in the field of fluid mechanics and heat transfer. Solving the Navier Stokes equation for viscous fluids is of particular importance in this field.
The Lattice Boltzmann Method is an alternative to the commonly used discretization principle for solving the Navier Stokes equation. It deals with the properties of fluid particles on a micro scale and subsequently uses them to generate the model of the entire flow domain on a macro scale. The flow domain is broken up into lattices inhabited by finitely many fluid particles. Governed by rules of streaming and collision, all the particles together generate the model of the entire flow.
A detailed study is performed on the LBM model and its underlying principles. A program for its execution is written in MATLAB environment. Simulations are run on a simple geometry involving steady flow through a circular pipe. The results obtained are analysed and verified against the expected results given by mathematical models and actual experimentation.
finite difference Method, For Numerical analysis. working matlab code. numeric analysis finite difference method. MATLAB provides tools to solve math. Using linear programing techniques we can easily solve system of equations. This file provides a running code of Finite difference matlab code
FINITE DIFFERENCE MODELLING FOR HEAT TRANSFER PROBLEMSroymeister007
This report provides a practical overview of numerical solutions to the heat equation using the finite difference method (FDM). The forward time, centered space (FTCS), the backward time, centered space (BTCS), and Crank-Nicolson schemes are developed, and applied to a simple problem in1volving the one-dimensional heat equation. Complete, working Matlab and FORTRAN codes for each program are presented. The results of running the codes on finer (one-dimensional) meshes, and with smaller time steps are demonstrated. These sample calculations show that the schemes realize theoretical predictions of how their truncation errors depend on mesh spacing and time step. The Matlab codes are straightforward and allow us to see the differences in implementation between explicit method (FTCS) and implicit methods (BTCS). The codes also allow us to experiment with the stability limit of the FTCS scheme.
This presentation gives a brief introduction to the concept of coupled CFD-DEM Modeling.
Link to file: https://drive.google.com/open?id=1nO2n49BwhzBtT6NnvpxADG5WsC9uMJ-i
The derivation of the equation of motion for various fluids is similar to the d derivation of Eular’s equation. However ,the tangential stresses arise during the motion of a real viscous fluid, must be considered
The phase eld model is a general name for a class of diuse interface models used to study a wide variety of materials phenomena. It has several advantages over other interface tracking approaches, therefore it has been used to model general multi-phase systems with features much larger than the real interface thicknesses.
The phase-eld method, as presented here, grows out of the work of Cahn, Hilliard and Allen. It is used for two general purposes:
• to model systems in which the diuse nature of interfaces is essential to the problem, such as spinodal decomposition and solute trapping during rapid phase boundary motion;
• as a front tracking technique to model general multi-phase systems.
Generally, we speak about two types of phase eld models. In the rst, called Cahn-Hilliard, the phase is uniquely determined by the value of a conserved eld variable, such as the concentration C, e.g. if C ≤ C1, then we are in one phase, if C ≥ C2 then the other. These models were rst applied to understand spin- odal decomposition, and are now used for a wide range of phenomena. In the second, called Allen-Cahn, the phase is not uniquely determined by concentra- tion, temperature, pressure, etc., so we add one or more extra eld variable(s) sometimes called the order parameter φ which determines the local phase. This class of models is widely used to study solidication and solid-state phase trans- formations in metals.
The phase-eld method is a xed-grid method; it diers from other methods in that the interface is diuse in a physical rather than numerical sense. Thus, it is also known as the diuse-interface model. More precisely, the diuse inter- face is introduced through an energetic variational procedure that results in a thermodynamic consistent coupling system. The basic idea was derived from the consideration that the two components, though nominally immiscible, does mix in reality within a narrow interfacial region. A phase-eld variable φ can be thought of as the volume fraction, to demarcate the two species and indicate the location of the interface. A mixing energy is dened based on φ which, through
a convection-diusion equation, governs the evolution of the interfacial prole. The phase-eld method can be viewed as a physically motivated level-set method. When the thickness of the interface approaches zero, the diuse-interface model becomes asymptotically identical to a sharp-interface level-set formulation. It also reduces properly to the classical sharp-interface model in general.
From the statistical (phase eld approach) point of view, the interface represents a continuous, but steep change of the properties (density, viscosity, etc.) of two uids. Within this "thin" transitional region, the uid is mixed. The mixing is determined by molecular interactions between the two species, and can be described by a stored mixing energy, which represents the balance between the competing phobic/philic relation
FINITE DIFFERENCE MODELLING FOR HEAT TRANSFER PROBLEMSroymeister007
This report provides a practical overview of numerical solutions to the heat equation using the finite difference method (FDM). The forward time, centered space (FTCS), the backward time, centered space (BTCS), and Crank-Nicolson schemes are developed, and applied to a simple problem in1volving the one-dimensional heat equation. Complete, working Matlab and FORTRAN codes for each program are presented. The results of running the codes on finer (one-dimensional) meshes, and with smaller time steps are demonstrated. These sample calculations show that the schemes realize theoretical predictions of how their truncation errors depend on mesh spacing and time step. The Matlab codes are straightforward and allow us to see the differences in implementation between explicit method (FTCS) and implicit methods (BTCS). The codes also allow us to experiment with the stability limit of the FTCS scheme.
This presentation gives a brief introduction to the concept of coupled CFD-DEM Modeling.
Link to file: https://drive.google.com/open?id=1nO2n49BwhzBtT6NnvpxADG5WsC9uMJ-i
The derivation of the equation of motion for various fluids is similar to the d derivation of Eular’s equation. However ,the tangential stresses arise during the motion of a real viscous fluid, must be considered
The phase eld model is a general name for a class of diuse interface models used to study a wide variety of materials phenomena. It has several advantages over other interface tracking approaches, therefore it has been used to model general multi-phase systems with features much larger than the real interface thicknesses.
The phase-eld method, as presented here, grows out of the work of Cahn, Hilliard and Allen. It is used for two general purposes:
• to model systems in which the diuse nature of interfaces is essential to the problem, such as spinodal decomposition and solute trapping during rapid phase boundary motion;
• as a front tracking technique to model general multi-phase systems.
Generally, we speak about two types of phase eld models. In the rst, called Cahn-Hilliard, the phase is uniquely determined by the value of a conserved eld variable, such as the concentration C, e.g. if C ≤ C1, then we are in one phase, if C ≥ C2 then the other. These models were rst applied to understand spin- odal decomposition, and are now used for a wide range of phenomena. In the second, called Allen-Cahn, the phase is not uniquely determined by concentra- tion, temperature, pressure, etc., so we add one or more extra eld variable(s) sometimes called the order parameter φ which determines the local phase. This class of models is widely used to study solidication and solid-state phase trans- formations in metals.
The phase-eld method is a xed-grid method; it diers from other methods in that the interface is diuse in a physical rather than numerical sense. Thus, it is also known as the diuse-interface model. More precisely, the diuse inter- face is introduced through an energetic variational procedure that results in a thermodynamic consistent coupling system. The basic idea was derived from the consideration that the two components, though nominally immiscible, does mix in reality within a narrow interfacial region. A phase-eld variable φ can be thought of as the volume fraction, to demarcate the two species and indicate the location of the interface. A mixing energy is dened based on φ which, through
a convection-diusion equation, governs the evolution of the interfacial prole. The phase-eld method can be viewed as a physically motivated level-set method. When the thickness of the interface approaches zero, the diuse-interface model becomes asymptotically identical to a sharp-interface level-set formulation. It also reduces properly to the classical sharp-interface model in general.
From the statistical (phase eld approach) point of view, the interface represents a continuous, but steep change of the properties (density, viscosity, etc.) of two uids. Within this "thin" transitional region, the uid is mixed. The mixing is determined by molecular interactions between the two species, and can be described by a stored mixing energy, which represents the balance between the competing phobic/philic relation
Chaos Suppression and Stabilization of Generalized Liu Chaotic Control Systemijtsrd
In this paper, the concept of generalized stabilization for nonlinear systems is introduced and the stabilization of the generalized Liu chaotic control system is explored. Based on the time-domain approach with differential inequalities, a suitable control is presented such that the generalized stabilization for a class of Liu chaotic system can be achieved. Meanwhile, not only the guaranteed exponential convergence rate can be arbitrarily pre-specified but also the critical time can be correctly estimated. Finally, some numerical simulations are given to demonstrate the feasibility and effectiveness of the obtained results. Yeong-Jeu Sun | Jer-Guang Hsieh "Chaos Suppression and Stabilization of Generalized Liu Chaotic Control System" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-1 , December 2018, URL: http://www.ijtsrd.com/papers/ijtsrd20195.pdf
http://www.ijtsrd.com/engineering/electrical-engineering/20195/chaos-suppression-and-stabilization-of-generalized-liu-chaotic-control-system/yeong-jeu-sun
HEATED WIND PARTICLE’S BEHAVIOURAL STUDY BY THE CONTINUOUS WAVELET TRANSFORM ...cscpconf
Nowadays Continuous Wavelet Transform (CWT) as well as Fractal analysis is generally used for the Signal and Image processing application purpose. Our current work extends the field of application in case of CWT as well as Fractal analysis by applying it in case of the agitated wind particle’s behavioral study. In this current work in case of the agitated wind particle, we have mathematically showed that the wind particle’s movement exhibits the “Uncorrelated” characteristics during the convectional flow of it. It is also demonstrated here by the Continuous Wavelet Transform (CWT) as well as the Fractal analysis with matlab 7.12 version
Exercise Package 2 Systems and its properties (Tip Alwa.docxelbanglis
Exercise Package 2:
Systems and its properties: (Tip: Always use the components symbols, C, RS, KT, etc., in the derivation of
transfer function and only plug in component values at the last step. Show your steps and tell me a complete
story.)
1) Consider a 100mH inductor with v-i relationship in passive device labeling convention:
a. Find transfer function H(s) with current flowing through the inductor as the input, i(t),
and voltage across the inductor as the output, v(t), (in the unit of Ohms).
b. Find the same input-output relationship in the expression of differential equation.
c. Find v1(t) with input i1(t)=2sin(100t) (mA) and v2(t) with input i2(t)=0.4cos(500t) (mA)
respectively.
d. Show time invariant such that v(t)=v1(t−τ) as i(t)=i1(t−τ)=2sin(100t−0.9) (mA).
e. Show linearity using superposition such that v(t)=v1(t)+v2(t) as i(t)=i1(t)+i2(t).
2) Given following, a practical integrator, circuit, where Rf=100KΩ, R1=9.1KΩ, RS=100Ω, C=0.1µF,
and the OpAmp is an ideal operational amplifier:
a. Find the transfer function in between the output VO(t) and input VS(t), VO(t)=H(s){VS(t)}.
b. Find the same input-output relationship in the expression of differential equation.
c. Find VO1(t) (sinusoidal steady state response) with input VS1(t)=0.2sin(100t) (V) and VO2(t)
with input VS2(t)=0.4cos(5000t) (V) respectively.
d. Show time invariant such that VO(t)= VO1(t−τ) as VS(t)= VS1(t−τ)=0.2sin(100t−0.9) (V).
e. Show linearity using superposition such that VO(t)= VO1(t)+VO2(t) with VS(t)=VS1(t)+ VS2(t).
3) Here is a typical coupling network in electronics where coupling capacitor, selected, C=0.022µF,
input impedance, Zi=5.7KΩ, and input source resistor, RS=520Ω:
a. Find the transfer function, H(s), Vout(t)=H(s){Vin(t)}.
b. Find the same input-output relationship in the expression of differential equation.
c. Find VOut(t) (sinusoidal steady state response) with input Vin1(t)=2sin(50t+0.4) (V) and
Vin2(t) with input Vin2(t)=4cos(10000t) (V) respectively.
4) Here is a typical bypass network in electronics where bypass capacitor, selected, C=10µF, and
the equivalent (Thevenin) resistor of circuit to be bypassed, Req=376Ω:
Vcc+
Vcc-
Vo
Vs
Rf
R1Rs
C
Vin Vout
CRs
Zi
a. Find the transfer function, H(s), VS(t)=H(s){IS(t)} (note: the unit is ohm).
b. Find the same input-output relationship in the expression of differential equation.
c. Find VS1(t) (sinusoidal steady state response) with input Is1(t)=0.2cos(10t+0.3) (A) and
VS2(t) with input IS2(t)=0.5cos(10000t) (A) respectively.
5) The following circuit is an active filter (2nd order Butterworth low-pass filter), with the selected
values: R=10KΩ, C=8200pF, Rf=68KΩ, and R1=120KΩ.
a. Derive the transfer function, H(s), Vout(t)=H(s){Vin(t)}. (Tip: the selected R is much greater
than RS such that RS can be ignored in the derivation. Label extraordinary nodes and use
node voltage method. OpAmp is considered ideal.)
b. Show that th ...
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
HEAP SORT ILLUSTRATED WITH HEAPIFY, BUILD HEAP FOR DYNAMIC ARRAYS.
Heap sort is a comparison-based sorting technique based on Binary Heap data structure. It is similar to the selection sort where we first find the minimum element and place the minimum element at the beginning. Repeat the same process for the remaining elements.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
plant. If a power plant could recover and reuse a portion of this
moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
from flue gas is to use a condensing heat exchanger. The power
plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
Water billing management system project report.pdfKamal Acharya
Our project entitled “Water Billing Management System” aims is to generate Water bill with all the charges and penalty. Manual system that is employed is extremely laborious and quite inadequate. It only makes the process more difficult and hard.
The aim of our project is to develop a system that is meant to partially computerize the work performed in the Water Board like generating monthly Water bill, record of consuming unit of water, store record of the customer and previous unpaid record.
We used HTML/PHP as front end and MYSQL as back end for developing our project. HTML is primarily a visual design environment. We can create a android application by designing the form and that make up the user interface. Adding android application code to the form and the objects such as buttons and text boxes on them and adding any required support code in additional modular.
MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software. It is a stable ,reliable and the powerful solution with the advanced features and advantages which are as follows: Data Security.MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software.
Water billing management system project report.pdf
Lattice Boltzmann methhod slides
1. Lattice Boltzmann Method and ItsLattice Boltzmann Method and Its
Applications in Multiphase FlowsApplications in Multiphase Flows
Xiaoyi He
Air Products and Chemicals, Inc.
April 21, 2004
3. A Brief History of LatticeA Brief History of Lattice
Boltzmann MethodBoltzmann Method
Lattice Gas Automaton (Frisch, Hasslacher,
Pomeau,, 1987)
Lattice Boltzmann model (McNamara and Zanetti
(1988)
Lattice Boltzmann BGK model (Chen et al 1992
and Qian et al 1992)
Relation to kinetic theory (He and Luo, 1997)
4. Lattice Boltzmann BGK ModelLattice Boltzmann BGK Model
τ
δδ
eq
aa
aaa
ff
txftttexf
−
−=−++ ),(),(
• fa: density distribution function;
• τ: relaxation parameter
• f
eq
: equilibrium distribution
∑∑ ==
−
⋅
+
⋅
+=
a
aa
a
a
aa
a
eq
a
efuf
RT
u
RT
ue
RT
ue
f
ρρ
ρω
,
2)(2
)(
1
2
2
2
5. Kinetic Theory of Multiphase FlowKinetic Theory of Multiphase Flow
BBGKY hierarchy
functionondistributiparticle-two
potentialularintermolec
functionondistributiparticle-single
:)r,,r,(
:)(
:
)()()(
2211
)2(
12
2212
1
)2(
1 111
ξξ
ξ
ξ
ξ ξξ
f
rV
f
drdrV
f
fFf
t
f
r ∇⋅
∂
∂
=∇⋅+∇⋅+
∂
∂
∫∫
8. Model for Intermolecular AttractionModel for Intermolecular Attraction
For D2 (attraction tail), by assuming
fVdrdrV
f
I m
D
1
2
1 2212
1
)2(
2 )( ξξ ξ
ξ
∇⋅∇=∇⋅
∂
∂
= ∫∫
)r,()r,()r,,r,( 22112211
)2(
ξξξξ fff =
We have
9. Model for Intermolecular AttractionModel for Intermolecular Attraction
Vm is the mean-field potential of intermolecular attraction
∫
∫
>
>
−=
−=
dr
dr
drrVr
drrVa
)(
6
1
)(
2
1
2
κ
ρκρ 2
2 ∇−−= aVm
where
Control phase transition
Control surface tension
For small density variation:
∫>
=
dr
m drrVrV )()(ρ
10. Kinetic Model for Multiphase FlowKinetic Model for Multiphase Flow
Boltzmann equation for non-ideal gas / dense fluid
functionondistributiparticle-single:
)()( 1
f
fVIfFf
t
f
m ξξξ ∇⋅∇+=∇⋅+∇⋅+
∂
∂
11. Kinetic Model for Multiphase FlowKinetic Model for Multiphase Flow
Mass transport equation
0)( =⋅∇+
∂
∂
u
t
ρ
ρ
2
0
22
0
)1(),(
2
),(
)()(
ρρχρρ
ρ
κ
ρκρρ
ρρκρρ
ρ
abRTTp
Tpp
pFuu
t
u
−+=
−∇−=
∇∇⋅∇+Π⋅∇+∇−=⋅∇+
∂
∂
Momentum transport equation
Chapman-Enskog expansion leads to the following macroscopic
transport equations:
12. Kinetic Model for Multiphase FlowKinetic Model for Multiphase Flow
Comments on momentum transport equation
1. Correct equation of state
2. Thermodynamically consistent surface tension
drT∫
+=Ψ
2
2
),( ρ
κ
ρψ
3. Thermodynamically consistent free energy
(Cahn and Hillary, 1958)
interfaceinenergyfreeexcess:)(2 )W(dW ρρρκσ ∫=
13. Kinetic Model for Multiphase FlowKinetic Model for Multiphase Flow
Energy transport equation
ρρκρ
κ
ρκρρ
ρρρρκ
λ
∇∇+−∇−=
∇⋅∇−∇∇∇+
Π∇+∇⋅∇+∇−=⋅∇+
∂
∂
ITpP
u
uTuPue
t
e
]
2
),([
)](
2
1
)([:
:)(:)(
22
0
14. Kinetic Model for Multiphase FlowKinetic Model for Multiphase Flow
Comments on energy transport equation
1.Total energy needs include both kinetic and potential
energies, otherwise the pressure work becomes:
2. Last term is due to surface tension and it is
consistent with existing literature (Irving and
Kirkwood, 1950)
upubRT ⋅∇≠⋅∇+ )1( ρχρ
15. LBM Multiphase Model Based onLBM Multiphase Model Based on
Kinetic TheoryKinetic Theory
Temperature variations in lattice Boltzmann models;
Discretization of velocity space;
Discretization of physical space;
Discretization of temporal space.
16. Temperature in Lattice BoltzmannTemperature in Lattice Boltzmann
MethodMethod
Non-isothermal model model is still a challenge
– Small temperature variations can be modeled
– Need for high-order velocity discretization
Isothermal model is well developed
−
⋅
+
⋅
+−+=
+=
0
2
2
0
2
00
2
0
2)(2
)(
)
2
3
2
(1
)1(
RT
u
RT
u
RT
u
RT
f
TT
a
eq ξξ
θ
ξ
ω
θ
17. Isothermal Boltzmann Equation forIsothermal Boltzmann Equation for
Multiphase FlowMultiphase Flow
)
2
)(
exp(
)2(
)(
)()(
2
RT
u
RT
f
fV
RT
uff
fFf
t
f
D
eq
eq
m
eq
−
−=
∇⋅
−
+
−
−=∇⋅+∇⋅+
∂
∂
ξ
π
ρ
ξ
τ
ξ ξ
18. Discretization in Velocity SpaceDiscretization in Velocity Space
Constraint for velocity stencil
Further expansion of f eq
32,1,0,nfor, ==∫ exactdf eqn
ξξ
−
⋅
+
⋅
+−=
RT
u
RT
u
RT
u
RT
f eq
2)(2
)(
1)
2
exp(
2
2
22
ξξξ
ρ
5...,1,0,nfor,)
2
exp(
2
==−∫ exactd
RT
n
ξ
ξ
ξ
19. Discretization in Velocity SpaceDiscretization in Velocity Space
−
⋅
+
⋅
+=
RT
u
RT
ue
RT
ue
f aa
a
eq
a
2)(2
)(
1
2
2
2
ρω
9-speed model 7-speed model
ωa: weight coefficients
20. Discretization in Physical andDiscretization in Physical and
Temporal SpacesTemporal Spaces
Integrate Boltzmann equation
eq
am
a
eq
aa
aaa fV
RT
tue
t
ff
txftttexf ∇⋅
−
+
−
−=−++
δ
δτ
δδ
)(
/
),(),(
• Discretizations in velocity, physical and temporal spaces are
independent in principle;
• Synchronization simplifies computation but requires
• Regular lattice
• Time-step constraint: RTtx 3/ =δδ
21. Further Simplification for NearlyFurther Simplification for Nearly
Incompressible FlowIncompressible Flow
Introduce an index function φ:
)()(
)(
),(),( u
RT
ueff
txftttexf a
eq
aa
aaa Γ∇⋅
−
+
−
−=−++ φψ
τ
δδ
)]())0()(())(([
)(),(),(
ρψ
τ
δδ
∇Γ−Γ−+Γ
⋅−+
−
−=−++
uGFu
ue
gg
txgtttexg
s
a
eq
aa
aaa
∑
∑
∑
++=
−∇⋅−=
=
)(
2
)(
2
1
GF
RT
geRTu
RTpugp
f
saa
a
a
ρ
ρ
φ
31. Other ApplicationsOther Applications
Multiphase flow in porous media (Rothman 1990,
Gunstensen and Rothman 1993);
Amphiphilic fluids (Chen et al, 2000)
Bubbly flows (Sankaranarayanan et al, 2001);
Hele-Shaw flow (Langaas and Yeomans, 2000).
Boiling flows (Kato et al, 1997);
Drop break-up (Halliday et al 1996);
32. Challenges in Lattice BoltzmannChallenges in Lattice Boltzmann
MethodMethod
Need for better thermal models;
Need for better model for multiphase flow
with high density ratio;
Need for better mode for highly
compressible flows;
Engineering applications …
33. ConclusionsConclusions
Lattice Boltzmann method is a useful tool
for studying multiphase flows;
Lattice Boltzmann model can be derived
form kinetic theory;
It is easy to incorporate microscopic physics
in lattice Boltzmann models;
Lattice Boltzmann method is easy to
program for parallel computing.