2014 - modelos físico-matemáticos de fluidos
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Work in English about lattice-boltzmann methodsTRANSCRIPT
Arthur Vinicius Secato Rodrigues
Fisico-Mathematical Models of Fluids
From Mesoscopic to Continuum
Florianópolis
2014
Arthur Vinicius Secato Rodrigues
Modelos Físico-Matemáticos de Fluidos:
Do Mesoscópico ao Contínuo
Monografia submetida ao
Programa de Graduação em Engenharia
de Produção da Universidade Federal
de Santa Catarina para a obtenção do
Grau de Engenheiro Mecânico c/
Habilitação em Engenharia de
Produção.
Orientador: Prof. Dr. Sérgio F. Mayerle
Coorientador: Prof. Dr. Paulo C. Philippi
.
Florianópolis
2014
Ficha de identificação da obra elaborada pelo autor
através do Programa de Geração Automática da Biblioteca
Universitária da UFSC.
A ficha de identificação é elaborada pelo próprio autor
Maiores informações em:
http://portalbu.ufsc.br/ficha
Arthur Vinicius Secato Rodrigues
Modelos Físico-Matemáticos de Fluidos:
Do Mesoscópico ao Contínuo
Este Trabalho de Conclusão de Curso foi julgado adequado para
obtenção do Título de Engenheiro Mecânico c/ habilitação em Engenharia
de Produção, e aprovado em sua forma final pelo Departamento de
Engenharia de Produção e Sistemas.
Florianópolis, 1 de dezembro de 2014.
Prof.ª Mônica Maria Mendes Luna, Dr.ª
Coordenadora do Curso
Banca Examinadora:
Prof. Sérgio Fernando Mayerle, Dr.
Orientador
Universidade Federal de Santa Catarina
Prof. Paulo Cesar Philippi, Dr.
Coorientador
Universidade Federal de Santa Catarina
Prof. Osmar Possamai, Dr.
Universidade Federal de Santa Catarina
To the Cosmos.
ACKNOWLEDGEMENTS
I want to begin to thank Prof. Philippi and Prof. Mayerle for
trusting in this work. I appreciated it a lot. Special thanks to LMPT team
and ex-team for all conversations, laughs and daily working.
Thanks to UFSC for being a public university and the place
where most of my life was spent in the last few years, implying my debt
to the Brazilian people who finance this university and its aid to students
like me, with their daily work.
Thanks to the pirate community, especially to The Pirate Bay,
for sharing with me and with the world, information and knowledge that
some want to keep locked in their copyrights.
I thank my parents and my wonderful family for the opportunity
to exist and for the freedom to choose. I am the developing result of their
love and kindness.
I thank all my friends, ALL OF THEM. Their friendship is my
biggest treasure. Even those who are distant in space or time, they are
constituent parts of my being, inspiring me with the memories of their
smiles and words.
Finally, I thank the Gods for given me some order, stars in the
sky, light, life and love; and I thank the Demons for giving me chaos, sex,
music, wine, science and friends.
1
1 O disbelievers!
I do not worship what you worship;
Nor are you worshippers of what I worship;
Nor will I be a worshipper of what you worship;
Nor will you be worshippers of what I worship;
For you is your truth, and for me is my truth.
Surat Al –Kāfirūn (free author’s translation)
RESUMO
A descrição de fluidos na natureza é um desafio para físicos,
matemáticos e engenheiros. O conceito de fluidos permeia os estados da
matéria que compreendem os líquidos, os gases e os plasmas, sendo
portanto, o comportamento físico-cinemático mais abundante no
universo, justificando assim sua importância. A complexidade dessa
descrição é abordada através de idealizações matemáticas pertinentes,
normalmente restringindo-a a fluidos newtonianos incompressíveis,
onde, mesmo assim, carece de soluções analíticas gerais. Os fluidos,
protagonistas deste trabalho, são contextualizados em sua dimensão
econômica a nível nacional e internacional; sua modelagem clássica,
baseada na hipótese do contínuo, é revisada e aprofundada, focando-se na
interpretação de seus termos e na compreensão de sua linguagem. Na
sequência será introduzido os conceitos básicos da teoria cinética, de onde
emerge uma nova abordagem para fluidos baseada na equação de
Boltzmann, uma equação fenomenológica que descreve a matéria
considerando-a um sistema de partículas. Dentro desta nova abordagem
deriva-se modelos numéricos, comumente chamados de Lattice-
Boltzmann Method (LBM). Com estes modelos, simulações são
conduzidas e os resultados discutidos..
Palavras-chave: Mecânica dos fluidos. LBM. Lattice-Boltzmann.
ABSTRACT
The description of fluids is a challenge for physicists,
mathematicians and engineers. The concept of fluids permeates most of
states of matter which includes liquids gases and plasmas, being therefore,
the most abundant physic-kinematic behavior in the universe and thus
justifying its multidimensional importance. The complexity of this
description is managed with pertinent mathematical idealizations,
normally restricting it to incompressible Newtonian fluids which, even
though, lacks general solutions. Fluids, the protagonists of this work, are
contextualized in its economical dimension at national and international
level. Its classical modelling based on the continuum hypothesis is
revisited, emphasizing the interpretation of its terms and comprehension
of the language. In the sequence, basic topics on kinetic theory will be
introduced, from where a new approach for dealing with fluids emerges,
based on the Boltzmann equation, a phenomenological equation that
describes matter considering it as a system of particles. With this new
approach numerical methods will be derived. They are called Lattice-
Boltzmann Methods (LBM). With these methods some simulations will
be carried out and its results discussed.
Keywords: Fluid dynamics. LBM. Lattice-Boltzmann.
LIST OF FIGURES
Figure 1 – Water fishbones. ...................................................................32 Figure 2 - Brazil’s GDP growth in recent years. ....................................38 Figure 3 – GDP in 2013 .........................................................................39 Figure 4 – Common gases at 300K ........................................................58 Figure 5 – Velocity vectors in the D2Q9 model: ...................................73 Figure 6 – Streaming step with halfway bounce-back ...........................79 Figure 7 – Periodic topological manifolds for a 2D domain ..................80 Figure 8 – Three unknown populations..................................................81 Figure 9 - Framework for LBM models. ................................................86 Figure 10 – Coalescence cascade of a drop. ...........................................87 Figure 11 - Comparison between analytical and simulated velocity. .....96 Figure 12 – Transient to periodic evolution of the forces. .....................99 Figure 13 – Normalized Fast Fourier Transform (FFT) .......................100 Figure 14 - Magnification of some vortices of Fig. 15.........................101 Figure 15 – Flow past a cylinder: vorticity field. .................................102 Figure 16 - Flow past a cylinder: velocity field. ..................................103 Figure 17 – Cangaceiro’s original picture. ...........................................104 Figure 18 – The immiscible cangaceiro after 50 time steps ................105 Figure 19 - The immiscible cangaceiro after 250 time steps ...............106 Figure 20 – Immiscible cangaceiro after 30 000 time steps .................108 Figure 21 – EOS curves for some attraction forces. .............................110 Figure 22 – The initial configuration ...................................................110 Figure 23 – Above is shown the system after 100 time steps ..............111 Figure 24 – System in equilibrium after 50 000 time steps. .................112 Figure 25 – Symmetrical flow field during the first time steps ............114 Figure 26 - Asymmetrical flow field during the first time steps ..........115
LIST OF TABLES
Table 1 – LBGK models ....................................................................... 70
Table 2 – Parameters for the flow past a cylinder. ................................ 93
Table 3 – Main parameters of the immiscible cangaceiro. .................. 103
Table 4 – Parameters for the SCMP simulation. ................................. 105
ACRONYMS
ABNT – Associação Brasileira de Normas Técnicas
IBGE – Instituto Brasileiro de Geografia e Estatística
IPEA – Instituto de Pesquisa Econômica Aplicada
IMO – International Maritime Organization
UNCTAD – United Nations Conference on Trade and Development
SCN – Sistema de Contas Nacionais
GDP – Gross domestic product
Eq. – Equation
Dist. – Distribution
NS – Navier-Stokes
BE – Boltzmann equation
LGA – Lattice-Gas Automaton
LBGK – LBM with BGK collision operator.
LBM – Lattice-Boltzmann method
LBE – Lattice-Boltzmann equation
EOS – Equation of state
BC – Boundary condition
SCMP – Single component multiphase
MCMP – Multicomponent multiphase
LIST OF SYMBOLS
SI Units
m Meter (lenght) [m]
s Second (time) [s]
kg Kilogram (mass) [kg]
c Meter per second (velocity) [c]
Pa Pascal (pressure, stress) [Pa]
N Newton (force) [N]
J Joule (energy, work) [J]
K Kelvin (temperature)
[K]
Greek Letters
ψ Intensive tensor property
Ψ Extensive tensor property
𝜌 Specific mass/density [kg/m³]
𝚷,Π𝛼𝛽 Momentum flux tensor [kg/(m.s²)]
Π𝛼𝛽𝑠𝑡𝑟 Strees tensor [kg/(m.s²)]
Π𝛼𝛽𝑣𝑖𝑠𝑐 Deviatoric viscous stress tensor [kg/(m.s²)]
𝜇 Dynamic viscosity [kg/(m.s)]
𝜇𝑛 n-th moment of a function
𝜆 Second viscosity [kg/(m.s)]
𝜈 Kynematic viscosity [m²/s]
𝜉 (𝜇 + 𝜆) [kg/(m.s)]
Ω, Ω𝑖 Collision operator [kg/(c³m³s²)]
𝛷 Positive unknown function
Φ Collision invariants vector
𝜃 Lattice sound speed squared
𝜔 Relaxation frequency parameter
𝜔+ Symmetric part of 𝜔
𝜔− Amtisymmetric part of 𝜔
Roman Letters
A Surface area [m²]
𝑐 Mean particles velocity modulus [m/s]
𝑐𝛼 Mean particles velocity vector [m/s]
𝒄, 𝑐𝛼, 𝒄𝒊 Particle velocity vector [m/s]
𝑐𝑠 Sound speed [m/s]
𝑐𝑟 Lattice reference speed
𝐶1, 𝐶2, 𝐶3 Arbitrary constants
ℯ(𝐫, 𝑡) Massic energy function [J/kg]
ℯ𝑝 Mean peculiar massic kinetic energy [J/kg]
𝒇 Body force [m/s²]
𝑓, 𝑓𝑖 Mass probability distribution [kg/(c³m³s)]
[𝑓′∗, 𝑓′] Bullet and target dist. before collision [kg/(c³m³s)]
[𝑓∗, 𝑓] Bullet and target dist. after collision [kg/(c³m³s)]
F Normalized probability distribution [1/(c³m³s)]
𝐹𝑐(𝑐) Equilibrium distribution (maxwellian) [1/c³]
𝐹(𝑐) Equilibrium dist. (maxwell-boltzmann) [1/c³]
𝑓(𝒄)𝑒𝑞 Mass Equil. dist. (maxwell-boltzmann) [kg/c³]
Ӻ, Ӻ𝛼 Force vector [(kg.m)/s²]
𝒈, 𝑔𝛼 Body force vector [m/s²]
ℋ Hamiltonian [J]
𝐻(𝑓) Boltzmann H-function
𝑘𝑏 Boltzmann constant [J/K]
𝐾 Kinetic energy [(kg.m²)/s²]
𝐾𝑐 Bulk modulus of compressibility [kg/(m.s²)]
Ӄ Bulk viscosity coefficient [kg/(m.s)]
𝐿 Characteristic length; Length [m]
𝑚 mass [kg]
𝑀 Molar mass [kg/mol]
𝐧 Unit normal vector [m]
𝑛 Number of moles; Number density
N Number of particles
𝑁𝐴 Avogadro’s number. [1/mol]
𝑝 Mean pressure [kg/(m.s²)]
𝐩, p𝛼 , 𝐩𝐢 Momentum vector [(kg.m)/s]
Mean mechanical pressure [kg/(m.s²)]
q Position vector in phase space ℙ
𝔔𝛽 Kinetic energy transport vector [(kg.m³)/s³]
𝒓, 𝑟𝛼, 𝒓𝒊 Position vector [m]
𝑅 Ideal gas constant [J/(K.mol)]
𝑆𝛾𝛿 Strain rate tensor [1/s]
𝑡 Time [s]
𝑇 Temperature [K]
𝒖, 𝑢𝛼 Velocity vector [m/s]
u Bulk flow velocity [m/s]
𝑈 Characteristic velocity [m/s]
𝑣 Peculiar velocity [m/s]
V Volume [m³]
𝑤𝑖 Discrete weights
Other Symbols
Ӂ𝛼𝛽𝛾𝛿 Viscosity tensor [kg/(m.s)]
Ϣ𝑐𝛼𝑐𝛽 ...𝑐𝜔 Generic higher order moments
Ç Number of replicates
Щ Collision operator model [kg/(c³m³s²)]
Щ𝐵𝐺𝐾 BGK collision operator [kg/(c³m³s²)]
Dimensionless Numbers
𝑀𝑎 Mach number
𝑅𝑒 Reynolds number
𝑆𝑡 Strouhal number
Mathematical Symbols
𝛿𝛼𝛽 Krocecker’s delta
𝐷𝑡 Material derivative
∫ (∙)𝑑𝑎∂𝑣
Surface integral [m²]
∫ (∙)𝑣
𝑑𝑣 Volume integral [m³]
𝜕𝑡 Time derivative [1/s]
𝜕𝛽 Space derivative [1/m]
𝛁 Nabla operator [1/m]
∇𝒓 Nabla operator over spatial space [1/m]
∇𝒄 Nabla operator over velocities space [1/(m/s)]
∇2 Laplacian operator [1/m²]
⊗ Tensor product
dq Infinitesimal phase space volume [m³.(m/s)³]
𝒪2 Second to higher order terms
ℙ, 𝔹 Phase space
Continuum body א
CONTENTS
1 INTRODUCTION ................................................................. 31
1.1 PHYLOSOPHICAL AND HISTORICAL CONSIDERATIONS….33
1.2 OBJECTIVES .................................................................................. 37
1.3 FIELDS’ RELEVANCE .................................................................. 37
1.3.1 Brazil: a fluid based industrial economy ................................... 38
2 MATHEMATICAL MODELLING .................................... 43
2.1 MACROSCOPIC MODELS ............................................................ 43
2.1.1 Reynolds Transport Theorem .................................................... 44
2.1.2 Transport Equations in Continuum .......................................... 45
2.1.3 Newtonian fluids .......................................................................... 47
2.1.4 Incompressibility hypothesis ...................................................... 50
2.1.5 Dynamic similarity ...................................................................... 51
2.2 MESOSCOPIC MODELS ............................................................... 52
2.2.1 Ideal gases (pressure and temperature) .................................... 54
2.2.2 The equilibrium distribution ...................................................... 55
2.2.3 The local equilibrium distribution ............................................. 59
2.2.4 The Boltzmann equation (BE) ................................................... 59
2.2.5 Macroscopic equations recovery ................................................ 63
2.2.6 The BGK kinetic model .............................................................. 66
3 THE LATTICE-BOLTZMANN METHOD ....................... 71
3.1 LBGK SCHEMES ........................................................................... 72
3.1.1 Physical LBGK simulations ...................................................... 75
3.1.2 External force implementation .................................................. 77
3.2 BOUNDARY CONDITIONS (BC) ................................................. 79
3.2.1 Non-slip at walls .......................................................................... 79
3.2.2 Periodic BC .................................................................................. 80
3.2.3 Constant flux BC (Von Neumann) ............................................. 81
3.2.4 Constant pressure/density BC (Dirichlet) ................................. 82
3.2.5 Zero derivative at boundaries .................................................... 83
3.3 COLLISION OPERATOR ALTERNATIVES ................................ 83
3.3.1 Two-time relaxation collision operator (TRT) ......................... 83
3.4 MULTIPHASE AND MULTICOMPONENT MODEL ................. 85
3.4.1 Interparticle potential model ..................................................... 88
4 SIMULATIONS ..................................................................... 93
4.1 HAGEN-POISEUILLE FLOW ....................................................... 95
4.2 FLOW PAST A CYLINDER .......................................................... 97
4.3 THE IMMISCIBLE CANGACEIRO (MCMP) .............................. 104
4.4 PHASE TRANSITION (SCMP) .................................................... 109
4.5 BIDIMENSIONAL SWALLOW FLIGHT ................................... 113
5. FINAL CONSIDERATIONS ............................................. 117
REFERENCES ........................................................................ 121
APPENDIX: Code examples .................................................. 127
31
1 INTRODUCTION
Fluid flows are ubiquitous in nature and are experienced
everywhere by those who observe their environment attentively. They are
vital to all those transport processes that make our tiny Earth like it is and
even life like we know it. The entire homeostasis of the planet can be
thought as a gracious balance of heat, mass and momentum transport that
shape its atmosphere, rivers, oceans and even life.
For life, fluids act as an important sculptor of its ontogeny, being
the cradle of its early development. This importance might be seen in an
interesting article from CARTWRIGHT et al. (2009) which exposes the
relevance of the field:
“It is becoming increasingly clear that
the number of genes in the genome of a typical organism is not sufficient to specify the
minutiae of all features of its ontogeny. Instead, genetics often acts as a
choreographer, guiding development but
leaving some aspects to be controlled by
physical and chemical means. Fluids are
ubiquitous in biological systems, so it is not surprising that fluid dynamics should play an
important role in the physical and chemical
processes shaping ontogeny.”
Furthermore and beyond applied sciences, fluids are also
embedded with intrinsic beauty that tickles our minds, being also motifs
of art, photography and contemplation, showing lovely patterns like
clouds, vortexes and like the water fishbones shown in Figure 1.
In matter of science, their models are on the edge of physical-
mathematical research. From one side, solution’s smoothness and
existence of the Navier-Stokes Equations, the equations of the continuum
approach modelling, is still one of the Millenium Prize Problems
announced from The Clay Mathematics Institute2, which gives a million
dollars prize for its solution. Although these nonlinear partial differential
equations were well stablished in the 19th century, until these days it is a
benchmark problem for mathematicians and engineers, who often adopt
a numerical approach to approximate solutions of the equation in many
applications.
2 http://www.claymath.org
32
On the other hand there is a relatively new approach for modelling
fluid dynamics and transport phenomena. It is based on the Boltzmann
equation, which got big attention since such kinetic models could be
verified with the always advancing computing resources.
Figure 1 – Water fishbones.
Source: (BUSH; HASHA, 2004)
Such models offer a more flexible way to describe transport
phenomena, which are vital to engineering, since it can be easily adapted.
From fluid dynamics to drop coalescence, immiscible
displacement, thermal and chemical diffusion and even relativistic flows.
There is a big list of applications where the solutions of this kinetic
approach are valid. Its mathematics is a quite a hard job to comprehend,
since most of it lies on statistical mechanics’ tools, which are not normally
taught in ordinary engineering courses. Nevertheless they are an
enjoyable mind exercise. From the abstraction of colliding particles to
computer simulations recovering observed physical behavior. It is
simultaneously scary and delightful to think on the possibilities of the
human mind with its incommensurable creativity and abstraction power. In this work, the essentials behind both approaches will be
described through a deeper review on the main topics. More emphasis on
the kinetic approach will be given, since it is a relatively new method. It
does not have any pretension to put or propose anything new. This work
33
just expresses the passion from this author for the subject and represents
the results of its intention to have a deeper comprehension on the subject,
both physically and mathematically, while linking that knowledge with
the physical world, where the author thinks he lives in, but is not really
sure.
1.1 PHYLOSOPHICAL AND HISTORICAL CONSIDERATIONS
Imre Lakatos, a philosopher of mathematics and science wrote
once: “Philosophy of science without history of science is empty, history
of science without philosophy of science is blind.”
From an engineering perspective, an analogue thought could be
constructed: engineering without its own history is blind and dangerous.
The age of science is also the age of the political and financial
manipulation of its activity to stablish and/or maintain cultural and
economical hegemony. An engineering act is always a political act, since
it has a defined impact in space-time of political societies. Mankind lives
in a world that still uses most of its engineering for selfish and coercive
purposes. Humans are now the victims of the technology sold with
massive propaganda. Victims of the pollution from cars, industries.
Victims of the contamination of its environment, agriculture. From the
everywhere spraying of pesticides, to the chemicals in the more and more
industrialized food, which many times could look like everything but
food. Not to mention one of the biggest industrial markets in the world:
military.
History is essential for the formation of a critical thinker as it is for
the formation of a critical engineer. This spirit is beautifully caught by
Alexis de Tocqueville: “When the past no longer illuminates the future, the spirit walks in darkness.”
This is not a work on philosophy of science or history, but the
opportunity for a brief reflection on it couldn't have been missed, since it
is of great value for its author.
With that in mind, we shall present a brief history of fluid
mechanics poetically, highlighting that it is an ancient issue in human
mind and culture. This poem, called The Turbulent History of Fluid Mechanics, was written by Naomi Tsafnat and published in May 17,
1999. It gives a lovely list of the main historical events that marks this
branch of human knowledge.
It all started with Archimedes, way back in BC, Who was faced with an interesting problem, you see...
34
The king came to me, and this story he told:
I am not sure if my crown is pure gold.
You are a wise man, or so it is said, Tell me: is it real, or is it just lead?
I paced and I thought, and I scratched my head, But the answer eluded me, to my dread.
I sat in my bath, and pondered and tried, And then...”Eureka! Eureka! I found it!” I cried.
As I sat in my tub and the water was splashing, I knew suddenly that a force had been acting.
On me in the tub, it’s proportional, see,
To the water that was where now there is me.
Of course, Archimedes caused quite a sensation But not because of his great revelation;
As he was running through the streets of Syracuse
He didn’t notice he was wearing only his shoes.
The great Leonardo –oh what a fellow... No, not diCaprio, DaVinci I tell you!
He did more than just paint the lovely Mona,
He also studied fluid transport phenomena.
Then came Pascal, who clarified with agility,
Basic concepts of pressure transmissibility. Everyone knows how a barometer looks,
But he figured out just how it works.
How can we talk about great scientists,
Without mentioning one of the best: Sir Isaac Newton, the genius of mathematics,
Also contributed to fluid mechanics.
One thing he found, and it’s easy as pie,
Is that shear stress, τ, equals μ dv/dy. His other work, though, was not as successful;
His studies on drag were not all that useful.
He thought he knew how fast sound is sent, But he was way off, by about twenty percent.
35
And then there was Pitot, with his wonderful tubes,
Which measure how fast an airplane moves. Poiseuille, d’Alembert, Lagrange and Venturi –
Through his throats – fluid pass in a hurry.
Here is another hero of fluid mechanics,
In fact, he invented the word “hydrodynamics”. It would take a book to tell you about him fully,
But here is the short tale of Daniel Bernoulli:
Everyone thinks is just one Bernoulli...
It is not so! There are many of us, truly.
My family is big, many scientists in this house, With father Johan, nephew Jacob and brother Nicolaus.
But the famous principle is mine, you know,
It tells of the relationship of fluid flow,
To pressure, velocity, and density too.
I also invented the manometer – out of the blue!
Yes, Bernoulli did much for fluids, you bet!
He even proposed the use of a jet.
There were others too, all wonderful folks, Like Lagrange, Laplace, Navier and Stokes.
Here is another well - known name, A mathematician and scientist of great fame:
He is Leonard Euler, I’m sure you all know,
His equations are basis for inviscid flow.
He did more than introduce the symbols π, I, e, He also derived the equation of continuity.
And with much thought and keen derivation, He published the famous momentum equation.
Those wonderful equations and diagrams you see?
They are all thanks to Moody, Weisbach and Darcy.
Then there was Mach, and the road that he paves, After studying the shocking field of shock waves.
36
Rayleigh studied wave motion, and jet instability,
How bubbles collapse, and dynamic similarity. He was also the first to correctly explain.
Why the sky is blue – except when it rains.
Osborne Reynolds, whose number we know, Found out all about turbulent flow.
He also examined with much persistence,
Cavitation, viscous flow, and pipe resistance.
In the discovery of the boundary layer
Prandtl was the major player. It’s no wonder that all the scientists say,
He’s the father of Modern Fluid Mechanics, hooray!
It is because of Prandtl that today we all can
Describe the lift and drag of wings of finite span.
If it weren’t for him, then the brothers Wright
Would probably never have taken flight.
And so we come to the end of this story,
But it’s not the end of the tales of glory! The list goes on, and it will grow too
Maybe the next pioneer will be you?
Naomi Tsafnat
37
1.2 OBJECTIVES
The main objective of this work is to make a generalized review
on the subject of fluid dynamics, focusing on a detailed derivation of its
mathematical models and on the physical interpretation of its
implications, providing a deeper mathematical and physical
understanding on the subject. This theoretical understanding is vital for
those who want to make further research, contributing scientifically on
the field as for those who want to use benchmark numeric models for
engineering applications, optimizing and/or predicting the behavior of
turbomachinery, productive systems and so on.
A further objective is to explore some recently developed numeric
mesoscopic models, which are based on the Boltzmann equation. This
implies a review on the main topics of kinetic theory, while giving some
examples and applications. Moreover, the results of some simulations will
be analyzed and discussed.
As a secondary objective is the highlighting of the theme in an
economical perspective, exploring its importance in modern productive
systems and specially its importance for the Brazilian industrial economy.
1.3 FIELDS’ RELEVANCE
The understanding of the underlying physics in fluid dynamical
phenomena has a huge impact in the current globalized context of human
economy and its technical systems. Airplanes, cars, trains, pumps,
spaceships, weather forecast, sports and so on. The list of applications is
endless. From microfluidics in cells, blood vessels or porous media to
macro phenomena like weather forecasting and stellar relativistic flows.
From optimizing the technique of swimming, allowing competitors to be
faster, to the shape optimization of turbines, allowing them to extract
more power from the same water or wind flow. Fluids are ubiquitous in
our daily life. The whole earth is immersed in a fluidic atmosphere. Every
movement between the surface and the outer space from one point to
another, by any means, is a fluid dynamical motion. Likewise, more than
2/3 of the Earths’ surface is covered with liquid water, through which over
90% of world’s economic trade flow (UNITED NATIONS; IMO3, 2014).
According to UNCTAD4 statistics a fleet of around 1,5 million ships
transported circa 600 million containers in 2012 (UNITED NATIONS,
UNCTAD, 2014).
3 International Maritime Organization, 4 United Nations Conference on Trade and Development.
38
The importance of advancing knowledge in the field is evident.
Imagine, for example, the inherent economical cost associated to the lack
of knowledge in turbulence alone, i.e., all the aggregate cost to society of
our limited turbulence prediction abilities which result in necessity of
adopting big safety factors, depending on empiricism and
experimentation for designing fluid-thermal systems, from heat
exchangers to hypersonic planes. Not to mention the cost of inaccurate
weather forecasting (GEORGE, 1990). For instance, turbulence is one of
the biggest open research fields in fluid dynamics. A fundamental
mathematical modelling is still unknown.
1.3.1 Brazil: a fluid based industrial economy
In recent years, Brazil has seen many improvements in its socio-
economic scenario. According to the WORLD BANK (2014), poverty
(people living with US$2 per day) has fallen from 21% of the population
in 2003 to 9% in 2012. Extreme poverty (people living with less than
US$1.25 per day) also dropped from 10% in 2004 to 2.2% in 2009. This
social achievements were also accompanied by its economical
counterpart. GDP grew consistently in the last years and is shown in
Figure 1.
Figure 2 - Brazil’s GDP growth in recent years.
Source: Author
R$ 0,00
R$ 5,00
R$ 10,00
R$ 15,00
R$ 20,00
R$ 25,00
R$ 0,00
R$ 1.000,00
R$ 2.000,00
R$ 3.000,00
R$ 4.000,00
R$ 5.000,00
1990 1993 1996 1999 2002 2005 2008 2011
Brazil's GDP Growth
Services - value added - R$ (billions) - Reference 2000 (IBGE/SCN 2000)
Industry - value added - R$ (billions) - Reference 2000 (IBGE/SCN 2000)
Agriculture - value added - R$ (billions) - Reference 2000 (IBGE/SCN 2000)
PIB per capita - R$ (Thousands) - IPEA - Reference 2000.
39
Despite all this promising scenario there is still a huge social
inequality in Brazil. The 20% richest had almost 60% of the total
country’s income in 2009 (WORLD BANK, 2014). It is getting
constantly better, that is true, but there is still much work to do.
Something interesting to note in Figure 2 is the proportion of the
economic sectors in the total GDP. Many people in Brazil believe that
agriculture is a very important economic activity that sustains the
country’s economic health. Agriculture should not be underestimated but
the fact is that the agricultural sector is responsible for less than 6% of
GDP.
Figure 2 shows the proportion of economy sectors in GDP in 2013.
Just one quarter of it is based on industrial activities. It is not much since
around half of it is based on the primary sector (FIESP, 2013).
Figure 3 – GDP in 2013
Source: Author
Now a curious fact: The sector of oil and gas grew from 3% in
2000 to 13% of GDP in 2014 (NUNES, 2014). This number will reach
around 20% in 2020 (NOVAES, 2012). The main responsible for such
numbers is Petrobras, which has an investment plan of U$220,6 billion
for the 2014-2018 period. It includes 28 perforation underwater probes,
32 oil production platforms, 154 large support ships and 81 tank-ships.
All this production will take place here, in Brazil (NOVAES, 2012).
Therefore, this sector represents 52% of the Brazilian industrial GDP and
5,7%
25,0%
69,3%
GDP 2013
Agriculture - percentage of GDP( IBGE 2013)
Industry - percentage of GDP( IBGE 2013)
Services - percentage of GDP( IBGE 2013)
40
may grow to 80% of it, if its industrial sector remains with the same 25%
of the total economy.
So, analyzing this data is clear that the sector of oil and gas is and
will be the most strategic activity in Brazil. It pulls many other productive
sectors around it such as the navy and steel industry. It is of great
importance in development of science and technical systems, fomenting
universities and research centers.
The Petroleum National Agency, ANP (2013), estimates
investments of U$20 to U$30 billion in R&D for the next 35 years for the
Campo de Libra alone, one of the Pre-Salt5 reserves.
Likewise, the recently sanctioned PNE, Plan of National
Education, among other goals, wants to increase the educational budget
to 10% of GDP until 2024 (BRASIL, 2014). The current number is around
5,5%. It is widely known that most of this investment has as its primary
source the oil royalties that come, and will come from the Pre-Salt
extraction.
The strategic dimension of the sector for Brazil’s future is
undeniable. Education, economy, science, technology, among others, will
have these prehistoric fluids as its main source of foment.
Oil. The black fluid. Thousands, maybe millions of years of sun’s
light energy captured and stored. All that energy that once irradiated over
those places is now there, available to drive the wheels of modern
capitalistic societies, which are thirsty for energy, especially for oil.
Ships, pipelines, gas, oil, reservoir rocks, sea water… this world is a world
of fluids. Improvements of 1% in efficiency to extract, pump, transport,
etc… may have an impact of billions of dollars. They want those dollars.
We want those dollars too. But to improve, technology is needed. For
technology, models are needed. Not only some kind of models, but always
changing-to-better models. Models that become more and more
sophisticated. Models that are able to describe and recover the essential
physics in the phenomena of interest.
5 The Brazil’s Pre-salt is a sequence of sedimentary rocks formed more
than 100 million years ago by the separation of the current American and African
continents. Its huge gas and oil reserves pushed Brazil into the international
scenario of this sector. This reserves, however, are allocated deep under the sea
level, between 7 and 8 kilometers, from where circa 6 kilometers are post-salt and
salt rocks (FOLHA, 2010). This technological challenge pushes the investments
from Petrobras in R&D, in order to make the extraction economically viable.
41
Physics is mathematical not because we
know so much about the physical world, but because
we know so little; it is only its mathematical
properties that we can discover.
Bertrand Russell
42
43
2 MATHEMATICAL MODELLING
Now we enter in the world where abstract thoughts become
symbols. These symbols become models and these models describe their
own structure in an ontological sense. The philosophical foundation of
mathematics is still a mystery. Some theories say it deals with real objects,
being the language of an intrinsic reality. Other theories say that
mathematics is only a human mind construct, nothing more than games
derived from some set of rules.
Real or not, it is effective. The success of mathematical modelling
in physics is enormous. We appreciate it daily using modern machinery
created with applied physical description of things. Behind any physical
description lies a model, and often, a mathematical model.
In fluid dynamical phenomena modelling there are two main
approaches, macroscopic and mesoscopic. The latter came first, being the
standard approach for teaching fluid mechanics in engineering
graduation. The former came as a translation of the microscopic models
to match the latter.
2.1 MACROSCOPIC MODELS6
Macroscopic approach was known at least since the Ancient
Greece. The work published by Archimedes of Syracuse, On Floating Bodies (250 BC), is considered the first major work on fluid mechanics.
It studies fluid statics and buoyancy based on macroscopic observations.
Archimedes’ Principle is still widely used in shipbuilding and
submarines’ buoyancy.
Since then, many improvements have been made in the design of
ships, canals and flow systems, most of them based on empiric
observations and practice. Advances in flow analysis though really began
with the birth of Renaissance.
The essence behind the macroscopic approach for analysis lies
on the continuum hypothesis. It treats matter as a continuum medium that
completely fills the space it occupies. Thermodynamic properties like
specific mass7 ρ, pressure p and temperature T can be then defined as
continuous functions of space and time.
Such approach was assumed by Euler in his inviscid equations of
hydrodynamics and was later develop to a formal mathematical branch by
the French mathematician Augustin-Louis Cauchy in the 19th century
6 Ref. (LANDAU; LIFSHITZ; 1987), (MATTILA, 2010), (Wikipedia) 7 Density and specific mass will be considered synonyms along the text,
so that they will be used interchangeably.
44
which is called continuum mechanics. It deals with the mechanical
behavior of deformable media (continua), describing their kinematics,
and represents the basis upon which classical disciplines of fluid
dynamics and strength of materials are modeled.
By applying the physical conservation laws of mass, momentum
and energy to these kinematic models, differential equations arise, which
describe the transport behavior of those quantities in such a continuum.
This conservation laws might be applied to a system or a control volume. A system is a fixed quantity of mass that can be described by a
Lagrangian reference frame, also called material coordinates. A control
volume, however, is a defined and fixed region in the domain and is
described by an Eulerian reference frame. Given that is harder to track a
fixed quantity of mass in a fluid flow, the Eulerian reference frame is
almost always used in the description of fluid dynamics. Lagrangian
reference frame is mostly used in strength of materials, also known as
solid mechanics.
The link between these two descriptions is called Reynolds
Transport theorem.
2.1.1 Reynolds Transport Theorem8
Let Ψ(x,t) be an extensive tensor9 property with its respective
intensive tensor property ψ(𝐱, t) in a continuum body א. An extensive
property is that property which is proportional to the mass (or extension)
of the system, while an intensive property, is not.
In a system, it can be called with
Ψ(𝐱, t) = ∫ ψ(𝐱, t)𝜌(𝐱, t)
𝑣
𝑑𝑣10
8 Ref. (ARIS, 1989) 9 A Tensor is a generalization of vectors in a vector space. Considering
the Euclidian vector space ℝ3 with which we are working here, a n-th order tensor
will have 3𝑛 components. Therefore, a 0th order tensor is a scalar; a 1st order tensor
is a vector and so on… 10 Intensive properties are usually given per mass unit. E.g.: specific heat
𝑐𝑝 is an intensive property with dimensions of [kJ/(kg.K)]. The value of 𝑐𝑝 is only
applicable to points in a non-uniform field. By performing this integral one
recovers the extensive property with dimensions [kJ/K], which is the amount of
energy absorbed/released by the considered extension of the system (integrated
volume) by changing its temperature in one Kelvin.
45
Now, the theorem states that the material rate of an extensive
tensor property associate to a continuum body א is equal to the local rate
of such property in a control volume 𝑣 plus the efflux of the respective
intensive property across its control surface ∂ 𝑣, hence,
𝐷𝑡Ψ(𝐱, t) = 𝜕𝑡Ψ(𝐱, t) + ∫ ψ(𝐱, t)𝜌(𝐱, t)
∂𝑣
𝒖 ∙ 𝐧 𝑑𝑎,
where 𝒖 is the velocity vector, n the unit normal, 𝑑𝑎 a surface element
and 𝐷𝑡 the material derivative.
2.1.2 Transport Equations in Continuum Applying the Reynolds Transport Theorem with the conservation
laws of mass and momentum yields to the general conservation equations
of continuum bodies in an Eulerian reference frame:
𝜕𝑡𝜌 + 𝛁 ∙ (𝜌𝒖) = 0
𝜕𝑡(𝜌𝒖) + 𝛁 ∙ 𝚷 − 𝜌𝒈 = 0
where 𝜌, 𝒖, 𝒈,𝚷 are specific mass, velocity vector, body force and a
momentum flux tensor, respectively. These both equations can be written
in index notation as
𝜕𝑡𝜌 + 𝜕𝛼𝜌𝑢𝛼 = 0 (1)
𝜕𝑡𝜌𝑢𝛼 + 𝜕𝛽Π𝛼𝛽 − 𝜌𝑔𝛼 = 0 (2)
The result expressed in equation (1) is easy to catch. If mass
appears or disappears from a continuum body with time, it must have
crossed its boundaries. This is just the conservation of mass expressed
mathematically for a continuum differential element in a Cartesian
coordinate system.
Interpretation of equation (2) is identic. The crossing momentum through the boundaries is caused by the second term, which will be
explored next, plus the term of the body force, e.g. gravity, which acts in
the positive direction of the axis. Likewise, this is just the conservation
of momentum expressed mathematically for a continuum element.
46
The momentum flux tensor 𝚷 = Π𝛼𝛽 gives the 𝛼 component of the
momentum in the 𝛽 direction, remember that the 𝛼 component is
associated with the 𝛼 plane of the differential cube which is orthogonal to
the 𝛼 direction of the coordinate system. Since we know that momentum
is transferred through forces, the term 𝜕𝛽Π𝛼𝛽 is interpreted as the net
force, or stresses, on the element due to its neighbors.
So now, another good picture from equation (2) can be made: if
momentum is growing inside the continuum element (first term will be
positive in Eq. (2)), it must be caused by net force due to its neighbors
(the divergence in the second term of Eq. (2) will be negative) plus the
body force which acts in the positive directions of the coordinate system,
transferring momentum to the continuum element.
This tensor Π𝛼𝛽 is made out of two parts,
Π𝛼𝛽 = 𝜌𝑢𝛼𝑢𝛽 − Π𝛼𝛽𝑠𝑡𝑟 (3)
The first term on the right-hand side of equation (3) is the
convective momentum flux tensor and represents the 𝛼 component of the
momentum being transported in the 𝛽 direction. Second term of equation
(3) is the stress tensor of the continuum, which might be split11 into
contributions from two tensors, an isotropic mean pressure tensor −𝑝𝛿𝛼𝛽
and a deviatoric viscous stress tensor Π𝛼𝛽𝑣𝑖𝑠𝑐, hence
Π𝛼𝛽𝑠𝑡𝑟 = −𝑝𝛿𝛼𝛽 + Π𝛼𝛽
𝑣𝑖𝑠𝑐 (1)
Remember that the mean pressure tensor12 is related with
stretching and squeezing deformations, which maintain the edge angles
in the fluid element, while the deviatoric viscous stress tensor is related
with distortion of those angles. If incompressibility is not assumed, these
deformations may cause volume change, otherwise volume is kept
constant.
Now, using relations (3) and (4), equation (2) can be better
analyzed when written in the following form,
11 Any given second-rank tensor can be split into its isotropic and
deviatoric part. 12 Which is equivalent to the thermodynamic pressure given by an
equation of state.
47
𝜕𝑡𝜌𝑢𝛼 = −(𝜕𝛽𝜌𝑢𝛼𝑢𝛽 + 𝜕𝛽𝑝𝛿𝛼𝛽) + 𝜕𝛽Π𝛼𝛽𝑣𝑖𝑠𝑐 + 𝜌𝑔𝛼
which is nothing more than the expression of Newton’s Second Law in a
continuum body.
𝜕𝑡p𝛼 = Ӻ𝛼
where p𝛼 and Ӻ𝛼 is the momentum and force in the α direction,
respectively. The term in parenthesis is negative, because from one side
pressure acts on the opposite direction of its gradient (compressive) and
the divergence is negative for a positive momentum inflow in the
continuum element.
2.1.3 Newtonian fluids
It is important to note that absolutely nothing was said about
fluids. Until now only classical principles of mechanics was applied in
the mathematical abstraction of continuum media. To associate the
mathematical model with the actual behavior of matter some constitutive
relation is needed, which relates physical measurable properties between
each other. For some fluids, Sir. Isaac Newton discovered that the viscous
stress tensor is linearly proportional to the strain rate tensor, defined as
𝑆𝛾𝛿 =
(𝜕𝛿𝑢𝛾 + 𝜕𝛾𝑢𝛿)
2
(5)
hence,
Π𝛼𝛽𝑣𝑖𝑠 = Ӂ𝛼𝛽𝛾𝛿𝑆𝛾𝛿 (6)
where the fourth-rank tensor Ӂ𝛼𝛽𝛾𝛿 is some thermophysical property of
the fluid which represents the viscosity of the medium. Viscosity, thus,
characterizes how fluids react to strain rate and measures its internal
friction. Temperature has a strong effect and pressure a moderate if not
negligible effect on viscosity (WHITE, 2003). For isothermal flows,
therefore, Ӂ𝛼𝛽𝛾𝛿 is assumed to be made of constant coefficients whereas
pressure influence is neglected, a common approximation. Fluids which
can be good modeled with this assumptions are called Newtonian fluids.
Tensor Ӂ𝛼𝛽𝛾𝛿 has 81 components, a huge number for practical
purposes. For this reasons another simplification is made, restricting the
model only for isotropic media. That means that the thermophysical
48
property which relates the strain rate tensor to the stress tensor - viscosity
- is constant over all directions. Mathematically it means that the tensor
Ӂ𝛼𝛽𝛾𝛿 must be isotropic13. A fourth-rank isotropic tensor is given
generally as
Ӂ𝛼𝛽𝛾𝛿 = 𝐶1𝛿𝛼𝛽𝛿𝛾𝛿 + 𝐶2𝛿𝛼𝛾𝛿𝛽𝛿 + 𝐶3𝛿𝛼𝛿𝛿𝛽𝛾 (7)
where 𝐶1, 𝐶2 and 𝐶3 are constants. Putting relation (7) on equation (6) and
using (5) yields to
Π𝛼𝛽𝑣𝑖𝑠𝑐 = (𝐶2 + 𝐶3)𝑆𝛼𝛽 + 𝐶1𝛿𝛼𝛽𝑆𝛾𝛾
therefore,
Π𝛼𝛽𝑣𝑖𝑠𝑐 = 𝜇(𝜕𝛼𝑢𝛽 + 𝜕𝛽𝑢𝛼) + 𝜆𝛿𝛼𝛽𝜕𝛾𝑢𝛾 (8)
The divergent 𝜕𝛾𝑢𝛾 in equation (8) is related to the dilatation or
compression of the fluid element, so that coefficient 𝐶1 = 𝜆 is the a
compression viscosity coefficient, also called second viscosity coefficient.
Furthermore, (𝐶2 + 𝐶3)/2 = 𝜇 is the first viscosity coefficient, also
known as dynamic viscosity, which represents the shear stress coefficient
and is related to distortion.
The isotropy consideration made the 81 components of Ӂ𝛼𝛽𝛾𝛿
collapse in only two constants, i.e., 2𝜇 and 𝜆. It is easy to imagine that
any work with non-isotropic fluids, measuring its 81 components
experimentally, since these coefficients are empirical from a macroscopic
approach, must be an infernal task.
The final result is the general Navier-Stokes equation for
Newtonian, isotropic and compressible fluids.
𝜕𝑡𝜌𝑢𝛼 + 𝜕𝛽𝜌𝑢𝛼𝑢𝛽 = −𝜕𝛽𝑝𝛿𝛼𝛽 + 𝜇𝜕𝛽𝜕𝛽𝑢𝛼
+(𝜇 + 𝜆)𝜕𝛼𝜕𝛾𝑢𝛾 + 𝜌𝑔𝛼
13 An isotropic tensor is a tensor whose components are unchanged in all
Cartesian coordinate systems.
49
Or using a more familiar vector notation:
𝜕𝑡(𝜌𝒖) + 𝛁 ∙ (𝜌𝒖 ⊗ 𝒖) = −𝛁𝑝 + 𝜇∇2𝒖 + 𝜉 𝛁 ∙ 𝛁𝒖 + 𝜌𝒈14 (9)
where 𝜉 = (𝜇 + 𝜆). This equation can be written in many forms,
especially the convective term on the left side, which can be expressed as
𝛁 ∙ (𝜌𝒖 ⊗ 𝒖) = (𝛁𝜌𝒖) ∙ 𝒖 = (𝛁 ∙ 𝜌𝒖)𝒖
If we define the mean mechanical pressure as the negative one-
third of the sum of the three normal stresses of Π𝛼𝛽𝑠𝑡𝑟, i.e., the negative one-
third of the trace (a tensor invariant) of Π𝛼𝛽𝑠𝑡𝑟, yields to
=
−Π𝛼𝛼𝑠𝑡𝑟
3=
−(−𝑝𝛿𝛼𝛼 + 𝜇(𝜕𝛼𝑢𝛼 + 𝜕𝛼𝑢𝛼) + 𝜆𝛿𝛼𝛼𝜕𝛼𝑢𝛼)
3
which results in
= 𝑝 − (𝜆 +
2
3𝜇) 𝜕𝛼𝑢𝛼
(10)
The factor Ӄ = (𝜆 +2
3𝜇) is called the bulk viscosity coefficient,
although many textbooks mistakenly reserve this designation to 𝜆 itself.
It represents the mechanical dissipation related to any volume change at
finite rate, acting as a dumping factor of volumetric vibrations such as
sound absorption. Eq. (10) also reveals that unless either Ӄ or 𝜕𝛼𝑢𝛼 is
zero, the mean mechanical pressure in a deforming viscous fluid is not
equal to the thermodynamic one (MOHAMED, 1995). Moreover, Second
Law of Thermodynamics enforces this factor to be positive.
In 1845 Stokes simply assumed Ӄ = 0, which now is known as
Stokes’ hypothesis. This relation between the viscosity coefficients is
frequently used, but has not yet been definitely confirmed as a proper approximation.
14 The gradient operator 𝛁(∗) adds one dimension to the tensor.
The divergent operator 𝛁 ∙ (∗) subtracts one dimension from the tensor.
The laplacian operator ∇2(∗) = Δ(∗) is a scalar sum operation.
50
2.1.4 Incompressibility hypothesis
So far the equations derived here describe isotropic-newtonian
fluids. A further simplification is made with the incompressibility
hypothesis. Its validity is measured by the Mach number, which is a
measure of compressibility in the fluid flow. The Mach number is defined
as
𝑀𝑎 =
𝑈
𝑐𝑠
where U is the characteristic fluid flow velocity and 𝑐𝑠 the speed of sound
in the fluid medium. The speed of sound in a fluid is a thermodynamic
property and is given generally as
𝑐𝑠 = (
𝜕𝑝
𝜕𝜌 |
𝑠
)
1/2
meaning a derivative taken in an isentropic process. For liquids and solids
is common to define the bulk modulus of compressibility 𝐾𝑐
𝐾𝑐 = 𝜌
𝜕𝑝
𝜕𝜌 |
𝑠
𝐾𝑐 measures to resistance of the medium to uniform compression,
i.e. the pressure increase needed to a relative decrease in volume. Thus,
another relation for the sound speed can be given with
𝑐𝑠 = √𝐾𝑐
𝜌
For fluid flows with 𝑀𝑎 ≤ 0.3 density variations are negligible,
therefore, the incompressibility hypothesis for these flows becomes valid.
A statement of incompressibility, is mathematically equivalent to
𝜌 = 𝑐𝑡𝑒 → 𝜕𝑡𝜌 = 0. From Eq. (1) follows that with this consideration the divergent of velocity must vanish,
𝛁 ∙ 𝒖 = 0 (11)
51
This yields to the vanishing of the compressibility term in Eq. (9),
hence,
𝜕𝑡(𝜌𝒖) + 𝛁 ∙ (𝜌𝒖 ⊗ 𝒖) = −𝛁𝑝 + 𝜇∇2𝒖 + 𝜌𝒈
and dividing per 𝜌 yields to
𝜕𝑡𝒖 + (𝛁 ∙ 𝒖)𝒖 = −
1
𝜌𝛁𝑝 + 𝜈∇2𝒖 + 𝒈
(12)
Eq. (11) and Eq. (12) form the most widely used equations in fluid
dynamics for engineering purposes. They offer a set of four equations
solvable for four variables. Three components of velocity and pressure.
Kinematic viscosity 𝜈 = 𝜇/𝜌 and acceleration 𝒈 are input parameters.
Since there is no equation of state (EOS) for incompressible
substances, thermodynamic pressure 𝑝 cannot be defined. Pressure 𝑝 in
this case is interpreted as the mean mechanical pressure , as shown in
Eq. (10), and remembering that 𝜕𝛼𝑢𝛼 = 0 for incompressible fluids.
2.1.5 Dynamic similarity
It is useful to turn Eq. (12) dimensionless to see which
parameters remain in the equation. Defining a set of dimensionless
variables a
𝒓∗ =
𝒓
𝐿 𝒖∗ =
𝒖
𝑈 𝑡∗ = 𝑡
𝑈
𝐿 𝑝∗ =
𝑝
𝜌𝑈2 𝒇∗ = 𝒇
𝐿
𝑈2
where 𝐿 is a characteristic length of the flow, and 𝑈 a characteristic fluid
flow velocity. 𝐿 might be the diameter of a cylinder, the size of a wing or
any length that characterizes the flow under analysis. 𝑈 is normally taken
as the velocity of the free flow for external flows problems. Both are
considered constant. It is important to remember that also the derivatives
have to be turned dimensionless
𝜕𝛼 =𝜕
𝜕𝑟𝛼=
𝜕
𝜕𝑟𝛼∗𝐿
=1
L∇∗
52
Inserting the above dimensionless relations into Eq. (12) one
finds the incompressible NS equations in its most generic form
𝜕𝑡
∗𝒖∗ + (𝛁∗ ∙ 𝒖∗)𝒖∗ = −𝛁∗𝑝∗ +
1
𝑅𝑒∇∗
2𝒖∗ + 𝒇∗ ; 𝑅𝑒 =𝑈𝐿
𝜈
This result is of huge importance. It expresses that for a given
dimensionless body force, which might be null, the dynamic behavior of
velocity and pressure field will be identic for any flow with the same
Reynolds number. The Reynolds number is the only parameter which
really characterizes the evolution of the flow field. This enables
engineers, as an example, to know the flow conditions around an airplane,
by constructing a small model of it, and putting this model to the same
Reynolds regime that the real plane would have in the sky. This is the
power of dynamic similarity. Flows sharing the same geometrical
relations and Reynolds number are dynamically similar.
2.2 MESOSCOPIC MODELS
The mesoscopic scale lies between the atomic-molecular and the
macroscopic scale. It is a bridge that connects these two levels of
“reality”. The essence behind a mesoscopic model is the assumption that
by correct averaging the molecular behavior of matter, one should be able
to recover the macroscopic manifestations of it. Such averaging is made
with statistical tools specially developed to deal with it. It is today a whole
branch in theoretical physics and it is called statistical mechanics.
Obviously, statistical mechanics is based on the atomic theory of
matter, which although as old as at least ancient Greece, was not a widely
accept theory until the beginning of the 20th century.
Its derivation starts at the micro scale. Suppose a system of N
particles of mass 𝑚𝒊 in 3D space. Each particle could be tracked
instantaneously by its position 𝒓𝒊 and its momentum 𝐩𝐢 , 𝑖 = 1, 2,… , 𝑁.
By doing so, the state of the particle is well defined and if the state of all
N particles is defined so it is the state of the system. Note that in a
microscopic system, pressure, temperature and specific mass have no
meaning. The system is defined only based on the mechanical state of its
particles. Now, three components of position plus three components of
velocity, then multiplied by N particles. 6N is the total number of degrees
of freedom of that system. The instantaneous state of this system could be
then plotted in a phase space ℙ with 6N mutually orthogonal axis and a
53
parametric hyper dimensional line would appear as the system evolves in
time, following the basic laws of mechanics. So the phase space ℙ is a
space in which all possible states of a mechanic system are representable
and each state corresponds to a unique point represented by a vector q.
This dynamical system is called a Hamiltonian system since it can be
described by Hamilton’s equations15
𝜕𝑡𝐩 = −
𝜕ℋ
𝜕𝒓
𝜕𝑡𝐫 =
𝜕ℋ
𝜕𝒑
where ℋ = ℋ(𝒓, 𝐩, 𝑡) is the Hamiltonian16.
Now consider Ç replicates of that system of N particles. They are
all macroscopically equivalent between each other but microscopically
different. One could think in this, for analogy, as Ç hourglasses with N
sand grains each. They all have the same macroscopic property of flowing
all the sand down in a determined time, although those N sand grains are
not identically flowing in same speeds in each hourglass. In other words,
for every macroscopic system S there will be a set of Ç distinct
microscopic systems which are all macroscopically equivalent.
Plotting those Ç replicates as points in the phase space ℙ, these
points become dense enough as Ç→∞ to enable the definition of a
continuous function, called the normalized probability density function F(q,t), where t is time and q is the position vector of a point in the phase
space ℙ, i.e., q has 6N components, 3 components of 𝒓𝒊 and 3 from 𝐩𝐢 for
each particle i=1,2,…,N. Hence, F(q,t)dq represents the expected fraction
of the total Ç replicates points which lies inside the volume dq around
point q in the phase space ℙ. In other words, F(q,t)dq represents the
probability of a macroscopic system be in a specific microscopic state
interval.
Remember that N is normally a huge number. If the system is, for
example, 18 grams of water, there would be 𝑁𝐴 ≅ 6.0221422 x 1023
water molecules in the system, which is the Avogadro’s number. It is just
15 A mathematical formalism in analytical mechanics. 16 It corresponds to the total energy of the system. For a closed system,
the Hamiltonian is the sum of the kinetic and potential energy.
54
an insane number for any practical purpose. If someone write down the
whole sequence of coordinates 𝒓𝒊 𝐩𝐢 for each particle from 1 to 𝑁𝐴 in a
writing rate of six coordinates per second, it will be needed more than a
quadrillion years to finish the job. By the way, the age of the universe is
estimated in 14 billion years. This person would need almost a million
reincarnations of the current universe to accomplish this herculean task.
Facing this impossibility to track such a system considering each
individual particle it is natural to put the first efforts to treat a new
problem with the most basic physical situation. This situation is an ideal
gas in thermodynamic equilibrium.
2.2.1 Ideal gases (pressure and temperature)
Imagine a flying particle in a cubic container of side L, colliding
elastically (no dissipation of kinetic energy) against its inner surfaces with
velocity c and flying over just one dimension α. The force Ӻ exerted on
the walls is given by the derivative of momentum: 𝜕𝑡p𝛼 = Ӻ𝛼 . Integrating
it over the time interval ∆t between two collisions, Ӻ∆𝑡 = 𝑚𝑐𝛼—𝑚𝑐𝛼 =2𝑚𝑐𝛼, where m is the mass of the particle. Now, time between hits is
2L/𝑐𝛼. Hence, Ӻ = 𝑚𝑐𝛼2/𝐿 and the force that N particles would exert in
that box would be Ӻ = 𝑁𝑚(𝑐𝛼12 + 𝑐𝛼2
2 + …+ 𝑐𝛼𝑁2)/𝐿.
It is much easier to take the mean square velocity 𝑐𝛼 2of the
particles then the individual velocity of every particle. It can be then
generalized to 3D thinking that since there are a huge number of particles,
the relation 𝑐2= 𝑐𝛼
2 + 𝑐2 + 𝑐
2 → 𝑐2= 3 𝑐𝛼
2 is a good hypothesis.
The total force exerted on the walls of a container from N particles in
space is therefore Ӻ = 𝑁𝑚𝑐2/3𝐿. Pressure p is Ӻ/𝐴, where A is the inner
surface area of the walls. But the product LA is the volume V of the
container, thus,
𝑝 =
𝑚𝑐2
2
2𝑁
3𝑉
This equation links pressure, which is a macroscopic property
with kinetic energy of constituent particles. Defining temperature T as the
macroscopic manifestation of thermal motion, which is proportional to
the kinetic energy 𝐾 and comparing with the widely known empiric
equation of state for ideal gases
𝑝𝑉 = 𝑛𝑅𝑇 (13)
55
which describes with precision gases far from its critical point, yields
𝑛𝑅𝑇 =𝑚𝑐
2
2
2
3𝑛𝑁𝐴, where n is the number of moles of the gas and R the
ideal gas constant. The Boltzmann constant can be then defined as, 𝑘𝑏 =𝑅/𝑁𝐴 and the 𝐾 given per particle is
𝐾 =
𝑚𝑐2
2=
3
2𝑘𝑏𝑇
So now there is a microscopic correspondent to the macroscopic
ideal gas law, namely,
𝑝𝑉 = 𝑁𝑘𝑏𝑇 (14)
The Boltzmann constant 𝑘𝑏 = 1.38065 x 10−23J/K is therefore
the microscopic equivalent of the ideal gas constant 𝑅 = 8.314472 J/(K.mol) and relates energy at the particle level with temperature at the
bulk level, having the same units as entropy. They are interrelated with
𝑘𝑏 = 𝑅/𝑁𝐴 .
Relation (9) can be rewritten in a very illuminative form,
𝑝 =
𝑅
𝑀𝜌𝑇
(15)
where M is the molar mass of the constituent particles.
Eq. (15) relates explicitly pressure, density 𝜌 = 𝑛𝑀/𝑉 and
temperature. Remember that 𝑛𝑀 is the total mass of the system.
The ideal gas law whether in its macro- (Eq. (13)) or microscopic
(Eq. (14)) form is just the mathematical manifestation of the ideal gas model, being a constitutive equation which provides a relation between
the state variables.
2.2.2 The equilibrium distribution17
Consider the normalized probability density function F for an
ideal gas in thermal equilibrium inside a container. If it is in
thermodynamic equilibrium, the function F does not depend on time t as
long as no disturbance is performed on the gas. Likewise, the function
17 Also called maxwellian or maxwell-boltzmann distribution.
56
does not depend on space neither, since in equilibrium a gas fills the space
it occupies uniformly. That means that the macroscopic system will have
no preference to any microscopic configuration so that statistically, the
function F is constant throughout the space, depending only on
momentum. Hence, F(q,t)= F(r,p,t) → F(p).
If it is assumed that the mass of each particle is identic and does
not change in time F(p) →F(c), where c is the velocity vector of each
point in a molecular velocity space 𝕧. Now, two hypothesis are important about the F(c) function. They
were made by Maxwell in his first derivation of the equilibrium solution,
(Maxwell, 1860).
1. It is isotropic.
2. The velocities of a particle in orthogonal direction are
uncorrelated (it depends only on the modulus).
So, integrating over the whole 3D space and over all the possible
velocities, i.e., ∭−∞
∞𝑁𝐹(𝑐𝛼)𝐹(𝑐𝛽)𝐹(𝑐𝛾)𝑑𝑐𝛼𝑑𝑐𝛽𝑑𝑐𝛾 yields the number
of particles in the system, N. Hence,
∭−∞
∞𝐹(𝑐𝛼)𝐹(𝑐𝛽)𝐹(𝑐𝛾)𝑑𝑐𝛼𝑑𝑐𝛽𝑑𝑐𝛾 = 1
Since directions are arbitrary, function F should not depend on
direction but only on the distance of the origin, which is equivalent to
depend on the square of the particle speed (modulus). This leads to,
𝐹(𝑐𝛼)𝐹(𝑐𝛽)𝐹(𝑐𝛾) = 𝛷(𝑐𝛼² + 𝑐𝛽² + 𝑐𝛾²) (16)
where Φ is a positive unknown function which represents the combined
probability in the three directions of space. The mathematical behavior
shown in Eq. (16) is found in logarithmic and exponential functions, but
only a special exponential function can satisfy the two hypothesis given.
It is a Gaussian function of the following form
𝐹(𝑐) = 𝐴3𝑒−𝐵𝑐2
The probability of particles lying in a spherical shell with velocities
between c + dc will be then 𝐹(𝑐)𝑑𝑐 = 4𝜋𝑐²𝐴3𝑒−𝐵𝑐2𝑑𝑐.
57
Constants A and B are found by integrating F over all possible
speeds to find the total number of particles N and their energy E, yielding
to
𝐹𝑐(𝑐) = 4𝜋𝑐2 (
𝑚
2𝜋𝑘𝑏𝑇)3/2
𝑒−
𝑚𝑐2
2𝑘𝑏𝑇 (17)
This probability density function gives the probability of finding
a particle within an infinitesimal spherical shell of radius c. This shell
represents the equidistant particles from the origin, i.e., the particles that
have the same velocity modulus. Therefore, this is the distribution used
to represent molecular velocities of an ideal gas. Distribution (17)
increases parabolically from zero for low speeds, reaching its maximum
and then decreases exponentially. When temperature T increase, the curve
is shifted to the right, meaning a distribution with higher velocities. An
example of such a distribution is given in Fig. 3 for common gases at 300
Kelvin. While high temperatures tend to shifts the curve to the right,
making the exponential decrease slower, the mass of each molecule tend
to shift the curve to the left, speeding up the exponential decrease. These
are the two competing parameters that shape the equilibrium distribution
for a given gas.
Dividing Eq. (17) by the surface of the sphere yields to
𝐹(𝑐) = (
𝑚
2𝜋𝑘𝑏𝑇)3/2
𝑒−
𝑚𝑐2
2𝑘𝑏𝑇
which is the probability of finding a particle inside an infinitesimal
Cartesian element dc with speeds around c.
If the ideal gas has a bulk velocity 𝑢 relative to a reference frame,
a peculiar velocity is defined as 𝑣 = 𝑐 − 𝑢, which is the velocity of a
particle with respect to the flow.
Now let 𝑓(𝑐)𝑒𝑞 = 𝑚𝑁𝐹(𝑐) be the mass of particles expected to be
found in the element dc, around c. Since 𝐹(𝑐) does not depend on r, this
also represents the mass of particles expected in a dcdr volume in phase
space, thus mN is equivalent to the specific mass 𝜌. Therefore,
𝑓(𝒄)𝑒𝑞 = 𝜌 (
𝑚
2𝜋𝑘𝑏𝑇)3/2
𝑒−𝑚(𝒄−𝒖)2
2𝑘𝑏𝑇 (18)
58
Figure 4 – Common gases at 300K.
Source: Author
An ideal gas in thermodynamical equilibrium is governed by Dist.
(18). This distribution makes the pressure tensor hydrostatic and the
energy flux vanishes. The corresponding fields of internal energy, density
and pressure are everywhere constant. (Truesdell, Muncaster, 1980)
The equilibrium distribution was first derived by Maxwell in 1860
with some abstract arguments, namely the two hypothesis assumed for the
distribution function’s nature. In 1867 he extended the analysis trying to
justify his arguments. He used the consideration that the distribution
should be stationary at thermodynamical equilibrium, i.e. it should not
change its shape as a result of the continual collisions between the
particles. This would involve a more sophisticated analysis on the nature
of collision but it was still an essentially mathematical analysis, with no
persuasive physical arguments to fundament why atoms should behave
like that (MAXWELL, 1860, 1867; LINDLEY, 2001; UFFINK, 2014).
In 1868 Boltzmann verified Maxwell’s arguments in a variety of
models, including gases in a static external force field. He then replaced
the logic-mathematical assumptions made by Maxwell with a more
59
physical set of arguments, deriving the same distribution from the ergodic
hypothesis18, using a Hamiltonian system. These results suggests that the
distribution of the molecular velocities for an isolated mechanical system
in a stationary state will always tend to the maxwellian distribution as the
number of particles approaches infinity. They dispense any assumptions
about the collisions or the state of matter. They had the power of
generality. (LINDLEY, 2001; UFFINK, 2014).
For all these reasons, Dist. (18) is also called the Maxwell-Boltzmann distribution.
2.2.3 The local equilibrium distribution
If not further mentioned, a stationary system is in a global
thermodynamic equilibrium, which is a state where all intensive
properties are homogeneous throughout the system vanishing all kind of
macroscopic fluxes. A local thermodynamical equilibrium however,
means that intensive properties do vary with space, but so slowly that in
the neighborhood of any given point it can be assumed thermodynamic
equilibrium and thus, a local equilibrium distribution can be defined just
replacing the constants of Dist. (18) by its counterpart functions of space
and time. Therefore,
𝑓(𝐫, 𝒄, 𝑡)𝑒𝑞 = 𝜌(𝑟, 𝑡) (
𝑚
2𝜋𝑘𝑏𝑇(𝑟, 𝑡))3/2
𝑒−𝑚(𝒄−𝒖(𝑟,𝑡))2
2𝑘𝑏𝑇(𝑟,𝑡) (19)
2.2.4 The Boltzmann equation (BE)
Let 𝑓(𝐫, 𝒄, 𝑡)∆r∆c = f(𝐪, t)∆𝐪 be the number of particles at time
𝑡 expected to be found in the hypercube ∆𝐪 in the phase space 𝔹. They
have coordinates between (𝐪 + ∆𝐪) = (𝐫 + ∆𝐫, 𝒄 + ∆𝐜). A Taylor
expansion can be used to infer the expected value of 𝑓 for a small
displacement of its variables, hence
𝑓(𝐫 + ∆𝐫, 𝒄 + ∆𝐜, 𝑡 + ∆t) = 𝑓(𝐫, 𝒄, 𝑡) (20)
18 By the Hamiltonian equations of motion, a point in phase space
representing a system evolves in time, and thus describes a trajectory 𝒒𝑡. This
trajectory is constrained to lie on a given energy hypersurface ℋ(𝒒𝑡) = 𝐸, where
ℋ denotes the Hamiltonian function and E the associated energy. Now, the
ergodic hypothesis states that all dynamic states associated to the energy E are
equiprobable, implying that a dynamical system will pass over all these states in
sufficient long periods of time.
60
+𝜕𝑓
𝜕𝒓∆𝐫 +
𝜕𝑓
𝜕𝒄∆𝐜 +
𝜕𝑓
𝜕𝑡∆t + 𝒪2
where 𝒪2 represents the higher order terms. Rearranging Eq. (20) and
dividing by ∆t,
𝑓(𝐫 + ∆𝐫, 𝒄 + ∆𝐜, 𝑡 + ∆t) − 𝑓(𝐫, 𝒄, 𝑡)
∆t
=𝜕𝑓
𝜕𝒓
∆𝐫
∆t+
𝜕𝑓
𝜕𝒄
∆𝐜
∆t+
𝜕𝑓
𝜕𝑡
∆t
∆t+
𝒪2
∆t
Then, taking the limit when ∆𝐭 0 yields to
𝐷𝑓
𝐷𝑡=
𝜕𝑓
𝜕𝑡+ 𝐜
𝜕𝑓
𝜕𝒓+ 𝐠
𝜕𝑓
𝜕𝒄 ,
(21)
which is just the convective derivative19 of 𝑓 derived from a Taylor
expansion.
But if 𝑓 is being transported from(𝐪, t) to (𝐪 + ∆𝐪, 𝑡 + ∆t), the
convective derivative is zero, because it represents f traveling in a
hypercube through the phase space 𝔹 . Changes inside this hypercube is
zero except if collisions occur.
If collisions occur, it shall be then represented by a collision
operator Ω , hence
𝜕𝑓
𝜕𝑡+ 𝐜
𝜕𝑓
𝜕𝒓+ 𝐠
𝜕𝑓
𝜕𝒄= Ω
(21)
where 𝐠 is an acceleration due to external forces acting on the particles.
The collision process can bring in, or expel out particles from the
infinitesimal hypercube. Particles in the spatial coordinates’ interval (𝐫 +
∆𝐫) may acquire, or loose the velocities in the range (𝒄 + ∆𝐜) during time
interval ∆t. In Boltzmann’s model, particles behave like hard spheres colliding
elastically. Because particles (material points) do not have dimensions,
19 Also called material, substantial and total derivative. This is a more
generalized definition, whereas in fluid mechanics textbooks the third right-hand
term of Eq. (21) is suppressed from the definition, because 𝑓 in the NS equation
is the velocity vector and this derivative vanishes.
61
they are modelled as points with a force field around them. Furthermore,
his model assumes some premises:
i. Binary collisions: the gas is sufficiently dilute, so that the
particles take most of time travelling in a straight line and eventually
encounters another particle (collision). Three or more particle-collision
would be so improbable, that they would not affect results.
ii. Conservation laws: collisions conserve momentum, mass and
kinetic energy (elastic collisions).
iii. Molecular chaos20: velocities of two particles about to collide
are uncorrelated and independent of position.
From the reference frame of a target particle with distribution 𝑓′
before collision, a bullet particle with distribution 𝑓∗′ comes about to
collide. The relative velocity before collision is given by ‖𝒄′∗ − 𝒄′‖, and [𝑓∗, 𝑓] are the distribution after collision for bullet and target particles,
which acquire velocities [𝒄∗, 𝒄], respectively.
The net balance inside the hypercube dq due to collisions of these
particles is described by the collision operator Ω, given by Boltzmann as
Ω = ∭‖𝒄′∗ − 𝒄′‖(𝑓∗
′𝑓′ − 𝑓∗𝑓)𝑠𝑑𝑠𝑑𝜖𝒅𝒄 (22)
where 𝑠𝑑𝑠𝑑𝜖 are scattering parameters.21
The revolutionary BE is then:
𝜕𝑓
𝜕𝑡+ 𝐜
𝜕𝑓
𝜕𝒓+ 𝐠
𝜕𝑓
𝜕𝒄= ∭‖𝒄′∗ − 𝒄′‖(𝑓∗
′𝑓′ − 𝑓∗𝑓)𝑠𝑑𝑠𝑑𝜖𝒅𝒄 (23)
or in equivalent notation
𝜕𝑡𝑓 + 𝒄 ∙ ∇𝒓𝑓 + 𝐠 ∙ ∇𝐜𝑓 = Ω,
20 Also known as Stoβzahlansatz (SZA). It is an important assumption,
because it introduces time asymmetry in the modelling, as stated by the
Loschmidt’s paradox in 1874, in which it should not be possible to derive an
irreversible model from time-symmetric dynamics. 21 Details on the derivation of the Boltzmann equation can be found in
Ref. (PHILIPPI; BOLTZMANN, 1896, 1872; LIBOFF, 2003; CERCIGNANI,
1988)
62
The left-hand side of (23) is the convective derivative of 𝑓,i.e., a
linear transport operator. The right-hand side is the non-linear collision
operator.
The BE was first published in 1872 and forms the basis for the
kinetic theory22 of gases (BOLTZMANN, 1872). It belongs to the
fundamental equations of physics since its birth and although derived for
the context of dilute gases, its validity stretches from transport processes
and hydrodynamics all the way to cosmology applications (ALEXEEV,
2014; SUCCI, 2001).
In equilibrium conditions, the right-hand side of Eq. (23) must
vanish, otherwise 𝑓 would vary with time. This implies 𝑓∗′𝑓′ = 𝑓∗𝑓, which
also yields to the Maxwell-Boltzmann distribution 𝑓(𝒄)𝑒𝑞.
An important result derived from the BE is called the H-theorem.
Assuming that Eq. (23) is valid for all times, it is possible to define a
function
𝐻(𝑓) = ∫𝑓 𝑙𝑛𝑓 𝒅𝒄
and demonstrate that this function always decrease over time, except
when 𝑓 = 𝑓(𝒄)𝑒𝑞, satisfying the relation
𝜕𝐻(𝑓)
𝜕𝑡≤ 0
The H-theorem is interpreted as the molecular counterpart of the
Second Law of thermodynamics, in which a property called entropy can
only increase over time for closed systems. This theorem is the result of
the time asymmetry introduced by the assumption of molecular chaos,
indicating irreversibility in time, which is also a property of the
thermodynamic entropy.
The BE can be discretized in a finite set of velocities Φ𝑛 =𝒄0, 𝒄1, 𝒄2, … , 𝒄𝑛−1 to cut its dependency on the three velocity variables,
yielding to the discrete Boltzmann equation
22 “Kinetic theory is the branch of statistical physics dealing with
dynamics of non-equilibrium processes and their relaxation to thermodynamic
equilibrium.” (SUCCI, 2001, p. 3)
63
𝜕𝑓𝑖𝜕𝑡
+ 𝐜𝑖
𝜕𝑓𝑖𝜕𝒓
+ 𝐠𝜕𝑓𝑖𝜕𝒄
= Ω𝑖(𝑓 ) , 𝑖 = 0,1,… , 𝑛 − 1.
Remember, velocities range from negative to positive infinity.
Now 𝑓𝑖 = 𝑓(𝒓, 𝐜𝑖 , 𝑡) and the vector 𝑓 = (𝑓0, 𝑓1, … , 𝑓𝑛−1). The final result
is that the discrete BE simplified Eq. (23) by turning its dependency on
seven variables of 𝑓 in set of 𝑛 coupled equations for the unknown
function 𝑓𝑖, which depends on four variables instead of seven. It is also
important to note that the integrals in the collision operator was
substituted by summations in its discrete counterpart, representing a
substantial simplification in the most complex term of the BE.
2.2.5 Macroscopic equations recovery
Moments of a function is a common mathematical concept used
over continuous functions, especially for probability density functions.
The 𝑛𝑡ℎ moment of a real-valued continuous function ℎ(𝑥) about 𝑥0 is
defined as:
𝜇𝑛 = ∫ (𝑥 − 𝑥0)
𝑛ℎ(x)𝑑𝑥∞
−∞
If 𝑥0=0 and ℎ(𝑥) is a probability density function, the 𝑛𝑡ℎ
moment represents the most probable value of 𝑥𝑛, or its expectancy.
Since distribution 𝑓(𝐫, 𝒄, 𝑡) is a function with physical meaning, its
moments over the velocities 𝒄 are interpreted as the hydrodynamic
functions of the continuum description. So, the zeroth moment of 𝑓 is just
its integration over all possible velocities, giving rise to a function that
represents the expected mass in a differential element about 𝐫 at time 𝑡,
corresponding to the macroscopic specific mass function 𝜌(𝐫, 𝑡),
therefore,
𝜌(𝐫, 𝑡) = ∫ 𝑓(𝐫, 𝒄, 𝑡)𝒅𝒄
∞
−∞
(24)
The first moment recovers the momentum density function, from
where the macroscopic velocity 𝒖(𝐫, 𝑡) can be determined:
64
𝜌(𝐫, 𝑡)𝒖(𝐫, 𝑡) = ∫ 𝒄𝑓(𝐫, 𝒄, 𝑡)𝒅𝒄
∞
−∞
(25)
Moments of any order can be defined arbitrarily,
Ϣ𝑐𝛼𝑐𝛽 ...𝑐𝜔
= ∫ 𝑐𝛼𝑐𝛽 . . . 𝑐𝜔𝑓(𝐫, 𝒄, 𝑡)𝒅𝒄∞
−∞
but only low order moments have physical meaning. Second order
moments are related to momentum transfer and energy. There are two of
them. The first one is a scalar second moment, which recovers from the
physical underworld, the kinetic energy associated with the particles’
speeds. The second one, is the second-rank momentum flux tensor Π𝛼𝛽 . A monoatomic gas has only energy associated to the translational
motion, thus, the total energy of a particle is its kinetic energy
K = mc²/2 , therefore,
𝜌(𝐫, 𝑡)ℯ(𝐫, 𝑡) =
1
2∫ 𝑐𝛼𝑐𝛼 𝑓(𝐫, 𝒄, 𝑡)𝒅𝒄
∞
−∞
is the energy density in a monoatomic gas, remembering that 𝑓 = 𝑚𝑁𝐹. Substituting the peculiar velocity 𝑣𝛼 = 𝑐𝛼 − 𝑢𝛼 in the equation
above yields to
𝜌(𝐫, 𝑡)ℯ(𝐫, 𝑡) =
1
2∫ 𝑣𝛼𝑣𝛼 𝑓(𝐫, 𝒄, 𝑡)𝒅𝒄
∞
−∞
+1
2𝜌(𝐫, 𝑡)(𝑢(𝐫, 𝑡))
2
(
(26)
where the second right-hand term is associated with the macroscopic
kinetic energy and the equation can be expressed as (suppressing the
arguments):
𝜌ℯ = 𝜌ℯ𝑝 +
1
2𝜌𝑢2
The term 𝜌ℯ𝑝 is called the mean peculiar kinetic energy of the
particles. This energy is associated with the molecular motion of the
65
particles, being independent from macroscopic velocity 𝑢 and vanishes
only at absolute zero. It is proportional to the mean kinetic energy per
particle, thus, this energy sums to
𝜌ℯ𝑝 = 𝑛 (
3
2𝑘𝑏𝑇)
where the number density 𝑛 is given by
𝑛(𝐫, 𝑡) =
1
𝑚∫ 𝑓(𝐫, 𝒄, 𝑡)𝒅𝒄
∞
−∞
The momentum flux tensor Π𝛼𝛽 can be recovered by relation
Π𝛼𝛽 = ∫ 𝑐𝛼𝑐𝛽 𝑓(𝐫, 𝒄, 𝑡)𝒅𝒄
∞
−∞
And inserting the peculiar velocity,
Π𝛼𝛽 = 𝜌𝑢𝛼𝑢𝛽 + ∫ 𝑣𝛼𝑣𝛽 𝑓(𝐫, 𝒄, 𝑡)𝒅𝒄
∞
−∞
= 𝜌𝑢𝛼𝑢𝛽 − Π𝛼𝛽𝑠𝑡𝑟
which is just exactly the result derived in the macroscopic approach.
As before, the mechanical pressure can be defined as =−Π𝛼𝛼
𝑠𝑡𝑟
3
(see page 47), so (𝐫, 𝑡) =1
3∫ 𝑣𝛼𝑣𝛼 𝑓(𝐫, 𝒄, 𝑡)𝒅𝒄
∞
−∞. For ideal gases =
𝑝 and since from Eq. (26) 1
2∫ 𝑣𝛼𝑣𝛼 𝑓(𝐫, 𝒄, 𝑡)𝒅𝒄
∞
−∞= 𝜌ℯ𝑝, it yields to
𝑝 =
2
3𝜌ℯ𝑝 = 𝑛𝑘𝑏𝑇
and integrating over a volume 𝑉 = ∆𝒓 where intensive properties are held
constant
∫ 𝑝𝒅𝒓
𝑉
= ∫ 𝑛(𝐫, 𝑡)𝑘𝑏𝑇𝒅𝒓
𝑉
→→→ 𝑝𝑉 = 𝑁𝑘𝑏𝑇
66
which is just the microscopic ideal gas law Eq. (14)!
The third order moment is a vector, and represents the transport, or
flux of the kinetic energy.
𝔔𝛽 =
1
2∫ 𝑐𝛼𝑐𝛼𝑐𝛽 𝑓(𝐫, 𝒄, 𝑡)𝒅𝒄
∞
−∞
2.2.6 The BGK kinetic model Examining Eq. (23) for a possible solution, one finds that it
actually deals with a non-linear partial stochastic integro-differential
equation! Now, that sounds really scary, and it is. Solving the Boltzmann
equation for any application is a challenging task. It includes
• Solving the seven fold, space (six) + time (one), nonlinear
integro-differential equation given by Eq. (23).
• Computing the collision operator (22) by performing its integrals.
• Computing specific mass and momentum given by Eq. (24) and
Eq. (25).
So, it becomes natural trying to simplify this equation while
retaining the essential physics in it. Most of this complexity is due to the
non-linear collision operator Ω(𝑓, 𝑓), which is replaced by simpler linear
ones, forming approximating models to the BE, called kinetic models.
Whatever might be the proposed simplifying collision operator, it
must satisfy three basic properties of the continuous, original one. It might
be denoted as Щ(𝑓) and the three properties are:
i. Щ(𝑓𝑒𝑞) = 0
ii. ∫Щ(𝑓)𝒅𝒄 = ∫c𝛼Щ(𝑓)𝒅𝒄 = ∫c²Щ(𝑓)𝒅𝒄 = 0
iii. ∫log (𝑓) Щ(𝑓)𝒅𝒄 ≤ 0
67
meaning: it vanishes for a Maxwell-Boltzmann distribution 𝑓𝑒𝑞. It must
conserve mass, momentum and kinetic energy and, it must satisfy the H-
theorem, and vanishes only if 𝑓 = 𝑓𝑒𝑞.
Property iii expresses a tendency of the gas to the equilibrium
distribution 𝑓𝑒𝑞. This feature can be taken into account considering that
the average effect of collisions on f will be proportional to the departure
of f from 𝑓𝑒𝑞,i.e., the collisions will relax f toward 𝑓𝑒𝑞 proportionally to
a collision frequency parameter 𝜔23. So, if the collision frequency 𝜔 is
constant relative to velocity 𝒄, the BGK model is introduced as
Щ𝐵𝐺𝐾(𝑓) = 𝜔(𝑓𝑒𝑞(𝒄) − 𝑓(𝒄)) (27)
The main advantage of BGK collision model is the ability to give
integral equations for macroscopic variables 𝜌, 𝒖, 𝑇, which, although non-
linear, are easily solvable in fast computers. On the other hand, BGK’s
non-linearity is much worse than that of the original operator. While the
latter is only quadratic in 𝑓, the former contains 𝑓 in the numerator and
denominator of an exponential (u and T in 𝑓𝑒𝑞 are functions of 𝑓). (CERCIGNANI, 1988). The final kinetic model is then:
𝜕𝑡𝑓 + 𝒄 ∙ ∇𝒓𝑓 + 𝐠 ∙ ∇𝐜𝑓 = 𝜔(𝑓𝑒𝑞(𝒄) − 𝑓(𝒄)) (28)
The purpose of a kinetic model is not to solve the full BE but to
retrieve the macroscopic equations which describe the physical behavior
of a system, since it is the primary goal of most of applications.
One important aspect of the BGK model is a result derived from
property ii. This might be formulated as
∫ Φ Щ(𝑓)𝒅𝒄 = 0
(29)
where the vector Φ represents the collisional invariants ,i.e., the
quantities which must be conserved during collisions, given as
23 A correspondent parameter is often used in LBM literature; 𝜏 = 1/𝜔,
where 𝜏 is interpreted as the mean free flight time between collisions.
(MATTILA, 2010)
68
Φ = [
1𝒄𝒄²
]
Replacing the BGK operator Eq. (27) into Eq. (29) yields to the
following result: (BRESOLIN, 2012)
𝜌(𝐫, 𝑡) = ∫ 𝑓(𝐫, 𝒄, 𝑡)𝒅𝒄∞
−∞
= ∫ 𝑓𝑒𝑞(𝐫, 𝒄, 𝑡)𝒅𝒄∞
−∞
𝜌(𝐫, 𝑡)𝒖(𝐫, 𝑡) = ∫ 𝒄𝑓(𝐫, 𝒄, 𝑡)𝒅𝒄∞
−∞
= ∫ 𝒄𝑓𝑒𝑞(𝐫, 𝒄, 𝑡)𝒅𝒄∞
−∞
𝜌ℯ𝑝 =1
2∫ 𝑣𝛼𝑣𝛼 𝑓(𝐫, 𝒄, 𝑡)𝒅𝒄
∞
−∞
=1
2∫ 𝑣𝛼𝑣𝛼 𝑓
𝑒𝑞(𝐫, 𝒄, 𝑡)𝒅𝒄∞
−∞
remembering that the peculiar velocity 𝑣𝛼 = 𝑐𝛼 − 𝑢𝛼 is the velocity of
the particle in a reference frame that moves with the fluid macroscopic
velocity and 𝜌ℯ𝑝 the mean peculiar kinetic energy.
This is an important advantage in the BGK model, since the
hydrodynamic equation can be recovered from the moments of the
equilibrium distribution 𝑓𝑒𝑞, which is a well known function.
69
We adore chaos because we love to produce order.
M. C. Escher*
70
* 24
24 Maurits Cornelius Escher (1898-1972) was a Dutch graphic artist,
known for his mathematically inspired works, exploring impossible
constructions, infinity and symmetry.
Circle Limit - Woodcut in black and ocre. Escher, 1960.
71
3 THE LATTICE-BOLTZMANN METHOD
Discrete simulations for fluid flow really began to advance in the
80’s with a modified version of Lattice-Gas Automaton (LGA)25 for the
Navier-Stokes equation. Limitations of this Boolean26 model were soon
recognized and the Boolean variables were further replaced by real ones.
This idea, introduced by McNamara and Zanetti in 1988, is widely
considered as the birth of LBM. Since then, scientific research grew
rapidly around this method, which is now a prolific theme with an
uncountable number of models, variations of those models, and
applications of them.
The Lattice-Boltzmann method (LBM) is the general name of the
discrete models based on the Lattice-Boltzmann equation (LBE), given as
𝑓𝑖(𝒓 + ∆𝑡𝒄𝒊, 𝑡 + ∆𝑡) = 𝑓𝑖(𝒓, 𝑡) − Ω𝑖(𝒓, 𝑡)
where the subscript 𝑖 means the set of prescribed velocities chosen to
represent the whole velocity space. The LBE can be seen as the fully
discrete counterpart of the BE. Remember that in the discrete BE only the
velocity space has been discretized. In the LBE, both space and time are
also discretized, yielding to a fully discrete scheme.
It is important to stress that standard LBE methods, or LBM, are
weakly compressible approximations of the incompressible NS equations,
intended to recover its macroscopic physics with the mesoscopic
approach (SUCCI, 2001).
Numerically, the method consists in a stream-collide algorithm,
which can be detailed in steps by the following scheme:
1. Set up distributions in the domain for the initial time
accordingly to the initial conditions.
2. Compute collision term Ω𝑖(𝒓, 𝑡) for every point in the
domain.
3. Compute the new distributions for every point.
4. Stream the distributions.
5. Apply boundary conditions.
6. Return to step 2.
25 A more detailed description of this LGA model and its evolution to
LBM is given by (SUCCI, 2001). 26 In Boolean logic variables have only two possible states, often
represented as true/false, or, 0/1.
72
The loop that represents steps (2-3-4-5-6-2) can also be done with
steps (4-5-6-2-3-4), where the initial distributions are considered to be
given after a full colliding step.
3.1 LBGK SCHEMES
A further step for LBM occurred in 1991 and 1992, when three
different parties proposed the replacement of the collision term as a
simple relaxation process involving a single parameter. (MATTILA,
2010).
Ω𝑖(𝒓, 𝑡) = 𝜔(𝑓𝑖(𝒓, 𝑡) − 𝑓𝑖𝑒𝑞
(𝒓, 𝑡))
These models were called LBGK and can be seen as a discrete
counterpart of the BGK kinetic model Eq. (28). A direct derivation from
the BE has been first stablished by HE; LUO (1997), who demonstrated
that LBM can be seen as a finite-difference approximation of the BE. A
systematically discretization approach from the BGK kinetic model to
higher order schemes is shown by PHILIPPI et al. (2006) and SHAN;
YUAN; CHEN (2006).
The LBGK general representation is therefore
𝑓𝑖(𝒓 + ∆𝑡𝒄𝒊, 𝑡 + ∆𝑡) = 𝑓𝑖(𝒓, 𝑡) − 𝜔(𝑓𝑖(𝒓, 𝑡) − 𝑓𝑖𝑒𝑞
(𝒓, 𝑡))
QIAN et al. (1992) proposed a family of BGK schemes that share
a common discrete equilibrium function
𝑓𝑖𝑒𝑞(𝜌, 𝒖) = 𝑤𝑖𝜌 (1 +
𝑐𝑖𝛼𝑢𝛼
𝜃+
𝑐𝑖𝛼𝑢𝛼𝑐𝑖𝛽𝑢𝛽
2𝜃2−
𝑢𝛼𝑢𝛼
2𝜃)
(
30)
The discrete weights 𝑤𝑖 and the parameter 𝜃 are model dependent.
The arguments of 𝑓𝑖𝑒𝑞
are local, and represent the hydrodynamic variables
recovered with the local moments of the discrete distribution 𝑓𝑖:
𝜌(𝐫, 𝑡) = ∑𝑓𝑖(𝒓, 𝑡)
𝑖
𝜌(𝐫, 𝑡)𝑢𝛼(𝐫, 𝑡) = ∑𝑐𝑖𝛼𝑓𝑖(𝒓, 𝑡)
𝑖
73
Note the similarity with the equations above to their continuous
counterpart Eq. (24) and Eq. (25). The integrals were replaced by
summations and this means that the dimensionality of 𝑓 and 𝑓𝑖 are
different. While the former has a dimensionality of [𝑘𝑔/𝑟³𝑐³], where
𝑟³𝑐³, is a hypercube in the phase space, the later has it as [𝑘𝑔/𝑟³]. Each of the models proposed by QIAN et al. (1992) are defined
when a set D of d dimensions in space, and a set Q of q velocities are
specified. They called each model of DqQq. For these models, a uniform
lattice discretization is used.
As an example, the D2Q9 model is a model which operates in a
uniform two dimensional lattice with a set of nine velocities. Eight
velocities go towards the neighboring lattice points, and one velocity goes
nowhere. It stays still due to zero speed. There are only three possible
speeds in this model: 0, 𝑐𝑟, √2𝑐𝑟, where 𝑐𝑟 is the reference speed in the
Cartesian axes. This scheme can be visualized in Fig. 5.
Figure 5 – Velocity vectors in the D2Q9 model: the distributions 𝒇𝒊 are
transported with three different speeds along the 𝒊 different directions,
represented by the nine velocity vectors. The speeds are just the modulus of these
vectors. In this model only three speeds are possible: 0, 𝒄𝒓, √𝟐𝒄𝒓. The lattice
spacing ∆𝒓 and the discrete time step are related with the reference speed by ∆𝒓 =∆𝒕𝒄𝒓.
Source: (MATTILA, 2010)
74
The discrete weights 𝑤𝑖 are assumed to be speed dependent. For
the D2Q9 model, 𝑤0 = 𝑊0 is the weight for speed zero. 𝑤𝑖 =𝑊1, 𝑓𝑜𝑟 𝑖 = 1,2,3,4; because these speeds are identical, and similarly,
𝑤𝑖 = 𝑊2, 𝑓𝑜𝑟 𝑖 = 5,6,7,8. Therefore, for the D2Q9 model there are only
three different weights 𝑊𝑖 which represent the set of nine velocities in Eq.
(20).
The parameter 𝜃 is given by
𝜃 =
𝑐𝑟²
3
(4)
and is related to the speed of sound through a Chapman-Enskog analysis27, which derives an appropriate equation of state for these LBGK
family of schemes, given by
𝑝 = 𝜃𝜌 (31)
Remembering from section 2.1.4 that the sound speed is given as
𝑐𝑠 = (𝜕𝑝
𝜕𝜌 |
𝑠
)
1/2
The lattice sound speed is then defined for analogy as 𝑐𝑠 = √𝜃.
Furthermore, the LBGK models share a common expression for the
kinematic viscosity:
𝜈 = 𝜃 (
1
𝜔−
1
2)Δ𝑡
Table 1 shows the LBGK models proposed by QIAN et al. (1992)
with their respective weights. The most common models are the two
dimensional D2Q9 and the three dimensional D3Q19.
27 A multiple scale analysis which recover the Navier-Stokes equation and
the macroscopic transport coefficients from the LBM theory.
75
Table 1 – LBGK models
Model 𝑾𝟎 𝑾𝟏 𝑾𝟐 𝑾𝟑
D1Q3 2/3 1/6 0 0
D2Q9 4/9 1/9 1/36 0
D3Q15 2/9 1/9 0 1/72
D3Q19 1/3 1/18 1/36 0
D4Q25 1/3 1/36 0 0 Source: (QIAN et al., 1992)
3.1.1 Physical LBGK simulations The best way to simulate physical flows is utilizing the dynamic
similarity discussed in section 2.1.5. This powerful feature allow us to
forget our worries with quantitatives aspect of some variables. All that
matters is the Reynolds number of the physical problem to be simulated.
This number is then implemented in a LBGK in its lattice-dimensional form.
In order to work with this lattice-dimensional form, a set of lattice-
parameters must be set equal to unity, remembering the relation ∆𝑟𝑙 =∆𝑡𝑙𝑐𝑟𝑙
which bounds them together, where subscript 𝑙 indicates lattice
variables28.
So we have
𝑐𝑟𝑙= 𝜌𝑙 = ∆𝑟𝑙 = ∆𝑡𝑙 = 1
Now, any characteristic length 𝐿 follows the relation 𝐿 = 𝐿𝑙∆𝑟,
where 𝐿𝑙 will be the number of lattice-spacing ∆𝑟𝑙 (=1) which composes
the 𝐿 representation. As an elucidative example, imagine a characteristic
length 𝐿 = 1 𝑚𝑒𝑡𝑒𝑟. We decide to represent it with 11 pixels, or 11
lattice-nodes (or points). It is easy to conclude that in 11 lattice-nodes,
there will be 10 lattice-spacing ∆𝑟𝑙, so 𝐿𝑙 = 10. We conclude that every
lattice-spacing ∆𝑟𝑙 represents in fact a real discrete-spacing
∆𝑟 = 0.1𝑚𝑒𝑡𝑒𝑟
𝑙𝑎𝑡𝑡𝑖𝑐𝑒−𝑠𝑝𝑎𝑐𝑖𝑛𝑔 .
The other parameters follow analogous relations. This lattice-
variables allow to construct also a lattice-viscosity and lattice-sound-speed
28 Which are identical to dimensionless variables. Subscript 𝑙 and
superscript * are used interchangeably.
76
𝑐𝑠𝑙2 = 𝜃𝑙 =
1
3 → 𝜈𝑙 =
1
3(1
𝜔−
1
2)
These lattice-variables are also related to as dimensionless
variables, because they do not refer to a dimensional world but instead, to
the computing world. The reader should note that thinking in terms of
lattice-units helps to understand the problems dimensionally, but that
dimensionless units or lattice-units are totally correspondent. Changing
now subscript 𝑙 for lattice-units to superscript * for dimensionless
variables, the whole LBGK method will be given as
𝑓𝑖∗(𝒓∗ + ∆𝑡∗𝒄𝒊
∗, 𝑡∗ + ∆𝑡∗) = 𝑓𝑖∗(𝒓∗, 𝑡∗) − 𝜔 (𝑓𝑖
∗(𝒓∗, 𝑡∗) − 𝑓𝑖𝑒𝑞∗
(𝒓∗, 𝑡∗))
𝑓𝑖
𝑒𝑞∗(𝜌∗, 𝒖∗) = 𝑤𝑖𝜌
∗ (1 + 3𝑐𝑖𝛼∗ 𝑢𝛼
∗ + 𝑐𝑖𝛼∗ 𝑢𝛼
∗ 𝑐𝑖𝛽∗ 𝑢𝛽
∗ −3
2𝑢𝛼
∗ 𝑢𝛼∗ )
𝜌∗(𝒓∗, 𝑡∗) = ∑𝑓𝑖∗(𝒓∗, 𝑡∗)
𝑖
𝜌∗(𝒓∗, 𝑡∗)𝑢𝛼∗ (𝒓∗, 𝑡∗) = ∑𝑐𝑖𝛼
∗ 𝑓𝑖∗(𝒓∗, 𝑡∗)
𝑖
And to recover the dimensional forms, the following relations are
called:
𝑐𝑖𝛼 = 𝑐𝑖𝛼∗ 𝑐𝑟; 𝑢𝛼 = 𝑢𝛼
∗ 𝑐𝑟; 𝑓𝑖 = 𝜌𝑓𝑖∗; 𝑓𝑖
𝑒𝑞= 𝜌𝑓𝑖
𝑒𝑞∗
where 𝜌 is a physical reference specific mass.
Now the Reynolds number can be simply set as
𝑅𝑒𝑙 =
𝐶∗𝐿∗
𝜈∗≡
𝑈𝐿
𝜈= 𝑅𝑒
𝐶∗ is a dimensionless characteristic lattice-speed. Likewise, 𝐿∗ is a
dimensionless characteristic lattice length counted in lattice-spacings and
𝜈∗ the lattice viscosity. The relation above relates the Reynolds number
77
in the lattice world with this number in the physical world. If they are
equal, the discrete dynamics in the computer will be dynamically similar
to the real world dynamics, provided the limitations of the LBGK are not
overlooked.
Recover from section 2.1.4 that the incompressibility hypothesis is
generally valid if the Mach number is small enough. This limit is
normally accepted for 𝑀𝑎 ≤ 0.3.
Since LBGK is derived from the BE, which deals with ideal gases,
the model is naturally compressible, as can be seen in Eq. (31), which is
an ideal gas equation of state. Pressure and density are linearly dependent
on the parameter 𝜃, which can be thought as equivalent to temperature T.
Our dimensionless lattice sound speed is 𝑐𝑠∗ =
1
√3, so the lattice
Mach number is then 𝑀𝑎 = 𝐶𝑚𝑎𝑥∗ /𝑐𝑠
∗, where 𝐶𝑚𝑎𝑥∗ is the maximum speed
(modulus) found in the lattice. SUCCI, (2001) states that in order to avoid
compressibility errors, the Mach number should be held under control,
this would mean
𝑀𝑎2 < ~0.1
and therefore the maximum velocity should be held under
𝐶𝑚𝑎𝑥∗ < ~ 0.182
Being more precise than this limit, like SUKOP; THORNE,
(2007), state that 𝐶𝑚𝑎𝑥∗ should remain under approximately 0.1, thus
𝐶𝑚𝑎𝑥∗ <≈ 0.1
From this point on, we will always use the dimensionless form
of the LBM equations and relations. We will suppress all the
superscripts * for the sake of notation clarity and simplicity.
3.1.2 External force implementation
Recover the BE:
𝜕𝑓
𝜕𝑡+ 𝐜
𝜕𝑓
𝜕𝒓+ 𝐠
𝜕𝑓
𝜕𝒄= Ω .
78
The third term on the left hand side of the BE is related to external
forces acting on the particles. The standard LBE is derived without this
term and assumes the general form:
𝑓𝑖(𝒓 + ∆𝑡𝒄𝒊, 𝑡 + ∆𝑡) = 𝑓𝑖(𝒓, 𝑡) − Ω𝑖(𝒓, 𝑡)
Many systems have internal or external body forces acting on the
particles. These forces might be electric forces, magnetic forces, gravity
forces, etc… If these forces are present, this external force term in the BE
must be incorporated into the model.
The effect of such forces can be seen physically as being injections
of momentum into the fluid particles. Therefore, an external force term is
added to the LBE, resulting in
𝑓𝑖(𝒓 + ∆𝑡𝒄𝒊, 𝑡 + ∆𝑡) = 𝑓𝑖(𝒓, 𝑡) − Ω𝑖(𝑠𝑑)(𝒓, 𝑡) + Ω𝑖
(𝑒)(𝒓, 𝑡)
where (sd) stands for short distance and (e) for external. The short
distance operator is related with the original collision operator Ω in the
BE and can be modelled with the BGK approximation. The new LBGK
model with external forces Ω𝑖(𝑒)29 is then:
𝑓𝑖(𝒓 + ∆𝑡𝒄𝒊, 𝑡 + ∆𝑡) = 𝑓𝑖 − 𝜔(𝑓𝑖 − 𝑓𝑖𝑒𝑞
) +𝑤𝑖
𝑐𝑠2𝑔𝑖𝑐𝑖
Arguments (𝒓, 𝑡) of 𝑓𝑖 and 𝑓𝑖𝑒𝑞
in the right hand side were
suppressed for simplicity. The body force in the 𝑖-th direction 𝑔𝑖
contributes to the probability of populations being transported with
molecular velocity 𝑐𝑖 through an internal product (𝑔𝑖𝑐𝑖) and is weighted
by the weight factors 𝑤𝑖.
Alternatively, for each time step, the net momentum induced by a
body force 𝑔𝛼 can be added to the macroscopic velocity with the
expression (SHAN; CHEN, 2003):
𝜌(𝐫)𝑢𝛼(𝐫) = 𝑐𝑖𝛼𝑓𝑖(𝒓) + (𝑔𝛼
𝜔)
29 Dimensionally, Ω𝑖(𝑒) is given as: Ω𝑖
(𝑒) =𝑤𝑖Δ𝑡
𝑐𝑠2 𝑔𝑖𝑐𝑖 .
79
where the term (𝑔𝛼
𝜔) 30 represents an increase of momentum at a lattice
node by an amount of Δ𝑡𝑔𝛼 during each time step without changing the
specific mass 𝜌 (ZHANG, 2011).
A numerical technique to achieve stability improvement consists
in splitting the momentum contribution of the external force in two
halves. One half is added in the collision step and another half is added to
the macroscopic velocity.
3.2 BOUNDARY CONDITIONS (BC)
3.2.1 Non-slip at walls From the boundary layer theory it is known that a fluid particle
adjacent to a solid wall has exactly its speed, meaning that they are
“glued” to each other and no relative slip occurs.
This BC is numerically implemented with the halfway bounce-
back. It is a very simple idea: the walls are put between two nodes
(halfway). All distributions that point towards the wall are reversed in its
direction after a streaming step. This can be seen in Fig. 6. It shows as an
example, the halfway bounce-back for a D2Q9 scheme. The three
distributions that point towards the wall are just reversed at the same
origin point in the streaming step, while the other distributions, pointing
towards the fluid domain, stream normally.
Figure 6 – Streaming step with halfway bounce-back. The walls are
allocated between two adjacent nodes. The distributions that point towards the
wall are just reversed at the same origin point in the streaming step, while the
other distributions, pointing towards the fluid domain, stream normally.
Source: (RIIKILÄ, 2012)
30 This term is given dimensionally as (Δ𝑡𝑔𝛼
𝜔).
80
This is mathematically stated as
𝑓𝑖𝑏(𝒓, 𝑡 + ∆𝑡) = 𝑓−𝑖
𝑏(𝒓, 𝑡)
meaning that all populations going in directions "𝑖" towards the boundary
"𝑏" will remain after streaming "𝑡 + ∆𝑡" at the same site "𝒓", but with
reversed directions " − 𝑖".
3.2.2 Periodic BC
This is the simplest boundary condition which can be applied at the
boundaries of the domain. It consists in topologically connect its
boundaries, forming a continuum domain in one or more directions. As
an example, a 2D computational domain can be converted in two different
topological manifolds with periodic boundary conditions. It will be either
a cylinder or a torus. This can be seen in Fig. 7.
Figure 7 – Periodic topological manifolds for a 2D domain. Black lines
shown where the extremities are “glued” together to form a continuum in either
one or both directions of the domain.
Source: (RIIKILÄ, 2012)
81
3.2.3 Constant flux BC (Von Neumann)
When there is a desired velocity at the boundaries in order to
constrain a specific flux on them, there will be some unknown populations
coming from outside the domain that must be specified. For a D2Q9
model, there will be three unknown populations after the streaming step,
as can be seen in Figure 8. It shows a north boundary but it can be easily
understood for other boundaries by analogy.
Figure 8 – Three unknown populations (7,4,8) coming from outside the
computational domain, after the streaming step at the north boundary.
Source: (SUKOP; THORNE, 2007)
Let us consider a macroscopic velocity applied to the north boundary
of Fig. 7, i.e., a velocity vector 𝒖 = (𝑢𝑥 , 𝑢𝑦), then we have to solve for
this three distributions plus the density 𝜌. Therefore, four equations are
needed. The first three equations come from the moments of 𝑓𝑖.
𝜌(𝐫, 𝑡) = ∑𝑓𝑖(𝒓, 𝑡)
𝑖
𝜌(𝐫, 𝑡)𝑢𝛼(𝐫, 𝑡) = ∑𝑐𝑖𝛼𝑓𝑖(𝒓, 𝑡)
𝑖
Remembering that the second moment expression gives two
equations, one for each direction of the coordinate axis.
82
The fourth equation can be found by assuming that the halfway
bounce-back condition holds for the non-equilibrium31 distributions in the
normal direction to the boundary, as proposed by ZOU; HE (2007), thus
𝑓2𝑛𝑒𝑞
= 𝑓4𝑛𝑒𝑞
→ 𝑓2 − 𝑓2𝑒𝑞
= 𝑓4 − 𝑓4𝑒𝑞
.
With some algebra it is possible to solve the unknowns with the
following expressions:
𝜌 =𝑓0 + 𝑓1 + 𝑓3 + 2(𝑓2 + 𝑓5 + 𝑓6)
1 + 𝑢𝑦
𝑓4 = 𝑓2 −2
3 𝜌𝑢𝑦
𝑓7 = 𝑓5 +(𝑓1 − 𝑓3)
2−
1
2𝜌𝑢𝑥 −
1
6𝜌𝑢𝑦
𝑓8 = 𝑓6 −(𝑓1 − 𝑓3)
2+
1
2𝜌𝑢𝑥 −
1
6𝜌𝑢𝑦
3.2.4 Constant pressure/density BC (Dirichlet)
Analogous to the Von Neumann BC, here, specific mass 𝜌0 is
specified, from where velocity is computed. Note that in standard LBGK
models density is related to pressure by the EOS Eq. (31). Therefore,
setting density is equivalent to setting pressure.
Using Fig. 7 again as an example and considering 𝑢𝑥 = 0 we need
to solve for the three unknown distributions plus velocity 𝑢𝑦. Using the
same equations as before and using the bounce-back of the non-
equilibrium distribution as proposed by ZOU; HE (2007), we get after
similar algebraic manipulations also similar expressions for the
unknowns:
31 The non-equilibrium distribution is defined as 𝑓𝑖
𝑛𝑒𝑞(𝒓, 𝑡) = 𝑓𝑖(𝒓, 𝑡) −
𝑓𝑖𝑒𝑞(𝒓, 𝑡), i.e., it is the deviation of 𝑓𝑖 from the equilibrium distribution 𝑓𝑖
𝑒𝑞.
83
𝑢𝑦 =𝑓0 + 𝑓1 + 𝑓3 + 2(𝑓2 + 𝑓5 + 𝑓6)
𝜌0− 1
𝑓4 = 𝑓2 −2
3 𝜌0𝑢𝑦
𝑓7 = 𝑓5 +(𝑓1 − 𝑓3)
2−
1
6𝜌0𝑢𝑦
𝑓8 = 𝑓6 −(𝑓1 − 𝑓3)
2−
1
6𝜌0𝑢𝑦
3.2.5 Zero derivative at boundaries
This is a simpler and faster variation of the Von Neumann
boundary condition. It is suitable for unsteady flows at the outlets. It
consists of prescribing to the boundary nodes the same macroscopic
velocity from its nearest neighbors, which are in the normal directions to
the boundary, in such a way that the spatial derivative of macroscopic
velocities are zero over these directions, i.e., 𝜕𝑥𝑢𝑥 = 0 for west and east
boundaries and 𝜕𝑦𝑢𝑦 = 0 for south and north boundaries. This can be
done by copying all the distributions from the neighbors to the boundary
nodes. However, This BC might lead to unphysical results for some
simulations, as it was found in a flow past a cylinder. A better variation
is to copy only the unknown distributions from the neighbors. Using
Figure 7 as an example for the D2Q9 model, only Dist. 7, 4 and 8 would
be copied from the south neighbor site for this BC. This strategy gave
good results.
3.3 COLLISION OPERATOR ALTERNATIVES
3.3.1 Two-time relaxation collision operator (TRT)
The TRT is based on the decomposition of the collision operator
in its symmetric and anti-symmetric components, having each component
its own relaxation-time parameter (GINZBURG; D’HUMIÈRES, 2003).
Denoting index “+” for the symmetric part and “–” for the antisymmetric
one, the TRT is defined as
Ω𝑖 = Ω𝑖+(𝒓, 𝑡) + Ω𝑖
−(𝒓, 𝑡)
84
Ω𝑖±(𝒓, 𝑡) = 𝜔± [𝑓𝑖
±(𝒓, 𝑡) − 𝑓𝑖𝑒𝑞±
(𝒓, 𝑡)]
𝑓𝑖
±(𝒓, 𝑡) =𝑓𝑖(𝒓, 𝑡) ± 𝑓−𝑖(𝒓, 𝑡)
2
𝑓𝑖𝑒𝑞±
(𝜌, 𝒖) =𝑓𝑖
𝑒𝑞(𝜌, 𝒖) ± 𝑓−𝑖𝑒𝑞(𝜌, 𝒖)
2
and the model is now
𝑓𝑖 = 𝑓𝑖 − 𝜔+ [𝑓𝑖+(𝒓, 𝑡) − 𝑓𝑖
𝑒𝑞+(𝒓, 𝑡)] − 𝜔−[𝑓𝑖
−(𝒓, 𝑡) − 𝑓𝑖𝑒𝑞−
(𝒓, 𝑡)]
yielding to
𝑓𝑖 = 𝑓𝑖 − 𝜔+ [𝑓𝑖 + 𝑓−𝑖
2−
𝑓𝑖𝑒𝑞
+ 𝑓−𝑖𝑒𝑞
2] − 𝜔− [
𝑓𝑖−𝑓−𝑖
2−
𝑓𝑖𝑒𝑞
− 𝑓−𝑖𝑒𝑞
2]
Again, 𝑓𝑖 , 𝑓−𝑖 and 𝑓𝑖𝑒𝑞
, 𝑓−𝑖𝑒𝑞
denote populations in opposite
directions.
The main reason to use the TRT is the gain in stability it provides.
This comes from the extra degree of freedom provided by the
antisymmetric relaxation parameter 𝜔−. To understand this is important
to remember that the decomposition of the distributions fulfil the
following relations
𝑓𝑖 = 𝑓𝑖+ + 𝑓𝑖
− 𝑓−𝑖 = 𝑓𝑖+ − 𝑓𝑖
− 𝑓𝑖+ = 𝑓−𝑖
+ 𝑓𝑖− = −𝑓−𝑖
−
the last two expressions exposes the fundamental property in TRT: odd
moments of the symmetric function vanishes32 as the even moments of
the antisymmetric function. The relaxation parameter 𝜔+ is therefore
32 Remember that the integral over the whole real line of an odd function
vanishes. The symmetric distribution 𝑓𝑖+
is even and becomes odd when
multiplied by odd functions like 𝑐𝛼 , 𝑐𝛼𝑐𝛼𝑐𝛽 , 𝑒𝑡𝑐.; The odd distribution
𝑓𝑖−
remains odd when multiplied by even functions, vanishing in the integral.
85
coupled with the even moments, tuning viscosity, while relaxation
parameter 𝜔− is coupled with the odd moments, giving a further degree
of freedom. It is chosen to minimize the viscosity dependence of the slip
velocity, a numerical peculiarity of LBM (MATTILA, 2010).
The kinematic viscosity can be determined as before with 𝜔+:
𝜈 =1
3(
1
𝜔+−
1
2)
By choosing a “magic” proportion we can defined a 𝜔− which
minimizes this viscosity dependency of the slip velocity. The super-
convergent proportion for a D2Q9 Hagen-Poiseuille flow is given by
DUBOIS; LALLEMAND; TEKITEK (2010) as:
𝜔− =
8(2 − 𝜔+)
8 − 𝜔+
Note that if 𝜔− = 𝜔+ the TRT collision model
reduces to the standard single relaxation time BGK collision
model and this value is
𝜔± = 8 − 4√3
3.4 MULTIPHASE AND MULTICOMPONENT MODEL
The standard LBGK model is suitable for simulating fluid
dynamics in low Reynolds regimes. The true strength of LBM, however,
is its ability to simulate multiple fluids and phases. These models in LBM
are called multicomponent and multiphase models (MCMP). They are
very important for many applications, since they offer solutions for
physical phenomena as surface tension, evaporation, condensation,
cavitation, immiscible displacement, contact angle and others. Such
phenomena are mainly driven from microscopic forces and interactions
between molecules. Since LBM is based on a mesoscopic scale, it is able
to provide a bridge between this microscopic interactions and its bulk
result in the macroscopic world.
Fig. 10 shows an interesting example of such complex phenomena.
It is called “coalescence cascade”. It happens in fractions of a second
when dropping a small drop of ultra-purified deionized water in its own
86
surface. The frames shown in Fig. 10 were taken with a high speed
camera.
MCMP models are particularly important for the petroleum
industry. Oil is often found with water, and/or displaced with water, in
reservoir rocks. This displacement process occurs in porous media, which
is an extremely intricate labyrinth of small pores down to the micrometer
scale. MCMP models are able to simulate such behavior, enlightening its
comprehension.
These models might be separate in two families: single component
multiphase models (SCMP), in which a single component fluid is
governed with a Van der Waals like EOS, instead of an ideal gas EOS.
This make possible phase transition.
The other family is the MCMP, where many non-ideal fluids can
coexist and interact with each other. Fig. 9 gives an overview of the LBM
models. It shows how components, interactions and parallelism are
interrelated.
Figure 9 - Framework for LBM models.
Source: (SUKOP; THORNE, 2007)
87
Figure 10 – Coalescence cascade of a drop.
Source: Author
88
3.4.1 Interparticle potential model
The model proposed by SHAN; CHEN (1993) is probably the most
common multiphase and multicomponent model due to its simplicity. It
is based on the idea of a pairwise intermolecular forces between the
conceptual particles in LBM. This force is considered to be relevant only
between a first neighborhood range, i.e., only between nearest adjacent
populations in the lattice. If a system is multicomponent, there must exist
a distribution function for every component 𝜎, therefore we get the
following LBM equation
𝑓𝑖𝜎(𝒓 + ∆𝑡𝒄𝒊, 𝑡 + ∆𝑡) = 𝑓𝑖
𝜎(𝒓, 𝑡) − Ω𝑖𝜎(𝒓, 𝑡)
representing that this equality holds in every direction 𝑖 of the lattice for
each component 𝜎 that constitutes the system.
The collision operator Ω𝑖𝜎(𝒓, 𝑡) is the single relaxation time BGK
Ω𝑖𝜎(𝒓, 𝑡) = 𝜔𝜎(𝑓𝑖(𝒓, 𝑡) − 𝑓𝑖
𝑒𝑞(𝒓, 𝑡))
SHAN; CHEN (1993) define an interaction potential between
particles as
𝑉(𝒓, 𝒓′) = 𝐺𝜎(𝒓, 𝒓′)𝜙𝜎(𝒓)𝜙(𝒓′)
and consider only nearest neighbors interactions, i.e.
𝐺𝜎(𝒓 − 𝒓𝒊
′) = 0, 𝑖𝑓 | 𝒓 − 𝒓𝒊
′| > 𝑐𝑖
𝐺𝜎 , 𝑖𝑓 | 𝒓 − 𝒓𝒊′| = 𝑐𝑖
where 𝑐𝑖 is the velocity modulus33 in the direction of the velocity vector
𝒄𝒊. 𝐺𝜎(𝒓, 𝒓′) is a Green’s function34 where its magnitude controls the
33 Which is equal to the lattice distance (𝒓 − 𝒓𝒊
′) in a dimensionless LBM
scheme. 34 “In mathematics, a Green's function is the impulse response of an
inhomogeneous differential equation defined on a domain, with specified initial
conditions or boundary conditions. Via the superposition principle, the
convolution of a Green's function with an arbitrary function f(x) on that domain
89
interaction strength between two different components and, its sign, if this
interaction is rather attractive or repulsive. The quantity 𝜙𝜎(𝒓) =𝜙𝜎(𝑓𝜎(𝒓)) is a function of the density distribution 𝑓𝜎(𝒓) and acts as the
effective number density for component 𝜎. It must increase
monotonically and be bounded.
With the interaction potential, it is possible to calculate the net
momentum change in a site for each time step, which is equivalent to the
net force, as
Ӻ 𝜎(𝒓) = −𝜙𝜎(𝒓)∑ 𝐺𝜎 ∑𝜙(𝒓 + 𝒄𝒊)
𝑛
𝑖=1
𝑆
=1
𝒄𝒊
where S is the total number of components in the system. Therefore, after
collision, the new net momentum at site 𝐫 for the 𝜎𝑡ℎ component is
𝜌𝜎(𝐫)𝑢𝛼𝜎(𝐫) = 𝑐𝑖𝛼𝑓𝑖
𝜎(𝒓) + (Ӻ𝛼
𝜎
𝜔𝜎)
The above relation indicates a non-conservative momentum
equation for a site. However, it can be demonstrated that the total
momentum within the system is conserved, provided 𝐺𝜎 is a symmetric
matrix with dimensions 𝑆 x 𝑆.
It is possible to show that the equation of state for a D2Q9 model
is
𝑝 =𝜌
3+
𝐺
6𝜙2
The second term from the right hand side is the non-ideal
attractive forces contribution from molecules and leads to a reduction in
pressure when 𝐺 < 0. This leads to a non linear 𝑝- 𝜌 plot in which a
substance can coexist with two different densities (phases) at the same
pressure.
is the solution to the inhomogeneous differential equation for f(x).”
(WIKIPEDIA, 2014)
90
91
Alles vergängliche
Ist nur ein Gleichniss!
Goethe*
92
* 35
35 Boltzmann’s epigraph for his “Vorlesungen über Gastheorie”.
(BOLTZMANN, 1986, 1898), meaning everything temporal is only a likeness,
or all that is transitory is only a metaphor.
93
4 SIMULATIONS
One of the problems faced by engineers when dealing with
numerical methods is often the need to learn a low level language and its
extension to parallelization. Low level programing languages like C and
FORTRAN and its extensions to parallelization, like OpenMP, take time
to master. The possibility to create an optimized code for a given problem
is their main advantage, extracting the biggest possible efficiency from
the given computational resources.
However, this might cost a precious time of an engineer, who
instead of concentrating his efforts to comprehend the physical problem
and interpreting it, must learn and explore a low level programming
language up to a reasonable level, considering many aspects from the field
of computational science. Although this kind of knowledge is desirable
and welcome it might also discourage engineers when the construction of
an algorithm for a simple problem demands a deeper understanding of a
low level programming language and much time.
In this context a family of software come in scene to help in the
task of scientific computing. They work in a high level environment
where many built-in functions are available for the users. Furthermore,
they work with a matrix logic, being very versatile when manipulating
matrices. Matlab, Scilab, GNU Octave and Mathematica are some
examples of high level softwares. The environment used to learn and
explore LBM in this work was Matlab. Matlab is a powerful software. It
is possible to program parallelized codes without any knowledge of
parallelization at low level. Its built in vectorized functions are already
parallelized in its source code. Of course, for being a high level language
it still has not the same efficiency as a full low level parallelized code.
However, this author believe that the future of scientific computing will
be such software, with which an engineer will not need anymore to care
so much with the deep numerics. As an example to show how powerful
Matlab is and its ability to work in a logic of multidimensional matrices,
let us consider an example:
The streaming step consists of transporting the net resulting
populations after the collision step to their respective neighbors. In a low
level language this would normally be made with a loop over all the
dimension of the domain plus a loop over the velocities. In a D2Q9 model,
for example, there would be loops in the two directions of the
bidimensional domain, visiting each point of it and for each point, there
will be 9 velocities directions where the functions should be streamed.
In Matlab, this step can be done recalling a single function within
a single loop over the directions.
94
% STREAMING STEP FLUID A AND B
for i=1:9
fIn(i,:,:) = circshift(fOut(i,:,:), [0,cx(i),cy(i)]);
end
where cx and cy are the components of the velocity directions
cx = [ 0, 1, 0, -1, 0, 1, -1, -1, 1]; cy = [ 0, 0, 1, 0, -1, 1, 1, -1, -1];
As can be seen, fIn is a 3 dimensional matrix. The above code
says: We shift all the values of every “i” bidimensional slice, which
represents the domain points for each velocity direction “i”, to its
nearest neighbors located at [0,cx(i),cy(i)].
The circshift function is a vectorized built-in Matlab function
which automatically parallelizes the computing. It is important to note
that with this function periodic boundary condition are naturally built in.
Matlab possesses many such vectorized built-in function and suggests
users should make use of them to enhance their code efficiency.
We see such software as the future of scientific computing. While
computational technology is growing very fast in the last decades, high
level software are becoming more and more sophisticated. We believe
that soon the trade of between the complexity of programming and the
efficiency of computation will not be a big question anymore.
95
4.1 HAGEN-POISEUILLE FLOW
A Hagen-Poiseuille Flow is one of the simplest flow
configurations to validate results. It consists of a steady one dimensional
laminar viscous flow in a pipe or between two flat plates. This model is
successfully used in blood flows inside veins and capillaries, air flow in
lung alveoli, flow in hypodermic needles, flows in oil and gas pipelines,
etc… Short, it is a model for any Newtonian incompressible viscous fluid
flowing through a pipe with constant cross-section that is much smaller
than its length.
The flow velocity profile was independently derived by Gotthilf
Heinrich Ludwig Hagen and Jean Léonard Marie Poiseuille between
1838 and 1839. It is axisymmetric and depends on coordinate 𝑟 from the
center only. For a flow between flat plates, the velocity profile is given
by
𝑢(𝑟) = −
1
2𝜇𝜕𝑥𝑝(𝑅2 − 𝑟2)
where p is pressure, x the direction of the flow, R half of the channel width
and 𝜇 the dynamic viscosity of the fluid. If some bodyforce is acting on
the fluid, in the direction of the motion, e.g. gravity, the derivative of
pressure is given from hydrostatics as
𝜕𝑥𝑝 = −𝜌𝑔
Where 𝜌 is the fluid specific mass and 𝑔 the gravity acceleration.
Hence,
𝑢(𝑟) =𝑔
2𝜈(𝑅2 − 𝑟2)
or with dimensionless parameters,
𝑢(𝑙) =
3𝑔
2 (𝜏 −12)
(𝐿2 − 𝑙2)
Results for different channel widths are shown in Fig. 11. Solid
lines are the analytical curves, while the plotted points are the simulated
ones. Simulations show very good agreement with the analytical model.
96
Figure 11 - Comparison between analytical and simulated velocity.
Source: Author
97
4.2 FLOW PAST A CYLINDER
Flow past a cylinder is also a classical and interesting problem.
As the Reynolds number increases, instabilities in the laminar wake
develop to periodic vortex shedding36, which induce a periodic force on
the cylinder driven by the pressure gradients. The frequency of this
vortices and thus, the forces, can match the natural frequency of the
cylinder. This would lead to resonance if the cylinder is not rigidly
mounted and the structure would vibrate harmonically with the flow’s
energy, emitting sound. This vortex shedding occurs in the range 102 <𝑅𝑒 < 107 and its frequency is related by the Strouhal37 number
𝑆𝑡 =
𝑓𝑞𝐷
𝑈≅ 0.198 (1 −
19.7
𝑅𝑒)
where 𝐷,𝑈 are the cylinder diameter and the free flow velocity,
respectively.
A LBGK simulation was conducted on this problem. The main
parameters are shown
Table 2 – Parameters for the flow past a cylinder.
LBGK model D2Q9
BC at inlet Von Neumann
BC at outlet Von Neumann
BC at the side walls Periodic
Cylinder diameter 34
Domain 410 x 1022
Free stream velocity 0.3
Relaxation parameter 𝝎 10/7
Reynolds number 153 Source: Author
36 They are called von Kármán vortex street, named after Theodore von
Kármán, who explained it theoretically in 1912. (WHITE, 2011 ) 37 Named after a German physicist, who in the late 19th century
experimented with wires singing in the wind.
98
The dimensionless Strouhal number for lattice variables can be
found with
𝑆𝑡 =
𝑓𝑞∗𝐷∗
𝑈∗≅ 0.198 (1 −
19.7
𝑅𝑒)
where 𝑓𝑞∗, 𝐷∗, 𝑈∗ are the frequency per time step, the diameter of the
cylinder in lattice nodes and the free flow lattice velocity, respectively.
The simulation was conducted from rest until a periodic steady
state was reached using a small body force to accelerate the field. A
balance of momentum in a volume control was made in both directions,
as shown in Fig. 12, where the evolution from rest of the lift and drag
force is given until periodicity is reached. As expected, the drag force
grows with the flow velocity while it is laminar. A critical value38 is
reached (around -30), which represents the minimal angle of flow
separation. Instabilities on the wake begin and the flow separation angle
increases, driving to the periodic vortex shedding.
From the Strouhal number for lattice units, the frequency39 of the
shadings is predicted to be 𝒇𝒒∗ ≅ 𝟏. 𝟓𝟐 × 𝟏𝟎−𝟑. To compare this
expected result, the Fast Fourier Transform (FFT) from a periodic interval
of the lift force was taken, which gives the main frequencies of the signal.
This is given in Fig.13. The FFT gives a main frequency as 𝒇𝒒∗ ≅ 𝟏. 𝟕 ×
𝟏𝟎−𝟑, which means that the simulated value was underpredicted by
around 12%. This is a good result.
38 Not shown in Fig.7 because of scaling. 39 Means a frequency per time step.
99
Figure 12 – Transient to periodic evolution of the forces.
Source: Author
100
Figure 13 – Normalized Fast Fourier Transform (FFT) of the periodic lift force.
Source: Author
101
Another check might be done be a relation given as: (DOUGLAS
et al., 2001 apud SUKOP; THORNE, 2007)
ℎ
𝑙≅ 0.281
The parameter ℎ
𝑙 is the ratio between the vertical distance ℎ of
two lanes and the distance 𝑙 of two vortices in the same lane. For this
purpose, a view of the vorticity and velocity field is prepared in Fig. 15
and Fig. 16, where the lovely von Kármán vortex street can be seen. From
the velocity field we take a closer look into some vortices in the wake,
which is depicted in Fig. 14. By taking the position of the pixels in the
center of the vortices both parameters can be estimated. It was found
ℎ
𝑙=
44.5
156≅ 0.285
(8)
which is a great result, diverging less than 2% of the relation given above.
Figure 14 - Magnification of some vortices of Fig. 15.
Source: Author
102
Figure 15 – Flow past a cylinder: vorticity field.
Source: Author
103
Figure 16 - Flow past a cylinder: velocity field.
Source: Author
104
4.3 THE IMMISCIBLE CANGACEIRO40 (MCMP)
We might now apply the multicomponent interparticle potential
model in an example to illustrate the behavior of such a model. In this
simulation, two different components were considered. We shall call
them component A and component B. The initial condition was taken
from a picture shown in Fig. 17.
Figure 17 – Cangaceiro’s original picture.
Source: CINEMOTION41
This picture was converted into an 8 bits matrix, whose values
range from 0 to 255, representing a gray scale, 0 for black to 255 for white.
Thereafter a normalization was made dividing all the values by 255 so
40 Cangaceiro is the name given to the warriors of cangaço, a social
banditry movement in northeast Brazil in late 19th and early 20th centuries. 41 From the 1964 film directed and written by Glauber Rocha: Deus e o
Diabo na Terra do Sol.
105
that they all are in an interval [0,1]. We set the the density distribution for
component A as [1 + NormalizedMatrix] and for component B as [1 −
NormalizedMatrix].
It is used periodic boundary conditions in the boundaries. The first
steps of the simulation can be seen in Fig. 18 and Fig. 19, which shows
the real nature of our cangaceiro.
Figure 18 – The immiscible cangaceiro after 50 time steps. The initial
density for Fluids A and B are in the interval [0,1], [1,2], respectively. After 50
time steps concentrations raised considerably due to the repulsive interparticle
force between Fluid A and Fluid B. They tend to concentrate in some regions of
the domain. The column on the side indicates density for Fluid B.
Source: Author
106
The main parameters utilized in this simulation are shown in
Table 3. 𝑮𝐴𝐴, 𝑮𝐵𝐵, 𝑮𝐴𝐵 are the interparticle forces between the fluids in
the subscript.
Figure 19 - The immiscible cangaceiro after 250 time steps. Colorbar
shows the density for Fluid B.
Source: Author
107
Table 3 – Main parameters of the immiscible cangaceiro.
LBGK model D2Q9 MCMP
BC Periodic
𝑮𝑨𝑨 0
𝑮𝑩𝑩 0
𝑮𝑨𝑩 0.7
Domain 600 x 555
Relaxation parameter 𝝎𝑨 1
Relaxation parameter 𝝎𝑩 1
Referencial Density 1 (for both Fluids) Source: Author
We can see that Fluid A, indicated with the darker zones, does not
want to mix with Fluid B, indicated by the white zones. They are
immiscible and that is why our unlucky cangaceiro is falling apart, he is
becoming a system of drops. We might think of this as an emulsion of
water/oil or any two immiscible fluids, which are put at an initial
concentration field as the original pixel scale of Figure 13. Concentration
rises since the fluids do not want to share the same space.
This diffusion process continues until an equilibrium configuration
is reached. This equilibrium configuration will be a single bubble of a
fluid inside the other. But this requires a lot of time. Figure 20 shows the
system after 30 thousand time steps and it is still far from this equilibrium
situation.
108
Figure 20 – Immiscible cangaceiro after 30 000 time steps. Colorbar
shows the density for Fluid B.
Source: Author
109
4.4 PHASE TRANSITION (SCMP)
A single component multiphase simulation was conducted in order
to show the ability of LBM in modelling phase transition. The simulation
was conducted with fully periodic BC and an appropriate function 𝜙 was
used as shown in Table 4.
Table 4 – Parameters for the SCMP simulation.
LBGK model D2Q9 SCMP
BC Periodic
𝑮𝑨𝑨 -120
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏 𝝓 𝜙(𝝆(𝒓)) = 𝟒𝑒−200/𝜌(𝒓)
Domain 256 x 256
Relaxation parameter 𝝎 1
Referential Density 200 Source: Author
With this function 𝜙 defining the interaction potential, and, taking
the reference density, the EOS curves of this component can be plotted
for many magnitudes of G, as can be seen in Fig. 21..
It is easy to note in Figure 21 that the modulus of G must be
beyond a value, in order to get phase transition, i.e., two different densities
for a same pressure.
Figure 22 represents our initial condition for the distributions. The
reference density was applied with a random perturbance.
Fig. 23 shows the state of the system after 100 (above) and 500
(below) time steps. The colorbar in the right side indicates the density
values. It can be noted that vapor has a density around 90 while liquid a
density around 520. We can then find the correspondent pressure in Fig.
21, which is around 28.
110
Figure 21 – EOS curves for some attraction forces.
Source: Author
Figure 22 – The initial configuration with a reference density of 200 plus
a random perturbation in the interval [0,1].
Source: Author
111
Figure 23 – Above is shown the system after 100 time steps. Below, after
500 time steps. The colorbar indicates the density, where it can be seen that black
represents the liquid state, while white the gaseous state.
Source: Author
112
After 50 thousands time steps the system is in perfect
equilibrium, just a bubble of fluid immersed in its gas, as shown in Fig.
24.
Figure 24 – System in equilibrium after 50 000 time steps.
Source: Author
113
4.5 BIDIMENSIONAL SWALLOW FLIGHT
Another example simulated was a bidimensional flow over
swallows in its classic triangular formation. Here, the picture of a
silhouette showing five swallows arranged triangularly was binarized, in
order to be used as the domain of the simulation. Figure 25 (above) shows
the velocity flow field with its respective streamlines in the first time
steps. Darker zones indicate low velocities. The below figure shows the
correspondent vorticity field, representing clockwise vorticity with darker
tons.
It can be seen the symmetry of the flow field in these first time
steps, which develops to a periodic asymmetrical flow that is shown in
Figure 26. As in Fig. 25, Fig. 26 also shows the velocity field with
streamlines and vorticity field.
The main parameters for this simulation are listed in Table 7.
Table 5 – Main simulation’s parameters used for the swallow flight.
LBGK model D2Q9
BC at inlet Von Neumann
BC at outlet Von Neumann
BC at the side walls Von Neumann
Characteristic Length 435
Domain 877 x 1205
Free stream velocity 0.0753
Relaxation parameter 𝝎 10/6
Reynolds number 982
Source: Author
114
Figure 25 – Symmetrical flow field during the first time steps.
Velocity field with streamlines (above). Vorticity field.
Source: Author
115
Figure 26 - Asymmetrical flow field during the first time steps.
Velocity field with streamlines (above). Vorticity field (below).
Source: Author
116
117
5. FINAL CONSIDERATIONS
In this work we have explored the mathematical modelling which
describes the behavior of fluids, beginning with the classical continuum
approach and then introducing the mesoscopic models, which recover
with proper mathematical treatment the fundamental macroscopic
equations. The mesoscopic insight gives new possibilities for the
descriptions of fluids. This description have shown that more complex
phenomena are easier to implement than its classical counterpart. LBM is
a powerful method in fully development. We cited few models and gave
examples of them. However, the literature in LBM is getting constantly
bigger. There are other models available for the same problems. The
objective is always to turn our models as sophisticated as possible in order
to advance with human technical possibilities. Of course, these
technological advances and possibilities should come to help mankind
achieve what it really wants, a life of completeness. Why should we care
about helping some corporations to extract and pump oil, while taking the
bio-ecological risk of an oil spill or environmental catastrophe, if this does
not help people to live better? As before, we emphasize that science is
just a tool. Our free will decides what we make out of it.
Fluid dynamical phenomena are not only useful to understand
practical problems from the physical world and advance or technologies,
they are also intrinsically beautiful as some of our simulations show. The
approximation of nature’s behavior through mathematical modelling
makes possible the reproduction of it. As in Mr. Escher words, we adore chaos because we love to produce order, and indeed is a lovely feeling to
see through our simulations the manifestations of order, movements
towards equilibrium states, that come from the chaotic collision of
molecules, symbols, abstractions all the way to the electric pulses in
microprocessors that compute millions of operations in a second.
Finally, we hope that this work helps some in the path of discrete
fluid dynamics simulation, showing doors and directions for those who
want to begin to work with it but has little knowledge.
118
119
Man muss noch Chaos in sich
haben, um einen tanzenden Stern gebären
zu können.
Nietzsche*
120
* 42
42 One must have interior chaos, in order to give birth to a dancing star.
121
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127
APPENDIX: Code examples
I – LBGK FLOW PAST A CYLINDER
%%%%%%%%%%%%%%%%%%%%% %%% LBGK TRT D2Q9 %%%%%% %%%%% FLOW PAST A CYLINDER %%%% function [ fIn Domain ux uy MOMENTUMX MOMENTUMY] = ... cylinder(Gx,tau,D,time, fIn) inputt = input('Type "1" for velocity field or "2" for the vorticity field: '); p=0; % CREATING DOMAIN SIZE WITH 2 GHOST LAYERS ly=D*6; lx=D*18; Domain = logical(zeros(ly+2,lx+2)); % INITIAL VELOCITY FIELD CONDITIONS ux=0.1; uy=0; % INITIAL REFERENCE DENSITY rho0=1; % DECLARING BASIC VARIABLES % use of "single" just to save memory omega1 = 1/tau; omega2 = (16-8*omega1)/(8-omega1); tPlot = 20; ux = single (ones (ly+2, lx+2) * ux); uy = single (ones (ly+2, lx+2) * uy); rho = single (ones (ly+2, lx+2) * rho0); fEq = single (zeros (9, ly+2, lx+2)); % CREATING THE CENTER POINT OF THE CYLINDER qy=(ly+3.0)/2; qx=(lx/8); % CREATING CYLINDER for i=2:lx+1 for j=2:ly+1 if (i-qx)^2+(j-qy)^2 <= (D/2.0)^2 && (i-qx)^2+(j-qy)^2 >= ((D-3)/2.0)^2 Domain(j,i)= 1; else end end end % LISTING SOLID NODES SolidRegion=find(Domain); % CREATING OPPOSITION ARRAY
128
opp = [1 3 2 7 9 8 4 6 5]; % CREATING ARRAY OF VELOCITIES cx=[0 0 0 1 1 1 -1 -1 -1]; cy=[0 1 -1 0 1 -1 0 1 -1]; % CREATING ARRAY OF WEIGHTS w = [16/36 4/36 4/36 4/36 1/36 1/36 4/36 1/36 1/36]; % SETTING INITIAL CONDITIONS FOR INPUT DISTRIBUTION FUNTION EQUAL TO ZERO if fIn==0 for i=1:9 cu = 3*(cx(i)*ux+cy(i)*uy); fEq(i,:,:) = rho0 .* w(i) .*... ( 1 + cu + 1/2*(cu.*cu) - 3/2*(ux.^2+uy.^2) ); end fIn=fEq; fOut=fIn; % SETTING INITIAL CONDITIONS FOR A GIVEN INPUT DISTRIBUTION FUNTION else fOut=fIn; for i=1:9 cuInicial=3*cx(i)*UxInicial; UxEntrada(i,:,1) = rho0.* w(i) .*... ( 1 + cuInicial + 1/2*(cuInicial.*cuInicial) - 3/2*(UxInicial.^2) ); end end for t=1:time % MACROSCOPIC VARIABLES rho = sum(fIn,1); ux = reshape ( cx*(reshape(fIn,9,(ly+2)*(lx+2))),1, ly+2,lx+2) ; ux = (ux + Gx/2)./rho; uy = reshape (cy* (reshape(fIn,9,(ly+2)*(lx+2))),1, ly+2,lx+2) ./rho; % COLISION STEP for i=1:9 cu = 3*(cx(i)*ux+cy(i)*uy); fEq(i,:,:) = rho .* w(i) .*( 1 + cu + 1/2*(cu.*cu) - 3/2*(ux.^2+uy.^2) ); end for i=1:9 BodyForce = rho*w(i)*3 * cx(i)*Gx; fIn(i,:,:) = BodyForce/2 + fIn(i,:,:) - ... omega1.*((fIn(i,:,:) + fIn(opp(i),:,:))./2 - (fEq(i,:,:)+fEq(opp(i),:,:))./2) ... - omega2.*((fIn(i,:,:) - fIn(opp(i),:,:))./2 - (fEq(i,:,:) - fEq(opp(i),:,:))./2); end
129
% STREAMING STEP for i=1:9 fOut(i,:,: ) = circshift(fIn(i,:,: ), [0, cy(i),cx(i)]); end fIn=fOut; % BOUNCE BACK for i=1:9 fIn(i,SolidRegion) = fOut(opp(i),SolidRegion); end % OUTLET B.C. for k=7:9 fIn(k,:,lx+2)= fIn(k,:,lx+1); end % VISUALIZATION OF VELOCITY FIELD if inputt == 1 if (mod(t,tPlot)==0) p=p+1; %MOMENTUM BALANCE IN CONTROL VOLUME UUY = (+sum(uy(1,ceil(qy-D),ceil(qx-D): ceil(qx+D)))... + sum(uy(1,ceil(qy+D),ceil(qx-D): ceil(qx+D))))*rho0; UUX = (-sum(ux(1,ceil(qy-D):ceil(qy+D),ceil(qx-D)))... + sum(ux(1,ceil(qy-D):ceil(qy+D),ceil(qx+D))))*rho0; MOMENTUMX(p)=UUX; MOMENTUMY(p)=UUY; u =sqrt(ux.^2+uy.^2); LP(:,:)=u(1,:,:); LP(Domain)=nan; imagesc(LP); colormap(hot(128)) colorbar axis equal off; drawnow t end % VISUALIZATION OF VORTICITY FIELD elseif inputt==2 if (mod(t,tPlot)==0); uyy=circshift(uy(:,:,:),[0, 0,-1]); uxx=circshift(ux(:,:,:),[0,-1,0]); Vorticity =(uy - uyy) - (ux - uxx); Vorticity = reshape(Vorticity,ly+2, lx+2); Vorticity(Domain)=nan; Vorticity(:,1)=0;
130
imagesc(Vorticity, [-0.01 0.01]); colormap(hot(128)) colorbar axis equal off; drawnow t end end end
131
II – SINGLE COMPONENT MULTIPHASE MODEL
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% SINGLE COMPONENT MULTIPHASE MODEL %%% %%%%% INTERPARTICLE POTENTIAL MODEL %%%% %%% LBGK D2Q9 %%% function [fOut]=scmp(tau,G) %DOMAIN lx=200; ly=200; % TIME STEPS maxT = 80000; rho0=200; % PLOTS FOR EVERY TPLOT ITERATIONS tPlot = 10; % D2Q9 LATTICE CONSTANTS w = [4/9, 1/9,1/9,1/9,1/9, 1/36,1/36,1/36,1/36]; cx = [ 0, 1, 0, -1, 0, 1, -1, -1, 1]; cy = [ 0, 0, 1, 0, -1, 1, 1, -1, -1]; opp = [ 1, 4, 5, 2, 3, 8, 9, 6, 7]; % MOMENTUM CONTRIBUTION'S CONSTANTS omega=1/tau; Gtau = G/omega; % INITIAL CONDITION FOR BOTH DISTRIBUTION FUNCTIONS: perturbation=rand(lx,ly); for i=1:9 fIn(i,1:lx,1:ly) = w(i).*(rho0+perturbation); end % GETTING INITIAL FRAMES for i=1:50 rho = reshape(sum(fIn),lx,ly); imagesc(rho); colormap(flipud(gray(256))); axis equal off; drawnow end % MAIN LOOP (TIME CYCLES) for cycle = 1:maxT % MACROSCOPIC VARIABLES rho = sum(fIn); phi = 4*exp(-rho0/rho); MomentumX = reshape ( (cx * reshape(fIn,9,lx*ly)), 1,lx,ly); MomentumY = reshape ( (cy * reshape(fIn,9,lx*ly)), 1,lx,ly); % FORCE INDUCED BY RHO
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rhoFX = 0; rhoFY = 0; for i=2:9 rhoFX = rhoFX + circshift(phi*w(i), [0,cx(i),cy(i)])*cx(opp(i)); rhoFY = rhoFY + circshift(phi*w(i), [0,cx(i),cy(i)])*cy(opp(i)); end %POTENTIAL CONTRIBUTION MomentumTotX = MomentumX - phi.*rhoFX*Gtau; MomentumTotY = MomentumY - phi.*rhoFY*Gtau; Ux = MomentumTotX./rho; Uy = MomentumTotY./rho; % COLLISION STEP for i=1:9 cu = 3*(cx(i)*Ux+cy(i)*Uy); fEq(i,:,:) = rho .* w(i) .* ... ( 1 + cu + 0.5*(cu.*cu) - 1.5*(Ux.*Ux + Uy.*Uy) ); fOut(i,:,:) = fIn(i,:,:) - omega.*(fIn(i,:,:) ... - fEq(i,:,:)); end % STREAMING STEP FLUID A AND B for i=1:9 fIn(i,:,:) = circshift(fOut(i,:,:), [0,cx(i),cy(i)]); end % VISUALIZATION if(mod(cycle,tPlot)==0) rho = reshape(rho,lx,ly); imagesc(rho); colorbar colormap(flipud(gray(256))); title('Fluid 1 density'); axis equal off; drawnow cycle end end
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III – MULTI COMPONENT MULTIPHASE MODEL
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% MULTI COMPONENT MULTIPHASE MODEL %%% %%%%% INTERPARTICLE POTENTIAL MODEL %%%% %%% LBGK D2Q9 %%% function [fOut]=MCMP(tauA, tauB,GAA, GBB, GAB) %DOMAIN lx=200; ly=200; % TIME STEPS maxT = 80000; rho0A=200; rho0B=100; % PLOTS FOR EVERY TPLOT ITERATIONS tPlot = 10; % D2Q9 LATTICE CONSTANTS w = [4/9, 1/9,1/9,1/9,1/9, 1/36,1/36,1/36,1/36]; cx = [ 0, 1, 0, -1, 0, 1, -1, -1, 1]; cy = [ 0, 0, 1, 0, -1, 1, 1, -1, -1]; opp = [ 1, 4, 5, 2, 3, 8, 9, 6, 7]; % MOMENTUM CONTRIBUTION'S CONSTANTS omegaA=1/tauA; omegaB=1/tauB; GtauBA = GAB/omegaA; GtauAB = GAB/omegaB; GtauAA = GAA/omegaA; GtauBB = GBB/omegaB; % INITIAL CONDITION FOR BOTH DISTRIBUTION FUNCTIONS: perturbation=rand(lx,ly); for i=1:9 fIn(i,1:lx,1:ly) = w(i).*(rho0A+perturbation); gIn(i,1:lx,1:ly) = w(i).*(rho0B-perturbation); end % GETTING INITIAL FRAMES for i=1:50 rhoA = reshape(sum(fIn),lx,ly); imagesc(rhoA); colormap(flipud(gray(256))); axis equal off; drawnow end % MAIN LOOP (TIME CYCLES)
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for cycle = 1:maxT % MACROSCOPIC VARIABLES rhoA = sum(fIn); rhoB = sum(gIn); phiA = 4*exp(-rho0A/rhoA); phiB = 4*exp(-rho0B/rhoB); MomentumXA = reshape ( (cx * reshape(fIn,9,lx*ly)), 1,lx,ly); MomentumYA = reshape ( (cy * reshape(fIn,9,lx*ly)), 1,lx,ly); MomentumXB = reshape ( (cx * reshape(gIn,9,lx*ly)), 1,lx,ly); MomentumYB = reshape ( (cy * reshape(gIn,9,lx*ly)), 1,lx,ly); rhoMedio = rhoA*omegaA + rhoB*omegaB; UxMedio = (MomentumXA*omegaA+MomentumXB*omegaB) ./ rhoMedio; UyMedio = (MomentumYA*omegaA+MomentumYB*omegaB) ./ rhoMedio; % FORCE INDUCED BY RHO rhoFXA = 0; rhoFYA = 0; rhoFXB = 0; rhoFYB = 0; for i=2:9 rhoFXA = rhoFXA + circshift(phiA*w(i), [0,cx(i),cy(i)])*cx(opp(i)); rhoFYA = rhoFYA + circshift(phiA*w(i), [0,cx(i),cy(i)])*cy(opp(i)); rhoFXB = rhoFXB + circshift(phiB*w(i), [0,cx(i),cy(i)])*cx(opp(i)); rhoFYB = rhoFYB + circshift(phiB*w(i), [0,cx(i),cy(i)])*cy(opp(i)); end %POTENTIAL CONTRIBUTION FROM FLUID A to A MomentumTotXA = rhoA.*UxMedio - phiA.*rhoFXA*GtauAA; MomentumTotYA = rhoA.*UyMedio - phiA.*rhoFYA*GtauAA; %POTENTIAL CONTRIBUTION FROM FLUID B to A MomentumTotXA = MomentumTotXA - phiA.*rhoFXB*GtauBA; MomentumTotYA = MomentumTotYA - phiA.*rhoFYB*GtauBA; UxA = MomentumTotXA./rhoA; UyA = MomentumTotYA./rhoA; %POTENTIAL CONTRIBUTION FROM FLUID B to B MomentumTotXB = rhoB.*UxMedio - phiB.*rhoFXB*GtauBB; MomentumTotYB = rhoB.*UyMedio - phiB.*rhoFYB*GtauBB; %POTENTIAL CONTRIBUTION FROM FLUID A to B MomentumTotXB = MomentumTotXB - phiB.*rhoFXA*GtauAB; MomentumTotYB = MomentumTotYB - phiB.*rhoFYA*GtauAB; UxB = MomentumTotXB./rhoB; UyB = MomentumTotYB./rhoB;
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% COLLISION STEP for i=1:9 cuA = 3*(cx(i)*UxA+cy(i)*UyA); cuB = 3*(cx(i)*UxB+cy(i)*UyB); fEq(i,:,:) = rhoA .* w(i) .* ... ( 1 + cuA + 0.5*(cuA.*cuA) - 1.5*(UxA.*UxA + UyA.*UyA) ); gEq(i,:,:) = rhoB .* w(i) .* ... ( 1 + cuB + 0.5*(cuB.*cuB) - 1.5*(UxB.*UxB+ UyB.*UyB) ); fOut(i,:,:) = fIn(i,:,:) - omegaA.*(fIn(i,:,:) - fEq(i,:,:)); gOut(i,:,:) = gIn(i,:,:) - omegaB .* (gIn(i,:,:)-gEq(i,:,:)); end % STREAMING STEP FLUID A AND B for i=1:9 fIn(i,:,:) = circshift(fOut(i,:,:), [0,cx(i),cy(i)]); gIn(i,:,:) = circshift(gOut(i,:,:), [0,cx(i),cy(i)]); end % VISUALIZATION if(mod(cycle,tPlot)==0) rhoA = reshape(rhoA,lx,ly); imagesc(rhoA); colorbar colormap(flipud(gray(256))); title('Fluid 1 density'); axis equal off; drawnow cycle end end
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