The Lattice Boltzmann Method with Applications in Acoustics

108 Pages·2009·1.51 MB·English

by  Erlend Magnus Viggen

The Lattice Boltzmann Method with
Applications in Acoustics
Erlend Magnus Viggen
Department of Physics — NTNU 2009

Thesis description
Student: Erlend Magnus Viggen
Written at: Department of Physics, NTNU
Period: January 20 – June 16, 2009
Internal supervisor: Alex Hansen, Department of Physics, NTNU
External supervisor: Ulf Kristiansen, Department of Electronics and
Telecommunications, NTNU
English title: The Lattice Boltzmann Method with Applications
in Acoustics
Norwegian title: LatticeBoltzmann-MetodenmedAnvendelserinnen
Akustikk
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Abstract in English
An introduction is given to the lattice Boltzmann method and its background, with a
view towards acoustic applications of the method. To make a larger range of acoustic
applications possible, a point source method is proposed. This point source is applied
to simulate cylindrical waves and plane waves, and is shown to give a very good nu-
merical result compared with analytic solutions of viscously damped cylindrical and
plane waves. Good results are found for simulations of Doppler effect, diffraction, and
viscously damped standing waves.
It is concluded that the lattice Boltzmann method could be suitable for simulating
acoustics in complex flows, at ultrasound frequencies and very small spatial scales. The
lattice Boltzmann method is shown to be unfeasible at lower frequencies or for larger
systems.
Abstract in Norwegian
En introduksjon gis til lattice Boltzmann-metoden og dens bakgrunn, med henblikk
p˚a akustiske anvendelser av metoden. For˚a muliggjøre flere forskjellige akustiske an-
vendelser foresl˚as en punktkildemetode. Denne punktkilden anvendes for ˚a simulere
sylinder- og planbølger, og det vises at den gir et svært bra numerisk resultat sammen-
lignet med analytiske løsninger av viskøst dempede sylinder- og planbølger. Gode resul-
tater finnes for simuleringer av Dopplereffekt, diffraksjon og viskøst dempede st˚aende
bølger.
DetkonkluderesmedatlatticeBoltzmann-metodenkanværepassendefor˚asimulere
akustikk i komplekse strømninger, ved ultralydfrekvenser og svært sm˚a romlige skalaer.
Lattice Boltzmann-metoden vises˚a være upraktisk ved lavere frekvenser og for større
systemer.

Preface
The thought of writing my master’s thesis on the possibility of acoustics simulations in
thelatticeBoltzmannmethodcameaboutwhenIwasdiscussingpossiblethesissubjects
with my advisor, Ulf Kristiansen. We had already agreed that I would write my master
thesis on something in the field of numerical acoustics, but we had not decided on a
final subject.
While we were both familiar with lattice gases, none of us were aware that the field
of lattice gases had morphed into the field of lattice Boltzmann some time ago. When
we discovered this, we decided that it would be interesting to see how well this method
for fluid simulations would be suited to perform acoustics simulations. After all, sound
waves are special cases of fluid behaviour, right?
I had not touched fluid mechanics since the introductory course I took on it three
years ago. I hadforgotten quite a lotsince then, butthis thesis gave me the opportunity
to relearn a lot of interesting fluid mechanics theory.
As mentioned, I had no previous knowledge of the lattice Boltzmann method before
starting with this thesis, but I quickly discovered that it was a very interesting field.
One of the advantages of the method is its relative simplicity, which meant that I was
up and running in a reasonably short time. Within the first week of learning about the
method, I had working lattice Boltzmann code.
Still, I wasted some time in the beginning barking up the wrong tree. For instance, I
was confused by several papers stating that the lattice Boltzmann method gives incom-
pressible flow behaviour. If the method was restricted to incompressible flow, it would
mean that simulations of sound wave propagation would be impossible, as sound is a
phenomenon of compressible flow. It turned out I had misconstrued the statement —
themethodgivesabehaviourconsistentwithcompressibleflow, ofwhichincompressible
flow is a special case. Incompressible flow is merely the most popular special case of the
lattice Boltzmann method.
In writing this master’s thesis, I have tried to comply with the guidelines from my
department. Specifically, I have tried to make this text accessible to my peers, meaning
other fifth-year university students of physics. The unfortunate side-effect of this is
that I spend a lot of time explaining theory which should be familiar for people with
particular experience with the subjects discussed here.
I would like to thank all the people who have helped me in this work. First, I would
like to thank Joris Verschaeve, who is a PhD student at the fluids engineering group at
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NTNU’s Department of Energy and Process Engineering. Joris has been very helpful in
explaining the finer points of the lattice Boltzmann method to me.
I would like to thank the community at LBMethod.org, in particular Jonas L¨att and
Orestis Malaspinas. This site has been a very useful repository of knowledge for me,
and I have received help at its forums when it was needed. Jonas has also been very
helpful with questions about his PhD thesis, which has been one of my main sources.
Finally, I would like to thank my advisors, for discussions and follow-up questions,
and for letting me write about such an interesting subject.
Contents
1 Introduction 1
1.1 Available resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Thesis structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Theory 5
2.1 Tensors and index notation . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Useful examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Fluid mechanics and Navier-Stokes . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Bulk viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Navier-Stokes and the wave equation . . . . . . . . . . . . . . . . . . . . 9
2.4.1 Solutions to the wave equation . . . . . . . . . . . . . . . . . . . 11
3 Lattice gas automata 13
3.1 The HPP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.1 Advantages and disadvantages . . . . . . . . . . . . . . . . . . . 16
3.2 The FHP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.1 Advantages and disadvantages . . . . . . . . . . . . . . . . . . . 19
3.3 Lattice isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Macroscopic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 The lattice Boltzmann method 23
4.1 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 The BGK operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Lattice isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.4 The equilibrium distribution . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.5 Simple boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.6.1 Example simulation . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.7 Unit conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.7.1 Direct conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.7.2 Dimensionless formulation . . . . . . . . . . . . . . . . . . . . . . 39
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5 Lattice Boltzmann boundary conditions 41
5.1 Numerical accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 Zou-He pressure and density boundaries . . . . . . . . . . . . . . . . . . 42
5.3 Regularized boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.4 Acoustic point source . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.4.1 Single point source . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.4.2 Line of point sources . . . . . . . . . . . . . . . . . . . . . . . . . 50
6 Simulations 53
6.1 Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.2 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.3 Standing waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7 Alternative lattice Boltzmann models 61
7.1 Chopard-Luthi wave model . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.2 Yan wave model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7.3 External forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.4 Adjustable bulk viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.5 Regularized model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.6 Multiphase and multicomponent models . . . . . . . . . . . . . . . . . . 65
7.7 Thermal models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
8 Discussion 67
8.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
8.2 Potential and problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8.3 Looking forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
9 Conclusion 71
A Derivation of Navier-Stokes from LBM dynamics 73
A.1 Multi-scale Chapman-Enskog expansion . . . . . . . . . . . . . . . . . . 73
A.2 Applying conservation properties . . . . . . . . . . . . . . . . . . . . . . 75
A.3 Resolving 2nd and 3rd order moments . . . . . . . . . . . . . . . . . . . 78
A.4 Finding Navier-Stokes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
A.5 Final notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
B Miscellaneous derivations 85
B.1 From one Navier-Stokes formulation to another . . . . . . . . . . . . . . 85
B.1.1 Viscosity errors in lattice Boltzmann . . . . . . . . . . . . . . . . 86
B.2 Viscosity-damped waves from an infinite line source . . . . . . . . . . . 87
C Code 89
C.1 Verification of LGA lattice isotropy . . . . . . . . . . . . . . . . . . . . . 90
C.2 Verification of LBM lattice isotropy. . . . . . . . . . . . . . . . . . . . . 92
Bibliography 95