Asymptotic analysis of Lattice Boltzmann method for fluid-structure interaction problems [Elektronische Ressource] / Alfonso Caiazzo
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Asymptotic analysis of Lattice Boltzmann method for fluid-structure interaction problems [Elektronische Ressource] / Alfonso Caiazzo

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Published 01 January 2007
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Department of MathematicsScuola Normale Superiore
PISA
Asymptotic Analysis of lattice Boltzmann
method for Fluid-Structure interaction problems
ALFONSO CAIAZZO
Vom Fachbereich Mathematik
der Universit at KaiserslauternTesi di Perfezionamento in Matematica
zur Verleihung des akademischen Gradesper la Tecnologia e l’Industria
Doktor der Naturwissenschaften
(Dr. rer. nat.) genehmigte Dissertation
REFEREES:
Prof. Dr. Michael JUNK
Prof. Dr. Li-Shi LUO
Prof. Dr. Sauro SUCCI
Prof. Dr. Marcello ANILE
Prof. Dr. Axel KLAR
Datum der Disputation: 06.02.2007Acknowledgements
I am more than grateful to Prof. Michael Junk, with whom I learned a lot, and
not only about the \Asymptotic Analysis" and the \lattice Boltzmann method".
Thanks for all the time he spent with me in several discussions, held in di eren t
o ces, di eren t trains and through di eren t telephones.
The work has been nancially and technically supported by the "Scuola Nor-
male Superiore" of Pisa, the Technische Universit at of Kaiserslautern and the
Fraunhofer "Institut fur Techno- und Wirtschaftsmathematik" (ITWM) of Kaiser-
slautern. Particular thanks to Prof. Marcello Anile and Prof. Helmut Neunzert,
who proposed me to complete the PhD in Germany, and to Prof. Axel Klar, who
supervised me in Kaiserslautern.
My best gratitude goes also to the SKS department, in particular (in order
of appearance) to Prof. Konrad Steiner, Dr. Dirk Kehrwald, and Dr. Guido
Th ommes.
I would like to thank Prof. Li-Shi Luo, for having joint with me some interesting
ideas and Prof. Sauro Succi, who helped me a lot at the beginning of the work.
I thank also Martin Rheinl ander and Zhaoxia Yang, for the cooperation and the
useful discussion we had during these years.
However, the strongest support came from a lot of friends, whose complete list
would probably be longer than the thesis itself. Special thanks to Anna and
Monika, for the useful excursions far from work. I want to thank my parents,
my sister and my family in Italy, for the logistic connections with Germany they
set up for me, all the roommates and the friends I have met in Kaiserslautern,
for everything we shared, Natalia, who has been my very personal sound track
throughout the work, and the part of family I have in Colombia.i
Abstract
The lattice Boltzmann method (LBM) is a numerical solver for the Navier-Stokes
equations, based on an underlying molecular dynamic model. Recently, it has
been extended towards the simulation of complex uids.
We use the asymptotic expansion technique to investigate the standard scheme,
the initialization problem and possible developments towards moving boundary
and uid-structur e interaction problems. At the same time, it will be shown how
the mathematical analysis can be used to understand and improve the algorithm.
First of all, we elaborate the tool \asymptotic analysis", proposing a general
formulation of the technique and explaining the methods and the strategy we
use for the investigation. A rst standard application to the LBM is described,
which leads to the approximation of the Navier-Stokes solution starting from the
lattice Boltzmann equation.
As next, we extend the analysis to investigate origin and dynamics of initial layers.
A class of initialization algorithms to generate accurate initial values within the
LB framework is described in detail. Starting from existing routines, we will be
able to improve the schemes in term of e ciency and accuracy.
Then we study the features of a simple moving boundary LBM. In particular,
we concentrate on the initialization of new uid nodes created by the variations
of the computational uid domain. An overview of existing possible choices is
presented. Performing a careful analysis of the problem we propose a modi ed
algorithm, which produces satisfactory results.
Finally, to set up an LBM for uid structure interaction, e cien t routines to
evaluate forces are required. We describe the Momentum Exchange algorithm
(MEA). Precise accuracy estimates are derived, and the analysis leads to the
construction of an improved method to evaluate the interface stresses.
In conclusion, we test the de ned code and validate the results of the analysis on
several simple benchmarks.
From the theoretical point of view, in the thesis we have developed a general
formulation of the asymptotic expansion, which is expected to o er a more exible
tool in the investigation of numerical methods. The main practical contribution
o ered by this work is the detailed analysis of the numerical method. It allows to
understand and improve the algorithms, and construct new routines, which can
be considered as starting points for future researches.Table of Contents
Introduction 1
1 Asymptotic Analysis of LBE 9
1.1 The Lattice Boltzmann Method . . . . . . . . . . . . . . . . . . . 9
1.1.1 Discrete kinetic framework . . . . . . . . . . . . . . . . . . 10
1.1.2 Two-step dynamics and compact notation . . . . . . . . . 12
1.1.3 BGK approximation and basic algorithm . . . . . . . . . . 14
1.2 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.1 Formal asymptotic: general background . . . . . . . . . . . 18
1.2.2 Asymptotic analysis as optimization problem . . . . . . . . 21
1.2.3 Ordering the approximations . . . . . . . . . . . . . . . . . 24
1.2.4 The heuristic algorithm . . . . . . . . . . . . . . . . . . . . 33
1.3 A regular case: from the LBE to Navier-Stokes . . . . . . . . . . . 41
1.3.1 The regular expansion . . . . . . . . . . . . . . . . . . . . 43
1.3.2 Preparing the regular ansatz for the LBM . . . . . . . . . 46
1.3.3 Asymptotic of the initialization . . . . . . . . . . . . . . . 54
1.3.4 Prediction of the LB solution . . . . . . . . . . . . . . . . 57
1.3.5 Outline of the asymptotic expansion technique . . . . . . . 62
2 Initial layers and Multiscale expansion 67
2.1 Need of irregular expansions . . . . . . . . . . . . . . . . . . . . . 68
2.1.1 Equilibrium initial conditions . . . . . . . . . . . . . . . . 68
2.1.2 Extending the regular ansatz . . . . . . . . . . . . . . . . 69
2.2 Two-scales expansion . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.2.1 Projected algorithm . . . . . . . . . . . . . . . . . . . . . . 72
2.2.2 Preparing the ansatz . . . . . . . . . . . . . . . . . . . . . 75
2.2.3 Initial layer equations . . . . . . . . . . . . . . . . . . . . . 76
2.2.4 Prediction of initial layers . . . . . . . . . . . . . . . . . . 83
2.3 Discrete time scale . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.3.1 Lack of non equilibrium . . . . . . . . . . . . . . . . . . . 87
iiiiv TABLE OF CONTENTS
2.3.2 TRT models . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3 Analysis of initialization algorithms 97
3.1 The initial layer problem . . . . . . . . . . . . . . . . . . . . . . . 98
3.1.1 Test problems . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.1.2 Lattice Boltzmann initial conditions . . . . . . . . . . . . . 99
3.2 LB Initialization routines (periodic case) . . . . . . . . . . . . . . 100
3.2.1 A rst initialization algorithm . . . . . . . . . . . . . . . . 100
3.2.2 Accelerated . . . . . . . . . . . . . . . . . . . 105
3.3 Initialization for boundary value problems . . . . . . . . . . . . . 108
3.3.1 The Neumann condition for pressure . . . . . . . . . . . . 108
3.3.2 LB-boundary algorithms . . . . . . . . . . . . . 109
3.3.3 Corrected initialization . . . . . . . . . . . . . . . . . . . . 112
3.3.4 Accelerated routines . . . . . . . . . . . . . . . . . . . . . 116
3.4 Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.4.1 Overfrozen LBM . . . . . . . . . . . . . . . . . . . . . . . 119
3.4.2 Higher order initialization . . . . . . . . . . . . . . . . . . 123
4 Moving boundary problems 125
4.1 A Moving boundary lattice Boltzmann . . . . . . . . . . . . . . . 125
4.1.1 Fixed lattice and moving boundaries . . . . . . . . . . . . 125
4.1.2 Description and analysis of the algorithm . . . . . . . . . . 128
4.2 Re ll methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.2.1 Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.2.2 Equilibrium re ll (EQ) . . . . . . . . . . . . . . . . . . . . 138
4.2.3 Interpolation + Advection re ll (IA) . . . . . . . . . . . . 140
4.3 A compromise: Equilibrium + Non equilibrium re ll . . . . . . . 144
4.3.1 Numerical simulations . . . . . . . . . . . . . . . . . . . . 145
5 Fluid structure interaction 147
5.1 The o w model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.1.1 Benchmark: the cylinder-in- ow . . . . . . . . . . . . . . . 149
5.2 Extrapolation approaches . . . . . . . . . . . . . . . . . . . . . . 151
5.3 Momentum exchange algorithm . . . . . . . . . . . . . . . . . . . 157
5.3.1 Numerical tests and asymptotic analysis . . . . . . . . . . 158
5.3.2 Corrected and averaged momentum . . . . . . . . . . . . . 161
5.4 The ME-stress extraction . . . . . . . . . . . . . . . . . . . . . . . 164
5.4.1 Improving the stress evaluation . . . . . . . . . . . . . . . 165
5.4.2 Numerical tests and comparisons . . . . . . . . . . . . . . 169TABLE OF CONTENTS v
5.5 Ultimate numerical simulations . . . . . . . . . . . . . . . . . . . 169
5.5.1 Moving cylinder . . . . . . . . . . . . . . . . . . . . . . . . 169
5.5.2 Periodic elastic b er . . . . . . . . . . . . . . . . . . . . . 172
6 Conclusions 183
6.1 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . 183
6.2 Further ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
A List of principal notations 187
B Accuracy properties of the MEA 189
B.1 Preliminary de nitions . . . . . . . . . . . . . . . . . . . . . . . . 189
B.2 Straight boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . 191
B.2.1 Averaged momentum exchanged . . . . . . . . . . . . . . . 194
B.3 Generalization for curved boundaries . . . . . . . . . . . . . . . . 200
B.3.1 Interface as smooth function of x . . . . . . . . . . . . . . 200
B.3.2 Averaged inclination . . . . . . . . . . . . . . . . . . . . . 201
B.3.3 Smooth interfaces . . . . . . . . . . . . . . . . . . . . . . . 208
B.3.4 Statement for the original MEA . . . . . . . . . . . . . . . 208
B.4 Global force evaluation . . . . . . . . . . . . . . . . . . . . . . . . 210
List of Figures 219
Bibliography 223