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Réduction de modèles pour la simulation, l'estimation et le côntrole d`écoulements, Low order modelling for flow simulation, estimation and control

De
163 pages
Sous la direction de Angelo Iollo, Maria Vittoria Salvetti
Thèse soutenue le 03 février 2010: Università di Pisa, Bordeaux 1
L’objectif est de développer et de tester des instruments peu côuteux pour la simulation, l’estimation et le contrôle d’écoulements. La décomposition orthogonale aux valeurs propres (POD) et une projection de Galerkin des équations sur les modes POD sont utilisées pour construire des modèles d’ordre réduit des équations de Navier-Stokes incompressibles. Dans ce travail, un écoulement autour d’un cylindre carré est considéré en configuration bidimensionnelle et tridimensionnelle. Des actionneurs de soufflage/aspiration sont placés sur la surface du cylindre. Quelques techniques de calibration sont appliquées, fournissant des modèles précis, même pour les écoulements tridimensionnels avec des structures tourbillonaires compliquées. Une méthode d’estimation d’état, impliquant des mesures, est ensuite mise au point pour des écoulements instationnaires. Une calibration multi-dynamique et des techniques d’échantillonnage efficaces sont appliquées, visant à construire des modèles robustes à des variations des paramètres de contrôle. Nous amorçons une analyse de stabilité linéaire en utilisant des modèles d’ordre réduit linéarisés autour d’un état d’équilibre contrôlé. Les techniques présentées sont appliquées à écoulements autour du cylindre carré à des nombres de Reynolds compris entre Re = 40 et Re = 300.
-POD
-Contrôle
-Modélisation d’ordre réduit
-Modèles robustes
The aim is to develop and to test tools having a low computational cost for flow simulation, estimation and control applications. The proper orthogonal decomposition (POD) and a Galerkin projection of the equations onto the POD modes are used to build low order models of the incompressible Navier-Stokes equations. In this work a flow past a square cylinder is considered in two-dimensional and three-dimensional configurations. Two blowing/suction actuators are placed on the surface of the cylinder. Calibration techniques are applied, providing stable and rather accurate models, even for three-dimensional wake flows with complicated patterns. A state estimation method, involving flow measurements, is then developed for unsteady flows. Multi-dynamic calibrations and efficient sampling techniques are applied to build models that are robust to variations of the control parameters. A linear stability analysis by using linearized low order models around a controlled steady state is briefly addressed. The presented techniques are applied to the square cylinder configuration at Reynolds numbers that range between Re = 40 and Re = 300.
-POD
-Robust low-order modelling
-Estimation
-Control
Source: http://www.theses.fr/2010BOR14000/document
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◦N d’ordre : 4000
`THESE
en co-tutelle entre
´L’UNIVERSITE BORDEAUX I
´ ´ECOLE DOCTORALE DE MATHEMATIQUES ET INFORMATIQUE
et
`L’UNIVERSITA DI PISA
SCUOLA DI DOTTORATO IN INGEGNERIA AEROSPAZIALE
pr´esent´ee `a
´L’UNIVERSITE BORDEAUX I
par Edoardo LOMBARDI
pour obtenir le grade de
DOCTEUR
´ ´SPECIALITE : Math´ematiques Appliqu´ees et Calcul Scientifique
CURRICULUM : Fluidodinamica
LOW ORDER MODELLING FOR FLOW SIMULATION,
ESTIMATION AND CONTROL
Th`ese dirig´ee par M. Angelo IOLLO, Professeur
Mme Maria Vittoria SALVETTI, Professeur
commission d’examen :
M. Michel BERGMANN Charg´e de Recherche - INRIA Bordeaux Sud-Ouest
M. Eric BLAYO Professeur - Universit´e Joseph Fourier Rapporteur
M. Simone CAMARRI Chercheur - Universita` di Pisa
M. Thierry COLIN Professeur - Universit´e Bordeaux I
M. Flavio GIANNETTI Chercheur - Universit`a degli studi di Salerno
M. Angelo IOLLO Professeur - Universit´e Bordeaux I Directeur
M. Laurent JACQUIN Directeur de Recherche - DAFE/ONERA Rapporteur
Mme Maria Vittoria SALVETTI Professeur - Universit`a di Pisa DirecteurVerily, this vichyssoise of verbiage veers most verbose...Acknowledgements
Without the support, guidance and patience of a lot of people this thesis would not
have been accomplished.
IwouldliketoexpressmydeepestgratitudetomyadvisorAngeloIollo,forhisguidance,
patience, and most importantly, his friendship during these years. I am deeply grateful
to him for the long discussions on my work and for his continuous presence during the
entire period of my research.
I thank my cotutelle advisor, Maria Vittoria Salvetti, for her guidance and for the fact
to be always there to listen and give advice. I am grateful to her for giving me the
possibility to work in the Departement of Aerospace Engineering in Pise.
I wish to thank for the review and the advices the referees Eric Blayo and Laurent
Jacquin as well as the other members of committee Thierry Colin, Michel Bergmann,
Simone Camarri, Flavio Giannetti.
A very special thank to my colleegues and mainly great friends Manlio and Lido with
whom I am really pleased to work and to spend great days and evenings in Bordeaux
and in Italy.
I am deeply indebted to Mario and I would like to thank for his friendship and his help
especially during the “last dinner”.
A big thank to Jessie for her help and patience in our collaboration and for the infinite
discussions during the coffee breaks.
Despitehisopinionaboutfootball, IamgratefultoMichelforhisprofessionalanduseful
support, in addition to the fun time spent in company.
I would also like to thank Valerio for his ideas and with whom is a great pleasure to
work.
A great thanks the people with I spend a lot of time during these years in Bordeaux,
Luca and his cooking, Poncherello for his good mood, Gatto and Haysam for precioushelp and wonderful welcome in Turin; a thank to the MC2 team in particular Heloise
and her friends of intelligence service, Lisl for nice breaks and conversations, Yannick,
Thomas and Victor for learning quickly my teaching in pool and for hosting me; I would
like to thank Cecile for infecting me with the board game mania.
Iwouldliketothanksl’allemandandilCosimiforthefinesharingofachaoticapartement
in town.
Thanks to Marco who has in the hands the destiny of this final draft.
I would like to thank my friends Toro, Gessi, Barza and Lunino to make me feel Pratese
everytime I came back in Italy.
A very special thanks goes to my family for all their support, encouragement and their
constant presence.
I most want to thank Lisa for keeping herself close, even when far away.This work was funded by the MENRT and INRIA. High performance computational
resources were provided by IDRIS (Orsay, France) and DRIMM/M3PEC (Bordeaux,
France). Local computer facilities were provided mostly by the Laboratory of Applied
Mathematics (MAB) at Universit´e Bordeaux I, France.R´eduction de mod`eles pour la simulation, l’estimation et le cˆontrole
d’´ecoulements
L’objectif est de d´evelopper et de tester des instruments peu couˆteux pour la sim-
ulation, l’estimation et le contrˆole d’´ecoulements. La d´ecomposition orthogonale aux
valeurs propres (POD) et une projection de Galerkin des ´equations sur les modes POD
sont utilis´ees pour construire des mod`eles d’ordre reduit des ´equations de Navier-Stokes
incompressibles. Dans ce travail, un ´ecoulement autour d’un cylindre carr´e est con-
sid´er´e en configuration bidimensionnelle et tridimensionnelle. Des actionneurs de souf-
flage/aspirationsontplac´essurlasurfaceducylindre. Quelquestechniquesdecalibration
sontappliqu´ees,fournissantdesmod`elespr´ecis,mˆemepourles´ecoulementstridimension-
nels avec des structures tourbillonaires compliqu´ees. Une m´ethode d’estimation d’´etat,
impliquant des mesures, est ensuite mise au point pour des ´ecoulements instationnaires.
Une calibration multi-dynamique et des techniques d’´echantillonnage efficaces sont ap-
pliqu´ees, visant a` construire des mod`eles robustes a` des variations des param`etres de
contrˆole. Nousamorc¸onsuneanalysedestabilit´elin´eaireenutilisantdesmod`elesd’ordre
r´eduitlin´earis´esautourd’un´etatd’´equilibrecontrˆol´e. Lestechniquespr´esent´eessontap-
pliqu´eesa`´ecoulementautourducylindrecarr´ea`desnombresdeReynoldscompriscentre
Re= 40 et Re=300.
Motscl´es: POD,mod´elisationd’ordrer´eduit,estimation,mod`elesrobustes,contrˆole
Low order modelling for flow simulation, estimation and control
The aim is to develop and to test tools having a low computational cost for flow
simulation, estimation and control applications. The proper orthogonal decomposition
(POD) and a Galerkin projection of the equations onto the POD modes are used to
build low order models of the incompressible Navier-Stokes equations. In this work a
flow past a square cylinder is considered in two-dimensional and three-dimensional con-
figurations. Two blowing/suction actuators are placed on the surface of the cylinder.
Calibration techniques are applied, providing stable and rather accurate models, even
forthree-dimensionalwakeflowswithcomplicatedpatterns. Astateestimationmethod,
involving flow measurements, is then developed for unsteady flows. Multi-dynamic cal-
ibrations and efficient sampling techniques are applied to build models that are robust
to variations of the control parameters. A linear stability analysis by using linearized
low order models around a controlled steady state is briefly addressed. The presented
techniques are applied to the square cylinder configuration at Reynolds numbers that
range between Re=40 and Re=300.
Key words: POD, low-order modelling, estimation, robust models, controlContents
Contents i
List of Figures iii
1 Introduction 1
2 Background on employed techniques 9
2.1 Low-order modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Proper orthogonal decomposition (POD) . . . . . . . . . . . . . . 9
2.1.2 Low order model of Navier-Stokes equations . . . . . . . . . . . . . 12
2.2 Linear and Quadratic stochastic estimation . . . . . . . . . . . . . . . . . 18
2.3 Spectral stochastic estimation . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 General least-square technique . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Centroidal Voronoi Tessellation . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Flow setup and numerical simulation 25
4 A non-linear observer for an unsteady three-dimensional flow 31
4.1 Flow set up and low order model . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Non-linear observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3.1 Two-dimensional case: Re=150 . . . . . . . . . . . . . . . . . . . 38
4.3.2 Three-dimensional case: Re=300 . . . . . . . . . . . . . . . . . . 41
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5 Robust POD modeling of actuated vortex wake 55
5.1 Flow setup and low order model . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Robust low order models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3.1 Divergence of a classical Reduced Order Model . . . . . . . . . . . 64
5.3.2 Testing model robustness: Re=60 and Re=150 . . . . . . . . . . 66
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
iCONTENTS
6 A Residual based strategy to sample POD database 77
6.1 Reynolds dependent pressure extended reduced order model . . . . . . . 78
6.1.1 Reynolds adaptive pressure extended reduced order model . . . . 78
6.1.2 Calibration procedure . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2 Improvement of the model robustness . . . . . . . . . . . . . . . . . . . . 81
6.2.1 Effect of the Reynolds number variations onto the projection error 83
6.2.2 A residuals based error estimator . . . . . . . . . . . . . . . . . . . 83
6.2.3 A residual based sampling method . . . . . . . . . . . . . . . . . . 85
6.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7 Linearized low-order model of actuated transient flow 93
7.1 POD-basedmodelofthelinearizedNavier-Stokesequationswithfeedback
control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.2 Linearized low-order feedback model . . . . . . . . . . . . . . . . . . . . . 95
7.2.1 Results Re=85 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.3 Design of a control strategy based on the linear model . . . . . . . . . . . 100
7.3.1 Results Re=85 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.3.2 Results Re=150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8 Experimental signal analysis by POD 109
8.1 Application of POD for one-dimensional signals . . . . . . . . . . . . . . . 111
8.2 Application of POD for experimental fluid dynamics signals . . . . . . . . 121
8.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Conclusions 129
Conclusions 133
List of publications 137
Bibliography 139
ii