Visualization of complex three-dimensional flow structures [Elektronische Ressource] / von Christoph Garth
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Visualization of complex three-dimensional flow structures [Elektronische Ressource] / von Christoph Garth

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Visulization of ComplexThree-Dimensional Flow StructuresFachbereich Informatik der Technischen Universit¨at KaiserslauternZur Erlangung des akademischen GradesDoktor der Naturwissenschaften (Dr. rer. nat.)eingereichte Dissertation vonChristoph GarthKaiserslautern, 27. Juni 2007Datum der wissenschaftlichen Aussprache: 19. Oktober 2007Dekan: Prof. Dr. Reinhard GotzheinVorsitzender der Prufufungsk¨ ommission: Prof. Dr. Klaus MadlenerErster Berichterstatter: Prof. Dr. Hans HagenZweiter Berichterstatter: Prof. Dr. Gerik ScheuermannAbstractThe visualization of numerical fluid flow datasets is essential to the engineeringprocesses that motivate their computational simulation. To address the need forvisual representations that convey meaningful relations and enable a deep under-standingofflowstructures,thedisciplineofFlowVisualizationhasproducedmanymethodsandschemesthataretailoredtoavarietyofvisualizationtasks. Theeverincreasing complexity of modern flow simulations, however, puts an enormous de-mand on these methods. The study of vortex breakdown, for example, which isa highly transient and inherently three-dimensional flow pattern with substantialimpact wherever it appears, has driven current techniques to their limits. In thisthesis, we propose several novel visualization methods that significantly advancethe state of the art in the visualization of complex flow structures.

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Published 01 January 2007
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Visulization of Complex
Three-Dimensional Flow Structures
Fachbereich Informatik der Technischen Universit¨at Kaiserslautern
Zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften (Dr. rer. nat.)
eingereichte Dissertation von
Christoph Garth
Kaiserslautern, 27. Juni 2007
Datum der wissenschaftlichen Aussprache: 19. Oktober 2007
Dekan: Prof. Dr. Reinhard Gotzhein
Vorsitzender der Prufufungsk¨ ommission: Prof. Dr. Klaus Madlener
Erster Berichterstatter: Prof. Dr. Hans Hagen
Zweiter Berichterstatter: Prof. Dr. Gerik ScheuermannAbstract
The visualization of numerical fluid flow datasets is essential to the engineering
processes that motivate their computational simulation. To address the need for
visual representations that convey meaningful relations and enable a deep under-
standingofflowstructures,thedisciplineofFlowVisualizationhasproducedmany
methodsandschemesthataretailoredtoavarietyofvisualizationtasks. Theever
increasing complexity of modern flow simulations, however, puts an enormous de-
mand on these methods. The study of vortex breakdown, for example, which is
a highly transient and inherently three-dimensional flow pattern with substantial
impact wherever it appears, has driven current techniques to their limits. In this
thesis, we propose several novel visualization methods that significantly advance
the state of the art in the visualization of complex flow structures.
First, we propose a novel scheme for the construction of stream surfaces from
the trajectories of particles embedded in a flow. These surfaces are extremely
useful since they naturally exploit coherence between neighboring trajectories and
are highly illustrative in nature. We overcome the limitations of existing stream
surface algorithms that yield poor results in complex flows, and show how the
resulting surfaces can be used a building blocks for advanced flow visualization
techniques.
Moreover, we present a visualization method that is based on moving section
planes that travel through a dataset and sample the flow. By considering the
changes to the flow topology on the plane as it moves, we obtain a method of
visualizingtopologicalstructuresinthree-dimensionalflowsthatarenotaccessible
byconventionaltopologicalmethods. Onthesamealgorithmicbasis,weconstruct
an algorithm for the tracking of critical points in such flows, thereby enabling the
treatment of time-dependent datasets.
Last, weaddresssomeproblemswiththerecentlyintroducedLagrangiantech-
niques. While conceptually elegant and generally applicable, they suffer from an
enormous computational cost that we significantly use by developing an adaptive
approximation algorithm. This allows the application of such methods on very
large and complex numerical simulations.
Throughout this thesis, we will be concerned with flow visualization aspect of
general practical significance but we will particularly emphasize the remarkably
challenging visualization of the vortex breakdown phenomenon.Acknowledgements
I am grateful to have the chance to thank all those people who have helped and
supported me working on this dissertation. First of all, I would like to express my
sincerethankfulnesstoHansHagenforhisadviceandsupportduringmydoctoral
endeavor. As my doctoral advisor, he provided me with valuable suggestions and
comments in both scientific and non-scientific matters. I would also like to thank
myco-advisorGerikScheuermannforprovidingongoingsupportandmuchneeded
perspective.
AmongthemostpleasingaspectsofscientificresearchIconsidertheintellectual
exchange with collaborating researchers. First and foremost, I wish to express my
gratitude to Xavier Tricoche for the intense intellectual exchange he provided and
themanyminutedetailsofscientificpracticeIwasabletolearnfromhim, inspite
ofgeographicalseparationbytheAtlanticOcean. Furthermore,Iamalsoindebted
to my friends and colleagues at both the University of Kaiserslautern and the
University of Leipzig, most notably Inga Scheler and Tom Bobach, for providing
the friendly, collegial and supportive atmosphere that carried me throughout the
genesis of this work. I also wish to thank Markus Rut¨ ten, who, by providing me
with datasets and discussion, made possible much of this thesis.
Last-butcertainlynotleast-Iwishtothankmyfamily: myparents, fortheir
unwavering yet unobtrusive support, and my wife Leonie and my son Alexander,
for enduring many moods and working nights, for their ongoing encouragement
and confidence, and for showing me that anything is possible.Contents
1 A Brief Introduction to Flow Visualization 5
1.1 Modeling and Simulation . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 The Navier-Stokes Equation . . . . . . . . . . . . . . . . . . 6
1.1.2 Basic Equation . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Numerical Representation . . . . . . . . . . . . . . . . . . . 8
1.1.4 Lagrangian and Eulerian Perspectives . . . . . . . . . . . . . 9
1.1.5 Flow Features . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Feature-Based Visualization . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.2 Separation and Attachment Lines . . . . . . . . . . . . . . . 12
1.3 Topology-Based Visualization . . . . . . . . . . . . . . . . . . . . . 14
1.4 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.1 Delta Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.2 Rotating Lid Cylinder . . . . . . . . . . . . . . . . . . . . . 15
1.4.3 High-Speed Train . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.4 K´arm´an vortex street . . . . . . . . . . . . . . . . . . . . . . 16
2 Vector Fields and Dynamical Systems 17
2.1 Basic Definitions and Fundamental Properties . . . . . . . . . . . . 17
2.2 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Orbits and Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.1 Linear Vector Fields . . . . . . . . . . . . . . . . . . . . . . 23
2.4.2 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Topological Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6 Poincar´e Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.7 Parameter-dependent Dynamical Systems. . . . . . . . . . . . . . . 33
2.8 A Note on Divergence-Free Vector Fields . . . . . . . . . . . . . . . 37
2.9 Lyapunov Exponents and Chaos . . . . . . . . . . . . . . . . . . . . 37
2.10 Numerical Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.10.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.10.2 Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . 40
i3 Stream Surfaces 43
3.1 Formal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 An Overview of Computational Algorithms . . . . . . . . . . . . . . 45
3.3 Advancing Front Methods . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.1 Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.2 Front Resolution Control . . . . . . . . . . . . . . . . . . . . 48
3.3.3 Hultquist’s Implementation . . . . . . . . . . . . . . . . . . 48
3.4 High-Quality Stream Surface Computation . . . . . . . . . . . . . . 50
3.4.1 Improved Streamline Integration. . . . . . . . . . . . . . . . 51
3.4.2 Arc Length Sampling . . . . . . . . . . . . . . . . . . . . . . 51
3.4.3 Front Resolution Control . . . . . . . . . . . . . . . . . . . . 53
3.4.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.5 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.6 Comparison to State Of The Art . . . . . . . . . . . . . . . 57
3.5 Stream Surfaces as
Visualization Building Blocks . . . . . . . . . . . . . . . . . . . . . 60
3.5.1 Enhanced Color Mapping . . . . . . . . . . . . . . . . . . . 60
3.5.2 Geometric Vortex Extraction and Verification . . . . . . . . 62
3.5.3 Miscellaneous Applications . . . . . . . . . . . . . . . . . . . 65
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Topology-Based Methods for Three-Dimensional Datasets 67
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Topology Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.1 Linear Interpolation . . . . . . . . . . . . . . . . . . . . . . 69
4.3.2 Critical Point Paths . . . . . . . . . . . . . . . . . . . . . . 70
4.3.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.4 Practical Concerns . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 Tracking Extremal Values of a Scalar Field . . . . . . . . . . . . . . 74
4.5 Moving Section Planes . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5.1 Plane Trajectory . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5.2 Planar Resampling and Projection . . . . . . . . . . . . . . 76
4.5.3 Section Plane Tracking . . . . . . . . . . . . . . . . . . . . . 76
4.5.4 Typical Trajectories . . . . . . . . . . . . . . . . . . . . . . 77
4.5.5 Plane Orientation . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5.6 In-Context Visualization using Volume Rendering . . . . . . 79
4.6 Visualization of Vortices and Vortex Breakdown . . . . . . . . . . . 80
4.6.1 Rotating Lid Cylinder . . . . . . . . . . . . . . . . . . . . . 80
4.6.2 Delta Wing dataset . . . . . . . . . . . . . . . . . . . . . . . 82
4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
ii5 Lagrangian Techniques 89
5.1 The Finite Time Lyapunov Exponent . . . . . . . . . . . . . . . . . 90
5.2 Lagrangian Coherent Structures . . . . . . . . . . . . . . . . . . . . 91
5.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.4 Numerical Realization . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.4.1 Incremental Approximation of Maps . . . . . . . . . . . . . 93
5.4.2 Efficient FTLE Computation . . . . . . . . . . . . . . . . . 96
5.4.3 Ridge Extraction . . . . . . . . . . . . . . . . . . . . . . . . 97
5.5 Visualization of Coherent Flow Structures . . . . . . . . . . . . . . 98
5.5.1 Three-dimensional FTLE . . . . . . . . . . . . . . . . . . . . 99
5.5.2 FTLE on Planar Sections . . . . . . . . . . . . . . . . . . . 100
5.6 Visualization Results . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.6.1 Choice of Parameters . . . . . . . . . . . . . . . . . . . . . . 102
5.7 Performance Analysis and Discussion . . . . . . . . . . . . . . . . . 105
5.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
A Pseudocode 111
A.1 TheDopri5 Integration Scheme . . . . . . . . . . . . . . . . . . . 111
A.2 Improved Stream Surface Algorithm. . . . . . . . . . . . . . . . . . 118
Bibliography 127
Curriculum Vitae 137
iiiiv