Implicit finite element schemes for compressible gas and particle-laden gas flows [Elektronische Ressource] / vorgelegt von Marcel Gurris

Sujets

• Mathematics

Informations

• Publié par: technischen_universitat_dortmund
• Publié le: 01 janvier 2010
• Nombre de lectures: 18
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

ImplicitFiniteElementSchemesfor
CompressibleGasandParticle-Laden
GasFlows
Dissertation
zur Erlangung des Grades eines
Doktors der Naturwissenschaften
Der Fakultat¨ fur¨ Mathematik
der Technischen Universitat¨ Dortmund
vorgelegt von
Diplom-Mathematiker Marcel GurrisImplicit Finite Element Schemes for Compressible Gas and Particle-
LadenGasFlows
Marcel Gurris
Gutachter:
Prof. Dr. D. Kuzmin (1. Gutachter, Betreuer)
Prof. Dr. S. Turek (2. Gutachter)
Dissertation eingereicht am: 15.12.2009To my parents
Marion and Franz-Josef
and my love
Kim VerenaAcknowledgments
Many people contributed directly or indirectly to the preparation of this thesis and it is
impossible to thank everyone individually. First of all, I have to express my most sin-
cere gratitude to my first supervisor Professor Dmitri Kuzmin for his invaluable sup-
port and guidance of my graduate and undergraduate study during the recent years.
Without his commitment, I would never have finished my thesis yet.
At the same time I am deeply indebted to my second supervisor Professor Stefan Turek
for offering me the opportunity to work in the research project SFB 708, his academic
advice, and sharing his experience in numerical mathematics with me.
I am grateful to all my colleagues at ’LS III’. You were a great help in all topics corre-
sponding to our finite element toolbox FEATFLOW. I would like to thank Dr. Matthias
Moller¨ for fruitful discussions on topics related to the Euler equations. My sincerest
appreciation goes to Dr. Shu-Ren Hysing for proofreading my manuscripts and giving
valuable feedback. The assistance with mesh generation of Jens F. Acker was a great
help for the simulation of flows in complex geometries with curved boundaries.
Moreover, I would like to thank the DFG (German Research Association, SFB 708, TP
B7) for the financial support.
My deepest gratitude goes to the members of my family, in particular my parents Mar-
ion and Franz-Josef. Their unimaginable support and backing during my whole life
and their encouragement and patience during the time of my education were impor-
tant prerequisites along my way to advanced topics in computational fluid dynamics.
At the same time my gratefulness goes to my partner in life Kim Verena for her commit-
ment, patience, and unconditional kindness. I am very grateful for the love you offer to
me.
Dortmund, December 2009
Marcel GurrisviContents
0 Introduction 1
0.1 Research Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
I Modeling of Compressible Gas and Particle-Laden Gas Flows 7
1 The Euler Equations: Modeling of a Pure Compressible Gas 9
1.1 Mathematical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.1 Definitions and Equations of State . . . . . . . . . . . . . . . . . . 10
1.1.2 Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.3 Homogeneity Property . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Nondimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 The Two-Fluid Model: Modeling of a Particle-Laden Gas 15
2.1 Averaging of the Continuity Equation . . . . . . . . . . . . . . . . . . . . 18
2.2 A of the Momentum Equations . . . . . . . . . . . . . . . . . . . 19
2.3 Averaging of the Energy Equation . . . . . . . . . . . . . . . . . . . . . . 19
2.4 The Computational Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.1 Remarks on Conservation . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.2 Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.3 Interfacial Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.4 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4.5 Interfacial Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.5.1 Drag Force . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.5.2 Virtual Mass Force . . . . . . . . . . . . . . . . . . . . . . 26
2.4.5.3 Basset Force . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.5.4 Lift Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.5.5 Pertinence of Interfacial Forces . . . . . . . . . . . . . . . 27
2.4.6 Interfacial Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.7 Summary of the Equations . . . . . . . . . . . . . . . . . . . . . . 31
II Numerical Methods for Compressible Gas and Particle-Laden
Gas Flows 33
3 Scalar Conservation Laws 35
3.1 Physical and Numerical Criteria . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.1 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.2 Total Variation Diminishing Schemes . . . . . . . . . . . . . . . . 39viii Contents
3.1.3 Local Extremum Diminishing Schemes . . . . . . . . . . . . . . . 40
3.1.4 Positivity Preservation . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.5 Conservation Property . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 High-Resolution Schemes Based on Algebraic Flux Correction . . . . . . 42
3.2.1 High-Order Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.1.1 Discrete Conservation . . . . . . . . . . . . . . . . . . . . 46
3.2.1.2 Conservative Flux Decomposition . . . . . . . . . . . . . 47
3.2.2 Discrete Diffusion Operators . . . . . . . . . . . . . . . . . . . . . 48
3.2.3 Low-Order Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.4 Upwind-Biased Flux Limiting of TVD Type . . . . . . . . . . . . . 50
4 Coupled Transport Equations 53
4.1 Mathematical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Discretization and Stabilization . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.1 High-Order Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.2 Low-Order and Algebraic Flux Correction . . . . . . . . 56
5 Euler Equations 59
5.1 Vectorial High-Resolution Scheme Based on Algebraic Flux Correction . 60
5.1.1 High-Order Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.1.2 Characteristic Variables . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1.3 Roe Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1.4 LED Criterion . . . . . . . . . . . . . . . . . . . . . 63
5.1.5 Discrete Diffusion Tensors and Low-Order scheme . . . . . . . . 64
5.1.6 Characteristic TVD Type Flux Limiting for the Euler Equations . 67
6 Treatment of Source Terms 71
6.1 Operator Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.1.1 Yanenko Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.1.2 Douglas-Rachford Splitting . . . . . . . . . . . . . . . . . . . . . . 72
6.1.2.1 Source Term Update . . . . . . . . . . . . . . . . . . . . . 73
6.2 Finite Element Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 74
7 Boundary Conditions 77
7.1 Ghost Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.2 Weak Imposition of Boundary Conditions . . . . . . . . . . . . . . . . . . 79
7.3 Definition of the Unit Outer Normal and Tangent . . . . . . . . . . . . . 81
7.4 Evaluation of the Boundary Integral . . . . . . . . . . . . . . . . . . . . . 82
7.5 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.5.1 Riemann Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.5.2 Inflow and Outflow Boundary conditions . . . . . . . . . . . . . . 85
7.5.3 Supersonic Inflow and Outflow Boundary Conditions . . . . . . 85
7.5.4 Subsonic Inflow and . . . . . . . . 85
7.5.5 Free Stream Boundary Conditions . . . . . . . . . . . . . . . . . . 85
7.5.6 Pressure Outlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.5.7 Pressure-Density Inlet . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.5.8 Wall Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 88
7.5.9 Solution of the Boundary Riemann Problem . . . . . . . . . . . . 89
7.6 Two-Fluid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Contents ix
7.6.1 Inlet and Outlet Boundary Conditions . . . . . . . . . . . . . . . . 91
7.6.2 Wall Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 92
8 Fully Coupled Implicit Time Integration 95
8.1 Linearized Backward Euler Scheme . . . . . . . . . . . . . . . . . . . . . . 96
8.2 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.3 Iterative Solution of Stationary Equations . . . . . . . . . . . . . . . . . . 98
8.3.1 Underrelaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.3.2 The Linear Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8.3.3 Computation of the Initial Guess . . . . . . . . . . . . . . . . . . . 100
8.4 The Approximate Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.4.1 Edge-Based Approximate Interior Flux Jacobian . . . . . . . . . . 102
8.4.2 Approximate Boundary Flux Jacobian . . . . . . . . . . . . . . . . 103
8.4.3 Appr Source Term . . . . . . . . . . . . . . . . . 103
8.4.4 Flux Jacobian Structure and Assembly . . . . . . . . . . . . . . . . 104
8.4.5 The Two-Fluid Model Approximate Jacobian . . . . . . . . . . . . 105