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Unstructured h-adaptive finite-volume schemes for compressible viscous fluid flow [Elektronische Ressource] / vorgelegt von Frank Dieter Bramkamp

178 pages
Unstructured h-AdaptiveFinite-Volume Schemes forCompressible Viscous Fluid FlowVon der Fakultat¨ fur¨ Mathematik, Informatik und Naturwissenschaftender Rheinisch-Westfalischen¨ Technischen Hochschule Aachenzur Erlangung des akademischen Grades einesDoktors der Ingenieurwissenschaften genehmigte Dissertationvorgelegt vonDiplom-IngenieurFrank Dieter Bramkampaus Hattingen / DeutschlandBerichter: Universitatsprofessor¨ Dr.-Ing. J. BallmannUniversit¨ Dr. rer. nat. W. DahmenTag der mundlichen¨ Prufung:¨ 26. Juli 2003Diese Dissertation ist auf den Internetseitender Hochschulbibliothek online verfugbar¨ .IIIPrefaceThe present thesis grew out of my work as a research scientist at the Lehr- und Forschungsgebietfur¨ Mechanik, Aachen University of Technology (RWTH Aachen). It was sponsored by the Deut-sche Forschungsgemeinschaft within the frame of the Collaborative Research Center SFB 401 “FlowModulation and Fluid-Structure Interaction at Airplane Wings”.I would like to express my deep gratitude to Prof. Dr.-Ing. Josef Ballmann for his competent guidanceduring the entire period of this research and for giving me the opportunity to work within this fasci-nating area of CFD. I thank Prof. Dr. rer. nat. Wolfgang Dahmen for accepting to be the co-advisorof the thesis.Special thanks go to my colleagues Priv.-Doz. Dr. Siegfried Muller¨ and Dipl.-Math.
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Unstructured h-Adaptive
Finite-Volume Schemes for
Compressible Viscous Fluid Flow
Von der Fakultat¨ fur¨ Mathematik, Informatik und Naturwissenschaften
der Rheinisch-Westfalischen¨ Technischen Hochschule Aachen
zur Erlangung des akademischen Grades eines
Doktors der Ingenieurwissenschaften genehmigte Dissertation
vorgelegt von
Diplom-Ingenieur
Frank Dieter Bramkamp
aus Hattingen / Deutschland
Berichter: Universitatsprofessor¨ Dr.-Ing. J. Ballmann
Universit¨ Dr. rer. nat. W. Dahmen
Tag der mundlichen¨ Prufung:¨ 26. Juli 2003
Diese Dissertation ist auf den Internetseiten
der Hochschulbibliothek online verfugbar¨ .III
Preface
The present thesis grew out of my work as a research scientist at the Lehr- und Forschungsgebiet
fur¨ Mechanik, Aachen University of Technology (RWTH Aachen). It was sponsored by the Deut-
sche Forschungsgemeinschaft within the frame of the Collaborative Research Center SFB 401 “Flow
Modulation and Fluid-Structure Interaction at Airplane Wings”.
I would like to express my deep gratitude to Prof. Dr.-Ing. Josef Ballmann for his competent guidance
during the entire period of this research and for giving me the opportunity to work within this fasci-
nating area of CFD. I thank Prof. Dr. rer. nat. Wolfgang Dahmen for accepting to be the co-advisor
of the thesis.
Special thanks go to my colleagues Priv.-Doz. Dr. Siegfried Muller¨ and Dipl.-Math. Philipp Lamby
for the excellent cooperation during the entire period of the research, many suggestions and countless
hours of work devoted to the QUADFLOW project.
I am indebted to Dr. Ralf Massjung and Dipl.-Ing. Carsten Braun for fruitful discussions on general
CFD-matters, to Dipl.-Math. Michael Hesse for providing several computational grids, to Dipl.-Ing.
Willy Hofmann for the excellent computer support and Sauria Ray, MSc, for programming assistan-
ce. The guidance of Dr.-Ing. Ingo Grotowsky during the early stages of the research is gratefully
acknowledged. I particularly thank Dr. David Zingg for his kind hospitality at the Aerospace Depart-
ment of the University of Toronto.
Finally, I thank all members of the Lehr- und Forschungsgebiet fur¨ Mechanik for their support and
the enjoyable work environment.Contents V
Contents
List of Symbols IX
1 Introduction 1
1.1 Scope of the Thesis . .................................. 3
2 Description of Fluid Motion 7
2.1 Governing Equations 7
2.2 Material Properties . 8
2.3 Dimensionless Form of the Governing Equations . .................. 9
2.4 The Closure Problem 1
2.4.1 Averaging Techniques . . . .......................... 12
2.4.2 Closure Approach . .............................. 13
2.5 The Spalart-Allmaras Turbulence Model . . ...................... 14
2.6 Boundary Conditions .................................. 16
3 Numerical Method 19
3.1 The QUADFLOW Program System .......................... 19
3.2 Adaptation Criteria . 21
3.3 Mesh Generation . . .................................. 2
3.3.1 Parametric Meshes . .............................. 2
3.3.2 B-Spline Representation . . 23
3.3.3 Multiblock Concept 24
3.4 Finite Volume Discretization 25
3.5 Data Structures . . . .................................. 27
3.5.1 Concept of Polyhedral Elements . . ...................... 27
3.5.2 Classification of Grid Objects . . . 28
3.5.3 Interconnection between Grid Objects . . . .................. 29
3.6 Discretization of Convective Fluxes .......................... 30VI Contents
3.6.1 Upwind Methods . . .............................. 30
3.6.1.1 HLLC Flux-Difference Splitting .................. 31
3.6.1.2 Flux–Vector Splitting (van Leer, Hanel/Schw¨ ane) . . . ...... 32
3.6.1.3 AUSMDV(P) (Edwards/Liou) . . 3
3.6.2 Higher Order Method in Space . . . ...................... 36
3.6.2.1 Gradient Approximation for Higher Order Methods . . ...... 37
3.6.2.2 Monotonicity Enforcement . . . .................. 39
3.7 Discretization of Diffusive Fluxes . .......................... 42
3.7.1 Tools for the Analysis of Diffusive Flux Formulae .............. 42
3.7.2 Gradient Estimation for Diffusive Fluxes . 43
3.8 Numerical Treatment of Boundary Conditions . . . .................. 46
3.8.1 Impermeable Slip Wall . . . 46
3.8.2 No-Slip Wall . 46
3.8.3 Far-Field Boundaries .............................. 47
3.9 Time Integration . . . .................................. 51
3.9.1 Local Preconditioning . . . .......................... 51
3.9.2 Survey of Time Integration Methods ...................... 53
3.9.3 Explicit Runge-Kutta Scheme . . . 53
3.9.4 Implicit Time Integration . . 54
3.9.4.1 Jacobian Evaluation . . . 56
3.9.4.2 Iterative Solution of Linear Systems . . .............. 62
3.9.5 Selection of the Time Step . .......................... 63
4 Numerical Results 65
4.1 Aerodynamic Coefficients . .............................. 6
4.2 Convergence Study of Newton-Krylov Solver . . . .................. 67
4.2.1 Subsonic NACA0012 airfoil 68
4.2.2 SFB 401 Cruise Configuration in the Transonic Regime . .......... 71
4.2.3 Turbulent Flow about RAE2822 Airfoil . . 7
4.3 Two-Dimensional Inviscid Flows . . .......................... 80
4.3.1 Ringleb Flow .................................. 80Contents VII
4.3.2 Sod’s Shock Tube Problem . .......................... 84
4.3.3 Two-Dimensional Wave Interaction ...................... 86
4.3.4 Transonic Channel Flow . . 88
4.3.5 Subsonic NACA0012 Airfoil 91
4.3.6 Transonic NACA0012 Airfoil . . . 92
4.3.7 Transonic SFB 401 Cruise Configuration . .................. 97
4.3.8 Hypersonic Flow over Double-Ellipse . . .101
4.3.9 Oscillating NACA0012 Airfoil . . . ......................103
4.3.10 Low Mach Number Flows . ..........................107
4.3.10.1 Channel Flow . .107
4.3.10.2 NACA0012 Airfoil . . .109
4.4 Three-Dimensional Inviscid Flows .14
4.4.1 Swept Wing in Channel . .14
4.4.2 ONERA-M6 Wing . ..............................17
4.4.3 Generic F15 Aircraft Fighter ..........................120
4.5 Two-Dimensional Viscous Flows . .123
4.5.1 Laminar Boundary Layer . .123
4.5.2 Laminar Flow over NACA0012 Airfoil . . ..................127
4.5.3 Turbulent Boundary Layer .129
4.5.4 Turbulent Flow over RAE 2822 Airfoil . .132
4.5.5 Three-Element Airfoil System . . . ......................136
5 Conclusions 145List of Symbols IX
List of Symbols
Scalar Variables
A : Surface
a : Speed of sound
α : Angle of attack
α : Runge-Kutta coefficienti
b : Span
c : Chord length
c : Skin friction coefficientf
c : von Karm` an` constantκ
c : Specific heat at constant pressure, pressure coefficientp
c : heat at volumev
C : Aerodynamic drag coefficientD
C : lift coefficientL
C : Aerodynamic moment coefficientM
CFL : Courant-Friedrichs-Levy number
d : Wall distance
δ : Boundary-layer thickness
δ : Kronecker deltaik
e : Specific internal energy
e : total energytot
: Threshold of multiscale analysis
: Constant of Venkatakrishnan limiterV
γ : Specific heat ratio
Γ : Circulation
h : Specific enthalpy
h : total enthalpytot
k : Specific kinetic energy of turbulent fluctuations
κ : Thermal conductivity
L : Level of grid adaptation
l : Length
λ : Eigenvalue
µ : Molecular viscosity
tµ : Eddy viscosity
µ : VolumeV
N(Ω ) : Stencil of cell Ωi iX List of Symbols
n : Component of the normal unit vector ni
ν : Kinematic viscosity
tν : eddy viscosity
∗ν˜ : Intermediate variable of Spalart-Allmaras turbulence model
p : Pressure
φ : Limiter
q : Component of the heat flux vectori
tq : of the turbulent heat flux vectori
R : Perfect gas constant
ρ : Density
kR : Reconstruction operator
N(Ω )
i
ω : Absolute value of vorticity
S : Component of mean strain-rate tensorij
T : Temperature
T : Sutherland constants
t : Time
τ : Component of Reynolds-stress tensorij
U : Reference velocity of low Mach number preconditionerr
V : Volume of cell Ω
∂Ω : Boundary of cell Ω
u, v, w, v : Cartesian velocity componenti
x, y, z, x : coordinatesi
ix : General
+u : Dimensionless scaled velocity
+y : scaled wall distance
Vector notations
c c
F , F : Convective fluxesi
d d
F , F : Diffusive fluxesi
g : Covariant base vectork
k
g : Contravariant base vector
n : Normal unit vector
Q : Source term vector
q : Heat flux vector
R : Residual
u(x,t) : Vector of conservative variables
v : Instantaneous velocity vector
x : Position vector
ω : Vorticity, angular velocity vector

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