A domain decomposition method for the efficient direct simulation of aeroacoustic problems [Elektronische Ressource] / by Jens Utzmann

Jens Utzmann

Informations

• Publié par: universitat_stuttgart
• Publié le: 01 janvier 2009
• Nombre de lectures: 18
• Langue: English
• Poids de l'ouvrage: 16 Mo

Extrait

A Domain Decomposition Method
for the Efficient Direct Simulation
of Aeroacoustic Problems
A thesis accepted by the
Faculty of Aerospace Engineering and Geodesy
of the Universit¨at Stuttgart
in partial fulfilment of the requirements for the degree of
Doctor of Engineering Sciences (Dr.-Ing.)
by
Jens Utzmann
born in Bamberg
Main-referee : Prof. Dr. Claus-Dieter Munz
Co-referee : Prof. Dr. Eric Sonnendru¨cker
Date of defence : 01.12.2008
Institut fu¨r Aerodynamik und Gasdynamik
Universit¨at Stuttgart
2008”In den Wissenschaften ist viel Gewisses, sobald man sich von den
Ausnahmen nicht irre machen l¨aßt und die Probleme zu ehren weiß.”
Johann Wolfgang von Goethe
”D’OH!”
Homer J. Simpson
iiiPreface
IwouldliketothankmydoctoralsupervisorProf. Dr. Claus-DieterMunz,who
managed to spark my interest in the field of numerical methods already in my
years of study. He supported and arranged my stay in Ann Arbor, Michigan,
USA, where I wrote my diploma thesis under theinspiring supervision of Prof.
Philip L. Roe. After that, I received a lot of support and thought-provoking
impulses from him during my time as a Ph.D. candidate at the IAG. There,
especially the informal and creative working environment left much space for
the realization of own ideas.
Also many thanks to my colleagues at the IAG, they were always available
for discussions, offered a lot of input and made my time at the institute a very
enjoyableone. SpecialthankstoHaraldKlimachfrom theHLRS:Hisexpertise
and humor helped me out more than once during debugging sessions.
This work was financedbytheDeutscheForschungsgemeinschaft (DFG) in the
framework of the project C8 in the SFB 404 ”Multifield Problems in Solid
and Fluid Mechanics” and the German-French DFG-CNRS Research Group
FOR 508, Noise Generation in Turbulent Flows. Thank you for supporting me
in these projects, which opened the doors to a very interesting community of
researchers.
stStuttgart, 1 of October 2008
Jens Utzmann
ivContents
Symbols vii
Abbreviations x
Kurzfassung xi
Abstract xiii
1 Introduction 1
1.1 Domain Decomposition Methods: State of the Art . . . . . . . 2
1.2 Heterogeneous Domain Decomposition by Schwartzkopff . . . . 6
1.3 Aims of this Work . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Domain Decomposition 11
2.1 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 ADER Discontinuous Galerkin Schemes . . . . . . . . . 13
2.1.2 ADER Finite Volume Schemes . . . . . . . . . . . . . . 16
2.1.3 ADER and Taylor Finite Difference Schemes . . . . . . 20
2.1.4 The Space-Time Expansion Finite Volume Method . . . 23
2.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.2.1 Linearized Euler Equations . . . . . . . . . . . . . . . . 56
2.2.2 Euler Equations . . . . . . . . . . . . . . . . . . . . . . 56
2.2.3 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . 57
2.2.4 State Vector Conversion . . . . . . . . . . . . . . . . . . 58
2.3 The Coupling between Grids . . . . . . . . . . . . . . . . . . . 60
2.3.1 Grid Configurations . . . . . . . . . . . . . . . . . . . . 60
2.3.2 Interpolation Points . . . . . . . . . . . . . . . . . . . . 63
2.3.3 Structured Source Domains . . . . . . . . . . . . . . . . 67
2.3.4 Unstructured Source Domains. . . . . . . . . . . . . . . 72
2.3.5 Interpolation Stencil Symmetry . . . . . . . . . . . . . . 75
2.3.6 Setting the Ghost Elements . . . . . . . . . . . . . . . . 79
v2.3.7 Pre-Integration . . . . . . . . . . . . . . . . . . . . . . . 81
2.4 The Coupling of Different Time Steps . . . . . . . . . . . . . . 85
2.5 The Implementation of the KOP3D Framework . . . . . . . . . 88
2.6 The Coupling of Different System Architectures . . . . . . . . . 91
2.6.1 Connection to External Codes . . . . . . . . . . . . . . 91
2.6.2 Code-Internal Distribution onto Different Systems . . . 100
2.7 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
2.7.1 Global Convergence . . . . . . . . . . . . . . . . . . . . 101
2.7.2 High-Frequency Perturbations. . . . . . . . . . . . . . . 108
2.7.3 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.7.4 Efficiency Estimates . . . . . . . . . . . . . . . . . . . . 125
3 Numerical Examples 127
3.1 Multiple Cylinder Scattering . . . . . . . . . . . . . . . . . . . 127
3.2 Von Karman Vortex Street . . . . . . . . . . . . . . . . . . . . 135
3.3 Sphere Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 143
3.4 Supersonic Free Jet . . . . . . . . . . . . . . . . . . . . . . . . . 150
4 Conclusions 157
A Convergence Tables 159
Bibliography 173
List of Tables 185
List of Figures 187
Lebenslauf 190
viSymbols
Symbols
O Order of the numerical scheme in space and time
a Maximum convection speed
c Speed of sound
c , c Specific heatsp v
d Dimension
f Frequency / function
~f Flux in x-, y- and z-direction of a scalar variable
g Numerical flux of a variable in normal direction
i Spatial Gauss point index in a ghost elementGP
i Stencil element indexS
~k, k Wave number and wave number vector
l Minimum inner circle or sphere radiusmin
n Refraction index
n Number of degrees of freedomDOF
n Number of spatial Gauss points in a ghost elementGP
n Number of spatial GPs in a domain elementλ
n Number of temporal GPs in a domain elementξ
n Polynomial degreePoly
n Number of stencil cellsS
n Number of state vector variablesVar
n Number of different interpolation stencilsW
~n Normal vector
p Pressure
r Radius
s Sponge power
t Time
u, v, w Velocity component in x-, y- and z-direction
u General scalar state
v Group velocityG
~v Velocity vector
w Characteristic variable
viiSymbols
w Reconstructed WENO polynomialWENO
A Amplitude
A,B,C Jacobians in x-, y- and z-direction
C , C Lift and Drag coefficientL D
CK CK procedure for the kth time derivativek
D Diameter
~ ~ ~F, G, H Flux vectors in x-, y- and z-direction
I , I , I Intensity of impinging, reflected, transmitted waveI R T
J Jacobian of the transformation matrix
L Lagrangian polynomial / sponge thickness
L , L , L Error norms1 2 ∞
M Number of Gauss integration points in 1D
Ma Mach number
N Maximum degree of the basis functions
N Number of mesh elements in x-directionΔh
Pr Prandtl number
R Specific gas constant or reflection index
Re Reynolds number
S Smoothing factorF
~S Right-hand side source term
T Temperature / period length / transmission index
~U State vector
V Volume
~X Vector of spatial coordinates
~X Coordinate of Gauss point ii GPGP
T~X Target coordinate
X Transposed vector of monomials
T
γ Ratio of specific heats
λ Wave length
λ , ǫ, r WENO parametersC
~λ Vector of Eigenvalues
Dynamic viscosity
ν Kinematic viscosity
ω Angular velocity
viiiSymbols
ω Weight of Gauss point ii GPGP
ω , ω˜ Normalized and non-norm. nonlin. WENO weighti i
ω Weight of spatial Gauss point λλ
ω Weight of temporal Gauss point ξξ
ρ Density
ρE Total energy per mass unit
σ Sponge parameter
σ WENO oscillation indicatori
nτ Relative time with respect to time level t
τ Viscous stress tensor
ξ, η, ζ Coordinates in the reference coordinate system
Δh Characteristic element size
Δt Time step
Δx, Δy, Δz Size of a rectangular element in x-, y-, z-direction
Γ Coupling boundary
Ω Domain volume / control volume
∂Ω Domain boundary / control volume boundary
Φ Basis and test function
() Ambient flow variable∞
() Mean flow variable0
′() Perturbation variable
() , () , () , () Time and space derivatives of a variablet x y z
±() Value at the left (-) / right (+) side of an interface
() , () Value at the left / right cell interface (x-index i)1 1i− i+2 2
() Value for an element with the struct. indices i,j,kijk
n c() , () Value at time level n and at current time level
ˆ() Degree of freedom (DG) / amplitude
T() Transposed
() TargetT
~(), () Vector
() Matrix
ixAbbreviations
Abbreviations
ADER Arbitrary high order using derivatives
CAA Computational aeroacoustics
CFD Computational fluid dynamics
CFL Courant-Friedrichs-Levy number
CK Cauchy-Kovalevskaja
CPU Central processing unit
DMR Double Mach reflection
DNC Direct noise computation
DNS Direct numerical simulation
DG Discontinuous Galerkin
DOF Degree of freedom
DRP Dispersion relation preserving
EE Euler equations
ENO Essentially non-oscillatory
FD Finite difference
Flop Floating point operations
FV Finite volume
GP Gauss integration point
GRP Generalized Riemann problem
HLLE Harten, Lax, Van Leer, Einfeldt
LEE Linearized Euler equations
LTS Local time stepping
N.-S. Navier-Stokes
PDE Partial differential equation
PPW Points per wavelength
QF Quadrature-free
Rec-FV Reconstructed finite volume
RK Runge-Kutta
RMS Root mean square
RP Riemann problem
STE Space-time expansion
TVD Total variation diminishing
WENO Weighted essentially non-oscillatory
x