Development of an unstructured cartesian flow solver [Elektronische Ressource] / vorgelegt von Khalid Mohammad Sultan

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2005
• Nombre de lectures: 18
• Langue: English
• Poids de l'ouvrage: 4 Mo

Extrait

DEVELOPMENT OF AN UNSTRUCTURED
CARTESIAN FLOW SOLVER
Von der Fakult˜ at fur˜ Maschinenwesen der Rheinisch-Wesf˜ alischen Technischen
Hochschule Aachen zur Erlangung des akademischen Grades eines Doktors der
Ingenieurwissenschaften genehmigte Dissertation
vorgelegt von
Khalid Mohammad Sultan
aus Benghazi, Libyen
Berichter: Universit˜ atsprofessor Dr.-Ing. Wolfgang Schr˜ oder
Univ Dieter H˜ anel
Tag der munlic˜ hen Prufung:˜ 02.12.2005
Diese Dissertation ist auf der Internetseiten der Hochschulbibliothek online verfugbar˜Acknowledgements
This thesis was written during my work as a scholarship holder at the Chair of
Fluid Mechanics and Institute of Aerodynamics AIA at the RWTH -AACHEN Uni-
versity, in Germany.
First, I would like to express my appreciation and gratitude to my advisor Pro-
fessor Dr.-Ing. Wolfgang Schr˜ oder for his guidance, proposals, and advice. I would
like also to thank Professor Dr.-Ing. Dieter H˜ anel and Professor Dr.-Ing. Herbert
Olivier for accepting to be in the examination committee.
I would like also to thank Dr.-Ing. Matthias Meinke and Dr.-Ing. Andreas Henze for
the comments during the development stage of this work. Special sincere gratitude
and thanks due to my former colleagues Dr.-Ing. Ehab Fares and Dipl.-Ing. Jens
Kr˜ omer for the scientiflc support and the continuous encouragement.
Last but not least, I would like to thank my wife and my children for their con-
tinuous support, endless patience, and for providing the comfort, warmness, and the
suitable environment at home.
Aachen, December 2005 Khalid M. SultanContents
Abstract iii
Nomenclature v
1 Introduction 1
2 Data Structure and Grid Generation 5
2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Grid Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Geometry -Based Automatic Grid Adaptation . . . . . . . . . . . . . 17
3 Mathematical Model 19
3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Boundary Conditions at the Fluid -Body interface . . . . . . . . . . . 23
3.2.1 Inviscid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.2 Viscous Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Far Field Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 27
3.4 Initial Conditions and a CFL Cut-Back
Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Method of Solution 31
4.1 The Reconstruction of the Variables . . . . . . . . . . . . . . . . . . . 32
4.2 Discretization of the Inviscid Terms:
+The AUSM and the AUSM ¡ au Schemes . . . . . . . . . . . . . 35
4.3 of the Viscous Terms . . . . . . . . . . . . . . . . . . . 38
4.4 Time Stepping Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5 Solution-Based-Automatic Grid Adaptation . . . . . . . . . . . . . . 41
4.5.1 Wavelets as a Data Processing Tool . . . . . . . . . . . . . . . 42
4.5.2 Formulation of the Adaptation Sensor . . . . . . . . . . . . . 45
5 Results 47
5.1 Inviscid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.1.1 NACA0012 Aerofoil . . . . . . . . . . . . . . . . . . . . . . . . 48
5.1.2 BAC3 Aerofoil . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Viscous Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2.1 Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2.2 Lid -Driven Cavity Flow . . . . . . . . . . . . . . . . . . . . . 62ii Contents
6 Conclusions 65
Bibliography 66
Appendices 73
A Reference variables 73
B LSM applied to three dimensions 75
+C AUSM ¡ au scheme 77
D Cell classiflcation: Three dimensional bodies 81Abstract
n anisotropic unstructured cartesian grid generator and a o w solver wereA developed for the compressible viscous o w. The grid generator assumes
a body conflguration given as a set of data points and generates automatically a
computational cartesian grid with geometry adaptation. The cartesian o w solver
employs an upwind high resolution shock capturing scheme with linear reconstruc-
tion to achieve globally second order accuracy. In order to render the o w solver
capable of solving supersonic o ws, a limiter function was incorporated in the re-
construction algorithm. A solution -based grid adaptation algorithm that employs
the discrete wavelet transform was built in the developed o w solver and tested for
several standard and non-standard test cases.
˜Ubersicht
in anisotroper unstrukturierter kartesischer Gittergenerator undE Str˜ omungsl˜ oser sind zur L˜ osung kompressibler reibungsbehafteter
Str˜ omungen entwikelt worden. Der Basis fur˜ den wird der
K˜ orper mit einer Reihe von Datenpunkten beschrieben, wobei nach dem Ein-
lesen der Datenpunkte automatisch ein kartesisches Gitter mit geometrischen
Gitteradaptierung generiert wird. Der kartesische Str˜ omungsl˜ oser verwendet ein
hochau ˜osendes Upwind-Schema mit linearer Rekonstruktion der Variablen, um
eine allgemeine Genauigkeit von zweiter Ordnung zu erhalten. Ein Limiter ist
implementiert worden, so dass der Str˜ omungsl˜ oser auch zur Berechenung von
ub˜ erschall Str˜ omungen herausgezogen werden kann. Die Gitteradaption w˜ ahrend
der L˜ osung wird mit Hilfe von der diskreten Wavelet-Transformation ausgefuhrt.˜
Zur Validierung des Gittergenerators und des Str˜ omungsl˜ oser werden mehre
Testf˜ alle berechnet.Nomenclature
Latin Letters
H ux vector .
a~ split speed of sound.L=R
~ ~ ~E, F, and G ux vectors in x, y, and z directions,respectively.
~L (s) parametric representation of a line in space.
~P (r;t) representation of a plane.
~Q vector of the conservative variables.
~q heat ux vector.
~v velocity vector.
~W vector of the primitive variables.
~dr position vector.
dCFL temporary CFL deflned by equation (3.32).
A surface area of the control volume.
a numerical speed of sound.1=2
a data points that compromise the wavelet window.i;i=0;:::;3
a left or right speed of sound at the cell face according to the up-L=R
stream direction.
B BERNSTEIN Polynomials.i;n
C , C , and C characteristics.o ¡ +
C pressure coe–cient.p
C speciflc heat at constant volume.v
CFL maximum COURANT number allowed by the time advancing scheme.
CFL cut -back CFL deflned by equation (3.31).cutBvi Nomenclature
d distance between center of cell number one and the face, see flgure1
(4.5).
d distance between center of cell number two and the face, see flgure2
(4.5).
D detail of the wavelet transform.i
+D pressure difiusion term in AUSM ¡au scheme.p
dA difierential element of the area.
e speciflc total energy.
f pirimitive variable to be reconstructed.
g wavelet coe–cients.i;i=0;:::;3
h wavelet coe–cients.i;i=0;:::;3
I identity tensor.
k thermal conductivity.
Ld length of the square cavity.
+M left split MACH number.L
¡M right split MACH number.R
§ +M second order polynomial of MACH number used in AUSM ¡au(2)
scheme.
M interface MACH number.1=2
p=m
M interface split MACH numbers.1=2
Mean statistical mean.
Median median.
n degree of the BEZIER curve.
n number of cells contributing in the slope calculation by the least
squares method.
n , n , and n normal vector in x, y, and z directions, respectively.x y z
p pressure.
P (t) point coordinates as represented by BEZIER curve.
§ +p split pressures used in AUSM ¡au scheme.
P outlet plane.1
P ;P ;P ;:::;P set of a given data points that represent the body geometry.1 2 3 n
+p left split pressure.L