Table Of ContentErnst Heinrich Hirschel
Wilhelm Kordulla
Shear Flow in
Surface-Oriented
Coordinate
Notes on Numerical Fluid Mechanics
Volume 4
Volume 1 Boundary Algorithms for Multidimensionallnviscid
Hyberbolic Flows (Karl Forster, Ed.)
Volume 2 Third GAMM-Conference on Numerical Methods in
Fluid Mechanics (Ernst Heinrich Hirschel, Ed.)
Volume 3 Numerical Methods for the Computation of I n
viscid Transonic Flows with Shock Waves (Arthur
Rizzi/Henri Viviand, Eds.)
Volume 4 Shear Flow in Surface-Oriented Coordinate (Ernst
Heinrich Hirschel/Wilhelm Kordulla)
Menuscripts should be _II over 1.00 peges. As they will be reproduced foto
mechanicelly they should be typed with utmost cere on speciel stationary which
will be supplied on request. In print, the size will be reduced linearly to approxi
metely 75 'It.. Figurll end diagrems should be lenered accordingly so as to produce
letters not smeller then 2 mm in print. The same is valid for handwritten formulae.
Menuscripts lin English) or proposals should be sent to Prof. Dr. K. Forster, Insti
tut fiir Aerodynemik und Gesdynemik, Pfaffenwaldring 21, 0-7000 Stungan 80.
Ernst Heinrich Hirschel
Wilhelm Kordulla
Shear Flow
in Surface-Oriented
Coordinate
With 40 Figures
Friedr. Vieweg & Sohn Braunschweig/Wlesbaden
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Hirschel. Ernst H.:
Shear flow in surface-oriented coordinate/Ernst
Heinrich Hirschel; Wilhelm Kordulla. - 1. Aufl. -
Braunschweig. Wiesbaden: Vieweg. 1981.
(Notes on numerical fluid mechanics; Vol. 4'
NE: Kordulla. Wilhelm; GT
All rights reserved
© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH. Braunschweig 1981
No part of this publication may be reproduced. stored in a retrieval system or
transmitted. mechanical. photocopying. recording or otherwise. without prior
permission of the copyright holder.
Produced by IDV. Industrie-und Verlagsdruck. Walluf b. Wiesbaden
ISBN 978-3-663-05277-7 ISBN 978-3-663-05276-0 (eBook)
DOI 10.1007/978-3-663-05276-0
Preface
Having worked on problems of three-dimensional boundary-layer flow
for more than a decade, the authors decided to publish some of
their results and ideas as a book. They believe this is worthwhile
to do in spite of the availability of other books on boundary-layer
theory. These books have their emphasis mainly on two-dimensional
flow, on the modeling of turbulent flow, or on prediction techniques.
The present book, on the other hand, is dedicated to the descrip
tion of three-dimensional shear flow past realistic configurations
and to the provision of formulations for prediction methods. To
this end several sets of governing equations for the treatment of
high-Reynolds number viscous flows are being discussed, together
with approximations of the geometry of general configurations.
Thus the book , which gives a uniform representation of the for
mulation of the many problems, and not so much a review of the work
done so far at different places, is aimed at the perspective in
vestigator of three-dimensional viscous-flow problems.
The reader should not allow himself to ~e scared off'by the use of
tensorial concepts and of the index notation, both not too familiar
in classic fluid mechanics. The authors made the experience that
the use of these concepts alleviates much the work on realistic
flow problems. In the book they have emphasized the use of these
concepts and not their mathematical proofs.
The book was written while the first author was head of the Ab
teilung fUr Theorie und Numerik (Department for Theoretical and
Numerical Fluid Mechanics) at the Institut fUr Angewandte Gasdyna
mik der Deutschen Forschungs- und Versuchsanstalt fUr Luft- und
Raumfahrt (DFVLR) in K6ln-Porz (Institute for Applied Gasdynamics
of the German Research Establishment for Aerospace Engineering),
and later of the Abteilung fUr Grenzschichten at the new DFVLR
-Institut fUr Theoretische Str6mungsmechanik (Dep,rtment for
Boundary-Layer Research at the DFVLR-Institute for Theoretical
Fluid Mechanics), G6ttingen. The second author worked at the same
places, and is still with the latter institute.
The authors thank their colleagues from the DFVLR for the many
v
discussions and for the contributions which went into the book.
Those who should be mentioned explicitly are K. ROBERT and his co
-author R. GRUNDMANN, whose papers form a large part of the basis
of the book, Mrs. V. JAWTUSCH, who performed many computations in
early stages of the work, and D. SCHWAMBORN, who checked through
some of the chapters, and contributed directly to parts of chapter
12. The authors also thank numerous colleagues from outside the
DFVLR, in particular J. P. F. LINDHOUT, NLR for fruitful and en
couraging discussions. Last not least they thank the ladies who
type-wrote the manuscript and produced the Indian-ink versions of
the figures.
VI
Contents
Preface
Page
1 • In troduction .••.••.•.•.•..••.•.•••••••••.•.•••.••...•••
2. First-Order Boundary-Layer Equations ••••••••••••••••••• 10
3. Boundary-Layer Parameters •••••••••••••••••••••••••••••• 19
3.1 Shear Stress •.•.............•....•....•••....... 19
3.2 Heat Flux ••.••..•••.....•.•••..••.•••••••••••..• 20
3.3 Displacement Thicknesses •••••••••••••.•••••••••• 21
3.3.1 Mass-Flow Displacement Thickness •••••••••••••••• 21
3.3.2 Momentum-Flow Displacement Thickness •••••••••••• 23
3.3.3 Energy-Flow Displacement Thickness •••••••••••••• 25
4. Stagnation-Point Solution ••••••••••••••••••••••.••••••• 26
4.1 Boundary-Layer Equations •••••••••••••••••••••••• 26
4.2 Boundary-Layer Parameters •••••••••••••••.••••••• 30
4.3 Start of the Field Computation •••••••••••••••••• 32
5. Quasi-Two-Dimensional Boundary Layers •••••••••••••••••• 34
5.1 Plane-af-Symmetry Flow •••••••••••••••••••••••••• 35
5.2 Infinite-Swept-Wing Flow 42
5.2.1 Leading-Edge Oriented Orthogonal Coordinates •••• 44
5.2.2 Leading-Edge Oriented Non-Orthogonal Coordinates. 48
5.2.3 Locally-Infinite-Swept-Wing Concept ••••••••••••• 52
5.2.4 Initial Data for the Locally-Infinite-Swept-
-Wing Solution .••..••.•.••••••.••••.•..••••..•.. 56
6. Initial Conditions for the Prediction of Boundary Layers
on Wings and Fuselages 62
7. Higher-Order Boundary-Layer Equations •••••••••••••••••• 70
8. Thin-Layer Approximations of the Navier-Stokes Equations 80
8. 1 Introduction ................................... . 80
VII
Page
8.2 Thin-Layer Approximations 91
8.2.1 Boundary-Layer Like Flows 91
8.2.2 Corner-Layer Like Flows •••••••••••••••••••••••• 94
8.3 Integral Properties •••••••••••••••••••••••••••• 96
9. On the Influence of Surface Curvature ••••••••••••••••• 100
9.1 Hierarchy of Governing Equations ••••••••••••••• 100
9.2 Boundary-Layer Flows at the Leading Edge of
Swept Wings ...................................................................... 103
9.2.1 A Criterion for the Validity of First-Order
Theory for Laminar Incompressible Flows •••••••• 103
9.2.2 Results for Zero- and First-Order Theory ••••••• 113
9.3 Some Remarks on General Cases •••••••••••••••••• 116
10. Samples of Surface-Oriented Coordinate Systems •••••••• 117
10.1 Surface-Oriented Wing Coordinate Systems ••••••• 118
10.1.1 General Wing Coordinate Systems •••••••••••••••• 118
10.1.2 Infinite-Swept-Wing Coordinate Systems ••••••••• 128
10.2 Surface-Oriented Body Coordinate Systems ••••••• 129
10.2.1 Pure Cross-Section Coordinate System ••••••••••• 130
10.2.2 Pure Surface-Oriented Coordinate System •••••••• 139
10.3 Coordinate System of BLOTTNER and ELLIS •••••••• 143
11. Conditions of Compatibility at the Body Surface ••••••• 146
12. Local Topology of Three-Dimensional Separation and
Attachment Lines ............................................................................ 156
12.1 General Remarks •••••••••••••••••••••••••••••••• 156
12.2 Local Topology of Separation Lines ••••••••••••• 158
12.3 Local Topology of Attachment Lines ••••••••••••• 172
12.3.1 External Inviscid Flow ••••••••••••••••••••••••• 172
12.3.2 Viscous Flow (Boundary Layer) •••••••••••••••••• 175
13. Similarity-Type Transformation of Boundary-Layer
Equations in Contravariant Form ••••••••••••••••••••••• 180
13.1 General Remarks •••••••••••••••••••••••••••••••• 180
VIII
Page
13.2 General Three-Dimensional Flow •••••••••••••••• 184
13.3 Stagnation-Point Flow ••••••.•••••••••••••••.•• 190
13.4 Plane-af-Symmetry Flow ••••••••••••••••••.•.•••. 193
13.5 Locally-Infinite-Swept-Wing Concept ••••••••••• 195
13.6 Initial Data for the Locally-Infinite-
-swept-Wing solution .••••••••••••••••••••••••• 196
Appendix AI Basic Geometrical Relations ••••••••••••••••••• 199
A.1 General Coordinates ••••••••••••••••••••••••••• 199
A.2 Surface-Oriented Locally Monoclinic Coordinates 200
A.3 Covariant Base Vectors of Surface-Oriented
Coordinate Systems •••••••••••••••••••••••••••• 203
A.4 Transformation of a Vector ••.••••••••••.••..•• 205
A.5 Partial Derivatives 207
A.6 The Covariant Metric Tensor for Surface-
-Oriented Coordinates ••••••••••••••••••••••••• 208
A.7 The Covariant Curvature Tensor baa of the
Surface ••.•••••••••.•••.•.•••••••••••.•.••••.• 210
A.8 The Two Principal Curvatures of the Surface ••• 210
A.9 Directions of the Two Principal Curvatures of
the Surface ••••••••••••••••••••••••••••••••••• 211
A.10 The Contravariant Metric Tensor aaa of the
Surface ••••••••••••••••••••••••••••••••••••••• 211
A.11 The Mixed-Variant Curvature Tensor ba of the
a
Surface ••••.•••••••••••••••..••••••••••.••.•.. 212
A.12 The Metric Tensor gaa and the Curvature
Properties of the Surface ••••••••••••••••••••• 213
A.13 Christoffel Symbols and Metric Factors ••••••••• 215
A.14 Shifters •••••.••••.•••..••.•.•.•••••••••..•... 218
Appendix BI Series Expansion of Vector Quantities in
General Coordinates 221
Appendix CI The Vorticity Vector •••••••••••••••••••••••••• 225
Appendix D: Dimensions and Boundary-Layer Stretching •••••• 228
IX
Page
Appendix E: Calculation of Streamlines, Surface and
Volume Element 236
List of Symbols •..••.••••.••.••••.•..••••••.•.•.•......•..•• 240
References ••••...••.••••.••••..•••.•.•••.•••••.......••••..• 248
Subject Index ..•••.•••••...•.•••••.•...•.•.........•••...••. 259
x
