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Elements of Vorticity Aerodynamics James C. Wu

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Elements of Vorticity Aerodynamics James C. Wu

Preface

Helmholtz’ ground breaking vortex theorems in the mid-nineteenth century provided
the tools for the momentous discoveries in theoretical aerodynamics by Prandtl and
others nearly a century ago. Since then, studies of vorticity dynamics have received
continual impetus from diverse applications in engineering, physics, and mathematics.
A number of books on vortex theory and vortex dynamics addressing a broad
range of topics of theoretical and practical interests are presently available. In
comparison with these books, the present monograph has a narrower focus. It is
aimed at sharing my understanding of theoretical aerodynamics with the reader
interested in the classical circulation theory and in the role of modern vorticity
dynamics in theoretical aerodynamics involving unsteady and non-streamlined
flows.
My lectures in a short course at the Tsinghua University and at the Second
Biennial Retreat on Vorticity Aerodynamics, both scheduled for September in
Beijing, presented an ideal occasion for preparing my notes: an audience and a firm
target date. As it turned out, however, my initial estimate of required time was
overly optimistic. In the end, to meet target dates, certain compromises had to be
made.
One major compromise is the omission of a chapter discussing vorticity-based
flow computations. While computational aerodynamics is one of my favorite
subjects, time limitations prevented the inclusion of this subject as a component of
this monograph. Topics discussed in the present work, however, form the core of
vorticity-based computation methods. Detailed discussions of these methods are
available in some of the references quoted in Chaps. 1 and 3.
Other compromises involve editorial issues such as curtailing redundancies and
adding figures, exercises, and more sample problems. Chapters of my notes were
prepared more or less as independent articles, each with its own themes, references,
and introductory discussions, intermittently over an extended period of time. Efforts
to tie the chapters together and to implement the obviously desirable improvements
were ultimately limited by available time.
The present monograph is based essentially on my lecture notes completed in
August of 2004. It is my intent to prepare a “Version 2.0” of the monograph in the
reasonable future. It is my hope that my colleagues would kindly provide
commentaries and critiques about this initial version. One issue of special concern
during my preparation of the present version is the proper discussion of certain
viewpoints and strategies about vorticity aerodynamics, acquired and used over the
years in my research and teaching. These viewpoints and strategies are obviously
not the only ones that work; they are by no means a panacea for all applications of
vorticity dynamics. I am, however, convinced that they are consistent, rational, and
very effective within the perimeters defined in this monograph. Advocating these
viewpoints and strategies is not meant to underplay the merits of alternative
viewpoints and strategies, especially the classical ones. In this regard, I wish to
acknowledge the special help of Prof. J.Z. Wu, who kindly reviewed my draft
manuscripts on very short notices.
During my teaching and research career, I had the good fortune of associations
with many brilliant and marvelous individuals—teachers, colleagues, and former
students—who provided indispensable inspiration for my work. I wish to take this
opportunity to express my gratitude for their contribution to my understanding of
vorticity aerodynamics.
I would be remiss not to mention again the love and support of my wife
Mei-Ying Wu, especially during the past year, as the preparation of my notes took
up more and more, eventually virtually all, of the time at my disposal.

Contents

1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Differential Equations, Initial and Boundary Conditions . . . . . . . . . 3
1.3 Major Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Vorticity-Dynamic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Theorems of Helmholtz and Kelvin . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Kinematic-Kinetic Partition of Flow Problems . . . . . . . . . . . . . . . . 18
2.3 Kinematic Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Helmholtz’ First Vortex Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Vorticity Loops. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Kelvin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.7 Helmholtz’ Second Vortex Theorem. . . . . . . . . . . . . . . . . . . . . . . . 30
2.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Vorticity Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1 Differential Equations of Vorticity Kinematics . . . . . . . . . . . . . . . . 35
3.2 Vector Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Poisson’s Equation and Integral Representations . . . . . . . . . . . . . . 40
3.4 Poisson’s Equation and Boundary Conditions . . . . . . . . . . . . . . . . 43
3.5 Law of Biot–Savart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.6 Generalized Law of Biot–Savart. . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.7 Uniqueness of Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.8 Boundary Conditions in Vorticity Kinematics . . . . . . . . . . . . . . . . 52
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4 Vorticity Kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 Navier–Stokes Momentum Equation. . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Convection and Material Derivative . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Viscous Force and Diffusion of Momentum . . . . . . . . . . . . . . . . . . 62
4.4 Vorticity and Flow Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Appearance and Transport of Vorticity. . . . . . . . . . . . . . . . . . . . . . 66
4.6 Speeds of Vorticity Transport. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.7 Boundary Conditions in Vorticity Kinetics . . . . . . . . . . . . . . . . . . . 69
4.8 Approaching Steady Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5 Vorticity-Moment Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Total Vorticity Conservation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3 Asymptotic Behavior of Velocity Field . . . . . . . . . . . . . . . . . . . . . 79
5.4 Total Velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.5 Aerodynamic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.6 Moment of Aerodynamic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.7 Two-Dimensional Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.8 Observations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6 Classical Aerodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.1 Vorticity-Moment Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2 Kutta–Joukowski’s Theorem and Vortical Flow Zones. . . . . . . . . . 96
6.3 Non-Streamlined Flow and Karman Vortex Street . . . . . . . . . . . . . 100
6.4 Vorticity Loop and Vorticity Moment . . . . . . . . . . . . . . . . . . . . . . 103
6.5 Lifting-Line Theory: Uniform Circulation Lift . . . . . . . . . . . . . . . . 105
6.6 Lift on Finite Wing with Varying Circulation. . . . . . . . . . . . . . . . . 110
6.7 Induced Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.8 Wake-Integral Expressions for Lift and Drag . . . . . . . . . . . . . . . . . 113
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7 Unsteady Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.1 Aircraft and Streamlined Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.2 Animal Flight and Flapping Wings. . . . . . . . . . . . . . . . . . . . . . . . . 120
7.3 Classical Versus Vorticity-Dynamic Approach . . . . . . . . . . . . . . . . 123
7.4 Apparent Mass of Sphere and Circular Disk. . . . . . . . . . . . . . . . . . 127
7.5 Apparent Mass Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139