Waves and Compressible Flow

198 Pages·2004·0.888 MB·English

by  Hilary Ockendon, John R. Ockendon (auth.)

47
Texts in Applied Mathematics
Editors
J.E.Marsden
L.Sirovich
S.S.Antman
Advisors
G.Iooss
P.Holmes
D.Barkley
M.Dellnitz
P.Newton
Springer
NewYork
Berlin
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Hilary Ockendon John R. Ockendon
Waves and
Compressible Flow
With60Figures
1 3
HilaryOckendon JohnR.Ockendon
OxfordCentreforIndustrialand OxfordCentreforIndustrialand
AppliedMathematics AppliedMathematics
24–29St.Giles 24–29St.Giles
OxfordOX13LB OxfordOX13LB
UK UK
[email protected] [email protected]
SeriesEditors
J.E.Marsden L.Sirovich
ControlandDynamicalSystems,107–81 DivisionofAppliedMathematics
CaliforniaInstituteofTechnology BrownUniversity
Pasadena,CA91125 Providence,RI02912
USA USA
[email protected] [email protected]
S.S.Antman
DepartmentofMathematics
and
InstituteofPhysicalScience
andTechnology
UniversityofMaryland
CollegePark,MD20742-4015
USA
[email protected]
MathematicsSubjectClassification(2000):76-02,76Nxx,76Bxx
LibraryofCongressCataloging-in-PublicationData
Ockendon,Hilary.
Wavesandcompressibleflow/HilaryOckendon,JohnR.Ockendon.
p.cm.—(Textsinappliedmathematics;v.47)
Includesbibliographicalreferencesandindex.
ISBN0-387-40399-X(alk.paper)
1.Wavemotion,Theoryof. 2.Fluiddynamics. 3.Compressibility. I.Title. II.Textsin
appliedmathematics;47.
QA927.O252003
(cid:1)
532.0535—dc21 2003054314
ISBN0-387-40399-X Printedonacid-freepaper.
(cid:1)c
2004Springer-VerlagNewYork,Inc.
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Series Preface
Mathematicsisplayinganevermoreimportantroleinthephysicalandbiolog-
ical sciences, provoking a blurring of boundaries between scientific disciplines
and a resurgence of interest in the modern as well as the classical techniques
of applied mathematics. This renewal of interest, both in research and teach-
ing, has led to the establishment of the series Texts in Applied Mathematics
(TAM).
The development of new courses is a natural consequence of a high level
of excitement on the research frontier as newer techniques, such as numerical
andsymboliccomputersystems,dynamicalsystems,andchaos,mixwithand
reinforce the traditional methods of applied mathematics. Thus, the purpose
ofthistextbookseriesistomeetthecurrentandfutureneedsoftheseadvances
and to encourage the teaching of new courses.
TAM will publish textbooks suitable for use in advanced undergraduate
and beginning graduate courses, and will complement the Applied Mathe-
matical Sciences (AMS) series, which will focus on advanced textbooks and
research-level monographs.
Pasadena, California J.E. Marsden
Providence, Rhode Island L. Sirovich
College Park, Maryland S.S. Antman
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Contents
The starred sections are self-contained and may be omitted at a first reading.
Series Preface ................................................. v
1 Introduction............................................... 1
2 The Equations of Inviscid Compressible Flow.............. 5
2.1 The Field Equations ..................................... 5
2.2 Initial and Boundary Conditions .......................... 13
2.3 Vorticity and Irrotationality .............................. 14
2.3.1 Homentropic Flow................................. 14
2.3.2 Incompressible Flow ............................... 17
Exercises ................................................... 18
3 Models for Linear Wave Propagation ...................... 21
3.1 Acoustics............................................... 21
3.2 Surface Gravity Waves in Incompressible Flow .............. 24
3.3 Inertial Waves .......................................... 26
3.4 Waves in Rotating Incompressible Flows.................... 29
3.5 Isotropic Electromagnetic and Elastic Waves ................ 30
Exercises ................................................... 33
4 Theories for Linear Waves ................................. 41
4.1 Wave Equations and Hyperbolicity ........................ 41
4.2 Fourier Series, Eigenvalues, and Resonance ................. 43
4.3 Fourier Integrals and the Method of Stationary Phase........ 47
4.4 *Dispersion and Group Velocity ........................... 52
4.4.1 Dispersion Relations ............................... 52
4.4.2 Other Approaches to Group Velocity................. 55
4.5 The Frequency Domain .................................. 57
4.5.1 Homogeneous Media ............................... 57
4.5.2 Scattering Problems in Homogeneous Media .......... 59
viii Contents
4.5.3 Inhomogeneous Media ............................. 62
4.6 Stationary Waves........................................ 64
4.6.1 Stationary Surface Waves on a Running Stream ....... 65
4.6.2 Steady Flow in Slender Nozzles ..................... 66
4.6.3 Compressible Flow past Thin Wings ................. 68
4.6.4 Compressible Flow past Slender Bodies .............. 73
4.7 High-frequency Waves.................................... 75
4.7.1 The Eikonal Equation.............................. 75
4.7.2 *Ray Theory ..................................... 77
4.8 *Dimensionality and the Wave Equation.................... 81
Exercises ................................................... 84
5 Nonlinear Waves in Fluids................................. 99
5.1 Introduction ............................................ 99
5.2 Models for Nonlinear Waves ..............................101
5.2.1 One-dimensional Unsteady Gasdynamics .............101
5.2.2 Two-dimensional Steady Homentropic Gasdynamics ...102
5.2.3 Shallow Water Theory .............................104
5.2.4 *Nonlinearity and Dispersion .......................106
5.3 Smooth Solutions for Nonlinear Waves .....................114
5.3.1 The Piston Problem for One-dimensional Unsteady
Gasdynamics .....................................114
5.3.2 Prandtl–Meyer Flow ...............................117
5.3.3 The Dam Break Problem...........................120
5.4 *The Hodograph Transformation ..........................121
Exercises ...................................................123
6 Shock Waves ..............................................135
6.1 Discontinuous Solutions ..................................135
6.1.1 Introduction to Weak Solutions .....................136
6.1.2 Rankine–Hugoniot Shock Conditions.................142
6.1.3 Shocks in Two-dimensional Steady Flow..............144
6.1.4 Jump Conditions in Shallow Water ..................150
6.2 Other Flows involving Shock Waves........................153
6.2.1 Shock Tubes......................................153
6.2.2 Oblique Shock Interactions .........................154
6.2.3 Steady Quasi-one-dimensional Gas Flow..............157
6.2.4 Shock Waves with Chemical Reactions ...............159
6.2.5 Open Channel Flow ...............................160
6.3 *Further Limitations of Linearized Gasdynamics.............162
6.3.1 Transonic Flow ...................................162
6.3.2 The Far Field for Flow past a Thin Wing.............163
6.3.3 Non-equilibrium Effects ............................165
6.3.4 Hypersonic Flow ..................................166
Exercises ...................................................170
Contents ix
7 Epilogue...................................................181
References.....................................................183
Index..........................................................185