Lecture Notes in Physics

Transcription

1 Lecture Notes in Physics Editorial Board R. Beig, Wien, Austria W. Beiglböck, Heidelberg, Germany W. Domcke, Garching, Germany B.-G. Englert, Singapore U. Frisch, Nice, France P. Hänggi, Augsburg, Germany G. Hasinger, Garching, Germany K. Hepp, Zürich, Switzerland W. Hillebrandt, Garching, Germany D. Imboden, Zürich, Switzerland R. L. Jaffe, Cambridge, MA, USA R. Lipowsky, Golm, Germany H. v. Löhneysen, Karlsruhe, Germany I. Ojima, Kyoto, Japan D. Sornette, Nice, France, and Los Angeles, CA, USA S. Theisen, Golm, Germany W. Weise, Garching, Germany J. Wess, München, Germany J. Zittartz, Köln, Germany

2 The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research to serve the following purposes: to be a compact and modern up-to-date source of reference on a well-defined topic; to serve as an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas; to be a source of advanced teaching material for specialized seminars, courses and schools. Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Proceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive is available at springerlink.com. The series content is indexed, abstracted and referenced by many abstracting and information services, bibliographic networks, subscription agencies, library networks, and consortia. Proposals should be sent to a member of the Editorial Board, or directly to the managing editor at Springer: Dr. Christian Caron Springer Heidelberg Physics Editorial Department I Tiergartenstrasse Heidelberg/Germany christian.caron@springer-sbm.com

3 Radyadour Kh. Zeytounian Topics in Hyposonic Flow Theory ABC

4 Author Professor Radyadour Kh. Zeytounian 12 Rue St Fiacre Paris France Radyadour Kh. Zeytounian, Topics in Hyposonic Flow Theory, Lect. Notes Phys. 672 (Springer, Berlin Heidelberg 2006), DOI /b Library of Congress Control Number: ISSN ISBN Springer Berlin Heidelberg New York ISBN Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the author and TechBooks using a Springer LATEX macro package Printed on acid-free paper SPIN: /TechBooks

5 ...after subsonic, transonic, supersonic and hypersonic flows, now the fluid dynamicists must cope with a new challenge: the theory of hyposonic fluid flows

6

7 Preface It is well known that the characteristic Mach number (M), the ratio of the reference constant velocity to the constant magnitude of the speed of sound, is the main dimensionless parameter which characterizes the effects of compressibility in the problem of the flow of a Newtonian fluid. Indeed, in the investigation of the flow of a compressible fluid, much of the analysis is considered in terms of the Mach number, and four cases are usually studied, namely subsonic (M < 1), transonic (M 1), supersonic (M > 1) and hypersonic (M 1). The main purpose of this book is to present various facets of the unsteady, very slow flow of a fluid at low Mach number which are strongly related to a fifth category of fluid flow problems, called hyposonic here (M 1). It seems that the word hyposonic, in relation to fluid mechanics, was mentioned for the first time in a short paper of mine ( Aerodynamics of hyposonic speeds ) written at the end of 1981 and published in the Russian journal Applied Mechanics and Technological Physics (vol. 2, pp , 1983), published by the Siberian branch of the Russian Academy of Sciences. That paper was, in fact, a syllabus for a future book, which is the present book. Low-Mach-number, unsteady, very slow hyposonic motion plays a dominant role in various situations, even if this has not actually been established in such a firm way as is the case for flows with a high or low Reynolds number. If hypersonic fluid flow problems are closely linked to space fluid dynamics, hyposonic fluid flow problems, in contrast, are mainly linked to the fluid dynamics of the terrestrial environmental. More precisely, we shall emphasize in the seven chapters of this book that at least five areas of Newtonian hyposonic fluid dynamics are concerned. Namely, slightly compressible unsteady external and internal aerodynamics, combustion, nonlinear acoustics, slow atmospheric motions and thermal convection are important parts of the general subject area related to hyposonic flow theory. The first three chapters of this book play the role of an enlarged introduction. The next three Chaps. 4, 5 and 6, are devoted to three main areas where the theory of low-mach-number flows has various applications, namely external and internal hyposonic aerodynamics, and slow atmospheric motions. The last Chap. 7, is a miscellany of

8 VIII Preface low-mach-number fluid problems and motions (related, for instance, to nonlinear acoustics and thermal convection). At the end of each chapter there is a list of references. To tell the truth, the smallness of the Mach number in hyposonic flow theory is particularly significant when it occurs simultaneously with the smallness of one or several other basic dimensionless parameters in a problem of the flow of a Newtonian fluid governed by the unsteady Navier Stokes Fourier (NSF) equations for a viscous, compressible, heat-conducting fluid. In such cases, a fundamental role is played by similarity rules between M 1 and the other small dimensionless parameters (for instance, the inverses of the Reynolds and Strouhal numbers, the Rossby/Kibel number, and the Froude/Boussinesq number). In particular, via these similarity rules, we also obtain various criteria for the validity and consistency of the hyposonic models derived. A very difficult problem in hyposonic flow theory is closely linked to the filtering of fast acoustic waves in low-mach-number unsteady flows. This filtering of parasite waves is very well justified from a physical point of view for very slow motions, but changes strongly the mathematical nature of the unsteady evolution problem considered (in particular, the well-posedness is lost), and for this reason one encounters two basic problems. First, for an external flow, we must ask what initial conditions one may prescribe for the derived model (for instance, this concerns the Navier Fourier model for a divergenceless, temperature-dependent flow). Second, for an internal flow with combustion in a time-dependent domain, we must elucidate the role of the interplay with fast acoustics, since the averaged functions in the model equations are coupled to high-frequency oscillations produced by the motion of the boundary through the linear acoustic system and the averaged equations. Matching is possible in the case of an external flow, between the main Navier incompressible region and the two singular far-field and initial-time acoustic regions; unfortunately, this is not the case for an internal flow with combustion, where a family of fast times nonlinearly related to the slow evolution time of the boundary velocity must be introduced in the framework of a multiple-timescale analysis. It is clear that the main practical goal in hyposonic flow theory is to derive various simplified, approximate but significant model problems which can be solved with less numerical and financial effort than the exact full problem governed by the NSF or Euler equations for a slightly compressible fluid flow often, time-dependent numerical compressible-flow schemes are ineffective at low Mach numbers. Concerning the rigorous mathematical theory (initiated in France by P.-L. Lions in 1993), the reader can find, in Sect. 2.6, a short comment with some recent references. In that section, we have also mentioned various references concerning numerical computations and simulations. Many problems remain unsolved (some of these problems are mentioned in the three first chapters of this book), in spite of the fact that their applications are vital to the understanding of our environment. I hope that the present book will provide a motivation for fluid dynamicists to investigate these unsolved hyposonic flow problems in depth later on. It would be a good (but difficult!)

9 Preface IX exercise for postgraduate students to work through the various parts of this book and attempt to understand it for the purpose of applying the theory, and also to refer to the many original papers quoted. This book gives essentially an analytical view of hyposonic flow problems, but I think (and I hope) that, on the basis of the analytical results derived, it will be possible to consider various practical problems in combustion theory, in road and rail traffic aerodynamics, and in meteorology. Many friends and colleagues have provided stimulation and encouragement that have made this book possible, but I wish particularly to thank Professor Jean-Pierre Guiraud, who has played an important role, over a period of twenty years, in the formulation and asymptotic modelling of the various low- Mach-number flow problems presented in this book. The reader should take into account that the present book is the first book devoted to hyposonic flow theory, and it is the author s hope that the various unavoidable deficiencies in this book will encourage others to write new and improved books on a similar theme. After subsonic, transonic, supersonic and hypersonic flows, now the fluid dynamicists must cope with a new challenge, the theory of hyposonic fluid flows. Finally, I thank Dr Christian Caron, Physics Editor, and the members of the Springer Physics Editorial Department, where the camera-ready manuscript was produced in LaTeX and reread by a native English speaker for publication in the Lecture Notes in Physics Series with the agreement of the LNP Board. Yport/Paris January 2005 R. Kh. Zeytounian

10

11 Contents 1 Aerothermodynamics, Nonlinear Acoustics and the Meteorological Equations The Dynamics of a Newtonian Fluid The Dimensionless NSF Equations From the Euler Incompressible Equations to the Navier Equations The Euler Incompressible, Non-Viscous Equations The Navier Equation From the Euler Compressible Equations to the Navier Stokes Equations The Euler Compressible Equations The Navier Stokes Equations The Navier Stokes Fourier Equations The Eulerian Case: Re Initial and Boundary Conditions The Equations of Nonlinear Acoustics The Meteorological Equations The Eulerian Inviscid, Adiabatic Meteorological Equations The f 0 -Plane Approximation Equations for Atmospheric Inviscid Motion The Kibel Primitive Equations in the Pressure Coordinates System The Hydrostatic Dissipative System of Equations References The Many Faces of the Asymptotics of Low-Mach-Number Flows External Aerodynamics Far Field Initial-Time Region and Adjustment Problem

12 XII Contents Navier Fourier Model Problem Burgers Equation Vanishing-Viscosity Case Internal Aerodynamics Time-Dependent Domains Application to Combustion Atmospheric Motions NonlinearAcoustics SomeSpecialProblems Shallow-Convection Equations Compressible Stokes Oseen Exterior Problem Rigorous Mathematical Results and Numerical Simulations References A First Approach to the Asymptotics of Low-Mach-Number Flows Constant-DensityEulerInviscidFlow Incompressible (Isochoric) Eulerian Fluid Flow From Adiabaticity to Isochoricity The Fedorchenko Approach Small-Mach-Number Non-Viscous (Eulerian) Models Euler Compressible Equations and the Limit M Small-M Non-Viscous Euler Equations Various Small-Mach-Number Systems in a Time-Dependent Bounded Container Ω(t) The Problem of Adjustment to the Initial Conditions in a Time-Dependent Bounded Container Small-Mach-Number Viscous and Heat-Conducting (NSF) Models The Case of a Time-Dependent Bounded Domain The Case of an External Dissipative, Unsteady Flow TheWeaknessofOurFirstApproach References Some Aspects of Low-Mach-Number External Flows The Navier Fourier Initial Boundary-Value Model Problem The Incompressible (Navier) Main Limit The Far-Field (Lighthill) Local Limit The Initial-Time (Acoustic) Local Limit The Navier Initial Boundary-Value Model Problem The Role of Thermal Effects The Navier Fourier Quasi-Compressible Model Problem The Second-Order Acoustic Equations and the Initial Condition for T

13 Contents XIII 4.2 From the NSF Equations to Burgers Model Equation A Simple Derivation of Burgers Equation Burgers Equation as a Dissipative Model Transport Equation in a Thin Region Close toanacousticwavefront The Blasius Problem for a Slightly Compressible Flow Basic Dimensionless Equations and Conditions The Limiting Euler Equations for M The Limiting Prandtl Equations for M Flow Due to Displacement Thickness Boundary-Layer Equations with Weak Compressibility Self-Similar Solution Compressible Flow at Low Reynolds Number in Low-Mach-Number Asymptotics The Stokes Limiting Case and the Steady, Compressible Stokes Equations The Oseen Limiting Case and the Steady, Compressible Oseen Equations The Rarefied-Gas Point of View for Small Knudsen Number References Some Aspects of Low-Mach-Number Internal Flows Acoustic Waves Inside a Cavity with a Rigid Wall The Solution of the Wave Equation for Free Acoustic Oscillations The Solution of the Wave Equation for Forced Acoustic Oscillations Damping of Acoustic Waves by Viscosity and Heat Conduction Inside a Cavity With a Rigid Wall Solution for U Solution for U Solution for U Further Considerations Long-Time Evolution of Acoustic Waves and the Concept of Incompressibility: Inviscid Perfect Gas in a Cavity with a Boundary that is Deformable as a Function of Time Persistence of the Oscillations The Second-Order Approximation Some Comments Low-Mach-Number Flows in a Time-Dependent Cavity: the Dissipative Case

14 XIV Contents The Case Re = O(1) and Pr = O(1) Fixed and M 0. The Navier Fourier Average Model Problem The Case Re 1andM 1. Viscous Damping of Acoustic Oscillations Comments References Slow Atmospheric Motion as a Low-Mach-Number Flow Some Comments on the Boussinesq Approximation and the Derivation of the Boussinesq Equations Asymptotic Approaches Some Comments An Asymptotic, Consistent Justification of the Boussinesq Approximation for Lee Wave Phenomena in the Atmosphere Asymptotic Derivation of Boussinesq Lee-Waves- Model Inviscid Equations When M The Case of Steady Two-Dimensional Flow andthelongproblem The Free-Circulation (Breeze) Problem More Concerning the Hydrostatic Dissipative Equations The Boussinesq Free -Circulation Problem Complementary Remarks Relating to Derived Boussinesq Model Equations The Problem of Initial Conditions The Problem of the Upper Condition intheboussinesqcase Non-Boussinesq Effects: Isochoric and Deep-Convection Approximate Equations Low-Mach-Number (M 1) Asymptotics of the Hydrostatic Model Equations, With Ki = O(1) The Monin Charney Quasi-Non-Divergent Model The Second-Order Low-Mach-Number Model Adjustment to Quasi-Non-Divergent Model Equations The Investigations of Guiraud and Zeytounian Some Remarks Low-Mach-Number (M 1) Asymptotics of the Hydrostatic Model Equations, with Ki The Hydrostatic Dissipative Model Equations with the β-effect The Quasi-Geostrophic Approximation The Second-Order Ageostrophic Model of Guiraud and Zeytounian Conclusion References

15 Contents XV 7 Miscellaneous: Various Low-Mach-Number Fluid Problems and Motions An Asymptotic Derivation of the Kuznetsov Zabolotskaya Khokhlov (KZK) Model Equation Asymptotic Approach The Leading-Order System for U The Second-Order System for U The KZK Model Equation as a Compatibility Non-Secularity Condition From the Bénard Problem to the Rayleigh Bénard Rigid Free ModelProblem Some Physical Aspects of the BénardProblem A Mathematical Formulation of the Bénard Convection Leading-Order Problem The RB Rigid Free Shallow-Convection Model Problem Some Complementary Remarks Flow Over Relief When M Tends to Zero The Primitive Model Models Derived From the Euler Equations when M The Howarth Initial-Stage Equation for M 1andRe The Dominant NSF Equations Close to the Initial Time and Near the Wall The Results of Howarth (1951) and Hanin (1960) Nearly Incompressible Hydrodynamics Hydrodynamic NI Turbulence The Equations of NI Hydrodynamics (Non-Viscous Case with Conduction) The Equations of NI Hydrodynamics (Viscous case with Conduction) References Epilogue Index...281