Fluid Mechanics E-Book

Jean-Laurent Puebe

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 714

Veröffentlichungsjahr: 2013

Beschreibung

This book examines the phenomena of fluid flow and transfer as governed by mechanics and thermodynamics. Part 1 concentrates on equations coming from balance laws and also discusses transportation phenomena and propagation of shock waves. Part 2 explains the basic methods of metrology, signal processing, and system modeling, using a selection of examples of fluid and thermal mechanics.

Leseprobe

Table of Contents

Preface

Chapter 1. Thermodynamics of Discrete Systems

1.1. The representational bases of a material system

1.2. Axioms of thermostatics

1.3. Consequences of the axioms of thermostatics

1.4. Out-of-equilibrium states

Chapter 2. Thermodynamics of Continuous Media

2.1. Thermostatics of continuous media

2.2. Fluid statics

2.3. Heat conduction

2.4. Diffusion

Chapter 3. Physics of Energetic Systems in Flow

3.1. Dynamics of a material point

3.2. Mechanical material system

3.3. Kinematics of continuous media

3.4. Phenomenological laws of viscosity

Chapter 4. Fluid Dynamics Equations

4.1. Local balance equations

4.2. Mass balance

4.3. Balance of mechanical and thermodynamic quantities

4.4. Boundary conditions

4.5. Global form of the balance equations

4.6. Similarity and non-dimensional parameters

Chapter 5. Transport and Propagation

5.1. General considerations

5.2. First order quasi-linear partial differential equations

5.3. Systems of first order partial differential equations

5.4. Second order partial differential equations

5.5. Discontinuities: shock waves

5.6. Some comments on methods of numerical solution

Chapter 6. General Properties of Flows

6.1. Dynamics of vorticity

6.2. Potential flows

6.3. Orders of magnitude

6.4. Small parameters and perturbation phenomena

6.5. Quasi-1D flows

6.6. Unsteady flows and steady flows

Chapter 7. Measurement, Representation and Analysis of Temporal Signals

7.1. Introduction and position of the problem

7.2. Measurement and experimental data in flows

7.3. Representation of signals

7.4. Choice of representation and obtaining pertinent information

Chapter 8. Thermal Systems and Models

8.1. Overview of models

8.2. Thermodynamics and state representation

8.3. Modeling linear invariant thermal systems

8.4. External representation of linear invariant systems

8.5. Parametric models

8.6. Model reduction

8.7. Application in fluid mechanics and transfer in flows

Appendix 1. Laplace Transform

A1.1. Definition

A1.2. Properties

A1.3. Some Laplace transforms

A1.4. Application to the solution of constant coefficient differential equations

Appendix 2. Hilbert Transform

Appendix 3. Cepstral Analysis

A3.1. Introduction

A3.2. Definitions

A3.3. Example of echo suppression

A3.4. General case

Appendix 4. Eigenfunctions of an Operator

A4.1. Eigenfunctions of an operator

A4.2. Self-adjoint operator

Bibliography

Index

First published in France in 2006 by Hermes Science/Lavoisier entitled Physique des écoulements et des transferts in volumes 1 and 2 © LAVOISIER, 2006

First published in Great Britain and the United States in 2009 by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE LtdJohn Wiley & Sons, Inc.27-37 St George’s Road111 River StreetLondon SW19 4EUHoboken, NJ 07030UK www.iste.co.ukUSA www.wiley.com

& ISTE Ltd, 2009

The rights of Jean-Laurent Peube to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Peube, J. L.

[Physique des écoulements et des transferts. English]

Fundamentals of fluid mechanics and transport phenomena / Jean-Laurent Peube.

   p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-065-3

1. Fluid mechanics. 2. Transport theory. I. Title.

TA357.P49813 2008

620.1'06--dc22

2008036553

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-065-3

Preface

The study of fluid mechanics and transfer phenomena in flows involves the association of difficulties which are encountered in different disciplines: thermodynamics, mechanics, thermal conduction, diffusion, chemical reactions, etc. This book is not intended to be an encyclopaedia, and we will thus not endeavour to cover all of the aforementioned disciplines in a detailed fashion. The main objective of the text is to present the study of the movement of fluids and the main consequences in terms of the transfer of mass and heat. The book is the result of many years of teaching and research, both theoretical and applied, in scientific domains which are often considered separately. In effect, the development of new disciplines which are at the same time specialized and universal was very much a characteristic of science in the 20th century. Thus, signal processing, system analysis, numerical analysis, etc. are all autonomous disciplines and indispensable means for students, engineers or researchers working in the domain of fluid mechanics and energetics. In the same way, various domains such as the design of chemical reactors, the study of the stars and meteorology require a solid knowledge of fluid mechanics in addition to that of their specific topics.

This book is primarily aimed at students, engineers and researchers in fluid mechanics and energetics. However, we feel that it can be useful for people working in other disciplines, even if the reading of some of the more theoretical and specialized chapters may be dispensable in this case. The science and technology of the first half of the 20th century was heavily rooted in classical mechanics, with concepts and methods which relied on algebra and differential and integral calculus, these terms being taken into account in the sense they were used at that time. Furthermore, scientific thought was fundamentally deterministic during this period, even if the existence of games of chance using mechanical devices (dice, roulette, etc.) seemed far from the philosophy of science or Cauchy’s theorem. Each time has its concepts, which are based on the current state of knowledge, and the science of fluid mechanics was reduced for the most part to semi-empirical engineering formulae and to particular analytical solutions. Between the 1920s and the 1950s, our ideas on boundary layers and hydrodynamic stability were progressively elucidated. Studies of turbulence, which began in the 1920s from a conceptual statistical point of view, have really only made further progress in the 1970s, with the writing of the balance equations using turbulence models with a physical basis. This progress remains quite modest, however, considering the immensity of the task which remains.

It should be noted that certain disciplines have seen a spectacular renewal since the 1970s for two main reasons: on the one hand, the development of information technology has provided formidable computation and experimental methods, and on the other hand, multidisciplinary problems have arisen from industrial necessities. Acoustics is a typical example: many problems of propagation had been solved in the 1950s-1960s and those which were not made only very slow progress. Physics focused on other fundamental, more promising sectors (semiconductors, properties of matter, etc.). However, in the face of a need to provide practical solutions to industrial problems (sound generated by fluid flow, the development of ultra-sound equipment, etc.), acoustics became an engineering science in the 1970s. Acoustics is indeed a domain of compressible fluid mechanics and it will constitute an integral part of our treatment of the subject.

Parallel to this, systems became an object of study in themselves (automatic control) and the possibilities of study and understanding of the complexity progressed (signal processing, modeling of systems with large numbers of variables, etc.). Determinism itself is now seen in a more modest light: it suffices to remember the variable level of our ambitions with regard to meteorological prediction in the last 30 years to see that we have not yet arrived at a point where we have a definite set of concepts. Meteorological phenomena are largely governed by fluid mechanics.

The conception of this book results from the preceding observations. The author refuses to get into the argument which consists of saying that the time of analytical solutions has passed and that numerical simulation will solve all our problems. The reality is clearly more subtle than this: analytical solution in the broad sense, that is, the obtaining of results derived from reasoning and mathematical concepts, is the basis of physical concepts. Computations performed by computers by themselves cannot provide any more insight than an experiment, although both must be performed with great care. The state of knowledge and of understanding of mechanisms varies depending on the domain studied. In particular, the science of turbulence is still at a somewhat embryonic stage, and the mystery of turbulent solutions of the Navier-Stokes equations is far from being thoroughly cleared up. We are still at the stage of Galileo who attempted to understand mechanics without the ideas of differential calculus. Nobody can today say precisely what are the difficulties to be solved, and the time which will be required for their resolution (10 years, a century or 10 centuries). We will therefore present the state of our knowledge in the current scientific context by also considering some of the accompanying disciplines (thermodynamics, ideas related to partial differential equations, signal processing, system analysis) which are directly useful to the concepts, modeling, experiments and applications in fluid mechanics and energetics of flows. We will not cover specific combustion phenomena, limiting ourselves to a few simplified cases of physico-chemical reactions.

This book covers the necessary fundamentals for the study and understanding of the specific concepts and general properties of flows: the establishment and discussion of the balance equations of extensive quantities in fluid motions, the transport of these quantities by convection, wave-propagation or diffusion. These physical concepts are issued from the comprehension of theoretical notions associated with equations, such as characteristic curves or surfaces, perturbation methods, modal developments (Fourier series, etc.) and integral transforms, model reduction, etc. These mathematical aspects are either consequences of properties of partial differential equations or derived from other disciplines such as signal processing and system analysis, whose impact is important in every scientific or technological domain. They are discussed and illustrated by some elementary problems of fluid mechanics and thermal conduction, including measurement methods and experimental data processing This book is an introduction to the study of more specialized topics of fluid flow and transfer phenomena encountered in different domains of application: incompressible or compressible flow, dynamic and thermal boundary layers, natural or mixed convection, 3D boundary layers, physicochemical reactions in flows, acoustics in flows, aerodynamic sound, thermoacoustics, etc.

Chapter 1 is devoted to a synthetic presentation of thermodynamics. After recalling the basics of the representation of material systems, thermostatics is covered in an axiomatic fashion which avoids the use of differential formulations and which allows for a simplified presentation of classical results. Taking entropy dynamics as a starting point, the thermodynamics of non-equilibrium states is then discussed using simple examples with phenomenological laws of linear thermodynamics.

The continuous medium at rest is obtained by taking the limit of discrete systems in Chapter 2. The exchange of extensive quantities is modeled by means of flux densities, and irreversible thermodynamics leads to the diffusion equations. Some reminders of fluid statics are given. We then discuss the difficulties specific to the diffusion of matter.

The association of mechanical phenomena with thermodynamics is briefly developed in Chapter 3 along with the formalism used for the description of the motion of continuous media. The elementary properties of viscosity are then discussed.

Chapter 4 is dedicated to the writing of the general equations of the dynamics of fluid and transfer. The integration of local equations in a domain enables the separation of sources and fluxes of extensive quantities, these fluxes being transfer phenomena involving definition of input-output mechanisms for that domain, considered as a system. The energy equation explicitly expresses the interactions between thermodynamics and the movement of matter. The main usual boundary conditions and similarity and its consequences are then discussed.

Chapter 5 discusses the classification of partial differential equations in fluid mechanics. The mathematical aspects at the basis of physical concepts are well understood, but unfortunately rarely taught. These are very important, both for the numerical solution of equations and for the understanding of physical phenomena. We will present them here without providing any thorough demonstrations. The reader who struggles with this chapter should nonetheless try to assimilate its content while leaving aside the details of certain calculations.

Chapter 6 is dedicated in the main to the influence of diffusion in the convection of linear or angular momentum. It firstly covers vortex dynamics, the transposition to continuous media of concepts used in solid body rotation. Vorticity often results from transitional processes which may be more or less viscous, but its transport is very often governed by the equations for an inviscid fluid. Lagrange’s theorem introduces the idea of conservation of circulation of velocity which allows the rotation to be treated as a frozen material field. Elementary solutions of the 2D incompressible potential flows are quickly discussed. We then look at the quasi-1D approximation, which is particularly important in fluid mechanics, either for pipes or for flows in the vicinity of walls when a non-dimensional quantity becomes large. This last circumstance corresponds to a singular perturbation problem in the form of a boundary layer, which corresponds to the effects of viscous diffusion from the walls. The discussion of the boundary-layer equations reveals the separation mechanisms which are associated with the non-linear terms in steady flow equations.

The measurement of flow and transfer phenomena presents difficulties which are outlined in Chapter 7. The recent evolution of techniques based on the digitization of measurements, signal processing, analysis and reduction of models are naturally suited to applications in fluid mechanics and energetics. These methods have led to a renewal of progress in disciplines where unsteady phenomena are encountered, and in particular in the study of acoustic phenomena and turbulent flows. Improvements in computing have of course also led to considerable progress in the modeling of phenomena. The use of these methods requires specialized techniques whose treatment is beyond the scope of this book. The elements of signal processing and system analysis which we provide are only intended to alert the reader to the possibilities and utility of these methods, but also to show their limits. The idea that computers will allow the resolution of all our problems remains too ubiquitous. Computers only provide a tool to help us find the solutions we seek. These recent methods, signal processing or system analysis, are also useful for the identification of physical concepts associated with phenomena and the representation of solutions.

In Chapter 7, we also indicate in a synthetic manner the essential ideas necessary for measurement and signal processing procedures which are most useful in the domains studied. The possibility of large computations in modeling and experimental data processing leads us to evoke the idea of conditioning of linear systems, which is a generalization of elementary calculations of errors and uncertainties.

Chapter 8 is dedicated to modeling which provides a general context for the study of the evolution of physical systems. However, automatic control is reasoning in a general way on models without taking account of the laws of thermodynamics. These are essential for the disciplines studied in this book. We will present a few points of view and methods developed in automatic control, directly applied to the balance equations of basic problems of thermal conduction. The approximation procedures for the balance equations are far from being equivalent depending on the way in which we proceed. In order to simplify the presentation and to clearly separate the difficulties, we will mainly limit ourselves here to the state representation which is derived from thermodynamic modeling, leaving aside models derived from the approximation of solutions which do not exactly satisfy the balance equations.

NOTE.

We have chosen to respect the usual notation of physical quantities in each discussed scientific domain, while trying to have consistent notations whenever possible.

At the same time, the notations for derivatives are different, depending on the domain covered (thermodynamics, mechanics or more mathematical developments) and the size of equations. They all are usual and well known:

- For functions y (x) of one variable, they are marked y′ (x), y″ (x), y′″ (x), y′″ (x),…, y(n)(x).

- When discussing mechanical questions, the two first temporal derivatives of x(t) are written with dots: (t) and (t).

- The symbol is used only for material (Lagrangian) derivatives, which are indeed derivatives with respect to time of compound functions in Euler variables; this is equivalent to the other usual notation .

- For functions f(x, y) of several variables, the two following notations are used according circumstances: either with symbol or with indices marking the variables with respect of which derivations are performed: fx, fy, fxy, fxxy.

Integrals are always indicated by a simple integration sign, as the nature of this (single, double, triple, etc.) should be clear from the integration domain indicated and the differential element.

When tensor notation is used, vectors or matrices are denoted using upper case letters, their components being written in lower case letters. The convention of summation over repeated indices (Einstein’s convention) will systematically be used.

Chapter 1

Thermodynamics of Discrete Systems

The general objective of thermodynamics is to describe the properties of matter. After recalling the representational bases of material systems, thermostatics is dealt with by postulating the existence of a general equation of state which relates the extensive quantities. In this way we can forgo the need to delve into principles related to differential forms, and thereby simplify the presentation of traditional results. Then the thermodynamics of out of equilibrium systems are considered in terms of entropy dynamics, and discussed using simple examples. Finally, the phenomenological laws of linear thermodynamics are then considered.

1.1. The representational bases of a material system

1.1.1. Introduction

1.1.1.1. Geometric Euclidean space and physical quantities

The object of the physical sciences is the study of matter, for which the formulation of physical laws is necessary. However prior to the formulation of any such laws it is clearly necessary to characterize matter in terms of the various physical quantities which we can directly or indirectly measure. Matter is present all around us, and in a first instance we will limit ourselves to considering it in a static way, at a given instant which we can identify (this supposes a minimal definition of time); we perform geometric measurements in a 3D Cartesian coordinates system in order to identify the position and/or dimension of material elements. Measuring length presents no particular difficulty, excepting the choice of units. We will observe material elements in a geometric Euclidean space.

The geometric description of space is independent of the presence of matter; in other words the metric tensor does not depend on any physical quantity. This is not true for certain astrophysical phenomena which require us to place ourselves in the context of general relativity where geometric properties of space are no longer independent of the presence of matter. Simplistically put, the length of a meter depends on the mass found in its vicinity, which considerably complicates matters. In the following we exclude such phenomena, as they only become important at scales which greatly exceed those of our terrestrial physics.

We thus postulate (Axiom 1) the existence of a geometric space whose structure is independent of the properties of matter and the associated physical phenomena (gravitation, force fields, etc.).

We also admit (Axiom 2) that this space is homogenous and isotropic, which leads us to a traditional geometric Euclidean description of space R3 with its associated notions of length, surface and volume, whose scalar values are independent of the particular geometric frame of reference we choose to consider. This property of homogenity and isotropy will have important consequences for the expression of physical laws, which must not favor any given point or physical spatial direction. In particular, physical laws should neither favor any particular point in the universe, nor change as a result of a change in reference frame.

Finally, we suppose (Axiom 3) that matter can be characterized by physical quantities which are measurable at each instant in time, and not by mathematical entities (wavefunctions etc.) which allow, via mathematical operations, access to information of a probabilistic kind with regard to a physical quantity. This hypothesis of the possibility of directly measuring physical quantities supposes that the measure does not change the physical quantities of the material element considered. We therefore exclude microscopic phenomena relevant to quantum mechanics from our field of study, and we suppose the smallest material elements studied to contain a number of atoms or molecules sufficient for the neglect of statistical microscopic fluctuations to be justified.

1.1.1.2. The existence of isolated systems and the definition of time

The study of physical phenomena presupposes their reproducibility; the same effects should be observed under identical conditions. The establishment of physical laws thus supposes the definition of a time with the property of homogenity: in particular, quantifiable and reproducible observations of the evolution of a given material system must be possible.

The definition of time should thus be appropriately chosen. Previously associated with the length of the day, the definition of time has varied considerably between different individuals and epochs. For example, during the Roman period the lengths of the day and the night were respectively divided into seven and four parts, the Babylonians 2,000 years beforehand divided the day and the night each into 12 hours, which were clearly of unequal duration and varied according to the seasons. The Chinese and the Japanese divided each of the two cycles, from dawn to dusk and from dusk to dawn, into six periods. Japan only adopted the occidental system in 1873, but this did not prevent Japanese clockmakers from making mechanical clocks as early as the 17th century, these having quite complex mechanisms in order to accommodate the variable length of their hour.

The definition and measurement of time are thus not automatic operations for human beings. The relatively old notion of regular time (homogenous in the physical sense) is related to the use of indefinitely reproducible phenomena; this notion dates from the end of antiquity, the early Middle Ages and the invention of the clock (clepsydras, mechanical clocks, hourglass).

We will thus postulate (Axiom 4) that , regardless of when an experiment is performed. Any evolutionary phenomenon which is considered reproducible will allow a time unit to be defined. A temporal dimension can be constructed simply by virtue of the reproducibility of a phenomenon, which amounts to admitting that time is homogenous, i.e. no instant in the universe is given any special privilege. This homogenity of time does not really exist in cosmological problems, and in particular during the time of the initial big bang. We exclude these kinds of problem.