Peube J-L. Fundamentals of fluid mechanics and transport phenomena
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Table of Contents ix
Chapter 7. Measurement, Representation and Analysis of
Temporal Signals .................................... 339
7.1. Introduction and position of the problem .................. 339
7.2. Measurement and experimental data in flows................ 340
7.2.1. Introduction................................. 340
7.2.2. Measurement of pressure......................... 341
7.2.3. Anemometric measurements....................... 342
7.2.4. Temperature measurements ....................... 346
7.2.5. Measurements of concentration..................... 347
7.2.6. Fields of quantities and global measurements............. 347
7.2.7. Errors and uncertainties of measurements............... 351
7.3. Representation of signals............................ 357
7.3.1. Objectives of continuous signal representation ............ 357
7.3.2. Analytical representation......................... 360
7.3.3. Signal decomposition on the basis of functions; series and
elementary solutions ............................... 361
7.3.4. Integral transforms............................. 363
7.3.5. Time-frequency (or timescale) representations ............ 374
7.3.6. Discretized signals............................. 381
7.3.7. Data compression............................. 385
7.4. Choice of representation and obtaining pertinent information...... 389
7.4.1. Introduction................................. 389
7.4.2. An example: analysis of sound ..................... 390
7.4.3. Analysis of musical signals ....................... 393
7.4.4. Signal analysis in aero-energetics.................... 402
Chapter 8. Thermal Systems and Models ..................... 405
8.1. Overview of models............................... 405
8.1.1. Introduction and definitions....................... 405
8.1.2. Modeling by state representation and choice of variables...... 408
8.1.3. External representation.......................... 410
8.1.4. Command models............................. 411
8.2. Thermodynamics and state representation.................. 412
8.2.1. General principles of modeling..................... 412
8.2.2. Linear time-invariant system (LTIS).................. 420
8.3. Modeling linear invariant thermal systems................. 422
8.3.1. Modeling discrete systems........................ 422
8.3.2. Thermal models in continuous media.................. 431
8.4. External representation of linear invariant systems ............ 446
8.4.1. Overview.................................. 446
8.4.2. External description of linear invariant systems ........... 446
8.5. Parametric models................................ 451
8.5.1. Definition of model parameters ..................... 451
8.5.2. Established regimes of linear invariant systems ........... 453
x Fundamentals of Fluid Mechanics and Transport Phenomena
8.5.3. Established regimes in continuous media............... 458
8.6. Model reduction................................. 465
8.6.1. Overview.................................. 465
8.6.2. Model reduction of discrete systems.................. 466
8.7. Application in fluid mechanics and transfer in flows........... 474
Appendix 1. Laplace Transform........................... 477
A1.1. Definition .................................... 477
A1.2. Properties.................................... 477
A1.3. Some Laplace transforms .......................... 478
A1.4. Application to the solution of constant coefficient
differential equations................................. 479
Appendix 2. Hilbert Transform ........................... 481
Appendix 3. Cepstral Analysis............................ 483
A3.1. Introduction .................................. 483
A3.2. Definitions................................... 483
A3.3. Example of echo suppression........................ 484
A3.4. General case.................................. 485
Appendix 4. Eigenfunctions of an Operator.................... 487
A4.1. Eigenfunctions of an operator........................ 487
A4.2. Self-adjoint operator ............................. 487
A4.2.1. Eigenfunctions.............................. 487
A4.2.2. Expression of a function of f using an eigenfunction basis-set. . 488
Bibliography ....................................... 489
Index ............................................ 497
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 20
th
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 20
th
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
xii Fundamentals of Fluid Mechanics and Transport Phenomena
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.
Preface xiii
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, physico-
chemical 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.
xiv Fundamentals of Fluid Mechanics and Transport Phenomena
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.
Preface xv
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:
)(tx
and
)(tx
.
xvi Fundamentals of Fluid Mechanics and Transport Phenomena
The symbol
dt
d
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
Dt
D
.
For functions
f
(
x
,
y
) of several variables, the two following notations are used
according circumstances: either with symbol
w
(
yx
f
yx
f
y
f
x
f
ww
w
ww
w
w
w
w
w
2
32
,,, ,...) or with
indices marking the variables with respect of which derivations are performed: f
x
,
f
y
, f
xy
, f
xxy
.
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.
2 Fundamentals of Fluid Mechanics and Transport Phenomena
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 R
3
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