alinik
۱۲ مهر ۱۳۸۸, ۱۹:۳۵
The invention of the digital computer and its introduction in the world of science and
technology has led to the development, and increased awareness, of the concept of
approximation. This concerns the theory of the numerical approximation of a set of
equations, taken as a mathematical model of a physical system. But it concerns also
the notion of the approximation involved in the definition of this mathematical model
with respect to the complexity of the physical world.
We are concerned here with physical systems for which it is assumed that the
basic equations describing their behavior are known theoretically, but for which no
analytical solutions exist, and consequently an approximate numerical solution will
be sought instead.
For various reasons, the first of these being the great complexity, it is often not
practically possible to describe completely the evolution of the system in its full
complexity. Of course, the definition of these limits is relative to a given time and
environment and these are being extended with the evolution of the computer technology.
But taken at a given period, it is necessary to define mathematical models that
will reduce the complexity of the original basic equations and make them tractable
within fixed limits. Actually, the first level to be defined is the ‘scale of reality’ level.
Physicists propose various levels of description of our physical world, ranging from
subatomic, atomic or molecular, microscopic, macroscopic (defined roughly as the
scale of classical mechanics) up to the astronomical scale. As is well known, in the
statistical description of a gas, the motion of the individual atoms or molecules are
taken into consideration and the behavior is ruled by the Boltzmann equation. This
description leads for instance to the definition of temperature as a measure of the mean
kinetic energy of the gas molecules; to a definition of pressure as a result of the impulse
of molecules on the walls of the body containing the gas; to a definition of viscosity
connected to the momentum exchange due to the thermal molecular motion, and so on.
At this molecular level of description the fundamental variables are molecule velocities,
number of particles per volume and other variables defining the motion of the
individual molecules, while pressure, temperature, viscosity e.g. are mean propertieswhich are deduced from other variables, more basic at this level of reality. Hence,
we may consider that each level of reality can be associated with a set of fundamental
variables, from which other variables can be defined as measures of certain mean
properties. Continuing in the line of this examplewe have, beyond the molecular level
of statistical mechanics, the atomic level, the nuclear and the subnuclear level that we
do not plan to discuss here, since they are fully outside the domain of definition of
a fluid.
Actually, fluid dynamics starts to exist as soon as the interaction between a sufficiently
high number of particles affects and dominates, at least partly, the motion
of each individual particle. Hence, fluid dynamics is essentially the study of the
interactive motion and behavior of large number of individual elements.
The limit between individual motions of isolated particles or elements and their
interactive motion is of significance in the study of rarefied gases. It is known that
the interaction between the particles becomes negligible if the mean free path length
attains a magnitude of the order of the length scale of the considered system. The ratio
of the mean free path length to the reference length scale is called the Knudsen number.
For higher values of the mean free path length, or of the Knudsen number, the
particles behave essentially as individual elements. These limit situations will not
be considered here since they are outside the field of classical fluid dynamics. Note
however that the intermediate range between the continuum and the rarefied gas
approximations is of practical significance for the prediction of the re-entry phase of
a Space Shuttle. When re-entering the earth atmosphere from space, the Space Shuttle
crosses the atmosphere from very high altitudes, where it cannot be considered as
a continuum, through an intermediate range that evolutes with reducing altitude to
a continuum fluid. We need therefore special models, intermediate between the
Navier–Stokes and the Boltzmann equations to handle these situations, which are
extremely critical for the safety of the return phase of the Shuttle.
We will focus, in the following, on the level of reality in which the density of
elements is high enough, so that we can make the approximation of considering the
system of interacting elements as a continuum. This expresses that continuity or closeness
exists between the elements such that their mutual interaction dominates over
the individual motions, although these are not suppressed. What actually happens, is
that a collective motion is superimposed on the motion of the isolated elements as a
consequence of the large number of these elements coexisting within the same domain.
From this point of view, we understand easily why the concepts of fluid mechanics
can be applied to a variety of systems consisting of a large number of interacting
individual elements.
This is the case for the current fluids and gases where the individual ‘element’,
or fluid particle is actually not a single molecule, but consists of a large number
of molecules occupying a small region with respect to the scale of the considered
domain, but still sufficiently large in order to be able to define a meaningful and nonambiguous
average of the velocities and others properties of the individual molecules
and atoms occupying this volume. It implies that this elementary volume contains a
sufficiently high number of molecules, with for instance a well defined mean velocity,
mean kinetic energy, allowing to define velocity, temperature, pressure, entropy and
so on, at each point. Hence, associated fields, which will become basic variables for
the description of the system, can be defined although the temperature, or pressure, or
entropy of an individual atom or molecule is not defined and generally meaninglessenters into a stochastic motion and in defining mean turbulent variables, such as a
mean turbulent velocity field, an average is performed, in this case an average in time,
over the motion of the fluid particles themselves.
A still higher level of averaging occurs in the description of flows through porous
media such as soils. In the description of groundwater flows an ‘element’ is the set
of fluid particles, as defined above, contained in a volume large enough as to contain
a great number of soil particles and fluid particles such that a meaningful average
can be performed, but still small with respect to the dimensions of the region to be
analyzed. Such a volume is considered as a ‘point’ at this level of description, and the
fields are attached to these points, implying that groundwater flow theories do study
the behavior of collection of fluid particles.
Following this line, the movement or overall displacement of crowds at exit of railway
stations during rush hours, or of a football stadium, can be analyzed with fluid
mechanical concepts. In this case, an ‘element’ is the set of persons contained in a
region small with regard to the dimensions of the station for instance, but still containing
a sufficiently high number of individuals in order to define non-ambiguous average
values, such as velocity and other variables. In this description, the displacement of
an individual is not considered, but only the motion of groups of individuals.
A similar analysis can be defined for heavy traffic studies, where an ‘element’ is
defined as a set of cars (in the one-dimensional space formed by the road). Obviously
in a light traffic, the isolated car behaves as a single particle but collective motion
comes in when a certain intensity of traffic has been reached such that the speed of
an individual car is influenced by the presence of the other cars. This is actually to be
considered as the onset of a ‘fluid mechanical’ description.
Finally, at a still larger scale, astrophysical fluid dynamics can be defined for the
study of the interstellar medium or for the study of the formation and evolution of
galaxies. In this latter case for instance, an ‘element’consists of a set of stellar objects,
including one or several solar systems and the dimensions of a ‘point’ can be of the
order of light years.
In conclusion of these considerations, we can say that fluid mechanics is essentially
the study of the behavior of averaged quantities and properties of a large number of
interacting elements. The same is true for another domain of scientific knowledge,
namely thermodynamics, which is also the study of systems of large numbers of interacting
elements. It is therefore no wonder that thermodynamics is, with the exception
of incompressible isothermal media, tightly interconnected with fluid mechanics and
plays an important role in the description of the evolution of ‘fluid mechanical’systems
as mentioned above.
An essential step in fluid dynamics is therefore the averaging process. We have
to decide, in front of a given system, which level of averaging will be performed
in function of the quantities to be predicted, in function of the significant variables
which can be defined in a meaningful way, in function of the precision and degree of
accuracy to be achieved in the description of the system’s behavior.
technology has led to the development, and increased awareness, of the concept of
approximation. This concerns the theory of the numerical approximation of a set of
equations, taken as a mathematical model of a physical system. But it concerns also
the notion of the approximation involved in the definition of this mathematical model
with respect to the complexity of the physical world.
We are concerned here with physical systems for which it is assumed that the
basic equations describing their behavior are known theoretically, but for which no
analytical solutions exist, and consequently an approximate numerical solution will
be sought instead.
For various reasons, the first of these being the great complexity, it is often not
practically possible to describe completely the evolution of the system in its full
complexity. Of course, the definition of these limits is relative to a given time and
environment and these are being extended with the evolution of the computer technology.
But taken at a given period, it is necessary to define mathematical models that
will reduce the complexity of the original basic equations and make them tractable
within fixed limits. Actually, the first level to be defined is the ‘scale of reality’ level.
Physicists propose various levels of description of our physical world, ranging from
subatomic, atomic or molecular, microscopic, macroscopic (defined roughly as the
scale of classical mechanics) up to the astronomical scale. As is well known, in the
statistical description of a gas, the motion of the individual atoms or molecules are
taken into consideration and the behavior is ruled by the Boltzmann equation. This
description leads for instance to the definition of temperature as a measure of the mean
kinetic energy of the gas molecules; to a definition of pressure as a result of the impulse
of molecules on the walls of the body containing the gas; to a definition of viscosity
connected to the momentum exchange due to the thermal molecular motion, and so on.
At this molecular level of description the fundamental variables are molecule velocities,
number of particles per volume and other variables defining the motion of the
individual molecules, while pressure, temperature, viscosity e.g. are mean propertieswhich are deduced from other variables, more basic at this level of reality. Hence,
we may consider that each level of reality can be associated with a set of fundamental
variables, from which other variables can be defined as measures of certain mean
properties. Continuing in the line of this examplewe have, beyond the molecular level
of statistical mechanics, the atomic level, the nuclear and the subnuclear level that we
do not plan to discuss here, since they are fully outside the domain of definition of
a fluid.
Actually, fluid dynamics starts to exist as soon as the interaction between a sufficiently
high number of particles affects and dominates, at least partly, the motion
of each individual particle. Hence, fluid dynamics is essentially the study of the
interactive motion and behavior of large number of individual elements.
The limit between individual motions of isolated particles or elements and their
interactive motion is of significance in the study of rarefied gases. It is known that
the interaction between the particles becomes negligible if the mean free path length
attains a magnitude of the order of the length scale of the considered system. The ratio
of the mean free path length to the reference length scale is called the Knudsen number.
For higher values of the mean free path length, or of the Knudsen number, the
particles behave essentially as individual elements. These limit situations will not
be considered here since they are outside the field of classical fluid dynamics. Note
however that the intermediate range between the continuum and the rarefied gas
approximations is of practical significance for the prediction of the re-entry phase of
a Space Shuttle. When re-entering the earth atmosphere from space, the Space Shuttle
crosses the atmosphere from very high altitudes, where it cannot be considered as
a continuum, through an intermediate range that evolutes with reducing altitude to
a continuum fluid. We need therefore special models, intermediate between the
Navier–Stokes and the Boltzmann equations to handle these situations, which are
extremely critical for the safety of the return phase of the Shuttle.
We will focus, in the following, on the level of reality in which the density of
elements is high enough, so that we can make the approximation of considering the
system of interacting elements as a continuum. This expresses that continuity or closeness
exists between the elements such that their mutual interaction dominates over
the individual motions, although these are not suppressed. What actually happens, is
that a collective motion is superimposed on the motion of the isolated elements as a
consequence of the large number of these elements coexisting within the same domain.
From this point of view, we understand easily why the concepts of fluid mechanics
can be applied to a variety of systems consisting of a large number of interacting
individual elements.
This is the case for the current fluids and gases where the individual ‘element’,
or fluid particle is actually not a single molecule, but consists of a large number
of molecules occupying a small region with respect to the scale of the considered
domain, but still sufficiently large in order to be able to define a meaningful and nonambiguous
average of the velocities and others properties of the individual molecules
and atoms occupying this volume. It implies that this elementary volume contains a
sufficiently high number of molecules, with for instance a well defined mean velocity,
mean kinetic energy, allowing to define velocity, temperature, pressure, entropy and
so on, at each point. Hence, associated fields, which will become basic variables for
the description of the system, can be defined although the temperature, or pressure, or
entropy of an individual atom or molecule is not defined and generally meaninglessenters into a stochastic motion and in defining mean turbulent variables, such as a
mean turbulent velocity field, an average is performed, in this case an average in time,
over the motion of the fluid particles themselves.
A still higher level of averaging occurs in the description of flows through porous
media such as soils. In the description of groundwater flows an ‘element’ is the set
of fluid particles, as defined above, contained in a volume large enough as to contain
a great number of soil particles and fluid particles such that a meaningful average
can be performed, but still small with respect to the dimensions of the region to be
analyzed. Such a volume is considered as a ‘point’ at this level of description, and the
fields are attached to these points, implying that groundwater flow theories do study
the behavior of collection of fluid particles.
Following this line, the movement or overall displacement of crowds at exit of railway
stations during rush hours, or of a football stadium, can be analyzed with fluid
mechanical concepts. In this case, an ‘element’ is the set of persons contained in a
region small with regard to the dimensions of the station for instance, but still containing
a sufficiently high number of individuals in order to define non-ambiguous average
values, such as velocity and other variables. In this description, the displacement of
an individual is not considered, but only the motion of groups of individuals.
A similar analysis can be defined for heavy traffic studies, where an ‘element’ is
defined as a set of cars (in the one-dimensional space formed by the road). Obviously
in a light traffic, the isolated car behaves as a single particle but collective motion
comes in when a certain intensity of traffic has been reached such that the speed of
an individual car is influenced by the presence of the other cars. This is actually to be
considered as the onset of a ‘fluid mechanical’ description.
Finally, at a still larger scale, astrophysical fluid dynamics can be defined for the
study of the interstellar medium or for the study of the formation and evolution of
galaxies. In this latter case for instance, an ‘element’consists of a set of stellar objects,
including one or several solar systems and the dimensions of a ‘point’ can be of the
order of light years.
In conclusion of these considerations, we can say that fluid mechanics is essentially
the study of the behavior of averaged quantities and properties of a large number of
interacting elements. The same is true for another domain of scientific knowledge,
namely thermodynamics, which is also the study of systems of large numbers of interacting
elements. It is therefore no wonder that thermodynamics is, with the exception
of incompressible isothermal media, tightly interconnected with fluid mechanics and
plays an important role in the description of the evolution of ‘fluid mechanical’systems
as mentioned above.
An essential step in fluid dynamics is therefore the averaging process. We have
to decide, in front of a given system, which level of averaging will be performed
in function of the quantities to be predicted, in function of the significant variables
which can be defined in a meaningful way, in function of the precision and degree of
accuracy to be achieved in the description of the system’s behavior.