alinik
۱۲ مهر ۱۳۸۸, ۱۹:۳۷
This first chapter will introduce the initial step in defining a CFD application, namely
the selection of the model to be discretized. In order to guide you through the complexities
of fluid mechanics, which we alluded to in the introduction to this Volume,
we need to establish the basic laws governing fluid flows.
You certainly have seen many flow situations and you have certainly recognized
that they can be very complex, with phenomena such as turbulence, which is a global
instability of a flow, as a dominant element of most of the flows encountered in nature
and in technology.
In addition, applications to CFD have led to a new approach and a new way of
looking at the laws of fluid mechanics. Although they can be written in many different
mathematical forms, CFD has led us to put forward a specific form of these laws,
through the concept of conservation and of conservation laws. This concept will be
central to most of this chapter.
In Section 1.1 we develop and present the most general form of a conservation law,
without specifying the nature of the ‘conserved’ quantity. To achieve this, we have
to define first what conservation means and how we recognize an equation written
in conservative form. We will see that this is a fundamental concept for CFD in
many chapters later on, but the main reason for the privileged conservative form
is connected to the requirement that, after the equations are discretized, essential
quantities such as mass or energy will be conserved at the discrete level. This is
certainly essential, as you can imagine, since a numerical simulation wherein mass or
energy would be lost because of numerical artifacts, would be totally useless and not
reliable.
A conservation law is strongly associated to the concept of fluxes and we will
introduce in Section 1.1.2 the extremely important distinction between convective and
diffusive fluxes. This distinction is central to thewhole of fluid mechanics and of CFD.
With the basis obtained in Section 1.1, we are ready to apply the general conservation
laws to the three quantities that define uniquely the laws of fluid mechanics; mass,
momentum and energy, described and developed in detail in Sections 1.2, 1.3 and 1.4.
The flow chart in Figure 1.0.1 illustrates the links and the structure of this chapter.
We strongly suggest that you refer regularly, while progressing through the material,
to this chart as a guide for the order and relative importance of the various topics.
In addition, the part in gray indicates the sections containing more advanced material
that can form the basis of a more advanced CFD course. Of course, any instructor
can make his/her own ‘cocktail’ between the various topics, according to the level of
the students.
These Advanced sections cover a few important topics, when dealing with CFD
applications to rotating systems or moving grids, which often occur in practice,in large areas of industry, such as rotating machines, flow around helicopters,
or with moving grids as encountered in fluid–structure interactions with vibrating
surfaces.
We have also added a short section, which is a straightforward extension of the
general integral form of the momentum equation, in presence of solid bodies, providing
the formulation to post-process CFD data in order to extract the forces exerted
on a body by the flow, such as lift and drag.
1.1 GENERAL FORM OF A CONSERVATION LAW
As mentioned in the introduction, the conservation law is the fundamental concept
behind the laws of fluid mechanics.
But what is a conservation law?
It is altogether very simple in its basic logic, but can become complicated by its
internal content. Conservation means that the variation of a conserved (intensive)
flow quantity within a given volume is due to the net effect of some internal sources
and of the amount of that quantitywhich is crossing the boundary surface. This amount
is called the flux and its expression results from the mechanical and thermodynamic
properties of the fluid. It will be defined more precisely in the next section. Similarly,
the sources attached to a given flow quantity are also assumed to be known from
basic studies. The fluxes and the sources are in general dependent on the space–time
coordinates, as well as on the fluid motion. The associated fluxes are vectors for a
scalar quantity and tensors for a vector quantity like momentum.
We can state the conservation law for a quantity U as the following logical
consistency rule:
The variation of the total amount of a quantityUinside a given domain is equal
to the balance between the amount of that quantity entering and leaving the
considered domain, plus the contributions from eventual sources generating
that quantity.
Hence, we are looking at the rate of change of the quantityU during the flowevolution,
as a flow is a moving and continuously changing system.
Although we will write the conservation law for an undefined quantity U, it should
be mentioned at this stage that not all flow quantities obey a conservation law. The
identification of the quantities that obey a conservation law is defined by the study
of the physical properties of a fluid flow system. It is known today that the laws
describing the evolution of fluid flows (this is what we call fluid dynamics) are totally
defined by the conservation of the following three quantities:
1. Mass
2. Momentum
3. Energy.
This represents in total five equations, as the momentum, defined as the product of
density and velocity, is a vector with three components in space.On the other hand, it is essential to keep in mind that other quantities, such as
pressure, temperature, entropy, for instance, do not satisfy a conservation law. This
does not mean that we cannot write an equation for these quantities, it just means that
this equation will not be in the form of a conservation law.
1.1.1 Scalar Conservation Law
Let us consider a scalar quantity per unit volumeU, defined as a flowrelated property.
We now consider an arbitrary volume , fixed in space, bounded by a closed
surface S (see Figure 1.1.1) crossed by the fluid flow.
The surface S is arbitrary and is called a control surface, while the volume is
called a control volume.
Our goal here is to write the fundamental lawin its most general form, by expressing
the balance of the variation ofU, for a totally arbitrary domain. This control volume
can be anywhere in the flow domain and can be of arbitrary shape and size.
To apply the conservation lawas defined above, we have to translate mathematically
the quantities involved. The first one is the ‘total amount of a quantity U inside a
given domain’. If we consider the domain of volume , the total amount of U in
is given by
U d
and the variation per unit time of the quantity U within the volume is given by
∂
∂t
U dI wish here to draw your attention to the interpretation of this mathematical expression.
The above relation would be read by a mathematician as ‘the partial derivative
with respect to time of the volume integral of U over ’. However, I would like you to
‘translate’ this mathematical language in its physical meaning, by reading this relation
instead as: ‘the variation (∂) per unit of time (./∂t) of the total amount of U in ’.
Try therefore, whenever appropriate, to always ‘read’ a mathematical expression by
its translation of a physical property.
Coming back to the conservation law, we have now to translate mathematically
‘the amount of that quantity U entering and leaving the considered domain’.
This is where the physics comes in: we know from the study of the laws of physics
that the local intensity of U will vary through the effect of quantities called fluxes,
which express the contribution from the surrounding points to the local value of U,
describing how the quantity U is transported by the flow.
The flux is a fundamental quantity associated to a conserved flow variable U,
and is defined as the amount of U crossing the unit of surface per unit of time.
It is therefore a directional quantity, with a direction and an amplitude, so that it
can be represented as a vector. If this vector is locally parallel to the surface, then
nothing will enter the domain. Consequently, only the component of the flux in the
direction of the normal to the surface will enter the domain and contribute to the rate
of change of U. So, the amount of U crossing the surface element d S per unit of
time is defined by the scalar product of the flux and the local surface element (see
Figure 1.1.1),
Fn dS = F · dS
with the surface element vector dS pointing along the outward normal.
The net total contribution from the incoming fluxes is the sum over all surface
elements dS of the closed surface S, and is given by
−S F · dS
The minus sign is introduced because we consider the flux contribution as positive
when it enters the domain. With the outward normal as positive, the scalar product
will be negative for an entering flux, as seen from Figure 1.1.1. Hence the
need to add the minus sign. If we had defined the inward normal as positive, we
would not have added the minus sign. However, the generally accepted convention
is to define as positive the outward normal, so that the minus sign is of current
acceptance.
To finalize the balance accounts, we have to add contributions from the sources of
the quantity U.
These sources can be divided into volume and surface sources, QV and QS and the
total contribution is
QV d + S QS · dS
the selection of the model to be discretized. In order to guide you through the complexities
of fluid mechanics, which we alluded to in the introduction to this Volume,
we need to establish the basic laws governing fluid flows.
You certainly have seen many flow situations and you have certainly recognized
that they can be very complex, with phenomena such as turbulence, which is a global
instability of a flow, as a dominant element of most of the flows encountered in nature
and in technology.
In addition, applications to CFD have led to a new approach and a new way of
looking at the laws of fluid mechanics. Although they can be written in many different
mathematical forms, CFD has led us to put forward a specific form of these laws,
through the concept of conservation and of conservation laws. This concept will be
central to most of this chapter.
In Section 1.1 we develop and present the most general form of a conservation law,
without specifying the nature of the ‘conserved’ quantity. To achieve this, we have
to define first what conservation means and how we recognize an equation written
in conservative form. We will see that this is a fundamental concept for CFD in
many chapters later on, but the main reason for the privileged conservative form
is connected to the requirement that, after the equations are discretized, essential
quantities such as mass or energy will be conserved at the discrete level. This is
certainly essential, as you can imagine, since a numerical simulation wherein mass or
energy would be lost because of numerical artifacts, would be totally useless and not
reliable.
A conservation law is strongly associated to the concept of fluxes and we will
introduce in Section 1.1.2 the extremely important distinction between convective and
diffusive fluxes. This distinction is central to thewhole of fluid mechanics and of CFD.
With the basis obtained in Section 1.1, we are ready to apply the general conservation
laws to the three quantities that define uniquely the laws of fluid mechanics; mass,
momentum and energy, described and developed in detail in Sections 1.2, 1.3 and 1.4.
The flow chart in Figure 1.0.1 illustrates the links and the structure of this chapter.
We strongly suggest that you refer regularly, while progressing through the material,
to this chart as a guide for the order and relative importance of the various topics.
In addition, the part in gray indicates the sections containing more advanced material
that can form the basis of a more advanced CFD course. Of course, any instructor
can make his/her own ‘cocktail’ between the various topics, according to the level of
the students.
These Advanced sections cover a few important topics, when dealing with CFD
applications to rotating systems or moving grids, which often occur in practice,in large areas of industry, such as rotating machines, flow around helicopters,
or with moving grids as encountered in fluid–structure interactions with vibrating
surfaces.
We have also added a short section, which is a straightforward extension of the
general integral form of the momentum equation, in presence of solid bodies, providing
the formulation to post-process CFD data in order to extract the forces exerted
on a body by the flow, such as lift and drag.
1.1 GENERAL FORM OF A CONSERVATION LAW
As mentioned in the introduction, the conservation law is the fundamental concept
behind the laws of fluid mechanics.
But what is a conservation law?
It is altogether very simple in its basic logic, but can become complicated by its
internal content. Conservation means that the variation of a conserved (intensive)
flow quantity within a given volume is due to the net effect of some internal sources
and of the amount of that quantitywhich is crossing the boundary surface. This amount
is called the flux and its expression results from the mechanical and thermodynamic
properties of the fluid. It will be defined more precisely in the next section. Similarly,
the sources attached to a given flow quantity are also assumed to be known from
basic studies. The fluxes and the sources are in general dependent on the space–time
coordinates, as well as on the fluid motion. The associated fluxes are vectors for a
scalar quantity and tensors for a vector quantity like momentum.
We can state the conservation law for a quantity U as the following logical
consistency rule:
The variation of the total amount of a quantityUinside a given domain is equal
to the balance between the amount of that quantity entering and leaving the
considered domain, plus the contributions from eventual sources generating
that quantity.
Hence, we are looking at the rate of change of the quantityU during the flowevolution,
as a flow is a moving and continuously changing system.
Although we will write the conservation law for an undefined quantity U, it should
be mentioned at this stage that not all flow quantities obey a conservation law. The
identification of the quantities that obey a conservation law is defined by the study
of the physical properties of a fluid flow system. It is known today that the laws
describing the evolution of fluid flows (this is what we call fluid dynamics) are totally
defined by the conservation of the following three quantities:
1. Mass
2. Momentum
3. Energy.
This represents in total five equations, as the momentum, defined as the product of
density and velocity, is a vector with three components in space.On the other hand, it is essential to keep in mind that other quantities, such as
pressure, temperature, entropy, for instance, do not satisfy a conservation law. This
does not mean that we cannot write an equation for these quantities, it just means that
this equation will not be in the form of a conservation law.
1.1.1 Scalar Conservation Law
Let us consider a scalar quantity per unit volumeU, defined as a flowrelated property.
We now consider an arbitrary volume , fixed in space, bounded by a closed
surface S (see Figure 1.1.1) crossed by the fluid flow.
The surface S is arbitrary and is called a control surface, while the volume is
called a control volume.
Our goal here is to write the fundamental lawin its most general form, by expressing
the balance of the variation ofU, for a totally arbitrary domain. This control volume
can be anywhere in the flow domain and can be of arbitrary shape and size.
To apply the conservation lawas defined above, we have to translate mathematically
the quantities involved. The first one is the ‘total amount of a quantity U inside a
given domain’. If we consider the domain of volume , the total amount of U in
is given by
U d
and the variation per unit time of the quantity U within the volume is given by
∂
∂t
U dI wish here to draw your attention to the interpretation of this mathematical expression.
The above relation would be read by a mathematician as ‘the partial derivative
with respect to time of the volume integral of U over ’. However, I would like you to
‘translate’ this mathematical language in its physical meaning, by reading this relation
instead as: ‘the variation (∂) per unit of time (./∂t) of the total amount of U in ’.
Try therefore, whenever appropriate, to always ‘read’ a mathematical expression by
its translation of a physical property.
Coming back to the conservation law, we have now to translate mathematically
‘the amount of that quantity U entering and leaving the considered domain’.
This is where the physics comes in: we know from the study of the laws of physics
that the local intensity of U will vary through the effect of quantities called fluxes,
which express the contribution from the surrounding points to the local value of U,
describing how the quantity U is transported by the flow.
The flux is a fundamental quantity associated to a conserved flow variable U,
and is defined as the amount of U crossing the unit of surface per unit of time.
It is therefore a directional quantity, with a direction and an amplitude, so that it
can be represented as a vector. If this vector is locally parallel to the surface, then
nothing will enter the domain. Consequently, only the component of the flux in the
direction of the normal to the surface will enter the domain and contribute to the rate
of change of U. So, the amount of U crossing the surface element d S per unit of
time is defined by the scalar product of the flux and the local surface element (see
Figure 1.1.1),
Fn dS = F · dS
with the surface element vector dS pointing along the outward normal.
The net total contribution from the incoming fluxes is the sum over all surface
elements dS of the closed surface S, and is given by
−S F · dS
The minus sign is introduced because we consider the flux contribution as positive
when it enters the domain. With the outward normal as positive, the scalar product
will be negative for an entering flux, as seen from Figure 1.1.1. Hence the
need to add the minus sign. If we had defined the inward normal as positive, we
would not have added the minus sign. However, the generally accepted convention
is to define as positive the outward normal, so that the minus sign is of current
acceptance.
To finalize the balance accounts, we have to add contributions from the sources of
the quantity U.
These sources can be divided into volume and surface sources, QV and QS and the
total contribution is
QV d + S QS · dS