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1. Fundamental Equations for Newtonian Fluids
1.1 Introduction
In this chapter we recall the standard form of the classical
fluid dynamics
equations in eulerian form. For the evolution of a fluid in 1N
spatial dimensions, the description involves ( )2N + fields. These
are the mass density, the velocity field and the energy. The
fundamental equations of
fluid dynamics are derived following the next three universal
conservation
laws:
1. Conservation of Mass.
2. Conservation of Momentum or the Newtons Second Law.
3. Conservation of Energy or the First Law of
Thermodynamics.
The resulting equations are the continuity equation, the
momentum
equations and the energy equation, respectively. The system of
governing
equations is closed by:
1. Statements characterizing the thermodynamic behavior of the
fluid.
These are called state equations. The state equations do not
depend
on the fluid motion and represent relationships between the
thermodynamic variable: pressure (p), density ( ), temperature
(T), specific internal energy per unit mass (e) and total enthalpy
per unit
mass (h). The microscopic transport properties like the
dynamic
viscosity ( ) and the thermal conductivity (k) are also related
to the thermodynamic variables.
2. Constitutive relations, which are postulated relationships
between
stress and rate of strain and heat flux and temperature
gradient. The
previously mentioned microscopic transport properties appear in
the
constitutive relations as proportionality factors (between
stress and
rate of strain and heat flux and temperature gradient) and must
be
determined experimentally.
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The flow is considered as known if, at any instant in time, the
velocity field
and a precise number of thermodynamic variables are known at any
point.
This number equals two for a real compressible fluid in
thermodynamic
equilibrium (for example, the pressure and the density).
There is some freedom in choosing a set of variables to describe
the fluid
flow. A possible choice is the set of primitive or physical
variables,
namely the density, pressure and velocity components along the
three
directions of a reference frame, i.e. the set ( ), , , ,p u v w
. Another choice is the set of the so-called conservative variables
which are the density, the
three momentum components and the total energy per unit mass,
i.e. the set
( ), , , ,u v w E . The set of conservative variables results
naturally from the application of the above fundamental laws of
conservation. From a
computational point of view, there are some advantages in
expressing the
governing equations in terms of the conservative variables. We
will call
conservative methods the class of numerical methods based on
the
formulation of the governing equations in conservative
variable.
The primitive and conservative variables are dependent
variables. They are
depending on time and space coordinates, i.e. on the
independent
variables. For example, the density is ( ), , ,t x z y = .
Recall that the number of the spatial coordinates is N.
The reader is invited here to notice an important difference
between the
governing equations and the closure conditions. The governing
equations
are exact mathematical conditions that must be satisfied by the
flow
variables. The state equations and the constitutive relations
are only
practical models and they depend on the degree of knowledge the
user has
about the fluid and flow under consideration. Thus, they
represent sources
of uncertainties that are introduced in the mathematical model
of the flow.
The derivation of the fundamental equations mentioned above will
not be
presented here. Almost all of the books devoted to fluid
mechanics or to
CFD have chapters devoted to this issue. For example, more
details on the
equations can be found in (Hirsch, 1990), (Ferziger and Peric,
1999) and
(Warsi, 1999).
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1.2 Preliminary Concepts
Before proceeding to the derivation of the equations in integral
form review
some preliminary concepts that are needed. Consider a scalar
field function
),,,( tzyx , then the time rate of change of as registered by an
observer moving with the fluid velocity ( . , )u v w=V is given by
the substantial derivative (or material derivative) defined as:
D
gradDt t
= +
V (1.1)
The first term t
denotes the partial derivative of with respect to time
and represents the local rate of change of while the second term
is the convective rate of change. If the scalar field is replaced
by a vector field, the
operator Dt
Dcan also be applied in a component-wise manner. In
particular,
we can obtain the substantial derivatives of ( . , )u v w=V ,
that is:
D
gradDt t
= +
V V
V V (1.2)
which in full becomes:
( , , )
T TDu u u v w
Dt t x x x
Dv v u v wu v w
Dt t u y y
Dw w u v w
Dt t z z z
= +
(1.3)
The left hand side of (1.3) is the acceleration vector of an
element of moving
fluid. Let us now consider the new scalar field
( ) ( , , , )V
t x y z t dV = (1.4) where the volume of integration V is
enclosed by a piece-wise smooth
boundary surface A V= that moves with the material under
consideration. It can be shown that the material derivative of is
given by:
( )V AD
dV dADt t
= +
n V (1.5)
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where 1 2 3( , , )n n n=n is the outward pointing unit vector
normal to the
surface A. Expression (1.5) can be generalized to vectors ( , ,
, )x y z t as
follows:
( )V AD
dV dADt t
= +
n V (1.6)
The surface integrals in the last equalities may be transformed
to a volume
integral by virtue of Gausss theorem. This states that for any
differentiable
vector field 1 2 3( , , ) = and a volume V with smooth bounding
surface A the following identity holds:
( )A V
dA div dV = n (1.7) The above identity also applies to
differentiable scalar and tensor fields. If
we use it, for example, to transform the right hand side of
equation (1.5) into
a volume integral, one obtains:
( )VD
div dVDt t
= + V (1.8)
Please notice that in the above relations the volume V is
arbitrary and
moving with the fluid particles. Let J be the jacobian of the
transform that
links two successive positions of the volume V. There is a
theorem due to
Euler which states that the time derivative of the jacobian is
(Lions, 1996):
D J
J divDt
= V (1.9)
From (1.9) we remark that mathematically, the incompressibility
means:
0div =V (1.10)
1.3 Integral Form of the Governing Equations
1.3.1 Conservation of Mass
The law conservation of mass can now be stated in integral form
by
identifying the scalar in (1.4) as the density . In this case
)(t in (1.5) becomes the total mass in the volume V. By assuming
that no mass is
generated or annihilated within the material control volume V,
we have
0=Dt
D, or:
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( ) 0V AdV dAt
+ =
n V (1.11) This is the integral form of the law of the
conservation of mass. This
integral conservation law may be generalised to include sources
of mass ,
which will then appear as additional integral terms. A useful
reinterpretation
of the integral form result if we rewrite it as
( )V A
dV dAt
=
n V (1.12) If now V is a fixed control volume then the left hand
side of (1.12) becomes:
V V
ddV dV
t dt
=
(1.13) and thus is the time-rate of change of the mass enclosed
by the volume V.
The right hand side of (1.12) shows that the mass enclosed by
the control
volume V, in the absence of sources or skins, can only change by
virtue of
mass flow through the boundary of the control volume V. One
obtains from
(1.8) and (1.12):
( ) 0V
div dVt
+ = V (1.14)
As V is arbitrary it follows that the integrand must vanish,
that is:
( ) ( ) ( ) 0t x yu v w + + + = (1.15) which is the differential
form of the continuity equation.
We notice here that the differential and integral forms (1.15)
and (1.14) are
not valid in general. Under the assumption of inviscid fluids,
the governing
equations admit discontinuities in the solution, such as shock
waves and
contract surfaces. The integral form (1.11), however, remains
valid.
1.3.2 Conservation of Momentum
As done for the mass equation, we now provide the foundations
for the law
of conservation of momentum, derive its integral form in quite
general
terms and show that under appropriate smoothness assumptions
the
differential form is implied by the integral form. A control
volume V with
bounding surface A is chosen and the total momentum in V is
given by:
( )V
t dV= V (1.16) The law of conservation of momentum results from
the direct application of
Newtons law: the time rate of change of the momentum in V is
equal to the
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total force acting on the volume V. The total force is divided
into surface
forces fs and volume fv given by:
, vA VdA dV= = sf S f g (1.17)
Here g is the specific volume-force vector and may account for
inertial
forces, gravitational forces, electromagnetic forces and so on.
S is the stress
vector, which is given in terms of a stress tensor as = S n .
The stress tensor can be split into a spherical symmetric part due
to pressure p, and
a viscous part :
p= I (1.18) Application of Newtons Law gives:
( ) ( ) sV A
dV dAt
= + + vV V n V f f (1.19)
Regarding V as a fixed volume in space, independent of time, we
write:
( ) ( )V A
ddV dA
dt = + + s vV V n V f f ,
We interpret the above as saying that the time rate of change of
momentum
within the fixed control volume V is due to the net momentum
inflow over
momentum outflow, given by the first term in (1.19), plus
surface and
volume forces.
Substituting the stress tensor from (1.18) into (1.19) and
writing all surface
terms into a single integral we have:
[ ]( ) ( ) .V A V
ddV p dA dV
dt = + + V V n V n n g (1.20)
This is a general statement and is valid even for the case of
discontinuous
solutions.
The differential conservation law can now be derived from (1.20)
under
assumption that the integrand in the surface integral is
sufficiently smooth
so that Gausss theorem may be invoked. The first term of the
integrand of
the surface integral can be rewritten as:
( ) , = V n V n V V where the dyadic (tensor) product VV is a
tensor. Its components are easily determined from the next
equality. The three columns of the left hand
side are:
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2
2
2
( ) , , ,
( ) , , ,
( ) , , .
T
T
T
u u uv uw
v uv v vw
w uw vw w
=
=
=
n V n
n V n
n V n
Now one can apply the Gausss theorem to each of the surface
terms:
[ ]( ) ( ) .V V V
div gradp div dV dVt
= + + V V V g (1.21)
As this is valid for any arbitrary volume V the intregrand must
vanish, i.e.:
[ ]( ) div pt
+ + =
V V V I g (1.22)
This is the differential conservative form of the momentum
equation,
including a source term due to volume forces. In order to use it
for a given
fluid flow the components of the viscous stress tensor must be
provided
through a constitutive relation.
When the viscous stresses are identically zero, and the volume
forces are
neglected, we obtain the so-called compressible Euler equation.
If the
viscous stresses are given by a constitutive relation under the
Newtonian
assumption we obtain the compressible Navier-Stokes equation
in
differential conservation law form.
1.3.3 Conservation of Energy
As done for mass and momentum we now consider the total energy
in a
control volume V, that is:
( ) .V
t EdV = (1.23) where E is the total energy per unit volume:
( )2 2 2 21 12 2
E e e u v w = + = + + +
V (1.24)
Here e is the specific internal energy and the second term
represents the
specific kinetic energy.
Before we discuss the equations in more detail, let us make a
physical
comment: the derivation of the energy equation relies upon the
assumption
that, in a fluid in motion, the fluctuation around thermodynamic
equilibrium
are sufficiently weak so that the classical thermodynamics
results hold at
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every point and at all times. In particular, the thermodynamical
state of the
fluid is determined by the same state variables as in
classical
thermodynamics, and these variables are determined by the same
state
equations (see the devoted paragraph in this chapter).
The energy conservation law states that the time rate of change
of total
energy )(t is equal to the work done, per unit time, by all
forces acting on the volume plus the influx of energy per unit time
into the volume. The
surface and volume forces in (1.17) respectively give rise to
the following
terms:
( ) ( )surfA A
E p dA dA= + V n V n , ( ) .vol
VE dV= V g
The first term corresponds to the work done by the pressure
while the
second term corresponds to the work done by the viscous
stresses. volE is
the work done by the volume force g. To account for the influx
of energy
into the volume we denote the energy flux vector by 1 2 3( , ,
)q q q=Q . The flow of energy per unit time across a surface dA is
given by the flux
( )dA n Q . This gives the total influx of energy to be included
in the equation of balance of energy:
inf ( )A
E dA= n Q , Thus, one can write that:
infsurf volD
E E EDt
= + + (1.25)
The left hand side of (1.25) can be transformed into:
( )
( )V A
D tEdV E dA
Dt t
= +
n V (1.26) As done for the laws of conservation of mass momentum
we now reinterpret
the volume V as fixed in space and independent of time. This
leads to:
[ ]( ) ( ( )V A V
dEdV E p dA dV
dt= + + + n V V Q V n V g (1.27)
The differential form of the conservation of energy law can now
be derived
by assuming sufficient smoothness and applying Gausss theorem to
all
surface integrals of (1.27). Direct application of Gausss
theorem gives:
( ) ( )A V
p dA div p dV = n V V
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.A V
dA div dV = n Q Q The second term of (1.27) can be transformed
via Gausss theorem by first
observing that ( ) ( ). = V n n V This follows from the symmetry
of the viscous stress tensor . Hence:
( ) ( )A V
dA div dV = V n V Substitution of these volume integrals into
the integral form of the law of
conservation of energy (1.84) gives:
( ){ }( ) 0tVE div E p dV+ + + = V V Q V g (1.28)
Since the volume V is arbitrary the integrand must vanish
identically, that
is:
( ) ( ) 0tE div E p + + + = V V Q V g (1.29) This is the
differential conservative form of the law of conservation of
energy with a source term accounting for effect of body forces;
if these are
neglected we obtain the homogeneous energy equation
corresponding to the
Navier-Stokes equation. The energy flux vector Q must be
specified through
a constitutive relation.
Further, when viscous and heat conduction effects are neglected
we obtain
the energy equation corresponding to the compressible Euler
equations.
1.3.4 The Role of the Integral Form of the Equations
The previous derivation of the governing equations, such as
the
compressible Euler and Navier-Stokes equations stated earlier,
is based on
integral relations on control volume and their boundaries.
The differential form of the equations results from further
assumptions on
the flow variables (smoothness). In the case of the
Navier-Stokes equations,
the smoothness of the solution is naturally assured by the
diffusion due to
the viscosity and heat conductivity. On the contrary, in the
absence of
viscous diffusion and heat conduction one obtains the Euler
equations.
These admit discontinuous solutions and the smoothness
assumption that
leads to the differential form no longer holds true. Thus, one
must return to
the more fundamental integral from involving integrals over
control
volumes and their boundaries. However, for most of the
theoretical
developments the differential form of the equations is
preferred.
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Form a computational point of view there is another good reason
for
returning to the integral form of the equations. Discretised
domains result
naturally in finite control volumes or computational cells.
Local
enforcement of the fundamental equations in these volumes leads
to Finite
Volume numerical methods.
1.4 Equations of State
1.4.1 Generalities
The governing equations expressed in integral or differential
form are
describing the dynamics of a compressible material. However,
these
equations are insufficient to completely describe the physical
processes
involved during a flow. There are more unknowns than equations
and thus
closure conditions are required. For example, if we take as
unknowns the
five conservative variables ( ), ,E V , it is easy to see that
there are terms in the integral or differential form of the
governing equations which must be
expressed in terms of the unknowns. Such terms are the pressure
and the
specific internal energy. One therefore requires another
relation defining e
and p in terms of quantities already given, such as the
conservative
variables. For a number of applications, due to the complexity
of the
physical phenomena other variables, such as temperature T or
entropy s for
instance, may need to be introduced.
The formulation of the closure conditions we have referred to
previously is
provided by Thermodynamics under the form of state equations.
The
specific internal energy e has an important role in the First
Law of
Thermodynamics, while the entropy s is involved with the Second
Law of
Thermodynamics. We mention here that the entropy plays an
important not
just in establishing the governing equations but also at the
level of their
mathematical properties and the designing of numerical methods
to solve
them. Details can be found in (Lions, 1996) and (Toro,
1997).
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1.4.2 Equations of State for Systems in Thermodynamic
Equilibrium
A system in thermodynamic equilibrium can be completely
described by the
basic thermodynamic variables pressure p and the specific volume
. The specific volume is defined as
1
= (1.30)
A family of states in thermodynamic equilibrium may be described
by a
curve in the p plane each characterized by a particular value of
a variable temperature T. There are physical situations that
require additional
variables. Here we are only interested in p T systems. In these,
one can relate the variables via the thermal equation of state:
( ),T T p = (1.31) The p T relationship changes from substance
to substance. For thermally ideal gases one has the simple
expression:
p p
TR R
= = (1.32)
where R is a constant which depends on the particular gas
under
consideration.
The First Law of Thermodynamics states that for a non-adiabatic
system
the change e in internal energy e in a process will be given by
e=W
+Q, where W is the work done on the system and Q is the heat
transmitted to the system. Taking the work done as dW = - pdv
one may
write:
dQ de pd= + (1.33) The internal energy e can also be related to
the pressure and specific volume
through a caloric equation of state:
( ),e e p = (1.34) The relations (1.31) and (1.34) are
invertible.
For a calorically ideal gas one has the simple expression or
equation of
state (EOS):
( 1)
pe
=
(1.35)
where is a constant that depend on the particular gas under
consideration.
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The thermal and caloric equation of state for a given material
are closely
related. Both are necessary for a complete description of
the
thermodynamics of a system. Choosing a thermal EOS does restrict
the
choice of a caloric EOS but does not determinate it. Note that
for the Euler
equations one requires a caloric EOS , e.g. ( ),p p e= unless
temperature T is needed for some other purpose, in which case a
thermal EOS needs to
be given explicitly.
1.4.3 Other Thermodynamic Variables and Relations
The entropy s results as follows. We first introduce an
integrating factor 1/T
so that that expression
e e
de pd p d dpv p
+ = + +
(1.36)
becomes an exact differential. Then for the Second Law of
Thermodynamics one introduces the new variable s, called
entropy, via the
relation:
Tds de pd= + (1.37)
For any physical process the change in entropy is due to the
entropy carried
into the system through the boundaries of the system, 0s , and
to the
entropy generated in the system during the process, is .
Examples of entropy-generating mechanism are heat transfer, and
viscosity such as may
operate within the internal structure of shock waves.
The importance of the entropy in fluid dynamics and not only
resides from
the Second Law of Thermodynamics. This states that
0is (1.38) in any physical process. If the process is reversible
then the entropy does not
change, i.e. 0is = . An equivalent but more useful form of the
equation (1.38) is the next one:
( ) ( ) 0t
s div s + V (1.39) As long as shocks do not form during the
flow, the entropy equation is valid
and if initially s is constant, it remains constant. This is a
very strong quality
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criterion for the evaluation of different numerical schemes
developed for the
Euler equations and not only.
Another variable of interest is the specific enthalpy h. This is
defined in
terms of other thermodynamic variables, namely:
h e p= + (1.40)
One can also establish other relationship amongst the basic
thermodynamic
variable already defined. For instance one may choose to express
the
internal energy e in terms of the variables appearing in the
differentials, i.e.
).,( vsee =
Also, taking the differential of (1.40) we have ,vdppdvdedh ++=
which becomes ,vdpTdsdh += and thus we can choose to define h in
terms of s and p, i.e. ).,( pshh =
We call the last two relations canonical equations of state and,
unlike the
thermal and caloric equations of states (1.31) and (1.34), each
of these
provides a complete description of the Thermodynamics.
Two more quantities can be defined if a thermal EOS ( ),p T = .
These are the volume expansivity (or expansion coefficient) and the
isothermal
conpressibility , namely:
pT
v
v
=1
and
Tp
v
v
=1
.
The heat capacity at constant pressure cp and the heat capacity
at
constant volume cv (specific heat capacities) are now
introduced. In
general, when an addition of heat dQ changes the temperature by
dT the
ratio dTdQc /= is called the heat capacity of the system. For a
thermodynamic process at constant pressure one obtains:
,),( dhvpddedQ =+=
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14
where the definition of the specific enthalpy has been used.
Assuming
),( pThh = , the heat capacity cp at constant pressure
becomes:
pp p
h sc T
T T
= = (1.41)
The heat capacity cv at constant volume may be written,
following a similar
argument, as:
vv v
e sc T
T T
= = (1.42)
The speed of sound is another variable of fundamental interest.
For flows
in which particles undergo unconstrained thermodynamic
equilibrium one
defines a new state variable a, called the equilibrium speed of
sound or
just speed of sound. Given a caloric equation of state such as
),,( spp = one defines the speed of sound a as:
s
pa
= (1.43)
This basic definition can be transformed in various ways using
established
thermodynamic relations. For instance, given a caloric EOS in
the form
),,( phh = we can also write:
+
+=
+
dss
pd
pTdsd
hdp
p
h
spp
1
Settings ds=0 and using definition (1.43) we obtain
successively:
1
2
=
p
h
h
ap
and for a thermally ideal gas characterized by the EOS of the
form h= h(T) :
1
2
=
p
pp
h
h
a
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From pp
ch
=
and for a thermally ideal gas (i.e., if the thermal EOS is
acceptable), we obtain the widely used relation for the speed of
sound:
( )P
a R
= = (1.44)
The reader should notice the dependence on the temperature of
the
coefficient .
For a general material the caloric EOS is a functional
relationship involving
the variable p--e. The derived expression for the speed of sound
a depends
on the choice of dependent variables. Two possible choices and
their
respective expression for a are:
2
2
( , ),
( , ),
e
p p
pp p e a p p
epe e p a
e e
= = +
= =
(1.45)
where subscripts denote partial derivates.
1.4.4 Ideal Gases
We consider in what follows gases obeying the ideal thermal EOS
of the
form:
pV nRT= (1.46) where V is the volume, R=8.134 x 10
3 J kilomole
-1, called the Universal
Gas Constant, and T is the absolute temperature measured in
degrees
Kelvin (K). Recall that a mole of a substance is numerically
equal to gram
and contains 6.02x1023 particles of that substance, where is the
relative
atomic mass or relative molecular mass; 1 kilomole = kg. One
kilomole of
substance contains NA = 6.02 x 1026 particles of that substance.
The constant
NA is called the Avogadro Number. Sometimes this number is given
in
terms of one mole. The Boltzmann constant k is k = RNA. The
number of
kilomoles in volume V is n = N/NA and N is the number of
molecules. On
division by the mass m = n have:
,p RT R
= = (1.47)
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or
p
RT
= (1.48)
where R is called the Gas constant. Equation (1.48) is known as
Mariottes
law. Further, we obtain from (1.47) the ideal gas thermal
equation of state
as:
.),(p
RTpTvv ==
The volume expansivity and the isothermal compressibility
defined
previously become 1 1and
T p = = , respectively. After some
manipulations, one deduce that:
0T
e
v
= (1.49)
This means that if the ideal thermal EOS is assumed, then it
follows that the
internal energy e is a function of temperature alone, that is
e=e(T). A
remarkable particular case is Joules law, that is one in
which:
ve c T= (1.50) where the specific heat capacity cv is a
constant. In this case one speaks of
a calorically ideal gas, or a polytropic gas.
It is now possible to relate cp and cv via the general
expression 2
.p vT
c c
= + For a thermally ideal gas equations this expression
simplifies to:
p vc c R = (1.51)
We mention here that, a necessary condition for thermal
stability is cv>0 and
a condition for mechanical stability is >0, see (Toro, 1997).
From (1.51) the
following inequalities result 0p vc c> > .
The ratio of specific heats , or adiabatic exponent, is defined
as:
p
v
c
c = (1.52)
which if used in conjuntion with (1.51) gives:
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, .1 1
p v
R Rc c
= =
(1.53)
In order to determinate the caloric EOS (1.50) we need to
determine the
specific heat capacities, cv in particular. Molecular Theory and
the principle
of equipartition of energy (Toro, 1997) can also provide an
expression for
the specific internal energy of molecule. In general a molecule,
however
complex, has M degrees of freedom, of which three are
translational. Other
possible degrees of freedom are rotational and vibrational. From
Molecular
Theory it can be shown that if the energy associated with any
degree of
freedom is a quadratic function in the appropriate variable
expressing that
degree of freedom, then the mean value of the energy is kT2
1where k is the
Boltzmann constant. Moreover, from the principle of
equipartition of energy
this is the same for each degree of freedom. Therefore, the mean
total
energy of a molecule is 1
2e MkT= and for N molecule we have
1
2Ne MNkT=
From which the specific internal energy is
.2
1MRTe =
From the above considerations we obtain successively:
,2
1MR
T
ec
v
v =
=
and
.2
2R
Mcp
+=
Thus, the ratio of specific heat becomes can be expressed in
terms of the
degrees of freedom M:
.2
M
M +=
And hence:
1
2
=
M
From the thermal EOS for ideal gases we have:
-
18
.2
1Mpve =
And
( ) ( )1 1pv p
ep
= =
(1.54)
Thus, we have found once again the expression for the specific
internal
energy.
These theoretical expressions for the specific heats and their
ratio in terms
of R and M are found to be very accurate for monatomic gases,
when M = 3
(three translational degrees of freedom). For polyatomic gases
rotational and
vibrational degrees of freedom contribute to M but now the
expression
might be rather inaccurate when compared with experimental data.
A strong
dependence on T is observed. However, the inequality ,3
51
-
19
The covolume EOS can be further corrected to account for the
forces of
attraction between molecules, the so-called van der Waal forces.
These are
neglected in both the ideal and covolume equations of state.
Accounting for
such forces results in a reduction of the pressure by an amount
of c/v2,
where c is a quantity that depend on the particular gas under
consideration.
Thus from (1.49) the pressure is corrected as
.2v
c
bv
RTp
=
Then we can write:
2( )
cp v b RT
v
+ =
(1.56)
This is generally known as the van der Waals equation of state
for real
gases.
So far, we have presented the governing equations for the
dynamics of a
compressible medium along with some EOS so that a closed system
is
obtained.
1.5 Viscous Stresses and Heat Conduction
The stresses in a fluid, given by a tensor , are due to the
effects of the
thermodynamic pressure p and the viscous stresses. Thus the
stress tensor
can be written as
p= + I (1.57) where pI is the spherically tensor due to p, I is
the unit tensor and is the
viscous stress tensor. It is desirable to express in terms of
flow variables
already defined. For the pressure contribution this has already
been achieved
by defining p in terms of other thermodynamic variables via an
equation of
state. Recall that equations of state are approximate statements
about
the nature of a material. In defining the viscous stress
contribution one
may resort to the Newtonian approximation whereby is related to
the
derivatives of the velocity field ( ), ,u v w=V via the
deformation tensor:
-
20
1 1( ) ( )
2 2
1 1( ) ( )
2 2
1 1( ) ( )
2 2
x x y x z
x y y y z
x z y z z
u v u w u
v u v w v
w u w v w
+ + = + + + +
D (1.58)
The Newtonian assumption is an idealisation in which the
relationship
between and D is linear and homogeneous, that is will vanish
only if
D vanishes, and the medium is isotropic with respect to this
relation. An
isotropic medium is that in which there are no preferred
directions. By
denoting the stress tensor by:
xx xy xz
yx yy yz
zx zy zz
=
(1.59)
the Newtonian approximation becomes:
2
2 ( )3
b div = +
D V I (1.60)
In full we have:
( )
( )
( )
4 2,
3 3
4 2,
3 3
4 2,
3 3
( ),
( ),
( ).
xx
x y x b
yy
y z x b
zz
z x y b
xy yx
y x
yz zy
z y
zx xz
x z
u v w divV
v w u divV
w u v divV
u v
v w
w u
= + + = + +
= + + = = += = +
= = +
(1.61)
In the Newtonian relationship (1.60) there are two scalar
quantities that are
still undetermined. These are the coefficient of shear viscosity
and the
coefficient of bulk viscosity b. Approximate expressions for
there are
obtained mainly from experiments.
-
21
In particular, for monatomic gases Molecular Theory gives for
the
coefficient of bulk viscosity ,0=b which is found to agree well
with experiment. For polyatomic gases 0b and appropriate values for
b are to be obtained experimentally.
Concerning the coefficient of shear viscosity , it is observed
that, as long as
temperature are not too high, depends strongly on temperature
and only
slightly on pressure. A relatively accurate relation between and
T is the
Sutherland formula:
1
21 1
CC T
T
= +
(1.62)
where C1 and C2 are two experimentally adjustable constants.
When T is
measured in Kelvin degrees, for case of air one has:
.112,1046.1 26
1 KCxC ==
Sutherlands formula describes the dependence of on T rather well
for a
wide range of temperatures, provided no dissociation or
ionisation take
place. These phenomena occur at very high temperatures where
the
dependence of on pressure p, in addition to temperature T,
cannot be
neglected.
In summary, the Navier-Stokes equations (momentum equation) can
now be
written in differential conservation law form as:
( ) ( ) 0t
p + + =V V V I (1.63) where is given by (for monatomic
gases):
2
2 ( )3
div = D V I (1.64)
The Euler equations written in differential conservation form
are:
( ) ( ) 0t
p + + =V V V I (1.65)
Influx of energy contributes to the rate of change of total
energy E. We
denoted by ( )1 2 3, ,T
q q q=Q the energy flux vector, which results from: i)
heat flow due to temperature gradients, ii) diffusion processes
in gas
mixture and (iii) radiation.
In what follows we only consider the effect of the heat flux due
to
temperature gradients. Thus, Q is identical to the heat flux
vector caused by
-
22
temperature gradients. In a similar manner to that it which
viscous stresses
were related to gradient of the velocity vector V, one can
assume that the
fluid is isotropic and thus one can relate Q to gradients of
temperature T via
Fouriers heat conduction law:
k T= Q (1.66) where k is a positive scalar quantity already
called the coefficient of
thermal conductivity, and is yet to be determined.
Note the analogy between the two microscopic transport
properties of the
fluid, and k. This analogy between and k goes further in that k,
just as ,
depends on T but only slightly on pressure p.
In fact, Molecular Theory says that k is directly proportional
to . Under the
assumption that the specific heat at constant pressure cp is
constant, the
dimensionless quantity called the Prandtl number:
p
r
cP
k
= (1.67)
is a constant. For monatomic gases Pr is very nearly constant.
For air in the
temperature range KTK 100200 Pr differs only slightly from its
mean value of 0.7.
There is also a formula attributed to Eucken, see (Toro, 1997),
that relates Pr
to the ratio of specific heats via
59
4
=
rP
to account for departures from calorically ideal gas
behaviour.
1.6 Differential Form of the Governing Equations
We have determined previously the integral form of the
governing
equations, as well as the conservative differential form of the
equations.
Here we summarize the governing equations written in
conservative
differential form. We notice that the name Euler and/or
Navier-Stokes
originally given to the momentum equations (or equations of
motion)
transfers to the entire system of equations.
We summarize below the general laws of conservation of mass,
momentum
and total energy, written both in differential and integral
form:
-
23
a) the continuity equation:
( ) 0t div + =V or
( ) 0V A
ddV dA
dt + = n V
b) the momentum eqution:
[ ]( ) ,t div p + + =V V V I g or
[ ]( ) ( ) .V V V
div gradp div dV dVt
+ + = V V V g
c) the energy equation
[ ]( ) ( ).tE div E p + + + = V V Q V g or
[ ]( ) ( ( )V A V
dEdV E p dA dV
dt+ + + = n V V Q V n V g
where ),,( 321 gggg = is a body force vector.
At this point, all the terms, such as the viscous stress tensor,
the heat flux,
the pressure, the temperature and the specific internal energy
are expressed
as functions of the conservative variables. For practical
reasons, it is useful
to write the previous equations in a more compact form, which is
called
conservation law form.
1.6.1 The Euler Equations
In this section we consider the time-dependent Euler equations.
These are a
system of non-linear hyperbolic conservation laws that govern
the dynamics
of a compressible material, such as gases or liquids at high
pressure.
When body forces are included via a source term vector but
viscous and
heat conduction effects are neglected we have the Euler
equations:
( ) ( ) ( ) ( )t x y z+ + + =U F U G U H U S U (1.68)
where the vector of conservative variables is U and the
convective fluxes
are F,G,H in the x, y and z directions. Their explicit forms
are:
-
24
2
2
2
, ,
( )
,
( ) ( ).
u
u u p
v uv
w puw
E u E p
wv
uwuv
vwv p
vw w p
v E p w E p
+ = = +
= =+ + + +
U F
G H
(1.69)
First, it is important to note that the fluxes are nonlinear
functions of the
conserved variable vector. Any set of partial differential
equations written in
the form (1.68) is called a system of conservation law (in
differential
formulation). Recall that the differential form assumes smooth
solutions,
that is, partial derivatives are assumed to exist. It is also
clear now that the
integral form of the equations is an alternative way of
expressing the
conservation laws in which the smoothness assumption is relaxed
so that to
include discontinuous solutions.
Here ( )=S S U is a source or forcing term. Due to the presence
of the source term, the equations are said to be inhomogeneous.
There are several
physical effects that can be included in the forcing term: body
forces such as
gravity, injection of mass, momentum and/or energy. Usually, (
)S U is a
prescribed algebraic function of the flow variables and does not
involve
derivatives of these, but there are exceptions. When ( ) 0=S U
one speaks of homogeneous equations. We also mention here that
there are situations in
which source terms arise as a consequence of approximating
the
homogeneous equations to model situations with particular
geometric
features (axy-symmetric flows, for instance). In this case the
source term is
of geometric character, but we shall still call it a source
term.
Sometimes it is convenient to express the equations in term of
the primitive
or physical variables , u, v, w and p. By expanding derivatives
in the
conservation law form and using the mass equation into the
momentum
-
25
equations and in turn using these into the energy equation one
can re-write
the thee-dimensional Euler equations for ideal gases with a
body-force
source term as:
1
2
3
( ) 0
1,
1,
1,
( ) 0
t x y z x y z
t x y z x
t x y z y
t x y z z
t x y z x y z
u v w u v w
u uu vu wu p g
v uv vv wv p g
w uw vw ww p g
p up vp wp p u v w
+ + + + + + =
+ + + + =
+ + + + =
+ + + + =
+ + + + + + =
(1.70)
For computational purposes it is the conservation law form
(1.68) that is
most useful. The formulation in primitive variables is more
convenient to
theoretical developments.
1.6.2 The Navier-Stokes Equations
When the effects of viscosity and heat conduction are added to
the basic
Euler equations one has the Navier-Stokes equations with heat
conduction
in conservation law form:
c c c d d dt x y z x y z+ + + = + +U F G H F G H (1.71)
where U is once again the vector of conserved variables, the
flux vectors Fc,
Gc and H
c are the inviscid fluxes (c stand for convection) for the
Euler
equations as given previously and the respective flux vectors
Fd, G
d and H
d
(d stands for diffusion) due to viscosity and heat conduction
are:
-
26
1 2
3
0 0
, '
0
yxxx
yyd dxy
yzxz
xx xy xz yx yy yz
zx
zyd
zz
zx zy zz
u v w q u v w q
u v w q
= = + + + +
= + +
F G
H
(1.72)
The form of the equations given by (1.71) split the effect of
convection on
the left-hand side from those of viscous diffusion and heat
conduction on
the right-hand side. For numerical purpose, the particular form
of the
equations adopted depends largely on the numerical technique to
be used to
solve the equations. One possible approach is to split the
convection effects
from those of viscous diffusion and heat conduction during a
small time
interval t, in which case the above from is perfectly adequate.
An
alternative form is obtained by combining the fluxes due to
convection,
viscous diffusion and heat conduction into new fluxes so that
the governing
equations look formally like a homogeneous system (zero
right-hand side)
of conservation laws:
0,
with
, ,
t x y z
c d c d c d
+ + + =
= = =
U F G H
F F F G G G H H H
(1.73)
This form is only justified if the numerical method employed
actually
exploits the coupling of convection, viscosity and heat
conduction when
defining numerical approximations to the flux vectors F, G and H
in (1.73).
-
27
1.7 Conclusion
We have presented in this chapter the time-dependent Euler and
Navier-
Stokes equations of fluid dynamics. The equations are
accompanied by
equations of state and by constitutive relations. Both models
can be used for
homogeneous gases and/or liquids at high pressure. We appreciate
that for
usual purposes the Euler and Navier-Stokes are the
representative models.
We now conclude this chapter with a brief discussion of initial
and
boundary conditions. Our goal in this book is to present
numerical schemes
that can be used to solve the Cauchy problems for the models
derived and
described above. In other words, we solve the systems of
equations for 0t prescribing the values of all the unknowns at 0t =
. The question of the boundary condition is much more delicate.
Simply said, one impose
physical boundary condition at the solid boundaries for some of
the
variables, and these can be of Dirichlet or Neumann type
(depending on the
variable type). Since the boundary conditions are replacing the
governing
equations on the boundaries, the rest of the variables are
determined via
numerical techniques. These are strongly related to the
numerical solvers
and thus we will discuss the imposition of the boundary
conditions in the
context of numerical solvers.
A short comment on the Navier-Stokes model is necessary. The
equations
presented in this chapter can be used for the numerical
simulation of laminar
flows only. At practical Reynolds numbers (see the
introduction), the effects
of the turbulence must be taken into account. The Reynolds
Averaged
Navier-Stokes (RANS) equations and the Large Eddy Simulation
(LES)
equations have formally the same mathematical aspect, with
minor
differences. The huge difference comes from the necessity of
modeling the
turbulence effects, mainly in the vicinity of solid boundaries.
As we already
mentioned, it is beyond the purposes of this book to discuss
about
turbulence modeling and turbulence models.
More details about the derivation of these and other equations
can be found
in (Anderson et. all, 1984), (Danaila and Berbente, 2003),
(Hirsch, 1990),
(Warsi, 1999). A very rigorous discussion of the models and
their solutions
can be found in (Lions, 1996).
-
28
Two remarks we wish to make before at the end of this first
chapter. The
first is that the Navier-Stokes and Euler equations are the
cornerstones for
the development of practical CFD codes. From a mathematical
point of
view, these models are systems of partial differential
equations. Since with a
few exceptions we cannot find exact solutions of these equations
for
practical problems, the only way is to solve them is to use
numerical
techniques. It is generally accepted that the discretization
techniques must
be based not only on the underlying physics bust also on the
mathematical
properties of the partial differential equations. It is
therefore useful to begin
with the analysis of the mathematical nature of the governing
equations of
the flow before trying to solve them numerically.
The second remark is that the equations of compressible fluid
flow reduce to
hyperbolic conservation laws (i.e. the Euler system) when the
effects of
viscosity and heat conduction are neglected. Furthermore, the
hyperbolic
part represents the convection and pressure gradient effects and
it can be
identified in the equations even the above physical effects are
not negligible.
This is the reason why we allocate a special chapter to study
the hyperbolic
partial differential equations.
-
29
2. Approximated and Simplified Models
2.1 Generalities
In this chapter we wish to consider successively simplified
version, or
submodels, of the governing equations and their closure
conditions. There
are many reasons and many ways to use and derive simplified
models. On
one hand, it is clear that any simplification leads to the
reduction of the
generality of the equations. On the other hand, a rational
simplification may
lead to a significant reduction of the computational effort
without penalties
on the quality of the solution. There are some possibilities
that can be
exploited to derive simplified models, and perhaps the most
common of
them are the reduction of the dimensionality of the problem and
the
incompressibility assumption.
One can also augment the basic equations by source terms to
account for
additional physics. This is not exactly a simplification of the
basic model;
on the contrary, adding some semi-empirical source terms may
enlarge the
area of practical applications that can be solved with that
model.
In the previous chapter we already done a significant
simplification of the
Navier-Stokes equations. We have neglected the viscosity and the
heat
conduction effects and we have arrived to the Euler equations.
In this
chapter we discuss further simplifications of both the Euler and
Navier-
Stokes equations.
Compressible submodels will include flows with variation; flows
with axial
symmetry; flows with cylindrical and spherical symmetry; plane
one-
dimensional flow and further simplifications of this to include
linearised and
scalar submodels. Incompressible submodels will include
various
formulations of the incompressible Navier-Stokes equations.
-
30
2.2 Compressible Submodels
2.2.1 Flow with Area Variation
Flows with area variation arise naturally in the study of fluid
flow
phenomena in ducts, pipes, shock tubes and nozzles. One may
start from the
two dimensional homogeneous version of Euler equations to
produce, under
the assumption of smooth area variations, a quasi-two
dimensional system
with a geometric source term ( )S U , namely:
( ) ( )t x+ =U F U S U (2.1) where
2 21
, ,
( ) ( )
x t
x
x
u uA A
u u p pu AA
E u E p u E p A
= = + = + +
U F S (2.2)
Here x denotes distance along the tube, nozzle, etc.; A is the
cross-sectional
area and in general is a function of both space and time, that
is ),( txAA = . The most common case is that in which A depends on
x only, for which one
cam write governing equations in the more convenient form:
( ) ( )t x S+ =U F U U (2.3) where
2
0
, ( ) ,
( ) 0
x
A A u
A u A u p pA
AE Au E p
= = + = +
U F S
This is a convenient form in that by defining area weighted
values for and
p, equation (2.3) may be interpreted as the usual
one-dimensional equations
of Gas Dynamics plus a simple source term S.
2.2.2 Axi-Symmetric Flows
Here we consider domains that are symmetric around a coordinate
direction.
We choose this coordinate to be the z-axis and is called the
axial direction.
The second coordinate is r, which measures distance from the
axis of
symmetry z and is called the radial direction. There are two
component of
velocity, namely ),( zru and ),( zrv . These are respectively
the radial (r)
and axial (z) components of velocity. Then the three
dimensional
-
31
(inhomogeneous) conservation laws are approximated by a two
dimensional
problem with geometric source terms )(US , namely:
( ) ( ) ( )t r z+ + =U F U G U S U (2.4) where
22
2
1, , ,
( )( ) ( )
u v u
uvu uu p
v uvruv u p
E u E pu E p v E p
+ = = = = +
++ +
U F G S (2.5)
2.2.3 Cylindrical and Spherical Symmetry
Cylindrical and spherical symmetric wave motion arises naturally
in the
theory of explosion waves in water, air and other media. In
these situations
the multidimensional equation may be reduced to essentially
one-
dimensional equations with a geometrical source term vector S(U)
to
account for the second and third spatial dimensions. We
write:
( ) )t r+ =U F U S(U (2.6) where
2 2, ,
( ) ( )
u u
u u p ur
E u E p u E p
= = + = + +
U F S (2.7)
Here r is the radial distance from the origin and u is the
radial velocity.
When 0= we have plane one-dimensional flow; when 1= we have
cylindrically symmetric flow, an approximation to two-dimensional
flow.
This is a special case of the axy-symmetric equations when no
axial
variations are present )0( =v . For 2= we have spherically
symmetric flow, an approximation to three-dimensional flow.
Approximations of this
kind can easily be solved numerically to a high degree of
accuracy by a
good one-dimensional numerical method. These accurate
one-dimensional
solutions can then be very useful in partially validating two
and three
dimensional numerical solutions of the full models.
-
32
2.2.4 Plain One-Dimensional Flow
We first consider the one-dimensional time depend case:
) 0t x+ =U F(U (2.8) where
2,
( )
u
u u p
E u E p
= = + +
U F
These equations also result from the previous equations, for
example from
(2.3). They are useful for solving shock-tube type problems.
Further, under
suitable physical assumptions they produce even simpler
mathematical
models. In all the submodels studied so far we have assumed
some
thermodynamic closure condition given by an Equation of State
(EOS).
The isentropic equations result under the assumption that the
entropy s is
constant everywhere, which is a simplification of the
thermodynamics. Now
the EOS becomes ( )p p C = , where C = constant. This makes the
energy equation redundant and we have 2x2 system:
2
) 0,
,
t x
u
u u p
+ =
= = +
U F(U
U F (2.9)
with the pressure p given by the above simple EOS. Notice please
that even
in the isentropic assumption, the equation are still
non-linear.
The isothermal equations are even a simpler model then the
isentropic
equations, still non-linear. These may be viewed as resulting
from the
isentropic equations (2.9) with a simpler EOS, that is 2)( app =
, where a is a non-zero constant and represents the propagation
speed of
sound. Thus, the isothermal equations are:
2 2
0,
,
t x
u
u u a
+ =
= = +
U F (U)
U F (2.10)
More submodels may be further obtained by writing the isentropic
equations
as:
-
33
0
1
t x x
t x x
u u
u uu p
+ + = + =
(2.11)
The inviscid Burgers equation is a scalar (single equation) non
linear
equation given by:
0t xu uu+ = (2.12) and can be obtained from the above momentum
equation (2.11) by neglecting
density and thus pressure variations. In conservative form
equation (1.115)
reads:
2
02
t
x
uu
+ =
(2.13)
The Linearised Equations of Gas Dynamics are obtain from (2.11)
by
considering small disturbances ,u to a motionless gas. Set u u=
and
0 = + , where 0 is a constant density value. Recall that )(pp =
and neglecting products of small quantities we have:
0 0( ) ( )p
p p
= +
that is, 20p p a = + with )( 00 pp = , and constant.)( 02 =
= p
a
Substituting into (2.11) and neglecting squares of small
quantities we obtain
the linear equations:
0
2
0
0
0
t x
t x
u
au
+ =
+ =
(2.14)
In matrix form system (1.118)-(1.119) reads:
02
0
0,
0
,0
t x
au
+ = = =
W AW
W A (2.15)
where bars have been dropped. The coefficient matrix A is now
constant
and thus the system (2.15) is a linear system with constant
coefficients, the
linearised equations of gas dynamics.
-
34
The linear advection equation, sometimes called linear
convection,
equation is:
0t xu au+ = (2.16) where a is a constant speed of wave
propagation. This is also known as the
one-way wave equation and plays a major role in the designing,
analysing
and testing of numerical methods for wave propagation
problems.
2.2.5 Steady Flow
The steady, or time-independent, homogeneous version of the
three-
dimensional Euler equations are:
0x y z+ + =F G H (2.17)
In the steady regime it is important to identify subsonic and
supersonic flow
regions. To this end we recall the definition of Mach number
M:
( ) 21
2
222
++=
a
wvuM
where a is the speed of sound. Supersonic flow requires M>1,
while for
subsonic flow we have M
-
35
Steady linearised models can be further obtained from the steady
Euler
equations. An interesting submodel is the small perturbation,
two-
dimensional steady supersonic system of equations:
2 0
0
x y
x x
u a v
v u
+ =
= (2.21)
with
1
12
2
=
Ma
and where M denotes the constant free-stream Mach number and
),(),,( yxvyxu are small perturbations of the x and y velocity
components.
In matrix form these equations read
0x y+ =W AW (2.22)
with 20
,1 0
u a
v
= =
W A .
2.2.6 Viscous Scalar Equations
From the viscous compressible Navier-Stokes equations one can
deduce,
after some manipulations, two representative scalar equations.
The first is
the viscous Burgers equation, which is the viscous version of
(2.13). It
reads:
2
2t xx
x
uu u
+ =
(2.23)
where is a coefficient of viscosity. A linearised form of this
is the linear advection-diffusion equation:
t x xxu au u+ = (2.24) wich is the viscous version of
(2.16).
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36
2.3 Incompressible Submodels
2.3.1 About Incompressible Models and Low Mach Number
Expansions
In the field of CFD, the incompressibility assumption is very
important for
applications since many common fluids (liquids) are
incompressible or only
very slightly compressible. Mathematically, the
incompressibility
condition means:
0div =V (2.25) Therefore, the volume occupied by a group of
fluid particles at the initial
time remains constant during the flow. The continuity equation
written as:
( ) 0t x y z x y zu v w u v w + + + + + + = leads to
0tD
Dt
= + =V (2.26)
This means that if the density is constant initially and on the
boundaries
from where the fluid comes inside the domain under consideration
it
remains so. This is equivalent to say that the fluid is
homogeneous.
Further, having in mind the previous derivation of the
compressible models
and their EOS and constitutive models, it could be useful to
make a
distinction between models obtained by:
1. Incompressibility hypothesis.
2. Low Mach number expansions.
We emphasize here that the incompressibility hypothesis does not
impose
an explicit restriction on the magnitude of the velocity.
Moreover, the
incompressible models aim to describe liquids where compression
effects
are neglected and the density is taken as constant. For example,
it is not
rational to expect from an incompressible model to describe
accurately the
propagation of acoustic or pressure waves through liquids. This
is due the
fact that the incompressibility condition (2.25) is normally
associated with
other working hypothesis made on the EOS and on the fluid
transport
properties. Further, the energy equation is firmly decoupled
from the
continuity and momentum equations, by stating that the
temperature field
can be calculated separately, after the velocity and pressure
fields have been
determined.
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37
On the other side, the compressible models presented in the
previous
chapter are valid for gases. Starting from the compressible
models it is
rational to discuss about the low Mach number expansions. The
precise
definition of the Mach number is: ( )'M p = V . Therefore,
letting M go to zero means that, keeping constant values for the
density and the
temperature, the magnitude of the velocity is a small parameter.
An
asymptotic analysis starting from the Navier-Stokes equation
derived for a
compressible ideal gas shows that there is a lack of consistency
between
compressible models and incompressible submodels, see (Lions,
1996), in
the presence of heat conduction. A possible physical explanation
is the
following assertion: compressible models are valid for gases and
the low
Mach number limit yields particular incompressible submodels.
These
particular submodels are definitively determined by the EOS and
transport
properties chosen for the gas. Further, such an asymptotic
analysis reveals
that the incompressible submodel is very sensitive to the errors
in the
pressure calculation.
In what follows, we assume the fluid to be incompressible,
homogeneous,
non-heat conducting and viscous, with constant coefficient of
viscosity . Body forces are also neglected, only for simplicity. We
study three
mathematical formulations of the governing equations in
Cartesian
coordinates and restrict our attention to the two-dimensional
case.
2.3.2 The Incompressible Navier-Stokes Equations in
Primitive
Variable Form
The primitive variable formulation of the incompressible two
dimensional
Navier-Stokes equations is given by:
0
1
1
x y
t x y x xx yy
t x y y xx yy
u v
u uu vu p v u u
v uv vv p v v v
+ =
+ + + = +
+ + + = +
(2.27)
where the kinematic viscosity is:
v
= (2.28)
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38
Recall that is the coefficient of shear viscosity. We have a set
of three equations for the three unknowns u, v, p, the primitive
variables. This is a
mixed elliptic-parabolic system. Due to the mixed nature of
the
mathematical model, the solution cannot be obtained directly via
time-
marching algorithms. In principal, given a domain along with
initial and
boundary conditions for the equations one should be able to
solve this
problem using the primitive variable formulation.
2.3.3 The Incompressible Navier-Stokes Equations in
Stream-Function
Vorticity Form
The stream-function vorticity formulation is another way of
expressing the
incompressible Navier-Stokes equations. This formulation is
attractive for
the two-dimensional case but not so much in three dimensions, in
which the
role of a stream function is replaced by that of a vector
potential. The
magnitude of the vorticity vector can be written as:
x yv u = (2.29) Introducing a stream function we have for the
velocity components:
,y xu v = = . By combining the momentum equations so as to
eliminate the pressure p,
and using (2.29) we obtain the vorticity transport equation:
t x y xx yyu v v + + = + (2.30)
This is an advection-diffusion equation of parabolic type. In
order to
solve it, one requires the solution for the stream function ,
which is in turn related to the vorticity via: xx yy + = (2.31)
This is called the Poisson equation and is of elliptic type. There
are
numerical schemes to solve (2.30)-(2.31) using the apparent
decoupling of the
parabolic-elliptic problem to transform it into the parabolic
equation vor the
vorticity and the elliptic equation for the stream function.. A
relevant
observation, from the numerical point of view, is that the
convection terms
of the left hand side of equation (2.30) can be written in
conservative form
and hence we have:
( ) ( )t xx yyx yu v v + + = + (2.32) This follows from the fact
that 0=+ yx vu , which was also used to obtain.
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39
2.3.4 The Incompressible Navier-Stokes Equations in
Artificial
Compressibility Form
The artificial compressibility formulation is yet another
approach to
formulate the incompressible Navier-Stokes equations and was
originally
put forward by Chorin, see (Chorin, 1968) for the steady case.
Let us
consider the two-dimensional incompressible Navier-Stokes
equations
written in non-dimensional form:
0x yu v+ = (2.33)
t x y x xx yy
t x y y xx yy
u uu vu p u u
v uv vv p v v
+ + + = +
+ + + = + (2.34)
where the following non-dimensionalisation has been used:
.,1
'',
,,,2
==
V
LVR
R
L
tVt
L
yy
L
xx
V
pp
V
vv
V
uu
eL
eL
Multiplying (2.33) by the non-zero parameter 2c and adding an
artificial
compressibility term tp the first equations reads:
( ) ( )2 2 0t x yp uc vc+ + = (2.35) By using equation (2.33)
the convective terms in the momentum equations
can be written in conservative form, so that the modified system
becomes:
( ) ( )2 22
2
0
( ) ( )
( ) ( )
t x y
t x y xx yy
t x x xx yy
p uc vc
u u p uv u u
v uv v p v v
+ + =
+ + + = +
+ + + = +
(2.36)
The equations can be written in compact form as
t x y+ + =U F (U) G (U) S(U) (2.37)
where the vectors of unknowns, fluxes and source terms are:
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40
2 2
2
2
0
, , , ( )
( )
xx yy
xx yy
p c u c v
u u p uv u u
v uv v p v v
= = + = = + + +
U F G S (2.38)
The above equations are called the artificial compressibility
equations. Here
c2 is the artificial compressibility factor, usually taken as a
constant
parameter. The source term vector in this case is a function of
second
derivatives. Note that the modified equations are equivalent to
the original
equations in the steady state limit only. The left-hand side of
the artificial
compressibility equations form a non-linear hyperbolic
system.
More recently, new formulations have been proposed for the
solution of
steady and unsteady incompressible Navier-Stokes equations.
Since time-
marching methods cannot be applied directly, the system (2.36)
must be
transformed into a more convenient one. The dual time approach
requires
the addition of derivatives of a fictitious pseudo-time to each
of the three equations to give:
( ) ( )22
2
10
( ) ( )
( ) ( )
t x y
t x y xx yy
t x x xx yy
p p u v
u u u p uv u u
v v uv v p v v
+ + + =
+ + + + = +
+ + + + = +
(2.39)
where is a parameter and the term added to the continuity
equation has the same form as the basic artificial compressibility
method. A steady-state
solution in pseudo-time ( ), , 0p u v corresponds to an
instantaneous unsteady solution in real time. A recommended value
for the
parameter in the case the governing equations are written in
dimensionless form is ( )1 . The convective part of the system
(2.39) is of hyperbolic type and therefore a time-marching solution
procedure is
possible, (Gaitonde, 1998).