1. No category

Fluid Mechanics
Chapter 4 – Differential analysis of fluid flows
last edited May 22, 2015
These lecture notes are based on textbooks by White [4], Çengel & al.[6], and Munson & al.[8].
4.1
Motivation
In this chapter we assign ourselves with the —admittedly daunting— task of
describing the movement of fluids in an extensive manner. We wish to obtain, or
at least characterize, the pressure and velocity fields for any given flow.
The method we develop here makes use of advanced mathematical concepts, yet it
will only allow us to obtain an analytical solution for a few simple cases. Despite
this, it has two extremely important uses:
•
it allows us to formally describe and relate the key parameters that regulate fluid flow, and so is the key to developing an understanding of any
phenomenon even when solutions cannot be derived;
•
it is the backbone for computational fluid dynamics in which approximate
solutions are obtained using numerical procedures.
4.2
The fluid particle
We wish to describe the velocity and pressure fields of a fluid with the highest
possible resolution, yet without ever reducing our scale so much that we can
distinguish molecules (whose trajectories are much more complex than those of
the fluid).
We will therefore describe the properties of fluid particles, which we define as
groups containing a fixed number of molecules occupying a very small volume.
A particle is small enough to have homogeneous properties (only one pressure,
one velocity, one temperature etc.) but large enough to contain a large number of
molecules whose behavior can be treated as one. (fig. 4.1).
One can conceptualize a particle in an approximate manner as being a cube whose
edge is less than 0,1 mm (this is enough to contain many molecules!). As time
passes, a given particle will deform, and its molecules will progressively defuse.
59
Figure 4.1: Measurement of the average value of a property inside a given volume. As
the volume shrinks towards zero, the fluid can no longer be treated as a continuum; and
property measurements will oscillate wildly.
figure CC-0 o.c.
4.3
Eulerian description of fluid flow
4.3.1
Problem description
Newton’s second law allows us to quantify how the speed vector of a particle
varies with time. If we know all of the forces to which it is subjected, we can
thus obtain a speed vector V~particle = (u, v, w) = f (x , y, z, t) as it moves through
our area of interest. This description amounts to describing the speed vector as
a function of initial conditions only (V~particle = f (x 0 , y0 , z 0 , t 0 )). We always use
this process in solid mechanics when we wish to describe the movement of an
object (e.g. a satellite).
This method, however, is poorly suited to the description of fluid flow, for three
reasons:
60
•
Firstly, in order to describe a given fluid flow (e.g. air flow around a car external mirror), we would need a large number of initial points (x 0 , y0 , z 0 , t 0 ),
and we would then obtain as many trajectories V~particle . It then becomes
very difficult to describe to study and describe a problem that is local in
space (e.g. the wake immediately behind the car mirror), because this requires finding out where the particles of interest originated, and accounting
for the trajectories of each of them.
•
Secondly, the concept of a “fluid particle” has less and less meaning as a
particle travels along a flow. Particles indeed not only deform and rotate,
but also diffuse into the surrounding particles, “blurring” themselves one
into another.
•
Finally, the speed and other properties of a given particle very strongly
depend on the properties of the surrounding particles. We thus have to
resolve simultaneously the movement equations of all of the particles. A
space-based description of properties —one in which we describe properties
at a chosen fixed point of coordinates x , y, z, t— can therefore be more
useful than a particle-based description which depends on departure points
x 0 , y0 , z 0 , t 0 to predict how a flow behaves.
We will therefore be looking for a description of the velocity fields that is expressed in terms of a fixed observation point V~point = (u, v, w) = f (x point , ypoint , z point , t)
(through which particles of many different origins may originate) and then generalize this description for all the points within our study zone, therefore obtaining
a velocity field V~point that is a function of time.
4.3.2
Substantial derivative
Let us imagine a canal flowing at constant speed u and filled with water whose
temperature is steady in time, but not homogeneous (fig. 4.2). If we use a temperature probe at a point fixed in space, the temperature measured will vary in time.
The time rate of change of temperature will be:
dTprobe
dTwater
=
u water
dt
dx
(4/1)
Figure 4.2: A one-dimensional water flow, for example in a canal. The water has a nonuniform temperature, which, even if it is constant in time, translates in a temperature
time rate change at the probe.
figure CC-0 o.c.
61
If the water temperature is not simply heterogeneous, but also unsteady —so that
it varies in time with a rate ∂T
∂t — then that rate will also affect the rate which is
measured at the probe:
dTprobe
∂Twater ∂Twater
=
+
u water
dt
∂t
∂x
(4/2)
We keep in mind that the rate ∂T
∂t can itself be a function of time and space; in this
equation, it is its value at the position of the probe and at the time of measurement
which is taken into account.
This line of thought can be generalized for three dimensions and for any property A of the fluid (including vector properties). The property A of one individual
particle can vary as it is moving, so that it has a distribution A = f (x , y, z, t)
within the fluid. The time rate change of A measured at a point fixed in space is
named the substantial derivative of A and written DA
Dt :
∂
∂
∂
∂
D
≡
+u
+v
+w
Dt
∂t
∂x
∂y
∂z
(4/3)
∂A
∂A
∂A
∂A
DA
=
+u
+v
+w
Dt
∂t
∂x
∂y
∂z
(4/4)
In eq. eq. (4/3), we now wish to simplify the writing of the last three terms. We
~
know the coordinates of the speed vector V~ and of the operator gradient ∇:
V~ = ~i u + ~j v + k~ w
(4/5)
~ ≡ ~i ∂ + ~j ∂ + k~ ∂
∇
∂x
∂y
∂z
(4/6)
~
We thus define the advective operator, (V~ · ∇):
~ = u ∂ +v ∂ +w ∂
V~ · ∇
∂x
∂y
∂z
(4/7)
We can now rewrite eqs. 4/3 and 4/4 in a more concise way:
D
∂
~
≡
+ (V~ · ∇)
Dt
∂t
DA
∂A ~ ~
=
+ (V · ∇)A
Dt
∂t
(4/8)
(4/9)
As we explored in eq. (4/1), the substantial derivative of a property can be nonnull even if the flow is perfectly steady; similarly, the substantial derivative
of a property can be zero (e.g. when a stationary probe provides a constant
62
measurement) even if the properties of the particles have a non-zero time change
(e.g. a fluid with falling temperature, but with a non-uniform temperature).
4.4
Conservation of mass
Let us consider a fluid particle of volume dV, at a given moment (fig. 4.3). We
can reproduce our analysis from chapter 3 by quantifying the time change of
mass within an arbitrary volume.
In the present case, the control volume is stationary and the particle (our system)
is flowing through it.
dm particle
d
= 0 =
dt
dt
$
"
ρ dV +
CV
ρ(V~rel · n~ ) dA
(4/10)
CS
The first of these two integrals can be rewritten using the Leibniz integral rule:
$
"
$
∂ρ
d
ρVS dA
ρ dV =
dV +
dt
CS
CV
$CV ∂t
∂ρ
=
dV
(4/11)
CV ∂t
where VS is the speed!of the control volume wall;
and where the term CS ρVS dA is simply zero because we chose a fixed
control volume.
!
Now we turn to the second term of eq. (4/10), CS ρ(V~rel · n~ ) dA, which represents
the mass flow m˙ net flowing through the control volume.
Figure 4.3:
the x-direction, a mass flow
! Conservation of mass within a fluid particle. In !
m˙ 1 =
ρ1u 1 dz dy is flowing in, and a mass flow m˙ 2 =
ρ2u 2 dz dy is flowing out.
These two flows may not be equal, since mass may also flow in the two other directions.
figure CC-0 o.c.
63
In the direction x, the mass flow m˙ net x flowing through our control volume can
be expressed as:
"
"
m˙ net x =
−ρ1 |u 1 | dA +
ρ2 |u 2 | dA
CS
CS
"
∂
=
(ρu) dx dA
CS ∂x
$
∂
=
(ρu) dV
(4/12)
CV ∂x
The same applies for directions y and z, so that we can write:
"
ρ(V~rel · n~ ) dA = m˙ net = m˙ net x + m˙ net y + m˙ net z
CS
#
$ "
∂
∂
∂
(ρu) + (ρv) + (ρw) dV
=
∂x
∂y
∂z
CV
$
~ · (ρV~ ) dV
=
∇
(4/13)
CV
Now, with these two equations 4/11 and 4/13, we can come back to eq. (4/10),
which becomes:
"
$
dm particle
d
ρ(V~rel · n~ ) dA
= 0 =
ρ dV +
dt
dt
CS
CV
$
$
∂ρ
~ · (ρV~ ) dV
=
dV +
∇
∂t
CV
CV
(4/14)
Dropping the integrals, we obtain:
0 =
∂ρ ~
+ ∇ · (ρV~ )
∂t
(4/15)
for all flows, with all fluids.
This eq. (4/15) is named continuity equation and is of crucial importance in
fluid mechanics. All fluid flows, in all conditions and at all times, obey this
law. In itself, the continuity equation gives us insufficient information to solve
the majority of problems in fluid mechanics; it behaves more as a kinematic
constraint to solutions. We shall always make sure that solutions to our flow
obey this equation, either fully or (in the case of numerical computations) with
an acceptably small error.
The most assiduous readers will have no difficulty reading the following equation,
∂ρ ~
∂ρ ~ ~
~ · V~ ) = 0
+ ∇ · (ρV~ ) =
+ V · ∇ρ + ρ(∇
∂t
∂t
64
(4/16)
which allows us to re-write eq. (4/15) like so:
1 Dρ ~ ~
+ ∇ ·V = 0
ρ Dt
(4/17)
Therefore, we can see that for an incompressible flow, the volumetric dilatation
~ · V~ is zero:
rate ∇
~ · V~ = 0
∇
(4/18)
for any incompressible flow.
In this case, eq. (4/18) can simply be re-expressed as:
∂u ∂v ∂w
+
+
= 0
∂x ∂y ∂z
(4/19)
for any incompressible flow.
65
4.5
Conservation of linear momentum
We now turn to the crux of analytical fluid mechanics: we wish to relate the
velocity field to the forces that are applied on the fluid. In order to do this, we
turn to Newton’s second law, which we are going to apply to a fluid particle.
4.5.1
The Cauchy equation
We start by writing Newton’s second law (1/11) as it applies to a fluid particle of
mass m particle (fig. 4.4):
m particle
dV~
= F~weight + F~net, pressure + F~net, shear
dt
(4/20)
If we now re-write this equation for a control volume CV of infinitesimal volume
dV, for which we use a Eulerian description, we obtain:
ρ
1 ~
1 ~
DV~
= ρ~
д+
F net, pressure +
F net, shear
Dt
dV
dV
(4/21)
Figure 4.4: In our study of fluid mechanics, we consider only forces due to gravity, shear,
or pressure.
figure CC-0 o.c.
66
Let us now account for pressure and shear efforts applying on each of the faces
of the fluid particle. For pressure, we obtain:
F net, pressure,x = dy dz [px1 − px 2 ]
#
"
∂p
= dy dz − dx
∂x
−∂p
∂x
−∂p
= dV
∂y
−∂p
= dV
∂z
F net, pressure,x = dV
(4/22)
F net, pressure,y
(4/23)
F net, pressure,z
(4/24)
This allows us to write:
1 ~
~
F net, pressure = −∇p
dV
(4/25)
We proceed in a similar fashion with shear, expressing the force resulting from
shear efforts like so:


 τxx τxy τxz 


1 ~
~ · τi j
~ ·  τyx τyy τyz  = ∇
(4/26)
F net, shear = ∇


dV
τzx τzy τzz
Here the term τi j is the shear tensor, a tensor of order 2 expressing the entire
shear efforts exerting on the fluid particle (fig. 4.5). The divergent of this tensor,
~ · τi j , is a vector expressing the net vector sum of the shear forces.
∇
Figure 4.5: Shear efforts on a cubic fluid particle. The shear tensor τi j has three members
of three components each: it is a tensor of order 2.
figure CC-0 o.c.
67
Inserting eqs. (4/25) and (4/26) in eq. (4/21), we obtain the Cauchy equation:
ρ
DV~
~ +∇
~ · τi j
= ρ~
д − ∇p
Dt
(4/27)
for all flows, with all fluids.
This equation 4/27 is an expression of Newton’s second law applied to a fluid
particle. It expresses that the time change of the speed vector measured at a fixed
point is equal to the effect of gravity, minus the space change of pressure, plus
the space change of shear.
Unfortunately, this equation is not detailed enough for us, for it lacks a practical
description for the space change of the shear tensor τi j . Ideally, this term should
be expressed solely as a function of the flow’s other properties. Obtaining such
an expression is what we set ourselves to in §4.5.2.
4.5.2
The Navier-Stokes equation for incompressible flow
The Navier-Stokes equation expands on the Cauchy equation, eq. (4/27), by adding
a viscosity model. From here on, we restrict ourselves to the study of Newtonian
fluids (§1.4.3), for which shear efforts are simply proportional to the spatial change
of speed.
In this case, we have:
~ · V~ )
τi j = µ(∇
(4/28)
or, as expressed on each of three directions:
τix
τiy
τiz
!
∂u
∂u
∂u
~ = µ
= µ ∇u
+
+
∂x ∂y ∂z
!
∂v
∂v
∂v
~ = µ
= µ ∇v
+
+
∂x ∂y ∂z
!
∂w
∂w
∂w
~ = µ
= µ ∇w
+
+
∂x ∂y ∂z
(4/29)
(4/30)
(4/31)
for any Newtonian fluid.
From here on, we further restrict ourselves to incompressible flows (those for
which ρ remains constant). In that case, the normal stresses τii vanish, and the
68
forces due to shear on a fluid particle can be expressed as:
h
i
F net, shear,x = dx dz −τyx + (τyx + δτyx ) + dx dy [−τzx + (τzx + δτzx )]
#
"
#
"
∂τyx
∂τzx
dy + dx dy
dz
= dx dz
∂y
∂z
"
#
i
i
h
h
∂τyx ∂τzx
~ · (µ ∇u)
~
~ · τix = dV ∇
= dV
+
= dV ∇
∂y
∂z
~ · (∇u)
~
= dV µ ∇
(4/32)
With an identical derivation for the two other directions, we obtain:
~ 2u
F net, shear,x = dV µ ∇
~ 2v
F net, shear,y = dV µ ∇
~ 2w
F net, shear,z = dV µ ∇
1 ~
~ 2V~
F net, shear = µ ∇
dV
(4/33)
(4/34)
(4/35)
(4/36)
for incompressible flow of a Newtonian fluid.
~2 ≡ ∇
~ · ∇,
~ which represents the spatial
Here we have used the laplacian operator, ∇
variation of the spatial variation of a scalar or a vector.
With this new expression, we are within reach of a much more powerful expression than eq. (4/27), which relates Newton’s second law to the velocity field. It is
the glorious Navier-Stokes equation for incompressible flow:
ρ
DV~
~ + µ∇
~ 2V~
= ρ~
д − ∇p
Dt
(4/37)
for all incompressible flows of a Newtonian fluid.
This formidable equation describes the property fields of all incompressible flows
of Newtonian fluids. The solutions we look for in eq. (4/37) are the velocity
(vector) field V~ = (u, v, w) = f 1 (x , y, z, t) and the pressure field p = f 2 (x , y, z, t),
given a set of constraints to represent the problem at hand. These constraints,
called boundary conditions, may be expressed in terms of velocities (e.g. the
presence of a fixed solid body is expressed as a region for which V~ = ~0) or pressure
(e.g. a discharge into an open atmosphere may be expressed as a region of known
constant pressure).
The combination of eqs. (4/15) and (4/37) (continuity & Navier-Stokes) may not
be enough to predict the solution of a given problem, most especially if large
energy transfers take place within the flow. In that case, the addition of an
69
energy equation and an expression of the second law, both in a form suitable for
derivative analysis, may be needed to provide as many equations as there are
unknowns.
Though it is without doubt charming, eq. (4/37) should be remembered for what
it is really: a three-dimensional system of coupled equations. In Cartesian
coordinates this complexity is more apparent:
" 2
#
∂p
∂u
∂2u
∂2u
= ρдx −
+µ
+
+
∂x
(∂x)2 (∂y)2 (∂z)2
(4/38)
"
#
" 2
#
2
2
∂p
∂v
∂v
∂v
∂v
∂v
∂v
∂v
ρ
+u
+v
+w
= ρдy −
+µ
+
+
2
2
∂t
∂x
∂y
∂z
∂y
(∂x )
(∂y)
(∂z)2
(4/39)
"
#
" 2
#
2
2
∂p
∂w
∂w
∂w
∂w
∂w
∂w
∂w
ρ
+u
+v
+w
= ρдz −
+µ
+
+
2
2
∂t
∂x
∂y
∂z
∂z
(∂x )
(∂y)
(∂z)2
(4/40)
∂u
∂u
∂u
∂u
ρ
+u
+v
+w
∂t
∂x
∂y
∂z
"
#
Indeed in the 150 years since it was first written, no general expression has been
found for velocity or pressure fields that would solve this equation in the general
case. Nevertheless, in this course we will use it directly:
•
to evaluate the relative importance of flow parameters affecting given fluid
flows;
•
to find analytical solutions to flows in a few selected cases.
After this course, the reader might also engage into Computational Fluid Dynamics (cfd) a discipline entirely architectured around this equation, and to which it
purposes to find solutions as fields of discrete values.
As a finishing remark, we note that when the flow is strictly two-dimensional, the
Navier-Stokes equation is considerably simplified, shrinking down to the system:
#
#
" 2
∂p
∂u
∂u
∂u
∂u
∂2u
ρ
= ρдx −
+
+u
+v
+µ
∂t
∂x
∂y
∂x
(∂x)2 (∂y)2
#
"
#
" 2
∂p
∂v
∂v
∂v
∂v
∂2v
ρ
+u
+v
= ρдy −
+µ
+
∂t
∂x
∂y
∂y
(∂x )2 (∂y)2
"
(4/41)
(4/42)
This simplification unfortunately does not allow us to account for turbulence,
which is a strongly three-dimensional phenomenon.
4.6
Conservation of angular momentum
There is no differential equation for conservation of angular momentum in fluid
flow. Indeed, as the size of our particle tends to zero, it loses its ability to possess
70
(or exchange) angular momentum about its center of gravity. Infinitesimal fluid
particles are not able to rotate about themselves, only about an external point. So,
in this chapter, no analysis is carried out using the concept of angular momentum.
4.7
Conservation of energy
and increase in entropy
The last two key principles used in fluid flow analysis can be written together in
a differential equation similar to the Navier-Stokes equation.
Once again, we start from the analysis of transfers on an infinitesimal control
volume. We are going to relate three energy terms in the following form, naming
them A, B and C:
the rate of change
of energy inside
the net flux
=
the fluid element
A
of heat into
the rate of work done
+
the element
=
B
on the element due to
body and surface forces
+
C
(4/43)
Let us first evaluate term C. In the x-direction, the net power received by the
system as work is simply the net force in the x-direction multiplied by the velocity
in the x-direction. We have already expressed the net force due to pressure in
terms of the pressure gradient as eq. (4/22):
F net, pressure,x = dV
−∂p
∂x
and the net force due to shear in terms of the divergent of the x-direction component of the shear tensor in eq. (4/33),
h
i
~ · τix
F net, shear,x = dV ∇
so that now term Cx can be expressed as:
Cx = u(F weight, x + F net, pressure,x + F net, shear,x )
i!
−∂p h~
= u dV ρдx +
+ ∇ · τix
∂x
(4/44)
(4/45)
This can be generalized to three dimensions as follows:
~ +∇
~ · τi j dV
C = Cx + Cy + Cz = V~ · ρ~
д − ∇p
(4/46)
71
for the case of a Newtonian fluid, this expression can be re-written as:
~ + µ∇
~ 2V~ dV
C = V~ · ρ~
д − ∇p
(4/47)
We now turn to term B, the net flux of heat into the element. We attribute this flux
to two contributions, the first named Q˙ radiation from the emission or absorption of
radiation, and the second, named Q˙ conduction to thermal conduction through the
faces of the element. We make no attempt to quantify Q˙ radiation , simply stating
that
Q˙ radiation = ρq˙radiation dV
(4/48)
in which q˙radiation is the local power per unit mass (in J kg−1 s−1 ) transfered
to the element, to be determined from the boundary conditions and flow
temperature distribution.
In the x-direction, thermal conduction through the faces of the element causes a
net flow of heat Q˙ conduction,x expressed as
Q˙ conduction,x
!#
"
∂q˙x
= q˙x − q˙x +
dx dy dz
∂x
∂q˙x
= −
dx dy dz
∂x
∂q˙x
dV
= −
∂x
(4/49)
(4/50)
Summing contributions from all three directions, we obtain:
Q˙ conduction = Q˙ conduction,x + Q˙ conduction,y + Q˙ conduction,z
"
#
∂q˙x ∂q˙y ∂q˙z
= −
+
+
dV
∂x
∂y
∂z
(4/51)
So we finally obtain an expression for B:
B = Q˙ radiation + Q˙ conduction
"
#!
∂q˙x ∂q˙y ∂q˙z
= ρq˙radiation −
+
+
dV
∂x
∂y
∂z
(4/52)
Finally, term A, the rate of change of energy inside the fluid element, can be
expressed as a function of the specific kinetic energy and specific internal energy:
A = ρ
72
D 1 2
i + V dV
Dt
2
(4/53)
We are therefore able to relate the properties of a fluid particle to the principle of
energy conservation as follows:
"
#!
∂q˙x ∂q˙y ∂q˙z
D 1 2
~ + µ∇
~ 2V~
= ρq˙radiation −
+
+
+ V~ · ρ~
д − ∇p
ρ
i+ V
Dt
2
∂x
∂y
∂z
(4/54)
This is a scalar equation — it only has one dimension, and involves the length of
h
i1
the velocity vector, V ≡ u 2 + v 2 + w 2 2 .
In an interesting hack, we are able to incorporate an expression for the second
principle of thermodynamics in this equation simply by expressing the fluxes qi
as a function of the temperature gradients. Indeed, expressing a heat flux as the
result of a difference in temperature according to the Fourier law,
qi = −k
∂T
∂i
(4/55)
we constrain the direction of the heat fluxes and thus ensure that all dissipative
terms resulting in temperature increases cannot be fed back into other energy
terms, thus increasing the overall entropy.
Equation (4/55) inserted into eq. 4/54 yields:
D 1 2
ρ
i+ V
=
Dt
2
∂2T
∂2T
∂2T
ρq˙radiation − k
+
+
(∂x)2 (∂y)2 (∂z)2
"
#!
~ + µ∇
~ 2V~
+ V~ · ρ~
д − ∇p
Making use of a laplacian operator, we come to:
D 1 2
~ 2T + V~ · ρ~
~ + µ∇
~ 2V~
i+ V
= ρq˙radiation − k ∇
д − ∇p
ρ
Dt
2
(4/56)
This equation has several shortcomings, most importantly because the term
q radiation is not expressed in terms of fluid properties, and because µ is typically
not independent of the temperature T . Nevertheless, it in principle brings closure
to our system of continuity and momentum equations, and these two influences
may be either neglected, or modeled numerically.
4.8
The Navier-Stokes equations in practice
In our analysis of fluid flow from a derivative perspective, our five physical
principles from §1.5 have been condensed into three equations (often loosely
referred together to as the Navier-Stokes equations). Out of these, the first two,
for conservation of mass and linear momentum, are often enough to characterize
73
most free flows:
0 =
ρ
∂ρ ~
+ ∇ · (ρV~ )
∂t
DV~
~ + µ∇
~ 2V~
= ρ~
д − ∇p
Dt
(4/15)
(4/37)
In these two equations, the primary unknown is the velocity field V~ . In practice the analytical and numerical procedures used to obtain this field are very
involving, and we shall conclude with three remarks.
First, we must be precise when we state that “the solution to these equations is
unknown”. The solutions to millions of particular flows are indeed known (either
fully, and we shall describe several such flows in the upcoming chapters, or as
satisfactory numerical approximations); what is lacking though is one general
solution: one that would encompass all of those. For example, in solid mechanics
we have long understood that all pure free fall movements can be described with
the solution x = x 0 + u 0t and y = y0 + v 0t + 12 дt 2 , regardless of the particularities
of each fall. In fluid mechanics, even though our analysis was carried out in the
same manner, we have yet to find such one general solution — or even to prove
that one exists at all.
Second, the structure of conservation equations makes them quite unforgiving
to approximation. In many practical cases, we are either not interested in an
exhaustive description of the velocity field, or unable to compute its variations
in space and time to the fullest detail, and so we might be tempted to search
for solutions for the average flow (u, v, w). The reader may take a little time to
re-write eqs. 4/15 and 4/37 above, replacing each velocity component ui by the
sum of a time-average value ui and of an instantaneous fluctuation ui0. In doing so,
they will find that while the mass conservation equation remains unaffected (i.e.
it can be split in two independent problems, one for the average and the other for
the instantaneous velocity), the momentum conservation equation changes shape
significantly. In other words, the time-average of a real flow cannot be obtained
by solving for the time-average velocities. This is a tremendous burden in the
study of turbulent flows; in numerical studies it often has to be addressed with
the help of approximations named turbulence models.
Finally, a great unknown in our sets of equations remains pressure. The conservation equations, in principle, yield an expression for the time-change of the
velocity field as a function of the velocity and pressure fields; but they say nothing
about the time-change of pressure. Thus, in practice we are bound to developing
elaborate (and sometimes uncomfortable) mathematical and physical schemes to
evaluate this crucial part of the solution.
74
These three remarks, therefore, should hopefully hint at the fact that an understanding of the mathematical nature of the differential conservation equations is
of great practical importance in fluid dynamics. It is for that reason that we shall
dedicate the exercises of this chapter solely to playing with the mathematics of
our two main equations.
75