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Cylindrical and cartesian coordinates

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In this chapter we develop the mathematical basis for a comprehensive general-purpose model of ﬂuid ﬂow and heat transfer from the basic prin-ciples of conservation of mass, momentum and energy. This leads to the governing equations of ﬂuid ﬂow and a discussion of the necessary auxiliaryconditions – initial and boundary conditions. The main issues covered in thiscontext are:ãDerivation of the system of partial differential equations (PDEs) thatgovern ﬂows in Cartesian (
x
,
y
,
z
) co-ordinatesãThermodynamic equations of stateãNewtonian model of viscous stresses leading to the Navier–StokesequationsãCommonalities between the governing PDEs and the deﬁnition of thetransport equationãIntegrated forms of the transport equation over a ﬁnite time interval anda ﬁnite control volumeãClassiﬁcation of physical behaviours into three categories: elliptic,parabolic and hyperbolicãAppropriate boundary conditions for each categoryãClassiﬁcation of ﬂuid ﬂowsãAuxiliary conditions for viscous ﬂuid ﬂowsãProblems with boundary condition speciﬁcation in high Reynoldsnumber and high Mach number ﬂowsThe governing equations of ﬂuid ﬂow represent mathematical statements of the
conservation laws of physics
:ãThe mass of a ﬂuid is conservedãThe rate of change of momentum equals the sum of the forces on a ﬂuidparticle (Newton’s second law)ãThe rate of change of energy is equal to the sum of the rate of heataddition to and the rate of work done on a ﬂuid particle (ﬁrst law of thermodynamics)The ﬂuid will be regarded as a continuum. For the analysis of ﬂuid ﬂows at macroscopic length scales (say 1
µ
m and larger) the molecular structure of matter and molecular motions may be ignored. We describe the behaviourof the ﬂuid in terms of macroscopic properties, such as velocity, pressure,density and temperature, and their space and time derivatives. These may
Chapter two
Conservation laws of fluid motion and boundary conditions
Governing equations of fluid flow and heat transfer
2.1
OTONBARA LIBRARY
10
CHAPTER 2CONSERVATION LAWS OF FLUID MOTION
Figure 2.1
Fluid element forconservation laws
be thought of as averages over suitably large numbers of molecules. A ﬂuidparticle or point in a ﬂuid is then the smallest possible element of ﬂuid whosemacroscopic properties are not inﬂuenced by individual molecules.We consider such a small element of ﬂuid with sides
δ
x
,
δ
y
and
δ
z
(Figure 2.1).The six faces are labelled
N
,
S
,
E
,
W
,
T
and
B
, which stands for North,South, East, West, Top and Bottom. The positive directions along the co-ordinate axes are also given. The centre of the element is located at position(
x
,
y
,
z
). A systematic account of changes in the mass, momentum and energyof the ﬂuid element due to ﬂuid ﬂow across its boundaries and, where appro-priate, due to the action of sources inside the element, leads to the ﬂuid ﬂowequations.All ﬂuid properties are functions of space and time so we would strictlyneed to write
ρ
(
x
,
y
,
z
,
t
),
p
(
x
,
y
,
z
,
t
),
T
(
x
,
y
,
z
,
t
) and
u
(
x
,
y
,
z
,
t
) for thedensity, pressure, temperature and the velocity vector respectively. To avoidunduly cumbersome notation we will not explicitly state the dependence onspace co-ordinates and time. For instance, the density at the centre (
x
,
y
,
z
)of a ﬂuid element at time
t
is denoted by
ρ
and the
x
-derivative of, say, pres-sure
p
at (
x
,
y
,
z
) and time
t
by
∂
p
/
∂
x
. This practice will also be followed forall other ﬂuid properties.The element under consideration is so small that ﬂuid properties at thefaces can be expressed accurately enough by means of the ﬁrst two terms of a Taylor series expansion. So, for example, the pressure at the
W
and
E
faces, which are both at a distance of
1
–
2
δ
x
from the element centre, can beexpressed as
p
−
δ
x
and
p
+
δ
x
2.1.1Mass conservation in three dimensions
The ﬁrst step in the derivation of the mass conservation equation is to writedown a mass balance for the ﬂuid element:Rate of increaseNet rate of flow of mass in fluid
=
of mass into elementfluid element12
∂
p
∂
x
12
∂
p
∂
x
2.1GOVERNING EQUATIONS OF FLUID FLOW AND HEAT TRANSFER
11
Figure 2.2
Mass ﬂows in andout of ﬂuid element
The rate of increase of mass in the ﬂuid element is(
ρδ
x
δ
y
δ
z
)
=
δ
x
δ
y
δ
z
(2.1)Next we need to account for the mass ﬂow rate across a face of the element,which is given by the product of density, area and the velocity componentnormal to the face. From Figure 2.2 it can be seen that the net rate of ﬂow of mass into the element across its boundaries is given by
ρ
u
−
δ
x
δ
y
δ
z
−
ρ
u
+
δ
x
δ
y
δ
z
+
ρ
v
−
δ
y
δ
x
δ
z
−
ρ
v
+
δ
y
δ
x
δ
z
+
ρ
w
−
δ
z
δ
x
δ
y
−
ρ
w
+
δ
z
δ
x
δ
y
(2.2)Flows which are directed into the element produce an increase of mass in theelement and get a positive sign and those ﬂows that are leaving the elementare given a negative sign.
D E F
12
∂
(
ρ
w
)
∂
z
A BC D E F
12
∂
(
ρ
w
)
∂
z
A BC D E F
12
∂
(
ρ
v
)
∂
y
A BC D E F
12
∂
(
ρ
v
)
∂
y
A BC D E F
12
∂
(
ρ
u
)
∂
x
A BC D E F
12
∂
(
ρ
u
)
∂
x
A BC
∂ρ ∂
t
∂ ∂
t
The rate of increase of mass inside the element (2.1) is now equated to thenet rate of ﬂow of mass into the element across its faces (2.2). All terms of theresulting mass balance are arranged on the left hand side of the equals signand the expression is divided by the element volume
δ
x
δ
y
δ
z
. This yields
+ + + =
0(2.3)or in more compact vector notation
+
div(
ρ
u
)
=
0(2.4)Equation (2.4) is the
unsteady, three-dimensional mass conservationor continuity equation
at a point in a
compressible ﬂuid
. The ﬁrst term
∂ρ ∂
t
∂
(
ρ
w
)
∂
z
∂
(
ρ
v
)
∂
y
∂
(
ρ
u
)
∂
x
∂ρ ∂
t
on the left hand side is the rate of change in time of the density (mass per unitvolume). The second term describes the net ﬂow of mass out of the elementacross its boundaries and is called the convective term.For an
incompressible ﬂuid
(i.e. a liquid) the density
ρ
is constant andequation (2.4) becomesdiv
u
=
0(2.5)or in longhand notation
+ + =
0(2.6)
2.1.2Rates of change following a fluid particle and for a fluidelement
The momentum and energy conservation laws make statements regardingchanges of properties of a ﬂuid particle. This is termed the Lagrangianapproach. Each property of such a particle is a function of the position (
x
,
y
,
z
) of the particle and time
t
. Let the value of a property per unit massbe denoted by
φ
. The total or substantive derivative of
φ
with respect to timefollowing a ﬂuid particle, written as
D
φ
/
Dt
, is
= + + +
A ﬂuid particle follows the ﬂow, so d
x
/d
t
=
u
, d
y
/d
t
=
v
and d
z
/d
t
=
w
.Hence the substantive derivative of
φ
is given by
= +
u
+
v
+
w
= +
u
. grad
φ
(2.7)
D
φ
/
Dt
deﬁnes rate of change of property
φ
per unit mass. It is possible to develop numerical methods for ﬂuid ﬂow calculations based on theLagrangian approach, i.e. by tracking the motion and computing the rates of change of conserved properties
φ
for collections of ﬂuid particles. However,it is far more common to develop equations for collections of ﬂuid elementsmaking up a region ﬁxed in space, for example a region deﬁned by a duct, apump, a furnace or similar piece of engineering equipment. This is termedthe Eulerian approach. As in the case of the mass conservation equation, we are interested indeveloping equations for rates of change per unit volume. The rate of changeof property
φ
per unit volume for a ﬂuid particle is given by the product of
D
φ
/
Dt
and density
ρ
, hence
ρ
=
ρ
+
u
. grad
φ
(2.8)The most useful forms of the conservation laws for ﬂuid ﬂow computationare concerned with changes of a ﬂow property for a ﬂuid element that is stationary in space. The relationship between the substantive derivative of
φ
,which follows a ﬂuid particle, and rate of change of
φ
for a ﬂuid element isnow developed.
D E F
∂φ ∂
t
A BC
D
φ
Dt
∂φ ∂
t
∂φ ∂
z
∂φ ∂
y
∂φ ∂
x
∂φ ∂
t D
φ
Dt dzdt
∂φ ∂
zdydt
∂φ ∂
ydxdt
∂φ ∂
x
∂φ ∂
t D
φ
Dt
∂
w
∂
z
∂
v
∂
y
∂
u
∂
x
12
CHAPTER 2CONSERVATION LAWS OF FLUID MOTION

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