76493_09.pdf

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Drag
and
lift
In Chapters
7
and
8
our study concerned ‘internal flow’ enclosed by solid
walls. Now, how shall we consider such cases as the Aight of a baseball or
golf ball, the movement of an automobile or when an aircraft flies in the air,
or where a submarine moves under the water? Here, flows outside such solid
walls, i.e. ‘external flows’, are discussed.
Generally speaking, flow around a body placed in a uniform flow develops a
thin layer along the body surface with largely changing velocity, Le. the
boundary layer, due to the viscosity of the fluid. Furthermore, the flow
separates behind the body, discharging a wake with eddies. Figure 9.1 shows
the flows around a cylinder and a flat plate. The flow from an upstream point
a
is
stopped at point b on the body surface with its velocity decreasing to
zero; b is called a stagnation point. The flow divides into the upper and lower
flows at point b. For a cylinder, the flow separates at point c producing a
wake with eddies.
Let the pressure upstream at a, which is not affected by the body, be
pbo,
the flow velocity be
U
and the pressure at the stagnation point be
p o .
Then
P
u2
Po=Poa+T
(9.1)
Fig.
9.1
Flow
around
a
body
The drag
of
a body
149
Whenever a body is placed in
a
flow, the body is subject to a force from the
surrounding fluid. When a flat plate is placed in the flow direction, it is only
subject to a force in the downstream direction.
A
wing, however, is subject to
the force
R
inclined to the flow as shown in Fig. 9.2. In general, the force
R
acting on a body is resolved into a component
D
in the flow direction
U
and
the component
L
in
a
direction normal to
U.
The former is called drag and
the latter lift.
Drag and lift develop in the following manner. In Fig.
9.3,
let the pressure
of fluid acting on a given minute area dA on the body surface be
p,
and the
friction force per unit area be
z.
The force pdA due to the pressure
p
acts
normal to dA, while the force due to the friction stress
z
acts tangentially.
The drag
D,,
which is the integration over the whole body surface of the
component in the direction of the flow velocity
U
of this force
p
dA,
is
called
form drag or pressure drag. The drag
D/
is the similar integration of zdA
and is called the friction drag.
D,
and
D/
are shown as follows in the form of
equations:
D,=
D
-
4,
J,
pdAcos8
zdAsin8
(9.2)
(9.3)
The drag
D
on a body is the sum
of
the pressure drag
D,
and friction drag
Or,
whose proportions vary with the shape
of
the body. Table
9.1
shows the
contributions of
D,
and
D,
for various shapes. By integrating the component
7
,
of pd.4 and ~ d . 4
normal to
1
the lift
L
is obtained.
Fig. 9.2
Drag and
lift
Fig.
9.3
Force acting on
body
9.3.1 Drag
coefficient
The drag
D
of a body placed in the uniform flow
U
can be obtained from eqns
(9.2)
and
(9.3).
This theoretical computation, however, is generally difficult
except for bodies of simple shape and for a limited range of velocity.
150
Drag
and
lift
Table
9.1
Contributions
of
&and
Df
for
various shaoes
Therefore, there is no other way but to rely on experiments. In general, drag
D
is expressed as follows:
PU‘
D
=
CDA-
2
(9.4)
where A is the projected area of the body on the plane vertical to the
direction of the uniform flow and
C,
is a non-dimensional number called
the drag coefficient. Values of
C,
for bodies
of
various shape are given in
Table
9.2.
9.3.2
Drag for a cylinder
Ideal
fluid
Let
us
theoretically study (neglecting the viscosity of fluid) a cylinder placed
in a flow. The flow around a cylinder placed at right angles to the flow
U
of
an ideal fluid is as shown in Fig. 9.4. The velocity
uo
at a given point on the
cylinder surface
is
as follows (see Section
12.5.2):
v8
=
2U
sin0
(9.5)
Putting the pressure of the parallel flow as pm, and the pressure at
a
given
point
on
the cylinder surface as
p,
Bernoulli’s equation produces the
following result:
P
u2
pm
-
=
p
+
2
Pd
+
-
2
P-Pm=
--
’-Po,
p(U2
-
Ui)
--
2
(1
-4sin28)
-
pu’
2
-
1
-4sinz0
PU2/2
(9.6)
The
drag
of
a
body
151
Table
9.2
Drag
coefficients
for
various
bodies
152
Orag
and
lift
Fig.
9.4
Flow
around
a
cylinder
Fig.
9.5
Pressure distribution around cylinder:
A,
Re
=
1
.I
x
10’
<
Re,;
B,
Re
=
6.7
x 1
O5
>
Re,;
C,
Re
=
8.4
x
lo6
>
Re,
This pressure distribution is illustrated in Fig.
9.5,
where there is left and
right symmetry to the centre line at right angles to the flow. Consequently the
pressure resistance obtained by integrating this pressure distribution turns
out to be zero, i.e. no force at all acts on the cylinder. Since this phenomenon
is
contrary to actual
flow,
it
is
called d’alembert’s paradox, after the French
physicist
(1717-83).

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