fluid_mechanics

464 Chapter 9 ■ Flow over Immersed Bodies
y
pdA
θ
τ w dA
U
dA
θ
x
F I G U R E 9.4 Pressure and shear
forces on a small element of the surface of a
body.
integrated or resultant effects of these distributions are needed. The resultant force in the direction
of the upstream velocity is termed the drag, d, and the resultant force normal to the upstream velocity
is termed the lift, l, as is indicated in Fig. 9.3c. For some three-dimensional bodies there
may also be a side force that is perpendicular to the plane containing d and l.
The resultant of the shear stress and pressure distributions can be obtained by integrating the
effect of these two quantities on the body surface as is indicated in Fig. 9.4. The x and y components
of the fluid force on the small area element dA are
and
dF x 1 p dA2 cos u 1t w dA2 sin u
dF y 1 p dA2 sin u 1t w dA2 cos u
Thus, the net x and y components of the force on the object are
d dF x p cos u dA t w sin u dA
(9.1)
and
l dF y p sin u dA t w cos u dA
(9.2)
Lift and drag on a
section of a body
depend on the orientation
of the surface.
Of course, to carry out the integrations and determine the lift and drag, we must know the body
shape 1i.e., u as a function of location along the body2 and the distribution of t w and p along the
surface. These distributions are often extremely difficult to obtain, either experimentally or theoretically.
The pressure distribution can be obtained experimentally by use of a series of static pressure
taps along the body surface. On the other hand, it is usually quite difficult to measure the wall
shear stress distribution.
F l u i d s i n t h e N e w s
Pressure-sensitive paint For many years, the conventional
method for measuring surface pressure has been to use static
pressure taps consisting of small holes on the surface connected
by hoses from the holes to a pressure measuring device. Pressuresensitive
paint (PSP) is now gaining acceptance as an alternative
to the static surface pressure ports. The PSP material is typically
a luminescent compound that is sensitive to the pressure on it and
can be excited by an appropriate light which is captured by special
video imaging equipment. Thus, it provides a quantitative
measure of the surface pressure. One of the biggest advantages of
PSP is that it is a global measurement technique, measuring pressure
over the entire surface, as opposed to discrete points. PSP
also has the advantage of being nonintrusive to the flow field. Although
static pressure port holes are small, they do alter the surface
and can slightly alter the flow, thus affecting downstream
ports. In addition, the use of PSP eliminates the need for a large
number of pressure taps and connecting tubes. This allows pressure
measurements to be made in less time and at a lower cost.
It is seen that both the shear stress and pressure force contribute to the lift and drag, since
for an arbitrary body u is neither zero nor 90° along the entire body. The exception is a flat plate
aligned either parallel to the upstream flow 1u 90°2 or normal to the upstream flow 1u 02 as
is discussed in Example 9.1.