142
ANSWERS TO PROBLEMS REFERRED TO IN THE TEXT OF
CHAPTER 15
Please note: There are two problems that are described within the text of Chapter 15; these are
not specifically itemized at the end of the chapter. In this Solutions Manual, they will be referred
to by page numbers.
15.1 (See page 562 in the text.)
Statement of Problem: Using newtonian theory, show that maximum
c
for a flat plate occurs at
= 54.7.
SOLUTION From Eq. (15.26)
c
143
15.2 (See pages 563 and 564 in the text.)
Statement of Problem: Show from newtonian theory that cd =
4
3
for a cylinder of infinite span
and cD = 1 for a sphere.
SOLUTION
144
D = q R
o
2
Cp cos d (2)
Since is measured between the freestream direction and the normal to the body surface, the
newtonian “sine–squared” law is written as
Cp = 2 cos2
Over the body surface, we have
Cp = 2 cos2 for –
2
2
Cp = 0 for –
2
3
2
Hence, Eq. (2) becomes
D = q R
−
2
2
2 cos3 d
Define the drag coefficient as
Cd =
D
q R
( )2
(3)
Combining Eqs. (2) and (3),
Cd =
−
2
2
cos3 d =
1
3
[sin (cos2 + 2]
−
2
2
=
1
3
[2 – (-1)(2)]
Thus
Cd =
4
3
For the case of a sphere, consider the following sketch:
145
Consider the circular strip (shaded) with radius R sin , circumference 2Rsin, and elemental
height ds = Rd. The pressure, p, is constant over this strip. Hence, the drag is given by
2
2
146
then