u
u
CHAPTER 9
9.1 For small perturbations, u/V, v/V and w/V << 1. Consequently, in Eq. (9.4), the
squares of these terms (second-order terms) can be ignored in comparison to the first-order
terms. For example
 
++−+
1
1
2
2 2
u
v w
u
83
065
.
9.3 (a) CL =
C
M
Lo
12
−
=
−
−
0 9
1 06 2
.
( . )
= -1.125
− + + −
−
1 06 06
1 1 06
09
2
22
2
( . ) ( . )
( . )
( . )
9.4 Consider a flat plate of length (chord) c at an angle of attack, , to the supersonic freestream,
as sketched in the Figure below.
From both shock-expansion theory and linear theory, we know the pressure distributions over the
top and bottom surfaces are uniform, but are of different magnitudes, as sketched in the above
CM
pL=−
2
1
2
(3)
1
4
4
9.5
Recall: A positive moment is a pitch-up moment, i.e., one that increases the angle of attack.
c
c
‘/
4
87
we have
Since
−
2
2
4
=−
−
 
1
1 1
2 2 2
M M M
9.6 (a) From incompressible flow theory for flow over a flat plate, with in radians,
4
M
2
1−
88
9.7 Consider again the diamond-shaped airfoil sketched in Figure 4.35. In Section (4.14), the
shock-expansion theory was applied to obtain the drag on this airfoil. Here, we will apply linear
theory.
CM
p2
2
1
2
=−
(1)
and
CM
p3
2
1
2
=−
−
(2)
In addition, from the definition of cp,
( )
( )
p p
2 3
2 3
 
89
1
2
M−
c
From Eq. (7), the drag coefficient is
D
4
2
t
where is a function of the airfoil geometry.
9.8
90
Let be the angle between the tangent to the surface and the chord line. is + when measured
above the chord direction, and is – when measured below the chord direction.
Consider two coordinate systems, (x,y) and (,) as sketched above. On both the upper
and lower surfaces.
dy
q c
cc
LE
p
From linearized theory,
dy
dx
2
2
−
2
4
TE
TE
4
TE
92
p p
−
p p
−
p
(degrees) (rad)
Cp3
Cp2
0 0 0 0
p32
p22
T3 T2
(degrees) (N/m2) (N/m2) (K) (K)
30 0 3.37 x 105 0 381
Note: When p3 and T3 are calculated as negative numbers, zeros are inserted instead, because
negative pressures and temperatures do not exist on an absolute scale.
4
L D L/D
(deg.) (rad) (N/M) (N/m) _____
0 0 0 0 –
The results are plotted as the dashed lines on the graphs for Problem 4.17. Note that:
94
9.10 Supersonic thin airfoil theory predicts that the wave drag coefficient will decrease with
increasing Mach number. What does this say about the drag force itself? Does it increase or
decrease with M? Indeed, intuition might say that a body undergoing a continuous increase in
Where the constant k3 = 4k2(
2
)p. From Eq. (4) we already begin to suspect that for values of
M near one, the denominator of Eq. (4) may dominate the drag, and cause it to decrease with an
or,
96
Return for a moment to Section 1.1, and the history-making flight of Chuck Yeager in the
Bell XS-1 on October 14, 1947. On that day, for the first time in history, a piloted airplane flew
faster than the speed of sound, reaching a maximum Mach number of 1.06. But from our above
discussion, Yeager should have entered a region of decreasing drag, and should have had no
trouble sliding up to Mach 1.4. The result is misleading. In reality, as we know from shock-
expansion theory, as M increases above one, the shock strength increases continuously, and
hence the wave drag must increase monotonically above M = 1.
9.11
M Cp Cp Cp
CR Prandtl-Glauert Karman-Tsien
___ From Eq. (9.55) (Eq. 9.36) (Eq. 9.40)__