FLIGHT MECHANICS
Exercise Problems
CHAPTER 4
FLIGHT MECHANICS
Exercise Problems
CHAPTER 4
Problem 4.1
• Consider the incompressible flow of
water through a divergent duct. The
inlet velocity and area are 5 ft/s and
10 ft
2
, respectively. If the exit area is
4 times the inlet area, calculate the
water flow velocity at the exit.
Solution 4.1
s
ft
A
A
V
V
V
A
V
A
m
/
25
.
1
4
1
5
2
1
1
2
2
2
2
1
1
1
Problem 4.2
• 4.2 In the above problem calculate
the pressure difference between the
exit and the inlet. The density of
water is 62.4 Ibm/ft
3
.
Solution 4.2
2
2
2
1
2
3
2
2
2
1
1
2
/
7
.
22
2
25
.
1
5
94
.
1
/
94
.
1
2
.
32
4
.
62
2
0
2
1
2
1
ft
lb
p
p
ft
slug
V
V
p
p
VdV
dp
v
v
p
p
Problem 4.3
• Consider an airplane flying with a
velocity of 60 m/s at a standard
altitude of 3 km. At a point on the
wing, the airflow velocity is 70 m/s.
Calculate the pressure at this point.
Assume incompressible flow.
Solution 4.3
H.W.
Problem 4.4
• An instrument used to measure the airspeed
on many early low-speed airplanes, principally
during 1919 to 1930, was the venturi tube.
This simple device is a convergent - divergent
duct (The front section's cross-sectional area
A decreases in the flow direction, and the back
section's cross-sectional area increases in the
flow direction. Somewhere in between the
inlet and exit of the duct, there is a minimum
area, called the throat.) Let A
1
and A
2
denote
the inlet and throat areas, respectively. Let p
1
and p
2
be the pressures at the inlet and
throat, respectively.
The venturi tube is mounted at a specific
location on the airplane (generally on the
wing or near the front of the fuselage),
where the inlet velocity V, is essentially the
same as the freestream velocity that is,
the velocity of the airplane through the air.
With a knowledge of the area ratio A
2
/A
1
(a
fixed design feature) and a measurement
of the pressure difference p
1
- p
2
the
airplane's velocity can be determined. For
example, assume A
2
/A
1
=1/4 and p
1
- p
2
=
80 Ib/ft
2
. If the airplane is flying at standard
sea level, what is its velocity?
Solution 4.4
H.W.
Problem 4.5
Consider the flow of air through a
convergent-divergent duct, such as the
venturi described in Prob. 4.4. The inlet,
throat, and exit areas are 3, 1.5, and 2 m
2
respectively. The inlet and exit pressures
are 1.02 x 10
5
and 1.00 x 10
5
N/m
2
,
respectively. Calculate the flow velocity
at the throat. Assume incompressible
flow with standard sea-level density.
Solution 4.5
s
m
V
A
A
V
A
A
p
p
V
A
A
V
V
V
p
V
p
/
22
.
102
1
2
3
225
.
1
10
)
00
.
1
02
.
1
(
2
5
.
1
3
1
)
(
2
2
2
2
5
1
2
1
2
2
3
1
3
1
1
3
1
1
3
2
3
3
2
1
1
Note that only a
pressure change of
0.02 atm produce
this high speed
Problem 4.6
An airplane is flying at a velocity of 130
mi/h at a standard altitude of 5000 ft. At
a point on the wing, the pressure is
1750.0 Ib/ft
2
. Calculate the velocity at
that point assuming incompressible flow.
Solution 4.6
s
ft
V
V
p
p
V
V
p
V
p
s
ft
mph
V
/
8
.
216
7
.
190
0020482
.
0
1750
9
.
1760
2
2
/
7
.
197
60
88
130
130
2
4
2
1
2
1
2
2
2
2
2
2
1
1
1
Problem 4.7
Imagine that you have designed a low-speed
airplane with a maximum velocity at sea level
of 90 m/s. For your airspeed instrument, you
plan to use a venturi tube with a 1.3 : 1 area
ratio. Inside the cockpit is an airspeed
indicator—a dial that is connected to a
pressure gauge sensing the venturi tube
pressure difference p
1
- p
2
and properly
calibrated in terms of velocity. What is the
maximum pressure difference you would
expect the gauge to experience?
Solution 4.7
2
2
2
2
1
2
2
1
2
1
2
1
2
1
1
2
2
2
2
2
1
1
/
3423
1
3
.
1
2
90
225
.
1
1
2
2
2
m
N
p
p
A
A
V
p
p
A
A
V
V
V
p
V
p
Maximum when
maximum velocity 90 m/s
and sea level density;
however better design for
over speed during diving
Problem 4.8
A supersonic nozzle is also a
convergent-divergent duct, which is fed
by a large reservoir at the inlet to the
nozzle. In the reservoir of the nozzle, the
pressure and temperature are 10 atm
and 300 K, respectively. At the nozzle
exit, the pressure is 1 atm. Calculate the
temperature and density of the flow at
the exit. Assume the flow is isentropic
and, of course, compressible.
Solution 4.8
H.W.
Problem 4.9
Derive an expression for the exit
velocity of a supersonic nozzle in
terms of the pressure ratio between
the reservoir and exit p
o
/p
e
and the
reservoir temperature To.
Solution 4.9
1
0
e
0
1
0
e
0
e
0
2
2
2
0
1
2
)
(
2
2
1
2
1
2
1
p
p
T
c
V
p
p
T
T
T
T
c
V
V
h
h
V
T
c
V
T
c
p
e
e
p
e
e
e
o
e
e
p
o
p
Note that the velocity
increases as T
o
goes
up or pressure ratio
goes down; used for
rocket engine
performance analysis
Problem 4.10
Consider an airplane flying at a standard
altitude of 5 km with a velocity of 270
m/s. At a point on the wing of the
airplane, the velocity is 330 m/s.
Calculate the pressure at this point.
Solution 4.10
H.W.
Problem 4.11
The mass flow of air through a
supersonic nozzle is 1.5 Ibm/s. The
exit velocity is 1500 ft/s, and the
reservoir temperature and pressure
are 1000°R and 7 atm, respectively.
Calculate the area of the nozzle exit.
For air, Cp = 6000 ft • lb/(slug)(°R).
Solution 4.11
2
1
4
.
1
1
1
1
o
e
0
0
0
0
2
2
0
2
0
0061
.
0
1500
0051
.
0
2
.
32
5
.
1
2
.
32
5
.
1
0051
.
0
1000
5
.
812
0086
.
0
0086
.
0
1000
1716
2116
7
5
.
812
6000
2
1500
1000
2
2
1
ft
V
m
A
V
A
m
T
T
RT
p
R
c
V
T
T
V
T
c
T
c
e
e
e
e
e
e
e
p
e
e
e
e
p
p
Energy eq.
Continuity
eq.
No shock wave,
isentropic
relationship
Problem 4.12
A supersonic transport is flying at a
velocity of 1500 mi/h at a standard
altitude of 50,000 ft. The temperature
at a point in the flow over the wing is
793.32°R. Calculate the flow velocity at
that point.
Solution 4.12
s
ft
V
V
s
ft
s
ft
h
mi
V
V
T
T
c
V
V
T
c
V
T
c
p
p
p
/
3
.
6
2200
32
.
7993
99
.
389
6000
2
/
2200
/
60
88
1500
/
1500
2
2
1
2
1
2
2
2
2
1
2
1
2
1
2
2
2
2
2
2
1
1
Very low value, almost a stagnant point
Problem 4.13
For the airplane in Prob. 4.12, the total cross-
sectional area of the inlet to the jet engines is
20 ft
2
. Assume that the flow properties of the
air entering the inlet are those of the
freestream ahead of the airplane. Fuel is
injected inside the engine at a rate of 0.05 Ib
of fuel for every pound of air flowing through
the engine (i.e., the fuel-air ratio by mass is
0.05). Calculate the mass flow (in slugs/per
second) that comes out the exit of the
engine.
Solution 4.13
H.W.
Problem 4.14
Calculate the Mach number at the
exit of the nozzle in Prob. 4.11.
Solution 4.14
07
.
1
1397
1500
/
1397
5
.
812
1716
4
.
1
5
.
812
/
1500
e
e
a
V
M
s
ft
RT
a
R
T
s
ft
V
e
e
e
e
e
Problem 4.15
A Boeing 747 is cruising at a velocity
of 250 m/s at a standard altitude of
13 km. What is its Mach number?
Solution 4.15
H.W.
Problem 4.16
A high-speed missile is traveling at
Mach 3 at standard sea level. What
is its velocity in miles per hour?
Solution 4.16
H.W.
Problem 4.17
Calculate the flight Mach number for the
supersonic transport in Prob. 4.12.
Solution 4.17
27
.
2
94
.
967
2200
/
94
.
967
99
.
389
1716
4
.
1
/
2200
a
V
M
s
ft
RT
a
s
ft
V
Problem 4.18
Consider a low-speed subsonic wind
tunnel with a nozzle contraction ratio of
1 : 20. One side of a mercury
manometer is connected to the settling
chamber, and the other side to the test
section. The pressure and temperature
in the test section are 1 atm and 300 K,
respectively. What is the height
difference between the two columns of
mercury when the test section velocity is
80 m/s?
Solution 4.18
cm
m
A
A
V
h
h
h
p
p
A
A
V
p
p
m
kg
RT
p
8
.
2
028
.
0
20
1
1
2
80
10
*
33
.
1
173
.
1
1
2
10
*
33
.
1
1
2
/
173
.
1
300
287
10
*
01
.
1
2
2
5
2
1
2
2
2
5
2
1
2
1
2
2
2
2
1
3
5
Manometer reading
Problem 4.19
We wish to operate a low-speed subsonic wind
tunnel so that the flow in the test section has a
velocity of 200 mi/h at standard sea-level
conditions. Consider two different types of
wind tunnels: (a) a nozzle and a constant-area test
section, where the flow at the exit of the test
section simply dumps out to the surrounding
atmosphere, that is, there is no diffuser, and
(b) a conventional arrangement of nozzle, test
section, and diffuser, where the flow at the exit of
the diffuser dumps out to the surrounding
atmosphere. For both wind tunnels (a) and (b)
calculate the pressure differences across the entire
wind tunnel required to operate them so as to have
the given flow conditions in the test section.
For tunnel (a) the cross-sectional area
of the entrance is 20 ft
2
, and the cross-
sectional area of the test section is 4
ft
2
. For tunnel (b) a diffuser is added to
(a) with a diffuser area of 18 ft
2
. After
completing your calculations, examine
and compare your answers for tunnels
(a) and (b). Which requires the smaller
overall pressure difference? What does
this say about the value of a diffuser
on a subsonic wind tunnel?
Solution 4.19 (a)
2
2
2
2
1
2
1
2
2
2
2
1
1
2
2
1
2
2
2
2
1
1
/
15
.
98
20
4
1
2
3
.
293
002377
.
0
1
2
2
2
ft
lb
p
p
A
A
V
p
p
A
A
V
V
V
p
V
p
Solution 4.19 (b)
2
2
2
2
2
1
2
1
2
2
3
2
2
2
3
1
3
2
2
3
1
2
2
1
2
3
3
2
1
1
/
959
.
0
20
4
18
4
2
3
.
293
002377
.
0
2
,
2
2
ft
lb
p
p
A
A
A
A
V
p
p
A
A
V
V
A
A
V
V
V
p
V
p
Economical to use diffuser (running
compressor or vacuum pump)
Problem 4.20
A Pitot tube is mounted in the test
section of a low-speed subsonic wind
tunnel. The flow in the test section
has a velocity, static pressure, and
temperature of 150 mi/h, 1 atm, and
70°F, respectively. Calculate the
pressure measured by the Pitot tube.
Solution 4.20
2
2
0
2
0
2
0
3
/
2172
220
2
00233
.
0
2116
60
88
*
150
2
00233
.
0
2116
2
/
00233
.
0
460
70
1716
2116
ft
lb
p
p
V
p
p
ft
slug
RT
p
Problem 4.21
The altimeter on a low-speed Piper
Aztec reads 8000 ft. A Pitot tube
mounted on the wing tip
measures a pressure of 1650 Ib/ft
2
. If
the outside air temperature is 500°R,
what is the true velocity of the
airplane? What is the equivalent
airspeed?
Solution 4.21
H.W.
Problem 4.22
The altimeter on a low-speed
airplane reads 2 km. The airspeed
indicator reads 50 m/s. If the outside
air temperature is 280 K, what is the
true velocity of the airplane?
s
m
V
V
V
m
kg
RT
p
true
eq
true
/
56
989
.
0
225
.
1
50
/
989
.
0
280
287
10
*
95
.
7
0
3
4
Solution 4.22
Problem 4.23
A Pitot tube is mounted in the test
section of a high-speed subsonic
wind tunnel. The pressure and
temperature of the airflow are 1 atm
and 270 K, respectively. If the flow
velocity is 250 m/s, what is the
pressure measured by the Pitot tube?
Solution 4.23
5
5
0
1
4
.
1
4
.
1
2
1
2
0
10
*
48
.
1
10
*
01
.
1
*
47
.
1
47
.
1
47
.
1
2
76
.
0
)
1
4
.
1
(
1
2
)
1
(
1
76
.
0
329
250
/
329
270
*
287
*
4
.
1
p
p
M
p
p
a
V
M
s
m
RT
a
Problem 4.24
A high-speed subsonic Boeing 777
airliner is flying at a pressure
altitude of 12 km. A Pitot tube on
the vertical tail measures a pressure
of 2.96 x 10
4
N/m
2
. At what Mach
number is the airplane flying?
Solution 4.24
801
.
0
N/m
10
*
94
.
1
p
km,
12
altitude
at
note;
1
10
*
94
.
1
10
*
96
.
2
1
4
.
1
2
1
1
2
10
*
94
.
1
1
2
4
4
.
1
1
4
.
1
4
4
1
1
0
2
1
4
M
p
p
M
p
Problem 4.25
A high-speed subsonic airplane is flying
at Mach 0.65. A Pitot tube on the wing
tip measures a pressure of 2339
Ib/ft
2
. What is the altitude reading on the
altimeter?
Solution 4.25
1761
328
.
1
2339
328
.
1
328
.
1
2
65
.
0
)
1
4
.
1
(
1
2
)
1
(
1
0
1
4
.
1
4
.
1
2
1
2
0
p
p
M
p
p
Appendix B, pressure altitude reads 5000 ft
Problem 4.26
A high-performance F-16 fighter is
flying at Mach 0.96 at sea level. What is
the air temperature at the
stagnation point at the leading edge of
the wing?
Solution 4.26
H.W.
Problem 4.27
An airplane is flying at a pressure
altitude of 10 km with a velocity of 596
m/s. The outside air
temperature is 220 K. What is the
pressure measured by a Pitot tube
mounted on the nose of the airplane?
Solution 4.27
2
5
4
02
4
1
2
1
4
.
1
4
.
1
2
2
2
2
1
1
2
1
2
1
2
1
02
1
1
1
1
/
10
*
49
.
1
10
*
65
.
2
*
64
.
5
10
*
65
.
2
64
.
5
1
4
.
1
2
*
4
.
1
*
2
4
.
1
1
)
1
4
.
1
(
2
2
*
4
.
1
*
4
2
)
1
4
.
1
(
1
2
1
)
1
(
2
4
)
1
(
0
.
2
297
596
/
297
220
*
287
*
4
.
1
m
N
p
p
as
M
M
M
p
p
a
V
M
s
m
RT
a
Use Rayleigh Pitot tube formula
Problem 4.28
The dynamic pressure is defined as
q = 0.5V
2
. For high-speed flows,
where Mach number is used
frequently, it is convenient to
express q in terms of pressure p and
Mach number M rather than and V.
Derive an equation for q = q(p,M).
Solution 4.28
2
2
2
2
1
2
2
2
2
2
2
2
2
2
1
2
1
M
p
a
V
p
V
p
p
q
p
c
d
c
d
d
dp
a
V
p
p
V
p
p
V
q
so
as
Problem 4.29
After completing its mission in orbit
around the earth, the Space Shuttle
enters the earth's atmosphere at very
high Mach number and, under the
influence of aerodynamic drag, slows as
it penetrates more deeply into the
atmosphere. (These matters are
discussed in Chap. 8.) During its
atmospheric entry, assume that the
shuttle is flying at Mach number M
corresponding to the altitudes h:
Calculate the corresponding values of the
freestream dynamic pressure at each one
of these flight path points. Suggestion:
Use the result from Prob. 4.28. Examine
and comment on the variation of q
∞
as
the shuttle enters the atmosphere.
h,
km
60
50
40
30
20
M
17
9.5
5.5
3
1
Solution 4.29
2
2
M
p
q
h, km
60
50
40
30
20
p
∞
25.6
87.9
299.8
1.19*10
3
5.53*1
0
3
M
17
9.5
5.5
3
1
q
∞
5.2*10
3
5.6*10
3
6.3*10
3
7.5*10
3
3.9*10
3
Problem 4.30
Consider a Mach 2 airstream at
standard sea-level conditions.
Calculate the total pressure of this
flow. Compare this result with (a) the
stagnation pressure that would exist
at the nose of a blunt body in the
flow and (b) the erroneous result
given by Bernoulli's equation, which
of course does not apply here.
Solution 4.30
16560
2116
824
.
7
824
.
7
824
.
7
2
2
)
1
4
.
1
(
1
2
)
1
(
1
0
1
4
.
1
4
.
1
2
1
2
0
p
p
M
p
p
Total pressure when the flow is isentropically stopped (true
for supersonic and subsonic)
2
4
02
2
1
4
.
1
4
.
1
2
2
2
2
1
1
2
1
2
1
2
1
02
/
10
*
193
.
1
116
.
2
*
64
.
5
64
.
5
1
4
.
1
2
*
4
.
1
*
2
4
.
1
1
)
1
4
.
1
(
2
2
*
4
.
1
*
4
2
)
1
4
.
1
(
1
2
1
)
1
(
2
4
)
1
(
ft
lb
p
M
M
M
p
p
But there must be a shockwave at the nose (at the
stagnation point)
2
4
2
0
2
2
0
/
10
*
804
.
0
2
*
116
.
2
*
2
4
.
1
116
.
2
2
2
ft
lb
p
M
p
p
V
p
p
If Bernoulli’s equation is used accidentally
51% error
Problem 4.31
Consider the flow of air through a
supersonic nozzle. The reservoir
pressure and temperature are 5 atm
and 500 K, respectively. If the Mach
number at the nozzle exit is 3,
calculate the exit pressure,
temperature, and density.
Solution 4.31
3
4
0
0
0
1
2
0
4
1
4
.
1
4
.
1
2
5
1
2
0
/
267
.
0
6
.
178
287
10
*
37
.
1
6
.
178
357
.
0
*
500
2
)
1
(
1
10
*
37
.
1
2
3
)
1
4
.
1
(
1
10
*
01
.
1
*
5
2
)
1
(
1
m
kg
RT
p
K
M
T
T
M
p
p
e
e
e
e
Problem 4.32
• Consider a supersonic nozzle across
which the pressure ratio is p
e
/p
o
=
0.2. Calculate the ratio of exit area
to throat area.
Solution 4.32
35
.
1
71
.
1
2
1
4
.
1
1
1
4
.
1
2
71
.
1
1
2
1
1
1
2
1
71
.
1
92
.
2
1
2
.
0
5
1
)
1
(
2
2
)
1
(
1
1
4
.
1
1
4
.
1
2
2
1
1
2
2
286
.
0
1
0
2
1
2
0
e
e
t
e
e
e
e
e
e
M
M
A
A
M
p
p
M
M
p
p
Problem 4.33
• Consider the expansion of air through a
convergent-divergent supersonic nozzle.
The Mach number varies from essentially
zero in the reservoir to Mach 2.0 at the exit.
Plot on graph paper the variation of the
ratio of dynamic pressure to total pressure
as a function of Mach number; that is, plot
q/ p
o
versus M from M = 0 to M = 2.0.
Solution 4.33
5
.
3
2
2
1
2
2
2
2
2
2
2
2
.
0
1
7
.
0
2
1
1
2
2
2
2
2
1
M
M
p
q
M
M
p
p
M
p
q
M
p
a
V
p
V
q
The graph shows that the local
dynamic pressure has a peak value
at M=1.4
Problem 4.34
The wing of the Fairchild Republic A-10A
twin-jet close-support airplane is
approximately rectangular with a wingspan
(the length perpendicular to the flow
direction) of 17.5 m and a chord (the length
parallel to the flow direction) of 3 m. The
airplane is flying at standard sea level with
a velocity of 200 m/s. If the flow is
considered to be completely laminar,
calculate the boundary layer thickness at
the trailing edge and the total skin friction
drag. Assume the wing is approximated by
a flat plate. Assume incompressible flow.
Solution 4.34
H.W.
Problem 4.35
In Prob. 4.34, assume the flow is
completely turbulent. Calculate the
boundary layer thickness at the
trailing edge and the total skin friction
drag. Compare these turbulent results
with the above laminar results.
Solution 4.35
N
N
D
bottom
and
top
N
SC
q
D
C
cm
m
L
f
f
f
L
f
lar
turb
L
5660
2830
*
2
2830
0022
.
0
*
5
.
17
*
3
*
10
*
45
.
2
0022
.
0
10
*
10
.
4
0074
.
0
Re
0074
.
0
75
.
13
24
.
0
3
.
3
3
.
3
033
.
0
10
*
10
.
4
3
*
37
.
0
Re
37
.
0
4
2
.
0
7
2
.
0
2
.
0
7
2
.
0
10.5 times larger than laminar
flow assumption
Problem 4.36
• If the critical Reynolds number for
transition is 10
6
, calculate the skin
friction drag for the wing in Prob. 4.34.
Laminar Flow
A
Turbulent Flow B
X
cr
Solution 4.36
N
D
m
m
S
m
N
V
q
S
q
S
q
SC
q
D
m
V
x
x
V
turb
f
cr
f
turb
f
cr
cr
cr
cr
146
5
.
17
*
10
*
3
.
7
/
10
*
45
.
2
200
*
225
.
1
2
1
2
1
10
074
.
0
Re
074
.
0
10
*
3
.
7
200
*
225
.
1
10
*
7894
.
1
*
10
Re
Re
2
2
4
2
2
2
.
0
6
2
.
0
2
5
6
Drag of one side
Calculate
drag force if
the laminar
flow portion
A were
turbulent
flow
N
N
N
D
N
S
q
SC
q
D
N
D
D
D
N
D
f
cr
f
A
f
A
f
total
f
B
f
turbulent
total
f
turb
5452
2684
42
42
5
.
17
*
10
*
3
.
7
10
*
45
.
2
10
135
Re
1328
2684
146
2830
2830
2
4
2
.
0
6
5
.
0
laminar
On the wing, it is mostly turbulent
flow
Problem 4.37
Let us reflect back to the fundamental
equations of fluid motion discussed in the
early sections of this chapter. Sometimes
these equations were expressed in terms of
differential equations, but for the most pan
we obtained algebraic relations by integrating
the differential equations. However, it is
useful to think of the differential forms as
relations that govern the change in flowfield
variables in an infinitesimally small region
around a point in the flow.
(a) Consider a point in an inviscid flow,
where the local density is 1.1 kg/m
3
. As
a fluid element sweeps through this
point, it is experiencing a spatial change
in velocity of two percent per millimeter.
Calculate the corresponding spatial
change in pressure per millimeter at this
point if the velocity at the point is 100
m/sec. (b) Repeat the calculation for the
case when the velocity at the point is
1000 m/sec. What can you conclude by
comparing your results for the low-
speed flow in part (a) with the results for
the high-speed flow part (b).
Solution 4.37
mm
m
N
ds
dp
mm
m
N
ds
dp
mm
ds
V
dV
ds
V
dV
V
ds
dV
V
ds
dp
VdV
dp
.
/
22000
02
.
0
1000
1
.
1
.
/
220
02
.
0
100
1
.
1
/
02
.
0
2
2
2
2
2
It requires a much larger pressure gradient
in a high-speed flow
Problem 4.38
The type of calculation in Problem 4.3 is a
classic one for low-speed, incompressible
flow, i.e., given the freestream pressure and
velocity, and the velocity at some other
point in the flow, calculate the pressure at
that point. In a high-speed compressible
flow, Mach number is more fundamental
than velocity. Consider an airplane flying at
Mach 0.7 at a standard altitude of 3 km. At
a point on the wing, the airflow Mach
number is 1.1. Calculate the pressure at this
point. Assume an isentropic flow.
Solution 4.38
4
4
0
0
1
4
.
1
4
.
1
2
1
2
0
1
4
.
1
4
.
1
2
1
2
0
10
*
555
.
4
10
*
0121
.
7
*
65
.
0
135
.
2
387
.
1
135
.
2
2
1
.
1
)
1
4
.
1
(
1
2
)
1
(
1
387
.
1
2
7
.
0
)
1
4
.
1
(
1
2
)
1
(
1
p
p
p
p
p
p
p
M
p
p
M
p
p
Pressure at 3 km
altitude
Problem 4.39
• Consider an airplane flying at a
standard altitude of 25,000 ft at a
velocity of 800 ft/sec. To experience
the same dynamic pressure at sea
level, how fast must the airplane be
flying?
Solution 4.39
s
ft
V
V
V
e
e
/
8
.
535
10
*
3769
.
2
10
*
0663
.
1
800
3
3
0
Problem 4.40
In Section 4.9, we defined hypersonic
flow as that flow where the Mach number
is five or greater. Wind tunnels with a
test section Mach number of five or
greater are called hypersonic wind
tunnels. From Eq. (4.88), the exit-to-
throat area ratio for supersonic exit
Mach numbers increases as the exit
Mach number increases. For hypersonic
Mach numbers, the exit-to-throat ratio
becomes extremely large, so hypersonic
wind tunnels are designed with long,
high-expansion ratio nozzles.
In this and the following problems, let us
examine some special characteristics of
hypersonic wind tunnels. Assume we wish
to design a Mach 10 hypersonic wind
tunnel using air as the test medium. We
want the static pressure and temperature
in the test stream to be that for a
standard altitude of 55 km. Calculate: (a)
the exit-to-throat area ratio, (b) the
required reservoir pressure (in atm), and
(c) the required reservoir temperature.
Examine these results. What do they tell
you about the special (and sometimes
severe) operating requirements for a
hypersonic wind tunnel.
Solution 4.40
K
M
T
T
atm
p
M
p
p
M
M
A
A
e
e
o
o
e
e
o
e
e
t
e
5791
2
10
)
1
(
1
78
.
275
2
)
1
(
1
\
3
.
20
10
*
053
.
2
373
.
48
10
*
224
.
4
10
*
224
.
4
2
10
)
1
4
.
1
(
1
2
)
1
(
1
9
.
535
10
2
1
4
.
1
1
1
4
.
1
2
10
1
2
1
1
1
2
1
2
2
6
4
4
5
.
3
2
1
2
1
4
.
1
1
4
.
1
2
2
1
1
2
2
The surface of the sun is about 6000k;
sacrifice accuracy because of temperature
Problem 4.41
• Calculate the exit velocity of the
hypersonic tunnel in Problem 4.40.
Solution 4.41
s
m
a
M
V
s
m
RT
a
e
e
e
e
e
/
3329
9
.
332
10
/
9
.
332
78
.
275
287
4
.
1
Problem 4.42
Let us double the exit Mach number of
the tunnel in Problem 4.40 simply by
adding a longer nozzle section with the
requisite expansion ratio. Keep the
reservoir properties the same as those
in Problem 4.40. Then we have a Mach
20 wind tunnel, with test section
pressure and temperature considerably
lower than in Problem 4.40, i.e., the test
section flow no longer corresponds to
conditions at a standard altitude of 55
km. Be that as it may, we have at least
doubled the Mach number of the tunnel.
• Calculate: (a) the exit-to-throat area
ratio of the Mach 20 nozzle, (b) the exit
velocity. Compare these values with
those for the Mach 10 tunnel in
Problems 4.40 and 4.41. What can you
say about the differences? In particular,
note the exit velocities for the Mach 10
and Mach 20 tunnels. You will see that
they are not much different. What is
then giving the big increase in exit
Mach number?
Solution 4.42
s
m
RT
M
a
M
V
K
M
T
T
M
M
A
A
e
e
e
e
e
e
e
e
e
t
e
/
3390
5
.
71
287
4
.
1
20
5
.
71
2
20
)
1
(
1
5791
2
)
1
(
1
15377
20
2
1
4
.
1
1
1
4
.
1
2
20
1
2
1
1
1
2
1
1
2
1
2
0
1
4
.
1
1
4
.
1
2
2
1
1
2
2
Not much increase in velocity
28.7 times
increase of exit
area