Solution 9.146
(a) At the initial condition Ma1 = 2.0, from Table B.5 read
1 = 26.38. The first turn is 10, so
Problem 9.147
A converging-diverging nozzle with a 4:1 exit-area ratio and p0 = 500 kPa operates in an
underexpanded condition (case 1 of Fig. 9.12b) as in Fig. P9.147. The receiver pressure is
pa = 10 kPa, which is less than the exit pressure, so that expansion waves form outside the exit. For
the given conditions, what will the Mach number Ma2 and the angle
of the edge of the jet be?
Assume k = 1.4 as usual.
Solution 9.147
Get the Mach number in the exit and then execute a Prandtl-Meyer expansion:
Problem 9.148
Air flows supersonically over a circular-arc surface as in Fig. P9.148. Estimate (a) the Mach
number Ma2 and (b) the pressure p2 as the flow leaves the circular surface.
Solution 9.148
(a) At the initial condition Ma1 = 2.5, from Table B.5 read
1 = 39.12.
Problem 9.149
Air flows at Ma = 3.0 past a doubly symmetric diamond airfoil whose front and rear included
angles are both 24°. For zero angle of attack, compute the drag coefficient with shock-expansion
theory and compare with Ackeret theory.
Solution 9.149
The airfoil and its front and rear pressures are shown below.
Ma2 = 2.406
Problem 9.150
A flat plate airfoil with C = 1.2 m is to have a lift of 30 kN/m when flying at 5000-m standard
altitude with U = 641 m/s. Using Ackeret theory, estimate (a) the angle of attack; and (b) the
drag force in N/m.
Solution 9.150
At 5000 m,
= 0.7361 kg/m3, T = 256 K, and p = 54008 Pa. Compute Ma:
Problem 9.151
Air flows at Ma = 2.5 past a half-wedge airfoil whose angles are 4, as in Fig. P9.151. Compute
the lift and drag coefficients at
equal to (a) 0; and (b) 6.
Solution 9.151
Let’s use Ackeret theory here:
Problem 9.152
The X-43 model A scramjet aircraft in Fig. C9.8 is small, W = 3000 lbf, and unmanned, only
12.33 ft long and 5.5 ft wide. The aerodynamics of a slender arrowhead-shaped hypersonic
vehicle is beyond our scope. Instead, let us assume it is a flat plate airfoil of area 2.0 m2. Let
Ma = 7 at 12,000 m standard altitude. Estimate the drag, by shock-expansion theory. [HINT: Use
Ackeret theory to estimate the angle of attack.]
Solution 9.152
From Table B.6 at 12,000 m,
= 1
Problem 9.153
A supersonic transport has a mass of 65 Mg and cruises at 11-km standard altitude at a Mach
number of 2.25. If the angle of attack is 2 and its wings can be approximated by flat plates,
estimate (a) the required wing area in m2 and (b) the thrust required in N.
Solution 9.153
At 11 km (Table B.6), take p = 22612 Pa. (a) Use linearized theory:
Problem 9.154
The F-22 supersonic fighter cruises at 11,000 m altitude, with a weight of 50,000 lbf and thrust
of 10,000 lbf. Its wing area is 840 ft2. Assume the wing is a 6 percent-thick diamond shape and
provides all lift and thrust. Use Ackeret theory to estimate the resulting Mach number.
Solution 9.154
At 11,000 m altitude, from Table B.6, p∞ = 22,612 Pa. Convert the data to SI units:
W = 222,400 N, thrust F = 44,500 N, Ap = 78 m2. Then the lift and drag relations are:
Problem 9.155*
The F-35 airplane in Fig. 9.30 has a wingspan of 10 m and a wing area of 41.8 m2. It cruises at
about 10 km altitude with a gross weight of about 200 kN. At that altitude, the engine develops a
thrust of about 50 kN. Assume the wing has a symmetric diamond airfoil with a thickness of
8 percent, and accounts for all lift and drag. Estimate the cruise Mach number of the airplane.
For extra credit, explain why there are two solutions.
Solution 9.155
At 10 km standard altitude, from Table B.6,
= 0.4125 kg/m3, and atmospheric pressure is
p = 26416 Pa. We need to match lift and drag to the given weight/thrust data:
Problem 9.156
Consider a flat-plate airfoil at an angle of attack of 6°. The Mach number is Ma∞ = 3.2 and the
stream pressure p∞ is unspecified. Calculate the predict lift and drag coefficients, by (a) shock-
expansion theory; and (b) Ackeret theory.
Solution 9.156
(a) We need a 5º shock turn on the lower surface and a 5º expansion turn on the upper surface.
The results obtain by the writer are shown here:
Problem 9.157
The Ackeret airfoil theory of Eq. (9.104) is meant for moderate supersonic speeds,
1.2 < Ma < 4. How does it fare for hypersonic speeds? To illustrate, calculate (a) CL and (b) CD
for a flat-plate airfoil at
= 5, with Ma = 8.0, using shock-expansion theory, and compare with
Ackeret theory. Comment.
Solution 9.157
Like Example 9.19, we need only calculate a (steep) shock wave on the bottom and a (steep)
expansion wave fan on the top. The results are shown in the figure below.