Problem 8.66*
The inviscid velocity along the wedge in Prob. 8.65 has the analytic form U(x)
=
Cxm, where
m
=
n − 1 and n is the exponent in Eq. (8.53). Show that, for any C and n, computation of the
laminar boundary layer by Thwaite’s method, Eqs. (7.53) and (7.54), leads to a unique value of
the Thwaites parameter
. Thus wedge flows are called similar [15, p. 244].
Problem 8.65
Potential flow past a wedge of half-angle
leads to an important application of laminar-
boundary-layer theory called the Falkner-Skan flows [15, pp. 239-245]. Let x denote distance
along the wedge wall, as in Fig. P8.65, and let
= 10. Use Eq. (8.53) to find the variation of
surface velocity U(x) along the wall. Is the pressure gradient adverse or favorable?
Solution 8.66
The momentum thickness is computed by Eq. (7.54), assuming
o = 0:
Problem 8.67
Investigate the complex potential function f(z) = U(z + a2z), and interpret the flow pattern.
Solution 8.67
This represents flow past a circular cylinder of radius a, with stream function and velocity
Problem 8.68
Investigate the complex potential function f(z) = Uz + m ln[(z + a)(z − a)], and interpret the flow
pattern.
Solution 8.68
Problem 8.69
Investigate the complex potential function f(z) = A cosh(
za), and plot the streamlines inside
the region shown in Fig. P8.69. What hyphenated French word might describe such a flow
pattern?
Solution 8.69
This potential splits into
Problem 8.70
Show that the complex potential f(z) = U[z + (a4) coth(
za)] represents flow past an oval
shape placed midway between two parallel walls y = a2. What is a practical application?
Solution 8.70
The stream function of this flow is
Problem 8.71
Figure P8.71 shows the streamlines and potential lines of flow over a thin-plate weir as
computed by the complex potential method. Compare qualitatively with Fig. 10.16a. State the
proper boundary conditions at all boundaries. The velocity potential has equally spaced values.
Why do the flow-net “squares” become smaller in the overflow jet?
Solution 8.71
Solve Laplace’s equation for either
or
(or both), find the velocities u =
x, v =
y,
Problem 8.72
Use the method of images to construct the flow pattern for a source +m near two walls, as shown
in Fig. P8.72. Sketch the velocity distribution along the lower wall (y = 0). Is there any danger
of flow separation along this wall?
Solution 8.72
This pattern is the same as that of Prob. 8.28. It is created by placing four identical sources
Problem 8.73
Set up an image system to compute the flow of a source at unequal distances from two walls, as
shown in Fig. P8.73. Find the point of maximum velocity on the y-axis.
Solution 8.73
Similar to Prob. 8.72 , we place identical sources (+m) at the symmetric (but non-square) positions
Problem 8.74
A positive line vortex K is trapped in a corner, as in Fig. P8.74. Compute the total induced
velocity at point B, (x, y) = (2a, a), and compare with the induced velocity when no walls are
present.
Solution 8.74
The two walls are created by placing vortices, as shown at right, at (x, y) = (a, 2a). With only
one vortex (a), the induced velocity Va would be
Problem 8.75
Using the four-source image pattern needed to construct the flow near a corner shown in
Fig. P8.72, find the value of the source strength m which will induce a wall velocity of 4.0 m/s at
the point (x, y) = (a, 0) just below the source shown, if a = 50 cm.
Solution 8.75
The flow pattern is formed by four equal sources m in the 4 quadrants, as in the figure at right.
Problem 8.76
Use the method of images to approximate the flow pattern past a cylinder at distance 4a from the
wall, as in Fig. P8.76. To illustrate the effect of the wall, compute the velocities at corresponding
points A, B, C, and D, comparing with a cylinder flow in an infinite expanse of fluid.
Solution 8.76
Let doublet #1 be above the wall, as shown, and let image doublet #2 be below the wall, at
(x, y) = (0, −5a). Then, at any point on the y-axis, the total velocity is
Problem 8.77
Discuss how the flow pattern of Prob. 8.58 might be interpreted to be an image system
construction for circular walls. Why are there two images instead of one?
Problem 8.58
Plot the streamlines due to the combined flow of a line sink –m at the origin, plus line sources +m at
(a, 0) and (4a, 0). [Hint: A cylinder of radius 2a appears.]
Solution 8.77
The missing “image sink” in this problem is at y = + so is not shown. If the source is placed
Problem 8.78*
Indicate the system of images needed to construct the flow of a uniform stream past a Rankine
half-body constrained between two parallel walls, as in Fig. P8.78. For the particular dimensions
shown in this figure, estimate the position of the nose of the resulting half-body.
Solution 8.78
A body between two walls is created by an infinite array of sources, as shown at right. The
Problem 8.79
Indicate the system of images needed to simulate the flow of a line source placed
unsymmetrically between two parallel walls, as in Fig. P8.79. Compute the velocity on the
lower wall at x = a. How many images are needed to establish this velocity to within
1 percent?
Solution 8.79
To form a wall at y = 0 and also at y = 3a, with the source located at (0, a), one needs an infinite
Problem 8.80*
The beautiful expression for lift of a two-dimensional airfoil, Eq. (8.59), arose from applying the
Joukowski transformation,
=+2
z /z,a
where
z x iy=+
and
=
+ i
The constant a is a
length scale. The theory transforms a certain circle in the z plane into an airfoil in the
plane.
Taking a = 1 unit for convenience, show that (a) a circle with center at the origin and radius >1
will become an ellipse in the
plane, and (b) a circle with center at
1, 0,xy
= − − =
and
radius
(1 )
+
will become an airfoil shape in the
plane. [Hint: The Excel spreadsheet is
excellent for solving this problem.]
Solution 8.80
Introduce z = x + iy into the transformation and find real and imaginary parts:
Problem 8.81*
Given an airplane of weight W, wing area A, aspect ratio AR, and flying at an altitude where the
density is
. Assume all drag and lift is due to the wing, which has an infinite-span drag
coefficient CD. Further assume sufficient thrust to balance whatever drag is calculated.
(a) Find an algebraic expression for the best cruise velocity Vb, which occurs when the ratio of
drag to speed is a minimum. (b) Apply your formula to the data in Prob. P7.119, for which a
laborious graphing procedure gave an answer Vb 180 m/s.
Problem 7.119
A transport plane has a mass of 45,000 kg, a wing area of 160 m2, and an aspect ratio of 7.
Assume all lift and drag due to the wing alone, with CD = 0.020 and CL,max = 1.5. If the aircraft
flies at 9,000 m standard altitude, make a plot of drag (in N) versus speed (from stall to 240 m/s)
and determine the optimum cruise velocity (minimum drag per unit speed).
Solution 8.81*
The drag force, for a finite aspect ratio, is given by
2
2
2
2
( ) and
2
L
DL
CW
D C V A C VA
= + =
AR
D
Problem 8.82
The ultralight plane Gossamer Condor in 1977 was the first to complete the Kremer Prize figure-
eight course under human power. Its wingspan was 29 m, with Cav = 2.3 m and a total mass of
95 kg. Its drag coefficient was approximately 0.05. The pilot was able to deliver 1/4 horsepower
to propel the plane. Assuming two-dimensional flow at sea level, estimate (a) the cruise speed
attained, (b) the lift coefficient; and (c) the horsepower required to achieve a speed of 15 knots.
Solution 8.82
For sea-level air, take
= 1.225 kg/m3. With CD known, we may compute V:
Problem 8.83
The world’s largest plane, the Airbus A380, has a maximum weight of 1,200,000 lbf, wing area
of 9100 ft2, wingspan of 262 ft, and CDo = 0.026. When cruising at maximum weight at
35,000 ft, the four engines each provide 70,000 lbf of thrust. Assuming all lift and drag are due
to the wing, estimate the (a) cruise velocity, in mi/h. and (b) the Mach number.
Solution 8.83
At 35,000 ft = 10,668 m, from Table A.6, ρ ≈ 0.379 kg/m3 = 0.000736 slug/ft3. Stay in BG units.
The aspect ratio = (262)2/9100 = 7.54. Write formulas for the lift, and the drag = thrust:
Problem 8.84
Reference 12 contains inviscid theory calculations for the upper and lower surface velocity
distributions V(x) over an airfoil, where x is the chordwise coordinate. A typical result for small
angle of attack is as follows.
x/c
V/U (upper)
V/U (lower)
0.0
0.0
0.0
0.025
0.97
0.82
0.05
1.23
0.98
0.1
1.28
1.05
0.2
1.29
1.13
0.3
1.29
1.16
0.4
1.24
1.16
0.6
1.14
1.08
0.8
0.99
0.95
1.0
0.82
0.82
Use these data, plus Bernoulli’s equation, to estimate (a) the lift coefficient; and (b) the angle of
attack if the airfoil is symmetric.
Solution 8.84
From Bernoulli’s equation, the surface pressures may be computed, whence the lift coefficient
then follows from an integral of the pressure difference:
Problem 8.85
A wing of 2 percent camber, 5-in chord, and 30-in span is tested at a certain angle of attack in a
wind tunnel with sea-level standard air at 200 ft/s and is found to have lift of 30 lbf and drag of
1.5 lbf. Estimate from wing theory (a) the angle of attack, (b) the minimum drag of the wing and
the angle of attack at which it occurs, and (c) the maximum lift-to-drag ratio.
Solution 8.85
For sea-level air take
= 0.00238 slug/ft3. Establish the lift coefficient first:
Problem 8.86
An airplane has a mass of 20,000 kg and flies at 175 m/s at 5000-m standard altitude. Its
rectangular wing has a 3-m chord and a symmetric airfoil at 2.5 angle of attack. Estimate (a) the
wing span; (b) the aspect ratio; and (c) the induced drag.
Solution 8.86
For air at 5000-m altitude, take
= 0.736 kg/m3. We know W, find b:
Problem 8.87
A freshwater boat of mass 400 kg is supported by a rectangular hydrofoil of aspect ratio 8,
2 percent camber, and 12% thickness. If the boat travels at 7 m/s and
= 2.5°, estimate (a) the
chord length; (b) the power required if CD = 0.01, and (c) the top speed if the boat is refitted
with an engine which delivers 20 hp to the water.
Solution 8.87
For fresh water take
= 998 kg/m3. (a) Use Eq. (8.82) to estimate the chord length:
3
22
2.5
2 sin 2(0.02) 998 kg/m
57.3 (7 m/s) (8 ) ,
2 1 2/8 2
L
Lift C V bC C C
+
==
+
LD
C 0.0526, C 0.0101, 1.69 , . (c)Ans
= = = − max 19.8 m/ s 44 mi / h
=V
Problem 8.88
The Boeing 787-8 Dreamliner has a maximum weight of 502,500 lbf, a wingspan of 197 ft, a
wing area of 3501 ft2, and cruises at 567 mi/h at 35,000 ft altitude. When cruising, its overall
drag coefficient, based on wing area, is about 0.027. Estimate (a) the aspect ratio; (b) the lift
coefficient; and (c) the engine thrust needed when cruising.
Solution 8.88
At 35,000 ft = 10,668 m, from Table A.6, ρ ≈ 0.379 kg/m3 = 0.000736 slug/ft3. Stay in BG units.
Convert 567 mi/h = 831.6 ft/s.
Problem 8.89
The Beechcraft T-34C airplane has a gross weight of 5500 lbf, a wing area of 60 ft2, and flies at
322 mi/h at 10000 feet standard altitude. It is driven by a propeller which delivers 300 hp to the
air. Assume for this problem that its airfoil is the NACA 2412 section described in Figs. 8.23
and 8.24 and neglect all drag except the wing. What is the appropriate aspect ratio for the wing?
Solution 8.89
At 10000 ft = 3048 m,
0.00176 slug/ft3. Convert 322 mi/h = 472 ft/s. From the weight and
power we can compute the lift and drag coefficients:
Problem 8.90
NASA is developing a swing-wing airplane called the Bird of Prey [37]. As shown in Fig. P8.90,
the wings pivot like a pocketknife blade: forward (a), straight (b), or backward (c). Discuss a
possible advantage for each of these wing positions. If you can’t think of any, read the article
[37] and report to the class.