Solution 8.90
Each configuration has a different advantage: (a) highly maneuverable but unstable, needs
Problem 8.91
If
(r,
) in axisymmetric flow is defined by Eq. (8.72) and the coordinates are given in
Fig. 8.28, determine what partial differential equation is satisfied by
.
Solution 8.91
The velocities are related to
by Eq. (8.87), and direct substitution gives
Problem 8.92
A point source with volume flow Q = 30 m3/s is immersed in a uniform stream of speed 4 m/s.
A Rankine half-body of revolution results. Compute (a) the distance from the source to the
stagnation point; and (b) the two points (r,
) on the body surface where the local velocity
equals 4.5 m/s.
Solution 8.92
The properties of the Rankine half-body follow from Eqs. (8.89) and (8.94):
Problem 8.93
The Rankine body of revolution of Fig. 8.30 could simulate the shape of a pitot-static tube
(Fig. 6.30). According to inviscid theory, how far downstream from the nose should the static
pressure holes be placed so that the local velocity is within 0.5 percent of U∞? Compare your
answer with the recommendation x 8D in Fig. 6.30.
Solution 8.93
We search iteratively along the surface until we find V = 1.005U:
Problem 8.94
Determine whether the Stokes streamlines from Eq. (8.73) are everywhere orthogonal to the
Stokes potential lines from Eq. (8.74), as is the case for cartesian and plan polar coordinates.
Solution 8.94
Compare the ratio of velocity components for lines of constant
:
Problem 8.95
Show that the axisymmetric potential flow formed by superposition of a point source +m at
(x,y) = (–a, 0), a point sink-−m at (+a, 0), and a stream U∞ in the x direction forms a Rankine
body of revolution as in Fig. P8.95. Find analytic expressions for determining the length 2L and
diameter 2R of the body in terms of m, U∞ and a.
Solution 8.95
The stream function for this three-part superposition is given below, and the body shape (the
Rankine ovoid) is given by
= 0.
Some numerical values of length and diameter are as follows:
m/Ua2:
0.01
0.1
1.0
10.0
100.0
L/a:
1.100
1.313
1.947
3.607
7.458
R/a:
0.198
0.587
1.492
3.372
7.348
L/R:
5.553
2.236
1.305
1.070
1.015
As m/Ua2 increases, the ovoid approaches a large spherical shape, L/R 1.0.
Problem 8.96
Consider inviscid flow along the streamline approaching the front stagnation point of a sphere, as
in Fig. 8.31. Find (a) the maximum fluid deceleration along this streamline, and (b) its position.
Solution 8.96
Along the stagnation streamline, the flow is purely radial, hence v
= 0. Thus
Problem 8.97
The Rankine body of revolution in Fig. P8.97 is 60 cm long and 30 cm in diameter. When it is
immersed in the low-pressure water tunnel as shown, cavitation may appear at point A. Compute
the stream velocity U, neglecting surface wave formation, for which cavitation occurs.
Solution 8.97
For water at 20C, take
= 998 kg/m3 and pv = 2337 Pa. For an ovoid of ratio
L/R = 60/30 = 2.0, we may interpolate in the Table of Prob. 8.95 to find
Problem 8.98
We have studied the point source (sink) and the line source (sink) of infinite depth into the paper.
Does it make any sense to define a finite-length line sink (source) as in Fig. P8.98? If so, how
would you establish the mathematical properties of such a finite line sink? When combined
with a uniform stream and a point source of equivalent strength as in Fig. P8.98, should a closed-
body shape be formed? Make a guess and sketch some of these possible shapes for various
values of the dimensionless parameter m/(U∞L2).
Solution 8.98
Yes, the “sheet” sink makes good sense and will create a body with a sharper trailing edge. If
q(x) is the local sink strength, then m = q(x) dx, and the body shape is a teardrop which
Problem 8.99*
Consider air flowing past a hemi-sphere resting on a flat surface, as in Fig. P8.99. If the
internal pressure is pi, find an expression for the pressure force on the hemisphere. By
analogy with Prob. 8.49 at what point A on the hemisphere should a hole be cut so that the
pressure force will be zero according to inviscid theory?
Problem 8.49
In strong winds the force in Prob. 8.48 above can be quite large. Suppose that a hole is
introduced in the hut roof at point A to make pi equal to the surface pressure there. At what
angle
should hole A be placed to make the net wind force zero?
Solution 8.99
Recall from Eq. (8.100) that the velocity along the sphere surface is
Problem 8.100
A 1-m-diameter sphere is being towed at speed V in fresh water at 20C as shown in Fig. P8.100.
Assuming inviscid theory with an undistorted free surface, estimate the speed V in m/s at which
cavitation will first appear on the sphere surface. Where will cavitation appear? For this
condition, what will be the pressure at point A on the sphere which is 45 up from the direction
of travel?
Solution 8.100
For water at 20C, take
= 998 kg/m3 and pv = 2337 Pa. Cavitation will occur at the lowest-
pressure point, which is point B on the top of the cylinder, at
= 90:
Problem 8.101
Consider a steel sphere (SG = 7.85) of diameter 2 cm, dropped from rest in water at 20C.
Assume a constant drag coefficient CD = 0.47. Accounting for the sphere’s hydrodynamic mass,
estimate (a) its terminal velocity; and (b) the time to reach 99 percent of terminal velocity.
Compare these to the results when hydrodynamic mass is neglected, Vterminal 1.95 m/s and t99%
0.605 s, and discuss.
Solution 8.101
For water take
= 998 kg/m3. Add hydrodynamic mass to the differential equation:
Problem 8.102
A golf ball weighs 0.102 lbf and has a diameter of 1.7 in. A professional golfer strikes the ball at
an initial velocity of 250 ft/s, an upward angle of 20, and a backspin (front of the ball rotating
upward). Assume that the lift coefficient on the ball (based on frontal area) follows Fig. P7.108. If the
ground is level and drag is neglected, make a simple analysis to predict the impact point (a)
without spin and (b) with backspin of 7500 r/min.
Solution 8.102
For sea-level air, take
= 0.00238 slug/ft3. (a) If we neglect drag and spin, we just use classical
Problem 8.103
Consider inviscid flow past a sphere, as in Fig. 8.31. Find (a) the point on the front surface
where the fluid acceleration amax is maximum; and (b) the magnitude of amax. (c) If the stream
velocity is 1 m/s, find the sphere diameter for which amax is 10 times the acceleration of gravity.
Comment.
Solution 8.103
Along the sphere surface, the flow is purely tangential, hence vr
= 0. Thus
Problem 8.104
Consider a cylinder of radius a moving at speed U through a still fluid, as in Fig. P8.104. Plot
the streamlines relative to the cylinder by modifying Eq. (8.32) to give the relative flow with
K = 0. Integrate to find the total relative kinetic energy, and verify the hydrodynamic mass of a
cylinder from Eq. (8.91).