fluid_mechanics

326 Chapter 6 ■ Differential Analysis of Fluid Flow
Section 6.6 Superposition of Basic, Plane
Potential Flows
6.59 Obtain a photograph/image of a situation that mimics the superposition
of potential flows (see Ex. 6.7). Print this photo and
write a brief paragraph that describes the situation involved.
6.60 Potential flow against a flat plate 1Fig. P6.60a2 can be described
with the stream function
c Axy
where A is a constant. This type of flow is commonly called a “stagnation
point” flow since it can be used to describe the flow in the
vicinity of the stagnation point at O. By adding a source of strength
m at O, stagnation point flow against a flat plate with a “bump” is obtained
as illustrated in Fig. P6.60b. Determine the relationship between
the bump height, h, the constant, A, and the source strength, m.
6.63 One end of a pond has a shoreline that resembles a half-body
as shown in Fig. P6.63. A vertical porous pipe is located near the
end of the pond so that water can be pumped out. When water is
pumped at the rate of 0.08 m 3 s through a 3-m-long pipe, what will
be the velocity at point A? Hint: Consider the flow inside a halfbody.
(See Video V6.5.)
Pipe
A
y
5 m
15 m
F I G U R E P6.63
(a)
y
O
x
6.64 Two free vortices of equal strength, but opposite direction
of rotation, are superimposed with a uniform flow as shown in
Fig. P6.64. The stream functions for these two vorticies are
c 3;12p24 ln r. (a) Develop an equation for the x-component
of velocity, u, at point P1x,y2 in terms of Cartesian coordinates x
and y. (b) Compute the x-component of velocity at point A and
show that it depends on the ratio H.
U
y
P(x, y)
H
h
x
A
H
x
Source
(b)
F I G U R E P6.60
6.61 The combination of a uniform flow and a source can be used
to describe flow around a streamlined body called a half-body. (See
Video V6.5.) Assume that a certain body has the shape of a halfbody
with a thickness of 0.5 m. If this body is placed in an airstream
moving at 15 m/s, what source strength is required to simulate flow
around the body?
6.62 A vehicle windshield is to be shaped as a portion of a halfbody
with the dimensions shown in Fig. P6.62. (a) Make a scale
drawing of the windshield shape. (b) For a free stream velocity of
55 mph, determine the velocity of the air at points A and B.
y
B
Windshield
r
1.5 ft
U = 55 mph
θ
x
A 2.0 ft
F I G U R E P6.62
F I G U R E P6.64
6.65 A Rankine oval is formed by combining a source–sink pair,
each having a strength of 36 ft 2 s and separated by a distance of 12 ft
along the x axis, with a uniform velocity of 10 fts 1in the positive x direction2.
Determine the length and thickness of the oval.
*6.66 Make use of Eqs. 6.107 and 6.109 to construct a table showing
how /a, ha, and /h for Rankine ovals depend on the parameter
pUam. Plot /h versus pUam and describe how this plot could
be used to obtain the required values of m and a for a Rankine oval
having a specific value of / and h when placed in a uniform fluid
stream of velocity, U.
6.67 An ideal fluid flows past an infinitely long, semicircular
“hump” located along a plane boundary, as shown in Fig. P6.67.
Far from the hump the velocity field is uniform, and the pressure is
p 0 . (a) Determine expressions for the maximum and minimum values
of the pressure along the hump, and indicate where these points
are located. Express your answer in terms of r, U, and p 0 . (b) If
the solid surface is the c 0 streamline, determine the equation of
the streamline passing through the point u p2, r 2a.