fluid_mechanics

TABLE 6.1
Summary of Basic, Plane Potential Flows
6.6 Superposition of Basic, Plane Potential Flows 295
Description of
Velocity
Flow Field Velocity Potential Stream Function Components a
Uniform flow at
angle a with the x
axis 1see Fig.
6.16b2
f U1x cos a y sin a2
c U1 y cos a x sin a2
u U cos a
v U sin a
Source or sink
1see Fig. 6.172
m 7 0 source
m 6 0 sink
Free vortex
1see Fig. 6.182
7 0
counterclockwise
motion
6 0
clockwise motion
Doublet
1see Fig. 6.232
f m c m v r
m
2p ln r
2p u
2pr
f
2p u c
2p ln r
v r 0
v u 0
f K cos u
r
c K sin u
r
v u
2pr
v r K cos u
r 2
v u K sin u
r 2
a Velocity components are related to the velocity potential and stream function through the relationships:
u 0f
.
0x 0c
v 0f
0y 0y 0c v
0x r 0f
0r 1 0c
v
r 0u u 1 0f
r 0u 0c 0r
provides a useful representation of some flow fields of practical interest. For example, we will
determine in Section 6.6.3 that the combination of a uniform flow and a doublet can be used to
represent the flow around a circular cylinder. Table 6.1 provides a summary of the pertinent
equations for the basic, plane potential flows considered in the preceding sections.
6.6 Superposition of Basic, Plane Potential Flows
As was discussed in the previous section, potential flows are governed by Laplace’s equation, which
is a linear partial differential equation. It therefore follows that the various basic velocity potentials
and stream functions can be combined to form new potentials and stream functions. 1Why is this
true?2 Whether such combinations yield useful results remains to be seen. It is to be noted that any
streamline in an inviscid flow field can be considered as a solid boundary, since the conditions
along a solid boundary and a streamline are the same—that is, there is no flow through the boundary
or the streamline. Thus, if we can combine some of the basic velocity potentials or stream functions
to yield a streamline that corresponds to a particular body shape of interest, that combination can
be used to describe in detail the flow around that body. This method of solving some interesting
flow problems, commonly called the method of superposition, is illustrated in the following three
sections.
Flow around a
half-body is
obtained by the
addition of a source
to a uniform flow.
6.6.1 Source in a Uniform Stream—Half-Body
Consider the superposition of a source and a uniform flow as shown in Fig. 6.24a. The resulting
stream function is
c c uniform flow c source
Ur sin u m 2p u
(6.97)