fluid_mechanics

292 Chapter 6 ■ Differential Analysis of Fluid Flow
Arbitrary
curve C
ds
V
F I G U R E 6.20 The notation
for determining circulation around closed
curve C.
A mathematical concept commonly associated with vortex motion is that of circulation. The
circulation, , is defined as the line integral of the tangential component of the velocity taken
around a closed curve in the flow field. In equation form, can be expressed as
≠ ˇ C
V ds
(6.89)
where the integral sign means that the integration is taken around a closed curve, C, in the
counterclockwise direction, and ds is a differential length along the curve as is illustrated in Fig. 6.20.
For an irrotational flow, V f so that V ds f ds df and, therefore,
≠ ˇ C
df 0
This result indicates that for an irrotational flow the circulation will generally be zero. (Chapter 9
has further discussion of circulation in real flows.) However, if there are singularities enclosed
within the curve the circulation may not be zero. For example, for the free vortex with u Kr
the circulation around the circular path of radius r shown in Fig. 6.21 is
The numerical value
of the circulation
may depend on the
particular closed
path considered.
V6.4 Vortex in a
beaker
which shows that the circulation is nonzero and the constant K 2p. However, for irrotational
flows the circulation around any path that does not include a singular point will be zero. This can
be easily confirmed for the closed path ABCD of Fig. 6.21 by evaluating the circulation around
that path.
The velocity potential and stream function for the free vortex are commonly expressed in
terms of the circulation as
and
≠
2p
0
v
K
1r du2 2pK
r
f
2p u
c
2p ln r
(6.90)
(6.91)
The concept of circulation is often useful when evaluating the forces developed on bodies immersed
in moving fluids. This application will be considered in Section 6.6.3.
v θ
ds
B
C
D
A
dθ
θ
r
F I G U R E 6.21
in a free vortex.
Circulation around various paths