fluid_mechanics

300 Chapter 6 ■ Differential Analysis of Fluid Flow
Potential Flow
Viscous Flow
A doublet combined
with a uniform flow
can be used to represent
flow around
a circular cylinder.
V6.6 Circular
cylinder
around a long slender body is described, whereas for small values of the parameter, flow around a
more blunt shape is obtained. Downstream from the point of maximum body width the surface pressure
increases with distance along the surface. This condition 1called an adverse pressure gradient2 typically
leads to separation of the flow from the surface, resulting in a large low pressure wake on the downstream
side of the body. Separation is not predicted by potential theory 1which simply indicates a symmetrical
flow2. This is illustrated by the figure in the margin for an extreme blunt shape. Therefore, the potential
solution for the Rankine ovals will give a reasonable approximation of the velocity outside the thin,
viscous boundary layer and the pressure distribution on the front part of the body only.
6.6.3 Flow around a Circular Cylinder
As was noted in the previous section, when the distance between the source–sink pair approaches zero,
the shape of the Rankine oval becomes more blunt and in fact approaches a circular shape. Since the
doublet described in Section 6.5.4 was developed by letting a source–sink pair approach one another,
it might be expected that a uniform flow in the positive x direction combined with a doublet could be
used to represent flow around a circular cylinder. This combination gives for the stream function
and for the velocity potential
c Ur sin u K sin u
r
f Ur cos u K cos u
r
(6.110)
(6.111)
In order for the stream function to represent flow around a circular cylinder it is necessary that
c constant for r a, where a is the radius of the cylinder. Since Eq. 6.110 can be written as
c aU K r 2b r sin u
it follows that c 0 for r a if
V6.7 Ellipse
υ θs
U
2
1
0
0
± π 2
θ
± π
which indicates that the doublet strength, K, must be equal to Ua 2 . Thus, the stream function for
c Ur a1 a2
r 2b sin u (6.112)
flow around a circular cylinder can be expressed as
and the corresponding velocity potential is
A sketch of the streamlines for this flow field is shown in Fig. 6.26.
The velocity components can be obtained from either Eq. 6.112 or 6.113 as
and
v r 0f
0r 1 0c
r 0u
v u 1 r
f Ur a1 a2
r 2b cos u
0f
0u 0c 0r
U K a 2 0
a2
U a1
r 2b cos u
a2
U a1
r 2b sin u
On the surface of the cylinder 1r a2 it follows from Eq. 6.114 and 6.115 that v r 0 and
v us 2U sin u
(6.113)
(6.114)
(6.115)
As shown by the figure in the margin, the maximum velocity occurs at the top and bottom of
the cylinder 1u p22 and has a magnitude of twice the upstream velocity, U. As we move
away from the cylinder along the ray u p2 the velocity varies, as is illustrated in Fig. 6.26.