fluid_mechanics

6.6 Superposition of Basic, Plane Potential Flows 303
SOLUTION
(a) The velocity at the stagnation point is zero so the Bernoulli
equation written between a point on the stagnation streamline upstream
from the cylinder and the stagnation point gives
Thus,
(Ans)
COMMENT A measurement of the difference between the
pressure at the stagnation point and the upstream pressure can
be used to measure the approach velocity. This is, of course, the
same result that was obtained in Section 3.5 for Pitot-static tubes.
(b) If the direction of the fluid approaching the cylinder is not
known precisely, it is possible that the cylinder is misaligned by
some angle, a. In this instance the pressure actually measured,
p a , will be different from the stagnation pressure, but if the misalignment
is not recognized the predicted approach velocity, U¿,
would still be calculated as
Thus,
p 0
g U 2
2g p stag
g
U c 2 12
r 1 p stag p 0 2d
U¿ c 2 12
r 1 p a p 0 2d
U1true2
U¿1predicted2 ap stag p 12 0
b
p a p 0
The velocity on the surface of the cylinder, v u , where r a, is obtained
from Eq. 6.115 as
v u 2U sin u
(1)
If we now write the Bernoulli equation between a point upstream
of the cylinder and the point on the cylinder where r a, u a,
it follows that
p 0 1 2 rU 2 p a 1 2 r12U sin a22
and, therefore,
p a p 0 1 2 rU 2 11 4 sin 2 a2
Since p stag p 0 1 2 rU 2 it follows from Eqs. 1 and 2 that
U1true2
U¿1predicted2 11 4 sin2 a2 1 2
(2)
(Ans)
This velocity ratio is plotted as a function of the misalignment angle
a in Fig. E6.8c.
COMMENT It is clear from these results that significant errors
can arise if the stagnation pressure tap is not aligned with the
stagnation streamline. As is discussed in Section 3.5, if two additional,
symmetrically located holes are drilled on the cylinder, as
are illustrated in Fig. E6.8d, the correct orientation of the cylinder
can be determined. The cylinder is rotated until the pressures in
the two symmetrically placed holes are equal, thus indicating that
the center hole coincides with the stagnation streamline. For
b 30° the pressure at the two holes theoretically corresponds to
the upstream pressure, p 0 . With this orientation a measurement of
the difference in pressure between the center hole and the side
holes can be used to determine U.
An additional, interesting potential flow can be developed by adding a free vortex to the
stream function or velocity potential for the flow around a cylinder. In this case
c Ur a1 a2
r 2b sin u
2p ln r
(6.119)
and
f Ur a1 a2
r
2b cos u
2p u
(6.120)
Flow around a rotating
cylinder is
approximated by
the addition of a
free vortex.
where is the circulation. We note that the circle r a will still be a streamline 1and thus can be
replaced with a solid cylinder2, since the streamlines for the added free vortex are all circular. However,
the tangential velocity, v u , on the surface of the cylinder 1r a2 now becomes
v us 0c
0r ` 2U sin u
ra
2pa
(6.121)
This type of flow field could be approximately created by placing a rotating cylinder in a uniform
stream. Because of the presence of viscosity in any real fluid, the fluid in contact with the rotating
cylinder would rotate with the same velocity as the cylinder, and the resulting flow field would resemble
that developed by the combination of a uniform flow past a cylinder and a free vortex.