Page 148 - Aerodynamics for Engineering Students

Potential flow 131

way upstream, to flow around the circle and again to revert to uniform flow down-
stream. That inside the circle is from the doublet. This is confined within the circle
and does not mingle with the horizontal stream at all. This inside flow pattern is
usually neglected. This combination is consequently a mathematical device for giving
expression to the ideal two-dimensional flow around a circular cylinder.
The velocity potential due to this combination is that corresponding to a uniform
stream flowing parallel to the Ox axis, superimposed on that of a doublet at the
origin. Putting x = r cos e:

(3.41)
where a = d m is the radius of the streamline $J = 0.
The streamlines can be obtained directly by plotting using the superposition
method outlined in previous cases. Rewriting Eqn (3.39) in polar coordinates

P
$=-sine- Ursine
27rr
and rearranging, this becomes
$J = usine(- P - r)
27rr U
and with p/(27ru> = u2 a constant (a = radius of the circle$ = 0)

= usine($- r) (3.42)

Differentiating this partially with respect to r and 8 in turn will give expressions for
the velocity everywhere, i.e.:

(3.43)
a$
qt = --= Usin8
dr
Putting r = u (the cylinder radius) in Eqns (3.43) gives:
(i) qn = U cos 8 [l - 11 = 0 which is expected since the velocity must be parallel to
the surface everywhere, and
(ii) qt = Usin€J[l + 11 = 2Usine.
Therefore the velocity on the surface is 2U sin e and it is important to note that the
velocity at the surface is independent of the radius of the cylinder.

The pressure distribution around a cylinder
If a long circular cylinder is set in a uniform flow the motion around it will, ideally,
be given by the expression (3.42) above, and the velocity anywhere on the surface by
the formula

q = 2Usin13 (3.44)