Page 153 - Aerodynamics for Engineering Students

136 Aerodynamics for Engineering Students

respectively, occurring on the horizontal axis. As r is increased positively a small
amount the stagnation points move down below the horizontal axis.
Since from the equation for the velocity anywhere on the surface
r
qt = 2U sin 0 + - 0 at the stagnation points
=
27ra
8 = arc sin(-r/47raU)

which is negative. As r is further increased a limiting condition occurs when
0 = -42, i.e. I' = 47raU, the stagnation points merge at the bottom of the cylinder.
When I' is greater than 47raU the stagnation point (S) leaves the cylinder. The cylinder
continues to rotate within the closed loop of the stagnation streamline, carrying
round with it a region of fluid confined within the loop.

3.3.1 1 Bernoulli's equation for rotational flow

Consider fluid moving in a circular path. Higher pressure must be exerted from the
outside, towards the centre of rotation, in order to provide the centripetal force. That
is, some outside pressure force must be available to prevent the particles moving in a
straight line. This suggests that the pressure is growing in magnitude as the radius
increases, and a corollary is that the velocity of flow must fall as the distance from the
centre increases.
With a segmental element at P(r, 0) where the velocity is qt only and the pressurep,
the pressures on the sides will be shown as in Fig. 3.26 and the resultant pressure
thrust inwards is

(. + % g) (. + $) se - (. - % $) (r - $) se -p sr se

which reduces to

(3.53)
This must provide the centripetal force = mass x centripetal acceleration
= pr sr se &Ip (3.54)

Fig. 3.26