Page 150 - Aerodynamics for Engineering Students

Potential flow 133

3.3.10 A spinning cylinder in a uniform flow
This is given by the stream function due to a doublet, in a uniform horizontal flow,
with a line vortex superimposed at the origin. By adding these cases
P I’r
$=-sine- Uy--1n-
2rr 2r ro
Converting to homogeneous coordinates

but from the previous case d m = a, the radius of the cylinder.
Also since the cylinder periphery marks the inner limit of the vortex flow, ro = a;
therefore the stream function becomes:
(3.46)

and differentiating partially with respect to r and 0 the velocity components of the
flow anywhere on or outside the cylinder become, respectively:
@
qt =--= Usin0
ar (3.47)
1w
q ---=ucos0
n-r 8tI
and

4 = dq; + 4:

On the surface of the spinning cylinder r = a. Therefore,
qn = 0
r
qt = 2U sin 0 + - (3.48)
27ra
Therefore
r
q = qt = 2U sin 0 + -
2ra
and applying Bernoulli’s equation between a point a long way upstream and a point
on the cylinder where the static pressure is p:

Therefore

1- 2sin0f-
p-po=-pU 2[ ( 2rua >’ (3.49)
2