Page 151 - Aerodynamics for Engineering Students

134 Aerodynamics for Engineering Students

This equation differs from that of the non-spinning cylinder in a uniform stream of
the previous section by the addition of the term (r/(2nUu)) = B (a constant), in the
squared bracket. This has the effect of altering the symmetry of the pressure dis-
tribution about a horizontal axis. This is indicated by considering the extreme top
and bottom of the cylinder and denoting the pressures there by p~ and p~ respect-
ively. At the top p = p~ when 8 = 7r/2 and sin 8 = 1. Then Eqn (3.49) becomes
1
PT -PO =-pU2(1 - [2+B]’)
2
1
= --pU2(3+4B+BZ) (3.50)
2
At the bottom p = p~ when 8 = -n/2 and sin O = - 1 :
1
PB -PO = --pU2(3 -4B+BZ) (3.51)
2
Clearly (3.50) does not equal (3.51) which shows that a pressure difference exists
between the top and bottom of the cylinder equal in magnitude to

which suggests that if the pressure distribution is integrated round the cylinder then a
resultant force would be found normal to the direction of motion.

The normal force on a spinning circular cylinder in a uniform stream
Consider a surface element of cylinder of unit span and radius a (Fig. 3.24). The area
of the element = a68 x 1, the static pressure acting on element = p, resultant
force = (p - po)a 68, vertical component = (p - po)a Sf3 sin 6.
4
Substituting for (p - po) from Eqn (3.49) and retaining the notation B = I? 27rUa, the
vertical component of force acting on the element = 4 pU2[ 1 - (2 sin 8 + B) ]a 66 sin 8.
The total vertical force per unit span by integration is (Zpositive upwards):

Z=12T-fpU2a[l - (2~in8+B)~]sinOdO

which becomes
2 I”
I=--pU a [sin8(1-BZ)-4Bsin28-4sin38]d0

Fig. 3.24 The pressure and velocity on the surface of unit length of a cylinder of radius a