Page 189 - Wind Energy Handbook

DEFINITION OF LIFT AND ITS RELATIONSHIP TO CIRCULATION 163

A3.6 Definition of Lift and its Relationship to
Circulation

The lift on a body immersed is defined as the force on the body in a direction
normal to the flow direction. Lift will only be present if the flow incorporates a
circulatory flow about the body such as that which exists about a spinning circular
cylinder. If the fluid also has a uniform velocity U past the cylinder, the resulting
flow field is as shown in Figure A3.12. The velocity above the cylinder is increased,
and so the static pressure there is reduced. Conversely, the velocity beneath is
slowed down, giving an increase in static pressure. There is clearly a normal force
upwards on the cylinder, a lift force.
The phenomenon is known as the Magnus effect after its original discoverer and
explains, for example, why spinning tennis balls veer in flight. The circulatory flow,
shown in Figure A3.13, is generated by skin friction and has the same structure as
that of a vortex.
The lift force is given by the Kutta-Joukowski theorem called after the two
pioneering aerodynamicists who, independently, realized that this was the key to
the understanding of the phenomenon of lift:

L ¼ r(ˆ 3 U) (A3:5)

where ˆ is the circulation, or vortex strength, around the cylinder, defined as the
integral
ð
ˆ ¼ v ds (A3:6)

around any path enclosing the cylinder and v is the velocity tangential to the path s.
For convenience choosing a circular path of radius r around the cylinder, and

U L

Ω

Figure A3.12 Flow Past a Rotating Cylinder