Page 89 - Wind Energy Handbook

ROTOR BLADE THEORY 63

It is convenient to put

C l cos ö þ C d sin ö ¼ C x

and

C l sin ö C d cos ö ¼ C y

Solving Equations 3.48 and (3.50a) to obtain values for the flow induction factors a
and a9 using two-dimensional aerofoil characteristics requires an iterative process.
The following equations, derived from (3.48) and (3.50a), are convenient in which
the right-hand sides are evaluated using existing values of the flow induction
factors yielding simple equations for the next iteration of the flow induction factors.

a ó r ó r 2
¼ (C x ) C y (3:51)
2
1 a 4 sin ö 2 4 sin ö

a ó r C y
¼ (3:52)
1 þ a9 4 sin ö cos ö

Blade solidity ó is defined as total blade area divided by the rotor disc area and is a
primary parameter in determining rotor performance. Chord solidity ó r is defined
as the total blade chord length at a given radius divided by the circumferential
length at that radius.
N c N c
ó r ¼ ¼ (3:53)
2ð r 2ðì R

It is argued by Wilson and Lissaman (1974) that the drag coefficient should not be
included in Equations (3.51) and (3.52) because the velocity deficit caused by drag is
confined to the narrow wake which flows from the trailing edge of the aerofoil.
Furthermore, Wilson and Lissaman reason, the drag-based velocity deficit is only a
feature of the wake and does not contribute to the velocity deficit upstream of the
rotor disc. The basis of the argument for excluding drag in the determination of the
flow induction factors is that, for attached flow, drag is caused only by skin friction
and does not affect the pressure drop across the rotor. Clearly, in stalled flow the
drag is overwhelmingly caused by pressure. In attached flow it has been shown by
Young and Squire (1938) that the modification to the inviscid pressure distribution
around an aerofoil caused by the boundary layer has an affect both on lift and drag.
The ratio of pressure drag to total drag at zero angle of attack is approximately the
same as the thickness to chord ratio of the aerofoil and increases as the angle of
attack increases.
One last point about the BEM theory: the theory is strictly only applicable if the
blades have uniform circulation, i.e., if a is uniform. For non-uniform circulation
there is a radial interaction and exchange of momentum between flows through
adjacent elemental annular rings. It cannot be stated that the only axial force acting