Page 126 - Wind Energy Handbook

100 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

supports the weight of the aircraft and the horizontal component constitutes drag.
In horizontal flight the vertical component of the lift does no work but the drag
does do work.
The vector triangles of Figure 3.49 show that

D u
¼ (3:93)
L V
In the wake of the aircraft the induced velocity u w caused by the trailing vortices
is greater than that at the rotor. A certain mass flow rate of air, rVS, where S is an
area, normal to velocity V, yet to be determined, undergoes a downward change in
velocity u w in the far wake. By the momentum theory the rate of change of
downward momentum is equal to the lift, therefore

L ¼ rVSu w (3:94)
The rate of work done by the drag DV must be equal to the rate at which kinetic
1
2
energy is created in the wake ru VS, because the ambient static pressure in the
2 w
wake of the aircraft is the same as the pressure ahead of the aircraft.
1 2
DV ¼ ru VS (3:95)
w
2
Combining Equations (3.93), (3.94) and (3.95) gives
L 2
D ¼ (3:96)
2
2rV S
and

u w ¼ 2u (3:97)

Equation (3.97) should look familiar. Combining Equation (3.92), the lifting line
theory’s assessment of the induced velocity at the rotor, with Equation (3.93) gives

2L 2
D ¼ (3:98)
2
rV ð(2R) 2
Comparing Equations (3.96) and (3.98) leads to an estimate of the required area S

S ¼ ðR 2 (3:99)

S has the same area as the rotor disc but is normal to the flight direction.
Note that the above analysis has been simplified by assuming that the angle of
attack is small. Actually, the trailing vortices from the rotor are influenced by their
own induced velocity and so trail downwards behind the rotor. The induced
velocity must therefore have a forward component, which means that the air