Page 168 - Wind Energy Handbook

142 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

dynamic flow conditions at the rotor disc. It could be assumed that the induced
velocity remains fixed at the level determined by the average wind speed blowing
over a set period of time, that may be quite short. The wake remains frozen while
the unsteady component of the wind passes through the rotor disc unattenuated by
the presence of the rotor. The unsteady forces, that would impose zero net force on
the rotor, would be determined by the blade element theory. Alternatively, the
induced velocity through the rotor disc could be determined from the instantaneous
wind velocity as if that velocity was steady. The induced velocity will change as the
wind speed changes but it must be assumed that the entire wake changes instanta-
neously to remain in step. Equilibrium in the wake is maintained at all times. The
truth lies somewhere between the two scenarios given above, both of which rely on
simple assumptions about the state of the wake.
The acceleration potential method avoids reference to the wake and allows the
flow conditions at the rotor disc to be determined by the upwind flow field, which
is much simpler to determine than that of the wake.
In steady-flow conditions the velocity at a fixed point in the upwind flow field is
constant; acceleration of the flow from point to point takes place (e.g., U 1 (@u=@x)in
the x-direction, assuming u, the induced velocity, is much smaller than U 1 ) but no
rate of change of velocity with time (e.g., @u=@t) occurs at a single point. In
unsteady flow, conditions at a fixed point do change with time and the total
acceleration in the x-direction is then (@u=@t) þ U 1 (@u=@x). The additional accelera-
tion requires an additional inertia force the reaction to which will change the force
on the rotor disc. The additional force is often termed the added mass force because
if the unsteadiness in the relative flow past a blade can be attributed not to flow
turbulence but to an unsteady motion of the blade itself some of the air will be
forced to move (accelerate) with the blade, effectively adding to the mass of the
blade.

3.13.2 Adaptation of the acceleration potential method to unsteady
flow

If the unsteady acceleration terms are added to Equations (3.145), which are
simplified to account for the induced velocities being very much smaller than the
wind velocity, then those equations become

@u @u @ p
r þ U 1 ¼
@t @x @x

@v @v @ p
r þ U 1 ¼ (3:191)
@t @x @ y

@w @w @ p
r þ U 1 ¼
@t @x @z
As before, differentiating each equation with respect to its particular direction and
adding together the results gives