Page 173 - Wind Energy Handbook

UNSTEADY FLOW – DYNAMIC INFLOW 147

the acceleration is independent of yaw angle. The mean acceleration is zero and so
there is no coupling between the cases.
The relationship between accelerations and force coefficients is therefore
2 32 3
16 @a 0
0 0 2 3
6 3ð 76 @ô 7
6 76 7 C T
6 32 76 7 6 7
6 0 0 76 @a c 7 ¼ 4 C my 5 (3:207a)
6 76 7
6 45ð 76 @ô 7
4 54 5
32 @a s C mz D
0 0
45ð @ô

@a
[M] ¼fCg D (3:207b)
@ô
The complete equation of motion combines Equation (3.207) and the steady yaw
Equation (3.188). The combination is achieved by adding the corresponding force
coefficients, which means that both equations must be inverted.

@a 1
[M] þ [L] fag¼fCg D þfCg S (3:208)
@ô
The right-hand side of Equation (3.208) can also be determined from blade element
theory and will be a time-dependent function of the inflow factor. The blade forces
will vary in a manner determined by the time-varying velocity of the oncoming
wind and consequent dynamic structural deflections of the necessarily elastic rotor.
Equation (3.208) applies to the whole rotor disc and the blade element forces need
to be integrated along the blade lengths.
Numerical solutions to Equation (3.208) require a procedure for dealing with
first-order differential equations and the tried and tested fourth-order Runge-Kutta
method is recommended. Starting with a steady-state solution the progress in time
of the induced velocity as an unsteady flow passes through the rotor can be tracked.
However, non-dimensionalizing with respect to wind speed is not very useful if
wind speed is changing dynamically and it is common to work directly in terms of
induced velocity rather than flow factors.
Equation (3.208) really applies to the whole rotor and the only spatial variation of
the induced velocity and acceleration that is permitted is as defined in Equations
(3.182) and (3.202). However, a relaxation of the strict approach has been adopted
by several workers (see, for example, Schepers and Snel, 1995) where the induced
velocities are determined for separate annular rings, as described in Section 3.10.8.
The added mass term for an annular ring can be taken as a proportion of the whole
added mass according to the appropriate acceleration distribution, Equations
(3.198), (3.204) and (3.206).
Figure 3.76 shows measured and calculated flap-wise (out of the rotor plane)
blade root bending moments for the Tjæreborg turbine caused by a pitch change
from 0:0708 to 3:7168 with the reversed change 30 seconds later. The turbine was not
in yaw and the wind speed was 8.7 m=s. The calculated results were made
according to the equilibrium wake method and with a differential equation method