Page 171 - Wind Energy Handbook

UNSTEADY FLOW – DYNAMIC INFLOW 145

By means of Equation (3.195), at the rotor plane, where ç ¼ 0, the pressure
gradient is found to be

@ p 1 15ð 2
¼ C TD (9v 7)
@x0 R 64

Therefore, in terms of parameter ì, from Equation (3.192)

@u @ p 1 2 1 15ð 2 1 2
r ¼ rU 1 ¼ C TD (9ì 2) rU 1 (3:198)
@t @x0 2 R 64 2

It should be noted that the axial acceleration distribution is axi-symmetric and
independent of the yaw angle.
The mean value of axial acceleration over the area of the disc is

@u o 75ð U 2 1
¼ C TD (3:199)
@t 256 R
The non-dimensional form of the acceleration can be expressed as

@a o ¼ R @u o ¼ 75ð
@ô U 2 1 @t 256 C TD (3:200)

where a o ¼ u o =U 1 , axial flow factor and ô ¼ tR=U 1 which is called non-dimen-
sional time.
The axial force on the disc is

1 2 2
F x ¼ rU ðR C TD
1
2
Substituting for C TD from Equation (3.199) gives

128 3 @u o
F x ¼ rR (3:201)
75 @t

3
The added mass is, therefore, (128=75)rR .
The added mass term associated with a solid disc is 8=3 (Tuckerman, 1925),
compared with 128=75 given in Equation (3.201), is in agreement with the value that
is given by the uncorrected axi-symmetric Kinner pressure distribution of Equa-
tions (3.160). Although Pitt and Peters (1981) determine the value 128=75 in subse-
quent papers by Peters and other workers the value 8=3 is recommended and has
come to be generally accepted (see Schepers and Snel, 1995). The use of the so-called
‘corrected’ pressure distribution for wind turbines has already been questioned in
Section 3.11.3; there is no need to impose a zero pressure difference on the rotor
disc at the rotation axis.