Page 153 - Wind Energy Handbook

THE METHOD OF ACCELERATION POTENTIAL 127

0
(x}, y , z), as defined in Figure 3.56, are transformed to, what is termed, an
ellipsoidal co-ordinate system (í, ç, ł), ł is the azimuth angle.

x0 y9 p ffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffi z p ffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffi
2
¼ vç, ¼ 1 v 2 1 þ ç sin ł and ¼ 1 v 2 1 þ ç cos ł (3:147)
2
R R R
p ffiffiffiffiffiffiffiffiffiffiffiffiffi
2
On the surface of the rotor disc ç ¼ 0and r=R ¼ ì ¼ 1 v or, conversely,
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v ¼ 1 ì 2
The transformation separates the variables and allows the pressure field to be
expressed as the product of three functions
p(v, ç, ł) ¼ Ö 1 (v)Ö 2 (ç)Ö 3 (ł) (3:148)
each separate function being the solution of a separate, ordinary differential equa-
tion,

d 2 d m 2
(1 v ) Ö 1 (v) þ n(n þ 1) Ö 1 (v) ¼ 0 (3:149a)
dv dv 1 v 2
" #
d 2 d m 2
(1 ç ) Ö 2 (ç) þ n(n þ 1) Ö 2 (ç) ¼ 0 (3:149b)
dç dç 1 ç 2
d 2 2
dł 2 Ö 3 (ł) þ m Ö 3 (ł) ¼ 0 (3:149c)

where m and n are positive integers.
Equations (3.149a) and (3.149b) have the form of Legendre’s associated differen-
tial equations which has solutions which are called associated Legendre polyno-
mials of the first and second kinds, respectively (see van Bussel, 1995).
If m ¼ 0 then Equations (3.149a) and (3.149b) are reduced to Legendre’s differ-
ential equations the solutions for which are
Ö 1 (v) ¼ P n (v) (3:150a)

and

Ö 1 (v) ¼ Q n (v) (3:150b)

where P n (v) is a Legendre polynomial of the first kind and Q n (ç) is a Legendre
polynomial of the second kind.

1 d n
2
P n (v) ¼ (v 1) n (3:151)
n
2 n! dv n
2
Although the polynomials extend beyond the range v < 1, over that interval the
polynomials are mutually orthogonal