Page 170 - Wind Energy Handbook

144 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

for example, and so the complete Jacobian can be determined:
2 3 2 32 3
@ @x0 @ y9 @z @
6 @v 7 6 @v @v @v 76 7
6 7 6 76 @x0 7
6 7 6 76 7
6 @ 7 6 @x0 @ y9 @z 76 @ 7
6 7 ¼ 6 76 7 (3:193)
6
6 @ç 7 6 @ç @ç @ç 7 @ y9 7
6 7 6 76 7
4 @ 5 4 @x0 @ y9 @z 54 @ 5
@ł @ł @ł @ł @z
the inverse of which is what is required:
2 3 2 32 3
@ @v @ç @ł @
6 @x0 7 6 @x0 @x0 @x0 76 @v 7
6 7 6 76 7
6 7 6 @ç 76 7
6 @ 7 6 @v @ł 76 @ 7
6 7 ¼ 6 76 7 (3:194)
6
6 @ y9 7 6 @ y9 @ y9 @ y9 7 @ç 7
6 7 6 76 7
4 5 4 54 5
@ @v @ç @ł @
@z @z @z @z @ł
The Jacobian matrix of Equation (3.193) can be determined algebraically from
Equations (3.147) and this can then be inverted algebraically to give the inverse
Jacobian of Equation (3.194). From the inverse Jacobian it is found that
2
2
@ ç(1 v ) @ v(1 þ ç ) @
¼ þ (3:195)
2
2
2
2
@x0 R(ç þ v ) @v R(ç þ v ) @ç
However, only the acceleration at the rotor disc itself is required and there the value
of ç is zero so
@ 1 @
¼ (3:196)
@x0 Rv @ç
If the corrected axi-symmetric pressure drop distribution of Equation (3.166) is
chosen, to conform with the steady flow case then, for the whole flow field,
2 1 3
2
7ç tan 1 þ 4(1 v )
6 ç 7
6 7
6 7
15 6 1 7
6
2 2
p(v, ç) ¼ vC TD þ 15v ç ç tan 1 1 7 (3:197)
6
7
32 6 ç 7
6 7
4 1 5
2
2
þ 9çç þ (v ç )tan 1
ç
2
in which the pressure is normalized by (1=2)rU . The term C TD is the contribution
1
to the total thrust coefficient of the dynamic acceleration @u=@t. Note that, as
explained at the end of Section 3.11.1, the pressure level just upwind of the rotor
disc, as given by Equation (3.197), is half the magnitude of the pressure drop across
the disc given by Equation (3.166).