Page 155 - Wind Energy Handbook

THE METHOD OF ACCELERATION POTENTIAL 129

d m
m
2 m=2
Q (v) ¼ (1 v ) Q n (v)
n m
dv
m
2
But, from Equation (3.153), Q (v) ¼1 at v ¼ 1 which is physically inapplicable
n
and so these functions are excluded from the solution.
For solutions of Equation (3.149b) for non-zero values of m
d m
2 m=2
m
P (iç) ¼ (1 þ ç ) diç m P n (iç)
n
m
Inspection reveals that P (iç) !1 as ç !1 which means that pressure would be
n
infinite in the far field which is not physically acceptable therefore these terms will
also not be included.
d m
m
2 m=2
Q (iç) ¼ (1 þ ç ) Q n (iç) (3:157)
n diç m
Equations (3.156) and (3.157) are known as associated Legendre polynomials.
The solution to differential Equation (3.149c) is more straight forward than for the
other two governing equations
Ö 3 (ł) ¼ cos mł, sin mł (3:158)

The complete solution, therefore, for the pressure field surrounding a rotor disc is

N
X X
M
m
m
m
m
p(v, ç, ł) ¼ P (v)Q (iç)(C cos mł þ D sin mł) (3:159)
n n n n
m¼0 n¼m
The upper limits M and N can have any positive integer value.
m
The polynomial Q (iç) is imaginary for odd values of m and real for even values
n
m
m
therefore the arbitrary constants C and D must be real or imaginary, accordingly,
n
n
in order that the pressure field be real.
Any combination of terms in Equation (3.159) can be used, whatever suits the
conditions. For there to be a pressure discontinuity across the disc, but continuously
varying pressure elsewhere the solutions must be restricted to those for which
n þ m is odd. Of course, limiting the number of terms, other than has been
described, may result in an approximate solution.
The pressure discontinuity across the rotor disc will be as shown in Figure 3.2.
The magnitude of the step in pressure will be twice the pressure level (above the far
field level) that occurs just upstream of the disc. The pressure gradient, however,
normal to the rotor disc, will be continuous.
3.11.3 The axi-symmetric pressure distributions
For the wind turbine rotor disc the simplest situation is for m ¼ 0 which means that
the pressure distribution is axisymmetric. The permitted values of n must be odd.