Page 152 - Wind Energy Handbook

126 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

@v @ p
rU 1 ¼ (3:145b)
@w @ y
and

@w @ p
rU 1 ¼ (3:145c)
@x @z
Differentiating each momentum equation with respect to its particular direction
and adding together the results gives
!
2
2
2
@ @u @v @w @ p @ p @ p
rU 1 þ þ ¼ þ þ
@x @x @ y @z @x 2 @ y 2 @z 2
but, for continuity of the flow,
@u @v @w
þ þ ¼ 0,
@x @ y @z
therefore

2
2
2
@ p @ p @ p
þ þ ¼ 0 (3:146)
@x 2 @ y 2 @z 2
which is the Laplace equation governing the pressure field on and surrounding the
actuator disc. Given the boundary conditions at the actuator disc Equation (3.146)
can be solved for the pressure field and, in particular, the pressure distribution at
the disc. The pressure is continuous everywhere except across the disc surfaces
where there is the usual pressure discontinuity, or pressure drop, in the wind
turbine case.
In Coleman’s analysis (1945) the pressure drop distribution across the disc is
uniform (it is only as a result of combining the theory with the blade element theory
that a non-uniform pressure distribution can be achieved) but falls to zero, abruptly,
at the disc edge. Kinner assumes that the pressure drop is zero at the disc edge and
changes in a continuous manner as radius decreases.
The simplified Euler Equations (3.145) allow pressure to be regarded as the
potential field from which the acceleration field can be obtained, by differentiation,
and thence the velocity field, by integration. Commencing upstream where the
known free-stream conditions apply the velocity components can be determined by
progressive integration towards the disc.

3.11.2 The general pressure distribution theory of Kinner

Kinner’s solution (1937) is mathematically complex and is achieved by means of a
co-ordinate transformation. The Cartesian co-ordinates centred in the rotor plane