Page 169 - Wind Energy Handbook

UNSTEADY FLOW – DYNAMIC INFLOW 143

2 2 2 !
@ @u @v @w @ @u @v @w @ p @ p @ p
r þ þ þ U 1 þ þ ¼ þ þ
@t @x @ y @z @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 the condition
2
2
2
@ p þ @ p þ @ p ¼ 0
@x 2 @ y 2 @z 2
applies together with the Kinner pressure distributions of Section 3.11.2.
The accelerations

@u @v @w
, ,
@t @t @t
can be determined directly from Equations (3.191) without integration being
necessary but it is only the component @u=@t that is required because it is normal to
the rotor disc and so will give rise to a normal force.
The solutions for

@u @ p
rU 1 ¼
@x @x
have already been obtained in Sections 3.11.2–3.11.4 and so it remains to determine
the solutions for
@u @ p
r ¼ (3:192)
@t @x

Equation (3.192) cannot be solved for the velocity u because that is the solution of
the complete equation that is the first of Equations (3.191). What can be determined
from Equation (3.192) is the acceleration @u=@t for which it is necessary to differ-
entiate the chosen pressure distribution. The Kinner pressure distributions, that are
solutions of Equations (3.149), are given as functions of the ellipsoidal co-ordinates
(í, ç, ł) so to obtain the derivative with respect to x a co-ordinate transformation is
required. The relationships between the ellipsoidal co-ordinates and the Cartesian
co-ordinates (x, y, z) are given in Equations (3.157) from which can be obtained the
derivatives @x=@v, @x=@ç, @x=@ł, etc., but what are really needed are the inverses
of these derivatives.
We can find by appropriate differentiations of Equations (3.147)
@ @x0 @ @ y9 @ @z @
¼ þ þ
@v @v @x0 @v @ y9 @v @z