Winds in Complex Terrain - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 81

This implies that the relative speed-up Du/u ? over a 100 m high ridge with a half-
width of 1,000 m is about 10 % or:

Du
u 1

H
L

ð 4 : 8 Þ

4.2.2 Modifications to the Potential Flow: Addition
of an Inner Layer

As said above, the potential flow solution is unrealistic when approaching the
surface, because it produces a solution which is symmetrical to the crest line. The
potential flow solution is valid in an outer layer only. The decrease of the wind
speed towards zero speed at the surface (non-slip condition) takes place in an inner
layer with depth l within which the surface friction dominates. This has led to the
idea of a two-layer model (Jackson and Hunt 1975 ). The depth of the inner layer
depends on the half-width L again. Jackson and Hunt ( 1975 ) derived the following
implicit relation for l:

¼ 2j 2 L

l
z 0

l ln ð 4 : 9a Þ

with the surface roughness length z 0 . Jensen et al ( 1984 ), Mason ( 1986 ), and Hoff
( 1987 ) derived a similar but slightly different relation:

¼ 2j 2 L

l
z 0

l ln 2 ð 4 : 9b Þ

For large values of L/z 0 , the height of the inner layer calculated from Eq. 4.9b
is much smaller than calculated from ( 4.9a ) (see Fig. 4.5 ). Roughly spoken, the
inner layer depth from ( 4.9a ) is of the order of 3-6 % of the half-width of the
ridge (Fig. 4.3 right), or—from Eq. ( 4.9b )—in the order of 1-2 % of the half-
width of the ridge. Experimental data from Taylor et al. ( 1987 ) and Frank et al.
( 1993 ) support the latter formulation ( 4.9b ).
As stated above after Eq. ( 4.6 ) the potential flow solution is unrealistic when
directly approaching the surface. The true wind profile can be described by
matching the potential flow profile ( 4.2 ) for the outer layer above l with the
logarithmic profile ( 3.6 ) for the inner layer:

¼ u 1 ð z Þþ Du ð x ; z\l Þ

u ð x ; z\l Þ¼ u 1 ð z Þþ u 1 ð z Þ ln z 0
ln

H
L r L ; z

x

ð 4 : 10 Þ

l
z 0

L