Winds in Complex Terrain - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 79

( 1987 ). Some of the linear approaches are quite old and date back to work of, e.g.,
Jackson and Hunt ( 1975 ). These analytical approaches have always been accom-
panied by numerical efforts, see e.g., the work of Taylor ( 1977 ). Also the well-
known WAsP model is based on such linear analytical approaches (Troen and
Petersen 1989 ).

4.2.1 Potential Flow

The simplest case for a description of flow over a hill is frictionless potential flow.
This implies a laminar flow of a non-viscous fluid with no surface friction. It is
presented here in order to present an analytical model that shows first-order effects
of flow over hills. The main feature is the speed-up of the wind speed over the hill,
a slight wind speed reduction upstream of the hill and a considerable reduction of
the wind speed over the downwind slope of the hill.
For a flow perpendicular to a two-dimensional ridge (i.e. a ridge which is
infinitely long in the direction perpendicular to the flow), the speed-up of the
potential flow over the hill can be described using the thin airfoil theory (Hoff
1987 ):

Du pot ð x ; z Þ¼ u 1 ð L Þ H

L ; z

x

L r

ð 4 : 1 Þ

L

where x is the direction perpendicular to the ridge, z is the vertical coordinate, H is
the height of the ridge, L is the half-width of the ridge (the distance from the crest to
the place where the height is H/2), u ? (L) is the scaling wind speed in the undis-
turbed flow at height L. Therefore, all heights in this simple model scale with L. r is
the form function of the ridge cross-section. H/L is the aspect ratio of the ridge and
describes the magnitude of the slope. Adding ( 4.1 ) to the undisturbed flow,
u ? (z) yields for the wind profile in the potential flow over the ridge:
u pot ð x ; z Þ¼ u 1 ð z Þþ u 1 ð L Þ H

x

L ; z

L r

ð 4 : 2 Þ

L

In contrast to all wind profile relations given in Chap. 3 , the wind profile
relation ( 4.2 ) does not only depend on the vertical coordinate but also contains a
horizontal coordinate. The form function r can be given analytically as long as the
ridge cross-section h(x) can be described by the inverse polynom (see Fig. 4.4 ):

¼

x
L

1
1 þ L 2

h ð 4 : 3 Þ

The associated form function r for this ridge cross-section ( 4.3 ) reads: