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Fig. 4.4 Ridge function h(x/L)(full line) and form function r(x/L,0)(dashed line). Ridge height
H and half-width
L are indicated as well
1 þ L 2 L 2
1 þ L
¼
L ; z
x
r
ð 4 : 4 Þ
2
2 þ L 2
L
For the position of the ridge crest (x = 0) we obtain the following special
relation:
¼
r 0 ; z
L
1
1 þ L
ð 4 : 5 Þ
2
Equation 4.5 describes the decrease of the form function with height that is a
function of the half-width of the ridge only. The wider the ridge the higher up the
hill influences the flow. The vertical profile of the potential flow speed over the
ridge crest is thus:
u pot ð 0 ; z Þ¼ u 1 ð z Þþ u 1 ð L Þ H
L
1
1 þ L
ð 4 : 6 Þ
2
This vertical wind profile function ( 4.6 ) is unrealistic when approaching the
surface, because potential flow is without friction and therefore the flow speed in
potential flow does not vanish at the surface. Rather, the contrary is the case and
the potential flow speed is at its maximum at the ridge crest. There we have
(x = 0, z = 0) r = 1 and
u pot ð 0 ; 0 Þ¼ u 1 ð 0 Þþ u 1 ð L Þ H
L
ð 4 : 7 Þ
Equation 4.7 means that the speed-up of the wind speed over a ridge crest is
proportional to the slope of the flanks of the ridge. The form function ( 4.4 ) cannot
be given analytically for a Gaussian-shaped hill. Numerical integration yields a
slightly lower value than for the function given in ( 4.4 ) with r Gauss (0,0) = 0.939.
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