Journal of Wind Engineering and Industrial Aerodynamics, 15 (1983) 243--251
243
Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
ESCARPMENT
INDUCED FLOW PERTURBATIONS,
A COMPARISON OF MEASUREMENTS
AND
THEORY
N.O. JENSEN Meteorology
Section,
Ris~, 4000 Roskilde,
Denmark
SUMMARY Observations of surface-layer wind profiles over a ramp-like escarpment were compared with the Jackson and Hunt analytic theory. In agreement with earlier claims, the maximum perturbations agree with the theory, but it was found that the vertical and downwind variations do not agree as well. This conclusion is supported by previously published wind tunnel results. A reason for the observed discrepancy is suggested.
NOTATION h L AS u ~u u, x z z°
escarpment height (see Fig. i) inner layer depth (see equation (3)) escarpment length scale (see Fig. I) fractional speedup (Au/u) mean wind speed, far upstream perturbation on u friction velocity horizontal distance (see Fig. i) height above ground aerodynamic surface roughness length shape function (see equations (3) and (4))
Other symbols are defined as they are encountered
in the text
i. INTRODUCTION The paper considers perturbations as it encounters flow. height,
a two-dimensional
The idealized situation h, and horizontal
to the neutral
surface layer mean flow
ramp-like escarpment
is depicted
perpendicular
to the
in Fig. i, which also defines the
length scale, L, of an escarpment. Z
~ U
i --
h
.~-~-.~///////////
..~r.f77X//// ,','l.....
", ~ 7//////,4<,,r ~ / / , " "'. i "1 " l .. L
, ,'/ll
x Z
I a-I
I Fig. i. Idealized ramp-like escarpment length scales.
016%6105/83/$03o00
showing definitions
© 1983 Elsevier Science Publishers BoV.
of height and
244
Results from measurements Peterson et al.
in the atmospheric
surface layer reported by
[1,2] are analyzed and compared to the analytic solution of
Jackson and Hunt
[3] and Jackson
[4].
The conclusions
supported by results from wind tunnel measurements Previous analyses by Bowen and Pearse Hunt prediction
[5].
[6] have shown that the Jackson-
for the maximum perturbation
taken just above crests.
drawn are further
by Bowen and Lindley
velocity agrees with measurements
The present analyses confirm this, but do, on the
other hand, show that the theory is not as good for predicting
overspeeding
away from this point. The comparisons (defined below) up factor, theory,
between observations
instead of absolute values of the perturbation,
AS = Au/u.
Au, or speed-
The reason for this choice is that the o-factor,
is independent
roughness,
and theory are kept in terms of ~
of local conditions
and absolute
such as the ramp slope,
in
terrain
size of the ramp; o can be considered as a scaled
speedup factor.
2. THEORETICAL
RESULTS
The well-known Jackson-Hunt
model consists of an outer inviscid laye!r and
an inner layer of depth £ in which the turbulence
is in local equilibrium
(a mixing length closure is used in the latter layer). layers,
the velocity perturbation
Au(z) at height z
By matching the two
of order £ ,
relative
to the undisturbed
upstream wind speed u(z) at the same height,
becomes
(their eq.
)
(3.23b)
h I£nL/Zo] 2 As
=
L~n--r~-~oJ
o~
where z
,
(J)
is the aerodynamic
roughness
length,
and where ~ is a function
that
o depends on the shape of the escarpment curved function)
and the horizontal
0
=
i ~
(whether it is a ramp or another more
but not on the actual value of h/L. coordinate
(x/L+½)2
+ (z/L)2
(x/L-½)2
+ (z/L)2
x.
For a ramp Jackson
o also depends on z [4] gives (2)
~n
The depth of the inner layer is given implicitly gn £/z
=
aL
as (3)
,
O
where a is a constant of order one, given in Jackson and Hunt as 2 K 2 = 0.32 (the von Karman constant to be interpreted very rigorously
< = 0.40), but which is not supposed
[4].
The above results are only valid for quite gentle slopes. slopes the speedup ceases to depend on h/L or the slope. Lindley
[5], in a wind tunnel investigation
For steep
Thus Bowen and
using h/L = ~, i, ½, h, got,
245
in all four cases, a maximum fractional speedup factor (above the crest) of about 0.7.
This might be explained by the presence of an effective minimum
slope introduced by the separation stream line on the windward side of the crest. In another way, the slope and L lose their importance as factors determining the local speedup when sufficiently far downstream of the escarpment. For x > L ~> z equation L -~x
=
(2) reduces to (4)
,
which, as pointed out by Jensen and Peterson x replaces L in equation AS
~
-
[7], has the consequence that
(i) in determining the effective slope,
whereby
h x
(5)
-
irrespective of the shape of the escarpment.
3. RIS~ DATA During 1974 and 1975, observations were taken of the wind profile development along the flow inland from a stretch of the coastline on the Ris~ peninsula, situation
in what was thought to be a simple roughness change
(Fig. 2).
Analyses of these data are reported in [i] and further
in [7] with special emphasis on the influence of the coastal escarpment. Typical profiles are shown in Fig. 3.
N ROSKILDE FJORD
/ Most
~, ~Oj
Fig. 2. Map of Ris~ showing the positions of Mast I, 2 and 3, after Peterson et al. [i].
246
RIS~ SITE 17 OCTOBER 1974 0505 - 0645
/
3.5
~ ~e"
4.0
/ t
1 2
5.0
MEAN
WIND
AT BEACH 50m INLANO
6.0 SPEED
70
(ms -I )
Fig. 3. T y p i c a l w i n d profile for flow from the N N W along the line of masts, o b s e r v e d d u r i n g 1974-75. This p a r t i c u l a r o b s e r v a t i o n is a l O 0 - m i n a v e r a g e b e g i n n i n g at 0505, 17 October 1974. R i c h a r d s o n n u m b e r s m e a s u r e d b e t w e e n the 2 and 12 m levels at m a s t i, 2, and 3, r e s p e c t i v e l y , are 0.010, -0.013, -0.016. F r o m P e t e r s o n et al. [2].
The speedup given in [7], however, being calculated with reference
is about a factor of 2 noo high,
to the c o a s t l i n e mast
(#i); an that time
was not a p p r e c i a t e d that the mast 1 profile was subject
no a reduction
speed due to the a d v e r s e p r e s s u r e g r a d i e n t u p s t r e a m of the hill;
Lt
in
i.e., the
r e v e r s e of the s i t u a t i o n w h i c h causes speedup at the cresn. In a later e x p e r i m e n t on the same site in 1978 the m a s t c o n f i g u r a t l o n was s l i g h t l y changed: mast
m a s t 3 w a s abandoned, mast
(#0) was placed in the fjord about
2 was m a d e taller and a
I00 m u p s t r e a m of mast i.
Typical
p r o f i l e s f r o m this c o n f i g u r a t i o n are shown in Fig. 4. The terrain of the site r e s e m b l e s the idealized ramp m e n t i o n e d above with h = 1.6 m and L = 50 m.
D o w n s t r e a m of mast 2 the ground is level.
The
s i m i l a r i t y w i t h the m o d e l is not perfect though, as we are d e a l i n g w i t h real terrain
(see Fig. 5).
W e believe,
however,
that the "small scale"
u n d u l a t i o n s on the slope h a v e little effect on the flow a b o v e a height of 5 m (the a v e r a g e l e n g t h of u n d u l a t i o n s ) .
Furthermore,
the e f f e c t i v e
a m p l i t u d e of the u n d u l a t i o n s was decreased by the p r e s e n c e of v e g e t a t i o n of v a r i o u s heights,
in all tending to make the slope quite even.
However, w e
limit d i s c u s s i o n to the region above 5 m for other reasons e x p l a i n e d below. A m o r e serious d e f i c i e n c y of the present data w o u l d seem to be the s i m u l t a n e o u s p r e s e n c e of a r o u g h n e s s change.
We believe, however,
that this
can b e n e g l e c t e d at h e i g h t s larger than the intemnal b o u n d a r y layer induced
247
I
I
I
RIS~ SITE
/
IC
7 I-I(.9 LLI
"1-
/1./ ~,,~f
/
35
I
•
4.0
-
•
/
MAST 0
100 m OFFSHORE
I
I
5.0
6.0
7.0
MEAN WIND SPEED (ms -1)
Fig. 4. Typical wind profile for flow from NNW along the line of masts observed during summer 1978. This particular observation is a 100-min average beginning at 1135 on i0 July 1978. Slightly unstable flow (Ri ~ -10 -2 at mast 0). From Peterson et al. [2].
by the roughness
change
(i.e., above the profile kink w h i c h at mast 2 is at
a height of about 4 m). and in Pearse et al.
Support
for this can be found in Britter
et al.
[8]
[9].
I i
Mast
I
Most
30¢m
2
/
/ to 2 O~ ~keig~t(m)
30cm
06 O8 ~0 I.Z ~4
'~
de~ess~ worn m o s t base
2
1.6 Oislance from center of most
2 (m)
Fig. 5. Cross section of the Ris~ escarpment. It should be stressed that the contour shown is just one cross section, and, furthermore, that the contours were smoothed by the presence of vegetation.
248
An explanation for this is that a roughness change only produces a very small pressure perturbation which is insignificant compared to other terms in the equation of motion 6, is such that £n 6/z
[2,10,11] when the internal boundary layer height,
Thus a roughness change (from smooth to rough) o will not cause a speedup in u at higher levels in compensation for the slowdown near the ground.
> I.
Rather, a weak vertical motion,
typically of the
order of u,, is induced. In all we believe that the Ris~ situation warrants a direct comparison with theory as outlined above, and certainly is sufficiently similar to the ideal situation dealt with by the theory for a qualitative judgement to be made of the model performance.
In this connection we shall use data from
the 5 m level and above. Table 1 gives profiles corresponding to Fig. 4 and speedup according to Un(Z) - u ° ( z ) u (z) o
AS
where subscripts at
sufficient
values
(b) '
refer
to mast number.
distanceupstream
Fortunately, but because a choice.
computation
It
roughness
sensitive
however,
to two,
are
choice
computed.
values
If
the other
further
gives
h a s t o b e made r e g a r d i n g
(at
situation
least
i s made ( i . e .
(Zo),
we h a d t o make
in the near
of
on w h i c h b a s i s
would be reduced by a factor
~.
to the boundary condition
in the definition
((~nL/Zo)/(£nE/Zo)) 2 equal
is measured
o
The t a b l e
change in the present
seems reasonable,
choose the upstream
u
(1).
an a s s u m p t i o n
~ is not particularly of the roughness
assume that
to be unperturbed.
of o computed from equation
To do t h e l a t t e r
We t h u s
field)
to
~, a nd i n s o d o i n g we g e t the
~ values
in Table
1
the downstream roughness)
the
of 0.8.
4. DISCUSSION The f i r s t
thing
5 and 8 m levels values The
to note
from T a b l e 1 i s m o d e r a t e l y
between the
theoretical
of o both above the crest variation
of o with
and symmetry predicted
height,
by theory.
predications
and a b o v e t h e
foot
however, differs At t h e f o o t
good a g r e e m e n t
at
the
and the measured of the ramp. both
in the magnitude
of the hill,
the
speed
perturbation decreases with height faster than theory, while the opposite is true at the crest.
Thus at the 12 m level above the foot the observed value
of a is more than a factor of 3 smaller than the theoretical, whereas at the crest it is almost independent of height to the top level of 26 m, where the theoretical figure has decreased by a factor of 3. We believe that this deficiency of the Jackson and Nunt model is caused by the poor turbulence closure used,
in which advection and divergence of
turbulence transport are neglected, e.g.
~(p---~-rw')/~z,where p' is the
deviation from average pressure and w' is vertical velocity.
Such terms
249
generally have a smoothing effect on the field and also introduce a measure of sluggishness in the flow response.
TABLE 1 Wind profiles corresponding to Fig. 4 (i h average), as well as values of slowdown (AS 1 at mast i) and speedup (AS 2 at mast 2) relative to mast 0. Wind speeds in parenthesis are found by least square fit of u = gn(z) to the four measured points at mast 0, which gave u = 0.57 ~n(z) + 4.71, corresponding to u, = 0.23 m/s (friction velocity) and z ° = 0.26 mm. The table gives corresponding values of the shape function, o, calculated from equation (i), as well as theoretical values from equation (2) denoted oj.
z
u (m/s)
(m) 1 2 3 5 8 12 18 26
AS 1
#0
#i
#2
4.73 (5.10) 5.32 5.59 (5.89) 6.14 (6.36) (6.57)
4.47 4.82 5.02 5.35 5.73 6.09
3.58 4.62 5.28 5.78 6.07 6.35 6.61 6.74
AS 2
oI
o2
%
-4.29 -2.72 -0.81
3.40 3.06 3.42 3.93 2.59
-0.67 -0.42 -0.13
0.53 0.48 0.53 0.61 0.40
o J (+ or -)
0.74 0.59 0.46 0.35 0.26
A qualitative argument can be suggested along these lines to explain the difference between theory and observation.
At the base of the escarpment,
the term, p'w', would tend to be positive because of the increase in static pressure
(slowdown effect) and the vertical velocity induced by the slope.
Due to the limited vertical range of this effect the vertical gradient of p'w' would be negative. for example,
From the turbulence kinetic energy budget,
[ii]) this would mean production of turbulence.
(see,
The rate of
increase of turbulent energy in a parcel flowing past this point would therefore become larger than expected from local equilibrium length)
theory.
(mixing
As a consequence, mixing would be larger than expected,
inducing more momentum in the lower layers and thus reducing the amount of slowdown.
At the crest diffusion caused by the extra turbulence would cause
the speedup region to be extended in the vertical direction.
This
asymmetry in slowdown and speedup is also seen in the wind tunnel measurements of Bowen and Lindley
[5], of which an example is shown in Fig. 6.
Another difference between theory and measurements is found in the downwind behaviour of the amount of overspeeding.
According to equation (2) or
(4), o should be reduced to 0.32, 0.16 and 0.ii at distances of 25, 75, and 125 m downwind of mast 2; and between mast 2 and mast 3 the decrease in o should be a factor of about four.
250
1I
//
t
/
1
~o5 F
/
,
J
i
0195
_m_z ,
1:7
0
5
4
3
2
1
1
2
3
5
6
0
(D)
7
8
9
10
4 :1 SLOPE
Fig. 6. Contours of equal amplification factor, (u + A u ) / u = I + A S , where AS = &U/Uo, over a 1:4 escarpment slope. After Bow~n and L~ndley [5].
This is in contrast to the observations:
Fig. 3 does not even show a
decline in the speedup at mast 3 relative to mast 2 (remember that we have to stay above the internal roughness-change boundary layer, which at mast 3 seems to be about i0 m).
Again this observation is in agreement with the
wind tunnel measurements of Bowen and Lindley
[5].
In the latter case the
effect could be caused by blockage in the wind tunnel
[4]; however, we do
not believe that this effect is significant enough to explain the observations.
Bowen and Lindley further make the interesting observation that
the prolonged "&u-wake" propagated upwards. 10h
The observations extend 5 to
downwind, at which stage the Au-maximum has risen to z ~ h.
This
casts doubt on the claim that the inner layer depth scale, ~, should be a constant
[12], but rather that,
in case of an escarpment,
to the "effective length scale" x, such that
~ should respond
~ ~n~/z ° ~ x (replacing
equation (3)) analogous to internal boundary layer growth [13].
5. CONCLUSION The above analysis suggests that the Jackson and Hunt theory, which has had quite a strong influence on the general understanding of flow over topography, does not give an entirely satisfactory explanation of observed flow perturbations.
For the case of flow over a ramp-like escarpment,
theory fails to predict correctly the vertical and downwind variation of the velocity perturbation.
REFERENCES i. E.W. Peterson, L. Kristensen and C.C. Su, Some observations and analysis of wind over non-uniform terrain, Quart. J. Roy. Met. Soc., 102 (1976) 857-869.
251
2.
3. 4.
5.
6.
7. 8.
9.
i0.
Ii.
12.
13.
E.W. Peterson, P.A° Taylor, J. H~jstrup, N.O. Jensen, L. Kristensen and E.L. Petersen, Ris~ 1978, Further investigations into the effect of local terrain irregularities on tower-measured wind profiles, BoundaryLayer Meteorol. 19 (1980) 303-313. P.S. Jackson and J.C.R. Hunt, Turbulent wind flow over a low hill, Quar~ J. Roy. Met. Soc., i01 (1975) 929-955. P.S. Jackson, The influence of local terrain features on the site selection for wind energy generating systems, Report BLWT-I-I079, Faculty of Engineering Science, The University of Western Ontario, London, Ontario, Canada, 1979. A.J. Bowen and D. Lindley, A wind tunnel investigation of the wind speed and turbulence characteristics close to the ground over various escarpment shapes, Boundary-Layer Meteorol., 12 (1977) 259-271. A.J. Bowen and J.R. Pearse, Wind flow over two-dimensional hills: a comparison of experimental data with theory, Proc. Construire avec le vent, CSTB, Nantes, Paper I-2, 1981. N.O. Jensen and E.W. Peterson, On the escarpment wind profile, Quart. J. Roy. Met. Soc., 104 (1978) 719-728. R.E. Britter, J.C.R. Hunt and K.J. Richards, Air flow over a twodimensional hill: Studies of velocity speedup, roughness effects and turbulence, Quart. J. Roy. Met. Soc., 106 (1980) 91-110. J.R. Pearse, D. Lindley and D.C. Stevenson, Wind flow over ridges in simulated atmospheric boundary layers, Boundary-Layer Meteorol., 21 (1981) 77-92. N.O. Jensen, Studies of the atmospheric surface layer during change in surface conditions, Proc. Construire avec le vent, CSTB, Nantes, paper I-4, 1981. E.W. Peterson, Relative importance of terms in the turbulent energy and momentum equations as applied to the problem of a surface roughness change, J. Atmos. Sci., 29 (1972) 1470-1476. J.C.R. Hunt and J.E. Simpson, Atmospheric boundary layers over nonhomogeneous terrain, Engineering Meteorology, E. Plate, Ed., Elsevier, 1982. H.A Panofsky, Tower Micrometeorology, Workshop on Micrometeorology, D.A. Haugen, Ed., Am. Meteorol. Soc., 1973.