Escarpment induced flow perturbations, a comparison of measurements and theory

Escarpment induced flow perturbations, a comparison of measurements and theory

Journal of Wind Engineering and Industrial Aerodynamics, 15 (1983) 243--251 243 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Nether...

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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.