Www.risoe.dk High Meteorology: Wind throughout the boundary-layer Sven-Erik Gryning, Hans E Jørgensen, Poul Astrup, Lars Landberg Wind Energy Department.

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High Meteorology: Wind throughout the boundary-layer

Sven-Erik Gryning, Hans E Jørgensen, Poul Astrup, Lars Landberg

Wind Energy Department

Risø National Laboratory, Denmark

[email protected]

EWEC 06, Athens

Wind profiles over flat homogeneous terrainMap of the Høvsøre site at the west coast of Jutland with measuring sector shown

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30 degrees 60 degrees90 degrees

views from the mast

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Measured wind profiles, sector 30 to 90 deg.

8 12 16 20 24 28wind-speed/ustar10 (d im ensionless)

1

10

100

Hei

ght

(m)

-50 to -200 m eters M O-200 to -500 m eters M O-500 to 500 m etres M O500 to 200 m etres M O

H øvsøre - w ind sector 30 to 90 deg.

EWEC 06, Athens

MOL

zbzz

u

zu0

0*

ln1

MOL

z

z

z

u

zu

)ln(1)(

00*

2

arctan22

1ln2

2

1ln

2

x

xx

L

z

MO

4/1161 MOLzx

.

)/ln()( 00* zzu

zu

Commonly used expression for the wind profile

Neutral atmosphere

Stable atmosphere (nighttime)

Unstable atmosphere (daytime)

with the standard stability correction (Businger)based on measuremets at small masts (Kansas experiment):

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Monin-Obukhov wind profilesplanetary boundary layer only, constant flux and based on Businger (-1/4 power)

10 20 30 40 50wind-speed/ustar10 (d im ensionless)

10

100

20

50

200

Hei

ght

(m)

-50 to -100 m eters; M O =-72m-100 to -200 m etres: M O =-148m-200 to -500 m eters; M O =-317m-500 to 500 m etres; M O ~neutra l500 to 200 m etres; M O =329m200 to 50 m etres; M O =107m50 to 10 m etres; M O =28

H øvsøre - w ind sector 30 < d ir < 90 degreesU m in=3 m /s

M O w ith Businger; -1 /4 unstab le fi function

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MOL

z

z

z

u

zu

)ln(1)(

00*

33

21arctan3

3

1ln

2

3 2

xxx

L

z

MO

3/1121 MOLzx

actually, the theoretically correct correction for convective conditions reads:

but for unstable conditions

2

arctan22

1ln2

2

1ln

2

x

xx

L

z

MO

4/1161 MOLzx

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Monin-Obukhov wind profilesplanetary boundary layer only, constant flux and based on convective scaling (-1/3 power)

10 20 30 40 50wind-speed/ustar10 (d im ensionless)

10

100

20

50

200

Hei

ght

(m)

-50 to -100 m eters; M O =-72m-100 to -200 m etres: M O =-148m-200 to -500 m eters; M O =-317m-500 to 500 m etres; M O ~9000m500 to 200 m etres; M O =329m200 to 50 m etres; M O =107m50 to 10 m etres; M O =28

H øvsøre - w ind sector 30 < d ir < 90 degreesU m in=3 m /s

M O w ith -1 /3 fi function

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Wind profile, common knowledgeThe wind profile for the boundary layer can be expressed as:

where is the local friction velocity (proportional to the square root of the local Reynolds stress). The length scale is denoted it is a function of the state of the atmosphere and height

l

u

dz

du *

*ul

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Length scales

The behaviour of the length scale is modelled by inverse summation of the three terms   .

which can be written   .

UBLMBLNsl llll

1111

,

zdzLzl iMBL

1111

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Length scales

0 20 40 60 80Length scale (m )

0

200

400

600

800

1000H

eigh

t (m

)d=0.8d=1

l=0.4z ; M O one term

three term s

tw o term s

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In the atmospheric surface layer (not influenced by and ) the above expression reduces to the logarithmic wind profile: 

MBLl iz

0

0* lnz

zuzu

surface layer

and in the lower part of the boundary layer (not influenced by ) 

iz

MBLL

zzz

uzu 0

0* ln

lower part of the boundary layer

and for the entire boundary layer  

MBLiMBLMO L

zdd

z

z

L

z

L

z

z

z

u

zu

21)ln(

1)(

00*

Neutral

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Stability correctionThe effect of atmospheric stability will be derived as a correction to the wind profile in neutral conditions.

UBLMBLMO llL

zb

zl

111

11

UBLMBLNsl

MOsl

lll

Lzf

l

111

,

neglecting the (unknown) stability dependence on and

MBLl

UBLl

For atmospheric stable conditions, Businger et al. (1971)

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Wind profile - unstable

For atmospheric unstable conditions ( negative )  

MOL

pMOsbl Lzaf /1

where Businger et al. (1971) suggested and   and the theoretical correct value for convective conditions is p= -1/3 and a= -12.  Then the length scale can be expressed as: 

4/1p

   16a

UBLMBLMO llL

z

zl

11121

113/1

MBLiMBLMO

MO

iMO L

zdd

z

z

L

z

Lz

Lz

z

zd

L

z

z

z

u

zu

21

8

11211)ln(

1)( 3/2

00*

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Monin-Obukhov wind profilesplanetary boundary layer only, constant flux and based onconvective scaling (-1/3 power)and constant length scale in the middle layer

10 20 30 40 50wind-speed/ustar10 (dim ensionless)

10

100

20

50

200

Hei

ght

(m)

-50 to -100 m eters; M O =-72m ; M BL=inf.-100 to -200 m etres: M O =-148m ; M BL=800m-200 to -500 m eters; M O =-342m ; M BL=400m-500 to 500 m etres; M O ~4000m ; M BL=160m500 to 200 m etres; M O =329m ; M BL=300m200 to 50 m etres; M O =107m ; M BL=inf.50 to 10 m etres; M O =28; M BL=inf.

H øvsøre - w ind sector 30 < d ir < 90 degreesU m in=3 m /s

M O w ith -1/3 fi function + M BL correction m inus z i

Term s I& II m inus z i

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Conclusion: the wind profile in the lower part of the boundary-layer over homogeneous terrain in near neutral conditions

160

7.4)/ln( 0

0* z

L

zzz

uzu

MO for

0MOL for

Both expressions reduces for neutral condtions, to 

160ln 0

0* zzz

uzu

0MOL

160)ln(

1)(

00*

z

L

z

z

z

u

zu

MO

LMBL(m) 22 107 329 neutral -342 -148 -72

L (m) ∞ ∞ 300 160 400 800 ∞

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0 0.4 0.8 1.2

(u*(z)/u

*(20))2

0

100

200

300

He

ight

(m

)

L=25 m

0 0.4 0.8 1.2N orm alized kinem atic heat flux (m Ks -1)

0

100

200

300

He

ight

(m

)

L=25 m

0 0.4 0.8 1.2

(u*(z)/u

*(20))2

0

100

200

300

He

ight

(m

)

L=97 m

0 0.4 0.8 1.2N orm alized kinem atic heat flux (m Ks -1)

0

100

200

300

He

ight

(m

)

L=97 m

0 0.4 0.8 1.2

(u*(z)/u

*(20))2

0

100

200

300

He

ight

(m

)

N eutra l

Profiles of momentum (left)and kinematic heat flux (below), to determine the boundary layer height

Stable conditions(nighttime, sometimes daytime winter)

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0 0.4 0.8 1.2

(u*(z)/u

*(20))2

0

100

200

300

He

ight

(m

)L=-280 m

0 0.4 0.8 1.2N orm alized kinem atic heat flux (m Ks -1)

0

100

200

300

He

ight

(m

)

L=-280 m

0 0.4 0.8 1.2

(u*(z)/u

*(20))2

0

100

200

300

He

ight

(m

)

L=-139 m

0 0.4 0.8 1.2N orm alized kinem atic heat flux (m Ks -1)

0

100

200

300

He

ight

(m

)

L=-139 m

0 0.4 0.8 1.2

(u*(z)/u

*(20))2

0

100

200

300

He

ight

(m

)

L=-75 m

0 0.4 0.8 1.2N orm alized kinem atic heat flux (m Ks -1)

0

100

200

300

He

ight

(m

)

L=-75 m

Profiles of momentum (far left)and kinematic heat flux (less left), to determine the boundary layer height

Unstable conditions (daytime)

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Boundary layer height estimated from the measured

profiles of momentum and kinematic heat fluxes.Momentum flux profile Kinematic heat flux profile

Monin-Obukhov length, L (m)

Exp(m)

Eq. (X)Power

(m)

Eq. (X)Exp(m)

Power(m)

Mean(m)

25 215 270 138 161 190

97 202 258 154 184 190

362 192 252 210 282 240

Neutral 289 418 300

-280 653 1676 246 312 500

-139 387 734 299 521 500

-75 489 1033 267 423 500

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Which one is the better?

10 20 30 40 50wind-speed/ustar10 (dim ensionless)

10

100

20

50

200H

eigh

t (m

)

-50 to -100 m eters; M O =-72m ; M BL=inf.-100 to -200 m etres: M O =-148m ; M BL=500m-200 to -500 m eters; M O =-317m ; M BL=250m-500 to 500 m etres; M O ~9000 m ; M BL=110m500 to 200 m etres; M O =329 m ; M BL=70m200 to 50 m etres; M O =107m ; M BL=100m50 to 10 m etres; M O =28; M BL=inf.

H øvsøre - w ind sector 30 < d ir < 90 degreesU m in=3 m /s

M O w ith -1 /3 fi function + M BL correction p lus z i

term s I& II p lus zi

10 20 30 40 50wind-speed/ustar10 (dim ensionless)

10

100

20

50

200

Hei

ght

(m)

-50 to -100 m eters; M O =-72m ; M BL=inf.-100 to -200 m etres: M O =-148m ; M BL=1000m-200 to -500 m eters; M O =-342m ; M BL=500m-500 to 500 m etres; M O ~4000m ; M BL=150m500 to 200 m etres; M O =329m ; M BL=120m200 to 50 m etres; M O =107m ; M BL=200m50 to 10 m etres; M O =28; M BL=inf.

H øvsøre - w ind sector 30 < d ir < 90 degreesU m in=3 m /s

M O w ith -1 /3 fi function + M BL correction

term s I& II& III p lus z i

It is clear that the height of the boundary layer is important for thestable cases where the height is about200 metres. But it is not clear how to parameterizethe length scale close to its top.

0 20 40 60 80Length scale (m )

0

200

400

600

800

1000

Hei

ght

(m)

d=1

term s I& II& III term s I& II

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• Above the surface boundary layer the neutral wind profile deviates from logarithmic. It can be argued to be caused by the length scale notbeing proportional to height (as in the surface layer) but approaching a constant value.• Under very convective conditions use of a formulation for the stability correction that fulfills the theoretical requirements for the convective limit is seen to perform better than the commonly used Businger formulation.• Inclusion of the boundary layer height improves the wind profile, the effect was clearly seen during atmospheric the stable conditions where the boundary layer height was only slightly higher than the maximum measuring height. The effect is less well seen during unstable and neutral conditions where the boundary layer height is much higher than he measuring height.• The behaviour of the length scale near the top of the boundary layer is not clear.

Conclusions on wind profiles

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Conclusions on measurements

• The measurements at 160 meters height were of decisive importance for the interpretation of the wind profiles. A 200 metre mast seems appropiate and wishful thinking for the national test station for large wind turbines

• Measurements of the height of the boundary-layer are missing and should be added. Research on how to achieve this parameter should be initiated.