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www.risoe.dk
High Meteorology: Wind throughout the boundary-layer
Sven-Erik Gryning, Hans E Jørgensen, Poul Astrup, Lars Landberg
Wind Energy Department
Risø National Laboratory, Denmark
EWEC 06, Athens
Wind profiles over flat homogeneous terrainMap of the Høvsøre site at the west coast of Jutland with measuring sector shown
EWEC 06, Athens
30 degrees 60 degrees90 degrees
views from the mast
EWEC 06, Athens
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):
EWEC 06, Athens
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
EWEC 06, Athens
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
EWEC 06, Athens
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
EWEC 06, Athens
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
EWEC 06, Athens
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
EWEC 06, Athens
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
EWEC 06, Athens
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
EWEC 06, Athens
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)
EWEC 06, Athens
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*
EWEC 06, Athens
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
EWEC 06, Athens
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 ∞
EWEC 06, Athens
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)
EWEC 06, Athens
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)
EWEC 06, Athens
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
EWEC 06, Athens
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
EWEC 06, Athens
• 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
EWEC 06, Athens
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.