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Anisotropic turbulence in the atmospheric surface layer
JOURNAL OF
windengineering ELSEVIER
Journal of Wind Engineering and Industrial Aerodynamics 69 71 (1997) 903-913
~ l ~ i ~
Anisotropic turbulence in the atmospheric surface layer P.J. R i c h a r d s a'*, S. F o n g a, R . P . H o x e y b aDepartment of Mechanical Engineering, The University of Auckland, Private Bag 92019, Auckland, New Zealand bAnimal Science and Engineering Division, Silsoe Research Institute, Wrest Park, Silsoe, BedJord MK45 4HS, UK
Abstract Within the atmospheric surface layer the turbulence that effects low rise buildings tends to be highly anisotropic. It is shown that while low level spectra match the expected form in the inertial subrange there also exists a range of frequencies within the energy containing range where the horizontal spectra can be matched by a power law. The gradients of the log-log graphs of spectra in this range tend to diminish with decreased height. This power law range seems to be a bridge between a very low frequency range where horizontal spectra scale in proportion to the square of the mean velocity and frequencies a decade below the low frequency end of the inertial subrange. This power law range is quite narrow at higher levels but broadens at low levels. The level of the normalised power spectral density in the inertial subrange is observed to be similar at all heights and does not vary in the manner suggested by Harris and Deaves. Turbulence spectra obtained in the University of Auckland wind tunnel are shown to exhibit similar patterns to those obtained in full scale. The variation in the size of eddies involved in the generation of turbulence at various heights is illustrated by considering the cospectra. A functional form for the cospectra is proposed. Keywords. Atmospheric turbulence; Spectra
1. Introduction W h i l e investigating the w i n d l o a d s on a low-rise building, the Silsoe Structures Building, s t r e a m w i s e a n d transverse t u r b u l e n c e s p e c t r a were m e a s u r e d using direct i o n a l p i t o t tubes at v a r i o u s heights between 0.32 a n d 25 m, see [1]. It was o b s e r v e d that in the lowest 10 m the t u r b u l e n c e s p e c t r a c h a n g e r a p i d l y with height a n d t h a t the
* Corresponding author. E-mail: [email protected]. 0167-6105/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PII S0 1 67-61 0 5 ( 9 7 ) 0 0 2 1 6-X
904
P.J. Richards et al./'.~ Wind Eng. Ind. Aerodyn. 69 71 (1997) 903 913
measured spectra exhibit significant frequency ranges where the spectral density ean be matched by a power law function of the form S,,.(n) = C , , n
(1)
~
but with the exponent ~ less than the ~ value expected in the inertial subrange and with its value decreasing as the ground is approached. Further investigations using three component sonic anemometers and a two component hot wire anemometer at heights between 0.115 and 10 m [2, 3] have confirmed these observations. This paper discusses in more detail the sonic and hot wire anemometer measurements and illustrates the similarities with turbulence spectra measured in the University of Auckland wind tunnel [4].
2. The atmospheric surface layer The atmospheric surface layer is characterised b___ya logarithmic velocity profile and an almost constant Reynolds shear stress ( The velocity profile at the Silsoe Research Institute has been measured at various times and as illustrated in Fig. 1 all of the recent measurements (since 1988) are well matched by a simple logarithmic profile of the form
puw).
U(z)=~ln(z/zo)
(2)
with a roughness length z0 = 0.01 m. The data used to construct this figure include directional pitot tube measurements at heights between 0.32 and 2 5 m [1],
1.4 1.2 1 0.8 ~" 0.6
5"
0.4 0.2
1E-2
1E-1
1E+0
1E+1
Height z (m)
Fig. 1. Velocity profile at the Silsoe Research Institute.
1E+2
P.J. Richards et al./J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 903 913
905
instrumented kites flown at heights between 25 and 100 m and more recent measurements using a pair of sonic anemometers at heights of0.115, 1.01 and 10 m [2]. In each case two simultaneous measurements were made with one instrument located at the reference height of 10 m and hence the velocities are plotted as a ratio of the velocity at the particular height to that recorded at I0 m during the same time period. On all occasions the mean wind speed at 10 m exceeded 7 m/s and on occasions when spectra were recorded exceeded 9 m/s. The turbulence spectra presented in this paper are therefore assumed to be for neutral atmospheric stability. The Reynolds stresses measured by the sonic anemometers at heights of 1 and 10 m were in good agreement with that estimated by squaring the friction velocity u, deduced from the velocity profile [2]. The Reynolds stress measured at 0.115 m was lower than expected but this can be partially explained by the fact that the instrument was incapable of measuring all relevant frequencies at this height and that the measurements were affected by attenuation at high frequencies.
3. Turbulence spectra at Silsoe Three component turbulence spectra and the uw cospectra were measured at the Silsoe Research Institute at heights of 0.115, 1.01 and 10 m using two 3-component ultrasonic anemometers (Gill Instruments Ltd). The instrument has a path length of 150 m m and transmits digital information at a frequency of 20.8 Hz. The three hour runs were processed as 26.25 min records, each consisting of 32 768 data points. The results presented are the averages of six non-overlapping records. It should be noted that the results are affected by the limiting response of the anemometers at high frequencies. It has been estimated that this attenuation, which is dependent on wind speed, is a 20% loss at 10 Hz at 10 m, a 20% loss at 5.5 Hz at 1.01 m and a 20% loss at 2.5 Hz at 0.115 m. This attenuation is most evident in the rapid fall off of the spectra for 0.115 m at high frequencies. The resulting three component turbulence spectra at the highest and lowest heights are shown in Figs. 2 and 3. The general shape of the 10 m spectra, Fig. 2, is quite normal with local isotropy being achieved at frequencies above 1 Hz and the slope of the inertial subrange (n > 1 Hz) matching the Kolm o g o r o v law with ~ = 2. At frequencies below 1 Hz the turbulence becomes increasingly anisotropic with Suu > Svv >> Sww. In contrast the spectra at 0.115 m shows that the turbulence is anisotropic across the complete range of frequencies measured, however there does exist a significant range of frequencies (0.01 < n < 1 Hz) where both the horizontal spectra can be matched by a power law with an exponent of = 0.83. It may also be noted that in Figs. 2 and 3 the spectral densities have been divided by the local mean velocity squared. This has been done in order to illustrate that at frequencies below 0.01 Hz, where the turbulence is almost two-dimensional and the spacial extent of turbulent fluctuations is of the order of kilometres, the turbulence is affected by the boundary condition at the ground which requires even fluctuating velocity components to be zero at the rough wall. If the ground was a smooth frictionless boundary then only the vertical components would be constrained by the
P.J. Richards et al./J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 903 913
906
I
102
s(r~:*4 10 ~.
s(n:v) • s(n:w)
,°°] 10"
10 ~v]
~
10'-
10 t
10 <
10
10 ~
10 2
10
10 L,
I
10'
Frequency n (Hz) F i g . 2. F u l l - s c a l e t u r b u l e n c e s p e c t r a at z = 10 m.
10 ~ -
s(~:~) s(~:v) 6'(~:~o)
101-
----....
10 °I
........................... '"..
10-'"
10 ~I r./3
"i
C-3
10 3.
10-4I
10 5. 10 '
10 ~
10 "
10
I
10 c'
10'
Frequency n (Hz) F i g . 3. F u l l - s c a l e t u r b u l e n c e s p e c t r a a t z = 0.115 m.
presence of the wall, b u t since the g r o u n d is a rough wall all three c o m p o n e n t s m u s t be zero at the effective g r o u n d plane. As a result both the u a n d v spectra tend to scale such that Saa(n : Z) OC ( l n ( z / z o ) ) z,
a = u or v, low n a n d low z
while the vertical spectra decrease m o r e rapidly.
(3)
P.J. Richards et al./J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 903-913
907
By using a combination of a sonic anemometer and a two component cross hot wire anemometer it has been possible to measure full scale spectra across a broader range of frequencies [-3]. Fig. 4 shows the resulting u and w spectra and uw cospectra at a height of 1 m. It may be noted that while the two instruments have good frequency responses over different ranges the results are very similar in the overlap region. From this figure it is possible to identify three regions: (I) A low frequency region below 0.01 Hz which is sensitive to non-stationary effects and is difficult to define accurately. However both the u and v components scale in proportion to the square of the local mean velocity if they are simultaneously recorded at two heights. (II) A mid-frequency range 0.01 < n < U/z which contains most of the power and is the region of importance for Reynolds stress. Within this region there appears to exist a subrange 0.01 < n < O.14U/z where the u and v spectra can be matched by a power law with 7 < 5/3. (III) The inertial subrange for n > U/z where the turbulence tends to be isotropic and the slope of the log-log spectral density graph is - ~. The power law subrange within region II appears to extend from a frequency of 0.01 Hz up to one decade below the apparent onset ofisotropy. The low frequency end of these ranges as shown by Figs. 2-4 and supported by other data tend towards a c o m m o n point which for the streamwise spectra is
S..(n = 0.01) = U(z) 2.
(4)
The high frequency end of the range can be more easily identified, as shown in Fig. 5 for the streamwise spectra, if the spectra are plotted in terms of the reduced frequency
10
~.
s.o (n)
x
10°
x_..._t~
10-'-
- Cu.(n)
hot wire hot wire
S,~(n)
hot wire
---
sonic
Suu (n)
sonic
- Cu..,,(n)
10_: -
..
. ~,:,....~, ),.~
.
S~(n)
sonic
~.~-, 1 0 ~ -
,
10
-~
%,,,,
.
10_~ rm
10-~. REGI)N
I
R "-'GION
10-'1 0 -~
10 -~
10 2
10-' Frequency
II
REGII
10 °
10'
N III
10 2
n (Hz)
F i g . 4. S o n i c a n d c r o s s h o t w i r e a n e m o m e t e r
spectra at z = 1 m.
10 ~
P.,L Richards et al./J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 903 913
908
f = nz/U(z). In this form the spectra converge on a point at f = 0.14 where
S,,(./= 0.14) = 7u2,.
(5)
Noting that S,,(n) = zS, u(f)/U(z).
(6)
Table 1 summarizes the extent and slope of the anisotropic power law ranges while Fig. 5 shows the limit points and slopes of these ranges. Another point arising from Fig. 5 is the level of the spectra in the high frequency inertial subrange which exists f o r f > 1.0, that is region Ill. As discussed by Tieleman [5], for a neutrally stable boundary layer with turbulence production balanced by dissipation, we may expect S,,(f)/u2, = A , f
5/3 for./> 1.0,
(7)
10 4 -
I
i
S,='~(J')
10 r ~ ......... S~.~(D 1.01 r n
10 ~.
-
-c~= -0.83
10 2 .
J
O. 1 1 5 r a
3~(.f)
,-~n,-~ -a = -1.05
10'-
\
10 %
10-;-
10 -2.,
10 -3_ 10 -s
10-4
10 -s
10 -~
10o
10-'
10'
Reduced frequency f = nz/U Fig. 5. Streamwise spectra in reduced frequency form, showing the limit points and slopes of the anisottopic power law ranges. Table 1 Anisotropic power law ranges for U(10) = 10 m/'s Low frequency limit
High frequency limit
Height z(m)
n
S.o(n)/U z f
SuL,(f)/u2. n
Sou(n)/U 2 f
Su,,(f)//u~ E x p o n e n t - . z ~
0.115 1.01 10
0.01 0.01 0.01
1.0 1.0 1.0
1141 880 298
0.0064 0.0085 0.024
7.0 7.0 7.0
0.00032 0.0015 0.01
4,3 0,93 0.14
0.14 0.14 0.14
0.83 - 1.05 1.4
P.J. Richards et al./J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 903-913
909
where Tieleman gives Au = 0.27 and Av = Aw = 0.36. However Harris and Deaves [6] and E S D U data item 85020 [7] both suggest that these constants should be a function of height at low levels. Harris and Deaves effectively give Au = 0.346 B(z)
(8)
B(z)=l
(9)
where forz>zc,
B(z) = (1 -- (1 -- z/zc)2) 1/2 for z < z~.
(10)
For the conditions at Silsoe zc = 66 m and so at z = 10 m Eq. (8) gives Au = 0.143 and at z = 1.01 m, Au = 0.047. The results shown in Fig. 5 appear to show that at f = 1.0, Suu(f)/u2, = 0.27 for all heights and hence match the values given by Tieleman and do not exhibit the trends given in Eq. (8)~10).
4. Wind tunnel turbulence spectra In order to shed further light on the Silsoe observations a wind tunnel study [4] has been conducted at the University of Auckland. Spires, roughness blocks and a rough floor were used to create a turbulent boundary layer in the 1.8 m × 1.1 m working section. This boundary layer exhibited a logarithmic velocity profile and had an almost constant Reynolds shear stress in the lowest 200 mm. The roughness length was 0.1 m m and hence represents a 1/100th scale model of the Silsoe full scale situation. Turbulence spectra were obtained by using TSI model 1241-T1.5 cross wire probes connected to a TSI model IFA 100 anemometer system. The 8.7 min runs were processed as 32 s records, each containing 65 536 data points. The results presented are the averages of 16 non-overlapping records. The analogue signals were low pass filtered at 1 kHz and sampled at 2 kHz. Some attenuation due to filtering is apparent in the highest decade of the spectra. The similarity between the full sale and wind tunnel spectra is illustrated in Fig. 6. Where the u and w spectra and the uw cospectra obtained at 1.01 m at full scale is superimposed on the wind tunnel spectra at 10 ram. The graphs are presented in reduced frequency form in order to remove the need for scaling. There is generally good agreement between the two sets of spectra with the most noticeable difference occurring at low reduced frequencies where the presence of the tunnel walls suppresses large scale turbulence. The variation in spectral shape, within the energy containing range, with height was also observed in the wind tunnel. This was most obvious when the spectra were plotted in wavenumber form as shown in Fig. 7.
5. Cospectra One feature of the energy containing range is the generation of turbulence. In a turbulent boundary layer the frequency distribution of this generation can be related
P.J. Richards et al./.~ Wind Eng. Ind. Aerodyn. 69 71 (1997) 903 913
910 1E+4
Suu(f)/u'. ~
1E+3
Sww(f)/u'.
1E+2
o
-Cuw(OM.
-
"~
1E+1 1E*0
1E-1 ~)
\
1E-2 1E-3
i
1 E-5
1E-4
1E-3
1 E-2
i
i
F J ~11
1E-1
1E+O
1E+I
Reduced frequency f = nz/U Fig. 6. S t r e a m w i s e and vertical spectra and uw cospectra. Silsoe full scale d a t a at t.01 m and wind tunnel d a t a at 10 mm.
IE+I H e i g h t
z(rrlrrl)
3 1E+O
10
30 100 1E-1
==
1 E-2
--
1E-3
1E-4
i
1E-3
i
i
i illl
i
1 E-2
i
i
IiiiJ
i
1 E-1
~
I
illll
i
1 E+O
~
i
iii
p
i
1 E* 1
i
ii1~1
1 E+2
1E+3
Wavenumber k = n/U (m -1) Fig. 7. W i n d tunnel s t r e a m w i s e spectra at low heights,
to the cospectrum. Fig. 8 illustrates how in full scale the cospectra vary with height. This figure has been plotted in wavenumber form in order to show how at low heights the generation is shifted to higher and higher wavenumbers. That is, at low levels most of the generation is associated with small eddies while the generation associated with larger eddies at higher heights is suppressed by the ground plane.
P.J. Richards et al./J. Wind Eng. Ind. Aerodyn. 69-71 (1997) 903-913
0.6 -¸
_ _ 1 _ _
911
1 I
- c , , , 4 k ) I.o~ ,~ f-c,,,,(~) o.f ~6,~ I
0.50.4-
,li
O.31
L)
/
O.2. 0.1 r
.
"~ '~i[.,~ ,D~I' ' , i,i-:.~l' .....""
0.O
10 -s
10-'
10 -~
I
'7i~,+.
` +
'+'% .~I
, ,~ r " ,
I .........#"
10 -2
'~"~,---,
10-'
10'
10 o
Wavenumber k = nAJ (m 4) Fig, 8. uw cospectra from the Silsoe full scale d a t a in w a v e n u m b e r form.
0,5 -
I
I
0.4-
-c,.,.(.f)
to m
li II 2 Y - ~::~I'°' o.,,5 +,,,
!
l
0,3"
"i o.2.
(.3
•..
0.1'
l
'D;','++'
0.0
10 -s
10-'
10 -~
10 -2
10-'
10 0
10'
Reduced frequency f = nz/U Fig. 9. uw c o s p e c t r a from the S i l s o e f u l l scale d a t a in reduced f f e q u e n c y f o r m .
If the cospectra are plotted in reduced frequency form then, as illustrated in Figs. 9 and 10, both the full scale and wind tunnel cospectra tend to form a single curve. Both of these figures can be approximately m a t c h e d by a function of the form -- Cuw(f) 2
u,
1.0 foE1 + (3f/4fo)] ~/~
(ll)
P.J. Richards et al./J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 903 913
912
0.4
H e i g h t z(rln*11 ) 3 10
0.3
30 100
N~
0.2 0 x
0.1
!%~Lf.~. t.y,r.rT
0.0
1E-5
IE-4
1E-3
1E-2
1E-1
1E+O
1E+l
1E+2
Reduced frequency f = nz/U Fig. 10. uw cospectra from the Auckland wind tunnel data in reduced frequency form.
with f0 = 0.05. This function has a m a x i m u m of 0.27 at f = j 0 and when integrated from f = 0 to o~ returns a value of unity. It also has the correct high frequency form for the inertial subrange.
6. Conclusions Within the atmospheric surface layer the turbulence that effects low rise buildings tends to be highly anisotropic. It has been shown that while low level spectra match the expected form in the inertial subrange there also exists a range of frequencies within the energy containing range where the horizontal spectra can be matched by a power law. The gradients of the log-log graphs of spectra in this range tend to diminish with decreased height. This power law range seems to be a bridge between a very low frequency range where horizontal spectra scale in proportion to the square of the mean velocity and frequencies a decade below the low frequency end of the inertial subrange. This power law range is quite narrow at higher levels but broadens at low levels. The level of the normalised power spectral density in the inertial subrange is observed to be similar at all heights and does not vary in the manner suggested by Harris and Deaves. Turbulence spectra obtained in the University of Auckland wind tunnel are shown to exhibit similar patterns to those observed in full scale. The variation in the size of eddies involved in the generation of turbulence at various heights has been illustrated by considering the cospectra, A functional form for the cospectra was proposed.
P.J. Richards et aL /,L Wind Eng. Ind. Aerodyn. 69-71 (1997) 903 913
913
References l-l] R.P. Hoxey, P.J. Richards, Structure of the atmospheric boundary layer below 25 m and implications to wind loading on low-rise buildings, J. Wind Eng. Ind. Aerodyn. 4 1 4 4 (1992) 317 327. [2] R.P. Hoxey, P.J. Richards, Spectral characteristics of the atmospheric boundary layer near the ground, in: UK Wind Engineering Conf., Cambridge, 1992. [3] R.P. Hoxey, P.J. Richards, Full-scale wind load measurements point the way forward, J. Wind Eng. Ind. Aerodyn. 57 (1995) 215 224. [4] S. Fong, The structure of equilibrium turbulent boundary layers, Master of Engineering Thesis, University of Auckland, New Zealand, 1995. 1-5] H.W. Tieleman, Universality of velocity spectra, J. Wind Eng. Ind. Aerodyn. 56 (1995) 55. 1-6] R.I. Harris, D.M. Deaves, The structure of strong winds, wind engineering in the eighties, in: Proc. CIRIA Conf., 1980. 1-7] ESDU, Characteristics of atmospheric turbulence near the ground, part II: single point data for strong winds (neutral atmosphere), Item 85020, 1985.
windengineering ELSEVIER
Journal of Wind Engineering and Industrial Aerodynamics 69 71 (1997) 903-913
~ l ~ i ~
Anisotropic turbulence in the atmospheric surface layer P.J. R i c h a r d s a'*, S. F o n g a, R . P . H o x e y b aDepartment of Mechanical Engineering, The University of Auckland, Private Bag 92019, Auckland, New Zealand bAnimal Science and Engineering Division, Silsoe Research Institute, Wrest Park, Silsoe, BedJord MK45 4HS, UK
Abstract Within the atmospheric surface layer the turbulence that effects low rise buildings tends to be highly anisotropic. It is shown that while low level spectra match the expected form in the inertial subrange there also exists a range of frequencies within the energy containing range where the horizontal spectra can be matched by a power law. The gradients of the log-log graphs of spectra in this range tend to diminish with decreased height. This power law range seems to be a bridge between a very low frequency range where horizontal spectra scale in proportion to the square of the mean velocity and frequencies a decade below the low frequency end of the inertial subrange. This power law range is quite narrow at higher levels but broadens at low levels. The level of the normalised power spectral density in the inertial subrange is observed to be similar at all heights and does not vary in the manner suggested by Harris and Deaves. Turbulence spectra obtained in the University of Auckland wind tunnel are shown to exhibit similar patterns to those obtained in full scale. The variation in the size of eddies involved in the generation of turbulence at various heights is illustrated by considering the cospectra. A functional form for the cospectra is proposed. Keywords. Atmospheric turbulence; Spectra
1. Introduction W h i l e investigating the w i n d l o a d s on a low-rise building, the Silsoe Structures Building, s t r e a m w i s e a n d transverse t u r b u l e n c e s p e c t r a were m e a s u r e d using direct i o n a l p i t o t tubes at v a r i o u s heights between 0.32 a n d 25 m, see [1]. It was o b s e r v e d that in the lowest 10 m the t u r b u l e n c e s p e c t r a c h a n g e r a p i d l y with height a n d t h a t the
* Corresponding author. E-mail: [email protected]. 0167-6105/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PII S0 1 67-61 0 5 ( 9 7 ) 0 0 2 1 6-X
904
P.J. Richards et al./'.~ Wind Eng. Ind. Aerodyn. 69 71 (1997) 903 913
measured spectra exhibit significant frequency ranges where the spectral density ean be matched by a power law function of the form S,,.(n) = C , , n
(1)
~
but with the exponent ~ less than the ~ value expected in the inertial subrange and with its value decreasing as the ground is approached. Further investigations using three component sonic anemometers and a two component hot wire anemometer at heights between 0.115 and 10 m [2, 3] have confirmed these observations. This paper discusses in more detail the sonic and hot wire anemometer measurements and illustrates the similarities with turbulence spectra measured in the University of Auckland wind tunnel [4].
2. The atmospheric surface layer The atmospheric surface layer is characterised b___ya logarithmic velocity profile and an almost constant Reynolds shear stress ( The velocity profile at the Silsoe Research Institute has been measured at various times and as illustrated in Fig. 1 all of the recent measurements (since 1988) are well matched by a simple logarithmic profile of the form
puw).
U(z)=~ln(z/zo)
(2)
with a roughness length z0 = 0.01 m. The data used to construct this figure include directional pitot tube measurements at heights between 0.32 and 2 5 m [1],
1.4 1.2 1 0.8 ~" 0.6
5"
0.4 0.2
1E-2
1E-1
1E+0
1E+1
Height z (m)
Fig. 1. Velocity profile at the Silsoe Research Institute.
1E+2
P.J. Richards et al./J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 903 913
905
instrumented kites flown at heights between 25 and 100 m and more recent measurements using a pair of sonic anemometers at heights of0.115, 1.01 and 10 m [2]. In each case two simultaneous measurements were made with one instrument located at the reference height of 10 m and hence the velocities are plotted as a ratio of the velocity at the particular height to that recorded at I0 m during the same time period. On all occasions the mean wind speed at 10 m exceeded 7 m/s and on occasions when spectra were recorded exceeded 9 m/s. The turbulence spectra presented in this paper are therefore assumed to be for neutral atmospheric stability. The Reynolds stresses measured by the sonic anemometers at heights of 1 and 10 m were in good agreement with that estimated by squaring the friction velocity u, deduced from the velocity profile [2]. The Reynolds stress measured at 0.115 m was lower than expected but this can be partially explained by the fact that the instrument was incapable of measuring all relevant frequencies at this height and that the measurements were affected by attenuation at high frequencies.
3. Turbulence spectra at Silsoe Three component turbulence spectra and the uw cospectra were measured at the Silsoe Research Institute at heights of 0.115, 1.01 and 10 m using two 3-component ultrasonic anemometers (Gill Instruments Ltd). The instrument has a path length of 150 m m and transmits digital information at a frequency of 20.8 Hz. The three hour runs were processed as 26.25 min records, each consisting of 32 768 data points. The results presented are the averages of six non-overlapping records. It should be noted that the results are affected by the limiting response of the anemometers at high frequencies. It has been estimated that this attenuation, which is dependent on wind speed, is a 20% loss at 10 Hz at 10 m, a 20% loss at 5.5 Hz at 1.01 m and a 20% loss at 2.5 Hz at 0.115 m. This attenuation is most evident in the rapid fall off of the spectra for 0.115 m at high frequencies. The resulting three component turbulence spectra at the highest and lowest heights are shown in Figs. 2 and 3. The general shape of the 10 m spectra, Fig. 2, is quite normal with local isotropy being achieved at frequencies above 1 Hz and the slope of the inertial subrange (n > 1 Hz) matching the Kolm o g o r o v law with ~ = 2. At frequencies below 1 Hz the turbulence becomes increasingly anisotropic with Suu > Svv >> Sww. In contrast the spectra at 0.115 m shows that the turbulence is anisotropic across the complete range of frequencies measured, however there does exist a significant range of frequencies (0.01 < n < 1 Hz) where both the horizontal spectra can be matched by a power law with an exponent of = 0.83. It may also be noted that in Figs. 2 and 3 the spectral densities have been divided by the local mean velocity squared. This has been done in order to illustrate that at frequencies below 0.01 Hz, where the turbulence is almost two-dimensional and the spacial extent of turbulent fluctuations is of the order of kilometres, the turbulence is affected by the boundary condition at the ground which requires even fluctuating velocity components to be zero at the rough wall. If the ground was a smooth frictionless boundary then only the vertical components would be constrained by the
P.J. Richards et al./J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 903 913
906
I
102
s(r~:*4 10 ~.
s(n:v) • s(n:w)
,°°] 10"
10 ~v]
~
10'-
10 t
10 <
10
10 ~
10 2
10
10 L,
I
10'
Frequency n (Hz) F i g . 2. F u l l - s c a l e t u r b u l e n c e s p e c t r a at z = 10 m.
10 ~ -
s(~:~) s(~:v) 6'(~:~o)
101-
----....
10 °I
........................... '"..
10-'"
10 ~I r./3
"i
C-3
10 3.
10-4I
10 5. 10 '
10 ~
10 "
10
I
10 c'
10'
Frequency n (Hz) F i g . 3. F u l l - s c a l e t u r b u l e n c e s p e c t r a a t z = 0.115 m.
presence of the wall, b u t since the g r o u n d is a rough wall all three c o m p o n e n t s m u s t be zero at the effective g r o u n d plane. As a result both the u a n d v spectra tend to scale such that Saa(n : Z) OC ( l n ( z / z o ) ) z,
a = u or v, low n a n d low z
while the vertical spectra decrease m o r e rapidly.
(3)
P.J. Richards et al./J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 903-913
907
By using a combination of a sonic anemometer and a two component cross hot wire anemometer it has been possible to measure full scale spectra across a broader range of frequencies [-3]. Fig. 4 shows the resulting u and w spectra and uw cospectra at a height of 1 m. It may be noted that while the two instruments have good frequency responses over different ranges the results are very similar in the overlap region. From this figure it is possible to identify three regions: (I) A low frequency region below 0.01 Hz which is sensitive to non-stationary effects and is difficult to define accurately. However both the u and v components scale in proportion to the square of the local mean velocity if they are simultaneously recorded at two heights. (II) A mid-frequency range 0.01 < n < U/z which contains most of the power and is the region of importance for Reynolds stress. Within this region there appears to exist a subrange 0.01 < n < O.14U/z where the u and v spectra can be matched by a power law with 7 < 5/3. (III) The inertial subrange for n > U/z where the turbulence tends to be isotropic and the slope of the log-log spectral density graph is - ~. The power law subrange within region II appears to extend from a frequency of 0.01 Hz up to one decade below the apparent onset ofisotropy. The low frequency end of these ranges as shown by Figs. 2-4 and supported by other data tend towards a c o m m o n point which for the streamwise spectra is
S..(n = 0.01) = U(z) 2.
(4)
The high frequency end of the range can be more easily identified, as shown in Fig. 5 for the streamwise spectra, if the spectra are plotted in terms of the reduced frequency
10
~.
s.o (n)
x
10°
x_..._t~
10-'-
- Cu.(n)
hot wire hot wire
S,~(n)
hot wire
---
sonic
Suu (n)
sonic
- Cu..,,(n)
10_: -
..
. ~,:,....~, ),.~
.
S~(n)
sonic
~.~-, 1 0 ~ -
,
10
-~
%,,,,
.
10_~ rm
10-~. REGI)N
I
R "-'GION
10-'1 0 -~
10 -~
10 2
10-' Frequency
II
REGII
10 °
10'
N III
10 2
n (Hz)
F i g . 4. S o n i c a n d c r o s s h o t w i r e a n e m o m e t e r
spectra at z = 1 m.
10 ~
P.,L Richards et al./J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 903 913
908
f = nz/U(z). In this form the spectra converge on a point at f = 0.14 where
S,,(./= 0.14) = 7u2,.
(5)
Noting that S,,(n) = zS, u(f)/U(z).
(6)
Table 1 summarizes the extent and slope of the anisotropic power law ranges while Fig. 5 shows the limit points and slopes of these ranges. Another point arising from Fig. 5 is the level of the spectra in the high frequency inertial subrange which exists f o r f > 1.0, that is region Ill. As discussed by Tieleman [5], for a neutrally stable boundary layer with turbulence production balanced by dissipation, we may expect S,,(f)/u2, = A , f
5/3 for./> 1.0,
(7)
10 4 -
I
i
S,='~(J')
10 r ~ ......... S~.~(D 1.01 r n
10 ~.
-
-c~= -0.83
10 2 .
J
O. 1 1 5 r a
3~(.f)
,-~n,-~ -a = -1.05
10'-
\
10 %
10-;-
10 -2.,
10 -3_ 10 -s
10-4
10 -s
10 -~
10o
10-'
10'
Reduced frequency f = nz/U Fig. 5. Streamwise spectra in reduced frequency form, showing the limit points and slopes of the anisottopic power law ranges. Table 1 Anisotropic power law ranges for U(10) = 10 m/'s Low frequency limit
High frequency limit
Height z(m)
n
S.o(n)/U z f
SuL,(f)/u2. n
Sou(n)/U 2 f
Su,,(f)//u~ E x p o n e n t - . z ~
0.115 1.01 10
0.01 0.01 0.01
1.0 1.0 1.0
1141 880 298
0.0064 0.0085 0.024
7.0 7.0 7.0
0.00032 0.0015 0.01
4,3 0,93 0.14
0.14 0.14 0.14
0.83 - 1.05 1.4
P.J. Richards et al./J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 903-913
909
where Tieleman gives Au = 0.27 and Av = Aw = 0.36. However Harris and Deaves [6] and E S D U data item 85020 [7] both suggest that these constants should be a function of height at low levels. Harris and Deaves effectively give Au = 0.346 B(z)
(8)
B(z)=l
(9)
where forz>zc,
B(z) = (1 -- (1 -- z/zc)2) 1/2 for z < z~.
(10)
For the conditions at Silsoe zc = 66 m and so at z = 10 m Eq. (8) gives Au = 0.143 and at z = 1.01 m, Au = 0.047. The results shown in Fig. 5 appear to show that at f = 1.0, Suu(f)/u2, = 0.27 for all heights and hence match the values given by Tieleman and do not exhibit the trends given in Eq. (8)~10).
4. Wind tunnel turbulence spectra In order to shed further light on the Silsoe observations a wind tunnel study [4] has been conducted at the University of Auckland. Spires, roughness blocks and a rough floor were used to create a turbulent boundary layer in the 1.8 m × 1.1 m working section. This boundary layer exhibited a logarithmic velocity profile and had an almost constant Reynolds shear stress in the lowest 200 mm. The roughness length was 0.1 m m and hence represents a 1/100th scale model of the Silsoe full scale situation. Turbulence spectra were obtained by using TSI model 1241-T1.5 cross wire probes connected to a TSI model IFA 100 anemometer system. The 8.7 min runs were processed as 32 s records, each containing 65 536 data points. The results presented are the averages of 16 non-overlapping records. The analogue signals were low pass filtered at 1 kHz and sampled at 2 kHz. Some attenuation due to filtering is apparent in the highest decade of the spectra. The similarity between the full sale and wind tunnel spectra is illustrated in Fig. 6. Where the u and w spectra and the uw cospectra obtained at 1.01 m at full scale is superimposed on the wind tunnel spectra at 10 ram. The graphs are presented in reduced frequency form in order to remove the need for scaling. There is generally good agreement between the two sets of spectra with the most noticeable difference occurring at low reduced frequencies where the presence of the tunnel walls suppresses large scale turbulence. The variation in spectral shape, within the energy containing range, with height was also observed in the wind tunnel. This was most obvious when the spectra were plotted in wavenumber form as shown in Fig. 7.
5. Cospectra One feature of the energy containing range is the generation of turbulence. In a turbulent boundary layer the frequency distribution of this generation can be related
P.J. Richards et al./.~ Wind Eng. Ind. Aerodyn. 69 71 (1997) 903 913
910 1E+4
Suu(f)/u'. ~
1E+3
Sww(f)/u'.
1E+2
o
-Cuw(OM.
-
"~
1E+1 1E*0
1E-1 ~)
\
1E-2 1E-3
i
1 E-5
1E-4
1E-3
1 E-2
i
i
F J ~11
1E-1
1E+O
1E+I
Reduced frequency f = nz/U Fig. 6. S t r e a m w i s e and vertical spectra and uw cospectra. Silsoe full scale d a t a at t.01 m and wind tunnel d a t a at 10 mm.
IE+I H e i g h t
z(rrlrrl)
3 1E+O
10
30 100 1E-1
==
1 E-2
--
1E-3
1E-4
i
1E-3
i
i
i illl
i
1 E-2
i
i
IiiiJ
i
1 E-1
~
I
illll
i
1 E+O
~
i
iii
p
i
1 E* 1
i
ii1~1
1 E+2
1E+3
Wavenumber k = n/U (m -1) Fig. 7. W i n d tunnel s t r e a m w i s e spectra at low heights,
to the cospectrum. Fig. 8 illustrates how in full scale the cospectra vary with height. This figure has been plotted in wavenumber form in order to show how at low heights the generation is shifted to higher and higher wavenumbers. That is, at low levels most of the generation is associated with small eddies while the generation associated with larger eddies at higher heights is suppressed by the ground plane.
P.J. Richards et al./J. Wind Eng. Ind. Aerodyn. 69-71 (1997) 903-913
0.6 -¸
_ _ 1 _ _
911
1 I
- c , , , 4 k ) I.o~ ,~ f-c,,,,(~) o.f ~6,~ I
0.50.4-
,li
O.31
L)
/
O.2. 0.1 r
.
"~ '~i[.,~ ,D~I' ' , i,i-:.~l' .....""
0.O
10 -s
10-'
10 -~
I
'7i~,+.
` +
'+'% .~I
, ,~ r " ,
I .........#"
10 -2
'~"~,---,
10-'
10'
10 o
Wavenumber k = nAJ (m 4) Fig, 8. uw cospectra from the Silsoe full scale d a t a in w a v e n u m b e r form.
0,5 -
I
I
0.4-
-c,.,.(.f)
to m
li II 2 Y - ~::~I'°' o.,,5 +,,,
!
l
0,3"
"i o.2.
(.3
•..
0.1'
l
'D;','++'
0.0
10 -s
10-'
10 -~
10 -2
10-'
10 0
10'
Reduced frequency f = nz/U Fig. 9. uw c o s p e c t r a from the S i l s o e f u l l scale d a t a in reduced f f e q u e n c y f o r m .
If the cospectra are plotted in reduced frequency form then, as illustrated in Figs. 9 and 10, both the full scale and wind tunnel cospectra tend to form a single curve. Both of these figures can be approximately m a t c h e d by a function of the form -- Cuw(f) 2
u,
1.0 foE1 + (3f/4fo)] ~/~
(ll)
P.J. Richards et al./J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 903 913
912
0.4
H e i g h t z(rln*11 ) 3 10
0.3
30 100
N~
0.2 0 x
0.1
!%~Lf.~. t.y,r.rT
0.0
1E-5
IE-4
1E-3
1E-2
1E-1
1E+O
1E+l
1E+2
Reduced frequency f = nz/U Fig. 10. uw cospectra from the Auckland wind tunnel data in reduced frequency form.
with f0 = 0.05. This function has a m a x i m u m of 0.27 at f = j 0 and when integrated from f = 0 to o~ returns a value of unity. It also has the correct high frequency form for the inertial subrange.
6. Conclusions Within the atmospheric surface layer the turbulence that effects low rise buildings tends to be highly anisotropic. It has been shown that while low level spectra match the expected form in the inertial subrange there also exists a range of frequencies within the energy containing range where the horizontal spectra can be matched by a power law. The gradients of the log-log graphs of spectra in this range tend to diminish with decreased height. This power law range seems to be a bridge between a very low frequency range where horizontal spectra scale in proportion to the square of the mean velocity and frequencies a decade below the low frequency end of the inertial subrange. This power law range is quite narrow at higher levels but broadens at low levels. The level of the normalised power spectral density in the inertial subrange is observed to be similar at all heights and does not vary in the manner suggested by Harris and Deaves. Turbulence spectra obtained in the University of Auckland wind tunnel are shown to exhibit similar patterns to those observed in full scale. The variation in the size of eddies involved in the generation of turbulence at various heights has been illustrated by considering the cospectra, A functional form for the cospectra was proposed.
P.J. Richards et aL /,L Wind Eng. Ind. Aerodyn. 69-71 (1997) 903 913
913
References l-l] R.P. Hoxey, P.J. Richards, Structure of the atmospheric boundary layer below 25 m and implications to wind loading on low-rise buildings, J. Wind Eng. Ind. Aerodyn. 4 1 4 4 (1992) 317 327. [2] R.P. Hoxey, P.J. Richards, Spectral characteristics of the atmospheric boundary layer near the ground, in: UK Wind Engineering Conf., Cambridge, 1992. [3] R.P. Hoxey, P.J. Richards, Full-scale wind load measurements point the way forward, J. Wind Eng. Ind. Aerodyn. 57 (1995) 215 224. [4] S. Fong, The structure of equilibrium turbulent boundary layers, Master of Engineering Thesis, University of Auckland, New Zealand, 1995. 1-5] H.W. Tieleman, Universality of velocity spectra, J. Wind Eng. Ind. Aerodyn. 56 (1995) 55. 1-6] R.I. Harris, D.M. Deaves, The structure of strong winds, wind engineering in the eighties, in: Proc. CIRIA Conf., 1980. 1-7] ESDU, Characteristics of atmospheric turbulence near the ground, part II: single point data for strong winds (neutral atmosphere), Item 85020, 1985.