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Journal of Wind Engineering
and Industrial Aerodynamics 90 (2002) 1855–1866
Unsteady flow on the sides of a 6 m cube
P.J. Richardsa,*, R.P. Hoxeyb
a Department of Mechanical Engineering, University of Auckland, Private Bag 92019, Auckland,
New Zealand b
Bio-Engineering Division, Silsoe Research Institute, Silsoe, Bedford MK45 4HS, UK
Abstract
Four ultrasonic anemometers have been used to measure flow velocities at two groups of
positions at mid-height on a 6 m cube. One other ultrasonic anemometer, located upstream at
cube height, provided reference wind data. The results obtained provide a picture of the mean
and fluctuating parts of the flow. Mean velocity results indicate that with the wind
perpendicular to one face, the flow detaches at the windward edge but is reattached to the sides
by about x=h ¼ 0:83: However, probability analysis shows that the velocity at this point isreversed for 54% of the time. In addition it is shown that while some of the fluctuations can be
attributed to a quasi-steady response to variations in wind direction, the influence of building-
induced turbulence is also very apparent. These results illustrate the highly turbulent and
unsteady nature of flow on the sides of the cube.
r 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Cube; Unsteady flow; Full-scale testing
1. Introduction
In order to provide a facility for fundamental studies of the interactions between
the wind and a structure, a 6 m cube has been constructed at Silsoe, UK, in an ‘open
country’ exposed position (Fig. 1). Surface pressure measurements have been made
on a vertical and on a horizontal centre line section with additional tapping points
on the roof. Measurements have also been made of wind velocity in the region
around the cube using ultrasonic anemometers.
In earlier papers [1,2] it has been shown that with the wind perpendicular to one
face of the cube the suction pressures measured in the centre of the roof are
significantly more negative than many of those measured in wind tunnels [3–5]. In
*Corresponding author. Tel.: +64-9-373-7599; fax: +64-9-373-7479.
E-mail address: [email protected] (P.J. Richards).
0167-6105/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 1 6 7 - 6 1 0 5 ( 0 2 ) 0 0 2 9 3 - 3
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addition, it was found that the suction pressures measured at mid-height on the
sidewalls showed a similar discrepancy between full scale and wind tunnel. The
unsteady flow over the roof of the cube has been discussed by Richards and Hoxey
[6]. In this paper, the nature of the flow around the sides of the cube will beconsidered.
2. The experimental facility
The velocity profile at the site has been measured at various times. Recent
measurements are well matched by a simple logarithmic profile with a roughness
length z0 ¼ 0:00620:01 m. This means that the cube has a Jensen number (h=z0) of 600–1000. The longitudinal turbulence intensity at roof height is typically 20%. As
an example measured values for one 12-h period are given in Table 1: in this periodthere was no significant trend in wind speed or direction. Included in Table 1 is the
streamwise turbulence length scale, xLu; derived from the magnitude of the powerspectral density in the inertial sub-range [7]. Estimates of the Monin–Obukhov
length L; have been obtained by using the fluctuating temperature derived from theultrasonic anemometer’s measurement of the speed of sound. These estimates show
that the data in Table 1 are associated with near neutral atmospheric stability
(0oz=Lo0:015 at z ¼ 6 m). Atmospheric stability analysis has not been carried outon all the data sets that make up the results presented in this paper.
The 6 m cube has a plain smooth surface finish and has surface tapping points
around a vertical and horizontal centre line section, with additional points on onequarter of the roof. Simultaneous measurements have been made of up to 32
pressures.
Fig. 1. The 6 m cube.
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Velocity measurements have been made using five Gill Instruments Ultrasonic
anemometers. With the cube oriented perpendicular to the prevailing south-west
wind, the reference anemometer was located at cube height (h), 3.48h windward of the
cube and 1.04h north-west of the centreline plane. The output from this reference
anemometer was logged at 20.8 samples/s. The remaining four anemometers were
logged at 10 samples/s. These were at various times located near the centre of the four
vertical faces (0.1h from the surface), positions W, N, E and S as shown in Fig. 2, oralong the sides, positions A, B, C and D, where the anemometers were 0.01 h from the
surface. The vertical location of the anemometers was intended to be at mid-height
but due to a measurement error, only detected after testing, the actual height was
0.515h. This small error is not thought to have had any significant effect on the
results. Primary analysis involved the calculation of 5 min mean values of all velocity
components. Further probability analysis was carried out on selected 15 min blocks.
3. 5 min mean data
Fig. 3a shows the 5 min mean velocity coefficients (defined as the ratio of the local
wind component to the 5 min mean reference wind speed) against reference mean
Table 1
Properties of the approach flow
z (m) U (m/s) Iu Iv Iw xLu (m) uw (m/s)2
1 6.97 0.243 0.196 0.077 11 0.281
3 8.65 0.208 0.166 0.072 33 0.270
6 9.52 0.193 0.150 0.078 53 0.251
10 10.13 0.186 0.151 0.083 62 0.343
A
0.833
Reference Mast
(1.0h high,
1.04h to the
side of cube
centre)
X
Y
0.167
0.5
0.1
0.01
0.1
3.4
0.1
W
N
D
E
B
S
C
Wind
Direction
θ = tan–1
(V/U)
Fig. 2. Plan view of the cube showing the measuring positions at mid-height.
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wind direction for measuring position W (0.1h off the centre of the west surface). In
constructing this figure, data from the four measuring points W, N, E and S have
been used, but they have been transformed to give equivalent data for position W at
appropriate angles. In addition the multiple planes of symmetry have been used. The
axes are oriented such that the U velocity component is normal to the face, the V
component horizontal and parallel to the face and the W component vertical
(positive upwards). Fig. 3a shows that when the wind is perpendicular to the west
face (y ¼ 0), the U component is small and positive, the V component is zero and theW component indicates flow down the face, showing that the stagnation point is
above mid-height. As the wind angle increases, the flow across the face increases
−1
0
1
0 30 60 90 120 150 180 210 240 270 300 330 360
Reference Wind Direction, Theta (degrees)
M e a n
V e
l o c
i t y
/ R e
f e r e n c e
W i n d S p e e
d Uw/Ref Sp
Vw/Ref Sp
Ww/Ref Sp
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 30 60 90 120 150 180 210 240 270 300 330 360
Reference Wind Direction, Theta (degrees)
r m s
V e
l o c
i t y
/ R e
f e r e n c e
W i n d S p e e
d Urms/Ref Sp
Vrms/Ref Sp
Wrms/Ref Sp
(b)
(a)
Fig. 3. Velocity components at measuring point W, 0.1 h from the centre of the west face: (a) 5-min mean
velocity coefficients and (b) rms velocity coefficients.
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with the wind perpendicular to the west face (y ¼ 01) the mean horizontal flow is
reversed at points A, B and D but is in the streamwise direction at point C. This
indicates that the mean flow detaches at the windward edge and on average
reattaches between points B and C (0:5ox=ho0:833). As the wind angle increasesand the wind is more onto the face associated with measuring points A, B and C, the
velocities become more positive and reach a maximum positive value at about 401. It
can be observed that the positive velocities are stronger at the more leeward
locations. In addition, it may be noted that an angle less than about 201 is required in
order to reverse the flow at the windward point (A), whereas an angle less than 101 is
required before the flow at the mid-face (point B) is reversed. This trend indicates
that the region of separated and reversed flow grows as the angle decreases and even
includes point C when the mean flow direction in less than about –51. The data from
the other side of the cube (point D) show that on this side the flow is reversed for all
angles between 01 and 901. Fig. 4 also includes a mirror image of the data from point
D, which forms a continuous trend with the data from point B. The data from points
B and D have been combined and reflected in order to provide a 3601 data set for
point B as shown in Fig. 5. Also shown in Fig. 5 is a Fourier series of order 10 which
has been fitted to the data by using a least-squares method (curve fit). The Fourier
series takes the form:
%C U ð% yÞ ¼X10
k ¼0
%ak cosðk % yÞ þ %bk sinðk % yÞ; ð1Þ
where %C U ð% yÞ is the 5 min mean horizontal velocity coefficient and % y thecorresponding 5 min mean wind direction. It should be noted that if it is assumed
that this curve also represents an approximation to the relationship between the
instantaneous velocity and the instantaneous wind direction, then the 5-min averages
would not necessarily lie on the curve. The modified curve fit shown in Fig. 5 is a
−1
−0.8
−0.6
-0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180
Reference Wind Direction, Theta (degrees)
V e l o c i t y C o e f f i c i e n t
Data
Curve fit
Modified Curve Fit
Fig. 5. 5-min mean velocity coefficients for position B constructed by using data from positions B and D.
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better estimate of the expected quasi-steady relationship which, as derived in an
earlier paper [9], is given by
C U ðyÞ ¼X10
k ¼0
ak cosðk yÞ þ bk sinðk yÞ; ð2Þ
where ak ¼ %ak expð12 k 2s2
yÞ and bk ¼ %bk expð
12
k 2s2yÞ:
In this relationship sy is the standard deviation of wind direction and is typically
101 (0.175 rad). It can be shown that if the wind directions are normally distributed
and the instantaneous velocity responds to the instantaneous wind direction in the
manner given by Eq. (2), then the 5-min average will lie on the original fitted curve
(Eq. (1)). This quasi-steady relationship will be discussed further in Section 5.
4. Probability analysis
Fig. 6 shows a probability analysis of the X-direction sidewall velocities for one
15-min period during which the mean wind direction was near 01. As expected, with
this mean wind direction, the statistical results for points B and D are very similar.
That is not to say that the same thing is occurring simultaneously on the two sides of
the building, but rather that the flow behaviour is similar. Certainly the time series
show that the velocity on one side often becomes positive while the velocity on the
other becomes more negative. From the cumulative distribution function, Fig. 6b, itcan be seen that while the velocities at points A, B and D are reversed for over 80%
of the time, there are occasions when these velocities are in the positive direction. On
the other hand, the velocity at point C is positive for about 46% of the time. Fig. 6a
also indicates the range of velocities occurring at each point. This shows that the
reversed flow is stronger near the centre of the wall than at the windward point A. In
addition it shows that the range of positive velocities at point C is greater than the
range of negative velocities. As a result, although the velocity at this point is positive
for only 46% of the time, the mean velocity is positive and not negative.
These observations appear to be supported by the time series shown in Fig. 7. In
this figure the U velocities from points A, B and C are shown alongside theinstantaneous reference wind direction (note that the wind direction is plotted
against the secondary Y -axis in order to make the situations clearer). It can be seen
that, while the wind direction is near 01, the velocity at point C is negative for most
of the time but is occasionally strongly positive. This observation is similar to that
made by Kawai [10], who when studying the surface pressures near the reattachment
point on a square prism commented ‘‘the fluctuating pressure is strongly
intermittentyandyalternates between slow fluctuation of low intensity and rapid
fluctuation of high intensity’’.
There is an interesting sequence of events during the first 15 s of Fig. 7 (490–505 s).
During this period the wind direction becomes very positive for a period of 5 s and asa result all the three velocities become significantly positive for a time. Then, several
seconds after the direction has returned to being near zero, the velocities at points A,
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B and C reverse in order. This suggests that a separation bubble has grown from the
windward edge and has progressively engulfed these points. It may be noted that the
reference anemometer is well upstream of the cube and so there is a time delay
between the change in direction at the reference anemometer and the associated
changes in flow at the cube. A general study of the time series shows that some of the
variation in wind speeds near the cube can be attributed to changes in the wind
direction; however, it is also apparent that not all of the variations could be
accounted for in this manner. In an attempt to identify what variations areassociated with changes in wind direction and what variations are caused by other
effects, a quasi-steady analysis has been applied to the results for point B.
0
0.5
1
1.5
2
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
p d f
UA
UB
UCUD
0
0.2
0.4
0.6
0.8
1
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
X Direction Wind Component / Reference Mean Wind Speed
c d f
UA
UB
UC
UD
(a)
(b)
X Direction Wind Component / Reference Mean Wind Speed
Fig. 6. Statistical distributions for the X-direction sidewall velocities when the mean wind angle is 01: (a)
probability density function and (b) cumulative distribution function.
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5. Quasi-steady analysis
A number of researchers (including Refs. [9–11]) have considered the validity of
using a quasi-steady method for relating the surface pressures on buildings tovariations in the approaching wind. Kawai [10] initially makes use of a linear
relationship between pressure and the longitudinal and transverse velocity
fluctuations and then modifies this relationship to include quadratic terms involving
longitudinal turbulence. Letchford et al. [11] take a similar approach and find that
preserving the non-linear terms gives a better agreement between measured and
predicted quasi-steady probability density functions (pdf’s). In addition Letchford
et al. include linear terms that account for the effects of vertical turbulence on roof
pressures. In both of these studies the effect of variations in wind direction are
accounted for through terms involving only the first derivative of the pressure
coefficient with respect to wind direction. This effectively means that it is assumedthat the pressure coefficient–direction function is nearly linear over the range of
directions of interest. However, if the standard deviation of wind directions is
typically around 101 then, as illustrated in Fig. 8, the direction pdf will cover a range
of at least 7301 around the mean and it is unlikely that any coefficient–direction
relationship will be linear over such a broad range of angles. It was for this reason
that the authors [9] suggested using a Fourier series to more accurately model the
coefficient–direction relationship. In all these studies it has been found that a quasi-
steady model can account for some effects, but not all of them.
While it is recognised that a quasi-steady model is imperfect, it is useful in order
to identify what effects can be directly related to the onset flow and what effectsresult from other phenomena. Fig. 9 shows a time series of X-direction velocities
measured at point B and the corresponding quasi-steady model. In order to generate
−20
−10
0
10
490 495 500 505 510 515 520 525 530
Time (s)
U V e l o c i t y ( m / s )
−60
−30
0
30
60
90
120
R e
f e r e n c e
W i n d D i r e c
t i o n ,
T h e
t a ( D e g
r e e s
)
UA
UB
UC
Theta
Fig. 7. A selected section of one time series for points A, B and C.
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the quasi-steady model it has been assumed that the relationship between the
instantaneous velocity at point B and that which occurred at the reference mast
slightly earlier can be modelled by the modified curve fit in Fig. 5. The quasi-steady
time series was generated by using the measured reference wind speeds and
directions. At each time step the measured direction was combined with Eq. (2) and
the associated coefficients to give a velocity coefficient, which was then multiplied bythe current wind speed to give the quasi-steady velocities for point B. Recognising
that there is a time delay between events at the mast and at the cube, correlation
0
0.01
0.02
0.03
0.04
0.05
−80 −60 −40 −20 0 20 40 60 80
Reference Wind angle (degrees)
p d f
Ref Theta
Normal
Fig. 8. Probability density function of reference wind direction for one 15-min period.
−1.5
0.0
1.5
0 100 200 300 400
Time (s)
U V e l o c i t y ( m / s )
Point B
Quasi-steady
Fig. 9. Instantaneous velocities as measured at point B and as predicted by a quasi-steady model.
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analysis was carried out between the measured and quasi-steady time series with a
variable time lag; this showed that the highest correlation could be obtained by
allowing a 6.6 s time delay. It can be seen that both records exhibit similar low
frequency variations and that the quasi-steady curve is a reasonable match to the
centre of the band of actually measurements. However, it appears that superimposedon top of the quasi-steady variations, the actual measurements include a large
number of short-duration excursions, both in the positive and negative directions,
but most often in the negative direction. This observation can also be seen in the
pdf’s of Fig. 10, where the measured pdf exhibits much higher probabilities of large
negative velocities than predicted by the quasi-steady model. The quasi-steady model
shows a very low probability of velocity coefficients lower than –0.6. This occurs
because the quasi-steady coefficient for point B, as seen in Fig. 5, is always greater
than –0.4 in the vicinity of 01 wind direction. In addition analysis of the wind speeds
shows that the peak wind gust is of the order of 1.65 times the mean. Combining
these means that values of the instantaneous velocity coefficients for point B lowerthan –0.66 will not be predicted by the quasi-steady model. It is thought that the
more negative velocities observed are the result of vortex structures which roll up,
become quite intense for a short time and are then shed into the flow. Comparison of
the rms velocities shows that up to 70% can be attributed to quasi-effects but the
additional 30% is the result of local building-induced turbulence.
6. Conclusions
Four ultrasonic anemometers have been used to measure flow velocities at twogroups of positions at mid-height on a 6 m cube. One other ultrasonic anemometer,
located upstream at cube height, provided the reference wind data. The results
0
0.5
1
1.5
2
2.5
3
3.5
−
2 −
1.5 −
1 −
0.5 0 0.5 1 1.5 2X Direction Wind Component / Reference Mean Wind Speed
p d f
UB
Quasi-steady
Fig. 10. Probability density functions for the velocity at point B and as predicted by quasi-steady
methods. Mean wind direction 01.
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obtained provide a picture of the mean and fluctuating parts of the flow. In
particular, the highly unsteady nature of the flow on the side of the cube has been
illustrated. It has been shown that while some of the fluctuations can be attributed to
a quasi-steady response to variations in wind direction and speed, the influence of building-induced turbulence is also very apparent.
References
[1] R.P. Hoxey, P.J. Richards, J.L. Short, A 6 m cube in an atmospheric boundary layer flow: Part 1,
Full-scale and wind-tunnel results, J. Wind Struct. 5 (2–4) (2002) 165–176.
[2] P.J. Richards, R.P. Hoxey, J.L. Short, Wind pressures on a 6 m cube, J. Wind Eng. Ind. Aerodyn. 89
(14–15) (2001) 1553–1564.
[3] I.P. Castro, A.G. Robins, The flow around a surface-mounted cube in uniform and turbulent streams,
J. Fluid Mech. 79 (2) (1977) 307–335.
[4] A. Hunt, Wind-tunnel measurements of surface pressures on cubic building models at several scales,
J. Wind Eng. Ind. Aerodyn. 10 (1982) 137–163.
[5] N. H .olscher, H-J. Niemann, Towards quality assurance for wind tunnel tests: a comparative testing
program of the Windtechnologische Gesellschaft, J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 599–608.
[6] P.J. Richards, R.P. Hoxey, Unsteady flow on the roof of a 6 m cube, Paper to be Presented at the
Third European and African Conference on Wind Engineering, Eindhoven, Netherlands, July 2001.
[7] ESDU, Characteristics of atmospheric turbulence near the ground, ESDU data item 85020, 1985 with
amendments to 1990.
[8] N. Steggel, I.P. Castro, Effects of stable stratification on flow and dispersion around a cube,
Proceedings of the 10th International Conference on Wind Engineering, Copenhagen, Denmark, June
1999.[9] P.J. Richards, R.P. Hoxey, B.S. Wanigaratne, The effect of directional variations on the observed
mean and rms pressure coefficients, J. Wind Eng. Ind. Aerodyn. 54/55 (1998) 599–608.
[10] H. Kawai, Pressure fluctuations on square prisms—applicability of strip and quasi-steady theories,
J. Wind Eng. Ind. Aerodyn. 13 (1983) 197–208.
[11] C.W. Letchford, R.E. Iverson, J.R. McDonald, The application of the quasi-steady theory to full-
scale measurements on the Texas Tech Building, J. Wind Eng. Ind. Aerodyn. 48 (1993) 111–132.
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