Richards 2002

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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.

P.J. Richards, R.P. Hoxey / J. Wind Eng. Ind. Aerodyn. 90 (2002) 1855–1866 1856

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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


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


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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