Richards 2006 Journal of Wind Engineering and Industrial Aerodynamics

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Journal of Wind Engineering

and Industrial Aerodynamics 94 (2006) 77–99

0167-6105/$ -

doi:10.1016/j

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Flow reattachment on the roof of a 6m cube

P.J. Richardsa,�, R.P. Hoxeyb

aDepartment of Mechanical Engineering, University of Auckland, Auckland, New ZealandbBio-Engineering Division, Silsoe Research Institute, Silsoe, Bedford, MK45 4HS, UK

Received 7 July 2004; received in revised form 29 September 2005; accepted 6 December 2005

Abstract

Five ultrasonic anemometers are used to measure flow velocities above the roof of a 6m cube and

at a reference point upstream. Various analysis techniques are applied to the data in order to

illustrate the differences between: the mean reattachment position when the mean wind is normal to

the windward face (01), which is at x/hE0.6; the median instantaneous reattachment position at

x=h ¼ 0:66 and the reattachment position that would occur if the wind direction was held at a

constant 01, which is at x=h ¼ 0:75 and is also the position of zero conditionally averaged u velocity

at instants when the v velocity is zero. It is also shown that the flow is highly unsteady and that the

reattachment length varies from negligible separation to no reattachment, which occurs for 20% of

the time. Some of these variations are related to fluctuations in the onset wind speed and direction,

but they are also influenced by the dynamic response of the separation vortex system. The formation

and shedding of vortices means that certain frequencies in the turbulence spectrum, around 1Hz, are

slightly amplified, whereas frequencies above 10Hz are filtered out as a result of the inertia of the

vortex system. The effects of reattachment length on the pressure distribution are briefly considered

but it is shown that these do not account for the differences between the Silsoe field data and typical

wind-tunnel results. It is suggested that the pressure differences may be related to Reynolds number,

but it appears that this is not associated with changes in reattachment length.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Cube; Field measurements; Reattachment

see front matter r 2005 Elsevier Ltd. All rights reserved.

.jweia.2005.12.002

nding author. Tel.: +649 3737999; fax: +64 9 3737479.

dress: [email protected] (P.J. Richards).

www.elsevier.com/locate/jweia

ARTICLE IN PRESSP.J. Richards, R.P. Hoxey / J. Wind Eng. Ind. Aerodyn. 94 (2006) 77–9978

1. Introduction

In both wind tunnel and computational modelling of wind engineering flows aroundbluff bodies the correct representation of the flow separation and reattachment is essentialin order to obtain an accurate prediction of the pressure field. One situation where this isimportant is with the flow over the roof of a cubic building with the wind direction almostnormal to one face. Castro and Robins [1] present wind-tunnel data relating to the flowover a cube in a variety of boundary layer situations. Their results show considerabledifferences between the roof pressures from smooth flow tests, where the pressure is almostuniform across the entire roof, to those in highly turbulent flow where there are highsuctions near the windward edge and a much lower suction towards the leeward edge. Thepressure coefficient at the leeward edge of the roof appeared to depend on the turbulenceintensity in the approach flow at half cube height. Their velocity measurements also clearlyshowed that in smooth conditions the flow over the roof separated at the windward edgebut did not reattach, whereas in more turbulent conditions the flow appeared to reattach ata location approximately 0.3H across the roof. Similar differences have been observed byRichards et al. [2], where they compare three Computational Fluid Dynamic (CFD)models of the turbulent rural wind flow over the Silsoe 6m cube. Although there are anumber of differences between the CFD models, the differences in the results are primarilyattributed to the turbulence model used in each case. It is shown that with the MMKturbulence models the flow did not reattach on the cube roof and so the pressures wererelatively uniform, in a manner similar to Castro and Robins’ smooth flow wind-tunnelresults, as shown in Fig. 1. In contrast with the k–e turbulence model the flow over the rooffailed to separate at the windward edge and as a result a high suction occurred at this edgeand then decreased very rapidly with distance across the roof. It was only with the RNGturbulence model that the flow separated and reattached and created a roof pressuredistribution similar to the turbulent boundary layer results of Castro and Robins.

-1.5

-1

-0.5

0

1 2Position

Pre

ssu

re C

oef

fici

ent

CP

Castro & Robins Smooth

Castro & Robins Turbulent

Richards et al. MMK

Richards et al. k-Epsilon

Richards et al. RNG

03

2

1

Fig. 1. Vertical central section mean pressure coefficients across the roof of a cube with wind normal to one

face (01).

ARTICLE IN PRESSP.J. Richards, R.P. Hoxey / J. Wind Eng. Ind. Aerodyn. 94 (2006) 77–99 79

Nevertheless, to simply classify the flows as ‘separated’, ‘no separation’ or ‘reattached’,is an over-simplification. In this paper, the flow over a 6m cube in a rural atmosphericboundary layer will be considered in some detail. In addition to the mean flowcharacteristics, the unsteady nature of this flow will be considered.

2. The experimental facility

In order to provide a facility for fundamental studies of the interactions between thewind and a structure, a 6m cube has been constructed at Silsoe in an ‘open country’exposed position (Fig. 2).

Surface pressure measurements have been made on a vertical and on a horizontalcentreline section with additional tapping points on the roof. Measurements have also beenmade of wind velocity in the region around the cube using ultrasonic anemometers. Thecube is mounted on a turntable so that it may be rotated in order create a variety of anglesbetween the tap locations and the prevailing southwest winds. In addition the rear can bejacked up so that the cube is tilted up to 51 into the wind.

The velocity profile at the Silsoe Research Institute site has been measured at varioustimes and the recent measurements are well matched by a simple logarithmic profile with aroughness length z0 ¼ 0:006 to 0.01m. This means that the cube has a Jensen number(h/z0) of 600–1000. Table 1 shows the typical wind speeds and turbulence intensities at thereference mast (height 6m) that occurred during the data collection for various purposes.

In all cases, the turbulence intensities are slightly higher than might be expected for thesite roughness length. This is probably due to the existence of rougher terrain, consisting ofvillages, hedges and trees, some 600m upstream of the site. These rougher terrain elementsintroduce large-scale turbulence structures that persist for some distance. It is also clear

Fig. 2. The Silsoe 6m cube.

ARTICLE IN PRESS

Table 1

Typical properties of the approach flow at a height of 6m

Averaging period (min) Speed (m/s) lu lv lw

Site profile 60 9.52 0.19 0.15 0.08

Velocities at positions A–D 5 5.54 0.23 0.20 0.10

Velocities at positions E–H 5 4.36 0.29 0.23 0.14

Pressure coefficients 12 7.60 0.21 0.16 Not recorded

EReference Mast (1.04h to the side)

X

Z

0.167

0.24

0.41

0.5

0.59

0.76

0.833

0.1

0.01

0.1

3.48

GFDBA

HC

Fig. 3. Measuring positions for the sonic anemometers in the vicinity of the cube roof, with distances given as

fractions of the cube height h.

P.J. Richards, R.P. Hoxey / J. Wind Eng. Ind. Aerodyn. 94 (2006) 77–9980

that the turbulence intensities increased during lower wind speeds. This is probably due toatmospheric instability.Velocity measurements have been made using five Gill Instruments Ultrasonic

anemometers. With the cube oriented perpendicular to the prevailing southwest wind,the reference anemometer was located at cube height (h), 3.48h windward of the cube and1.04h northwest of the centreline plane. The output from this reference anemometer waslogged at 20.8 samples/s. The remaining four anemometers were logged at 10 samples/sand were at various times located in the centreline vertical plane, either on the cube roof(sensing velocity 0.01h off the surface) at positions A, B, C and D or positions F, G and H,0.1h above the roof, together with position E, level with the roof and 0.1h windward of theeaves. The various measuring positions are illustrated in Fig. 3. For positions A–D thebodies of the sonic anemometers were horizontal and oriented parallel to the windwardface in alternating directions. From the geometry it is estimated that there was minimalinterference for wind directions in the range 7201, but it is recognised that there was somesheltering of the more leeward anemometers with more oblique wind directions. Forpositions F–H the stands used angled the anemometers upwards as well as across the roofin alternating directions. It is unlikely that there was significant interference for thesepositions. In all cases, the measured velocities were resolved into standardised componentswith the u component perpendicular to the windward face in the downwind direction, the w

component vertically upwards and the v component horizontal and fitting a right-handedaxis convention. Primary analysis involved the calculation of 5-min mean values of allvelocity components. Further probability analysis was carried out on selected 15-minblocks.


3. Mean reattachment length

Fig. 4 shows the 5-min mean u velocity coefficients against reference mean winddirection for the four measuring positions A–D on the cube roof (0.01h off the surface).The velocity coefficient is defined as the ratio of the 5-min mean local wind component tothe 5-min mean reference wind speed. These results show that, for a broad range of angles,the flow on the windward half of the roof (positions A and B at x=h ¼ 0:24 and 0.41) isreversed (negative coefficients). On the other hand, at position D (x=h ¼ 0:76), thecoefficient is consistently positive, indicating that the mean flow has reattached upwind ofthis position. Further, at position C (x=h ¼ 0:59), the coefficients are near zero for small

1

0

-1-20 -15 -10 -5 0 5 10 15 20

Reference Wind Direction (degrees)

u V

elo

city

Co

effi

cien

t

A, x/h = 0.24B, x/h = 0.41C, x/h = 0.59D, x/h = 0.79

Fig. 4. Five-minute mean u velocity coefficients (ratio of local mean to reference mean) at various positions across

the cube roof.

Table 2

Characteristics of the full-scale and wind-tunnel tests and the associated reattachment lengths

Typical

Re no.

Comment Size h/z0 lu(h) Reattachment

length at 01

Castro & Robins Smooth Flow [1] 105 Wind-tunnel 60mm 0.005 No

reattachment

Ogawa, Oikawa & Uehara (R0) [3] 3.5� 104 Wind-tunnel no

roughness

80mm 10500 0.067 No

reattachment

Murakami & Mochida [5] 7� 104 Wind-tunnel 170 0.165 E0.7h

Silsoe [2] 3� 106 Field 6m 750 0.23 E0.6h

Ogawa, Oikawa & Uehara [4] 3.5� 105 Field 1.8m 480 0.22 E0.55h

Minson, Wood & Belcher [6] 6.3� 104 LDA

measurements

200mm 60 0.3 0.4h

Castro & Robins Turbulent BL [1] 105 Wind-tunnel 200mm 50 0.27 E0.3h

Ogawa, Oikawa & Uehara (R6) [3] 1.9� 104 Wind-tunnel

6 cm roughness

80mm 106 0.265 No separation

Ranked in order of longest to shortest reattachment length.


wind angles and become positive for larger wind angles. These results indicate that at windangles around 01, the mean reattachment length is around 0.6h and that this decreases withincreased wind angle. This result is consistent with other full-scale and wind-tunnelobservations, as summarised in Table 2, where the reattachment length ranges from noreattachment with very low turbulence levels in the onset flow to no separation with highonset flow turbulence.The results in Table 2 have been ranked in order of longest to shortest reattachment

length. In general this corresponds to the order for increasing turbulence intensity (at cubeheight) in the onset flow although the last three results do not exactly fit this pattern, butare the three cases with the highest turbulence intensity. With turbulence levels similar toSilsoe, the observed reattachment lengths are also of the order of 0.6h.Further, Ogawa et al. [4] observed that the flow reversal, in the centre of their field-tested

cube roof, only occurred for wind angles less than 171 with low turbulence levels or o111with higher turbulence. A similar result can be deduced from Fig. 4, where flow reversal atthe centre of the roof (mid-way between positions B at x=h ¼ 0:41 and C at x=h ¼ 0:59) isonly indicated for angles less than about 121.Similar analysis of the vertical plane velocity components at all eight measuring

positions has been carried out. Fitting curves to these data provides the 5-min mean flowvectors for a mean wind direction normal to the windward face (01) as shown in Fig. 5.These are all consistent with the mean flow separating at the windward eaves andreattaching at x=h ¼ 0:6.Although Fig. 4 shows a fairly consistent pattern, it should be noted that with the

parabolic nature of the velocity coefficient–direction relationships then it is expected thatthe mean observation will be related to the variance of wind directions occurring duringthe averaging period. This can be analysed further by using quasi-steady theory. A numberof researchers [7–11] have considered the application of quasi-steady theories to thepressures observed on buildings and a similar, but simpler, approach is used here. It isinitially assumed that each velocity component is related to the instantaneous approachwind speed (vref) and direction (y) through functions of the form

u ¼ f ðvref ; yÞ. (1)

It has been shown by Letchford and Marwood [9] that the wind speed and direction aregenerally uncorrelated and hence,

u ¼ f ðvref Þ gðyÞ , (2)

where the overbar indicates time averaging.

Referenceunit vector

Fig. 5. Five-minute mean velocity vectors on the central vertical plane for a wind direction normal to the

windward face (01).


Further, it may be assumed that the local velocity components are linearly related to theapproach wind speed and so

u ¼ vrefgðyÞ or U ¼ V refgðyÞ, (3)

where capitals are also used to indicate mean values. As a result, the u velocity coefficient

CU ¼U

V ref¼ gðyÞ. (4)

The shape of Fig. 4 suggests a quadratic form for g(y), such as

gðyÞ ¼ aþ by2. (5)

Hence the observed mean velocity coefficient is expected to be

CU ¼ aþ bðyþ y0Þ2 ¼ aþ by2þ bs2y, (6)

where sy2 is the variance of wind directions (rad2) during the observation period.

Fig. 6 shows the u velocity coefficients, from runs where the mean wind direction waswithin 731 of normal to the windward face ðy

251Þ, in relation to the measured direction

variance. It is clear that all these follow the linear relationship of Eq. (6) and hence the y

intercept and gradient give estimates of the ‘a’ and ‘b’ parameters in that equation.Fig. 7 shows the data previously presented in Fig. 4 but corrected to give the expected

observations that might occur with a steady wind direction. These have been calculated byusing the gradients in Fig. 6 such that

CU ¼ aþ by2¼ CU � bs2y. (7)

In Fig. 7 symmetry has been assumed, about a reference wind direction of zero, and boththe calculated points and their mirror image included.

Comparing Figs. 4 and 7 shows that accounting for the changes in the variance of winddirection reduces the scatter in the data. In addition, it shows that variations in direction

1

0

-10 0.04 0.08

Variance of wind Direction (radians^2))

u V

eloc

ity C

oeffi

cien

t

y = 4.91x + 0.02

y = 4.90x - 0.23

y = 0.71x - 0.41y = 0.71x - 0.40

ACLinear (A) Linear (B)

Linear (D)Linear (C)

BD

Fig. 6. Five-minute mean u velocity coefficients at positions A–D for runs with mean wind directions �3�oyo3�

in relation to the variance of wind direction.

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1

0

-1-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Reference Wind Direction (radians)

u V

elo

city

Co

effi

cien

t y = 4.51x2 + 0.02

y = 4.63x2 - 0.23

y = 2.89x2 - 0.41

y = 0.78x2 - 0.40

A, x/h = 0.24 B, x/h = 0.41C, x/h = 0.59 D, x/h = 0.76Poly. (A, x/h = 0.24) Poly. (B, x/h = 0.41)Poly. (C, x/h = 0.59) Poly. (D, x/h = 0.76)

Fig. 7. Five-minute mean u velocity coefficients at various positions across the cube roof expected to occur if the

wind direction was steady.


do significantly alter the apparent position of the reattachment point when the wind isnormal to the building (01). In Fig. 4 it appears that with the mean wind normal to thebuilding the mean velocity at both points A and B are definitely reversed (negative velocitycoefficients), whereas at point D the mean velocity coefficient is consistently positive. It isonly at point C that the observations vary from positive to negative and hence it appearsthat the reattachment is near this point. However, Fig. 6 shows that the mean velocitycoefficient at point C has a near linear dependency on the variance of direction. The linearfit shows that if the wind direction was almost constant (low variance), then the velocitycoefficient would be near to �0.23 and it is the linear fit for point D which extrapolates to avalue near zero. Hence, if the wind direction was almost constant the reattachment pointwould be closer to point D, at x=h ¼ 0:76, than point C, at x=h ¼ 0:59. This point isreinforced by Fig. 7, where the effects of fluctuating wind direction have been removed.The differences between Figs. 4 and 7 indicate the sort of effects that may be observed

when comparing wind-tunnel and full-scale data. Due to width constraints, manyboundary layer wind tunnels exhibit lower lateral turbulence intensities than observed infull scale. As a result, the variance of wind directions may be considerably smaller. Forexample, Richards et al. [19] have modelled the Silsoe Cube in the University of Aucklandwind tunnel and obtained similar mean pressure distributions, even though the standarddeviation of wind directions was only 5.61, in comparison with the full-scale average of 101,which means that the variance of wind direction was only about one quarter of that in fullscale. In such cases, wind-tunnel observations of reattachment may be expected to becloser to Fig. 7 than Fig. 4. A similar situation may also occur with CFD modelling wherethe turbulence models used may not adequately reflect the systematic affects that lead tothe differences between Figs. 4 and 7.The coefficients of the curve fits in both Figs. 6 and 7 can be used to give independent

estimates of the parameters ‘a’ and ‘b’ in the quasi-steady function, Eq. (5). These aresummarised in Table 3 where there is close agreement on the value of ‘a’ and only slightdifferences in the ‘b’ estimates.

ARTICLE IN PRESS

Table 3

Estimates of the quasi-steady function coefficients of Eq. (5)

Position a b

Fig. 6 Fig. 7 Fig. 6 Fig. 7

A �0.40 �0.40 0.71 0.78

B �0.41 �0.41 3.27 2.89

C �0.23 �0.23 4.90 4.63

D 0.02 0.02 4.91 4.51

P.J. Richards, R.P. Hoxey / J. Wind Eng. Ind. Aerodyn. 94 (2006) 77–99 85

4. Unsteady flow behaviour

Although the reattachment point derived from mean velocities is significant, it does notfully represent the true situation. The mean velocity vectors shown in Fig. 5 are 5-minaverages; however, smoke visualisation has shown that this flow field is highly unsteadyand it is unusual to see a flow pattern, at any instant in time, which matches all thesevectors. The variability in the flow is illustrated in Fig. 8, which shows the scatter plots ofvector tip locations which occurred during two 15-min runs (one for positions A–D (run412) and one for E–H (run 604)). Both of these had a 15-min mean wind direction within 11of normal to the windward face. The vector scale in Fig. 8 is considerably smaller than inFig. 5 in order to minimise overlaps. The vectors shown on Fig. 8 are the 15-min averagesfrom these two runs, which are very similar to those shown in Fig. 5.

The data in Fig. 8 show that the flow at the eaves is relatively steady and does not reversecompletely at any time. In contrast, the flow at the other seven positions shows a mixtureof forward and reversed flow. The flows at the more elevated locations (F–H) areparticularly unsteady. Table 4 summarises the velocity component properties from thesetwo runs.

In Table 4, u, v, w are the instantaneous components, U, V, W the 15-min mean valuesand Vref the 15-min mean wind speed at the reference mast.

The local mean velocity magnitude coefficient (C|V|) and turbulent kinetic energy (TKE)are given by

CjV j ¼ðU2 þ V 2 þW 2Þ

0:5

V ref, (8)

TKE

V2ref

¼0:5 ðu�UÞ2 þ ðv� V Þ2 þ ðw�W Þ2� �

V 2ref

. (9)

The data in Table 4 show that while the mean transverse (v) velocities are all close tozero, their instantaneous values typically range from �1.5Vref to 1.5Vref, with position Ghaving an even broader range. Similarly, the u velocity components have ranges thatalways include both positive and negative values. At position A the maximum positivevalue is only 0.54Vref but the negative range extends to �1.63Vref. Positions B–D all have atotal range of about 3Vref but progressively the range becomes more positive with therange at D being primarily positive. The broadest ranges and the highest turbulence levelsare at positions F–H where although the ranges are principally positive, all ranges extend

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Table 4

Velocity coefficient properties at the eight measuring positions from two 15-minute runs

u/Vref v/Vref w/Vref General mean values

Mean Max Min St. dev. Mean Max Min St. dev. Mean Max Min St. dev. C|V| Elev (1) TKE/Vref2

Run 412

Ref 1.00 1.82 0.28 0.24 �0.05 0.62 �0.77 0.19 0 0.44 �0.39 0.1 1.00 0.00 0.05

A �0.38 0.54 �1.63 0.24 0.01 1.31 �1.48 0.27 0.00 0.43 �0.56 0.09 0.38 �0.17 0.07

B �0.32 1.14 �1.42 0.30 �0.01 1.29 �1.40 0.36 0.01 0.58 �0.70 0.11 0.32 �1.48 0.12

C �0.09 1.50 �1.47 0.37 �0.03 1.42 �1.55 0.42 0.00 0.56 �0.67 0.11 0.09 0.65 0.16

D 0.16 1.72 �1.14 0.38 0.00 1.33 �1.21 0.40 0.00 0.48 �0.48 0.10 0.16 �1.65 0.16

Run 604

Ref 1.00 2.02 0.07 0.26 �0.01 1.13 �0.88 0.22 0.01 0.51 �0.50 0.12 1.00 0.57 0.07

E 0.58 1.50 �0.22 0.22 0.03 1.62 �0.82 0.30 0.56 1.36 �0.18 0.17 0.81 43.99 0.09

F 0.38 2.79 �1.02 0.59 0.03 1.53 �1.25 0.30 0.11 1.10 �1.10 0.22 0.40 15.48 0.24

G 0.18 2.18 �1.33 0.54 0.08 2.03 �1.97 0.42 �0.08 1.07 �1.23 0.29 0.21 �25.25 0.28

H 0.52 2.31 �1.11 0.51 0.08 1.84 �1.39 0.45 �0.09 1.02 �1.35 0.25 0.53 �10.39 0.27

Fig. 8. Fifteen-minute velocity vector scatter diagrams on the central vertical plane for runs with a mean wind

direction nearly normal to the windward face (�11oyo11).


below �1Vref. With the vertical velocity components the mean values are near zero atpositions A–D and due to the proximity of the roof the range and standard deviations arealso quite small. At the more elevated positions F–H, the range is broader, at typically�1Vref to +1Vref, but is still much smaller than the other two components. Murakami etal. [12] show that the lack of separation with CFD simulations using the standard k–eturbulence model is caused by the very high turbulence levels near the windward eaves.Their wind-tunnel data showed that the highest turbulence levels occurred above the centreof the cube roof, where the level was about three times that in the approach flow, and notat the windward eaves. Table 4 also shows the highest turbulence level occurs at positionG, 0.1h above the centre of the roof, where the turbulence level is about four times that inthe approach flow. The TKE level at the windward eaves (position E) is only marginallyhigher than at the reference mast, with most of this coming from increased transverse (v)fluctuations. Of the three turbulence models reported by Richards et al. [2] and depicted inFig. 1, the k–e model over-estimated the turbulence level at the windward eaves, the MMKmodel under-estimated it and the RNG model approximately matched the Silsoe fieldresults. However, all of these CFD turbulence models underestimated the turbulence levelsabove the roof.


While all components contribute to the turbulent kinetic energy levels, it is the u velocitycomponent that is most obviously linked to flow separation or reattachment. Thedistributions of u velocities at positions E–H are illustrated in Fig. 9. This illustrates thegreater ranges and skewed distribution of u velocities at positions F–H in comparison withthe almost normally distributed velocities at the windward eaves (E). At point G the flow isreversed approximately 50% of the time, although when combined with the longer positivetail this still results in a positive mean value.

Observation of typical time histories, such as that shown in Fig. 10, shows that thevelocities at positions F–H are reasonably correlated. There are clearly times when all threevelocities are either significantly positive or even negative. Smoke visualisation of the flowover the roof reveals the formation and then shedding of vortices. At times, these vorticesgrow to a large size and cause reversed flow to some depth over the entire roof. It issuggested that time 180 s in Fig. 10 is one such time as the u component at positions F, Gand H are all negative. At other times the vortex disappears and the flow appears to bemomentarily not separating.

0

1

2

-2 -1 0 2u Velcoity Coefficient

pd

f

E- EavesFG- Roof centre

E- Eaves

G- Roof centre

H

(a) (b)

0

0.2

0.4

0.6

0.8

1

-2 -1 0 2u Velocity Coefficient

cdf

F

H

1 3 31

Fig. 9. u Velocity components at positions E–H during run 604, (a) probability density function (pdf) and (b)

cumulative distribution function (cdf).

2

0

-2100 120 140 160 180 200

Time (seconds)

u V

elo

city

Co

effi

cien

t

F G H

Fig. 10. Sample time history for positions F–H from run 604.

ARTICLE IN PRESS

-1.5

0

1.5

0 60 120 180 240 300 360

Wind Direction (degrees)

Vel

oci

ty C

oef

fici

ent

UGWGFitted Mean UGFitted Mean WGInstantaneous UGInstantaneous WG

Fig. 11. Five-minute mean u and v velocity component coefficients at position G, above the roof centre, and the

corresponding fitted mean and derived instantaneous functions.


It was shown in Section 3 that some of the processes affecting the velocity at thesepositions can be accounted for by treating the flow as a quasi-steady processes, that isassuming that changes in velocity are directly caused by variations in the wind speed anddirection. However, the simple quadratic model used in Section 3 is limited and so a morecomplex model is used in the following analysis. It may be noted that with position G,above the centre of the roof, there are multiple planes of symmetry and the v componentmeasurements are equivalent to u component measurements at an angle 901 lower. Inaddition, measurements were undertaken with the cube oriented normal to the prevailingwind and at 7451. This gave the equivalent of 846 5-min averages within each 901 sector.The resulting data are shown in Fig. 11, where in order to reduce clutter only the u and w

components are shown.Fourier series of the form

CiðyÞ ¼X8k¼0

aik cosðkyÞ þ bik sinðkyÞ, (10)

where i ¼ u, v or w indicates the particular component, have been fitted to the 5-min meanvelocity coefficient data by using a least-squares method. However, it must be recognisedthat a data point lying on this fitted line represents the average of data created by a rangeof instantaneous wind directions. Analysis of wind records at the site show that thedirection variations are approximately normally distributed about the mean angle suchthat during each 5-min block:

PðyÞ ¼1

syffiffiffiffiffiffi2pp exp

�ðy� yÞ2

2s2y

� �. (11)

During the runs the standard deviation of wind directions sy, ranged from 10.71 to 231,with an average of 13.51. Richards et al. [8] have shown that there can exist acorresponding instantaneous function which when combined with normally distributed


wind directions would generate the fitted mean function. The instantaneous function takesa similar form:

CiðyÞ ¼X8k¼0

aik cosðkyÞ þ bik sinðkyÞ (12)

with aik ¼ aik expð12k2s2yÞ, bik ¼ bik expð12k

2s2yÞ and sy ¼ 0:236 radians (13.51). Fig. 11shows both the fitted mean and instantaneous functions for the u and w velocitycomponents for position G.

Data from position E showed that for wind directions in the range 2901(�701) to 701 thev velocity coefficient was linearly proportional to the wind angle. In addition, the u and w

components had zero derivatives and small second derivatives with respect to direction at01. As a result these coefficients are not significantly affected by directional variations.Hence, the instantaneous values at position E have been used as indicators of the variationin wind speed and direction. These have been combined with the instantaneous functionsshown in Fig. 11 to give the corresponding quasi-steady variations in all velocitiescomponents at position G. These are shown in Fig. 12 along with the measured values. Itshould be noted that the quasi-steady model only accounts for variations in wind speedand direction and does not account for effects caused by vertical fluctuations, since thedependence of the flow at position G on the vertical fluctuations is unknown.

Comparison of Figs. 12(a) and (c) reveals that the transverse velocity componentprimarily reflects the variations in wind direction with the most obvious features being the

Fig. 12. (a) Instantaneous velocity coefficient and direction derived from measurements at position E, (b)–(d) u, v

and w measured and quasi-steady velocities at position G. All graphs show the first 400 s of run 604.

ARTICLE IN PRESS

0

0.5

1

1.5

0.1 10

Frequency (Hz)

Sp

ectr

al R

atio

Measured/Q-S

SDOF

1

Fig. 13. Spectral ratio of the measured u spectrum to the quasi-steady u spectrum for position G. Also shown

(solid line) is the spectral ratio for a simple single degree of freedom system with a natural frequency of 2Hz and a

damping factor of 0.5.


excursions to negative angles between times 100 and 200 s and the excursions to positiveangles between 300 and 400 s. Figs. 12(b) and (d) show that during both of these periodsthe u velocity component is more positive and the w velocity component consistently morenegative than at other times. These behaviours occur in both the measured and quasi-steady data. The highest correlation between the quasi-steady and measured data occurswith the v component, with the peak correlation coefficient of 0.57 occurring with a timedelay of 0.7 s between positions E and G. This delay is approximately equal to the distancebetween them divided by the mean wind speed.The u and w components are less well correlated with correlation coefficient, of 0.4 and

0.1, respectively. It is clear that the measured u component is often more negative than thatgiven by the quasi-steady model and that the vertical fluctuation has a much broader rangethan the model suggests. It appears that in general, the quasi-steady model correctlycaptures the longer duration fluctuations but that there are short-duration processes thatare not included. Spectral analysis of the u component showed significant coherencebetween the measured and quasi-steady time histories for frequencies less than 0.1Hz. Inaddition, comparing the measured u spectrum with the quasi-steady u spectrum in the formof the spectral ratio (Fig. 13), showed similar levels at low frequencies, enhanced levelsaround 1Hz and diminished levels at the highest frequencies. This behaviour is similar tothat of a single degree of freedom system as illustrated. It is suggested that this occursbecause the separation vortex responds in a dynamic manner to the fluctuating stimuli butalso has some inertia and so does not respond at the highest frequencies. This concept willbe discussed further in Section 6.

5. Instantaneous reattachment point

With the flow normal to the windward face the flow at positions F–H is only reversedoccasionally, however the flow at positions A–C is reversed most of the time and just under40% of the time at position D. If a negative instantaneous u velocity is taken as indicating


that the reattachment length reaches at least to that measuring position, then the data inFig. 14 may be interpreted as indicating that the reattachment length is 40.24h 95%,40.41h 87%, 40.59h 64% and 40.76h 38.5% of the time. These results are consistentwith the 5-min mean data which suggested a mean reattachment length of about 0.6h, sinceinterpolating between the cumulative distributions for positions C and D gives the positionwhere the flow is reversed 50% of the time as x=h ¼ 0:67. However, it may be noted thatthe velocity at position A, at only x=h ¼ 0:24, is positive 5% of the time, and that atposition D the velocity is negative 38.5% of the time. Hence, the instantaneousreattachment length is at times less than 0.24h and is quite often greater than 0.76h.

This suggests that the instantaneous flow pattern ranges from little or no separationthrough to no reattachment. This concept is reinforced by a study of the time histories,such as those in Fig. 15, where between times 110 and 120 s there are several instanceswhen all four velocities are positive and then between 140 and 150 s there are two periodswhen all four velocities are negative.

It was shown in Fig. 4 that as the wind direction moved away from normal the 5-minmean u velocities at positions A–D all became more positive. Fig. 16 shows thecorresponding changes in the probability distributions for position C taken from three

0

1

2

-2 -1 0u Velocity Coefficient u Velocity Coefficient

pd

f

A,0deg

B,0deg

C,0deg

D,0deg

0

0.2

0.4

0.6

0.8

1

-2 -1 0 2

cdf

(a) (b)21 1

A,0deg

B,0deg

C,0deg

D,0deg

Fig. 14. u Velocity components at positions A–D during run 412, (a) probability density function (pdf) and (b)

cumulative distribution function (cdf).

2

0

-2100 120 140 160 180 200

Time (seconds)

u V

elo

city

Co

effi

cien

t

A B C D

Fig. 15. Sample time history for positions A–D from run 412.

ARTICLE IN PRESS

0

0.5

1

0 0.2 0.4 0.6 0.8 1Position of reattachment from leading edge

(Fraction of roof width)

cdf

0deg

-11.6deg

6.9deg

Fig. 17. Cumulative distribution functions for the reattachment position during three 15-min runs with various

mean wind directions.

0

1

2

-2 -1 0 1 2u Velocity Coefficient u Velocity Coefficient

pd

f

C, 0deg

C, 6.9deg

C, -11.6deg

0

0.2

0.4

0.6

0.8

1

-2 -1 0 1 2

cdf

C, 0deg

C, 6.9deg

C, -11.6deg

(a) (b)

Fig. 16. u Velocity components at positions C during three 15-min periods with various mean wind directions

(runs 412(01), 356(6.91) and 383(�11.61)), (a) probability density function (pdf) and (b) cumulative distribution

function (cdf).


15-min runs. As the wind direction becomes more oblique the complete distribution movestowards more positive values. Further analysis of these three runs was carried out in orderto estimate the instantaneous reattachment length. This was carried out by interpolating,and extrapolating, between the four instantaneous velocities in order to find the position ofzero u velocity at each instant. The resulting cumulative distributions of reattachmentlengths are shown in Fig. 17.With the mean wind normal to the windward face (01) the median reattachment position

is at x=h ¼ 0:66, but it appears that the flow does not reattach at all for 20% of the time. Itmay be noted that at x=h ¼ 0:67 the flow is equally likely to be in the forward or reverseddirection; however, due to the skewed velocity pdf the mean velocity will be positive. In themore oblique flow cases, the reattachment lengths are shorter, with the median near thecentre of the roof with a mean wind direction of �11.61, which is consistent with the meanflow observations.

6. Inertia of the flow system

Figs. 16 and 17 illustrate the changes in flow behaviour as the 15-min mean flowdirection becomes more oblique. Some of these effects are apparent with much shorter


duration directional variations. Fig. 18 illustrates in two ways the joint probabilitydistribution function for position D during a 15-min period when the mean wind directionwas near zero. The approximate symmetry of the run can be seen in the similarity ofsections at v velocities of equal magnitude but opposite sign (Fig. 18(b)). These sectionsalso show the shift towards more positive u velocities as the v component increases.

From Fig. 4 it appeared that when the 5-min mean wind direction is near 01 then the 5-min mean u velocity coefficient for position D is approximately 0.2; however, Fig. 7showed that if the direction was held steady then one might expect the value to be only justpositive. Fig. 18 shows a similar result, but goes one step further in that if theinstantaneous wind direction is near 01, as indicated by a near zero v velocity, thenthe peak of the joint pdf is about �0.1. In spite of this, with the skewed distribution theaverage of the u velocity coefficients, at times when the v velocity coefficient is near zero,results in a conditional mean near zero. However, in reality the wind direction does not

1

0.75

0.5

0.25

0

-0.25

-0.5

-0.75

-1-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

v Velocity Coefficient

v Velocity Coefficient

v VelocityCoefficient

Join

t P

rob

abili

ty

u V

elo

city

Co

effi

cien

t

1.2-1.4

1.-1.2

0.8-1

0.6-0.8

0.4-0.6

0.2-0.4

0.0.2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-0.75

-0.5

-0.25

0.25

0.5

0.75

0

(a)

(b)

1.4

1.2

1

0.8

0.6

0.4

0.2

0

Fig. 18. (a) Joint probability distribution function of the u and v velocity components at position D during run

412 and (b) selected sections through the function.


remain near zero and so when all the directions, and corresponding v values, are includedthen the resulting 5-min average for this run is about 0.15.These observations clearly illustrate the difficulty in defining the reattachment position

when the wind is perpendicular to the windward face. From 5-min mean data it appearsthat position D, three-quarters of the way across the roof, is definitely beyond thereattachment point as in all observations the mean u velocity is positive. However, itappears that for instants when the velocity at position D is aligned with the vertical centralplane (v ¼ 0) then on average D appears to be at the reattachment point. But even this isnot the complete picture since the condition most likely to occur at such instants, the peakof the uv joint probability, is one where the flow is reversed and so the reattachment isbeyond position D.The data contained in the joint probability distribution can also be used as a link with

the quasi-steady function, Eq. (5), introduced in Section 3. In Fig. 19 the various lines arethe result of fitting quadratic equations to the 5-min mean data in Fig. 4 and the quasi-steady data in Fig. 7. Diagrams similar to Figs. 4 and 7 can be created for the v velocitycomponent but over this range of angles these are simply linear relationships withgradients of 0.0262 and 0.229/degree for positions C and D, respectively. Since the secondderivative is zero for the v component then the corresponding quasi-steady function havethe same gradients. These gradients have been used so that the u velocity functions can beplotted against the corresponding v velocity coefficients in order to allow comparison withconditional averages. The conditionally averaged u velocities in Fig. 19 are simply theaverage value of the u component at all instants during a 15-min period when the v velocitycoefficient lies within a narrow band (70.0625) around particular selected values (�0.5,�0.375, �0.25, etc.). Also shown in Fig. 19 are the three 5-min averages that made up each15-min period.Fig. 19(a) shows data from a period when the mean direction was near 01. For both

positions C and D the 5-min means (open symbols) lie close to the mean line while theconditional averages (closed symbols) are close to the quasi-steady line around 01 but

-0.4

-0.2

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.5 -0.3 -0.1 0.1 0.3 0.5 -0.5 -0.3 -0.1 0.1 0.3 0.5

V Velocity Coefficient

V Velocity Coefficient

U V

elo

city

Co

effi

cien

t

U V

elo

city

Co

effi

cien

t

C Fit to meansD Fit to meansC Quasi-steadyD Quasi-steadyC Cond Run 412D Cond Run 412C Mean Run 412D Mean Run 412

0

0.2

0.4

0.6

0.8

1D Fit to meansD Quasi-steadyD Cond Run 412D Cond Run 356D Cond Run 383D Mean Run 412D Mean Run 356D Mean Run 383

(a) (b)

Fig. 19. Mean, quasi-steady and conditionally averaged functions for the u velocity component: (a) at positions C

and D during a period when the wind direction is near normal (run 412) and (b) for position D during normal and

more oblique periods (runs 412(01), 356(6.91) and 383(�11.61)).


deviate away from this with more oblique flows. Similar behaviour has been observed byBanks and Meroney [10] during their wind-tunnel study of conical vortices on the roof of alow-rise building during cornering winds. They describe this phenomenon as a form ofhysteresis, where momentary excursions fail to establish the vortex structure that wouldoccur if a particular wind direction occurred for a longer period. The results in Fig. 19(b)suggest that this might be better described as inertia of the flow system. In this figure thedata from three 15-min periods, with differing mean wind directions, is given. In each casethe wind direction tends to dwell around the mean direction and make occasionalexcursions on either side of this. In all three runs the conditionally average values are closeto the quasi-steady function around the mean wind direction but deviate away from it onthe extremes. It does appear that time is needed in order to establish flow structures andthat short duration directional excursions only partially influence the form of the flow.

7. The effect of reattachment length on pressure distribution

In earlier papers [2,13], and as illustrated in Fig. 20, it has been shown that with the windperpendicular to the windward face the pressures measured on the Silsoe cube (Silsoe 01)are similar in character to those obtained in wind tunnels with turbulent boundary layers[1,5,14–16]. The pressures on the windward wall are all very similar; however, the roofpressures are more variable. All of these have the shape associated with reattaching flowand it is probable that the differences are due to a combination of differences in velocityprofile, turbulence structure and possibly Reynolds number.

Comparisons between published data show that turbulence levels do have an effect onflow reattachment and hence on the pressure distribution. In addition, the results presentedin this paper show how the flow over the roof behaves in a manner that is a combination ofquasi-steady effects and dynamic effects, both in terms of introducing fluctuations atcertain frequencies and not responding to short duration directional variations. Hence, the

-2

-1.5

-1

-0.5

0

0.5

1

0 2Position

Mea

n P

ress

ure

Co

effi

cien

t C

P

1 3

Silsoe 0°

Castro & Robins TurbulentHolscher & NiemannBainesMurakami & MochidaHunt 200,180B

3

2

0

Fig. 20. Pressure distributions on the central vertical plane from the Silsoe field study and various wind tunnel

studies for the wind normal to the windward face.

ARTICLE IN PRESS

y = -5E-08x - 0.3939

y = -8E-08x - 0.854

-1.5

-1

-0.5

00.E+00 1.E+06 2.E+06 3.E+06 4.E+06 5.E+06

Reynolds Number

Mea

n P

ress

ure

Co

effi

cien

t C

p

Tap 7, x/h =0.25

Tap 10, x/h=0.75

Linear (Tap 10, x/h=0.75)

Linear (Tap 7, x/h =0.25)

Fig. 21. Pressure coefficient for Taps 7 and 10 in relation to the Reynolds number from runs with mean wind

directions within 31 from normal to the windward face.


observed flow and pressures will be sensitive to the lateral as well as the longitudinalturbulence.It is generally assumed in wind engineering that bluff body flows, such as that on the

roof of the cube, are not affected by changes in Reynolds number. Richards et al. [17],however, presented some evidence from the Silsoe field results that showed some Reynoldsnumber dependency. Further, Fig. 21 shows the pressures at Taps 7 and 10, approximatelybelow positions A and D, from runs with wind directions within 31 of normal to thewindward face in relation to the Reynolds number. Plotting the same data against eitherlongitudinal or transverse turbulence intensities revealed no apparent pattern and so itdoes appear that there may be a Reynolds number effect.It appears from Fig. 21 that the pressure at both of these taps become about 20% more

negative between Reynolds numbers of 2� 106 and 4� 106. Extrapolation of these linesdown to typical wind-tunnel Reynolds numbers in the range 104–105 would similarlysuggest that the wind tunnel might show pressures that are 20% lower than observed atSilsoe. This could partially explain the differences between the pressures from Murakamiand Mochida [5] and Silsoe, where the turbulence intensity levels are fairly similar but thewind-tunnel Reynolds number is much lower. What is not clear is exactly how the flowchanges. For the other wind-tunnel studies, shown in Fig. 20, both the turbulence intensityand the Reynolds numbers are different from those at Silsoe and so it is not possible todiscriminate between these two effects.One possible process is through changes in the reattachment length. This has been

partially investigated by tilting the cube into the wind. In such a situation, it may beexpected that the reattachment length will be reduced. Although the velocities on the roofhave not been measured with the cube tilted, the velocities close to the sidewall have undervarious wind directions. Richards and Hoxey [18] show that the reattachment point atmid-height on the sidewall moves forward from x=h ¼ 0:83 at 01, to 0.5 at 81 and to 0.17 at161. If a similar behaviour occurs on the roof then tilting the cube 51 forward may shortenthe reattachment length by about 0.2 h. The effects of tilting the cube on the pressuredistribution are shown in Fig. 22 at both 2.51 and 51 pitch.Fig. 22 shows that with the cube tilted forwards the pressures at all positions on the roof

reduce and that with 51 pitch the Silsoe cube distribution is very close to that measured by


Murakami and Mochida [5]. It therefore appears that if the reattachment length is longerat high Reynolds numbers then this could explain some of the differences between theSilsoe field pressure data and typical wind-tunnel data. However, this would be incontradiction to the observed reattachment lengths, since Murakami and Mochida show amean reattachment length of 0.7h.

An analysis has therefore been conducted to look at the changes in velocity coefficientwith Reynolds number. The process used is similar to that for Fig. 21 except that in orderto remove the observed effects of direction variances the data plotted in Fig. 7 has beenused rather than that in Fig. 4. The results are shown in Fig. 23.

-2

-1.5

-1

-0.5

0

0.5

1

0Position

Pre

ssu

re C

oef

fici

ent

CP

Castro & Robins Turbulent

Murakami & Mochida

Silsoe zero Pitch

Silsoe 2.5° pitch

Silsoe 5° pitch

03

21

321

Fig. 22. Pressure distributions on the central vertical plane from the tilted Silsoe field study and two wind tunnel

studies for the wind normal to the windward face.

1

0

-10.E+00 1.E+06 2.E+06 3.E+06

Reynolds Number

u V

elo

city

Co

effi

cien

t

A, x/h = 0.24

y = 1.40E-08x - 1.12E-02

y = 1.96E-08x - 2.70E-01y = 2.16E-08x - 4.48E-01y = 2.03E-08x - 4.56E-01

C, x/h = 0.59

Linear (A, x/h = 0.24)

Linear (C, x/h = 0.59)

B, x/h = 0.41

D, x/h = 0.76

Linear (B, x/h = 0.41)

Linear (D, x/h = 0.76)

Fig. 23. Five-minute mean u velocities coefficients, corrected to remove directional variance effects, at positions

A–D from runs with a mean wind direction within 31 of normal to the windward face as a function of Reynolds

number.


Although the results in Fig. 23 do show a consistent pattern, in that all four velocitycoefficients become more positive with increased Reynolds numbers, these changes are toosmall and are in the opposite direction to that suggested by the pressure changes in Fig. 21.It appears from these results that as the Reynolds number increases the quasi-steadyreattachment length reduces from 0.76h at a Reynolds number of 1� 106 to perhaps 0.74h

at 3� 106. The differences between wind-tunnel and field pressure distributions thereforeremain unexplained.

8. Conclusions

Five ultrasonic anemometers have been used to measure flow velocities above the roof ofa 6m cube and at a reference point. The results show that for a wind direction normal toone face (01) the mean reattachment point on the cube centreline is near x=h ¼ 0:6, basedon an interpolation of where the mean near roof streamwise velocity is zero. However, it isshown that this observation is influenced by the variations in wind direction and that thequasi-steady reattachment point is nearer to x=h ¼ 0:75, that is the position ofreattachment that would occur if the wind direction was 01 and constant. It is clear thatthe mean reattachment length depends upon the onset flow direction and decreases as thedirection becomes more oblique.It is also noted that the flow over the roof is highly unsteady, with the velocities at all

measuring points above the roof in the forward or reversed directions for part of the time.The flow statistics show that, with the mean wind normal to the windward face, aninstantaneous interpolated estimate of the reattachment point has a median value ofx=h ¼ 0:66, and ranges from negligible separation to no reattachment for 20% of the time.Comparison between the measured data and a quasi-steady model shows that some ofthese variations are related to fluctuations in the onset wind speed and direction, but theyare also influenced by the dynamic response of the separation vortex system. This meantthat certain frequencies of the wind spectrum, around 1Hz, were slightly amplified whereasfrequencies above 10Hz were filtered out as a result of the inertia of the vortex system.By analysing the joint probability distribution for u and v velocities at a position x=h ¼

0:76 it is shown that there are significant differences between the overall mean u velocity,the conditional mean at instants when the v velocity is zero and the mode under the sameconditions. For this position the mean u velocity coefficient, during a 15-min period withmean wind direction normal to the windward face, was 0.15. Whereas the mean value, atinstants when the v velocity was zero, was nearer to zero, the peak of the joint probabilitydistribution on the v ¼ 0 line was near �0.1. These results suggest that this position iseither outside, on the edge or inside the separation vortex depending on the form ofanalysis. In reality, the unsteadiness of the flow means that all three conditions do occur, tovarying degrees, during the observation period. Conditional averaging of the u velocitiesaround a range of v values resulted in an instantaneous behaviour similar to that expectedfrom a quasi-steady approach. However, the flow also exhibited a form of inertia, in thatthe conditional mean at a particular angle, which would only occur for a short time duringa run, differed from that which was observed in another run where the particular angle wasnear the mean direction.The effects of reattachment length on the pressure distribution have been briefly

considered but it has been shown that these do not account for the differences between theSilsoe field data and typical wind-tunnel results. It is suggested that this may be related to


Reynolds number but it does not appear that this occurs through changes in reattachmentlength.

References

[1] I.P. Castro, A.G. Robins, The flow around a surface-mounted cube in uniform and turbulent streams,

J. Fluid Mech. 79 (Part 2) (1997) 307–335.

[2] P.J. Richards, A.D. Quinn, S. Parker, A 6m cube in an atmospheric boundary layer flow, Part 2.

Computational solutions, J. Wind Struct. 5 (2–4) (2002) 177–192.

[3] Y. Ogawa, S. Oikawa, K. Uehara, Field and wind tunnel studies of the flow and diffusion around a model

cube (a)—1. Flow measurements, Atmos. Environ. 17 (6) (1983) 1145–1159.

[4] Y. Ogawa, S. Oikawa, K. Uehara, Field and wind tunnel studies of the flow and diffusion around a model

cube II. Nearfield and cube surface flow and concentration patterns, Atmos. Environ. 17 (6) (1983)

1161–1171.

[5] S. Murakami, A. Mochida, 3-D numerical simulation of airflow around a cubic model by means of the k–emodel, J. Wind Eng. Ind. Aerodyn. 31 (1988) 283–303.

[6] A.J. Minson, C.J. Wood, R.E. Belcher, Experimental velocity measurements for CFD validation, J. Wind

Eng. Ind. Aerodyn. 58 (1995) 205–215.

[7] H. Kawai, Pressure fluctuations on square prisms—application of strip and quasi-steady theories, J. Wind

Eng. Ind. Aerodyn. 13 (1983) 197–208.

[8] 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 (1995) 359–367.

[9] C.W. Letchford, R. Marwood, On the influence of v and w component turbulence on roof pressures beneath

conical vortices, J. Wind Eng. Ind. Aerodyn. 69–71 (1997) 567–577.

[10] D. Banks, R.N. Meroney, The applicability of quasi-steady theory to pressure statistics beneath roof-top

vortices, J. Wind Eng. Ind. Aerodyn. 89 (2001) 569–598.

[11] P.J. Richards, R.P. Hoxey, Quasi-steady theory and point pressures on a cubic building, J. Wind Eng. Ind.

Aerodyn. 92 (14–15) (2004) 1173–1190.

[12] S. Murakami, A. Mochida, Y. Hayashi, S. Sakamoto, Numerical study of velocity-pressure field and wind

forces for bluff bodies by k–e, ASM and LES, J. Wind Eng. Ind. Aerodyn. 41–44 (1992) 2841–2852.

[13] R.P. Hoxey, P.J. Richards, J.L. Short, A 6m cube in an atmospheric boundary layer flow: Part 1. Full-scale

and wind-tunnel results, J. Wind Struct. 5 (2–4) (2002) 165–176.

[14] W.D. Baines, Effects of velocity distribution on wind loads and flow patterns on buildings, in: Proceedings of

the Symposium on Wind Effects on Buildings and Structures, vol. 16, NPL, UK, 1963.

[15] A. Hunt, Wind-tunnel measurements of surface pressures on cubic building models at several scales, J. Wind

Eng. Ind. Aerodyn. 10 (1982) 137–163.

[16] N. Holscher, 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.

[17] P.J. Richards, R.P. Hoxey, J.L. Short, Wind pressures on a 6m cube, J. Wind Eng. Ind. Aerodyn. 80 (2001)

1553–1564.

[18] P.J. Richards, R.P. Hoxey, Unsteady flow on the sides of a 6m cube, J. Wind Eng. Ind. Aerodyn. 90 (12)

(2002) 1855–1866.

[19] P.J. Richards, R.P. Hoxey, B.D. Connell, D.P. Lander, Wind tunnel modelling of the Silsoe Cube, in: Fourth

European and African Conference on Wind Engineering, Prague, Czech Republic, 11–15 July 2005.

Richards 2006 Journal of Wind Engineering and Industrial Aerodynamics

Documents

wind tunnel

mean wind

mean reattachment position

flow reattachment

wind direction almostnormal

effects of reattachment

present windtunnel data

pressure differences