characteristics of alongwind loads on rectangular...

BBAA VI International Colloquium on: Bluff Bodies Aerodynamics & Applications

Milano, Italy, July, 20-24 2008

CHARACTERISTICS OF ALONGWIND LOADS ON RECTANGULAR CYLINDERS IN TURBULENT BOUNDARY LAYER FLOWS

Chii-Ming Cheng∗, Ming-Shu Tsai†, Jenmu Wang＃

∗ Department of Civil Engineering, Tamkang University 151 Ying-chuan Road Tamsui,Taipei County Taiwan 25137

∗ e-mail: [email protected], †[email protected], ＃[email protected]

Keywords: Alongwind load, Rectangular Cylinder, Wind Tunnel, Boundary Layer, Spatial Coherence.

1 INTRODUCTION The alongwind loads of rectangular cylinders in turbulent boundary layers play important

role in most building wind codes. This is an important topic which has been studied by many researchers, either by wind tunnel experiment, numerical simulation and analytical methods. Most of the previous experimental investigations were based on high frequency force balance (HFFB) or limited number of pressure sensors. Among them, Kareem[1,1990], Wang and Cheng [2, 2003], Gu et al. [3, 2004] used HFFB technique to construct tall building’s aerody-namic database.. Although the total force of the rectangular cylinder could be obtained through these procedures, details of the wind loads were lost due to the limitation of instru-mentations. In the past ten years or so, high speed electronic pressure scanner system gradu-ally becomes affordable and standard laboratory equipment. The high speed, multi-channel sampling capability can produce complete records of fluctuating wind pressure over the entire structural surface. Not only the conventional local wind pressure data or the global wind force characteristics can be obtained, the detailed load pattern and spatial correlations can also be acquired with little effort. This experimental procedure has been adopted by several research-ers, Kareem [4, 1997], Lin et al. [5, 2005], to establish buildings aerodynamic database.

This article presents partial results of the experimental part of a research project on the tall building design wind loads. The semi-empirical formulation of the alongwind equivalent static design wind load was proposed, Cheng [6,2005], in which, several parameters such as wind load coefficients on windward and leeward faces of the building, the longitudinal, lateral and span-wise spatial correlations. These parameters need to be obtained in turbulent bound-ary layers that can properly representing wind fields in urban, suburban and open country en-vironments.

1

2 EXPERIMENT SETUP In order to investigate wind load characteristics of rectangular shaped tall buildings, pres-

sure models were built and tested in a wind tunnel with test section of 17m(L) × 2m(W) × 1.5m(H). Three turbulent boundary layer flows, designated by BL-A, BL-B, BL-C, with power

Chii-Ming Cheng, Ming-Shu Tsai, Jenmu Wang

law index α=0.32, 0.25, 0.15, respectively, were generated to represent wind profiles over ur-ban, suburban and open country terrains, shown in Figure 1. The geometry variations of the pressure models are: aspect ratio H /

Fig. 1. Mean wind velocity and turbulence intensity profiles of BL-A, BL-B and BL-C.

0.0

0.2

1.0

0.0 0.2 0.4 0.6 0.8 1.0

U(z)/U(zg)

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 30

Iu(%)

z/z g

Experiment(BL-A) Experiment(BL-A)Power Law α=0.32 Experiment(BL-B)Experiment(BL-B) Experiment(BL-C)Power Law α=0.25

0.4

0.6

0.8

z/z g

Experiment(BL-C)Power Law α=0.15

BD = 3, 4, 5, 6, 7; side ratios D/B= 1/5, 1/4, 1/3, 1/2, 1/1, 2/1, 3/1, 4/1, 5/1, shown in Figure 2. For model with aspect ratio of 7, 380 pressure taps were installed on 15 levels along the model height; and 9 levels, 230 pressure taps for the model with aspect ratio /H BD = 3. The sampling rate was 200Hz and the sample length was 287 seconds. Besides the conventional mean and RMS base drag force coefficients

and , the windward and leeward drag coefficients,

& and & ' are calculated, where C

dC 'dC

dwC dlC 'dwC dlCd = Cdw +

Cdl. The definition of drag coefficients are defined in the following paragraph.

3 EFFECTS OF MODEL GEOMETRY ON WIND LOADS

3.1 Mean base shear coefficients Shown in Figure 3(a) to 3(c) are the mean base drag coefficients of building with various

aspect ratio and cross-sectional side ratio in three different flow fields, urban (BL-A), subur-ban(BL-B) and open country(BL-C). The mean base drag coefficients are, divided into wind-ward side and leeward side, and defined as:

2

Fig2. Setup of pressure ports and Geometry of test model.

D/B=1 D/B=1/2 D/B=1/3 D/B=1/4 D/B=1/5

WINDTop View

SideView


20.5d

dH

FCU BHρ

= 20.5

dwdw

H

FCU BHρ

= 20.5

dldl

H

FCU BHρ

= (1)

In which, dF is the mean drag force at model base； dwF and dlF are the mean base drag

force on windward and leeward side, respectively; HU is the mean wind speed at building height; ρ is the air density; H and B are building height and breadth.

The plotted result indicate that in BL-C the windward face drag, Cdw, increases slightly with aspect ratio, but remains almost invariant with respect to side ratio. In BL-A and BL-B, however, Cdw remains almost invariant with respect to both aspect ratio and side ratio.

For side ratio D/B=0.5~1.0 in BL-C, the drag on the leeward side, Cdl, decreases slightly with increase of side ratio, then increases with side ratio afterwards. This side ratio effect can be attributed to the relative location of wake vortices to the leeward face and degree of reat-tachment. Cdl decreases with increase of aspect ratio due to lower turbulence intensities for model with higher aspect ratio. As the result, the mean base drag coefficient, Cd, shows an opposite trend with Cdl, i.e., for small side ratio models, D/B<0.5~1.0, Cd increases slightly with increase of side ratio, then decreases with side ratio afterwards. Incident turbulence tends to weaken the wake structure and causes higher leeward pressure and lower drag, therefore, lower drag for lower aspect ratio. Similar trend of Cdl can be observed in both BL-A and BL-B flow fields.

3.2 RMS base shear coefficients Shown in Figure 4(a) to 4(b) are the RMS base drag coefficients of building with various

aspect ratio and cross-sectional side ratio in different flow fields. The RMS base drag coeffi-cients are, divided into windward side and leeward side, and defined as:

3

Fig.3 Variation of mean base drag force coefficients with side ratio. /H BD =○:3, □:4, △:5, ●:6, ■:7

(c) Open Terrain (a) Urban Terrain

-1.0

-0.5

0.0

0.5

1.0

1.5

0.1 1 10D/B

Cd ,

Cdw

, Cdl

Cd

Cdl

Cdw

(b) Suburban Terrain

-1.0

-0.5

0.0

0.5

1.0

1.5

0.1 1 10D/B

Cd ,

Cdw

, Cdl

Cd

Cdl

Cdw

-1.0

-0.5

0.0

0.5

1.0

1.5

0.1 1 10D/B

Cd ,

Cdw

, Cdl

Cd

Cdl

Cdw


2'0.5

dd

H

CU BHσρ

= 20.5

dwdw

H

CU BHσρ

= 20.5

dldl

H

CU BHσρ

= (2)

In which, dσ is the RMS drag force at model base； dwσ and dlσ are the RMS base drag force on windward and leeward side.

It can be observed that the windward face RMS drag coefficient, Cdw, which reflecting the incident turbulence level, decreases with aspect ratio, but remains almost invariant with re-spect to side ratio.

The RMS drag coefficient of leeward side, C’dl, represents the intensity of wake vortices. In BL-A, the C’dl exhibits a peak value at D/B=0.5. It decreases with increase of side ratio for side ratio D/B＞0.5 and decrease with decrease of side ratio for side ratio D/B＜0.5. In BL-C, the peak value of C’dl is at D/B=1.0. It decreases with side ratio for side ratio D/B＞1.0 and decreases slightly for D/B＜1.0. For model with small aspect ratio, /H BD =3, the C’dl re-mains almost invariant for D/B≦1.0. In BL-B, the trend of C’dl with respect to side ratio is in-between BL-A and BL-C.

(a) Urban Terrain

0.0

0.1

0.2

0.3

0.4

0.5

0.1 1 10D/B

C'dw

0.0

0.1

0.2

0.3

0.4

0.5

0.1 1 10D/B

C'dl

0.0

0.1

0.2

0.3

0.4

0.5

0.1 1 10D/B

Cd '

(b) Open Terrain The plotted data also indicate C’dl decreases with increase of aspect ratio for all side ratio

case in BL-A. In flow fields BL-B and BL-C, C’dl decreases with increase of aspect ratio for

Fig.4 Variation of rms base drag force coefficients with side ratio. /H BD =○:3, □:4, △:5, ●:6, ■:7

0.0

0.1

0.2

0.3

0.4

0.5

0.1 1 10D/B

C'd

0.0

0.1

0.2

0.3

0.4

0.5

0.1 1 10D/B

C'dw

0.0

0.1

0.2

0.3

0.4

0.5

0.1 1 10D/B

C'dl

4


side ratio D/B＜1.0; then, it remains almost invariant with respect to aspect ratio for D/B≧1.0. Generally speaking, C’dl has greater value at D/B＜1.0 than at D/B＞1.0. As the result, the RMS base drag coefficient, C’d, shows an combined trend of C’dw and C’dl, i.e., there is a peak of C’d at side ratio D/B=0.5~1.0 and C’d decreases with aspect ratio for all side ratio cases.

3.3 Reduced base moment spectra The effects of side ratio on the spectral characteristics of base drag in three different flow

fields, BL-A BL-B and BL-C, are shown in Figure 5(a) to 5(c). The reduced base moment spectrum is defined as:

2 2

( )ˆ ( )( )

dmdm

H

fS fS fq BH

= (3)

In which, is the base moment spectrum；f is the frequency; ( )dmS f 20.5H Hq

5

Uρ= is the velocity pressure at building height; HU is the mean wind speed at building height; ρ is the air density; H and B are building height and breadth.

Generally speaking, the cross-sectional side ratio, D/B, casts similar influences on the

spectral characteristics of base drag to the RMS drag coefficients discussed above. D/B=0.5 is the critical side ratio for models tested BL-A flow field at which building model exhibits high-est spectral value; in BL-B and BL-C, this critical side ratio that produces highest spectral value, becomes D/B=1.0. For models with side ratio less than critical value, spectral estimates increase with side ratio; whereas, when D/B greater than the critical value, spectral estimates decrease with side ratio. For models with D/B≧2.0, a higher energy contents in the frequency range of / Hf BD U = 0.2~0.5 can be observed which can be attributed to the occurrence of reattachment phenomenon. Similar but milder phenomenon can be found in both urban and suburban flow fields.

Shown in Figure 6(a) to 6(c) are the reduced drag spectra with different aspect ratio in dif-ferent flow fields. As the aspect ratio increasing, building model extended into higher eleva-tion and lower turbulence intensity region, the spectral energy contents decrease noticeably. In BL-C, the spectral uplift in region of / Hf BD U = 0.2~0.5 occurs for models with side ratio D/B≧2.0; and it increases with aspect ratio. The drag spectra of square shape model, D/B=1.0, exhibit distinct spectral peak at twice of the Strouhal frequency for aspect ratio /H BD = 5~7. The bandwidth of this spectral peak becomes narrower as the aspect ratio increases and model extends into a less turbulent flow field. Model with aspect ratio of 7 exhibits the clearest spec-tral peak.


6

0.0001 0.001 0.01 0.1 1f(BD)0.5/UH

H/(BD)0.5=7

0.0001 0.001 0.01 0.1f(BD)0.5/UH

H/(BD)0.5=5

0.0001 0.001 0.01 0.1f(BD)0.5/UH

1x10-6

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

fS(f)

/ (q

BH2 )2

dmH

H/(BD)0.5=3

0.0001 0.001 0.01 0.1 1f(BD)0.5/UH

H/(BD)0.5=7

0.0001 0.001 0.01 0.1f(BD)0.5/UH

H/(BD)0.5=5

0.0001 0.001 0.01 0.1f(BD)0.5/UH

1x10-6

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

fS(f)

/ (q

BH2 )2

dmH

H/(BD)0.5=3

Fig.5 Variation of along-wind base moment spectra with side ratio. D/B=○:1/5, □:1/4, △:1/3, ◇:1/2, 1,●:2/1, ■:3/1, ▲:1/4, ◆:1/5 －:1/

0.0001 0.001 0.01 0.1 1f(BD)0.5/UH

H/(BD)0.5=7

0.0001 0.001 0.01 0.1f(BD)0.5/UH

H/(BD)0.5=5

0.0001 0.001 0.01 0.1f(BD)0.5/UH

1x10-6

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

H/(BD)0.5=3

(a)Urban Terrain(BL-A)

(c)Open Terrain (BL-C)

(b)Suburban Terrain (BL-B)

fSdm

(f) /

(qHBH

2 )2


(a) Urban Terrain (BL-A)

4 EFFECTS OF BOUNDARY LAYER ON WIND LOADS

4.1 Mean base shear coefficients Shown in Figure 7(a) to 7(c) are the mean drag coefficients of building models in different

boundary layer flows. For side ratio D/B≦0.5, flow field condition does not have definite trend of influence on drag force. However, for D/B≧1.0, urban flow field generally has the lowest absolute value of mean drag especially on the leeward face of more slender shaped buildings, e.g., for buildings with /H BD ≥ 5 . Open country flow field exhibits slightly higher mean drag than suburban flow field. The most significant effect of flow field occurs at B/D=1.0 with aspect ratio of 5~7. This trend disappears when aspect ratio less than 5.

0.0001 0.001 0.01 0.11x10-6

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

f(BD) /UH0.5

fSdm

(f) /

(qHBH

2 )2

D/B=1/5

0.0001 0.001 0.01 0.1f(BD)0.5/UH

D/B=1/1

0.0001 0.001 0.01 0.1f(BD)0.5/UH

D/B=2/1 D/B=5/1

0.0001 0.001 0.01 0.1 1f(BD)0.5/UH

(b) Urban Terrain (BL-B)

Fig.6 Variation of along-wind base moment spectra with aspect ratio. /H BD =○:3, □:4, △:5, ●:6, ■:7

(c) Open Terrain (BL-C)

0.0001 0.001 0.01 0.1f(BD)0.5/UH

1x10-6

10-5

10-4

10-3

10-2

10-1

1x

1x

1x

1x

1x

fSdm

(f) /

(qHBH

2 )2

D/B=1/5

0.0001 0.001 0.01 0.1f(BD)0.5/UH

D/B=1/1

0.0001 0.001 0.01 0.1 1f(BD)0.5/UH

D/B=2/1 D/B=5/1

0.0001 0.001 0.01 0.1 1f(BD)0.5/UH

7

0.0001 0.001 0.01 0.1f(BD)0.5/UH

1x10-6

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

fSdm

(f) /

(qHBH

2 )2

D/B=1/5

0.0001 0.001 0.01 0.1f(BD)0.5/UH

D/B=1/1

0.0001 0.001 0.01 0.1f(BD)0.5/UH

D/B=2/1

0.0001 0.001 0.01 0.1 1f(BD)0.5/UH

D/B=5/1


(a)H/(BD)0.5=3 (c)H/(BD)0.5=7 (b)H/(BD)0.5=5

Fig.7 Variation of base drag force coefficients with side ratio and flow. ○:Urban, □:Suburban, △:Open country

-1.0

-0.5

0.0

0.5

1.0

1.5

0.1 1 10D/B

Cd ,

Cdw

, Cdl

4.2 RMS base shear coefficients Shown in Figure 8(a) to 8(b) are the RMS base drag coefficients of building with various

flow fields.

Cd

dl

dw

C

C

-1.0

-0.5

0.0

0.5

1.0

1.5

0.1 1 10D/B

Cd ,

Cdw

, Cdl

Cd

Cdl

Cdw

-1.0

-0.5

0.0

0.5

1.0

1.5

0.1 1 10D/B

Cd ,

Cdw

, Cdl

Cd

Cdl

Cdw

8

(a)H/(BD)0.5=3

Fig.8 Variation of fluctuating base drag force coefficients with side ratio and flow. ○:Urban, □:Suburban, △:Open country

(b)H/(BD)0.5=7

0.0

0.1

0.2

0.3

0.4

0.5

0.1 1 10D/B

C'd

0.0

0.1

0.2

0.3

0.4

0.5

0.1 1 10D/B

C'dw

0.0

0.1

0.2

0.3

0.4

0.5

0.1 1 10D/B

C'dl

0.0

0.1

0.2

0.3

0.4

0.5

0.1 1 10D/B

C'd

0.0

0.1

0.2

0.3

0.4

0.5

0.1 1 10D/B

C'dw

0.0

0.1

0.2

0.3

0.4

0.5

0.1 1 10D/B

C'dl


Both windward and leeward RMS drag coefficients, C’dw and C’dl, increase with the turbu-lence intensity of approaching flow. The windward RMS drag coefficient, Cdw, remains al-most invariant with respect to side ratio. In BL-A, the maximum value of C’dl always occurs at side ratio D/B=0.5. In flow fields BL-B and BL-C, however, the maximum value of C’dl oc-curs in-between D/B=0.5-1.0. For buildings with aspect ratio of 6 and 7, the maximum C’dl occurs when side ratio D/B=1.0. The variation trend of RMS base drag coefficient, C’d is similar to the leeward RMS drag coefficients C’dl.

4.3 Reduced base moment spectra

9

Fig.9 Variation of along-wind base moment spectra with flow. ○:Urban, □:Suburban, △:Open country

0.0001 0.001 0.01 0.1f(BD)0.5/UH

1x10-6

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

fSdm

(f) /

(qHBH

2 )2

D/B=1/5, H/(BD)0.5=3

0.0001 0.001 0.01 0.1f(BD)0.5/UH

H/(BD)0.5=4

0.0001 0.001 0.01 0.1f(BD)0.5/UH

H/(BD)0.5=5

0.0001 0.001 0.01 0.1f(BD)0.5/UH

H/(BD)0.5=6

0.0001 0.001 0.01 0.1 1f(BD)0.5/UH

H/(BD)0.5=7

0.0001 0.001 0.01 0.1f(BD)0.5/UH

1x10-6

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

fSdm

(f) /

(qHBH

2 )2

D/B=1/1, H/(BD)0.5=3

0.0001 0.001 0.01 0.1f(BD)0.5/UH

H/(BD)0.5=4

0.0001 0.001 0.01 0.1f(BD)0.5/UH

H/(BD)0.5=5

0.0001 0.001 0.01 0.1f(BD)0.5/UH

H/(BD)0.5=6

0.0001 0.001 0.01 0.1 1f(BD)0.5/UH

H/(BD)0.5=7

0.0001 0.001 0.01 0.1f(BD)0.5/UH

1x10-6

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

fSdm

(f) /

(qHBH

2 )2

D/B=2/1, H/(BD)0.5=3

0.0001 0.001 0.01 0.1f(BD)0.5/UH

H/(BD)0.5=4

0.0001 0.001 0.01 0.1f(BD)0.5/UH

H/(BD)0.5=5

0.0001 0.001 0.01 0.1f(BD)0.5/UH

H/(BD)0.5=6

0.0001 0.001 0.01 0.1 1f(BD)0.5/UH

BL ABL BBL C

H/(BD)0.5=7

(a) D/B=1/5

(b) D/B=1/1

(c) D/B=2/1


Shown in figures 9(a) to 9(c) are the reduced drag spectra in different flow fields. There exists significant difference of spectral energy, especially in the energy contain region, among three different boundary layers. The fluctuating drag basically reflects the turbulence level of each flow field. A spectral peak at twice of the Strouhal frequency can be observed in the case of square shape model in the open terrain flow field; whereas it disappears completely in the urban terrain flow field. The spectral uplift in the higher frequency region caused by reat-tachment phenomenon can be observed in the cases with side ratio D/B＞1.

5 CHARACTERISTICS OF LOCAL WIND LOADS

5.1 Local wind force coefficients

Shown in Figure 10 to 11 are the local wind force coefficients of buildings with a fixed as-pect ratio, /H BD = 7 , and various side ratio, in urban (BL-A) and open country flow fields (BL-C). The mean and RMS local wind force coefficients are, divided into windward side and leeward side, and defined as:

2

( )( )0.5

dwpw

z z

F zC zU Aρ

= 2

( )' ( )

( )dwF

pwz u z

zC z

U I z Aσ

ρ=

2

( )( )0.5

dlpl

H z

F zC zU Aρ

= 2

( )' ( )

( )dlF

plH u z

zC z

U I H Aσ

ρ= (4)

in which, ( )dwF z , ( )

dwF zσ are the mean and RMS wind load at height z on windward side；

( )dlF z , (

10

)Fl zσ are the mean and RMS wind load on leeward side; HU , ( )uI H is the mean wind speed and turbulence intensity at building height; , zU ( )uI z is the mean wind speed and turbulence intensity at height z; is the projection area at level z; ρ is the air density and H is building height. It should be noted that mean wind speed at the elevation of each level, , was used as the reference wind speed in normalization for windward side; whereas the mean wind speed at building height,

zA

zU

HU , was used for the leeward side. In urban terrain (BL-A), the mean and RMS local wind force coefficients on windward

face remains almost invariant with respect to the building height, z/H. Generally speaking, the values for both and are between 0.7 ~1.0. In open terrain (BL-C), the distribution of mean local wind force coefficients show the same trend as in BL-A; however, the RMS co-efficient increases slightly with height in the lower half of the building and then remains con-stant over the upper half. These results indicate that the quasi-steady and strip theories are valid for the windward face.

( )pwC z ' ( )pwC z

On the leeward face, for both and open terrain urban terrain flow fields, BL-A and BL-C, the mean and RMS local wind force coefficients, and , generally exhibit constant values with respect to building height, z/H. Only in the case of the buildings with side ratio less than i.e., D/B<1.0, the mean coefficient in BL-C and the RMS coefficient in BL-A show relatively low value at 1/2 to 2/3 of building height. The results show that the pressure field in the building wake is quite uniform and quasi-steady and strip theories are not valid in leeward face.

( )plC z ' ( )plC z


(b) Open Terrain (BL-C) (a) Urban Terrain (BL-A)

Fig.10 Variation of windward local wind force coefficients with aspect ratio ( /H BD =7). D/B=○:1/5, □:1/2, △:1/1, ●:2/1, ■:5/1

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5

Cpw(z)

z /

H

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5

C'pw (z)

z /

H

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5

Cpw (z)z

/ H

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5

C'pw (z)

z /

H

(a) Urban Terrain (BL-A (b) Open Terrain (BL-C) )

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5

C'pl(z)

z /

H

0.0

0.2

0.4

0.6

0.8

1.0

-1.5 -1.0 -0.5 0.0Cpl(z)

z /

H

0.0

0.2

0.4

0.6

0.8

1.0

-1.5 -1.0 -0.5 0.0Cpl(z)

z /

H

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5C'pl(z)

z /

H

Fig.11 Variation of leeward local wind force coefficients with aspect ratio ( /H BD =7). D/B=○:1/5, □:1/2, △:1/1, ●:2 ■:5/1, /1

6 CHARACTERISTICS OF SPATIAL COHERENCE OF WIND LOADS

6.1 Chord-wise coherence of wind pressure Shown in Figure 12(a) and 12(b) are the chord-wise coherences, Rx,w(△x, f) and Rx,l(△x, f),

of windward face and leeward face pressure data, obtained at 2/3H model height, in flow field BL-B. All the chord-wise coherences exhibit the exponential decay form with respect to the reduced frequency, f△x/UH and the windward face show better spatial correlation than the leeward side. High correlation due to shedding frequency can be observed in most of the cases on the leeward face and the coherences of two opposite edge points on the windward face. However, it should noted that, base on the drag force spectra shown in Figure 12, these high

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correlations of local pressure field at shedding frequency bear no significant influence on the total wind load; therefore, can be neglected in the empirical model. The commonly known exponential decay form of coherence function, which was initially proposed by Davenport [7] for the spatial coherence of atmospheric turbulence and then used for the spatial coherence of wind loads by Solari [9,10] and others, was used for the empirical model:

, 1 2 , 1 2( , , ) exp( / ) , , x k x k HR x x f C f x x U k w l= − − = (5)

The representing exponential decay coefficients, Cxw and Cxl, were determined by letting the force variance calculated by the covariance integration of pressure spectra equal to the direct measurements; i.e.,

( )22

1 2 1 1 2 2 1 20 0 0

2 / 3 ( )

( , , ) ( , ) ( , )

p ii

B B

x p p

z H p t

R x x f S x f S x f dx dx df

σ

∞

= =

=

∑

∫ ∫ ∫ (6)

In Equation (6), 2

pσ is the variance of the wind load on either windward face or leeward face, at 2/3 of building height; pi(t) is the pressure data measured at ith pressure tap; is the wind pressure spectrum at i

( , )pi iS x fth pressure tap. Cxw and Cxl were found by minimize the dif-

ferences between two approaches. The best fitted Cxw and Cxl of all models are plotted in Figure 13(a) and 13(b). Figure 13(a)

indicates that the windward exponential decay coefficients, Cxw, remains almost invariant with respect to both side ratio and aspect ratio i.e., it would be reasonable to use a single value co-efficient in the empirical chord-wise coherence function on the windward side. However, the exponential decay coefficient on the leeward side, Cxl, does not show good uniformity over geometric shapes as shown in Figure 13(b). In two different Cxl increases with side ratio in two different regions, D/B=1/5 ~ 1/1 and D/B=2/1 ~ 5/1. Most of the Cxl are in the range of 17~32.

0.0

0.2

0.4

0.6

0.8

1.0

0 0.05 0.1 0.15 0.2fΔx/UH

Coh

eren

ce (x

-asi

s,w

indw

ard)

Fig.12 Chord-wise coherences of windward and leeward face pressure data (2/3 model height, BL-B). △x/B= ○:0.15B, □:0.20B, △:0.35B, ◇:0.40B, ●:0.55B, ■:0.60B, ▲:0.75B, ◆:0.90B,

(b) leeward face(a) windward face

0.0

0.2

0.4

0.6

0.8

1.0

0 0.05 0.1 0.15 0.2fΔx/UH

Coh

eren

ce(x

-axi

s,le

ewar

d)

—:Cxw= 9.8

12


Fig.13 The best fitted Cxw and Cxl of all models (2/3 model height, BL-B). /H BD =○:3, □:4, △:5, ●:6, ■:7

(a) windward face (b) leeward face

0

10

20

30

40

0.1 1 10D/B

C

0

10

20

30

40

0.1 1 10D/B

Cxlxw

6.2 Coherence between windward and leeward faces

Shown in Figure 14(a) and 14(b) are the coherence function, , and the empirical co-efficient, C

( )N fN, of wind load between windward and leeward faces at 2/3 of building height, ob-

tained in suburban terrain flow field (BL-B). The commonly accepted form of coherence function was adopted and plotted against the experimental data, first proposed by Vellozzi and Cohen [8,1968]:

( )22

1 1( ) 1 , 2

n

H

C f BDN f eU

η ηη η

−= − − = (7)

The value of empirical coefficient, CN, is again by minimizing the differences between the force variance calculated by the covariance integration and the direct measurements;

( ) ( )

( )

22

0

2 / 3 ( ) ( )

( ) ( ) 2 ( ) ( ) ( )

Dz Dw Dl

Dw Dl Dw Dl

z H F t F t

S f S f N f S f S f df

σ∞

= = +

= + +∫ (8)

In which, 2

Dzσ is the variance of the drag at 2/3 of building height; FDw(t), FDl(t) are instanta-neous drag of the windward face and leeward face and , are drag spectra of windward and leeward faces at 2/3 of building height, respectively.

( )DwS f ( )DlS f

The experimental result, shown in Figure 14(a), indicates that coherence between wind-ward and leeward drag decreases rapidly in low reduced frequency region, / Hf BD U <0.03, and then stays around the value of 0.3. The chosen empirical coherence, N(f), basically is not a good representation of the experimental data. Similar results can be observed in the cases of urban and open country flow fields. The best fitted CN of all models are plotted in Figure 14(b). It shows that for side ratio D/B<1, CN decreases with side ratio and increases with as-pect ratio. For side ratio D/B=2~5, CN shows only minor variation with respect to both side ratio and aspect ratio.

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Fig.14 Coherence function and CN of local wind force in y direction between windward and leeward face in suburban terrain (BL-B).

0

10

20

30

40

50

60

0.1 1 10D/B

CN

○ : H/(BD)0.5=3□ : H/(BD)0.5=4△ : H/(BD)0.5=5● : H/(BD)0.5=6■ : H/(BD)0.5=7

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5f(BD)0.5/UH

Coh

eren

ce (y

-axi

s)○ : Experiment

(a) (b)

CN = 16― :

6.3 Span-wise coherence

Shown in Figure 15(a) and 15(b) are the span-wise coherence function, 1 2( , , )zR z z f , and the corresponding empirical coefficient, CZ, of building’s drag force, obtained in suburban terrain flow field (BL-B). The exponential decay form of coherence was used for the empirical model [7,1968]:

1 2 1 2( , , ) exp( / )Z Z HR z z f C f z z U= − − (9) The value of empirical coefficient, CZ, is by minimizing the differences between the force variance calculated by the covariance integration and the direct measurements;

22

1 2 1 1 2 2 1 20 0 0

( )

( , , ) ( , ) ( , )

D Dii

H H

Z D D

F t

R z z f S z f S z f dz dz df

σ

∞

=

=

∑

∫ ∫ ∫ (10)

In which, 2Dσ is the variance of base shear; FDi(t) is the instantaneous drag at ith level, which is

the sum of all pressure taps on ith level and is the i( , )Di iS z f th level drag force spectrum.

0.0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4fΔz/UH

Coh

eren

ce (z

-axi

s)

0

10

20

30

40

50

60

0.1 1 10D/B

Cz

○ : H/(BD)0.5=3

□ : H/(BD)0.5=4

△ : H/(BD)0.5=5

● : H/(BD)0.5=6

■ : H/(BD)0.5=7

Fig.15 Coherence function and Cz of local drag force in z direction in suburban terrain (BL-B).

(a) (b)

△z/H ○: 0.09H □: 0.20H△: 0.36H ◇: 0.46H●: 0.56H ■: 0.67H▲: 0.77H ◆: 0.87H—: C

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z= 11.5


The experimental span-wise coherence, shown in Figure 15(a), exhibits the distinct nature of an exponential decay function. Therefore, the empirical equation shows good representa-tion of the wind tunnel data. The best fitted CZ of all models, plotted in Figure 15(b), indicate that CZ increases with side ratio and shows only insignificant variation with respect to aspect ratio. It is likely that a relatively simple function can be found to represent CZ for all building geometry.

7 CONCLUSIONS A series of wind tunnel aerodynamic tests were performed on rectangular shaped prisms to

study the characteristics of wind loads on tall buildings. On the global sense of wind load, i.e., base shear of the entire building, following summaries can be made from the experimental data. (1) The windward face drag coefficient, Cdw, remains almost invariant with respect to both

aspect ratio and side ratio. For side ratio D/B=0.5~1.0, the drag on the leeward side, Cdl, decreases slightly with increase of side ratio, then increases with side ratio afterwards. The aspect ratio has only minor effects on Cdl. Therefore, CD increases slightly with increase of side ratio, for side ratio D/B=0.5~1.0, then decreases afterwards.

(2) The windward face RMS drag coefficient, Cdw, which reflecting the incident turbulence level, decreases with aspect ratio, but remains almost invariant with respect to side ratio. The RMS drag coefficient of leeward side, C’dl, represents the intensity of wake vortices. C’dl exhibits a peak value at D/B=0.5 in BL-A; at D/B=1.0 in BL-C. Then it decreases with increase of side ratio. C’dl decreases with increase of aspect ratio in BL-A; aspect ratio has only weak influence on C’dl in flow fields BL-B and BL-C. Generally speaking, C’dl has greater value at D/B＜1.0 than at D/B＞1.0. As the result, there is a peak of C’d at side ratio D/B=0.5~1.0 and C’d decreases with aspect ratio for all side ratio cases.

(3) Spectral peak at twice of the Strouhal frequency can be observed in open terrain flow field. For models with D/B≧2.0, a higher energy contents in the frequency range of / Hf BD U = 0.2~0.5 can be observed which can be attributed to the occurrence of reattachment phe-nomenon.

Some conclusions can also be made on the local drag force of rectangular shaped buildings. (4) When windward force normalized with respect to local wind pressure, the mean and RMS

local wind force coefficients on windward face remains almost invariant along building height in most cases. When leeward side normalized with respect to wind pressure at building height, the mean and RMS local leeward coefficients, and , also ex-hibit constant values along building height.

( )plC z ' ( )plC z

(5) The chord-wise coherences, Rx,w(△x, f) and Rx,l(△x, f), of windward face and leeward face pressure data, obtained at 2/3H model height, exhibit the exponential decay form with respect to the reduced frequency, f△x/UH and the windward face show better spatial cor-relation than the leeward side. The windward exponential decay coefficients, Cxw, remains almost invariant with respect to both side ratio and aspect ratio i.e., it would be reasonable to use a single value coefficient in the empirical chord-wise coherence function on the windward side.

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(6) The span-wise coherence function, 1 2( , , )zR z z f , also show exponential decay form with respect to the reduced frequency. The corresponding empirical coefficient, CZ, increases with side ratio and shows only insignificant variation with respect to aspect ratio. It is likely that a relatively simple function can be found to represent CZ for all building ge-ometry.


(7) The experimental result of the coherence between windward and leeward drag decreases rapidly and then stays around the value of 0.3. The commonly used empirical coherence, N(f), basically is not a good representation of the experimental data.

ACKNOWLEDGEMENTS Partial financial support for this research project from Architecture and Building Research Institute (Grand No. 097301070000G1014) is gratefully acknowledged.

REFERENCES [1] Ahsan Kareem, (1990), ‘‘Measurements of pressure and force fields on building models

in simulated atmospheric flows.’’ Journal of Wind Engineering and Industrial Aerody-namics 36, 589–599.

[2] Jenmu Wang, Chii-Ming Cheng, (2003), “Knowledge Mangement In A Wind Tunnel Laboratory: The WERC-TKU Knowledge Project.”, The International Wind Engineer-ing Symposium, November 17-18, 2003, Tamsui, Taipei county, Taiwan.

[3] M. Gu, Y. Quan, (2004), “Across-wind load of typical tall buildings.”, Journal of Wind Engineering and Industrial Aerodynamics 92, 1147–1165.

[4] A. Kareem, (1997), “Correlation structure of random pressure fields.”, Journal of Wind Engineering and Industrial Aerodynamics 69-71, 507–516.

[5] N. Lin, C. Letchford, Y. Tamura, B. Liang, O. Nakamura, (2005), “Characteristics of wind forces acting on tall buildings.”, Journal of Wind Engineering and Industrial Aerodynamics 93, 217–242.

[6] C.M. Cheng, (2005), “Modifications on the alongwind design wind load”, The Fourth Africa-Europe Conference on Wind Engineering, Prague.

[7] A. G. Davenport,(1968), “The dependence of wind load upon meteorological parame-ters.”, in proceedings of the international research seminar on wind effects on buildings and structures, University of Toronto Press, Toronto, 19-82.

[8] J. Vellozzi and E. Cohen, (1968),“Gust response factors.”, J. Struct. Div., ASCE, 94, no. ST6, Proc. Paper 5980, 1295-1313.

[9] G. Solari, (1988), “Equivalent wind spectrum technique: theory and applications.”, Journal of Structural Engineering, vol.114, no. 6, 1303-1323.

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[10] G. Solrai, (1993), “Gust Buffeting. II: Dynamic alongwind response”, Journal of Struc-tural Engineering, vol.119, no. 2, 383-398.

characteristics of alongwind loads on rectangular...

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