- Email: [email protected]
Probability distribution of forces and pressures on a square cylinder
Journal of Wind Engineering and Industrial Aerodynamics, 41-44 (1992) 813-824
813
Elsevier
PROBABILITY DISTRIBUTION OF FORCES AND PRESSURES ON A SQUARE CYLINDER
J.L.D.
Ribeiro
and
J.
Blessmann
Laborat6rio de Aerodin~mica das ConstruqSes, UFRGS Av. Oswaldo Aranha, 99 - 90210 Porto Alegre, Brazil
Abstract Wind tunnel experiments on a three-dimensional square cylinder model revealed that, except perhaps on the frontal face, the probability distribution of pressures, forces or overturning moments are not normal (gaussian). In fact, on points where the effect of the Karman vortices is pronounced the probability distribution of pressures shows a significant skewness to the left and kurtosis (peakdeness). This means that the negative pressure peaks are more intense than the positive pressure peaks and the crest factor is considerably larger than 3.0 or 4.0. The present paper discusses this subject and shows results regarding smooth or turbulent shear flow. Besides probability distribution curves, parameters such as mean value, rms value, crest factor, and skewness and kurtosis coefficients (for pressures on several points, for forces on individual faces, and for global forces) are also presented.
NOTATION
~/B1 ,B2 C Cp] d,h Fx Fy F,R G,L
Mx My
Re U,ut x x' o~
P
1.
skewness coefficient, kurtosis coefficient denotes a force or moment coefficient pressure coefficient on point j lateral dimension and height of the square model (m} drag force, lateral (cross-stream) force(N) frontal face, rear face right lateral face, ieft lateral face streamwise and cross-stream overturning moments(N.m) Reynolds number free-stream velocity (m/s), same at the model top height denotes a mean value, denotes an rms value significance level for statistical tests density of free-stream (Kg/m 3)
INTRODUCTION
There are several applications where just the knowledge of the mean and rms value of the pressures and forces induced by the wind is not enough. Frequently information regarding the crest factor or regarding the probability distribution function (PDF] is desirable. This is particularly true when dealing with the problem of estimatinz parameters from observational data through confidence intervals.
0167-6105/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
814
I
normal distrib,
normal
/ <8,=o
I
I
/ -3
/
,-2~'~.skewed to the "i'B,
,,/",\/left.
'\
DEVIATION(RMS) 3
-3
distrib.
.
t-~ /strong kur tosis. B2>3
,// ~/
J),
DEVIATION [Ri~iS)
/i $
Figure I. Nonnormal distributions due to skewness or kurtosis.
In fact information about the probability distribution of pressures and forces arising on buildings exposed to the wind can be useful for the improvement of technical codes and also for the development of numerical procedures to compute the response of buildings exposed to the wind. The skewness coefficient, VBI, and the kurtosis coefficient, B2, are useful parameters to characterize a PDF curve. The normal (Gauss) distribution is a symmetric curve, so that it presents ~BI = O. Nonnormal distributions that are asymmetric present ~BI different from zero; ~BI > 0 corresponds to skewness to the right, while VBI < 0 to the left. The ~ord "kurtosis" means curvature. The curvature of the normal distribution l.s such that B2 = 3. Nonnormal distributions peaked at the center and with thicker tails than normal present Bg > 3, while nonnormal distributions with broader peak at the center and lighter tails than normal present B2 < 3. An illustration of nonnormal distributions can be seen in Fig. l. (For further discussion of skewness and kurtosis, reader~ are referred to D'Agostino eL al, [I]),
2. EXPERIMENTAL PROGRAM In order to obtain experimental data on the probabilistic distribution of pressure and forces, a three-dimensional square cylinder model 0.36 m high and 0.06 x 0.06 m wide was tested in the wind tunnel of the "Universidade Federal do Rio Grande do Sul". This is a closed-return wind tunnel, known as TV-2, which presents a test section measuring 1290 mm (width) x 900 am (height). The model was tested in a uniform smooth flow as well as in a condition of turbulent shear flow (power law exponent of the mean wind velocity equal to 0.23, corresponding to suburban terrain, see Fig,2). Both flows show velocity fluctuations normally distributed around the mean, The measurements of" forces and pressures were made at the maximum velocitLes reached in TV-2, that is, 42 m/s in the tests with smooth flow and 36 m/~ in the tests with turbulent shear flow (velocity at the mudel top height). The corresponding Reynolds number are respectively 168000 and 144000.
815 SO0
i
L'
!
450 Z Imm| 30O
/
150
0
\ O,S
0
1.0
0
2.
streamwise
Velocity turbulence
/
\
profile, length
/
I
i,
0
U'(ZIIUIZ|
O(z) I O(4sO)
Figure
• --
200 X
-
Lu (Z)
streamwise scale
40O (mini
turbulence
Lx o f t h e s h e a r
intensity
flow used
u'/U,
in the
and
tests.
U
The tests included measurements of PDF for pressures on several points of a cross-section (positioned at 2/3 of the model height), for forces on individual faces, and for global forces (drag, cross-stream force and overturning moments). Fig. 3 shows the positions of the pressure taps for PDF measurements of pressure, force or overturning moments fluctuations.
3.
MEASUREMENT TECHNIQUE
The forces on individual faces were obtained through the process of pneumatic averages. With the aid of restrictors conveniently placed in the tube system, a flat response up to a frequency of 200 Hz was obtained. Plastic tubes 250 am long and with 0.7 mm ID were used in the tube system. The pressure or force fluctuations were measured with "Endevco" (type 8510-2) transducers. The transducer signals were recorded by a "Hewlet Packard" (type 3960) tape-recorder. The signals were then analyzed in two ways: I. using an A/D converter and an XT personal computer, and 2. using a "Bruel & kjaer '° (model 2034) dual-channel frequency analyzer. The sampling rate of the personal computer was fixed at 800 measurements per second. Based on the Strouhal cycle period, this means a minimum of ten measurements per cycle. The sampling rate of the model 2034 was superior, 2048 measurements per second. In spite of the differences on sampling rate, the results from both analysis were almost identical. Thus, ten measurements per cycle seems an acceptable minimum to represent adequately the fluctuations measured. All results were computed for a period of time equivalent to approximately 3000 Strouhal cycles, and due to the data sa,,L~ling, the formulae Jsed was the following:
816 x = ~ xi /
mean value: rms v a l u e
:
Cl)
n
x' = [~ (x i- ~)2/
C2)
(n - I)] 1/2
^
positive
crest factor:
x = max
(x i- x) / x'
C3)
negative
crest
x" = m i n
" "^ i ~
(4)
factor: Z
skewness
:
~B
1
[~ (Xl_ Z
:
kurtosis
where
B
2
~)3i
Cx i -
=
=
(X|-
x)
/
x'
n
~)2/
n
~)4/
n
(S)
]3/2
(6)
[~ Cxl_ ~)2/ n ]2
x
r e p r e s e n t s a zeneric value of pressure, force or moment and n is i the total of the collected data in each case. The c o r r e s p o n d i n g m e a n and rms (pressure, force or moment) c o e f f i c i e n t s were o b t a i n e d as:
2p
/
(p u~)
;
c' = 2p' p
/
(p u 2) t
(7)
C- = 2 F
/
(p u 2 h d)
;
C'
/
Co u 2 h d)
(8)
;
C' : 4M' / (p u 2 h 2 d)
C- = p F
t
C- = 4M / (p u 2 h 2 d) M
where
u
t
is
F
t
the
blockage was small,
free-stream
= 2F'
H
at
the
model
Lop
OVERTURNING MOMENTS
$ PRESSURE MEASUREMENTS
f ro.tal face
I 2
S _ 7 3 4 5 G lateral
rear fq¢o
i
o
,
f
@
0 Nm
t
, m
0
face
0 ~p d,W r q f ~ f
Figure 3. P o s i t i o n of the p r e s s u r e
height.
for blockaEe were attempted.
FORCES
FLOW
(9)
t
velocity
no c o r r e c t i o n s
t
rJ
0m
'I"
taps for PDF measurements.
As
the
817
4. RESULTS AND DISCUSSION 4.1. Mean and rms measurements Table 1 presents the results measured in TV-2. Regarding mean and rms values, Matsumoto [2] and Sun, Lin and Gu [3] presented values measured in smooth flow tests; while Akins et al. [4], Cheung [5], Sakamoto [6], Kronke and Sockel [7], Tanaka and Lawen [8], Matsumoto [2], Golinger and Milford [9] and Sakamoto and Haniu [I0] presented values measured in turbulent shear flow. Since the experimental set-ups were not too different, the results reported by these authors can be compared with those collected in TV-2. Taking into account differences in blockage, aspect ratio or characteristics of the generated flow, the mean and rms results measured in TV-2 compare fairly well with those reported in the literature.
Table
I. Mean values, rms values, B2 measured in TV-2.
GOEF.
crest factors,
SMOOTH FLOW
Mean
rms
Crest (+)
Factor (-) t"B I
skewness ¢BI and Kurtosis
TURBULENT FLOW
Bz
Norma lity
Mean
rms
Crest (+)
~actor (-) I"BI
B2
Norma lity
Cpi Cpa Cp3 Cp, Cp6 Cpe Cp7 Cp8
.91 .69 -.72 -.74 -.75 ".78 ".1'0 ".69
.017 3.7 3.7 .023 3.4 4.6 .089 3.6 7.3 .099 2.9 6.0 .090 3,6 8.3 .12" ' 3 . 5 I 0 . 0 .119 5.2 8.0 .115 5.9 6 . 5
-0.01 -0.47 -0.82 -0,92 -0.96 "1.42 "0.61 "0.24
3.12 3.66 5. II 5.01 5.39 6.94 5.74 5.17
Y N N N N N N N
.60 .42 -.74 " .60 ".73 ".71 ",51 -.45
.133 .122 .276 .298 .285 .290 .180 .145
3.6 3.3 2.7 2.9 3.1 3.9 3.8 3.6
3.7 4.2 6.9 6.0 6.6 6.9 6.7 7.5
0.04 -0.05 -0~92 -I.06 -I.10 -0.75 -I.08 -0.96
3.01 2.91 4.17 4.65 5,01 4.29 5.11 5.94
Y Y N N N N H N
CFXF CFXR CFX CMXF CMXR GMX
.72 ".57 1.2g .76 " . 62 1.38
.022 .052 .050 .020 .063 .056
3.5 3.8 4.1 3.6 3.6 4.4
3.6 4.3 3.5 4.4 4.8 3.7
-0.14 -0.39 0.27 -0.33 -0.49 0.26
3.12 3.70 3.33 3.21 3.75 3.55
N N N N N N
.44 ".38 .82 .50 -.40 ,90
.089 .076 .140 .069 .074 .139
3.3 3.0 4.2 3.3 3.2 4.4
3.5 4.7 3.5 4.1 4.6 3,7
O.OI -0,37 0.15 -0.16 -0.27 0.16
2.98 3,24 3,05 3.01 3.31 3.21
Y N N N N N
CFys CFyL CFy CMyQ CMW" CidY
".66 - . 66 .00 -.72 -.71 .00
.076 .073 ,106 .098 .095 .125
3.1 3.2 6.5 3.0 2.9 5.5
6.4 6.6 6 .5 5.4 5.6 5.4
"1.11 -I.13 -0.02 -I.03 -I.08 0.01
5.61 5.69 6.69 5.25 5.34 6.01
N N N N N N
" .56 -.55 .00 -.57 -.59 .00
.120 .120 .198 .125 .125 .203
2.9 2.B 4.2 2.9 2.9 4.0
5.0 5.2 4.3 4.~, 4.4 4.0
-O.50 -0.59 -0.04 -0.57 -0.55 -0.02
3.55 3.~7 ~-.2~:J ~.44 ~ .51 3.20
N N N N N N
818
4.2. Crest factor measurements Few authors reported crest factor results measured on three-dimensional square cy]inders. Golinger and Milford [9] presented some results of peak pressures measured at a section positioned at 2h/3 on the CAARC standard tall building model. Their results for the BLI flow (which is similar to the turbulent shear wind generated in TV-2) indicate a negative crest factor around 4.0 at a point on the frontal face, around 6.0 at a point on the lateral face, and around 7.0 at a point on the rear face. These values were computed for the approaching flow perpendicular to one of the faces and they compare with corresponding values measured in TV-2 (see Table I). Kronke and Sockel [7] reported results of extreme drag coefficients. On tests with model #3 and boundary layer type II (same model shape and similar turbulent shear flow as those used in TV-2). They measured mean CFX = 0.84 and mode of extreme values of CFX = 1.36. These values are in agreement with those measured in TV-2, i.e., 0.82 (mean) and 1.30-1.40 (extreme values). Tanaka and Lawen [8] conducted measurements in turbulent shear flow on a three-dimensional cylinder of rectangular cross-section (2:3) and aspect ratio = 4. They reported that the crest factor at zones of positive mean pressure were generally found to be within the range 3.6-4.1; whereas at zones of negative mean pressures sometimes very high values (around 6.0 or 7.0) were observed for negative crest factors. Again the values are in agreement with the values measured in TV-2. 4.3. Skewness and kurtosis measurements Kanda and Ohkuma [11], based on available results of full-scale wind pressure measurements on sixteen tall building in Japan presented some information regarding PDF measurements. According to these authors, in most studies the PDF of negative pressures on leeward faces were found slightly skewed to the left, but there were divergences about the skewness of the PDF of positive pressures (on windward faces). One of these full-scale wind pressure measurements was performed on Chiba Port Tower, and Kawabata et al. [ 1 2 ] presented results concerning PDF measurements. Chiba Port Tower is not a square cylinder (but presents a diamond shape cross-section). In spite of this fact, as will be seen later, the main observations reported by these authors are in agreement with the measurements made in TV-2. Kawabata et al, [12] reported that the PDF of the pressures on windward face is similar to the PDF of the wind speed fluctuations and no skewness was founU, The PDF of the pressures on side face, however, shows a stron E skewness and kurtosis and is distinct from the normal distribution. The PDF of the pressures on leeward face still shows a strong kurtosis, but not as strong as observed on side faces. Returning to the results of TV-2, Table I also presents the parameters ~BI (skewness) and B2 (kurtosis) which characterize the corresponding PDF curves. The measured values of ~Bt and B2 were tested to verify if the hypothesis of normal distribution could be accepted. The tests employed follow the procedure proposed by D'Agostino et al. [I]. Actually two tests were performed on every PDF curve: (I) test of skewness, where the null hypothesis is Ho: normality, while the alternative is HI: nonnormality due to skewness; and (2) test of kurtosis, where the null hypothesis is Ho: normality, while the alternative is HI: nonnormality due to nonnormal kurtosis, To be accepted as normal, a distribution must pass both tests. As usual, the statistical level of significance adopted for the tests was ~ = 5~,
819
which means that the Ho hypothesis are discarded when the probability of the sample proceeding from a normal distribution is less than 5~. Figs. 4, 5 and 6 present (as measured in smooth and turbulent shear flow), the PDF curves of pressure fluctuations, while Fig. 7 presents the PDF curves of the force fluctuations on entire individual faces and Fig. 8 presents the PDF curves of the global force fluctuations. Figs. 4 to 8 complement the information included in Table I. For visual aid the normal [Gauss) distribution is also depicted in these figures. It can be said that the probability distribution of pressure fluctuations measured from position I to 8 (from frontal face to rear face) document in the statistical point of view the way the turbulence present in the approaching flow is distorted by the body. 4.3.1. Smooth flow results As can be seen in Table I, regarding smooth flow, the hypothesis of normality is not discarded only for the PDF of the pressures on point 1 (frontal face). On the frontal face it is possible to delimit a central (vertical) zone d/2 wide where the pressure fluctuations practically mirror the velocity fluctuations of the flow and, therefore, show a normal distribution (see Fig. 4). Outside this zone the pressure fluctuations are significantly affected by the Karman vortices and the corresponding PDF are distorted in such a way that they depart from normal (see Fig. 5). The effect of the Karman vortices is stronger on the lateral faces, where the PDF of the pressures appear significantly skewed to the left and peaked. On the rear face the skewness diminishes, but a strong kurtosis remains (see Fig. 6).
50
- 40
I
,
SMOOTH FLOW . •. . | . |
TURBULENT |
•
PI
|
•
•
~
FLOW lu
•
|
Pl
|=o I
0
. . . . .
-
. . . . .
P2 3O
I0 o
.
,
.,
OEVIATION
(RM$)
Figure 4. PDF of pressure fluctuations on frontal face.
|
820 TURBULENT FLOW
SMOOTH FLOW 50
p;
.
'
.
40 30
7
20
~.'".....
10
•
b
•
P4
m 40
•
•
•
A
•
,
•
•
•
•
,
P4 iI
I I 30
,.,'/i
at
20
,'/
I I I
N 10
&%
Z W
Q
•
L
•
0
~: 4o -1
P5
P5
I
#
I 9[ 30
f
I
@~o a_
I #
,/
10
0--~
40"
P6
P6
3o
/
,Y
2o IO.
-4
I
-3
-2
-I
0
I
2 3 4 -3 -;~ -I DEVIATION | RMS)
Figure 5. PDF of pressure fluctuations
on
0
lateral
I
2
3
4
face.
In smooth flow the pressures are quite well correlated on the frontal face and on the lateral faces. Thus the PDF of the forces on these faces are approximately the average of the corresponding PDF of the pressures. The PDF of the forces on the frontal face is not normal, but is not too different from normal, while on the lateral faces the PDF of the forces shows a strong skewness and kurtosis (see Fig. 7).
821
so
.
.SMOOTH , ~ .FLOW . . . . . . JF
.
:
|
I
I
,'! ~o
TURBULENT ^~ FLOW
I
t
V :
:
'
®~~4°' 30 ; P
~
:
P8
,,,""
:
• ":,.'-.-:
""'.
n_
I0 0
.
-4-g-~-~
o
i
z
i
4-g-2-I
o
I
2
i
4
DEVIATION (RMS)
Figure 6. PDF of pressure fluctuatlons on rear face.
On the rear face, however, the pressures are not well correlated. Strong pressure peaks (badly correlated} do not generate strong force peaks. Thus the resulting PDF of the forces on the rear face is not as skewed or peaked as the PDF of the local pressures on thls face. Regarding global forces, It Is interesting to observe that the PDF of the drag shows a positive skewness (see Flg. 8}. Thls happens because the suction peaks on the rear face are translated as positive contributions to the drag. Meanwhile, the skewness of the PDF of the global cross-stream forces Is virtually zero. Thls happens because while suction peaks on one lateral face contribute reducing Fy, the same suction peaks on the other lateral face contribute increasing Fy. In spite of the symmetry, the PDF of the global cross-stream force In smooth flow Is quite different from the normal distribution because it is quite peaked. On the other hand, the PDF of the global drag, although not acceptable as normal for a level of significance a = 5Z, in fact Is a distribution that is not quite different from the normal. The PDF of the overturning moments In smooth flow show small differences when compared to the PDF of the corresponding forces. The PDF of the streamwise overturning moment appears more skewed and peaked than the PDF of the drag. While the PDF of the cross-stream overturning moment appear~ less skewed and peaked than the PDF of the cross-stream force. It Is possible that these experimental observations are due to the top vortices which roll along the upper horizontal edge. These vortices; affect the zone close to the model top, which provides the major contribution on determining the overturning moments. As a last observation concerning smooth flow, the data collected in TV-2 indicate that taking the fluctuations of streamwise forces or moments as
822
50
i
40 30 ~0
z I0 In
o
|
SMOOTHFLOW ~
i
~
/,,\
FRONTAL
.c,
•
•
FRONTAL F
l
I
I
I
I
-
•
A
OAU,>l _/'.-.
"k__ '
40 z 30 20
FLOW
TURBULENT I
•
'
,
•
I
I
i
•
~
>"
I--
;J k ,,/ ~,
b,I
),I--
"°'
"; I 0 m 0 n,,
0
REAR 30 FACE et 20
REAR FACE~
~, 4 0
,:/
I0
0
\~
-~
-:,
-I
o
•
\~
\,
~,~.'... -4
•
',~
a
.%
z ~ 4 - ~ - z - i OeVlA'r,ON ( R M S )
.
. o
.
%._ z $
4
Figure 7. PDF of force fluctuations on entire individual faces.
as normally distributed generally results in an acceptable error. Nevertheless the assumption that the fluctuations of cross-stream forces or moments are normally distributed would lead to larger errors.
4.3.2. Turbulent shear flow results Turning back to Table I, the results measured in turbulent shear flow indicate that the hypothesis of normal distribution can not be discarded for the fluctuations of the pressures on points I and 2 (frontal face). In turbulent shear flow the Karman vortices are not as intense as in smooth flow and, therefore, do not affect significantly the pressure fluctuations anywhere on the frontal face (see Fig. 4). On the other hand, on the lateral face and on the rear face the pressure fluctuations are affected by the Karman vortices and the corresponding PDF appear significantly skewed to the left and peaked (see Figs. 5 and 6). The skewness and kurtosis are of such an order that the negative crest factor on these faces results around 6.0 OF 7.0 . Although the PDF of the pressures on the lateral faces and on the Fear face present a pronounced skewness and kurtosis, the effect is not percepti-
823
SMOOTH FLOW . . . . , .
50 40
DRAG
.
.
.
.
TURBULENT FLOW , ,, , ,
,
A
,o .
r,
.
/..X STREAM
¢L 2 0
-4
'
~'- "~
GAUSS,~,,,I
,o o
"
,
// -~
-z
-I
,
o
I
2 ~ 4 DEVIATION
-'~ -2 { RMS }
-{
o
i
~
~
4
F i g u r e 8. PDF of g l o b a l f o r c e f l u c t u a t i o n s .
ble in the PDF of the forces on these faces (see Fig. 7). In turbulent shear flow the correlation of the pressure fluctuations is poor. Strong suction peaks (badly correlated) do not generate strong force peaks on the entire face. Thus on the lateral faces and on the rear face the crest factor of the forces remain around 4.0 or 5.0 . The poor pressure correlation in turbulent shear flow is reflected in a still more dramatic way over the global forces. The PDF of the global forces seem to converge towards the normal distribution. Although neither the drag nor the crossstream force can be accepted as normally distributed for a level of significance ~ = 5~, actually their distributions are not quite different from the normal distribution (see Fig. 8). The PDF of the overturning moments in turbulent shear wind show small differences when compared to the PDF of the corresponding forces. The PDF of the streamwise moment is similar to the PDF of the drag, but the former is slightly more skewed and peaked. Meanwhile, the PDF of the cross-stream overturning moment is similar to the PDF of the cross-stream force, but it is slightly less skewed and peaked. As a last observation concerning turbulent shear flow, the data collected in TV-2 indicate that taking the pressure fluctuations as normally distributed would result in a large error, but in several applications taking the fluctuations of global forces (drag, cross-stream force and overturning moments) as normally distributed would be reasonable, resulting in an acceptable error. It is worth mentioning that the turbulent shear flow used in the tests here described corresponds to a suburban terrain. It is possible that tests conducted in a flow corresponding to an urban terrain (high turbulence) disclose global forces that are in fact normally distributed.
824
5. CONCLUSIONS Wind tunnel tests with a three-dimensional square cylinder model (aspect ratio = 6.0) revealed that except perhaps on points of the frontal face, the probability distribution function (PDF) of the pressures is not normal (gaussian). On the lateral face and on the rear face, where the effect of the Karman vortices is significant, the PDF of the pressures shows a strong skewness to the left and kurtosis (peakdeness). In fact, negative crest factors around 7.0 were measured on these faces. In smooth flow the pressure fluctuations over an entire face show a good correlation and the PDF of the forces is similar to the PDF of the pressures on the corresponding face. In turbulent shear flow, however, the pressure fluctuations over an entire face show a poor correlation and the PDF of the forces is not as skewed and peaked as the PDF of the pressures on the corresponding face. In turbulent shear flow taking pressure fluctuations on a three-dimensional cylinder as normally distributed would result in a considerable error; however, in several applications considering global forces (drag, crossstream force and overturning moments) as normally distributed could result in an acceptable error. Actually, for a statistical level of significance = 5Z, the global forces can not be accepted as normally distributed, but in turbulent shear flow the probability distribution of the global forces is not too different from the normal distribution.
REFERENCES 1 2 3
4 5 6
7 8 9
10
11 12
R.B. D'Agostino, A suggestion for using powerful and informative tests of normality, The American Statistician, 44-4 (1990) 316-321. T. Matsumoto, On the across-wind oscillation of tall buildings, J. Wind Eng. Ind. Aerodyn., 24 (1986) 69-85. T . F . Sun, S.T. Lin and Z.F. Gu, Mean wind loadings acting on two rectangular cylinders of side-by-side arrangements, Prec. 7th Int. Conf. on Wind Engineerlng, Aachen, West Germany, 1987, Vol.2, pp 21-30. R . E . Akins, J.A. Peterka and J.E. Cermak, Mean force and moment coeff. for buildings in turbulent boundary layers, J. Ind. Aerodyn, 2 (1977) 195. S.Y. Cheung, The interference effect on tall building, ES400 Project Report presented to the University of Western Ontario, {983. H. Sakamoto, Aerodynamic forces act|ng on a rectangular prism placed vertically in a turbulent boundary layer, J. Wind Eng. Ind. Aerodyn., 18
(1985) 131-151. I. Kronke and H. S o c k e l , Measurements of e x t r e m e d r a g c o e f f i c i e n t s of b u i l d i n g models, J, Wind Eng. Ind, Aerodyn., 23 (1986) 149-163. H. Tanaka and N. Lawen, T e s t on the CAARC s t a n d a r d t a l l b u i l d i n g model whit a l e n g t h s c a l e 1:1000, J. Wind Eng. Ind. Aerodyn., 25 (1986) 15-30. A,H. G o l i n g e r and R.V. M i l f o r d , S e n s i t i v i t y of t h e CAA~C s t a n d a r d b u i l d i n g model to g e o m e t r i c s c a l e and t u r b u l e n c e , J. Wind Eng. Ind. Aerodyn, 31 (1988) 105-123. H. Sakamoto and H. Haniu, Aerodynamic f o r c e s a c t i n g on two s q u a r e p r i s m s placed vertically in a t u r b u l e n t boundary l a y e r , J. Wind Eng. Ind. Aerodyn., 31 (1988) 41-66. J, Kanda and T. Ohkuma, R e c e n t d e v e l o p a e n t s i n f u l l - s c a l e wind p r e s s u r e s measurements in Japan, J. Wind Eng. Ind. A e r o d y n . , 33 (1990) 243-252. S. Kawabata, T, Ohkuma, J. Kanda, H. Kitamura and K. Ohtake, Chiba P o r t Tower: f u l l - s c a l e measurement . . . . J.Wind Eng. Ind.Aerodyn, 33 (1990) 253.
813
Elsevier
PROBABILITY DISTRIBUTION OF FORCES AND PRESSURES ON A SQUARE CYLINDER
J.L.D.
Ribeiro
and
J.
Blessmann
Laborat6rio de Aerodin~mica das ConstruqSes, UFRGS Av. Oswaldo Aranha, 99 - 90210 Porto Alegre, Brazil
Abstract Wind tunnel experiments on a three-dimensional square cylinder model revealed that, except perhaps on the frontal face, the probability distribution of pressures, forces or overturning moments are not normal (gaussian). In fact, on points where the effect of the Karman vortices is pronounced the probability distribution of pressures shows a significant skewness to the left and kurtosis (peakdeness). This means that the negative pressure peaks are more intense than the positive pressure peaks and the crest factor is considerably larger than 3.0 or 4.0. The present paper discusses this subject and shows results regarding smooth or turbulent shear flow. Besides probability distribution curves, parameters such as mean value, rms value, crest factor, and skewness and kurtosis coefficients (for pressures on several points, for forces on individual faces, and for global forces) are also presented.
NOTATION
~/B1 ,B2 C Cp] d,h Fx Fy F,R G,L
Mx My
Re U,ut x x' o~
P
1.
skewness coefficient, kurtosis coefficient denotes a force or moment coefficient pressure coefficient on point j lateral dimension and height of the square model (m} drag force, lateral (cross-stream) force(N) frontal face, rear face right lateral face, ieft lateral face streamwise and cross-stream overturning moments(N.m) Reynolds number free-stream velocity (m/s), same at the model top height denotes a mean value, denotes an rms value significance level for statistical tests density of free-stream (Kg/m 3)
INTRODUCTION
There are several applications where just the knowledge of the mean and rms value of the pressures and forces induced by the wind is not enough. Frequently information regarding the crest factor or regarding the probability distribution function (PDF] is desirable. This is particularly true when dealing with the problem of estimatinz parameters from observational data through confidence intervals.
0167-6105/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
814
I
normal distrib,
normal
/ <8,=o
I
I
/ -3
/
,-2~'~.skewed to the "i'B,
,,/",\/left.
'\
DEVIATION(RMS) 3
-3
distrib.
.
t-~ /strong kur tosis. B2>3
,// ~/
J),
DEVIATION [Ri~iS)
/i $
Figure I. Nonnormal distributions due to skewness or kurtosis.
In fact information about the probability distribution of pressures and forces arising on buildings exposed to the wind can be useful for the improvement of technical codes and also for the development of numerical procedures to compute the response of buildings exposed to the wind. The skewness coefficient, VBI, and the kurtosis coefficient, B2, are useful parameters to characterize a PDF curve. The normal (Gauss) distribution is a symmetric curve, so that it presents ~BI = O. Nonnormal distributions that are asymmetric present ~BI different from zero; ~BI > 0 corresponds to skewness to the right, while VBI < 0 to the left. The ~ord "kurtosis" means curvature. The curvature of the normal distribution l.s such that B2 = 3. Nonnormal distributions peaked at the center and with thicker tails than normal present Bg > 3, while nonnormal distributions with broader peak at the center and lighter tails than normal present B2 < 3. An illustration of nonnormal distributions can be seen in Fig. l. (For further discussion of skewness and kurtosis, reader~ are referred to D'Agostino eL al, [I]),
2. EXPERIMENTAL PROGRAM In order to obtain experimental data on the probabilistic distribution of pressure and forces, a three-dimensional square cylinder model 0.36 m high and 0.06 x 0.06 m wide was tested in the wind tunnel of the "Universidade Federal do Rio Grande do Sul". This is a closed-return wind tunnel, known as TV-2, which presents a test section measuring 1290 mm (width) x 900 am (height). The model was tested in a uniform smooth flow as well as in a condition of turbulent shear flow (power law exponent of the mean wind velocity equal to 0.23, corresponding to suburban terrain, see Fig,2). Both flows show velocity fluctuations normally distributed around the mean, The measurements of" forces and pressures were made at the maximum velocitLes reached in TV-2, that is, 42 m/s in the tests with smooth flow and 36 m/~ in the tests with turbulent shear flow (velocity at the mudel top height). The corresponding Reynolds number are respectively 168000 and 144000.
815 SO0
i
L'
!
450 Z Imm| 30O
/
150
0
\ O,S
0
1.0
0
2.
streamwise
Velocity turbulence
/
\
profile, length
/
I
i,
0
U'(ZIIUIZ|
O(z) I O(4sO)
Figure
• --
200 X
-
Lu (Z)
streamwise scale
40O (mini
turbulence
Lx o f t h e s h e a r
intensity
flow used
u'/U,
in the
and
tests.
U
The tests included measurements of PDF for pressures on several points of a cross-section (positioned at 2/3 of the model height), for forces on individual faces, and for global forces (drag, cross-stream force and overturning moments). Fig. 3 shows the positions of the pressure taps for PDF measurements of pressure, force or overturning moments fluctuations.
3.
MEASUREMENT TECHNIQUE
The forces on individual faces were obtained through the process of pneumatic averages. With the aid of restrictors conveniently placed in the tube system, a flat response up to a frequency of 200 Hz was obtained. Plastic tubes 250 am long and with 0.7 mm ID were used in the tube system. The pressure or force fluctuations were measured with "Endevco" (type 8510-2) transducers. The transducer signals were recorded by a "Hewlet Packard" (type 3960) tape-recorder. The signals were then analyzed in two ways: I. using an A/D converter and an XT personal computer, and 2. using a "Bruel & kjaer '° (model 2034) dual-channel frequency analyzer. The sampling rate of the personal computer was fixed at 800 measurements per second. Based on the Strouhal cycle period, this means a minimum of ten measurements per cycle. The sampling rate of the model 2034 was superior, 2048 measurements per second. In spite of the differences on sampling rate, the results from both analysis were almost identical. Thus, ten measurements per cycle seems an acceptable minimum to represent adequately the fluctuations measured. All results were computed for a period of time equivalent to approximately 3000 Strouhal cycles, and due to the data sa,,L~ling, the formulae Jsed was the following:
816 x = ~ xi /
mean value: rms v a l u e
:
Cl)
n
x' = [~ (x i- ~)2/
C2)
(n - I)] 1/2
^
positive
crest factor:
x = max
(x i- x) / x'
C3)
negative
crest
x" = m i n
" "^ i ~
(4)
factor: Z
skewness
:
~B
1
[~ (Xl_ Z
:
kurtosis
where
B
2
~)3i
Cx i -
=
=
(X|-
x)
/
x'
n
~)2/
n
~)4/
n
(S)
]3/2
(6)
[~ Cxl_ ~)2/ n ]2
x
r e p r e s e n t s a zeneric value of pressure, force or moment and n is i the total of the collected data in each case. The c o r r e s p o n d i n g m e a n and rms (pressure, force or moment) c o e f f i c i e n t s were o b t a i n e d as:
2p
/
(p u~)
;
c' = 2p' p
/
(p u 2) t
(7)
C- = 2 F
/
(p u 2 h d)
;
C'
/
Co u 2 h d)
(8)
;
C' : 4M' / (p u 2 h 2 d)
C- = p F
t
C- = 4M / (p u 2 h 2 d) M
where
u
t
is
F
t
the
blockage was small,
free-stream
= 2F'
H
at
the
model
Lop
OVERTURNING MOMENTS
$ PRESSURE MEASUREMENTS
f ro.tal face
I 2
S _ 7 3 4 5 G lateral
rear fq¢o
i
o
,
f
@
0 Nm
t
, m
0
face
0 ~p d,W r q f ~ f
Figure 3. P o s i t i o n of the p r e s s u r e
height.
for blockaEe were attempted.
FORCES
FLOW
(9)
t
velocity
no c o r r e c t i o n s
t
rJ
0m
'I"
taps for PDF measurements.
As
the
817
4. RESULTS AND DISCUSSION 4.1. Mean and rms measurements Table 1 presents the results measured in TV-2. Regarding mean and rms values, Matsumoto [2] and Sun, Lin and Gu [3] presented values measured in smooth flow tests; while Akins et al. [4], Cheung [5], Sakamoto [6], Kronke and Sockel [7], Tanaka and Lawen [8], Matsumoto [2], Golinger and Milford [9] and Sakamoto and Haniu [I0] presented values measured in turbulent shear flow. Since the experimental set-ups were not too different, the results reported by these authors can be compared with those collected in TV-2. Taking into account differences in blockage, aspect ratio or characteristics of the generated flow, the mean and rms results measured in TV-2 compare fairly well with those reported in the literature.
Table
I. Mean values, rms values, B2 measured in TV-2.
GOEF.
crest factors,
SMOOTH FLOW
Mean
rms
Crest (+)
Factor (-) t"B I
skewness ¢BI and Kurtosis
TURBULENT FLOW
Bz
Norma lity
Mean
rms
Crest (+)
~actor (-) I"BI
B2
Norma lity
Cpi Cpa Cp3 Cp, Cp6 Cpe Cp7 Cp8
.91 .69 -.72 -.74 -.75 ".78 ".1'0 ".69
.017 3.7 3.7 .023 3.4 4.6 .089 3.6 7.3 .099 2.9 6.0 .090 3,6 8.3 .12" ' 3 . 5 I 0 . 0 .119 5.2 8.0 .115 5.9 6 . 5
-0.01 -0.47 -0.82 -0,92 -0.96 "1.42 "0.61 "0.24
3.12 3.66 5. II 5.01 5.39 6.94 5.74 5.17
Y N N N N N N N
.60 .42 -.74 " .60 ".73 ".71 ",51 -.45
.133 .122 .276 .298 .285 .290 .180 .145
3.6 3.3 2.7 2.9 3.1 3.9 3.8 3.6
3.7 4.2 6.9 6.0 6.6 6.9 6.7 7.5
0.04 -0.05 -0~92 -I.06 -I.10 -0.75 -I.08 -0.96
3.01 2.91 4.17 4.65 5,01 4.29 5.11 5.94
Y Y N N N N H N
CFXF CFXR CFX CMXF CMXR GMX
.72 ".57 1.2g .76 " . 62 1.38
.022 .052 .050 .020 .063 .056
3.5 3.8 4.1 3.6 3.6 4.4
3.6 4.3 3.5 4.4 4.8 3.7
-0.14 -0.39 0.27 -0.33 -0.49 0.26
3.12 3.70 3.33 3.21 3.75 3.55
N N N N N N
.44 ".38 .82 .50 -.40 ,90
.089 .076 .140 .069 .074 .139
3.3 3.0 4.2 3.3 3.2 4.4
3.5 4.7 3.5 4.1 4.6 3,7
O.OI -0,37 0.15 -0.16 -0.27 0.16
2.98 3,24 3,05 3.01 3.31 3.21
Y N N N N N
CFys CFyL CFy CMyQ CMW" CidY
".66 - . 66 .00 -.72 -.71 .00
.076 .073 ,106 .098 .095 .125
3.1 3.2 6.5 3.0 2.9 5.5
6.4 6.6 6 .5 5.4 5.6 5.4
"1.11 -I.13 -0.02 -I.03 -I.08 0.01
5.61 5.69 6.69 5.25 5.34 6.01
N N N N N N
" .56 -.55 .00 -.57 -.59 .00
.120 .120 .198 .125 .125 .203
2.9 2.B 4.2 2.9 2.9 4.0
5.0 5.2 4.3 4.~, 4.4 4.0
-O.50 -0.59 -0.04 -0.57 -0.55 -0.02
3.55 3.~7 ~-.2~:J ~.44 ~ .51 3.20
N N N N N N
818
4.2. Crest factor measurements Few authors reported crest factor results measured on three-dimensional square cy]inders. Golinger and Milford [9] presented some results of peak pressures measured at a section positioned at 2h/3 on the CAARC standard tall building model. Their results for the BLI flow (which is similar to the turbulent shear wind generated in TV-2) indicate a negative crest factor around 4.0 at a point on the frontal face, around 6.0 at a point on the lateral face, and around 7.0 at a point on the rear face. These values were computed for the approaching flow perpendicular to one of the faces and they compare with corresponding values measured in TV-2 (see Table I). Kronke and Sockel [7] reported results of extreme drag coefficients. On tests with model #3 and boundary layer type II (same model shape and similar turbulent shear flow as those used in TV-2). They measured mean CFX = 0.84 and mode of extreme values of CFX = 1.36. These values are in agreement with those measured in TV-2, i.e., 0.82 (mean) and 1.30-1.40 (extreme values). Tanaka and Lawen [8] conducted measurements in turbulent shear flow on a three-dimensional cylinder of rectangular cross-section (2:3) and aspect ratio = 4. They reported that the crest factor at zones of positive mean pressure were generally found to be within the range 3.6-4.1; whereas at zones of negative mean pressures sometimes very high values (around 6.0 or 7.0) were observed for negative crest factors. Again the values are in agreement with the values measured in TV-2. 4.3. Skewness and kurtosis measurements Kanda and Ohkuma [11], based on available results of full-scale wind pressure measurements on sixteen tall building in Japan presented some information regarding PDF measurements. According to these authors, in most studies the PDF of negative pressures on leeward faces were found slightly skewed to the left, but there were divergences about the skewness of the PDF of positive pressures (on windward faces). One of these full-scale wind pressure measurements was performed on Chiba Port Tower, and Kawabata et al. [ 1 2 ] presented results concerning PDF measurements. Chiba Port Tower is not a square cylinder (but presents a diamond shape cross-section). In spite of this fact, as will be seen later, the main observations reported by these authors are in agreement with the measurements made in TV-2. Kawabata et al, [12] reported that the PDF of the pressures on windward face is similar to the PDF of the wind speed fluctuations and no skewness was founU, The PDF of the pressures on side face, however, shows a stron E skewness and kurtosis and is distinct from the normal distribution. The PDF of the pressures on leeward face still shows a strong kurtosis, but not as strong as observed on side faces. Returning to the results of TV-2, Table I also presents the parameters ~BI (skewness) and B2 (kurtosis) which characterize the corresponding PDF curves. The measured values of ~Bt and B2 were tested to verify if the hypothesis of normal distribution could be accepted. The tests employed follow the procedure proposed by D'Agostino et al. [I]. Actually two tests were performed on every PDF curve: (I) test of skewness, where the null hypothesis is Ho: normality, while the alternative is HI: nonnormality due to skewness; and (2) test of kurtosis, where the null hypothesis is Ho: normality, while the alternative is HI: nonnormality due to nonnormal kurtosis, To be accepted as normal, a distribution must pass both tests. As usual, the statistical level of significance adopted for the tests was ~ = 5~,
819
which means that the Ho hypothesis are discarded when the probability of the sample proceeding from a normal distribution is less than 5~. Figs. 4, 5 and 6 present (as measured in smooth and turbulent shear flow), the PDF curves of pressure fluctuations, while Fig. 7 presents the PDF curves of the force fluctuations on entire individual faces and Fig. 8 presents the PDF curves of the global force fluctuations. Figs. 4 to 8 complement the information included in Table I. For visual aid the normal [Gauss) distribution is also depicted in these figures. It can be said that the probability distribution of pressure fluctuations measured from position I to 8 (from frontal face to rear face) document in the statistical point of view the way the turbulence present in the approaching flow is distorted by the body. 4.3.1. Smooth flow results As can be seen in Table I, regarding smooth flow, the hypothesis of normality is not discarded only for the PDF of the pressures on point 1 (frontal face). On the frontal face it is possible to delimit a central (vertical) zone d/2 wide where the pressure fluctuations practically mirror the velocity fluctuations of the flow and, therefore, show a normal distribution (see Fig. 4). Outside this zone the pressure fluctuations are significantly affected by the Karman vortices and the corresponding PDF are distorted in such a way that they depart from normal (see Fig. 5). The effect of the Karman vortices is stronger on the lateral faces, where the PDF of the pressures appear significantly skewed to the left and peaked. On the rear face the skewness diminishes, but a strong kurtosis remains (see Fig. 6).
50
- 40
I
,
SMOOTH FLOW . •. . | . |
TURBULENT |
•
PI
|
•
•
~
FLOW lu
•
|
Pl
|=o I
0
. . . . .
-
. . . . .
P2 3O
I0 o
.
,
.,
OEVIATION
(RM$)
Figure 4. PDF of pressure fluctuations on frontal face.
|
820 TURBULENT FLOW
SMOOTH FLOW 50
p;
.
'
.
40 30
7
20
~.'".....
10
•
b
•
P4
m 40
•
•
•
A
•
,
•
•
•
•
,
P4 iI
I I 30
,.,'/i
at
20
,'/
I I I
N 10
&%
Z W
Q
•
L
•
0
~: 4o -1
P5
P5
I
#
I 9[ 30
f
I
@~o a_
I #
,/
10
0--~
40"
P6
P6
3o
/
,Y
2o IO.
-4
I
-3
-2
-I
0
I
2 3 4 -3 -;~ -I DEVIATION | RMS)
Figure 5. PDF of pressure fluctuations
on
0
lateral
I
2
3
4
face.
In smooth flow the pressures are quite well correlated on the frontal face and on the lateral faces. Thus the PDF of the forces on these faces are approximately the average of the corresponding PDF of the pressures. The PDF of the forces on the frontal face is not normal, but is not too different from normal, while on the lateral faces the PDF of the forces shows a strong skewness and kurtosis (see Fig. 7).
821
so
.
.SMOOTH , ~ .FLOW . . . . . . JF
.
:
|
I
I
,'! ~o
TURBULENT ^~ FLOW
I
t
V :
:
'
®~~4°' 30 ; P
~
:
P8
,,,""
:
• ":,.'-.-:
""'.
n_
I0 0
.
-4-g-~-~
o
i
z
i
4-g-2-I
o
I
2
i
4
DEVIATION (RMS)
Figure 6. PDF of pressure fluctuatlons on rear face.
On the rear face, however, the pressures are not well correlated. Strong pressure peaks (badly correlated} do not generate strong force peaks. Thus the resulting PDF of the forces on the rear face is not as skewed or peaked as the PDF of the local pressures on thls face. Regarding global forces, It Is interesting to observe that the PDF of the drag shows a positive skewness (see Flg. 8}. Thls happens because the suction peaks on the rear face are translated as positive contributions to the drag. Meanwhile, the skewness of the PDF of the global cross-stream forces Is virtually zero. Thls happens because while suction peaks on one lateral face contribute reducing Fy, the same suction peaks on the other lateral face contribute increasing Fy. In spite of the symmetry, the PDF of the global cross-stream force In smooth flow Is quite different from the normal distribution because it is quite peaked. On the other hand, the PDF of the global drag, although not acceptable as normal for a level of significance a = 5Z, in fact Is a distribution that is not quite different from the normal. The PDF of the overturning moments In smooth flow show small differences when compared to the PDF of the corresponding forces. The PDF of the streamwise overturning moment appears more skewed and peaked than the PDF of the drag. While the PDF of the cross-stream overturning moment appear~ less skewed and peaked than the PDF of the cross-stream force. It Is possible that these experimental observations are due to the top vortices which roll along the upper horizontal edge. These vortices; affect the zone close to the model top, which provides the major contribution on determining the overturning moments. As a last observation concerning smooth flow, the data collected in TV-2 indicate that taking the fluctuations of streamwise forces or moments as
822
50
i
40 30 ~0
z I0 In
o
|
SMOOTHFLOW ~
i
~
/,,\
FRONTAL
.c,
•
•
FRONTAL F
l
I
I
I
I
-
•
A
OAU,>l _/'.-.
"k__ '
40 z 30 20
FLOW
TURBULENT I
•
'
,
•
I
I
i
•
~
>"
I--
;J k ,,/ ~,
b,I
),I--
"°'
"; I 0 m 0 n,,
0
REAR 30 FACE et 20
REAR FACE~
~, 4 0
,:/
I0
0
\~
-~
-:,
-I
o
•
\~
\,
~,~.'... -4
•
',~
a
.%
z ~ 4 - ~ - z - i OeVlA'r,ON ( R M S )
.
. o
.
%._ z $
4
Figure 7. PDF of force fluctuations on entire individual faces.
as normally distributed generally results in an acceptable error. Nevertheless the assumption that the fluctuations of cross-stream forces or moments are normally distributed would lead to larger errors.
4.3.2. Turbulent shear flow results Turning back to Table I, the results measured in turbulent shear flow indicate that the hypothesis of normal distribution can not be discarded for the fluctuations of the pressures on points I and 2 (frontal face). In turbulent shear flow the Karman vortices are not as intense as in smooth flow and, therefore, do not affect significantly the pressure fluctuations anywhere on the frontal face (see Fig. 4). On the other hand, on the lateral face and on the rear face the pressure fluctuations are affected by the Karman vortices and the corresponding PDF appear significantly skewed to the left and peaked (see Figs. 5 and 6). The skewness and kurtosis are of such an order that the negative crest factor on these faces results around 6.0 OF 7.0 . Although the PDF of the pressures on the lateral faces and on the Fear face present a pronounced skewness and kurtosis, the effect is not percepti-
823
SMOOTH FLOW . . . . , .
50 40
DRAG
.
.
.
.
TURBULENT FLOW , ,, , ,
,
A
,o .
r,
.
/..X STREAM
¢L 2 0
-4
'
~'- "~
GAUSS,~,,,I
,o o
"
,
// -~
-z
-I
,
o
I
2 ~ 4 DEVIATION
-'~ -2 { RMS }
-{
o
i
~
~
4
F i g u r e 8. PDF of g l o b a l f o r c e f l u c t u a t i o n s .
ble in the PDF of the forces on these faces (see Fig. 7). In turbulent shear flow the correlation of the pressure fluctuations is poor. Strong suction peaks (badly correlated) do not generate strong force peaks on the entire face. Thus on the lateral faces and on the rear face the crest factor of the forces remain around 4.0 or 5.0 . The poor pressure correlation in turbulent shear flow is reflected in a still more dramatic way over the global forces. The PDF of the global forces seem to converge towards the normal distribution. Although neither the drag nor the crossstream force can be accepted as normally distributed for a level of significance ~ = 5~, actually their distributions are not quite different from the normal distribution (see Fig. 8). The PDF of the overturning moments in turbulent shear wind show small differences when compared to the PDF of the corresponding forces. The PDF of the streamwise moment is similar to the PDF of the drag, but the former is slightly more skewed and peaked. Meanwhile, the PDF of the cross-stream overturning moment is similar to the PDF of the cross-stream force, but it is slightly less skewed and peaked. As a last observation concerning turbulent shear flow, the data collected in TV-2 indicate that taking the pressure fluctuations as normally distributed would result in a large error, but in several applications taking the fluctuations of global forces (drag, cross-stream force and overturning moments) as normally distributed would be reasonable, resulting in an acceptable error. It is worth mentioning that the turbulent shear flow used in the tests here described corresponds to a suburban terrain. It is possible that tests conducted in a flow corresponding to an urban terrain (high turbulence) disclose global forces that are in fact normally distributed.
824
5. CONCLUSIONS Wind tunnel tests with a three-dimensional square cylinder model (aspect ratio = 6.0) revealed that except perhaps on points of the frontal face, the probability distribution function (PDF) of the pressures is not normal (gaussian). On the lateral face and on the rear face, where the effect of the Karman vortices is significant, the PDF of the pressures shows a strong skewness to the left and kurtosis (peakdeness). In fact, negative crest factors around 7.0 were measured on these faces. In smooth flow the pressure fluctuations over an entire face show a good correlation and the PDF of the forces is similar to the PDF of the pressures on the corresponding face. In turbulent shear flow, however, the pressure fluctuations over an entire face show a poor correlation and the PDF of the forces is not as skewed and peaked as the PDF of the pressures on the corresponding face. In turbulent shear flow taking pressure fluctuations on a three-dimensional cylinder as normally distributed would result in a considerable error; however, in several applications considering global forces (drag, crossstream force and overturning moments) as normally distributed could result in an acceptable error. Actually, for a statistical level of significance = 5Z, the global forces can not be accepted as normally distributed, but in turbulent shear flow the probability distribution of the global forces is not too different from the normal distribution.
REFERENCES 1 2 3
4 5 6
7 8 9
10
11 12
R.B. D'Agostino, A suggestion for using powerful and informative tests of normality, The American Statistician, 44-4 (1990) 316-321. T. Matsumoto, On the across-wind oscillation of tall buildings, J. Wind Eng. Ind. Aerodyn., 24 (1986) 69-85. T . F . Sun, S.T. Lin and Z.F. Gu, Mean wind loadings acting on two rectangular cylinders of side-by-side arrangements, Prec. 7th Int. Conf. on Wind Engineerlng, Aachen, West Germany, 1987, Vol.2, pp 21-30. R . E . Akins, J.A. Peterka and J.E. Cermak, Mean force and moment coeff. for buildings in turbulent boundary layers, J. Ind. Aerodyn, 2 (1977) 195. S.Y. Cheung, The interference effect on tall building, ES400 Project Report presented to the University of Western Ontario, {983. H. Sakamoto, Aerodynamic forces act|ng on a rectangular prism placed vertically in a turbulent boundary layer, J. Wind Eng. Ind. Aerodyn., 18
(1985) 131-151. I. Kronke and H. S o c k e l , Measurements of e x t r e m e d r a g c o e f f i c i e n t s of b u i l d i n g models, J, Wind Eng. Ind, Aerodyn., 23 (1986) 149-163. H. Tanaka and N. Lawen, T e s t on the CAARC s t a n d a r d t a l l b u i l d i n g model whit a l e n g t h s c a l e 1:1000, J. Wind Eng. Ind. Aerodyn., 25 (1986) 15-30. A,H. G o l i n g e r and R.V. M i l f o r d , S e n s i t i v i t y of t h e CAA~C s t a n d a r d b u i l d i n g model to g e o m e t r i c s c a l e and t u r b u l e n c e , J. Wind Eng. Ind. Aerodyn, 31 (1988) 105-123. H. Sakamoto and H. Haniu, Aerodynamic f o r c e s a c t i n g on two s q u a r e p r i s m s placed vertically in a t u r b u l e n t boundary l a y e r , J. Wind Eng. Ind. Aerodyn., 31 (1988) 41-66. J, Kanda and T. Ohkuma, R e c e n t d e v e l o p a e n t s i n f u l l - s c a l e wind p r e s s u r e s measurements in Japan, J. Wind Eng. Ind. A e r o d y n . , 33 (1990) 243-252. S. Kawabata, T, Ohkuma, J. Kanda, H. Kitamura and K. Ohtake, Chiba P o r t Tower: f u l l - s c a l e measurement . . . . J.Wind Eng. Ind.Aerodyn, 33 (1990) 253.