characteristic of wind load on a hemispherical dome in smooth flow...

1

BBAA VI International Colloquium on: Bluff Bodies Aerodynamics & Applications

Milano, Italy, July, 20-24 2008

CHARACTERISTIC OF WIND LOAD ON A HEMISPHERICAL DOME IN SMOOTH FLOW AND TUBULENT BOUNDARY LAYER FLOW

C. M. Cheng ∗, C. L. Fu † , Y.Y. Lin#

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

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

Keywords: wind load, hemispherical dome, wind tunnel, smooth flow, turbulent boundary layer, Reynolds number

1 INTRODUCTION Due to the structural efficiency and economic benefit, the hemispherical dome is a common

structural geometry shape for large span sports stadium or for storage purposes. The curve shape makes the accurate estimation of the wind pressure fluctuations on a hemispherical dome a difficult task due to the Reynolds number effects. In the past years, there have been reports of collapse of curve shaped storage domes during strong wind. The wind induced structural fail-ure could be attributed to inadequate wind resistant design and/or poor quality construction. In the past two decades, several innovative procedures were proposed by Kasperski and Niemann [1], Holmes [2], Katsumura et al. [3] to incorporate the structural responses into the formation of design wind loads. On the other hand, there exist only few works on the aerodynamics of hemispherical dome in either smooth or turbulent flows. Among them, Maher[4] presented mean pressure distribution of hemispherical dome at Reynolds number of 0.92×106 ~1.84×106; the drag coefficient becomes invariant when Reynolds number exceeds 1.4×106. Taylor [5] took measurements of both mean and fluctuating pressure of domes in turbulent boundary layer flows; his work confirmed the earlier finding that when Reynolds numbers exceeds 2.0×105 and turbulence intensity exceeds 4%, the pressure distribution becomes Reynolds number independent. Ogawa et al. [6] study the mean, RMS and spectral characteristics of wind pres-sure on domes; and indicates that for Reynolds numbers ranging from 1.2×105 ~2.1×105, the level of turbulence in turbulent boundary layer has little effect upon the mean pressure distri-bution. Toy et al. [7] showed that with increase of turbulence intensity, separation region and reattachment point move downstream. Letchford and Sarkar [8] investigated the effect of sur-face roughness to the pressure distribution of dome. In light of the past works on this issue, it is believed worthy to accumulate more detailed data on the aerodynamics of hemispherical domes.

This article is the first of a series of a systematic investigation on the wind load characteris-tics of curve shaped domes in turbulent boundary layer flow and smooth flow. The present in-vestigation focuses on the Reynolds number effects on pressure distribution and the wind load pattern of hemisphere dome in smooth flow and turbulent boundary layer flows. Besides to

C. M. Cheng, C. L. Fu, Y.Y. Lin

2

study the difference of wind load characteristics of hemispherical dome in different flow con-ditions, one important aim of this study is to find the load pattern under the Reynolds number independent flow condition to be used as the reference of future investigations.

2 EXPERIMENT DESIGN The majority of the wind tunnel tests of this project were conducted in a boundary layer

wind tunnel with a 24m(length)×4m(width)×2.6m(height) test section; Both smooth flow and turbulent boundary layer were used. Base plate elevated from floor was used in the case of smooth flow to minimize the boundary layer developed over base floor. The turbulent bound-ary layer has the characteristics of suburban flow field with power law index α=0.27 and turbu-lence intensity varies from 25% to 18% in the region of the model heights. Three acrylic hemispherical pressure models with diameters of 120 cm, 50 cm, and 20 cm, namely Dome L, M, and S, were used in this investigation; the corresponding Reynolds number varies from 5.3 × 104 to 2.0 × 106. Pressure taps were installed along the meridian and full circular of the lati-tude on the 120cm and 50 cm models; whereas over 200 pressure taps distributed on 8 levels of concentric circles were installed on the 20 cm model to investigate the pressure pattern in dif-ferent testing conditions. Shown in Figure 3 is the coordinate system of the pressure taps on hemisphere dome surface, in which, ψdenotes the angle of longitude and θ is the angle of center meridian ; for contours of pressure coefficients , define ω as the angle of latitude. In-stantaneous wind pressures were sampled simultaneously at sampling frequency of 300 Hz, through ZOC pressure scanner system. The blockage ratios of the tests were 0.0015 for Dome S to 0.054 for Dome L. Results shown in this paper are not corrected for the blockage effect. The wind tunnel testing cases are listed in Table 1.

dome in boundary layer flow dome in smooth flow

Reynolds number Diameter Smooth flow Boundary layer

flow

Dome S 20cm 6.6×10 4 -3.4×10 5 5.3×10 4 - 2.0×10 5

Dome M 50 cm 1.6×10 5 - 8.7×10 5 2.0×10 5 - 6.0×10 5

Dome L 120 cm 5.3×10 5 -2.0×10 6 5.8×10 5 - 1.7×10 6

Fig.2 Wind velocity and turbulence intensity pro-files (comparing with dome models).

Table.1 Wind tunnel testing conditions

Fig.1 hemispherical pressure models in smooth and boundary flow

0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

wind velocity

8 12 16 20 24

tubulence intensity

Turbulence intensity(%)

Mean wind velocity (U(z)/U( ))δ

Hei

ght

z/(

δ)

dome_S

dome_M

dome_L


3

0

20

40

6080100

120

140

160

180

200

220

240260 280

300

320

340

f

Figure 3. coordinate system of pressure taps on hemisphere models (a) angle of center meridian θ and angle of latitude ω , (b) angle of longitude φ

3 PRESSURE DISTRIBUTION IN SMOOTH FLOW

3.1 Mean and RMS pressure coefficients in smooth flow For validating the consistency of different model at the same Reynolds number, the com-

parison of the distributions of mean pressure coefficients along the center meridian of different model at the same Reynolds number in smooth flow condition is shown in Figure 4. As shown in Figure 4(a), the pressure distributions of Dome S and Dome M agree very well. The pres-sure distributions of Dome L and Dome M in Figure 4(b) show slight differences. Since Dome L is significantly larger than Dome S & M, in order to have a better blockage ratio, the base plate was descended 5 cm into the upper part of the boundary layer developed over the tunnel floor. The discrepancy of pressure distribution is probably due to the combined effects of the boundary layer effects and the higher blockage ratio for the Dome L.

The distributions of mean and RMS pressure coefficients along the center meridian at vari-ous Reynolds number in smooth flow condition is shown in Figure 5 and 6. As shown in Fig-ure 5(a), the maximum mean pressure occurs at 10°~20° and the minimum mean pressure locates at 70°~90°. For Reynolds number between 6.6 × 104 ~3.1 × 105, the negative pressure at dome apex increases with Reynolds number, whereas the negative pressure in the wake re-gion decreases. At Re=3.1 × 105, separation of surface shear layer occurs around 130°. How-ever, for Reynolds number between 3.0×105 ~2.0×106 as shown in Figure 5(b) and 5(c) , the wake suction increases slightly with Reynolds number whereas the suction near apex remains nearly invariant; the point of separation moves up about 10° upstream, from 130° to 120°.

-1.5

-1

-0.5

0

0.5

1

1.5

0° 20° 40° 60° 80° 100° 120° 140° 160° 180°

Cp

S -dome Re=3.0E+05

M-dome Re=3.0E+05

-1.5

-1

-0.5

0

0.5

1

1.5

0° 20° 40° 60° 80° 100° 120° 140° 160° 180°

Cp

M-dome Re=7.5E+05

L -dome Re=7.5E+05

Fig.4 the comparison of the distributions of mean pressure coefficients of different model

θ ，ω

base floor

(a) 0°≦θ≦180°,angle of center meridian 0°≦ω≦90°,angle of latitude

(b) 0°≦φ≦360°

(a) (b)


4

-1.5

-1

-0.5

0

0.5

1

1.5

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

Cp

6.6E+04 8.0E+04

9.2E+04 1.2E+05

1.3E+05 1.4E+05

1.5E+05 1.7E+05

1.8E+05 1.9E+05

2.1E+05 2.2E+05

2.3E+05 2.4E+05

2.7E+05 2.8E+05

2.9E+05 3.1E+05

-1.5

-1

-0.5

0

0.5

1

1.5

0 ° 20 ° 40 ° 60 ° 80 ° 100 ° 120 ° 140 ° 160 ° 180 ° 200 °

Cp

3.0E+05

3.3E+05

3.6E+05

3.9E+05

4.2E+05

4.6E+05

4.9E+05

5.2E+05

5.5E+05

5.8E+05

6.2E+05

6.5E+05

6.8E+05

7.1E+05

7.4E+05

8.1E+05 -1.5

-1

-0.5

0

0.5

1

1.5

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

Cp

8.2E+05

8.9E+05

9.6E+05

1.0E+06

1.1E+06

1.2E+06

1.3E+06

1.4E+06

1.5E+06

1.6E+06

1.7E+06

1.8E+06

1.9E+06

2.0E+06

Shown in Figure 6(a) to 6(c) are the distributions of RMS pressure coefficients along the

center meridian at various Reynolds number in smooth flow condition. The significant discrep-ancy of RMS pressure in the front stagnation zone shown in Figure 6(b) and 6(c) is most likely due to the effect of thin boundary layer flow developed over base plate onto different size models. This effect of model dimension diminished when θ> 45°. The experimental data indi-cate that Reynolds number casts significant influence on the distribution of RMS pressure co-efficients in the negative pressure zone: for Re<1.8×105, there is one distinct and narrow peak of PC′ locate at 80°~90° and a broader lump around 140°, suggesting the separation and reat-tachment. When Reynolds number increases to 1.8×105 ~ 3.0×105, a second distinct and nar-row peak appears at 110°~120°. The second peak value enhances with Reynolds number while the first peak value gradually diminishes with Re. when Reynolds number greater than 3.0×105, there remains only one peak of PC′ in the separation zone, near 120° and another one at very down-stream. As Reynolds number increases from 3.0×105~2.0×106 this separation related PC′ peak moves back and forth between 110°~120°.

0

0.05

0.1

0.15

0.2

0.25

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

Cp'

6.6E+04 8.0E+04

9.2E+04 1.1E+05

1.2E+05 1.3E+05

1.4E+05 1.5E+05

1.7E+05 1.8E+05

1.9E+05 2.1E+05

2.2E+05 2.3E+05

2.4E+05 2.6E+05

2.7E+05 2.8E+05

2.9E+05 3.1E+05

Fig.5 The distributions of mean pressure in smooth flow. (a)Dome_S, 6.6×104<Re<3.1×105 (b)Dome_M, 3.0×105<Re<8.1×105 (c)Dome_L, 8.2×105<Re<2.0×106

(a)

(b) (c)

(a)


5

0

0.05

0.1

0.15

0.2

0.25

0° 20° 40° 60° 80° 100° 120° 140° 160° 180°

Cp'

3.0E+05

3.3E+05

3.6E+05

3.9E+05

4.2E+05

4.6E+05

4.9E+05

5.2E+05

5.5E+05

5.8E+05

6.2E+05

6.5E+05

6.8E+05

7.1E+05

7.4E+05

8.1E+05 0

0.05

0.1

0.15

0.2

0.25

0° 20° 40° 60° 80° 100° 120° 140° 160° 180°

Cp'

8.2E+05

8.9E+05

9.6E+05

1.0E+06

1.1E+06

1.2E+06

1.3E+06

1.4E+06

1.5E+06

1.6E+06

1.8E+06

1.9E+06

2.0E+06

2.1E+06

3.2 Correlation length of pressure fluctuations in smooth flow In order to better interpreting the experimental data, a set of non-dimensional correlation

lengths of the fluctuating pressure were introduced in this paper. Let iλ− denotes the upstream

correlation length of pressure fluctuations at θi. In which,

( )

0

1 ,i

i iR dr

θλ θ θ θ− = ∫ (1)

( ) ( ) ( )

i j

, , ,

=correlation coefficient between pressure fluctuations at and i j i j pi pjR E p t p tθ θ θ θ σ σ

θ θ

⎡ ⎤= ⎣ ⎦ (2)

r = radius of dome.

Therefore, represents a correlation length of the fluctuating pressure considering only the upstream of the reference point θi. Similarly, let iλ

+ denotes the downstream correlation length that considering only the pressure taps downstream of the reference point θi; in which,

( )1 ,i

i iR dr

π

θλ θ θ θ+ = ∫

A threshold of R(θi , θ)≧0.05 was adopted during integration to avoid the fluctuating na-ture of correlation coefficient at large spatial separation.

For a pressure tap located immediately upstream of the separation point, a relatively high upstream correlation length, iλ

− , and a low downstream correlation length, iλ+ , can be ex-

pected. On the other hand, for pressure tap located immediately downstream of separation point, we can expect to find low upstream correlation length and relatively high downstream correlation length. These upstream and downstream correlation lengths were calculated; the results of smooth flow cases are shown in Figure 7 (a) to 7 (c); the results of turbulent flow cases are shown in Figure 14 (a) to 14(c).

(a1)

0

0.2

0.4

0.6

0.8

1

1.2

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

upst

ream

cor

rela

tion

len

gth

9.2E+04 1.1E+05

1.2E+05 1.3E+05

1.4E+05 1.5E+05

1.7E+05 1.8E+05

1.9E+05 2.1E+05

2.2E+05 2.3E+05

2.4E+05 2.6E+05

2.7E+05 2.8E+05

2.9E+05 3.1E+05

3.2E+05 3.3E+05

(a2)

0

0.2

0.4

0.6

0.8

1

1.2

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

dow

nstr

eam

cor

rela

tion

len

gth

9.2E+04 1.1E+05

1.2E+05 1.3E+05

1.4E+05 1.5E+05

1.7E+05 1.8E+05

1.9E+05 2.1E+05

2.2E+05 2.3E+05

2.4E+05 2.6E+05

2.7E+05 2.8E+05

2.9E+05 3.1E+05

3.2E+05 3.3E+05

Fig. 6 The distributions of fluctuating pressure in smooth flow. (a)Dome_S, 6.6×104<Re<3.1×105 (b)Dome_M, 3.0×105<Re<8.1×105 (c)Dome_L, 8.2×105<Re<2.0×106

(b) (c)


6

(b 1)

0

0.2

0.4

0.6

0.8

1

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

upst

ream

cor

rela

tion

len

gth

2.35E+05 2.70E+05

2.98E+05 3.29E+05

3.61E+05 3.91E+05

4.24E+05 4.57E+05

4.88E+05 5.21E+05

5.52E+05 5.84E+05

6.16E+05 6.48E+05

6.79E+05 7.12E+05

7.43E+05 7.76E+05

8.09E+05 8.39E+05

8.72E+05

(b 2)

0

0.2

0.4

0.6

0.8

1

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

dow

nstr

eam

cor

rela

tion

len

gth

2.35E+05 2.70E+05

2.98E+05 3.29E+05

3.61E+05 3.91E+05

4.24E+05 4.57E+05

4.88E+05 5.21E+05

5.52E+05 5.84E+05

6.16E+05 6.48E+05

6.79E+05 7.12E+05

7.43E+05 7.76E+05

8.09E+05 8.39E+05

8.72E+05 (c 1)

0

0.2

0.4

0.6

0.8

1

0 ° 20 ° 40 ° 60 ° 80 ° 100 ° 120 ° 140 ° 160 ° 180 ° 200 °

upst

ream

cor

rela

tion

len

gth

6.8E+05 7.5E+05

8.2E+05 8.9E+05

9.6E+05 1.0E+06

1.1E+06 1.2E+06

1.3E+06 1.3E+06

1.4E+06 1.5E+06

1.5E+06 1.6E+06

1.7E+06 1.8E+06

1.8E+06 1.9E+06

2.0E+06 2.0E+06

(c 2)

0

0.2

0.4

0.6

0.8

1

0 ° 20 ° 40 ° 60 ° 80 ° 100 ° 120 ° 140 ° 160 ° 180 ° 200 °

dow

nstr

eam

cor

rela

tion

len

gth

6.8E+05 7.5E+05

8.2E+05 8.9E+05

9.6E+05 1.0E+06

1.1E+06 1.2E+06

1.3E+06 1.3E+06

1.4E+06 1.5E+06

1.5E+06 1.6E+06

1.7E+06 1.8E+06

1.8E+06 1.9E+06

2.0E+06 2.0E+06

Figure 7. Correlation length of the fluctuating pressure in smooth flow. (a1、a2)Dome_S, 9.2×104<Re<3.3×105 (b1、b2)Dome_M, 2.3×105<Re<8.7×105

(c1、c2)Dome_L, 6.8×105<Re<2.0×106 Show in Figure7 (a1) and 7(a2) are the upstream correlation length and downstream cor-

relation length, respectively, of hemispherical models tested in smooth flow and Reynolds number in-between 9.2×104 to 3.3×105. In this region of Reynolds number, the upstream cor-relation length, iλ

− , increases with the angle of latitude, θ, and reach the highest value of 1.0 near θ=100°; then drops drastically to the minimum value. The point of suddenly appeared minimum value, which indicates downstream of separation point, locates at θ=140° for Re=9.2×104 and gradually moves upstream to θ=120° at Re≒2.0×105. The downstream correlation length, iλ

+ , shown in Figure 7(a2), exhibits a similar trend as iλ− . The minimum value of

iλ+ occurs lightly ahead of iλ

− at θ=110°~ 115°, and this position is quite consistent for all Rey-nolds number in this region. Since the position of minimum iλ

+ indicates this particular pres-sure tap locates upstream of separation point. It is logical to observe the location of minimum iλ+ appears slightly upstream of iλ

− . Figure 7(b1) and 7(b2) show the upstream and downstream correlation length at Reynolds

number Re=2.3~8.7×105. In this region of Reynolds number, the locations of the minimum iλ− and iλ

+ show more variations than the previous Reynolds number region. The minimum value of upstream correlation length, iλ

− , locates at θ=120° for Re=2~3×105; then a dual-minimum pattern appears at Re≒4×105, whereas the first minimum appears at θ=110° and the second minimum at θ=130°. When Reynolds number further increases to Re=8.0×105, the lo-cation of minimum iλ

− settles at a downstream location of θ=135°. Similar characteristics of minimum downstream correlation length, iλ

+ , can be observed in Figure 7(b2). For Re=2~3×105, the minimum value of iλ

+ occurs at θ=120°. At Re≒4×105, dual-minimum pat-tern appears; the first minimum appears at θ=105~110° and the second minimum at θ=120~125°. When Reynolds number increases to Re=8.0×105, the location of single mini-mum iλ

+ occurs at θ≒120°. Again, it can be observed that the location of minimum iλ+ consis-

tently appears slightly upstream of iλ− .


7

Figure 7(c1) and 7(c2) show the upstream and downstream correlation length at higher Reynolds number Re=6.8×105~2.0×106. In this Reynolds number region, the locations of minimum iλ

− and iλ+ have only minor variations: the minimum upstream correlation length, iλ

− , locates a θ=135~140° and minimum downstream correlation length, iλ

+ , locates a θ=125~135°.

3.3 Contours of mean and RMS pressure distributions on dome in smooth flow The contours of mean and RMS pressure coefficients of three domes were plotted to study

the variation of the overall pressure distribution pattern as function of Reynolds number. It should be noted that, due to the different distribution of the pressure taps installed on the models, some deviation of pressure contour may exist even at similar Reynolds number. Shown in Figure 8 are the contours of mean pressure distribution in smooth flow. It is inter-esting that, for a surface of 3D curvature, almost all mean pressure contours exhibit a 2D-like characteristics, i.e., the variation of the pressure contour is similar to the mean pressure distri-bution of the center meridian. It also can be observed that, when the Reynolds number ex-ceeds 2.0×105, the overall mean pressure distribution pattern remains nearly invariant.

The contours of RMS pressure coefficient distribution are shown in Figure 9. For Re<2.0×105, the RMS of pressure fluctuations in the region of 60°<ωand 90°<φ<270°, the rear half of the dome in the separation wake, increases with the Reynolds number. Near Re=3.0×105, a few spots of local maximum RMS pressure can be observed at ω ≒20°~30° and φ≒±120°, in the low rear part of the dome,. When Re>3.0×105, the local fluctuating pressure decreases and the fluctuating pressure distribution pattern exhibits only minor varia-tions.

In order to give a quantitative index to the similarity of pressure distribution pattern, the correlation coefficient of two pressure contours are calculated and presented in Figure 10. Af-ter several tryout, the pressure contour at Re=3.0×105 and selected to be the reference pres-sure contour. In Figure 10, ρmean and ρrms denote correlation coefficients of mean and RMS pressure distributions arbitrary Reynolds number and Re=3.0×105. The result indicates that the correlation coefficient of mean pressure distribution, ρmean, approaches 1.0 near Re=2.0×105 and remains to be 1.0 afterwards. As for the correlation coefficient of RMS pres-sure distributions, ρrms decreases rapidly for Reynolds number less the reference case; the cor-relation coefficient can be as low as ρrms<0.5. In the cases of Reynolds number greater than the reference case, i.e., Re>3.0×105, the correlation coefficient has a much slower decaying trend and levels-up at ρrms =0.8 when the Reynolds number reaches 8.0×105.

Re. Number_65700 Re. Number_179000 Re. Number_205000

Re=6.6×104 Re=2.0×105 Re=1.7×105

°


8


Figure 8. Contours of mean pressure distribution of hemispherical model at various Reynolds number in smooth flow.



Figure 9. Contours of RMS pressure distribution of hemispherical model at various Reynolds number in smooth flow.

wind

wind Re=3.0×105 Re=8.7×105 Re=2.0×106

Re=6.6×104 Re=2.0×105

Re=3.0×105 Re=5.2×105 Re=2.0×106

0

0.1

0.2

0.3

0.4

0.5

-1.8

-1.4

-1

-0.6

-0.2

0.2

0.6

1

Re=1.4×105


9

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0.00E+00 1.00E+05 2.00E+05 3.00E+05 4.00E+05 5.00E+05 6.00E+05 7.00E+05 8.00E+05 9.00E+05 1.00E+06

Re

corr

elat

ion c

oef

fici

ent

S-dome_mean-ref_3E+05

S-dome_rms-ref_3E+05

M-dome_mean-ref_3E+05

M-dome_rms-ref_3E+05

3.4 Mean and RMS meridian drag coefficients in smooth flow

Integrating the pressure along the dome meridian of each testing case, a drag coefficient called the “meridian drag coefficients” can be obtained. The meridian drag coefficients are shown in Figure 11(a) &11(b). Figure 11(a) indicates that, for Re<3.0 × 105 , the mean drag coefficient decreases with the increase of the Reynolds number, and it recovers gradually thereafter. Shown in Figure 11(b), the maximum RMS drag coefficient occurs at Re≒1.5 ×105 , then decreases rapidly and exhibits a minimum value at Re≒3.0×105.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.00E+00 5.00E+05 1.00E+06 1.50E+06 2.00E+06 2.50E+06

Re

Cd

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.00E+00 5.00E+05 1.00E+06 1.50E+06 2.00E+06 2.50E+06

Re

Cd'

4 PRESSURE DISTRIBUTION IN TURBULENT BOUNDARY LAYER FLOW

4.1 Mean and RMS pressure coefficients in turbulent boundary layer flow

Shown in Figure 12 and 13 are the mean and RMS pressure coefficients measured along the dome meridian in turbulent boundary layer flow. As shown in Figure 12(a), for Reynolds number between 5.3 × 104 ~1.58 × 105, the negative pressure at dome apex increases, and de-creases in wake region. For Reynolds number greater than 1.58 × 105, both the negative pres-sure at apex and in the wake region show only minor change with respect to Reynolds number; and the separation point locates near θ=130°. For Re= 5.2 × 105 to 1.6 × 106, the point of sepa-ration moves up about 10° upstream, from 130° to 120°.

Fig. 11(a) Mean drag coefficient in smooth flow.

Fig.11(b) RMS drag coefficient in smooth flow.

Fig. 10 pressure distribution correlation coefficient in smooth flow


10

-1.5

-1

-0.5

0

0.5

1

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

Cp

5.28E+04

6.74E+04

8.11E+04

1.11E+05

1.26E+05

1.41E+05

1.58E+05

1.77E+05

1.83E+05

1.79E+05

1.96E+05

2.03E+05

-1.5

-1

-0.5

0

0.5

1

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°Cp

2.06E+05

2.29E+05

2.47E+05

2.73E+05

2.93E+05

3.16E+05

3.28E+05

3.53E+05

3.73E+05

3.93E+05

4.19E+05

4.33E+05

4.58E+05

4.90E+05

5.09E+05

5.48E+05 -1.5

-1

-0.5

0

0.5

1

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

Cp

5.19E+05

5.85E+05

6.58E+05

7.08E+05

7.70E+05

8.36E+05

9.65E+05

1.02E+06

1.15E+06

1.22E+06

1.28E+06

1.41E+06

1.54E+06

1.61E+06

1.65E+06

The distributions of RMS pressure coefficient are shown in Figure 13(a) to 13(c). It

should be noted that the RMS pressure coefficient is strongly influenced by the difference of incident turbulence level exerted onto the three different size dome models. The RMS pressure exhibits a local peak at the front stagnation zone, at 15°~20°. As for the RMS pressure coeffi-cient in the negative pressure zone: when Re<8.1×104 , there are two peaks of PC′ locate at 80°

~90°and 110°~120°; for Re>1.2×105, there is only a single peak of PC′ near 90°~100°. Pressure data suggest that transition of the separation flow occurs at lower Reynolds number in the tur-bulent boundary layer than the smooth flow. Carefully comparing all pressure measurements of the three different size models, it is concluded that, when Reynolds number is in between 1.2~1.5×105, both PC and PC′ approach the state of Reynolds number independent.

0

0.1

0.2

0.3

0.4

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

Cp

5.31E+04

6.63E+04

8.12E+04

1.12E+05

1.23E+05

1.40E+05

1.55E+05

1.77E+05

1.83E+05

1.96E+05

2.05E+05

0

0.1

0.2

0.3

0.4

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

Cp

2.06E+05

2.29E+05

2.47E+05

2.73E+05

2.93E+05

3.16E+05

3.28E+05

3.53E+05

3.73E+05

3.93E+05

4.19E+05

4.33E+05

4.58E+05

4.90E+05

5.09E+05

5.48E+05 0

0.1

0.2

0.3

0.4

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

Cp

5.19E+05

5.85E+05

6.58E+05

7.08E+05

7.70E+05

8.36E+05

9.65E+05

1.02E+06

1.15E+06

1.22E+06

1.28E+06

1.41E+06

1.54E+06

1.61E+06

1.65E+06

Fig .12 Mean pressure distribution in turbulent boundary layer. (a)Dome_S, 5.3×104<Re<2.0×105 (b)Dome_M, 2.0×105<Re<5.5×105 (c)Dome_L, 5.2×105<Re<1.6×106

Fig.13 RMS pressure distribution in turbulent boundary layer. (a)Dome S, 5.3×104<Re<2.0×105 (b)Dome M, 2.0×105<Re<5.5×105 (c)Dome L, 5.2×105<Re<1.6×106

(b) (c)

(a)

(b) (c)

(a)


11

4.2 Correlation length of pressure fluctuations in turbulent boundary layer flow

Shown in Figure 14 are the upstream and downstream correlation length obtained in turbulent flow field at various Reynolds number. Although the three different size models submerged in different turbulence intensity environments, the distributions of pressure correla-tion length of three models show good similarity for all testing Reynolds numbers, especially on the wind ward side of the dome; there exist some variations before and after the separation point. The correlation length obtain in turbulent boundary layer flow exhibits an additional minimum value near θ=60° which is not related to the separation phenomenon.

Figure 14(a1) and 14(a2) show the upstream and downstream correlation length ob-tained from Dome S with Reynolds number Re=6.7×104 to 2.0×105. In this Reynolds number region, the location of the minimum upstream correlation length, iλ

− moves upstream from θ=140° to θ≒130° at Re=1.6×105. There is not much difference on the distribution of down-stream correlation length, iλ

+ , the minimum value of iλ+ appears consistently at θ=120°. The

correlation lengths obtained from Dome M are shown in Figure 14(b1) and 14(b2). There ap-pears to having a sudden change of pattern around Re=3.5×105, which is the side effect of the selected correlation coefficient threshold; the location of minimum correlation lengths remain invariantly at θ=130° and θ=120°, for iλ

− and iλ+ , respectively. As for the correlation lengths

obtained from Dome L, shown in Figure 14(c1) and 14(c2), the minimum value of upstream correlation length, iλ

− , locates at θ=125~135°; whereas the minimum value of downstream cor-relation length, iλ

+ , appears at θ=125~130°.

(a 1)

0

0.2

0.4

0.6

0.8

1

1.2

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

upst

ream

cor

rela

tion

len

gth

6.74E+04

8.11E+04

9.69E+04

1.11E+05

1.26E+05

1.41E+05

1.37E+05

1.55E+05

1.55E+05

1.58E+05

1.77E+05

1.83E+05

1.79E+05

1.96E+05

2.03E+05

(a 2)

0

0.2

0.4

0.6

0.8

1

1.2

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

dow

nstr

eam

cor

rela

tion

len

gth

6.74E+04

8.11E+04

9.69E+04

1.11E+05

1.26E+05

1.41E+05

1.37E+05

1.55E+05

1.55E+05

1.58E+05

1.77E+05

1.83E+05

1.79E+05

1.96E+05

2.03E+05 (b 1)

0

0.2

0.4

0.6

0.8

1

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

upst

ream

cor

rela

tion

len

gth

1.85E+05 2.06E+05

2.29E+05 2.47E+05

2.73E+05 2.93E+05

3.16E+05 3.28E+05

3.53E+05 3.73E+05

3.93E+05 4.19E+05

4.33E+05 4.58E+05

4.90E+05 5.09E+05

5.22E+05 5.48E+05

5.67E+05 5.89E+05

6.06E+05

(b 2)

0

0.2

0.4

0.6

0.8

1

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

dow

nstr

eam

cor

rela

tion

len

gth

1.85E+05 2.06E+05

2.29E+05 2.47E+05

2.73E+05 2.93E+05

3.16E+05 3.28E+05

3.53E+05 3.73E+05

3.93E+05 4.19E+05

4.33E+05 4.58E+05

4.90E+05 5.09E+05

5.22E+05 5.48E+05

5.67E+05 5.89E+05

6.06E+05 (c 1)

0

0.2

0.4

0.6

0.8

1

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

upst

ream

cor

rela

tion

len

gth

5.85E+05 6.58E+05

7.08E+05 7.70E+05

8.36E+05 9.13E+05

9.65E+05 1.02E+06

1.09E+06 1.15E+06

1.22E+06 1.28E+06

1.34E+06 1.41E+06

1.48E+06 1.54E+06

1.61E+06 1.65E+06

(c 2)

0

0.2

0.4

0.6

0.8

1

0° 20° 40° 60° 80° 100° 120° 140° 160° 180° 200°

upst

ream

cor

rela

tion

len

gth

5.85E+05 6.58E+05

7.08E+05 7.70E+05

8.36E+05 9.13E+05

9.65E+05 1.02E+06

1.09E+06 1.15E+06

1.22E+06 1.28E+06

1.34E+06 1.41E+06

1.48E+06 1.54E+06

1.61E+06

Figure 14. Correlation length of the fluctuating pressure in turbulent boundary layer flow. (a1、a2)Dome_S,6.7×104<Re<2.0×105 (b1、b2)Dome_M, 1.8×105<Re<6.0×105

(c1、c2)Dome_L, 5.8×105<Re<1.6×106


12

4.3 Contours of mean and RMS pressure distributions on dome

Contours of mean and RMS pressure coefficient in boundary layer flow are shown in Fig-ures 15 and 16. As shown in Figure 15, in the apex region of dome, i.e., ω>80°, the mean pressure increases with Reynolds number up to Re=1.58×105; then the mean pressure distri-bution pattern would remain nearly invariant thereafter. The pressure contours in Figure 16 indicate that in a larger dome apex region, i.e., ω>60°, the RMS pressure coefficient distribu-tion increases with Reynolds number up to Re=2.0×105. When Reynolds number exceeds 2.0×105, the RMS pressure coefficient would then decrease. Even though the incident turbu-lence intensity level varies for different dome models in turbulent boundary layer flow; the fluctuating pressure distribution pattern exhibits only minor variation for the cases of Re> 2.0×105.

The correlation coefficients for the mean and RMS pressure distributions at different Reynolds numbers, ρmean and ρrms, are also calculated for the pressure measurements in boundary layer flow. However, in this case, the pressure contour at Re=2.0×105 was selected to be the reference pressure contour. It is clearly shown in Figure 17 that the correlation coef-ficient of mean pressure distribution, ρmean, approaches 1.0 near Re=1.0×105, slightly earlier than the smooth flow condition, and remains to be ρmean=1.0 afterwards.

As for the correlation coefficient of RMS pressure distributions, ρrms decreases rapidly for Reynolds number less Re=1.0×105. In the cases of Reynolds number greater than the refer-ence case, i.e., Re>2.0×105, the correlation coefficient shows a very mild decaying trend and ρrms =0.85~0.9 when the Reynolds number greater than 4.0×105.



Figure 15. Contours of mean pressure distribution of hemispherical model

at various Reynolds number in boundary layer flow.

Re=6.7×104 Re=1.2×105 Re=1.6×105

Re=5.2×105 Re=8.4×105 Re=1.6×106 wind

-1.8

-1.4

-1

-0.6

-0.2

0.2

0.6

1


13



Figure 16. Contours of RMS pressure distribution of hemispherical model

at various Reynolds number in boundary layer flow.

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0.00E+00 1.00E+05 2.00E+05 3.00E+05 4.00E+05 5.00E+05 6.00E+05 7.00E+05

Re

corr

elat

ion c

oeff

icie

nt

S-dome-mean-ref_2E+05

S-dome-rms-ref_2E+05

M-dome-mean-ref_2E+05

M-dome-rms-ref_2E+05

Fig.17 pressure distribution correlation coefficient in boundary layer flow

The mean pressure distributions of dome in turbulent boundary layer and ASCE7-05 are shown in Figure 18 for comparison. As shown in the figure, the pressure distribution of ASCE7-05 and wind tunnel measurement are almost the same on the windward side. On leeward side, the pressure coefficient from ASCE7-05 is slightly larger than the present work. The correlation coefficient of the entire dome surface pressure was found to be 0.88.

Re=6.7×104 Re=1.2×105

Re=2.0×105

Re=1.8×106

0

0.1

0.2

0.3

0.4

0.5

Re=5.2×105 Re=1.6×106

wind


14

This comparison suggests that ASCE7-05 provides a reasonably accurate and slightly con-servative mean pressure distribution for hemispherical domes.

Re. Number_1651521

Fig.18 Comparison of the mean pressure distribution between present work and ASCE7-05

4.4 Mean and RMS meridian drag coefficients in turbulent boundary layer flow

Shown in Figure 19(a) & 19(b) are the meridian drag coefficients of the domes tested in turbulent boundary layer flow. For Re<2 × 105 , The mean drag coefficient decreases with the increase of the Reynolds number; for Re>2 × 105, the mean drag coefficient remains nearly invariant. The open circle symbols in Figure 19(b) represent the original RMS drag coeffi-cients obtained from three dome models. The discrepancies are mostly due to the fact that the incident turbulence intensity levels are not constant at different height in a boundary layer flow as shown in Figure 1. The solid circle symbol represents the RMS drag coefficient further normalized with respect to the turbulence intensity at the dome height, /P P uC C I′= , where Iu is the turbulence intensity at dome height. It is noted that the possible effects of different turbu-lence level can not be eliminated this way. However, after adjustment for turbulence intensity differences, the RMS drag coefficients are in much better agreement, and the data suggest that the RMS drag coefficient becomes invariant when Reynolds number greater than 1.8× 105.

0

0.1

0.2

0.3

0.4

0.0E+00 5.0E+05 1.0E+06 1.5E+06 2.0E+06 2.5E+06

Re

Cd

Fig. 19(a) Mean drag coefficient in turbulent boundary layer.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.0E+00 5.0E+05 1.0E+06 1.5E+06 2.0E+06 2.5E+06

Re

C'd

origin

0

0.1

0.2

0.3

0.4

0.5

0.6

adjusted

Fig.19(b) RMS drag coefficient in turbulent boundary layer.

Experiment in Re=1.6×106 ASCE7-05


15

5 CONCLUSIONS

A series of wind tunnel pressure measurements were performed on hemispherical domes to study the Reynolds number effects on characteristics of wind loads. Following summaries can be made from the tests in the smooth flow condition. In smooth flow, the transition of separa-tion flow occurs near Re=3.0×105. The mean meridian drag coefficient decreases with Rey-nolds number for Re<3.0×105, and then increase monotonically up to Re=2.0×106; RMS meridian drag coefficient shows maximum and minimum values at Re≒1.5×105 and Re≒3.0×105, respectively. The correlation coefficients of mean and RMS pressure contours indicate that, the pressure distributions become relatively stable at Re=2.0~3.0×105.

Some conclusions can also be made on the pressure field characteristics in turbulent bound-ary layer flow. In turbulent flow, the transition of separation flow occurs at lower Reynolds number, Re=9.7×104~1.5×105, and both mean and RMS pressure distributions approach Reynolds number independent when Re=1.2~1.5×105; and the mean and RMS meridian drag coefficients, CD and DC′ , become invariant when Re>2×105. Study of Correlation length of pressure fluctuations also confirms that the separation point becomes quite stable near θ=130° when Re>1.8×105. The correlation coefficients of mean and RMS pressure contours indicate that, in turbulent boundary layer flow, the pressure distributions become Reynolds number in-dependent at Re=1.0~2.0×105.

ACKNOWLEDGEMENTS Financial supports for this research project from National Science Council (project No. NSC96-2221-E-032-022-MY3) is gratefully acknowledged.

REFERENCES [1] M. Kasperski, and H.-J., Niemann, The L.R.C.(load-response-correlation) method: a gen-

eral method of estimating unfavorable wind load distributions for linear and non-linear structural behavior, J. Wind Eng. Ind. Aerodyn. 41-44 (1992) 1753-1763

[2] J.D., Holmes, Effective static load distributions in wind engineering, J. Wind Eng. Ind. Aerodyn. 90 (2002) 91–109.

[3] A. Katsumura, Y. Tamura, O. Nakamura, Universal wind load distribution simultane-ously reproducing largest load effects in all subject members on large-span cantilevered roof , J. Wind Eng. Ind. Aerodyn. 95 (2007) 1145–1165.

[4] F.J. Maher, Wind loads on basic dome shapes, J. Struct. Div. ASCE ST3 (1965) 219-228.

[5] T.J. Taylor, Wind pressures on a hemispherical dome, J. Wind Eng. Ind. Aerodyn. 40 (1991) 199-213.

[6] T. Ogawa, M. Nakayama, S. Murayama, Y. Sasaki, Characteristics of wind pressures on basic structures with curved surfaces and their response in turbulent flow, J. Wind Eng. Ind. Aerodyn. 38 (1991) 427-438.

[7] N. Toy, W.D. Moss, E. Savory, Wind tunnel studies on a dome in turbulent boundary layers, J. Wind Eng. Ind. Aerodyn. 1 (1983) 201-212.


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[8] C.W. Letchford, P.P. Sarkar, Mean and fluctuating wind loads on rough and smooth parabolic domes, J. Wind Eng. Ind. Aerodyn. 88 (2000) 101–117.

characteristic of wind load on a hemispherical dome in smooth flow...

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