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Characteristic of wind loads on a hemispherical dome in smooth flow and turbulent boundary layer flow
ARTICLE IN PRESS J. Wind Eng. Ind. Aerodyn. 98 (2010) 328–344
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Characteristic of wind loads on a hemispherical dome in smooth flow and turbulent boundary layer flow C.M. Cheng , C.L. Fu Department of Civil Engineering, Tamkang University, 151 Ying-chuan Road Tamsui, Taipei 25137, Taiwan
a r t i c l e in f o
a b s t r a c t
Article history: Received 6 March 2009 Received in revised form 4 December 2009 Accepted 4 December 2009 Available online 31 December 2009
A series of wind tunnel tests was performed to investigate the effects of Reynolds number on the aerodynamic characteristics of hemispherical dome in smooth and turbulent boundary layer flows. Reynolds number of this study varies from 5.3 104 to 2.0 106. Instantaneous pressures were measured through high frequency electronic scanner system. Mean and RMS pressure coefficients on the center meridian and the overall pressure patterns of domes were calculated for comparative study. The results indicate that, in smooth flow, the transition of separation flow occurs in-between Re = 1.8 105–3.0 105 and pressure distributions become relatively stable after Re>3.0 105. In turbulent flow, the transition of separation flow occurs at lower Reynolds number, Re o 1.1 105, while the pressure distributions become Reynolds number independent at Re=1.0–2.0 105. & 2009 Elsevier Ltd. All rights reserved.
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 stadiums or for storage purposes. The curved shape makes the accurate estimation of the wind pressure fluctuations on a hemispherical dome a difficult task due to 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 failure 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 (1992), Holmes (2002), Katsumura et al. (2007) 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 (1965) 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 (1991) 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 Corresponding author. Tel.: + 886 2 26297827; fax: + 886 2 26251749.
E-mail addresses: [email protected] (C.M. Cheng), [email protected] (C.L. Fu). 0167-6105/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jweia.2009.12.002
exceeds 4%, the pressure distribution becomes Reynolds number independent. Ogawa et al. (1991) studied the mean, RMS and spectral characteristics of wind pressure on domes and indicated that for Reynolds numbers ranging from 1.2 105– 2.1 105, the level of turbulence in the turbulent boundary layer has little effect upon the mean pressure distribution. Toy et al. (1983) showed that with increase of turbulence intensity, separation region and reattachment point move downstream. Letchford and Sarkar (2000) investigated the effect of surface roughness to the pressure distribution of dome. Meroney et al. (2002) conducted numerical simulations to study the mean pressure distribution of an isolated dome and interference effects of two neighboring domes immersed in turbulent boundary layer flow. It was found that, for the smooth surface isolated dome, there is only insignificant difference in the mean pressure distribution between Reynolds number 185,000 and 1,440,000. 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 presents the results of a systematic investigation on the wind load characteristics of hemispherical domes in smooth and turbulent boundary layer flows. The present investigation focuses on the Reynolds number effects on pressure distribution and the wind load patterns. In addition to examining wind load characteristics of hemispherical domes in different flow conditions, another aim of this study is to find the load pattern under Reynolds number independent flow condition to be used as the reference of future investigations.
ARTICLE IN PRESS C.M. Cheng, C.L. Fu / J. Wind Eng. Ind. Aerodyn. 98 (2010) 328–344
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All pressure coefficients were non-dimensionalized with respect to mean dynamic pressure at the top of dome. Mean pressure coefficient :
Dp
Cp ¼
0:5rU R:M:S: pressure coefficient :
ð1Þ
2
Dprms
Cp0 ¼
0:5rU
ð2Þ
2
At the very beginning of this investigation, mean pressure distributions taken from different models at the same Reynolds number were compared to validate the consistency and continuity of test cases for various Reynolds numbers. As shown in Fig. 4(a), the mean pressure distributions measured from Domes S and M at Re =3.0 105 match very well. The pressure distributions of Domes L and M, as shown in Fig. 4(b), exhibits slight differences. Since the size of Dome L is significantly larger than Domes S and M, the base plate was lowered extra 5 cm into the upper part of the boundary layer developed over the tunnel floor to maintain a better blockage ratio for Dome L. The minor discrepancy of pressure distribution noted in Fig. 4(b) is probably due to the combined effects of the impinging boundary layer and the higher blockage ratio for the Dome L. Turbulence intensity(%) 8
12
16
20
24
0.8
1
1 tubulence intensity wind velocity
0.8
0.6 z/
All the wind tunnel tests of this project were conducted in a boundary layer wind tunnel with a 24 m(length) 4 m(width) 2.6 m(height) dimensions; Both smooth flow and turbulent boundary layer flow were used. The wind tunnel setups are shown in Fig. 1. A base plate elevated from floor was used in the case of smooth flow to minimize the boundary layer developed over the base floor. The turbulent boundary layer has the characteristics of suburban flow field with power law index a =0.27; turbulence intensity varies from 25% to 18% in the region of the model heights as shown in Fig. 2. Three acrylic hemispherical pressure models with diameters of 120, 50, and 20 cm, namely Domes L, M, and S, were used in this investigation; the corresponding Reynolds number varied from 5.3 104 to 2.0 106. Pressure taps were installed along the meridian and three horizontal rings on the 120 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 dome’s global pressure pattern in different test conditions. Shown in Fig. 3 is the coordinate system of the pressure taps on hemisphere dome surface, in which, j denotes the angle of longitude and y is the angle of center meridian; for the contours of pressure coefficients, o is used to denote the angle of latitude. Instantaneous wind pressures were sampled simultaneously by a ZOC pressure scanner system at frequency of 300 Hz. The blockage ratios of the tests were 0.0015 for Dome S, 0.009 for Dome M and 0.054 for Dome L. Results shown in this paper are not further corrected for the blockage effect. The wind tunnel test cases are listed in Table 1. Several wind tunnel test cases were repeated to ensure the repeatability of the pressure measurements. At low test wind speed where the larger errors would occur, the differences between the repeated measurements on Dome S were calculated. With respect to the mean stagnation pressure, the maximum difference of mean pressure is about 2.7%; the average difference is 0.8%. The maximum difference of R.M.S pressure is 0.7% of the mean stagnation pressure whereas the average difference is 0.3%.
Height
2. Experiment design
0.4
0.2
3. Pressure distribution in smooth flow 3.1. Mean and RMS pressure coefficients in smooth flow
0 0.2
The definitions of pressure coefficients used in this article are listed below. Dp is the instantaneous pressure difference between dome surface pressure and a reference pressure in wind tunnel.
Dome in boundary layer flow
0.4
0.6 Mean wind velocity (U(z)/U( ))
Fig. 2. Wind velocity and turbulence intensity profiles (compare with dome models).
Dome in smooth flow
Fig. 1. Hemispherical pressure models in smooth and boundary flow.
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300
280
260 240
320
220
340
200 180
0 20
base floor
160 40
140 60
120 80
100
Fig. 3. Coordinate system of pressure taps on hemisphere models: (a) angle of center meridian y and angle of latitude o (01%y%1801, angle of center meridian 01%o%901, angle of latitude) and (b) angle of longitude j (01%j%3601).
Table 1 Wind tunnel testing conditions. Dome
S M L
Diameter (cm)
20 50 120
Reynolds number Smooth flow
Boundary layer flow
6.6 104–3.4 105 1.6 105–8.7 105 5.3 105–2.0 106
5.3 104–2.0 105 2.0 105–6.0 105 5.8 105–1.7 106
Fig. 4. Comparison of the mean pressure distributions on center meridian of different models: (a) Domes S and M at Re = 3 105, and (b) Domes M and L at Re = 7.5 105.
ARTICLE IN PRESS C.M. Cheng, C.L. Fu / J. Wind Eng. Ind. Aerodyn. 98 (2010) 328–344
The distributions of mean and RMS pressure coefficients along the center meridian at various Reynolds number in smooth flow condition are shown in Figs. 5 and 6. In Fig. 5(a), at Re =6.6 104– 1.5 105, the minimum mean pressure occurs around 701, then the negative pressure gradually recovers and reaches a stable value at about 851 which indicates the point of separation and beginning of wake region. For Reynolds number between 1.9 105–2.5 105, the minimum mean pressure occurs at 901, the boundary of the wake region moves downstream to 1151. It can also be observed that, for all cases in this region of Reynolds number, Re = 6.6 104–3.1 105, the negative pressure at dome apex increases with Reynolds number, whereas the negative pressure in the wake region decreases.
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When Reynolds number further increases from 3.0 105 to 2.0 106, as shown in Fig. 5(b) and (c), the mean pressure coefficient near dome apex remains nearly invariant. However, the wake suction increases slightly with Reynolds number and the base pressure becomes more negative. The separation point and the boundary of wake region remain stable at about 1301 when Re o1.4 106, and then move 101 upstream, from 1301 to 1201, when Reynolds number increases from 1.4 106 to 2.0 106. Shown in Fig. 6(a)–(c) are the distributions of RMS pressure coefficients along the center meridian at various Reynolds number in smooth flow condition. The experimental data indicates that Reynolds number casts significant influence on the distribution of RMS pressure coefficients in the negative
Fig. 5. Mean pressure distributions on center meridian in smooth flow: (a) Dome S, 6.6 104 o Re o3.1 105, (b) Dome M, 3.0 105 o Re o8.1 105 and (c) Dome L, 8.2 105 o Re o 2.0 106.
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Fig. 6. R.M.S. pressure distributions on center meridian in smooth flow: (a) Dome S 6.6 104 o Re o 3.1 105, (b) Dome M 3.0 105 o Re o 8.1 105 and (c) Dome L 8.2 105 o Re o 2.0 106.
pressure zone, especially downstream of separation point. For Re o1.8 105, there is one distinct and narrow peak of CP0 at 801–901 and a broader lump around 1401. When Reynolds number increases to 1.8 105–3.0 105, a second distinct and narrow peak appears at 1101–1201. The height of second peak enhances with Reynolds number while the first peak gradually diminishes with Reynolds number. This dual-peaks phenomenon suggests that transition of separated free shear layer occurred and that a separation bubble is formed at this stage; the first and second peaks indicate the locations of fore-separation and the reattachment/rear-separation. When Reynolds number is 43.0 105, there remains only one peak of CP0 near 1201, at the separation zone, and another one at very down-stream. As Reynolds number
increases from 3.0 105 to 2.0 106, the CP0 peak moves upstream to about 1101.
3.2. Correlation coefficients of pressure fluctuations in smooth flow In order to better interpreting the experimental data, the correlation coefficients, R(yi,y), and a non-dimensional correlation length, li, of the fluctuating pressure are introduced in this paper. In following relationships yi is the location of the reference pressure tap; Rðyi ; yÞ ¼ E½pðyi ; tÞpðy; tÞ=si sy ;
y¼0p
ð3Þ
ARTICLE IN PRESS C.M. Cheng, C.L. Fu / J. Wind Eng. Ind. Aerodyn. 98 (2010) 328–344
li ¼
Z p 0
Rðyi ; yÞ dy
ð4Þ
After carefully examine the correlation coefficient plots, a threshold of R(yi,y)^0.4 was adopted during the above integration to avoid the fluctuating nature of correlation coefficient at large spatial separation. Shown in Fig. 7(a)–(c) are the distributions of correlation coefficients for Re =1.4 105, 2.5 105 and 3.5 105. Each plotted line represents the pressure correlation coefficients of pressure taps along the center meridian of the dome with respect to the indicated reference tap. Fig. 7(a) shows the distributions of correlation coefficients at Re =1.4 105. In this figure, a rapid decrease of correlation coefficient can be observed when the reference pressure tap located at small angles, yi%201. Based on the study of pressure distributions, it is established that the stagnation point locates at roughly y = 201. Before the stagnation point, the dome surface pressure is under the effect of the upstream horseshoe vortices, therefore, would exhibits low correlation with the downstream pressure taps. When location of the reference tap moves downstream, the dome surface boundary layer starts to develop and gradually becomes stable. Thus, the pressure correlation increases noticeably. When the reference pressure tap reaches 591 and beyond, the near-by correlation coefficients reach the highest value which suggests the surface boundary layer is well developed and stable. When the reference pressure tap moves to 871, although the near-by correlations are still quite good, slightly decrease of the upstream correlations can be observed. When the reference pressure tap moves further downstream to 981, the correlation plot shows an obvious decreasing slope in the upstream direction and even more rapid decrease in the downstream direction. For the reference tap located further
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downstream, all correlation plots exhibit a similarly symmetric and steeply decreasing pattern. For the reference tap located at nearby and upstream of the separation region, stable and high-value correlations are observed, especially towards the upstream of the reference tap. However, for reference tap located at nearby and downstream of separation point, some reduction of correlation is to be expected in the upstream direction. Following this reasoning, it can be deduced from Fig. 7(a) that the separation of surface flow occurs nearby 871. This observation is in agreement with the conclusions of previous study on mean and RMS pressure distributions. As for the symmetric and steeply decreasing pattern observed at further downstream, it should be the pressure correlation form of the wake region. Shown in Fig. 7(b) are the distributions of correlation coefficients at Re = 2.5 105. The pattern of the pressure distributions for various reference taps are similar to those in Fig. 7(a). At this Reynolds number, the correlation plot starts to show slightly decrease towards upstream direction when the reference tap located at 981 instead of 871 as in the previous case. In other words, the separation point moves downstream, from 871 at Re =1.4 105 to 981 at Re = 2.5 105. Shown in Fig. 7(c) are the distributions of correlation coefficients at Re =3.5 105. Comparing to 7(a) and (b), the correlation plots become less stable and have lower values. The best interpretation authors can make from this figure is that the separation occurs around 1111 while y ¼ 1283 is definitely located in the wake region. The volatility of correlation distribution in the pre-separation region is probably due to the effect that, near the separation point at this Reynolds number, the surface boundary layer has just transited from laminar to turbulent flow. The pressure correlation lengths integrated by Eq. (4) are plotted in Fig. 8. At Re =1.4 105, the correlation length
Fig. 7. (a) Correlation coefficient of pressure fluctuations of Dome S in smooth flow, Re =1.4E + 05, (b) Re = 2.5E+ 05 and (c) Re =3.5E +05.
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Fig. 8. Correlation length of pressure fluctuations on center meridian of Dome in smooth flow.
approaches a relatively high value when the y increases from 201 to 601, and remains stable up to around 901. This stable and high correlation region is likely to be the region where the surface boundary layer is laminar and well developed. When Reynolds number increases to Re = 2 106, at which the surface boundary layer has transited to turbulent flow, the correlation length decreases significantly in this region. The above study on the distributions of correlation coefficients indicates that there is only a minor reduction of upstream correlation at separation region. In other words, it is unlikely to identify the separation point on the correlation length plot. The distribution of correlation coefficients also indicates that the wake region would exhibit low value of correlation length. It is clearly shown in Fig. 8 that, at Re = 1.4 105, the correlation length decreases to the lowest value at y = 1151. At Re = 2.5 105, the position of lowest correlation length moves to 1201. At Re =3.5 105 the position of lowest correlation length moves further downstream to 1301. But at Re =1.5 106–2.0 106, the lowest correlation length position moves 51–101 upstream. This trend of lowest correlation length reflects the movement of the separation point; and it is in agreement with the earlier observations made on mean and RMS pressure distributions. Achenbach’s (1972) study on sphere indicates that, in the subcritical regime at Re =1.62 105, the laminar separation of a sphere occurs at 821. When Re 42.0 105, the separation point shifts downstream to about 951. With further increase of Reynolds number at Re =3.2 105, transition occurs in free shear layer from laminar to turbulent flow and an separation bubble formed follows the intermediate reattachment of the free shear layer. Based on the above pressure studies, it is found that the separation/reattach phenomenon of the hemispherical dome is surprisingly similar to a sphere, although the dome is mounted on the floor and sphere is submerged freely in the approaching flow. The difference between the two mean pressure distributions is that the hemispherical dome exhibits a more noticeable pressure
recovery in the rear part of the dome and a lower negative pressure in the wake region.
3.3. Contours of mean and RMS pressure distributions on dome in smooth flow The contours of mean and RMS pressure coefficients of the three domes were plotted to study the Reynolds number effects on the overall pattern of pressure distributions. 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 Fig. 9 are contours of mean pressure distribution in smooth flow. It is interesting to notice that, for the surface of 3-dimensional curvature, almost all mean pressure contours exhibit a 2-dimensional like characteristics, i.e., the variation of the pressure contour is similar to the mean pressure distribution on the center meridian. It also can be observed that, when the Reynolds number exceeds 2.0 105, the overall mean pressure distribution pattern remains nearly invariant. The contours of RMS pressure coefficient distribution are shown in Fig. 10. For Reo2.0 105, the RMS of pressure fluctuations in the region of 601o o and 901o j o2701, i.e., 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 o =201–301 and j = 71201, 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 variations. 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 Fig. 11. After several tries, the pressure contour at Re =3.0 105 was selected as the reference pressure contour. In Fig. 11, rmean and rrms denote
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Re=6.6 104
Re=2.0 105
335
Re=1.7 105
Re=3.0 105
wind
Re=8.7 105
Re=2.0 106 1 0.6 0.2 -0.2 -0.6 -1 -1.4 -1.8
Fig. 9. Contours of mean pressure distribution of domes at various Reynolds number in smooth flow.
correlation coefficients of mean and RMS pressure distributions between an arbitrary Reynolds number and Re = 3.0 105. The result indicates that the correlation coefficient of mean pressure distribution, rmean, approaches 1.0 near Re = 2.0 105 and remains 1.0 afterwards. As for the correlation coefficient of RMS pressure distributions, rrms decreases rapidly for Reynolds number less the reference case; the correlation coefficient can be as low as rrms o0.5. 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 rrms =0.8 when the Reynolds number reaches 8.0 105. 3.4. Mean and RMS meridian drag coefficients in smooth flow Integrating the pressure along the dome center meridian of each testing case, a drag coefficient called the ‘‘meridian drag coefficient’’ can be obtained. The meridian drag coefficients are shown in Figs. 12 and 13. Fig. 12 indicates that, for Re o3.0 105, the mean drag coefficient decreases with the increase of the
Reynolds number, and it recovers gradually thereafter. Fig. 13 shows that the maximum RMS drag coefficient occurs at Re =1.5 105, then decreases rapidly and exhibits a minimum value at Re = 3.0 105, and then gradually recovers and reaches a relatively stable value when Re 41.0 106.
4. Pressure distribution in turbulent boundary layer flow 4.1. Mean and RMS pressure coefficients in turbulent boundary layer flow Shown in Figs. 14 and 15 are the mean and RMS pressure coefficients measured along the dome meridian in turbulent boundary layer flow. As shown in Fig. 14(a), for Reynolds number between 5.3 104–1.58 105, the negative pressure at dome apex increases, and decreases in wake region. For Reynolds number 41.58 105, both the negative pressure at apex and in the wake region show only minor change with respect to Reynolds number;
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Re=6.6 104
Re=1.4 105
Re=2.0 105
Re=3.0 105
Re=5.2 105
Re=2.0 106
wind
0.5 0.4 0.3 0.2 0.1 0 Fig. 10. Contours of R.M.S. pressure distribution of domes at various Reynolds number in smooth flow.
while the separation point locates near y =1301. For Re =5.2 105 to 1.6 106, the point of separation moves about 101 upstream, from 1301 to 1201. The distributions of RMS pressure coefficient are shown in Fig. 15(a)–(c). It should be noted that the RMS pressure coefficients are 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 151–201. As for the RMS pressure coefficient in the negative pressure zone: when Re o1.1 105, there are two peaks of CP0 locate at 801–901 and 1101–1201; for Re 41.2 105, there is only a single peak of CP0 near 901–1001. Pressure data suggest that transition of the separation flow occurs at lower Reynolds number in the turbulent boundary layer than the smooth flow. Carefully comparing all cases of mean and RMS pressure measurements on the center meridian of the three different size models, it is concluded that, when Reynolds number is in between 1.2 and 1.5 105, both CP and CP0 approach the state of Reynolds number independence.
4.2. Pressure correlation length in turbulent boundary layer flow Shown in Fig. 16(a)–(c) are the distributions of correlation coefficients at Re = 6.7 104, 1.4 105 and 2.0 105 in turbulent boundary layer flow. Each plotted line represents the pressure correlation coefficients of pressure taps along the center meridian of the dome with respect to the indicated reference tap. Fig. 16(a) shows the distributions of correlation coefficients at Re= 6.7 104. In this figure, high correlations exist at small angle, the correlation decreases rapidly when the location of reference pressure tap moves downstream, yi Z 313 . The high value of correlation at small angle is the effect of the large scale turbulence in the approaching flow. When the location of reference tap moves downstream, the incident turbulence effect diminishing, pressure correlation decreases and approaches the lowest value at yi C 603 . When location of the reference tap moves further downstream, where the dome surface boundary layer starts to develop and
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Fig. 11. Correlation coefficient of pressure distributions at various Reynolds number in smooth flow, Re,ref =3 105.
Fig. 12. Mean center meridian drag coefficients of domes in smooth flow.
Fig. 13. R.M.S. meridian drag coefficient of domes in smooth flow.
gradually becomes stable, the pressure correlation recovers. When the reference pressure tap reaches 871, the near-by correlation coefficients reach the highest value which suggests the surface boundary layer is well developed and stable. When the reference tap moves downstream to 981, slightly decrease of
the upstream correlations can be observed. Based on the discussion for the smooth flow condition, it is an indication of separation point. At y ¼ 1103 , pressure correlation plot shows clear decrease in both upstream and downstream directions. In other words, this location is well into the wake region.
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Fig. 14. Mean pressure distributions in turbulent boundary layer: (a) Dome S, 5.3 104 o Re o 2.0 105, (b) Dome M, 2.0 105 o Re o5.5 105 and (c) Dome L, 5.2 105 o Re o 1.6 106.
Shown in Fig. 16(b) are the distributions of correlation coefficients at Re = 1.4 105. The pattern of the pressure distributions for various reference taps are similar to those in Fig. 16(a). The difference is that, at this Reynolds number, the correlation plot starts to show slightly decrease towards upstream direction when the reference tap locates at 1101 instead of 981 as in the previous case. In other words, the separation point moves downstream, from 981 at Re = 6.7 104 to 1101 at Re =1.4 105. Shown in Fig. 7(c) are the distributions of correlation coefficients at Re = 2.0 105. The distributions of pressure correlation are almost identical to those in Fig. 16(b). In other words, the pressure field on a hemispherical dome in turbulent boundary layer flow would approach a stable pattern when Re^1.4 105. Shown in Figs. 17–19 are the pressure correlation length obtained from Domes S, M and L, respectively. Fig. 16 indicates
that, at Re = 6.7 104, pressure correlation length remains constant when y increase from 201 to 401. It then decreases and reaches the lowest value at 601. The pressure correlation length will increase again and reach the highest value at y C 803 . It then remains stable up to y C 1003 . In this region, the surface boundary layer becomes well developed and stable. When y moves further downstream, pressure correlation length starts to decrease. The lowest pressure correlation occurs at y C 1203 . At different Reynolds numbers, the distribution of pressure correlation length remains almost invariant. Except the location of lowest correlation length moves to 1301 when Re 41.4 105. For the pressure correlation length obtained from Domes M and L, shown in Figs. 18 and 19, the distribution patterns are almost identical to those in Fig. 17. Since the Reynolds number of all these cases are 42.0 105, the location of lowest correlation length keeps invariant at y C 1303 .
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Fig. 15. R.M.S. pressure distributions in turbulent boundary layer: (a) Dome S, 5.3 104 oRe o 2.0 105, (b) Dome M, 2.0 105 o Re o 5.5 105 and (c) Dome L, 5.2 105 o Re o 1.6 106.
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 Figs. 20 and 21. As shown in Fig. 20, in the apex region of dome, i.e., o 4801, the mean pressure increases with Reynolds number up to Re =1.58 105; then the mean pressure distribution pattern would remain nearly invariant thereafter. The pressure contours in Fig. 21 indicate that in a larger dome apex region, i.e., o 4601, the RMS pressure coefficient distribution obtained from Dome S increases with Reynolds number up to Re =1.8 105, although the RMS pressure contours of three dome models are different due to the variations of pressure tap arrangements and incident turbulence intensities. The RMS pressure contours exhibit only minor variation for Dome M at Re =2.0 105–5.7 105, and Dome L at Re =5.2 105– 1.6 106, i.e., if we compare the results from the same dome model.
The correlation coefficients for the mean and RMS pressure distributions at different Reynolds numbers, rmean and rrms, are also calculated for the pressure measurements in boundary layer flow. However, in this case, the pressure contour at Re = 2.0 105 was used as the reference pressure contour. It is clearly shown in Fig. 22 that the correlation coefficient of mean pressure distribution, rmean, approaches 1.0 near Re = 1.0 105, slightly earlier than the smooth flow condition, and remains to be rmean = 1.0 afterwards. As for the correlation coefficient of RMS pressure distributions, rrms decreases rapidly for Reynolds number less Re = 1.0 105. In the cases of Reynolds number greater than the reference case, i.e., Re 42.0 105, the correlation coefficient shows a very mild decaying trend and rrms = 0.85–0.9 when the Reynolds number 44.0 105. The mean pressure distributions of dome submerged in turbulent boundary layer is then compared with the pressure coefficients
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Fig. 16. (a) Correlation coefficient of pressure fluctuations of Dome S in turbulent boundary layer, Re =6.7E + 04, (b) Re = 1.4E+ 05 and (c) Re =2.0E+ 05.
Fig. 17. Correlation length of pressure fluctuations on center meridian of Dome S in turbulent boundary layer.
Fig. 18. Correlation length of pressure fluctuations on center meridian of Dome M in turbulent boundary layer.
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Fig. 19. Correlation length of pressure fluctuations on center meridian of Dome L in turbulent boundary layer.
Re=6.7 104
Re=1.6 105
Re=1.2 105
Re=5.2 105
wind
Re=8.4 105
Re=1.6 106 1 0.6 0.2 -0.2 -0.6 -1 -1.4 -1.8
Fig. 20. Contours of mean pressure distribution of domes at various Reynolds number in boundary layer flow.
recommended in ASCE7-05 (2005), as shown in Fig. 23. On the windward side, the pressure distribution of current investigation is similar to the ASCE7-05 recommendations. On leeward side, the
negative 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. This comparison suggests
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a
Re=6.7 104
b
Re=1.4 105
Re=2.0 105
Re=3.7 105
Re=2.0 105
Re=5.7 105
wind
c
Re=5.2 105
Re=1.0 106
Re=1.6 106 0.5 0.4 0.3 0.2 0.1 0
Fig. 21. Contours of R.M.S. pressure distribution of domes at various Reynolds number in boundary layer flow.
Fig. 22. Correlation coefficient of pressure distributions at various Reynolds number in turbulent boundary layer flow, Re, ref =2 105.
that ASCE7-05 provides a reasonably accurate and slightly conservative mean pressure distribution for hemispherical domes.
4.4. Mean and RMS meridian drag coefficients in turbulent boundary layer flow Shown in Figs. 24 and 25 are the meridian drag coefficients of the domes tested in turbulent boundary layer flow. For
Reo2 105, the mean drag coefficient decreases with the increase of the Reynolds number; for Re 42 105, the mean drag coefficient remains nearly invariant. The open circle symbols in Fig. 25 represent the original RMS drag coefficients 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 Fig. 2. The solid circle symbol represents the RMS drag coefficient further normalized with respect to the turbulence intensity at the dome
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Pressure work at Re=1.6 106
ASCE7-05
Fig. 23. Comparison of the mean pressure distribution between present work and ASCE7-05 recommendations.
Fig. 24. Mean center meridian drag coefficient of domes in turbulent boundary layer.
Fig. 25. R.M.S. center meridian drag coefficient of domes in turbulent boundary layer.
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height, C~ D ¼ CD0 =Iu , where Iu is the turbulence intensity at dome height. It is noticed that the possible effects of different turbulence level can not be entirely 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 is 41.8 105.
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 separation flow occurs near Re = 1.8 105–3.0 105, separation/reattachment occur in this Reynolds number region. The pressure distributions become relatively stable after Re 43.0 105. The mean meridian drag coefficient decreases with Reynolds number for Reo3.0 105, 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 of domes in turbulent boundary layer flow. In turbulent flow, the transition of separation flow occurs at lower Reynolds number, Re o1.1 105, and both mean and RMS pressure distributions approach Reynolds number independent when Re = 1.2–1.5 105. The mean and RMS meridian drag coefficients, CD and CD0 , become invariant when Re42 105. The correlation coefficients of mean and RMS pressure contours indicate that, in turbulent boundary layer flow, the pressure
distributions become Reynolds number independent at Re= 1.0– 2.0 105.
Acknowledgments Financial supports for this research project from National Science Council (Project no. NSC96-2221-E-032-022-MY3) is gratefully acknowledged.
References Achenbach, E., 1972. Experiments on the flow past spheres at very high Reynolds Numbers. J. Fluid Mech. 54 (part 3), 565–575. American Society of Civil Engineers (ASCE), ASCE Standard, 2005. Minimum Design Loads for Buildings and Other Structures, ASCE/SEI 7-05, ASCE, RESTON, VA, USA. Holmes, J.D., 2002. Effective static load distributions in wind engineering. J. Wind Eng. Ind. Aerodyn. 90, 91–109. Kasperski, M., Niemann, H.-J., 1992. The L.R.C. (load-response-correlation) method: a general method of estimating unfavorable wind load distributions for linear and non-linear structural behavior. J. Wind Eng. Ind. Aerodyn. 41–44, 1753–1763. Katsumura, A., Tamura, Y., Nakamura, O., 2007. Universal wind load distribution simultaneously reproducing largest load effects in all subject members on large-span cantilevered roof. J. Wind Eng. Ind. Aerodyn. 95, 1145–1165. Letchford, C.W., Sarkar, P.P., 2000. Mean and fluctuating wind loads on rough and smooth parabolic domes. J. Wind Eng. Ind. Aerodyn. 88, 101–117. Maher, F.J., 1965. Wind loads on basic dome shapes. J. Struct. Div. ASCE ST3, 219–228. Meroney, R.N., Letchford, C.W., Sarkar, P.P., 2002. Comparison of numerical and wind tunnel simulation of wind loads on smooth, rough and dual domes immersed in a boundary layer. Wind Struc. 5 (2), 347–358. Ogawa, T., Nakayama, M., Murayama, S., Sasaki, Y., 1991. Characteristics of wind pressures on basic structures with curved surfaces and their response in turbulent flow. J. Wind Eng. Ind. Aerodyn. 38, 427–438. Taylor, T.J., 1991. Wind pressures on a hemispherical dome. J. Wind Eng. Ind. Aerodyn. 40, 199–213. Toy, N., Moss, W.D., Savory, E., 1983. Wind tunnel studies on a dome in turbulent boundary layers. J. Wind Eng. Ind. Aerodyn. 1, 201–212.
Contents lists available at ScienceDirect
Journal of Wind Engineering and Industrial Aerodynamics journal homepage: www.elsevier.com/locate/jweia
Characteristic of wind loads on a hemispherical dome in smooth flow and turbulent boundary layer flow C.M. Cheng , C.L. Fu Department of Civil Engineering, Tamkang University, 151 Ying-chuan Road Tamsui, Taipei 25137, Taiwan
a r t i c l e in f o
a b s t r a c t
Article history: Received 6 March 2009 Received in revised form 4 December 2009 Accepted 4 December 2009 Available online 31 December 2009
A series of wind tunnel tests was performed to investigate the effects of Reynolds number on the aerodynamic characteristics of hemispherical dome in smooth and turbulent boundary layer flows. Reynolds number of this study varies from 5.3 104 to 2.0 106. Instantaneous pressures were measured through high frequency electronic scanner system. Mean and RMS pressure coefficients on the center meridian and the overall pressure patterns of domes were calculated for comparative study. The results indicate that, in smooth flow, the transition of separation flow occurs in-between Re = 1.8 105–3.0 105 and pressure distributions become relatively stable after Re>3.0 105. In turbulent flow, the transition of separation flow occurs at lower Reynolds number, Re o 1.1 105, while the pressure distributions become Reynolds number independent at Re=1.0–2.0 105. & 2009 Elsevier Ltd. All rights reserved.
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 stadiums or for storage purposes. The curved shape makes the accurate estimation of the wind pressure fluctuations on a hemispherical dome a difficult task due to 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 failure 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 (1992), Holmes (2002), Katsumura et al. (2007) 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 (1965) 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 (1991) 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 Corresponding author. Tel.: + 886 2 26297827; fax: + 886 2 26251749.
E-mail addresses: [email protected] (C.M. Cheng), [email protected] (C.L. Fu). 0167-6105/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jweia.2009.12.002
exceeds 4%, the pressure distribution becomes Reynolds number independent. Ogawa et al. (1991) studied the mean, RMS and spectral characteristics of wind pressure on domes and indicated that for Reynolds numbers ranging from 1.2 105– 2.1 105, the level of turbulence in the turbulent boundary layer has little effect upon the mean pressure distribution. Toy et al. (1983) showed that with increase of turbulence intensity, separation region and reattachment point move downstream. Letchford and Sarkar (2000) investigated the effect of surface roughness to the pressure distribution of dome. Meroney et al. (2002) conducted numerical simulations to study the mean pressure distribution of an isolated dome and interference effects of two neighboring domes immersed in turbulent boundary layer flow. It was found that, for the smooth surface isolated dome, there is only insignificant difference in the mean pressure distribution between Reynolds number 185,000 and 1,440,000. 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 presents the results of a systematic investigation on the wind load characteristics of hemispherical domes in smooth and turbulent boundary layer flows. The present investigation focuses on the Reynolds number effects on pressure distribution and the wind load patterns. In addition to examining wind load characteristics of hemispherical domes in different flow conditions, another aim of this study is to find the load pattern under Reynolds number independent flow condition to be used as the reference of future investigations.
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All pressure coefficients were non-dimensionalized with respect to mean dynamic pressure at the top of dome. Mean pressure coefficient :
Dp
Cp ¼
0:5rU R:M:S: pressure coefficient :
ð1Þ
2
Dprms
Cp0 ¼
0:5rU
ð2Þ
2
At the very beginning of this investigation, mean pressure distributions taken from different models at the same Reynolds number were compared to validate the consistency and continuity of test cases for various Reynolds numbers. As shown in Fig. 4(a), the mean pressure distributions measured from Domes S and M at Re =3.0 105 match very well. The pressure distributions of Domes L and M, as shown in Fig. 4(b), exhibits slight differences. Since the size of Dome L is significantly larger than Domes S and M, the base plate was lowered extra 5 cm into the upper part of the boundary layer developed over the tunnel floor to maintain a better blockage ratio for Dome L. The minor discrepancy of pressure distribution noted in Fig. 4(b) is probably due to the combined effects of the impinging boundary layer and the higher blockage ratio for the Dome L. Turbulence intensity(%) 8
12
16
20
24
0.8
1
1 tubulence intensity wind velocity
0.8
0.6 z/
All the wind tunnel tests of this project were conducted in a boundary layer wind tunnel with a 24 m(length) 4 m(width) 2.6 m(height) dimensions; Both smooth flow and turbulent boundary layer flow were used. The wind tunnel setups are shown in Fig. 1. A base plate elevated from floor was used in the case of smooth flow to minimize the boundary layer developed over the base floor. The turbulent boundary layer has the characteristics of suburban flow field with power law index a =0.27; turbulence intensity varies from 25% to 18% in the region of the model heights as shown in Fig. 2. Three acrylic hemispherical pressure models with diameters of 120, 50, and 20 cm, namely Domes L, M, and S, were used in this investigation; the corresponding Reynolds number varied from 5.3 104 to 2.0 106. Pressure taps were installed along the meridian and three horizontal rings on the 120 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 dome’s global pressure pattern in different test conditions. Shown in Fig. 3 is the coordinate system of the pressure taps on hemisphere dome surface, in which, j denotes the angle of longitude and y is the angle of center meridian; for the contours of pressure coefficients, o is used to denote the angle of latitude. Instantaneous wind pressures were sampled simultaneously by a ZOC pressure scanner system at frequency of 300 Hz. The blockage ratios of the tests were 0.0015 for Dome S, 0.009 for Dome M and 0.054 for Dome L. Results shown in this paper are not further corrected for the blockage effect. The wind tunnel test cases are listed in Table 1. Several wind tunnel test cases were repeated to ensure the repeatability of the pressure measurements. At low test wind speed where the larger errors would occur, the differences between the repeated measurements on Dome S were calculated. With respect to the mean stagnation pressure, the maximum difference of mean pressure is about 2.7%; the average difference is 0.8%. The maximum difference of R.M.S pressure is 0.7% of the mean stagnation pressure whereas the average difference is 0.3%.
Height
2. Experiment design
0.4
0.2
3. Pressure distribution in smooth flow 3.1. Mean and RMS pressure coefficients in smooth flow
0 0.2
The definitions of pressure coefficients used in this article are listed below. Dp is the instantaneous pressure difference between dome surface pressure and a reference pressure in wind tunnel.
Dome in boundary layer flow
0.4
0.6 Mean wind velocity (U(z)/U( ))
Fig. 2. Wind velocity and turbulence intensity profiles (compare with dome models).
Dome in smooth flow
Fig. 1. Hemispherical pressure models in smooth and boundary flow.
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300
280
260 240
320
220
340
200 180
0 20
base floor
160 40
140 60
120 80
100
Fig. 3. Coordinate system of pressure taps on hemisphere models: (a) angle of center meridian y and angle of latitude o (01%y%1801, angle of center meridian 01%o%901, angle of latitude) and (b) angle of longitude j (01%j%3601).
Table 1 Wind tunnel testing conditions. Dome
S M L
Diameter (cm)
20 50 120
Reynolds number Smooth flow
Boundary layer flow
6.6 104–3.4 105 1.6 105–8.7 105 5.3 105–2.0 106
5.3 104–2.0 105 2.0 105–6.0 105 5.8 105–1.7 106
Fig. 4. Comparison of the mean pressure distributions on center meridian of different models: (a) Domes S and M at Re = 3 105, and (b) Domes M and L at Re = 7.5 105.
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The distributions of mean and RMS pressure coefficients along the center meridian at various Reynolds number in smooth flow condition are shown in Figs. 5 and 6. In Fig. 5(a), at Re =6.6 104– 1.5 105, the minimum mean pressure occurs around 701, then the negative pressure gradually recovers and reaches a stable value at about 851 which indicates the point of separation and beginning of wake region. For Reynolds number between 1.9 105–2.5 105, the minimum mean pressure occurs at 901, the boundary of the wake region moves downstream to 1151. It can also be observed that, for all cases in this region of Reynolds number, Re = 6.6 104–3.1 105, the negative pressure at dome apex increases with Reynolds number, whereas the negative pressure in the wake region decreases.
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When Reynolds number further increases from 3.0 105 to 2.0 106, as shown in Fig. 5(b) and (c), the mean pressure coefficient near dome apex remains nearly invariant. However, the wake suction increases slightly with Reynolds number and the base pressure becomes more negative. The separation point and the boundary of wake region remain stable at about 1301 when Re o1.4 106, and then move 101 upstream, from 1301 to 1201, when Reynolds number increases from 1.4 106 to 2.0 106. Shown in Fig. 6(a)–(c) are the distributions of RMS pressure coefficients along the center meridian at various Reynolds number in smooth flow condition. The experimental data indicates that Reynolds number casts significant influence on the distribution of RMS pressure coefficients in the negative
Fig. 5. Mean pressure distributions on center meridian in smooth flow: (a) Dome S, 6.6 104 o Re o3.1 105, (b) Dome M, 3.0 105 o Re o8.1 105 and (c) Dome L, 8.2 105 o Re o 2.0 106.
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Fig. 6. R.M.S. pressure distributions on center meridian in smooth flow: (a) Dome S 6.6 104 o Re o 3.1 105, (b) Dome M 3.0 105 o Re o 8.1 105 and (c) Dome L 8.2 105 o Re o 2.0 106.
pressure zone, especially downstream of separation point. For Re o1.8 105, there is one distinct and narrow peak of CP0 at 801–901 and a broader lump around 1401. When Reynolds number increases to 1.8 105–3.0 105, a second distinct and narrow peak appears at 1101–1201. The height of second peak enhances with Reynolds number while the first peak gradually diminishes with Reynolds number. This dual-peaks phenomenon suggests that transition of separated free shear layer occurred and that a separation bubble is formed at this stage; the first and second peaks indicate the locations of fore-separation and the reattachment/rear-separation. When Reynolds number is 43.0 105, there remains only one peak of CP0 near 1201, at the separation zone, and another one at very down-stream. As Reynolds number
increases from 3.0 105 to 2.0 106, the CP0 peak moves upstream to about 1101.
3.2. Correlation coefficients of pressure fluctuations in smooth flow In order to better interpreting the experimental data, the correlation coefficients, R(yi,y), and a non-dimensional correlation length, li, of the fluctuating pressure are introduced in this paper. In following relationships yi is the location of the reference pressure tap; Rðyi ; yÞ ¼ E½pðyi ; tÞpðy; tÞ=si sy ;
y¼0p
ð3Þ
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li ¼
Z p 0
Rðyi ; yÞ dy
ð4Þ
After carefully examine the correlation coefficient plots, a threshold of R(yi,y)^0.4 was adopted during the above integration to avoid the fluctuating nature of correlation coefficient at large spatial separation. Shown in Fig. 7(a)–(c) are the distributions of correlation coefficients for Re =1.4 105, 2.5 105 and 3.5 105. Each plotted line represents the pressure correlation coefficients of pressure taps along the center meridian of the dome with respect to the indicated reference tap. Fig. 7(a) shows the distributions of correlation coefficients at Re =1.4 105. In this figure, a rapid decrease of correlation coefficient can be observed when the reference pressure tap located at small angles, yi%201. Based on the study of pressure distributions, it is established that the stagnation point locates at roughly y = 201. Before the stagnation point, the dome surface pressure is under the effect of the upstream horseshoe vortices, therefore, would exhibits low correlation with the downstream pressure taps. When location of the reference tap moves downstream, the dome surface boundary layer starts to develop and gradually becomes stable. Thus, the pressure correlation increases noticeably. When the reference pressure tap reaches 591 and beyond, the near-by correlation coefficients reach the highest value which suggests the surface boundary layer is well developed and stable. When the reference pressure tap moves to 871, although the near-by correlations are still quite good, slightly decrease of the upstream correlations can be observed. When the reference pressure tap moves further downstream to 981, the correlation plot shows an obvious decreasing slope in the upstream direction and even more rapid decrease in the downstream direction. For the reference tap located further
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downstream, all correlation plots exhibit a similarly symmetric and steeply decreasing pattern. For the reference tap located at nearby and upstream of the separation region, stable and high-value correlations are observed, especially towards the upstream of the reference tap. However, for reference tap located at nearby and downstream of separation point, some reduction of correlation is to be expected in the upstream direction. Following this reasoning, it can be deduced from Fig. 7(a) that the separation of surface flow occurs nearby 871. This observation is in agreement with the conclusions of previous study on mean and RMS pressure distributions. As for the symmetric and steeply decreasing pattern observed at further downstream, it should be the pressure correlation form of the wake region. Shown in Fig. 7(b) are the distributions of correlation coefficients at Re = 2.5 105. The pattern of the pressure distributions for various reference taps are similar to those in Fig. 7(a). At this Reynolds number, the correlation plot starts to show slightly decrease towards upstream direction when the reference tap located at 981 instead of 871 as in the previous case. In other words, the separation point moves downstream, from 871 at Re =1.4 105 to 981 at Re = 2.5 105. Shown in Fig. 7(c) are the distributions of correlation coefficients at Re =3.5 105. Comparing to 7(a) and (b), the correlation plots become less stable and have lower values. The best interpretation authors can make from this figure is that the separation occurs around 1111 while y ¼ 1283 is definitely located in the wake region. The volatility of correlation distribution in the pre-separation region is probably due to the effect that, near the separation point at this Reynolds number, the surface boundary layer has just transited from laminar to turbulent flow. The pressure correlation lengths integrated by Eq. (4) are plotted in Fig. 8. At Re =1.4 105, the correlation length
Fig. 7. (a) Correlation coefficient of pressure fluctuations of Dome S in smooth flow, Re =1.4E + 05, (b) Re = 2.5E+ 05 and (c) Re =3.5E +05.
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Fig. 8. Correlation length of pressure fluctuations on center meridian of Dome in smooth flow.
approaches a relatively high value when the y increases from 201 to 601, and remains stable up to around 901. This stable and high correlation region is likely to be the region where the surface boundary layer is laminar and well developed. When Reynolds number increases to Re = 2 106, at which the surface boundary layer has transited to turbulent flow, the correlation length decreases significantly in this region. The above study on the distributions of correlation coefficients indicates that there is only a minor reduction of upstream correlation at separation region. In other words, it is unlikely to identify the separation point on the correlation length plot. The distribution of correlation coefficients also indicates that the wake region would exhibit low value of correlation length. It is clearly shown in Fig. 8 that, at Re = 1.4 105, the correlation length decreases to the lowest value at y = 1151. At Re = 2.5 105, the position of lowest correlation length moves to 1201. At Re =3.5 105 the position of lowest correlation length moves further downstream to 1301. But at Re =1.5 106–2.0 106, the lowest correlation length position moves 51–101 upstream. This trend of lowest correlation length reflects the movement of the separation point; and it is in agreement with the earlier observations made on mean and RMS pressure distributions. Achenbach’s (1972) study on sphere indicates that, in the subcritical regime at Re =1.62 105, the laminar separation of a sphere occurs at 821. When Re 42.0 105, the separation point shifts downstream to about 951. With further increase of Reynolds number at Re =3.2 105, transition occurs in free shear layer from laminar to turbulent flow and an separation bubble formed follows the intermediate reattachment of the free shear layer. Based on the above pressure studies, it is found that the separation/reattach phenomenon of the hemispherical dome is surprisingly similar to a sphere, although the dome is mounted on the floor and sphere is submerged freely in the approaching flow. The difference between the two mean pressure distributions is that the hemispherical dome exhibits a more noticeable pressure
recovery in the rear part of the dome and a lower negative pressure in the wake region.
3.3. Contours of mean and RMS pressure distributions on dome in smooth flow The contours of mean and RMS pressure coefficients of the three domes were plotted to study the Reynolds number effects on the overall pattern of pressure distributions. 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 Fig. 9 are contours of mean pressure distribution in smooth flow. It is interesting to notice that, for the surface of 3-dimensional curvature, almost all mean pressure contours exhibit a 2-dimensional like characteristics, i.e., the variation of the pressure contour is similar to the mean pressure distribution on the center meridian. It also can be observed that, when the Reynolds number exceeds 2.0 105, the overall mean pressure distribution pattern remains nearly invariant. The contours of RMS pressure coefficient distribution are shown in Fig. 10. For Reo2.0 105, the RMS of pressure fluctuations in the region of 601o o and 901o j o2701, i.e., 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 o =201–301 and j = 71201, 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 variations. 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 Fig. 11. After several tries, the pressure contour at Re =3.0 105 was selected as the reference pressure contour. In Fig. 11, rmean and rrms denote
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Re=6.6 104
Re=2.0 105
335
Re=1.7 105
Re=3.0 105
wind
Re=8.7 105
Re=2.0 106 1 0.6 0.2 -0.2 -0.6 -1 -1.4 -1.8
Fig. 9. Contours of mean pressure distribution of domes at various Reynolds number in smooth flow.
correlation coefficients of mean and RMS pressure distributions between an arbitrary Reynolds number and Re = 3.0 105. The result indicates that the correlation coefficient of mean pressure distribution, rmean, approaches 1.0 near Re = 2.0 105 and remains 1.0 afterwards. As for the correlation coefficient of RMS pressure distributions, rrms decreases rapidly for Reynolds number less the reference case; the correlation coefficient can be as low as rrms o0.5. 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 rrms =0.8 when the Reynolds number reaches 8.0 105. 3.4. Mean and RMS meridian drag coefficients in smooth flow Integrating the pressure along the dome center meridian of each testing case, a drag coefficient called the ‘‘meridian drag coefficient’’ can be obtained. The meridian drag coefficients are shown in Figs. 12 and 13. Fig. 12 indicates that, for Re o3.0 105, the mean drag coefficient decreases with the increase of the
Reynolds number, and it recovers gradually thereafter. Fig. 13 shows that the maximum RMS drag coefficient occurs at Re =1.5 105, then decreases rapidly and exhibits a minimum value at Re = 3.0 105, and then gradually recovers and reaches a relatively stable value when Re 41.0 106.
4. Pressure distribution in turbulent boundary layer flow 4.1. Mean and RMS pressure coefficients in turbulent boundary layer flow Shown in Figs. 14 and 15 are the mean and RMS pressure coefficients measured along the dome meridian in turbulent boundary layer flow. As shown in Fig. 14(a), for Reynolds number between 5.3 104–1.58 105, the negative pressure at dome apex increases, and decreases in wake region. For Reynolds number 41.58 105, both the negative pressure at apex and in the wake region show only minor change with respect to Reynolds number;
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Re=6.6 104
Re=1.4 105
Re=2.0 105
Re=3.0 105
Re=5.2 105
Re=2.0 106
wind
0.5 0.4 0.3 0.2 0.1 0 Fig. 10. Contours of R.M.S. pressure distribution of domes at various Reynolds number in smooth flow.
while the separation point locates near y =1301. For Re =5.2 105 to 1.6 106, the point of separation moves about 101 upstream, from 1301 to 1201. The distributions of RMS pressure coefficient are shown in Fig. 15(a)–(c). It should be noted that the RMS pressure coefficients are 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 151–201. As for the RMS pressure coefficient in the negative pressure zone: when Re o1.1 105, there are two peaks of CP0 locate at 801–901 and 1101–1201; for Re 41.2 105, there is only a single peak of CP0 near 901–1001. Pressure data suggest that transition of the separation flow occurs at lower Reynolds number in the turbulent boundary layer than the smooth flow. Carefully comparing all cases of mean and RMS pressure measurements on the center meridian of the three different size models, it is concluded that, when Reynolds number is in between 1.2 and 1.5 105, both CP and CP0 approach the state of Reynolds number independence.
4.2. Pressure correlation length in turbulent boundary layer flow Shown in Fig. 16(a)–(c) are the distributions of correlation coefficients at Re = 6.7 104, 1.4 105 and 2.0 105 in turbulent boundary layer flow. Each plotted line represents the pressure correlation coefficients of pressure taps along the center meridian of the dome with respect to the indicated reference tap. Fig. 16(a) shows the distributions of correlation coefficients at Re= 6.7 104. In this figure, high correlations exist at small angle, the correlation decreases rapidly when the location of reference pressure tap moves downstream, yi Z 313 . The high value of correlation at small angle is the effect of the large scale turbulence in the approaching flow. When the location of reference tap moves downstream, the incident turbulence effect diminishing, pressure correlation decreases and approaches the lowest value at yi C 603 . When location of the reference tap moves further downstream, where the dome surface boundary layer starts to develop and
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Fig. 11. Correlation coefficient of pressure distributions at various Reynolds number in smooth flow, Re,ref =3 105.
Fig. 12. Mean center meridian drag coefficients of domes in smooth flow.
Fig. 13. R.M.S. meridian drag coefficient of domes in smooth flow.
gradually becomes stable, the pressure correlation recovers. When the reference pressure tap reaches 871, the near-by correlation coefficients reach the highest value which suggests the surface boundary layer is well developed and stable. When the reference tap moves downstream to 981, slightly decrease of
the upstream correlations can be observed. Based on the discussion for the smooth flow condition, it is an indication of separation point. At y ¼ 1103 , pressure correlation plot shows clear decrease in both upstream and downstream directions. In other words, this location is well into the wake region.
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Fig. 14. Mean pressure distributions in turbulent boundary layer: (a) Dome S, 5.3 104 o Re o 2.0 105, (b) Dome M, 2.0 105 o Re o5.5 105 and (c) Dome L, 5.2 105 o Re o 1.6 106.
Shown in Fig. 16(b) are the distributions of correlation coefficients at Re = 1.4 105. The pattern of the pressure distributions for various reference taps are similar to those in Fig. 16(a). The difference is that, at this Reynolds number, the correlation plot starts to show slightly decrease towards upstream direction when the reference tap locates at 1101 instead of 981 as in the previous case. In other words, the separation point moves downstream, from 981 at Re = 6.7 104 to 1101 at Re =1.4 105. Shown in Fig. 7(c) are the distributions of correlation coefficients at Re = 2.0 105. The distributions of pressure correlation are almost identical to those in Fig. 16(b). In other words, the pressure field on a hemispherical dome in turbulent boundary layer flow would approach a stable pattern when Re^1.4 105. Shown in Figs. 17–19 are the pressure correlation length obtained from Domes S, M and L, respectively. Fig. 16 indicates
that, at Re = 6.7 104, pressure correlation length remains constant when y increase from 201 to 401. It then decreases and reaches the lowest value at 601. The pressure correlation length will increase again and reach the highest value at y C 803 . It then remains stable up to y C 1003 . In this region, the surface boundary layer becomes well developed and stable. When y moves further downstream, pressure correlation length starts to decrease. The lowest pressure correlation occurs at y C 1203 . At different Reynolds numbers, the distribution of pressure correlation length remains almost invariant. Except the location of lowest correlation length moves to 1301 when Re 41.4 105. For the pressure correlation length obtained from Domes M and L, shown in Figs. 18 and 19, the distribution patterns are almost identical to those in Fig. 17. Since the Reynolds number of all these cases are 42.0 105, the location of lowest correlation length keeps invariant at y C 1303 .
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Fig. 15. R.M.S. pressure distributions in turbulent boundary layer: (a) Dome S, 5.3 104 oRe o 2.0 105, (b) Dome M, 2.0 105 o Re o 5.5 105 and (c) Dome L, 5.2 105 o Re o 1.6 106.
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 Figs. 20 and 21. As shown in Fig. 20, in the apex region of dome, i.e., o 4801, the mean pressure increases with Reynolds number up to Re =1.58 105; then the mean pressure distribution pattern would remain nearly invariant thereafter. The pressure contours in Fig. 21 indicate that in a larger dome apex region, i.e., o 4601, the RMS pressure coefficient distribution obtained from Dome S increases with Reynolds number up to Re =1.8 105, although the RMS pressure contours of three dome models are different due to the variations of pressure tap arrangements and incident turbulence intensities. The RMS pressure contours exhibit only minor variation for Dome M at Re =2.0 105–5.7 105, and Dome L at Re =5.2 105– 1.6 106, i.e., if we compare the results from the same dome model.
The correlation coefficients for the mean and RMS pressure distributions at different Reynolds numbers, rmean and rrms, are also calculated for the pressure measurements in boundary layer flow. However, in this case, the pressure contour at Re = 2.0 105 was used as the reference pressure contour. It is clearly shown in Fig. 22 that the correlation coefficient of mean pressure distribution, rmean, approaches 1.0 near Re = 1.0 105, slightly earlier than the smooth flow condition, and remains to be rmean = 1.0 afterwards. As for the correlation coefficient of RMS pressure distributions, rrms decreases rapidly for Reynolds number less Re = 1.0 105. In the cases of Reynolds number greater than the reference case, i.e., Re 42.0 105, the correlation coefficient shows a very mild decaying trend and rrms = 0.85–0.9 when the Reynolds number 44.0 105. The mean pressure distributions of dome submerged in turbulent boundary layer is then compared with the pressure coefficients
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Fig. 16. (a) Correlation coefficient of pressure fluctuations of Dome S in turbulent boundary layer, Re =6.7E + 04, (b) Re = 1.4E+ 05 and (c) Re =2.0E+ 05.
Fig. 17. Correlation length of pressure fluctuations on center meridian of Dome S in turbulent boundary layer.
Fig. 18. Correlation length of pressure fluctuations on center meridian of Dome M in turbulent boundary layer.
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Fig. 19. Correlation length of pressure fluctuations on center meridian of Dome L in turbulent boundary layer.
Re=6.7 104
Re=1.6 105
Re=1.2 105
Re=5.2 105
wind
Re=8.4 105
Re=1.6 106 1 0.6 0.2 -0.2 -0.6 -1 -1.4 -1.8
Fig. 20. Contours of mean pressure distribution of domes at various Reynolds number in boundary layer flow.
recommended in ASCE7-05 (2005), as shown in Fig. 23. On the windward side, the pressure distribution of current investigation is similar to the ASCE7-05 recommendations. On leeward side, the
negative 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. This comparison suggests
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a
Re=6.7 104
b
Re=1.4 105
Re=2.0 105
Re=3.7 105
Re=2.0 105
Re=5.7 105
wind
c
Re=5.2 105
Re=1.0 106
Re=1.6 106 0.5 0.4 0.3 0.2 0.1 0
Fig. 21. Contours of R.M.S. pressure distribution of domes at various Reynolds number in boundary layer flow.
Fig. 22. Correlation coefficient of pressure distributions at various Reynolds number in turbulent boundary layer flow, Re, ref =2 105.
that ASCE7-05 provides a reasonably accurate and slightly conservative mean pressure distribution for hemispherical domes.
4.4. Mean and RMS meridian drag coefficients in turbulent boundary layer flow Shown in Figs. 24 and 25 are the meridian drag coefficients of the domes tested in turbulent boundary layer flow. For
Reo2 105, the mean drag coefficient decreases with the increase of the Reynolds number; for Re 42 105, the mean drag coefficient remains nearly invariant. The open circle symbols in Fig. 25 represent the original RMS drag coefficients 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 Fig. 2. The solid circle symbol represents the RMS drag coefficient further normalized with respect to the turbulence intensity at the dome
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Pressure work at Re=1.6 106
ASCE7-05
Fig. 23. Comparison of the mean pressure distribution between present work and ASCE7-05 recommendations.
Fig. 24. Mean center meridian drag coefficient of domes in turbulent boundary layer.
Fig. 25. R.M.S. center meridian drag coefficient of domes in turbulent boundary layer.
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height, C~ D ¼ CD0 =Iu , where Iu is the turbulence intensity at dome height. It is noticed that the possible effects of different turbulence level can not be entirely 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 is 41.8 105.
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 separation flow occurs near Re = 1.8 105–3.0 105, separation/reattachment occur in this Reynolds number region. The pressure distributions become relatively stable after Re 43.0 105. The mean meridian drag coefficient decreases with Reynolds number for Reo3.0 105, 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 of domes in turbulent boundary layer flow. In turbulent flow, the transition of separation flow occurs at lower Reynolds number, Re o1.1 105, and both mean and RMS pressure distributions approach Reynolds number independent when Re = 1.2–1.5 105. The mean and RMS meridian drag coefficients, CD and CD0 , become invariant when Re42 105. The correlation coefficients of mean and RMS pressure contours indicate that, in turbulent boundary layer flow, the pressure
distributions become Reynolds number independent at Re= 1.0– 2.0 105.
Acknowledgments Financial supports for this research project from National Science Council (Project no. NSC96-2221-E-032-022-MY3) is gratefully acknowledged.
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