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Cross-flow pressure measurement on a 2D square cylinder
Journal of Wind Engineering and Industrial Aerodynamics 89 (2001) 1459–1470
Cross-flow pressure measurement on a 2D square cylinder Y.F. Li*, P.J. Richards, R.G.J. Flay Department of Mechanical Engineering, The University of Auckland, Private Bag 92019, Auckland, New Zealand
Abstract This paper presents and discusses the results of wind tunnel measurements of the pressure distributions on a 2D square cylinder in cross-flows at 3 different turbulence levels. Differential pressures across the lateral sides of the model were sampled simultaneously to give sets of signals at 5 different distances from the front surface. Power spectra, cross-spectra, coherence and normalised co-spectra of these signals were calculated. It was found that increasing turbulence in the flow caused a reduction in the normalised co-spectra at low frequencies and a reduction in the peak of power spectra at the shedding frequency. r 2001 Elsevier Science Ltd. All rights reserved. Keywords: 2D square cylinder; Pressure measurement; Frequency domain analysis
1. Introduction Flow induced vibration are very important for some structures. In wind engineering applications, the cross-wind response for tall buildings often dominates over the along-wind response. This study focuses on the frequency domain analysis of the pressure differences across the sides of a 2D square cylinder, and is part of an overall study aimed at finding methods of mitigating cross-wind excitation. A summary of the findings has been presented in Ref. [1]
*Corresponding author. Tel.: +64-9-373-7599; fax: +64-9-373-7479. E-mail addresses: [email protected] (Y.F. Li), [email protected] (P.J. Richards) 0167-6105/01/$ - see front matter r 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 0 1 ) 0 0 1 3 6 - 2
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2. Experimental set-up 2.1. Wind tunnel and model The experiments were carried out in the low speed working section (1.8 mW 1.1 mH 8 mL) of the de Bray wind tunnel at The University of Auckland. The model was a rigid square cylinder 200 mm 200 mm 1100 mm, with 1 face normal to the wind. On each of the lateral surfaces 5 columns of 4 pressure taps were located symmetrically about the mid-span, at 0.1, 0.3, 0.5, 0.7 and 0.9 model-widths from the front surface. The centre-to-centre distance of the pressure taps in both the span-wise and depth direction was 40 mm. The whole series of tests were carried out at a Reynolds number of about 1 105. 2.2. Tubing systems Differential pressures across the lateral sides of the model were sampled simultaneously with 20 tapping pairs through 4:1 manifolded tubing systems. These connected pairs of signals from columns of pressure taps at 5 different distances from the front surface across their corresponding pressure transducers. Because both sides of the pressure transducers were used, in addition to the ordinary requirements for an optimal tubing system, the two sets of tubing systems have similar gain and phase shift. The two sets of tubing systems were designed and tested as outlined by Gumley [2]. Both gave a near unity gain up to 150 Hz and a linear phase shift up to 250 Hz. Figs. 1 and 2 show the gain and phase shift of these tubing systems, respectively, and Figs. 3 and 4 are their combined gain and phase shift performance, respectively. 2.3. Turbulence Homogeneous turbulence was generated using grids with a mesh ratio of 4. The integral length scales were determined by fitting a von Karman spectrum to the wind 500 Front
Rear
Phase Shift (deg)
400
300
200
100
0 0
50
100
150
200
-100 Frequency (Hz)
Fig. 1. Gain of the tubing systems.
250
300
Y.F. Li et al. / J. Wind Eng. Ind. Aerodyn. 89 (2001) 1459–1470 1.6 1.4 1.2
Gain
1 0.8 0.6 0.4 0.2 Front
Rear
0 0
50
100
150
200
250
300
250
300
Frequency (Hz)
Fig. 2. Phase shift of the tubing systems.
1.2 1
Gain
0.8 0.6 0.4 0.2
Rear / Front
0 0
50
100
150
200
Frequency (Hz)
Fig. 3. Ratio of gains of the tubing systems.
400
Phase Shift (deg)
350 300 250 200 150 100 50 Rear-Front
0 0
50
100
150
200
250
Frequency (Hz)
Fig. 4. Difference in phase shifts of the tubing systems.
300
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Table 1 Turbulence characteristics at 500 mm Configuration
Mean speed (m/s)
Turbulence intensity (%)
Longitudinal length scale ratio Lx =D
Smooth Low Turb. Mod. Turb.
7.0 7.0 7.1
1.0 5.8 10.6
F 1.25 0.55
spectra. Details of the turbulence characteristics are illustrated in Table 1 where D is the width of the cylinder. Measurements of the transverse velocity component were not made during this initial set of measurements due to equipment limitation, but measurements of transverse turbulence intensity and length scale are in progress. It is useful to have an unifying turbulence parameter to combine the effect of Iu and Lx =D: Bearman [3] used Iu ðL2x =AÞ for plates and Iu ðD=Ly Þ1=5 for spheres and circular cylinders. At present there is insufficient data regarding square prisms to identity a suitable unifying parameter. However, Laneville et al. [4] have measured oscillations on a square prism under similar turbulence conditions and found that turbulence scale had no effect on the oscillation phenomena. Hence the turbulence intensity is believed to be the more important parameter in these sets of measurements. 2.4. Data acquisition and processing Differential pressure data were sampled using a PC via an A/D converter. Sixteen blocks of 8192 data points were sampled from each of the 5 channels at 1000 Hz. Signals were low pass filtered at 500 Hz before the A/D operation. Power-, crossspectra, coherence and normalised co-spectra were calculated offline using a Pentium PC. Ensemble averaging in the frequency domain with varying ensemble sizes was performed to obtain the results presented in this paper. All results were treated with the same ensemble averaging process to ensure they can be cross-compared. The results have not been corrected for blockage, which was about 10%.
3. Results and discussion 3.1. Smooth flow Power spectra results for the smooth flow are shown in Fig. 5. These results indicate that the Strouhal number is 0.133. At very low frequencies where it is believed that pressure behaves in a quasi-steady manner, the spectra for the 4 taps closest to the leading edge (x=D ¼ 0:1 to 0.7) were almost identical. On the other hand, the spectra at x=D ¼ 0:9 showed a significant deviation from the others. This deviation suggests that the signal at the trailing edge was weaker than the rest of the surface. Data from Barriga et al. [5] and Lee [6] showed that there are significant differences in the response of differential pressure to angle of attack at different
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F(f)/q^2 (Arbitrary scale)
100
S(f)/q^2
10 1 0.1 0.01
x/D=0.1 x/D=0.3 x/D=0.5 x/D=0.7 x/D=0.9 F(f)
0.001 0.0001 0.001
0.01
0.1
1
10
f=nD/u
Fig. 5. Pressure-spectra for smooth flow results.
0.2
0.1
Differential pressure coefficient
0 0
2
4
6
8
-0.2
-0.4
-0.6
-0.8
-1
x/D=0.1 x/D=0.3 x/D=0.5 x/D=0.7 x/D=0.9
10
12
Differential pressure coefficient
0.05 0 -0.05
(a)
2
4
6
8
10
12
-0.15 -0.2 -0.25 -0.3 -0.35
Angle of attack (degree)
0
-0.1
-0.4
x/D=0.1 x/D=0.3 x/D=0.5 x/D=0.7 x/D=0.9 Angle of Attack (degree)
(b)
Fig. 6. Effect of angle of attack on mean differential pressure in (a) Iu ¼ 0:75%; (b) Iu ¼ 10% (Ref. [5]).
streamwise locations, possibly due to reattachment of separated streamlines. The variation of differential pressure with respect to angle of attack using data from Ref. [5] is shown in Fig. 6(a) which shows similar sensitivity to flow angle at x=D ¼ 0:1 to 0:7; but a much lower sensitivity at x=D ¼ 0:9: All 5 pressure spectra showed a peak at the shedding frequency. The magnitudes of the peaks were about 3 decades higher than the corresponding low frequency parts, and all peaks were of similar magnitude regardless of their positions on the model. One of the most obvious differences between the pressures at different locations was the strength of model induced fluctuations at reduced frequencies above the shedding frequency. A steady increase in signal strength can be seen from the leading
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edge to the trailing edge, which is possibly caused by the increase in shear layer thickness, and an increase in shear layer turbulence which drives the surface pressures. Also apparent in the spectrum at x=D ¼ 0:9 is a strong peak with some broad-band signal centred at 3 times the shedding frequency. These existed in other locations, although they were less obvious. Similar observations have been reported by Surry and Djakovich [7] and Kareem [8] where pressure fluctuation at frequencies higher than Strouhal frequency were more pronounced near the trailing corner. To investigate the resultant forces from individual pressures, one should look at the normalised co-spectra of the individual pressure signals. In direct frequency terms, the normalised co-spectra of two signals can be thought of as the average product of the two signals within a narrow frequency interval, divided by the frequency interval and the square root of the product of the power spectra of the two signals [9]. A value of 1.0 shows two perfectly in-phase signals while a value of 1.0 shows two perfectly out-of-phase signals. Normalised co-spectra were used instead of coherences because the signals must be in phase with each other if they are to combine and yield a resultant force. Normalised co-spectra for the smooth flow results are shown in Fig. 7. In the low frequency region, the normalised co-spectra can be divided into 2 groups. The first group has a value of almost 1.0 in this low frequency region, while the second group has values of about 0.6. The second group of normalised co-spectra consists of all the combinations involving the tap at x=D ¼ 0:9: This observation is consistent with the power spectra results which suggest that the cross-flow quasi-steady differential pressure signals are almost identical from the leading edge up to 70% of the model depth, after which, the differential pressures are significantly different from the rest of the surface.
1.2 1
Norm. Co-spectrum
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0.001
x/D=0.1x0.3 x/D=0.1x0.5 x/D=0.1x0.7 x/D=0.1x0.9 x/D=0.3x0.5 x/D=0.3x0.7 x/D=0.3x0.9 x/D=0.5x0.7 x/D=0.5x0.9 x/D=0.7x0.9 0.01
0.1
1
f=nD/u
Fig. 7. Normalised co-spectra for smooth flow results.
10
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All combinations showed a very high peak at the shedding frequency; The lowest peak among all 10 combinations was 0.98 for the combination of x=D ¼ 0:1 and 0.9, which was the most separated pair of taps in the test. A trough in each combination occurred above the shedding frequency. Most of the significant troughs occurred at about 3 times the shedding frequency, and the largest trough had a value of 0.78 from combination x=D ¼ 0:1 and 0.9. The occurrence of the troughs were at the same frequency where harmonic peaks were detected in the power spectra. As each pressure is representative of an area of the same size, the sum of all the individual pressure time histories reflect the total cross-wind force on the model as shown in Eq. (1). X X X FðtÞ ¼ Pi ðtÞAi ¼ A Pi ðtÞp Pi ðtÞ: ð1Þ Robson [10] gives a method for combining individual spectra into a single forcing spectrum. For this case, results from Robson were reduced to Eq. (2) where Coij ð f Þ is the co-spectrum of signal i and j; the real part of the cross spectrum. The high frequency part was computed using Eq. (2) and is shown in Fig. 5. X X SFF ðf ÞpSP ð f Þ ¼ SP ð f Þ þ CoP P ð f Þ note CoP P ð f Þ ¼ CoP P ð f Þ: Pi
i
i
i
j
i
j
j
i
i; j
ð2Þ Because of the peaks in the power spectra and the negative troughs in the cospectra at that particular frequency, there was no detectable peak in the power spectrum of the sum of the pressure time histories despite the peaks in the individual pressure spectra. Instead, pressure fluctuations at 3 times the shedding frequency will tend to twist or deform the model. It was observed that the rate of decline of the normalised co-spectra above the shedding peaks were lower for combinations with pairs closer to each other. For pairs with the same separation, pairs closer to the trailing edge generally declined faster. 3.2. Low turbulence flow The results of the low turbulence tests were similar to those obtained in smooth flow. As shown in Fig. 8, the Strouhal number is 0.136. Similar to smooth flow results, pressure spectra at very low frequencies had values of 0.3–0.4 except at x=D ¼ 0:9 where these fluctuations were significantly lower than those obtained at other locations. For locations up to x=D ¼ 0:7; the strength of vortex shedding was similar to the smooth flow results, at a level of 200. However, the trend of the low frequency signal at x=D ¼ 0:9 being weaker extended up to and included the shedding frequency, in contrast to the smooth flow result where the pressure signals were equal in strength over the whole lateral wall at the shedding frequency. At frequencies above the shedding frequency, the spectra were generally stronger than the smooth flow spectra for all locations. A steady increase in the strength of these high frequency signals from the leading edge to trailing edge was still observed.
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1 0.1 0.01
x/D=0.1 x/D=0.3 x/D=0.5 x/D=0.7 x/D=0.9
0.001 0.0001 0.001
0.01
0.1
1
10
f=nD/u
Fig. 8. Pressure-spectra for low turbulence test.
1.2 1
Norm. Co-spectrum
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0.001
x/D=0.1x0.3 x/D=0.1x0.5 x/D=0.1x0.7 x/D=0.1x0.9 x/D=0.3x0.5 x/D=0.3x0.7 x/D=0.3x0.9 x/D=0.5x0.7 x/D=0.5x0.9 x/D=0.7x0.9 0.01
0.1
1
10
f=nD/u
Fig. 9. Normalised co-spectra for low turbulence.
However, the peak at 3 times the shedding frequency was destroyed due to the introduction of the onset turbulence. Normalised co-spectra for this flow showed results consistent with the power spectra. Fig. 9 shows that combinations involving x=D ¼ 0:9 have a significantly lower value of about 0.5 in the low frequency region. This value is lower than 0.6
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obtained in smooth flow and is possibly due to earlier reattachment of the separated shear layer promoted by the onset turbulence. The comparatively lower energy level of the x=D ¼ 0:9 data at the shedding peak was reflected in a lower co-spectrum for combinations involving x=D ¼ 0:9: The lowest value at the shedding peak was 0.95 from the combination of x=D ¼ 0:1 and 0.9. The onset turbulence destroyed both the peaks at 3 times the shedding frequency in the spectra, and the troughs in the co-spectra at the same frequency, which were present in the smooth flow. Similar trends in the slopes of the curves above the shedding frequency were also observed in this test as for the smooth flow, with pairs which were closer together and pairs closer to leading edge being more coherent. 3.3. Moderate turbulence flow Power spectra from the moderately turbulent flow test are shown in Fig. 10. The Strouhal number for this test is 0.128. All the Strouhal numbers from this series of test were slightly higher than those of Lee [6], which could be caused by blockage and local flow acceleration around the model. At very low frequencies, the weakest signal was measured at x=D ¼ 0:9; with a relatively weak signal also at x=D ¼ 0:7; whereas signals from the other locations are at similar levels of 0.3–0.4. These observations are in line with the results in Fig. 6(b) from Barriga et al. [5] with 10% turbulence intensity which show similar sensitivity to angle of attack at x=D ¼ 0:1 and 0.3, slightly less sensitivity at x=D ¼ 0:5; and much lower sensitivities at x=D ¼ 0:7 and 0.9.
1000 100
S(f)/q^2
10 1 0.1 0.01
x/D=0.1 x/D=0.3 x/D=0.5 x/D=0.7 x/D=0.9
0.001 0.0001 0.001
0.01
0.1
1
f=nD/u
Fig. 10. Pressure-spectra for moderate turbulence test.
10
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The shedding peak, the spectral peak has a value of 25, which is significantly weaker than those obtained in the low turbulence test. Furthermore, a significantly weaker shedding strength was observed at x=D ¼ 0:9; hence completing the trend that shedding strength at the trailing edge is dependent on the onset turbulence level. Similar observations were also made by Lee [6]. At reduced frequencies higher than the Strouhal number, similar trends were observed in moderate turbulence flow as for the low turbulence flow, and pressure fluctuations become stronger further from the leading edge. In addition, the spectral peak at 3 times the shedding frequency is further broadened by the higher level of onset turbulence. Normalised co-spectra for this test are shown in Fig. 11. Generally lower correlations are found compared to the previous results. In contrast to the previous co-spectra plots where only 2 groups can be identified in the quasi-steady region, results in moderate turbulence flow can be divided into 4 groups. Combinations involving x=D ¼ 0:9 form one group, with a value as low as 0.0 or slightly negative. Combinations involving x=D ¼ 0:7 can be grouped into 2 sub-groups. The first group contains x=D ¼ 0:7 and 0.9, with a value of 0.4, and the second group contains x=D ¼ 0:7 with other locations, with a typical value of 0.8. The rest of the combinations form the last group, with co-spectra values of about 1.0. These observations are in agreement with the corresponding power spectra results. The zero or slightly negative co-spectra involving x=D ¼ 0:9; and the relatively high cospectra between x=D ¼ 0:7 and 0.9 are again in good agreement with the quasisteady differential pressure behaviour as shown in Fig. 6b [5]. All combinations reached their maxima at the shedding peak, with magnitudes ranging from 0.90 for combination x=D ¼ 0:3 and 0.9, up to 1.00 for combination
1.2 1
Norm. Co-spectrum
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0.001
x/D=0.1x0.3 x/D=0.1x0.5 x/D=0.1x0.7 x/D=0.1x0.9 x/D=0.3x0.5 x/D=0.3x0.7 x/D=0.3x0.9 x/D=0.5x0.7 x/D=0.5x0.9 x/D=0.7x0.9 0.01
0.1
1
f=nD/u
Fig. 11. Normalised co-spectra for moderate turbulence test.
10
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x=D ¼ 0:1 and 0.3. This result shows that at the shedding frequency, even in moderate turbulence flow, pressure fluctuations at the shedding frequency are highly correlated and contribute to a single excitation force at the shedding frequency. Negative peaks occurred at reduced frequencies from 0.4 to 1.8. The most negative value is 0.73 from the pair x=D ¼ 0:5 and 0.9. It appears that the negative peaks occur at higher frequencies for tap pairs with more physical separation. Among the pairs with the same separation, the pair closer to the trailing edge usually has a lower normalised co-spectrum value.
4. Conclusions *
*
*
*
Quasi-steady pressure fluctuations at the trailing edge were significantly weaker than the fluctuations in other areas, possibly due to local sensitivity to changes in angle of attack. For all flow configurations, fluctuations were strongest at the shedding frequency, and these fluctuations were highly correlated over the whole lateral surface of the model. Apart from the spectral peak at the shedding frequency, a harmonic peak at 3 times the shedding frequency was also observed. This peak was stronger in smooth flow than in turbulent flow, and it was stronger for taps near the trailing edge than for taps near the leading edge. It did not result in an excitation force due to the windward/leeward negative correlation at this frequency. High frequency pressure fluctuations were strongest near the trailing edge indicating that self-induced turbulence from the model increased. The fact that the separated shear layer curves back towards the side surface may also be partially responsible for this trend.
References [1] Y.F. Li, P.J. Richards, R.G.J. Flay, Cross flow pressure measurement on a 2D square cylinder, in: Proceedings of the Fourth International Colloquium on Bluff Body Aerodynamics and Applications, Bochum, Germany, 2000. [2] S.J. Gumley, Tubing system for pneumatic averaging of fluctuating pressures, J. Wind Eng. Ind. Aerodyn. 12 (1983) 189–228. [3] P.W. Bearman, Some effects of turbulence on the flow around bluff bodies. in: Proceedings of the Symposium on Wind Effects on Buildings and Structures, Loughborough University of Technology, 1968. [4] A. Laneville, I.S. Gartshore, G.V. Parkinson, An explanation of some effects of turbulence on bluff bodies, in: Proceedings of the Fourth International Conference on Wind Effects on Buildings and Structures, Cambridge University Press, Heathrow, 1975. [5] A.R. Barriga, C.T. Crowe, J.A. Roberson, Pressure distribution on a square cylinder at a small angle of attack in a turbulent cross flow, in: Proceedings of the Fourth International Conference on Wind Effects on Buildings and Structures, Cambridge University Press, London, 1975. [6] B.E. Lee, The effect of turbulence on the surface pressure field of a square prism, J. Fluid Mech. 69 (2) (1975) 263–282.
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[7] D. Surry, D. Djakovich, Fluctuating pressures on models of tall buildings, J. Wind Eng. Ind. Aerodyn. 58 (1995) 81–112. [8] A. Kareem, Fluctuating wind loads on buildings, J. Eng. Mech. Division 108 (1982) 1086–1102. [9] J.S. Bendat, A.G. Piersol, Random Data: Analysis and Measurement Procedures 1st Edition, Wiley, Los Angeles, 1971, p. 407. [10] J.D. Robson, An Introduction to Random Vibration, 1st Edition, Edinburgh University Press, Edinburgh, 1963, p. 144.
Cross-flow pressure measurement on a 2D square cylinder Y.F. Li*, P.J. Richards, R.G.J. Flay Department of Mechanical Engineering, The University of Auckland, Private Bag 92019, Auckland, New Zealand
Abstract This paper presents and discusses the results of wind tunnel measurements of the pressure distributions on a 2D square cylinder in cross-flows at 3 different turbulence levels. Differential pressures across the lateral sides of the model were sampled simultaneously to give sets of signals at 5 different distances from the front surface. Power spectra, cross-spectra, coherence and normalised co-spectra of these signals were calculated. It was found that increasing turbulence in the flow caused a reduction in the normalised co-spectra at low frequencies and a reduction in the peak of power spectra at the shedding frequency. r 2001 Elsevier Science Ltd. All rights reserved. Keywords: 2D square cylinder; Pressure measurement; Frequency domain analysis
1. Introduction Flow induced vibration are very important for some structures. In wind engineering applications, the cross-wind response for tall buildings often dominates over the along-wind response. This study focuses on the frequency domain analysis of the pressure differences across the sides of a 2D square cylinder, and is part of an overall study aimed at finding methods of mitigating cross-wind excitation. A summary of the findings has been presented in Ref. [1]
*Corresponding author. Tel.: +64-9-373-7599; fax: +64-9-373-7479. E-mail addresses: [email protected] (Y.F. Li), [email protected] (P.J. Richards) 0167-6105/01/$ - see front matter r 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 0 1 ) 0 0 1 3 6 - 2
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Y.F. Li et al. / J. Wind Eng. Ind. Aerodyn. 89 (2001) 1459–1470
2. Experimental set-up 2.1. Wind tunnel and model The experiments were carried out in the low speed working section (1.8 mW 1.1 mH 8 mL) of the de Bray wind tunnel at The University of Auckland. The model was a rigid square cylinder 200 mm 200 mm 1100 mm, with 1 face normal to the wind. On each of the lateral surfaces 5 columns of 4 pressure taps were located symmetrically about the mid-span, at 0.1, 0.3, 0.5, 0.7 and 0.9 model-widths from the front surface. The centre-to-centre distance of the pressure taps in both the span-wise and depth direction was 40 mm. The whole series of tests were carried out at a Reynolds number of about 1 105. 2.2. Tubing systems Differential pressures across the lateral sides of the model were sampled simultaneously with 20 tapping pairs through 4:1 manifolded tubing systems. These connected pairs of signals from columns of pressure taps at 5 different distances from the front surface across their corresponding pressure transducers. Because both sides of the pressure transducers were used, in addition to the ordinary requirements for an optimal tubing system, the two sets of tubing systems have similar gain and phase shift. The two sets of tubing systems were designed and tested as outlined by Gumley [2]. Both gave a near unity gain up to 150 Hz and a linear phase shift up to 250 Hz. Figs. 1 and 2 show the gain and phase shift of these tubing systems, respectively, and Figs. 3 and 4 are their combined gain and phase shift performance, respectively. 2.3. Turbulence Homogeneous turbulence was generated using grids with a mesh ratio of 4. The integral length scales were determined by fitting a von Karman spectrum to the wind 500 Front
Rear
Phase Shift (deg)
400
300
200
100
0 0
50
100
150
200
-100 Frequency (Hz)
Fig. 1. Gain of the tubing systems.
250
300
Y.F. Li et al. / J. Wind Eng. Ind. Aerodyn. 89 (2001) 1459–1470 1.6 1.4 1.2
Gain
1 0.8 0.6 0.4 0.2 Front
Rear
0 0
50
100
150
200
250
300
250
300
Frequency (Hz)
Fig. 2. Phase shift of the tubing systems.
1.2 1
Gain
0.8 0.6 0.4 0.2
Rear / Front
0 0
50
100
150
200
Frequency (Hz)
Fig. 3. Ratio of gains of the tubing systems.
400
Phase Shift (deg)
350 300 250 200 150 100 50 Rear-Front
0 0
50
100
150
200
250
Frequency (Hz)
Fig. 4. Difference in phase shifts of the tubing systems.
300
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Table 1 Turbulence characteristics at 500 mm Configuration
Mean speed (m/s)
Turbulence intensity (%)
Longitudinal length scale ratio Lx =D
Smooth Low Turb. Mod. Turb.
7.0 7.0 7.1
1.0 5.8 10.6
F 1.25 0.55
spectra. Details of the turbulence characteristics are illustrated in Table 1 where D is the width of the cylinder. Measurements of the transverse velocity component were not made during this initial set of measurements due to equipment limitation, but measurements of transverse turbulence intensity and length scale are in progress. It is useful to have an unifying turbulence parameter to combine the effect of Iu and Lx =D: Bearman [3] used Iu ðL2x =AÞ for plates and Iu ðD=Ly Þ1=5 for spheres and circular cylinders. At present there is insufficient data regarding square prisms to identity a suitable unifying parameter. However, Laneville et al. [4] have measured oscillations on a square prism under similar turbulence conditions and found that turbulence scale had no effect on the oscillation phenomena. Hence the turbulence intensity is believed to be the more important parameter in these sets of measurements. 2.4. Data acquisition and processing Differential pressure data were sampled using a PC via an A/D converter. Sixteen blocks of 8192 data points were sampled from each of the 5 channels at 1000 Hz. Signals were low pass filtered at 500 Hz before the A/D operation. Power-, crossspectra, coherence and normalised co-spectra were calculated offline using a Pentium PC. Ensemble averaging in the frequency domain with varying ensemble sizes was performed to obtain the results presented in this paper. All results were treated with the same ensemble averaging process to ensure they can be cross-compared. The results have not been corrected for blockage, which was about 10%.
3. Results and discussion 3.1. Smooth flow Power spectra results for the smooth flow are shown in Fig. 5. These results indicate that the Strouhal number is 0.133. At very low frequencies where it is believed that pressure behaves in a quasi-steady manner, the spectra for the 4 taps closest to the leading edge (x=D ¼ 0:1 to 0.7) were almost identical. On the other hand, the spectra at x=D ¼ 0:9 showed a significant deviation from the others. This deviation suggests that the signal at the trailing edge was weaker than the rest of the surface. Data from Barriga et al. [5] and Lee [6] showed that there are significant differences in the response of differential pressure to angle of attack at different
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F(f)/q^2 (Arbitrary scale)
100
S(f)/q^2
10 1 0.1 0.01
x/D=0.1 x/D=0.3 x/D=0.5 x/D=0.7 x/D=0.9 F(f)
0.001 0.0001 0.001
0.01
0.1
1
10
f=nD/u
Fig. 5. Pressure-spectra for smooth flow results.
0.2
0.1
Differential pressure coefficient
0 0
2
4
6
8
-0.2
-0.4
-0.6
-0.8
-1
x/D=0.1 x/D=0.3 x/D=0.5 x/D=0.7 x/D=0.9
10
12
Differential pressure coefficient
0.05 0 -0.05
(a)
2
4
6
8
10
12
-0.15 -0.2 -0.25 -0.3 -0.35
Angle of attack (degree)
0
-0.1
-0.4
x/D=0.1 x/D=0.3 x/D=0.5 x/D=0.7 x/D=0.9 Angle of Attack (degree)
(b)
Fig. 6. Effect of angle of attack on mean differential pressure in (a) Iu ¼ 0:75%; (b) Iu ¼ 10% (Ref. [5]).
streamwise locations, possibly due to reattachment of separated streamlines. The variation of differential pressure with respect to angle of attack using data from Ref. [5] is shown in Fig. 6(a) which shows similar sensitivity to flow angle at x=D ¼ 0:1 to 0:7; but a much lower sensitivity at x=D ¼ 0:9: All 5 pressure spectra showed a peak at the shedding frequency. The magnitudes of the peaks were about 3 decades higher than the corresponding low frequency parts, and all peaks were of similar magnitude regardless of their positions on the model. One of the most obvious differences between the pressures at different locations was the strength of model induced fluctuations at reduced frequencies above the shedding frequency. A steady increase in signal strength can be seen from the leading
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edge to the trailing edge, which is possibly caused by the increase in shear layer thickness, and an increase in shear layer turbulence which drives the surface pressures. Also apparent in the spectrum at x=D ¼ 0:9 is a strong peak with some broad-band signal centred at 3 times the shedding frequency. These existed in other locations, although they were less obvious. Similar observations have been reported by Surry and Djakovich [7] and Kareem [8] where pressure fluctuation at frequencies higher than Strouhal frequency were more pronounced near the trailing corner. To investigate the resultant forces from individual pressures, one should look at the normalised co-spectra of the individual pressure signals. In direct frequency terms, the normalised co-spectra of two signals can be thought of as the average product of the two signals within a narrow frequency interval, divided by the frequency interval and the square root of the product of the power spectra of the two signals [9]. A value of 1.0 shows two perfectly in-phase signals while a value of 1.0 shows two perfectly out-of-phase signals. Normalised co-spectra were used instead of coherences because the signals must be in phase with each other if they are to combine and yield a resultant force. Normalised co-spectra for the smooth flow results are shown in Fig. 7. In the low frequency region, the normalised co-spectra can be divided into 2 groups. The first group has a value of almost 1.0 in this low frequency region, while the second group has values of about 0.6. The second group of normalised co-spectra consists of all the combinations involving the tap at x=D ¼ 0:9: This observation is consistent with the power spectra results which suggest that the cross-flow quasi-steady differential pressure signals are almost identical from the leading edge up to 70% of the model depth, after which, the differential pressures are significantly different from the rest of the surface.
1.2 1
Norm. Co-spectrum
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0.001
x/D=0.1x0.3 x/D=0.1x0.5 x/D=0.1x0.7 x/D=0.1x0.9 x/D=0.3x0.5 x/D=0.3x0.7 x/D=0.3x0.9 x/D=0.5x0.7 x/D=0.5x0.9 x/D=0.7x0.9 0.01
0.1
1
f=nD/u
Fig. 7. Normalised co-spectra for smooth flow results.
10
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All combinations showed a very high peak at the shedding frequency; The lowest peak among all 10 combinations was 0.98 for the combination of x=D ¼ 0:1 and 0.9, which was the most separated pair of taps in the test. A trough in each combination occurred above the shedding frequency. Most of the significant troughs occurred at about 3 times the shedding frequency, and the largest trough had a value of 0.78 from combination x=D ¼ 0:1 and 0.9. The occurrence of the troughs were at the same frequency where harmonic peaks were detected in the power spectra. As each pressure is representative of an area of the same size, the sum of all the individual pressure time histories reflect the total cross-wind force on the model as shown in Eq. (1). X X X FðtÞ ¼ Pi ðtÞAi ¼ A Pi ðtÞp Pi ðtÞ: ð1Þ Robson [10] gives a method for combining individual spectra into a single forcing spectrum. For this case, results from Robson were reduced to Eq. (2) where Coij ð f Þ is the co-spectrum of signal i and j; the real part of the cross spectrum. The high frequency part was computed using Eq. (2) and is shown in Fig. 5. X X SFF ðf ÞpSP ð f Þ ¼ SP ð f Þ þ CoP P ð f Þ note CoP P ð f Þ ¼ CoP P ð f Þ: Pi
i
i
i
j
i
j
j
i
i; j
ð2Þ Because of the peaks in the power spectra and the negative troughs in the cospectra at that particular frequency, there was no detectable peak in the power spectrum of the sum of the pressure time histories despite the peaks in the individual pressure spectra. Instead, pressure fluctuations at 3 times the shedding frequency will tend to twist or deform the model. It was observed that the rate of decline of the normalised co-spectra above the shedding peaks were lower for combinations with pairs closer to each other. For pairs with the same separation, pairs closer to the trailing edge generally declined faster. 3.2. Low turbulence flow The results of the low turbulence tests were similar to those obtained in smooth flow. As shown in Fig. 8, the Strouhal number is 0.136. Similar to smooth flow results, pressure spectra at very low frequencies had values of 0.3–0.4 except at x=D ¼ 0:9 where these fluctuations were significantly lower than those obtained at other locations. For locations up to x=D ¼ 0:7; the strength of vortex shedding was similar to the smooth flow results, at a level of 200. However, the trend of the low frequency signal at x=D ¼ 0:9 being weaker extended up to and included the shedding frequency, in contrast to the smooth flow result where the pressure signals were equal in strength over the whole lateral wall at the shedding frequency. At frequencies above the shedding frequency, the spectra were generally stronger than the smooth flow spectra for all locations. A steady increase in the strength of these high frequency signals from the leading edge to trailing edge was still observed.
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1 0.1 0.01
x/D=0.1 x/D=0.3 x/D=0.5 x/D=0.7 x/D=0.9
0.001 0.0001 0.001
0.01
0.1
1
10
f=nD/u
Fig. 8. Pressure-spectra for low turbulence test.
1.2 1
Norm. Co-spectrum
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0.001
x/D=0.1x0.3 x/D=0.1x0.5 x/D=0.1x0.7 x/D=0.1x0.9 x/D=0.3x0.5 x/D=0.3x0.7 x/D=0.3x0.9 x/D=0.5x0.7 x/D=0.5x0.9 x/D=0.7x0.9 0.01
0.1
1
10
f=nD/u
Fig. 9. Normalised co-spectra for low turbulence.
However, the peak at 3 times the shedding frequency was destroyed due to the introduction of the onset turbulence. Normalised co-spectra for this flow showed results consistent with the power spectra. Fig. 9 shows that combinations involving x=D ¼ 0:9 have a significantly lower value of about 0.5 in the low frequency region. This value is lower than 0.6
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obtained in smooth flow and is possibly due to earlier reattachment of the separated shear layer promoted by the onset turbulence. The comparatively lower energy level of the x=D ¼ 0:9 data at the shedding peak was reflected in a lower co-spectrum for combinations involving x=D ¼ 0:9: The lowest value at the shedding peak was 0.95 from the combination of x=D ¼ 0:1 and 0.9. The onset turbulence destroyed both the peaks at 3 times the shedding frequency in the spectra, and the troughs in the co-spectra at the same frequency, which were present in the smooth flow. Similar trends in the slopes of the curves above the shedding frequency were also observed in this test as for the smooth flow, with pairs which were closer together and pairs closer to leading edge being more coherent. 3.3. Moderate turbulence flow Power spectra from the moderately turbulent flow test are shown in Fig. 10. The Strouhal number for this test is 0.128. All the Strouhal numbers from this series of test were slightly higher than those of Lee [6], which could be caused by blockage and local flow acceleration around the model. At very low frequencies, the weakest signal was measured at x=D ¼ 0:9; with a relatively weak signal also at x=D ¼ 0:7; whereas signals from the other locations are at similar levels of 0.3–0.4. These observations are in line with the results in Fig. 6(b) from Barriga et al. [5] with 10% turbulence intensity which show similar sensitivity to angle of attack at x=D ¼ 0:1 and 0.3, slightly less sensitivity at x=D ¼ 0:5; and much lower sensitivities at x=D ¼ 0:7 and 0.9.
1000 100
S(f)/q^2
10 1 0.1 0.01
x/D=0.1 x/D=0.3 x/D=0.5 x/D=0.7 x/D=0.9
0.001 0.0001 0.001
0.01
0.1
1
f=nD/u
Fig. 10. Pressure-spectra for moderate turbulence test.
10
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The shedding peak, the spectral peak has a value of 25, which is significantly weaker than those obtained in the low turbulence test. Furthermore, a significantly weaker shedding strength was observed at x=D ¼ 0:9; hence completing the trend that shedding strength at the trailing edge is dependent on the onset turbulence level. Similar observations were also made by Lee [6]. At reduced frequencies higher than the Strouhal number, similar trends were observed in moderate turbulence flow as for the low turbulence flow, and pressure fluctuations become stronger further from the leading edge. In addition, the spectral peak at 3 times the shedding frequency is further broadened by the higher level of onset turbulence. Normalised co-spectra for this test are shown in Fig. 11. Generally lower correlations are found compared to the previous results. In contrast to the previous co-spectra plots where only 2 groups can be identified in the quasi-steady region, results in moderate turbulence flow can be divided into 4 groups. Combinations involving x=D ¼ 0:9 form one group, with a value as low as 0.0 or slightly negative. Combinations involving x=D ¼ 0:7 can be grouped into 2 sub-groups. The first group contains x=D ¼ 0:7 and 0.9, with a value of 0.4, and the second group contains x=D ¼ 0:7 with other locations, with a typical value of 0.8. The rest of the combinations form the last group, with co-spectra values of about 1.0. These observations are in agreement with the corresponding power spectra results. The zero or slightly negative co-spectra involving x=D ¼ 0:9; and the relatively high cospectra between x=D ¼ 0:7 and 0.9 are again in good agreement with the quasisteady differential pressure behaviour as shown in Fig. 6b [5]. All combinations reached their maxima at the shedding peak, with magnitudes ranging from 0.90 for combination x=D ¼ 0:3 and 0.9, up to 1.00 for combination
1.2 1
Norm. Co-spectrum
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0.001
x/D=0.1x0.3 x/D=0.1x0.5 x/D=0.1x0.7 x/D=0.1x0.9 x/D=0.3x0.5 x/D=0.3x0.7 x/D=0.3x0.9 x/D=0.5x0.7 x/D=0.5x0.9 x/D=0.7x0.9 0.01
0.1
1
f=nD/u
Fig. 11. Normalised co-spectra for moderate turbulence test.
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x=D ¼ 0:1 and 0.3. This result shows that at the shedding frequency, even in moderate turbulence flow, pressure fluctuations at the shedding frequency are highly correlated and contribute to a single excitation force at the shedding frequency. Negative peaks occurred at reduced frequencies from 0.4 to 1.8. The most negative value is 0.73 from the pair x=D ¼ 0:5 and 0.9. It appears that the negative peaks occur at higher frequencies for tap pairs with more physical separation. Among the pairs with the same separation, the pair closer to the trailing edge usually has a lower normalised co-spectrum value.
4. Conclusions *
*
*
*
Quasi-steady pressure fluctuations at the trailing edge were significantly weaker than the fluctuations in other areas, possibly due to local sensitivity to changes in angle of attack. For all flow configurations, fluctuations were strongest at the shedding frequency, and these fluctuations were highly correlated over the whole lateral surface of the model. Apart from the spectral peak at the shedding frequency, a harmonic peak at 3 times the shedding frequency was also observed. This peak was stronger in smooth flow than in turbulent flow, and it was stronger for taps near the trailing edge than for taps near the leading edge. It did not result in an excitation force due to the windward/leeward negative correlation at this frequency. High frequency pressure fluctuations were strongest near the trailing edge indicating that self-induced turbulence from the model increased. The fact that the separated shear layer curves back towards the side surface may also be partially responsible for this trend.
References [1] Y.F. Li, P.J. Richards, R.G.J. Flay, Cross flow pressure measurement on a 2D square cylinder, in: Proceedings of the Fourth International Colloquium on Bluff Body Aerodynamics and Applications, Bochum, Germany, 2000. [2] S.J. Gumley, Tubing system for pneumatic averaging of fluctuating pressures, J. Wind Eng. Ind. Aerodyn. 12 (1983) 189–228. [3] P.W. Bearman, Some effects of turbulence on the flow around bluff bodies. in: Proceedings of the Symposium on Wind Effects on Buildings and Structures, Loughborough University of Technology, 1968. [4] A. Laneville, I.S. Gartshore, G.V. Parkinson, An explanation of some effects of turbulence on bluff bodies, in: Proceedings of the Fourth International Conference on Wind Effects on Buildings and Structures, Cambridge University Press, Heathrow, 1975. [5] A.R. Barriga, C.T. Crowe, J.A. Roberson, Pressure distribution on a square cylinder at a small angle of attack in a turbulent cross flow, in: Proceedings of the Fourth International Conference on Wind Effects on Buildings and Structures, Cambridge University Press, London, 1975. [6] B.E. Lee, The effect of turbulence on the surface pressure field of a square prism, J. Fluid Mech. 69 (2) (1975) 263–282.
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[7] D. Surry, D. Djakovich, Fluctuating pressures on models of tall buildings, J. Wind Eng. Ind. Aerodyn. 58 (1995) 81–112. [8] A. Kareem, Fluctuating wind loads on buildings, J. Eng. Mech. Division 108 (1982) 1086–1102. [9] J.S. Bendat, A.G. Piersol, Random Data: Analysis and Measurement Procedures 1st Edition, Wiley, Los Angeles, 1971, p. 407. [10] J.D. Robson, An Introduction to Random Vibration, 1st Edition, Edinburgh University Press, Edinburgh, 1963, p. 144.