On the fluctuating wall pressure field in tube banks

Nuclear Engineering and Design 203 (2001) 13 – 26 www.elsevier.com/locate/nucengdes

On the fluctuating wall pressure field in tube banks L.A.M. Endres, S.V. Mo¨ller * Programa de Po´s-Graduac¸a˜o em Engenharia Mecaˆnica, Uni6ersidade Federal do Rio Grande do Sul, Rua Sarmento Leite 425, 90050 -170 Porto Alegre, RS, Brazil Received 5 January 2000; received in revised form 12 May 2000; accepted 17 May 2000

Abstract This paper presents the experimental analysis of pressure and velocity fluctuations of the cross flow in tube banks, with triangular and square arrangements, and four different aspect ratios. Air is the working fluid, driven by a centrifugal blower, passed by a settling chamber and a set of honeycombs and screens, before reaching the tube bank at an incidence angle of 90°. Both triangular and square arrangements have pitch-to-diameter ratios P/D= 1.60, 1.26, 1.16 and 1.05, with Reynolds numbers, calculated with the tube diameter and the velocity of the flow in the narrow gap between the tubes, from Re = 4× 104 to 7 × 104. Velocity and velocity fluctuations were measured by a constant-temperature hot-wire anemometer, while pressure fluctuations were measured by a piezo-resistive pressure transducer mounted inside one of the tubes in the bank. Behavior of fluctuating quantities is described by means of dimensionless root mean square values and autospectral density functions, while their interdependence is discussed based on cross-correlation functions. © 2001 Elsevier Science B.V. All rights reserved.

1. Introduction Banks of tubes or rods are found in the nuclear and process industries, being the most common geometry used in heat exchangers. Attempts to increase heat exchange ratios in heat transfer equipment do not consider, as a priority of project criteria, structural effects caused by the turbulent fluid flow across the tube bank, unless failures occur (Paı¨doussis, 1982). By reducing the aspect ratio (pitch-to-diameter ratio) of a tube bank, to improve the heat transfer process, dynamic loads * Corresponding author. Tel.: +55-51-3163218; Fax: +5551-3163355. E-mail address: [email protected] (S.V. Mo¨ller).

will appear that are not associated to vortex shedding, as in large aspect ratios tube banks. The influence of the other hydrodynamic processes, like fluidelastic instabilities, turbulent buffeting and acoustic resonance, may also be present in tube banks of all aspect ratios. Therefore, new information about the hydrodynamic phenomena in tube banks is necessary for the design and development of new equipment. The generalization of the large amount of experimental data, through similarity studies, and the constant improvement of the existing experimental techniques in past decades allowed a better comprehension of thermal and hydrodynamic phenomena in tube banks, leading also to the need to know the features of the resulting flow and its effects on the

0029-5493/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 9 - 5 4 9 3 ( 0 0 ) 0 0 2 9 3 - 4

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L.A.M. Endres, S.V. Mo¨ller / Nuclear Engineering and Design 203 (2001) 13–26

walls of the tubes in the bank. In spite of the recent developments in numerical results in this area (for example, Ibrahim and Hassan, 1995), where large eddy simulation is applied, showing it to be a powerful technique, experimental results are necessary for the development and validation of new numerical models or methods. The cross flow passing a tube in a bank is strongly influenced by the presence of the neighboring tubes. In the narrow gap between two tubes in a row, the strong pressure gradient will influence not only the flow in that region, but the flow distribution downstream of this point, in the narrow gap between two tubes in the next row, and so on. Zukauskas (1972) compares the flow through tube banks with staggered arrays to the flow in a curved channel with periodically converging and diverging cross-sections. For in-line arrays, the comparison is made with a straight channel, being the velocity distribution strongly influenced by the velocity in the narrow gaps. Heat exchange coefficients follow, approximately, the same distribution as the velocity, increasing where velocity increases, and becoming almost uniform, at least in triangular arrays, as the aspect ratio is reduced (Mo¨ller, 1982). The heat exchange is also influenced by the viscous sublayer on the walls, where the heat is transferred mainly by conduction. Velocity fluctuations in the viscous layer will occur under the influence of the large eddies of the main flow. Therefore, to increase heat exchange ratios, the thickness of the viscous sublayer must be reduced, while turbulence intensity in the main flow must be increased. Knudsen and Katz (1958) emphasize that the presence of large eddies in the main flow would increase the energy consumption, leading to the need of the search of the equilibrium point between low pumping costs and high heat transfer efficiency or increase turbulence intensities and reduce pumping energy. It is clear that increasing pumping power, without influencing the viscous sublayer (to increase heat transfer) will lead to energy waste and prejudice equipment integrity. The concern about heat transfer equipment integrity is due to the close relationship between fluid flow around a solid surface or a structural element and the vibrations induced by the flow in

the structure. Different from large aspect ratio tube banks, where dynamic loads are mainly associated with the vortex shedding process, the turbulent flow in tube banks with small aspect ratios is characterized as broad band turbulence, without a defined shedding frequency (Blevins, 1990). Pressure fluctuations result from velocity fluctuations at several points of the flow field. The resulting pressure field is described by Poisson’s equation, obtained from the divergence of the Navier–Stokes equation (Willmarth, 1975). 92p= − r

( 2(ui uj ) (xi(xj

(1)

where r is the density of the fluid, ui and uj are velocity components, and xi and xj are spatial coordinates. By introducing in Eq. (1) the Reynolds statement, representing velocity components and pressure by their time average value and the fluctuating part, and rewriting the resulting equation in terms of pressure fluctuation, Eq. (1) becomes 92p%= − 2r

(ui (u%j ( 2(u%i u%j ) ( 2u%i u%j −r +r (xj (xi (xi(xj (xi(xj

(2)

Pressure fluctuations are, thus, produced by the interaction of velocity gradients with velocity fluctuations and Reynolds stresses (Rotta, 1972). According to Townsend (1976), the amplitude of the pressure fluctuations may be influenced by velocity fluctuations at a distance comparable with the wavelength of these fluctuations. The search of form and magnitude of pressure and velocity fluctuations, and the interdependence between these quantities, are necessary for the comprehension of the complex phenomena described in this paper, since the resulting forces applied to the tubes by the turbulent flow will be given by the integral of the pressure field around each tube in the bank. From the fluctuating pressure field, a fluctuating excitation force will result, which may induce vibration of the tube if its natural frequency is present in the excitation force. The natural frequency of the tubes is estimated from the fundamental natural frequency of a beam pinned at both ends, being a function of the distance between support grids (Blevins, 1990).

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The purpose of this paper is, therefore, to investigate the behavior of pressure and velocity fluctuations, and their interdependence, in tube banks, with triangular and square arrangements and several aspect ratios, as described in more detail in Endres (1997). 2. Test section and measurement technique The test section is the same as described in Endres et al. (1995); a 1370 mm long rectangular channel with 146 mm height and a maximal (adjustable) width of 193 mm. Air, at room temperature, is the working fluid, driven by a centrifugal blower, passed by a settling chamber and a set of honeycombs and screens, before reaching the tube bank at an incidence angle of 90° and about 2% turbulence intensity. The angle of incidence of the air on the tubes was 90°. The tubes were rigid and not heated. The flow rate, and thus the Reynolds number, was controlled with help of a gate valve. Before the tube bank, a Pitot tube was placed, at a fixed position to measure the reference velocity for the experiments. The tube banks were both triangular and square arrangements, five rows deep, for the measurements of fluctuating quantities. A scheme of the test section is shown in Fig. 1.

Fig. 1. Schematic view of the test section.

Fig. 2. Instrumented tube (schematic).

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The geometries investigated and presented in this paper had aspect ratios (P/D) of 1.60; 1.26; 1.16 and 1.05, with tubes of 32.1 mm diameter and corresponding Reynolds numbers (ReR), based on the tube diameter and the incidence velocity, of 1.7× 104, 1.2 × 104, 1.0 × 104 and 5.8× 103. By taking the velocity measured in the center of the narrow gap between the rods, the values of the Reynolds numbers (ReG) are, respectively, 5.1 ×104, 7.1 × 104, 8.5 × 104 and 8.2 × 104. Subscripts R and G denote reference velocity or the velocity in the narrow gaps between the tubes, respectively. For the measurement of velocity and velocity fluctuations between the tubes, DANTEC constant-temperature hot-wire anemometers were applied. Wall pressure fluctuations were measured by ENDEVCO piezo-resistive pressure transducers, mounted inside the tubes and connected to a pressure tap by plastic tubes. Prior measurements in pipe flow showed that this mounting technique was adequate for the measurements to be performed (Endres and Mo¨ller, 1994). Fig. 2 shows a scheme of the pressure transducer mounted inside the tube. The tube instrumented with the pressure transducer was the central one in the third row, as shown in Fig. 3. Measurements were performed at each 10° by turning the tube about its axis. These angles are measured clockwise between the direction of the main flow and the position of the pressure tap. Zero degrees (0°) corresponds to the position where the pressure tap faces the main flow. Velocity and velocity fluctuations were measured in three positions: in the center of the narrow gap between the tubes, directly in front of the pressure tap at position 90°, and at a half row distance upstream and downstream of that point. These positions are marked ‘gap’, ‘upstream’ and ‘downstream’, respectively, in the figures of results. Data acquisition of pressure and velocity fluctuations was performed simultaneously by a Keithley DAS-58 A/D-converter board controlled by a personal computer, which was also used for the evaluation of the results. Experimental results in this paper were characterized by root mean square (RMS) values of the wall pressure fluctuations, autospectral density

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&

Fig. 3. Schematic representation of the tube banks with the instrumented tube shaded.

functions of velocity and pressure fluctuations, and their interdependence was investigated with help of cross-correlation functions, described through Fourier analysis, since they have a random behavior. Fourier analysis is a valuable tool for the study of random phenomena being widely applied to turbulence studies. Usually, random data are presented in form of time series, representing a continuous (analog) function of time, sampled for digital analysis with a frequency, f, as a sequence of numbers at regular time intervals. The autospectral density function (or power spectrum) represents the rate of change of the mean square value of a certain time function, x(t), with the frequency f (Bendat and Piersol, 1986). 1 u 2 fxx ( f)= x ( f,B,t) dt (3) Bu 0 where u is an adequate integration (observation) time and B the bandwidth. In the Fourier space, the autospectral density function will be defined as the Fourier transform of the autocorrelation function Rxx (t), defined as the mean value of the product of this function at a time t, with its value at a time t + t. Let x(t) and y(t) be two generic functions of time, so that a correlation function of x(t) and y(t) can be written as

Rxy (t)=

1 u

u

x(t)y(t+t) dt

(4)

0

The function defined via Eq. (4) is called the cross-correlation function, normalized by the RMS values of x(t) and y(t), and will be noted as Cxy. The particular case of x(t)= y(t) is the autocorrelation function therefore, the autospectral density will be given by

&

Fig. 4. RMS values of the wall pressure fluctuations in square arrays, normalized with Umea (HWA) in the gaps. Continuous lines for visualization of the results only.

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fxx ( f)=

Fig. 5. RMS values of the wall pressure fluctuations in triangular arrays, normalized with Umea (HWA) in the gaps. Continuous lines for visualization of the results only.

&

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+

−

Rxx (t) e − i2pft dt

(5)

In this research work, time functions x(t) and y(t) are velocity and pressure fluctuations at different points of the flow. For the determination of autospectral density functions, the sampling frequency was 16.1 kHz, while the signals of the instruments were high pass filtered at 1 Hz and low pass filtered at 8.05 Hz. Cross-correlation functions were determined from data sampled at 3 kHz, and low pass filtered at 1 kHz. In this case, the high pass filter was set at the same frequency of 1 Hz. Analysis of uncertainties in the results have a contribution of 1.4% from the measurement equipment (including hot wire, pressure transducer and A/D converter). In the measurements of pressure fluctuations, tubings are responsible for 5% of the uncertainties, leading to a total value for the spectra of pressure fluctuations, up to 1000 Hz, of 6.4%.

Fig. 6. Autospectral densities of pressure fluctuations in several positions of a tube in a bank with P/D= 1.60 and square (a) and triangular (b) arrays.

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Fig. 7. Autospectral densities of pressure fluctuations in several positions of a tube in a bank with P/D= 1.26 and square (a) and triangular (b) arrays.

3. Results and discussion

3.1. RMS 6alues

Before starting the experiments, the flow distribution and the turbulence intensity in the test section were measured, applying the technique described in Vosa´hlo (1984), Mo¨ller (1988), showing a uniform velocity profile and a turbulence intensity of about 2%. This value was almost the same of the work of Zdravkovich and Stonebanks (1988). Measurements of vibrations of the test section (including the tubes of the tube bank) were also performed with help of a METRA accelerometer to identify possible influence produced by the blower or by its electrical motor. Two important resonance frequencies of the test section were detected, these being about 2 and 8 kHz.

Figs. 4 and 5 present RMS values of the wall pressure fluctuations, scaled with the dynamic pressure given by the velocities measured in the center of the narrow gaps between the tubes (Umea), as a function of the angular position for square and triangular arrays, respectively. There are two regions in Fig. 4 where local maxima of the RMS values of the pressure fluctuations in the banks with square arrangements can be observed. The first lies between 0 and 60°, the location of the second one being between 70 and 130°. As the P/D ratio is reduced, the position of the maxima in the curves tends to occur at smallest angles. By reducing the P/D ratio from 1.60 to 1.26, a significant increase in the local

L.A.M. Endres, S.V. Mo¨ller / Nuclear Engineering and Design 203 (2001) 13–26

maxima of the RMS values is observed. By further reducing the P/D ratio, the local maxima diminish until vanishing by the extreme geometry, P/D =1.05, which shows an almost flattened distribution, indicating a uniform distribution of the pressure fluctuation. In Fig. 5, on the other hand, there is a pattern in the form of the curves of the distribution of RMS values in the banks with triangular array and P/D ratios 1.26, 1.16 and 1.05, with decreasing values, although some superposition of the results is found in the results from P/D = 1.16 and 1.05. After 90°, the curves of P/D =1.60 and 1.26 have similar behavior with almost equal numerical values. The same is observed in the results of P/D = 1.16 and 1.05. The bank with the largest P/D ratio has increasing RMS values until the position at 60°, in contrast to the others which have decreasing values. The almost uniform distribution of the curves for P/D = 1.16 and 1.05 is similar to the behavior observed in Fig. 4 for the bank with the lowest P/D ratio.

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The influence of pressure gradients due to the converging–diverging channel effect produced by the tubes on the RMS values is not observed. In the two largest geometries, the influence of previous tube rows is very strong, increasing RMS values of pressure fluctuations at the positions where shedded vortices are expected to impinge. The small variations of the distribution of pressure fluctuations, as the P/D ratio decreases, indicate that turbulence distribution around the tubes is also uniform. This can also be reflected in the local heat transfer coefficient. The results of Mayinger and Shad, presented by Zukauskas (1972), show for a tube bank with square arrangement and P/D = 1.5, local maxima of the local heat transfer at 50 and 110° while, in the P/D = 1.6 test section studied in this work (Fig. 4), local maxima in the RMS value of pressure fluctuations distribution appear at 40 and 110°. For P/D = 1.05, the uniform distribution of pressure fluctuations coincide with the uniform

Fig. 8. Autospectral densities of pressure fluctuations in several positions of a tube in a bank with P/D= 1.16 and square (a) and triangular (b) arrays.

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Fig. 9. Autospectral densities of pressure fluctuations in several positions of a tube in a bank with P/D= 1.05 and square (a) and triangular (b) arrays.

local heat transfer distribution measured in a triangular pin arrangement (Mo¨ller, 1982).

3.2. Spectral analysis Figs. 6–9 show dimensionless autospectral densities of wall pressure fluctuations measured in several angular positions of a tube in the third row, as described in Fig. 3. Experimental results of spectra are presented as functions of the dimensionless frequency in form of the Strouhal number, defined with the tube diameter and the velocity in the narrow gaps between the rods. This velocity was measured with the hot wire probe except in the tube bank with P/D =1.05, where the gap was too narrow to place there the hot wire probe. In this case, a Pitot tube was used. In general, all curves for square arrays have the same decay pattern after a Strouhal number Str = 0.1, with dimensionless values of the autospectral densities of the same order. For Strouhal numbers below this value, the curves have lower decays,

mainly in the region of flow incidence for square arrays between 45 and 90°. This fact may indicate differentiated characteristics of the flow in the regions before and after the narrow gap, producing unbalanced forces over the tubes due to the different energy distribution with the frequency. This is evident in the spectra taken at 45 and 90° in the tube banks with P/D = 1.60 and 1.26 (Fig. 6a and Fig. 7a). The higher values of energy in the curves show this unbalanced force distribution, produced by the impinging flow coming from the narrow gap in the previous row. By reducing the P/D ratio (Fig. 8a and Fig. 9a), P/D = 1.16 and 1.06, pressure fluctuations have almost coinciding spectra, showing that the distribution of the pressure fluctuations around the tube tends to be uniform as the P/D is reduced. The results for triangular arrays do not have a defined pattern as in square arrays. In this case, the spectra of pressure fluctuations have different features for each P/D ratio analyzed. Fig. 6b shows an energy peak at Str= 0.68 at positions of 45, 135 and 180°, characterizing vortex shedding with a fre-

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quency of 398 Hz. This is in accordance with the results of Polak and Weaver (1994), obtained from anemometry and flow visualization, in a range of 14%. By reducing the P/D ratio, peaks of vortex shedding disappear and spectra show a more uniform decay as the Strouhal number increases. This also produces an increase in the energy in the positions of 0°, due to the impinging flow coming from the narrow gap of the preceding row. In the bank with lower P/D ratio (Fig. 9b), an energy peak is found at Str =0.74, at positions of 45, 135 and 180°, with a frequency of 589 Hz. This is in accordance with the results of Fitzhugh, presented in Blevins (1990), with a deviation of 4.4%. It is not clear if these peaks are produced by vortex shedding since no similar result appears in the spectra of the intermediate P/D ratios (Fig. 7b and Fig. 8b). All the curves show pronounced peaks at about Str = 2.0 and 9.0. These values correspond to the calculated resonance frequencies (Strasberg, 1963), inherent of the use of tubings to connect pressure transducer to pressure tap (Endres and Mo¨ller,

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1994) (Fig. 2). These peaks can be completely disregarded in this analysis. Axisa et al. (1988) suggest that the spectra of the lateral force in tube arrays are proportional to Str − 1/2 for 0.015 Str50.2 and to Str − 7/2 for 0.25 Str5 3. These are approximately the declivities found in the spectra of pressure fluctuations of the three smaller geometries, with both triangular and square arrangements (Figs. 7–9). The value of the Strouhal number where the exponent changes from − 1/2 to − 7/2 is about 1.0. In the tube banks with P/D =1.6 (Fig. 6), this behavior appears only in the frontal part of the tubes. After 45° in the square array and after 90° in the triangular array, the spectra show a declivity of − 7/2 starting at values of the Strouhal number about 0.2. Figs. 10–12 show spectra of velocity fluctuations in three positions of the tube banks investigated. In general, the curves have a very steep but uniform decay. As the P/D ratio is reduced, the decay in the spectra taken in the narrow gaps become less steep than those taken at positions

Fig. 10. Autospectral densities of velocity fluctuations in tube banks with P/D=1.60 and square (a) and triangular (b) arrays.

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Fig. 11. Autospectral densities of velocity fluctuations in tube banks with P/D=1.26 and square (a) and triangular (b) arrays.

upstream and downstream of the narrow gap. There, the energy in the low Strouhal numbers, i.e. low frequency ranges, is reduced, while in the high Strouhal numbers it is increased. This means that, in the narrow gaps between the tubes, the energy in the large eddies is reduced while the energy in the range of small eddies becomes higher. This fact appears first in the results of measurements in the banks with triangular arrays (Fig. 11b), P/D = 1.26. For square array, this effect appears only in the bank with P/D = 1.16 (Fig. 12a). In the bank with triangular array and P/D = 1.16, spectra of velocity in the upstream position are similar to the spectra in the gap with approximately the same magnitude. No measurements at a position ‘downstream’ in this geometry were performed, due to the difficulty of positioning the hot wire probe close to the tube wall in this P/D ratio. Zukauskas’ description of the flow through tube banks (Zukauskas, 1972) can explain this behavior: in square arrangements, the main flow behaves like a channel flow, and effects of reducing gap space will appear only with very small P/D

ratios; on the other hand, in triangular arrangements, the main flow is constantly submitted to changes in its direction, leading to an increase in the energy of the smaller eddies. Thus, in triangular arrays, the small passages and the constant changes in flow direction contribute to reduce the size of turbulent structures as the P/D ratio is reduced. Spectra of velocity fluctuations in both P/D = 1.60 show, in the positions ‘upstream’, the presence of peaks at Str= 0.21, corresponding to a frequency of 152 Hz, which coincides with the value expected in the case of a single cylinder. This can be observed in the spectra of pressure fluctuations only in the bank with triangular array (Fig. 6b). This value is not in accordance with Fitzhugh’s compilation, presented in Blevins (1990) (Str= 0.31), but is also present in the normalized crosscorrelations, Cvp, between velocity and pressure fluctuations (Fig. 13a), where a periodicity of about 6.6 ms, especially with low values, appears. While correlation values in the tube bank with P/D =1.60 are very low, they show relatively high negative values for P/D =1.26. As the gap width is reduced

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to P/D =1.16, values of the maxima of cross-correlation coefficients decrease. This confirms the reduction of the scales of the turbulent flow and the raise in the energy content of the smaller eddies. Therefore, the small scale does not carry the major responsibility of producing wall pressure fluctuations in the tube banks with the small aspect ratio investigated.

4. Concluding remarks This paper presents experimental results of pressure and velocity fluctuations in the turbulent cross flow through tube banks with triangular and square arrangements, and four different aspect ratios. The experimental technique are shown to be adequate for the experimental work performed. RMS values of the wall pressure fluctuations, scaled with the velocity measured in the center of the narrow gaps between the tubes, have approximately the same magnitudes for both triangular

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and square arrays, being differently influenced by the gap spacing. In triangular arrays, with the exception of the arrangement with P/D = 1.60, which shows a local maximum at 60°, the behavior of the arrangements with lower P/D ratios is almost the same, without the presence of this local maximum. In square arrays, the banks with the two largest P/D ratios present pronounced local maxima at locations at about 30 and 45°. In both triangular and square arrangements, the reduction of the gap spacing tends to make uniform the distribution of the wall pressure fluctuations, indicating that a uniform distribution of dynamic loads around the tubes is expected. This fact can also explain, or, at least contribute to, the observed uniform distribution of the local heat transfer coefficient in the extreme geometry P/D = 1.05 (Mo¨ller, 1982). Results of spectra of pressure fluctuations show, in general, that all curves for square arrays have the same decay pattern after a Strouhal number Str = 0.1, with dimensionless values of the autospectral

Fig. 12. Autospectral densities of velocity fluctuations in tube banks with P/D= 1.16 and square (a) and triangular (b) arrays. No ‘downstream’ data available in (b).

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Fig. 13. Normalized cross-correlation functions between velocity and pressure fluctuations in tube banks with square and triangular arrays: P/D=1.60 (a), P/D= 1.26 (b), P/D= 1.16 (c).

densities of the same order, while in the results for triangular arrays the spectra have different features for each P/D ratio analyzed. All results have almost uniform distributions, except the bank with the triangular arrangement and the smallest P/D ratio, where the highest values of pressure fluctuations were found at 0°, this being the point where the pressure tap faces the main flow. Spectra of velocity fluctuations in both P/D = 1.60 show the presence of small peaks at Str= 0.21, which coincides with the value expected for vortex shedding in the case of a single cylinder, but only in the tube bank with triangular array and P/D = 1.60 does a pronounced peak indicate the presence of this phenomenon. By reducing the P/D ratio, the turbulence structure will be reduced. In the range of 1.60 – 1.16, the energy of velocity fluctuations with small scales (large Strouhal numbers) is increased as P/D is reduced, while the

highest values of cross-correlations between velocity and pressure fluctuations were found at P/D = 1.26. This indicates that, although an increase in the energy is found by further reducing the P/D ratio, this does not affect the viscous sublayer, as observed in the measurements in the bank with P/D = 1.16 by the reduction of the correlation coefficient. This could not be confirmed in the present work, since correlation measurements in the tube bank with the smallest P/D ratio were not possible with the present technique. Therefore, the design of heat exchangers must consider that turbulence scale in tube banks have to be reduced, since small eddies can produce an efficient mixing, affecting the viscous sublayer on the tube walls, enhancing heat transfer, without producing high values of pressure fluctuations, which can induce vibrations of the structure. Even so, the natural frequency of the tubes must be out

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of this range, to avoid coupling due to turbulent buffeting as described by Paı¨doussis (1982).

Acknowledgements This research work was supported by the CNPq-Brazilian Scientific and Technological Council, under the grants 414216/90-3 and 400180/92-8, and by FAPERGS-Research Foundation of the State of Rio Grande do Sul, grant 90.1549-0.

Appendix A. Nomenclature

B Cxy (t) D e f Fxx ( f ) i L n P p P/D r Re0 RMS(x) Rxy (t) Str t t, u U, 6 ui,j U0 Umea xi,j x(t), y(t)

bandwidth (Hz) correlation coefficient diameter (m) base of natural logarithm frequency (Hz) autospectral density function, [x(t)]2.(s) i = −1 length (m) kinematic viscosity (m2 s−1) pitch (m) pressure (Pa) aspect ratio density (kg m−3) U D Reynolds number, Re= 0 n root mean square value, [x] correlation function, [x(t).y(t)] fD Strouhal number, Str= U time delay (s) time (s) velocity (m s−1) velocity components (m s−1) incidence velocity (m s−1) measured velocity (m s−1) spatial coordinates (m) generic time functions [x(t), y(t)]

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