Fluidelastic behaviour of a normal triangular array subject to cross-flow

Journal of Fluids and Structures (1991) 5, 259-278

FLUIDELASTIC B E H A V I O U R OF A N O R M A L T R I A N G U L A R A R R A Y SUBJECT TO CROSS-FLOW S. J. PRICE and M. L. ZAHN

Department of Mechanical Engineering, McGill University, Montreal, Qudbec, H3A 2K6 Canada (Received 6 June 1990 and in revised form 12 October 1990)

The flow-induced vibratory response of a single flexible cylinder in an otherwise rigid normal triangular array, with a pitch-to-diameter ratio of 1-375, and subject to air cross-flow, has been investigated experimentally. Fluidelastic instability is obtained with the flexible cylinder in each of the seven rows of the array. The instability is predominantly in the cross-flow direction, although in rows 4, 5 and 6 an in-flow instability can also exist. For the seventh, and final, row of the array, the instability is purely in the in-flow direction. Stability boundaries for each of the seven rows are presented. Characteristics of the interstitial flow are discussed, including the existence of periodicities within the flow. The nature of flexible cylinder vibrational resonances and acoustic resonances with these flow periodicities are also considered. Finally, unconventional flow and vibrational behaviour are discussed, namely a nonuniformity and asymmetry of the flow downstream of the array, and a fluidelastic instability in the first row which is apparently triggered by an acoustic resonance. 1. I N T R O D U C T I O N has joined heat transfer and pressure drop as one of the principal design criteria for tube-in-shell heat exchangers and related components such as steam generators, evaporators and condensers. This surge to a primary design parameter has been relatively fast, considering that prior to the mid-1960s the effects of unsteady fluid loads were not often considered in heat exchanger design. On the problem of flow-induced vibration, the TEMA Standards (Tubular Exchanger Manufacturers Association 1968) could offer little guidance, stating that "existing quantitative data are inadequate to ensure that designs will be vibration free". However, the financial implications of numerous failures in heat exchangers, particularly those associated with nuclear reactors, caused by flow-induced vibration over the past 30 years have made designers very aware of its significance. Examples of such failures have been reviewed by Paidoussis (1980). Cross-flowinduced vibrations are responsible for the vast majority of vibration-induced failures in heat exchangers. In comparison, axial flow causes very few vibration problems (Pa'idoussis 1980). Therefore, the bulk of recent research in this field, including the present work, has been concerned with cylinder arrays subjected to cross-flow. Although there is now a general awareness of the problem, flow-induced vibration failures continue to occur. Possibly the reason for this is that despite over 20 years of intensive research, uncertainties still remain in the fundamental nature of the vibration phenomena. Thus, to ensure the success of future designs, there is still a need for further fundamental research to give new insight into the nature of these mechanisms; this was the primary motivation behind the present study. IT IS ONLY IN RECENT YEARS THAT FLOW-INDUCED VIBRATION

0889-9746/91/030259 + 20 $03.00

© 1991 Academic Press Limited

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The vibrational response of a cylinder within an array subjected to cross-flow can be attributed to one or more of the following three excitation mechanisms: (a) turbulent buffeting, which causes low-amplitude vibration, and operates at essentially all flow velocities; (b) resonance with a flow periodicity, which occurs over the velocity range where the frequency of a periodicity within the flow, fp, coincides with the natural frequency of the cylinder, fn, causing vibratory response above that of turbulent buffeting; and (c) fluidelastic instability, which is a self-excited mechanism, and results in highamplitude vibration. The state of knowledge on these phenomena was reviewed by Pa'idoussis (1983), and more recently by Weaver & Fitzpatrick (1989), who emphasized developments between 1983 and 1987, and also by Chen (1987). The work presented in this paper, which is a continuation of an ongoing experimental research programme at McGill University (Price et al. 1986, 1987, 1989, 1990; Pa'idoussis et al. 1989), concentrates exclusively on the latter two phenomena for a single flexible cylinder in a normal triangular array with pitch-to-diameter ratio, P/d, of 1-375. However, some measurements of the turbulence-induced force spectra in this array in the in-flow and cross-flow directions are presented by Zahn (1989). While a single flexible cylinder in a rigid array is unquestionably a significant simplification of the real situation in a heat exchanger, important insights into the nature of flow-induced vibrations can nonetheless be gained by its study. Furthermore, the principal objective of this study was to investigate, in detail, the nature of the underlying excitation mechanisms; this can best be done by removing additional complications, such as multiple flexible cylinders, from the experiment. 2. APPARATUS Experiments were performed on a seven-row normal triangular array with P / d = 1.375 (see Figure 1). The array completely spanned the 0.91 m width of the wind tunnel, there being either 25 or 26 cylinders in each row of the array depending on which row the flexible cylinder was positioned in (this was because the flexible cylinder was located in a fixed position, and to change its row location, the rest of the array was moved with respect to it). Each cylinder had a diameter of 25-4 mm and a length of approximately 0.59 m--spanning the wind tunnel height. With this array of cylinders positioned in the wind tunnel, gaps existed between the cylinders and tunnel walls. To avoid the nonuniformities in flow which these gaps would undoubtedly have caused, the gaps were closed with styrofoam fillers. Thus, a homogeneous resistance to the flow was imposed across the total width of the wind tunnel. In the vibration experiments, all but one of the cylinders were rigid and fixed to the top and bottom surfaces of the wind tunnel; the one remaining cylinder was flexibly mounted. The apparatus has been described in detail previously (Price et al. 1987; Zahn 1989), and thus, for the sake of brevity, few details will be given here. Suffice to say that the natural frequency, fn, and logarithmic decrement, 6, of the flexible cylinder may be varied, as can the cylinder mass (which enabled different values of nondimensional mass, m / p d 2, to be obtained). The flexible cylinder mounting system permitted correction of the "blow-back", caused by the static drag force on the cylinder producing a mean deflection in the in-flow direction. At each flow velocity, before a measurement was made, the mounting system was adjusted such that both the top and bottom ends of the flexible cylinder were aligned with the rest of the array. This

ARRAY FLU1DELASTIC BEHAVIOUR IN CROSS-FLOW

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alignment was done using a feeler gauge, and it is estimated that it could be done to an accuracy of better than 1 mm. If this is not done, then the array geometry is distorted and the results obtained are not representative of this particular array. The cylinder vibration was monitored using two accelerometers, orthogonally mounted such that the acceleration components in the in-flow (parallel to the upstream flow) and cross-flow (normal to the upstream flow) directions were measured. In addition to the vibration measurements, flow measurements were made in the interstitial gaps between cylinders using a hot-wire anemometer, some flow measurements were also made downstream of the array. The cylinder acceleration and anemometer signals were analyzed on a HP 3562A dual channel Signal Analyzer coupled to a HP microcomputer. Typically 150 samples were averaged to produce the power spectra; the r.m.s, acceleration (and hence, displacement) at the dominant frequencies in the spectra could then be obtained using the Analyzer. To determine the upstream flow velocity, the difference in static pressure across the wind tunnel contraction, upstream of the array, was measured and a previously obtained calibration curve employed. Unless otherwise stated, all velocities quoted in this paper refer to the upstream velocity.

3. RESULTS Over 80 different vibration experiments were conducted in this array. The flexible cylinder was positioned in each of the seven rows and its response measured as a function of flow velocity for at least two different values of m / p d 2 and a range of iS. These vibration experiments are summarized in the following sections, and a more complete description is given by Zahn (1989). The values of m / p d 2 and t5 for the flexible cylinder varied in the ranges 279-2,480 and 0.007-0.95, respectively; f, varied

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between 5 and 40 Hz, although most of the experiments were done with f~ in the range 7-15 Hz. It was estimated that the errors in the damping and frequency measurements were less than 5.0% and 0.1%, respectively. In general, the flexible cylinder had the same damping and natural frequency in the in-flow and cross-flow directions. However, for 6 <0.01 there was sometimes a difference between the in-flow and cross-flow values of b; this is always indicated in the relevant figures. This difference between the in-flow and cross-flow (~ is thought to be due to the accelerometer leads producing extra damping in one direction because of the way they were located on the piano wires supporting the flexible cylinder.

3.1.

F L U I D E L A S T I C INSTABILITY

Fluidelastic instability occurred with the flexible cylinder located in all seven rows of the array. A typical instability, with the flexible cylinder positioned in row 3 of the array, is presented in Figure 2. At a nondimensional velocity (based on freestream 0"08

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A R R A Y F L U I D E L A S T I C B E H A V I O U R IN CROSS-FLOW

263

velocity), U/f,,d, of approximately 20, there is an abrupt increase in the cylinder cross-flow displacement, indicating a fluidelastic instability. Accompanying the fluidelastic instability is a gradual increase in the cross-flow fluidelastic frequency with increasing velocity; see Figure 2(b). As indicated in Figure 2(a), the instability in row 3 is purely in the cross-flow direction, with no in-flow motion; this was the case for all experiments with the flexible cylinder in rows 1, 2 and 3; however, as discussed later in this section, in the other rows an in-flow instability was also obtained. The results of Figure 2 show the cylinder vibrational response for both increasing and decreasing flow velocity; there is very little difference between these two sets of results, indicating that there are no "hard-oscillator" or hysteretic effects for this instability. This is consistent with the results of Lever & Rzentkowski (1988) who also found no hysteretic effects for a single flexible cylinder, with m~/pd2=2.5, in a rotated triangular array with P/d = 1-375; it should be noted that Lever & Rzentkowski did obtain substantial hysteretic effects in the same array with multiple flexible cylinders. Andjeli6 & Popp (1989), on the other hand, did observe a hysteresis for a single flexible cylinder, with m6/pd 2 in the range 11-2-13-3, in a triangular array with P/d = 1.25, and obtained different values of Uc/fnd depending on whether the damping was being increased or decreased. Somewhat surprisingly, in their experiments with multiple flexible cylinders Andjeli6 & Popp did not obtain any hysteretic effects. A further set of interesting results is presented in Figure 3 where the flexible cylinder, also located in row 3, initially becomes unstable in the "traditional" translational mode. However, as the velocity is further increased, the cylinder eventually becomes unstable in the rotational or "rocking mode". In the "idealized" translational mode the two ends of the flexible cylinder move in-phase with each other (with frequency fn) and have equal vibrational magnitudes; in the rocking mode the two ends move out-of-phase with each other (with frequency fnr). It should be appreciated that, because the exact point of rotation in the rocking mode was not known, it was not possible to calibrate the vibration measurement system for this mode, and thus the vibrational units in this case, unlike for the translational mode, are arbitrary. Although the cylinder motion is very different in these two modes, it will be shown later that the nondimensional critical flow velocities obtained collapse very well onto a single stability boundary, providing the correct modal damping, mass and frequency are employed--using f~ and f~r for the translational and rocking modes, respectively. (Unless otherwise stated, the instabilities in this paper correspond to the "traditional" translational motion). In addition to cross-flow instabilities, examples of which are presented in Figures 2 and 3, in-flow instabilities were also obtained with the flexible cylinder in rows 4, 5 and 6; the direction of the instability (in-flow or cross-flow) depending on the particular values of m/pd 2 and & In row 7 only in-flow instabilities were obtained. An example of a purely in-flow instability is shown in Figure 4(a) for a flexible cylinder (m/pd2=2,370, /~i=0-002 and ~x =0.008) in row 4.* Increasing the damping to dt = 0.014 (both directions now having the same damping) a combined in-flow and cross-flow instability is obtained at a slightly higher critical flow velocity, as shown in Figure 4(b). A further increase in 6 to 0.045 produces a sensibly pure cross-flow instability, as seen in Figure 4(c). Teh & Goyder (1988) also obtained an in-flow instability for this array, although in their experiments this was obtained with the flexible cylinder free to move in the in-flow direction only. t This figureshows an example of where two separate experimentswere done with the same parameters, the repeatabilityis representativeof that obtainedin the completeset of experimentsreported in this paper.

264

S. J. PRICE AND M. L. ZAHN

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For each row of the array it is possible to collapse the fluidelastic instability results onto a single curve of nondimensional critical flow velocity, Uc/fnd, versus massdamping parameter, m6/pd z. A typical example of this for the results of row 3 is shown in Figure 5. It should be appreciated that these results include different values of m/pd 2, f, and 6, as well as both translational and rocking motion of the cylinder; nevertheless, a good collapse of the data is obtained, and a stability boundary, of the conventional form, may be written as

U~/f~d = A(rna / pd2) ~, where, in this case, A = 1.6 and te = 0.55. This was done for the other six rows of the array and the results obtained are summarized in Table 1. As previously mentioned, in rows 4, 5 and 6 an in-flow instability was obtained in addition to the cross-flow instability; however, for the in-flow instability there are insufficient data points to obtain a correlation of the form presented in Table 1. For a fixed value of m6/pd z the least stable row of the array is the second; Uc/f,d for

265

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next least stable row, the third row, being approximately 50% greater. The stability boundaries for the fourth, fifth and sixth rows of the array are virtually the same, indicating that, as far as fluidelastic instability is concerned, the interstitial flow seems to have reached some sort of equilibrium by the fourth row. By far the most stable rows of the array are the first and last rows. It is interesting that the exponent on m r / p d 2 is greater for the fourth, fifth and sixth rows vis-?~-vis the second and third rows, which in turn have an exponent which is considerably greater than that obtained in the first or last rows of the array. In Figure 6 the results obtained in rows 2 to 6 are c o m p a r e d with other known data for normal triangular arrays; for the sake of clarity, the data for rows 4, 5 and 6, which have essentially the same stability boundary, are lumped together. Because the critical flow velocities obtained with the flexible cylinder in rows 1 and 7 are so much higher than for the other rows, these data are not included in Figure 6. As an a t t e m p t to collapse data from arrays with different values of pitch onto one curve, the results are presented i n terms of the "so-called" reduced gap-flow velocity defined by Chen (1984), vr =

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TABLE 1

Summary of stability boundaries in the form Uc/f,d = A ( m r / p d 2 ) ~ for the seven rows of the array; X and I indicate cross-flow or in-flow instabilities, respectively. Row

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where Up¢ = UcP/(P - d). As can be seen from Figure 6, the results for rows 4, 5 and 6 fall above the majority of the other data points; however, the results obtained for row 2 agree remarkably well. Bearing in mind that the majority of the other data points come from arrays of multiple flexible cylinders, this suggests that in this array for mO/pd2<200 (the highest value of m 6 / p d 2 for the other data) a single flexible cylinder in row 2 goes unstable at essentially the same velocity as an array with multiple flexible cylinders. This is in agreement with the results of Lever & Rzentkowski (1988) and Andjeli6 & Popp (1989), both of whom obtained substantially the same critical velocity with single and multiple flexible cylinders in this array, with m 6 / p d 2 in the ranges of 2.5-7-5 and 11-13, respectively (the P / d in Andjeli6 & Popp's array was 1.25, as compared with 1.375 for the present experiments and those of Lever & Rzentkowski). However, it should be born in mind that theoretical considerations (Price & Paidoussis 1984, 1985) suggest that for this array there will be substantial differences between the single- and multiple-flexible cylinder values of Uc/f,,d at higher values of m 6 / p d 2. 3.2. FLOW PERIODICITIES With the flexible cylinder replaced by a rigid one, hot-wire measurements were made at positions A - S in the array (see Figure 1), for Reynolds numbers, Re, (based on freestream velocity) in the range 2.2 × 103 to 1-74 × 104. Periodicities in the interstitial flow, with frequencies which varied approximately linearly with flow velocity, were detected in and behind the first two rows of the array, as well as a short distance upstream of the array. The most dominant of these periodicities had a Strouhal number, based on freestream velocity, of 2-40; this compares very well with a Strouhal number of 2-51 suggested by the correlation given by Zukauskas, Ulinskas & Katinas (1988) but is somewhat higher than the value of 1.54 suggested by Weaver & Fitzpatrick's (1988) correlation. A second flow periodicity, less dominant than the one described above, was observed in and behind the first row of cylinders; the average

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Strouhal number was 0-32 although it appeared to vary somewhat with Reynolds number, St ranging from 0.41 at Re = 2.2 x 103 to 0.30 at Re = 4.4 × 104. No such lower Strouhal number is reported in the correlations given by Weaver & Fitzpatrick (1988) or Zukauskas et al. (1988). Neither of these Strouhal peaks produced a resonant vibration with the flexible cylinder, for the simple reason that any such resonance would have occurred at a velocity lower than the minimum velocity of the wind tunnel. However, special vibration experiments were done with the frequency range of the FFT Analyzer extended above its usual maximum, and off-resonant peaks were detected in the vibration spectra which corresponded to St = 2.40. There were other significant resonance-type peaks in the vibrational response of the flexible cylinder (for some values of m / p d 2 and 0) when it was positioned in the downstream rows of the array; these were characterized by vibration amplitudes considerably greater than those caused by turbulent buffeting. See, for example, Figure 7, where the vibration response of a flexible cylinder in row 5 (m/pd2=2370, 0.002 and 0x = 0.008) is presented. A broad resonant peak is obtained in the nondimensional velocity range 12-22, with maximum response at U/f,,d = 18 corresponding approximately to St--0.056. A broad resonant peak at a similar Strouhal number was obtained with the flexible cylinder (m/pd2=2,370, 01=0.006 and 0x = 0-012) positioned in row 6 of the array, while a Strouhal number of approximately 0-12 (twice 0-056) was obtained with the flexible cylinder in rows 4 and 5 (m/pd z = 300; 0 = 0.081 and 0 = 0.086, respectively). As mentioned previously in this section, no flow periodicities corresponding to these Strouhal numbers were detected deep in the array, and thus, the origin of these broad resonant peaks is somewhat perturbing; at present the authors have no physical explanation for them. 01

=

3.3.

ACOUSTIC

RESONANCE

During the course of these experiments very loud acoustic resonances with sound pressure levels downstream of the array of order 120 dB (these resulted in the authors being "banned" from doing daytime experiments) were obtained with this array for 0.15

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ARRAY FLUIDELASTICBEHAVIOUR IN CROSS-FLOW

certain flow velocities. Further investigation of these resonances identified them as occurring at U = 8.0 and 10.2 m/s at frequencies of 733 Hz and 900 Hz, respectively; thus, it is immediately apparent that they were triggered by the S t = 2 . 4 0 flow periodicity. At these acoustic resonances, hot-wire traverses were done across the half-width of the wind tunnel test section upstream of the array; the results obtained are presented in Figure 8. Bearing in mind that a single hot-wire a n e m o m e t e r measures the magnitude of the flow only, and not its direction, it is apparent that the two acoustic resonances correspond to standing waves with mode numbers (the number of half-waves across the tunnel section) of 4 and 5. It is interesting to note that the standing waves for both acoustic resonances exhibited maxima in the acoustic velocityt at the walls of the tunnel. In a standing acoustic wave the maxima of acoustic velocity are separated from the pressure maxima by a quarter wavelength (Temkin 1981); this implies that the acoustic pressure at the wind tunnel walls is zero, which is contrary to what is usually obtained (Temkin 1981; Ziada et al. 1988a). The probable reason for this is that the wind tunnel walls are made

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S. J. PRICE, A N D M. L. Z A H N

of plexiglass and are compliant and capable of moving in phase with the transverse acoustic velocity at the walls. Recently Ziada et al. (1988b) proposed a method of predicting the occurrence of acoustic resonances in staggered cylinder arrays. They defined a resonance parameter, Gs, which is based on the array geometry and properties of the flow (the complete expression is given by Ziada et al.), and predicted that acoustic resonances will occur only if Gs is greater than a critical value; for this array this critical value of G~ being approximately 2.1 × 10 -4. For the fourth- and fifth-mode acoustic resonances which occurred in this array, G~ was approximately~ 2.7 x 10-4 and 3.0 x 10 -4, respectively, which are above the critical value, and hence, in agreement with the predictions of Ziada et al. A resonance of the third acoustic mode with the St = 2.40 flow periodicity would have occurred at a frequency and velocity of approximately 540 Hz and 5.7 m/s, respectively; the corresponding value of G~ is approximately 2.3 x 10 -4, which is very close to the critical value. In fact, in some experiments a very soft acoustic tone was noted near U = 5.7m/s, although it was not as intense as the other two acoustic resonances; nevertheless, this suggests that the array may have been close to an acoustic resonance at this velocity. All of these results suggest that the prediction method of Ziada et al. is very accurate. Apart from causing considerable time-tabling problems for these experiments, the acoustic resonances also produced some interesting interactions with both the array interstitial flow and flexible cylinder vibration behaviour; these interactions are discussed below. The first effect of the acoustic resonance was to organize the flow throughout the array. Normally, no periodicities were detected beyond the second row of the array; however, coincident with the acoustic resonance, flow periodicities, at the same frequency as the acoustic tone, were evident throughout the complete array. A second, and more puzzling, aspect of the acoustic resonances is that the 733 Hz tone produced a significant peak in the interstitial flow PSD, at a frequency of approximately 7.8 Hz, at locations N, L, J, H, F and D (see Figure 1). These locations correspond to the transverse ("diagonal") gaps between cylinders in different rows of the array. It is interesting to note that the periodicity was not detected at locations in the gaps between cylinders of the same row. There does not seem to be any obvious relationship between the frequency of this flow periodicity (7-8 Hz) and the acoustic resonance (733 Hz), and the authors have no explanation for the origin of this new interstitial flow periodicity. The third, and most puzzling, consequence of the acoustic resonance at 733 Hz concerns the cylinder vibrational behaviour. With the flexible cylinder positioned in rows 2 to 7 of the array, the acoustic resonances sometimes had a relatively minor effect on the cylinder vibrational response. Typically, there was a reduction or increase in the vibrational amplitude coincident with the acoustic resonance; examples of this are shown in Figure 4(a) and (b) where the 733 Hz resonance is marked as AR1 on the figure. However, when the flexible cylinder was positioned in the leading row of the array, the 733 Hz acoustic resonance sometimes had a much more pronounced effect, and appeared to be able to initiate a fluidelastic instability. An example of this acoustic resonance-induced instability is presented in Figure 9; large amplitude vibrations occurred at exactly the same instant as the acoustic resonance developed. However, if the velocity is increased, causing the acoustic resonance to cease, the large amplitude vibrations continue; indeed, even if the cylinder is manually brought to rest, upon releasing the cylinder the large vibrations reappear. Even more surprising, is that if 6 is increased from 0-019 to 0.1, the

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instability still occurs at the same flow velocity (still coincident with the acoustic resonance). A further increase o f / i to 0.22 still produced large amplitude vibrations at the same velocity, although in this Case if the velocity is further increased beyond the acoustic resonance and the vibrations are physically stopped they will not reappear. If the large amplitude vibrations were not physically stopped at velocities beyond the acoustic resonance they would continue indefinitely. To further investigate the effect of the acoustic resonance, experiments were done with four different cylinder natural frequencies; the results obtained are summarized in Table 2 (in all cases, the occurrence of fluidelastic instability was coincident with the acoustic resonance). As is immediately apparent from the results of Table 2, a four-fold increase in frequency has virtually no effect on the critical flow velocity. One subtle difference was noticed between the results obtained with f, = 10.9 and 16.8 Hz and those with f, =24.2 and 40-1Hz. For the lower fnS, the fluidelastic instability persisted when the velocity was increased past the acoustic resonance; even if the cylinder was manually brought to rest the fluidelastic instability would restart as soon as the cylinder was released. For the higher frequencies, the fluidelastic instability would continue if the velocity was increased beyond the acoustic resonance (even for a time duration greater than 5 rain), but if the cylinder was manually brought to rest the fluidelastic instability would not redevelop. Considering the large frequency difference between the acoustic resonance and the natural frequency of the cylinder, it is not clear how the acoustic resonance can have such a dramatic effect on the cylinder vibrational behaviour and, at present, the authors have no explanation for this interaction. TABLE 2 Effect of natural frequency on the critical flow velocity for fluidelastic instability, when triggered by an acoustic resonance, with the flexible cylinder in row 1. f, (Hz) t~ m/pd 2 Uc (m/s)

10.9 0.019 300 7-5

16.8 0.022 300 7.7

24-2 0.016 338 8.0

40.1 0.010 338 8-0

272 3.4.

S. J. PRICE AND M. L. ZAHN INTERSTITIAL AND DOWNSTREAM FLOW

Velocity measurements in the interstitial gaps were made over the velocity range of the wind tunnel at the locations labelled A - S in Figure 1. Typical distributions of the time-averaged flow velocity and turbulence intensity throughout the array, at Re = 1.0 x 104, are presented in Figure 10. The time-averaged flow velocity measured at locations in the gaps between cylinders in a row is relatively constant throughout the array at approximately 3.9U, while the time-averaged flow velocity at locations in the gaps between cylinders of different rows decreases in the first three rows of the array before reaching a relatively steady state value.t It is interesting to note that for the rows farther downstream in the array, the velocity measured at the locations between cylinders of different rows is much less than the corresponding flow velocity measured between cylinders in the same row, see Figure 10(a). Whereas, based on continuity arguments, it is expected that the average gap velocity for the flow between cylinders in 5.0

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ARRAY FI~UIDELASTICBEHAVIOURIN CROSS-FLOW

273

the same row should be approximately double the average gap velocity for the flow between cylinders in different rows. Also, as seen in Figure 10(b), the turbulence intensity between cylinders in the same row is much less than that obtained in the gaps between cylinders of different rows.t Both of these observations suggest that the position mid-way between cylinders of different rows is in the wake of the upstream cylinder; they also suggest that between this position and the surface of the downstream cylinder, the mean flow velocity must be considerably greater than the mean flow velocity in the gap between cylinders of a row. Thus, it appears that in the latter rows of the array the flow accelerates from the front stagnation point of the cylinder to the 30 ° point and then decelerates till, or possibly just past, the 90 ° position. These conclusions are qualitatively in agreement with the pressure measurements of Zdravkovich & N a m o r k (1979). Hot-wire traverses across the test section were also done at various distances downstream of the array, and these revealed the existence of an asymmetric and nonuniform downstream flow. Measurements were initially made at Re = 1-26 x 10 4 with the flexible cylinder located in the fourth row; in this configuration, the downstream row of the array consisted of 26 cylinders. The velocity distribution 0.660 m (26 d) downstream of the array [0-152 m(6 d) upstream of the test section exit plane] was significantly skewed, as shown in Figure l l ( a ) . Also shown in this figure is the velocity profile at this location measured when the flexible cylinder was replaced with a rigid one; the two velocity profiles are essentially identical, indicating that this behaviour is not a consequence of the flexible cylinder. This skewed velocity profile was f o u n d to exist along the entire height of the test section exit plane. Further measurements were made closer to the array at 6 d downstream of the array, and the mean velocity profile, shown in Figure l l ( b ) , exhibited significant humps at a distance of approximately 0.152 m (6 d) from the test section walls. Furthermore, a wool-tuft probe indicated frequent back-flow (flow in the upstream direction) in the wall regions, suggesting that the mean flow near the walls is less than that indicated in Figure l l ( b ) . Vertical hot-wire traverses made over the test section height revealed even more surprising flow behaviour, as shown in Figure l l ( c ) . While the vertical velocity profile along the horizontal centreline was v e r y uniform, the vertical velocity profile at one of the humps in the horizontal profile was skewed, displaying a significant velocity defect near the test section ceiling, This suggests that the flow directly downstream of the array is very three-dimensional. Similar velocity distributions were also obtained at a Reynolds number of 8.4 x 103. If the cylinder array was relocated such that the flexible cylinder was in the fifth row, the downstream row of the array had 25 cylinders, as opposed to 26 cylinders which was the case for the results of Figure 11. This seemingly insignificant change appeared to greatly influence the downstream flow pattern. The velocity profile 6 d behind the array was now relatively uniform, see Figure 12. Near the test section exit, 0.686 m (27 d) downstream of the array, the flow was no longer skewed, but relatively symmetrical. However, it was significantly nonuniform, with the maximum flow velocity surprisingly occurring near the walls. Vertical velocity profiles near the test section exit were relatively fiat, suggesting that the mean flow pattern at the exit was fairly two-dimensional. When the flexible cylinder was located in the sixth row, the overall array t The turbulence intensities measured at locations A, B, D, F, H, L and J are too high to be measured accurately with the present techniques. Hence, the values shown in Figure 10(b), at these locations, should be regarded as being indicative of the turbulence intensities and not exact values.

274

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configuration was identical to when this cylinder had been in the fourth row. The double-humped velocity profile was again evident 6 d downstream of the array, and at the test section exit the flow was again skewed, but this time towards the other side of the wind tunnel. Attempts to reverse the bias of the flow at the exit by twisting the array in alternate directions as much as the clearance between the end plates and the test section walls would allow were unsuccessful. However, to facilitate further vibration tests at a later date, the array was reassembled to relocate the flexible cylinder in row 6. Measurements indicated that this time the flow at the test section exit was consistently biased to the opposite side of that shown in Figure ll(b). Thus the nature of the mechanism which determines towards which side the flow will be biased is not clear. As expected, locating the flexible cylinder in row 7 gave a downstream flow pattern similar to that observed when the cylinder was in row 5. The two possible overall configurations of the 7-row array therefore gave two significantly different downstream flow patterns. When the sequence of the number of cylinders in each row was 26-25-26-25-26-25-26, the flow directly behind the array was characterized by a double-humped velocity profile, and far downstream by a skewed velocity profile. When the sequence was 25-26-25-26-25-26-25, the velocity profile far downstream was symmetric, but nonuniform, with the flow biased towards the walls of the test section. Zdravkovich & Stonebanks (1988) recently reviewed other researchers' observations of nonuniform flow in and behind uniformly spaced cylinders subjected to cross-flow, and also presented results of their own surface-pressure-measurement experiments on a single- and double-row of cylinders. They attributed nonuniformity of the flow behind single- and double-row arrays to the irregular coalescence of gap flows behind the final row, and postulated that such nonuniformity might exist behind the last row of a closely spaced array containing any number of rows. The observed nonuniform flow behind the present array supports this theory. One would expect the pattern of coalescence of gap flows to be affected by the number of cylinders (or gaps) in the last row of the array. The results of the present study also support this supposition in that the pattern of downstream flow was dependent on whether there were 25 or 26 cylinders in the final row of the seven-row array.

276

S . J . PRICE AND M. L. ZAHN

4. C O N C L U S I O N S Based on the results presented in this paper and other results given by Zahn (1989) the following conclusions can be made regarding the fluid-structure interaction of a single flexible cylinder in an otherwise rigid seven-row normal triangular array with P / d = 1.375. (i) Fluidelastic instability occurs with the flexible cylinder in all rows of the array. In rows 1 through 6, the initial, or principal, instability occurs in the cross-flow direction. In rows 4, 5 and 6, there also exists an in-flow fluidelastic instability at a much higher critical flow velocity. In row 7, the principal instability occurs in the in-flow direction. (ii) In the mass-damping parameter range 10 ~< m6/pd2<~ 500, the flexible cylinder is least stable in the second row. (iii) The cross-flow stability boundary for row 1 and the in-flow stability boundary for row 7 exhibit only a weak dependence on the mass-damping parameter. (iv) With respect to fluidelastic instability, the flow appears to achieve a regular repeating pattern by the fourth row, since the stability boundaries for rows 4, 5 and 6 are essentially the same. (v) Periodicities occurring at a Strouhal n u m b e r of 2.4 exist within the interstitial flow in and behind the first two rows of the array. Periodicities at a Strouhal number of approximately 0.32 occur behind the first row only. In and beyond the third row, no periodicities could be detected above the random turbulence of the flow. (vi) The periodicities associated with the Strouhal number of 2.4 are capable of causing acoustic resonances at 7 3 3 H z and 900Hz. Furthermore, the 733 Hz acoustic resonance is capable of causing a flexible cylinder in the first row of the array to go unstable, even though the acoustic frequency is more than 10 times greater than the cylinder natural frequency; the nature of this instability triggering mechanism is not understood. Both acoustic resonances can organize the flow throughout the array, causing peaks at the acoustic frequency to be visible in the flow spectra throughout the array. The 733 Hz acoustic resonance also produces a very significant peak at approximately 7.8 Hz in the spectra of flow velocity in the open transverse channels between the rows of cylinders. (vii) The flow downstream of the array is very much dependent on the number of cylinders in the final row of the array and can be very nonuniform and asymmetric.

ACKNOWLEDGEMENTS The authors gratefully acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada and Le Fonds F C A R of Qu6bec.

REFERENCES

ANDJELIC, A. • PoPP, K. 1989 Stability effects in a normal triangular cylinder array. Journal of Fluids and Structures 3, 165-186 CHEN, S. S. 1984 Guidelines for the instability flow velocity of tube arrays in cross-flow. Journal of Sound and Vibration 93, 439-455 CHEN, S. S. 1987 Flow-Induced Vibration of Circular Cylindrical Structures. Washington: Hemisphere Publishing

ARRAY FLUIDELASTIC BEHAVIOUR IN CROSS-FLOW

277

LEVER, J- H. & RZENTKOWSKI, G. 1988 A n investigation into the post-stable behaviour of a tube array in cross-flow. In Proceedings International Symposium on Flow-Induced Vibration and Noise: Volume 3. Flow Induced Vibration and Noise in Cylinder Arrays (eds M.P. Paidoussis, S. S. Chen & M. D. Bernstein), pp. 95-110. New York: A S M E pAiDOUSSlS, M. P. 1980 Flow-induced vibrations in nuclear reactors and heat exchangers: practical experiences and state of knowledge. In Proceedings I A H R / I U T A M Symposium on Practical Experiences with Flow-Induced Vibrations (eds E. Naudascher & D. Rockwell), pp. 1-81. Berlin: Springer-Verlag PAiDOUSSlS, M. P. 1983 A review of flow-induced vibrations in reactors and reactor components. Nuclear Engineering and Design 74, 31-60 PAiDOUSSIS, M. P., PRICE, S. J., NAKAMURA, T., MARK, B. & MUREITHI, W. N. 1989 Flow-induced vibrations and instabilities in a rotated-square cylinder array in cross-flow. Journal of Fluids and Structures 3, 229-254 PRICE, S. J. & PAiDOUSSIS, M. P. 1984 A n improved mathematical model for the stability of cylinder rows subject to cross-flow. Journal of Sound and Vibration 97, 615640 PRICE, S. J. & PAiDOUSSlS, M. P. 1985 Fluidelastic instability of a full array of flexible cylinders subject to cross-flow. In Proceedings A S M E Symposium on Fluid-Structure Interaction and Aerodynamic Damping (eds E. H. Dowell & M. K. Au-Yang), pp. 171-192. New York: ASME PRICE, S. J., MARK, B. & PA~DOUSSIS, M. P. 1986 A n experimental stability analysis of a single flexible cylinder positioned in an array of cylinders and subject to cross flow. Journal of Pressure Vessel Technology 108, 62-72 PRICE, S. J., PAiDOUSSIS, M. P., MACDONALD, R. & MARK, B. 1987 The flow-induced vibration of a single flexible cylinder in a rotated square array of rigid cylinders with pitch-to-diameter ratio of 2.12. Journal of Fluids and Structures 1, 359-378 PRICE, S. J. & PAiDOUSSIS, M. P. 1989 The flow-induced response of a single flexible cylinder in an in-line array of rigid cylinders. Journal of Fluids and Structures 3, 61-82 PRICE, S. J. & KURAN, S. 1990 Fluidelastic stability of a rotated square array, with multiple flexible cylinders, subject to cross-flow. In Proceedings A S M E Symposium on Flow-Induced Vibration (eds S. S. Chen, K. Fujita & M. K. Au-Yang), pp. 107-117. New York: ASME TEH, C. E. & GOYDER, H. G. D. 1988 Data for the fluidelastic instability of heat exchanger tube bundles. Proceedings International Symposium on Flow-Induced Vibration and Noise; Volume 3. Flow Induced Vibration and Noise in Cylinder Arrays (eds M. P. Pffidoussis, S. S. Chen & M. D. Bernstein), pp. 77-94. New York: A S M E TEMKIN, S. 1981 Elements of Acoustics. New York: J. Wiley & Sons TUBULAR EXCHANGER MANUFACTURERS ASSOCIATION 1968 Standards of Tubular Exchanger Manufacturers Association, Fifth Edition, New York WEAVER, D. S. & FITZPATRICK, J. A. 1988 A review of flow induced vibrations in h e a t exchangers. Journal of Fluids and Structures 2, 73-93. ZAHN, M. L. 1989 Flow-induced vibration of a single flexible cylinder in a normal triangular array. M. Eng. Thesis. McGill University ZIADA, S., OENGOREN, A. & BUHLMANN, E. T. 1989a On acoustical resonance in tube arrays-Part I: Experiments. Journal of Fluids and Structures 3, 293-314 ZIADA, S., OENGOREN, A. & B~]HLMANN, E. Z. 1989b On acoustical resonance in tube arrays-Part II: Damping criteria. Journal of Fluids and Structures 3, 315-324 ZDRAVKOVICH,M. M. & NAMORK, J. E. 1979 Structure of interstitial flow between closely spaced tubes in staggered array. In Flow-Induced Vibrations (eds S. S. Chen & M. D. Bernstein), pp. 41-46. New York: A S M E ZDRAVKOVICH, M. M. & STONEBANKS, K. L. 1988 Intrinsically non-uniform and metastable flow in and behind tube arrays, In International Symposium on Flow Induced Vibrations and

Noise; Volume 1. Flow-Induced Vibration in Cylindrical Structures: Solitary Cylinders and Arrays in Cross-Flow (eds M. P. Pa'idoussis, O. M. Griffin & C. Dalton), pp. 61-73. New York: A S M E ZUKAUSKAS, A. & KATINAS, V. 1980 Flow-induced vibration in heat-exchanger tube banks, In Practical Experiences with Flow-Induced Vibrations (eds E. Naudascher & D. Rockwell), pp. 188-196. Berlin; Springer-Verlag ~UKAUSKAS,A., ULINSKAS, R. & KATINAS, V. 1988 Fluid Dynamics and Flow-Induced Vibrations of Tube Banks. New York: Hemisphere Publishing

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APPENDIX A: NOMENCLATURE d fn

cylinder diameter natural frequency of oscillation for translational mode of the flexible cylinder in still air (Hz) fnr natural frequency of oscillation for the rocking mode of flexible cylinder in still air (Hz) fp frequency of a periodicity in the interstitial flow (Hz) m mass per unit length of the flexible cylinder P pitch between cylinders Re Reynolds number, U d / v St Strouhal number, fp d/U T transverse separation between cylinders U freestream velocity Uc critical value of U for fluidelastic instability Upc critical value of pitch velocity for fluidelastic instability, (Uc/f,,d)(P/(P - d)) Ur Reduced critical gap velocity, (Up¢/fnd)/[2.105 (T/d - 0.9)] 6 cylinder logarithmic decrement in still air; subscripts I and X indicate different values of 6 in the in-flow and cross-flow directions, respectively 6j value of 6 in the in-flow direction 6x value of 6 in the cross-flow direction p fluid density v kinematic viscosity of air