29 1999, D. W. Guillaume.pdf

Experiments in Fluids 27 (1999) 145d156 ( Springer-Verlag 1999

Investigation of the flopping regime with two-, three- and four-cylinder arrays D. W. Guillaume, J. C. LaRue

145 Abstract The variation of the base pressure coefficient (Cp), and the characteristics of the velocity power spectra for arrays of two-, three- and four- cylinders aligned normal to the flow are presented. For the two-cylinder array with s/d\0.750 (where s is the spacing between the top and bottom surfaces of adjacent cylinders and d is the diameter of the cylinder) and Re\2.5]103, peaks in the power spectra of 145 and 45 Hz which correspond to Strouhal numbers of 0.35 and 0.11 have been observed. For the three-cylinder array with Re\2.5]103, at 0.338Os/dO0.730, three quasi-stable modes are observed. For 0.730Os/dO0.850, flopping and one quasi-stable mode are observed. For 0.850Os/dO1.202, only one mode is observed. The hot-wire power spectra measured downstream of the cylinder array on the center plane between the top and center cylinder and, also on the plane at s/2 above the top cylinder has three relative peaks that correspond to a wake structure, i.e. a pattern of vortices. For the four-cylinder array, when 0.338Os/dO0.750, four quasi-stable modes are observed. First, a mode can be observed in which the average Cp value of the top cylinder is relatively high, the average Cp value of the bottom cylinder is relatively low, and the average Cp values of the two center cylinders are nearly equal. A second mode is sometimes observed that is similar to the first except that the relatively high and low average Cp values of the outer cylinders are interchanged. A third mode is observed in which the average Cp value of the upper—inner cylinder is relatively high,

Received: 19 February 1997/Accepted: 9 February 1999 D. W. Guillaume Department of Mechanical Engineering California State University, Los Angeles Los Angeles, CA 90032, USA J. C. LaRue Department of Mechanical and Aerospace Engineering University of California, Irvine Irvine, CA 92697, USA Correspondence to: D. W. Guillaume The authors gratefully acknowledge the valuable assistance of Graham W. Clark in constructing the apparatus and performing the tests. This project is part of a larger research project and we wish to acknowledge the helpful suggestions of Professor G. S. Samuelsen, Mr. S. R. Vilayanur, and Mr. S. W. Shaffar. This work is supported by the US Department of Energy, Morgantown Energy Technology Center, Contract Number DE-FC21—92MC29061 and the Southern California Gas Company, Contract SCG-19596.

the average Cp value of the lower—inner cylinder is relatively low, and the average Cp values of the outer cylinders are nearly equal. A fourth mode can be observed that is similar to the third mode except that the relative high and low average Cp values of the two inner cylinders are interchanged. For 0.750Os/dO1.202, only two of these modes are observed.

1 Introduction The flow downstream of a plane array of cylinders placed normal to the flow has many practical engineering applications. They include flows in heat exchanger tube banks and flows around closely spaced electrical power poles. For infinitely long cylinders, the important geometrical parameters associated with the flow structure downstream of the cylinders are the spacing between the top and bottom surfaces, s, of adjacent cylinders and the diameter of the cylinder, d. The flow pattern about a cylinder is characterized, in part, by the magnitude of the base pressure coefficient (Cp\(P[PR)/1/2oUR2), which is always negative (hereafter, for simplicity, only the magnitude of Cp is used when comparing the flow patterns of different cylinder array geometries). Hence, the phrase ‘‘higher Cp’’ refers to a more negative Cp magnitude while the phrase ‘‘lower Cp’’ refers to a less negative Cp magnitude. For a two-cylinder array, when the cylinder spacing to diameter ratio (s/d) is set to equal or greater than four, the near wake of each cylinder in the array is similar to that found downstream of an independent cylinder. Far downstream, the individual wakes amalgamate to form a single wake similar to that of a single cylinder (cf. Bearman and Wadcock 1973). Conversely, as the cylinder spacing is reduced to nearly zero (s/d+0) a single wake will be observed after a very short downstream distance. Previous studies of the transition from multiple independent wakes to a single wake have shown that, within a critical spacing ratio (0.1Os/dO1.3), the average Cp of each cylinder varies in time and takes on two different values. Kim and Durbin (1988) describe this as the ‘‘flopping regime.’’ They suggest that flopping occurs spontaneously when the wake behind each cylinder alternates between a wide wake with a low magnitude Cp, and a narrow wake with a high Cp. In an earlier study, Bearman and Wadcock (1973), for a two-cylinder array with the same s/d range and with a Reynolds number (Re) based on cylinder diameter of 2.5]104, also find that the average Cp of each cylinder has two different values. In their study, the Cp of a cylinder may switch between relatively high and low values spontaneously, but sometimes may require forcing by either stopping and starting

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the flow or by causing a large flow perturbation. Thus, either spontaneous flopping or a quasi-stable behavior is observed. Since both flopping and quasi-stable behavior of the average Cp values have been observed in previous studies, it is useful to define terms that describe each of the three types of behavior: quasi-stable behavior, spontaneous flopping, and forced flopping. — Quasi-stable behavior is the wake behavior observed downstream of the cylinders in which different Cp values exist behind each cylinder. The Cp values do not vary with time. A large amplitude flow perturbation can cause the average Cp values to change, but the Cp values remain at the new values until another large perturbation is applied. — Spontaneous flopping is the behavior observed downstream of the cylinders where the average Cp values are always observed to alternate over time between relatively high and low values, even when no large perturbation is applied to the flow field. — Forced flopping is the behavior observed downstream of the cylinders in which initially stable wakes exhibit flopping as a result of a large, one-time perturbation. After the initial large perturbation is applied and flopping occurs, there is no observable difference between forced and spontaneous flopping.

2 Background Bearman and Wadcock (1973) are the first to study quasistable behavior and flopping for a two-cylinder array as a function of cylinder spacing. Their specific goal is to locate the critical cylinder spacing where the wakes downstream of the two cylinders stop behaving as individual cylinder wakes and combine to form a single wake. They show that, when s/dP4.0, the cylinder wakes do not interact until after a significant distance downstream. The cylinders must nearly be touching for a single wake to be formed. In the intermediate range of spacing (0.1Os/dO1.0) and at a Reynolds number of 2.5]104, they find that either quasi-stable behavior or flopping can be induced by stopping and starting the windtunnel. Thus, they observe forced flopping, not spontaneous flopping. They further find that each mode generates peaks at different frequencies in the velocity power spectra, and therefore each mode has different and multiple Strouhal numbers. In addition, they report quasi-stable behavior in the flow downstream of two parallel flat plates for 0.1Os/dO2.0 at Re\4.4]104 where d is the thickness of the plate in the direction normal to the flow. Quasi-stable behavior is also observed by Ishigai and Nishikawa (1975) for five-cylinder arrays. They find quasistable behavior when the five cylinders are aligned in a plane normal to the flow with 0.2Os/dO1.5, over a Reynolds number range of 3.3]104—4.0]104. They do not observe forced or spontaneous flopping. They argue that the different average Cp values of the various cylinders is a result of the Coanda effect which deflects the wakes. Zdravkovich (1977) reviews both the Ishigai and Nishikawa (1975) and the Bearman and Wadcock (1973) studies. Zdravkovich states that, for forced or spontaneous flopping to be a result of the Coanda effect, flopping must be related to the movement of the separation point on the cylinders. However,

since the separation positions on plates are not affected by the Coanda effect, and Bearman and Wadcock show flopping with an array of two-flat plates, quasi-stable behavior cannot be explained by the Coanda effect. Kiya et al. (1980), with 0.4Os/dO1.0 and Re\1.58]104, find flopping when the plane of a two-cylinder array is aligned perpendicular to the flow. When the array is ten or more degrees out of alignment, no flopping is found. They do not explicitly state whether the flopping is forced or spontaneous. The first visualization of quasi-stable behavior for a twocylinder array is reported by Williamson (1985) for s/d\1.0 and 50OReO200. His streakline photographs show a relatively narrow and short wake behind one of the cylinders and a relatively wide and long wake behind the other cylinder. Because the size and shape of the wakes remain stable, this flow behavior is taken to correspond to quasi-stable behavior. Flopping behind two cylinders is also studied by Kim and Durbin (1988). For their study, s/d is varied between 0.1 and 1, and the Reynolds number is varied between 2.2]103 and 6.2]103. Only spontaneous flopping is observed. They present the first statistical analysis of the time interval for each period that the average Cp value remains relatively high combined with the time for each period that the average Cp value remains relatively low and they find that the probability density function for the time intervals has a zero event Poisson distribution. The average length of time between transitions decreases with increases in velocity. Specifically, for a Reynolds number of 200, they estimate that the average time interval between flops would be on the order of hours. As pointed out by Kim and Durbin (1988), this may explain why Williamson (1985) did not observe spontaneous flopping. Le Gal et al. (1990) perform studies using two cylinders with 0Os/dO6.5 and Re\110 and find wake interaction at separation ratios as high as 4.5. Probably because of their low Reynolds number, they state that flopping is difficult to observe. However, they do observe quasi-stable behavior and perform a flow visualization study which is in agreement with the large-wake, small-wake model of Williamson (1985). However, in their photographs, it can be seen that vortices sequentially appear on opposite sides of the cylinder center plane for the wide wake. In contrast, for the narrow wake, the vortices are found on the center plane of the cylinder from x/d\2 to 8. Farther downstream in the narrow wake, sequential vortices begin to move to opposite sides of the wake center plane. Therefore, the vortices in the wide wake should lead to a relatively low frequency fluctuation in the velocity while, for x/d\2 to 8, the vortices in the narrow wake, which are relatively closer together, should lead to a higher frequency fluctuation in the velocity. When the vortices formed at the inside surface of the wide wake interact with the vortices in the narrow wake, a fluctuation in the velocity may be created that has a higher frequency than that associated with the narrow wake. Peschard and Le Gal (1996) observe with flow visualization and numerically model the interaction of wakes downstream of a two-cylinder array with 1Os/dO6 and 90OReO150. Although they successfully model quasi-stable behavior, no spontaneous or forced flopping is observed. Eastop and Turner (1982) observe the flow behavior downstream of a three-cylinder array with 1.2Os/dO2.6 and 4.5]104OReO1.11]105 placed normal and parallel to the

freestream flow. Pressure distributions about the cylinders and Strouhal numbers downstream of the center cylinder are presented. They only observe quasi-stable behavior when s/d\1.375. No spontaneous nor forced flopping was observed. Zdravkovich and Stonebanks (1990) observed quasi-stable and forced flopping with seven- and eleven-cylinder arrays with 1.1Os/dO1.75 and 3.7]104OReO7.4]104. The forced flopping they produced by increasing the velocity rapidly from rest stops after 10—20 min. Pressure distributions and velocity power spectra are presented. Twenty-one cylinders in a planer array with 0.5Os/dO2.0 and Re\100 are used by Le Gal et al. (1996) in a study to determine how large numbers of cylinders interact. Along the array of cylinders, different average Cp values randomly occur and are stable until a significant velocity perturbation is applied, i.e., quasi-stable behavior is observed. Morretti and Cheng (1987) present flow visualization images and velocity measurements of the flowfield downstream of an array with an unspecified number of tubes with s/d\1.3 and Re\2500. Mizushima and Takemoto (1996) numerically solve and experimentally visualize the flow pattern downstream or an array of square bars. For 1.2Os/dO2.6 (d in this case is the length of a side) and 80OReO320, flow visualization shows that at specific Reynolds number and s/d combinations, both spontaneous flopping and quasi-stable behavior can exist downstream of the array. In summary, quasi-stable behavior, and forced and spontaneous flopping have been observed for two-cylinder arrays, and the effects of spacing, velocity, and angular alignment on flopping has been studied. It has also been suggested that only quasi-stable behavior occurs for cylinder arrays with three or more cylinders. However, few studies of the effect of spacing on flopping and quasi-stable behavior have been presented for three- and four-cylinder arrays. Thus, the study herein first explores two-cylinder arrays for comparison with results from previous studies to validate the test apparatus and experimental protocol. Next, the study investigates the types of modes and flopping that exist for different spacings for planer three- and four-cylinder arrays placed normal to the flow.

3 Facilities and approach The large UCI closed return wind tunnel is used for all the experiments. The test section has a length of 6.71 m, a crosssection of 61 by 91 cm, and is preceded by a contraction section with an area reduction from 5.15 to 0.55 m2 (a contraction ratio of 9.36). For the velocity range of 3—24 m/s, the mean velocity in the central portion of the test section (outside the wall boundary layer) is found to be constant to within 1%, and the freestream turbulence intensities at both the entrance and exit of the test section for a centerline velocity of 10 m/s are, respectively, 0.17 and 0.22%. The temperature in the wind tunnel is found to be within ^1.0°C for each data collection period. The test cylinders used in this study are 1.27 cm in diameter, 0.30 m in length, and are mounted at each end to identical 1.27 cm diameter, 0.62 m high rods that are attached to chemistry stands. The vertical rods are placed 0.29 m apart and mounted so that the rear of the test cylinder touches the

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Fig. 1. Schematic illustration of forced or spontaneous flopping. The other mode is the opposite in that the wide wake is on the bottom and the narrow wake is on the top

vertical rod. Each cylinder array is centered between the floor and the ceiling of the tunnel. The base of each stand has an approximate width of 27 cm, a length of 16 cm, and a height of 2.54 cm. The leading edge of the base is directly below the center line of the cylinders. The downstream coordinate, x, has its origin in the plane containing the cylinder axis (Fig. 1). A 0.01 mm diameter pressure port is located at the rear of each cylinder at the center plane of the array. The port intercepts a 0.48 cm diameter passage that is drilled along the centerline of the cylinder. The passage is blocked on one end of the cylinder and connected to a nipple on the other. This nipple is connected to a Setra (Model 339-1) differential pressure transducer by means of a 2.5 m long silicone tube with an inner diameter of 1.8 mm. The output of the Setra transducer is connected to a filter, to a Computer Boards Inc. (CBI) SSH-16 sample and hold board and a CBI CIO-AD16F 12 bit analogue to digital converter which are controlled by a PC clone. The pressure signal is sampled at 10 samples/s for a three-hour time period and filtered at 5 Hz. Flow visualization, using smoke which is illuminated with a laser light sheet, is used to obtain flow images. The nominal 0.8 mm thick laser sheet is produced by passing a 2 W argon-ion beam through a cylindrical lens. The smoke is produced using mineral and is injected at (1), the leading edge stagnation point, (2) 45° above the leading edge stagnation point and (3), 45° below the leading edge stagnation point on each cylinder. Images are collected with a Sony video camera (Model CCD-V101) at a shutter speed of 1/10000 s. The images are digitized with ‘‘frame grabbing’’ hardware and software (Snappy). The use of a video camera has two advantages: first, since the allowable shutter speed is much faster than traditional cameras, a continuous light source can be used; second, the images can be viewed immediately. The single hot-wire sensor, that is used to obtain the frequency spectra, is made by soldering a 2 mm long, 0.00508 mm diameter, Wollaston wire to a TSI-1210 sensor holder. The sensor holder is connected to a traverse that is located 45 cm downstream of the hot-wire sensor and has a resolution of 0.1 mm in the vertical direction. The hot-wire sensor is connected to a TSI Model 1050 constant temperature

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anemometer and then to the same acquisition system used to obtain the pressure data. The data are collected at a sample rate of 2000 samples/s and filtered at 1000 Hz for a time period of 5 mins. The sensor is positioned in the flow so that the hot-wire is parallel to the horizontal plane and perpendicular to the air flow. Three methods are used to apply external perturbations to the flow field. The first is to simply turn off and on the windtunnel fan. The second is to open and close a door in the test section of the windtunnel near the cylinder array. The third, and most effective means to perturb the flow, is to block the flow slightly upstream from the cylinders with a 25 cm wide and 15 cm long plate and then quickly remove it.

4 Results First, flopping is discussed for a two-cylinder array. Second, the flow characteristics of a three-cylinder array are presented, and lastly, the flow characteristics of a four-cylinder array are presented.

Fig. 2a, b. Base pressure coefficient, Cp, as a function of time for a two-cylinder array with U\6.8 m/s and s/d\0.75. Figure 1a corresponds to the top cylinder and Fig. 1b corresponds to the bottom cylinder

4.1 Two-cylinder array For Reynolds numbers greater than 2.2]103, Bearman and Wadcock (1973) observe forced flopping while Kim and Durbin (1988) observe spontaneous flopping. Since no clear explanation has been offered for these differences in behavior, the effect of cylinder supports and surrounding windtunnel geometry on flopping of two cylinder arrays is investigated for s/d\0.75 and Re\4.4]103. One type of support used consists of square edged plates that are 1.27 cm wide and extend 2.54 cm upstream of the centerline of the cylinders. Neither spontaneous nor forced flopping are observed with this support. When the square edged plates are replaced by rods, as described in the Facilities and Approach section, spontaneous flopping always occurs. Hence, end effects associated with the cylinder supports can prevent flopping. Further, when the bases of the chemistry stands are mounted to a 57 cm]39 cm plate that is 2 cm thick and placed on the windtunnel floor, spontaneous flopping never occurs but forced flopping does. When this plate is removed and the chemistry stands are placed directly on the windtunnel floor, spontaneous flopping is again observed. Lastly, when the traverse that holds the hot-wire is located 30 cm downstream of the cylinder array, spontaneous flopping occurs but the time duration of the transitions between the two Cp values becomes significantly longer than when the traverse is removed or placed more than 45 cm downstream of the cylinder array. There is no observable change in flopping behavior when the traverse is placed 45 cm downstream of the array or when the traverse is removed from the windtunnel. Thus, flopping behavior is sensitive to cylinder end supports, local area changes in the windtunnel geometry, and downstream flow field interference. With s/d\0.75 and Re\4.4]103, the pressure behind each cylinder is measured as a function of time, and is shown in Fig. 2. Each time the Cp value behind one of the cylinders rises, the Cp value behind the other cylinder falls. The flow visualization of Fig. 3a shows a wide wake downstream of the upper cylinder, which corresponds to a low Cp value, and a narrow wake downstream of the lower cylinder,

Fig. 3a, b. Smoke visualization showing spontaneous flopping of the wakes downstream of the two-cylinder array with U\5.2 m/s and s/d\0.75. a Wide wake downstream of the upper cylinder and a narrow wake downstream of the lower cylinder; b Shows the opposite

which corresponds to a high Cp value. Fig. 3b shows that after flopping has occurred the narrow wake is downstream of the upper cylinder and the wide wake is downstream of the lower cylinder. Thus, the change in Cp corresponds to a change in the

Fig. 4. Power spectra of an uncalibrated hotwire signal, E , for a two-cylinder array hw measured in the plane at s/2 above the upper cylinder with s/d\0.75. The solid line corresponds to U\5.2 m/s and the dotted line corresponds to U\10 m/s. Figures 2a, b, and c show the power spectra, respectively, at x/d\2, 3, and 4

wake size downstream of the cylinders. Since the two-cylinder results are similar to those observed by Kim and Durban (1988), the apparatus and procedure used in this current study is valid. Power spectra obtained using the signal from an uncalibrated hot-wire which is placed at x/d\2, 3, and 4 in the horizontal plane located at s/2 above the top cylinder, and in the horizontal plane midway between the cylinders as shown in Fig. 4. Above the cylinder and at x/d\2, a dominant peak is observed in the power spectra at 145 Hz at x/d\2 for U\5.2 m/s and at 271 Hz for U\10 m/s. At x/d\3 and U\5.2, along with the peak at 145 Hz, a 45 Hz peak in the power spectra is noticeable. Similarly, at x/d\3 and with U\10 m/s, along with the peak at 271, a 90 peak in the power spectra is observed. The 45 and 90 Hz peak become even stronger at x/d\4. Bearman and Wadcock (1973), at s/d\0.75, observe that frequencies of the vortices shed in the narrow and wide wakes

correspond, respectively, to Strouhal numbers of 0.32 and 0.13. Thus, since the 145 and 271 Hz peaks correspond, respectively, to Strouhal numbers of 0.32 and 0.31, they are attributed to the narrow wake vortices and since the 45 and 90 Hz peak correspond, respectively, to Strouhal numbers of 0.11 and 0.10, they are attributed to the wide wake vortices. Figure 5 shows the power spectra of the hot-wire measured on the midplane between the cylinders at x/d\2, 3 and 4. Similar to the observations at s/2 above the top cylinder at x/d\2 with U\5.2 and 10 m/s, corresponding peaks in the power spectra are observed respectively at 45 and 145 Hz and at 90 and 271 Hz. At x/d\3 with U\5.2 and 10 m/s, the corresponding peaks at 145 and 271 Hz are nearly negligible, and the peaks at 45 and 90 Hz are relatively larger. At x/d\4, with U\5.2 and 10 m/s, the corresponding peaks at 45 and 90 Hz have grown even larger and there are no observable peaks in the power spectra at 145 and 271 Hz. Therefore, on the midplane at x/dP3, only the vortices from the wide wake

Fig. 5. Power spectra of an uncalibrated hotwire signal, E , for a two-cylinder array hw measured on the centerplane between the cylinders with s/d\0.75. The solid line corresponds to U\5.2 m/s and the dotted line corresponds to 10 m/s. Figures 3a, b, and c show the power spectra, respectively, at x/d\2, 3, and 4

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a two-event Poisson distribution would be appropriate if either of two events may occur (e.g. change in base pressure coefficient from one value to one of two others). As shown in Fig. 6a, the measured probability density function for the time interval between Cp transitions is well described by the zeroevent Poisson distribution. Hence, consistent with the observation of Kim and Durban (1988), the duration of time intervals at both high and low Cp values follow a zero-event Poisson distribution. Figure 6b shows the pdf of the time intervals between one transition from low to high Cp value and the subsequent transition from low to high Cp value, which is a one-event process. The solid line represents the one-event Poisson distribution, P(t/T )+(t/T )e~t/Ti, where T is the flopping i i i period and is measured as 24.4 s with a 17.3 s rms. Clearly, the measured pdf and one-event Poisson distribution are dissimilar. Hence, flopping for a two-cylinder array is not a Poisson process.

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4.2 Three-cylinder array

Fig. 6. a Probability density function for the time duration of highaverage and low-average Cp values obtained for the spontaneous flopping of a two-cylinder array with U\5.2 m/s and s/d\0.75. The solid line corresponds to the zero event Poisson distribution (P(t/T)\e~t/T); j corresponds to the measured pdf for time durations at high average Cp values; and, (s) corresponds to the low average Cp values; b Probability density function for the time intervals between one transition from low to high Cp and the subsequent transition from low to high Cp for the spontaneous flopping of a two-cylinder array with U\5.2 m/s and s/d\0.75. The solid line corresponds to the one event Poisson distribution (P(t/T)\(t/T)e~t/T); and, (j) corresponds to the measured pdf of the time interval

make a significant contribution to the power spectra of the velocity fluctuations. In contrast, in the plane at s/2 above the top cylinder at x/d\3, the effect of vortices from the wide and narrow wakes can be observed in the power spectra. In order to determine the probability density function (pdf) of the time interval between Cp transitions, a threshold is set to delineate between the high and low Cp values. The mean time interval where the Cp magnitude remains relatively high is 9.8 s with a 9.4 s rms. whereas the mean time interval where the Cp remains low is 14.7 s with a 15.1 s rms. The length of time for each period that the average Cp magnitude remains relatively high or low for each cylinder is determined and the two probability density functions are shown on Fig. 6a. The solid line represents the zero-event Poisson distribution, P(t/T) +e~t/T, where T is the mean interval between transitions. A zero-event distribution is appropriate since no changes in the average value of Cp occurs during each measured period. A one-event Poisson distribution would be appropriate if an event occurs during the measured time period (e.g. change in base pressure coefficient from one value to another). Similarly,

For the studies that involve three cylinders, the spacing between each adjacent cylinder is kept the same. Figure 7 shows the Cp values as a function of time with s/d\0.75 and Re\4.4]103 for the top and middle cylinders after a strong flow perturbation is applied. The average Cp value behind the center cylinder remains relatively constant at a higher average Cp value than the highest average Cp value found behind either of the outer cylinders. Specifically, the Cp value behind the center cylinder averages [1.4 while the average Cp values for the outer cylinders are about [1.2 and [0.9. Thus, forced flopping can occur. It does not occur between two adjacent cylinders, but rather between the outer cylinders. The flow visualization of Fig. 8a shows a wide wake downstream of the upper cylinder, which corresponds to a low Cp value, an intermediate narrow wake downstream of the lower cylinder, which corresponds to a high Cp value. Figure 8b shows that after flopping has occurred the intermediate wake is downstream of the upper cylinder and the wide wake is downstream of the lower cylinder. In both images, the most narrow wake remains downstream of the center

Fig. 7a, b. Base pressure coefficient, Cp, as a function of time for a three-cylinder array with U\6.8 m/s and s/d\0.75. Figure 5a corresponds to the top cylinder and Fig. 5b corresponds to the middle cylinder

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Fig. 9. Smoke visualization showing quasi-stable behavior of the wakes downstream of the three-cylinder array with U\5.2 m/s and s/d\0.75

Fig. 8a, b. Smoke visualization showing forced flopping of the wakes downstream of the two-cylinder array with U\5.2 m/s and s/d\0.75. a Wide wake downstream of the upper cylinder and a narrower wake downstream of the lower cylinder; b Shows the opposite

cylinder and is nearly unaffected by flopping. Thus, the change in Cp corresponds to a change in the wake size downstream of the cylinders. It should be noted that after an application of a large perturbation, forced flopping may not always occur. When forced flopping does not occur, a quasi-stable mode is observed in which the two outer cylinders have average Cp values of [1.14, and the center cylinder has an average Cp value of [0.86. The flow visualization of Fig. 9 is consistent

with the Cp measurements and shows narrow wakes downstream of the upper and lower cylinders and a wide wake downstream of the center cylinder. A summary of the average values of Cp for 0.338Os/dO1.202 with Re\2.5]103 is shown in Table 1. Specifically, for 0.338Os/dO0.730, quasi-stable behavior is found, but forced or spontaneous flopping is not observed. Three quasi-stable modes are observed in this s/d range. For the first mode, the average Cp values of the two outer cylinders are relatively equal and high, and the average Cp value of the center cylinder is relatively low. For the second, the average Cp value of the top cylinder is relatively high, the average Cp value of bottom cylinder is relatively low, and the average Cp value of center cylinder is intermediate between those of the top and bottom cylinders. For the third mode, the average Cp value of the top cylinder is relatively low, the average Cp value of bottom cylinder is relatively high, and the average Cp value of center cylinder is intermediate between those of the top and bottom cylinders. For 0.730Os/dO0.850, forced flopping can be generated between the outer two cylinders with the average Cp value of the center cylinder remaining higher than the values for either of the outer cylinders. Only one quasi-stable mode is observed. It is similar to the first mode for 0.338Os/dO0.730, in that the average Cp values of the two outer cylinders are relatively equal and high, and the average Cp value of the center cylinder is

Table 1. Modes of stability of a three-cylinder array for various spacing ranges s/d Range 0.338Os/dO0.730 0.730Os/dO0.850 0.850Os/dO1.202

Mode description (In terms of average Cp values) Outer cylinders equal Outer cylinders opposite Outer cylinders opposite Outer cylinders equal Flopping between outer cylinders Outer cylinders equal

Cp Top cylinder

Middle cylinder

Bottom cylinder

[1.06 [1.08 [0.59 [1.06 [0.83 or [1.13 [1.06

[0.77 [0.98 [0.98 [0.77 [1.37 [0.77

[1.06 [0.59 [1.06 [1.06 [1.13 or [0.83 [1.06

152 Fig. 10. Power spectra of an uncalibrated hotwire signal, E , for a three-cylinder array hw measured in the plane at s/2 above the upper cylinder with U\5.2 m/s and s/d\0.75. Figures 6a, b, and c show the power spectra, respectively, at x/d\2, 3, and 4

relatively low. No other quasi-stable mode is observed in this spacing range. For 0.850Os/d O 1.202, neither forced nor spontaneous flopping is observed. Only one mode is observed where, after any applied perturbation, the average Cp values of the two outer cylinders remains relatively equal and high, and the average Cp value of the center cylinder remains relatively low. This mode is completely stable with respect to all applied perturbations. The power spectra of the uncalibrated hot-wire are shown in Fig. 10 for the three-cylinder array with s/d\0.75 and Re\2.5]103 with x/d\2, 3, and 4 on the plane located at s/2 above the top cylinder. At x/d\2, dominant peaks in the power spectra of 400, 200 and 80 Hz are observed. At x/d\3, the 80 Hz peak is more pronounced and the 400 and 200 Hz peaks are negligible. Finally, at x/d\4, the 80 Hz peak is present and a new peak is observed at 40 Hz.

Figure 11 shows the hot-wire power spectra for the same spacing and Reynolds number, but measured in the midplane between the top and center cylinders at x/d\2, 3, and 4. At x/d\2, four peaks in the power spectra are observed at 400, 200, 80 and 40 Hz. In contrast, at x/d\2 and s/2 above the top cylinder, peaks are only observed at 400, 200, and 80 Hz. On the midplane at x/d\3, the peaks at 200 and 80 Hz are relatively large, but the peak at 40 Hz is relatively small. In contrast, at x/d\3 and s/2 above the top cylinder, only the peak at 80 Hz is large and the other peaks are nearly negligible. On the midplane at x/d\4, peaks can be observed at 200, 80 and 40 Hz. In contrast, at x/d\4 and s/2 above the top cylinder only the peaks of 80 and 40 Hz are observable. Flopping of the wakes of the three-cylinder array, at s/d\0.75 and Re\4.4]103, leads to four peaks at 400, 200, 80, and 40 Hz that correspond, respectively, to Strouhal numbers

Fig. 11a+c. Power spectra of an uncalibrated hot-wire signal, E , for a three-cylinder hw array measured on the centerplane between the upper and middle cylinder with U\5.2 m/s and s/d\0.75. Figures 7a, b, and c show the power spectra, respectively, at x/d\2, 3, and 4

of 0.98, 0.49, 0.20, and 0.10. Based on the findings of Bearman and Wadcock (1973) and the two-cylinder study herein, the 0.10 and corresponding peak at 40 Hz are associated with the wide wake. A Strouhal number of 0.20^0.01 is a typical value of a Strouhal number reported for flow past a single cylinder for Re\4.4]103 (cf. White 1991). Thus, since the 80 Hz frequency is present in all six power spectra, and its value corresponds to the Strouhal number that is similar to the wake of an independent cylinder, the peak at 80 Hz is believed to correspond to vortices in the wake of the center cylinder. Since the peak at 400 Hz is believed to be a harmonic of the peak at 200 Hz, the peak at 200 Hz must correspond to the vortices associated with the narrow wake. This frequency corresponds to a Strouhal number of 0.49, and is higher than the higher Strouhal number of 0.34 found for the two-cylinder array. Thus, for a three-cylinder array, in addition to the vortical structures associated with the wide and narrow wakes, there appears to be a another wake type which has a vortical structure similar to the wake of a single cylinder. A probability density function for the time intervals that each average Cp magnitude remains relatively high or relatively low for each of the two outer cylinders of the three-cylinder array is created using the same data collection methods and analysis technique that are used for the corresponding data from the two-cylinder array. As shown in Fig. 12a, both the time lengths of the high Cp periods and the time lengths of the low Cp periods closely follow a zero-event Poisson distribution. The mean time duration that the Cp magnitude remains relatively high is 5.4 s with a 5.8 s rms whereas the mean time duration that the Cp remains low is 11.7 s with a 11.8 s rms. Thus, for the two- and three-cylinder arrays, the rms values for both the relatively high Cp and low Cp time intervals are very close in magnitude to their respective mean time values. However, the mean time interval for the high Cp of the two-cylinder array is approximately 80% longer than the corresponding time interval of the three-cylinder array. The mean time interval for the low Cp of the two-cylinder array is approximately 25% longer than the corresponding data from the three-cylinder array. Figure 12b shows the pdf of the time intervals between sequential transitions from low to high Cp values. The solid line represents the one-event Poisson distribution, P(t/T ) i +(t/T )e~t/Ti, where T is the flopping period and is equal to i i 11.1 s with a 7.7 s rms. Hence, flopping for a three-cylinder array is not a Poisson process. In comparison, the flopping period for the two-cylinder has an approximately 120% longer period and an 100% higher rms. Thus, the center cylinder decreases the stability of the wide or narrow wake which makes the wake more susceptible to natural perturbations in the flowfield and leads to more frequent transitions between wake sizes.

4.3 Four-cylinder array For a four-cylinder array, neither forced nor spontaneous flopping is observed. However, many different modes of quasi-stable behavior can be produced. Specifically, Fig. 13

153

Fig. 12a. Probability density function for the time duration of high-average and low-average Cp values obtained for the forced flopping of a three-cylinder array with U\5.2 m/s and s/d\0.75. The solid line corresponds to the Poisson distribution (P(t/T)\e~t/T); j corresponds to the measured pdf for time durations at high average Cp values; and, (s) corresponds to the low average Cp values; b Probability density function for the time intervals between one transition from low to high Cp and the subsequent transition from low to high Cp for the forced flopping of a three-cylinder array with U\5.2 m/s and s/d\0.75. The solid line corresponds to the one event Poisson distribution (P(t/T)\(t/T)e~t/T); and, (j) corresponds to the measured pdf of the time interval

shows the Cp values as a function of time with s/d\0.75 and Re\2.5]103 for all four cylinders in the array. When a first perturbation is applied at t\5.3 s, the average Cp value behind the upper—inner cylinder changes from approximately [0.6—[1.0, the average Cp value behind the bottom cylinder changes from approximately [1.0—[0.6, and the average Cp value behind the top and bottom lower—inner remain unchanged. When a second perturbation is applied at t\6.5 s, the average Cp value behind the upper—inner cylinder changes from approximately [1.0—[0.6, the average Cp value behind the bottom cylinder changes from approximately [0.6[1.0, and the average Cp value behind the top and bottom lower—inner remain unchanged. Finally, when a third perturbation is applied at t\8.3 s, the average Cp value behind the top cylinder changes from approximately [1.0—[0.5, the average Cp value behind the upper-inner cylinder changes from approximately [0.6—[1.0, and the average Cp value behind the lower—inner and bottom remain unchanged. The flow visualization of Fig. 14a shows a wide wake downstream of the upper—inner cylinder, a narrow wake

154

Fig. 13a+d. Base pressure coefficient, Cp, as a function of time for a four-cylinder array with U\6.8 m/s and s/d\0.75. a Corresponds to the top cylinder; b Corresponds to the upper-inner cylinder; c Corresponds to the lower-inner cylinder; d Corresponds to the bottom cylinder

Fig. 14a, b. Smoke visualization showing quasi-stable behavior of the wakes downstream of the four-cylinder array with U\5.2 m/s and s/d\0.75. a Wide wake downstream of the upper-inner cylinder and a narrower wake downstream of the lower-inner cylinder; b Shows the opposite

downstream of the lower—inner cylinder, and intermediate wakes downstream of both outer cylinders. Consistent with the time series of pressure coefficient shown in Fig. 13, a new mode can occur after a large perturbation is applied. The flow visualization of Fig. 14b shows a narrow wake downstream of the upper—inner cylinder, a wide wake downstream of the lower—inner cylinder, and intermediate wakes downstream of both outer cylinders. The flow visualization of Figs. 15a and b show examples of two other modes of quasi-stable behavior observed downstream of the four-cylinder array. In Fig. 15a, wide wakes are observed downstream of the top and bottom cylinders and narrow wakes are observed downstream of the upper—inner and lower—inner cylinders. In Fig. 15b, a wide wake is observed downstream of the top cylinder while narrow wakes are observed downstream of the upper—inner, lower—inner and bottom cylinders. Table 2 presents a summary of the Cp values and modes for 0.338Os/dO1.202 with Re\2.5]103 for a four-cylinder array where the spacing between each of the cylinders is the same. Quasi-stable behavior can be achieved within a critical s/d spacing but flopping is never observed. Specifically, when 0.338Os/dO0.750, four quasi-stable modes are observed. First, a mode can be observed in which the average Cp value of the top cylinder is relatively high, the average Cp value of the bottom cylinder is relatively low, and the average Cp values of the two center cylinders are nearly equal with values intermediate between those of the outer cylinders. A second mode is sometimes observed that is similar to the first except that the relatively high and low average Cp values of the outer cylinders are interchanged. Third, a mode is observed in which the average Cp value of the upper—inner cylinder is relatively high, the average Cp value of the lower—inner cylinder is relatively low, and the average Cp values of the outer cylinders are equal with values intermediate between those of the two inner cylinders. A fourth mode can be observed that is similar to the third mode except that the relative high and low average Cp values of the two inner cylinders are interchanged.

with 0.338Os/dO0.750 except that the relative high and low average Cp values of the two inner cylinders are interchanged. Neither spontaneous nor forced flopping is observed.

5 Summary

Fig. 15a, b. Smoke visualization showing quasi-stable behavior of the wakes downstream of the four-cylinder array with U\5.2 m/s and s/d\0.75. a Wide wakes downstream of the top and bottom cylinders and a narrower wakes downstream of the upper—inner and lower—inner cylinders; b Wide wake downstream of the upper cylinder and narrow wakes downstream of the remaining three cylinders

With the four-cylinder array and 0.750Os/dO1.202, only two modes are observed. First, similar to the third mode with 0.338Os/dO0.750, a mode is observed where the average Cp value of the upper—inner cylinder is relatively high, the average Cp value of the lower—inner cylinder is relatively low, and the average Cp values of the outer cylinders are nearly equal with values intermediate between those of the two inner cylinders. A second mode that is observed is similar to the second mode

This study confirms that when two cylinders are placed normal to the flow in close proximity, the pressure fields generated around them interact and cause the average magnitude of the Cp values to behave differently than for a single cylinder. Depending on the spacing between cylinders, this interaction results in flow field behavior that is either quasi-stable or flopping. For the two-cylinder array with s/d\0.750 and Re\2.5]103, peaks in the power spectra of 145 and 45 Hz which correspond to Strouhal numbers of 0.35 and 0.11 have been observed. The values are similar to those observed by Bearman and Wadcock (1973). The relative strengths of these peaks in the power spectra are dependent on the measurement location. For example, the peak at 45 Hz is largest when measured on the array midplane at x/d\3, while the peak at 145 Hz is largest when measured s/2 above the upper cylinder at x/d\4. While a pdf of the time duration that each average Cp value remains relatively high or relatively low closely matches a zero-event Poisson distribution, the pdf of the flopping period does not match the one-event Poisson distribution. Thus, flopping for a two-cylinder array is not a Poisson process. For the three-cylinder array with Re\2.5]103 and 0.338\s/d\0.730, three quasi-stable modes are observed. At 0.730Os/dO0.850, forced flopping and one quasi-stable mode are observed. At 0.85Os/dO1.202, only one stable mode is observed. The hot-wire power spectra measured downstream of the cylinders midway between the top and center cylinders, have three frequency peaks at 200, 80, and 40 Hz. The 200 and 40 Hz peaks correspond, respectively, to the narrow and wide wakes. The 80 Hz peak, which corresponds to a Strouhal number of 0.20, is caused by a wake that has a vortical structure similar to that of an independent cylinder (cf. White 1991). The strength of these peaks is dependent on the measurement location. For example, the peak at 40 Hz can only be detected at x/d\4 on the plane of s/2 above the upper cylinder. The 200 Hz peak, however, is only present in power spectra obtained on the midplane between the top and center cylinders. While a pdf of the time duration that each average Cp value, of the outer cylinders, remains relatively high or relatively low closely matches a zero-event Poisson distribution, the pdf of the flopping period does not match the one

Table 2. Modes of stability of a four-cylinder array for various spacing ranges s/d Range

Modes description (In terms of average Cp values)

0.338Os/dO0.750 Outer cylinders opposite Outer cylinders opposite Middle cylinders opposite Middle cylinders opposite 0.750Os/dO1.202 Middle cylinders opposite Middle cylinders opposite

Cp Top cylinder

Upper middle cylinder

Lower middle cylinder

Bottom cylinder

[0.53 [1.10 [0.95 [0.95 [0.99 [0.98

[1.01 [1.01 [0.62 [1.09 [1.22 [0.71

[1.01 [1.01 [1.09 [0.62 [0.77 [1.22

[1.09 [0.53 [0.95 [0.95 [0.98 [0.99

155

event Poisson distribution. Thus, flopping for a three-cylinder array is also not a Poisson process. For the four-cylinder array, with Re\2.5]103 and 0.338Os/dO0.750, four quasi-stable modes are observed. At 0.750Os/dO0.750 only two quasi-stable modes are observed. Neither spontaneous nor forced flopping is observed with the four-cylinder array. This quasi-stable behavior is similar to that found for arrays of 5 and 21 cylinders.

156

6 Conclusion The flow downstream of three- and four- cylinder arrays placed normal to the flow differs from that found downstream of a twocylinder array. For three-cylinder arrays, either forced flopping or multi-mode behavior occurs. In contrast to the two wake structures found for two-cylinder arrays, three-cylinder arrays have an additional downstream wake structure that behaves similarly to that of an independent cylinder. For four-cylinder arrays, only multi-mode behavior is observed.

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