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An experimental study on the cross-flow vibration of a flexible cylinder in cylinder arrays
Experimental Thermal and Fluid Science 29 (2005) 523–536 www.elsevier.com/locate/etfs
An experimental study on the cross-flow vibration of a flexible cylinder in cylinder arrays Tsun-kuo Lin, Ming-huei Yu
*
Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung 804, Taiwan, ROC Received 31 October 2003; accepted 26 June 2004
Abstract In the experiment, a monitored cylinder equipped with two accelerometers inside was flexibly mounted in a water tunnel, surrounded by one to six identical cylinders elastically mounted in rotated triangular pattern. The amplitude diagrams, spectra and orbits of the cylinder motion are used to examine the vibration behavior of the cylinder under the various test conditions of the free stream velocity, the number of the surrounding cylinders, and the cylinderÕs natural frequency. In the case of the monitored cylinder having the same natural frequency as the surrounding cylinders (22 Hz), amplitude response shows that fluid elastic instability occurs when the flow velocity is above a critical value for the cylinder in all the six-cylinder arrays. Above the critical velocity, the cylinder vibrates around an oval orbit with line-dominated spectrum, implying that the cylinder behaves like an oscillator with the streamwise and cross-stream responses have the same frequency but with a phase shift. By comparison of amplitude diagrams of the cylinder in the six bundles, it reveals that the upstream cylinders have significant influence on the amplitude response of the monitored cylinder–promote the fluid elastic instability of the monitored cylinder, and enhance the cylinder vibration above the critical velocity. The downstream cylinders could suppress the vibration amplitude while the number of the downstream cylinders has little effect on the amplitude response. In case of the monitored cylinder having different natural frequency from that of the surrounding cylinders, it is found that the difference in natural frequency of the cylinders has little effect on the critical velocity, but strong influence on the vibration amplitude above the critical velocity. 2004 Elsevier Inc. All rights reserved. Keywords: Flow-induced vibration; Circular cylinder; Cross flow
1. Introduction Bundles of circular cylinders are commonly used in engineering applications as in power transmission lines, pipelines in deep water, suspension bridges, heat exchangers, etc. Their potential vibrations induced by fluid flow have been extensively studied for the last three decades. Blevins [1] summarized the most important results of the flow-induced vibrations of circular cylinders. Later, researches on the flow-induced vibration of two *
Corresponding author. Tel.: +886 7 5252000x4238; fax: +886 7 5254299. E-mail address: [email protected] (M.-h. Yu). 0894-1777/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2004.06.004
identical circular cylinders was continued because of its fundamental importance, by Matsumoto et al. [2], and Dielen and Ruscheweyh [3], for instances. Meanwhile, many researchers, Pettigrew and Taylor [4] and Schroeder and Gelbe [5], among others, focused on the fluid elastic instability of cylinder rows and arrays because of their practical applications. While the flow-induced vibrations of two cylinders and cylinder arrays are the main focuses in the previous studies, the flow-induced vibrations from two cylinders to cylinder arrays are systematically investigated in the study, an effort to bridge the gap between the two cases in aspect of flow-induced vibration. For cylinder bundles in cross flow, fluid elastic instability is an excitation mechanism for flow-induced
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Nomenclature Arms Ax Ay c D E F fn fs m P Re
r.m.s. value of tube vibration amplitude (m) tube vibration amplitude in the streamwise (X) direction (m) tube vibration amplitude in the lateral (Y) direction (m) damping coefficient outer tube diameter (m) voltage output from the measurement system (Volt) force per unit length (N/m) the natural frequency of the monitored cylinder (Hz) the natural frequency of the surrounding cylinders (Hz) mass per unit length (kg/m) pitch (m) Reynolds number
vibrations, resulting in large-amplitude vibrations and even failure of the cylinder bundles. Fluid elastic instability, which may occur in multiple circular cylinders, is a result of the interaction between the cylinders and the flowing fluid. In cylinder bundles, the interactions of the cylinders with the fluid, and the coupling among the cylinders through the fluid are very complex. However, it is reasonable to suppose that the unsteady fluid dynamic forces on a cylinder are mainly induced by vibrations of the cylinder itself and its neighboring cylinders [6,7]. With this assumption, the vibration behavior of a cylinder in cylinder bundle should be most affected by its surrounding cylinders. For instance, a cylinder in twin-cylinder bundle may have different vibration behavior from the cylinder surrounded by more cylinders under otherwise the same condition. The paper is aimed to describe the flow-induced vibrations of a flexibly mounted cylinder surrounded by various numbers of cylinders in cross flow. The emphases of the study are the effects of the number of surrounding cylinders, and the natural frequency of the cylinder on the vibration behavior of the cylinder. For these purposes, the amplitudes, orbits, and spectra of the cylinder vibration are obtained by simultaneous measurements of the cylinder vibration in the streamwise (X) and lateral (Y) directions.
2. Experimental aspects 2.1. Water tunnel and instrumentation A low speed water tunnel was designed and built for this study, as shown schematically in Fig. 1. The length
St Ucrit Uin x xrms X, Y y yrms
Strouhal number critical velocity (m/s) inlet velocity (m/s) displacement in the streamwise (X) direction (m) the r.m.s. value of the cylinder displacement in the X-direction (m) rectangular Cartesian coordinates displacement in the lateral (Y) direction (m) the r.m.s. value of the cylinder displacement in the Y-direction (m)
Greek symbols f damping factor (dimensionless) l dynamic viscosity (kg/m s) m kinematic viscosity (m2/s) q density (kg/m3)
of the contraction section is approximately 0.6 m and the contraction ratio is 9:1. The test section, made up of bolted acrylic plates, has internal dimensions of 0.2 m · 0.2 m. A tank with 0.76 m3 capacity serves as an accumulator placed after the test section. The temperature of the water flow was at 20 C. In operating condition, water flow was driven by a variable speed pump. After passing through a filter, the water entered a diffuser, which provides a gradual transition from 10 cm diameter circular pipe to a rectangular channel of 0.6 m · 0.6 m. Before the 9:1 contraction, a layer of honeycomb and several screens were installed in the rectangular channel to reduce turbulence. At the entrance of the test section, a pitot tube was used to monitor the inlet flow velocity. The average water velocity in the test section of the facility was up to 0.74 m/s. The Reynolds number, based on the inlet flow velocity and the diameter of test cylinders is 1.1 · 104. The average upstream turbulence intensity over the flow range is about 1.2% measured by hot-film anemometry. The velocity distribution at the test section, excluding the boundary layer, is found to be flat within 1%. In the test section, stainless steel tubes were used to form cylinder bundles for testing. Six cylinder bundles were tested in the experiment, they all being in a 60 equilateral triangle pattern. As shown in Fig. 2(a), each bundle consists of a monitored cylinder (bold circles in the figure), equipped with two accelerators inside, and one to six surrounding cylinders. The cylinder bundles are named as Array I–VI, respectively, according to the number of the surrounding cylinders. The diameter of the entire cylinders is 16 mm, and the ratio of the cylinder center-to-center spacing to cylinder diameter P/D is 1.33. Fig. 2(b) is a side view of the test section to show
T.-k. Lin, M.-h. Yu / Experimental Thermal and Fluid Science 29 (2005) 523–536
Honeycomb Screen
525
Pitot tube
Filter
Tank Test section
Flow
10 h.p. motor
Regulator
Speed control unit
220 V a.c. supply
Fig. 1. Water tunnel facility.
how to mount a cylinder in the test section. Each stainless tube in the test array is suspended by two tensional steel wires at both ends. The tension in the steel wire can be adjusted to change the natural frequency of the cylinder. As shown in the figure, a stainless cylinder 198 mm long, 16 mm outside diameter and 1.5 mm wall thickness is mounted on a 1.2 mm diameter steel wire. The weight of each cylinder is 153 g within 1% deviation. The free length of the 1.2 mm diameter wire is 180 mm and the screw-nut adjustments at both ends of the wire provide fine tension adjustment for tuning the natural frequency. In the experiment, the natural frequencies of the cylinders surrounding the monitored cylinder were all fixed at fs = 22 Hz, and the natural frequency of the monitored cylinder, fn, was set at 22 Hz, 16.5 Hz (=3/4fs), 11 Hz (=1/2fs), and 27.5 Hz (=5/4fs) for tests with different natural frequencies. The instrumentation includes two accelerometers manufactured by Endevco (Model #25A) with sensitivity 4.693 mV/g and frequency response of 2–8 kHz. The two accelerometers were installed inside the tube at midspan. Their orientations were adjusted such that they were sensitive to tube vibrations only in the streamwise (X) and lateral (Y) directions, respectively. The output of each accelerometers was amplified by a B&K charge amplifier (Model #2693) and coupled to B&K double integrator circuit (Model #0788), as shown in Fig. 3. The double integrator circuit, consisting of an input filter and two operational amplifier integrators, was used to integrate an acceleration signal twice and provide an output signal proportional to displacement. The Xand Y displacement signals obtained from the double integrators were digitized simultaneously at a sampling rate of 512 data/s by a 16 bit IOtech IEEE 488 A/D converter (ADC488/8SA) and transmitted to a PC compu-
ter for data storage and further processing. An FFT analyzer, AND AD3524, was connected to the double integrators to monitor the amplitude spectra of the cylinder during the experiments. The voltage outputs from the data acquisition system were calibrated against the vibration amplitudes before vibration measurements. In calibration, the tube with the two accelerometers inside was given a circular motion. Typical time series of displacements in the X and Y direction are shown in Fig. 4(a). The amplitudes of the sinusoidal output voltages for the X- and Y-directions were recorded and then plotted against the vibration amplitude (the radius of the circular motion). As shown in Fig. 4(b), the calibration data for both the X- and Y-directions are fitted to a straight line E = aA + b, where A is the vibration amplitude and E is the voltage output. Curve-fitting gives the values of the coefficients, a = 0.10 and b = 0.007. It is thus shown that the voltage outputs can linearly represent the vibration amplitudes in the experimental range. 2.2. Uncertainty To characterize the magnitude of cylinder vibration, the r.m.s. value of vibration amplitude Arms is introduced in dimensionless form as, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Arms =D ¼ x2rms þ y 2rms =D ð1Þ where D is the diameter of the cylinder. xrms and yrms are the r.m.s. values of the cylinder displacements in the X- and Y-directions, respectively. The displacements were obtained by digitizing the output voltages from the measurement system that transforms the displacements into voltage signals. The
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The velocity profile in the center region of the test section, exclusive of the boundary layer region, is uniform within 5% variation at the maximum freestream velocity Uin = 0.74 m/s, and the turbulence intensity is 1.3%, measured by a Dantec hot-film anemometer. The tolerance in measuring the natural frequency is ±0.2 Hz in the range of 9–31 Hz. The natural frequency and damping factor of a cylinder were determined by measuring the amplitude response of the cylinder after given a perturbation in otherwise still water, as shown in Fig. 5 for a typical amplitude response in time domain. The damping factor can be estimated from the decay of the amplitude, and the natural frequency of the cylinder can be estimated by spectral analysis of the time trace. For each natural frequency, the average value from several measurements is adopted, and the uncertainty can be estimated by evaluating the standard deviation. Perturbations of various directions were applied in the measurements to make sure that the natural frequency is independent of cylinder vibration orientation in the experimental setup. For spectral analysis of the vibrations of the cylinder in cross flow, the software MATLAB was used to obtain spectra data. Additionally, a spectrum analyzer AND AD3524, with a setting of frequency resolution Df = 0.125 Hz, was connected to the voltage output to monitor the cylinder vibration during the experiments. The uncertainty of mass damping can be estimated by using MoffatÕs [8] uncertainty analysis. The uncertainties of the experimental parameters are summarized in Table 1.
3. Results and discussion 3.1. Vibration amplitude
Fig. 2. (a) Top view of the test cylinders. (b) Side view of the cylinders in the test section.
uncertainties of the displacement measurements could be due to the accuracy of the vibration measurement system, and of the data acquisition system. The resolution of the data acquisition used to digitize the voltage from the double integrator circuit is 4 · 104 V in a range of 5 V. The tolerance of the measured amplitude due to the measurement system is estimated to be 0.3 mm, based on the standard deviation of the calibration data from the fitting curve, shown in Fig. 4(b).
For the effect of surrounding cylinders on the vibration of the monitored cylinder, vibration measurements of the monitored cylinder were carried out with the surrounding cylinder added one by one, while the natural frequency of the entire cylinders was set at the same (22 Hz). Figs. 6(a–f) show the amplitude responses of the monitored cylinder in the six-cylinder arrays, respectively. In each of the figures, there are two curves representing the vibration amplitudes in the X- and Ydirections. It is seen that all the amplitude curves have similar trend as the flow velocity is increased. The vibration amplitudes are very small, slightly increasing with the flow velocity, as the velocity is below a critical value – Uin 0.5 m/s for all the test arrays except Uin 0.6 m/s for the two-cylinder array. These small-amplitude vibrations could be attributable to turbulence, considering that the monitored was in the wake region behind the upstream cylinders. Above the critical velocity, the vibration amplitudes increase significantly as the flow
T.-k. Lin, M.-h. Yu / Experimental Thermal and Fluid Science 29 (2005) 523–536
FFT Analyzer
Computer
Amplifier and Double Integrator Circuit
Data Acquisition System
X direction Acceleration Meter
527
Y direction Acceleration Meter Fig. 3. The data acquisition system for 2-channel vibration measurements.
2
20
Ax
10
1
-10
y/D (%)
Displacement(mm)
0
-20
0
20
Ay 10
-1
0 -10
-2 0
-20
1
2
3
4
5
t(sec) 0.0
0.5
1.5
2.0
2.5
Fig. 5. A typical time trace of the cylinder vibration after perturbation in otherwise still water.
Ey Ex
3.5
Ex,Ey(Volt)
1.0
Time(sec)
(a)
3.0 2.5 2.0 1.5 15
(b)
20
25
30
35
40
Vibration amplitude (mm)
Fig. 4. (a) Typical time series of displacements in the X and Y direction, (b) the calibration of the output voltages from the data acquisition system versus the vibration amplitudes, Ex for the X-direction, and Ey for the Y direction.
velocity is increased, with the amplitude in the Y-direction being larger than that in the X-direction for each
test case. The continuous increase of vibration amplitude with flow velocity implies that fluid elastic instability has occurred. The serious vibration above the critical velocity could be explained as follows. When the flow velocity is high enough, the fluid has sufficient energy to excite the cylinders to vibrate with certain amplitude at their natural frequency. Through the coupling of the fluid surrounding the cylinders, resonance occurs among the cylinders that have the same natural frequency, and as a result the vibration amplitude was amplified and thus serious vibration was observed. The explanation of the serious vibration due to fluid elastic instability in the viewpoint of resonance is consistent with the experimental findings that the dominant frequency of the fluid flow around the cylinders is the natural frequency of the cylinders. As shown in Fig. 7 are typical time history of the flow around the cylinders and the corresponding velocity spectrum, obtained by using a Dantec hot-film anemometer
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Table 1 Uncertainties of the natural frequency, damping factor and mass damping for the test cases fn (Hz) 22 ± 0.2 16.5 ± 0.2 11 ± 0.2
(fn = fs) (fn = 3/4fs) (fn = 1/2fs)
f
m/qD2
m(2pf)/qD2
0.0096 ± 0.0003 0.0104 ± 0.0004 0.0088 ± 0.0002
3.32 ± 0.02 3.32 ± 0.02 3.32 ± 0.02
0.19 ± 0.006 0.22 ± 0.008 0.18 ± 0.004
Uin(m/s)
(a)
xrms,yrms /D(%)
xrms,yrms /D(%)
Flow
24 Flow X direction 22 Y direction 20 fn= fs 18 16 14 12 10 8 6 4 2 0 -2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
24 X direction 22 Y direction 20 fn= fs 18 16 14 12 10 8 6 4 2 0 -2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Uin(m/s)
(b)
Flow
24 X direction 22 Y direction 20 fn= fs 18 16 14 12 10 8 6 4 2 0 -2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Uin(m/s)
(c)
xrms,yrms/D(%)
xrms,yrms/D(%)
Flow
24 X direction 22 Y direction 20 fn= fs 18 16 14 12 10 8 6 4 2 0 -2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Uin(m/s)
(d)
Flow
24 X direction 22 Y direction 20 fn= fs 18 16 14 12 10 8 6 4 2 0 -2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Uin(m/s)
(e) 0
0.5
1.0
Uin /fnD
xrms,yrms/D(%)
xrms,yrms/D(%)
Flow
24 X direction 22 Y direction 20 fn= fs 18 16 14 12 10 8 6 4 2 0 -2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Uin(m/s)
(f) 1.5
2.0
0
0.5
1.0
1.5
2.0
Uin /fnD
Fig. 6. Vibration amplitudes of the monitored cylinder with fn = fs in the X-direction (s), and in the Y-direction (n). The monitored cylinder in (a) Array I, (b) Array II, (c) Array III, (d) Array IV, (e) Array V, (f) Array VI.
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0.002
E(volt)
0.001 0.000 -0.001 -0.002 -0.003 -0.004 0.0
0.2
0.4
0.6
0.81
1.0
Time(sec)
(a) 0.004
Uin=0.5 m/s
E(volt)
0.002
0.000
-0.002 0
(b)
10
20
30
40
50
60
70
80
90
100
f(Hz)
Fig. 7. (a) Typical time history of the fluid velocity around the cylinder (b) the corresponding velocity spectrum.
(StreamLine) with a hot-film probe (type 55R11). Similar spectra were obtained regardless of the inlet flow velocity once the velocity is high enough. The velocity spectra indeed exhibit only one spectral peak at fn (22 Hz), indicating that the fluid flow oscillates at the natural frequency of the cylinders. The oscillation frequency is not the vortex shedding frequency that is linearly related to flow velocity. The Strouhal number of the vortex shedding of the two-cylinder configuration, for instance, is estimated to be 0.12 (by the figure in Ref. [1, p. 176]). The vortex shedding frequency in the flow velocity range, Uin = 0.6–0.7 m/s, is then about 5 Hz, not 22 Hz. Due to the mismatch of the two frequencies, vortex-induced instability is not likely responsible for the large-amplitude vibration in the velocity range. The cylinder vibration is identified as a result of fluid elastic instability, based on the character of the vibration amplitudes that increase rapidly once a critical cross-flow velocity is exceeded, as shown in Fig. 6. It is known that the vibration amplitude of fluid elastic vibration increases very rapidly with the flow velocity once a critical cross-flow velocity is exceeded, while the amplitude response of a vortex-induced vibration is typically a hump, or multiple humps if vibrations also occur at sub and super harmonics of the shedding frequency (see Chapters 3 and 5, by Blevins [1], for instance). With the fluid flow and the cylinders oscillate dominantly at the natural frequency, almost independent of the inlet
flow velocity, the resonance of the cylinders through the fluid coupling is evident for the case that all the cylinders have the same natural frequency. The effect of the natural frequency difference between the monitored cylinder and its surrounding cylinders on the cylinder vibration will be further discussed in Section 3.3. It is noted that before the amplitude response being used to determine the critical velocity, the amplitude– velocity curve should be checked if hysteresis exists. For this purpose, the vibration amplitude in both the X- and Y-directions were examined with the flow velocity being increased from zero to 0.74 m/s and then decreased to zero, as shown in Fig. 8 for the sevencylinder array, as an example. No obvious hysteresis was observed for all the cylinder arrays in the experimental range. The experimental results of vibration amplitude are also compared with the experimental result by Kassera and Strohmeier [9], as shown in Fig. 9. The test conditions and results of the two studies are summarized in Table 2. From the comparison, it is shown that the experimental results agree reasonably with previous studies at least in low velocity range. After these checks, the amplitude–velocity curves are then used to determine the critical velocity for the onset of the fluid elastic instability of the cylinder. The critical flow velocity here is defined as the flow velocity where
xrms /D(%)
0.003
24 22 20 18 16 14 12 10 8 6 4 2 0 -2 -0.1
velocity increase velocity decrease
Flow
fn=fs
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.5
0.6
0.7
0.8
Uin(m/s)
(a)
yrms/D(%)
0.004
529
24 22 20 18 16 14 12 10 8 6 4 2 0 -2 -0.1
velocity increase velocity decrease
fn=fs
0.0
0.1
0.2
0.3
0.4
Uin(m/s)
(b) 0
0.5
1.0
1.5
2.0
Uin /fnD
Fig. 8. Vibration amplitudes of the monitored cylinder in the sevencylinder array with flow increasing from 0 to 0.74 m/s and then decreasing.
T.-k. Lin, M.-h. Yu / Experimental Thermal and Fluid Science 29 (2005) 523–536 24 22 20 18 16 14 12 10 8 6 4 2 0 -2
present study(exp. data) Kassera and Strohmeier(exp. data)
Arms /D(%)
Arms /D (%)
530
0.0
0.5
1.0
1.5
2.0
2.5
24 22 20 18 16 14 12 10 8 6 4 2 0 -2 -0.1
cylinder array I cylinder array II cylinder array III cylinder array IV cylinder array V cylinder array VI
fn=fs
0.0
0.1
0.2
U in /fnD
0.3
0.4
0.5
0.6
0.7
0.8
Uin(m/s)
Fig. 9. Comparison of the experimental results of vibration amplitude with the experimental data by Kassera and Strohmeier [9]. The experimental and computational conditions are listed in Table 2.
0
0.5
1.0
1.5
2.0
Uin /fn D Fig. 10. Vibration amplitudes of the monitored cylinder in different cylinder arrays. In the case of fn = fs, i.e. all the cylinders have the same natural frequency.
Table 2 Test conditions of the present experiment, and the experiment by Kassera and Strohmeier [9] Investigators
The present authors
Kassera and Strohmeier [9]
P/D D (mm) Arrangement Tube location f m(2pf)/qD2 fn (Hz)
1.33 16 60 (rotated triangular) Central row 0.0096 0.19 22
1.2 16 30 (triangular) Central row 0.0145 0.225 27.2
the vibration amplitude exhibits sudden increase, following the criterion used by Weaver and El-Kashlan [10]. The curves of the dimensionless amplitude Arms/D as function of inlet velocity are shown in Fig. 10. With one upstream cylinder, the monitored cylinder in Array I has sudden increase of vibration amplitude at the free stream velocity Uin 0.6 m/s, equivalently the reduced velocity about 1.7. With one more upstream cylinder (Array II), the reduced velocity decreased to about 1.5. It is therefore illustrated that with two upstream neighboring cylinders, the cylinder in Array II becomes more elastically unstable than that in Array I. In Array I, there is no gap flow between two cylinders impinging on the monitored cylinder, the unique flow situation may be the reason that the flow velocity is required to be at a higher value for onset of fluid elastic instability for Array I compared to Array II, and other arrays. With three upstream cylinders (Array III), the critical velocity is approximately the same as Array II. However, the vibration amplitude above the critical velocity is larger in Array III than Array II, implying that more upstream cylinders provide more and better coupling
among the cylinders once above the critical velocity. The resonance among the cylinders is therefore enhanced, given the condition that all cylinders are at the same natural frequency. Thus significant vibration amplitude was detected for the cylinder in Array III. With one cylinder added downstream (Array IV), the critical velocity remains the same, but the vibration amplitude above the critical velocity is suppressed, back to the case of Array II approximately. With one or two more downstream cylinders (Array V, VI), the critical velocity and vibration amplitude beyond the critical velocity are basically the same. In summary, in case all the cylinders having the same natural frequency, the upstream cylinders have significant influence on the amplitude response of the monitored cylinder; either promote the instability by forming a gap flow impinging on the monitored cylinder, or enhance the cylinder vibration through better fluid coupling with more upstream cylinders. The downstream neighboring cylinders could suppress the vibration amplitude while the number of the downstream cylinders has little effect on the amplitude response for the present configuration. 3.2. Orbit of cylinder vibration The vibration behavior of the cylinder can be further understood by examining the orbit of the cylinder motion in the fluid flow. Fig. 11 shows the orbits of the monitored cylinder in the six test arrays. The orbits were plotted by tracing out the time-dependent displacements of the cylinder motion in the X- and Y-directions for duration of 16 s.
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531
25% D
(a)
25% D
Uin=0.28 m/s
0.55
0.60
0.64
0.68
0.74
(b)
Uin=0.28 m/s
0.60
0.64
0.68
0.74
0.53
0.55
0.57
0.60
0.64
0.55
0.60
0.64
0.68
0.55
0.60
0.64
0.68
0.55
0.60
0.64
0.55
(c)
Uin=0.28 m/s
(d)
Uin=0.28 m/s
0.74
(e)
Uin=0.28 m/s
0.74
(f)
Uin=0.28 m/s
0.68
0.74
Fig. 11. Orbits of the monitored cylinder in the six-cylinder arrays: (a) Array I, (b) Array II, (c) Array III, (d) Array IV, (e) Array V, (f) Array VI. For each array, orbits were obtained with varying inlet flow velocity Uin, which is labeled at the left-bottom of the corresponding orbit.
At the low velocity Uin = 0.28 m/s, less than the critical velocity, the cylinder in all the six arrays vibrates randomly around a ‘‘fixed point’’. The small amplitude and random nature of the orbits illustrate that the cylinder vibration in the low velocity range is due to turbu-
lence. At the flow velocity slightly above the critical value, such as Uin = 0.55, and 0.60 m/s, the orbits look like an ellipse. The elliptical orbits imply that the streamwise and cross-stream responses have the same frequency but with a phase shift. The cylinder obviously
T.-k. Lin, M.-h. Yu / Experimental Thermal and Fluid Science 29 (2005) 523–536
Flow
Arms/D(%)
vibrates in an organized, ‘‘periodic’’ manner above the critical velocity, compared to the random motion below the critical velocity. With this result, the onset of instability might be indicated as the streamwise and crossstream responses have a certain phase shift. As the velocity far beyond the critical velocity, for instance Uin = 0.74 m/s shown in the figure, the orbits tend to be a vertical ellipse, illustrating that the streamwise and cross-stream responses have a 90 phase shift. The effects of the neighboring cylinders on the cylinder vibration can also be discussed by examining the orbits. With one upstream cylinder, the orbits of the cylinder in Array I are elliptical shape as shown in Figs. 11(a). With two or three upstream cylinders (Figs. 11(b) and (c)), the streamwise vibration of the cylinder is obviously suppressed, while the cross-stream vibration is amplified especially in Array III (Fig. 11(c)). With one downstream cylinder added, the orbits maintain in oval shape above the critical velocity as shown in Fig. 11(d) (Uin = 0.55–0.68 m/s). However, at higher velocity (Uin = 0.74 m/s), the cylinder vibration becomes relatively wild—in oval shape generally but with certain deviation cycle by cycle. The large deviations cycle by cycle of the orbit illustrate that the down stream cylinder has already disturbed the original periodicity of the streamwise and cross-stream responses. With more downstream cylinders, the effect could happen at lower flow velocity; 0.68 m/s in Fig. 11(e) for Array V and 0.64 m/s in Fig. 11(f) for Array VI. The disturbance by the downstream cylinders accounts for the suppression of the vibration amplitude in Arrays IV, V, and VI, compared to in Array III.
fn=fs fn=3/4fs fn=1/2fs
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Uin(m/s) Flow
24 22 20 18 16 14 12 10 8 6 4 2 0 -2
fn=fs fn=3/4fs fn=1/2fs
-0.1
0.0
0.1
0.2
(b)
0.3
0.4
0.5
0.6
0.7
0.8
Uin(m/s) Flow
Arms/D(%)
3.3. Effects of the natural frequency As mentioned in Section 3.1, the serious vibration of the monitored cylinder in supercritical conditions is considered as a result of resonance with the surrounding cylinders of the same natural frequency through the coupling of the fluid flow. It is of interest to examine the response of the monitored cylinder to the cross flow when its natural frequency (fn) is not the same as that of the surrounding cylinders (fs). It is first noted that no hysteresis was detected for the case of fn 5 fs in the experimental range, following the same procedure for hysteresis check as described in the case of fn = fs (Fig. 8, for example). The vibration amplitudes of the cylinder are shown in Fig. 12 for the cases of fn = 3/4fs and 1/2fs, and fn = fs for comparison. It is shown that the critical velocities in the cases of fn 5 fs are approximately the same as in the case of fn = fs. In other words, the difference of the natural frequency is not a major factor on determining the critical velocity, if fluid elastic instability occurs—there may be no fluid elastic instability if the two natural frequencies, fn and fs, are far away. Once the flow velocity is above the critical value, the vibration amplitude of the
24 22 20 18 16 14 12 10 8 6 4 2 0 -2
(a)
Arms/D(%)
532
24 22 20 18 16 14 12 10 8 6 4 2 0 -2
fn=fs fn=3/4fs fn=1/2fs
-0.1
(c)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Uin(m/s)
Fig. 12. Vibration amplitudes of the monitored cylinder with fn = fs, 3/4fs and 1/2fs in the X- and Y-directions. (a) Cylinder array I, (b) Cylinder array III, (c) Cylinder array VI.
cylinder, however, depends on its natural frequency; i.e. how close the two natural frequencies, and also the value of the natural frequency that is an indication of stiffness of the cylinder. Regarding the effect of the cylinder stiffness on the cylinder vibration, comparing the vibration amplitudes for the cases of fn = 3/4fs and fn = fs shown in Fig. 12, the monitored cylinder with
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533
Flow
25% D
(a)
25% D
Uin=0.28 m/s
0.55
0.60
0.55
0.60
0.55
0.60
0.74
(b)
Uin=0.28 m/s
0.66
(c)
Uin =0.28 m/s
0.74
Fig. 13. Orbits of the monitored cylinder in the seven-cylinder array with different natural frequencies. (a) fn = fs, (b) fn = 3/4fs, (c) fn = 1/2fs. For each natural frequency, there are four orbits corresponding to four different inlet velocities.
lower structural stiffness (fn = 3/4fs) has a larger vibration amplitude when the flow velocity is just above the critical value. It is noted that at the high frequency fn = 5/4fs or beyond, the cylinder exhibited no obvious vibration in the high velocity range of the experiment, likely due to the more stiffness of cylinder structure. When fn = 1/2fs, though a sub-harmonics of fs but distant away from the fs, the vibration amplitudes above the critical velocity are not so large as in the other two cases. This could be a result of less degree of resonance when the natural frequency fn deviates from the fs. Especially, no obvious fluid elastic instability occurs in the case of the two-cylinder array as shown in Fig. 12(a) with fn = 1/2fs. The degree of resonance can be indicated by the characteristics of the orbits of the cylinder vibrations. As shown in Fig. 13 are the orbital paths of the monitored cylinder at the different natural frequencies in the sevencylinder array below, and above the critical velocity. The cylinders at fn = fs, and 3/4fs vibrate in a relatively organized orbit beyond the critical velocity. However, the cylinder at fn = 1/2fs, vibrates randomly, especially at high flow velocity. The randomness can be considered
as an indication of lack of resonance when fn and fs are far away, and thus an organized vibration cannot be produced. For the frequency content of the orbits, spectral data are further examined. 3.4. Spectrum of cylinder vibration When the inlet velocities are below the critical value, the spectra of the monitored cylinder are broadband, as shown in Fig. 14 for the case of Array III with Uin = 0.47 m/s. The broadband spectra reflect the random nature of the cylinder vibrations in the low velocity range.
Ay/D(%)
1
0.1 0.01 1E-3 1E-4 0
10
20
30
40
50
60
70
80
90
100
f(Hz)
Fig. 14. Amplitude spectrum of the monitored cylinder in the fourcylinder array, Uin = 0.47 m/s.
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Above the critical velocity, the spectra of the monitored cylinder exhibit various spectral peaks depending on its natural frequency. Figs. 15–17 are the spectra of the cylinder vibrations in Arrays I, III, and VI, respectively. In the experiment, the natural frequency of the cylinder in each test array was adjusted at fn = fs, 3/4fs and 1/2fs. Only the amplitude spectra of the cylinder vibration in the Y-direction are presented here, since under the same test condition the streamwise spectrum is similar in nature to the corresponding cross-stream spectrum; namely, both the spectra show either broadband or line-dominated. In the latter case, the spectral peaks for the X- and the Y-directions occur at the same frequencies, though the spectral peaks may have different magnitudes. In the case of fn = fs, as shown in Figs. 15(a,b)– 17(a,b) for cases of different inlet velocities and cylinder arrays, a clear spectral peak at the natural frequency fn is observed when the flow velocity is around the critical velocity (Uin = 0.53 m/s). As the Uin increases, more
spectral peaks at harmonics nfn appear, suggesting an organized oscillatory behavior. In the case of fn = 3/4fs, as shown in Figs. 15(c,d)– 17(c,d), spectral peak at fn is observed when the flow velocity is around the critical value, Uin = 0.53 m/s. As the velocity increases, besides the fn-peak, more spectral peaks appear at fs, and at the harmonics of fn and fs (Figs. 15(d)–17(d)). It is thus evident that the vibration of the monitored cylinder has two major components at fn and fs above the critical velocity. The former component is associated with the natural frequency of the monitored cylinder, and the latter with the natural frequency of the surrounding cylinders. It is therefore illustrated that the cylinder, induced by the fluid flow, vibrates initially only at its natural frequency around the onset of fluid elastic instability. As the fluid velocity is increased, the cylinder vibrations have the frequency component of the surrounding cylinders through the coupling of the fluid flow.
Flow
Flow
(a)
(a)
fn
fn
(b)
(b)
fn
Ay/D(%)
Ay/D(%)
(c) fn
(c)
fn fs
fs
(d)
(d)
(e) (e) 10
fn
1
10
fs
1 0.1
0.1 0.01
(f)
(f)
0.01 1E -3
1E-3
1E -4 0
10
20
30
40
50
60
70
80
90
100
f(Hz)
Fig. 15. Amplitude spectrum of the monitored cylinder with the natural frequency fn in the two-cylinder array in cross flow with velocity Uin. (a) fn = fs, Uin = 0.53 m/s, (b) fn = fs, Uin = 0.74 m/s, (c) fn = 3/4fs, Uin = 0.53 m/s, (d) fn = 3/4fs, Uin = 0.74 m/s, (e) fn = 1/2fs, Uin = 0.53 m/s, (f) fn = 1/2fs, Uin = 0.74 m/s.
0
10
20
30
40
50
60
70
80
90
100
f(Hz)
Fig. 16. Amplitude spectrum of the monitored cylinder with the natural frequency fn in the four-cylinder array in cross flow with velocity Uin. (a) fn = fs, Uin = 0.53 m/s, (b) fn = fs, Uin = 0.62 m/s, (c) fn = 3/4fs, Uin = 0.53 m/s, (d) fn = 3/4fs, Uin = 0.62 m/s, (e) fn = 1/2fs, Uin = 0.53 m/s, (f) fn = 1/2fs, Uin = 0.62 m/s.
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indicated that the resonance among the cylinders is not so well when the two natural frequencies of the monitored cylinder and the surrounding cylinders are not close enough. Consequently, the cylinder vibrates relatively randomly, as already shown in Fig. 13(c).
Flow
(a) fn
4. Conclusions (b)
fs fn Ay/D(%) (c)
(d) fn fs
(e) 10 1 0. 1
(f)
0.01 1E -3 1E -4 0
10
20
30
40
50
60
70
80
90
100
f(Hz)
Fig. 17. Amplitude spectrum of the monitored cylinder with the natural frequency fn in the seven-cylinder array in cross flow with velocity Uin. (a) fn = fs, Uin = 0.53 m/s, (b) fn = fs, Uin = 0.74 m/s, (c) fn = 3/4fs, Uin = 0.53 m/s, (d) fn = 3/4fs, Uin = 0.74 m/s, (e) fn = 1/2fs, Uin = 0.53 m/s, (f) fn = 1/2fs, Uin = 0.74 m/s.
Particularly, for the case of the seven-cylinder array, the spectra have the frequency component fs in addition to fn, even when the fluid velocity is just slightly above the critical value, Uin = 0.53 m/s shown in Fig. 17(c). The enhancement of the fs-vibration is related to the more surrounding cylinders downstream in the sevencylinder array compared to the four-cylinder array. As the Uin increases, more spectral peaks are observed at the harmonics of fn and fs (Fig. 17(d)). The line-dominated spectra suggest an organized oscillatory behavior, which is consistent with the observation of the orbits already shown in Fig. 13(b). In the case of fn = 1/2fs shown in Figs. 15(e,f)–17(e,f), it is illustrated again that the monitored cylinder vibrates initially at its natural frequency fn around the onset of fluid elastic instability, and the frequency fs as well beyond the critical velocity. However, at high velocity such as U = 0.74 m/s, no obvious spectral peaks at the harmonics of fn and fs are observed (Figs. 15(f)–17(f)). It is
The cross-flow vibration of a circular cylinder (the monitored cylinder) surrounded by one to six identical cylinders elastically mounted in rotated triangular pattern is investigated. Analysis of the vibration amplitude reveals that the upstream cylinders have significant influence on the amplitude response of the monitored cylinder, either promote the instability by forming a gap flow impinging on the monitored cylinder, or enhance the cylinder vibration through better fluid coupling with more upstream cylinders. The downstream neighboring cylinders could suppress the vibration amplitude while the number of the downstream cylinders has little effect on the amplitude response for the present configuration. By the orbits of the cylinder motion, it is illustrated that around the critical velocity, the streamwise and cross-stream responses have the same frequency but with a phase shift. The phase shift tends to be 90 as the flow velocity is increased. With two or three upstream cylinders, the streamwise vibration of the cylinder is suppressed, while the cross-stream vibration is amplified, compared to the vibration of the cylinder with only upstream cylinder. With downstream cylinders added in addition to three upstream cylinders, the organized periodicity of the cylinder motion is disturbed and hence the vibration amplitude is suppressed. The discrepancy of the natural frequencies between the monitored cylinder and the surrounding cylinders is not a factor on determining the critical velocity. However, the vibration amplitude of the cylinder above the critical velocity depends on the value of its natural frequency, and the discrepancy of the natural frequencies. The monitored cylinder at high natural frequency fn P 5/4fs has no obvious vibration due to the cylinder stiffness. Large discrepancy of the natural frequencies reduces the vibration amplitude due to lack of resonance among the cylinders. Spectral data show that the cylinder, induced by the fluid flow, vibrates initially only at its natural frequency around the onset of fluid elastic instability. As the fluid velocity is increased, the cylinder vibrations include the frequency component of the surrounding cylinders through the coupling of the fluid flow. If both the nature frequencies are close, for instances fn = 3/4fs, and fn = fs, the cylinder exhibits a very organized oscillatory behavior, evidenced by the line-dominated spectra with spectral peaks at the harmonics of fn and fs.
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Acknowledgement This work was supported by the National Science Council, Taiwan, R.O.C. Thanks are due to Prof. C.C. Cheng for use of his computer facility.
[5]
[6]
References [1] R.D. Blevins, Flow-Induced Vibration, Van Nostrand Reinhold, New York, 1990. [2] M. Matsumoto, N. Shiraishi, H. Shirato, Aerodynamic instabilities of twin circular cylinders, Journal of Wind Engineering and Industrial Aerodynamics 33 (1–2) (1990) 91–100. [3] B. Dielen, H. Ruscheweyh, Mechanism of interference galloping of two identical circular cylinders in cross flow, Journal of Wind Engineering and Industrial Aerodynamics 54–55 (1995) 289–300. [4] M.J. Pettigrew, C.E. Taylor, Fluid elastic instability of heat exchanger tube bundles. Review and design recommendations,
[7]
[8] [9]
[10]
Journal of Pressure Vessel Technology, Transactions of the ASME 113 (2) (1991) 242–256. K. Schroeder, H. Gelbe, Two- and three-dimensional CFD-simulation of flow-induced vibration excitation in tube bundles, Chemical Engineering and Processing 38 (4–6) (1999) 621–629. H. Tanaka, S. Takahara, Fluid elastic vibration of tube array in cross flow, Journal of Sound and Vibration 77 (1) (1981) 19–37. H. Tanaka, K. Tanaka, F. Shimizu, Fluid elastic analysis of tube bundle vibration in cross-flow, Journal of Fluids and Structures 16 (1) (2002) 93–112. R.J. Moffat, Describing the uncertainties in experimental results, Experimental Thermal and Fluid Science 1 (1988) 3–17. V. Kassera, K. Strohmeier, Experimental determination of tube bundle vibrations induced by cross-flow, in: M.K. Au-Yang (Ed.), Proceedings ASME Symposium on Flow-Induced Vibration, New York, PVP-273, ASME, New York, 1994, pp. 91–97. D.S. Weaver, M. El-Kashlan, On the number of tube rows required to study cross-flow induced vibrations in tube banks, Journal of Sound and Vibration 75 (1981) 265–273.
An experimental study on the cross-flow vibration of a flexible cylinder in cylinder arrays Tsun-kuo Lin, Ming-huei Yu
*
Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung 804, Taiwan, ROC Received 31 October 2003; accepted 26 June 2004
Abstract In the experiment, a monitored cylinder equipped with two accelerometers inside was flexibly mounted in a water tunnel, surrounded by one to six identical cylinders elastically mounted in rotated triangular pattern. The amplitude diagrams, spectra and orbits of the cylinder motion are used to examine the vibration behavior of the cylinder under the various test conditions of the free stream velocity, the number of the surrounding cylinders, and the cylinderÕs natural frequency. In the case of the monitored cylinder having the same natural frequency as the surrounding cylinders (22 Hz), amplitude response shows that fluid elastic instability occurs when the flow velocity is above a critical value for the cylinder in all the six-cylinder arrays. Above the critical velocity, the cylinder vibrates around an oval orbit with line-dominated spectrum, implying that the cylinder behaves like an oscillator with the streamwise and cross-stream responses have the same frequency but with a phase shift. By comparison of amplitude diagrams of the cylinder in the six bundles, it reveals that the upstream cylinders have significant influence on the amplitude response of the monitored cylinder–promote the fluid elastic instability of the monitored cylinder, and enhance the cylinder vibration above the critical velocity. The downstream cylinders could suppress the vibration amplitude while the number of the downstream cylinders has little effect on the amplitude response. In case of the monitored cylinder having different natural frequency from that of the surrounding cylinders, it is found that the difference in natural frequency of the cylinders has little effect on the critical velocity, but strong influence on the vibration amplitude above the critical velocity. 2004 Elsevier Inc. All rights reserved. Keywords: Flow-induced vibration; Circular cylinder; Cross flow
1. Introduction Bundles of circular cylinders are commonly used in engineering applications as in power transmission lines, pipelines in deep water, suspension bridges, heat exchangers, etc. Their potential vibrations induced by fluid flow have been extensively studied for the last three decades. Blevins [1] summarized the most important results of the flow-induced vibrations of circular cylinders. Later, researches on the flow-induced vibration of two *
Corresponding author. Tel.: +886 7 5252000x4238; fax: +886 7 5254299. E-mail address: [email protected] (M.-h. Yu). 0894-1777/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2004.06.004
identical circular cylinders was continued because of its fundamental importance, by Matsumoto et al. [2], and Dielen and Ruscheweyh [3], for instances. Meanwhile, many researchers, Pettigrew and Taylor [4] and Schroeder and Gelbe [5], among others, focused on the fluid elastic instability of cylinder rows and arrays because of their practical applications. While the flow-induced vibrations of two cylinders and cylinder arrays are the main focuses in the previous studies, the flow-induced vibrations from two cylinders to cylinder arrays are systematically investigated in the study, an effort to bridge the gap between the two cases in aspect of flow-induced vibration. For cylinder bundles in cross flow, fluid elastic instability is an excitation mechanism for flow-induced
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Nomenclature Arms Ax Ay c D E F fn fs m P Re
r.m.s. value of tube vibration amplitude (m) tube vibration amplitude in the streamwise (X) direction (m) tube vibration amplitude in the lateral (Y) direction (m) damping coefficient outer tube diameter (m) voltage output from the measurement system (Volt) force per unit length (N/m) the natural frequency of the monitored cylinder (Hz) the natural frequency of the surrounding cylinders (Hz) mass per unit length (kg/m) pitch (m) Reynolds number
vibrations, resulting in large-amplitude vibrations and even failure of the cylinder bundles. Fluid elastic instability, which may occur in multiple circular cylinders, is a result of the interaction between the cylinders and the flowing fluid. In cylinder bundles, the interactions of the cylinders with the fluid, and the coupling among the cylinders through the fluid are very complex. However, it is reasonable to suppose that the unsteady fluid dynamic forces on a cylinder are mainly induced by vibrations of the cylinder itself and its neighboring cylinders [6,7]. With this assumption, the vibration behavior of a cylinder in cylinder bundle should be most affected by its surrounding cylinders. For instance, a cylinder in twin-cylinder bundle may have different vibration behavior from the cylinder surrounded by more cylinders under otherwise the same condition. The paper is aimed to describe the flow-induced vibrations of a flexibly mounted cylinder surrounded by various numbers of cylinders in cross flow. The emphases of the study are the effects of the number of surrounding cylinders, and the natural frequency of the cylinder on the vibration behavior of the cylinder. For these purposes, the amplitudes, orbits, and spectra of the cylinder vibration are obtained by simultaneous measurements of the cylinder vibration in the streamwise (X) and lateral (Y) directions.
2. Experimental aspects 2.1. Water tunnel and instrumentation A low speed water tunnel was designed and built for this study, as shown schematically in Fig. 1. The length
St Ucrit Uin x xrms X, Y y yrms
Strouhal number critical velocity (m/s) inlet velocity (m/s) displacement in the streamwise (X) direction (m) the r.m.s. value of the cylinder displacement in the X-direction (m) rectangular Cartesian coordinates displacement in the lateral (Y) direction (m) the r.m.s. value of the cylinder displacement in the Y-direction (m)
Greek symbols f damping factor (dimensionless) l dynamic viscosity (kg/m s) m kinematic viscosity (m2/s) q density (kg/m3)
of the contraction section is approximately 0.6 m and the contraction ratio is 9:1. The test section, made up of bolted acrylic plates, has internal dimensions of 0.2 m · 0.2 m. A tank with 0.76 m3 capacity serves as an accumulator placed after the test section. The temperature of the water flow was at 20 C. In operating condition, water flow was driven by a variable speed pump. After passing through a filter, the water entered a diffuser, which provides a gradual transition from 10 cm diameter circular pipe to a rectangular channel of 0.6 m · 0.6 m. Before the 9:1 contraction, a layer of honeycomb and several screens were installed in the rectangular channel to reduce turbulence. At the entrance of the test section, a pitot tube was used to monitor the inlet flow velocity. The average water velocity in the test section of the facility was up to 0.74 m/s. The Reynolds number, based on the inlet flow velocity and the diameter of test cylinders is 1.1 · 104. The average upstream turbulence intensity over the flow range is about 1.2% measured by hot-film anemometry. The velocity distribution at the test section, excluding the boundary layer, is found to be flat within 1%. In the test section, stainless steel tubes were used to form cylinder bundles for testing. Six cylinder bundles were tested in the experiment, they all being in a 60 equilateral triangle pattern. As shown in Fig. 2(a), each bundle consists of a monitored cylinder (bold circles in the figure), equipped with two accelerators inside, and one to six surrounding cylinders. The cylinder bundles are named as Array I–VI, respectively, according to the number of the surrounding cylinders. The diameter of the entire cylinders is 16 mm, and the ratio of the cylinder center-to-center spacing to cylinder diameter P/D is 1.33. Fig. 2(b) is a side view of the test section to show
T.-k. Lin, M.-h. Yu / Experimental Thermal and Fluid Science 29 (2005) 523–536
Honeycomb Screen
525
Pitot tube
Filter
Tank Test section
Flow
10 h.p. motor
Regulator
Speed control unit
220 V a.c. supply
Fig. 1. Water tunnel facility.
how to mount a cylinder in the test section. Each stainless tube in the test array is suspended by two tensional steel wires at both ends. The tension in the steel wire can be adjusted to change the natural frequency of the cylinder. As shown in the figure, a stainless cylinder 198 mm long, 16 mm outside diameter and 1.5 mm wall thickness is mounted on a 1.2 mm diameter steel wire. The weight of each cylinder is 153 g within 1% deviation. The free length of the 1.2 mm diameter wire is 180 mm and the screw-nut adjustments at both ends of the wire provide fine tension adjustment for tuning the natural frequency. In the experiment, the natural frequencies of the cylinders surrounding the monitored cylinder were all fixed at fs = 22 Hz, and the natural frequency of the monitored cylinder, fn, was set at 22 Hz, 16.5 Hz (=3/4fs), 11 Hz (=1/2fs), and 27.5 Hz (=5/4fs) for tests with different natural frequencies. The instrumentation includes two accelerometers manufactured by Endevco (Model #25A) with sensitivity 4.693 mV/g and frequency response of 2–8 kHz. The two accelerometers were installed inside the tube at midspan. Their orientations were adjusted such that they were sensitive to tube vibrations only in the streamwise (X) and lateral (Y) directions, respectively. The output of each accelerometers was amplified by a B&K charge amplifier (Model #2693) and coupled to B&K double integrator circuit (Model #0788), as shown in Fig. 3. The double integrator circuit, consisting of an input filter and two operational amplifier integrators, was used to integrate an acceleration signal twice and provide an output signal proportional to displacement. The Xand Y displacement signals obtained from the double integrators were digitized simultaneously at a sampling rate of 512 data/s by a 16 bit IOtech IEEE 488 A/D converter (ADC488/8SA) and transmitted to a PC compu-
ter for data storage and further processing. An FFT analyzer, AND AD3524, was connected to the double integrators to monitor the amplitude spectra of the cylinder during the experiments. The voltage outputs from the data acquisition system were calibrated against the vibration amplitudes before vibration measurements. In calibration, the tube with the two accelerometers inside was given a circular motion. Typical time series of displacements in the X and Y direction are shown in Fig. 4(a). The amplitudes of the sinusoidal output voltages for the X- and Y-directions were recorded and then plotted against the vibration amplitude (the radius of the circular motion). As shown in Fig. 4(b), the calibration data for both the X- and Y-directions are fitted to a straight line E = aA + b, where A is the vibration amplitude and E is the voltage output. Curve-fitting gives the values of the coefficients, a = 0.10 and b = 0.007. It is thus shown that the voltage outputs can linearly represent the vibration amplitudes in the experimental range. 2.2. Uncertainty To characterize the magnitude of cylinder vibration, the r.m.s. value of vibration amplitude Arms is introduced in dimensionless form as, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Arms =D ¼ x2rms þ y 2rms =D ð1Þ where D is the diameter of the cylinder. xrms and yrms are the r.m.s. values of the cylinder displacements in the X- and Y-directions, respectively. The displacements were obtained by digitizing the output voltages from the measurement system that transforms the displacements into voltage signals. The
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The velocity profile in the center region of the test section, exclusive of the boundary layer region, is uniform within 5% variation at the maximum freestream velocity Uin = 0.74 m/s, and the turbulence intensity is 1.3%, measured by a Dantec hot-film anemometer. The tolerance in measuring the natural frequency is ±0.2 Hz in the range of 9–31 Hz. The natural frequency and damping factor of a cylinder were determined by measuring the amplitude response of the cylinder after given a perturbation in otherwise still water, as shown in Fig. 5 for a typical amplitude response in time domain. The damping factor can be estimated from the decay of the amplitude, and the natural frequency of the cylinder can be estimated by spectral analysis of the time trace. For each natural frequency, the average value from several measurements is adopted, and the uncertainty can be estimated by evaluating the standard deviation. Perturbations of various directions were applied in the measurements to make sure that the natural frequency is independent of cylinder vibration orientation in the experimental setup. For spectral analysis of the vibrations of the cylinder in cross flow, the software MATLAB was used to obtain spectra data. Additionally, a spectrum analyzer AND AD3524, with a setting of frequency resolution Df = 0.125 Hz, was connected to the voltage output to monitor the cylinder vibration during the experiments. The uncertainty of mass damping can be estimated by using MoffatÕs [8] uncertainty analysis. The uncertainties of the experimental parameters are summarized in Table 1.
3. Results and discussion 3.1. Vibration amplitude
Fig. 2. (a) Top view of the test cylinders. (b) Side view of the cylinders in the test section.
uncertainties of the displacement measurements could be due to the accuracy of the vibration measurement system, and of the data acquisition system. The resolution of the data acquisition used to digitize the voltage from the double integrator circuit is 4 · 104 V in a range of 5 V. The tolerance of the measured amplitude due to the measurement system is estimated to be 0.3 mm, based on the standard deviation of the calibration data from the fitting curve, shown in Fig. 4(b).
For the effect of surrounding cylinders on the vibration of the monitored cylinder, vibration measurements of the monitored cylinder were carried out with the surrounding cylinder added one by one, while the natural frequency of the entire cylinders was set at the same (22 Hz). Figs. 6(a–f) show the amplitude responses of the monitored cylinder in the six-cylinder arrays, respectively. In each of the figures, there are two curves representing the vibration amplitudes in the X- and Ydirections. It is seen that all the amplitude curves have similar trend as the flow velocity is increased. The vibration amplitudes are very small, slightly increasing with the flow velocity, as the velocity is below a critical value – Uin 0.5 m/s for all the test arrays except Uin 0.6 m/s for the two-cylinder array. These small-amplitude vibrations could be attributable to turbulence, considering that the monitored was in the wake region behind the upstream cylinders. Above the critical velocity, the vibration amplitudes increase significantly as the flow
T.-k. Lin, M.-h. Yu / Experimental Thermal and Fluid Science 29 (2005) 523–536
FFT Analyzer
Computer
Amplifier and Double Integrator Circuit
Data Acquisition System
X direction Acceleration Meter
527
Y direction Acceleration Meter Fig. 3. The data acquisition system for 2-channel vibration measurements.
2
20
Ax
10
1
-10
y/D (%)
Displacement(mm)
0
-20
0
20
Ay 10
-1
0 -10
-2 0
-20
1
2
3
4
5
t(sec) 0.0
0.5
1.5
2.0
2.5
Fig. 5. A typical time trace of the cylinder vibration after perturbation in otherwise still water.
Ey Ex
3.5
Ex,Ey(Volt)
1.0
Time(sec)
(a)
3.0 2.5 2.0 1.5 15
(b)
20
25
30
35
40
Vibration amplitude (mm)
Fig. 4. (a) Typical time series of displacements in the X and Y direction, (b) the calibration of the output voltages from the data acquisition system versus the vibration amplitudes, Ex for the X-direction, and Ey for the Y direction.
velocity is increased, with the amplitude in the Y-direction being larger than that in the X-direction for each
test case. The continuous increase of vibration amplitude with flow velocity implies that fluid elastic instability has occurred. The serious vibration above the critical velocity could be explained as follows. When the flow velocity is high enough, the fluid has sufficient energy to excite the cylinders to vibrate with certain amplitude at their natural frequency. Through the coupling of the fluid surrounding the cylinders, resonance occurs among the cylinders that have the same natural frequency, and as a result the vibration amplitude was amplified and thus serious vibration was observed. The explanation of the serious vibration due to fluid elastic instability in the viewpoint of resonance is consistent with the experimental findings that the dominant frequency of the fluid flow around the cylinders is the natural frequency of the cylinders. As shown in Fig. 7 are typical time history of the flow around the cylinders and the corresponding velocity spectrum, obtained by using a Dantec hot-film anemometer
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Table 1 Uncertainties of the natural frequency, damping factor and mass damping for the test cases fn (Hz) 22 ± 0.2 16.5 ± 0.2 11 ± 0.2
(fn = fs) (fn = 3/4fs) (fn = 1/2fs)
f
m/qD2
m(2pf)/qD2
0.0096 ± 0.0003 0.0104 ± 0.0004 0.0088 ± 0.0002
3.32 ± 0.02 3.32 ± 0.02 3.32 ± 0.02
0.19 ± 0.006 0.22 ± 0.008 0.18 ± 0.004
Uin(m/s)
(a)
xrms,yrms /D(%)
xrms,yrms /D(%)
Flow
24 Flow X direction 22 Y direction 20 fn= fs 18 16 14 12 10 8 6 4 2 0 -2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
24 X direction 22 Y direction 20 fn= fs 18 16 14 12 10 8 6 4 2 0 -2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Uin(m/s)
(b)
Flow
24 X direction 22 Y direction 20 fn= fs 18 16 14 12 10 8 6 4 2 0 -2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Uin(m/s)
(c)
xrms,yrms/D(%)
xrms,yrms/D(%)
Flow
24 X direction 22 Y direction 20 fn= fs 18 16 14 12 10 8 6 4 2 0 -2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Uin(m/s)
(d)
Flow
24 X direction 22 Y direction 20 fn= fs 18 16 14 12 10 8 6 4 2 0 -2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Uin(m/s)
(e) 0
0.5
1.0
Uin /fnD
xrms,yrms/D(%)
xrms,yrms/D(%)
Flow
24 X direction 22 Y direction 20 fn= fs 18 16 14 12 10 8 6 4 2 0 -2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Uin(m/s)
(f) 1.5
2.0
0
0.5
1.0
1.5
2.0
Uin /fnD
Fig. 6. Vibration amplitudes of the monitored cylinder with fn = fs in the X-direction (s), and in the Y-direction (n). The monitored cylinder in (a) Array I, (b) Array II, (c) Array III, (d) Array IV, (e) Array V, (f) Array VI.
T.-k. Lin, M.-h. Yu / Experimental Thermal and Fluid Science 29 (2005) 523–536
0.002
E(volt)
0.001 0.000 -0.001 -0.002 -0.003 -0.004 0.0
0.2
0.4
0.6
0.81
1.0
Time(sec)
(a) 0.004
Uin=0.5 m/s
E(volt)
0.002
0.000
-0.002 0
(b)
10
20
30
40
50
60
70
80
90
100
f(Hz)
Fig. 7. (a) Typical time history of the fluid velocity around the cylinder (b) the corresponding velocity spectrum.
(StreamLine) with a hot-film probe (type 55R11). Similar spectra were obtained regardless of the inlet flow velocity once the velocity is high enough. The velocity spectra indeed exhibit only one spectral peak at fn (22 Hz), indicating that the fluid flow oscillates at the natural frequency of the cylinders. The oscillation frequency is not the vortex shedding frequency that is linearly related to flow velocity. The Strouhal number of the vortex shedding of the two-cylinder configuration, for instance, is estimated to be 0.12 (by the figure in Ref. [1, p. 176]). The vortex shedding frequency in the flow velocity range, Uin = 0.6–0.7 m/s, is then about 5 Hz, not 22 Hz. Due to the mismatch of the two frequencies, vortex-induced instability is not likely responsible for the large-amplitude vibration in the velocity range. The cylinder vibration is identified as a result of fluid elastic instability, based on the character of the vibration amplitudes that increase rapidly once a critical cross-flow velocity is exceeded, as shown in Fig. 6. It is known that the vibration amplitude of fluid elastic vibration increases very rapidly with the flow velocity once a critical cross-flow velocity is exceeded, while the amplitude response of a vortex-induced vibration is typically a hump, or multiple humps if vibrations also occur at sub and super harmonics of the shedding frequency (see Chapters 3 and 5, by Blevins [1], for instance). With the fluid flow and the cylinders oscillate dominantly at the natural frequency, almost independent of the inlet
flow velocity, the resonance of the cylinders through the fluid coupling is evident for the case that all the cylinders have the same natural frequency. The effect of the natural frequency difference between the monitored cylinder and its surrounding cylinders on the cylinder vibration will be further discussed in Section 3.3. It is noted that before the amplitude response being used to determine the critical velocity, the amplitude– velocity curve should be checked if hysteresis exists. For this purpose, the vibration amplitude in both the X- and Y-directions were examined with the flow velocity being increased from zero to 0.74 m/s and then decreased to zero, as shown in Fig. 8 for the sevencylinder array, as an example. No obvious hysteresis was observed for all the cylinder arrays in the experimental range. The experimental results of vibration amplitude are also compared with the experimental result by Kassera and Strohmeier [9], as shown in Fig. 9. The test conditions and results of the two studies are summarized in Table 2. From the comparison, it is shown that the experimental results agree reasonably with previous studies at least in low velocity range. After these checks, the amplitude–velocity curves are then used to determine the critical velocity for the onset of the fluid elastic instability of the cylinder. The critical flow velocity here is defined as the flow velocity where
xrms /D(%)
0.003
24 22 20 18 16 14 12 10 8 6 4 2 0 -2 -0.1
velocity increase velocity decrease
Flow
fn=fs
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.5
0.6
0.7
0.8
Uin(m/s)
(a)
yrms/D(%)
0.004
529
24 22 20 18 16 14 12 10 8 6 4 2 0 -2 -0.1
velocity increase velocity decrease
fn=fs
0.0
0.1
0.2
0.3
0.4
Uin(m/s)
(b) 0
0.5
1.0
1.5
2.0
Uin /fnD
Fig. 8. Vibration amplitudes of the monitored cylinder in the sevencylinder array with flow increasing from 0 to 0.74 m/s and then decreasing.
T.-k. Lin, M.-h. Yu / Experimental Thermal and Fluid Science 29 (2005) 523–536 24 22 20 18 16 14 12 10 8 6 4 2 0 -2
present study(exp. data) Kassera and Strohmeier(exp. data)
Arms /D(%)
Arms /D (%)
530
0.0
0.5
1.0
1.5
2.0
2.5
24 22 20 18 16 14 12 10 8 6 4 2 0 -2 -0.1
cylinder array I cylinder array II cylinder array III cylinder array IV cylinder array V cylinder array VI
fn=fs
0.0
0.1
0.2
U in /fnD
0.3
0.4
0.5
0.6
0.7
0.8
Uin(m/s)
Fig. 9. Comparison of the experimental results of vibration amplitude with the experimental data by Kassera and Strohmeier [9]. The experimental and computational conditions are listed in Table 2.
0
0.5
1.0
1.5
2.0
Uin /fn D Fig. 10. Vibration amplitudes of the monitored cylinder in different cylinder arrays. In the case of fn = fs, i.e. all the cylinders have the same natural frequency.
Table 2 Test conditions of the present experiment, and the experiment by Kassera and Strohmeier [9] Investigators
The present authors
Kassera and Strohmeier [9]
P/D D (mm) Arrangement Tube location f m(2pf)/qD2 fn (Hz)
1.33 16 60 (rotated triangular) Central row 0.0096 0.19 22
1.2 16 30 (triangular) Central row 0.0145 0.225 27.2
the vibration amplitude exhibits sudden increase, following the criterion used by Weaver and El-Kashlan [10]. The curves of the dimensionless amplitude Arms/D as function of inlet velocity are shown in Fig. 10. With one upstream cylinder, the monitored cylinder in Array I has sudden increase of vibration amplitude at the free stream velocity Uin 0.6 m/s, equivalently the reduced velocity about 1.7. With one more upstream cylinder (Array II), the reduced velocity decreased to about 1.5. It is therefore illustrated that with two upstream neighboring cylinders, the cylinder in Array II becomes more elastically unstable than that in Array I. In Array I, there is no gap flow between two cylinders impinging on the monitored cylinder, the unique flow situation may be the reason that the flow velocity is required to be at a higher value for onset of fluid elastic instability for Array I compared to Array II, and other arrays. With three upstream cylinders (Array III), the critical velocity is approximately the same as Array II. However, the vibration amplitude above the critical velocity is larger in Array III than Array II, implying that more upstream cylinders provide more and better coupling
among the cylinders once above the critical velocity. The resonance among the cylinders is therefore enhanced, given the condition that all cylinders are at the same natural frequency. Thus significant vibration amplitude was detected for the cylinder in Array III. With one cylinder added downstream (Array IV), the critical velocity remains the same, but the vibration amplitude above the critical velocity is suppressed, back to the case of Array II approximately. With one or two more downstream cylinders (Array V, VI), the critical velocity and vibration amplitude beyond the critical velocity are basically the same. In summary, in case all the cylinders having the same natural frequency, the upstream cylinders have significant influence on the amplitude response of the monitored cylinder; either promote the instability by forming a gap flow impinging on the monitored cylinder, or enhance the cylinder vibration through better fluid coupling with more upstream cylinders. The downstream neighboring cylinders could suppress the vibration amplitude while the number of the downstream cylinders has little effect on the amplitude response for the present configuration. 3.2. Orbit of cylinder vibration The vibration behavior of the cylinder can be further understood by examining the orbit of the cylinder motion in the fluid flow. Fig. 11 shows the orbits of the monitored cylinder in the six test arrays. The orbits were plotted by tracing out the time-dependent displacements of the cylinder motion in the X- and Y-directions for duration of 16 s.
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531
25% D
(a)
25% D
Uin=0.28 m/s
0.55
0.60
0.64
0.68
0.74
(b)
Uin=0.28 m/s
0.60
0.64
0.68
0.74
0.53
0.55
0.57
0.60
0.64
0.55
0.60
0.64
0.68
0.55
0.60
0.64
0.68
0.55
0.60
0.64
0.55
(c)
Uin=0.28 m/s
(d)
Uin=0.28 m/s
0.74
(e)
Uin=0.28 m/s
0.74
(f)
Uin=0.28 m/s
0.68
0.74
Fig. 11. Orbits of the monitored cylinder in the six-cylinder arrays: (a) Array I, (b) Array II, (c) Array III, (d) Array IV, (e) Array V, (f) Array VI. For each array, orbits were obtained with varying inlet flow velocity Uin, which is labeled at the left-bottom of the corresponding orbit.
At the low velocity Uin = 0.28 m/s, less than the critical velocity, the cylinder in all the six arrays vibrates randomly around a ‘‘fixed point’’. The small amplitude and random nature of the orbits illustrate that the cylinder vibration in the low velocity range is due to turbu-
lence. At the flow velocity slightly above the critical value, such as Uin = 0.55, and 0.60 m/s, the orbits look like an ellipse. The elliptical orbits imply that the streamwise and cross-stream responses have the same frequency but with a phase shift. The cylinder obviously
T.-k. Lin, M.-h. Yu / Experimental Thermal and Fluid Science 29 (2005) 523–536
Flow
Arms/D(%)
vibrates in an organized, ‘‘periodic’’ manner above the critical velocity, compared to the random motion below the critical velocity. With this result, the onset of instability might be indicated as the streamwise and crossstream responses have a certain phase shift. As the velocity far beyond the critical velocity, for instance Uin = 0.74 m/s shown in the figure, the orbits tend to be a vertical ellipse, illustrating that the streamwise and cross-stream responses have a 90 phase shift. The effects of the neighboring cylinders on the cylinder vibration can also be discussed by examining the orbits. With one upstream cylinder, the orbits of the cylinder in Array I are elliptical shape as shown in Figs. 11(a). With two or three upstream cylinders (Figs. 11(b) and (c)), the streamwise vibration of the cylinder is obviously suppressed, while the cross-stream vibration is amplified especially in Array III (Fig. 11(c)). With one downstream cylinder added, the orbits maintain in oval shape above the critical velocity as shown in Fig. 11(d) (Uin = 0.55–0.68 m/s). However, at higher velocity (Uin = 0.74 m/s), the cylinder vibration becomes relatively wild—in oval shape generally but with certain deviation cycle by cycle. The large deviations cycle by cycle of the orbit illustrate that the down stream cylinder has already disturbed the original periodicity of the streamwise and cross-stream responses. With more downstream cylinders, the effect could happen at lower flow velocity; 0.68 m/s in Fig. 11(e) for Array V and 0.64 m/s in Fig. 11(f) for Array VI. The disturbance by the downstream cylinders accounts for the suppression of the vibration amplitude in Arrays IV, V, and VI, compared to in Array III.
fn=fs fn=3/4fs fn=1/2fs
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Uin(m/s) Flow
24 22 20 18 16 14 12 10 8 6 4 2 0 -2
fn=fs fn=3/4fs fn=1/2fs
-0.1
0.0
0.1
0.2
(b)
0.3
0.4
0.5
0.6
0.7
0.8
Uin(m/s) Flow
Arms/D(%)
3.3. Effects of the natural frequency As mentioned in Section 3.1, the serious vibration of the monitored cylinder in supercritical conditions is considered as a result of resonance with the surrounding cylinders of the same natural frequency through the coupling of the fluid flow. It is of interest to examine the response of the monitored cylinder to the cross flow when its natural frequency (fn) is not the same as that of the surrounding cylinders (fs). It is first noted that no hysteresis was detected for the case of fn 5 fs in the experimental range, following the same procedure for hysteresis check as described in the case of fn = fs (Fig. 8, for example). The vibration amplitudes of the cylinder are shown in Fig. 12 for the cases of fn = 3/4fs and 1/2fs, and fn = fs for comparison. It is shown that the critical velocities in the cases of fn 5 fs are approximately the same as in the case of fn = fs. In other words, the difference of the natural frequency is not a major factor on determining the critical velocity, if fluid elastic instability occurs—there may be no fluid elastic instability if the two natural frequencies, fn and fs, are far away. Once the flow velocity is above the critical value, the vibration amplitude of the
24 22 20 18 16 14 12 10 8 6 4 2 0 -2
(a)
Arms/D(%)
532
24 22 20 18 16 14 12 10 8 6 4 2 0 -2
fn=fs fn=3/4fs fn=1/2fs
-0.1
(c)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Uin(m/s)
Fig. 12. Vibration amplitudes of the monitored cylinder with fn = fs, 3/4fs and 1/2fs in the X- and Y-directions. (a) Cylinder array I, (b) Cylinder array III, (c) Cylinder array VI.
cylinder, however, depends on its natural frequency; i.e. how close the two natural frequencies, and also the value of the natural frequency that is an indication of stiffness of the cylinder. Regarding the effect of the cylinder stiffness on the cylinder vibration, comparing the vibration amplitudes for the cases of fn = 3/4fs and fn = fs shown in Fig. 12, the monitored cylinder with
T.-k. Lin, M.-h. Yu / Experimental Thermal and Fluid Science 29 (2005) 523–536
533
Flow
25% D
(a)
25% D
Uin=0.28 m/s
0.55
0.60
0.55
0.60
0.55
0.60
0.74
(b)
Uin=0.28 m/s
0.66
(c)
Uin =0.28 m/s
0.74
Fig. 13. Orbits of the monitored cylinder in the seven-cylinder array with different natural frequencies. (a) fn = fs, (b) fn = 3/4fs, (c) fn = 1/2fs. For each natural frequency, there are four orbits corresponding to four different inlet velocities.
lower structural stiffness (fn = 3/4fs) has a larger vibration amplitude when the flow velocity is just above the critical value. It is noted that at the high frequency fn = 5/4fs or beyond, the cylinder exhibited no obvious vibration in the high velocity range of the experiment, likely due to the more stiffness of cylinder structure. When fn = 1/2fs, though a sub-harmonics of fs but distant away from the fs, the vibration amplitudes above the critical velocity are not so large as in the other two cases. This could be a result of less degree of resonance when the natural frequency fn deviates from the fs. Especially, no obvious fluid elastic instability occurs in the case of the two-cylinder array as shown in Fig. 12(a) with fn = 1/2fs. The degree of resonance can be indicated by the characteristics of the orbits of the cylinder vibrations. As shown in Fig. 13 are the orbital paths of the monitored cylinder at the different natural frequencies in the sevencylinder array below, and above the critical velocity. The cylinders at fn = fs, and 3/4fs vibrate in a relatively organized orbit beyond the critical velocity. However, the cylinder at fn = 1/2fs, vibrates randomly, especially at high flow velocity. The randomness can be considered
as an indication of lack of resonance when fn and fs are far away, and thus an organized vibration cannot be produced. For the frequency content of the orbits, spectral data are further examined. 3.4. Spectrum of cylinder vibration When the inlet velocities are below the critical value, the spectra of the monitored cylinder are broadband, as shown in Fig. 14 for the case of Array III with Uin = 0.47 m/s. The broadband spectra reflect the random nature of the cylinder vibrations in the low velocity range.
Ay/D(%)
1
0.1 0.01 1E-3 1E-4 0
10
20
30
40
50
60
70
80
90
100
f(Hz)
Fig. 14. Amplitude spectrum of the monitored cylinder in the fourcylinder array, Uin = 0.47 m/s.
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T.-k. Lin, M.-h. Yu / Experimental Thermal and Fluid Science 29 (2005) 523–536
Above the critical velocity, the spectra of the monitored cylinder exhibit various spectral peaks depending on its natural frequency. Figs. 15–17 are the spectra of the cylinder vibrations in Arrays I, III, and VI, respectively. In the experiment, the natural frequency of the cylinder in each test array was adjusted at fn = fs, 3/4fs and 1/2fs. Only the amplitude spectra of the cylinder vibration in the Y-direction are presented here, since under the same test condition the streamwise spectrum is similar in nature to the corresponding cross-stream spectrum; namely, both the spectra show either broadband or line-dominated. In the latter case, the spectral peaks for the X- and the Y-directions occur at the same frequencies, though the spectral peaks may have different magnitudes. In the case of fn = fs, as shown in Figs. 15(a,b)– 17(a,b) for cases of different inlet velocities and cylinder arrays, a clear spectral peak at the natural frequency fn is observed when the flow velocity is around the critical velocity (Uin = 0.53 m/s). As the Uin increases, more
spectral peaks at harmonics nfn appear, suggesting an organized oscillatory behavior. In the case of fn = 3/4fs, as shown in Figs. 15(c,d)– 17(c,d), spectral peak at fn is observed when the flow velocity is around the critical value, Uin = 0.53 m/s. As the velocity increases, besides the fn-peak, more spectral peaks appear at fs, and at the harmonics of fn and fs (Figs. 15(d)–17(d)). It is thus evident that the vibration of the monitored cylinder has two major components at fn and fs above the critical velocity. The former component is associated with the natural frequency of the monitored cylinder, and the latter with the natural frequency of the surrounding cylinders. It is therefore illustrated that the cylinder, induced by the fluid flow, vibrates initially only at its natural frequency around the onset of fluid elastic instability. As the fluid velocity is increased, the cylinder vibrations have the frequency component of the surrounding cylinders through the coupling of the fluid flow.
Flow
Flow
(a)
(a)
fn
fn
(b)
(b)
fn
Ay/D(%)
Ay/D(%)
(c) fn
(c)
fn fs
fs
(d)
(d)
(e) (e) 10
fn
1
10
fs
1 0.1
0.1 0.01
(f)
(f)
0.01 1E -3
1E-3
1E -4 0
10
20
30
40
50
60
70
80
90
100
f(Hz)
Fig. 15. Amplitude spectrum of the monitored cylinder with the natural frequency fn in the two-cylinder array in cross flow with velocity Uin. (a) fn = fs, Uin = 0.53 m/s, (b) fn = fs, Uin = 0.74 m/s, (c) fn = 3/4fs, Uin = 0.53 m/s, (d) fn = 3/4fs, Uin = 0.74 m/s, (e) fn = 1/2fs, Uin = 0.53 m/s, (f) fn = 1/2fs, Uin = 0.74 m/s.
0
10
20
30
40
50
60
70
80
90
100
f(Hz)
Fig. 16. Amplitude spectrum of the monitored cylinder with the natural frequency fn in the four-cylinder array in cross flow with velocity Uin. (a) fn = fs, Uin = 0.53 m/s, (b) fn = fs, Uin = 0.62 m/s, (c) fn = 3/4fs, Uin = 0.53 m/s, (d) fn = 3/4fs, Uin = 0.62 m/s, (e) fn = 1/2fs, Uin = 0.53 m/s, (f) fn = 1/2fs, Uin = 0.62 m/s.
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535
indicated that the resonance among the cylinders is not so well when the two natural frequencies of the monitored cylinder and the surrounding cylinders are not close enough. Consequently, the cylinder vibrates relatively randomly, as already shown in Fig. 13(c).
Flow
(a) fn
4. Conclusions (b)
fs fn Ay/D(%) (c)
(d) fn fs
(e) 10 1 0. 1
(f)
0.01 1E -3 1E -4 0
10
20
30
40
50
60
70
80
90
100
f(Hz)
Fig. 17. Amplitude spectrum of the monitored cylinder with the natural frequency fn in the seven-cylinder array in cross flow with velocity Uin. (a) fn = fs, Uin = 0.53 m/s, (b) fn = fs, Uin = 0.74 m/s, (c) fn = 3/4fs, Uin = 0.53 m/s, (d) fn = 3/4fs, Uin = 0.74 m/s, (e) fn = 1/2fs, Uin = 0.53 m/s, (f) fn = 1/2fs, Uin = 0.74 m/s.
Particularly, for the case of the seven-cylinder array, the spectra have the frequency component fs in addition to fn, even when the fluid velocity is just slightly above the critical value, Uin = 0.53 m/s shown in Fig. 17(c). The enhancement of the fs-vibration is related to the more surrounding cylinders downstream in the sevencylinder array compared to the four-cylinder array. As the Uin increases, more spectral peaks are observed at the harmonics of fn and fs (Fig. 17(d)). The line-dominated spectra suggest an organized oscillatory behavior, which is consistent with the observation of the orbits already shown in Fig. 13(b). In the case of fn = 1/2fs shown in Figs. 15(e,f)–17(e,f), it is illustrated again that the monitored cylinder vibrates initially at its natural frequency fn around the onset of fluid elastic instability, and the frequency fs as well beyond the critical velocity. However, at high velocity such as U = 0.74 m/s, no obvious spectral peaks at the harmonics of fn and fs are observed (Figs. 15(f)–17(f)). It is
The cross-flow vibration of a circular cylinder (the monitored cylinder) surrounded by one to six identical cylinders elastically mounted in rotated triangular pattern is investigated. Analysis of the vibration amplitude reveals that the upstream cylinders have significant influence on the amplitude response of the monitored cylinder, either promote the instability by forming a gap flow impinging on the monitored cylinder, or enhance the cylinder vibration through better fluid coupling with more upstream cylinders. The downstream neighboring cylinders could suppress the vibration amplitude while the number of the downstream cylinders has little effect on the amplitude response for the present configuration. By the orbits of the cylinder motion, it is illustrated that around the critical velocity, the streamwise and cross-stream responses have the same frequency but with a phase shift. The phase shift tends to be 90 as the flow velocity is increased. With two or three upstream cylinders, the streamwise vibration of the cylinder is suppressed, while the cross-stream vibration is amplified, compared to the vibration of the cylinder with only upstream cylinder. With downstream cylinders added in addition to three upstream cylinders, the organized periodicity of the cylinder motion is disturbed and hence the vibration amplitude is suppressed. The discrepancy of the natural frequencies between the monitored cylinder and the surrounding cylinders is not a factor on determining the critical velocity. However, the vibration amplitude of the cylinder above the critical velocity depends on the value of its natural frequency, and the discrepancy of the natural frequencies. The monitored cylinder at high natural frequency fn P 5/4fs has no obvious vibration due to the cylinder stiffness. Large discrepancy of the natural frequencies reduces the vibration amplitude due to lack of resonance among the cylinders. Spectral data show that the cylinder, induced by the fluid flow, vibrates initially only at its natural frequency around the onset of fluid elastic instability. As the fluid velocity is increased, the cylinder vibrations include the frequency component of the surrounding cylinders through the coupling of the fluid flow. If both the nature frequencies are close, for instances fn = 3/4fs, and fn = fs, the cylinder exhibits a very organized oscillatory behavior, evidenced by the line-dominated spectra with spectral peaks at the harmonics of fn and fs.
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Acknowledgement This work was supported by the National Science Council, Taiwan, R.O.C. Thanks are due to Prof. C.C. Cheng for use of his computer facility.
[5]
[6]
References [1] R.D. Blevins, Flow-Induced Vibration, Van Nostrand Reinhold, New York, 1990. [2] M. Matsumoto, N. Shiraishi, H. Shirato, Aerodynamic instabilities of twin circular cylinders, Journal of Wind Engineering and Industrial Aerodynamics 33 (1–2) (1990) 91–100. [3] B. Dielen, H. Ruscheweyh, Mechanism of interference galloping of two identical circular cylinders in cross flow, Journal of Wind Engineering and Industrial Aerodynamics 54–55 (1995) 289–300. [4] M.J. Pettigrew, C.E. Taylor, Fluid elastic instability of heat exchanger tube bundles. Review and design recommendations,
[7]
[8] [9]
[10]
Journal of Pressure Vessel Technology, Transactions of the ASME 113 (2) (1991) 242–256. K. Schroeder, H. Gelbe, Two- and three-dimensional CFD-simulation of flow-induced vibration excitation in tube bundles, Chemical Engineering and Processing 38 (4–6) (1999) 621–629. H. Tanaka, S. Takahara, Fluid elastic vibration of tube array in cross flow, Journal of Sound and Vibration 77 (1) (1981) 19–37. H. Tanaka, K. Tanaka, F. Shimizu, Fluid elastic analysis of tube bundle vibration in cross-flow, Journal of Fluids and Structures 16 (1) (2002) 93–112. R.J. Moffat, Describing the uncertainties in experimental results, Experimental Thermal and Fluid Science 1 (1988) 3–17. V. Kassera, K. Strohmeier, Experimental determination of tube bundle vibrations induced by cross-flow, in: M.K. Au-Yang (Ed.), Proceedings ASME Symposium on Flow-Induced Vibration, New York, PVP-273, ASME, New York, 1994, pp. 91–97. D.S. Weaver, M. El-Kashlan, On the number of tube rows required to study cross-flow induced vibrations in tube banks, Journal of Sound and Vibration 75 (1981) 265–273.