The results are presented in four sections. The first section compares the non-dimensional displacement obtained from experiments with different space ratios L/D with the first or the second cylinder free to vibrate. The second section focus on the frequency ranges and patterns observed in the vibration of the cylinder with the change in the space ratio and flow velocity. In the third section, the analysis is related to the space ratio of 1.26, where the coherence between wake flow and cylinder vibration is investigated. The last section focuses on the case when both cylinders are free to vibrate, considering a fixed L/D = 1.26.
All the cases presented are compared to the results from the measurements with a single cylinder. The analysis with L/D = 1.26 was detailed due to the observed results comparing with higher space ratios.
3.1. Tandem Configurations—Non-Dimensional Amplitude and Main Frequencies
The vibration response with the cylinders is performed using non-dimensional amplitude and reduced velocity for cases with different values of L/D and the configurations first cylinder free to vibrate (FV), second cylinder free to vibrate (SV) and both cylinders free to vibrate (BV).
Figure 3 presents the cases tested and low amplitudes are observed due to the high mass ratio and the selected natural frequencies. The reduced velocities are higher than 6 and lower than 75. The cylinder was free to vibrate transversally to the flow; the mass and damping ratios and the natural frequencies are described in
Table 1.
The results for a single cylinder,
Figure 3, show non-dimensional amplitude RMS increasing with the increase in the reduced velocity, which is a consequence of the main flow velocity increase. This response is equivalent to the results presented in [
13]. Due to the high mass ratio and the high reduced velocities, the resulting amplitude values are low [
20].
The analysis with the first cylinder free to vibrate (FV),
Figure 3, presents higher values of non-dimensional displacement for L/D = 1.26 (Case 2), 1.4 (Case 4), and 1.6 (Case 6) than the single cylinder. In the case with L/D = 3.52 (Case 8), the first cylinder free to vibrate presents a similar non-dimensional displacement to the single cylinder. The higher non-dimensional amplitudes are observed in the L/D = 1.26 and decrease with the increase of the space between the cylinders. In the literature, the results presented by [
9] showed that the non-dimensional displacement was low for a reduced velocity higher than 10. In [
6] the high non-dimensional amplitudes for a cylinder with the same diameter and L/D higher than 5 was observed. From these analyses, it is expected that, in this configuration, the higher L/D will not influence the first cylinder wake formation, making the larger L/D to present an equivalent response with single cylinder.
The results with the second cylinder free to vibrate (SV),
Figure 3, show lower amplitudes than the single cylinder at L/D = 1.26 (Case 3), 1.4 (Case 5), and 1.6 (Case 7). In the case of L/D = 3.52 (Case 9), the second cylinder free to vibrate presents similar non-dimensional amplitudes with the single-cylinder and increases with higher values of reduced velocity. In this configuration, the low amplitudes can be related to the proximity of the cylinder and a non-interaction of the shear layers from the first cylinder with the second cylinder due to the close space ratio. The authors of [
12] stated that as the separation distance between the cylinders increases, the amplitudes of the oscillation of the second cylinder increase over a wide range of reduced velocities. This response can be related to wake-induced vibration and observed in
Figure 3.
The results from both cylinders free to vibrate are for L/D = 1.26 and indicate that the relation between the assembly characteristics of each cylinder is an important factor regarding the vibration response. In the results of Case 12, an increase in amplitudes on the second cylinder compared with the first cylinder can be observed. The results are equivalent in the initial reduced velocities and increase in C1 more than in C2. In this case, the first cylinder presents a lower natural frequency than the second cylinder. In the Case 13, the first cylinder presents higher amplitudes in the reduced velocities up to Vr = 12 that can be related to the vortex shedding process. After that the amplitude decrease. In Case 14, the response is closer but higher in the second cylinder; in this case both, cylinders present similar natural frequencies. The results for most cases remain between reduced velocities 20 and 72, but Case 12 (C2) and Case 13 (C1) start at a reduced velocity 6 due to the natural frequencies tested being higher than the additional cases tested.
In the literature, some studies observed similar results. The authors of [
9] studied two cylinders with a space ratio of 1.2 and observed higher amplitudes on the downstream cylinder for high reduced velocities; this was observed with both cylinders free to vibrate and with just the second cylinder free to vibrate. In L/D = 1.5, the first cylinder vibrating showed higher amplitudes than the second cylinder vibrating but with lower values than that observed for small L/D. For the two cylinders free to vibrate in a tandem arrangement, [
10] observed that the first cylinder free to vibrate increased the amplitudes for reduced velocities over 12 L/D = 1.57, while the second cylinder maintained low amplitudes.
Studies that investigated both cylinders free to vibrate include [
7,
9,
10,
21], or studies concerning the second cylinder free to vibrate are [
11,
22]. In all the cases, the results indicate that the increase in displacement occurs due to the influence of the first cylinder wake and due to the relation between the frequencies of the cylinders free to vibrate. The increase in displacement on the first cylinder with a close space, as observed in
Figure 3, is not common in the literature. Even with small amplitudes, the phenomena are relevant to applications such as tube banks, where the change in the fixation of the tubes can represent an application similar to the present configuration, FV or SV, and the close space is a characteristic observed in tube banks. To better understand the influence of the space ratio and the position of a cylinder free to vibrate on the dominant frequencies on a cylinder free to vibrate, the frequency spectrum, continuous wavelet, cross-correlation, and wavelet coherence were applied in the present analysis.
3.2. Frequency Ranges and Flows Patterns in Tandem Configurations
The comparison between the frequency response in the time domain and frequency–time domain for the tandem arrangement L/D = 1.26 (Case 2 and Case 3); L/D = 1.4 (Case 4 and Case 5); L/D = 1.6 (Case 6 and Case 7); and L/D = 3.52 (Case 8 and Case 9) are presented in
Figure 4. The results for the configuration with the first cylinder free to vibrate and the second cylinder free to vibrate are plotted. The results are presented for the reduced velocity of 72 for all the space ratios. The use of reduced velocities over 5, which is related to the vortex shedding, is due to the fluid elastic instability in tube banks that occur in reduced velocities over 30.
The results for L/D = 1.26 with the first cylinder free to vibrate are presented in
Figure 4a (Case 2), wherein the power spectrum of three distinct peaks can be observed. The two first peaks are associated with the natural frequencies and the third with the excitation of the flow and could be associated with the vortex shedding. The results for L/D = 1.26 with the second cylinder free to vibrate (Case 3) are presented in
Figure 4b. Two peaks linked to the natural frequencies are observed, the peak around 87.9 Hz can be linked to the excitations due to the increase in the flow velocity and the first cylinder wake. The response in the energy magnitudes and the distribution in the frequencies in
Figure 4a shows high energy levels encompassing the three peaks observed in the power spectrum. In
Figure 4b, the levels of energy decrease and spread in four main regions. The energy magnitudes are lower in the second cylinder than in the first cylinder. The higher non-dimensional amplitudes observed in
Figure 3 can be related to the conditions of the test that concentrate the energy in the main frequency.
The results for L/D = 1.4 with the first cylinder free to vibrate are shown in
Figure 4c, Case 4. The energy levels at the first and second peaks observed in the power spectrum can be associated with the natural frequencies. The third peak is observed at 33.2 Hz, which could be associated with the excitation of the flow. In the continuous wavelets, there is an increase in the energy spectrum associated with the first peak, mainly up to 15 s. The results for L/D = 1.4, with the second cylinder free to vibrate, are shown in
Figure 4d, and three peaks of energy are observed in the power spectrum results: the first two peaks are associated with the cylinder’s natural frequencies. The energy levels increased mainly with the third frequency peak, which can be associated with turbulence excitations and an increase in the kinetic energy.
In the literature, some studies presented similar results. The authors of [
9] studied two cylinders with a space ratio of 1.2 and observed higher amplitudes on the downstream cylinder for high reduced velocities; this was observed with both cylinders free to vibrate and with just the second cylinder free to vibrate. For L/D = 1.5, the first cylinder vibrating showed higher amplitudes than the second cylinder vibrating but with lower values than observed for small L/D. For the two cylinders free to vibrate in a tandem arrangement, [
10] observed that the first cylinder free to vibrate increased the amplitudes for reduced velocities over 12 and L/D = 1.57, while the second cylinder maintained low amplitudes.
The results for the configuration of tandem cylinders with L/D = 1.6 with the first cylinder free to vibrate are presented in
Figure 4e, Case 6. The main excitation frequencies remain associated with the natural frequencies range. The energy levels decrease and spread between the frequency ranges. The results for L/D = 1.6 with the second cylinder free to vibrate are presented in
Figure 4f (Case 7). The first peak is related to the second natural frequency and the second peak could be related to the electric power network but, in additional tests, this frequency increases with the flow velocity increase, since it is associated with the flow excitation
The results for the tandem configuration L/D = 3.52 with the first cylinder free to vibrate are presented in
Figure 4g (Case 8). Two peaks associated with the natural frequencies are observed. For L/D = 3.52, with the second cylinder free to vibrate (results are in
Figure 4h, Case 9), the energy increases and the third peak changes to 72.3 Hz and is associated with the increase in the velocity flow.
The results presented from L/D = 1.4,
Figure 4c,d, show similar responses to the results for L/D = 1.26,
Figure 4a,b. The main frequencies are in the same range, but the energy level is lower with the increase in the space ratio. For the first cylinder free to vibrate, it can be observed that the peak of frequency associated with the first natural frequency is higher and could be related to the mass-damping parameters and the pressure field that generates an excitation frequency close to the natural frequency.
The authors of [
2] presented an analysis of the mean pressure distribution around tandem fixed cylinders for several space ratios L/D. For the space ratio of L/D = 1.2, the author observed higher pressure coefficients on the first cylinder than in the second one. It was also observed that on the second cylinder, the pressure coefficients have asymmetric distribution along time due to the wake of the first cylinder. The increase in vibration of the first cylinder observed in this work could be related to the pressure distribution due to the second cylinder fixed and inside the wake.
In the results for L/D = 1.6,
Figure 4e, the energy level decrease compared with the smaller space ratios and the energy concentrates on the two natural frequencies. In
Figure 4f, the energy levels are lower than the ones observed for small space ratios, but the ranges of frequency remain the same. Comparing the results from L/D = 3.52 with the lower space ratios, it can be observed that for the first cylinder free to vibrate, energy levels must decrease significantly (
Figure 4g). For the second cylinder free to vibrate and L/D = 3.52, an increase in energy associated with the natural frequencies can be observed.
The authors of [
8] studied the wake-induced vibration in an L/D = 4 tandem cylinder set. In their numerical results for low Reynolds numbers, they observed a large amplitude of the downstream cylinder associated with a low-frequency component of the transversal load, which was closer to its natural frequency. The authors of [
7] tested both cylinders free to vibrate and observed that the second cylinder underwent a resonance-like response as the flow velocity increased and did not stabilize until the highest reduced velocity for small and moderate spacings (L/D = 3.5 to 10). The second cylinder’s oscillation frequency in the transverse direction was not affected by the spacing variations, which was a result of the interaction between fluid and the cylinder itself. The authors of [
9] tested cases with L/D = 3 and L/D = 4, with lower amplitudes than for the lower L/D tested in this paper.
For L/D = 3, [
2] observed that for tandem-fixed cylinders, higher pressure coefficients in the second cylinder and an asymmetric distribution compared with lower space ratios; due to the distance from the first cylinder, its wake impinged on the second cylinder. The authors of [
11] showed that by increasing the mass-damping parameter, the amplitude decreases and the range of lock-in becomes narrow. For small space ratios, lower amplitudes were observed.
The flow patterns observed in visualizations and the monitoring of pressure or force in the literature show similarities; a scheme of the flow over tandem cylinders with variation in the space ratio tested is shown in
Figure 5. The scheme derives from the results from fixed cylinders and the pressure distribution presented in the literature [
1,
2,
23,
24]. The discussion is linked to its impact on the vibration response.
For L/D = 1.26 (
Figure 5), it is observed that the wake from the first cylinder encompasses the second cylinder due to the close space, this characteristic generates a mean pressure field symmetric [
2] in the second cylinder and can be related to the observed increase in displacement observed in
Figure 3. The frequency peaks around the natural frequencies of the first cylinder can be associated with the small distance between the cylinders and non-interaction of the shear layers from the first cylinder with the second cylinder. On the second cylinder free to vibrate, the vortex shedding frequency is observed due to the changes in lift force during the vortex formation.
For the L/D = 1.4 arrangement, the flow pattern can allow the reattachment of the shear layer on the second cylinder creating an asymmetric pressure field. The distance between cylinders in the L/D = 3.52 configuration allows the formation of a vortex street between the cylinders that impinges on the second cylinder in the tandem configuration, generating an excitation frequency linked to the wake from the first cylinder.
The visualization presented in [
23] for L/D = 1.26 and [
24] for L/D = 2.5 and L/D = 5 showed the different flow pattern around tandem cylinders.
The study of the frequency ranges of excitation for all the conditions tested is presented in
Figure 6, where the results from the power spectrum analysis were compiled and plotted. The analysis is presented using the acceleration signals obtained in the cylinder free to vibrate. The natural frequencies are represented with vertical lines for 7.8 Hz and 21.5 Hz. The tested cases are presented for each reduced velocity with the energy level obtained from the power spectrum results. In the x-axis, the frequency and the corresponding Strouhal number are presented. The Strouhal number is determined with the reference velocity and the diameter of one cylinder.
In a general analysis, it can be observed that all the cases concentrate peaks around the natural frequencies, but results between 60 Hz and 90 Hz are also visible, depending on the reduced velocity. In
Figure 6a, the results obtained in reduced velocity, Vr, equal to 49, show the highest levels of energy linked to the second natural frequency for L/D = 1.26 and with the second cylinder free to vibrate. The second cylinder free to vibrate shows frequency peaks around 60 Hz to L/D = 1.26, 1.4, and 1.6 and can be associated with St = 0.16 and St = 0.17. In
Figure 6b, in the results for reduced velocity, Vr, around 60, the frequencies remain to concentrate around the natural frequencies. The second cylinder free to vibrate presents frequency peaks between 60 Hz and 80 Hz, generating a Strouhal number between 0.13 and 0.17. In
Figure 6c, the results with reduced velocity, Vr, around 70, show that the second cylinder free to vibrate remains with a peak of frequency between 60 Hz and 100 Hz, which is related to the increase in the flow velocity. The Strouhal numbers observed remain between St = 0.13 and St = 0.16.
The Strouhal numbers obtained from the first cylinder free to vibrate are lower than expected for the tandem configuration tested in [
2,
3]. The frequency peak observed in the power spectrum can be an excitation not directly associated with the vortex shedding. Due to the high interaction between the cylinders at a close space ratio, the flow excitation was able to maintain the response around the natural frequency once a non-clear excitation occurred in the first cylinder due to the presence of the second cylinder inside the wake. The flow excitation can also be influenced by the mass-damping parameter of the free to vibrate cylinder, altering the FIV-response.
The Strouhal numbers obtained from the second cylinder free to vibrate agree with the literature [
2,
3], being associated with the wake from the first cylinder. Even for the closest space tested cases, it can be observed the shear layer interact between the cylinders due to the vortex formation.
3.3. Tandem Configuration L/D = 1.26—Coherence between the Wake Velocity and the Acceleration
The results in
Figure 3 and
Figure 6 showed that the close space impacts the first cylinder vibration response, which is a different mechanism than the wake-induced vibration observed in higher space ratios, where the second cylinder present higher amplitudes. To understand this behavior, a study with hot wire and an accelerometer were utilized for L/D = 1.26, changing the cylinder free to vibrate position, the first cylinder free to vibrate (Case 10), and the second cylinder free to vibrate (Case 11). The Fourier spectrum of the wake velocity signal for both cases remain an equivalent frequency range, between 50 Hz and 100 Hz, as observed in
Figure 7a,b; but in Case 11, the adjacent regions present lower energy. In both cases, the hot wire anemometer was positioned after the second cylinder, as presented in
Figure 2c.
The acceleration spectra, in
Figure 7c,d, show a change in the frequency ranges where the energy level is higher. In the result from the first cylinder free to vibrate,
Figure 7c, the energy concentrates in the range under 50 Hz that encompasses the cylinder natural frequencies. In
Figure 7d, the levels of energy are low, but the region of excitation is between 50 Hz and 100 Hz, the same as observed in the wake velocity signals.
To investigate this relation, the coherence analyses were applied to the signals of wake velocity and acceleration for the Case 10 and Case 11 with the reduced velocity 63. The results are presented in
Figure 8a for the first cylinder free to vibrate (Case 10) and 8b for the second cylinder free to vibrate (Case 11). The coherence analysis applies the coherence function with values between zero, which represents no coherence to a value of 1, which represents a complete coherence between the signals. As the coherence values decrease, the results can be associated with non-linear systems [
16]. This analysis allows signals to be related and to identify whether they present a frequency range of direct relation.
In
Figure 8a, a region of high coherence, around 90%, between 80 Hz and 90 Hz is observed and a peak with around 10% coherence is observed around 30 Hz. A level in the phase angle of 120° can be observed between 20 Hz and 50 Hz in
Figure 7b, demonstrating that there is a relationship between the signals in the second natural frequency range. Between 70 Hz and 100 Hz, a region around 40° shows a stabilization tending to a level but not in a single-phase angle. The random phase angle response, over 100 Hz, can be linked to higher frequencies in the turbulence spectrum.
In
Figure 8c, the region of high coherence is between 90 Hz and 100 Hz and a second region with about 10% coherence is observed between 100 Hz and 110 Hz. The phase angle, in
Figure 7d, presents a level of around 10° between 60 Hz and 100 Hz and a random phase angle response over 110 Hz. The high coherence is linked to the flow excitation, and a Strouhal number of around 0.17. This response represents the relation between the signals but not the strength of this relation (levels of energy).
The wavelet coherence is applied due to the non-stationary characteristic of the signals. The wavelet coherence analysis can provide insights into how two responses have evolved in time and frequency. The results of wavelet coherence show the color pallet representing red for one (100% coherence) and blue for zero (no coherence). The arrows represent the phase angle between signals for coherence values higher than 0.75. The wavelet coherence is shown in
Figure 9a, representing the result of the wake velocity signal and the acceleration. A region of high coherence is observed between 90 Hz and 120 Hz. The regions between 8 and 16 Hz occasionally show high coherence (8 s, 12 s, and 28 s). The region between 16 Hz and 32 Hz shows high coherence in some positions.
In
Figure 9b, the coherence remains over 75% between 64 Hz and 128 Hz for all the time analyzed. The phase regions with lower frequencies show punctual regions as between 4 and 16 Hz around 15 s and between 16 and 32 Hz before 10 s, this region can be linked to the natural frequencies. The response observed in
Figure 8 and
Figure 9 shows that the shedding frequency is presented in both situations tested, but it is not the main contribution to the increase in amplitude for the L/D 1.26, because the high energy in those cases is related to the natural frequency, as observed in
Figure 4 and
Figure 7. The region of natural frequency shows a low coherence in the case with the first cylinder free to vibrate and can be a consequence of the movement observed in the first cylinder that is free to vibrate. In the second cylinder free to vibrate, the coherence between the vortex shedding and acceleration is close to the maximum due to the direct interactions of the cylinder movement and the wake; this response was expected due to the energy response observed in
Figure 4, where a region of excitation associated with the vortex shedding was also observed.
The response observed can be related to results presented by [
1], where the pressure distribution around cylinders in tandem with L/D = 1.2 was presented and in the first cylinder showed low-pressure coefficients after 90°; this is also observed between 0° and 50° in the second cylinder. The pressure fluctuations are high in the second cylinder between 60° and 120°. The region of low-pressure coefficients occurs between the cylinder and influences the fluctuations of the lift force. The fluctuations of pressure and force are low, creating a small movement in the cylinder around the natural frequencies. This force is not high enough to result in a significant amplitude but causes the cylinder to move at a frequency near to the natural, due to the close space.