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Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem
Journal of Fluids and Structures xxx (xxxx) xxx
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Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem ∗
F.J. Huera-Huarte a , , J.I. Jiménez-González b a b
Department of Mechanical Engineering, Universitat Rovira i Virgili, 43007 Tarragona, Spain Departamento de Ingeniería Mecánica y Minera, Universidad de Jaén, Campus de las Lagunillas, 23071 Jaén, Spain
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Article history: Received 5 October 2018 Received in revised form 11 April 2019 Accepted 15 April 2019 Available online xxxx Keywords: Vortex-induced vibrations VIV VIV sensitivity Tandem cylinders Vortex-shedding
a b s t r a c t We present an experimental study on the effects on the vortex-induced vibrations (VIV), of the relative size of two rigidly connected circular cylinders in a tandem arrangement, with a fixed centre-to-centre separation of 1.3D, where D represents the diameter of the upstream cylinder. This separation distance was selected after the results obtained by Jiménez-González and Huera-Huarte (2017), where a high VIV sensitivity region was identified at such location in the wake of a cylinder oscillating in cross-flow. The flowinduced response of the system, which has one degree of freedom, has been analysed for several values of the diameter ratio d/D ≤ 1, where d is the diameter of the downstream cylinder. As the value of d/D increases, the VIV response is attenuated, it being associated with important reductions in the transverse and in-line force coefficients. In addition, Digital Particle Image Velocimetry (DPIV) has allowed us to study the changes in the near wake and forcing introduced by the presence of different downstream cylinders. Thus, the flow dynamics is clearly driven by the relative size of the cylinders, and therefore, the size of the gap region. In particular, for small values of d/D, a moderate disruption in the vortex formation and shear layer interaction of the wake of the upstream cylinder takes place. For intermediate values of d/D, the downstream cylinder generates a strong local wake that results in the shedding of two co-rotating vortices at each side of the system, while markedly undermining the shedding from the upstream cylinder. In the limit of d/D = 1, a stagnant region develops in the gap between the cylinders, and the tandem behaves as a single bluff body, featuring a very attenuated flow-induced response. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction The study of the interaction between the wakes of bluff bodies has received a considerable amount of attention, since its analysis is crucial for the understanding of the response and forcing mechanisms in structures exposed to winds or water currents. Circular cylinders are widely used as a canonical idealisation of such structures. Most of the existing literature has dealt with stationary cylinders such as in the pioneering work by Igarashi (1982). Zdravkovich (1997) reviews in his book the wide variety of works that deal with the flow around stationary configurations of cylinders. Sumner (2010) more recently compiled a large number of works devoted to the analysis of the forcing and the flow around pairs of cylinders in different configurations, ranging from the side-by-side to the tandem ones. Nevertheless, the number of published works ∗ Corresponding author. E-mail addresses: [email protected] (F.J. Huera-Huarte), [email protected] (J.I. Jiménez-González). https://doi.org/10.1016/j.jfluidstructs.2019.04.006 0889-9746/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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is much more limited in the case of a non-stationary tandem arrangement of circular cylinders. The fact that one or both of the cylinders are allowed or forced to vibrate, changes considerably the flow physics around the bodies even if the flow regime is not modified, resulting in large changes in the separation points, formation lengths, formation of shear layers, etc. This is why we focus this brief introductory review to cases that deal with tandem arrangements in which motion is allowed or forced. Knisely and Kawagoe (1990) investigated the loading characteristics on a cylinder forced to move inside the wake of an stationary leading cylinder, revealing important timing relationships between vortex formation and lift forces. Yang et al. (2014) used a forced vibration set-up, to focus on the study of the different types of wake that appear behind a tandem arrangement of cylinders in which motions could be forced independently in each model. The work was limited to the analysis of the wake patterns that appeared behind the models after carrying out hydrogen bubble flow visualisation. The authors described mainly two modes of combined shedding depending on the imposed oscillation frequency, with a different number of vortices per cycle at each side of the wake. Going one step further in complexity, some authors have analysed the case of the cylinders being flexibly mounted, therefore leaving the possibility of fluid-elastic instabilities and self-excited phenomena to appear. See for example the pioneering works by Bokaian and Geoola (1984a,b), where the authors varied systematically the relative position of the cylinder models to measure dimensionless frequencies, excitation and response of the trailing cylinder, describing what they called galloping-like dynamics. Hover and Triantafyllou (2001) carried out experiments with a circular cylinder placed 4.75 diameters behind another stationary model, because at that distance (Zdravkovich, 1985) had previously found the largest motions in his experiments. Brika and Laneville (1997) conducted studies in a wind tunnel with flexible cylinders (although limited to move in their first mode of vibration), showing a clear hysteretic behaviour in their response depending on whether the upstream cylinder was fixed or allowed to vibrate. More work with free-to-oscillate cylinders in the wake of stationary ones has been carried out recently (Assi et al., 2006, 2010, 2013; Carmo et al., 2011; Qin et al., 2017). Some of these authors have pointed out the fact that the mechanisms that drive the wake-induced vibrations of the downstream cylinders, are not resonant but caused by the flow unsteadiness. This is in contrast to what happens in the case of the cylinders being elastic themselves, where the resonant characteristics of the structure dominate the response (Huera-Huarte et al., 2016). In an scenario such as this, the response and the forcing on a flexible cylinder immersed in the wake of a stationary one, is characterised by very rich dynamics. The case in which both cylinders in a tandem arrangement are free to move and subject to a cross-flow, has been barely studied as the variable space that needs to be covered is very large and the interactions are of extreme complexity (see e.g. Kim et al., 2009; Borazjani and Sotiropoulos, 2009; Huera-Huarte and Bearman, 2011; Huera-Huarte and Gharib, 2011; Griffith et al., 2017). In all these works it is emphasised that the driving physics are closely conditioned by the gap between the cylinders, that it is in fact dynamic and dependent on the flow conditions. In this work, we study the effects of the relative size, on the vortex-induced vibrations (VIV), of two rigidly connected circular cylinders in a tandem arrangement with a fixed gap. The separation distance (1.3 diameters) was selected after the results obtained by Jiménez-González and Huera-Huarte (2017), where a series of high VIV sensitivity regions were identified in the wake of a cylinder oscillating in a cross-flow. The study was conducted by rigidly connecting a small cylinder (3.2% in size when compared to the main one) in the near wake of an elastically mounted low mass-damping cylinder. An open question remained unanswered: what would be the effect of varying the size of the cylinder placed at these high sensitivity regions? The effect of size in symmetric configurations falling in high sensitivity regions was studied by Jiménez-González and Huera-Huarte (2018), and here we intend to clarify such issue for the case of a tandem arrangement, providing as well wake dynamics measurements. The works by Zhao (2013) and Zhao et al. (2015) are therefore relevant as the authors analysed tandem configurations formed by rigidly linking a pair of circular cylinders. The paper is organised as follows: the experimental methods are presented in Section 2; while results of dynamic response obtained from tests conducted at different reduced velocities for selected values of the downstream cylinder’s diameter are presented in Section 3. Measured force distributions and corresponding frequencies are first analysed, to subsequently describe flow visualisations and discuss how the near wake modifications introduced when the size of the downstream cylinder is modified affects the excitation mechanism. Finally, the main conclusions are drawn in Section 4. 2. Experimental methods The facility used to conduct the research is the free surface water channel (FSWC) at the Laboratory for Fluid–Structure Interactions (LIFE) of the Universitat Rovira i Virgili (URV) in Tarragona. The channel has a three-dimensional contraction after which a uniform profile is generated, in a 1 m × 1.1 m × 2.25 m translucent working section, especially designed to use optical measurement techniques. Periodic measurements carried out with a traverse system and an Acoustic Doppler Velocimetry (ADV) at several locations in the working section (usually 3 cross-sectional planes with 33 time series measurements in each) are used to confirm the high quality of the flow generated, with a spatio-temporal variability of the flow speed of less than 1.6%, with a turbulence intensity (root mean square of the velocity upon its mean) of 2.79% at the highest regime of the channel. Fig. 1 displays a lay-out of the experimental set-up and the tandem cylinder model arrangement used in the present investigation. As shown in Fig. 1(a), a support structure made of aluminium profiles was built on top of the channel, without touching it to avoid the transmission of any small vibrations due to the operating pumps, hanging from the Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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Fig. 1. Schematic view of the experimental set-up.
building structure. An air bearing rig was installed on the supporting structure, with its main beam perpendicular to the flow direction in the channel. The air bearing system consisted of a beam in which a carriage could move freely having no friction. The displacements of the carriage along the air bearing beam (y-coordinate) were measured using a non-intrusive laser displacement sensor (absolute resolution of 0.1 mm.). A six-axis load cell, with a measurement range in the in-line and transverse directions of 100 N (with an uncertainty rated at less than 0.15% of its full scale.), was installed below the carriage by means of a especially designed 3D printed solid part (100% density with Polylactic Acid PLA), to which the cylinder model was attached to. This load cell allowed the direct measurement of in-line and transverse forces acting on the cylinder. To validate the direct measurements, a second independent precision uni-axial load cell (measurement range of ±8.9 N with an uncertainty of 0.1% of its full scale) was used to redundantly measure the in-line or drag forces in specific cases. This was done for validation, because the observed forces in the experiments were in the 10% band of the full scale of the six-axis load cell. Moreover, forces were also obtained by using the displacement data and its time derivatives, fed to a one degree of freedom model equation. All three independent and redundant ways of measuring forces, showed differences of less than 3% at all times. More details and comparisons between these independent measurement methodologies can be seen in the recent work Huera-Huarte (2018), conducted with the same set-up and techniques. The circular cylinder model used for the experiments was made of a transparent acrylic tube with external diameter D=50 mm (inner diameter 40 mm), and once in position, penetrated the free surface having a submerged length L of 0.5 m. An acrylic plate was installed at lower end of the model to avoid having end effects. Two 3D printed solid parts (see Fig. 1b), one near the bottom end resting on top of the end plate, and another just above the free surface, provided support for the secondary (or tandem) cylinder model. The parts ensured a precise fixed location of the secondary cylinder behind the main cylinder model, at a radial distance r = 1.3D, whilst introducing minimal flow disturbances. Note that the distance r (Fig. 1c) was selected after the recent VIV sensitivity study by Jiménez-González and Huera-Huarte (2017), where it was seen that a local perturbation, with the form of a very small control cylinder, at certain centre-to-centre locations in tandem, had large effects on the VIV response of a cylinder. An omnidirectional system based on a wire mesh designed to suppress VIV has also shown great capabilities on modifying the dynamic response of a circular cylinder when similar radial distances are adopted (Huera-Huarte, 2017). Restoring forces for the carriage were provided by two springs that connected the cylinder to the external supporting structure. The trailing cylinder support parts allowed to install models with different diameters d in the selected fixed tandem position, so that the effect of the diameter ratio, d/D, could be studied in detail at this high VIV sensitivity fixed position. In particular, 6 different values of diameters ratio were analysed, namely d/D = [0.12, 0.2, 0.32, 0.4, 0.5, 1]. The study is limited to d/D ≤ 1, because with the fixed separation distance in the tandem (1.3D), as the cylinders start to have a similar size, the shear layers formed in the upstream one attach to the trailing one, in a single bluff-body like mode of shedding, and if further increased they may eventually become a single body. In addition, an in-house developed Planar Time-Resolved Digital Particle Image Velocimetry (DPIV) system, allowed the interrogation of the flow field around the rigidly coupled tandem arrangement of cylinders in specifically selected cases. The purpose was to study the global changes introduced in the wake of the leading cylinder by the control one, as its size was varied. Seeding in the channel consisted of 10 µm neutrally buoyant Polyamide seeding particles (PSP) that were illuminated using a 5 W continuous wave Diode Pumped Solid State (DPSS) green laser source. A 12 bit 1.3 Mpixel high speed camera equipped with a 20 mm fixed focal lens objective, provided images of the illuminated particles at a rate of 500 Hz, yielding time resolved flow fields with a ∆t of 2 ms. The image dynamic masking, PIV image processing and Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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post-processing of the velocity fields obtained, result from an in-house code that includes part of the work by Willert and Gharib (1991) and Thielicke and Stamhuis (2014). Image processing consisted of an initial step in which regions without seeding particles or illumination were masked. The position of the centre of the cylinder was also identified in each snapshot by image processing. A scaling factor of 3.31·10−4 m/pixel was obtained by imaging an illuminated rectangular target, placed at the plane defined by the laser sheet. A Fast Fourier Transform (FFT) cross-correlation algorithm was applied to the dynamically masked images, using an interrogation area of 24 × 24 pixels with a 50% of area overlapping, to produce the velocity vectors. The optimal window interrogation area was found by reducing progressively its size until the point at which the number of spurious vectors resulting from the processing was considered not acceptable. With the 24 × 24 pixels window, the number of spurious vectors obtained after the processing was very small and always less than 2% of the total. Outlier vectors output by the cross-correlation scheme, were identified after applying a threshold and replaced by new values obtained from averaging around the outlier. The field of view (FOV) was 265.1 × 232.0 mm (5.3×4.6D) and a total of 3705 (65 × 57) velocity vectors were obtained, implying a spatial resolution of 4.1 mm (1 velocity vector in a region of the measurement plane with size of 4.1 × 4.1 mm2 ). This resolution is the consequence of the need for accommodating a relatively large field of view, in the order of 5D, with the smaller structures emanating from the trailing cylinder with size d. Although √ this set-up cannot resolve quantities such as for example, the boundary layer on the cylinders (at this Re is ≈ O(D/ Re) ≈ 0.028D ≈ 1.4 mm) and other small scale structures in the flow, it is adequate to study qualitatively the overall changes in the wake of the system at scales in the order of D, resulting from the variations in the size of the trailing cylinder. An indication of the quality of the measurements obtained, apart from the very small number of spurious vectors obtained, is given by comparing the flow speed in the channel, measured with the PIV system and with another independent technique. As described above, the flow speed was measured using a high-resolution acoustic Doppler velocimeter at points very near the PIV FOV, without perturbing it, and was compared to the output given by the PIV (spatio-temporal average of a region inside the FOV), showing differences between both that were always smaller than 5%. 3. Results and discussion The results presented in this section consist of the dynamic response of the cylinder and the fluid excitation. These are related to the wake topology investigated in several selected experiments, for a detailed physical explanation. In all the plots of this section, the response of the isolated cylinder is included together with the rest of configurations for reference. Results for each configuration were obtained by stepping up the free stream velocity (U) in the channel in small increments and allowing the flow to become steady. Thus, the range of reduced velocity covered in the experiments is approximately U ∗ = U /fn D = [4, 14], where fn represents the natural frequency of system oscillations in still water. Therefore, this range corresponds to the following interval of the Reynolds number, Re = UD/ν = [7000, 28000], with ν being the fluid kinematic viscosity. Besides, in these experiments, as the trailing cylinder is changed to achieve the different diameter ratios investigated, the mass (and the added mass) of the moving system change depending on the specific set-up. This aspect has been taken into account in each set of results, by compensating the displaced mass by means of additional weight to keep a constant value of the mass-damping parameter m∗ ξ ≃ 0.009, where m∗ is the mass ratio, the ratio between system and displaced mass, and ξ corresponds to the structural damping in air (measured in freedecay tests). In addition, the response and the forcing were measured during at least 60 s, to ensure that a large quantity of oscillation cycles were acquired, while the signals sampling frequency was set to Fs = 1.5 kHz to properly resolve the system dynamics and characteristic frequencies. The amplitude of the oscillating response, A, was computed as the mean of the envelope (Hilbert transform) of the time series of displacement, as in Huera-Huarte (2017). According to the description of the experiment given in the previous section, the uncertainty in the dimensionless amplitudes (A∗ = A/D) is estimated to be in the order of 0.2% D. In the case of Re uncertainties are approximately 1.6%, as it directly derives from the free-stream velocity in the water channel. The uncertainty associated to the Fourier analysis of the signals, due to the available frequency resolution is 1.5%, if referred to the natural frequency of the system (0.002% of the Nyquist frequency in the experiments). All these, yield an uncertainty estimate for the reduced velocity with an upper bound of approximately 2.2%. The data set has been divided into two subsets, that have been arranged in columns inside the figures, for the sake of clarity. Circles (◦) are used in all the plots for referring to the isolated cylinder case. In that sense, the amplitude response of cylinders in dimensionless form A∗ is presented in the upper row of plots of Fig. 2, against the reduced velocity U ∗ . As it can be seen in the plots, the isolated cylinder exhibited the low mass-damping classical VIV response, with well defined initial (U ∗ < 6), upper (6 < U ∗ < 8) and lower (U ∗ > 8) branches, with results being close to those reported previously, using a similar set-up (Huera-Huarte, 2017; Jiménez-González and Huera-Huarte, 2017; Huera-Huarte, 2018). The maximum amplitude is over 1D at a reduced velocity near 6.5. Dominant frequencies, f , appear in the lower row of Fig. 2, divided upon the natural frequency of the model in water (f ∗ = f /fn ). For the isolated cylinder case, at the lowest reduced velocities, the frequency followed that imposed by the shedding, until reaching the natural frequency of the system, point at which it stabilised indicating a resonant condition. In the lower branch there is a plateau with frequencies that are slightly higher than the natural frequency of the system. When a second cylinder is rigidly coupled to the main cylinder, at the fixed distance of r = 1.3D in tandem position, the response varies considerably. In fact, it is highly dependent on the diameter ratio d/D between the cylinders. The Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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Fig. 2. Dimensionless amplitude (A∗ ) and frequency (f ∗ ) response of all configurations as a function of reduced velocity (U ∗ ) and Re. Inset plot in (b) depicts the maximum dimensionless amplitude A∗max against the diameter ratio d/D.
cases with d/D of 0.12 and 0.20 depict response trends that show the initial, upper and lower branches but with smaller amplitudes and narrower reduced velocity ranges. For the smallest diameter ratio investigated, the maximum amplitude is around a 10% smaller, although happening at the same reduced velocity near 6.5, as in the isolated case. These results confirm previous research conducted by Jiménez-González and Huera-Huarte (2017). The upper branch is narrower with a sharp peak that drops just after the maximum is reached. The amplitudes in the lower branch are again lower that in the isolated case. These two effects, are more pronounced when the diameter ratio is 0.2, with amplitudes even lower, a narrower upper branch and smaller amplitudes in the lower branch. Frequencies in these two cases do not practically vary when compared to the isolated case. Responses change dramatically at all the other diameter ratios investigated. In the cases with d/D of 0.32 and 0.4, the upper branch disappears and amplitudes do not surpass a value of 0.5D at any reduced velocity. Once the amplitude reaches that value at a reduced velocity around 6, it stays at that level going slightly down to values closer to 0.4D for U ∗ > 8. Although the frequencies for the case with a diameter ratio of 0.32 are almost identical to those of the isolated cylinder, for the case with d/D= 0.4, frequencies change becoming in general smaller at all U ∗ . A change in the diameter ratio to 0.5 yields larger amplitudes, near 0.7D, with the response peak at a reduced velocity of almost 8. Frequencies evolve parallel to those of the other arrangements but at smaller values. Finally, the case with the two cylinders in the tandem with the same diameter, shows practically no response for reduced velocities up to 7, and responses with amplitudes near 10% of the diameter at the rest of reduced velocities. In this case, there is a large jump in frequencies from the shedding expected ones to values that stay at f ∗ ≈ 0.75 for U ∗ > 6.5. This sudden change in frequency was not reported in the low Re number simulations by Zhao (2013), but appeared in the 2D RANS simulations by Zhao et al. (2015), conducted at a Re = 5000, associated to amplitudes in the order of 0.5D outside the lock-in range. The authors associated these phenomenon to a galloping like response. Maximum amplitudes A∗max as a function of d/D are shown in the inset plot of Fig. 2b), where it can be seen how increasing the diameter of the control cylinder means decreasing VIV amplitude, except for a small jump at a d/D = 0.5. Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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Fig. 3. Dimensionless standard deviation of the transverse force (C˜ y ) and phase difference between response and force (φ ) of all configurations as a function of reduced velocity (U ∗ ) and Re. Inset plot in (b) depicts the maximum transverse force C˜ ymax against the diameter ratio d/D.
Following the dynamic response of the system, in Fig. 3, the transverse excitation is presented. The upper row of plots is for the standard deviation of the dimensionless transverse force (C˜y ). The time series of the force coefficient were obtained from the measured transverse forces Fy (t) by applying the expression, Cy (t) =
Fy (t) 1 2
ρ U 2 DL
(1)
with ρ being the fluid density. The uncertainty estimated for the force coefficient, derived from that of the force measurements and free stream velocity, has an upper bound around 3%. In the lower row of plots in the figure, the phase difference between the transverse force and the displacement, φ , is included. Phase differences have been obtained as in Huera-Huarte and Bearman (2009), by using the analytical signal concept, using Hilbert transforms (Pikovsky et al., 2001). In the case of an isolated cylinder undergoing VIV, it is well known that transverse force ramps up quickly up to its highest values (near 1.8 in this case), that coincide with the response peak, at a U ∗ ≈ 6. Then it quickly descends, in this case to values near 0.4, to continue decreasing very slowly to values near 0.2, at the end of the lower branch. The forcing is in phase with the response in the initial and upper branches with the system being dominated by elastic forces, and changes suddenly at the end of the upper branch to an out-of-phase condition in which inertial forces are the dominant. The same behaviour, with variations in the force amplitudes, has been observed for diameter ratios of 0.12, 0.2 and 0.32 with peak C˜y in the order of 1.7, 1.7 and 1.1 respectively. Phase differences for these 3 cases are practically identical to those observed in the isolated case. When the cylinder with diameter d = 0.4D was installed in the tandem, a narrow forcing peak appeared but at higher U ∗ , reaching values around 1.4. After the peak, the forcing became very similar to that shown by the isolated cylinder. The phase jump is also delayed to larger reduced velocities. When the diameter ratio was further increased to a value of 0.5, the C˜y curve moved to higher reduced velocities and peak values near 1.5, with phases that implied a clear in-phase condition only when U ∗ < 9. For larger reduced velocities, phases seemed not to follow a Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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Fig. 4. Dimensionless time averaged in-line force (C x ) as a function of reduced velocity (U ∗ ) and dimensionless amplitude (A∗ ). Inset plot in (b) depicts the maximum time averaged in-line force coefficient C xmax against the diameter ratio d/D.
clear trend, with values in-between the in-phase and the out-of-phase conditions. Finally, the results for the one-to-one diameter ratio show very small lift coefficients that stay at all reduced velocities in-phase with the displacements. The inset plot in Fig. 3b) shows the evolution of the maximum transverse force coefficient (C˜ ymax ) as a function of the diameter ratio (d/D). If the time series of the in-line force Fx (t) is used in Eq. (1), instead of Fy (t), the drag coefficient is obtained. Fig. 4 presents the time averaged drag coefficients (C x ) as a function of U ∗ in the upper row of plots and as a function of the response amplitude (A∗ ) in the lower row. The so called drag amplification has been observed not only in the VIV of elastically mounted rigid cylinders as in here, but also in flexible ones (Chaplin et al., 2005; Trim et al., 2005; Huang et al., 2011), with large in-line forces correlated to large amplitude responses. Here, while the plots show a clear relationship between amplitude and drag for the isolated case and the cases with diameter ratio of 0.12 and 0.2, this is not true for the rest of cases. Diameter ratios of 0.32, 0.4 and 0.5 are characterised by similar wide inverted parabolic C x curves when plotted against reduced velocity, with values that stay between 1.2 and 1.5 at all reduced velocities, yielding drag coefficients that are smaller than those of the isolated case in the upper branch, and larger outside. The case of equal diameters, depicts a very different situation, with a large drag reduction at all reduced velocities with coefficient values that stay under 1. Again an inset plot in Fig. 4a) shows the maxima of the drag coefficient (C xmax ) at each diameter ratio. From the three previous figures it appears obvious that the classical VIV response has been completely altered when the trailing cylinder has the same diameter as the upstream one, when r = 1.3D. The fact that there is practically no amplitude response together with very small cross-flow and transverse excitations, indicates that a large modification of the classical vortex street has taken place. Fig. 5 shows the vorticity field around the rigidly linked tandem arrangement for a case with U ∗ ≈ 7.5. Vorticity appears in all plots phase averaged and in dimensionless form (ω∗ = ωz D/U), with ωz being the out-of-plane vorticity, and the red colour indicating clockwise direction of rotation and blue counter-clockwise. Phase averaged plots were obtained after identifying, using the cylinder position, several specific phases along the trajectory of the cylinder, using at least 10 oscillation cycles. The boundary layers that separate in the upstream cylinder reattach alternatively to the surface of the downstream one, suppressing the wake between the two bodies, that act as a single Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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Fig. 5. Phase averaged dimensionless vorticity (ω∗ ) for the case with diameter ratio d/D=1 and U ∗ ≈ 7.5.
Fig. 6. Averaged dimensionless velocity magnitude field (v 2x + v 2y )1/2 U −1 for the case with diameter ratio d/D=1 and U ∗ ≈ 7.5.
bluff body. As the system is practically stationary, the two cylinders behave as described for the case of stationary tandem arrangements with small separation distance (see e.g. Zdravkovich, 1997; Sumner, 2010). Fig. 6 is for the magnitude of the velocity field, computed as (v 2x + v 2y )1/2 U −1 , where v i indicates the time averaged velocity in the direction i. From both figures, it is clear how the downstream cylinder acts as an extension of the leading one, producing a practically stagnant region in the gap between the cylinders that yields the excitation described in Figs. 3 and 4. The low drag coefficients presented here, are in close agreement to those reported by Alam and Zhou (2008) for the case of two stationary cylinders. The lift coefficients are as well in close agreement to the ones presented by Zhao (2013) in his cases with equal diameter rigidly linked models with centre-to-centre separation of 1.5D, where the author attributed these results to the nonexistence of vortex shedding in the gap region and to the fact that the shear layer interaction region moved away from the models. Conversely, when the trailing cylinder had the smallest diameter investigated, i.e. d/D = 0.12, the flow resembled that typical of a cylinder undergoing VIV, as previously described using a similar set-up by Huera-Huarte (2017). Moreover, Jiménez-González and Huera-Huarte (2017) demonstrated the high sensitivity of the VIV amplitudes and frequencies caused by the small trailing cylinder at this position (r = 1.3D), and how although changes in the response and excitation took place, the large structures in the wake were barely modified. The previous plots in this section, depict Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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Fig. 7. Phase averaged dimensionless vorticity (ω∗ ) fields at 8 positions of an oscillation cycle, for the case with diameter ratio d/D=0.12 and U ∗ ≈ 6.5.
how these effects are minor when compared to that produced by cylinders of larger d. In Fig. 7, a wake consisting of two vortices per cycle fully develops behind the system even though vortices are altered in the near wake region because of the presence of the trailing cylinder. The case in the figure is for a U ∗ ≈ 6.5, falling in the upper branch where the amplitudes are the largest due to the locked-in conditions, as seen in Fig. 2. Eight phase averaged vorticity snapshots are shown in the figure, for a complete VIV oscillation cycle. When the cylinders are descending, in plots 7(a) to (c), the lower shear layer is perturbed by the downstream cylinder, generating local vortex structures of size ≈ d. The same process can be observed in the upper shear layer when the cylinder moves upwards in plots 7(f) to (h). As in one of the cases described by Jiménez-González and Huera-Huarte (2017), the effect of the smaller cylinder is to generate a wake that is narrower with vorticity that dissipates faster, resulting in reduced cross-flow and in-line excitations, as observed in Figs. 3 and 4. The mechanism that leads to the VIV response modifications observed in Fig. 2 can be clearly associated to the modification of the shear layer interaction region. The fact that frequencies in this case have been barely changed by the presence of the trailing cylinder, when compared to the isolated cylinder case, indicates only a moderate disruption in the vortex formation and interaction processes. Shear layers are able to roll up forming vortices that are shed during the Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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Fig. 8. Phase averaged dimensionless vorticity (ω∗ ) fields at 8 positions an oscillation cycle, for the case with diameter ratio d/D=0.5 and U ∗ =6.75.
oscillation and it is only once fully detached from the surface of the cylinder, that the perturbation caused by the trailing cylinder has an effect. Similar vortex dynamics take place for the case with d/D = 0.2. An intermediate situation such as the one given by the case with d/D = 0.5 allows the explanation of the differences observed as the trailing diameter is increased, from the case with d/D = 0.12 to the one with d/D = 1. This case, for a U ∗ =6.75, appears in Fig. 8. The phase averaged vorticity fields show a very different vortex formation if compared to that happening with d/D=0.12. The downstream cylinder generates a local wake that results in the shedding of two co-rotating vortices per cycle at each side of the system. In plots 8(a) to (d), the shear layer forming at the lower surface of the upstream cylinder is forced to bend to the gap region between the cylinders and interacts with the opposite signed vorticity shed from the upper part of the trailing cylinder. In plot 8(d), this weakened vortex is shed together with the co-rotating one formed at the lower surface of the trailing cylinder. The reduced size of the trailing cylinder in the previous case with d/D = 0.12, was not enough to promote such a large vortex modification. Although the case of two rigidly connected cylinders in tandem in cross-flow has been barely studied at the Re in the order of the ones proposed here, in the work by Zhao (2013), the author numerically simulated at low Re, some cases Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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that are relevant to ones presented here. In particular, Zhao (2013) studied cases in which the tandem was formed by two rigidly linked cylinders with equal diameter (d/D=1), and observed a large variation in the dynamic response of the system, as the separation between them was modified. Going from a centre-to-centre distance of 1.5 to 2D implied a large increase in the amplitude, that ended up being 3 times larger. In the case of a distance of 1.5D, the amplitudes he reported, always smaller than 0.2D with a very small lock-in region, are comparable to the ones reported here. The fact that amplitudes are massively changed either by modifying the separation distance with equally sized cylinders as in Zhao (2013), or the relative size of the cylinders, indicates the problem is clearly driven by the size of the gap region. Obviously as the size of the downstream cylinder is reduced its effect on the total force exerted on the system is decreased, and that is a major difference with the cases of equal size largely separated in Zhao (2013). 4. Conclusions We have presented an experimental study on the vortex-induced vibrations, of a one degree of freedom system that consists of two rigidly connected circular cylinders of different relative size, in a tandem configuration with a fixed gap of 1.3D, with D being the diameter of the upstream cylinder. As mentioned, such gap distance has been selected based on our previous results on VIV sensitivity (Jiménez-González and Huera-Huarte, 2017), where large VIV amplitude reductions were found when perturbations are introduced at such radial distance. Several values of the downstream cylinder diameter d have been tested, including the following diameter ratios d/D = 0.12, 0.2, 0.32, 0.4, 0.5 and 1. The oscillating amplitude of tandem displacement, A∗ , and transverse and drag forces coefficients, C˜ y and C x , have been measured to evaluate the effect of d/D on the flow-induced dynamics. Thus, important amplitude reductions are observed for all arrangements investigated (Fig. 2), especially for d/D > 0.2 and in particular, for d/D = 1 where the amplitude response barely represents a 15% of the maximum displacement for the isolated cylinder case. Attenuation of dynamic response is particularly relevant for low reduced velocities, i.e. U ∗ < 8, leading to the disappearance of the upper branch in the plots of A∗ (U ∗ ). Besides, oscillation frequencies do not change significantly upon increasing the value of d/D below 0.4, although above such threshold a clear decrease of f ∗ takes place for U ∗ > 6. This observation is especially remarkable for the tandem configuration of d/D = 1, for which the value of f ∗ drops suddenly at large values of reduced velocity, to reach a plateau of f ∗ < 1. A similar type of frequency was numerically reported by Zhao et al. (2015), who interpreted it in terms of galloping-like behaviour. In general, the mitigated VIV response is related to a weakening of the excitation within the upper branch (U ∗ < 8), which is quantified by means of the transverse force coefficient C˜ y , as shown in Fig. 3. Again, the qualitative trend displayed by the d/D = 1 tandem is significantly different, with the transverse force coefficient not exceeding the value of 0.3 within the range of U ∗ investigated. Important changes are also identified for d/D > 0.32 in the phase difference between displacement and forcing (Fig. 3). Interestingly, no shift in the phase value exists for d/D = 1. As far as the drag force coefficient, C x , is concerned, a relevant decay is observed within the upper branch interval for d/D < 1, providing with reductions of approximately 40% with respect to the isolated case. This drag reduction is more pronounced for the limit case of equal size cylinders, where C x remains barely constant for the whole range of U ∗ tested. The drag improvement has been shown to be linked to the weakening of A∗ as the size of the downstream cylinder increases in the C x (A∗ ) plots. Finally, flow visualisations performed by using Digital Particle Image Velocimetry have allowed to identify modifications introduced in the vortex shedding process by the presence of downstream cylinders of increasing size, and how such changes may alter the forcing of the flexibly mounted system. As discussed above in reference to Figs. 5–8, the flow dynamics is strongly driven by the relative size of the cylinders, and therefore, the size of the gap region. More specifically, we have first analysed the case of tandem with d/D = 1 by means of contours of phase averaged vorticity ω∗ and averaged velocity magnitude, in Figs. 5–6, to identify features of the forcing mechanism that mitigates the dynamic amplitude response. Thus, it has been shown that a practically stagnant, irrotational region develops in the gap between cylinders, since the boundary layers that separate in the upstream cylinder reattach alternatively to the surface of the downstream one. Hence, the wake between the two bodies is suppressed, and the whole tandem behaves as a single bluff body. On the other hand, for a small value of d/D, i.e. 0.12, a moderate disruption in the vortex formation and shear layers interaction of the wake of the upstream cylinder takes place (Fig. 7). However, when the value of d/D is increased up to 0.5 (Fig. 8), there is also an intense wake behind the downstream cylinder which leads to the shedding of two corotating vortices at each side of the system. Moreover, the interaction in the gap region between the upstream cylinder shear layer and the opposite signed vorticity shed from the trailing cylinder leads to weaker vortices, an therefore, to an attenuated excitation of the system, with the subsequent weaker response. Acknowledgements The authors would like to acknowledge the funding provided by the Spanish Ministerio de Economía y Competitividad, Spain through grants DPI2015-71645-P and DPI2017-89746-R. Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem ∗
F.J. Huera-Huarte a , , J.I. Jiménez-González b a b
Department of Mechanical Engineering, Universitat Rovira i Virgili, 43007 Tarragona, Spain Departamento de Ingeniería Mecánica y Minera, Universidad de Jaén, Campus de las Lagunillas, 23071 Jaén, Spain
article
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Article history: Received 5 October 2018 Received in revised form 11 April 2019 Accepted 15 April 2019 Available online xxxx Keywords: Vortex-induced vibrations VIV VIV sensitivity Tandem cylinders Vortex-shedding
a b s t r a c t We present an experimental study on the effects on the vortex-induced vibrations (VIV), of the relative size of two rigidly connected circular cylinders in a tandem arrangement, with a fixed centre-to-centre separation of 1.3D, where D represents the diameter of the upstream cylinder. This separation distance was selected after the results obtained by Jiménez-González and Huera-Huarte (2017), where a high VIV sensitivity region was identified at such location in the wake of a cylinder oscillating in cross-flow. The flowinduced response of the system, which has one degree of freedom, has been analysed for several values of the diameter ratio d/D ≤ 1, where d is the diameter of the downstream cylinder. As the value of d/D increases, the VIV response is attenuated, it being associated with important reductions in the transverse and in-line force coefficients. In addition, Digital Particle Image Velocimetry (DPIV) has allowed us to study the changes in the near wake and forcing introduced by the presence of different downstream cylinders. Thus, the flow dynamics is clearly driven by the relative size of the cylinders, and therefore, the size of the gap region. In particular, for small values of d/D, a moderate disruption in the vortex formation and shear layer interaction of the wake of the upstream cylinder takes place. For intermediate values of d/D, the downstream cylinder generates a strong local wake that results in the shedding of two co-rotating vortices at each side of the system, while markedly undermining the shedding from the upstream cylinder. In the limit of d/D = 1, a stagnant region develops in the gap between the cylinders, and the tandem behaves as a single bluff body, featuring a very attenuated flow-induced response. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction The study of the interaction between the wakes of bluff bodies has received a considerable amount of attention, since its analysis is crucial for the understanding of the response and forcing mechanisms in structures exposed to winds or water currents. Circular cylinders are widely used as a canonical idealisation of such structures. Most of the existing literature has dealt with stationary cylinders such as in the pioneering work by Igarashi (1982). Zdravkovich (1997) reviews in his book the wide variety of works that deal with the flow around stationary configurations of cylinders. Sumner (2010) more recently compiled a large number of works devoted to the analysis of the forcing and the flow around pairs of cylinders in different configurations, ranging from the side-by-side to the tandem ones. Nevertheless, the number of published works ∗ Corresponding author. E-mail addresses: [email protected] (F.J. Huera-Huarte), [email protected] (J.I. Jiménez-González). https://doi.org/10.1016/j.jfluidstructs.2019.04.006 0889-9746/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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is much more limited in the case of a non-stationary tandem arrangement of circular cylinders. The fact that one or both of the cylinders are allowed or forced to vibrate, changes considerably the flow physics around the bodies even if the flow regime is not modified, resulting in large changes in the separation points, formation lengths, formation of shear layers, etc. This is why we focus this brief introductory review to cases that deal with tandem arrangements in which motion is allowed or forced. Knisely and Kawagoe (1990) investigated the loading characteristics on a cylinder forced to move inside the wake of an stationary leading cylinder, revealing important timing relationships between vortex formation and lift forces. Yang et al. (2014) used a forced vibration set-up, to focus on the study of the different types of wake that appear behind a tandem arrangement of cylinders in which motions could be forced independently in each model. The work was limited to the analysis of the wake patterns that appeared behind the models after carrying out hydrogen bubble flow visualisation. The authors described mainly two modes of combined shedding depending on the imposed oscillation frequency, with a different number of vortices per cycle at each side of the wake. Going one step further in complexity, some authors have analysed the case of the cylinders being flexibly mounted, therefore leaving the possibility of fluid-elastic instabilities and self-excited phenomena to appear. See for example the pioneering works by Bokaian and Geoola (1984a,b), where the authors varied systematically the relative position of the cylinder models to measure dimensionless frequencies, excitation and response of the trailing cylinder, describing what they called galloping-like dynamics. Hover and Triantafyllou (2001) carried out experiments with a circular cylinder placed 4.75 diameters behind another stationary model, because at that distance (Zdravkovich, 1985) had previously found the largest motions in his experiments. Brika and Laneville (1997) conducted studies in a wind tunnel with flexible cylinders (although limited to move in their first mode of vibration), showing a clear hysteretic behaviour in their response depending on whether the upstream cylinder was fixed or allowed to vibrate. More work with free-to-oscillate cylinders in the wake of stationary ones has been carried out recently (Assi et al., 2006, 2010, 2013; Carmo et al., 2011; Qin et al., 2017). Some of these authors have pointed out the fact that the mechanisms that drive the wake-induced vibrations of the downstream cylinders, are not resonant but caused by the flow unsteadiness. This is in contrast to what happens in the case of the cylinders being elastic themselves, where the resonant characteristics of the structure dominate the response (Huera-Huarte et al., 2016). In an scenario such as this, the response and the forcing on a flexible cylinder immersed in the wake of a stationary one, is characterised by very rich dynamics. The case in which both cylinders in a tandem arrangement are free to move and subject to a cross-flow, has been barely studied as the variable space that needs to be covered is very large and the interactions are of extreme complexity (see e.g. Kim et al., 2009; Borazjani and Sotiropoulos, 2009; Huera-Huarte and Bearman, 2011; Huera-Huarte and Gharib, 2011; Griffith et al., 2017). In all these works it is emphasised that the driving physics are closely conditioned by the gap between the cylinders, that it is in fact dynamic and dependent on the flow conditions. In this work, we study the effects of the relative size, on the vortex-induced vibrations (VIV), of two rigidly connected circular cylinders in a tandem arrangement with a fixed gap. The separation distance (1.3 diameters) was selected after the results obtained by Jiménez-González and Huera-Huarte (2017), where a series of high VIV sensitivity regions were identified in the wake of a cylinder oscillating in a cross-flow. The study was conducted by rigidly connecting a small cylinder (3.2% in size when compared to the main one) in the near wake of an elastically mounted low mass-damping cylinder. An open question remained unanswered: what would be the effect of varying the size of the cylinder placed at these high sensitivity regions? The effect of size in symmetric configurations falling in high sensitivity regions was studied by Jiménez-González and Huera-Huarte (2018), and here we intend to clarify such issue for the case of a tandem arrangement, providing as well wake dynamics measurements. The works by Zhao (2013) and Zhao et al. (2015) are therefore relevant as the authors analysed tandem configurations formed by rigidly linking a pair of circular cylinders. The paper is organised as follows: the experimental methods are presented in Section 2; while results of dynamic response obtained from tests conducted at different reduced velocities for selected values of the downstream cylinder’s diameter are presented in Section 3. Measured force distributions and corresponding frequencies are first analysed, to subsequently describe flow visualisations and discuss how the near wake modifications introduced when the size of the downstream cylinder is modified affects the excitation mechanism. Finally, the main conclusions are drawn in Section 4. 2. Experimental methods The facility used to conduct the research is the free surface water channel (FSWC) at the Laboratory for Fluid–Structure Interactions (LIFE) of the Universitat Rovira i Virgili (URV) in Tarragona. The channel has a three-dimensional contraction after which a uniform profile is generated, in a 1 m × 1.1 m × 2.25 m translucent working section, especially designed to use optical measurement techniques. Periodic measurements carried out with a traverse system and an Acoustic Doppler Velocimetry (ADV) at several locations in the working section (usually 3 cross-sectional planes with 33 time series measurements in each) are used to confirm the high quality of the flow generated, with a spatio-temporal variability of the flow speed of less than 1.6%, with a turbulence intensity (root mean square of the velocity upon its mean) of 2.79% at the highest regime of the channel. Fig. 1 displays a lay-out of the experimental set-up and the tandem cylinder model arrangement used in the present investigation. As shown in Fig. 1(a), a support structure made of aluminium profiles was built on top of the channel, without touching it to avoid the transmission of any small vibrations due to the operating pumps, hanging from the Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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Fig. 1. Schematic view of the experimental set-up.
building structure. An air bearing rig was installed on the supporting structure, with its main beam perpendicular to the flow direction in the channel. The air bearing system consisted of a beam in which a carriage could move freely having no friction. The displacements of the carriage along the air bearing beam (y-coordinate) were measured using a non-intrusive laser displacement sensor (absolute resolution of 0.1 mm.). A six-axis load cell, with a measurement range in the in-line and transverse directions of 100 N (with an uncertainty rated at less than 0.15% of its full scale.), was installed below the carriage by means of a especially designed 3D printed solid part (100% density with Polylactic Acid PLA), to which the cylinder model was attached to. This load cell allowed the direct measurement of in-line and transverse forces acting on the cylinder. To validate the direct measurements, a second independent precision uni-axial load cell (measurement range of ±8.9 N with an uncertainty of 0.1% of its full scale) was used to redundantly measure the in-line or drag forces in specific cases. This was done for validation, because the observed forces in the experiments were in the 10% band of the full scale of the six-axis load cell. Moreover, forces were also obtained by using the displacement data and its time derivatives, fed to a one degree of freedom model equation. All three independent and redundant ways of measuring forces, showed differences of less than 3% at all times. More details and comparisons between these independent measurement methodologies can be seen in the recent work Huera-Huarte (2018), conducted with the same set-up and techniques. The circular cylinder model used for the experiments was made of a transparent acrylic tube with external diameter D=50 mm (inner diameter 40 mm), and once in position, penetrated the free surface having a submerged length L of 0.5 m. An acrylic plate was installed at lower end of the model to avoid having end effects. Two 3D printed solid parts (see Fig. 1b), one near the bottom end resting on top of the end plate, and another just above the free surface, provided support for the secondary (or tandem) cylinder model. The parts ensured a precise fixed location of the secondary cylinder behind the main cylinder model, at a radial distance r = 1.3D, whilst introducing minimal flow disturbances. Note that the distance r (Fig. 1c) was selected after the recent VIV sensitivity study by Jiménez-González and Huera-Huarte (2017), where it was seen that a local perturbation, with the form of a very small control cylinder, at certain centre-to-centre locations in tandem, had large effects on the VIV response of a cylinder. An omnidirectional system based on a wire mesh designed to suppress VIV has also shown great capabilities on modifying the dynamic response of a circular cylinder when similar radial distances are adopted (Huera-Huarte, 2017). Restoring forces for the carriage were provided by two springs that connected the cylinder to the external supporting structure. The trailing cylinder support parts allowed to install models with different diameters d in the selected fixed tandem position, so that the effect of the diameter ratio, d/D, could be studied in detail at this high VIV sensitivity fixed position. In particular, 6 different values of diameters ratio were analysed, namely d/D = [0.12, 0.2, 0.32, 0.4, 0.5, 1]. The study is limited to d/D ≤ 1, because with the fixed separation distance in the tandem (1.3D), as the cylinders start to have a similar size, the shear layers formed in the upstream one attach to the trailing one, in a single bluff-body like mode of shedding, and if further increased they may eventually become a single body. In addition, an in-house developed Planar Time-Resolved Digital Particle Image Velocimetry (DPIV) system, allowed the interrogation of the flow field around the rigidly coupled tandem arrangement of cylinders in specifically selected cases. The purpose was to study the global changes introduced in the wake of the leading cylinder by the control one, as its size was varied. Seeding in the channel consisted of 10 µm neutrally buoyant Polyamide seeding particles (PSP) that were illuminated using a 5 W continuous wave Diode Pumped Solid State (DPSS) green laser source. A 12 bit 1.3 Mpixel high speed camera equipped with a 20 mm fixed focal lens objective, provided images of the illuminated particles at a rate of 500 Hz, yielding time resolved flow fields with a ∆t of 2 ms. The image dynamic masking, PIV image processing and Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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post-processing of the velocity fields obtained, result from an in-house code that includes part of the work by Willert and Gharib (1991) and Thielicke and Stamhuis (2014). Image processing consisted of an initial step in which regions without seeding particles or illumination were masked. The position of the centre of the cylinder was also identified in each snapshot by image processing. A scaling factor of 3.31·10−4 m/pixel was obtained by imaging an illuminated rectangular target, placed at the plane defined by the laser sheet. A Fast Fourier Transform (FFT) cross-correlation algorithm was applied to the dynamically masked images, using an interrogation area of 24 × 24 pixels with a 50% of area overlapping, to produce the velocity vectors. The optimal window interrogation area was found by reducing progressively its size until the point at which the number of spurious vectors resulting from the processing was considered not acceptable. With the 24 × 24 pixels window, the number of spurious vectors obtained after the processing was very small and always less than 2% of the total. Outlier vectors output by the cross-correlation scheme, were identified after applying a threshold and replaced by new values obtained from averaging around the outlier. The field of view (FOV) was 265.1 × 232.0 mm (5.3×4.6D) and a total of 3705 (65 × 57) velocity vectors were obtained, implying a spatial resolution of 4.1 mm (1 velocity vector in a region of the measurement plane with size of 4.1 × 4.1 mm2 ). This resolution is the consequence of the need for accommodating a relatively large field of view, in the order of 5D, with the smaller structures emanating from the trailing cylinder with size d. Although √ this set-up cannot resolve quantities such as for example, the boundary layer on the cylinders (at this Re is ≈ O(D/ Re) ≈ 0.028D ≈ 1.4 mm) and other small scale structures in the flow, it is adequate to study qualitatively the overall changes in the wake of the system at scales in the order of D, resulting from the variations in the size of the trailing cylinder. An indication of the quality of the measurements obtained, apart from the very small number of spurious vectors obtained, is given by comparing the flow speed in the channel, measured with the PIV system and with another independent technique. As described above, the flow speed was measured using a high-resolution acoustic Doppler velocimeter at points very near the PIV FOV, without perturbing it, and was compared to the output given by the PIV (spatio-temporal average of a region inside the FOV), showing differences between both that were always smaller than 5%. 3. Results and discussion The results presented in this section consist of the dynamic response of the cylinder and the fluid excitation. These are related to the wake topology investigated in several selected experiments, for a detailed physical explanation. In all the plots of this section, the response of the isolated cylinder is included together with the rest of configurations for reference. Results for each configuration were obtained by stepping up the free stream velocity (U) in the channel in small increments and allowing the flow to become steady. Thus, the range of reduced velocity covered in the experiments is approximately U ∗ = U /fn D = [4, 14], where fn represents the natural frequency of system oscillations in still water. Therefore, this range corresponds to the following interval of the Reynolds number, Re = UD/ν = [7000, 28000], with ν being the fluid kinematic viscosity. Besides, in these experiments, as the trailing cylinder is changed to achieve the different diameter ratios investigated, the mass (and the added mass) of the moving system change depending on the specific set-up. This aspect has been taken into account in each set of results, by compensating the displaced mass by means of additional weight to keep a constant value of the mass-damping parameter m∗ ξ ≃ 0.009, where m∗ is the mass ratio, the ratio between system and displaced mass, and ξ corresponds to the structural damping in air (measured in freedecay tests). In addition, the response and the forcing were measured during at least 60 s, to ensure that a large quantity of oscillation cycles were acquired, while the signals sampling frequency was set to Fs = 1.5 kHz to properly resolve the system dynamics and characteristic frequencies. The amplitude of the oscillating response, A, was computed as the mean of the envelope (Hilbert transform) of the time series of displacement, as in Huera-Huarte (2017). According to the description of the experiment given in the previous section, the uncertainty in the dimensionless amplitudes (A∗ = A/D) is estimated to be in the order of 0.2% D. In the case of Re uncertainties are approximately 1.6%, as it directly derives from the free-stream velocity in the water channel. The uncertainty associated to the Fourier analysis of the signals, due to the available frequency resolution is 1.5%, if referred to the natural frequency of the system (0.002% of the Nyquist frequency in the experiments). All these, yield an uncertainty estimate for the reduced velocity with an upper bound of approximately 2.2%. The data set has been divided into two subsets, that have been arranged in columns inside the figures, for the sake of clarity. Circles (◦) are used in all the plots for referring to the isolated cylinder case. In that sense, the amplitude response of cylinders in dimensionless form A∗ is presented in the upper row of plots of Fig. 2, against the reduced velocity U ∗ . As it can be seen in the plots, the isolated cylinder exhibited the low mass-damping classical VIV response, with well defined initial (U ∗ < 6), upper (6 < U ∗ < 8) and lower (U ∗ > 8) branches, with results being close to those reported previously, using a similar set-up (Huera-Huarte, 2017; Jiménez-González and Huera-Huarte, 2017; Huera-Huarte, 2018). The maximum amplitude is over 1D at a reduced velocity near 6.5. Dominant frequencies, f , appear in the lower row of Fig. 2, divided upon the natural frequency of the model in water (f ∗ = f /fn ). For the isolated cylinder case, at the lowest reduced velocities, the frequency followed that imposed by the shedding, until reaching the natural frequency of the system, point at which it stabilised indicating a resonant condition. In the lower branch there is a plateau with frequencies that are slightly higher than the natural frequency of the system. When a second cylinder is rigidly coupled to the main cylinder, at the fixed distance of r = 1.3D in tandem position, the response varies considerably. In fact, it is highly dependent on the diameter ratio d/D between the cylinders. The Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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Fig. 2. Dimensionless amplitude (A∗ ) and frequency (f ∗ ) response of all configurations as a function of reduced velocity (U ∗ ) and Re. Inset plot in (b) depicts the maximum dimensionless amplitude A∗max against the diameter ratio d/D.
cases with d/D of 0.12 and 0.20 depict response trends that show the initial, upper and lower branches but with smaller amplitudes and narrower reduced velocity ranges. For the smallest diameter ratio investigated, the maximum amplitude is around a 10% smaller, although happening at the same reduced velocity near 6.5, as in the isolated case. These results confirm previous research conducted by Jiménez-González and Huera-Huarte (2017). The upper branch is narrower with a sharp peak that drops just after the maximum is reached. The amplitudes in the lower branch are again lower that in the isolated case. These two effects, are more pronounced when the diameter ratio is 0.2, with amplitudes even lower, a narrower upper branch and smaller amplitudes in the lower branch. Frequencies in these two cases do not practically vary when compared to the isolated case. Responses change dramatically at all the other diameter ratios investigated. In the cases with d/D of 0.32 and 0.4, the upper branch disappears and amplitudes do not surpass a value of 0.5D at any reduced velocity. Once the amplitude reaches that value at a reduced velocity around 6, it stays at that level going slightly down to values closer to 0.4D for U ∗ > 8. Although the frequencies for the case with a diameter ratio of 0.32 are almost identical to those of the isolated cylinder, for the case with d/D= 0.4, frequencies change becoming in general smaller at all U ∗ . A change in the diameter ratio to 0.5 yields larger amplitudes, near 0.7D, with the response peak at a reduced velocity of almost 8. Frequencies evolve parallel to those of the other arrangements but at smaller values. Finally, the case with the two cylinders in the tandem with the same diameter, shows practically no response for reduced velocities up to 7, and responses with amplitudes near 10% of the diameter at the rest of reduced velocities. In this case, there is a large jump in frequencies from the shedding expected ones to values that stay at f ∗ ≈ 0.75 for U ∗ > 6.5. This sudden change in frequency was not reported in the low Re number simulations by Zhao (2013), but appeared in the 2D RANS simulations by Zhao et al. (2015), conducted at a Re = 5000, associated to amplitudes in the order of 0.5D outside the lock-in range. The authors associated these phenomenon to a galloping like response. Maximum amplitudes A∗max as a function of d/D are shown in the inset plot of Fig. 2b), where it can be seen how increasing the diameter of the control cylinder means decreasing VIV amplitude, except for a small jump at a d/D = 0.5. Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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Fig. 3. Dimensionless standard deviation of the transverse force (C˜ y ) and phase difference between response and force (φ ) of all configurations as a function of reduced velocity (U ∗ ) and Re. Inset plot in (b) depicts the maximum transverse force C˜ ymax against the diameter ratio d/D.
Following the dynamic response of the system, in Fig. 3, the transverse excitation is presented. The upper row of plots is for the standard deviation of the dimensionless transverse force (C˜y ). The time series of the force coefficient were obtained from the measured transverse forces Fy (t) by applying the expression, Cy (t) =
Fy (t) 1 2
ρ U 2 DL
(1)
with ρ being the fluid density. The uncertainty estimated for the force coefficient, derived from that of the force measurements and free stream velocity, has an upper bound around 3%. In the lower row of plots in the figure, the phase difference between the transverse force and the displacement, φ , is included. Phase differences have been obtained as in Huera-Huarte and Bearman (2009), by using the analytical signal concept, using Hilbert transforms (Pikovsky et al., 2001). In the case of an isolated cylinder undergoing VIV, it is well known that transverse force ramps up quickly up to its highest values (near 1.8 in this case), that coincide with the response peak, at a U ∗ ≈ 6. Then it quickly descends, in this case to values near 0.4, to continue decreasing very slowly to values near 0.2, at the end of the lower branch. The forcing is in phase with the response in the initial and upper branches with the system being dominated by elastic forces, and changes suddenly at the end of the upper branch to an out-of-phase condition in which inertial forces are the dominant. The same behaviour, with variations in the force amplitudes, has been observed for diameter ratios of 0.12, 0.2 and 0.32 with peak C˜y in the order of 1.7, 1.7 and 1.1 respectively. Phase differences for these 3 cases are practically identical to those observed in the isolated case. When the cylinder with diameter d = 0.4D was installed in the tandem, a narrow forcing peak appeared but at higher U ∗ , reaching values around 1.4. After the peak, the forcing became very similar to that shown by the isolated cylinder. The phase jump is also delayed to larger reduced velocities. When the diameter ratio was further increased to a value of 0.5, the C˜y curve moved to higher reduced velocities and peak values near 1.5, with phases that implied a clear in-phase condition only when U ∗ < 9. For larger reduced velocities, phases seemed not to follow a Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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Fig. 4. Dimensionless time averaged in-line force (C x ) as a function of reduced velocity (U ∗ ) and dimensionless amplitude (A∗ ). Inset plot in (b) depicts the maximum time averaged in-line force coefficient C xmax against the diameter ratio d/D.
clear trend, with values in-between the in-phase and the out-of-phase conditions. Finally, the results for the one-to-one diameter ratio show very small lift coefficients that stay at all reduced velocities in-phase with the displacements. The inset plot in Fig. 3b) shows the evolution of the maximum transverse force coefficient (C˜ ymax ) as a function of the diameter ratio (d/D). If the time series of the in-line force Fx (t) is used in Eq. (1), instead of Fy (t), the drag coefficient is obtained. Fig. 4 presents the time averaged drag coefficients (C x ) as a function of U ∗ in the upper row of plots and as a function of the response amplitude (A∗ ) in the lower row. The so called drag amplification has been observed not only in the VIV of elastically mounted rigid cylinders as in here, but also in flexible ones (Chaplin et al., 2005; Trim et al., 2005; Huang et al., 2011), with large in-line forces correlated to large amplitude responses. Here, while the plots show a clear relationship between amplitude and drag for the isolated case and the cases with diameter ratio of 0.12 and 0.2, this is not true for the rest of cases. Diameter ratios of 0.32, 0.4 and 0.5 are characterised by similar wide inverted parabolic C x curves when plotted against reduced velocity, with values that stay between 1.2 and 1.5 at all reduced velocities, yielding drag coefficients that are smaller than those of the isolated case in the upper branch, and larger outside. The case of equal diameters, depicts a very different situation, with a large drag reduction at all reduced velocities with coefficient values that stay under 1. Again an inset plot in Fig. 4a) shows the maxima of the drag coefficient (C xmax ) at each diameter ratio. From the three previous figures it appears obvious that the classical VIV response has been completely altered when the trailing cylinder has the same diameter as the upstream one, when r = 1.3D. The fact that there is practically no amplitude response together with very small cross-flow and transverse excitations, indicates that a large modification of the classical vortex street has taken place. Fig. 5 shows the vorticity field around the rigidly linked tandem arrangement for a case with U ∗ ≈ 7.5. Vorticity appears in all plots phase averaged and in dimensionless form (ω∗ = ωz D/U), with ωz being the out-of-plane vorticity, and the red colour indicating clockwise direction of rotation and blue counter-clockwise. Phase averaged plots were obtained after identifying, using the cylinder position, several specific phases along the trajectory of the cylinder, using at least 10 oscillation cycles. The boundary layers that separate in the upstream cylinder reattach alternatively to the surface of the downstream one, suppressing the wake between the two bodies, that act as a single Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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Fig. 5. Phase averaged dimensionless vorticity (ω∗ ) for the case with diameter ratio d/D=1 and U ∗ ≈ 7.5.
Fig. 6. Averaged dimensionless velocity magnitude field (v 2x + v 2y )1/2 U −1 for the case with diameter ratio d/D=1 and U ∗ ≈ 7.5.
bluff body. As the system is practically stationary, the two cylinders behave as described for the case of stationary tandem arrangements with small separation distance (see e.g. Zdravkovich, 1997; Sumner, 2010). Fig. 6 is for the magnitude of the velocity field, computed as (v 2x + v 2y )1/2 U −1 , where v i indicates the time averaged velocity in the direction i. From both figures, it is clear how the downstream cylinder acts as an extension of the leading one, producing a practically stagnant region in the gap between the cylinders that yields the excitation described in Figs. 3 and 4. The low drag coefficients presented here, are in close agreement to those reported by Alam and Zhou (2008) for the case of two stationary cylinders. The lift coefficients are as well in close agreement to the ones presented by Zhao (2013) in his cases with equal diameter rigidly linked models with centre-to-centre separation of 1.5D, where the author attributed these results to the nonexistence of vortex shedding in the gap region and to the fact that the shear layer interaction region moved away from the models. Conversely, when the trailing cylinder had the smallest diameter investigated, i.e. d/D = 0.12, the flow resembled that typical of a cylinder undergoing VIV, as previously described using a similar set-up by Huera-Huarte (2017). Moreover, Jiménez-González and Huera-Huarte (2017) demonstrated the high sensitivity of the VIV amplitudes and frequencies caused by the small trailing cylinder at this position (r = 1.3D), and how although changes in the response and excitation took place, the large structures in the wake were barely modified. The previous plots in this section, depict Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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Fig. 7. Phase averaged dimensionless vorticity (ω∗ ) fields at 8 positions of an oscillation cycle, for the case with diameter ratio d/D=0.12 and U ∗ ≈ 6.5.
how these effects are minor when compared to that produced by cylinders of larger d. In Fig. 7, a wake consisting of two vortices per cycle fully develops behind the system even though vortices are altered in the near wake region because of the presence of the trailing cylinder. The case in the figure is for a U ∗ ≈ 6.5, falling in the upper branch where the amplitudes are the largest due to the locked-in conditions, as seen in Fig. 2. Eight phase averaged vorticity snapshots are shown in the figure, for a complete VIV oscillation cycle. When the cylinders are descending, in plots 7(a) to (c), the lower shear layer is perturbed by the downstream cylinder, generating local vortex structures of size ≈ d. The same process can be observed in the upper shear layer when the cylinder moves upwards in plots 7(f) to (h). As in one of the cases described by Jiménez-González and Huera-Huarte (2017), the effect of the smaller cylinder is to generate a wake that is narrower with vorticity that dissipates faster, resulting in reduced cross-flow and in-line excitations, as observed in Figs. 3 and 4. The mechanism that leads to the VIV response modifications observed in Fig. 2 can be clearly associated to the modification of the shear layer interaction region. The fact that frequencies in this case have been barely changed by the presence of the trailing cylinder, when compared to the isolated cylinder case, indicates only a moderate disruption in the vortex formation and interaction processes. Shear layers are able to roll up forming vortices that are shed during the Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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Fig. 8. Phase averaged dimensionless vorticity (ω∗ ) fields at 8 positions an oscillation cycle, for the case with diameter ratio d/D=0.5 and U ∗ =6.75.
oscillation and it is only once fully detached from the surface of the cylinder, that the perturbation caused by the trailing cylinder has an effect. Similar vortex dynamics take place for the case with d/D = 0.2. An intermediate situation such as the one given by the case with d/D = 0.5 allows the explanation of the differences observed as the trailing diameter is increased, from the case with d/D = 0.12 to the one with d/D = 1. This case, for a U ∗ =6.75, appears in Fig. 8. The phase averaged vorticity fields show a very different vortex formation if compared to that happening with d/D=0.12. The downstream cylinder generates a local wake that results in the shedding of two co-rotating vortices per cycle at each side of the system. In plots 8(a) to (d), the shear layer forming at the lower surface of the upstream cylinder is forced to bend to the gap region between the cylinders and interacts with the opposite signed vorticity shed from the upper part of the trailing cylinder. In plot 8(d), this weakened vortex is shed together with the co-rotating one formed at the lower surface of the trailing cylinder. The reduced size of the trailing cylinder in the previous case with d/D = 0.12, was not enough to promote such a large vortex modification. Although the case of two rigidly connected cylinders in tandem in cross-flow has been barely studied at the Re in the order of the ones proposed here, in the work by Zhao (2013), the author numerically simulated at low Re, some cases Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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that are relevant to ones presented here. In particular, Zhao (2013) studied cases in which the tandem was formed by two rigidly linked cylinders with equal diameter (d/D=1), and observed a large variation in the dynamic response of the system, as the separation between them was modified. Going from a centre-to-centre distance of 1.5 to 2D implied a large increase in the amplitude, that ended up being 3 times larger. In the case of a distance of 1.5D, the amplitudes he reported, always smaller than 0.2D with a very small lock-in region, are comparable to the ones reported here. The fact that amplitudes are massively changed either by modifying the separation distance with equally sized cylinders as in Zhao (2013), or the relative size of the cylinders, indicates the problem is clearly driven by the size of the gap region. Obviously as the size of the downstream cylinder is reduced its effect on the total force exerted on the system is decreased, and that is a major difference with the cases of equal size largely separated in Zhao (2013). 4. Conclusions We have presented an experimental study on the vortex-induced vibrations, of a one degree of freedom system that consists of two rigidly connected circular cylinders of different relative size, in a tandem configuration with a fixed gap of 1.3D, with D being the diameter of the upstream cylinder. As mentioned, such gap distance has been selected based on our previous results on VIV sensitivity (Jiménez-González and Huera-Huarte, 2017), where large VIV amplitude reductions were found when perturbations are introduced at such radial distance. Several values of the downstream cylinder diameter d have been tested, including the following diameter ratios d/D = 0.12, 0.2, 0.32, 0.4, 0.5 and 1. The oscillating amplitude of tandem displacement, A∗ , and transverse and drag forces coefficients, C˜ y and C x , have been measured to evaluate the effect of d/D on the flow-induced dynamics. Thus, important amplitude reductions are observed for all arrangements investigated (Fig. 2), especially for d/D > 0.2 and in particular, for d/D = 1 where the amplitude response barely represents a 15% of the maximum displacement for the isolated cylinder case. Attenuation of dynamic response is particularly relevant for low reduced velocities, i.e. U ∗ < 8, leading to the disappearance of the upper branch in the plots of A∗ (U ∗ ). Besides, oscillation frequencies do not change significantly upon increasing the value of d/D below 0.4, although above such threshold a clear decrease of f ∗ takes place for U ∗ > 6. This observation is especially remarkable for the tandem configuration of d/D = 1, for which the value of f ∗ drops suddenly at large values of reduced velocity, to reach a plateau of f ∗ < 1. A similar type of frequency was numerically reported by Zhao et al. (2015), who interpreted it in terms of galloping-like behaviour. In general, the mitigated VIV response is related to a weakening of the excitation within the upper branch (U ∗ < 8), which is quantified by means of the transverse force coefficient C˜ y , as shown in Fig. 3. Again, the qualitative trend displayed by the d/D = 1 tandem is significantly different, with the transverse force coefficient not exceeding the value of 0.3 within the range of U ∗ investigated. Important changes are also identified for d/D > 0.32 in the phase difference between displacement and forcing (Fig. 3). Interestingly, no shift in the phase value exists for d/D = 1. As far as the drag force coefficient, C x , is concerned, a relevant decay is observed within the upper branch interval for d/D < 1, providing with reductions of approximately 40% with respect to the isolated case. This drag reduction is more pronounced for the limit case of equal size cylinders, where C x remains barely constant for the whole range of U ∗ tested. The drag improvement has been shown to be linked to the weakening of A∗ as the size of the downstream cylinder increases in the C x (A∗ ) plots. Finally, flow visualisations performed by using Digital Particle Image Velocimetry have allowed to identify modifications introduced in the vortex shedding process by the presence of downstream cylinders of increasing size, and how such changes may alter the forcing of the flexibly mounted system. As discussed above in reference to Figs. 5–8, the flow dynamics is strongly driven by the relative size of the cylinders, and therefore, the size of the gap region. More specifically, we have first analysed the case of tandem with d/D = 1 by means of contours of phase averaged vorticity ω∗ and averaged velocity magnitude, in Figs. 5–6, to identify features of the forcing mechanism that mitigates the dynamic amplitude response. Thus, it has been shown that a practically stagnant, irrotational region develops in the gap between cylinders, since the boundary layers that separate in the upstream cylinder reattach alternatively to the surface of the downstream one. Hence, the wake between the two bodies is suppressed, and the whole tandem behaves as a single bluff body. On the other hand, for a small value of d/D, i.e. 0.12, a moderate disruption in the vortex formation and shear layers interaction of the wake of the upstream cylinder takes place (Fig. 7). However, when the value of d/D is increased up to 0.5 (Fig. 8), there is also an intense wake behind the downstream cylinder which leads to the shedding of two corotating vortices at each side of the system. Moreover, the interaction in the gap region between the upstream cylinder shear layer and the opposite signed vorticity shed from the trailing cylinder leads to weaker vortices, an therefore, to an attenuated excitation of the system, with the subsequent weaker response. Acknowledgements The authors would like to acknowledge the funding provided by the Spanish Ministerio de Economía y Competitividad, Spain through grants DPI2015-71645-P and DPI2017-89746-R. Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.
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Please cite this article as: F.J. Huera-Huarte and J.I. Jiménez-González, Effect of diameter ratio on the flow-induced vibrations of two rigidly coupled circular cylinders in tandem. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.04.006.