The Effect of Reynolds Numbers on Flow-Induced Vibrations: A Numerical Study of a Cylinder on Elastic Supports

Abstract

In the field of fluid dynamics, the Reynolds number is a key parameter that influences the flow characteristics around bluff bodies. While its impact on flow around stationary cylinders has been extensively studied, systematic research into flow-induced vibrations (FIVs) under these conditions remains limited. This study utilizes numerical simulations to explore the FIV characteristics of smooth cylinders and passive turbulence control (PTC) cylinders supported elastically within a Reynolds number range from 0.8 × 10⁴ to 1.1 × 10⁵. By comparing the vibration responses, lift coefficients, and wake structures of these cylinders across various Reynolds numbers, this paper aims to elucidate how Reynolds numbers affect the flow and vibration characteristics of these structures. The research employs images of instantaneous lift changes and vortex shedding across multiple sections to visually demonstrate the dynamic changes in flow states. The findings are expected to provide theoretical support for optimizing structural design and vibration control strategies in high-Reynolds-number environments, emphasizing the importance of considering Reynolds numbers in structural safety and design optimization.

1. Introduction

The Reynolds number is a key parameter for analyzing flow characteristics around a cylinder [1,2,3]. Schlichting et al. [4] categorized the flow around a cylinder based on the Reynolds number, broadly divided into the following regions: creeping or laminar ($\mathrm{R}\mathrm{e}<5$), vortex shedding ($5<\mathrm{R}\mathrm{e}<40$), laminar separation ($40<\mathrm{R}\mathrm{e}<300$), subcritical ($300<\mathrm{R}\mathrm{e}<1.4\times {10}^5$), critical ($1.4\times {10}^5<\mathrm{R}\mathrm{e}<1\times {10}^6$), supercritical ($1\times {10}^6<\mathrm{R}\mathrm{e}<4\times {10}^6$), and transcritical ($\mathrm{R}\mathrm{e}>4\times {10}^6)$. Furthermore, the subcritical regime is further differentiated based on the characteristics of the boundary layer and separation shear layer on the cylinder surface into the wake transition zone (TrW) and the shear layer transition zone (TrSL). Within the TrSL, based on variations in the lift coefficient $C_l$, three distinct sub-regions are identified: TrSL1 where $C_l$ gradually increases ($350\le \mathrm{R}\mathrm{e}\le 2\times {10}^3$), TrSL2 where $C_l$ rapidly rises ($2\times {10}^3\le \mathrm{R}\mathrm{e}\le 4\times {10}^4$), and TrSL3 where $C_l$ remains relatively stable ($4\times {10}^4\le \mathrm{R}\mathrm{e}\le 2\times {10}^5$).

Most research on flow-induced vibrations (FIVs) focuses on the TrSL1 and TrSL2 regions, including comprehensive studies on vortex-induced vibrations (VIVs). Since Williamson’s observation of three branches (initial, upper, and lower) in low mass–damping parameter $m^{\mathrm{*}}\zeta$ conditions, scholars have concentrated on the highest amplitude ratio $A_{max}^{\mathrm{*}}$ that VIVs can excite. Griffin et al. [5] proposed the use of logarithmic coordinates to create the Griffin plot, which highlights the trend of $A_{max}^{\mathrm{*}}$ by reducing data dispersion. This $A_{max}^{\mathrm{*}}$ trend has been validated by many researchers [6,7], with the Reynolds numbers involved in the Griffin plot in the order of ${10}^3$. In this area, numerous scholars have conducted thorough research. The subcritical regime, as a complex stage, has generated significant research interest. Jiang et al. [8], through their study on vortex-induced vibrations (VIVs) of a cylinder at a Reynolds number of 10,000, established a correlation between $V_r$ and the lift coefficient. Cicolin et al. [9] sought to reduce VIVs in the subcritical stage by installing Ventilated Trousers (VTs) on the cylinder. Modir et al. [10] conducted an investigation into vortex-induced vibrations (VIVs) within a water tank, exploring a Reynolds number range of approximately $4\times {10}^3\le \mathrm{R}\mathrm{e}\le 1\times {10}^5$. Notably, their study utilized springs with varying stiffness, which may have influenced the results. The primary focus of their research was the VIVACE technology, which seeks to capture hydrokinetic energy from ocean currents by amplifying VIVs. However, as research on Reynolds numbers has expanded, Bernitsas et al. [11] observed $A_{max}^{\mathrm{*}}$ values up to 1.96 in VIV experiments conducted between Reynolds numbers of $4\times {10}^4<\mathrm{R}\mathrm{e}<1.5\times {10}^5$, and they found that the lower branch was increasingly replaced by the rising upper branch, suggesting that the Griffin plot may no longer be suitable for VIVs at high Reynolds numbers [11,12]. Subsequently, Govardhan and Williamson [13], and Narendran et al. [14], proposed new equations for the maximum amplitude ratio $A_{max}^{\mathrm{*}}$ that include Reynolds numbers to more accurately describe the characteristics of VIVs. The Bernitsas team [11] speculated that the higher $A_{max}^{\mathrm{*}}$ at high Reynolds numbers might be related to changes in the lift coefficient $C_l$ across different TrSL regions, but their experiments primarily focused on the energy conversion of VIVs, so this conclusion has not been further confirmed.

In studies on galloping, another significant phenomenon of FIVs, the focus is generally on Reynolds numbers below the TrSL2 region. Galloping typically manifests in non-streamlined sectional columns [15,16], and though its vibration phenomena are less complex than those of VIVs, it frequently results in substantial oscillations that can easily damage structures [15,17]. Consequently, researchers are keenly interested in strategies to prevent galloping [18,19]. Recently, Bernitsas et al. [20] proposed exploiting passive turbulence control (PTC) to induce galloping for energy harvesting, drawing increased scrutiny on galloping at higher Reynolds numbers [21,22,23]. Ma et al. [24,25] explored the mechanisms of PTC cylinders by installing symmetrical strips on either side of a cylinder. Their numerical simulations provided a detailed analysis of the flow field, revealing that the vibration response of the PTC cylinder, unlike that of smooth cylinders, encompasses the initial branch, passive upper branch, transition region, and galloping zone. Nevertheless, the influence of Reynolds numbers on galloping remains underexplored.

Building on the foundational understanding of Reynolds number effects, this study seeks to advance our knowledge of flow-induced vibrations by systematically investigating the impact of varying Reynolds numbers on the dynamic responses of both smooth and passive turbulence control (PTC) cylinders. Specifically, the objectives are as follows:

(1)

Compare the vibration responses of smooth and PTC cylinders under varying Reynolds numbers to detail changes in flow dynamics.

(2)

Analyze key vibration characteristics—amplitude ratios, lift coefficients, and phase differences—across different flow regimes.

(3)

Investigate the mechanisms affecting vibration responses and wake structures, emphasizing the influence of Reynolds numbers.

(4)

Focus on the galloping phenomena in PTC cylinders at high Reynolds numbers, an area currently underexplored.

This paper investigates the impact of Reynolds numbers on the FIVs of cylinders on elastic supports by varying the natural frequency. It begins with an introduction to the numerical method and computational details, followed by a validation of the solver. Subsequent analyses focus on the vibration response of both smooth cylinders and PTC cylinders under elastic support, discussing in detail their vibration responses, lift coefficients, phase differences, lift correlations, and wake structures. This approach aims to explore the effects and underlying mechanisms of Reynolds numbers in FIVs.

2. Numerical Method and Computational Details

In this study, the fluid–structure solver used, developed within the OpenFOAM open-source framework, has been previously validated across diverse applications in ocean engineering and hydrodynamics [26,27,28]. For the FIV analysis, the solver incorporates various modules: input handling, flow field computation, high Reynolds number solutions, mooring systems, 6DoF (six degrees of freedom) motion, and grid movement and updating [29].

2.1. Governing Equations and Turbulence Modeling

The continuity and momentum equations for incompressible viscous flow are expressed as follows:

$$\nabla \cdot \mathbf{U}=0,$$

(1)

$${\displaystyle \frac{\partial \mathbf{U}}{\partial t}}+\nabla \cdot \left(\mathbf{U}-{\mathbf{U}}_g\right)\mathbf{U}=\nabla \cdot \left({\nu }_{eff}\nabla \mathbf{U}\right)+\left(\nabla \mathbf{U}\right)\cdot \nabla {\nu }_{eff}-{\displaystyle \frac{1}{\rho }}\nabla p,$$

(2)

where $\mathbf{U}$ and $p$ denote the fluid velocity and pressure, respectively, $\rho$ represents the fluid density, ${\mathbf{U}}_g$ indicates the grid velocity, and ${\nu }_{eff}$ combines the molecular viscosity $\nu$ and turbulent eddy viscosity ${\nu }_t$.

The numerical simulations leverage the SST-DDES approach, founded on the framework outlined by Gritskevich et al. [30]. The turbulence model equations for kinetic energy $k$ and specific dissipation rate $\omega$ are as follows:

$${\displaystyle \frac{\partial k}{\partial t}}+\nabla \cdot [(\mathbf{U}-{\mathbf{U}}_g)k]=\nabla \cdot \left[\left(\nu +{\alpha }_k{\nu }_t\right)\nabla k\right]+\tilde{G}-{\displaystyle \frac{\sqrt{k^3}}{l_{DDES}}},$$

(3)

$${\displaystyle \frac{\partial \omega }{\partial t}}+\nabla \cdot [(\mathbf{U}-{\mathbf{U}}_g)\omega ]=\nabla \cdot \left[\left(\nu +{\alpha }_{\omega }{\nu }_t\right)\nabla \omega \right]+2\left(1-F_1\right){\alpha }_{\omega 2}{\displaystyle \frac{\nabla k\cdot \nabla \omega }{\omega }}+{\displaystyle \frac{\gamma }{{\nu }_t}}\tilde{G}-\beta {\omega }^2,$$

(4)

Here, all constants, ${\alpha }_k,{\alpha }_{\omega },\gamma , \mathrm{a}\mathrm{n}\mathrm{d} \beta$, are interpolated as specified by Gritskevich et al. [30] and Zhao and Wan [31].

The simulations are performed using the finite volume approach. Temporal elements in the momentum and turbulence transport equations utilize a second-order backward differencing scheme for discretization. For convective terms, the momentum equation employs a second-order linear-upwind stabilized transport (LUST) scheme, while the turbulent transport equation uses a second-order limited linear scheme. Diffusion terms are addressed through a second-order linear scheme. Pressure–velocity coupling is managed using the PIMPLE algorithm, which integrates PISO and SIMPLE elements, configured with three outer and two inner corrector loops. To maintain the Courant–Friedrichs–Lewy (CFL) number close to 1, the time step is adjusted based on the inflow velocities in different scenarios.

2.2. Computational Details

The computational domain for this study is depicted in Figure 1 as a rectangular prism of dimensions $40D\times 20D\times \pi D$, with $D$ representing the cylinder’s diameter ($D=0.0889 \mathrm{m}$) and an aspect ratio $L/D=\pi$. The cylinder, supported laterally by two springs perpendicular to the inflow direction (x-axis), is centered at the coordinate origin, positioned $10D$ downstream from the inlet boundary. The y-axis is perpendicular to both the cylinder and the inflow, and the z-axis is along the span of the cylinder. The boundary conditions are as follows: At the inlet, the velocity is set to a uniform inflow $U_{\mathrm{\infty }}$ with zero normal gradient for pressure. The cylinder surface employs a no-slip solid wall condition, while the front, rear, top, and bottom boundaries are set as symmetric in the solver. At the outlet, both velocity and pressure gradients are set to zero.

This study explores different vibration systems by varying the linear spring stiffness $K=30,60,120,240\mathrm{N}/\mathrm{m}$, where the spring force is linearly related to the stiffness, and each stiffness value corresponds to a fixed natural frequency ($f_n=0.39,0.55,0.78,1.1 \mathrm{H}\mathrm{z}$). Sun et al. [21] studied the effect of the mass–damping ratio, damping, and stiffness on the optimal hydrokinetic energy conversion of a single rough cylinder in flow-induced motions. In their article, the range of K values was 400–1200 $\mathrm{N}/\mathrm{m}$, with $L=10D$, corresponding to a length-to-diameter ratio of 10. In contrast, in our article, we selected $L=\pi D$, resulting in a length-to-diameter ratio of π. Therefore, we chose a K range of 30–240 $\mathrm{N}/\mathrm{m}$. This study does not involve changes in cylinder geometry or the damping ratio $\zeta$, with a constant mass–damping ratio for all spring stiffnesses ($m^{*}\zeta =0.063$). By adjusting the inflow velocity $U_{\infty }$, the present study aims to capture the branch responses of FIVs for both smooth and PTC cylinders, reflecting practical scenarios and highlighting the influence of Reynolds numbers. The parameters of the symmetric strips on the PTC cylinder here are consistent with previous studies, with the leading edge located 60 degrees from the forward stagnation point, covering 20 degrees, and with a height equivalent to 5% of the cylinder diameter, which can induce galloping.

Figure 2 illustrates the structured grid of the current study’s model, featuring an overall view, and a close-up of the smooth cylinder and the PTC cylinder. In the present FIV study, the smooth cylinder is configured with 196 nodes circumferentially, 131 radially, and 31 axially, totaling $1.10\times {10}^6$ mesh elements. Given the wide range of Reynolds numbers in this study, it is crucial to ensure that the dimensionless spacing of the first grid layer near the cylinder surface meets the requirement of $y^{+}<5$ for all flow velocities; thus, the minimum grid distance to the cylinder surface is set to $∆r/D=3.7\times {10}^{-4}$ to satisfy this criterion. Additionally, to compare the smooth cylinder and observe the impact of passive turbulence control on FIVs without the computational errors that might arise from remeshing, the grid for the smooth cylinder is designed with reserved spaces for PTC features, and the PTC cylinder grid contains $1.09\times {10}^6$ elements.

2.3. Validation

To further validate the accuracy of the solver and numerical methods used, this study conducted numerical simulations on a cylinder with a single degree of freedom in lateral vibration perpendicular to the flow direction. The simulation parameters were aligned with the setups of Williamson et al. and Han et al. [32,33], conducted using a circular cylinder with a very low mass–damping ratio of $m^{\mathrm{*}}\zeta =0.013$, where $\zeta$ represents the damping ratio, $\zeta =C/\left(2\sqrt{mk}\right)$.

Figure 3 displays the amplitude ratio $A^{\mathrm{*}}$ and frequency ratio $f^{\mathrm{*}}$ response of a smooth cylinder, with the reduced velocity $U^{\mathrm{*}}$ and corresponding Reynolds number plotted on the horizontal axis. The reduced velocity $U^{\mathrm{*}}$ is defined as $U^{\mathrm{*}}=U_{\mathrm{\infty }}/{Df}_n$, where $f_n$ is the natural frequency of the cylinder in water. $U^{\mathrm{*}}$ links the flow velocity to the natural frequency of the cylinder, thus representing the non-dimensional parameter critical for characterizing the synchronization or lock-in phenomena observed in VIVs. The Reynolds number $\mathrm{R}\mathrm{e}$ is given by $\mathrm{R}\mathrm{e}=D{U}_{\mathrm{\infty }}/\nu$, where $D$ is the cylinder diameter and $\nu$ is the kinematic viscosity of the fluid. The amplitude ratio $A^{\mathrm{*}}$ is $A^{\mathrm{*}}=A/D$, where $A^{\mathrm{*}}$ is the average of the maximum displacement over 60 vibration cycles. The frequency ratio $f^{\mathrm{*}}=f_{osc}/f_n$ is $f^{\mathrm{*}}=f_{osc}/f_n$, where $f_{osc}$ is the structural vibration frequency obtained from the vibration displacement through Fast Fourier Transform (FFT). The red color in the graph indicates the results from the current numerical simulations, while the black triangles represent the experimental results from the literature [32], offering a comparison to validate our numerical model. The amplitude and frequency response of the smooth cylinder in the current numerical simulations show typical FIV characteristics. The initial branch is excited around $U^{\mathrm{*}}=3$. The upper branch covers the reduced velocity range of ${4.6<U}^{\mathrm{*}}<7$, and the lower branch occurs in the range of ${7<U}^{\mathrm{*}}<12$, with the desynchronization area appearing when $U^{\mathrm{*}}>12$. Figure 3 demonstrates typical FIV characteristics such as the initial, upper, and lower branches of the vibration responses, highlighting the onset and cessation of synchronization with flow velocity changes, which is critical for understanding the dynamic stability of structures exposed to fluid flow.

Figure 4 presents the Lissajous figures that map the relationship between the lift and vibration displacement across different phases of vibration, illustrating changes in the phase angle during the transition through the initial, upper, and lower branches of VIVs. The results from the current numerical simulations align closely with experimental data [32]. For the initial branch, $U^{\mathrm{*}}=3.7$ and the phase angle is near 0°, indicating in-phase lift and displacement, characteristic of stable and predictable interactions between the fluid and the cylinder. For the upper branch, $U^{\mathrm{*}}=6.3$ and the phase exhibits instability, intermittently shifting between nearly 0° and 180° as the amplitude changes. This behavior suggests a transitional dynamic, indicative of nonlinear fluid–structure interactions and varying energy transfer efficiencies. For the lower branch, $U^{\mathrm{*}}=10.1$ and the phase stabilizes near 180°, showing out-of-phase lift and displacement, which are typical at higher energy states, leading to increased damping and potential structural stresses. These observations of the phase transitions demonstrate the system’s complex response to increasing flow velocities, with a notable hysteretic behavior moving from the initial to the upper branch.

Thus, the comparison with experimental data in this section confirms the accuracy and effectiveness of the solver used in these numerical simulations.

3. Reynolds Number Effects on FIVs in Smooth Cylinders

The FIV of the elastically supported smooth cylinder primarily manifests as a VIV, as evidenced in the solver validations discussed above. This section examines the impact of the Reynolds number on the FIV of the smooth cylinder, with Reynolds numbers in the range of $0.8\times {10}^4\le \mathrm{R}\mathrm{e}\le 1.1\times {10}^5$, covering the TrSL2 and TrSL3 regions.

3.1. Vibration Responses of Smooth Cylinders

The VIV response of the smooth cylinder across different Reynolds number ranges is depicted in Figure 5. For ease of description, four types of spring stiffness are used to designate the different cases, named K30-smooth, K60-smooth, K120-smooth, and K240-smooth. The horizontal axis in the graph represents the reduced velocity $U^{\mathrm{*}}$, and the vertical axis shows the amplitude ratio $A^{\mathrm{*}}$ and frequency ratio $f^{\mathrm{*}}$. It is important to note that synchronization or lock-in phenomena are not discussed here; therefore, the frequency response graph only displays the dominant frequency, and the corresponding $f^{\mathrm{*}}$ is the ratio of the vibration frequency to the natural frequency of the system in air.

From the amplitude ratio $A^{\mathrm{*}}$ presented in Figure 5, it is apparent that the initial branches of all four cases largely overlap, with discernible changes commencing at the start of the upper branch. The amplitude responses of K30-smooth and K60-smooth distinctly demonstrate the VIV initial, upper, and lower branches, with K60-smooth’s lower branch reaching an $A^{\mathrm{*}}=0.8$. Conversely, for K120-smooth and K240-smooth, the lower branches become progressively less pronounced. Consequently, it can be inferred that as the Reynolds numbers transition from TrSL2 to TrSL3, the amplitude of the VIV’s lower branch gradually increases, reducing the distinction from the upper branch, until their transitions become indistinguishable, aligning with experimental observations [14]. Furthermore, as the Reynolds numbers rise, there is an observed increase in the maximum amplitudes, as depicted in Figure 6. On the horizontal axis, $\alpha =(m^{*}+C_A)\zeta$, and the formula in the figure is $Y_{max}=-0.4435\left[{\log }_{10}\alpha /\mathrm{R}\mathrm{e}\right]-1.5$.$The{A}_{max}^{\mathrm{*}}$ across varying Reynolds numbers closely conforms to the fitting curve proposed by Narendran [14], thus providing robust data from the numerical simulations for the exploration of $A_{max}^{\mathrm{*}}$ in VIVs at elevated Reynolds numbers. Nevertheless, the $A_{max}^{\mathrm{*}}$ in the VIVs within TrSL3 falls short of reaching the experimental values of $A_{max}^{\mathrm{*}}=1.97$ reported by Raghavan [21,34]. This discrepancy suggests that the high amplitudes observed may be due to the experimental setup, where the cylinder is buoyed by fluid dynamics caused by friction between the cylinder ends and the tank walls, resulting in greater amplitudes. This is also a key aspect that we will explore through numerical simulations in our future research.

From the frequency ratio $f^{\mathrm{*}}$ depicted in Figure 5, it is observable that up to $U^{\mathrm{*}}<7$, the vibration frequency ratios for all four cases coincide perfectly, subsequently diverging at different reduced velocities, indicating the transition into the lower branch. As $U^{\mathrm{*}}$ increases, $f^{\mathrm{*}}$ stabilizes and tends to approach 1 with the increase in the natural frequency of the vibration system. Govardhan and Williamson [13] have discussed that due to the effect of added mass, the $f^{\mathrm{*}}$ in the lower branch of VIVs for cylinders with a high $m^{\mathrm{*}}\zeta$ tends to be closer to 1, whereas it exceeds 1 for a low $m^{\mathrm{*}}\zeta$. However, in the current study, with $m^{\mathrm{*}}\zeta =0.065$, it appears that changes in the Reynolds numbers are driving the variations in $f^{\mathrm{*}}$. Additionally, at high Reynolds numbers, the corresponding range of $U^{\mathrm{*}}$ for the VIVs narrows, consistent with phenomena observed in experiments [14,34].

However, as the Reynolds numbers increase, the amplitude of the lower branch enlarges and the distinction from the upper branch becomes less defined. Could this be due to higher lift coefficients in the TrSL3 region? This will be further analyzed in the subsequent examination of lift variations.

3.2. Lift Coefficients and Phase Differences of Smooth Cylinders

Figure 7 illustrates the relationship between the root mean square lift coefficient $C_{l,rms}$ and the lift displacement phase difference $\theta$ with the reduced velocity $U^{\mathrm{*}}$ for smooth cylinders. Across all Reynolds numbers, the peak of the lift coefficient for smooth cylinders consistently occurs at $U^{\mathrm{*}}\approx 4$, where $C_{l,rms}\approx 2$, marking the beginning of the upper branch. During this phase, the lift coefficient differences across various Reynolds numbers are minimal, especially when $U^{\mathrm{*}}>6$. At this point, the curves for K30-smooth and K60-smooth overlap, while those for for K120-smooth and for K240-smooth within the TrSL3 region are lower.

The phase diagrams show a jump from in-phase to out-of-phase during the transition from the upper to the lower branch of the VIVs. Notably, for K240-smooth, the lift and displacement do not fully shift to out-of-phase ($\theta =180°$), but maintain around $\theta =150°$. This shift may be influenced by the increased Reynolds number, which impacts the flow around the moving cylinder and affects the lower-branch vibrations.

Thus, the differences in the FIV between TrSL2 and TrSL3 cannot be solely attributed to changes in lift. The shear layer changes around the vibrating cylinder may not specifically occur within TrSL2 or TrSL3. Changes in fluid flow around moving bodies are always delayed compared to stationary objects. This suggests that significant variations in lift coefficients may only become apparent with further increases in Reynolds numbers.

3.3. End Lift Correlation of Smooth Cylinders

The lift correlation of a cylinder can reflect the stability of structural vibrations and the three-dimensional effects of flow. Initially, the characteristics of each branch of the FIV through the end lift correlation are analyzed. Figure 8 presents the end lift correlation diagram for smooth cylinders. For the initial branch, each cylinder with different spring stiffnesses exhibits strong correlation. As the VIV transitions into the upper branch, all R curves display a distinct “notch”.

Significant differences occur after $U^{\mathrm{*}}>7$, as the cylinders sequentially enter the lower branch of the VIVs. For K30-smooth, the lift correlation increases to $R\approx 0.8$, and for K60-smooth, $R\approx 0.7$. The lift correlations for K120-smooth and K240-smooth are, respectively, $R\approx 0.4$ and $R\approx 0.4$. It is evident that with increasing Reynolds numbers, K120-smooth and K240-smooth do not regain strong correlations in the lower branch of the VIVs. This lack of strong correlation is associated with their phase transitions in the lift, seen in Figure 7, which also impacts the less distinct performance of the lower branch.

3.4. Spanwise Lift Correlation and Wake Structure of Smooth Cylinders

This section examines the flow states at different stages under varying Reynolds numbers through the observation of instantaneous lift changes and vortex shedding from different sections of the cylinder. According to the VIV behavior of smooth cylinders, the vibration responses are categorized into the initial branch, upper branch, lower branch, and desynchronization phase. Due to the commonality in the effects of lift correlation across different spring stiffnesses, a representative spring stiffness is selected for each branch.

Figure 9 presents 100 instantaneous lift variation contour maps for the cylinder across different branches. For the initial branch of K30-smooth at $U^{\mathrm{*}}=3.6$, as shown in Figure 9a, the vibration amplitude is very low, and the three-dimensional flow along the span of the cylinder is not prominent, with the lift displaying consistent color changes and a strong correlation at different instants. As vibrations enter the upper branch at $U^{\mathrm{*}}=6.6$, regardless of the Reynolds number, the end lift correlation drops to around zero, and the contour in Figure 9b becomes increasingly chaotic. For $t/T=40-50$, the lift contour along the span shows completely inverse colors. As the reduced velocity increases further, the VIV progresses to the lower branch, where the three-dimensional flow becomes less pronounced compared to the upper branch. For K120-smooth, now within the TrSL3 range, Figure 9c illustrates that at $U^{\mathrm{*}}=8.7$, although the instantaneous lift variation shows some recovery, it is not as strong as at lower Reynolds numbers. When vibrations enter the desynchronization phase, as depicted in Figure 9d at $U^{\mathrm{*}}=10.7$, the vortex shedding frequency is high, the instantaneous lift is disordered, and nonlinear flow is evident.

Overall, Figure 9 vividly illustrates the process of lift variations experienced by smooth cylinders under four different elastic supports, with the results consistent with those shown in Figure 8. It can be concluded that in TrSL3, the trend of VIVs with changes in reduced velocity is similar to that in TrSL2, but the impact of three-dimensional flow on the vibrations increasingly dominates.

Figure 10 illustrates the wake structures at different cross-sections of the cylinder, corresponding to the moments shown in Figure 9. The cross-sections were chosen to be close to the ends of the cylinder while avoiding the effects of the end boundaries (at $z/L=0.18$ and $z/L=0.82$); the instances of vibration occur as the cylinder moves upward near the x-axis, with a continuing upward trend. For the initial branch vibration at $U^{\mathrm{*}}=3.6$ [Figure 10a], the vortices shed in a similar manner at both cross-sections, identifiable as a 2S mode. For K60-smooth, as shown in Figure 10b, the wake vortices at different cross-sections exhibit entirely distinct forms. At $z/L=0.18$, a pair of vortices shedding simultaneously can be observed, displaying the classic 2P mode. However, at $z/L=0.82$, the wake pattern is indeterminate, confirming asynchronous vortex shedding along the span of the cylinder. This is consistent with the weak lift correlation observed in Figure 9b. For K120-smooth, which is in the lower branch with its Reynolds number fully within the TrSL3 region, distinct but identifiable 2P modes can be observed at different cross-sections along the span, as shown in Figure 10c. This conforms to the traditional definition of the lower branch, but exhibits weaker spanwise correlation compared to the Reynolds numbers in the TrSL2 region. At $U^{*}=10.7$, the wake vortices of K240-smooth, shown in Figure 10d, shed asynchronously along the span. Near the cylinder, the sequential formation of vortices on both cross-sections is evident, indicating a weak spanwise lift correlation and strong nonlinear flow during the minimal amplitude desynchronization phase.

The above discusses the impact of Reynolds numbers on FIVs in smooth cylinders, including analyses of vibration responses, lift coefficients, and wake structures. However, the FIV studied here includes only a VIV. Next, PTC cylinders will be used as the subject of study to explore the effects of Reynolds numbers on a broader range of FIV phenomena.

4. Reynolds Number Effects on FIVs in PTC Cylinders

The FIV of elastically supported PTC cylinders predominantly manifests as a VIV, transitional phases, and galloping, conclusions drawn from studies within the TrSL2 range [24]. This section discusses the impact of Reynolds numbers on FIVs in PTC cylinders, with Reynolds numbers in the range of $0.8\times {10}^4\le \mathrm{R}\mathrm{e}\le 1.1\times {10}^5$, encompassing both the TrSL2 and TrSL3 regions.

4.1. Vibration Responses of PTC Cylinders

Figure 11 illustrates the vibrational responses of PTC cylinders across different Reynolds number ranges, similarly categorized using four types of spring stiffness: K30-PTC, K60-PTC, K120-PTC, and K240-PTC. The vibration responses depicted in Figure 11 are similar to those studied within the TrSL2 region, where passive turbulence control strips alter the cylinder’s vibration response, triggering four types of FIV phenomena: VIVs with initial and passive upper branches, transitional stages from VIVs to galloping, and pure galloping.

Two notable observations from the amplitude responses in the graph are as follows: firstly, as the Reynolds numbers increase, the passive upper branches exhibit higher amplitudes; secondly, the exact onset of galloping in PTC cylinders is indeterminate. It was found that in the study by Sun et al. [15], the PTC cylinder also induced galloping, which is consistent with our experimental findings. With increasing Reynolds numbers, galloping can be initiated at lower reduced velocities $U^{\mathrm{*}}$. Consequently, at the same $U^{\mathrm{*}}$, cylinders at higher Reynolds numbers display greater amplitude ratios and frequency ratios during galloping.

Additionally, as depicted in Figure 11, across all branches of the FIV, the frequency response $f^{*}$ remains below 1, consistent with the response of passive control strips on stationary cylinders, regardless of the Reynolds number range. In the VIV phases of PTC cylinders (including the initial and passive upper branches) and during transitional stages, the changes in the frequency ratio among the cylinders are minimal. The difference becomes more pronounced as the PTC cylinders enter the galloping phase, where the decrease in $f^{*}$ diminishes with increasing Reynolds numbers. This illustrates that as the flow conditions intensify, the influence of Reynolds numbers on the frequency response during galloping becomes less pronounced.

4.2. Lift Coefficients and Phase Differences of PTC Cylinders

Figure 12 displays the relationship between the root mean square lift coefficient $C_{l,rms}$ and the lift displacement phase difference $\theta$ with the reduced velocity $U^{\mathrm{*}}$ for PTC cylinders. Overall, the lift coefficients across all four cylinders show no significant differences. The initial branch $C_{l,rms}$ is approximately 1, higher than that of smooth cylinders, and peaks around $U^{\mathrm{*}}\approx 4$ at $C_{l,rms}\approx 1.8$, slightly lower than the corresponding lift coefficient for smooth cylinders. It then rapidly decreases to around $C_{l,rms}\approx 0.5$ at $U^{\mathrm{*}}\approx 7$, covering the transitional and galloping phases.

In the phase difference graph for PTC cylinders, a fluctuation of about $30°$ is observed for ${4<U}^{\mathrm{*}}<12$, which correlates with the passive upper branch and the transitional phase. Ma et al. [24], in their study of FIVs in PTC cylinders within TrSL2, highlighted that symmetric control strips suppress the modal competition that would otherwise occur, delaying it to the transitional phase. At higher Reynolds numbers, such suppression still causes significant fluctuations due to enforced flow separation and the impacts of fluid vortices at the rear of the cylinder, which can induce flutter. Consequently, the effectiveness of symmetric control strips in suppressing nonlinear flow in the passive upper branch may be less at high Reynolds numbers compared to lower ones.

4.3. End Lift Correlation of PTC Cylinders

Figure 13 illustrates the end lift correlation for PTC cylinders, revealing two distinct degrees of “notches” within the $5<U^{\mathrm{*}}<10$ range, which can be divided into two parts. The first part, $5<U^{\mathrm{*}}<7$, falls within the passive upper branch range. Here, the correlation decreases from an initial $R\approx 0.9$ to about $R\approx 0.5$, significantly maintaining a 50% higher correlation compared to the weak correlation $R\approx 0$ of the smooth cylinder’s upper branch in the same range. Although slightly poorer than the performance at lower TrSL2 Reynolds numbers, this does not negate the role of passive control strips in preventing phase changes and enhancing lift correlation.

The second distinct “notch” occurs at $7<U^{\mathrm{*}}<10$, corresponding to the transitional phase from VIVs to galloping. This period of weak correlation is linked to the instability caused by flow separation and flutter, signaling the unstable modal competition between different types of FIVs. The end lift correlation of PTC cylinders tends to improve as vibrations transition into galloping. However, at higher Reynolds numbers, this improvement is not as pronounced, particularly for K240-PTC during the galloping phase, where $R\approx 0.3$. This is related to the unsteady flow dynamics, high-frequency vortex shedding, and significant structural vibrations typical at high Reynolds numbers.

4.4. Spanwise Lift Correlation and Wake Structure of PTC Cylinders

This section investigates the flow dynamics at various stages across different Reynolds numbers by tracking instantaneous lift variations and vortex shedding at different cross-sections of PTC cylinders. The vibrational response of these cylinders is categorized into four distinct phases based on their FIV characteristics: initial branch, passive upper branch, transitional phase, and galloping. Due to the uniform impact of spring stiffness on lift correlation across these phases, one representative stiffness value is chosen for each phase for in-depth analysis.

Figure 14 displays the contour plots that capture 100 snapshots of lift variation across different branches for a cylinder. In the initial branch with K30-PTC at $U^{\mathrm{*}}=3.6$, as shown in Figure 14a, the lift demonstrates consistent changes across the span, indicating a strong spanwise correlation. When the vibrations progress to the passive upper branch at $U^{\mathrm{*}}=6.6$, the frequency of the lift changes increases, as seen in Figure 14b, slightly diminishing the correlation, which is still noticeably stronger than that observed in the upper branch of a smooth cylinder. As the system transitions from VIVs to galloping, depicted in Figure 14c, the spanwise lift correlation initially weakens from the passive upper branch level and then strengthens upon reaching the galloping.

Figure 14 vividly illustrates the process of lift variation in PTC cylinders supported by four different types of elasticity, with the results consistent with those shown in Figure 13. Similar conclusions can be drawn as with smooth cylinders; in TrSL3, the trend of FIV changes with a reduced velocity in PTC cylinders is similar to that in TrSL2, but the impact of three-dimensional flow on the vibrations increasingly dominates.

Figure 15 displays the instantaneous vortex shedding diagrams at different cross-sections of PTC cylinders, with section selection and instantaneous states analogous to those of the smooth cylinders shown in Figure 10.

For K30-PTC, as shown in Figure 15a, the presence of PTC results in more disordered vortex shedding compared to K30-smooth, yet recognizable vortex shedding patterns can still be identified at different spanwise positions, correlating with the lift spanwise correlation analysis in Figure 14.

For K60-PTC, as depicted in Figure 15b, more consistent vortex shedding patterns are evident at cross-sections $z/L=0.18$ and $z/L=0.82$. Although it is currently challenging to definitively categorize the vortex shedding pattern from the diagram, this phenomenon confirms that passive control strips have reduced the cylinder’s three-dimensional flow along the span.

During the transition from VIVs to galloping in K120-PTC, where the spanwise lift correlation $R\approx 0$, entirely different vortex patterns are observed at the two cross-sections, as shown in Figure 15c: at $z/L=0.18$, vortices on both the upper and lower sides of the cylinder shed simultaneously, potentially forming a 2P mode, while at $z/L=0.82$, the asynchronous shedding suggests a possible 2S mode. While varying instantaneous vortices do not conclusively prove stronger nonlinear flow at this stage, in conjunction with discussions on the end lift and instantaneous lift correlation in Figure 13 and Figure 14, it can be inferred that robust three-dimensional flow exists during the transitional phase of FIVs in PTC cylinders.

For the galloping phase of K240-PTC, shown in Figure 15d, relatively regular synchronized vortex shedding is observed near the rear of the cylinder across both cross-sections; although significant variations appear downstream, the spanwise correlation remains superior to that of K240-smooth at this stage. Additionally, asymmetrical wake patterns downstream indicate increased vortex shedding frequency, with up to 12 or more vortices shedding during a single cylinder vibration cycle, which then rapidly mix with the inflow in the downstream area, causing differences in the wake across different cross-sections.

5. Conclusions

This paper conducts a thorough analysis and comparison of FIV phenomena in smooth and passive turbulence control (PTC) cylinders supported elastically across a range of Reynolds numbers ($0.8\times {10}^4\le Re\le 1.1\times {10}^5$). The cylinders have an aspect ratio ($L/D$) of 3.14 and a mass–damping parameter ($m^{\mathrm{*}}\zeta$) of 0.063. This study focuses on changes in key parameters such as vibration response and lift coefficient, utilizing images of instantaneous lift variations and vortex shedding across multiple cross-sections to observe differences in flow states. The main conclusions drawn are as follows:

• For smooth cylinders, the peak amplitude ratio ($A^{\mathrm{*}}$) of VIV upper branches increases with increases in the Reynolds number, consistent with the experimental fitting formulas. When Reynolds numbers fall into the TrSL3 range, it becomes difficult to distinguish between the upper and lower branches, with the entire lower branch not vibrating at consistent amplitudes, as seen in TrSL2.
• Passive turbulence control strips facilitate four types of FIV phenomena across various Reynolds numbers: initial branch, passive upper branch, transitional phase, and galloping. As Reynolds numbers increase, galloping occurs at lower reduced velocities.
• Although FIV responses show some consistency during the initial and upper branches of vibration, as Reynolds numbers increase, particularly during the transitional and galloping phases, the variability in responses increases. This suggests that higher Reynolds numbers augment the complexity and nonlinearity of the flow field, impacting the predictability and control of vibration behavior.
• Changes in lift coefficients across different Reynolds numbers are not significant, indicating that variations in lift coefficients are not the primary cause of differences in FIV responses between TrSL2 and TrSL3. While passive turbulence control strips do mitigate nonlinear flows to some extent, their effectiveness diminishes with increasing Reynolds numbers.

In summary, Reynolds numbers are a critical parameter influencing FIV phenomena, particularly when considering structural safety and design optimization. This study provides insights into how Reynolds numbers influence flow fields and structural responses to control FIVs, offering a theoretical basis for designing more effective vibration mitigation strategies at varying Reynolds numbers. This study’s findings are pertinent to engineering fields concerned with fluid–structure interactions, particularly in marine and civil engineering where understanding flow-induced vibrations is crucial for the structural safety of offshore platforms and bridges. Additionally, the insights have implications for energy-harvesting technologies using flow dynamics. Future work could focus on a broader range of Reynolds numbers and different structural configurations to reveal more details about the mechanisms of FIVs.