2. Formulation and numerical method

A sketch of the physical system is presented in figure 1(a). The circular cylinder of diameter $D$ and mass per unit length $M_c$ is parallel to the $z$ axis. It is mounted on an elastic support of structural stiffness per unit length $k$ , and damping per unit length $d$ , and is free to translate along the $y$ axis within a quiescent fluid of density $\rho _f$ and viscosity ${\mu}$ . The cylinder is subjected to a forced, anticlockwise, steady rotation of angular velocity $\varOmega$ , about its axis. All the physical variables are non-dimensionalised by $\rho _f$ , $D$ , and the body surface velocity $U=\varOmega D/2$ . The cylinder displacement, non-dimensionalised by $D$ , is denoted by $\zeta$ . The tangential force coefficient is defined as $C=2F/(\rho _f D U^2)$ , where $F$ is the dimensional fluid force per unit length aligned with the direction of motion ( $y$ axis). The dynamics of the cylinder is governed by the equation

(2.1) \begin{equation} \ddot \zeta + \dfrac {4\unicode{x03C0} \xi }{U^\star } \dot \zeta + \left (\dfrac {2\unicode{x03C0} }{U^\star }\right )^2 \zeta = \dfrac {2 C}{\unicode{x03C0} m^\star }, \end{equation}

where $\dot {\phantom {a}}$ designates the non-dimensional time derivative. The structure to displaced fluid mass ratio, structural damping ratio, non-dimensional natural frequency and associated reduced velocity are defined as $m^\star =4 M_c/(\unicode{x03C0} \rho _f D^2)$ , $\xi =d/(2\sqrt {k M_c})$ , $f_n=[D/(2\unicode{x03C0} U)]\sqrt {k/M_c}$ and $U^\star =1/f_n$ , respectively.

Figure 1.

(a) Sketch of the physical system. (b) Peak value of the tangential force coefficient exerted on a non-rotating cylinder forced to oscillate in quiescent fluid, for different values of $\textit{Re}$ and $\textit{KC}$ . (c) Phase-averaged amplitude of the displacement of an elastically mounted, non-rotating cylinder, subjected to an oscillatory flow / a forced oscillation in the perpendicular direction, for $(\textit{Re},m^\star ,U^\star )=(150,2,5)$ and $\textit{KC}\in \{125,500\}$ . For the validation cases, the peak velocity attained during the forced oscillation is used to normalise fluid force, and to define $\textit{Re}$ , $\textit{KC}$ and $U^\star$ . The present results are compared to those reported by Elston, Blackburn & Sheridan (Reference Elston, Blackburn and Sheridan2006) and Koehler et al. (Reference Koehler, Beran, Vanella and Balaras2015) in (b), and Dorogi et al. (Reference Dorogi, Baranyi and Konstantinidis2023) in (c).

The mass ratio ranges from $0.1$ to $1000$ , and the reduced velocity is varied up to $250$ . Structural damping is introduced in specified cases, and set to zero otherwise. The Reynolds number $\textit{Re}=\rho _f U D/{\rm \mu}$ is kept below or equal to $100$ , which ensures that the flow remains two-dimensional over most of the parameter space. The three-dimensional transition is found to occur in the higher- $\textit{Re}$ range, typically above $\textit{Re}=75$ . In the present work, as a first step, the flow dynamics is predicted by the two-dimensional Navier–Stokes equations throughout the parameter space. Three-dimensional simulation results indicate that the principal features of the system behaviour described hereafter persist beyond the transition.

The numerical method is the same as in previous studies concerning comparable systems (Bourguet Reference Bourguet2020, Reference Bourguet2023, Reference Bourguet2025). It is briefly summarised, and some additional validation results are presented. The coupled flow–structure equations are solved by the parallelised code Nektar, which is based on the spectral/hp element method (Karniadakis & Sherwin Reference Karniadakis and Sherwin1999). The cylinder is placed at the centre of a large square computational domain (side length $600D$ ), which is discretised into $3975$ spectral elements. A no-slip condition is applied on the cylinder surface, and a Neumann-type boundary condition is used on the external boundaries. Convergence studies were carried out to set the polynomial order to $4$ and the non-dimensional time step to $0.005$ .

In order to validate the method in physical configurations close to the present one, comparisons with prior results concerning a non-rotating cylinder, either forced to oscillate in quiescent fluid (Elston et al. Reference Elston, Blackburn and Sheridan2006; Koehler et al. Reference Koehler, Beran, Vanella and Balaras2015), or elastically mounted in the direction normal to an oscillatory flow (Dorogi et al. Reference Dorogi, Baranyi and Konstantinidis2023), are proposed in figure 1(b,c). In the latter configuration, the oscillatory flow is emulated by a forced oscillation of the cylinder along the $x$ axis, while the body is free to move along the $y$ axis. For these validation cases, the peak velocity attained during the sinusoidal oscillation is used as reference to normalise fluid force and to define the Reynolds number, Keulegan–Carpenter number ( $\textit{KC}$ ) and reduced velocity. The prescribed non-dimensional amplitude and frequency are equal to $\textit{KC}/(2\unicode{x03C0} )$ and $1/{\textit{KC}}$ , respectively. The peak value of the tangential force coefficient and the phase-averaged amplitude of the displacement match those reported in previous studies, which confirms the validity of the present numerical method.

Each simulation is initialised with the established flow past a rigidly mounted, rotating cylinder. Then the body is released with an initial velocity $\dot \zeta =0.1$ . The analysis is based on long time series collected after the initial transient dies out.

3. Fluid–body system behaviour

The rotating cylinder is found to vibrate over a wide region of the parameter space. Typical examples of RIV are visualised in figure 2, via time series of the displacement and associated frequency spectra. The selected cases, which are representative of the different regimes of the system, as shown in the following, depict a variety of behaviours in terms of regularity, amplitude and frequency content.

Figure 2.

(a) Selected time series of the cylinder displacement and tangential force coefficient, and (b) frequency spectrum of the displacement, for (i) $(\textit{Re},m^\star ,U^\star )=(100,1,20)$ (periodic regime I), (ii) $(\textit{Re},m^\star ,U^\star )=(25,1,70)$ (periodic regime II), (iii) $(\textit{Re},m^\star ,U^\star )=(50,1,33)$ (aperiodic regime, transition region), and (iv) $(\textit{Re},m^\star ,U^\star )=(32,1,240)$ (aperiodic regime). In (b), the spectral amplitude is normalised by its maximum value; blue dashed lines denote the vibration frequency and its odd harmonics, and a green dash-dotted line represents the oscillator natural frequency. In (b–iii), the incommensurable component associated with the low-frequency modulation of the displacement is indicated by a red dotted line. The spectra are coloured according to the periodic (yellow) or aperiodic (grey) nature of the response.

Figure 3.

(a) Fluid–body system regime, (b) vibration amplitude, (c) vibration frequency normalised by the oscillator natural frequency, and (d) sinusoidal motion criterion ( $\Delta R$ ), as functions of the Reynolds number and reduced velocity, for $m^\star =1$ . The vibration region and, within this region, the zones where the system behaviour is periodic, are delimited by solid and dashed lines, respectively. The dotted area denotes the region of aperiodic behaviour where the response remains close to sinusoidal, with a slight low-frequency modulation. In (a), periodic and aperiodic behaviour regions are identified by yellow and grey background colours. Blue stars indicate the cases examined in figure 2.

The properties of RIV are examined in the $(\textit{Re},U^\star )$ domain for $m^\star =1$ in figure 3; it is recalled that no structural damping is considered ( $\xi =0$ ), unless stated otherwise. In each plot, the region where the body vibrates is delineated by a solid line. It extends down to $\textit{Re}\approx 4$ and $U^\star \approx 7$ , and up to the largest values under study, $\textit{Re}=100$ and $U^\star =250$ . The vibration region is composed of two distinct zones of periodic responses (delimited by dashed lines and coloured in yellow in figure 3 a), a first one (I) along the frontier of the vibration region, and a second one (II) that emerges further inside, and a vast area of aperiodic responses (coloured in grey). The same colour code is used in the spectra of figure 2 to specify the periodic/aperiodic nature of the responses. The cases selected in this figure are indicated by blue stars in figure 3(a). More generally, beyond vibration regularity, figure 3(a) maps the areas of periodic and aperiodic behaviours of the fluid–body system. This can be assessed by monitoring fluid velocity or force signals, as illustrated in figure 2(a), where $C$ time series are also represented.

The time-averaged value of the displacement vanishes in all cases. The vibration amplitude is quantified by the averaged top $10\,\%$ of $\zeta$ (denoted by $\zeta _{10}$ ). It tends to regularly increase with $U^\star$ (figure 3 b). Some fluctuations can be noted as $\textit{Re}$ is varied, but the amplitudes reached in the higher- $U^\star$ range, of the order of $30D$ , are much larger than VIV amplitudes (Williamson & Govardhan Reference Williamson and Govardhan2004), and are comparable to those reported for the saccades observed without structural restoring force (Bourguet Reference Bourguet2025). The vibration frequency ( $f_\zeta$ ) refers to the frequency of the dominant component of the $\zeta$ spectrum, which is also its fundamental component in all periodic response cases. As shown in figure 3(c), the vibration frequency remains lower than the oscillator natural frequency throughout the vibration region, and the ratio $f_\zeta /f_n$ may attain very low values. To visualise this deviation, $f_n$ is indicated in the spectra of figure 2 (green dash-dotted line). Such low frequency ratios also contrast with VIV typical trends. In the absence of structural damping and for the periodic responses, the condition $f_\zeta \lt f_n$ implies that the components of $C$ and $\zeta$ at $f_\zeta$ are in phase. This phasing state is found to persist in the aperiodic response cases (figure 2 a).

The departure of the displacement signal from a harmonic oscillation is measured via the unsigned relative difference of the ratio $R=\zeta _{10}/\zeta _{\textit{RMS}}$ with respect to the value expected for a sinusoidal response, denoted by $\Delta R$ , where $\zeta _{\textit{RMS}}$ designates the root mean square value of $\zeta$ . The departure from harmonic evolution is particularly pronounced in the aperiodic response region (figure 3 d). This criterion also highlights the distinct shapes of the vibrations occurring in each periodic response region: close to sinusoidal with higher harmonic contributions lower than $5\,\%$ of the fundamental component amplitude in region I, versus multi-harmonic with substantial higher harmonic contributions, typically third harmonic contributions larger than $25\,\%$ of the fundamental component amplitude, in region II. The examples in figure 2(a–i,a–ii,b–i,b–ii) depict these two forms of periodic responses. In all cases, only odd harmonic components (blue dashed lines in the spectra) contribute to the periodic responses. This symmetry of the oscillation betrays the symmetry of the organisation of the flow, which is discussed hereafter.

The strong sinusoidal nature of the vibrations in the periodic response region I persists beyond the edge of the aperiodic response region, in a transition zone denoted by a dotted area in the plots: the oscillation, still dominated by a single spectral component, is modulated by a low-frequency, incommensurable component (red dotted line in figure 2 b–iii). Beyond this transition zone, additional components progressively appear in the response spectrum, and the oscillation becomes increasingly erratic. Even though it is less clearly defined, a narrow zone of modulation can also be identified during the transition between the second periodic response region (II) and the aperiodic response region.

To recapitulate, RIV are found to develop through three regimes: two periodic regimes whose responses differ by their spectral contents, i.e. sinusoidal (periodic I) versus multi-harmonic (periodic II), and an aperiodic regime.

Figure 4.

Selected time series of the cylinder displacement and instantaneous iso-contours of spanwise vorticity for (a) $(\textit{Re},m^\star ,U^\star )=(100,1,20)$ (periodic regime I, $\omega _z\in [-0.075,0.075]$ ), (b) $(\textit{Re},m^\star ,U^\star )=(25,1,70)$ (periodic regime II, $\omega _z\in [-0.01,0.01]$ ) and (c) $(\textit{Re},m^\star ,U^\star )=(32,1,240)$ (aperiodic regime, $\omega _z\in [-0.02,0.02]$ ). Positive/negative vorticity values are plotted in red/blue. The successive instants visualised in the snapshots are indicated by red lines in the time series. In (a,b), the positive vortices are numbered in their order of formation; number $1$ corresponds to the first vortex formed during the sampling period.

When the rotating cylinder does not vibrate, the flow surrounding it is steady. Body oscillation and flow unsteadiness arise simultaneously. The vibrations reflect the unstable nature of the fluid–body system, and the existence of oscillatory states where the equilibrium of the energy transfer between the fluid and the body is reached over each cycle ( $1/f_\zeta$ , periodic regimes), or over a longer time interval (aperiodic regime). The increasing trend of the amplitude with $U^\star$ is reminiscent of the quasi-steady mechanism driving, for example, the galloping oscillations of non-axisymmetric bodies (Modarres-Sadeghi Reference Modarres-Sadeghi2022). In this case, the time scales of the flow and moving body are decoupled, and each step of body oscillation is seen as a steady configuration. As also observed for the galloping-like responses of a rotating cylinder in cross-current (Bourguet Reference Bourguet2020, Reference Bourguet2023), a quasi-steady modelling of fluid forcing, i.e. $C$ replaced by its quasi-steady model in (2.1), fails to predict the present vibrations. This suggests that the interplay of the rotating cylinder and flow unsteadiness is important for the emergence of RIV.

The radial and azimuthal velocity components of the axisymmetric steady flow surrounding the non-vibrating cylinder are $0$ and $D/(2\sqrt {x^2+y^2})$ , respectively, i.e. circular Couette flow. Typical flows encountered in the different oscillatory regimes are visualised in figure 4 via series of snapshots of instantaneous spanwise vorticity. Flow dynamics essentially consists of an undulation, synchronised with body motion, of the vorticity layers wrapped around the cylinder. As the body oscillates, elongated zones of vorticity form in its wake, with the detachment of distinct vortices. The size of the vortical structures tends to increase as $\textit{Re}$ is decreased, of the order of $D$ at $\textit{Re}=100$ (figure 4 a), and one order of magnitude larger at $\textit{Re}=25$ (figure 4 b).

In the absence of cross-current, the vortices are not convected away. They remain in the vicinity of the body until their diffusion, and may thus interact/collide with it during its subsequent oscillation cycles. The interaction with previously formed vortices may explain the development of irregular responses (aperiodic regime; figure 4 c). This phenomenon was reported in oscillatory flow studies (Anagnostopoulos & Iliadis Reference Anagnostopoulos and Iliadis1998; Dorogi et al. Reference Dorogi, Baranyi and Konstantinidis2023). Despite the above-mentioned interaction, body motion and flow organisation can combine in such a way that the system dynamics is periodic. The number of vortices present close to the cylinder, and their arrangement, are expected to be important factors for the appearance of such dynamics. The periodic regimes exhibit persistent patterns: a single positive vortex is shed per half-cycle in the periodic regime I, versus two positive vortices of unequal sizes and magnitudes in the periodic regime II. The successively formed vortices are numbered in figure 4(a,b). The symmetry of these patterns during each half-cycle coincides with the symmetry of the cylinder displacement (odd harmonics only in the $\zeta$ spectrum).

Figure 5.

(a) Fluid–body system regime and (b) vibration amplitude, as functions of the mass ratio and reduced velocity, at $\textit{Re}=50$ . The vibration region and, within this region, the zones where the system behaviour is periodic, are delimited by solid and dashed lines, respectively. The dotted area denotes the region of aperiodic behaviour where the response remains close to sinusoidal, with a slight low-frequency modulation. In (a), periodic and aperiodic behaviour regions are identified by yellow and grey background colours. (c) Vibration amplitude as a function of the vibration period ( $1/f_\zeta$ ), for selected mass ratios ( $\textit{Re}=50$ ). A range of reduced velocities is considered for each $m^\star$ . The symbols are coloured according to the periodic or aperiodic nature of the system behaviour. The two forms of periodic behaviours (I and II) are specified in the plot. The results examined to study the impact of structural damping for $m^\star =1$ are represented by red dots/circles ( $\xi \in \{0\,\%,10\,\%,50\,\%\}$ ); the selected values of $U^\star$ are indicated in red.

To generalise the analysis, in figure 5, the behaviour of the system is examined over a range of mass ratios, at $\textit{Re}=50$ . The distribution of the regimes previously identified for $m^\star =1$ , and the evolution of the vibration amplitude, across the $(m^\star ,U^\star )$ domain, are plotted in figure 5(a,b). The vibration region reaches the upper limit of the $U^\star$ range until $m^\star \approx 10$ . It tends to shrink beyond, and the periodic regime II vanishes. The aperiodic regime, which dominates the $U^\star$ range for low $m^\star$ , is observed until $m^\star \approx 40$ . For higher $m^\star$ values, RIV develop for $U^\star \in [20,130]$ , through the periodic regime I. The peak amplitude versus $U^\star$ (close to $5D$ ) and, overall, the system behaviour, do not substantially vary with $m^\star$ any more. From a general perspective, low- $m^\star$ RIV are aperiodic and grow regularly with $U^\star$ , while high- $m^\star$ RIV are mainly sinusoids of bounded magnitude. The vibration frequency (not plotted) remains lower than $f_n$ but tends towards it as $m^\star$ is increased, e.g. $f_\zeta /f_n\in{} ]0.99,1[$ for $m^\star =1000$ .

A complementary visualisation of the response domain visited by the system is proposed in figure 5(c), which represents the vibration amplitude as a function of the vibration period ( $1/f_\zeta$ ), over a range of $U^\star$ values, for selected mass ratios. The results obtained over four orders of magnitude of $m^\star$ follow comparable increasing trends, and tend to collapse over a broad response range. The responses are particularly indiscernible in the periodic regime I, where body motion is close to sinusoidal. Such a coincidence is expected since under a sinusoidal oscillation assumption, a solution of the dynamics equation (2.1) for $(m^\star ,U^\star )$ is also a solution of this equation for any other mass ratio, via an adjustment of the reduced velocity (Bourguet Reference Bourguet2025).

To further extend the investigation, structural damping is introduced for $m^\star =1$ and selected reduced velocities (red circles in figure 5 c). An increase of the damping ratio from $0\,\%$ to $10\,\%$ and $50\,\%$ causes a reduction of the vibration amplitude. This reduction can be accompanied by a transition of regime, e.g. from the periodic regime II to the aperiodic regime for $U^\star =50$ , and the inverse for $U^\star =80$ . Yet the three regimes are found to persist for damped systems, and the trend of the responses in the amplitude/frequency domain remains globally the same as in the undamped cases.

The above observations illustrate the robustness of the phenomena uncovered in this work: RIV, and the distinct regimes through which they occur, exist over wide ranges of the different physical parameters.