3.1.1. Time-averaged displacement and reconfiguration

The concave and convex configurations are characterized by a cross-flow symmetry about the $x$ axis, which suggests that the time-averaged position of the cylinder in flowing fluid should match its equilibrium position in quiescent fluid, i.e. $\bar {\theta }=0^\circ$. A shift from this equilibrium position may however occur, under the effect of mean fluid forcing. Such a shift results in a reconfiguration of the oscillator that breaks the cross-flow symmetry of the system since it introduces an asymmetry in the cylinder trajectory.

The shift due to mean fluid forcing can be estimated a priori, based on the only knowledge of the time-averaged force exerted on a stationary cylinder, by considering a static version of (2.2), i.e.

(3.1)\begin{equation} \dfrac{8{\rm \pi}^2 m r}{U^{{\star} 2}} \theta_{eq} ={-} \bar{C}^{f}_x \sin\left(\theta_{eq}+\theta_0\right), \end{equation}

where $\theta _{eq}$ designates the angular position of the predicted equilibrium and $C^{f}_x$ denotes the in-line force coefficient in the fixed body case ($\bar {C}^{f}_x=1.32$ at $Re=100$). In the absence of vibration, $\theta _{eq}$ is equal to $\bar {\theta }$; a deviation appears when the body vibrates, as the right-hand side of (3.1) departs from $\bar {C}$. Based on (3.1), no shift is expected in the concave configuration ($\theta _0=0^\circ$), while a shift is predicted in the convex configuration ($\theta _0=180^\circ$) when

(3.2)\begin{equation} U^\star{>}\sqrt{\dfrac{8{\rm \pi}^2 m r}{\bar{C}^{f}_x}}\approx 24.45 \sqrt{r}. \end{equation}

It is recalled that the mass ratio $m$ is equal to $10$.

The time-averaged position issued from the flow–body system simulation is plotted in figures 4 (concave configuration) and 5 (convex configuration) as a function of the reduced velocity, over a range of path radii (black dots). In these figures panels (a) represent the results obtained in the cross-flow, rectilinear motion configuration. In the other panels, body position is reported in terms of curvilinear (left axis) and angular (right axis) displacements. The equilibrium position predicted via (3.1) is denoted by a black dashed line. In the concave configuration no shift of the time-averaged position is observed relative to the position in quiescent fluid, as indicated by the above static analysis. A shift may occur in the convex configuration. The critical value of $U^\star$ beyond which the reconfiguration arises and the trend of the time-averaged position with $U^\star$ are globally captured by the static analysis. Yet, deviations appear, for example, close to the boundaries of the orange areas in figure 5(h–j), where irregular evolutions are not predicted. They are associated with the emergence of significant vibrations of the body, as shown in § 3.1.2. The background colours in figures 4 and 5 denote the different regimes of the flow–body system; the colour code will be explicited later in the paper.

Figure 4. Time-averaged, maximum and minimum values of the body curvilinear (left axis) and angular (right axis) displacements in the concave configuration, as functions of the reduced velocity, over a range of path radii. The values of path radius and signed curvature are specified in each panel. For comparison purposes, the displacements observed in the cross-flow, rectilinear motion configuration are reported in panel (a). In each case, the dark grey area depicts the displacement range swept by the body. The equilibrium position predicted by (3.1) is represented by a black dashed line (from panel (b)). Black dotted lines indicate $\theta =\pm 180^\circ$ (from panel (g)). The background colours denote the different regimes of the flow–body system; the regimes are described in § 3.2 and the colour code is explicited in figure 9.

Figure 5. Same as figure 4 in the convex configuration.

A remarkable feature is that the oscillator may recover a time-averaged position that corresponds to its position in quiescent fluid, beyond the onset of reconfiguration (red areas in figure 5l–o). This phenomenon, called symmetry recovery in the following in reference to the cross-flow symmetry of the system before reconfiguration, is not captured by the above analysis that predicts that reconfiguration should occur. Among the cases selected to illustrate the system behaviour in figure 3, three are expected to be subjected to reconfiguration (based on (3.1)); they are depicted in figure 3(c–h). Reconfiguration is actually observed in the first two cases, where $\bar {\theta }=162.37^\circ$ and $\bar {\theta }=121.25^\circ$, respectively, while the third one exhibits symmetry recovery, i.e. $\bar {\theta }=0^\circ$ vs $\theta _{eq}=153.6^\circ$.

It can be noted that, for low path radii and large reduced velocities, the time-averaged position of the reconfigured oscillator tends towards $180^\circ$ and the arrangement is then close to the concave configuration.

3.1.2. Vibration amplitude

The maximum and minimum values of the body curvilinear and angular displacements are reported in figures 4 and 5 (blue and green triangles) for the concave and convex configurations, respectively. As previously mentioned, panels (a) represent the results in the rectilinear motion configuration, in order to better visualize the impact of path curvature. The displacement range swept by the body, i.e. $[\zeta _{min},\zeta _{max}]$ or $[\theta _{min},\theta _{max}]$, where the subscripts $\phantom {}_{min}$ and $\phantom {}_{max}$ denote the minimum and maximum values, is indicated by a dark grey area. The response amplitude values reported hereafter refer to the maximum fluctuation about the time-averaged value ($\tilde {\zeta }_{max}$, $\tilde {\theta }_{max}$).

The cylinder is found to vibrate throughout the parameter space investigated, with distinct regions of large-amplitude responses. In the concave configuration (figure 4) the cylinder oscillates about its position in quiescent fluid ($\bar {\theta }=0^\circ$, no reconfiguration), i.e. along a path that is symmetrical relative to the $x$ axis. The magnitudes of its minimum and maximum displacements are identical, as illustrated by the example depicted in figure 3(a). The typical bell-shaped evolution of the vibration amplitude as a function of $U^\star$, observed in the rectilinear path configuration, is progressively distorted as the radius of curvature is reduced. Several elements can be noted. A comparison of figure 4(a) (rectilinear path) and figure 4(b) ($r=10$) shows that the introduction of a slight curvature of the trajectory has only an imperceptible influence on the response amplitude. The peak value of curvilinear displacement amplitude reached over the $U^\star$ range decreases with $r$, from $0.56$ diameters for $r=10$ (and the rectilinear path case) to $0.1$ diameters for $r=0.05$. Simultaneously, the peak value of angular displacement amplitude increases, up to $120^\circ \unicode{x2013}130^\circ$ below $r=0.1$. The value of $U^\star$ associated with the onset of the large-amplitude responses and the value associated with the peak amplitude tend to increase as $r$ is reduced, while the bell shape of the response amplitude curve widens. The shift of the peak amplitude along the $U^\star$ range as $r$ is varied was previously reported for pivoted cylinders (Malefaki & Konstantinidis Reference Malefaki and Konstantinidis2018). For low path radii, substantial vibrations are encountered until $U^\star =30$, i.e. the largest value examined here, vs $U^\star =8.5$ in the rectilinear path configuration. A kink can be identified in the evolution of the response amplitude with $U^\star$, at the boundary between the yellow and grey striped areas (figure 4e–j). This phenomenon will be connected to a change of interaction regime.

The bell-shaped amplitude region, typical of the rectilinear motion configuration, where the body vibrates along a symmetrical path about its position in quiescent fluid, is also found to persist in the convex configuration (first yellow areas close to $U^\star =5$ in figure 5). However, contrary to what was observed in the concave configuration, this region tends to shrink and slightly shift towards lower $U^\star$ values when $r$ is reduced. As in the concave configuration, the peak value of curvilinear displacement amplitude in this region decreases, down to $0.08$ diameters for $r=0.05$, while angular displacements close to $100^\circ$ are reached for low path radii. Two large-amplitude vibration regions emerge for $r<0.2$, in the higher range of $U^\star$ values. A first region, encountered around $r=0.17$, is characterized by oscillations of curvilinear and angular amplitudes close to $0.3$ body diameters and $100^\circ$, respectively. These oscillations develop about a reconfigured position, with $\bar {\theta }\approx 120^\circ$ (orange areas in figure 5h–j). An example of such responses is depicted in figure 3(e). A second region appears below $r=0.13$, where the cylinder exhibits a wide range of vibration amplitudes. In this region the curvilinear amplitude reaches $0.58$ diameters and is thus slightly larger than in the rectilinear path configuration, while the angular amplitude exceeds $280^\circ$. The vibrations may occur about a reconfigured position (second yellow and grey striped areas in figure 5m–r) or about the quiescent-fluid position, after symmetry recovery (red areas in figure 5l–o). The cases presented in figure 3(c,g) illustrate these distinct behaviours. In the absence of reconfiguration, or after symmetry recovery, the magnitudes of the maximum and minimum displacements about the time-averaged position are the same, as previously observed in the concave configuration. When the cross-flow symmetry of the trajectory is broken by the reconfiguration, differences exist, for example, around $U^\star =20$ for $r=0.175$ (figure 5i). As also mentioned for the concave configuration, the jumps in the evolution of the response amplitude are related to switches between interaction regimes, that will be clarified later in the paper.

For low path radii and large reduced velocities, typically below $r=0.075$ and beyond $U^\star =20$, the reconfiguration tends to transform the convex configuration into a concave arrangement ($\bar {\theta }\approx 180^\circ$). As a result, the vibration amplitudes are comparable for both configurations in this region.

To summarize the above observations, a global vision of the vibration amplitude across the parameter space is proposed in figure 6, which represents the maximum fluctuation of the curvilinear displacement about its time-averaged value, as a function of the signed curvature and reduced velocity. It is recalled that the signed curvature $\kappa$ is the inverse of path radius affected with a positive sign in the concave configuration and a negative sign in the convex configuration; $\kappa =0$ designates the rectilinear path configuration, which corresponds to the transition between the concave and convex configurations. All the cases examined in figures 4 and 5 are gathered in figure 6(a). A map, which provides a complementary and more continuous visualization of the vibration amplitude, is plotted in figure 6(b). The cases depicted in figure 3 are indicated by blue points in the map.

Figure 6. Curvilinear displacement amplitude as a function of the signed curvature and reduced velocity: (a) three-dimensional view of the cases depicted in figures 4 and 5, and (b) isocontours. In panel (b) white dashed lines delimit the significant vibration regions ($\tilde \zeta _{max}\geq 0.05$), which are designated by Roman numerals (I, II, III). The dotted area indicates the region where the oscillator is subjected to reconfiguration, i.e. $\bar {\zeta }\ne 0$. The cases considered in figure 3 are denoted by blue points.

The three regions of the parameter space where the curvilinear displacement amplitude is larger than or equal to $0.05$ diameters are delineated by white dashed lines in figure 6(b). These regions are identified by Roman numerals (I, II, III) and referred to as the significant vibration regions in the following. The vibration region I extends across the entire range of curvatures investigated. Its evolution with $\kappa$ highlights the distortion of the typical bell-shaped amplitude curve associated with rectilinear vibrations ($\kappa =0$). The two other significant vibration regions arise in the convex configuration. The area where the oscillator is subjected to reconfiguration, which includes the vibration region II and a portion of region III, is indicated by white dots in the map. The peak-amplitude part of region III represents an island of symmetry recovery in the reconfiguration area.

3.1.3. Vibration frequency

The vibrations are periodic or close to periodic in all studied cases. As illustrated by the examples selected in figure 3, the vibration spectrum is generally dominated by a single frequency, denoted by $f_\zeta$ and referred to as the vibration frequency. The possible emergence of higher harmonics or incommensurable components is discussed at the end of this subsection.

In figures 7 and 8 the vibration frequency normalized by the natural frequency in vacuum is plotted as a function of the reduced velocity over the range of path radii examined in figures 4 and 5, for the concave and convex configurations. To ease interpretation, different symbols are employed within and outside the significant vibration regions identified in figure 6(b), i.e. $\tilde {\zeta }_{max}\geq 0.05$ vs $\tilde {\zeta }_{max} < 0.05$. Outside these regions, the vibration frequency (green triangles) is close to the Strouhal frequency ($St=0.164$, black dotted line), as also noted in previous studies concerning rectilinear oscillations at comparable Re (e.g. Shiels et al. Reference Shiels, Leonard and Roshko2001; Bourguet & Lo Jacono Reference Bourguet and Lo Jacono2014). Within the significant vibration regions, the vibration frequency (blue squares) may substantially depart from St. It remains relatively close to the natural frequency in vacuum for $r>0.2$. Yet, major deviations can be observed for lower path radii, in both configurations, with frequency ratios larger than $4$.

Figure 7. Dominant frequencies of body vibration and wake fluctuation in the concave configuration, as functions of the reduced velocity, over a range of path radii. The values of path radius and signed curvature are specified in each panel. For comparison purposes, the frequencies observed in the cross-flow, rectilinear motion configuration are reported in panel (a). The frequencies are normalized by the natural frequency of the oscillator in vacuum. Distinct symbols are used to designate the vibration frequency within and outside the significant vibration regions ($\tilde {\zeta }_{max}\geq 0.05$ vs $\tilde {\zeta }_{max} < 0.05$). The vortex shedding frequency in the rigidly mounted body case (Strouhal frequency, $St=0.164$) and the modified natural frequency taking into account the mean drag ((3.3), from panel (b)) are indicated by a black dotted line and a black dashed line, respectively. The background colours denote the different regimes of the flow–body system; the regimes are described in § 3.2 and the colour code is explicited in figure 9.

Figure 8. Same as figure 7 in the convex configuration. In panels (h–j) and (l–o), the $`\times$’ and $`+$’ symbols designate $1/2$ and $1/3$ of the wake fluctuation frequency, respectively.

Prior works concerning pivoted cylinders suggested to take into account the effect of the mean in-line force to modify the expression of the natural frequency of the oscillator in vacuum (Arionfard & Nishi Reference Arionfard and Nishi2017; Malefaki & Konstantinidis Reference Malefaki and Konstantinidis2018). By considering small oscillations about the equilibrium position predicted by the static analysis (3.1), a modified natural frequency can be defined as

(3.3)\begin{equation} f'_n=\sqrt{f^2_n+\dfrac{\bar{C}^{f}_x}{8{\rm \pi}^2 m r}\cos(\theta_{eq}+\theta_0)}. \end{equation}

The derivation of $f'_n$ is explained in an appendix dedicated to a quasi-steady analysis of fluid forcing (Appendix A). Equation (3.3) indicates that the influence of the mean drag on the natural frequency should be more pronounced for low path radii and tends to vanish as the trajectory gets closer to rectilinear. In the absence of reconfiguration ($\theta _{eq}=0^\circ$), it predicts that the mean force increases the natural frequency in the concave configuration ($\theta _0=0^\circ$) and reduces it in the convex configuration ($\theta _0=180^\circ$). Under the assumption that a peak of vibration occurs when the natural frequency $f'_n$ coincides with St, a shift of this peak towards higher (lower, respectively) $U^\star$ values is expected in the concave (convex, respectively) configuration, compared with the rectilinear path configuration. This phenomenon, also reported by Malefaki & Konstantinidis (Reference Malefaki and Konstantinidis2018), is actually observed in region I (figure 6b). After reconfiguration, the effect of the mean drag on the natural frequency depends on the equilibrium position.

In the significant vibration regions, $f_\zeta$ is found to globally follow the trend of $f'_n$, which is represented by a black dashed line in figures 7 and 8, and by a grey dashed line in the spectra of figure 3. An exception can however be noted in the red areas in figure 8. In this part of the parameter space, the oscillator is subjected to symmetry recovery, which is not captured by the above analysis, and the vibration frequency is found to be close to $f_n$.

Another element can be noted. The vibrations observed in the orange and red areas in figure 8 occur at frequencies relatively close to $St/2$ and $St/3$, respectively. Such a coincidence suggests that these vibrations could develop under a subharmonic synchronization with the wake, i.e. at a submultiple of wake unsteadiness frequency. This aspect will be clarified in the following.

Some higher harmonic contributions may emerge in the vibration spectrum. Their magnitudes are small in all cases, typically lower than $15\,\%$ of the fundamental component amplitude, but they reflect the symmetry of the vibration. For periodic responses in the absence of reconfiguration, or after symmetry recovery, only odd harmonics are encountered and the displacement is thus symmetrical about the time-averaged position (figure 3g,h). Once the cross-flow symmetry of the trajectory is altered by the reconfiguration, even harmonics may also appear (figure 3f). In certain regions of the parameter space, incommensurable components of low magnitudes arise in the vibration spectrum (figure 3j). They break the periodicity of the oscillation. They also break its strict cross-flow symmetry, even though the magnitudes of the maximum and minimum displacements measured over a large number of cycles are identical.

The above observations concerning the structural response raise the question of the nature of the interaction between the flow and the vibrating body. This is the object of the next subsection.

3.2.1. Flow–body synchronization – regime identification

For each case depicted in figure 3, flow unsteadiness is represented by a time series of the cross-flow component of flow velocity fluctuation ($\tilde {v}$), sampled $10$ diameters downstream of the cylinder, at $(x,y)=(10+r\cos (\theta _0),0)$. A comparison of these signals with the time series of body displacement, and the associated spectra, suggest that distinct interaction regimes may develop.

In the parameter space under study, flow unsteadiness, quantified via $\tilde {v}$ time series, is generally dominated by a single frequency and the contributions of the other spectral components remain marginal. The dominant frequency of flow unsteadiness, denoted by $f_v$ and referred to as flow frequency in the following, is superimposed on the vibration frequency plots in figures 7 and 8 (black dots; the frequencies are normalized by $f_n$ in these plots). Outside the significant vibration regions ($\tilde {\zeta }_{max} < 0.05$), $f_v$ matches the vibration frequency and is always close to the Strouhal frequency. Once significant structural oscillations occur ($\tilde {\zeta }_{max} \geq 0.05$), the vibration frequency may substantially deviate from St, as previously noted. However, the condition of synchronization where $f_\zeta =f_v$ is found to persist (yellow areas in the plots). This condition represents the typical lock-in mechanism, usually observed in cross-flow VIV of circular cylinders (Williamson & Govardhan Reference Williamson and Govardhan2004). Examples of such synchronization, in the concave configuration and in the convex configuration in a case where the oscillator is subjected to reconfiguration, are depicted in figures 3(a,b) and 3(c,d), respectively.

Two other forms of flow–body synchronization are uncovered within the significant vibration regions, in the convex configuration. First, the vibration frequency can coincide with $f_v/2$, which is specified by the ‘$\times$’ symbols in figure 8h–j) (orange areas). Second, the vibration frequency can be equal to $f_v/3$, indicated by the ‘$+$’ symbols in figure 8(l–o) (red areas). The deviations of the dominant frequency of the flow from St appear to be smaller than those encountered when $f_\zeta =f_v$. Examples of these two additional forms of synchronization are presented in figures 3(e, f) and 3(g,h), respectively. The time series and spectra show that, in spite of some low-amplitude modulations at $f_\zeta$, flow velocity fluctuation is essentially determined by the dominant component at $f_v$, with distinct frequency ratios, $f_v/f_\zeta =2$ and $f_v/f_\zeta =3$. Flow–body synchronization where the structure oscillates at a submultiple of flow dominant frequency is referred to as subharmonic synchronization hereafter. It is not observed when the circular cylinder is restrained to rectilinear motion, but it was reported in this case for asymmetrical bodies, e.g. a square prism in Zhao et al. (Reference Zhao, Leontini, Lo Jacono and Sheridan2014). Subharmonic synchronization was not detected in previous works concerning elastically mounted, pivoted cylinders; it is recalled that the range of low path radii where such synchronization appears was not explored in these prior studies.

Under flow–body synchronization, regardless of the frequency ratio, the system behaviour is periodic. As previously noted for the structural response, in the absence of reconfiguration, only odd harmonic contributions appear in the flow velocity spectrum. This reflects the strict anti-symmetrical organization of the wake, which will be addressed in § 3.2.2.

In addition to the three forms of flow–body synchronization, a desynchronized state where the body and the flow oscillate at incommensurable frequencies is also encountered in the significant vibration regions, both in the concave and convex configurations (grey striped areas in figures 7d–j and 8m–p). In this desynchronized condition, the vibration frequency follows the modified natural frequency ($f'_n$), while the dominant frequency of flow unsteadiness is close to St. A typical example is presented in figure 3(i,j). Such a condition resembles the desynchronization or decoherence usually reported for VIV at higher Re, when $U^\star$ is increased beyond the lock-in range (e.g. Khalak & Williamson Reference Khalak and Williamson1999). It should however be mentioned that this condition does not occur at $Re~=100$ when the body moves along a rectilinear path (figure 7a) and its appearance here is thus due to path curvature. Despite their limited amplitudes, the emergence of incommensurable components, at $f_v$ in the vibration spectrum and at $f_\zeta$ in the flow velocity spectrum, results in an aperiodic dynamics of the system, which contrasts with the periodic behaviours encountered under flow–body synchronization.

Based on the different forms of synchronization or desynchronization between the flow and the moving body, determined via the frequencies $f_v$ and $f_\zeta$, four distinct regimes of interaction can be identified within the parameter space investigated. A first visualization of these regimes is proposed in figure 9, which is used to introduce the nomenclature employed to designate them. This figure represents, for all studied cases, the amplitude of the body curvilinear displacement as a function of the ratio between the flow frequency and vibration frequency. Different symbols are used to distinguish the concave and convex configuration cases. The threshold of the significant vibration regions ($\tilde {\zeta }_{max}= 0.05$) is specified by a dark grey dashed line. The three regimes where the flow and the body are synchronized with an integer frequency ratio, $f_v/f_\zeta \in \{1,2,3\}$, are referred to as the locked regimes. These regimes are denoted by plain background colours and called locked 1 : 1, locked 2 : 1 and locked 3 : 1, in reference to flow/body frequency ratios. The locked 1 : 1 regime is encountered in the concave and convex configurations, and associated with a wide range of vibration amplitudes, from the lowest amplitudes detected to $0.56$ body diameters. This is the only regime observed outside the significant vibration regions identified in figure 6(b), i.e. $\tilde {\zeta }_{max} < 0.05$, below the dark grey dashed line in figure 9. It is denoted by a yellow background colour, with a lighter tone outside the significant vibration regions. In contrast, the locked 2 : 1 and locked 3 : 1 regimes, which only develop in the convex configuration, are associated with specific ranges of response amplitudes: intermediate amplitudes around $0.3$ diameters in the locked 2 : 1 regime versus large amplitudes close to $0.5$ diameters in the locked 3 : 1 regime, which is the regime where the largest amplitude is measured ($0.58$). These regimes are indicated by orange and red background colours, respectively. The regime where the flow and the body exhibit incommensurable frequencies, i.e. they are desynchronized, is referred to as the unlocked regime and identified by a grey striped background. It is observed in the concave and convex configurations, and involves vibration of low amplitudes, typically around $0.1$ diameters, and flow/body frequency ratios ranging from $1.1$ to $1.4$.

Figure 9. Curvilinear displacement amplitude as a function of the ratio between the flow frequency and vibration frequency. Distinct symbols are used to designate the concave and convex configuration cases. A dark grey dashed line represents the threshold of the significant vibration regions ($\tilde {\zeta }_{max}= 0.05$). The integer values of the frequency ratio are specified by black dashed-dotted lines. Plain background colours denote the three regimes where the flow and the body are synchronized, with a frequency ratio of $1$ (locked 1 : 1 regime; yellow/light yellow within/outside the significant vibration regions), $2$ (locked 2 : 1 regime; orange) and $3$ (locked 3 : 1 regime; red). The unlocked regime where the flow and the body are desynchronized is denoted by a grey striped area.

The colour code introduced in figure 9 is used in the backgrounds of figures 4, 5, 7 and 8, to track the different regimes. As in figure 9, a lighter yellow colour is employed to designate the locked 1 : 1 regime out of the significant vibration regions. This continuous monitoring of the interaction regime shows that, within a significant vibration region, the kinks in the evolution of structural response properties, especially vibration amplitude, are generally linked to the passage from one regime to the other. This phenomenon is illustrated by the transition between the locked 1 : 1 and unlocked regimes in figure 4(e–j).

The distribution of the interaction regimes in the $(\kappa,U^\star )$ parameter space is visualized in figure 10. In this map, the areas associated with the different regimes are delimited by plain black lines and the colour code follows the nomenclature introduced in figure 9. The boundaries of the three regions of significant vibrations identified in figure 6(b) are indicated by dark grey dashed lines. Within these regions, to ease description, the areas associated with distinct regimes are specifically designated (Ia, Ib, IIIa, IIIb and IIIc). The regime names are also mentioned. Two regimes are encountered in region I (locked 1 : 1 and unlocked), a single in region II (locked 2 : 1) and three in region III (locked 1 : 1, locked 3 : 1 and unlocked). This map highlights the pronounced asymmetry of regime distribution relative to the $\kappa =0$ axis, i.e. concave ($\kappa >0$) versus convex ($\kappa <0$) configurations. Some symmetry can however be noted for large curvature magnitudes, typically $|\kappa |>13$, beyond $U^\star =20$. As mentioned in § 3.1, the reconfiguration of the oscillator (denoted by black dots in the map) tends to transform the convex configuration into a concave arrangement in this region ($\bar {\theta }\approx 180^\circ$), which leads to comparable behaviours for $\kappa >0$ and $\kappa <0$. The locked 1 : 1 regime is found to develop over the entire curvature and $U^\star$ ranges investigated. As previously noted, this is the only regime encountered out of the significant vibration regions. The locked 2 : 1 and 3 : 1 regimes appear close to $\kappa =-6$ ($r\approx 0.17$) and $\kappa =-10$ ($r\approx 0.1$), for $U^\star \in [17,26]$ and $U^\star \in [15,23]$, respectively. The unlocked regime occurs for $U^\star >9$, and is not observed beyond $|\kappa |\approx 15$ (below $r \approx 0.07$).

Figure 10. Flow–body system regime as a function of the signed curvature and reduced velocity. The areas associated with distinct regimes are separated by plain black lines and the regime names are specified. The colour code used to denote the different regimes is the same as in figure 9. Dark grey dashed lines delimit the significant vibration regions ($\tilde {\zeta }_{max}\geq 0.05$), which are designated by Roman numerals (I, II, III). The dotted area represents the region where the oscillator is subjected to reconfiguration, i.e. $\bar {\zeta }\ne 0$. The cases considered in figure 3 are indicated by blue points.

The locked 1 : 1 and unlocked regimes arise both in the absence of, or after reconfiguration. This is not the case for the other regimes. The locked 2 : 1 regime is systematically associated with a reconfigured arrangement, while the locked 3 : 1 regime is found to coincide with the island of symmetry recovery detected in the peak-amplitude part of region III (figure 6b). A corollary aspect is that the periodic responses appearing in the locked 3 : 1 regime are strictly symmetrical (odd harmonics only) whereas those developing in the locked 2 : 1 regime are asymmetrical (odd and even harmonics). Both asymmetrical and symmetrical periodic oscillations are encountered in the locked 1 : 1 regime, depending on whether the oscillator is subjected to reconfiguration or not. In the unlocked regime the presence of incommensurable frequency components breaks the oscillation symmetry, regardless of the occurrence of reconfiguration.

The examples selected in figure 3 are localized by blue points in the map. They cover the different regimes observed in the significant vibration regions, including the locked 1 : 1 regime with and without reconfiguration.

The vibrations encountered in the locked 2 : 1 and locked 3 : 1 regimes present similarities with the galloping oscillations observed for non-axisymmetric bodies (Païdoussis et al. Reference Païdoussis, Price and de Langre2010), in particular their relatively large amplitudes, low frequencies compared with flow unsteadiness and the high values of $U^\star$ where they arise. Some substantial differences can however be noted. The present vibrations develop over finite intervals of $U^\star$ and their amplitudes exhibit bell-shaped evolutions, which contrasts with the typical unbounded growth of galloping oscillation amplitudes. The local occurrence of flow–body synchronization was shown to induce kinks in the evolution of galloping response amplitudes (e.g. Zhao et al. Reference Zhao, Leontini, Lo Jacono and Sheridan2014). Yet, such synchronization is not required for galloping oscillations, since they are driven by a quasi-steady mechanism, decoupled from flow unsteadiness. Here, flow–body synchronization is found to persist, at a subharmonic level, throughout the locked 2 : 1 and locked 3 : 1 regimes and a quasi-steady modelling of fluid forcing fails to predict the present responses, as discussed in Appendix A.

The organization of the wakes associated with the different interaction regimes is investigated hereafter.

3.2.2. Wake organization

In order to shed some additional light on the interaction regimes identified in § 3.2.1, focus is now placed on the flow patterns encountered downstream of the vibrating body. Throughout the parameter space, regardless of the interaction regime, the dominant frequency of flow unsteadiness ($f_v$) is associated with the formation of a pair of counter-rotating vortices, as in the fixed body configuration. The typical flow patterns observed in the three synchronized regimes, i.e. locked 1 : 1, 2 : 1 and 3 : 1 regimes, are visualized in figures 11, 12 and 13, through instantaneous isocontours of spanwise vorticity. For each selected case, a general view of the wake is presented in the upper panel to show the global structure of the flow and snapshots of the near-wake region, collected every $1/(4 f_v)$ over one period of body oscillation ($1/f_\zeta$), are plotted in the lower panels, to track vortex formation process. Body trajectory and instantaneous position are indicated in each panel. The vortical pattern formed during one oscillation cycle is enclosed by a dashed grey line in the upper panel and the vortices shed over each half-cycle are separated by a black dashed-dotted line.

Figure 11. Instantaneous isocontours of spanwise vorticity for $(r,\kappa,U^\star )=(0.5,2,6.5)$ (locked 1 : 1 regime; case considered in figure 3a,b): (a) general visualization of the wake ($\omega _z\in [-0.5,0.5]$); (b–e) visualization of the near-wake region ($\omega _z\in [-2,2]$) at four instants over one period of body oscillation, i.e. one period of vortex shedding. The trajectory of the cylinder centre is indicated by a black line. Positive/negative vorticity values are plotted in red/blue. In panel (a) a dashed grey line encloses the vortical pattern formed over one oscillation period and a black dashed-dotted line separates the vortices shed over each half-period. Part of the computational domain is shown.

Figure 12. Same as figure 11 for $(r,\kappa,U^\star )=(0.175,-5.714,22)$ (locked 2 : 1 regime; case considered in figure 3e, f). In panels (b–i) the near-wake region is visualized at eight instants over one period of body oscillation, i.e. two periods of vortex shedding.

Figure 13. Same as figure 11 for $(r,\kappa,U^\star )=(0.111,-9.001,20)$ (locked 3 : 1 regime; case considered in figure 3g,h). In panels (b–m) the near-wake region is visualized at twelve instants over one period of body oscillation, i.e. three periods of vortex shedding.

One pair of alternating vortices is formed per oscillation cycle in the locked 1 : 1 regime, versus two and three pairs in the locked 2 : 1 and 3 : 1 regimes, respectively. In the locked 1 : 1 regime without reconfiguration (figure 11) and in the locked 3 : 1 regime (figure 13), the vortices shed over each half-period of oscillation exhibit anti-symmetrical structures, i.e. a reflection symmetry about the wake centreline, similarly to the von Kármán street observed downstream of a fixed cylinder. The anti-symmetry of wake topology is marked by the absence of even harmonic components in the spectrum of the cross-flow component of flow velocity sampled on the wake centreline, as illustrated in figure 3(b,h). Such organization of the wake coincides with the cross-flow symmetry of cylinder vibrations, i.e. odd harmonics only in the displacement spectrum. The strict anti-symmetry of the locked 1 : 1 regime wakes is perturbed when the body oscillates along an asymmetrical path, due to reconfiguration: slight differences can be noted in the magnitudes and shapes of the positive and negative vortices, even though the overall topology remains close to anti-symmetrical. The six vortices shed over one period of oscillation in the locked 3 : 1 regime tend to regroup into two triplets, which closely resemble the $2{\rm P}+2{\rm S}$ pattern reported by Williamson & Roshko (Reference Williamson and Roshko1988) under forced rectilinear oscillations near $St/3$. In the locked 2 : 1 regime, two comparable pairs of counter-rotating vortices are formed, one over each half-period of oscillation (figure 12). In contrast to the patterns described above, the vortices shed over each half-period do not present anti-symmetrical structures. In this regime, the body vibrates about a reconfigured position, along an asymmetrical path, and, therefore, no specific symmetry of the wake is expected. It can be noted that the asymmetrical responses (locked 1 : 1 regime with reconfiguration, locked 2 : 1 regime) are not accompanied by major distortion or inclination of the wake, as illustrated by figure 12(a) where the vortices remain globally aligned with the $x$ axis. In the subharmonic synchronization regimes (figures 12 and 13), the rows of the lower panels place side-by-side the two/three successive periods of vortex shedding occurring during one cycle of body oscillation. Even if the vortex formation process is comparable from one shedding period to the other, some subtle differences can be identified, in relation with body motion. For example, in the case depicted in figure 12, the red shear layer developing as the positive vortex is formed appears to be longer in the first period as the body moves upstream (figure 12d,e) than in the second period as it moves downstream (figure 12h,i).

In the unlocked regime, vortex formation and body motion are not synchronized. Based on the flow/body frequency ratio ($f_v/f_\zeta$), between $2.2$ and $2.8$ vortices are shed per oscillation cycle. The wake resembles that observed downstream of a fixed cylinder, except that its anti-symmetrical organization is slightly altered, as expected in the presence of incommensurable vibrations of the body.

The description of the spatial organization of the wake shows that the typical patterns associated with the different interaction regimes are essentially variations about the von Kármán street occurring in the fixed body configuration. In particular, as previously mentioned, the dominant frequency $f_v$ always coincides with the shedding of a pair of counter-rotating vortices. Figures 7 and 8 indicate that the shedding frequency can clearly depart from the Strouhal frequency once the body is subjected to significant vibrations and that the magnitude of this departure depends on the interaction regime. To further examine this aspect, the amplitude of the body curvilinear displacement is plotted in figure 14 as a function of the relative deviation of $f_v$ from St, for all studied cases. Distinct symbols are used to distinguish the four interaction regimes and the colour code employed is the same as in figure 9. The limit of the significant vibration regions is denoted by a dark grey dashed line. In the locked 1 : 1 regime, the relative deviation of the shedding frequency is found to vary from $-25\,\%$ to $+20\,\%$, approximately. A pronounced departure from St can be observed both in the concave and convex configurations. A monitoring of the deviation versus path curvature shows that the largest differences are encountered for $|\kappa |<10$ ($r>0.1$). The frequency range tends to widen as the vibration amplitude increases, up to $0.2$ diameters. It follows a global V shape that roughly matches the boundaries of the wake synchronization region identified by Koopmann (Reference Koopmann1967) for forced rectilinear oscillations, at the same Re (green dotted lines). On the other hand, the locked 2 : 1 and 3 : 1 regimes are associated with specific, narrow ranges of vortex shedding frequencies, slightly lower than St, typically $-2\,\%$, and larger than St, around $+7\,\%$, in the former and latter regimes, respectively. In the unlocked regime the shedding frequency is also restrained to a limited range, with a typical deviation of $-4\,\%$ relative to St. Prior studies concerning forced oscillations have reported comparable deviations towards values lower than St in the absence of synchronization (e.g. Cheng & Moretti Reference Cheng and Moretti1991). This phenomenon could be attributed to the larger apparent diameter of the body seen by the flow, when the cylinder vibrates. A scaling by the effective transverse length covered by the body, instead of $D$, results indeed in a reduction of the frequency but the values are lower than those actually measured, for example, close to $0.85$ St ($-15\,\%$) in the case depicted in figure 3(i,j).

Figure 14. Curvilinear displacement amplitude as a function of the relative deviation of the flow frequency from the Strouhal frequency. A dark grey dashed line represents the threshold of the significant vibration regions ($\tilde {\zeta }_{max}= 0.05$). A black dashed-dotted line denotes the absence of deviation ($f_v=St$). Distinct symbols are employed to designate the interaction regimes, with the colour code introduced in figure 9. Green dotted lines delimit the region of synchronization reported by Koopmann (Reference Koopmann1967) under forced, cross-flow oscillations.

The system exhibits four distinct regimes characterized by different forms of synchronized or unsynchronized interaction between the flow and the vibrating cylinder. Each regime has been associated with typical properties of the structural response and wake organization. Another facet of the interaction relates to fluid forcing, which is addressed in the next subsection.

3.3.1. Time-averaged in-line force

The time-averaged value of $C_x$ across the $(\kappa,U^\star )$ parameter space is represented in figure 15(a). The range of $\bar {C}_x$ values is indicated on the right axis of the colourbar and the corresponding range of relative deviations from the fixed body case value ($\bar {C}^{f}_x=1.32$) is specified on the left axis. The limits of the significant vibration regions and distinct regime areas are denoted by white dashed and plain lines, respectively. The different zones of the parameter space are identified as in figure 10. The cases considered in figure 3 are localized by blue points.

Figure 15. (a) Time-averaged value of the in-line force coefficient as a function of the signed curvature and reduced velocity. The range of $\bar {C}_x$ values is indicated on the right axis of the colourbar and the associated range of relative deviations from the fixed body case value is specified on the left axis. White dashed lines delimit the significant vibration regions and the areas associated with distinct regimes are separated by plain white lines; the area names are those introduced in figure 10. The cases considered in figure 3 are identified by blue points. (b) Relative deviation of the time-averaged value of the in-line force coefficient as a function of the curvilinear displacement amplitude. A dark grey dashed line represents the threshold of the significant vibration regions. Distinct symbols are employed to designate the interaction regimes, with the same colour code as in figure 9. The areas of the significant vibration regions associated with each regime are indicated in the legend. Open blue symbols, with the same shapes as those reported in the legend, represent the results issued from quasi-steady modelling ((A2) in Appendix A).

The time-averaged drag exhibits a peak in region Ia, where the locked 1 : 1 regime is established. The peak appears in the area of large path radii, with a shift towards positive curvatures (concave configurations). In this region, $\bar {C}_x$ is amplified by $55\,\%$ compared with the fixed body case value. The location of the peak closely coincides with the peak of vibration amplitude depicted in figure 6(b). Such a coincidence was often emphasized in previous works (e.g. Khalak & Williamson Reference Khalak and Williamson1999). To visualize this connection, the deviation from the fixed body case value is plotted as a function of the curvilinear displacement amplitude in figure 15(b). The significant vibration region limit is indicated by a dark grey dashed line and distinct symbols are employed to designate the different regimes (same colour code as in figure 9). The time-averaged in-line force is also enhanced in the locked 2 : 1 and 3 : 1 regimes (regions II and IIIc), even though to a lesser extent, typically around $+6\,\%$ and $+10\,\%$, respectively. It can be noted that the amplification is much less pronounced in the locked 3 : 1 regime than in the locked 1 : 1 regime (region Ia), whereas the response amplitudes are comparable and even larger in the former regime. In spite of the dispersion observed between the different regimes, it appears that for a given locked regime, $\bar {C}_x$ tends to globally increase with vibration amplitude. A slight reduction, down to $-5\,\%$, is however encountered in some regions of low to moderate vibration amplitudes in the locked 1 : 1 regime. A minor reduction of $\bar {C}_x$ is also noted in the unlocked regime.

The above observations indicate that $\bar {C}_x$ alteration is not strictly determined by the response amplitude. It should be mentioned that the dispersion persists if the cross-flow projection of the displacement is considered, instead of $\zeta$. Prior studies have shown that the response frequency may also play a role (e.g. Carberry, Sheridan & Rockwell Reference Carberry, Sheridan and Rockwell2005). Here, a slightly better collapse of $\bar {C}_x$ can be achieved by considering the product of the response amplitude and frequency, i.e. a typical velocity of the moving cylinder, or the time-averaged relative velocity seen by the body.

For comparison purposes, the time-averaged values issued from the quasi-steady modelling of $C_x$ ((A2) in Appendix A), based on the structural responses obtained via the unsteady simulation approach, are plotted in figure 15(b) (open blue symbols). The quasi-steady modelling predicts an increase of $\bar {C}_x$ with vibration magnitude, with lower amplifications in the locked 2 : 1 and 3 : 1 regimes than in the locked 1 : 1 regime, as also noted in the unsteady simulation results. The enhancement of the time-averaged force is however not captured, as the maximum amplification only reaches $+10\,\%$ compared with the fixed body case value.

3.3.2. Tangential force, phasing with displacement and energy transfer

The reconfiguration of the oscillator in the locked 1 : 1, locked 2 : 1 and unlocked regimes is accompanied by the emergence of a time-averaged component of the tangential force, which vanishes otherwise. In the locked regimes the tangential force is periodic and synchronized with body motion/flow unsteadiness. Several components may arise in its spectrum but they are all harmonics of the vibration frequency $f_\zeta$, which represents the fundamental component. Only odd harmonics are encountered when the cylinder vibrates along a symmetrical path. In particular, the projection of the in-line force, which is composed of even harmonics, on the tangential direction (2.1), results in odd harmonics only. Both odd and even harmonics appear once path symmetry is broken by the reconfiguration. The higher harmonic contributions represent a moderate fraction of the first harmonic magnitude in the locked 1 : 1 regime, typically around $20\,\%$, except near the force–displacement phase difference jump where the first harmonic vanishes and the relative magnitude of the higher harmonics is consequently very large; this point will be discussed later in this subsection. They represent a significant fraction of the first harmonic magnitude in the locked 2 : 1 regime, around $70\,\%$, and are even larger in the locked 3 : 1 regime, from $2$ to $10$ times the first harmonic magnitude. In the unlocked regime both vibration and flow frequency components, which are incommensurable, appear in the spectrum of the aperiodic tangential force, as well as their higher harmonics. The higher harmonic contributions are small in this regime, typically lower than $5\,\%$ of the fundamental component magnitudes. These different elements are visualized in figure 3, where the time series of $C$ and corresponding spectra are plotted for each selected case.

The statistics of the tangential force are further examined in figure 16. The r.m.s. value of $C$ fluctuation and its relative deviation from the r.m.s. value of $C_y$ fluctuation in the fixed body case ($(\tilde {C}^{f}_y)_{rms}=0.23$) are represented over the parameter space in figure 16(a). The cross-flow force coefficient is chosen as reference since $C=\pm C_y$ in the absence of vibration and reconfiguration. A major amplification of the tangential force fluctuation is observed in all the significant vibration regions (delimited by white dashed lines), not only in the locked regimes but also in the unlocked regime (regions Ib and IIIb). Peak values are encountered in region Ia (locked 1 : 1 regime), where the fluctuation may become six times larger than in the fixed body case. They occur for convex configurations and do not coincide with the peak of the time-averaged drag identified in figure 15(a). A complementary visualization is proposed in figure 16(b) where the deviation from the fixed body case value is plotted as a function of the vibration frequency, normalized by the natural frequency of the oscillator in vacuum. This plot depicts more precisely the typical ranges of force fluctuations associated with the different regimes: a wide dispersion in the locked 1 : 1 and unlocked regimes where the deviation ranges from $-100\,\%$ to $650\,\%$ and from $0\,\%$ to $250\,\%$, respectively, a narrower dispersion in the locked 2 : 1 regime, from $150\,\%$ to $300\,\%$, and a concentration around $300\,\%$ in the locked 3 : 1 regime.

Figure 16. (a) Same as figure 15(a) for the r.m.s. value of the tangential force coefficient fluctuation and its relative deviation from the r.m.s. value of $C_y$ fluctuation in the fixed body case. The values of the force–displacement phase difference ($\varphi =0^\circ$ or $\varphi =180^\circ$) are specified in grey and grey dotted lines denote the phase difference jumps. (b) Relative deviation of the r.m.s. value of the tangential force coefficient fluctuation as a function of the vibration frequency normalized by the natural frequency in vacuum. A grey dashed-dotted line denotes the frequency ratio of $1$. The values of the force–displacement phase difference are indicated on each side of this line. Distinct symbols are employed to designate the interaction regimes, with the same colour code as in figure 9. The areas of the significant vibration regions associated with each regime are mentioned in the legend. A light green area delimited by green dotted lines depicts the drop of force fluctuation amplitude occurring close to the phase difference jump in the locked 1 : 1 regime. Open red triangular symbols represent the contribution of the first harmonic of the force (i.e. at $f_\zeta$) in the locked 3 : 1 regime. Part of the parameter space is shown to ease visualization of the phase difference jump region.

Figure 16(b) shows that the vibration frequency crosses the natural frequency of the oscillator in vacuum in the locked 1 : 1 and locked 3 : 1 regimes, while it remains larger than $f_n$ in the locked 2 : 1 and unlocked regimes. The frequency ratio $f_\zeta /f_n$ is closely linked to the phasing between force and displacement, and to the relative contributions of the different harmonics in the force spectrum (e.g. Gsell, Bourguet & Braza Reference Gsell, Bourguet and Braza2016). The phase difference between the components of $C$ and $\zeta$ occurring at the dominant vibration frequency $f_\zeta$ is denoted by $\varphi$. For periodic responses in the absence of structural damping, the system may exhibit two possible states where the force is either in phase with the displacement ($\varphi =0^\circ$), when $f_\zeta < f_n$, or in phase opposition ($\varphi =180^\circ$), when $f_\zeta >f_n$. The phase difference jump occurring during the transition between these two states is accompanied by the disappearance of the force component at vibration frequency. Such a binary behaviour is verified in the three locked regimes, which are periodic: both phasing states are encountered in the locked 1 : 1 and locked 3 : 1 regimes, and $\varphi =180^\circ$ in the locked 2 : 1 regime. In the aperiodic, unlocked regime the phase difference is always equal to $180^\circ$, as illustrated in figure 3(i).

The locations of the force–displacement phase difference jumps as well as the values of the phase difference are indicated in the map of figure 16(a). A jump crosses the area of the parameter space associated with the locked 1 : 1 regime and another delineates a small portion of region IIIc, associated with the locked 3 : 1 regime, where force and displacement are in phase. In figure 16(a) a drop in the magnitude of the tangential force fluctuation can be noted near the phase difference jump in the locked 1 : 1 regime, especially in region Ia close to $\kappa =0$. In contrast, no drop appears in region IIIc, in the locked 3 : 1 regime. These distinct behaviours are visualized in figure 16(b), where the frequency ratio of $1$, which coincides with the phase difference jumps, is denoted by a grey dashed-dotted line; the values of $\varphi$ are specified on each side of this line. The contrasted trends noted in the locked 1 : 1 and locked 3 : 1 regimes are connected to the relative contributions of the higher harmonics in force fluctuation. In the locked 1 : 1 regime, even if higher harmonics exist in the force spectrum, their magnitudes remain small and the first harmonic (fundamental component at $f_\zeta$) generally dominates. Therefore, when the first harmonic contribution decreases near the phase difference jump, the magnitude of force fluctuation also drops. This drop is depicted by the light green area in figure 16(b). In the locked 3 : 1 regime the contributions of the higher harmonics dominate. This is emphasized in figure 16(b), where the contribution of the first harmonic is indicated by open red triangular symbols. The first harmonic contribution indeed vanishes near $f_\zeta =f_n$, but its evolution has only a negligible impact on the force fluctuation magnitude, which does not substantially change during the jump of the force–displacement phase difference.

The energy transfer between the flow and the moving cylinder is quantified by the power coefficient $e=C\dot {\zeta }$, as previously defined. The time series and spectra of $e$ in figure 3 illustrate the typical evolutions observed in the different interaction regimes. In the time series the intervals over which the flow excites/damps body motion, i.e. positive/negative values of $e$, are denoted by yellow/grey areas. For regular vibrations, as those reported here and in the absence of structural damping, the time-averaged power coefficient vanishes. More precisely, the power coefficient averaged over one oscillation cycle is equal to zero in the locked regimes (periodic responses). Zero averaged energy transfer is only reached over a number of oscillation cycles in the unlocked regime (aperiodic responses). As expected from its definition, the spectrum of $e$ is composed of even harmonics of the fundamental vibration component ($f_\zeta$) in the locked regimes without reconfiguration, essentially the second and fourth harmonics in the present cases (figure 3b,h). Odd and even harmonics participate in the energy transfer after reconfiguration (figure 3d, f). In the unlocked regime the spectral components occur at $2f_\zeta$, $2f_v$, $f_v \pm f_\zeta$, and other linear combinations of the fundamental components of body response and flow unsteadiness (figure 3j). Once the permanent behaviour of the system is reached, except for large radii close to the rectilinear path configuration, the energy transfer is dominated by the contribution of the time-averaged in-line force, $\bar {C}_x \dot \zeta _x$, where $\dot \zeta _x=-\dot \zeta \sin (\theta +\theta _0)$ is the in-line projection of body velocity (the contributions of the in-line and cross-flow forces to the tangential force are discussed in Appendix B). This explains why body excitation tends to occur at a comparable phase of the oscillation cycle, i.e. when the body moves downstream, regardless of the interaction regime at play.

This subsection has shed some light on the general properties of fluid forces and emphasized specific features associated with each regime, concerning their amplification, frequency content and phasing. The principal findings of this work are summarized in the next section.