2. Flow–structure system and numerical method

The physical system is schematized in figure 2(a). The circular cylinder of diameter $D$ and mass per unit length $M_c$ is parallel to the $z$ axis and placed in an incompressible uniform cross-current of velocity $U$, density $\rho _f$, viscosity $\mu$, aligned with the $x$ axis. The Reynolds number $Re =\rho _f U D/\mu$ is kept below or equal to $100$, which ensures that the flow is two-dimensional across the parameter space investigated. This point has been verified in a preliminary phase of the present work, via three-dimensional simulations initialized with three-dimensional flow fields. The two-dimensional Navier–Stokes equations are employed to predict the flow dynamics. The cylinder is free to translate along a circular path of radius $R$, parallel to the $(x,y)$ plane and centred at the origin of the $(x,y,z)$ frame. The cylinder position is tracked by the angle $\theta$ relative to the $x$ axis, referred to as the angular displacement. The physical variables are non-dimensionalized by $D$, $U$ and $\rho _f$. The non-dimensional, curvilinear displacement of the cylinder along the circular path is expressed as $\zeta =\theta /\kappa$, where $\kappa =D/R$ is the non-dimensional curvature magnitude. The transverse rectilinear motion configuration corresponds to the limiting case where $R$ tends to infinity ($\kappa =0$). In this configuration, $\zeta$ is the non-dimensional displacement aligned with the $y$ axis. The in-line, transverse and tangential force coefficients are defined as $C_x=2 F_x /(\rho _f D U^2)$, $C_y=2 F_y /(\rho _f D U^2)$ and $C=2 F /(\rho _f D U^2)$, where $F_x$, $F_y$ and $F$ are the dimensional fluid forces per unit length, parallel to the $x$ and $y$ axes, and to the direction of body motion, respectively. The motion of the cylinder is governed by the equation

(2.1)\begin{equation} \ddot{\zeta} = \dfrac{2C}{{\rm \pi} m^\star}, \quad {\rm with}\ C={-}C_x \sin(\kappa\zeta) + C_y \cos(\kappa\zeta).\end{equation}

The $^{\cdot}$ symbol designates the non-dimensional time derivative. The structure to displaced fluid mass ratio is defined as $m^\star =4 M_c/({\rm \pi} \rho _f D^2)$. The system behaviour is examined for $\kappa \in [0,20]$ and $m^\star \in [0.05,10]$.

Figure 2.

(a) Sketch of the physical system. (b) Curvilinear displacement amplitude (maximum value of signal fluctuation) and frequency as functions of the polynomial order for $(Re,\kappa,m^\star )=(100,0.5,0.05)$.

Assuming a decoupling of the flow and moving cylinder time scales, a quasi-steady model of $C$, identified by the superscript $qs$, and its first-order approximation about $\zeta =\dot \zeta =0$ can be expressed as (Bourguet Reference Bourguet2023b)

(2.2)\begin{equation} C^{qs}={-}\bar{C}^{f}_x (\dot\zeta+\sin(\kappa\zeta))\sqrt{\dot\zeta^2 +2\dot\zeta\sin(\kappa\zeta)+1} \approx{-}\bar{C}^{f}_x ( \dot\zeta + \kappa\zeta ),\end{equation}

where $\bar {C}^{f}_x$ is the mean in-line force (or drag) coefficient in the fixed body case. The $\bar {\phantom {a}}$ symbol denotes the time-averaged value. The first term of the approximation (relative to $\dot \zeta$) acts as a damping term through which the force tends to oppose body motion. This suggests that no vibration should develop when $C$ is replaced by $C^{qs}$ in the dynamics equation (2.1); this is indeed the case. The second term (relative to $\zeta$) is used to derive a non-dimensional natural frequency

(2.3)\begin{equation} f_n=\sqrt{\dfrac{\bar{C}^{f}_x \kappa}{2 {\rm \pi}^3 m^\star}}, \text{ and the corresponding reduced velocity } U^\star{=}\frac{1}{f_n}.\end{equation}

Even if the present system behaviour generally departs from the quasi-steady assumption, such modelling may shed some light on the effect of mean fluid forcing, and the above natural frequency will be used in the analysis of the results.

The numerical method is the same as in previous works concerning comparable systems (Bourguet Reference Bourguet2023a,Reference Bourguetb). It is briefly summarized here, and some additional convergence/validation results are presented. The coupled flow–structure equations are solved by the parallelized code Nektar, which is based on the spectral/$hp$ element method (Karniadakis & Sherwin Reference Karniadakis and Sherwin1999). A large rectangular computational domain is considered ($350D$ downstream, and $250D$ in front, above and below the cylinder) to avoid any blockage effects. It is discretized in $3975$ spectral elements. A no-slip condition is applied on the cylinder surface. A convergence study in the region of peak amplitude vibrations at $Re =100$ is presented in figure 2(b). The evolutions of the displacement amplitude and frequency ($\,f_\zeta$), as functions of the spectral element polynomial order, show that an increase from order $4$ to $5$ has no impact on the results. A polynomial order $4$ was selected. A similar procedure was employed to set the non-dimensional time step to $0.0025$. In figure 1, the evolution of the displacement amplitude with $m^\star$ for $\kappa =0$ is compared to the evolutions reported by Shiels et al. (Reference Shiels, Leonard and Roshko2001), Ryan et al. (Reference Ryan, Thompson and Hourigan2005) and Navrose & Mittal (Reference Navrose and Mittal2017) at $Re =100$. Some slight differences, which may be attributed to the distinct simulation strategies, can be noted in the location of the amplitude jump. The global trend of the displacement amplitude and the peak values reached in the low-$m^\star$ range are, however, comparable to prior results. This confirms the validity of the present numerical method.

Each simulation is initialized with the established flow past a fixed body at the selected $Re$. Then the body is released with an initial velocity $\dot \zeta =0.1$. The analysis is based on time series collected after convergence, over $30$ oscillation cycles in the unsteady cases.

3. Impact of path curvature on the system behaviour

The emergence of a vibration region in the $(\kappa,m^\star )$ domain, and its shape and evolution with $Re$, are examined in § 3.1, together with response amplitude and frequency. Flow–structure interaction mechanisms, including the synchronization regimes, wake patterns and some salient features of fluid forcing, are discussed in § 3.2.

3.1. Vibration region, amplitude and frequency

Within the $(\kappa,m^\star )$ domain investigated, the present simulations show that the onset of vibrations and thus flow unsteadiness may be shifted down to $Re \approx 19.5$, versus $Re \approx 31$ in the rectilinear path configuration (figure 1). As a result, the critical $Re$ value typically reported for an elastically mounted cylinder, close to $20$ (e.g. Kou et al. Reference Kou, Zhang, Liu and Li2017), can be reached in the absence of structural restoring force, under the effect of path curvature.

Figure 3(a) represents the curvilinear displacement amplitude in the $(\kappa,m^\star )$ domain at $Re =20$, i.e. close to vibration onset; $\bar {\zeta }=0$ in all cases, and the amplitude is measured as the maximum value of the displacement signal. The vibration region is delimited by dashed lines. Outside this region, the flow–structure system is steady at subcritical $Re$. As shown in figure 3(e), the vibration region tends to follow the line where the natural frequency induced by the mean drag (2.3) coincides with the Strouhal frequency ($St$, frequency of flow unsteadiness for a fixed body), or equivalently $U^\star =1/St$. At subcritical $Re$, the $St$ values are those determined by Kou et al. (Reference Kou, Zhang, Liu and Li2017) by triggering the flow: $0.10$, $0.11$ and $0.115$, at $Re =20$, $Re =30$ and $Re =40$, respectively. This trend persists as $Re$ is increased to higher subcritical values, while the vibration region expands continuously (figures 3b,c,f,g). At postcritical $Re$ ($Re =100$ in figures 3d,h), the cylinder oscillates and the flow is unsteady throughout the parameter space (shaded background). In this case, the region delineated by dashed lines is the area where $\zeta _{max}\ge 0.05$, and the terms ‘vibrations’ and ‘vibration region’ designate the corresponding responses and area. The vibration region remains aligned with the isoline $U^\star =1/St$ ($St=0.164$ at $Re =100$). This persistent orientation of the vibration region points out the close link between body and flow dynamics, which is addressed in § 3.2. The isolines $U^\star \in \{5,12\}$ locate the typical $U^\star$ range where VIV occur for an elastically mounted body (Williamson & Govardhan Reference Williamson and Govardhan2004). They do not capture the actual limits of the vibration region, but may provide an estimate of its global triangular shape.

Figure 3.

(a–d) Curvilinear displacement amplitude and (e–h) vibration region, as functions of $\kappa$ and $m^\star$, at (a,e) $Re =20$, (b,f) $Re =30$, (c,g) $Re =40$ and (d,h) $Re =100$. Dashed lines delimit the area where $\zeta _{max} > 0$ in (a–c,e–g) and $\zeta _{max}\ge 0.05$ in (d,h). In (a–d), the colour levels are the same for all $Re$. In (d), the location of the peak amplitude as a function of $m^\star$ is indicated by a black dotted line. The cases visualized in figure 8 are denoted by blue stars. In (e–h), the shaded area denotes the region where the cylinder exhibits oscillations of any amplitude. The isolines $U^\star =1/St$, $U^\star =5$ and $U^\star =12$ are represented by green dash-dotted, blue dotted and solid lines, respectively. At subcritical $Re$ (e–g), $St$ values are those determined by Kou et al. (Reference Kou, Zhang, Liu and Li2017).

Vibrations are encountered over much wider ranges of $m^\star$ when the path is curved, compared to the narrow ranges depicted in figure 1 for $\kappa =0$. Two elements can be noted. First, for any $m^\star$, there always exists an interval of $\kappa$ where vibrations develop. Second, an increase of $\kappa$ tends to shift the $m^\star$ range towards higher values and simultaneously widens it. This widening is visualized in figures 4(a,b), where the curvilinear and angular displacement amplitudes are represented as functions of $m^\star$ for selected $\kappa$, at $Re =100$. The amplitudes measured for $\kappa =0$ are also plotted, for comparison (grey diamonds). For $\kappa =5$, the vibration region encompasses most of the $m^\star$ range under study.

Figure 4.

Displacement amplitude (upper plots) and frequency (lower plots) as functions of $m^\star$ for (a) $\kappa =2$ and (b) $\kappa =5$, and functions of $\kappa$ for (c) $m^\star =1$ and (d) $m^\star =7$, at $Re =100$. In the upper plots, both curvilinear (left-hand axis) and angular (right-hand axis) displacement amplitudes are represented. In the lower plots, the displacement frequency is plotted together with flow unsteadiness frequency, the natural frequency (2.3) and $St$; a normalization by $St$ is proposed on the right-hand axis. In (a,b), $\zeta _{max}$ for $\kappa =0$ is recalled for comparison (grey diamonds). In (d), blue arrows indicate local peaks of the displacement amplitude. Yellow and grey background colours denote the locked and unlocked regimes, respectively.

The influence of path curvature on vibration amplitude is not monotonic, as illustrated in figures 4(c,d) for selected $m^\star$. However, the peak amplitude at a given $m^\star$ is systematically larger for $\kappa >0$ than the rectilinear response amplitude, including in the low-$m^\star$ range. For an elastically mounted body, the increasing evolution of the displacement frequency with $\kappa$ denotes a reduction of the effective added mass, which may become negative for peak amplitude responses (figure 7 in Bourguet Reference Bourguet2023b). This suggests that the subset of responses reached once the elastic support is removed should include the peak amplitude vibrations (Govardhan & Williamson Reference Govardhan and Williamson2002). Such a phenomenon is actually observed: for a fixed $\kappa$ value larger than a threshold that depends on $Re$ (e.g. $\kappa \approx 0.7$ at $Re =100$), the peak amplitude attained over the $m^\star$ range is the same as that reported for an elastically mounted body. It should be mentioned that this trend, i.e. reduction of the effective added mass as $\kappa$ is increased, may also explain the reduction of the critical $Re$ for vibration onset down to values encountered in the elastically mounted body case. The maximum amplitude detected in the present $(\kappa,m^\star )$ domain, $0.55D$ at $Re =100$ ($+45\,\%$ compared to the $\kappa =0$ configuration, where the peak amplitude is $0.38D$), is similar to that measured with an elastic support. A jump in the evolution of the peak amplitude location versus $m^\star$ can be noted at $Re =100$ in figure 3(d) (black dotted line). The two local maxima observed close to the jump are indicated by blue arrows in figure 4(d). This jump reflects the coexistence of two distinct interaction regimes.

When the body oscillates, the displacement spectrum is dominated by a single frequency, $f_\zeta$. A general view of $f_\zeta$ over the $(\kappa,m^\star )$ domain, at $Re \in \{20,30,40,100\}$, is presented in figures 5(a–d). For $\kappa =0$, when vibrations develop ($Re =40$ and $100$), their frequency is restrained to a narrow range and is always lower than $St$. The frequency range explored by the body once it vibrates along a curved path is considerably extended and includes $St$ (blue dotted line), for example at $Re =100$, $f_\zeta \in [0.11,0.198]$ versus $f_\zeta \in [0.132,0.161]$ for $\kappa =0$. The $f_\zeta$ range tends to increase with $Re$. A distinct region of low frequency relative to $St$ can be identified close to the lower edge of the vibration region at $Re =100$ (white dots in figure 5d). As discussed in the next subsection, this region relates to the emergence of a specific regime of the flow–structure system, under the influence of path curvature.

Figure 5.

(a–d) Displacement frequency as a function of $\kappa$ and $m^\star$, and (e–h) curvilinear displacement amplitude as a function of the ratio between the flow unsteadiness and body displacement frequencies, at (a,e) $Re =20$, (b,f) $Re =30$, (c,g) $Re =40$ and (d,h) $Re =100$. In (a–d), the isoline $f_\zeta =St$ is represented by a blue dotted line. In (d), white dashed lines delimit the area where $\zeta _{max}\ge 0.05$. The unlocked regime region is indicated by white dots and delineated by a thin white line. The cases visualized in figure 8 are denoted by blue stars. In (e–h), the frequency ratio $1$ is specified by a black dash-dotted line. In (g,h), the results obtained for $\kappa =0$ are represented by black crosses; $\zeta =0$ for $\kappa =0$ at $Re \in \{20,30\}$.

3.2. Flow–structure interaction mechanisms

In order to visualize the connection between cylinder and flow dynamics, the displacement frequency is represented in figure 4, together with the dominant frequency of flow unsteadiness, $f_v$, issued from the time series of the transverse component of flow velocity ($v$) sampled $10D$ downstream of the body. These plots depict two states of flow–body synchronization, where $f_\zeta$ and $f_v$ are either equal or incommensurable. They are the two possible unsteady states encountered in the parameter space, as illustrated in figures 5(e–h), where the response amplitude is plotted as a function of the flow/body frequency ratio $f_v/f_\zeta$, for all simulated cases, at $Re \in \{20,30,40,100\}$. The associated regimes are named locked and unlocked, and denoted by yellow and grey background colours, respectively, in figure 4.

The locked regime ($\,f_\zeta =f_v$) corresponds to the lock-in condition usually reported for VIV of elastically mounted cylinders (Williamson & Govardhan Reference Williamson and Govardhan2004). In the present configuration and up to the highest $Re$ considered ($Re =100$), this is the only unsteady regime encountered for $\kappa =0$ (black crosses in figures 5(g,h); $\zeta =0$ for $\kappa =0$ at $Re =20$ and $30$). At subcritical $Re$, this is also the only unsteady regime appearing for $\kappa >0$. The locked regime may be associated with deviations from $f_n$ and $St$ (figure 4). Such deviations are quantified over the $(\kappa,m^\star )$ domain in figures 6 and 7, respectively, via amplitude versus frequency ratio plots. The large-amplitude responses often develop relatively close to $f_n$, but substantial departures, mainly oriented towards lower frequencies ($\,f_\zeta < f_n$), can occur. The frequency range may vary from $-20\,\%$ to $+35\,\%$ of $St$, depending on $Re$. At $Re =100$, it follows approximately the boundaries of the wake synchronization region determined by Koopmann (Reference Koopmann1967) for forced rectilinear oscillations (red dotted lines in figure 7d). As previously mentioned, the system frequency is lower than $St$ for $\kappa =0$ (black crosses).

Figure 6.

Curvilinear displacement amplitude as a function of the ratio between the body displacement and natural frequencies at (a) $Re =20$, (b) $Re =30$, (c) $Re =40$ and (d) $Re =100$. The frequency ratio $1$ is specified by a black dash-dotted line. Distinct symbols are used to designate the locked and unlocked regime cases.

Figure 7.

Curvilinear displacement amplitude as a function of the ratio between the body displacement/flow unsteadiness and Strouhal frequencies at (a) $Re =20$, (b) $Re =30$, (c) $Re =40$ and (d) $Re =100$. The frequency ratio $1$ is specified by a black dash-dotted line. In (c,d), the results obtained for $\kappa =0$ are represented by black crosses; $\zeta =0$ for $\kappa =0$ at $Re \in \{20,30\}$. In (d), a blue dashed line visualizes the threshold below which flow–body synchronization ceases, i.e. the unlocked regime; $f_\zeta$ and $f_v$ are plotted in this case. Red dotted lines delimit the synchronization region reported by Koopmann (Reference Koopmann1967) under forced transverse oscillations at $Re =100$.

The unlocked regime ($\,f_\zeta$ and $f_v$ incommensurable) resembles the desynchronized condition observed for VIV at higher $Re$, beyond the lock-in range. It does not appear at subcritical $Re$. At $Re =100$, it emerges for $\kappa >1$. Its location in the $(\kappa,m^\star )$ domain is indicated by white dots in figure 5(d); the locked regime is established over the rest of the domain. The unlocked regime is associated with a range of moderate response amplitudes, $\zeta _{max}\in [0.05,0.2]$, and flow/body frequency ratios varying from $1.15$ to $1.45$ (figure 5h). The frequency plots in figure 4 suggest that in the unlocked regime, the body oscillates very close to $f_n$, possibly far from $St$ compared to $f_v$. These observations are confirmed in figures 6(d) and 7(d). In particular, the displacement frequency is found to remain lower than a threshold located at approximately $0.8 St$ (blue dashed line in figure 7d).

The dynamics of the flow–structure system is periodic in the locked regime. In the unlocked regime, the body displacement and flow velocity signals are close to periodic. Yet the presence of incommensurable components, at $f_v$ in the $\zeta$ spectrum and at $f_\zeta$ in the $v$ spectrum, results, despite their limited contributions, in modulations of the temporal evolutions. Selected time series in cases representative of each regime are plotted in figure 8, together with instantaneous visualizations of the wake. Across the entire parameter space, and regardless of $Re$, $f_v$ corresponds to the shedding of a pair of counter-rotating vortices. Therefore, a pair of vortices is formed per oscillation cycle in the locked regime. In the unlocked regime, based on flow/body frequency ratio ($\,f_v/f_\zeta \in [1.15,1.45]$), between $2.3$ and $2.9$ vortices are shed per oscillation cycle. In all cases, the organization of the wake remains close to the von Kármán street occurring in the fixed body configuration. The aperiodic dynamics of the system in the unlocked regime is, however, betrayed by subtle variations in the streamwise distance between consecutive vortices and in their alignment. It is recalled that the unlocked regime does not develop for $\kappa =0$: the reported alteration of wake regularity is an effect of path curvature.

Figure 8.

Selected time series of the curvilinear displacement, transverse component of flow velocity, tangential force coefficient and mean drag contribution to the tangential force coefficient, and instantaneous isocontours of spanwise vorticity ($\omega _z\in [-0.8,0.8]$), for (a) $(\kappa,m^\star )=(1,1)$ (locked regime) and (b) $(\kappa,m^\star )=(5,7)$ (unlocked regime), at $Re =100$. The time instant visualized in the right-hand plot is indicated by a red dot in the $\zeta$ time series. The circular path and trajectory of the cylinder centre are represented by black dashed and solid lines. The sampling point of flow velocity is denoted by a green cross. Positive/negative vorticity values are plotted in red/blue.

The amplification of the mean drag associated with the emergence of vibrations is enhanced by path curvature. For example, at $Re =100$, it reaches $+65\,\%$ relative to $\bar {C}^{f}_x=1.32$ close to $\kappa =1.5$, versus $+15\,\%$ for $\kappa =0$. In the absence of structural restoring force and damping, for periodic responses such as those encountered in the locked regime, the components of $C$ and $\zeta$ at $f_\zeta$ are in phase opposition. This phasing state persists in the unlocked regime, as shown in figure 8, where $C$ time series are represented for both regimes. For $\kappa =0$, the drag is not involved in $C$ and is thus disconnected from body excitation. In contrast, it may play a significant role once the path is curved. The mean drag contribution ($-\bar {C}_x\sin (\kappa \zeta )$) actually becomes predominant over most of the vibration regions identified in figure 3, as illustrated by its proximity with $C$ in figure 8. As a result, body excitation, i.e. positive $C\dot \zeta \approx - \bar {C}_x \dot \zeta \sin (\kappa \zeta ) = \bar {C}_x \dot \zeta _x$, where $\dot \zeta _x$ is the in-line velocity of the body, occurs mainly during downstream motion.