2.1 Incompressible flow equations

To investigate the interaction between the fluid and the oscillating cylinder, linear stability analysis of the governing equations is carried out in a moving frame attached to the cylinder. Such an approach has been used earlier to study vortex-induced vibration of a cylinder (Étienne & Pelletier Reference Étienne and Pelletier2012; Lu & Papadakis Reference Lu and Papadakis2014). Let $\boldsymbol{x}$ and $t$ denote the spatial and temporal coordinates respectively. The spatial domain is represented by ${\it\Omega}\subset \mathbb{R}^{2}$ . ${\it\Gamma}$ represents the boundary of ${\it\Omega}$ . The temporal domain is represented by $(0,T)$ . The Navier–Stokes equations governing an incompressible flow in the absence of body forces are

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\rho}\left(\frac{\partial \boldsymbol{u}}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{u}+\dot{\boldsymbol{U}}_{b}\right)-\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }{\bf\sigma}=\mathbf{0}\quad \text{on}~{\it\Omega}\times (0,T), & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}=0\quad \text{on}~{\it\Omega}\times (0,T). & \displaystyle\end{eqnarray}$$

Here, ${\it\rho}$ , $\boldsymbol{u}$ and ${\bf\sigma}$ are the density, velocity and stress tensor, respectively. $\boldsymbol{U}_{b}$ denotes the instantaneous velocity of the cylinder. For a Newtonian fluid the stress tensor is given as ${\bf\sigma}=-p\boldsymbol{I}+2{\it\mu}{\bf\varepsilon}(\boldsymbol{u})$ , where, ${\bf\varepsilon}$ is the strain rate given as ${\bf\varepsilon}(\boldsymbol{u})=((\boldsymbol{{\rm\nabla}}\boldsymbol{u})+(\boldsymbol{{\rm\nabla}}\boldsymbol{u})^{\text{T}})/2$ and, $p$ and ${\it\mu}$ are the pressure and coefficient of dynamic viscosity, respectively.

Unlike the LSA, the DTI of the governing equations is carried out in an inertial frame of reference. Therefore, the third term in (2.1) is dropped. More details on the same can be found in our earlier work (Prasanth & Mittal Reference Prasanth and Mittal2008). The computational domain along with the boundary conditions are shown in figure 2. No-slip condition is applied on the velocity at the surface of the cylinder. The location of the cylinder and its velocity are updated at each nonlinear iteration of the solution to the flow equations.

Figure 2.

Flow past a freely vibrating cylinder: (a) finite element mesh that has been employed for the computations in this work, (b) close-up view of the mesh near the surface of the cylinder. The mesh consists of 12 814 nodes and 12 540 quadrilateral elements. The location of the various boundaries and boundary conditions are also shown.

2.2 The equations of motion for the structure

The motion of the cylinder mounted on elastic supports, in the two directions along the Cartesian axes, is governed by the following system of equations:

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{\boldsymbol{U}}_{b}+4{\rm\pi}F_{N}{\it\zeta}\boldsymbol{U}_{b}+(2{\rm\pi}F_{N})^{2}\boldsymbol{X}=\frac{2\boldsymbol{C}_{\boldsymbol{F}}}{{\rm\pi}m^{\ast }}\quad \text{for}~(0,T), & \displaystyle\end{eqnarray}$$

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{\boldsymbol{X}}=\boldsymbol{U}_{b}\quad \text{for}~(0,T). & \displaystyle\end{eqnarray}$$

Here, $F_{N}$ is the reduced natural frequency of the oscillator, ${\it\zeta}$ the structural damping ratio and $m^{\ast }$ the non-dimensional mass of the body. $\boldsymbol{C}_{\boldsymbol{F}}$ is the instantaneous coefficient of fluid force acting on the cylinder. The fluid force is computed by integrating stress on the boundary of the cylinder. $\boldsymbol{X}$ denotes the normalized displacement. The displacement and velocity are normalized by the diameter of the cylinder, $D$ , and the free-stream speed $U$ , respectively. $F_{N}$ is related to the natural frequency of the spring mass system ( $f_{N}$ ) as $F_{N}=f_{N}D/U$ . Another related parameter is the reduced velocity, $U^{\ast }$ . It is defined as $U^{\ast }=U/(f_{N}D)=1/F_{N}$ . Although the formulation presented here is general, results are presented for transverse-only oscillation of the structure.

2.3 Global LSA

We carry out a global linear stability analysis of the fluid–structure system represented by (2.1)–(2.4). A similar approach was employed by Cossu & Morino (Reference Cossu and Morino2000). The unsteady flow variables, $(\boldsymbol{u},p)$ are decomposed as a combination of the steady and the disturbance field: $\boldsymbol{u}=\boldsymbol{U}+\boldsymbol{u}^{\prime }$ and $p=P+p^{\prime }$ . $(\boldsymbol{U},P)$ , the steady flow past a stationary cylinder, is obtained by dropping the unsteady terms in (2.1) and (2.2). $\boldsymbol{u}^{\prime }$ and $p^{\prime }$ are the perturbation fields of the velocity and pressure, respectively. In the steady state the cylinder is at rest at its equilibrium position. Therefore, $\boldsymbol{U}_{b}$ is also the disturbance in the velocity of the cylinder. In the equilibrium state, the spring is in a stretched position, $\boldsymbol{X}_{\boldsymbol{e}\boldsymbol{q}}$ , due to the steady force acting on the cylinder. In general, for a circular cylinder in uniform flow, this force is along the streamwise direction. The displacement of the cylinder, therefore, can be decomposed as $\boldsymbol{X}=\boldsymbol{X}_{\boldsymbol{e}\boldsymbol{q}}+\boldsymbol{x}^{\prime }$ , where $\boldsymbol{x}^{\prime }$ represents the disturbance field of the cylinder displacement. Substituting for the decomposition of flow and the structural variables in (2.1)–(2.4), and subtracting from them, the equations for steady flow, the equations for the evolution of the disturbance field can be obtained. We assume that the disturbances are small and drop the nonlinear terms. We also assume that the system is devoid of any structural damping. This leads to the following linearized equations of the disturbance field:

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\rho}\left(\frac{\partial \boldsymbol{u}^{\prime }}{\partial t}+\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{U}+\boldsymbol{U}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{\prime }+\dot{\boldsymbol{U}}_{b}\right)-\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }{\bf\sigma}^{\prime }=\mathbf{0}, & \displaystyle\end{eqnarray}$$

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }=0, & \displaystyle\end{eqnarray}$$

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{\boldsymbol{U}}_{b}+(2{\rm\pi}F_{N})^{2}\boldsymbol{x}^{\prime }=\frac{2\left(\displaystyle \frac{\partial \boldsymbol{C}_{\boldsymbol{F}}}{\partial \boldsymbol{U}_{\boldsymbol{b}}}\boldsymbol{\cdot }\boldsymbol{U}_{\boldsymbol{b}}+\displaystyle \frac{\partial \boldsymbol{C}_{\boldsymbol{F}}}{\partial p^{\prime }}p^{\prime }\right)}{{\rm\pi}m^{\ast }}, & \displaystyle\end{eqnarray}$$

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{\boldsymbol{x}}^{\prime }=\boldsymbol{U}_{\boldsymbol{b}}. & \displaystyle\end{eqnarray}$$

In the equation above, ${\bf\sigma}^{\prime }$ represents the stress tensor due to disturbance field of the flow. The variation in non-dimensional forces because of change in the velocity of the cylinder with respect to the equilibrium state is denoted by the matrix $\partial \boldsymbol{C}_{\boldsymbol{F}}/\partial \boldsymbol{U}_{\boldsymbol{b}}$ . Similarly, the variation in forces because of the change in the pressure on the surface of the cylinder is $\partial \boldsymbol{C}_{\boldsymbol{F}}/\partial p^{\prime }$ . For conducting the global LSA of the combined fluid–cylinder system, we assume the disturbance field of the following form:

(2.9a−d ) $$\begin{eqnarray}\displaystyle \boldsymbol{u}^{\prime }(\boldsymbol{x},t)=\hat{\boldsymbol{u}}(\boldsymbol{x})\text{e}^{{\it\lambda}t},\quad p^{\prime }(\boldsymbol{x},t)=\hat{p}(\boldsymbol{x})\text{e}^{{\it\lambda}t},\quad \boldsymbol{U}_{\boldsymbol{b}}=\hat{\boldsymbol{U}}_{b}\text{e}^{{\it\lambda}t},\quad \boldsymbol{x}^{\prime }=\hat{\boldsymbol{x}}\text{e}^{{\it\lambda}t}. & & \displaystyle\end{eqnarray}$$

Substituting this form of the disturbance in (2.5)–(2.8) we obtain:

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\rho}({\it\lambda}(\hat{\boldsymbol{u}}+\hat{\boldsymbol{U}}_{b})+\hat{\boldsymbol{u}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{U}+\boldsymbol{U}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\hat{\boldsymbol{u}})-\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\hat{{\bf\sigma}}=0, & \displaystyle\end{eqnarray}$$

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\hat{\boldsymbol{u}}=0, & \displaystyle\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\lambda}\hat{\boldsymbol{U}}_{b}+(2{\rm\pi}F_{N})^{2}\hat{\boldsymbol{x}}=\frac{2\left(\displaystyle \frac{\partial \boldsymbol{C}_{\boldsymbol{F}}}{\partial \hat{\boldsymbol{U}}_{b}}\boldsymbol{\cdot }\hat{\boldsymbol{U}}_{b}+\displaystyle \frac{\partial \boldsymbol{C}_{\boldsymbol{F}}}{\partial \hat{p}}p^{\prime }\right)}{{\rm\pi}m^{\ast }}, & \displaystyle\end{eqnarray}$$

(2.13) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\lambda}\hat{\boldsymbol{x}}=\hat{\boldsymbol{U}}_{b}. & \displaystyle\end{eqnarray}$$

Here, ${\it\lambda}$ is the eigenvalue of the coupled fluid–structure system and governs its stability. In general, ${\it\lambda}={\it\lambda}_{r}+\text{i}{\it\lambda}_{i}$ , where ${\it\lambda}_{r}$ and ${\it\lambda}_{i}$ are the real and imaginary parts, respectively. We define the non-dimensional frequency of the eigenmode as $F_{LSA}={\it\lambda}_{i}/2{\rm\pi}$ . The corresponding reduced velocity is represented by $U_{LSA}^{\ast }=2{\rm\pi}/{\it\lambda}_{i}=1/F_{LSA}$ . In this study, we restrict the motion of the cylinder in the direction transverse to the free-stream flow. This is commonly referred to as $Y$ -only oscillations. A typical global eigenmode of the fluid–structure system is thus represented as ( $\hat{\boldsymbol{u}}(\boldsymbol{x})$ , $\hat{p}(\boldsymbol{x})$ , $\hat{{\dot{Y}}}$ , ${\hat{Y}}$ ).

4.1 Classification of modes

Compared to the fluid alone, LSA of the fluid–structure model is associated with two additional eigenmodes. Cossu & Morino (Reference Cossu and Morino2000) and Meliga & Chomaz (Reference Meliga and Chomaz2011) presented a classification of the eigenmodes based on their characteristics in the limit of very large $m^{\ast }$ . To identify the class to which an eigenmode belongs for a given value of $Re$ and $U^{\ast }$ , the eigenvalue has to be followed up to large values of $m^{\ast }$ . Most studies on VIV that explore the characteristics of the fluid–structure system in lock-in regime consider its variation with reduced speed while holding the mass ratio fixed. A different perspective towards mode classification is adopted in this work wherein $m^{\ast }$ is fixed and the eigenmodes are computed over a wide range of $U^{\ast }$ . The present classification makes it relatively easier to compare the characteristics of fluid–structure system in the lock-in regime with the properties of the unstable modes. It is observed that for systems with large $m^{\ast }$ , the modes corresponding to two leading eigenvalues are quite distinct and maintain this distinction for all $U^{\ast }$ . That is, in terms of variation with $U^{\ast }$ , the two leading eigenmodes are decoupled. We refer to the two modes as fluid mode and elastic mode. The frequency of FM remains nearly constant with $U^{\ast }$ . The frequency of EM, on the other hand, varies inversely with $U^{\ast }$ . For low $m^{\ast }$ , the two leading eigenmodes do not exhibit a clear distinction in terms of their affiliation to being either class elastic or class fluid. We refer to such modes as coupled modes and denote them by fluid–elastic mode I (FEMI) and fluid–elastic mode II (FEMII). For small $U^{\ast }$ , FEMI resembles fluid mode and FEMII is similar to elastic mode. However, for large $U^{\ast }$ , the characteristics of FEMI and FEMII resemble EM and FM respectively. A detailed description of decoupled and coupled modes for the $Re=60$ flow is presented for two values of mass ratio, $m^{\ast }=20$ and $5$ . The modes are decoupled for $m^{\ast }=20$ , while they are coupled for $m^{\ast }=5$ . In the discussion that follows, we refer to the mode responsible for the onset of global instability in the flow past a stationary cylinder as the stationary wake mode.

Figure 3.

Linear stability analysis of the steady flow past a fluid–structure system: stability characteristics for (a,c)  $(Re,m^{\ast })=(60,20)$ , (b,d)  $(Re,m^{\ast })=(60,5)$ . ${\it\lambda}_{r}$ represents the growth rate and $F_{LSA}={\it\lambda}_{i}/2{\rm\pi}$ represents the non-dimensional frequency.

Figure 3(a) shows the variation of the growth rate of the fluid and the elastic mode with $U^{\ast }$ for $(Re,m^{\ast })=(60,20)$ . The fluid mode is unstable for all $U^{\ast }$ . It achieves maximum growth rate for $U^{\ast }=8.0$ . The growth rate of FM approaches the growth rate of the stationary wake mode for $Re=60$ for very small and very large values of $U^{\ast }$ . The elastic mode is unstable for $5.8<U^{\ast }<7.3$ . This mode attains highest growth rate for $U^{\ast }=7.0$ . Figure 3(c) shows the variation of non-dimensional frequency, $F_{LSA}={\it\lambda}_{i}/2{\rm\pi}$ , with $U^{\ast }$ . The frequency of the fluid mode remains close to the frequency of the stationary wake mode for all $U^{\ast }$ . It is noted that the growth rate of both modes achieve maximum within the range of $U^{\ast }$ where the frequencies of the two modes are close to each other. Figure 4 shows the vorticity field for the real and imaginary parts of the unstable eigenmode for the steady flow past a stationary cylinder at $Re=60$ . Figure 5 shows the vorticity field for FM and EM for different $U^{\ast }$ at $(Re,m^{\ast })=(60,20)$ . It is observed that the shape of the eigenmode is closely related to its frequency ( $F_{LSA}$ ). If the frequency associated with the eigenmode is relatively larger, then the region of large perturbation associated with the mode is closer to the cylinder. As the frequency of the eigenmode decreases, the large perturbation region shifts downstream with respect to the cylinder. The frequency of FM remains close to that of the $Re=60$ stationary wake mode for all values of $U^{\ast }$ . Therefore, the mode shape of FM resembles the stationary wake mode (figure 4) for all reduced speeds. On the other hand, the frequency of EM follows the natural frequency of the structure-only system over a wide range of $U^{\ast }$ . As a result, with increase in $U^{\ast }$ the region of large perturbation associated with EM moves downstream to the cylinder. For $U^{\ast }$ values where the frequency of FM and EM are close to each other, no perceptible difference is observed in the structure of the two modes.

Figure 4.

Linear stability analysis of the $Re=60$ steady flow past a circular cylinder: vorticity field for the (a) real and (b) imaginary parts of the eigenmode corresponding to the stationary wake mode. The flow is from left to right.

Figure 5.

Linear stability analysis of the steady flow past a fluid–structure system for $(Re,m^{\ast })=(60,20)$ : spanwise vorticity field of the EM (a,c,e) and the FM (b,d,f) for (a,b)  $U^{\ast }=5.5$ , (c,d)  $U^{\ast }=7.0$ and (e,f)  $U^{\ast }=8.0$ . The flow is from left to right.

Figure 3(b,d) shows the stability characteristics of FEMI and FEMII for $(Re,m^{\ast })=(60,5)$ . The vorticity field of the eigenmodes for different values of $U^{\ast }$ are shown in figure 6. The growth rate, frequency and mode shape of FEMI resemble the stationary wake mode for low $U^{\ast }$ . In the same $U^{\ast }$ regime, FEMII is stable and its frequency follows the structure-only system. Concomitantly, the region of large perturbation associated with FEMII moves downstream to the cylinder with increase in reduced speed. The growth rate of FEMI decreases with increase in $U^{\ast }$ and the mode becomes stable beyond $U^{\ast }=6.0$ . FEMII is unstable for $U^{\ast }\geqslant 5.0$ . Towards the higher- $U^{\ast }$ end, growth rate, frequency and mode shape of FEMII resemble the stationary wake mode. For FEMI, at large values of $U^{\ast }$ , the large perturbation region associated with the eigenmode moves downstream with increase in $U^{\ast }$ . For $6.0\leqslant U^{\ast }\leqslant 9.0$ , the frequencies of the two modes are close to each other and their variations with $U^{\ast }$ are also similar. The growth rate of the unstable FEMII achieves maximum within this range.

Figure 6.

Linear stability analysis of the steady flow past a fluid–structure system for $(Re,m^{\ast })=(60,5)$ : spanwise vorticity field of FEMI (a,c,e) and FEMII (b,d,f) for (a,b)  $U^{\ast }=4.0$ , (c,d)  $U^{\ast }=6.8$ and (e,f)  $U^{\ast }=10.0$ . The flow is from left to right.

Figure 7.

Linear stability analysis of the steady flow past a fluid–structure system: variation of $U_{LSA}^{\ast }=2{\rm\pi}/{\it\lambda}_{i}=1/F$ with $U^{\ast }$ for (a)  $(Re,m^{\ast })=(60,20)$ and (b)  $(Re,m^{\ast })=(60,5)$ . The point corresponding to stationary wake mode is shown in bull’s eye symbol.

To further bring out the distinction between the coupled and decoupled modes we replot the results shown in figure 3 on the ${\it\lambda}_{r}{-}U_{LSA}^{\ast }$ plane; $U_{LSA}^{\ast }(=1/F_{LSA})$ is the reduced velocity based on the frequency of the eigenmode. Figure 7 shows the variation of the growth rate with $U_{LSA}^{\ast }$ for $(Re,m^{\ast })=(60,20)$ and $(60,5)$ . The end points of the curves are tagged with $A{-}B$ for FM, $C{-}D$ for EM, $P{-}Q$ for FEMI and $R{-}S$ for FEMII. The location of these points in figure 3 is also marked. In the case of decoupled modes, the end points of the fluid mode, at the low (point  $A$ ) and high end of $U^{\ast }$ (point  $B$ ), approach each other with increase in the range of $U^{\ast }$ . For the limiting case when the reduced velocity spans from zero to infinity, FM would appear as a closed ring, separate from EM, on ${\it\lambda}_{r}{-}U_{LSA}^{\ast }$ plane. The point where the end points of the ring merge, corresponds to that for the stationary wake mode (shown as bull’s eye symbol in figure 3). For the case of coupled modes, with increase in the range of $U^{\ast }$ , the end point of FEMII (point  $S$ ) and the beginning of FEMI (point  $P$ ) approach the point corresponding to the stationary wake mode. In the limiting case when the reduced velocity spans from zero to infinity, the two modes would merge at the point corresponding to stationary wake mode on ${\it\lambda}_{r}{-}U_{LSA}^{\ast }$ plane and appear as a single curve. Figure 7 clearly brings out the distinction between the coupled and decoupled modes and the identification of the two branches.

4.2 Regime of coupled and decoupled modes

In general, whether the modes are coupled or decoupled depends on $m^{\ast }$ and $Re$ . At a given $Re$ , the modes are coupled for relatively low $m^{\ast }$ and get decoupled to EM and FM at larger $m^{\ast }$ . We define critical mass ratio, $m_{c}^{\ast }$ , as the largest mass ratio for which modes are coupled at a certain $Re$ . We also define $Re_{o}$ as the Reynolds number beyond which the stationary wake mode becomes unstable. The value of $Re_{o}$ estimated with the mesh employed in the present study is $Re_{o}=47.5$ . The value of $Re_{o}$ from the study reported by Kumar & Mittal (Reference Kumar and Mittal2006), independent of mesh and blockage effects, is $Re_{o}=47$ . The two values are in very good agreement. The small difference between the two studies is attributed to the slightly lower spatial resolution in the present study. Compared to the flow past stationary cylinder, the coupled fluid–structure system has additional degrees of freedom corresponding to cylinder displacement and velocity. In addition, these degrees of freedom are strongly coupled with the flow variables at the surface of the cylinder. Therefore, LSA of the coupled fluid–structure problem is relatively more demanding on computational resources as compared to the fluid-only and structure-only problems. One of the factors in choosing the finite element mesh used in the present study is that with the computational resources at our disposal, we should be able to carry out LSA over a wide range of the various parameters of interest ( $Re$ , $U^{\ast }$ , $m^{\ast }$ ). A mesh convergence study is carried out to check the adequacy of the mesh against a mesh of finer spatial resolution. The details of the mesh convergence study are presented in appendix A. The results from the two meshes are found to be in very good agreement. The effect of $m^{\ast }$ on (de-)coupling of modes is presented with $Re$ . To account for the offset in the value of $Re_{o}$ compared to that reported by Kumar & Mittal (Reference Kumar and Mittal2006), we additionally present the results with respect to $Re/Re_{o}$ . Marais et al. (Reference Marais, Godoy-Diana, Barkley and Wesfreid2011) used a similar approach to present their results on stability analysis. They recorded a value of $Re_{o}=64$ in their experiments because of confinement and blockage effects.

Figure 8(a) shows the regions in $Re{-}m^{\ast }$ plane where the modes are coupled or decoupled. We describe the procedure used to generate this figure. For several values of $Re$ , computations are carried out for many values of $m^{\ast }$ that are separated by a step size of ${\rm\Delta}m^{\ast }=5.0$ . At each of these $(Re,m^{\ast })$ values, LSA is carried out over a wide range of reduced speed. The variation of ${\it\lambda}_{r}$ with $U_{LSA}^{\ast }$ is utilized to identify coupled/decoupled modes. In the range of $m^{\ast }$ where the system transits between coupled and decoupled modes, computations are carried out for more $m^{\ast }$ values that are separated by a finer step size of ${\rm\Delta}m^{\ast }=1.0$ . These are utilized to estimate the critical mass ratio, $m_{c}^{\ast }$ , beyond which the modes are decoupled for a given $Re$ . These values are shown in figure 8(a) by solid circles. We note that $m_{c}^{\ast }$ decreases as one moves away from $Re_{o}$ . Figure 8(b–d) shows the variation of ${\it\lambda}_{r}$ with $U_{LSA}^{\ast }$ for $m^{\ast }=75$ at $Re=37.5$ , 47 and 56.5, respectively. For $Re=37.5$ and 56.5 the modes are decoupled. While the fluid mode is stable for former, it is unstable in the latter case. For $Re=47.0$ ( $Re/Re_{o}=1.0$ ), the modes are coupled. The value of ${\it\lambda}_{r}$ at the junction, where the branches corresponding to modes FEMI and FEMII meet, is equal to zero. The transition from the regime of coupled modes to decoupled modes with increase in mass ratio is demonstrated in figure 9 for $Re=42$ and 52 by the evolution of the topology of the curves.

Figure 8.

Linear stability analysis of the steady flow past a fluid–cylinder system: (a) the regions of mode coupling/decoupling. The variation of growth rate of modes with $U_{LSA}^{\ast }$ for $m^{\ast }=75$ at (b)  $Re=37.5$ , (c)  $Re=Re_{o}=47$ and (d)  $Re=56.5$ .

Figure 9.

Linear stability analysis of the steady flow past a fluid–cylinder system: variation of growth rate with $U_{LSA}^{\ast }$ of modes for for (a)  $(Re,m^{\ast })=(42,10)$ , (c)  $(Re,m^{\ast })=(42,25)$ , (e)  $(Re,m^{\ast })=(42,50)$ , (b)  $(Re,m^{\ast })=(52,10)$ , (d)  $(Re,m^{\ast })=(52,25)$ , (f) $(Re,m^{\ast })=(52,40)$ .

The term critical mass ratio has been used in earlier studies on free vibrations in different contexts. For example, Govardhan & Williamson (Reference Govardhan and Williamson2000) introduced a critical mass ratio, $m_{crit}^{\ast }$ , below which synchronization is observed for values of $U^{\ast }$ up to infinity. Prasanth et al. (Reference Prasanth, Premchandran and Mittal2011) introduced a critical mass ratio, $m_{cr}^{\ast }$ , below which hysteresis between the initial and lower branch, at low $Re$ and in an unbounded domain, disappears. The term critical mass ratio, in the present work, is not related to these two phenomena. In the following sections most of the results and figures are presented for the case of coupled modes. The general argument can be extended to the case of coupled modes as well.

6.1 Energy ratio

We consider the disturbance field given by (2.9). In general, the eigenvalues/eigenmodes are complex. The total energy of the fluid–oscillator system associated with the disturbance field is the sum of the structural energy, $E_{s}$ , and the kinetic energy of the fluid, $E_{f}$ . The two components are defined as:

(6.1) $$\begin{eqnarray}\displaystyle & \displaystyle E_{s}(t)=\textstyle \frac{1}{2}m^{\ast }{\dot{Y}}\overline{{\dot{Y}}}+\textstyle \frac{1}{2}(2{\rm\pi}F_{n})^{2}m^{\ast }Y\overline{Y}, & \displaystyle\end{eqnarray}$$

(6.2) $$\begin{eqnarray}\displaystyle & \displaystyle E_{f}(t)=\frac{1}{2}\int _{{\it\Omega}}\boldsymbol{u}\boldsymbol{\cdot }\overline{\boldsymbol{u}}\,\text{d}{\it\Omega}. & \displaystyle\end{eqnarray}$$

In (6.1) the first term on the right-hand side is the contribution of the kinetic energy of the cylinder while the second term arises from the potential energy of the spring. Equations (6.1) and (6.2) can be rewritten as

(6.3) $$\begin{eqnarray}\displaystyle & \displaystyle E_{s}(t)=\hat{E}_{s}\text{e}^{2{\it\lambda}_{r}t}, & \displaystyle\end{eqnarray}$$

(6.4) $$\begin{eqnarray}\displaystyle & \displaystyle E_{f}(t)=\hat{E}_{f}\text{e}^{2{\it\lambda}_{r}t}, & \displaystyle\end{eqnarray}$$

where $\hat{E}_{s}$ and $\hat{E}_{f}$ are given as,

(6.5) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{E}_{s}=\textstyle \frac{1}{2}m^{\ast }\hat{{\dot{Y}}}\overline{\hat{{\dot{Y}}}}+\textstyle \frac{1}{2}(2{\rm\pi}F_{n})^{2}m^{\ast }{\hat{Y}}\overline{{\hat{Y}}}, & \displaystyle\end{eqnarray}$$

(6.6) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{E}_{f}=\frac{1}{2}\int _{{\it\Omega}}\hat{\boldsymbol{u}}\boldsymbol{\cdot }\overline{\hat{\boldsymbol{u}}}\,\text{d}{\it\Omega}. & \displaystyle\end{eqnarray}$$

We define the energy ratio as $E_{r}=E_{s}(t)/(E_{s}(t)+E_{f}(t))$ . It denotes the fraction of the total energy of the disturbance field contained in the structure; $(1-E_{r})$ is the fraction of the total energy of the disturbance due to the motion of the fluid. We note that $E_{r}$ is constant with respect to time and can also be computed via $E_{r}=\hat{E}_{s}/(\hat{E}_{s}+\hat{E}_{f})$ . The evolution of $E_{s}(t)$ , therefore, depends on $E_{r}$ and ${\it\lambda}_{r}$ . If ${\it\lambda}_{r}$ is positive, $E_{s}$ , grows with time. However, the nonlinear terms become significant when the disturbance field becomes relatively large. As a consequence, the fluid–structure system tends towards a limit cycle. The saturated oscillation amplitude of the spring mass oscillator significantly depends on $E_{r}$ of the unstable mode. We will show later in the paper that systems with relatively large $E_{r}$ , coupled with positive growth rate, attain large amplitude oscillations resulting in synchronization. The threshold value of $E_{r}$ beyond which the unstable eigenmodes lead to lock-in state depends on $Re$ and $m^{\ast }$ . For example for $(Re,m^{\ast })=(60,20)$ , all unstable eigenmodes that are associated with $E_{r}>2\times 10^{-4}$ lead to lock-in. Figure 13(a) shows the variation of $E_{r}$ with $U^{\ast }$ for the elastic and fluid mode for $(Re,m^{\ast })=(60,20)$ . The same for the case of coupled modes at $(Re,m^{\ast })=(60,5)$ is shown in figure 13(b). To clearly bring out the features in the variation of $E_{r}$ with $U^{\ast }$ and the relative behaviour of the two eigenmodes, the vertical axis is shown in log scale. Also shown in the figures, via shading, is the lock-in regime obtained from DTI. The stable and the unstable eigenmodes are denoted by hollow and solid symbols, respectively. $E_{r}$ of the unstable eigenmodes is relatively low outside the lock-in regime. Within the lock-in regime, for each $U^{\ast }$ , the energy ratio of at least one unstable eigenmode has a large value. For $(Re,m^{\ast })=(60,20)$ , EM has high $E_{r}$ for $U^{\ast }$ up to 7.3 approximately. Beyond $U^{\ast }\sim 7.3$ , EM is stable and is not expected to play any role in lock-in. In this flow regime, though, $E_{r}$ for the fluid mode is large and is expected to drive the system towards lock-in. For $(Re,m^{\ast })=(60,5)$ , FEMII has a significantly large values of $E_{r}$ over the entire lock-in regime. Lock-in in this case, therefore, is expected via FEMII.

Figure 13.

Linear stability analysis of the steady flow past a fluid–cylinder system: variation of energy ratio of the eigenmodes with $U^{\ast }$ for (a)  $(Re,m^{\ast })=(60,20)$ and (b)  $(Re,m^{\ast })=(60,5)$ . The stable and the unstable eigenmodes are shown as hollow and solid symbols, respectively. The lock-in regime from DTI, in the presence of nonlinear effects, is shaded in the figures.

6.2 Energy transfer per oscillation cycle

The key to self-excited nature of free vibrations is the net energy transfer over a cycle from the fluid to the oscillator (Williamson & Roshko Reference Williamson and Roshko1988; Morse & Williamson Reference Morse and Williamson2009). The time evolution of cylinder displacement and lift coefficient corresponding to an eigenmode ( $\hat{\boldsymbol{u}},\hat{p},\hat{{\dot{Y}}},{\hat{Y}}$ ) is given as:

(6.7) $$\begin{eqnarray}\displaystyle & \displaystyle Y(t)=\frac{({\hat{Y}}\text{e}^{{\it\lambda}t}+\text{c.c.})}{2}, & \displaystyle\end{eqnarray}$$

(6.8) $$\begin{eqnarray}\displaystyle & \displaystyle C_{L}(t)=\frac{({\hat{C}}_{L}\text{e}^{{\it\lambda}t}+\text{c.c.})}{2}. & \displaystyle\end{eqnarray}$$

Here, ${\hat{C}}_{L}$ is the lift coefficient calculated by integrating the stress corresponding to the eigenmode at the boundary of the cylinder. In general, ${\hat{Y}}$ and ${\hat{C}}_{L}$ are complex and c.c. represents the complex conjugate. Let $|{\hat{Y}}|$ and $|{\hat{C}}_{L}|$ represent the magnitude of ${\hat{Y}}$ and ${\hat{C}}_{L}$ , respectively. Further, let ${\it\theta}$ and ${\it\theta}+{\it\phi}$ denote the argument of ${\hat{Y}}$ and ${\hat{C}}_{L}$ , respectively, where ${\it\phi}$ is the phase angle between the cylinder displacement and lift coefficient. We recall that ${\it\lambda}={\it\lambda}_{r}+\text{i}{\it\lambda}_{i}$ . Equations (6.7) and (6.8) can be rearranged and written as:

(6.9a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle Y(t)=Y_{o}(t)\text{e}^{{\it\lambda}_{r}t};\quad Y_{o}(t)=|{\hat{Y}}|\cos ({\it\lambda}_{i}t+{\it\theta}), & \displaystyle\end{eqnarray}$$

(6.10a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle C_{L}(t)=C_{L_{o}}(t)\text{e}^{{\it\lambda}_{r}t};\quad C_{L_{o}}(t)=|{\hat{C}}_{L}|\cos ({\it\lambda}_{i}t+{\it\theta}+{\it\phi}). & \displaystyle\end{eqnarray}$$

The non-dimensional energy transfer from the fluid to the oscillator over a time interval, $T_{1}$ – $T_{2}$ is defined as:

(6.11) $$\begin{eqnarray}\displaystyle E_{\mathit{transfer}}(T_{1},T_{2})=\int _{T_{1}}^{T_{2}}C_{L}(t){\dot{Y}}(t)\,\text{d}t. & & \displaystyle\end{eqnarray}$$

We now consider the non-dimensional energy transfer over a time period of cylinder oscillation from that part of $C_{L}$ and $Y$ that excludes the exponential growth/decay. The time period of cylinder oscillation is related to ${\it\lambda}_{i}$ as $T_{c}=2{\rm\pi}/{\it\lambda}_{i}$ . We define $E_{c}$ , energy transfer per oscillation cycle, as:

(6.12) $$\begin{eqnarray}\displaystyle E_{c}=\int _{0}^{T_{c}}C_{L_{o}}(t)\dot{Y_{o}}(t)\,\text{d}t. & & \displaystyle\end{eqnarray}$$

$E_{c}$ and $E_{\mathit{transfer}}$ are related as

(6.13) $$\begin{eqnarray}\displaystyle & \displaystyle E_{\mathit{transfer}}(t,t+T_{c})=E_{c}\int _{t}^{t+T_{c}}\text{e}^{2{\it\lambda}_{r}t}\,\text{d}t. & \displaystyle\end{eqnarray}$$

Plugging in the expressions of $C_{L_{o}}$ and $Y_{o}$ from (6.9) and (6.10) in (6.12), it can be shown that

(6.14) $$\begin{eqnarray}\displaystyle & \displaystyle E_{c}={\rm\pi}|{\hat{C}}_{L}||{\hat{Y}}|\sin {\it\phi}. & \displaystyle\end{eqnarray}$$

It is noted that $E_{c}$ is directly proportional to $\sin {\it\phi}$ . For $0^{\circ }<{\it\phi}<180^{\circ }$ , $E_{c}>0$ and the energy transfer per cycle of oscillation is from the flow to the oscillator. In this case, the amplitude of cylinder oscillation grows with time. Instead, for $-180^{\circ }<{\it\phi}<0^{\circ }$ , $E_{c}$ is negative and the energy is transferred to the fluid by the oscillator. As a result the oscillation of cylinder decays with time leading to a steady state. Figure 14(a) shows the variation of $E_{c}$ with $U^{\ast }$ for $(Re,m^{\ast })=(60,20)$ . The lock-in regime based on DTI is also shaded in the figure. $E_{c}$ can be written in terms of the growth rate and frequency of the eigenmode (details in appendix C) as:

(6.15) $$\begin{eqnarray}\displaystyle & \displaystyle E_{c}=2{\rm\pi}^{2}|{\hat{Y}}|^{2}m^{\ast }{\it\lambda}_{r}{\it\lambda}_{i}. & \displaystyle\end{eqnarray}$$

Equation (6.15) shows that the sign of $E_{c}$ is determined by the sign of ${\it\lambda}_{r}$ . Therefore, $E_{c}$ is positive for unstable eigenmode and negative for stable eigenmodes. If an eigenmode is unstable ( ${\it\lambda}_{r}>0$ ), a net energy transfer takes place from the base flow to the fluid–oscillator system. A part of this energy, determined by the energy ratio $E_{r}$ , is transferred to the structure. This energy transfer is via the work done by the fluid force in displacing the oscillator. The energy transferred to the oscillator adds up to its kinetic energy as well as the potential energy of the spring. We note that the variation of $E_{c}$ with $U^{\ast }$ can be quite different than the variation of ${\it\lambda}_{r}$ with $U^{\ast }$ , even though they are of the same sign. Therefore, the $U^{\ast }$ for largest growth rate may not necessarily coincide with that for maximum energy transfer. It is noted that $E_{c}$ of the EM is generally higher than that of the fluid mode in the regime of instability of the elastic mode ( $5.8<U^{\ast }<7.3$ ). However, we recall here that the growth rate of FM is higher than the elastic mode for all $U^{\ast }$ (figure 11 c). This suggests that for $5.8<U^{\ast }<7.3$ initially FM would dominate the temporal evolution of cylinder displacement. However, as time progresses, EM is expected to drive the structural response towards lock-in. This is confirmed by direct time integration in § 8.3. The variation of $E_{c}$ with $U^{\ast }$ for the case of coupled modes is shown in figure 14(b). FEMII is expected to dominate the dynamics of the fluid–structure system in this case.

Figure 14.

Linear stability analysis of the steady flow past a fluid–cylinder system: variation of energy transfer per cycle of cylinder oscillation with $U^{\ast }$ for (a)  $(Re,m^{\ast })=(60,20)$ and (b)  $(Re,m^{\ast })=(60,5)$ . The stable and the unstable eigenmodes are shown as hollow and solid symbols, respectively. The lock-in regime from DTI, in the presence of nonlinear effects, is shaded in the figures.

8.1 When do nonlinear terms become significant?

In the preceding sections we have shown that there are differences between LSA and DTI with respect to the frequency, phase and the extent of lock-in regime. These differences are attributed to the nonlinear terms in DTI of the governing equations. We recall that the oscillator used in the present study is associated with linear spring and damper. Therefore, the nonlinearity in the fluid–oscillator system results entirely from the convection terms in (2.1). We assume that the nonlinear terms involving the disturbance field become relatively significant when their magnitude exceeds 10 % of the size of the other terms, i.e.  $O(|\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{\prime }|)\sim 0.1O(|\boldsymbol{U}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{U}|)$ . Based on this assumption, the time at which the nonlinear effects are expected to become significant is estimated to be:

(8.1) $$\begin{eqnarray}\displaystyle & \displaystyle T_{nl}\sim \left.\left(\ln \frac{0.33|U_{\infty }|}{|\hat{\boldsymbol{u}}_{max}|}\right)\right/\!{\it\lambda}_{r}. & \displaystyle\end{eqnarray}$$

The details for the derivation of $T_{nl}$ are presented in appendix B. We now present an estimate of $T_{nl}$ for $(Re,m^{\ast })=(60,20)$ for two values of reduced velocity. Figure 16 shows the estimate of $T_{nl}$ for $U^{\ast }=5.5$ and $8.0$ . While $U^{\ast }=5.5$ lies before the onset of synchronization, the oscillator experiences lock-in for $U^{\ast }=8.0$ (see figure 11 a). From figure 16 we observe that up to $t=T_{nl}$ the fluid–structure system indeed retains its linear behaviour. This is most evident from the time variation of the structural energy, $E_{s}$ . The structural energy at any instant is the sum of the kinetic energy and potential energy of the oscillator. As seen from (6.3), the slope of $\ln (E_{s})$ versus $t$ is $2{\it\lambda}_{r}$ in the linear regime. The match between the two slopes, from DTI and the estimate via LSA, is excellent. Beyond $t=T_{nl}$ , the nonlinear terms cause saturation in the oscillation amplitude and $E_{s}$ , leading to a limit cycle. The saturation in the Strouhal number and the added mass coefficient is also observed for $t>T_{nl}$ .

We note that much of the increase in the amplitude of transverse vibration takes place in the linear regime ( $t<T_{nl}$ ). For example, for $U^{\ast }=5.5$ , the amplitude of cylinder response increases eightfold during $0\leqslant t\leqslant T_{nl}$ and only by a factor of 1.5 beyond $T_{nl}$ up to the attainment of limit cycle. These numbers for $U^{\ast }=8.0$ are 17 and 3, approximately.

Figure 17.

Flow past a freely vibrating cylinder: distribution of energy ratio with the growth rate of the eigenmodes for (a)  $(Re,m^{\ast })=(60,20)$ , and (b)  $(Re,m^{\ast })=(60,5)$ . All unstable eigenmodes that lead to lock-in with respect to the fully developed state are shown in solid symbols. These eigenmodes are characterized by energy ratio that is greater than a threshold value, $E_{rt}$ .

8.2 What causes lock-in?

The observations from the preceding section motivate us to propose a hypothesis to as to why lock-in occurs for a certain range of $U^{\ast }$ , and not for others. We believe that $E_{r}$ (fraction of energy in the structural part of the eigenmode) determines if a certain disturbance field will lead to attainment of lock-in. To demonstrate the relevance of the energy ratio, $E_{r}$ , we consider the case of $(Re,m^{\ast })=(60,20)$ . The two modes that, in general, govern the aeroelastic instability of the fluid–structure system for various $U^{\ast }$ are EM and FM. Figure 17(a) shows the points for the two eigenmodes on the $E_{r}{-}{\it\lambda}_{r}$ plane, for various $U^{\ast }$ . DTI is carried out for each case. The initial condition is the steady flow superimposed with the eigenmode:

(8.2) $$\begin{eqnarray}\displaystyle & \displaystyle (\boldsymbol{u}(\boldsymbol{x}),p(\boldsymbol{x}),{\dot{Y}},Y)^{\text{T}}=(\boldsymbol{U}(\boldsymbol{x}),P(\boldsymbol{x}),0,0)^{\text{T}}+s\ast (\hat{\boldsymbol{u}}(\boldsymbol{x}),\hat{p}(\boldsymbol{x}),\hat{{\dot{Y}}},{\hat{Y}})^{\text{T}}. & \displaystyle\end{eqnarray}$$

$s$ is a scalar and is used to control the magnitude of the initial disturbance. Unless stated otherwise, a value of 1.0 is assigned to $s$ for all the cases presented in this work. In figure 17(a) the eigenmodes that lead to lock-in are marked in solid symbols and the ones that do not cause lock-in are marked in hollow symbol. It is noted that all unstable eigenmodes, irrespective of their growth rate, that have an $E_{r}$ beyond a threshold value of $2\times 10^{-4}$ lead to lock-in. Interestingly, the case that leads to maximum oscillation amplitude is associated with smallest positive value of ${\it\lambda}_{r}$ , but the largest value of $E_{r}$ . Similarly, modes that are associated with relatively large ${\it\lambda}_{r}$ , but with $E_{r}$ below the threshold value, do not lead to lock-in. The region in the $E_{r}{-}{\it\lambda}_{r}$ plane that leads to lock-in is marked in figure 17(a). Figure 17(b) shows the same for $(Re,m^{\ast })=(60,5)$ . Similar observations hold for this figure as well. The value of $E_{rt}$ in this case is $1\times 10^{-3}$ . We note that the threshold value of $E_{r}$ , required for lock-in depends on $m^{\ast }$ , and possibly on  $Re$ .

8.3 Eigenmodes responsible for lock-in

For subcritical $Re$ ( $Re\leqslant Re_{o}$ ), the instability of the fluid–structure system, if any, is caused by only one unstable eigenmode. In the scenario of coupled modes, FEMII is responsible for lock-in. For large mass ratios, where the modes are decoupled, lock-in is attributed to the growth of EM. For supercritical $Re$ ( $Re>Re_{o}$ ), the situation becomes complicated as LSA predicts lock-in for all $U^{\ast }$ . Further, in certain flow regimes multiple eigenmodes are unstable. We now study in more detail the mechanism of how an unstable eigenmode leads to lock-in for $Re>Re_{o}$ . We carry out DTI for $(Re,m^{\ast })=(60,20)$ for various values of $U^{\ast }$ in the lock-in regime. As shown in figure 11(c), for $(Re,m^{\ast })=(60,20)$ , LSA predicts the instability of the FM for the fluid–structure system for all values of $U^{\ast }$ . The EM is unstable for $5.8<U^{\ast }<7.3$ . The growth rate of FM is higher than that of EM in the entire $U^{\ast }$ regime. However, the energy ratio, $E_{r}$ , that has been shown to be very relevant for determining the amplitude of cylinder oscillation in the limit cycle, is larger for EM than for FM, and above the threshold value for $5.8<U^{\ast }<7.2$ . This is shown in figure 13(a). Figure 17(a) shows that lock-in occurs only when $E_{r}$ for the concerned mode exceeds the threshold value. The range for lock-in for $(Re,m^{\ast })=(60,20)$ is $5.8<U^{\ast }<9.3$ .

8.3.2 Lock-in via elastic mode

For $(Re,m^{\ast })=(60,20)$ , both the elastic and the fluid mode are unstable for $5.8<U^{\ast }<7.3$ . In this regime $E_{r}$ of the elastic mode is above the threshold energy for lock-in (figure 17 a). The fluid mode, though unstable, has energy ratio lower than $E_{rt}$ . Hence, lock-in over this regime results from the instability of EM. To elucidate this, we carry out computations for $U^{\ast }=6.0$ with two initial conditions. In the first case, the simulation is initiated with the elastic mode superimposed on the steady solution. The value of $s$ (8.2) used for this case is 10.0. Figure 18 shows the time evolution of $Y/D$ and $E_{s}$ . It is observed that the amplitude of cylinder oscillation increases with time up to $t=T_{nl}$ with a rate that matches the growth rate of EM. Concomitantly, the structural energy grows with twice the growth rate of EM over this time interval. Beyond $t=T_{nl}$ , the response of the fluid–structure system approaches a limit cycle where the cylinder exhibits relatively large amplitude of oscillation with frequency that is close to the natural frequency of the oscillator in vacuum. The fluid–structure system is in a state of lock-in in the limit cycle. Simulations with lower value of $s$ (for example, $s=1$ ) lead to an interesting evolution. The numerical inaccuracies in the eigenmodes act as noise and excite other modes, including FM. Since FM has a higher growth rate than EM, for this case, the FM grows very rapidly compared to EM. Therefore, in the linear regime for $s=1.0$ , the growth of amplitude of the cylinder exhibits the growth rate of EM at low $t$ followed by a growth rate corresponding to that of FM up to the end of the linear regime. On the other hand, with $s=10.0$ , even though the FM grows faster than the EM, the growth is dominated by EM owing to its significantly larger initial strength.

Figure 18.

Flow past a freely vibrating cylinder for $(Re,m^{\ast },U^{\ast })=(60,20,6.0)$ initiated with EM: time evolution of (a) cylinder displacement, Strouhal number based on frequency of cylinder oscillation and (b) structural energy. The time instant beyond which the nonlinear terms become significant is marked in dash-dotted lines. The line with slope two times the growth rate of the elastic mode is shown in green colour.

Figure 19.

Flow past a freely vibrating cylinder for $(Re,m^{\ast },U^{\ast })=(60,20,6.0)$ initiated with FM: time evolution of (a) cylinder displacement, Strouhal number based on frequency of cylinder oscillation and (b) structural energy. The time instant beyond which the nonlinear terms become significant is marked in dash-dotted lines. The different flow regions are demarcated by broken lines. The line with slope two times the growth rate of the elastic mode is shown in green colour. Also shown in inset is the evolution of structural energy in the linear regime. The line with slope two times the growth rate of the fluid mode is shown in blue colour.

In the second case, the computations are initiated with the fluid mode superimposed on the steady solution. Figure 19 shows the time history of $Y/D$ and $E_{s}$ for this case. It is observed that the initial growth of the amplitude of cylinder oscillation is governed by the fluid mode. This is shown as region I in the figure (also shown in the inset). In this region, the contribution of nonlinear terms based on the analysis, described earlier, is negligible. In region II, the oscillations achieve a state where the amplitude and the structural energy are nearly constant. It is further observed that $Y/D$ and $E_{s}$ in region II are comparable to that observed for $U^{\ast }=5.5$ in the fully developed state; the latter corresponds to a lock-in state. Beyond region II, the response of the cylinder exhibit beats. The frequency spectrum of $Y/D$ show two prominent peaks: one at 0.136 and the other at 0.157. The former is equal to the frequency of oscillation in region II, while the latter matches the frequency of the elastic mode for $U^{\ast }=6.0$ . It is noted that the structural energy, in region III, grows almost linearly with twice the growth rate of the elastic mode. The linear growth becomes more clear in region IV. Further the frequency spectrum in region IV is dominated by the elastic mode. These observations suggest that the dynamics of the fluid–structure system in regions III and IV is governed by the elastic mode. The transition of the system from region II to region IV may be attributed to transfer of the energy from FM to EM due to nonlinear effects which become significant beyond $T_{nl}$ . We recall that the energy transfer per cycle of cylinder oscillation ( $E_{c}$ ) for EM is quite large (figure 14). As a result, the structural energy of the system increases significantly in region IV. At the end of region IV the oscillation amplitude and structural energy are similar to the values observed at the end of the linear regime for computations started with the elastic mode. Finally, in region $V$ , the oscillations achieve a limit cycle with amplitude $Y/D\sim 0.5$ . In summary, the computations initiated with the FM drive the system to an intermediate state corresponding to no lock-in. The system then transitions to a state of lock-in via the elastic mode.