Bifurcation analysis of the primary instability in the flow around a flexibly mounted circular cylinder

Abstract

The nonlinear character of the primary bifurcation is investigated for the flow around a flexibly mounted circular cylinder. We have considered the cases in which the cylinder can oscillate in the transverse direction only and in both transverse and in-line directions. Low and high values of mass ratio ($m^{\ast }=5$ and 50) were studied, and reduced velocity ($V_{r}$) values are chosen inside ($V_{r}=9$) and outside ($V_{r}=5$ and $V_{r}=13$) the lock-in range for low Reynolds numbers. For each combination of $m^{\ast }$ and $V_{r}$, a global linear stability analysis was applied to find the critical Reynolds number $Re_{c}$ of the fluid–structure system. For $V_{r}$ in the lock-in range, the values of $Re_{c}$ were noticeably less than the critical Reynolds number of the flow around a fixed circular cylinder ($Re_{c_{0}}\cong 47$). On the other hand, for $V_{r}$ outside the lock-in range, the values of $Re_{c}$ were close to $Re_{c_{0}}$. Next, nonlinear analyses were performed in the vicinity of $Re_{c}$ for each case. Subcritical character (with hysteresis) was observed for $V_{r}$ in the lock-in range, while for $V_{r}$ outside the lock-in region the bifurcations were found to be supercritical (without hysteresis). This shows that when the coupling between the structure and flow is strong, due to the proximity of the natural frequencies of the isolated systems, it significantly changes both the linear and nonlinear responses observed.

Vortex-induced vibrations of a flexible cylinder at subcritical Reynolds number

“…The maximum amplitude of CF vibration along the span for , and is plotted at and in figure 3( a ). Vibrations arise between and at and between and at ; these values of are close to those reported for rigid-cylinder VIV (Dolci & Carmo 2019). Considering the possible influence of the other structural parameters, no attempt was made to determine the value of for the onset of vibration for all .…”

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Figure 1.Vortex-induced vibrations of a rigid cylinder at : ( a ) maximum amplitude of cross-flow displacement as a function of the reduced velocity; ( b , c ) instantaneous isocontours of spanwise vorticity (), for ( b ) and ( c ) . In ( a ), the present simulation results are compared to the amplitudes reported by Mittal & Singh (2005) and Dolci & Carmo (2019).

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“…If the rigid cylinder is elastically mounted, previous works have shown that VIV may occur in the subcritical-Re range, i.e. for Re < Re c , possibly down to Re ≈ 20, depending on the system parameters (Cossu & Morino 2000;Mittal & Singh 2005;Kou et al 2017;Dolci & Carmo 2019). A typical case of rigid-cylinder VIV in the subcritical-Re range (Re = 33) is presented in figure 1.…”

“…plants and trees in wind, or cables and mooring lines exposed to ocean currents. Owing to their impact on engineering structures and their fundamental interest as a paradigm of fluid-structure interaction involving synchronization, or lock-in, between body motion and flow unsteadiness, VIV In (a), the present simulation results are compared to the amplitudes reported by Mittal & Singh (2005) and Dolci & Carmo (2019).…”

Whistling of deep cavities subject to turbulent grazing flow: intermittently unstable aeroacoustic feedback

“…In figure 7(d), the excellent match between the peak frequency of the overlayed power spectral density and the frequency of the least-stable eigenvalue indicates that the present coupled oscillators model performs very well. Moreover, in contrast with the phenomenon of frequency lock-in in flow-induced vibration problems, which are also modelled as coupled oscillators (De Langre 2006;Mohany et al 2014;Shoshani 2018;Dolci & Carmo 2019), the frequencies of the eigenvalues Im(λ) do not merge in the range of L and U for which the present aeroacoustic system is linearly unstable. Indeed, the black lines corresponding to the root loci are repelled from the intersection point of the natural frequencies of the two oscillators (dashed black lines).…”