Phase-locking of laminar wake to periodic vibrations of a circular cylinder

Phase synchronization between the vortex shedding behind a two-dimensional circular cylinder and its vibrations is investigated using the phase-reduction analysis. Leveraging this approach enables the development of a one-dimensional, linear model with respect to the limit-cycle attractor of the laminar wake, which accurately describes the phase dynamics of the high-dimensional, nonlinear fluid flow and its response to rotational, transverse, and longitudinal vibrations of the cylinder. This phase-based model is derived by assessing the phase response and sensitivity of the wake dynamics to impulse perturbations of the cylinder, which can be performed in simulations and experiments. The resulting model in turn yields the theoretical conditions required for phase-locking between the cylinder vibrations and the wake. We furthermore show that this synchronization mechanism can be employed to stabilize the wake and subsequently reduce drag. We also uncover the circumstances under which the concurrent occurrence of different vibrational motions can be used to promote or impede synchronization. These findings provide valuable insights for the study of vortex-induced body oscillations, the enhancement of aerodynamic performance of flyers, and the mitigation of structural vibrations by synchronizing or desynchronizing the oscillatory motions of a body to the periodic wake.

Figure 1

Schematic of a uniform flow past a 2D circular cylinder, when the cylinder oscillates in a (a) rotary, (b) streamwise, or (c) cross-flow fashion, with the respective tangential, streamwise, or cross-flow velocities $U_{\theta }$, $U_x$, and $U_y$. All these velocities are given by $U_{f}sin\left({\omega }_{f}t\right)$, with $U_f$ and ${\omega }_f$ indicating the velocity oscillation amplitude and frequency, respectively.

Figure 2

(a) Phase sensitivity $\mathit{Z}$ and (b) phase-coupling functions $\mathrm{\Gamma }$ for different types of cylinder oscillations. Note that presented $\mathrm{\Gamma }$ for the rotary and cross-flow oscillations are the first-harmonic phase-coupling functions, while that of the streamwise oscillation is calculated at the second harmonic.

Figure 3

Top: Theoretical and numerical boundaries of synchronization for (a) rotary, (b) cross-flow, and (c) streamwise oscillations in the frequency-amplitude plane, indicated by black solid lines and blue crosses, respectively. Middle: Time-averaged drag coefficients ${\overline{C}}_D$ for the synchronized cases of (d) rotary, (e) cross-flow, and (f) streamwise oscillations. Bottom: Same as the middle, but for maximum lifts $C_L^{\max }$. Subscripts $f$ and $s$ refer to the oscillatory and stationary cylinder cases, respectively. Horizontal dotted lines represent zero values, or when the cylinder is at rest. Red circles in (a)–(c) show the numerical results of [28] at $\mathrm{Re}=110$, the experimental observations of [3] at $\mathrm{Re}=100$, and the findings of [26] at $\mathrm{Re}=80$, respectively.

Figure 4

(a)–(c) The time-averaged vorticity fields of the synchronized cases with the lowest mean drag coefficient ${\overline{C}}_D$ ($\overline{\omega }$), substracted from that of the stationary cylinder ${\overline{\omega }}_{\mathrm{ref}}$ shown in (d) to illustrate the wake elongations for the three cases. (d) Time-averaged vorticity field of the base flow.

Figure 5

(a) Synchronizability as a function of the weight $\alpha$ and phase difference ${\varphi }_f$ of concurrent rotary and cross-flow oscillations. (b) Theoretical (solid lines) and numerical (blue crosses) Arnold tongues corresponding to the green point in (a) for which $\alpha =0.5$ and ${\varphi }_f=0$. (c) Same as (b), but for the blue point in (a) at which synchronizability is minimized.