Unsteady dynamics in the streamwise-oscillating cylinder wake for forcing frequencies below lock-on

The flow around a cylinder oscillating in the streamwise direction with a frequency, $f_f$, much lower than the shedding frequency, $f_s$, has been relatively less studied than the case when these frequencies have the same order of magnitude or the transverse oscillation configuration. In this study, particle image velocimetry and the dynamic mode decomposition (DMD) are used to investigate the streamwise-oscillating cylinder wake for forcing frequencies $f_f/f_s\sim 0.04$–0.2 and mean Reynolds number, ${\mathrm{Re}}_0=900$. The amplitude of oscillation is such that the instantaneous Reynolds number remains above the critical value for vortex shedding at all times. Characterization of the wake reveals a range of phenomena associated with the interaction of the two frequencies, including modulation of both the amplitude and frequency of the wake structure by the forcing. DMD reveals a frequency spreading of dynamic modes. A scaling parameter and associated transformation are developed to relate the unsteady, or forced, dynamics of a system to that of an unforced system. For the streamwise-oscillating cylinder, it is shown that this transformation leads to a dynamic mode decomposition similar to that of the unforced system.

Figure 1

(a) The captive trajectory system mounted on top of the NOAH water channel's test section. Flow is from left to right. (b) Simplified diagram of experimental setup.

Figure 2

(a) Instantaneous Reynolds number, $\mathrm{Re}\left(t\right)$, corresponding to experimental forcing trajectory. (b) Phase portrait showing change in $\tau$ and $\dot{\tau }$ during forcing cycle. Linewidth is proportional to speed in phase space. (c) Quasisteadiness parameter plotted for streamwise forcing trajectory. (d) Variation in scaled time, $\tilde{t}$, with laboratory time, $t$.

Figure 3

Experimental sampling rate in laboratory time (top) and corresponding scaled time (bottom). Vertical lines correspond to snapshot acquisition in the experiment. For the sake of clarity, the number of samples shown has been significantly decreased relative to the actual experiment.

Figure 4

Snapshots of streamwise velocity (a), transverse velocity (b), and vorticity (c) for the stationary cylinder. The cross section of the cylinder at the measurement plane is represented by $\oplus$ and the shaded circle corresponds to an obstruction in the field of view due to the bottom of the cylinder. The location of the interrogation point is denoted by $\times$. Flow is from right to left.

Figure 5

Time trace of transverse velocity, $v/U_{\infty }$, at interrogation point. The dashed line is proportional to the instantaneous Reynolds number, $\mathrm{Re}\left(t\right)$. Top: Stationary cylinder; middle: ${\mathrm{Re}}_0=900,$ ${\mathrm{Re}}_q/{\mathrm{Re}}_0=0.35,$ ${\mathrm{St}}_f/{\mathrm{St}}_0=0.18$; bottom: ${\mathrm{Re}}_0=900,$ ${\mathrm{Re}}_q/{\mathrm{Re}}_0=0.35,$ ${\mathrm{St}}_f/{\mathrm{St}}_0=0.036$.

Figure 6

DMD spectrum (a) and modes (b) for stationary cylinder, ${\mathrm{Re}}_0=900$. With the exception of the mean mode ($\mathrm{St}=0$), the colored circles on the spectrum represent shedding modes. The colored mode labels correspond to the circles of the same color on the spectrum. The black dot-dashed line represents the stationary shedding frequency.

Figure 7

DMD eigenvalues for stationary cylinder, ${\mathrm{Re}}_0=900$. The colored symbols correspond to the modes indicated in Fig. 6.

Figure 8

Flow around a streamwise oscillating cylinder, ${\mathrm{Re}}_0=900,{\mathrm{Re}}_q/{\mathrm{Re}}_0=0.35,{\mathrm{St}}_f/{\mathrm{St}}_0=0.18$. The top row corresponds to the position of the cylinder in the forcing cycle. The second row corresponds to visualization of the flow using fluorescent dye. The third row corresponds to vorticity ${\omega }_z$. The fourth and fifth rows correspond to the transverse and streamwise components of velocity, $\frac{v}{U_{\infty }}$ and $\frac{u}{U_{\infty }}$, respectively. From left to right, the columns correspond to $t/{\tau }_f=0,0.25,0.50$, and 0.75, respectively. The $⨁$ symbol represents the cross section of the cylinder at the measurement plane while the shaded circle represents an obstruction in the field of view due to the bottom of the cylinder. Flow is from right to left.

Figure 9

DMD spectrum (a) and modes (b) for oscillating cylinder, ${\mathrm{Re}}_0=900,$ ${\mathrm{Re}}_q/{\mathrm{Re}}_0=0.35,$ ${\mathrm{St}}_f/{\mathrm{St}}_0=0.18$. The colored mode labels correspond to the circles of the same color on the spectrum. The black dot-dashed line represents the stationary shedding frequency while the purple dashed line denotes the forcing frequency. The additional colored symbols on the spectrum indicate shedding modes.

Figure 10

DMD spectrum (a) and modes (b) for oscillating cylinder, ${\mathrm{Re}}_0=900,$ ${\mathrm{Re}}_q/{\mathrm{Re}}_0=0.35,$ ${\mathrm{St}}_f/{\mathrm{St}}_0=0.036$.

Figure 11

Phase-averaged vorticity (top) and transverse velocity (bottom), ${\mathrm{Re}}_0=900,$ ${\mathrm{Re}}_q/{\mathrm{Re}}_0=0.35,$ ${\mathrm{St}}_f/{\mathrm{St}}_0=0.18$. (a) $t/{\tau }_f=0$; (b) $t/{\tau }_f=0.25$; (c) $t/{\tau }_f=0.50$; (d) $t/{\tau }_f=0.75$.

Figure 12

Streamlines for phase-averaged velocity field: Before formation of starting vortex, $\frac{t}{{\tau }_f}=0.4$ (a); during formation of starting vortex, $\frac{t}{{\tau }_f}=0.5$ (b). ${\mathrm{Re}}_0=900,$ ${\mathrm{Re}}_q/{\mathrm{Re}}_0=0.35,$ ${\mathrm{St}}_f/{\mathrm{St}}_0=0.18$.

Figure 13

Dye flow visualization of starting vortex generation. (a) $t/{\tau }_f=0.50$; (b) $t/{\tau }_f=0.55$; (c) $t/{\tau }_f=0.60$; (d) $t/{\tau }_f=0.65$.

Figure 14

DMD spectrum (a) and modes (b) for phase-averaged time series, ${\mathrm{Re}}_0=900,$ ${\mathrm{Re}}_q/{\mathrm{Re}}_0=0.35,$ ${\mathrm{St}}_f/{\mathrm{St}}_0=0.18$.

Figure 15

Time trace of transverse velocity, $v/U_{\infty }$, at interrogation point with quasisteady timescaling. Top: ${\mathrm{Re}}_0=900,$ ${\mathrm{Re}}_q/{\mathrm{Re}}_0=0.35,$ ${\mathrm{St}}_f/{\mathrm{St}}_0=0.18$; bottom: ${\mathrm{Re}}_0=900,$ ${\mathrm{Re}}_q/{\mathrm{Re}}_0=0.35,$ ${\mathrm{St}}_f/{\mathrm{St}}_0=0.036$.

Figure 16

DMD spectrum (a) and modes (b) for oscillating cylinder with quasisteady timescaling, ${\mathrm{Re}}_0=900,$ ${\mathrm{Re}}_q/{\mathrm{Re}}_0=0.35,$ ${\mathrm{St}}_f/{\mathrm{St}}_0=0.18$.

Figure 17

DMD spectrum (a) and modes (b) for oscillating cylinder with quasisteady timescaling, ${\mathrm{Re}}_0=900,$ ${\mathrm{Re}}_q/{\mathrm{Re}}_0=0.35,$ ${\mathrm{St}}_f/{\mathrm{St}}_0=0.036$.

Figure 18

Comparison of predicted and observed dominant frequencies, ${\mathrm{Re}}_0=900,$ ${\mathrm{St}}_f/{\mathrm{St}}_0=0.18$. The solid lines represent the predicted dominant frequencies while the $\times$ symbols represent the actual dominant frequencies observed in experiments.

Figure 19

Comparison of interrogation point power spectra before (left) and after (right) application of quasisteady timescaling.