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Vortex shedding patterns, their competition, and chaos in flow past inline oscillating rectangular cylinders
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View: Figures


Image of FIG. 1.
FIG. 1.

(Color online) (a) Schematic of the domain used in the simulations. Time signals are stored at the monitor points A, B, and C, located at (2 h, 0), (h, D) and (2.25D, 0.5D), respectively, (not to scale). (b) The grid used.

Image of FIG. 2.
FIG. 2.

(Color online) Comparison between the spectra obtained from Minewitsch et al. (Ref. 14) (dashed line) and the present code (solid line) for Re = 200, A/D = 0.25, and f e  = 0.32U /D. The present spectrum is obtained from the time signal of u y at monitor point A in Fig. 1(a). The power spectrum of Minewitsch et al. is that of their lift coefficient and has been arbitrarily scaled.

Image of FIG. 3.
FIG. 3.

(Color online) Qualitative comparison of shedding modes for the square cylinder. Vorticity fields are shown. (a) f e /f o  = 2 and A/D = 0.175. This mode is similar to the one obtained by Couder and Basdevant. (Ref. 17) (b) f e /f o  = 1.73 and A/D = 0.5. This is S-II mode of shedding, similar to that seen in the experiment of Xu et al. (Ref. 6).

Image of FIG. 4.
FIG. 4.

(Color online) Vorticity fields at a typical time for A/D = 0.1 at various excitation frequencies for a cylinder of aspect ratio 4. (a) f e /f o  = 2, (b) f e /f o  = 3, (c) f e /f o  = 4, and (d) f e /f o  = 5.

Image of FIG. 5.
FIG. 5.

Flow patterns in the wake of an incline oscillating rectangular cylinder at Re = 200 and A/D = 0.175. Circles: antisymmetric shedding, squares: symmetric shedding. The solid squares indicate the S-II mode, the open squares stand for the S-I mode, while the patterned square indicates an S-III shedding. Triangles: mixed mode, where the shedding is symmetric but the vortices arrange themselves into an antisymmetric pattern downstream. Stars: chaotic flow, single solid diamond: the Couder-Basdevant mode.

Image of FIG. 6.
FIG. 6.

(Color online) Mixed mode in the case of square cylinder, f e /f o  = 4, A/D = 0.175.

Image of FIG. 7.
FIG. 7.

(Color online) Two of the modes of shedding at A/D = 0.175 on a body of aspect ratio 4. (a) The S-III mode at f e /f o  = 2.15. Three pairs of binary vortices are shed. (b) The S-II mode at f e /f o  = 4. In this mode, two binary vortices are shed during each time period.

Image of FIG. 8.
FIG. 8.

(Color online) Power spectra at the monitor point A for D/h = 4 and A/D = 0.175. (a) Subharmonic shedding at f e /f o  = 2. (b) The shedding is harmonic (symmetric) for f e /f o  = 2.15.

Image of FIG. 9.
FIG. 9.

(Color online) Phase information for the S-II mode for D/h = 8. Solid line: the vorticity ω at monitor point B. Dashed line: inlet velocity U total . Here, f e /f o  = 2 and A/D = 0.175. The circles indicate the phases at which the vorticity field is shown in Figures 10(a)–10(d).

Image of FIG. 10.
FIG. 10.

(Color online) The S-II mode of vortex shedding; D/h = 8, f e /f o  = 2, and A/D = 0.175. Time has been non-dimensionalized using the convective time-scale, D/U . (a) Attached primary vortices are growing. (b) Vorticity is generated on the lee side. (c) Close to the cylinder, the flow is from right to left. The primary vortices are pushed apart and the secondary vortices are “stretched.” (d) The secondary vortices cut off the supply to primary.

Image of FIG. 11.
FIG. 11.

(Color online) The S-III mode of vortex shedding; D/h = 4, f e /f o  = 2.15, and A/D = 0.175. In both figures, the cylinder is moving towards the left (upstream). (a) The stretching of secondary vortices has begun, and they are about to be dislodged from the cylinder by the next pair of primary vortices. (b) The central part of the secondary vortices has thinned, creating a pinch off of an extra pair vortices which then move along the centreline.

Image of FIG. 12.
FIG. 12.

(Color online) Delay plots for D/h = 4, f e /fo = 2, and 2.085. The oscillation amplitude A/D = 0.175. The monitor point is behind the cylinder at (0.5D, 0.35D). The delay plot (a) consists of closed curves, indicating periodicity, whereas (b) is characteristic of an aperiodic time signal. The delay time used is τ*≡τ D/U=2.

Image of FIG. 13.
FIG. 13.

(Color online) Chaotic window for D/h = 4. (a) f e /f o  = 2, the shedding here is antisymmetric. (b) f e /f o  = 2.085, the shedding is chaotic. (c) f e /f o  = 2.15, S-III mode of symmetric shedding.

Image of FIG. 14.
FIG. 14.

(Color online) D/h = 4. Variation of S with Δt for different cases. Squares: f e /f o  = 2, circles: f e /f o  = 2.085, and diamonds: f e /f o  = 2.15. The first and third curves exhibit periodicity and have sharp dips at Δt = nT for any integer n, whereas the second curve is aperiodic.

Image of FIG. 15.
FIG. 15.

(Color online) D/h = 8, f e /f o  = 2.5, and A/D = 0.175. (a) The spectrum is broadband, and (b) the arrangement of vortices is unordered, showing that the flow is chaotic.

Image of FIG. 16.
FIG. 16.

(Color online) D/h = 8. Variation of S with Δt in a region x/D = 2.5 − 8.75. Diamonds: f e /f o  = 2, and is repetitive indicating periodicity. Circles: f e /f o  = 2.5, and shows the aperiodic nature of the vorticity field in that region.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Vortex shedding patterns, their competition, and chaos in flow past inline oscillating rectangular cylinders