^{1,a)}and Kunihiko Taira

^{1,b)}

### Abstract

We numerically investigate the influence of sinusoidal flow control on the von Kármán vortex shedding behind a circular cylinder in two-dimensional flow. Actuator location, direction, frequency, and amplitude are varied to examine their effects on the wake and the corresponding change in drag on the cylinder. We place focus on the conditions for which the cylinder wake locks onto the actuation frequency. The lock-on region is found to be consistent with stability horns observed in oscillator dynamics. Under certain conditions, the actuation reduces drag by elongating the wake structure to appear more streamlined than the wake without flow control. In other cases, the use of actuation led to less streamlined wakes, resulting in no significant drag reduction or for some instances in a drag increase. Purely steady and oscillatory actuation components are examined to highlight their individual influence on the lock-on and drag characteristics. We also note that low frequency oscillations are observed for cases in the neighborhood of the lock-on boundaries due to the competition between low and high-drag states.

We would like to thank the reviewers for the helpful comments. Part of this research was supported by the First Year Assistant Professor Award from the Florida State University.

I. INTRODUCTION

II. SIMULATION AND CONTROL SETUP

III. BASELINE FLOW

IV. ACTUATED FLOW

A. Lock-on

B. Effect of actuation direction

C. Effect of actuation position

D. Effect of actuation frequency and amplitude

E. Effect of steady actuation

F. Behavior near lock-on boundaries

V. SUMMARY

### Key Topics

- Drag reduction
- 22.0
- Rotating flows
- 14.0
- Flow control
- 12.0
- Flow instabilities
- 9.0
- Reynolds stress modeling
- 7.0

## Figures

Computational setup for the present study. (a) Control setup for separated flow around a circular cylinder. (b) Computational domain setup with 5 different multi domains. The inner most domain (domain 1) is of size (x, y) ∈ [−1, 3] × [−2, 2], and the largest one (domain 5) is [−31, 33] × [−32, 32]. Each domain has 200 × 200 cells.

Computational setup for the present study. (a) Control setup for separated flow around a circular cylinder. (b) Computational domain setup with 5 different multi domains. The inner most domain (domain 1) is of size (x, y) ∈ [−1, 3] × [−2, 2], and the largest one (domain 5) is [−31, 33] × [−32, 32]. Each domain has 200 × 200 cells.

Steady and oscillatory momentum coefficients ( and ) for tangential non-zero-mean sinusoidal actuation (Eq. (4) ) with varied forcing amplitude a.

Steady and oscillatory momentum coefficients ( and ) for tangential non-zero-mean sinusoidal actuation (Eq. (4) ) with varied forcing amplitude a.

Representative cases with (left, f a /f n = 1.05) and without (right, f a /f n = 0.3) lock-on, for a = 0.05. Shown on top are phase diagrams with (C D , ) and on the bottom are the frequency contents of the drag coefficient time series.

Representative cases with (left, f a /f n = 1.05) and without (right, f a /f n = 0.3) lock-on, for a = 0.05. Shown on top are phase diagrams with (C D , ) and on the bottom are the frequency contents of the drag coefficient time series.

Time-average coefficient of drag ( ) shown for different forcing amplitudes over a range of actuation frequency (f a /f n ). The dashed line (−− −) corresponds to the average baseline drag. Actuation is applied normal (left) and tangential (right) to the surface at the separation point (θ = 58°).

Time-average coefficient of drag ( ) shown for different forcing amplitudes over a range of actuation frequency (f a /f n ). The dashed line (−− −) corresponds to the average baseline drag. Actuation is applied normal (left) and tangential (right) to the surface at the separation point (θ = 58°).

Contour plots for the change in time-average drag ( ) resulting from normal and tangential forcing for varied actuation amplitude (a) and frequency (f a /f n ). Actuators located at θ = 58°. Solid lines represent the boundaries between cases with (○) and without (•) lock-on.

Contour plots for the change in time-average drag ( ) resulting from normal and tangential forcing for varied actuation amplitude (a) and frequency (f a /f n ). Actuators located at θ = 58°. Solid lines represent the boundaries between cases with (○) and without (•) lock-on.

Contour plots of the flow without actuation and with normal sinusoidal actuation with Eq. (4) and a = 0.50.

Contour plots of the flow without actuation and with normal sinusoidal actuation with Eq. (4) and a = 0.50.

Comparison of the time-average drag coefficient for different actuator positions on the cylinder. All cases are for tangential forcing and a = 0.06, using Eq. (4) . Results for zero-average forcing with Eq. (8) is also shown.

Different lock-on profiles observed for a = 0.05 with non-zero-mean forcing, Eq. (4) . Cases (i)-(v) are described in text.

Different lock-on profiles observed for a = 0.05 with non-zero-mean forcing, Eq. (4) . Cases (i)-(v) are described in text.

Contour plots of flows with non-zero-mean tangential sinusoidal actuation and a = 0.05 (Eq. (4) ). All cases are related to Fig. 8 .

The left figure shows the reduction in for increasing actuator strength a with steady blowing, Eq. (7) . The dashed line (−− −) corresponds to the average drag from the case without actuation. The right plot shows the corresponding maximum amplitude in the frequency spectrum for each amplitude.

The left figure shows the reduction in for increasing actuator strength a with steady blowing, Eq. (7) . The dashed line (−− −) corresponds to the average drag from the case without actuation. The right plot shows the corresponding maximum amplitude in the frequency spectrum for each amplitude.

Contour plots of flows with steady tangential actuation for a = 0.05 and 0.09 with Eq. (7) .

Contour plots of flows with steady tangential actuation for a = 0.05 and 0.09 with Eq. (7) .

The change in and lock-on characteristics for zero-mean tangential actuation. The left figure shows for different forcing amplitudes over a range of actuation frequency (f a /f n ). The dashed line (−−−) corresponds to the average baseline drag. The right plot represents the corresponding lock-on characteristics with the change in shown by the counters for varied actuation amplitude (a) and frequency (f a /f n ). Solid lines represent the boundaries between cases with (○) and without (•) lock-on.

The change in and lock-on characteristics for zero-mean tangential actuation. The left figure shows for different forcing amplitudes over a range of actuation frequency (f a /f n ). The dashed line (−−−) corresponds to the average baseline drag. The right plot represents the corresponding lock-on characteristics with the change in shown by the counters for varied actuation amplitude (a) and frequency (f a /f n ). Solid lines represent the boundaries between cases with (○) and without (•) lock-on.

Time history of the drag coefficient for Cases (I)-(III) (see text) using non-zero-mean sinusoidal forcing with Eq. (4) and a = 0.03. Shown by the dashed line (−−−) is the baseline average drag . Inserted contour plots represent the vorticity fields for Case (III).

Time history of the drag coefficient for Cases (I)-(III) (see text) using non-zero-mean sinusoidal forcing with Eq. (4) and a = 0.03. Shown by the dashed line (−−−) is the baseline average drag . Inserted contour plots represent the vorticity fields for Case (III).

## Tables

Lift and drag coefficients and Strouhal number for flow over a circular cylinder at Re = 100.

Lift and drag coefficients and Strouhal number for flow over a circular cylinder at Re = 100.

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