### Abstract

In this study, numerical experiments are carried out to control the vortex shedding of a circular cylinder by utilizing an oscillating foil. The thin foil of elliptic shape undergoes prescribed harmonic oscillations in the streamwise direction in the near wake region. This simplified model is intended to study how wake dynamics are modified via localized wake disturbance, and then to stabilize the global wake instability. The results show that, at proper gap spacing, the oscillating foil can completely suppress the wake unsteadiness and recover the recirculating bubble type flow. The global instability suppression is then established on the imposition of local symmetry into the reversed flow behind the cylinder. It is revealed that the dynamic interaction between the main shears layer and oscillatory boundary layers is responsible for the wake stabilization mechanism. In addition, the kinematic/dynamic parameters related to foil motions and flow properties are widely discussed to reveal their effects on the performance of wake stabilization and drag reduction.

This work is supported by the National Science Foundation of China (NSFC) (Grant Nos. 10921202 and 11225209) and CPSFFP (Grant No. 2012M510003).

I. INTRODUCTION

II. PROBLEM DEFINITION

III. NUMERICAL METHOD

A. Governing equations and global flow quantities

B. Flow domain and boundary condition

C. Numerical method and grid system

IV. FLOW OVER A CYLINDER WITH A NON-OSCILLATING FOIL

V. FLOW OVER A CYLINDER WITH A STREAMWISE OSCILLATING FOIL

A. Flow pattern modification

B. Hydrodynamic force characteristics

C. Effects of oscillation frequency, amplitude, and vertical distance

D. Effect of Reynolds number

E. Effectiveness of drag reduction

F. Wake control mechanism behind streamwise oscillating foil

VI. SUMMARY

### Key Topics

- Rotating flows
- 40.0
- Flow instabilities
- 27.0
- Drag reduction
- 20.0
- Vortex dynamics
- 20.0
- Kinematics
- 13.0

## Figures

Geometry configuration of the flow over a stationary cylinder with a streamwise oscillating foil.

Geometry configuration of the flow over a stationary cylinder with a streamwise oscillating foil.

Computational domain and boundary conditions for the simulation.

Computational domain and boundary conditions for the simulation.

Time-dependent drag and lift coefficients of the upstream cylinder calculated under different grid densities at S/D = 3.

Time-dependent drag and lift coefficients of the upstream cylinder calculated under different grid densities at S/D = 3.

Close-up view of the unstructured mesh for the cylinder-foil system at S/D = 3.0.

Close-up view of the unstructured mesh for the cylinder-foil system at S/D = 3.0.

Flow pattern for the stationary cylinder with non-oscillating foil at different gap spacings (Re = 100): (a) S/D = 0.5; (b) S/D = 2.0; (c) S/D = 3.0; and (d) S/D = 4.0.

Flow pattern for the stationary cylinder with non-oscillating foil at different gap spacings (Re = 100): (a) S/D = 0.5; (b) S/D = 2.0; (c) S/D = 3.0; and (d) S/D = 4.0.

Flow regimes in the phase diagram of gap spacing ratio versus oscillation amplitude (Re = 100). (red circle): flow regime I; (green square): flow regime II; (blue diamond): flow regime III.

Flow regimes in the phase diagram of gap spacing ratio versus oscillation amplitude (Re = 100). (red circle): flow regime I; (green square): flow regime II; (blue diamond): flow regime III.

Instantaneous vorticity contours for the different flow regimes (Re = 100): (a) flow regime I (A/D = 0.3 and S/D = 1.0); (b) flow regime II (A/D = 0.8 and S/D = 3.0); and (c) flow regime III (A/D = 0.8 and S/D = 4.0).

Instantaneous vorticity contours for the different flow regimes (Re = 100): (a) flow regime I (A/D = 0.3 and S/D = 1.0); (b) flow regime II (A/D = 0.8 and S/D = 3.0); and (c) flow regime III (A/D = 0.8 and S/D = 4.0).

The comparison of the time-averaged flows for the cases of A/D = 0.3 (left) and A/D = 0.8 (right) at S/D = 3.0 (Re = 100): (a) and (b) streamwise velocity contour; (c) and (d) vorticity contour; and (e) and (f) velocity profiles in the wake region. The flow field is averaged over a full motion cycle.

The comparison of the time-averaged flows for the cases of A/D = 0.3 (left) and A/D = 0.8 (right) at S/D = 3.0 (Re = 100): (a) and (b) streamwise velocity contour; (c) and (d) vorticity contour; and (e) and (f) velocity profiles in the wake region. The flow field is averaged over a full motion cycle.

Time series of the hydrodynamic force coefficients at different gap spacing and oscillating amplitude (Re = 100): (a) drag coefficient of the cylinder; (b) drag coefficient of the foil; (c) lift coefficient of the cylinder; and (d) lift coefficient of the foil.

Time series of the hydrodynamic force coefficients at different gap spacing and oscillating amplitude (Re = 100): (a) drag coefficient of the cylinder; (b) drag coefficient of the foil; (c) lift coefficient of the cylinder; and (d) lift coefficient of the foil.

Statistical parameters of the hydrodynamic forces exerted on the cylinder and the foil (Re = 100): (a) mean drag coefficient, cylinder; (b) mean drag coefficient, foil; (c) r.m.s. value of the drag coefficient, cylinder; (d) r.m.s. value of the drag coefficient, foil; (e) r.m.s. value of the lift coefficient, cylinder; and (f) r.m.s. value of the lift coefficient, foil. In the plot, the corresponding results of the isolated cylinder and the cylinder-stationary system (A/D = 0.0) are also included.

Statistical parameters of the hydrodynamic forces exerted on the cylinder and the foil (Re = 100): (a) mean drag coefficient, cylinder; (b) mean drag coefficient, foil; (c) r.m.s. value of the drag coefficient, cylinder; (d) r.m.s. value of the drag coefficient, foil; (e) r.m.s. value of the lift coefficient, cylinder; and (f) r.m.s. value of the lift coefficient, foil. In the plot, the corresponding results of the isolated cylinder and the cylinder-stationary system (A/D = 0.0) are also included.

Instantaneous variation of the vorticity contours against frequency ratio at S/D = 1.0 and A/D = 0.3 (Re = 100): (a) f o /f n = 1.0; (b) f o /f n = 2.0; (c) f o /f n = 4.0; and (d) f o /f n = 5.0.

Instantaneous variation of the vorticity contours against frequency ratio at S/D = 1.0 and A/D = 0.3 (Re = 100): (a) f o /f n = 1.0; (b) f o /f n = 2.0; (c) f o /f n = 4.0; and (d) f o /f n = 5.0.

Time series of lift coefficient for the cylinder with different frequency ratios (S/D = 1.0, A/D = 0.3) at Re = 100.

Time series of lift coefficient for the cylinder with different frequency ratios (S/D = 1.0, A/D = 0.3) at Re = 100.

Mean and r.m.s. drag coefficient and r.m.s. lift coefficient as functions of the frequency ratio (Re = 100).

Mean and r.m.s. drag coefficient and r.m.s. lift coefficient as functions of the frequency ratio (Re = 100).

Instantaneous variation of the vorticity contours against oscillation amplitude at S/D = 3.0 and f o /f n = 1.0 (Re = 100): (a) A/D = 0.1; (b) A/D = 0.3; (c) A/D = 0.4; (d) A/D = 0.6; (e) A/D = 0.7; and (f) A/D = 0.8, respectively.

Instantaneous variation of the vorticity contours against oscillation amplitude at S/D = 3.0 and f o /f n = 1.0 (Re = 100): (a) A/D = 0.1; (b) A/D = 0.3; (c) A/D = 0.4; (d) A/D = 0.6; (e) A/D = 0.7; and (f) A/D = 0.8, respectively.

Instantaneous vorticity contours at different offset distances for the typical case A/D = 0.8 and S/D = 3.0 (Re = 100): (a) d/D = 0.0; (b) d/D = 0.1; (c) d/D = 0.2; (d) d/D = 0.3; (e) d/D = 0.4; and (f) d/D = 0.5, respectively.

Instantaneous vorticity contours at different offset distances for the typical case A/D = 0.8 and S/D = 3.0 (Re = 100): (a) d/D = 0.0; (b) d/D = 0.1; (c) d/D = 0.2; (d) d/D = 0.3; (e) d/D = 0.4; and (f) d/D = 0.5, respectively.

Instantaneous variation of the vorticity contours against Reynolds number at A/D = 0.8 and S/D = 3.0: (a) Re = 100; (b) Re = 120; (c) Re = 180; and (d) Re = 200.

Instantaneous variation of the vorticity contours against Reynolds number at A/D = 0.8 and S/D = 3.0: (a) Re = 100; (b) Re = 120; (c) Re = 180; and (d) Re = 200.

Mean and r.m.s. drag coefficient and r.m.s. lift coefficient as functions of Reynolds number.

Mean and r.m.s. drag coefficient and r.m.s. lift coefficient as functions of Reynolds number.

The variation of the effectiveness of drag reduction as a function of (a) the oscillation amplitude at different gap ratios (Re = 100) and (b) Re for the case with A/D = 0.8 and S/D = 3.0.

The variation of the effectiveness of drag reduction as a function of (a) the oscillation amplitude at different gap ratios (Re = 100) and (b) Re for the case with A/D = 0.8 and S/D = 3.0.

Instantaneous streamline pattern (left), streamwise velocity contour (middle), and vorticity contour (right) at four phases in a cycle (A/D = 0.8 and S/D = 3.0 at Re = 100).

Instantaneous streamline pattern (left), streamwise velocity contour (middle), and vorticity contour (right) at four phases in a cycle (A/D = 0.8 and S/D = 3.0 at Re = 100).

Time series of hydrodynamic coefficients for the cases of (a) A/D = 0.8 and S/D = 3.0, and (b) A/D = 0.8 and S/D = 2.0 at Re = 100. The initial field used for the simulation is a fully saturated unsteady flow obtained from corresponding stationary cylinder-foil system.

Time series of hydrodynamic coefficients for the cases of (a) A/D = 0.8 and S/D = 3.0, and (b) A/D = 0.8 and S/D = 2.0 at Re = 100. The initial field used for the simulation is a fully saturated unsteady flow obtained from corresponding stationary cylinder-foil system.

Comparison of the instantaneous vorticity contours at Re = 100: (a) t = 5.0 (A/D = 0.8 and S/D = 3.0); (b) t = 500.0 (A/D = 0.8 and S/D = 3.0); (c) t = 5.0 (A/D = 0.8 and S/D = 2.0); and (d) t = 1200.0 (A/D = 0.8 and S/D = 2.0). The initial field used for the simulation is a fully saturated unsteady flow obtained from corresponding cylinder-stationary foil system.

Comparison of the instantaneous vorticity contours at Re = 100: (a) t = 5.0 (A/D = 0.8 and S/D = 3.0); (b) t = 500.0 (A/D = 0.8 and S/D = 3.0); (c) t = 5.0 (A/D = 0.8 and S/D = 2.0); and (d) t = 1200.0 (A/D = 0.8 and S/D = 2.0). The initial field used for the simulation is a fully saturated unsteady flow obtained from corresponding cylinder-stationary foil system.

## Tables

Mesh generation parameters for the two grid systems.

Mesh generation parameters for the two grid systems.

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