^{1}, D. Lo Jacono

^{1}, M. C. Thompson

^{1}and J. Sheridan

^{1}

### Abstract

Previous two-dimensional numerical studies have shown that a circular cylinder undergoing both oscillatory rotational and translational motions can generate thrust so that it will actually self-propel through a stationary fluid. Although a cylinder undergoing a single oscillation has been thoroughly studied, the combination of the two oscillations has not received much attention until now. The current research reported here extends the numerical study of Blackburn *et al.* [Phys. Fluids11, L4 (1999)] both experimentally and numerically, recording detailed vorticity fields in the wake and using these to elucidate the underlying physics, examining the three-dimensional wake development experimentally, and determining the three-dimensional stability of the wake through Floquet stability analysis. Experiments conducted in the laboratory are presented for a given parameter range, confirming the early results from Blackburn *et al.* [Phys. Fluids11, L4 (1999)]. In particular, we confirm the thrust generation ability of a circular cylinder undergoing combined oscillatory motions. Importantly, we also find that the wake undergoes three-dimensional transition at low Reynolds numbers to an instability mode with a wavelength of about two cylinder diameters. The stability analysis indicates that the base flow is also unstable to another mode at slightly higher Reynolds numbers, broadly analogous to the three-dimensional wake transition mode for a circular cylinder, despite the distinct differences in wake/mode topology. The stability of these flows was confirmed by experimental measurements.

M.N. would like to acknowledge the support of a Monash Graduate Scholarship (MGS) and a Monash International Postgraduate Research Scholarship (MIPRS). D.L. acknowledges support from ARC Discovery under Grant No. DP0774525 and computing time from the Australian Partnership for Advanced Computing (APAC).

I. INTRODUCTION

II. PROBLEM DEFINITION

III. METHODS AND TECHNIQUES

A. Experimental setup

B. Experimental techniques

C. Numerical formulation

IV. RESULTS AND DISCUSSION

A. Wake profile in the streamwise direction

B. Wake profile in the spanwise direction

V. CONCLUSIONS

### Key Topics

- Reynolds stress modeling
- 21.0
- Flow instabilities
- 15.0
- Rotating flows
- 15.0
- Vortex dynamics
- 15.0
- Kinematics
- 7.0

## Figures

Contours of vorticity around the cylinder undergoing combined translational and rotational oscillation at , , and (a) , (b) , (c) , taken at from numerical simulation. The dashed lines (enclosing blue) correspond to clockwise direction of vorticity (negative). The solid lines (enclosing red) correspond to counterclockwise direction of vorticity (positive).

Contours of vorticity around the cylinder undergoing combined translational and rotational oscillation at , , and (a) , (b) , (c) , taken at from numerical simulation. The dashed lines (enclosing blue) correspond to clockwise direction of vorticity (negative). The solid lines (enclosing red) correspond to counterclockwise direction of vorticity (positive).

Schematic of the problem geometry and important parameters relevant to the combined forcing. Left: the two-dimensional overview (-plane) of the cylinder. Right: spanwise view of the cylinder (-plane) with end plate and field of view (PIV).

Schematic of the problem geometry and important parameters relevant to the combined forcing. Left: the two-dimensional overview (-plane) of the cylinder. Right: spanwise view of the cylinder (-plane) with end plate and field of view (PIV).

Comparison of the velocity components at three cross sections of the translationally oscillating cylinder in a quiescent fluid at constant values. The values from the top to bottom rows are −0.6, 0.6, and 1.2, respectively. The measurements are taken at and , similar to previous numerical and experimental results (Ref. 8). The left column depicts the and the right column . The solid lines (red) show the present experiment, the dashed lines (black) the present numerical simulation and the filled circle points (blue) the experimental results of Dütsch *et al.* (Ref. 8).

Comparison of the velocity components at three cross sections of the translationally oscillating cylinder in a quiescent fluid at constant values. The values from the top to bottom rows are −0.6, 0.6, and 1.2, respectively. The measurements are taken at and , similar to previous numerical and experimental results (Ref. 8). The left column depicts the and the right column . The solid lines (red) show the present experiment, the dashed lines (black) the present numerical simulation and the filled circle points (blue) the experimental results of Dütsch *et al.* (Ref. 8).

Floquet multipliers for for the three-dimensional instability of regime B at . The current results (solid line) are compared to those from Elston *et al.* (Ref. 16) (squares).

Floquet multipliers for for the three-dimensional instability of regime B at . The current results (solid line) are compared to those from Elston *et al.* (Ref. 16) (squares).

Vorticity contours around the cylinder undergoing combined translational and rotational oscillation at and . The numerical result is presented at the left, and the experimental result at the right. The experimental result is a phase-average of ten successive cycles. The phase shown corresponds to . The dashed lines (enclosing blue) correspond to clockwise direction of vorticity (negative), and the solid lines (enclosing red) corresponds to counterclockwise direction of vorticity (positive).

Vorticity contours around the cylinder undergoing combined translational and rotational oscillation at and . The numerical result is presented at the left, and the experimental result at the right. The experimental result is a phase-average of ten successive cycles. The phase shown corresponds to . The dashed lines (enclosing blue) correspond to clockwise direction of vorticity (negative), and the solid lines (enclosing red) corresponds to counterclockwise direction of vorticity (positive).

Flow produced by a cylinder with combined oscillatory translation and rotation. This figure shows the sequence and development of the vorticity for one complete cycle, (a) to (h) at and , where is the period of oscillation. The radial line shows the rotational displacement of the cylinder. The dashed lines (enclosing blue) correspond to clockwise direction of vorticity (negative), and the solid lines (enclosing red) correspond to counterclockwise vorticity (positive).

Flow produced by a cylinder with combined oscillatory translation and rotation. This figure shows the sequence and development of the vorticity for one complete cycle, (a) to (h) at and , where is the period of oscillation. The radial line shows the rotational displacement of the cylinder. The dashed lines (enclosing blue) correspond to clockwise direction of vorticity (negative), and the solid lines (enclosing red) correspond to counterclockwise vorticity (positive).

Experimental results of the spanwise distribution of flow for two values of at : (a) , streamlines; (b) , velocity isocontours; (c) , vorticity isocontours; the dashed lines (enclosing blue) correspond to clockwise direction of vorticity (negative), and the solid lines (enclosing red) correspond to counterclockwise vorticity (positive).

Experimental results of the spanwise distribution of flow for two values of at : (a) , streamlines; (b) , velocity isocontours; (c) , vorticity isocontours; the dashed lines (enclosing blue) correspond to clockwise direction of vorticity (negative), and the solid lines (enclosing red) correspond to counterclockwise vorticity (positive).

Floquet multipliers for several Reynolds numbers for the spanwise instability at . (a) Floquet multipliers as a function of spanwise wavelength. The open (closed) symbols represent the shortest (longest) wavelength mode. The circles and black line represent results for ; the squares and blue line represent results for ; the diamonds and red line represent results for ; the lower triangles and magenta line represent results for . (b) Comparison of the Floquet analysis predicted wavelength values as a function of Reynolds number with experimental measurements. The blue dashed line corresponds to the wavelength range of the longest spanwise wavelength mode. The red dashed dotted line corresponds to the extent of the shortest spanwise wavelength mode. The thick blue and red lines correspond to the predicted values of the long and short wavelengths, respectively. The black circles represent the present experimental measurements with error bars reporting the standard deviation of the measurements.

Floquet multipliers for several Reynolds numbers for the spanwise instability at . (a) Floquet multipliers as a function of spanwise wavelength. The open (closed) symbols represent the shortest (longest) wavelength mode. The circles and black line represent results for ; the squares and blue line represent results for ; the diamonds and red line represent results for ; the lower triangles and magenta line represent results for . (b) Comparison of the Floquet analysis predicted wavelength values as a function of Reynolds number with experimental measurements. The blue dashed line corresponds to the wavelength range of the longest spanwise wavelength mode. The red dashed dotted line corresponds to the extent of the shortest spanwise wavelength mode. The thick blue and red lines correspond to the predicted values of the long and short wavelengths, respectively. The black circles represent the present experimental measurements with error bars reporting the standard deviation of the measurements.

Contours of streamwise perturbation vorticity taken at and and when the cylinder is at for the following: (left) the long wavelength; (right) the short wavelength. The dashed lines correspond to the base flow clockwise direction of vorticity (negative), and the solid lines correspond to the counterclockwise vorticity (positive).

Contours of streamwise perturbation vorticity taken at and and when the cylinder is at for the following: (left) the long wavelength; (right) the short wavelength. The dashed lines correspond to the base flow clockwise direction of vorticity (negative), and the solid lines correspond to the counterclockwise vorticity (positive).

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