^{1,a)}, A. Katz

^{1}and D. Greenblatt

^{1}

### Abstract

The coupling between shear layer, near wake dynamics, and structural oscillations downstream of a tethered spherical pendulum undergoing vortex induced vibrations (VIV) has been experimentally investigated using time-resolved particle image velocimetry in a wind tunnel. One quarter of the sphere was imaged in the field of view (spatial resolution 0.043D) that extended to 1.17D from the sphere's center (D is the sphere diameter). Reynolds numbers based on D, ranged between 493 ≤ Re ≤ 2218 and reduced velocities between 3.18 ≤ U * ≤ 14.1, covering a non-oscillating sphere, periodic oscillations, and the onset of non-stationary sphere oscillations. After the first Hopf bifurcation, the sphere exhibited large amplitude periodic oscillations and the near-wake vortices periodically interacted with the sphere and flapping shear layer. At U * = 5.97, a “secondary” counterclockwise rotating vortex seemed to facilitate shear layer pinch-off. In agreement with the onset of shear layer instabilities for a stationary sphere, only at Re = 2218 power spectra of velocity fluctuations inside the shear layer indicated a weak, broad frequency peak centered at 15 Hz similar as those measured for stationary cylinders and spheres. This peak was consistent with the results of linear instability theory indicating that despite the inherent three-dimensionality of the shear layer, its instability characteristics (at least for the Re investigated here) can be considered to be quasi-two-dimensional. Small-scale, near-wake structures were observed in the instantaneous swirling strength maps at all U * and it is conjectured here that their interaction with the sphere and separating shear layer is the feedback mechanism through which VIV occurs and is sustained.

This work was supported in part by the Edmund J. Safra Philanthropic Foundation, the Wolfson Family Charitable Trust, and the Technion Fund for Promotion of Research.

I. INTRODUCTION

II. EXPERIMENTAL SETUP AND METHODS

III. SPHERE DYNAMICS

IV. SHEAR LAYER AND NEAR WAKE DYNAMICS

A. First bifurcation region: Stationary sphere

B. Second bifurcation region: Periodic sphere oscillations

C. Near wake dynamics and shear layer instabilities at condition IV

V. SUMMARY AND DISCUSSION

### Key Topics

- Rotating flows
- 76.0
- Vortex dynamics
- 33.0
- Shear flow instabilities
- 27.0
- Velocimetry
- 19.0
- Flow instabilities
- 16.0

## Figures

Schematic layout of experimental setup and employed coordinate system. (a) Cross section, (b) top view.

Schematic layout of experimental setup and employed coordinate system. (a) Cross section, (b) top view.

Example of collapsed sequence of 21 PIV images centered at t/T = 0.25 (Table I , U * = 4.84, see Fig. 7 ).

Normalized, mean transverse sphere oscillation amplitude as a function of reduced velocity. Present results, M * = 0.182: • PIV, ○ sphere tracking; Literature results: □ m * = 2.8, M * = 0.026, 28 ◊ m * = 28, M * = 0.228, 28 × m * = 7.87, M * = 0.167. 31

Time series of sphere transverse centroid positions (left column), phase diagrams and Poincaré points (middle column), and power spectra of sphere motion (right column). (a) U * = 3.18, (b) U * = 4.84, (c) U * = 5.97, (d) U * = 14.1.

Time series of sphere transverse centroid positions (left column), phase diagrams and Poincaré points (middle column), and power spectra of sphere motion (right column). (a) U * = 3.18, (b) U * = 4.84, (c) U * = 5.97, (d) U * = 14.1.

Instantaneous sequence of (a) vector maps, (b) vorticity, (c) swirling strength, (d) and (e) Galilean decomposed parts of vector map (full vector resolution) showing V1 and V2, respectively; subtracted in (d) U c = (−0.025, 0.005) and (e) U c = (−0.025, 0.007) in [m/s]. U * = 3.18. Lightest gray (red online) vectors are reference vectors, in (a) reference vector = 0.2 [m/s]. Dashed vertical lines in (c) indicate sampling lines (see Fig. 6 ).

Instantaneous sequence of (a) vector maps, (b) vorticity, (c) swirling strength, (d) and (e) Galilean decomposed parts of vector map (full vector resolution) showing V1 and V2, respectively; subtracted in (d) U c = (−0.025, 0.005) and (e) U c = (−0.025, 0.007) in [m/s]. U * = 3.18. Lightest gray (red online) vectors are reference vectors, in (a) reference vector = 0.2 [m/s]. Dashed vertical lines in (c) indicate sampling lines (see Fig. 6 ).

Spatiotemporal swirling strength characteristics for U * = 3.18 sampled at (a) x 1/D = 0.5, (b) x 1/D = 0.7. Sampling lines are indicated as vertical dashed lines in Fig. 5 (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812181.1]doi: 10.1063/1.4812181.1.

Spatiotemporal swirling strength characteristics for U * = 3.18 sampled at (a) x 1/D = 0.5, (b) x 1/D = 0.7. Sampling lines are indicated as vertical dashed lines in Fig. 5 (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812181.1]doi: 10.1063/1.4812181.1.

Instantaneous sequence of (a) vector maps, (b) vorticity, (c) swirling strength. U * = 4.84. Vector maps were Galilean decomposed by subtracting U c = (0.02, −0.08), (−0.008, 0.01), (0.0015, 0.01), (0,0) in [m/s] for t/T = 0, 0.25, 0.5, and 0.75, respectively. Lightest gray (red online) vector in (a) denotes reference vector: 0.2 [m/s].

Instantaneous sequence of (a) vector maps, (b) vorticity, (c) swirling strength. U * = 4.84. Vector maps were Galilean decomposed by subtracting U c = (0.02, −0.08), (−0.008, 0.01), (0.0015, 0.01), (0,0) in [m/s] for t/T = 0, 0.25, 0.5, and 0.75, respectively. Lightest gray (red online) vector in (a) denotes reference vector: 0.2 [m/s].

Instantaneous sequence of (a) vector maps, (b) vorticity, (c) swirling strength. U * = 5.97. Vector maps were Galilean decomposed by subtracting U c = (0.0859, 0), (0.003, 0), (0, 0), (−0,0962, 0) in [m/s] for t/T = 0, 0.25, 0.5, and 0.75, respectively. Lightest gray (red online) vector in (a) denotes reference vector: 0.2 [m/s]. Vertical dashed line denotes sampling position (see Fig. 10 ).

Instantaneous sequence of (a) vector maps, (b) vorticity, (c) swirling strength. U * = 5.97. Vector maps were Galilean decomposed by subtracting U c = (0.0859, 0), (0.003, 0), (0, 0), (−0,0962, 0) in [m/s] for t/T = 0, 0.25, 0.5, and 0.75, respectively. Lightest gray (red online) vector in (a) denotes reference vector: 0.2 [m/s]. Vertical dashed line denotes sampling position (see Fig. 10 ).

Transverse locations of shear layer and sphere center as a function of time. (a) U * = 4.84, (b) U * = 5.97. Solid and dashed vertical lines indicate zero angle shear layer inclination. ▵ x 1/D = 0.88, □ x 1/D = 0.52, ◯ x 1/D = 0.16, — sphere transverse motion.

Transverse locations of shear layer and sphere center as a function of time. (a) U * = 4.84, (b) U * = 5.97. Solid and dashed vertical lines indicate zero angle shear layer inclination. ▵ x 1/D = 0.88, □ x 1/D = 0.52, ◯ x 1/D = 0.16, — sphere transverse motion.

Spatiotemporal characteristics sampled at x 1/D = 0.6, U * = 4.84. (a) Vorticity, (b) swirling strength (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812181.2]doi: 10.1063/1.4812181.2.

Spatiotemporal characteristics sampled at x 1/D = 0.6, U * = 4.84. (a) Vorticity, (b) swirling strength (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812181.2]doi: 10.1063/1.4812181.2.

Spatiotemporal characteristics sampled at x 1/D = 0.7, U * = 5.97. (a) Vorticity, (b) swirling strength (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812181.3]doi: 10.1063/1.4812181.3.

Spatiotemporal characteristics sampled at x 1/D = 0.7, U * = 5.97. (a) Vorticity, (b) swirling strength (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812181.3]doi: 10.1063/1.4812181.3.

Instantaneous sequence of vorticity (left column) and swirling strength (right column) maps for U * = 4.84 showing the pinch-off of V4. (a) t/T = 0.374, (b) t/T = 0.435, (c) t/T = 0.523, (d) t/T = 0.611. Time between frames, Δt = 40/700 s.

Instantaneous sequence of vorticity (left column) and swirling strength (right column) maps for U * = 4.84 showing the pinch-off of V4. (a) t/T = 0.374, (b) t/T = 0.435, (c) t/T = 0.523, (d) t/T = 0.611. Time between frames, Δt = 40/700 s.

Instantaneous sequence of vorticity and swirling strength maps showing vortex-shear layer interaction and shear layer pinch-off for U * = 5.97 at t/T = (a) 0.692, (b) 0.736, (c) 0.780, (d) 0.824, (e) 0.868, (f) 0.912, (g) 0.956, (h) 1.0. Time between frames: Δt = 20/700 s.

Instantaneous sequence of vorticity and swirling strength maps showing vortex-shear layer interaction and shear layer pinch-off for U * = 5.97 at t/T = (a) 0.692, (b) 0.736, (c) 0.780, (d) 0.824, (e) 0.868, (f) 0.912, (g) 0.956, (h) 1.0. Time between frames: Δt = 20/700 s.

Instantaneous sequence of (a) vector maps, (b) vorticity, (c) swirling strength. U * = 14.1. Vector maps were Galilean decomposed by subtracting U c = (0.16, 0), (0.07, −0.04), (−0.05, 0.06), (0, 0) in [m/s] for t/T = 0, 0.25, 0.5, and 0.75, respectively. Lightest gray (red online) vector in (a) denotes reference vector: 0.2 [m/s]. Vertical dashed lines denote sampling positions (see Fig. 15 ).

Instantaneous sequence of (a) vector maps, (b) vorticity, (c) swirling strength. U * = 14.1. Vector maps were Galilean decomposed by subtracting U c = (0.16, 0), (0.07, −0.04), (−0.05, 0.06), (0, 0) in [m/s] for t/T = 0, 0.25, 0.5, and 0.75, respectively. Lightest gray (red online) vector in (a) denotes reference vector: 0.2 [m/s]. Vertical dashed lines denote sampling positions (see Fig. 15 ).

Spatiotemporal swirling strength characteristics for U * = 14.1 sampled at (a) x 1/D = 0.7, (b) x 1/D = 1.0. The sampling lines are depicted as dashed lines in Fig. 14 (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812181.4]doi: 10.1063/1.4812181.4.

Spatiotemporal swirling strength characteristics for U * = 14.1 sampled at (a) x 1/D = 0.7, (b) x 1/D = 1.0. The sampling lines are depicted as dashed lines in Fig. 14 (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812181.4]doi: 10.1063/1.4812181.4.

Power spectrum of transverse velocity fluctuations in the shear layer at U * = 14.1. Dashed rectangle and horizontal bar indicate measured broad peak and predicted range of f sl (Eq. (1)). + Frequencies estimated from Fig. 15 .

Power spectrum of transverse velocity fluctuations in the shear layer at U * = 14.1. Dashed rectangle and horizontal bar indicate measured broad peak and predicted range of f sl (Eq. (1)). + Frequencies estimated from Fig. 15 .

## Tables

Experimental parameters of TR-PIV measurements. Conditions I–III contain three data sets, condition IV, two data sets.

Experimental parameters of TR-PIV measurements. Conditions I–III contain three data sets, condition IV, two data sets.

## Multimedia

Article metrics loading...

Full text loading...

Commenting has been disabled for this content