# Computational Study of Three-Dimensional Flow Past an Oscillating Cylinder Following a Figure Eight Trajectory

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## Abstract

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## 1. Introduction

## 2. Formulation and Numerical Method

_{y}and A

_{x}in the transverse and streamwise direction, respectively, and corresponding oscillation frequencies f

_{y}and f

_{x}. The frequency of in-line direction is twice the transverse frequency (f

_{x}= 2f

_{y}). The instantaneous displacement of the cylinder in the y-direction and x-direction is defined as:

_{y}/D), the amplitude ratio, $\mathsf{\epsilon}={A}_{x}/{A}_{y}$, and the frequency ratio $\mathrm{F}={f}_{y}/{f}_{s}$, where f

_{s}is the non-dimensional vortex shedding frequency in flow past a stationary cylinder (Strouhal frequency). For Re = 400, f

_{s}= 0.204 [22].

_{x}, η

_{y}, 0) is now the non-dimensional cylinder displacement, and $\overrightarrow{i}$, $\overrightarrow{j}$, $\overrightarrow{k}$ are the unit vectors in the three directions. To avoid reconstructing the computational grid at each time step, we use a frame of reference fixed on the cylinder. Let $\overrightarrow{u}$ now express the relative velocity with respect to the moving cylinder. The incompressibility equation remains unchanged, while the momentum equation (Equation (7)) becomes:

## 3. Results

#### 3.1. Power Transfer and Hydrodynamic Forces

_{y}/D, are presented. Although power transfer is essentially uniform along the cylinder span in the present problem of flow past a rigid cylinder, variation along the span (corresponding to different oscillation patterns) has been reported for the case of flow past a long flexible cylinder in shear flow [31].

_{y}/D, reaches a maximum, and decreases for non-dimensional oscillation amplitude higher than about 0.4; negative values of P are obtained for A

_{y}/D values higher than about 0.5, in agreement with the results of Leontini et al. [9] and Peppa et al. [32] for two-dimensional flow at lower Reynolds numbers. For the case of above-resonant forcing (F = 1.1), among the amplitudes considered, a positive value of P is only obtained for A

_{y}/D = 0.5, accompanied by a drastic increase in time-average drag coefficient, <C

_{D}> (Figure 2b). The drag coefficient is monotonically increasing for resonant forcing (F = 1). For F < 1, increasing <C

_{D}> values are obtained after a threshold value of oscillation amplitude, and this should be associated with lock-in of the vortex street to the forcing frequency.

_{y}/D = 0.6. Consequently, VIV in terms of a counter-clockwise should only occur at higher oscillation amplitude. For the clockwise mode, for F ≤ 1, P is nearly zero at low oscillation amplitude, and is negative and decreasing for A

_{y}/D values higher than about 0.3; negative values, characterized by a monotonic decrease with A

_{y}/D, are attained for F = 1.1. Thus, the appearance of a clockwise mode is less likely in VIV.

_{L,RMS}, with oscillation amplitude, for the different values of transverse oscillation frequency, is presented in Figure 4a,b, for the counter-clockwise and the clockwise mode, respectively. Interestingly, for the counter-clockwise oscillation at F = 1.0, the lift fluctuation curve remains flat for A

_{y}/D ≥ 0.2. For F = 1.1, a significant variation is found around A

_{y}/D = 0.30, which is in correspondence with the sharp decrease in the value of power transfer parameter (Figure 3a). For clockwise motion, a monotonic increase of C

_{L,RMS}with oscillation amplitude is found (Figure 3b), with the values attained in the high end of oscillation amplitude being higher than those for counter-clockwise motion by a factor of two.

_{y}/D ≥ 0.2. In all cases, the spectra demonstrate the absence of Strouhal frequency, and the presence of the excitation frequency and its higher harmonics (lock-in). For F = 0.9, the spectra of both modes exhibit a strong peak at the third harmonic (Figure 7a,b), while they also show the next higher odd harmonic; even harmonics exhibit a strong presence in the case of the clockwise mode (Figure 7b), and are absent from the counter-clockwise mode spectra (Figure 7a). The results presented for F = 1.0 verify the presence of both odd and even harmonics (Figure 9), similarly to the results obtained for F = 0.9. For F = 1.1, the lift spectra of the counter-clockwise mode show the strong presence of the third harmonic, as well as the absence of even harmonics (Figure 8a); for clockwise oscillation, the spectra are populated by only odd harmonics at low and moderate oscillation amplitude, and by both odd and even harmonics at high amplitude (Figure 8b).

#### 3.2. Visualization of the Flow in the Wake

_{y}/D = η

_{x}/D = 0. For resontant forcing, Peppa et al. [22] have shown that, for the counter-clockwise mode, the flow three-dimensionality is reduced at low oscillation amplitude, with the flow becoming increasingly more complex at higher amplitudes, while always maintaining the 2S shedding mode.

_{y}/D = 0.30–0.40), commonly referred to as the “S+P” wake mode. Interestingly, a full return to two-dimensional flow is attained at the transverse oscillation amplitude of 0.40. The shedding mode characterized by the presence of vortex splitting has first been reported in [33], and characterized as “partial S+P” mode. These flow visualizations demonstrate that the “S+P” mode is the cause of the even harmonic frequency components identified in the lift spectrum (Figure 7b).

_{y}/D = 0.20. For the same cases, Figure 15 presents a visualization of the wake in a plane of constant z. The presence of a two-dimensional wake is evident for A

_{y}/D ≥ 0.20, as well as the presence of the “S+P” mode for A

_{y}/D = 0.30. We note that these transitions do not have a marked effect on the variation of power transfer and force coefficients (Figure 3, Figure 4, Figure 5 and Figure 6). This should be associated with the fact that in all cases shedding is initiated as a 2S pattern, which may by modified into a “S+P” mode farther downstream. Finally, the “S+P” mode is also verified as the cause for the existence of even harmonics in the lift spectrum, in the present case of F = 1.0 (Figure 9).

_{y}/D = 0.20) a nearly two-dimensional vortex street is initiated, and modified farther downstream by vortex pairing, which gives rise to a stronger flow three-dimensionality. For A

_{y}/D = 0.30, the 2S structure persists farther donwstream, giving a narrow wake, which is in accordance with the drop in the mean drag coefficient value (Figure 5a). Finally, for A

_{y}/D = 0.40, a vortex pairing process is identified, leading to vortex dipoles.

_{y}/D = 0.30, as well as the presence of a “S+P” mode at A

_{y}/D = 0.60, thus bearing some similarities with the case of resonant forcing. However, for F = 1.1, vortex pairing and dipole formation is present several diameters downstream the cylinder, upon return to two-dimensional flow (A

_{y}/D = 0.30). Finally, the “S+P” mode becomes rather complex farther downstream (A

_{y}/D = 0.60). Again, the direct relation between the “S+P” mode and the even harmonics in the lift spectrum is verified (Figure 8b).

## 4. Discussion and Conclusions

_{y}/D = 0.20, while, for F = 0.9 and F = 1.1, the flow has been found to be two-dimensional at A

_{y}/D = 0.40 and 0.30, respectively. No return to two-dimensionality has been found for the counter-clockwise oscillation mode; nonetheless, a weaker three-dimensionality of the wake has been identified for moderate values of oscillation amplitude.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Sketch of cylinder oscillation with respect to a uniform stream: (

**a.a**) counter-clockwise mode, (

**a**.

**b**) clockwise mode. (

**b**) Spectral element grid for three-dimensional flow past a circular cylinder. The origin of axes is at the cylinder center.

**Figure 2.**Variation of (

**a**) power transfer parameter, P, and (

**b**) mean drag coefficient, <C

_{D}>, versus the non-dimensional amplitude, A

_{y}/D, for transverse-only oscillation, for frequency ratio F = 0.8, 0.9, 1.0 and 1.1.

**Figure 3.**Variation of the power transfer parameter, P, for ε = 0.2, versus the non-dimensional amplitude A

_{y}/D, for frequency ratio F = 0.8, 0.9, 1.0, 1.1: (

**a**) counter-clockwise cylinder oscillation, (

**b**) clockwise cylinder oscillation.

**Figure 4.**Variation of the RMS of the lift coefficient, C

_{L,RMS}, for ε = 0.2 versus the non-dimensional amplitude A

_{y}/D, for frequency ratio F = 0.8, 0.9, 1.0, 1.1: (

**a**) counter-clockwise cylinder oscillation, (

**b**) clockwise cylinder oscillation.

**Figure 5.**Variation of the mean drag coefficient, <C

_{D}>, for ε = 0.2, versus the non-dimensional amplitude A

_{y}/D, for frequency ratio F = 0.8, 0.9, 1.0, 1.1: (

**a**) counter-clockwise cylinder oscillation, (

**b**) clockwise cylinder oscillation.

**Figure 6.**Variation of the RMS of the drag coefficient, C

_{D,RMS}, for ε = 0.2, versus the non-dimensional amplitude A

_{y}/D, for frequency ratio F = 0.8, 0.9, 1.0, 1.1: (

**a**) counter-clockwise cylinder oscillation, (

**b**) clockwise cylinder oscillation.

**Figure 7.**Spectra of the lift force coefficient for F = 0.9: (

**a**) counter-clockwise cylinder oscillation, (

**b**) clockwise cylinder oscillation.

**Figure 8.**Spectra of the lift force coefficient for F = 1.1: (

**a**) counter-clockwise cylinder oscillation, (

**b**) clockwise cylinder oscillation.

**Figure 9.**Spectra of the lift force coefficient for F = 1.0 and A

_{y}/D = 0.20, 0.30, 0.40, for the clockwise cylinder oscillation.

**Figure 10.**Counter-clockwise cylinder oscillation: Instantaneous vorticity isosurfaces (top view) for F = 0.9 and (

**a**) A

_{y}/D = 0.20, (

**b**) A

_{y}/D = 0.30, (

**c**) A

_{y}/D = 0.60.

**Figure 11.**Counter-clockwise cylinder oscillation: Instantaneous isocontours of spanwise vorticity for the plane z = 3 for F = 0.9 and (

**a**) A

_{y}/D = 0.20, (

**b**) A

_{y}/D = 0.30, (

**c**) A

_{y}/D = 0.60.

**Figure 12.**Clockwise cylinder oscillation: Instantaneous vorticity isosurfaces (top view) for F = 0.9 and (

**a**) A

_{y}/D = 0.20, (

**b**) A

_{y}/D = 0.30, (

**c**) A

_{y}/D = 0.40.

**Figure 13.**Clockwise cylinder oscillation: Instantaneous isocontours of spanwise vorticity for the plane z = 3 for F = 0.9 and (

**a**) A

_{y}/D = 0.20, (

**b**) A

_{y}/D = 0.30, (

**c**) A

_{y}/D = 0.40.

**Figure 14.**Clockwise cylinder oscillation: Instantaneous vorticity isosurfaces (top view) for F = 1.0 and (

**a**) A

_{y}/D = 0.10, (

**b**) A

_{y}/D = 0.20, (

**c**) A

_{y}/D = 0.30.

**Figure 15.**Clockwise cylinder oscillation: Instantaneous isocontours of spanwise vorticity for the plane z = 3 for F = 1.0 and (

**a**) A

_{y}/D = 0.10, (

**b**) A

_{y}/D = 0.20, (

**c**) A

_{y}/D = 0.30.

**Figure 16.**Counter-clockwise cylinder oscillation: Instantaneous vorticity isosurfaces (top view) for F = 1.1 and (

**a**) Ay/D = 0.20, (

**b**), Ay/D = 0.30, (

**c**) Ay/D = 0.40.

**Figure 17.**Counter-clockwise cylinder oscillation: Instantaneous isocontours of spanwise vorticity for the plane z = 3 for F = 1.1 and (

**a**) A

_{y}/D = 0.20, (

**b**), A

_{y}/D = 0.30, (

**c**) A

_{y}/D = 0.40.

**Figure 18.**Clockwise cylinder oscillation: Instantaneous vorticity isosurfaces (top view) for F = 1.1 and (

**a**) Ay/D = 0.10, (

**b**), Ay/D = 0.30, (

**c**) Ay/D = 0.60.

**Figure 19.**Clockwise cylinder oscillation: Instantaneous isocontours of spanwise vorticity for the plane z = 3 for F = 1.1 and (

**a**) A

_{y}/D = 0.10, (

**b**), A

_{y}/D = 0.30, (

**c**) A

_{y}/D = 0.60.

**Figure 20.**Counter-clockwise cylinder oscillation: Instantaneous vorticity isosurfaces (top view) for F = 0.8 and (

**a**) A

_{y}/D = 0.10, (

**b**), A

_{y}/D = 0.30, (

**c**) A

_{y}/D = 0.40.

**Figure 21.**Counter-clockwise cylinder oscillation: Instantaneous isocontours of spanwise vorticity for the plane z = 3 for F = 0.8 and (

**a**) A

_{y}/D = 0.10, (

**b**), A

_{y}/D = 0.30, (

**c**) A

_{y}/D = 0.40.

**Figure 22.**Clockwise cylinder oscillation: Instantaneous vorticity isosurfaces (top view) for F = 0.8 and (a) A

_{y}/D = 0.10, (b), A

_{y}/D = 0.30, (c) A

_{y}/D = 0.40.

**Figure 23.**Clockwise cylinder oscillation: Instantaneous isocontours of spanwise vorticity for the plane z = 3 for F = 0.8 and (

**a**) A

_{y}/D = 0.10, (

**b**) A

_{y}/D = 0.30, (

**c**) A

_{y}/D = 0.40.

**Figure 24.**Instantaneous isocontours of spanwise vorticity for the plane z = 0, with the cylinder at its mean position (η

_{y}/D = η

_{x}/D = 0): (

**a**) F = 0.9, A

_{y}/D = 0.40, (

**b**) F = 1.1, A

_{y}/D = 0.30. The left column corresponds to counter-clockwise cylinder oscillation, and the right column to clockwise oscillation.

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**MDPI and ACS Style**

Peppa, S.; Kaiktsis, L.; Frouzakis, C.E.; Triantafyllou, G.S.
Computational Study of Three-Dimensional Flow Past an Oscillating Cylinder Following a Figure Eight Trajectory. *Fluids* **2021**, *6*, 107.
https://doi.org/10.3390/fluids6030107

**AMA Style**

Peppa S, Kaiktsis L, Frouzakis CE, Triantafyllou GS.
Computational Study of Three-Dimensional Flow Past an Oscillating Cylinder Following a Figure Eight Trajectory. *Fluids*. 2021; 6(3):107.
https://doi.org/10.3390/fluids6030107

**Chicago/Turabian Style**

Peppa, Sofia, Lambros Kaiktsis, Christos E. Frouzakis, and George S. Triantafyllou.
2021. "Computational Study of Three-Dimensional Flow Past an Oscillating Cylinder Following a Figure Eight Trajectory" *Fluids* 6, no. 3: 107.
https://doi.org/10.3390/fluids6030107