# Spanwise wake development of a pivoted cylinder undergoing vortex-induced vibrations with elliptic trajectories

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### Abstract

The wake development of a pivoted circular cylinder undergoing vortex-induced vibrations with elliptical trajectories is examined experimentally at a fixed Reynolds number of 3027 and mass ratio of 10.8. Simultaneous cylinder displacement measurements and time-resolved, two-component particle image velocimetry in multiple horizontal and vertical planes are used to quantify the structural response and wake development. The selected test cases pertain to \(U^*=U_0/f_\mathrm{n} D=5.48\) and 7.08, and exhibit different orientations of elliptical cylinder trajectory, both with a clockwise direction of orbiting. Three-dimensional reconstructions of the phase-averaged wake velocity measurements reveal 2S shedding along the span of a stationary cylinder and hybrid shedding for the two vibrating cylinder cases, with planar wake topology transitioning from 2S to P+S to 2S for \(U^*=5.48\), and 2S to P+S for 7.08. The observed wake topologies show significant deviation from predictions based on the Morse and Williamson (J Fluids Struct 25(4):697–712, 2009) shedding map. Vortex identification and strength quantification are used to provide insight into vortex dynamics and to propose a model of the dislocations. Examination of the time averaged wake characteristics shows the formation length, wake half-width, and maximum velocity deficit exhibit distinct spanwise trends aligning with the regions associated with specific shedding regimes.

## List of symbols

- AR
Aspect ratio,

*L*/*D*- \(A_x, A_y\)
Half of peak-to-peak amplitude of streamwise and transverse vibrations, respectively

- \(A_x^*, A_y^*\)
Normalized amplitude of streamwise and transverse vibrations, \(A_x/D,~A_y/D\), respectively

- \(a_{i}\)
Temporal POD coefficients

*D*Cylinder diameter

- \(d_{\mathrm{wake}}\)
Wake half-width

- \(f_\mathrm{n}\)
Natural frequency in quiescent water

- \(f_\mathrm{s}\)
Vortex shedding frequency of a stationary cylinder

- \(f_u,f_v\)
Frequency of the streamwise and transverse velocity signal, respectively

- \(f_x, f_y\)
Frequency of streamwise and transverse vibrations, respectively

*I*Moment of inertia of the cylinder about the pivot point

- \(I_\mathrm{d}\)
Moment of inertia of the displaced fluid about the pivot point

- \(I^*\)
Moment of inertia ratio, \(I/I_\mathrm{d}\)

- \(\hat{i}, \hat{j}, \hat{k}\)
Unit vectors in

*x*,*y*,*z*directions, respectively*k*Spring stiffness coefficient

*L*Length of cylinder

- \(L_\mathrm{f}\)
Formation length

*m*Mass of the cylinder

- \(m_\mathrm{d}\)
Mass of displaced fluid

- \(m^*\)
Mass ratio, \(m/m_\mathrm{d}\)

- PSD
Power spectrum density

*Re*Reynolds number, \(Re = U_0D/\nu\)

- \(\mathbf {U}\)
Mean velocity field, \(\mathbf {U}=U\hat{i}+V\hat{j}+W\hat{k}\)

- \(U_0\)
Free stream velocity

- \(U_\mathrm{d}\)
Local velocity deficit

- \(U_\mathrm{e}\)
Velocity at the transverse extent of the wake measurements

- \(U^*\)
Reduced velocity, \(U_0/f_\mathrm{n}D\)

- \(\mathbf {u}\)
Velocity field, \(\mathbf {u}=u\hat{i}+v\hat{j}+w\hat{k}\)

- \(\mathbf {u}_{\mathrm{RMS}}\)
Root-mean-square (RMS) velocity field, \(\mathbf {u}_{\mathrm{RMS}}=u_{\mathrm{RMS}}\hat{i}+v_{\mathrm{RMS}}\hat{j}+w_{\mathrm{RMS}}\hat{k}\)

*x*,*y*,*z*Streamwise, transverse and spanwise directions, respectively

- \(\varGamma\)
Circulation

- \(\varDelta \theta\)
Phase bin size

- \(\zeta\)
Damping ratio

- \(\theta\)
Phase angle of the cylinder’s elliptic orbit

- \(\lambda _{i}\)
POD mode energy

- \(\nu\)
Kinematic viscosity of water

- \(\mathbf {\phi _i}\)
Spatial POD modes, \(\mathbf {\phi _i} = \phi _{ix}\hat{i} + \phi _{iy}\hat{j}\)

- \(\psi\)
Phase angle between streamwise and transverse motion

- \(\omega _z\)
Spanwise vorticity

## Notes

### Acknowledgements

The authors gratefully acknowledge the Natural Sciences and Engineering Research Council of Canada (NSERC) (RGPIN-2017-04222) for funding this work.

## References

- Bearman PW (1965) Investigation of the flow behind a two-dimensional model with a blunt trailing edge and fitted with splitter plates. J Fluid Mech 21(02):241CrossRefGoogle Scholar
- Bearman PW (1967) On vortex street wakes. J Fluid Mech 28(04):625–641CrossRefGoogle Scholar
- Bearman PW (1984) Vortex shedding from oscillating bluff bodies. Annu Rev Fluid Mech 16(1):195–222CrossRefGoogle Scholar
- Bernitsas MM, Raghavan K, Ben-Simon Y, Garcia EMH (2008) VIVACE (vortex induced vibration aquatic clean energy): a new concept in generation of clean and renewable energy from fluid flow. J Offshore Mech Arct Eng 130(4):041101CrossRefGoogle Scholar
- Blevins RD, Coughran CS (2009) Experimental investigation of vortex-induced vibration in one and two dimensions with variable mass, damping, and Reynolds number. J Fluids Eng 131(10):101202CrossRefGoogle Scholar
- Cagney N, Balabani S (2014) Streamwise vortex-induced vibrations of cylinders with one and two degrees of freedom. J Fluid Mech 758:702–727CrossRefGoogle Scholar
- Chong MS, Perry AE, Cantwell BJ (1990) A general classification of three dimensional flow fields. Phys Fluids A Fluid Dyn 2(5):765–777MathSciNetCrossRefGoogle Scholar
- Dailey JE, Weidler JB, Hanna SY, Zedan MF, Yeung JY (1987) Pile fatigue failures. 3: motions in seas. J Waterw Port Coast Ocean Eng 113(3):233–250CrossRefGoogle Scholar
- Every M, King R, Weaver D (1982) Vortex-excited vibrations of cylinders and cables and their suppression. Ocean Eng 9(2):135–157CrossRefGoogle Scholar
- Feng CC (1968) The measurement of vortex induced effects in flow past stationary and oscillating circular and D-section cylinders. Masters in applied science thesis, University of British ColumbiaGoogle Scholar
- Ferré JA, Giralt F (1989) Pattern-recognition analysis of the velocity field in plane turbulent wakes. J Fluid Mech 198:27MathSciNetCrossRefGoogle Scholar
- Flemming F, Williamson C (2005) Vortex-induced vibrations of a pivoted cylinder. J Fluid Mech 522:215–252CrossRefGoogle Scholar
- Gao Y, Fu S, Xiong Y, Zhao Y, Liu L (2016) Experimental study on response performance of vortex-induced vibration on a flexible cylinder. Ships Offshore Struct 12(1):116–134CrossRefGoogle Scholar
- Gerrard JH (1966) The mechanics of the formation region of vortices behind bluff bodies. J Fluid Mech 25:401–413CrossRefGoogle Scholar
- Gharib MR (1999) Vortex-induced vibration, absence of lock-in and fluid force deduction. Phd, California Institute of TechnologyGoogle Scholar
- Govardhan R, Williamson CHK (2000) Modes of vortex formation and frequency response of a freely vibrating cylinder. J Fluid Mech 420:85–130MathSciNetCrossRefGoogle Scholar
- Graftieaux L, Michard M, Grosjean N (2001) Combining PIV, POD and vortex identification algorithms for the study of unsteady turbulent swirling flows. Meas Sci Technol 12(9):1422–1429CrossRefGoogle Scholar
- Griffin OM (1995) A note on bluff body vortex formation. J Fluid Mech 284:217–224CrossRefGoogle Scholar
- Huera-Huarte F, Bearman P (2009) Wake structures and vortex-induced vibrations of a long flexible cylinder-part 1: dynamic response. J Fluids Struct 25(6):969–990CrossRefGoogle Scholar
- Hunt JC, Wray AA, Moin P (1988) Eddies, streams, and convergence zones in turbulent flows. In: Proceedings of the Summer Program, Center for Turbulence Research, pp 193–208Google Scholar
- Huse E, Kleiven G, Nielsen F (1998) Large scale model testing of deep sea risers. In: offshore technology conference, offshore technology conference, pp 1–9Google Scholar
- Jauvtis N, Williamson CHK (2004) The effect of two degrees of freedom on vortex-induced vibration at low mass and damping. J Fluid Mech 509:23–62CrossRefGoogle Scholar
- Jeon D, Gharib M (2001) On circular cylinders undergoing two-degree-of-freedom forced motions. J Fluids Struct 15(3–4):533–541CrossRefGoogle Scholar
- Jeong J, Hussain F (1995) On the identification of a vortex. J Fluid Mech 285:69–94MathSciNetCrossRefGoogle Scholar
- Khalak A, Williamson C (1997) Fluid forces and dynamics of a hydroelastic structure with very low mass and damping. J Fluids Struct 11(8):973–982CrossRefGoogle Scholar
- Khalak A, Williamson C (1999) Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping. J Fluids Struct 13(7–8):813–851CrossRefGoogle Scholar
- Kheirkhah S, Yarusevych S, Narasimhan S (2012) Orbiting response in vortex-induced vibrations of a two-degree-of-freedom pivoted circular cylinder. J Fluids Struct 28:343–358CrossRefGoogle Scholar
- Kheirkhah S, Yarusevych S, Narasimhan S (2016) Wake topology of a cylinder undergoing vortex-induced vibrations with elliptic trajectories. J Fluids Eng 138(5):054501CrossRefGoogle Scholar
- Kim J, Moin P, Moser R (1987) Turbulence statistics in fully developed channel flow at low Reynolds number. J Fluid Mech 177:133–166CrossRefGoogle Scholar
- King R (1974) Vortex excited structural oscillations of a circular cylinder in flowing water. Phd, Loughborough UniversityGoogle Scholar
- Leong C, Wei T (2008) Two-degree-of-freedom vortex-induced vibration of a pivoted cylinder below critical mass ratio. Proc R Soc A Math Phys Eng Sci 464(2099):2907–2927CrossRefGoogle Scholar
- Marble E, Morton C, Yarusevych S (2018) Vortex dynamics in the wake of a pivoted cylinder undergoing vortex-induced vibrations with elliptic trajectories. Exp Fluids 59(5):16CrossRefGoogle Scholar
- Moe G, Wu Z (1990) The lift force on a cylinder vibrating in a current. J Offshore Mech Arct Eng 112(4):297CrossRefGoogle Scholar
- Moin P, Kim J (1985) The structure of the vorticity field in turbulent channel flow. Part 1. Analysis of instantaneous fields and statistical correlations. J luid Mech 155:441–464Google Scholar
- Morse T, Williamson C (2009) Fluid forcing, wake modes, and transitions for a cylinder undergoing controlled oscillations. J Fluids Struct 25(4):697–712CrossRefGoogle Scholar
- Morton BR (1984) The generation and decay of vorticity. Geophys Astrophys Fluid Dyn 28(3–4):277–308CrossRefGoogle Scholar
- Norberg C (2001) Flow around a circular cylinder: aspects of fluctuating lift. J Fluids Struct 15(3–4):459–469CrossRefGoogle Scholar
- Oviedo-Tolentino F, Pérez-Gutiérrez F, Romero-Méndez R, Hernández-Guerrero A (2014) Vortex-induced vibration of a bottom fixed flexible circular beam. Ocean Eng 88:463–471CrossRefGoogle Scholar
- Perrin R, Braza M, Cid E, Cazin S, Barthet A, Sevrain A, Mockett C, Thiele F (2007) Obtaining phase averaged turbulence properties in the near wake of a circular cylinder at high Reynolds number using POD. Exp Fluids 43(2–3):341–355CrossRefGoogle Scholar
- Roshko A (1955) On the wake and drag of bluff bodies. J Aeronaut Sci 22(2):124–132CrossRefGoogle Scholar
- Roshko A (1993) Perspectives on bluff body aerodynamics. J Wind Eng Ind Aerodyn 49(1–3):79–100CrossRefGoogle Scholar
- Sanchis A, Sælevik G, Grue J (2008) Two-degree-of-freedom vortex-induced vibrations of a spring-mounted rigid cylinder with low mass ratio. J Fluids Struct 24(6):907–919CrossRefGoogle Scholar
- Sarpkaya T (1995) Hydrodynamic damping, flow-induced oscillations, and biharmonic response. J Offshore Mech Arct Eng 117(4):232CrossRefGoogle Scholar
- Sarpkaya T (2004) A critical review of the intrinsic nature of vortex-induced vibrations. J Fluids Struct 19(4):389–447CrossRefGoogle Scholar
- Schaefer JW, Eskinazi S (1959) An analysis of the vortex street generated in a viscous fluid. J Fluid Mech 6(02):241MathSciNetCrossRefGoogle Scholar
- Song Jn LuL, Teng B, Hi Park, Tang Gq WuH (2011) Laboratory tests of vortex-induced vibrations of a long flexible riser pipe subjected to uniform flow. Ocean Eng 38(11–12):1308–1322CrossRefGoogle Scholar
- Stephen N (2006) On energy harvesting from ambient vibration. J Sound Vib 293(1–2):409–425CrossRefGoogle Scholar
- Techet AH, Hover FS, Triantafyllou MS (1998) Vortical patterns behind a tapered cylinder oscillating transversely to a uniform flow. J Fluid Mech 363:79–96CrossRefGoogle Scholar
- van Oudheusden BW, Scarano F, van Hinsberg NP, Watt DW (2005) Phase-resolved characterization of vortex shedding in the near wake of a square-section cylinder at incidence. Exp Fluids 39(1):86–98CrossRefGoogle Scholar
- Vandiver JK, Jaiswal V, Jhingran V (2009) Insights on vortex-induced, traveling waves on long risers. J Fluids Struct 25(4):641–653CrossRefGoogle Scholar
- Vickery B, Watkins R (1964) Flow-induced vibrations of cylindrical structures. In: Hydraulics and fluid mechanics. Proceedings of the First Australasian Conference Held at the University of Western Australia, 6th to 13th December 1962. Elsevier, Pergamon, pp 213–241. https://doi.org/10.1016/B978-0-08-010291-7.50018-5 CrossRefGoogle Scholar
- Welch P (1967) The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans Audio Electroacoust 15(2):70–73CrossRefGoogle Scholar
- Wieneke B (2015) PIV uncertainty quantification from correlation statistics. Meas Sci Technol 26(7):074002CrossRefGoogle Scholar
- Williamson CHK (1996) Vortex dynamics in the cylinder wake. Annu Rev Fluid Mech 28(1):477–539MathSciNetCrossRefGoogle Scholar
- Williamson C, Govardhan R (2004) Vortex-induced vibrations. Annu Rev Fluid Mech 36(1):413–455MathSciNetCrossRefGoogle Scholar
- Williamson C, Roshko A (1988) Vortex formation in the wake of an oscillating cylinder. J Fluids Struct 2(4):355–381CrossRefGoogle Scholar
- Zdravkovich M (1981) Review and classification of various aerodynamic and hydrodynamic means for suppressing vortex shedding. J Wind Eng Ind Aerodyn 7(2):145–189CrossRefGoogle Scholar
- Zhou J, Adrian RJ, Balachandar S, Kendall TM (1999) Mechanisms for generating coherent packets of hairpin vortices in channel flow. J Fluid Mech 387:353–396MathSciNetCrossRefGoogle Scholar