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

  • Erik Marble
  • Chris Morton
  • Serhiy YarusevychEmail author
Research Article
  • 54 Downloads

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 xyz 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, \(\mathbf {u}_{\mathrm{RMS}}=u_{\mathrm{RMS}}\hat{i}+v_{\mathrm{RMS}}\hat{j}+w_{\mathrm{RMS}}\hat{k}\)

xyz

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.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mechanical and Mechatronics EngineeringUniversity of WaterlooWaterlooCanada
  2. 2.Department of Mechanical and Manufacturing EngineeringUniversity of CalgaryCalgaryCanada

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