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

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


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


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


Temporal POD coefficients


Cylinder diameter


Wake half-width


Natural frequency in quiescent water


Vortex shedding frequency of a stationary cylinder


Frequency of the streamwise and transverse velocity signal, respectively

\(f_x, f_y\)

Frequency of streamwise and transverse vibrations, respectively


Moment of inertia of the cylinder about the pivot point


Moment of inertia of the displaced fluid about the pivot point


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

\(\hat{i}, \hat{j}, \hat{k}\)

Unit vectors in xyz directions, respectively


Spring stiffness coefficient


Length of cylinder


Formation length


Mass of the cylinder


Mass of displaced fluid


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


Power spectrum density


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

\(\mathbf {U}\)

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


Free stream velocity


Local velocity deficit


Velocity at the transverse extent of the wake measurements


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


Streamwise, transverse and spanwise directions, respectively



\(\varDelta \theta\)

Phase bin size


Damping ratio


Phase angle of the cylinder’s elliptic orbit

\(\lambda _{i}\)

POD mode energy


Kinematic viscosity of water

\(\mathbf {\phi _i}\)

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


Phase angle between streamwise and transverse motion

\(\omega _z\)

Spanwise vorticity



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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