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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.

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Abbreviations

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

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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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Authors and Affiliations

  1. Department of Mechanical and Mechatronics Engineering, University of Waterloo, 200 University Ave W, Waterloo, ON, N2L 3G1, Canada

    Erik Marble & Serhiy Yarusevych

  2. Department of Mechanical and Manufacturing Engineering, University of Calgary, 2500 University Dr NW, Calgary, AB, T2N 1N4, Canada

    Chris Morton

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  1. Erik Marble
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Correspondence to Serhiy Yarusevych.

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Marble, E., Morton, C. & Yarusevych, S. Spanwise wake development of a pivoted cylinder undergoing vortex-induced vibrations with elliptic trajectories. Exp Fluids 60, 81 (2019). https://doi.org/10.1007/s00348-019-2725-2

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