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Bifurcations of vortex-induced vibrations of a fixed membrane wing at Re \(\le \) 1000

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Abstract

Vortex-induced vibrations (VIVs) of a fixed two-dimensional perimeter-reinforced (PR) membrane wing at \(0\le \alpha \) Re (Reynolds number) \(\le \) 1000 and \(0^\circ \le \alpha \) (angle of attack) \(\le \) 30\(^{\circ }\) are investigated using fluid–structure interaction simulations. By employing very fine increments for Re and \(\alpha \), bifurcation boundaries of the dynamic response of the membrane wing in the Re–\(\alpha \) plane are captured. With increase in Re and/or \(\alpha \), it is found that the VIV state of a fixed PR membrane wing will change progressively from static state to period 1 via a Hopf bifurcation and then from period 1 to multiple period and chaos via a succession of period-doubling bifurcations. The Hopf bifurcation is triggered by the shedding of the leading- and/or trailing-edge vortices, while the period-doubling bifurcations are induced by the appearance and evolution of the secondary vortices on the upper surface of the membrane wing at higher Re and \(\alpha \). With an increase in the structure rigidity or pre-strain, the overall responses of the membrane wing are not changed much in the Re–\(\alpha \) plane except that the period 1 response near \(700\le Re\le 1000\) and \(14^{\circ }\le \alpha \le 16^{\circ }\) is destroyed, due to the significant change of the shedding process of the leading-edge vortices. Moreover, it is also found that unsteady responses of the PR membrane wing at \(\alpha =0^{\circ }\) can be suppressed by small pre-strain.

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Abbreviations

\(\alpha \) :

Angle of attack of the membrane wing

c :

Chord length of the membrane wing

\(C_\mathrm{d} \) :

Structural damping normalized by \(u_\infty \)

\(\bar{C}_\mathrm{L}\) :

Mean lift coefficient

\(\bar{C}_\mathrm{D}\) :

Mean drag coefficient

\(\delta _0\) :

Membrane pre-strain

\(\Delta p\) :

Pressure difference between the lower and upper surfaces of the membrane normalized by \(\rho _\infty u_\infty ^2\)

\(\Delta t\) :

Time step normalized by \(c/{u_\infty }\)

\(\xi \) :

Coordinate of the local coordinate system on the flexible membrane normalized by c

E :

Elastic modulus of the membrane normalized by \(\rho _\infty u_\infty ^2\)

f :

Frequency normalized by \({u_\infty }/c\)

h :

Membrane thickness normalized by c

L :

Membrane length before deforming

\({L}'\) :

Membrane length before deforming normalized by c

\(L_\mathrm{S}\) :

Membrane length after deforming normalized by c

\(n_\mathrm{P} \) :

Total number of grid nodes in the flow domain

\(n_\mathrm{E} \) :

Total number of grid elements in the flow domain

\(n_\mathrm{M}\) :

Total number of grid elements on the flexible membrane

p :

Pressure normalized by \(\rho _\infty u_\infty ^2 \)

Re :

Reynolds number with respect to the chord c and velocity of the free stream \(u_\infty \)

\(\rho _\infty \) :

Density of the incompressible flow

\(\rho _\mathrm{S}\) :

Membrane density per unit length normalized by \(\rho _\infty \)

t :

Time normalized by \(c/{u_\infty }\)

T :

Membrane tension normalized by \({\rho _\infty u_\infty ^2 }/c\)

\(u_\infty \) :

Velocity component of the free stream in x direction

\(u_\mathrm{x}, u_\mathrm{y}\) :

Velocity components of the flow field in x and y directions normalized by \(u_\infty \)

v :

Membrane velocity normalized by \(u_\infty \)

\(_{x,\,y}\) :

Coordinate components of the flow domain normalized by c

z :

Membrane displacement normalized by c

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 51506224), Opening Fund of State Key Laboratory of Nonlinear Mechanics and Science Foundation of China University of Petroleum-Beijing (No. C201602). The author would like to thank for the kindly support of these foundations.

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Authors and Affiliations

  1. National Engineering Laboratory for Pipeline Safety/MOE Key Laboratory of Petroleum Engineering, China University of Petroleum-Beijing, Beijing, 102249, China

    Xu Sun & Ze-Hua Ye

  2. The State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190, China

    Shi-Zhao Wang

  3. School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, 710049, China

    Jia-Zhong Zhang

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  1. Xu Sun
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Correspondence to Xu Sun.

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Sun, X., Wang, SZ., Zhang, JZ. et al. Bifurcations of vortex-induced vibrations of a fixed membrane wing at Re \(\le \) 1000. Nonlinear Dyn 91, 2097–2112 (2018). https://doi.org/10.1007/s11071-017-4004-1

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