An asymptotic expansion for the vortex-induced vibrations of a circular cylinder

Abstract

This paper investigates the vortex-induced vibrations (VIV) of a spring-mounted circular cylinder. We compute analytically the leading-order equations describing the nonlinear interaction of the fluid and structure modes by carrying out an asymptotic analysis of the Navier–Stokes equations close to the threshold of instability of the fluid-only system. We show that vortex-shedding can occur at subcritical Reynolds numbers as a result of the coupled system being linearly unstable to the structure mode. We also show that resonance occurs when the frequency of the nonlinear limit cycle matches the natural frequency of the cylinder, the displacement being then in phase with the flow-induced lift fluctuations. Using an extension of this model meant to encompass the effect of the low-order added-mass and damping forces induced by the displaced fluid, we show that the amount of energy that can be extracted from the flow can be optimized by an appropriate choice of the structural parameters. Finally, we suggest a possible connection between the present ‘exact’ model and the empirical wake oscillator model used to study VIV at high Reynolds numbers. We show that for the low Reynolds numbers considered here, the effect of the structure on the fluid can be represented by a first coupling term proportional to the cylinder acceleration in the fluid equation, and by a second term of lower magnitude, which can stem either from an integral term or from a term proportional to the third derivative of the cylinder position.