Nonlinear perturbation developments in flow around a vibrating cylinder

We investigate the flow around a circular cylinder vibrating with a prescribed magnitude and frequency at Reynolds number 230. The Navier-stokes equations and the adjoint equations are solved to search for the 3-D nonlinear optimal initial perturbation, which is amplified by the attached shear via the Orr mechanism and then activates wake instabilities. The energy growth and the spanwise distributions of the nonlinear optimal perturbation vary significantly with the initial phase to introduce the perturbation. As the initial perturbation energy increases, linear amplification, nonlinear saturation and nonlinear growth are sequently observed. The last one is owing to the generation of a spanwise homogeneous mode, which modifies the pressure distributions on the cylinder surface and induces an extra lift force. Interestingly, at all the initial phases, this force is opposite to the acceleration of the cylinder, illustrating the potential to reduce the magnitude of vibration.