A self-consistent model for the saturation dynamics of the vortex shedding around the mean flow in the unstable cylinder wake

The supercritical instability leading to the Benard-von Karman vortex street in a cylinder wake is a well known example of supercritical Hopf bifurcation: the steady solution becomes linearly unstable and saturates into a periodic limit cycle. Nonetheless, a simplified physical formulation accurately predicting the transition dynamics of the saturation process is lacking. Building upon our previous work, we present here a simple self-consistent model that provides a clear description of the saturation mechanism in a quasi-steady manner by means of coupling the instantaneous mean flow with its most unstable eigenmode and its instantaneous amplitude through the Reynolds stress. The system is coupled for different oscillation amplitudes, providing an instantaneous mean flow as function of an equivalent time. A transient physical picture is described, wherein a harmonic perturbation grows and changes in amplitude, frequency, and structure due to the modification of the mean flow by the Reynolds stress forcing, saturating when the flow is marginally stable. Comparisons with direct numerical simulations show an accurate prediction of the instantaneous amplitude, frequency, and growth rate, as well as the saturated mean flow, the oscillation amplitude, frequency, and the resulting mean Reynolds stresses. (C) 2015 AIP Publishing LLC.