A Seven Mode Truncation of the Kolmogorov Flow with Drag: Analysis and Control

Abstract

The transition from laminar to chaotic motions in a viscous °uid °ow is in- vestigated by analyzing a seven dimensional dynamical system obtained by a truncation of the Fourier modes for the Kolmogorov °ow with drag friction. An- alytical expressions of the Hopf bifurcation curves are obtained and a sequence of period doubling bifurcations are numerically observed as the Reynolds num- ber is increased for ¯xed values of the drag parameter. An adaptive stabilization of the system trajectories to an equilibrium point or to a periodic orbit is ob- tained through a model reference approach which makes the control global. Finally, the e®ectiveness of this control strategy is numerically illustrated.