STABILITY THEORY FOR THREE - DIMENSIONAL WAKETRANSITIONDwight

A nonlinear stability theory is presented for three-dimensional transition in the wake of a circular cylinder up to Reynolds number 300. The primary input to the theory is the linear and the nonlinear stability results of Barkley and Hen-derson. Based on symmetric bifurcation theory, normal form equations are proposed to describe (to lowest order) the nonlin-ear interaction of three-dimensional shedding modes. It is shown that the nonlinear interaction naturally leads to the transition scenario observed experimentally by Williamson et al. NOMENCLATURE Re Reynolds number deened by Re U 1 d==, where U 1 is the free-stream velocity far from the cylinder, d is the cylinder diameter, and is the kinematic viscosity. U Two-dimensional, time-periodic wake ow with components (U; V). ~ u Floquet mode. Spanwise wavenumber of three-dimensional modes. Spanwise wavelength of three-dimensional modes. Floquet multiplier, with jj > 1 corresponding to linear instability. A n Amplitude of Mode A at shedding cycle n. B n Amplitude of Mode B at shedding cycle n.