Global mode interaction and pattern selection in the wake of a disk: a weakly nonlinear expansion

Direct numerical simulations (DNS) of the wake of a circular disk placed normal to a uniform flow show that, as the Reynolds number is increased, the flow undergoes a sequence of successive bifurcations, each state being characterized by specific time and space symmetry breaking or recovering (Fabre, Auguste & Magnaudet, Phys. Fluids, vol. 20 (5), 2008, p. 1). To explain this bifurcation scenario, we investigate the stability of the axisymmetric steady wake in the framework of the global stability theory. Both the direct and adjoint eigenvalue problems are solved. The threshold Reynolds numbers Re and characteristics of the destabilizing modes agree with the study of Natarajan & Acrivos (J. Fluid Mech., vol. 254, 1993, p. 323): the first destabilization occurs for a stationary mode of azimuthal wavenumber m = 1 at Re-c(A) = 116.9, and the second destabilization of the axisymmetric flow occurs for two oscillating modes of azimuthal wavenumbers m +/- 1 at Re-c(B) = 125.3. Since these critical Reynolds numbers are close to one another, we use a multiple time scale expansion to compute analytically the leading-order equations that describe the nonlinear interaction of these three leading eigenmodes. This set of equations is given by imposing, at third order in the expansion, a Fredholm alternative to avoid any secular term. It turns out to be identical to the normal form predicted by symmetry arguments. Though, all coefficients of the normal form are here analytically computed as the scalar product of an adjoint global mode with a resonant third-order forcing term, arising from the second-order base flow modification and harmonics generation. We show that all nonlinear interactions between modes take place in the recirculation bubble, as the contribution to the scalar product of regions located outside the recirculation bubble is zero. The normal form accurately predicts the sequence of bifurcations, the associated thresholds and symmetry properties observed in the DNS calculations.