Analysis of transitions between fluttering, tumbling and steady descent of falling cards

We study the dynamics of a rigid card falling in air using direct numerical simulations of the two-dimensional Navier-Stokes equation and a fluid force model based on ordinary differential equations derived from recent experiments and simulations. The system depends on three non-dimensional parameters, i.e. the thickness-to-width ratio, the dimensionless moment of inertia, and the Reynolds number. By increasing the thickness-to-width ratio in the direct numerical simulations and thereby the non-dimensional moment of inertia we observe a transition from periodic fluttering to periodic tumbling with a wide transition region in which the cards flutter periodically but tumble once between consecutive turning points. In the transition region the period of oscillation diverges and the cards fall vertically for distances of up to 50 times the card width. We analyse the transition between fluttering and tumbling in the ODE model and find a heteroclinic bifurcation which leads to a logarithmic divergence of the period of oscillation at the bifurcation point. We further discuss the bifurcation scenario of the ODE model in relation to our direct numerical simulations and the phase diagrams measured by Willmarth, Hawk & Harvey (1964) and Belmonte, Eisenberg & Moses (1998).