Experimental investigation of the second oscillatory bifurcation past a stationary sphere

We report experimental results on the low-frequency dynamics in the wake of a stationary sphere and the corresponding bifurcation. This low frequency is the second frequency appearing past a sphere when the Reynolds number is increased. We measure the velocity field using particle image velocimetry in transverse planes past a sphere placed in a water channel, and obtain the streamwise vorticity. We compute the position of the barycentre of the absolute value of the vorticity and projected the vorticity on azimuthal mode 1. We deduce from periodograms of these signals the Strouhal numbers St1 and St2 associated with the first ( f1) and second ( f2) frequencies. The frequency ratio f2/ f1 decreases with the Reynolds number, and is close to 2/7 near the threshold. The two frequencies are generally incommensurate. In the periodograms, many combinations of f1 and f2 are present, in particular f1 + f2 and f1 − f2, consistent with a T 2 torus dynamics. We also obtain the squared magnitude associated with f1 and f2. The linear variation of this squared magnitude above the threshold Re1 and the variation of the frequency f1 confirm that this first oscillatory bifurcation is a supercritical Hopf bifurcation. Similar variations for f2 above the corresponding threshold Re2 show that the second oscillatory bifurcation is a supercritical secondary Hopf bifurcation (supercritical Neimark–Sacker bifurcation). The component at f2 of the vorticity field remains significant even 9 diameters downstream
of the sphere

Guy-Jean Michon, Jose Eduardo Wesfreid and Benoit Semin, Journal of Fluid Mechanics , Volume 1039 , 25 July 2026 , A3, DOI : https://doi.org/10.1017/jfm.2026.11768