Experimental and numerical investigation of the competition between axisymmetric time-periodic modes in an enclosed swirling flow

Time-periodic vortex flows in an enclosed circular cylinder driven by the rotation of one endwall are investigated experimentally and numerically. This work is motivated partly by the linear stability analysis of Gelfgat [J. Fluid Mech. 438, 363 (2001)], which showed the existence of an axisymmetric double Hopf bifurcation, and the purpose of the experiment is to see if the nonlinear dynamics associated with this double Hopf bifurcation can be captured under laboratory conditions. A glycerin/water mixture was used in a cylinder with variable height-to-radius ratios between Γ =1.67 and 1.81, and Reynolds numbers between Re=2600 and 2800 (i.e., in the neighborhood of the double Hopf). Hot-film measurements provide, for the first time, experimental evidence of the existence of an axisymmetric double Hopf bifurcation, involving the competition between two stable coexisting axisymmetric limit cycles with periods (nondimensionalized by the rotation rate of the endwall) of approximately 31 and 22. The dynamics is also captured in our nonlinear computations, which clearly identify the double Hopf bifurcation as "type I simple," with the characteristic signatures that the two Hopf bifurcations are supercritical and that there is a wedge-shaped region in (Γ, Re) parameter space where both limit cycles are stable, delimited by Neimark-Sacker bifurcation curves.