Mode-locking and quasi-periodicity in the bifurcation behaviour of a vortex shedding model

Vortex-shedding at low Reynolds numbers in the wake of an oscillating cylinder has been shown to exhibit characteristics universal to nonlinear dynamical systems with competing frequencies, generically referred to as quasi-periodic systems, The (1:1) lock-on phenomenon is the best known and, to date, of most interest in engineering applications. At lock-in, frequency entrainment occurs so that the characteristic frequencies of the two coupled oscillators coincide. The 1:1 lock-in is one in a potentially large (infinite) number of possible m:n (frequency ratio) mode-lockings for nonlinear systems with competing frequencies. In this paper, the lock-in behavior of an analytical model for vortex-shedding at high Reynolds number is studied, An analysis of the model shows that the frequency characteristics of a cylinder under forced excitation are dominated by m:n (rational frequency ratio) mode-locking as the flow velocity (shedding frequency) is varied. Hence, when the cylinder is free to oscillate, vortex shedding is no longer expressed by a constant Strouhal number. Instead the shedding frequency increases in discontinuous steps. Each step marks a jump from one mode-locked m:n state to the next. Although the model corresponds to tube excitation in the lift direction, it is anticipated that the observed behavior is also reflected in the drag direction in view of the fact that vortex shedding interacts with the cylinder in both directions simultaneously. The results may explain the observation by King (1977) that "... the ratio of cylinder frequency to wake frequency apparently 'slips' into convenient numbers ....7:2, 3:1, 7:3, 13:6 ......." in the drag direction, The results are practically significant in view of a recent temperature probe failure in a Japanese fast breeder reactor caused by drag-direction vortex excitation. The failure was attributed to a mode locked resonance (symmetric vortex shedding) which commences at a 4:1 frequency ratio.