Synchronization phenomenon of temperature oscillation in rotating fluid annulus and optimal waveforms of external forcing

The synchronization phenomena observed in traveling and oscillating thermal convection within a rotating fluid annulus are investigated using three-dimensional direct numerical simulations. The numerical simulations using the direct method calculate the phase-sensitivity and phase-coupling functions, thereby revealing that the waveforms followed sinusoidal patterns. This finding indicates that a similar synchronization state is achieved from any initial condition, as there are only one local maximum and minimum value. The theoretical synchronization criteria provide accurate predictions of the synchronization region. Furthermore, an individual comparing the period difference between the forcing oscillation period τf and the effective oscillation period τe of the system is found to be insufficient to determine the synchronization state. An accurate assessment requires considering the mean values and magnitude of the standard deviation in the oscillation period of the system. Three optimal waveforms-each optimized with respect to the entrainment range, the entrainment speed, and the duty cycle-are calculated. The waveforms obtained within the entrainment range and the entrainment speed also approximately exhibit a sinusoidal pattern owing to roughly a sinusoidal phase-sensitivity function of the system. Consequently, the synchronization region for both methods exhibits minimal extension. However, the maximum entrainment range is theoretically obtained for an optimal duty cycle of 50%, thereby resulting in an entrainment range that is 12% larger than that for a 100% duty cycle. Numerical experiments confirm that the optimal waveform enlarges the synchronization region.