Lock-in phenomenon of vortex shedding in flows excited with two commensurate frequencies: a theoretical investigation pertaining to combustion instability

Abstract

An analytical investigation is performed to study the dynamics of vortex shedding behaviour during two commensurate frequency velocity excitations, with an emphasis on the phenomenon of lock-in. We attempt to theoretically study the dynamical features of lock-in under two-frequency excitations and contrast the behaviour with single-frequency excitation. We employ an existing low-order model to characterise the vortex shedding process behind a step/bluff-body. The continuous-time domain model is transformed to a nonlinear dynamical map that relates time instances of successive vortex shedding. Further, these time instances are converted to phase instances, involving which criteria for a generic p : q phase lock-in is obtained. Four parameters are involved: amplitude and frequency of the two excitation components termed as primary and secondary. Bifurcations occurring are investigated using return maps. The inclusion of secondary excitation leads to the existence of two orders of lock-in within a single lock-in boundary. Furthermore, our results indicate that secondary excitation can be used as a control in order to tailor the 1:1 lock-in region formed by the primary excitation. Finally, analytical expressions are obtained to identify lock-in boundaries and their salient geometrical features. Interesting features such as the occurrence of bistability and change in the order of lock-in are observed, which can be explored further with future experiments.