Energy stability analysis for impulsively decelerating swirl flows

The onset of Taylor–Görtler vortices in impulsively decelerating swirl flows is analyzed by using the recently developed, relative stability model. This model takes the growth rate of the kinetic energy of the base state and also that of disturbances into consideration. In the present system the primary transient swirl flow is laminar but for the Reynolds number $Re>Rec$ secondary motion sets in at a certain time. The present model yields the critical Reynolds number of $Rec=153$⁠. This value is larger than that from the strong stability model, but smaller than that from the propagation theory. For $Re>Rec$ the dimensionless critical time to mark the onset of vortex instabilities, $τc$⁠, is presented as a function of Re. It is found that the predicted $τc$ value is much smaller than experimental detection time of first observable secondary motion. Therefore, it seems evident that small disturbances initiated at $τc$ require some growth period until they are detected experimentally. Since the present system is a rather simple one, the present results will be helpful in comparing available stability models.