Structure and Branching of Unstable Modes in a Swirling Flow

Swirling has a significant effect on the main characteristics of flow and can lead to its fundamental restructuring. On the flow axis, a stagnation point with zero velocity is possible, behind which a return flow zone is formed. The apparent instability leads to the formation of secondary vortex motions and can also be the cause of vortex breakdown. In the paper, a swirling flow with a velocity profile of the Batchelor vortex type has been studied on the basis of the linear hydrodynamic stability theory. An effective numerical method for solving the spectral problem has been developed. This method includes the asymptotic solutions at artificial and irregular singular points. The stability of flows was considered for the values of the Reynolds number in the range 10≤Re≤5×106. The calculations were carried out for the value of the azimuthal wavenumber parameter n=−1. As a result of the analysis of the solutions, the existence of up to eight simultaneously occurring unstable modes has been shown. The paper presents a classification of the detected modes. The critical parameters are calculated for each mode. For fixed values of the Reynolds numbers 60≤Re≤5000, the curves of neutral stability are plotted. Branching points of unstable modes are found. The maximum growth rates for each mode are determined. A new viscous instability mode is found. The performed calculations reveal the instability of the Batchelor vortex at large values of the swirl parameter for long-wave disturbances.