Abstract:
The nonlinear stability of columnar swirling flows in a finite constant-area pipe with periodic boundary conditions imposed at the inlet and outlet of the pipe is investigated by the Arnold energy-Casimir method. Szeri and Holmes [“Nonlinear stability of axisymmetric swirling flows,” Philos. Trans. R. Soc. London, Ser. A 326, 327 (1988) ] applied the Arnold energy-Casimir method to the swirling flow and developed a general form of Arnold function, which forms the basis for the current investigation. It has been shown that for the base flow which has a uniform axial velocity profile, a reduced form of the Arnold function can be used to obtain nonlinear stability. This essentially extends Rayleigh’s linear stability theory to the nonlinear theory which is applicable to finite-amplitude disturbances, and leads to several nonlinear stability criteria which can readily be applied to swirling flows with various configurations. The nonlinear stability criteria are then applied to the Lamb–Oseen vortex and the Taylor–Couette flow, and crucial information about the disturbance’s size in terms of its kinetic energy has been obtained. A notable feature of the nonlinear stability is that it is applicable to a base flow which is linearly unstable, which was demonstrated by the linearly unstable Taylor–Couette flow.
