Energy transfer mechanism of the instability of an axisymmetric swirling flow in a finite-length pipe

Abstract

The rate of change of the perturbation's kinetic energy E of a perturbed inviscid, incompressible, axisymmetric, columnar and near-critical swirling flow in a finite-length, straight, circular pipe with periodic and non-periodic inlet–outlet conditions is studied using the Reynolds–Orr equation. The perturbation's mode shape and growth rate are computed from the linear-stability eigenvalue problem using a novel asymptotic solution in the case of a flow in a long pipe. This solution technique is general and can be applied to any vortex flow profile, in a range of swirl levels around the critical level, and for various boundary conditions. The solutions are used to analytically estimate the production (or loss) of E at the pipe boundaries and inside the domain and to shed new light on the Wang–Rusak mechanism of exchange of global stability around the critical swirl, that is leading to the vortex breakdown process. It is shown that the production of E inside the domain is modulated by the base flow strain-rate tensor. For the special case of a solid-body rotating flow, this term vanishes and the stability is determined only by the asymmetric transfer of E at the boundaries. For a general base flow, the dominant perturbation's mode shape develops deviations in response to the non-periodic inlet–outlet conditions. These deviations couple with the base flow strain-rate tensor to generate production or loss of E in the bulk. Together with the asymmetric transfer of E at the boundaries, they form a critical balance of production of E and determine the flow stability around the critical state. This behaviour is demonstrated for the Lamb–Oseen and Q vortex models. This analysis reveals a more complicated, as well more realistic, interaction between the perturbed flow in the domain and at the boundaries that dominates vortex flow dynamics.