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 ͑1988͔͒ 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.

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The three-dimensional, inviscid and viscous flow instability modes that appear on a solid-body rotation flow in a finite-length straight, circular pipe are analysed. This study is a direct extension of the Wang & Rusak (Phys. Fluids, vol. 8 (4), 1996a, pp. 1007–1016) analysis of axisymmetric instabilities on inviscid swirling flows in a pipe. The linear stability equations are the same as those derived by Kelvin (Phil. Mag., vol. 10, 1880, pp. 155–168). However, we study a general mode of perturbation that satisfies the inlet, outlet and wall conditions of a flow in a finite-length pipe with a fixed in time and in space vortex generator ahead of it. This mode is different from the classical normal mode of perturbations. The eigenvalue problem for the growth rate and the shape of the perturbations for any azimuthal wavenumber $m$ consists of a linear system of partial differential equations in terms of the axial and radial coordinates ($x,r$). The stability problem is solved numerically for all azimuthal wavenumbers $m$. The computed growth rates and the related shapes of the various perturbation modes that appear in sequence as a function of the base flow swirl ratio (${\it\omega}$) and pipe length ($L$) are presented. In the inviscid flow case, the $m=1$ modes are the first to become unstable as the swirl ratio is increased and dominate the perturbation’s growth in a certain range of swirl levels. The $m=1$ instability modes compete with the axisymmetric ($m=0$) instability modes as the swirl ratio is further increased. In the viscous flow case, the viscous damping effects reduce the modes’ growth rates. The neutral stability line is presented in a Reynolds number ($Re$) versus swirl ratio (${\it\omega}$) diagram and can be used to predict the first appearance of axisymmetric or spiral instabilities as a function of $Re$ and $L$. We use the Reynolds–Orr equation to analyse the various production terms of the perturbation’s kinetic energy and establish the elimination of the flow axial homogeneity at high swirl levels as the underlying physical mechanism that leads to flow exchange of stability and to the appearance of both spiral and axisymmetric instabilities. The viscous effects in the bulk have only a passive influence on the modes’ shapes and growth rates. These effects decrease with the increase of $Re$. We show that the inviscid flow stability results are the inviscid-limit stability results of high-$Re$ rotating flows.

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The incompressible, inviscid and axisymmetric dynamics of perturbations on a solid-body rotation flow with a uniform axial velocity in a rotating, finite-length, straight, circular pipe are studied via global analysis techniques and numerical simulations. The investigation establishes the coexistence of both axisymmetric wall-separation and vortex-breakdown zones above a critical swirl level, ω 1 . We first describe the bifurcation diagram of steady-state solutions of the flow problem as a function of the swirl ratio ω. We prove that the base columnar flow is a unique steady-state solution when ω is below ω 1 . This state is asymptotically stable and a global attractor of the flow dynamics. However, when ω > ω 1 , we reveal, in addition to the base columnar flow, the coexistence of states that describe swirling flows around either centreline stagnant breakdown zones or wall quasi-stagnant zones, where both the axial and radial velocities vanish. We demonstrate that when ω > ω 1 , the base columnar flow is a min-max point of an energy functional that governs the problem, while the swirling flows around the quasi-stagnant and stagnant zones are global and local minimizer states and become attractors of the flow dynamics. We also find additional min-max states that are transient attractors of the flow dynamics. Numerical simulations describe the evolution of perturbations on above-critical columnar states to either the breakdown or the wall-separation states. The growth of perturbations in both cases is composed of a linear stage of the evolution, with growth rates accurately predicted by the analysis of Wang & Rusak (Phys. Fluids, vol. 8, 1996a, pp. 1007-1016, followed by a stage of saturation to either one of the separation zone states. The wall-separation states have the same chance of appearing as that of vortex-breakdown states and there is no hysteresis loop between them. This is strikingly different from the dynamics of vortices with medium or narrow vortical core size in a pipe.