On the stability of an axisymmetric rotating flow in a pipe

Abstract

The linear stability of an inviscid, axisymmetric and rotating columnar flow in a finite length pipe is studied. A well posed model of the unsteady motion of swirling flows with compatible boundary conditions that may reflect the physical situation is formulated. A linearized set of equations for the development of infinitesimal axially-symmetric disturbances imposed on a base rotating columnar flow is derived. Then, a general mode of axisymmetric disturbances, that is not limited to the axial-Fourier mode, is introduced and an eigenvalue problem is obtained. Benjamin’s critical state concept is extended to the case of a rotating flow in a finite length pipe. It is found that a neutral mode of disturbance exists at the critical state. In the case of a solid body rotating flow with a uniform axial velocity component, analytical solution of the eigenvalue problem is found. It is demonstrated that the flow changes its stability characteristics as the swirl changes around the critical level. When the flow is supercritical an asymptotically stable mode is found, and when the flow is subcritical, an unstable mode of disturbance may develop. This result cannot be predicted by Rayleigh’s classical stability criterion. In the case of a general columnar swirling flow in a pipe, the asymptotic solution of the eigenvalue problem around the critical state is also studied. It is shown that the critical swirl ratio is a point of exchange of stability for any swirling flow in a finite length pipe. This result reveals an unknown instability mechanism of swirling flows that cannot be detected by previous stability analyses and sheds new light on the relation between stability of vortex flows and the vortex breakdown phenomenon.

Vortical flow. Part 1. Flow through a constant-diameter pipe

“…Figure 23(b) shows the Rossby number Ro = w 1 /r 1 ω introduced by Spall, Gatski & Grosch (1987), where w 1 is the axial velocity at the radial position of maximum Rusak, Whiting & Wang (1998b), where v max is the maximum azimuthal velocity and w ctr is the axial velocity at the axis of symmetry. There are two critical values (v max /w ctr ) 0 and (v max /w ctr ) 1 which follow from the rigorous theoretical study of inviscid axisymmetric vortex breakdown by Wang & Rusak (1996a, b, 1997b. For v max /w ctr < (v max /w ctr ) 0 , only one steady-state solution is possible, which is columnar.…”

“…If breakdown had occurred (the transients associated with starting the apparatus are extremely large), it would have been located downstream of the working section and therefore close to the orifice. Wang & Rusak (1996a, b, 1997b use a boundary condition which is not compatible with the converging flow close to an orifice. On the other hand, it is possible that had profiles been available further upstream, these might have explicitly predicted the absence of vortex breakdown.…”

Control of axisymmetric vortex breakdown in a constricted pipe: Nonlinear steady states and weakly nonlinear asymptotic expansions

“…[23][24][25] These authors proposed an inviscid mechanism relying on the upstream propagation of disturbance waves in the pipe and on their interaction with the inlet boundary, and further treated the case of weak viscosity and weak variations in the pipe geometry as perturbations from this "ideal" case. [26][27][28][29] Such an approach has provided relevant models consistent with previous and later experimental and numerical results, helping to understand the occurrence of subcritical vortex breakdown (see for instance the recent experiments by Mattner et al 30 ).…”

Novel liquid blanket configurations and their hydrodynamic analyses for innovative confinement concepts

“…There are studies performed for high velocity flow over concave surfaces and swirl chambers (which is totally filled with fluid for combustion studies) separately [8][9][10][11]. The experimental information obtained from these studies suggests that the radial distribution of the tangential flow is divided into two regions, i.e.…”