A comparison between the experimental visualization and numerical simulations of the occurrence of vortex breakdown in laminar swirling flows produced by a rotating endwall is presented. The experimental visualizations of Escudier (1984) were the first to detect the presence of multiple recirculation zones and the numerical model presented here, consisting of a numerical solution of the unsteady axisymmetric Navier-Stokes equations, faithfully reproduces these phenomena and all other observed characteristics of the flow. Further, the numerical calculations elucidate the onset of oscillatory flow, an aspect of the flow that was not clearly resolved by the flow visualization experiments. Part 2 of the paper examines the underlying physics of these vortex flows.

The physical mechanisms for vortex breakdown which, it is proposed here, rely on the production of a negative azimuthal component of vorticity, are elucidated with the aid of a simple, steady, inviscid, axisymmetric equation of motion. Most studies of vortex breakdown use as a starting point an equation for the azimuthal vorticity (Squire 1960), but a departure in the present study is that it is explored directly and not through perturbations of an initial stream function. The inviscid equation of motion that is derived leads to a criterion for vortex breakdown based on the generation of negative azimuthal vorticity on some stream surfaces. Inviscid predictions are tested against results from numerical calculations of the NavierStokes equations for which breakdown occurs.

The bifurcation structure is presented for an axisymmetric swirling flow in a constricted pipe, using the pipe geometry of Beran and Culick [J. Fluid Mech. 242, 491 (1992)]. The flow considered has been restricted to a two-dimensional parameter space comprising the Reynolds number Re and the relative swirl V, of the incoming swirling flow. The bifurcation diagram is constructed by solving the time-dependent axisymmetric Navier-Stokes equations. The stability of the steady results presented by Beran and Culick, obtained from a steady axisymmetric Navier-Stokes code, has been confirmed. Further, the steady solution branch has also been extended to much larger V, values. At larger V,,, a stable unsteady solution branch has been identified. This unsteady branch coexists with the previously found stable steady solution branch and originates via a turning point bifurcation. The bifurcation diagram is of the type described by Benjamin [Proc. R. Sot. London Ser. A 359, 1 (1978)] as the canonical unfolding of a pitchfork bifurcation. This type of bifurcation structure in the two-dimensional parameter space (Re,V& suggests the possibility of hysteresis behavior over some part of parameter space, and this is observed in the present study. The implications of this on the theoretical description of vortex breakdown and the search for a criterion for its onset are discussed.

A number of two-dimensional time-periodic flows, for example the Kármán street wake of a symmetrical bluff body such as a circular cylinder, possess a spatio-temporal symmetry: a combination of evolution by half a period in time and a spatial reflection leaves the solution invariant. Floquet analyses for the stability of these flows to three-dimensional perturbations have in the past been based on the Poincaré map, without attempting to exploit the spatio-temporal symmetry. Here, Floquet analysis based on the half-periodflip map provides a comprehensive interpretation of the symmetry breaking bifurcations.

Previous studies dealing with Floquet secondary stability analysis of the wakes of circular and square cross-section cylinders have shown that there are two synchronous instability modes, with long (mode A) and short (mode B) spanwise wavelengths. At intermediate wavelengths another mode arises, which reaches criticality at Reynolds numbers higher than modes A or B. Here we concentrate on these intermediate-wavenumber modes for the wakes of circular and square cylinders. It is found in both cases that the modes arise with complex-conjugate pair Floquet multipliers, and can be combined to produce either standing or travelling waves. Both these states are quasi-periodic.