J. M. Lopez

Axisymmetric Vortex Breakdown. Part 2. Physical Mechanisms

Abstract : Numerical solutions of the axisymmetric Navier Stokes equations are presented and compared with results from experiments for a confined cylindrical flow. The details of the vortex breakdown phenomenon are calculated with a high degree of accuracy. From solutions over a range of parameters the essential features of the flow are obtained. These solutions also provide flow quantities such as the vorticity and the pressure throughout the volume which would be difficult to obtain from experiments. The solutions are explored and the essential physical mechanisms of vortex breakdown in this particular geometry are identified. These mechanisms, which rely on the production of a negative azimuthal component of vorticity as a result of the stretching and tilting of the predominantly axially directed vorticity vector, are elucidated with the aid of a simple, steady, inviscid, axisymmetric equation of motion. This equation has been a starting point for most studies of vortex breakdown but a departure in the present study is that it is explored directly and not through perturbations of an initial stream function. The findings are then generalised to the case of vortex breakdown in swirling pipe flows. Australia.

AXISYMMETRIC VORTEX BREAKDOWN, PART 1. CONFINED SWIRLING FLOW

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.

Axisymmetric vortex breakdown. Part 3. Onset of periodic flow and chaotic advection

When the fluid inside a completely filled cylinder is set in motion by the rotation of one endwall, steady and unsteady axisymmetric vortex breakdown is possible. Nonlinear dynamical systems theory is used to describe the changing kinematics of the flow as the speed of the rotating endwall is increased. Two distinct modes of oscillation have been found in the unsteady regime and the chaotic advection caused by the oscillations has been investigated. The results of this study are used to describe the filling and emptying processes of the vortex breakdown bubbles observed in flow visualization experiments.

ON THE BIFURCATION STRUCTURE OF AXISYMMETRIC VORTEX BREAKDOWN IN A CONSTRICTED PIPE

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 V0 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 V0 values. At larger V0, 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. Soc. London Ser. A 359, 1 (1978)] as the canonical unfolding of a pit...

On three-dimensional quasiperiodic Floquet instabilities of two-dimensional bluff body wakes

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-wave number modes for the wakes of circular and square cylinders. It is found that in both cases these modes possess complex-conjugate pair Floquet multipliers, and can be combined to produce either standing or traveling waves. Both these states are quasiperiodic.

Symmetry breaking of two-dimensional time-periodic wakes

A number of two-dimensional time-periodic flows, for example the Karman 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 Poincare map, without attempting to exploit the spatio-temporal symmetry. Here, Floquet analysis based on the half-period- flip map provides a comprehensive interpretation of the symmetry breaking bifurcations.

Oscillatory flow states in an enclosed cylinder with a rotating endwall

A combined experimental and numerical investigation is presented of the multiple oscillatory states that exist in the flows produced in a completely filled, enclosed, circular cylinder driven by the constant rotation of one of its endwalls. The flow in a cylinder of height to radius ratio 2.5 is interrogated experimentally using flow visualization and digitized images to extract quantitative temporal information. Numerical solutions of the axisymmetric Navier–Stokes equations are used to study the same flow over a range of Reynolds numbers where the flow is observed to remain axisymmetric. Three oscillatory states have been identified, two of them are periodic and the third is quasi-periodic with a modulation frequency much smaller than the base frequency. The range of Reynolds numbers for which the quasi-periodic flow exists brackets the switch between the two periodic states. The results from the combined experimental and numerical study agree both qualitatively and quantitatively, providing unambiguous evidence of the existence and robustness of these multiple time-dependent states.

Mode interactions in an enclosed swirling flow: a double Hopf bifurcation between azimuthal wavenumbers 0 and 2

A double Hopf bifurcation has been found of the flow in a cylinder driven by the rotation of an endwall. A detailed analysis of the multiple solutions in a large region of parameter space, computed with an ecient and accurate three-dimensional Navier{ Stokes solver, is presented. At the double Hopf point, an axisymmetric limit cycle and a rotating wave bifurcate simultaneously. The corresponding mode interaction generates an unstable two-torus modulated rotating wave solution and gives a wedgeshaped region in parameter space where the two periodic solutions are both stable. By exploring in detail the three-dimensional structure of the flow, we have identied the two mechanisms that compete in the neighbourhood of the double Hopf point. Both are associated with the jet that is formed when the Ekman layer on the rotating endwall is turned by the stationary sidewall.

Symmetry breaking of the flow in a cylinder driven by a rotating end wall

The flow driven by a rotating end wall in a cylindrical container with aspect ratio H/R=2.5 is time dependent for Reynolds numbers Re=ΩR2/ν>2700. For Reynolds numbers up to 4000 three solution branches have been identified, and we examine a solution on each one. At Re=3000, the flow is axisymmetric and time periodic. At Re=3500, the flow is quasiperiodic with a low-frequency modulation and supports a modulated rotating wave with azimuthal wave number k=5. At Re=4000, the flow is time periodic with a qualitatively different mode of oscillation to that at Re=3500. It also supports a modulated rotating wave, with k=6. The peak kinetic energy of the nonaxisymmetric modes is associated with the jet-like azimuthal flow in the interior.

Instability and mode interactions in a differentially driven rotating cylinder

The flow in a completely filled rotating cylinder driven by the counter-rotation of the top endwall is investigated both numerically and experimentally. The basic state of this system is steady and axisymmetric, but has a rich structure in the radial and axial directions. The most striking feature, when the counter-rotation is sufficiently large, is the separation of the Ekman layer on the top endwall, producing a free shear layer that separates regions of flow with opposite senses of azimuthal velocity. This shear layer is unstable to azimuthal disturbances and a supercritical symmetry-breaking Hopf bifurcation to a rotating wave state results. For height-to-radius ratio of 0.5 and Reynolds number (based on cylinder radius and base rotation) of 1000, rotating waves with azimuthal wavenumbers 4 and 5 co-exist and are stable over an extensive range of the ratio of top to base rotation. Mixed modes and period doublings are also found, and a bifurcation diagram is determined. The agreement between the Navier–Stokes computations and the experimental measurements is excellent. The simulations not only capture the qualitative features of the multiple states observed in the laboratory, but also quantitatively replicate the parameter values over which they are stable, and produce accurate precession frequencies of the various rotating waves.