The dynamics of a swirling flow in a pipe and transition to axisymmetric vortex breakdown

This paper provides a new study of the axisymmetric vortex breakdown phenomenon. Our approach is based on a thorough investigation of the axisymmetric unsteady Euler equations which describe the dynamics of a swirling flow in a finite-length constant-area pipe. We study the stability characteristics as well as the time-asymptotic behaviour of the flow as it relates to the steady-state solutions. The results are established through a rigorous mathematical analysis and provide a solid theoretical understanding of the dynamics of an axisymmetric swirling flow. The stability and steady-state analyses suggest a consistent explanation of the mechanism leading to the axisymmetric vortex breakdown phenomenon in high-Reynolds-number swirling flows in a pipe. It is an evolution from an initial columnar swirling flow to another relatively stable equilibrium state which represents a flow around a separation zone. This evolution is the result of the loss of stability of the base columnar state when the swirl ratio of the incoming flow is near or above the critical level.

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The evolution of a perturbed vortex in a pipe to axisymmetric vortex breakdown

Journal of Fluid Mechanics ◽

10.1017/s0022112098001396 ◽

1998 ◽

Vol 366 ◽

pp. 211-237 ◽

Cited By ~ 59

Author(s):

Z. RUSAK ◽

S. WANG ◽

C. H. WHITING

Keyword(s):

Vortex Breakdown ◽

Swirling Flow ◽

Nonlinear Ordinary Differential Equation ◽

Swirling Flows ◽

Axisymmetric Vortex ◽

Burgers Vortex ◽

Theoretical Predictions ◽

Necessary And Sufficient Criteria ◽

High Reynolds ◽

Necessary And Sufficient

The evolution of a perturbed vortex in a pipe to axisymmetric vortex breakdown is studied through numerical computations. These unique simulations are guided by a recent rigorous theory on this subject presented by Wang & Rusak (1997a). Using the unsteady and axisymmetric Euler equations, the nonlinear dynamics of both small- and large-amplitude disturbances in a swirling flow are described and the transition to axisymmetric breakdown is demonstrated. The simulations clarify the relation between our linear stability analyses of swirling flows (Wang & Rusak 1996a, b) and the time-asymptotic behaviour of the flow as described by steady-state solutions of the problem presented in Wang & Rusak (1997a). The numerical calculations support the theoretical predictions and shed light on the mechanism leading to the breakdown process in swirling flows. It has also been demonstrated that the fundamental characteristics which lead to vortex instability and breakdown in high-Reynolds-number flows may be calculated from considerations of a single, reduced-order, nonlinear ordinary differential equation, representing a columnar flow problem. Necessary and sufficient criteria for the onset of vortex breakdown in a Burgers vortex are presented.

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Structure and Branching of Unstable Modes in a Swirling Flow

Mathematics ◽

10.3390/math10010099 ◽

2021 ◽

Vol 10 (1) ◽

pp. 99

Author(s):

Vadim Akhmetov

Keyword(s):

Vortex Breakdown ◽

Swirling Flow ◽

Maximum Growth ◽

Critical Parameters ◽

Return Flow ◽

Neutral Stability ◽

Wave Disturbances ◽

Branching Points ◽

Hydrodynamic Stability Theory ◽

The Stability

Swirling has a significant effect on the main characteristics of flow and can lead to its fundamental restructuring. On the flow axis, a stagnation point with zero velocity is possible, behind which a return flow zone is formed. The apparent instability leads to the formation of secondary vortex motions and can also be the cause of vortex breakdown. In the paper, a swirling flow with a velocity profile of the Batchelor vortex type has been studied on the basis of the linear hydrodynamic stability theory. An effective numerical method for solving the spectral problem has been developed. This method includes the asymptotic solutions at artificial and irregular singular points. The stability of flows was considered for the values of the Reynolds number in the range 10≤Re≤5×106. The calculations were carried out for the value of the azimuthal wavenumber parameter n=−1. As a result of the analysis of the solutions, the existence of up to eight simultaneously occurring unstable modes has been shown. The paper presents a classification of the detected modes. The critical parameters are calculated for each mode. For fixed values of the Reynolds numbers 60≤Re≤5000, the curves of neutral stability are plotted. Branching points of unstable modes are found. The maximum growth rates for each mode are determined. A new viscous instability mode is found. The performed calculations reveal the instability of the Batchelor vortex at large values of the swirl parameter for long-wave disturbances.

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The influence of swirl and confinement on the stability of counterflowing streams

Journal of Fluid Mechanics ◽

10.1017/s0022112093003362 ◽

1993 ◽

Vol 251 ◽

pp. 149-172 ◽

Cited By ~ 1

Author(s):

J. D. Goddard ◽

A. K. Didwania ◽

C.-Y. Wu

Keyword(s):

Mass Transfer ◽

Stability Analysis ◽

Linear Stability ◽

Swirling Flow ◽

Hydrodynamic Instability ◽

Aqueous System ◽

Inviscid Limit ◽

Angular Velocities ◽

High Reynolds ◽

The Stability

A linear stability analysis of laterally confined swirling flow is given, of the type described by Long's equation in the inviscid limit or by the von Kármán similarity equations in the absence of lateral confinement. The flow of interest involves identical counterflowing fluid streams injected with equal velocity W0 through opposing porous disks, rotating with angular velocities Ω and ±Ω, respectively, about a common normal axis. By means of mass transfer experiments on an aqueous system of this type we have detected an apparent hydrodynamic instability having the appearance of an inviscid supercritical bifurcation at a certain |Ω| > 0. As an attempt to elucidate this phenomenon, linear stability analyses are performed on several idealized flows, by means of a numerical Galerkin technique. An analysis of high-Reynolds-number similarity flow predicts oscillatory instability for all non-zero Ω. The spatial structure of the most unstable modes suggests that finite container geometry, as represented by the confining cylindrical sidewalls, may have a strong influence on flow stability. This is borne out by an inviscid stability analysis of a confined flow described by Long's equation. This analysis suggests a novel bifurcation of the inviscid variety, which serves qualitatively to explain the results of our mass transfer experiments.

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Review of recent studies on the axisymmetric vortex breakdown phenomenon

10.2514/6.2000-2529 ◽

2000 ◽

Cited By ~ 12

Author(s):

Z. Rusak

Keyword(s):

Vortex Breakdown ◽

Axisymmetric Vortex ◽

Breakdown Phenomenon

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Numerical Simulation of Confined Swirling Flows of Oldroyd Fluids

Volume 1: Symposia, Parts A, B and C ◽

10.1115/fedsm2009-78340 ◽

2009 ◽

Author(s):

Sahand Majidi ◽

Ashkan Javadzadegan

Keyword(s):

Numerical Simulation ◽

Vortex Breakdown ◽

Swirling Flow ◽

Maxwell Model ◽

Reynolds Numbers ◽

Weissenberg Number ◽

Swirling Flows ◽

Breakdown Phenomenon ◽

Volume Method ◽

Upper Convected Maxwell Model

The effect of a fluid’s elasticity has been investigated on the vortex breakdown phenomenon in confined swirling flow. Assuming that the fluid obeys upper-convected Maxwell model as its constitutive equation, the finite volume method together with a collocated mesh was used to calculate the velocity profiles and streamline pattern inside a typical lid-driven swirling flow at different Reynolds and Weissenberg numbers. The flow was to be steady and axisymmetric. Based on the results obtained in this work, it can be concluded that fluid’s elasticity has a strong effect on the secondary flow completely reversing its direction of rotation depending on the Weissenberg number. Even in swirling flows with low ratio of elasticity to inertia, vortex breakdown is postponed to higher Reynolds numbers. Also, the effect of retardation ratio on the flow structure of viscoelastic fluid with the Weissenberg number being constant was surveyed. Based on our results, by decreasing the retardation ratio the flow becomes Newtonian like.

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Axisymmetric vortex breakdown Part 1. Confined swirling flow

Journal of Fluid Mechanics ◽

10.1017/s0022112090003664 ◽

1990 ◽

Vol 221 ◽

pp. 533-552 ◽

Cited By ~ 213

Author(s):

J. M. Lopez

Keyword(s):

Oscillatory Flow ◽

Stokes Equations ◽

Vortex Breakdown ◽

Swirling Flow ◽

Swirling Flows ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Axisymmetric Vortex ◽

Recirculation Zones ◽

Visualization Experiments

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.

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Similarities in the structure of swirling and buoyancy-driven flows

Journal of Fluid Mechanics ◽

10.1017/s0022112093003799 ◽

1993 ◽

Vol 252 ◽

pp. 357-382 ◽

Cited By ~ 6

Author(s):

P. A. Davidson

Keyword(s):

Angular Momentum ◽

Steady State ◽

Energy Minimization ◽

High Speed ◽

Swirling Flow ◽

Buoyancy Driven Flows ◽

Minimization Technique ◽

Steady Solutions ◽

The Stability ◽

Momentum Integral

We look at two classes of contained flow: swirling flow and buoyancy-driven flow. We note that the strong links between these arise from the way in which vorticity is generated and propagated within each. We take advantage of this shared behaviour to investigate the structure of steady-state solutions of the governing equations. First, we look at flows with a small but finite viscosity. Here we find that, Batchelor regions apart, the steady state for each type of flow must consist of a quiescent stratified core, bounded by high-speed wall jets. (In the case of swirling flow, this is a radial stratification of angular momentum.) We then give a general, if approximate, method for finding these steady-state flow fields. This employs a momentum-integral technique for handling the boundary layers. The resulting predictions compare favourably with numerical experiments. Finally, we address the problem of inviscid steady states, where there is a well-known class of steady solutions, but where the question of the stability of these solutions remains unresolved. Starting with swirling flow, we use an energy minimization technique to show that stable solutions of arbitrary net azimuthal vorticity do indeed exist. However, the analogy with buoyancy-driven flow suggests that these solutions are all of a degenerate, stratified form. If this is so, then the energy minimization technique, which conserves vortical invariants, may mimic the stratification of temperature or angular momentum in a turbulent flow.

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