Showing papers on "Vortex published in 1986"

Coherent structures and turbulence.

Abstract: This is a personal statement on the present state of understanding of coherent structures, in particular their spatial details and dynamical significance. The characteristic measures of coherent structures are discussed, emphasizing coherent vorticity as the crucial property. We present here a general scheme for educing structures in any transitional or fully turbulent flow. From smoothed vorticity maps in convenient flow planes, this scheme recognizes patterns of the same mode and parameter size, and then phase-aligns and ensemble-averages them to obtain coherent structure measures. The departure of individual realizations from the ensemble average denotes incoherent turbulence. This robust scheme has been used to educe structures from velocity data using a rake of hot wires as well as direct numerical simulations and can educe structures using newer measurement techniques such as digital image processing. Our recent studies of coherent structures in several free shear flows are briefly reviewed. Detailed data in circular and elliptic jets, mixing layers, and a plane wake reveal that incoherent turbulence is produced at the ‘saddles’ and then advected to the ‘centres’ of the structures. The mechanism of production of turbulence in shear layers is the stretching of longitudinal vortices or ‘ribs’ which connect the predominantly spanwise ‘rolls’; the ribs induce spanwise contortions of rolls and cause mixing and dissipation, mostly at points where they connect with rolls. We also briefly discuss the role of coherent structures in aerodynamic noise generation and argue that the structure breakdown process, rather than vortex pairing, is the dominant mechanism of noise generation. The ‘cut-and-connect’ interaction of coherent structures is proposed as a specific mechanism of aerodynamic noise generation, and a simple analytical model of it shows that it can provide acceptable predictions of jet noise. The coherent-structures approach to turbulence, apart from explaining flow physics, has also enabled turbulence management via control of structure evolution and interactions. We also discuss some new ideas under investigation: in particular, helicity as a characteristic property of coherent structures.

Flow regimes in a circular Couette system with independently rotating cylinders

Abstract: Our flow-visualization and spectral studies of flow between concentric independently rotating cylinders have revealed a surprisingly large variety of different flow states. (The system studied has radius ratio 0.883, aspect ratios ranging from 20 to 48, and the end boundaries were attached to the outer cylinder.) Different states were distinguished by their symmetry under rotation and reflection, by their azimuthal and axial wavenumbers, and by the rotation frequencies of the azimuthal travelling waves. Transitions between states were determined as functions of the inner- and outer-cylinder Reynolds numbers, Ri and Ro, respectively. The transitions were located by fixing Ro and slowly increasing Ri. Observed states include Taylor vortices, wavy vortices, modulated wavy vortices, vortices with wavy outflow boundaries, vortices with wavy inflow boundaries, vortices with flat boundaries and internal waves (twists), laminar spirals, interpenetrating spirals, waves on interpenetrating spirals, spiral turbulence, a flow with intermittent turbulent spots, turbulent Taylor vortices, a turbulent flow with no large-scale features, and various combinations of these flows. Some of these flow states have not been previously described, and even for those states that were previously described the present work provides the first coherent characterization of the states and the transitions between them. These flow states are all stable to small perturbations, and the transition boundaries between the states are reproducible. These observations can serve as a challenge and test for future analytic and numerical studies, and the map of the transitions provides several possible codimension-2 bifurcations that warrant further study.

Streamwise vortex structure in plane mixing layers

Abstract: The development of three-dimensional motions in a plane mixing layer was investigated experimentally. It is shown that superimposed on the primary, spanwise vortex structure there is a secondary, steamwise vortex structure. Three aspects of this secondary structure were studied. First, the spanwise vortex instability that generates the secondary structure was characterized by measurements of the critical Reynolds number and the spanwise wavelength at several flow conditions. While the critical Reynolds number was found to depend on the velocity ratio, density ratio and initial shear-layer-profile shape, the mean normalized wavelength is independent of these parameters. Secondly, flow visualization in water was used to obtain cross-sectional views of the secondary structure associated with the streamwise counter-rotating vortices. A model is proposed in which those vortices are part of a single vortex line winding back and forth between the high-speed side of a primary vortex and the low-speed side of the following one. Finally, the effect of the secondary structure on the spanwise concentration field was measured in a helium-nitrogen mixing layer. The spatial organization of the secondary structure produces a well-defined spanwise entrainment pattern in which fluid from each stream is preferentially entrained at different spanwise locations. These measurements show that the spanwise scale of the secondary structure increases with downstream distance.

Transient cavities near boundaries. Part 1. Rigid boundary

Abstract: The growth and collapse of transient vapour cavities near a rigid boundary in the presence of buoyancy forces and an incident stagnation-point flow are modelled via a boundary-integral method. Bubble shapes, particle pathlines and pressure contours are used to illustrate the results of the numerical solutions. Migration of the collapsing bubble, and subsequent jet formation, may be directed either towards or away from the rigid boundary, depending on the relative magnitude of the physical parameters. For appropriate parameter ranges in stagnation-point flow, unusual ‘hour-glass’ shaped bubbles are formed towards the end of the collapse of the bubble. It is postulated that the final collapsed state of the bubble may be two toroidal bubbles/ring vortices of opposite circulation. For buoyant vapour cavities the Kelvin impulse is used to obtain criteria which determine the direction of migration and subsequent jet formation in the collapsing bubble.

A study of singularity formation in a vortex sheet by the point-vortex approximation

Abstract: The initial-value problem for perturbations of a flat, constant-strength vortex sheet is linearly ill posed in the sense of Hadamard, owing to Kelvin-Helmholtz instability. Previous numerical studies of this problem have experienced difficulty in converging when the mesh was refined. The present work examines Rosenhead’s point-vortex approximation and seeks to understand better the source of this difficulty. Using discrete Fourier analysis, it is shown that perturbations introduced spuriously by computer roundoff error are responsible for the irregular point-vortex motion that occurs at a smaller time as the number of points is increased. This source of computational error is controlled here by using either higher precision arithmetic or a new filtering technique. Computations are presented which use a linear-theory growing eigenfunction of small amplitude/wavelength ratio as the initial perturbation. The results indicate the formation of a singularity in the vortex sheet at a finite time as previously found for other initial data by Moore and Meiron, Baker & Orszag using different techniques of analysis. Numerical evidence suggests that the point vortex approximation converges up to but not beyond the time of singularity formation in the vortex sheet. For large enough initial amplitude, two singularities appear along the sheet at the critical time.

Experimental study of the two-dimensional inverse energy cascade in a square box

Abstract: A quantitative experimental study of the two-dimensional inverse energy cascade is presented. The flow is electrically driven in a horizontal layer of mercury and three-dimensional perturbations are suppressed by means of a uniform magnetic field, so that the flow can be well approximated by a two-dimensional Navier–Stokes equation with a steady forcing term and a linear friction due to the Hartmann layer. Turbulence is produced by the instability of a periodic square network of 36 electrically driven alternating vortices. The inverse cascade is limited at large scales, either by the linear friction or by the finite size of the domain, depending on the experimental parameters. In the first case, spectra are measured and the corresponding two-dimensional Kolmogorov constant is in the range 3–7. In the second case, a condensation of the turbulent energy in the lowest mode, corresponding to a spontaneous mean global rotation, is observed. Such a condensation was predicted by Kraichnan (1967) from statistical thermodynamics arguments, but without the symmetry breaking. Random reversals of the rotation sense, owing to turbulent fluctuations, are more and more sparse as friction is decreased. The lowest mode fluctuations and the small scales are statistically independent.

On the large-scale structures in two-dimensional, small-deficit, turbulent wakes

Abstract: A systematic study of two-dimensional, turbulent, small-deficit wakes was carried out to determine their structure and the universality of their self-preserving states. Various wake generators, including circular cylinders, a symmetrical airfoil, a flat plate, and an assortment of screens of varying solidity, were studied for a wide range of downstream distances. Most of the generators were tailored so that their drag coefficients, and therefore their momentum thicknesses, were identical, permitting comparison at identical Reynolds numbers and aspect ratios. The flat plate and airfoil had a small, trailing-edge flap which could be externally driven to introduce forced sinuous oscillations into the wake. The results indicate that the normalized characteristic velocity and length scales depend on the initial conditions, while the shape of the normalized mean velocity profile is independent of these conditions or the nature of the generator. The normalized distributions of the longitudinal turbulence intensity, however, are dependent on the initial conditions.Linear inviscid stability theory, in which the divergence of the mean flow is taken into account, predicts quite well the amplification and the transverse distributions of amplitudes and phases of externally imposed sinuous waves on a fully developed turbulent wake generated by a flat plate. There is a strong indication that the large structures observed in the unforced wake are related to the two-dimensional instability modes and therefore can be modelled by linear stability theory. Furthermore, the interaction of the two possible modes of instability may be responsible for the vortex street-type pattern observed visually in the small-deficit, turbulent wake.

Universal short-wave instability of two-dimensional eddies in an inviscid fluid.

Abstract: It is shown that a broad class of two-dimensional vortices occurring in the flow of an incompressible, inviscid fluid are unstable to three-dimensional perturbations. At short wavelengths along the vortex axis the growth rate becomes independent of wavelength, and the eigenmode becomes concentrated near the center of the vortex.

The nonlinear development of Gortler vortices in growing boundary layers

Abstract: The Growth of Gortler vortices in boundary layers on concave walls is investigated. It is shown that for vortices of wavelength comparable to the boundary-layer thickness the appropriate linear stability equations cannot be reduced to ordinary differential equations. The partial differential equations governing the linear stability of the flow are solved numerically, and neutral stability is defined by the condition that a dimensionless energy function associated with the flow should have a maximum or minimum when plotted as a function of the downstream variable X. The position of neutral stability is found to depend on how and where the boundary layer is perturbed, so that the concept of a unique neutral curve so familiar in hydrodynamic-stability theory is not tenable in the Gortler problem, except for asymptotically small wavelengths. The results obtained are compared with previous parallel-flow theories and the small-wavelength asymptotic results of Hall (1982a, b), which are found to be reasonably accurate even for moderate values of the wavelength. The parallel-flow theories of the growth of Gortler vortices are found to be irrelevant except for the small-wavelength limit. The main deficiency of the parallel-flow theories is shown to arise from the inability of any ordinary differential approximation to the full partial differential stability equations to describe adequately the decay of the vortex at the edge of the boundary layer. This deficiency becomes intensified as the wavelength of the vortices increases and is the cause of the wide spread of the neutral curves predicted by parallel-flow theories. It is found that for a wall of constant radius of curvature a given vortex imposed on the flow can grow for at most a finite range of values of X. This result is entirely consistent with, and is explicable by the asymptotic results of, Hall (1982a).

Experimental and numerical study of vortex couples in two-dimensional flows

Abstract: Two-dimensional turbulence is investigated experimentally in thin liquid films. This study shows the spontaneous formation of couples of opposite-sign vortices in von Karman wakes. The structure of these couples, their behaviour and their role in turbulent flows is then studied using both a numerical simulation and laboratory results.

An approximate model of vortex decay in the atmosphere

Abstract: An approximate analysis of atmospheric effects on wake vortex motion and decay is presented. The effects of density stratification, turbulence, and Reynolds number are combined in a single model so that the relative importance of different parameters can be determined. Predicted wake motion is shown to be in good agreement with limited data from both ground facility and flight test measurements taken under low turbulence conditions. Wake decay was found to depend strongly on both density stratification and turbulence. For typical levels of turbulence, wake decay was found to result from the 'Crow instability' except under strongly stratified conditions.

Secondary vortices in the wake of circular cylinders

Abstract: Using both the hydrogen-bubble flow-visualization technique and hot-wire-anemometry measurements, secondary vortices have been detected in the near-wake of circular cylinders and their characteristics studied over a Reynolds-number range of 1200–11000. The vortex-shedding characteristics of these secondary vortices clearly indicate that ‘transition waves’, detected in the cylinder near wake by Bloor (1964), and secondary vortices are identical phenomena. It is established that the non-dimensional shedding frequency of the secondary vortices demonstrates a 0.87 power-law relationship relative to Reynolds number, contrary to the 0.5 power-law reported by Bloor. The results suggest that the vortices result from a near-wake, free-shear instability which causes the separated cylinder boundary layer to roll up into the secondary vortices. Visual observations indicate that immediately following their formation the vortices undergo a strong three-dimensional distortion, which may provide the mechanism for the transition from laminar to turbulent Strouhal vortices.

On the origin and evolution of streamwise vortical structures in a plane, free shear layer

Abstract: A plane, isothermal, chemically reacting mixing layer has been experimentally investigated to analyse the origin and the development of three-dimensional stream-wise vorticity. The results show that early in its evolution, the plane, free shear layer is composed of counter-rotating pairs of streamwise vortices superimposed upon the spanwise ones. This coherent, streamwise vortical structure was found to be the result of the unstable response of the layer to three-dimensional perturbations in the upstream conditions. Depending on the magnitude and location of the upstream disturbances, the location of the transition to three-dimensionality varied. However, the concentrated streamwise vorticity was always seen to form first on the braids between consecutive spanwise vortices and then to propagate into their cores.For the low and moderate Reynolds numbers of this study, it was found that the onset of the so-called ‘mixing transition’ does not necessarily coincide with that of the formation of concentrated streamwise vorticity. These vortices were observed to have a scale, as measured by the size of their cores, somewhat smaller than but comparable with that of the spanwise ones, thus contributing substantially to the entrainment process in the early stages of mixing-layer development.

Nonlinear Response of Atmospheric Vortices to Heating by Organized Cumulus Convection

Abstract: Using an axisymmetric primitive tropical cyclone model, we first illustrate the way in which nonlinear processes contribute to the development of an atmospheric vortex. These numerical experiment show that nonlinearities allow a given diabatic beat source to induce larger tangential wind (and kinetic energy) changes as the vortex develops and the inertial stability becomes large. In an attempt to gain a deeper theoretical understanding of this process, we consider the energy cycle in the balanced vortex equations of Eliassen. The temporal behavior of the total potential energy P is governed by dP/dt=H−C where H is the rate of generation of total potential energy by diabatic heating, and C is the rate of conversion to kinetic energy. We define a time-dependent system efficiency parameter as η¯(t)=C/H. Then, using the dynamical simplifications of balanced vortex theory, we express η¯(t) as a weighted average of a dynamic efficiency factor η(r, z, t). The dynamic efficiency factor is a measure of th...

The neutral curve for stationary disturbances in rotating-disk flow

Abstract: The neutral curve for stationary vortex disturbances in rotating-disk flow is computed up to a Reynolds number of 10 to the 7th using the sixth-order system of linear stability equations which includes the effects of streamline curvature and Coriolis force. It is found that the neutral curve has two minima: one at R = 285.36 (upper branch) and the other at R = 440.88 (lower branch). At large Reynolds numbers, the upper branch tends to Stuart's asymptotic solution while the lower branch tends to a solution that is associated with the wave angle corresponding to the direction of zero mean wall shear.

Free‐Surface Air Core Vortex

Abstract: A Rankine vortex model is used as the basis for an equation for critical submergence of intakes. The equation relates critical submergence to Froude number, circulation number, Reynolds number, and Weber number. The Froude and circulation numbers are the major controlling parameters. The model accounts for the effect of eddy viscosity on vortex-core characteristics. The equation for critical submergence is in agreement with experimental results. Criteria for neglect of surface tension and kinematic viscosity are established.

Vortex shedding and lock-on of a circular cylinder in oscillatory flow

Abstract: An experimental study has been made of a circular cylinder in steady and oscillatory flow with non-zero mean velocity up to a Reynolds number of 40000. The results for the stationary cylinder are in close agreement with previously published data. Skin-friction measurements revealed the amplitude of fluctuation of the boundary layer for different angular locations. It has been universally accepted that bluff bodies shed vortices at their natural frequency of shedding (Strouhal frequency), or, when synchronized with an external unsteadiness, at the frequency of the disturbance or half of it, depending of the direction of the unsteadiness. Our findings, instead, indicate that the shedding frequency may vary smoothly with the driving frequency before locking on its subharmonic. Moreover, the present results indicate that, at the lowest frequency limit of lock-on, vortices are shed simultaneously on both sides of the model. A more traditional alternate pattern of vortex shedding is then recovered at higher driving frequencies.

On the vorticity dynamics of a turbulent jet in a crossflow

Abstract: We present numerical solutions of the fully three-dimensional flow of a round, turbulent jet emitted normal to a uniform free stream. Comparisons with available laboratory data and comparison between different numerical grid resolutions are used to demonstrate the quality of the simulation. Examination of the detailed flow pattern within a computational domain, which extends 15 jet diameters from the source allows us to follow the vorticity dynamics in the transition from an initially vertical jet to a wake with a vortex pair essentially aligned with the free stream. The transition is presented as a function of the ratio of the jet exit velocity to free stream velocity. For large velocity ratios, the source of the streamwise vorticity in the vortex pair can be readily traced back to the original streamwise vorticity in the sides of the vertical jet.

Flow characteristics associated with abrupt changes in geometry in the case of highly elastic liquids

Abstract: Consideration is given to the influence of small changes in corner geometry on flow characteristics in the case of elastic liquids flowing in various contraction geometries. Both Boger fluids and aqueous solutions of polyacrylamide are used to show that it is very difficult to make generalizations from one geometry to another or from one type of non-Newtonian fluid to another. Of major importance is the observation that the precise mechanism of vortex enhancement varies with the contraction ratio. For large contraction ratios, a re-entrant corner vortex (not present in a 4:1 contraction) is shown to have an important influence on the development of vortex enhancement.

Behavior of Tip Leakage Flow Behind an Axial Compressor Rotor

Abstract: Performance testing and detailed flow measurements were made in an axial compressor rotor with various tip clearances. The experiments were conducted on the condition of the same incidence angle at midspan. Thus, the effect of tip clearance distinguished from that of incidence angle was investigated on the overall performance, work-done factor, blockage factor, and increases in displacement, momentum, and blade-force-deficit thicknesses of the casing wall boundary layer, The phase-locked flow patterns obtained by the multisampling technique show clear evidence of a leakage vortex core behind the rotor. Behavior of the leakage vortex was clarified for various tip clearances by examining loci of the vortex center, decay characteristics of the vorticity at the center, and the total amount of vorticity shed from the blade tip. These results were compared with the leakage vortex model presented by Lakshminarayana.

Experimental study of convective structures in rotating fluids

Abstract: We describe a series of laboratory experiments to study convective structures in rotating fluids (distilled water) in ranges of Rayleigh flux number Raf from 106 to 2 × 1011 and of Taylor number Ta from 106 to 1012. An intermediate quasi-stationary ring pattern of convection was found to arise from the interaction of the onset of convection with the fluid spin-up, for which we determined the times of origin and destruction, the distances between the rings, and the diameter of the central ring in terms of Raf and Ta. The ring structure evolves into a vortex grid which can be regular or irregular. In terms of Raf and Ta the regular grid exists in the linear regime, when the number of vortices N is in accord with the linear theory, when , or in the nonlinear regime when N ∝ h−2Ta½Raf−⅙ ∝ where Ω is the angular velocity and h is the fluid depth. In the irregular regime we always have N ∝ Ω. The transition from the regular regime to the irregular one is rather gradual and is determined by the value of the ordinary Rayleigh number, which we found to be greater than the first critical number Ra ∝ Ta2/3 by a factor about 25–40. In the transition region vortex interactions are observed, which start with rotation of two adjacent vortices around a common axis, then the vortices come closer and rotation accelerates, following which the vortices form a double helix and then coalesce into one stronger vortex.Some other qualitative experiments show that if the rotating vessel with the convective fluid is inclined to the horizontal, the vortex grid is formed along the rotation axis in accordance with the Proudman–Taylor theorem.

On the formation of vortex streets behind stationary cylinders

Abstract: The formation of vortex streets behind stationary cylinders is found to be caused by an absolute instability in the wake immediately behind the cylinder. The inviscid Orr–Sommerfeld equation is used together with measured profiles at Reynolds numbers of (a) Re = 56 when the absolute instability provides a Strouhal number of 0.13; and (b) Re = 140000 providing a Strouhal number of 0.21, both in agreement with experimental values. At the subcritical Re = 34 the instability is of the convective type; i.e. the disturbance decays, being convected away once the external disturbance is removed, in agreement with experimental observations. Finally, the instability of the mode which causes a symmetric array of vortices is shown to be always of the convective type.

The structure of the vorticity field in turbulent channel flow. Part 2. Study of ensemble-averaged fields

Abstract: Several conditional sampling techniques are applied to a data base generated by large-eddy simulation of turbulent channel flow. It is shown that the bursting process is associated with well-organized horseshoe vortices inclined at about 45 deg. to the wall. These vortical structures are identified by examining the vortex lines of three-dimensional, ensemble averaged vorticity fields. Two distinct horseshoe-shaped vortices corresponding to the sweep and ejection events are detected. These vortices are associated with high Reynolds shear stress and hence make a significant contribution to turbulent energy production. The dependency of the ensemble averaged vortical structures on the detection criteria, and the question of whether this ensemble-averaged structure is an artifact of the ensemble averaging process are examined. The ensemble-averaged pattern of these vortical structures that emerge from the analysis could provide the basis for a hypothetical model of the organized structures of wall-bounded shear flows.

The nonlinear evolution of rotating configurations of uniform vorticity

Abstract: The nonlinear evolution of perturbed equilibrium configurations of constant-vorticity vortices is calculated To illustrate a variety of nonlinear behaviour, we consider the following relatively simple configurations: the corotating configurations of N vortices whose linear stability has been treated in a previous study; the elliptical vortex; and the annular vortex Our calculations test for nonlinear stability as well as categorize the possible forms of stability and instability The energy ideas announced in the previous study are found to greatly constrain vortex evolution In particular, we show that two vortices and an elliptical vortex may evolve into each other, and that an annular vortex may break cleanly into five co-rotating vortices

Further observations of elastic effects in tubular entry flows

Abstract: The current understanding of tubular entry flows for Newtonian fluids and the influence of elasticity on these flows in the absence of shear thinning and significant fluid inertia is reviewed. While tubular entry flow for Newtonian fluids both with and without fluid inertia is now a solved problem, no single worker has been able to predict any of the significant elastic effects observed for constant viscosity elastic liquids in such flows. New elastic flow phenomena are reported which complicate the problem even more. Two distinctly different flow patterns are observed in a four to one contraction for two constant viscosity elastic fluids with essentially the same characteristics times. Tubular entry flows of elastic liquids develop (with increasing λγ) in the same way only if the contraction ratio is high enough. At sufficiently low contraction ratio, a lip vortex (at the tube entrance) and a corner vortex (in the upstream tube corner) can both be present and interacting with the lip vortex dominating the flow when vortex enhancement and growth occurs. The entry flow problem for elastic liquids is far more complex than originally imagined and is indeed a challenge for numerical simulation and selection of constutitive equations.

A Theory for the Maximum Windspeeds in Tornado-like Vortices

Abstract: We have developed a physical theory for the finding that the most intense laboratory vortex occurs when it is in the form of an end-wall vortex We argue that the end-wall vortex allows no standing centrifugal waves (ie, it is supercritical), and therefore, disturbances cannot propagate down from aloft This allows the low central pressure of the end-wall vortex at the level of maximum azimuthal velocity to be balanced by a central axial jet which jet which accelerates from the lower end wall to this level This supercritical, end-wall vortex undergoes a transition to a subcritical vortex aloft through a vortex breakdown We construct a model for the maximum intensity of these vortices by developing a model for the end-wall vortex and by finding the criterion for a vortex breakdown to be in steady suspension above the lower end wall The model agrees well with previous experimental simulations of tornado-like vortices in the Purdue tornado vortex chamber a steady end-wall vortex adjacent to the

Modification of vortex interactions in a reattaching separated flow

Abstract: Recent experimental observations have shown that large-scale organized vortices are produced in reattaching separated flows. Interactions between these vortices are important in the development of these flows downstream. Experimental studies from a downstream-fac ing step flow are presented to demonstrate that substantial changes in a reattaching flow can be produced by controlled forcing techniques. The forcing apparently works by affecting the vortex merging process in a fashion similar to that observed in forced mixinglayer experiments. The separated mean flow spreading rate could be increased most effectively by forcing at a nondimensional frequency (based on step height and freestream velocity) between 0.2 and 0.4. This result was found to be relatively independent of step Reynolds numbers over the range (26,000-76,000) studied. A significant decrease in the reattachment length accompanied the increased growth of the separated shear layer. Considerable changes in the turbulence energy and the Reynolds stress levels were also observed for the forced flows.

The instability of barotropic circular vortices

Abstract: The linear, normal mode instability of barotropic circular vortices with zero circulation is examined in the f-plane quasigeostrophic equations. Equivalents of Rayleigh's and Fjortoft's criteria and the semicircle theorem for parallel shear flow are given, and the energy equation shows the instability to be barotropic. A new result is that the fastest growing perturbation is often an internal instability, having a finite vertical scale, but may also be an external instability, having no vertical structure. For parallel shear flow the fastest growing perturbation is always an external instability; this is Squire's theorem. Whether the fastest growing perturbation is internal or external depends upon the profile: for mean flow streamfunction profiles which monotonically decrease with radius, the instability is internal for less steep profiles with a broad velocity extremum and external for steep profiles with a narrow velocity extremum. Finite amplitude, numerical model calculations show that this ...