Predicting the aftermath of vortex breakup in rotating flow

This paper demonstrates the applicability of a two-dimensional eigenvalue problem approach to the study of linear instability of analytically constructed and numerically calculated models of trailing-vortex systems. Chebyshev collocation is used in the 2D eigenvalue problem solution in order to discretize two spatial directions on which non-axisymmetric vorticity distributions are defined, while the third, axial spatial direction is taken to be homogeneous and is resolved by a Fourier expansion. The leading eigenvalues of the matrix discretizing the equations which govern small-amplitude perturbations superimposed upon such a vorticity distribution are obtained by Arnoldi iteration. The present approach has been validated by comparison of its results on the problem of instability of an isolated Batchelor vortex. Here benchmark computations exist, employing classic instability analysis, in which the azimuthal direction is also treated as homogeneous. Subsequently, the proposed methodology has been shown to be able to recover the classic long-(Crow) and short-wavelength instabilities of a counter-rotating vortex-pair basic flow obtained by direct numerical simulation. Finally, the effect on the eigenspectrum of the isolated Batchelor vortex is documented, when the basic flow consists of a linear superposition of such vortices. The modifications of the eigenspectrum of a single vortex point to the potential pitfalls of drawing conclusions on the instability characteristics of a trailing-vortex system by monitoring the constituent vortices in isolation.

2001

The aim of this work is to identify current developments in the field of vortex breakdown modelling in order to initiate the development of a numerical model for the simulation of F/A-18 empennage buffet. Some attempts at using vortex models for prediction of vortex breakdown were found in the open literature. Their advantages and shortcomings are discussed, allowing the evaluation of their feasibility for the computation of LEX vortex burst. It was concluded that vortex methods are able to simulate an onset of breakdown in simple vortical flows. However, their predictive abilities require further improvement to provide accurate modelling of the temporal and spatial characteristics of the unsteady pressure field past a burst LEX vortex.

Journal of Fluid Mechanics, 2006

This paper is motivated by an observation: in the nascent state of vortex breakdown before it develops into a full-grown radial expansion, an initially straight vortex core first swells, and does so even in a straight pipe for no apparent reason. Although this initial swelling may be explained in many ways according to the perspectives chosen, we offer our own interpretation framed solely within vorticity dynamics: the radial swelling as well as the subsequent growth are induced by the azimuthal vorticity gradient decreasing downstream. The negative azimuthal vorticity gradient first appears at start-up and moves eventually into the region where the circulation reaches its steady-state value. The vorticity gradient can become negative without necessarily being accompanied by a sign-switch of the azimuthal vorticity itself. The key point-that the negative azimuthal vorticity gradient induces initial radial swelling and its growth-is demonstrated in two analyses. First, a kinematic analysis results in an equation for the radial velocity where the azimuthal vorticity gradient appears as a source term. Its solution shows, in general and explicitly, that the negative azimuthal vorticity gradient does induce radially outward velocity. Two heuristic examples serve to illustrate this point further. In the second analysis, using the equation of motion in the streamline coordinates, the negative azimuthal vorticity gradient is shown to diverge the meridional streamlines radially. A numerical simulation using a modified vortex filament method not only corroborates this role of the azimuthal vorticity gradient in initiating and promoting the radial expansion, but also adds details to track the formation process. Both analyses and simulation support our interpretation that the initial radial swelling and its subsequent growth are induced by the negative azimuthal vorticity gradient.

Geophysical & Astrophysical Fluid Dynamics, 2002

We present results from a new series of experiments on the geophysically important issue of the instability of anticyclonic columnar vortices in a rotating fluid in circumstances such that the Rossby number exceeds unity. The vortex pair consisting of a cyclonic and an anticyclonic vortex is induced by a rotating flap in a fluid which is itself initially in a state of solid-body rotation. The anticyclonic vortex is then subject to either centrifugal or elliptical instability, depending on whether its initial ellipticity is small or large, while the cyclone always remains stable. The experimental results demonstrate that the perturbations due to centrifugal instability have a typical form of toroidal vortices of alternating sign (rib vortices). The perturbations due to elliptical instability are of the form of sinuous deformation of the vortex filament in the plane of maximal stretching which corresponds to the plane of symmetry for the vortex pair. The initial perturbations in both cases are characterized by a definite wave number in the vertical direction. The characteristics of the unstable anticyclone are determined by the main nondimensional parameter of the flow -the Rossby number. The appearance of both centrifugal and elliptical instabilities are in accord with the predictions of theoretical criteria for these cases.

Journal of Fluid Mechanics, 2002

Time-dependent swirling flows inside an enclosed cylindrical rotor–stator cavity with aspect ratio H/R = 4, larger than the ones usually considered in the literature, are studied. Within a certain range of governing parameters, vortex breakdown phenomena can arise along the axis. Very recent papers exhibiting some particular three-dimensional effects have stimulated new interest in this topic. The study is carried out by a numerical resolution of the three-dimensional Navier–Stokes equations, based on high-order spectral approximations in order to ensure very high accuracy of the solutions.The first transition to an oscillatory regime occurs through an axisymmetric bifurcation (a supercritical Hopf bifurcation) at Re = 3500. The oscillatory regime is caused by an axisymmetric mode of centrifugal instability of the vertical boundary layer and the vortex breakdown is axisymmetric, being composed of two stationary bubbles. For Reynolds numbers up to Re = 3500, different three-dimension...

Journal of Fluid Mechanics, 2010

Journal of Fluid Mechanics, 2003

A rapidly growing instability is observed to develop between unequal-strength counterrotating vortex pairs. The vortex pairs are generated in a towing tank in the wakes of wings with outboard triangular flaps. The vortices from the wing tip and the inboard tip of the flap form the counter-rotating vortex pair on each side of the wing. The flow fields are studied using flow visualization and particle image velocimetry. Both chordbased and circulation-based Reynolds numbers are of O(10 5). The circulation strength ratios of the flap-to tip-vortex pairs range from −0.4 to −0.7. The initial sinuous stage of the instability of the weaker flap vortex has a wavelength of order one wing span and becomes observable in about 15 wing spans downstream of the wing. The nearly straight vortex filaments first form loops around the stronger wing-tip vortices. The loops soon detach and form rings and move in the wake under self-induction. These vortex rings can move to the other side of the wake. The subsequent development of the instability makes the nearly quasi-steady and two-dimensional wakes unsteady and three-dimensional over a distance of 50 to 100 wing spans. A rectangular wing is also used to generate the classical wake vortex pair with the circulation ratio of −1.0, which serves as a reference flow. This counter-rotating vortex pair, under similar experimental conditions, takes over 200 spans to develop visible deformations. Velocity, vorticity and enstrophy measurements in a fixed plane, in conjuction with the flow observations, are used to quantify the behaviour of the vortex pairs. The vortices in a pair initially orbit around their vorticity centroid, which takes the pair out of the path of the wing. Once the three-dimensional interactions develop, two-dimensional kinetic energy and enstrophy drop, and enstrophy dispersion radius increases sharply. This rapid transformation of the wake into a highly three-dimensional one offers a possible way of alleviating the hazard posed by the vortex wake of transport aircraft.

Laboratory experiments have shown that monopolar isolated vortices in a rotating flow undergo instabilities that result in the formation of multipolar vortex states such as dipoles and tripoles. In some cases the instability is entirely two-dimensional, with the vortices taking the form of vortex columns aligned along the direction of the ambient rotation at all times. In other cases, the vortex first passes through a highly turbulent three-dimensional state before eventually reorganizing into vortex columns. Through a series of three-dimensional numerical simulations, the roles that centrifugal instability, barotropic instability, and the bottom Ekman boundary layer play in these instabilities are investigated. Evidence is presented that the centrifugal instability can trigger the barotropic instabilities by the enhancement of vorticity gradients. It is shown that the bottom Ekman layer is not essential to these instabilities but can strongly modify their evolution.

Swirling flows with jet-like axial velocities are studied by means of direct numerical simulations for various control parameters. The numerical code solves the low-Mach number approximation of the Navier-Stokes equations in cylindrical coordinates which allows the incorporation of compressible and variable-density effects without the adverse effect of acoustic waves on the numerical time-step. The governing parameters, such as swirl and coflow, are varied, and their effect on the breakdown type and breakdown location is studied. The nonlinear direct numerical simulation traces the steady states of the swirling flow, and the structure of the bifurcation is described as a function of the swirl parameter, the Reynolds number and the coflow parameter. In particular, unstable branches are computed and their physical relevance is discussed. The analysis gives new insight into the prevalence of coherent states and their controllability.

AIAA Journal, 2012

and the bifurcation to a global spiral mode in turbulent swirling jets. The different flow states that evolve at incrementally increasing swirl are characterized by means of time-resolved stereo PIV measurements in conjunction with post-processing tools, including Fourier analysis and proper orthogonal decomposition. Vortex breakdown occurs first intermittently, accompanied by strong axial oscillations. By further increasing the swirl, vortex breakdown stabilizes and a region of reversed flow appears in the mean flow. This region grows linearly with increasing swirl until the flow undergoes a supercritical Hopf bifurcation to a global single-helical mode and vortex breakdown becomes spiral shaped. The appearance of an internal stagnation point is accompanied by a supercritical-to-subcritical transition of the inflow profiles, in accordance to Benjamin's inviscid theory [1]. This critical swirl number is found to be smaller than the one for the supercritical Hopf bifurcation. The observed mean flow sequence compares well with the transient formation of spiral vortex breakdown in laminar swirling jets as reported by Brücker & Althaus[2], Liang & Maxworthy[3] and Ruith et al.[4]. The present experiment is properly scaled by the swirl number based on the axial momentum flux when omitting the commonly used boundary layer approximations.