Short-wave instabilities in two-vortex systems (original) (raw)
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Influence of an elliptic instability on the merging of a co-rotating vortex pair
2007
We study the nonlinear evolution of the elliptic instability and its influence on the merging process of two corotating Batchelor vortices using a spectral DNS approach. First, we analyse the nonlinear saturation of the elliptic instability for a single strained vortex, with and without axial jet, for moderate Reynolds numbers (Re = Γ/ν ≈ 12500, where Γ is the circulation and ν the kinematic viscosity). We show that the vortex deformation induced by the instability remains limited to the vortex core region. The second part of our work focuses on the influence of the elliptic instability on the merging process. We compare three cases : no instability (2D), elliptic instability without axial jet, and elliptic instability with axial jet, the latter case being relevant to aircraft wakes. Qualitative and quantitative differences between the three different cases are pointed out and discussed in the context of aircraft vortices.
Experimental study of the instability of unequal-strength counter-rotating vortex pairs
Journal of Fluid Mechanics, 2003
A rapidly growing instability is observed to develop between unequal-strength counter- rotating 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 chord- based and circulation-based Reynolds numbers are of O(105). 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...
2004 - On instability characteristics of isolated vortices and models of trailing-vortex systems
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.
Absolute/Convective Instabilities and Spatial Growth in a Vortex Pair
Lecture Notes in Physics, 2000
Airplane trailing vortices have a destabilizing e ect on ensuing aircrafts. Security spacings, related to the trailing vortices \lifetime" are actually enforced between take-o s and landings. This spacing limits the maximum take-o and landing frequency in saturated airports 19]. A number of studies have been devoted to the understanding of vortex wake dynamics, usually modeled by a pair of counterrotating vortices. Two types of vortex pair three-dimensional instabilities have been identi ed in the past: a long-wave instability (of the order of the spacing b between the two vortices) and a short-wave instability (of the order of the vortex core radius a) have been rst considered respectively by Crow 4] and by Moore & Sa man 15] and Tsai & Widnall 20]. These two mechanisms, which are thought to participate in the vortex wake dissipation, have been observed in recent experiments 12]. One possible technique to accelerate the dissipation of aircraft wakes is to force these instabilities by on-board control devices 2,21,22]. Until now only temporal vortex pair instability analyses are available. If one wants to force these instabilities, however, it would be more appropriate to analyze their spatial stability in the airplane reference frame. As the spatial analysis makes sense only when the instabilities are convective, an absolute/convective stability analysis is required. The results presented consist of the absolute/convective and spatial stability analyses of both long{ and short{wave instabilities.
Elliptic instability of a co-rotating vortex pair
Journal of Fluid Mechanics, 2005
In this paper, we report experimental results concerning a three-dimensional shortwave instability observed in a pair of equal co-rotating vortices. The pair is generated in water by impulsively started plates, and is analysed through dye visualizations and detailed quantitative measurements using particle image velocimetry. The instability mode, which is found to be stationary in the rotating frame of reference of the two-vortex system, consists of internal deformations of the vortex cores, which are characteristic of the elliptic instability occurring in strained vortical flows. Measurements of the spatial structure, wavelengths and growth rates are presented, as functions of Reynolds number and non-dimensional core size. The self-induced rotation of the vortex pair, which is not a background rotation of the entire flow, is found to lead to a shift of the unstable wavelength band to higher values, as well as to higher growth rates. In addition, a dramatic increase in the width of the unstable bands for large values of the rescaled core radius is found. Comparisons with recent theoretical results by Le concerning elliptic instability of co-rotating vortices show very good agreement.
Three-dimensional stability of a vortex pair
2001
This paper investigates the three-dimensional stability of the Lamb-Chaplygin vortex pair. Short-wavelength instabilities, both symmetric and antisymmetric, are found. The antisymmetric mode possesses the largest growth rate and is indeed the one reported in a recent experimental study ͓J. Fluid Mech. 360, 85 ͑1998͔͒. The growth rates, wave numbers of maximum amplification, and spatial eigenmodes of these short-wavelength instabilities are in good agreement with the predictions from elliptic instability theory. A long-wavelength symmetric instability similar to the Crow instability of a pair of vortex filaments is also recovered. Oscillatory bulging instabilities, both symmetric and antisymmetric, are identified albeit their growth rates are lower than for the short-wavelength instabilities. Their behavior and eigenmodes resemble those of the oscillatory bulging instability occurring in the mixing layer.
Spatio-temporal development of the long and short-wave vortex-pair instabilities
Physics of Fluids, 2000
We consider the spatio-temporal development of the long-wave and shortwave instabilities in a pair of counter-rotating vortices in the presence of a uniform axial advection velocity. The stability properties depend upon the aspect ratio a=b of the vortex pair, where a is the core radius of the vortices and b their separation, and upon W 0 =U 0 the ratio between the self-induced velocity of the pair and the axial advection velocity. For su ciently small W 0 =U 0 , the instabilities are convective, but an
Experiments on the stability of vortex pairs are described. The vortices (ratio of length to core diameter L / c of up to 300) were generated at the edge of a flat plate rotating about a horizontal axis in water. The vortex pairs were found to be unstable, displaying two distinct modes of instability. For the first time, as far as it is known to the authors, a long-wave as well as a short-wave mode of instability were observed to develop simultaneously on such a vortex pair. Experiments involving single vortices show that these do not develop any instability whatsoever. The wavelengths of the developing instability modes on the investigated vortex pairs are compared to theoretical predictions. Observed long wavelengths are in good agreement with the classic symmetric long-wave bending mode identified by . The developing short waves, on the other hand, appear to be less accurately described by the theoretical results predicted, for example, by .
Three-dimensional instabilities and transient growth of a counter-rotating vortex pair
Physics of Fluids, 2009
This paper investigates the three-dimensional instabilities and the transient growth of perturbations on a counter-rotating vortex pair. The two dimensional base flow is obtained by a direct numerical simulation initialized by two Lamb-Oseen vortices that quickly adjust to a flow with elliptic vortices. In the present study, the Reynolds number, Re ⌫ = ⌫ / , with ⌫ the circulation of one vortex and the kinematic viscosity, is taken large enough for the quasi steady assumption to be valid. Both the direct linearized Navier-Stokes equation and its adjoint are solved numerically and used to investigate transient and long time dynamics. The transient dynamics is led by different regions of the flow, depending on the optimal time considered. At very short times compared to the advection time of the dipole, the dynamics is concentrated on the points of maximal strain of the base flow, located at the periphery of the vortex core. At intermediate times, depending on the symmetry of the perturbation, one of the hyperbolic stagnation points provides the optimal amplification by stretching of the perturbation vorticity as in the classical hyperbolic instability. The growth of both short time and intermediate time transient perturbations are non-or weakly dependent of the axial wavenumber whereas the long time behavior strongly selects narrow bands of wavenumbers. We show that, for all unstable spanwise wavenumbers, the transient dynamics last until the nondimensional time t = 2, during which the dipole has traveled twice the separation distance between vortices b. During that time, all the wavenumbers exhibit a transient growth of energy by a factor of 50, for the Reynolds number Re ⌫ = 2000. For time larger than t = 2, energy starts growing at a rate given by the standard temporal stability theory. For all wavenumbers and two Reynolds numbers, Re ⌫ = 2000 and Re ⌫ =10 5 , different instability branches have been computed using a high resolution Krylov method. At large Reynolds number, the computed Crow and elliptic instability branches are in excellent agreement with the inviscid theory ͓S.
Controlled interaction of co-rotating vortices
The two-and three-dimensional interactions of a pair of co-rotating vortices, representing a simplified model of flows found in the extended near wake behind aircraft wings, are analyzed using water tank experiments and numerical simulations. At low Reynolds numbers (Re), the vortices undergo two-dimensional merging, when the core size exceeds a certain fraction of the vortex separation distance. The time it takes to reach this limit increases with Re. At higher Re, a three-dimensional instability is observed, showing the characteristics of an elliptic instability of the vortex cores. The spatial structure of the amplified unstable modes, as well as their growth rate as function of axial wavelength are given and compared to theoretical predictions, showing excellent agreement between the three approaches. The instability is found to rapidly generate small-scale motion, initiating merging for smaller core sizes and producing a turbulent final vortex.