On the three‐dimensional instability of elliptical vortex subjected to stretching (original) (raw)
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The three-dimensional instability of elliptical vortices in a viscoelastic fluid
Journal of Non-newtonian Fluid Mechanics, 1993
A linear stability analysis is presented for an upper-convected Maxwell fluid undergoing unbounded two-dimensional flows with elliptical streamlines and uniform vorticity. The flows are found to exhibit a Floquet-type instability to a plane-wave disturbance whose wave vector is periodically distorted in time. The elasticity of the fluid is found to exert a destabilizing influence.
Stretched vortices – the sinews of turbulence; large-Reynolds-number asymptotics
Journal of Fluid Mechanics, 1994
A large-Reynolds-number asymptotic theory is presented for the problem of a vortex tube of finite circulation [Gcy ] subjected to uniform non-axisymmetric irrotational strain, and aligned along an axis of positive rate of strain. It is shown that at leading order the vorticity field is determined by a solvability condition at first-order in ε = 1/R[Gcy ] where R[gcy ] = [gcy ]/ν. The first-order problem is solved completely, and contours of constant rate of energy dissipation are obtained and compared with the family of contour maps obtained in a previous numerical study of the problem. It is found that the region of large dissipation does not overlap the region of large enstrophy; in fact, the dissipation rate is maximal at a distance from the vortex axis at which the enstrophy has fallen to only 2.8% of its maximum value. The correlation between enstrophy and dissipation fields is found to be 0.19 + O(ε2). The solution reveals that the stretched vortex can survive for a long time ...
Three-dimensional instability of Burgers and Lamb–Oseen vortices in a strain field
Journal of Fluid Mechanics, 1999
The linear stability of Burgers and Lamb-Oseen vortices is addressed when the vortex of circulation Γ and radius δ is subjected to an additional strain field of rate s perpendicular to the vorticity axis. The resulting non-axisymmetric vortex is analysed in the limit of large Reynolds number R Γ = Γ /ν and small strain s Γ /δ 2 by considering the approximations obtained by and for each case respectively. For both vortices, the TWMS instability ) is shown to be active, i.e. stationary helical Kelvin waves of azimuthal wavenumbers m = 1 and m = −1 resonate and are amplified by the external strain in the neighbourhood of critical axial wavenumbers which are computed. The additional effects of diffusion for the Lamb-Oseen vortex and stretching for the Burgers vortex are proved to limit in time the resonance. The transient growth of the helical waves is analysed in detail for the distinguished scaling s ∼ Γ /(δ 2 R 1/2 Γ ). An amplitude equation describing the resonance is obtained and the maximum gain of the wave amplitudes is calculated. The effect of the vorticity profile on the instability characteristic as well as of a time-varying stretching rate are analysed. In particular the stretching rate maximizing the instability is calculated. The results are also discussed in the light of recent observations in experiments and numerical simulations. It is argued that the Kelvin waves resonance mechanism could explain various dynamical behaviours of vortex filaments in turbulence.
Experimental evidence for a new type of stretched vortex
European Journal of Mechanics - B/Fluids, 1998
Experimental measurements performed under conditions which reproduce most of the dynamical characteristics of the natural evolution of vorticity filaments in turbulent flows are presented here. Strong deviations from the Burgers vortex (which is a non-confined stretched vortex model) are observed and analyzed, 0 Elsevier, Paris
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.
Stability of Quasi-Two-Dimensional Vortices
Lecture Notes in Physics, 2010
Large-scale coherent vortices are ubiquitous features of geophysical flows. They have been observed as well at the surface of the ocean as a result of meandering of surface currents but also in the deep ocean where, for example, water flowing out of the Mediterranean sea sinks to about 1000 m deep into the Atlantic ocean and forms long-lived vortices named Meddies (Mediterranean eddies). As described by Armi et al. , these vortices are shallow (or pancake): they stretch out over several kilometers and are about 100 m deep. Vortices are also commonly observed in the Earth or in other planetary atmospheres. The Jovian red spot has fascinated astronomers since the 17th century and recent pictures from space exploration show that mostly anticyclonic long-lived vortices seem to be the rule rather than the exception. For the pleasure of our eyes, the association of motions induced by the vortices and a yet quite mysterious chemistry exhibits colorful paintings never matched by the smartest laboratory flow visualization (see .1). Besides this decorative role, these vortices are believed to structure the surrounding turbulent flow. In all these cases, the vortices are large scale in the horizontal direction and shallow in the vertical. The underlying dynamics is generally believed to be two-dimensional (2D) in first approximation. Indeed both the planetary rotation and the vertical strong stratification constrain the motion to be horizontal. The motion tends to be uniform in the vertical in the presence of rotation effects but not in the presence of stratification. In some cases the shallowness of the fluid layer also favors the two-dimensionalization of the vortex motion. In the present contribution, we address the following question: Are such coherent structures really 2D? In order to do so, we discuss the stability of such structures to three-dimensional (3D) perturbations paying particular attention to the timescale and the length scale on which they develop. Five instability mechanisms will be discussed, all having received renewed attention in the past few years. The shear instability and the generalized centrifugal instability apply to isolated vortices. Elliptic and hyperbolic instability involve an extra straining effect due to surrounding vortices or to mean shear. The newly discovered zigzag instability
Fluid Dynamics Research, 1997
The linear instability of a steady elliptical vortex in a stably stratified rotating fluid is investigated, using the quasi-geostrophic, f-plane approximation. The vortex is embedded in a uniform background straining field e with uniform vorticity 27 (the Moore-Saffman vortices). An elliptical vortex in an irrotational strain field (7 = 0) is shown to be unstable to long wave quasi-three-dimensional disturbances of azimuthal wavenumber m = 1 (bending wave). The long wave instability has a two-dimensional origin. An elliptical vortex in a simple shear, whose major axis is parallel to the shear streamlines (e = 7), is stable against any disturbance. In contrast, an ellipse in a simple shear, whose major axis is perpendicular to the shear streamlines (e =-7), is unstable to quasi-three-dimensional bending modes irrespective of a/b. Short wave quasi-three-dimensional disturbances grow faster than two-dimensional instability modes in the parameter range-0.5 < 7 < 0. The origin of short wave instability is attributed to resonance between inertial waves and the imposed strain field e.
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.
Dynamically stretched vortices as solutions of the 3D Navier–Stokes equations
1999
A well known limitation with stretched vortex solutions of the 3D Navier–Stokes (and Euler) equations, such as those of Burgers type, is that they possess uni-directional vorticity which is stretched by a strain field that is decoupled from them. It is shown here that these drawbacks can be partially circumvented by considering a class of velocity fields of the type u=(u1 (x, y, t), u2 (x, y, t), γ (x, y, t) z+ W (x, y, t)) where u1, u2, γ and W are functions of x, y and t but not z.
Instability of strained vortex layers and vortex tube formation in homogeneous turbulence
Journal of Fluid Mechanics, 1995
A modulational perturbation analysis is presented which shows that when a strained vortex layer becomes unstable, vorticity concentrates into steady tubular structures with finite amplitude, in quantitative agreement with the numerical simulations of Lin & Corcos (1984). Elaborated three-dimensional visualizations suggest that this process, due to a combination of compression and self-induced rotation of the layer, is at the origin of intense and long-lived vortex tubes observed in direct numerical simulations of homogeneous turbulence.