Stéphane Le Dizès

Theoretical predictions for the elliptical instability in a two-vortex flow

Two parallel Gaussian vortices of circulations Γ 1 and Γ 2 radii a 1 and a 2 , separated by a distance b may become unstable by the elliptical instability due the elliptic deformation of their cores. The goal of the paper is to analyse this occurrence theoretically in a general framework. An explicit formula for the temporal growth rate of the elliptical instability in each vortex is obtained as a function of the above global parameters of the system, the Reynolds number Γ 1 / v and the non-dimensionalized axial wavenumber k z b of the perturbation. This formula is based on a known asymptotic expression for the local instability growth rate at an elliptical stagnation point which depends on the local characteristics of the elliptical flow and the inclination angle of the local perturbation wavevector at this point. The elliptical flow characteristics are estimated by considering each Gaussian vortex alone in a weak uniform external strain field whose properties are provided by a point vortex modelling of the vortex pair. The inclination angle is obtained from the dispersion relation for the Gaussian vortex normal modes and the local expression near each vortex centre for the two helical modes of azimuthal wavenumber m = 1 and m = −1 which constitute the elliptical instability global mode. Both the final formula and the hypotheses made for its derivation are tested and validated by direct numerical simulations and large-eddy simulations.

Viscous interactions of two co-rotating vortices before merging

The viscous evolution of two co-rotating vortices is analysed using direct two-dimensional numerical simulations of the Navier–Stokes equations. The article focuses on vortex interaction regimes before merging. Two parameters are varied: a steepness parameter n which measures the steepness of the initial vorticity profiles in a given family of profiles, and the Reynolds number Re (between 500 and 16 000). Two distinct relaxation processes are identified. The first one is non-viscous and corresponds to a rapid adaptation of each vortex to the external (strain) field generated by the other vortex. This adaptation process, which is profile dependent, is described and explained using the damped Kelvin modes of each vortex. The second relaxation process is a slow diffusion phenomenon. It is similar to the relaxation of any non-Gaussian axisymmetrical vortex towards the Gaussian. The quasi-stationary solution evolves on a viscous-time scale toward a single attractive solution which corresponds to the evolution from two initially Gaussian vortices. The attractive solution is analysed in detail up to the merging threshold a / b ≈ 0.22 where a and b are the vortex radius and the separation distance respectively. The vortex core deformations are quantified and compared to those induced by a single vortex in a rotating strain field. A good agreement with the asymptotic predictions is demonstrated for the eccentricity of vortex core streamlines. A weak anomalous Reynolds number dependence of the solution is also identified. This dependence is attributed to the advection–diffusion of vorticity towards the hyperbolic points of the system and across the separatrix connecting these points. A Re 1/3 scaling for the vorticity at the central hyperbolic point is obtained. These findings are discussed in the context of a vortex merging criterion.

Three-dimensional instability of Burgers and Lamb–Oseen vortices in a strain field

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 / v and small strain s [Lt ] Γ /δ 2 by considering the approximations obtained by Moffatt et al . (1994) and Jimenez et al . (1996) for each case respectively. For both vortices, the TWMS instability (Tsai & Widnall 1976; Moore & Saffman 1975) 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.

Elliptical instability in a rotating spheroid

Summary This study concerns the elliptical instability of a flow in a rotating deformed sphere. The aim of our work is to observe and measure the characterics of this instability in experiments and to compare them with theorical predictions. For this purpose, an elastic and transparent hollow sphere has been moulded. The flow is visualised using Kalliroscope flakes as the sphere is set into rotation and compressed by two rollers. The elliptical instability occurs by the appearance of the so-called ’spin-over’ mode whose growth rates and saturations are measured for different Eckman numbers by video image analysis. These growth rates compare avantageously to theorical calculations which are performed using classical asymptotic expansions. The linear analysis is then completed by a non linear model which predicts correctly the asymptotic regimes for high Eckman numbers. Some results that concern the elliptic instability in a rotating deformed spherical shell or the triangular instability will also be presented.

Elliptic and triangular instabilities in rotating cylinders

In this article, the multipolar vortex instability of the flow in a finite cylinder is addressed. The experimental study uses a rotating elastic deformable tube filled with water which is elliptically or triangularly deformed by two or three rollers. The experimental control parameters are the cylinder aspect ratio and the Reynolds number based on the angular frequency. For Reynolds numbers close to threshold, different instability modes are visualized using anisotropic particles, according to the value of the aspect ratio. These modes are compared with those predicted by an asymptotic stability theory in the limit of small deformations and large Reynolds numbers. A very good agreement is obtained which confirms the instability mechanism; for both elliptic and triangular configurations, the instability is due to the resonance of two normal modes (Kelvin modes) of the underlying rotating flow with the deformation field. At least four different elliptic instability modes, including combinations of Kelvin modes with azimuthal wavenumbers m = 0 and m = 2 and Kelvin modes m = 1 and m = 3 are visualized. Two different triangular instability modes which are a combination of Kelvin modes m = −1 and m = 2 and a combination of Kelvin modes m = 0 and m = 3 are also evidenced. The nonlinear dynamics of a particular elliptic instability mode, which corresponds to the combination of two stationary Kelvin modes m = −1 and m = 1, is examined in more detail using particle image velocimetry (PIV). The dynamics of the phase and amplitude of the instability mode is shown to be predicted well by the weakly nonlinear analysis for moderate Reynolds numbers. For larger Reynolds number, a secondary instability is observed. Below a Reynolds number threshold, the amplitude of this instability mode saturates and its frequency is shown to agree with the predictions of Kerswell (1999). Above this threshold, a more complex dynamic develops which is only sustained during a finite time. Eventually, the two-dimensional stationary elliptic flow is reestablished and the destabilization process starts again.

Stability of the Rankine vortex in a multipolar strain field

In this paper, the linear stability of a Rankine vortex in an n-fold multipolar strain field is addressed. The flow geometry is characterized by two parameters: the degree of azimuthal symmetry n which is an integer and the strain strength e which is assumed to be small. For n=2, 3 and 4 (dipolar, tripolar and quadrupolar strain fields, respectively), it is shown that the flow is subject to a three-dimensional instability which can be described by the resonance mechanism of Moore and Saffman [Proc. R. Soc. London, Ser. A 346, 413 (1975)]. In each case, two normal modes (Kelvin modes), with the azimuthal wave numbers separated by n, resonate and interact with the multipolar strain field when their axial wave numbers and frequencies are identical. The inviscid growth rate of each resonant Kelvin mode combination is computed and compared to the asymptotic values obtained in the large wave numbers limits. The instability is also interpreted as a vorticity stretching mechanism. It is shown that the inviscid gr...

Elliptic instability in a strained Batchelor vortex

The elliptic instability of a Batchelor vortex subject to a stationary strain field is considered by theoretical and numerical means in the regime of large Reynolds number and small axial flow. In the theory, the elliptic instability is described as a resonant coupling of two quasi-neutral normal modes (Kelvin modes) of the Batchelor vortex of azimuthal wavenumbers m and m + 2 with the underlying strain field. The growth rate associated with these resonances is computed for different values of the azimuthal wavenumbers as the axial flow parameter is varied. We demonstrate that the resonant Kelvin modes m = 1 and in =-1 which are the most unstable in the absence of axial flow become damped as the axial flow is increased. This is shown to be due to the appearance of a critical layer which damps one of the resonant Kelvin modes. However, the elliptic instability does not disappear. Other combinations of Kelvin modes m=-2 and m=0, then in = -3 and in = -1 are shown to become progressively unstable for increasing axial flow. A complete instability diagram is obtained as a function of the axial flow parameter for several values of the strain rate and Reynolds number. The numerical study considers a system of two counter-rotating Batchelor vortices in which the strain field felt by each vortex is due to the other vortex. The characteristics of the most unstable linear modes developing on the frozen base flow are computed by direct numerical simulations for two axial flow parameters and compared to the theory

Non-axisymmetric vortices in two-dimensional flows

Slightly non-axisymmetric vortices are analysed by asymptotic methods in the context of incompressible large-Reynolds-number two-dimensional flows. The structure of the non-axisymmetric correction generated by an external rotating multipolar strain field to a vortex with a Gaussian vorticity profile is first studied. It is shown that when the angular frequency w of the external field is in the range of the angular velocity of the vortex, the non-axisymmetric correction exhibits a critical-point singularity which requires the introduction of viscosity or nonlinearity to be smoothed. The nature of the critical layer, which depends on the parameter h = 1/( Re e 3/2 ), where e is the amplitude of the non-axisymmetric correction and Re the Reynolds number based on the circulation of the vortex, is found to govern the entire structure of the correction. Numerous properties are analysed as w and h vary for a multipolar strain field of order n = 2, 3, 4 and 5. In the second part of the paper, the problem of the existence of a non-axisymmetric correction which can survive without external field due to the presence of a nonlinear critical layer is addressed. For a family of vorticity profiles ranging from Gaussian to top-hat, such a correction is shown to exist for particular values of the angular frequency. The resulting non-axisymmetric vortices are analysed in detail and compared to recent computations by Rossi, Lingevitch & Bernoff (1997) and Dritschel (1998) of non-axisymmetric vortices. The results are also discussed in the context of electron columns where similar non-axisymmetric structures were observed (Driscoll & Fine 1990).

Three-dimensional instability of a multipolar vortex in a rotating flow

In this paper, the elliptic instability is generalized to account for Coriolis effects and higher order symmetries. We consider, in a frame rotating at the angular frequency Ω, a stationary vortex which is described near its center r=0 by the stream function written in polar coordinates Ψ=−(r2/2)+p(rn/n)cos(nθ), where the integer n is the order of the azimuthal symmetry, and p is a small positive parameter which measures the strength of the nonaxisymmetric field. Based on the Lifschitz and Hameiri [Phys. Fluids A 3, 2644–2651 (1991)] theory, the local stability analysis of the streamline Ψ=−1/2 is performed in the limit of small p. As for the elliptic instability [Bayly, Phys. Rev. Lett. 57, 2160–2163 (1986)], the instability is shown to be due to a parametric resonance of inertial waves when the inclination angle ξ of their wave vector with respect to the rotation axis takes a particular value given by cos ξ=±4/(n(1+Ω)). An explicit formula for the maximum growth rate of the inertial wave is obtained for...

An asymptotic description of vortex Kelvin modes

A large-axial-wavenumber asymptotic analysis of inviscid normal modes in an axisymmetric vortex with a weak axial flow is performed in this work. Using a WKBJ approach, general conditions for the existence of regular neutral modes are obtained. Dispersion relations are derived for neutral modes confined in the vortex core (‘core modes’) or in a ring (‘ring modes’). Results are applied to a vortex with Gaussian vorticity and axial velocity profiles, and a good agreement with numerical results is observed for almost all values of k . The theory is also extended to deal with singular modes possessing a critical point singularity. We demonstrate that the characteristics for vanishing viscosity of viscous damped normal modes can also be obtained. Known viscous damped eigenfrequencies for the Gaussian vortex without axial flow are, in particular, shown to be predicted well by our estimates. The theory is also shown to provide explanations for a few of their peculiar properties.