Two-dimensional instabilities of a viscous vortex dipole

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.

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.

Physics of Fluids, 2006

Evolution of a two-dimensional flow induced by a jet ejected from a nozzle of finite size is studied experimentally. Vortex dipole forms at the front of the developing flow while a trailing jet establishes behind the dipole. The dynamics of the flow is discussed on the basis of detailed measurements of vorticity and velocity fields which are obtained using particle image velocimetry. It is found that dipoles do not separate ͑pinch-off͒ from the trailing jet for values of the stroke ratio up to 15, which fact can be contrasted with the behavior of vortex rings reported previously by other authors. A characteristic time scale that is defined differently from the formation time of vortex rings can be introduced. This time scale ͑startup time͒ indicates the moment when the dipole starts translating after an initial period when it mainly grows absorbing the jet from the nozzle. A simple model that considers the competing effects of expansion and translation is developed to obtain an estimate of the dimensionless startup time. The dynamics of a dipole after the formation is characterized by a reduced flux of vorticity from the jet. The dipole moves forward with constant speed such that a value of the ratio of the speed of propagation of the dipole to the mean velocity of the jet is found to be 0.5. A universality of this ratio is explained in the framework of a model based on conservation of mass and momentum for the moving dipole.

Regular and Chaotic Dynamics, 2013

Point vortices have been extensively studied in vortex dynamics. The generalization to higher singularities, starting with vortex dipoles, is not so well understood. We obtain a family of equations of motion for inviscid vortex dipoles and discuss limitations of the concept. We then investigate viscous vortex dipoles, using two different formulations to obtain their propagation velocity. We also derive an integro-differential for the motion of a viscous vortex dipole parallel to a straight boundary.

Dynamical arguments based on the structure of the Euler equations suggest the possibility of rapid amplification of vorticity in which the vorticity and the rate of strain grow proportionately. During such growth, the vorticity is expected to amplify as a ͑t s − t͒ −1 power-law in time. This behavior is difficult to demonstrate numerically, in part, because initial transients tend to obscure it. Lamb dipoles are used here to construct the initial vorticity. This helps to avoid these transients and results in a flow exhibiting the expected power-law vorticity amplification for a period of time. The spatial region where the vorticity growth rate is maximal is investigated in detail using a decomposition of the vorticity along the principal axes of the rate-of-strain tensor. It is demonstrated that the vorticity and strain rate in one direction in this decomposition are proportional during the period of rapid vorticity growth. These findings suggest that, during this period, the Euler equations can be reduced to a one-dimensional model equation for vorticity in the rate-of-strain coordinate system.

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.

2008

Journal of Fluid Mechanics, 2000

This paper shows that a long vertical columnar vortex pair created by a double flap apparatus in a strongly stratified fluid is subjected to an instability distinct from the Crow and short-wavelength instabilities known to occur in homogeneous fluid. This new instability, which we name zigzag instability, is antisymmetric with respect to the plane separating the vortices. It is characterized by a vertically modulated twisting and bending of the whole vortex pair with almost no change of the dipole's cross- sectional structure. No saturation is observed and, ultimately, the vortex pair is sliced into thin horizontal layers of independent pancake dipoles. For the largest Brunt–Väisälä frequency N = 1.75 rad s−1 that may be achieved in the experiments, the zigzag instability is observed only in the range of Froude numbers: 0.13 < Fh0 < 0.21 (Fh0 = U0/NR, where U0 and R are the initial dipole travelling velocity and radius). When Fh0 > 0.21, the elliptic instability develop...

Experiments in Fluids, 2010

Experimental results on the dynamics of a vortex dipole evolving in a shallow fluid layer are presented. In particular, the generation of a spanwise vortex at the front of the dipole is observed in agreement with previous experiments at larger Reynolds numbers. The results show that this secondary vortex is of comparable strength to the dipole. The present physical analysis suggests that the origin of this structure involves the stretching induced by the dipole of the boundary-layer vorticity generated by the dipole’s advection over the no-slip bottom.

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