Vortex-induced instabilities and accelerated collapse due to inertial effects of density stratification (original) (raw)

Instabilities due a vortex at a density interface: gravitational and centrifugal effects

IUTAM Bookseries, 2009

A vortex placed at an initially straight density interface winds it into an ever-tightening spiral. This flow then displays rich dynamics, due to inertial effects caused by density stratification (non-Boussinesq effects), and gravitational effects. In the absence of gravity we showed recently that the flow is subject to centrifugal Rayleigh-Taylor and spiral Kelvin-Helmholtz instabilities. The latter grows slightly faster than exponentially. In this paper we present computations including gravity with and without and with inertial effects. Gravity modifies the spiralling process and contributes to the breakdown of the vortex. When both effects are allowed to operate together, the resulting flow has a complex radial character, with small-scale structures near the vortex core attributed to non-Boussinesq effects, and large scale roll-up due to gravity followed by breakdown.

Stability of a vortex in radial density stratification: role of wave interactions

Journal of Fluid Mechanics, 2011

We study the stability of a vortex in an axisymmetric density distribution. It is shown that a light-cored vortex can be unstable in spite of the ‘stable stratification’ of density. Using a model flow consisting of step jumps in vorticity and density, we show that a wave interaction mediated by shear is the mechanism for the instability. The requirement is for the density gradient to be placed outside the vortex core but within the critical radius of the Kelvin mode. Conversely, a heavy-cored vortex, found in other studies to be unstable in the centrifugal Rayleigh–Taylor sense, is stabilized when the density jump is placed in this region. Asymptotic solutions at small Atwood number At show growth rates scaling as At1/3 close to the critical radius, and At1/2 further away. By considering a family of vorticity and density profiles of progressively increasing smoothness, going from a step to a Gaussian, it is shown that sharp gradients are necessary for the instability of the light-co...

The Rayleigh–Taylor instability of two-dimensional high-density vortices

Journal of Fluid Mechanics, 2005

We investigate the stability of variable-density two-dimensional isolated vortices in the frame of incompressible mixing under negligible gravity. The focus on a single vortex flow stands as a first step towards vortex interactions and turbulent mixing. From heuristic arguments developed on a perturbed barotropic vortex, we find that high-density vortices are subject to a Rayleigh-Taylor instability. The basic mechanism relies on baroclinic vorticity generation when the density gradient is misaligned with the centripetal acceleration field. For Gaussian radial distributions of vorticity and density, the intensity of the baroclinic torque due to isopycnic deformation is shown to increase with the ratio δ/δ ρ of the vorticity radius to the density radius. Concentration of mass near the vortex core is confirmed to promote the instability by the use of an inviscid linear stability analysis. We measure the amplification rate for the favoured azimuthal wavenumbers m = 2, 3 on the whole range of positive density contrasts between the core and the surroundings. The separate influence of the density-contrast and the radius ratio is detailed for modes up to m = 6. For growing azimuthal wavenumbers the two-dimensional structure of the eigen mode concentrates on a ring of narrowing radial extent centered on the radius of maximum density gradient. The instability of the isolated high-density vortex is then explored beyond the linear stage based on high Reynolds number numerical simulations for modes m = 2, 3 and a moderate density contrast C ρ = 0.5. Secondary roll-ups are seen to emerge from the non-linear evolution of the vorticity and density fields. The transition towards m smaller vortices involves vorticity exchange between initially-rotating dense fluid particles and the irrotational less-dense medium. It is shown that baroclinic enstrophy production is associated with the centrifugal mass ejection away from the vortex center.

Mixing induced by Rayleigh–Taylor instability in a vortex

Physics of Fluids, 2005

The direct numerical simulation ͑DNS͒ of a two-dimensional Lamb-Oseen vortex with a heavy internal core has been performed. Linear stability theory predicts the existence of Rayleigh-Taylor ͑RT͒ instabilities due to the destabilizing effect of the centrifugal force on the radial flow nonhomogeneities. The DNS first exhibits wavy azimuthal perturbations which are nonlinearly distorted into bubble-like patterns, characteristic of the standard development of the RT instabilities, i.e., instabilities obtained in a planar nonhomogeneous flow in the presence of gravity. Nevertheless, important differences may be observed in the late stage development of the instability: contrary to the standard case, the bubbles are then stretched in the azimuthal direction leading to a strong radial filamentation of the flow.

Inertio-elastic instability of a vortex column

arXiv (Cornell University), 2021

We analyze the instability of a vortex column in a dilute polymer solution at large Re and De with El = De/Re, the elasticity number, being finite. Here, Re = Ω 0 a 2 /ν s and De = Ω 0 τ are, respectively, the Reynolds and Deborah numbers based on the core angular velocity (Ω 0), the radius of the column (a), the solvent-based kinematic viscosity (ν s = µ s /ρ), and the polymeric relaxation time (τ). The stability of small-amplitude perturbations in this distinguished limit is governed by the elastic Rayleigh equation whose spectrum is parameterized by E = El (1 − β), β being the ratio of the solvent to the solution viscosity. The neglect of the relaxation terms, in the said limit, implies that the polymer solution supports undamped elastic shear waves propagating relative to the base-state flow. The existence of these shear waves leads to multiple (three) continuous spectra associated with the elastic Rayleigh equation in contrast to just one for the original Rayleigh equation. Further, unlike the neutrally stable inviscid case, an instability of the vortex column arises for finite E due to a pair of elastic shear waves being driven into a resonant interaction under the differential convection by the irrotational shearing flow outside the core. An asymptotic analysis for the Rankine profile shows the absence of an elastic threshold; although, for small E, the growth rate of the unstable discrete mode is transcendentally small, being O(E 2 e −1/E 1 2). An accompanying numerical investigation shows that the instability persists for smooth vorticity profiles, provided the radial extent of the transition region (from the rotational core to the irrotational exterior) is less than a certain E-dependent threshold.

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.

Core dynamics of a strained vortex: instability and transition

Journal of Fluid Mechanics, 2001

We study the instability of a laminar vortex column (in an external orthogonal strain field) to an axisymmetric core size perturbation, and the resulting transition to fine-scale turbulence. The perturbation, which evolves as a standing wave oscillation (i.e. core dynamics, CD), is inviscidly amplified by the external strain. Analysis of a weakly strained Rankine vortex explains the physical mechanism of instability: resonant interaction between the perturbation – the azimuthal wavenumber m = 0 wave – and m = ±2 waves. The CD instability (CDI) – a type of elliptic instability – experiences the fastest growth when the CD oscillation frequency equals vortex column's fluid angular velocity, such matching occurring only at specific discrete values of the axial wavenumber k. At this resonant frequency, the net effect of the swirl-induced tilting of perturbation vorticity and the CD-induced tilting of base flow vorticity is such that perturbation vorticity is continually aligned with ...

Stability of a vortex with a heavy core

Journal of Fluid Mechanics, 2005

This paper examines the stability of swirling flows in a non-homogeneous fluid. Density gradients are shown to produce two distinct kinds of instability. The first is the centrifugal instability (CTI) which mainly affects axisymmetric, short-axialwavelength eigenmodes. The second is the Rayleigh-Taylor instability (RTI) which mainly affects non-axisymmetric, two-dimensional eigenmodes. These instabilities are described for a family of model flows for which the velocity law V (r) corresponds to a Gaussian vortex with radius 1, and the density law R(r) corresponds to a Gaussian distribution characterized by a density contrast C and a characteristic radius b. A full map in the (C, b)-plane is given for the amplification rate and the structure of the most amplified eigenmode. For small density contrasts (C < 0.5), the CTI occurs only for b > 1 and the RTI for b . 0.8. On the other hand, for high density contrasts (C > 0.5), a competition between the two kinds of instabilities is observed. From a fundamental point of view, the nature of the instability depends on the local values of G 2 = −r −1 V 2 R −1 dR/dr and the Rayleigh discriminant Φ = r −3 d(r 2 V 2 )/dr. CTI occurs whenever G 2 > Φ somewhere in the flow. For RTI, a necessary condition is that G 2 > 0 somewhere in the flow. By an asymptotic analysis, we show that this condition is also sufficient in the limit b → 0, C → 0. This asymptotic analysis also confirms that shear has a stabilizing effect on RTI and that this instability is strictly analogous to the standard RTI obtained in the case where light fluid is situated below heavier fluid in the presence of gravity. † Present address: IMFT, Allée du Professeur Camille Soula,