Laurent Lacaze

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 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

Magnetic field induced by elliptical instability in a rotating spheroid

The tidal or the elliptical instability of the rotating fluid flows is generated by the resonant interaction of the inertial waves. In a slightly elliptically deformed rotating sphere, the most unstable linear mode is called the spin-over mode, and is a solid body rotation versus an axis aligned with the maximum strain direction. In the non-viscous case, this instability corresponds to the median moment of the inertial instability of the solid rotating bodies. This analogy is furthermore illustrated by an elliptical top experiment, which shows the expected inviscid heteroclinic behaviour. In geophysics, the elliptical instability may appear in the molten liquid cores of the rotating planets, which are slightly deformed by the tidal gravitational effects of the close bodies. It may then participate in the general outer core dynamics and possibly the geodynamo process. In this context, Kerswell and Malkus (Kerswell, R.R. and Malkus, W.V.R., Tidal instability as the source for Ios magnetic signature. Geophys. Res. Lett., 1998, 25, 603–606) showed that the puzzling magnetic field of the Jovian satellite Io may indeed be induced by the elliptically unstable motions of its liquid core that deflect the Jupiters magnetic field. Our magnetohydrodynamics (MHD) experiment is a toy-experiment of this geophysical situation and demonstrates for the first time the possibility of an induction of a magnetic field by the flow motions due to the elliptical instability. A full analytical calculation of the magnetic dipole induced by the spin-over is presented. Finally, exponential growths of this induced magnetic field in a slightly deformed rotating sphere filled with galinstan liquid metal are measured for different rotating rates. Their growth rates compare well with the theoretical predictions in the limit of a vanishing Lorentz force.

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.

Elliptical instability of the flow in a rotating shell

Abstract A theoretical and experimental study of the spin-over mode induced by the elliptical instability of a flow contained in a slightly deformed rotating spherical shell is presented. This geometrical configuration mimics the liquid rotating cores of planets when deformed by tides coming from neighboring gravitational bodies. Theoretical estimations for the growth rates and for the non linear amplitude saturations of the unstable mode are obtained and compared to experimental data obtained from Laser Doppler anemometry measurements. Visualizations and descriptions of the various characteristics of the instability are given as functions of the flow parameters.

Elliptic instability in a Rankine vortex with axial flow

The elliptic instability of a Rankine vortex with axial flow subject to a weak strain field perpendicular to its axis is analyzed by asymptotic methods in the limit of small strain rate. General unstable modes associated with resonant Kelvin modes of arbitrary azimuthal wavenumbers are considered. Both the effects of axial flow and viscosity are analyzed in details.

A three-dimensional experimental investigation of the structure of the spanwise vortex generated by a shallow vortex dipole

The three-dimensional dynamics of shallow vortex dipoles is investigated by means of an innovative three-dimensional, three-component (3D-3C) scanning PIV technique. In particular, the three-dimensional structure of a frontal spanwise vortex is characterized. The technique allows the computation of the three-dimensional pressure field and the planar (x, y) distribution of the wall shear stress, which are not available using standard 2D PIV measurements. The influence of such a complex vortex structure on mass transport is discussed in the context of the available pressure and wall shear stress fields.

On the existence and evolution of a spanwise vortex in laminar shallow water dipoles

The present work investigates the existence and evolution of a spanwise vortex at the front of shallow dipolar vortices. The vortex dipoles are experimentally generated using a double flap apparatus. Particle image velocimetry measurements are performed in a horizontal plane and in the vertical symmetry plane of the flow. The dynamics of such vortical structures is investigated through a parametric study in which both the Reynolds number Re=U0D0/ν∈[90,470] and the aspect ratio α=h/D0∈[0.075,0.7], associated with the shallowness of the flow, are varied, where U0 is the initial velocity of the vortex dipole, D0 is the initial diameter, h is the water depth, and ν is the kinematic viscosity of the fluid. The present experiments confirm the numerical results obtained in a companion paper by Duran-Matute et al. [Phys. Fluids 22, 116606 (2010)], namely that the flow remains quasi parallel with negligible vertical motions below a critical value of the parameter α2Re. By contrast, for large values of α2Re and α≲0...

Simulation of an Avalanche in a Fluid with a Soft-sphere/Immersed Boundary Method Including a Lubrication Force

The present work aims at reproducing the local dynamics of a dense granular media evolving in a viscous fluid from the grain scale to that of thousands of grains, encountered in environmental multiphase flows. To this end a soft-sphere collision/immersed-boundary method is developed. The methods are validated alone through various standard configurations including static and dynamical situations. Then, simulations of binary wall-particle collisions in a fluid are performed for a wide range of Stokes number ranging in [10-1, 104]. Good agreement with available experimental data is found provided that a local lubrication model is used. Finally, three- dimensional simulations of gravity/shear-driven dense granular flows in a viscous fluid are presented. The results open the way for a parametric study in the parameter space initial aspect ratio-initial packing.

Magnetic field induced by elliptical instability in a rotating tidally-distorted sphere

It is usually believed that the geo-dynamo of the Earth or more generally of other planets, is created by the convective fluid motions inside their molten cores. An alternative to this thermal or compositional convection can however be found in the inertial waves resonances generated by the eventual precession of these planets or by the possible tidal distorsions of their liquid cores. We will review in this paper some of our experimental works devoted to the elliptical instability and present some new results when the experimental fluid is a liquid metal. We show in particular that an imposed magnetic field is distorted by the spin- over mode generated by the elliptical instability. In our experiment, the field is weak (20 Gauss) and the Lorenz force is negligible compared to the inertial forces, therefore the magnetic field does not modify the fluid flow and the pure hydrodynamics growth rates of the instability are recovered through magnetic measurements.