Internal shear layers in a stratified and rotating fluid

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Thermo-elliptical instability in a rotating cylindrical shell

Journal of Fluid Mechanics (in press, 2006)The linear stability of a rotating flow in an elliptically deformed cylindrical shell with an imposed radial temperature contrast is studied using local and global approaches. We demonstrate that (i) a stabilising temperature profile can either increase or decrease the growth rate of the elliptical instability depending on the selected mode and on the strength of the radial buoyancy force; (ii) when the temperature profile is destabilising, the elliptical instability coexists with 2D convective instabilities at relatively small values of the Rayleigh number, the fastest growing mode depending on the relative values of the Rayleigh number and of the eccentricity; (iii) the elliptical instability totally disappears for larger values of the Rayleigh number. We argue that thermal effects have to be taken into account when looking for the occurrence and influence of inertial instabilities in geophysical and astrophysical systems, especially in planetary cores

L'instabilité elliptique en géophysique

L'étude de l'instabilité elliptique a été motivée par des problèmes rencontrés en aérodynamique (instabiltés secondaires diverses, vortex de bout d'ailes, turbulence), mais un autre intérêt suscité par cette instabilité elliptique relève de la géophysique. En effet, lorsqu'une planète tourne autour d'un soleil, et (ou) qu'une lune tourne autour de la planète, le noyau liquide de celle-ci subit une déformation elliptique (une marée) causée par le champ de gravitation. La rotation de la planète sur son axe pourrait alors faire résonner des ondes de Kelvin dans le noyau liquide ce qui résulterait en l'apparition de l'instabilité elliptique. Nos résultats expérimentaux et théoriques montrent qu'en effet, le mode dit de "spin-over" apparaît au seuil de l' instabilité dans la géométrie sphérique et qu'une dynamique intermittente existe à plus haut nombre de Reynolds. Finalement, une expérience sous champ magnétique et utilisant un métal liquide, met en évidence la génération d'un champ magnétique directement induit par l'instabilité

Libration-induced mean flow in a spherical shell

Accepted for publication in Journal of Fluid MechanicsInternational audienceWe investigate the flow in a spherical shell subject to a time harmonic oscillation of its rotation rate, also called longitudinal libration, when the oscillation frequency is larger than twice the mean rotation rate. In this frequency regime, no inertial waves are directly excited by harmonic forcing. We show however that it can generate through non-linear interactions in the Ekman layers a strong mean zonal flow in the interior. An analytical theory is developed using a perturbative approach in the limit of small libration amplitude $\epsilon$ and small Ekman number $E$. The mean flow is found to be at leading order an azimuthal flow which scales as the square of the libration amplitude and only depends on the cylindrical-radius coordinate. The mean flow also exhibits a discontinuity across the cylinder tangent to the inner sphere. We show that this discontinuity can be smoothed through multi-scale Stewartson layers. The mean flow is also found to possess a weak axial flow which scales as $O(\epsilon^2 E^{5/42})$ in the Stewartson layers. The analytical solution is compared to axisymmetric numerical simulations and a good agreement is demonstrated

Two-dimensional instability of smooth compressible vortices

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Generalized helical vortex pairs

International audienceNew solutions describing the interaction of helical pairs of counter-rotating vortices are obtained using a vortex filament approach. The vortices are assumed to have a small core size allowing the calculation of the self-induced velocities from the Biot-Savart law using the cutoff theory. These new vortex structures do not possess any helical symmetry but they exhibit a spatial periodicity and are stationary in a rotating and translating frame. Their properties, such as radial deformation, frame velocity and induced flow, are provided as a function of the four geometric parameters characterizing each solution. Approximate solutions are also obtained when the mutual interaction is weak. This allows us to provide explicit expressions for the rotation and translation velocities of the structure in this limit. First-order corrections describing helix deformation are also calculated and used for comparison with the numerical results. The variation of the vortex core size induced by the helix deformation is also analysed. We show that these variations have a weak effect on the shape and characteristics of the solutions, for the range of parameters that we have considered. The results are finally applied to rotor wakes. It is explained how these solutions could possibly describe the far wake of an helicopter rotor in vertical flight

Physics of vortex merging

20 p.This article deals with the interaction of co-rotating vortices, in configurations similar to those found in the extended near-wake of typical transport aircraft. The fundamental process of vortex merging is analyzed and modeled in detail in a two-dimensional context, giving insight into the conditions for merging and its physical origin, and yielding predictions for the resulting flow. Three-dimensional effects, in the form an elliptic short-wave instability arising in the initial co-rotating vortex flow, are described and analyzed theoretically. They are found to cause significant changes in the merging process, like earlier merging and larger final vortex cores. Illustrations from recent experimental, numerical and theoretical studies are given, and the relevance of the results for applications to real aircraft wakes is discussed

Inviscid instability of a stably stratified boundary layer

International audienceThe three-dimensional stability of an inflection-free boundary layer flow of length scale L and maximum velocity U0 in a stably stratified fluid of constant Brunt-Väisälä frequency N is examined in an inviscid Boussinesq framework. The plane of the boundary layer is assumed to be inclined with an angle θ with respect to the vertical direction of stratification. The stability analysis is performed using both numerical and theoretical methods for all the values of θ and Froude number F = U0 /(LN ). The boundary layer flow is found to be unstable whatever F as soon as θ ̸= 0. The growth rate of the most unstable mode is shown to increase with the inclination angle to reach its maximum for a vertical boundary layer θ = π/2. The unstable modes are 3D in nature. The mechanism of instability is shown to be associated with internal gravity wave emission. In the weakly stratified limit, both the oscillation frequency and the growth rate are found to scale with the Brunt-Väisälä frequency N. In the strongly stratified limit, frequency and growth rate become independent of the Froude number and proportional to the sine of the inclination angle (as long as F / sin θ ≪ 1)

Fluid flows in a librating cylinder

International audienceThe flow in a cylinder driven by time harmonic oscillations of the rotation rate, called longitudinal librations, is investigated. Using a theoretical approach and axisymmetric numerical simulations, we study two distinct phenomena appearing in this librating flow. First, we investigate the occurrence of a centrifugal instability near the oscillating boundary, leading to the so-called Taylor-Görtler vortices. A viscous stability criterion is derived and compared to numerical results obtained for various libration frequencies and Ekman numbers. The strongly nonlinear regime well above the instability threshold is also documented. We show that a new mechanism of spontaneous generation of inertial waves in the bulk could exist when the sidewall boundary layer becomes turbulent. Then, we analyse the librating flow below the instability threshold and characterize the mean zonal flow correction induced by the nonlinear interaction of the boundary layer flow with itself. In the frequency regime where inertial modes are not excited, we show that the mean flow correction in the bulk is a uniform rotation, independent of the Ekman number and cylinder aspect ratio, in perfect agreement with the analytical results of Wang [J. Fluid. Mech., 41, pp. 581 - 592, 1970]. When inertial modes are resonantly excited, the mean flow correction is found to have a more complex structure. Its amplitude still scales as the square of the libration amplitude but now depends on the Ekman number