Patrice Le Gal

A numerical and theoretical study of the first Hopf bifurcation in a cylinder wake

The first Hopf bifurcation of the infinite cylinder wake is analysed theoretically and by direct simulation. It is shown that a decomposition into a series of harmonics is a convenient theoretical and practical tool for this investigation. Two basic properties of the instability allowing the use and truncation of the series of harmonics are identified: the lock-in of frequencies in the flow and separation of the rapid timescale of the periodicity from the slow timescale of the non-periodic behaviour. The Landau model is investigated under weak assumptions allowing strong nonlinearities and transition to saturation of amplitudes. It is found to be rather well satisfied locally at a fixed position of the flow until saturation. It is shown, however, that no truncated expansion into a series of powers of amplitude can account correctly for this fact. The validity of the local Landau model is found to be related to the variation of the form of the unstable mode substantially slower than its amplification. Physically relevant characteristics of the Hopf bifurcation under the assumption of separation of three timescales – those of the periodicity, amplification and deformation of the mode – are suggested.

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.

Hysteretic mode exchange in the wake of two circular cylinders in tandem

Our experimental study is devoted to the analysis of the flow past two tandem circular cylinders near the vortex shedding threshold. A recent bidimensional numerical analysis of this flow [Mizushima and Suehiro, Phys. Fluids 17, 104107 (2005)] has predicted that the bifurcation diagram should become complex in the vicinity of the instability threshold. Subcritical and saddle node bifurcations that lead to hysteretic exchanges between two different modes of vortex shedding were detected for particular distances of separation of the cylinders. We present here visualizations and velocity measurements of this flow in a water channel that prove the robustness of the complexity of the bifurcation diagram in real flows.

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.

Coriolis effects on the elliptical instability in cylindrical and spherical rotating containers

The effects of Coriolis force on the elliptical instability are studied experimentally in cylindrical and spherical rotating containers embarked on a table rotating at a fixed rate

Elliptical instability in hot Jupiter systems

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Flow-induced vibrations of high mass ratio flexible filaments freely hanging in a flow

. For a given set-up, changing the ratio

The universal aspect ratio of vortices in rotating stratified flows: experiments and observations

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Revisiting the two first instabilities of the flow in an annular rotor-stator cavity

of global rotation