Maurice Rossi

Information stored in Faraday waves: the origin of a path memory

On a vertically vibrating fluid interface, a droplet can remain bouncing indefinitely.When approaching the Faraday instability onset, the droplet couples to the wave itgenerates and starts propagatinghorizontally.Theresultingwave–particleassociation,called a walker, was shown previously to have remarkable dynamical properties,reminiscent of quantum behaviours. In the present article, the nature of a walker’swave field is investigated experimentally, numerically and theoretically. It is shownto result from the superposition of waves emitted by the droplet collisions with theinterface.Asingleimpactisstudiedexperimentallyandinafluidmechanicstheoreticalapproach. It is shown that each shock emits a radial travelling wave, leaving behinda localized mode of slowly decaying Faraday standing waves. As it moves, the walkerkeeps generating waves and the global structure of the wave field results from thelinear superposition of the waves generated along the recent trajectory. For rectilineartrajectories, this results in a Fresnel interference pattern of the global wave field. Sincethe droplet moves due to its interaction with the distorted interface, this means that itis guided by a pilot wave that contains a path memory. Through this wave-mediatedmemory, the past as well as the environment determines the walker’s present motion.Key words: drops, Faraday waves, pattern formation

Optimal linear growth in magnetohydrodynamic duct flow

Transient linear growth in laminar magnetohydrodynamic duct flow is analysed. The duct is straight with rectangular cross-section and electrically insulating walls. The applied uniform magnetic field is oriented perpendicular to the mean flow direction and parallel to one of the walls. Optimal perturbations and their maximum amplifications over finite time intervals are computed. The optimal perturbations are increasingly damped by the magnetic field, localized in the boundary layers parallel to the magnetic field irrespective of the duct aspect ratio. Typically, the optimal perturbations have non-vanishing streamwise wavenumber as found in magnetohydrodynamic channel flow with spanwise magnetic field. The Hartmann boundary layers perpendicular to the magnetic field do not contribute to the transient growth.

Optimal growth and transition to turbulence in channel flow with spanwise magnetic field

Instability and transition to turbulence in a magnetohydrodynamic channel flow are studied numerically for the case of a uniform magnetic field imposed along the spanwise direction. Optimal perturbations and their maximum amplifications over finite time intervals are computed in the framework of the linear problem using an iterative scheme based on direct and adjoint governing equations. It is shown that, at sufficiently strong magnetic field, the maximum amplification is no longer provided by classical streamwise rolls, but rather by rolls oriented at an oblique angle to the basic flow direction. The angle grows with the Hartmann number Ha and reaches the limit corresponding to purely spanwise rolls at Ha between 50 and 100 depending on the Reynolds number. Direct numerical simulations are applied to investigate the transition to turbulence at a single subcritical Reynolds number Re = 5000 and various Hartmann numbers. The transition is caused by the transient growth and subsequent breakdown of optimal perturbations, which take the form of one or two symmetric optimal modes (streamwise, oblique or spanwise modes depending on Ha) with low-amplitude three-dimensional noise added at the moment of strongest energy amplification. A sufficiently strong magnetic field (Ha larger than approximately 30) is found to completely suppress the instability. At smaller Hartmann numbers, the transition is observed but it is modified in comparison with the pure hydrodynamic case.

Nonlinear evolution of a swirling jet instability

The nonlinear evolution of the three-dimensional instability of a viscous unsteady swirling jet, namely, the Batchelor vortex, is addressed. Two types of initial perturbations are considered: a single unstable normal mode with given azimuthal symmetry and white noise. Three different scenarios have been put into evidence according to the value of the swirl number q selected in the unstable range q<1.5. When helical symmetry is present, the dynamics can be interpreted in a two-dimensional framework. More specifically, the process is viewed as the simultaneous action of a destabilizing “instantaneous” swirling jet instability and a stabilizing accelerated viscous diffusion induced by differential rotation. For swirl numbers close to the critical value (1<q<1.5), this latter effect dominates and leads the vortex to relaminarize in the nonlinear regime. For intermediate values of swirl (q∼0.8), it breaks into an array of equal sign vortices containing most of the initial circulation and surrounding a central ...

Viscous instability of a sheared liquid-gas interface: Dependence on fluid properties and basic velocity profile

In the framework of linear stability theory, we analyze how a liquid-gas mixing layer is affected by several parameters: viscosity ratio, density ratio, and several length scales. These scales reflect the presence of a velocity defect induced by the wake behind the splitter plate and the presence of boundary layers which develop ahead of the plate trailing edge. Incorporating such effects, we compute the various temporal and spatial instability modes and identify their driving instability mechanism based on their Reynolds number dependence, spatial structure, and energy budget. It is examined how the velocity defect modifies the temporal and the spatial stability properties. In addition, the transition from convective to absolute instability occurs at lower velocity contrast between gas and liquid free streams when a defect is present. This transition is also promoted by surface tension. Compared to inviscid stability computations, our spatial stability analysis displays a better agreement with measured g...

Three-dimensional stability of a Burgers vortex

The evolution of infinitesimal three-dimensional perturbations superimposed on a Burgers vortex is studied. By a sequence of variable transformations and scalings this linear evolution problem is reduced to a time-dependent system which is nearly identical to the stability equations governing a Lamb–Oseen vortex. The maximum amplification reached by perturbations over a finite time interval is computed through an iterative scheme based on the direct and adjoint governing equations, and results on the asymptotic stability of the Burgers vortex are deduced. The Burgers vortex is shown to be asymptotically stable, although significant short-term amplification may occur.

Of Vortices and Vortical Layers: An Overview

A theoretical overview of local flow models such as hyperbolic point flows or localized vorticity structures is presented. Vortex layers and tubes are particularly emphazised. Various exact Navier-Stokes or Euler solutions are introduced to analyse generic features of vorticity dynamics: vorticity gradients, vorticity stretching, interplay between axial and azimuthal vorticity, effect of a large scale strain rate or the existence of a helical symmetry. The linear stability of some of these basic flows is considered.

Identification of parameters in amplitude equations describing coupled wakes

Abstract We study the flow behind an array of equally spaced parallel cylinders. A system of Stuart-Landau equations with complex parameters is used to model the oscillating wakes. Our purpose is to identify a set of six scalar parameters which accurately reproduces the experimental data of Chauve and Le Gal [Physica D 58 (1992) 407–413]. To do so, we perform a computational search for the minimum of a distance J . We define J as the sum-square difference of the experimental data and reconstructed data obtained from our model. The differences are computed over a “training window” (a finite-time interval). The initial conditions are obtained from observations at te beginning of the interval. Boundary conditions are zero amplitude oscillations just outside the domain. The search algorithm uses a backpropagation technique for the computation of the descent direction. Using the parameters computed via the backpropagation method, the coupled Stuart-Landau equations accurately predicted the experimental data from Chauve and Le Gal over several wake oscillations and reproduced the qualitative features of the dynamics. A relatively large degree of uncertainty remains on the parameters controlling the coupling between adjacent cylinders. This ambiguity does not prevent short-time predictions, which indicates that a family of parameter values are nearly equivalent for the description of the system. Synthetic data were obtained using the model system of equation together with a model of the noisy measurement process. These data allow to reproduce the uncertainty in parameter identification. The “horizon time” after which forecasts of the system behavior loose their accuracy has been measured. We show that as expected the horizon time increases slightly with the length of the data series but seems to reach a limit. Interestingly, there is also an optimal length for the training window that yields the largest horizon time.

The dynamics of a viscous vortex dipole

The structure of a two-dimensional viscous dipole is accurately analyzed using both numerical simulations and theoretical analyses. First, a model is proposed, which computes the dipole velocity and the vortex ellipticity based on a heuristic relation between a vortex patch and a vortex with distributed vorticity profile. Second, during the stage where vortices are close to each other, a generalized self-similar solution is postulated to describe the vorticity profiles observed during the viscous spreading of the dipole. Numerical as well as theoretical considerations are given, which demonstrate the adequacy of such a hypothesis. Finally the structure of the tail that is generated behind the dipole is given in an analytical form, which favorably compares to numerical results.

Transient growth and instability in rotating boundary layers

The three-dimensional temporal instability of rotating boundary layer flows is investigated by computing classical normal modes as well as by evaluating the transient growth of optimal disturbances. The flows examined are the rotating Blasius (RB) and the rotating asymptotic suction layers (RAS), with the rotation axis normal to the basic flow plane. In agreement with an inviscid criterion, streamwise unstable modes are found in both flow cases for anticyclonic rotation: at high Reynolds numbers, one obtains the Rossby number unstable range 0<1/Ro<0.57 for RB, or 0<1/Ro<1 for RAS. Critical Reynolds and Rossby numbers are also determined in both instances. Moreover, the dependence of transient growth with respect to wavenumbers, Rossby and Reynolds numbers is presented for both cyclonic and anticyclonic regimes. In particular, the peak transient growth is computed for a wide range of parameter values within the cyclonic regime and is shown to be reduced by rotation. A scaling analysis with respect to the R...