Marc Rabaud

On the hydrodynamics of soap films

Abstract Several experiments aiming at the exploration of the hydrodynamical properties of soap films are presented. Their interpretation takes into account the very specific equation of state of these films. It is shown that on short time scales each element of the film moves as a whole so that the film can be considered as a two-dimensional fluid with a local density proportional to its thickness. When set horizontally, quasi two-dimensional turbulent flows can be obtained. The film behaves as an incompressible fluid whenever the motions occur at velocities small compared to the velocity of its elastic waves. An estimate of the role of air friction is given. The static quasi equilibrium of a film when set vertical is discussed. Phenomena equivalent to the rise of buoyant bubbles can be obtained. It is shown that lee waves can also be generated confirming that a vertical soap film has the dynamical properties of a two-dimensional density stratified fluid.

Decaying grid-generated turbulence in a rotating tank

The decay of initially three-dimensional homogeneous turbulence in a rotating frame is experimentally investigated. Turbulence is generated by rapidly towing a grid in a rotating water tank, and the velocity field in a plane perpendicular to the rotation axis is measured by means of particle image velocimetry. During the decay, strong cyclonic coherent vortices emerge, as the result of enhanced stretching of the cyclonic vorticity by the background rotation, and the selective instability of the anticyclonic vorticity by the Coriolis force. This asymmetry towards cyclonic vorticity grows on a time scale Ω−1 (Ω is the rotation rate), until the friction from the Ekman layers becomes dominant. The energy spectrum perpendicular to the rotation axis becomes steeper as the instantaneous Rossby number Roω=ω′∕2Ω decreases below the value 2±0.5 (ω′ is the root-mean square of the vertical vorticity). The spectral exponent increases in time from its classical Kolmogorov value 5∕3 up to values larger than 2. Below the...

Granular avalanches in fluids.

Three regimes of granular avalanches in fluids are put in light depending on the Stokes number St which prescribes the relative importance of grain inertia and fluid viscous effects and on the grain/fluid density ratio r. In gas (r>>1 and St>1, e.g., the dry case), the amplitude and time duration of avalanches do not depend on any fluid effect. In liquids (r approximately 1), for decreasing St, the amplitude decreases and the time duration increases, exploring an inertial regime and a viscous regime. These three regimes are described by the analysis of the elementary motion of one grain.

Axisymmetric propagating vortices in the flow between a stationary and a rotating disk enclosed by a cylinder

The destabilization of the stationary basic flow occurring between two disks enclosed by a cylinder is studied experimentally when the radius of the disks is large compared to the spacing. In the explored range of the cell aspect ratio, when one disk only is rotating, circular vortices propagating to the centre are observed above a critical angular velocity. These structures occur naturally but can also be forced by small modulations of the angular velocity of the disk. For each rotation rate the dispersion relation of the instability is experimentally reconstructed from visualizations and it is shown that this dispersion relation can be scaled by the boundary layer thickness measured over the disk at rest. The bifurcation is found to be of supercritical nature. The effect of the forcing amplitude is in favour of a linear convective nature of this instability of the non-parallel inward flow existing above the stationary disk. The most unstable temporal frequency is found to be about four times the frequency of the rotating disk. The evolution of the threshold of this primary instability is described for different aspect ratios of the cell. Finally, two sets of experiments made under transient conditions are presented: one in order to investigate further a possible convective/absolute transition for the instability, and the other to compare with the impulsive spin-down-to-rest experiments of Savas (1983).

Shear instability of two-fluid parallel flow in a Hele-Shaw cell

We study experimentally the parallel flow in a Hele–Shaw cell of two immiscible fluids, a gas and a viscous liquid, driven by a given pressure gradient. We observe that the interface is destabilized above a critical value of the gas flow and that waves grow and propagate along the cell. The experimental threshold corresponds to a velocity difference of the two fluids in good agreement with the inviscid Kelvin–Helmholtz instability, while the wave velocity corresponds to a pure viscous theory deriving from Darcy’s law. We report our experimental results and analyze this instability by the study of a new equation where the viscous effects are added to the Euler equation through a unique drag term. The predictions made from the linear stability analysis of this equation agree with the experimental measurements.

Experimental and numerical investigation of a forced circular shear layer

In a previous article we introduced a dissipative circular geometry in which stationary states of the shear flow instability were obtained. We show here that the dynamical behaviour of this flow depends strongly on the aspect ratio of the cell. In large cells, where the number of vortices is large, transitions from a mode with m vortices to a mode with ( m −1) vortices occur through localized processes. In contrast to that situation, in small cells, transition takes place after a series of bifurcations which correspond to the successive breaking of all the symmetries of the flow. We show that, provided an adequate forcing term is introduced, a two-dimensional numerical simulation of this flow is sufficient to recover all the dynamical processes which characterize the experimental flow.

A shear-flow instability in a circular geometry

A circular shear zone is created in a thin layer of fluid. The Kelvin-Helmholtz instability induces regular, steady patterns of m vortices. The experimental conditions are such that neither the centrifugal nor the Coriolis forces play a role in the motion. The state of the flow is defined by a Reynolds number, the value of which is controlled by the imposed velocities. The pattern of vortices can be characterized by its wavevector k or by m , the order of its symmetry. As k is quantized, its evolution, due to an increase or a decrease of the controlled stress, leads to transitions between patterns of different m. The transitory states between different symmetries are investigated. The experiments are performed with a soap film which provides a new type of visualization of an air flow.

Ship Wakes: Kelvin or Mach Angle?

From the analysis of a set of airborne images of ship wakes, we show that the wake angles decrease as U(-1) at large velocities, in a way similar to the Mach cone for supersonic airplanes. This previously unnoticed Mach-like regime is in contradiction with the celebrated Kelvin prediction of a constant angle of 19.47° independent of the ships speed. We propose here a model, confirmed by numerical simulations, in which the finite size of the disturbance explains this transition between the Kelvin and Mach regimes at a Froude number Fr=U/√[gL]~/=0.5, where L is the hull ship length.

Instabilities in the flow between co- and counter-rotating disks

The flow between two rotating disks (radius to heigh ratio of 20.9), enclosed by a rotating cylinder, is investigated experimentally in the cases of both co- and counter-rotation. This flow gives rise to a large gallery of instability patterns. A regime diagram of these patterns is presented in the ( Re b , Re t )-plane, where Re b,t is the Reynolds number associated with each disk. The co-rotation case and the weak counter-rotation case are very similar to the rotor–stator case, both for the basic flow and the instability patterns: the basic flow consists of two boundary layers near each disk and the instability patterns are the axisymmetric vortices and the positive spirals described in the rotor–stator experiments of Gauthier, Gondret & Rabaud (1999), Schouveiler, Le Gal & Chauve (2001), and the numerical study of Serre, Crespo del Arco & Bontoux (2001). The counter-rotation case with higher rotation ratio is more complex: above a given rotation ratio, the recirculation flow becomes organized into a two-cell structure with the appearance of a stagnation circle on the slower disk. A new kind of instability pattern is observed, called negative spirals. Measurements of the main characteristics of this pattern are presented, including growth times, critical modes and phase velocities.

Experimental and numerical study of the shear layer instability between two counter-rotating disks

The shear layer instability in the flow between two counter-rotating disks enclosed by a cylinder is investigated experimentally and numerically, for radius-to-height ratio Γ=R/h between 2 and 21. For sufficiently large rotation ratio, the internal shear layer that separates two regions of opposite azimuthal velocities is prone to an azimuthal symmetry breaking, which is investigated experimentally by means of visualization and particle image velocimetry. The associated pattern is a combination of a sharp-cornered polygonal pattern, as observed by Lopez et al. (2002) for low aspect ratio, surrounded by a set of spiral arms, first described by Gauthier et al. (2002) for high aspect ratio. The spiral arms result from the interaction of the shear layer instability with the Ekman boundary layer over the faster rotating disk. Stability curves and critical modes are experimentally measured for the whole range of aspect ratios, and are found to compare well with numerical simulations of the three-dimensional time-dependent Navier–Stokes equations over an extensive range of parameters. Measurements of a local Reynolds number based on the shear layer thickness confirm that a shear layer instability, with only weak curvature effect, is responsible for the observed patterns. This scenario is supported by the observed onset modes, which scale as the shear layer radius, and by the measured phase velocities.