laboratoire de physique statistique
laboratoire de physique statistique




Turbulent 2.5-dimensional dynamos - Seshasayanan, K. and Alexakis, A.

Abstract : We study the linear stage of the dynamo instability of a turbulent two-dimensional flow with three components (u(x, y, t), v(x, y, t), w(x, y, t)) that is sometimes referred to as a 2.5-dimensional (2.5-D) flow. The flow evolves based on the two-dimensional Navier-Stokes equations in the presence of a large-scale drag force that leads to the steady state of a turbulent inverse cascade. These flows provide an approximation to very fast rotating flows often observed in nature. The low dimensionality of the system allows for the realization of a large number of numerical simulations and thus the investigation of a wide range of fluid Reynolds numbers Re, magnetic Reynolds numbers Rm and forcing length scales. This allows for the examination of dynamo properties at different limits that cannot be achieved with three-dimensional simulations. We examine dynamos for both large and small magnetic Prandtl-number turbulent flows Pm = Rm/Re, close to and away from the dynamo onset, as well as dynamos in the presence of scale separation. In particular, we determine the properties of the dynamo onset as a function of Re and the asymptotic behaviour in the large Rm limit. We are thus able to give a complete description of the dynamo properties of these turbulent 2.5-D flows.
Rotating Taylor-Green flow - Alexakis, A.

Abstract : The steady state of a forced Taylor-Green flow is investigated in a rotating frame of reference. The investigation involves the results of 184 numerical simulations for different Reynolds numbers ReF and Rossby numbers RoF. The large number of examined runs allows a systematic study that enables the mapping of the different behaviours observed to the parameter space (ReF; RoF), and the examination of different limiting procedures for approaching the large ReF small RoF limit. Four distinctly different states were identified: laminar, intermittent bursts, quasi-twodimensional condensates and weakly rotating turbulence. These four different states are separated by power-law boundaries R-OF proportional to Re-F(-gamma) in the small RoF limit. In this limit, the predictions of asymptotic expansions can be directly compared with the results of the direct numerical simulations. While the first-order expansion is in good agreement with the results of the linear stability theory, it fails to reproduce the dynamical behaviour of the quasi-two-dimensional part of the flow in the nonlinear regime, indicating that higher-order terms in the expansion need to be taken into account. The large number of simulations allows also to investigate the scaling that relates the amplitude of the fluctuations with the energy dissipation rate and the control parameters of the system for the different states of the flow. Different scaling was observed for different states of the flow, that are discussed in detail. The present results clearly demonstrate that the limits of small Rossby and large Reynolds numbers do not commute and it is important to specify the order in which they are taken.
Role of the basin boundary conditions in gravity wave turbulence - Deike, L. and Miquel, B. and Gutierrez, P. and Jamin, T. and Samin, B. and Berhanu, M. and Falcon, E. and Bonnefoy, F.

Abstract : Gravity wave turbulence is investigated experimentally in a large wave basin in which irregular waves are generated unidirectionally. The roles of the basin boundary conditions (absorbing or reflecting) and of the forcing properties are investigated. To that purpose, an absorbing sloping beach opposite the wavemaker can be replaced by a reflecting vertical wall. We observe that the wave field properties depend strongly on these boundary conditions. A quasi-one-dimensional field of nonlinear waves propagates towards the beach, where they are damped whereas a more multidirectional wave field is observed with the wall. In both cases, the wave spectrum scales as a frequency power law with an exponent that increases continuously with the forcing amplitude up to a value close to -4. The physical mechanisms involved most likely differ with the boundary condition used, but cannot be easily discriminated with only temporal measurements. We also studied freely decaying gravity wave turbulence in the closed basin. No self-similar decay of the spectrum is observed, whereas its Fourier modes decay first as a time power law due to nonlinear mechanisms, and then exponentially due to linear viscous damping. We estimate the linear, nonlinear and dissipative time scales to test the time scale separation that highlights the important role of a large-scale Fourier mode. By estimation of the mean energy flux from the initial decay of wave energy, the Kolmogorov-Zakharov constant of the weak turbulence theory is evaluated and found to be compatible with a recently obtained theoretical value.
Faraday instability in floating liquid lenses: the spontaneous mutual adaptation due to radiation pressure - Pucci, G. and Ben Amar, M. and Couder, Y.

Abstract : Fluid dynamics instabilities are usually investigated in two types of situations, either confined in cells with fixed boundaries, or free to grow in open space. In this article we study the Faraday instability triggered in a floating liquid lens. This is an intermediate situation in which a hydrodynamical instability develops in a domain with flexible boundaries. The instability is observed to be initially disordered with fluctuations of both the wave field and the lens boundaries. However, a slow dynamics takes place, leading to a mutual adaptation so that a steady regime is reached with a stable wave field in a stable lens contour. The most recurrent equilibrium lens shape is elongated with the Faraday wave vector along the main axis. In this self-organized situation an equilibrium is reached between the radiation pressure exerted by Faraday waves on the borders and their capillary response. The elongated shape is obtained theoretically as the exact solution of a Riccati equation with a unique control parameter and compared with the experiment.
Spatial variations of magnetic permeability as a source of dynamo action - Gallet, B. and Petrelis, F. and Fauve, S.

Abstract : We investigate dynamo action for a parallel flow of an electrically conducting fluid located over a boundary with spatially varying magnetic permeability. We first compute the dynamo threshold numerically. Then we perform an asymptotic expansion in the limit of small permeability modulation, which gives accurate results even for moderate modulation. We present in detail the mechanism at work for this dynamo It is an interplay between shear (an omega-effect) and a new conversion mechanism that originates from the non-uniform magnetic boundary. We illustrate how a similar mechanism leads to dynamo action in the case of spatially modulated electrical conductivity, a problem studied by Busse \& Wicht (Geophys. Astrophys. Fluid Dyn., vol. 64, 1992, pp. 135-144). Finally, we discuss the relevance of this effect to experimental dynamos and present ways to increase the dynamo efficiency and reduce the instability threshold.
Global bifurcations to subcritical magnetorotational dynamo action in Keplerian shear flow - Riols, A. and Rincon, F. and Cossu, C. and Lesur, G. and Longaretti, P. -Y. and Ogilvie, G. I. and Herault, J.

Abstract : Magnetorotational dynamo action in Keplerian shear flow is a three-dimensional nonlinear magnetohydrodynamic process, the study of which is relevant to the understanding of accretion processes and magnetic field generation in astrophysics. Transition to this form of dynamo action is subcritical and shares many characteristics with transition to turbulence in non-rotating hydrodynamic shear flows. This suggests that these different fluid systems become active through similar generic bifurcation mechanisms, which in both cases have eluded detailed understanding so far. In this paper, we build on recent work on the two problems to investigate numerically the bifurcation mechanisms at work in the incompressible Keplerian magnetorotational dynamo problem in the shearing box framework. Using numerical techniques imported from dynamical systems research, we show that the onset of chaotic dynamo action at magnetic Prandtl numbers larger than unity is primarily associated with global homoclinic and heteroclinic bifurcations of nonlinear magnetorotational dynamo cycles born out of saddle-node bifurcations. These global bifurcations are found to be supplemented by local bifurcations of cycles marking the beginning of period-doubling cascades. The results suggest that nonlinear magnetorotational dynamo cycles provide the pathway to injection of both kinetic and magnetic energy for the problem of transition to turbulence and dynamo action in incompressible magnetohydrodynamic Keplerian shear flow in the absence of an externally imposed magnetic field. Studying the nonlinear physics and bifurcations of these cycles in different regimes and configurations may subsequently help to understand better the physical conditions of excitation of magnetohydrodynamic turbulence and instability-driven dynamos in a variety of astrophysical systems and laboratory experiments. The detailed characterization of global bifurcations provided for this three-dimensional subcriticalfluid dynamics problem may also prove useful for the problem of transition to turbulence in hydrodynamic shear flows.
Dynamic interfacial tension effects in the rupture of liquid necks - Saint Vincent, M. Robert de and Petit, J. and Aytouna, M. and Delville, J. P. and Bonn, D. and Kelly, H.

Abstract : By examining the rupture of fluid necks during droplet formation of surfactant-laden liquids, we observe deviations from expected behaviour for the pinch-off of such necks. We suggest that these deviations are due to the presence of a dynamic (time-varying) interfacial tension at the minimum neck location and extract this quantity from our measurements on a variety of systems. The presence of such dynamic interfacial tension effects should change the rupture process drastically. However, our measurements show that a simple ansatz, which incorporates the temporal change of the interfacial tension, allows us to understand the dynamics of thinning. This shows that this dynamics is largely independent of the exact details of what happens far from the breakup location, pointing to the local nature of the thinning dynamics.
Linear and nonlinear stability of floating viscous sheets - Pfingstag, G. and Audoly, B. and Boudaoud, A.

Abstract : We study the stability of a thin, Newtonian viscous sheet floating on a bath of denser fluid. We first derive a general set of equations governing the evolution of a nearly flat sheet, accounting for geometrical nonlinearities associated with moderate rotations. We extend two classical models by considering arbitrary external body and surface forces; these two models follow from different scaling assumptions, and are derived in a unified way. The equations capture two modes of deformation, namely viscous bending and stretching, and describe the evolution of thickness, mid-surface and in-plane velocity as functions of two-dimensional coordinates. These general equations are applied to a floating viscous sheet, considering gravity, buoyancy and surface tension. We investigate the stability of the flat configuration when subjected to arbitrary in-plane strain. Two unstable modes can be found in the presence of compression. The first one combines undulations of the centre-surface and modulations of the thickness, with a wavevector perpendicular to the direction of maximum applied compression. The second one is a buckling mode; it is purely undulatory and has a wavevector along the direction of maximum compression. A nonlinear analysis yields the long-time evolution of the undulatory mode.
Bounding the scalar dissipation scale for mixing flows in the presence of sources - Alexakis, A. and Tzella, A.

Abstract : We investigate the mixing properties of scalars stirred by spatially smooth, divergence-free flows and maintained by a steady source sink distribution. We focus on the spatial variation of the scalar field, described by the dissipation wavenumber, k(d), that we define as a function of the mean variance of the scalar and its gradient. We derive a set of upper bounds that for large Peclet number (Pe >> 1) yield four distinct regimes for the scaling behaviour of kd, one of which corresponds to the Batchelor regime. The transition between these regimes is controlled by the value of Pe and the ratio rho = l(u)/l(s), where l(e) and l(s) are, respectively, the characteristic length scales of the velocity and source fields. A fifth regime is revealed by homogenization theory. These regimes reflect the balance between different processes: scalar injection, molecular diffusion, stirring and bulk transport from the sources to the sinks. We verify the relevance of these bounds by numerical simulations for a two-dimensional, chaotically mixing example flow and discuss their relation to previous bounds. Finally, we note some implications for three-dimensional turbulent flows.
Bistability between a stationary and an oscillatory dynamo in a turbulent flow of liquid sodium - Berhanu, M. and Gallet, B. and Monchaux, R. and Bourgoin, M. and Odier, Ph. and Pinton, J. -F. and Plihon, N. and Volk, R. and Fauve, S. and Mordant, N. and Petrelis, F. and Aumaitre, S. and Chiffaudel, A. and Daviaud, F. and Dubrulle, B. and Ravelet, F.

Abstract : We report the first experimental observation of a bistable dynamo regime. A turbulent flow of liquid sodium is generated between two disks ill the von karman geometry (VKS experiment). When one disk is kept at rest, bistability is observed between a stationary and an oscillatory magnetic field. The stationary and oscillatory branches occur in the vicinity of a codimension-two bifurcation that results from the coupling between two modes of magnetic field. We present an experimental study or the two regimes and study in detail the region of bistability that we understand ill terms of dynamical system theory. Despite the very turbulent nature of the flow, the bifurcations of the magnetic field are correctly described by a low-dimensional model. In addition, the different regimes are robust; i.e. turbulent fluctuations do not drive any transition between the oscillatory and stationary states in the region of bistability.
Quasi-static rheology of foams. Part 2. Continuous shear flow - Kabla, Alexandre and Scheibert, Julien and Debregeas, Georges

Abstract : The evolution of a bidimensional foam submitted to continuous quasi-static shearing is investigated both experimentally and numerically. We extract, from the images of the sheared foam, the plastic flow profiles as well as the local statistical properties of the stress field. When the imposed strain becomes larger than the yield strain, the plastic events develop large spatial and temporal correlations, and the plastic flow becomes confined to a narrow shear band. This transition and the steady-state regime of flow are investigated by first focusing on the elastic deformation produced by an elementary plastic event. This allows us to understand (i) the appearance of long-lived spatial heterogeneities of the stress field, which we believe are at the origin of the shear-banding transition, and (ii) the statistics of the dynamic fluctuations of the stress field induced by plastic rearrangements in the steady-state regime. Movies are available with the online version of the paper.
Quasi-static rheology of foams. Part 1. Oscillating strain - Kabla, Alexandre and Debregeas, Georges

Abstract : A quasi-static simulation is used to study the mechanical response of a disordered bidimensional aqueous foam submitted to an oscillating shear strain. The application of shear progressively extends the elastic domain, i.e. the strain range within which no plastic process occurs. It is associated with the development of an irreversible normal stress difference, and a decrease in the shear modulus, which are both signatures of the appearance of anisotropy in the film network. Beyond this mechanical measurement, the evolution of the structural properties of the foam is investigated. We focus in particular on the energy E-0 defined as the minimum line-length energy under zero shear stress. For strain amplitude less than similar to 0.5, this quantity is found to decay with the number of applied cycles as a result of the curing of topological defects. However, for higher strain amplitude, plastic events appear to increase the structural disorder and tend to gather near the shearing walls. This process is a precursor of the shear-banding transition observed in fully developed flows, which will be studied in the companion paper. Movies are available with the online version of the paper.
Viscosity of a dense suspension in Couette flow - Huang, Nicolas and Bonn, Daniel

Abstract : We study the rheology of a granular paste, i.e. a dense suspension of non-Brownian particles, quantitatively at steady state, in a cylindrical Couette cell. Previous studies have shown a discrepancy between local and global measurements of the viscosity for these materials, making it impossible to predict their resistance to flow. Using both MRI investigation techniques and classical rheology studies, we show that agreement between local and global measurements can be obtained, provided the migration of particles inside the gap is taken into account. As found by Leighton \& Acrivos (J. Fluid Mech. vol. 181, 1987, p. 415), the migration leads to a particle density gradient in the flow, the highly sheared regions being less dense in particles. Here, by comparing the local viscosity and particle density measurements from MRI with the macroscopic relation between viscosity and the volume fraction, it is shown that global and local measurements agree with each other. This consequently allows us to define a viscosity for dense suspensions.