Convection de Rayleigh Bénard dans l’He3 au voisinage du point critique

On étudie les scenarii de transition pour l’écoulement 2D de type Rayleigh Bénard dans l’He3 au voisinage de son point critique. Il s’agit d’une approche : à la fois théorique de stabilité linéaire et par simulation numérique directe. Pour le numérique on utilise une approximation de type anélastique des équations de Navier-Stokes compressibles et une loi d’état paramétrique. Quatre scenarii possibles de transition et la structuration spatiale des modes d’instabilité sont déterminés

Natural convection for large temperature gradients around a square solid body within a rectangular cavity

International audienceA numerical study of natural convection in cavity filled with air has been carried out under large temperature gradient. The flows under study are generated by a heated solid body located close to the bottom wall in a rectangular cavity with cold vertical walls and insulated horizontal walls. They have been investigated by direct simulations using a two-dimensional finite volume numerical code solving the time-dependent Navier-Stokes equations under the low Mach number approximation. This model permits to take into account large temperature variations unlike the classical Boussinesq model which is valid only for small temperature differences. We were particularly interested in the first transitions which occur when the Rayleigh number is increased for flows in cavities of aspect ratio A = 1, 2, 4. Starting from a steady state, the results obtained for A = 1 and A = 4 show that the first transition occurs through a supercritical Hopf bifurcation. The induced disturbances determined for weakly supercritical regimes indicate the existence of two instability types driven by different physical mechanisms: shear and buoyancy-driven instabilities, according to whether the flow develops in a square or in a tall cavity. For A = 2, the flow undergoes a pitchfork bifurcation leading to an asymmetric steady state which in turn becomes periodic via a supercritical Hopf bifurcation point. In both cases, the flow is found to be strongly deflected towards one vertical wall and instabilities are found to be of shear layers type

A finite difference method for 3D incompressible flows in cylindrical coordinates

International audienceIn this work, a finite difference method to solve the incompressible Navier-Stokes equations in cylindrical geometries is presented. It is based upon the use of mimetic discrete first-order operators (divergence, gradient, curl), i.e. operators which satisfy in a discrete sense most of the usual properties of vector analysis in the continuum case. In particular the discrete divergence and gradient operators are negative adjoint with respect to suitable inner products. The axis r = 0 is dealt with within this framework and is therefore no longer considered as a singularity. Results concerning the stability with respect to 3D perturbations of steady axisymmetric flows in cylindrical cavities with one rotating lid, are presented

Rayleigh-Bénard convection in 3He near its critical point

International audienceWe study by direct numerical simulation the Rayleigh-Bénard convection in supercritical 3He using an anelastic approximation. The latter is based on scaling analysis associated with asymptotic expansions of the full governing equations which has an important consequence for numerical integration of the model. The approximate equations are supplemented by a parametric state equation and are solved numerically by a finite difference method coupled with a projection method. Numerical results giving time evolutions of the temperature difference between the horizontal walls are calculated and compared to experimental data. The convection onset is examined for the studied configuration

Solutions multiples pour les écoulements tridimensionnels en rotation

The symmetry-breaking in a swirling flow generated inside a cylindrical tank of aspect ratio h = 1 by the rotation of one lid is studied numerically. Beyond a critical Reynolds number, the flow undergoes a bifurcation to three-dimensional solutions. The spatial and temporal behaviour on these branches are examined.La rupture de l'axisymétrie des écoulements entraînés par la rotation d'un disque au fond d'une cuve cylindrique de rapport de forme h=1 est étudiée numériquement. Au delà d'un nombre de Reynolds critique, l'écoulement bifurque vers des solutions tridimensionnelles. Les caractéristiques spatiales et temporelles des branches atteintes sont analysées

Numerical investigation of the first bifurcation for the flow in a rotor-stator cavity of radial aspect ratio 10

International audienceThe nature of transition to unsteadiness of rotor–stator disk flows of large radial aspect ratio is investigatedby means of several numerical tools which consist in computing the base flow even when unstable,performing linearized or non-linear time integrations starting from initial conditions of different amplitudesand computing the spectrum of the Jacobian using the ARPACK library. From these numerical experiments we conclude that,in a cavity of radial aspect ratio 10,the transition to unsteadiness occurs through a subcritical Hopf bifurcation. In addition these calculations show the existence of a large amplitude chaotic branch for values of the Reynolds number far below the linear stability threshold,and onto which the solutions are attracted for large subcritical values due to the strong non-normality of the Jacobian of the evolution operator

Etude numérique des écoulements tridimensionnels dans des cavités rotor- stator cylindriques

L'écoulement d'un fluide visqueux imcompressible dans une cuve cylindrique rotor-stator est étudié. Ces écoulements ont fait l'objet de récents travaux numériques tridimensionnels pour des rapports d'aspect h > 1.75. On propose une étude numérique de la rupture de l'axisymétrie de l'écoulement dans une cavité de rapport d'aspect réduit (h = 1 et 1.5) et des nombres de Reynolds Re < 8500. Les équations de Navier-Stokes sont résolues par un code vitesse-pression en coordonnées cylindriques qui repose sur des méthodes aux différences finies du second ordre. Deux approches numériques sont mises en oeuvre pour mener cette investigation. L'une consiste à linéariser les équations autour d'un écoulement de base asymétrique stationnaire. Les conditions initiales résultent de la superposition de lécoulement de base et d'une perturbation aléatoire. Cette analyse de stabilité linéaire permet de déterminer le premier seuil de criticalité et met en évidence le mode azimutal Kc le plus instable. L'autre consiste à intégrer les équations du mouvement avec différentes conditions initiales : (i) soit un écoulement de base axisymétrique stationnaire perturbé aléatoirement, (ii) soit un état où le fluide est au repos, (iii) soit un régime établi trouvé pour une autre valeur de Re. L'analyse de stabilité non-linéaire (i) montre que le développement d'un mode Kc est responsable d'une bifurcation de Hopf super-critique lorsque Re dépasse une première valeur critique. L'écoulement de base bascule alors vers des solutions instationnaires de période T, dont les caractéristiques spatiales dépendent de la valeur de kc. L'axisymétrie de l'écoulement n'est brisée que lorsque Kc [différent de] 0. Dans ce cas, les solutions super-critiques sont des ondes qui exhibent une invariance par rotation d'angle 2p/kc autour de l'axe et qui tournent avec une période azimutale T [indice RWKc] = Kc x T. L'écoulement subit ensuite d'autres bifurcations et les caractéristiques spatio-temporelles des ondes tournantes bifurquées sont présentées en détails. L'utilisation d'autres conditions initiales (ii)-(iii) met en évidence des branches de solutions multiples : Il apparaît que la convergence vers l'une ou l'autre des solutions possibles dépend de la façon dont est répartie l'énergie cinétique de départ.EVRY-BU (912282101) / SudocSudocFranceF

DNS of flows with helical symmetry

International audienceSome flows such as the wakes of rotating devices often display helical symmetry. We present an original DNS code for the dynamics of such helically symmetric systems. We show that, by enforcing helical symmetry, the three-dimensional Navier–Stokes equations can be reduced to a two-dimensional unsteady problem. The numerical method is a generalisation of the vorticity/streamfunction formulation in a circular domain, with finite differences in the radial direction and spectral decomposition along the azimuth. When compared to a standard three-dimensional code, this allows us to reach larger Reynolds numbers and to compute quasi-steady patterns. We illustrate the importance of helical pitch by some physical cases: the dynamics of several helical vortices, and a quasi-steady vortex flow. We also study the linear dynamics and nonlinear saturation in the Batchelor vortex basic flow, a paradigmatic example of trailing vortex instability. We retrieve the behaviour of inviscid modes and present new results concerning the saturation of viscous centre modes

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

International audienceThe 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

Bifurcations et solutions multiples en cavité 3D différentiellement chauffée

In this paper, a differentially heated square/cubic cavity is studied by performing three-dimensional direct numerical simulations. The first bifurcation observed at Ra ≈ 3.2 × 107 is due to the 3D vortex structures generated at the end regions of vertical boundary layers near the median plane. The main results of this Note are that the flow returns to a steady state for higher values of the Rayleigh number R a (7 × 107 and 108 for example) still exhibiting these 3D vortex structures, and that multiple steady flows which differ by their symmetry properties, are obtained for Ra = 10 8. However, the flow reverts to unsteadiness for Ra = 3 × 108. In this latter case, the instability is due to the vertical boundary layers