H. Ben Hadid

On the onset of convective instabilities in cylindrical cavities heated from below. I. Pure thermal case

Three-dimensional steady flows are simulated in a circular cylindrical cavity of aspect ratio A=H/D, where H is the height and D the diameter of the cavity. The cavity is heated from below and its sidewalls are considered to be adiabatic. The effect of the geometry of the cavity on the onset of convection and on the structure and symmetries of the flow is analyzed. The nonlinear evolution of the convection beyond its onset is presented through bifurcation diagrams for two typical aspect ratios A=0.5 and A=1. Axisymmetric (m=0) and asymmetric (m=1 and m=2) azimuthal modes [exp (imφ)] are observed. For A=0.5, the axisymmetric solution loses its stability to a three-dimensional solution at a secondary bifurcation point. Better understanding of the mechanisms leading to this instability is obtained by analyzing the energy transfer between the basic state and the critical mode. To study the influence of the Prandtl number on the flow pattern and on the secondary bifurcation, three values of the Prandtl number ...

Magnetohydrodynamic convection in molten gallium

We present the results of an experimental and numerical study of the effects of a steady magnetic field on sidewall convection in molten gallium. The magnetic field is applied in a direction which is orthogonal to the main flow which reduces the convection and good agreement is found for the scaling of this effect with the relevant parameters. Moreover, qualitatively similar changes in the structure of the bulk of the flow are observed in the experiment and the numerical simulations. In particular, the flow is restricted to two dimensions by the magnetic field, but it remains different to that found in two-dimensional free convection calculations. We also show that oscillations found at even greater temperature gradients can be suppressed by the magnetic field.

Three-dimensional free convection in molten gallium

Convective flow of molten gallium is studied in a small-aspect-ratio rectangular, dierentially heated enclosure. The three-dimensional nature of the steady flow is clearly demonstrated by quantitative comparison between experimental temperature measurements, which give an indication of the strength of the convective flow, and the results of numerical simulations. The three-dimensional flow structure is characterized by cross-flows which are an order of magnitude smaller than the main circulation, and spread from the endwall regions to the entire enclosure when the Grashof number is increased beyond Gr =1 0 4 . The mergence of these eects in the centre of the enclosure leads to a complex central divergent flow structure which underpins the observed transition to oscillatory convection.

Interface curvature and convection related macrosegregation in the vertical Bridgman configuration

The interference between natural convection and macrosegregation is investigated numerically over a wide range of relevant parameters for the vertical Bridgman growth configuration. Our code is validated with respect to analytical, numerical and experimental data. In parallel, we propose scaling laws controlling segregation for the different convective regimes that allow for a discussion of the physical mechanisms governing mass transport in the melt. This work is of interest for the practician since an estimation of both radial and longitudinal segregation can be given from the obtained scaling laws.

Macrosegregation and convection in the horizontal Bridgman configuration I. Dilute alloys

Abstract Numerical studies of macrosegregation in the horizontal Bridgman configuration for a rectangular cavity filled with a dilute low Prandtl alloy are performed by investigating the flow structures and the species distributions under various conditions. Two configurations are studied: buoyancy-driven convection in confined cavity (rigid-rigid case) and surface tension-driven flow in a cavity where the upper boundary is a free surface (rigid-free case). The effects of the Grashof, the Reynolds Marangoni and the Schmidt numbers are described for an aspect ratio A = 4 ( A = L / H , where L is the length and H is the height of the cavity), a segregation coefficient k = 0.087 (corresponding to an alloy of Ge-Ga or GaAs-In), a growth rate V f = 2.7 10 -5 m/s and a Prandtl number Pr = 0.015. All the results are shown when half of the cavity is solidified. Particular attention is focused on understanding the influence of a transverse magnetic field on the flow structure and on the dopant distribution. One important feature of the action of the magnetic field is that strong magnetic field (Ha ≥ 200) has a significant effect in reducing radial segregation while intermediate values enhance the segregation effect.

Numerical simulation of convective three-dimensional flows in a horizontal cylinder under the action of a constant magnetic field

The effects of a constant magnetic field on electrically conducting liquid-metal flows in a cylindrical cavity corresponding to horizontal Bridgman growth are investigated numerically by solving the Navier-Stokes and Ohm equations for three-dimensional flows. The increase in the strength of the applied magnetic field leads to several fundamental changes in the properties of the thermal buoyant convection. The convective circulation progressively loses its intensity and is reorganized specifically depending on the direction (vertical, longitudinal or transversal) of the applied magnetic field. The characteristic behaviours observed correspond to the appearance of specific velocity profiles, of Hartmann layers and of parallel layers, and to the tendency towards two dimensionality. These structural changes are in fact closely connected to the repartition of the induced electric current inside the cavity.

Scaling and dimensional analysis of acoustic streaming jets

This paper focuses on acoustic streaming free jets. This is to say that progressive acoustic waves are used to generate a steady flow far from any wall. The derivation of the governing equations under the form of a nonlinear hydrodynamics problem coupled with an acoustic propagation problem is made on the basis of a time scale discrimination approach. This approach is preferred to the usually invoked amplitude perturbations expansion since it is consistent with experimental observations of acoustic streaming flows featuring hydrodynamic nonlinearities and turbulence. Experimental results obtained with a plane transducer in water are also presented together with a review of the former experimental investigations using similar configurations. A comparison of the shape of the acoustic field with the shape of the velocity field shows that diffraction is a key ingredient in the problem though it is rarely accounted for in the literature. A scaling analysis is made and leads to two scaling laws for the typical velocity level in acoustic streaming free jets; these are both observed in our setup and in former studies by other teams. We also perform a dimensional analysis of this problem: a set of seven dimensionless groups is required to describe a typical acoustic experiment. We find that a full similarity is usually not possible between two acoustic streaming experiments featuring different fluids. We then choose to relax the similarity with respect to sound attenuation and to focus on the case of a scaled water experiment representing an acoustic streaming application in liquid metals, in particular, in liquid silicon and in liquid sodium. We show that small acoustic powers can yield relatively high Reynolds numbers and velocity levels; this could be a virtue for heat and mass transfer applications, but a drawback for ultrasonic velocimetry.

Stability of convection in a horizontal channel subjected to a longitudinal temperature gradient. Part 1. Effect of aspect ratio and Prandtl number

The paper deals with the numerical investigation of the steady convective flow in a horizontal channel of rectangular cross-section subjected to a uniform longitudinal temperature gradient imposed at the walls. It is shown that at zero Prandtl number the solution of the problem corresponds to a plane-parallel flow along the channel axis. In this case, the fluid moves in the direction of the imposed temperature gradient in the upper part of the channel and in the opposite direction in the lower part. At non-zero values of the Prandtl number, such solution does not exist. At any small values of Pr all three components of the flow velocity differ from zero and in the channel cross-section four vortices develop. The direction of these vortices is such that the fluid moves from the centre to the periphery in the vertical direction and returns to the centre in the horizontal direction. The stability of these convective flows (uniform along the channel axis) with regard to small three-dimensional perturbations periodical in the direction of the channel axis is studied. It is shown that at low values of the Prandtl number the basic state loses its stability due to the steady hydrodynamic mode related to the development of vortices at the boundary of the counter flows. The growth of the Prandtl number results in the strong stabilization of this instability mode and, beyond a certain value of the Prandtl number depending on the cross-section aspect ratio, a new steady hydrodynamic instability mode becomes the most dangerous. This mode is characterized by the localization of the perturbations near the sidewalls of the channel. At still higher values of the Prandtl number, the spiral perturbations (rolls with axis parallel to the temperature gradient) become the most dangerous modes, at first the oscillatory spiral perturbations and then the Rayleigh-type steady spiral perturbations. The influence of the channel width on these different instabilities is also emphasized.

On the onset of oscillatory convection in molten gallium

The results of experimental and numerical investigations of the onset of oscillatory convection in a sidewall heated rectangular cavity of molten gallium are reported. Detailed comparisons are made between experimental observations and calculations from numerical simulations of a three-dimensional Boussinesq model. The onset of time-dependence takes place through supercritical Hopf bifurcations and the loci of critical points in the (Gr, Pr)-plane are qualitatively similar with excellent agreement between the frequencies of the oscillatory motion. This provides a severe test of the control of the experiment since the mode of oscillation is extremely sensitive to imperfections. Detailed numerical investigations reveal that there are a pair of Hopf bifurcations which exist on two asymmetric states which themselves arise at a subcritical pitchfork from the symmetric state. There is no evidence for this in the experiment and this qualitative difference is attributed to non-Boussinesq perturbations which increase with Gr. However, the antisymmetric spatial structure of the oscillatory state is robust and is present in both the experiment and the numerical model. Moreover, the detailed analysis of the numerical results reveals the origins of the oscillatory instability.

On the onset of convective instabilities in cylindrical cavities heated from below. II. Effect of a magnetic field

The effect of a constant and uniform magnetic field on electrically conducting liquid-metal flow, in cylindrical cavities heated from below, is numerically analyzed by using a spectral element method to solve the three-dimensional Navier–Stokes and Ohm equations. The cavity is characterized by its aspect ratio defined as A=H/D. The lateral surfaces are adiabatic and all the boundaries are electrically insulating. The flow with a vertical magnetic field has the same symmetries as that without a magnetic field, so that similar convective modes (m=0, m=1, and m=2) occur, but they are not equally stabilized. Here m is the azimuthal wave number. For A=0.5, for sufficiently large values of the Hartmann number Ha, the mode m=2 becomes the critical mode in place of m=0. The horizontal magnetic field breaks some symmetries of the flow. The axisymmetric mode disappears giving an asymmetric mode m=02, i.e., a combination of the m=0 and m=2 modes, whereas the asymmetric modes (m=1 and m=2), which were invariant by az...