Hamda Ben Hadid

Numerical study of convection in the horizontal Bridgman configuration under the action of a constant magnetic field. Part 1. Two-dimensional flow

Studies of convection in the horizontal Bridgman configuration were performed to investigate the flow structures and the nature of the convective regimes in a rectangular cavity filled with an electrically conducting liquid metal when it is subjected to a constant vertical magnetic field. Under some assumptions analytical solutions were obtained for the central region and for the turning flow region. The validity of the solutions was checked by comparison with the solutions obtained by direct numerical simulations. The main effects of the magnetic field are first to decrease the strength of the convective flow and then to cause a progressive modification of the flow structure followed by the appearance of Hartmann layers in the vicinity of the rigid walls. When the Hartmann number is large enough, Ha > 10, the decrease in the velocity asymptotically approaches a power-law dependence on Hartmann number. All these features are dependent on the dynamic boundary conditions, e.g. confined cavity or cavity with a free upper surface, and on the type of driving force, e.g. buoyancy and/or thermocapillary forces. From this study we generate scaling laws that govern the influence of applied magnetic fields on convection. Thus, the influence of various flow parameters are isolated, and succinct relationships for the influence of magnetic field on convection are obtained. A linear stability analysis was carried out in the case of an infinite horizontal layer with upper free surface. The results show essentially that the vertical magnetic field stabilizes the flow by increasing the values of the critical Grashof number at which the system becomes unstable and modifies the nature of the instability. In fact, the range of Prandtl number over which transverse oscillatory modes prevail shrinks progressively as the Hartmann number is increased from zero to 5. Therefore, longitudinal oscillatory modes become the preferred modes over a large range of Prandtl number.

Buoyancy- and thermocapillary-driven flows in differentially heated cavities for low-Prandtl-number fluids

The influence of thermocapillary forces on buoyancy-driven convection is numerically simulated for shallow open cavities with differentially heated endwalls and filled with low-Prandtl-number fluid. Calculations are carried out by solving two-dimensional Navier-Stokes equations coupled to the energy equation, for three aspects ratios A = (length/height) = 4, 12.5 and 25, and several values of the Grashof number (up to 6 × 104) and Reynolds number (|Re| ≤ 1.67 × 104). Thermocapillarity can have a quite significant effect on the stability of a primarily buoyancy-driven flow. The result of the combination of the two basic mechanisms (thermo-capillarity) and buoyancy) depends on whether their effects are additive (positive Re) or opposing (negative Re); counter-acting mechanisms yield more complex flow patterns. The critical Grashof number Grc for the onset of the unsteady regime is found to decrease substantially within a small range of negative Re, and to increase for positive Re (and also for large negative Re). For Gr = 4 × 104, A = 4 and small negative Reynolds numbers, −2.4 × 103 ≤ Re < 0, mono-periodic and bi- or quasi-periodic regimes are shown to exist successively, followed by a reverse transition. The development of the instabilities from an initial steady-state regime has been investigated by varying Re for Gr = 1.5 × 104 (below Grc at Re = 0); the onset of buoyant instabilities is enhanced in a narrow range of Re only (-1200 < Re < -200). It is also noteworthy that for small enough Grashof numbers (e.g. Gr = 3 × 103), a steady-state solution prevails over the whole range of Reynolds numbers investigated. This means that a critical Grashof number exists below which the effect of the thermocapillary forces is no longer destabilizing.

Three-dimensional flow transitions under a rotating magnetic field

The present paper deals with the flows occurring in isothermal metallic liquids confined in cylindrical cavities and subjected to a rotating magnetic field. We analyze the influence of the main parameters: aspect ratio, Hartmann number and rotating Reynolds number on the structure and the nature of the melt flow. We also give the scaling laws for characteristic quantities such as azimuthal and meridional velocities.

Low-order dynamical model for low-Prandtl number fluid flow in a laterally heated cavity

By applying proper orthogonal decomposition (method of snapshots) to low Prandtl number fluid flow in a laterally heated cavity of dimensions 4×2×1 in length×width×height, characteristic basic modes have been extracted. Using Galerkin projection of the governing equations on these basic modes, a low-dimensional dynamical model (set of ordinary differential equations) was constructed. Some results obtained from the low-order model are presented and compared with those calculated by direct numerical simulation (DNS). The factors influencing the reliability of the low-order model such as the length of the reference signal, the snapshot density, the number of modes chosen for Galerkin projection, the characteristic velocity, and the chosen expansions for velocity and temperature are discussed. It is found that the low-order model can exactly reproduce the results obtained by DNS at the design conditions (i.e., for the Grashof and Prandtl numbers at which the basic modes have been obtained). The model can also fairly well approach the DNS results in a domain around these conditions. Nevertheless, it seems that such models have to be used with care and that, in any case, they can qualitatively predict the DNS results only in a not very large range around the design conditions.

Onset of Oscillatory Convection in Horizontal Layers of Low-Prandtl-Number Melts

This study is devoted to horizontal layers of low-Prandtl-number, Pr, fluids subjected to buoyancy forces in a long rectangular cavity which vertical endwalls are maintained at different temperatures. Our main motivation is to study temperature fluctuations occurring during the growth of metals and semi-conductor crystals (like GaAs) in horizontal-boats (e.g. by Bridgman technique).

Linear temporal and spatio-temporal stability analysis of a binary liquid film flowing down an inclined uniformly heated plate

Temporal and spatio-temporal instabilities of binary liquid films flowing down an inclined uniformly heated plate with Soret effect are investigated by using the Chebyshev collocation method to solve the full system of linear stability equations. Seven dimensionless parameters, i.e. the Kapitza, Galileo, Prandtl, Lewis, Soret, Marangoni, and Biot numbers (Ka, G, Pr, L, X, M, B), as well as the inclination angle (beta) are used to control the flow system. In the case of pure spanwise perturbations, thermocapillary S- and P-modes are obtained. It is found that the most dangerous modes are stationary for positive Soret numbers (chi >= 0), and oscillatory for chi 0 and even merges with the long-wave S-mode. In the case of streamwise perturbations, a long-wave surface mode (H-mode) is also obtained. From the neutral curves, it is found that larger Soret numbers make the film flow more unstable as do larger Marangoni numbers. The increase of these parameters leads to the merging of the long-wave H- and S-modes, making the situation long-wave unstable for any Galileo number. It also strongly influences the short-wave P-mode which becomes the most critical for large enough Galileo numbers. Furthermore, from the boundary curves between absolute and convective instabilities (AI/CI) calculated for both the long-wave instability (S- and H-modes) and the short-wave instability (P-mode), it is shown that for small Galileo numbers the AI/CI boundary curves are determined by the long-wave instability, while for large Galileo numbers they are determined by the short-wave instability.

Oscillating acoustic streaming jet

The present paper provides the first experimental investigation of an oscillating acoustic streaming jet. The observations are performed in the far field of a 2 MHz circular plane ultrasound transducer introduced in a rectangular cavity filled with water. Measurements are made by Particle Image Velocimetry (PIV) in horizontal and vertical planes near the end of the cavity. Oscillations of the jet appear in this zone, for a sufficiently high Reynolds number, as an intermittent phenomenon on an otherwise straight jet fluctuating in intensity. The observed perturbation pattern is similar to that of former theoretical studies. This intermittently oscillatory behavior is the first step to the transition to turbulence.

Effet de l'orientation d'un champ magnétique horizontal sur la stabilité de l'écoulement de Hadley

Effet of the orientation of a horizontal magnetic field on the stability of Hadley flow. A numerical study based on the linear stability analysis is undertaken, in order to determine the influence of a horizontal magnetic field on the marginal modes occuring in a fluid layer subjected to a horizontal temperature gradient. A particular interest is devoted to the influence of the magnetic field orientation on both nature and critical values of the unstable modes. Calculations show, that when it is subjected to such a magnetic field, this type of flow, known as Hadley flow, can present oblique waves, hitherto non-existent when no magnetic field is applied and even when a vertical, a transverse or a longitudinal magnetic field is imposed. A new asymptotic behavior is also observed for the stabilizing effects. To cite this article: S. Kaddeche et al., C. R. Mecanique 331 (2003).  2003 Academie des sciences. Publie par Editions scientifiques et medicales Elsevier SAS. Tous droits reserves.

Y-shaped jets driven by an ultrasonic beam reflecting on a wall.

This paper presents an original experimental and numerical investigation of acoustic streaming driven by an acoustic beam reflecting on a wall. The water experiment features a 2 MHz acoustic beam totally reflecting on one of the tank glass walls. The velocity field in the plane containing the incident and reflected beam axes is investigated using Particle Image Velocimetry (PIV). It exhibits an original y-shaped structure: the impinging jet driven by the incident beam is continued by a wall jet, and a second jet is driven by the reflected beam, making an angle with the impinging jet. The flow is also numerically modeled as that of an incompressible fluid undergoing a volumetric acoustic force. This is a classical approach, but the complexity of the acoustic field in the reflection zone, however, makes it difficult to derive an exact force field in this area. Several approximations are thus tested; we show that the observed velocity field only weakly depends on the approximation used in this small region. The numerical model results are in good agreement with the experimental results. The spreading of the jets around their impingement points and the creeping of the wall jets along the walls are observed to allow the interaction of the flow with a large wall surface, which can even extend to the corners of the tank; this could be an interesting feature for applications requiring efficient heat and mass transfer at the wall. More fundamentally, the velocity field is shown to have both similarities and differences with the velocity field in a classical centered acoustic streaming jet. In particular its magnitude exhibits a fairly good agreement with a formerly derived scaling law based on the balance of the acoustic forcing with the inertia due to the flow acceleration along the beam axis.

Acoustic Streaming Jets in Liquids

Theoretical and experimental investigations of acoustic streaming jets in water are described. The jet is produced by a plane circular ultrasonic transducer in a cavity inside a water tank, either in the near field or in the far field of the acoustic beam. The approach combines an experimental characterization of both the acoustic field and the obtained acoustic streaming velocity field on one hand, with both scaling analysis and CFD using an incompressible Navier-Stokes solver on the other hand. It is shown that good comparisons between experimental and numerical results can be obtained with a theoretical model based on a linear acoustic propagation model accounting for diffraction coupled to a hydrodynamic model including inertia effects. The coupling is obtained by the introduction of a momentum source term, the acoustic streaming force, in the hydrodynamic model. Both experimentally and numerically, the shape of the flow is thus found to be directly affected by both the overall shape of the acoustic beam and the local variations in acoustic pressure amplitude, in particular in the acoustic near field. Through scaling analysis, two scaling laws featuring linear or square root variations of the streaming velocity level with the acoustic power have been found. These scaling laws are shown to apply with a reasonable agreement to our numerical and experimental data, as well as to other former experimental investigations found in the literature.Copyright