Combined free and forced convection with two-dimensional solidification over a horizontal semi-infinite plate (original) (raw)
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Oscillatory Regimes for the Horizontal Thermal Convection in Solidification Problems
1998
A numerical study of the buoyancy driven natural convection of liquid materials is presented. There is experimental evidence that at low values of the Grashof number (Gr) the ow is laminar and steady while is periodic and generally time dependent as the value of Gr increases. This behaviour a ects the products in many applications, for instance the quality of semi-conductor crystals grown by the Czochralski and Bridgman techniques. The bifurcation pattern of this ow has been almost completely identi ed in the two dimensional case whereas in three dimensions we cannot mention any published result due to the inadequacy of the computational resources. In this work we shall present the ow con gurations occuring in the case of a shallow (4 1 1) box where the temperature gradient is applied at the smaller vertical faces. The uid considered has Prandtl number equal to 0.015. The vorticity-velocity formulation of the Navier-Stokes equations has been integrated by means of a fully implicit nite di erence scheme. By starting from a steady ow con guration as Gr increases we shall describe the transition to a periodic ow, and then from this one the transition to a non-periodic timedependent ow.
International Journal for Numerical Methods in Engineering, 1986
A new numerical technique was developed for the analysis of two-dimensional transient solidification processes in the presence of time dependent natural convection in the melt. The method can cope with irregular, transient morphologies of the solid-liquid interface using a new Galerkin formulation for the energy balance on the solid-liquid interface. The finite element solution to the Galerkin formulation yields the displacement of individual nodes on the solid-liquid interface. The displacement of the nodes is expressed by uncoupled components in the x and y direction. The fluid flow problem was solved using a "penalty" formulation. Numerical experiments were performed for Rayleigh numbers as high as 10 6 to demonstrate the method and to indicate the effect of natural convection on the solid-liquid interface morphology. '. ' .
International Journal of Heat and Mass Transfer, 2012
The natural convection and solidification in an annular enclosure has been studied experimentally and theoretically. Here the inner cylinder of the annular enclosure was cooled below the solidification temperature of the water, while the outer cylinder was kept on a uniform temperature well above 0°C. The problem of the unsteady growth of the ice-layer on the inner cold cylindrical surface is studied theoretically and an approximate solution has been obtained for the quasi-steady development of the ice-layer thickness. In addition, it has been found that the influence of the contact layer between the frozen layer and the cold surface is of significant importance for the solidification process. Results are presented and compared between the experimental and the analytical investigation.
Solidification in the presence of natural convection
International Communications in Heat and Mass Transfer, 1990
A method is proposed to calculate the time-dependent heat flux arising from natural convection during solidification of a liquid medium. This flux is incorporated in a solidification model which is then solved numerically using a time-stepping procedure. The predicted results are compared with experimentally obtained values, and the findings rationalized on the basis of heat transfer mechanisms operating in the system.
Journal of Crystal Growth, 1995
As a metallic melt solidifies, the viscosity variation may dramatically influence the convection induced by solidification as well as the microstructure of the resultant casting. In the present paper, we study the effect due to viscosity variation on the onset of double-diffusive convection occurring during the directional solidification of a binary solution cooling from below. Results show that as the viscosity contrast, denoted by y in Eq. (14), increases the stability of convection is enhanced and the mode of convection can change from the mushy-layer mode into the boundary-layer mode. The critical Rayleigh number R~m, for both the mushy-layer and boundary-layer modes, increases exponentially with 3' for all the parameter ranges considered, in contrast to the thermal convection case in which the critical Rayleigh number increases linearly with y. This difference emerges mainly from the length scale of the convection layer, which is of solutal boundary-layer thickness in the present study and is of thermal boundary-layer thickness in the thermal convection case.
A quasi-two-dimensional solidification benchmark, with controlled thermal boundary conditions, is proposed. Experiment have been performed with Sn-3wt.% Pb alloy. The experiment consists in solidifying a rectangular sample using two lateral heat exchangers which allow to control independently, the horizontal temperature gradient and cooling rate temperature. In this study, the temperature difference (DT) between the two lateral sides may vary from 0 to 40K and the cooling rate (CR) equal to 0.03K/s. This slow-cooling condition has been used in order to promote the formation of segregation. The forced convection is driven by a travelling magnetic field induced by a linear inductor. An array of fifty thermocouples is placed on the lateral wall in order to determine the instantaneous temperature distribution. The time evolution of the temperature field are recorded and analyzed. This allows us to evaluate the evolution due to the natural and forced convection, as well as its influence on the initial conditions, the solidification macrostructure and segregation behavior. Post mortem patterns of the macrosegregations have been obtained both by Xray radiographies and solute distribution is carried out by ICP analysis. Our objective in this study is to provide useful quantitative data for the validation of numerical models, in order to study how the segregation characteristics in the mushy zone are influenced by the natural and forced convection. It is shown that it modify the macrostructure and the segregation pattern, especially the freckle distribution.
Journal of Thermophysics and Heat Transfer, 2005
An experimental and numerical study of natural convection and solidification in a two-dimensional cavity driven by constant and oscillating temperature gradients is presented. Finite element models are developed to predict the flowfield, the temperature distribution, and the solid-liquid interphase shapes during solidification. Both the fixedgrid and moving-grid methods are applied in the numerical simulations, using the former to illustrate the oscillating thermal gradient conditions and using the latter to illustrate the constant temperature gradient conditions. An experimental system is set up where succinonitrile is used as a working fluid. The flow pattern, the velocity field, and the solidification interface shape are measured using the laser-based particle-image-velocimetry system. Numerical simulations and experiments are conducted for various configurations and different thermal gradients. In most cases, convection is dominated by one recirculating loop. With an inverted temperature gradient, however, multiple convection loops are observed. Both the convective flow pattern and the velocity strongly affect the solid-liquid interface shapes during solidification. In a majority of the cases studied, the model predictions are in good agreement with the experimental measurements.
Rayleigh-benard convection during solidification of an eutectic solution cooled from the top
Metallurgical and Materials Transactions B-process Metallurgy and Materials Processing Science, 2002
The interaction between laminar Rayleigh-Benard convection and directional solidification is studied for the case of an eutectic solution kept in a rectangular cavity cooled from the top. Experiments and numerical simulations are carried out using an NH4Cl-H2O solution as the model fluid. The flow is visualized using a sheet of laser light scattered by neutrally buoyant, hollow-glass spheres seeded in the fluid. The numerical modeling is performed using a pressure-based finite-volume method according to the SIMPLER algorithm. The present configuration enables us to visualize flow vortices in the presence of a continuously evolving solid/liquid interface. Clear visualization of the Rayleigh-Benard convective cells and their interaction with the solidification front are obtained. It is observed that the convective cells are characterized by zones of up-flow and down-flow, resulting in the development of a nonplanar interface. Because of the continuous advancement of the solid/liquid interface, the effective liquid height of the cavity keeps decreasing. Once the height of the fluid layer falls below a critical value, the convective cells become weaker and eventually die out, leading to the growth of a planar solidification front. Results of flow visualization and temperature measurement are compared with those from the numerical simulation, and a good agreement is found.
Thermosolutal convection during directional solidification
Metallurgical Transactions A, 1984
During solidification of a binary alloy at constant velocity vertically upward, thermosolutal convection can occur if the solute rejected at the crystal-melt interface decreases the density of the melt. We assume that the crystal-melt interface remains planar and that the flow field is periodic in the horizontal direction. The time-dependent nonlinear differential equations for fluid flow, concentration, and temperature are solved numerically in two spatial dimensions for small Prandtl numbers and moderately large Schmidt numbers. For slow solidification velocities, the thermal field has an important stabilizing influence: near the onset of instability the flow is confined to the vicinity of the crystal-melt interface. Further, for slow velocities, as the concentration increases, the horizontal wavelength of the flow decreases rapidly--a phenomenon also indicated by linear stability analysis. The lateral inhomogeneity in solute concentration due to convection is obtained from the calculations. For a narrow range of solutal Rayleigh numbers and wavelengths, the flow is periodic in time.
Two-dimensional solidification and melting in potential flows
Journal of Fluid …, 1999
The problem of solidification or melting under the action of a forced hydrodynamic flow is considered. In the appropriate parameter régime, the problem admits a formulation in terms of analytic functions. It is shown that a crystal with parabolic tip propagates without change of shape at a steady velocity. Some novel explicit solutions are presented.