Numerical Simulation of Dendritic Solidification with Convection: Two-Dimensional Geometry (original) (raw)
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International Journal of Heat and Mass Transfer, 2003
We present and validate a numerical technique for computing dendritic growth of crystals from pure melts in the presence of forced convection. The Navier-Stokes equations are solved on a fixed Cartesian mesh and a mixed Eulerian-Lagrangian framework is used to treat the immersed phase boundary as a sharp solid-fluid interface. A conservative finite-volume discretization is employed which allows the boundary conditions to be applied exactly at the moving surface. Results are presented for a range of the growth parameters, namely crystalline anisotropy, flow Reynolds number and Prandtl number. Direct comparisons are made between the present results and those obtained with phase-field methods and excellent agreement is obtained. Values for the tip characteristics are tabulated to serve as benchmarks for computations of two-dimensional dendritic growth with convection.
IOP Conference Series: Materials Science and Engineering, 2012
A two-dimensional model is built and used to study thermo-solutal (double diffusive) convection generated during the solidification of a binary mixture in a rectangular enclosure cooled from bottom and side walls. In order to catch the smallest solute plumes scale the solidification model simulates directly the envelope of the columnar dendrites with a cellular automaton model. The mushy interior of the dendrite is modelled with a volume averaging method. The model predicts the occurrence of several flow regimes during solidification, such as turbulent, stratified and meandering flows. As a result or origin of the two dimensional meandering flow pattern, the concentration field is found to be organised in horizontal layers of uniform concentration. Those layers are separated by very thin boundary layers so that the concentration varies vertically in staircases.
Sharp-interface simulation of dendritic solidification of solutions
International Journal of Heat and Mass Transfer, 2002
A numerical method is developed for the simulation of solidification of solutions/alloys. The heat and species transport equations are solved with appropriate interface conditions. The interface shape and thermal and solutal fields are calculated in a fully coupled manner. The effects of capillarity are included in the interfacial dynamics. The present mixed Eulerian–Lagrangian framework treats the immersed phase boundary as a sharp solid–fluid interface and a conservative finite-volume formulation allows boundary conditions at the moving surface to be exactly applied. We first compare the planar growth results with published one-dimensional numerical results. We then show that the method can compute the breakdown of the solid–liquid interface due to the Mullins–Sekerka instability. The dendritic growth of the crystals under various growth parameters is computed.
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Journal of Crystal Growth, 2005
Three-dimensional free dendritic growth of a pure material into an undercooled melt in the presence of fluid flow is investigated numerically using the phase-field method. Such computations are made possible by solving the Navier-Stokes equations for the flow and the energy equation for the heat transport on a grid that is twice as coarse as the grid for the phase-field equation. The effect of the flow on the upstream growing dendrite tip velocity and radius of curvature is investigated as a function of the imposed flow velocity, undercooling, crystalline anisotropy, and Prandtl number. The results are compared to available theories of dendritic growth with and without convection. The predicted growth Pe´clet numbers as a function of the flow Pe´clet number are in reasonable agreement with the theoretical predictions. The dendrite tip selection parameter is essentially independent of the flow velocity within the range studied, which is also in accordance with theory. The three-dimensional dendrite tip shape is found to be well fitted by the same universal scaling relation as without flow. r 2005 Published by Elsevier B.V.
NUMERICAL MODELING OF DENDRITIC GROWTH IN ALLOY SOLIDIFICATION WITH FORCED CONVECTION
A two dimensional (2D) cellular automaton (CA) -lattice Boltzmann (LB) model is presented to investigate the effects of forced melt convection on the solutal dendritic growth. In the model, the CA approach of simulating the dendritic growth is incorporated with the kinetic-based lattice Boltzmann method (LBM) for numerically solving the melt flow and solute transport. Two sets of distribution functions are used in the LBM to model the convective-diffusion phenomena during dendritic growth. After validating the model by comparing the numerical results with the theoretical solutions, it is applied to simulate the single and multi dendritic growth of Al-Cu alloys without and with a forced convection. The typical asymmetric growth features of convective dendrite are reproduced and the dendritic morphology is strongly influenced by melt convection. The simulated convective multi dendritic features by the present model are also compared with that by the CA-NS model. The present model is found to be more computationally efficient and numerically stable than the CA-NS model.
Model of Dendritic Solidification
Dendritic structures are one of the most frequent patterns in nature, that appear in crystalline systems (such as metals), advanced ceramics and neural systems of the living organisms. Embedded in the core of many industrially important processes such as casting, there is a strong relationship between the material properties and the solidified dendritic microstructures. In our proposed study microstructural simulations were performed using MICRESS® code, which implements a multi-phase-field approach and essentially provides a mathematical solution to the coupled partial differential equations of phase-field and diffusion. In the context of the study, MICRESS® was used to simulate the morphological transitions and the solute segregations during directional solidification under a wide spectrum of temperature gradients, solidification rates and chemical compositions. The proposed model is integrated with an in-situ experimental study of solidification in a high-temperature laser-scanning confocal microscope. Such in-situ studies are able to provide an experimental mean to measure some of the crucial physical properties of the studied alloys, resulting in more refined microstructural simulations of such phenomena. For instance we demonstrated a real-time method for the calculation of Gibbs-Thomson coefficient and the solid-liquid interfacial energy of the studied alloys by analysing the shape of the grain-boundary grooves at the solid-liquid interface. The interface mobility parameter was calibrated so to achieve a realistic interface velocity aligned with the regime experimentally observed in real-time in the confocal microscopy.
Simulation of dendritic crystal growth with thermal convection
Interfaces and Free Boundaries, 2000
The dendritic growth of crystals under gravity influence shows a strong dependence on convection in the liquid. The situation is modelled by the Stefan problem with a Gibbs-Thomson condition coupled with the Navier-Stokes equations in the liquid phase. A finite element method for the numerical simulation of dendritic crystal growth including convection effects is presented. It consists of a parametric finite element method for the evolution of the interface, coupled with finite element solvers for the heat equation and Navier-Stokes equations in a time dependent domain. Results from numerical simulations in two space dimensions with Dirichlet and transparent boundary conditions are included.
Macroscopic modeling of columnar dendritic solidification
Computational & Applied Mathematics, 2004
This paper deals with the derivation of a macroscopic model for columnar dendritic solidification of binary mixtures using the volume averaging method with closure. The main originalities of the model are first related to the explicit description of evolving heterogeneities of the dendritic structures and their consequences on the derivation of averaged conservation equations, where additional terms involving porosity gradients are present, and on the determination of effective transport properties. These average properties are defined by the associated closure problems taking into account the geometry of the dendrites and the local intensity of the flow. The macroscopic solute transport is obtained by considering macroscale non-equilibrium giving rise to macroscopic dispersion and interfacial exchange phenomena. Mass exchange coefficients are accurately explicited as a function of the local geometry. Mathematical subject classification: 76S05, 76T05, 76R05.
Modelling Columnar Dendritic Growth into an Undercooled Metallic Melt in the Presence of Convection
MATERIALS TRANSACTIONS, 2005
A front-tracking technique on a fixed Cartesian grid, based on the kinetics of dendritic growth, is used to model the progress of an undercooled columnar dendritic front in non-equilibrium 2D solidification controlled by conduction and thermal natural convection. The effect of the alloy latent heat of fusion is included in this single-domain model through a careful definition of source terms in the energy conservation equation to account for both the advance of solidification front and subsequent thickening of the mushy zone within a control volume. The model is compared with the enthalpy approach showing its superiority in the detection of the undercooled liquid zone and, thus, in potentially modelling of columnar/equiaxed grain structures. It is used to predict the influence of both alloy composition and convective heat transfer coefficient on the size of the undercooled liquid zone in front of columnar dendrite tips during solidification of Al-Cu in a square mould. The predictions obtained confirm that natural convection in the melt reduces local temperature gradients and thus widens the undercooled liquid zone ahead of a curve joining columnar dendrite tips, increasing the potential for growth of equiaxed grains.