Simulation of dendritic crystal growth with thermal convection (original) (raw)
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Sharp-interface simulation of dendritic growth with convection: benchmarks
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.
Numerical Simulation of Dendritic Solidification with Convection: Two-Dimensional Geometry
Journal of Computational Physics, 2002
A front tracking method is presented for simulations of dendritic growth of pure substances in the presence of flow. The liquid-solid interface is explicitly tracked and the latent heat released during solidification is calculated using the normal temperature gradient near the interface. A projection method is used to solve the Navier-Stokes equations. The no-slip condition on the interface is enforced by setting the velocities in the solid phase to zero. The method is validated through a comparison with an exact solution for a Stefan problem, a grid refinement test, and a comparison with a solution obtained by a boundary integral method. Three sets of two-dimensional simulations are presented: a comparison with the simulations of Beckermann et al. (J. Comput. Phys. 154, 468, 1999); a study of the effect of different flow velocities; and a study of the effect of the Prandtl number on the growth of a group of dendrites growing together. The simulations show that on the upstream side the dendrite tip velocity is increased due to the increase in the temperature gradient and the formation of side branches is promoted. The flow has the opposite effect on the downstream side. The results are in good qualitative agreement with published experimental results,even though only the two-dimensional aspects are examined here.
Three-dimensional phase-field simulations of the effect of convection on free dendritic growth
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.
Phase-field simulations of dendritic crystal growth in a forced flow
2001
Convective effects on free dendritic crystal growth into a supercooled melt in two dimensions are investigated using the phase-field method. The phase-field model incorporates both melt convection and thermal noise. A multigrid method is used to solve the conservation equations for flow. To fully resolve the diffuse interface region and the interactions of dendritic growth with flow, both the phase-field and flow equations are solved on a highly refined grid where up to 2.1 million control volumes are employed. A multiple time-step algorithm is developed that uses a large time step for the flow-field calculations while reserving a fine time step for the phase-field evolution. The operating state ͑velocity and shape͒ of a dendrite tip in a uniform axial flow is found to be in quantitative agreement with the prediction of the Oseen-Ivantsov transport theory if a tip radius based on a parabolic fit is used. Furthermore, using this parabolic tip radius, the ratio of the selection parameters without and with flow is shown to be close to unity, which is in agreement with linearized solvability theory for the ranges of the parameters considered. Dendritic sidebranching in a forced flow is also quantitatively studied. Compared to a dendrite growing at the same supercooling in a diffusive environment, convection is found to increase the amplitude and frequency of the sidebranches. The phase-field results for the scaled sidebranch amplitude and wavelength variations with distance from the tip are compared to linear Wentzel-Kramers-Brillouin theory. It is also shown that the asymmetric sidebranch growth on the upstream and downstream sides of a dendrite arm growing at an angle with respect to the flow can be explained by the differences in the mean shapes of the two sides of the arm.
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.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2018
Motivated by important applications in materials science and geophysics, we consider the steady-state growth of anisotropic needle-like dendrites in undercooled binary mixtures with a forced convective flow. We analyse the stable mode of dendritic evolution in the case of small anisotropies of growth kinetics and surface energy for arbitrary Péclet numbers and n -fold symmetry of dendritic crystals. On the basis of solvability and stability theories, we formulate a selection criterion giving a stable combination between dendrite tip diameter and tip velocity. A set of nonlinear equations consisting of the solvability criterion and undercooling balance is solved analytically for the tip velocity V and tip diameter ρ of dendrites with n -fold symmetry in the absence of convective flow. The case of convective heat and mass transfer mechanisms in a binary mixture occurring as a result of intensive flows in the liquid phase is detailed. A selection criterion that describes such solidific...
Numerical Heat Transfer Part B-fundamentals, 2002
We present and validate a numerical technique for computing dendritic growth of crystals from pure melts. The solidification process is computed in the diffusion-driven limit. The governing 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. The results from our calculations are compared with two-dimensional microscopic solvability theory. It is shown that the method predicts dendrite tip characteristics in good agreement with the theory. The sharp interface treatment allows discontinuous material property variation at the solid±liquid interface. Calculations with such discontinuities are also shown to produce results in agreement with solvability and with other sharp interface simulations.
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.
A volume of fluid approach for crystal growth simulation
Journal of Computational Physics, 2010
A new approach to simulating the dendritic growth of pure metals, based on a recent volume of fluid (VOF) method with PLIC (piecewise linear interface calculation) reconstruction of the interface, is presented. The energy equation is solved using a diffuse-interface method, which avoids the need to apply the thermal boundary conditions directly at the solid front. The thermal gradients at both sides of the interface, which are needed to obtain the front velocity, are calculated with the aid of a distance function to the reconstructed interface. The advection equation of a discretized solid fraction function is solved using the unsplit VOF advection method proposed by López et al. [J. Comput. Phys. 195 (2004) 718-742] (extended to three dimensions by Hernández et al. [Int. J. Numer. Methods Fluids 58 (2008) 897-921]), and the interface curvature is computed using an improved height function technique, which provides second-order accuracy. The proposed methodology is assessed by comparing the numerical results with analytical solutions and with results obtained by different authors for the formation of complex dendritic structures in two and three dimensions.