Shiyan Pan

Lattice Boltzmann modeling of dendritic growth in forced and natural convection

A two-dimensional (2D) coupled model is developed for the simulation of dendritic growth during alloy solidification in the presence of forced and natural convection. Instead of conventional continuum-based Navier-Stokes (NS) solvers, the present model adopts a kinetic-based lattice Boltzmann method (LBM), which describes flow dynamics by the evolution of distribution functions of moving pseudo-particles, for the numerical computations of flow dynamics as well as thermal and solutal transport. The dendritic growth is modeled using a solutal equilibrium approach previously proposed by Zhu and Stefanescu (ZS), in which the evolution of the solid/liquid interface is driven by the difference between the local equilibrium composition and the local actual liquid composition. The local equilibrium composition is calculated from the local temperature and curvature. The local temperature and actual liquid composition, controlled by both diffusion and convection, are obtained by solving the LB equations using the lattice Bhatnagar-Gross-Krook (LBGK) scheme. Detailed model validation is performed by comparing the simulations with analytical predictions, which demonstrates the quantitative capability of the proposed model. Furthermore, the convective dendritic growth features predicted by the present model are compared with those obtained from the Zhu-Stefanescu and Navier-Stokes (ZS-NS) model, in which the fluid flow is calculated using an NS solver. It is found that the evolution of the solid fraction of dendritic growth calculated by both models coincides well. However, the present model has the significant advantages of numerical stability and computational efficiency for the simulation of dendritic growth with melt convection.

Modelling of dendritic growth during alloy solidification under natural convection

A two-dimensional (2D) lattice Boltzmann method (LBM)-cellular automaton model is presented to investigate the dendritic growth of binary alloys in the presence of natural convection. The kinetic-based LBM is adopted to calculate the transport phenomena by the evolution of distribution functions of moving pseudo-particles. To numerically solve natural convection thermal and solute transport simultaneously, three sets of distribution functions are employed in conjunction with the lattice Bhatnagar–Gross–Krook scheme. Based on the LBM calculated local temperature and concentration at the solid/liquid interface, the kinetics of dendritic growth is determined according to a local solute equilibrium approach. Thus, the physics of a complete time-dependent interaction of natural convection, thermal and solutal transport, and dendritic growth during alloy solidification is embedded in the model. Model validation is performed by comparing the simulated results with literature data and analytical predictions. The model is applied to simulate dendritic growth in binary alloys under the influence of natural convection. The effects of Rayleigh numbers and initial undercooling on dendrite growth are investigated. The results show that natural buoyancy flow, induced by thermal and solutal gradients under gravity, transports the heat and solute from the lower region to the upper region. The dendritic growth is thus accelerated in the downward direction, whereas it is inhibited in the upward direction, yielding asymmetrical dendrite patterns. Increasing the Rayleigh number and undercooling will enhance and reduce, respectively, the influence of natural flow on the dendritic growth.

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.

Modeling of Microstructure Evolution during Alloy Solidification

In recent years, considerable advances have been achieved in the numerical modeling of microstructure evolution during solidification. This paper presents the models based on the cellular automaton (CA) technique and lattice Boltzmann method (LBM), which can reproduce a wide variety of solidification microstructure features observed experimentally with an acceptable computational efficiency. The capabilities of the models are addressed by presenting representative examples encompassing a broad variety of issues, such as the evolution of dendritic structure and microsegregation in two and three dimensions, dendritic growth in the presence of convection, divorced eutectic solidification of spheroidal graphite irons, and gas porosity formation. The simulations offer insights into the underlying physics of microstructure formation during alloy solidification.

Modeling of Ferrite‐Austenite Phase Transformation

A two-dimensional cellular automaton model is adopted to simulate the ferrite (α)-austenite (γ) transformation in low-carbon steels. The preferential nucleation sites of austenite, the driving force of phase transformation, carbon redistribution at α/γ interfaces, and carbon diffusion in both the α and γ phases are considered. The model is applied to simulate the phase transformation and carbon diffusion during heating at 815°C, and subsequent cooling at 5°C/s to room temperature and tempering at 300°C-500°C. The process of heating at 600°C after cooling from 815°C, but prior to cooling to room temperature, is also simulated to compare to the tempering process. The results show that during the isothermal heating at 815°C, the carbon distribution becomes uniform gradually in both the α and γ phases. The subsequent cooling to room temperature at 5°C/s results in a non-uniform carbon distribution, while the uniformity increases with tempering temperature. During the 600°C heating, the carbon distribution is uniform within 1 min. The simulation results are used to understand the processing-microstructure-property relationships of an enameling steel.

Phase-field modeling of liquid droplet migration in a temperature gradient

The migration of liquid droplets in a solid phase caused by temperature gradient zone melting (TGZM) is simulated by employing a quantitative phase-field (PF) model proposed by Echebarria et al. The PF simulation results are compared with the predictions of an analytical model that describes the droplet migration for both static and dynamic conditions, allowing the direct solution of the time dependent migration velocity of a liquid droplet that is initially located at an arbitrary position in the mushy zone. For dynamic conditions as e.g. during directional solidification, criteria for the critical pulling velocity and critical droplet position are suggested and validated by the PF simulations. When the pulling velocity is lower than the critical pulling velocity, the droplet will migrate through the moving liquidus into the bulk liquid. The droplet velocity gradually increases as it is approaching liquidus. On the other hand, when a pulling velocity higher than the critical pulling velocity is imposed, the droplet will travel through the moving solidus into the fully solid region while the droplet velocity decreases with time. The droplets initially located above the critical position migrate toward liquidus, while the others sink into the bulk solid. The effect of the temperature gradient on the droplet migration kinetics is investigated by both PF simulations and analytical predictions. The results confirm that the upward droplet migration velocity increases, while the time needed for a liquid droplet to move through the entire mushy zone decreases with increasing temperature gradient. The PF simulation results compare well with the analytical predictions.

Cellular automaton simulation of ferrite-austenite transformation in low-carbon steels

A two-dimensional (2-D) cellular automaton (CA) model is adopted to simulate ferrite (α)-austenite (γ) transformation in low-carbon steels. In the model, the preferential nucleation sites of austenite, the driving force of phase transformation coupled with thermodynamic parameters, solute partition at the α/γ interface, and carbon diffusion in both the α and γ phases are taken into consideration. The model is able to simulate the ferrite-to- austenite transformation during isothermal heating in the ferrite-austenite two-phase region, and the austenite-to-ferrite transformation during continuous cooling. The influences of cooling rate and α grain size on the final carbon concentration field are investigated. The results show that after isothermal heating at 815°C for 300 s, the carbon concentration in both the α and γ phases reach the respective equilibrium values. The simulated microstructures compare well with the SEM images. After cooling to room temperature, the carbon distribution is more uniform when cooled at 1.2°C/s than at 7°C/s. The carbon distributions for different α grain sizes cooled at 1.2°C/s are similar. The simulation results are used to understand the mechanisms of the experimental phenomena of an enameling steel.