Lamellar eutectic growth under forced convection: A phase-field lattice-Boltzmann study based on a modified Jackson-Hunt theory

Effect of forced convection on the lamellar eutectic growth was investigated by combining a modified Jackson-Hunt theory and a phase-field lattice-Boltzmann approach. Under the consideration of the asymmetrical tilting pattern and curvature effect, the classical eutectic Jackson-Hunt theory [Jackson and Hunt, Trans. Metall. Soc. AIME 236, 1129 (1966)] was modified to better understand the physical mechanism driving the eutectic growth with convection. Results showed that the eutectic growth velocity increased linearly with increasing undercooling and exhibited a parabola trend versus the inverse of initial lamellar spacing. The flow induced by a horizontal external force tilted the eutectic lamellae by altering the solute distribution near the interface. Under weak convection, the phase-field lattice-Boltzmann simulation results agreed well with those predicted by the modified Jackson-Hunt theory. But under strong convection, the consistency of the two results was largely dependent on the alloy parameters and convection intensity.

Figure 1

Typical three-level hierarchical data structure superimposed on the cloud picture of the solute field.

Figure 2

Schematic diagram of lamellar eutectic growth under forced convection.

Figure 3

(a) Simulated cloud picture of the solute field. The arrows denote the velocity vector. (b) Simulated velocity isolines superimposed on the solute concentration isolines. The cloud picture of the flow field is inserted at the upper-right corner. (c) Simulated fluid velocity versus the distances along different directions, i.e., the directions I–VII as designated by the arrows in the inserted cloud picture of the flow field superimposed on the solute field.

Figure 4

Evolution of solute field under (a) forced convection and (b) nonconvection conditions.

Figure 5

Distribution of solute concentration along the given directions, i.e., $H_1$ and $H_2$ for the horizontal direction, and $V_1$ and $V_2$ for the longitudinal direction, respectively. The insets are the grayscale contour maps of the solute field with and without convection, in which the dark color favors the growth of the $\alpha$ phase.

Figure 6

(a)–(e) Simulated solute field with different computational domain sizes. (f) Simulated fluid velocity vs the distance away from the $S\text{−}L$ interface. The insets are the corresponding flow fields.

Figure 7

(a),(b) Schematic diagrams of the eutectic growth and lamellar tilt. (c) Illustration for the effect of the forced convection on concentration isolines.

Figure 8

(a) Illustration for the effect of the forced convection on interface morphology. (b) Growth velocity including $V$ and $V_d$ and the maximum fluid velocity vs the convection intensity.

Figure 9

Evolution of the eutectic lamellae under forced convection. The boundary conditions are zero-Neumann boundary condition for the solute and phase fields, and no-slip boundary condition for the flow field, respectively.

Figure 10

Eutectic lamellar growth under different undercoolings. (a) Typical solute fields under four different undercoolings (i.e., 0.8, 1.2, 1.4, and 1.8 K, respectively). (b) Interfacial lamellar width and lamellar tilting angle vs the undercooling.

Figure 11

Comparisons between the simulation results using the phase-field lattice-Boltzmann model (PFLBM) and those predicted by the modified JH theory (mJHt) under (a) different undercoolings and (b) different initial lamellar spacings. The numbers in the legend denote the magnification of the convection intensity.

Figure 12

Eutectic lamellar growth under different initial lamellar spacings. (a) Typical solute fields under four different initial lamellar spacings (i.e., 6.4, 12.8, 19.2, and 25.6 μm, respectively). (b) Interfacial lamellar width and lamellar tilting angle vs the initial lamellar spacing.

Figure 13

Comparisons between the simulation results using the phase-field lattice-Boltzmann model (PFLBM) and those predicted by the modified JH theory (mJHt) under (a) different undercoolings and (b) different initial lamellar spacings for the $\mathrm{CB}{\mathrm{r}}_4\text{−}{\mathrm{C}}_2\mathrm{C}{\mathrm{l}}_6$ alloy. The numbers in the legend denote the magnification of the convection intensity.

Figure 14

Schematic diagram of contact angles at the triple point of tilted eutectic lamellae.

Figure 15

Distribution of the solute concentration along the $y$ axis (see $V_1$ and $V_2$) at the initial time and the steady state for the ideal alloy (the first row) and Al-Cu alloy (the second row). Panels (b) and (d) are the local enlarged images of the solute concentration near the interface for the ideal alloy and Al-Cu alloy, respectively.

Figure 16

Liquid solute concentration vs the distance along the direction perpendicular to the solid-liquid interface for the Al-Cu alloy. (b) Local enlarged image of the solute concentration near the interface. The domain height is eight times larger than the domain width.