Fig. 1. Schematic phase diagram for a typical dilute binary alloy. Blue line: liquidus, red line: solidus. l1 = Tm + mc∞ - T, l2 = Tl-s. cl, cs: equilibrium concentrations in liquid and solid at temperature T.In Eq. (1)

Fig. 2. Evolution of F due to dendritic growth in a time step dt. Dot line: advancing F by equation without splitting (Eq. (22)); solid line: advancing F by “split equation 1” (Eq. (23)) and “split equation 2” (Eq. (24)); F*: intermediate solution.

Fig. 3. Viscosity variation through s-l interface in linear and exponential way. It is shown in logarithmic coordinate and μs = 1010μl.

Fig. 4. Two-level adaptive time stepping scheme. The step size used in each operation is indicated beside them. “Y”: yes, “N”: no.

Fig. 5. Comparison between flow fields computed by our FSI model and Fluent: (a) The configuration of the curved channel (unit: mm); (b, d) phase field and inflow velocity profile; (c, e) flow field at t = 1 s. Dashed lines in (c) indicate the s-l interface.

Fig. 6. Velocity distribution along line X1 = 0.25 mm in Fig. 5: (a) velocity component in X1 direction; (b) velocity component in X2 direction.

Fig. 7. Evolution of s-l interface of a growing circular grain: (a) moving grain driven by uniformly imposed flow field v = (1, 1) W0/τ0 (when t < 128 τ0) and -v (when t > 128 τ0); (b) motionless grain; (c) comparison between the final interfaces in two cases. The domain was l × l with l = 256 W0 and the initial radius was 25 W0 (blue interface: moving grain; red interface: motionless grain). The crosses indicate where the grain should have been centered. O1 : (0.25, 0.25) l, O2 : (0.75, 0.75) l.

Fig. 8. (a) Snapshots of s-l interface of a rotational dendrite (red: t = 300 τ0, black: t = 1800 τ0, green: t = 3300 τ0); (b) comparison with motionless dendrite at t = 3000 τ0 (red: with rotation, black: without rotation); (c, d) dimensionless temperature field, phase and flow field of rotational dendrite at t = 3000 τ0. The lines in (a) indicate the orientation this dendrite should have at each moment. In (b), the rotational dendrite has been rotated back to α = 0 for comparison.

Fig. 9. (a) Snapshots of s-l interface of the settling dendrite; (b) drop and growth velocity of the settling dendrite over time.

Fig. 10. The growth and settlement of a single dendrite (t = 1.53 s): (a) velocity distribution and s-l interface; (b) solute distribution; (c) concentration isolines after zooming in (from 0.9 to 5.7 wt.% at an interval of 0.4 wt.%).

Fig. 11. (a) Initial positions and orientations of solid seeds: (1) (0.14, 0.2) mm, 0.19π, (2) (0.3, 0.94) mm, 0.08π, (3) (0.48, 1.8) mm, 0.04π, (4) (0.62, 1.04) mm, -0.23π, (5) (0.84, 0.24) mm, -0.2π; (b, c) Velocity (with s-l interfaces) and solute distribution of settling dendrites at t = 1.46 s.

Fig. 12. Phase field (a), concentration (b), orientation (c) field and mesh partition (d) at t = 3.17 s. In (d), each color identifies the subdomain owned by the corresponding processor, with 380 thousand nodes in all.

Fig. 13. (a) A close-up image of adaptive mesh after zooming to the box in Fig. 12(a); (b) vector-valued phase field F in the box in (a); (c) relationship between F and (?, α).

Fig. 14. Makeup of run time when simulation was ended at t = 3.17 s. Overall run time is 115.75 h.