*4.1. Single Dendrite Movement*

As shown in Figure 6, the dendrite arm at the front end of the dendrite grows faster in the moving state, and a secondary dendrite arm is generated. The upper dendrite arm has a lower speed and almost stops growing. The reason is that the dendrite movement compresses the lower solute boundary layer, the concentration gradient becomes larger, and the temperature gradient becomes lower; the upper solute boundary layer is stretched, the concentration gradient becomes smaller, and the temperature gradient increases. Therefore, the growth of the dendrite arms at the upstream side will be further promoted. As the falling speed increases, this asymmetric growth phenomenon becomes more and more obvious. In actual solidification, when the dendrite grows asymmetrically, the dendrite will rotate. From Figure 6c2, it can be seen that compared with pure falling dendrites (Figure 6b2), the former dendrite arms have prominent tips, asymmetric growth is more obvious, and the latter dendrites are more uniform, which is consistent with the literature [6].

The solute distribution is also closely related to the flow of the solution around the dendrites. It can be seen from Figure 7 that the flow direction of the solution is different in the three movement states. The solution near the stationary dendrite flows upward, and the solution far away from the dendrite flows downward. For a moving dendrite, due to the high viscosity of the solution, a downward pulling force will be generated on the surrounding solution during the drop of the dendrite. This causes

the solution near the dendrite to flow downwards and the solution away from the dendrites to flow upwards. In addition, for moving dendrites, two vortices will be generated behind it.

**Figure 6.** Effects of different motion states on dendrite growth. (**a1**), (**b1**), (**c1**) the growth morphology diagram of dendrite under the condition of stationary, translational falling without rotation, and rotational falling; (**a2**), (**b2**), (**c2**) the solute distribution diagram of dendrite under the condition of stationary, translational falling without rotation, and rotational falling.

**Figure 7.** Flow field distribution in the process of dendrite falling; (**a**) stationary dendrites; (**b**) falling dendrites; (**c**) falling and rotating dendrites.

It can be seen from Figure 8 that the solution flows faster around the moving dendrites. When the solidification time is 0.5 s, the flow velocity of the solution around the dendrite in the stationary, pure falling, and rotating falling states is 0.008 mm/s, −0.01 mm/s, −0.009 mm/s. This is because the rotation of the dendrite makes the contact surface larger between the lower end and the solution, resulting in an increase in the resistance of dendrites, which reduces the falling speed of the rotating dendrite to a certain extent.

**Figure 8.** Flow velocity in the process of dendrite falling. (**a**) Stationary dendrites; (**b**) falling dendrites; (**c**) falling and rotating dendrites.

#### *4.2. Multi-Dendrite Movement*

As the dendrites grow, the solute boundary layers around the dendrites come into contact and fuse with each other to form a high-concentration solute domain. The growth of dendrite arms in this domain is inhibited, and the growth of dendrite arms away from this domain is promoted. The asymmetry of the dendrite is aggravated, and the rotation speed of the dendrite is increased.

The rotation of dendrites is different from that of single dendrites during solidification. See Table 2 for the change of preferential dendrite growth angle. The initial preferred growth angle of No. 4 dendrite is 0.1 rad. The solutes between the dendrites are close to each other and form a high concentration solute domain, therefore the growth of the right dendrite arm of No. 4 dendrite is inhibited, and the growth of the left dendrite arm is promoted. This causes the mass on the left side of the dendrite to be greater than the mass on the right side, so the dendrite is subjected to a counterclockwise torque. When the solidification time is 0.4 s, the dendrite rotates 0.3 rad in a counterclockwise direction. For the No. 1 dendrite, the lower part of the simulated domain is less affected by the solute field. It starts to rotate at 0.2 s and falls to the bottom at 0.5 s. At this time, the preferred growth direction of the dendrite is −0.5 rad. For the No. 9 dendrite, the dendrite arm at the lower end of the dendrite extends to the high-concentration domain, and the growth is inhibited. Therefore, the dendrite is more symmetrical on the left and right, and the torque is less. During its movement, the dendrite only rotated 0.2 rad. The No. 6 dendrite is located in the center of the simulation area. The uncertainty of its growth behavior leads to the complexity of its movement behavior. It can be seen that the dendrite rotates 0.3 rad clockwise when the solidification time is 0.4 s. At 0.5 s, the dendrite rotates counterclockwise by 0.1 rad, and the dendrite rotates left and right.


**Table 2.** Preferred growth angle of dendrites at different times.

Multi-dendritic effects are also manifested in interactions between fluids. As shown in Figure 9a, as the dendrites grow, the fluid vortices begin to merge, forming a strong convection between the dendrites. This strong convection hinders the vertical drop of the dendrite and has a lateral force on the dendrite. The dendrite starts to move laterally, and the grains on both sides have a tendency of centrifugal movement. And the movement of the grains in the central domain shows a trend of swinging left and right.

**Figure 9.** The evolution and concentration distribution of multi-falling-dendrites. (**a**) t1=200; (**b**) t2=500; (**c**) t3=800; (**d**) t4=1200.

Figure 10 shows the falling speed of the four dendrites. Each dendrite undergoes a process of acceleration and deceleration, and their absolute speeds are lower than those of the single dendrites. It is shown that during the growth of multi-dendritic, due to the interaction between the dendrites, a part of the gravity is offset and the dendrite's moving speed is reduced to a certain extent.

**Figure 10.** The falling velocity of dendrite 1, dendrite 4, dendrite 6, dendrite 9.

Compared with the growth of multiple dendrites in the moving state, the growth mode of the dendrites in the stationary state is much simpler. The growth of the dendrites in the middle of the simulated domain is suppressed, and the growth of dendrites near the boundary is promoted. (As shown in Figure 11)

**Figure 11.** The evolution and concentration distribution of dendrites without motion. (**a**) t1 = 200; (**b**) t2 = 500; (**c**) t3 = 800; (**d**) t4 = 1000.

From the above analysis, the movement behavior of multi-dendrites is related to the melt convection and solute overlap between the dendrites. In response to this phenomenon, scholars proposed to apply an external force field (such as an electromagnetic field) to the dendrite to offset gravity, thereby changing the movement state of the dendrite, and then controlling the solute segregation of the casting [20,21].

#### **5. Conclusions**

In this work, a CA-LBM-Ladd coupling model for calculating multi-dendritic motion was established, and an extrapolated distribution method for calculating the solute distribution around the dendrite under the state of dendritic motion in the melt was proposed, which realizes the calculation of the solute field in the real state of dendrite movement. The CA-LBM-Ladd coupling model was verified, and then the motion of single and multiple dendrites was simulated using this model. The simulation results show that: 1) the superposition of the flow fields between the multiple dendrites causes the movement state to change; 2) the superposition of solute field results in the change of concentration gradient, changes the growth mode of dendrite, and then changes the movement state. This is quite different from the growth mode of single dendrite.

**Author Contributions:** Conceptualization, Y.B.; data curation, Y.B.; formal analysis, Y.B.; investigation, Y.B.; methodology, S.Z.; software, Y.W.; validation, Y.B.; visualization, Q.W. project administration, R.L.; Funding acquisition, R.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Natural Science Foundation of China, (No. 51475138 and 51975182)

**Acknowledgments:** This work was supported by a grant from the National Natural Science Foundation of China (No. 51475138). Y.B. would like to thank Ri Li (Hebei University of Technology) for providing academic guidance.

**Conflicts of Interest:** The authors declare no conflict of interest.
