Numerical simulation of dendritic growth of magnesium alloys using phase-field method under forced flow

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1 Vol.0 No.8 The Chinese Journal of Nonferrous Metals Aug ( ( Wheeler hcp m/s % 53.7% TG44 A Numerical simulation of dendritic growth of magnesium alloys using phase-field method under forced flow YUAN Xun-feng, DING Yu-tian, GUO Ting-biao, HU Yong (State Key Laboratory of Gansu Advanced Non-ferrous Metal Materials, Lanzhou University of Technology, Lanzhou , China Abstract: Based on the diffusion Wheeler model which couples the temperature field, concentration field and flow field, the phase-field model for the magnesium alloys with hcp structure was established to simulate both of the single and multi-grain dendritic growths of magnesium alloys during solidification in a forced flow. The results show that when the flow velocity is m/s, the growths of three directions on the upstream side are much faster than those of three directions on the downstream side, the tip velocity at steady state on the upstream is increased by about 78.0% compared with the case without flow, the tip velocity at the steady state on the downstream is decreased by about 53.7% compared with the case without flow. There exists much difference in the length of dendrite arm, so that the six-fold symmetry of the dendrite morphology is severely damaged. Under the case of the multi-grain, the grains grown face to face influence mutually and grow competitively. The dendrite arms of different dendrite are mutually inhibited, ultimately, form asymmetric dendrite morphology. Key words: magnesium alloy; phase-field; forced flow; dendritic growth; numerical simulation [1 3] [3 10] [11 1] Navier-Stokes dingyt@lut.cn

2 Boltzmann [13 14] fcc hcp fcc hcp hcp EIKEN [15] [16] [17] [18] WHEELER [19] hcp Ф Ф=0 Ф=1 / Ф 0~1 Ф 1 Ф WHEELER [19] ε Φ 1 = Φ (1 Φ [ Φ + 30 εαω u Φ (1 Φ ] + m t Φ ε ( η ( θ Φ ε ( η( θ η ( θ + X Y Φ ε ( η( θ η ( θ (1 Y X = i+ j u x y t X Y u=(t T m /(T T 0 t=t /(ω /κ X=X /ω Y=Y /ω Ω=c ( T+m L (c c 0 /L α=( ωl / (1cσT m m=µσt m /(κl ε =δ/ω T T m δ T 0 T X Y ω t κ µ m L c 0 L σ x 0 ( x ( η(θ η(θ=1+γcos(6θ θ X γ θ θ i θ i θ i =K i π/+arctan(ф Y /Ф X (i=1 3 n ( i Ф Y Ф X X Y n K i Navier-Stokes ( ΦV = 0 (3 ( ΦV + ΦV V = Φ p/ ρ+ [ ν ( ΦV] + M t d l (4 ρ p υ V M d l = υ(1 Ф ФhV/ M d l ε (Ф 0 0 (Ф 1 h BECKERMANN [0] u 1 Φ + ΦV u+ p ( Φ = u t Ω t c Φc + ( V = t Φ + k (1 Φ 0 (5 ( k0 1 c D [ c + Φ ] (6 Φ + k (1 Φ 0

3 p (Ф p(ф=ф 3 (10 15Ф+6Ф Ф X +Y R Ф=0 u=0 V X =0 V Y =0 x=x 0 D D s D l X +Y R Ф=1 u= 1 V X =0 V Y =vω/κ x=x 0 k 0 D ΦD + 0 (1 + Φ l k D D = Φ + k (1 + Φ 0 s Neumann 3.1 AZ91D [18] T m =868.5 K L=675 J/cm 3 c p =.353 J/(K cm 3 x 0 =8.179 σ= J/cm υ= cm /s κ=0.361 cm /s D l = cm /s D s = cm /s k 0 =0.4 m L =.1 µ=00 cm/(k s ω= cm α=115 m=0.034 ε =0.005 γ=0.05. (5 (1 (3 (4 (6 ( X t < 4mk (7 m =max(d l m Pr k 1~ Pr=υ/κ X= Y=0.005 t= R v hcp x 0 =8.17% T=100 K v=17.19 m/s (a AZ91D [1] 3(a 3(b (h 3(b (b [] 1 Fig.1 Schematic diagram of computational domain model 3. 4

4 Fig. Evolution for phase field morphologies of single grains without flow ((a, (b, (c, (d and with flow ((e, (f, (g, (h at different times: (a, (e t=5 000 t; (b, (f t= t; (c, (g t=0 000 t; (d, (h t= t 3 AZ91D [1] Fig.3 As-cast microstructures of sand mould AZ91D under polarized light [1] : (a Multi-grain; (b Magnification of single grain shown in Fig.3(a 4 Fig.4 Distribution of solvent for single grains without flow ((a, (b, (c, (d and with flow ((e, (f, (g, (h at different times: (a, (e t=5 000 t; (b, (f t= t; (c, (g t=0 000 t; (d, (h t= t

5 [3] T m L (c c % 53.7% 56.6% 9.% 47.4% 17.6% 5 Fig.5 Relationships among tip velocity(a, tip temperature(b and tip concentration(c and solidification time for case of single-grain dendritic growth without and with flow

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