Phase-Field Simulations of Columnar
Dendritic Structure in Forced Flow
T. Takaki*, H. Kashima, T. Fukui, K. Morinishi
Mechanical and System Engineering,
Kyoto Institute of Technology, JAPAN* takaki@kit.ac.jp http://www.cis.kit.ac.jp/~takaki/

INTRODUCTIONSIMULATIONS AND RESULTS

The typical microstructure of ingot consists of the outer chill zone, the intermediate
columnar zone and the central equiaxed zone. Although the microstructure is thought tobe largely affected by the melt flow, the detail studies have not been reported so far.
In this study, we investigate the effect of the forced flow on the columnar denditic
structure which grow from the multi nuclei on the mold wall with arbitrary crystalorientation. Here, we employ the coupling model of phase-field method and lattice
Boltzmann method.

We perform the dendrite solidification simulations of Al-10%Si supersaturated
alloy in forced flow. Some nuclei with arbitrary crystal orientation are putted on thebottom of computational domain in which the constant fluid flow in x-direction andthe constant concentration are set as initial condition. The effects of forced flow onthe columnar dendritic structure that grows from the mold wall are investigated.
ux0 = const. (top)

MODEL
Coupling simulations using phase-field equation, concentration equation and lattice
Boltzmann equation are performed to investigate the solidification microstructure ofbinary alloy in forced flow.
The interface migration is
Phase-field
simulated by the phase-field equaiton
Equationwhere the thermodynamic driving
force is calculated from the local
concentration. The concentration
φ c
φdistribution is calculated by the ⎧⎨u x = (1 − φ )u x
diffusion equation of solute with ⎩u y = (1 − φ )u y
advective term by melt flow. The
velocities of melt flow is simulatedConcentration
LBby the lattice Boltzmann scheme. TheEquationux, uyEquation
coupling of above three equations are
Advection of mass by fluid flow
schematically shown in the rightfigure.

Phase-Field Equation⎡∂φ∂ ⎛ ∂a ∂φ ⎞ ∂ ⎛ ∂a ∂φ ⎞⎟ + ⎜a= M φ ⎢∇(a 2∇φ ) − ⎜⎜ a⎟∂t∂
x ⎝ ∂θ ∂y ⎟⎠ ∂y ⎝ ∂θ ∂x ⎠⎣a(θ ) = a [1 + ς cos{k (θ − θ 0 )}]Concentration Equation⎡(1 − k )c ∇φ ⎤∂c ⎞∂c ⎛ ∂c+ u y ⎟⎟ = ∇ ⋅ D ⎢∇c ++ ⎜⎜ u x⎥
1 − φ + kφ∂y ⎠∂t ⎝ ∂x⎣
⎦Pressure distribution functionpα (x + eα Δt , t + Δt ) = pα (x, t ) −PressureVelocity
88p = ∑ pα ,u = ∑ pα eα

α =0α =0p0⎧c = φcS + (1 − φ )cL⎨⎩k = c S c Lc : concentrationcL, cS : concentration in liquid and solid
k : partition coefficient
DS : diffusion coefficient in solid
DL : diffusion coefficient in liquidux, uy : fluid velocities1τ{p (x, t ) − p (x, t )}
eq

α

α1⎛2⎝

τ = ⎜⎜ 6ν⎞+ 1⎟⎟⎠Δtτ : collision relaxation timewα : weight coefficient= 4/9 (α = 0)
= 1/9 (α = 1-4)
= 1/36 (α = 5-8)
2 ⎫ ν : kinetic viscosity⎤(Δx )2Equilibrium distribution function

Nine sets of simulations by changing fluid velocity and the number of nuclei areperformed. The following figures are the solidification morphologies (solid blackline), concentration (color contour) and fluid velocities (vector) at 130,000 step. It
is observed that the flow velocity changes the dendritic morphology dramaticallyespecially for high velocity and the number of nuclei changes the solidification
morphology at chill zoen and dendrite shape.
ux0 = 0.01 m/s
ux0 = 0.1 m/s
ux0 = 0 m/s

concentration

1-nucleus

0.00

mol frac.

2ux0
fluid velocity

Anisotropy equation for interface energy

Lattice Boltzmann Equation

Effects of the flow velocity and number of nuclei

5-nuclei=1 in solid, = 0 in liquid
T : temperature in domain
Tm : temperature on liquidus line
χ : noiseδ : interface thickness
γ : interface energy
M : interface mobilityζ : strength of anisotropy
k : anisotropy modeq : angle of interface normal
θ0 : crystal orientation

3δγ
6γb2W
, W=, Mφ =Mδ
b6a1−φD = DS + (DL − DS )1 − φ + kφ2000 x 2000 FD lattices (Δx = 50 nm)

020-nuclei

φ : phase-field variable

Phase-field parameters and material constants

Diffusion coefficient

Nuclei (φ = 1) with various orientation

0.13

⎛
⎞⎤
1 15 L(T − Tm )− 4Wφ (1 − φ )⎜⎜ φ − −φ (1 − φ ) + χ ⎟⎟⎥Tm2 2W
⎝⎠⎦
a=ux0 = 0 (bottom)

Initial conditions•Constant fluid flow ux = 0, 0.01, 0.1 m/s and uy = 0 m/s•Constant concentration c = 0.1 mol frac.
•Constant temperature T = 855 K•Liquid condition φ = 0
Boundary conditions•Periodic in x-direction for all variables•Zero Nouman on top and bottom for φ and c

Growth processes with and without fluid flow
Without fluid flow, the dendrites grow almost straightly to the preferred growthdirection. On the other hand, with fluid velocity, the dendrite changes the growth
direction depending on the distance to the nearest-neighbor dendrites.
Without fluid flow⎧⎪⎡ 3(e ⋅ u ) 9(e ⋅ u )2 3u ⎪pαeq (x, t ) = wα ⎨ p + p0 ⎢ α2 + α 4 − 2 ⎥ ⎬2c2c ⎦ ⎪⎭⎪⎩⎣ cD2Q9 model
2635e6e2e3

Fictive fluid particle

0e77e5e0e1 1e4e8

e 0 = {0 0}TParticle velocities

Te1 = {c 0}e 5 = {c c}e 2 = {0 c}Δx8Δxc=50,000 step

100,000 step

Te 6 = {− c c}

T4ΔxΔxTe 3 = {− c 0} e 7 = {− c − c}TTΔxΔt

e 4 = {0 − c} e8 = {c − c}TTWith fluid flow : ux0 = 0.1 m/s

150,000 step

200,000 step