1. A numerical study comparing effects of forced and
buoyancy convection on columnar dendrite growth
using lattice Boltzmann modeling
Sergio Felicelli1
, Mohsen Eshraghi2
, Amirreza Hashemi1
, Mohammad
Hashemi1
1
Department of Mechanical Engineering, The University of Akron
2
Department of Mechanical Engineering, California State University
Los Angeles
2. Dendritic growth is the primary form of crystal growth observed
in most metallic alloys.
The material properties strongly depend on the shape, size,
orientation and composition of the dendritic matrix formed
during solidification.
Forced and natural convection have a significant influence on
the shape, orientation and size of dendrite growth.
The channel-like macrosegregation defects, also known as
freckles, are often observed during directional solidification of
metallic alloys.
the buoyancy-induced mechanism of micro-segregation defects
leading to formation of freckles in solidification of binary alloys.
The channel-like defects that form during solidification have a
significant influence on mechanical properties of the cast
products
Motivation
3. Modeling Microstructural Evolution
Interface tracking methods, e.g. Cellular Automaton
Interface non-tracking methods. e.g. Phase Field
Cellular Automaton
MESH Defining neighborhood
STATE Solid (S) / Liquid (L) / Interface (I)
RULES Transition rules from one state to another
Von Neumann Moore
4. Lattice Boltzmann Method (LBM)
A Computational Fluid Dynamics (CFD) technique for solving fluid
and thermal problems
LBM relies on the solution of a minimal form of Boltzmann kinetic
equation for a group of fictive particle in a discretized domain
The fictive particles stream across the lattice along the links
connecting neighboring lattice sites, and then undergo collisions
upon arrival at a lattice site
Simple implementation
Capability for simulating highly complex geometries and boundaries
Computational efficiency
Local calculations and inherent parallel-processing structure.
Advantages:
5. LBM Basics
0 1
2
3
4
56
7 8
D2Q9
e1
e2
e3
e4
e5e6
e7
e8
f1
f2
f3
f4
f5f6
f7 f8
f0
Histogram view of the
distribution function, f.
f1
f5
1 2 5 a
f
f2
f3
f4
3 4 6 7 8
f6 f7 f8
D3Q15
6. Single relaxation time BGK (Bhatnagar-
Gross-Krook) approximation
( ) ( ) ( ) ( )[ ]tftftftttef eq
iiiii ,,,, xxxx −−=∆+∆+ ω
Streaming Collision (i.e., relaxation towards local equilibrium)
StreamingCollision
�� =
4/9 � = 0
1/9 � = 1 − 4
1/36 � = 5 − 8
( ) ( )
−
•
+
•
+= 2
2
4
2
2
2
3
2
9
31)(
ccc
wf ii
i
eq
i
uueue
xx ρ
D3Q15
It is proved that the following equation can reproduce the continuum equation
using the Chapman-Enskog expansion.
Discrete Velocities: ei
Directional Densities: fi
Macroscopic Density:
D2Q9
7. Challenges
3D simulations:3D simulations:
Not many 3D studies
Growth kinetics is different in 2D and 3D simulations
Mass conservation can not be satisfied in the situations with the
effect of natural convection for 2D simulation
Previous works have been mostly focused on a single
dendrite and pure metals
3D simulation are computationally expensive
2D 3D
Yuan & Lee, 2010
Nestler & Choudhury, 2011
8. Lattice Boltzman + Cellular Automaton
Same calculation scale
Local structure Good for parallel processing
Promising for simulating large physical domains
9. Modeling Dendrite Growth using LBM
Medvedev et al. (2005) and Miller et al. (2006): LB in combination
with the phase-field; alloy solidification with fluid flow (2D)
Sun et al. (2009): LB-CA for 2D dendrite growth; solute transport +
fluid convection
Yin et al. (2010): LB-CA for 2D dendrite growth; solute transport +
fluid convection + heat transfer
Present Authors (2012): LB-CA model for 3D solutal dendrite growth
No 3D LB model for dendrite growth under convection
10. Validating the Diffusion Model
D = 3.0×10-9
(m2
s-1
)
μ = 0.0024 (N s m-2
)
U0 = 10 mm/s
Ci = 4 wt%
t = 2 ms
11. Solidification Model
The increase solid fraction:
ls CkC ⋅=Solid/Liquid interface :
l
eq
l
0l
.
m
wmcΓTT
CC
+−
+=
∗
∗
Interface equilibrium composition:
Gibbs-Thomson coefficient
interface equilibrium T
partition coefficient
equilibrium liquidus T at C0
liquidus slope
Weighted Mean Curvature:
Weighted Mean Curvature
anisotropy function
principal curvatures
principal directions
anisotropy parameter
local actual composition
local equilibrium composition
12. Validating the Natural Convection Model
For Heated Cavity
2 2
2 2
( ) ( )
( )
x
u uu uv p u u
Ra
t x y x x y
θ
∂ ∂ ∂ −∂ ∂ ∂
+ + = + + +
∂ ∂ ∂ ∂ ∂ ∂
2 2
2 2
( ) ( )
( )
y
u uv vv p v v
Ra
t x y y x y
θ
∂ ∂ ∂ −∂ ∂ ∂
+ + = + + +
∂ ∂ ∂ ∂ ∂ ∂
2 2
2 2
( ) ( ) 1
RePr
u v
t x y x y
θ θ θ θ θ ∂ ∂ ∂ ∂ ∂
+ + = +
∂ ∂ ∂ ∂ ∂
cold
hot cold
T T
T T
θ
−
=
−
Navier-Stokes equations for x
direction with natural convection
Navier-Stokes equations for y
direction with natural convection
Energy equation in x and y
directions
Velocity in
direction x
Velocity in
direction y
Pressure
3
g TH
Ra
β
αν
∆
=
Geometry and boundary conditions
13. Lattice Boltzmann equations (LBEs) for
heated cavity
( , ) ( , )
( , ) ( , )
eq
i i
i i i i
f x t f x t
f x c t t t f x t tF
τ
−
+ ∆ + ∆ − = − + ∆
LBE for fluid flow with effect of
natural convection
LBE for temperature
( , ) ( , )
( , ) ( , )
eq
i i
i i i
h x t h x t
h x c t t t h x t
τ
−
+ ∆ + ∆ − = −
Force term in LBE 2
3 . /i i iF w c F cρ= −
Buoyancy force 0 ( )T refF g T Tρ β= −
14. Validating the natural convection model
for heated cavity
(a) (b)
(c)
(d)
Steady state temperature profiles obtained by (a) LBM for Ra=10^3 and (c)
LBM for Ra=10^4 respectively. (b) OpenFOAM and Fluent for Ra=10^3
and(d) OpenFOAM and Fluent for Ra=10^4.
15. Ra 10^3 10^4 Velocity ratio
Umax@y(Present) 0.81 0.83 4.667
Umax @y(CIP-
LBM)
0.82 0.82 4.488
Vmax@x(Present) 0.19 0.13 5.374
Vmax@x(CIP-LBM) 0.19 0.13 5.369
Validating the natural convection model
for heated cavity
(a) Comparison of present simulation results with CIP-LBM results for vertical
velocities at mid width for Ra=10^3. (b) Comparison of present simulation results
with CIP-LBM results for vertical velocities at mid width for Ra=10^4.
(a) (b)
Velocity ratio states the ratio of maximum values at and . CIP[7] (Cubic-interpolated-
pseudo-Particle) is a numerical method for solving advection and it has this capability
to be coupled with LBM.
16. equiaxed dendrite growth
(a) (b)
(c) (d)
Simulated morphologies of an equiaxed dendrite freely growing in an undercooled
melt (∆T = 0.8 K) without convection: (a) and (c), and with natural convection: (b)
and (d). Here, (a) and (b) show the solutal field, and (c) and (d) show the thermal
field. The velocity vector plots indicate the strength and direction of natural flow.
Simulation Parameter Value
Initial concentration 0.4% Solute
Initial temperature 329.51 K
Undercooled melt (∆T) 0.8 K
Rayleigh number 5×10^4
Mesh size 0.3 µm
17. Columnar dendrite Growth
(b)(a)
Natural convection can stop the tip’s growth of some dendrites
The dendrites which are close to the wall grow faster
Formation of Vortexes help or retard the growth of dendrites
Simulation parameter Value
Initial concentration 3%
Temperature 921.2862 k
Rayleigh number 10^4
Mesh size 0.3µm
Material Al-Cu 3%
18. Why 3D Simulation
Simulated morphologies of columnar dendrites freely
growing under the effect of natural convection at
different time steps: (a) 6.67 ms, (b) 27.34 ms, and (c)
37.32 ms.
(b)(a) (c)
2D simulations are not capable of capturing the correct physical
phenomena, since the fluid regimes and growth kinetics are
completely different in 3D
Mass conservation can not be satisfied in 2D models with effect of
natural convection
20. Simulation Setup
Al-3%Cu
ΔT = 4.5°C
ε = 0.04
Δx = 0.3 μm
Boundary Conditions:
Boundary walls are insolated against solute
transport
No diffusion in the solid phase
Bounce back for solid/liquid interface
Uniform flow in x-direction entering from left side
for forced convection case and bounce back for
natural convection case as solid walls
Density, ρ
(kg m-3
)
Diffusion
Coefficient, D
(m2
s-1
)
Viscosity, μ
(N s m-2
)
Liquidus
Slope, m
(°C wt%-1
)
Partition
Coefficient, k
Gibbs-
Thomson
Coefficient, Г
(m °C)
Degree of
anisotropy
, ε
2475.0 3.0×10-9
0.0024 -2.6 0.17 2.4×10-7
0.04
24. Equiaxed dendrite growth with the effect
of natural convection
(c) High or lower solute
concentration can be
observed at upward and
downward tip respectively
(c)
(b)
(a)
(b) it can be observed that
upward and downward
tips grow with slower and
faster speed in
comparison with other
dendrites respectively
25. Columnar dendrite growth with the effect
of natural convection
This contour is related to
middle row of dendrites. We
observe high solute
composition around dendrite
tips
26. (a) (b)
(c) (d)
(a) and (b) shows 3D simulation of columnar dendrite growth with
and without effect of natural convection. With a slice of contour at
the mid-width of cubic domain. (c) and (d) show the counters of
those slice clearly.
It can be observed that
because natural
convection flow which
carry high solute
composition
Columnar dendrite growth with the effect of
natural convection
27. Conclusion
A numerical model combining LB and CA methods model is introduced for
simulating dendritic growth under forced convection.
The fluid flow, solute diffusion and dendrite growth models are validated against
analytical solutions.
Results show that transverse arms are not much affected by the flow.
3D dendrites grow faster than 2D and show more side branching.
Mass conservation can not be satisfied in 2D method
Increasing the degree of undercooling accelerates the growth rate in all directions.
Increasing the magnitude of flow velocity intensifies the convection effects. The
influence on the downstream arm seems more significant.
The growth rate decreases in all branches when the alloy contains a higher solute
concentration.
The size ratio of the upstream arm to the downstream arm grows by increasing inlet
velocity and solute content, and decreasing undercooling.
The size ratio of the upward arm to the downward arm grows by increasing Rayleigh
number .
Considering the special capabilities of LBM, like simplicity, stability, accuracy, local
characteristic, and inherent parallel structure, the proposed model offers a great
potential for simulating large domain solidification problems with good
computational efficiency.
