1. A Lattice Boltzmann Model for
Dendritic Growth Under Natural
Convection
Mohammad Hashemi1
, Mohsen Eshraghi2
, Sergio Felicelli1
1
Department of Mechanical Engineering, The University of Akron
2
Department of Mechanical Engineering, California State University
Los Angeles
TMS 2015
144th
annual meeting and exhibition
March 15-19, 2015, Walt Disney, Orlando, Florida, USA

2.  Natural convection has a significant influence on the
shape, orientation and size of dendrite growth.
 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.
 A comprehensive and coupled model, in fine scale,
necessarily is needed to have clear understanding of
freckling.
Motivation
Heat transfer
Flow fieldMass transfer
Felicelli et al.

3. 3
Freckle formation
b
X-ray Image sequence showing the dendrite growth and the
formation of segregation freckles
Shevchenko et al. 2013
Contours of
Mixture Concentration
of Sn
c
Felicelli et al.
a
Freckle
in ingot
Giamei, 1970

4.  Same calculation scale
 Local structure Good for parallel processing
 Promising for simulating large physical domains
StreamingCollision
Moore
D2Q9
Lattice Boltzman + Cellular Automaton

5. (b)
 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.
 3D simulation are computationally expensive.
Nestler & Choudhury, 2011
Challenges
(a)
Eshraghi, not published

6.  J. C. Ramirez, and C. Beckermann, “Evaluation of a
Rayleigh-number-based freckle criterion for Pb-Sn alloys and
Ni-base superalloys,” Metallurgical and Materials
Transactions A, 34(7) (2003), 1525-1536.
 S. D. Felicelli, J. C. Heinrich, and D. R. Poirier, ”Simulation of
freckles during vertical solidification of binary
alloys,” Metallurgical Transactions B, 22(6) (1991), 847-859.
 S.D. Felicelli, D. R. Poirier, and J. C. Heinrich, “Modeling
freckle formation in three dimensions during solidification of
multicomponent alloys,” Metallurgical and Materials
Transactions B, 29 (4) (1998), 847-855.
 L. Yuan, and P. D. Lee, “A new mechanism for freckle
initiation based on microstructural level simulation,” Acta
Materialia, 60(12) (2012), 4917-4926.
 No 3D LB model for dendrite growth under convection
Literature

7. Validating the natural convection model
for heated cavity
2 2
2 2
( ) ( )
( )
y
u uv vv p v v
Ra
t x y y x y
θ
 ∂ ∂ ∂ −∂ ∂ ∂
+ + = + + + 
∂ ∂ ∂ ∂ ∂ ∂ 
Navier-Stokes equations for y
direction with natural convection
Pressure
2 2
2 2
( ) ( )
( )
x
u uu uv p u u
Ra
t x y x x y
θ
 ∂ ∂ ∂ −∂ ∂ ∂
+ + = + + + 
∂ ∂ ∂ ∂ ∂ ∂ 
cold
hot cold
T T
T T
θ
−
=
−
Navier-Stokes equations for x
direction with natural convection
Velocity in
direction x
Velocity in
direction y
3
g TH
Ra
β
αν
∆
=
2 2
2 2
( ) ( ) 1
RePr
u v
t x y x y
θ θ θ θ θ ∂ ∂ ∂ ∂ ∂
+ + = + 
∂ ∂ ∂ ∂ ∂ 
Energy equation in x and y
directions
Geometry and boundary conditions

8. 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= −

9. Validating the natural convection model
for heated cavity
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.
(c)
(a) (b)
(d)

10. Governing equation
Fluid flow
Solute transport
Heat transfer
2 2
2 2
.( ) .( ) s
p
fT T T L
uT
t x y c t
α
∂∂ ∂ ∂
+ ∇ = + +
∂ ∂ ∂ ∂
r
2
.( ) . .(1 )i s
i i i i
C f
uC D C C k
t t
∂ ∂
+ ∇ = ∇ + −
∂ ∂
r
.( ) 0u∇ =
r
. .[ ( )] convection
u
u u P u F
t
ρ ρ µ
∂
+ ∇ = −∇ + ∇ ∇ +
∂
r
r r r

11. 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

12. Equiaxed dendrite growth
(c) High or lower solute
concentration can be observed
around upward and downward
tips respectively
(c)
(b) it can be observed that
upward and downward tips
grow with slower and faster
speed in comparison with
other tips respectively
(b)
(a)

13. Columnar dendrite growth
 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
(b)(a)
Simulation parameter Value
Initial concentration 3%
Temperature 921.2862 k
Rayleigh number 10^4
Mesh size 0.3µm
Material Al-Cu 3%

14. Micro-Segregated channel
18

15. Freckling
19
(a) (b) (d)(c)
20µm
%

16. Flow field in freckling
Flow field in columnar dendrite growth during formation of micro segregated
channels: (a) contour plot of concentration profile and the stream tracers of
flow (b) the stream tracers of flow and contour plot of velocity component in
Z direction
(a) (b)

17. Conclusions
 In the case of columnar dendrites, channels with high
solute composition form during solidification. A buoyancy-
induced vortex stream may form during solidification.
 Convective flow alter redistribution of solute and energy,
and significantly affect the kinetics of dendrite growth.
 2D simulations are not capable of capturing the correct
physical phenomena, since the fluid regimes and growth
kinetics are completely different in 3D.
 Investigating the temperature profile to see how channels
survive.
 Increasing the efficiency of model for the higher amount of
computation.
 Finding practical approach to avoid freckling during
directional solidification.
Future

18. Thank you