Application of the Enthalpy Method: From Crystal Growth to Sedimentary Basins
Grain Growth in Metal SolidificationFrom W.J. Boettinger
m
10km
“growth” of sediment delta into oceanGanges-Brahmaputra Delta
Vaughan VollerUniversity of Minnesota
The problem—simulate the growth of a crystal into an undercooled melt contained in an insulated cavity
How does solidificationproceed?
Why do we get a dendritic shape?
Solid-liquid interfacewith time
The Classic Stefan Problem-with curvature dependentPhase change temperature
seed
T0 < 0H = cT + fL
Initial liquid at a temperature below equilibrium solidification temperature T = 0seeded with solid at solidification temperature
H = 0 + fL, 0 < f < 1
Liquid layer adjacent to seed uses latent heat to heat up to T = 0
T = T0 < 0
Negative gradient into liquid removesresidual latent heat and drives solidification
If a solute is present the equilibrium tempand gradient slope will be lower—resultingin a slower advance for the solidification
How does solidification proceed?
Why do we get a dendritic shape?
Initial seed with radius
0*
om
m* T)CC(m
L
T)(TT
anisotropicsurface energy
liquidus slope
To < Tm
Surface of seed is under-cooled dueto curvature (Gibbs-Thompson) andsolute (Not Shown kinetic)
)C1(MC)4cos151(dT 0o Capillary length~10-9 for metal
With dimensionless numbers L
cmM,
C
CC,
c/L
mCTTT
0
*0f
*
Angle between normal and x-axis
)4cos151(
0.25
0.25 1
Anisotropic term makes under cooling less in preferred growth directions
As crystal growsthe sharper tempgrad at tip drives solharder BUT the increased tip curvature holds it backA steady tip operatingVelocity is reached
Alain Karma and Wouter-Jan Rappel
Phase Filed
Current Approaches (Pure Melt)
Thermodynamic equation –minimizing free energy across a diffusive interface
-1 < phase marker < 1
Heat equation with source
H. S. Udaykumar, R. Mittal, Wei Shyy
Interface Tracking
Juric Tryggvason
Zhao and HeinrichKim, Goldenfeld, DantzigAnd Chen, Merriman, Osher,andSmereka
Level Set
Solve for level setUsing speed functionfrom Stefan Cond.
Maintaindistance functionproperties by re-in
Solve heat con.Use level setto mod FD at interface
Enthalpy Method-First Proposed by Eyres et al 1947
For Crystal Growth by K H Tacke, 1988
)TK(t
H
fTH
A function of f if 0 < f < 1(f determines curvature)
In this work: use iterative sol. Include anisotropy and solute
0.60.70.3
0.0
1.0
liquid fraction0 in solid1 in liquid-a physical level set
Governing EquationsWith additional dimensional numbers
l
le2
*l
*
DL,
tt,
xx
Governing equations are
)TK(t
H
)VD(t
C
gTH
sl C)f1(fCC
k)k1(f
CV
l
s)f1(fK
l
s
D
D)f1(kf
Le
1D
)V1(MC
)4cos151(dT
0
o
Chemical potential
Continuous at interface
il
is kCC
concentration
If 0 < g < 1
2/32y
2x
yy2xxyyxxy
2y
)gg(
ggggg2gg
)
g
g(tan
y
x1
In a time step Solve for H and C (explicit time integration)
Calculate curvature and orientation from current nodal g field
Calculate interface undercooling
If 0 < g < 1 thenUpdate f from enthalpy as
Check that calculated liquid fraction is in [0,1]
Update
Iterate until
At end of time step—in cells that have just become all solidintroduce very small solid seed in ALL neighboring cells.Required to advance the solidification
Numerical Solution
xUse square finite difference grid, set length scale to
)V1(MC)4cos151(dT 0oi
k)k1(f
CV
)fTH(ff i
fHT
tol)fTH( i
Very Simple—Calculations can be done on regular PC
Initial condition—Circle r = 2.5do
Typical gridSize 200x00¼ geometry
ON A FIXEDUNIFORM GRID
Verification 1 Looks Right!!
k = 0 (pure), = 0.05, T0 = -0.65, x = 3.333d0
Enthalpy Calculation
Dimensionless time = 0 (1000) 60002
odtk
k = 0 (pure), = 0.05, T0 = -0.55, x = d0
Level Set Kim, Goldenfeld and Dantzig
Dimensionless time = 37,600
Verification 2
Verify solution coupling by Comparing with one-d solidification of an under-cooled binary alloy
Constant Ti, Ci
k = 0.1, Mc = 0.1, T0 = -.5, Le = 1.0
Compare with Analytical Similarity Solution—Rubinstein Carslaw and Jaeger combo.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 10 20 30 40x
C
T/T0k = 0.1
MC0 = 0.1
T0 = -0.5
Le = 1.0
Concentration and Temperatureat dimensionless time t =100
Symbol-numeric sol.
0
2
4
6
8
10
12
0 100 200 300
time
psoi
tion
Front Movement
Red-lineNumeric sol.Covers analytical
Verification 3
Compare calculated dimensionless tip velocity withSteady state operating state calculated from the microscopic solvability theory
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0 1000 2000 3000 4000 5000 6000
time
velocity
5do
3.333do
2.5do
x
dvv ogridtip
Verification 4
Check for grid anisotropy
)V1(MC))4cos(151(dT 0oi
Solve with
4 fold symmetry twisted 45o
Then Twist solution back
x = .36do
x = .38do
Dimensionless time = 6000
0d5.2x
0Le
10Le
1Le
2.Le
k = 0.15, Mc = 0.1, T0 = -.65
= 0.05, x = 3.333d0
Result: Effect of Lewis Number
l
le DL
All predictions atDimensionless time =6000
0
0.5
1
1.5
2
2.5
3
3.5
4
0 50 100 150 200 250 300 350 400 450 500
distance
Concentration
k = 0.15, Mc = 0.1, T0 = -.55, Le = 20.0
= 0.02, x = 2.5d0
Dimensionless time = 30,000
Result: Prediction of Concentration
2
odtk
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0 1000 2000 3000 4000 5000 6000
time
velocity
5do
3.333do
2.5do
Conclusion –Score card for Dendritic Growth Enthalpy Method (extension of original work byTacke)
Ease of Coding Excellent
CPU Very Good (runs shown here took between 30 minutes and 2 hours on a regular PC)
Convergence to known analytical sol.
Excellent
Convergence to known operating state
Good (further study with finer grid required)
Grid Anisotropy Reasonable (further study with finer grid required)
Flexibility to add more physics
Excellent (adding solute only required 10% more lines)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 10 20 30 40x
C
T/T0k = 0.1
MC0 = 0.1
T0 = -0.5
Le = 1.0
Fans Toes Shoreline
Two Sedimentary Moving Boundary Problems of Interest
Moving Boundaries in Sediment Transport
NCED’s purpose:to catalyze development of an integrated, predictive science of the processes shaping the surface of the Earth, in order to transform management of ecosystems, resources, and land use
The surface is the environment!
Research fields• Geomorphology• Hydrology• Sedimentary geology• Ecology• Civil engineering• Environmental economics• Biogeochemistry
Who we are: 19 Principal Investigators at 9 institutions across
the U.S.
Lead institution: University of Minnesota
1km
Examples of Sediment FansMoving Boundary
How does sediment-basement interfaceevolve
Badwater Deathvalley
Sediment mass balance gives
Sediment transported and deposited over fan surfaceby fluvial processes
xxt
From a momentum balance anddrag law it can be shown thatthe diffusion coefficient is a function of a drag coefficientand the bed shear stress
when flow is channelized = cont.
Convex shape
sediment
h(x,t)
x = u(t)
0q
bed-rock
ocean
x
shoreline
x = s(t)
land surface
A Sedimentary Ocean Basin
An Ocean Basin
Melting vs. Shoreline movement
pressurizedwater reservoir
to water supply
solenoidvalve
stainless steelcone
to gravel recycling
transport surface
gravel basement
rubber membrane
experimental deposit
Experimental validation of shoreline boundary model
~3m
Base level
Measured and Numerical results ( calculated from 1st principles)
1-D finite difference deforming grid vs. experiment
xxt+Shoreline balance
An Ocean Basin
Melting vs. Shoreline movement
Limit Conditions: A Fixed Slope Ocean
q=1
s(t)
similarity solution
q
2
)(erfe
)(erf1,t2s
22/1
21
21
2
21
21
0
5
10
15
20
25
0 100 200 300
Time
sh
ore
line
0if),x(LH
2
2
xt
H
Enthalpy Sol.
A Melting Problem driven by a fixed flux with SPACE DEPENDENT
Latent Heat L = s
dt
dss
x)t(sx0,
xt s2
2
s Depth at toe
)TK(t
H
L)(TH
h(x,y,t)
q
bed-rock
ocean
y
shoreline
x = s(t)
land surface
(x,y,t)
A 2-D Front -Limit of Cliff face Shorefront But Account of Subsidence and relative ocean level
0hif),t,y,x(LhH
)h(t
H
Enthalpy Sol.
xy
]L/H,[MINfrac 1
Solve on fixed gridin plan view
Track Boundary by calculating in each cell
0
5
10
15
20
25
30
0 100 200 300 400 500
time
shor
elie
n po
sitio
n
s(t)
0
5
10
15
20
25
30
0 500 1000 1500 2000
time
shor
elin
e po
sitio
n
numerical
steady state
s(t)
Hinged subsidence
q
2
)(erfe
)(erf1,t2s
22/1
21
21
2
21
21
A 2-D problem Sediment input into an oceanwith an evolving trench driven By hinged subsidence
With Trench
No Trench Trench
Plan view movement of fronts
Stratigraphy and Shoreline
-30-20-10
0102030405060
0 10 20 30
Models can predict stratigraphy“sand pockets” = OIL
The Po
Shoreline position is signature of channels
WHY Build a model
Shoreline Tracking Model has been
)t,y,x(LhH
)h(tH
Latent Heat a function of Space and time
Enthalpy methods can generate models for distinct moving boundary systems
)TK(tH
L)(TH
Crystal Growth in undercooled pure melt
Phase change temperature a function of interface
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