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This work was performed under the auspices of the U.S. Department of Energy by University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.
NUMERICAL MODELLING OF EXPLOSIONS IN UNDERGROUND CHAMBERS
USING INTERFACE TRACKING AND MATERIAL MIXING
Benjamin T. Liu and Ilya Lomov
Energy and Environment Directorate
Lawrence Livermore National Laboratory
Numerical Methods for Multi-Material Fluid Flows
September 5th-8th, 2005
St. Catherine’s College, Oxford, UK
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2UCRL-PRES-214999
Introduction: Sharp and Diffusive Interfaces
Diffusive
“Interface”
Sharp
Interfaces
Gas-phase mixing Droplets or bubbles
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3UCRL-PRES-214999
Outline
• Treatment of sharp interfaces
• Treatment of diffusive interfaces
• Simulations combining sharp and diffusive interfaces
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4UCRL-PRES-214999
GEODYN
• High-order Godunov Eulerian code – Able to model large deformations– Able to capture shocks– Treatment of interfaces is important
• Structured rectangular grids with adaptive mesh refinement
• Multi-material with a fully integrated stress tensor– Characteristic tracing of stress tensor– Acoustic approximation for shear waves
• Flexible material library• Analytic and tabular EOS• Wide range of constitutive models
– Especially designed to model the response of geophysical media
– Includes a variety of yield strength models
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5UCRL-PRES-214999
Outline: Treatment of Sharp Interfaces
• Treatment of sharp interfaces– Standard treatment
– Hybrid energy update
– Stress equilibration
• Treatment of diffusive interfaces
• Simulations combining sharp and diffusive interfaces
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6UCRL-PRES-214999
Standard Treatment of Sharp Interfaces
volume fraction of fluid
bulk modulus of fluid
1
f ff K
t K
f
K
K f K
v v
• Volume-of-fluid approach
• High order interface reconstruction– used to calculate transport volumes
– preserves linear interface during translation
• Thermodynamics based equations for the mixed cell update
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7UCRL-PRES-214999
Standard Pressure Relaxation Scheme
• Iterative adjustment of volume fractions
• - bulk modulus
• - numerically or physically based limiter
f p Kp
f K
f f p p K
K
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8UCRL-PRES-214999
Problems with the Standard Treatment
• Conservative energy update is not robust for mixed materials
– Materials with drastically different properties (, K, etc)
– Most severe when kinetic energy is large relative to internal energy
• Pressure relaxation unsuitable when strength in mixed cells is important
– Relaxation scheme ignores strength
– Effective strength for material in mixed cells with fluid is zero
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9UCRL-PRES-214999
Hybrid Energy Update
• Conservative equation
• Non-conservative equation
• Hybrid (conserves energy)
vvTvvvvv
1212
1
K
KfT
ff
t
f nn
vvTv
1
K
KfT
ff
t
f nn
c
ncc
wheref ,max
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10UCRL-PRES-214999
Hybrid Energy Update Test: Aluminum Flyer Plate (3 km/s) in Air
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0 100 200 300 400 500 600 700
ConservativeNonconservativeHybrid
-5-4-3-2-1 0 1 2 3 4 5
0 100 200 300 400 500 600 700
Normal stress Internal energy
Position of flyer plate
GPa GPa
mm mm
Conservative Non-conservative
Hybrid
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11UCRL-PRES-214999
Pressure Equilibration in Mixed Cells with Strength
• Pressure relaxation ignores strength
• Problem in mixed cells with solid and fluid– Solid w/strength and fluid w/o strength
– Pressures in solid and fluid are equal
• Mixed cells containing fluid have no strength– Material is weaker near interfaces
– Introduces strong mesh dependence
• Results in cells containing differing solids w/strength are also wrong
P
fluid no strength
solid w/ strength
no
strength
no
strength
no
strength
no
strength
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12UCRL-PRES-214999
Normal Stress Equilibration in Mixed Cells with Strength
• Equilibrate normal stress instead of pressure
• Information within mixed cell insufficient– Need to calculate T’nn
– Requires elastic hoop strain (ett)
• Solution: Use properties from single-material
cells in the vicinity of the mixed cell
• Consistency conditions:– Stress normal to interface is continuous
– Elastic strains in transverse direction taken from single-material cells
– Interfacial shear stress can be calculated using a friction law
• Fall back to pressure relaxation scheme when:– No single-material cells in the direction of normal
– More then 2 materials in the cells
Tnn
ett1
ett2
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13UCRL-PRES-214999
Stress Relaxation
• Elastic hoop strain in the single material cell:
• Normal component of the stress deviator in the mixed cell:
Relax total normal stress in each material to the average across the cell:
nn
nn
nn nn
f T KT
f K
f f T T K
Constraint modulusConstraint modulus
GKC 34
S
S
SS
nnS
tS
tStt G
CG
TTT
e2
2121
43
21
CG
C
GpStt
nn
eGT
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14UCRL-PRES-214999
Stress Relaxation Test: Aluminum Flyer Plate (3 km/s) in Air
-0.002
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0 100 200 300 400 500 600 700 800 900 1000
N=2000
Pressure
Normal stress
v=3km/s v=3km/s
-0.002
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0 100 200 300 400 500 600 700 800 900 1000
N=2000Pressure RelaxationStress Relaxation
Pressure
Normal Stress (-Tnn)
Pressure
Normal Stress (-Tnn)
)(3
2
34 nnnn TTGK
KP
for the aluminum plate
For elastic 1D strain:
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15UCRL-PRES-214999
Test Problem - Cylindrical Cavity Expansion
Radial Stress
+ 1 bar
0
-1 bar
Aluminum
Air
1 bar
Pressure Relaxation ResultsVacuum
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16UCRL-PRES-214999
Pressure RelaxationR
ad
ial S
tre
ssH
oo
p S
tres
sStress Relaxation
+ 1 bar
0
-1 bar
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17UCRL-PRES-214999
Problems Requiring Stress Equilibration
• Quasi-static solution after initial waves have passed– Cavity expansion
– Blast or impact loading of deeply buried structures
• Overall response driven by deformation in the mixed zones – Fast moving solids undergoing “slow” deformation
• Void nucleation and growth under positive pressure– Pressure relaxation will cause voids to immediately close
– Strength in the material surrounding voids is important
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18UCRL-PRES-214999
Outline: Treatment of Diffusive Interfaces
• Treatment of sharp interfaces– Standard treatment
– Hybrid energy update
– Stress equilibration
• Treatment of diffusive interfaces– Track mass fractions of components
– Use effective mixture gamma
– Iterate for real materials
• Simulations combining sharp and diffusive interfaces
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19UCRL-PRES-214999
• Consider materials that diffuse into one another– Separate components within a single computational “material”
– Mass fractions (with total , ) sufficient to reconstruct mixture state variables
• Should enforce pressure and temperature equilibrium between components
Diffusive Material Interface Treatment
0)()(
v
mt
m
m
),(),(
),(),(
111
111
TT
pp
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20UCRL-PRES-214999
Ideal Gas Mixture
• Internal energy
• Effective molecular weight
• Effective gamma
11
1
ii
i
w
mw
i
i
w
mw
1
1,ii
iiviii w
mRTTcmm
Ideal Gas Mixing
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21UCRL-PRES-214999
Ideal Gas Pressure
w
RT
w
mRT
w
RTm
w
RTpp
i
i
i
i
i
ii
w
RTp
w
RTp
w
wm
i
ii
i
ii
•Pressure for ideal gas mixture independent of spatial component distribution
iii pww
p 1
i: fraction of mixture volume occupied by component i
Molecular mixture
1ii
ii
m
Droplets or bubbles
Ideal Gas Pressure Calculation
For an ideal gas:
Enforcing pressure equilibrium:
Applying Dalton’s Law:
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22UCRL-PRES-214999
• Define an effective (component) gamma:– a constant for ideal gases
– a relatively slowly varying parameter for a wide range of densities and temperatures for many real materials
• Calculate pressure based on mixture gamma:
• Similarly calculate temperature:
• Zeroth order approximation: i = imi– Yields correct averages for ideal gases
Non-Ideal Equations of State
ii
iiii
p
),(ˆ
1
),(ˆ)1(1
1 2
iiii
i
ii
i
pw
mw
w
mw
p
Non-Ideal Equations of State
),(ˆ
12
iiii
i
pwm
w
p
),(ˆ
1
iii
i
T
mT
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23UCRL-PRES-214999
• Initial guess: i = imi
• Iterate on component densities and energies– Iterative estimate for energy
– Pressure relaxation scheme for density
• Two-phase region may be singular and non-convergent– Solution has oscillations
• Saurel & Abgrall (1999), Karni (1994), et al
• Zeroth order approximation good when gamma is changing slowly
Non-Ideal Equations of State
1
1
ii
i w
w
Iterative Refinement for Non-Ideal Gases
i
ii
i
iii
m
K
pp
),(ˆ
12
iiii
i
pwm
w
p
),(ˆ
1
iii
i
T
mT
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24UCRL-PRES-214999
Outline: Simulations
• Treatment of sharp interfaces– Standard treatment
– Hybrid energy update
– Stress equilibration
• Treatment of diffusive interfaces– Track mass fractions of components
– Use effective mixture gamma
– Iterate for real materials
• Simulations combining sharp and diffusive interfaces– Mixing and heating in underground chambers
– 2D simulation
– Large-scale 3D simulation
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25UCRL-PRES-214999
Explosions in Underground Chambers
• Fundamental study of multi-material mixing and heating – Demonstrate combination of diffuse and sharp interfaces– No explicit subgrid model
• Turbulence implicitly modeled by truncation errors • Monotone Integrated Large Eddy Simulation (MILES) [J. Boris, 1992]• Physical rationale by L. Margolin and W. Rider in 2002
• Examine heating of water contained in underground chambers– Consider different modes of heating after an explosion
• Shock heating (PdV work)• Convective mixing
– Measure degree of heating by fraction of water above 650K• Critical point for water• Vapor and liquid indistinguishable
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26UCRL-PRES-214999
2D Problem Setup
1.5mm steel liner
167 GJ source
4 tons water
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27UCRL-PRES-214999
Density/Temperature Profiles
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28UCRL-PRES-214999
Temperature Distribution
0.0
0.5
1.0
0 20 40 60 80time (ms)
wa
ter
ma
ss
fra
cti
on
T < 650 K
650 K T < 2600 K
T 2600 K
Shock heatingShock heating
Expansion and coolingExpansion and cooling
Convective mixing dominates heat transfer
Convective mixing dominates heat transfer
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29UCRL-PRES-214999
3D Calculation
60 m x 10 m x 10 m
chamber
0.5 m DOB
• Run on LLNL’s Thunder supercomputer– Utilized 960 nodes (3840 Itanium CPU’s)– Used almost 1 TB of total memory
• Largest problem of its kind to date– Two levels of refinement– 16.8 million zones (6 cm resolution) on the coarse
level– ~160 million zones (1.5 cm resolution) on the fine
level
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30UCRL-PRES-214999
8.4 TJ200 tons water
60 m x 10 m x 10 m
chamber
0.5 m roof
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31UCRL-PRES-214999
Conclusions
• Improved treatment of sharp interfaces– Hybrid energy update robustly captures shocks while conserving energy
– Stress equilibration improves modelling of material with strength
• Implemented simple treatment of diffusive interfaces– Store mass fractions and calculate an effective gamma
– Zeroth order approximation sufficient for many applications
• Successfully simulated problems including sharp and diffusive interfaces
– Performed both 2D and 3D simulations
– Examined mixing and heating of explosions in bunkers