this work was performed under the auspices of the u.s. department of energy

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

2UCRL-PRES-214999

Introduction: Sharp and Diffusive Interfaces

Diffusive

“Interface”

Sharp

Interfaces

Gas-phase mixing Droplets or bubbles

3UCRL-PRES-214999

Outline

• Treatment of sharp interfaces

• Treatment of diffusive interfaces

• Simulations combining sharp and diffusive interfaces

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

5UCRL-PRES-214999

Outline: Treatment of Sharp Interfaces

• Treatment of sharp interfaces– Standard treatment

– Hybrid energy update

– Stress equilibration

• Treatment of diffusive interfaces


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

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

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

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

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

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

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

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

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:

15UCRL-PRES-214999

Test Problem - Cylindrical Cavity Expansion

Radial Stress

+ 1 bar

0

-1 bar

Aluminum

Air

1 bar

Pressure Relaxation ResultsVacuum

16UCRL-PRES-214999

Pressure RelaxationR

ad

ial S

tre

ssH

oo

p S

tres

sStress Relaxation

+ 1 bar

0

-1 bar

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

18UCRL-PRES-214999

Outline: Treatment of Diffusive Interfaces




• Treatment of diffusive interfaces– Track mass fractions of components

– Use effective mixture gamma

– Iterate for real materials


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

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

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:

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


),(ˆ

12

iiii

i

pwm

w

p

),(ˆ

1

iii

i

T

mT

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


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

24UCRL-PRES-214999

Outline: Simulations




• 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

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

26UCRL-PRES-214999

2D Problem Setup

1.5mm steel liner

167 GJ source

4 tons water

27UCRL-PRES-214999

Density/Temperature Profiles

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

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

30UCRL-PRES-214999

8.4 TJ200 tons water

60 m x 10 m x 10 m

chamber

0.5 m roof

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

this work was performed under the auspices of the u.s. department of energy

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