This work was performed under the auspices of the U.S. Department of Energy

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NUMERICAL MODELLING OF EXPLOSIONS IN UNDERGROUND CHAMBERS USING INTERFACE TRACKING AND MATERIAL MIXING

Numerical Methods for Multi-Material Fluid Flows

September 5th-8th, 2005

St. Catherine’s College, Oxford, UK

Benjamin T. Liu and Ilya Lomov

Energy and Environment Directorate

Lawrence Livermore National Laboratory

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.

Gas-phase mixing

Droplets or bubbles

Sharp

Interfaces

Diffusive

“Interface”

- Treatment of sharp interfaces
- Treatment of diffusive interfaces
- Simulations combining sharp and diffusive interfaces

- 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

- Treatment of sharp interfaces
- Standard treatment
- Hybrid energy update
- Stress equilibration

- Treatment of diffusive interfaces
- Simulations combining sharp and diffusive interfaces

- 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

- Iterative adjustment of volume fractions
- - bulk modulus
- - numerically or physically based limiter

- Conservative energy update is not robust for mixed materials
- Materials with drastically different properties (r, 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

- Conservative equation
- Non-conservative equation
- Hybrid (conserves energy)

GPa

GPa

ConservativeNon-conservativeHybrid

mm

mm

Position of flyer plate

no

strength

no

strength

no

strength

no

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

fluid

no strength

P

solid

w/ 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

ett2

Tnn

ett1

- 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:

Constraint modulus

Pressure Relaxation

Stress Relaxation

Pressure

Normal Stress (-Tnn)

Pressure

Normal Stress (-Tnn)

For elastic 1D strain:

for the aluminum plate

Pressure Relaxation Results

Vacuum

Aluminum

+ 1 bar

0

-1 bar

Air

1 bar

Radial Stress

Stress Relaxation

Pressure Relaxation

Radial Stress

+ 1 bar

0

-1 bar

Hoop Stress

- 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

- 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

- Consider materials that diffuse into one another
- Separate components within a single computational “material”
- Mass fractions (with total r, e) sufficient to reconstruct mixture state variables

- Should enforce pressure and temperature equilibrium between components

Ideal Gas Mixture

- Internal energy
- Effective molecular weight
- Effective gamma

Molecular mixture

Droplets or bubbles

fi: fraction of mixture volume occupied by component i

Ideal Gas Pressure

Applying Dalton’s Law:

For an ideal gas:

Enforcing pressure equilibrium:

Pressure for ideal gas mixture independent of spatial component distribution

Non-Ideal Equations of State

- 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: ei = e, ri = mir
- Yields correct averages for ideal gases

Non-Ideal Equations of State

- Initial guess: ei = e, ri = mir
- 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

- Solution has oscillations
- Zeroth order approximation good when gamma is changing slowly

- 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

- 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

- Consider different modes of heating after an explosion

167 GJ source

4 tons water

1.5mm

steel liner

Convective mixing dominates heat transfer

Expansion and cooling

Shock heating

T < 650 K

650 K T < 2600 K

T 2600 K

- 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

0.5 m DOB

60 m x 10 m x 10 m

chamber

0.5 m roof

8.4 TJ

60 m x 10 m x 10 m

chamber

200 tons water

- 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