Applications of the Multi-Material DEM Model

Applications of the Multi-Material DEM Model

Presented by:

David Stevens, Jaroslaw Knap, Timothy Dunn

September 2007

Lawrence Livermore National Laboratory

This work was performed under the auspices of the U.S. Department of Energy by the University of CaliforniaLawrence Livermore National Laboratory under Contract No. W-7405-Eng-48.

IntroductionIntroduction

Multiphase flow is important in many shock-driven applications

This talk will describe several ways in which we have adapted the multiphase DEM method of Chinnayya et al to handle: Complex geometries Curvi-Linear coordinates Deviatoric strength Interface reconstruction Adaptive mesh refinement.

Multiphase flow model developmentMultiphase flow model development

The multiphase model is built upon an Eulerian treatment of each phase.

This treatment uses a hierarchy of flux-nozzling pairs defined by the number of waves used in the numerical solver.

Saurell and Abgrall (1999) are the basis for the flux-nozzling pairs for the Rusanov (1 wave) and HLL (2 waves) solvers.

The DEM method of Chinnayya et al. (2003) allows the use of HLLC.

intintintint ,,0,

,,,1

,,,

uPPuH

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HFUS

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i

i

t

nii

nii

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ni

ni

ni HH

x

tFF

x

tUU 2/1,2/1,2/12/1

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DEM GeneralizationsDEM Generalizations

The following formulation is not limited to the Euler Equations

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i

xt

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0

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lag

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t

t

xx

t

t

Upon Integration by Parts and simplifying

Discrete Equation MethodDiscrete Equation Method

The most promising model is “DEM” or Discrete Equation Method and is based on an acoustic Riemann Solver.

DEM allows the use of very sophisticated single phase solvers in a multiphase context.

Flows with deviatoric stresses can easily be incorporated.

The following results were presented at the latest International Detonation Symposium in 2006.

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DEM

AUSFAUSF

AUSF (Sun and Takayama, JCP, 2003) is the Riemann solver used in our version of DEM.

Extends easily to unstructured meshes

Extends easily to arbitrary equations of state.

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Coordinate Rotation and

Wave Selection

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In 1D this method is identical to HLLC.

AUSF ValidationAUSF Validation

Sod Problem Sedov Problem

We have compared AUSF on a number of analytic planar and spherical test problems

The Rogue Shock TubeThe Rogue Shock Tube

At right is Figure 15 from “Experimental and numerical investigation of the shock-induced fluidization of a particle bed, Shock Waves, 8, 29-45, 1998.

Numerical results agree well with the experimental data.

The simulated fluidized bed is slightly ahead of the experimental observations.

Numerical Method ComparisonNumerical Method Comparison

Fan Zhang et. al., “Explosive Dispersal of Solid Particles”, Shock Waves, 11, 431-443, 2001 has proven to be a useful data set for model validation.

DEM appears to best match the experimental data.

Reactive Multiphase flowsReactive Multiphase flows

Long mixing time scales are often the rate limiting step for many turbulent reacting flows. Air and fuel products can compete as oxidizers in cases when

combustion of embedded metals is present. Mixing effects on reaction rates:

Pre-initiation Enhancement of slower rate reactions Extended reaction times Long-lived non-reacted fuel parcels.

These mixing effects require a model for subgrid-scale heterogeneity.

Prescribed PDF Methods for Reactive flowsPrescribed PDF Methods for Reactive flows

iN

iiz

z

ZP

dPz

1

1

0

Beta PDF distribution

15.02 z

01.02 z

dPN

j jZ

z0

Mixture fraction and reaction progress variables are powerful tools for simplifying reaction chemistry.

Several widely used closures result from assuming a prescribed beta PDF for the mixture fraction. Simple covariance closures Flamelet models Infinitely fast chemistry Binned PDF for use with an equilibrium chemistry package. Finite rate kinetics

Infinite Rate ChemistryInfinite Rate Chemistry

O2

N2 F

P

N2 F

O2

Single Reaction model loosely based on Kuhl, Howard, and Fried, Proceedings 34th Int.. ICT conference Energetic Materials: Reactions of Propellants, Explosives, and Pyrotechnics, 2003.

Detonation Products (F) mix with air (A) to completely combust to product (P).

Figure 4. Le Chatelier diagram for combustion of TNT explosion products in air.

Equilibrium Chemistry

The mean and varience of the mixture fraction PDF is tracked in this simulation.

The mixture fraction is defined as 1 inside the charge and 0 outside.

Only the adiabat for the pure fuel stage at high energies is needed.

Upon mixing at lower energies, an entire plane of equilibrium calculations are needed.

Charge Position

4 inch L/D=1 TNT cylinder detonation in Air

Equilibrium chemistry calculations as a function of T,P and Mixture fraction.

Adaptive Mesh Refinement (AMR)Adaptive Mesh Refinement (AMR)

Our approach to AMR combines block structured AMR and an unstructured local discretization.

The unstructured local discretization handles reduced and enhanced connectivity in the form of multi-block meshes with no modification to the numerics.

Block Structured AMR is used to handle inter-domain communication via rotations and translations. All O(n2) operations have been eliminated.

Zone-centered data for all state variables dramatically simplifies computer science requirements.

SummarySummary

We have describes ways in which we have adapted DEM to incorporate additional physics and geometrical complexity:

turbulent and kinetic effects are being incorporated by the

use of assumed subgrid PDF methods: Particle size distributions. Burned as Mixed model. Equilibrium Chemistry. Progress variables for finite rate kinetics.

AMR, deviatoric strength interface reconstruction.