DEBRIS FLOWS & MUD SLIDES: A Lagrangian method for two-

phase flow simulation

Matthias Preisig and Thomas Zimmermann, Swiss Federal Institute of Technology

Lausanne, Switzerland

Funded by the Swiss National Science Foundation

Goal: Modeling debris flows

Large displacements

Free surface flow

Two-phase material(soil-water)

La Conchita, CA, January 2005 © by AP

Initiation of flow

Transport

Deposition

Being able to: Predict flow path (danger zone) Obtain design parameters for protection

devices (depth, quantity, energy)

Why model debris flows?

WSL WSL

Outline

Governing equations of 2-phase flow

Computation of volume fractions

Lagrangian update and remapping

Numerical examples

Two-Phase Flow

Flow of two viscous fluids (solid phase is regarded as fluid)

Phases occupy same control volume in space (no phase interfaces)

Momentum exchange via drag force

Fluid phase: Cf

Solid phase: Cs

Concentrations:

Cf = 1

Cs = 1Cf = 1

Cf + Cs = 1

Governing equations

Mass balance

Momentum balance

Constitutive model

Momentum exchange (drag force)

Post-calculation of volume fractions

Knowing the velocities, compute volume fractions:

Mass balance

Remesh:Remesh:

Lagrangian update algorithmSolve for vsn+1, vfn+1 and pn+1Solve for vsn+1, vfn+1 and pn+1

Find free surfaceFind free surface

Update nodesUpdate nodes

Solve for Csn+1 and Cf

n+1Solve for Cs

n+1 and Cfn+1

Remesh inside boundary → n+1Remesh inside boundary → n+1

Map vsn+1, vfn+1, pn+1,Csn+1 and Cf

n+1 on n+1Map vsn+1, vfn+1, pn+1,Csn+1

and Cfn+1 on n+1

n

dsn+1

dfn+1

Numerical method

Meshless (NEM – natural neighbor based, Sukumar et al.) Unique interpolation for a given nodal

distribution Less sensitive to uneven nodal distribution than

FEM

u = 1u = 0

support of shape function

NEM – FEM: Automatic Remeshing

FEM: nodal connectivity using Delaunay triangulation

NEM: connectivity depends only on point location

Dam break

Releasing horizontal BC’s on right side

Automatic remeshing prevents excessive element distortion

Triangles in above picture represent integration domains, no elemental connectivity!

PREVIOUSLY

Eulerian Description (Frenette &Zimmermann)

Solitary wave propagation

Drop of heavy fluid in light fluid

C1 = 0.9

C1 = 0.1

Density: 1 = 2 2

High momentum exchange coefficient Kdrag

( )

Free surface

Drop of heavy fluid in light fluid

Vol

ume

frac

tion

of d

ense

r flu

id

Drop of heavy fluid in light fluid

C1 = 0.9

C1 = 0.1

Density: 1 = 10 2

Low momentum exchange coefficient Kdrag

( )

Drop of heavy fluid in light fluid

Vol

ume

frac

tion

of d

ense

r flu

id

NEM – FEM: Pro’s and Con’s

NEM FEM (lin. triangles)

Irregular point distribution

++ --

Regular grid + -

Locking Stabilization required

Numerical integration

- +

Implementation - +

Incompressible Elasticity (Stokes)

(i)

(iia)

(iib)

Find u, p such that:

Stabilization (Laplacian Pressure Operator Scheme) after Brezzi & Pitkäranta (1984)

Conclusions

Updated Lagrangian algorithm for two-phase flow Only material domain is modeled Definition of free surface straightforward No stabilization of convective terms required

Most general continuum model for two-phase flows