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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

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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

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Being able to: Predict flow path (danger zone) Obtain design parameters for protection devices (depth, quantity, energy) Why model debris flows? WSL

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Outline Governing equations of 2-phase flow Computation of volume fractions Lagrangian update and remapping Numerical examples

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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: C f Solid phase: C s Concentrations: C f = 1 C s = 1 C f = 1 C f + C s = 1

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Governing equations Mass balance Momentum balance Constitutive model Momentum exchange (drag force)

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Post-calculation of volume fractions Knowing the velocities, compute volume fractions: Mass balance

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Remesh: Lagrangian update algorithm Solve for v s n+1, v f n+1 and p n+1 Find free surface Update nodes Solve for C s n+1 and C f n+1 Remesh inside boundary → n+1 Map v s n+1, v f n+1, p n+1,C s n+1 and C f n+1 on n+1 nn d s n+1 d f n+1

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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 = 1 u = 0 support of shape function

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NEM – FEM: Automatic Remeshing FEM: nodal connectivity using Delaunay triangulation NEM: connectivity depends only on point location

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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!

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PREVIOUSLY Eulerian Description (Frenette &Zimmermann)

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Solitary wave propagation

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Drop of heavy fluid in light fluid C 1 = 0.9 C 1 = 0.1 Density: 1 = 2 2 High momentum exchange coefficient K drag ( ) Free surface

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Drop of heavy fluid in light fluid Volume fraction of denser fluid

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Drop of heavy fluid in light fluid C 1 = 0.9 C 1 = 0.1 Density: 1 = 10 2 Low momentum exchange coefficient K drag ( )

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Drop of heavy fluid in light fluid Volume fraction of denser fluid

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NEM – FEM: Pro’s and Con’s NEMFEM (lin. triangles) Irregular point distribution ++-- Regular grid+- LockingStabilization required Numerical integration -+ Implementation-+

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Incompressible Elasticity (Stokes) (i) (iia) (iib) Find u, p such that: Stabilization (Laplacian Pressure Operator Scheme) after Brezzi & Pitkäranta (1984)

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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

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