1. 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

2. 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

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

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

5. 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

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

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

8. 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 nn  d s n+1  d f n+1

9. 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

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

11. 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!

12. PREVIOUSLY Eulerian Description (Frenette &Zimmermann)

13. Solitary wave propagation

14. 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

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

16. 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 ( )

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

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

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

20. 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