Direct Numerical Simulation of SolidLiquid Flow P LAURE

Direct Numerical Simulation of Solid-Liquid Flow P. LAURE 1 , G. BEAUME 1, 2 and T. COUPEZ 2 ( A. Megally 2 , Thesis July 2005) 1 2 3

Framework Software Numerical library : CIMLIB (H. Digonnet, L. Silva, J. Bruchon) Hardware • Finite Element Solveur EII (C. Torrin) • C++ , MPI (parallelized) • 3 clusters with 64 PC • mesh partitionning • Meca. Grid Poject • r-adaptation and h-adaptation (C. Gruau) Monolithic, Multi-Domain, Finite Element methods

Solid-Liquid flows Solid domain rigid motion

Characteristic Functions (1) Description of several domains in a single mesh j = fluid or solid (fibers) Fluid domain Fiber domain

Characteristic Functions (2) Update with voxelisation method P 0 approximation (VOF) Computation of characteristic function Voxelisation Pixels in K Mesh cavity Swith on pixels Fiber domain

New Formulation in Solid Domain Rigid motion Boundary conditions Stress tensor Penalization factor Lagrange multiplier

Weak Formulation Rigid motion constraint Simplifications : • only penalization • no inertial effect Mixing relation : • no gravity

Computation of Velocity field Penalization ~ 103 l Periodic boundary conditions ux uy Shear rate

Update solid domain (1) Transport Equation : discontinuous Galerkin method • numerical difusion • need r-adaptation • need more elements

Update fiber position and orientation Particle method Langrangian updating X 2 p O X i ( t + Dt ) = X i + VX Dt VX 2 i V X 1 VX 1 Rigid motion p= X 2 X 1 O Vector orientation VO O ( t + Dt ) = O + VO Dt Advantages § Perfect rigid motion of each fiber Fiber center § Conservation of the length § No mumerical diffusion 10

Numerical Procedure Voi Velocity field computation Update particle positions finite element method multi-domain approach ( 1 W ) Particle method Boundary conditions and collisions Hydrodynamic Interaction 1 W j § Numerical approach similar to Glowinski & Joseph’s modelling [Glowinski 1999] (Fictitious domain method for particulate flows) 11

Single Fiber Motion in shear flow Isoline of characteristic function 2 D computed Isoline of characteristic function 3 D Shear flow, Periodic Cell Equivalent aspect ratio bref Find Jeffery's orbit for a shear flow Periodic motion - Jeffery Evolution of vs time 1 2 p T = ( b + ). ref bref g 12

Hydrodynamical Interactions It is not necessary to have an explicit form (as in [Yamane 1994] , [Fan 1998]) - drag forces - lubrication forces (short range interactions) The central particle moves due to hydrodynamic interactions Sphericals particles in Couette Flow The period of rotation changes 13

Spheres – short-range hydr. forces (moderate concentration) Lubrication approximation : repulsion force exerted by j on i // Fij as Oi Oj • Modifies ui and moves Oi in the nij direction • Accurate computation needs a small region between two spheres depends on mesh

Examples with Spheres Particule update : Prevent the overlapping i j dij < 0 a a =. 005 t = 27 t = 30 15

Examples with fibers No collision strategy 16

3 D Computations Initial time Random orientation Fiber orientation in the shear direction

Perspectives § Lagrange multiplier for rigid motion constraint improve matrix conditioning § Lagrange multiplier for boundary condition (computations for homogenization studies) impose shear rate instead velocity on boundary § add repulsive forces in the weak formulation § use P 1 approximation ( « level set » ) instead of P 0 approximation (VOF) for characteristic functions better description of the solid-liquid interface

Other Examples (1) Interface capturing (O. Basset) : • • Navier-Stokes and Levelset + Continuous Galerkin + reinitialization Hamilton-Jacobi Mesh : 1, 499, 405 nodes and 8, 740, 205 elements – 600 time steps – 2 linear systems per increment of 6 millions of unknowns, and of 1. 5 millions, respectively : a total of 4 billion and 500 million of unknowns – CPU time : 5 days and 6 hours on a computational grid with 33 processors

Other Examples (2) Falling Sphere in air and liquid : (O. Basset, L. Silva, R. Valette) NS + Level set + Penalization rs/rl ~ 8 Air Liquid

Other Examples (3) Solid objects in an oven ( C. Gruau) NS + Temperature (convection and diffusion) air Solid

Temperature evolution Hot air exit

Other Examples (4) Flow motion induces by moving bodies : (R. Valette, B. Hiroux) Stoke + VOF + Penalization Shear rate between the two screws