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October 14, 2008

October 14, 2008. Scaling Up for Flow in Porous Media, October 13-18, 2008, Dubrovnik. ON NUMERICAL UPSCALING FOR STOKES AND STOKES-BRINKMAN FLOWS. Oleg Iliev , Z.Lakdawala, J.Willems, Fraunhofer Institute for Industrial Mathematics, Kaiserslautern, Germany V.Starikovicius ,

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October 14, 2008

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  1. October 14, 2008 Scaling Up for Flow in Porous Media, October 13-18, 2008, Dubrovnik ON NUMERICAL UPSCALING FOR STOKES AND STOKES-BRINKMAN FLOWS Oleg Iliev, Z.Lakdawala, J.Willems, Fraunhofer Institute for Industrial Mathematics, Kaiserslautern, Germany V.Starikovicius, Vilnius Gediminas Technical University, Lithuania P.Popov, Inst. Scientific Computation, Texas A&M University, USA

  2. Content Motivation and aims Basic solver 3. Multiple scales. Subgrid approach 4. Computer simulations 5. Perspectives

  3. 1. Motivation and aims Motivation and aims

  4. CFD simulations for filtration Main criteria determining the performance of a filter element: Pressure drop – flow rate ratio; Dirt storage capacity; Size of penetrating particles. depend on: microscale (e.g. fibrous geometry, local deposition of particles, etc), and macroscale (e.g., filter element geometry, pressure, velocity distribution, etc.)

  5. Challenges to CFD simulations • Multiple scales (particles, fibres, pleats, ribs, housing,…); • Time-dependent performance; • Shortening the design time and Needs for new design ideas; • Virtual filter element design; • Extensive computational time; • Parameters measurement or calculation (permeability, deposition rate,..) • Validation of the numerical simulation results; • …

  6. (Navier-)Stokes-Brinkmann in fluid and in porous media coupled with concentration of particles (Navier-)Stokes in pore space coupled with stochastic ODE for particles, …. Multiple scales in filtration Particles level Filter components Filter element Complete system Nano scale Micro scaleMillimeterCentimeter Meter Particles-Fiberinteraction Dirt loading ofthe filtering medium Pleats in cartrigefilters Flow within Filter element Filter installation

  7. 2. Basic solver Basic solver

  8. Basic CFD Solver: SuFiS • Grids: Cartesian grid • Finite volume discretization on cell-centred collocated grid • Chorin projection method with implicit treatment of Darcy term • Proper treatment of discontinuous coefficients in pressure- correction equation • Subgridapproach incorporated • Specialized for filtration applications • Paralleization

  9. K can be fixed, or can change due to loading of the filtering medium Macro scale: Flow through fluid and porous regions Momentum Equations Continuity Equation

  10. 3. Multiple scales. Subgrid approach Multiple scales. Subgrid approach

  11. 3. Multiple scales. Subgrid approach • State of the art (Stokes to Darcy; Darcy to Darcy; two-level DD for multiscale) • Microscale to mesoscale upscaling (Stokes to Darcy or to Brinkman) • Mesoscale to macroscale upscaling (Brinkman to Brinkman)

  12. 3. Multiple scales. Known: Upscaling Stokes to Darcy +boundary conditions: • periodic (Sanchez Palencia) • const. velocity (Allaire) • engineering approach

  13. 3. Multiple scales. Known: Darcy to Darcy +boundary conditions: • periodic • linear • presure drop+oscilatory • presure drop+Neumann Note: Some results available for Macroheterogeneous case (block permeability, e.g., Wu, Efendiev,Hou)

  14. 3. Multiple scales. Brinkman to Darcy or Brinkman

  15. Multiple scales. Subgrid approach • Choose a basic grid on which the simulations are possible; • Provide information about the fine geometrical details; • For each grid cell check if it overlaps unresolved fine geometry details • In marked cells (or their agglomeration) solve auxiliary problems on fine grid, and calculate effective permeability tensor; • Solve the modified equations on the chosen grid (the fine details are accounted via the effective permeability). Example of selected location for which effective permeability is calculated

  16. Multiple scales. Subgrid approach Usage of the subgrid approach: • Upscale and solve upscaled equations; • Upscale, solve upscaled equations and prolong the solution to the fine scale; • Iterate over scales (two-level DD with upscaling-based coarse scale operator). Open problems: • No theory for upscaling blocks containing solid, • porous and fluid; • No theory for macroheterogeneous case; • …..

  17. 4. Computer simulations Computer simulations using subgrid approach

  18. 4. Computer simulations Pleated filter, simulations with subgrid approach

  19. 4. Computer simulations

  20. Macro scale: Flow through fluid and porous regions

  21. 5. Perspectives Perspectives

  22. Microscale Macroscale Permeability Particles motion and deposition Navier-Stokes-Brinkman Electrical filed Upscaling Downscaling Stokes Particles concentration Filter elements design Rate of deposition filtration (life time) filtration (clogging) Multiscale Microstructures: www.geodict.com

  23. Thank you www.itwm.fhg.de Fraunhofer ITWM www.dasmod.de Dependable Adaptive Systems and Mathematical Modeling, TU Kaiserslautern

  24. On the Picture: Mass flux through the upper surface of the porous filtering medium • Observation: Inhomogeneus mass flux distribution --> Non-uniform loading of the filter • An aim of simulations: Optimization of the performance of the filter Simulation of 3-D flow through oil filters Simulation of Flow through a Filter: Flow Rate

  25. CFD simulation for filtration Filter Lower part of a filter housing Upper part of a filter housing • Inlet Outlet Filtering medium

  26. CFD simulation for filtration Visualization tool based on PV-4D (ITWM):

  27. Microscale Macroscale efficient solvers filtration (life time) Permeability Stokes filtration (clogging) Elasticity Effective mech. properties Microstructures: • image of samples (tomography) • Design virtual microstructure www.geodict.com • Navier-Stokes-Brinkman • Darcy • Biot (poroelasticity) • Particles concentration • Non-isothermal processes UPSCALING Heat Eqn Thermalconductivity Particles motion Filter elements design Electrical filed Rate of absorption Multiscale

  28. Mathematical model No F Note, permeability, capturing rate, etc.,may depend on time, loading, etc.: Here C is mass concentration of particles, Q is the amount of uploaded particles in the filtering medium etc.

  29. Mathematical model No F Note, permeability, capturing rate, etc.,may depend on time, loading, etc.: Here C is mass concentration of particles, Q is the amount of uploaded particles in the filtering medium etc.

  30. Microscale efficient solvers Permeability Stokes Elasticity Effective mech. properties Microstructures: • image of samples (tomography) • Design virtual microstructure www.geodict.com Homogenization theory for periodic and statistically homogeneous media: Solve cell problem at microscale and use its solution to calculate effective coefficients of the macroscopic equation UPSCALING Heat Eqn Clogged nonwoven Thermalconductivity Fine-to-coarse upscaling

  31. Benefits of CFD simulations

  32. Before: K was constant over all time iterations Macro scale: Flow through fluid and porous regimes Momentum Equations Continuity Equation Now: K changes according to the loading of the porous medium

  33. Filter design simulation • Simulation of complete filter system • Efficient numerics for the coupled system: Navier-Stokes and Darcy/Brinkman • Calculation of pressure distribution, velocities and particle concentrations in the complete filter including the filter media • SuFiS: Filter design software for transmission filters (SPX/IBS Filtran) • Optimization of filter housing geometry and position of stabilizing ribs and filter medium

  34. Macro scale: Flow through fluid and porous regions

  35. Macro scale: Flow through fluid and porous regions

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