ON NUMERICAL UPSCALING FOR STOKES AND ON NUMERICAL UPSCALING FOR - - PowerPoint PPT Presentation

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ON NUMERICAL UPSCALING FOR STOKES AND ON NUMERICAL UPSCALING FOR - - PowerPoint PPT Presentation

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Scaling Up for Flow in Porous Media, October 13-18, 2008, Dubrovnik ON NUMERICAL UPSCALING FOR STOKES AND ON NUMERICAL UPSCALING FOR STOKES AND STOKES- -BRINKMAN FLOWS BRINKMAN FLOWS STOKES Oleg Iliev , Z.Lakdawala, J.Willems, Fraunhofer

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SLIDE 1

October 14, 2008

ON NUMERICAL UPSCALING FOR STOKES AND ON NUMERICAL UPSCALING FOR STOKES AND STOKES STOKES-

  • BRINKMAN FLOWS

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

Scaling Up for Flow in Porous Media, October 13-18, 2008, Dubrovnik

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Content Content

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

Motivation and aims Motivation and aims

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SLIDE 4

Main criteria determining the performance of a filter element: Main criteria determining the performance of a filter element: Main criteria determining the performance of a filter element: Main criteria determining the performance of a filter element: Main criteria determining the performance of a filter element: Main criteria determining the performance of a filter element: Main criteria determining the performance of a filter element: Main criteria determining the performance of a filter element: 1) 1) 1) 1) Pressure drop Pressure drop Pressure drop Pressure drop – – – – flow rate ratio; flow rate ratio; flow rate ratio; flow rate ratio; 1) 1) 1) 1) Dirt storage capacity; Dirt storage capacity; Dirt storage capacity; Dirt storage capacity; 1) 1) 1) 1) Size of penetrating particles. Size of penetrating particles. Size of penetrating particles. Size of penetrating particles.

depend on: microscale microscale microscale microscale (e.g. fibrous geometry local deposition of particles, etc), and macroscale macroscale macroscale macroscale (e.g., filter element geometry, pressure, velocity distribution, etc.)

CFD simulations for filtration CFD simulations for filtration

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SLIDE 5

Challenges to CFD simulations Challenges to CFD simulations

  • Multiple scales (particles, fibres, pleats, ribs, housing,

Multiple scales (particles, fibres, pleats, ribs, housing,… …); );

  • Time

Time-

  • dependent performance;

dependent performance;

  • Shortening the design time and Needs for new design ideas;

Shortening the design time and Needs for new design ideas;

  • Virtual filter element design;

Virtual filter element design;

  • Extensive computational time;

Extensive computational time;

  • Parameters measurement or calculation (permeability, deposition

Parameters measurement or calculation (permeability, deposition rate,..) rate,..)

  • Validation of the numerical simulation results;

Validation of the numerical simulation results;

  • …

…

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SLIDE 6

Multiple scales in filtration

Particles level Filter components Filter element Complete system Nano scale Micro scale Millimeter Centimeter Meter

(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, ….

Filter installation Particles-Fiber interaction Dirt loading of the filtering medium Flow within Filter element Pleats in cartrige filters

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SLIDE 7
  • 2. Basic solver
  • 2. Basic solver

Basic solver Basic solver

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SLIDE 8

Basic CFD solver: Basic CFD solver: SuFiS 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 Subgrid approach incorporated Specialized for filtration applications

Paralleization

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SLIDE 9

Macro scale: Flow through fluid and porous regions

1

( ) ( , )

Darcy Navier Stokes

u u u u K u p f t µ ρ µ

− −

∂ −∇⋅ ∇ + ∇ + + ∇ = ∂

  • ɶ
  • u

∇⋅ =

  • Momentum

Equations Continuity Equation K can be fixed, or can change due to loading of the filtering medium

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SLIDE 10
  • 3. Multiple scales.
  • 3. Multiple scales. Subgrid

Subgrid approach approach

Multiple scales. Multiple scales. Subgrid Subgrid approach approach

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SLIDE 11
  • 3. Multiple scales.
  • 3. Multiple scales. Subgrid

Subgrid approach approach

  • State of the art (Stokes to Darcy; Darcy to Darcy; two

State of the art (Stokes to Darcy; Darcy to Darcy; two-

  • level DD for

level DD for multiscale multiscale) )

  • Microscale

Microscale to to mesoscale mesoscale upscaling upscaling (Stokes to Darcy or to Brinkman (Stokes to Darcy or to Brinkman

  • Mesoscale

Mesoscale to to macroscale macroscale upscaling upscaling (Brinkman to Brinkman) (Brinkman to Brinkman)

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SLIDE 12

u p u ν − ∆ = ∇ ∇ = i

1

K u p u µ

−

− = ∇ ∇ = i

  • 3. Multiple scales. Known:
  • 3. Multiple scales. Known: Upscaling

Upscaling Stokes to Darcy Stokes to Darcy +boundary conditions: +boundary conditions:

  • periodic (Sanchez Palencia)

periodic (Sanchez Palencia)

  • const. velocity (
  • const. velocity (Allaire

Allaire) )

  • engineering approach

engineering approach

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SLIDE 13

1

K u p u µ

−

− = ∇ ∇ = ɶ i

1

K u p u µ

−

− = ∇ ∇ = i

  • 3. Multiple scales. Known: Darcy to Darcy
  • 3. Multiple scales. Known: Darcy to Darcy

+boundary conditions: +boundary conditions:

  • periodic

periodic

  • linear

linear

  • presure

presure drop+oscilatory drop+oscilatory

  • presure

presure drop+Neumann drop+Neumann Note: Note: Some results available for Some results available for Macroheterogeneous Macroheterogeneous case case (block permeability, (block permeability, e.g., Wu, e.g., Wu, Efendiev,Hou Efendiev,Hou) )

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SLIDE 14
  • 3. Multiple scales. Brinkman to Darcy or Brinkman
  • 3. Multiple scales. Brinkman to Darcy or Brinkman
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Multiple scales. Multiple scales. Subgrid Subgrid approach approach

  • Choose a basic grid on which the simulations are possible;

Choose a basic grid on which the simulations are possible;

  • Provide information about the fine geometrical details;

Provide information about the fine geometrical details;

  • For each grid cell check if it overlaps unresolved fine geometr

For each grid cell check if it overlaps unresolved fine geometry y details details

  • In marked cells (or their agglomeration) solve auxiliary proble

In marked cells (or their agglomeration) solve auxiliary problems ms

  • n fine grid, and calculate effective permeability tensor;
  • n fine grid, and calculate effective permeability tensor;
  • Solve the modified equations on the chosen grid (the fine detai

Solve the modified equations on the chosen grid (the fine details ls are accounted via the effective permeability). are accounted via the effective permeability). Example of selected location for which effective permeability is calculated

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Multiple scales. Multiple scales. Subgrid Subgrid approach approach Usage of the Usage of the subgrid subgrid approach: approach:

  • Upscale and solve

Upscale and solve upscaled upscaled equations; equations;

  • Upscale, solve

Upscale, solve upscaled upscaled equations and prolong the solution to equations and prolong the solution to the fine scale; the fine scale;

  • Iterate over scales (two

Iterate over scales (two-

  • level DD with

level DD with upscaling upscaling-

  • based coarse

based coarse scale operator). scale operator). Open problems: Open problems:

  • No theory for

No theory for upscaling upscaling blocks containing solid, blocks containing solid, porous and fluid; porous and fluid;

  • No theory for

No theory for macroheterogeneous macroheterogeneous case; case;

  • …

….. ..

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SLIDE 17
  • 4. Computer simulations
  • 4. Computer simulations

Computer simulations using Computer simulations using subgrid subgrid approach approach

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SLIDE 18
  • 4. Computer simulations
  • 4. Computer simulations

Pleated filter, simulations with Pleated filter, simulations with subgrid subgrid approach approach

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SLIDE 19
  • 4. Computer simulations
  • 4. Computer simulations
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SLIDE 20

Macro scale: Flow through fluid and porous regions

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SLIDE 21
  • 5. Perspectives
  • 5. Perspectives

Perspectives Perspectives

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SLIDE 22

Multiscale

Macroscale

Microstructures: www.geodict.com

filtration (life time) Stokes Particles motion and deposition Electrical filed Permeability Rate of deposition

Navier-Stokes-Brinkman

filtration (clogging) Filter elements design

Particles concentration

Upscaling Upscaling Downscaling Downscaling

Microscale

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SLIDE 23

Thank you Thank you

www.itwm.fhg.de

Fraunhofer ITWM

www.dasmod.de

Dependable Adaptive Systems and Mathematical Modeling, TU Kaiserslautern