Fluid structure interaction for fluid flow normal to deformable porous media [Elektronische Ressource] / Sabine Muntz

Sujets

• Mathematics
• Physik

Informations

• Publié par: technische_universitat_kaiserslautern
• Publié le: 01 janvier 2009
• Nombre de lectures: 18
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Fluid structure interaction
for fluid flow normal to
deformable porous media
Sabine Muntz
Vom Fachbereich Mathematik der Technischen Universita¨t
Kaiserslautern zur Verleihung des akademischen Grades
Doktor der Naturwissenschaften
(Doctor rerum naturalium, Dr. rer. nat.)
genehmigte Dissertation.
Gutachter:
Prof. Dr. habil. Oleg Iliev
Prof. D.Sc. Svetozar Margenov
Datum der Disputation: 02.12.2008
D 386Acknowledgements
Many of the things you can count, don’t count.
Many of the things you can’t count, really count.
Albert Einstein
At this point I would like to thank all people who contributed to my work in uncountable ways.
First of all, I thank Prof. Dr. habil. Oleg Iliev and Priv.-Doz. Dr. Heiko Andra¨ of the Fraunhofer
1ITWM for supervising my PhD thesis.
I thank Oleg Iliev for introducing me to the interesting field of my research. I am very grateful
for all fruitful discussions and useful advice that helped me gain deeper understanding of the topic
and led me to new ideas. Many thanks to Heiko Andra¨ for productive conversations, for his help
with various problems and especially for the enrichment of my research by his point of view as an
expert in mechanical engineering.
I owe a lot of gratitude to Dr. Anna Naumovich and Dr. Dariusz Niedziela. Many scientific dis-
cussions and helpful advice, regarding the software, helped to accelerate my work.
Furthermore, I thank all my colleagues of the department “Flow and Material Simulation” of the
Fraunhofer ITWM. I appreciated the friendly atmosphere very much.
For the financial support of this work I thank the Fraunhofer Gesellschaft.
2 3Moreover, I would like to express my deepest gratitude to Prof. Quentin Fisher , Prof. Rob Knipe
3and Dr. Simon Harris for introducing me to another interesting application of my PhD research
3topic while I was doing a research fellowship at RDR and the Centre for Computational Fluid
4 5 5Dynamics in Leeds. Many thanks also to Dr. Lionel Elliot , Prof. Derek Ingham and Jayne
3Harnett for taking so good care of me.
This fellowship has been funded by the Marie Curie programme of the European Union.
Last but not least, I would like to thank my family and my friends Martina, Katrin and Emanuel for
their understanding, neverending motivation and for always believing in me. I am very grateful to
my boyfriend Peter for his love, encouragement and endless patience with me, especially during
the last months of my PhD.
1Fraunhofer Institut fu¨r Techno- und Wirtschaftsmathematik, Kaiserslautern, Germany
2School of Earth and Environment, University of Leeds, UK
3Rock Deformation Research Ltd., Leeds, UK
4Faculty of Engineering, University of Leeds, UK
5Department of Applied Mathematics, University of Leeds, UK
3Table of Contents
Introduction and outline 1
1 Governing equations 7
1.1 Equations in free fluid flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Linear elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Equations in porous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Darcy’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 The Kozeny-Carman equation . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.3 The Navier-Stokes-Brinkman system . . . . . . . . . . . . . . . . . . . 13
1.3.4 Total stress in porous media and the concept of effective stress by Terzaghi 14
1.3.5 The computation of effective elastic moduli . . . . . . . . . . . . . . . . 15
1.3.6 The Biot poroelasticity equations . . . . . . . . . . . . . . . . . . . . . 16
1.4 Poroelasticity versus elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 The transmission conditions 19
2.1 The model geometry for fluid flow normal to porous media . . . . . . . . . . . . 19
2.2 Derivation of the transmission conditions . . . . . . . . . . . . . . . . . . . . . 21
2.2.1 The transmission conditions in three dimensions . . . . . . . . . . . . . 24
2.2.2 The transmission conditions in one dimension . . . . . . . . . . . . . . . 27
2.3 The continuous coupled formulation in three dimensions . . . . . . . . . . . . . 28
3 The coupled formulation in one dimension 31
3.1 Stokes and Biot equations in one dimension . . . . . . . . . . . . . . . . . . . . 31
3.2 The continuous coupled formulation . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Consideration of extreme cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 The discrete coupled Biot-Stokes system on matching node-based grids 35
4.1 Discretisation of the Biot poroelasticity system on a node-based grid . . . . . . . 37
4.2 Discretisation of the Stokes equations on a node-based grid . . . . . . . . . . . . 38
4.3 Discretisation of the transmission conditions . . . . . . . . . . . . . . . . . . . . 38
4.4 The discrete coupled Biot-Stokes system . . . . . . . . . . . . . . . . . . . . . . 40
III TABLE OF CONTENTS
5 The discrete coupled Biot-Stokes system on non-matching grids 51
5.1 Discretisation of the Biot poroelasticity system on a staggered grid . . . . . . . . 51
5.2 Discretisation of the Stokes equations on a cell-centred grid . . . . . . . . . . . . 54
5.3 Discretisation of the transmission conditions on non-matching grids . . . . . . . 54
5.4 The discrete coupled Biot-Stokes system . . . . . . . . . . . . . . . . . . . . . . 61
6 Numerical solution for the discrete coupled Biot-Stokes system on non-matching
grids 69
6.1 Domain decomposition methods . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 The iterative solution algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.3 The software for the solution of the separate problems in the subdomains . . . . . 75
6.4 Steady state examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
36.4.1 Example 1: Filter of dimension 2× 3× 3 mm . . . . . . . . . . . . . 76
36.4.2 Example 2: Filter of dimension 1× 5× 5 mm . . . . . . . . . . . . . 82
6.5 Depth filtration examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
36.5.1 Filter of dimension 2× 3× 3 mm . . . . . . . . . . . . . . . . . . . . 88
36.5.2 Filter 1× 5× 5 mm . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.6 Cake filtration examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
36.6.1 Filter of dimension 2× 3× 3 mm (thickness of cake 1 mm) . . . . . . 98
36.6.2 Filter of dimension 1× 5× 5 mm (thickness of cake 1 mm) . . . . . . 98
Summary 103
Notation 105
Bibliography 111Introduction and outline
The interaction of fluid with deformable porous media can be found in many industrial and envi-
ronmental applications, for example in filtration processes, oil production, geomechanics, flooding
simulation, waste water treatment, breakwater design or production of food and beverages. We are
mainly interested in applications in filtration. Filtration processes are mostly characterised by the
performance and the life time of the filter medium. During the filtration process, the properties
of the filter medium change: efficiency, pressure drop, permeability, porosity, mass flux, stresses
and so on. These modifications might lead for example to a clogging of the filter or even to a
destruction. Especially for ceramic filters it happens very often that they break when the stresses
are too high. Of course, this has to be avoided and therefore, it is desirable to predict the filtration
process by simulations, such that the interactions of the fluid flow and the solid skeleton are taken
into account.
The simulation of the interaction of fluid flow with deformable porous media is a challenging task.
There are different physical phenomena that have to be combined: complicated problems like
fluid-structure interaction, flow in deformable porous media and coupling of free fluid and porous
media flow are only subtasks for the complete problem. During the last years, efficient solvers
for the subtasks, mentioned above, have been developed in the department ”Flow and Material
6Simulation” of the Fraunhofer ITWM . Readers, interested in fluid-structure interaction, we refer
to [1], [24], [34] and [44]. Algorithms for the coupling of free flow with rigid porous media and
applications in oil filtration are described in [32], [33], [40] and [68]. A description of a flow
solver for various applications, e.g. non-Newtonian fluids, granular flows, can be found in [47],
[69] and [70]. We refer to [21], [46], [45] and [65] for reports on an efficient algorithm for flows
in deformable porous media.
Domain decomposition (DD) methods are well-established tools for the numerical solution of
3such problems. The main idea is the division of the problem domain Ω ⊂R into finitely many
smaller subdomains Ω ,...,Ω , Ω ⊂ Ω,i = 1,...,n ∈N, where the subproblems can be solved1 n i
independently and to relate the subdomains to each other