COMPUTATION OF THE PERMEABILITY OF MULTI-SCALE POROUS … · 2010. 11. 6. · Computation of the...

s KATHOLIEKE UNIVERSITEIT LEUVEN

FACULTEIT INGENIEURSWETENSCHAPPEN

DEPARTEMENT COMPUTERWETENSCHAPPEN

AFDELING NUMERIEKE ANALYSE EN

TOEGEPASTE WISKUNDE

Celestijnenlaan 200A – B-3001 Leuven

COMPUTATION OF THE PERMEABILITY OF

MULTI-SCALE POROUS MEDIA WITH APPLICATION TO

TECHNICAL TEXTILES

Promotoren:

Prof. dr. ir. D. Roose

Prof. dr. S.V. Lomov

Proefschrift voorgedragen tot

het behalen van het doctoraat

in de ingenieurswetenschappen

door

Bart VERLEYE

Maart 2008

s KATHOLIEKE UNIVERSITEIT LEUVEN

FACULTEIT INGENIEURSWETENSCHAPPEN

DEPARTEMENT COMPUTERWETENSCHAPPEN

AFDELING NUMERIEKE ANALYSE EN

TOEGEPASTE WISKUNDE

Celestijnenlaan 200A – B-3001 Leuven

BEREKENING VAN DE PERMEABILITEIT VAN

MEERSCHALIGE POREUZE MATERIE TOEGEPAST OP

TECHNISCH TEXTIEL

Jury:

Prof. dr. ir. E. Aernoudt, voorzitter

Prof. dr. ir. D. Roose, promotor

Prof. dr. S.V. Lomov, promotor

Prof. dr. ir. M. Baelmans

Prof. dr. M. Griebel (Univ. Bonn)

Prof. dr. A. Long (Univ. of Nottingham)

Prof. dr. ir. S. Vandewalle

Proefschrift voorgedragen tot

het behalen van het doctoraat

in de ingenieurswetenschappen

door

Bart VERLEYE

U.D.C. 681.3*I6

Maart 2008

c© Katholieke Universiteit Leuven — Faculteit IngenieurswetenschappenArenbergkasteel, B-3001 Leuven, Belgie

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D/2008/7515/22

ISBN 978-90-5682-913-1

Computation of the permeability of

multi-scale porous media with application totechnical textiles

Bart Verleye

K.U.Leuven, Departement Computerwetenschappen

Celestijnenlaan 200A, B-3001 Leuven, Belgie

Abstract

Technical textiles are used as reinforcement in composite materials. Thepermeability of the textiles is an important input parameter for the simula-tion of the impregnation stage of the Liquid Composite Moulding process,an often used production technique for composite materials.

Textile reinforcements are multi-scale porous structures and the permeabil-ity is considered on the different scales. The fibres inside the yarns determinethe micro-scale properties of the textile. On the meso-scale level, we con-sider a unit cell of the textile, which has the same average permeability asthe macro-scale textile layer.

We present a fast and accurate method to compute the permeability oftextile reinforcements, based on the finite difference discretisation of theStokes equations. The input for the CFD simulations (Computational FluidDynamics), is a unit cell of the textile model, provided by textile modellingsoftware like WiseTex and TexGen. If intra-yarn flow is taken into account,the Stokes equations are locally extended with a penalty term, resulting inthe Brinkman equation. The penalty is computed analytically based on theproperties of the fibres inside the yarn. We compare different formulas tocompute this penalty.

iv

We validate the results of our simulations with permeability values obtainedby experiments. Results for woven fabrics, random structures and non-crimp fabrics are presented. Moreover, experimental and computationalresults are compared for a structure that is designed to minimise the exper-imental errors. The influence of shear and nesting on the permeability oftextiles is discussed.

To avoid the 3D-simulation problem, different methods that reduce the di-mension of the problem are presented in literature. The Grid2D methodreduces the 3D Stokes problem to a 2D Darcy problem. We explore thismethod, and compare the results with our CFD computations.

A second model-reduction method is the pore-network method. Thismethod describes the porous medium as a network of pores, and computesthe overall permeability via the law of conservation. The Stokes solver isapplied to compute the conductivity of the pores of the network. We explainhow our Stokes solver is used for this method. We also compare the resultsof the method with CFD computed values for samples of porous rock.

Berekening van de permeabiliteit van

meerschalige poreuze materie toegepast optechnisch textiel

Bart Verleye

K.U.Leuven, Departement Computerwetenschappen

Celestijnenlaan 200A, B-3001 Leuven, Belgie

Korte samenvatting

Technisch textiel worden gebruikt als versterking in composietmaterialen.De doorlaatbaarheid of permeabiliteit van dit textiel is een belangrijke inputvoor de simulatie van het impregnatie onderdeel van het Vloeibaar Compo-sietmouleren (Liquid Composite Moulding), een veel gebruikte productie-techniek voor composietmaterialen.

Technisch textiel is een meerschalige poreuze structuur, en dus bekijken wede permeabiliteit op verschillende niveaus. De vezels van een draad bepalende structuur op het microscopisch niveau. Een eenheidscel van het textiel,een deel van het textiel dat dezelfde permeabiliteit heeft als het volledigestuk, bepaalt het middenniveau.

In dit werk introduceren we een snelle en nauwkeurige methode voor de be-rekening van de permeabiliteit van textiel. De methode is gebaseerd op deeindige differentie discretisatie van de Stokes vergelijkingen. De input voordeze RV-simulaties (Rekenkundige Vloeistofdynamica), is een eenheidscelvan het textielmodel dat gemodelleerd is met programma’s om textielmo-dellen te ontwerpen zoals bijvoorbeeld WiseTex of TexGen. Als ook destroming in de draden zelf in rekening moet gebracht worden, dan worden

vi

de Stokes vergelijkingen lokaal uitgebreid met een strafterm, wat resulteertin de Brinkman vergelijkingen. De grootte van de straf kan analytisch be-paald worden aan de hand van de lokale eigenschappen van de vezels. Wevergelijken in dit werk verschillende analytische formules voor de berekeningvan deze strafterm.

We vergelijken de resultaten van onze berekeningen met experimenteel be-paalde permeabiliteitswaarden, voor geweven textiel, willekeurige-structuurtextiel en textiel met ongebogen draad. Bovendien stellen we de resultatenvoor van berekeningen op een speciaal ontworpen structuur. Experimentenop deze structuur hebben minder last van spreiding op de resultaten dan ex-perimenten op textiel. We bespreken ook nog de invloed van vervormingenzoals afschuiving en nesten van verschillende lagen, op de permeabiliteit.

Er bestaan verschillende methodes die de permeabiliteit van poreus mate-riaal schatten, zonder een 3D-RV simulatie. De Grid2D methode bijvoor-beeld, reduceert het 3D Stokes probleem tot een 2D Darcy probleem. Indeze thesis onderzoeken we deze methode, en vergelijken de resultaten metonze RV-berekeningen.

Een tweede methode die de probleemstelling reduceert, is de porie-netwerkmethode. Deze methode beschrijft het poreus materiaal als een netwerk vanporieen, en berekent dan de permeabiliteit aan de hand van de wetten vanbehoud. We leggen uit hoe ons Stokes programma gebruikt wordt in dezecontext aan de hand van de berekening van de permeabiliteit van poreusgesteente.

Woord voorafVorwortPreface

De voorbije viereneenhalf jaar had ik het geluk met vele personen samente kunnen werken. Dit doctoraat is niet het resultaat van eenzaam onder-zoek, maar van een vruchtbare samenwerking tussen personen met een ver-schillende achtergrond en uit verscheidene onderzoeksgroepen. Deze samen-werking was voor mij een groot plezier, en zonder deze waren de resultatenbeschreven in dit proefschrift minder talrijk geweest. Een woord van dankis dus op zijn plaats.

Dirk, dank je wel om mij vier jaar geleden de kans te geven om aan eendoctoraatsproject te beginnen en de vele kansen die je me hebt gegeventijdens het onderzoek. Begeleiding, conferenties, buitenlandse verblijven,bijscholingen, . . . , het was nooit een probleem, intengendeel.

Stepan, bedankt voor de interessante discussies, inhoudelijke inbreng enaangename momenten tijdens de conferenties.

Die Monate die ich in Bonn verbracht habe, sind fur mich und meineForschung sehr wichtig gewesen. Michael Griebel, ich mochte mich gernebedanken fur die haufigen Einladungen um gemeisam mit Ihnen, in Bonn,im Institut fur Numerische Simutation zu arbeiten. Dabei hatte ich auchdie Moglichkeit zusammen mit Roberto Croce, Martin Engel und vor allemMargrit Klitz arbeiten zu konnen. Margrit, danke fur die interessante, an-genehme und hauffige Zusammenarbeit.

I would like to express my gratitude to Andrew Long, for the opportu-nity to spend three months conducting research in the Polymer CompositesGroup at the University of Nottingham under his direction, and for manyinteresting discussions.

I would also like thank Jon & Rachel, Martin and Sophie to make my stayin Nottingham enjoyable.

De voltallige jury, waarvan ik Stefan Vandewalle en Tine Baelmans nog nietheb vernoemd, bedank ik voor het nauwkeurig lezen van het proefschrift;eventuele toekomstige lezers en ikzelf hebben er veel aan. Ook dank ik devoorzitter Etienne Aernoudt.

Ook in Vlaanderen en Nederlanstalig Brussel heb ik met mensen samen-gewerkt. Frederik Desplentere bracht mij de beginselen van de varieteitbij. Gerd Morren en Philippe Van Marcke zijn echte aanraders als collega,mannen, merci he.

Duvel is het beste bier van de wereld, en al wie dat beaamt is – zeker naer enkele gedronken te hebben – een vriend. En zo zijn er vele, te veel omallemaal op te noemen. Maar ze hebben allemaal bijgedragen aan dit doc-toraat, want zonder de leuke momenten samen had ik de serieuze momentenniet kunnen verwerken.

Enkele ‘groepen’ van mensen wil ik toch speciaal vernoemen. Er zijn deToffe Mensen en de Strandvolleyballers. Er zijn de koffiedrinkcollega’s Ares,Dirk, Yves, Sven, Steven. Er zijn aangenamebureaugenoten Hendrik, Kris.Er zijn de samenleefmensen Joris, Jan, Adriaan, Wim, Dirk, Jan Frederik,Johan, Stefan, Roel, Christophe, Pieter. Es gibt die Deutschen Andi, Woj-cieh, Gunar, Sabrina, David, Daniel. Er is Liesbet. Buiten categorie.

Mama, papa, Els & Pieter, Leen, Ann, bedankt voor de onvoorwaardelijkesteun en positieve invloed op mij. Wouter en Ruben, bedankt om zo’n leukeneefjes te zijn.

Bart (bartje)Maart 2008

Acknowledgement

This research is part of the IWT-GBOU-project (Flemish government, Bel-gium): Predictive tools for permeability, mechanical and electro-magnetic

properties of fibrous assemblies: modelling, simulations and experimental

verification.

My visits in Bonn were supported in part by the Sonderforschungsbe-reich 611 Singulare Phanomene und Skalierung in Mathematischen Modellen

sponsored by the Deutsche Forschungsgemeinschaft.

ix

Contents

Woord vooraf vii

Acknowledgement ix

Contents xi

List of symbols xv

Nederlandse samenvatting NL.1

1 Introduction 1

1.1 Multi-scale modelling . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Problem statement and goals . . . . . . . . . . . . . . . . . . 4

1.3.1 Permeability of textile reinforcements . . . . . . . . . 5

1.3.2 Permeability of porous rock . . . . . . . . . . . . . . . 5

1.4 Input for the simulations . . . . . . . . . . . . . . . . . . . . . 7

1.4.1 Textile modelling . . . . . . . . . . . . . . . . . . . . . 7

1.4.2 X-ray computed tomography . . . . . . . . . . . . . . 8

1.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Permeability of porous media: definition 13

xi

xii CONTENTS

2.1 Definition of a porous medium . . . . . . . . . . . . . . . . . 13

2.2 Permeability at different length scales . . . . . . . . . . . . . 15

2.3 Darcy’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Mathematical homogenisation . . . . . . . . . . . . . . . . . . 18

2.5 Volume averaging . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6 Properties of the permeability tensor . . . . . . . . . . . . . . 23

2.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Solution of the Navier-Stokes and the Stokes equations 31

3.1 The (Navier-)Stokes equations . . . . . . . . . . . . . . . . . . 32

3.2 Dimension analysis . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Requirements for the solver . . . . . . . . . . . . . . . . . . . 36

3.4 Direct solution . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4.1 The Navier-Stokes solver . . . . . . . . . . . . . . . . . 38

3.4.2 The Stokes solver . . . . . . . . . . . . . . . . . . . . . 44

3.5 Boundary conditions for sheared unit cells . . . . . . . . . . . 45

3.6 Solution with a lattice Boltzmann simulation . . . . . . . . . 48

3.7 Calculation of the permeability . . . . . . . . . . . . . . . . . 51

3.8 Developed software package . . . . . . . . . . . . . . . . . . . 51

3.9 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . 52

3.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Computation of the intra-yarn flow 59

4.1 Representation of the yarns . . . . . . . . . . . . . . . . . . . 60

4.2 Computation of the local permeability . . . . . . . . . . . . . 61

4.3 Stokes-Darcy-Brinkman . . . . . . . . . . . . . . . . . . . . . 63

4.4 Homogenisation . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.5 Influence on the meso-scale permeability . . . . . . . . . . . . 68

xiii

4.6 Influence on the implementation and the computation . . . . 70

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 Experimental validation with textile reinforcements 73

5.1 Resin Transfer Moulding . . . . . . . . . . . . . . . . . . . . . 74

5.2 Experimental set-ups . . . . . . . . . . . . . . . . . . . . . . . 75

5.3 Interpretation of permeability values . . . . . . . . . . . . . . 77

5.3.1 Experimental errors . . . . . . . . . . . . . . . . . . . 77

5.3.2 Modelling issues . . . . . . . . . . . . . . . . . . . . . 78

5.3.3 Flow model and numerical errors . . . . . . . . . . . . 79

5.4 Stereo-Lithographic specimen . . . . . . . . . . . . . . . . . . 80

5.4.1 The stereo-lithographic production process . . . . . . 81

5.4.2 The design . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4.3 The results . . . . . . . . . . . . . . . . . . . . . . . . 83

5.5 Validation on textiles . . . . . . . . . . . . . . . . . . . . . . . 88

5.5.1 Basket woven monofilament fabric . . . . . . . . . . . 88

5.5.2 Plain-Woven Textile . . . . . . . . . . . . . . . . . . . 92

5.5.3 Non-Crimp Fabrics . . . . . . . . . . . . . . . . . . . . 94

5.5.4 Non-woven structures - Random mats . . . . . . . . . 96

5.6 Influence of Shear . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.7 Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.7.1 Influence of the variability on the meso-scale perme-ability . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.7.2 Correlation . . . . . . . . . . . . . . . . . . . . . . . . 104

5.7.3 Influence of the variability on the macro-scale . . . . . 105

5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6 Assessment of the Grid2D method 109

6.1 The methodology . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2 Calculation of the local permeability . . . . . . . . . . . . . . 111

xiv CONTENTS

6.3 Solution of Darcy’s law . . . . . . . . . . . . . . . . . . . . . . 114

6.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.4.1 Artificial structures . . . . . . . . . . . . . . . . . . . . 115

6.4.2 Textiles . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.4.3 Stereo-Lithographic specimen . . . . . . . . . . . . . . 117

6.4.4 Sheared textiles . . . . . . . . . . . . . . . . . . . . . . 119

6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7 Adaptation of the pore network model 125

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.2 The pore network model . . . . . . . . . . . . . . . . . . . . . 126

7.2.1 Construction of the pore network . . . . . . . . . . . . 126

7.2.2 Computing the local conductivities in the network . . 129

7.3 Validation and conclusion . . . . . . . . . . . . . . . . . . . . 135

8 Conclusions 137

A FlowTex GUI 143

B Volume averaging of the momentum equation 145

C Data Mono-Filament Fabric 151

D Data of the non-crimp fabrics 153

Bibliography 159

Curriculum Vitae 169

List of symbols

This list is not a complete list of the symbols used in this text, as some sym-bols are only used locally. These symbols are not mentioned here. Upper-case boldface letters are tensors, small-case boldface letters are vectors,small-case letters are scalars.

〈〉 volume average, p. 18/ quotient set|Y | volume of Y: Frobenius inner product· scalar productα anisotropy, p. 88aǫ micro-scale obstacle size, p. 14β principal direction, p. 24

∆f Laplacian of f: ∂2f∂x2 + ∂2f

∂y2 + ∂2f∂z2

x discrete step length in the X-directionε characteristic length of the micro-scale, p. 14εp precision of the solution of linear systems, p. 45f external body force, p. 17Fr Froude number, p. 32h steplength, otherwise hydraulic head, p. 17κ stabilisation parameter, p. 44Kerror stop criterion for the permeability

K permeability tensor

Kxx Kxy Kxz

Kyx Kyy Kyz

Kzx Kzy Kzz

l characteristic length of the meso-scaleL characteristic length of the computational domain, p. 32Lm characteristic length of the macro-scale, p. 15λ segmentation parameter, p. 129

xv

xvi LIST OF SYMBOLS

µ dynamic viscosityN(ε) number of cells in the macroscopic domain, p. 14n outward pointing unit vector

∇f gradient of f:[

∂f∂x ,

∂f∂y ,

∂f∂z

]

∇u

∂u1

∂x∂u2

∂x∂u3

∂x∂u1

∂y∂u2

∂y∂u3

∂y∂u1

∂z∂u2

∂z∂u3

∂z

∇ · u divergence of u: ∂u1

∂x + ∂u2

∂y + ∂u3

∂z

Ω macroscopic domain, p. 14Ωε macroscopic fluid domain, p. 14p pressurep∞ constant related to the pressure, p. 32φ porosity, p. 15q fluxr radiusρ densityρ∞ constant related to the density, p. 32Re Reynolds number, p. 32τ shear stress exerted by the fluid, p. 32θ shear angle, p. 45Γ1 fluid-solid boundary in a unit cell, p. 15Γ2 unit cell boundary, p. 15u velocityu∞ constant related to the velocity, p. 32Vf fluid volume, otherwise volume fraction, p. 61Y a scaled unit cell of the porous medium, p. 14Y ǫ

i a unit cell of the porous medium, p. 14YS the solid part of Y , p. 14YF the fluid part of Y , p. 14Y ǫ

Sithe solid part of Y ǫ

i , p. 14Y ǫ

Fithe fluid part of Y ǫ

i , p. 14x position vector

Nederlandse samenvatting

Berekening van de permeabiliteit vanmeerschalige poreuze materie toegepast op

technisch textiel

NL.1

Inhoudsopgave

1 Inleiding NL.3

2 Definitie van de permeabiliteit van een poreus medium NL.6

2.1 De wet van Darcy . . . . . . . . . . . . . . . . . . . . . . NL.7

2.2 Eigenschappen van de permeabiliteitstensor . . . . . . . . NL.9

3 Het oplossen van de Navier-Stokes- en de Stokesvergelij-kingen NL.9

3.1 Vereisten van de oplosser . . . . . . . . . . . . . . . . . . NL.10

3.2 De Stokesoplosser . . . . . . . . . . . . . . . . . . . . . . . NL.10

3.3 Resultaten . . . . . . . . . . . . . . . . . . . . . . . . . . . NL.11

4 Intra-draadstroming NL.11

5 Experimentele validatie NL.13

6 De 2D-roostermethode NL.14

7 De porie-netwerkmethode NL.15

8 Besluit NL.18

NL.3

1 Inleiding

Deze thesis behandelt de berekening van de doorlaatbaarheid of permea-biliteit van een poreus medium. Permeabiliteit is de constante verhoudingtussen de gemiddelde snelheid van een vloeistof die stroomt doorheen hetporeuze medium, en de kracht die de stroming veroorzaakt. Deze constantewordt beschreven door de wet van Darcy. De wet van Darcy is een gehomo-geniseerde vergelijking, die de eigenschappen van het poreuze medium opmicroniveau middelt.

Met dit onderzoek introduceren we een snelle en algemeen toepasbare me-thode voor de berekening van de permeabiliteit van een poreus medium, metals invoer een eerste orde benadering van de geometrie van het medium. Webekijken de snelheid van onze methode, en vergelijken de berekende perme-abiliteit met de resultaten van experimenten voor technische textiel. Voorde berekening van de permeabiliteit van gesteente leggen we uit hoe onzemethode werd ingebed in de implementatie van een porie-netwerkmethode.

Stand van zaken Tabel 1 geeft een overzicht van de huidige stand van hetonderzoek in dit domein. Er zijn methodes die een 3D stromingssimulatievermijden door het probleem eenvoudiger te formuleren. Veel van dezemethodes zijn echter nog niet voldoende gevalideerd, of zijn onbevredigendvoor reele, complexe structuren. Nauwkeurige simulaties zijn mogelijk, metbijvoorbeeld eindige elementen (EE) of eindige volume (EV) methodes ofmet een rooster-Boltzmannsimulatie.

Het nadeel van een EE- of EV-simulatie op een ongestructureerd rooster, isdat deze methodes een hoge kwaliteit eisen van het rooster. Het genererenvan een voldoende kwalitatief rooster is moeilijk en vraagt interactie vande gebruiker, zeker voor compacte textielgeometrien waarbij er veel scherpehoeken tussen de draden aanwezig zijn. Het is echter ons uitdrukkelijkdoel om een volledig automatische methode te ontwikkelen. Een rooster-Boltzmannoplosser die rekent op een regelmatig rooster, om de moeilijkeroostergeneratie te vermijden, is beschikbaar, maar onvoldoende snel vooronze toepassingen.

Probleem- en doelstelling De bedoeling van het onderzoek beschrevenin dit proefschrift is de ontwikkeling van een snel en automatisch programmavoor de berekening van de permeabiliteit van poreuze materie. Daarenbo-ven moet het programma gebruikt kunnen worden voor alle verschillendesoorten textiel. De methode die we daarvoor gebruiken is gebaseerd op deeindige differentie van de Stokesvergelijkingen op een regelmatig, rechthoe-

NL.4 NEDERLANDSE SAMENVATTING

Tabel 1: Overzicht van de bestaande methodes voor de berekening vanpermeabiliteit.

Methode Ref. Opmerking

Theoretische formules[15, 20, 39,85]

Onnauwkeurig voortextiel

EE/EV model met ongestruc-tureerde roosters

[53, 87, 95]Roostergeneratie ismoeilijk

Rooster-Boltzmannmodel [14]Weinig beschikbareversnellingstechnieken

2D-Roostermethode [110, 111]Weinig tot geen valida-tie

Porie-netwerk model[17, 27, 31,34]

Geen bevredigende va-lidatie

Methode van het willekeurigpad

[47, 91]Geen bevredigende va-lidatie

ED modellering[Dit proef-schrift]

kig rooster.

De permeabiliteit van textiel is belangrijk als invoer voor de simulatie vanhet vloeibaar composietmouleren (Liquid Composite Moulding), een pro-ductieproces voor composietmaterialen. Deze simulaties gebeuren met zo-genaamde Darcyoplossers, zoals bijvoorbeeld PAM-RMT of LIMS [4, 5]. Depermeabiliteit is sterk afhankelijk van kleine variaties in de textielgeometrie,en de Darcy oplossers eisen een correcte invoer van deze afhankelijk. Daar-om is het belangrijk dat de oplosser snel is, zodat er berekeningen gedaankunnen worden op verschillende variaties van de gemiddelde eenheidscel,en ook dat de oplosser nauwkeurig is, zodat de verandering van de per-meabiliteit in functie van de verandering van de geometrie juist berekendwordt.

In ingenieurswetenschappen zoals de exploratie van gas, of de bouwkunde,is er grote interesse in de permeabiliteit van gesteente. Aangezien het reken-kundige domein van een staal van het gesteente veel groter is dan een staalvan textiel, maar de details van de geometrie even klein, passen we hier eenandere methode toe om de permeabiliteit te berekenen. Voor dit probleemwordt een porie-netwerkmethode gebruikt, waar onze Stokesoplosser deel

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van uitmaakt.

Brinkman punt (Navier-)Stokes punt

Figuur 1: Een textielmodel gemodelleerd met WiseTex (links), en een 2Ddoorsnede samen met een (grove) eerste orde benadering of voxelbeschrij-ving.

Invoer voor de simulaties De invoer voor de berekening van de per-meabiliteit van textiel wordt aangemaakt met textielmodelleringssoftwarezoals WiseTex [103] en TexGen [87] (figuur 1). Textiel wordt als een perio-dieke structuur beschouwd, en dus volstaat een eenheidscel van de geome-trie. Aangezien we berekeningen uitvoeren op een regelmatig, rechthoekigrooster, hebben we enkel een eerste orde beschrijving nodig van het tex-tiel. Daarom exporteren textielmodelleringsprogramma’s de textielgeome-trie naar een zogenaamde voxelbeschrijving. Dit houdt in dat er een 3Drooster gedefinieerd wordt in de eenheidscel, en er voor elk punt bijgehou-den wordt of het punt in een draad of in de vrije ruimte tussen de dradenligt (figuur 1).

Een tweede manier om een voxelbeschrijving van een poreus medium tebekomen is computertomografie, waarbij X-stralen gebruikt worden om eenbeeld van de geometrie te maken.

Overzicht van het proefschrift In hoofdstuk twee gaan we dieper inop de vergelijkingen die opgelost worden om de permeabiliteit te berekenen.Daarbij beschouwen we drie verschillende niveaus, nl. het microscopische,mesoscopische en het macroscopische. De vergelijkingen die gelden op hetmicroscopische niveau worden gemiddeld en gehomogeniseerd tot vergelij-kingen op het macroscopische niveau.

Hoofdstuk drie beschrijft de oplossingsmethode van de Navier-Stokes enStokes vergelijkingen, waarbij de stroming in de draden zelf verwaarloosd


ε

εaε

Ωε

ε

Ω

Figuur 2: Een wiskundige beschrijving van een poreus medium [50].

wordt. Hoofdstuk vier handelt over een eenvoudige manier om deze intra-draadstroming mee in de berekeningen te betrekken.

Hoofdstuk vijf geeft een overzicht van de experimentele validatie van onzenumerieke methode. De resultaten van simulaties worden vergeleken metexperimenteel bepaalde permeabiliteiten, en dat voor verschillende soortentextiel.

Hoofdstukken zes en zeven behandelen twee verschillende methodes die de3D stromingssimulatie omzetten in een eenvoudiger probleem om tijdroven-de berekeningen te vermijden.

De besluiten en resultaten staan beschreven in hoofdstuk acht.

2 Definitie van de permeabiliteit van een po-

reus medium

Een wiskundige beschrijving van een poreus medium wordt voorgesteld infiguur 2. Het domein Ω is opgedeeld in eenheidscellen met afmeting ǫ, enelke cel bevat een obstakel met afmeting aǫ. De vrije ruimte in Ω, de ruimtetussen de obstakels waar de vloeistof kan stromen, is Ωǫ.

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2.1 De wet van Darcy

De wet van Darcy zegt dat de gemiddelde flux q van een vloeistof die stroomtdoorheen een poreus medium, een constante verhouding heeft tot de krachtdie de vloeistof doet stromen [24],

q =K

µ(f −∇p)

[m

s

]

=

[m2]

[kg

m s

]

[kg

m2 s2

]

. (1)

Deze constante wordt de permeabiliteit K genoemd. In deze vergelijkingbetekent µ de dynamische viscositeit, p is de druk, en f is een algemenedrijvende kracht. De wet van Darcy is empirisch vastgesteld, door op temerken dat de snelheid van water dat stroomt doorheen een zandbak inverhouding staat tot het hoogteverschil van de zandbak [56].

De wet van Darcy kan ook wiskundige gefundeerd worden. In de volgendeparagrafen bespreken we daartoe twee technieken, namelijk het homogenise-ren [50, 74] en het volumetrisch middelen [109]. Beide methodes vertrekkenvan een beschrijving van de stroming op het microscopische niveau, om dande details op dat niveau te doen verdwijnen zodat er een vergelijking ophet macroscopisch niveau ontstaat die de details niet bevat, maar wel deeigenschappen van het medium beschrijft.

Wiskundige homogenisering We gaan ervan uit dat de stroming ophet microscopische niveau voldoet aan de Stokes vergelijkingen,

Pǫ

∇pǫ − µ∆uǫ = f in Ωǫ

∇ · uǫ = 0 in Ωǫ

uǫ = 0 op ∂Ωǫ\∂Ω

een randvoorwaarde op ∂Ω.

. (2)

Deze vergelijkingen hangen af van de parameter ǫ (figuur 2). Door dezeparameter te varieren krijgen we dus een set van vergelijkingen Pǫ die geldenop domeinen Ωǫ. Wiskundige homogenisering bekijkt nu de limiet voor ǫgaande naar nul, waarbij de cellen, en dus ook de obstakels, in het domeinΩ steeds kleiner worden, maar hun aantal groeit. In de limiet verdwijnt hethoogfrequente microscopische niveau, en bekomen we een macroscopischevergelijking. Het besluit van het homogenisatie proces is volgende stelling:

Beschouw (uǫ, pǫ), de oplossingen van de het Stokes probleem (2), uitgebreidop het volledige domein Ω. De snelheid uǫ convergeert zwak in L2(Ω)n naaru, en pǫ convergeert sterk in L2(Ω)/R naar p, waarbij (u, p) de oplossing is


van de gehomogeniseerde vergelijking, namelijk de volgende wet van Darcy:

P0 =

q =1

µK(f −∇p) in Ω

∇ · u = 0 in Ω

een randvoorwaarde op ∂Ω.

(3)

De permeabiliteitstensor K is gedefinieerd als

Kij =1

|Y |

∫

YF

∇wi : ∇wj dy for 1 ≤ i, j ≤ n, (4)

met wi voor i gaande van 1 tot n een deel van de oplossing van de volgendeeenheidscelproblemen.

Voor 1 ≤ i, j ≤ n bepaal (wi, πi) ∈ H1per(Y )n × L2(Y ) zodat

−∆wi + ∇πi = ei in YF

∇ ·wi = 0 in YF

wi = 0 on ∂YF \∂Ywi, πi Y − periodic,

(5)

waarbij ei de vector met componenten ei

j = δij is, en H1per(Y )n de Sobolev

ruimte H1,2(Y )n beperkt tot Y -periodieke functies.

Volumetrisch middelen Ook hier wordt vertrokken vanaf het microsco-pische niveau en de Stokes vergelijkingen. Het concept van volumetrischmiddelen (volume averaging) is om het gemiddelde van deze vergelijkingenover een voldoende groot volume te nemen. Met voldoende groot wordtbedoeld dat een kleine toename van het volume slechts resulteert in eenkleine verandering van het gemiddelde over het volume. Het volumetrischgemiddelde van de snelheid u is gedefinieerd als

〈u〉 =1

V

∫

Vf

u dV. (6)

Voor de definitie van het volumetrisch gemiddelde van de Stokes vergelij-kingen, en de daaruitvolgende Darcy vergelijking, wordt de lezer verwezennaar Whitaker [109]. Het gemiddelde van de divergentie is ook samengevatweergegeven in sectie 2.5 van het Engelstalige deel van dit proefschrift, deuitwerking van het gemiddelde van de momentvergelijking is gekopieerd inappendix B.

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2.2 Eigenschappen van de permeabiliteitstensor

• Uit vergelijking (4) volgt onmiddellijk dat de tensor K symmetrischis.

• Uit het vorige volgt dat er is een coordinatensysteem bestaat waarinde permeabiliteit een diagonaalmatrix is. De assen van dit systeemgeven de hoofdrichtingen aan van de stroming, en de permeabiliteitin dit systeem wordt de principale permeabiliteit genoemd (principalpermeability).

• Indien een materiaal isotroop is, kan de permeabiliteitstensor met eencijfer voorgesteld worden in het systeem volgens de hoofdrichtingen.

• Er geldt dat

Kij =1

|Y |

∫

YF

∇wi : ∇wj dy =1

|Y |

∫

YF

wij dy, (7)

voor 1 ≤ i, j ≤ n [74].

3 Het oplossen van de Navier-Stokes- en de

Stokesvergelijkingen

De Navier-Stokesvergelijkingen zijn een set van partiele differentiaalver-gelijkingen die de stroming van vloeistoffen beschrijven. Indien we eenrotatieloze, isotherme, onsamendrukbare stroming beschouwen van eenNewtoniaanse vloeistof, dan zijn de Navier-Stokesvergelijkingen (in dimen-sieloze vorm) gelijk aan

∂u

∂t+ (u · ∇)u − 1

Re∆u = −∇p+

1

Frf

∇ · u = 0.(8)

Hierbij staat Re voor het Reynolds getal en Fr voor het Froude nummervan de stroming, u staat voor de snelheid, t voor de tijd, f voor een externedrijvende kracht en p is de druk.

Voor onze toepassing zijn we geınteresseerd in de evenwichtsoplossing, envoor een laag Reynolds getal domineert de convectie term de diffusie, watresulteert in de eenvoudigere, lineaire en stationaire Stokesvergelijkingen,

∆u = Re ∇p− Re

Frf

∇ · u = 0.(9)


Voor onze berekeningen is het Reynolds getal van grootteorde 10−2, watvoldoende klein is om de convectie ter verwaarlozen.

Aangezien we hier met de dimensieloze vergelijkingen werken, passen weook de Darcyvergelijking (1) aan,

q =Re

L2K

(f

Fr−∇p

)

. (10)

De afmeting L staat voor een karakteristieke lengte van het probleem, bij-voorbeeld de gemiddelde grootte van een obstakel.

3.1 Vereisten van de oplosser

Een oplosser voor de stromingsvergelijkingen die kan toegepast worden opde moeilijke en dichte geometrie van textiel, moet aan specifieke eisen vol-doen. Om te beginnen moet de oplosser snel en nauwkeurig zijn, aangezienwe een groot aantal berekeningen willen doen op structuren die onderlingweinig verschillen om zo de invloed van die variaties te bepalen. Kleine va-riaties kunnen een grote invloed hebben op de permeabiliteit, en het is dusnoodzakelijk dat de oplossingsmethode deze kleine variaties in de geometrievoldoende nauwkeurig in rekening brengt.

Alhoewel dit niet nodig is voor een goede stromingssimulatie, willen we eenzelfstandige oplosser die geen interactie van de gebruiker vraagt. Deze eisheeft twee redenen: enerzijds zijn de gebruikers meestal geen experten inhet simuleren van stroming, anderzijds willen we het programma gebruikenals module in een groter softwarepakket.

Aangezien we de stroming simuleren in een eenheidscel van de textielgeo-metrie, moet de mogelijkheid bestaan om periodieke randvoorwaarden opte leggen. Deze randvoorwaarden moeten aangepast worden voor de simu-latie in een eenheidscel van afgeschoven textiel. Voor de simulatie van destroming in een porie van het porie-netwerkmodel, leggen we symmetrischerandvoorwaarden op.

3.2 De Stokesoplosser

Een eerste stap in het numeriek oplossen van de partiele differentiaalver-gelijkingen (9) is de discretisatie van de veranderlijken. Roostergenera-tieproblemen worden vermeden door te discretiseren op een regelmatig,rechthoekig rooster. Om oscillaties in de oplossing van de drukverdeling

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te voorkomen, worden de veranderlijken vaak gediscretiseerd op een niet-samenvallend rooster (staggered grid) aangezien dit leidt tot een goede kop-peling tussen de verschillende veranderlijken [42]. Het gebruik van dit soortrooster heeft echter verschillende nadelen [section 3.4.1], en daarom discre-tiseren we hier de veranderlijken op een samenvallend rooster (collocatedgrid).

Een stabiele oplossing op een samenvallend rooster wordt bekomen door eenstabilisatieterm van de vorm

κ

(

x2 ∂2p

∂x2+ y2 ∂

2p

∂y2+ z2∂

2p

∂z2

)

(11)

toe te voegen aan de divergentievergelijking.

De stokesoplosser gebruikt de PETSc bibliotheek [11] om het uiteindelijkediscrete stelsel op te lossen. PETSc is een numerieke bibliotheek die ver-schillende numerieke methodes bevat, waarvan GMRES(m) [89] de besteresultaten geeft voor ons probleem.

3.3 Resultaten

Voor een geweven textiel waarvan de geometrische details in appendix Cvermeld zijn, tonen figuren 3 en 4 een vergelijking tussen de resultaten vande Stokesoplosser, en de resultaten van een aangepaste versie van de Navier-Stokesoplosser NaSt3DGP [1, 42]. De Stokesoplosser is aanzienlijk snellerdan de Navier-Stokesoplosser, en berekent dezelfde permeabiliteit indien dediscretisatiefouten vermeden worden door op een zeer fijn rooster te reke-nen. Dit toont numeriek aan dat de convectie inderdaad mag verwaarloosdworden. De stopcriteria voor de resultaten van deze figuren en voor beideoplossers zijn zo gekozen dat een minimale rekentijd nodig is, zonder dat hetresultaat minder nauwkeurig is in vergelijking met strengere stopcriteria.

4 Intra-draadstroming

De draden van een textiel bestaan uit vezels, en zijn (meestal) zelf vloei-stofdoorlatend, wat tot hiertoe werd verwaarloosd. Indien we deze stromingmee in rekening willen brengen, stellen er zich twee problemen: hoe stellenwe de vezels in de draad voor, en hoe kan de stroming op dit microsco-pische niveau met zeer fijne geometrie berekend worden zonder al te veelcomputerkracht.


Figuur 3: De rekentijd van de Stokes- en de Navier-Stokesoplosser in functievan de roosterfijnheid.

De geometrie van de vezels nauwkeurig bepalen is vrijwel onmogelijk, aan-gezien de vezels volledig willekeurig in de draad liggen. En zelfs als ditmogelijk zou zijn, dan nog is het een quasi onmogelijke opdracht om be-rekeningen te doen in deze fijne geometrie die zelf in een eenheidcel ligt,aangezien dit zou leiden tot een zeer groot rooster, en dus enorme reken-tijden. We zullen daarom de vezels benaderen door een gestructureerdegeometrie van parallelle cilinders, en de draad lokaal als een homogeen po-rous medium beschouwen. De lokale permeabiliteit van de draad Klokaal

kan dan meegenomen worden in de stromingsvergelijkingen als strafterm,wat resulteert in het oplossen van de Brinkmanvergelijking,

∆u = Re∇p− Re

Frf − K−1

lokaalu

∇ · u = 0.(12)

De Brinkmanvergelijking geldt op het volledige domein, met Klokaal onein-dig in de ruimte tussen de draden, en 10−7 ≪ ||Klokaal|| ≪ 10−4 in dedraden. Als Klokaal = ∞, is de Brinkmanvergelijking gelijk aan de Stokes-vergelijking, en als Klokaal voldoende klein is, is de Brinkmanvergelijkingeen goede benadering voor de wet van Darcy (1), gecombineerd met de wetvan behoud van massa.

Aangezien we de structuur van de vezels benaderen door een serie van paral-lelle cilinders, kunnen we de lokale permeabiliteit Klokaal analytisch bere-

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Figuur 4: De berekende permeabiliteit met de Stokes- en de Navier-Stokesoplosser in functie van de roosterfijnheid.

kenen. In de literatuur bestaan daarvoor verschillende formules, zoals dezevan Kozeny-Carman [20], Berdichevsky en Cai [15] en Gebart [39]. We ver-gelijken deze verschillende formules met resultaten van numerieke simulaties[101]. Daaruit blijkt dat er formules zijn die goed aansluiten bij de numerie-ke resultaten, maar ook dat de keuze van de formule niet veel invloed heeftop de permeabiliteit van de totale eenheidscel.

5 Experimentele validatie

Figuur 5 geeft een overzicht van experimenteel en numeriek bepaalde per-meabiliteitswaarden voor vier verschillende materialen. De modellen vande ongeweven materialen zijn gemaakt met NoWoTex [section 5.5.4]. Deexperimentele waarden komen uit Feser [38] en Hoes [44]. Geometrischedetails van de ongeweven materialen worden er echter niet vermeld, en demodellen zijn gebaseerd op basis van de volumefractie en foto’s waaruit wede diameter van de vezels geschat hebben. De informatie en experimentelewaarden van de geweven materialen kan gevonden worden in Hoes [44]; degeometrische data is samengevat in appendix C voor het monovezel textielen op pagina 93 voor het met rechte steek geweven textiel.

Alhoewel er een factor twee verschil is tussen sommige experimentele en nu-


Figuur 5: Overzicht van de experimenteel (exp.) en numeriek (num.)bepaalde permeabiliteitswaarden. MFF staat voor het monovezel textiel,PWF voor het met rechte steek geweven textiel.

merieke waarden, zijn deze resultaten beter dan wat tot nu in de literatuurgevonden wordt, en zijn ze ook aanvaardbaar: als input in de Darcyoplossersis niet alleen de absolute waarde van de permeabiliteit van belang. Min-stens even belangrijk is de invloed op de permeabiliteit van vervormingenzoals afschuiving, wrijving en buiging of van vervormingen onder invloedvan spanning of druk. De numerieke resultaten volgen het verloop van deexperimentele waarden in functie van de volumefractie, en dat duidt eropdat we de invloed van deze vervormingen goed kunnen inschatten, aangeziende modellen met hogere volume fractie bekomen werden door het simulerenvan het in elkaar nesten van verschillende lagen textiel en de invloed vandruk.

6 De 2D-roostermethode

De onderzoeksgroep Polymer Composites van de universiteit Nottingham,onder leiding van Prof. A. Long, ontwikkelde een methode voor de bere-kening van de permeabiliteit van textiel, gebaseerd op de vereenvoudigingvan de 3D stromingssimulatie tot een 2D Darcyprobleem. De implemen-tatie is gebaseerd op input van het textielmodelleringsprogramma TexGen[87]. Voor dit proefschrift ontwikkelden wij een versie waarvoor de voxel-

NL.15

beschrijving van het textielmodel als invoer fungeert, zodat de methodeook op modellen van WiseTex kan toegepast worden en een uitgebreiderevalidatie van de methode mogelijk is. De validatie van deze methode be-schreven in dit proefschrift, geeft meer inzicht in de toepasbaarheid van de2D-roostermethode.

De methode kan samengevat worden in vijf stappen (figuur 6 voor een 2Dnaar 1D voorbeeld):

1. modelleren van het 3D textiel;

2. zoeken van de basisvolumes van het model en definieren van een 2Drooster;

3. berekenen van een lokale permeabiliteit voor elke punt van het 2Drooster, gebaseerd op de eigenschappen van de basisvolumes onderdat punt;

4. oplossen van de Darcyvergelijking, gecombineerd met de wet van be-houd van massa, op het 2D rooster;

5. berekenen van de permeabiliteit K door middel van de wet van Darcy.

De lokale permeabiliteit van een punt op het 2D rooster (bv. K4 in figuur 6)wordt berekend als het gewogen gemiddelde van de permeabiliteit van debasisvolumes onder dat punt (K4,1,K4,2,K4,3 in figuur 6). De permeabiliteitvan een leeg basisvolume wordt benaderd door de permeabiliteit van een leegkanaal met dezelfde hoogte als het volume, en is gelijk aan de permeabiliteitvan de draad indien het basisvolume gevuld is met vezels.

De numerieke experimenten staan beschreven in [section 6.4]; in deze sa-menvatting vermelden we enkel het besluit. De 2D-roostermethode is eensnelle methode voor de berekening van de permeabiliteit van textiel, maaris minder nauwkeurig dan een 3D stromingssimulatie, en de methode werktbeter beter voor textiel met platte draden, dat wil zeggen voor textiel metdraden waarvan de hoogte veel kleiner is dan de breedte.

7 De porie-netwerkmethode

Net zoals de 2D-roostermethode, is de porie-netwerkmethode (PNM) eenprobleemreducerende methode voor de berekening van de permeabiliteit vaneen poreus medium. Deze methode kan samengevat worden in vier stappen:

1. opdelen van het poreus materiaal in porien;


Poreuze draad

K_4

K_4,1

K_4,2

K_4,3

Poreuze draad

Poreuze draad

Figuur 6: De eerste drie stappen van de 2D-roostermethode, voor een re-ductie van 2D naar 1D.

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Figuur 7: Links: 2D computertomogram van een poreus materiaal, de wittepixels zijn de porien, de zwarte pixels de vaste stof; de twee andere figu-ren tonen een verschillende opdeling van het tomogram in porien, het isonduidelijk en niet gedefinieerd welke de beste is.

2. berekening van de permeabiliteit van de porien;

3. opstellen van een netwerk met de porien als knooppunten en met deberekende permeabiliteit als afstandsfunctie voor de takken;

4. berekenen van de permeabiliteit van het netwerk door het oplossenvan de wet van behoud van massa.

De PNM werd geıntroduceerd door Fatt [34, 35, 36], en uitgebreid dooronder andere Delerue [26], die de methode ook gebruikte voor de permea-biliteit van textiel. De resultaten zijn tot nu toe echter niet bevredigend.Daarom stellen Van Marcke e.a. [72] een verbetering voor van de bestaandePNM’s, waarbij onze Stokesoplosser gebruikt wordt voor de berekening vande permeabiliteit van de porien (stap 2). Deze verbetering werd ontwikkeldvoor de berekening van de permeabiliteit van poreus gesteente.

De input voor de berekeningen is een computertomogram, dat in de eerstestap van het proces moet opgedeeld worden in porien. Dat is niet vanzelf-sprekend, aangezien deze porien niet eenduidig gedefinieerd zijn. Figuur 7 iseen computertomogram van een poreus materiaal, samen met twee verschil-lende opdelingen in porien. Het is daarbij niet eenduidig te bepalen welkevan de twee opdelingen de beste is. In de hier voorgestelde methode hangtde beslissing of een grens tussen twee porien al dan niet getrokken wordt, afvan de parameter λ: indien de doorgang op een bepaald punt klein genoegis in verhouding tot de grootte van de aangrenzende porien wordt een grensgetrokken, en het is de opgegeven parameter λ die bepaalt wat klein genoegis. Een andere keuze van λ leidt tot een andere opdeling van het poreuzemedium.


Voor de tweede stap in het proces gebruiken we de eerder vermelde Stokes-oplosser, wat eenvoudig kan, aangezien de geometrische informatie van deporien beschikbaar is in pixel/voxel formaat, en dus rechtstreeks naar hetrekenkundig rooster kan omgezet worden.

Eenmaal de permeabiliteit van de porien berekend is, kan de globale perme-abiliteit berekend worden door een drukverschil op het netwerk op te leggen,en voor elke porie de wet van behoud van massa uit te schrijven.

Figuur 8 toont de resultaten van berekeningen met de Stokesoplosser en deporie-netwerkmethode op dertien stalen van poreus gesteente. Gezien debeperkingen van de Stokesoplosser, zijn de stalen beperkt tot een groottevan 400x400x100 roosterpunten; de porie-netwerkmethode kan in principeop grotere stalen toegepast worden. Uit de figuur kunnen we twee beslui-ten trekken, namelijk dat de resultaten van de porie-netwerkmethode sterkafhangen van de paramter λ, maar ook dat voor een goede keuze van λ deresultaten van de Stokesoplosser en de porie-netwerkmethode van dezelfdegrootteorde zijn.

8 Besluit

• De Stokesoplosser kan de invloed van vervorming op de permeabiliteitvan textiel nauwkeurig berekenen; de waarde van de permeabiliteit zelfwijkt tot een factor twee af van experimentele resultaten, wat mindergoed is dan gewenst, maar beter is dan wat tot nu in de literatuurgevonden wordt.

• De 2D-roostermethode is een snelle methode voor de berekeningvan de permeabiliteit van textiel, maar is minder nauwkeurigdan een Stokessimulatie, waarbij de nauwkeurigheid van de 2D-roostermethode afhangt van de geometrische eigenschappen van dedraden van het textiel.

• De porie-netwerkmethode is een snelle methode voor de berekeningvan de permeabiliteit van poreus gesteente, maar is minder nauwkeurigdan een Stokessimulatie, waarbij de nauwkeurigheid sterk afhangt vaneen empirische parameter.

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Figuur 8: Vergelijking tussen resultaten berekend met de Stokesoplosser ende porie-netwerkmethode. De parameter λ heeft een aanzienlijke invloed opde resultaten van de porie-netwerkmethode.

Chapter 1

Introduction

The work described in this thesis is concerned with the problem of comput-ing the permeability of a porous medium. The permeability is a measure ofthe ability of a fluid to flow through a porous medium when subjected toan external force. Mathematically, the permeability is defined by Darcy’slaw, a macroscopic equation that averages or homogenises the micro-scaleproperties of the porous medium.

In particular, we propose a generally applicable and fast method basedon the solution of the Stokes equations on a regular rectangular grid, i.e.on a first order approximation of the geometry. The performance of themethod is explored and validated for the simulation of the impregnationwith resin of textile reinforcements to produce composite materials. Also,the computation of the permeability of porous rock is investigated.

1.1 Multi-scale modelling

In permeability modelling, we distinguish between different length scales(figure 1.1). A composite piece (macro-scale) consists of different layersof textile reinforcements. In the ideal case, textile has a periodic patternand the study of a unit cell of the layer (meso-scale) reveals all relevantinformation of the layer. The yarns of a textile unit cell are made of fibres(micro-scale), whose architecture determines the yarn properties. Similarly,the pores of rock (micro-scale) and their connectivity determine the prop-erties, particularly the permeability, of a sample of a rock (meso-scale).

In this thesis we concentrate on the meso-scale permeability, i.e. the perme-

1

2 CHAPTER 1. INTRODUCTION

ability of a unit cell textile model and the permeability of a representativesample of a rock. However, to compute the meso-scale permeability, themicro-scale geometry must be taken into account correctly. Therefore, wealso consider the intra-yarn permeability and the pore structure of the rocks.

The meso-scale permeability on its turn is input for flow simulations onthe macro-scale. In reality, textiles are not perfectly periodic structures.The variation on the meso-scale has an important influence on the macro-scale permeability. It is important that these variations are captured in asufficiently accurate way, in order to be able to perform reliable macro-scalesimulations.

1.2 State of the art

For simple structures, the permeability can be determined analytically [15,20, 39, 85]. Most analytical formulas give correct results, but they are onlyvalid for elementary, structured geometries. Porous media on the other handare typically stochastic, highly irregular and dense. Thus, more advancedmethods are required for the computation of the permeability of realisticporous media.

Various authors propose to reduce the 3D flow simulation to the solution ofa pore network model [17, 27, 31, 34]. The idea is to construct a networkthat represents the pore structure, in which every pore of the medium isassigned a certain conductivity. Then, the overall permeability can be com-puted using the laws of conservation. The concept of pore network mod-elling is explained in more detail in chapter 7. The pore network methodwas validated for textile geometries and porous rock, but the results areunsatisfactory for the geometries we are dealing with.

A second model reduction method is the Grid2D method, which reducesthe 3D problem to a 2D problem [110, 111]. In contrast with the previousmethod, this method was especially designed for the permeability of textiles.Extensive experimental validation of this method is not available. TheGrid2D method is explained in more detail in chapter 6.

A completely different approach is based on the Monte Carlo method. Themain idea of this approach is that random paths through the porous sampleare generated, and that the average path length is related to the permeabil-ity of the sample [47, 91]. These methods have not been validated yet forcomplex porous media, and it is not sure whether they are faster than otherknown methods.

Accurate predictions can be obtained by solving the 3D Navier-Stokes or

1.2. STATE OF THE ART 3

MA

CRO

MESO

MIC

RO

km

m

mm

mm

km

m

mm

mm

Figure 1.1: Permeability modelling involves different scales.


Table 1.1: Several methods for the prediction of the permeability of textiles.

Method Ref. Comment

Theoretical formulas[15, 20, 39,85]

Inaccurate for realistictextiles

FE/FV modelling with irreg-ular meshes

[53, 87, 95] Cumbersome meshing

Lattice Boltzmann modelling [14] Difficult acceleration

Grid2D [110, 111]No validation for differ-ent kinds of textile

Pore network model[17, 27, 31,34]

No satisfactory validation

Random walk methods [47, 91] No satisfactory validation

FD modelling This thesis

Stokes equations, or by solving an equivalent lattice Boltzmann model. Sim-ulation tools based on a lattice Boltzmann model use a regular grid andavoid the difficult mesh generation. However, in order to be useful for pa-rameter studies, the calculation of the permeability must be accurate andfast, which is not always the case with the available lattice Boltzmann soft-ware [14]. Direct numerical solution of the Navier-Stokes or Stokes equationscan be performed by a Finite Element (FE), a Finite Volume (FV) or a Fi-nite Difference (FD) approach. FE/FV simulation tools which work on anon-structured mesh have the advantage that the geometry can be meshedaccurately, but the disadvantage that they are not suited for automatic per-meability computations since these solvers require the mesh generation ofthe fluid phase of the textile model. Authors do not mention problems inthat regard, however, we are not aware of publications with results for re-alistic volume fractions, where the textile model has sharp edges and thusfor which the generation of an appropriate mesh is difficult [53, 87, 95].

1.3 Problem statement and goals

We aim at fast, accurate and automatic permeability computing softwarethat is applicable to all porous media. The above-mentioned methods, sum-marised in table 1.1, do not fulfil one or more of these requirements.

1.3. PROBLEM STATEMENT AND GOALS 5

In this thesis we develop a method based on the finite difference discretisa-tion of the Navier-Stokes and of the Stokes equations. We demonstrate the-oretically and show numerically that this method performs well for complexstructures like textile geometries. We compare our method with analyticalformulas for simple geometries and with the Grid2D method for complextextile geometries. An improved pore network model is suggested, whichuses the finite difference solver to compute the conductivities of the pores.

1.3.1 Permeability of textile reinforcements

Composite materials are widely used in industries like aerospace and trans-portation for their light weight, good mechanical properties and chemicalresistance. The involved composite materials consist of a textile reinforce-ment embedded in a matrix of hardened resin. To produce composites witha complex structure, Liquid Composite Moulding (LCM) is the most of-ten used production technology [84, 88]. One variety of LCM is the ResinTransfer Moulding (RTM) technique, which is explained in more detail inchapter 5. To simulate the impregnation stage of the LCM process, thepermeability of the textile reinforcement is a crucial input parameter. Ifperiodicity of the textile geometry is assumed, the permeability is computedon a meso-scale unit cell of the textile structure. However, this provides uswith an average permeability value only, as a realistic porous medium isnot perfectly periodic. The yarn dimensions and the spacing in between theyarns, are variable because of the production process, transportation andother treatments (figure 1.2). Moreover, nesting of different layers of thetextile reinforcement and shear are inevitable during the composite manu-facturing. The variability has an important influence on the formation ofdry, and therefore weak, spots in the manufactured part. So, the litmus testfor a permeability computing method is whether it captures the influenceon the permeability of small changes in the geometry accurately. Differenttypes of textile reinforcements exist (figure 1.3), all of which have their owndifficulties to create a correct model and to allow for accurate permeabil-ity prediction. The particularities of the different textiles are discussed inchapter 5.

1.3.2 Permeability of porous rock

The computation of the permeability of porous rock is crucial for engineeringdisciplines such as the exploration of gas and oil from a petroleum reservoir,hydrology and hydro-geology, building physics and environmental studies.A unit cell of a porous rock has dimensions which are orders of magnitude


ShearTension

Bending

Compression

Friction

Figure 1.2: Textile layers are draped into the desired shape (middle). Withthat, different sorts of distortion occur.

1.4. INPUT FOR THE SIMULATIONS 7

Figure 1.3: Examples of models of textile reinforcements: 2D woven; 2Dwoven laminate; 3D woven; UD laminate; 2-axial braid; 3-axial braid; weft-knitted; Non Crimp Fabric.

larger than the dimensions of a textile unit cell, but the geometry inside isof the same order of magnitude. A full CFD simulation on such a sample re-quires too much computation time, and therefore an improved pore networkmodel is suggested [72]. Our FD Stokes solver is used for the computationof the conductivity of the pores in the network.

1.4 Input for the simulations

1.4.1 Textile modelling

A crucial task in textile permeability modelling is the characterisation ofthe reinforcement. For the creation of a single layer model of a reinforce-ment, we use the WiseTex software [65, 66, 103], which implements a gener-alised description of the internal structure of textile reinforcements on theunit cell level. The description integrates mechanical models of the relaxedand deformed state of 2D and 3D woven [62, 63, 64], two- and three-axialbraided [67], weft-knitted [76] and non-crimp warp-knit stitched [60] fabrics(NCF) and laminates [70] (figure 1.3). Non-woven structures are modelledwith the NoWoTex software [100].

All these models, including the models of deformed fabrics, use a unifieddescription format of the geometry of the reinforcement unit cell. Thisformat allows for the calculation of physical and mechanical parametersof the fibres near an arbitrary point in the unit cell, as well as the fibrevolume fraction and the direction of the fibres. The models can deal withthe calculation of the change of the internal geometry of the reinforcement


Brinkman point (Navier-)Stokes point

Figure 1.4: A textile model (left) and a 2D cut of its first order approxima-tion on the grid (middle); 3D voxel geometry(right).

in shear, tension and compression, accounting for local variations of thepreform in the mould.

It is also possible to assess the non-uniformity of the textile structure, cre-ating a sampling of models with randomly perturbed parameters [29, 30].

Once a textile model is made, the model is exported to the so-called voxeldescription (figure 1.4). The fabric repeat is mapped into an orthorhombic,regular grid of cells (= voxels), and for every grid point it is decided whetherit lies inside a yarn (intra-yarn or Brinkman point) or in between the yarns(inter-yarn or Navier-Stokes point). The voxel description only providesus with a first order geometry definition of the textile geometry. However,we use a finite difference discretisation on a regular grid, with a first orderapproximation of the boundary conditions to solve the flow equations. Thus,a first order definition of the geometry is sufficient and can be used as directinput for the flow solver. The complete procedure, from textile model tothe permeability, is summarised in figure 1.5.

1.4.2 X-ray computed tomography

A suitable technique to acquire an image of the pore space of rock is X-ray computed tomography. X-ray computed tomography (CT) allows tovisualise the internal structure of objects in a non-destructive way. A CTimage of a textile could also be used as input for permeability computations,however we have not used such input data in this thesis. After havingacquired a CT image of a sample, the pore space of that sample is extractedfrom the CT image by thresholding. Thresholding is a technique to segmentimages into binary images and in this context thresholding converts thegreyscale CT image into a binary or voxel image, which is direct input forthe fluid solver.

1.4. INPUT FOR THE SIMULATIONS 9

K

Transform Input

Solution of the

equations

Navier-Stokes: time-stepping

Stokes: linear iterative or

direct solver

Discretisation

of the

(Navier-)Stokes

equations

Voxel

description

Textile

modelling:

vector

description

Textile

modelling:

vector

description

Textile

modelling:

vector

description

K-predicting

software

Homogenisation

Figure 1.5: Flowchart for the computation of the textile permeability withthe finite difference discretisation of the (Navier-)Stokes equations. Thedotted lines indicate that no intervention of the user is required.


1.5 Outline of the thesis

Chapter 2 At the macro-level, Darcy’s law holds, which states that thepermeability is the ratio of the average fluid velocity and the external forceimposed on the medium. In this chapter, we explore Darcy’s law in detail,supported by the homogenisation and volume averaging theory.

Chapter 3 Introduction of the Navier-Stokes and Stokes equations. A so-lution method based on a finite difference discretisation of the equations ispresented. The complex structure of textile models impose special require-ments to the simulations software. In this chapter the requirements and theresulting decisions are explained.

Chapter 4 The yarns of a textile are porous themselves. Chapter 4 dealswith the intra-yarn flow and its influence on the meso-scale permeability.

Chapter 5 Papers on permeability computing methods often lack exper-imental validation. Chapter 5 gives the results of experimental validationon different textiles and structures.

Chapter 6 The Grid2D method is explored in detail and validated againstthe Stokes solver.

Chapter 7 The Stokes solver is used to compute the conductivity of thepores of a pore network.

Chapter 8 Conclusions and outlook.

1.6 Summary of results

• A fast and accurate computation of the permeability of textiles can beachieved with a finite difference discretisation of the Stokes equations.

• The dense geometry of a textile model, and the narrow channels thatare formed between the yarns, have an important influence on theimplementation aspects of the Stokes solver.

• The intra-yarn flow can be taken into account straightforwardly bysolving the Brinkman equations. These equations are the Stokes equa-tions extended with a penalty term. The penalty is computed withanalytical formulas. Different analytical formulas exist, however, thechoice of the formula has almost no influence on the meso-scale per-meability.

• For different kind of textiles, we obtain permeability results in good

1.6. SUMMARY OF RESULTS 11

agreement with experimental values. In absolute value there is still adifference between the experimental and computational results. How-ever, to our knowledge, we present the best experimental validationuntil now. Also, the influence of nesting, changing volume fractionand shear is captured.

• The Grid2D method is a permeability estimating method that requiresconsiderably less computation resources than a full CFD simulation.However, the computed results are less accurate and the accuracy ofthe results depends on the textile structure.

• The Stokes solver is well suited to be included into the implementationof a pore network solver.

Chapter 2

Permeability of porousmedia: definition

Darcy’s law states that the ratio of the average velocity of a fluid flowthrough a porous medium and the applied force on the fluid is a constant.This constant is called the permeability. Darcy established his law exper-imentally by observing a linear coupling between the water flux through asand bed and the height difference of the two ends of the bed. AlthoughDarcy’s law suffices itself to formulate a method to compute the permeabil-ity of porous media, homogenisation theory and volume averaging provideus with more insight in the definition and properties of the permeabilitytensor.

In this chapter we first define a porous medium. Then, the definition ofthe permeability via Darcy’s law will be investigated and compared withthe definition arising from the homogenisation theory and from the volumeaveraging technique. Additionally, some properties of the permeabilitytensor are discussed.

2.1 Definition of a porous medium

According to the dictionary, porous means having minute spaces throughwhich liquid or air may pass. The aim of this thesis is to investigate howfast a fluid passes through a porous medium, given a certain external force.This definition leaves one issue: what is minute?

13

14 CHAPTER 2. DEFINITION OF PERMEABILITY

ε

εaε

Ωεε

YεFi

YεSi

Figure 2.1: A mathematical description of a periodic porous medium [50].

Accordingly, in order to derive the porous media flow equations, we firstneed a mathematical description of a porous medium and a condition forthe pore size for which Darcy’s law is valid. Let Ω ⊂ Rn, n = 2, 3, bea smooth, bounded, connected domain representing a periodic structure(figure 2.1). Suppose the set Ω is covered by a regular mesh of size ǫ. Inthis mesh, each cell is denoted by Y ǫ

i with 1 ≤ i ≤ N(ǫ), where N(ǫ) denotesthe number of cells. At the centre of each cell that is entirely included in Ω,spherical obstacles Y ǫ

Siwith size aǫ represent the solid part of the medium.

The symbol Y ǫFi

denotes Y ǫi \Y ǫ

Si. Then the fluid domain Ωǫ ⊂ Ω is obtained

from Ω by removing these periodically distributed obstacles

Ωǫ = Ω \ ∪N(ǫ)i=1 Y ǫ

Si. (2.1)

Next, we define the associated unit cell or periodic cell Y =]0, 1[n, n = 2, 3 ofthe porous medium. We scale the cells Y ǫ

i to a unit square, i.e. Y is obtainedby a linear homeomorphism Πǫ

i of each cell Y ǫi with ratio of magnification

1/ǫ. Hence,

YS = Πǫi(Y

ǫSi

) and YF = Πǫi(Y

ǫFi

) (2.2)

denote the solid and the fluid part of the unit cell. Further, we assume thatthe fluid part YF repeated by Y -periodicity is a smooth, connected, openset of Rn and that YS is a smooth closed set. For the boundary of the fluidpart in the unit cell ∂YF we distinguish the boundary Γ1 of the obstaclesand the boundary Γ2 of the cell itself, where ∂YF = Γ1 ∪ Γ2 (figure 2.2).Note that in 2D these quite restrictive conditions imply that the solid partis strictly contained in the period and that the union of solid parts does notform a connected body. This is different in the typical 3D porous media

2.2. PERMEABILITY AT DIFFERENT LENGTH SCALES 15

YF

Γ1

Γ2YS

Figure 2.2: Unit cell of a porous medium with definition of the bound-aries [50].

cell, where the parts YS are typically connected, but the fluid domain is stillconnected.

We now have a mathematical description of a porous medium, however stillno condition for the pore size. In the next three sections we discuss threederivations for the permeability tensor K and check whether they provideus with conditions for aε.

2.2 Permeability at different length scales

Figure 1.1 gives an overview of the different length scales for two porousmedia that will be considered in this thesis: porous rocks and textiles.The different length scales have to be considered in the definition of thepermeability tensor. On the micro-scale, with characteristic length ε, wetalk about the pores of the medium. The average pore size and the waythey are interconnected determine the properties of the porous medium.Unless we deal with a designed, structured medium, one pore does notcontain information about the porous medium at the macro-scale with acharacteristic length Lm. However, if we consider for example the porosity φof the medium, there exists a volume, smaller than the whole volume of themedium, which has the same average porosity as the medium. The smallestvolume for which this is true, is the unit cell or representative volume (figure2.3).

The meso-scale is a connection between the micro-scale and the macro-scale.A meso-scale volume has the same averaged properties as a macro-scalevolume, but it contains also the micro-scale information. A lower boundfor the meso-scale is the unit cell as defined before. There is no preciseupper bound for the meso-scale. As we want to perform computations onthe meso-scale level, the lower bound is more useful for us, as it defines our


Figure 2.3: The average of the porosity φ depends on the volume V overwhich the average is taken. As the volume increases, the average becomesless oscillatory. For this example, a representative meso-scale volume hassize 30 and the average of φ related to V is approximately 0.7.

computational domain. Figure 1.1 shows that the definition of the differentscales is open to more than one interpretation. If the aeroplane is the macro-scale, a textile layer is the meso-scale. But if we consider the textile layeras macro-scale, a unit cell of the layer is the meso-scale. In this thesis themeso-scale refers to the unit cell level.

The permeability is a homogenised, macroscopic property of the medium.Darcy describes the permeability by considering macroscopic data only, theaverage velocity and applied force. The permeability however, depends onthe internal structure, the micro scale, of the medium. Sections 2.4 and2.5 deal with the theoretical definition of the permeability. The equationswhich hold on the micro scale are scaled up (homogenisation) or averaged(volume averaging) to the macro scale.

2.3 Darcy’s law

Henry Darcy (1803-1858) built a pressurised pipe system that was 28kmlong, and worked with the gravity as the unique driving force [56]. In anappendix of the publication with the description of the water system, LesFontaines Publiques de la Ville de Dijon [24], Darcy wrote his now famouslaw

q = −k dh

dz(2.3)

2.3. DARCY’S LAW 17

with q the water flux, k the permeability or hydraulic conductivity, h thehydraulic head (see further) and z the height. In this definition, the per-meability depends on the fluid viscosity: the more viscous the fluid, thesmaller k will be for a constant q and dh/dz. To avoid this dependence, onegenerally uses the intrinsic permeability

K = cd2p (2.4)

with dp the averaged diameter of the pores of the porous medium and ca dimensionless factor which contains information about the form of themedium. The permeability is related to the intrinsic permeability as

k =Kρg

µ(2.5)

with g the gravity constant, µ the dynamic viscosity and ρ the fluid’s den-sity. The units of the intrinsic permeability are [m2]. This explains whyK is a quadratic function of the pore diameter. By definition, the intrinsicpermeability can only depend on properties of the geometry. The charac-teristic property of the geometry is the (average) pore diameter. And thus,via dimension analyses, the intrinsic permeability must be a square functionof the pore diameter.

In this thesis, the intrinsic permeability K will be further denoted as thepermeability. The notation k will not be used further.

We further note that the hydraulic head is the sum of the hydraulic pressureand height

h =p

ρg+ z. (2.6)

Differentation of h and conversion to 3D results in

∇h =∇pρg

+ ∇z. (2.7)

Equations (2.3), (2.5) and (2.7) yield

q =K

µ(−ρg∇z −∇p) . (2.8)

To generalise the driving force, we write −ρg∇z, that could be read as thegravity force only, as f ,

q =K

µ(f −∇p)

[m

s

]

=

[m2]

[kg

m s

]

[kg

m2 s2

]

. (2.9)


The flux q is related to the velocity u as

q =1

V

∫

Vf

u dV = 〈u〉 . (2.10)

Here, V denotes the volume of the sample, Vf the fluid part of the volumeand u = [ux,uy,uz] = u(x, y, z) the velocity vector. Equations (2.9) and(2.10) give a direct connection between the permeability K, a driving force f ,∇p and the velocity u.

From literature we know that the permeability ranges from 10−7 m2 forporous gravel to 10−18 m2 for granite [12]. The permeability of the textilereinforcements mentioned in this thesis, ranges from 10−7 m2 to 10−11 m2

(chapter 5). These data do not determine a condition for the average poresize, however they show that Darcy’s law is used for a large range of mate-rials and pore sizes.

2.4 Mathematical homogenisation

Consider again the domain Ωε, with the micro-scale dimensions ε (fig-ure 2.1). Mathematical homogenisation is the upscaling of the equationsthat hold at the micro-scale by considering the limiting case for ε going tozero. This section is based on [50].

We suppose that fluid flow at the micro-scale is stationary and viscous andcan be modelled by the Stokes equations. In chapter 3 we explain the Stokesequations and their validity in more detail. For now it is sufficient to knowthat the equations, referred to by Pǫ, are:

Pǫ

∇pǫ − µ∆uǫ = f in Ωǫ

∇ · uǫ = 0 in Ωǫ

uǫ = 0 on ∂Ωǫ\∂Ω

boundary condition (b.c.) on ∂Ω.

(2.11)

Here, uǫ denotes the velocity and pǫ the pressure in a porous medium witha unit cell of the periodic structure with size ǫ (figure 2.1). In the space

Wǫ = z ∈ H1(Ωǫ)n, z = 0 on ∂Ωǫ\∂Ω, b.c. on ∂Ω

the variational problem associated with (2.11) is to find uǫ ∈ Wǫ, ∇ · uǫ =0 in Ωǫ and pǫ ∈ L2(Ωǫ) such that

∫

Ωǫ

pǫ ∇·w dx−µ

∫

Ωǫ

∇uǫ : ∇w dx =

∫

Ωǫ

f ·w dx ∀w ∈Wǫ. (2.12)

2.4. MATHEMATICAL HOMOGENISATION 19

The Frobenius inner product : of two matrices A and B is∑

i

∑

j AijBij .

If the force term is assumed to have the usual regularity f ∈ L2(Ωǫ)n, then

elliptic variational theory gives the existence of a unique velocity field whichsolves (2.12) [74]. Uniqueness of the pressure field can be shown up to anadditive constant, which is usually fixed by imposing

∫

Ωǫpǫ dx = 0.

Note that equations (2.11) do not consist of a single problem, but of awhole series of problems Pǫ with ǫ converging to zero. The purpose of thehomogenisation process is now to prove that the solutions u0 and p0 arethe solution of the Darcy equation. The sequence of solutions (uǫ, pǫ) is notdefined in a fixed domain independent of ǫ but in varying sets Ωǫ. However,for homogenisation purposes convergence proofs in fixed Sobolev spaces onΩ are required, which is why the first step in the up-scaling of (2.11) mustbe to extend (uǫ, pǫ) on the whole domain Ω.

The velocity can be extended to zero in Ω\Ωǫ. This extension is not onlycompatible with the velocity’s no-slip boundary condition on ∂Ωǫ\∂Ω, alsoLq and H1,q

0 norms are preserved for 1 ≤ q ≤ ∞. Thus we define theextension

uǫ =

uǫ in Ωǫ

0 in Ω\Ωǫ.(2.13)

The extension of the pressure is not so obvious, as the extension of the pres-sure gradient in the functional space H−1 with zero, is not necessarily thegradient of some L2-function. Therefore, Mikelic [74] extends the pressureto the whole domain Ω with

pǫ =

pǫ in Ωǫ

1˛

˛

˛Y ǫ

Fi

˛

˛

˛

∫

Y ǫFi

pǫ dy in Y ǫSi

for 1 ≤ i ≤ N(ǫ). (2.14)

In the following we will always use the extended versions of pressure andvelocity and we will omit the tilde symbol. The next step in the homogeni-sation process is to find estimates for the velocity and the pressure. Theextensions (uǫ, pǫ) of the solution of the Stokes problem (2.11) satisfy theestimates

1

ǫ2‖uǫ‖L2(Ω)n +

1

ǫ‖∇uǫ‖L2(Ω)n×n ≤ C (2.15)

‖pǫ‖L2(Ω)/R ≤ C, (2.16)

where the constant C does not depend on ǫ. The proof can be found in [74].Until now, we have not yet given explicit boundary conditions on the outerboundary ∂Ω for the Stokes equations. Mikelic handles the problem with


periodic boundary conditions in a rectangular domain Ω = [0, L]n, i.e. hespecifies in (2.11)

uǫ, pǫ is Ω − periodic. (2.17)

Homogenisation of the Stokes equations with other boundary conditions on∂Ω can be found in Allaire [45] or Sanchez-Palencia [90].

The main theorem in the homogenisation process states that the solutionsu0 and p0 converge to the solution of the Darcy equation for ε→ 0.

Main theorem Let (uǫ, pǫ) denote the extension (2.13) and (2.14) of thesolution of the Stokes problem (2.11). Then uǫ converges weakly in L2(Ω)n

to u, and pǫ converges strongly in L2(Ω)/R to p, where (u, p) is the solutionof the respective homogenised problem, which is a Darcy’s Law

P0 =

q =1

µK(f −∇p) in Ω

∇ · u = 0 in Ω

b.c. on ∂Ω.

(2.18)

The permeability tensor K is denoted by

Kij =1

|Y |

∫

YF

∇wi : ∇wj dy for 1 ≤ i, j ≤ n, (2.19)

where wi for 1 ≤ i ≤ n is part of the solutions (wi, πi) of the following UnitCell Problems:

For 1 ≤ i, j ≤ n find (wi, πi) ∈ H1per(Y )n × L2(Y ) such that

−∆wi + ∇πi = ei in YF

∇ ·wi = 0 in YF

wi = 0 on ∂YF \∂Ywi, πi Y − periodic.

(2.20)

where ei denotes the vector with components ei

j = δij , and H1per(Y )n is the

Sobolev space H1,2(Y )n restricted to Y -periodic functions. Note that theseUnit Cell Problems resemble the Stokes equations with Reynolds numberRe = 1 (chapter 3). Also, this denotation implies the symmetry of K.

The proof of this theorem is given in two steps. First, existence and unique-ness of the limit equations are shown. Then, the convergence of (uǫ, pǫ) to-wards Darcy’s Law is addressed with the two-scale convergence method [74].

2.5. VOLUME AVERAGING 21

We will not repeat the proof here. However, one important assumption ismade in the proof:

limε→0

aε

ε= C with C /∈ 0,∞. (2.21)

The obstable size aε is defined on figure 2.1 on page 14. This is a naturalassumption. If C would be equal to zero, there is no obstacle left in thelimit case. If C would be ∞, the solid phase will be such, that almost nofluid flow is possible. In the next chapter we address these unnatural cases,as the mathematical solution of the homogenisation process in these caseshelps us to model the flow inside the yarns.

Until now no method to compute the permeability has been addressed.The homogenisation theory however does provide us with a computationalmethod: the numerical solution of the unit cell problem (2.20) results in thepermeability value of the porous medium. The numerical solution of (2.20),together with other methods to compute K, is discussed in chapter 3.

2.5 Volume averaging

An older method to confirm the law of Darcy theoretically is volume av-eraging. Also starting from the Stokes equations on micro-level, the flowequations are now scaled up to the macro level by taking the average of theequations over a representative volume. The discussion and formulation ofthis section is based on Whitaker [109].

The volume average of the velocity u is defined as

〈u〉 =1

V

∫

Vf

u dV. (2.22)

Here, V is the volume of the considered part over which the average is takenand Vf is the fluid part of V . As for this theory periodicity of the structureis not a constraint, we do not use the symbol Y here as before. The velocityu is defined in the fluid part Vf only, but the integral must be divided bythe whole volume V to obtain the average. The average is a point quantityitself. We associate with every point of the porous medium a volume V(e.g. a sphere with the point as centre), and the average over that volumeis assigned to that point.

The average is defined on the whole volume – not only on the fluid region –and the average of the average must be equal to the average:

1

V

∫

V

〈u〉 dV = 〈〈u〉〉 = 〈u〉 . (2.23)


This restriction is not without consequences, and results in the restriction

l ≪ Lm

with l the characteristic length of the volume V and Lm the characteristiclength of the macro-scale [109].

The Stokes equations that hold on the micro-scale can be written as

∇p− µ ∆u = f (2.24a)

∇ · u = 0. (2.24b)

To obtain the equations that hold on the macro-scale, the volume averageof the Stokes equations is considered. The volume average of the continuityequation is

1

V

∫

Vf

∇ · u dV = 0. (2.25)

Consider now the theorem of Slattery [92]

∇ ·∫

Vf

u dV =

∫

Ae

u · n dA (2.26)

and the divergence theorem

∫

Vf

∇ · u dV =

∫

Ae

u · n dA+

∫

Ai

u · n dA. (2.27)

Here, Ae is the area of entrances and exits and Ai the area of the solid-fluidinterface contained within Af , the total surface of V (Af = Ae +Ai). Thecombination of (2.26) and (2.27) yields

∫

Vf

∇ · u dV = ∇ ·∫

Vf

u dV +

∫

Ai

u · n dA. (2.28)

Using (2.22) and by division by V we can write (2.28) as

〈∇ · u 〉 = ∇ · 〈u 〉 +1

V

∫

Ai

u · n dA. (2.29)

Equation (2.29) provides us with the link between the two scales, as theaverage of the divergence of a micro-scale variable is expressed in terms ofthe divergence of the average which is a macro-scale variable.

2.6. PROPERTIES OF THE PERMEABILITY TENSOR 23

With this, equation (2.25) can be written as

∇ · 〈u〉 +1

V

∫

Ai

u · n dA = 0. (2.30)

If we consider no-slip boundary conditions, (2.30) simplifies to:

∇ · 〈u〉 = 0.

The reader is referred to Whitaker [109] for the technical details of thevolume averaging of the momentum equation. The proof is copied in ap-pendix B. The volume averaging technique yields Darcy’s law (2.9) with

K−1 =1

V

∫

Ae

mn dA.

The vector m is defined in the proof, and depends on the velocity field andthe geometry of the porous medium.

2.6 Properties of the permeability tensor

3D-tensor In three dimensions equation (2.9) becomes

〈ux〉〈uy〉〈uz〉

=1

µ

Kxx Kxy Kxz

Kyx Kyy Kyz

Kzx Kzy Kzz

fx − ∂p∂x

fy − ∂p∂y

fz − ∂p∂z

, (2.31)

with Kij the permeability in the direction i under an external force in direc-tion j. Consequently, nine unknowns have to be determined to compute thepermeability tensor. However, one numerical CFD simulation to compute uwill only result in three equations. Therefore, three numerical simulations,with different f , have to be performed for the computation of K. This canalso be seen from equation (2.19), where Kij has to be computed with wi

for 1 ≤ i ≤ 3, and thus for three different Unit Cell Problems (2.20). If sym-metry is accounted for (see further), there are only six unknowns. However,again three experiments have to be performed as the equations resultingfrom two experiments are not linear independent.

For the simplicity of the formulas and figures, we will often neglect theZ-direction in the discussion of the permeability tensor. However, all theresults presented are results of 3D fluid simulations (chapter 3).


Principal direction The permeability tensor is a characteristic of theporous medium, and from a physical point of view only depends on theinternal structure of the medium. Yet, as the permeability is a tensor, itsvalues relate to the coordinate system in which the permeability is defined.The permeability tensor is a symmetric tensor, and thus there must be a co-ordinate system in which the permeability is a diagonal matrix. This systemis named the principal system of the porous medium, and the permeabil-ity in the principal system is the principal permeability. In the principalsystem, Kxx has a maximum value, Kyy a minimum.

Isotropy Materials can have an isotropic permeability tensor, i.e. the per-meability does not depend on the direction. For such materials the perme-ability tensor in the principal system can be represented by one number.

Computation Once the solution for the Unit Cell Problem (2.20) is com-puted, the permeability is determined with formula (2.19). This formula cantransformed to a more practical formulation [74].

Theorem

Kij =1

|Y |

∫

YF

∇wi : ∇wj dy =1

|Y |

∫

YF

wij dy (2.32)

holds for 1 ≤ i, j ≤ n. The proof starts with following lemma.

Lemma

∫

YF

wi · ∇πj = 0 (2.33)

Proof

∫

YF

wi · ∇πjdy =

∫

YF

∇ ·(πjwi

)dy −

∫

YF

πj∇ ·widy

Gauss theorem and (2.20)

=

∫

∂YF

πjwi · n dS + 0

= 0.

The integral over ∂YF vanishes because the no-slip condition holds on thefluid-surface interface Γ1 (figure 2.2), and periodic boundary conditions holdon Γ2.

2.7. EXAMPLE 25

Proof of (2.32)

For 1 ≤ i, j ≤ n

Kij =1

|Y |

∫

YF

∇wi : ∇wj dy

Integration by parts and (2.20)

= − 1

|Y |

∫

YF

wi · ∆wj dy

Insert (2.20)

= − 1

|Y |

∫

YF

wi · (∇πj − ej) dy

= − 1

|Y |

∫

YF

wi · ∇πj dy

︸︷︷︸

=0 (2.33)

+1

|Y |

∫

YF

wi · ej dy

=1

|Y |

∫

YF

wij dy.

Whitaker [109] shows that the tensor is symmetric, starting from the defi-nition from the volume averaging theory. The proof is rather extensive andwe will not repeat it here.

2.7 Example

We have now a definition of the permeability, and, via homogenisation the-ory we also have equations (2.20) to compute the permeability numerically.To demonstrate the properties of the permeability tensor, two examples areconsidered. First, the properties are introduced intuitively, based on thebasic structure of a gap between two walls (figure 2.4). Second, we intro-duce the model of a monofilament basket woven textile (figure 2.6). Thismodel will be used to demonstrate definitions and numerical properties ofthe permeability and the applied methods throughout this thesis. Notethat we do not validate the methods here, the structures are only used as asteppingstone for the example. Validation is given in chapters 5, 6 and 7.

Figure 2.4 shows a 2D-cut of the gap geometry, the unit cell coordinatesystem XY and the principal axis X’Y’. The Z-direction is the directionperpendicular to the XY-plane. The gap has a dimensionless height of 0.5,


Table 2.1: Computed velocity values of the channel.

Applied force Average velocity

fx fy fz 〈ux〉〈uy〉〈uz〉0.01 -2.5 E-07 -2.5 E-07 4.2 E-04

0.01 2.2 E-04 2.1 E-04 -1.8 E-08

0.01 0.01 4.3 E-04 4.3 E-04 -8.4 E-08

and is rotated 45 with respect to the coordinate axis. Table 2.1 presentsthe computed average velocities for three different experiments on the gap.Periodic boundary conditions were imposed in the three directions.

Figure 2.5 shows the entries of the permeabilities K = RK0R−1 as function

of the rotation angle ψ, with

R =

[

cosψ − sinψ

sinψ cosψ

]

. (2.34)

The tensor K0 is the tensor that was computed in the system with ψ equalszero. The permeability in the Z-direction does not depend on a rotationof the axis system in the XY-plane. First we notice that in every systemKyx=Kxy. Our numerical approach results in a symmetric permeabilitytensor, which corresponds to the statement in section 2.4. Next, for ψ = 45,Kxx reaches a maximum of 0.042, the same permeability as computed in thefirst experiment in the Z-direction. This agrees with our expectation thatthe permeability in the Z-direction is the same as in the ψ = 45-direction.In the principal system, Kyy is zero, as no fluid goes through the solid walls.Moreover, in this principal system, Kyx and Kxy indeed are zero.

The same holds for the more complex structure of the monofilament textile(figure 2.7). The geometrical data of the geometry can be found in ap-pendix C. From the figure we can also see that the textile has an almostisotropic permeability tensor.

2.8 Conclusion

The permeability of a porous medium is defined by Darcy’s law, which statesthat the permeability is the ratio of the average fluid velocity and the ex-ternal force imposed on the medium. Darcy’s law is a macroscopic law and

2.8. CONCLUSION 27

averages out the microscopic details of the geometry of the porous medium.The law can also be analytically derived for incompressible Newtonian fluidsby means of mathematical homogenisation or by means of averaging the-ory. This way, the permeability of a porous medium is determined by thegeometry of the unit cell.

The permeability is a 3D tensor, and three numerical experiments have tobe performed to compute the full tensor. If stationary flow of a Newtonianfluid is considered on the micro-scale, the simulations are the solution ofthe Stokes equations. The system in which the permeability is a diagonaltensor, is the principal system of the porous medium.


Y

X

Y’

X’

y

Figure 2.4: Channel under an angle ψ = 45 angle, the unit cell coordinatesystem XY and the principal system X ′Y ′; the black region represents thewalls, the white region is the gap.

Figure 2.5: Permeability values of a channel under an angle of 45 with theXY -plane, as function of the rotation angle ψ of the coordinate system.

2.8. CONCLUSION 29

Figure 2.6: The textile model which will be used as example input.

Figure 2.7: Permeability values of the monofilament textile as function ofthe rotation angle ψ of the coordinate system.

Chapter 3

Solution of theNavier-Stokes and theStokes equations

The nonlinear Navier-Stokes partial differential equations (PDEs) describethe motion of fluids (liquids or gases). The linear Stokes equations are asimplification of the Navier-Stokes equations and are not valid for the samevariety of flows, but are valid for the flow through the porous media we aredealing with. In order to predict the permeability of a textile, we can solvethe Navier-Stokes or the Stokes equations in a unit cell of the textile model.

To avoid meshing problems, we have chosen to discretise the system of equa-tions on a regular rectangular grid using a finite difference discretisation.An open-source 3D finite difference Navier-Stokes solver, NaSt3DGP, hasbeen developed by the research group of Prof. M. Griebel (University ofBonn) [1, 42]. We have used this software, and we have made several ex-tensions to this code in order to speed up the computations. This was donein close cooperation with the research group at the University of Bonn, inparticular with R. Croce, M. Engel and M. Klitz.

In this chapter we present the Navier-Stokes and Stokes equations, and thefinite difference solvers. We discuss the differences between the solvers andcompare them with a lattice Boltzmann code.

Note that we treat the yarns of the textile as solid material in this chapter.The case of permeable yarns is treated in chapter 4.

31

32 CHAPTER 3. THE (NAVIER-)STOKES EQUATIONS

3.1 The (Navier-)Stokes equations

We consider irrotational flows, i.e.

∇× u = 0, (3.1)

of Newtonian fluids, for which the components of the stress tensor τ satisfy

τij = µ

(∂ui

∂xj+∂uj

∂xi

)

. (3.2)

Here, u is the velocity vector, µ the dynamic viscosity and x is the positionvector. In addition, we consider isothermal flow of an incompressible fluid,i.e. the temperature and the density are constant. With these limitations,the Navier-Stokes equations are

ρ

(∂u

∂t+ u · ∇u

)

− µ∆u = f −∇p (3.3a)

∇ · u = 0, (3.3b)

with ρ the density of fluid, p the pressure and f an external body force.Equation (3.3a) is the momentum equation and describes the movement ofthe fluid due to an external force f and/or a pressure gradient ∇p. Thisequation is derived from the laws of Newton applied on a fluid particle. Thesecond equation reflects the conservation of mass.

If we introduce the dimensionless variables

u∗ ≡ u

u∞, x∗ ≡ x

L, t∗ ≡ u∞t

L, p∗ ≡ p− p∞

ρ∞u2∞, f∗ ≡ f

ρ∞||f ||

and the dimensionless Reynolds and Froude numbers

Re ≡ ρ∞u∞L

µand Fr ≡ u2

∞

L‖f‖ , (3.4)

with L, u∞, p∞, ρ∞ as explained below, we obtain the dimensionless Navier-Stokes equations

∂u∗

∂t∗+ u∗ · ∇∗u∗ − 1

Re∆∗u∗ = −∇∗p∗ +

1

Frf∗ (3.5a)

∇∗ · u∗ = 0. (3.5b)

The operators ∇∗ and ∆∗ refer to x∗, hence

∇∗ = L∇ and ∆∗ = L2∆. (3.6)

3.1. THE (NAVIER-)STOKES EQUATIONS 33

Periodic boundaryconditions

Wall boundaryconditions

f

ps

Figure 3.1: A textile unit cell with possible boundary conditions and drivingforces.

For an incompressible fluid, ρ∞ is the constant density. The constant u∞is normally the mean fluid velocity, p∞ is some reference pressure. L is acharacteristic length of the geometry, for example the average length of anobstacle or the length of a unit cell. The Reynolds number of slow, creepingflow is small. For such flows, the diffusive term will dominate the convectiveterm in equations (3.5). Moreover, as we are only interested in the steadystate solution, we can also omit the time derivative from the equations. Thisresults in the linear, steady Stokes equations,

∆∗u∗ = Re ∇∗p∗ − Re

Frf∗ (3.7a)

∇∗ · u∗ = 0. (3.7b)

As a result, also the Darcy equation (2.9) must be transformed to the di-


mensionless form to interpret the computed permeability correctly,

q =K

µ(f −∇p)

Insert q∗ ≡ q

u∞

q∗u∞ =ReK

ρ∞u∞L

(ρ∞f∗||f || − ∇

(p∗ρ∞u

2∞ + p∞

))

q∗ =Re

L2K

(f∗

Fr−∇∗p∗

)

.

(3.8)

If we compute the permeability via the dimensionless equations, we obtainthe dimensionless permeability K∗ = K/L2. However, we are interested inthe physical permeability K. Thus, the computational result K∗ is scaledwith L2.

For the remainder of this thesis, the asterisk will be omitted. If physicalresults are given, the scaled values are used and the dimensions are added.

Boundary conditions For both equations, two sets of boundary condi-tions are required: the boundary conditions on the borders of the unit cell ofthe porous medium, and the boundary conditions on the border of the fluidregion and the solid part. If we consider a textile unit cell, then on the unitcell boundary, periodic boundary conditions for the velocity are the mostobvious in the X- and Y-direction as textile has a periodic pattern. In theZ-direction one can choose between periodic or wall boundary conditions(figure 3.1).

At the boundary of the fluid and solid region Γ1, we impose no-slip boundaryconditions

u = 0 on Γ1. (3.9)

Unlike the boundary condition for the velocity, the boundary condition forthe pressure is not straightforward. A zero pressure gradient in the normaldirection of the boundary is acceptable, as there is no fluid flow through theboundary. This supports the statement that in that direction no pressuredifference is present. The often used boundary condition for the pressure iswritten as

n · ∇p ≡ ∂p

∂n= 0, (3.10)

with n the outward-pointing unit normal vector. Let γ be the unit tangent

3.2. DIMENSION ANALYSIS 35

vector, then on the boundary it holds

∆ =∂2

∂x2+

∂2

∂y2=

∂2

∂n2+

∂2

∂γ2

and

u · ∇ = un∂

∂n+ uγ

∂

∂γ≡ ux

∂

∂x+ uy

∂

∂y.

If we apply the momentum equation (3.5a) with f = 0, on the boundaryitself and project in the normal direction, we get

∂p

∂n=

1

Re∆un −

(∂un

∂t+ u · ∇un

)

(3.11)

for the Navier-Stokes equations or

∂p

∂n=

1

Re∆un (3.12)

for the Stokes equations [41]. If f 6= 0, the component of f in the normaldirection divided by the Froude number, must be added to the right handside of these equations. For high Reynolds numbers, condition (3.10) is agood approximation for the boundary condition (3.12), but this is not thecase for the low Reynolds numbers for which the Stokes equations hold. Inthe Navier-Stokes solver the zero pressure gradient (3.10) was implemented.Condition (3.12) is used in the Stokes solver.

Unit cell problem In chapter 2, we discussed the macroscopic law ofDarcy, both from experimental point of view and in the context of ho-mogenisation.

If we scale the unit cell dimensions to unity and use the three unit vectorsin the X-,Y- and Z-direction as right hand side, the Stokes system withRe = 1 is equal to the unit cell problem from the homogenisation theory.The solution of the Stokes equations (3.7) in the unit cell provides us withthe velocity and pressure to include into the Darcy equation. To computethe 3D permeability tensor, also three Stokes simulations (with different f)have to be performed.

3.2 Dimension analysis

Before we start with the description of the solution methods, we first givean example of the dimenions of a typical unit cell and flow in the RTM


process. The analysis was done by Moesen [75] who used the data fromAstrom [10].

Based on the data from Astrom, these numbers are acceptable for the flowof the RTM process:

u∞ = 0.001 m/s

L = 0.005 m

ρ = 103 kg/m3

µ = 0.1 Pa s.

(3.13)

The characteristic length is chosen in between the length of a typical unitcell (1 cm) and the height (2 mm). The velocity is based on the RTM fillingtime of 10 to 20 minutes for a piece of 1m. This results in a Reynoldsnumber Re = 0.05.

This calculation is based on the average velocity and average length scale,not on the velocity inside the pores and the pore size. If we estimate thevelocity inside the pores as u∞/φ, the average velocity divided by the poros-ity, and the pore size as half of the height, the Reynolds number stays thesame for φ = 0.5.

3.3 Requirements for the solver

The goal of this thesis is the development of a numerical solver for the com-putation of the permeability of porous media that can be applied to textilemodels, and also to other materials such as porous rocks. These applica-tions impose some requirements and restrictions on the solution method andimplementation strategy.

Fast and accurate The solver has to be fast and accurate. The non-periodic and heterogeneous structure of a realistic porous medium makesthis requirement stringent for the application domain we concentrate on.As input for the Darcy solvers, not only the permeability of an undeformedtextile is required, but also the influence on the permeability of shear, pres-sure, nesting and tension. This results in a large number of samples onwhich simulations have to be performed.

The solver will also be used in a geological application to compute the localpermeabilities in a network-model of the porous rock. For a representativenetwork, 100 to 1000 simulations are required.

Automatic Once a textile model has been made, the permeability mustbe computed with a single click. Textile engineers are often not experts in

3.3. REQUIREMENTS FOR THE SOLVER 37

CFD, and prefer a black box solver, with the textile model as input andthe permeability tensor as output. For the computation of the permeabilityof rocks, the Stokes solver is part of a larger programme, and one doesnot want to intervene in the flow simulation. Also for the computation ofthe permeability of textiles, it would be convenient if the CFD-solver wereintegrated in the Darcy-solver. At present, this is not yet the case.

This requirement might seem a bit peculiar, however it has an importantinfluence on the choice of our solution method, for this requirement rules outthe use of unstructured meshes. Meshing is a cumbersome task we want toavoid. Moreover, textile models have sharp angles where two yarns overlapor nest. This makes it unlikely that meshers will provide us with a highquality mesh without any intervention of the user.

Periodic boundary conditions We simulate the flow on the meso-scalelevel, and assume for every simulation that the structure has a periodicpattern. Therefore, our solver must be able to impose periodic boundaryconditions for the velocity and the pressure. On top of the periodic bound-ary conditions, we want to be able to impose a pressure gradient on theunit cell as driving force. Our experience with available meshers generatingunstructured meshes, indicates that meshers have difficulties with periodicboundary conditions.

Symmetric boundary conditions The computation of the permeabilitiesof the nodes of the network-model requires symmetric boundary conditions.Symmetric boundary conditions correspond to the physical assumption thaton both sides of the boundary the same processes exist. The boundary canbe seen as a mirror. Sections 3.5 and 7.2.2 provide a more detailed discussionof this topic.

Narrow channels and fine geometries The solver must be able to sim-ulate the flow in narrow channels. This is for example necessary for woventextiles, that often have narrow fluid channels, certainly when sheared orcompressed.

On the other hand, random structures can be made of fibres with a diameterless than a tenth of the unit cell dimensions. The fine geometry of the fibreshas to be captured without too much effort.

Both requirements have had an influence on decisions that were made whileselecting and implementing the solution method. We come back to thisin the sections dealing with the choice of the spatial discretisation of the(Navier-)Stokes equations.

Sheared unit cells If textiles are deformed and put inside a 3D-shapedmould, shear occurs. To determine the influence of shear on the permeabil-


ity, the flow field inside a unit cell of a sheared textile has to be simulated.Unlike the straight textile model, a unit cell of a sheared model cannot berepeated along orthogonal axis. The code was extended to compute thepermeability of sheared textiles. Section 3.5 explains how.

3.4 Direct solution

Most available CFD packages, both commercial and open source, typicallydiscretise the (Navier-)Stokes equations on an irregular grid and solve theequations with a finite element or finite volume methodology. Here, we havechosen for a finite difference discretisation of the Navier-Stokes and Stokesequations on a regular grid. An irregular discretisation of the fluid regionof a porous medium is cumbersome. In the region where yarns of a textilecross each other, often sharp angles occur between two yarns, which resultsin meshes with a high aspect ratio. This high aspect ratio can be removedmanually by the user, but we are explicitly aiming at a fast and automaticsimulation tool. Automatic meshers have been developed and improvedmuch during the last years. However, our hands-on experience with differentmeshers, discussions with other researchers and the lack of published resultsfor simulations in textiles with unstructured meshes supports our choice forstructured meshes.

In the next section, the strategy of the NaSt3DGP solver and the adap-tations that were made are explained. Although the Navier-Stokes solverand the Stokes solver share some aspects, a fast solution of the resultinglinear system requires different approaches, which will be explained in para-graph 3.4.2.

3.4.1 The Navier-Stokes solver

As we are only interested in the steady state solution, we could opt for aniterative solver for the steady Navier-Stokes equations. Solvers based onNewton iteration have been established ([94], and references therein). Asecond option is solving the Navier-Stokes equations with a time steppingprocedure.

Discretisation of the time derivative NaSt3DGP employs the so-called Chorin projection method [22]. If we omit for the moment the exter-nal forces f , and apply the explicit Euler time discretisation, we write the

3.4. DIRECT SOLUTION 39

Navier-Stokes equations (3.5) as

un+1 − un

t + ∇pn = − [u · ∇u]n

+1

Re∆un in Ω (3.14a)

∇ · un+1 = 0 in Ω (3.14b)

un+1 = 0 on ∂Ωǫ\∂Ω. (3.14c)

Here, un is the velocity at time step n and t is the time step. The Chorinprojection method is given by two successive steps.

Step 1:Solve the momentum equations for an intermediate velocity field u∗

u∗ − un

t + [u · ∇u]n =1

Re∆un (3.15)

with boundary conditions

u∗ = 0 on ∂Ωǫ\∂Ω. (3.16)

Step 2:Project the vector field u∗ on a divergence-free vector field un+1

un+1 = u∗ −t∇p (3.17a)

∇ · un+1 = 0. (3.17b)

Application of the divergence operator to (3.17a) results in a Poisson equa-tion for the pressure from which we can obtain the pressure implicitly

1

t∇ · u∗ = ∆p. (3.18)

Then, we compute the velocity field un+1 of the next time step by equa-tion (3.17a). To solve the Poisson equation, boundary conditions for thepressure are required. By the projection of equation (3.17a) onto the outerunit normal of the domain’s boundary

∂pn+1

∂n=

u∗ − un+1

t · n

homogeneous Neumann boundary conditions for the pressure are induced,if we set for the intermediate velocity field

u∗ = 0 on ∂Ωǫ\∂Ω (3.19)

as implied in equation (3.16).


For the time discretisation, the NaSt3DGP code provides explicit Euleras well as second order Adams-Bashforth or Runge-Kutta schemes. Thepressure Poisson equation (3.18) is solved iteratively by methods like SOR,Red-Black Gauss-Seidel or BiCGStab [23]. To be sure that the Poissonequation with the homogeneous boundary conditions has a solution, thepressure values of the boundary cells are copied into the ghost cells priorto each iteration step [42]. Numerical experiments have shown that thisovercomes the singularity. The systems are solved with precision εp, i.e.the residual must be smaller than εp. The requested precision εp has aninfluence on the computational cost. A numerical example of the influenceof εp is given in section 3.9.

Semi-implicit time stepping Equation (3.14a) describes the explicitdiscretisation of the time derivative. To obtain a stable solution, a timestep restriction is imposed on t.The time step restriction for the convective terms is generally known as theCourant-Friedrich-Levy (CFL) condition. The CFL condition ensures thatconvection can take effect only on one further grid cell per time step. Forexplicit Euler discretisation, the CFL condition is

t ≤ minΩ

x|ux|max

,y

|uy|max,

z|uz|max

, (3.20)

with | · |max the absolute maximum over the whole computational domain.In the same manner, we restrict the diffusion to act no further than onegrid cell per time step

t ≤[

1

Re

(2

(x)2 +2

(y)2 +2

(z)2)]−1

. (3.21)

Volume forces can be included in the stability constraint for the convectiveterms (3.20) [23]. The linear approximation |ux|max + |fx|t is an upperbound for the velocity component in the X-direction with the volume forceincluded. The X-component of (3.20) is now

t ≤( |ux|max + |fx|t

x

)−1

, (3.22)

or solved for t

t ≤ 2

|ux|max

x +

√( |ux|max

x

)2

+4|fx|x

−1

. (3.23)


uy;i,j,k

ux;i−1,j,k

uy;i,j−1,k

uz;i,j,k

uz;i,j,k−1

ux;i,j,k

Figure 3.2: A cell from the staggered grid. The pressure is discretised inthe centre of the cell [50].

In permeability computations for porous media, we usually deal withReynolds numbers of Re ≈ 1 or smaller. In this low Reynolds numberregime, the CFL condition for the convective terms (3.23) usually allows fora much larger time step than (3.21), since the time step restriction for thediffusive terms depends strongly on the magnitude of the Reynolds number.Therefore, M. Klitz extended the NaSt3DGP solver with a semi-implicittime stepping procedure in her master thesis [50]. Although per time stepmore work is involved, a substantial speed-up in comparison with the ex-plicit method is achieved. For the technical details of this implementationthe reader is referred to Klitz [50].

Discretisation of the spatial derivatives For the discretisation inspace, the Navier-Stokes equations are discretised using a staggered grid.In the staggered grid approach, the pressure is discretised at the centre ofthe cells, while the velocities are discretised on the side surfaces (figure 3.2).This discretisation leads to a strong coupling between pressure and veloc-ities, and therefore avoids the occurrence of spurious oscillations in thepressure. From a numerical point of view, boundary conditions between thefluid and the solid region can be implemented in two ways:

• boundary values are set explicitly in the solid cells which are borderedby fluid cells;

• the boundary conditions are included in the equation to be solved inthe boundary points in the fluid region.


S

F

F

ux;i−1,j,k

S

uy;i,j+1,k

ux;i,j+1,k

ux;i,j,k

ux;i+1,j,k

ux;i,j−1,k

uy;i,j−2,k

Figure 3.3: A solid cell (S) cannot be bordered by two fluid cells (F) atopposite sides. To impose zero velocity at the boundary, the velocity com-ponent ux;i,j,k must be equal to −ux,i,j−1,k, but also to −ux,i,j+1,k in thiscase.

NaSt3DGP sets the boundary values explicitly. This leads to the require-ment that a solid point may not be bordered at two opposite sides by fluidpoints (figure 3.3). When the solid region forms very fine structures, as isthe case for random fibre assemblies, e.g. non-woven textiles, this constraintleads to a very fine mesh (finer than required to capture the geometry itselfand to obtain a sufficiently accurate solution).

Cell boundary conditions On the boundary of the unit cell, boundaryvalues for the velocity and pressure are required in the discrete stencils.Therefore extra cells, ghost cells, are added to the grid in which the valuescan be set explicitly. On figure 3.4, cell -1 and N+1 are the ghost cells.These values can be derived from several conditions:

• inflow,

• outflow,

• wall, with slip or no-slip conditions,


−1 0 1 ... N + 12 NN − 1

Figure 3.4: A one dimensional staggered grid showing the values that haveto be transferred if periodic boundary conditions are imposed. The cir-cles represent the discretisation of the pressure values, the squares are thevelocity values [50].

• symmetric boundary conditions,

• periodic boundary conditions.

In our applications no-slip or periodic boundary conditions are imposed onthe cell boundary (figure 3.1). No-slip boundary conditions do not requireextra attention, they are no different from the conditions on an obstacle in-side the unit cell. Periodic boundary conditions are imposed by transferringthe computed values from one ghost cell to the opposite ghost cell, and thisat every time step (figure 3.4).

If a body force f is used as driving force, the imposed force has no influenceon the boundary conditions. An imposed pressure gradient does have aninfluence, p±p instead of p has to be transferred to the opposite boundarycells of the direction in which ∇p is imposed. From a mathematical point ofview, the driving forces ∇p and f are equal. However, via condition (3.23),f has influence on the discrete time step, ∇p has no influence.

Stopping criterion Every 10 time steps, the permeability of the solutionat that time is determined, and compared with the previously computedpermeability. If the difference drops under a given Kerror, it is assumedthat a satisfactory convergence is reached and the iteration is stopped.


3.4.2 The Stokes solver

Discretisation of the time derivative Also for the Stokes equations atime stepping procedure can be used. As this only requires a small changein the Navier-Stokes solver, it is an option to consider. However, the timestep restriction is determined by the diffusive term, which is also present inthe Stokes equations. Hence, this would not lead to a speed-up of the com-putations, which is the only reason why one would like to solve the Stokesequations instead of the Navier-Stokes equations. The Stokes equations arelinear equations; thus a direct or iterative linear solver is more appropriatethan time steppping.

Discretisation of the spatial derivatives As mentioned before, theNavier-Stokes solver does not allow that a solid point is bordered at twoopposite sides by fluid points (figure 3.3). To avoid this disadvantage, theStokes equations are discretised on a non-staggered or collocated grid andthe boundary conditions are included in the equation that is solved at theboundary. If one solves the Stokes equations on a collocated grid, spuriousoscillations in the solution of the pressure field occur. The occurrence ofthese oscillations is indeed the reason to use a staggered grid. However,several techniques exist to obtain a stable solution on a collocated grid.One possible solution is to use the discretisation formulas of Rhie and Chow[86]. It has been shown that the Rhie-Chow interpolation is the same asadding a pressure term to the momentum equation, which is proportionalto a third derivative of the pressure. In the continuity equation the addedterm is proportional to a fourth-order derivative [25].

These formulas use a 13-point stencil instead of the 7-point stencil, i.e. fourneighbours in every direction. For the applications we are dealing with,often very narrow channels exist with only two or three neighbouring dis-cretisation points in the fluid region (unless a very fine discretisation isused). Therefore we have chosen to discretise with the basic central differ-ence schemes, and to add a stabilisation term to the continuity equation. Weadopted this approach from the finite element discretisation methods [32].We have chosen to add the term

κ

(

x2 ∂2p

∂x2+ y2 ∂

2p

∂y2+ z2∂

2p

∂z2

)

(3.24)

to the continuity equation. The choice of the parameter κ is not withoutconsequences: a large κ clearly introduces an error in the continuity equationand results in a non-divergence free flow field. On the other hand, a smallκ results in a slow convergence of the iterative solvers. If κ is chosen too

3.5. BOUNDARY CONDITIONS FOR SHEARED UNIT CELLS 45

small, the stabilisation effect disappears and unwanted pressure oscillationsare again present in the solution.

The driving force of the system is the body force f . Unlike for the Navier-Stokes solver, f does not influence a time step restriction, and if an imposedpressure gradient is set, special care has to be taken of the equations nearthe pressure jump.

All in all, the discretisation of (3.7) together with (3.24) yields

∂2ux

∂x2+∂2ux

∂y2+∂2ux

∂z2− Re

∂p

∂x= −Re

Frfx

∂2uy

∂x2+∂2uy

∂y2+∂2uy

∂z2− Re

∂p

∂y= −Re

Frfy

∂2uz

∂x2+∂2uz

∂y2+∂2uz

∂z2− Re

∂p

∂z= −Re

Frfz

∂ux

∂x+∂uy

∂y+∂uz

∂z+ κ

(

x2 ∂2p

∂x2+ y2 ∂

2p

∂y2+ z2∂

2p

∂z2

)

= 0,

(3.25)

with

u = 0 and∂p

∂n=

1

Re∆un on ∂Ωǫ\∂Ω. (3.26)

We repeat that ∂Ωǫ\∂Ω is the boundary between the fluid and the solidregion inside the domain Ω (figure 2.1). The discretised linear system issolved with the PETSc package [11]. The PETSc package offers a choice ofvarious linear, nonlinear, iterative and direct solvers. For the system (3.25)the Generalised Minimal Residual (GMRES) solver [89] with restart afterm iterations (GMRES(m)) is chosen.

Stopping criterion If a direct solver of the PETSc package is chosen, forexample the LU-solver, the solution is obtained in one single step. Iterativesolvers like the GMRES(m) solver, proceed until a residual smaller than thegiven εp is reached.

3.5 Boundary conditions for sheared unitcells

The unit cell of an undeformed fabric is orthogonal, i.e. the planes that formthe boundaries in the X-,Y- and Z-direction are orthogonal to each other.On the contrary, a unit cell of a sheared fabric is sheared itself, as the angle


Figure 3.5: A unit cell of a sheared fabric, with the orthogonal and thesheared unit cell boundaries.

between the boundary planes is now the complement of the shear angle(figure 3.5). As we solve the (Navier-)Stokes equations on a regular gridwith orthogonal boundaries, the boundary conditions have to be adaptedfor the sheared models. On figure 3.6 we see that if the unit cell is repeatedwith the normal periodicity, a shift occurs in the model, and fluid channelsare blocked.

A possible solution is to impose symmetric boundary conditions instead ofperiodic conditions. This is the approach B. Laine uses for his finite elementsimulations [[53],personal communication]. On figure 3.7 we see that thesymmetric boundary conditions do not block the flow paths like periodicboundary conditions do, however the model does not resemble reality.

An alternative approach is as follows. Consider a grid with dimensionsM,N,O, points xijk, and with m,n, o points in the X-,Y-, and Z-direction.If normal periodic boundary conditions are applied, the stencil of a pointxi1k on the lower boundary in the Y-direction, needs the pressure and ve-locity values from point xink. If the yarns in the direction of the Y-axis arenow sheared with a shear angle θ, then point xi1k needs the values at pointxsnk, with

s = Modulo(i+ shearOffset,m)

and

shearOffset =

⌊N tan(θ)

m

⌋

.

(3.27)

3.5. BOUNDARY CONDITIONS FOR SHEARED UNIT CELLS 47

Figure 3.6: If a sheared model is repeated along the orthogonal axes, a shiftin the model occurs, and fluid channels are unnaturally blocked (top). Thebottom figure shows how the unit cell must be repeated.


Figure 3.7: The resulting textile model if symmetric boundary conditionsare applied.

The value of s is the number of voxels a point at the boundary has beenshifted by the shear.

Figure 3.8 shows two velocity fields of a simulation on a sheared model:one with the normal periodic boundary conditions, and one with the shear-periodic boundary conditions. The figure demonstrates that in order toobtain a correct flow field, the adaptation of the boundary conditions is in-dispensable. Note that this straightforward implementation of the sheared-periodic boundary conditions is an extra argument to use a structured mesh.

3.6 Solution with a lattice Boltzmann simu-

lation

As part of the WiseTex family software, a lattice Boltzmann solver (LBsolver) for the permeability of textiles is available [14]. The lattice Boltz-mann model (LBM) is a meso-scopic approach to fluid dynamics. It hasbeen shown that the LBM can be used to solve the Navier-Stokes equations(3.5) [93]. The LB solver implements the permeability model based on theLBM D3Q19. Here, ”Q19” describes the connectivity pattern of the 3D lat-

3.6. SOLUTION WITH A LATTICE BOLTZMANN SIMULATION 49

X

Y

Z

Figure 3.8: Velocity ux in a sheared model with θ = −10, fx = fy = 0.01.Top: computation with normal periodic boundary conditions; Bottom: withshear-periodic boundary conditions.


Table 3.1: Comparison of the lattice Boltzmann solver and the Stokes solver;h is the constant lattice step in every direction.

Solver h (mm) Kxx(mm2)

LB 0.04 4.3E-4

0.02 3.4E-4

Stokes 0.04 3.5E-4

0.02 3.2E-4

Reference solution: Stokes 0.007 3.2E-4

tice (”D3”): every cell is connected with its neighbour and its next-nearestneighbours. The discretisation step h is taken constant in every direction,as this is required by the available LB solver.

On a grid with ±250000 points of the example textile (figure 2.6), theavailable LB solver took about 17min. to compute the permeability withKerror = 0.01; our Stokes solver only required 32 sec., with εp = 10−3f .Also, the Stokes solver converges faster with respect to the discretisationstep h, see for example table 3.1.

We do not give a more detailed comparison between the LB solver and theStokes solver, and do not claim that the above is valid for all LB solversand Stokes solvers. The LB software is compiled with the Borland C++6.0 compiler, the Stokes solver uses the PETSc package, compiled witha Microsoft Visual Studio 2003 compiler. Also, the LB solver solves theNavier-Stokes equations instead of the Stokes equations. Thus, we comparetwo different available solvers, rather than two methodologies. For a moredetailed description of the LBM and a comparison with a finite differencediscretisation method, we refer to [71, 81].

As stated, the LB solver requires the same step size h in the three direc-tions, in contrast with our Stokes solver. For simulations on samples of e.g.sandstone, this is not a prohibitive disadvantage. On the contrary, for tex-tile unit cells, the required discretisation step in the Z-direction is typicallyone order of magnitude smaller than in the other two directions. Recently,LBMs for a different step size in the three directions have been introduced[43, 49], but we did not implement them and cannot discuss their applica-bility.

3.7. CALCULATION OF THE PERMEABILITY 51

An other issue with the LBM, is its dependence on parameters that arenot directly connected with the physics. This makes it rather difficult toimplement an accurate LB solver. Without minimising the merits of theLB method, we have not found a reason to use the LB method instead ofa finite difference discretisation of the Stokes equations to solve the fluidproblem we are dealing with.

3.7 Calculation of the permeability

Once the flow field and the pressure distribution is computed, we have forevery point ijk on the grid four values: the three components of the velocityvector uijk, and the pressure value in that point pijk. If a grid point belongsto the solid region, these values are zero.

If the external force is a unity gravity force in the X-direction (fx) andRe = 1, then we can use the definition from the homogenisation theoryto compute Kxx. Let m,n, o the number of grid points in respectively theX-,Y- and Z-direction, then the volume Vijk of every grid cell is |Y |/(mno).

Kxx =1

|Y |

∫

YF

∇ux : ∇ux dy

(2.32)

=1

|Y |

∫

YF

ux dy

≈ 1

|Y |

i=m,j=n,k=o∑

i=1,j=1,k=1

ux,ijkVijk

=1

mno

i=m,j=n,k=o∑

i=1,j=1,k=1

ux,ijk

(3.28)

Mutatis mutandis, this also holds for the other components of K.

3.8 Developed software package

Input The input for the flow simulations is a voxel data file (figure 3.9).The textile modelling software packages WiseTex and TexGen can exportthe textile model to such a file format. Also Computed Tomography resultsin a voxel description of the visualised structure, which can easily be writtenout in the voxel file format.


Simulation software Our software for the computation of the perme-ability of textile models is available on different platforms. Windows userscan choose between a graphical user interface (GUI), named FlowTex, anda command line version. The strength of the GUI is its simplicity as only afew parameters have to be set and a single click is sufficient to compute thepermeability (appendix A). The user can choose between lattice Boltzmann,Navier-Stokes and Stokes simulations.

A command line version of the (Navier-)Stokes solvers simplifies batch runson different textile models, for example to perform parameter studies. TheGUI is developed with the Borland Builder C++ 6.0. However, as the Bor-land compiler was not sufficiently optimal1, the GUI calls the command lineprogram that is compiled with the Microsoft Visual Studio 2003 compiler.

The NaSt3DGP code works completely in parallel on MPI [3] platforms,both on Windows and Linux. The command line interface of the open-source version of NaSt3DGP, is extended to allow direct voxel file input.The solver was used both on single processor Linux machines and highperformance clusters of the K.U.Leuven and the University of Bonn. TheStokes solver shares the user interface with the NaSt3DGP solver, and usesthe PETSc package to solve the linear system of equations.

Output The result of a flow simulation is the permeability. The interfaceof the NaSt3DGP that was retained in our implementation, allows for theexport of the flow field solution to different file formats, e.g. VTK or Matlab.This is a very useful feature, not only to produce nice pictures, but also tocheck the correctness of the flow simulation during the development of thecode. For the user of the software, the visualisation of the flow field helpsunderstanding the filling process. The visualisation also helps tracing errorsin the textile model that result in an unnatural flow simulation.

3.9 Numerical example

As an example geometry, the same textile geometry as in the previous chap-ter is used (figure 2.6).

For an irregular geometry, it is not possible to refine the grid without chang-ing the problem that is being solved. On a finer grid, the geometry is betterapproximated, so a different problem is solved. If we check the convergenceas function of the grid resolution, we check the convergence as function of

1The Borland compiler cannot treat while-loops in inline functions. Such functions

are often used in the implementation of the Navier-Stokes solver.

3.9. NUMERICAL EXAMPLE 53

Figure 3.9: WiseTex and TexGen provide the input data via a voxel filefor the flow simulations with FlowTex. The voxel files contains for everypoint the letter S (solid) or F (fluid). The result of the simulations is thepermeability and data files with the flow field.


Figure 3.10: Convergence of the GMRES(m) solver to solve the linear sys-tem (3.25) with different settings.

two refinements at the same time: a finer grid, and a better approximatedgeometry. Tests have been performed on regular geometries to check theconvergence of the solvers as function of the grid refinement solely. How-ever, we do not present the results here, since these are only simple tests tocheck the algorithms and the software.

The GMRES(m) method, applied in the Stokes solver, can be used withdifferent restart values m, and with or without preconditioning. Figure 3.10and table 3.2 show that ILU preconditioning decreases the computation timesubstantially. Also, a higher restart value m results in less iterations anda faster computation time. The computations have been performed on anode of the K.U.Leuven HPC cluster, with a 2 Ghz Opteron processor. Thegrid consisted of about 90k fluid cells. It must be noted that this is a ratheracademic example as computations with a higher m need more memory.Also, there is an optimum in the choice of m, as each iteration step requiresmore and more work until the restart.

As mentioned, both solvers solve the systems of linear equations up to aresidal εp. The Stokes solver only solves one system to that accuracy, theNavier-Stokes solver checks the convergence of the permeability after everytime step. From figures 3.11 and 3.12 we can conclude that the parameterεp has a large influence on the computation times, but not on the computedpermeability. Note that εp denotes the residual of the solution of all the lin-

3.9. NUMERICAL EXAMPLE 55

Table 3.2: Computation time for the GMRES(m) solver to solve the linearsystem (3.25) with a requested residual of 10−4||f ||.

Preconditioner Restart Time (s)

ILU 250 188

ILU 100 214

None 150 297

ear systems. For the Stokes solver, this is also the global stopping criterion,for the Navier-Stokes solver this is only a stopping criterion for the solutionof the systems at every time step. For the Navier-Stokes solver a stoppingcriterion of Kerror ≤ 0.001 was used. For all the experiments, the appliedforce was f=[0.01,0,0].

Figure 3.12 shows a difference between the permeability results of theNavier-Stokes and the Stokes solver. A possible source of this differenceis that the Stokes solver does not take into account convection. However,the convection term of the solution obtained with the Navier-Stokes solveris zero (up to machine precision). Thus, for our application, the Stokesequations are valid and the convection is not the source of the differencebetween the two solvers. Another possibility is the influence of the cellsthat are deleted by the Navier-Stokes solver so that no obstacle cell is bor-dered by two opposite fluid cells. Figure 3.12 also shows the results of Stokescomputations on the geometry without these inadmissible cells. On a coarsegrid, this is the source of the difference between the solvers. However, onfiner grids less cells are deleted, and the difference has another reason.

The two solvers discretise the equations on different grids (staggered ver-sus collocated), and implement the boundary conditions differently. Thisresults in different discretisation errors, which explains the differences onfigure 3.12. Note that the difference decreases for finer grids.

It must be mentioned that the example textile is a rather academic case.Computations on other textile structures have shown that εp has a largerinfluence than can be seen for this example. From the figures, one couldconclude that for the Stokes solver εp = 10−2||f || is sufficient. However,other experiments have shown that εp = 10−4||f || is a safer option. Thesame holds for the Navier-Stokes solver, for which we suggest εp = 10−8||f ||.


Figure 3.11: Computation time for the Stokes and the Navier-Stokes solveras function of the mesh size for different accuracies εp (for the textile offigure 2.6).

3.10 Conclusion

In this chapter the Navier-Stokes and Stokes equations for the mathematicaldescription of fluid flow were discussed. For low Reynolds numbers, thenonlinear convection term of the Navier-Stokes equations can be neglectedand the Stokes equations are sufficiently accurate. The results of a Navier-Stokes and a Stokes simulation showed that for computing the permeabilityof textiles, the Reynolds number is low enough to neglect the convection.This corresponds with the conclusion of the first chapter which showedtheoretically that the solution of the Stokes equations provides us with allnecessary information to compute the permeability.

A solution method for the simulation of the fluid flow through porous mediahas to satisfy some specific requirements, which influence the choice of themesh and the discretisation method. We presented the development of aNavier-Stokes solver which uses a time-stepping method on a staggered grid,and a Stokes solver which solves the linear system with an iterative methodon a collocated grid. The iterative solution of the linear Stokes solver issubstantially faster than the time-stepping method applied for the non-linear Navier-Stokes equations. Moreover, the discretisation of the Stokesequations on a regular, collocated grid made it possible to meet the require-ments of a solver for the permeability of textiles, such as the ability to be

3.10. CONCLUSION 57

Figure 3.12: Computed permeability as function of the mesh size for thetextile of figure 2.6. For different εp the same permeability value is ob-tained. Also the result for Stokes computations on the geometry withoutthe inadmissible cells is shown.

used on structures with a fine geometry.

Special attention was paid to the boundary conditions for models of shearedtextiles.

Chapter 4

Computation of theintra-yarn flow

In the previous chapter, the computation of the permeability on the meso-scale was discussed. We considered a unit cell of a textile model, assumingthat the unit cell contained only a solid and a fluid region, and we simulatedthe fluid flow in it. However, in a textile, the yarns are actually permeablethemselves.

A possibility to compute the intra-yarn flow, is to model the fibre structure,and to solve the Stokes equations on the whole domain. However, thefibres’ dimensions are at least one order of magnitude smaller than the yarndimensions. To capture such a detailed geometry, the grid should be refinedby at least a factor 10 in every direction, resulting in a prohibitive numberof grid points for a fast permeability prediction.

We implement a feasible computational method to include the intra-yarnflow into the simulations: a penalisation term is added to the Stokes equa-tions. The value of the penalisation is computed theoretically as a functionof the volume fraction and the direction of the fibres. We discuss differentformulas to compute the penalisation and their influence on the meso-scalepermeability.

59

60 CHAPTER 4. INTRA-YARN FLOW

Figure 4.1: A representation of the random packing of the fibres inside ayarn. Compare with figure 1.1, where a picture of real fibres is shown.

4.1 Representation of the yarns

Figure 4.1 shows a 2D cut of the packing of the fibres at different locationsin the same yarn. Although the fibres have a round profile and a cylindricalshape, they are not straight and are randomly packed. The latter makes itimpossible to model the geometry of the fibres inside a yarn deterministi-cally. Therefore, we model the fibres as a regular packing of cylinders withthe same volume fraction and diameter as the real fibres. The cylinderslocally have the direction of the yarns.

We can choose between different regular packings of cylinders; the squareand hexagonal packing are discussed here (figure 4.2). Various formulas havebeen derived to compute the permeability of these regular structures. In thenext section we compare three formulas with the results of our numericalsimulations.

Figure 4.2: Unit cell of a square and hexagonal packing of an array ofcylinders.

4.2. COMPUTATION OF THE LOCAL PERMEABILITY 61

4.2 Computation of the local permeability

A first formula is the Kozeny-Carman equation [20], which relates the per-meability to the volume fraction of the structure:

K =r2

4kk

(1 − Vf )3

V 2f

, (4.1)

with r the radius of the cylinders, kk the Kozeny-Carman constant, and Vf

the volume fraction of the packing, defined as

Vf =fluid domain volume

total volume. (4.2)

The formula was not designed for the special structure of a cylinder packing,but for the permeability of granular beds. This explains some peculiaritiesof the formula. Firstly, the formula assumes isotropic permeability (unlesskk is changed for every direction), which is not the case for the cylinderpacking. Secondly, the permeability is larger than zero for volume fractionsat which the transverse flow is blocked. Still, the biggest disadvantage of theformula is that the parameter kk, the Kozeny-Carman constant, must bedetermined empirically. And yet, the Kozeny-Carman equation is frequentlycited in papers on textile permeability and is often used by textile engineersas a rule of thumb.

A second approach is presented by Berdichevsky and Cai, who derived theirformulas with the self-consistent method [15]. This method assumes a typ-ical basic element of a heterogeneous medium being embedded in an equiv-alent homogeneous medium whose properties are unknown and to be deter-mined. Alternatively, consider a homogeneous porous medium, in which theflow follows Darcy’s law, with a permeability K. If a region of the homoge-neous medium is now replaced by a heterogeneous cell, without disturbingthe overall flow, the heterogeneous part must have the same permeabilityK on average. This approach results in

Kalong =r2

8Vf

[

ln1

V 2f

− (3 − Vf ) (1 − Vf )

]

Ktrans =r2

8Vf

(

ln1

Vf−

1 − V 2f

1 + V 2f

)

,

(4.3)

with Kalong the permeability in the direction along the cylinders’ axis andKtrans the permeability in the orthogonal direction (transversal flow). Con-trary to (4.1), these formulas are different for the two directions of flow.


However, the formulas do not take into account the kind of packing of thecylinders. The permeability of an array of cylinders depends nevertheless onthat. Therefore, the authors present formulas based on a curve fitting of theresults obtained with a finite element simulation. For the square packingthe formulas are

Kalong =e(B(Vm)+C(Vm)Vf )

Vm(Vm)f

r2

Ktrans = A(Vm)

(

1 −√

Vf

Vm

) 52

(√Vf

Vm

)n(Vm)r2,

(4.4)

where

A(Vm) = 0.244 + 2 (0.907− Vm)1.299

B(Vm) = 5.43 − 18.5 Vm + 10.7 V 2m

C(Vm) = −4.27 + 6.16 Vm − 7.1 V 2m

m(Vm) = −1.74 + 7.46 Vm − 3.72 V 2m

n(Vm) = 2.051 + 0.381 V 4.472m .

(4.5)

Here, Vm is the maximal possible volume fraction without merging of thecylinders (table 4.1).

The third approach that we consider is the one from Gebart [39]. Gebartobtained formulas for the permeability transverse to the fibre direction bytreating the cylinder packing in that direction as a channel with slowlyvarying height. For the permeability along the cylinders, the friction factorλ, function of the pressure drop and average velocity, is considered. Sub-stitution of the geometric parameters into the formula for λ results in thefollowing formulas for the permeability

Kalong =8r2

c

(1 − Vf )3

V 2f

Ktrans = C1

(√

Vm

Vf− 1

) 52

r2,

(4.6)

with C1, Vm and c the values from table 4.1. Note the similarity between theKozeny-Carman equation (4.1) and the formula of Gebart for the perme-ability in the direction of the cylinders. The corresponding Kozeny-Carmanconstant for the two different packings is also mentioned in table 4.1.

4.3. STOKES-DARCY-BRINKMAN 63

Table 4.1: Numerical values of the parameters in equations (4.4) and (4.6),and the Kozeny-Carman constant kk of the structure for longitudinal flow.

Packing C1 Vm c kk

Quadratic 169π

√2

π4 ≈ 0.78 57 1.78

Hexagonal 169π

√6

π2√

3≈ 0.90 53 1.66

Figures 4.3 and 4.4 compare the formulas of Berdichevsky and Gebart withnumerically computed values. From the figures we conclude that

• the formulas (4.4) of Berdichevsky agree well with the numerical re-sults for both the flow along and transverse to the cylinders. Thus,we have good formulas for the computation of the permeability of thepacking. Moreover, our numerical simulations give the same values asthe simulations of Berdichevsky.

• also Gebart’s formulas (4.6) are in good agreement with the numericalvalues, although we suggest c = 100 instead of 57 for the lower volumefractions.

In realistic structures, the yarns do not lie in a structured manner, parallel tothe unit cell axis. Thus neither do the fibre bundles. The presented formulasgive the permeability along and orthogonal to the direction of the cylinderaxis. To be useful in the meso-scale computations, these permeability valuesare first projected into the unit cell coordinate axis system.

4.3 Stokes-Darcy-Brinkman

Assuming that the local permeability in the yarns is known, the fluid flowinside the whole unit cell can now be simulated. The yarns form a porousmedium themselves, so the Darcy regime holds. But in between the yarns,the Stokes equations hold. To simulate the fluid flow on the meso-scale, bothequations have to be solved in their domain, yielding a boundary conditionproblem on the porous-fluid interface.

A possible solution to tackle the boundary problem is to impose explicitboundary conditions. However, Darcy’s equation and the Stokes equationhave a different order of differentials and the velocity in Darcy’s law is an


Figure 4.3: Permeability values of a parallel array of cylinders with squarepacking; comparison between our CFD results (chapter 3) and the formulasof Berdichevsky (4.3) and (4.4).

averaged velocity, which is not the case in the Stokes equations. Amico andLekakou discuss the boundary problem for a simple geometry and assumethat the flow front through a unidirectional fibre mat only changes in twodirections [9]. The boundary condition is obtained following the concept ofBeavers and Joseph [13], stating that the slope of the inter-yarn velocityprofile at the interface should be proportional to the difference betweenthe interface velocity ub, and the mean intra-yarn velocity umi. In thei-direction, this boundary condition is written as

dui

dxi=

α√K

(umi − ub), (4.7)

with K the permeability of the porous medium, umi the velocity in theporous part, ub the velocity at the boundary and α an empirical constantdepending on the material. Note that isotropy is assumed, and that thepermeability of the porous medium can thus be denoted with the scalar K.Although Amico and Lekakou present good agreement with experimentalresults, it is not clear how to determine α in an unambiguous way.

For a simplified model of Stokes and Darcy flow, Simacek and Advani [107]derive boundary conditions based on physical assumptions. However, the

4.4. HOMOGENISATION 65

Figure 4.4: Permeability values of a parallel array of cylinders withsquare packing, comparison between the CFD results and the formulas ofGebart (4.6).

simplified model only works for “flat” structures and the authors do notmention experimental validation.

Brinkman proposes an easier solution [19]. Adding a penalty term to theStokes equations (3.7) to account for the resistance of the obstacles leads to

∆u = Re∇p− Re

Frf − K−1

localu (4.8a)

∇ · u = 0, (4.8b)

with Klocal the local permeability tensor, finite in the yarns, infinite in thefluid region. Note that if Klocal is infinite, equations (4.8) are the Stokesequations. This equation has been used successfully before, for example inBhatt and Nirmal [16] and Iliev and Laptev [48].

4.4 Homogenisation

In chapter 2, Darcy’s law has been derived by homogenising the Stokesequations from micro-level to meso-scale. Also the Brinkman law can be


derived via homogenisation. This section is based on [50].

For the derivation of Darcy’s law, we imposed the condition

limε→0

aε

ε= C with C /∈ 0,∞. (4.9)

In the following we assume that for our porous medium, the restriction (4.9)is not valid. We assume that aǫ goes faster to zero than ǫ,

limǫ→0

aǫ

ǫ= 0. (4.10)

Depending on the scaling of aǫ, there are three different limit flow regimesrepresented by three different macroscopic equations. For a so-called criticalsize acrit

ǫ , the homogenised problem is a Brinkman’s law. For larger obsta-cles we find again Darcy’s law, but a considerably different one than wehave obtained previously. If the obstacles are very small, the macroscopicproblem reduces to the initial Stokes equations. In the derivation of thesemacroscopic equations, we will follow Allaire’s works [7] and [8] to whichwe also refer for the proof of equations (4.11-4.13).

Let us begin with the definition of the critical obstacle size acritǫ .

Critical Size of the Obstacles The so-called critical obstacle size acritǫ

is defined by

acritǫ ≡ C · ǫ n

n−2 for n ≥ 3

acritǫ ≡ e−

C

ǫ2 for n = 2,

where C is a strictly positive constant. For our application, the dimensionn = 3. In this context, n ≥ 3 is only of mathematical importance. We alsodefine σǫ,

σǫ ≡(

ǫn

an−2ǫ

) 12

for n ≥ 3

σǫ ≡ ǫ ·∣∣log

(aǫ

ǫ

)∣∣12 for n = 2.

Homogenised Stokes Equations for Small Obstacles There are threedifferent limit flow regimes depending on the size of the obstacles.

1. If limǫ→0 σǫ = +∞, then (uǫ, pǫ) converges strongly to (u, p) in[H1

0 (Ω)]n × [L2(Ω)/R], where (u, p) is the unique solution of the (di-

4.4. HOMOGENISATION 67

mensionalised) Stokes equations

∇p− µ∆u = f in Ω

∇ · u = 0 in Ω

u · n = 0 on ∂Ω.

(4.11)

2. If limǫ→0 σǫ = σ > 0, then (uǫ, pǫ) converges weakly to (u, p) in[H1

0 (Ω)]n × [L2(Ω)/R], where (u, p) is the unique solution of theBrinkman-type equations

∇p− µ∆u − µ

σ2Mu = f in Ω

∇ · u = 0 in Ω

u · n = 0 on ∂Ω,

(4.12)

where M−1 is some kind of permeability tensor, however not a physicalone (see further).

3. If limǫ→0 σǫ = 0, then (uǫ

σ2ǫ, pǫ) converges strongly to (u, p) in

[L2(Ω)]n× [L2(Ω)/R], where (u, p) is the unique solution of the Darcyequations

u =M−1

µ(f −∇p) in Ω

∇ · u = 0 in Ω

u · n = 0 on ∂Ω.

(4.13)

In (4.11) the obstacles are too small and the limit flow regime is again Stokesflow. If the size of the obstacles is critical as in (4.12), a supplementary termhas to be taken into account in the Stokes equations. In (4.13), the obstaclesare comparably large and Stokes flow degenerates into Darcy flow.Note that Darcy’s law in (2.18) has little to do with the one derived herein (4.12-4.13). The permeability tensors K and M−1 take in both casesdifferent values and are computed by different Unit Cell Problems aroundthe model obstacle YS . For the computation of M, the local problem occursin the entire space around the obstacle, and no longer only in the unitperiod of the porous medium with a periodic boundary condition, when theobstacle and the period have the same size ǫ. We refer to [7] for a definitionof the local problems and of M.

We have now two different permeability tensors K and M−1. The tensorK is valid for porous media for which restriction (4.9) holds, i.e. for a largerange of obstacle size smaller than ε. However, if the obstacle size is equal


to ε, abruptly M is valid. This is not compatible with the physical pointof view that expects some continuity. Accordingly, Allaire shows in [6] thatactually there is a continuous behaviour of the permeability tensor.

Apart from two tensors, we also have four limit flow regimes. The limitflow regime, i.e. the equation that holds on the macro-scale, are the Stokesequations (4.11), the Brinkman equations (4.12), Darcy’s law (4.13) or theother Darcy law (2.18). In the previous two chapters, we have assumed thatfor our application Darcy’s law (2.18) is valid on the macro-scale, and thuscomputed the permeability via (2.18). This is a correct assumption. Laptevshows that the limit equations (4.12-4.13) are a good approximation, butonly if the additional constraint aε ≪ ε is made [54]. This means that theBrinkman equation holds for porous media with extremely high porosity(φ > 0.95) ([50], and references therein).

However, in this thesis, we do not use the Brinkman equation as a macro-scale law. The Brinkman equation is applied here to avoid the difficultboundary conditions between a region with free fluid flow and a regionwith porous material. In the fluid region, the Brinkman equation (4.8)with infinite permeability is equal to the Stokes equations. In the porousregion, the local permeability is finite, and for the yarns of a textile typically10−7 ≪ ||Klocal|| ≪ 10−4. As the porosity of yarns is smaller than thelimit 0.95 stated above, the fluid regime satisfies Darcy’s law inside theyarns. Since the permeability of the yarns is low, the penalty term in theBrinkman equations dominates the Laplacian term, and we obtain a goodapproximation for Darcy’s law. For a more profound error analysis in thisregard, we refer to Klitz [50].

4.5 Influence on the meso-scale permeability

For an elementary structure as the parallel array of cylinders, the formulas ofGebart and Berdichevsky give different results. For some volume fractionsthe results even differ by an order of magnitude. However, in the scopeof this thesis, the permeability of the yarns themselves is not our primaryconcern, but the meso-scale permeability is. To investigate the influence ofthe applied formula on the meso-level, consider the example textile model(figure 2.6), of which the yarns are made virtually permeable. For differentdensities of the yarns, figure 4.5 shows the computed permeabilities withthe different formulas. Only for highly porous yarns, the formula used hasan influence on the meso-scale permeability.

If one of the diagonal elements of Klocal is smaller than 10−14, the cell wasassumed solid. This explains why the permeability values computed with

4.5. INFLUENCE ON THE MESO-SCALE PERMEABILITY 69

Figure 4.5: The permeability of the example textile with (virtually) perme-able yarns, showing the influence of the formula used for the computationof the local permeability.

the formula for a square packing, is equal to the permeability of the unitcell with impermeable yarns for lower volume fractions. For the three cases,the permeability values drop abruptly to the solid-yarn value. This cannotbe explained by a sudden lack of fluid flow inside the yarns, as on expectsthat this would result in a smooth convergence. The sudden drop of thepermeability value can be explained by the change in boundary conditions.If the intra-yarn flow is accounted for in the computations, the fluid isallowed to move on the boundary between the fluid and the porous region.This corresponds to slip boundary conditions. On the contrary, between thefluid and the solid region, we impose no-slip boundary conditions, whichresults in a lower computed permeability.

Note that this is a numerical exercise only, as in reality the yarns of thistextile are impermeable, and also that we deal with a textile model whichhas relatively large fluid region in between the yarns.


4.6 Influence on the implementation and thecomputation

Equation (4.8) does not differ much from the Stokes equations (3.7). Con-sider the discretised momentum equation in the X-direction at point i

Repi+1 − pi−1

2∆x=ui−1 − 2ui + ui+1

∆x2−(

K−1

local,xx

)

iui. (4.14)

The added penalisation term only influences the coefficient of ui. As theoff diagonal elements of K−1

localare an order of magnitude smaller than the

diagonal elements, these elements are neglected. This makes the additionto the finite difference (Navier-)Stokes code quite straightforward, both forthe implicit and explicit Navier-Stokes solver, as well as for the linear Stokessolver.

If the Brinkman equations are solved on the whole domain, with finite orinfinite local permeability, every cell is considered as a fluid cell. As the localpermeability tensor of the yarns differs from point to point, for every pointinside the yarn, the tensor information has to be stored. This influences

• the program memory requirements. Apart from the matrices for thevelocity in the three directions and the pressure, also three matricesfor the diagonal elements of the local tensor must be stored;

• the same accounts for the voxel file size;

• the computation time. Both the Navier-Stokes and the Stokes solverignore the solid points in their computations. In this case howeverno solid points are present, and thus the computations have to beperformed on every point. For a porous medium with a porosity of30%, this results in 230% more points to include in the computations.However, on the former boundaries between the fluid and solid region,now a much smoother fluid profile is computed. This results in a betterconvergence of the solvers.

4.7 Conclusion

The computation of the permeability of a porous medium is a multi-scaleproblem. More specific in the application field of textiles three levels ofpermeability exist

• the macro-level, or the permeability of a textile layer;

4.7. CONCLUSION 71

• the meso-level, or the permeability of a textile unit cell;

• the micro-level, or the permeability of a yarn of the textile.

To capture the micro-level on a meso-scale level, a prohibitively fine compu-tational grid is required. Therefore, the geometry properties on the micro-level are scaled to the meso-level via a local permeability term. This localpermeability is then used to solve the Brinkman equations on the wholedomain. It was shown that the solution of the Brinkman equation is astraightforward method to tackle the boundary problem, that would ariseif we would use the coupling of a Stokes and a Darcy solver.

To estimate the local permeability, or local penalisation, the geometryof the fibres in the yarns is approximated by a regular packing of cylin-ders. The permeability of the regular structure can then be calculated withan analytical formula. Several analytical formulas have been compared.Berdichevsky’s interpolation formula and Gebart’s formulas are in goodagreement with our numerical results.

Although the analytical examination of the permeability of a regular pack-ing of cylinders is an interesting topic, the influence of the used formula onthe meso-level is minor. Only for yarns with an unrealistically high poros-ity, the formula used on micro-level influences the meso-scale permeabilitysignificantly, at least for the model used in this chapter.

Chapter 5

Experimental validationwith textilereinforcements

Composite materials consist of a textile reinforcement filled with hardenedresin. The manufacturing of a composite part with the Resin TransferMoulding (RTM) technique involves the impregnation with resin of the re-inforcement, which is placed in a mould of the desired shape. This impreg-nation process can be simulated with Darcy solvers. However, they needthe permeability of the reinforcement as crucial input.

In the previous chapters, a permeability prediction tool was presented, basedon the finite difference discretisation of the Stokes equations. The strengthof this approach is its general applicability. In this chapter we show that themethod can indeed be applied on different textile types, and we compare thecomputed permeability values with experimental results. The influence ofthe nesting of layers and the influence of shear is treated. In the last sectionof this chapter we discuss the influence of the variations at the meso-scaleon the macro-scale permeability.

73

74 CHAPTER 5. VALIDATION

Figure 5.1: The RTM process (figure from [21]).

5.1 Resin Transfer Moulding

The RTM process takes place in several steps (figure 5.1):

1. the design and manufacturing of the mould;

2. putting the textile reinforcement into the mould and closing themould;

3. resin flow through the textile;

4. hardening of the resin;

5. opening the mould and taking out the finished part.

For our application, we are mostly interested in the third stage, the resinflow. The set-up of figure 5.1 has one inlet on top and two outlets on theleft and right side. Another option would be to put one inlet on the leftand one outlet on the right. However, the impregnation process would thentake twice as long. For this simple mould, two possible locations of thein- and outlets make sense. One can imagine that for a complex mouldthe positioning of the in- and outlets is a difficult task, not only for theproduction time, but also for the product quality. If the design of themould is not well considered, dry, and thus weak, spots occur in the finalproduct (figure 5.2).

To avoid dry spots and to optimise the production process, the impregnationprocess is simulated with different mould models before a prototype mould

5.2. EXPERIMENTAL SET-UPS 75

Figure 5.2: A poor mould design results in dry spots in the finished part.The picture shows an automotive part, made of composite materials butwith dry spots [2]. Left is the result of a macro-simulation.

is made. The moulds are first designed with CAD software, and then theimpregnation process is simulated using Darcy solvers like PAM-RTM [4]or LIMS [5]. Darcy solvers use Darcy’s law (5.1) in a different way thanexplained in the previous chapters: instead of simulating the flow to findK, the Darcy law

q =Re

L2K

(f

Fr−∇p

)

(5.1)

is solved to simulate the fluid flow with a given K. Therefore, the Darcysolvers need the permeability of the textile reinforcements as input param-eter. Unfortunately, the permeability tensor of a single layer of the straighttextile is not enough as input for the simulation of the impregnation process.If several layers are put on top of each other in the mould, the layers nest.Nesting refers to the phenomenon of two layers that penetrate into eachother in order to obtain an optimal configuration (figure 5.5). This nestinghas a considerable influence on the permeability. Next to that, shearing ofthe textile layers is inevitable, and has also an influence on the textile per-meability. We will show that the latter makes it also hard to experimentallydetermine the permeability.

5.2 Experimental set-ups

To validate our method and models, we compare computed permeabilityvalues with experimental results. The experimental values presented in thisthesis are provided by different researchers and institutions. It appears thatpermeability values obtained using different set-ups can differ significantly.


Figure 5.3: The PIERS set-up and one of the sensor plates with 60 sensors.

Our main partner for the experimental validation is the Vrije UniversiteitBrussel; the majority of the results have been published in Hoes [44]. Forthe development of a new reference specimen, we worked together withG. Morren [80]. The experimental prediction of the permeability is per-formed at the V.U.B. with a highly automated central injection rig, calledPIERS set-up. This PIERS (Permeability Identification using Electrical Re-sistance Sensors) set-up consists of a mould cavity with two sensor plates,each containing 60 electrical sensors (figure 5.3) [78]. After placing the re-inforcement and closing the mould, the test fluid can be injected. This isdone centrally in the reinforcement through a hole in the middle of the lowersensor plate. While the flow front propagates through the reinforcement,the fluid flow makes contact with the electrical sensors. Since an electri-cally conductive fluid is used, the wetting of these DC-resistance sensors willchange their electrical resistance. This variation is registered and hence anarrival time for the sensors can be stored. From this data, the experimen-tally determined permeability is computed with an inverse method [78, 79].

As mentioned, different set-ups exist. For example, at the K.U.Leuven,the plates of the mould have no sensors, however, the top of the mouldis transparent (figure 5.4). With a camera pictures of the flow front aretaken with a constant time interval during the impregnation. Given thesepictures, the average speed of the flow front can be determined.

5.3. INTERPRETATION OF PERMEABILITY VALUES 77

Figure 5.4: The K.U.Leuven test RTM set-up.

In the next section we discuss some of the issues experiments suffer from.It will explain why we develop a permeability simulation tool and also whywe will not have exact similarity between the computed and experimentalpermeability values.

5.3 Interpretation of permeability values

5.3.1 Experimental errors

Hoes [44] and Gommer [40] describe in a precise way the difficulties andsources of errors which occur with the experimental permeability estimation.We will only discuss the problems which are independent of the used set-up. Before comparing experimental and numerical results, some propertiesof the setup and experiments should be discussed.

• Viscosity of the fluid Once the arrival times are detected, the per-meability can be computed. However, it depends on the viscosity ofthe fluid. Measurements of Morren showed that the viscosity provided


by the supplier of the resin is not always correct. Moreover, Gommerfound out that the fluid he used, a fluid used in numerous previoustests, was not Newtonian.

• Cavity thickness In order to determine the volume fraction of tex-tile inside the mould, the cavity thickness has to be measured. Wewill show that the volume fraction has an important influence on thepermeability validation. Gommer mentions that because of the highpressure put on the mould, the transparent plate deflects during theexperiments. The PIERS set-up suffers less from deflection. However,both cavities have a minimal thickness, which makes it impossible toperform experiments on one layer of the textile.

• Nesting Because of the foregoing, several layers of textile have to beput inside the mould to obtain a certain volume fraction. With that,nesting of the layers occurs. Figure 5.5 shows the example textile withthe different possible amounts of nesting. This nesting is randomlydistributed inside the mould and the nesting will be different fromexperiment to experiment.

All this makes it impossible to obtain one experimental value for thepermeability of a textile. Scatter in the results is observed and anaverage value has to be taken. This average value will in most casesdiffer from set-up to set-up with which the experiments are performed.

5.3.2 Modelling issues

The input for the textile modelling software (for woven textiles) is

1. the weave pattern;

2. the yarn dimensions, in warp and weft direction;

3. the spacing between the yarns (3D);

4. the yarn’s bending properties.

The first parameter is not difficult to determine. The latter three howeverare. The textile’s manufacturer normally provides the linear density or thefibre count. Based on this data, the dimensions must be estimated [68].However, the provided data does not always agree with the actual measure-ments on the textile. Moreover, the shape of the yarns and the spacing willchange because of the compression in the mould.

5.3. INTERPRETATION OF PERMEABILITY VALUES 79

Figure 5.5: Micrograph of actual composite material (left) and computersimulation of random stacking of multiple layers (right) [70].

Next to that, the textile is modelled in a unit cell, as the undeformed (ideal)textile has a periodic pattern. The repeated unit cell should represent themacro textile, but in reality on the macro-scale differences between themeso-scale parts exist. In section 5.7 we explain in more detail the influenceof the variability on meso-scale on the macro-scale.

In addition to these two issues, every type of textile has its own difficultiesin modelling. We present results for woven, non-crimp, and random fab-rics. The modelling parameters and difficulties of obtaining them will beexplained for every textile in the concerning paragraph.

5.3.3 Flow model and numerical errors

In chapter 3, the numerical methods for the solution of the (Navier-)Stokesequations are discussed in detail. The equations are solved with a finitedifference discretisation which results in discretisation errors. Other possiblesources of errors are the follwing.

• The Stokes equations are only valid for Newtonian fluids. In reality,no perfect Newtonian fluids exist. Experiments are performed withalmost Newtonian fluids, or a correction factor is applied. Thus, tocompare the computed permeability values with the experiments, the


Newtonian assumption is acceptable. If for the manufacturing of realparts highly non-Newtonian fluids are used, corrections have to beapplied.

• At the boundary, the pressure gradient is approximated to be zero(Navier-Stokes) or the pressure is stabilised with an extra term in thecontinuity equation (Stokes).

• The discretisation is done on a regular grid, which results in a firstorder approximation of the geometry of the textile model.

• We assume laminar single-phase flow, and consequently take theStokes equations as mathematical model for the flow through the unitcell. We neglect some physical aspects however, which we believe tohave an negligible influence on the overall permeability. For example,the capillary effect inside the yarns is not taken into account in ourmodel.

The latter is not self-evident. In engineering, a rule of thumb says thatthe flow inside a porous medium is governed by the capillary effect if thecapillary number is less than 10−5. The capillary number Ca relates theviscosity µ and velocity u∞ to the fluid surface tension γ

Ca =µu∞

γ. (5.2)

Rudd et al. [88] discuss in detail the influence of Ca on the void forma-tion in the composite piece. Capillary numbers of the order of magnitudefrom 10−5 to 10−3 are mentioned for the flow in LCM processes. As a lowcapillary number results in a high void formation, one will try to applyhigher pressures to increase Ca. For the simulation of these processes, wecan neglect the capillarity. If macro-simulations are performed for low Ca,the capillary effect must be compensated for.

5.4 Stereo-Lithographic specimen

As mentioned in the previous section, the experimentally determined per-meability values suffer from a large scatter and the textile models are notexactly the same as the real textile.

In principle, the sources of scatter are not a substantial problem, as theyalso appear in the effective mould in which the composite parts are made.It is even interesting to have a certain error prediction on the average per-meability value. However, for the Darcy simulations it is also important

5.4. STEREO-LITHOGRAPHIC SPECIMEN 81

to be able to interpret the experimental value correctly. If several institu-tions and several set-ups yield different values, a standardisation is essen-tial. Therefore, at the V.U.B., Morren designed a specimen produced withstereo-lithography (SL specimen) to eliminate the mentioned issues withthe experimental determination of the permeability and the problems tocompare the results with the numerical values [78, 79, 80, 102].

5.4.1 The stereo-lithographic production process

Stereo-lithography (SL), also known as 3D layering or 3D printing, is usedto create three-dimensional objects from liquid photosensitive polymers thatsolidify when exposed to ultraviolet light. The stereo-lithography machinehas generally four important parts

• a tank filled with a liquid photopolymer that is sensitive to ultravioletlight;

• a perforated platform immersed in the tank. The platform can moveup and down in the tank as the printing process proceeds;

• an ultraviolet laser;

• a computer that drives the laser and the platform.

At first, a 3D model of the requested object has to be created in a CADprogram. Dedicated software chops this CAD model up into thin layers.Then, the 3D printer laser paints one of the layers, exposing the liquid plasticin the tank and hardening it. Subsequently the platform drops down intothe tank a fraction of a millimetre and the laser paints the next layer. Thisprocess repeats, layer by layer, until the whole model is complete. Finallythe object has to be rinsed with a solvent and can then be post-cured in anultraviolet oven.

5.4.2 The design

The SL test specimen has to satisfy several requirements. The goal is toobtain a test specimen for a 2D central injection rig that allows flow frontmeasurements in about 30 seconds using a moderate pressure of 1.5 bar.Therefore, the magnitude of the permeability has to correspond to thosereported in literature [44]. It has to be noted that the obtainable magnitudeof the permeability is obviously limited by the SL production techniquebecause e.g. the holes in the unit cell could not be made smaller than 0.5mm.


Figure 5.6: The CAD design of a unit cell of the SL specimen [80].

The design of the structure is composed of five layers, each having a thick-ness of 0.6mm, which are joined to form the unit cell, so that the totalthickness of the structure (3mm) corresponds to the mould cavity of the2D central injection rig (figure 5.6). Furthermore, the structure has to beorthotropic so that a sufficient anisotropy ratio is present. In materials sci-ence, an orthotropic material is one that has different materials propertiesin the three orthogonal directions. The anisoptropy is the ratio between thepermeability in two orthogonal directions

α =Kyy

Kxx. (5.3)

Since the geometry design of the SL specimen is based on the repetitionof a small unit cell, these requirements can easily be verified before pro-ducing the SL specimen, using the numerical flow simulation software thatwe developed. This was necessary considering the rather high productioncost. Figure 5.7 shows the first design of the five layers, which howeverwas isotropic (Table 5.1). On the right in figure 5.7 and in figure 5.8, theadapted design is shown, which is more anisotropic. This is a first applica-tion of our simulation software which led to cost saving, as only one designof the specimen reached the final production stage.

Although the SL process is very accurate for producing 3D parts, differencesbetween the model and the final part are ascertained. A first specimen wasproduced with epoxy material (figure 5.9). Measurement on this specimentsreveal a difference of more than 0.25mm in one direction, and 0.5mm inthe other direction. As these errors are average measured errors, these aretoo large errors to allow for a precise validation. (A constant error would


Figure 5.7: The layers of the isotropic (left) and anisotropic (right) design.

only require a new model.)

A second specimen was produced with Nanoform material. This specimenhas a maximal error of 0.1mm in the one direction, and of 0.1mm in theother. This second specimen is much more precise than the first one. Twomodels of these specimen were made: one with the maximal difference incomparison with the original model (figure 5.10), and one with the averagedifference of 0.05mm (figure 5.11).

5.4.3 The results

The computed permeability values of the five different SL specimen mod-els, together with the experimental values of the Nanoform specimen, areshown in table 5.1 We notice that the differences between the computed re-sults is significant, even between the two Nanoform models that only differminimally.

To compare the computational results and the experimental results, weconsider the model based on the measurement on the specimen producedwith the Nanoform material and with the 0.05mm error. Although manysources of errors in the experiments and the modelling are avoided by usingthis specimen (table 5.2), a difference between the experimental and com-putational values is present. In the Y-direction there is better agreementthan in the X-direction.

This exercise shows that small differences in the geometry have a significantinfluence on the permeability. Also, it gives us an estimate on the accuracywe can expect of permeability computations on textile models which havean irregular geometry.


Figure 5.8: The model of the designed, anisotropic SL specimen. The dif-ferent shading shows the parts that are present in the different layers. Thetop and bottom layers contain both shadings [77].


Figure 5.9: The model of the SL specimen produced with epoxy [77].


Figure 5.10: The model of the SL specimen produced with nanoform andthe maximal error in the vertical direction [77].


Figure 5.11: The model of the SL specimen produced with nanoform andthe average error in the vertical direction [77].


[th]

Table 5.1: The computed permeability values on the SL specimen models,and the experimental result. The discretisation step is taken constant inthe three directions x = y = z.Model x (mm) Kxx(mm2) Kyy(mm2) α

Isotropic model 0.06250 8.9 E-3 7.3 E-3 0.82

0.03125 8.6 E-3 6.9 E-3 0.82

Anisotropic model 0.06250 7.6 E-3 3.0 E-3 0.39

0.03125 7.5 E-3 2.9 E-3 0.40

Produced with epoxy 0.06250 4.8 E-3 1.4 E-3 0.29

0.03125 4.2 E-3 1.2 E-3 0.29

Produced with Nanoform,0.10mm error

0.06250 6.3 E-3 2.2 E-3 0.35

0.03125 6.4 E-3 2.3 E-3 0.35

Produced with Nanoform,0.05mm error

0.06250 7.0 E-3 2.6 E-3 0.37

0.03125 6.8E-3 2.5 E-3 0.37

Experiments on Nanoform 5.5 E-3 2.4 E-3 0.43

5.5 Validation on textiles

5.5.1 Basket woven monofilament fabric

The monofilament fabric named Natte 2115 (figure 5.12) is a filter fabric,originally not produced for RTM purposes. This basket woven fabric isan engineered material with some useful properties for permeability valida-tions. First, the glass fibres are PVC coated, which eliminates the microflows inside the yarns. Second, the textile is heated after the weaving pro-cess, so that the PVC coating melts and fuses the crossing yarns together.As a result, a stable structure is obtained, which can be reused in severalexperimental tests. The fused yarns and the absence of the intra-yarn floware supposed to eliminate the scatter on the experimental results. However,the scatter does not differ much of the scatter on other results presented.Therefore, the Natte 2115 textile was used to search for an explanation forthe scatter in the experimental results. Extra experiments were performedon one layer of the specimen, stitched and fused layers of the specimen (toeliminate nesting) and also on maximal nested layers. In that way, Hoes

5.5. VALIDATION ON TEXTILES 89

Table 5.2: Issues of experimental permeability prediction and the solutionof the SL specimen.

Issue Solution

Nesting of layers The SL specimen consists of onelayer that fits perfectly into themould.

Deformation of the textile, e.g.shear

The SL specimen is a solid, rigidstructure, no deformation can oc-cur.

Tests are performed on differenttextile specimens

The SL specimen can be cleanedand reused.

The macro permeability is notequal to the meso-scale permeabil-ity as no perfect periodicity existsin the mould

The SL specimen is perfectly pe-riodic in the XY-directions.

Textile geometry data is oftenmissing or incomplete

The CAD design of the SL speci-men is available. However, the ge-ometry of the produced specimendiffers slightly from the design.

The determination of the volumefraction

The determination of volume frac-tion of the SL specimen is no issue.


Figure 5.12: A single and two-layer model of the monofilament fabric.

showed that the different nesting of the layers is the main source of scat-ter [44]. In the following, we will only discuss the average result on theaverage nested layers.

From figure 5.5 we learn that nesting layers form a kind of wall betweenthe other layers. This explains why we have an overestimation if we useperiodic boundary conditions in the three directions. If we use a wall condi-tion in the Z-direction, we have a small underestimation. So for this textile,imposing a wall condition in the Z-direction does not yield a better result.On the contrary, for other textiles periodic boundary conditions in the threedirections result in a large overestimation of the permeability. To obtainthe model with the higher volume fraction (69%), we made a 3-layered,maximally nested model with the LamTex software [70]. As the LamTexsoftware does not provide us with a periodic model in the Z-direction (fig-ure 5.12), we did not perform the computations on the whole model. Themain fluid flow occurs on top and bottom of the textile model (figure 5.13).Therefore, to obtain a good result, only the middle layer of three-layeredmodel (the layer where the nesting actually happens), was used as input forthe computations.

The results of the computations and the experiments are presented in fig-ure 5.14. The error bars on the figure show the standard deviation of themeasured permeability values. A relative difference of about 40% is stated.This difference is larger than what we hoped for. However, if we take intoaccount the experimental and modelling issues (section 5.3), this is an ac-ceptable result. Moreover, the computational results capture the change inpermeability as function of the volume fraction correctly.


Figure 5.13: A 2D cut of the velocity profile in a two-layer model of themonofilament fabric textile; the grey region are the yarns. The pictureshows ux of the fluid field obtained with a force in the X-direction. Themaximum velocity occurs on top and bottom of the textile layer.

Figure 5.14: Comparison of the numerical (num) and experimental (exp)permeability values for the monofilament fabric. Numerical results for bothperiodic boundary conditions in the three direction and for periodic condi-tions in the X- and Y-direction are shown. The error bars show the standarddeviation.


Figure 5.15: The computed permeability values for the plain woven textile.

5.5.2 Plain-Woven Textile

Two models of the plain-woven textile, produced by Syncoglass, are de-picted in figure 5.16. One model is based on the data provided by themanufacturer, the second is based on actual measurements on the textile(before compression). At first sight the models (after compression) do notdiffer much. However, the numerical experiments on both models result indifferent permeability values (table 5.3). Moreover, the volume fraction Vf

of the two models differ some percentages. As can be seen in figure 5.15,the change in volume fraction makes that the the difference between theexperiments and computational results for both textile models is about thesame. For the computations we imposed periodic boundary conditions inthe X- and Y-direction and wall conditions in the Z-direction. If periodicityin all directions is imposed, the computed permeability is in the order of10−3 mm2, which is a large overestimation.

The experimental data of figure 5.15 was obtained from Hoes [44]. For avolume fraction of about 45%, similar results are found in Liu et al. [55].Themodel with Vf = 45% was made as described in the previous section. Themodel with the highest volume fraction (54.5%) was obtained in three steps

1. on the single layer model, a pressure of 0.003MPa was applied. Mod-elling the influence of pressure on a textile layer is a feature of theWiseTex software;


2. a three layered model of the compressed layer with maximal nestingwas made;

3. top and bottom layer of the 3-layered model were merged 10%, andthen the middle layer was used for the simulations.

The last step is not a natural procedure. However, to obtain such a highvolume fraction in the mould, many layers of the textile are put inside themould and compressed together. The yarns of the textile will move andshear to find the maximal possible nesting and with that, they block freechannels. All this cannot be modelled correctly, and thus we simulate theeffect by merging the layers. The merging cannot be done unlimited, asthe volume fraction is a limiting factor. This all makes that we have atextile model with a high, but correct, volume fraction that results in agood estimation of the influence of the volume fraction on the permeability.

Table 5.3: Geometrical data of the single layer plain woven textiles.

Parameter Measurements Manufacturer

Width warp yarns (mm) 2.21 2.50

Gap warp (mm) 0.58 0.58

Width fill yarns (mm) 2.79 2.50

Gap fill (mm) 0.35 0.35

Areal density (g/m2) 420 NA

Specific density (kg/m3) 2520 NA

Yarn tex warp (g/km) 580 NA

Yarn tex weft (g/km) 600 NA

Vf (%) 38.4 42.3

Kxx(mm2) 4.29E-04 2.83E-4

Kyy(mm2) 4.38E-3 3.24E-4

α 1.02 1.14


Figure 5.16: Two models for the plain woven textile: one based on themanufacturer data (left); one based on measurements (right). At first sightthey look equal, but small differences are present.

5.5.3 Non-Crimp Fabrics

Non-crimp Fabrics (NCFs) are plies of aligned fibres, stitched together [58,59, 61, 69, 97, 98]. The plies are normally turned with angles of 45 or90 in relation to each other. Figure 5.17 is an example of a unidirectionalnon-crimp fabric, on which the alignment of the fibres and the stitching canbe seen. Although the stitching yarns are an order of magnitude smallerthan the fibre bundles they join, they have an important influence on thepermeability as empty spaces are formed around the stitching.

Several authors have investigated the influence of structural parameters onthe permeability of NCFs, e.g. Nordlund [82, 83] and Loendersloot [57].These works give good insight in the structural properties of NCFs and theinfluence of the structure parameters on the permeability. For the compu-tation of the permeability, Loendersloot approximates the geometry of theNCF with a resistance network. Nordlund uses a CFD package for the fluidsimulations, however on simplified models of the NCFs. Loendersloot doesnot mention experimental values for the absolute permeability. Nordlunddoes, but unfortunately no detailed description of the textile is given, so thevalues cannot be compared with the results of our method. Here, we firstgive an experimental validation of our method for non-crimp fabrics.

We have experimental results of three different non-crimp fabrics, from dif-ferent institutions. The geometrical data of the two Bi-Axial non-crimpfabrics (B1 and B2) and the Quadri-Axial fabric (Q) is summarised in ap-pendix D. For the computations on the NCFs, periodic boundary conditionsin the all directions are imposed, as not much nesting will occur because of


the structure properties of the textiles. Moreover, non-crimp fabrics have adense structure, with not much uninterrupted free channels. Therefore, theintra-yarn flow was included into the simulations.

The computed permeability underestimates the experimental data, certainlyfor the bi-axial structures (figure 5.19). Although the models are made withgreat care, and with accurate data, the underestimation probably meansthat the size of the openings is slightly underestimated. From the data inappendix D, we learn that the size of the openings has a large influence onthe permeability. Moreover, from all parameters the opening has the largestinfluence.

Figure 5.17: Bottom and top view on a non-crimp fabric with a chain stitchpattern [57].

Figure 5.18: Two non-crimp fabric models, a bi-axial (left) and a quadri-axial (right).


Figure 5.19: Comparison of the experimental and numerical permeabilityvalues of non-crimp fabrics.

5.5.4 Non-woven structures - Random mats

Random mats are commonly used as reinforcement for composite materialsas they are cheap and easy to fabricate. In this section we present exper-mental and numerical results for two random structures. Unfortunately,detailed information of the used textiles is not available, however, we willapproximate the structures as well as we can with the given data.

Geometry model The models of the random structures are producedwith the software NoWoTex (NOn WOven TEXtiles). NoWoTex was de-veloped at the Department of Metallurgy and Materials Engineering of theK.U.Leuven. The geometrical description of the internal structure of non-woven material is based on the following data:

• fibre volume fraction of the material;

• fibre geometrical and mechanical properties;

• distribution of the fibre length;

• fibre orientation distribution, given as a 2nd order orientation tensor;

• fibre waviness, expressed as a random combination of two harmonics.


The geometrical model uses these parameters of the individual fibres andcreates a random fibrous assembly via a hierarchy of modelled objects:

• straight fibre, characterised by fibre properties, length and orientation;

• curved fibre, modelled as consisting of several straight intervals, andcharacterised by fibre properties, total length, averaged orientation ofthe intervals and shape of the fibre, generated using the given wavinessparameters;

• random realisation of an assembly of a given number of (curved orstraight) fibres. The boundaries of the unit cell of the material (whereall the centres of gravity of the fibres are randomly placed) are cal-culated based on the given fibre volume fraction and thickness of thenon-woven fabric. The given number of fibres are randomly generatedby the model according to the given distributions of length, orienta-tion and waviness of the fibres. In the case of non-woven fabric (asopposed to bulk material) the orientation is corrected to fit all thefibres inside the given thickness of the fabric. The random realisationof the non-woven assembly is considered to be periodic; the degree ofrandomness included in the description is regulated by the number offibres in the assembly.

Results The first textile is taken from Hoes [44]. Unfortunately, no de-tailed material or manufacturer data is available. The only informationavailable is mentioned in table 5.4. One important parameter is missing,namely the radius (or diameter) of the fibres. From the picture 5.20, wecan only determine the fibre radius approximately. Therefore, we computedthe permeability for three different possible fibre radii, 160, 170 and 180µm.The best results are obtained with a fibre radius of 170µm, and those resultsare shown in figure 5.22.

A second structure for which experimental data are available was found in[38]. Also for this structure the only available relevant data are a picture ofthe textile (figure 5.20) and the volume fraction. The radius of the fibresof this specimen is one order of magnitude smaller than the fibres of therandom mat of Hoes, which results in a significantly lower permeability. Forfibres with radius of 15µm, the computed results are in agreement with theexperimental results.


Table 5.4: Material data of the non-woven structure from [44].

Areal density [g/m2] 580

Specific density [kg/m3] 2520

Yarn tex [g/km] NA

Figure 5.20: Structure of the continuous fibre random mat of Hoes (left)and Feser.

Figure 5.21: Unit cell of the model of a random mat.

5.6. INFLUENCE OF SHEAR 99

Figure 5.22: Results of the permeability computations and experimentalresults for the non-woven, random structures.

5.6 Influence of Shear

WiseTex allows the modelling of shear on the textile [68]. The shearedunit cell has, however, no orthogonal repeat. This means that the periodicboundary conditions of the simulation software have to be put parallel tothe direction of shear, and not, as in the original case, parallel to the or-thogonal system axes. We have adapted our simulation software in orderto compute the permeability in a sheared unit cell. To validate the results,we compare the computations with analytical formulas from Lai and Young[52]. The authors also present experimental validation for their formulas,unfortunately they do not give detailed geometrical information of the usedtextiles, so we cannot compare their experimental results with calculations.

Figure 5.23 illustrates the nomenclature: IJ are the principal axis of thestraight textile, XY of the sheared textile, θ is the shear angle, β the anglebetween IJ and XY. The ellipse represents the flow front in case of thesheared textile. If it is assumed that the flow rates in warp and weft directionof the deformed textile are kept at the same ratio, then

(b

a

)2

≈ α =Kyy

Kxx, (5.4)

with K the principal permeability tensor of the non-deformed textile.


Figure 5.23: Description of the shear angle and the deformed fabric [52].

The ellipse of figure 5.23 has equation

AI2 +BIJ + J2 +D = 0 (5.5)

and passes through (a cos θ, a sin θ) and (0, b). If we assume that also forthe deformed fabric the tangent of the flow front in the warp direction isparallel to the weft direction (and vice versa), then the slope at (0, b) is

(dJ

dI

)

b

= tan θ. (5.6)

However, this assumption may be valid for unidirectional mats, it is not forwoven fabrics as the weft roving only has partial barrier effect on the flow inthe warp direction. Therefore, the authors introduce a correction factor m:

(dJ

dI

)

b

= m tan θ. (5.7)

With equation (5.7) and the points (a cos θ, a sin θ) and (0, b), A,B and Dfrom (5.5) can be determined, resulting in the following equations for the

5.6. INFLUENCE OF SHEAR 101

Figure 5.24: The principal direction as function of the shear angle.

principal direction and permeability ratio:

β =1

2tan−1

(m sin(2θ)

1 − α− 2m sin2(θ)

)

α =sin2(β − θ) − α cos2(β)

α sin2(β) − cos2(β − θ),

(5.8)

with β the direction of the principal axes of the sheared textile, θ the angleof shear, α the permeability ratio of the original textile, α the permeabilityratio of the sheared textile and m a correction factor depending on thestructure of the fibre network.

In chapter 2, it was explained how the principal direction of a unit cell ofa porous medium can be computed with two fluid simulations. Figure 5.24compares, for different shear angles θ, the numerically computed principaldirection with the one computed using the formula of Lai and Young. Fig-ure 5.25 shows the numerical and analytical permeability ratios. We canconclude that the theoretical results agree well with the numerical values,and that the assumptions that have been made by Lai and Young are valid.However, the formulas have the disadvantage that they contain the correc-tion factor m, which is not known beforehand.


Figure 5.25: The permeability ratio as function of the shear angle.

5.7 The variability on the meso-scale: influ-ence on the macro-scale

To obtain the meso-scale permeability, we started from the geometry at themicro-level. By solving the Stokes equations and taking the average of thefluid velocity, the variations on the micro-scale are homogenised into onemeso-scale parameter, the permeability. However, we also mentioned thatvariations on the meso-level exist. A macroscopic part exists of differentmeso-scale unit cells, but with varying yarn diameters and spacing betweenthe yarns’ central axes (and thus varying volume fractions). This is a resultof different rates of tension, shear, nesting and so forth.

For the macro-simulations, every meso-scale element has to be assigneda permeability. If all meso-scale elements are assigned the same averagepermeability value, the macro-simulation is not realistic [28, 30, 33]. More-over, if statistically determined permeability values are assigned randomlyto the elements, the result depends too much on the number of points inthe macro-mesh and the numerical implementation. Therefore, Desplentereworked out a method to correlate the unit cell properties to its position inthe macro-model [28].

5.7. VARIABILITY 103

Table 5.5: Variational tests on the plain woven textile, performed on 3x20models. The coefficient of variation is mentioned between brackets.

Exp 1 Exp 2 Exp 3

Yarn spacing (mm) 2.7 (5%) NA 2.67 (5%)

Yarn width (mm) NA 2.21 (13%) 2.22 (13%)

Gap width (mm) 0.49 (26%) 0.44 (64%) 0.45 (55%)

Permeability (mm2) 8.5E-4 (11%) 9.5E-4 (32%) 8.3E-4 (28%)

5.7.1 Influence of the variability on the meso-scale per-meability

The order of variability is expressed with the coefficient of variation (CV)

CV =standard deviation

average value.

Measurements on the plain-woven fabric resulted in the conclusion that theyarn dimensions of a single layer of textile have a CV of 15%, while the CVfor the yarn spacings is about 5%. The variability of the volume fraction ofthe meso-scale part corresponds to the variability of the spacing between theyarns (based on the information of 500 WiseTex models of the plain woventextile). This is an interesting property, as the permeability is normallyexpressed as function of the volume fraction.

To investigate the influence of the yarn dimensions and spacing on themeso-permeability, three runs of 20 simulations were performed. One withvariation on the yarn spacing, a second with variation on the yarn widthand a third one with both parameters varied. Table 5.5 gives an overviewof the results, and the results of the third experiment are displayed in figure5.26. The yarn dimensions are less informative for the permeability valuethan the dimensions of the gap between the yarns. A CV of 5% for the yarnwidth and 13% for the spacing between the yarns results in a CV of 55%for the gap width, and 28% for the permeability. The CV of the gap width(micro-level) is smaller than the CV of the permeability (meso-level).


Figure 5.26: The dimensions of the plain woven textile models and the corre-sponding permeability values. The gap between the yarns is the determiningparameter for the permeability.

5.7.2 Correlation

If the textile layer is not cut for the draping, the gap dimensions do notchange abruptly. The gap dimensions are a continuous and smooth functionof the location in the macro-part, and a correlation between the permeabilityof two meso-scale cells exist.

Consider one direction s in the macro-part and the distance is betweenthe centres of two cells in that direction. The discrete function K(si) is thepermeability along the direction s at position si. The discrete correlationfunction for the permeability RKK is given by

RKK(is) =VK,K(is)VK,K(0)

(5.9)

5.7. VARIABILITY 105

with the variance function

VK,K(is) =1

M

M−i∑

j=1

[K(sj) − K

] [(K(sj+i) − K

]. (5.10)

Here, M is the number of measurements for K, and K is the overall av-erage permeability. To fit the discrete correlation function, an exponentialfunction is used

RK(si, sj) = exp

(

−|si − sj |A

)

. (5.11)

The unknown parameter A is the correlation length, and can be determinedby means of our software. WiseTex allows the modelling of a set of unitcells with varying gap widths corresponding to the measured gap widths andtaking into account the measured correlation length of the gap dimensions.The permeability of these models is then computed with our software, andfinally the correlation length for the permeability is computed.

5.7.3 Influence of the variability on the macro-scale

In this thesis, we concentrate on the permeability of a meso-scale unit cell.The investigation of the influence of the meso-scale level is beyond the scopeof this thesis. Nevertheless, we briefly discuss the macro-level in order tohave an idea about the required accuracy of the meso-scale models andcomputations for a realistic macro-simulation.

Consider first a macro-simulation without scatter on the meso-scale. On themacro-level Darcy’s law (5.12) is solved, and an error in the permeabilityinput has a linear influence on the computed velocity field and fill time,

q =Re

L2K

(f

Fr−∇p

)

. (5.12)

More than the average permeability, the variability of the permeability in-side the macro-part influences the fill time and dry spot formation [28, 33].Endruweit et al. investigated the influence of a stochastic fibre angle of anon-crimp fabric. The meso-scale elements were assigned randomly a dif-ferent fibre angle, which influences the permeability of the local element.Table 5.6 presents the computed fill times for different standard deviationsof the assigned fibre angles. The variation on the fibre angle has a largeinfluence on the final fill time.

With different case studies, Desplentere demonstrated the necessity of accu-rate meso-scale permeability simulations. Not so much an accurate average


Table 5.6: Computed fill times as function of the standard deviation of thefibre angle of a non-crimp fabric [33].

standard deviation () fill time (s)

0 58.6

10 61.6

20 74.1

30 110

40 183

permeability is needed, but a meso-scale solver must provide us with ac-curate data on the variability coming from shear, changing gaps betweenthe yarns, nesting and so forth. We refer to the work of Desplentere [28],Endruweit et al. [33], Markicevic et al. [73] and Simacek and Advani [108]for more detailed information on the macro-simulations and case studies.

Note that we have shown in this section that our method and solver is suitedfor an accurate meso-scale permeability computation, the prediction of theinfluences of changing geometry included.

5.8 Conclusions

Our permeability computing software has been applied for the computationof the permeability of various textile models. The software was used tocompute the permeability of a reference specimen, woven fabrics, non-crimpfabrics and non-woven fabrics. For all these materials, the computed resultsare compared with experimentally obtained permeability values. For a largerange of volume fractions, the computed values show good agreement withthe experiments, in comparison with results found in literature (figure 5.27).This is an important difference with other permeability simulation tools,for which often experimental validation lacks or results for higher volumefractions are not reported. Moreover, several of these methods are onlyapplicable to one specific type of textile.

On a unit cell of a sheared textile, the correct boundary conditions have tobe imposed, as shown in section 3.5. The results of the computations ontextile models with different shear angles are reported and are comparedwith analytical formulas.

5.8. CONCLUSIONS 107

Figure 5.27: The computed permeability and experimental values for twonon-woven and two woven fabrics. MFF is the monofilament fabric, PWFis the plain woven fabric.

The permeability of the meso-scale unit cells is input for Darcy-solvers thatperform macroscopic simulations. A macro-part consists of many meso-scaleelements, however, with a certain variability on for example the dimensionsof the gap between the yarns and thus a variability on the permeability. Toobtain a realistic macro-simulation, a correct prediction of this variabilityis indispensable. The software that we developed is capable of this.

Chapter 6

Assessment of the Grid2Dmethod

At the University of Nottingham, the research group of Prof. A. Long hasdeveloped a homogenisation method to compute the permeability of textiles.This Grid2D method reduces the 3D Stokes/Brinkman problem to a 2DDarcy problem. Instead of solving the 3D Stokes/Brinkman equations, thedomain is discretised into a regular grid in the XY -plane. Then, everypoint on the 2D grid is assigned a local permeability value depending onthe textile geometry at that point. Finally, the Darcy equation, combinedwith the mass conservation law, is solved on the 2D grid. The aim of this3D to 2D reduction is to predict the textile permeability with a substantialdecrease in computation time in comparison with a full 3D Stokes solver.

In this chapter the Grid2D method is first explained in detail. Then, valida-tion for some artificial setups will be given and discussed. It will be shownthat the Grid2D method does not always perform well on the prediction ofthe permeability of artificial structures. However, under certain limitations,it is a good and fast alternative for the full Stokes/Brinkman solver.

6.1 The methodology

Basically, the Grid2D method reduces the 3D CFD problem to a 2D prob-lem, by locally joining the properties of the geometry textile into one param-eter, namely a local permeability. In the Grid2D approach, the domain isfirst divided into basic volumes consisting of the open channels and porous

109

110 CHAPTER 6. ASSESSMENT OF THE GRID2D METHOD

Porous tow Porous tow

Porous tow

K_4

K_4,1

K_4,2

K_4,3

Figure 6.1: Example of the basic volumes and the reduced grid with localpermeabilities for a 2D to 1D example.

tows (figure 6.1). The domain is then discretised into a regular rectangulargrid in the XY- plane. For every point on the XY-plane, the permeabilitiesof the involved volumes are computed. Then, using the law of arithmeticmean formation, these permeabilities are used to compute the local per-meability Kij . The computation of this local permeability is explained inmore detail in section 6.2. As now for every point on the XY-plane a localpermeability term Kij is known, the Darcy law combined with the law ofconservation (6.1) can be solved on the 2D-plane to predict 〈u〉 and 〈∇p〉.Finally, the unit cell permeability is calculated, with the original Darcy’slaw (3.8).

u =Re

L2K

(f

Fr−∇p

)

(6.1a)

∇ · u = 0 (6.1b)

Note that in equations (6.1), we omitted the volume averaging. Here, weactually assume that Darcy’s law holds on the micro-scale.

To summarise, the method consists of five steps (figures 6.1 and 6.2):

6.2. CALCULATION OF THE LOCAL PERMEABILITY 111

• define the 3D geometry of the textile;

• find the basic volumes of the geometry and define a 2D grid in theXY-direction;

• for every point on the 2D grid, calculate a local permeability, basedon the properties of the basic volumes at that point;

• solve the Darcy equation combined with the law of mass conservationon the 2D grid;

• Calculate K by averaging the velocity and pressure gradient on the2D grid, and using Darcy’s law.

6.2 Calculation of the local permeability

In his PhD thesis [110] C.C. Wong describes his implementation of theGrid2D method, which imports the textile geometry from TexGen [87] (fur-ther referred to as the vector solver). TexGen is software for modelling thegeometry of any textile fabric in a generic way and is developed at the Uni-versity of Nottingham. The geometric models obtained from TexGen canalso be used for various numerical analysis techniques. We have developeda second implementation, based on the input of a voxel description of thetextile model (further referred to as the voxel solver). Basically, the methodof the implementations and the results are the same, however some differ-ence exists in the accuracy of the computation of the height of the basicvolumes.

The calculation of the local permeability is explained for a 2D to 1D reducingmethod, as this facilitates the understanding of the formulas. An extensionto the 3D to 2D method is straightforward. A 2D XZ-cut of a textile model isshown in figure 6.3. The figure also shows a very coarse voxel representationof the textile model. The black points are in the centre of a solid or porouscell, the white points are at the centre of a cell in the fluid domain.

For every point in the X-direction we need a value Ki that represents thelocal permeability of the structure at that point. We differentiate betweentwo cases: solid yarns and permeable yarns.

Solid yarns For solid yarns, Ki is calculated as

Ki =1

H

n−1∑

m=0

hmKi,m (6.2)


Figure 6.2: The consecutive steps of the Grid2D method. The textile modelis the monofilament fabric, here the result of the import into TexGen isshown.

6.2. CALCULATION OF THE LOCAL PERMEABILITY 113

Brinkman point (Navier-)Stokes point

Figure 6.3: 2D cut of a textile model and its coarse voxel representation.

with H the thickness or height of the total unit cell, n the number of basicvolumes at point i, hm the height of basic volume m and Ki,m the perme-ability of the basic volume. A basic volume in the fluid domain is locallyapproximated as a straight channel with height hm, and thus

Ki,m =h2

m

12. (6.3)

Basic volumes in the solid region haveKi,m = 0. As hm is determined basedon the voxel description, it suffers from a discretisation error. Here lies themain difference with the implementation based on the vector description ofthe textiles, as with a vector description the height of the channel can becalculated more precisely.

As an example, take the second point in the X-direction in figure 6.3. Withx the cell width and height, we have

H = 4xh0 = 2xh1 = xh2 = x

K2,0 =h2

0

12K2,1 = 0

K2,2 =h2

2

12

Thus, for this example

K2 =1

4x

(

2x4x2

12+ xx

2

12

)

. (6.4)


In the Z-direction wall conditions are applied, as otherwise hm is not definedfor the top and bottom volumes. Moreover, in chapter 5 it was shownthat wall conditions in the Z-direction are a better representation of theconditions in a mould than periodic boundary conditions are.

To solve Darcy’s equation in 2D we actually need 4 permeability values:Kxx, Kyy, Kxy and Kyx. However, in a fluid basic volume it holds thatKxx=Kyy, and both Kxy and Kyx are neglected in this implementation asnumerical experiments show that these terms have almost no influence onthe meso-scale permeability.

Permeable yarns The procedure is similar as in the solid case. If we takethe same example, but with permeable yarns, only K2,1 must be calculateddifferently.

Similar to the description in chapter 4, yarns are considered as a regularpacking of cylinders. For regular arrays analytical permeability values areavailable (chapter 4).

6.3 Solution of Darcy’s law

Once the local permeability values on the 2D-grid are known, Darcy’s law,coupled with the continuity equation for incompressible flows (6.1), is usedto compute the velocity field and pressure distribution.

[

ux

uy

]

= − 1

µ

[

Kxx Kxy

Kyx Kyy

][∂p∂x∂p∂y

]

(6.5a)

∂ux

∂x+∂uy

∂y= 0. (6.5b)

As we neglect the off-diagonal elements of K substitution of (6.5a) into(6.5b) yields

∂

∂x

(

Kxx∂p

∂x

)

+∂

∂y

(

Kyy∂p

∂y

)

= 0. (6.6)

Equation (6.6) is discretised with a finite difference discretisation schemeand then solved with a linear solver from the PETSc package [11]. Periodicboundary conditions are imposed in the X- and Y-direction, except for onedirection in which periodic boundary conditions with a pressure difference

6.4. VALIDATION 115

is imposed. Note that a constant body force disappears by including themomentum equation into the continuity equation. Thus, in this case wecannot choose between the two driving forces.

6.4 Validation

6.4.1 Artificial structures

Channel

For an empty unit cell, 1x1x1mm, with periodic boundary conditions inthe X- and Y-direction and a wall in the Z-direction, both the Stokes solverand Grid2D result in a value Kxx = Kyy = 0.083 mm2. This is exactly thetheoretical permeability value for a channel with height 1mm.

Parallel array of cylinders

In chapter 4 the permeability of a parallel array of cylinders was discussedbecause the fibres inside a yarn are modelled as such a structure. Here,the results of the Stokes calculations on a similar structure are comparedwith the Grid2D results. As the Grid2D does not capture periodicity in theZ-direction, a wall condition is imposed on top and bottom of the unit cellof figure 6.4. Therefore, we cannot compare the results with the formulasof Gebart (4.6), as they are only valid for a periodic hexagonal packing ofcylinders.

Figure 6.5 shows the computed results for the permeability along andtransversal to the cylinders, and for different volume fractions. We makethe following observations.

• Also for this structure, the graph has a typical flat S-shape, howeverless outspoken than the periodic structure (figure 4.4).

• The Grid2D method overestimates the Stokes results, and the resultsdo not have the typical S-shape.

• For almost all volume fractions, the vector solver and voxel solver forthe Grid2D method yield the same results. However, for the highestvolume fractions, the computed values of the vector solver deviatefrom the descending trend. The voxel solver yields better results.The reason for this difference is not clear.


Figure 6.4: Unit cell of a hexagonal array of cylinders.

6.4.2 Textiles

Mono-filament fabric (Natte)

Figure 6.6 gives the computed permeability values for different monofila-ment (MFF) textile models. Only textile number 7 is the real textile. Thefirst textile model (number 0), is a two layered model with maximal nest-ing. Numbers 2-6 are textiles with the same structure, but the yarns havedifferent diameters. Table 6.1 contains the parameters which set the modelsapart. The difference between model 4 and 5 is the spacing between theyarns (figure 6.7).

Although the Grid2D method gives an overestimation, for the majorityof the models, the values are in good agreement with the Stokes results.However, the results for model 4 and 5 attract attention, because of a largeror a smaller difference. The smaller error agrees with what we will discussin the next section: the Grid2D method performs better on flat textiles.

The Grid2D method is developed as a fast homogeniser for the computationof textiles. In comparison with the Stokes solver on a fine grid, the Grid2Dmethod is indeed much faster. If for both methods a grid with x =0.02mm is used, the Grid2D method needs 1-2 seconds, the Stokes solverabout 2 minutes. However, if a grid with x = 0.04mm is used as input forthe Stokes solver, the solver only needs about 3-4 seconds, and from figure6.6 we can conclude that we also obtain accurate results on this rathercoarse grid.

6.4. VALIDATION 117

Figure 6.5: Computational results for the flow along a hexagonal packingof cylinders.

Plain Woven Fabric (Syncoglass)

The plain woven fabric (PWF) has a more flat and dense structure thanthe Natte textile. The permeability values of both methods agree betterthan in the previous case. The results of the computations are shown infigure 6.8. Models 0-5 are models based on the real model 6 (figure 5.16),but with changing diameter of the yarns and spacing between the yarns.Models 7-12 are two layered models with random nesting. From figure 6.8we can conclude that the Grid2D method is suitable for parametric studieswith textile models of the same kind as Syncoglass.

6.4.3 Stereo-Lithographic specimen

In section 5.4 we presented the SL specimen and the results of numericalpermeability predictions as well as experimental determined values. The SLspecimen was designed and produced to compare different experimental set-ups with each other, however, also to analyse and validate several existingsimulation methods and tools.


Table 6.1: Model parameters of the different monofilament textile models.

Parameter 1 2 3 4 5 6 7

yarn diameter 1 (mm) 20 25 22 20 20 15 20

yarn diameter 2 (mm) 43 43 53 40 40 43 43

volume fraction (%) 66.4 51.4 57.7 64.8 63.0 82.9 63.7

Figure 6.6: Computational results for the parametric study with themonofilament fabric.

Figure 6.7: Top view on model 4 and 5 of figure 6.6.

6.4. VALIDATION 119

Figure 6.8: Computational results for the Syncoglass parametric study. Theerror bar shows the standard deviation of the experimental measurements.

For the SL specimen, figure 6.9 shows the 2D planes with the local per-meabilities, and for a driving force in the Y-direction, the resulting pres-sure field and velocity. The results of the computations are depicted infigure 6.10. For the three different SL models, the permeability is overes-timated, but the Grid2D method does capture the differences. From thepicture of the local permeabilities, we see that the 2D approximation of thestructure does not capture the vertical distances the fluid is obliged to take.This is a general disadvantage of the method, and explains also why theresults for the rather flat PWF textile are better than for the MFF textilewith circular yarns.

6.4.4 Sheared textiles

It was shown that the Grid2D method is trustworthy for textiles like thePWF. The method does not perform equally good for artificial structures orthe MFF, however, one could argue that the results are still satisfying for afast permeability simulation and that the precise value of the permeability


Figure 6.9: The local permeabilities (top), and for a driving force in theY-direction, the resulting pressure field (middle) and the velocity in theY-direction (bottom).

6.4. VALIDATION 121

Figure 6.10: A comparison of the computed values with the Stokes solverand the Grid2D method.

of a textile model is not important as such, as long the order of magnitudeis correct. Nevertheless, it is important that the permeability simulationscapture the influence of deformation of the structure, e.g. shear.

In section 5.6, the influence of shear on the permeability was discussed. Itwas shown that the results of the Stokes solver on models of sheared textilesand the formulas of Lai [52] correspond (figures 5.25 and 5.24, p. 102). Here,we compare the results of the Stokes solver with the Grid2D method results.For both methods, figure 6.11 shows the computed permeability values asfunction of the shear angle. Kxx is the permeability in the principal di-rection β, Kyy in the direction perpendicular to β. Figure 6.12 gives anoverview of the computed local permeabilities, the pressure field and veloc-ity field for both an applied pressure gradient in the X- and Y-direction, andthat for a shear angle θ of 15. Figures 6.14 and 6.13 show the computedratio and principal angle as function of the shear angle. For these computa-tions, wall conditions in the Z-direction were used, in contrast to the resultsof figures 5.25 and 5.24. First, we note that the formulas of Lai and Youngare better suited for the prediction of the influence of shear with periodic


Figure 6.11: The computed permeabilities of the sheared monofilament tex-tile as function of the shear angle.

boundary conditions. Also, the values of α and m had to be adapted to fitthe Stokes results with wall conditions in the Z-direction. Concerning theGrid2D method, we can conclude that the method captures the trend of theStokes computations, but with less accuracy.

6.5 Conclusion

In this chapter the Grid2D method was discussed and compared with theStokes solver. The Grid2D method reduces the 3D CFD-problem to a 2DDarcy problem, which results in faster computations, however at the cost ofaccuracy. We have shown that although the method cannot be used for anaccurate prediction of all artificial structures, it does perform well on sometypes of textiles.

6.5. CONCLUSION 123

Figure 6.12: The sheared model, the local permeability and computed pres-sure and velocity for a sheared monofilament fabric with θ = 15.


Figure 6.13: The principal direction as function of the shear angle.

Figure 6.14: The permeability ratio as function of the shear angle.

Chapter 7

Adaptation of the porenetwork model

The fluid flow through a porous sample is mainly governed by the poregeometry, i.e. the pore size distribution and the connectivity of the pores.The pore geometry and the fluid flow in the pore space can be related usinga pore network model. A pore network is a simplification of the geometryof the pore space: the pores are replaced by nodes which are connectedby links. Instead of computing or modelling the fluid flow in the wholepore space, these computations can be done on the network. By doingso, a drastic reduction in computational power and time can be achievedcompared to the accurate solution of the Stokes equations.

Available pore network models simplify the micro-scale geometry of a poreto a basic geometry, e.g. a sphere. In this way, the computation of theconductivity of the pore can be done analytically. In this section, we showthat our Stokes solver can be used to compute the conductivity betweentwo nodes in the network more accurately.

7.1 Introduction

The use of a network model to compute the permeability of a porousmedium, was introduced by Fatt [34, 35, 36]. He applies a regular 2D latticeof pores and throats and assigns to each throat a hydraulic conductivity.In that way, curves for drainage can be computed. A major shortcoming ofFatt’s model is the use of a 2D regular network to describe a 3D pore space

125

126 CHAPTER 7. THE PORE NETWORK MODEL

with a very irregular and complex geometry. Also 3D versions of Fatt’smodel have been constructed, but they mostly fail to grasp the statisticsof the pore structures. This yields the conclusion that the pore space itselfshould be analysed to be able to capture the real complexity of the porestructure.

Thovert et al. [96] reconstruct the porous medium using spheres with radiidetermined by the probability density of the covering radius. Vogel quan-tifies the complex pore geometry in terms of the pore size distribution andthe pore connectivity [105]. Both characteristics are obtained from serial2D sections through an impregnated sample [104, 106]. A regular network isthen adjusted to be consistent with the measured pore size distribution andconnectivity. Valvatne and Blunt generate geologically realistic networks byadjusting the pore size distribution of their network to match experimentallyobtained capillary pressures [99].

A 3D image of the pore space of samples can be acquired with X-ray com-puted tomography (CT). Delerue uses CT images and generates a skeletonof the pore space using a Voronoi approach [26, 27]. Subsequently non-overlapping spheres are placed on the skeleton and are inflated to fill thewhole pore space. Replacing these spheres by nodes and connecting neigh-bouring nodes by links yields a network representation of the pore space.

The drawback of such an approach is that skeletonisation algorithms areoften computationally expensive, sensitive to noise and not very robust. Aminor adaptation to the pore space can give rise to a different skeleton anda different segmentation. Figure 7.1 illustrates this for a simple example, forwhich a small change in the pore space results in a segmentation with threezones instead of two. Therefore, Van Marcke et al. developed a differentapproach where the sidestep of first simplifying the pore space to a skeletonand subsequently rebuilding this pore space is avoided [72]. This will proveto be a much more robust approach in most cases and is computationallyless expensive.

7.2 The pore network model

7.2.1 Construction of the pore network

To construct the pore network, the pore space has to be segmented into aset of individual pores. However, this segmentation is not uniquely defined.The approach developed here is based on the rather intuitive definitionthat pores are separated from each other at the bottlenecks in the pore

7.2. THE PORE NETWORK MODEL 127

(a)

(f)(e)(d)

(c)(b)

Figure 7.1: (a) 2D image of a part of a pore space (full lines) with skeleton(dashed line). (b) The biggest possible non-overlapping balls are placed onthe skeleton. (c) The balls are inflated to fill the whole pore space. (d) Thesame part of the pore space is considered, but a small deviation is addedto the image. This alters the skeleton as a branch. (e) The branch in theskeleton results in an extra ball. (f) After inflation of the balls, a differentsegmentation of the pore space is achieved.

space. Figure 7.2 (left) shows a 2D image of a pore space. Dividing thispore space into a set of pores cannot unambiguously be done. Figures 7.2(middle) and 7.2 (right) illustrate two different segmentations of this porespace. Both segmented images might seem logical and one cannot say whichone is the best segmentation. When the pore space in figure 7.2 (left) isconsidered, one might intuitively segment this pore space into the poresshown in figure 7.2 (middle). Every pore is separated from the other onesat the bottlenecks between the pores. One could also intuitively remark thatthe distinction between pores A and B is much more obvious than betweenpores A and C.

The new approach divides the pore space at what can be called throats:narrow places in the pore space. This has the great advantage that no sim-plification or skeletonisation is necessary which makes this approach lesssensitive to adaptations in the pore space geometry. The approach is de-veloped and applicable in 3D, but will be explained by using 2D images asthey make it easier to explain and comprehend.

The first step is the construction of what is called an aperture map. Theaperture map contains for each pore voxel its distance to the nearest borderof the pore space (figure 7.3 middle). The aperture map can also be plottedas a topological map (figure 7.3 right). The local maxima of the aperturemap are determined and assigned a number. These maxima make up thepores in the segmented pore space and are expanded contour by contour,


Figure 7.2: Left: 2D image of a pore space where the white pixels representthe pore space and the black pixels the matrix. Middle and Right: Thepictures show two possible segmentations of this pore space into individualpores. This illustrates the ambiguity in the segmentation of the pore space.

Figure 7.3: Left: 2D image of a pore space. Again, the white pixels representthe pore space and the black pixels the matrix. Middle: The aperture mapof this pore space contains for each pore voxel its distance to the border ofthe pore space. Right: Topological representation of the aperture map.


starting from the biggest maximum.

When two expanding maxima touch (figure 7.5 left), they either merge ora boundary between them is constructed. Different criteria can be definedto decide whether two expanding pores should be merged or not. Here, thecriterion

Ap(C) > λ min [Ap(M1), Ap(M2)] (7.1)

is used, with λ a user defined percentage, Ap(C) the aperture of the contactpoint between the two expanding maxima and Ap(M1) and Ap(M2) are theapertures of these maxima. The parameters are shown in figure 7.5 (right).If the criterium (7.1) is fulfilled, the expanding maxima are merged. Inthe other case, the expanding maxima are considered as different poresand a boundary between them is constructed. This boundary follows thehighest gradient in the aperture map. Expanding maxima cannot cross thisboundary.

That way, the maxima are expanded until the whole pore space is filled andsegmented into different pores (figure 7.6 left). This segmented image of thepore space is then converted into a pore network. Every pore is representedby a node in the network which is placed at the voxel with the maximumaperture in the pore. Neighbouring pores are connected by links betweentheir nodes (figure 7.6 right).

The network representation is used to compute the permeability of the sam-ple. Nodes that are not connected with the inlet and outlet of the sampleare removed from the network as they do not affect the permeability. Thesame applies for dead-end nodes. To be able to compute the permeabilityof the network, the conductivities of the links must now be determined.

7.2.2 Computing the local conductivities in the net-work

Most pore network approaches estimate the conductivities by simplifyingthe links to simple geometrical shapes, e.g. squares [17, 18, 31, 37], trian-gles [46] or even a network consisting of stars shaped links [51]. Delerueconsiders them to be cylindrical, and thus computes the conductivity be-tween two nodes as a serial connection of two cylinders. The length of thesecylinders is the distance from the centre of the pore to the interface betweenthe pores. The radius is the radius of the inflated spheres that made up thenetwork (figure 7.7).

It is however a rough approximation to simplify the irregular shapeof the pore space to uniform ducts with regular cross-sectional shapes.


Figure 7.4: The local maxima in the aperture map are determined andassigned a different number. The local maxima are then expanded by addingthe contours of the aperture map. This expansion is executed starting fromthe biggest local maxima and gradually adding more and more contours.


Figure 7.5: Left: When two expanding maxima touch, they will be mergedor a boundary between them is constructed. An eventual boundary willfollow the highest gradient in the aperture map starting from the contactpoint between the two expanding maxima. Right: The decision whether ornot to merge is based on a criterium (7.1) which uses the aperture Ap(C)at the contact point and the apertures of the expanding maxima Ap(M1)and Ap(M2).

Figure 7.6: Left: The resulting segmentation of the pore space that wasshown in figure 1a. The different gray scales represent the different pores.Right: Each pore is represented by a node placed in the pixel with thebiggest aperture in that pore. Neighbouring pores are connected by links.Nodes and links make up the pore network.


Figure 7.7: The network is constructed by placing non-overlapping balls onthe skeleton of the pore space. The centre of these balls will be the nodes inthe network representation of the pore space. The conductivity of the linksbetween the nodes is computed by considering them as a serial connectionof cylinders. The length of the cylinders is the distance from the centre ofthe pores to the interface between the pores. The radius is the radius of theballs.

Stokes solvers compute the permeability accurately, but are too time con-suming to use on the whole sample. However, the sample is now dividedinto a large number of small pores, for which the Stokes solver can be used.

We have chosen to compute the conductivity of the links with the methodpresented in chapter 3. In figure 7.8 two linked pores are drawn. A pressuredifference ∇ijp between the two nodes i, j results in a flux qij from the onenode to the other. This fluid flux is described by Darcy’s law (cf. (3.8)),

qij =Re

L2Kij

(f

Fr−∇ijp

)

. (7.2)

Again, the Stokes equations have to be solved to obtain qij for a given ∇ijpor a given f .

To solve the Stokes equations, the same approach as for the computationof the permeability of textiles is used. There is however a difference in theapplied boundary conditions at the boundary of the cell in the X-direction.The pores are not periodic, and if periodic boundary conditions are applied,the fluid flow will be blocked unnaturally (figure 7.9). Therefore it waschosen to apply symmetric boundary conditions. The symmetric boundaryconditions are not the real conditions either, however, we assume that theporous medium has a sufficiently smooth course and that the symmetricconditions are a good approximation for the reality.

Once the local conductivities in the network are computed, its global con-ductivity can be determined. Figure 7.10 shows a schematic representation


p_2p_1

q

Figure 7.8: Two neighbouring pores which are modelled as nodes connectedby a link in the network representation. The local conductivity of thislink relates the fluid flow between the nodes which results from a pressuredifference between them.

Figure 7.9: If periodic boundary conditions are used, fluid flow is unnatu-rally blocked (top). Symmetric boundary conditions are a better approxi-mation of reality (bottom).


Figure 7.10: Schematic representation of a pore network. The dots representnodes while the lines are the connections between these nodes. The nodesare numbered so that the system matrix has a minimal bandwidth.

of the pore network where all the local conductivities are known. The fluidflow through the global sample is determined for a given pressure differenceover the sample. The sample is assumed to be saturated by the fluid andthe flow is supposed to be laminar. As the flux q through the sample iscomputed for the given pressure difference, the global permeability K canbe calculated by Darcy’s law (3.8). Then the local pressures in each nodeare calculated by writing a mass conservation equation in each node i, i.e.by stating that the fluxes entering a pore are equal to the fluxes leaving thispore

∑

j

qij =∑

j

Kij

µ

(pi − pj

)= 0. (7.3)

In the case of a network with n nodes, a system of n equations with nunknown pressures pi is obtained. This system can be written down in amatrix equation

1

µAP = B,

with, for the example of figure 7.10,

7.3. VALIDATION AND CONCLUSION 135

A =

Kin,1 + K

12−K

12 0 . . . 0

−K21

Kin,2

K12 + K

23 + K24

−K23 . . . 0

0 −K32

K32 + K

35 . . . 0

.

.

.

.

.

.

.

.

.

...

.

.

.

0 0 0 . . . Kn,out + K

n(n−1)

P =

p1

p2

p3

.

.

.

Pn

B =

Kin,1

· pin

Kin,2

· pin

0

.

.

.

Kn,out

· pout

When the local pressures in the network are known, the fluxes entering andleaving the network can be calculated by summing the fluxes entering theinlet pores or leaving the outlet pores:

q =∑

j∈ inlet

Kij

µ

(pi − pinlet

)=∑

j∈out

Kij

µ

(pout − pi

)(7.4)

The global flux through the network is now known for a given pressuredifference and equation (7.2) allows to compute the overall conductivity ofthe network.

7.3 Validation and conclusion

The permeability of 13 samples computed with the presented pore networkmethod is compared with the values computed with finite difference Stokessolver. Comparing these results allows to estimate the effect of using anetwork to compute the permeability. As the finite difference approachrequires more computation time and memory than the pore network model,the size of the samples is limited to 400x400x100 voxels. The results of thecomputations are shown in figure 7.11.

First, we notice that the parameter λ (cf. equation (7.1)) has a significantinfluence on the results of the pore network method. This is clearly animportant disadvantage of the applied segmentation method. Van Marckeet al. [72] explain how λ can be estimated. Second, for an appropriate choice


Figure 7.11: Comparison between the permeability values computed withthe Stokes solver and the pore network solver. The parameter λ has asignificant influence on the results of the pore network solver.

of λ, the pore network computes the permeability values in the same order ofmagnitude of the values obtained by Stokes simulations. The pore networksolver allows for an estimation of the permeability considerably faster thanthe Stokes solver. The pore network solver needs about 2 hours on a single 2Ghz Opteron processor. The Stokes computations required at least 5 hourson 2 processors.

Chapter 8

Conclusions

In this thesis the development of a finite difference Stokes solver for thecomputation of the meso-scale permeability of textiles is described. A fullcomparison with other methods to obtain permeability values is workedout, with particular attention to a comparison with experimentally obtainedvalues. We also explain the use of the Stokes solver as part of a pore-networkmethod to compute the permeability of porous rock.

The RTM process to produce composite parts is simulated with Darcysolvers that solve Darcy’s law on the macroscopic level. For that, the solversrequire the input of the meso-scale permeability of the textile and a mea-sure for the influence of distortions and changing volume fractions on thepermeability. The finite difference Stokes solver, combined with the textilemodels from WiseTex, provides the required input, automatically, fast andaccurately.

Our method allows for both the computation of the inter-yarn flow andthe intra-yarn flow. To include the intra-yarn flow in the simulations, theBrinkman equations are solved on the whole computational domain. Thesolution of the Brinkman equations requires the assignment of a local per-meability value to every grid point that lies inside a yarn. The local perme-ability is computed analytically, based on the volume fraction and directionof the fibres at that point. Various analytical formulas to calculate the localpermeability were validated, and the formulas of Berdichevsky are similar tovalues obtained with our CFD software. However, the choice of the formulahas little effect on the meso-scale permeability for realistic fibre structures.

The computed results agree well with experimental results for a variety ofstructures: woven textiles, non-crimp fabrics, non-woven structures and a

137

138 CHAPTER 8. CONCLUSIONS

specimen produced by stereolithography. To obtain the good results, com-pressed models, multi-layered models with nesting of the layers and shearedunit cells were used. Compression and nesting yield models with narrowchannels and sharp edges, which however do not influence the accuracy ofour software. In addition, for the sheared unit cells special boundary condi-tions have to be applied, which can also be done automatically on a regulargrid. Figure 8.1 gives an overview of the experimental validation for twonon-wovens and two woven fabrics. Not to overload the figure, we left outthe results for the non-crimp fabrics and the SL-specimen, which can befound in chapter 5.

Apart from the experimental validation, mathematical homogenisation andvolume averaging theory support our approach.

Figure 8.1: The computed permeability and experimental values for twonon-woven and two woven fabrics.

A Grid2D solver that is compatible with the voxel description of textilesis developed and validated. The Grid2D solver is faster than the Stokessolver, but at the cost of accuracy. The method is well suited for textileswith a more flat structure, and for fast, basic parameter studies.

Existing pore-network models are not sufficiently accurate to capture thecomplexity of a porous medium, in particular for the computation of thepermeability of rock. These models estimate the conductivity of the pores byapproximating the pore geometry with simple volumes for which analyticalsolutions are available. A new pore-network method is presented, which

139

uses the Stokes solver to compute the conductivity of the pores. This newapproach yields a considerable improvement of the accuracy of the pore-network solver.

Contributions

• The existing NaSt3DGP Navier-Stokes solver has been extended toenable the solution of the Brinkman equation and the use of a pres-sure difference as driving force. The interface of NaSt3DGP has beenextended to automatically read the voxel input from WiseTex.

• A Stokes solver has been developed, giving special attention to all con-straints that are imposed by the complex, dense geometry of porousmedia. The solver can handle sheared unit cells.

• The simulation of the intra-yarn flow has been investigated, and dif-ferent analytical formulas have been compared with the results of ourCFD simulations.

• Our method and solver have been validated extensively using experi-mental data, and available models were refined.

• The Grid2D method has been re-implemented to be compatible withthe voxel description of the textiles. This second implementation al-lows for a validation of the existing software as well as for an extensivecomparison between the Stokes and the Grid2D solver.

• Symmetric boundary conditions have been implemented and the in-terface of the Stokes solver has been adapted, to make it suitable forthe computation of the conductivity of the pores of a network-model.

Recommendations for further work

Numerical methods and textile modelling

The core of the thesis is the development of a Stokes solver based on thefinite difference discretisation on a regular grid. A disadvantage of themethod is that fine grids are required to obtain an accurate solution. Thesolution on a locally refined, structured mesh could be a solution for this.As the grid is still structured, the meshing can be done automatically, andthe grid is only refined if necessary. A finite volume approach would thenfacilitate the equations.


If more accurate permeability computation on the meso-scale are desirable,this will be at the cost of computation time. In the RTM process, fluid-solidand fluid-fluid interactions are present. In literature, many publications onthe topic of fluid-solid and two-phase flow modelling can be found. Thiscan be integrated in our software package, however, the computation timewill rise drastically.

The Grid2D method is a fast method, but at the cost of accuracy. A de-tailed investigation of the reason for the loss of accuracy could yield animprovement. For example an extra penalty could be added which capturesthe crimp of the yarns.

Macro-simulations

The final goal of permeability predictions is to perform reliable macro-simulations. The macro-simulations are used to predict dry spot formationsand to optimise the filling time. We have addressed the variability brieflyin this work, but in that domain still interesting research can be performed.What are the most important parameters for the dry spot formation and fill-ing time, and how can they be optimised? A combination of the meso-scaleand macro-scale tools can also help to produce new textile reinforcementswith optimal properties for the LCM process.

Partners

The research presented in this thesis is the result of close cooperation withresearchers from the Universities of Bonn, Nottingham and Brussels (figure8.2). The experimental results of the non-crimp fabrics are made availableby several institutions.

141

WiseTex

TexGen

Textile Modelling

K.U.Leuven

Textile Modelling

Univ.

Nottingham

__________

Grid2D

K-prediction

Univ. Nottingham

NaSt3DGP

3D Finite Difference Navier-Stokessolver

Universitaet Bonn

Experiments

Vrije Univ. Brussel TECABS-project(Ecole des Mines,EPFL Lausanne,

K.U.Leuven)

K

Input

FlowTex

K-prediction

K.U.Leuven

Figure 8.2: The scientific partners

Appendix A

FlowTex GUI

De Stokes solver (chapter 3) has been integrated in a graphical user interface(GUI) that facilitates the permeability computations and the data flow withWiseTex. The GUI is suited for user with little knowledge on CFD.

The user first loads a textile model designed with the WiseTex software.The image and data of the fabric appear in the Fabric Data tab. Before thecomputations can start, a voxel file has to be made. The user chooses thediscretisation step in every direction and presses the Create file button. Ifa voxel file has been made before, it can be loaded via the Browse button.

The default options for the flow simulation are

1. the computation of flow in the X-direction;

2. periodic boundary conditions in the three directions;

3. solution of the Stokes equations, neglecting intra-yarn flow;

4. a required residue of 10−4;

5. a restart of the GMRES(m) method after 300 steps;

6. a maximal number of 300 iterations.

The magnitude of the applied force cannot be chosen and is ||f || = 0.01. Thefirst three options are important for the fluid flow that will be simulated andthe resulting permeability value. The final three are numerical options, thatnormally do not have to be changed. Note that this is the only influencethe user can have or should have on the numerical core of the permeability

143

144 APPENDIX A. FLOWTEX GUI

Figure A.1: The FlowTex graphical user interface.

computation. Three numerical solvers are available in the GUI, the Stokessolver, the Navier-Stokes solver and the lattice Boltzmann solver. Theyhave similar options to set.

The output is the flux in every direction, and the flux divided by the appliedfor force for the direction of the applied force. If three simulations areperformed, enough data is available to compute the permeability tensorand the principal direction. The latter is not included in the GUI.

For the more advanced user, a command line version of the Stokes solver isavailable. This facilitates batch runs on more than one voxel file.

Appendix B

Volume averaging of themomentum equation

In section 2.5 the derivation of the volume average of the divergence of thefluid is given. In this appendix we give the derivation of the volume averageof the momentum equation of the Stokes equations. This is a literal copyfrom [109]. (Symbols may have another denotation as before.)

Before attacking the equations of motion we define a piezometric pressure,P , as

P = p− p0 + ρψ (B.1)

where p0 is some reference pressure and ψ is the gravitational potentialfunction which must satisfy the condition

g = −∇ψ. (B.2)

Equation (B.2) specifies ψ to within an arbitrary constant, and we willchoose this constant so that ψ = 0 when p = p0. In term of equation (B.1),the equations of motion can be written as

0 = −∇P + µ∆u. (B.3)

Again, it should be kept in mind that u is the velocity relative to thevelocity of the porous medium (if the medium is moving itself.) Taking thedivergence of (B.3) and making use of the continuity equation show that Pmust satisfy the Laplace equation

∆P = 0. (B.4)

145

146 APPENDIX B. VOLUME AVERAGING

If we now operate on equation (B.3) with the Laplacian, the velocity mustsatisfy the biharmonic equation

∇4u = 0. (B.5)

In addition, the velocity must satisfy the no-slip condition at the fluid-solidinterface.

u = 0 on Ai. (B.6)

If we were to specify u over the entrances and exits of any averaging volumeV , i.e.

u = f(x) on Ae (B.7)

a unique solution for u would result in the region V1. It follows that equa-tions (B.5),(B.6) and (B.7) are sufficient to determine 〈u〉 for the particularregion in question. Thus, these three equations lead to

〈u〉 = g(ro) (B.8)

where r0 is the point with which the averaging volume is associated.

Now we are going to assume that this problem can be, in effect, turnedaround and stated in the following way:

Given the governing differential equation (B.5), the no-slipcondition (B.6) and the average velocity (B.8), the velocity fieldis uniquely defined.

(B.A1)One can prove this for a number of flows. For example, the one-dimensionalflow in a capillary tube is uniquely determined by the equations of motion,the no-slip condition at the walls of the capillary, and the average velocity.Flow past a sphere is uniquely determined by the equations of motion, theno-slip boundary condition, and the average velocity, provided the velocityis uniform at infinity. (...)

Since u and 〈u〉 are continuous and defined everywhere in the region underconsideration, we can map 〈u〉 into u, i.e.

u = M · 〈u〉 (B.9)

where M is the transformation which maps the average velocity into thepoint velocity. By the assumption B.A1, the mapping is unique although

1O.A. Ladyzhenskaia. The mathematical theory of viscous incompressible flow. Gor-

don and Breach, New York, 1963.

147

not necessarily linear. If we substitute the above expression for u intoequation (B.5), we obtain

(∇4M

)· 〈u〉 + M ·

(∇4 〈u〉

)= 0. (B.10)

From previous order of magnitude arguments (see original paper), it is clearthat ∇4 〈vu〉 is negligible and the transformation matrix is determined by

(∇4M

)= 0. (B.11)

We must keep in mind here that equation (B.11) follows rigorously fromequation (B.10) if 〈u〉 is a constant vector or depends on the spatial coor-dinates to the third order or less. For the more general case, we need onlyremember that significant variations in 〈u〉 take place over a distance L andthat significant variations in M take place over a distance d where d≪≪ L.The boundary conditions for M are

M = 0 on Ai (B.12a)

〈M〉 = U at r0, (B.12b)

where (B.12b) is a logical extension of assumption B.A1. From equations(B.11)-(B.12b) we conclude that M is independent of 〈u〉, thus the pointvelocity is a linear vector function of the average velocity. From Halmos2

we know that a linear transformation is invertible if, and only if, M ·〈u〉 = 0implies that 〈u〉 = 0. However, u = 0 everywhere in the solid regardless ofthe value of 〈u〉. Thus, we cannot say that M has an inverse.

Returning now to equation (B.3), we form the scalar product with λ

0 = −λ · ∇P + µλ · ∆u (B.13)

where λ is a tangent vector to some arbitrary curve lying wholly withinthe fluid region. Letting s be the arc length measured along this curve andsubstituting equation (B.9) for u allow us to write

dP

ds= µλ · (∆M) · 〈u〉 . (B.14)

Here we have used the relation λ · ∇ = d/ds and used the same orderof magnitude arguments that were imposed between equation (B.10) and(B.11). (This step and several subsequent ones are all rigourous if 〈u〉 is aconstant vector or is a linear function of the spatial coordinates. For the

2P.R. Halmos. Finite-Dimensional vector spaces, 2nd edition, D. van Nostrand Co.,

Princeton, 1958.


general case, we constantly invoke the argument that the spatial variationsin 〈u〉 are small compared to those of M.) Integrating from s = 0 whereP = P0 to any point on the curve yields

P (r) = P0 + µ

∫ η=s(r)

η=0

(λ · ∆M) dη

· 〈u〉

The term in braces depends only on the geometry through equation (B.11)and on the position vector r through the upper limit of integration. Some-what more simply we can state that P is determined to within an arbitraryconstant by the expression

P = −µm · 〈u〉

where m depends on the position vector r in addition to the detailed struc-ture of the porous media. (The minus sign is included so that the perme-ability for isotropic porous media will be positive.)

Returning now to equation (2.26); if we replace u by the vectors [ψ, 0, 0],[0, ψ, 0] and [0, 0, ψ], this leads to a series of equations which can be sum-marised as

∇∫

Vf

ψ dV =

∫

Ae

ψn dA. (B.15)

We can substitute P for ψ on the left-hand side and −µm · 〈u〉 for ψ on theright-hand side and divide by sides by V to obtain

∇[

1

V

∫

Vf

PdV

]

= − 1

V

∫

Ae

µm · 〈u〉ndA. (B.16)

This can be expressed as

∇〈P 〉 = −µK−1 · 〈u〉 (B.17)

where K−1 is given by

K−1 =1

V

∫

Ae

nm dA

= − 1

V

∫

Ae

n

∫ η=s(r)

η=0

(λ · ∆M) dηdA.

(B.18)

We are now confronted with the question as to whether K−1 has an inverse.We know that if P is a constant, the pressure variation is hydrostatic, thepoint velocity is zero, and 〈u〉 = 0. The same line of reasoning does notnecessarily follow if 〈P 〉 is a constant; however, it is intuitively appealing,and we will make the assumption

149

∇〈P 〉 = 0 implies 〈u〉 = 0.

(B.A2)

The theorem of Halmos then indicates that K−1 has an inverse which wedesignate by K so that equation (B.16) takes the form

〈u〉 = − 1

µK · ∇ 〈P 〉 (B.19)

Appendix C

Data Mono-FilamentFabric

Spacing 2

Spacing 1

Heig

ht

Parameter Value

Weave Basket

Width yarns (mm) 0.43

Height yarns 0.20

Gap fill (mm)

Areal density (g/m2) 429.6

Spacing Warp direction 0.43/0.65

Spacing Weft direction 0.43/0.70

Porosity 0.37

151

152 APPENDIX C. DATA MONO-FILAMENT FABRIC

Appendix D

Data of the non-crimpfabrics

Width

1/2 L

ength

Spacing 1Spacing 1Spacing 1

Spacing 2

Figure D.1: Stitching yarn and ply 4 of the Q 0-45-90-45 fabric, showingthe geometrical parameters.

153

154 APPENDIX D. DATA OF THE NON-CRIMP FABRICS

Textile tag Part Property Value

Q 0-45-90-45 Ply Overall Vf 41.2 %Knitting pattern tricot+chainSpacing 2.74/5.07mmAreal density 667.9 g/(sq.m.)Ply Vf 45 %Fibre orientation 0/45/9045

Stitching yarn d1/d2 0.07/0.14mmVf 46.3 %

Openings Face factor 7.47Back factor 5.48Reduction after shear 1Width/length ply 1 0.66/NAmmWidth/length ply 2 0.18/2.6mmWidth/length ply 3 0.18/2.6mmWidth/length ply 4 0.48/7.3mm

Kxx 8.83 E-04mm2

Kyy 1.28 E-03mm2

B2 0-90 Ply Overall Vf 38.3 %Knitting pattern tricot+chainSpacing 1.71/4.94mmAreal density 308.7 g/(sq.m.)Ply Vf 47 %Fibre orientation 0/90


Openings Face factor 2.05Back factor 4.5Reduction after shear 0.5Width/length ply 1 0.18/NAmmWidth/length ply 2 0.40/NAmm

Kxx 4.09 E-4mm2

Kyy 8.60 E-4mm2

155


B1 45-45 Ply Overall Vf 38.5 %Knitting pattern tricot closedSpacing 1.71/4.94mmAreal density 309.7 g/(sq.m.)Ply Vf 47 %Fibre orientation 45/-45


Openings Face factor 3.13Back factor 5.52Reduction after shear 0.5Width/length ply 1 0.28/4.10mmWidth/length ply 2 0.46/6.90mm

Kxx 5.04 E-4mm2

Kyy 4.77 E-4mm2

Q tricot Ply Overall Vf 41.2 %Knitting pattern tricotSpacing 2.74/5.07mmAreal density 668.5 g/(sq.m.)Ply Vf 45 %Fibre orientation 0/45/9045



Kxx 8.83 E-04mm2

Kyy 1.28 E-03mm2

156 APPENDIX D. DATA OF THE NON-CRIMP FABRICS


Q crack7-5 Ply Overall Vf 41.2 %Knitting pattern tricot+chainSpacing 2.74/5.07mmAreal density 667.9 g/(sq.m.)Ply Vf 45 %Fibre orientation 0/45/9045



Kxx 8.10 E-04mm2

Kyy 1.24 E-03mm2

Q crack8-6 Ply Overall Vf 41.2 %Knitting pattern tricot+chainSpacing 2.74/5.07mmAreal density 667.9 g/(sq.m.)Ply Vf 45 %Fibre orientation 0/45/9045



Kxx 9.60 E-04mm2

Kyy 1.33 E-03mm2

157


Q yarn02 Ply Overall Vf 43.4 %Knitting pattern tricot+chainSpacing 2.74/5.07mmAreal density 667.9 g/(sq.m.)Ply Vf 45 %Fibre orientation 0/45/9045

Stitching yarn d1/d2 0.05/0.07mmVf 100 %


Kxx 1.75 E-03mm2

Kyy 2.39 E-03mm2

Q chain Ply Overall Vf 43.6 %Knitting pattern chainSpacing 2.74/5.07mmAreal density 666.8 g/(sq.m.)Ply Vf 45 %Fibre orientation 0/45/9045



Kxx 8.88 E-04mm2

Kyy 1.28 E-03mm2

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ites part A: applied science and manufacturing, 36:1207–1221, 2005.

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Curriculum Vitae

Opleiding en onderzoeksloopbaan

• 1998-2003Burgerlijk Ingenieur Computerwetenschappen, K.U.Leuven

• 09/2003- 05/2008Wetenschappelijk medewerker K.U.Leuven,Technisch-Wetenschappelijk Rekenen,K.U.Leuven

Publicaties

Hoofdstuk in een boek

• B. Verleye, M. Klitz, R. Croce, D. Roose, S. Lomov, and I. Verpoest.Computation of permeability of textile with experimental validation for

monofilament and non crimp fabrics, volume 55 of Studies in Compu-

tational Intelligence, chapter 6, pages 93–110. Springer, May 2007.

Internationale tijdschriften

• G. Morren, J. Gu, H. Sol, B. Verleye, and S. Lomov. Stereolithogra-phy specimen to calibrate permeability measurements for RTM flowsimulations. Advanced Composites Letters, 15(4):119–125, 2006.

• B. Verleye, R. Croce, M. Griebel, M.Klitz, S.V. Lomov, G. Morren,H. Sol, I. Verpoest, and D. Roose. Permeability of textile reinforce-ments: Simulation; influence of shear, nesting and boundary condi-tions; validation. Composites Science and Technology, submitted.

169

170 CURRICULUM VITAE

• B. Verleye, G. Morren, S.V. Lomov, H. Sol, and D. Roose. User-friendly permeability predicting software for technical textiles. Re-

search Journal of Textile and Apparel , submitted.

• P. Van Marcke and B. Verleye et al. Pore-network model for thecomputaton of the permeability of porous rock. in preparation.

Proceedings van internationale conferenties

• B. Verleye, S.V. Lomov, A. Long, C.C. Wong, and D. Roose. Per-meability of textile reinforcements: efficient prediction and validation.In K. Kageyama, T. Ishikawa, N. Takeda, M. Hojo, S. Sugimoto, andT. Ogasawara, editors, Proceedings of the 6th international conference

on composite materials, pages 220–221. Japan Society for CompositeMaterials, 2007.

• B. Verleye, G. Morren, S.V. Lomov, H. Sol, and D. Roose. User-friendly permeability predicting software for technical textiles. InJ. Ottjes and H. Veeke, editors, Proc. Industrial Simulation Confer-

ence 2007, pages 455–458, 2007.

• B. Verleye, R. Croce, M. Griebel, M. Klitz, S. V. Lomov, I. Verpoest,and D. Roose. Finite difference computation of the permeability oftextile reinforcements with a fast Stokes solver and new validationexamples. In E. Cueto and F. Chinesta, editors, American Institute

of Physics Conference Series, volume 907 of American Institute of

Physics Conference Series, pages 945–950, April 2007.

• B. Verleye, D. Roose, S. Lomov, I. Verpoest, G. Morren, and H. Sol.Computation of permeability of textile reinforcements. In N. Justerand A. Rosochowski, editors, The 9th International Conference on

Material Forming ESAFORM 2006, pages 735–738, 2006.

• S.V. Lomov, A.Prodromou, I. Verpoest, B. Verleye, D. Roose,T. Peeters, and B. Laine. Voxel representation of a unit cell of textilecomposite: Mechanical properties and permeability. In Proceedings

of ECCM, 2006.

• B. Verleye, M. Klitz, R. Croce, M. Griebel, S.V. Lomov, D. Roose,and I. Verpoest. Predicting the permeability of textile reinforcementsvia a hybrid Navier-Stokes/Brinkman solver. In 8th International

conference on flow processes in composite materials, Douai, France,July 2006.

CURRICULUM VITAE171

• S.V. Lomov, I. Verpoest, B. Verleye, B. Lane, and F. Boust. Wisetexmodels of permeability of textiles. In 8th International conference on

flow processes in composite materials, pages 57–63, 2006.

• G. Morren, J. Gu, H. Sol, B. Verleye, and S. Lomov. Permeabilityidentification of textile structures by inverse methods. In Proceed-

ings of the 7th national congress on theoretical and applied Mechanics,pages 1–10, 2006.

• G. Morren, J. Gu, H. Sol, B. Verleye, and S. Lomov. Permeabilityidentification of textile structures by inverse methods. In Proceedings

of the 2006 SEM Annual Conference and Exposition on Experimental

and Applied Mechanics, pages 1–10, 2006.

• S.V. Lomov, I. Verpoest, E. Bernal, F. Boust, V. Carvelli, J.-F.Delerue, P. De Luca, L. Dufort, S. Hirosawa, G. Huysmans, S. Kon-dratiev, B. Laine, T. Mikolanda, H. Nakai, C. Poggi, D. Roose,F. Tumer, B. Van Den Broucke, B. Verleye, and M. Zako. Virtual tex-tile composites software wisetex: integration with micro-mechanical,permeability and structural analysis. In Proc. ICCM-15, page 10,Durban, South Africa, July 2005. CD-edtion.

• B. Verleye, M. Klitz, R. Croce, D. Roose, S. Lomov, , and I. Ver-poest. Computation of permeability of textile reinforcements. InProc. IMACS2005, page 6.

Technische rapporten

• B. Verleye, M. Klitz, R. Croce, D. Roose, S. Lomov, and I. Verpoest.Computation of permeability of textile reinforcements. TechnicalReport SFB 228, Universitat Bonn, 2005.

• B. Verleye, M. Klitz, R. Croce, D. Roose, S. Lomov, and I. Verpoest.Computation of the permeability of textiles. Technical Report TW455, K.U.Leuven, Department of Computer Science, April 2006.

172 CURRICULUM VITAE

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