The stability and behaviour of flows in stenotic geometries Martin Griffith To cite this version: Martin Griffith. The stability and behaviour of flows in stenotic geometries. Fluid Dynamics [physics.flu-dyn]. Université de Provence - Aix-Marseille I, 2007. English. tel-00437443 HAL Id: tel-00437443 https://tel.archives-ouvertes.fr/tel-00437443 Submitted on 30 Nov 2009 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITÉ DE PROVENCE AIX-MARSEILLE I Institut de Recherche sur les Phénomènes Hors Équilibre et MONASH UNIVERSITY, MELBOURNE, AUSTRALIE Department of Mechanical Engineering ______________ THÈSE présentée pour obtenir le grade de DOCTEUR DE L’UNIVERSITÉ DE PROVENCE Discipline: Mécanique et Physique des Fluides Ecole Doctorale: Physique, Modélisation et Sciences pour l’Ingénieur par Martin D. GRIFFITH le 3 décembre 2007 Stabilité et dynamique des écoulements en géométrie de sténose ________________ Directeurs de thèse : M. Thomas LEWEKE M. Kerry HOURIGAN M. Mark C. THOMPSON ________________ JURY : M. Kerry HOURIGAN M. Thomas LEWEKE M. GertJan F. VAN HEIJST Rapporteur M. Peter WITT M. Sanjay MITTAL Rapporteur The Stability and Behaviour of Flows in Stenotic Geometries by Martin D. Griffith BA/BEng(Hons) A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosopy (Cotutelle) Department of Mechanical Engineering, Monash University and Institut de Recherche sur les Ph´enom`enes Hors Equilibre (IRPHE), Universit´e de Provence Aix-Marseille I September 2007 Contents 1 Introduction 1 1.1 Project aims . 1 1.2 Overview of the study . 2 1.3 Structure of the thesis . 3 2 Literature Review 5 2.1 Fluid mechanics in blood flow . 5 2.1.1 Modelling blood flow . 5 2.1.1.1 Analytical solution to pulsatile flow . 6 2.1.1.2 Lumped parameter models . 9 2.1.1.3 One-dimensional models . 9 2.1.1.4 Localised cardiovascular phenomena . 10 2.1.2 Biological response to fluid mechanical characteristics . 11 2.2 Steady flow in occluded paths . 14 2.2.1 The backward-facing step . 15 2.2.1.1 Paths to instability . 16 2.2.1.2 Expansion ratio effects in backward-facing step type flows 20 2.2.1.3 Present work (chapter 4) . 22 2.2.2 Stenotic geometries { steady flow . 22 2.2.2.1 Present work (chapter 5) . 27 2.3 Pulsatile flows . 28 2.3.1 Two-dimensional geometries . 28 2.3.2 Stenotic geometries { pulsatile flows . 29 2.3.2.1 Direct stenotic flow modelling . 30 2.3.2.2 Idealised stenosis modelling . 31 2.3.2.3 Present work (chapter 6) . 40 2.4 Chapter summary . 40 3 Methodology 43 3.1 Numerical method: simulating the flow field . 43 iii 3.1.1 The spectral-element method . 44 3.1.1.1 Temporal discretisation . 44 3.1.1.2 Coordinate systems . 45 3.1.2 Floquet stability analysis . 45 3.2 Creating the flow models . 49 3.2.1 The two-dimensional channel . 49 3.2.1.1 Boundary conditions . 51 3.2.1.2 Mesh layout and resolution . 51 3.2.2 Axisymmetric problem definition: steady inlet flow . 54 3.2.2.1 Boundary conditions . 55 3.2.2.2 Mesh layout and resolution . 56 3.2.3 Numerical pulsatile flow . 57 3.2.3.1 Pulsatile boundary condition: Womersley's solution . 58 3.2.4 Mesh layout and validation: pulsatile flow . 60 3.3 Experimental method . 62 3.3.1 Description of the apparatus . 62 3.3.2 Diagnostic techniques . 68 3.3.2.1 Coloured dye visualisation . 68 3.3.2.2 Particle image velocimetry (PIV) . 70 3.3.3 Optimisation of the apparatus . 73 3.4 Chapter summary . 74 4 Wake Behaviour Through a Constricted Channel 77 4.1 Introduction . 77 4.2 Flow recirculation length . 79 4.3 Wake behaviour . 82 4.4 Three-dimensional behaviour . 86 4.4.1 Instability growth rates . 86 4.4.2 Structure of the instability . 89 4.4.3 Instability mechanism - centrifugal . 91 4.4.4 Instability mechanism - elliptic . 95 4.5 Chapter summary and discussion . 98 5 Steady Flow Through an Axisymmetric Stenotic Geometry 101 5.1 Introduction . 101 5.2 Flow characteristics . 102 5.2.1 Recirculation zone lengths . 104 5.2.2 Similarity . 109 5.3 Instability . 112 5.3.1 Linear stability analysis . 112 5.3.1.1 Mode structures . 112 5.3.2 Instability: experimental . 115 5.3.2.1 Shear-layer oscillations . 119 5.3.2.2 Periodically-forced flows: experimental . 123 5.3.2.3 Periodically-forced flows: numerical . 125 5.4 Chapter summary and discussion . 133 6 Pulsatile Flow in Axisymmetric Stenotic Geometries 137 6.1 Introduction . 137 6.2 Flow characteristics . 138 6.2.1 Vortex ring behaviour . 141 6.2.2 Wall shear stresses . 147 6.3 Floquet stability analysis: blockage ratio effect . 152 6.3.1 Stability analysis: T = 2:5 . 152 6.3.2 Stability analysis: T = 5:0 . 156 6.3.3 Stability analysis: T = 1:0 . 159 6.3.4 Stability analysis summary: A = 0:75 . 165 6.3.5 Effect of amplitude on absolute stability . 166 6.4 Experimental results . 173 6.4.1 Experimental validation . 173 6.4.2 Stability of the experimental flow . 176 6.4.2.1 Instability: b = 0:50 . 180 6.4.2.2 Instability: b = 0:75 . 182 6.4.2.3 Instability: b = 0:90 . 189 6.5 Chapter summary and discussion . 193 7 Conclusions 199 7.1 Summary of conclusions . 199 7.2 Possible future work . 204 Abstract Steady and pulsatile flows in two blockage geometries are investigated. The geometries serve as idealised models of arterial stenosis, a common condition whereby a localised constriction grows within an artery, reducing the blood flow and pressure downstream and potentially causing stroke. The geometries are simplified models of the physiological reality of diseased arteries; the aim is to isolate and describe the fundamental fluid mechanical characteristics of stenotic flows. An emphasis is placed on the effect on the flow and its stability of changes in blockage ratio, or stenosis severity. The first part of the thesis presents a numerical investigation of the steady inlet flow through a two-dimensional channel with a semi-circular blockage on one side of the channel, for blockage ratios of 0.05 to 0.9 and Reynolds numbers up to 3000. Variations in the wake behaviour and stability with Reynolds number and blockage are reported. The leading instability mode of the flow is described and its underlying mechanism determined to be elliptical. A geometry consisting of a straight tube with an axisymmetric blockage is the focus of the second part of the thesis. Flows of both steady and pulsatile inlet velocity are modelled numerically and experimentally, for blockage ratios of 0.20 to 0.95 and Reynolds numbers up to 2400 for steady flow and 700 for pulsatile flow. The changes in wake behaviour with the various flow parameters are described. Numerically, different linear modes of instability are identified for different blockage ratios. Experimentally, the action of convective shear layer instability is found to be important. For steady flow, comparisons are made between the periodic behaviour of the convective instability and the response of stable flows to periodic forcing. For pulsatile flow, some evidence is found of the action of linear instability modes experimentally, although convective shear layer instability dominates. The variation of both forms of instability with Reynolds number and blockage ratio is characterised. Les ´ecoulements stationnaires et puls´es sont examin´es dans deux g´eom´etries bloqu´es. Les g´eom´etries utilis´ees sont des mod`eles d'art`eres st´enotiques, une condition com- mune ou une constriction localis´ee grandit dans une art`ere, r´eduisant ainsi l'´ecoulement du sang et la pression en aval et pouvant conduire a` la formation d'un caillot. Les g´eom´etries sont des mod`eles simplifi´es de la r´ealit´e physiologique d'une vrai st´enose, l'intention de l'´etude ´etant avant tout d'isoler et de d´ecrire les caract´eristiques fonda- mentales des ´ecoulements st´enotiques. L'´etude se concentre sur l’effet sur l'´ecoulement et la stabilit´e de la variation du taux constriction, autrement dit la s´ev´erit´e du st´enose. La premi`ere partie de la th`ese pr´esente une investigation de l'´ecoulement station- naire a` travers une conduite bidimensionelle, avec une constriction asymm´etrique semi- circulaire, pour des taux de constriction de 0.05 a` 0.9 et des nombres de Reynolds jusqu’`a 3000. Les changements de la stabilit´e et du comportement du sillage avec le nombre de Reynolds et le taux de constriction sont r´eport´es. Le mode primaire de l'instabilit´e est d´ecrit et son m´ecanisme d´etermin´e analytiquement comme ´etant ´elliptique. La deuxi`eme partie concerne une g´eom´etrie cylindrique avec une constriction ax- isymm´etrique. Les ´ecoulements stationnaires et puls´es sont examin´es num´eriquement et exp´erimentalement, pour des taux de constriction de 0.20 a` 0.95 et des nombres de Reynolds jusqu’`a 2400 pour l'´ecoulement stationnaire et 700 pour l'´ecoulement puls´e.