Fluid structure interaction in flexible vessels

Fluid structure interaction in flexible vessels Christina Grigoria Giannopapa Thesis submitted for the Degree of Doctor of Philosophy of the University of London King’s College London 2004

Dedicated to my grandmother Mrs Christina Katrivanou, to my parents and to my cousin George.

SUPERVISORS Dr. G. Papadakis Dr. M.C.M Rutten (Eindhoven University of Technology) Dr. K. Lee

c 2004 by Christina G. Giannopapa Copyright All rights are reserved. No part of this publication may be reproduced, stored in retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the author. This research was conducted in King’s College London (UK) and in Eindhoven University of Technology (The Netherlands). Financially support was provided by EPSRC, King’s College London and Marie Curie Fellowships, European Commission.

Abstract The thesis is concerned with the study of fluid-structure interaction in flexible tubes both from the modelling as well as the experimental point of view. More specifically, it presents the first stage of development and testing of a novel unified solution method suitable for fluid-structure interaction problems. In the conventional approach for modelling such problems, the fluid and solid components are treated separately, information is exchanged at their interface and different solution algorithms are used for the two components. The equations for solids are solved for displacement and stress and, the ones for fluids are solved for velocity and pressure. The exchange of information between two solution methods that solve for different quantities is not a trivial task and has also known drawbacks such as high computational cost and potential numerical instabilities, especially for very flexible structures. In the new method presented in the thesis, a single set of equations is used to describe both fluid and solid, while the interface between them is contained within the solution domain itself. This is achieved by reformulating the solid equations to contain the same primitive variables used in fluids i.e. velocity and pressure. The PISO algorithm is used to handle the velocity-pressure coupling. The method proposed is fully tested for solids on a structural dynamic problem (beam bending) and the results compared successfully with the classical structural analysis. In order to quantify the dissipation characteristics of the numerical integration technique, a stability eigenvalue analysis of the proposed time marching and spatial discretisation scheme is performed in one dimension but the conclusions of this analysis were also in agreement with the results of the beam bending. The new formulation for solids is found to be stable and robust, thus it can be used in the next stage of testing in full fluid-structure-interaction problems. The new algorithm can be validated against the results obtained during the experimental phase of the work, which is focused on wave propagation in flexible vessels. This experimental study is also motivated by the need to understand arterial blood flow. Although the general principles governing the arterial hemodynamics are well known, the assessment of non-linearities arising from wall thickness variation and geometric tapering, naturally present in the arterial tree morphology, have not been fully investigated. To this end, a complete experimental data set on wave propagation was collected for six flexible tubes with different wall thickness and geometric tapering. i

ii A special manufacturing methodology was used to produce the tubes. They were manufactured in such a way that pairs of tubes had the same wave speed according to the linear pulse wave propagation theory. Any discrepancy in the wave propagation characteristics thus indicates the importance of the non-linearities. The measured quantities were pressure and pressure gradient using two pressure wires, flow rate using a ultrasound flow probe, and wall distension using ultrasound. The geometric tapering was found to be of great importance as it alters the shape of the pressure signal. The experimental measurements of the straight tubes are compared with the linear theory and highly encouraging levels of agreement are found when the viscoelastic properties of the wall are taken into account.

Acknowledgements I would like to express my sincere gratitude to my supervisors: Dr. G. Papadakis, Dr. M.C.M. Rutten (Eindhoven University of Technology) and Dr. K.C. Lee, for their continuous interest, support and guidance during this study. Equally I would like to thank Dr. A.S. Tijsseling (Eindhoven University of Technology) who has been my supervisor under the European Commission Marie Curie Fellowship grant. I am indebted to my colleagues and friends in the groups of Prof. M. Yianneskis, Prof. R.M.M. Mattheij (Eindhoven University of Technology) and Prof. F.N. van de Vosse (Eindhoven University of Technology), as well as the Professors themselves. In particular I would like to thank Dr. M.E. Verbeek (Eindhoven University of Technology) for his numerous valuable comments. I am grateful to Mr. M.W. Wijlaars (Eindhoven University of Technology) for his help and guidance in the laboratory, Mrs E.R.H. van Dijk (Eindhoven University of Technology) and Mr. J. Greenberg for the arrangement of many administrative matters. I would like to thank Dr. C.J. Greenshields for helping me during the first year to aquire the background knowledge needed to develop the unified solution method and for initially stimulating my interest in the field; and Mr. H. Weller from Nabla Ltd. for his initial assistance on technical issues related to the finite volume C++ library. I am sincerely greatful to Dr. S. Balabani for being my guardian angel during my entire studies in King’s College London; I am in debt to her for life. Finally, I would like to thank Mr. J.D. Malo and Mr. R.J. Smits from Research DG, European Commission for allowing me to allocate time in writing up this thesis while working for them. I would also like to thank Mr. G. Papageorgiou and Mr. P. Keraudren for their advice and support in related administrative maters. The financial support provided by the EPSRC (Engineering and Physical Sciences Research Council) under the GR/N65769 grant, by the King’s College London top up grant and by the Marie Curie Fellowships supported by the European Commission is greatfully acknowledged.

iii

iv

Contents

Contents Contents

iv

List of Figures

ix

List of Tables

xv

Nomenclature

xvi

1 Introduction and literature survey

1

1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2 Morphology of arteries . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2.1

Wall layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.2

Wall dimensions . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3 Computational Methods for fluid structure interaction

. . . . . . . .

4

1.4 Wave propagation in flexible vessels . . . . . . . . . . . . . . . . . . . 10 1.4.1

Theoretical models on straight tubes . . . . . . . . . . . . . . 10

1.4.2

Experimental models on straight tubes . . . . . . . . . . . . . 13

1.4.3

Theoretical models on tapered tubes . . . . . . . . . . . . . . 16

1.4.4

Experimental models on tapered tubes . . . . . . . . . . . . . 17

1.4.5

Concluding summary . . . . . . . . . . . . . . . . . . . . . . . 18

1.5 Objectives of this study . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.6 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 Mathematical formulation of a unified framework for fluids and solids 25 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.1

Standard stress analysis for linear elastic (or Hookean) solid . 28

2.3.2

Velocity based formulation for linear elastic (or Hookean) solid 28

2.3.3

Velocity and Pressure based formulation for linear elastic (or Hookean) solid . . . . . . . . . . . . . . . . . . . . . . . . . . 29 v

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2.4 Comparison of the new velocity-pressure formulation for solids with the fluids formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.6 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3 Numerical solution method 3.1 Introduction

35

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Discretisation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2.1

Determination the face value φ f . . . . . . . . . . . . . . . . . 39

3.2.2

Discretisation of the gradient . . . . . . . . . . . . . . . . . . 40

3.2.3

Discretisation of the divergence . . . . . . . . . . . . . . . . . 41

3.2.4

Discretisation of the Laplacian term

3.2.5

Laplacian versus Divergence-Grad

3.2.6

Temporal Discretisation . . . . . . . . . . . . . . . . . . . . . 43

3.2.7

Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 46

. . . . . . . . . . . . . . 41 . . . . . . . . . . . . . . . 41

3.3 Final form of equations and discretisation of the transient term

. . . 48

3.3.1

Reformulation in order to increase convergence rate . . . . . . 48

3.3.2

Temporal discretisation approaches . . . . . . . . . . . . . . . 49

3.4 Iterative solution methods of governing equations . . . . . . . . . . . 52 3.4.1

Governing equations . . . . . . . . . . . . . . . . . . . . . . . 52

3.4.2

Non-linearity and pressure/velocity coupling . . . . . . . . . . 53

3.4.3

Derivation of pressure equation . . . . . . . . . . . . . . . . . 54

3.4.4

Velocity-Pressure coupling algorithms . . . . . . . . . . . . . . 56

3.5 Investigation of boundary conditions for fluids . . . . . . . . . . . . . 59 3.6 Boundary condition for solids for the unified solution method

. . . . 63

3.6.1

Boundary conditions for the displacement formulation . . . . . 64

3.6.2

Boundary conditions for the velocity formulation . . . . . . . 64

3.6.3

Boundary conditions for the velocity-pressure formulation . . . 65

3.6.4

3.6.3.1

Boundary conditions for velocity . . . . . . . . . . . 65

3.6.3.2

Boundary condition types for pressure . . . . . . . . 66

Optimal choice of boundary conditions . . . . . . . . . . . . . 67

3.7 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.7.1

Wave equation (1D) . . . . . . . . . . . . . . . . . . . . . . . 70

3.7.2

Velocity formulation for linear elastic Hookean solid (1D) . . . 72

3.8 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4 Validation of the new formulation for solids

81

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2 Case Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3 Analytical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

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4.4.1

Displacement calculated using the standard stress analysis . . 85

4.4.2

Discretisation error analysis for the new formulations . . . . . 87 4.4.2.1

Calculation of the accumulated term . . . . . . . . . 88

4.4.2.2

Temporal term discretisation . . . . . . . . . . . . . 91

4.4.2.3

Mesh quality . . . . . . . . . . . . . . . . . . . . . . 96

4.4.3

Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 97

4.4.4

Other cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.4.4.1

Analytical solution

. . . . . . . . . . . . . . . . . . 98

4.4.4.2

Numerical solution . . . . . . . . . . . . . . . . . . . 98

4.5 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5 Wave propagation experiments in flexible vessels with wall thickness variation and geometric tapering

105

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.2 The Tube Models Methodology . . . . . . . . . . . . . . . . . . . . . 105 5.2.1

The vessels design and specifications . . . . . . . . . . . . . . 106

5.2.2

Manufacturing Method . . . . . . . . . . . . . . . . . . . . . . 109

5.3 Material Properties of the Tubes

. . . . . . . . . . . . . . . . . . . . 109

5.4 Measurement Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.4.1

Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . 112

5.4.2

Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.4.3

Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.4.4

Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.5.1

Static pressure - initial diameter relation . . . . . . . . . . . . 115

5.5.2

Standard deviation of measurements . . . . . . . . . . . . . . 117

5.5.3

Fluid motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.5.4

Wall motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.6 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6 Comparison of experimental results with linear wave propagation methods

137

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2 Linear Theory of Wave Propagation in Flexible Vessels . . . . . . . . 137 6.2.1

Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.2.2

Wave propagation speeds . . . . . . . . . . . . . . . . . . . . . 140

6.2.3

Wave reflections through discrete transitions . . . . . . . . . . 142

6.3 Implementation of the continuous linear model . . . . . . . . . . . . . 144 6.4 Comparisons with Linear Model for Elastic Material . . . . . . . . . . 144 6.5 Comparisons with Linear Model for Viscoelastic Material . . . . . . . 145 6.6 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

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Contents

7 Conclusions 7.1 Overview

159 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.2 Main achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 7.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 7.3.1 Mathematical modelling . . . . . . . . . . . . . . . . . . . . . 162 7.3.2

Experimental work . . . . . . . . . . . . . . . . . . . . . . . . 164

Bibliography

165

A The Tube Models Manufacturing Methodology A.1 The vessels design and specifications . . . . . . . . . . . . . . . . . . A.2 Manufacturing set-up . . . . . . . . . . . . . . . . . . . . . . . . . . .

i i iii

A.3 Equations for manufacturing . . . . . . . . . . . . . . . . . . . . . . . A.4 Straight tube manufacturing . . . . . . . . . . . . . . . . . . . . . . .

iii v

A.4.1 Constant thickness . . . . . . . . . . . . . . . . . . . . . . . . vi A.4.2 Variable thickness . . . . . . . . . . . . . . . . . . . . . . . . . vi A.5 Tapered tube manufacturing . . . . . . . . . . . . . . . . . . . . . . . viii A.5.1 Constant thickness . . . . . . . . . . . . . . . . . . . . . . . . viii A.5.2 Variable thickness . . . . . . . . . . . . . . . . . . . . . . . . . xi A.6 Wall thickness accuracy . . . . . . . . . . . . . . . . . . . . . . . . .

xi

List of Figures 1.1 FSI categories.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2 Cross sections of the arterial wall (not to scale). . . . . . . . . . . . .

3

1.3 Solution procedure of several FSI methods. . . . . . . . . . . . . . . .

6

1.4 FSI methods conventional terminology. . . . . . . . . . . . . . . . . .

7

2.1 The velocity integral from [t0,t + ∆t] . . . . . . . . . . . . . . . . . . . 28 3.1 Cell based structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 Evaluation of the face value φ f from cell centre values φP and φN assuming linear interpolation. . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Cells involved in the evaluation of the Laplacian operator at cell with cell centre denoted as P. . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4 Cells involved in the evaluation of the Divergence- Gradient operator at cell with centre denoted as P. . . . . . . . . . . . . . . . . . . . . . 43 3.5 PISO algorithm flow chart for compressible flow (for one time step). . 57 3.6 Shortest resolvable wave. . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.7 Stencil for the 1D hyperbolic finite difference equation (3.90).

. . . . 70

3.8 Accuracy portrait of the amplification factor G for the 1D hyperbolic equation (3.96). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.9 Stencil for the 1D system of equations that is equivalent to the 1D velocity formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.10 Amplitude portrait of the 1D velocity formulation in comparison with the wave equation (displacement formulation). . . . . . . . . . . . . . 77 4.1 Beam bending test case. . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.2 Analytical calculations for the vibration eigenvalues, eigenmodes and frequency of oscilation using a 1D approximation for the solution of a cantilever beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3 End displacement (m) versus time (s) (standard stress analysis). . . . 86 4.4 Standard stress analysis (envelope of displacement). . . . . . . . . . . 87 4.5 Total power comparison for the ∇2 and the ∇ • ∇ operators in the accumulated term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.6 Total power comparison for different tolerances:10e-6, 10e-7, 10e-8. . . 91 ix

x

List of Figures

4.7 Comparison of displacement formulation and velocity-based formulation for the Euler Implicit discretisation scheme (envelope of displacement).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.8 Comparison of Euler Implicit and Backward Differencing discretisation scheme (envelope of displacement). . . . . . . . . . . . . . . . . . 93 4.9 Comparison of different time step sizes: 1e-4, 1e-5, 1e-6 s for the first time derivative Euler Implicit. . . . . . . . . . . . . . . . . . . . . . . 94 4.10 Comparison of Euler Implicit using time step size 1e-5 s against Backward differencing using time step size of 1e-4 s (envelope of displacement). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.11 Mesh resolution comparison for meshes: 40x10, 60x20 and 200x50 cells. Time step size used is 1e-4 and temporal term discretisation scheme is Backward differencing (envelope of displacement). . . . . . 96 4.12 Comparison of different boundary conditions for pressure in the fully implicit velocity-pressure formulation. . . . . . . . . . . . . . . . . . . 97 4.13 Beam with size 10mx5m. No of cells used for the mesh is 20x10cells , time step size used is 1e-4 and temporal term discretisation scheme is Backward differencing. . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.14 Beam with size 40mx5m. No of cells used for the mesh is 80x10cells , time step size used is 1e-4 s and temporal term discretisation scheme is Backward differencing. . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.15 Beam with size 20mx5m, with applied end shear τ = 5e5 Pa. No of cells used for the mesh is 40x10cells , time step size used is 1e-4 s and temporal term discretisation scheme is Backward differencing. . . . . 101 5.1 Wall thickness variation for tubes C and F. . . . . . . . . . . . . . . . 108 5.2 Typical relaxation test curve for Polyurethane specimen (3% elongation). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.3 Experimental set-up for wave propagation experiments (TU/e). . . . 113 5.4 Static pressure-initial diameter relation of the straight tube (Type B). 116 5.5 A typical result at a location showing the mean of 16 measurements and the standard deviation from the mean. . . . . . . . . . . . . . . 117 5.6 Normalised pressure measurements every 50 mm along the length of the tube against scaled time for straight tubes: types A,B,C (A: straight tube with constant wall thickness of 0.1 mm; B: straight tube with constant wall thickness of 0.05 mm; C: straight tube with variable wall thickness of 0.05-0.1 mm). . . . . . . . . . . . . . . . . . 118

List of Figures

xi

5.7 Normalised pressure measurements every 50 mm along the length of the tube against time for tapered tubes: types D,E,F (D: tapered tube with constant wall thickness of 0.1 mm; E: tapered tube with constant wall thickness of 0.05 mm; F:tapered tube with variable wall thickness of 0.1-0.05 mm). . . . . . . . . . . . . . . . . . . . . . . . . 119 5.8 Normalised pressure measurements every 50 mm along the length of the tube against time for tube types A and F (A: straight tube with constant wall thickness of 0.1 mm; F: tapered tube with variable wall thickness of 0.1-0.05 mm). . . . . . . . . . . . . . . . . . . . . . . . . 120 5.9 Normalised pressure measurements every 50 mm along the length of the tube against time for tubes types C and E (C: straight tube with variable wall thickness 0.05-0.1 mm; E: tapered tube with constant wall thickness of 0.05 mm). . . . . . . . . . . . . . . . . . . . . . . . 121 5.10 Normalised flow rate measurements every 50 mm along the length of the tube against scaled time for straight tubes: types A, B, C (A: straight tube with constant wall thickness of 0.1 mm; B: straight tube with constant wall thickness of 0.05 mm; C: straight tube with variable wall thickness of 0.05-0.1 mm). . . . . . . . . . . . . . . . . . 123 5.11 Normalised flow rate measurements every 50 mm along the length of the tube against scaled time for straight tubes: types D, E, F (D: tapered tube with constant wall thickness of 0.1 mm; E: tapered tube with constant wall thickness of 0.05 mm; F: tapered tube with variable wall thickness of 0.1-0.05 mm). . . . . . . . . . . . . . . . . . 124 5.12 Normalised flow rate measurements every 50 mm along the length of the tube against time for tubes types A and F (A: straight tube with constant wall thickness of 0.1 mm; F: tapered tube with variable wall thickness of 0.1-0.05 mm). . . . . . . . . . . . . . . . . . . . . . . . . 125 5.13 Normalised flow rate measurements every 50 mm along the length of the tube against time for tubes types C and E (C: straight tube with variable wall thickness 0.05-0.1 mm; E: tapered tube with constant wall thickness of 0.05 mm). . . . . . . . . . . . . . . . . . . . . . . . 126 5.14 Normalised pressure gradient measurements every 50 mm along the length of the tube against time for tubes types A, B, C (A: straight tube with constant wall thickness of 0.1 mm; B: straight tube with constant wall thickness of 0.05 mm; C: straight tube with variable wall thickness of 0.05-0.1 mm). . . . . . . . . . . . . . . . . . . . . . 128

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List of Figures

5.15 Normalised pressure gradient measurements every 50 mm along the length of the tube against time for tubes types D, E, F (D: tapered tube with constant wall thickness of 0.1 mm; E: tapered tube with constant wall thickness of 0.05 mm; F: tapered tube with variable wall thickness of 0.1-0.05 mm). . . . . . . . . . . . . . . . . . . . . . 129 5.16 Normalised pressure gradient measurements every 50 mm along the length of the tube against time for tubes types A and F (A: straight tube with constant wall thickness of 0.1 mm; F: tapered tube with variable wall thickness of 0.1-0.05 mm). . . . . . . . . . . . . . . . . . 130 5.17 Normalised pressure gradient measurements every 50 mm along the length of the tube against time for tubes types C and E (C: straight tube with variable wall thickness 0.05-0.1 mm; E: tapered tube with constant wall thickness of 0.05 mm). . . . . . . . . . . . . . . . . . . 131 5.18 Normalised wall motion measurements every 50 mm along the length of the tube against time for tubes types A, B, C (A: straight tube with constant wall thickness of 0.1 mm; B: straight tube with constant wall thickness of 0.05 mm; C: straight tube with variable wall thickness of 0.05-0.1 mm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.19 Normalised wall motion measurements every 50 mm along the length of the tube against time for tubes types D, E, F (D: tapered tube with constant wall thickness of 0.1 mm; E: tapered tube with constant wall thickness of 0.05 mm; F: tapered tube with variable wall thickness of 0.1-0.05 mm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.20 Normalised wall motion measurements every 50 mm along the length of the tube against time for tubes types A and F (A: straight tube with constant wall thickness of 0.1 mm; F: tapered tube with variable wall thickness of 0.1-0.05 mm). . . . . . . . . . . . . . . . . . . . . . 134 5.21 Normalised wall motion measurements every 50 mm along the length of the tube against time for tubes types C and E (C: straight tube with variable wall thickness 0.05-0.1 mm; E: tapered tube with constant wall thickness of 0.05 mm). . . . . . . . . . . . . . . . . . . . . . . . . 135 6.1 Tube motion variables. Point P(z, r) on the surface of the wall at rest displaces to position P’(z + ζ, r + ξ) . . . . . . . . . . . . . . . . . . . 138 6.2 Discrete transitions between segments. . . . . . . . . . . . . . . . . . 142 6.3 Properties used for the calculations. . . . . . . . . . . . . . . . . . . . 144 6.4 Comparison of pressure experimental measurements of the straight tube with constant wall thickness of 0.1 mm with linear analytical model foran elastic tube. . . . . . . . . . . . . . . . . . . . . . . . . . 146

List of Figures

xiii

6.5 Comparison of the experimental measurements of the flow on a straight tube with constant wall thickness of 0.1 mm with linear analytical model foran elastic tube. . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.6 Comparison of the experimental measurements of the wall distension on a straight tube with constant wall thickness of 0.1 mm with linear analytical model foran elastic tube. . . . . . . . . . . . . . . . . . . . 148 6.7 Comparison of pressure experimental measurements of the straight tube with constant wall thickness of 0.05 mm with linear analytical model foran elastic tube. . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.8 Comparison of the experimental measurements of the flow on a straight tube with constant wall thickness of 0.05 mm with linear analytical model foran elastic tube. . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.9 Comparison of the experimental measurements of the wall distension on a straight tube with constant wall thickness of 0.05 mm with linear analytical model foran elastic tube. . . . . . . . . . . . . . . . . . . . 151 6.10 Comparison of the experimental measurements of the pressure on a straight tube with constant wall thickness of 0.1 mm with linear analytical model fora viscoelastic tube. . . . . . . . . . . . . . . . . . 152 6.11 Comparison of the experimental measurements of the flow on a straight tube with constant wall thickness of 0.1 mm with linear analytical model fora viscoelastic tube. . . . . . . . . . . . . . . . . . . . . . . 153 6.12 Comparison of the experimental measurements of the wall distension on a straight tube with constant wall thickness of 0.1 mm with linear analytical model fora viscoelastic tube. . . . . . . . . . . . . . . . . 154 6.13 Comparison of the experimental measurements of the pressure on a straight tube with constant wall thickness of 0.05 mm with linear analytical model fora viscoelastic tube. . . . . . . . . . . . . . . . . . 155 6.14 Comparison of the experimental measurements of the flow on a straight tube with constant wall thickness of 0.05 mm with linear analytical model fora viscoelastic tube. . . . . . . . . . . . . . . . . . . . . . . 156 6.15 Comparison of the experimental measurements of the wall distension on a straight tube with constant wall thickness of 0.1 mm with linear analytical model fora viscoelastic tube. . . . . . . . . . . . . . . . . 157 7.1 The different properties distribution in the single mesh for solving fluid structure interaction problems with the unified solution method. 163 A.1 Spin coating set-up ( TU/e). . . . . . . . . . . . . . . . . . . . . . . .

iv

A.2 Spin coating process of a tube. . . . . . . . . . . . . . . . . . . . . . .

iv

A.3 Straight tube steel rod dimensions. . . . . . . . . . . . . . . . . . . .

vi

xiv

List of Figures

A.4 Translational velocity, rotational velocity, tube wall thickness and tube diameter versus the tube length for tube C. . . . . . . . . . . . . vii A.5 Tapered tube steel rod dimensions. . . . . . . . . . . . . . . . . . . . viii A.6 Translational velocity, rotational velocity, tube wall thickness and tube diameter versus the tube length for tube E. . . . . . . . . . . . . x A.7 Translational velocity, rotational velocity, tube wall thickness and tube diameter versus the tube length for tube F. . . . . . . . . . . . . xii

List of Tables 1.1 Modelling assumptions for the fluid and solid component as found in the literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Assumptions for the fluid-solid components for straight tubes as found in the literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3 Fluid-solid assumptions for tapered tubes as found in the literature. . 20 3.1 Fourier series forms for time level n, n − 1, n − 2 and grid points j − 1, j, j + 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.1 Material properties and dimentions of the beam. . . . . . . . . . . . . 81 4.2 Computational calculations for the vibration eigenfrequencies of vibration using for the two dimensional beam bending case using the ANSYS finite element commercial package. . . . . . . . . . . . . . . 95 4.3 Comparison between analytical and computational solution for beams with different size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.1 Aorta anatomical data (Westerhof et al., 1969). . . . . . . . . . . . . 106 5.3 Geometrical parameters of tubes manufactured. . . . . . . . . . . . . 107 5.4 Physical properties of polyurethane. . . . . . . . . . . . . . . . . . . . 109 6.1 Values of coefficient ψ describing different longitudinal support conditions for thin- and thick-wall tubes. . . . . . . . . . . . . . . . . . . 141 A.1 Aorta anatomical data (Westerhof et al., 1969). . . . . . . . . . . . . A.3 Geometrical parameters of tubes manufactured. . . . . . . . . . . . .

i ii

A.5 Straight tube specifications. . . . . . . . . . . . . . . . . . . . . . . . v A.7 Tapered tube specifications. . . . . . . . . . . . . . . . . . . . . . . . viii

xv

xvi

Nomenclature

Nomenclature General Character s

Explanation scalar

a T

vector second order tensor

Operators and functions Character

Explanation

∇ ∇• ∇2

gradient operator divergence operator laplacian operator

∂s ∂t

∆s

time derivative of s discrete increment of s

|a| TT trT

absolute value of a transpose of T trace of T

dev T

deviatoric part of T

xvii

xviii

Nomenclature

Latin symbols Character

Unit

A C

analytical solution m/sec characteristic velocity of fluid or solid

Co d D

m m

Courant number vector from P to N cell centre displacement

W Hz

truncation error external power frequency

E EP f F gb

Explanation

fluid fixed gradient at the boundary

h I K

m Pa

height of the beam unit tensor bulk modulus

KP l

W m

kinetic power length of the beam

m n N

kg

mass unit vector normal to a control volume face neighbour cell centre

p P

Pa

pressure present cell centre

S S Sf

m2 m2

solid closed surface face area vector

SP t

W sec

strain power time

TP U V

W total power m/sec velocity vector m3 volume

x y

m m

position in x direction position in y direction

Nomenclature

xix

Greek symbols Character

Unit

Explanation

m

any spatial operator end displacement

η

Pa

strain tensor rate of deformation tensor dynamic viscocity

λ µ

sec Pa Pa

Lamé’s coefficient Lamé’s coefficient

kg/m3

Poison’s ratio density

A δ ε ε˙

ν ρ σ τ φ

Pa Pa

Cauchy stress tensor applied end shear any property (scalar, vector or tensor)

ϒ ω

Pa Hz

Young’s modulus frequency of undamped oscillation

ωN

weighting factor form P to N cell centre

Superscripts Character

Explanation

o

old values old old values new values

oo n ∗

spatial discretisation

xx

Nomenclature

Subscripts Character

Explanation

0

reference situation face value

f N P

value at neighbour cell value at present cell

Abbreviations Characters

Explanation

A BD

analytical backward differencing

CD CFD CSM

central differencing computational fluid dynamics computational solid mechanics

CV EP

control volume external power

EI FE FV

Euler implicit finite element finite volume

FSI KP LCR

fluid structure interaction kinetic power inductance-capacitance-resistance circuit

N NM

numerical not mentioned

PDE SP UD

partial differential equation strain power upwind differencing

Chapter 1 Introduction and literature survey 1.1

General Introduction

The term fluid-structure interaction (FSI) is a general term used to describe certain physical phenomena. Let us first define the meaning of the term, since it is sometimes misused. The important aspect is that there must be a genuine interaction between a fluid and a solid component. This implies that, at the interface, a property of the fluid influences a property of the solid and, crucially, vica versa. This project is concerned with FSI, using the term in its most common sense, that is interaction of forces and the corresponding movement of the interface (momentum interaction) rather than thermal interaction. The movement of the solid because of momentum exchange with the fluid can occur in one of two ways (Figure 1.1): by a local deformation of the solid body, or by rigid body motion. The term FSI is commonly used in flow of liquids in pipes to describe the effect of pressure on rigid body motion on complete pipe structures. Extensive reviews by Tijsseling (1996) and Tijsseling and Wiggert (2001) describe the work performed in this areas. However, this project investigates the interaction between the local deformation of flexible tubes and liquid pressure, in particular its effect on the propagation of pressure waves. Waveforms are highly dependent on the geometry of the tube. Fluid structure interaction becomes particularly important when the liquid is almost incompressible and deformation on the solid can not be neglected (Korteweg, 1878). Prediction of pressure waves is particularly important in liquid filled vessels in areas such as arterial flow, impact of filled vessels and pipelines. The study of the wave propagation phenomenon in fluid filled flexible tubes is often motivated by the need to understand arterial blood flow. The arterial flow is almost unique in that it is driven by pressure waves that initiate from the contraction of the cardiac muscle. Pulse propagation phenomena in the arteries are governed by the interaction of the blood with the elastic arterial wall. Many investigators have tried to analyse the wave propagation phenomena and 1

2

Chapter 1. Introduction and literature survey

FSI

Momentum interaction

Rigid body motion

Thermal interaction

Local deformation

Figure 1.1: FSI categories.

in particular in the cardiovascular system, the resulting blood flow and and pressure wave forms. The methods used vary from the simple windkessel model to highly complicated multidimensional mathematical and computational models. This is not trivial due to the fact that the hemodynamics of blood circulation is affected by many factors such as: vessel geometry, pulsatility, flow rates, bifurcations in branches, nonNewtonian behaviour of the blood as well as compliance of the vessel walls. For the validation of these models there is a need for in-vivo measurements as well as in vitro laboratory experiments in mechanically and constitutively well-defined systems. In Section 1.2 the morphology of the arteries is described. The literature survey is separated in two parts: the different methods used for handling the fluid-structure coupling is described in Section 1.3 while the experimental and analytical work in wave propagation in straight and tapered vessels are discussed in Section 1.4. Finally, the objectives of this study is outlined in Section 1.5 .

1.2

Morphology of arteries

The blood vessels form a closed network that carries the blood away from the heart and back. This vessel network consists of arteries, arterioles, capillaries, venules and vains. The arteries and arterioles transfer the blood away from the heart in order to deliver oxygen to the tissues and organs. The arteries are large vessels that are very strong and elastic and they deform as the blood flows away from the heart under hight pressure. They subdivide progressively to thinner and thinner tubes and eventually end up to the finest branched arterioles. Therefore according to their diameter can be grouped to: elastic arteries (aorta, brachiocephalic trunk and carotid arteries), muscular arteries (all others with diameter > 0.1 mm ) and arterioles 10 − 100 µm Roades and Tanner (1995); Levick (2000).

3

1.2. Morphology of arteries

THREE LAYERS Endothelium Tunica Intima Connective tissue

Tunica media

Tunica adventitia C.G.Giannopapa

Figure 1.2: Cross sections of the arterial wall (not to scale).

1.2.1

Wall layers

The wall of the artery consists of three distinct layers or tunics, shown in Figure 1.2, which, from inside to outside are called: tunica interna or intima, tunica media and tunica externa or adventitia.

Tunica intima The tunica intima or internal consists of a layer of a simple squamous epithelium called endothelium, that rests on a connective tissue membrane that is rich in elastic and collagenous fibres.

Tunica media In the muscular arteries the tunica media makes up the bulk of the arterial wall. It includes small muscle fibres that encircle the tube and a thick layer of elastic connective tissue. The connective tissue gives to the artery a tough elasticity to withstand the blood pressure force and at the same time stretch in order to accommodate the sudden increase of blood volume that accompanies the opening of the heart valve due to the ventricular contraction of the cardiac muscle.

4


Tunica adventitia Tunica adventitia or externa is a thin layer and mainly consists of connective tissue with irregular elastic collagenous fibres. This layer attaches the artery to the surrounding tissues. It also contains minute vessels (vasa vasorum) that give rise to capillaries and provide blood to the most external cells of the artery wall.

1.2.2

Wall dimensions

The measurement of wall thickness of the blood vessels is not a trivial task. This is due to the fact that there is not a clear line separating the adventitia from the surrounding tissues. This means that the dissection process may influence the results. Another factor that may influence the measurements is that the vessels shrink when removed from the body, so in order to have reliable data, they must be stretched to their natural length before measurement. The first measurements of wall thickness have been done under the microscope, which has the obvious problem of maintaining the vessel in normal length and pressure. Another problem of this method is the fact that the chemicals used for fixation alter significantly the dimensions of the vessel. Another method used in the past was based on Archimedes’ principle, which gives more accurate results. Nowadays, there is an option of non-invasive measurement of the wall thickness using ultrasound. This method is though limited to measurement of thickness of intima-media because the outer boundary of the adventitia can not be distinguished from the surrounding tissue, as mentioned before (Hoeks et al., 1997). One of the most referenced sources on vessel dimensions is the paper of Westerhof et al. (1969). The morphological data presented in his work has been used as a guidance for the design of the tubes used in this work in Chapter 5 (Table A.1). Information about the research conducted to define the mechanical behaviour of the blood vessels can be found in the data book of Abé et al. (1996), where a summarised collection of papers published in the area until 1996 is presented.

1.3

Computational Methods for fluid structure interaction

Typically in FSI, the fluid and solid components are modeled using different techniques to different levels of complexity, ranging from simple analytical solutions to 3-dimensional numerical schemes with advanced physical models. In addition to the range of techniques available for modelling the individual fluid and solid components, there is also the question of exchanging information, typically in the form of boundary conditions, at the interface. The options here are limited and can be

1.3. Computational Methods for fluid structure interaction

5

classified on the basis of the level of coupling between fluid and solid, as shown in Figure 1.3. • The most basic approach is non-iterative over all time (method 1). In literature it can also be found under the name uncoupled approach. The fluid and solid equations are solved separately for the whole time domain. The fluid is solved first to obtain velocity and pressure and the pressure at the interface is specified as a time-varying boundary condition for the solution of the solid equations. • The second method is iterative over all time (method 2). It is similar to the non-iterative approach except that the solution for the solid, i.e. displacements or velocities, is used as a time-varying boundary condition on the fluid. The process is repeated by solving for the fluid, passing the pressure boundary condition to the solid, solving for the solid etc. The process can be repeated until it converges to a point where the solutions are the same, to within a prescribed tolerance, from one simulation to the next (i.e. from fluid to solid and vice versa). • The third method can be named non-iterative over each time step (method 3a). In this case, boundary conditions are passed between fluid and solid at the end of individual time steps, but no iterations from fluid to solid solutions take place within the time step. The time steps need not be the same for both fluid and solid in which case, the exchange of boundary data can not occur after each time step. This case may be referred to as non-iterative over unequal time steps (method 3b). • The fourth method is iterative over each time step (method 4). In this approach, the fluid equations are solved for a single time step and the pressure solution becomes the boundary condition for the solid equations. The solid

equations are solved for the same time step and the solution obtained is returned as a boundary condition for the fluid which is again solved for the same time step. The process is repeated for that particular time step until the system of both fluid and solid equations has converged to within a prescribed tolerance. Only then the procedure advances into the next time step. In the case of non-iterative over all time (method 1), non-iterative over time step (method 3a), non-iterative over unequal time steps (method 3b), the fluid solution preceeds the solid one; so, data transfer is one-way only, i.e. from fluid to solid. When FSI is taken into account, fully coupled methods should be adopted. Both fluid and solid equations should be solved simultaneously and two-way data transfer should be performed, like in methods: iterative over all time (method 2) and iterative

6


Non-iterative over all time METHOD 1

F

Iterative over all time METHOD 2

F

1 2

p

S 135 . . .

Non-iterative over time step METHOD 3a

S

F S

Non-iterative over uniqual time steps METHOD 3b

F

Iterative over time step METHOD 4

F

S

U p 246 . . . 1 2

3 p

4 p

U

U

5 6 p

U

7 8 p

U

∆t ′ ∆t ′ ∆t ′ ∆t ′ 1 2 3 4 6 7 8 9 11 12 13 14 16 17 18 19 p p p p 5 U 10 U 15 U 20 U

135 . . . U p

U p

S

U p

U p

246 . . . Implicit single solution METHOD 5

F

1

2

3

4

∆t

∆t

∆t

∆t

S

Start Time

End Time

NOTE: The numbers in italics are counters of the computational time step. The straight dashed arrow represents the transfer of information of the denoted variable from one medium to the other. The curved dashed arrow represents the iterative procedure. Figure 1.3: Solution procedure of several FSI methods.

7


S S

S

F

F

monolithic method

single solution method

F partitioned method

Figure 1.4: FSI methods conventional terminology.

over time step (method 4). In order to get a realistic simulation, the exchange of information should be done at least once in each time step. In the discretisation process there are two issues involved, the treatment in time and space. Detailed discussion about the choice of discretisation methods used to solve the partial differential equations describing the problem is presented in Section 3.1. Looking at the time treatment of the fluid and solid, according to the conventional terminology found in the literature, current numerical methods can be grouped in two major categories: Partitioned method s and monolithic method s (Figure 1.4). The partitioned methods are based on partitioning the fluid and the solid solution, the fluid and structural equations are solved alternately and the enforcement of kinematic and dynamic interface conditions is asynchronous. It is typical for these methods that two separate software packages are used for modelling the solid and the fluid. The integration of two software codes is possible in principle, but the complexity and size of the software make this approach quite unattractive. Furthermore, the computational overhead to run such codes is quite exorbitant as information has to pass from one code to the other in each time step, adding to the total overhead (Belytschko et al., 1986). Data transfer usually requires an extra program that acts as an interface between the other two codes, thus sacrifices the modularity of the method. In the fluid structure interaction community, some researchers have focused in utilising a modular approach of the interface program for the exchange of information between two codes (Farhat et al., 1998, 2001; Raveh, 2000). Such an approach is often called modular approach. An overview of the benefits and disadvantages of using these methods can be found in Felippa et al. (2001). Partitioning leads inherently to loss of conservation of properties of the continua (fluid and structure). The energy increase in the system leads to instability which is the major drawback of this method. The monolithic methods use two separate sets of equations for fluid and solid and couple the fluid dynamics and structural dymamics implicitly and solve them

8


syncronously at the their common interface (Tallec and Mouro, 2001; H¨ ubner et al., 2004; Bloom, 1998; Alonso and Jameson, 1994; Rifai et al., 1998). The discretised equations are solved by subiteration until convergence within one time step. These methods can be unconditionaly stable and energy conservative (van Brummelen et al., 2003) when the modified Osher scheme is used for the fluid elements (van Brummelen and Koren, 2003). These methods are quite complex and computationaly expensive due to the subiteration. The single solution method proposed in this thesis is quite different from the partitioned and the monolithic methods. Figure 1.4 assists the reader with the conseptual and computational understanding of this novel approach and its differences from the conventional methods. The single solution solution methods treats both fluid and solid as a continium, thus the whole computational domain is a single entity in a single grid. Its behaviour is described by a single set of equations and is solved fully implicitly. There is no explicit exchange of information between the fluid and solid interface as it is inherently implicit. In this way, the computational expence of the subiterations of the monolithic approach is expected to be avoided. The difficulty that lies with this method is the conceptual understanding of using a single set of equations to describe both fluid and solid, the choice of this single set of equations and the choice of appropriate boundary conditions. The creation this single set of equations can be done in one of two ways: use the solid as the prime model and reformulate the equations of the fluid to match the ones for the solid or the other way around. In this thesis the later approach is chosen as it was considered to be more natural for flexible vessels. In a single solution method, the distinction between the state of the continium (fluid or solid) is associated with different coefficients in a single set of equations (Section 2.4). Early studies on wave propagation of incompressible fluids in elastic tubes, like rubber hose and blood vessels can be found in Young (1808) and for compressible fluids in Korteweg (1878). Even though the basic equations and the first theories date back to the 19th century, only in 1970s, with the introduction of computers, could the basic FSI equations be solved. Nowadays with the continuous advancement of computer power, special-purpose commercial, as well as ’in-house’, codes exist in the area of FSI. Reuderink et al. (1989) were amongst the first researchers to compute pulsatile flow in elastic arteries based on one dimensional wave propagation. They applied both linear and non-linear theory in blood vessels and compared them with experimental data. It was found that the linear model seemed to be more appropriate, since damping of the wave can be accurately described in the linear model. Nonetheless the non-linear terms in mass and momentum conservation equation may be significant.


9

Perktold and Rappitsch (1995) used an iterative approach for the same flow field examined by Reuderink et al. (1989). The boundary conditions of the flow problem, the inlet and the outlet pressure, were obtained from experimental data. They compared the results from models using rigid and distensible wall and they found that the distensible wall model gave more realistic results. Steinman and Ethier (1994) adopted a similar approach to Perktold and Rappitsch (1995). They used an analytical approach to study the effect of wall distensibility of a flow on end-to-side anastomosis. The outlet pressure was obtained by wave theory. Comparing their results with rigid-wall simulations, they found moderate changes in the wall shear stress. According to them, models that neglect the wall distensibility are less useful for predicting the behaviour of local pressure gradient fields as well as velocity profiles. Henry and Collins (1993a,b) were concerned with the prediction of wall movement in elastic tubes using an iterative approach as a coupling method. The inlet and the outlet pressures were fixed to a certain value. The model was validated against analytical solutions. Taylor et al. (1998) used a numerical method to model only the fluid of a pulsating flow in straight arteries. For boundary conditions of the fluid-solid interface, they used zero wall motion. The numerical method was validated against Womersley (1957) analytical solution. They were concerned that the methods available for FSI produced enormous amount of data and took a considerable amount of computational time. In their opinion, these should be reduced and better engineered codes should be adopted. Bathe and Kamm (1999) used the ”iterative over time step” coupling approach in modelling pulsatile flow in stenotic arteries. Boundary conditions at the inlet and outlet were obtained from experimental data. Their model was compared with other mathematical models and was validated against experimental data. They compared arteries with different degrees of stenoses. They found that the inviscid predictions were naturally lower than the computed pressure drops due the fact that the viscous losses are neglected. They found that the bulk of the pressure drop into the stenosis is due to the convective acceleration of the flow. König et al. (1999) modeled only the fluid using a moving boundary. Inlet and outlet pressures were fixed to reference values. Their model was validated against experimental data. They compared high and low viscosity models and obtained better results with the high viscosity model. Tang et al. (1999a,b) studied stenotic arteries by using both thick and thin wall models. They noticed that the stenotic severity and asymmetry in thick wall models changed not only the wall geometry, but also the stiffness of the tube wall and this affected the wall deformation. The maximum shear stress from the thick wall asymmetric stenotic tube was considerably lower than that from thin wall model

10


due to increased stiffness of asymmetric stenosis. They came to the conclusion that arteries have a complex structure and should not be treated as a homogenous material. Zhao et al. (1998) and Xu et al. (1999) used both thin and thick wall models and showed that the thick wall model provides more realistic results. The computational model is compared with data obtained from Magnetic Resonance Imaging (MRI) scanning of real patients. They state that it is difficult to make a direct comparison because of the large variations in anatomy of the patients. The model takes into account neither the compliant behaviour of the vessel wall nor the nonNewtonian behaviour of the blood, as the authors consider these to be of a secondary importance. Greenshields et al. (1999) presented a finite volume (FV) method for solving three dimensional equations for both fluids and solids. They used the iterative coupling using unequal time steps (method 3b). The exchange of information at the interface was done in an explicit manner which is the main limitation of their model. The method was capable of predicting in detail the start of a propagation pressure wave accounting for two dimensional and pipe resonance effects. It was potentially unstable for extremely flexible structures such as arterial walls. The assumptions used in the literature to model the fluid and the solid components have been identified and are summurised in a tabular form in Table 1.1.

1.4

Wave propagation in flexible vessels

The main focus of this section of the literature review is wave propagation in flexible vessels from a theoretical as well as experimental point of view. The literature review is separated in four parts: theoretical wave propagation in straight tubes; experimental wave propagation in straight tubes; theoretical wave propagation in tapered tubes; and experimental wave propagation in tapered tubes.

1.4.1

Theoretical models on straight tubes

Young (1808) was the first investigator interested in understanding the transient motion of fluids in pipes, elastic tubes, conical vessels and blood circulation. He proposed a formula for the velocity of pressure waves in an elastic tube with thin, homogenous and isotropic wall, filled with an incompressible fluid. Witzig (1914) has also investigated the wave propagation by modelling thinwalled flexible tube by solving two dimensional linearised Navier-Stokes equations. He was the first one to show the effects of viscosity of the fluid and present fluid velocity profiles. The work of Womersley (1957) is the most referenced one in the literature and has

11

1.4. Wave propagation in flexible vessels

× × √

× × × √ × A

FE

FE

× × √ √

FV Fluids × × × √ √

FE

FV

FE

√

× × × √ √

× × √ × × × A × √ × √ × A

Bathe and Kamm (1999)

NM NM NM NM NM NM NM

NM NM NM NM NM NM NM

× √

× × × × √ √

FE

FE

× × × × √

× × × × √

× × × × √

× × × × √

N

N

FE

FV

√ √ × √

Xu et al. (1999)

× × × × √ √

Steinman and Ethier (1994)

Solids × × × × √ √

Taylor et al. (1998)

× × × A

× √

Tang et al. (1999a,b)

√

König et al. (1999)

Non Newtonian Compressible Turbulent Transient 3 Dimensional Method

√ √ √

Henry and Collins (1993a,b)

Non linear Viscoelastic Compressible Large strain Thick wall 3 Dimensional Method

Perktold and Rappitsch (1995)

CHARACTERISTIC

Reuderink et al. (1989)

REFERENCE

√ NOTE: The symbol denotes that the characteristic in the left column has been taken into consideration, whereas the × means that is has not. Table 1.1: Modelling assumptions for the fluid and solid component as found in the literature.

12


been extensively compared against other theoretical models and further extended. Womersley (1957) solved the two-dimensional linearised Navier-Stokes equations for thin-walled isotropic infinitely long elastic tubes filled with viscous Newtonian fluid. He studied both unrestrained tubes and tubes constrained in the axial direction. An extensive overview of the work performed in this area can be found in McDonald (1968); Cox (1969); Pedley (1980); Tijsseling (1996); Wood (1999) . Atabek and Chang (1961) studied analytically the unsteady flow near the entry of a circular tube and showed that the entry length varied with the time through the cycle, as do the boundary layers which determinate it. Their findings were assessed computationally and extended by Ku et al. (1990). Klip et al. (1968) studied non-axisymmetric wave propagation in compressible fluids using a thick wall viscoelastic tube. Atabek and Lew (1966) extended the Womersley theory to initially stressed thin walled tubes in the axial and circumferential direction. They mention the existence of two waves: radial and longitudinal that can be found with the Womersley theory even though he did not mention this himself. Using the continuity and momentum equation the frequency equation can be obtained. The two roots of this equation will give the velocity of the propagation of the two waves. Mirsky (1968) used the Womersley models with longitudinal tethering and extended it to include tubes with orthotropic walls. Cox (1969) reviewed the work performed in this area until then by dividing it in three categories: thin-wall with no constraint; thin-wall with longitudinal constraint and thick walled tubes. He presented a table comparing the different theoretical models developed by that time. Atabek (1968) continued the work using the membrane theory of shells on orthotropic tubes. He found that the propagation properties of the slower waves are very slightly affected by the degree of anisotropy of the wall. For the faster waves the velocity of propagation decreases as the ratio of the longitudinal modulus of elasticity to circumferential modulus decreases. When tethering is used, the faster waves are completely attenuated, while the slower ones are hardly affected. His findings were in good agreement with the Womersley theory and the work of Mirsky (1968). He pointed out that in order for the theory to be complete and realistic for use in an arterial system there is a need to include taper, branching and the viscoelastic properties of the wall. The theories should be validated against well defined experimental data that were lacking at the time. Ling and Atabek (1972) introduced the nonlinear terms of the Navier-Stokes equations as well as the nonlinear behaviour and large deformations of the arterial wall. They also performed experiments. From the comparison of the experimental data with the linear and non-linear model, they concluded that their non-linear theory predicts the velocity profiles much better than the linear one. The wave of the wall shear predicted by the linear theory is very close to the one predicted by the non-linear theory. Their model was assessed computationally by Dutta et al.


13

(1992). Blood circulation has also been studied by comparing it with other physical models employing hydro-dynamic and electrical analogies. A review of such models can be found in Westerhof et al. (1969). Westerhof et al. (1969) modelled the entire arterial tree, discarding the viscous behaviour of the vessel, using an electrical analogue and compared it with clinical measurements. He concluded that reflections occur at all branch points and play a major role in determining the behaviour of the system. He showed how the nature of the input impedance and wave traveling pattern can be explained in terms of these reflections. He also published full data of human tree physiological parameters.

1.4.2

Experimental models on straight tubes

There is a vast literature involving in-vivo measurements in animals and humans, using open-chest measurements or using other techniques such as MRI scanning, but since they are beyond the scope of this project, they are not be mentioned here. The interest of the investigation is focused on experiments with flexible tubes. Rubber-like materials have been quite popular in modelling arteries, as the modulus is similar to that of human arteries. von Kries (1883) was interested in measuring the pressure pulse in human bodies. He performed experiments on a rubber hose in order to validate his theory. He used a 4 to 5 m long, thin-walled rubber hose of 5 mm diameter supplied with water through a constant-head reservoir. Klip (1962), realising that propagation velocity and damping of pressure waves in arterial systems can be used for diagnostic purposes, performed a series of experiments on tethered tubes of great length. He used a homogeneous, isotropic, viscoelastic tube of more than 60 m long. A piston was used to initiate a pressure wave. For about 4 m after the piston the tube was kept straight and the rest was wound up in a spiral. No reflections were present. The tube was filled with different water-glycerine solutions. Pressure was measured with a manometer and phase differences with an electric phasemeter. He considered both thick wall and thin wall tubes. He compared his data with other methods of calculation for the phase velocity and he found that they were in good agreement with Womersley’s results as well as with Moes-Korterweg predictions. Discrepancies were present for damping, however. Ling and Atabek (1972) were interested in simulating blood flow in dogs with an experimental rig. They used a composite straight structure of silicon rubber and corrugated nylon fibres. The tube diameter was appropriate for a medium sized dog and the thickness of the tube was 1 mm with ±0.1 mm variations. They used a glycerin-water mixture as a fluid. The pressure and pressure gradient were measured

14


using two pressure transducers at a 50 mm distance from each other. Velocity profiles were measured using a hot-film velocity probe. Wall shear stress was measured as well. The pressure-radius relation was obtained by photographing simultaneously the inflation of the vessel and the pressure signal, using an 8mm cine camera equipped with high power photography. Nerem et al. (1971) investigated the transition to turbulence in the aorta and related the results to equivalent steady flow ones in which the similarity parameters were the wave number and the Reynolds number. Liepsch and Moravec (1984) prepared a rubber replica of the femoral artery and performed experiments of pulsatile flow. Deters et al. (1986) made a silicon rubber cast from luminal mould of an aortic bifurcation. They measured phase fluid velocity by LDV at a single point close to the wall. The motion of the wall was obtained by integrating the velocity. The shear rate at the wall was estimated by dividing the fluid velocity by the distance from the velocity measurement point to the wall. Up to that time the Womersley theory had been tested only for tethered tubes. Gerrard (1985) was interested to determine the behaviour of infinitely long tubes, where the longitudinal motion was present. In his set up, he used isotropic latex rubber tubes, with small viscoelasticity. He glued together two tubes of 15m length, inner diameter of 6.2mm and thickness of 1.8mm. The tube was filled with water. A wave was initiated by a piston at one end and the other end was closed. The free motion of the tube was obtained by suspending it from the ceiling with cotton sewing threads 100mm apart. This tube behaved like a semi-infinite one over almost all its length. No reflections were present. From the comparison of the experimental data with Womersley theory for an infinite tube with no constraint it was concluded that the experimental data were in good agreement beyond the entrance length. There were some discrepancies though, near the end of the tube. That was an indication that there may be an end effect at the closed end far from the piston, which considerably reduces the amplitude calculated from the infinite-tube theory. He also performed experiments on tethered tubes of 30m long and found that his measurements were in good agreement with those of Klip (1962). van Steenhoven and van Dongen (1986) were interested, apart from wave propagation phenomena, in aortic valve closure. They performed experiments on water filled latex tube 0.6m long with 18mm inner-diameter and thickness 0.2mm. Transmural pressure was applied at one end of the tube. Pressure was measured using two catheter-tip manometers. Wall deflection was measured using a photonic sensor and the flow volume was measured electromagnetically. The fluid was suddenly stopped locally starting from steady flow. The measurements describing the wall behaviour were in good agreement with those of Gerrard (1985). From their measurements they obtained the viscoelastic properties of the tube. The compared their experimental data with the one dimensional non-linear theory for the wall shear


15

stress, that was solved numerically using the method of characteristics. They wall was treated as viscoelastic and wave reflections were also taken into account. From the experiments they concluded that the wall viscoelasticity is a dominant factor in the gradual flattening of the waveform. They also mention that the local change in compliance generates expected wave reflections and has strong influence on the rise-time of the wave front. The most important consequence is that the pressure jump of the wavefront decays while propagating upstream.

Horsten et al. (1989) used for their experiment the same experimental setup and the same tube as van Steenhoven and van Dongen (1986) but 0.9m long to simulate wave propagation. They compared their experimental data with one dimensional linear theory with focus on the viscous phenomena of the fluid and tube wall and found them in good agreement for small pulsed shape waves. They compared and assessed different linear models on their performance in describing the wall behaviour and it was found that there were no major deviations amongst them. They concluded that the one dimensional Womersley linear theory, where the fluid is treated as incompressible, describes fairly well the propagation phenomena. The wave velocity, though, was underestimated and the damping was overestimated. The discrepancies between experimental and analytical data are partially explained by the non-linearities. The rigid support of the tube could be another explanation of the discrepancies.

Reuderink et al. (1989) also focused on assessing the one dimensional linear and non-linear theory describing the pulse wave propagation in a uniform viscoelastic tube. A 1m long latex rubber tube filled with a salt solution was used. A pneumatically driven piston was used for the pulse initiation. A catheter tip manometer was used for measuring the pressure at different positions along the tube. They attempted to measure pulsatile diameter changes using an ultrasonic transit-time technique but they stated that the influence on the wall motion was present, even though minimised. The experimental data showed that the pressure vs cross-sectional area relation was nonlinear for the pressure changes. By comparison of the experimental data with the linear and non-linear models they came to the conclusion that in spite of the nonlinearity of the system, the linear viscoelastic Womersley model described the pulse wave propagation satisfactorily. They explained that the discrepancies between the experimental findings and the prediction of the non-linear model are due to the fact that frictional losses due to the wall viscoelasticity are neglected and due to fluid viscosity are underestimated. Therefore, non-linear models predict small damping and formation of shock waves, which were not observed experimentally.

16


1.4.3

Theoretical models on tapered tubes

The need to capture the nature of the arterial tree and define its physical properties in order to use them as the correct parameters in modelling, have lead to investigation of the geometrical tapering of the tubes. Young (1808), was one of the first to mention possible effects in the blood circulation. Taylor (1965) was concerned with wave propagation in a non-uniform transition line. Wemple and Mockros (1972) solved a one dimensional non-linear mathematical model by the the method of characteristics. Their non-linear model included geometric and elastic taper of the flexible tube. They compared their model with data measured in humans. The elastic taper theoretically affects the wave transmision and reflection in the same way as to that of geometric taper. The degree of the elastic taper is small compared to that of the geometric taper, therefore they concluded that the elimination of the elastic taper does not have significant effects on the model. The geometric tapering on the other hand is quite important for the presence of reflection waves. They concluded that the system behaves in a linear way for the lower frequencies, while for the higher frequencies the non-linearities are important. The linear theory is unable to deal with tapered tube if the pressure pulse is high. Belardinelli and Cavalcanti (1992) used a two dimensional non-linear model. They point out that the natural tapering of the arteries should be taken into account as it has been indicated from in-vivo measurements. Their model encompasses the motion of a pulse-driven viscous fluid in a geometrically tapered flexible tube. They make the assumption of uniform pressure in a cross section. Their results show that the tapering does not influence the wave velocity but it influences the waves’ attenuation rate. They used infinite extremity impedances to maximally enhance the reflections so that the overall attenuation is only due to arterial properties and in particular the natural tapering. The natural tapering causes a continuous increase in the pulse amplitude as it moves from one side of the tube to the other. In a 0.6m long tube with taper angle of 0.1 the pulse amplitude at the end of the tube is more than twice the input pulse. The reflected pulse is greatly damped and its shape is quite different from that of the direct pulse. Einav et al. (1988) used an LCR (inductance-capacitance-resistance circuit) electrical analogue to study wave propagation in exponentially tapered tubes with main interest in reflections at bifurcations. Their model was compared with the one of Westerhof et al. (1969). They concluded that the input impedance is low for high frequencies. Therefore, blocked branches in the vicinity of the heart do not significantly contribute to the input impedance. More distal bifurcation, such as the ileac bifurcation, can affect the input impedance at low frequencies. From their reflection condition they conclude that in order to maintain continuity in a junction, the characteristic impedance and peripheral impedance are doubled and the cross-section of


17

the branches is 15% larger than the main branch. Chang et al. (1994) used the electrical analogue in which they included the non-uniform properties of the tube, as well as the geometric and elastic tapering. They compared their model with in-vivo measurements in dogs. They found good agreement between their impedance parameters derived by their non-uniform model and the ones measured in the animals. Comments about their work can be found in Burattini et al. (1996). Fogliardi et al. (1997) used an exponentially tapered electrical analogue to model descending aortic circulation. In their model they used five parameters to characterise the input impedance: the characteristic impedance, the compliance of the tube, the tube length, the tapering, the time constant of the load and the peripheral resistance. They performed open chest in-vivo measurements in dogs to obtain pressure and flow measurements. From the comparison of their model with the in-vivo data they found that the tapered tube models showed a slightly closer matching with the experimental flow and the reproduction of the input impedance.

1.4.4

Experimental models on tapered tubes

After a thorough survey of experiments performed with tapered tubes, the author found a vast amount of literature of in-vivo measurements in humans and animals but only two papers studying wave propagation in geometrically tapered elastic tubes: von Kries (1892) and Reuderink et al. (1988). von Kries (1892) was interested in understanding blood pressure waves. He was the first one to perform experiments on a rubber tapered tube. He had two straight tubes of 22 mm and 5.5 mm diameter connected to each other by a 140 mm long conical part. His interest was to use tapering to eliminate the wave reflections of a pressure wave form when transmitted from a tapered tube to a straight one. Reuderink et al. (1988) used a uniform latex tube 0.5m long with 12.73mm outer diameter and thickness 0.14mm with a variation of ±0.01mm for the straight tube; the tapered one varied from 15.88 to 9.45mm outer diameter (46 degrees taper) with horizontal to vertical slope of 0.008 and thickness 0.13mm ±0.01mm. The tubes were manufactured by dumping the mould in latex rubber. The working fluids used were salt solutions of different concentration and glycerine solution. The salt solutions were used in order to be able to measure electromagnetically the flow. A pneumatically driven piston was used for the pulse initiation. Impulse or sine waves were used for the excitation. The sine wave did not produce a steady flow component. A catheter tip manometer was used for measuring the pressure at different positions along the tube. They compared the real part of the true propagation coefficient with the apparent damping and the damping coefficient calculated from Womersley theory. They

18


also compared the true phase velocity with the measured apparent phase velocity, foot-to-foot velocity, and calculations of phase velocity parameters using the Womersley’s theory and the Moens-Korteweg equation. From their comparison they conclude that the three point method used to obtain the propagation coefficient is in agreement with all other estimate for a uniform tube. For a tapered tube the three point method causes an error estimation of the propagation coefficient. They state that in their experiments tapering only cannot take account for the differences between in-vivo measurements of the propagation coefficient using the three-point method and calculations based on the Womersley’s theory since in their results taper caused a discrepancy only at some frequencies, and at these frequencies the damping was largely underestimated instead of overestimated.

1.4.5

Concluding summary

The complexity of the physical phenomena and the simultaneous interaction of various effects make a complete analysis of blood flow almost impossible. Certain assumptions are necessary but they have to be verified. Validation of the theoretical models and assumptions can be done through comparison with in-vivo and in vitro measurements. In-vivo measurements have obvious limitations like: handling of the subject, conditions of measurements, law restrictions etc. On the experimental side, one is usually limited to measure only a small fraction of the quantities of interest and even then they can be sampled only at a few times and special locations, with a limited degree of accuracy. Therefore it is important that well defined experiments are carried out. The most dominant theoretical model with numerous extentions in wave propagation in flexible vessels is that of Womersley. The theory of Womersley for infinitely long tubes, with or without tethering, has been experimentally validated by the work of Klip (1962)(longitudinal constraint) and Gerrard (1985)(no longitudinal constraint). They both verified its validity beyond the entrance length. For tubes with finite length the theory of Womersley and its extentions has been validated thoroughly against other theoretical models and experimental data. Nevertheless, in wave propagation, there is a number of quantitative questions that have not been answered satisfactorily yet. There is a lack of quantitative agreement between measurements and theoretical models. For the prediction of pressure wave velocity, the linear theory gives good agreement with experimental and in-vivo measurements. On the other hand, accurate predictions of the attenuation rate with distance along a given named vessel of the arterial pulse is doubtful. The measured pulse consists of forward-going components and reflected components, due to the closed end in finite tubes. Incorrect modelling of reflections leads to

19


× × × √ 1 A

2 A

Ling and Atabek (1972)

× × √

× × √

E

× × A&E

× × 1 A

× 1 A&E

× × × √

× × × √

× × × √

× × × √

E

A&E

1 A

1 A&E

Klip (1962)

Gerrard (1985) E

Atabek (1968)Atabek

Non Newtonian Compresible Turbulent Transient 3 Dimentional Method

× × 1 A

Solids × × × × √ × × × × × 2 A E Fluids √ × × × × × √ √

Westerhof et al. (1969)

Non linear Viscoelastic Compresible Large strain Thik wall Dimentions Method

√ √ √

Steinman and Ethier (1994)

CHARACTERISTIC


REFERENCE

× × × × √

√ √ × √

√ NOTE: The symbol denotes that the characteristic in the left column has been taken into consideration, × means that is has not and − means that it is not mentioned. Table 1.2: Assumptions for the fluid-solid components for straight tubes as found in the literature.

20


√ √ √ × × 1 A × × √ √ 2 A

√


Wemple and Mockros (1972)

Belardinelli and Cavalcanti (1992)

CHARACTERISTIC Solids Non linear × Viscoelastic × √ Compresible Large strain × Thik wall × Dimentions 1 Method A Fluids Non Newtonian × Compresible × Turbulent × √ Transient 3 Dimentional 1 Method A

van Steenhoven and van Dongen (1986)

Horsten et al. (1989)

REFERENCE

√ √ √

√

× √ 1 A

× × E

× × √

× × × √

× × × √

2 A

1 A

E

× × × × 2 A √

× √

√ NOTE: The symbol denotes that the characteristic in the left column has been taken into consideration, × means that it has not and − means that it is not mentioned. Table 1.3: Fluid-solid assumptions for tapered tubes as found in the literature.

1.5. Objectives of this study

21

incorrect calculation of the reflection coefficient. The incorrect characterisation of the tethering of the tube is also one of the reasons that these discrepancies between theoretical, experimental measurements and in-vivo measurements may occur. Another reason for these discrepancies is the presence of physical non-linearities, which are modelled incorrectly. The arterial system is geometrically and thermodynamically non-uniform (Pedley, 1980; McDonald, 1968). It has continuous variations in cross-sectional area and distensibility (compliance), as well as repeated branching. Non-linearities are introduced in the system due to the dependence of pressure on the above factors. There is no question that geometric and elastic tapering are significant aspects in the arterial system. Due to the tapering, the local compliance of blood vessels decreases with distance from the heart, whereas the characteristic impedance increases. Wemple and Mockros (1972) state that in spite of the numerous non-linearities in the system, it behaves in somewhat linear fashion for lower frequency components (at 80 beats per second). At high shear rates, however, non-linearities are important. A quantitative agreement of in-vivo measurements, experiments and analytical models has to be achieved in order to check the importance of these physical nonlinearities of the arterial system and before one decides whether to neglect them or not. This can only be achieved by producing reliable data through well defined experiments. There is a number of theoretical models taking into account the non-linearities of the arteries. The validation of these models is limited to in-vivo measurements, the accuracy and the conditions which are quite difficult to asses. It is explicitly stated in the literature that there is insufficient data for non-linear tubes. Only the work of (von Kries, 1892; Reuderink et al., 1988) was found by the author to be concerned with experiments taking into account geometric non-linearities simultaneously with flexibility. The elastic taper affects wave transition and reflection in a manner that is theoretically similar to that of geometric taper. The degree of elastic taper in the system is small relatively compared to the geometric taper. Thus, it is important to investigate geometric taper in wave propagation. Therefore, one of the objectives of this work is to obtain reliable experimental data on geometrically tapered tubes that would help the further development and validation of theoretical and computational models.

1.5

Objectives of this study

The objectives of this study are both computational and experimental and aim at filling existing gaps in the literature. The use of two separate solution methods for solving FSI problems leads to casespecific codes and to problems regarding the efficiency of the coupling of the two

22


methods, as already explained. There is a need for general purpose codes that will be better engineered, more flexible, and be able to solve the equations for both the fluid and the solid components simultaneously. In order to obtain a robust FSI modelling method, suitable for general applications, the use of a single solution environment for Fluid and Solid needs to be adopted (method 5) (see Figures 1.3 and 1.4). In the context presented above, the first objective of the thesis is to contribute towards the development of such a unified approach by reformulating the equations for solids as to contain the same unknown variables as the ones for fluids, namely velocity and pressure. In this way the solution at the interface can be obtained in an implicit manner, thus the fluid-structure domain can be considered as a single entity described by a single set of equations. It is expected that the new formulation for solids will be suitable for modelling a variety of FSI applications such as blood flow in deforming arteries, container impact, pipeline surge etc. The second objective is to develop and test a stable and robust numerical method for the discretisation and solution of the reformulated equations for solids. The method should be compatible with the one used for the solution of the fluids equations. The third objective is to test the accuracy of the developed method for dynamic structural problems. Finally, the fourth objective is to collect a detailed experimental data set that can be used for the next step of the validation of the unified approach in fluid-structure interaction problems. The experimental work is also motivated by the need to understand further arterial blood flow. Although the general principles governing the arterial hemodynamics are well known (McDonald, 1968; Pedley, 1980), there are some questions that have not yet been satisfactory answered. Amongst these is the assessment of non-linearities arising from wall thickness variation and geometric tapering that are naturally present in the arterial tree morphology. The main reason for this are the apparent limitations of in-vivo measurements in combination with the lack of well defined laboratory experiments in the literature, as explained in the previous Section 1.4. Thus, there is a need for such experiments so as to help the validation and further development of theoretical and numerical models. Thus, the experimental part of this work aims to cover this gap and to assess the linear theory which is widely used in wave propagation phenomena. The main interest is to investigate the effect of geometric tapering and wall thickness variation of flexible vessels. The experimental data can be used for the assessment of computational methods to check whether they can cope with the anatomical non-linearities.

1.6. Outline of the thesis

1.6

23

Outline of the thesis

In Chapter 2 the mathematical framework for a single solution method for fluid structure interaction problems is developed and presented. In Chapter 3, general information about the discretisation method used for the solution of the mathematical model is presented. In this chapter a stability analysis of the single solution method is also presented in order to check theoretically the amount of dissipation that the method introduces. In Chapter 4, the validation case used for the mathematical model is described and the results obtained from the code developed are presented and discussed. In Chapter 5, the experimental methods of the wave propagation experiments are described and the measurements obtained are presented. The tube manufacturing, the experimental set up and the protocols are also explained. In Chapter 6 a comparison between the experimental measurements for straight tubes with the linear methods is presented. In Chapter 7, the conclusions of the completed work according to the project objectives are outlined. Suggestions for extending the present work are also highlighted.

24


Chapter 2 Mathematical formulation of a unified framework for fluids and solids 2.1

Introduction

The equations describing the behaviour of a Hookean solid and a Newtonian fluid (Section 2.2) are typically solved for displacement and for velocity and pressure respectively. This is due to the fact that the stress tensor in solids is defined in terms of displacement while, in fluids in terms of velocity and pressure. In order to obtain a single solution method, both fluid and solid equations should be solved for the same variables. The convective nature of displacement and the nature of pressure in fluids leads to the decision of altering the solid formulation so as to contain as unknown variables velocity and pressure. In the mathematical model described here, the constitutive equations for solids are reformulated by introducing first velocity instead of displacement and second the hydrostatic pressure, in order to express the stress tensor (Section 2.3). In the following subsections, the governing equations are presented as well as the developed mathematical model. For the basic background of tensor mathematics one can refer to Adams (2003); Aris (1962) and for continuum mechanics to Chadwick (1976).

2.2

Governing Equations

Solids and fluids are both continua, whose behaviour can be described by the same continuity and momentum equations. There are no simplifying assumptions in the momentum and continuity equations for fluids and solids and both are treated as compressible. Only the constitutive laws are different. Therefore, thes will be pre25

26

Chapter 2. Mathematical formulation of a unified framework for fluids and solids

sented separately. Details can be found in most continuum mechanics test books, such as Malvern (1969) and Segel (1977). The constitutive law for solids presented here assumes a linear elastic (or Hookean) solid and provides the stress-strain relationship. The constitutive law for the fluid assumes a linear viscous (or Newtonian) fluid and provides a relation between stress, thermodynamic pressure p and rate of deformation tensor (ε˙ ). Continuity equation or mass conservation ∂ρ + ∇ • (ρU) = 0 ∂t

(2.1)

Momentum equation (neglecting body forces) ∂ρU + ∇ • (ρUU) = ∇ • σ ∂t

(2.2)

Constitutive equations for linear elastic or Hookean solid A linearly elastic solid is considered and so there is a one to one relationship between the state of stress and the rate of strain. If the material is elastically isotropic, i.e the elastic constants are the same for all possible choices of Cartesian coordinates, then the generalised Hooke’s law is obtained: σ = 2µε + λtr (ε)I

(2.3)

where µ and λ are Lame’s coefficients, which are related to Young’s modulus of elasticity and Poison’s ratio ν, by the following equations: µ=

ϒ 2(1 + ν)

(2.4)

and λ=

νϒ (1+ν)(1−ν) νϒ (1+ν)(1−2ν)

for plain stress for plain strain and 3D

(2.5)

Constitutive equations for linear viscous or Newtonian fluid For a viscous Newtonian fluid the stress tensor can be determined by the pressure and the deformation rate tensor with the following linear relationship: σ = 2ηε˙ + ξtr (ε˙ ) − pI

(2.6)

where the viscosity coefficients ξ and η (dynamic viscosity) are related to the bulk viscosity k = ξ + 2/3η. The deformation rate tensor is the symmetric part of

27

2.3. Mathematical Model

the velocity gradient tensor i.e. ε˙ ≡ sym(∇U ) ≡ 1/2 ∇U + (∇U)T . Thus, Equation 2.6 reads: σ = η∇U + η(∇U)T + ξtr (∇U) − pI

(2.7)

For flow analysis we usually make the Stokes condition assumption k = 0, thus Equation 2.7 reads: 2 σ = η∇U + η(∇U)T − ηtr (∇U) − pI 3

(2.8)

σ = 2ηdev (sym(∇U)) − pI

(2.9)

or

Barotropic relationship When interested in the wave propagation in a material, it is important to define the equation of state for a barotropic fluid and a compressible solid, showing the relationship between the density and the thermodynamic pressure in the fluid or the solid. This relationship can be derived by the definition of the bulk modulus K in the material: K=ρ

∂p ∂ρ ρ ⇔ = ∂ρ ∂p K

(2.10)

For small variations of density about a reference density ρ0 , we can assume that ρ ≃ ρ0 , so Equation 2.10, can be linearised giving the linearised form of the barotropic relationship: p − p0 ρ ≈ ρo 1 + K

(2.11)

where p0 is the reference pressure and ρ0 is the initial density for which ρ(p0 ) = ρ0 .

2.3

Mathematical Model

Generally speaking, for fluids there is an interest in the velocity of the flow and the pressure of the fluid, whereas in structures there is an interest in the resulting stress and deformation that the structure undergoes. For the new stress analysis formulation, velocity takes the place of displacement which is used in the standard formulation, and finally a velocity and pressure formulation is obtained. The standard stress analysis is first examined, followed by the new formulation.

28


U Un U0 ∆t 2

Do

t0

t − ∆t

t

[Un + Uo ]

t

Figure 2.1: The velocity integral from [t0,t + ∆t]

2.3.1

Standard stress analysis for linear elastic (or Hookean) solid

For small strain, the strain tensor is the symmetric part of the displacement gradient: ε = symm(∇D) =

1 ∇D + (∇D)T 2

(2.12)

and thus the stress tensor is written as:

σ = µ∇D + µ(∇D)T + λtr (∇D)I

(2.13)

Using the displacement formulation of the stress tensor Equation 2.13, and since U=

∂D ∂t ,

Equation 2.2 becomes: ∂ρ

h

∂D ∂t

∂t

i

+ ∇ • (ρUU) = ∇ • µ∇D + µ(∇D)T + λtr (∇D)I

(2.14)

It should be mentioned that if the deformations concerned in solids are sufficiently small, the convection term ∇ • (ρUU) on the left hand side of the momentum equation can be ignored. For the generality of the derivation of the unified solution method for fluids and solids, the convection term is not omitted here but in the computations for the validation of the model it is discarded to decrease computational time.

2.3.2

Velocity based formulation for linear elastic (or Hookean) solid

The velocity based formulation for solids can be obtained by expressing the displacement as a function of the velocity and substitute it in the governing equations. The displacement is the area under the curve of the velocity against time as seen

29

2.3. Mathematical Model

in Figure 2.1. The time domain is split to a finite number of time steps ∆t with starting time t0 . At any given time t, the displacement can be evaluated from the integral of the velocity from t0 to t:

Z

Z

t

t0

Z

t−∆t

Udt =

D(t) =

t

Udt + t0

Udt

(2.15)

t−∆t

The integral form in Equation 2.15 can be discretised in various ways. When the trapezoidal rule is chosen, the new displacement at t is approximated as: Dn = Do +

∆t n [U + Uo ] 2

(2.16)

where Uo is the value of velocity calculated from previous time step (old value) and Un is the value of velocity calculated at present time step (new value). From now on, the superscript n will not be used when there is reference to the new values evaluated at t, i.e. U ≡ Un . Then the stress tensor can be written as:

∆t µ∇U + µ(∇U)T + λtr (∇U)I 2

(2.17)

∆t µ∇Uo + µ(∇Uo )T + λtr (∇Uo )I 2

(2.18)

σ = Σ+ +

where sigma plus (Σ+ ) is given by Equation 2.18 and Σ is the accumulated stress from previous time steps. Σ+ = Σ +

Thus, the momentum Equation 2.2 over the time interval [t0 , t] , becomes: ∆t ∂ρU ∇ • [µ∇U] + ∇ • µ(∇U)T + λ∇ • [tr (∇U)I] + ∇ • Σ+ + ∇ • (ρUU) = ∂t 2

(2.19)

where ∇ • Σ+ represents the divergence of the accumulated stress tensor up to time t − ∆t (old values). It should be mentioned here that this is not the final equation that is discretised. As explained in Section 3.3, Equation 2.19 is written in a slightly different form, which is more suitable from a numerical point of view, because it increases the stability of the algorithm.

2.3.3

Velocity and Pressure based formulation for linear elastic (or Hookean) solid

In order to obtain a velocity and pressure based formulation, so as to have the same variables used for fluids, the stress tensor has to be split into its deviatoric and hydrostatic parts. The deviatoric part is responsible for changes in shape while the hydrostatic part is responsible for changes in volume.

30


1 σ = dev σ + tr (σ)I = dev σ − pI 3

(2.20)

After some tensor manipulation, the stress tensor can be written as: ∆t 2 T σ = dev Σ + µ∇U + µ(∇U) − µtr(∇U)I − pI 2 3 +

(2.21)

where dev Σ+ consists of the accumulated deviatoric part of the stress tensor Σ plus the terms including the old values of velocity and is given by the Equation 2.22: ∆t 2 o o T o dev Σ = dev Σ + µ∇U + µ(∇U ) − µtr(∇U )I 2 3 +

(2.22)

In the same way, the momentum Equation 2.2 over the time interval [t0, t] using the new velocity and pressure formulation is given by the following Equation 2.23. 2 ∂ρU ∆t T ∇ • [µ∇U] + ∇ • µ(∇U) − µ∇ • [tr (∇U)I] + ∇ • dev Σ+ − ∇p + ∇ • (ρUU) = ∂t 2 3 (2.23) where now on the right hand side, the −∇p term, appears as in the momentum equation for fluids. The continuity equation for solid (Equation 2.1) already contains velocities, so no modification is required. For the solution of the momentum equation, an equation for pressure is needed. There are two ways of solving the momentum equation: (a) by solving for velocity and calculating the pressure using the value obtained by the velocity (pressure explicit), or (b) by solving for both velocity and pressure implicitly (pressure implicit). In the latter case velocity and pressure are solved fully coupled. This is equivalent to solving the Navier-Stokes equations for fluids and will lead to a unified formulation for solving fluid-structure interaction problems. Pressure explicit In order to derive the equation used for the pressure calculation, let us consider the definition of pressure shown in equation 1 p = − tr (σ) 3

(2.24)

Substituting Equations 2.13 and 2.16 in 2.24, the equation used for evaluation of pressure can is obtained: p = p+ − K

∆t tr (∇U) 2

where K is the solid bulk modulus and is given by:

(2.25)

2.4. Comparison of the new velocity-pressure formulation for solids with the fluids formulation

K=

ϒ 3(1 − 2ν)

31

(2.26)

The pressure is also accumulated every time step. The accumulated pressure p+ , contains the old values of pressure up to time t and is given by Equation 2.27.

Uo p = −Ktr ∇(D + ∆t) 2 +

o

(2.27)

The above expression for pressure is an explicit expression.

Pressure implicit An implicit expression can be obtained by using the continuity equation. The continuity equation, does not have a dominant variable in incompressible flows; it acts as a kinematic constraint on the velocity field. Therefore, a pressure field can be constructed so as to guarantee the satisfaction of the continuity equation. Based on this assumption the pressure equation can be derived, both for incompressible as well as for compressible materials. If we substitute the Barotropic relationship described by Equation 2.11 in the continuity equation (Equation 2.1), the following equation is obtained for the pressure: ∂ψp + ∇ • [(ρ0 − ψp0 )U] + ∇ • [ψpU] = 0 ∂t where p0 is the reference pressure for which ρ(p0 ) = ρ0 and ψ =

(2.28) ρ0 K.

The above

expression is derived in the same way that it is derived in solving the Navier-Stokes equations for fluid dynamic problems. In the standard dispacement formulation, pressure can also be evaluated implicitly or explicitly. But in the case of incompressible solids, the role of pressure is similar to that in incompressible fluids, i.e. to enforce a divergence free velocity and displacement field (Bathe, 1996; Hughes, 1987).

2.4

Comparison of the new velocity-pressure formulation for solids with the fluids formulation

In the previous section, we have obtained a new formulation for linear elastic (or Hookean) solids and expressed the momentum equation with velocity and pressure as primitive variables. At this paragraph the new formulation is compared with the momentum equation for a linear viscous fluid which after substituting Equation 2.8 or 2.9 in the Equation 2.1, can be written as:

32


or

2 ∂ρU + ∇ • (ρUU) = ∇ • [η∇U] + ∇ • η(∇U)T − ηtr (∇U)I − ∇p ∂t 3 ∂ρU + ∇ • (ρUU) = 2ηdev(sym(∇U)) − ∇p ∂t

If we set α = reads:

∆t 2µ

(2.29)

(2.30)

then the momentum Equation 2.23 for the linear elastic solid

2 ∂ρU + ∇ • (ρUU) = ∇ • [α∇U] + ∇ • α(∇U)T − α∇ • [tr(∇U)I] + ∇ • dev Σ+ − ∇p ∂t 3 (2.31) or ∂ρU + ∇ • (ρUU) = 2αdev(sym(∇U)) + ∇ • dev Σ+ − ∇p (2.32) ∂t It can be seen from Equations 2.29 and 2.31 (or 2.30 and 2.31) that a unified mathematical expression of the same form for both fluids and solids has been obtained. The difference between the two lies in the coefficient a used in the solids (instead of the η in the fluids) and the additional term ∇ • dev Σ+ , which represents the accumulated history of the diviatoric component plus an explicit part associated with the old values of the velocity (Equation 2.22).

2.5

Boundary conditions

In order to derive a unique solution to any system of PDE’s, a set of conditions needs to be specified at the boundary of the solution domain. The boundary condition type used for the displacement and the velocity can be either fixed value or fixed gradient. The appropriate equation can be obtained by prescribing a force balance at the boundary which is described by the following Equation: n • σ = t − npext

(2.33)

where pext is the external pressure applied at the boundary, and t is the external traction. The appropriate expression for the stress tensor is thereafter substituted in the force balance equation i.e. Equation 2.13 for the displacement based formulation, Equation 2.17 for the velocity based formulation and Equation 2.21 for the velocitypressure based formulation. The final forms of the boundary conditions for the displacement, the velocity and the velocity-pressure formulation are presented in Section 3.6 as stability issues are involved in the derivation of these expressions.

33

2.6. Closure

The chosen boundary conditions for pressure are either fixed value or fixed gradient. There are three relationships that can give a boundary condition for pressure: using the definition of pressure, applying the force balance relationship at the boundary or using the momentum equation. pressure definition A possible boundary condition for pressure can be obtained using the definition of pressure: ∆t tr (∇U + ∇U0 ) (2.34) 2 where p0 are the old values obtained at the end of the previous time step. p = p0 − K

applying force balance

The boundary condition for pressure in this case is

derived in the same manner as for the velocity.

p = −t • n + pext + n • dev Σ

+•

n − αn •

2 ∇U + (∇U) − tr (∇U)I • n 3 T

(2.35)

applying momentum The momentum equation can be projected to the unit normal vector at the boundary and solved for the pressure gradient: 2 ∂ρU T • • • • • −n ∇ (ρUU) + ∇ [α∇U] + ∇ α(∇U) − α∇ [tr (∇U)I] +n • ∇ • dev Σ+ n∇p = − ∂t 3 (2.36) Using the momentum equation to derive the boundary condition for pressure is the most appropriate choice, as it leads to a well posed problem for solving the fully implicit velocity-pressure system of equations for solids. The reasons behind this choice are explained in detail in Chapter 3.

2.6

Closure

In classical solid mechanics, a linear solid is typically solved for the displacement components while in fluid dynamics, the fluids are solved for the velocity components and pressure. As we are interested in creating a single mathematical framework for solving fluids and solids, both of them are looked at as a continuum. In this case, they are both described by the same momentum and the continuity equations. The only difference lies in the constitutive equations of the stress tensor. In this chapter, we have reformulated the equation of state for a linear elastic (or Hookean) solid to have as primitive variables velocity and pressure as in fluids. Thereafter, a common expression for the momentum equation can be obtained for fluids and solids where, for both of them, the primitive variables are velocity and pressure. In

34


these unified expressions the fluid and the solid state can be distinguished by the different coefficients that appear. Thus, in this manner the fluid-solid interface in the solution domain is internal and no extra attention needs to be drawn. Appropriate boundary conditions need to be found only for the solid as it will have external boundaries in the solution domain. Possible boundary condition expressions have been presented for velocity and pressure. It has been mentioned that the most appropriate one for solids can be derived by applying force balance at the boundary. For pressure, the most appropriate condition can be derived by projecting the momentum equation at the unit vector normal to the boundary. The reasons that lead to this choice are presented in the following chapter. The mathematical representation presented in this chapter is standard for the fluids and is typicaly used in CFD to solve the Navier-Stokes equations and so there is no need for investigation or validation. On the other hand, as it has never been used before for solids, it needs to be investigated and validated. This investigation is presented in Chapter 4. If this mathematical representation proves to be able to solve classic solid mechanics problems, then the unified solution method will have been shown to work and can be used to solve FSI problems.

Chapter 3 Numerical solution method 3.1

Introduction

Different techniques can be applied for the discretisation of the governing mathematical equations presented in the previous Chapter. There are three main discretisation methods: Finite Element Method (FE), Finite Differencing Method (FD) and Finite Volume Method (FV). FE method was born by the work of Turner et al. (1956) and was developed mainly to solve problems in the area of structural analysis. On the other hand the FV method was developed from the FD method and is more recent. Initially is was designed to solve problems in the area of fluid flow and heat transfer. Over the last twenty years there have been intensive attempts to use the FE method in the area of Computational Fluid Dynamics (CFD) (Zienkiewcz and Taylor, 1989; Girault and Raviart, 1986; Bathe, 1996; Gresho and Sani, 2000), among others. The use of FV methods in the field of Computational Solid Mechanics (CSM) has been developing mainly for the last ten years. In the area of structural analysis FV method appears to have been introduced by Wilkins (1964). The governing equations of fluid flow and solid body stress analysis are of similar form, indicating that the FV method is also applicable in CSM as demonstrated by Demirdzić and Martinović (1993). So far the FV method has successfully been applied to elastic, elastoplastic and viscoplastic problems, as well as geometrically non-linear stress analysis (Demirdzić and Martinović, 1993; Demirdzić and Muzaferija, 1994, 1995). This shows that the barriers for the use of these methods are not clear. Over the past ten years these two methods are getting closer to each other and according to Zienkiewcz and Taylor (1991) the FV method appears to be a particular case of FE with non-Galerkin weight. Nowadays there is an emerging need to simulate multi-physics processes such as FSI that are governed by a number of interactive physical phenomena. In modelling an FSI application, it seems that the FE method is more popular when both Fluid and Solid are modeled, whereas the FV method is used when only the fluid is modeled. Another quite common alternative is to couple two different codes: a FV 35

36

Chapter 3. Numerical solution method

code for solving the fluid and a FE code for solving the solid, possibly using two different meshes. The exchange of information between the two codes is performed by a third program that acts as an interface between them. As long as the dominant effects in the process can be classified either as fluid or solid and the interaction is weak, these methods and their algorithms are suitable as process modelling tools. In the case where strong coupling is needed at the fluid and solid interface, such method is time consuming and leads to large errors in the analysis (Bailey et al., 1999). This dictates the need for further development and expansion of the FV method in areas such as FSI, as an alternative to the traditional coupled FV-FE methods or FE method. Let us now examine several characteristics of the FE and FV methods, which give them different advantages and disadvantages. Since in this project there is a system that comprises both a fluid and a solid, these differences have to be kept in consideration, in order to obtain the most realistic solution with the least possible approximations and the minimum computational effort. The FE method uses predefined shape functions, depending on the element, and can be extended to higher order discretisation. For the solution of the PDEs, the FE method produces large matrices and relies mainly on direct solvers. On the other hand, the FV discretisation method is based on the integral form of the PDEs equations and, using Gauss’s theorem, the three dimensional volume integrals are transformed to two dimensional surface integrals. Usually this method uses segregated solvers, i.e the equations are solved sequentially one after the other, until convergence for the whole system is achieved. Due to the nature of a direct solver (i.e large memory and time requirements), the FE method is most suitable for static problems and also for cases where the matrix size is relatively small. In contrast, the FV method with the use of a segregated solver, has a particular advantage in transient problems. The coupling terms are treated explicitly and this may lead to convergence problems, especially if these terms carry a lot of information. So, the choice of a direct solver over an iterative solver and vice versa lies mainly on a trade-off between high expense of the direct solver for large matrices and cheaper iterative solvers with the necessary iterations over the explicit cross component coupling (Jasak and Weller, 2000). The FV method has become popular because of its ability to conserve physical quantities locally as well as globally. The FE method is still preferable over the FV method in cases were the material is linearly elastic. In cases were the material concerned is non-linear or viscoelastic, however, resulting in constant changes in the material properties, the FV will have an advantage over the FE method. This is mainly due to high requirement on CPU time and data storage for these cases (Demirdzić and Martinović, 1993). Another reason for the popularity of the FV methods is that they can model

3.2. Discretisation Procedure

37

easily highly non-linear phenomena in a computationally efficient manner. Due to the high non-linearity of the governing equations for fluid flow, the FE community has more difficulties modelling it, due to the matrix complexity. In contrast, when the equations are linear and the solution matrix is simple, the use of direct solution with the FE method is significantly faster. The FV method can handle easily the convection of fluxes across a cell boundary, since values are defined at the cell faces. On the other hand, handling of fluxes across a boundary does not come ’naturally’ for the FE method, because of the way it is designed as a discretisation method. In the FE values are specified at points, therefore the calculation of flux across an element face is not an easy task. In the case where the model consists of an incompressible material, the FE method has a serious drawback. Incompressibility comprises strong coupling between the continuity and momentum equations. The combination of these equations would result in a big and complicated matrix which would involve massive computational time to solve directly using the FE method. Thus, usually in FE the material is treated as compressible but with a very high bulk modulus. In FV the problem gets solved iteratively in a segregated manner, using PISO or SIMPLE algorithms (which enforces the incompressibility condition on the velocities). Another important improvement of FV techniques is their capability to handle complex boundary conditions, especially when heat transfer, fluid flow and solid structure deformation are coupled and their interaction is important. Moving boundaries and free surfaces, as well as other boundary conditions, have been developed and applied during the past decade in FV solvers (Bailey et al., 1999). For this project, the selected discretisation method used for the modelling is the FV method. The mathematical model was implemented into the FOAM (Field Operation and Manipulation) C++ Finite Volume library (Weller et al., 1998; Nabla, 2002). In the following Section 3.2 of this chapter, only the general principles of this discretisation method of the partial differential equations are presented. Other good sources of information about the FV methods are the books of Ferziger and Peric (1996); Versteeg and Malalasekera (1995) and Caughey and Hafez (1994). In Section 3.4, the choice for the boundary conditions used for the mathematical model presented in Chapter 2 is presented. Specific numerical considerations that need to be taken into account for the numerical solution are described in Section 3.3.1. Finally the one dimensional stability analysis of the numerical solution of the new formulation for solids is presented in Section 3.7.

3.2

Discretisation Procedure

An engineering problem can be described by a set of partial differential equations (PDEs). In order to solve the problem, the PDEs are discretised and expressed as

38


cell volume cell centre

face f

Sf

d P

face area vector N

| dN | |d|

Figure 3.1: Cell based structure.

a set of equivalent algebraic equations in a matrix form. The equations are solved computationally to obtain the solution of a certain variable at discrete points in space and time. The discretisation involves two parts: the discretisation of the computational domain and the equation discretisation. The FV method discretises the integral form of the PDEs.

Discretisation of the computational domain The discretisation of the computational domain involves the time discretisation and the space discretisation. For time discretisation, the time domain is broken down into a finite number of time steps. The size of the time step is specified and can be either constant or variable. Typically, space discretisation in FV method concerns the division of the spatial domain into a finite number of continuous non-overlapping control volumes (CV) known as cells. Every cell is constructed by a finite number of faces enclosing the CV. Every cell face is constructed by a list of spatial points. Five bits of information characterise the cell description: the two adjacent cells on either side of the face, the cell area, the centre to centre distance of two adjacent cells (d), the face area vector (S f ) and the weighting factor(ωN ). Those cell faces that have no neighbour cells are the boundary faces f . The cells constitute the FV mesh. The boundaries of the mesh are constructed by grouping the relevant cell faces into patches. These patches form the boundaries of the domain.

39


Discretisation of the equation The discretisation of the equation is performed by discretising each individual term. The procedure is as follows: every term of the PDE is integrated over the cell volume, then using the Gauss divergence theorem the volume integral is transformed to a surface integral, and then by using different schemes the resulting equations are converted into a set of algebraic equations. Most terms in a PDE comprise one or more of the three main operators: gradient, divergence and Laplacian. In the following subsection, the way in which each operator is discretised is described. The temporal term of a PDE is discussed in a separate section. The description of the discretisation procedure is general, so a general variable φ is assumed and the discretisation of each operator for this property φ is presented. Before proceeding to the description of the discretisation of the three main operations, let us consider how we can obtain the value of φ on the face between two cells.

3.2.1

Determination the face value φ f

Three different discretisation methods can be used in order to determine the value of a variable φ on the face of the two adjacent cells (with cell centres denoted N and P). The face value is evaluated from the cell centre values (φP , φN ) of the adjacent cells (see Figure 3.1). Central differencing (CD) Assuming linear variation of φ between the cell centres P and N the face values are calculated as (see Figure 3.2): φ f = ωN φP + (1 − ωN )φN

(3.1)

The weighting factor is determined as the ratio of the distances | dN | and | d |: ωN =

| dN | |d|

(3.2)

The central differencing scheme is second order accurate but can cause nonphysical oscillations in the solution. The oscillations can appear in the case where there is a steep of gradient of φ and can be reduced by mesh refinement. Upwind Differencing (UD) The face value of φ is determined according to the direction of the flow. ( φP for F ≥ 0 φf = (3.3) φN for F < 0 where F represents the mass flux passing through the face: F = S f • (ρU) f

(3.4)

40


φ φP

φN φf

d P

N

f dN

Figure 3.2: Evaluation of the face value φ f from cell centre values φP and φN assuming linear interpolation.

With this method the solution is bounded but at the expense of accuracy (first order accuracy). Blending Differencing (BD) This method is a linear combination of UD and CD. The face value is given by: φ f = (1 − γ) φ f

+γ φf UD

CD

(3.5)

were 0 ≤ γ ≤ 1 is the blending factor and determines the amount of numerical

diffusion introduced. This amount is evaluated in such a way as to remove the oscillations produced by the CD scheme. When γ = 0 this sceme reduces to the use of UD, whereas for γ = 1, it reduces to CD.

3.2.2

Discretisation of the gradient

The integral of the gradient term can be evaluated explicitly by Gauss integration. The way the gradient normal to the phase is evaluated is different, so it is explained separately. Gauss integration The discretisation is performed using the standard method of applying Gauss’s theorem to the volume integral, when keeping in mind that a CV is bounded by a series of faces. Thus, Z Z (3.6) ∇φ dV = dS φ = ∑ S f φ f V

S

f

41


Surface normal gradient The gradient normal to a surface n f • (∇φ) f for orthogonal mesh can be evaluated at cell faces using the scheme n f • (∇φ) f =

3.2.3

φN − φP |d|

(3.7)

Discretisation of the divergence

The integral of the divergence term is also evaluated explicitly. Note that the property φ can not be scalar (it has to be at least a first order rank tensor, i.e vector, or higher). The term is integrated over a control volume as follows: Z Z • ∇ φ dV = dS • φ = ∑ S f • φ f V

3.2.4

S

(3.8)

f

Discretisation of the Laplacian term

The Laplacian term is integrated over a control volume as follows: Z Z • ∇ (Γ∇φ) dV = dS • (Γ∇φ) = ∑ Γ f S f • (∇φ) f V

S

(3.9)

f

The treatment of the Laplacian term in a PDE can be either implicit or explicit. The internal product S f • (∇φ) f of Equation 3.9 is calculated using the values of φ at the centroids of the cells on either side of the face f . If the mesh is orthogonal, then:

3.2.5

φN − φP S f • (∇φ) f = S f |d|

(3.10)

Laplacian versus Divergence-Grad

To facilitate the presentation, the coefficient Γ in the equations that follow is dropped out, but, if one wants to include it, the principal idea is the same. So, the Laplacian operator as described in Section 3.2.4 is integrated over a control volume and is linearised as follows:

Z V

∇ φ dV = 2

Z S

dS • (∇φ) = ∑ S f • (∇φ) f

(3.11)

f

Let us assume there is a need to calculate ∇2 φ at the cell with centre denoted as P (see Figure 3.3). According to Equation 3.11, S f • (∇φ) f should be evaluated at each one of the faces of the cell P and then summed. At each face it is evaluated directly from the cell centre values of the adjacent cells, using the scheme described in Equation 3.10. In the 2D case of a cartesian mesh five cells are involved in the process with this method. For general orthogonal meshes in 2D the number of cells involved would be n + 1, where n is the number of cell faces.

42


N

φW

φN

∇2 φ W

E

P

S

φE

φS

NOTE: × denotes the location where S f • (∇φ) f is evaluated directly. Figure 3.3: Cells involved in the evaluation of the Laplacian operator at cell with cell centre denoted as P.

The divergence-gradient and the Laplacian operators are the same mathematically. However, their discretisation is different, thus, different discretisation errors may be introduced. In order to calculate the divergence-gradient operator a two step procedure is used instead of one used for the Laplacian. First ∇φ is calculated as described in Section 3.2.2: Z V

∇φ dV =

Z S

dS φ = ∑ S f φ f

(3.12)

f

In the second stage, the divergence is applied on the ∇φ calculated before. So, Z Z (3.13) ∇ • (∇φ) dV = dS • (∇φ) = ∑ S f • (∇φ f ) V

S

f

where now (∇φ) f is obtained by linear interpolation of (∇φ) from the adjacent cell centroids. In this process the number of cells involved in a 2D case for the evaluation of the divergence-gradient operator at the cell centre, is n ∗ [n − (n − 3)] + 1, where n is again the number of cells faces. In the case of a tetrahedron with cell centre P, thirteen cells are involved (see Figure 3.4). Thus, in order to calculate ∇ • (∇φP ), the (∇φ) f at the cell centres of the four adjacent cells (N,W, E, S) is needed, which means that in order to calculate ∇φ at the cell centres N,W, E, S, the cell centre values of φ of the adjacent cells for each one of them needs to be used. So with the use of the Laplacian operator ∇φ is not evaluated at cell centres it is only the S∇φ which is evaluated directly at cell faces in contrast to the divergence gradient operator. The Laplacian and divergence-gradient are computed in different ways, one being a one step procedure and the other a two step procedure. This results in the

43


K φK M φM L

∇φN

W

φL

N

∇φW G φG

∇ • ∇φ P S ∇φS B

T φT E

F

∇φE

φF

H φH

φB NOTE: × denotes the location where S f • (∇φ) f is evaluated by linear interpolation. Figure 3.4: Cells involved in the evaluation of the DivergenceGradient operator at cell with centre denoted as P.

introduction of different discretisation errors in the system, so special care should be taken in their use. Clearly the stencil of the Laplace operator is smaller.

3.2.6

Temporal Discretisation

Before proceeding to the description of the temporal terms, let us consider the Taylor series expansion in order to obtain the order of the truncation error of the time-advancing methods.

Order of accuracy using Taylor series Let us consider the Taylor Series and then apply it to the different schemes in order to examine the errors involved. The Taylor polynomial expansion of φo about φn , where φn is the value of the variable φ at time t + ∆t and φo is the value t is: n′

φ = φ − ∆tφ + ∆t o

n

2φ

n′′

2!

+ ...

(3.14)

Applying it in first order derivative of property φ for Euler Implicit gives: ′′

′ φn φn − φo ∂φ = φn = + ∆t + ... ∂t ∆t 2!

(3.15)

Euler Implicit uses two time levels and from equation 3.16 the truncation error

44


is first order accurate: E = ∆t

φn′′ + ... 2!

(3.16)

Applying it in first order derivative for Backward Differencing (also referred to in literature as a three level scheme) gives: ∂φ 3φn − 4φo + φoo 1 2 n′′ = φn′ = + ∆t φ + ... ∂t 2∆t 3

(3.17)

Backward Differencing involves three time levels and, from equation 3.17 the truncation error is second order accurate: 1 E = ∆t 2 φn′′′ + ... 3

(3.18)

In the same way, when Taylor polynomial expansion is applied for the second order derivative using Euler Implicit, it gives: ∂2 φ φn − 2φo + φoo 2 n′′ = φ = + ∆tφn′′′ + ... ∂t 2 ∆t 2 3!

(3.19)

The truncation error is of first order: E=

2 ∆tφn′′′ + ... 3!

(3.20)

First order time derivative Assuming that the volume does not change with time, the first order time derivative ∂/∂t is integrated over a control volume as follows:

Z V

∂ ∂ ρφ dV = ∂t ∂t

Z V

ρφ dV

(3.21)

The term is discretised by simple differencing in time using: new values φn ≡ φ(t + ∆t) at the next time step solved for; old values φo ≡ φ(t) that were stored from the previous time step; old-old values φoo ≡ φ(t − ∆t) stored from a time step previous to the last. First order time derivative can be evaluated either implicitly or explicitly in the FOAM C++ library used. There are two discretisation schemes: Euler implicit and backward differencing. If the time derivative is used in the source term then it is treated explicitly, while in the matrix calculation it is treated implicitly. The latter treatment is used in the present work.

45


Euler implicit scheme, that is first order accurate in time:

Z

∂ ∂t

V

ρφ dV =

(ρP φPV )n − (ρP φPV )o + O(∆t) ∆t

(3.22)

Backward differencing scheme, that is second order accurate in time by storing the old-old values and therefore with a larger overhead in data storage than Euler implicit: ∂ ∂t

Z V

ρφ dV =

3 (ρP φPV )n − 4 (ρP φPV )o + (ρP φPV )oo + O(∆t 2 ) 2∆t

(3.23)

Second order time derivative Euler implicit The approximation used for the second order time derivative is first order accurate. The integration over a control volume is given by: ∂ ∂t

Z

∂φ (ρP φPV )n − 2 (ρP φPV )o + (ρP φPV )oo ρ dV = + O(∆t) ∆t 2 V ∂t

(3.24)

Treatment of spatial terms After the description of the discretisation of the temporal derivatives, the spatial derivatives will now be considered. If all the spatial terms are denoted as A φ where

A is any spatial operator, e.g. Laplacian, then a transient PDE can be expressed in integral form as

Z

t+∆t

t

Z

∂ ∂t

V

ρφ dV +

Z V

A φ dV dt = 0

(3.25)

Using the Euler implicit method, the first term of Equation 3.25 can be expressed as Z t+∆t Z Z t+∆t ∂ (ρP φPV )n − (ρP φPV )o dt = (ρP φPV )n − (ρP φPV )o ρφ dV dt = ∂t ∆t t V t (3.26) The second term can be expressed as

Z t

t+∆t

Z

V

A φ dV dt =

Z t

t+∆t

A ∗ φ dt

(3.27)

where A ∗ represents the spatial discretisation of A . The time integral can be discretised in three ways: Euler implicit uses implicit discretisation of the spatial terms. Thus the values of

46


φ at the n-th time instant are used:

Z

t+∆t

t

A ∗ φ dt = A ∗ φn ∆t

(3.28)

This is first order accurate in time, is unconditionally stable and guarantees boundedness. Euler explicit uses explicit discretisation of the spatial terms, thereby the values of φ at the old-time instant are used:

Z

t+∆t

t

A ∗ φ dt = A ∗ φo ∆t

(3.29)

This is first order accurate in time and is unstable if the Courant number Co is greater than a threshold value. The Courant number is defined as Co =

C • d ∗ ∆t |d|2

(3.30)

where C is a characteristic velocity, e.g. velocity of a wave front in solids, velocity of flow in fluids. Crank Nicholson uses the trapezoid rule to discretise the spatial terms. Thereby taking a mean of current values φn and old values φo .

Z t

t+∆t

A φ dt = A ∗

∗

φn + φo ∆t 2

(3.31)

This is second order accurate in time and it is unconditionally stable but it does not guarantee boundedness.

3.2.7

Boundary Conditions

In order to fully specify a problem, a set of boundary conditions around the boundary cell faces (patches) has to be specified. The type of numerical conditions applied at the boundary should correspond to the physical conditions of the surrounding environment. There are two types of numerical boundary conditions. The following description assumes orthogonal mesh. Dirichlet the value φ is fixed along the boundary, also called fixed value boundary condition. Neumann the normal gradient of φ (∇φ • n) is fixed to the boundary, also called fixed gradient boundary condition. The boundary condition can take a form of algebraic equations that are solved at the boundary.

47


Fixed value A fixed value at the boundary φb is specified • In cases where the discretisation requires the value on a boundary face φ f , φb can be simply substituted.

• In cases where the face gradient (∇φ) f is required, it is calculated using the boundary face value and cell centre value,

φb − φP S f • (∇φ) f = S f |d|

(3.32)

Fixed gradient The fixed gradient boundary condition gb is specified as the inner product of the gradient and the unit normal to the boundary: gb =

S • ∇φ |S|

(3.33) b

• When the discretisation requires the value on a boundary face φ f , the cell centre value must be extrapolated to the boundary by φ f = φP + d • (∇φ)b = φP + |d| gb

(3.34)

• gb can be directly substituted in cases where the discretisation requires the face gradient to be evaluated, S f • (∇φ) f = S f gb

(3.35)

48


3.3

Final form of equations and discretisation of the transient term

In this section, certain solution procedures regarding the discretisation of the equations presented in Section 2.3 are discussed. It should be mentioned that for solids, if the deformations concerned are sufficiently small, the convection term ∇ • (ρUU) on the left hand side of the momentum Equation 2.2 can be ignored. Thus it is omitted in the following discussion as it is mainly concerned with the validation of the formulation for solids.

3.3.1

Reformulation in order to increase convergence rate

Equations 2.14, 2.19 and 2.23 can be split in the implicit part containing the temporal term and the Laplacian term, and the explicit part containing all the other terms. Such a discretisation is only marginally convergent as found by Jasak and Weller (2000). The reason behind this behaviour is the fact that the explicit term contains a significant amount of information and therefore the convergence can be achieved only with under-relaxation which slows down the procedure. An alternative way is mentioned in the paper of Jasak and Weller (2000), which gives an improved convergence rate. The contribution of the most implicit div-grad term of the equation is included by the coefficient 2µ + λ. If this is taken into consideration the following expressions can be rewritten as: Displacement based formulation ∂ρ

h

∂D ∂t

∂t

i

= ∇ • [(2µ + λ)∇D] + ∇ • {z } | | implicit

µ(∇D)T + [λtr (∇D)I] − [(µ + λ)∇D] {z }

(3.36)

explicit (i.e source term)

One can see that the term ∇ • [(µ + λ)∇D] has been added and subtracted on the right hand side. The implicit part of Equation 3.36 is the maximum consistent implicit contribution to component-wise discretisation. In this way, the system is over-relaxed. It includes the term, which could nominally be discretised implicitly only under mesh alignment, were all CVs of the computational mesh are cubes aligned with the co-ordinate system. If this is not the case, the additional terms are taken out in an explicit manner. In this way aP and aN coefficients are identical for all components of D. aP = ∑ aK where K=E, W, N, S K

(3.37)

49

3.3. Final form of equations and discretisation of the transient term

and aK = (2µ + λ)

|S f | |d|

(3.38)

Where aP and aK are the diagonal and off-diagonal coefficients respectively of the sparse matrix of the discretised form of the PDE. In the same way, the other forms of momentum Equation 2.19 and 2.23 can be rewritten as: Velocity based formulation ∂ρU ∆t ∆t = ∇ • [(2µ + λ)∇U] + ∇ • µ(∇U)T + λtr(∇U) − (µ + λ)∇U + ∇ • Σ+ ∂t |2 {z } |2 {z } implicit

explicit

(3.39)

Velocity and explicit pressure based formulation

∆t 2 ∂ρU ∆t T • • = ∇ [(2µ + λ)∇U] + ∇ µ(∇U) − µtr (∇U) − (µ + λ)∇U + ∇ • dev Σ+ − ∇p ∂t 3 {z } |2 |2 {z } implicit

explicit

(3.40)

Velocity and implicit pressure based formulation

∆t 2 ∂ρU ∆t = ∇ • [(2µ + λ)∇U] + ∇ • µ(∇U)T − µtr (∇U) − (µ + λ)∇U + ∇ • dev Σ+ − ∇p |{z} ∂t 3 {z } |2 |2 {z } implicit implicit

explicit

(3.41)

At this point it should be noted that the implicit terms on the right hand sides of Equations 3.36, 3.39, 3.40 and 3.41 use the discretisation procedure for the Laplacian operator (i.e. compact stencil) rather than the one for the divergence-gradient operator (i.e. enlarged stencil) (Section 3.2.5).

3.3.2

Temporal discretisation approaches

There are two issues involved with the discretisation of momentum equations (Equations 3.36, 3.39 and 3.40): the treatment of the temporal term on the right hand side and the treatment of the spatial terms in transient problems.

Displacement based formulation The temporal term ∂∂tρD 2 can be discretised in one of two ways. One way is by using the Euler implicit discretisation scheme , involving two old-time levels: 2

∂2 ρD ρDn − 2ρDo + ρDoo = ∂t 2 ∆t 2

(3.42)

50


where Dn ≡ D(t + ∆t), Do ≡ D(t) and Doo ≡ D(t − ∆t) .

This discretisation is bounded but causes a certain amount of dissipation since it is only first order accurate depending on the Co (Jasak and Weller, 2000). An other alternative is to use Backward differencing discretisation scheme, using three old-time levels: ∂2 ρD 2ρDn − 5ρDo + 4ρDoo − ρDooo = ∂t 2 ∆t 2

(3.43)

where Doo ≡ D(t − 2∆t).

Although this is second order accurate in time and therefore more accurate than Euler implicit, it does not guarantee boundedness of the results. So, the first order accurate temporal discretisation Euler implicit is preferred (Jasak and Weller, 2000). It should be mentioned that the Backward differencing scheme for second order derivatives is not available at the moment in the Foam C++ library used for this project. For the treatment of the spatial terms in transient problems, the Euler implicit method has been used. This uses an implicit treatment of all the spatial terms, so the new values of D at time n are used on the right hand side of the momentum Equation 3.36. With this method, the system is unconditionally stable and guarantees boundedness but it is only first order accurate. This will give us all together a first order accurate time discretisation.

Velocity based formulation The temporal term

∂ρU ∂t

can be discretised by two ways. The first way is by using

the first order accurate Euler implicit method, using two time levels: ∂ρU ρUn − ρUo = (3.44) ∂t ∆t The second way is by using the second order accurate Backward differencing using three time levels: ∂ρU 3ρUn − 4ρUo + ρUoo = (3.45) ∂t 2∆t Both of this methods have been implemented and the results obtained are presented in Chapter 4. The velocity based formulation is in a way equivalent to performing the discretisation of the displacement formulation in two steps. In the first step (i.e. ∂D ∂t = U) the

discretisation has been done using the theta method for θ = 1/2. The theta methods are linear combinations of explicit and implicit Euler scheme. In such schemes the parameter θ is used to optimise the accuracy and stability of the schemes (Equations

3.44 and 3.45). For θ = 1/2 the scheme is called Crank-Nicolson and it is unconditionally stable. The theta method is in general first order accurate in time and

3.3. Final form of equations and discretisation of the transient term

51

second order accurate in space. For θ = 1/2 the scheme is second order accurate in time (Mattheij et al., 2005; Higham, 2000). The second step of integration is performed using either Euler implicit or Backward differencing. When the Euler implicit scheme is used, the method is first order accurate is time and second order accurate in space, whereas when Backward differencing is used, the discretisation is overall second order accurate.

52


3.4

Iterative solution methods of governing equations

In order to create a unified approach for fluid-structure analysis of fluid transients in flexible vessels, the equations of both fluid and solid need to be solved for velocity and pressure. There are two ways of treting a velocity/pressure formulation: only the velocity is evaluated implicitly and the pressure is calculated explicitly from the definition using the velocity values, or both velocity and pressure are solved implicitly. In the case were the Poisson’s ratio approaches the incompressible limit ν → 1/2, there is no velocity pressure explicit link and the pressure represents an additional unknown, that enforces continuity of displacement. The algorithm used for the coupling is the PISO (Pressure Implicit with Splitting Operators) algorithm developed by Issa (1986). The PISO algorithm is typically used for the solution of Navier-Stokes equations for fluids and to the best of the author’s knowledge it has not been used in structural analysis before.

3.4.1

Governing equations

Only the necessary equations for the present discussion are presented here. More details can be found in Giannopapa (2002). The fundamental laws that can be applied for both fluid and solids when treated as continua are:

Continuity equation or mass conservation ∂ρ + ∇ • (ρU) = 0 ∂t

(3.46)

Momentum equation (neglecting body forces) ∂ρU + ∇ • (ρUU) = ∇ • σ ∂t

(3.47)

The momentum equation for a linear elastic or Hookean solid after substituting the constitutive equation, ignoring the convection term ∇ • (ρUU) and reformulating it in order to have as primitive variables velocity and pressure: i 2 h ∂ρU ∆t T ∇ • [µ∇U] + ∇ • µ (∇U) − µ∇ • [tr (∇U) I] + ∇ • devΣ+ − ∇p = ∂t 2 3

(3.48)

The momentum equation (Equation3.47) for a linear viscous or Newtonian fluid reads:

53

3.4. Iterative solution methods of governing equations

2 ∂ρU + ∇ • (ρUU) = ∇ • [η∇U] + ∇ • η(∇U)T − ηtr (∇U)I − ∇p ∂t 3

(3.49)

Barotropic relationship When interested in the wave propagation in a material, it is important to define the equation of state for a barotropic fluid and a compressible solid. This equation establishes a relationship between the density and the thermodynamic pressure in the fluid or the solid. This relationship can be derived by the definition of the bulk modulus K in the material and for small density variations after linearisation is given by: p − p0 ρ ≈ ρo 1 + K

(3.50)

where p0 is the reference pressure and ρ0 is the initial density for which ρ(p0 ) = ρ0 .

3.4.2

Non-linearity and pressure/velocity coupling

In fluids the typical system of equations that has to be solved is the Navier Stokes equations and the continuity equations. For this system of equations the primitive variables that need to be evaluated are the three velocity components and the pressure. The solution of these equations is complicated because they are highly coupled since each velocity component appears in each equation and because of the lack of an independent equation for the pressure, whose gradient appears in the momentum equations. The solution of the equation set (Equations 3.49 and 3.46) presents two problems: • non-linearity of momentum equations (Equation 3.49) and • velocity-pressure coupling. The non linearity of momentum equations is introduced by the convection term ∇ • (ρUU). This leads to a quadratic discretised form in terms of velocity resulting in a non-linear algebraic system of equations. The preferred way to overcome such a problem is to linearise the convection term. As described in Giannopapa (2002), the convection term for a property φ = U can be described as follows:

Z V

∇ • (ρUU)dV ≈

Z S

ρUUdS ≈ ∑ S f • (ρU f )U f = ∑ Ff U f f

(3.51)

f

where fluxes F are defined by : F = S f • (ρU f )

(3.52)

54


The convection term can be linearised by treating only the U f term in Equation 3.51 implicitly and using the existing fluxes calculated by the previous time step. It is important that the fluxes satisfy the continuity equation (Equation3.46). The non-linearities in the system of equations and the velocity-pressure coupling can be treated by adopting an iterative solution strategy. The SIMPLE algorithm (Patankar and Spalding, 1972) or its revised versions like: SIMPLER (Patankar, 1980) and SIMPLEC (Doormal et al., 1987) using a staggered grid are the most commonly adopted algorithms used to handle the velocity-pressure coupling in steady state problems. For transient flow the SIMPLE algorithm is not very suitable since it does not converge rapidly and its performance depends greatly on the size of the time step. The PISO algorithm, introduced by Issa (1986), is the most suitable for transient problems. The algorithm was initially proposed for non-iterative solution of incompressible Navier-Stokes system of equations using a staggered grid and it has been successfully adopted for iterative methods. Generalisation to compressible and transonic flows can be found in Demirdˇzić and ˇ Z.Lilek (1993). This algorithm is used in the present study as well; to the best of the author’s knowledge it has never been used before for solving the solid solutions. The PISO algorithm involves one predictor step and two corrector steps and may be seen as an extention of SIMPLE with a further corrector step. PISO does not necessarily require iterations within a time level so it is less expensive than SIMPLE. In Section 3.4.3 the pressure equation for compressible fluids is derived and in Section 3.4.4 the PISO algorithm is presented.

3.4.3

Derivation of pressure equation

As mentioned above, one of the complexities in the solution of the Navier-Stokes and continuity equations is the lack of an independent equation for pressure. Before proceeding in the derivation of a pressure equation, certain things should be mentioned about the system of equations. The continuity equation does not have a dominant variable in incompressible flows and it acts as a kinematic constraint on the velocity field. Therefore one way to overcome this is to construct a pressure field so as to guarantee the satisfaction of the continuity equation. Based on this assumption, the pressure equation can be derived, both for incompressible and compressible flows. Let us consider the momentum equation (Equation 3.49). If both velocity and pressure are defined at the cell centre, a periodic non-uniform pressure field with period 2∆x will act as a uniform field in the discretised momentum equations, in which only the phase values appear. However, such a pressure field is non-physical. Typically a staggered grid is adopted to overcome this problem. This method is difficult to extend to unstructured meshes, therefore the Rie-Chow interpolation method (Rhie and Chow, 1982) that can detect and correct such non-uniform pressure fields

55


has been adopted. All dependent variables are stored at the cell centre using one control volume, for which the face values of velocities have been interpolated using the Rie-Chow interpolation method. So the semi-descretised form of the momentum equation is: αP UP + ∑ αK UK = S(U) − ∇p

(3.53)

f

where U, p are the values from the present time step; αP are the diagonal elements of the coefficient matrix; αK are the off-diagonal elements associated with the cell neighbours K; S(U) is the source term containing all the terms that are explicitly computed (for example see Equation 3.41). An iterative method is used to update S(U) at every iteration within a time step, so at convergence all terms are calculated at the new time step. This form is semi-descritised because the term ∇p is not discretised It should be noted that this form of the non-linear algebraic equations is identical to the one derived for solids. Equation 3.53 can be rewritten as: αP UP = H(U) − ∇p

(3.54)

H(U) = − ∑ αK UK + S(U)

(3.55)

where

f

So, the H(U) contains the diffusion, convection and temporal terms associated with cell neighbours as well as the source term calculated explicitly, except from pressure gradient. Equation 3.54 can be solved for UP , giving: UP =

H(U) 1 − ∇p αP αP

(3.56)

Now let us consider the barotropic relationship (Equation 3.50). Using ψ to denote ρ0 /K: ρ = ρ0 + ψ(p − p0 )

(3.57)

Substituting Equation 3.56 and Equation 3.57 into the continuity equation (Equation 3.46) and assuming that K, ρ0, p0 are constant in time and space, the equation for pressure (that substitutes the continuity equation) can be obtained: H(U) ρ H(U) ∂ψp + ∇ • ψp −∇• ∇p = 0 + ∇ • (ρ0 − ψp0 ) ∂t αP αP αP |{z} | {z } | {z }

implicit

explicit

implicit

(3.58)

56


The first and the fourth term of the equation are implicit, whereas the second and third are explicit. To summarise the final form of the system of Navier-Stokes equations for a compressible fluid is Equation 3.54 and 3.58.

3.4.4

Velocity-Pressure coupling algorithms

The system of equations (3.54 and 3.58), as already mentioned in Section 3.4.2 , is highly coupled. There are two options for solving this system of equations: using a direct solver or using an iterative solver. The direct (or simultaneous) solver, solves the system of equations containing all dependent variables (i.e. velocity and pressure) simultaneously over the whole solution domain. In this method, all equations are considered as part of a single matrix. The solution of a coupled system of equations is a generalisation of the method used for a single equation. When the computational grid is fine and the number of equations is large, this solution method is computationally very expensive in terms of memory requirement and is very slow. The option of a segregated iterative approach is more appealing. It is based on the idea of solving a decoupled system for each independent variable, by temporarily treating all the other variables as known (initially guessing them or taking the values obtained from the previous iteration or time step). The equations are solved in turn iteratively until convergence i.e. all equations are satisfied within each time step. The PISO algorithm uses such an approach for velocity-pressure coupling. The system of equations is solved using the Biconjugate Gradient method (Hageman and Young, 1981).

PISO (Pressure Implicit with Splitting of Operators) The PISO algorithm was initially developed by Issa (1986) for non-iterative computation of incompressible flows using a staggered grid. It was later extended by ˇ Demirdˇzić and Z.Lilek (1993) for non-iterative computation of compressible and transonic flows. PISO has been adopted successfully for the iterative segregated solution for the Navier-Stokes system of equations and can be implemented to a collocated grid arrangment, with the Rhie-Chow face interpolation method (Rhie and Chow, 1982). When the algorithm was first published, it was solving for pressure corrections. In the present implementation, the algorithm is used to solve directly for pressure rather than its corrections. The flow chart of the PISO algorithm implemented for the present study can be seen in Figure 3.5. The PISO algorithm can be described as follows: • STEP1. Momentum predictor: Momentum equation (3.53) is solved in order to obtain the predicted values for the velocity U∗ field at the new time step.

57


PRESSURE VELOCITY LOOP

START

Momentum Predictor

Solve momentum αP UP = H(U) − ∇p

PISO LOOP

U∗ Calculate H(U∗ ) H(U∗ ) = − ∑ αK U∗K + S(U0 ) H(U∗) Pressure Equation

Solve pressure equation h h i i i h H(U∗ ) H(U∗ ) ρ∇p ∂ψp • ψp • • (ρ − ψp0 ) + ∇ − ∇ + ∇ αP aP αP = 0 ∂t p∗

Velocity Correction & Density Correction

Correct velocity U∗∗ P =

H(U∗ ) αP

−

Correct density

1 ∗ αP ∇p

ρ = ρ0 [1 + ψ(p∗ − p0 )] ρ

U∗∗

Calculate flux i h (∇p∗ ) H(U∗∗ ) • F = S f ( αP ) f − ( αP ) f F is continuity satisfied ?

NO U∗ = U∗∗ p = p∗

YES is residual less than prescribed ?

NO

YES END NOTE: One asterisk “∗” denotes first estimation. Douple asterisk “∗∗” denotes second estimation. Figure 3.5: PISO algorithm flow chart for compressible flow (for one time step).

58


The pressure gradient is treated explicitly using the pressure gradient value obtained from the previous time step. If it is the first time step, the momentum is solved with guessed values for pressure obtained from the initial conditions. This step is performed before entering the PISO loop. • STEP 2: This is an intermediate step where the term H(U∗ ) is constructed

using the predicted velocity values using Equation 3.55. This term is going to be used for the solution of the pressure equation in STEP 3. The values for the source term are taken from the previous time step or iteration.

• STEP 3. Pressure equation: The pressure equation (3.58) is solved using the H(U∗ ) term and the new estimated pressure field p∗ is obtained. This pressure field is not completely correct before convergence is reached after a couple of iterations in the PISO loop, therefore it is denoted with an asterisk “∗ ”. • STEP 4. Velocity correction: Using the new estimated pressure field p∗ , the velocity field is updated U∗∗ (double asterisk “∗∗ ” denotes second estimation). The velocity correction is done explicitly using the new pressure field p∗ and the first velocity prediction in the H(U∗ ) term. It is assumed that the entire velocity error comes from the error in the pressure term and the error from H(U∗ ) is neglected. Even thought this is not true initially it is corrected since several PISO loops are executed, so as to make sure that H(U∗ ) is calculated using the velocities that satisfy continuity. • STEP 5. Density correction: The density is also updated using the new estimated pressure field p∗ in order to be used in the next loop. • STEP 6. Calculate flux: Using the velocity correction H(U∗∗ ) and the new pressure field p∗ the new fluxes are evaluated. The fluxes are evaluated by using the Rhie-Chow interpolation method at the cell fases of the velocities obtained from Equation 3.56 and substituting it in Equation 3.52. These fluxes are used in the next time step for the linearisation of the convection term in the momentum equations. These fluxes should satisfy the continuity equation. Checking if the fluxes satisfy the continuity equation, within a predefined tolerance, is decided in the decision box for exiting the PISO loop. If this requirement is not fulfilled, the algorithm returns to STEP 2 and the process is repeated. So, the PISO loop is from STEP 2 to STEP 6. Since a new set of fluxes is obtained it would be possible to recalculate the H(U) term. • STEP 7: This step is again a decision box that checks whether the momentum equation has been solved within a specified tolerance. If this requirement is not satisfied, the program returns to STEP 1 and repeats the loop. If it is fulfilled, the program moves to the next time step.

3.5. Investigation of boundary conditions for fluids

59

So, the PISO algorithm for compressible fluids consists of one implicit momentum predictor (STEP 1) followed by a series of pressure solutions (STEP 3), explicit velocity (STEP4) and density corrections (STEP 5). This series of corrections (STEP 3, 4, 5) is repeated until convergence is reached within the predefined tolerance.

3.5

Investigation of boundary conditions for fluids

In order to derive a unique solution to any system of PDEs, a set of conditions needs to be specified at the boundary of the computational domain. These boundary values have either to be known or be expressed as a combination of internal values and boundary data. These approximations have to be derived by internal value differences or extrapolations. Generally as mentioned in (Giannopapa, 2002) the boundary condition types for any property can be: fixed value (or Dirichlet); fixed gradient (or Neumann); or mixed boundary condition, which is a linear combination of the other two. In this and the next subsections, boundary condition types and their derivations for velocity and pressure for the Navier-Stokes equations for compressible and incompressible flows is examined. The main focus is the pressure boundary condition. In order to specify the correct condition needed for the solution of the NavierStokes equation for compressible and incompressible flow, the different roles of pressure in these equations should be considered and therefore are discussed below. Incompressible flow In incompressible flow, there is no equation of state. Therefore, pressure is not a thermodynamic variable. The pressure propagates at an infinite speed in order to establish an incompressible flow and its role is to force the time varying velocity field to remain divergence free at all times. In terms of computation, the Navier-Stokes equations are solved and an initial predicted velocity field is calculated. This velocity field is corrected using the pressure values derived by the solution of a Poison pressure equation and should be as close as possible to the initial predicted ones. It can be proved (Ferziger and Peric, 1996) that the pressure can be seen as a Lagrange multiplier used to minimise the functional 1 R= 2

Z V

[U∗∗ (r) − U∗ (r)]2 dV

(3.59)

where r is the position vector, U∗ is the original velocity field and U∗∗ is the corrected velocity field.

60


For the solution of the Navier-Stokes equations, boundary conditions have to be specified for both velocity and pressure. It is typical to apply a fixed value (i.e. Dirichlet) boundary condition for the velocity at the boundary. The Navier-Stokes equations require that all the components of the velocity vector should be specified on the boundary. For a wall, no-slip boundary conditions are specified. This means that the velocity of the fluid is equal to the velocity of the wall. It should also be mentioned that the normal viscous stress is zero at the wall due to the continuity equation. For a symmetry plane, the velocity component parallel to the surface of the symmetry plane has zero normal gradient. A gradient (usually zero) of all quantities is specified on the outflow surface. It is important to point out that the Navier-Stokes equations require no a priori knowledge of the boundary conditions of pressure. The velocity boundary conditions applied to the momentum equations are sufficient to allow the determination of body velocity and pressure. Since only the first time derivative is present in Equation 3.47, it is sufficient to prescribe the initial velocity field at t = 0. Of course this velocity field must satisfy the incompressibility condition (O.Ladyshenkaya, 1998). The boundary condition for pressure though, is one that has received the most debate in the literature. Orszag and Israeli (1974) conclude that either the normal or the tangential components of the (vector) momentum Navier-Stokes equation is permissible as a boundary condition for the pressure Poison equation. This raises a serious dilemma, since the former leads directly to a fixed gradient (or Neuman) boundary condition and the later indirectly to a fixed value (or Dirichlet) boundary condition. Moin and Kim (1980) stated that the fixed value and fixed gradient problems for pressure, if properly derived form a well-posed Navier-Stokes problem, will have the same solution at least for t > 0. According to Gustafsson and Sundstr˝om (1978) the boundary conditions for pressure equation are obtained by applying the momentum equation (normal component) and the continuity equation at the wall. Gresho and Sani (1987) agree with their general idea. Therefore, they rederive the equations again and they answer to the question of Gustafsson and Sundstr˝om (1978) paper by stating that the divergence-free condition is of the utmost importance for theoretical and computational fluid dynamics. Gresho and Sani (1987) demonstrate that for the solution of the pressure Poison equation, the fixed value (or Dirichlet) boundary condition for pressure is only appropriate for t > 0 and it often does not apply for t = 0. Only the fixed gradient (or Neumann) BC is always appropriate and provides a unique solution for t ≥ 0. Any consistent discrete approximation of the original Navier-Stokes equations contains a built-in boundary condition for the discrete pressure Poison equation that is fixed gradient (or Neuman) boundary condition for t ≥ 0. This does not obviously

61

3.5. Investigation of boundary conditions for fluids

satisfy the fixed value (or Dirichlet) boundary condition, however. The converged numerical solution will also satisfy the Dirichlet boundary condition, but for t > 0, complementing the conclusion of Moin and Kim (1980). Let us consider the momentum equation ∂U + ∇ • (UU) − ∇ • (η∇U) = −∇p ∂t

(3.60)

where η is the kinematic viscosity and the continuity equation for incompressible flow: ∇•U = 0

(3.61)

Gresho and Sani (1987) derive two pressure Poison equations and therefore two equations for fixed gradient (or Neumann) boundary condition. One is the simplified form and is derived by including the continuity equation into the momentum. The other one is the consistent form where continuity is not included in momentum and from which we can derive the following boundary condition equation. n • ∇p = n • (η∇U − U∇U) −

∂U ∂t

(3.62)

The conclusion from their paper is that the correct boundary condition is the fixed gradient (or Neumann) and is obtained by applying the normal component of the momentum equation at the boundary. It should be mentioned that the solution for pressure computed using Equation 3.62 also satisfies the fixed value (or Dirichlet) boundary condition, which emerges by projecting the equation of motion onto a tangential vector and then integrating it with respect to the tangential arc length.The equivalent fixed gradient (or Neumann) boundary condition for compressible fluids will be derived and presented in Section 3.6 and will be tested in Chapter 4. Deng and Tang (2002) solve the incompressible Navier-Stokes equations and they are also concerned with finding the correct boundary conditions for the solution of these equations. In their solution approach they use the SIMPLE algorithm with pressure corrector for the velocity and pressure coupling. From their investigation they conclude that when the velocity boundary conditions are Dirichlet, the boundary conditions for the pressure correction should be Neumann; but when the velocity boundary conditions are Neumann, the boundary conditions for the pressure correction should be Dirichlet. Compressible flow For compressible flow, the boundary conditions are different from the ones used for incompressible equations, since the compressible equations are hyperbolic in character. A compressible fluid can support sound and shock waves and it is not surprising

62


that these equations have essentially hyperbolic character. Hyperbolic flows have characteristics that are real and distinct. Information propagates in two sets of directions. The equations for viscous-compressible flow are still more complicated. Their characteristics are a mixture of elements that do not fit well into the classification scheme and numerical methods for them are difficult to construct. Therefore special care should be taken in the specification of the boundary conditions. According to Ferziger and Peric (1996) for incompressible flow, the following boundary conditions can be applied: • Inflow boundaries: prescribed velocity and temperature on inflow boundaries. • Symmetry planes: zero gradient normal to the boundary for all scalar quantities and the velocity component parallel to the surface on a symmetry plane; zero velocity normal to such a surface. • Solid surface: non-slip (zero relative velocity) conditions, zero normal stress and prescribed temperature or heat flux on a solid surface. • Outflow boundaries: rescribed gradient (usually zero) of all quantities on an outflow surface. These boundary conditions also hold for compressible flow and are treated in the same way as in incompressible flows. However, in compressible flow there are further boundary conditions. • prescribed total pressure (at the inflow) • prescribed total temperature (at the inflow) • prescribed static pressure (at the outflow) • at a supersonic outflow boundary, zero gradient of all quantities are usually specified.

In order to define the total pressure at the inflow, the equation of state is usually used and the direction of the flow must be specified. It should be mentioned that the implementation of Ferziger and Peric (1996) is for pressure correction. The static pressure specified at the outflow boundary is again implemented by taking into consideration pressure and velocity correction at the boundary. For computational reasons an artificial boundary is usually introduced. For purely hyperbolic problems, it is well known that enforcing these boundary conditions through the characteristic variables leads to a stable approximation however for dissipative wave problems this procedure is considerably more complicated. Hesthaven and Gottlieb (1996) and Hesthaven (1997) are interested in dissipative, wave dominated problems and they derive stable open boundary conditions

3.6. Boundary condition for solids for the unified solution method

63

ensuring that the continuous problem is well-posed. The proposed boundary conditions are applied through the penalty procedure. Once the form of the boundary conditions is known, the way to implement them is to solve the equation in the interior points of the computational domain and then to enforce the boundary conditions at the boundary points. However, this approach does not take into account the fact that the equation should be satisfied arbitrarily close to the open boundary. Therefore, the penalty method is used to enforce the boundary condition, as well as taking into account the equation at the boundary. Gustafsson and Sundstr˝om (1978) and Olivier and Sundstr˝om (1978) use the energy method to obtain boundary conditions for the linearised constant coefficient Navier-Stokes equations in the primitive variable formulation. Dutt (1988) introduced an entropy function which allowed him to derive boundary conditions for non-linear problems ensuring that the solution remains bounded in an entropy norm.

3.6

Boundary condition for solids for the unified solution method

In order to derive a unique solution for the equations of interest, a set of conditions needs to be specified at the boundary of the solution domain. The boundary condition investigation in Section 3.4 has guided the choice of the boundary conditions for solids that were used for the solution of the mathematical models described in Chapter 2 and are described in detail here. Every time the momentum equation (Equation 3.36, 3.39 and 3.40) is solved, the values of displacement (for Equation 3.36) or velocity (for Equation 3.39 or 3.40) need to be updated at the boundary. The solution of momentum and pressure at the interior point and the satisfaction of boundary conditions is achieved through an iterative process, as already explained. This process is repeated until convergence is reached and the equations are solved to a specified residual. In the first couple of time steps, it takes more iterations to reach the required residual. In order to speed up the process, the program moves to the next time step when the number of iterations has reached 50. This number was specified by trial and error. In the subsequent time steps, convergence is achieved within usually 10-15 iterations. Since, the deformations concerned in the present study are very small the convection term ∇ • (ρUU) on the left hand side of the momentum Equation 2.2 is negligible. So, momentum equation can be rewritten as: ∂ρU = ∇•σ ∂t

(3.63)

64


3.6.1

Boundary conditions for the displacement formulation

The appropriate boundary condition for the displacement can be obtained by applying the force balance at the boundary. This relationship can be described by: n • σ = t − npext

(3.64)

where pext is the external pressure applied at the boundary and t is the external traction. Fixed gradient using force balance This boundary condition can be obtained by substituting stress from Equation 2.13 in Equation 3.64 and solving for n • ∇D. The resulting equation is afterwards reformulated in order to be consistent with the form of the momentum equation as presented in Equation 3.36. The final form of the displacement gradient normal to the boundary is: t − npext − n • (−µ − λ)∇D + µ∇DT − nλtr (∇D) n • ∇D = (2µ + λ) Fixed value using force balance

(3.65)

This fixed normal gradient boundary con-

dition can be substituted to: nb • (∇D)b =

Db − DN |dN |

(3.66)

Therefore, from Equation 3.65 and 3.66 the fixed value boundary condition expression for the displacement can be obtained as t − npext − n • (−µ − λ)∇D + µ∇DT − nλtr (∇D) Db = DN + | dN | (2µ + λ)

3.6.2

Boundary conditions for the velocity formulation

The boundary condition types that were tried for velocity were fixed value and fixed gradient and were obtained by applying force balance at the boundary (Equation 3.64). Fixed gradient using force balance

This boundary condition is obtained

by substituting the stress from Equation 2.17 in Equation 3.64 and solving for n • ∇U. The resulting equation is afterwards reformulated in order to be consistent with the form of the momentum equation presented in Equation 3.39. The final form of the velocity gradient normal to the boundary is:

n • ∇U =

2 ∆t

[t − npext − nΣ+ ] + n • (−µ − λ)∇U + µ∇UT − nλtr (∇U) (2µ + λ)

(3.67)


65

where pext is the external pressure applied at the boundary and t is the external traction. Fixed value using force balance The fixed value boundary condition is obtained from the fixed gradient by simply substituting the face normal gradient to: nb • (∇U)b =

Ub − UN |dN |

(3.68)

Therefore, from Equation 3.67 and 3.68, we can derive the fixed value boundary condition expression for the velocity as

Ub = UN + | dN |

3.6.3

2 ∆t

[t − npext − nΣ+ ] + n • (−µ − λ)∇U + µ∇UT − nλtr (∇U) (3.69) (2µ + λ)

Boundary conditions for the velocity-pressure formulation

There are two ways that the velocity/pressure formulation is solved: pressure explicit and pressure implicit. When the pressure is specified explicitly, no partial differential equation is solved so no boundary condition is needed. The pressure p is linearly extrapolated from the internal field to the boundary. Only in the case where the pressure is solved implicitly is there a need to specify an appropriate condition for the solution of the pressure at the boundary. 3.6.3.1

Boundary conditions for velocity

Fixed gradient using force balance The same way it was obtained in the displacement formulation, this boundary condition is obtained by substituting stress from Equation 2.21 in Equation 3.64 and solving for n • ∇U. The resulting equation is afterwards reformulated in order to be consistent with the expression of momentum presented in Equation 3.40 (or 3.41). The final form of the velocity gradient at the boundary is:

n • ∇U =

2 ∆t

[t − npext − n(devΣ+ ) + np] + n • (−µ − λ)∇U + µ∇UT + n 23 µtr(∇U) (2µ + λ) (3.70)

Fixed value using force balance

The fixed value boundary condition for

the velocity is obtained as in the previous section by simply substituting the face normal gradient. This expression is:

66


Ub =| UN | +dN

3.6.3.2

2 ∆t

[t − npext − n(dev Σ+ ) + np] + n • (−µ − λ)∇U + µ∇UT + n 23 µtr(∇U) (2µ + λ) (3.71)

Boundary condition types for pressure

The boundary condition expressions for pressure can be obtained in one of three ways: from the definition of pressure or from applying the force balance relation at the boundary or by projecting the momentum equation at the unit vector normal to the boundary.

This boundary condition is

Fixed value using the definition of pressure given by p = p0 − K

∆t tr (∇U + ∇U0 ) 2

(3.72)

where p0 is the old value obtained at the end of the previous time step.

Fixed value using using force balance This boundary condition type used for the pressure is fixed value and has been derived by applying force balance (Equation 3.64) at the boundary and solving for pressure, in the same way the equation for velocity at the boundary was obtained. So, the value of pressure at the boundary is given by:

p = −t • n + pext + n • dev Σ where is α =

+•

n − αn •

2 ∇U + (∇U) − tr(∇U)I 3 T

•

n

(3.73)

∆t 2.

Fixed gradient using momentum

This boundary condition type is fixed

gradient and an equation for its value is derived by projecting the momentum equation at the unit vector normal to the boundary and solving for n • ∇p. So we get: 2 ∂ρU ∆t T n • ∇p = n • ∇ • [µ∇U] + ∇ • µ(∇U) − µ∇ • [tr ∇UI] + n • ∇ • dev Σ+ − 2 3 ∂t (3.74) This formulation has been derived according to the paper of Gresho and Sani (1987) but for a compressible material.


67

Fixed value using momentum This boundary condition is obtained from the gradient boundary condition (Equation 3.74) by simply substituting the face normal gradient at nb • (∇p)b =

pb − pN |dN |

(3.75)

Therefore, from Equation 3.74 and 3.75 we can derive the fixed value boundary condition expression for the displacement as 2 ∂ρU ∆t T pb = pN + | dN | n • ∇ • [µ∇U] + ∇ • µ(∇U) − µ∇ • [tr∇UI] +n • ∇ • dev Σ+ − 2 3 ∂t (3.76)

3.6.4

Optimal choice of boundary conditions

In this section the type of boundary conditions that give the best results and are used for obtaining the results in the following chapter was presented. For the standard displacement formulation, a fixed gradient boundary conditions was used which was obtained by applying the force balance relation at the boundary (Equation 3.65). For the velocity formulation a fixed gradient boundary condition was used and was obtained again using the force balance relation (Equation 3.67). In the velocity-pressure explicit formulation there is no need to specify a separate boundary condition for the pressure at the boundary. The pressure is linearly extrapolated from the internal fields to the boundary. For the velocity a fixed gradient boundary condition is applied and the expression is obtained again from the force balance relation (Equation 3.70). In the velocity-pressure implicit formulation a fixed value condition is applied for the pressure. The expression is obtained by projecting the momentum equation to the unit vector normal to the boundary and solving for the pressure gradient n • ∇p. The value of pressure at the boundary is then obtained from the gradient with linear interpolation from the internal values (Equation 3.76). For the velocity, again a fixed gradient boundary condition is chosen by applying force balance (Equation 3.70), as it was found to have worked quite well for the other cases. It should be noted that the boundary condition type for velocity and pressure when using an implicit iterative solution method should be fully reversible i.e. when a fixed gradient boundary condition is used for the velocity a fixed value boundary condition is used for pressure and vice versa (Deng and Tang, 2002).

68


L=2∆x

∆x Discrete grid points Shortest resolvable wave with highest resolvable wave number

Figure 3.6: Shortest resolvable wave.

3.7

Stability Analysis

A number of methods exist to investigate the stability limits of a finite difference scheme. One such a method is the Fourier or Von Neuman analysis (Mattheij et al., 2005; Hirsch, 1988; Anderson et al., 1984; Abbott and Basco, 1989). This method will be described here and will be used to investigate the stability of the numerical method for the solution of displacement equations used in the standard stress analysis and the velocity equation developed and used in this project. Suppose that the solution of any finite difference scheme at point j at time level n can be written as a Fourier series in complex, exponential form: Dnj =

kk

∑ bnk eiα j

(3.77)

k=1

or alternatively kk

D(x,t) =

∑ bk (t)eikmx

(3.78)

k=1

The index n is the time level index; j is the grid point index; k = 1, 2, 3..., kk is the wave number index; i is the imaginary unit; bnk is the Fourier coefficient (amplitude) for wave number k at time level n; km is the wave number index and is equal to km = 2π L k, where L is the wave length; and α is the dimensionless wave number

2π which is equal to α = km∆x = 2π L k∆x = k N (0 ≤ α ≤ π), were N is the number of grid intervals over one wavelength. In Figure 3.6 the graphical representation of the shortest resolvable wave with the highest resolvable wave number can be seen. It

is apparent that the number of grid integrals in one wavelength is ∞ > N ≥ 2. It is also obvious that x j = j∆x. The Fourier analysis method determines how each Fourier coefficient behaves

69

3.7. Stability Analysis

(grows, decays, or stays constant) in time for any wave number index k. For example for k = 1 Dnj = bn1 eiα j

(3.79)

is a solution of the finite difference scheme. Note that n is not a power but a time index. This equation can be used to obtain the solution at any point in space and time. For example at the n + 1 time instance at location j: Dn+1 = bn+1 eiα j j

(3.80)

and at j + 1 point at the n-th instance it gives: Dnj+1 = bn eiα( j+1)

(3.81)

where the wave number index k is dropped. When the terms from the Equations 3.80 and 3.81 and the corresponding ones from every other time instant and point that appear in the finite difference equation are substituted into the discretised equation, the resulting expressions are rearranged to take the form shown by bn+1 = G bn

(3.82)

where G is called the amplification or growth factor. For a particular numerical method, the amplification factor depends upon the mesh size and the wave number or frequency. For hyperbolic problems, like the ones ∆t , where c is the concerned in this thesis, G depends on the Courant number, Co = c ∆x velocity of the propagating wave. To have a stable FD scheme, the Fourier coefficient must not grow without bound i.e. the magnitude of the Fourier coefficient of each and every wave number should not increase in time. So the stability condition is given by |G| ≤ 1

(3.83)

The stability analysis can also be used to determine the amplitude and the phase accuracy for all possible α (or alternative, for all grid intervals per wave length, N). It is assumed that for a given equation the amplification factor has been obtained G(α,Co). This can be used to calculate the amplitude of the response module |G|and

the phase response Q. The celerity ratio Q is defined as shown in Equation 3.84.

Q=

−arg (G (α)) = Co α

−tan−1

Im(G) Re(G)

Co α

(3.84)

70


∆x

n ∆t n-1

n-2 t

j-1

j

j+1

Figure 3.7: Stencil for the 1D hyperbolic finite difference equation (3.90).

3.7.1

Wave equation (1D)

The standard stress analysis equation in 1D with the assumption of constant density ρ is a hyperbolic equation known as the wave equation and can be written as: ∂2 Dx ∂2 Dx = (2µ + λ) , tε [0, ∞) , xε [0, L] ∂t 2 ∂x2 The wave velocity for plain strain is given by Equation 3.86. ρ

c1 =

s

2µ + λ = ρ

s

1−ν ϒ (1 + ν) (1 − 2ν) ρ

(3.85)

(3.86)

Longitudinal waves q in uniform bars with uniform cross section are given by Equation 3.87, where c = ϒρ . The wave velocity c is lower than the wave velocity c1 and their ratio depends on ν. For example, for ν = 0.3, the ratio is c1 /c = 1.16. 2 ∂2 D 2∂ D = c ∂t 2 ∂x2

(3.87)

Using the first order Euler implicit difference approximation to approximate the 2 second order time derivative ∂∂tD2 and the second order central approximation for the

71

3.7. Stability Analysis n iα j φnj = ∑kk k=1 bk e

n iα( j+1) φnj+1 = ∑kk k=1 bk e

n−1 iα j φn−1 = ∑kk k=1 bk e j

n iα j φnj = ∑kk k=1 bk e

n−2 iα j φn−2 = ∑kk j k=1 bk e

n iα( j−1) φnj−1 = ∑kk k=1 bk e

Table 3.1: Fourier series forms for time level n, n − 1, n − 2 and grid points j − 1, j, j + 1.

space derivatives

∂2 D , ∂x2

the following expression is obtained: n−1 n−2 n ∂2 D D j − 2D j + D j = + O(∆t) ∂t 2 ∆t 2

(3.88)

n n n ∂2 D D j−1 − 2D j + D j+1 = + O(∆x2 ) ∂x2 ∆x2

(3.89)

By substituting these in Equation 3.87, one obtains: Dnj − 2Dn−1 + Dn−2 = Co2 Dnj−1 − 2Dnj + Dnj+1 j j

(3.90)

∆t where Co = c ∆x . The stencil of this Euler implicit scheme is shown in Figure 3.7.

Suppose that the solution of the finite difference scheme can be written as a Fourier series in complex, exponential form for any time level, n. Each term appearing in Equation 3.90 can be found in Table 3.1. After substitution of these terms in Equation 3.90; factorisation and cancellation of the common term eiα j ; and division by the term bnk leads to: kk bn−2 bn−1 2 k k + ] = C [e−iα − 2 + eiα ] [1 − 2 ∑ ∑ o n n b b k k k=1 k=1 kk

The ratio

bn−2 k bk

(3.91)

can be written as: bn−1 bn−2 bn−2 1 1 k k = n−1 k n = n−1 n n bk bk G G bk

(3.92)

and substitution together with the identities e−iα + eiα = 2 cos α and cos α − 1 = −2 sin2 α2 in Equation 3.91 yields: kk

1

1

kk

α

∑ [( Gn−1 − 2) Gn ] = −4Co2 ∑ (sin2 2 ) − 1

k=1

(3.93)

k=1

The equation is now considered for the wave number k=1. This number is arbitrary, as any wave number can be used since the equation is linear and solving

72


for the amplification factor Equation 3.94 is obtained, where α = 2π/N and Nε [2, ∞). n

G =

1 2 − Gn−1

(3.94)

1 + 4Co2 sin2 α2

In Equation 3.91 for a single mode a solution of the form bn = λn can be tried, where in b the use of n is time index and in λ power. After substitution the characteristic or dispersion Equation 3.95 is obtained. It can be seen that time level is not included so assuming that the amplification factor between consecutive time steps is the same Gn = Gn−1 = G = λ gives . α (1 + 4Co2 sin2 )λ2 − 2λ + 1 = 0 2

(3.95)

This equation has two solutions. Therefore two amplification factors exist that must satisfy the stability condition (Equation 3.83), although the exact solution has a single value of the amplification. The solution with the positive sign corresponds to the physical solution, whereas the one with the negative sign propagates in the other direction. The solutions of Equation 3.95 are: G=

1 1 ± i2Co sin α2

(3.96)

The amplitude and phase portrait of G at different Co numbers Co = 41 , 21 , 34 , 1, are shown in Figure 3.8.

5 4

The scheme used for the discretisation of the Equation 3.85 is unconditionally stable for all Co , but it is dissipative. This means that the amplitude of the wave will suffer an attenuation of some magnitude at each time step. This numerical damping is well known as numerical viscosity (or dissipation). The numerical dissipation gets smaller by reducing ∆x (or increasing N). From the phase portrait it can be seen that there is a phase shift of the travelling wave which can be improved by increasing the number of grid points per wave length N.

3.7.2

Velocity formulation for linear elastic Hookean solid (1D)

The equation of the velocity formulation in 1D with the assumption of constant density ρ can be written in the form bellow, where m = 1, 2, ..., n is the time step index.

ρ

n ∂2U m ∂2Uxm−1 ∂Ux ∆t = (2µ + λ) ∑ [ 2x + ] ∂t 2 ∂x2 m=1 ∂x

, tε [0, ∞) , xε [0, L]

(3.97)

73


1

0.9

0.8

|G|

0.7

0.6

0.5

Co=1/4 Co=1/2 Co=3/4 Co=1 Co=5/4 Co=2

0.4

0.3

0.2

0

5

10

15

20

25

N

(a) Amplitude portrait 1

0.9

0.8

Q

0.7

0.6

0.5

Co=1/4 Co=1/2 Co=3/4 Co=1 Co=5/4 Co=2

0.4

0.3

0.2

0

5

10

15

20

25

N

(b) Phase portrait

Figure 3.8: Accuracy portrait of the amplification factor G for the 1D hyperbolic equation (3.96).

74


p By substituting the wave velocity c = ϒ/ρ in Equation 3.97 the 1D velocity formulation can be written as Equation 3.98. ∂Ux ∆t 2 n ∂2Uxm ∂2Uxm−1 = c ∑[ 2 + ] ∂t 2 m=1 ∂x ∂x2

, tε [0, ∞) , xε [0, L]

(3.98)

The above expression is equivalent to the following system of first order ordinary differential equations ∂Ux ∂t

= c2 ∂∂xD2x 2

(3.99)

∂Dx = Ux (3.100) ∂t The velocity formulation expression that contains the summation (Equation 3.98) is not in a form suitable for perform a stability analysis. Thus, using the main principle of the von-Neuman analysis and extending it to a system of equations, the stability analysis can be performed on the system of Equations 3.99 and 3.100 instead. Using the first order Euler implicit difference approximation to approximate the first order time derivative ∂U ∂t and the second order central approximation for space ∂2 D derivatives ∂x2 for Equation 3.99 the following is obtained: n−1 n ∂U U j −U j = + O(∆t) ∂t ∆t

(3.101)

n n n ∂2 D D j−1 − 2D j + D j+1 = + O(∆x2 ) (3.102) 2 2 ∂x ∆x The trapezoidal rule is used for the approximation of the integral in Equation

3.100 (as described in Section 2.3.2) , thus ∆t n n−1 U j +U j 2 Thus, Equation 3.98 can be equivalently written as: Dnj

U jn −U jn−1 ∆t

= c2

= Dn−1 + j

n−1 Dn−1 + Dn−1 j+1 − 2D j j−1

∆x2

(3.103)

! n−1 n−1 n−1 n n n ∆t U j+1 − 2U j +U j−1 +U j+1 − 2U j +U j−1 + 2 ∆x2 (3.104)

The stencil of this Equation can be found in Figure 3.9. The treatment of the spatial term is equivalent to theta-method for θ = 1/2. Thus the stencil is that of the theta-method with θ = 1/2. The theta methods are linear combinations of the explicit and implicit Euler schemes. In such schemes, the parameter θ is used to optimise the accuracy and/or the stability of a scheme. For θ = 1/2 the scheme is a Crank-Nicolson scheme and it is unconditionally stable. The theta method

75


∆x

n ∆t n-1

n-2

t

j-1

j

j+1

Figure 3.9: Stencil for the 1D system of equations that is equivalent to the 1D velocity formulation.

76


is in general first order accurate in the time and second order accurate in space; for θ = 1/2 it is also second order accurate in ∆t (Mattheij et al., 2005; Higham, 2000). Thus, it is expected from the stability analysis that the equivalent system of equations to be unconditionally stable and compared with the wave equation (Section 3.7.1) to be less dissipative. The system of Equations 3.99 and 3.100 after substitution of the time and space C2 approximations and setting ζ = ∆t2 and ξ = ∆to give: Dnj − Dn−1 j

n n−1 = ζ U j +U j

U jn −U jn−1 = ξ Dnj−1 − 2Dnj + Dnj+1

(3.105)

(3.106)

Suppose that the solution of any finite difference scheme can be written as a Fourier series in complex, exponential form for any time level, n. Each term appearing in Equations 3.105 and 3.106 can be found in Table 3.1. After substitution of these terms in the system of equations, factorisation and cancellation of the common term eiα j ; and using the identities e−iα + eiα = 2 cos α and cos α − 1 = 2 sin2 α2 , the

following equations can be obtained: kk

kk

k=1

k=1

n n−1 ∑ [bnk,D − bn−1 k,D ] = ζ ∑ [bk,U + bk,U ]

(3.107)

kk α n−1 n − b ] = 4ξ [b ∑ [sin2 2 bnk,D] ∑ k,U k,U k=1 k=1

(3.108)

kk

The second index i.e. D or U in the subscript of the Fourier coefficient is used to denote the variable that this coefficient belongs to, i.e. displacement and velocity respectively. For a single mode, for example k = 1, the system of equations can take the matrix form as: "

bn+1 D n+1 bU

#

=A

"

bnD n bU

#

(3.109)

where A is the matrix 

A=

1−2Co2 sin2 2a 1+2Co2 sin2 2a ∆t 1+2Co2 sin2 2a

C2

−4 ∆to sin2 2a 2 1+2Co2 sin2 2a 1 1+2Co2 sin2 2a

 

The eigenvalues of the 2x2 matrix A will give the characteristic or dispersion equation, which reads: λ2 −

2(1 −Co2 sin2 α2 ) 1 + 2Co2 sin2 α2

λ+

1 =0 1 − 2Co2 sin2 α2

(3.110)

77

3.7. Stability Analysis 1

1 Co=1/4

Co=1/2 |G|

0.9

|G|

0.95 0.9 0.85

0.8

wave equation velocity formulation 0

5

10

15

20

0.7

25


5

10

N 1

20

25

1 Co=1

Co=3/4 |G|

0.8

|G|

0.8 0.6 0.4

0.6


5

10

15

20

0.4

25


5

10

N

20

25

1

0.8

0.8 |G|

Co=5/4

0.6 0.4 0.2

15 N

1

|G|

15 N

0.4


5

10

15

20

Co=2

0.6

25

0.2


5

N

10

15

20

25

N

Figure 3.10: Amplitude portrait of the 1D velocity formulation in comparison with the wave equation (displacement formulation).

Setting ϑ = −2Co2 sin2 α2 in Equation 3.110 and from the solution of the quadratic equations the two values for the amplification factor G = λ can be obtained. These values must satisfy the stability condition (Equation 3.83)

λ=G=

2 p −2 − ϑ ± ϑ(ϑ + 8)

(3.111)

where α = 2π/N and Nε [2, ∞). The solution with the positive sign in Equation 3.111 corresponds to the physical solution, whereas the one with the negative sign propagates in the other direction. The accuracy amplitude portraits for different Courant numbers Co = 14 , 21 , 34 , 1, 45 in comparison with these of the displacement equation are shown in Figure 3.10. The scheme used for the discretisation of the Equation 3.87 is unconditionally stable for all Co and the numerical damping is smaller than the scheme used for the wave equation. Thus this discretisation scheme is more accurate.

78


3.8

Closure

The finite volume method has been applied for the discretisation of the governing mathematical equations presented in Chapter 2. This method has a long tradition in computational fluid dynamics. As the reformulated equations for solids have velocity and pressure as primitive variables, the finite volume method seems to be the natural choice. Only the basic principles of the FV method have been presented in this chapter. Specific attention has been drawn to the discretisation of the Laplacian operator versus the divergence-gradient operator. The two operators, even though mathematically the same, are discretised differently. Thus, care should be taken when used as they introduce different discretisation errors. Practical issues involving the solution procedure with emphasis to the convergence rate have been addressed. The equations have been reformulated to their most implicit part in order to increase convergence rate according to Jasak and Weller (2000) paper. When velocity and pressure are both solved implicitly in the unified formulation for fluid structure interaction problems, the solution is complicated mainly due to the fact that there is no independent equation for pressure and each one of the velocity components appears in all equations creating a highly coupled system. The PISO algorithm has been adopted to solve iteratively the coupled system. A decoupled sub-system for each independent variable is solved by temporarily treating all the other variables as known in an iterative segregated manner. In the present implementation, the PISO algorithm solves for velocity and pressure, rather than their corrections. The PISO algorithm is typically used for the solution of the NavierStokes equations for fluid and, to the best of the author’s knowledge, it has never been used before for structural analysis. In order to derive a unique solution for the momentum equations whether they are used to solve fluid dynamic problems or solid mechanics problems or fluid-structure interaction problems, a set of conditions must be specified at the boundary of the computational domain. The boundary values can either be known or evaluated by descritising the boundary conditions using the internal cell values. In order to find the appropriate boundary conditions for solids, a thorough literature investigation was performed to see what are the most appropriate boundary conditions for fluids. The conclusion from this investigation was that the choice of boundary conditions for pressure has received the most intense debate in the literature. The best choice for deriving an expression for pressure at the boundary is to solve the momentum equation for the normal component of the pressure gradient. Based on this, the appropriate boundary conditions were derived for solving the solids with the new unified solution method that is consistent with a one for fluids.

3.8. Closure

79

For the velocity, a fixed gradient boundary condition can be obtained by applying the force balance relation at the boundary. For the pressure, a fixed value boundary expression can be derived by solving for the normal component of the pressure gradient in the momentum equation and then calculating the pressure value from the gradient. In the last part of this chapter, a stability analysis has been presented for the discretisation of the velocity based formulation in comparison with the equivalent standard stress analysis formulation for solids. The discretisation scheme for the new velocity formulation is unconditionally stable for all Courant numbers and is less dissipative than the one used to discretise the displacement equation. This means that the solution of the amplitude of the wave will suffer less attenuation at each time step in comparison with the standard formulation.

80


Chapter 4 Validation of the new formulation for solids 4.1

Introduction

The new unified formulation with velocity and pressure as primitive variables presented in Chapter 2 is standard for compressible or incompressible fluid modelling but is new for solids and therefore, needs to be tested. A beam bending case was chosen to validate the method. The interest in such a case stems from the need to use a difficult case that comprises, apart from normal stress, shear stress as well. The effect of shear is of great importance in wave propagation, so such a case would be a good validation tool.

4.2

Case Description

A narrow cantilivered beam was considered as shown in Figure 4.1, loaded at its free end by a concentrated force of such magnitude that the weight of the beam can be neglected. The material properties of the beam and its dimensions are shown in Table 4.1. Property Modulus E Poisson’s ratio ν Density ρs Length l Height h Depth w

Value 4 × 109 Pa 0.3 1450 kg/m3 20 m 5m 1m

Table 4.1: Material properties and dimentions of the beam.

81

82

Chapter 4. Validation of the new formulation for solids

x

1 0 0 1 0 1 0 1 0 1 0 1 0 1

l h

τ y

Figure 4.1: Beam bending test case.

The beam has the following physical boundary conditions: the left face is a fixed end , the right face has an applied end shear of τ = 106 Pa and the upper and lower faces are traction-free. The situation described may be regarded as a plain stress case, provide that the beam thickness w is small relative to the beam length. In our case it is w = 1 m. To decrease computational time the problem is solved in 2D. The mesh of the beam is constructed from 400 (40x10x1) square cells. Each cells size is 0.5x0.5x1 m3 . The analytical solution of the case is presented in the following section and is used for validation of the new method. The comparison between analytical and computational data is presented in Section 4.4.1.

4.3

Analytical solution

The one dimensional and two dimensional theory of beam bending cases can be found in many engineering books such as Dym and Shames (1970); Timoshenko and Goodier (1970); Géradin and Rixen (1997); Ugural and K.Fenster (2003). In order to calculate the main frequency of the oscillation of the beam, a one dimensional approximation is used for which an analytic solution is available. Unfortunately a two dimensional solution for the frequency has not been found. Thus, the 1-D solution is used only as a rough reference guide to validate the computational results. However a two dimensional analytic solution for the steady state is available. The distribution of stress in the beam is given by (Timoshenko and Goodier, 1970): σxx = 12τ

σyy = 0

xy h2

(4.1)

(4.2)

83

4.3. Analytical solution

1 y 2 σxy = 6τ − 4 h

(4.3)

The beam displacements in the horizontal and vertical direction respectively are given by: Dx =

2τ 2 3(l − x2 )y + (2 + ν)y3 3 ϒh

# " 2 h 12τ x3 l 3 x 2 (1 + ν)(l − x) + + (νy − l) + Dy = 3 ϒh 6 3 2 2

(4.4)

(4.5)

The maximum deflection of the beam at x = 0 is found by solving Equation 4.5 and reads as: " 2 # 4τ l 3 3 h δ= 1 + (1 + ν) 2 ϒh 4 l

(4.6)

The term in brackets in Equation 4.6 is a two dimetional correction; in the one 2 2 dimensional solution, this term is omitted. The term 34 (1 + ν) hl ≃ 2h is the 4

ratio of the shear deflection to the bending deflection at x = 0 and provides a measure of the beam slenderness. For a slender beam, h ≪ l, it is mainly due to bending. In vibration at higher modes and in wave propagation, the effect of shear is of great importance in slender as well as in other beams. Using the values from Table 4.1, the maximum deflection for the beam is δ = 0.340 m. The speed of propagation of the stress wave through the beam, for this particular material is: s ϒ = 1660 m/sec (4.7) C= ρ In the case of a one dimensional solution of a uniform cantilivered beam with no pre-stress, with bending stifness ϒI , where the second moment of area is I = h3 /12 and the mass per unit lengh m remains constant over the beam length, the eigen frequencies can be written as: ωn = µ2n

s

ϒh2 12ρsl 4

(4.8)

where µn is the eigenvalue at mode n. In Figure 4.2, the eigenvalues and the frequencies of oscillation can be seen in the two graphs (a) and (b). For the fundamental eigenvalue, µ1 = 1.875 the frequency of the undamped oscillation is: ω2 = 1.8754

ϒh2 12ρs l 4

and the main frequency of oscillation of the beam: f = ω/2π = 3.35 Hz.

(4.9)

84


6

5

10

10

5

4

10

Frequeny of oscillation [Hz]

Eigen frequencies

10

4

10

3

10

2

3

10

2

10

1

10

10

1

0

10 0 10

1

2

10

10

3

10

10 0 10

1

10 no of modes

Eigen values

2

10

(a) Eigenfrequencies ωn versus constants µn (b) Frequency of oscilation f against modes for no of modes n = 1 : 100. n = 1 : 100

n µn ωn fn

1 1.875 21.070 3.354

2 4.694 132.054 21.017

3 7.855 369.792 58.854

>3 (2n − 1) π2

(approx.) 724.660, 1197.788, ... 115.333, 190.634, ...

Figure 4.2: Analytical calculations for the vibration eigenvalues, eigenmodes and frequency of oscilation using a 1D approximation for the solution of a cantilever beam.

4.4. Results

4.4

85

Results

The beam bending case was used for testing the validity of the model described in Section 2.3. The mathematical model was implemented in the FOAM finite volume C++ library. The beam bending case was first run using the standard stress analysis model described in Section 2.3.1 and the results obtained are shown in Section 4.4.1. Within this chapter, whenever there is a reference to the standard stress analysis model, the expression displacement formulation is implied. In Chapter 2, in order to create the final unified solution method, where velocity and pressure are solved fully implicit, intermediate steps were presented. First the displacement formulation was altered to have the velocity as a primitive variable (velocity formulation). Then the pressure was introduced in the formulation, but it was solved explicitly (velocity-pressure explicit formulation). Finally the unified solution method was presented where both velocity and pressure are solved implicitly (velocity-pressure implicit formulation). Each one of these three cases has been run separately on the beam bending case and as all three of them give the same results. Only the velocity-pressure implicit formulation results are presented here and, for brevity, they are described as the velocity-pressure formulation. The main interest of the discussion appart from the accuracy of the method, is numerical dissipation and the issues involved with the discretisation error as well as the term accumulation in the mathematical model. In Section 3.7, the stability analysis of the displacement formulation was compared with the one from the velocity formulation and the results of this analysis will help with the interpretation of the results of the present chapter. Thereafter, the effects of discretisation discretisation scheme, time step, mesh resolution and dissipation are examined for the velocity pressure formulation. Further we present the effect of applying different boundary conditions in the velocity-pressure formulation. This illustration is given in order to stress the importance of making the correct choice when velocity and pressure are solved fully implicitly. This stems from the investigation presented in Section 3.4. Finally in Section 4.5 the conclusions gathered from this investigation are presented .

4.4.1

Displacement calculated using the standard stress analysis

The beam bending case has been used for testing the validity of the numerical model described in Section 2.3. The standard stress analysis code formulation that solves for displacement has been used in order to compare it with the velocity-pressure formulation. The end displacement versus time is shown in Figure 4.3. The time step used was ∆t = 1e − 4 s and the Co = 0.33 < 1 therefore, it is expected that the

86

Chapter 4. Validation of the new formulation for solids 0.7 0.6

Displacement d

0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

1 Time t (a)

1.2

1.4

1.6

1.8

2

Figure 4.3: End displacement (m) versus time (s) (standard stress analysis).

energy loss should be minimal. The discretisation method used for the temporal term discretisation (second order time derivative) is Euler implicit (Section 3.3.2). The stability analysis of the one dimensional displacement based formulation using this discretisation method has been presented in Section 3.7.1. The beam oscillates with a frequency of 3.32 Hz and has a maximum deflection of 0.62 m. The frequency of the oscillation is in quite a good agreement with the one dimensional analytical solution for the main frequency of oscillation presented in Section 4.3. It should be mentioned that the comparison with the analytical solution can give only an indication about the frequency of the oscillation as it has been calculated from the equivalent one dimensional problem while the numerical solution presented here is two dimensional. In order to have an exact comparison a two dimensional analytical solution should be used. However it is quite complicated to be solved analytically and thus such a solution for transient problems could not be found by the author. From a steady state analysis performed on the beam using the displacement based formulation, it was found that the beam has a maximum vertical deflection at 0.31 m from its original (horizontal) position. This value is in close agreement with the two dimensional analytical solution presented in Section 4.3, namely 0.34 m. The calculations were performed over a long period and Figure 4.4 presents the envelope of the displacement graph i.e. only the minimum and the maximum values. As it can be seen, the system dissipates after about 17 s (170500 time

87

4.4. Results 0.7 0.6

Displacement D (m)

0.5 0.4 0.3 0.2 0.1 0

0

2

4

6

8 10 Time t (sec)

12

14

16

18

Figure 4.4: Standard stress analysis (envelope of displacement).

steps); that is after around 56 beam oscillations. This is due to the fact that the discretisation method used is first order accurate in time. This introduces a certain amount of numerical dissipation depending on the Co number as provided in the previous chapter. One can also use a second order accurate expression, which is nominally more accurate, but does not preserve the boundeness of the solution. It may cause unphysical stress peaks and even solution instabilities. This is the reason why the first order accurate solution was preferred (Jasak and Weller, 2000) as mentioned in Chapter 3.3. There are other discretisation methods such as the Newmark method, that can provide better stability and higher accuracy than the analysis presented here for the displacement based formulation. Such an implementation for dynamic solid mechanics tested in a beam bending case can be found in Slone et al. (2003). However, such methods were not available in the FOAM C++ library and therefore were not used in the present investigation. Nevertheless, this does not affect the main contribution of this study, which is to demonstrate the validity of the velocity-pressure formulation for solids.

4.4.2

Discretisation error analysis for the new formulations

In stress analysis codes for solids, it is very important to check that the numerical errors are small in order to obtain realistic results. Typically, numerical errors depend on the accuracy of the equation discretisation method and the discretisation of the computational domain. The discretisation errors introduced by the term dis-

88


cretisation and time step size variations are studied in the following subsections. In the first subsection, the calculation of the accumulated term of momentum equation and its effect in the overall behaviour of the system are presented. In the second part, the effect of different discretisation schemes on the temporal term of momentum equation using different time steps are presented and discussed. Finally the effect of the mesh resolution will be examined. 4.4.2.1

Calculation of the accumulated term

The expressions for the momentum equation (Equation 2.19 and 2.23), after taking into consideration the discussion in Section 3.3.1 take the from of Equations 3.39, 3.40 and 3.41. The discretisation of the momentum equation can be performed in two ways, depending on how the term ∇ • Σ+ or ∇ • dev Σ+ is discretised. This term is given by the formula:

Σ+ = Σ +

∇ • Σ+ = ∇ • Σ +

∆t (2µ + λ)∇Uo + µ(∇Uo )T − λtr(∇Uo )I − (µ + λ)∇Uo 2

(4.10)

∆t ∇ • [(2µ + λ)∇Uo ] + ∇ • µ(∇Uo )T − λ∇ • tr (∇Uo )I − (µ + λ)∇ • ∇Uo 2 (4.11)

or 2 ∆t o o T o o (2µ + λ)∇U + µ(∇U ) − tr (∇U )I − (µ + λ)∇U dev Σ = dev Σ + 2 3 +

∆t ∇ dev Σ = ∇ dev Σ + 2 •

+

•

(4.12)

2 o o o o T • • • • ∇ [(2µ + λ)∇U ] + ∇ µ(∇U ) − ∇ tr (∇U )I − (µ + λ)∇ ∇U 3 (4.13)

Note that in Equations 3.39, 3.40 and 3.41, the term ∇ • [(2µ + λ)∇U] is used to calculate the matrix coefficient (implicit formulation using the Laplacian discretisation scheme in a compact stencil as described in Section 3.2.5). The same term ∇ • [(2µ + λ)∇Uo ] also appears in the evaluation of the divergence of the accumulated stress as shown in the previous equations (however the operator now acts on the old time step). This now is a source term and is evaluated explicitly, i.e. its contribution goes to the right hand side of the linear system of equations. As mentioned in Section 3.2.5, this term can be discretised either with the compact stencil (Laplace discretisation), or using a wider stencil (div-grad discretisation). Using two different discretisation techniques to discretise the same term, would introduce different discretisation errors. This inconsistency would result in higher dissipation. In Section 4.4.1, the variation of the displacement with time was used

89

4.4. Results

as a means to monitor dissipation. An alternative way to displacement would be the monitoring of the total power. Power dissipation presents a more involved physical understanding of a dissipative system and will give a better indication of the nature of the dissipation, whether it is physical or numerical. In a closed system, where there are no losses due to friction and other external factors, the total power should be equal to zero. In the beam bending case the powers applied in the system are: external power (due to shear force applied at the end of the beam), kinetic power (due to the oscillating movement of the beam); and strain power (due to its change of position during the oscillation). If the discretisation of the momentum equation is not consistent, then further numerical errors would be introduced that will result in energy dissipation of the system. The power formulation will be derived directly from momentum Equation 3.63. If the dot product with with velocity is taken in both sides of the momentum equation, then: U•

∂ρU = U•∇•σ ∂t

(4.14)

Using the following identity σ •• ∇U = ∇ • (σ • U) − U • ∇ • σ

(4.15)

and the momentum Equation 3.63 can be transformed to ∂ρU = ∇ • (σ • U) − σ •• ∇U (4.16) ∂t In Equation 4.16 the term on the left hand side denotes the kinetic power of the U•

system, the first term on the right hand side denotes the external power applied at the end of the beam and the second term is the strain power. The different types of power derived from momentum equation are presented below. External power EP = ∇ • (σ • U)

(4.17)

Kinetic power KP =

1 ∂ρUU 2 ∂t

(4.18)

Strain power SP = σ •• ∇U

(4.19)

In Equation 4.19 the double dot product ( •• ) operator is not conservative and this will create a discontinuity at the boundary. Using Equation 4.15 in Equation 4.19

90

Chapter 4. Validation of the new formulation for solids 5

2

x 10

Laplacian Divergence−Grad

Total Power (W)= External − Strain − Kinetic

1.5

1

0.5

0

−0.5

−1

−1.5

−2

0

0.1

0.2

0.3

0.4

0.5 time (s)

0.6

0.7

0.8

0.9

1

Figure 4.5: Total power comparison for the ∇2 and the ∇ • ∇ operators in the accumulated term.

the following formulation is obtained. SP = ∇ • (σ • U) − U • ∇ • σ

(4.20)

TP = EP − SP − KP

(4.21)

Total power

The total power of the system can be presented in Equation 4.21 and should be equal to zero if the discretisation conserves energy, i.e. if there is no artificial dissipation into the system. Figure 4.5 compares the total power against time for the inconsistent discretisation of the ∇ • [(2µ +λ)∇U] term and the consistent discretisation. In the inconsistent discretisation, the term is evaluated implicitly using the Laplacian operator (compact stencil), whereas when evaluated in the accumulated term (source term) the div-grad discretisation (wide stencil) is used. It can be seen from the figure that in this case the total power is highly erratic and non zero. In contrast, in the consistent discretisation, this term is discretised using the Laplacian operator. In this case, the total power is around zero. Thus, it is important to be consistent in the way the terms are discretised in order to avoid erratic behaviour and to get more accurate

91

4.4. Results Comparison of tolerences 10e−6, 10e−7, 10e−8 60000 tolerence 10e−7 tolerence 10e−6 toterence 10e−8

Total power (W)

40000 20000 0 −20000 −40000 −60000

0

0.1

0.2

0.3

0.4

0.5 0.6 Time t (sec)

0.7

0.8

0.9

1

Figure 4.6: Total power comparison for different tolerances:10e-6, 10e-7, 10e-8.

results. The total power represents the energy residual of the solution. Therefore, by solving the momentum equation in a tighter tolerance, the total power would approach even closer to zero. Using the consistent discretisation, different values of tolerance to which the momentum equation is solved can be compared in Figure 4.6. The tolerances compared are 10e-6, 10e-7, 10e-8. When the 10e-8 tolerance is used, the deviation from zero of the total power is the smallest. 4.4.2.2

Temporal term discretisation

Here two issues are examined: The first one is a comparison of the numerical accuracy of the displacement and the velocity based formulation of the governing equations. The second issue is a comparison of two different schemes applied for the discretisation of the temporal term: the Euler Implicit and the Backward Differencing schemes for the velocity-pressure formulation. It should be noted that in the following figures only the envelope of the displacement (i.e. only the minimum and the maximum values) is plotted. Comparison between the displacement and velocity based formulations In this section, the numerical dissipation of the standard displacement formulation against the velocity based formulation is examined and the results are interpreted along the lines of the stability analysis presented in Section 3.7. The displacement formulation has a temporal term of second order and the ve-

92

Chapter 4. Validation of the new formulation for solids 0.7 displacement formulation velocity based formulation 0.6

Displacement D (m)

0.5

0.4

0.3

0.2

0.1

0

0

5

10

15

20

25

30

Time t (sec)

Figure 4.7: Comparison of displacement formulation and velocity-based formulation for the Euler Implicit discretisation scheme (envelope of displacement).

locity formulation has a temporal term of first order and both of them are discretised using the Euler implicit difference approximation. The discretisation method, for the treatment of the spatial terms, applied in both cases, is second order central. Figure 4.7 compares the velocity-based formulation with the displacement formulation over a period of thirty seconds. It can be seen that the displacement obtained from the displacement formulation has dissipated after 56 oscillations (170,500 time steps) (Section 4.4.1) while the displacement calculated using the velocity-pressure formulation has dissipated by 14.7% over a period of 30 sec (300,000 time steps). This shows an important advantage of a velocity based formulation over a displacement formulation. The reason behind this behaviour can be explained from the conclusions obtained from the stability analysis and the comparison of the one dimensional displacement formulation and the one dimensional velocity based formulation. In Figure 3.10, it was shown that the velocity formulation is less dissipative compared to the displacement formulation for all Courant numbers. The way the velocity is integrated is equivalent to a two step integration, where the first step is performed using the trapezoidal rule which is second order accurate in time (Section 3.3.2) and the second step is performed using Euler implicit scheme

93

4.4. Results Comparison of Euler Implicit and Backward Differencing descretistion scemes for 1e−4 time step 0.7 Euler Implicit Backward Differencing 0.6

Displacement d

0.5 0.4 0.3 0.2 0.1 0

0

5

10

15 Time t

20

25

30

Figure 4.8: Comparison of Euler Implicit and Backward Differencing discretisation scheme (envelope of displacement).

which is first order accurate. Thus, just by using a velocity based formulation the accuracy of the computation increases without using a more accurate time integration scheme such as Newmark. Velocity-pressure formulation: Euler Implicit versus Backward Differencing In this subsection, a comparison of the first order time derivative (velocity-pressure formulation) using different discretisation schemes, for the treatment of the temporal term will be presented. The discretisation schemes compared are: Euler Implicit and Backward Differencing. From Figure 4.8 one can see the effect of the discretisation scheme for the first order time derivative. It can be seen that the Backward Differencing scheme is more accurate than the Euler implicit since the first is second order accurate while the later is only first order accurate. Over a 30 sec period (300,000 time steps) the Euler Implicit dissipates about 14.7% and over a 100 sec (1,000,000 time steps) about 33.3%. On the other hand the Backward Differencing over a 30 sec period has much smaller dissipation. In terms of computational overhead the Backward Differencing takes longer since it requires three time levels for the computations. The accuracy of the first time derivative Euler Implicit can be improved further with the decrease of the time step size. Figure 4.9 compares different time step sizes for the Euler Implicit discretisation scheme. It can be seen that when the time step decreases from 1e-4 s to 1e-5 s (Co = 0.033) the accuracy over a 30 s period improves

94

Chapter 4. Validation of the new formulation for solids Displacement vs Time 0.7 1e−4EI 1e−5EI 1e−6EI

0.6

Displacement d

0.5 0.4 0.3 0.2 0.1 0

0

2

4

6

8 Time t

10

12

14

Figure 4.9: Comparison of different time step sizes: 1e-4, 1e-5, 1e-6 s for the first time derivative Euler Implicit.

about 7.5%. When the time step decreases from 1e-5 s to 1e-6 s (Co = 0.003) there is no significant change, only 0.62% improvement, but the computational overhead is quite substantial. The improvement of the accuracy with the decrease of the time step of the first order accurate Euler Implicit scheme can be compared with the second order accurate Backward Differencing discretisation scheme. Figure 4.10 illustrates that for a period of 30 s the results of Backward Differencing with 1e-4 s time step and Euler Implicit with time step 1e-5 s are almost the same. The Backward Differencing scheme with 1e-4 s time step is 1.2% less dissipative than the Euler Implicit with 1e-5 s time step. In cases where a solution is needed for a short time, the Euler Implicit would give relatively realistic results and in a shorter computational time. In cases where a reliable solution is needed for longer periods, at least a second order accurate discretisation scheme should be used to obtain a realistic solution even though the computational time would be sufficiently longer. As it can be seen from Figures 4.8 , 4.9 and 4.10 the envelope of Backward Differencing scheme exhibits repeatable beats independent of the time step size. The existence of these beats is also indicated in the Euler Implicit scheme from t=0 to t=10 s, but it is not as vivid due to the high numerical dissipation. If these beats are physical they can only represent one eigenmode of the vibration. In order to investigate whether these beats are physical or numerical, the two dimensional beam bending case was run using the ANSYS finite element commercial package. From the standard stress analysis, the first four eigenfrequencies were

95

4.4. Results

Comparison of Euler Implicit for 1e−5 time step and Backward Differencing for 1e−4 time step 0.7 EI 1e−5 BD 1e−4 0.6

Displacement d

0.5 0.4 0.3 0.2 0.1 0

0

5

10

15 Time t

20

25

30

Figure 4.10: Comparison of Euler Implicit using time step size 1e-5 s against Backward differencing using time step size of 1e-4 s (envelope of displacement).

n fn

1 0.677

2 3.2102

3 4.2015

4 5.0791

Table 4.2: Computational calculations for the vibration eigenfrequencies of vibration using for the two dimensional beam bending case using the ANSYS finite element commercial package.

96

Chapter 4. Validation of the new formulation for solids Mesh resolution comparison 40x10, 60x20BD, 200x50BD 0.7 40x10 60x20BD 200x50BD

Displacement d (m)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

5

10

15 Time t (sec)

20

25

30

Figure 4.11: Mesh resolution comparison for meshes: 40x10, 60x20 and 200x50 cells. Time step size used is 1e-4 and temporal term discretisation scheme is Backward differencing (envelope of displacement).

obtained (Table 4.2 ). It should be mentioned that the first four eigenmodes are fundamental and the rest of the frequencies are combination of the first four. The beats appearing in the envelope of displacement of our discretisation have a frequency of 0.2369 Hz, where the frequency of the first mode found by ANSYS is 0.677 Hz. Thus, it can be concluded that their appearance is of numerical nature. These errors relate to the mesh quality as it is illustrated in the following section. 4.4.2.3

Mesh quality

In this section, the numerical errors introduced due to mesh quality are examined. Up to now, the grid was 40x10 cells . In the third dimension there is always one cell. As this is a two dimensional investigation, displacement and velocity are not computed in the third direction. For the other meshes used, the time step is kept to ∆t = 1e − 4 s (300,000 time

steps) resulting in a Co = 0.33 and the discretisation scheme for the temporal term is Backward differencing. The different mesh resolutions applied were 60x20 and 200x50 cells in x and y direction respectively. The results are presented in Figure 4.11. All cases run for 30 s. As it can be seen from Figure 4.11 there is very small dissipation. The displacement envelope beats appears in all three cases but the number of beats is reduced with the increase of mesh resolution. The frequency of these beats in the 40x10 cells mesh is 0.1148 Hz, while for 60x20 reduces to 0.1309

97

4.4. Results 0.7 Dirichlet using momentum Dirichlet using pressure definition 0.6

displacement (m)

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5 time (s)

0.6

0.7

0.8

0.9

1

Figure 4.12: Comparison of different boundary conditions for pressure in the fully implicit velocity-pressure formulation.

Hz and for 200x50 drops down to 0.0782 Hz. Thus, for 300,000 time steps for a 200x50 cells mesh, the occurence of the beats is almost dissapearing (only two beats appear). It should be stressed here that a solution of 300,000 time steps corresponds to 100 oscillations and the occurence of the beats was no longer examined. Thus, we need not to conceder further the occurence of the beats. The important finding from this investigation is that the method proposed in this thesis has minimal dissipation even if one decides to run the case for very long time.

4.4.3

Boundary conditions

The most difficult part in creating a unified solution method for solving fluidstructure interaction problems lies in the choice of appropriate boundary conditions. In the unified method where both velocity and pressure are implicitly solved using the PISO algorithm for the coupling, a set of boundary conditions is needed for the velocity and pressure. In Section 2.5 and 3.6 some possible boundary conditions for the velocity-pressure formulation were derived. There are two issues involved with the choise of the correct boundary conditions at the free boundary: (a) the equation that will give a relationship about the behaviour of the variables concerned at the boundary and (b) the type of the condition i.e. fixed value or fixed gradient.

98


In Section 3.4, an extensive literature review is presented on boundary conditions for incompressible fluids that lead us to the choice of the appropriate boundary conditions for compressible fluid-structure interaction problems using the unified solution method. The appropriate boundary conditions for velocity is fixed gradient and is derived from the force balance equation at the boundary (Equation 3.70) and for the pressure is fixed value derived from the momentum (Equation 3.76). As far as the first issue is concerned the derivation of the optimal pressure boundary condition is according to the paper of Gresho and Sani (1987) but for a compressible material. In Figure 4.12 it is demonstrated the accurate solution of the fully implicit velocity pressure formulation using boundary conditions for pressure according to Gresho and Sani (1987) in contrast to the suboptimal choice such as the use of the definition of pressure at the boundary.

4.4.4

Other cases

In this section variations of the beam bending case are presented. These cases were run in order to get another validation of the results and the behaviour of the code. The length of the beam was varied and the following two cases were examined: half and double the initial length. Another case was selected to run was to use the same beam length but with the applied end force to be half the initial one. 4.4.4.1

Analytical solution

For the beam with half the original length (5mx10m) using the analytical solution, the end displacement is found to be 0.0995 m and the frequency of the beam oscillation is 13.413 Hz. For the beam with double the length (40mx10m) the end displacement is 5.198 m and the frequency should be 0.8384 Hz. For the beam where the end shear applied is τ = 5e5 Pa, the end displacement is 0.34 m and the frequency of the beam oscillation is 3.35 Hz. 4.4.4.2

Numerical solution

In all cases the time step used was ∆t = 1e − 4 s, the temporal discretisation scheme applied was Backward differencing and the mesh resolution was kept equivalent with the beam length i.e. constant ∆x. For the case with half the beam length (10mx5m) the mesh used was 20x10 cells. The end displacement found was 0.087 m and the frequency was 11.64 Hz (Figure 4.13). The percentage difference for the frequency between the analytical solution and the numerical solution is 13.29 % and for the maximum displacement is 12.96 % (Table 4.3). It must be noted that as the analytical solution is 1D, the shorter the beam is in relation to its height, the less accurate the solution would be. This can explain the 12.96 % difference with the numerical solution.

99

4.4. Results

Variable

Analytical Beam size: 10mx5m; end Max Displacement [m] 0.0995 Frequency [Hz] 13.41 Beam size: 20mx5m; end Max Displacement [m] 0.68 Frequency [Hz] 3.35 Beam size: 40mx5m; end Max Displacement [m] 5.2 Frequency [Hz] 0.84 Beam size: 20mx5m; end shear: 5e5 Max Displacement [m] 0.34 Frequency [Hz] 3.35

Predicted shear: 1e6 0.0866 11.64 shear: 1e6 0.62 3.32 shear: 1e6 4.72 0.86 Pa 0.313 3.29

% Difference Pa 12.96 13.29 Pa 8.82 0.9 Pa 9.21 2.05 7.94 1.79

Table 4.3: Comparison between analytical and computational solution for beams with different size.

0.09

0.08

0.07

Displacement (m)

0.06

0.05

0.04

0.03

0.02

0.01

0

0

0.1

0.2

0.3

0.4

0.5 Time (sec)

0.6

0.7

0.8

0.9

Figure 4.13: Beam with size 10mx5m. No of cells used for the mesh is 20x10cells , time step size used is 1e-4 and temporal term discretisation scheme is Backward differencing.

1

100

Chapter 4. Validation of the new formulation for solids 5

4.5

4

Displacement (m)

3.5

3

2.5

2

1.5

1

0.5

0

0

0.5

1

1.5

2

2.5 Time (sec)

3

3.5

4

4.5

5

Figure 4.14: Beam with size 40mx5m. No of cells used for the mesh is 80x10cells , time step size used is 1e-4 s and temporal term discretisation scheme is Backward differencing.

For the case with double beam length (40mx5m) the mesh used was 80x10 cells (Figure 4.14). The end displacement was 4.72 m and the frequency was 0.859 Hz . The percentage difference between the analytical and the numerical solution is 2.05 % and for the displacement 9.21% (Table 4.3). In the case where the applied end shear was halved τ = 5e5 Pa, the time step used was ∆t = 1e − 4 s, the temporal discretisation scheme used was Backward differenc-

ing and the mesh resolution was 40x10 cells, the same as the one in the standard validation case. Figure 4.15 presents the results. The maximum end displacement is 0.313 m and the frequency of oscillation of the beam is 0.29 Hz. The percentage difference for the frequency between the analytical solution and the numerical solution is 1.79 % and for the maximum displacement is 7.94 % (Table 4.3).

4.5

Closure

In Chapter 2 the derivation of the unified solution method for solving fluid-structure interaction problems was presented. As the method is standard for solving fluids validation is needed only for solids. In this Chapter, a two dimensional beam bending case was chosen for the validation, which is more difficult to solve as it comprises,

101

4.5. Closure

0.35

0.3

Displacement (m)

0.25

0.2

0.15

0.1

0.05

0

0

0.1

0.2

0.3

0.4

0.5 Time (sec)

0.6

0.7

0.8

0.9

1

Figure 4.15: Beam with size 20mx5m, with applied end shear τ = 5e5 Pa. No of cells used for the mesh is 40x10cells , time step size used is 1e-4 s and temporal term discretisation scheme is Backward differencing.

102


shear stress as well, rather than normal stress only. The effect of shear is of great importance in wave propagation in fluid structure interaction problems. In Chapter 3, several discretisation issues were raised that have been examined here. The way the terms are discretised in the momentum equation affects the behaviour of the system. If the discretisation of the accumulated terms is done in an inconsistent way, different discretisation errors are induced that lead to inaccurate results. Consistent discretisation removes the erratic behaviour of the net energy. In Section 3.7, a stability analysis was performed for the displacement based formulation and compared with the velocity-based formulation. In this Chapter, we have illustrated this behaviour with a numerical example. A velocity-based formulation for solids where the trapezoid rule has been used, for the introduction of the velocity instead of displacement in the stress tensor, results in an accurate system without the need to use more accurate schemes such as Newark. The displacement based formulation, which is first order accurate, completely dissipates after 170,000 time steps whereas the velocity formulation after 300,000 time steps has dissipated by only 14.7%. In both cases, the Euler implicit scheme was used. The comparison of using backward differencing over Euler implicit for the discretisation of the temporal term in the velocity-pressure formulation has also been illustrated. The use of Backward Differencing produces a more accurate behaviour of the numerical model. The decrease of the time step size by a factor of 10 for first order time derivative Euler Implicit improves the accuracy and brings the results close to Backward Differencing. When the beam bending problem needs to be solved for a short period of time, the Euler Implicit discretisation method will produce relatively good results in a short period of calculation time. On the other hand, when there is a need for longer time solution, at least a second order accurate discretisation method should be used or the time step size should be decreased significantly. In both cases, the increase of the accuracy is accompanied by an increase of the computational overhead. In this Chapter, we have illustrated that the PISO algorithm that to the best of the author’s knowledge has never been used before for structural analysis can be used successfully for the pressure-velocity coupling in solids. The difficulty in its correct implementation in a fully implicit velocity-pressure coupling in solids (as well as in fluids), when a free boundary is used, lies in the choice of appropriate boundary conditions. The choice involves two issues: the first issue is the choice of the type (i.e. Dirichlet or Neumann) and the second is the choice of the correct condition that describes the behaviour of the variable at the boundary. Here, both issues have been illustrated. The appropriate boundary conditions for solving a fully implicit fluidstructure interaction problem with the unified solution method are: for the velocity, a fixed gradient boundary condition, that is obtained by applying the force balance relation at the boundary and for the pressure a fixed value boundary expression

4.5. Closure

103

that is derived by solving for the normal component of the pressure gradient in the momentum equation and calculating the value from the gradient. In Chapter 2, it was concluded that if the mathematical representation of the unified solution method proves to be able to solve classic solid mechanics problems, then the unified method probes to work and can be used for solving FSI problems. In this Chapter as an answer to that question, it was demonstrated that the method can indeed solve solid mechanics problems accurately. Thus, it has been proved that a unified solution method can be considered in solving fluid-structure interaction problems. The next step for the continuation of this project would be to use this method to solve a full FSI problem. The way it can be used is explained in Chapter 7 which is related to future work.

104


Chapter 5 Wave propagation experiments in flexible vessels with wall thickness variation and geometric tapering 5.1

Introduction

The study of wave propagation in fluid-filled tubes is often motivated by the need to understand arterial blood flow. Even though the general principles gathering wave propagation in flexible vessels are known (McDonald, 1968; Pedley, 1980; Fung, 1997), there is lack in the literature (Section 1.4) of well defined experiments taking into consideration the wall thickness variation and the geometric tapering that characterises the human vessels i.e. the aorta. In vitro laboratory experiments in mechanically and constitutively well-defined systems are needed for the validation of numerical and analytical models. To bridge this gap, a set of tubes was designed and manufactured to assess the role of geometric tapering and wall thickness variation in flexible vessels. The tubes were manufactured according to aortic specifications. They were designed such that the wave speed of the travelling wave would be equivalent to that of the aorta. The experiments were performed for small deformations.

5.2

The Tube Models Methodology

In Section 1.4, it was concluded that there is lack in the literature of well defined experiments assessing the non-linearities of flexible vessels i.e. wall thickness variation and geometric tapering. In order to obtain a complete set of experimental data assessing these variations, a set of flexible tubes was manufactured. 105

106

5.2.1

Chapter 5. Wave propagation experiments in flexible vessels with wall thickness variation and geometric tapering

The vessels design and specifications

The tubes were designed to be model analogues of the human aorta. One of the most referenced sources of arterial dimensions is the one from Westerhof et al. (1969). In Table 5.1, the data mentioned in his work are presented. In this work, they are used as a guidance for the design of the tubes. Variable Top internal radius [mm] Bottom internal radius[mm] Length [mm] Slope ϒ ∗ h/D [MPa]

Aorta Thoracalis 20 11 315 -0.014 0.02-0.04

ϒ : Youngs modulus, h: wall thickness, D: internal diameter Table 5.1: Aorta anatomical data (Westerhof et al., 1969).

To be able to assess the effects of morphological variations in wave propagation velocity c, six tubes were manufactured: three straight ones and three tapered ones. The geometrical parameters of these tubes are summarised in Table A.3. It should be mentioned that the tube of Type E has the same ϒ ∗ h/D and wave speed c as the aorta according to Westerhof et al. (1969) (See Table 5.1).

To separate effects due to geometric tapering, two pairs of tubes were manufactured such that they would have the same wave speed throughout according to linear wave propagation theory (Lighthill, 1975). The first pair consists of a straight tube with constant wall thickness (Type A) and a tapered tube with variable wall thickness (Type F). The variable wall thickness of the tapered tube was chosen such that according to linear theory the wave speed throughout its length is the same as for the straight tube with constant wall thickness. In this way the variable wall thickness of the tapered tube according to the linear theory will counterbalance the effect of geometric tapering. The second pair consists of a geometrically tapered tube with constant wall thickness (Type E) and a straight tube with variable wall thickness (Type C). The variable wall thickness of the tube was designed such that according to linear theory the wave speed variation along the length of this straight tube is the same as the for tapered one with constant wall thickness. The wall thickness variation for tubes C and F can be seen in Figure 5.1. For reference, a homogeneneous thick walled straight tube (type B) and a tapered, homogeneously thin-walled tube are made as well.

107

5.2. The Tube Models Methodology

D[mm]

h±0.002[mm]

L[mm]

z

ϒ ∗ h/D[MPa]

c[m/s]

A

25

0.1

446

0

0.04

6.3

B

25

0.05

446

0

0.02

4.5

C

25

0.05-0.1

446

0

0.02-0.04

4.5-6.3

D

25-12.5

0.1

446

-0.014

0.04-0.08

6.3-8.9

E

25-12.5

0.05

446

-0.014

0.02-0.04

4.5-6.3

F

25-12.5

0.1-0.05

446

-0.014

0.04

6.3

Type

ϒ: Young’s modulus, h: wall thickness, D: diameter Table 5.3: Geometrical parameters of tubes manufactured.


108

−3

x 10

Internal diameter of tube C [cm]

Thickness of tube C [cm]

10 9 8 7 6 5

0

10

20 30 Tube length [cm]

3.5

3

2.5

2

1.5

40

0

10


40

0

10


40

−3

x 10

Internal diameter of tube F [cm]

Thickness of tube F [cm]

10 9 8 7 6 5

0

10


40

3

2.5

2

1.5

1

Figure 5.1: Wall thickness variation for tubes C and F.

109

5.3. Material Properties of the Tubes

5.2.2

Manufacturing Method

The tubes were manufactured by the method of spin coating. The tube takes the shape of a steel rod that can rotate along its length axis through a servomotor (xservomotor). For the straight tubes, this rod is straight, with 25mm diameter and a length of 500mm. For the tapered tubes a rod with maximum diameter 25 mm and a minimum diameter of 12.5 mm was used. The length of the taper is 440mm. The liquid used in the spin coating process, polyurethane (PU, Besmopan 588, Bayer, Germany) dissolved in tetrahydrofurane (THF, BASF, Germany) is delivered at a constant flow rate through a nozzle, by a perfusion pump (Harvard medical sytems, USA). The nozzle is attached to a trolley that can translate along the length of the rod through by a rotating ball screw rod connected to a second servomotor (y-servomotor). The two servomotors are operated simultaneously by a computer-driven servocontroller. With a given concentration of the PU-solution, flow rate and a required geometry, the spin-coating device is programmed to generate the proper wall thickness. After evaporation of the solvent, the remaining tube is removed from the rod and is ready for use. The process generates tubes with prescribed wall thicknesses (either constant or variable) with an accuracy of 2 µm. A detailed description of the manufacturing process is given in Appendix A.

5.3

Material Properties of the Tubes

The physical properties of PU are given in Table 5.4. The solution used had concentrations varying from α = 17 − 22.73% of PU in THF solvent. Physical properties Density(kg/m3 ) Ultimate tensile strength(MPa) Elongation at break (%) Tear propagation resistance (kN/m)

Polyurothane (PU) 880 30 500 55

Table 5.4: Physical properties of polyurethane.

A little solvent remains in the tube. This solvent remnant causes the tube to have viscoelastic properties. To measure these properties, relaxation tests were performed. Tensile force at constant strain over time was determined in a uniaxial tensile testing machine (Zwick Z010, Germany). The specimens measured had a wall thickness that was double the average of wall thickness of all the tubes, thus


110

1

10

0

Tensile Force [N]

10

−1

10

−2

10

−3

10

−2

10

−1

10

0

1

10

10

2

10

time [s]

Figure 5.2: Typical relaxation test curve for Polyurethane specimen (3% elongation).

3

10

111

5.3. Material Properties of the Tubes

it was 0.14 mm. The width of the specimens was 37.3 mm. The specimens were strained 1%, 3% and 10% to check possible non-linear mechanical behaviour. For each strain the experiment was repeated six times each on a new specimen. In Figure 5.2 a typical relaxation test curve is shown. The graph shows two parts: the behaviour of the specimen under loading until it reaches the strain target and the specimen relaxed under constant strain. In linearly viscoelastic materials, it is straightforward to develop a relationship for the relaxation response ϒ(t). The stress and relaxation modulus relationship is given by the following Boltzmann integral, where τ is the time variable of integration: σ(t) =

Z

t

0

ϒ(t − τ)

dε(τ) dτ dτ

(5.1)

In the integral form, the time scale is considered just prior to time zero, so that step function load histories beginning at zero may be accomplished. As the Figure 5.2 shows, the stress in the material decays with time. A power law model is suitable to describe this material phenomenologically: ϒ(t) = ct −n

(5.2)

The integral of Equation 5.1 using the derivative theorem and the convolution theorem, with s the transformation variable, can be transformed to: σ(s) = sϒ(s)ε(s)

(5.3)

It should be noted that the relaxation modulus is complex in the frequency domain. Taking the Laplace transform of ϒ(t) and recognising Γ, as the gamma function, defined as follows: Γ(x) =

Z

∞ 0

t x−1 e−t dt

(5.4)

the complex Young’s modulus then is expressed as: b = c Γ(1 − n)b ϒ(s) sn

(5.5)

The complex viscoelastic modulus can be written as a function of the angular frequency as follows: nπ b ϒ(ω) = c Γ(1 − n)ωn ei 2

(5.6)

The values of c and n can be obtained from data fitting of the relaxation test data using the power law (Equation 5.2). This is done by using a standard Nelder-Mead minimisation scheme, as implemented in Matlab 6.5 (The MathWorks, Natick MA, USA)

112


The following values of c and n are obtained: 2.3 · 106 [Pa s−n ] and 0.065 [−] respectively. The real part of the complex modulus, the storage modulus relates to the elastic behaviour of the material and defines the stiffness of the material. The imaginary part is the loss modulus and relates to the materials viscous behaviour and defines the energy dissipative ability of the material. This values result in a loss modulus less than 10% of the value of the storage modulus. Therefore for modelling purposes the tubes can be modelled as purely elastic. The Young’s modulus of Polyurethane can be obtained by Rutten (1998): 1 N ϒ= ∑ Re[ϒi(s)] N + 1 i=0

(5.7)

where N is the harmonic number corresponding to the bandwidth of the excitation signal. In our case it is 40. The Youngs modulus was calculated to be 1.72 MPa.

5.4 5.4.1

Measurement Methods Experimental set-up

A schematic diagram of the experimental set-up used to carry out wave propagation experiments in flexible vessels is shown in Figure 5.3. The apparatus consists of a tube marked as (F) in the schematic diagram, placed in horizontal position inside an open container (E) filled with water. The water depth above the tube prescribes the pressure outside the vessel. The tube is pre-strained axially to 3% in order keep it in a straight after it is filled with water. The tube is fixed on both sides and can expand freely in the radial direction along its length. On one side, the tube has a closed end and, on the other side, it is connected to a three way solenoid valve (B) operated by a PC. The valve is connected at one side to a closed tank (C) and at the other side to a two way manually operated valve (A). The closed tank (C) is maintained at a constant pressurise of about 1 bar. The two way valve (A) is connected to an open tank (D). The system is filled with water. When the solenoid valve is not engaged the water column level inside the open tank (D) prescribes the pressure inside the tube. A block shaped pulse can be initiated through the PC. The duration of the opening of the valve which initiates the pulse was set to be 0.05 s. It is essential that the duration of the opening of the valve is as short as possible because the wavelength of the waves should be as short as possible, to enable distinction between forward and backward travelling waves. Furthermore, as little as possible liquid should be injected into the tube, to keep the stationary pressure rise during the experiment as low as possible. The flow rate meter (Q) and the ultrasound probe (W) were held stationary by retort-stand and clip. The ultrasound scanner was positioned so that the ultrasound beam is sent perpendicular to the surface of the tube. Each one of

5.4. Measurement Methods

113

the pressure catheters (P1,P2) was introduced via a junction beyond the closed end of the bath in which the tube is fixed.

A: two way manual operated valve, B: three way solenoid valve operated by a PC, C: closed tank pressurised at about 1 bar, D: open tank, E: open container filled with water, F: tube, P1 and P2: pressure catheters, Q: volumetric flow rate meter, W: wall motion ultrasound scanner. Figure 5.3: Experimental set-up for wave propagation experiments (TU/e).

5.4.2

Instrumentation

Pressure and pressure gradient Two pressure-wire sensors (Radi Medical Systems 12000XT ) were used to measure the pressure simultaneously at two points along the tube, 17 mm apart. The pressure wires were of 0.36 mm diameter, typically used for clinical measurements. Each pressure wire was connected to a Radi Medical Systems interface box. The interfaces introduce a time-delay in the signal due to the internal processing. This time delay was determined using a real-time analogue pressure measurement with Beckton Dickinson pressure sensor (PZ10E) in combination with a Peekel CA253 bridge amplifier. The time-shift between the pressures as simultaneously measured by the Radi pressure-wire and the BD PZ10E was determined by cross-correlation

114


of the two signals and turned out to be 10 ms. The time-shift was accounted for in the subsequent data processing. Flow rate The fluid flow rate was measured using a perivascular flow rate sensor (type MC28AX, Transonic, the Netherlands), with an inside diameter of 28 mm and a bandwidth of 160 Hz. The sensor is suitable for measurements of vessels with 22-28 mm outer diameter allowing maximum distension of the tubes without them touching at the surface of the probe. The probe was connected to its interface box and the signal was passed to a PC. Wall motion The wall motion was measured using an ultrasound wall track system (Brands et al., 1999). This single-beam ultrasound system acquires the RF-echo signal at a pulse repetition Frequency of 1000 Hz and is stored in the computer memory during acquisition. After the measurement (typical duration 4 seconds) the RFmatrix is stored to the hard disk for further processing. The wall displacement data are extracted from the RF-matrix by cross-correlation using the filtering technique described in Brands et al. (1999). This yields a spatial resolution of 250 µm and a temporal resolution of 1/200 s.

5.4.3

Protocol

For each one of the six tubes four instantaneous time variables were measured: Pressure, pressure gradient, flow rate and wall distension. The measurements were taken at 10 positions (z) along the tube length, each 50 mm apart. The positioning of the flow probe and the ultrasound scanner was accommodated by a ruler. The two pressure wires were placed at the two edges of the flow probes width, in order to have flow rate and pressure gradient measured at the same location.The distance between the two wires was 17 mm. For every measurement this distance was kept constant by accurately positioning the two sensors using a stereo microscope. From the two pressure measurements, the pressure gradient was obtained. Before each measurement, the pressure wires were calibrated to zero against the hydrostatic pressure imposed to the tube by the open air tank. During the measurements, the two way valve (A) connected to the open air tank was kept closed in order to preserve the volume of the water induced in the tube by the opening of the solenoid valve. The signals for pressure, pressure gradient and flow rate for each measurement 1000 samples/s were taken using LabView software with National Instruments hardware. To avoid any loss of signal, all measurements were

5.5. Results

115

taken with no extra filtering. For the wall movement measurements, the RF-signal received by the echo scanner was obtained and stored for 2 s. At each position, the measurements were repeated 16 times in order to obtain the mean for each variable. The standard deviation at 16 measurements is at about the noise level of the pressure sensors, so more measurement would not have increased the accuracy.

5.4.4

Data processing

Pressure and pressure gradient The data processing of the digitised pressure measurements starts with phase shift correction between the two signals introduced by the different Radi Medical System pressure wire interfaces. The signals were fast-Fourier transformed and, by examining the signal spectrum, undesired noise peaks were identified and filtered without inducing any phase shift.

Wall distension Once the RF signal (reflected and scattered) has been recorded and transferred to the computer, the digitised RF signal as a function of depth is displayed on the computer screen. The tube lumen and wall interface can be identified by the shape of the signal. Indicator markers have to be manually placed on the reflections of the anterior and posterior tube walls to indicate the initial search area for the walldetection algorithm. The algorithm then tracks the position of the walls over time. This renders the wall positions and therefore the lumen diameter as a function of time.

5.5 5.5.1

Results Static pressure - initial diameter relation

In order to ensure that the the tubes undergo small deformations during the experiments, the pressure-initial diameter relatio was measured for the thin straight tube (Tube B) by applying different pressures inside the tube. The different levels of pressure were defined by the different water levels in the open tank. As can be seen from Figure 5.4, the tube’s behaviour is linear throughout the pressure range considered. The corresponding circumferential strains are less than than 3% and may therefore be considered small. The initial pressure inside the tube was set to 2.94 kPa.


116

4

2.56

x 10

2.54

2.52

Initial Diameter [um]

2.5

2.48

2.46

2.44

2.42

2.4

2.38

1

1.5

2

2.5 3 Static pressure [kPa]

3.5

4

Figure 5.4: Static pressure-initial diameter relation of the straight tube (Type B).

4.5

117

5.5. Results

5.5.2

Standard deviation of measurements

For each tube, the four instantaneous time variables measured at 10 locations along the tube length are shown in Sections 5.5.3 and 5.5.4. For each tube at every location, the mean of 16 measurements and the standard deviation from the mean were calculated to assess the reliability and repeatability of the results for each one of the measured variables. The standard deviation is calculated from: σ=

s

2 ∑ni=1 (xi − x) n−1

(5.8)

Tube type B axial location along its length: 100 [mm] 15 mean mean+std mean−std 10

5

0

500 Wall distention [um]

Pressure P [mmHg]

A typical result can be seen in Figure 5.5.

mean mean+std mean−std

400 300 200 100 0

−5

0

0.2

0.4 0.6 Time t [s]

0.8

1

−100

8

0.4 0.6 Time t [s]

0.8

1

mean mean+std mean−std

2 Pressure gradient

Flow rate Q [l/min]

0.2

3 mean mean+std mean−std

6 4 2 0 −2 −4

0

1 0 −1 −2

0

0.2

0.4 0.6 Time t [s]

0.8

1

−3

0

0.2

0.4 0.6 Time t [s]

0.8

1

Figure 5.5: A typical result at a location showing the mean of 16 measurements and the standard deviation from the mean.

5.5.3

Fluid motion

The three measurements related to the fluid motion were: the pressure, the flow rate and the pressure gradient. It should be mentioned that only the mean value of the 16 measurements at each location is presented. In the following sub-sections in


118

order to be able to draw comparative conclusions the results presented have been scaled in amplitude so as the initial pulse has the same amplitude in all tubes. The time is also scaled using the peak-to-peak value of the first reflection. Pressure The normalised mean pressure measurements for various axial positions along the length of the tube is plotted against scaled time. The three straight tubes can be seen in Figure 5.6 and the three tapered ones in Figure 5.7. In Figures 5.8, the pressure propagation for tubes A and F is compared and, in 5.9, the propagation for tubes C and E is compared. Pressure propagation

Axial position

C A B

0

0.05

0.1

0.15

0.2

0.25 0.3 Scaled time

0.35

0.4

0.45

0.5

Figure 5.6: Normalised pressure measurements every 50 mm along the length of the tube against scaled time for straight tubes: types A,B,C (A: straight tube with constant wall thickness of 0.1 mm; B: straight tube with constant wall thickness of 0.05 mm; C: straight tube with variable wall thickness of 0.05-0.1 mm).

Figure 5.6 shows that the straight tube with variable wall thickness has slightly higher amplitude than the other straight ones. The shape of the pulse in all three of them is similar.

119

5.5. Results

Pressure propagation

Axial position

F D E

0

0.05

0.1

0.15

0.2


0.35

0.4

0.45

0.5

Figure 5.7: Normalised pressure measurements every 50 mm along the length of the tube against time for tapered tubes: types D,E,F (D: tapered tube with constant wall thickness of 0.1 mm; E: tapered tube with constant wall thickness of 0.05 mm; F:tapered tube with variable wall thickness of 0.1-0.05 mm).


120


Axial position

A F

0

0.05

0.1

0.15

0.2


0.35

0.4

0.45

0.5

Figure 5.8: Normalised pressure measurements every 50 mm along the length of the tube against time for tube types A and F (A: straight tube with constant wall thickness of 0.1 mm; F: tapered tube with variable wall thickness of 0.1-0.05 mm).

121

5.5. Results


Axial position

C E

0

0.05

0.1

0.15

0.2


0.35

0.4

0.45

0.5

Figure 5.9: Normalised pressure measurements every 50 mm along the length of the tube against time for tubes types C and E (C: straight tube with variable wall thickness 0.05-0.1 mm; E: tapered tube with constant wall thickness of 0.05 mm).

122


From Figure 5.7 it can be seen that the pressure wave of the two tubes with constant wall thickness (D and E) matches closely in amplitude and shape. The tube with the variable wall thickness (F) has slightly lower amplitude at the distal end, than the other two. The shape of the pulse between the first and the second reflection at the distal end is slightly less steep. In Figure 5.8 Tube F has been manufactured with wall thickness variation, to have the same wave velocity according to the linear theory along its length as the straight with constant wall thickness (Tube A). Thus, the wall thickness variation was expected to counterbalance for the tubes tapering. However, it is evident in Figure 5.8 that the tapering increases the pressure amplitude towards the distal end while the wave velocity also increases. At the distal end between the two peak-topeak values it can be seen that the tapering affects the shape of the pulse as it rises faster, where as in the straight one it remains flat until the next reflection. In Figure 5.9, the straight Tube C was manufactured with wall thickness variation, to have the same wave velocity according to the linear theory as the tapered one with constant wall thickness (Type E). Thus, the wall thickness variation is expected to give a similar effect in the propagation as the tapering. However, this is not the case. It can be seen from the graph that the wall thickness variation cannot accommodate the non-linear effects introduced by the geometric tapering. The pressure amplitude is significantly higher due to the tapering and the shape of the pulse is again different at the distal end and the rise of the pulse between the first and the second peak. Thus, the results suggest that for the pressure wave the tapering effects are strong and cannot be counterbalanced with the wall thickness variation. The tapering leads to higher pressure amplitude and the shape of the pressure pulse is different due to the tapering. Flow rate The normalised mean flow rate measurements for the various axial positions along the length of the tube is plotted against scaled time. The results from the three straight tubes can be seen in Figure 5.10 and from the three tapered ones in Figure 5.11. In Figures 5.12, the measurements from tubes A and F and, in Figure 5.13, from tubes C and E are presented. From Figure 5.10, it can be seen that the straight tubes have about the same wave form shape and the same amplitude. The tube with the wall thickness variation has slightly lower amplitude. From Figure 5.11, it can be seen that the three tapered tubes have the same wave form shape. The amplitude of the flow rate for the tube with the variable wall thickness (Tube F), is slightly higher than the other two. In Figure 5.12, where the results from the tube pair A and F are compared, it is seen that the amplitude of the flow rate wave reduces. This is expected as it has

123

5.5. Results

Flow propagation

Axial position

C A B

0

0.05

0.1

0.15

0.2


0.35

0.4

0.45

0.5

Figure 5.10: Normalised flow rate measurements every 50 mm along the length of the tube against scaled time for straight tubes: types A, B, C (A: straight tube with constant wall thickness of 0.1 mm; B: straight tube with constant wall thickness of 0.05 mm; C: straight tube with variable wall thickness of 0.05-0.1 mm).


124

Flow propagation

Axial position

F D E

0

0.05

0.1

0.15

0.2


0.35

0.4

0.45

0.5

Figure 5.11: Normalised flow rate measurements every 50 mm along the length of the tube against scaled time for straight tubes: types D, E, F (D: tapered tube with constant wall thickness of 0.1 mm; E: tapered tube with constant wall thickness of 0.05 mm; F: tapered tube with variable wall thickness of 0.1-0.05 mm).

125

5.5. Results

Flow propagation

Axial position

A F

0

0.05

0.1

0.15

0.2


0.35

0.4

0.45

0.5

Figure 5.12: Normalised flow rate measurements every 50 mm along the length of the tube against time for tubes types A and F (A: straight tube with constant wall thickness of 0.1 mm; F: tapered tube with variable wall thickness of 0.1-0.05 mm).


126

Flow propagation

Axial position

C E

0

0.05

0.1

0.15

0.2


0.35

0.4

0.45

0.5

Figure 5.13: Normalised flow rate measurements every 50 mm along the length of the tube against time for tubes types C and E (C: straight tube with variable wall thickness 0.05-0.1 mm; E: tapered tube with constant wall thickness of 0.05 mm).

5.5. Results

127

been previously observed that the equivalent pressure signal had increased due to the tapering. The same observations apply for the tube pair C and E as shown in Figure 5.13. These observations indicate that the geometric tapering strongly decreases the amplitude of the wave signal. The wall thickness variation on the other hand does not play any significant role apart from slightly reducing the amplitude compared to the constant thickness tubes. Pressure Gradient The normalised mean pressure gradient measurements for the various axial positions along the length of the tube is plotted against scaled time. The result from the three straight tubes can be seen in Figure 5.14 and the three tapered ones in Figure 5.15. In Figure 5.16 the results from tubes A and F are shown and in Figure 5.17 the results from tubes C and E are shown. In Figure 5.14 the results from the straight tubes are compared. The shape of the pressure gradient at the entrance of the tube for the tube Type C (with wall thickness variation), is the same as the one Type B. Throughout the propagation towards the distal end their behaviour is quite close. In Figure 5.15, where all the tapered tubes are compared, it can be seen that the tubes with the same wall thickness at the entrance of the wave i.e. Type D and F have the same shape for the pressure gradient. From the graphs it is suggested that the shape of the pressure gradient at the entrance of of the wave depends on the wall thickness of the tube at the entrance. From Figures 5.16 and 5.17 it can be observed that the geometric tapering has an affect on the amplitude of the pressure gradient, in particular it increases towards the distal end. If the results of the two pairs of tubes (A and F versus C and E) are compared together, it is clear that for each pair the shape is comparable but the two pairs themselves have different shape. This leads to the conclusion that the shape of the pulse at the entrance depends on the thickness of the wall at the entrance of the tube. Thus, the wall thickness variation plays a role in the shape of the pressure gradient. The geometric tapering affects the amplitude of the signal and forces it to rise significantly compared to the straight one.

5.5.4

Wall motion

The normalised mean wall distension measurements for the various axial positions along the length of the tube is plotted against scaled time. The measurements for the three straight tubes are shown in Figure 5.18 and the three tapered ones in Figure 5.19. In Figures 5.20 and 5.21, the two pairs are presented.


128

Pressure gradient propagation

Axial position

C A B

0

0.05

0.1

0.15

0.2


0.35

0.4

0.45

0.5

Figure 5.14: Normalised pressure gradient measurements every 50 mm along the length of the tube against time for tubes types A, B, C (A: straight tube with constant wall thickness of 0.1 mm; B: straight tube with constant wall thickness of 0.05 mm; C: straight tube with variable wall thickness of 0.05-0.1 mm).

129

5.5. Results


Axial position

F D E

0

0.05

0.1

0.15

0.2


0.35

0.4

0.45

0.5

Figure 5.15: Normalised pressure gradient measurements every 50 mm along the length of the tube against time for tubes types D, E, F (D: tapered tube with constant wall thickness of 0.1 mm; E: tapered tube with constant wall thickness of 0.05 mm; F: tapered tube with variable wall thickness of 0.1-0.05 mm).


130


Axial position

A F

0

0.05

0.1

0.15

0.2


0.35

0.4

0.45

0.5

Figure 5.16: Normalised pressure gradient measurements every 50 mm along the length of the tube against time for tubes types A and F (A: straight tube with constant wall thickness of 0.1 mm; F: tapered tube with variable wall thickness of 0.1-0.05 mm).

131

5.5. Results


Axial position

C E

0

0.05

0.1

0.15

0.2


0.35

0.4

0.45

0.5

Figure 5.17: Normalised pressure gradient measurements every 50 mm along the length of the tube against time for tubes types C and E (C: straight tube with variable wall thickness 0.05-0.1 mm; E: tapered tube with constant wall thickness of 0.05 mm).


132

Wall distension propagation

Axial position

C A B

0

0.05

0.1

0.15

0.2


0.35

0.4

0.45

0.5

Figure 5.18: Normalised wall motion measurements every 50 mm along the length of the tube against time for tubes types A, B, C (A: straight tube with constant wall thickness of 0.1 mm; B: straight tube with constant wall thickness of 0.05 mm; C: straight tube with variable wall thickness of 0.05-0.1 mm).

133

5.5. Results


Axial position

F D E

0

0.05

0.1

0.15

0.2


0.35

0.4

0.45

0.5

Figure 5.19: Normalised wall motion measurements every 50 mm along the length of the tube against time for tubes types D, E, F (D: tapered tube with constant wall thickness of 0.1 mm; E: tapered tube with constant wall thickness of 0.05 mm; F: tapered tube with variable wall thickness of 0.1-0.05 mm).


134


Axial position

A F

0

0.05

0.1

0.15

0.2


0.35

0.4

0.45

0.5

Figure 5.20: Normalised wall motion measurements every 50 mm along the length of the tube against time for tubes types A and F (A: straight tube with constant wall thickness of 0.1 mm; F: tapered tube with variable wall thickness of 0.1-0.05 mm).

135

5.5. Results


Axial position

C E

0

0.05

0.1

0.15

0.2


0.35

0.4

0.45

0.5

Figure 5.21: Normalised wall motion measurements every 50 mm along the length of the tube against time for tubes types C and E (C: straight tube with variable wall thickness 0.05-0.1 mm; E: tapered tube with constant wall thickness of 0.05 mm).

136


From Figure 5.18, it can be seen that the wall thickness variation significantly reduces the amplitude of the wall distension as the tube gets thicker towards its distal end. Comparing tube Type F with D and E, in Figure 5.19, it is indicated that the wall thickness variation affects again the amplitude of the wave. The more the tube becomes thinner towards the distal end the more the wall distension increases. In Figures 5.20 and 5.21, the two pairs are compared. It is evident that the geometric tapering decreases the amplitude of the wall motion towards the distal end. Thus, the wall thickness variation can reduce the amplitude of the wave. This effect is expected from linear theory. The geometric tapering reduces the amplitude of the wall distension signal, which is expected from linear theory (Laplace’s law).

5.6

Closure

From the comparison of the experimental data, it is concluded that, for the pressure wave, the tapering effects are strong and cannot be counterbalanced with the wall thickness variation. The tapering leads to higher pressure amplitude and the shape of the pressure pulse is different due to the tapering. This is in agreement with the findings of Belardinelli and Cavalcanti (1992), who studied the effect of tapering in wave propagation using a two-dimensional non-linear theory. The geometric tapering strongly decreases the amplitude of the flow wave which is expected to have the opposite effect to the pressure. The wall thickness variation does not play any significant role in the flow rate apart from slightly reducing the amplitude of the signal. The shape of the pressure gradient at the entrance of the wave depends on the wall thickness of the tube at the entrance. The geometric tapering affects the amplitude of the gradient and makes it rise compared to the straight one. The wall thickness variation can reduce the amplitude of the signal, but this effect is expected from the linear theory. The geometric tapering reduces the amplitude of the wall distension signal, which is expected from linear theory. The fact the the shape of the pressure pulse is changed by the geometric tapering effect is a very important observation in cardiovascular research as pressure is often used as a tool for diagnosis. Thus, for the correct evaluation of the pressure in the aorta the geometric tapering has to be taken into account in the computational models. This directly implies that non-linear theory needs to be incorporated in modelling the aorta. The wall thickness variation on the other hand, does not have any significant effect apparent from slightly increasing the pressure amplitude and this can be corrected by using the correct wave velocity using linear methods.

Chapter 6 Comparison of experimental results with linear wave propagation methods 6.1

Introduction

In Chapter 5, a complete experimental data set was presented on the role of geometric tapering and wall thickness variation in flexible vessels. Some of the tubes measured were designed in pairs according to linear theory in order to have the same wave propagation velocity. It was concluded that the linear theory cannot predict the amplitude and the shape of the pulse that alters due the constant reflections from the tapered wall. In this chapter, the quality of the measurements only for the straight tubes is tested by comparing them with the linear wave theory. Two cases are examined; one with purely elastic wall and the second with viscoelastic wall. It is expected that there will be good agreement between the measurements and the predictions using the linear theory. In the first part of this chapter, the main points of the linear theory are outlined and, in the second part, comparisons between experiments and predictions using purely elastic or viscoelastic material are presented. The chapter concludes with a discussion of the main findings from the comparisons.

6.2

Linear Theory of Wave Propagation in Flexible Vessels

The theoretical investigation of the propagation of pressure disturbances in distensible tubes containing an inviscid fluid has been performed by many researchers (Section 1.4). The basic theory for a circular uniform flexible tube filled with a viscous fluid is often referred to as “Womersley theory” (Womersley, 1957). This 137

138

Chapter 6. Comparison of experimental results with linear wave propagation methods

P’ ξ = ξ(z,t) P ζ = ζ(z,t)

h r

r z θ

Figure 6.1: Tube motion variables. Point P(z, r) on the surface of the wall at rest displaces to position P’(z + ζ, r + ξ) .

theory can be found in many text books such as McDonald (1968); Pedley (1980); Lighthill (1978) and is outlined here.

6.2.1

Basic theory

The momentum equation (Equation 6.1) and continuity equation (Equation 6.2) for incompressible Newtonian fluids in a uniform elastic tube of finite length under the assumption that the flow is axi-symmetric, can be solved in the frequency domain after linearisation. The equations are: ρ

∂U + ρ∇ • (UU) = ∇ • (η∇U) − ∇p ∂t ∇•U = 0

(6.1)

(6.2)

The wave length λ (λ = 2π ωc ) of the disturbance of interest is assumed to be long

compared to the diameter 2ro of the tube ( 2rλ0 ≫ 1 ). It is convenient to make the ′ ′ the Navier-Stokes equations non-dimensional. Therefore Uz and Ur are considered as typical velocities in the axial (z) and radial (r) directions respectively. The ratio ′

between the two velocities is defined by κ = Ur′ , and in the following treatment κ is Uz ′ considered to be small and L/2r0 ≫ 1. Provided that the Mach numbers Ucr ≪ 1 ′ U and cz ≪ 1, the convective terms plus all velocity derivatives in the z direction in Equation 6.1 can be neglected (Barnard et al., 1966; Reuderink et al., 1993).

The non-dimensional form of the Navier-Stokes equations in cylindrical coordi-

nates under axisymmetric conditions (θ direction neglected) can therefore be reduced to:

6.2. Linear Theory of Wave Propagation in Flexible Vessels

139

2 ∂ Ur 1 ∂Ur Ur ∂Ur ∂p + =η + − 2 ρ ∂t ∂r ∂r2 r ∂r r

(6.3)

2 ∂Uz ∂p ∂ Uz 1 ∂Uz ρ + =η + ∂t ∂z ∂r2 r ∂r

(6.4)

1 ∂(rUr ) ∂Uz + =0 r ∂r ∂z

(6.5)

where η is the kinematic viscosity. In order to be able to integrate over a tube cross section, appropriate boundary conditions must be specified. At the wall r = D/2 = r0 , the no-slip and no-leak conditions apply. It is assumed that there is no axial movement, a hypothesis which also has an in-vivo relevance in blood flow (Pedley, 1980). Thus, Uz |r=r0 =

∂ζ =0 ∂t

(6.6)

Symmetry requires, ∂Uz =0 Ur = 0, ∂r r=0

(6.7)

In linear theory, it can be assumed that the wave solution can be expressed as a combination of harmonics with angular frequency ω and a wave number k .Therefore, the wave solutions of ϕ which can be p, Ur, Uz is of the form bei(ωt−kz) ϕ=ϕ

(6.8)

In an elastic tube, the propagation constants are functions of the non-dimensional frequency only. This non-dimensional frequency is called the Womersley number α (Equation 6.9) which is also known as Stokes number. It is defined as the ratio of inertia forces and the viscous forces. α=r

r

ω η

(6.9)

The combination of the Navier Stokes equations and the equation of motion for the solid including its constitutive equation give a dispersion equation otherwise called a frequency equation. The solution of the frequency equation determines the wave number or propagation coefficient k as a function of the mechanical and geometrical properties of the tube, the density, the viscosity of the fluid and the Womersley number α. ω k(ω) = ± c0

r

1 1 − F10

(6.10)

where c0 is the wave speed given by Equation 6.18 and F10 is a function of the

140


Womersley parameter α and the Bessel functions of the first kind of order 1 and 0, J1 and J0 . It is given by: F10 =

2J1 (αi3/2 ) αi3/2 J0 (αi3/2 )

(6.11)

The two roots given by Equation 6.10 are complex numbers and therefore the propagation coefficient k can be expressed as k = ℜ(k) + iℑ(k), where the root is chosen such that ℜ(k) > 0. The real part ℜ(k) is the damping coefficient and the imaginary part ℑ(k) the phase coefficient. Using this expression for the wave number the wave speed and the attenuation constant can be defined as follows. The wave speed can be expressed as a function of the classical Moens-Korteweg wave speed and the real part of the propagation coefficient as: c0 ℜ(k)

(6.12)

−2πℑ(k) ℜ(k)

(6.13)

c= The attenuation constant is given by: γ=

It should be mentioned here for clarity that the solution of the dispersion equation can be expressed in the general form bI± ei(ωt±kI z) + ϕ bII± ei(ωt±kII z) ϕ=ϕ

(6.14)

where the two complex roots give the velocity of propagation of two distinct outgoing waves. In the original publication of Womersley, only one root was mentioned even though both were predicted (Atabek and Lew, 1966). The two waves are: a pulse wave in which the wall motion are principally radial, denoted as I, which can be found in the literature under various names such as: pressure wave, radial wave, Young wave; and another were the wall motion is principally longitudinal, denoted as II, found in the literature under the names: shear wave, secondary wave, wave of distortion. The propagation coefficients are kI and kII respectively and were predicted by Womersley in the particular case where the wave length is long compared to the tube diameter. The pressure wave propagates slower than the shear wave. In tethered tubes the faster waves are completely attenuated (Atabek, 1968).

6.2.2

Wave propagation speeds

When a disturbance occurs in a fluid-filled tube, it will propagate as a wave. The wave speed in the fluid is given by the Korteweg Equation (Korteweg, 1878).

141

6.2. Linear Theory of Wave Propagation in Flexible Vessels ψ for thin-wall tube

Type of tube anchoring

1 − ν2 1 − ν2

At its upstream only Throughout against axial movement

1

With expansion joints throughout

ψ for thick-wall tube t r

0 (1 + ν) + 2r2r0 +h 1 + ν2 2r0 (1+ν2 ) h (1 + ν) + r0 2r0 +h 2r0 h (1 + ν) + r0 2r0 +h

Table 6.1: Values of coefficient ψ describing different longitudinal support conditions for thin- and thick-wall tubes.

c=

s

K ρf

2r0 K −1 1+ψ h ϒ

(6.15)

In Equation 6.15, ψ is a coefficient that accounts for different longitudinal support conditions for thin-walled and thick-walled tubes. The thin wall assumption holds h h 1 wall thickness when inner diameter ≡ 2r0 ≪ 1 (say 2r0 < 20 ). The value of these coefficients can be

found in Table 6.1 (Wylie and Streeter, 1993). In deriving the values of ψ, tube wall inertia is neglected. The constant ν is the Poisson ratio if the wall material is isotropic, while for general materials it is a coefficient relating the circumferential and the tensile stress (Lighthill, 1978). Equation 6.15 can be written as 1 1 1 = 2+ 2 2 cf c0 c1

(6.16)

where

c1 =

s

K ρf

(6.17)

ϒh 2r0 ρ f

(6.18)

and

c0 =

s

ψ−1

In the case when the tube wall is very stiff, ϒ ≫ K and c f = c1 . This gives the speed of sound in an unconfined liquid, e.g. 1480 m/s for water at room temperature. When the tube wall is very flexible, K ≫ ϒ and c f = c0 . Examples of such cases are waves in rubber hoses and human arteries with typical speeds of about 5-10 m/s. The linear theory leading to c0 (Equation 6.18) for ψ = 1 was first performed by Young (1808), but is more widely known as Moens-Korteweg wave speed after two Dutch scientists who rediscovered it in 1878 (Moens, 1878; Korteweg, 1878) .

142


r

transmitted (part 1) z

j

incomming wave reflected wave n

m

k

transmitted (part 2)

L

L+1

L+2

Figure 6.2: Discrete transitions between segments.

6.2.3

Wave reflections through discrete transitions

In Section 6.2.1, the basics of wave propagation are described for the case where the wave is transmitted in a cylindrical, infinitely long tube filled with a fluid with uniform properties. In practice, however, a tube has a finite length with two ends. This section is concerned with what happens when the wave travels through a sudden or gradual change of properties (transitions). Such transitions can be geometric tapering, wall thickness variation, bifurcations, different fluid, closed end etc. These changes in linear theory as generally modelled as a sequence of transition line model segments (Streeter and Wylie, 1979; Pedley, 1980; Lighthill, 1978). In Figure 6.2, a general example is used to demonstrate the principle of transitions through junctions. When the transition between segments is small compared to the wavelength, there is a discrete transition. A transition between segments at junction n is considered as illustrated in Figure 6.2. A traveling wave named incoming or incident wave approaches along in the tube segment [m, n]. One part of the incoming wave i.e. of pressure or flow rate (pI , Q˙ I ) is reflected by the junction (pR , Q˙ R ) and another part goes through and it is partially transmitted in part 1 (pT1 , Q˙ T1 ) and part 2 (pT2 , Q˙ T2 ). The transition of a wave through a junction depends on the cross-sectional area A, the density ρ and the wave speed c of the two sections. This relationship is expressed by the characteristic impedance of the tube which is defined as: Zc =

ρc A

At any junction, two conditions hold: the pressure is a single valued function and the flow must be continuous. The relationship between the flow and the pressure is

6.2. Linear Theory of Wave Propagation in Flexible Vessels

143

given by A Q˙ = ± p ρc

(6.19)

At transition n where z = L holds that: pIm (ω, L,t) + pRm (ω, L,t) = pTm1 (ω, L,t) = pTm2 (ω, L,t) Q˙ Imn (ω, L,t) − Q˙ Rmn(ω, L,t) = Q˙ Tn1j (ω, L,t) + Q˙ Tnk2 (ω, L,t)

(6.20)

The reflection coefficient Γ is the ratio of the amplitude of the reflected and the incoming wave at the transition between two segments. For the nodal point m at level L, it can be defined as: Γm (ω) =

pbRm (ω, L) pbIm (ω, L)

(6.21)

In the case concerned in this chapter the transition as defined previously is a closed end. When the wave hits the closed end, it is completely reflected. As the tube is closed at both ends, the condition Γm = Γn = 1 holds and for brevity, it is denoted simply as Γ. For the ingoing pressure wave pI = pbI (ω, 0)ei(ω,z,t) in the segment with a transition at z = L, the pressure wave in that segment at any location z < L is equal to : p(ω, z,t) = pI (ω, z,t) + pR(ω, z,t)

The flow rate is similar

= pbI (ω, 0)e−ikz[1 + Γe−2ik(L−z) ]eiωt

˙ Q(ω, z,t) = Q˙ I (ω, z,t) − Q˙ R(ω, z,t) b˙ I (ω, 0)e−ikz [1 − Γe−2ik(L−z) ]eiωt =Q

(6.22)

(6.23)

The input impedance of the system is given by: ZI =

1 + Γe−2ik(L−z) pbI (ω, 0)e−ikz [1 + Γe−2ik(L−z) ]eiωt = Zc b˙ 1 − Γe−2ik(L−z) Q(ω, 0)e−ikz[1 − Γe−2ik(L−z) ]eiωt

(6.24)

As the tube is closed at the two ends, multiple reflections will occur. When N multiple reflections are taken into account, the pressure and the flow rate can be determined recursively as I

−ikz

p(ω, z,t) = pb (ω, 0)e

N

λ

∏ [1 + Γe(−1) 2ik(L−z) ]eiωt

(6.25)

λ=1 N

λ b˙ I (ω, 0)e−ikz ˙ Q(ω, z,t) = Q ∏ [1 − Γe(−1) 2ik(L−z) ]eiωt

λ=1

(6.26)

144


Properties ρf ηf ν c n Ys bs ϒ

Units kg/m3 N · s/m2 Pa S−n MPa MPa

Value 998 1e-3 0.5 1.3*10e6 0.065 1.72 nπ cΓ(1 − n)ωn ei 2

Figure 6.3: Properties used for the calculations.

6.3

Implementation of the continuous linear model

The properties used for the simulation are shown in Table 6.3. For tubes A (with a diameter of 25 mm, wall thickness of 0.1 mm and length of 446 mm) and B (with diameter of 25 mm, wall thickness of 0.05 mm and length of 446 mm) the first 8 ms of the pressure signal at the first location of the measurement (nearest to the valve) were used as the incoming wave for the simulation. During the performance of the experiments, the tubes were pre-strained by 3%. This elongation of the tube was taken into consideration for the calculations. The signal was decomposed into harmonics by a standard fast-Fourier-transform. Calculations were performed using the Youngs modulus for the elastic material and the frequency dependent complex modulus for the viscoelastic material. These data were obtained from the relaxation tests discussed in Section 5.3. In order to simulate the closed wall, the reflection coefficient Γ = 1. The wave that hits the wall gets fully reflected. For the modelling of the wall distensions, the principle used for the pressure was also used in order to compute its harmonic components. The pressure, the flow and the wall distension ware calculated at 10 locations along the length the tube, 50 mm apart. The two pressures p0 and p1 were 17 mm apart from each other at every location along the length of the tube, with the pressure denoted as p0 leading.

6.4

Comparisons with Linear Model for Elastic Material

The two straight tubes compared with the linear theory were tube A with a diameter of 25 mm, wall thickness of 0.1 mm and length of 446 mm and tube B with diameter of 25 mm, wall thickness of 0.05 mm and length of 446 mm. The pressure, the flow and the wall distension ware calculated at 10 locations along the length the tube 50 mm apart. The two pressures p0 and p1 where 17 mm apart from each other at

6.5. Comparisons with Linear Model for Viscoelastic Material

145

every location along the length of the tube they, with pressure denote as p0 leading. From the Figures 6.4, 6.5 and 6.6, it can be seen that the predicted velocities of propagation of the travelling wave are in close agreement between the experimental data and the linear model. The peak-to-peak values of the reflected wave are occurring at about the same time. However, the amplitude of the reflected wave is highly overestimated as there is only damping from the liquid in the elastic model.

6.5

Comparisons with Linear Model for Viscoelastic Material

The simulations for tubes A and B where repeated including the viscoelastic properties of the material. For the same tubes as in Section 6.4, the analytical simulations were performed and compared with the experimental measurements (Figure 6.10, 6.11, 6.12, 6.14 and 6.15). From the graphs in this section, it is suggested that the results from the experimental measurements and the linear theory are in good agreement. Thus, the wave propagation in the straight tube with constant wall thickness can be well predicted from the linear theory when the viscoelasticity of the material is included. The damping of the reflected waves computed from the linear theory is slightly underestimated. This is related to the accuracy with which the material properties of the wall are known. All figures suggest that the thinner the wall, the better the matching with the theory is. The best agreement between experimental and computational data is obtained for the pressure measurements. In Figures 6.12 and 6.15, the comparison of the experimental wave and the calculated wave at the entrance of the tube suggests that the experimental measurement is shifted. After the first peak, the wall distension value does not return to a small value but it remains at about a value closer to the first peak. This is due to incorrect data acquisition related to the sensitivity of the measurement close to the entrance point. In a closer examination, it is seen that peak-to-peak time of the reflected pulse and its shape is at the same location as the one predicted from the linear theory. Overall the figures suggest that the wall distension behaves according to the linear theory.

6.6

Closure

In this chapter, the classic wave propagation theory has been used for the analytical simulation of wave propagation characterisation of the straight tubes with constant diameter and constant wall thickness. The simulations have been performed for two cases depending on the wall treatment: modelling the material as elastic or as

Pressure P0 (Type A)

Axial position

Theoretical Experimental

0

0.05

0.1

0.15

0.2

0.25 Time [s]

0.3

0.35

0.4

0.45

0.5

(a) Pressure P1 (Type A) Theoretical Experimental

Axial position

146


0

0.05

0.1

0.15

0.2

0.25 Time [s]

0.3

0.35

0.4

0.45

0.5

(b)

Figure 6.4: Comparison of pressure experimental measurements of the straight tube with constant wall thickness of 0.1 mm with linear analytical model foran elastic tube.

147

6.6. Closure

Flow Q (Type A)

Axial position


0

0.05

0.1

0.15

0.2 0.25 Time [s]

0.3

0.35

0.4

0.45

Figure 6.5: Comparison of the experimental measurements of the flow on a straight tube with constant wall thickness of 0.1 mm with linear analytical model foran elastic tube.


148

Wall distention (Type A)

Axial position


0

0.05

0.1

0.15

0.2

0.25 Time [s]

0.3

0.35

0.4

0.45

0.5

Figure 6.6: Comparison of the experimental measurements of the wall distension on a straight tube with constant wall thickness of 0.1 mm with linear analytical model foran elastic tube.

149

6.6. Closure

Pressure P0 (Type B)

Axial position


0

0.05

0.1

0.15

0.2

0.25 Time [s]

0.3

0.35

0.4

0.45

0.5

(a) Pressure P1 (Type B)

Axial position


0

0.05

0.1

0.15

0.2

0.25 Time [s]

0.3

0.35

0.4

0.45

0.5

(b)

Figure 6.7: Comparison of pressure experimental measurements of the straight tube with constant wall thickness of 0.05 mm with linear analytical model foran elastic tube.


150

Flow Q (Type B)

Axial position


0

0.05

0.1

0.15

0.2

0.25 Time [s]

0.3

0.35

0.4

0.45

0.5

Figure 6.8: Comparison of the experimental measurements of the flow on a straight tube with constant wall thickness of 0.05 mm with linear analytical model foran elastic tube.

151

6.6. Closure

Wall distention (Type B)

Axial position


0

0.05

0.1

0.15

0.2

0.25 Time [s]

0.3

0.35

0.4

0.45

0.5

Figure 6.9: Comparison of the experimental measurements of the wall distension on a straight tube with constant wall thickness of 0.05 mm with linear analytical model foran elastic tube.

Pressure P0 (Type A)

Axial position


0

0.05

0.1

0.15

0.2

0.25 Time [s]

0.3

0.35

0.4

0.45

0.5

(a) Pressure P1 (Type A) Theoretical Experimental

Axial position

152


0

0.05

0.1

0.15

0.2

0.25 Time [s]

0.3

0.35

0.4

0.45

0.5

(b)

Figure 6.10: Comparison of the experimental measurements of the pressure on a straight tube with constant wall thickness of 0.1 mm with linear analytical model fora viscoelastic tube.

153

6.6. Closure

Flow Q (Type A)

Axial position


0

0.05

0.1

0.15

0.2 0.25 Time [s]

0.3

0.35

0.4

0.45

Figure 6.11: Comparison of the experimental measurements of the flow on a straight tube with constant wall thickness of 0.1 mm with linear analytical model fora viscoelastic tube.


154

Wall distention (Type A)

Axial position


0

0.05

0.1

0.15

0.2

0.25 Time [s]

0.3

0.35

0.4

0.45

0.5

Figure 6.12: Comparison of the experimental measurements of the wall distension on a straight tube with constant wall thickness of 0.1 mm with linear analytical model fora viscoelastic tube.

155

6.6. Closure Pressure P0 (Type B)

Axial position


0

0.05

0.1

0.15

0.2

0.25 Time [s]

0.3

0.35

0.4

0.45

0.5

(a) Pressure P1 (Type B)

Axial position


0

0.05

0.1

0.15

0.2

0.25 Time [s]

0.3

0.35

0.4

0.45

0.5

(b)

Figure 6.13: Comparison of the experimental measurements of the pressure on a straight tube with constant wall thickness of 0.05 mm with linear analytical model fora viscoelastic tube.


156

Flow Q (Type B)

Axial position


0

0.05

0.1

0.15

0.2

0.25 Time [s]

0.3

0.35

0.4

0.45

0.5

Figure 6.14: Comparison of the experimental measurements of the flow on a straight tube with constant wall thickness of 0.05 mm with linear analytical model fora viscoelastic tube.

157

6.6. Closure

Wall distention (Type B)

Axial position


0

0.05

0.1

0.15

0.2

0.25 Time [s]

0.3

0.35

0.4

0.45

0.5

Figure 6.15: Comparison of the experimental measurements of the wall distension on a straight tube with constant wall thickness of 0.1 mm with linear analytical model fora viscoelastic tube.

158


viscoelastic. The experimental waveforms for pressure, flow and wall distension have been compared with the theory. The pressure gradient has not been investigated as it is expected to be in good agreement if the pressure comparisons are good. The linear theory without viscoelasticity included cannot predict the pulse behaviour. The wave speed is well predicted by the linear theory as it can be seen from the peak-to-peak comparisons of the experimental and computational data graphs. However, the shape of the pulse after the first reflection at the distal end, is very different. The amplitude of the pulse is highly overestimated as there is damping only from the liquid. Viscoelasticity is the most important parameter for the attenuation of the waves. When the viscoelasticity of the material is included in the linear model, the matching of the experimental measurements and the theoretical predictions is very good. Therefore, we can conclude that that the tube is behaving in accordance to the linear theory and the quality of the measurements is good. As a result it can be verified that the findings of the previous chapter are valid.

Chapter 7 Conclusions 7.1

Overview

This thesis is concerned with the study of fluid-stucture interaction in flexible tubes both from the modelling as well as the experimental point of view. More specifically, it presents the first stage of development and testing of a novel unified solution method suitable for fluid-structure interaction problems. In Chapter 2, the mathematical description of the single solution method, was derived. Fluid and solid were treated at as a continuum with different constitutive equations for the stress tensor. The constitutive equation for a linear viscous Newtonian fluid has as primitivevariables the velocity and the pressure, whereas for a linear elastic solid the displacement. In order to develop a unified formulation, the equation for the solid was altered in order to have velocity and pressure as primitive variables. Thus, a single set of equations was obtained, with primitive variables velocity and pressure. In the unified expression, the states of fluid and solid were distinguished by different coefficients in the same equation. Thus, the fluid-solid interface in the solution domain is internal and does not need special conditions for the exchange of information between the two media, as it is inherently implicit. In Chapter 3, the numerical method used for the solution of the equations was described. Issues involving the convergence rate were addressed and taken into account in the solution of the equations. The PISO velocity-pressure coupling algorithm used was described. However, instead of solving for pressure corrections, it is used to solve directly for pressure. This algorithm was used and tested for the first time in solid dynamics. An extensive boundary condition investigation was presented that led to the derivation of the optimal boundary conditions in a fully implicit velocitypressure solution of the Navier Stokes equations for a compressible material. A one dimensional stability analysis was performed on the finite difference approximations used for the displacement formulation and for the new velocity-based formulation. In Chapter 4, the solution method for the reformulated equations for solids was validated against the structural dynamic problem of beam bending, a case which 159

160

Chapter 7. Conclusions

incluses not only normal but also shear stresses. The results were compared against the standard displacement formulation as well as with analytical solutions. For this particular problem the analytical solutions have been obtained using simplifying assumptions and this has to be taken into account when comparing analytical with the numerical results. The conservation of the total energy of the numerical implementation was also tested. The numerical accuracy of the standard displacement formulation was compared against the velocity based formulation. The dissipation characteristics of the numerical integration technique were also in agreement with the conclusions obtained from the one dimensional stability analysis. Different discretisation schemes were compared and the effect of the mesh resolution was investigated. For the fully implicit velocity-pressure coupling, the successful use of the optimal boundary conditions was illustrated. The optimal boundary conditions were obtained for velocity by applying force balance at the free boundary and for the pressure by projection of the momentum equation on the unit vector normal to the boundary and solving for the normal pressure gradient. The novel solution method was also validated for other beam bending cases with different dimensions. In Chapter 5, experimental measurements in flexible vessels were presented that can be used for the detailed validation of the unified approach. Six tubes were manufactured: three straight and three tapered ones. One straight tube had wall thickness variation such that the wave speed according to the linear theory would be the same as in one of the tapered tubes. One tapered tube had variable wall thickness such that the wave speed according to the linear theory would be the same as in one of the straight tubes. The material properties of the tubes were measured by relaxation tests. In the experiments performed, a pressure wave was initiated by the opening of a valve. Pressure, pressure gradient, flow rate and wall distension were measured simultaneously. The results from the different types of tubes were compared against each other and the importance of the geometric tapering and wall thickness variationin wave propagation were assessed. Finally, in Chapter 6, the experimental data measured in the straight tubes with constant wall thickness was compared with the one dimensional linear wave propagation theory. Calculations for both elastic and viscoelastic material were performed. The pressure, volumetric flow rate and wall distension propagation where well reproduced by the linear theory when the viscoelastic properties of the wall were taken into account.

7.2. Main achievements

7.2

161

Main achievements

This research has resulted in the following specific contributions: • The idea of a unified solution methodology for solving fluid-structure interaction problems is presented. It is believed that the idea is general and can be used to handle interaction between other continua which are described by different constitutive equations. The interaction problems can be described by a single set of equations in a single grid and, in this way the interface is internal to the domain. • The mathematical framework of the new method for fluid-structure interaction problems has been developed. The method is fully implicit, three dimensional and without simplifying assumptions. It is suitable for modelling a variety of FSI applications such as pulse wave propagation in flexible tubes, container impact etc. • A numerical solution method for the discretisation and solution of the reformulated equations was also developed. The method is fully compatible with the one currently used widely for the solution of the equations for fluids. More specifically, for discretisation the finite volume approach is used and for pressure-velocity coupling the PISO algorithm. It is the first time that this algorithm is used for solving structural dynamic problems. The fact that the reformulated equations have the same unknown variables as the ones for fluids while the same numerical method can be used for their solution, greatly facilitates the implementation of a unified methodology for coupled FSI problems. • Appropriate boundary conditions for the pressure equation were found for the free boundary for compressible materials when a fully implicit velocity-pressure method is used. • The reformulated equations and solution method were successfully tested for a structural dynamic problem (beam bending) that comprises both normal as well as shear stresses. • A complete experimental data set was produced that can be used for the next step of the testing of the unified FSI approach. More specifically, the experimental work focused on the effect of geometric tapering and wall thickness variation on pulse wave propagation in flexible vessels. The experimental measurements indicate that the tapering leads to higher pressure amplitude and alters the shape of the pressure pulse. • The classic wave propagation theory was used to simulate the aforementioned experimental measurements for straight tubes with constant diameter and wall

162


thickness. It was found that when the viscoelastic properties of the wall material are included the predictions match well the experiments.

7.3

Future work

This thesis has covered a wide range of aspects in the area of fluid structure interaction such as mathematical modelling and experiments. Therefore, the recommendations for future work will be split into these categories and will be discussed separately.

7.3.1

Mathematical modelling

Modelling an elastic material is quite restrictive as the materials in nature exhibit viscoelastic properties. The mathematical model developed can be extended to include a simple viscoelastic model for the solid. Large deformations are also very common in engineering practice, thus a arbitrary-Lagrangian-Eulerian large strain formulation for the continuum can be introduced to take account for the large deformations typically occurring in the vessel walls. The mathematical model presented here is quite general without approximations. Thus it can be used for a number of disciplines. The important issue is that there should be a genuine interaction between the fluid and the solid. Otherwise separate solution methods would be more suitable. The numerical model has been successfully validated against analytical solutions to dynamic structural problems. The next step is to validate it for wave propagation in tubes against the one dimensional linear wave theory presented in Chapter 6 of this thesis. Validation of the new methodology on wave propagation in a flexible tube was performed during the duration of this project and the results were very promising. However, due to lack of time they were not investigated deeply enough to be presented in this thesis. However, it will be described here how the unified solution method can solve a fluid-structure interaction problem. The single set of equations (momentum and continuity) are describing the fluid-structure continuum are Equations 7.1 and 7.2. ∂ρ + ∇ • (ρU) = 0 ∂t

(7.1)

∂ρU + ∇ • (ρUU) = 2αdev(sym(∇U)) + φ∇ • dev Σ+ − ∇p ∂t

(7.2)

and

The state of the continuum is distinguished by different values the density ρ, the constant α and phase constant φ (Equations 7.37.47.5).

163

7.3. Future work ρ = ρs α = µ ∆t2 φ=1 ρ = ρs α = µ ∆t2 φ=1

ρ = ρs α = µ ∆t2 φ=1 ρ = ρs α = µ ∆t2 φ=1

ρ = ρs α = µ ∆t2 φ=1 ρ = ρs α = µ ∆t2 φ=1

ρ = ρs α = µ ∆t2 φ=1 ρ = ρs α = µ ∆t2 φ=1

ρ = ρs α = µ ∆t2 φ=1 ρ = ρs α = µ ∆t2 φ=1

ρ = ρf α=η φ=0

ρ = ρf α=η φ=0

ρ = ρf α=η φ=0

ρ = ρf α=η φ=0

ρ = ρf α=η φ=0

ρ = ρf α=η φ=0

ρ = ρf α=η φ=0

ρ = ρf α=η φ=0

ρ = ρf α=η φ=0

ρ = ρf α=η φ=0

ρ = ρf α=η φ=0

ρ = ρf α=η φ=0

ρ = ρf α=η φ=0

ρ = ρf α=η φ=0

ρ = ρf α=η φ=0

NOTE: The coloured cells are in a solid state and the white cells are in a fluid state. Figure 7.1: The different properties distribution in the single mesh for solving fluid structure interaction problems with the unified solution method.

ρ=

(

ρs f or solid ρ f f or f luid

α=

(

µ ∆t2 f or solid

φ=

(

η f or f luid 1 f or solid

0 f or f luid

(7.3)

(7.4)

(7.5)

In Figure 7.1 the schematic of how this is applied in a single grid is seen. There is no exchange of information at the phase boundary between the two continua; the interface is inherently implicit. It would be very interesting to compare the unified solution method for solving fluid structure interaction problems against monolithic methods to see if there is reduction of computational time as it is expected. The solution accuracy comparison of the two method will give a good guide for the future development. This new way of solving solids in the same way as fluids can also be investigated further by comparisons with standard stress analysis codes that use accurate schemes of the discretisation. An error analysis together with a computational time comparison could be performed to see if there are any benefits in using a velocitypressure formulation over a displacement formulation for solving solids outside the fluid-structure interaction context. When viscoelasticity is included in the mathematical model, the models can be

164


validated both against the one dimensional theory with viscoelasticity included and against the experimental data with real pressure boundary conditions at the entrance taken from the measurements as presented in Chapter 6.

7.3.2

Experimental work

From the work presented Chapter 5 it was suggested that geometric tapering is of great importance, as the constant reflections from the tapered wall change the shape of the propagating wave, which cannot be predicted by the linear theory. Comparison of this data with non linear theory would be of great interest. The experimental measurements presented in Chapter 5 were obtained for small deformations. The next step would be to repeat the same measurements with large deformations and compare the results. This would further assist the understanding of wave propagation in flexible vessels and would provide more validation data for theoretical and numerical studies in the field. The ultrasound wall tracking system can also measure shear rate. Thus, this measurement can be included. The experiments presented in this thesis were obtained for a single type of initial pulse. It would be interesting to conduct measurements with different types of pulses. The most interesting would be to use a pulse that replicates the pulse from the heart.

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Appendix A The Tube Models Manufacturing Methodology In Section 1.4 it was presented the lack in the literature of well defined experiments assessing the non-linearities of flexible vessels i.e. wall thickness variation and geometric tapering. In order to obtain a complete set of experimental data assessing these variations, a set of flexible tubes was manufactured.

A.1

The vessels design and specifications

The tubes were designed to be model analogues of the human aorta. One of the most referenced sources of arterial dimensions is the one from Westerhof et al. (1969). In Table A.1, the data mentioned in his work are presented and is used as a guidance for the design of the tubes used in this work. Variable Top internal radius [mm] Bottom internal radius[mm] Length [mm] Slope ϒ ∗ h/D [MPa]

Aorta Thoracalis 20 11 315 -0.014 0.02-0.04

ϒ : Youngs modulus, h: wall thickness, D: internal diameter Table A.1: Aorta anatomical data (Westerhof et al., 1969).

In order to be able to assess the effects of morphological variations in wave propagation, six tubes were manufactured: three straight ones and three tapered ones. The geometrical parameters of these tubes are summarised in Table A.3. It should be mentioned that the tube of Type E has the same ϒ ∗ h/D as the aorta according to Westerhof et al. (1969) (See Table A.1). i

ii

A.1. The vessels design and specifications

In order to be able to separate effects due to geometric tapered from the six tubes, two pairs of tubes were manufactured in a specific manner, so that they would have the same wave speed throughout according to linear theory. The first pair consists of a geometrically tapered tube with constant wall thickness (Type E) and a straight tube with variable wall thickness (Type C). The variable wall thickness of the tube was designed so as according to linear theory the wave speed throughout this straight tube will be the same as the tapered one with constant wall thickness. The second pair consists of a straight tube with constant wall thickness (Type A) and a tapered tube with variable wall thickness (Type F). The variable wall thickness of the tapered tube was chosen such that according to linear theory the wave speed throughout its length is the same as for a tapered tube with constant wall thickness. In this way the variable wall thickness of the tapered tube according to the linear theory will counterbalance the effect of geometric tapering.

D[mm]

h±0.002[mm]

L[mm]

z

ϒ ∗ h/D[MPa]

c[m/s]

A

25

0.1

446

0

0.04

6.3

B

25

0.05

446

0

0.02

4.5

C

25

0.05-0.1

446

0

0.02-0.04

4.5-6.3

D

25-12.5

0.1

446

-0.014

0.04-0.08

6.3-8.9

E

25-12.5

0.05

446

-0.014

0.02-0.04

4.5-6.3

F

25-12.5

0.1-0.05

446

-0.014

0.04

6.3

Type

ϒ: Young’s modulus, h: wall thickness, D: diameter Table A.3: Geometrical parameters of tubes manufactured.

A.2. Manufacturing set-up

A.2

iii

Manufacturing set-up

The tubes were manufactured by the method of spin coating. The set-up is shown in Figure A.1. The tube will take the shape of a steel rod that can rotate along its length axis through a servomotor (x-servomotor). The process can be seen in Figure A.2. The liquid used in the spin coating process is delivered through a nozzle that has cross sectional area of 7mm2 (π∗(3mm/2)2 ) and is injected by a pump. The pump can operate only on constant flow rate throughout the process. The nozzle is attached on a trolley that can translate along the length of the rod through by a rotating ball screw rod connected to a second servomotor (y-servomotor). Different nozzles designs were tried and optimal shapes were found by trial and error. As the trolley translates the nozzle places a spiral stripe of resin of about 3 mm thickness on the rotating rod. Under infrared light the liquid stripes will blend into each other and solidify, creating a tube of certain thickness. The thickness of the tube is marginally small, therefore two things are important for the consecutive spiral stripes to merge: the positioning of the nozzle and the distance ξ between the stripes that should not exceed 3 mm. The x-servomotor rotates at Crot = 2000 counts/revolution and has a 1-1 relationship with the connected rotating beam. The y-servomotor rotates at Ctrans = 4000 counts/revolution and the ball screw pitch is λ = 2.5 mm. Thus, the rotational movement of 1600 counts the y-servomotor can be translated to translational movement of 1 mm of the trolley. The translational movement of the y-servomotor is responsible for the thickness h of the tube and the rotational movement of the x-servomotor is responsible for the spacing ξ between consequent raisin spirals delivered by the nozzle. The movement of the servo-motors can be controlled via an DMC-630 Galil controller connected to a PC. In the following three sections the equations used for the programming of the servomotors movement will be derived.

A.3

Equations for manufacturing

The following equations describing the behaviour of the machine regarding the tube specifications have been written out and have been used for the programming of the microcontroller.

Volume of straight tube VL = πD(x) ∗ X ∗ h ∗ 100/α , xε [0, X ]

(A.1)

where D is the diameter of the steel rod, h is the thickness of the tube, X is the

iv

A.3. Equations for manufacturing

infrared light

steel rod

trolley with nozzle x-servomotor y-servomotor

λ

Figure A.1: Spin coating set-up ( TU/e).

ξ

Figure A.2: Spin coating process of a tube.

v

A.4. Straight tube manufacturing

length of the rod and α is the concentration of the solution.

Volume of a tapered tube (Rade and Westergren, 1990) V1 =

V2 =

π∗X D(0)2 + D(0) ∗ D(X ) + D(X )2 , xε [0, X ] 12

(A.2)

π∗X D(0) + d)2 + (D(0) + d) ∗ (D(X ) + d) + (D(X ) + d)2 , xε [0, X ] 12

(A.3)

VL = (V 2 −V 1) ∗ 100/α

(A.4)

The horizontal to vertical slope of the cone is given by Equation A.5. z=

1 (D(X ) − D(0)) 2X

(A.5)

VL t

(A.6)

Flow rate FL =

Frequency of servomotor rotation U(x) (A.7) ∆x From the definition of frequency a relationship between the speed and the number of counts per revolution (C) for the servo-motors can be obtained: f (x) =

U(x) = C ∗ f (x)

(A.8)

For the x-servomotor Crot = 2000 counts/revolution and for y-servomotor Ctrans = 4000 counts /revolution.

A.4

Straight tube manufacturing

The specifications of the steel rod used as a manufacturing mould can be seen in Figure A.3, and the straight tube specifications can be seen in Table A.5). Variable Diameter[mm] Length [mm] Thicknes [mm]

Abbreviation D X h(x)

Value 25 500 0.05-0.1

Table A.5: Straight tube specifications.

vi

A.4. Straight tube manufacturing 2.5 [cm]

50 [cm]

Figure A.3: Straight tube steel rod dimensions.

A.4.1

Constant thickness

The equations necessary for the manufacturing of the straight tube are Equations A.1, A.6 and A.8. In order for the spiral liquid stripe of the solution to merge and solidify homogeneously the distance between the spiral lines should be 0.2 ≤ ξ ≤ 0.3 [mm]. The rotational rotor is responsible for the spacing distance ξ, therefore in Equation A.7 should have ∆x = ξ. So, its velocity in counts/s is: Urot =

FL ∗ α/100 Crot h∗π∗D ξ

(A.9)

The equation giving the axial velocity of the y-servomotor in counts/sec is given by: Uax =

FL ∗ α/100 Cax h∗π∗D λ

(A.10)

The relationship between the velocities of the two servo-motors is linear and always valid regardless of the type of tube to be manufactured. Therefore it is written in a general form at any point x along the length of the rod: Urot (x) = Uax (x)

A.4.2

Crot λ Cax ξ

(A.11)

Variable thickness

A tube with variable thickness is needed as described in Section A.1. The thickness of the tube will be varying in a way that the wave velocity inside the tube at all time will be the same as the wave velocity in a tapered tube with constant wall thickness according to Moens Korteweg equation. Considering a tapered tube with constant wall thickness h∗ , diameter varying with the the rod length D∗ (x) and varying wave speed c∗ (x). The wave velocity is give by c∗ (x) =

s

ϒ h∗ ρ D∗ (x)

(A.12)

vii

A.4. Straight tube manufacturing

The straight tube with constant diameter D will have a varying wall thickness with the rod length h(x) and a varying wave speed c(x). Thus, the wave velocity is given by

c(x) =

s

ϒ h(x) ρ D

(A.13)

In order to have at all points along the length of the two tubes the same wave velocity the two wave speeds should be the same at all times c(x) = c∗ (x). Substituting Equations A.12 and A.13 the relationship according to which the wall thickness should be varying is obtained. h(x) =

h∗ ∗ D D∗ (x)

(A.14)

The velocities of the x-servomotor and the y-servomotor can be given by sub-

Rotational Velocity [counts/sec]

7000

6000

5000

4000

3000

0

10

20 30 Rod length [cm]

40

4500 4000 3500 3000 2500 2000

0

10


40

0

10


40

−3

10

x 10

3.5

Rod diameter [cm]

9 Thickness [cm]

Translational Velocity [counts/sec]

stituting Equation A.14 in Equations A.9 and A.10. In Figure A.4 one can see the wall thickness variation in relation to the tube length.

8 7 6 5

0

10


40

3

2.5

2

1.5

Figure A.4: Translational velocity, rotational velocity, tube wall thickness and tube diameter versus the tube length for tube C.

viii

A.5. Tapered tube manufacturing

A.5

Tapered tube manufacturing

For the manufacturing of the tapered tube a tapered steel bar was constructed (FigureA.5) with dimensions shown in Table A.7. The equations describing this requirements can be derived as follows.

A.5.1

Constant thickness

Using Equation A.5 the radius at any point x along the tapered bar is increasing according to Equation A.15. D(x) = D(0) − 2x ∗ z , xε [0, X ]

(A.15)

Rewriting Equation A.15 in ∆x increments, Equation A.16 is obtained. D(x) = D(x − ∆x) − 2∆x ∗ z , xε [0, X ]

(A.16) 2.5 [cm]

1.25 [cm]

4 [cm]

44.6 [cm]

2.4 [cm]

Figure A.5: Tapered tube steel rod dimensions.

Variable Diameter at bottom [mm] Diamerter at top [mm] Length [mm] Horizontal to vertical slope Thickness [mm]

Abbreviation D(0) D(X ) X z h

Value 12.5 25 446 0.014 0.05-0.1

Table A.7: Tapered tube specifications.

In the same way the equations giving the change in volume of the tube for every increment ∆x is given by EquationsA.17, A.18, A.19 and A.20.

∆V1 (x) =

π ∗ ∆x D(x)2 + D(x) ∗ D(x + ∆x) + D(x + ∆x)2 , xε [0, X ] 12

(A.17)

ix


∆V2 (x) =

π ∗ ∆x (D(x) + h)2 + (D(x) + h) ∗ (D(x + ∆x) + h) + (D(x + ∆x) + h)2 , xε [0, X ] 12 (A.18) ∆VL (x) = [∆V2 (x) − ∆V1 (x)] ∗ 100/a

(A.19)

x=X

VL (x) =

∑ ∆VL(x) ∗ 100/α

(A.20)

x=0

Substituting for time increments ∆t = ∆x/U(x) in EquationA.6, the velocity of the trolley and the velocity of the rotating beam is given by: Urot (x) =

FL ∗ ∆x Crot ∆VL (x) ξ

(A.21)

Uax (x) =

FL ∗ ∆x Cax ∆VL (x) λ

(A.22)

From Equations A.22, A.21 and EquationA.6 it can be seen that the relationship between the distance along the bar and the translational velocity is non linear of hyperbolic form. The way of calculating the volume of the liquid described above is accurate using the theory for a cone cylinder and can be used to calculate accurately the total volume of the liquid that needs to be injected by the pump. For the programming of the microcontroller and the calculation of the speed of the x, y-servo-motors the volume of the stripe can be approximated by Equation A.1, where the diameter is given by Equation A.15: ∆VL (x) = π ∗ D(x) ∗ ∆x ∗ h ∗ 100/α

(A.23)

Therefore, by substituting Equation A.23 in Equations A.21 and A.22 the rotational and axial velocities in counts/sec are give by: Urot =

FL ∗ α/100 Crot h ∗ π ∗ D(x) ξ

(A.24)

Uax =

FL ∗ α/100 Cax h ∗ π ∗ D(x) λ

(A.25)

For the update of the diameter a relationship is needed between ∆x and ∆t is needed and can be obtained by substituting Equation A.23 in Equation A.6. One of the two will have to be fixed and chosen arbitrary and the other will get calculated. In Figure A.6 one can see the servo-motors hyperbolic velocity variation according to the diameter change of the tube.

x




9000 8000 7000 6000 5000 0

10


40

6000 5500 5000 4500 4000 3500 3000 2500

0.01

0

10


40

0

10


40

2.8

Rod diameter [cm]

Tube thickness [cm]

2.6 0.008 0.006 0.004 0.002

2.4 2.2 2 1.8 1.6 1.4

0

0

10


40

Figure A.6: Translational velocity, rotational velocity, tube wall thickness and tube diameter versus the tube length for tube E.

xi

A.6. Wall thickness accuracy

A.5.2

Variable thickness

A tapered tube with variable thickness is needed as described in Section A.1 in order to distinguish which effects are due to the geometric tapering. The thickness of the tube will be varying in a way that the wave velocity inside the tube at all time will be constant as it is in the straight tube with constant wall thickness according to Moens Korteweg equation. Considering a straight tube with constant wall thickness h∗ , diameter D∗ and constant wave speed c∗ . The wave velocity is given by c∗ =

s

ϒ h∗ ρ D∗

(A.26)

Considering a tapered tube with variable wall thickness h(x) , diameter varying with the the rod length 12.5 ≤ D(x) ≤ 25 mm and constant wave speed c. The wave velocity is give by c=

s

ϒ h(x) ρ D(x)

(A.27)

In order to have at all points along the length of the two tubes the same constant wave velocity the two wave speeds should be the same c = c∗ . Thus, from Equations A.26 and A.13 the relationship according to which the wall thickness should be varying is obtained: h∗ D(x) (A.28) D∗ In Figure A.7 one can see the wall thickness variation 0.05 ≤ h(x) ≤ 0.1 mm in relation to the tube length which is a linear relationship. The velocities of the x and y-servo-motors can be obtained by substituting Equation A.28 in Equations A.24 h(x) =

and A.25.

A.6

Wall thickness accuracy

The wall thickness of the tubes manufactured was measured every 10 mm along the tube length with a micrometer and it was found that the wall thickness accuracy was ±2µm. This variation is small. Thus, for modelling purposes can be neglected.

xii


16000 14000 12000 10000 8000 6000 4000 2000

0

10


40

10000

8000

6000

4000

2000

0

10


40

0

10


40

−3

10

x 10

2.8 2.6 Rod diameter [cm]

9 Thickness [cm]


A.6. Wall thickness accuracy

8 7 6

2.4 2.2 2 1.8 1.6 1.4

5

0

10


40

Figure A.7: Translational velocity, rotational velocity, tube wall thickness and tube diameter versus the tube length for tube F.

Fluid structure interaction in flexible vessels

Fluid structure interaction in flexible vessels

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