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

Copyright c© 2004 by Christina G. Giannopapa

All rights are reserved. No part of this publication may be reproduced, stored in re-

trieval 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 Commis-

sion.

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 so-

lution 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 equa-

tions 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 Sci-

ences 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 Com-

mission 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

vi Contents

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 35

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . 41

3.2.5 Laplacian versus Divergence-Grad . . . . . . . . . . . . . . . 41

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

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

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

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

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

3.6.4 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

Contents vii

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 thick-

ness 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

viii Contents

7 Conclusions 159

7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 i

A.1 The vessels design and specifications . . . . . . . . . . . . . . . . . . i

A.2 Manufacturing set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

A.3 Equations for manufacturing . . . . . . . . . . . . . . . . . . . . . . . iii

A.4 Straight tube manufacturing . . . . . . . . . . . . . . . . . . . . . . . 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 formula-

tion for the Euler Implicit discretisation scheme (envelope of displace-

ment). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.8 Comparison of Euler Implicit and Backward Differencing discretisa-

tion 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 Back-

ward differencing using time step size of 1e-4 s (envelope of displace-

ment). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 = 5e5Pa. 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% elonga-

tion). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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


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


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


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


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


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

xii List of Figures


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



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



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


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


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



of the tube against time for tubes types C and E (C: straight tube with


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



6.6 Comparison of the experimental measurements of the wall distension

on a straight tube with constant wall thickness of 0.1 mmwith linear

analytical model foran elastic tube. . . . . . . . . . . . . . . . . . . . 148

6.7 Comparison of pressure experimental measurements of the straight








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



model fora viscoelastic tube. . . . . . . . . . . . . . . . . . . . . . . 153



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



model fora viscoelastic tube. . . . . . . . . . . . . . . . . . . . . . . 156



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


tube diameter versus the tube length for tube E. . . . . . . . . . . . . x


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 vi-

bration 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 con-

ditions for thin- and thick-wall tubes. . . . . . . . . . . . . . . . . . . 141

A.1 Aorta anatomical data (Westerhof et al., 1969). . . . . . . . . . . . . i

A.3 Geometrical parameters of tubes manufactured. . . . . . . . . . . . . ii

A.5 Straight tube specifications. . . . . . . . . . . . . . . . . . . . . . . . v

A.7 Tapered tube specifications. . . . . . . . . . . . . . . . . . . . . . . . viii

xv

xvi Nomenclature

Nomenclature

General

Character Explanation

s scalar

a vector

T second order tensor

Operators and functions


∇ gradient operator

∇ • divergence operator

∇2 laplacian operator∂s∂t time derivative of s

∆s discrete increment of s

|a| absolute value of a

TT transpose of T

trT trace of T

devT deviatoric part of T

xvii

xviii Nomenclature

Latin symbols

Character Unit Explanation

A analytical solution

C m/sec characteristic velocity of fluid or solid

Co Courant number

d m vector from P to N cell centre

D m displacement

E truncation error

EP W external power

f Hz frequency

F fluid

gb fixed gradient at the boundary

h m height of the beam

I unit tensor

K Pa bulk modulus

KP W kinetic power

l m length of the beam

m kg mass

n unit vector normal to a control volume face

N neighbour cell centre

p Pa pressure

P present cell centre

S solid

S m2 closed surface

S f m2 face area vector

SP W strain power

t sec time

TP W total power

U m/sec velocity vector

V m3 volume

x m position in x direction

y m position in y direction

Nomenclature xix

Greek symbols

Character Unit Explanation

A any spatial operator

δ m end displacement

ε strain tensor

ε rate of deformation tensor

η Pa

sec

dynamic viscocity

λ Pa Lame’s coefficient

µ Pa Lame’s coefficient

ν Poison’s ratio

ρ kg/m3 density

σ Pa Cauchy stress tensor

τ Pa applied end shear

φ any property (scalar, vector or tensor)

ϒ Pa Young’s modulus

ω Hz frequency of undamped oscillation

ωN weighting factor form P to N cell centre

Superscripts


o old values

oo old old values

n new values

∗ spatial discretisation

xx Nomenclature

Subscripts


0 reference situation

f face value

N value at neighbour cell

P value at present cell

Abbreviations

Characters Explanation

A analytical

BD backward differencing

CD central differencing

CFD computational fluid dynamics

CSM computational solid mechanics

CV control volume

EP external power

EI Euler implicit

FE finite element

FV finite volume

FSI fluid structure interaction

KP kinetic power

LCR inductance-capacitance-resistance circuit

N numerical

NM not mentioned

PDE partial differential equation

SP strain power

UD 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

Local deformationRigid body motion

Momentum interaction Thermal interaction

FSI

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, non-

Newtonian 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 > 0.1mm ) and

arterioles 10−100µmRoades and Tanner (1995); Levick (2000).

1.2. Morphology of arteries 3

Endothelium

Connective tissue

Tunica Intima

Tunica media

Tunica adventitia

THREE LAYERS

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 connec-

tive 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.


Tunica adventitia

Tunica adventitia or externa is a thin layer and mainly consists of connective tis-

sue 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 sur-

rounding 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 pres-

sure. 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 ultra-

sound. 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 Abe et al. (1996), where a summarised

collection of papers published in the area until 1996 is presented.

1.3 Computational Methods for fluid structure in-

teraction

Typically in FSI, the fluid and solid components are modeled using different tech-

niques 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 com-

ponents, 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 litera-

ture 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 ap-

proach, 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 re-

turned 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


31 5 . . .

42 6 . . .

31 5 . . .

42 6 . . .

∆t ∆t ∆t∆t

Time TimeEndStart

METHOD 4over time stepIterative

Iterativeover all time

METHOD 2

Up

p p p p

1

2F

S

F

S

4 678

1 3 5

S

F

S

F

1 2 3 4

S

F

S

F

∆t ′

21 3 4 6 7 8 9 11 12 13 14

5 10 15

16 17 18 19

20

2

∆t ′ ∆t ′ ∆t ′

METHOD 5

Implicitsingle solution

p

p

p p

U

U U U U

U U U U

pU U U

METHOD 3btime stepsover uniqual

METHOD 3aover time step

Non-iterative

p p pp

over all time

METHOD 1

Non-iterative

Non-iterative

NOTE: The numbers in italics are counters of the computational time step. Thestraight dashed arrow represents the transfer of information of the denoted variablefrom one medium to the other. The curved dashed arrow represents the iterativeprocedure.

Figure 1.3: Solution procedure of several FSI methods.


F

SS S

F F

monolithicmethod

single solutionmethod

partitionedmethod

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 methods and monolithic methods

(Figure 1.4).

The partitioned methods are based on partitioning the fluid and the solid solu-

tion, 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 com-

plexity 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 informa-

tion 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


syncronously at the their common interface (Tallec and Mouro, 2001; Hubner 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 computation-

aly 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 con-

septual 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 experi-

mental 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 signifi-

cant.


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 Rap-

pitsch (1995). They used an analytical approach to study the effect of wall dis-

tensibility 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 move-

ment 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 pul-

sating 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 computa-

tional 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.

Konig 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


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 compu-

tational 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 non-

Newtonian 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 compo-

nents 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 thin-

walled 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

1.4. Wave propagation in flexible vessels 11

REFERENCE

CHARACTERISTIC Reu

der

ink

etal

.(1

989)

Per

kto

ldan

dR

appit

sch

(199

5)

Hen

ryan

dC

ollins

(199

3a,b

)

Tan

get

al.(1

999a

,b)

Ste

inm

anan

dE

thie

r(1

994)

Kon

iget

al.(1

999)

Tay

lor

etal

.(1

998)

Bat

he

and

Kam

m(1

999)

Xu

etal

.(1

999)

SolidsNon linear

√ √ × × × NM NM√ ×

Viscoelastic√ × × × × NM NM

√ ×Compressible

√ √ × × √NM NM × ×

Large strain × × × × × NM NM√ ×

Thick wall × × √ √ × NM NM × √

3 Dimensional × √ √ √ × NM NM√ √

Method A FE FV FE A NM NM FE FEFluids

Non Newtonian × √ × × × × × × ×Compressible × × × × √ × × × ×

Turbulent × × × × × × × × ×Transient

√ √ √ √ √ × × × ×3 Dimensional × √ √ √ × √ √ √ √

Method A FE FV FE A N N FE FV

NOTE: The symbol√

denotes that the characteristic in the left column has beentaken into consideration, whereas the × means that is has not.

Table 1.1: Modelling assumptions for the fluid and solidcomponent as found in the literature.


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 ex-

tended 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 pre-

sented a table comparing the different theoretical models developed by that time.

Atabek (1968) continued the work using the membrane theory of shells on or-

thotropic 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.


(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 ex-

periments 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 pres-

sure 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 1mmwith ±0.1mmvariations. They used a

glycerin-water mixture as a fluid. The pressure and pressure gradient were measured


using two pressure transducers at a 50mmdistance from each other. Velocity pro-

files were measured using a hot-film velocity probe. Wall shear stress was measured

as well. The pressure-radius relation was obtained by photographing simultane-

ously 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 prop-

agation phenomena, in aortic valve closure. They performed experiments on water

filled latex tube 0.6m long with 18mm inner-diameter and thickness 0.2mm. Trans-

mural pressure was applied at one end of the tube. Pressure was measured using

two catheter-tip manometers. Wall deflection was measured using a photonic sen-

sor 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 mea-

surements they obtained the viscoelastic properties of the tube. The compared their

experimental data with the one dimensional non-linear theory for the wall shear


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 di-

mensional 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 com-

pared 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 pneu-

matically 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 at-

tempted to measure pulsatile diameter changes using an ultrasonic transit-time tech-

nique 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 be-

tween 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.


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 investiga-

tion 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 geo-

metric 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 con-

cluded 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) elec-

trical 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 signifi-

cantly 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 char-

acteristic impedance and peripheral impedance are doubled and the cross-section of


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 charac-

terise 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 pres-

sure 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 appar-

ent damping and the damping coefficient calculated from Womersley theory. They


also compared the true phase velocity with the measured apparent phase velocity,

foot-to-foot velocity, and calculations of phase velocity parameters using the Wom-

ersley’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 var-

ious 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 prop-

agation in flexible vessels is that of Womersley. The theory of Womersley for infi-

nitely 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


REFERENCE

CHARACTERISTIC Reu

der

ink

etal

.(1

989)

Ste

inm

anan

dE

thie

r(1

994)

Ger

rard

(198

5)

Klip

(196

2)

Wes

terh

ofet

al.(1

969)

Ata

bek

(196

8)A

tabek

Lin

gan

dA

tabek

(197

2)

SolidsNon linear

√ × × × × × √

Viscoelastic√ × × × × × √

Compresible√ √ × × √ √ ×

Large strain × × × × × × √

Thik wall × × × √ × × ×Dimentions 1 2 - - - 1 1

Method A A E E A&E A A&EFluids

Non Newtonian × √ × × × × ×Compresible × × × × × × ×Turbulent × × × × × × ×Transient

√ √ √ √ √ √ √

3 Dimentional 1 2 - - - 1 1Method A A E E A&E A A&E

NOTE: The symbol√

denotes that the characteristic in the left column has beentaken into consideration, × means that is has not and − means that it is not men-tioned.

Table 1.2: Assumptions for the fluid-solid components for straighttubes as found in the literature.


REFERENCE

CHARACTERISTIC Hor

sten

etal

.(1

989)

van

Ste

enhov

enan

dva

nD

onge

n(1

986)

Bel

ardin

elli

and

Cav

alca

nti

(199

2)

Wem

ple

and

Mock

ros

(197

2)

Reu

der

ink

etal

.(1

988)

SolidsNon linear × √ √ √ √

Viscoelastic × √ × √ ×Compresible

√ √ × √ √

Large strain × × × × ×Thik wall × × × √ ×

Dimentions 1 1 2 1 -Method A A A A E

FluidsNon Newtonian × × √ × ×

Compresible × × × × ×Turbulent × √ × × ×Transient

√ √ √ √ √

3 Dimentional 1 2 2 1 -Method A A A A E

NOTE: The symbol√

denotes that the characteristic in the left column has beentaken into consideration, × means that it has not and − means that it is not men-tioned.

Table 1.3: Fluid-solid assumptions for tapered tubes as found inthe 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 thermody-

namically 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 characteris-

tic 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 non-

linearities 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 de-

fined 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 au-

thor 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 theo-

retically 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 inves-

tigate 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 case-

specific codes and to problems regarding the efficiency of the coupling of the two


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 equa-

tions.

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 hemody-

namics 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 Sec-

tion 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 inter-

est 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 computa-

tional methods to check whether they can cope with the anatomical non-linearities.

1.6. Outline of the thesis 23

1.6 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 in-

formation 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.

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 pre-

25

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 rela-

tionship. 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∂t

+∇ •(ρU) = 0 (2.1)

Momentum equation (neglecting body forces)

∂ρU∂t

+∇ • (ρUU) = ∇ •σ (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) for plain stressνϒ

(1+ν)(1−2ν) 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

2.3. Mathematical Model 27

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:

σ = η∇U +η(∇U)T− 23

ηtr(∇U)− pI (2.8)

or

σ = 2ηdev(sym(∇U))− pI (2.9)

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∂p

=ρK

(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:

ρ ≈ ρo

[1+

p− p0

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 stan-

dard stress analysis is first examined, followed by the new formulation.


t

∆t2 [Un+Uo]

Un

U0

U

t0

Do

tt −∆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) =12

[∇D+(∇D)T] (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[

∂D∂t

]

∂t+∇ •(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 dis-

placement 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

2.3. Mathematical Model 29

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:

D(t) =Z t

t0Udt =

Z t−∆t

t0Udt+

Z t

t−∆tUdt (2.15)

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+∆t2

[Un +Uo] (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:

σ = Σ+ +∆t2

[µ∇U+µ(∇U)T +λtr(∇U)I

](2.17)

where sigma plus (Σ+) is given by Equation 2.18 and Σ is the accumulated stress

from previous time steps.

Σ+ = Σ+∆t2

[µ∇Uo+µ(∇Uo)T +λtr(∇Uo)I

](2.18)

Thus, the momentum Equation 2.2 over the time interval [t0, t] , becomes:

∂ρU∂t

+∇ •(ρUU) =∆t2

[∇ • [µ∇U]+∇ •

[µ(∇U)T]+λ∇ • [tr(∇U)I]

]+∇ •Σ+ (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.


σ = devσ+13

tr(σ)I = devσ− pI (2.20)

After some tensor manipulation, the stress tensor can be written as:

σ = devΣ+ +∆t2

[µ∇U+µ(∇U)T− 2

3µtr(∇U)I

]− pI (2.21)

where devΣ+ 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:

devΣ+ = devΣ+∆t2

[µ∇Uo+µ(∇Uo)T − 2

3µtr(∇Uo)I

](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.

∂ρU∂t

+∇ •(ρUU) =∆t2

[∇ • [µ∇U]+∇ •

[µ(∇U)T]− 2

3µ∇ • [tr(∇U)I]

]+∇ •devΣ+−∇p

(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 ex-

plicit), 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

p = −13

tr(σ) (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∆t2

tr(∇U) (2.25)

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

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

formulation 31

K =ϒ

3(1−2ν)(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.

p+ = −Ktr

[∇(Do+

Uo

2∆t)

](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 con-

tinuity 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∂t

+∇ • [(ρ0−ψp0)U]+∇ • [ψpU] = 0 (2.28)

where p0 is the reference pressure for which ρ(p0) = ρ0 and ψ = ρ0K . 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 implic-

itly 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 for-

mulation 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:


∂ρU∂t

+∇ •(ρUU) = ∇ • [η∇U]+∇ •[η(∇U)T]− 2

3ηtr(∇U)I−∇p (2.29)

or

∂ρU∂t

+∇ •(ρUU) = 2ηdev(sym(∇U))−∇p (2.30)

If we set α = ∆t2 µ then the momentum Equation 2.23 for the linear elastic solid

reads:

∂ρU∂t

+∇ •(ρUU) = ∇ • [α∇U]+∇ •[α(∇U)T]− 2

3α∇ • [tr(∇U)I]+∇ •devΣ+−∇p

(2.31)

or

∂ρU∂t

+∇ •(ρUU) = 2αdev(sym(∇U))+∇ •devΣ+−∇p (2.32)

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 ob-

tained. 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 ∇ •devΣ+, 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 velocity-

pressure 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.

2.6. Closure 33

The chosen boundary conditions for pressure are either fixed value or fixed gra-

dient. There are three relationships that can give a boundary condition for pressure:

using the definition of pressure, applying the force balance relationship at the bound-

ary or using the momentum equation.

pressure definition A possible boundary condition for pressure can be ob-

tained using the definition of pressure:

p = p0−K∆t2

tr(∇U+∇U0) (2.34)

where p0 are the old values obtained at the end of the previous time step.

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 •devΣ+ •n−αn •

[∇U+(∇U)T− 2

3tr(∇U)I

]•n (2.35)

applying momentum The momentum equation can be projected to the unit

normal vector at the boundary and solved for the pressure gradient:

n∇p=−∂ρU∂t

−n •

{∇ •(ρUU)+∇ • [α∇U]+∇ •

[α(∇U)T]− 2

3α∇ • [tr(∇U)I]

}+n •∇ •devΣ+

(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


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 ex-

pressions 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 mathemat-

ical 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 Demirdzic and

Martinovic (1993). So far the FV method has successfully been applied to elastic,

elastoplastic and viscoplastic problems, as well as geometrically non-linear stress

analysis (Demirdzic and Martinovic, 1993; Demirdzic 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

(Demirdzic and Martinovic, 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 be-

tween the continuity and momentum equations. The combination of these equations

would result in a big and complicated matrix which would involve massive computa-

tional 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 han-

dle complex boundary conditions, especially when heat transfer, fluid flow and solid

structure deformation are coupled and their interaction is important. Moving bound-

aries 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


cellvolume

face

cellcentre

P N

d

fface areavector

S f

| d |

| dN |

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.


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: gradi-

ent, 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 prop-

erty 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 cen-

tres P and N the face values are calculated as (see Figure 3.2):

φ f = ωNφP +(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 non-

physical 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.

φ f =

{φP for F ≥ 0

φN for F < 0(3.3)

where F represents the mass flux passing through the face:

F = S f •(ρU) f (3.4)


P N

dN

d

φP

φN

φ

f

φ f

Figure 3.2: Evaluation of the face value φ f from cell centre valuesφP 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)UD + γ

(φ f)

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,ZV

∇φ dV =Z

SdSφ = ∑

f

S f φ f (3.6)


Surface normal gradient The gradient normal to a surface n f • (∇φ) f for orthog-

onal mesh can be evaluated at cell faces using the scheme

n f • (∇φ) f =φN −φP

|d| (3.7)

3.2.3 Discretisation of the divergence

The integral of the divergence term is also evaluated explicitly. Note that the prop-

erty 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:ZV

∇ •φ dV =

ZSdS •φ = ∑

f

S f •φ f (3.8)

3.2.4 Discretisation of the Laplacian term

The Laplacian term is integrated over a control volume as follows:ZV

∇ •(Γ∇φ) dV =

ZSdS • (Γ∇φ) = ∑

f

Γ f S f • (∇φ) f (3.9)

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:

S f •(∇φ) f =∣∣Sf∣∣ φN −φP

|d| (3.10)

3.2.5 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: ZV

∇2φ dV =

ZSdS • (∇φ) = ∑

f

S f •(∇φ) f (3.11)

Let us assume there is a need to calculate ∇2φ 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.


P

∇2φ

N

E

S

W

φS

φN

φEφW

NOTE: × denotes the location where S f •(∇φ) f is evaluated directly.

Figure 3.3: Cells involved in the evaluation of the Laplacianoperator at cell with cell centre denoted as P.

The divergence-gradient and the Laplacian operators are the same mathemati-

cally. 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:ZV

∇φ dV =

ZSdSφ = ∑

f

S f φ f (3.12)

In the second stage, the divergence is applied on the ∇φ calculated before. So,ZV

∇ •(∇φ) dV =

ZSdS • (∇φ) = ∑

f

S f •(∇φ f ) (3.13)

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 be-

ing a one step procedure and the other a two step procedure. This results in the


P

HG

K

M

F

φK

φM

φG φH

φF

N

∇φN

T

W

∇φW ∇φE

φT

E∇ •∇φ

S

∇φS

B

φB

L

φL

NOTE: × denotes the location where S f •(∇φ) f is evaluated by linear interpolation.

Figure 3.4: Cells involved in the evaluation of the Divergence-Gradient 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:

φo = φn−∆tφn′ +∆t2φn′′

2!+ ... (3.14)

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

∂φ∂t

= φn′ =φn−φo

∆t+∆t

φn′′

2!+ ... (3.15)

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


is first order accurate:

E = ∆tφn′′

2!+ ... (3.16)

Applying it in first order derivative for Backward Differencing (also referred to

in literature as a three level scheme) gives:

∂φ∂t

= φn′ =3φn−4φo+φoo

2∆t+

13

∆t2φn′′+ ... (3.17)

Backward Differencing involves three time levels and, from equation 3.17 the

truncation error is second order accurate:

E =13

∆t2φn′′′+ ... (3.18)

In the same way, when Taylor polynomial expansion is applied for the second

order derivative using Euler Implicit, it gives:

∂2φ∂t2 = φn′′ =

φn−2φo+φoo

∆t2 +23!

∆tφn′′′+ ... (3.19)

The truncation error is of first order:

E =23!

∆tφn′′′+ ... (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:ZV

∂∂t

ρφ dV =∂∂t

ZV

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.


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

∂∂t

ZV

ρφ dV =(ρPφPV)n− (ρPφPV)o

∆t+O(∆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

ZV

ρφ dV =3(ρPφPV)n−4(ρPφPV)o+(ρPφPV)oo

2∆t+O(∆t2) (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

ZV

ρ∂φ∂t

dV =(ρPφPV)n−2(ρPφPV)o +(ρPφPV)oo

∆t2 +O(∆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

[∂∂t

ZV

ρφ dV +Z

VA φ dV

]dt = 0 (3.25)

Using the Euler implicit method, the first term of Equation 3.25 can be expressed

asZ t+∆t

t

[∂∂t

ZV

ρφ dV

]dt =

Z t+∆t

t

(ρPφPV)n− (ρPφPV)o

∆tdt = (ρPφPV)n− (ρPφPV)o

(3.26)

The second term can be expressed asZ t+∆t

t

[ZVA φ dV

]dt =

Z t+∆t

tA

∗φ dt (3.27)

where A ∗ represents the spatial discretisation of A . The time integral can be dis-

cretised in three ways:

Euler implicit uses implicit discretisation of the spatial terms. Thus the values of


φ at the n-th time instant are used:Z t+∆t

tA

∗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

tA

∗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

tA

∗φ dt = A ∗(

φn+φo

2

)∆t (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.


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,

S f •(∇φ) f =∣∣Sf∣∣ φb−φP

|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|S|

•∇φ)

b(3.33)

• 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 =∣∣Sf∣∣gb (3.35)


3.3 Final form of equations and discretisation of

the transient term

In this section, certain solution procedures regarding the discretisation of the equa-

tions 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[

∂D∂t

]

∂t= ∇ • [(2µ+λ)∇D]︸︷︷︸

implicit

+∇ •{[

µ(∇D)T]+[λtr(∇D)I]− [(µ+λ)∇D]}

︸︷︷︸explicit (i.e source term)

(3.36)

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 = ∑K

aK where K=E, W, N, S (3.37)

3.3. Final form of equations and discretisation of the transient term 49

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

=∆t2

∇ • [(2µ+λ)∇U]︸︷︷︸

implicit

+∆t2

∇ •{

µ(∇U)T +λtr(∇U)− (µ+λ)∇U}

+∇ •Σ+

︸︷︷︸explicit

(3.39)

Velocity and explicit pressure based formulation

∂ρU∂t

=∆t2

∇ • [(2µ+λ)∇U]︸︷︷︸

implicit

+∆t2

∇ •

{µ(∇U)T− 2

3µtr(∇U)− (µ+λ)∇U

}+∇ •devΣ+−∇p


(3.40)

Velocity and implicit pressure based formulation

∂ρU∂t

=∆t2

∇ • [(2µ+ λ)∇U]︸︷︷︸

implicit

+∆t2

∇ •

{µ(∇U)T − 2

3µtr (∇U)− (µ+ λ)∇U

}+ ∇ •devΣ+


− ∇p︸︷︷︸implicit

(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 Lapla-

cian 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 (Equa-

tions 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 ∂2ρD∂t2 can be discretised in one of two ways. One way is by using

the Euler implicit discretisation scheme , involving two old-time levels:

∂2ρD∂t2 =

ρDn−2ρDo+ρDoo

∆t2 (3.42)


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:

∂2ρD∂t2 =

2ρDn−5ρDo+4ρDoo−ρDooo

∆t2 (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 guaran-

tees 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∂t

=ρUn−ρUo

∆t(3.44)

The second way is by using the second order accurate Backward differencing

using three time levels:∂ρU∂t

=3ρUn−4ρUo+ρUoo

2∆t(3.45)

Both of this methods have been implemented and the results obtained are pre-

sented in Chapter 4.

The velocity based formulation is in a way equivalent to performing the discreti-

sation 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 uncon-

ditionally 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.


3.4 Iterative solution methods of governing equa-

tions

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∂t

+∇ •(ρU) = 0 (3.46)

Momentum equation (neglecting body forces)

∂ρU∂t

+∇ • (ρUU) = ∇ •σ (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:

∂ρU∂t

=∆t2

[∇ • [µ∇U]+∇ •

[µ(∇U)T

]− 2

3µ∇ • [tr (∇U)I]

]+∇ •devΣ+−∇p (3.48)

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

reads:

3.4. Iterative solution methods of governing equations 53

∂ρU∂t

+∇ • (ρUU) = ∇ • [η∇U]+∇ •[η(∇U)T]− 2

3ηtr(∇U)I−∇p (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:

ρ ≈ ρo

[1+

p− p0

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 pres-

sure. 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:ZV

∇ •(ρUU)dV ≈Z

SρUUdS ≈ ∑

f

S f •(ρU f )U f = ∑f

Ff U f (3.51)

where fluxes F are defined by :

F = S f •(ρU f ) (3.52)


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 pro-

posed 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 Demirdzic and

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 pro-

ceeding 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 diffi-

cult 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


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:

αPUP +∑f

αKUK = S(U)−∇p (3.53)

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:

αPUP = H(U)−∇p (3.54)

where

H(U) = −∑f

αKUK +S(U) (3.55)

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)

αP− 1

αP∇p (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 (Equa-

tion 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:

∂ψp∂t︸︷︷︸

implicit

+∇ •

[(ρ0−ψp0)

H(U)

αP

]+∇ •

[ψp

H(U)

αP

]


−∇ •

[ρ

αP∇p

]

︸︷︷︸implicit

= 0 (3.58)


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 com-

putation of incompressible flows using a staggered grid. It was later extended by

Demirdzic 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 col-

located 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 im-

plemented 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.


END

PressureEquation

VELOCITYPRESSURE

PISOLOOP

START

Calculate flux

Correct velocity

Solve pressure equation

Calculate H(U∗)

Solve momentum

αPUP = H(U)−∇p

YES

YES

VelocityCorrection

F

U∗

U∗∗

p∗

H(U∗)

NO

NO

Momentum

U∗ = U∗∗

p = p∗

Predictor

∂ψp∂t +∇ •

[(ρ−ψp0)

H(U∗)αP

]+∇ •

[ψpH(U∗)

aP

]−∇ •

[ρ∇pαP

]= 0

DensityCorrection

&ρ

LOOP

satisfied ?is continuity

is residual lessthan prescribed ?

Correct density

H(U∗) = −∑αKU∗K +S(U0)

F = S f •[(

H(U∗∗)αP

) f − ((∇p∗)

αP) f

]

U∗∗P =

H(U∗)αP

− 1αP

∇p∗ ρ = ρ0 [1+ψ(p∗− p0)]

NOTE: One asterisk “∗” denotes first estimation. Douple asterisk “∗∗” denotes sec-ond estimation.

Figure 3.5: PISO algorithm flow chart for compressible flow (forone time step).


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 esti-

mated 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 continu-

ity 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 ve-

locity (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 flu-

ids

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 combina-

tion of internal values and boundary data. These approximations have to be derived

by internal value differences or extrapolations. Generally as mentioned in (Gi-

annopapa, 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 in-

compressible flows is examined. The main focus is the pressure boundary condition.

In order to specify the correct condition needed for the solution of the Navier-

Stokes equation for compressible and incompressible flow, the different roles of pres-

sure 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 pres-

sure 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

R=12

ZV

[U∗∗(r)−U∗(r)]2dV (3.59)

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

corrected velocity field.


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 ve-

locity 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 sym-

metry 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 com-

ponents of the (vector) momentum Navier-Stokes equation is permissible as a bound-

ary 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 Sundstrom (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 Sundstrom

(1978) paper by stating that the divergence-free condition is of the utmost impor-

tance for theoretical and computational fluid dynamics.

Gresho and Sani (1987) demonstrate that for the solution of the pressure Poi-

son 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 con-

tains 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

3.5. Investigation of boundary conditions for fluids 61

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∂t

+∇ •(UU)−∇ •(η∇U) = −∇p (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 tan-

gential 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 bound-

ary conditions for the pressure correction should be Neumann; but when the velocity

boundary conditions are Neumann, the boundary conditions for the pressure correc-

tion 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 charac-

ter. A compressible fluid can support sound and shock waves and it is not surprising


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 classifi-

cation 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 quanti-

ties 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 con-

ditions 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 dissipa-

tive, 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 con-

ditions 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 con-

ditions 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 Sundstrom (1978) and Olivier and Sundstrom (1978) use the

energy method to obtain boundary conditions for the linearised constant coeffi-

cient 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 condi-

tions 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 condi-

tions 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 equa-

tion (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 convec-

tion term ∇ •(ρUU) on the left hand side of the momentum Equation 2.2 is negligible.

So, momentum equation can be rewritten as:

∂ρU∂t

= ∇ •σ (3.63)


3.6.1 Boundary conditions for the displacement formulation

The appropriate boundary condition for the displacement can be obtained by ap-

plying 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 ob-

tained 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:

n •∇D =t−npext−n •

[(−µ−λ)∇D+µ∇DT

]−nλtr (∇D)

(2µ+λ)(3.65)

Fixed value using force balance 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 ex-

pression for the displacement can be obtained as

Db = DN+ | dN | t−npext−n •[(−µ−λ)∇D+µ∇DT

]−nλtr(∇D)

(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)


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 |2∆t [t−npext−nΣ+]+n •


]−nλtr(∇U)

(2µ+λ)(3.69)

3.6.3 Boundary conditions for the velocity-pressure formu-

lation

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 •


]+n2

3µtr(∇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:


Ub =|UN |+dN

2∆t [t−npext−n(devΣ+)+np]+n •


]+n2

3µtr(∇U)

(2µ+λ)(3.71)

3.6.3.2 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.

Fixed value using the definition of pressure This boundary condition is

given by

p = p0−K∆t2

tr(∇U+∇U0) (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 (Equa-

tion 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 •devΣ+ •n−αn •

[∇U+(∇U)T− 2

3tr(∇U)I

]•n (3.73)

where is α = ∆t2 .

Fixed gradient using momentum This boundary condition type is fixed

gradient and an equation for its value is derived by projecting the momentum equa-

tion at the unit vector normal to the boundary and solving for n •∇p. So we get:

n •∇p =∆t2

n •

[∇ • [µ∇U]+∇ •

[µ(∇U)T]− 2

3µ∇ • [tr∇UI]

]+n •∇ •devΣ+− ∂ρU

∂t(3.74)

This formulation has been derived according to the paper of Gresho and Sani

(1987) but for a compressible material.


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

pb = pN+ | dN | ∆t2

n •

[∇ • [µ∇U]+∇ •

[µ(∇U)T]− 2

3µ∇ • [tr∇UI]

]+n •∇ •devΣ+− ∂ρU

∂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 stan-

dard 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 bound-

ary 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).


Discrete grid pointsShortest resolvable wave with highestresolvable wave number

∆x

L=2∆x

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

∑k=1

bnkeiα j (3.77)

or alternatively

D(x, t) =kk

∑k=1

bk(t)eikmx (3.78)

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= 2πL k, where L is the wave length; and α is the dimensionless wave number

which is equal to α = km∆x = 2πL k∆x = k2π

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

3.7. Stability Analysis 69

(grows, decays, or stays constant) in time for any wave number index k. For example

for k = 1

Dnj = bn

1eiα 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+1j = bn+1eiα j (3.80)

and at j +1 point at the n-th instance it gives:

Dnj+1 = bneiα( 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 = Gbn (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

concerned in this thesis, G depends on the Courant number, Co = c∆t∆x, where c is the

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))

Coα=

−tan−1(

Im(G)Re(G)

)

Coα(3.84)


t

n

n-1

n-2

j j+1j-1

∆x

∆t

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:

ρ∂2Dx

∂t2 = (2µ+λ)∂2Dx

∂x2 , tε [0,∞) , xε [0,L] (3.85)

The wave velocity for plain strain is given by Equation 3.86.

c1 =

√2µ+λ

ρ=

√1−ν

(1+ν)(1−2ν)

ϒρ

(3.86)

Longitudinal waves in uniform bars with uniform cross section are given by Equa-

tion 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.

∂2D∂t2 = c2∂2D

∂x2 (3.87)

Using the first order Euler implicit difference approximation to approximate the

second order time derivative ∂2D∂t2 and the second order central approximation for the


φnj = ∑kk

k=1bnkeiα j φn

j+1 = ∑kkk=1bn

keiα( j+1)

φn−1j = ∑kk

k=1bn−1k eiα j φn

j = ∑kkk=1bn

keiα j

φn−2j = ∑kk

k=1bn−2k eiα j φn

j−1 = ∑kkk=1bn

keiα( j−1)

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

space derivatives ∂2D∂x2 , the following expression is obtained:

∂2D∂t2 =

Dnj −2Dn−1

j +Dn−2j

∆t2 +O(∆t) (3.88)

∂2D∂x2 =

Dnj−1−2Dn

j +Dnj+1

∆x2 +O(∆x2) (3.89)

By substituting these in Equation 3.87, one obtains:

Dnj −2Dn−1

j +Dn−2j = C2

o

(Dn

j−1−2Dnj +Dn

j+1

)(3.90)

where Co = c∆t∆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 appear-

ing 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

∑k=1

[1−2bn−1

k

bnk

+bn−2

k

bnk

] = C2o

kk

∑k=1

[e−iα −2+eiα] (3.91)

The ratiobn−2

kbk

can be written as:

bn−2k

bnk

=bn−2

k

bn−1k

bn−1k

bnk

=1

Gn−1

1Gn (3.92)

and substitution together with the identities e−iα +eiα = 2cosα and cosα−1 =

−2sin2 α2 in Equation 3.91 yields:

kk

∑k=1

[(1

Gn−1 −2)1

Gn ] = −4C2o

kk

∑k=1

(sin2 α2)−1 (3.93)

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


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

Gn =2− 1

Gn−1

1+4C2o sin2 α

2

(3.94)

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 charac-

teristic 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 .

(1+4C2o sin2 α

2)λ2−2λ+1 = 0 (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± i2Cosinα2

(3.96)

The amplitude and phase portrait of G at different Co numbers Co = 14, 1

2, 34, 1, 5

4

are shown in Figure 3.8.

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∂Ux

∂t=

∆t2

(2µ+λ)n

∑m=1

[∂2Um

x

∂x2 +∂2Um−1

x

∂x2 ] , tε [0,∞) , xε [0,L] (3.97)


0 5 10 15 20 250.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

N

|G|

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

(a) Amplitude portrait

0 5 10 15 20 250.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

N

Q

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

(b) Phase portrait

Figure 3.8: Accuracy portrait of the amplification factor G for the1D hyperbolic equation (3.96).


By substituting the wave velocity c =√

ϒ/ρ in Equation 3.97 the 1D velocity for-

mulation can be written as Equation 3.98.

∂Ux

∂t=

∆t2

c2n

∑m=1

[∂2Um

x

∂x2 +∂2Um−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∂2Dx

∂x2 (3.99)

∂Dx

∂t= Ux (3.100)

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

derivatives ∂2D∂x2 for Equation 3.99 the following is obtained:

∂U∂t

=Un

j −Un−1j

∆t+O(∆t) (3.101)

∂2D∂x2 =

Dnj−1−2Dn

j +Dnj+1

∆x2 +O(∆x2) (3.102)

The trapezoidal rule is used for the approximation of the integral in Equation

3.100 (as described in Section 2.3.2) , thus

Dnj = Dn−1

j +∆t2

(Un

j +Un−1j

)(3.103)

Thus, Equation 3.98 can be equivalently written as:

Unj −Un−1

j

∆t= c2

(Dn−1

j+1 −2Dn−1j +Dn−1

j−1

∆x2 +∆t2

Unj+1−2Un

j +Unj−1+Un−1

j+1 −2Un−1j +Un−1

j−1

∆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


t

n

n-1

n-2

j j+1j-1

∆x

∆t

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


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

approximations and setting ζ = ∆t2 and ξ =

C2o

∆t give:

Dnj −Dn−1

j = ζ(Un

j +Un−1j

)(3.105)

Unj −Un−1

j = ξ(Dn

j−1−2Dnj +Dn

j+1

)(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 appear-

ing 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 = 2cosα and cosα−1 = 2sin2 α2 , the

following equations can be obtained:

kk

∑k=1

[bnk,D−bn−1

k,D ] = ζkk

∑k=1

[bnk,U +bn−1

k,U ] (3.107)

kk

∑k=1

[bnk,U −bn−1

k,U ] = 4ξkk

∑k=1

[sin2 α2

bnk,D] (3.108)

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

bn+1U

]

= A

[bn

D

bnU

]

(3.109)

where A is the matrix

A =

1−2C2

osin2 a2

1+2C2osin2 a

2

−4C2o

∆t sin2 a2

2

1+2C2osin2 a

2∆t

1+2C2osin2 a

2

11+2C2

osin2 a2

The eigenvalues of the 2x2 matrix A will give the characteristic or dispersion

equation, which reads:

λ2− 2(1−C2o sin2 α

2)

1+2C2o sin2 α

2

λ+1

1−2C2o sin2 α

2

= 0 (3.110)


0 5 10 15 20 250.85

0.9

0.95

1

N

|G|

Co=1/4

wave equationvelocity formulation

0 5 10 15 20 250.7

0.8

0.9

1

N

|G|

Co=1/2


0 5 10 15 20 250.4

0.6

0.8

1

N

|G|

Co=3/4


0 5 10 15 20 250.4

0.6

0.8

1

N

|G|

Co=1


0 5 10 15 20 250.2

0.4

0.6

0.8

1

N

|G|

Co=5/4


0 5 10 15 20 250.2

0.4

0.6

0.8

1

N

|G|

Co=2


Figure 3.10: Amplitude portrait of the 1D velocity formulation incomparison with the wave equation (displacementformulation).

Setting ϑ =−2C2o sin2 α

2 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

−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, 1

2, 34, 1, 5

4

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.


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 mathe-

matically 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 conver-

gence 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 ve-

locity components appears in all equations creating a highly coupled system. The

PISO algorithm has been adopted to solve iteratively the coupled system. A decou-

pled 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 im-

plementation, the PISO algorithm solves for velocity and pressure, rather than their

corrections. The PISO algorithm is typically used for the solution of the Navier-

Stokes 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 liter-

ature 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 litera-

ture. 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.

Chapter 4

Validation of the new formulation

for solids

4.1 Introduction

The new unified formulation with velocity and pressure as primitive variables pre-

sented 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 Value

Modulus E 4×109 PaPoisson’s ratio ν 0.3Density ρs 1450 kg/m3

Length l 20 mHeight h 5 mDepth w 1 m

Table 4.1: Material properties and dimentions of the beam.

81

82 Chapter 4. Validation of the new formulation for solids

��

��

h

τ

l

y

x

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 = 106Pa 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= 1m. 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); Geradin 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. Unfor-

tunately 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 avail-

able. The distribution of stress in the beam is given by (Timoshenko and Goodier,

1970):

σxx = 12τxyh2 (4.1)

σyy = 0 (4.2)

4.3. Analytical solution 83

σxy = 6τ[

14−(y

h

)2]

(4.3)

The beam displacements in the horizontal and vertical direction respectively are

given by:

Dx =2τ

ϒh3

[3(l2−x2)y+(2+ν)y3] (4.4)

Dy =12τϒh3

[x3

6+

l3

3+

x2(νy2− l)+

(h2

)2

(1+ν)(l −x)

]

(4.5)

The maximum deflection of the beam at x = 0 is found by solving Equation 4.5

and reads as:

δ =4τϒ

l3

h2

[

1+34(1+ν)

(hl

)2]

(4.6)

The term in brackets in Equation 4.6 is a two dimetional correction; in the one

dimensional solution, this term is omitted. The term 34(1+ ν)

(hl

)2 ≃(2h

4

)2is the

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 = 0.340m.

The speed of propagation of the stress wave through the beam, for this particular

material is:

C =

√ϒρ

= 1660m/sec (4.7)

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

√ϒh2

12ρsl4 (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 funda-

mental eigenvalue, µ1 = 1.875 the frequency of the undamped oscillation is:

ω2 = 1.8754 ϒh2

12ρsl4 (4.9)

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


100

101

102

103

101

102

103

104

105

106

Eigen values

Eig

en fr

eque

ncie

s

(a) Eigenfrequencies ωn versus constants µn

for no of modes n = 1 : 100.

100

101

102

100

101

102

103

104

105

no of modes

Fre

quen

y of

osc

illat

ion

[Hz]

(b) Frequency of oscilation f against modesn = 1 : 100

n 1 2 3 >3

µn 1.875 4.694 7.855 (2n−1)π2 (approx.)

ωn 21.070 132.054 369.792 724.660, 1197.788, ...

fn 3.354 21.017 58.854 115.333, 190.634, ...

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

4.4. Results 85

4.4 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 ve-

locity formulation and the results of this analysis will help with the interpretation

of the results of the present chapter. Thereafter, the effects of discretisation dis-

cretisation 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 im-

portance 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 analy-

sis

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


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Dis

plac

emen

t d

Time t

(a)

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

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

4.4. Results 87

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14 16 18

Dis

plac

emen

t D (

m)

Time t (sec)

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 me-

chanics 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 con-

tribution 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 de-

pend on the accuracy of the equation discretisation method and the discretisation

of the computational domain. The discretisation errors introduced by the term dis-


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 momen-

tum 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 ∇ •devΣ+ is discretised. This term

is given by the formula:

Σ+ = Σ+∆t2

[(2µ+λ)∇Uo+µ(∇Uo)T −λtr(∇Uo)I− (µ+λ)∇Uo] (4.10)

∇ •Σ+ = ∇ •Σ+∆t2

{∇ • [(2µ+ λ)∇Uo]+ ∇ •

[µ(∇Uo)T]−λ∇ • tr (∇Uo)I− (µ+ λ)∇ •∇Uo}

(4.11)

or

devΣ+ = devΣ+∆t2

[(2µ+λ)∇Uo+µ(∇Uo)T− 2

3tr(∇Uo)I− (µ+λ)∇Uo

](4.12)

∇ •devΣ+ = ∇ •devΣ+∆t2

{∇ • [(2µ+ λ)∇Uo]+ ∇ •

[µ(∇Uo)T]− 2

3∇ • tr(∇Uo)I− (µ+ λ)∇ •∇Uo

}

(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 discreti-

sation 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 contribu-

tion 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

4.4. Results 89

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 sys-

tem. 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 equa-

tion, then:

U •∂ρU∂t

= U •∇ •σ (4.14)

Using the following identity

σ ••∇U = ∇ •(σ •U)−U •∇ •σ (4.15)

and the momentum Equation 3.63 can be transformed to

U •∂ρU∂t

= ∇ •(σ •U)−σ ••∇U (4.16)

In Equation 4.16 the term on the left hand side denotes the kinetic power of the

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 =12

∂ρUU∂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


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

5

time (s)

Tot

al P

ower

(W

)= E

xter

nal −

Str

ain

− K

inet

ic

LaplacianDivergence−Grad

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)

Total power

TP = EP−SP−KP (4.21)

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 discretisa-

tion of the ∇ • [(2µ+λ)∇U] term and the consistent discretisation. In the inconsistent

discretisation, the term is evaluated implicitly using the Laplacian operator (com-

pact 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

4.4. Results 91

−60000

−40000

−20000

0

20000

40000

60000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tot

al p

ower

(W

)

Time t (sec)

Comparison of tolerences 10e−6, 10e−7, 10e−8

tolerence 10e−7tolerence 10e−6toterence 10e−8

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 ap-

proach 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 ac-

curacy 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 Differenc-

ing schemes for the velocity-pressure formulation.

It should be noted that in the following figures only the envelope of the displace-

ment (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-


0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time t (sec)

Dis

plac

emen

t D (

m)

displacement formulationvelocity based formulation

Figure 4.7: Comparison of displacement formulation andvelocity-based formulation for the Euler Implicitdiscretisation 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 formu-

lation 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 displace-

ment 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 displace-

ment 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

4.4. Results 93

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20 25 30

Dis

plac

emen

t d

Time t

Comparison of Euler Implicit and Backward Differencing descretistion scemes for 1e−4 time step

Euler ImplicitBackward Differencing

Figure 4.8: Comparison of Euler Implicit and BackwardDifferencing discretisation scheme (envelope ofdisplacement).

which is first order accurate. Thus, just by using a velocity based formulation the ac-

curacy of the computation increases without using a more accurate time integration

scheme such as Newmark.

Velocity-pressure formulation: Euler Implicit versus Backward Differenc-

ing

In this subsection, a comparison of the first order time derivative (velocity-pressure

formulation) using different discretisation schemes, for the treatment of the tempo-

ral 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 discreti-

sation 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 secperiod has much smaller dissipation. In terms of computational over-

head 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


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14

Dis

plac

emen

t d

Time t

Displacement vs Time

1e−4EI1e−5EI1e−6EI

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 commer-

cial package. From the standard stress analysis, the first four eigenfrequencies were

4.4. Results 95

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20 25 30

Dis

plac

emen

t d

Time t

Comparison of Euler Implicit for 1e−5 time step and Backward Differencing for 1e−4 time step

EI 1e−5BD 1e−4

Figure 4.10: Comparison of Euler Implicit using time step size1e-5 s against Backward differencing using time stepsize of 1e-4 s (envelope of displacement).

n 1 2 3 4

fn 0.677 3.2102 4.2015 5.0791

Table 4.2: Computational calculations for the vibrationeigenfrequencies of vibration using for the twodimensional beam bending case using the ANSYS finiteelement commercial package.


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20 25 30

Dis

plac

emen

t d (

m)

Time t (sec)

Mesh resolution comparison 40x10, 60x20BD, 200x50BD

40x1060x20BD

200x50BD

Figure 4.11: Mesh resolution comparison for meshes: 40x10, 60x20and 200x50 cells. Time step size used is 1e-4 andtemporal term discretisation scheme is Backwarddifferencing (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 fre-

quency 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 Fig-

ure 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

4.4. Results 97

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time (s)

disp

lace

men

t (m

)

Dirichlet using momentumDirichlet using pressure definition

Figure 4.12: Comparison of different boundary conditions forpressure in the fully implicit velocity-pressureformulation.

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 fluid-

structure 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.


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 bound-

ary condition is according to the paper of Gresho and Sani (1987) but for a com-

pressible 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 os-

cillation 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 = 5e5Pa, 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.

4.4. Results 99

Variable Analytical Predicted % DifferenceBeam size: 10mx5m; end shear: 1e6 Pa

Max Displacement [m] 0.0995 0.0866 12.96Frequency [Hz] 13.41 11.64 13.29

Beam size: 20mx5m; end shear: 1e6 PaMax Displacement [m] 0.68 0.62 8.82Frequency [Hz] 3.35 3.32 0.9

Beam size: 40mx5m; end shear: 1e6 PaMax Displacement [m] 5.2 4.72 9.21Frequency [Hz] 0.84 0.86 2.05Beam size: 20mx5m; end shear: 5e5 PaMax Displacement [m] 0.34 0.313 7.94Frequency [Hz] 3.35 3.29 1.79

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

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Dis

plac

emen

t (m

)

Time (sec)

Figure 4.13: Beam with size 10mx5m. No of cells used for themesh is 20x10cells , time step size used is 1e-4 andtemporal term discretisation scheme is Backwarddifferencing.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Dis

plac

emen

t (m

)

Time (sec)

Figure 4.14: Beam with size 40mx5m. No of cells used for themesh is 80x10cells , time step size used is 1e-4 s andtemporal term discretisation scheme is Backwarddifferencing.

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 = 5e5Pa, 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 dif-

ference 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,

4.5. Closure 101

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (sec)

Dis

plac

emen

t (m

)

Figure 4.15: Beam with size 20mx5m, with applied end shearτ = 5e5Pa. No of cells used for the mesh is 40x10cells, time step size used is 1e-4 s and temporal termdiscretisation scheme is Backward differencing.


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 dis-

cretisation 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 fluid-

structure 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.

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 tak-

ing 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 varia-

tion and geometric tapering. In order to obtain a complete set of experimental data

assessing these variations, a set of flexible tubes was manufactured.

105

106Chapter 5. Wave propagation experiments in flexible vessels with wall thickness


5.2.1 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 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 Aorta ThoracalisTop internal radius [mm] 20

Bottom internal radius[mm] 11Length [mm] 315

Slope -0.014ϒ∗h/D [MPa] 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 manu-

factured 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 ta-

pered, homogeneously thin-walled tube are made as well.

5.2. The Tube Models Methodology 107

Type 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

ϒ: Young’s modulus, h: wall thickness, D: diameter

Table 5.3: Geometrical parameters of tubes manufactured.



0 10 20 30 405

6

7

8

9

10x 10

−3

Tube length [cm]

Thi

ckne

ss o

f tub

e F

[cm

]

0 10 20 30 401

1.5

2

2.5

3

Tube length [cm]

Inte

rnal

dia

met

er o

f tub

e F

[cm

]

0 10 20 30 405

6

7

8

9

10x 10

−3

Tube length [cm]

Thi

ckne

ss o

f tub

e C

[cm

]

0 10 20 30 401.5

2

2.5

3

3.5

Tube length [cm]

Inte

rnal

dia

met

er o

f tub

e C

[cm

]

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

5.3. Material Properties of the Tubes 109

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 (x-

servomotor). 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 servo-

controller. 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 thick-

ness. 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 con-

centrations varying from α = 17−22.73%of PU in THF solvent.

Physical properties Polyurothane (PU)

Density(kg/m3) 880Ultimate tensile strength(MPa) 30

Elongation at break (%) 500Tear propagation resistance (kN/m) 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



10−2

10−1

100

101

102

103

10−3

10−2

10−1

100

101

time [s]

Ten

sile

For

ce [N

]

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

5.3. Material Properties of the Tubes 111

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

0ϒ(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 ∞

0tx−1e−tdt (5.4)

the complex Young’s modulus then is expressed as:

ϒ(s) = cΓ(1−n)sn (5.5)

The complex viscoelastic modulus can be written as a function of the angular

frequency as follows:

ϒ(ω) = cΓ(1−n)ωnei nπ2 (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)



The following values of c and n are obtained: 2.3 · 106 [Pas−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):

ϒ =1

N+1

N

∑i=0

Re[ϒi(s)] (5.7)

where N is the harmonic number corresponding to the bandwidth of the excita-

tion signal. In our case it is 40. The Youngs modulus was calculated to be 1.72MPa.

5.4 Measurement Methods

5.4.1 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 withwater, 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 mea-

sure 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



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 mmouter

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 RF-

matrix 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 examin-

ing 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 wall-

detection 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 Results

5.5.1 Static pressure - initial diameter relation

In order to ensure that the the tubes undergo small deformations during the ex-

periments, 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.



1 1.5 2 2.5 3 3.5 4 4.52.38

2.4

2.42

2.44

2.46

2.48

2.5

2.52

2.54

2.56x 10

4

Static pressure [kPa]

Initi

al D

iam

eter

[um

]

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

5.5. Results 117

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 =

√∑n

i=1(xi −x)2

n−1(5.8)

A typical result can be seen in Figure 5.5.

0 0.2 0.4 0.6 0.8 1−4

−2

0

2

4

6

8

Flo

w r

ate

Q [l

/min

]

Time t [s]

meanmean+stdmean−std

0 0.2 0.4 0.6 0.8 1−5

0

5

10

15

Pre

ssur

e P

[mm

Hg]

Time t [s]

Tube type B axial location along its length: 100 [mm]


0 0.2 0.4 0.6 0.8 1−100

0

100

200

300

400

500

Wal

l dis

tent

ion

[um

]

Time t [s]


0 0.2 0.4 0.6 0.8 1−3

−2

−1

0

1

2

3

Pre

ssur

e gr

adie

nt

Time t [s]


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

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



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.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Scaled time

Axi

al p

ositi

on

Pressure propagation

CAB

Figure 5.6: Normalised pressure measurements every 50 mm alongthe length of the tube against scaled time for straighttubes: types A,B,C (A: straight tube with constantwall thickness of 0.1 mm; B: straight tube withconstant wall thickness of 0.05 mm; C: straight tubewith 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.

5.5. Results 119

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Scaled time

Axi

al p

ositi

on


FDE

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



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Scaled time

Axi

al p

ositi

on


AF

Figure 5.8: Normalised pressure measurements every 50 mm alongthe length of the tube against time for tube types Aand F (A: straight tube with constant wall thickness of0.1 mm; F: tapered tube with variable wall thicknessof 0.1-0.05 mm).

5.5. Results 121

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Scaled time

Axi

al p

ositi

on


CE

Figure 5.9: Normalised pressure measurements every 50 mm alongthe length of the tube against time for tubes types Cand E (C: straight tube with variable wall thickness0.05-0.1 mm; E: tapered tube with constant wallthickness of 0.05 mm).



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-to-

peak 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 varia-

tion, 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

5.5. Results 123

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Scaled time

Axi

al p

ositi

on

Flow propagation

CAB

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



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Scaled time

Axi

al p

ositi

on

Flow propagation

FDE

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

5.5. Results 125

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Scaled time

Axi

al p

ositi

on

Flow propagation

AF

Figure 5.12: Normalised flow rate measurements every 50 mmalong the length of the tube against time for tubestypes A and F (A: straight tube with constant wallthickness of 0.1 mm; F: tapered tube with variablewall thickness of 0.1-0.05 mm).



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Scaled time

Axi

al p

ositi

on

Flow propagation

CE

Figure 5.13: Normalised flow rate measurements every 50 mmalong the length of the tube against time for tubestypes C and E (C: straight tube with variable wallthickness 0.05-0.1 mm; E: tapered tube with constantwall 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 gradi-

ent. 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.



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Scaled time

Axi

al p

ositi

on

Pressure gradient propagation

CAB

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

5.5. Results 129

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Scaled time

Axi

al p

ositi

on


FDE

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



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Scaled time

Axi

al p

ositi

on


AF

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

5.5. Results 131

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Scaled time

Axi

al p

ositi

on


CE

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



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Scaled time

Axi

al p

ositi

on

Wall distension propagation

CAB

Figure 5.18: Normalised wall motion measurements every 50 mmalong the length of the tube against time for tubestypes A, B, C (A: straight tube with constant wallthickness of 0.1 mm; B: straight tube with constantwall thickness of 0.05 mm; C: straight tube withvariable wall thickness of 0.05-0.1 mm).

5.5. Results 133

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Scaled time

Axi

al p

ositi

on


FDE

Figure 5.19: Normalised wall motion measurements every 50 mmalong the length of the tube against time for tubestypes D, E, F (D: tapered tube with constant wallthickness of 0.1 mm; E: tapered tube with constantwall thickness of 0.05 mm; F: tapered tube withvariable wall thickness of 0.1-0.05 mm).



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Scaled time

Axi

al p

ositi

on


AF

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

5.5. Results 135

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Scaled time

Axi

al p

ositi

on


CE

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



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 geo-

metric 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 Flexi-

ble Vessels

The theoretical investigation of the propagation of pressure disturbances in disten-

sible 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

138Chapter 6. Comparison of experimental results with linear wave propagation

methods

r

z

ξ = ξ(z, t)

ζ = ζ(z, t)

θ

h

r

P

P’

Figure 6.1: Tube motion variables. Point P(z, r) on the surface ofthe 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∂t

+ρ∇ •(UU) = ∇ •(η∇U)−∇p (6.1)

∇ •U = 0 (6.2)

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

compared to the diameter 2ro of the tube ( λ2r0

≫ 1 ). It is convenient to make the

the Navier-Stokes equations non-dimensional. Therefore U′z and U

′r are considered

as typical velocities in the axial (z) and radial (r) directions respectively. The ratio

between the two velocities is defined by κ = U′r

U ′z, and in the following treatment κ is

considered to be small and L/2r0 ≫ 1. Provided that the Mach numbers

∣∣∣∣U

′r

c

∣∣∣∣≪ 1

and

∣∣∣∣U

′z

c

∣∣∣∣≪ 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

ρ∂Ur

∂t+

∂p∂r

= η[

∂2Ur

∂r2 +1r

∂Ur

∂r−Ur

r2

](6.3)

ρ∂Uz

∂t+

∂p∂z

= η[

∂2Uz

∂r2 +1r

∂Uz

∂r

](6.4)

1r

∂(rUr)

∂r+

∂Uz

∂z= 0 (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∂t

= 0 (6.6)

Symmetry requires,

Ur = 0,∂Uz

∂r

∣∣∣∣r=0

= 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

ϕ = ϕei(ωt−kz) (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

√ωη

(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.

k(ω) = ±ωc0

√1

1−F10(6.10)

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


methods

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)

αi3/2J0(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:

c =c0

ℜ(k)(6.12)

The attenuation constant is given by:

γ =−2πℑ(k)

ℜ(k)(6.13)

It should be mentioned here for clarity that the solution of the dispersion equation

can be expressed in the general form

ϕ = ϕI±ei(ωt±kI z) + ϕII±ei(ωt±kII z) (6.14)

where the two complex roots give the velocity of propagation of two distinct out-

going 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 pre-

dicted 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).


Type of tube anchoring ψ for thin-wall tube ψ for thick-wall tube

At its upstream only 1− ν2

tr (1+ν)+ 2r0

2r0+h

(1+ ν

2

)

Throughout against axial movement 1−ν2 hr0

(1+ν)+2r0(1+ν2)

2r0+h

With expansion joints throughout 1 hr0

(1+ν)+ 2r02r0+h

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

c =

√Kρ f

(1+ψ

2r0

hKϒ

)−1

(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

when wall thicknessinner diameter≡ h

2r0≪ 1 (say h

2r0< 1

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

c2f

=1

c20

+1

c21

(6.16)

where

c1 =

√Kρ f

(6.17)

and

c0 =

√

ψ−1 ϒh2r0ρ f

(6.18)

In the case when the tube wall is very stiff, ϒ ≫ K and cf = 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 cf = 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) .


methods

reflected wave

incomming wave

transmitted (part 2)

transmitted (part 1)

L L+1 L+2

z

r

m k

j

n

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 transi-

tions 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 in-

coming 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 , QI ) is reflected by the junction

(pR, QR) and another part goes through and it is partially transmitted in part 1

(pT1, QT1) and part 2 (pT2, QT2). The transition of a wave through a junction de-

pends 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 =ρcA

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


given by

Q = ± Aρc

p (6.19)

At transition n where z= L holds that:

pIm(ω,L, t)+ pR

m(ω,L, t) = pT1m(ω,L, t) = pT2

m(ω,L, t)

QImn(ω,L, t)− QR

mn(ω,L, t) = QT1n j(ω,L, t)+ QT2

nk(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) =pR

m(ω,L)

pIm(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 = pI (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)

= pI (ω,0)e−ikz[1+Γe−2ik(L−z)]eiωt(6.22)

The flow rate is similar

Q(ω,z, t) = QI (ω,z, t)− QR(ω,z, t)

= QI(ω,0)e−ikz[1−Γe−2ik(L−z)]eiωt

(6.23)

The input impedance of the system is given by:

ZI =pI (ω,0)e−ikz[1+Γe−2ik(L−z)]eiωt

Q(ω,0)e−ikz[1−Γe−2ik(L−z)]eiωt= Zc

1+Γe−2ik(L−z)

1−Γe−2ik(L−z)(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

p(ω,z, t) = pI (ω,0)e−ikzN

∏λ=1

[1+Γe(−1)λ2ik(L−z)]eiωt (6.25)

Q(ω,z, t) = QI(ω,0)e−ikz

N

∏λ=1

[1−Γe(−1)λ2ik(L−z)]eiωt (6.26)


methods

Properties Units Value

ρ f kg/m3 998

η f N ·s/m2 1e-3ν - 0.5c Pa S−n 1.3*10e6n - 0.065Ys MPa 1.72

ϒs MPa cΓ(1−n)ωnei nπ2

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 mmand 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 mmapart 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 mmand length of 446 mmand tube B with diameter

of 25 mm, wall thickness of 0.05 mmand length of 446 mm. The pressure, the flow

and the wall distension ware calculated at 10 locations along the length the tube

50 mmapart. The two pressures p0 and p1 where 17 mmapart 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 occur-

ring 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 Viscoelas-

tic Material

The simulations for tubes A and B where repeated including the viscoelastic proper-

ties 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 agree-

ment. 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 calcu-

lated 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


methods

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Time [s]

Axi

al p

ositi

on

Pressure P0 (Type A)

TheoreticalExperimental

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Time [s]

Axi

al p

ositi

on



(b)

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

6.6. Closure 147

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45Time [s]

Axi

al p

ositi

on

Flow Q (Type A)


Figure 6.5: Comparison of the experimental measurements of theflow on a straight tube with constant wall thickness of0.1 mmwith linear analytical model foran elastic tube.


methods

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Time [s]

Axi

al p

ositi

on

Wall distention (Type A)


Figure 6.6: Comparison of the experimental measurements of thewall distension on a straight tube with constant wallthickness of 0.1 mmwith linear analytical model foranelastic tube.

6.6. Closure 149

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Time [s]

Axi

al p

ositi

on

Pressure P0 (Type B)


(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Time [s]

Axi

al p

ositi

on



(b)

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


methods

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Time [s]

Axi

al p

ositi

on

Flow Q (Type B)


Figure 6.8: Comparison of the experimental measurements of theflow on a straight tube with constant wall thickness of0.05 mmwith linear analytical model foran elastictube.

6.6. Closure 151

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Time [s]

Axi

al p

ositi

on

Wall distention (Type B)


Figure 6.9: Comparison of the experimental measurements of thewall distension on a straight tube with constant wallthickness of 0.05 mmwith linear analytical model foranelastic tube.


methods

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Time [s]

Axi

al p

ositi

on



(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Time [s]

Axi

al p

ositi

on



(b)

Figure 6.10: Comparison of the experimental measurements of thepressure on a straight tube with constant wallthickness of 0.1 mmwith linear analytical model foraviscoelastic tube.

6.6. Closure 153

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45Time [s]

Axi

al p

ositi

on

Flow Q (Type A)


Figure 6.11: Comparison of the experimental measurements of theflow on a straight tube with constant wall thicknessof 0.1 mmwith linear analytical model foraviscoelastic tube.


methods

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Time [s]

Axi

al p

ositi

on

Wall distention (Type A)


Figure 6.12: Comparison of the experimental measurements of thewall distension on a straight tube with constant wallthickness of 0.1 mmwith linear analytical model foraviscoelastic tube.

6.6. Closure 155

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Time [s]

Axi

al p

ositi

on



(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Time [s]

Axi

al p

ositi

on



(b)

Figure 6.13: Comparison of the experimental measurements of thepressure on a straight tube with constant wallthickness of 0.05 mmwith linear analytical model foraviscoelastic tube.


methods

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Time [s]

Axi

al p

ositi

on

Flow Q (Type B)


Figure 6.14: Comparison of the experimental measurements of theflow on a straight tube with constant wall thicknessof 0.05 mmwith linear analytical model foraviscoelastic tube.

6.6. Closure 157

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Time [s]

Axi

al p

ositi

on

Wall distention (Type B)


Figure 6.15: Comparison of the experimental measurements of thewall distension on a straight tube with constant wallthickness of 0.1 mmwith linear analytical model foraviscoelastic tube.


methods

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 be-

haviour. 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 match-

ing 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 New-

tonian 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 de-

scribed. 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 velocity-

pressure 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 as-

sumptions and this has to be taken into account when comparing analytical with

the numerical results. The conservation of the total energy of the numerical imple-

mentation was also tested. The numerical accuracy of the standard displacement

formulation was compared against the velocity based formulation. The dissipa-

tion characteristics of the numerical integration technique were also in agreement

with the conclusions obtained from the one dimensional stability analysis. Differ-

ent 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 161

7.2 Main achievements

This research has resulted in the following specific contributions:

• The idea of a unified solution methodology for solving fluid-structure inter-

action 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 re-

formulated 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 facil-

itates 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 exper-

imental work focused on the effect of geometric tapering and wall thickness

variation on pulse wave propagation in flexible vessels. The experimental mea-

surements 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


thickness. It was found that when the viscoelastic properties of the wall ma-

terial 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 in-

teraction such as mathematical modelling and experiments. Therefore, the recom-

mendations 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 in-

clude 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 de-

formations 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∂t

+∇ •(ρU) = 0 (7.1)

and

∂ρU∂t

+∇ •(ρUU) = 2αdev(sym(∇U))+φ∇ •devΣ+−∇p (7.2)

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).

7.3. Future work 163

φ = 1φ = 1 φ = 1φ = 1φ = 1

φ = 0φ = 0φ = 0φ = 0φ = 0

α = ηα = ηα = ηα = ηα = ηρ = ρ fρ = ρ fρ = ρ fρ = ρ fρ = ρ f

ρ = ρ f ρ = ρ f ρ = ρ f ρ = ρ f ρ = ρ fα = η α = η α = η α = η α = η

φ = 0φ = 0φ = 0φ = 0φ = 0

φ = 0 φ = 0 φ = 0 φ = 0 φ = 0

φ = 1 φ = 1 φ = 1 φ = 1φ = 1

ρ = ρs ρ = ρs ρ = ρs ρ = ρs ρ = ρs

ρ = ρsρ = ρsρ = ρsρ = ρsρ = ρs

ρ = ρ f ρ = ρ f ρ = ρ f ρ = ρ f ρ = ρ fα = η α = η α = η α = η α = η

α = µ∆t2α = µ∆t

2α = µ∆t2α = µ∆t

2α = µ∆t2

α = µ∆t2 α = µ∆t

2 α = µ∆t2 α = µ∆t

2 α = µ∆t2

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 meshfor solving fluid structure interaction problems withthe unified solution method.

ρ =

{ρs f or solid

ρ f f or f luid(7.3)

α =

{µ∆t

2 f or solid

η f or f luid(7.4)

φ =

{1 f or solid

0 f or f luid(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 investi-

gated 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 velocity-

pressure 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


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 geo-

metric 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 Aorta Thoracalis

Top internal radius [mm] 20Bottom internal radius[mm] 11

Length [mm] 315Slope -0.014

ϒ∗h/D [MPa] 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.

Type 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

ϒ: Young’s modulus, h: wall thickness, D: diameter

Table A.3: Geometrical parameters of tubes manufactured.

A.2. Manufacturing set-up iii

A.2 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 = 2000counts/revolution and has a 1-1 relation-

ship with the connected rotating beam. The y-servomotor rotates at Ctrans = 4000

counts/revolution and the ball screw pitch is λ = 2.5mm. Thus, the rotational move-

ment 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 responsi-

ble 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

x-servomotornozzletrolley with

infraredlight

steel rod

y-servomotor

λ

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

ξ

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

A.4. Straight tube manufacturing v

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

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

V1 =π∗X12

[D(0)2+D(0)∗D(X)+D(X)2] , xε [0,X] (A.2)

V2 =π∗X12

[D(0)+d)2 +(D(0)+d)∗ (D(X)+d)+ (D(X)+d)2] , xε [0,X] (A.3)

VL = (V2−V1)∗100/α (A.4)

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

z=1

2X(D(X)−D(0)) (A.5)

Flow rate

FL =VL

t(A.6)

Frequency of servomotor rotation

f (x) =U(x)∆x

(A.7)

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:

U(x) = C∗ f (x) (A.8)

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

4000counts /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 Abbreviation Value

Diameter[mm] D 25Length [mm] X 500

Thicknes [mm] h(x) 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/100h∗π∗D

Crot

ξ(A.9)

The equation giving the axial velocity of the y-servomotor in counts/secis given

by:

Uax =FL ∗α/100h∗π∗D

Cax

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)Crot

Cax

λξ

(A.11)

A.4.2 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) =

√ϒρ

h∗

D∗(x)(A.12)

A.4. Straight tube manufacturing vii

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) =

√ϒρ

h(x)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). Substitut-

ing Equations A.12 and A.13 the relationship according to which the wall thickness

should be varying is obtained.

h(x) =h∗ ∗DD∗(x)

(A.14)

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

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.

0 10 20 30 403000

4000

5000

6000

7000

Rod length [cm]

Tra

nsla

tiona

l Vel

ocity

[cou

nts/

sec]

0 10 20 30 402000

2500

3000

3500

4000

4500

Rod length [cm]

Rot

atio

nal V

eloc

ity [c

ount

s/se

c]

0 10 20 30 405

6

7

8

9

10x 10

−3

Rod length [cm]

Thi

ckne

ss [c

m]

0 10 20 30 401.5

2

2.5

3

3.5

Rod length [cm]

Rod

dia

met

er [c

m]

Figure A.4: Translational velocity, rotational velocity, tube wallthickness and tube diameter versus the tube lengthfor 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)

1.25 [cm]

4 [cm] 44.6 [cm] 2.4 [cm]

2.5 [cm]

Figure A.5: Tapered tube steel rod dimensions.

Variable Abbreviation Value

Diameter at bottom [mm] D(0) 12.5Diamerter at top [mm] D(X) 25

Length [mm] X 446Horizontal to vertical slope z 0.014

Thickness [mm] h 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

12

[D(x)2+D(x)∗D(x+∆x)+D(x+∆x)2] , xε [0,X] (A.17)

A.5. Tapered tube manufacturing ix

∆V2(x) =π∗∆x

12

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

(A.18)

∆VL(x) = [∆V2(x)−∆V1(x)]∗100/a (A.19)

VL(x) =x=X

∑x=0

∆VL(x)∗100/α (A.20)

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∆VL(x)

Crot

ξ(A.21)

Uax(x) =FL ∗∆x∆VL(x)

Cax

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 rota-

tional and axial velocities in counts/secare give by:

Urot =FL ∗α/100h∗π∗D(x)

Crot

ξ(A.24)

Uax =FL ∗α/100h∗π∗D(x)

Cax

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 A.5. Tapered tube manufacturing

0 10 20 30 40

5000

6000

7000

8000

9000

Rod length [cm]

Tra

nsla

tiona

l Vel

ocity

[cou

nts/

sec]

0 10 20 30 402500

3000

3500

4000

4500

5000

5500

6000

Rod length [cm]

Rot

atio

nal V

eloc

ity [c

ount

s/se

c]

0 10 20 30 400

0.002

0.004

0.006

0.008

0.01

Rod length [cm]

Tub

e th

ickn

ess

[cm

]

0 10 20 30 40

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Rod length [cm]

Rod

dia

met

er [c

m]

Figure A.6: Translational velocity, rotational velocity, tube wallthickness and tube diameter versus the tube lengthfor tube E.

A.6. Wall thickness accuracy xi

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∗ =

√ϒρ

h∗

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 =

√ϒρ

h(x)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(x) =h∗

D∗D(x) (A.28)

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

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 A.6. Wall thickness accuracy

0 10 20 30 402000

4000

6000

8000

10000

12000

14000

16000

Rod length [cm]

Tra

nsla

tiona

l Vel

ocity

[cou

nts/

sec]

0 10 20 30 402000

4000

6000

8000

10000

Rod length [cm]

Rot

atio

nal V

eloc

ity [c

ount

s/se

c]

0 10 20 30 405

6

7

8

9

10x 10

−3

Rod length [cm]

Thi

ckne

ss [c

m]

0 10 20 30 40

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Rod length [cm]

Rod

dia

met

er [c

m]

Figure A.7: Translational velocity, rotational velocity, tube wallthickness and tube diameter versus the tube lengthfor tube F.

Fluid structure interaction in flexible vessels

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