PULSATILE FLOW IN A TUBE WITH A MOVING CONSTRICTION JI LIN ( B.Sc, FUDAN ) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2006 ACKNOWLEDGEMENTS ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my Supervisors, Assoc Prof H.T Low and Prof Y.T Chew for their valuable guidance, suggestions and support throughout the course of this research project Their advice and criticism has contributed much towards the formation and completion of the dissertation I would also like to express my gratitude to all the staff members in the Fluid Mechanics Laboratory for their constant assistance in the software and hardware support for the numerical work I also appreciate the technical advices and helpful encouragements from Assoc Prof S.X Xu of Fudan University, China He has corresponded with me through e-mail Financial support was sponsored through NUS Research Scholarship This support enables me to pursue the Ph.D program in the National University of Singapore My deepest appreciation is extended to my parents and aunt, whose many sacrifices made it possible for me to attempt and complete this contribution i TABLE OF CONTENTS TABLE OF CONTENTS Page ACKNOWLEDGEMENTS i TABLE OF CONTENTS ii SUMMARY vi NOMENCLATURE ix LIST OF FIGURES xi CHAPTER 1.1 INTRODUCTION Physiological Background 1.1.1 1.1.2 1.2 Physiological Flows Clinical and Bioengineering Applications Literature Review 1.2.1 1.2.2 1.3 Stationary Constriction Moving Constriction 15 Objectives of Present Study 21 1.3.1 Motivations 21 1.3.2 Objectives 22 1.3.3 Scope 23 ii TABLE OF CONTENTS CHAPTER METHODOLOGY 2.1 Problem Description 24 2.2 Analytical Approach 25 2.3 Numerical Method 33 2.3.1 Arbitrary-Lagrangian-Eulerian Finite Element Method 33 2.3.2 Governing Equations and Boundary Conditions 35 2.3.3 Numerical Procedures 38 2.3.4 Finite Element Discretization 40 CHAPTER VALIDATION OF NUMERICAL METHOD 3.1 Pulsatile Flow in a Circular Tube with a Stationary Stenosis 44 3.2 Flow in a 2-D Channel with a Moving Indentation on One Wall 46 3.3 Comparison between Analytical and Numerical Methods: Low Reynolds Number Pulsatile Flow in a Tube with a Radially-Oscillating Constriction CHAPTER 4.1 50 RESULTS AND DISCUSSION Analytical Study of Pulsatile Flow through a Radially-Oscillating Constriction 52 4.1.1 52 4.1.2 Flow Characteristics 54 4.1.3 4.2 Problem Definition Effect of Constriction Oscillation Amplitude ε 59 Numerical Study of Pulsatile Flow through a Radially-Oscillating Constriction 60 iii TABLE OF CONTENTS 4.2.1 60 4.2.2 Description of the Basic Flow 62 4.2.3 Effect of Constriction Oscillation Amplitude ε 67 4.2.4 Effect of Phase Lag θ 68 4.2.5 Effect of Reynolds Number Re 70 4.2.6 4.3 Problem Definition Effect of Womersley Number α 72 Numerical Study of Pulsatile Flow through an Axially-Oscillating Constriction 74 4.3.1 Problem Definition 74 4.3.2 Description of the Basic Flow 76 4.3.3 Effect of Constriction Ratio ε 80 4.3.4 Effect of Phase Lag θ 81 4.3.5 Effect of Reynolds Number Re 83 4.3.6 Effect of Womersley Number α 85 CHAPTER 5.1 Conclusions 5.1.1 5.1.2 5.1.3 5.2 CONCLUSIONS AND RECOMMENDATIONS 88 Analytical Study of Pulsatile Flow through a Radially-Oscillating Constriction 88 Numerical Study of Pulsatile Flow through a Radially-Oscillating Constriction 89 Numerical Study of Pulsatile Flow through an Axially-Oscillating Constriction 91 Recommendations 93 iv TABLE OF CONTENTS REFERENCES FIGURES 94 103 v SUMMARY SUMMARY In a diseased artery, the stenosis may vibrate with the pulsatile blood flow, mainly radially and to a smaller extent, axially In massage therapy, either by hand or mechanical devices, the artery wall is compressed and the resulting constriction may move radially and axially In a roller pump, having a radially or axially moving constriction on the tube wall may enhance the flow pulsation, which has been shown to improve vital-organ recovery after hypothermic cardiopulmonary bypass In order to study the mechanism of the above physiological/bioengineering phenomena, in the present study such constriction motion was modeled by two modes separately, i.e by imposing a radially-oscillating or axially-oscillating wave on a tube wall subjected to a pulsatile incoming flow A linear analytical approach was first developed to study a radially-oscillating axisymmetric constriction in a tube subjected to a low Reynolds number pulsatile flow An analytical form of the pressure-gradient versus velocity relationship was derived The results show that the fluctuations of pressure gradient, axial velocity and wall vorticity increase rapidly as the constriction oscillation amplitude increases The fluctuations due to the incoming pulsatile flow are amplified by the constriction motion If the constriction does not oscillate but remains at its mean position, the fluctuation in the downstream flow, due to the incoming pulsatile flow, is smaller The vi SUMMARY analysis may be used for mildly oscillating constrictions without complications of flow separation and non-linearity The analytical solution may also be useful as a means of validating numerical models of oscillating constrictions with large amplitudes Next, a numerical model was developed to solve pulsatile flow through a tube with a radially-oscillating axisymmetric constriction The moving boundary of the large amplitude oscillation was solved by an Arbitrary-Lagrangian-Eulerian (ALE) finite element method The effects of constriction oscillation amplitude, phase lag between the constriction motion and incoming flow pulsation, Reynolds number and Womersley number were considered The basic features observed are the flow fluctuation amplification and wavy flow pattern with complicated vortices development for large Womersley number (α = 10) However, the effects induced by the constriction radial oscillation are less obvious at large Reynolds number, for example Re = 1000, as the flow is dominated by the large convective inertia The results also show that a stationary constriction assumption may overestimate the wall shear stress in the stenosed arteries Finally, a numerical model was developed to solve pulsatile flow through an axiallyoscillating axisymmetric constriction The effects of constriction ratio, phase lag between the constriction motion and incoming flow pulsation, Reynolds number and Womersley number were considered The main findings are that the downstreammoving constriction reduces the wall vorticity and pressure loss across the vii SUMMARY constriction; and vice versa for the upstream-moving constriction Other observations include the flow unsteadiness amplification, wavy flow pattern and complicated vortices development when the Womersley number is large (α = 10) These observed effects are less obvious at high Reynolds number, where the flow unsteadiness induced by the constriction motion may be somehow overshadowed by the large convective inertia of the incoming flow viii NOMENCLATURE NOMENCLATURE 2πfρ α Womersley number = R0 ε Dimensionless oscillation amplitude of a radially-oscillating constriction, or dimensionless constriction ratio of an axiallyoscillating constriction θ Phase lag between the constriction motion and incoming flow pulsation µ Fluid dynamic viscosity ρ Fluid density σ Ratio of tube radius versus constriction length = D Constriction axial oscillation range for an axially-oscillating constriction j Unit imaginary number J0 Zeroth order Bessel function J1 First order Bessel function L Constriction length for a radially-oscillating constriction r Radial Coordinate R Radius of deformed tube R0 Radius of undeformed tube t Time T Period of the constriction oscillation motion and incoming flow µ R0 L ix FIGURES (a) t = 0.05 (b) t = 0.10 (c) t = 0.15 (d) t = 0.20 (e) t = 0.25 (f) t = 0.30 (g) t = 0.35 (h) t = 0.40 (i) t = 0.45 (j) t = 0.50 176 FIGURES (k) t = 0.55 (m) t = 0.65 (n) t = 0.70 (o) t = 0.75 (p) t = 0.80 (q) t = 0.85 (r) t = 0.90 (s) t = 0.95 Figure 4.51 (l) t = 0.60 (t) t = 1.0 Instantaneous streamline contours for θ = 270˚ (ε = 0.50, Re = 391, α = 3.34) 177 FIGURES Figure 4.52 Comparison of throat wall vorticity variations for θ = 0˚, 90˚, 180˚ and 270˚ (ε = 0.50, Re = 391, α = 3.34) and the stationary constriction case (ε = 0.50, Re = 391, α = 3.34) 178 FIGURES (a) t = 0.05 (b) t = 0.10 (c) t = 0.15 (d) t = 0.20 (e) t = 0.25 (f) t = 0.30 (g) t = 0.35 (h) t = 0.40 (i) t = 0.45 (j) t = 0.50 179 FIGURES (k) t = 0.55 (m) t = 0.65 (n) t = 0.70 (o) t = 0.75 (p) t = 0.80 (q) t = 0.85 (r) t = 0.90 (s) t = 0.95 Figure 4.53 (l) t = 0.60 (t) t = 1.0 Instantaneous streamline contours for Re = 100 (ε = 0.50, θ = 0˚, α = 3.34) 180 FIGURES (a) t = 0.05 (b) t = 0.10 (c) t = 0.15 (d) t = 0.20 (e) t = 0.25 (f) t = 0.30 (g) t = 0.35 (h) t = 0.40 (i) t = 0.45 (j) t = 0.50 181 FIGURES (k) t = 0.55 (m) t = 0.65 (n) t = 0.70 (o) t = 0.75 (p) t = 0.80 (q) t = 0.85 (r) t = 0.90 (s) t = 0.95 Figure 4.54 (l) t = 0.60 (t) t = 1.0 Instantaneous streamline contours for Re = 200 (ε = 0.50, θ = 0˚, α = 3.34) 182 FIGURES Figure 4.55 Comparison of throat wall vorticity variations for Re = 100, 200 and 391 (ε = 0.50, θ = 0°, α = 3.34) 183 FIGURES Figure 4.56a Comparison of throat wall vorticity variations between the axiallyoscillating constriction case (ε = 0.50, θ = 0˚, α = 3.34) and the stationary constriction case (ε = 0.50, α = 3.34) for Re = 100 Figure 4.56b Comparison of throat wall vorticity variations between the axiallyoscillating constriction case (ε = 0.50, θ = 0˚, α = 3.34) and the stationary constriction case (ε = 0.50, α = 3.34) for Re = 391 184 FIGURES Figure 4.57a Comparison of wall pressure distributions at t = 0.25 for Re = 100, 200 and 391 (ε = 0.50, θ = 0°, α = 3.34) Figure 4.57b Comparison of wall pressure distributions at t = 0.75 for Re = 100, 200 and 391 (ε = 0.50, θ = 0°, α = 3.34) 185 FIGURES (a) t = 0.05 (b) t = 0.10 (c) t = 0.15 (d) t = 0.20 (e) t = 0.25 (f) t = 0.30 (g) t = 0.35 (h) t = 0.40 (i) t = 0.45 (j) t = 0.50 186 FIGURES (k) t = 0.55 (m) t = 0.65 (n) t = 0.70 (o) t = 0.75 (p) t = 0.80 (q) t = 0.85 (r) t = 0.90 (s) t = 0.95 Figure 4.58 (l) t = 0.60 (t) t = 1.0 Instantaneous streamline contours for α = 10 (ε = 0.50, θ = 0˚, Re = 391) 187 FIGURES Figure 4.59a Comparison of wall vorticity distributions at t = 0.25 for α = 3.34 and 10 (ε = 0.50, θ = 0˚, Re = 391) Figure 4.59b Comparison of wall vorticity distributions at t = 0.75 for α = 3.34 and 10 (ε = 0.50, θ = 0˚, Re = 391) 188 FIGURES Figure 4.60a Comparison of throat wall vorticity variations between the axiallyoscillating constriction case (ε = 0.50, θ = 0˚, Re = 391) and the stationary constriction case (ε = 0.50, Re = 391) for α = 3.34 Figure 4.60b Comparison of throat wall vorticity variations between the axiallyoscillating constriction case (ε = 0.50, θ = 0˚, Re = 391) and the stationary constriction case (ε = 0.50, Re = 391) for α = 10 189 FIGURES Figure 4.61a Comparison of wall pressure distributions at t = 0.25 for α = 3.34 and 10 (ε = 0.50, θ = 0˚, Re = 391) Figure 4.61b Comparison of wall pressure distributions at t = 0.75 for α = 3.34 and 10 (ε = 0.50, θ = 0˚, Re = 391) 190 ... imposing a radially-oscillating or axially-oscillating wave on a tube wall subjected to a pulsatile incoming flow A linear analytical approach was first developed to study a radially-oscillating axisymmetric... was derived Non-Newtonian effect on flow behavior in a tube with constriction was also analytically studied by Santabrata and Chakravarty (1987) The flow was assumed to be characterized by a. .. effects, and rheological properties of blood (Dooren 1978; Jayaraman et al 1983; Jain and Jayaraman 1990) Among such studies, Ramachandra Rao (1983) presented an analytical work on the pulsatile flow