Analysis of Velocity Profiles in Curved Tubes Anna Catharina Verkaik February 20, 2008 BMTE 08.14 Committee: prof. dr. ir. F.N. van de Vosse prof. dr. ir. G.J.F. van Heijst dr. ir. A.A.F. van de Ven dr. ir. N.A.W. van Riel dr. ir. M.C.M. Rutten ir. B.W.A.M.M. Beulen Eindhoven University of Technology Department of Biomedical Engineering Division of Cardiovascular Biomechanics 2 Abstract Cardiovascular disease is responsible for nearly half of all deaths in Europe. The progress of cardiovascular disease can be monitored by ultrasound mea- surements to investigate non-invasively the arteries and the blood flow within the arteries. One of the objectives of the research project where this study contributes to, is to measure the blood velocity simultaneously with the wall displacement of the vessel in order to derive blood volume flow. This requires the development of a new velocity measurement method for the ultrasound scanner to calculate the blood flow through an artery. A simple Poiseuille approximation is not sufficient for curved arteries, therefore, the objective of this study is: to determine a method to assess the volume flow through a curved tube from a given asymmetric axial velocity profile, measured with an ultrasound sys- tem at a single line through the tube. Three steps were made to achieve the objective of this study. First an in- vestigation was made of the existing analytical approximation methods for fully developed flow in curved tubes as presented in literature. Secondly computational fluid dynamics (CFD) models were developed to investigate flow in curved tubes for higher Dean numbers and different curvature ratios. Finally, ultrasound measurements were performed in an experimental set-up to investigate whether it is possible to measure correctly the axial velocity profiles of a flow in a curved tube and to validate the CFD models. The analytical, computational and experimental results were compared with each other and to data obtained from literature and showed good agreement. From these results it is derived that the analytical approximation methods are valid for Dn 50, the axial velocity profiles can be measured with ≤ a good accuracy and the CFD model was validated with the experiments. It is concluded that for steady flow it is possible to determine the volume flow through a curved tube from an axial velocity profile measured at the centerline with an ultrasound scanner. CONTENTS 3 Contents 1 Introduction 6 1.1 Theory of Steady Flow in a Curved Tube . 7 1.2 HistoricalOverview. .. .. .. .. .. .. 8 1.3 Flow Measurement with Ultrasound [1] . 10 1.4 Objective ............................. 11 2 Stationary Flow in Curved Tubes 13 2.1 Scaling............................... 14 2.1.1 Scalingmethod1. .. .. .. .. .. 14 2.1.2 Scalingmethod2. .. .. .. .. .. 16 2.2 Analytical Approximations . 20 2.2.1 Dean1927[2]....................... 20 2.2.2 Dean1928[3]....................... 21 2.2.3 Topakoglu 1967 [4] . 24 2.2.4 Siggers and Waters 2005 [5] . 27 2.3 Results of the Analytical Approximations . 33 2.4 Discussion............................. 35 2.5 Conclusion ............................ 37 3 Computational Fluid Dynamics 39 3.1 Method .............................. 39 3.1.1 FiniteTube........................ 41 3.1.2 InfiniteTube ....................... 43 3.2 Results............................... 44 3.2.1 FiniteTube........................ 46 3.2.2 Finite tube versus Infinite tube . 49 3.2.3 InfiniteTube ....................... 53 3.3 Discussion ............................ 56 3.3.1 Simulations vs Analytical approximation Methods . 62 CONTENTS 4 3.4 Conclusion ............................ 63 4 Ultrasound measurements 68 4.1 Materials and Method . 68 4.1.1 Preparation of the Polyurethane Tubes . 68 4.1.2 Blood Mimicking fluid (BMF) . 69 4.1.3 Experimental Set-up . 70 4.1.4 Postprocessing using Cross Correlation [6] . 73 4.1.5 Ultrasound Measurements . 74 4.1.6 Ultrasound Measurements vs Infinite Tube Model . 75 4.2 Results............................... 76 4.2.1 Control experiment Agarose gel . 77 4.2.2 Newtonian BMF . 78 4.2.3 ShearThinningBMF . 78 4.2.4 Ultrasound Measurements vs Infinite Tube Model . 79 4.3 Discussion............................. 81 4.4 Conclusion ............................ 84 4.4.1 Recommendations . 86 5 Conclusions 87 5.1 Analytical Approximation Methods . 87 5.2 CFDmodels............................ 87 5.3 Ultrasound Measurements . 88 A The Navier Stokes equations in Toroidal Coordinates 90 A.1 Orthogonal curvilinear coordinates . 90 A.1.1 general derivation [7] . 91 A.2 Derivation of the Toroidal Coordinates . 94 A.2.1 The Navier-Stokes equations in toroidal coordinates . 98 A.3 Another way of choosing the coordinates . 99 CONTENTS 5 B Derivation of Stationary Flow in Curved Tubes 100 B.1 Introduction............................ 100 B.2 Stationaryflow .......................... 101 1 INTRODUCTION 6 1 Introduction Cardiovascular disease is the number one cause of death in the Western So- ciety; it is responsible for nearly half (49%) of all deaths in Europe [8]. The progress of cardiovascular disease is usually monitored by investigation of the pressure-flow relation at a specific area of the blood circulation. Ultra- sound measurements are a frequently used diagnostic method to investigate the arteries and the blood flow within the arteries. The velocity of blood is determined non-invasively with Doppler measurements and with that infor- mation an estimation of the blood flow through the artery can be made. For example the carotid artery may be investigated with an ultrasound scanner to assess the blood flow velocity in the arteries, that supply blood from the heart through the neck to the brain, see the left picture of Figure 1. Figure 1: An example of an ultrasound measurement with on the left a schematic picture of the ultrasound Doppler measurement of blood velocity in the carotid artery. In the right picture the information visualized by the ultrasound scanner is shown. In the upper half ultrasound signal is visualized in the B-mode, which results in a 2 dimensional cross section of the (com- mon) carotid artery. In the lower half the Doppler measurement is shown, which is measured at the line represented in the B-mode image [9]. The calculation of blood flow from the velocity in the artery can be based on the assumption of a Poiseuille profile, for steady flow, or Womersley pro- files, for instationary flow. These assumptions are appropriate for straight arteries. Since most arteries are tapered, curved, bifurcating and have side branches, which affects the velocity distribution in the artery, the straight 1 INTRODUCTION 7 tube assumptions are not valid. This study will focus on the effect of curva- ture on the axial velocity profile for steady flow through a curved tube and the way this can be measured with ultrasound. In this introduction, first a historical overview of experimental, analytical and computational research on steady flow in curved tubes as presented in literature is given. Secondly, the basic principles of the ultrasound scanner are explained. Finally the aims of this study are stated. 1.1 Theory of Steady Flow in a Curved Tube When a fluid flows from a straight tube into a curved tube, a change in the flow direction is imposed on the fluid. The fluid near the axis of the tube has the highest velocity and experiences therefore a larger centrifugal force ρw2 ( R , where w is the axial velocity, ρ the density and R the curvature radius) compared to the fluid near the walls of the tube. So the fluid in the center of the tube will be forced to the outside of the curve. The fluid at the walls on the outer side of the curve will be forced inwards along the walls of the tube, because the pressure is lower at the inside of the curve. This will result in a secondary flow, which influences the axial velocity distribution (Figure 2). The maximum velocity measured in the plane of symmetry of the curved tube will be forced to the outside of the tube. Even a small curvature in a tube gives a considerable increase in resistance in comparison to a straight tube. The secondary flow causes extra viscous dissipation, because the fluid is constantly moving from the axis of the tube, where it has a high velocity, towards the wall, where the velocity is low, and vice versa. Figure 2: An example of the axial velocity distribution in a curved tube on the left and on the right the secondary velocity profile. 1 INTRODUCTION 8 1.2 Historical Overview In the beginning of the twentieth century Eustice proved the existence of the secondary circulation by performing an experiment where he injected ink into water flowing through a curved tube [10, 11]. He noticed an in- crease in flow resistance through curved tubes, which can be calculated by Q n V n C ( △Q ) = ( △V ) = R , where n and C are constants, Q is the flow, V is the velocity and R is the radius of curvature. He also observed a total ab- sence of the ‘critical velocity’ region in the coiled tube experiments, while a ‘critical velocity’ region was observed in straight tubes under the same circumstances. Above this critical velocity the flow becomes turbulent. In 1927, Dean was the first to find an analytical solution describing the steady flow of an incompressible fluid in curved tubes with a small curva- ture [2]. This analytical solution was based on the assumption that the secondary flow is just a small disturbance of the Poiseuille flow in a straight tube. A first order series solution was used to find this analytical solution. However, in 1928 Dean published another article, because he was not satis- fied with the first [3]. He did not like his first approximation, which failed to show that the relation between the pressure gradient and the flow rate through a curved tube depends on the curvature.