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Experimental Analysis of Flow in a Curved Tube Using Particle Image Velocimetry

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Experimental Analysis of Flow in a Curved Tube Using Particle Image Velocimetry Experimental analysis of flow in a curved tube using particle image velocimetry Nick Jaensson BMTE 10.30 Internship report November 2010 Supervisors: prof. dr. ir. F.N. van de Vosse (TU/e) dr. ir. A.C.B. Bogaerds (TU/e) Eindhoven University of Technology Department of Biomedical Engineering Contents 1 Introduction 2 1.1 Theory of flow in curved tubes . 2 1.1.1 The Dean number . 4 1.1.2 Vorticity . 5 2 Materials and methods 6 2.1 Particle Image Velocimetry . 6 2.2 The experimental setup . 7 2.2.1 The model . 7 2.2.2 The fluid . 7 2.2.3 Flow setup . 8 2.2.4 Optical setup . 9 2.3 Post-processing . 9 3 Results 11 3.1 Start of the curve . 11 3.2 Middle of the curve . 13 3.3 End of the curve . 15 3.4 Vorticity versus Dean number . 17 4 Conclusion and discussion 20 1 Chapter 1 Introduction Cardiovascular disease causes nearly half of all deaths in Europe and is the main cause of the financial burden of disease [1]. A lot of research has been done in order to prevent, diagnose and treat cardiovascular disease, but there are still a lot of gaps in our knowledge about car- diovascular disease. One of the major problems is the complexity of the cardiovascular system. The vessels in the cardiovascular system bifurcate, curve and taper which makes the analysis far from straightforward. A thorough knowledge about the way that blood flows through the cardiovascular system could help understand the mechanisms underlying cardiovascular dis- ease. For example, flow patterns could influence the transport of molecules in the blood. Also, the hemodynamic shear stress is known to play a large role in the formation of atherosclerotic plaques [2]. One of the best known examples of a curved vessel in the human body is the aorta, which makes a 180 degrees turn right after exiting the heart. Apart from the aorta, many examples exist of curved vessels in the human body. Table 1.1 shows an overview of several vessels with the vessel radius a and the radius of the curve R0. Table 1.1: Radius of the vessel and radius of the curve for several vessels in the human body [3] Vessel Vessel radius a [cm] Curve radius R0[cm] Aortic Arch [4] 1.5 6.8 Left main coronary artery [5] 0.425 4.25 Left anterior descending coronary artery [5] 0.17 2.07 Right coronary artery [6] 0.097 4.04 A major difference between the flow in straight tubes and the flow in curved tubes is that the flow in straight tubes generally is axisymmetric, while the flow in curved tubes is not. The curve gives rise to a secondary motion in the plain perpendicular to the flow direction (secondary flow) and therefore a secondary shear stress component at the wall. The first goal of this research is to visualize the secondary flow field in a curved tube using particle image velocimetry (PIV). The second goal is to find a quantitative description for the measured secondary flow and to find a relationship with the magnitude of the flow. 1.1 Theory of flow in curved tubes For the analysis of the flow in a curved tube, a cylindrical coordinate system with coordinates R, Z and φ is used to define the axis of the tube. Furthermore, a toroidal coordinate system 2 with coordinates r, θ and φ is used to describe the position within the tube (see figure 1.1). In the tube three velocity components are defined. Vφ is is axial velocity, Vr the radial velocity and Vθ the circumferential velocity. Figure 1.1: Cylindrical coordinate system using R, z and φ and toroidal coordinate system using r, θ and φ It is assumed that before entering the curved section of the tube, the flow is fully developed and has a velocity of zero near the wall of the tube. These assumptions give rise to a parabolic velocity profile entering the curve (see figure 1.2). Figure 1.2: Parabolic velocity profile entering the curve 2 Once the flow enters the curve, the centrifugal force ρVφ /R starts to play a role. As can be seen in figure 1.2, the velocity near the center of the tube is larger and will therefore experience a centrifugal force that is larger than the pressure gradient in radial direction (@p/@R). This imbalance in forces pushes the fluid in the center of the tube outward. Because the axial velocity of the fluid is smaller in the boundary layer, the opposite will happen and the fluid is pushed inward in the boundary layer. The secondary (the primary motion is in the axial direction) motion of the flow gives rise to two symmetric vortices in the plane perpendicular to the tube axis (see figure 1.3). 3 Figure 1.3: Secondary motion of the fluid in the curve The secondary motion of the fluid also influences the velocity profile of the axial flow. Since particles with a larger axial velocity are moving outward, the maximum axial velocity will move outward due to convection. Also, particles with a smaller axial velocity are moving inwards and are thus decreasing the axial velocity on the inside of the curve. The secondary motion of the flow gives rise to a C-shaped axial velocity profile with the maximum shifted to the outer curve and a minimum near the inner curve (see figure 1.4). For large Reynolds numbers or large curvatures, even negative axial velocities can occur near the inner wall [7]. Figure 1.4: The maximum axial velocity shifts towards the outer part of the curve In three dimensions, the combination of the primary and secondary motion of the fluid results in a spiraling motion of the particles. 1.1.1 The Dean number The flow in a curved tube can be described using the Navier-Stokes equations in toroidal coordinates [8]. Two important dimensionless parameters that follow from the Navier-Stokes equation are the curvature ratio δ and the Reynolds number Re defined as: 4 a 2aV δ = Re = (1.1) R0 ν with a the radius of the tube, R0 the radius of the curve in the tube, V the characteristic velocity and ν the kinematic viscosity. For a small curvature, another dimensionless parameter, the Dean number can be defined as [7]: Dn = δ1=2Re (1.2) The Dean number, in combination with the Reynolds number Re and curvature ratio δ, is an important parameter in the analysis of flow through curves. In table 1.2 some characteristic values for these parameters in the human body can be found. Table 1.2: Values of the curvature ratio and Dean number for human arteries [3] Vessel δ = a/R Remean De Aortic Arch [4] 0.22 1500 707 Left main coronary artery [5] 0.10 150 47.4 Left anterior descending coronary artery [5] 0.082 80 22.9 Right coronary artery [6] 0.024 233 36 1.1.2 Vorticity The secondary motion of the flow in a curved tube is investigated. As can be seen in section 1.1, two symmetric vortices are expected. To quantify these vortices, the vorticity −!! of the flow is determined. The vorticity of the flow is defined as the curl of the flow field −!v (x,y) : −! −!! = r × −!v (1.3) The vorticity −!! consists of a vector field for which every vector is perpendicular to the flow field in the xy-plane. The component of the vector in the z-direction gives an indication of the magnitude and direction of the rotation of the fluid at that point. To be able to compare the vorticity to the vorticity found in other experiments, the vorticity is made dimensionless by: −! L !0 = −!! (1.4) V −! where !0 is the dimensionless vorticity, L is the characteristic length and V the characteristic velocity. To quantify the vorticity in the entire flow field, the absolute vorticity is integrated over the area of the cross section. Z −!0 !total = jj! jjdA (1.5) A The total vorticity is also made dimensionless by dividing it by the area over which it is integrated, yielding the average vorticity. ! !0 = total = ! (1.6) total A average 5 Chapter 2 Materials and methods 2.1 Particle Image Velocimetry Particle image velocimetry (PIV) is a technique which can be used to determine the velocity field of a flow in one plane. PIV works by seeding a flow with light scattering particles, and visualizing these particles using a laser sheet (see figure 2.1). The particles that are used to seed the fluid must have three important properties: they must be small enough so that they do not influence the flow, they must not interact with each other and they must reflect enough light so the camera can capture them. Figure 2.1: The setup for a PIV experiment [9] A high-speed camera is used to capture the scattered light from the particles in the laser sheet. By correlating two subsequent images, the general direction and distance these particles traveled between the two frames can be determined. Since the time between two subsequent images is known, the velocity field can be determined. For more information about PIV the reader is referred to [10]. 6 2.2 The experimental setup 2.2.1 The model In order to perform experiments using PIV, a transparent model was created. The tube diameter a of the model is 4 mm and the radius of the curve R0 is 40 mm (see figure 2.2).
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