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 −→ω of the flow is determined. The vorticity of the flow is defined as the curl of the flow field −→v (x,y) : −→ −→ω = ∇ × −→v (1.3) The vorticity −→ω 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 ω0 = −→ω (1.4) V −→ where ω0 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 ωtotal = ||ω ||dA (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. ω ω0 = total = ω (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). From equation 1.1 follows that the curvature ratio δ of the model equals 0.1.
Figure 2.2: The model that was used for the experiments
This model was created using a lost-wax method. First, the model was made out of two perspex plates which can be joined to form a whole. With the two plates bolted together, Woods metal, an alloy with a very low melting point of about 70oC, was poured into the model. After the Woods metal had cooled down and had solidified, the plates were removed. The Woods metal now formed a negative cast of the model. After polishing the Woods metal, the negative cast was placed in a container which was filled with a silicone elastomer. After the silicone had settled and hardened, the model could be removed from the container. By placing the model in boiling water, the Woods metal was melted and could be poured out. After cleaning the inside of the model with a small, soft brush, tube couplings were pushed into the model in order to attach the tubes.
2.2.2 The fluid The silicone model has a refractive index of n = 1.413. In order to obtain undisturbed images, the medium in which the model is placed and the medium that flows through the model must have the same refractive index. A fluid with a refractive index of n = 1.413 is created by making a 30%wt solution of CaCl2 in water [11]. The refractive index of the fluid is checked using a refractometer to make sure it equals the refractive index of the silicone model. Furthermore, the fluid that flows through the model is seeded with particles that scatter the light of the laser sheet. For these experiments, silver coated hollow glass spheres with a diameter of 10 µ from Dantec were used. The particles are mixed with the fluid by adding about 0.3 g to 250 ml of the CaCl2 solution. After mixing, this will result in a cloudy solution. The solution is filtered using an ordinary coffee filter, yielding a clear solution which can be used for the PIV measurements. The kinematic viscosty of the 30%wt CaCl2 solution is 2.6 · 10−6 m2s−1 [11].
7 2.2.3 Flow setup The experimental setup consists of a stationary pump (Cole and Palmar Mo. 75211-5), a flow meter, the model and a fluid reservoir in series.
Figure 2.3: The flow setup consists of a stationary pump, a flow meter, the model and the fluid reservoir in series
The model is placed in a plexiglass container that is filled with the CaCl2 solution. This is done to avoid disturbances in the image because of differences in the refractive index. The camera has to be positioned perpendicular to the plexiglass to minimize reflections. Therefore, measurements can only be done at three positions in the curve: at the end of the curve, at the middle of the curve and at the start of the curve (see figure 2.4).
Figure 2.4: Measurements are done at three positions: at the end of the curve, at the middle of the curve and at the start of the curve
The pump is capable of achieving flow rates between 0 and 500 ml/min. The maximum Reynolds number that can be obtained in the flow setup is 523 for a flow of 500 ml/min. From equation 1.3 follows that the maximum Dean number equals 165. The mean velocity of the flow at a flow rate of 300 ml/min was taken as the characteristic velocity used to make the vorticity dimensionless. At this flow rate the mean velocity is about 10 cm/s. For the characteristic length the tube radius a was used.
8 In the analysis of flow in a curved tube in section 1.1 it is assumed that the flow entering the curve is fully developed. However, it takes some length for the flow to become fully developed. This entrance length Le is given by [3]:
Le = 0.113aRe (2.1) From equation 2.1 follows that the entrance length for this setup is around 24 cm. During the experiment this was ensured by making sure the tubes were connected straight to the model with a total length of at least 24 cm.
2.2.4 Optical setup Since the aim of the experiments is to visualize the secondary flow, the laser sheet had to be vertical. The vertical laser sheet was created using three different lenses. The function of the first lens is to stretch the laser beam into a diverging sheet in the vertical plane. The function of the second lens was to create a sheet with a constant vertical width and to converge the sheet in the horizontal plane. The function of the third lens was to create a sheet with constant thickness (see figure 2.5). The theoretical thickness of the sheet is 0.22 mm [11].
Figure 2.5: The three lenses produce a vertical laser sheet with constant width and thickness
While most PIV experiments are performed to visualize the primary velocity, the goal of this research is to visualize the secondary velocity. A problem that could arise when trying to visualize the secondary flow is that the particles move through the sheet too fast due to their primary velocity and will not appear on two consecutive frames. During these experiments, the frame rate of the camera was kept high enough to make sure particles appear on at least two consecutive frames. The width and thickness of the sheet are minimized in order to increase the intensity and to minimize averaging over the thickness of the sheet.
2.3 Post-processing
The PIV images are post-processed using GPIV [12]. GPIV is an open source program that can be used to correlate PIV images and produce flow fields. Matlab was used to pre-process
9 the data and to visualize the flow fields produced by GPIV. It should be noted that GPIV gives the velocity vector at a discrete number of points. The vorticity at every node was computed using the Matlab curl function. The dimensionless average vorticity introduced in equation 1.6 was calculated by summing the absolute values of the vorticity in the nodes in the cross section and dividing by the number of nodes:
N 1 X −→ ω = ||ω0|| (2.2) average N i i=1 in which N is the number of nodes in the cross section.
10 Chapter 3
Results
Measurements were performed at the start of the curve, at the middle of the curve end at the end of the curve for flow rates between 25 ml/min and 500 ml/min with steps of 25 ml/min. For every measurement, 200 frames were shot and the resulting velocity field is the average of the 199 velocity field obtained from the 200 frames. For every position, the secondary velocity field, the secondary velocity magnitude field and the vorticity field is shown for flow rates of 100 ml/min, 300 ml/min and 500 ml/min. Also, the maximum vorticity in the cross section and the average vorticity in the cross section are shown as a function of the Dean number. For every image the inner curve is located to the right of the image and the outer curve is located to the left of the image.
3.1 Start of the curve
The experiments at the start of the curve were performed at a frame rate of 1000 frames per second. Figure 3.1 shows the position of the camera with respect to the model.
11 Figure 3.1: Position of the camera for the experiments at the start of the curve. Note that the flow is reversed compared to the experiments in the middle and at the end of the curve.
In figures 3.2 to 3.4 the PIV results for the experiments at the start of the curve can be seen.
−5 x 10 8 5.5 6 5
4.5 4 4 2 3.5
3 0 2.5
−2 Vorticity [−] 2
1.5 −4 1 Magnitude of Velocity [mm/s] −6 0.5
0
Figure 3.2: The secondary velocity field (left), secondary velocity magnitude field (middle) and vorticity field (right) at the start of the curve for a flow of 100 ml/min
12 −4 x 10 3 18
16 2
14
12 1
10 0 8 Vorticity [−] 6 −1
4 Magnitude of Velocity [mm/s] 2 −2
0
Figure 3.3: The secondary velocity field (left), secondary velocity magnitude field (middle) and vorticity field (right) at the start of the curve for a flow of 300 ml/min
−4 x 10
30
6 25
4
20 2
15 0 Vorticity [−] 10 −2
5 Magnitude of Velocity [mm/s] −4
0
Figure 3.4: The secondary velocity field (left), secondary velocity magnitude field (middle) and vorticity field (right) at the start of the curve for a flow of 500 ml/min
Figures 3.2 to 3.4 show the velocity and vorticity fields at the start of the curve. The secondary flow field at the start of the curve looks similar for all flow rates. There is a common motion downwards and towards the inner curve. Most of the vorticity is located at the edge of the tube and is concentrated in smaller regions for higher flow rates. No vortices are observed.
3.2 Middle of the curve
The experiments at the middle of the curve were performed at a frame rate of 600 frames per second. Figure 3.5 shows the position of the camera with respect to the model.
13 Figure 3.5: Position of the camera for the experiments at the middle of the curve
In figures 3.6 to 3.8 the PIV results for the experiments at the middle of the curve can be seen.
−4 x 10 1.5 11
10 1 9
8 0.5 7
6 0
5
4 −0.5 Vorticity [−]
3 −1 2 Magnitude of Velocity [mm/s]
1 −1.5 0
Figure 3.6: The secondary velocity field (left), secondary velocity magnitude field (middle) and vorticity field (right) at the middle of the curve for a flow of 100 ml/min
14 −4 x 10 5 25 4
3 20 2
1 15 0
−1 10 Vorticity [−]
−2
5 −3 Magnitude of Velocity [mm/s]
−4
0
Figure 3.7: The secondary velocity field (left), secondary velocity magnitude field (middle) and vorticity field (right) at the middle of the curve for a flow of 300 ml/min
−4 x 10 10 45 8 40 6 35 4 30 2 25 0 20 −2 Vorticity [−] 15 −4 10
Magnitude of Velocity [mm/s] −6 5 −8 0
Figure 3.8: The secondary velocity field (left), secondary velocity magnitude field (middle) and vorticity field (right) at the middle of the curve for a flow of 500 ml/min
Figures 3.6 to 3.8 show the velocity and vorticity fields at the middle of the curve. At 100 ml/min two mirrored vortices can be seen. At higher flow rates the vortices are also present, but the secondary flow field changes slightly and there is some additional vertical movement as the fluid moves towards the outer curve. The vorticity is concentrated in smaller regions for higher flow rates.
3.3 End of the curve
The experiments at the end of the curve were performed at a frame rate of 1000 frames per second. Figure 3.9 shows the position of the camera with respect to the model.
15 Figure 3.9: Position of the camera for the experiments at the end of the curve
In figures 3.10 to 3.12 the PIV results for the experiments at the end of the curve can be seen.
−4 x 10 1.5 12
10 1
8 0.5
6 0 Vorticity [−] 4 −0.5
2 Magnitude of Velocity [mm/s] −1
0
Figure 3.10: The secondary velocity field (left), secondary velocity magnitude field (middle) and vorticity field (right) at the end of the curve for a flow of 100 ml/min
16 −4 x 10 6 50
45 4
40 2 35
30 0
25 −2 20 Vorticity [−]
15 −4 10 Magnitude of Velocity [mm/s] 5 −6
0
Figure 3.11: The secondary velocity field (left), secondary velocity magnitude field (middle) and vorticity field (right) at the end of the curve for a flow of 300 ml/min
−3 x 10 1 60
50 0.5
40
0 30 Vorticity [−] 20 −0.5
10 Magnitude of Velocity [mm/s] −1
0
Figure 3.12: The secondary velocity field (left), secondary velocity magnitude field (middle) and vorticity field (right) at the end of the curve for a flow of 500 ml/min
Figures 3.10 to 3.12 show the velocity and vorticity fields at the end of the curve. At all flow rates, two vortices are observed which are horizontally mirrored. The velocity is directed towards the outside of the curve in the middle of the tube and towards the inside of the curve along the edges of the tube. At higher flow rates, the vorticity increases, but is pushed more towards the wall and is concentrated in a c-shape.
3.4 Vorticity versus Dean number
In figures 3.13 and 3.14 the (dimensionless) vorticity is plotted as a function of the Dean number.
17 −3 x 10 1.4 Start of the curve Middle of the curve 1.2 End of the curve
1
0.8
0.6
0.4 Maximum Vorticity [−]
0.2
0 0 20 40 60 80 100 120 140 160 180 Dean number
Figure 3.13: The maximum vorticity in the cross section as a function of the Dean number
In figure 3.13 can be seen that the maximum vorticity increases linearly with increasing Dean number. At the start of the curve, the maximum vorticity is smaller than at the middle and end of the curve, especially for high Dean numbers. The maximum vorticity at the middle of the curve is comparable to the maximum vorticity at the end of the curve, but for the highest Dean numbers the difference becomes larger.
−4 x 10
Start of the curve Middle of the curve End of the curve
3
2
Average Vorticity [−] 1
0 0 20 40 60 80 100 120 140 160 180 Dean number
Figure 3.14: The average vorticity in the cross section as a function of the Dean number
In figure 3.14 can be seen that also the average vorticity increases linearly with increasing Dean number. At the start of the curve, the average vorticity is about one third the average
18 vorticity observed at the end of the curve. At the middle of the curve, the average vorticity is slightly higher than at the end of the curve for Dean numbers up to 60. For Dean numbers larger than 60, the average vorticity at the end of the curve is highest.
19 Chapter 4
Conclusion and discussion
In this research PIV was used to visualize the secondary motion of a fluid in a curve at physiologically realistic Dean numbers. Secondary flow fields were obtained for several flow rates and at three different positions in the curve. The vorticity in the cross section was determined and plotted as a function of the Dean number. It has been shown that PIV is a suitable way to visualize the steady state secondary flow in a curved tube.
The results for the experiments at the start of the curve indicate that there is a secondary motion at this location as can be seen in figures 3.2 to 3.4. The secondary flow looks completely different from the secondary flows observed at the middle and the end of the curve. The fluid moves as a whole downward and towards the inner curve. Most of the vorticity is located around the walls of the tube, where the fluid is forced to rotate. A significant difference between the secondary flow at the middle of the curve (figures 3.6 to 3.8) and the start of the curve is observed. At the middle of the curve, two symmetric vortices can be seen that are mirrored in the horizontal axis. At higher flow rates, additional regions of vorticity can be seen near the axis of the tube. The secondary velocity profile at the end of the curve agrees with the theory in section 1.1, with two symmetric vortices. It can be seen in figures 3.10 to 3.12 that the vortices are pushed outward as the flow is increased, concentrating most of the vorticity at the top and the bottom of the tube in a c-shaped region. The results in figures 3.13 and 3.14 suggest that there is a linear relationship between the Dean number and the average and maximum vorticity of the secondary flow. The vorticity is comparable at the end and middle of the curve for Dean numbers up to 60, but for Dean numbers higher than 60 there is significant more vorticity at the end of the curve. The average vorticity does, however, say little about the distribution of the vorticity. The results show that the distribution of vorticity is completely different for the three positions.
Since vorticity is present at the start of the tube it can not be concluded that the vorticity seen in the middle and the end of the curve is completely caused by the curvature. A possible explanation for these results is that the curve that is located upstream does have some effect on the secondary motion of the fluid before the curve. Also, the entrance length for the flow to become fully developed as given in section 2.2.3 may be too small. The results suggest that the vorticity and velocity of the secondary flow increase as the secondary flow develops.
The results of this research help to gain a better insight into the flow in curved vessels, especially with respect to vorticity. Future research should focus on minimizing the vorticity before the fluid enters the curve to isolate the effect of the curve. Furthermore, the vorticity
20 should be determined at more positions in the curve to determine how the secondary flow develops in the curve. In order to obtain physiologically realistic results, pulsatile flow should be used to find the time dependent behavior of the vorticity. The linear relationship between the Dean number and the vorticity should be verified by performing experiments on models with different curvature ratio’s. The dimensionless vorticity could be a helpful tool to compare results obtained from different models.
21 References
[1] European Heart Network,“"CVD Statistics".” http://www.ehnheart.org/ cdv-statistics.html, October 2010.
[2] E. Cecchi et al., “Role of hemodynamic shear stress in cardiovascular disease,” Atheroscle- rosis. Article in Press.
[3] G. Truskey, F. Yuan, and D. Katz, Transport phenomena in biological systems. Pearson Prentice Hall, 2004.
[4] W. Nichols and M. O’Rourke, McDonalds’s Blood FLow in Arteries. Arnold and Oxford University Press, 1998.
[5] X. He and D. Ku, “Pulse flow in the human left coronary artery bifurcation: average conditions,” J. Biomech. Eng., vol. 118, pp. 74–82, 1996.
[6] J. Myers, J. Moore, M. Ojha, K. Johnston, and C. Ethier, “Factors influencing blood flow patterns in the human right coronary artery,” Ann. Biomed. Eng, vol. 29, pp. 109–120, 2001.
[7] F. van de Vosse, Cardiovascular Fluid Mechanics Lecture Notes. Eindhoven University of Technology, 2009.
[8] A. Ward-Smith, Internal Fluid Fow. Oxford University Press, 1980.
[9] Tampere University of Technology. http://www.tut.fi.
[10] R. Adrian, “Particle-imaging techniques for experimental fluid mechanics,” Annu. Rev. Fluid Mech., vol. 23, pp. 261–304, 1991.
[11] R. van der Burgt, “"PIV and video-densitometry in cerebral aneurysms: validation of and boundary conditions for CFD models, MSc Thesis,” 2009. Eindhoven University of Technology.
[12] G. van der Graaf, “Gpiv.” http://gpiv.sourceforge.net/.
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