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. He noticed that when the
fluid motion is slow the reduction in flow rate due to the2 curvature of the 2Re a tube depends on the single variable K defined by K= R , where Re is the Reynolds number, a the radius of the tube and R the curvature radius of the tube. This time Dean derived a higher order series solution to describe the flow analytically in a tube with a small K-number, he stated that his approximation is valid up to a K-number of 576. In 1929 Taylor proved experimentally that the flow in a curved tube can be- come turbulent, but only at much higher Reynolds numbers than in straight a tubes [12]. He showed that for a curvature ratio of δ = R = 0.0313 the flow motion stays steady until Re = 5010. At higher Reynolds numbers a transition region develops until Re = 6350, after which the flow motion becomes completely turbulent. Taylor’s experiments were inspired by the article written by White, who investigated the streamlines of flow in curved tubes and already noticed that turbulent flow through a curved tube does exist, but probably at higher Reynolds numbers as for a straight tube [13]. ρvd d White was the first to use the ’Dean’s criterion’ term: µ D , where ρ is the density and µ the dynamic viscosity of the fluid, v the meanq velocity, d the tube diameter and D the diameter of the curvature of the tube. In 1968, McConalogue and Srivastava made an extension to the work of 1 INTRODUCTION 9
Dean [14]. They derived a set of equations and solved the equations numer- ically with Fourier series for 96 < Dn< 600. The Dean number is defined as:
2a 2a3 Ga2 Dn = 4Re( )1/2 = ( ) . (1) R s ν2L µ
The Dean number is based on the K-number proposed by Dean, with Dn = 4√K and so a Dean number of 96 corresponds to a K-number of 576, where G is the mean pressure gradient, ν the kinematic viscosity and µ the dynamic viscosity coefficient. The Dean number, based on the K-number of Dean, is only useful for the smaller values of the K-number, because then the mean velocity is still related to the mean pressure gradient G, like in Poiseuille flow. For higher values it is better to use the Dean number based on the 2a3 Ga2 mean pressure gradient (Dn = ( ν2L ) µ ) because the mean velocity can not be estimated correctly in advanceq as it deviates too much from the Poiseuille flow theory. McConalogue and Srivastava showed that for Dn = 600, the position of the maximum axial velocity is reached at a distance less than 0.38 times the radius from the outer boundary and that the flow is reduced with 28% in comparison to a straight tube. In 1969, Ito derived a solution for stationary flow in curved tubes for higher a K values, but for a small δ = R [15]. He divided the flow into a core flow and a boundary layer flow and then followed the Pohlhausen’s approximate method to solve the equations numerically. Unfortunately Ito does not give the derived solutions for the velocities, he was much more interested in the friction and flow factors. Collins and Dennis obtained numerical solutions for the range of 96 1.3 Flow Measurement with Ultrasound [1] For more than fifty years, ultrasound systems are clinically used to assess blood flow velocity in patients non-invasively. Ultrasound measurements are frequently used because the method is harmless for patients and various hemodynamic variables can be obtained, such as vessel diameter, wall thick- ness, shear stress and velocity profiles. The probe of the ultrasound scanner used in this study consists of an array of 128 piezoelectric crystal elements, these crystals convert an electric signal into an pulsatile or continuous ul- trasound wave. The produced wave is a high frequent signal (2-10 MHz), which is harmless for the tissue at diagnostic energy level. Inside the body the waves can be returned to the probe by either reflection or scattering of the transmitted ultrasound wave. Ultrasound waves will be partly reflected when the ultrasound wave passes along a transition in acoustic impedance which size is greater than the wave- length of the signal, such as interfaces between different tissue structures. The waves will always be reflected with an angle equal to the angle of inci- dence. Sometimes an ultrasound wave will reverberate between the acoustic boundaries resulting in secondary reflections. Scattering of the ultrasound waves occurs when the acoustic boundary is small compared to the wavelength of the ultrasound signal. This happens for example with red blood cells, they have a diameter of about 6-8 µm and the interspacing between them is much smaller than the wavelength λ of the ultrasound signal (with a frequency of 6 MHz the wavelength is typically 250 µm). The ultrasound waves will be scattered in all directions. The returned ultrasound waves will cause a pressure difference on the piezo- electric crystals, who will transform the signal into an electric radio frequent (RF) signal. The RF-signal contains information about the reflections, re- verberations, scattering and noise of the measurement. The power of the reflections is about 20 dB higher than the reverberation power and about 40 dB higher than the scattering power. The power difference between the reflected signal and the scattered signal depends on the number of scattering particles, center frequency, depth and the power of the ultrasound pulse. The ultrasound scanner can visualize the returned RF-signal in different ways. An example is the (fast) B-mode visualization, which displays the real time changes in acoustic impedance in a 2 dimensional image. This results in a cross section of the object of interest, see the right part of Fig- ure 1. A frequently used method to determine blood flow velocity in the arteries is based on the Doppler effect (Figure 1). The following equation describes 1 INTRODUCTION 11 the relation between the Doppler frequency fD and the velocity of blood v 2v cos(θ) f = f . (2) D c E In this equation c is the sound wave propagation speed, fE the emitted ultrasound frequency and θ the angle between the emitted signal and the velocity direction. An ultrasound Doppler measurement of the common carotid artery is shown in the lower right picture of Figure 1. The velocity in the common carotid artery can be deduce from this measurement, which varies roughly between 1.0 m/s and 0.1 m/s. 1.4 Objective From the above historical overview of the research on steady flow in curved tubes it is clear that during the last years not much progress has been made. Most authors are interested in the friction factors, where they compare the flow in a curved tube to the flow in a straight tube. Such a comparison will not be possible in clinical measurements of flow in curved arteries. Ultrasound Doppler measurements are currently used in clinical settings to determine flow in arteries. This method is accurate to determine flow in a straight tube with only axial velocities. Flow in a curved tube will contain transversal velocity components, which will distort the Doppler measure- ment. Another disadvantage of this method is the incapability to mea- sure the wall displacement simultaneously with the velocity measurement. Knowledge about the wall positions is essential for integration of the axial velocity profile to calculate the flow and from the wall displacement, the pressure can be deduced. One of the objectives of the research project where this study contributes to, is to measure the blood flow simultaneously with the wall displacement of the vessel to derive the pressure in the artery. To achieve this goal the ultrasound probe has to be positioned perpendicular to the blood vessel, which makes it impossible to perform an ultrasound Doppler measurement. A new velocity measurement method is being developed for the ultrasound scanner to calculate the blood flow through an artery. 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. Several assumptions are made to investigate this objective. This study is 1 INTRODUCTION 12 restricted to steady flow in tubes with a small curvature ratio. The common carotid artery is taken as an example. The mean axial velocity is roughly 0.2 m/s, the radius is about 4 mm and the curvature ratio is about 0.16 [20]. This gives a Dean number of 580, with ν = 3.145 10 6 m2s 1. Therefore, · − − the main region of interest is 100 Dn 1000. ≤ ≤ Three steps are made to achieve the objective of this study. First an inves- tigation is made of the existing analytical approximation methods for fully developed flow in curved tubes as presented in literature. As the ultrasound scanner has to be able to calculate the flow from a given asymmetric axial velocity profile, the needed calculation time has to be as small as possible. Another ground for examining the analytical approximation methods is to gain more knowledge about the physical processes of flow in a curved tube. Secondly computational fluid dynamics (CFD) models are developed to in- vestigate flow in curved tubes for higher Dean numbers and different cur- vature ratios, so ranges where the analytical solutions do not exist. The results of the analytical approximation methods are compared to the results of the CFD models, to validate the analytical approximation methods and to determine their region of validity. The results of the CFD models are also compared to computational and experimental data from literature. Finally, ultrasound measurements are performed in an experimental set-up. The first objective is to investigate whether it is possible to measure cor- rectly the axial velocity profiles of a flow through a curved tube. The second objective is to compare the measured axial velocity profiles with the results from the CFD models, to validate the CFD models. Eventually, these three steps will enable us to evaluate possible ways to assess volume flow through a curved tube from a single line velocity mea- surement. This report is structured as follows: In Chapter 2 an overview of the analyt- ical approximation methods will be presented. The computational methods will be described in Chapter 3, where also a comparison with the analytical approximation method will be made. In Chapter 4 the ultrasound measure- ments on steady flow through curved tubes will be explained and discussed together with the results from the computational models with the same parameters. Finally, Chapter 5 contains an overview of the conclusions. 2 STATIONARY FLOW IN CURVED TUBES 13 2 Stationary Flow in Curved Tubes Flow in a curved tube can be described in a toroidal coordinate system with the coordinates (r, θ, z) see Figure 3, where the z-coordinate is defined as z = R0ϕ. In this system u is the velocity in the r-direction, v is the velocity in the θ-direction and perpendicular to u. The velocity in the z-direction is w, which is perpendicular to both u and v. r R0 q O j z Figure 3: The toroidal coordinate system, which is used to describe the flow in a curved tube. In Appendix A the Navier-Stokes equations are derived for the toroidal coor- dinate system and in Appendix B the equations are simplified for stationary ∂ ∂ flow ( ∂t = 0 and ∂z = 0). This results in the following equations for the description of steady flow in a curved tube: continuity equation: ∂u 1 ∂v u u cos θ v sin θ + + + − = 0, (3) ∂r r ∂θ r R0 + r cos θ r-direction: ∂u v ∂u v2 w2 cos θ u + = (4) ∂r r ∂θ − r − R0 + r cos θ 1 ∂p 1 ∂ sin θ ∂v v 1 ∂u ν + . −ρ ∂r − r ∂θ − R0 + r cos θ ∂r r − r ∂θ θ-direction: ∂v v ∂v uv w2 sin θ u + + + = (5) ∂r r ∂θ r (R0 + r cos θ) 2 STATIONARY FLOW IN CURVED TUBES 14 1 ∂p ∂ cos θ ∂v v 1 ∂u + ν + + . −rρ ∂θ ∂r R0 + r cos θ ∂r r − r ∂θ h i z-direction: ∂w v ∂w w(u cos θ v sin θ) u + + − = (6) ∂r r ∂θ R0 + r cos θ 1 R ∂p 1 ∂ ∂w rw cos θ 1 ∂ 1 ∂w w sin θ − 0 + ν r + + . ρ R + r cos θ ∂z r ∂r ∂r R + r cos θ r ∂θ r ∂θ R + r cos θ 0 0 − 0 2.1 Scaling Scaling is a technique used for the estimation of the order of magnitude of the various terms in equations. It is useful to determine which terms are the most important for a specific problem. Scaling results in dimensionless groups, which say something about the importance of the specific terms in relation to other terms in the equation. Finally the equations can be simplified to get a better understanding of the problem and the equations may be used to derive a simplified model of the problem. 2.1.1 Scaling method 1 When the final equations from the previous section are scaled according to: r p u v w r∗ = , p∗ = , u∗ = , v∗ = , w∗ = . (7) a ρU 2 U U U With a the radius of the tube and R0 the curvature radius of the tube. The result for the continuity equation will be: U ∂u U ∂v U u Uu cos θ Uv sin θ ∗ + ∗ + ∗ + ∗ − ∗ = 0. (8) a ∂r∗ ar∗ ∂θ a r∗ R0 + ar∗ cos θ 1 U R0 After division by a and multiplication the last term with 1 the result is: R0 a a ∂u∗ 1 ∂v∗ u∗ R0 u∗ cos θ R0 v∗ sin θ + + + a − = 0. (9) ∂r∗ r∗ ∂θ r∗ 1+ R0 r∗ cos θ 2 STATIONARY FLOW IN CURVED TUBES 15 After dropping of the asterisk and the introduction of the dimensionless group for the curvature ratio δ as a δ = , (10) R0 the continuity equation becomes: ∂u 1 ∂v u δu cos θ δv sin θ + + + − = 0. (11) ∂r r ∂θ r 1+ δr cos θ In more or less the same way the momentum equation in the r-direction can be scaled: 2 2 2 2 2 2 U ∂u∗ U v∗ ∂u∗ U v∗ U w∗ cos θ u∗ + = (12) a ∂r∗ a r∗ ∂θ − a r∗ − R0 + ar∗ cos θ U 2 ∂p 1 ∂ sin θ U ∂v U v U ∂u ∗ ν ∗ + ∗ ∗ . a ∂r ar ∂θ R + ar cos θ a ∂r a r ar ∂θ − ∗ − ∗ − 0 ∗ ∗ ∗ − ∗ 2 After division by U and multiplying the last term on the left side of the 1 a R0 equation with 1 the result is: R0 2 a 2 ∂u∗ v∗ ∂u∗ v∗ R0 w∗ cos θ u∗ + a = (13) ∂r∗ r∗ ∂θ − r∗ − 1+ R0 r∗ cos θ ∂p ν 1 ∂ a sin θ ∂v v 1 ∂u ∗ R0 ∗ + ∗ ∗ . −∂r − Ua r ∂θ − 1+ a r cos θ ∂r r − r ∂θ ∗ " ∗ R0 ∗ ! ∗ ∗ ∗ # After dropping again the asterisk and the introduction of the dimensionless groups the curvature ratio δ and the Reynolds number: Ua Re = , (14) ν this finally gives: ∂u v ∂u v2 δw2 cos θ u + = (15) ∂r r ∂θ − r − 1+ δr cos θ 2 STATIONARY FLOW IN CURVED TUBES 16 ∂p 1 1 ∂ δ sin θ ∂v v 1 ∂u + . −∂r − Re r ∂θ − 1+ δr cos θ ∂r r − r ∂θ The same procedure can be followed to obtain the dimensionless equations for the θ-direction and z-direction. θ-direction: ∂v v ∂v uv δw2 sin θ u + + + = (16) ∂r r ∂θ r (1 + δr cos θ) 1 ∂p 1 ∂ δ cos θ ∂v v 1 ∂u + + + , −r ∂θ Re ∂r 1+ δr cos θ ∂r r − r ∂θ z-direction: ∂w v ∂w δw(u cos θ v sin θ) u + + − = (17) ∂r r ∂θ 1+ δr cos θ a ∂p 1 1 ∂ r∂w δrw cos θ 1 ∂ 1 ∂w δw sin θ − + + + . 1+ δr cos θ ∂z Re r ∂r ∂r 1+ δr cos θ r ∂θ r ∂θ − 1+ δr cos θ 2.1.2 Scaling method 2 Another method of scaling results from looking at the plane of symmetry of the momentum equation in r-direction [21]. This method assumes that the centrifugal force terms are of the same order of magnitude as the viscous forces. At the plane of symmetry v = 0and θ = 0or π and so cos θ = 1 and ± sin θ = 0, with these values the momentum equation in r-direction becomes: ∂u w2 ∂p 1 1 ∂ ∂v 1 ∂u u δ ± = (18) ∂r − 1 δr −∂r − Re r ∂θ ∂r − r ∂θ ± In the above equation w = O(1) and the radius of the tube, r, is scaled between 0 and 1. If the curvature ratio is small, δ << 1, then the order ∂u 2 of magnitude for the u ∂r term becomes equal to O(δw ) = O(δ) and thus O(u) = δ1/2. The velocities u and v scale in a same manner, O(u)= O(v), meaning that O(v)= δ1/2, this can be deduced from the continuity equation [21]. It should be noticed that this method only holds for δ << 1. For fully developed flow in a curved tube, the pressure can be described as follows: z p(r, θ, z)= p (r, θ)+ P . (19) 0 1 R 2 STATIONARY FLOW IN CURVED TUBES 17 Where p0(r, θ) is the pressure in the cross section of the tube, which is P1 caused by the centrifugal forces and R is a pressure gradient constant in the axial direction, comparable to the Poiseuille flow. The second possibility for scaling of the continuity equation and the momentum equations becomes: r p0 u v w r∗ = , p∗ = , u∗ = , v∗ = , w∗ = (20) a 0 δρU 2 δ1/2U δ1/2U U The rescaling of the velocities is necessary to make sure that the centrifugal- force terms in the r- and θ-direction are of the same order of magnitude as the inertia and viscous terms. This makes sense because the secondary flow is driven by the centrifugal forces. ∂p P1 The pressure gradient in z-direction is equal to ∂z = R , where P1 has an ηU order of magnitude of aδ if the pressure gradient is related to the Poiseuille flow. The pressure gradient can be scaled according to: 2 a P1 aP1 P ∗ = = (21) 1 ηUR ηUδ A similar procedure is followed as with the other scaling method. First the new scaling parameters are inserted in the original equations, then the U 2δ1/2 continuity equation is divided by a . After dropping of the asterisk this results in continuity equation: ∂u 1 ∂v u δu cos θ δv sin θ + + + − = 0. (22) ∂r r ∂θ r 1+ δr cos θ The new scaling parameters are also inserted in the momentum equations. U 2δ After division of a and leaving the asterisk away the following equations will remain r-direction: ∂u v ∂u v2 w2 cos θ u + = (23) ∂r r ∂θ − r − 1+ δr cos θ ∂p 1 1 ∂ δ sin θ ∂v v 1 ∂u + , ∂r 1/2 r ∂θ 1+ δr cos θ ∂r r r ∂θ − − δ Re − − θ-direction: ∂v v ∂v uv w2 sin θ u + + + = (24) ∂r r ∂θ r (1 + δr cos θ) 2 STATIONARY FLOW IN CURVED TUBES 18 1 ∂p 1 ∂ δ cos θ ∂v v 1 ∂u + 1/2 + + . −r ∂θ δ Re ∂r 1+ δr cos θ ∂r r − r ∂θ The z-direction of the momentum equation scales in a bit different way than U 2δ1/2 the other two directions of the momentum equation and is divided by a , which results in: z-direction: ∂w v ∂w δw(u cos θ v sin θ) u + + − = (25) ∂r r ∂θ 1+ δr cos θ 1 1 1 1 ∂ r∂w δrw cos θ 1 ∂ 1 ∂w δw sin θ 1/2 P1 + 1/2 + + δ Re 1+ δr cos θ δ Re r ∂r ∂r 1+ δr cos θ r ∂θ r ∂θ − 1+ δr cos θ From these last equations it can be seen that in this case where δ << 1, another dimensionless group can be derived, namely the Dean number: 1/2 a 1/2 aU Dn∗ = δ Re = ( ) . (26) R ν This definition of the Dean number is slightly different from the Introduc- tion, the missing factor (4 (2)) results from the relation with the pressure gradient. p The centrifugal forces and their interaction with primarily the viscous forces induces the secondary flow, thus the Dean number is a measure of the mag- a nitude of the secondary flow. The curvature ratio parameter δ = R is a measure of the geometry effect and the extent to which the centrifugal forces vary on the cross section, δ effects the balance of inertia, viscous forces and centrifugal forces, so it is an important factor of the flow in curved tubes [18]. Under the assumption that δ << 1 the following equations result: continuity equation: ∂u 1 ∂v u + + = 0, (27) ∂r r ∂θ r r-direction: ∂u v ∂u v2 u + w2 cos θ = (28) ∂r r ∂θ − r − ∂p 1 1 ∂ ∂v v 1 ∂u + , ∂r 1/2 r ∂θ ∂r r r ∂θ − − δ Re − 2 STATIONARY FLOW IN CURVED TUBES 19 θ-direction: ∂v v ∂v uv u + + + w2 sin θ = (29) ∂r r ∂θ r 1 ∂p 1 ∂ ∂v v 1 ∂u + + , r ∂θ 1/2 ∂r ∂r r r ∂θ − δ Re − z-direction: ∂w v ∂w u + + δw(u cos θ v sin θ) = (30) ∂r r ∂θ − 1 1 1 ∂ r∂w 1 ∂ 1 ∂w P + + δrw cos θ + δw sin θ . 1/2 1 1/2 r ∂r ∂r r ∂θ r ∂θ δ Re δ Re − 2 STATIONARY FLOW IN CURVED TUBES 20 2.2 Analytical Approximations During the last century a few articles were written about analytical ap- proximations for stationary flow in a curved tube. A short summary of the analytical approximation methods derived by Dean, Topakoglu and Siggers and Waters are given. Their results are compared with each other. In this section the notation in each summary is based on the notation used by the authors of the respective article. 2.2.1 Dean 1927 [2] In 1927 Dean was the first who published an article about analytical so- lutions for stationary flow in a curved tube. He based his solutions on the experiments conducted by Eustice ([10, 11]) and also compared his solutions with the experimental results. Dean started his derivation with the equa- tions of Appendix B.2, but he used a slightly different coordinate system (see Appendix A.3). He made the assumption that as the curvature of the pipe is small, the flow in the different directions is just a small disturbance of the Poiseuille flow in a curved tube. He used the following definitions for the velocity and pressure components. U = u, V = v, W = A(a2 r2)+ w, P/ρ = Cz + p/ρ(31) − For which u, v, w and p are defined as being small and in the order of magnitude of δ and C is a constant. Dean ignored the terms of the equations which have an order of magnitude equal to δ2 and started with a long derivation, which finally resulted in equations for the velocity components: U na 2 2 2 = sin θ(1 r′ ) (4 r′ ) (32) W0 288R − − V na 2 2 4 = cos θ(1 r′ )(4 23r′ + 7r′ ) (33) W0 288R − − 2 W 2 3r sin θ n r sin θ 2 4 6 = (1 r′ )[1 + (19 21r′ + 9r′ r′ )] (34) W0 − − 4R 11520R − − 3 Aa aW0 Where n = ν = ν , also known as the Reynolds number, W0 is the r velocity at the center of the tube and r′ = a , with a the radius of the tube. 2 STATIONARY FLOW IN CURVED TUBES 21 Dean remarked in his article that Eustice stated that to cause a given flow rate a larger pressure gradient is needed in a curved tube in comparison to a straight tube. A considerable difference should exist even with a small curvature ratio. But in Deans equations the mean pressure gradient is the same as for flow in a straight tube. So his derivation did not give a satisfying solution to this phenomenon. But luckily Deans solutions (see his Figure 3) did explain the experiments with the colored dyes at different levels around the central plane of the tube. It was observed by Eustice that the colored matter of the line in the central plane divided into two parts, one going along the upper boundary and one along the lower boundary. But the experiment with a colored line at a sufficient distance above the central plane shows that the line stays in the upper half of the tube, where it forms a helix and is not dispersed into two parts. Unfortunately, Deans solutions could not explain why the shape of these helixes depends on the velocity. 2.2.2 Dean 1928 [3] Dean was not satisfied with his first approximation method, mainly because ‘it failed to show that the relation between the pressure gradient and the rate of flow through a curved tube depends on the curvature ratio of the tube’. He worked out another approximation, where he started also with the equations a of section B.2. Now he assumed again that R is small, but he used this to ∂ cos θ ∂ replace for example (R+r cos θ) by R and ∂r + R+r cos θ by ∂r . The equations became simplified and after introduction of the streamline function f: ∂f ∂f rU = , V = (35) − ∂θ ∂r and by using the following substitutions to get non-dimensional equations: f = νψ, W = W0w, r = ar′ (36) he derived the two constants K and C: 2W 2a3 K = 0 (37) ν2R Ga2 C = (38) µW0 2 STATIONARY FLOW IN CURVED TUBES 22 aW0 With for slow motion ν nearly equal to the Reynolds number which is dv¯ n = ( ν ), with d the diameter andv ¯ the mean velocity. So for slow motion 2n2a it holds that K = R . This approximation of the equations for a curved W 2 tube has as only an extra centrifugal force term R working in the θ = 0 direction, as the result for having a curvature. After this derivation Dean realized that ‘there is no theoretical ground for supposing that if a/R is small the motion in a curved pipe must approximate to that in a straight pipe: the far more stringent condition that K must be small has to be fulfilled’. That is the reason why he started a series solution expanded to K instead of δ, which he used in his first derivation [2]. He used the following series expansion to solve the equations: 2 w = w0 + Kw1 + K w2 + ..... (39) 2 ψ = Kψ1 + K ψ2 + ..... (40) where w0, w1, w2, ψ1, ψ2, .... are only functions of r’ and θ. Finally he gave the following solutions for the series expansion. 2 w = 1 r′ , (41) 0 − 5 7 9 cos θ 19r′ 3 3r′ r′ r′ w = ( r′ + + ). (42) 1 576 40 − 4 − 4 40 Afterwards he calculated that the last term needed a correction factor of: 3ar (1 r 2) cos θ − ′ − ′ . 4RK The first order term of the streamline function was: 2 sin θ 2 2 r′ ′ ψ = − r′(1 r′ ) (1 ). (43) 1 144 − − 4 The results of these equations for the different velocity terms are visualized in Figure 4 and Figure 5. After these derivations he looked at a relation to describe the flow in a curved tube in comparison to the flow in a similar straight tube. Because 2 STATIONARY FLOW IN CURVED TUBES 23 −3 −3 −4 x 10 w0 x 10 w1 x 10 2 4 0.9 4 3 0.8 3 2 0.7 2 1 1 0.6 1 0 0.5 0 0 −1 0.4 −1 −2 0.3 −2 −1 −3 0.2 −3 0.1 −4 −4 −2 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 −3 −3 x 10 x 10 (a) Result for w0 (b) Result for w1 −3 Ψ −3 x 10 1 x 10 4 1.5 3 1 2 0.5 1 0 0 −1 −0.5 −2 −1 −3 −4 −1.5 −4 −3 −2 −1 0 1 2 3 4 −3 x 10 (c) Result for ψ1 Figure 4: The derived solutions of Dean’s analytical approximation are shown in the different figures with (a) showing w0, (b) showing w1 and (c) showing ψ1 the calculations are long and the equations become more and more difficult for higher order approximations, he only gave the final result in his article: F K K c = 1 0.03058( )2 + 0.01195( )4. (44) Fs − 276 576 2 φW0a Where Fc is the flow through a curved tube and Fs = 2 the flow through a straight tube. Dean stated that this equation holds without a serious error until K=576, which gives a reduction caused by the curvature of 1.9%. Van Dyke proved in his article, about friction ratios in curved tubes, the parameters found by Dean in his flow equation and he extended the flow equation with another 24 terms [22]. He also proved that the flow equation of Dean is valid until K = 585.78878. 2 STATIONARY FLOW IN CURVED TUBES 24 −3 Dean 1928 total axial velocity x 10 -3 x 10 Dean 1928 streamfunction Ψ 4 0.9 4 0.25 3 0.2 0.8 3 0.15 2 0.7 2 0.1 1 0.6 1 0.05 0 0.5 0 0 -0.05 −1 0.4 -1 -0.1 −2 0.3 -2 -0.15 0.2 −3 -3 -0.2 −4 0.1 -4 -0.25 −4 −3 −2 −1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 −3 -3 x 10 x 10 (a) Result for w0 (b) Result for ψ1 Figure 5: The derived solutions of Dean’s analytical approximation are shown for Dn = 50 and δ = 0.02 with (a) showing the approximation for the axial velocity and (b) showing the approximation for the streamfunction 2.2.3 Topakoglu 1967 [4] The second approximation method for steady, incompressible laminar flow in curved tubes was derived by Topakoglu. He performed a similar deriva- tion as discussed in section B.2, although he utilized a different notation method. He did also define a stream function and made use of a power series expansion in δ (like Dean 1927) to find the solution for his nonlinear differential equation system. By insertion of: 2 w = w0 + δw1 + δ w2 + ..... (45) and 2 ψ = δψ1 + δ ψ2 + ....., (46) with: 1 ∂ψ 1 ∂ψ u = , v = (47) r ∂r −r ∂θ and w the velocity in axial direction and ψ the streamfunction. Topakoglu calculated the following equations: w = f = Re(1 r2) (48) 0 0 − 2 STATIONARY FLOW IN CURVED TUBES 25 ψ1 = g1 sin θ (49) where 1 g = f 2(4 r2)r (50) 1 −288 0 − w1 = f1 cos θ (51) where 3 1 f = f [1 Re2(19 21r2 + 9r4 + r6)]r (52) 1 −4 0 − 8640 − ψ2 = g2 sin 2θ (53) where 1 1 g = f 2[16 7r2 Re2(995.8 558.4r2+155.4r4 26.8r6+r8)]r2(54) 2 5760 0 − −16128 − − w2 = f20 + f22 cos 2θ (55) where 1 1 1 f = − f 3 11r2 + Re2 148 + 43r2 132r4 + 68r6 7r8 + Re2 20 32 0 7200 3225.6 − − − (823.8 3432.2r2 + 5835.8r4 5252.2r6 + 2713.8r8 803r10 + 121r12 7r14) − − − − io (56) and 1 1 1 f = 2.5 Re2 46.3 61.3r2 + 29.6r4 4r6 Re2 (1456.9 22 8 3456 42336 − − − − − 2402.06r2 + 1746.49r4 705.47r6 + 191.23r8 28.01r10 + 1.6r12 r2 − − io (57) The results of the these equations are shown in Figure 6 and Figure 7. 2 STATIONARY FLOW IN CURVED TUBES 26 −3 w −3 w x 10 0 x 10 1 120 4 4 800 3 3 600 100 2 2 400 80 1 1 200 0 0 0 60 −1 −1 −200 40 −2 −2 −400 −3 −3 −600 20 −4 −4 −800 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 −3 −3 x 10 x 10 (a) Result for w0 (b) Result for w1 −3 Ψ −3 Ψ x 10 1 x 10 2 300 4 50 4 40 3 3 200 30 2 2 20 100 1 1 10 0 0 0 0 −10 −1 −1 −20 −100 −2 −2 −30 −3 −3 −200 −40 −4 −50 −4 −300 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 −3 −3 x 10 x 10 (c) Result for ψ1 (d) Result for ψ2 Figure 6: The derived solutions of Topakoglu’s analytical approximation method for Dn = 50 and δ = 0.02 are shown in the different figures with (a) showing w0, (b) showing w1, (c) showing ψ1 and (d) showing ψ2 Topakoglu derived a formula for the normalised flow rate through the curved tube in comparison with flow through a straight tube, under the same con- ditions. Q 1 1 1.541 2 = 1 2 ( n + 1.1n 1) (58) Q0 − 48 σ 67.2 − Re 2 R 1 with n = ( 6 ) and σ = a = δ . 2 STATIONARY FLOW IN CURVED TUBES 27 −3 x 10 total axial velocity according to Topakoglu -3 Topakoglu streamfunction Ψ x 10 4 55 4 0.25 3 50 0.2 3 45 0.15 2 2 40 0.1 1 1 35 0.05 0 30 0 0 −1 25 -0.05 -1 20 -0.1 −2 -2 15 -0.15 −3 -3 10 -0.2 −4 5 -4 -0.25 −4 −3 −2 −1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 −3 -3 x 10 x 10 (a) Result for w (b) Result for ψ Figure 7: The derived solutions of Topakoglu’s analytical approximation are shown for Dn = 50 and δ = 0.02 with (a) showing the approximation for the axial velocity and (b) showing the approximation for the streamfunction 2.2.4 Siggers and Waters 2005 [5] Before Siggers and Waters started their derivation, they made some sim- plifications. First they scaled the velocities with U and the pressure with 2 ν ρU p, where they choose U = a as a suitable velocity scale. The steady ρν2 axial pressure gradient, which drives the flow, is give by G a3 , with G = Umaxa − 4Rey = 4 ν (notice that the Reynolds number becomes 1 with the chosen velocity scale). Because they consider a fully developed flow the derivatives with respect to the z-direction are zero. They rescaled the velocity in the z-direction with w w to get the centrifugal force terms into the equa- → √2δ tions in the same order of magnitude as the viscous and inertial terms in the limit of δ 0. With these simplifications and scaling they get slightly → different equations than the equations from section B.2. Continuity equation: ∂u 1 ∂v u δu cos θ δv sin θ + + + = 0, (59) ∂r r ∂θ r h − h r-direction: ∂u v ∂u v2 δw2 cos θ u + = (60) ∂r r ∂θ − r − 2h 2 STATIONARY FLOW IN CURVED TUBES 28 ∂p 1 ∂ h ∂v ∂u v + r , −∂r − hr ∂θ r ∂r − ∂θ h i θ-direction: ∂v v ∂v uv δw2 sin θ u + + + = (61) ∂r r ∂θ r (2h) 1 ∂p 1 ∂ h ∂v ∂u + v + r −r ∂θ h ∂r r ∂r − ∂θ h i z-direction: ∂w v ∂w δw(u cos θ v sin θ) u + + − = (62) ∂r r ∂θ h Dn 1 ∂ 1 ∂w 1 ∂ r ∂w + ( h δr sin θw h δ cos θw . − h r ∂θ hr ∂θ − − r ∂r h − ∂r − h i h i With Dn = √2δG the Dean number and h(r, θ)=1= δr cos θ. Siggers and Waters introduce the streamfunction ψ, derived as a solution of the continuity equation, as follows: 1 ∂hψ u = (63) hr ∂θ and 1 ∂hψ v = (64) −h ∂r Usage of the streamfunction and elimination of the pressure from the mo- mentum equations in the r- and θ-direction results in: 1 Lψ w J( , hψ) Hw = L2ψ (65) r h − h and the z-direction of the momentum equation becomes: 1 δw D J(w, hψ)+ Hψ = + Lw (66) hr h h 2 STATIONARY FLOW IN CURVED TUBES 29 In these equations the following definitions hold: 1 ∂ r ∂(hf) 1 ∂ 1 ∂(hf) Lf = ( )+ ( ) (67) r ∂r h ∂r r2 ∂θ h ∂θ ∂f ∂g ∂f ∂g J(f, g)= (68) ∂r ∂θ − ∂θ ∂r and ∂f cos θ ∂f Hf = sin θ + (69) ∂r r ∂θ For their analytical approximation for small Dean number and small curva- ture, Siggers and Waters used the series solution for w and ψ expanded in Dn, where wk and ψk are allowed to depend on δ. ∞ 2k w = Dn Dn wk (70) X0 2 ∞ 2k ψ = Dn Dn ψk (71) X0 with ∞ j 2 wk = δ wkj = wk0 + δwk1 + δ wk2 + ..... (72) X0 ∞ j 2 ψk = δ ψkj = ψk0 + δψk1 + δ ψk2 + ..... (73) X0 By inserting these series expansions in the momentum equations above the following results for the different terms in the series expansion can be ob- tained. 1 w = (1 r2) (74) 00 4 − 3 w = r(1 r2) cos θ (75) 01 −16 − 2 STATIONARY FLOW IN CURVED TUBES 30 1 w = (1 r2)( 3 + 11r2 + 10r2 cos 2θ) (76) 02 128 − − 1 ψ = r(1 r2)2(4 r2)sin θ (77) 00 210 32 − − · 1 ψ = r2(1 r2)2(56 17r2) sin 2θ (78) 01 −212 32 5 − − · · 1 ψ = r(1 r2)2[ (133 976r2+327r4)sin θ+2r2(499 172r2) sin 3θ](79) 02 217 32 5 − − − − · · 3 1 2 1 1 1 To get the O(Dn ) solution they set w1 = w1+w1, with w1 = w10+δw11+..... 1 2 2 and w2 = w10 + δw11 + ....., with: 1 w1 = r(1 r2)(19 21r2 + 9r4 r6) cos θ (80) 10 215 32 5 − − − · · 1 w1 = (1 r2)[6(109 586r2 + 689r4 311r6 + 39r8) (81) 11 218 33 52 − − − · · 5r2(163 193r2 + 86r4 10r6) cos 2θ] − − − and 2 w10 = 0 (82) 1 w2 = (1 r2)[ (257 543r2 + 557r4 243r6 + 32r8)(83) 11 217 32 52 − − − − · · 25r2(10 14r2 + 7r4 r6) cos 2θ] − − − Some more equations for higher order derivations are shown, but the explicit solutions are not written down. This analytical higher order series approxi- mation will be valid for Dn 96, above this value the solution will diverge. ≤ The different solutions are shown in Figure 8, 9 and 10. Siggers and Waters have calculated the axial flow rate, which is in according to their calculations given by: 1 1 11 1541 Q = πDn( + δ2 Dn2δ Dn4+O(δ4,Dn2δ3, .....)).(84) 8 27 3 −215 33 5 −228 36 52 7 · · · · · · 2 STATIONARY FLOW IN CURVED TUBES 31 −3 w −3 w x 10 00 x 10 01 4 4 0.06 0.22 3 0.2 3 0.04 2 0.18 2 0.16 0.02 1 1 0.14 0 0 0 0.12 −1 0.1 −1 −0.02 0.08 −2 −2 0.06 −0.04 −3 −3 0.04 −4 0.02 −4 −0.06 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 −3 −3 x 10 x 10 (a) Result for w00 (b) Result for w01 −3 w −3 Ψ −4 x 10 02 x 10 00 x 10 1 4 0.025 4 3 0.02 3 2 0.015 2 0.01 1 1 0.005 0 0 0 0 −1 −1 −0.005 −2 −2 −0.01 −3 −3 −0.015 −4 −4 −1 −0.02 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 −3 −3 x 10 x 10 (c) Result for w02 (d) Result for ψ00 −3 Ψ −5 −3 Ψ −5 x 10 01 x 10 x 10 02 x 10 4 4 1.5 3 3 3 1 2 2 2 0.5 1 1 1 0 0 0 0 −1 −1 −1 −0.5 −2 −2 −2 −1 −3 −3 −3 −4 −4 −1.5 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 −3 −3 x 10 x 10 (e) Result for ψ01 (f) Result for ψ02 Figure 8: The different solutions of the derived analytical approximation of Siggers and Waters are shown in the different figures with (a) showing w00, (b) showing w01, (c) showing w02, (d) showing ψ00, (e) showing ψ01 and (f) showing ψ02 2 STATIONARY FLOW IN CURVED TUBES 32 −3 w1 −6 −3 w1 −6 x 10 10 x 10 x 10 11 x 10 4 3 4 3 3 3 2 2.5 2 2 2 1 1 1 1.5 0 0 0 1 −1 −1 0.5 −1 −2 −2 0 −2 −3 −3 −0.5 −4 −3 −4 −1 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 −3 −3 x 10 x 10 (a) Result for w110 (b) Result for w111 −3 w2 −6 −3 −6 x 10 11 x 10 x 10 w1 total x 10 4 4 −0.5 −1 −1 3 3 −2 −1.5 2 2 −3 −2 1 1 −2.5 −4 0 0 −3 −5 −1 −1 −3.5 −6 −2 −2 −4 −4.5 −3 −7 −3 −5 −4 −8 −4 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 −3 −3 x 10 x 10 (c) Result for w211 (d) Result for w1 totaal Figure 9: The separate solutions of the derived analytical approximation for w1 are shown in the above figures for Dn = 50 and δ = 0.02 One should note that Siggers and Waters used a special scaling and the equation is derived for the dimensionless flow, the dimensional flow rate has aνQ to be calculated with (2δ)1/2 . In their article they also discuss an asymptotic analytical solution for ‘Large Dean number and finite curvature’. They tried to derive an analytical solu- tion by division of the flow into an inviscid core and a boundary layer. This method did not result in applicable equations. 2 STATIONARY FLOW IN CURVED TUBES 33 −3 x 10 Siggers&Waters axial velocity w −3 Ψ x 10 Siggers&Waters streamfunction 4 11 4 0.1 3 10 3 9 2 2 8 0.05 1 1 7 0 6 0 0 −1 5 −1 4 −0.05 −2 −2 3 −3 −3 2 −0.1 −4 1 −4 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 −3 −3 x 10 x 10 (a) Result for the axial velocity (b) Result for the streamfunction Figure 10: The derived solutions of Siggers and Waters’ analytical approx- imation are shown for Dn = 50 and δ = 0.02 with (a) showing the ap- proximation for the axial velocity and (b) showing the approximation for the streamfunction 2.3 Results of the Analytical Approximations Dean, Topakoglu and Siggers and Waters derived their analytical solutions using a series expansion, see Table 1, to solve the Navier-Stokes equations with the assumption that δ 1 and K 1 or Dn 1. The authors all use ≪ ≪ ≪ a different scaling method. Therefore, the results are normalized to make a comparison between the three different analytical solutions. The derived Table 1: The different series expansions used by the authors to derive their analytical approximations. Author Series expansion to: 2 2 Dn Dean (1928) K = 2Re δ = 16 Topakoglu δ Siggers & Waters Dn & δ flow ratios, Qcurved , of the different approximations are plotted in figure 11. Qstraight The solution derived by Dean does not depend on δ, but only on K, so it does not change for different curvature ratios. While the solutions of Topakoglu and Siggers and Waters do change for different curvature ratios. 2 STATIONARY FLOW IN CURVED TUBES 34 Around Dn = 60 Dean’s solution starts to deviate from the other solutions, it even increases for Dn > 100. The flowratios derived by Topakoglu and Siggers and Waters give nearly the same result. They keep on decreasing and become negative for Dn >220, which is not visible in Figure 11. Flowratios 1.05 1 0.95 flowratio Dean1928 δ=0.01 Dean1928 δ=0.16 Topakoglu δ=0.01 Topakoglu δ=0.16 Siggers & Waters δ=0.01 0.9 Siggers & Waters δ=0.16 0.85 0 20 40 60 80 100 120 Dn number Figure 11: The derived flowratios of the analytical approximations of Dean, Topakoglu and Siggers and Waters. The dotted line at Dn=96 marks the converging limit of the analytical solutions. Figures 12 through 16 show the normalized axial velocity profiles of the plane of symmetry, calculated with the three analytical solutions. The figures show the results for different Dn numbers and in each figure the axial velocity profile is given for two curvature ratios, δ = 0.01 and δ = 0.16. With increasing Dn number the top of the axial velocity profiles (the position of the maximum velocity) moves to the outside of the curve (to the positive x-axis). If the Dn or K is kept constant the solution of Dean does not change for different curvature ratios, while the solutions of Topakoglu and Siggers and Waters do change. A higher δ causes the position of the maximum velocity to move to the inside of the curve (to the negative x-axis). For Dn < 50 and δ = 0.16, this results in a maximum velocity at the inside of the curve. 2 STATIONARY FLOW IN CURVED TUBES 35 analytical approximations Dn=1 analytical approximations Dn=1 1 0.9 1 0.8 0.7 0.6 0.5 0.9 0.4 Dean δ=0.01 Dean δ=0.01 Dean δ=0.16 Dean δ=0.16 0.3 Topakoglu δ=0.01 Topakoglu δ=0.01 Topakoglu δ=0.16 Topakoglu δ=0.16 Siggers & Waters δ=0.01 Siggers & Waters δ=0.01 0.2 Siggers & Waters δ=0.16 Siggers & Waters δ=0.16 0.1 0 0.8 −5 −4 −3 −2 −1 0 1 2 3 4 5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −3 −3 x−axis x 10 x−axis x 10 (a) Dn = 1 (b) Dn = 1 magnification Figure 12: The normalized axial velocity profiles calculated from the analyt- ical solutions (a) Dn = 1 and δ = 0.01 or δ = 0.16 (b) magnification of Dn = 1 and δ = 0.01 or δ = 0.16 analytical approximations Dn=10 analytical approximations Dn=10 1 0.9 1 0.8 0.7 0.6 0.5 0.9 0.4 Dean δ=0.01 Dean δ=0.01 Dean δ=0.16 Dean δ=0.16 0.3 Topakoglu δ=0.01 Topakoglu δ=0.01 Topakoglu δ=0.16 Topakoglu δ=0.16 Siggers & Waters δ=0.01 Siggers & Waters δ=0.01 0.2 Siggers & Waters δ=0.16 Siggers & Waters δ=0.16 0.1 0 0.8 −5 −4 −3 −2 −1 0 1 2 3 4 5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −3 −3 x−axis x 10 x−axis x 10 (a) Dn = 10 (b) Dn = 10 magnification Figure 13: The normalized axial velocity profiles calculated from the analyt- ical solutions (a) Dn = 10 and δ = 0.01 or δ = 0.16 (b) magnification of Dn = 10 and δ = 0.01 or δ = 0.16 2.4 Discussion All analytical approximation methods are derived for Dn or K 1 and ≪ δ 1, however, the results seem very promising at least for Dn 50. ≪ ≤ The derived flow ratios show already that Dean’s analytical solution does 2 STATIONARY FLOW IN CURVED TUBES 36 analytical approximations Dn=25 analytical approximations Dn=25 1 0.9 1 0.8 0.7 0.6 0.5 0.9 0.4 Dean δ=0.01 Dean δ=0.01 Dean δ=0.16 Dean δ=0.16 0.3 Topakoglu δ=0.01 Topakoglu δ=0.01 Topakoglu δ=0.16 Topakoglu δ=0.16 Siggers & Waters δ=0.01 Siggers & Waters δ=0.01 0.2 Siggers & Waters δ=0.16 Siggers & Waters δ=0.16 0.1 0 0.8 −5 −4 −3 −2 −1 0 1 2 3 4 5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −3 −3 x−axis x 10 x−axis x 10 (a) Dn = 25 (b) Dn = 25 magnification Figure 14: The normalized axial velocity profiles calculated from the analyt- ical solutions (a) Dn = 25 and δ = 0.01 or δ = 0.16 (b) magnification of Dn = 25 and δ = 0.01 or δ = 0.16 analytical approximations Dn=50 analytical approximations Dn=50 1 0.9 1 0.8 0.7 0.6 0.5 0.9 0.4 Dean δ=0.01 Dean δ=0.01 Dean δ=0.16 Dean δ=0.16 0.3 Topakoglu δ=0.01 Topakoglu δ=0.01 Topakoglu δ=0.16 Topakoglu δ=0.16 Siggers & Waters δ=0.01 Siggers & Waters δ=0.01 0.2 Siggers & Waters δ=0.16 Siggers & Waters δ=0.16 0.1 0 0.8 −5 −4 −3 −2 −1 0 1 2 3 4 5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −3 −3 x−axis x 10 x−axis x 10 (a) Dn = 50 (b) Dn = 50 magnification Figure 15: The normalized axial velocity profiles calculated from the analyt- ical solutions (a) Dn = 100 and δ = 0.01 or δ = 0.16 (b) magnification of Dn = 50 and δ = 0.01 or δ = 0.16 not depend on δ for a constant K or Dn. His solution becomes earlier unrealistic, the flow ratio increases for Dn > 100. The solutions are very similar to each other. Investigation of the axial velocity profiles results in more or less the same observations. The three approximation methods give the same results for 2 STATIONARY FLOW IN CURVED TUBES 37 analytical approximations Dn=100 analytical approximations Dn=100 1 0.9 1 0.8 0.7 0.6 0.5 0.9 0.4 Dean δ=0.01 Dean δ=0.01 Dean δ=0.16 Dean δ=0.16 Topakoglu δ=0.01 Topakoglu δ=0.01 0.3 Topakoglu δ=0.16 Topakoglu δ=0.16 Siggers & Waters δ=0.01 Siggers & Waters δ=0.01 Siggers & Waters δ=0.16 Siggers & Waters δ=0.16 0.2 0.1 0 0.8 −5 −4 −3 −2 −1 0 1 2 3 4 5 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −3 −3 x−axis x 10 x−axis x 10 (a) Dn = 100 (b) Dn = 100 magnification Figure 16: The normalized axial velocity profiles calculated from the analyt- ical solutions (a) Dn = 100 and δ = 0.01 or δ = 0.16 (b) magnification of Dn = 100 and δ = 0.01 or δ = 0.16 δ = 0.01. An interesting effect is the movement of the maximum velocity to the inside of the curvature for higher curvature ratios of the analytical solutions derived by Topakoglu and Siggers and Waters. Topakoglu did not mentioned this effect in his paper. While Siggers and Waters did notice that their equation for w01 causes the maximum velocity to move towards the inside of the curvature for increasing δ. In general Siggers and Waters give an better overview of their derivations. They derived that their approximation method converges until Dn = 96, which corresponds to the value K=576 given by Dean. This value seems to be reasonable as the axial velocity profiles for 1 Dn 100 do not show ≤ ≤ any irregularities. Figure 17 shows the axial velocity profiles for Dn = 200 and δ = 0.01, these profiles start to deviate from a realistic solution. 2.5 Conclusion The analytical approximation methods derived by Dean, Topakoglu and Siggers and Waters are investigated in order to find an analytical solution, which is able to calculate the axial velocity distribution for a given curvature ratio and Dean number. First general scaling methods were discussed, before the three approximation methods were summarized. The solutions of the approximation methods were used to compare the normalized results of the flow ratios and the axial velocity profiles. 2 STATIONARY FLOW IN CURVED TUBES 38 analytical approximations Dn = 200 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Dean δ=0.01 0.2 Topakoglu δ=0.01 Siggers & Waters δ=0.01 0.1 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 −3 x−axis x 10 Figure 17: The derived normalized axial velocity profiles calculated from the analytical solutions with Dn = 200 and δ = 0.01. The solutions derived by Topakoglu and Siggers and Waters adapt to the curvature ratio and the solution of Dean not. The flow ratio derived by Dean starts earlier to become unrealistic in comparison to the flow ratios of Topakoglu and Siggers and Waters. The solutions derived by Topakoglu and Siggers and Waters give similar results. The analytical approximation method derived by Siggers and Waters will be used to compare the analytical results with the results of the computational fluid dynamics models (see section 3.3.1). The reason for this choice is that they explained their method clearer and their method is in general a bit more stable than the analytical approximation of Topakoglu. 3 COMPUTATIONAL FLUID DYNAMICS 39 3 Computational Fluid Dynamics Computational fluid dynamics (CFD) is frequently used to investigate car- diovascular fluid dynamics [23]. CFD has several advantages in comparison to experiments. In contrast to experimental studies most of the important parameters that describe the analyzed fluid dynamical problem can be con- trolled, independently from each other, in CFD models. CFD makes it easy to investigate the influence of these parameters. In addition, CFD models can simulate a wide range of different situations, which saves money and time compared to experimental studies. The aim of the simulations performed in this study is to calculate the axial velocity distribution of steady, fully developed flow in curved tubes. Though much more information can be obtained from the computational data, such as pressure difference, secondary flow and stress distribution, this study will focus on the axial velocity profiles of fully developed flow. The final ob- jective is to validate the results of the simulations with the velocity profile measurements of an ultrasound probe obtained in an experimental set-up. The simulations are performed in the finite element package SEPRAN. Within SEPRAN two different three dimensional finite element models have been constructed to describe the flow in curved tubes. The first model is called ‘finite tube’ and the second ‘infinite tube’. In this section both models will first be introduced, then an overview of the results will be given. In the discussion both simulation methods will be com- pared with each other and with analytical, computational and experimental results from literature. 3.1 Method The motion of flow is described by the mass and momentum equations, the conservation laws. It is assumed that the fluid is incompressible, which reduces the mass equation to the continuity equation ∂ρ + (ρv)= v = 0. (85) ∂t ∇ · ∇ · The momentum equation is then given by ∂v ρ + ρ(v v)= σ + f, (86) ∂t ·∇ ∇ · 3 COMPUTATIONAL FLUID DYNAMICS 40 with v the velocity field of the fluid, ρ the density of the fluid, t the time, σ the Cauchy stress tensor and f the body forces per unit of volume, e.g. gravity. The Cauchy stress tensor can be subdivided into the hydrostatic pressure p and the extra stress tensor τ σ = pI + τ . (87) − The extra stress tensor τ is: τ = 2η(γ ˙ )Dv, (88) when an inelastic generalized Newtonian behavior of the fluid is assumed [24]. In this equation η(γ ˙ ) is the viscosity, dependent on the shear rate, and Dv is the rate of deformation tensor, which is defined as 1 T Dv = [ v + ( v) ]. (89) 2 ∇ ∇ For incompressible fluids the shear rate parameterγ ˙ , defined in terms of the second invariant of the rate of deformation tensor Dv, is γ˙ = 2Dv : Dv = IIDv . (90) p q Blood is known to be a non-Newtonian fluid, with a shear thinning behavior. This behavior can be described by using the Carreau-Yasuda model ([25]): −1 η(γ ˙ ) η a n − ∞ = [1 + (λγ˙ ) ] a , (91) η0 η − ∞ where η0 is the viscosity at a low shear rate, η the viscosity at a high ∞ shear rate, λ a time constant, n the power-law constant and a determines the transition between the low shear rate region and the power law region [21]. Ordinary Newtonian behavior can be described by η(γ ˙ ) = η0, which corresponds to a time constant of λ = 0 in the Carreau-Yasuda model. Both computational fluid dynamics models are composed of isoparamet- ric hexahedral volume elements with 27 points. The elements are of the tri-quadratic hexahedron Crouzeix-Raviart type, with a discontinuous pres- sure over the element boundaries. A first order Euler-implicit discretization scheme is used for temporal discretization of the equations. To linearize the convective term, the Newton-Raphson method is chosen. The BI-CGstab 3 COMPUTATIONAL FLUID DYNAMICS 41 iterative solution method, with an incomplete LU decomposition precondi- tioner, solves the linearized set of equations. For the continuity equation the integrated or coupled approach is used. The model parameters used in the CFD models are based on the carotid artery, for this reason the curved tube models will have a radius of 4 mm. The fluid parameters chosen are based on blood properties. In the simu- lations of this section blood is assumed to be an Newtonian fluid with a 3 3 3 density of ρ = 1.132 10 kgm− and a dynamic viscosity of η = 3.56 10− 1 1 · η · 6 kgm− s− , this results in a kinematic viscosity of ν = ρ = 3.145 10− 2 1 · m s− . Sensitivity of the axial velocity profiles for measurement position: In a clin- ical set-up the exact position of the ultrasound probe with respect to the artery is unknown. Luckily it is still possible to obtain a rough idea of the measurement position. For the measurements in the experimental set-up it is assumed that that the probe is positioned in the symmetry plane and measures the profile along the centerline. However, the exact position is not precisely known. It is therefore relevant to investigate the sensitivity of the axial velocity profile for variations in probe position. 3.1.1 Finite Tube The mesh of the first CFD model consists of 80 mesh elements in the axial direction, 5 elements in the radial direction and 6 elements in the circumfer- ential direction, see Figure 18 and Figure 19. The length of the tube in axial direction is the radius of the tube times the number of mesh elements. In radial direction the tube consists of a straight inlet section of 10 elements, followed by a curved section of 60 elements and a straight outlet section of 10 elements (so 80 4 mm= 320 mm). · A no-slip boundary condition is imposed at the tube wall. The outlet flow is stress free and at the inlet of the tube the velocity profile is prescribed in axial direction: 2 tπ r 2 Vmax sin 2T 1 a for t 1; vz(r, t)= 2 − ≤ (92) V 1 r for 1 Figure 18: The mesh of the finite tube model in axial direction with a cur- vature ratio of δ = 0.02. Figure 19: The mesh of the finite tube model in radial and circumferential direction. tional converging problems of the solution, which especially occurs at high velocities (high Reynolds numbers). The equations are solved time depen- dently (instationary), the times steps are t = 0.1 seconds and the total △ simulation time ttot = 3 seconds. All the input parameters used for simulations performed with the finite tube model are stated in Table 2. 3 COMPUTATIONAL FLUID DYNAMICS 43 Table 2: The input parameters for simulations of the infinite tube model. Dean number δ Vmax (m/s) 50 0.02 4.91 10 2 · − 100 0.02 9.83 10 2 · − 200 0.02 1.97 10 1 · − 400 0.02 3.93 10 1 · − 600 0.02 5.90 10 1 · − 800 0.02 7.86 10 1 · − 1000 0.02 9.83 10 1 · − 3.1.2 Infinite Tube The infinite tube model is developed to obtain relatively quick fully devel- oped flow profiles. This will save calculation time, but no realistic informa- tion can be obtained about the development of the velocity distribution in time or space. The mesh of the infinite tube model contains only 6 axial elements with a total length of 4 times the radius (4 4 mm=16 mm). The mesh consists of 9 · elements in radial direction and 12 elements in the circumferential direction, see Figure 20. A no-slip boundary condition is imposed at the tube wall and a stress free outlet is prescribed. In the first 20 time steps the inlet velocity profile in axial direction is prescribed as being a Poiseuille distribution r 2 v (r)= V 1 . (93) z max a − ! In the subsequent time steps the velocity in all directions is taken from the midplane of the tube. At the outlet a stress free boundary condition is prescribed, while flow in a curved tube is not stress free. This leads to an inappropriate solution at the outlet of the tube. This effect is negligible and has no upstream influence on the velocity distribution of the midplane. This velocity distribution needs to be multiplied with a rotation matrix in order to correct for the curvature, before it is prescribed for the next time step at the inlet of the tube (see Figure 21 for a schematic overview). The time step is chosen to be t = 0.01 seconds and the total simulation △ time is ttot = 2 seconds. 3 COMPUTATIONAL FLUID DYNAMICS 44 Figure 20: The mesh of an infinite tube model with a curvature ratio of δ = 0.16. Table 3 gives an overview of the different parameter sets used for the simu- lations performed with the infinite tube model. 3.2 Results First the results of the finite tube will be shown to investigate the flow development in time and in space. Next the fully developed axial veloc- ity distribution of the finite tube model will be compared to the solutions obtained with the infinite tube model. If the infinite tube model predicts correctly the fully developed axial velocity distribution, many simulations can be performed to obtain more insight about fully developed flow in curved tubes. 3 COMPUTATIONAL FLUID DYNAMICS 45 t>1s rotation transformation Figure 21: A schematic overview of the method used to calculate the fully developed axial velocity distribution with the infinite tube model. Table 3: Input parameters for simulations of the infinite tube model, with the determined Vmax (m/s) in the table. The * means that this simulation did not converge. Dean number δ=0.01 δ=0.02 δ=0.04 δ=0.08 δ=0.10 δ=0.16 1 1.39 10 3 9.83 10 4 6.95 10 4 4.91 10 4 4.40 10 4 3.47 10 4 · − · − · − · − · − · − 10 1.39 10 2 9.83 10 3 6.95 10 3 4.91 10 3 4.40 10 3 3.47 10 3 · − · − · − · − · − · − 25 3.47 10 2 2.46 10 2 1.74 10 2 1.23 10 2 1.10 10 2 8.69 10 3 · − · − · − · − · − · − 50 6.95 10 2 4.91 10 2 3.48 10 2 2.46 10 2 2.20 10 2 1.74 10 2 · − · − · − · − · − · − 100 1.39 10 1 9.83 10 2 6.95 10 2 4.91 10 2 4.40 10 2 3.47 10 2 · − · − · − · − · − · − 200 2.78 10 1 1.97 10 1 1.39 10 1 9.83 10 2 8.79 10 2 6.95 10 2 · − · − · − · − · − · − 400 5.56 10 1 3.93 10 1 2.78 10 1 1.97 10 1 1.76 10 1 1.39 10 1 · − · − · − · − · − · − 600 8.34 10 1 5.90 10 1 4.17 10 1 2.95 10 1 2.64 10 1 2.08 10 1 · − · − · − · − · − · − 800 1.11 7.86 10 1 5.56 10 1 3.93 10 1 3.52 10 1 2.78 10 1 · − · − · − · − · − 1000 1.39* 9.83 10 1 6.95 10 1 4.91 10 1 4.40 10 1 3.47 10 1 · − · − · − · − · − 3 COMPUTATIONAL FLUID DYNAMICS 46 3.2.1 Finite Tube Development in Time: The flow development as a function of the time can be seen in Figures 22 and 23. In these figures the results are shown of a simulation with the following parameters: Dn = 800, δ = 0.02 and V max = 0.786 m/s. It should be mentioned that the flow is gradually increased during the first second by means of a sin2(t)-function, this explains the low velocities at t=0.1s and t=0.5s. -3 -3 x 10 time=0.1s x 10 time=0.5s 5 5 350 40 300 250 30 0 0 200 y-as [m] 20 y-as [m] 150 100 10 50 -5 0 -5 0 -5 0 5 -5 0 5 -3 -3 x-as [m] x 10 x-as [m] x 10 -3 -3 x 10 time=1.0s x 10 time=1.5s 5 5 600 600 500 500 400 400 0 0 300 300 y-as [m] y-as [m] 200 200 100 100 -5 0 -5 0 -5 0 5 -5 0 5 -3 -3 x-as [m] x 10 x-as [m] x 10 -3 -3 x 10 time=2.0s x 10 time=2.9s 5 5 600 600 500 500 400 400 0 0 300 300 y-as [m] y-as [m] 200 200 100 100 -5 0 -5 0 -5 0 5 -5 0 5 -3 -3 x-as [m] x 10 x-as [m] x 10 Figure 22: The contour plots of the axial velocity profile for different times at 60 a . The results are taken from a simulation with the following · curved input parameters: Dn=800 and δ = 0.02, with the velocity in mm/s. 3 COMPUTATIONAL FLUID DYNAMICS 47 Velocity profiles in the symmetrical plane Velocity profiles in the anti-symmetrical plane 700 600 t=0.1s t=0.1s t=0.5s t=0.5s 600 t=1.0s 500 t=1.0s t=1.5s t=1.5s 500 t=2.0s t=2.0s t=2.9s 400 t=2.9s 400 300 v (mm/s) 300 v (mm/s) 200 200 100 100 0 0 -5 0 5 -5 0 5 -3 -3 r (mm) along the y-axis x 10 r (mm) along the x-axis x 10 Figure 23: The axial velocity profiles for different times at 60 a of a · curved simulation with Dn=800 and δ = 0.02, with the velocity in mm/s. From Figures 22 and 23 it is derived that the flow profile is fully developed in time for t> 1.5s. In Figure 22 the characteristic C-shaped contour plots of a curved tube for the axial velocity profiles are observed. The results are shown of a simulation with a relatively high velocity and thus a high Dean number. Simulations with a lower maximum velocity will develop faster, so by looking at t = 2.9 seconds the results obtained from all the other simu- lations are also fully developed in time. Development in Space: The flow development in axial direction of the finite tube model can be seen in Figures 24 and 25. Here the results at various planes are shown for the last time step (t = 2.9s) of a simulation with: Dn = 1000, δ = 0.02. The axial velocity profiles are calculated on the 161 planes of the tube, because of the 80 elements in axial direction. The ax- ial velocity distribution for the straight part gives a poiseuille profile. At 10 a , with a the radius of the tube, a shift of the maximum ve- · curved curved locity to the outside of the tube can be seen. The position of the maximum velocity oscillates between 20 a through 50 a around the final · curved · curved solution at 60 a . At the last plane the axial velocity profile is already · curved moving back to become a Poiseuille profile, because the tube is straight again. Sensitivity of the axial velocity profiles for measurement position: To visu- 3 COMPUTATIONAL FLUID DYNAMICS 48 -3 -3 10*a x 10 straight inlet x 10 curved 5 5 800 800 600 600 0 0 400 400 y-as [m] y-as [m] 200 200 -5 0 -5 0 -5 0 5 -5 0 5 -3 -3 x-as [m] x 10 x-as [m] x 10 -3 20*a -3 30*a x 10 curved x 10 curved 5 5 800 800 600 600 0 0 400 400 y-as [m] y-as [m] 200 200 -5 0 -5 0 -5 0 5 -5 0 5 -3 -3 x-as [m] x 10 x-as [m] x 10 -3 40*a -3 50*a x 10 curved x 10 curved 5 5 800 800 600 600 0 400 0 400 y-as [m] y-as [m] 200 200 -5 -5 0 -5 0 5 -5 0 5 -3 -3 x-as [m] x 10 x-as [m] x 10 -3 60*a -3 x 10 curved x 10 straight outlet 5 800 5 800 600 600 0 400 0 400 y-as [m] y-as [m] 200 200 -5 0 -5 0 -5 0 5 -5 0 5 -3 -3 x-as [m] x 10 x-as [m] x 10 Figure 24: The contour plots of the axial velocity profile at different planes (time=2.9s) of a simulation with Dn=1000 and δ = 0.02, with the velocity in mm/s. alize the influence of the ultrasound probe position, different axial velocity profiles are shown for a change of position in angle or in y-direction from the centerline of the plane of symmetry. The axial velocity profiles at different angles are shown in Figure 26 and the axial velocity profiles at different positions parallel to the centreline are shown in Figure 27. Figure 26 shows that the axial velocity profile will not differ so much from the optimal mea- 3 COMPUTATIONAL FLUID DYNAMICS 49 Profile in the symmetrical plane Profile in the anti−symmetrical plane 1000 1000 straight inlet 10*a 900 curved 900 20*a curved 30*a 800 curved 800 40*a curved 50*a 700 curved 700 60*a curved 600 straight outlet 600 500 500 v (mm/s) v (mm/s) 400 400 300 300 200 200 100 100 0 0 −5 0 5 −5 0 5 −3 −3 r (mm) along the y−axis x 10 r (mm) along the x−axis x 10 Figure 25: The axial velocity profiles for different planes (time=2.9s) of a simulation with Dn=1000 and δ = 0.02, with the velocity in mm/s. surement position, until an angle of 20 degrees. From Figure 27 it can be deduced that a position difference of y = 1.0 mm does not result in serious △ change in the axial velocity profile. 3.2.2 Finite tube versus Infinite tube The axial velocity distributions of the infinite tube model are compared to the axial velocity distributions of the finite tube model to determine the accuracy of the infinite tube model. An error norm (ǫN )is defined to obtain a quantitative result for how accurate the infinite tube model calculates the axial velocity of the finite tube for the different simulations: v (i) v (i) N ft − it ǫ = | Vmax |. (94) N N Xi=1 3 COMPUTATIONAL FLUID DYNAMICS 50 Axial Velocity profile for different angles 900 800 700 600 500 400 0 degrees Velocity (mm/s) 10 degrees 300 20 degrees 200 30 degrees 40 degrees 100 90 degrees 0 −4 −3 −2 −1 0 1 2 3 4 r (mm) Figure 26: The axial velocity profiles when measured at one line through the contour plot, but with different angles. Dn=1000 and δ = 0.02, with the velocity in mm/s. -3 Axial velocity profiles at different positions Profile at different y-positions x 10 700 5 y=0.0mm y=-1.0mm 4 600 600 y=-2.0mm 3 y=-3.0mm 500 y=-3.5mm 500 2 1 400 400 0 y-as [m] 300 v (mm/s) 300 -1 -2 200 200 -3 100 100 -4 -5 0 -5 0 5 -5 0 5 x-as [m] -3 -3 x 10 r (mm) along x-as x 10 Figure 27: The axial velocity profiles for positions at 60 a (time=2.9s) · curved of a simulation with Dn=800 and δ = 0.02, with the velocity in mm/s. In this equation i = 1.....N are the nodes in the cross sectional plane, vft is the velocity of the finite tube model, vit is the velocity of the infinite tube model and Vmax is the maximum velocity based on the prescribed Poiseuille profile. When the calculated norm is smaller than 0.005 the flow is assumed to be fully developed. The development of the velocity distribution over the length of the finite tube model is investigated by performing a comparison with the final solution of the infinite tube model. The level of the velocity development is expressed in the error norm, see Figures 28, 29 and 30 for the results. 3 COMPUTATIONAL FLUID DYNAMICS 51 -3 Axial velocity infinite tube (t=1.99s) x 10 Axial velocity finite tube (plane=130) 5 800 5 800 600 600 0 400 0 400 y-axis [m] y-axis [mm] 200 200 -5 0 -5 0 -5 0 5 -5 0 5 -3 x-axis [mm] x-axis [m] x 10 symmetry plane anti-symmetry plane 1000 800 infinite tube finite tube 700 800 600 600 500 400 v (mm/s) 400 v (mm/s) 300 infinite tube 200 200 finite tube 100 0 0 -5 0 5 -5 0 5 r (mm) along x-axis r (mm) along y-axis norm value 0.1 Difference of axial velocity 5 norm 15 0.08 10 5 0.06 0 0 -5 0.04 norm value y-axis [mm] -10 -15 0.02 -5 -5 0 5 x-axis [mm] 0 0 50 100 150 plane number of the finite tube Figure 28: An example of the comparison between axial velocity distribution of the infinite and finite tube model at plane=130, Dn=1000 and δ = 0.02. Even for high Dean numbers, the axial velocities as calculated with the finite or the infinite tube model are the nearly exactly the same, see Figure 28, which shows the results for Dn = 1000. All simulations from Table 2 show similar results. The norm plots as function of the plane number for different curvature ratios are shown in Figure 29. In this figure a small increase of the norm value can be seen just after the beginning of the curved part, for all the different Dean numbers. The maximum velocity is slightly moving to the inside of the tube before it moves to the outside of the tube. The norm value is not going gradually to its lowest value. The axial velocity distribution fluctuates around the final distribution, before it reaches a really 3 COMPUTATIONAL FLUID DYNAMICS 52 norm value as function of plane 0.1 Dn=50 Dn=100 0.09 Dn=200 Dn=400 0.08 Dn=600 0.07 Dn=800 Dn=1000 0.06 threshold 0.05 norm value 0.04 0.03 0.02 0.01 0 0 20 40 60 80 100 120 140 160 180 plane number Figure 29: The norm value as function of the plane number of the finite tube simulation for different Dean numbers, with a δ = 0.02. stable norm value. This effect is larger for higher Dean numbers. norm value as function of plane 0.012 Dn=50 Dn=100 Dn=200 0.01 Dn=400 Dn=600 Dn=800 0.008 Dn=1000 threshold 0.006 norm value 0.004 0.002 0 20 40 60 80 100 120 140 160 plane number Figure 30: A magnification of Figure 29 to have a better look at the region of interest. Figure 30 shows a magnification of the lower norm values of Figure 29 to clarify from which plane on the axial velocity of the finite tube model is called fully developed. Table 4 shows from which plane on the axial velocity 3 COMPUTATIONAL FLUID DYNAMICS 53 is fully developed for the different Dean numbers. From this plane number (Npl) the angle of the tube is calculated, with a (Npl 20) (N 20) δ θ = − · 2 360o = pl − · 360o. (95) fd 2πR · 4π · The lowest norm value for all the simulations is obtained around 57 a . · curved The norm value increases again at the end of the curved geometry. All simulations show a little decrease around plane 145, in the straight part, probably due to overcompensation. Except for the simulation with Dn=50, all simulations seem to follow nearly the same ‘recovery’ path in the straight outlet part of the tube. Table 4: The plane, number of curved radii and angle at which fully devel- oped flow is reached in the finite tube model for different Dean numbers, based on the infinite tube model o Dean number plane # curved radii θfd[ ] 50 26 3 3.4 100 40 10 11.5 200 63 21.5 24.6 400 77 28.5 32.7 600 108 44 50.4 800 112 46 52.7 1000 116 48 55.0 3.2.3 Infinite Tube The results from the last section showed that the infinite tube model pre- dicts correctly the fully developed axial velocity distribution of the finite tube model. Now a wide range of simulations can be performed to investi- gate if characteristic trends for flow in a curved tube can be observed. One of the characteristics of a curved tube is the (relative) position of the max- imum velocity at the symmetry plane. Simulations are performed with the parameters from Table 3 and results are shown in Figures 31, 32 and 33. Figure 31 shows an example of the axial velocity profiles for different Dean numbers of a tube with a curvature ratio of δ = 0.16. In the normalized figures the shape of the velocity profiles is more clear. From these profiles 3 COMPUTATIONAL FLUID DYNAMICS 54 δ = 0.16 δ = 0.16 0.4 1 1 10 25 0.3 50 100 max 200 (m/s) 0.2 /v 0.5 400 600 ax ax 800 v v 0.1 1000 0 0 -4 -2 0 2 4 -3 -1 0 1 r (m) x 10 r/a0 δ = 0.16 δ = 0.16 0.4 1 0.3 max (m/s) 0.2 /v 0.5 ax ax v v 0.1 0 0 -5 0 5 -1 0 1 r (m) -3 r/a x 10 0 Figure 31: The axial velocity profiles for different Dean numbers of δ = 0.16, with the velocity in the left figures in m/s and in the right figures the normalized velocity profiles for the symmetry plane and the anti-symmetry plane. the relative position of the maximum velocity is determined. It is obvious that for a higher Dean number and so a higher Vmax, the position of the maximum velocity shifts more to the outside of the curve. Figure 32 shows the position of the maximum velocity as function of the curvature ratio (δ) for different values of Dean. From this figure there seems to be a linear relation between the position of the maximum and (δ), but this linear relation is not the same for different Dean numbers. A least squares fit is made to obtain the slope and offset to describe the linear polynomials for the different Dean numbers, see Table 5 for the results. In the other figure, Figure 33, the relative position of the maximum velocity is plotted against the Dean number for different curvature ratios. For low Dean numbers the position of the maximum velocity shifts to the inside of the curve, this effect is larger for higher curvature ratios. From Dn = 50 3 COMPUTATIONAL FLUID DYNAMICS 55 0.1 0 ) 0 Dean 1 −0.1 Dean 10 Dean 25 −0.2 Dean 50 Dean 100 −0.3 Dean 200 Dean 400 −0.4 Dean 600 Dean 800 −0.5 Dean 1000 Position maximum (r/a −0.6 −0.7 0 0.05 0.1 0.15 0.2 δ Figure 32: The normalized position of the maximum velocity as function of δ for different Dean numbers. Table 5: Linear relation between the relative position of the maximum ve- locity and δ for different Dean numbers (P osV = a δ + b) max · Dean number a b 1 0.38 5.2 10 4 · − 10 0.38 -3.0 10 3 · − 25 0.37 -1.7 10 2 · − 50 0.34 -6.1 10 2 · − 100 0.32 -2.2 10 1 · − 200 0.32 -4.6 10 1 · − 400 0.19 -5.7 10 1 · − 600 0.16 -6.2 10 1 · − 800 0.14 -6.7 10 1 · − 1000 0.10 -6.9 10 1 · − and higher the position of the maximum velocity is always shifted to the outside of the curve. The higher the Dean number the more the position is shifted. It seems that for really high Dean numbers (Dn > 1000), the shift 3 COMPUTATIONAL FLUID DYNAMICS 56 0.1 δ = 0.01 0 δ = 0.02 ) δ 0 = 0.04 −0.1 δ = 0.08 δ = 0.10 −0.2 δ = 0.16 −0.3 −0.4 −0.5 Position of maximum (r/a −0.6 −0.7 0 200 400 600 800 1000 Dean Figure 33: The normalized position of the maximum velocity as function of Dean numbers for different δ. of the position of the maximum velocity will be limited. The differences in the position of the maximum velocity for different curvature ratios but with the same Dean number become less for higher Dean numbers. The relative position of the maximum axial velocity as function of Reynolds number is shown in Figure 34 and Figure 35. Around Re = 21 all curves pass the symmetry point (zero). For smaller Reynolds numbers the posi- tion of the maximum is shifted to the inside of the curvature and for higher Reynolds numbers the position is shifted to the outside of the curvature. 3.3 Discussion The infinite tube model gives a good prediction of the fully developed flow profiles which are also obtained with the finite tube simulation with t=2.9 seconds and for the positions between 48 a and 60 a . The infinite · curved · curved tube model takes about 20 times less calculation time in comparison to the finite tube model. The infinite tube model is thus a time saving solution to calculate fully developed flow profiles. However, if the development of the flow profiles is of interest, the infinite tube model can not be used. The obtained fully developed flow profiles with both finite element models seem 3 COMPUTATIONAL FLUID DYNAMICS 57 0.1 δ = 0.01 0 δ = 0.02 ) δ 0 = 0.04 −0.1 δ = 0.08 δ = 0.10 −0.2 δ = 0.16 −0.3 −0.4 −0.5 Position of maximum (r/a −0.6 −0.7 0 500 1000 1500 Reynolds number Figure 34: The normalized position of the maximum velocity as function of Reynolds number for different δ. 0.08 δ = 0.01 δ = 0.02 ) 0.06 δ 0 = 0.04 δ = 0.08 0.04 δ = 0.10 δ 0.02 = 0.16 0 −0.02 −0.04 Position of maximum (r/a −0.06 −0.08 0 20 40 60 80 Reynolds number Figure 35: The normalized position of the maximum velocity as function of Reynolds number for different δ. 3 COMPUTATIONAL FLUID DYNAMICS 58 to correspond with results from literature, which were obtained numerically and experimentally [16, 14]. However, it is difficult to compare the results exactly. Often only the Dean number is given sometimes in combination with the value of the (scaled) maximum axial velocity, but nothing is known about the exact values for the curvature ratio, diameter, viscosity parame- ters etc. While a good description of a flow problem in a curved tube should be given in at least δ&Re or δ&Dn or Dn&Re. Most research is focused on the flow ratio of flow through a straight tube in comparison with flow in a curved tube driven by the same pressure gradient. In the simulations of this section a certain flow was imposed and no attention has been paid to the pressure gradient, because the pressure gradient is not known when a velocity profile is measured with an ultrasound system. Besides a qualitative comparison between the contour plots of the axial ve- locities, another more quantitative comparison can be made by observing the relative position of the maximum velocity. Although this is just one of the characteristics of curved tube flow, it gives some well defined results and it is the only quantitative comparison possible as all other necessary information is missing in literature. McConalogue and Srivastava used a Fourier-series development method to solve the momentum and continuity equation in the toroidal system numer- ically [14]. They published their resulting contour plots of the axial velocity for different values of Dean number between Dn=96 and Dn=605.72. From these contour plots the relative position of the maximum velocity can be de- duced. In Figure 36 the results are shown in the same graph as the maximum positions computed with the numerical simulations of the infinite tube. The figure shows a great resemblance between the results from McConalogue and Srivastava and the results from the infinite tube model. As McConalogue and Scrivastava state that they assume δ to be small, one should expect their results should correspond to the δ = 0.01 solutions of the infinite tube simulations. As can be seen in Figure 36 their results do coincide with the δ = 0.01 results, except two data points around Dn=200. Though part of the difference may be the result of digitizing errors during the scanning of the results from McConalogue and Scrivastava. Siggers and Waters derived in their article an analytical approximation for curved tubes using series expansion and by assuming that Dn 1 and δ so ≪ small that δ 0, but it still has some curvature, see Section 2.2 [5]. They ↓ related the relative position of the maximum velocity rV max to the Dean 3 COMPUTATIONAL FLUID DYNAMICS 59 0.1 δ = 0.01 0 δ = 0.02 ) δ 0 = 0.04 −0.1 δ = 0.08 δ = 0.10 −0.2 δ = 0.16 McConalogue&Srivastava −0.3 −0.4 −0.5 Position of maximum (r/a −0.6 −0.7 0 200 400 600 800 1000 Dean Figure 36: The normalized position of the maximum velocity as function of Dean number for different δ’s with the results of [14] number and curvature ratio: 19Dn2 3δ r = + O(δ3,Dn2δ2,Dn4δ,Dn6). (96) V max 214 32 5 − 8 · · Figure 37 shows this relative position as function of Dean number for dif- ferent values of δ in comparison with the results obtained with the infinite tube model (see also Figure 33). In Figure 38 the normalized position of the maximum velocity as function of δ is given for both methods. The analytical solution predicts the position of the maximum velocity well for Dn 50. Despite the assumptions made for the approximation, the ≤ analytical solution coincides well with the results from the infinite tube sim- ulations for higher curvature ratios, like δ = 0.16. The computational method presented in this study and the analytical method of Siggers and Waters independently predict the phenomenon that for low Dean numbers the maximum position is shifted to the inside of the curve. This effect is larger for a larger curvature ratio. It seems to be an odd effect, because the axial velocity profiles under consideration are totally developed in space and time, so it cannot be explained by inflow ‘artefacts’ [20]. Siggers and Waters notice this effect too, but do not give a physical explanation. A 3 COMPUTATIONAL FLUID DYNAMICS 60 δ 0.05 = 0.01 δ = 0.02 ) δ 0 = 0.04 0 δ = 0.08 δ = 0.10 −0.05 δ = 0.16 −0.1 −0.15 −0.2 Position of maximum (r/a −0.25 0 50 100 150 Dean Figure 37: The normalized position of the maximum velocity as function of Dean numbers for different δ compared with the analytical derived function of Siggers and Waters [5]. possible explanation might be that for a low Dean numbers and especially for low Dean numbers with a large curvature ratio, the Reynolds number is low e.g., around 1. A low Reynolds number means that the inertia forces are relatively weak (these forces push the fluid to the outside of the curve) and so a relatively small secondary stream profile will develop. This results in a negligible pressure gradient in the radial direction which is comparable to the pressure distribution in a straight tube. But the geometry of the tube is still curved, so the fluid that flows through the tube will choose the shortest way possible through the tube, following the highest pressure gradient in axial direction, which is at the inside of the curve. This implies the shifting to the inside of the tube is a pure geometry driven effect. To examine this hypothesis, the pressure plots of some simulations of the infinite tube model are compared. As example a combination of simulations is chosen with a relatively low cur- vature ratio (δ=0.02) versus high curvature ratio (δ=0.16) and a high Dean number (Dn=1000) versus a low Dean number (Dn=1). Figure 39 shows the input plane and midplane pressure plots for the different simulations. A problem, visible from the pressure contour plots, is that the pressure values 3 COMPUTATIONAL FLUID DYNAMICS 61 0.1 0.05 ) 0 0 −0.05 Dean 1 −0.1 Dean 10 Dean 25 −0.15 Dean 50 Dean 100 −0.2 Position maximum (r/a −0.25 −0.3 0 0.05 0.1 0.15 0.2 δ Figure 38: The normalized position of the maximum velocity as function of δ for different Dean numbers. The infinite tube model results (dotted lines) are compared with the analytical derived function of Siggers and Waters (the solid lines with *) [5]. are really low for simulations with Dn=1 causing irregularities in which the mesh distribution can be seen. Table 6 gives the pressure ratio between the calculated radial pressure gradient and the axial pressure gradient. The pressure ratio is defined as follows: p p = △ radial (97) △ ratio p △ axial 3 COMPUTATIONAL FLUID DYNAMICS 62 Table 6: Pressure ratio between the axial and radial pressure gradient for several simulations Dn δ p [P a/m] p [P a/m] p [P a/m] △ radial △ axial △ ratio 1 0.02 2.2 10 3 8.7 10 1 2.6 10 3 · − · − · − 1 0.16 2.2 10 3 2.8 10 1 9.8 10 3 · − · − · − 1000 0.02 1.8 103 1.7 103 1.1 · · 1000 0.16 1.7 103 5.4 102 3.1 · · 3.3.1 Simulations vs Analytical approximation Methods The normalized axial velocity profiles of the analytical solution derived by Siggers and Waters compared to the profiles calculated with the infinite tube model are shown in Figure 40 through Figure 44. These figures show that the analytical solution corresponds very well for Dn = 1 until Dn = 50, for the whole range of simulated δ’s. For Dn = 100 the analytical approximation deviates from the normalized axial velocity profile derived with the infinite tube model, which will become worse for higher Dean numbers. The same results will be obtained by looking at the normalized axial velocity profiles calculated from the analytical approximation method of Topakoglu. As his results gives nearly the same results as the approximation method from Siggers and Waters. The analytical approximation method of Dean will only be correct for δ = 0.01. 3 COMPUTATIONAL FLUID DYNAMICS 63 -6 -5 pressure midplane Dn=1 δ=0.02 x 10 presure inputplane Dn=1 δ=0.02 x 10 5 5 6.755 1.374 6.75 1.372 6.745 0 0 6.74 1.37 y-axis [mm] 6.735 y-axis [mm] 1.368 6.73 -5 -5 1.366 -5 0 5 -5 0 5 x-axis [mm] x-axis [mm] -6 -5 pressure midplane Dn=1 δ=0.16 x 10 pressure inputplane Dn=1 δ=0.16 x 10 5 5 2.12 1.374 2.115 1.372 0 0 2.11 1.37 y-axis [mm] y-axis [mm] 2.105 1.368 -5 -5 1.366 -5 0 5 -5 0 5 x-axis [mm] x-axis [mm] pressure midplane Dn=1000 δ=0.02 pressure inputplane Dn=1000 δ=0.02 5 5 0.038 0.022 0.036 0.02 0.034 0.018 0.032 0 0 0.016 0.03 y-axis [mm] y-axis [mm] 0.028 0.014 0.026 0.012 0.024 -5 -5 -5 0 5 -5 0 5 x-axis [mm] x-axis [mm] -3 pressure midplane Dn=1000 δ=0.16 x 10 pressure inputplane Dn=1000 δ=0.16 5 5 14 0.018 12 0.016 10 0.014 0 0 8 0.012 y-axis [mm] 6 y-axis [mm] 0.01 4 0.008 -5 2 -5 0.006 -5 0 5 -5 0 5 x-axis [mm] x-axis [mm] Figure 39: The contour plots of the pressure (kPa) at the midplane (left figures) and the inflow plane (right figures) for the different. 3.4 Conclusion The aim of the simulations was to calculate the axial velocity profiles of fully developed flow in curved tubes, with a focus on flows which can be characterized by 100 Dn 1000 for different curvature ratios. ≤ ≤ 3 COMPUTATIONAL FLUID DYNAMICS 64 simulation vs analytical approximation Dn = 1 1 simulation vs analytical approximation Dn = 1 1 0.9 0.8 0.95 0.7 0.6 0.9 0.5 0.4 δ 0.85 Infinite Tube = 0.01 Infinite Tube δ = 0.01 δ Siggers and Waters = 0.01 Siggers and Waters δ = 0.01 0.3 Infinite Tube δ = 0.16 Infinite Tube δ = 0.16 Siggers and Waters δ = 0.16 Siggers and Waters δ = 0.16 0.2 0.8 0.1 0 0.75 −5 −4 −3 −2 −1 0 1 2 3 4 5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 x−axis −3 −3 x 10 x−axis x 10 (a) Dn = 1 (b) Dn = 1 magnified Figure 40: The normalized axial velocity profiles of the analytical solution versus derived by Siggers and Waters the infinite tube simulation (a) Dn = 1 and δ = 0.01 or δ = 0.16 (b) magnification of Dn = 1 and δ = 0.01 or δ = 0.16 simulation vs analytical approximation Dn = 10 1 simulation vs analytical approximation Dn = 10 1 0.9 0.8 0.95 0.7 0.6 0.9 0.5 0.4 δ 0.85 Infinite Tube = 0.01 Infinite Tube δ = 0.01 δ Siggers and Waters = 0.01 Siggers and Waters δ = 0.01 δ 0.3 Infinite Tube = 0.16 Infinite Tube δ = 0.16 δ Siggers and Waters = 0.16 Siggers and Waters δ = 0.16 0.2 0.8 0.1 0 0.75 −5 −4 −3 −2 −1 0 1 2 3 4 5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 x−axis −3 −3 x 10 x−axis x 10 (a) Dn = 10 (b) Dn = 10 magnification Figure 41: The normalized axial velocity profiles calculated from the analyti- cal solution derived by Siggers and Waters versus the infinite tube simulation (a) Dn = 10 and δ = 0.01 or δ = 0.16 (b) magnification of Dn = 10 and δ = 0.01 or δ = 0.16 The simulations of the finite tube model show that after a simulated time of 1.5 seconds the axial velocity profile is already nearly fully developed in time, so the total simulation time of 3 seconds is sufficient. From the 3 COMPUTATIONAL FLUID DYNAMICS 65 simulation vs analytical approximation Dn = 25 1 simulation vs analytical approximation Dn = 25 1 0.9 0.8 0.95 0.7 0.6 0.9 0.5 0.4 δ 0.85 Infinite Tube = 0.01 Infinite Tube δ = 0.01 δ Siggers and Waters = 0.01 Siggers and Waters δ = 0.01 0.3 Infinite Tube δ = 0.16 Infinite Tube δ = 0.16 Siggers and Waters δ = 0.16 Siggers and Waters δ = 0.16 0.2 0.8 0.1 0 0.75 −5 −4 −3 −2 −1 0 1 2 3 4 5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 x−axis −3 −3 x 10 x−axis x 10 (a) Dn = 25 (b) Dn = 25 magnification Figure 42: The normalized axial velocity profiles calculated from the analyti- cal solution derived by Siggers and Waters versus the infinite tube simulation (a) Dn = 25 and δ = 0.01 or δ = 0.16 (b) magnification of Dn = 25 and δ = 0.01 or δ = 0.16 simulation vs analytical approximation Dn = 50 1 simulation vs analytical approximation Dn = 50 1 0.9 0.8 0.95 0.7 0.6 0.9 0.5 0.4 0.85 Infinite Tube δ = 0.01 Infinite Tube δ = 0.01 Siggers and Waters δ = 0.01 Siggers and Waters δ = 0.01 0.3 Infinite Tube δ = 0.16 Infinite Tube δ = 0.16 Siggers and Waters δ = 0.16 Siggers and Waters δ = 0.16 0.2 0.8 0.1 0 0.75 −5 −4 −3 −2 −1 0 1 2 3 4 5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 x−axis −3 −3 x 10 x−axis x 10 (a) Dn = 50 (b) Dn = 50 magnification Figure 43: The normalized axial velocity profiles calculated from the analyti- cal solution derived by Siggers and Waters versus the infinite tube simulation (a) Dn = 50 and δ = 0.01 or δ = 0.16 (b) magnification of Dn = 50 and δ = 0.01 or δ = 0.16 comparison with the infinite tube model it can be concluded that between 48 a and 60 a all simulations are fully developed in space. The · curved · curved infinite tube model gives a good prediction of the fully developed velocity 3 COMPUTATIONAL FLUID DYNAMICS 66 simulation vs analytical approximation Dn = 100 simulation vs analytical approximation Dn = 100 1 1 0.9 0.8 0.95 0.7 0.6 0.9 0.5 0.4 0.85 Infinite Tube δ = 0.01 Infinite Tube δ = 0.01 Siggers and Waters δ = 0.01 Siggers and Waters δ = 0.01 0.3 Infinite Tube δ = 0.16 Infinite Tube δ = 0.16 Siggers and Waters δ = 0.16 Siggers and Waters δ = 0.16 0.2 0.8 0.1 0 0.75 −5 −4 −3 −2 −1 0 1 2 3 4 5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −3 −3 x−axis x 10 x−axis x 10 (a) Dn = 100 (b) Dn = 100 magnification Figure 44: The normalized axial velocity profiles calculated from the analyti- cal solution derived by Siggers and Waters versus the infinite tube simulation (a) Dn = 100 and δ = 0.01 or δ = 0.16 (b) magnification of Dn = 100 and δ = 0.01 or δ = 0.16 profiles, while it takes less time to perform the simulation. The results of the simulations are needed to give a prediction of the axial velocity profiles which can be measured with an ultrasound system. To inves- tigate the influence of the ultrasound probe position, axial velocity profiles were calculated laterally shifted or rotated with respect to the centerline. When the lateral shift is less than 1 mm or the rotation is less than 20◦ the measured axial velocity are still acceptable. The axial flow profiles obtained from the computational fluid dynamics sim- ulations show a great resemblance with results presented in literature, al- though it is sometimes hard to compare the results exactly with each other. Usually not so much information is given about the parameters used [14, 16]. Three different analytical approximation methods ([3, 4, 5]) were compared to the results of the simulations. All three methods seem to predict the velocity profiles very well until Dn 50 and not bad until Dn 100, but ≤ ≤ for higher Dean numbers no proper analytical approximation method exists. In addition to their analytical approximation method, Siggers and Waters derived an equation for the relative position of the maximum velocity as function of Dn and δ. The equation predicts nearly exact the positions ob- tained with the computational method for the full range of δ, used in this section, and for Dn 50. The same conclusion applies to the analytical ap- ≤ proximation method of Topakoglu. While for Dean’s approximation method 3 COMPUTATIONAL FLUID DYNAMICS 67 the results will only correspond to the simulations with δ = 0.01. From these results it is concluded that the analytical approximation methods are valid for Dn 50, so unfortunately these methods are not applicable for the ≤ range of interest of this research, 100 Dn 1000. ≤ ≤ Both methods show that for low Dean numbers the position of the maximum velocity is moving to the inside of the curve. This phenomenon can possi- bly be explained by the relatively low pressure gradient in radial direction in comparison to the axial pressure gradient, causing the fluid to take the shortest route by following the path with the highest axial pressure gradient. 4 ULTRASOUND MEASUREMENTS 68 4 Ultrasound measurements The main objective of this study is to assess the flow through a curved tube, from a given asymmetric axial velocity profile obtained from ultrasound measurements. An experimental set-up is designed to investigate whether this objective is feasible for clinical practice. In the experimental set-up an ultrasound scanner, which is used in a clinical practice, measures the axial velocity profiles of a blood mimicking fluid flowing through a curved tube. The first goal of the ultrasound measurements is to investigate whether it is possible to correctly measure the axial velocity profiles in a curved tube. The data obtained must be accurate enough to determine the position of the maximum axial velocity, to derive the flow characteristics (Section 3.3). The axial velocity profiles are calculated with a newly developed method, based on cross correlation. This method allows a simultaneous measurement of the wall displacement and the axial velocity. The second goal is to validate the infinite tube model. Therefore the mea- sured axial velocity profiles are compared with the results from the com- putational fluid dynamics models. In the experimental set-up the flow and curvature are known beforehand. This information can be applied in the CFD tube model to validate the simulations experimentally. 4.1 Materials and Method In this section first an explanation is given about the preparation of the curved tubes and the blood mimicking fluids. Secondly the experimental set- up and the special settings of the ultrasound scanner are described. Finally additional information is given concerning the simulations with the infinite tube model. 4.1.1 Preparation of the Polyurethane Tubes In the experiments polyurethane tubes with an inner radius of 4 mm, a wall thickness of 0.2 mm and a specified curvature ratio (δ =0.01 and 0.02) are used. The polyurethane tubes are produced with the spin coating method [26]. First a silicone tube with an inner diameter of 6 mm and an outer diameter of 8 mm is placed on a straight steel rod with a diameter of 5 mm. The liquid used for the spin coating process consists of 17 % polyurethane by weight (Desmopan 588, Bayer, Germany) dissolved in tetrahydrofurane (THF, BASF, Germany). This liquid is deposited on the rotating silicone 4 ULTRASOUND MEASUREMENTS 69 tube with a constant flow rate through a nozzle. The nozzle moves along the length of the steel rod, with a constant velocity to produce a tube with a certain length. One minute after finishing this process, the silicone tube is shifted from the straight steel rod onto a curved steel rod. After one day nearly all the tetrahydrofurane has evaporated and the curved polyurethane tube segment is removed from the silicone tube. This process is repeated until enough tube segments are made. Finally, the tube segments are placed ‘head on’ on a HMPE (teflon) cilinder ( 8 mm) and some TMF-PU mixture ∅ is applied to create a smooth coupling of the segments to create one single tube (Figure 46). 4.1.2 Blood Mimicking fluid (BMF) Blood is a Non-Newtonian fluid with a shear thinning behavior, when the shear rate increases the viscosity decreases. Xanthan Gum has been used previously to succesfully mimic the mechanical behavior of blood [27]. For ultrasound purposes the liquid needs to have blood-analog ultrasound scat- tering properties as well. In blood this scattering is caused by the red blood cells. To imitate these viscous and acoustic properties in the experimental set-up a fluid is developed by Beulen [6] based on Ramnarine [28]. The shear thinning BMF consists of Xanthan Gum (Fluka, 95465) (0.5 g/L) dissolved in water (97.3 mass percent), where 1.8 mass percent Ultrafine Polyamide Particles (Orgasol, ELF Atochem, Paris, France) is added to- gether with 0.9 mass percent Synperonic NP10 (Fluka 86208). The Xanthan Gum is added to mimic the shear thinning behavior of blood. The Orgasol particles have a diameter of 5 µm, which is close to the diameter of red blood cells. The concentration of the particles used, has been proven to result in a blood-similar backscattering of the ultrasound waves [28]. Synperonic NP10 is used as a surfactant to prevent coagulation of the Orgasol particles in the Xanthan Gum solution. To determine the exact shear thinning behavior of the fluid, the kinematic viscosity is measured as function of the shear rate with a Couette rheometer (RFS 2, Rheometrics Scientific). The viscosity is determined for shear rates 1 1 between 0.1 s− until 1000 s− with 10 measurement points per decade (Fig- ure 45). The viscosity of the BMF is measured daily to test for the stability of the mixture and for reproducibility of the protocol. A Newtonian fluid similar to the shear thinning BMF is made as a control fluid. This fluid consists of 97.3 mass percent water, with 1.8 mass percent Orgasol particles and 0.9 mass percent Synperonic NP10. The kinematic 4 ULTRASOUND MEASUREMENTS 70 Viscosity Measurement −1 10 ) −1 s −1 −2 10 (kgm η −3 10 −1 0 1 2 3 10 10 10 10 10 γ˙ (s−1 ) Figure 45: The viscosity measurement of the shear thinning blood mimicking fluid. viscosity of this fluid is determined with the Couette rheometer to be η = 3 kg 3 1.3 0.2 10− ms . This is close to the viscosity of water η = 1.0 10− kg ± · · ms , so from now on it is assumed that the Newtonian BMF has the same properties of water. 4.1.3 Experimental Set-up The experimental set-up consists of a large water filled container. The curved polyurethane tube is placed in the water and is supported by small cilinders to fixate the tube. Together with some connective tubes a closed circulation is made (Figure 46 and Figure 47). The Blood Mimicking Fluid is first placed in a reservoir, from which it is steadily pumped into the cir- culation system. The BMF enters the straight part of the tube where the flow probe (Transonic, 10PAA) is positioned. Then the BMF flows into the curved part of the tube. At 270◦ the axial velocity profile is measured by 4 ULTRASOUND MEASUREMENTS 71 the ultrasound probe. The ultrasound probe is fixed on a 6-dimensional traversing device, to enable accurate and reproducible positioning. After the curved part, the fluid enters a straight tube. At the end of the straight tube, the BMF flows into a small reservoir with an adjustable resistance, which controls the flow in the circulation system. Finally the fluid flows back to the reservoir. pump BMF flowprobe R PC Labview Ultrasound Art. Lab Figure 46: A semi-schematic overview of the experimental set-up used to measure velocity profiles in curved polyurethane tubes with a curvature ratio of δ=0.01 and 0.02 for different flow rates. BMF is the reservoir with the blood mimicking fluid. The BMF is pumped first trough the straight part of the tube, where also the flow probe is placed. Then the BMF flows through the curved tube, at 180◦ the ultrasound probe is placed to measure the axial velocity profile. After the curved part, the tube is straight again and flows back to the reservoir. Meanwhile it passes the resistor (R), which is used to regulate the flow. The ultrasound probe is assumed to measure the axial velocity profile of the curved tube in the plane of symmetry. To achieve this position as good as possible a calibration device is used to align the probe, see Figure 48. This alignement is based on the visualized ultrasound data. The ultrasound probe consists of an array of 128 piezoelectric crystal ele- ments with a center frequency of 7.5 MHz. The ultrasound probe transmits 4 ULTRASOUND MEASUREMENTS 72 BMF flow probe pump BMF curved tube ultrasound probe Figure 47: A picture of the experimental set-up, with the Blood Mimicking Fluid being white (because of the orgasol particles). The curved tube on the picture has a δ = 0.04. the ultrasound waves with a Pulse Repetition Frequency (PRF) of 10 kHz. The ultrasound scanner acquires the RF-data of the ultrasound probe with a sampling rate of 33 MHz, with a sound wave velocity of 1540 m/s this results in a sample size of x = 2.3 10 5 m. △ · − The ultrasound scanner (Picus, ESAOTE) saves the raw RF-data for offline processing later on. In this research the measurements are performed in the fast B-mode, this mode displays the real time changes in acoustic impedance in a 2 dimensional image, resulting in a cross section of the tube. The ul- trasound probe uses for the fast B-mode only its 14 central elements, which results in a frame rate of 710 frames per second. The measurement depth is set to 50 mm and with an element width of 0.315 mm this means that the Field Of View (FOV) is 4.41 mm by 50.0 mm. The voltage to activate the 4 ULTRASOUND MEASUREMENTS 73 Figure 48: A picture of the calibration device used to align the ultrasound probe. piezoelectric crystal elements is reduced from 50 V, used in the clinic, to 10 V, to prevent saturation of the signal. The maximum measurement time is limited to 5 seconds, because of the system memory limitations. 4.1.4 Postprocessing using Cross Correlation [6] The processed RF-data from the ultrasound scanner contains not only the scattering information of the Orgasol particles, from which the axial velocity profiles will be determined. The RF-data is also composed of reflections from the walls, reverberations, the DC component and noise. Filters are used to remove these disturbing signals from the raw RF-data. In radial direction the DC-component and the noise are removed with a fourth order Butterworth band pass (4.2 MHz and 12.5 MHz) filter. A high pass filter (10 Hz) in temporal direction is applied to remove the static and slow moving objects, such as the walls and reverberations. The axial velocity profile of the fluid in the curved tube in the plane of the ultrasound scanner measurement is determined with a newly developed cross correlation based technique, similar to the technique used for Particle Imaging Velocimetry (PIV) (Figure 49) [29]. First, interrogation windows are applied in radial direction to the processed RF-data. The shift between two corresponding interrogation windows from subsequent time frames is determined by performing a cross correlation. The axial shift can be converted into axial velocity by multiplication of the transducer element width and the frame rate (710 Hz). 4 ULTRASOUND MEASUREMENTS 74 Figure 49: A schematic overview of the RF-data postprocessing by applying a cross correlation technique similar to PIV in temporal direction [6]. 4.1.5 Ultrasound Measurements Control experiment: The settings of the ultrasound scanner, described above, are checked by a control experiment, where the Orgasol particles are dissolved in a agarose gel. After solidification of the gel it is directly connected to a steel rod, which is moved in axial direction by a linear actuator. The linear actuator is controlled by a computer. A displacement, x(t), is prescribed, according to: x(t)= A cos(2πft), (98) in which A is the amplitude, f the frequency and t the time in seconds. The gel is placed in water parallel to the ultrasound probe, see Figure 50 The actual velocity of the agarose gel is calculated by differentiation of the equation which prescribed the displacement of the agarose gel: v(t)= 2πfA sin(2πft). (99) − Ultrasound measurements on curved tubes: First the axial velocity profiles of the Newtonian BMF in a curved tube with δ = 0.02 are measured with the ultrasound scanner. The measurements are performed at an angle of 270◦ and for different flow rates. Then the axial velocity profiles of the shear thinning BMF are measured, for different flow 4 ULTRASOUND MEASUREMENTS 75 Figure 50: A picture of the control experiment with Orgasol particles dis- solved into Agarose gel, to validate the settings of the ultrasound scanner. ratios in curved tubes with δ = 0.01 and 0.02. The ultrasound measurements of the Newtonian BMF, with δ = 0.02, and the shear thinning BMF, with δ = 0.01, are used to validate the infinite tube model. Therefore, the ultrasound measurements of the shear thinning BMF are repeated on the straight part of the δ = 0.01 tube, with the same flow conditions to be able to validate the shear thinning behavior. 4.1.6 Ultrasound Measurements vs Infinite Tube Model The second goal of the ultrasound measurements is to provide data that can be used to compare the measured axial velocity profiles with the calculated velocity distribution of the infinite tube model (Section 3). The input pa- rameters for the CFD model are based on the experimental parameters to obtain a realistic and comparable result. The density is assumed to be equal to water, ρ = 1.0 103 kg/m3 and the viscosity of the fluid in the infinite · tube model will be adapted to the BMF used. The flow is set equal to the flow measured with the flow probe and the radius of the tube will be based on the measured axial velocity profiles. For the shear thinning blood mimicking fluid a least squares fit is applied to the viscosity measurements to determine the parameters of the Carreau- Yasuda model (see Section 3.1). The viscosity measurement and the fit re- 2 kg 3 sulting from the determined parameters η0 = 4.2 10− , η = 1.6 10− · ms ∞ · kg , n = 4.6 10 1, a = 1.3 and λ = 6.1 10 1 are shown in Figure 51. These ms · − · − parameters are applied to the CFD model. During the experiments it became clear that the flow probe did not work properly. The input flow for the CFD model will, therefore, be based on an 4 ULTRASOUND MEASUREMENTS 76 Carreay-Yasuda model -1 10 Fit: eta0=4.2e-02 eta8=1.6e-03 a=1.3e+00 n=4.6e-01 labda=6.1e-01 Measurement ) -1 s -1 -2 10 (kgm η -3 10 -1 0 1 2 3 10 10 10 10 10 γ˙ (s−1 ) Figure 51: The viscosity measurement of the shear thinning blood mimicking fluid. approximation, in which the flow is determined from the asymmetric profile based on Leguy [30]. For the approximation, the axial velocity profile is divided into two parts. Through each part a profile is fitted and this profile is integrated over the whole surface of the cross section of the tube: 2π a v(r)+ v( r) q¯ = − rdrdθ. (100) 2 Z0 Z0 4.2 Results First the results obtained with agarose gel will be shown, which are used to validate the standard settings of the ultrasound scanner. Then the measured axial velocity profiles will be shown of the Newtonian BMF and the shear thinning BMF. The axial velocity profiles of the ultrasound measurements will be compared to the calculations of the infinite tube model. 4 ULTRASOUND MEASUREMENTS 77 4.2.1 Control experiment Agarose gel The velocity measurement of the control experiment with the agarose gel for A = 2 10 2m and f = 2Hz is shown in Figure 52. In this figure · − the derived velocity is plotted together with the measured velocity of the ultrasound scanner, calculated with the cross correlation technique. The measured velocities show a good agreement with the imposed velocity. A clear sine-function is visible with an amplitude of 0.23 m/s and a frequency of 2 Hz. The measured velocity shows a small DC-component and distortion of the maxima and minima. Control Measurement vs Calculated Velocity 0.3 Ultrasound Measurement Setpoint Velocity 0.2 0.1 0 (m/s) ax v −0.1 −0.2 −0.3 −0.4 0 0.5 1 1.5 2 2.5 3 3.5 4 t(s) Figure 52: The ultrasound velocity measurement of the agarose gel performed with the ultrasound scanner calculated with the cross correlation technique, compared to the setpoint velocity. 4 ULTRASOUND MEASUREMENTS 78 4.2.2 Newtonian BMF The axial velocity profiles of the Newtonian BMF through a curved tube with δ = 0.02 and for different flow rates are measured with the ultrasound scanner, see Figure 53. For increasing flows the maximum axial velocity increases and the position of the maximum axial velocity is shifted more to the outside of the curve. The higher the flow, the more concave the right part of the axial velocity profile seems to become. Newtonian BMF δ = 0.02 Q =0.39 L/min probe Q =0.60 L/min probe Q =0.70 L/min 0.5 probe Q =0.90 L/min probe Q =1.00 L/min probe Q =1.10 L/min 0.4 probe std 0.3 v(m/s) 0.2 0.1 0 −4 −3 −2 −1 0 1 2 3 4 −3 r(m) x 10 Figure 53: The velocity measurement of the Newtonian BMF in a curved tube with δ = 0.02. 4.2.3 Shear Thinning BMF The axial velocity profiles of the shear thinning BMF are shown in Figure 54 for a curved tube with δ = 0.01 and in Figure 55 for a curved tube with δ = 0.02. For higher flows the maximum axial velocity is higher and the position of the maximum shifts to the outside of the curve. The axial velocity 4 ULTRASOUND MEASUREMENTS 79 profiles seem to be more convex in comparison to the Newtonian BMF axial velocity profiles. Comparing the shear thinning BMF axial velocity profiles to the Newtonian BMF profiles for δ = 0.02 shows that the position of the maximum axial velocity of the Newtonian BMF is shifted more to the outside of the curve. δ = 0.01 Q =0.29 L/min probe Q =0.47 L/min probe Q =0.69 L/min 0.5 probe Q =0.88 L/min probe Q =1.08 L/min probe Q =1.26 L/min 0.4 probe std 0.3 v(m/s) 0.2 0.1 0 −4 −3 −2 −1 0 1 2 3 4 −3 r(m) x 10 Figure 54: The velocity measurement of the shear thinning BMF in a curved tube with δ = 0.01. 4.2.4 Ultrasound Measurements vs Infinite Tube Model The flows used for the infinite tube model are based on an estimation of the flow determined from the measured axial velocity profiles from the ul- trasound scanner. The Qprobe in the legend gives the flow measured by the flow probe and the Qcalc gives the calculated flow based on the axial flow profiles, see Figures 56 through 58. The measured axial velocity profiles of the Newtonian BMF in a curved tube with δ = 0.02 show a great resemblance with the calculated axial flow 4 ULTRASOUND MEASUREMENTS 80 δ =0.02 0.8 Q =0.14 probe Q =0.31 probe 0.7 Q =0.55 probe Q =0.74 probe 0.6 Q =0.93 probe Q =1.21 probe 0.5 Q =1.50 probe std 0.4 v(m/s) 0.3 0.2 0.1 0 −4 −3 −2 −1 0 1 2 3 4 −3 r(m) x 10 Figure 55: The velocity measurement of the shear thinning BMF in a curved tube with δ = 0.02 profiles of the infinite tube model, see Figure 56. Almost every point of the three calculated axial velocity profiles is within one standard deviation of the measured axial velocity profiles. The calculated axial velocity profiles do have the same concave contour as the measured profiles. The axial velocity profiles in a straight part of the tube are measured to com- pare them with the calculated profiles of the infinite tube model, but with δ = 0.00, to investigate whether the Carreau-Yasuda model can describe the shear thinning properties. The axial velocity profiles for the 4 lower flows are predicted within the standard deviation of the ultrasound measurement, see Figure 57. However, for the two higher flows the measured axial velocity profiles are more flattened than the calculated axial velocity profiles. The axial velocity profiles of the shear thinning BMF in a curved tube with δ = 0.01 are compared to the infinite tube model with the shear thinning fluid properties. The axial velocity profiles correspond within one standard deviation of the measurements for the four lower flows as can be seen in 4 ULTRASOUND MEASUREMENTS 81 Newtonian BMF δ = 0.02 0.5 Q =0.39 L/min probe Q =0.70 L/min 0.45 probe Q =0.90 L/min probe 0.4 Q =0.26 L/min calc Q =0.50 L/min calc 0.35 Q =0.65 L/min calc 0.3 std 0.25 v(m/s) 0.2 0.15 0.1 0.05 0 −4 −3 −2 −1 0 1 2 3 4 −3 r(m) x 10 Figure 56: The velocity measurement of the Newtonian BMF versus the simulation of the infinite tube model for δ = 0.02. Figure 58. The two calculated axial velocity profiles of the highest flows deviate from the measured profiles. 4.3 Discussion The settings of the ultrasound scanner were validated with the agarose gel control experiment. A displacement of the gel was imposed by the linear actuator and the actual velocity of the agarose gel is calculated by differ- entiation of the equation which prescribed the displacement of the agarose gel. Overall the velocities measured by the ultrasound scanner and calcu- lated with the cross correlation method agree with the imposed velocities. The inaccurate values of the maximum velocities are probably caused by the linear actuator. The same experiment was performed for lower and higher frequencies and it resulted also in imprecise measurements of the maximum and minimum velocities only. 4 ULTRASOUND MEASUREMENTS 82 Shear Thinning BMF straight tube Q =0.29 L/min probe Q =0.47 L/min probe Q =0.68 L/min 0.5 probe Q =0.89 L/min probe Q =1.08 L/min probe Q =1.26 L/min 0.4 probe Q =0.16 L/min calc Q =0.24 L/min calc 0.3 Q =0.36 L/min calc v(m/s) Q =0.48 L/min calc Q =0.61 L/min calc 0.2 Q =0.72 L/min calc std 0.1 0 −4 −3 −2 −1 0 1 2 3 4 −3 r(m) x 10 Figure 57: The velocity measurement of the shear thinning BMF versus the simulation of the infinite tube model for a straight tube. The axial velocity profiles can be measured with a reasonable accuracy, the standard deviation is in general small. The axial velocity profiles measured with the ultrasound scanner of the BMF in curved tubes do show the ex- pected behavior based on literature and CFD models. For higher flows the maximum axial velocity is higher and the position of the maximum velocity is shifted to the outside of the curve. A difference in axial velocity profiles between Newtonian BMF and shear thinning BMF is clearly visible. The axial velocity profiles of the shear thinning BMF are more convex and flat- tened than the Newtonian BMF. The position of the maximum velocity for the Newtonian BMF is much more shifted to the outside of the curve. The main cause of this effect is the lower viscosity, resulting in a higher Dean number for equal flows of the Newtonian BMF versus the shear thinning BMF. A fair comparison between the Newtonian and shear thinning BMF can be made if a characteristic shear rate of the shear thinning BMF is de- termined, but more ultrasound measurements are needed [27]. 4 ULTRASOUND MEASUREMENTS 83 δ = 0.01 Q =0.29 L/min probe Q =0.47 L/min probe Q =0.69 L/min 0.5 probe Q =0.88 L/min probe Q =1.08 L/min probe Q =1.26 L/min 0.4 probe Q =0.16 L/min calc Q =0.28 L/min calc 0.3 Q =0.42 L/min calc v(m/s) Q =0.52 L/min calc Q =0.66 L/min calc 0.2 Q =0.77 L/min calc std 0.1 0 −4 −3 −2 −1 0 1 2 3 4 −3 r(m) x 10 Figure 58: The velocity measurement of the shear thinning BMF versus the simulation of the infinite tube model for δ = 0.01. To be able to make a comparison between the ultrasound measurements and the infinite tube model, the flow is calculated from the measured axial flow profile based on a method for clinical measurements proposed by Leguy [30]. It should be noticed that this determination of the flow contains some errors and is biased. Despite these inaccuracies, the measured axial velocity pro- files of the Newtonian BMF agree very well with the calculated axial velocity profiles of the infinite tube model. The calculated axial velocity profiles are within one standard deviation of the measured axial velocity profiles. The flow estimation method proposed by Leguy may be very useful in clinical practice. This may be explained by the analytical approximation methods of Dean, Topakoglu and Siggers and Waters (Section 2.2). All three meth- ods show that the first correction term on the Poiseuille component of the axial velocity depends on cos(θ), for a fixed r. Schematically this can be 4 ULTRASOUND MEASUREMENTS 84 explained in the following equation: v(r, θ)+ v(r, θ + π) v¯(r)= = (101) 2 v (r)+ A cos(θ)+ v (r)+ A cos(θ + π) offset offset = v (r). 2 offset Where voffset(r) is very close to the velocity at the same r in the straight tube. This method is probably only accurate for lower Dean numbers. In future research the estimation error should be calculated from the results with the infinite tube model. The relative position of the maximum velocity is determined from the ultra- sound measurements and compared to the results obtained with the infinite tube model (Figure 59). The first two measured positions correspond to the predicted positions of the infinite tube model. The other four points agree with the trend of the figure. The three ultrasound measurements with the highest flow rate may not be fully developed, as they have a relatively high Dean number. Additional simulations with the infinite tube model need to be performed to verify that. The shear thinning BMF axial velocity profiles for lower flows are also com- parable with the calculated profiles. The calculated profiles are within one standard deviation of the measured axial velocity profiles. The axial veloc- ity profiles of the higher flows contain a larger error. Probably the Carreau- Yasuda fit is less suitable for the higher flows, because a deviation in axial velocity profile is observed in the straight tube (see Figure 57). For higher flows, the shear rate is higher and the Carreau-Yasuda fit seems to deviate from the viscosity measurements at higher shear rates. The ultrasound measurements are very sensitive to a wide range of inaccura- cies, which may explain the minor deviations between the measured profiles and the calculated profiles. Examples are the position of the ultrasound probe with respect to the centerline of the curved tube, irregularities of the tube in curvature but also in the circular cross section and the reflections of the ultrasound signal through the whole reservoir filled with water. 4.4 Conclusion The aim of the ultrasound measurements was to investigate whether it is possible to measure the axial velocity profiles with a reasonable accuracy 4 ULTRASOUND MEASUREMENTS 85 0.1 δ = 0.01 0 δ = 0.02 ) δ 0 = 0.04 −0.1 δ = 0.08 δ −0.2 = 0.10 δ = 0.16 −0.3 Newtonian BMF −0.4 −0.5 −0.6 Position of maximum (r/a −0.7 −0.8 0 500 1000 1500 2000 Dean Figure 59: The normalized position of the maximum velocity as function of Dean number for different δ’s with the results of the ultrasound measure- ments of the Newtonian BMF for δ = 0.02. and validate the simulations of the infinite tube model. To achieve this, an experimental set-up was designed to measure, with an ultrasound scanner, the axial velocity profiles of a Newtonian and a shear thinning BMF flowing in a curved tube. Measurements were compared with the calculated axial velocity profiles, with the input flow based on the measured axial velocity profiles. The agarose gel control experiment proved that the cross correlation method works accurate enough to determine the velocity of the gel from the scat- tering signal of the orgasol particles. The results indicate, that it is possible to measure the axial velocity profiles of a BMF in a curved tube with an ultrasound scanner with a reasonable accuracy . The relative position of the maximum velocity for the Newtonian BMF ultrasound measurements could even be deduced. They show a good resemblance with the results obtained earlier with the infinite tube model. The measured and calculated axial velocity profiles of both the Newtonian and the shear thinning BMF show a good resemblance with each other. The difference is usually less than one standard deviation. Probably a better fit of the Carreau-Yasuda model for higher shear rates will give better results. 4 ULTRASOUND MEASUREMENTS 86 The results show that the infinite tube model is validated with the ultra- sound experiments. The objective of this study is to determine the volume flow through a curved tube from a given asymmetric axial velocity profile, measured with an ultra- sound system at one single line through the tube. The results of the ultra- sound measurements in the experimental set-up show that this is possible with a reasonable accuracy by performing the volume flow approximation of Leguy [30]. This approximation method can be explained by the solu- tions derived with analytical approximation methods. Though experiments with a properly operating flow measurement device are needed. Then an indepent control measurement is possible to quantify the accuracy of this volume flow approximation method and to perform unbiased simulations with the infinite tube model. It should be noted that in this study several simplifications were made, such as steady flow and that the axial velocity profile is measured at the centerline only. 4.4.1 Recommendations A problem observed during the experiments was the incorrect measurements of the flow measurement device. The flow measurement device was even specially aligned to measure correctly the flow of the shear thinning blood mimicking fluid. However, the flow measurements did not seem to corre- spond with the axial velocity profiles obtained from the ultrasound scanner. In order to know the correction factor, the flow was measured manually af- ter the ultrasound measurements. Unfortunately, the correction factor was not stable and it sometimes even changed during the measurements. The flow measurement should work as a reference value to be able to reproduce the ultrasound measurements and it should provide the flow input for the comparison with the infinite tube model, but it failed. An unbiased com- parison with the infinite tube model was not possible, while an independent flow measurement device is a prerequisite to conduct and interpret these experiments correctly. 5 CONCLUSIONS 87 5 Conclusions The aim of this study was to determine the flow through a curved tube from a given asymmetric flow profile, measured with an ultrasound scanner at one line through the tube. The study was restricted to steady flow in tubes with a small curvature ratio. To reach the objective analytical approximation methods were investigated, two computational fluid dynamics models were developed and an experimental set-up was built to measure velocity profiles with an ultrasound system in curved tubes. For the determination of the flow characteristics the common carotid artery was taken as an example, this resulted in a main region of interest of 100 Dn 1000. ≤ ≤ 5.1 Analytical Approximation Methods First analytical approximation methods available in literature were inves- tigated, to gain more insight in the physical processes of flow in a curved tube and to obtain a relatively quick method to determine the flow from an axial velocity profile. The analytical approximation methods derived by Dean, Topakoglu and Siggers and Waters were discussed. The methods of Topakoglu and Siggers and Waters give comparable results, both adapt to the curvature ratio, while Deans method is independent of the curvature ratio and is only valid for smaller curvature ratio’s (δ < 0.01). All methods give unrealistic results for Dn 100. ≥ 5.2 CFD models Next, two CFD models were developed, the finite tube model and the infinite tube model, to investigate the flow in curved tubes for different Dean num- bers (1 Dn 1000) and curvature ratios (0.01 δ 0.16). The infinite ≤ ≤ ≤ ≤ tube model is capable of calculating the fully developed velocity distribu- tions in curved tubes. The axial velocity profiles obtained were compared to the analytical approximation method of Siggers and Waters. The analytical approximation method predicts the normalized axial velocity profiles very well for Dn 50 and not bad for Dn 100, for all the investigated curvature ≤ ≤ ratios. The relative position of the maximum velocity as function of Dean number was compared to the results of McConalogue and Srivastava and the approximation derived by Siggers and Waters. A great resemblance is observed between the different studies. The phenomenon that the relative position of the maximum velocity shifts to the inside of the curve for low 5 CONCLUSIONS 88 Dean numbers, is also predicted by Siggers and Waters. 5.3 Ultrasound Measurements Finally, ultrasound measurements were performed on curved tubes in an ex- perimental set-up with a Newtonian and a shear thinning Blood Mimicking Fluid. The first objective was to investigate whether the axial velocity pro- files could be measured with a reasonable accuracy, by using the ultrasound scanner and the newly developed method, based on cross correlation, to cal- culate the velocity of the fluid. The agarose control experiment proved that the cross correlation method worked. The measured axial velocity profiles of the Newtonian BMF and the shear thinning BMF have a small standard deviation. The second objective was to validate the infinite tube model with the ultra- sound measurements. The flow in the experimental set-up was determined by an integration method and used as input for the CFD models. The cal- culated axial velocity profiles show a good agreement with the measured axial velocity profiles. So, the infinite tube model seems to be accurate in calculating the axial velocity profiles for fluids with both an Newtonian and a shear thinning behavior and the volume flow integration method seems to work. A control flow measurement is needed to make an unbiased quan- tification of the accuracies of the infinite tube model and the volume flow integration method. The relative position of the maximum velocity of the Newtonian BMF was determined and compared to the results obtained earlier with the finite tube model and analytical and computational results obtained from liter- ature (Figure 60). The experimental results agree very well with the data presented in literature and the infinite tube model, but simulations with higher Dean numbers have to be preformed for a complete comparison. The objective of this study was to determine the volume flow through a curved tube from a given asymmetric axial velocity profile, measured with an ultrasound system at one single line through the tube. The results pre- sented in this study show that this is possible with the volume flow integra- tion method of Leguy, for axial velocity profiles measured at the centerline and for steady flows in curved tubes with a relatively small curvature ratio. 5 CONCLUSIONS 89 0.1 δ = 0.01 0 δ = 0.02 ) δ 0 = 0.04 −0.1 δ = 0.08 δ −0.2 = 0.10 δ = 0.16 −0.3 Newtonian BMF McConalogue&Srivastava −0.4 Siggers and Waters −0.5 −0.6 Position of maximum (r/a −0.7 −0.8 0 500 1000 1500 2000 Dean Figure 60: The normalized position of the maximum velocity as function of Dean number for different δ’s calculated with the infinite tube model and compared with the results of (1) the ultrasound measurements of the Newto- nian BMF, (2) the results derived numerically by McConalogue and Scrivas- tava and (3) the analytical approximation of Siggers and Waters A THE NAVIER STOKES EQUATIONS IN TOROIDAL COORDINATES 90 A The Navier Stokes equations in Toroidal Coor- dinates In figure 61 a schematic overview is given of the coordinates used to describe the flow in a curved tube. This curved cylindrical system is also known as a toroidal coordinate system with the coordinates (r, θ, z), where the z- coordinate is defined as z = R0ϕ. In this system the velocity components are chosen as follows, u is the velocity in the r-dirtection, v is the velocity in the θ-direction and perpendicular to u. Finally w is the velocity in the z-direction and perpendicular to both u and v. r R0 q O j z Figure 61: The coordinate system used for the flow in a curved tube. A.1 Orthogonal curvilinear coordinates The toroidal coordinate system belongs to the orthogonal curvilinear co- ordinates group, together with for example the cylindrical and spherical coordinate systems. The curvilinear coordinate system is composed of in- tersecting surfaces, if the intersections of the surfaces are all perpendicular (90 degrees angles) then it is called an orthogonal curvilinear coordinate system, see figure 62 for a schematic overview. The general derivation of the Navier-Stokes equations for orthogonal curvilinear coordinates can be followed to derive the Navier-Stokes equations for the toroidal coordinates, the general derivation as written down in [7] Chapter A2. ‘Orthogonal Curvilinear Coordinates’ is followed. A THE NAVIER STOKES EQUATIONS IN TOROIDAL COORDINATES 91 g b a Figure 62: A schematic overview of the orthogonal curvilinear coordinate system. A.1.1 general derivation [7] The system of orthogonal curvilinear coordinates consists of the general co- ordinates α, β and γ. The elements of length at α, β, γ in the direction of increasing α, β, γ are h1dα, h2dβ, h3dγ, where h1, h2, h3 can be functions of α, β, γ. In the α, β, γ direction the velocity components will be given by re- spectively v1, v2, v3 and the vorticity components by respectively ω1,ω2,ω3. With these definitions the equations below apply. The arc length is given by: 2 2 2 2 2 2 2 (ds) = h1(dα) + h2(dβ) + h3(dγ) (102) 1 ∂ 1 ∂ 1 ∂ The components of the gradient will become h1 ∂α , h2 ∂β , h3 ∂γ . The general continuity equation is defined as: ∂ρ + (ρ~v) = 0. (103) ∂t ∇ · With the divergence for orthogonal curvilinear coordinates working on ∇· the velocity vector, the continuity equation for a unsteady flow in a com- pressible fluid is (see [31] for the derivation): ∂ρ 1 ∂ ∂ ∂ + [ (h2h3ρv1)+ (h3h1ρv2)+ (h1h2ρv3)] = 0 (104) ∂t h1h2h3 ∂α ∂β ∂γ A THE NAVIER STOKES EQUATIONS IN TOROIDAL COORDINATES 92 The momentum equation in vector notation is ∂V 1 + (V )V = F + σ (105) ∂t ·∇ ρ∇ · with V the velocity vector and F the body force vector and σ the stress tensor. With the following definition (V )V (1/2 V 2) V ω (106) ·∇ ≡∇ | | − × where ω = V is the vorticity vector, the components of (V )V in the ∇× ·∇ directions of α, β, γ are: 1 ∂ 2 2 2 (v1 + v2 + v3) (v2ω3 v3ω2) (107) 2h1 ∂α − − 1 ∂ 2 2 2 (v1 + v2 + v3) (v3ω1 v1ω3) (108) 2h2 ∂β − − 1 ∂ 2 2 2 (v1 + v2 + v3) (v1ω2 v2ω1) (109) 2h3 ∂γ − − The σ components in α, β, γ direction are respectively given by: ∇ · 1 ∂ ∂ ∂ [ (h2h3σαα)+ (h3h1ταβ)+ (h1h2τγα)] (110) h1h2h3 ∂α ∂β ∂γ 1 ∂h1 1 ∂h1 1 ∂h2 1 ∂h3 +ταβ + τγα σββ σγγ h1h2 ∂β h3h1 ∂γ − h1h2 ∂α − h3h1 ∂α 1 ∂ ∂ ∂ [ (h2h3ταβ)+ (h3h1σββ)+ (h1h2τβγ)] (111) h1h2h3 ∂α ∂β ∂γ 1 ∂h2 1 ∂h2 1 ∂h3 1 ∂h1 +τβγ + ταβ σγγ σαα h2h3 ∂γ h2h1 ∂α − h2h3 ∂β − h1h2 ∂β 1 ∂ ∂ ∂ [ (h2h3τγα)+ (h3h1τβγ)+ (h1h2σγγ )] (112) h1h2h3 ∂α ∂β ∂γ 1 ∂h3 1 ∂h3 1 ∂h1 1 ∂h2 +τγα + τβγ σαα σββ h3h1 ∂α h2h3 ∂β − h3h1 ∂γ − h2h3 ∂γ A THE NAVIER STOKES EQUATIONS IN TOROIDAL COORDINATES 93 The Laplacian operator is defined as 2(f)= ( f) [32], for orthogonal ∇ ∇ · ∇ curvilinear coordinates this results in: 1 ∂ h h ∂ ∂ h h ∂ ∂ h h ∂ 2 = [ ( 2 3 )+ ( 3 1 )+ ( 1 2 )] (113) ∇ h1h2h3 ∂α h1 ∂α ∂β h2 ∂β ∂γ h3 ∂γ In the equations above the definitions of the vorticity components and the stress tensor components are needed. The vorticity components in the re- spectively α, β, γ direction are: 1 ∂ ∂ ω1 = [ (h3v3) (h2v2)] (114) h2h3 ∂β − ∂γ 1 ∂ ∂ ω2 = [ (h1v1) (h3v3)] (115) h3h1 ∂γ − ∂α 1 ∂ ∂ ω3 = [ (h2v2) (h1v1)] (116) h1h2 ∂α − ∂β The components of the stress tensor σ are: σ = p + τ = p + µe + (1/2µ′ 1/3µ)(e + e + e )(117) αα − αα − αα − αα ββ γγ σ = p + τ = p + µe + (1/2µ′ 1/3µ)(e + e + e )(118) ββ − ββ − ββ − αα ββ γγ σ = p + τ = p + µe + (1/2µ′ 1/3µ)(e + e + e ) (119) γγ − γγ − γγ − αα ββ γγ ταβ = τβα = µeαβ (120) τβγ = τγβ = µeβγ (121) τγα = ταγ = µeγα (122) In the fluid dynamics of a continuous medium the bulk viscosity µ′ = 0 or µ µ and practical experience has shown that the bulk viscosity µ can be ′ ≪ ′ neglected [7] page 28. In the equations for the components of the stress vector the strain compo- nents need to be known. The relation between the strain components and the gradients of the velocity components are defined as: 1 ∂v1 v2 ∂h1 v3 ∂h1 eαα = 2[ + + ] (123) h1 ∂α h1h2 ∂β h3h1 ∂γ A THE NAVIER STOKES EQUATIONS IN TOROIDAL COORDINATES 94 1 ∂v2 v3 ∂h2 v1 ∂h2 eββ = 2[ + + ] (124) h2 ∂β h2h3 ∂γ h1h2 ∂α 1 ∂v3 v1 ∂h3 v2 ∂h3 eγγ = 2[ + + ] (125) h3 ∂γ h3h1 ∂α h2h3 ∂β h2 ∂ v2 h1 ∂ v1 eαβ = ( )+ ( ) (126) h1 ∂α h2 h2 ∂β h1 h3 ∂ v3 h2 ∂ v2 eβγ = ( )+ ( ) (127) h2 ∂β h3 h3 ∂γ h2 h1 ∂ v1 h3 ∂ v3 eγα = ( )+ ( ) (128) h3 ∂γ h1 h1 ∂α h3 A.2 Derivation of the Toroidal Coordinates The defined coordinate system in figure 61 is used to derive the Navier- Stokes equations for the toroidal coordinates. In the chosen system the arc length is: r (ds)2 = [1+ cos θ]2(dz)2 + (dr)2 + r2(dθ)2 (129) R0 From the arc length and by the definition of the coordinate system with (r, θ, z) and the corresponding velocities (u, v, w) it can be stated that: α = z β = r γ = θ v1 = w v2 = u v3 = v r h1 =1+ cos θ h2 = 1 h3 = r R0 These parameters will be inserted in the general Navier-Stokes equations (equations 104 to 128) together with the assumption that the fluid under consideration is incompressible (ρ = constant). This results in the following equations. Continuity equation: R ∂w u ∂u 1 ∂v u cos θ v sin θ 0 + + + + + − = 0(130) R0 + r cos θ ∂z r ∂r r ∂θ R0 + r cos θ R0 + r cos θ A THE NAVIER STOKES EQUATIONS IN TOROIDAL COORDINATES 95 The Laplacian operator: 2 2 2 2 2 R0 ∂ ∂ 1 ∂ cos θ ∂ sin θ ∂ 1 ∂ = 2 2 + 2 + + + − + 2 2 (131) ∇ (R0 + r cos θ) ∂z ∂r r ∂r R0 + r cos θ ∂r R0 + r cos θ ∂θ r ∂θ The vorticity components in respectively z,r,θ direction: v ∂v 1 ∂u ω = + (132) 1 r ∂r − r ∂θ w sin θ 1 ∂w R0 ∂v ω2 = − + (133) R0 + r cos θ r ∂θ − R0 + r cos θ ∂z R0 ∂u w cos θ ∂w ω3 = (134) R0 + r cos θ ∂z − R0 + r cos θ − ∂r With the vorticity components, the components of (V )V in the directions ·∇ of z,r,θ can be derived. z-direction: wR ∂w uw cos θ vw sin θ ∂w v ∂w 0 + + u + (135) R0 + r cos θ ∂z R0 + r cos θ − R0 + r cos θ ∂r r ∂θ r-direction: ∂u v2 v ∂u R w ∂u w2 cos θ u + + 0 (136) ∂r − r r ∂θ R0 + r cos θ ∂z − R0 + r cos θ θ-direction: v ∂v w2 sin θ R w ∂v v ∂v + + 0 + u + u (137) r ∂θ R0 + r cos θ R0 + r cos θ ∂z r ∂r To calculate the right side of the impuls equations the equations for the strain components have to be known. R0 ∂w u cos θ v sin θ ezz = 2[ + + − ] (138) R0 + r cos θ ∂z R0 + r cos θ R0 + r cos θ ∂u e = 2 (139) rr ∂r A THE NAVIER STOKES EQUATIONS IN TOROIDAL COORDINATES 96 1 ∂v u e = 2[ + ] (140) θθ r ∂θ r R0 ∂u ∂w w cos θ ezr = + + − (141) R0 + r cos θ ∂z ∂r R0 + r cos θ ∂v v 1 ∂u e = + − + (142) rθ ∂r r r ∂θ 1 ∂w w sin θ R0 ∂v eθz = + + (143) r ∂θ R0 + r cos θ R0 + r cos θ ∂z With the strains the components of the stress tensor can be calculated. z-direction: 1 σ = p + τ = p + µe µ(e + e + e ) (144) zz − zz − zz − 3 zz rr θθ 4 R ∂w u cos θ v sin θ 2 ∂u 1 ∂v u = p + µ[ 0 + + − ] µ[ + + ] − 3 R0 + r cos θ ∂z R0 + r cos θ R0 + r cos θ − 3 ∂r r ∂θ r r-direction: 1 σ = p + τ = p + µe µ(e + e + e ) (145) rr − rr − rr − 3 zz rr θθ 4 ∂u 2 R ∂w u cos θ v sin θ 1 ∂v u = p + µ[ ] µ[ 0 + + − + + ] − 3 ∂r − 3 R0 + r cos θ ∂z R0 + r cos θ R0 + r cos θ r ∂θ r θ-direction: 1 σ = p + τ = p + µe µ(e + e + e ) (146) θθ − θθ − θθ − 3 zz rr θθ 4 1 ∂v u 2 R ∂w u cos θ v sin θ ∂u = p + µ[ + ] µ[ 0 + + − + ] − 3 r ∂θ r − 3 R0 + r cos θ ∂z R0 + r cos θ R0 + r cos θ ∂r And the shear stresses are given by: R0 ∂u ∂w w cos θ τzr = τrz = µ[ + + − 2 ] (147) R0 + r cos θ ∂z ∂r (R0 + r cos θ) ∂v v 1 ∂u τ = τ = µ[ + − + ] (148) rθ θr ∂r r r ∂θ 1 ∂w w sin θ R0 ∂v τθz = τzθ = µ[ + 2 + ] (149) r ∂θ (R0 + r cos θ) R0 + r cos θ ∂z A THE NAVIER STOKES EQUATIONS IN TOROIDAL COORDINATES 97 This will finally (after a lot of paperwork) result in the following components for the σ. ∇ · z-direction: 2 2 2 2 R0 ∂p R0 ∂ w ∂ w 1 ∂ w w ( )+µ 2 2 + 2 + 2 2 2 +(150) R0 + r cos θ −∂z (R0 + r cos θ) ∂z ∂r r ∂θ −(R0 + r cos θ) n o 1 ∂w 1 ∂w sin θ ∂w 2R0 ∂v ∂u µ + [cos θ ]+ 2 [ sin θ + cos θ ] r ∂r R0 + r cos θ ∂r − r ∂θ (R0 + r cos θ) − ∂z ∂z n o r-direction: 2 2 2 2 ∂p R0 ∂ u ∂ u 1 ∂u u 1 ∂ u 2 ∂v +µ 2 2 + 2 + 2 + 2 2 2 +(151) −∂r (R0 + r cos θ) ∂z ∂r r ∂r −r r ∂θ −r ∂θ n o 1 ∂u sin θ ∂u v sin θ 2R0 sin θ ∂w µ [cos θ + ]+ 2 R0 + r cos θ ∂r − r ∂θ r (R0 + r cos θ) ∂z − n cos θ 2 [u cos θ v sin θ] (R0 + r cos θ) − o θ-direction: 2 2 2 2 1 ∂p R0 ∂ v ∂ v 1 ∂v v 1 ∂ v 2 ∂u +µ 2 2 + 2 + 2 + 2 2 + 2 +(152) −r ∂θ (R0 + r cos θ) ∂z ∂r r ∂r −r r ∂θ r ∂θ n o cos θ ∂v sin θ 1 ∂v u 2R0 sin θ ∂w µ [ + ]+ 2 + R0 + r cos θ ∂r − R0 + r cos θ r ∂θ r (R0 + r cos θ) ∂z n sin θ 2 [u cos θ v sin θ] (R0 + r cos θ) − o A THE NAVIER STOKES EQUATIONS IN TOROIDAL COORDINATES 98 A.2.1 The Navier-Stokes equations in toroidal coordinates Now the derived equations of above can be combined to give the overview of the continuity and momentum (Navier-Stokes) equations in toroidal co- ordinates. continuity equation: R ∂w ∂u 1 ∂v u u cos θ v sin θ 0 + + + + − = 0 (153) (R0 + r cos θ) ∂z ∂r r ∂θ r R0 + r cos θ r-direction: ∂u wR ∂u ∂u v ∂u v2 w2 cos θ + 0 + u + = (154) ∂t R0 + r cos θ ∂z ∂r r ∂θ − r − R0 + r cos θ 2 2 2 2 ∂ p R0 ∂ u ∂ u 1 ∂u u 1 ∂ u 2 ∂v ( )+ ν 2 2 + 2 + 2 + 2 2 2 + −∂r ρ (R0 + r cos θ) ∂z ∂r r ∂r − r r ∂θ − r ∂θ n o 1 ∂u sin θ ∂u v sin θ 2R0 sin θ ∂w ν [cos θ + ]+ 2 R0 + r cos θ ∂r − r ∂θ r (R0 + r cos θ) ∂z − n cos θ 2 [u cos θ v sin θ] (R0 + r cos θ) − o θ-direction: ∂v wR ∂v ∂v v ∂v uv w2 sin θ + 0 + u + + + = (155) ∂t R0 + r cos θ ∂z ∂r r ∂θ r (R0 + r cos θ) 2 2 2 2 1 ∂p R0 ∂ v ∂ v 1 ∂v v 1 ∂ v 2 ∂u + ν 2 2 + 2 + 2 + 2 2 + 2 + −rρ ∂θ (R0 + r cos θ) ∂z ∂r r ∂r − r r ∂θ r ∂θ n o cos θ ∂v sin θ 1 ∂v u 2R0 sin θ ∂w ν [ + ]+ 2 + R0 + r cos θ ∂r − R0 + r cos θ r ∂θ r (R0 + r cos θ) ∂z n sin θ 2 [u cos θ v sin θ] (R0 + r cos θ) − o z-direction: ∂w R ∂w ∂w v ∂w w(u cos θ v sin θ) + 0 w + u + + − = (156) ∂t R0 + r cos θ ∂z ∂r r ∂θ R0 + r cos θ A THE NAVIER STOKES EQUATIONS IN TOROIDAL COORDINATES 99 2 2 2 2 1 R0 ∂p R0 ∂ w ∂ w 1 ∂ w w ( )+ ν 2 2 + 2 + 2 2 2 + ρ R0 + r cos θ −∂z (R0 + r cos θ) ∂z ∂r r ∂θ − (R0 + r cos θ) n o 1 ∂w 1 ∂w sin θ ∂w 2R0 ∂v ∂u ν + [cos θ ]+ 2 [ sin θ + cos θ ] r ∂r R0 + r cos θ ∂r − r ∂θ (R0 + r cos θ) − ∂z ∂z n o A.3 Another way of choosing the coordinates In the literature on curved tubes [2, 3, 4, 5, 7, 18, 17], different definitions for the coordinate system are used. Most authors choose the toroidal coordinate system, but sometimes with slight differences which results of course in some differences in their derivations of the Navier-Stokes equations. For example Dean [2, 3] and Ward-Smith [7] used the toroidal coordinate system as shown in figure 63. When this system is compared with the system from figure 61, q r R0 O j z Figure 63: The toroidal coordinate system used by Dean and Ward-Smith. which is also used by Berger [18], Pedley [17] and Siggers and Waters [5], it can be noticed that the θ-coordinate is chosen in a different way. This results in a different definition for the arc length for this system, namely: r (ds)2 = [1+ cos θ]2(dz)2 + (dr)2 + r2(dθ)2 (157) R0 Finally the Navier-Stokes equations for this system will have cos θ instead of sin θ and sin θ instead of cos θ compared to the system defined in figure 61 − and derived above. B DERIVATION OF STATIONARY FLOW IN CURVED TUBES 100 B Derivation of Stationary Flow in Curved Tubes B.1 Introduction In Appendix A the Navier-Stokes equations were derived for the toroidal coordinate system, this resulted in the following equations continuity equation: R ∂w ∂u 1 ∂v u u cos θ v sin θ 0 + + + + − = 0, (158) (R0 + r cos θ) ∂z ∂r r ∂θ r R0 + r cos θ r-direction: ∂u wR ∂u ∂u v ∂u v2 w2 cos θ + 0 + u + = (159) ∂t R0 + r cos θ ∂z ∂r r ∂θ − r − R0 + r cos θ 2 2 2 2 ∂ p R0 ∂ u ∂ u 1 ∂u u 1 ∂ u 2 ∂v ( )+ ν 2 2 + 2 + 2 + 2 2 2 + −∂r ρ (R0 + r cos θ) ∂z ∂r r ∂r − r r ∂θ − r ∂θ n o 1 ∂u sin θ ∂u v sin θ 2R0 sin θ ∂w ν [cos θ + ]+ 2 R0 + r cos θ ∂r − r ∂θ r (R0 + r cos θ) ∂z − n cos θ 2 [u cos θ v sin θ] , (R0 + r cos θ) − o θ-direction: ∂v wR ∂v ∂v v ∂v uv w2 sin θ + 0 + u + + + = (160) ∂t R0 + r cos θ ∂z ∂r r ∂θ r (R0 + r cos θ) 2 2 2 2 1 ∂p R0 ∂ v ∂ v 1 ∂v v 1 ∂ v 2 ∂u + ν 2 2 + 2 + 2 + 2 2 + 2 + −rρ ∂θ (R0 + r cos θ) ∂z ∂r r ∂r − r r ∂θ r ∂θ n o cos θ ∂v sin θ 1 ∂v u 2R0 sin θ ∂w ν [ + ]+ 2 + R0 + r cos θ ∂r − R0 + r cos θ r ∂θ r (R0 + r cos θ) ∂z n sin θ 2 [u cos θ v sin θ] , (R0 + r cos θ) − o B DERIVATION OF STATIONARY FLOW IN CURVED TUBES 101 z-direction: ∂w R ∂w ∂w v ∂w w(u cos θ v sin θ) + 0 w + u + + − = (161) ∂t R0 + r cos θ ∂z ∂r r ∂θ R0 + r cos θ 2 2 2 2 1 R0 ∂p R0 ∂ w ∂ w 1 ∂ w w ( )+ ν 2 2 + 2 + 2 2 2 + ρ R0 + r cos θ −∂z (R0 + r cos θ) ∂z ∂r r ∂θ − (R0 + r cos θ) n o 1 ∂w 1 ∂w sin θ ∂w 2R0 ∂v ∂u ν + [cos θ ]+ 2 [ sin θ + cos θ ] . r ∂r R0 + r cos θ ∂r − r ∂θ (R0 + r cos θ) − ∂z ∂z n o B.2 Stationary flow The Navier-Stokes equations can be simplified when the problem under con- ∂ sideration is stationary ( ∂t = 0) and by assuming a fully developed flow, ∂ then the derivatives with respect to the z-coordinate become zero ( ∂z = 0), ∂p except for the pressure ∂z . After omitting the derivatives with respect to the time and the z-coordinate, the Navier-Stokes equations in toroidal co- ordinates (r, θ, z) are as follows: continuity equation: ∂u 1 ∂v u u cos θ v sin θ + + + − = 0, (162) ∂r r ∂θ r R0 + r cos θ r-direction: ∂u v ∂u v2 w2 cos θ u + = (163) ∂r r ∂θ − r − R0 + r cos θ 1 ∂p ∂2u 1 ∂u u 1 ∂2u 2 ∂v + ν + + + −ρ ∂r ∂r2 r ∂r − r2 r2 ∂θ2 − r2 ∂θ n o 1 ∂u sin θ ∂u v sin θ cos θ ν [cos θ + ] 2 [u cos θ v sin θ] , R0 + r cos θ ∂r − r ∂θ r − (R0 + r cos θ) − n o θ-direction: ∂v v ∂v uv w2 sin θ u + + + = (164) ∂r r ∂θ r (R0 + r cos θ) B DERIVATION OF STATIONARY FLOW IN CURVED TUBES 102 1 ∂p ∂2v 1 ∂v v 1 ∂2v 2 ∂u + ν + + + + −rρ ∂θ ∂r2 r ∂r − r2 r2 ∂θ2 r2 ∂θ n o cos θ ∂v sin θ 1 ∂v u sin θ ν [ + ]+ 2 [u cos θ v sin θ] , R0 + r cos θ ∂r − R0 + r cos θ r ∂θ r (R0 + r cos θ) − n o z-direction: ∂w v ∂w w(u cos θ v sin θ) u + + − = (165) ∂r r ∂θ R0 + r cos θ 2 2 1 R0 ∂p ∂ w 1 ∂ w w + ν 2 + 2 2 2 + −ρ R0 + r cos θ ∂z ∂r r ∂θ − (R0 + r cos θ) n o 1 ∂w 1 ∂w sin θ ∂w ν + [cos θ ] . r ∂r R0 + r cos θ ∂r − r ∂θ n o These equations can be rewritten in a simplified form. The momentum equation in the r-direction can be simplified by inserting the derivative to r ∂ ( ∂r ) of the continuity equation into the momentum equation. The derivative of the continuity equation is: ∂ ∂u 1 ∂v u u cos θ v sin θ + + + − = (166) ∂r ∂r r ∂θ r R0 + r cos θ n o ∂2u 1 ∂u u cos θ ∂u cos θu cos θ 2 + 2 + + − 2 + ∂r r ∂r − r R0 + r cos θ ∂r (R0 + r cos θ) 1 ∂v 1 ∂2v ∂v sin θ v cos θ sin θ −2 + + − 2 r ∂θ r ∂r∂θ − ∂r R0 + r cos θ (R0 + r cos θ) = 0. This equation is inserted in the momentum equation in the r-direction and will result in the following equation: ∂u v ∂u v2 w2 cos θ u + = (167) ∂r r ∂θ − r − R0 + r cos θ 1 ∂p 1 ∂ sin θ ∂v v 1 ∂u ν ( )( + ) . ρ ∂r r ∂θ R + r cos θ ∂r r r ∂θ − − − 0 − B DERIVATION OF STATIONARY FLOW IN CURVED TUBES 103 In the same way the momentum equation in the θ-direction can be simplified, ∂ but now the derivative to θ ( ∂θ ) is taken of the continuity equation and this 1 equation is multiplied with r this will result in 1 ∂ ∂u 1 ∂v u u cos θ v sin θ + + + − = (168) r ∂θ ∂r r ∂θ r R0 + r cos θ h n oi ∂2u 1 ∂u ∂u cos θ u sin θ cos θ sin θ 1 ∂2v + 2 + + u 2 + 2 2 r∂θ∂r r ∂θ r∂θ R0 + r cos θ − r R0 + r cos θ (R0 + r cos θ) r ∂θ − ∂v sin θ v cos θ sin θ sin θ + − v 2 = 0. r∂θ R0 + r cos θ r R0 + r cos θ − (R0 + cos θ) Inserting this result in the momentum equation in the θ-direction gives ∂v v ∂v uv w2 sin θ u + + + = (169) ∂r r ∂θ r (R0 + r cos θ) 1 ∂p ∂ cos θ ∂v v 1 ∂u + ν ( + )( + ) . −rρ ∂θ ∂r R0 + r cos θ ∂r r − r ∂θ h i The momentum equation in z-direction can be simplified by a simple reor- ganisation, resulting in: ∂w v ∂w w(u cos θ v sin θ) u + + − = (170) ∂r r ∂θ R0 + r cos θ 1 R ∂p 1 ∂ ∂w rw cos θ 1 ∂ 1 ∂w w sin θ − 0 ( )+ ν ( )(r + )+ ( ) . ρ R + r cos θ ∂z r ∂r ∂r R + r cos θ r ∂θ r ∂θ R + r cos θ 0 0 − 0 REFERENCES 104 References [1] P.J. Brands. Non-invasive Methods for the Assesment of Wall Shear Rate and Arterial Impedance. PhD thesis, University of Maastricht, 1996. [2] W.R. Dean. Note on the motion of fluid in a curved pipe. Philosophical Magazine, 4(20):208–222, July 1927. [3] W.R. Dean. The stream-line motion of fluid in a curved pipe. Phil. Mag. S., 5(30):673–695, April 1928. [4] H.C. Topakoglu. Steady laminar flows of an incompressible viscous fluid in curved pipes. Journal of Mathematics and Mechanics, 16(12):1321– 1337, 1967. [5] J.H. Siggers and S.L. Waters. Steady flows in pipes with finite curva- ture. Physics of Fluids, 17(077102), 2005. [6] B.W.A.M.M. Beulen. Personal communication, 2007-2008. [7] A.J. Ward-Smith. Internal Fluid Flow. Oxford University Press, 1980. [8] S. Petersen, V. Peta, M. Rayner, J.Leal, R. Luengo-Fernandez, and A. Cray. European cardiovascular disease statistics. Technical report, University of Oxford, 2005. [9] W.J.G. van de Wassenberg. Evaluation of a method to assess wall shear stress in the common carotid artery with ultrasound. Master’s thesis, Eindhoven University of Technology, 2004. [10] J. Eustice. Flow of water in curved pipes. Proceedings of the Royal Society of London. Series A, 84:107–118, 1910. [11] J. Eustice. Experiments of streamline motion in curved pipes. Proceed- ings of the Royal Society of London. Series A, 85:119–131, 1911. [12] G.I. Taylor. The criterion for turbulence in curved pipes. Proceed- ings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 124(794):243–249, June 1929. [13] C.M. White. Streamline flow through curved pipes. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathemat- ical and Physical Character, 123(792):645–663, April 1929. REFERENCES 105 [14] D.J. McConalogue and R.S. Srivastava. Motion of a fluid in a curved tube. Proceedings of the Royal Society of London. Series A, 307:37–53, 1968. [15] H. Ito. Laminar flow in curved pipes. Z.AMM, pages 653–663, 1969. [16] W.M. Collins and S.C.R. Dennis. The steady motion of a viscous fluid in a curved tube. Q.Jl Mech. appl. Math., XXVIII, 1975. [17] T.J. Pedley. The Fluid Mechanics of Large Blood Vessels. Cambridge University Press, 1980. [18] S.A. Berger and L. Talbot. Flow in curved pipes. Ann. Rev. Fluid Mech., 15:461–512, 1983. [19] J. Larrain and C.F. Bonilla. Theoretical analysis of pressure drop in the laminar flow of fluid in a coiled pipe. The Society of Rheology, Inc., 14(2):135–147, 1970. [20] P.H.M. Bovendeerd, A.A. van Steenhoven, F.N. van de Vosse, and G. Vossers. Steady entry flow in a curved pipe. Journal of Fluid Me- chanics, 177:233–246, 1987. [21] F.N. van de Vosse and M.E.H. van Dongen. Cardiovascular fluid me- chanics -lecture notes-. Technical report, Eindhoven University of Tech- nology, 1998. [22] M. Van Dyke. Extended stokes series: Laminar flow throug a loosely coiled pipe. J. Fluid Mech, 86(1):129–145, 1978. [23] A. Quarteroni, M. Tuveri, and A. Veneziani. Computational vascu- lar fluid dynamics: Problems, models and methods. Computing and Visualization in Science, 2:163–197, 2000. [24] F.N. van de Vosse, J. de Hart, C.H.G.A. van Oijen, D. Bessems, T.W.M. Gunther, A. Segal, B.J.B.M.Wolters, J.M.A. Stijnen, and F.P.T. Baai- jens. Finite-element-based computational methods for cardiovascu- lar fluid-structure interaction. Journal of Engineering Mathematics, 47:335–368, 2003. [25] F.J.H. Gijsen, E. Allanic, F.N. van de Vosse, and J. D. Janssen. The influence of the non-newtonian properties of blood on the flow in large arteries: unsteady flow in a 90◦ curved tube. Journal of Biomechanics, 32:705–713, 1999. REFERENCES 106 [26] C.G. Giannopapa. Fluid Structure Interaction in Flexible Vessels. PhD thesis, King’s College London, 2004. [27] F.J.H. Gijsen. Modelling of Wall Shear Stress in Large Arteries. PhD thesis, Eindhoven University of Technology, 1998. [28] K.V. Ramnarine, D.K. Nassiri, P.R. Hoskins, and J. Lubbers. Valida- tion of a new blood-mimicking fluid for use in doppler flow test objects. Ultrasound in Medicine and Biology, 24:451–459, 1998. [29] R.J. Adrian. Twenty years of particle image velocimetry. Experiments in Fluids, 39:159–169, 2005. [30] C. Leguy. Personal communication, 2007-2008. [31] Z.U.A. Warsi. Fluid Dynamics: Theoretical and Computational Ap- proaches. CRS Press, 2 edition, 1998. [32] E. Kreyszig. Advanced Engineering Mathematics. John Wiley and Sons, inc, 8 edition, 1999.