# A Comparison of Analytical and Finite Element Solutions for Laminar Flow Conditions Near Gaussian Constrictions

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1 A Comparison of Analytical and Finite Element Solutions for Laminar Flow Conditions Near Gaussian Constrictions by Laura Noelle Race An Engineering Project Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute in Partial Fulfillment of the Requirements for the degree of MASTER OF ENGINEERING Major Subject: Mechanical Engineering Approved: Professor Ernesto Gutierrez-Miravete, Project Adviser Rensselaer Polytechnic Institute Hartford, CT December, 2014 i

2 Copyright 2014 by Laura Noelle Race All Rights Reserved ii

3 CONTENTS LIST OF TABLES AND CHARTS... v LIST OF FIGURES... vi LIST OF SYMBOLS... vii ACKNOWLEDGMENT... viii ABSTRACT... ix 1. Introduction Fluid Flow Near Constrictions and Expansions in Capillary Tubes Prior Work Methodology Problem Description Analytical Setup Mathematical Theory of Flow Near Gaussian Constrictions Final Analytical Equations Fluent Setup Geometry Mesh Fluent Solutions No Constriction Case Case Case Case Case Case Case Case iii

4 3.10 Case Results and Discussion Conclusions References APPENDIX A Curve Coordinates APPENDIX B Mesh Settings APPENDIX C Maple Worksheet For Analytical Solutions iv

5 LIST OF TABLES AND CHARTS Table 1 Description of Geometry Cases... 3 Table 2 Description of Fluid Parameters and Flow Cases... 4 Table 3 Maple Input Parameters for Constriction Amplitude and Width Table 4 Analytical Solutions for ΔP ex Table 5 Finite Element Solutions for ΔP ex *Figure out how to add charts in here* v

6 LIST OF FIGURES Figure 1 Hagen-Poiseuille Flow... 5 Figure 2 Gaussian Constriction Geometry... 5 Figure 3 Base Mesh (No Constriction)... 9 Figure mm Constriction Mesh, 5 mm Width... 9 Figure mm Constriction Mesh, 5 mm Width... 9 Figure mm Constriction Mesh, 5 mm Width Figure mm Constriction Mesh, Width 10 mm Figure mm Constriction Mesh, Width 10 mm Figure mm Constriction Mesh, Width 10 mm Figure 10 Pressure Contour, No Constriction Figure 11 Velocity Vectors, No Constriction Figure 12 Pressure Contour, Case Figure 13 Velocity Vectors, Case Figure 14 Pressure Contour, Case Figure 15 Velocity Vectors, Case Figure 16 Pressure Contour, Case Figure 17 Velocity Vectors, Case Figure 18 Pressure Contour, Case Figure 19 Velocity Vectors, Case Figure 20 Pressure Contour, Case Figure 21 Velocity Vectors, Case Figure 22 Pressure Contour, Case Figure 23 Velocity Vectors, Case Figure 24 Pressure Contour, Case Figure 25 Velocity Vectors, Case Figure 26 Pressure Contour, Case Figure 27 Velocity Vectors, Case Figure 28 Pressure Contour, Case Figure 29 Velocity Vectors, Case vi

7 LIST OF SYMBOLS Symbol Description Units g z Gravity (Axial Direction) m/s 2 p Pressure Pa Δp ex Excess Pressure Drop Pa R Constriction Radius m R o Initial Radius m r Radial Position m s Radial Position Variable [dimensionless] U o Axial Velocity Scale [dimensionless] u r Radial Velocity m/s u z Axial Velocity m/s Radial Velocity [dimensionless] Axial Velocity [dimensionless] V o Radial Velocity Scale [dimensionless] z Axial Position m Π Pressure Scale [dimensionless] ε Ratio of Constriction Radius to Flow Radius [dimensionless] ζ Axial Position Variable [dimensionless] µ Dynamic Viscosity Kg/(m-s) ρ Density Kg/m 3 vii

8 ACKNOWLEDGMENT I would like to thank my husband, Andy, for being supportive throughout my entire academic career and especially while I complete the last semester of my degree program. I would also like to thank my employer, The Lee Company, for financing my degree. Lastly, I would like to thank my adviser, Ernesto Gutierrez-Miravete for supplying the necessary guidance while I completed this report. viii

9 ABSTRACT This report evaluates the relationship between the analytical solution and the finite element solution of steady state laminar flow through a Gaussian constriction. The analytical solution of the excess pressure drop between Poiseuille Flow and the flow through a Gaussian constriction has been determined utilizing the continuity and momentum equations. The analytical solution is then compared with finite element solutions obtained using ANSYS Fluent. A variety of various cases for the size of constriction have been considered with water as the fluid. In general, it is found for a low Reynold s number of 2, the finite element solutions are within 10% of the analytical solutions. The lowest percentage difference for the actual pressure drop between the analytical and the finite element solutions are when the amplitude is its smallest at 25% of the tube radius. ix

10 1. Introduction 1.1 Fluid Flow Near Constrictions and Expansions in Capillary Tubes Fluid flow through a pipe with a constant internal radius and surface is expected in the theoretical world. However, many applications arise where the flow path is locally interrupted by some sort of constriction or expansion. In general, the larger the pipe or tube inner radius, the easier it is to disregard any internal surface inconsistencies. In the case of smaller inner diameter tubes, such as capillaries, the effect of such a constriction or expansion cannot be neglected so easily. Capillary tubes can be used in multiple applications ranging from the medical field to refrigeration and to plant life. A capillary tube depends on the phenomena of capillary action, which is when a fluid can be drawn up a tube against gravity without the need for help from external forces. Examples of real applications in which the inner diameter of a capillary tube varies include peristaltic pumps, viscometers, and the medical investigation of Aeterioscelerosis. A peristaltic pump is a medical device in which. Even though capillary action may not be the dominant phenomena, many analyses of constricting and expanding tubes can be applied. The example of a viscometer only relates to a tube in which an expansion takes place. Fluid is drawn up a capillary tube into a bulb (expansion) and then is allowed to flow to another bulb in the u-shaped capillary tube. Two marks are made on the capillary tube and the time that it takes for the known volume of fluid to pass through the two marks yields the kinematic viscosity. Lastly, the example in which the radius of a tube constricts is in the cardiovascular disease of Aeterioscelerosis. The disease causes fat to build up on the artery walls (constriction) and can cause serious health problems by preventing proper blood flow. 1.2 Prior Work Prior work on varying axial internal constricting radii has been investigated as far back as 1970, where Lee and Fung determined numerical techniques determining the fluid parameter distributions near the varied radius. It is possible that the investigation began 1

11 before Lee and Fung. It is shown that the most efficient way to perform an analytical solution to the flow regime is to assume a fixed shape of how the internal radius varies in relation to the initial radius. However, applying a fixed shape limits the analysis that can be completed. In 1971, M.J. Manton determined that the numerical techniques could be expanded to apply to an arbitrarily shaped constriction. The solution considers an internal radius that is slowly varying and is not shape dependent. As mentioned in Section 1.1, the application of radii varying axially is usually seen in the medical field where capillary tubes are of large use. Not only is the situation of Aeterioscelerosis considered, but additionally Expansions of the internal cross-section of the tube have practical applications in viscometric capillary tubes and peristaltic pumping. In the case of peristaltic pumping The most common shapes that are chosen appear to be that of normal (Gaussian) or sinusoidal curves. The Gaussian constriction has been adequately investigated and it will be used in this study. 2

12 2. Methodology 2.1 Problem Description The problem at hand is to compare the analytical solution to the finite element solution for the pressure drop through a tube with a Gaussian constriction. The analytical solution will be evaluated for varying cases of Reynold s numbers, but within the laminar flow regime. The geometry of the Gaussian constriction will be varied appropriately while the radius of the tube will be kept constant. The tube radius will be chosen as such to ensure that the flow path at r = 0 will be unaltered and the flow directly near the Gaussian constriction will be analyzed. The analysis will also include the size at which the Gaussian constriction needs to be as a percentage of the tube radius in order to affect the flow at r = 0. This project will also evaluate the flow path variation with multiple fluids and flow conditions. The fluid under consideration will be that of water. The fluid chosen has a practical application and is typically seen used with capillary tubes. The analytical solution to the Gaussian constriction should reveal approximate solutions to the velocity and pressure in local areas. These solutions will then be compared with the flow characteristics that are calculated utilizing finite element analysis modeled in Fluent. The constriction length will be varied between 5, 10, and 15 mm. For each length of constriction, the amplitude will be varied to be 0.25, 0.50, and 0.75 mm. Table 1 depicts the length and amplitude combination on how each case will be analyzed. Case Number Parameter Amplitude (mm) Width (mm) Table 1 Description of Geometry Cases Even though a capillary tube with a inch radius seems small, it is actually quite large compared to blood capillaries. Only one red blood cell is allowed to pass through the capillary at a time, leaving the capillary diameter at about 7 micrometers (about inch). At these diameters, fluid flow is quite slow and is practical for this report. 3

13 Because of approximate size of a capillary tube, each geometry case will then be analyzed with the fluid properties shown in Table 2. Parameter Water Density 1000 kg/m 3 Dynamic Viscosity kg/m-s Velocity m/s Reynold s Number 2 Table 2 Description of Fluid Parameters and Flow Cases A velocity of m/s, yielding a Reynold s Number of 2, is higher than what is typically seen in capillary tubes. A typical volumetric flow rate in a capillary tube is on the order of 1 x m 3 /s while the flow rate utilized for this report is approximately 3.14 x 10-9 m 3 /s. Utilizing a slightly faster velocity will check the ability of the analytical solution and the finite element solution to accurately predict the required system parameters. 4

14 2.2 Analytical Setup Mathematical Theory of Flow Near Gaussian Constrictions The most basic form of steady state laminar fluid flow through a tube has a parabolic velocity profile. The equation for the velocity profile is known as the Hagen-Poiseuille equation. The velocity profile typically will look similar to the profile shown in Figure 1. Figure 1 Hagen-Poiseuille Flow In theory, all flows through pipes and tubes would resemble the flow profile of Hagen- Poiseuille flow. However, the reality of all flows having a similar profile is impractical. The flow path may exhibit a constriction that disturbs local flow paths from the standard Hagen-Poiseuille flow. One example is defined as the Gaussian constriction, as shown in Figure 2. r =0 Figure 2 Gaussian Constriction Geometry A Gaussian constriction takes the form of a Gaussian, or normal distribution bell curve. In this report, the continuity and momentum equations will be solved on the basis 5

15 of a defined geometry change to the radius. Equation 1 shows how the radius varies in the case of the Gaussian constriction. (1) The parameters ε and λ represent the dimensionless amplitude of the curve at any point and a measure of the width of the constriction, respectively. The continuity and momentum equations will need to be simplified to include dimensionless parameters so that different approximate solutions do not need to be found for each value of ε and λ. The continuity and momentum equations for steady flow without dimensionless parameters are shown in Equation 2 through Equation 4. (2) (3) (4) Once the proper continuity and momentum equations are determined for the flow characteristics, the equations can now be simplified. The first step is to define dimensionless parameters for each variable and substitute those values into the original equations. Dimensionless parameters are defined for the axial and radial distances, axial and radial velocities, and the pressure. Equation 5 through Equation 7 exhibits the new continuity and momentum equations with the substitutions. (5) 6

16 (6) (7) These equations can then be simplified even further, with the differentiation of equations for low (Laminar Flow < 2000) and high (Turbulent Flow > 2000) Reynold s numbers. The final simplified continuity and momentum equations for low Reynold s numbers are shown in Equation 8 through Equation 10. (8) (9) (10) These equations now represent the system of equations that can be solved in order to obtain the analytical solution for flow near a Gaussian constriction Final Analytical Equations In order to complete the study, the analytical solutions to the continuity and momentum equations for low Reynolds numbers need to be determined. The analytical solution will been determined by integrating Equation 8 through Equation 10. Since no heat transfer is taking place, no energy equation is necessary. Boundary conditions need to be determined and the technique utilized for integration is similar to the integral boundary layer technique of von Karman and Pohlhausen. A perturbation method is utilized to find an equation for the excess pressure drop (Equation 11). 7

17 (11) The excess pressure drop is not defined as the pressure drop through the tube, but as the difference in pressure drop between a tube with no constriction (Poiseuille flow) and a tube with a constriction. Equation 12 shows how to determine the pressure drop in a tube with no constriction. (12) Equation 13 shows how to determine the pressure drop in a tube with a defined Gaussian constriction. (13) In the following sections, the finite element results will be obtained for the pressure drops and then compared to the analytical solutions. 2.3 Fluent Setup The model setup in Fluent will be similar to Figure 2, except that the area below r = 0 will not be modeled. The geometry will be treated as a symmetrical 2-d geometry in spherical coordinates except for the inclusion of the constriction. The constriction is expected to be modeled utilizing coordinate inputs as a curve into Fluent 2. The boundaries will consist of an inlet, outlet, and two non-permeable walls that simulate the tube. The mesh will be refined near the constriction so that accurate results are obtained. Additionally, mesh verification will be run to validate the mesh size in all other areas of the flow path. The length of the tube will be chosen as such to ensure that accurate fully developed flow is obtained prior to the flow reaching the constriction. This method will also ensure that is it known when the baseline flow is being affected by the constriction. The constriction will be varied by length and amplitude. The size of the Gaussian constriction will be varied both in the analytical solution and the finite element solution. 8

18 The length of the entire tube will be set 40 mm, with the appropriate geometry modifications as stated in Table Geometry The geometry setups utilizing the values in Table 1 are created by importing coordinates as a 3D curve into Fluent. Two different curves are needed in order to create the constriction as a surface. The coordinate inputs for each case s curve are listed in Appendix A. The two curves can then be used to create a Surface from Edges, which creates a surface body. A rectangle is then drawn that is meters long and meters wide. The curve ends up being positioned in the center of the tube. A surface body is then created with the rectangle using the option Surface from Sketches. The surface body of the curve can then be treated as an area that will be removed from the surface body of the rectangle. The option to complete this action is to select Body Operation and then select Cut Material. The curve area is selected and when the body operation is applied, the geometry leftover is the flow area for a tube with a constriction Mesh Using the geometries specified in the aforementioned sections, the appropriate meshes were generated. The mesh is defined as a number of divisions in the axial and radial directions. This fine mesh allows the velocity and pressure throughout the flow region to be modeled accurately with the set conditions. Examples of the meshes generated for the base case and Cases 1 through 3 are shown in Figure 3 through Figure 6. Figure 3 Base Mesh (No Constriction) Figure mm Constriction Mesh, 5 mm Width Figure mm Constriction Mesh, 5 mm Width 9

19 Figure mm Constriction Mesh, 5 mm Width The cases where the constriction is 10 mm wide or greater require different mesh settings to ensure that the area near the constriction is properly meshed. Sample meshes for Cases 4 through 6 are shown in Figure 7 through Figure 9. Figure mm Constriction Mesh, Width 10 mm Figure mm Constriction Mesh, Width 10 mm Figure mm Constriction Mesh, Width 10 mm The mesh settings that were utilized for each case can be found in Appendix B. 10

20 3. Fluent Solutions In the following sections, the finite element solution results are shown in terms of the pressure contours and velocity vectors near the constriction. 3.1 No Constriction To accurately evaluate the flow characteristics through a Gaussian constriction, it is necessary to understand what normal Poiseuille flow looks like in a capillary tube with no constriction. The pressure contour and the velocity vectors for a tube with no constriction can be seen in Figure 10 and Figure 11, respectively. Figure 10 Pressure Contour, No Constriction Figure 11 Velocity Vectors, No Constriction 11

21 3.2 Case 1 With a constriction amplitude of 0.25 mm, the flow path is not interrupted severely from the normal pressure gradient. Figure 12 shows the pressure drop through the constriction. Figure 12 Pressure Contour, Case 1 The pressure drop along the entire length of the tube is increased to approximately Pa, which equals an excess pressure drop over a tube with no constriction of Pa. Figure 13 shows the velocity vectors near the area of the constriction. Figure 13 Velocity Vectors, Case 1 It is shown that the flow path varies only slightly as it passes through the constriction. There is a slight increase of velocity to approximately m/s for a short period, but it quickly returns to the constant velocity of m/s. 12

22 3.3 Case 2 Keeping the constriction width the same and increasing the amplitude to 0.50 mm yields a slightly higher pressure drop. Figure 14 shows the pressure drop through the constriction. Figure 14 Pressure Contour, Case 2 The pressure drop along the entire length of the tube is increased to approximately Pa, which equals an excess pressure drop over a tube with no constriction of Pa. Figure 15 shows the velocity vectors near the area of the constriction. Figure 15 Velocity Vectors, Case 2 In Case 2, the velocity increases to approximately m/s as it passes through the constriction 13

23 3.4 Case 3 With an amplitude of 0.75 mm, the pressure drop is significantly higher than with Case 1 or Case 2. Figure 16 shows the pressure drop through the constriction. Figure 16 Pressure Contour, Case 3 The pressure drop along the entire length of the tube is increased to approximately Pa, which equals an excess pressure drop over a tube with no constriction of Pa. Figure 17 shows the velocity vectors near the area of the constriction. Figure 17 Velocity Vectors, Case 3 14

24 3.5 Case 4 Figure 18 Pressure Contour, Case 4 The pressure drop along the entire length of the tube is increased to approximately Pa, which equals an excess pressure drop over a tube with no constriction of Pa. Figure 19 shows the velocity vectors near the area of the constriction. Figure 19 Velocity Vectors, Case 4 15

25 3.6 Case 5 Pressure and velocity contours Case 5 Figure 20 Pressure Contour, Case 5 The pressure drop along the entire length of the tube is increased to approximately Pa, which equals an excess pressure drop over a tube with no constriction of Pa. Figure 21 shows the velocity vectors near the area of the constriction. Figure 21 Velocity Vectors, Case 5 16

26 3.7 Case 6 Figure 22 Pressure Contour, Case 6 The pressure drop along the entire length of the tube is increased to approximately Pa, which equals an excess pressure drop over a tube with no constriction of Pa. Figure 23 shows the velocity vectors near the area of the constriction. Figure 23 Velocity Vectors, Case 6 17

27 3.8 Case 7 Figure 24 Pressure Contour, Case 7 The pressure drop along the entire length of the tube is increased to approximately Pa, which equals an excess pressure drop over a tube with no constriction of Pa. Figure 25 shows the velocity vectors near the area of the constriction. Figure 25 Velocity Vectors, Case 7 18

28 3.9 Case 8 Figure 26 Pressure Contour, Case 8 The pressure drop along the entire length of the tube is increased to approximately Pa, which equals an excess pressure drop over a tube with no constriction of Pa. Figure 27 shows the velocity vectors near the area of the constriction. Figure 27 Velocity Vectors, Case 8 19

29 3.10 Case 9 Figure 28 Pressure Contour, Case 9 The pressure drop along the entire length of the tube is increased to approximately Pa, which equals an excess pressure drop over a tube with no constriction of Pa. Figure 29 shows the velocity vectors near the area of the constriction. Figure 29 Velocity Vectors, Case 9 20

30 21

31 4. Results and Discussion The analytical solutions were determined utilizing a Maple worksheet and plugging in the necessary parameters to obtain the pressure drop values. The Maple worksheet utilized to determine the analytical results (Case 1 shown) can be seen in Appendix C. The main parameters that need to be changed for each case are the amplitude and width of the constriction. The velocity, tube radius, fluid density and fluid kinematic viscosity are the same for each case. Table 4 shows the input parameters for the amplitude (a) and the width (b) for each case. Case # a b Table 3 Maple Input Parameters for Constriction Amplitude and Width The values of b were determined to yield the constriction width required based on Equation X. The actual analytical results for each case obtained from the Maple worksheet and utilizing the above parameters is shown in Table 4. Case Pressure Drop - No Pressure Drop - ΔP ex Number Constriction (Pa) Constriction (Pa) Table 4 Analytical Solutions for ΔP ex 22

32 Percentage Difference Case Pressure Drop - No Pressure Drop - ΔP ex Number Constriction (Pa) Constriction (Pa) Table 5 Finite Element Solutions for ΔP ex The pressure values utilized for comparison from Fluent were the static pressure values listed at the inlet of the centerline. This value does not take into account the rise in pressure that occurs as the fluid is entering the tube since the flow is not fully developed at this point. However, the flow is fully developed at approximately m from the inlet, allowing the point of fully developed flow to take place before the area of the constriction begins % Percentage Difference of Analytical vs. Finite Element Pressure Drop Values With Constriction 8.00% 6.00% 4.00% 2.00% 0.00% Case Number Chart 1 Percentage Difference of Analytical vs. Finite Element Pressure Drop Values With Constriction 23

33 Percentage Difference Percentage Difference Between Analytical and Finite Element Excess Pressure Drop Values 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% Case Number Chart 2 Percentage Difference Between Analytical and Finite Element Excess Pressure Drop Values 24

34 5. Conclusions In general, the analytical and the finite element solutions were fairly close to one another. It was found that for the geometry and fluid properties chosen, the percentage difference of the finite element solution was within 10% of the analytical solution at all times. Typically, the lowest percentage differences of the actual pressure drop through the constriction occurred when the constriction width was limited to 5 mm, despite the amplitude being up to 75% of the tube radius. 25

35 6. References *Need to still assign references to spots in the text and put in order of appearance. Additional references for some theory, etc. are still required. [1] Middleman, Stanley. "Modeling Axisymmetric Flows: Dynamics of Films, Jets, and Drops." Academic Press, n.d. [2] Lee, T. S. "Numerical Study of Fluid Flow through Double Bell-Shaped Constrictions in a Tube." International Journal of Numerical Methods for Heat & Fluid Flow 12.2 (2002): ProQuest. Web. 7 Sep [3] Manton, M.J. "Low Reynolds Number Flow in Slowly Varying Axisymmetric Tubes." Fluid Mechanics (1971): Document. [4] Haber Shimon, Clark Alys, Tawhai Merryn. Blood Flow in Capillaries of the Human Lung J Biomech Eng 135, (2013) (11 pages); Paper No: BIO ; doi: / [5] a.pdf&code=b34ee34ac92a df0a7aa58, accessed September 29, 2014 [6] accessed October 17, 2014 [7] w, accessed September XX,

36 APPENDIX A Curve Coordinates The coordinates for each curve were obtained from the following sample code in Maple: > > > > > > The equation for how a radius varies axially due to a constriction is defined (Equation 1). Then, the parameters are chosen based on the curve that is to be plotted. The program then provides twenty-one coordinates over the entire length of the 40 mm tube in order to build the curve. Since the program gives the coordinates for a length of meters to meters, the x values were adjusted accordingly to put the peak of the curve in the center of a length of 0 meters to meters. The following shows the adjusted coordinates that were used for plotting each curve in order to build the finite element flow domain, with the first coordinates being those that created the straight line curve used in all models. 27

37 Straight Line (Used For All Curves) #group #point #x_coord #y_coord #z_coord Amplitude 0.25 mm, Width 5 mm #group #point #x_coord #y_coord #z_coord

38 Amplitude 0.25 mm, Width 10 mm #group #point #x_coord #y_coord #z_coord Amplitude 0.25 mm, Width 15 mm #group #point #x_coord #y_coord #z_coord

39 Amplitude 0.5 mm, Width 5 mm #group #point #x_coord #y_coord #z_coord Amplitude 0.5 mm, Width 10 mm #group #point #x_coord #y_coord #z_coord

40 Amplitude 0.5 mm, Width 15 mm #group #point #x_coord #y_coord #z_coord Amplitude 0.75 mm, Width 5 mm #group #point #x_coord #y_coord #z_coord

41 Amplitude 0.75 mm, Width 10 mm #group #point #x_coord #y_coord #z_coord Amplitude 0.75 mm, Width 15 mm #group #point #x_coord #y_coord #z_coord

42 APPENDIX B Mesh Settings *Need to input mesh settings and also show a diagram of the geometry with corresponding named elements* Number of Divisions Case # Axial Centerline Pipewall

43 APPENDIX C Maple Worksheet For Analytical Solutions > > > > > > > > 34

44 > > > > > > 35

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