A Comparison of Analytical and Finite Volume Method Solutions for
Laminar Pipe Flow Conditions With 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
© Copyright 2014
by
Laura Noelle Race
All Rights Reserved
ii
CONTENTS
LIST OF TABLES AND CHARTS .................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
LIST OF SYMBOLS ....................................................................................................... vii
LIST OF KEYWORDS .................................................................................................. viii
ACKNOWLEDGMENT .................................................................................................. ix
ABSTRACT ...................................................................................................................... x
1. Introduction.................................................................................................................. 1
1.1
Laminar Fluid Flow in Capillary Tubes with Constrictions .............................. 1
1.2
Prior Work.......................................................................................................... 2
2. Methodology ................................................................................................................ 3
2.1
Problem Description........................................................................................... 3
2.2
Analytical Setup ................................................................................................. 5
2.3
2.2.1
Mathematical Theory of Flow Near Gaussian Constrictions ................. 5
2.2.2
Final Analytical Equations ..................................................................... 7
Fluent Setup ....................................................................................................... 8
2.3.1
Geometry ................................................................................................ 8
2.3.2
Mesh ....................................................................................................... 9
2.3.3
Solution Setup ...................................................................................... 10
3. Fluent Solutions ......................................................................................................... 11
3.1
No Constriction ................................................................................................ 11
3.2
Case 1 ............................................................................................................... 12
3.3
Case 2 ............................................................................................................... 13
3.4
Case 3 ............................................................................................................... 14
3.5
Case 4 ............................................................................................................... 15
3.6
Case 5 ............................................................................................................... 16
3.7
Case 6 ............................................................................................................... 17
iii
3.8
Case 7 ............................................................................................................... 18
3.9
Case 8 ............................................................................................................... 19
3.10 Case 9.. ............................................................................................................. 20
4. Results and Discussion .............................................................................................. 21
5. Conclusions................................................................................................................ 25
6. References.................................................................................................................. 26
APPENDIX A – Curve Coordinates................................................................................ 27
APPENDIX B – Mesh Settings ....................................................................................... 33
APPENDIX C – Maple Worksheet For Analytical Solutions ......................................... 34
iv
LIST OF TABLES AND CHARTS
Tables
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 .................... 21
Table 4 – Comparison of Analytical to Finite Volume Results....................................... 22
Charts
Chart 1 – Pressure Drop Across Centerline Length of Tube ........................................... 23
Chart 2 – Percentage Difference of Analytical vs. Finite Volume Pressure Drop Values
With Constriction............................................................................................................. 24
v
LIST OF FIGURES
Figure 1 – Hagen-Poiseuille Flow ..................................................................................... 5
Figure 2 – Gaussian Constriction Geometry ..................................................................... 5
Figure 3 – Base Mesh (No Constriction) ........................................................................... 9
Figure 4 – 0.25 mm Constriction Mesh, 5 mm Width ....................................................... 9
Figure 5 – 0.50 mm Constriction Mesh, 5 mm Width ....................................................... 9
Figure 6 – 0.75 mm Constriction Mesh, 5 mm Width ....................................................... 9
Figure 7 – 0.25 mm Constriction Mesh, 10 mm Width ................................................... 10
Figure 8 – 0.50 mm Constriction Mesh, 10 mm Width ................................................... 10
Figure 9 – 0.75 mm Constriction Mesh, 10 mm Width ................................................... 10
Figure 10 – Pressure Contour, No Constriction............................................................... 11
Figure 11 – Velocity Vectors, No Constriction ............................................................... 11
Figure 12 – Pressure Contour, Case 1 ............................................................................. 12
Figure 13 – Velocity Vectors, Case 1 .............................................................................. 12
Figure 14 – Pressure Contour, Case 2 ............................................................................. 13
Figure 15 – Velocity Vectors, Case 2 .............................................................................. 13
Figure 16 – Pressure Contour, Case 3 ............................................................................. 14
Figure 17 – Velocity Vectors, Case 3 .............................................................................. 14
Figure 18 – Pressure Contour, Case 4 ............................................................................. 15
Figure 19 – Velocity Vectors, Case 4 .............................................................................. 15
Figure 20 – Pressure Contour, Case 5 ............................................................................. 16
Figure 21 – Velocity Vectors, Case 5 .............................................................................. 16
Figure 22 – Pressure Contour, Case 6 ............................................................................. 17
Figure 23 – Velocity Vectors, Case 6 .............................................................................. 17
Figure 24 – Pressure Contour, Case 7 ............................................................................. 18
Figure 25 – Velocity Vectors, Case 7 .............................................................................. 18
Figure 26 – Pressure Contour, Case 8 ............................................................................. 19
Figure 27 – Velocity Vectors, Case 8 .............................................................................. 19
Figure 28 – Pressure Contour, Case 9 ............................................................................. 20
Figure 29 – Velocity Vectors, Case 9 .............................................................................. 20
vi
LIST OF SYMBOLS
Symbol
gz
p
Δpex
R
Ro
r
s
Uo
ur
uz
𝑢̃𝑟
𝑢̃𝑧
Vo
z
Π
ε
ζ
µ
ρ
b/λ
σ
Description
Gravity (Axial Direction)
Pressure
Excess Pressure Drop
Constriction Radius
Initial Radius
Radial Position
Radial Position Variable
Axial Velocity Scale
Radial Velocity
Axial Velocity
Radial Velocity
Axial Velocity
Radial Velocity Scale
Axial Position
Pressure Scale
Ratio of Constriction Radius to Flow Radius
Axial Position Variable
Dynamic Viscosity
Density
Measure of Constriction Width
Standard Deviation
vii
Units
m/s2
Pa
Pa
m
m
m
[dimensionless]
[dimensionless]
m/s
m/s
[dimensionless]
[dimensionless]
[dimensionless]
m
[dimensionless]
[dimensionless]
[dimensionless]
Kg/(m-s)
Kg/m3
[dimensionless]
[dimensionless]
LIST OF KEYWORDS

Gaussian Constriction

Excess Pressure Drop

Aeterioscelerosis

Finite Volume

Fluent

CFD
viii
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.
ix
ABSTRACT
This report evaluates the relationship between the analytical solution and the finite
volume solution of steady state laminar pipe 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 volume
solutions obtained using ANSYS Fluent. A variety of various cases for the size of
constriction have been considered with water as the fluid. The analysis utilized a tube
with a diameter of 2 mm to simulate a capillary. In general, it was found that for a low
Reynold’s number of 2, the finite volume solutions are within 6% of the analytical
solutions. The study also concluded that the lowest percentage difference between the
analytical and finite volume solution was when the amplitude was only 25% of the tube
radius (0.25 mm) and the constriction width was at its smallest (5 mm).
As the
amplitude was increased up to 0.75 mm, the percentage difference increased. All values
of the pressure drop have been compared to a base case in which there was no
constriction.
x
1. Introduction
1.1 Laminar Fluid Flow in Capillary Tubes with Constrictions
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 a set of rollers rotate
within a circumference of flexible tubing. This motion draws fluid out from the fluid
source with one roller and supplies it to the exit of the pump with the opposite roller [1].
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 a 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
[2].
1
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
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 [3].
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. Expansions and constrictions of
the internal cross-section of the tube have practical applications in viscometric capillary
tubes and peristaltic pumping. In viscometric capillary tubes the internal cross-section
expands and in the case of peristaltic pumping, the tube both constricts and expands.
Additional work has been completed on how a drop of fluid reacts to a constriction [4].
The situation of Aeterioscelerosis has been developed to even an analytical calculation
of the flow and pressure characteristics through a double constriction [5]. 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
2. Methodology
2.1 Problem Description
The problem at hand is to compare the analytical solution to the finite volume 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 volume 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.
Parameter
Amplitude (mm)
Width (mm)
Base
0
0
1
0.25
5
2
0.50
5
Case Number
3
4
5
0.75 0.25 0.50
5
10
10
6
0.75
10
7
0.25
15
8
0.50
15
9
0.75
15
Table 1 – Description of Geometry Cases
Even though a capillary tube with a 0.001 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
0.00028 inch). At these diameters, fluid flow is quite slow and is practical for this report.
3
Because of approximate size of a capillary tube, each geometry case will then be
analyzed with the fluid properties shown in Table 2.
Parameter
Density
Dynamic Viscosity
Velocity
Reynold’s Number
Water
1000 kg/m3
0.001 kg/m-s
0.001 m/s
2
Table 2 – Description of Fluid Parameters and Flow Cases
A velocity of 0.001 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 10-10 m3/s while the flow rate utilized for this report is approximately
3.14 x 10-9 m3/s [6]. Utilizing a slightly faster velocity will check the ability of the
analytical solution and the finite volume solution to accurately predict the required
system parameters.
4
2.2 Analytical Setup
2.2.1
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 HagenPoiseuille 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
of a defined geometry change to the radius based on research completed by Stanley
Middleman [7]. Equation 1 shows how the radius varies in the case of the Gaussian
constriction.
5
𝑅 = 𝑅0 {1 − 𝜀
𝑧 2
−(
)
𝜆𝑅
0 }
𝑒
(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.
𝜕𝑢𝑧 1 𝜕(𝑟𝑢𝑟 )
+ +
=0
𝜕𝑧
𝑟
𝜕𝑟
∂ur
∂uz
∂p
∂ 1∂
∂2 ur
(rur )] + 2 )
+ uz
) = − + μ( [
∂r
∂z
∂r
∂r r ∂r
∂z
(3)
∂uz
∂uz
∂p
1∂
∂
∂2 uz
+ uz
) = − +μ(
[r ] +
) + ρg z
∂r
∂z
∂z
r ∂r ∂r
∂z 2
(4)
ρ (ur
ρ (ur
(2)
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.
𝜀𝑈0 𝜕𝑢̃𝑧 1 𝜕(𝑠𝑢̃𝑟 )
(
)
+
=0
𝜆𝑉0 𝜕𝜁
𝑠 𝜕𝑠
(5)
̃ 1 ∂ ∂ũz
ε2 ρR 0 U0
∂ũz
∂ũz
ε2 R 0 Π ∂℘
ε2 ∂2 ũz
(ũr
+ ũz
)=−
+
(s
)+ 2 2
μλ
∂s
∂ζ
λμU0 ∂ζ s ∂s
∂s
λ ∂ζ
(6)
̃ ∂ 1∂
ε2 ρR 0 U0
∂ũr
∂ũr
λR 0 Π ∂℘
ε2 ∂2 ũr
(sũr )) + 2 2
(ũr
+ ũz
)=−
+ (
μλ
∂s
∂ζ
μU0 ∂ζ ∂s s ∂s
λ ∂ζ
(7)
6
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.
𝜕𝑢̃𝑧 1 𝜕(𝑠𝑢̃𝑟 )
+
=0
𝜕𝜁
𝑠 𝜕𝑠
0=−
(8)
̃ 1 ∂ ∂ũz
∂℘
+
(s
)
∂ζ s ∂s
∂s
(9)
̃
∂℘
∂ζ
(10)
0=−
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.
2.2.2
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).
Λ
𝜋𝑅03 ∆𝑃𝑒𝑥
2 −4
= 2 ∫ [(1 − 𝜀 𝑒 −𝜁 ) − 1] 𝑑𝜁
8𝜇𝑄𝜆
0
(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
7
tube with a constriction. Equation 12 shows how to determine the pressure drop in a tube
with no constriction.
∆𝑃 =
8𝜇𝑄
(𝑧 − 𝑧1 )
𝜋𝑅𝑜4 2
(12)
Equation 13 shows how to determine the pressure drop in a tube with a defined Gaussian
constriction.
z2
5432𝜌𝑄 2 𝑑𝑅 8𝜇𝑄
∆𝑃 = ∫ −
+
1575𝜋 2 𝑅5 𝑑𝑧 𝜋𝑅 4
𝑧1
(13)
In the following sections, the finite volume 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 [8]. 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 Fluent2.
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 volume solution.
The length of the entire tube will be set 40 mm, with the appropriate geometry
modifications as stated in Table 1.
2.3.1
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
8
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 0.040 meters long and 0.001
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.
2.3.2
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 4 – 0.25 mm Constriction Mesh, 5 mm Width
Figure 5 – 0.50 mm Constriction Mesh, 5 mm Width
Figure 6 – 0.75 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.
9
Figure 7 – 0.25 mm Constriction Mesh, 10 mm Width
Figure 8 – 0.50 mm Constriction Mesh, 10 mm Width
Figure 9 – 0.75 mm Constriction Mesh, 10 mm Width
The mesh settings that were utilized for each case can be found in Appendix B.
2.3.3
Solution Setup
Once the geometry and mesh have been loaded into Fluent, the mesh should be checked
to make sure that it can properly be used for the solution. The solver then needs to be
told that it is assuming that the geometry is of an axisymmetric nature. The model
utilized for the calculation should be “Viscous – Laminar”, with all other models turned
off. The fluid should be defined as water and the properties should be set to the values
shown in Table 2. It is pertinent to make sure that the surface body in the model is then
utilizing water as the fluid and not air, the default fluid for Fluent. The boundary
conditions can then be set by setting the Centerline to “Axis”, the Interior-Surface_Body
to “Interior”, the Outlet to “Pressure-Outlet” and the Pipewall to “Wall”. The Inlet then
needs to be set as “Velocity-Inlet” and the velocity of 0.001 m/s input as a magnitude
that is normal to the boundary. The solution methods utilized are then “Simple” for
Scheme, “Green-Gauss Cell Based” for Gradient, “PRESTO!” for Pressure, and “Second
Order Upwind” for Momentum. The residuals were all set to 1e-6 for convergence and
the solution was initialized from the inlet prior to running the final calculation.
10
3. Fluent Solutions
In the following sections, the finite volume 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 in Figure 10 shows that the pressure drop is linear
across the length of the tube.
Figure 10 – Pressure Contour, No Constriction
Figure 11 shows that the velocity vectors are fairly constant throughout the length of the
tube, which is a result reached quite quickly once the fluid has entered the tube.
Figure 11 – Velocity Vectors, No Constriction
11
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
0.3543 Pa, which equals an excess pressure drop over a tube with no constriction of
0.0414 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 0.00339 m/s for a short period, but
it quickly returns to the constant velocity of 0.001 m/s.
12
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
0.5384 Pa, which equals an excess pressure drop over a tube with no constriction of
0.2255 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 0.0076 m/s as it passes through the
constriction, nearly double that of Case 1.
13
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
2.5156 Pa, which equals an excess pressure drop over a tube with no constriction of
2.2027 Pa. Figure 17 shows the velocity vectors near the area of the constriction.
Figure 17 – Velocity Vectors, Case 3
In Case 3, the velocity increases to approximately 0.0305 m/s as it passes through the
constriction. This velocity is significantly higher than those achieved in Case 1 or 2.
14
3.5 Case 4
If the width of the constriction is increased from 5 mm to 10 mm, Figure 18 shows the
pressure drop for an amplitude of 0.25 mm.
Figure 18 – Pressure Contour, Case 4
The pressure drop along the entire length of the tube is increased to approximately
0.3805 Pa, which equals an excess pressure drop over a tube with no constriction of
0.0676 Pa. Figure 19 shows the velocity vectors near the area of the constriction.
Figure 19 – Velocity Vectors, Case 4
In Case 4, the velocity increases to approximately 0.003427 m/s as it passes through the
constriction. This value is only about 0.8% higher than that of Case 1.
15
3.6 Case 5
If the width of the constriction is increased from 5 mm to 10 mm, Figure 20 shows the
pressure drop for an amplitude of 0.50 mm.
Figure 20 – Pressure Contour, Case 5
The pressure drop along the entire length of the tube is increased to approximately
0.6553 Pa, which equals an excess pressure drop over a tube with no constriction of
0.3424 Pa. Figure 21 shows the velocity vectors near the area of the constriction.
Figure 21 – Velocity Vectors, Case 5
In Case 5, the velocity increases to approximately 0.00738 m/s as it passes through the
constriction. This value is about 3% lower than that of Case 2.
16
3.7 Case 6
If the width of the constriction is increased from 5 mm to 10 mm, Figure 22 shows the
pressure drop for an amplitude of 0.75 mm.
Figure 22 – Pressure Contour, Case 6
The pressure drop along the entire length of the tube is increased to approximately
3.7267 Pa, which equals an excess pressure drop over a tube with no constriction of
3.4138 Pa. Figure 23 shows the velocity vectors near the area of the constriction.
Figure 23 – Velocity Vectors, Case 6
In Case 6, the velocity increases to approximately 0.02953 m/s as it passes through the
constriction. This value is about 3.3% lower than that of Case 3.
17
3.8 Case 7
If the width of the constriction is increased from 10 mm to 15 mm, Figure 24 shows the
pressure drop for an amplitude of 0.25 mm.
Figure 24 – Pressure Contour, Case 7
The pressure drop along the entire length of the tube is increased to approximately
0.4094 Pa, which equals an excess pressure drop over a tube with no constriction of
0.0965 Pa. Figure 25 shows the velocity vectors near the area of the constriction.
Figure 25 – Velocity Vectors, Case 7
In Case 7, the velocity increases to approximately 0.003403 m/s as it passes through the
constriction. This value is about 0.7% lower than that of Case 4 and 0.4% higher than
Case 1.
18
3.9 Case 8
If the width of the constriction is increased from 10 mm to 15 mm, Figure 26 shows the
pressure drop for an amplitude of 0.50 mm.
Figure 26 – Pressure Contour, Case 8
The pressure drop along the entire length of the tube is increased to approximately
0.7989 Pa, which equals an excess pressure drop over a tube with no constriction of
0.4860 Pa. Figure 27 shows the velocity vectors near the area of the constriction.
Figure 27 – Velocity Vectors, Case 8
In Case 8, the velocity increases to approximately 0.007446 m/s as it passes through the
constriction. This value is about 0.9% lower than that of Case 5 and 2.1% lower than
Case 2.
19
3.10 Case 9
If the width of the constriction is increased from 10 mm to 15 mm, Figure 28 shows the
pressure drop for an amplitude of 0.75 mm.
Figure 28 – Pressure Contour, Case 9
The pressure drop along the entire length of the tube is increased to approximately
5.1738 Pa, which equals an excess pressure drop over a tube with no constriction of
4.8609 Pa. Figure 29 shows the velocity vectors near the area of the constriction.
Figure 29 – Velocity Vectors, Case 9
In Case 9, the velocity increases to approximately 0.02979 m/s as it passes through the
constriction. This value is about 0.9% higher than that of Case 6 and 2.3% lower than
Case 3.
20
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 3 shows the input parameters for the amplitude (a) and
the width (b) for each case.
Case #
a
b
1
0.25
1.75
2
0.5
1.75
3
0.75
1.75
4
0.25
3
5
0.5
3
6
0.75
3
7
0.25
4.25
8
0.5
4.25
9
0.75
4.25
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 14, where “b” is equal to λ.
𝜎=
𝑏𝑅𝑜
√2
(14)
Equation 14 is simply the equation for the standard deviation of a Gaussian, or normal,
bell curve. The parameter “b” is known to be a measure of the width of the curve in
which 99.7% of the curve falls. The actual analytical and finite volume results for each
case obtained from the Maple worksheet and utilizing the above parameters are shown in
Table 4.
21
Calculated
Case
#
Pressure
Drop w/o
Pressure Drop
Constriction w/Constriction
Fluent
Excess
Pressure
Drop
Pressure
Drop w/o
Pressure Drop
Constriction w/Constriction
Percentage Difference To Calculated
Excess
Pressure
Drop
Pressure
Drop w/o
Pressure Drop
Constriction w/Constriction
Excess
Pressure
Drop
1
0.32
0.3629
0.0429
0.3129
0.3543
0.0414
2.22%
2.38%
3.59%
2
0.32
0.5326
0.2126
0.3129
0.5384
0.2255
2.22%
1.09%
6.07%
3
0.32
2.438
2.118
0.3129
2.5156
2.2027
2.22%
3.18%
4.00%
4
0.32
0.3935
0.0735
0.3129
0.3805
0.0676
2.22%
3.30%
8.03%
5
0.32
0.6845
0.3645
0.3129
0.6553
0.3424
2.22%
4.27%
6.06%
6
0.32
3.9511
3.6311
0.3129
3.7267
3.4138
2.22%
5.68%
5.98%
7
0.32
0.4241
0.1041
0.3129
0.4094
0.0965
2.22%
3.47%
7.30%
8
0.32
0.8363
0.5163
0.3129
0.7989
0.4860
2.22%
4.47%
5.87%
9
0.32
5.4641
5.1441
0.3129
5.1738
4.8609
2.22%
5.31%
5.51%
Table 4 – Comparison of Analytical to Finite Volume Results
22
The pressure values utilized for comparison from Fluent were the maximum 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 0.002 m
from the inlet, allowing the point of fully developed flow to take place before the area of
the constriction begins. Chart 1 graphs the pressure drop that is seen along the length of
the tube.
Pressure Drop Across Centerline Length of
Tube
6,000000
5,000000
Case 1
Case 2
Pressure (Pa)
4,000000
Case 3
Case 4
3,000000
Case 5
Case 6
2,000000
Case 7
Case 8
1,000000
Case 9
0,000000
0,00000
Base
0,01000
0,02000
0,03000
0,04000
Length (m)
Chart 1 – Pressure Drop Across Centerline Length of Tube
It can be seen that as the constriction amplitude increases, the pressure drop across the
length of the tube increases. Most of the cases fall beneath a pressure drop of 1 Pa. These
cases have amplitudes of 0.25 mm to 0.50 mm and include the base case with no
constriction. For an amplitude of 0.75 mm, the pressure drop is at minimum over double
the pressure drop for even an amplitude of 0.50 mm, a significant increase. The point at
which a significant pressure drop occurs appears to happen at approximately the same
spot for each curve (just before 0.020 m). The slope of the pressure drop across the
23
constriction is much steeper for an amplitude of 0.075 mm then for 0.25 mm or 0.50
mm.
For the base case with no constriction, the analytical pressure drop value obtained was
0.32 Pascals. The finite volume pressure drop was 0.3129 Pascals; a 2.2% difference.
Even though for the case with no constriction the value should theoretically be fairly
accurate, there are multiple variables when calculated pressure drops in finite volume
software. It was expected that the percentage would have been lower, but it is not out of
the realm of the other percentages obtained for other pressure drop values. Chart 2
graphs the percentage difference between the analytical and finite volume pressure drop
values for Case 1 through Case 9.
Percentage Difference
Percentage Difference of Analytical vs. Finite
Volume Pressure Drop Values Through
Constriction
6,00%
5,00%
4,00%
3,00%
2,00%
1,00%
0,00%
1
2
3
4
5
6
7
8
9
Case Number
Chart 2 – Percentage Difference of Analytical vs. Finite Volume Pressure Drop Values With
Constriction
It is shown that as the constriction width increases, the percentage difference between
the three amplitude values increases. Additionally, with each amplitude increase, the
percentage difference also generally increases, with the amplitude of 0.75 mm having
the largest percentage difference in all constriction width cases.
24
5. Conclusions
In general, the analytical and the finite volume solutions were fairly close to one another.
It was found that for the geometry and fluid properties chosen, the percentage difference
of the finite volume solution was within 6% 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. The percentage difference for a
constriction width of 5 mm varied from 1.09% to 3.18%. As the constriction width was
increased to 15 mm, it was found that the percentage difference was higher ranging in
values from 3.47% to 5.31%. Even though the percentages at a constriction width of 15
mm were generally higher, the highest percentage difference at 5.68% was an amplitude
of 0.75 mm and a constriction width of 10 mm. In conclusion, it is possible to
analytically calculate the pressure drop through a Gaussian constriction within a
reasonable percentage tolerance for amplitudes less than 0.75 mm and widths less than
15 mm.
25
6. References
[1] http://arteriosclerotic.org/arteriosclerotic-cardiovascular-disease/, accessed October
17, 2014
[2] Manton, M.J. "Low Reynolds Number Flow in Slowly Varying Axisymmetric
Tubes." Fluid Mechanics (1971): 451-459. Document.
[3] 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): 258-89. ProQuest. Web. 7 Sep. 2014.
[4] http://en.wikipedia.org/wiki/Peristaltic_pump, accessed November 15, 2014
[5]http://journals.cambridge.org/download.php?file=%2FFLM%2FFLM274%2FS00221
12094002090a.pdf&code=b34ee34ac92a02604579326df0a7aa58, accessed September
29, 2014
[6] Haber Shimon, Clark Alys, Tawhai Merryn. Blood Flow in Capillaries of the Human
Lung J Biomech Eng 135, 101006 (2013) (11 pages); Paper No: BIO-12-1427;
doi:10.1115/1.4025092
[7] Middleman, Stanley. "Modeling Axisymmetric Flows: Dynamics of Films, Jets, and
Drops." Academic Press, n.d.
[8]https://confluence.cornell.edu/display/SIMULATION/FLUENT++Laminar+Pipe+Flo
w, accessed September 21, 2014
26
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 -0.020 meters
to 0.020 meters, the x values were adjusted accordingly to put the peak of the curve in
the center of a length of 0 meters to 0.040 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
Straight Line (Used For All Curves)
#group #point
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
11
1
12
1
13
1
14
1
15
1
16
1
17
1
18
1
19
1
20
1
21
#x_coord
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
0.022
0.024
0.026
0.028
0.030
0.032
0.034
0.036
0.038
0.040
#y_coord
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
#z_coord
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Amplitude 0.25 mm, Width 5 mm
#group #point
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
11
1
12
1
13
1
14
1
15
#x_coord
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
0.022
0.024
0.026
0.028
0.030
0.032
0.034
#y_coord
0.001
0.001
0.001
0.001
0.000999998
0.000998654
0.000932283
0.00075
0.000932283
0.000998654
0.000999998
0.001
0.001
0.001
0.001
#z_coord
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
28
Amplitude 0.25 mm, Width 10 mm
#group #point
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
11
1
12
1
13
1
14
1
15
#x_coord
#y_coord
#z_coord
0.00600000000
0.0009999999999
0.00800000000
0.0009999999719
0.01000000000
0.0009999962637
0.01200000000
0.0009997960030
0.01400000000
0.0009954210903
0.01600000000
0.0009577466712
0.01800000000
0.0008397049029
0.02000000000
0.00075
0.02200000000
0.0008397049029
0.02400000000
0.0009577466712
0.02600000000
0.0009954210903
0.02800000000
0.0009997960030
0.03000000000
0.0009999962637
0.03200000000
0.0009999999719
0.03400000000
0.0009999999999
Amplitude 0.25 mm, Width 15 mm
#group #point
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
11
1
12
1
13
1
14
1
15
1
16
1
17
#x_coord
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
0.022
0.024
0.026
0.028
0.030
0.032
0.034
0.036
#y_coord
0.001
0.000999995
0.000999914
0.000999015
0.00099277
0.000965931
0.000896905
0.000799662
0.00075
0.000799662
0.000896905
0.000965931
0.00099277
0.000999015
0.000999914
0.000999995
0.001
#z_coord
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
29
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Amplitude 0.5 mm, Width 5 mm
#group #point
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
11
1
12
1
13
1
14
1
15
1
16
#x_coord
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
0.022
0.024
0.026
0.028
0.030
0.032
0.034
0.036
#y_coord
0.001
0.001
0.001
0.001
0.000999996
0.000997308
0.000864566
0.0005
0.000864566
0.000997308
0.000999996
0.001
0.001
0.001
0.001
0.001
#z_coord
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Amplitude 0.5 mm, Width 10 mm
#group #point
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
11
1
12
1
13
1
14
1
15
#x_coord
#y_coord
#z_coord
0.00600000000
0.0009999999998
0.00800000000
0.0009999999437
0.01000000000
0.0009999925273
0.01200000000
0.0009995920061
0.01400000000
0.0009908421806
0.01600000000
0.0009154933423
0.01800000000
0.0006794098058
0.02000000000
0.0005
0.02200000000
0.0006794098058
0.02400000000
0.0009154933423
0.02600000000
0.0009908421806
0.02800000000
0.0009995920061
0.03000000000
0.0009999925273
0.03200000000
0.0009999999437
0.03400000000
0.0009999999998
30
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Amplitude 0.5 mm, Width 15 mm
#group #point
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
11
1
12
1
13
1
14
1
15
1
16
1
17
#x_coord
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
0.022
0.024
0.026
0.028
0.030
0.032
0.034
0.036
#y_coord
0.001
0.00099999
0.000999828
0.00099803
0.00098554
0.000931862
0.000793811
0.000599323
0.0005
0.000599323
0.000793811
0.000931862
0.00098554
0.00099803
0.000999828
0.00099999
0.001
#z_coord
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Amplitude 0.75 mm, Width 5 mm
#group #point
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
11
1
12
1
13
1
14
1
15
1
16
1
17
#x_coord
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
0.022
0.024
0.026
0.028
0.030
0.032
0.034
0.036
#y_coord
0.001
0.001
0.001
0.001
0.001
0.000999994
0.000995963
0.000796849
0.00025
0.000796849
0.000995963
0.000999994
0.001
0.001
0.001
0.001
0.001
#z_coord
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
31
Amplitude 0.75 mm, Width 10 mm
#group #point #x_coord
#y_coord
#z_coord
1
1
0.00600000000
0.0009999999997
1
2
0.00800000000
0.0009999999156
1
3
0.01000000000
0.0009999887910
1
4
0.01200000000
0.0009993880091
1
5
0.01400000000
0.0009862632708
1
6
0.01600000000
0.0008732400134
1
7
0.01800000000
0.0005191147086
1
8
0.0200000000
0.00025
1
9
0.02200000000
0.0005191147086
1
10
0.02400000000
0.0008732400134
1
11
0.02600000000
0.0009862632708
1
12
0.02800000000
0.0009993880091
1
13
0.03000000000
0.0009999887910
1
14
0.03200000000
0.0009999999156
1
15
0.03400000000
0.0009999999997
Amplitude 0.75 mm, Width 15 mm
#group #point
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
11
1
12
1
13
1
14
1
15
1
16
1
17
#x_coord
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
0.022
0.024
0.026
0.028
0.030
0.032
0.034
0.036
#y_coord
#z_coord
0.000999999
0
0.000999985
0
0.000999741
0
0.000997044
0
0.000978311
0
0.000897794
0
0.000690716
0
0.000398985
0
0.00025
0
0.000398985
0
0.000690716
0
0.000897794
0
0.000978311
0
0.000997044
0
0.000999741
0
0.000999985
0
0.000999999
0
32
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
APPENDIX B – Mesh Settings
To set the mesh parameters for the domain, each case was considered individually in
order to obtain the most accurate results. The number of divisions for each named
selection is shown in the below table:
Number of Divisions
Case #
Axial
Centerline
Pipewall
Base
5
100
100
1
10
150
150
2
25
360
360
3
55
572
572
4
15
250
250
5
25
250
250
6
45
450
450
7
15
250
250
8
25
250
250
9
45
450
450
The following figure shows which edges in green that were selected to contain the above
mesh settings:
A
x
i
a
l
Pipewall
Centerline
A
x
i
a
l
For Case 4 through Case 9, the number of divisions for the Pipewall was only applied in
the area of curve. This method allowed the area near the curve to exhibit more accurate
calculations.
A
x
i
a
l
Pipewall
Centerline
33
A
x
i
a
l
APPENDIX C – Maple Worksheet For Analytical Solutions
>
>
>
>
>
>
>
>
34
>
>
>
>
>
>
35