Project #5: Simulation of Fluid Flow in the ScreenBounded Channel in a Fiber Separator
By
Lana Sneath and Sandra Hernandez
3rd Year, Biomedical Engineering
University of Cincinnati
Faculty Mentor: Dr. Urmila Ghia
School of Dynamic Systems
University of Cincinnati
Sponsored By the National Science Foundation
Grant ID No.: DUE-0756921
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Abstract
The goal of this project is to classify asbestos fibers in large batches to use in toxicology
studies. Previous studies have shown that fiber toxicity depends on fiber length. In the
present study, the Bauer McNett Classifier (BMC) is modeled using CFD (Computational
Fluid Dynamics) to analyze its potential use to classify fibers based on length. The focus
of our study is to simulate the flow in the deep open channel of the BMC, with focus on
the screen (wire mesh). As a first step, the screen is replaced by a solid wall boundary to
facilitate the computations. The channel geometry is modeled in Gambit, and the
computational grid is imported into FLUENT to solve the three-dimensional NavierStokes equations governing the fluid flow. Turbulence in the channel is founded using the
Reynolds Stress Model (RSM). As a second step, a portion of a sidewall is replaced by a
porous boundary to represent the screen openings in order to determine the flow
behavior. The results conclude the out-plane angle of shear stress on the xy plane is
around 8 degrees for the solid wall model, inferring that the shear stress distribution of
the flow is parallel to the screen.
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Contents
Abstract ............................................................................................................................... 2
Contents .............................................................................................................................. 2
Chapter 1 ............................................................................................................................. 6
Introduction ..................................................................................................................... 6
1.1 Bauer McNett Fiber Classifier .............................................................................. 6
1.2 Objectives ............................................................................................................. 8
1.3 Materials ............................................................................................................... 9
1.4 Methods................................................................................................................. 9
Chapter 2 ........................................................................................................................... 10
Flow in the BMC Open Channel – Solid Side Walls.................................................... 10
2.1 Geometry and Computational Grid ..................................................................... 10
2.2 Boundary Conditions .......................................................................................... 11
2.3 Results and Discussion ....................................................................................... 12
Chapter 3 ........................................................................................................................... 18
Flow in the BMC Open Channel with a Screen on a Side Wall ................................... 18
3.1 Geometry and Computational Grid ..................................................................... 18
3.2 Boundary Conditions .......................................................................................... 19
Conclusion ........................................................................................................................ 24
3
Nomenclature
The following symbols have been used in this document:
μ: Viscosity of the fluid
Re: Reynolds number.
Fr: Froude number.
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d: Wire diameter.
K: Permeability
𝐶2 : Pressure-jump coefficient
Ui: Velocity.
Elent: Entrance length number.
lentrance: Length to fully developed velocity profile.
𝜌: density
𝜏𝑖𝑗 : Shear stress
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Chapter 1
Introduction
Asbestos fibers are naturally found in the environment and have been used in many
commercial products such as insulation, flooring, plaster, and cloth materials. The use of
asbestos is a problem because asbestos is a known carcinogen that can lead to one or
more disorders if inhaled [3]. It has been proposed that alveolar macrophages (AM) are
unable to completely engulf long fibers and oxidants and enzymes leak from the AM,
causing cell damage [12]. The effect of asbestos can be determined by various factors
including fiber length, concentration, and duration of exposure. Previous experiments
conducted by the National Institute of Occupational Safety and Health (NIOSH) have
shown that asbestos toxicity varies with fiber length; the longer the fibers the greater the
chance of it being toxic [1]. Currently, studies are limited due to having only small
quantities of fibers available at a time, so it is difficult to conduct a large-scale toxicology
study. The objective of this study is to determine the efficiency of the Bauer McNett
Fiber Classifier (BMC) as a fiber length-based classifier.
1.1 Bauer McNett Fiber Classifier
The Bauer McNett Fiber Classifier is a device commonly used for fiber classification
based on length. The BMC is a system with 5 elliptical tanks arranged in a cascade, as
shown in Figure 1. An agitator slowly circulates the water flow within the elliptical tanks.
The water then flows through the screen in each of the tanks, causing separation of fibers
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based on their length, as shown in Figure 2. This fiber separation occurs due to the cross
flow through the screen which allows fluid to escape through the square apertures of the
mesh, leaving behind the fibers that are too long to fit lengthwise through the aperture.
Fiber orientation upon passing through the screen is governed by the shear stress
distribution on the wire mesh screen. It is assumed that the fibers flowing in the fluid
align themselves in the direction of the shear stress on the boundary, and any change in
the direction of the shear stress vectors will result in a change in the fiber orientation [1].
For successful length-based separation, the fibers must be oriented parallel to the screen.
If the fiber orientation is parallel to the screen and the diameter is greater than the
opening, it is expected that fibers larger in length than the aperture size of the particular
screen size will be filtered.
= Region of
focused study
Figure 1. The Bauer McNett Fiber
Classifier [13]
Figure 2. Top View of a Single Tank of
the Bauer McNett Classifier
The flow within the tanks is governed due to gravity in the tank cascade. This
study concentrates on modeling the deep, open channel within the BMC as a porous
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boundary to determine the shear stress distribution on the wire mesh. The previous study
on length-based fiber orientation in the BMC apparatus found that the Reynolds number
of the deep open channel was greater than 4000, classifying the flow as turbulent. The
Reynolds number is a dimensionless number, which correlates the viscous behavior of
Newtonian fluids [6]. The flow was also found to be subcritical as determined from the
value of Froude number, which was about 0.18 [1].
1.2 Objectives
The motivation behind this study is to understand the behavior of the fluid flow
within the deep open channel in the BMC apparatus. The goal of the present study is to
numerically simulate the three-dimensional flow in the BMC deep open channel of aspect
ratio (H/B) 10. The BMC deep open channel is first modeled as a channel with both
vertical side boundaries as solid walls. Next, a portion of one of the solid walls is
replaced by a porous-jump boundary condition, which represents the screen in the BMC
apparatus. The focus of the experiment is to analyze the shear stress distribution on the
screen, modeled as a porous boundary, and determine the effectiveness of the BMC for
length-based separation of fibers.
The objectives of the study are to:
a) Learn the fundamentals of fluid dynamics.
b) Learn the fundamentals of solving fluid dynamic problems numerically.
c) Simulate and study the flow in the open channel of the BMC apparatus, modeling
the screen as a solid wall boundary (i)
d) Model the screen as a porous boundary (ii)
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1.3 Materials
Commercially available Computational Fluid Dynamics (CFD) tools FLUENT
and Gambit are used for the simulations in this study.
1.4 Methods
The goal of this study is to numerically simulate the fluid flow in the screenbounded channel within the BMC fiber separator. To numerically study the fluid flow,
Computational Fluid Dynamic (CFD) software FLUENT and Gambit are used.
Computational Fluid Dynamics is the “concepts, procedures, and applications of
computational methods in fluids and heat transfer” [9]. CFD tools apply the principles of
engineering to the modeling of fluid flow. [10]. Using the CFD tool FLUENT, the 3D,
unsteady, incompressible Reynolds-Averaged Navier-Stokes Equations (RANS) are
solved to determine the three-dimensional flow in the deep open channel of aspect ratio
10. The Semi-Implicit Pressure-Linked (SIMPLE) algorithm is used to achieve pressurevelocity coupling. The solution is deemed converged, when the residuals of the continuity
equation and the conservation of momentum equation reach 10e-6.
Continuity Equation (Equation of Mass Conservation):
(1)
Conservation of Momentum:
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(2)
Chapter 2
Flow in the BMC Open Channel – Solid Side Walls
2.1 Geometry and Computational Grid
The deep open channel geometry, within the BMC apparatus, is created and
meshed in Gambit and the mesh is later imported into FLUENT where the RANS
equations are solved to simulate fluid flow.
The channel geometry is created in gambit with dimensions of 0.217 x 0.02 x .2 m
in the x, z, and y directions, respectively, giving the channel geometry an aspect ratio of
10. A computational grid is created, with grid spacing of 50, 180, and 45 in the x, y, and z
directions, respectively. The first step size of the grid is 0.00005 m away from the
boundaries in the y direction, and 0.00007 m in the z direction. The small step sizes allow
for more computations to be taken along the boundary edges where the fluid flow has
greater variation. The grid spacing along the x direction has a successive ratio of 1,
meaning that the grid points are evenly spaced. The fluctuations in the fluid flow along
the distance of the channel are moderated as compared to the y- and z-directions, hence it
was not necessary to cluster the points around the edges. Figure 3 shows the
computational grid used for this study.
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Figure 3. Computational Grid [1]
Table 1. Computational Grid Spacing
Total Size
X
Y
Z
∆Y
∆X
405000
50
180
45
0.00005
0.0007
2.2 Boundary Conditions
The boundary conditions used for modeling the wire-mesh wall as a solid
boundary are shown in Figure 4. The two sidewalls and the bottom wall are specified as
no-slip stationary walls, where the values of the u, v, and w components of velocity are
zero. The average velocity at the inlet is specified to be 0.25 m/s. The Reynolds number
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for the BMC apparatus is equal to 9982, classifying it as a turbulent flow. Turbulent
flows contain fluctuations, whereas laminar fluid flows are smooth without many
irregularities. The fluid flow is computed in FLUENT at every discrete grid point, and the
Reynolds-stress model includes the effects of turbulence. The Reynolds-stress model
takes into account the fluid rotation, curvature, and rapid changes in strain rate more
rigorously than one or two-equation models and, therefore, is an ideal model to use when
analyzing the complex flow within the BMC apparatus [7]. The turbulence boundary
conditions were specified in the form of turbulence intensity and viscosity of 5% and 10,
respectively, for the inlet and outlet boundaries.
Free-Slip Wall, v=0, du/dy=0, dw/dy=0
No-Slip Wall, u = v = w = 0
Inlet, u = U, v = w =0
Outlet, pstat = 0
Figure 4. Boundary conditions applied on the BMC open channel
2.3 Results and Discussion
This study focuses on understanding the fluid behavior in the BMC deep open
channel in order to analyze the effectiveness of the system as a large-scale fiber
separator. Results examined in this study include the x-velocity contours, tangential
velocity vectors, x-vorticity, and the shear stress distribution.
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The non-dimensional entrance length, 𝐸𝑙𝑒𝑛𝑡 , for turbulent flow is expressed as
𝐸𝑙𝑒𝑛𝑡 = 4.4(𝑅𝑒)1/6
(3)
For a Re of 9844, the non-dimensional entrance length is:
𝐸𝑙𝑒𝑛𝑡 = 4.4(9844)1/6 = 20.37
𝐸𝑙𝑒𝑛𝑡 =
𝑙𝑒𝑛𝑡𝑟𝑎𝑛𝑐𝑒
𝐷ℎ
(4)
= 20.37
(5)
Therefore the entrance length for the flow is:
𝑙𝑒𝑛𝑡𝑟𝑎𝑛𝑐𝑒 = 20.37 ∗ 0.01 = 0.2 𝑚
(6)
Entrance length is defined as the length of the inlet to the point where the flow
becomes fully developed. It is assumed that the fluid does not undergo any further
changes in velocity along x, after the entrance length. The velocity and vorticity results
are analyzed in cross planes at x= 0.2 m, y = 0 to 20 m, and z= 0.02 m.
Figures 5 and 6 show the x-velocity contours on the cross plane of the open
channel near the free surface and the channel base, respectively. Velocity is lowest at the
edges of the channel due to friction of the stationary wall. The velocity contours are
bulged towards the corners near the free surface at the top of the channels, while
elsewhere along the vertical wall boundaries, the contours gradually shift towards the
center of the plane. The shift and bulge in the velocity contours are attributed to the
secondary flows [1]. The free surface at the top of the channel is modeled by a zero-shear
stress wall, which attracts the high-momentum fluid from the center of the open channel.
In order to preserve the channel continuity, low momentum fluid from the corner regions
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flows towards the center of the channel. This motion of fluid flow creates a circulatory
effect near the corners, which causes the bulge in velocity contours near the free surface.
Figure 5. Top and Bottom Halves of X-Velocity Contours
Figure 6 shows the x-vorticity on the cross plane of the BMC open channel.
Vorticity is defined as the tendency of a fluid to rotate [11]. The vortices represent the
secondary flows within the channel. The vortices are different near the free surface than
near the channel base. In the corners near the free surface, the vortex pair is not
symmetrical about the angle bisector. The vortex near the channel base is symmetrical
about the angle bisector. There are noticeably more vortices near the free surface than the
channel base. The variation in vorticity along the top and bottom of the channel is due to
the different boundary conditions which changes the behavior of the fluid flow.
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Figure 6. X-Vorticity Contours of the bottom and top halves of BMC open channel flow
Shear stress, as shown in Figure 8, was computed along the x-direction of the
channel. Total shear stress, including 𝜏𝑧𝑥 , 𝜏𝑧𝑦 , and 𝜏𝑧𝑧 , was calculated along the z=0.02 m
sidewall at x = 0 to 0.217 m, y = 0.1 m (mid-plane). The computations from FLUENT for
each partial derivative in respect to z were written out as a text file, and uploaded into
Microsoft Excel in order to calculate 𝜏 for each shear stress component on the sidewall.
Each of the three components was squared and added together. The square root of the
summation is taken to obtain the total shear stress along the x- direction acting on the
sidewall at y = 0.1m. The shear stress is highest at the inlet (x = 0m), reaching around 1.7
Pascal. Along the x-axis, the shear stress steeply decreases between 0 and ~ 0.02 m. After
around 0.02 m in the x-direction, the shear stress levels off and gradually approaches 0.2
pascals.
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Figure 8: Total of All X-Wall Shear Stress Components along the X-direction
Table 2 shows the calculated off-plane and in-plane angles. The off-plane angle is
the angle between resultant shear stress vectors and the xy plane, whereas the in-plane
angle is the angle between the resultant shear stress vectors and the x-axis. These results
show the shear stress angle between the free surface and the wall in the BMC channel
with respect to the x-position in the channel geometry. Several data points were selected
to showcase the shear stress angle trend, as the entire data set was rather large. From
these calculations, it is shown that the shear stress angle is greatest at the inlet, and
quickly drops down to around 8 degrees as the x-position increases. In the actual channel
in the BMC apparatus, the screen is on the z = 0.02 m wall and has one-inch margins
from the inlet, outlet, and bottom wall. The highest off-plane angle occurs at the inlet, x=
0m, and quickly drops down to 8.36 degrees around x = 0.0116 m. It can be inferred from
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this table that the shear stress is mostly aligned tangential to the fluid flow, which is the
ideal situation.
Table 2. Shear Stress Angles along the X-position at y = 0.1 m, z = 0.02 m
In the deep open channel within the BMC apparatus, shear stress distribution on
the wire mesh governs the orientation of the fibers [14]. It is expected that the fibers align
themselves in the direction of shear stress, and any changes or fluctuations of shear stress
will result in a change in orientation. The ideal shear stress distribution would be parallel
to the wire mesh so the fibers are properly filtered by size.
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Chapter 3
Flow in the BMC Open Channel with a Screen on a Side Wall
3.1 Geometry and Computational Grid
In order to further understand and analyze the fluid flow in the deep open channel
of the BMC apparatus, a portion of the side wall of the channel containing the screen was
replaced with a porous boundary. A porous boundary was chosen as a method to model
the screen within the BMC channel rather than a wire-screen to facilitate computation.
The porous boundary represents the screen in the BMC apparatus.
The porous boundary is modeled with 1 inch (0.0254 m) margins lengthwise and a 1 inch
margin from the bottom of the channel. The dimensions of the boundary are x= 0.1662 m,
y= 0.1746 m, and a wire diameter 0.0004572 m. The grid spacing for the channel with a
porous boundary maintains the same grid spacing and ratio of points per inch as the
previous mesh. Figure 9 shows the computational mesh that was used to analyze the fluid
flow in FLUENT.
Figure 9. Computational mesh with porous boundary on a side wall
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3.2 Boundary Conditions
The boundary conditions for all parameters except a portion of a side wall are the
same as the previous case with solid side walls, as seen in Figure 9. In Fluent, the sides of
the screen were specified as a porous-jump boundary condition, and the other as a
pressure-outlet. In order for FLUENT to solve the Reynolds Stress Equations for the fluid
flow in the channel with a porous boundary, the permeability, pressure-jump coefficient,
and the thickness of the porous boundary need to first be determined.
Calculating Permeability (K) and Pressure-Jump Coefficient (𝐶2 ):
The standard screen thickness for a 16 Mesh is 0.0009144 m (0.036 Inch) and the
wire diameter is 0.0004572m (0.0180 Inch). When evaluating through-plane flow
𝐾
through a 2D planar structure, the 𝑑2 value is given to be 0.0056 with F=0.118. The
equation used to calculate the permeability (K) and the pressure-jump coefficient are:
𝐶2 =
2𝐹
√𝐾
and
𝐾
𝐾
𝑑2
= 𝑑2
(7)
𝐾
To first solve for K, the given wire diameter (d) and the 𝑑2 value for a through-plance
flow through a planar 2D structure are imputed into the equation
𝐾
0.0046 = (0.0004572)2
(8)
𝐾 = 9.6154𝑒 −10 𝑚2
This means the face permeability (K) of the screen mesh is 9.6154𝑒 −10 𝑚2
The pressure-jump coefficient (𝐶2 ) is calculated by replacing F with the given 0.118
value and using the calculated K
𝐶2 =
2(0.118)
√(9.6154𝑒 −10 𝑚2 )
1
= 7610.739 𝑚
(9)
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These values are then entered into FLUENT along with the overall screen thickness of
0.0009144 m to analyze the flow in the 3D open channel with the screen modeled as a
porous-jump boundary.
Free-Slip Wall, u= v = w = 0
No-Slip Wall, u = v = w = 0
Inlet, u = U, v = w =0
Outlet, pstat = 0
Porous-Jump, K = 9.6e-10, C2=7610.7 1/m, thickness = 9e-4 m
−10 2
𝐾=
9.6154𝑒
𝑚 Conditions of Channel with Porous Boundary
Figure
9. Boundary
The turbulence boundary conditions were specified in the form of turbulence
intensity and viscosity of 0.5% and 1, respectively, for the inlet and outlet boundaries.
The edges surrounding the porous boundary are modeled as solid no-slip walls with u=
v= w=0. The thickness of the porous boundary is equal to 0.0004572 m, which is the set
value of the wire diameter for a 16 mesh.
3.3 Results and Discussion
Figures 10 shows the x-velocity contours on the cross plane of the open channel
near the free surface and the channel base, respectively. The velocity is similar to the two
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wall case with the velocity being highest towards the center and lowest at the edges of the
walls. The velocity contours are bulged towards the corners near the free surface at the
top of the channels, while elsewhere along the vertical wall boundaries, the contours
gradually shift towards the center of the plane. However, unlike the previous case, the
velocity contours along each wall are not symmetric. This can be attributed to the
addition of the porous boundary.
Figure 10. Top and Bottom Halves of X-Velocity Contours
Figure 11 shows the x-vorticity on the cross plane of the BMC open channel.
Similar to the results with the solid wall boundaries, the vorticies at the bottom corners
are symmetrical about the angle bisectors. The porous boundary is modeled from x=
0.0254 m to 0.1916 m. The vorticity contours were taken at a plane at x= 0.20m in line
with the outlet, right after the porous boundary ends. The addition of the porous
boundary greatly increased the vorticity along the side wall of the channel.
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Figure 11. Top and Bottom Halves of X-Vorcitity Contours
In Figure 12 shows the total shear stress profile for a deep open channel with a
porous boundary on one of the sidewalls to represent a screen. The total shear stress,
𝜏𝑧𝑥 , 𝜏𝑧𝑦 , and 𝜏𝑧𝑧 , was calculated along the z=0.02 m sidewall at x = 0 to 0.217 m, y = 0.1
m (mid-plane). It can be seen that the total shear stress profile at the beginning of the
channel is similar to that of the channel with non-porous sidewalls. Once the fluid flow
encounters the porous boundary, the total shear stress profile changes drastically. On
leaving the porous boundary wall, the shear stress reaches a value of approximately 0.22,
which is equal to the shear stress value in a non-porous wall.
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Figure 12. Total of All X-Wall Shear Stress Components along the X-direction
The off-plane and in-plane angles for the channel with a porous boundary are
displayed in Table 3. The shear stress angles at the beginning and end of the channel are
similar to those from the previous case. However, the off plane shear stress angle in
between is different because the porous boundary is present. The in plane shear stress
angle is relatively small, which means that the in plane shear stress angle is nearly
horizontal.
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Table 3. Shear Stress Angles along the X-position at y = 0.1 m, z = 0.0
Conclusion
The objective of this study was to numerically study the flow through the deep
open channel of the BMC with an aspect ratio of 10. The BMC channel geometry is
initially modeled as two sidewalls and the bottom wall that are no-slip stationary walls,
where the u, v, and w components of velocity are equal to zero. As a second step, the
BMC channel geometry is modified to have one of the side walls contain a porous
boundary condition. The channel geometry is created in the CFD software Gambit with
grid spacing of 50, 180, and 45 grids in the x, y, and z directions. To determine the fluid
flow, FLUENT calculates the fluid flow at each discrete point along the grid. The
Reynolds-stress model accounts for effects of fluid rotation, curvature, and rapid changes
in strain rate of the fluid within the BMC apparatus.
Initially it was found in the first model with solid sidewalls that the x-velocity
contours were highest in magnitude towards the center of the channel and lowest at the
stationary, non-porous sidewalls. In the second porous wall model, it was found that the
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x-velocity contours followed the same behavior as the previous case. However, once the
flow reaches the porous boundary, the x-velocity of the flow begins to fluctuate. In both
of the x-velocity contour plots, bulges are observed near the free surface.
The bulges in the x-velocity contour plots can be attributed to the presence of a
free surface attracting higher momentum fluid, pushing the lower momentum fluid
towards the center of the channel [1]. The shear stress distribution on the vertical
sidewalls near the free surface is affected by this circulatory effect. Furthermore, the
circulation of fluid is observed in the x-vorticiy contour plots, which show the highest
vorticity in the corners near the free surface.
Looking at the shear stress of the fluid flow in the solid side walls channel
geometry, the off-plane angle is greatest at the inlet and drops down and remains around
8 degrees as the x-position increases. These results indicate that the shear stress is mainly
aligned tangential to the fluid flow.
In the actual channel in the BMC apparatus, the screen is on the z = 0.02 m wall
and has one-inch margins from the inlet, outlet, and bottom wall. In the porous-wall case,
the screen is placed between x = 0.0254m and x = 0.1916m. The total shear stress
gradient at the beginning of the channel is similar in both the non-porous wall and the
porous wall cases. In the porous boundary case, once the flow encounters the porous
boundary, the shear stress changes drastically. On passing the porous boundary wall, the
shear stress nears a value of 0.22 pascals, which is similar to that observed in the solid
wall case.
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Future Work
The future work of this study is to further analyze the off-plane and in-plane shear stress
angles of the channel modeled with a porous boundary. Due to time constraints, this
study was not able to fully determine the effects of the porous boundary results. As a
further step, the results of both geometries will be compared with similar case studies.
Additionally, literature research will be performed to further investigate the fluid flow
through a porous boundary.
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Acknowledgements
We would like to thank our faculty mentor, Dr. Urmila Ghia. She has devoted
much of her time to have us fully understand the concepts behind this project. We came
into this project with no background in fluid mechanics and improved knowledge of this
subject. Her mentorship has been a valuable aspect of our research project. Dr. Ghia’s
graduate students have been great resources to us as well, as they have helped us with out
project on numerous occasions and assisted us in learning the CFD software. Thank you
to Chandrima, Deepak, and Santosh for taking time out of your busy schedule to teach us
the software and help us with our problems along the way.
We would also like to thank the National Science Foundation for sponsoring this
study. Without the program in place we would have not had this great opportunity for
part time research.
Throughout the length of this research project, our understanding of fluid
mechanics has grown exponentially. We both are studying biomedical engineering, and
began this project with absolutely zero background in fluid mechanics and dynamics.
Though the learning curve was steep, with the assistance of Dr. Ghia we were able to get
a solid understanding of the fluid dynamic properties and were able to conduct research
studying the fluid flow within a channel. Through this project we also learned how to use
CFD tools Gambit and FLUENT, as well as how to interpret the results from FLUENT.
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Bibliography
1. Jana, C. (2011), “Numerical Study of Three-Dimensional Flow Through a Deep Open
Channel-Including a Wire-Mesh Segment on One Side Wall.” M.S. Mechanical
Engineering Thesis, University of Cincinnati.
2. Dodson, R., Atkinson, M., Levin, J. (2003), “Asbestos Fiber Length as Related to
Potential Pathogenicity: A Critical Review.” American Journal of Industrial Medicine,
Vol. 44, p. 291-297
3. http://www.nlm.nih.gov/medlineplus/asbestos.html
4. Guo, J., Julien, PY. (2005). “Shear stress in smooth rectangular open-channel flows.”
American Society of Civil Engineers, 131(1), 30-37.
5. Janna, W. S. (2009) “Introduction to Fluid Mechanics”, CRC press, 4th Edition.
6. White, F. M. (2003) “ Fluid Mechanics”, McGraw-Hill, 5th Edition.
7. Fluent 6.3 User’s Guide.
8. Gambit 2.4 User’s Guide.
9. Chung, T. J. (2010) “Computational Fluid Dynamics”, Cambridge University Press,
2nd Edition.
10. Birchall, D. (2009) “Computational fluid dynamics”, The British Institute of
Radiology, 82, S1-S2.
11. “Vorticity.” Def. 1. Meriam Webster Online, Merriam Webster
12. Blake, T., Castranova, V., Baron, P., Schwegler-Berry, D., Deye, G.J., Li, C., and
Jones, W., (1998), “Effect of Fiber Length on Glass Microfiber Cytotoxicity,” Journal of
Toxicological Environmental Health, Part A, Vol. 54, Issue 4.
13. www.uceindia.com/fiber-classifier-bauer-mcnett-type.htm
28
14. Paschkewitz, J.S., Dubiel, Y., Shaqfeh, E., (2005), “The dynamic mechanism for
turbulent drag reduction using rigid fibers based on Lanrangian conditional statistics”,
American Institute of Physics, Vol. 17.
29