Porto_intern_report

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CEFT, Department of Mechanical Engineering, Faculdade de Engenharia da
Universidade do Porto
Summer Internship
Report
Supervisor: Prof. Fernando Tavares de Pinho
Centro de Estudos de Fenómenos de Transporte
Faculdade de Engenharia, Universidade do Porto
Rua Dr. Roberto Frias, s/n
4200-465 Porto
Portugal
Harsh Ranjan
Department of Mechanical Engineering
Indian Institute of Technology Guwahati
Guwahati, Assam
India
Laminar flow characteristics in a 2D elbow: a numerical investigation
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ACKNOWLEDGEMENTS
First and foremost, I would like to express my heartfelt gratitude to my supervisor, Prof.
Fernando Pinho for giving me the opportunity to work on this project and guiding me all the
way through. The fact that this was my first research experience makes this even more
special and I have no one else to thank more than my supervisor for being there to
supervise and guide me whenever I needed his help. Even before the start of my internship,
Prof. Pinho was instrumental in getting me abreast with the needs of my project and helping
me get ready for it by suggesting literature that I needed to study. His help and guidance in
the project and even outside are hugely appreciated.
At the same time, I would like to express my sincere thanks to my lab mates Dr. Francisco
Galindo J. Rosales, Dr. Laura Campo Deaño, Mr Mohamad Masudian, Miss Mohanna
Heibati and Mr Romeu Matos for helping me with my doubts in the lab and for being the
great friends that they were throughout my stay in Portugal. A special mention should be
made for Mr Pouya Samani, who along with my lab mates always pepped me up during
the course of my internship.
Try as I may to sum up the help of all these people during my internship and express my
gratitude to them in a couple of paragraphs, I know that mere words will never be enough
to do justice to the sheer magnitude of their invaluable help and guidance.
Harsh Ranjan
Guwahati, Assam
India
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PREFACE
This report documents the work done during my summer internship at CEFT (Centro de
Estudos de Fenómenos de Transporte/Transport Phenomena Research Centre), Department
of Mechanical Engineering, Faculdade de Engenharia da Universidade do Porto, Porto,
Portugal under the supervision of Prof. Fernando Tavares de Pinho from 8 May, 2013 to 7
July, 2013. The report shall first give an overview of the tasks completed during the period
of the internship with technical details. Then the results obtained shall be discussed and
analysed.
This report shall also elaborate on the future works which can be pursued as an
advancement of the current work.
I have tried my best to keep report simple yet technically correct. I hope I succeed in my
attempt.
Harsh Ranjan
Guwahati, Assam
India
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LIST OF CONTENTS
1. INTRODUCTION 1
2. NUMERICAL ANALYSIS 2
2.1 COMPUTATION SCHEMES 2
2.2 COURANT-FRIEDRICHS-LEWY CONDITION 7
2.3 LOCAL LOSS COEFFICIENT 7
2.4 KINETIC ENERGY CORRECTION FACTOR 10
3. VALIDATION OF CODE 11
3.1 LAMINAR FLOW IN A STRAIGHT CHANNEL 11
3.2 RESULTS OF VALIDATION PROCESS 14
4. ELBOW 16
4.1 NEWTONIAN FLOWS 16
4.1.1 RESULTS FOR DUCT ASPECT RATIO=1 16
4.1.2 RECIRCULATION 18
4.1.3 RESULTS FOR DUCT ASPECT RATIO=3 19
4.1.5 RESULTS FOR DUCT ASPECT RATIO=1/3 21
4.1.7 SUMMARY 22
4.2 NON-NEWTONIAN FLUIDS 23
4.2.1 GENERALISED NEWTONIAN FLUID 23
4.2.2 POWER LAW FLUID 24
4.2.3 RESULTS 26
5. CLOSURE 28
5.1 CONCLUSION 28
5.2 FUTURE WORK 28
6. REFERENCES 29
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1. INTRODUCTION
An elbow is a pipe fitting installed between two lengths of pipe or tubing to allow a change
of direction. This makes the elbow a very common device and hence an important geometry
in fluid mechanics. These days microfluidic devices for manipulating fluids are widespread
and finding uses in many scientific and industrial contexts. Microfluidics refers to devices
and methods for controlling and manipulating fluid flows with length scales less than a
millimetre. In microfluidics, more than circular pipes, those with a rectangular cross-section
are preferred because of their ease of construction. An elbow is a very common geometry
used in microfluidics and in my project elbows with a rectangular cross-section have been
considered.
The aim of my internship project was to study the laminar flow characteristics in a 2D elbow
by numerically investigating the laminar flow of Newtonian and non-Newtonian fluids
passing through it in order to characterise the local loss coefficient, the existence and size of
regions of separated flow and their variation as a function of relevant parameters like the
Reynolds number and duct aspect ratio.
Newtonian fluids are those for which the viscous stresses arising in the course of its flow, at
every point, are proportional to the local strain rate; the associated constant of
proportionality being the coefficient of viscosity. On the other hand, non-Newtonian fluids
are those whose flow properties differ in any way other than those of Newtonian fluids.

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2. NUMERICAL ANALYSIS
2.1 COMPUTATION SCHEMES
General transport equation for the conservation of property is,
+ ∇. ( ⃗) = ∇. (Г∇ ) +
↑ ↑ ↑ ↑
Г is the diffusion coefficient corresponding to and S is the source term. The above
equation transforms into certain famous equations for different ’s.
 = 1 Mass Conservation Equation
 = Momentum Conservation Equation
 = Energy Conservation Equation
A key concept in Finite Volume Method is the integration of the above differential equation
over the control volume.
∆
+ ( ⃗. ⃗ ) = ( ⃗. ∇⃗ )
∆∆
+
∆
The solution procedure involves the following three steps-
1. Grid generation
2. Discretization of differential equations into algebraic equations
3. Simultaneous solution of the algebraic equations
Grid Generation
 The domain is divided into discrete control volumes.
 Nodes are determined on the system boundaries.
 Boundaries of control volumes are placed midway between the adjacent nodes.
Source
Term
Diffusion
Term
Convection
Term
Unsteady
Term

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Discretization
The general form of the discretized equation is,
= +
Where,
 Σ indicates summation over the neighbouring nodes
 is the neighbouring node coefficient
 is the value of the property at the neighbouring node
 is a constant
Figure 2.1. Figure depicting the process of grid generation.
Source: Versteeg, H.K. and Malalasekera, W. (1995) An Introduction to
Computational Fluid Dynamics (The Finite Volume Method)
Figure 2.2
Source: Versteeg, H.K. and Malalasekera, W. (1995) An
Introduction to Computational Fluid Dynamics (The Finite Volume
Method)

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For the solutions to be physically realistic, a discretization scheme should satisfy the
following requirements:
 Conservativeness
There should be flux consistency at the CV faces, i.e., Flux of leaving a
control volume= Flux of entering through the same face.
 Boundedness
The discretized equations are solved by iterative methods. Sufficient
condition for a convergent iterative solution is,
∑| |
| |
≤ 1
Where,
is the net coefficient of the central node, i.e., − .
 Transportiveness
A non-dimensional Peclet number is defined to measure the relative
strengths of convection and diffusion.
= =
Г⁄
Where,
 is the convective mass flux per unit area
 is the fluid density
 Г is the diffusion coefficient
 is the conductance coefficient
 is the fluid flow velocity
 is the characteristic length
The relationship between the magnitude of the Peclet number and the
directionality of influencing, i.e., transportiveness is borne out of the
discretization scheme.
 For Pe=0, there is pure diffusion.
 As Pe increases, the contours become more elliptic.
 As Pe=∞, there is pure convection.
Spatial Discretization
Consider the steady convection and diffusion of a property in a one-dimensional flow field
u (refer to figure 2.3 below)

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( ) = (Г )
The flow must also satisfy the continuity equation,
( ) = 0
Integration of the transport equation over the control volume yields,
( ) − ( ) = Г − Г
Integration of the continuity equation yields,
( ) − ( ) = 0
Assuming = = and employing central differencing scheme approach to represent
the diffusion term, we get
− = ( − ) − ( − )
The integrated continuity equation is,
− = 0
Where,
= ( )
= ( )
=
Г
=
Г
There are various schemes available for calculating the value of :
 Central Differencing Scheme
Figure 2.3
Source: Versteeg, H.K. and Malalasekera, W. (1995) An Introduction to
Computational Fluid Dynamics (The Finite Volume Method)

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 First Order Upwind Scheme
 Exact Solution
 Exponential Solution
 Hybrid Scheme
 Power Law Scheme
 QUICK Scheme
For most of the simulations, the Linear Upwind Differencing Scheme (LUDS) was used but
for some initial simulations, the Upwind Differencing Scheme (UDS) was used.
In computational fluid dynamics, upwind schemes denote a class of numerical discretization
methods for solving partial differential equations. Upwind schemes use an adaptive or
solution-sensitive finite difference stencil to numerically simulate the direction of
propagation of information in a flow field. The upwind schemes attempt to discretize partial
differential equations by using differencing biased in the direction determined by the sign of
the characteristic speeds.
A brief explanation of the First Order Upwind Scheme is given below.
Here,
= , > 0
= , < 0
is defined in a similar fashion.
Now we can define the ⟦1⟧ operator as,
⟦ , ⟧ = ( , )
The final discretization equation is,
= +
where,
+ ⟦ , 0⟧ + ⟦− , 0⟧ − + ( − )
W w P e E
Figure 2.4 Grid generation for First Order Upwind Scheme

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The spatial accuracy of the first-order upwind scheme can be improved by including 3 data
points instead of just 2, which offers a more accurate finite difference stencil for the
approximation of spatial derivative. This is because this helps reduce the Taylor Series
truncation error. This is the second-order upwind scheme.
This scheme is less diffusive compared to the first-order accurate scheme and is called
Linear Upwind Differencing Scheme (LUDS).
2.2 COURANT–FRIEDRICHS–LEWY CONDITION
In mathematics, the Courant–Friedrichs–Lewy condition (CFL condition) is a necessary
condition for convergence while solving certain partial differential
equations (usually hyperbolic PDEs) numerically by the method of finite differences. It arises
in the numerical analysis of explicit time-marching schemes, when these are used for the
numerical solution. As a consequence, the time step must be less than a certain time in
many explicit time-marching computer simulations, otherwise the simulation will produce
incorrect results. The condition is named after Richard Courant, Kurt Friedrichs, and Hans
Lewy who described it in their 1928 paper.
The two results that I considered for my simulations are listed below-
=
× ∆
∆
=
× ∆ ×
∆
Where,
V= Average velocity of fluid flow
ν= Viscosity
∆t= Time step
D= Effective diameter of pipe
∆x= Length interval
In order to obtain convergence, the values for viscosity, velocity and the time step were
adjusted keeping the Reynolds number constant to have the Courant numbers less than or
close to 1.
2.3 LOCAL LOSS COEFFICIENT
The additional components attached to straight pipes add to the overall head loss of the
system. Pipe systems often include inlets, outlets, bends, and other pipe fittings in the flow
that create eddies resulting in head losses in addition to those due to pipe friction. Such

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losses are generally termed minor losses, with the apparent implication being that the
majority of the system loss is associated with the friction in the straight portions of the
pipes, the major losses or local losses. In many cases this is true. In other cases the minor
losses are greater than the major losses. The minor losses may be caused by-
1. Pipe entrance or exit
2. Sudden expansion or contraction
3. Bends, elbows, tees, and other ﬁttings
4. Valves (open or partially closed)
5. Gradual expansions or contractions
The local loss coefficient, K is dimensionless, and is a function of Reynolds number. In the
standard literature the head loss coefficient is not usually correlated with Reynolds number
and roughness but simply with its geometry and the diameter of the pipe, implicitly
assuming that the pipe flow is turbulent. Through the simulations, I tried to study the
dependence of the local loss coefficient (K) on the Reynolds number.
The local head loss produced by a device obstructing the pipe flow is characterized by the
local loss coefficient, K, usually expressed as the ratio of the head loss through the device,
ℎ to the velocity head, 2 .
=
ℎ
2
This can also be written as,
=
∆
2
Where,
ρ- Density of fluid
V- Average velocity of fluid flow
In channel flow (flow between two parallel plates) if the flow is fully developed then,
=
where, is a constant and is the friction factor, as proved earlier in Section. 3.1.
Here, the Reynolds number is defined by the wall to wall distance. So, we have-

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= or, =
However, the continuity forces ℎ = ℎ thus ensuring that in both channels of the
elbow, the friction factor remains unchanged.
Now, the head loss is represented as,
ℎ =
× 2
With the knowledge that = 2ℎ and = 2ℎ we have the following results-
=
×
and, =
×
Using the continuity equation with the above two results, we obtain,
ℎ
=
ℎ
(
ℎ
ℎ
)
With ℎ =
∆
, the above can be rewritten as,
∆
=
∆
( )
So, effectively what was plotted was vs. distance and × ( ) vs. distance for the fully
developed flow region. The first plot for ‘before the bend’ was extrapolated to the end of
the first arm, or in the other words, just before the beginning of the block constituting the
elbow’s bend. Similarly, the second plot for ‘after the bend’ was extrapolated backwards to
the start of the second arm of the elbow. At this point the value given by the second plot
was readjusted by dividing by ( ) to have
× ( )
( )
. The difference between
and
× ( )
( )
gave the pressure loss which was then used for computing the
local loss coefficient.

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2.4 KINETIC ENERGY CORRECTION FACTOR
For real flows the Bernoulli’s equation for energy is modified to the following-
+ + = + + + ∆ + ∆ + ∆
where, ∆ + ∆ is the net frictional loss.
So this gives,
∆ = ( − ∆ ) − ( + ∆ ) +
1
2
−
1
2
Hence, for duct aspect ratios other than 1 in the elbow there was another factor that
needed to be accounted for, that of the kinetic energy correction factor as the bulk velocity
was changing after the bend due to variance in the cross-section. For this, the kinetic energy
correction factor ( ) was computed for channel flow using the formula-
=
∫ 2
⃗. ⃗
̇ 2
This, on simplification leads to-
=
1
( )
Using, = 1.5 [1 − ( ) ] for channel flow as per Section 3.1, was found to be 54
35.

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3. VALIDATION OF CODE
3.1 LAMINAR FLOW IN A STRAIGHT CHANNEL
Consider steady flow between two infinitely broad parallel plates as shown in Figure 3.1.
Flow is independent of any variation in z direction, hence, z dependence is gotten rid of and the
Equation (3.1) becomes Equation (3.2),
= − + [ + ] (3.1)
(Eq. (3.1) is obtained from Navier Stokes Equation for incompressible flow in the x-direction
with the velocities in the other directions being zero)
= (3.2)
The boundary conditions are at y = b, u = 0; and y = -b, u = 0. Now, From Eq. (3.2), we can
write-
=
1
+
or, = + +
On applying the boundary conditions, we get,
= − ( − ) (3.3)
Figure 3.1 Laminar flow in a straight channel

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which implies that the velocity profile is parabolic.
Average Velocity and Maximum Velocity
To establish the relationship between the maximum velocity and average velocity in the
channel, we analyse as follows:
At y=0, = which yields,
= − (3.4a)
On the other hand, the average velocity,
=
2
=

=
1
2
Using Equation (3.3) in the above expression and integrating, we get,
= − (3.4b)
which in turn gives,
= (3.4c)
The shearing stress at the wall for the parallel flow in a channel can be determined from the
velocity gradient as follows:
( = ) = = = −2
Since the upper plate is a ‘minus y surface’, a negative stress acts in the positive x direction,
i.e. to the right.
The local friction coefficient, is defined by
=
( )
1
2
=
3 ⁄
1
2
= ( ) = (3.4d)

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where = (2 )/ is the Reynolds number of flow based on average velocity and the
channel height (2b).
While considering the problem on channel flow, I constructed meshes with varied
refinements and studied the flow after pre-setting the Reynolds number, an appropriate
entrance velocity and the corresponding viscosity using the formula for Reynolds number in
a pipe,
= = =
Where,
 is the hydraulic diameter of the pipe (m).
It was set to 50 mm in all the cases that were considered.
 is the volumetric flow rate (m3
/s).
 is the pipe cross-sectional area (m²).
 is the mean velocity of the fluid (m/s).
 is the dynamic viscosity of the fluid (Pa·s or N·s/m² or kg/(m·s)).
 is the kinematic viscosity ( = ⁄ ) (m²/s).
 is the density of the fluid (kg/m³).
The meshes for which the simulation was performed were all 1 m long and 50 mm wide (2
Dimensional). Meshes of three different refinements were produced:
 Mesh 1, having 10 cells in the direction and 15 cells in the direction
As the mesh was refined, the velocity profiles could be seen with greater accuracy and
detail.

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3.2 RESULTS OF VALIDATION PROCESS
Using Eq. 3.4d-
= ( ) =
The Poiseuille number ( × ) is found to be 12. In the numerical analysis performed for
the three meshes individually, the Poiseuille numbers were as follows:
 For mesh 1, Poiseuille number= 12.878
Using Eq. 3.4c-
=
So, the ratio of maximum velocity to the bulk velocity should be 1.5. In the numerical
analysis performed for the three meshes individually the ratio of the maximum velocity to
the bulk velocity in the fully developed regions were as follows:
0.00E+00
2.00E-02
4.00E-02
6.00E-02
8.00E-02
1.00E-01
1.20E-01
1.40E-01
1.60E-01
0.00E+00 1.00E-02 2.00E-02 3.00E-02 4.00E-02 5.00E-02 6.00E-02
Mesh 1
Mesh 2
Mesh 3
Figure 3.2. Plots of X component of velocity vs. distance for meshes
1, 2 and 3 comparing fully developed velocity profiles at x=0.9 m.

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 For mesh 1, = 1.4919
 For mesh 2, = 1.4980
 For mesh 3, = 1.4996
From the above data analysis, it was observed that the simulated results were indeed
approaching the verified analytical results as the mesh refinement was improved to allow
for a more accurate simulation.

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4. ELBOW
After having successfully validated the results for laminar flow between two parallel plates, I
proceeded on to performing a similar analysis, this time for the main geometry- the elbow.
The first task was to perform the numerical analysis for the geometry with varying
refinements for a particular Reynolds number (here, 50). I calculated the pressure gradient
from the plots of pressure vs. distance for each of the meshes under consideration and
subsequently found the local loss coefficient.
For a given Reynolds number and all other properties of the flow kept same a monotonous
trend was observed as the mesh refinement was improved. Due to the unavailability of an
analytical solution to the above problem, the monotonous trend with minor fluctuations in
the value of the local loss coefficient (K) for the elbow was taken as an indication of the
correctness of the simulation. Meshes of three different refinements were generated:
Mesh Cells K Value
1 210 × 21 + 21 × 21 + 210 × 21 0.404088
2 310 × 31 + 31 × 31 + 310 × 31 0.407584
3 410 × 41 + 41 × 41 + 410 × 41 0.409604
Thereafter, various other Reynolds numbers were considered for Mesh 2. While performing
the simulations, I ensured that the Courant numbers for convection and diffusion were less
than or close to 1 in order to obtain convergence. Also, it was observed that the closer the
Courant numbers were to 1, the lesser time it took for the simulation to complete.
4.1 NEWTONIAN FLOWS
4.1.1 RESULTS FOR DUCT ASPECT RATIO=1
The results have been tabulated briefly in Table 4.2 and the corresponding plot of Log10(Re)
vs. K*Re is shown in Figure 4.1.
Table 4.1. Table showing local loss coefficients for meshes of different refinements.

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Re K*Re
0.001 42.08
0.01 41.916
0.1 41.68
0.2 41.9
0.5 41.856
1 41.94
2 42
5 42.6248
20 49.28
50 68.3952
100 108.472
0
20
40
60
80
100
120
0 1 2 3 4 5 6
Log10(Re) vs. K*Re(for h2/h1=1)
K*Re
Table 4.2. Re and local loss coefficient × Re values for elbow with duct aspect ratio=1.
Figure 4.1. Plot of Log10(Re) vs. local loss coefficient × Re for elbow with duct aspect ratio=1.
(Note: Graph is shifted by 3 units along + direction.)

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4.1.2 RECIRCULATION
It was observed that at the outer corner of the bend and for some distance along the inner
corner, vortices were formed owing to perturbation caused by the abrupt change in
geometry. The recirculation lengths were calculated by observing the peculiarity of the
velocity vectors in such regions. Such effects were negligible for smaller Reynolds numbers
but became more pronounced as the Reynolds numbers became larger as is evident from
the following graphic.
Figure 4.2. Y component of velocity in the elbow (Duct Aspect Ratio=1) for Reynolds number=1 (top)
and Reynolds number=100 (bottom).
Legend values are in m/s.
(Arm Length=1m, Bend Dimensions=(100x100)mm2
; X/Y=1.0, X/Z=1.0)

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The information for recirculation observed for elbow with duct aspect ratio=1 for various
Reynolds numbers is shown in the Table 4.3.
Inner Corner Outer Corner
Re From(in m) To(in m) Up till(in m)
0.001 Nil Nil 9.84E-02
0.01 Nil Nil 9.84E-02
0.1 Nil Nil 9.84E-02
0.2 Nil Nil 9.84E-02
0.5 Nil Nil 9.84E-02
1 Nil Nil 9.84E-02
2 Nil Nil 9.84E-02
5 Nil Nil 9.84E-02
20 Nil Nil 9.84E-02
50 1.02E-01 1.73E-01 9.84E-02
100 1.02E-01 2.66E-01 9.52E-02
4.1.3 RESULTS FOR DUCT ASPECT RATIO=3
After this, the same investigation was carried out for varying duct aspect ratios- once with
the outlet being three times the inlet and another time with the inlet being thrice the outlet.
The results have been tabulated in Table 4.4 and the corresponding plot of Log10(Re) vs.
K*Re is shown in Fig. 4.6.
Re K*Re
0.001 14.55248
0.01 13.96883
0.1 13.76529
0.2 13.80243
0.5 14.20097
1 13.9418
2 13.99219
5 14.20728
20 15.75372
50 71.01884
100 106.5906
Table 4.3. Recirculation data for elbow with duct aspect ratio=1.
Table 4.4. Re and local loss coefficient × Re values for elbow with duct aspect ratio=3.

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4.1.4 RECIRCULATION
The information for recirculation observed for elbow with duct aspect ratio=3 for various
Reynolds numbers is shown in the Table 4.5:
RE From(in m) To(in m) Up till(in m)
0.001 Nil Nil 9.52E-02
0.01 Nil Nil 9.52E-02
0.1 Nil Nil 9.52E-02
0.2 Nil Nil 9.52E-02
0.5 Nil Nil 9.52E-02
1 Nil Nil 9.52E-02
2 Nil Nil 9.52E-02
5 Nil Nil 9.52E-02
20 1.02E-01 4.37E-01 9.52E-02
50 1.02E-01 8.63E-01 1.15E-01
100 1.02E-01 1.16E+00 1.66E-01
0
20
40
60
80
100
120
0 1 2 3 4 5 6
Log10(Re) vs. K*Re (for h2/h1=3)
K*Re
Figure 4.3. Plot of Log10(Re) vs. local loss coefficient × Re for elbow with duct aspect
ratio=3.
+
Table 4.5. Recirculation data for elbow with duct aspect ratio=3.
+

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4.1.5 RESULTS FOR DUCT ASPECT RATIO=1/3
For the inlet being thrice the outlet, the results have been tabulated briefly in Table 4.6 and
the corresponding plot of Log10(Re) vs. K*Re is shown in Figure. 4.4.
Re K*Re
0.001 -29.9586
0.01 -30.1463
0.1 -29.8229
0.2 -28.0857
0.5 -23.3543
1 -16.1206
2 -2.20314
5 42.09394
20 263.2686
50 726.0914
100 1245.143
-200
0
200
400
600
800
1000
1200
1400
0 1 2 3 4 5 6
Log10(Re) vs. K*Re (for h2/h1=1/3)
K*Re
Table 4.6. Re and local loss coefficient × Re values for elbow with duct aspect ratio=1/3.
Figure 4.4. Plot of Log10(Re) vs. local loss coefficient × Re for elbow with duct aspect ratio=1/3.
(Note: Graph is shifted by 3 units along + direction.)

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4.1.6 RECIRCULATION
The recirculation data is as follows:
RE From(in m) To(in m) Up till(in m)
0.001 Nil Nil 2.98E-01
0.01 Nil Nil 2.98E-01
0.1 Nil Nil 2.98E-01
0.2 Nil Nil 2.98E-01
0.5 Nil Nil 2.98E-01
1 Nil Nil 2.98E-01
2 Nil Nil 2.98E-01
5 Nil Nil 2.98E-01
20 Nil Nil 2.79E-01
50 3.02E-01 4.60E-01 2.98E-01
100 3.18E-01 4.85E-01 2.76E-01
4.1.7 SUMMARY
The dependence of the local loss coefficient on the Reynolds number is such that at low
Reynolds numbers (<10) the product of Reynolds number and local loss coefficient tends to
a constant value. However, normally after Reynolds number=10, the values deviate and
thereafter it is the local loss coefficient that tends to a constant. Thus, we see that all the
curves in the figure below intersect each other around this mark.
-200
0
200
400
600
800
1000
1200
1400
0 1 2 3 4 5 6
Log10(Re) vs. K*Re
h2/h1=1
h2/h1=3
h2/h1=1/3
Figure 4.5. Plots of Log10(Re) vs. local loss coefficient ×Re for elbows with duct
aspect ratios 1, 3 and 1/3. (Note: Graph is shifted by 7 units along + direction.)
Table 4.7. Recirculation data for elbow with duct aspect ratio=1/3.

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4.2 NON-NEWTONIAN FLOWS
After having done the numerical analysis for Newtonian flows, I started a similar analysis for
non-Newtonian flows. For this I used the Power Law Model.
4.2.1 GENERALISED NEWTONIAN FLUID
A generalized Newtonian fluid is an idealized fluid for which the shear stress, τ, is a function
of shear rate at the particular time, but not dependent upon the history of deformation.
= ( )
Here, is the shear rate or the velocity gradient perpendicular to the plane of shear
The quantity
= ( )/( )
represents an apparent or effective viscosity as a function of the shear rate.
0.00E+00
2.00E+00
4.00E+00
6.00E+00
8.00E+00
1.00E+01
1.20E+01
0 20 40 60 80 100 120
Re vs. (Recirculation Length/Inlet Width)
h2/h1=1
h2/h1=3
h2/h1=1/3
Figure 4.6. Plots of Re vs. Recirculation length/Inlet width for elbows with duct
aspect ratios equal to 1, 3 and 1/3.

24
The most commonly used types of generalized Newtonian fluids are:
 Power-law fluid
 Cross fluid
 Carreau fluid
 Second-order fluid
In all the simulations the Power Law Model was used. It is described in detail in the
following section.
4.2.2 POWER LAW FLUID
A Power-law fluid or the Ostwald–de Waele relationship, is a type of generalized Newtonian
fluid for which the shear stress, , is given by
= ( )
where:
 is the flow consistency index (SI units Pa.sn
),
 is the shear rate or the velocity gradient perpendicular to the plane of shear, and
 is the flow behaviour index or power index (dimensionless).
The quantity
= ( )
represents an apparent or effective viscosity as a function of the shear rate.
Also known as the Ostwald–de Waele power law this mathematical relationship is useful
because of its simplicity, but only approximately describes the behaviour of a real non-
Newtonian fluid. For example, if n were less than one, the power law predicts that the
effective viscosity would decrease with increasing shear rate indefinitely, requiring a fluid
with infinite viscosity at rest and zero viscosity as the shear rate approaches infinity, but a
real fluid has both a minimum and a maximum effective viscosity that depend on
the physical chemistry at the molecular level.
Therefore, the power law is only a good description of fluid behaviour across the range of
shear rates to which the coefficients were fitted. There are a number of other models that
better describe the entire flow behaviour of shear-dependent fluids, but they do so at the
expense of simplicity, so the power law is still used to describe fluid behaviour, permit
mathematical predictions, and correlate experimental data.

25
Power-law fluids can be subdivided into three different types of fluids based on the value of
their flow behaviour index:
 Pseudoplastic or shear thinning fluid (n<1)
 Newtonian fluid (n=1)
 Dilatant or shear thickening fluid (n>1)
A power law is a functional relationship between two quantities. For example, if the
frequency (with which an event occurs) varies as a power of some attribute of that event
(e.g. its size), the frequency is said to follow a power law.
0.00E+00
5.00E+00
1.00E+01
1.50E+01
2.00E+01
2.50E+01
0.00E+00 1.00E-01 2.00E-01 3.00E-01 4.00E-01
Shear Stress(s-1) vs. Viscosity(Pa.s)
Viscosity
Figure 4.7. An example power-law graph, being used to demonstrate ranking of popularity. To the
right is the long tail, and to the left are the few that dominate (also known as the 80–20 rule).
(Source: http://en.wikipedia.org/wiki/File:Long_tail.svg)
Figure 4.8. Shear stress vs. viscosity curve showing the power law dependence for a non-
Newtonian flow through an elbow with duct aspect ratio=1.
(Parameters of importance: Re=1, average velocity at inlet=0.01, power index=0.6)

26
4.2.3 RESULTS
For the generalised Newtonian fluids, just like in the analysis for Newtonian flows through
an elbow, many Reynolds numbers were tried with different power indices, viz. 0.4, 0.6, 0.8
and 1.2. Clearly, a greater emphasis was on studying the shear thinning fluids as such fluids
are more common that the shear thickening ones.
n=1.2 n=0.8 n=0.6 n=0.4
Ln(Re) K*Re Ln(Re) K*Re Ln(Re) K*Re Ln(Re) K*Re
-6.90776 38.2 -6.90776 46.2 -6.90776 50.96 -6.90776 56.96
-4.60517 38.112 -4.60517 46.128 -4.60517 50.896 -4.60517 56.816
-2.30259 38.2 -2.30259 46.2 -2.30259 51 -2.30259 56.76
-1.60944 37.38 -1.60944 45.8 -1.60944 50.38 -1.60944 56.54
-0.69315 38.072 -0.69315 45.984 -0.69315 50.992 -0.69315 56.776
0 38.14 0 45.988 0 50.904 0 56.372
0.693147 37.932 0.693147 45.352 0.693147 50.712 0.693147 56.632
1.609438 38.1728 1.609438 45.9328 1.609438 50.832 1.609438 56.5368
2.995732 42.812 2.995732 49.376 2.995732 53.73 2.995732 59.136
3.912023 59.2512 3.912023 69.544 3.912023 63.1592 3.912023 67.6864
4.60517 95.828 4.60517 87.656 4.60517 85.504 4.60517 88.34
Figure 4.9. Diagram of the various models and the ranges that they cover.
(Source: A Handbook of Elementary Rheology by Howard A. Barnes)
Table 4.8. Ln (Re) and local loss coefficient × Re values for elbow with power indices being
0.4, 0.6, 0.8 and 1.2.

27
For the above computations, the required formula for the Reynolds number needed to be
for a generalised Newtonian fluid. This formula is given below-
= [8(
6 + 2
) ]
Where,
 is density of the fluid
 is the effective diameter of the elbow
 is the average velocity of the fluid
 is the power index
 is equal to ( )
−1
specifically for the Power Law Model (refer to equation)
0
20
40
60
80
100
120
0 2 4 6 8 10 12 14
Ln(Re) vs. K*Re
n=1.2
n=0.8
n=0.6
n=0.4
Figure 4.10. Plots of Ln (Re) vs. K ×Re for various power indices for the
flow of a non-Newtonian fluid through an elbow.

28
5. CLOSURE
5.1 CONCLUSION
During the course of the internship the objective of numerically investigating Newtonian
and non-Newtonian flows through a 2D elbow was realised. For the Newtonian fluids, an
exhaustive analysis was done as the flows were compared with respect to variance in
parameters like the Reynolds number and duct aspect ratio of the elbow. Thus, the
dependence of the local loss coefficient of the elbow on the Reynolds number of the flow
was established.
For non-Newtonian fluids the flow was numerically analysed with respect to variance in the
power index of the Power Law Model along with the Reynolds number. In addition to this,
the recirculation in certain areas near the bend of the elbow was also studied and recorded.
5.2 FUTURE WORK
This project had me perform an investigative study on fluid flows through an elbow. The
Newtonian fluids were studied in some detail with varying duct aspect ratios and Reynolds
numbers. Subsequently the non-Newtonian fluids were studied with different Reynolds
numbers and power indices. Having said that, there was only a superficial analysis of such
flows as the practical scenario, that of 3-D elbows was still unexplored. This could provide
motivation for a detailed follow-up to the current work. Also, these studies could be
extended to study flows through various other geometries as the computational part would
remain pretty much the same except for minor alterations subject to various aspects of the
new geometry.
While the elbow had no analytical solution available to validate the results of the
computational analysis, it could be used to study and validate simpler cases like laminar
flow between parallel plates as demonstrated in the report to have a better understanding
of the things gained by reading associated literature.
Moreover, given the availability of appropriate resources (powerful computing system), it
could be used to generate detailed simulation of flows through more practical as well as
more complex geometries.

29
6. REFERENCES
[1]. Barnes, H.A., Hutton, J.F. and Walters, K. (1989) Rheology Series Vol. 3, An
Introduction to Rheology. Amsterdam (The Netherlands): Elsevier Science Publishers
[2]. Barnes, H.A. (2000) A Handbook of Elementary Rheology. Aberystwyth : The
University of Wales, Institute of Non-Newtonian Fluid Mechanics, Aberystwyth
[3]. Munson, B.R., Young, D.F., Okiishi, T.W. and Huebsch, W.W. (1990) Fundamentals of
Fluid Mechanics. 6th
Ed. Hoboken, N.J.: John Wiley and Sons, Inc.
[4]. Bird, B.R., Armstrong, R.C. and Hassager, O. (1987) Dynamics of Polymeric Liquids.
2nd
Ed. Hoboken, N.J.: John Wiley and Sons, Inc.
[5]. Versteeg, H.K. and Malalasekera, W. (1995) An Introduction to Computational Fluid
Dynamics (The Finite Volume Method). 2nd
Ed. Harlow: Pearson Education Ltd.
[6]. NPTEL, IIT Kanpur, Fluid Mechanics, Lecture 25
http://www.nptel.iitm.ac.in/courses/Webcourse-contents/IIT-KANPUR/FLUID-
MECHANICS/lecture-25/25-3_parallel_flow.htm [accessed 01/07/2013]
[7]. Cornell University Library, arXiv.org
http://arxiv.org/ftp/arxiv/papers/0912/0912.5249.pdf [accessed 01/07/2013]
[8]. Hydraulic losses in pipes, Kudela, H.
http://www.itcmp.pwr.wroc.pl/~znmp/dydaktyka/fundam_FM/Lecture11_12.pdf
[accessed 01/07/2013]
[9]. The University of Iowa, College of Engineering, IIHR resources
http://www.engineering.uiowa.edu/~cfd/pdfs/53-071/lab3.pdf [accessed
01/07/2013]

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