UoM_intern_report

The University of Manchester
Internship Report
May 12, 2014 to July 4, 2014
Multiple Solutions in Curved-Pipe Flow
Author:
Harsh Ranjan
Final year undergraduate
Department of Mechanical Engineering
Indian Institute of Technology Guwahati
Guwahati, Assam
India
Supervisor:
Dr. Andrew Hazel
Reader
School of Mathematics
The University of Manchester
Manchester
The United Kingdom
—————————

Disclosure Page
I hereby state and verify that I have reviewed this internship report. I hereby aﬃrm that the
report contains the actual project or assignment that I (or the company I work for) assigned to
this intern.
Supervisor: Dr. Andrew Hazel
Date: Signed:
—————————
1

Preface
I, Harsh Ranjan am a final year undergraduate student enrolled in the B.Tech programme
at Indian Institute of Technology Guwahati, Assam, India in the Department of Mechanical
Engineering.
As per my course requirements, I was supposed to undertake an internship after the com-
pletion of my third year of engineering studies at a reputed university, R&D facility or an
industrial establishment. To fulfil this requirement I undertook my internship at The Univer-
sity of Manchester in the School of Mathematics under the supervision of Dr. Andrew Hazel,
Reader, School of Mathematics, The University of Manchester.
The purpose of this report to provide a comprehensive presentation of the work done during
my internship (12 May, 2014 to 4 July, 2014). The report is organised such that it first begins
with an introduction of the research problem that was explored along with a historical overview
before going to the finer intricacies such as the methodology used to achieve the results, pa-
rameters that were focussed upon, etcetera, before ultimately presenting the results along with
thoughts on future scope of this work.
I have tried my best to keep the report simple yet technically correct. I hope I succeed in
my attempt.
2

Acknowledgements
First and foremost, I would like to express my heartfelt gratitude to my supervisor, Dr. An-
drew Hazel for giving me the opportunity to work on this project and guiding me all the way
through. The fact that this was my ﬁnal and of course, most important research internship at
the undergraduate level, 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, Dr. Hazel was instrumental in getting me abreast with the
needs of my project by 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 fellow interns Mr. Valentin
Foissac, Mr. Jordan Rosso and Miss Narjess Akriche for helping me with my doubts during the
internship and for being the great friends that they were throughout my stay in Manchester.
Even though I am trying 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
Department of Mechanical Engineering
Indian Institute of Technology Guwahati
Guwahati, Assam
India
3

Abstract
The internship dealt with the case of fluid flows in uniformly curved pipes driven by a steady
axial pressure gradient. With particular focus on blood flow in arteries, the curvature parameter
was appropriately set and the cross-section was examined for Dean vortices. The primary
solution involving the formation of two-vortices was found and validated with existing results
and then computations were performed to look for bifurcations.
Bifurcation was found for curved pipes with a square cross-section and attempts were made
to extend this new solution to the circular cross-section firstly, to obtain the four-vortex solution
for the circular cross-section, and secondly, to investigate if the additional solution branches of
different cross-sections were in any way connected to each other. In addition to this, various
parameters like the wall shear stresses, stream function and vorticity were computed for different
curvature ratios and Dean numbers and all of these have been documented in the report.
4

Contents
1 Introduction 6
1.1 Flow in a curved tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 History of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 A note on OOMPH-LIB 9
3 Objectives of the internship 9
4 Flow characterisation 10
4.1 Wall shear stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2 Axial vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.3 Stream function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5 Results and Validation 12
5.1 Axial wall shear stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.1.1 Dean number = 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.1.2 Dean number = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.1.3 Dean number = 2500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.1.4 Quantitative comparison for axial wall shear stress values . . . . . . . . . 13
5.2 Azimuthal wall shear stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2.1 Dean number = 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2.2 Dean number = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2.3 Dean number = 2500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2.4 Quantitative comparison for azimuthal wall shear stress values . . . . . . 14
5.3 Axial Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.4 Axial Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.5 Stream function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.6 Secondary ﬂows in the cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6 The case of square cross-section of torus 19
7 Conclusions and Future work 21
8 Appendix 22
8.1 The coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
8.2 Comparison of scaling scheme used in OOMPH-LIB with that of Siggers and
Waters (2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
List of ﬁgures 24
List of tables 24
5

1 Introduction
1.1 Flow in a curved tube
Fluid flow in a curved tube is distinctly different from that in a straight tube. In the former, due
to its curvature, the fluid near the axis of the tube, which has the highest velocity experiences
greater centrifugal force (ρw2
R , where w is the axial velocity, ρ the density and R the radius of
curvature) than the fluid nearer to the walls of the tube. As a result, the fluid in the center of
the tube is forced to the outside of the curvature. On the other hand, the fluid at the walls on
the outer side of the curve is forced inwards along the walls of the tube owing to the existence
of lower pressure at the inside of the curvature.This leads to the origin of secondary flows in a
curved tube in addition to the axial flow along the tube.
This, however, also affects the axial velocity as then the maximum velocity measured in the
plane of symmetry of the curved tube gets shifted off centre more towards the outside of the
tube. Therefore, in a curved tube, the fluid is constantly moving from the axis of the tube,
where it has higher velocity towards the wall, where the velocity is low, and vice versa, thus
leading to more viscous dissipation than in the case of a straight tube.
A critical parameter in the study of flows in curved tubes is the Dean number which provides
a measure of the importance of inertial and centrifugal forces relative to viscous forces, and
since secondary flows result from the interaction of centrifugal and viscous forces, the Dean
number provides an estimate of their strength. In the numerical investigations carried out, the
Dean number was varied which consequently changed the axial pressure gradient which was
responsible for driving the flow. Increasing this driving force caused the centrifugal force in the
tube to grow and this initiated the secondary flows in the system owing to the no-slip boundary
conditions.
For small values of Dean number the flow field is roughly symmetric about a line through
the centre of the duct parallel to the axis about which the tube is coiled (Figure 1(a)). As
Dean number is increased the axial velocity contours and the secondary flow streamlines tend
to become distorted and the locations of the maximum in axial velocity and of the centres of
the secondary flow vortices move towards the outer wall (Figures 2(a) and (b)). At even larger
values of D, the vortex centres move back toward the inner wall of the tube (Figure 1(b)), with
boundary layers developing near the walls of the pipe while the core appears to be inviscid.
(a) D = 10, δ = 0.3, ReO = 1 (b) D = 2500, δ = 0.3, ReO = 1
Figure 1: Visualisation of secondary flows in the cross-section of tube of torus for two different
sets of parameters.
From the perspective of application, research in this domain can potentially improve our
understanding of blood flow in arteries or air flow in lungs’ air flow vessels or otherwise, general
fluid flow through curved pipes in industries. As a matter of fact, the bulk of the literature
in this field focusses on blood flow in aorta with a focus on making the physical modelling of
6

(a) D = 10, δ = 0.3, ReO = 1 (b) D = 2500, δ = 0.3, ReO = 1
Figure 2: Axial velocity profiles for two different sets of parameters plotted against the radius
measured from the center of torus with the left edge representing the inner boundary and the
right edge representing the outer boundary of the torus’ tube.
the flow as realistic as possible. Since arteries are rarely straight, and have a curvature, it’s
sensible to consider curvature ratios close to the realistic curvature ratio of the aortic arch.
Early research for curved ducts had been conducted with the curvature ratio, δ → 0. However,
the aortic arch has finite curvature with δ normally around 1/4; and therefore most literature
consider δ values close to the same.
Another reason for special focus on the curvature sites of arteries lies in the observation that
in atherosclerosis, which involves the development of plaques on the inside walls of the arterial
circuit, much of the formation of plaques happens near the bends, exhibiting a close relation to
the variation of wall shear stress along the bend. Atherosclerotic plaques are known to develop
more at regions of low wall shear stress or where the wall shear stress changes direction during
the cardiac cycle as these mark the regions of increased residence time for the flow.
In engineering situations, laminar flow rarely occurs because Reynolds numbers are usually
too high, and for this reason research on laminar secondary flow in general, is relatively less.
But in cardiovascular system, flow is usually laminar. Hence the focus on laminar flow in related
literature.
To study the problem well, the modelling of the arterial system needs to be as realistic
as possible. Hence it is important that its geometric and physiological details are taken into
account as much as possible. These may include but are not limited to its curvature, taper,
branching, elasticity and also the pulsatile nature of blood flow.
1.2 Bifurcation
As discussed earlier, the centrifugal forces induced due to the curvature of the pipe lead to the
formation of two vortices which are symmetrical about the horizontal plane cutting through the
centreline of the cross-section of the tube. This two-vortex solution is stable. However, as the
Dean number increases for a given curvature ratio, there occurs a critical Dean number above
which the dual solutions are obtained and the system becomes unstable. The new solution has
four vortices instead of two in the cross-section. This multiplicity of solutions at higher Dean
number, that is higher axial pressure gradient can be credited to the strong nonlinearity of the
corresponding modified Navier Stokes equations. Also, this critical Dean number, Dc ≈ 956 for
circular cross-section for δ = 0. A second critical Dean number above which two more solution
branches appear occurs around 2494 [10].
As for this text, extensive calculations have been performed for δ ∈ {0.1, 0.3} and corre-
sponding results are presented in the subsequent sections. For δ = 0.3, Dc ≈ 1510 and for
δ = 0.1, Dc is slightly less [10].
7

1.3 History of the problem
The peculiarity of pressure-driven laminar flow of a Newtonian fluid through a curved tube was
first investigated theoretically by Dean (1927, 1928) which was essentially a follow-up to an
experimental investigation of the same problem carried by Eustice (1911) which pointed to the
existence of secondary flow in curved tubes. Thus, Dean’s work provided a theoretical backing
to Eustice’s observations and explained the secondary flows which distorted the velocity profile
of the primary flow in the curved tube.
Following Dean’s work, White (1929) further substantiated the theoretical evaluation and
his work helped arrive at how these secondary flows were leading to greater viscous dissipation
in the flow. White made use of Dean’s criterion term- ρwmeand
µ
d
D where ρ is the density of the
incompressible Newtonian fluid, wmean the mean axial velocity of flow, d the diameter of the
tube, µ the dynamic viscosity of the fluid and D the curvature diameter.
Later, Taylor (1929) experimentally investigated the transition of laminar flow to turbulent
flow at high Reynolds numbers, in curved tubes which had been concluded by White, earlier
in his work. Based on White’s work which had focussed on the study of streamlines of flow in
curved tubes, Taylor experimented by introducing coloured fluids through a small hole in the
side of a glass helix which had water running through it. He concluded that a considerably
higher flow rate was necessary than in the case of a straight pipe in order to attain turbulence
in a curved pipe thereby verifying White’s argument of the existence of such a critical flow rate
which had not been considered by Eustice and Dean in their work.
In 1968, McConalogue and Srivastava published their work which was an extension to Dean’s
contribution in the field of steady motion of an incompressible fluid through a curved tube of
circular cross-section. They too made use of the Dean number which they defined as D =
4R 2a
L where R is the Reynolds number, a the radius of cross-section of the tube and L the
curvature radius. In his work, Dean (1927) had shown that up to first order approximation the
relation between pressure gradient and rate of flow is not dependent on the curvature. Later, in
1928, in order to show its dependence he modified the analysis by including terms of higher order
and was able to show that the reduction in flow due to curvature depends on a single variable
K, equal to 2R2( a
L ) with R being the Reynolds number in Dean’s notation, a the radius of the
tube and L the radius of curvature of the bent tube. Dean (1928) showed that his analysis was
reasonably reliable for values of K up to 576. This work was built upon by McConalogue and
Srivastava who carried out investigations for flow having K values in excess of 576. They solved
equations using Fourier series expansions for D ∈ [96, 605.72]. The definition of Dean number
based on K, i.e., D = 4
√
K was used for the smaller values of K as then the mean velocity
was derivable from the mean axial pressure gradient on the lines of Poiseuille flow. However,
for larger values of K, there was considerable deviation from Poiseuille flow and dean number
was obtained directly from the mean axial pressure gradient-
D = (
2a3
ν2L
)
Ga2
µ
,
where G is the mean pressure gradient, ν is the kinematic viscosity and µ the dynamic viscosity.
In quantitative terms, for D = 605.72, McConalogue and Srivastava found that the position of
the maximum axial velocity is reached at a distance less than 0.38 times the radius from the
outer boundary and that the flow is reduced by 28% in comparison to a straight tube.
Collins and Dennis (1981) obtained numerical solutions for the range of D ∈ [96, 5000] and
validated their results with those obtained by McConalogue and Srivastava. Like McConalogue
and Srivastava, they too solved their equations by substituting Fourier series expansions of
stream function, axial velocity and vorticity along with the symmetry assumption,
w(r, −α) = w(r, α), φ(r, −α) = −φ(r, α), Ω(r, −α) = −Ω(r, α),
8

about the horizontal plane cutting across the cross-section where w is the axial velocity, φ is
the stream function and Ω the vorticity with r and α being the dimensionless polar coordinates.
Since then some more extensive studies have been done by Pedley (1980), Berger, Talbot and
Yao (1983), Ito (1987) and Hamakiotes (1986).
The two-vortex solution is the primary solution, while the four-vortex solution, as pointed
out earlier in (1.2), appears at a bifurcation point which, for ducts of circular cross-section with
δ → 0, occurs for D ≈ 956 [10].
The four-vortex flow has been observed and studied experimentally by flow visualization in
rectangular ducts by, notably, Cheng, Nakayama & Akiyama (1979) and in semicircular ducts
by Masliyah (1980). This feature was subsequently described for circular ducts by Dennis & Ng
(1982) and Nandakumar & Masliyah (1982) for values of D > 956; Cheng, Inaba & Akiyama
(1985) verified the numerical predictions experimentally by flow visualization. These studies
did not, however, resolve the issue of how the two- and four-vortex flows are related, although
Nandakumar, Masliyah & Law (1985), in a paper dealing with bifurcation in steady laminar
mixed convection flow in horizontal ducts, pointed out the similarities with the problem of flow
in curved pipes, and suggested that instead of one critical Dean number, there should be a
lower and an upper critical value of the flow parameter, the Dean number. This would define a
region of coexistence of the two solutions, with only a four-vortex flow pattern existing above
the upper critical value and only the two-vortex one below the lower critical value.
2 A note on OOMPH-LIB
OOMPH-LIB is an object-oriented, open-source finite-element library for the simulation of
multi-physics problems, developed and maintained by Prof. Matthias Heil and Dr. Andrew
Hazel of the School of Mathematics at The University of Manchester.
The main aim of the library is to provide an environment that facilitates the monolithic
discretisation of multi-physics problems while maximising the potential for code re-use. This
is achieved by the extensive use of object-oriented programming techniques, including multiple
inheritance, function overloading and template (generic) programming, which allow existing
objects to be (re-)used in many different ways without having to change their original imple-
mentation
OOMPH-LIB’s design is based on a (finite-)element-like framework in which the system
of non-linear algebraic equations arising from the fully coupled discretisation of multi-physics
problems is generated using an element-by-element assembly procedure. The library provides
fully-functional elements for a wide range of ‘classical‘ partial differential equations (the Poisson,
Advection-Diffusion, and the Navier-Stokes equations; the Principle of Virtual Displacements
(PVD) for solid mechanics; etc.) and it is easy to formulate new elements for other, more
‘exotic‘ problems. Furthermore, it is straightforward to combine existing single-physics elements
to create hybrid elements that can be used in multi-physics simulations.
In OOMPH-LIB, the Galerkin Method for weighted residuals is used to solve the equations
using the finite element method and iterations are performed using Newton’s method until the
residuals are sufficiently small.
3 Objectives of the internship
The internship’s objective was to study the bifurcation of flow in a curved pipe subject to rele-
vant parameters like the Reynolds number, dimensionless axial pressure gradient and curvature
ratio (δ). With appropriate values of these parameters, using OOMPH-LIB, various flow char-
acteristics like the wall shear stresses, stream function, axial vorticity and axial velocity in the
cross-section of the tube of torus were studied and also compared for validation whenever cor-
responding results were available in the form of past studies by other researchers whose papers
9

have been gratefully acknowledged in the references section. Most of the validation work was
done as per the paper, ’Steady flows in pipes with finite curvature’, by Siggers and Waters,
published in 2005. In order to be able to compare with their results, the Reynolds number was
set to unity and the dimensional axial pressure gradient was rescaled and varied along with the
Dean number and curvature ratio to obtain the various flow characteristics.
4 Flow characterisation
4.1 Wall shear stresses
Siggers and Waters focus their attention on the axial and azimuthal shear stresses at the walls
of the torus. In local polar coordinates of the cross-section of tube of torus in consideration,
the axial and azimuthal wall shear stress are given as-
(Note that the local polar coordinates in consideration are defined using ρ and θ for the radial
and angular positions respectively, whereas the cylindrical coordinates are defined using r, z and
φ for the radial, vertical, and azimuthal angular position for torus, respectively with the axial,
radial and tangential components of velocity for the former being w‘, u‘ and v‘ while the same
for the latter system being w, u, v. So, r − 1
δ = ρ cos θ and z = ρ sin θ and for the velocities:
w‘ = −w, v‘ = v cos θ − u sin θ, u‘ = u cos θ + v sin θ. This transformation has been illustrated
in Section 8.1).
τaxial = −
dw‘
dρ ρ=1
and τazimuthal = −
dv‘
dρ ρ=1
For the axial wall shear stress-
τaxial = −
dw‘
dρ ρ=1
(1)
=
dw
dρ ρ=1
(2)
=
dw
dr
dr
dρ
+
dw
dz
dz
dρ
(3)
=
dw
dr
cos θ +
dw
dz
sin θ (4)
For the azimuthal wall shear stress-
τazimuthal = −
dv‘
dρ ρ=1
(5)
= −
d(v cos θ − u sin θ)
dρ ρ=1
(6)
= −
dv
dρ
cos θ +
du
dρ
sin θ (7)
= −
dv
dr
dr
dρ
cos θ −
dv
dz
dz
dρ
cos θ +
du
dr
dr
dρ
sin θ +
du
dz
dz
dρ
sin θ (8)
= −
dv
dr
cos2
θ −
dv
dz
sin θ cos θ +
du
dr
sin θ cos θ +
du
dz
sin2
θ (9)
= −
dv
dr
cos2
θ + (
du
dr
−
dv
dz
) sin θ cos θ +
du
dz
sin2
θ (10)
4.2 Axial vorticity
Vorticity is a vector field that describes the local spinning motion of a fluid near some point,
as would be seen by an observer located at that point and travelling along with the fluid. The
10

axial vorticity as defined by Siggers and Waters in the polar coordinate system local to the
cross-section of torus’ tube is,
ζ = −
1
ρ
∂u‘
∂θ
−
∂(ρv‘)
∂ρ
(11)
After making appropriate substitutions for u‘ and v‘ and on changing the coordinate system we
have the following:
ζ = −
1
ρ
[
∂(u cos θ + v sin θ)
∂r
∂r
∂θ
+
∂(u cos θ + v sin θ)
∂z
∂z
∂θ
] −
1
ρ
[
∂(ρ(v cos θ − u sin θ))
∂r
∂r
∂ρ
+
∂(ρ(v cos θ − u sin θ))
∂z
∂z
∂ρ
] (12)
ζ = [(
∂u
∂r
cos θ +
∂v
∂r
sin θ) sin θ − (
∂u
∂z
cos θ +
∂v
∂z
sin θ) cos θ] −
1
ρ
[
∂(r − 1/δ)(v − u tan θ)
∂r
∂r
∂ρ
+
∂(z(v cot θ − u))
∂z
∂z
∂ρ
] (13)
ζ = [(
∂u
∂r
cos θ +
∂v
∂r
sin θ) sin θ − (
∂u
∂z
cos θ +
∂v
∂z
sin θ) cos θ] −
cos2 θ
(r − 1/δ)
(v − u tan θ) − cos2
θ(
∂v
∂r
−
∂u
∂r
tan θ)
− sin2
θ
v cot θ − u
z
+ z sin θ(
∂v
∂z
cot θ −
∂u
∂z
) (14)
4.3 Stream function
The stream function can be used to plot streamlines (lines for which the stream function is
a constant), which represent the trajectories of particles in a steady flow. It is defined for
incompressible (divergence-free) flows in two dimensions (Lagrange stream function), as well as
in three dimensions with axisymmetry (Stokes stream function).
Considering the particular case of fluid dynamics, the difference between the stream function
values at any two points gives the volumetric flow rate (or volumetric flux) through a line
connecting the two points.
Numerically, the stream function can be related to the vorticity as
2
φ = −ζ (15)
The usefulness of the stream function lies in the fact that the velocity components in the x- and
y- directions at a given point are given by the partial derivatives of the stream function at that
point.
In other words, the flow velocity components can be expressed as the derivatives of the
scalar stream function. In terms of the flow velocity components,
u‘
=
1
ρ
∂φ
∂θ
(16)
v‘
= −
∂φ
∂ρ
(17)
11

5 Results and Validation
The subsequent subsections compare the results for wall shear stresses obtained using OOMPH-
LIB with those obtained by Siggers and Waters [10]. Kindly note that the data for the tables
showing the quantitative comparison for the computations performed by Siggers and Waters
was obtained by a regular data-point extraction software from the plots provided by the authors
in their paper. Therefore the minor disagreements between the data provided by them and that
obtained using OOMPH-LIB can be attributed to inaccuracies during data extraction.
5.1 Axial wall shear stress
Axial wall shear stresses for Reynolds number being unity and the curvature ratio(δ) being
0.3(dashed), 0.1(dotted) and 0(solid), as obtained by Siggers and Waters [10] are shown in the
left column and the corresponding results obtained using OOMPH-LIB are shown in the right
column with red and green standing for δ = 0.3 and δ = 0.1, respectively.
5.1.1 Dean number = 10
Figure 3: Comparison of axial wall shear stresses for D = 10 obtained by Siggers and Wa-
ters(left) and OOMPH-LIB(right).
12

5.1.4 Quantitative comparison for axial wall shear stress values
Siggers and Waters (2005) OOMPH-LIB
Dean Number Delta=0.1 Delta=0.3 Delta=0.1 Delta=0.3
10 5.37 6.35 5.40 6.45
100 52.70 55.60 52.53 55.76
2500 1750 1390 1759.45 1383.64
Table 1: Comparison of maximum axial wall shear stresses
5.2 Azimuthal wall shear stress
Azimuthal wall shear stresses for Reynolds number being unity and the curvature ratio(δ) being
0.3(dashed), 0.1(dotted) and 0(solid), as obtained by Siggers and Waters [10] are shown in the
left column and the corresponding results obtained using OOMPH-LIB are shown in the right
column with red and green standing for δ = 0.3 and δ = 0.1, respectively.
Figure 6: Comparison of azimuthal wall shear stresses for D = 10 obtained by Siggers and
Waters(left) and OOMPH-LIB(right).
13

5.2.4 Quantitative comparison for azimuthal wall shear stress values
Siggers and Waters (2005) OOMPH-LIB
Dean Number Delta=0.1 Delta=0.3 Delta=0.1 Delta=0.3
10 0.26 0.28 0.27 0.28
100 23.10 21.75 23.25 21.84
2500 1385 1509 1388.89 1510.99
Table 2: Comparison of maximum azimuthal wall shear stresses
14

5.3 Axial Velocity
(a) D = 10, δ = 0.1, ReO = 1 (b) D = 10, δ = 0.3, ReO = 1
(c) D = 100, δ = 0.1, ReO = 1 (d) D = 100, δ = 0.3, ReO = 1
(e) D = 2500, δ = 0.1, ReO = 1 (f) D = 2500, δ = 0.3, ReO = 1
Figure 9: Contours of axial velocity for steady ﬂow in torus.
15

5.4 Axial Vorticity
(a) D = 10, δ = 0.1, ReO = 1 (b) D = 10, δ = 0.3, ReO = 1
(c) D = 100, δ = 0.1, ReO = 1 (d) D = 100, δ = 0.3, ReO = 1
(e) D = 2500, δ = 0.1, ReO = 1 (f) D = 2500, δ = 0.3, ReO = 1
Figure 10: Contours of vorticity for steady ﬂow in torus.
16

5.5 Stream function
(a) D = 10, δ = 0.1, ReO = 1 (b) D = 10, δ = 0.3, ReO = 1
(c) D = 100, δ = 0.1, ReO = 1 (d) D = 100, δ = 0.3, ReO = 1
(e) D = 2500, δ = 0.1, ReO = 1 (f) D = 2500, δ = 0.3, ReO = 1
Figure 11: Contours of stream function for steady ﬂow in torus.
17

5.6 Secondary ﬂows in the cross-section
(a) D = 10, δ = 0.1, ReO = 1 (b) D = 10, δ = 0.3, ReO = 1
(c) D = 100, δ = 0.1, ReO = 1 (d) D = 100, δ = 0.3, ReO = 1
(e) D = 2500, δ = 0.1, ReO = 1 (f) D = 2500, δ = 0.3, ReO = 1
Figure 12: Velocity vectors for secondary ﬂows in torus’ cross-section.
18

6 The case of square cross-section of torus
Following Nandakumar and Masliyah’s work [8] the square cross-section of torus was investigated
in an attempt to arrive at the four-vortex solution. Some key papers which were followed in
this area were those by Winters [12], Werner [5] apart from Nandakumar and Masliyah. The
common observation recorded by some of the papers was that the four-vortex solution was easier
to obtain for certain specific cross-sections than some other geometries. Those shapes which
favoured this new solution branch more than the circular cross-section include the semi-circular
cross-section and the square cross-section. This is because of the fact that the bifurcations in
the solutions for these ’easier’ cross-sections have been found to be connected to the primary
branch, therefore making it easier to arrive upon the additional solutions from the primary
branch unlike the circular cross-section where the additional branches have been found to be
disconnected from the primary branch. In order to make this transition from the primary
solution branch on to the new solution branches, appropriate perturbations in the form of say,
an external force were used.
So, upon constant failures to reach the four-vortex solution for the circular cross-section,
and motivated by the above conclusion put forward by former researches, the square shape was
investigated for the four-vortex solution. For validation, the results obtained by Werner [5] were
used.
The idea was to first arrive at the four-vortex solution for the square cross-section and then
use this solution to potentially arrive upon the same for the circular cross-section by forcing a
change in the geometry of the cross-section, iteratively.
Figure 13: The coordinate system for the case of the square cross-section.
The above figure (Figure 13) illustrates the case of the square cross-section of torus along
with the necessary geometric considerations. Werner performed computations for the particular
cases of γ(= B/A) = 1 and γ = 1.45 for rectangular cross-section and then moved on to compute
results for tori with elliptic cross-section. Here, only the case of the square cross-section was
considered, that is, for γ = 1. Then the result was validated with that of Werner’s by comparing
19

the primary solution branch through a plot of the central axial velocity versus the axial pressure
gradient (q) which was defined as five times the Dean number (Figure 14).
Figure 14: The bifurcation diagram for square cross-section of torus as obtained by Werner(left)
and the primary solution branch obtained using OOMPH-LIB for the same(right).
Thereafter, the solutions computed at various Dean numbers (or driving pressure gradients)
were checked for bifurcation using the continuation method. Here, the continuation method
offered a distinct advantage over the newton method as here the aim was to look for bifurcation
in the solution and not to find the solution for a pre-determined Dean number (or pressure
gradient). For the case of the continuation method, if the residuals are found to be blowing up,
then the steps are automatically adjusted and the corresponding residuals are checked again for
convergence which is not the case in a newton solver.
The various ranges for which a bifurcation was found were tested with a perturbation for
the possibility of a four-vortex solution. This perturbation was achieved by adding a parabolic
velocity profile in the vertical direction (blowing). The various bifurcation points do not all
necessarily imply the existence of a four-vortex solution. Therefore all the ranges had to be
tested with perturbations in the hope to push the solution on to the new branch emanating
from the bifurcation point and then results were checked for the existence of four vortices.
(a) γ = 1, ReO = 1 (without perturbation) (b) γ = 1, ReO = 1 (with perturbation)
Figure 15: (a)Primary solution branch obtained for the square cross-section and, (b)the new
solution branch obtained after adding an appropriate perturbation.
Also, the perturbations had to be varied as it could not be pre-determined as to what would
be the right level of perturbation that would be able to push the solution from the primary
branch on to the new branch. For this purpose, the amplitude of blowing was varied and the
results were checked to see if the solution had been able to reach the new branch and then of
20

course, if this new branch was indeed the four-vortex solution!
On performing the computations again, but this time with an appropraite perturbation, for
a particular range of Dean number (starting from ≈ 700) the solution was found to jump on to
a new solution branch which was then identified as the four-vortex solution. This new solution
branch is compared above with the primary solution branch which was achieved without any
perturbation (Figure 15).
Also, a graphical comparison of the recirculation pattern for the two-vortex solution is shown
below with that observed for a four-vortex solution which was obtained for a Dean number of
1363.16 which was slighlty higher up in this new solution branch (Figure 16).
(a) D = 518.822, γ = 1, ReO = 1 (b) D = 1363.293, γ = 1, ReO = 1
Figure 16: (a)The primary, two-vortex solution obtained for a Dean number of 518.822 for the
unperturbed case and, (b)the four-vortex solution obtained for a Dean number of 1363.293 for
the perturbed case.
7 Conclusions and Future work
The difficulty associated with finding bifurcations in the case of the circular cross-section is due
to the absence of any connection between the primary solution branch and the additional solu-
tion branches. Hence, the bifurcation trackers which look for changes in the sign of the jacobian
matrix of the system are unable to work in this case. In the case of the rectangular/square
cross-section, these solution branches have been found to be connceted and thus bifurcation
detection was possible.
So, after being unsuccesful in reaching the additional solution branches for the four-vortex
solution, the idea was to use the four-vortex solution obtained for the square cross-section and
then force a change in the shape of the cross-section and make it change to a circle, iteratively
in the hope of retaining the four vortices while the change in the shape was being made. Had
this method been found to yield expected results, this would have implied a possible connection
between the additional branches of solutions of two different cross-sections: a result which
hasn’t been discussed in earlier researches. Unfortunately, this idea did not materialise during
the internship and hence, remains an unverfied possibility. So, clearly, for future work, the
current level of investigation offers plenty of scope. It is not just the four-vortex solution for the
circular cross-section which remained elusive during the internship and which needs attention,
but also the peculiarity of the disconnected bifurcations in the case of the circular cross-section
is another area which begs for more careful studies. The various potential explanations that
could be put forward to explain the disconnected bifurcation in the circular cross-section and
sheer volume of interesting possibilities make this field an exciting domain for research and offer
immense scope for future investigation.
21

8 Appendix
8.1 The coordinate system
Figure 17: The curvilinear coordinate system
For validation of results obtained using OOMPH-LIB with those of Siggers and Waters [10],
it was important that the coordinate system defined in the OOMPH-LIB code be adjusted
accordingly.
The OOMPH-LIB code used the cylindrical coordinate system with its centre based at
the centre of torus while Siggers and Waters in their paper, ’Steady flows in pipes with finite
curvature’ used the toroidal coordinate system. Both systems have been shown in Figure 17.
Note that the radius of torus is defined as R while the radius of the cross-section of tube of
torus is defined as a which was taken as unity. The two coordinate systems were related as
follows:
r = a + ρ cos θ (18)
z = ρ sin θ (19)
φ = arctan(z/r−a) (20)
Also, the velocities were related as:
u‘
= u cos θ + v sin θ (21)
v‘
= v cos θ − u sin θ (22)
w‘
= −w (23)
22

8.2 Comparison of scaling scheme used in OOMPH-LIB with that of Siggers
and Waters (2005)
(Note the use of subscript SW as a reference to Siggers and Waters’ paper and the subscript O
for the equivalent term in OOMPB-LIB)Let the dimensional form of the body force be G∗. As
per the non-dimensionalisation scheme used by Siggers and Waters, its non-dimensional form
is-
GSW = −
G∗
ρU2
SW
a
(24)
Using USW = ν
a ,
GSW = −
G∗
ρν2
a3
(25)
⇒ GSW = −G∗ a3
ρν2
(26)
Following the deﬁnition of Reynolds number in OOMPH-LIB, we have
ReO =
ρUOa
µ
(27)
⇒ UO =
ReOµ
ρa
(28)
Now, using the non-dimensionalisation scheme of OOMPH-LIB, the non-dimensional body force
is given as
G = G∗ a2
µUO
R
r
(29)
where R is the radius of torus and r is the dimensional coordinate in the radial direction for
the polar coordinate system local to the cross-section of the tube. Substituting equation (5) in
equation (6) and non-dimensionalising the R and r terms individually, with a, the radius of the
torus’ tube-
G = G∗ a2
µReOµ
ρa
R/a
r/a
(30)
Since a
R = δ and deﬁning ˆr = r
a, that is, the non-dimensional radial coordinate in the polar
coordinate system local to the cross-section of the tube we can say,
G = G∗ a3ρ
µ2ReO
1
δˆr
(31)
⇒ G = G∗ a3
ρν2ReO
1
δˆr
(32)
On using equation (10) to substitute the value of G∗ in equation (3), we have the following:
G = −
GSW
ReOδˆr
(33)
Using the expression for Dean number in Siggers and Waters (2005), D = GSW
√
2δ, we have
G = −
D
ReOδˆr
√
2δ
(34)
In order to properly compare with the results obtained by Siggers and Waters (2005), all
computations were performed with ReO = 1 and so, the non-dimensional pressure gradient
varied as,
G = −
D
δˆr
√
2δ
. (35)
23

List of figures
1 Visualisation of secondary flows in the cross-section of tube of torus for two
different sets of parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Axial velocity profiles for two different sets of parameters plotted against the
radius measured from the center of torus with the left edge representing the
inner boundary and the right edge representing the outer boundary of the torus’
tube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Comparison of axial wall shear stresses for D = 10 obtained by Siggers and
Waters(left) and OOMPH-LIB(right). . . . . . . . . . . . . . . . . . . . . . . . . 12
6 Comparison of azimuthal wall shear stresses for D = 10 obtained by Siggers and
7 Comparison of azimuthal wall shear stresses for D = 100 obtained by Siggers and
8 Comparison of azimuthal wall shear stresses for D = 2500 obtained by Siggers
and Waters(left) and OOMPH-LIB(right). . . . . . . . . . . . . . . . . . . . . . . 14
9 Contours of axial velocity for steady flow in torus. . . . . . . . . . . . . . . . . . 15
10 Contours of vorticity for steady flow in torus. . . . . . . . . . . . . . . . . . . . . 16
11 Contours of stream function for steady flow in torus. . . . . . . . . . . . . . . . . 17
12 Velocity vectors for secondary flows in torus’ cross-section. . . . . . . . . . . . . . 18
13 The coordinate system for the case of the square cross-section. . . . . . . . . . . 19
14 The bifurcation diagram for square cross-section of torus as obtained by Werner(left)
and the primary solution branch obtained using OOMPH-LIB for the same(right). 20
15 (a)Primary solution branch obtained for the square cross-section and, (b)the new
solution branch obtained after adding an appropriate perturbation. . . . . . . . . 20
16 (a)The primary, two-vortex solution obtained for a Dean number of 518.822 for
the unperturbed case and, (b)the four-vortex solution obtained for a Dean num-
ber of 1363.293 for the perturbed case. . . . . . . . . . . . . . . . . . . . . . . . . 21
17 The curvilinear coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . 22
List of tables
1 Comparison of maximum axial wall shear stresses . . . . . . . . . . . . . . . . . . 13
2 Comparison of maximum azimuthal wall shear stresses . . . . . . . . . . . . . . . 14
24

References
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25

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