Extended Stokes Series for Dean Flow in Weakly Curved Pipes

EXTENDED STOKES SERIES FOR DEAN FLOW IN WEAKLY CURVED PIPES

A thesis presented for the degree of Doctor of Philosophy of Imperial College by

Florencia Amalia Tettamanti

Department of Mathematics Imperial College 180 Queen’s Gate, London SW7 2BZ

June 24, 2012 2

I certify that this thesis, and the research to which it refers, are the product of my own work, and that any ideas or quotations from the work of other people, published or otherwise, are fully acknowledged in accordance with the standard referencing practices of the discipline.

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Abstract

This thesis considers steady, fully-developed ﬂows through weakly curved pipes using the extended Stokes series method.

The Stokes series for pipes of circular cross-section is expanded in powers of the Dean number, K, to 196 terms by computer. Analysis shows that the convergence is limited by an imaginary conjugate pair of square-root singularities K = iK . ± c Contrary to previous analysis of this solution, analytic continuation of the series indicates that the ﬂux ratio in a weakly curved pipe does not vary asymptotically as K−1/10 for large K.

Using generalised Pad´eapproximants it is proposed that the singularity at iKc corresponds to a symmetry breaking bifurcation, at which three previously unreported complex branches are identiﬁed. The nature of the singularity is supported in part by numerical consideration of the governing equations for complex Dean number.

It is postulated that there exists a complex solution to the governing equations for which the azimuthal velocity varies asymptotically as K−1/2, and the streamfunction as K0 near K =0. This is supported by the results from the generalised Pad´e approximants.

Brief consideration is given to pipes of elliptic cross-section. The Stokes series for pipes of elliptic cross-section for various aspect ratios, λ, is expanded up to the K24 term by computer. For small K, it is found that the ﬂux ratio achieves a minimum for aspect ratio λ 1.75. This, and the behaviour of the total vorticity, is in agreement with previous ≈ studies which found that the effect of the curvature is reduced in the limit of small and large aspect ratios. The convergence of the series solution is found to be limited by an imaginary conjugate pair of square-root singularities K = iK (λ), which varies with λ. ± c 6

Acknowledgements

I would like to thank my supervisor, Jonathan Mestel, for introducing this interesting problem in ﬂuid dynamics to me. His insight, patience and guidance over the last 3 years have been invaluable. I would like to thank Jennifer Siggers for access to her code and guidance in its application. I am grateful also to Slobodan Radosavljevic for sharing his work on this problem with me.

Work aside, thanks to all my friends for providing much needed distractions and support throughout my thesis.

This research was supported by funding from the Mathematics Department of Imperial College London. 7

Table of contents

Abstract 5

1 Introduction 13 1.1 Motivation-thephysicalproblem ...... 13 1.2 Motivation-thetheoreticalproblem ...... 19 1.2.1 DeanEquations...... 20 1.2.1.1 Non-dimensionalisation & boundary conditions . . . .. 22 1.2.2 SolutionstotheDeanequations ...... 24 1.2.2.1 SmallDeannumber ...... 24 1.2.2.2 IntermediateDeannumber ...... 25 1.2.2.3 LargeDeannumber ...... 29 1.2.3 Alternativecross-sections ...... 31 1.3 Aimsofthethesis...... 32 1.4 Organisationofthethesis ...... 33

2 Extended Stokes series: circular cross-section 35 2.1 Computer-extendedseries...... 35 2.2 Radiusofconvergence ...... 40 2.2.1 Domb-Sykesanalysis...... 41 2.2.2 Results ...... 45 2.3 Analyticcontinuation ...... 50 2.3.1 Eulertransform...... 50 2.3.2 Results ...... 52 2.4 Comparisonwithnumerics ...... 61 2.4.1 Flux ratio, total vorticity and outer wall shear stress...... 62 2.4.2 Azimuthalvelocityandstreamfunction ...... 64 2.5 Summaryandconclusions ...... 66

3 Pade´ approximants 67 3.1 GeneralisedPad´eapproximants-procedure ...... 67 3.2 Results...... 70 3.2.1 Real K ...... 71 3.2.1.1 Anillustrativeaccount-theﬂuxratio ...... 71 3.2.1.2 Discussionof results and comparisonwith other techniques 74 3.2.2 Imaginary K ...... 76 8

3.2.2.1 Anillustrativeaccount-theﬂuxratio ...... 76 3.2.2.2 Discussionof the complexbifurcation diagram for imaginary K ...... 81 3.2.2.3 Comparison with numerics of Branch I and II ...... 86 3.2.2.4 Asymptotic analysis of Branch II ...... 86 3.3 Summaryandconclusion ...... 88

4 Numerical solution of the Dean equations for complex K 91 4.1 The complex K ﬁnitecurvatureDeanequations ...... 91 4.2 Fictitioustimescheme ...... 93 4.2.1 Overviewoftheprocedure ...... 94 4.2.2 Results ...... 95 4.3 Path-continuation ...... 97 4.3.1 Overviewoftheprocedure ...... 97 4.3.1.1 Simplecontinuation ...... 99 4.3.1.2 Pseudo-arclengthcontinuation ...... 100 4.3.2 Results ...... 102 4.4 Summaryandconclusion ...... 106

5 Extended Stokes series: elliptic cross-section 109 5.1 Governingequations ...... 109 5.2 Asymptotic behaviour for large and small λ ...... 113 5.3 Computer-extendedseries...... 115 5.4 Radiusofconvergence ...... 118 5.5 GeneralisedPad´eapproximants...... 119 5.6 DiscussionoftheextendedStokesseries ...... 120 5.6.1 Contours of azimuthal velocity and streamfunction ...... 120 5.6.2 Cross-sectional velocities for small λ ...... 121 5.6.3 Azimuthal velocity for large λ ...... 123 5.6.4 Totalvorticity...... 124 5.6.5 Fluxratio ...... 127 5.7 Summaryandconclusion ...... 129

6 Final Conclusions 131 6.1 Summaryofkeyresultsandpossibleextensions ...... 131 6.2 Conclusionsandfuturework ...... 133

References 145

A Introduction 146 A.1 Boundarylayerasymptotics ...... 146

B Extended Stokes series: circular cross-section 149 B.1 Expressionfortherecurrencerelation ...... 149 B.1.1 Expanding ωn (r, θ) and ψn (r, θ) ...... 151 B.2 CoefﬁcientsoftheEulertransformedseries ...... 157 9

B.3 Revisiting Van Dyke’s work as K ...... 157 B.3.1 Completingtheseries. . .→∞ ...... 158 B.3.2 Comparison with ln(1 x)α ...... 159 B.3.3 Critical-pointrenormalisation− ...... 159 B.4 ‘Conﬂuent-singularityanalysis’ ...... 160 B.5 Alternativeextrapolationtechniques ...... 162

C Pade´ approximants 166 C.1 LinearPadéapproximants ...... 166 C.1.1 Results ...... 167 C.2 QuadraticPadéapproximants...... 170 C.2.1 Results ...... 170 C.3 Generalised Padéapproximants: total vorticity Branch II ...... 172 C.4 Generalised Padéapproximants: path-continuation ...... 174

D Numerical solution of the Dean equations for complex K 178 D.1 Path-continuation ...... 178

E Extended Stokes series: elliptic cross-section 180 E.1 Derivation of the Dean equations for λ< 1 ...... 180 E.2 Dean equations for λ> 1 ...... 183 E.3 Computer-extendedseries...... 184 E.4 Calculationoftheﬂuxratio...... 188 E.5 Calculationofthetotalvorticity ...... 191 E.6 Cuming’sscaling ...... 192 10

List of Figures

1.1 Pictorial representation of pressure and ﬂow over the cardiaccycle...... 16 1.2 Sketch of streamlines of the azimuthal and secondary velocity in curved pipes...... 17 1.3 Sketch of thestreamlinesof thesecondary velocityin curved pipes as found byLyne[49]...... 18 1.4 Coordinate system for curved pipe with circular cross-section...... 20 1.5 Dean’s series solution: azimuthal velocity and secondary streamlines contours...... 25 1.6 Contour plots: two-vortex solution from Siggers & Waters[73]...... 26 1.7 Schematic representation of bifurcation diagram for real K ...... 27 1.8 Contour plots: four-vortex solution from Siggers & Waters[73]...... 28

2.1 Domb-Sykes plot K-series: flux ratio, friction ratio, outer wall shear stress andtotalvorticity ...... 48 2.2 GraphicalrepresentationoftheEulertransform ...... 50 2.3 Domb-Sykes plot ǫ-series: flux ratio, friction ratio, total vorticity and outer wallshearstress...... 54 2.4 Domb-Sykesplotformodelfunctions ...... 57 2.5 Domb-Sykes plot for flux ratio Q(ǫ) ...... 60 2.6 Domb-Sykesplotfor the coefficients of theseries expansion of (1 ǫ)−1/20Q (ǫ) 61 − 2.6 Comparison of the flux ratio, total vorticity and outer wall shear stress ǫ- serieswithnumerics ...... 63 2.7 Comparison of contour plots of ǫ-serieswithnumerics ...... 65

3.1 Schematic representation of bifurcation diagram for imaginary K ..... 70 3.2 Generalised Padéapproximant Branch I: example of spurious behaviour for flux ratio for real K ...... 73 3.3 Generalised Padéapproximant Branch I: flux ratio for real K ...... 73 3.4 Generalised Padéapproximant Branch I: Comparison with numerics and ǫ series...... 76 3.5 Generalised− Padéapproximant Branch I and II: example of spurious behaviour of flux ratio for imaginary K ...... 79 3.6 Generalised Padéapproximants Branch I and II: flux ratio for imaginary K 79 3.7 Generalised Padéapproximant Branch III : flux ratio for imaginary K ... 81 3.8 Generalised Padéapproximant Branch I and II: flux ratio, total vorticity, outer wall shear stress imaginary K ...... 84 LIST OF FIGURES 11

3.9 Generalised Padéapproximant Branch III and IV: flux ratio, total vorticity, outer wall shear stress for imaginary K ...... 85 3.10 Generalised Padéapproximant Branch I and II: comparison with numerics for imaginary K ...... 86 3.11 Generalised Padéapproximant Branch II asymptotics: log-log plot of flux ratio for imaginary K ...... 88

4.1 Flux ratio on Branch I: comparison of ﬁctitious time numerics (TNS) and generalised Pad´eapproximants (GPA) for real and imaginary K ...... 96 4.2 Depictionofsimplecontinuation ...... 99 4.3 Depiction of pseudo-arclength method for path-continuation ...... 101 4.4 Contour plotsof w, ψ and Ω on Branch I for K =0.4i M = 200, κ = 15 . 104 4.5 Contour plots of w, ψ and Ω on Branch II for K = 0.82921i, M = 200, κ = 15 ...... 105

5.1 Coordinate system for curved pipe with elliptical cross-section with small λ 1 ...... 112 5.2 Generalised≤ Pad´eapproximant: elliptic cross-section ...... 119 5.3 Contour plot of azimuthal velocity and streamfunction for λ =0.8 . . . . .120 5.4 Comparing secondary velocity on central axis for varying λ ...... 122 5.5 Comparing azimuthal velocity on center-line for varying λ ...... 123 5.6 Comparing the total vorticity calculated by Cuming [19] and the extended Stokesseries(ESS) ...... 125 5.7 Asymptotic behaviour of the total vorticity for varying λ and small K . . .126 5.8 Comparing the ﬂux ratio calculated by Srivastava [79] and the extended Stokesseries(ESS) ...... 128

B.1 Completingtheseries: ﬂuxandfrictionratio ...... 158

C.1 Linear Padéapproximants: estimate of singularities forthefluxratio . . . . 168 C.2 Linear Padéapproximants: estimate of asymptotic behaviour ...... 169 C.3 Quadratic Padéapproximant: plot of the flux ratio ...... 173 C.4 Generalised Padéapproximant Branch I and II: total vorticity for imaginary K ...... 174 C.5 Depictionofturningpointbifurcation ...... 175 C.6 Depictionoftranscriticalbifurcation ...... 175

D.1 Block matrix structure for Jacobian of Dean equations ...... 179

E.1 Coordinate system for curved pipe with elliptical cross-section with large λ 1 ...... 184 E.2 Cuming’s≥ coordinate system for curved pipe with ellipticcross-section . . . 192 12

List of Tables

2.1 Coefficients of K - series: flux ratio, friction ratio, total vorticity and outer wallshearstress ...... 46 −2 2.2 Neville table estimates of Kc : flux ratio, friction ratio, total vorticity and outerwallshearstress ...... 49 2.3 Bottom Row of the Neville Table for (reading left to right), the flux ratio and friction ratio ǫ-series ...... 53 2.4 Neville table estimates of the critical exponent of ǫc:fluxratio ...... 55 2.5 Dominant singularity and corresponding asymptotic behaviour of the ratio of the coefficients of model functions (2.50)- (2.53) ...... 56

3.1 Generalised Pad´eapproximant: estimates of Kc foundusingﬂuxratio . . . 77

5.1 Values of 0 and λ for which the extended Stokes series has been calculated. 116 5.2 Estimateof Kc for various λ ...... 118 5.3 Valueof x˜max for various λ ...... 121 B.1 Coefﬁcients of ǫ - series: ﬂux ratio, friction ratio, total vorticity and outer wallshearstress ...... 157

C.1 Quadratic Pad´eapproximants: estimate of nearest singularity using the ﬂux ratio,totalvorticityandouterwallshearstress ...... 171 13

Chapter 1

Introduction

The study of flow in curved pipes is a classical problem in fluid dynamics, motivated by its prevalence in many physiological and industrial systems. Herein, the method of computer- extended Stokes series is used to model steady, fully-developed flow through weakly curved pipes. By way of introduction, a brief overview of the physical and theoretical motivations for considering this problem and review of past work are given below. The chapter concludes with an overview of the structure of the thesis.

1.1 Motivation - the physical problem

The study of ﬂuid ﬂow in curved and coiled pipes has been driven by the practical importance of many industrial and physiological applications.

In industry, curved and coiled pipes can be found in heat engines, pipe-line transport, chemical reactors and other such industrial equipment. Particularly relevant effects of the curvature of the pipe on the flow include: (i) migration to the outside bend of the maximal azimuthal velocity (ii) an increase in the pressure drop as compared to that in a straight pipe of the same length [22, 29]; (iii) an increase in the heat-transfer rate [59, 60], which is important in cooling systems; (iv) an increase in the residence time distribution [55], which is important in chemical reactions and (v) a delay in the onset of turbulence compared to a straight-pipe [82]. In particular, in a coiled pipe with a radius of curvature 18 times that of 1.1 Motivation - the physical problem 14 its cross-section, the flow is laminar up to a Reynolds number∗ of 5830, that is almost three times the Reynolds’ criterion for instability in a straight pipe [82]. Whilst, accordingly, curved pipes can be used in industrial applications to improve flow characteristics, they are generally found in practical situations due to spatial constraints. For example, laminar flow through curved pipes occurs in the circulatory system of organisms. Of particular interest in physiological fluid dynamics is the modelling of arterial blood flow [67]. Recent studies have focused on understanding the effect of the blood flow in the development of cardiovascular disease, specifically arterial atherosclerosis [73, 74].

Atherosclerosis is a progressive disease, characterised by plaque formation within the arterial wall. In addition to narrowing the arterial lumen, plaques may rupture causing thrombo- embolism and distal tissue ischemia. Common examples include miocardial infarction and cerebral vascular accidents, which are the result of coronary and cerebral artery occlusion respectively. Coronary artery atherosclerosis alone is the cause of one third of all deaths in the Western world [92].

Though the mechanism by which atherosclerosis is formed is not fully understood, empirical studies have concluded that the plaques predominate in areas of signiﬁcant curvature and bifurcation within the arterial system, which are associated with ﬂow disturbance. Such areas include the aorta, coronary artery and carotid bifurcation [12, 31, 48]. It is widely accepted that the spatial distribution of atherosclerosis is correlated with areas of low mean wall shear stress and regions where the wall shear stress changes direction, both of which contribute to the stasis of atherogenic material [68]. Consequently, much interest has been given to modelling the aorta, and the cardiovascular system in general, with the aim of providing insight into the sites of plaque development.

There are two main approaches to modelling these flows. The first aims to replicate the system as accurately as possible, using empirical and anatomical data, and studies the fluid flow by full computational fluid dynamic simulations. This procedure has become particularly attractive in recent years due to advances in computational power, refinement of numerical schemes and medical imaging. Reviews of computational modelling in the cardiovascular system are given by Steinman [80], Steinman et al. [81] and more recently

∗Using the well-known Reynolds number aw/ν¯ , where a is the radius of the cross-section, w¯ is the mean velocity ﬂow and ν is the kinematic viscosity. Chapter 1. Introduction 15

Hoskings & Hardmand [41]. The second approach is to idealise the system by, for example, simplifying the geometry and form of the driving pressure gradient. This leads to a reduced set of equations that can be investigated, if not solved, by mathematical methods (which may of course include computational work). A review of this second approach is given by Pedley [68].

The idealised problem is studied in order to gain insight into the physical one. Accord- ingly, the simplifying assumptions made must have some physical basis. Returning to the discussion of the aortic arch, below its physical attributes and some simpliﬁcations of these are explored.

First, the rheology of aortic blood is considered. Strictly speaking, blood is an incompressible non-Newtonian fluid - its effective viscosity increases as the stress decreases. This is because blood is a suspension compromising of 40-50% deformable particles, including red blood cells (erythrocytes), white blood cells (leucocytes) and platelets (thrombocytes), and up to 55% of plasma, a colourless Newtonian fluid. In arteries however, where the particle size is appreciably small in comparison to the dimensions of the channel, this deviation is not so marked. Accordingly, arterial blood can be, and is usually, modelled as a Newto- nian fluid. Consequently, its motion can be described by the Navier-Stokes equations. Of course, it is also incompressible.

Furthermore, mathematical models of the arterial system generally neglect entrance effects. As arteries are long compared with their diameter and have a regular cross-section, the assumption of fully-developed flow is perhaps reasonable. In reality however, the flow in the aortic arch is affected by the profile of the flow as it leaves the left ventricle.

The flow in the cardiovascular system is driven by an unsteady pressure gradient, generated by the pumping of the heart. There are two stages to the cardiac cycle shown in Figure 1.1. In the first stage, known as systole, the heart contracts and ejects blood from the heart into the major arteries. During this stage the pressure in the system increases. In the second stage, known as diastole, the heart is relaxed and once more fills with blood. The flow in this final stage is approximately constant. In vivo measurements of the physiological pressure gradients can be used to calculate the flow waverform exactly [80]. However, more simply, the oscillatory behaviour of the cardiac cycle can be captured by a sinusoidal driving pressure gradient. Alternatively, a much simpler model would consider a constant 1.1 Motivation - the physical problem 16 driving pressure gradient. Though this final simplification is perhaps more suitable to areas further away from the heart than the aortic arch.

65 100 pressure 90 pressure (mmHg) flow (m/s) flow 0 flow 70

Systole (0.2 s) Diastole (0.4 s) Figure 1.1: Pictorial representation of pressure and ﬂow over the cardiac cycle.

The geometry of the vessel is also another important aspect of the flow. The curvature of the aortic arch is non-planar, with the radius of the vessel 0.3 times the size of its radius of curvature [11]. Mathematically, the simplest such curve is that of a helical pipe. Assuming the torsion to be small, one could consider the aortic arch as a curved pipe of finite curvature or even weak curvature. This latter assumption, provides a significant mathematical simplification though perhaps is not the most suitable, given that the aortic arch is highly curved. Furthermore, the walls of the vessel are compliant. However, as the diameter in large arteries only varies by up to 5% over the cardiac cycle [57], it is not unreasonable to model the aorta as a rigid pipe.†

It follows from the above considerations that suitable curved-pipe models for the aortic arch, of varying complexity, can be considered. Importantly, the analysis of the different idealised problems can provide physical insight into the underlying mechanisms and effects of individual components in the ﬂow. To conclude the discussion on the physical aspects of this problem a very brief overview of some of these is given. The focus of which is to highlight the essential differences of the characteristic ﬂow in each model. Much of the information for this is drawn from the review by Pedley [68].

Steady, fully-developed ﬂow in a rigid, weakly curved pipe. The ﬁrst theoretical study of

†A recent analytic study on compliant curved pipes was completed by Payvandi [66], which also includes an extensive review of previous analytic and numerical work. Chapter 1. Introduction 17

flow in a curved pipe was completed by Dean [21, 22]. He showed that the flow depends on 2 a single dimensionless parameter, known as the Dean number, K = (a/288L) (aw0/ν) where a is the radius of the pipe, L is the radius of curvature, ν is the kinematic viscosity and w0 is the maximal velocity in a straight pipe driven by the same pressure gradient. He expanded the velocity field in powers of the Dean number. Characteristically, in addition to the down-pipe stream there is a secondary flow in the cross-section of the pipe. This occurs as the pressure gradient induced to balance the centrifugal force generated by the curvature of the pipe is smaller near the wall. Consequently, the fluid in the core is swept to the outside bend and the fluid near the wall to the inside bend. Therefore, the secondary flow takes the form of two counter-rotating vortices. This coupled with the main down-pipe stream results in a helical motion of the fluid in the upper and lower halves of the pipe. The contour plot is depicted in Figure 1.2. Some of the effects of this secondary flow have been discussed previously, in the context of industrial applications. More details of this problem will be given in Section 1.2.

Figure 1.2: Sketch of streamlines of the azimuthal and secondary velocity in curved pipes.

Unsteady, fully-developed flow in a rigid, weakly curved pipe. Lyne [49], extended Dean’s work to flow driven by a high-frequency sinusoidal pressure gradient using matched asymptotics. There are three parameters which characterise unsteady flow in a curved pipe: the Dean number, as in the steady case, the Womersley number α = (ωa2/ν)1/2, where ω is the frequency of the oscillation and; the secondary streaming Reynolds number defined as 1.1 Motivation - the physical problem 18

2 Rs = w0a/Lων, which acts as a Reynolds number for the steady flows that are generated in the secondary circulation. Lyne considered a zero mean pressure gradient, so that the Dean number vanishes. In the limit of large α (> 12.9), for both small and large streaming Reynolds number, he found the flow to be composed of two regions: (i) the boundary layer, in which viscous effects dominate and the secondary flow is a vortex driving fluid from the outside to the inside of the bend, similar to vortex found in the flows driven by steady pressure gradient; (ii) the core region, which is essentially inviscid, where the secondary flow is a vortex rotating from the inside to the outside of the bend formed by the drag on the core fluid of the vortex in the boundary layer. This four-vortex structure is sketched in Figure 1.3.

Figure 1.3: Sketch of the streamlines of the secondary velocity in curved pipes as found by Lyne [49].

For α < 12.9, Lyne postulated that a two-vortex, Dean-like ﬂow would exist. This hypothesis appears to be conﬁrmed by the experimental work of Bertelsen [8] and Munson [61].

Other important analytical work on this problem are that of Blennerhasset [9] and Smith [75]. Blennerhasset considered the regime in which the magnitude of the secondary streaming found by Lyne is comparable to the secondary flow found by Dean, K R , in the ≤ s high Wormesley number limit. For small K he found the secondary flow was towards the inside wall. For large K he found the secondary flow was towards the outside wall. Smith studied several cases which arise when Rs is of order unity and one of the other parameters α and K takes a large or small value. Chapter 1. Introduction 19

Unsteady / steady, fully-developed flow in a rigid, finitely curved pipe. The studies discussed above considered weakly curved pipes. For pipes of finite curvature, Siggers & Wa- ters considered the flow driven by a steady [73] and an unsteady pressure gradient [74]. For small and large parameter values, curvature and Dean number, solutions were determined by asymptotic analysis, whilst numerical methods were employed for the intermediate parameter values. The features of this flow were qualitatively similar to the weak curvature limit however, the effects of finite curvature leads to significant quantitative difference. For instance, in the steady case, they found that there was a significant increase in the wall shear stress at the inner wall compared to the weakly curved pipe predictions.

Unsteady / steady, fully-developed flow in a rigid, helical pipe. Relatively recently, the effect of torsion has been considered by Zabielski & Mestel, who considered flow driven by a steady [97] and an unsteady pressure gradient [98] in a helical pipe. Notably, in the former study, they found that torsion caused a transition from a symmetric one-vortex secondary flow, to a symmetric two-vortex Dean-like flow. The latter study included a study of physiological pressure pulse, in addition to the more common sinusoidal pressure gradient of Lyne. Significantly, they found that torsion may result in higher shear stresses localised at the inner wall when compared with toroidal pipe flows. They suggest that the helical geometry may play a beneficial role in the macro-circulation and so inhibit the formation of atherosclerotic lesions.

This thesis is concerned with steady, fully-developed ﬂow through weakly curved pipes, driven by a constant pressure gradient, which, as discussed, can be considered the simplest model for the aortic blood ﬂow. Notwithstanding any reservations about its applicability in this respect, it is an interesting problem worthy of study in its own right. Indeed, the work presented within this thesis is primarily motivated by theoretical considerations which are now examined.

1.2 Motivation - the theoretical problem

Incompressible, steady, fully-developed ﬂow in curved pipes is a formidable problem. Over 90 years of study have highlighted some interesting aspects of the ﬂow, some of which are not yet entirely resolved. A review of previews work is given in this section, highlighting some of those problems which form the main incentive here. Much of the information for this section is drawn from the review by Berger [7]. 1.2 Motivation - the theoretical problem 20

1.2.1 Dean Equations

The governing equations for steady, fully-developed ﬂow in curved pipes were derived by Dean [21, 22] and now bear his name. For completeness, the details of this derivation are now presented.

Consider a circular pipe with ﬁxed radius a and radius of curvature L described by the orthogonal coordinate system (r,θ,φ), as shown in Figure 1.4, with:

x =(r sin θ + L) cos φ, y = r cos θ, z =(r sin θ + L) sin φ, (1.1) where r denotes the distance from the centre of the pipe, θ is the angle between the radius vector and the plane of symmetry and Lφ is is the down-pipe distance which increases in the direction of ﬂow.

The velocity u = (u,v,w); where w is the azimuthal velocity normal to the cross-section while u and v are the radial and angular velocity components in the pipe cross-section respectively (as shown in Figure 1.4).

Figure 1.4: Coordinate system for curved pipe with circular cross-section

The ﬂow is assumed steady and incompressible. The relevant continuity and Navier-Stokes equation are thus:

u =0, (1.2) ∇ p u u = + ν 2u, (1.3) ∇ −∇ ρ ∇ Chapter 1. Introduction 21 where p is the pressure, ρ is the density and ν is the kinematic viscosity. The ﬂow is fully developed, so that u is independent of φ. In the coordinate system above the continuity and momentum equations reduce to:

∂u u u sin θ 1 ∂v v cos θ continuity: + + + + =0, (1.4) ∂r r L + r sin θ r ∂θ L + r sin θ

∂u v ∂u v2 w2 sin θ r-momentum: u + ∂r r ∂θ − r − L + r sin θ ∂ p 1 ∂ cos(θ) ∂v v 1 ∂u = ν + + , (1.5) −∂r ρ − r ∂θ L + r sin θ ∂r r − r ∂θ

∂v v ∂v uv w2 cos θ θ-momentum: u + + ∂r r ∂θ r − L + r sin θ 1 ∂ p ∂ sin θ ∂v v 1 ∂u = + ν + + , (1.6) −r ∂θ ρ ∂r L + r sin θ ∂r r − r ∂θ

∂w v ∂w uw sin θ vw cos θ φ-momentum: u + + + ∂r r ∂θ L + r sin θ L + r sin θ 1 ∂ p ∂ 1 ∂w w sin θ = + ν + + −L + r sin θ ∂φ ρ ∂r r ∂r L + r sin θ 1 ∂ 1 ∂w w cos θ + + . (1.7) r ∂θ r ∂θ L + r sin θ

p As (u, v, w) are assumed independent of φ, it follows from (1.7) that ρ must be of the form φf1 (r, θ)+ f2 (r, θ) and then from (1.5) and (1.6) that f1 (r, θ) must be a constant.

The continuity equation (1.4) is satisﬁed by the streamfunction ψ (r, θ) such that:

1 ∂(hψ) 1 ∂(hψ) = u, = v, (1.8) hr ∂θ − h ∂r where h = L + r sin(θ).

The mean axial pressure gradient is

1 ∂ p G = , (1.9) −L ∂φ ρ ρ 1.2 Motivation - the theoretical problem 22 where G is a constant.

These identities are substituted into equations (1.5), (1.6) and (1.7) and the pressure terms are eliminated from equations (1.6) and (1.7) by cross-differentiation. The governing equations become:

∂(hψ) ∂ ∂(hψ) ∂ 2w ∂w ∂w νr 4ψ = 2ψ + sin θ r cos θ , ∇ ∂r ∂θ − ∂θ ∂r ∇ h ∂θ − ∂r 1 ∂(hψ) ∂w ∂(hψ) ∂w w sin θ ∂(hψ) r cos θ ∂(hψ) G ν 2w = , ∇ hr ∂r ∂θ − ∂θ ∂r − hr h ∂θ − h ∂r − ρ where

1 ∂ r ∂(h f) 1 ∂ 1 ∂(h f) 2f = + . ∇ r ∂r h ∂r r2 ∂θ h ∂θ 1.2.1.1 Non-dimensionalisation & boundary conditions

The following non-dimensional variables are introduced:

r ψ w a h rˆ = , ψˆ = , wˆ = , δ = , h = , a ν ω0 L L where ω0 is the maximum down-pipe velocity in a straight pipe with circular cross-section driven by the same pressure gradient and ν is the kinematic viscosity.

The governing equations are then reduced to (dropping the hat notation for convenience):

1 ∂(hψ) ∂ ∂(hψ) ∂ w sin θ ∂w ∂w 4ψ = 2ψ + 576K cos θ , ∇ r ∂r ∂θ − ∂θ ∂r ∇ h r ∂θ − ∂r 1 ∂ψ ∂w ∂ψ ∂w δw sin θ ∂(hψ) r cos θ ∂(hψ) 2w = C, ∇ r ∂r ∂θ − ∂θ ∂r − hr h ∂θ − h ∂r −

2 2 where C = Ga ρ is a constant, and K = 1 a aw0 is the non-dimensional parameter νω0 288 L ν ‡ known as the Dean number . The Dean number provides a measure of the ratio of the

‡This K differs from that of Dean by a factor of 1/576. In Chapter 2 the series is computer-extended to arbitrary precision and this normalisation will ensure that numbers are more manageable. Chapter 1. Introduction 23 inertial and centrifugal forces relative to the viscous forces. At K = 0 the equations are applicable to straight-pipe ﬂow, so C = 4. The governing equations for ﬁnite curvature − are thus

1 ∂(hψ) ∂ ∂(hψ) ∂ w sin θ ∂w ∂w 4ψ = 2ψ + 576K cos θ , (1.10) ∇ r ∂r ∂θ − ∂θ ∂r ∇ h r ∂θ − ∂r 1 ∂ψ ∂w ∂ψ ∂w δw sin θ ∂(hψ) r cos θ ∂(hψ) 2w = 4. (1.11) ∇ r ∂r ∂θ − ∂θ ∂r − hr h ∂θ − h ∂r − Of particular interest is the weakly curved pipe limit, for which h =1+ δr sin θ 1 as → δ 0. The governing equations in the small curvature limit, known as the Dean equations, → are:

1 ∂ψ ∂ ∂ψ ∂ sin θ ∂w ∂w 4ψ = 2ψ + 576Kw cos θ , (1.12) ∇ r ∂r ∂θ − ∂θ ∂r ∇ r ∂θ − ∂r

1 ∂ψ ∂w ∂ψ ∂w 2w = 4, (1.13) ∇ r ∂r ∂θ − ∂θ ∂r − where is in the limit δ 0: ∇ ∇ → ∂2 1 ∂ 1 ∂2 2 = + + . ∇ ∂r2 r ∂r r2 ∂θ2

Regularity is assumed at r = 0 and the no-slip condition is applied at the boundary, (u,v,w) = (0, 0, 0) at r = 1. It follows from the deﬁnition of the streamfunction (1.8) in the limit of small curvature, h =1, that at r =1:

1 ∂ψ ∂ψ =0, =0 (1.14) r ∂θ ∂r

Therefore, ψ(r, θ) is constant at r =1. Choosing the constant of integration to be zero, the no-slip condition becomes:

∂ψ w =0, =0, ψ =0. (1.15) ∂r 1.2 Motivation - the theoretical problem 24

1.2.2 Solutions to the Dean equations

Several techniques have been adopted to solve the Dean equations, the choice of which primarily depends on the Dean number regime of interest: small, intermediate or large. Accordingly, an overview of the main results and unresolved aspects in each of the three regimes is now presented. For all contour plots shown here, as in the rest of the thesis, the 3π symbol “I” is used to denote the inside of the pipe bend θ = 2 and “O” is used to denote π the outside of the bend θ = 2 . 1.2.2.1 Small Dean number

For small K, the Dean equations (1.12 ) and (1.13) were solved by Dean [21, 22]. He considered a Stokes series in K of the azimuthal velocity, w(r, θ) and the streamfunction ψ(r, θ): ∞ ∞ i i w = wi(r, θ)K , ψ = ψi(r, θ)K , (1.16) i=0 i=0 which he expanded up to order K [21]:

19 3 1 1 w(r, θ)=(1 r2)+ K r r3 + r5 r7 + r9 sin θ + O(K2). − 40 − 4 − 4 40 For K =0, the solution corresponds to straight-pipe Poiseuille ﬂow. It is characterised by an azimuthal velocity, with contours that are concentric circles.

For small K, a secondary flow arises. This is generated as the curvature-induced pressure gradient drives the slower-moving fluid near the wall inward, whilst the faster-moving fluid in the core is swept outwards. This secondary flow, ψ, takes the form of two symmetric counterrotating vortices in the upper and lower half of the pipe. Accordingly, the flow consists of a pair of counterrotating helical vortices, symmetrical about the plane of symmetry. The secondary flow causes the maximum azimuthal velocity to migrate towards the outer bend.

The contours of w and ψ for K =0 and 1 are shown in Figure 1.5. Chapter 1. Introduction 25

I O I O

K = 0 K = 1 Figure 1.5: Dean’s series solution: azimuthal velocity and secondary streamlines contours

Dean later extended the series (1.16) up to order K4 [22] . He found the ratio of the ﬂux through a curved pipe to that of the corresponding straight pipe under the same pressure gradient to be:

Q (K)=1 0.03058 (K)2 +0.01195 (K)4 + O(K6). −

Consequently, the secondary flow increases the resistance to the azimuthal flow and results in a decrease in flux.

Dean’s results are in agreement with early experimental work which demonstrated the migration of the maximum down-pipe flow [93], the existence of the secondary flow and reduced flux [29, 30] in curved pipes.

The Stokes series (1.16) is usually referred to as the Dean series. It is valid for only small values of K (. 1.017) [21, 88].

1.2.2.2 Intermediate Dean number

For intermediate Dean numbers, the Dean equations are fully non-linear. Consequently, previous authors have used various numerical techniques to study the Dean equations in this regime. Whilst some authors have opted for spectral and pseudo-spectral techniques [25, 56, 96, 95, 73], others have adopted ﬁnite-difference methods [2, 3, 14, 23, 24, 33, 45, 50, 63, 65, 85, 99]. The most recent of these techniques described by Siggers & Waters [73] yield results which are in close agreement with previous numerical ﬁndings.

Figure 1.6 shows the contour plots for the azimuthal velocity, streamfunction and the vor- 1.2 Motivation - the theoretical problem 26 ticity for K = 0.01, 1.08 and 678.17 as found by Siggers & Waters [73]. Of interest are the solid lines which correspond to the weakly curved pipe, δ = 0. As K increases the outward movement of the location of the maximal azimuthal velocity and the centre of the secondary vortices is evident. For larger K, Figure 1.3c, the maximal azimuthal velocity continues to migrate outward, whereas the centre of the vortices move to the inner wall. The azimuthal velocity is almost vertical in the core, indicating it is independent of the y coordinate, and the vortices are signiﬁcantly distorted. In addition, a boundary-layer is discernible.

w ψ 2ψ ∇

(a) K = 0.01 δ = 0, 0.1 0.3

w ψ 2ψ ∇

(b) K = 1.08 δ = 0, 0.1 0.3

w ψ 2ψ ∇

Figure 1.6: Contour plots of the two-vortex solution from Siggers & Waters [73]: δ = 0, δ = 0.1, δ = 0.3, shaded areas indicate regions where the functions are positive Chapter 1. Introduction 27

This two-vortex solution is not unique. Dennis & Ng [25] and Nandakumar & Masliyah [63] independently found another solution for values of K greater than some critical value

K1. Dennis & Ng [25] found K1 = 99.17, a value later conﬁrmed by Daskopoulos & Lehnhoff [20] and Yanase et al. [95]. Daskopoulos & Lehnhoff [20] and Yanase et al.

[95] found another solution for K >K1. Daskopoulos & Lehnhoff [20] also reported a further two solutions for K greater than another critical value K 675. All of these 2 ≈ secondary solutions are four-vortex in character near the bifurcation point K1. Daskopoulos & Lehnhoff [20], Yang & Keller [96] and, more recently, Machane [50] found that on Branch 2 the additional vortex pair vanishes with increasing Dean number.

The solutions branches differ in their stability. Daskopoulos & Lehnhoff [20] found the four-vortex solution of Dennis & Ng [25] to be stable with respect to symmetric distur- bances and all the other four-vortex solutions to be unstable. Alternatively, Yanase et al. [95] performed a more general stability test and found the four-vortex solution of Dennis & Ng [25] to be unstable. However, Cheng & Mok [13] were able to observe the four-vortex solution experimentally conﬁrming the original ﬁndings of Daskopolous & Lenhoff [20].

A schematic representation of the part of the bifurcation diagram constructed from these results is shown in Figure 1.7.

Branch 5: unstable four-vortex

Branch 4: unstable four-vortex

Branch 3: unstable four-vortex

Branch 2: symmetrically stable four-vortex

Branch 1: stable two-vortex

0 K Figure 1.7: Schematic representation of bifurcation diagram; solid lines indicate stable solutions. Branch 1-3 were obtained by several authors [25, 20, 95, 96, 73], Branch 4-5 in [20] 1.2 Motivation - the theoretical problem 28

Figure 1.8 shows the four-vortex solution on Branch 2 as calculated by Siggers & Waters [73].

w ψ 2ψ ∇

Figure 1.8: Contour plots: four-vortex solution from Siggers & Waters [73]

Yang & Keller [96] found Branch 2 to bifurcate from Branch 1 at K = K 68, 600. 3 ≈ They also reported six and eight-vortex structure solutions and suggested that there was an infinite number of solutions exhibiting this 2n-vortex structure. However no other study verified the bifurcation at K3 or the existence of the six and eight-vortex solutions they found. Yang & Keller [96] stressed that there was a need for improved mesh-refinement calculations of their work - their finest mesh was composed of only 20 spectral modes and 1 a radial discretisation of 60 . It is possible for insufficient mesh refinement to cause artificial bifurcations and solutions.

It may be that Dean flow is subcritical, analogous to the closely related straight-pipe flow. Mathematically, subcritical flows are linearly stable for all Reynolds numbers; that is all sufficiently small perturbations will decay. Nevertheless, experimentally these flows become turbulent. For example, in the case of the straight-pipe flow experimentally transition by sufficiently large perturbations has been triggered for Reynolds numbers in between 1760 and 2300 [47]. Experiments by Hof et al. [38] show that as the Reynolds number increases the critical threshold of the perturbation decreases. Consequently, for sufficiently large Reynolds numbers there may exist a branch of solutions which coalesce with the basic laminar solutions. Accordingly, in some sense, the infinite Reynolds number is regarded as the bifurcation point for this problem. It is therefore possible that for Dean flow the two-vortex solution does not bifurcate to the unstable four-vortex solution at finite Dean numbers. Chapter 1. Introduction 29

The complete bifurcation diagram for the Dean ﬂow problem remains unknown.

1.2.2.3 Large Dean number

Boundary layer methods are the main approach to solving the Dean equations for large Dean number [7]. Work on this includes that of Adler [1], Barua [6], Mori & Nakayama [60], It¯o[44], Smith [76] and Pedley [67]. The argument by Smith [76] is detailed in Appendix A.1. Though these studies differ on the boundary layer model they adopt, all agree that ψ K1/6 and w K−1/6 and the boundary layer has thickness O(K1/6). The ∼ ∼ ﬂux ratio is expected to scale like w, accordingly Q(K) K−1/6. However, this result ∼ disagrees with Van Dyke’s analysis [88].

In his work, Van Dyke [88] employed the extended Stokes series method. This method involves relegating the calculation of the higher order terms of the Stokes series to a computer and then analysing the coefﬁcients to determine the analytic structure of the solution [87, 89]. Van Dyke computer-extended Dean’s series (1.16) to 24 terms. Using Domb- Sykes analysis he estimated the radius of convergence of the series and concluded that it was limited by a conjugate pair of imaginary square-root singularities at K 1.017i. He ≈ ± analytically continued the series by means of an Euler transform and argued that the transformed series was convergent for all values of K. By Domb-Sykes analysis, he concluded that the ﬂux ratio behaved like K−1/10 for large K.

The variation between these two asymptotic theories is known as the ‘Flux Ratio Paradox’ [71, 46]. An overview of possible causes for this discrepancy are discussed below.

Boundary Layer Model. Van Dyke [88] argued that the boundary layer theory results were incorrect as they assumed an incorrect boundary layer model. Specifically, the boundary layer models all assumed the boundary layer thickness to be constant around the circumference of the circle. However, at the inner wall of the pipe it is widely accepted that a reentrant jet forms [1], which some studies suggest occurs in conjunction with a separation of the boundary layer [6] though others disagree [44]. The boundary layer structure of the problem remains unresolved. Despite Van Dyke’s objection, the boundary layer theories all agree on the asymptotic behaviour [7, 46]. Furthermore, it has been argued that such fine details of the boundary layer model should not affect the final expression for the circumferentially averaged flux ratio [46]. 1.2 Motivation - the theoretical problem 30

Numerical Calculations. Van Dyke [88] claimed that conclusions from early numerical schemes [3, 56] with regards to the asymptotic behaviour were subject to poor mesh reﬁne- ment. However, recent numerical work [23, 46, 73], in which the effect of grid reﬁnement was investigated, continue to agree with the asymptotic behaviour proposed by boundary layer theory and early numerical studies. It is unlikely that this is a cause for the discrepancy.

Experimental Data. Ramshankar & Sreenivasan [71] proposed that the experimental results of Hasson [35] and Srinivasan [78] did not agree with Van Dyke’s asymptotic model as they did not replicate the physical assumptions of the Dean equations (1.12) and (1.13). Accordingly, Ramshankar & Sreenivasan [71] designed an experiment which they claimed accounted for the effects of finite curvature. They concluded that their results corroborated Van Dyke’s asymptotic theory. However, early experimental work was found to be in agreement with later numerical studies on weakly curved pipes despite Ramshankar’s objection. In addition, more recent numerical and empirical experiments [45, 77] have shown that the curvature has a negligible effect on the behaviour of the flux ratio and Dean number. In particular Soh and Berger [77], showed that the flux ratio, for the same Dean number, varies by less than 5% when the curvature is changed from 0.01 to 0.2. Furthermore, Jayanti [46] noted that Ramshankar & Sreenivasan [71] calculated their flux ratio using experimental rather than theoretical values for the flux down the straight-pipe. He found that considering experimental calculated flux ratios, Ramshankar & Sreenivasan work results agreed with previous experimental studies and not Van Dyke’s asymptotic model.

Bifurcation of solution. Dennis & Ng [25], postulated that the analytically continued Stokes series may correspond to one of the multiple-vortex solutions. One objection to this hypothesis is the debate regarding the use of the flux ratio to distinguish between branches. The work of Dennis & Ng [25] suggests that the four-vortex solution also manifests the one-sixth power variation proposed by the boundary layer theory, indicating that the flux ratio cannot be used to distinguish between the solution branches. In comparison, the work of Yang & Keller [96] indicates that though the flux ratio is similar for the multiple solutions found, they are distinct. Whether or not the flux ratio is an appropriate function to use to identify the multiple solutions, there is no other evidence to suggest that the analytically continued Dean series is not a two-vortex solution. The Dean series, for low K, is the two-vortex solution. At the transition between a two and four-vortex structure the Chapter 1. Introduction 31 extended Dean series should exhibit a singularity [87], however Van Dyke identified the nearest singularity of the analytically continued series to be at infinity [88]. Only further consideration of this solution would indicate whether or not the branch is the same as the two-vortex solution.

Truncation of the Dean Series. Jayanti [46] suggested that the cause of the discrepancy lay in Van Dyke using only the ﬁrst 24 terms in the series to estimate the analytic structure for high values of the Dean number. Furthermore Van Dyke worked to double precision, and the rounding errors which ensue from the Euler transform technique may well be signiﬁcant at values of the Dean number for which bifurcations and other interesting bifurcations may occur.

The ‘Flux Ratio Paradox’ has yet to be resolved and over time it is the results of Van Dyke which have been called into question and it is not, apparently, supported by any other result.

1.2.3 Alternative cross-sections

The review above concerns only ﬂow through pipes of circular cross-section, but studies have also been carried out on elliptical cross-sections. This is of practical interest, as due to their construction curved pipes in industry tend to be elliptical rather than circular. Fur- thermore, it seems a natural extension of a circular pipe.

The earliest study on curved pipes of finite curvature with elliptic cross-section is that of Cuming [19]. In his work, he solved the resulting Dean equations by constructing a series expansion in the curvature of the pipe and the associated Reynolds number as far as the first power of the curvature. He found that for aspect ratios, λ = (length of the ellipse along the x-axis)/(length of the ellipse along the y-axis), away from one the total vorticity in the upper half of the pipe, Ω, diminished. For small aspect ratios Ω= O(λ2) and that for large aspect ratios Ω = O(λ−2). Srivastava [79], extended Cuming’s analysis to second order terms to calculate the flux ratio. In his paper he presented the flux ratio for only particular values of the aspect ratio which suggested that the flux ratio decreased with λ, but no further conclusions were drawn. Topakoglu [83], presents the same analysis as Cuming but in elliptic coordinates. This work plots the secondary-flow streamlines for particular values of the aspect ratio.

More recently Machane [50], completed a bifurcation and stability analysis of laminar ﬂow 1.3 Aims of the thesis 32 in curved duct, using a path continuation techniques similar to Yang & Keller [96]. Al- though the primary discussion is on rectangular cross-sections, he also considered elliptic cross-sections of aspect ratio λ = 1 and 1.45. The results in the former case are in agreement with previous results as discussed earlier. For the latter case he succeeded in ﬁnding three branches of symmetric solutions: Branchλ I is a two-vortex solution and Branchλ 2 - 3 are four-vortex solutions which bifurcate at a critical Dean number. Their behaviour is analogous to solutions Branch1 - 3 in the circular pipe. Notably, this bifurcation point occurs at a smaller Dean number than in the case for a circular pipe. Furthermore, whilst the a pair of the four-vortex solutions on Branchλ 2 vanish, the number of vortex pairs on

Branchλ 3 increases without running through a singular point.

Other cross-sections considered include: rectangular [20, 94], triangular [15, 16, 64, 84], semi-circular [54, 62], moon-shaped [63] and annular [40].

1.3 Aims of the thesis

The literature on steady, fully-developed ﬂow in weakly curved pipes is extensive, however the review above has revealed some areas in need of further study. The aim of this research is to: (i) address the ‘Flux Ratio Paradox’, by direct extension of Van Dyke’s work; (ii) complete a numerical study of the bifurcations of Dean ﬂow by analytical continuation of the extended Stokes series [28] and; (iii) to construct the extended Stokes series solution for pipes of elliptical cross-section.

In a broader context, the work also provides an assessment of the use of the extended Stokes series method in fluid dynamics. In particular, this technique has been used to solve the Navier-Stokes equations for several confined flows with varying levels of success. An account of which is given by Van Dyke [89]. Notably, the controversy between results of the extended Stokes series method and boundary layer techniques for Dean flow, also occurs in two related internal flow problems for which the boundary structure is not known; flow in a straight slowly rotating pipe [51] and flow in a heated horizontal pipe [90]. The former problem, prompted Van Dyke himself to remark “until this discrepancy is resolved, the series solution - and the method itself - lie under a cloud” [89], a view which the latter two only help to compact. Nevertheless, this technique is attractive for solving the Navier- Stokes equations at high Reynolds number, which is traditionally the regime of boundary layer methods, especially for problems such as these where the boundary layer is unclear Chapter 1. Introduction 33 or where there is separation. Consequently, its vindication seems worthwhile.

1.4 Organisation of the thesis

In Chapter 2 a resolution to the ‘Flux Ratio Paradox’ is sought by readdressing Van Dyke’s work [88]. Dean’s series (1.16) is extended to higher order and its coefficients are calculated to greater accuracy than Van Dyke (using MAPLE and C++, with an infinite precision package). Van Dyke’s analysis is checked and his conclusion regarding the asymptotic behaviour of the flow is corrected.

In Chapter 3, the generalised Pad´eapproximant technique developed by Drazin & Tourigny [28] is used to analytically continue the Dean series. This technique can investigate the existence of multiple solutions for real and complex Dean number. Particular attention is given to the singularities at K iK which were found to limit the Dean series. The ± c bifurcation in the complex plane at K iK has never been discussed before, despite the ± c prevalence of such singularities in a number of other related problems [51, 90].

In Chapter 4, the complex Dean equations are formally discussed. Details of numerical techniques used to solve the Dean equations for complex Dean number are presented. The results are used to corroborate some of the conclusions of Chapter 2 and 3.

In Chapter 5, the computer-extended Stokes solution for ﬂow in an elliptic cross-section is presented. The effect of the aspect ratio on the radius of convergence of the series is examined. Furthermore, the behaviour of the ﬂow for various aspect ratios is reported.

Finally in Chapter 6 the main results of the thesis are summarised and proposed extensions to the research are described.

Comment on notation. The symbol is used sometimes to mean ‘behaves like’ and some- ∼ times to mean ‘asymptotically equal to’ in a more rigorous sense. 34 35

Chapter 2

Extended Stokes series: circular cross-section

In this chapter the works of Dean [21] and Van Dyke [88], who used asymptotic regular expansions to solve the Dean equations for pipes with circular cross-section, are revisited. The details of the computer-extended Dean series are presented, for which the first 196 terms are calculated. The higher order terms are analysed to determine the analytic structure of the series. Following Van Dyke, the series is found to have a finite radius of convergence, limited by an imaginary conjugate pair of square-root singularities. Analytic continuation is completed by means of an Euler transform. Subsequent analysis finds Van Dyke’s claim regarding the asymptotic structure at large K to be unsubstantiated. The chapter concludes with the first full comparison of this series expansion solution with numerics. The work highlights the need for an alternative analytic continuation technique.

2.1 Computer-extended series

The Dean equations (1.12) and (1.13), derived in Chapter 1, were solved using the extended Stokes series method. This approach involves relegating the construction of the higher order terms of the Dean series (1.16) to a computer. The form of the coefﬁcients ψn(r, θ) and wn(r, θ) were derived and these were used to construct a recurrence relation to implement on the computer. The details of this are presented below.

The Dean series (1.16) are substituted into the Dean equations (1.12) and (1.13). By col- 2.1 Computer-extended series 36 lecting powers of K, the system is reduced to a sequence of successively more complicated linear problems:

∞ 1 ∂ψ (r, θ) ∂ ( 2ψ (r, θ)) ∂ψ (r, θ) ∂ ( 2ψ (r, θ)) 4ψ (r, θ)= i ∇ j i ∇ j ∇ n r ∂r ∂θ − ∂θ ∂r i,j:i+j=n ∞ sin θ ∂w (r, θ) ∂w (r, θ) + w (r, θ) j cos θ j , (2.1) i r ∂θ − ∂r i,j:i+j+1=n ∞ 1 ∂ψ (r, θ) ∂w (r, θ) ∂ψ (r, θ) ∂w (r, θ) 2w (r, θ)= i j i j , (2.2) ∇ n r ∂r ∂θ − ∂θ ∂r i,j:i+j=n which can be solved sequentially to ﬁnd the functions ψn (r, θ) and wn (r, θ).

It follows from equation (2.2) that in order to calculate ψn (r, θ), for arbitrary n, the set of functions (w (r, θ) , ψ (r, θ)) , i =0...n 1 are required. In comparison, to calculate { i i − } wn (r, θ) from (2.1) requires this set as well as ψn (r, θ). Therefore the equations must be solved successively starting with ψ1 (r, θ). For this calculation it is useful to note that for an arbitrary function f (r, θ)= rγ cos(λθ):

2f (r, θ)= γ2 λ2 rγ−2 cos(λθ) (2.3) ∇ − and 4f (r, θ)= γ2 λ2 (γ 2)2 λ2 rγ−4 cos(λθ) , (2.4) ∇ − − − which also holds when cos(λθ) is replaced by sin (λθ).

For n =1 equation (2.1) is:

4ψ (r, θ) = 1152r 1 r2 cos(θ) . (2.5) ∇ 1 − Solving for ψ1 (r, θ), using identity (2.4):

ψ (r, θ)= C r + C r3 +6r5 r7 cos(θ) , (2.6) 1 1,0,0 1,1,0 − 3 where C1,0,0 r and C1,1,0 r are complementary functions. The unknown coefﬁcients, C1,0,0 and C1,1,0, are found by satisfying boundary conditions (1.15), from which

ψ (r, θ)= 4r 9r3 +6r5 r7 cos(θ) . (2.7) 1 − − Chapter 2. Extended Stokes series: circular cross-section 37

Now w1 (r, θ) can be solved. From equation (2.2 ), for n =1:

2w (r, θ)= 2r 4r 9r3 +6r5 r7 sin (θ) . ∇ 1 − − −

From identity (2.3), with cos(θ) replaced by sin (θ):

3 1 1 w (r, θ)= E r r3 + r5 6r7 + r9 sin (θ) . 1 1,0,0 − 4 − 4 40

Again, the complementary function E1,0,0r is found by imposing the boundary conditions (1.15):

19 3 1 1 w (r, θ)= r r3 + r5 6r7 + r9 sin (θ) . 1 40 − 4 − 4 40 By iteration the general structure of the Dean series (1.16) is found to be:

In−1 Jn cos(2iθ)r2j n even wn = En,i,j , (2.8)  2j+1 i=0 j=0 sin((2i + 1)θ)r n odd   In−1 Jn−1 sin(2iθ)r2j n even ψn = Cn,i,j , (2.9)  2j+1 i=0 j=0 cos((2i + 1)θ)r n odd   where the coefﬁcients Enii, Cnii and Cni (i−1) of the complementary functions are found by imposing the boundary conditions (1.15) on ψn and wn individually. The index In is 1 the greatest integer less than or equal to 2 (n + 2) and Jn is the greatest integer less than or 1 equal to 2 (7n + 2).

Two programs have been written to calculate the coefficients Enij and Cnij. The first is a MAPLE program, which calculates all the desired coefficients exactly as rationals. The second is a C++ program which utilises GMP arbitrary precision package, and so sacrifices working with rationals for speed. The expressions used to construct the recurrence relation employed in these programs are presented in Appendix B.1.

The first 80 coefficients of the Dean series (1.16) have been calculated exactly using the MAPLE program and a further 116 coefficients have been found using the C++ program. 2.1 Computer-extended series 38

The analysis used all 196 terms, opting for exact values where available owing to the de- pendence of the results on their precision. The theory outlined below, may be applied to any quantity of interest. Herein, the series of the ﬂux ratio, friction ratio, total vorticity and outer wall shear stress are studied. The deﬁnition and notation for these is as follows:

Definition 2.1.1 The flux ratio Q (K) is the ratio of the flux in the curved pipe, Qc (K), to that in the straight pipe, Qs (K), of the same cross-section.

The ﬂux down a straight pipe with circular cross-section is:

2π 1 Qs (K)= w0(r, θ) drdθ 0 0 2π 1 = 1 r2 dr dθ − 0 0 π = . (2.10) 2

The ﬂux down a curved pipe with circular cross-section, using (2.8), is:

2π 1 Qc (K) = w (r, θ) rdrdθ 0 0 ∞ In−1 Jn 2π 1 = Kn cos(2iθ) r2j dr dθ neven i=0 j=0 0 0 ∞ In−1 Jn 2π 1 n 2j+1 + K Enij sin ((2i + 1) θ) r dr dθ i=0 j=0 0 0 n odd ∞ J2n 2E = K2n (2n) 0 j . (2.11) j +1 n=0 j=0

The odd terms do not contribute to the ﬂux down a curved pipe, so that the series expansion of Qc in K is even. From the above calculation, it follows that the ﬂux ratio is:

Q (K) Q (K)= c Qs(K) ∞ 2n = anK , (2.12) n=0 Chapter 2. Extended Stokes series: circular cross-section 39

Jn 2 2E(2n) 0 j where an = j=0 π j+1 . Deﬁnition 2.1.2 The friction ratio, Q(K), is the reciprocal of the ﬂux ratio. It is a measure of the resistance in the curved pipe divided by that in a straight pipe.

The friction ratio can be calculated using the series expansion for the ﬂux ratio (2.12):

1 Q(K)= Q (K) 1 1 = a 1+ ∞ an K2n 0 n=1 a0 m 1 ∞ ∞ a = n K2n . (2.13) a − a 0 m=0 n=1 0

From which it follows that the friction ratio is also even in K:

∞ 2n Q(K)= bnK . (2.14) n=0 Deﬁnition 2.1.3 The outer wall shear stress is the shear stress evaluated at the outermost π ∂w part of the wall (r =1, θ = 2 ), denoted τ(K). As u =0 at the wall, τ(K)= ∂r (r, θ).

Using (2.8):

∞ n τ(K)= (K) tn n=0 ∞ ∞ 2m e 2p+1 o = (K) tm + (K) tp, (2.15) m=0 p=0 where ∂w π ∂w π te = 2n 1, , to = 2n+1 1, . n ∂r 2 n ∂r 2 Deﬁnition 2.1.4 The total vorticity, Ω, in the upper half of the pipe, is the surface integral in the half-pipe of the vorticity, 2ψ, normalised by the ﬂux down a straight pipe with the ∇ same cross-section. 2.2 Radius of convergence 40

Using (2.9) the total vorticity in the upper half of the plane is

π 1 2 2 Ωc (K)= ψrdrdθ 0 − π ∇ 2 In−1 Jn−1 π 1 2 n 2 2j = K Cn,i,j (r sin(2iθ)) rdrdθ π ∇ n, even i=0 j=0 0 − 2 In−1 Jn−1 π 1 2 n 2 2j+1 + K Cn,i,j (r cos((2i + 1)θ)) rdrdθ π ∇ i=0 j=0 0 − 2 n, odd In−1 Jn−1 π 1 2 n 2 2j−1 = K Cn,i,j (2j)(2j 1)+2j (2i) r sin (2iθ) dr dθ 0 − π − − n, even i=0 j=0 2 In−1 Jn−1 π 1 2 n 2 2j + K Cn,i,j (2j + 1)(2j)+2j +1 (2i + 1) r cos ((2i + 1) θ) dr dθ 0 − π − n, odd i=0 j=0 2 In−1 Jn−1 (2j + 1)2 (2i + 1)2 = 2 Kn C − . − n,i,j (2j + 1)(2i + 1) i=0 j=0 n,odd

The even terms in K do not contribute to the total vorticity. The series expansion of Ωc is therefore odd in K. By Stokes’ theorem and the boundary conditions, this corresponds to the cross-pipe velocity integral along the centre-line.

The total vorticity is Ω (K) ∞ Ω(K)= c = d K2n+1, (2.16) Q (K) n s n where I n −1 J n −1 4 (2 +1) (2 +1) (2j + 1)2 (2i + 1)2 d = C − . n −π (2n+1),i,j (2j + 1)(2i + 1) i=0 j=0 2.2 Radius of convergence

The Dean series (2.8) and (2.9), is expected to have a ﬁnite radius of convergence as is usual for Stokes series. An estimate of this radius of convergence and the limiting singularity associated with it was found by Domb-Sykes analysis of the higher order terms [26, 87, 88]. The details of this method and its results are presented below. Chapter 2. Extended Stokes series: circular cross-section 41

2.2.1 Domb-Sykes analysis

The series expansion of a function f(z),

∞ n f(z)= cnz , n=0 can either have an inﬁnite, ﬁnite or, in the worst case, a zero radius of convergence, R. In theory, R can often be calculated by D’Alembert’s ratio test:

c R = lim n−1 , n→∞ c n if the limit exists. In practice, as only a ﬁnite number of coefﬁcients are known, this limit cannot be computed. It can however be estimated.

The distance of the closest singularity, zc, from the origin is equal to R. In physical problems these singularities are usually algebraic or algebro-logarithmic, such as:

z α ∞ A 1 α =0, 1,... (2.17a) 1 − z f(z)= c zn  c n ≈ z α z n=0  A1 1 ln 1 α =0, 1,... (2.17b) − z − z c c   where α is known as the critical exponent of the singularity, and A1 is known as the amplitude of the singularity. If α is an integer in (2.17a), then the function f(z) is not singular at zc. Consequently the presence of a logarithmic singularity, as in (2.17b), is inferred if there appears to be a singularity at zc with an integer critical exponent α.

From the Taylor expansion of the algebraic singularity (2.17a), it follows that:

α 1 c = A (2.18) n 1 n zn c α (α 1) ... (α n + 1) 1 = A − − . (2.19) 1 n! zn c

The inverse ratio of the coefﬁcients is therefore: 2.2 Radius of convergence 42

α 1 A1 n cn n zc = α 1 (2.20) cn−1 A1 n−1 n−1 zc 1 1+ α = 1 . (2.21) z − n c

A similar consideration of the Taylor expansion of the algebro-logarithmic singularity (2.17b) shows [27]: c 1 1+ α 1 n 1 + . (2.22) c ≈ z − n n log n n−1 c It follows that for large enough n, in the case of algebraic singularities (2.17a) and algebro- 1 logarithmic (2.17b) singularities the ratio of the coefﬁcients behaves linearly in n :

c 1 1+ α n 1 . (2.23) c ≈ z − n n−1 c

1 Therefore, the linear extrapolation of the plot of the ratio of the coefﬁcients against n , known as a Domb-Sykes plot [27], will provide an estimate of the radius of convergence, R = z from the reciprocal of the intercept at 1 = 0 and an estimate of α from the | c| n terminal slope.

More accurate estimates of the intercept (and so of R) can be found by ﬁtting higher-order 1 polynomial in powers of n through several ratios. In Domb-Sykes anlaysis the polynomial interpolants are usually presented in the form of a Neville table [87, 89]:

0 c0

0 1 c1 c1

0 1 2 c2 c2 c2 . . . ..

c0 c1 cN N N N where cj denotes the ith-order polynomial passing though the i-points: ( cn , 1 ), n = i { cn−1 n Chapter 2. Extended Stokes series: circular cross-section 43 j i...j . The polynomial interpolants cj are found recursively by the Neville algorithm − } i [70]: (i) cj−1 (i j) cj−1 cj = i − − i−1 , i j with c0 = ci , the ratio of the coefﬁcients of the series and c1 is the straight line passing i ci−1 i through the points ci , 1 and ci−1 , 1 . The reciprocal of the intercepts of the last ci−1 i ci−2 i−1 row in the Neville table, highlighted in grey above, give the most accurate sequence of estimates for R, whose convergence can be tested using standard techniques.

With a suitable estimate for R, estimates of the critical exponent α of the singularity zc are found by constructing a Neville table of the points:

c α0 = 1 n R (n) 1. n − c − n−1 This expression is obtained from a simple rearrangement of equation (2.23). Once estimates for α and R are known, estimates of the amplitude A1 are found by considering a Neville table of the points: 0 cn (A1) = . n ( z )−n α − c n The sign pattern of the ratio of the coefficients can classify zc further. Considering the Taylor expansion of the algebraic singularity (2.19), it follows that if the ratios are all eventually positive, i.e. the coefficients themselves have a fixed sign, then zc necessarily lies on the positive real axis. Alternatively, if the ratios are all eventually negative, i.e. the coefficients themselves alternate in sign, then zc necessarily lies on the negative real axis. This similarly holds for the algebro-logarithmic singularity (2.17b). More complicated sign patterns can also arise if zc is complex. For example, if f(z) has a conjugate pair of ±iθ complex singularities at zc = e :

1 1 f(z)= + (2.24) eiθ z e−iθ z − − cos θ z =2 − (2.25) 1 2z cos θ + z2 ∞ − =2 cos((n + 1)θ)zn. (2.26) n=0

Consequently, if θ = π the coefﬁcients have the sign pattern (+ +++ ...). 4 − − − 2.2 Radius of convergence 44

In general the Domb-Sykes plots are not perfectly linear. Indeed, the ratio of the coefﬁcients 1 of the algebro-logarithmic singularity (2.22) has a correction term of order O n log n to the model (2.23). In addition, algebraic singularities in the amplitude of the singularity, 1 A1 = A3(z), can result in correction terms of order O n2 in the ratio of the coefﬁcients of the singularities (2.17) [27]. Consequently, a small shift in n, typically n+1/2, is usually considered in order to account for these errors and straighten the Domb-Sykes plot [32].

Finally, the singularity can be either multiplicative, additive or both. For a function f(z) with an algebraic singularity identiﬁed at zc with critical exponent α these are of the form, respectively:

z α f(z) 1 A (z), (2.27) ∼ − z 3 c z α f(z) A 1 + A (z), (2.28) ∼ 1 − z 2 c z α f(z) 1 A (z)+ A (z), (2.29) ∼ − z 3 2 c where A2(z) and A3(z) have singularities further from the origin than zc or are entire. These three structures can be distinguished by considering algebraic extraction of the singularity, by subtraction and division, and also considering the analytic structure of the reciprocal of f(z).

Algebraic extraction. First, multiplicative extraction is considered. This involves multiplying f(z) by (z z)−α. If the series expansion of (z z)−αf(z) has a greater radius of c − c − convergence than f(z), then the singularity at zc of f(z) is purely multiplicative (2.27). In comparison, if (z z)−αf(z) has a singularity at z with exponent α, then f(z) have an c − c − additive term of the form (2.28) or (2.29).

Following this, the singularity structures (2.28) and (2.29) can be distinguished by additive extraction. Additive extraction involves subtraction A (z z)α, where A is the estimate 1 c − 1 of the amplitude of the singularity, from f(z). If the series expansion of f(z) A (z − 1 c − z)αf(z) has a greater radius of convergence than f(z), then f(z) must be of the form (2.28). Alternatively, if it has a singularity at z with exponent α and amplitude A , then f(z) it c − 1 must have a multiplicative and additive term (2.29).

Reciprocal of the series. An alternative way to distinguish the structure of the singularity Chapter 2. Extended Stokes series: circular cross-section 45

−1 1 of f(z) is to compare the critical behaviour of f(z) and its reciprocal f(z) = f(z) . If f(z) is of the form (2.27) then its reciprocal:

1 f(z)−1 = (2.30) f(z) 1 z −α = 1 . (2.31) A (z) − z 3 c So the series expansion of f(z)−1 has a singularity at z with exponent α. Otherwise the c − singularity has an additive term such as in (2.28) and (2.29) and these can be distinguished by additive extraction.

2.2.2 Results

The Domb-Sykes analysis was applied to the ﬂux ratio (2.12); friction ratio (2.14); total vorticity (2.16) and outer wall shear stress (2.15).

Table 2.1 shows the first twelve coefficients of the four series of interest to ten significant figures. The coefficients of Q(K) and Q(K) correspond exactly to the values presented by Van Dyke [88]. First, the sign pattern of the coefficients are considered. The coefficients of Q(K), Q(K), which are even in K, and Ω(K), which is odd in K, alternate in sign. Such a sign pattern is found in the following model function:

K2 −1 K2 K2 2 1+ =1 + .... (2.32) K2 − K2 K2 − c c c

This indicates the ﬂux ratio Q(K), the friction ratio Q(K) and the total vorticity Ω(K) have a singularity at K2 = K2. That is that the radius of convergence is limited by a − c conjugate pair of imaginary singularities iK . ± c Similarly, the outer wall shear stress τ(K), which has odd and even terms, is found to alternate in sign with a period of two. This is reminiscent of the following model function: 2.2 Radius of convergence 46

1+ K K K 2 K 3 Kc =1+ + ..., (2.33) 2 K Kc − Kc − Kc 1+ 2 Kc which indicates that the outer wall shear stress τ(K), also has a singularity at K2 = K2. − c That is its radius of convergence is limited by a conjugate pair of imaginary singularities iK . ± c 2n 2n 2n+1 n n an (K ) bn (K ) dn (K ) tn (K ) 0 1.0000000000 1.0000000000 2.6192356349e+00 -2.0000000000e+00 1 -3.05753968e-02 3.05753968e-02 -4.2026554736e-01 -3.0000000000e-01 2 1.19311827e-02 -1.09963278e-02 1.9434377809e-01 -3.6944444444e-02 3 -6.5846067e-03 5.8835890e-03 -1.1582824675e-01 5.3949712056e-02 4 4.2384991e-03 -3.7260799e-03 7.7894757373e-02 1.6606671608e-02 5 -2.9771923e-03 2.5910675e-03 -5.6283279275e-02 -2.3293124569e-02 6 2.2124224e-03 -1.9123651e-03 4.2660537966e-02 -9.3658126699e-03 7 -1.7101745e-03 1.4709324e-03 -3.3461299663e-02 1.334220183e-02 8 1.3610001e-03 -1.1662205e-03 2.6930263382e-02 6.0641804756e-03 9 -1.1076549e-03 9.463439e-04 -2.2113521455e-02 -8.7573334810e-03 10 9.176324e-04 -7.821394e-04 1.8453044033e-02 -4.2683754724e-03 11 -7.712689e-04 6.561071e-04 -1.5603151951e-02 6.2447915328e-03 12 6.560579e-04 -5.571890e-04 1.3339661214e-02 3.1745144407e-03

Table 2.1: Coefﬁcients for K series of ﬂux ratio (an ), friction ratio (bn), total vorticity (dn) and outer wall shear stress (tn)

The Domb-Sykes plot of tn shown in Figure 2.1a, suggests that the odd and even terms should be considered separately, as they alternate in sign. The Domb-Sykes plots of these, o e tn and tn, as well as an, bn and dn are shown in Figure 2.1b. Their behaviour, after approximately 10 terms, is identical, indicating the closest singularity to the origin has the same form for all four quantities.

The bottom row of the Neville table estimates for the radius of convergence are shown in Table 2.2. All are found to converge to:

(K )2 = (0.966858403134677743805629253147411348505494512902817)−1. (2.34) c − Chapter 2. Extended Stokes series: circular cross-section 47

The closest singularities, iK , are therefore the imaginary conjugate pair: ± c

iK = (1.01699440099353827824160962486653508113571791362423)i. (2.35) ± c ±

This is in agreement with the value found by Van Dyke and extends the accuracy by a further forty-three signiﬁcant ﬁgures. The Neville table of the terminal slope indicates a 1 critical exponent α = 2 . The asymptote

1 3/2 1 (2.36) −K2 − n +1/2 c suitably predicts the behaviour of the ratio of the coefﬁcients as shown in Figure 2.1

Attempts at extracting this singularity algebraically (as described in Section 2.2.1), have indicated that Q(K), Q(K), Ω(K) and τ(K) are of the form (2.29) . This is also conﬁrmed as Q(K) and its reciprocal Q(K) have the same nearest singularity and critical exponent. In summary, series Q(K), Q (K), Ω(K) and τ(K) are found to have the same radius of convergence and singularity structure:

1 2 2 2 Q (K)= K + Kc q1(K)+ q2(K) (2.37)

1 2 2 2 Q (K)= K + Kc q1(K)+ q2(K) (2.38) 1 2 2 2 Ω(K)= K + Kc ω1(K)+ ω2(K) (2.39)

1 2 2 2 τ (K)= K + Kc τ1(K)+ τ2(K) (2.40) where q1(K), q2(K), q1(K), q2(K), ω1(K), ω2(K), τ1(K) and τ2(K) have singularities 2 further from the origin than Kc or are entire. Consequently, the Dean Series (2.8) and (2.9) must also take this form. 2.2 Radius of convergence 48

0.5

0 n-1

/t -0.5 n t

-1

-1.5

-2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1/(n+1/2)

(a) outer wall shear stress (tn)

0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 an -0.6 b

ratio of coefficients n e -0.7 t n o -0.8 t n d -0.9 n Kc-2(1-3/2s) -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1/(n+1/2)

(b) Domb-Sykes plots of the K-series of the ﬂux ratio (an), friction ratio (bn), total vorticity (dn), even e o terms of the outer wall shear stress (tn), odd terms of the outer wall shear stress (tn)

Figure 2.1: Domb-Sykes plots of the K-series of the ﬂux ratio, friction ratio, total vorticity and outer wall shear stress Chapter 2. Extended Stokes series: circular cross-section 49 ), n d o n t ), total vorticity ( n b ) o n t e n ), friction ratio ( t n a he ﬂux ratio ( 2367 ...902817172738263 ...902817155665619 50465 ...90281717026779519870 ...902817159134025 ...902817171253185 ... 902817156286907 08322 ...90281717442197201117 ...902817156730318 ...90281717607476970433 ...902817158864123 ...90281717755055570718 ...902817161496793 ...90281717877685275744 ...902817164164741 ...90281717973503650668 ...902817166525566 ...90281718043874838582 ...902817168346855 ...90281718091505249780 ...902817169485680 ...902817181190244 2 − c n d K ), odd terms of the outer wall shear stress ( e n t n b -0.966858403134677743805692953147411348505494512... n a Bottom row of the Neville table for (reading left to right), t even terms of the outer wall shear stress ( n 87 ...90215168097330788 ...903073320325823 ...9021597897672889 ...8997234261 ...902887162535997 ...90335308936849890 ...902817585848549 ...90281083813 ...90278748319024391 ...9030686306 ...902773565309195 ...90281029892211592 ...9028276782 ...902815411101297 ...90281932064910293 ...9027971292 ...902820656542921 ...90281782084696994 ...9028155241 ...902817457585198 ...90281702654006895 ...9028187417 ...902816903335062 ...90281711714835796 ...9028173723 ...902817141842698 ...902817185452649 ...9028170559 ...902817196468681 ...9028171548 86 ...908451239423548 ...896357142452346 ...9013856009 Table 2.2: 2.3 Analytic continuation 50

2.3 Analytic continuation

As anticipated the Dean series, (2.8) and (2.9), were found to have a ﬁnite radius of convergence. The series was analytically continued with the aim of determining the large-K behaviour of the solution. Following Van Dyke [88], the Euler transform method was employed.

2.3.1 Euler transform

Due to the structure of equations (2.37)-(2.40), the square-root singularites at iK cannot ± c be extracted algebraically from the Dean series (2.8) and (2.9). However, as they are not physically signiﬁcant they can be mapped to inﬁnity using the Euler transform:

K2 (2.41) ǫ = 2 2 . K + Kc

The interesting high Dean number behaviour is then mapped close to ǫ =1 and the unphys- ical singularities K = iK are mapped to ǫ = . The origin, however, remains ﬁxed. A ± c ∞ graphical representation of this map is shown in Figure 2.2.

Figure 2.2: Graphical representation of the Euler transform Chapter 2. Extended Stokes series: circular cross-section 51

∞ n The K-series is recast in the new expansion variable ǫ. For some general series n=0 cnK ∞ n 2 the Euler transformed series is denoted n=0 cnǫ , where the coefﬁcients cn are found as follows: Even terms. An expansion of K2 in ǫ is found from the mapping (2.41):

ǫK2 K2 = c = ǫK2 1+ ǫ + ǫ2 + ... . (2.42) 1 ǫ c − ∞ n Substituting this expansion in the even terms of the series n=0 cnK :

∞ 2n 2 2 2 4 2 2 c2nK = c0 + c2 ǫKc 1+ ǫ + ǫ + ... + c4ǫ Kc 1+ ǫ + ǫ + ... n=0 3 6 2 4 + c6 ǫ Kc 1+ ǫ + ǫ + ... + ... 2 2 4 2 2 4 6 3 = c0 + c2Kc ǫ + c2Kc +c4Kc ǫ + c2Kc +2c4Kc + c6Kc ǫ + ... n = c2nǫ , (2.43) n where

n n 1 c = − c (K )2i . (2.44) 2n i 1 2i c i=1 − Odd terms. An expansion of K in ǫ is found from the mapping (2.41):

1 ǫ 2 Kc 1 1 3 3 2 2 K = 1 = ǫKc ǫ + ǫ + ǫ + ... . (2.45) (1 ǫ) 2 2 8 − ∞ n Substituting this expansion in the odd terms of the series n=0 cnK :

∞ ∞ 2n+1 2n c2n+1K = K c2n+1K n=0 n=0 1 1 3 3 2 2 =(ǫK ) ǫ 2 + ǫ + ǫ 2 + ... c + c ǫK 1+ ǫ + ǫ + ...... c 2 8 1 3 c ∞ n+1 = c2n+1ǫ 2 , (2.46) n=0 2.3 Analytic continuation 52 where,

∞ 2i+1 1 2 c2n+1 = (Kc) fi fn−i (2.47) n=0 with

j j 1 j 2i 1 f 2 = − c (K )2i and f 1 = i=2 − . (2.48) j i 1 2i+1 c j i!2i i=1 − As for the K-series the analytic structure of the ǫ-series can be found by Domb-Sykes analysis. If the ǫ-series has a radius of convergence of one, then the validity of the series has been extended to all the physically signiﬁcant K. In this case, the dominant singularity at ǫ = 1 will give the asymptotic behaviour of the solution for large values of K which is exactly the regime of interest.

Note that the Euler transform (2.41) can be applied to the Dean series (1.16), analytically continuing the entire solution. Then, the Euler transformed ﬂux ratio, friction ratio, outer wall shear stress and total voriticity, can be re-calculated from the Euler transformed Dean series. Alternatively, the Euler transform (2.41) can be applied directly to the series expansion of the ﬂux ratio (2.12), friction ratio (2.14), outer wall shear stress (2.15) and total vorticity (2.16) or any other quantity of interest. Both, clearly, give equivalent Euler transformed series.

2.3.2 Results

The series expansion of the flux ratio, friction ratio, outer wall shear stress, and total vorticity were recast in the variable ǫ. The respective notations for these series are: Q(ǫ) = n n e n o n n/2 anǫ ; Q(ǫ) = bnǫ ; τ(ǫ) = tnǫ + tnǫ ; Ω(ǫ) = dnǫ . The first twelve coefficients of these series are found in Appendix B.2. The subsequent conclusions from Van Dyke’s analysis were reconsidered in light of the further coefficients calculated and the results are presented below.

The Domb-Sykes plots for the quantities of interest are shown in Figure 2.3, where again the odd and even terms of τ(ǫ) are considered separately. All four plots suggest that the radius of convergence is close to one though, as expected, their terminal slopes are consid- Chapter 2. Extended Stokes series: circular cross-section 53 erably different. These plots exhibit two unusual behaviours, firstly that the odd coefficients τ(K) change sign at the 20th term and secondly that the remaining three Domb-Sykes plots display small oscillations and a leisurely curvature which cannot be removed by shifting n in the plot. This is further reflected in the misbehaviour of the Neville analysis.

The corresponding Neville table estimates of the limiting singularity for all three quantities exhibit oscillations. This is exempliﬁed in Table 2.3 which shows the Neville table of the intercept for the Euler transformed ﬂux ratio and friction ratio. Though it is tempting to conclude from this that the nearest singularity is at ǫ =1, the oscillation of these estimates suggest that such a conclusion is premature.

n ﬂux ratio friction ratio 8 0.9989359266 1.0001070895 9 0.9994970914 0.9991733543 10 0.9998438589 0.9997266248 11 1.0000813451 0.999179807 12 1.0002494590 1.0000978237 . . 92 0.9999859979 0.9998999966 93 0.9999865418 0.9999904018 94 0.9999870709 0.9999907973 95 0.9999875849 0.9999911826 96 0.9999880836 0.9999915573

Table 2.3: Bottom Row of the Neville Table for (reading left to right), the ﬂux ratio and friction ratio ǫ-series

Furthermore, assuming the nearest singularity is at ǫ = 1, the corresponding Neville table analysis of the critical exponent is found to be particularly irregular. As an example a sample of the Neville table for the ﬂux ratio is shown in Table 2.4. 2.3 Analytic continuation 54 0.7 0.4 0.35 0.6 0.3 0.5 0.25 0.4 d outer wall shear stress 0.2 1/(n+1/2) 1/(n+1/2) 0.3 0.15 total vorticity (b) 0.2 0.1 outer wall shear: odd terms (d) 0.1 0.05 0 0 2 1 0 1 3 -1 -2

2.5 1.5 0.5

0.9 0.8 0.7 0.6 0.5 0.4 0. -0.5 -1.5

n-1 n

t t

n-1 n /

/d o o 0.1 0.7 0.09 n n a b 0.6 0.08 0.5 0.07 -series of the ﬂux ratio, friction ratio, total vorticity an ǫ 0.06 0.4 1/(n+1/2) 0.05 1/(n+1/2) 0.3 0.04 ﬂux and friction ratio 0.2 0.03 (a) outer wall shear: even terms (c) Domb-Sykes plot for the 0.02 0.1 0.01 1 .9 0 0 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 1 3 0.9 0.8 0.7 0.6 0.5 0.4 0.

Figure 2.3:

n-1 n t /t

e e Chapter 2. Extended Stokes series: circular cross-section 55

1 -0.7726 2 -0.6258 -0.4056 3 -0.5227 -0.2649 -0.1593 4 -0.4460 -0.1776 -0.0684 -0.0230 5 -0.3865 -0.1190 -0.0166 0.0265 0.0452 . .

96 0.04299 0.0568 0.0224 -0.0096 -0.0351 . . . -0.0291 -0.0290

Table 2.4: Neville table estimates of the critical exponent at ǫ = 1 for the ﬂux ratio

The observations regarding the behaviour of the Domb-Sykes plot and the subsequent Neville analysis, are indicative of conﬂuent singularities of the form [27]:

ǫ α1 ǫ α2 A 1 + A 1 1 − ǫ 2 − ǫ c1 c2 or ǫ α1 ǫ α2 A 1 1 , (2.49) 1 − ǫ − ǫ c1 c2

where ǫc1 and ǫc2 are equal, nearly equal or the same distance from the origin. The behaviour of the Domb-Sykes plots for such series depends on the respective sizes of the amplitudes, A1 and A2, the critical exponents, α1 and α2, and the location of the singularities, ǫ1 and ǫ2. Where the singularities are equal, the Domb-Sykes plots of the lower order coefficients, may be inherently curved or possess a change in the sign of the coefficients as the weaker singularity effects the behaviour of the lower coefficients in the series. Alterna- tively, where the singularities are the same distance from the origin the Domb-Sykes plots possess oscillations [87]. Such behaviours are better understood by example. As such, the 2.3 Analytic continuation 56 following model functions are considered:

1 1 (1 ǫ) 2 0.5 (1 ǫ) 6 , (2.50) − − − 1 7 (1 ǫ) 2 + (1 ǫ) 15 (2.51) − − (1 ǫ)−2 +(1+ ǫ)−5 , (2.52) − and (1 ǫ)−2 (1 + ǫ)−5 . (2.53) −

The asymptotic behaviour of the ratio of the coefﬁcients, cn , of the functions (2.50) - (2.53), dictated by the dominant singularity in each, is shown in Table 2.5.

Model function Dominant singularity Asymptotic behaviour

1 Additive conﬂuent singularities (2.50) 0.5 (1 ǫ) 6 1 7 − − − 6n 7 22 Additive conﬂuent singularities (2.51) (1 ǫ) 15 1 − − 15n −5 4 Additive singularities at equal distance from (1 + ǫ) 1+ − n the origin (2.52) Multiplicative singularities at equal distance (1 + ǫ)−5 1+ 4 − n from the origin (2.53)

Table 2.5: Dominant singularity and corresponding asymptotic behaviour of the ratio of the coefﬁ- cients of model functions (2.50) - (2.53)

The Domb-Sykes plots of the first 100 terms of the Taylor expansion for the model functions (2.50) - (2.53) are shown in Figure 2.4. The lower order coefficients of the Taylor expansion of the additive confluent singularities (2.50), are all positive whereas the higher order coefficients are negative. This is reflected in its Domb-Sykes plot Figure 2.4a, and such behaviour can also be found in the Domb-Sykes plots of the odd terms of the coefficients of the outer wall shear stress Figure 2.3d. In comparison, the Domb-Sykes plot of the additive confluent singularities (2.51), shown in Figure 2.4b , exhibit a small curvature. This is also the case for the Domb-Sykes plot for the total vorticity, Figure 2.3b, and the even terms of the outer wall shear stress, Figure 2.3c. Finally, the additive and multiplicative singularities at equal distance from the origin, (2.52) and (2.53), both exhibit small oscillations in the Domb-Sykes plots, shown in Figure 2.4c and Figure 2.4d respectively. Chapter 2. Extended Stokes series: circular cross-section 57 ual 0.7 0.7 0.6 0.6 -(1+4s) (1-22/15s) 0.5 0.5 0.4 0.4 1/(n+1/2) 1/(n+1/2) 0.3 0.3 0.2 0.2 0.1 0.1 0 0 Domb-Sykes plot for additive confluent singularity (2.51) 1 0 -1 -2 -3 -4 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 -1.5 -2.5 -3.5

Domb-Sykes plot for two multiplicative singularities at eq distance from the origin (2.53)

n-1 n n-1 n /c /c c c (b) (d) 0.7 0.7 tance 0.6 0.6 -(1+4s) (1-7/6s) 0.5 0.5 Domb-Sykes plot for model functions (2.50) - (2.53) 0.4 0.4 1/(n+1/2) 1/(n+1/2) Figure 2.4: 0.3 0.3 0.2 0.2 0.1 0.1 0 0 4 3 2 1 0 -1 -1 -2 -3 -4 -5 -6 Domb-Sykes plot for additive conﬂuent singularity (2.50)

3.5 2.5 1.5 0.5 -0.5 -1.5 -2.5 -3.5 -4.5 -5.5

n-1 n-1 n n Domb-Sykes plot for two additivefrom singularities the at origin equal (2.52) dis (a) /c c /c c (c) 2.3 Analytic continuation 58

Though not as marked, this appears to be the case for the Domb-Sykes plot of the ﬂux ratio and friction ratio, Figure 2.3a.

Clearly, such singularity structures require significantly more coefficients to distinguish the critical exponent, or the location of the singularity by Domb-Sykes analysis. Attempts have been made to overcome this difficulty by considering the following:

‘Conﬂuent-singularity analysis’ due to Baker and Hunter [43]. • Alternative extrapolation methods to the Neville table, which may hasten the con- • vergence of the ratio of the coefﬁcients [27, 72]. This includes Wynn’s ǫ algorithm, Brezinki’s θ algorithm, Barber-Hamer algorithm, Levin U-T transform and Lubkin’s three-term transformation.

Neither of these approaches have proved successful. For reference, the results of these studies are given in more detail in Appendix B.4 and B.5 respectively.

The difficulties outlined above, are the same which faced Van Dyke [88]. This is surprising, as this study benefits from the calculation of further coefficients and a more accurate estimate of Kc. Importantly, it is in overcoming the difficulty in identifying the analytic structure of the Euler transformed series of the flux ratio that Van Dyke concludes that it − 1 − 1 behaves asymptotically like K 10 for large K [88] contrary to K 6 as proposed by boundary layer theory [76]. Consequently, below an overview of Van Dyke’s analysis of the Euler transformed series of the flux ratio is given and his conclusions are addressed in light of the further coefficients available in the present study.

Van Dyke, reported the closest singularity to the origin for the ﬂux ratio Q(ǫ) at ǫ =1 [88], despite the oscillations present in the corresponding Neville table. As discussed above, the calculation of the critical exponent α is more difﬁcult as the Neville table is completely irregular. As a result, he adopted the following techniques to provide an estimate of the critical exponent α:

Completing the series • Critical-point renormalisation • Chapter 2. Extended Stokes series: circular cross-section 59

Comparison of the series expansion for the logarithm of the flux ratio Q(ǫ) with • ln(1 x)α − Linear Padéapproximants of the K-series of the flux ratio •

1 He concluded that each of these techniques suggested that α = 20 . Therefore, he mul- 1 tiplicatively extracted the singularity (1 ǫ) 20 and found the resulting series to have a − logarithmic singularity (1 ǫ) ln(1 ǫ) by Domb-Sykes analysis. He concluded that the − − Euler transformed ﬂux ratio behaved as follows:

1 Q(ǫ)=1.0343(1 ǫ)1/20 1+ (1 ǫ) ln(1 ǫ)+ f(ǫ) , (2.54) − 40 − − where f(ǫ) is a function whose expansion in ǫ can be calculated.

Accordingly, the Euler transformation (2.41), implies :

2 − 1 − 1 Q(ǫ) (K ) 20 (K) 10 . (2.55) ∼ ∼

In comparison, boundary layer theory suggests [76]:

− 1 Q(K) K 6 , (2.56) ∼ which, using the Euler transform (2.41), implies:

1 Q(ǫ) (1 ǫ) 12 . (2.57) ∼ −

1 That is the Euler transformed series should have a critical exponent α = 12 , if boundary layer theory is correct.

Each of the techniques he adopted to find α, listed above, have been reapplied to the higher order coefficients available in this study and their results are given in detail in Appendix B.3.1, B.3.2, B.3.3 and C.1 respecitvely. In the first three instances, the estimates of the 1 exponent α no longer converge to α = 20 . Alternatively, the linear Padéapproximant method may suggest that Van Dyke’s conclusion is correct. However, one should be careful in drawing conclusions from these in the presence of additive square-root singularities such 2.3 Analytic continuation 60 as (2.37) [27] particularly as the other techniques suggest a complicated analytic structure. Therefore, there is evidence to discard Van Dyke’s results as each of the four techniques do not corroborate any particular estimate for α. Further evidence to support this claim is found in a closer study of the Domb-Sykes plots.

The Domb-Sykes plot of the flux ratio Q(ǫ), shown in Figure 2.5, exhibits an oscillation between the asymptote in accordance with Van Dykes analysis, 1 21 , and the cor- − 20n responding asymptote for the boundary layer theory, 1 13 . Notably, the first twelve − 12n coefficients available to Van Dyke do not exhibit this behavi our and in fact appear to converge to the asymptote 1 21 . − 20n 1 an

0.99

(1-13/12

(1 21/20 0.98

0.97

0.96

0.95 0.99

0.94 0.985

0.93 0.98 0.92 0.975 0.91

0.97 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.026 0.028 0.03 0.9 0 0.02 0.04 0.06 0.08 0.1 1/(n+1/2)

Figure 2.5: Domb-Sykes plot for ﬂux ratio Q(ǫ)

The Domb-Sykes plot of the coefﬁcients of the series expansion of (1 ǫ)−1/20Q (ǫ) is − shown in Figure 2.6. Though the plot is curved, it is apparent that the terminal gradient does not approach 1 which should be the case if a logarithmic singularity is present as in Van Dyke’s asymptotic model (2.54).

In conclusion, though no asymptotic model can be proposed, Van Dyke’s suggestion that the ﬂux behaves like K−1/20 is not borne out by more accurate calculations. It is proposed that the difﬁculties which arise are most likely due to a cluster of singularities close to ǫ = 1. In the case of the Dean equations, the existence of such coincident singularities Chapter 2. Extended Stokes series: circular cross-section 61

1 (1-2s)

0.8

0.6

0.4

0.2

0 0 0.1 0.2 0.3 0.4 0.5 0.6 1/(n+1/2)

Figure 2.6: Domb-Sykes plot for the coefficients of the series expansion of (1 ǫ)−1/20Q (ǫ) − can also be argued theoretically. The ǫ-series may be expected to possess singularities at the critical points of the Dean equations, several of which have been identified numerically for large K (see Chapter 1). These large values of K are mapped close to ǫ = 1, for example the saddle node bifurcations at K 99.17 and K 675 [73] would be mapped ≈ ≈ to ǫ 0.999895 and ǫ 0.999998 respectively. These critical points would then present ≈ ≈ as a cluster of additive and multiplicative singularities close to ǫ = 1 which provides an explanation as to why the above analysis proves difficult. Whilst further coefficients may enlighten our understanding of the behaviour of the ǫ-series the practicality of such an approach is questionable given the computational cost. An alternative analytic continuation technique which can distinguish the critical points effectively is presented in the next chapter. For now it is accepted that the Dean series solution has been analytically continued beyond its initial radius of convergence.

2.4 Comparison with numerics

The ‘Flux Ratio Paradox’ has sparked debate as to whether the analytically continued Dean series is the same solution branch as numerical and experimental results. However until now only the evaluation of the ﬂux has been presented for the Dean series. Therefore, in this section a comparison of the Euler transformed extended Stokes series with numerical 2.4 Comparison with numerics 62 results of Chapter 4 is given. In the ensuing discussion, the results will be labelled as follows:

ESS 1978/2011 - Euler transformed extended Stokes series • VDK 2011 - Van Dyke’s model (2.54) for the Euler transformed extended Stokes • series

NS - Numerical solution found by path-continuation presented in Chapter 4 •

2.4.1 Flux ratio, total vorticity and outer wall shear stress

The variation of the ﬂux ratio, total vorticity and outer wall shear stress with K are shown in Figure 2.6. The Euler transformed series and numerical results for the ﬂux ratio are in agreement for K 7, whereas for the total vorticity and outer wall shear stress they ≤ only agree up to K 5. The Euler transformed series, clearly displays a poor rate of ≤ convergence.

In Figure 2.7a Van Dyke’s model of the ﬂux ratio (2.54) is compared with the numerical results of Chapter 4 - these differ at K 100 as reported in previous studies [46, 53]. ≈ 1 ESS 1978 ESS 2011 VDK 2011 NS 0.9 0.66

0.65

0.64

0.8 0.63

0.62

0.61 Q(K)

0.6 0.7 0.59

0.58 100 120 140 160 180 200 220 240

0.6

0.5 0 50 100 150 200 250 K (a) ﬂux ratio Chapter 2. Extended Stokes series: circular cross-section 63

35 ESS 1978 ESS 2011 30 NS

20 (K) Ω 15

0 0 5 10 15 20 25 30 35 40 K (b) total vorticity

-2 ESS 1978 ESS 2012 NS -2.2

-2.4 (K) τ -2.6

-2.8

-3 0 5 10 15 20 25 30 35 40 K (c) outer wall shear stress

Figure 2.6: Comparison of the ﬂux ratio, total vorticity and outer wall shear stress ǫ-series with numerical results of Chapter 3 2.4 Comparison with numerics 64

2.4.2 Azimuthal velocity and streamfunction

The contour plots of the series solution and numerical solution are shown in Figure 2.7, on the left and right respectively. Their behaviour is in general agreement with what is found in previous studies of Dean ﬂow, detailed in Chapter 1.

Figures 2.7d and 2.7e show the contour plots for K = 1. The secondary stream function consists of two symmetric counter-rotating vortices. The axial-velocity consists of concentric circles, and the maximal central velocity has migrated to the outer wall. The two solutions appear to be in good agreement.

The contour plots are in qualitative agreement when K = 20 shown in Figures 2.7f and 2.7g. In both, the maximal axial-velocity migrates further to the outer wall and the down- pipe velocity appears independent in the y-direction (taken to be along the vertical). This is as predicted by asymptotic theory. The contours of the NS, however, present much greater distortions than the ESS.

At K = 150, shown in Figures 2.7h and 2.7i, the difference is even more apparent. Though in both, the centre of the vortices has moved closer to the outer wall, it is clear that the distortion of the axial velocity contours is not as pronounced in the ESS as in NS. Whilst the NS solution appears to exhibit the beginning of a boundary layer structure, clear from the increased incidence of contour lines on the outer wall indicating an increase in gradient change in this component, this is not the case for ESS. It is believed that the discrepancy lies in the slow convergence of the ǫ series. Indeed, the ESS plots in Figure 2.7f and 2.7h are virtually indistinguishable.

The point K = 150 is much beyond the bifurcation point K = 100 reported by numerical and experimental techniques. It is therefore unlikely that, as Dennis & Ng [23] suggested, the “Flux Ratio Paradox” might result from the extended series expansion solution being the reported four-vortex solution. Chapter 2. Extended Stokes series: circular cross-section 65

0.8

0.6

0.4

0.2 I O 0 I O

−0.2

−0.4

−0.6

−0.8

−1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (d) ESS K =1 (e) NS K =1

0.8

0.6

0.4

0.2 I O 0 I O

−0.2

−0.4

−0.6

−0.8

−1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (f) ESS K = 20 (g) NS K = 20

0.8

0.6

0.4

0.2 I O 0 I O

−0.2

−0.4

−0.6

−0.8

−1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (h) ESS K = 150 (i) NS K = 150

Figure 2.7: Contour plots of the azimuthal velocity and streamfunction as found by the ǫ-series (left) and numerical results of Chapter 4 (right) 2.5 Summary and conclusions 66

2.5 Summary and conclusions

In this chapter the extended Stokes series in the Dean number K for Dean ﬂow in a pipe of circular cross-section was considered, following closely the work of Van Dyke [88].

The Dean series (1.16) was computer-extended to the 196th term, (K196). This is more than eight times the number of coefﬁcients previously reported, of which 80 are found as rationals. Accordingly, the current solution for low Dean number ﬂow is more accurate than ever before.

Domb-Sykes analysis found the series to have a finite radius of convergence limited by an imaginary conjugate pair of square-root singularities at iK . The series was analytically ± c continued by means of an Euler transform by recasting it in the variable ǫ. Analysis of the ǫ-series indicated that the analytic continuation was successful. Though the location of the closest singularity from the origin in the ǫ-plane and its critical exponent proved difficult to determine. Whilst no asymptotic structure can be put forward for large K, re-evaluating Van Dyke’s analysis using the higher-order coefficients of the ǫ-series calculated here, no longer supports his asymptotic model (2.54). This may result from the complicated analytic structure of the solution, with known critical points, which correspond to the bifurcation to multiple-vortex solutions [20, 25, 73, 96], mapped close to ǫ = 1 in the ǫ-plane. In this instance it is possible that further coefficients may provide greater insight.

Interestingly, the Euler transformed series was found to produce convincing results for Dean numbers where the original series became unintelligible. As expected, the analytically continued series is a two-vortex solution. Though for intermediate values of K the Euler transformed solution agreed with numerical results of Chapter 3 for larger K the series solution has a slow rate of convergence.

In conclusion, the results of this chapter indicate that Van Dyke’s asymptotic model for the ﬂux ratio (2.54) is not correct. The study suggests that an alternative analytic continuation technique which can distinguish the critical points for large K may prove more effective. One such possibility is the generalised Pad´eapproximant, introduced in the next chapter. 67

Chapter 3

Pade´ approximants

In this chapter analytic continuation of the Dean series is achieved by Padéapproximants. Motivation for this originates from the analysis of Chapter 2 which revealed the need for an alternative analytic continuation technique to the Euler transform. Preliminary work with linear and quadratic approximants have shown that the complicated analytic structure of the solution limits their applicability (discussed in Appendix C.1 and C.2) and prompts the consideration of the generalised Padéapproximants introduced by Drazin & Tourigny [28]. These approximants are known to be effective in non-linear problems with multiple solutions not only as a means of analytic continuation but also in constructing the bifurcation diagram of the Dean equations (1.12) and (1.13). The chapter begins with a brief overview of the theory outlined in Drazin & Tourigny’s paper. The technique is then applied to the flux ratio, total vorticity and outer wall shear stress; with both real and imaginary K considered. For real K the approximants analytically continue the Dean series up to K = 150 and the extension is found to be in agreement with numerical results of Chapter 4 and not Van Dyke’s asymptotic model (2.54). For imaginary K, the technique identifies four solution branches which bifurcate at iKc. It is postulated that the singularity at iKc is symmetry breaking.

3.1 Generalised Pade´ approximants - procedure

This chapter focuses on the generalised Pad´eapproximant of Drazin & Tourigny [28] applied to the Dean series. A brief account of the theory outlined in their paper is given 3.1 Generalised Pade´ approximants - procedure 68 below.

For some function p(K), assumed to satisfy p(0) = 0 without loss of generality, the linear Pad´eapproximant ∗ is found by constructing a linear equation in p(K):

f (K) p (K) f (K)=0 (3.1) 1 − 0 f (K) = p (K)= 0 , (3.2) ⇒ f1 (K) where f0(K) and f1(K) are polynomials of order M and N respectively, sought so that the N n truncated Taylor expansion PN (K)= n=1 pnK is a solutionof (3.1) to order M +N +1. If p(K) is a meromorphic function then the approximant is an effective way by which to analytically continue PN (K) beyond its initial radius of convergence. In addition, the poles of p(K) can be estimated by the zeros of f1(K).

Solutions to non-linear problems are usually not, however, meromorphic. They generally possess many solution branches which arise at bifurcations. Accordingly, it is more reasonable to postulate that p(K) is an algebraic function satisfying some higher order polynomial F = F (K,p) of degree d 2 †: d d ≥

d l l−m m Fd(K,p)= fl−m,mK p =0, (3.3) m=0 l=1 for which p (K) is a solution up to order N +1,

F (K,P (K)) = O KN+1 as K 0. (3.4) d N → The root that vanishes at K =0 is made unique by imposing,

∂F d (0, 0)=1 (3.5) ∂p = f =1. (3.6) ⇒ 0,1

∗A brief overview of linear and quadratic approximants as well as relevant results for the current problem can be found in Appendix C.1 and C.2

† Unlike other higher order Pad´eapproximants [17], Drazin and Tourigny construct Fd such that its order in both K and p increases with d Chapter 3. Pade´ approximants 69

The system then has 1+ d (m +1) = 1 (d2 +3d 2) coefﬁcients (f ) to calculate, m=2 2 − i,j and N equations. It is made fully determined by taking:

1 N = d2 +3d 2 . (3.7) 2 − Once the coefﬁcients are found, the approximant is constructed by solving:

Fd (K,p)=0. (3.8)

Such an approximant can construct up to d solution branches from a suitably accurate Tay- lor expansion. Some of the solutions to (3.8) may be spurious, but these can be identiﬁed by their transient nature as d is increased. From a study of the solutions a reconstruction of the whole bifurcation diagram of p(K) can be made. In this way, Drazin & Tourigny have been able to expand the technique of Pad´eapproximants to more general functions.

The above technique was applied to the computer-extended K-series of the ﬂux ratio (2.12), the total vorticity (2.16) and the outer wall shear stress (2.15 ) found in Chapter 2. This is different to the Euler transformation of Chapter 2 which can be applied globally to the Dean series (1.16). The corresponding polynomials Fd(K,p)=0 considered were as follows.

98 2n For the ﬂux ratio, Q(K)= n=0 anK : F K2, Q(K) a , for d =2 ... 12. (3.9) d − 0 97 2n+1 For the total vorticity, Ω(K)= n=0 dnK : Ω(K) F K2, d , for d =2 ... 12. (3.10) d K − 0

196 n For the outer wall shear stress, τ(K)= n=0 tnK : F (K, τ(K) t ) , for d =2 ... 18. (3.11) d − 0

The maximum order of the polynomials, d, follow from (3.1). The transformations ensure that the assumption p(0) = 0 is satisﬁed. The results however, are presented for Q(K), Ω(K) and τ(K) with suitable inversion. In this way, there is no need to solve the Dean 3.2 Results 70 equations (1.12) and (1.13), entirely, but only construct the quantities of interest.

Initial solutions of (3.9)-(3.11) were found using MAPLE and the branches were continued using standard path-continuation methods summarised in Appendix C.4.

3.2 Results

The approximants were constructed for real and imaginary K, though they also apply for complex K. Whilst real K is of physical interest, the ﬂow for imaginary K, being completely complex, is not. Notwithstanding, the consideration of imaginary K is of mathematical interest, particularly as the Dean series (1.16) was found to have square-root singularities at iK (as shown in Chapter 2). The results for real and imaginary K are presented ± c below.

Discussion of the notation. Four solution branches of the Dean equations which will be discussed herein are labelled Branch I, II, III and IV. A schematic representation of the bifurcation diagram found is depicted in Figure 3.1. Branch I corresponds to the two-vortex solution, ﬁrst calculated by Dean [22] ‡. This branch is real for real K and becomes complex when K is imaginary. For imaginary K three further complex branches are detected at the bifurcation point iKc. Branch II exists when the imaginary part of K is less than Kc,

Branches III and IV when the imaginary part of K is greater than Kc. This notation will be used throughout this section.

Branch II (complex) Branch IV ( complex)

Branch III (complex) Branch I (real) Branch I (complex)

iK c 0 K Figure 3.1: Schematic representation of bifurcation diagram for imaginary K. The dashed line denotes K = 0. Solutions for real K are to the right of dashed line, and those of imaginary K to the left.

‡In the notation of Chapter 1 this corresponds to Branch 1 Chapter 3. Pade´ approximants 71

Where possible, comparison has also been made with the numerical results of Chapter 4. The notation is equivalent to Chapter 2:

VDK 2011 - Van Dyke’s model (2.54) for the Euler transformed extended Stokes • series

GPA - Generalised Pad´eapproximant • NS - Numerical solution found by path-continuation presented in Chapter 4 •

3.2.1 Real K

Branch I has been analytically continued for real K. A detailed account of all the approximants, d = 2 ... 12, will be given for the ﬂux ratio in order to illustrate the method. Then the results of the analysis for the ﬂux ratio, total vorticity and outer wall shear stress will be compared with numerical results of Chapter 4.

3.2.1.1 An illustrative account - the ﬂux ratio

As expected, Branch I is real for real K. The different order approximants of Branch I, for the ﬂux ratio, exhibit on of three behaviours which are shown in Figure 3.2.

Figure 3.2a shows approximants d = 5 and 8. These approximants to Branch I possess two turning points and become negative for finite K. The value of K at which the turning points occur increases with d. Beyond these turning points a secondary branch is detected which takes the form of an isolated isola. The position of this isola varies with d. This secondary branch is therefore spurious. Figure 3.2b shows approximants d = 2, 4, 7, 9 and 12. These approximants to Branch I become negative for finite K. The value of K at which this occurs increases with d. In addition to its transient nature, this behaviour can be disregarded on physical grounds; there is no reason to suspect the flux to become negative for any value of K > 0. Figure 3.2c shows approximants d = 3, 6, 10 and 11. These approximants to Branch I are characterised by a single turning point at which the branch tends to infinity. The value of K at which the turning point occurs increases with d. For these approximants a secondary branch is identified, which for small K was negative. This secondary branch has a turning point near the turning point of Branch I, after which 3.2 Results 72 it appears to follow the general behaviour of Branch I, but this secondary branch is also spurious.

2 d = 5

d = 8

1.5 spurious isolated isola ) 1 Q(K

Branch I (real) 0.5

0 0.1 1 10 100 K (a) Branch I and spurious branches for d =5, 8 1.2 d = 2 d = 4 d = 7 1 d = 9 d = 12

0.8

0.6 Q(K) Branch I (real)

0.4

0.2

0 0.1 1 10 100 K (b) Branch I and spurious branches for d =2, 4, 7, 9, 12 Chapter 3. Pade´ approximants 73

1 Branch I (real)

0.8 secondary spurious branches

0.6 Q(K)

0.4 d = 3 d = 6

0.2 d = 10 d = 11 0 0.1 1 10 100 K (c) Branch I and spurious branches for d =3, 6, 10, 11

Figure 3.2: Plot of the ﬂux ratio on Branch I for real K found using generalised Pad´eapproximants. Spurious branches are dotted.

The ‘true’ flux ratio is identified by comparing approximants d = 2 ... 12 together as in Figure 3.3. The approximants agree if they are in accordance up to three significant figures. This plot indicates the highest order approximants are in agreement up to K 150. The ≈ dashed lines indicate where the solutions of the lower order approximants differ from these.

2 d = 2 d = 3 d = 4 d = 5 1.5 Branch I (real) d = 6 d = 7 d = 8

(K) 1 Q d = 9 d = 10 d = 11 0.5 d = 12

0 0.1 1 10 100 K Figure 3.3: Plot of the ﬂux ratio on Branch I for real K found using the generalised Pad´eapproximants. Spurious behaviour is dashed. 3.2 Results 74

A similar analysis was completed for the total vorticity and outer wall shear stress, the details of this is not discussed as they are comparable to the ﬂux ratio these are not discussed.

3.2.1.2 Discussion of results and comparison with other techniques

The generalised Padéapproximants have analytically continued the flux ratio and outer wall shear stress on Branch I up to K 150. The total vorticity has been followed up to ≈ K 85 on this branch. Though numerical studies have identified four-vortex solutions ≈ which bifurcate away from Branch I at K 99.17 [25, 20, 95, 73], the approximants have ≈ not done so. In hindsight, this is not surprising as the generalised Padéapproximants can only detect branches which bifurcate from the analytically continued branch [28]. Yang & Keller [96] such a bifurcation from Branch I at K 68, 600, though many repute that such ≈ a bifurcation occurs for finite K.

In Figure 3.4a, the flux ratio on Branch I found by the generalised Padéapproximants is compared with Van Dyke’s asymptotic model (2.54) and the numerical results of Chapter 4. Notably, the results of the generalised Padéapproximants follow closely those of the numerical results beyond K 100, where both begin to differ from Van Dyke’s asymptotic ≈ model (2.54). This supports the claim in Chapter 2 that Van Dyke’s model is incorrect. Further continuation is required to assess the asymptotic behaviour for large K and so address the ‘Flux Ratio Paradox’. This is deferred for a future study in which the Dean series is computer-extended even further than here, which in turn allows the construction of the higher order generalised Padéapproximants.

The total vorticity and outer wall shear stress also compare well, to at least three significant ﬁgures, with the numerical results of Chapter 4 as depicted in Figure 3.4b and 3.4c respectively. Chapter 3. Pade´ approximants 75

1.1 VDK 2011 GPA NS

0.9 0.7

0.69

0.68 0.8 0.67 Q(K)

0.66

0.65 0.7 0.64

0.63

0.62 0.6 0.61

0.6 100

0.5 0.01 0.1 1 10 100 K (a) ﬂux ratio

35 GPA NS 30

20 (K) Ω 15

0 0.01 0.1 1 10 K (b) total vorticity 3.2 Results 76

-2 GPA NS

-2.2

-2.4 (K) τ

-2.6

-2.8

-3 0.01 0.1 1 10 100 K (c) outer wall shear stress

Figure 3.4: Comparison on Branch I of the ﬂux ratio, total vorticity and outer wall shear stress by generalised Pad´eapproximant with numerics from Chapter 4 and Euler transformed series of Chapter 2

3.2.2 Imaginary K

The generalised Pad´eapproximant has extended our understanding in the imaginary K regime, where physical intuition is absent. It has analytically continued Branch I for imaginary K and detected the singularity at iKc. At iKc three new branches, Branch II -IV, are found to bifurcate. This last branch is only detected by consideration of the outer wall shear stress, this is discussed below,. A detailed account of all the approximants, d = 2 ... 12, will be given for the ﬂux ratio in order to illustrate the method. This is followed by a discussion of Branch II-IV for the three functions of interest. Finally Branch II found here will be compared with numerical results of Chapter 4, for the three functions of interest.

3.2.2.1 An illustrative account - the ﬂux ratio

The approximants of Branch I of the flux ratio for imaginary K are shown in Figure 3.5a. On this branch the flux is real, though the flow itself is complex. The approximants of order greater than two are identical, characterised by a turning point bifurcation at iKc. Chapter 3. Pade´ approximants 77

Estimates of Kc were calculated, in accordance with standard bifurcation theory outlined in Appendix C.4, by solving

Fd(p,K)=0 (3.12) and

∂F (p,K) d =0, (3.13) ∂p where p = Q(K) a . The results are shown in Table 3.1 and agree with the estimates of − 0 the Domb-Sykes analysis (Table 2.2). This is expected, as Branch I corresponds to the K- series of the ﬂux ratio (2.12) which, as shown in Chapter 2, is convergent for these values of K.

d N Kc 2 4 0.998957008022677315028485100212427861805103991458096184055062 3 8 1.01701905066412623131055251239684887455546066744223945474432 4 13 1.01699441129209494700346618869725358576281789916711323751848 5 19 1.01699440099325825338681126141945814701521645335148471798984 6 26 1.01699440099353827746202240932317540755288209387897035350640 7 34 1.01699440099353827824160947031506305442282587496235889914201 8 43 1.01699440099353827824160962486655760278449003731803489772101 9 53 1.01699440099353827824160962486653508113512274805710525879170 10 64 1.01699440099353827824160962486653508113571791362453723562481 11 76 1.01699440099353827824160962486653508113571791362423320211180 12 89 1.01699440099353827824160962486653508113571791362423320211157

Table 3.1: Estimates of Kc using generalised Pad´eapproximants of the ﬂux ratio

At the turning point bifurcation iKc, Branch II and III are detected. Branch II has real flux ratio for imaginary K. The different order approximants of Branch II exhibit two behaviours and these are shown in Figure 3.2b and Figure 3.2c. Approximants, d =2, 5, 6 and 7, cross the Q(K)-axis again. The point at which this axis is crossed increases with d, indicating that the behaviour of these branches is spurious. Furthermore, one should note that no secondary solutions for K =0 exist due to uniqueness of the Poiseuille flow . Approximants d =3, 4, 8, 9, 10, 11 and 12, have a turning point. These critical points occur closer to the Q(K)-axis and for larger flux ratio as d is increased. It is postulated that on Branch II Q(K) as K 0 on the imaginary axis. A more detailed discussion of → ∞ → this is deferred to section 3.2.2.4. 3.2 Results 78

The approximants of Branch I and II, are superimposed in Figure 3.6 for comparison.

1.1 d = 2 d = 3 d = 4 (Q(iK c ),iK c ) 1.08 d = 5 d = 6 d = 7 d = 8 d = 9 1.06 d = 10 d = 11 d = 12 Q(K) 1.04

Branch I

1.02

1 0 0.2 0.4 0.6 0.8 1 Im(K) (a) Branch I for d =2 ... 12

5 d = 2 d = 5 4.5 d = 6 Branch II d = 7 4

3.5

Q(K) 3

2.5

(Q(iKc c), iKc c) 2

1.5

0 0.2 0.4 0.6 0.8 1 Im(K) (b) Branch II for d =2, 5, 6, 7 Chapter 3. Pade´ approximants 79

6 d = 3 5.5 d = 4 d = 8 5 d = 9 Branch II d = 10 d = 11 4.5 d = 12 4

3.5 Q(K) (Q(iK ),iK ) 3 c c

2.5

1.5

0 0.2 0.4 0.6 0.8 1 Im(K) (c) Branch II for d =3, 4, 8, 9, 10, 11, 12

Figure 3.5: Plot of the ﬂux ratio on Branch I and II for imaginary K found using generalised Pad´e approximants.

5 d = 2 d = 3 d = 4 4 Branch II d = 5 d = 6 d = 7 d = 8 d = 9 3 d = 10 d = 11 d = 12 Q(K) 2

1 Branch I

(Q(iK c),iK c) 0 0 0.2 0.4 0.6 0.8 1 Im(K) Figure 3.6: Plot of the ﬂux ratio on Branch I and II as found using generalised Pad´eapproximants. Spurious lines are dashed. 3.2 Results 80

Branch III has a complex flux ratio for imaginary K. Figure 3.7a and 3.7b show the real and imaginary parts, respectively, of the approximants d = 2 ... 12. The real part of approximants d = 2 and 3 become negative, whilst the real part of the approximant d = 4 tends to as purely imaginary K becomes large. Other approximants do not exhibit ∞ any interesting behaviour before they diverge. These branches are found to agree up to K i100. ≈ An equivalent analysis was carried out for the total vorticity and outer wall shear stress, the outcome of which was similar to that which has already been described in the discussion concerning the flux ratio. In addition to these branches, the study of the outer wall shear stress identified a further branch Branch IV. The results of this analysis discussed in the next section.

1.2 d = 2 d = 3 d = 4 1.1 d = 5 d = 6 (Q(iKc ), iKc ) d = 7 d = 8 1 d = 9 d = 10 Branch III d = 11 d = 12 0.9 Re(Q(K))

0.8

0.7

1 10 100 Im(K) (a) Real part of complex Branch III Chapter 3. Pade´ approximants 81

0 d = 2 d = 3 -0.02 d = 4 (Q(iKc ),iKc ) d = 5 -0.04 d = 6 d = 7 d = 8 -0.06 d = 9 d = 10 d = 11 -0.08 d = 12

Im(Q(K)) Branch III -0.1

-0.12

-0.14

-0.16 1 10 100 Im(K) (b) Imaginary part of complex Branch III

Figure 3.7: Plot of the ﬂux ratio on Branch III found using generalised Pad´eapproximants

3.2.2.2 Discussion of the complex bifurcation diagram for imaginary K

The generalised Pad´eapproximants have successfully identiﬁed four complex branches for imaginary K; Branch I - IV shown in Figures 3.8 and 3.9.

Branch I, is displayed in Figure 3.8. On this branch the outer wall shear stress is fully complex, whereas the ﬂux ratio and total vorticity are real and imaginary respectively. This behaviour is unsurprising as Branch I corresponds to the Dean series (1.16). Substituting K = iκ, it follows that the real part of the solution corresponds to the even terms in the expansion and the imaginary part to the odd terms. In particular, from their respective expansions, the outer wall shear stress (2.15) is complex

∞ ∞ τ(K)= τ(iκ)= ( 1)m (κ)2m te + i ( 1)p (κ)2p+1 to, (3.14) − m − p m=0 p=0 3.2 Results 82 the ﬂux ratio (2.12), is real

∞ Q(K)= Q(iκ)= a ( 1)n κ2n, (3.15) n − n=0 and the total vorticity (2.16) is imaginary

∞ Ω(K)=Ω(iκ)= i d ( 1)n κ2n+1. (3.16) n − n

The continuation of this branch identiﬁes a critical point at iKc from which Branch II-IV bifurcate.

Branch II, displayed in Figure 3.8, is detected for Im(K)

Branch III and IV are found for Im(K) >Kc.

Branch III, shown in Figure 3.9, has been followed up to K = 100i for all three functions. On this branch τ(K), Q(K) and Ω(K) are complex. It follows that at the bifurcation point iKc the symmetry of the Dean series (1.16), discussed above, is broken.

Branch IV, shown in Figures 3.9e and Figures 3.9f, has only been identiﬁed by the approximants of the outer wall shear stress τ(K). Onit τ(K) is complex. One explanation for the approximants of the total vorticity and ﬂux ratio not detecting Branch IV is that their values are the same as for Branch III.

The generalised Padéapproximant does not act globally on the Dean series (1.16), the full solution on the branches identified here is not known. Consequently, it cannot be used to test whether the solution on Branch II possesses the symmetry of Branch I as Chapter 3. Pade´ approximants 83 discussed above or whether the flux ratio and total vorticity are the same on Branch III and IV. However, the arguments outlined above seem the most plausible. Several of these conclusions are confirmed by the numerical study in the next chapter.

Solutions for the conjugate of K = K, have also been constructed. These are found to be the complex conjugate of the above branches. This follows as if w and ψ C are ∈ solutions to the Dean equations (1.12) for K C then their complex conjugates w and ψ ∈ are solutions for K. That is w and ψ C satisfy, ∈

1 ∂ψ ∂ ∂ψ ∂ ∂w ∂w 4ψ = 2ψ + 576Kw sin θ r cos θ , (3.17) ∇ r ∂r ∂θ − ∂θ ∂r ∇ ∂θ − ∂r

1 ∂ψ ∂w ∂ψ ∂w 2w = 4. (3.18) ∇ r ∂r ∂θ − ∂θ ∂r − Taking the complex conjugate of the equations above, as the complex conjugate of a sum of complex terms is equal to the sum of the conjugates:

1 ∂ψ ∂ ∂ψ ∂ ∂w ∂w 4ψ = 2ψ + 576Kw sin θ r cos θ , (3.19) ∇ r ∂r ∂θ − ∂θ ∂r ∇ ∂θ − ∂r

1 ∂ψ ∂w ∂ψ ∂w 2w = 4, (3.20) ∇ r ∂r ∂θ − ∂θ ∂r − from which, as the complex conjugate of a product of complex terms is equal to the product of the conjugates:

1 ∂ψ ∂ ∂ψ ∂ ∂w ∂w 4ψ = 2ψ + 576Kw sin θ r cos θ , (3.21) ∇ r ∂r ∂θ − ∂θ ∂r ∇ ∂θ − ∂r

1 ∂ψ ∂w ∂ψ ∂w 2w = 4. (3.22) ∇ r ∂r ∂θ − ∂θ ∂r − 3.2 Results 84 ) c 1 1 ) )),iK c c )),iK c Branch I .8 (Im(τ(iK 0 0.8 ( Im(Ω(iK Branch II 6 . 0 0.6 found using generalised Pad´eapproxi- ) ) ) K K Im(K) Im(K) ( K τ Ω( .4 Im( 0 0.4 (b) (d) Branch II for imaginary Branch I 2 . II 0 0.2 and 0 0 5 5 0 1 1 . . -1 -2 10 0

Branch I . 0.1

-0 -1

100

0 Im( (K)) Ω Im( (τ(K)) Im c 1 1 ) ress on c c ),iK c 8 (Q(iK (Re(τ(iK )),iK ) .8 0. 0 6 .6 0. 0 ) ) ) Im(K) K Im(K) ( K ( τ Branch II Q Re( 4 0.4 . (a) 0 (c) Branch I Branch II Branch I 0.2 2 . 0 0 5 5 5 5 5 5 4 3 2 1 0 0 4. 3. 2. 1. 0.

4 3 2 1 0 -1 -2

(K)) SK) Q( (τ Re Plot of the ﬂux ratio, total vorticity and outer wall shear st mants Figure 3.8: Chapter 3. Pade´ approximants 85

1.2 0

-0.02 1.1 (Im(Q(iK c)),iKc ) (Re(Q(iKc)),iKc ) -0.04 1 -0.06

0.9 -0.08 e(Q(K)) m(Q(K)) I R -0.1 Branch III 0.8 Branch III -0.12 0.7 -0.14

0.6 -0.16 1 10 100 1 10 100 Im(K) Im(K) (a) Real Q(K) (b) Imaginary Q(K)

0 14

-5 12 Branch III -10 10 (Re(Ω(iKc)),iKc ) -15 8 Branch III Ω(K))

-20 Ω(K)) ( e m( I

R 6 -25 (Im(Ω(iK )),iK ) 4 c c -30

-35 2

-40 0 1 10 100 1 10 100 Im(K) Im(K) (c) Real Ω(K) (d) Imaginary Ω(K)

-0.5 0.4 Branch III -1 Branch IV 0.2

-1.5 0

Branch IV -0.2 τ(K)) τ(K))

( -2

e (Re(τ(iKc)),iK )

c Im ( R Branch III -0.4 -2.5 -0.6 (Im(τ(iK )),iK ) -3 c c -0.8

-3.5 -1 1 10 100 1 10 100 Im(K) Im(K) (e) Real τ(K) (f) Imaginary τ(K)

Figure 3.9: Plot of the ﬂux ratio, total vorticity and outer wall shear stress on Branch III and of the outer wall shear stress on Branch IV for imaginary K found using generalised Pad´e approximants 3.2 Results 86

3.2.2.3 Comparison with numerics of Branch I and II

The numerical scheme of Chapter 4 has detected Branch I and II for imaginary K. The results for the generalised Padéapproximants and numerics are shown in Figure 3.10. The solutions from there are in agreement, to at least two significant figures, with the generalised Padéapproximants.

1.5 GPA GPA NS 14 NS 1.4 12 1.3 10 1.2 8 Q(K) Ω(K)) 1.1 ( m 6 I

1 4

0.9 2

0.8 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Im(K) Im(K) (a) Q(K) (b) Imaginary Ω(K)

-1 0 GPA GPA NS NS

-1.2

-0.2

-1.4

-1.6 -0.4 (K)) (K)) τ τ -1.8 Im( Re ( -0.6

-2

-2.2 -0.8

-2.4

-1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Im(K) Im(K) (c) Real τ(K) (d) Imaginary τ(K)

Figure 3.10: Comparison on Branch I and II of the ﬂux ratio, total vorticity and outer wall shear stress by generalised Pad´eapproximant with numerics from Chapter 4

3.2.2.4 Asymptotic analysis of Branch II

In this section an asymptotic structure for Branch II is postulated. In particular, the generalised Pad´eapproximants indicate that Q(K) as the imaginary part of K tends → ∞ to zero on Branch II. As w scales like Q(K), a solution to the Dean equations (1.12) and (1.13) is sought such that w as K 0 along the imaginary axis. →∞ → Chapter 3. Pade´ approximants 87

From (1.13) it follows that for such a scaling the driving pressure term is of lower order. Consequently, there must be a balance between the inertial and viscous terms for the azimuthal equation (1.13):

1 ∂ψ ∂w ∂ψ ∂w = 2w. (3.23) r ∂r ∂θ − ∂θ ∂r ∇

From equation (3.23), assuming that there is no boundary layer structure so that the length- scales are of order one,

ψw w, (3.24) 1 ∼ = ψ 1. (3.25) ⇒ ∼

In the vorticity equation (1.12), there is a balance between the inertial, centrifugal and viscous terms

1 ∂ψ ∂ ∂ψ ∂ sin θ ∂w ∂w 4ψ 2ψ Kw cos θ (3.26) ∇ ∼ r ∂r ∂θ − ∂θ ∂r ∇ ∼ r ∂θ − ∂r = ψ ψ2 Kw2. (3.27) ⇒ ∼ ∼

From (3.25),

1 1 Kw2 (3.28) ∼ ∼ = w K−1/2. (3.29) ⇒ ∼

Consequently, there exists a solution with inﬁnite ﬂux Q as K 0 provided there is a → complex solution to the unforced Dean equations,

1 ∂ψ ∂ ∂ψ ∂ sin θ ∂w ∂w 4ψ = 2ψ + iw cos θ , (3.30) ∇ r ∂r ∂θ − ∂θ ∂r ∇ r ∂θ − ∂r

1 ∂ψ ∂w ∂ψ ∂w 2w = , (3.31) ∇ r ∂r ∂θ − ∂θ ∂r satisfying the no-slip boundary conditions, where w has been rescaled with K−1/2. 3.3 Summary and conclusion 88

The results of the generalised Padéapproximants corroborate this to some degree. Given that the length-scales are of order one it is expected that, as stated earlier, the flux ratio scales like the azimuthal velocity. Figure 3.11 shows the log-log plot of Branch II for the flux ratio as found by the generalised Padéapproximants. Though the flux ratio is not followed to values very close to K =0 on this branch, its behaviour is consistent with the asymptotic structure Q(K) w K−1/2 as iK 0, as discussed above. ∼ ∼ → 0.85 GPA -1/2 log(Im(K)) 0.8

0.75

0.7 g(Q(K))

lo 0.65

0.6

0.55

0.5 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 log(Im(K)) Figure 3.11: Log-log plot of the ﬂux ratio on Branch II

3.3 Summary and conclusion

In this chapter the family of Pad´eapproximants has been investigated for the purpose of analytically continuing the Dean series (1.16) beyond its radius of convergence. The complicated analytic structure of the solution was found to limit the applicability of linear and quadratic approximants (discussed in Appendix C.1 and C.2) and prompted consideration of the generalised Pad´eapproximants introduced by Drazin & Tourigny [28]. This technique had the further attraction of being able to reconstruct the bifurcation diagram of the problem.

The generalised Pad´eapproximants were constructed for the ﬂux ratio (2.12) , total vorticity Chapter 3. Pade´ approximants 89

(2.16) and outer wall shear stress (2.15). These approximants are particularly suited to the solution of non-linear problems which are known to have multiple solutions, as is the case here. Accordingly, they have proved successful as a means of analytic continuation and also in constructing part of the bifurcation diagram of the Dean solutions in the complex K-plane.

Branch I was analytically continued for real K 150. Comparison of the approximants ≤ for the three functions agreed with the numerics of Chapter 4 where the latter shows disagreement with Van Dyke’s model (2.54). This indicates that the discrepancy between Van Dykes’s work and numerical results was caused by the inefficacy of employing an Euler transform as a means of analytic continuation for this problem; in agreement with the conclusions of Chapter 2. The four-vortex solution, widely accepted to exist at K = 100, was not detected as was initially hoped. As stated earlier, this is not surprising as the generalised Padéapproximants can only detect branches which bifurcate from the analytically continued branch [28]. As Yang & Keller [96] reported that the four-vortex solution does not bifurcate from Branch I until K 68, 600, this may explain the encountered difficulty. ≈ Construction of the bifurcation diagram for imaginary K however, was more successful.

The continuation of Branch I for imaginary K identified a bifurcation point at K = iKc. At this critical point, Branch II - IV were identified. Branch III and IV were continued up to K = 100i. Branch IV was only detected by the outer wall shear stress. One explanation for this is that Branch III and IV may have the same total vorticity and flux ratio.

These results have provided further understanding of the singularity at iKc. On Branch I, for imaginary K, whilst the outer wall shear stress is complex, the ﬂux ratio is real and the total vorticity is imaginary. This follows from the parity of the Dean series for these functions. Whilst this is also the case on Branch II, on Branch III the ﬂux ratio and total vorticity become fully complex. Therefore the bifurcation at iKc must break the symmetry of the Dean series of w(r, θ) and ψ(r, θ).

It has been postulated that on Branch II, the azimuthal velocity w K−1/2 and the stream- ∼ function ψ K0 as K 0 on the imaginary axis. The behaviour of the approximants ∼ → appear to agree well with this behaviour.

Consideration of the conjugate of iK, iK, has shown that the solution branches in this − 3.3 Summary and conclusion 90 regime are the complex conjugates of Branch I - IV. This is expected by considering the conjugation of the Dean equations which indicate that if w and ψ C are solutions for ∈ K C then their complex conjugates w and ψ are solutions for K. ∈ In conclusion, the generalised Pad´eapproximants presents an alternative analytic continuation technique to the Euler transformation of Chapter 2, and it fares well in comparison with the numerical solutions of Chapter 4. The technique has its limitations, as discussed it has yet to provide further understanding of the bifurcation diagram from real K, though it has proved successful for imaginary K. The study in this latter regime, does highlight one further drawback of this technique in that it does not analytically continue the entire Dean series solution (1.16). Nevertheless, studying carefully chosen quantities such as the ﬂux ratio, total vorticity and outer wall shear stress, one can use this technique to infer certain aspects of the structure of the solution for imaginary K. This is a particularly novel contribution of this work as little is known for the solutions in this regime. 91

Chapter 4

Numerical solution of the Dean equations for complex K

In this chapter two numerical schemes are developed to solve the Dean equations for complex K. Specific aims of this chapter are to provide an independent verification of the solution for real K and to gain a further understanding of the complex branches identified in Chapter 3. The chapter begins with the extension of the finite curvature Dean equations to complex K. The two numerical schemes used to solve the equations and the results from each are then presented. The first technique considered is based on the fictitious time- stepping scheme of Siggers & Waters [73] and the second on the path-continuation of Yang & Keller [96], each of which is extended to solve the Dean equations for complex K. Re- sults from these schemes corroborate the symmetry of Branch II postulated in Chapter 3 and further suggest that this branch, as well as Branch III-IV may be unstable.

4.1 The complex K ﬁnite curvature Dean equations

Though the results of the small curvature limit are of interest, the numerical schemes are developed for the Dean equations of ﬁnite curvature δ with complex K = KR + iKI . The following notation is introduced: 4.1 The complex K ﬁnite curvature Dean equations 92

∂f ∂g ∂f ∂g (f,g)= , (4.1) J ∂r ∂θ − ∂θ ∂r sin θ ∂f ∂f (f)= cos θ . (4.2) H r ∂θ − ∂r and the ﬁnite curvature Dean equations, (1.10) and (1.11) in Chapter 2 for complex K are:

1 w 4ψ = J hψ, 2ψ +576(K + iK ) (w), (4.3) ∇ r ∇ R I h H 1 δw 2w = (ψ,w) (hψ) 4, (4.4) ∇ rhJ − h H − where, as deﬁned in Chapter 2,

a h =1+ δr cos θ and δ = . (4.5) L

The down-pipe velocity and the streamfunction are fully complex,

w (r, θ)= wR (r, θ)+ iwI (r, θ) ,

ψ (r, θ)= ψR (r, θ)+ iψI (r, θ) , 2ψ (r, θ)= 2ψ (r, θ)+ i 2ψ (r, θ) . (4.6) ∇ ∇ R ∇ I

Two numerical schemes were used to solve this complex system. The ﬁrst is an alteration of the ﬁctitious time scheme of Siggers & Waters [73] and the second is based on the path- continuation technique of Yang & Keller [96]. The details of these schemes and the results from each are discussed in turn.

Notation. In accordance with Chapter 3 and 4 the following notation is used:

GPA - Generalised Pad´eapproximant solution • TNS - Numerical solution found by the ﬁctitious time scheme • NS - Numerical solution found by path-continuation • Chapter 4. Numerical solution of the Dean equations for complex K 93

4.2 Fictitious time scheme

In the ﬁctitious-time scheme steady solutions to the time-dependent version of the complex- iﬁed Dean equations (1.10) and (1.11) are sought by a pseudo-spectral technique. Identities (4.6) are substituted into the time-dependent equations and these are split into their real and imaginary parts. Accordingly, the number of equations and variables is doubled and the number of parameters is increased by one compared to its real counterpart. The real parts of the Dean equations are:

∂w 1 1 R + 2w = (w , hψ ) (w , hψ ) ∂t ∇ R hr J R R − hr J I I δwR δwI 4 + ψR ψI , (4.7) h H − h H − h 2 2 2 ∂ ψR 4 1 ψR 1 ψI ∇ + ψR = ∇ , hψR ∇ , hψI ∂t ∇ r J h − r J h K K + 576 R (w w w w ) 576 I (w w + w w ) h RH R − I H I − h RH I I H R (4.8) and the imaginary parts are:

∂w 1 1 I + 2w = (w , hψ )+ (w , hψ ) ∂t ∇ I hrJ R I hrJ I R δwR δwI + ψI + ψR, h H h H 2 2 2 ∂ ψI 4 1 ψR 1 ψI ∇ + ψI = ∇ , hψI + ∇ , hψR (4.9) ∂t ∇ r J h r J h K K + 576 R (w w + w w )+576 I (w w w w ) . h RH I IH R h RH R − I H I (4.10)

The above system is real. Presented below is an overview of the computational details of this scheme. The program is a direct alteration of that of Siggers & Waters [73], which was written in Fortran 95. The results and their implications are also discussed. 4.2 Fictitious time scheme 94

4.2.1 Overview of the procedure

0 0 0 0 0 For some suitable initial estimate X =(wR, wI , ψR, ψI ), at (KR,KI, δ), equations (4.10) are forward stepped in time until a steady solution is obtained.

The temporal discretisation, with time step dt, is achieved by the implicit Euler method, which is second order accurate. Denoting Xn as the nth temporal iterate, this is:

Xn+1 Xn − . (4.11) dt

In the r-direction uniform discretisation is used; r = j dr, dr = 1 for 0 j M +1. j (M+1) ≤ ≤ The derivatives are approximated by the central difference operator, D0 which is second order accurate. On some mesh function u(rj) this is deﬁned as:

u(r ) u(r ) D u(r )= j+1 − j−1 . (4.12) 0 j 2dr

In the azimuthal direction pseudo-spectral discretisation is performed by approximating the n n n n solutions (wR, wI , ψR, ψI ) as:

κ κ κ wn (r, θ)= wn(r)eikθ, ψn (r, θ)= f n(r)eikθ, 2ψn (r, θ)= gn(r)eikθ, R k R k ∇ R k k=−κ k=−κ k=−κ (4.13)

κ κ κ wn(r, θ)= W n(r)eikθ, ψn(r, θ)= F n(r)eikθ, 2ψn(r, θ)= Gn(r)eikθ, I k I k ∇ I k k=−κ k=−κ k=−κ (4.14) where the non-linear terms are evaluated in the real space and then transformed into the spectral space by fast Fourier transform. In this way, the need to solve a non-linear algebraic system by iteration is avoided [69]. Chapter 4. Numerical solution of the Dean equations for complex K 95

The convergence criteria for stability is:

gn(r ) gn−1(r ) wn(r ) wn−1(r ) j j err j j err (4.15) max n−1 < 1, max n−−1 < 2, (k,j)∈S1 g (r ) (k,j)∈S2 w (r ) j j G n(r ) Gn−1(r ) W n(r ) W n−1(r ) j j err j j err (4.16) max n−−1 < 3, max n−−1 < 4, (k,j)∈S3 G (r ) (k,j)∈S4 W (r ) j j where erri are chosen small positive constants and

S = (k, j): κ k κ, 0 j M +1 and gn−1(r ) >ε , (4.17) 1 { − ≤ ≥ ≤ ≥ | k j | 1} S = (k, j): κ k κ, 0 j M +1 and wn−1(r ) >ε , (4.18) 2 { − ≤ ≥ ≤ ≥ | k j | 2} S = (k, j): κ k κ, 0 j M +1 and Gn−1(r ) >ε , (4.19) 3 { − ≤ ≥ ≤ ≥ | k j | 3} S = (k, j): κ k κ, 0 j M +1 and W n−1(r ) >ε , (4.20) 4 { − ≤ ≥ ≤ ≥ | k j | 4} for chosen positive constants ε . For the results presented here err =1e 5. i i − The accuracy of the numerical solutions was determined by reﬁning the radial mesh, increasing the number of modes in the azimuthal direction and reducing the tolerance criteria.

4.2.2 Results

The numerical scheme was run in the limit of small curvature, δ =0. The KRe = KIm =0 solutions w (r, θ)=1 r2 and ψ = 0 are used as initial estimates of the solution to the 0 − 0 system. The results for the ﬂux ratio from the scheme are shown in Figure 4.1.

In this study Branch I was followed for real K up to 100, though Siggers & Waters [73] followed this solution up to K = 24, 400. However, K = 100 is sufficiently high to compare with the results from the generalised Padéapproximants of Chapter 3. Figure 4.1a shows that, as expected, the results for the flux ratio as calculated here compare well with the generalised Padéapproximant results.

For imaginary K, the scheme has successfully followed Branch I also. As shown in Figure 4.1b, the results for the ﬂux ratio are in agreement with generalised Pad´eapproximants of

Chapter 3. Near iKc the method no longer converges. Attempts to ‘jump’ onto Branch II, by perturbation of results from Branch I, have also proved unsuccessful. This suggests that Branch II is unstable in accordance with bifurcation theory. Similarly, the method does not converge for K beyond iKc, that is Branch III and IV are not detected. As a consequence, 4.2 Fictitious time scheme 96 an alternative technique is required to converge onto these branches.

1 GPA TNS

0.9

0.8 Q(K)

0.7

0.6

0.5 0.01 0.1 1 10 100

(a) K = KR

1.1 GPA TNS

1.08

1.06 Q(K)

1.04

1.02

1 0 0.2 0.4 0.6 0.8 1

(b) K = iKI

Figure 4.1: Flux ratio on Branch I: comparison of ﬁctitious time numerics (TNS) and generalised Pad´eapproximants (GPA) for real and imaginary K Chapter 4. Numerical solution of the Dean equations for complex K 97

4.3 Path-continuation

In this technique the steady complexiﬁed Dean equations, (1.10) and (1.11), are solved by path-continuation in parameters KR, KI and δ. The extension into the complex plane is achieved in light of Henderson’s work on complex bifurcation [36, 37].

Contrary to the ﬁctitious time-stepping method discussed previously, the azimuthal velocity and streamfunction are implicitly considered complex and the system is not split into its real and imaginary parts as in (4.10). The reason for this is two-fold. Firstly, the number of equations and variables is kept the same, with only the number of parameters increased by one compared to the Dean equations for real K. This is computationally advantageous as the scheme requires the inversion of the Jacobian of the discretised system. Secondly, the simple bifurcation points in the complex K-plane become multiple bifurcation points in the real system (4.10), which are more cumbersome to detect [36]. The formulation has the further advantage of only requiring small complex arithmetic considerations to the original scheme of Yang & Keller [96].

Therefore, the path-continuation technique is applied to the following rearrangement of (1.10) and (1.11):

1 w 4ψ J hψ, 2ψ 576(K + iK ) (w)=0, (4.21) ∇ − r ∇ − R I h H 1 δw 2w (hψ,w)+ (hψ)+4=0. (4.22) ∇ − rhJ h H An overview of the computational details of this scheme are given below. The program was written in Matlab. The results and their implications are also discussed.The results and their implications are also discussed.

4.3.1 Overview of the procedure

The system is written in vector equation form:

G (X; KR,KI, δ)=0, (4.23) 4.3 Path-continuation 98

where G (X; KR,KI , δ) is the ﬁnite approximations of the complexiﬁed Dean equations and X represents the solution to the equations w, ψ, 2ψ . In order to simplify the pro- ∇ ceeding discussion this is denoted by G∗. Whilst the non-linear terms are explicitly evaluated in the spectral space, the remaining spatial discretisation is similar to the time-stepping method.

In the r-direction uniform discretisation is used; r = j dr, dr = 1 for 0 j j (M+1) ≤ ≤ M +1. The derivatives are approximated by the forward (D+), backward (D−) and central difference (D0) operators, which are second order accurate. On some mesh function uj these are deﬁned as:

u u u u u u D u j+1 − j ,D u j − j−1 ,D u j+1 − j−1 . (4.24) + j ≡ dr − j ≡ dr 0 j ≡ 2dr

In the azimuthal direction spectral discretisation is performed by approximating the solutions w, ψ, 2ψ , which are now complex, as †: ∇ κ κ κ w(r, θ)= w (r)eikθ, ψ(r, θ)= f (r)eikθ, 2ψ(r, θ)= g (r)eikθ. (4.25) k k ∇ k k=−κ k=−κ k=−κ The discretised equations are arranged into the matrix G. The spectral coefﬁcients evaluated at the grid points r are w (r ), f (r ) and g (r ), for k = κ...κ and j = j k j k j k j − 0 ...M +1. These are arranged in the vector X. The order of the equations in G and the coefﬁcients in X are chosen to minimise the computational cost of this scheme. Full details of this can be found in Appendix D.1.

The solutions to (4.23) are found by path-continuation. This involves ﬁnding a solution at

(KR,KI + ds,δ) from a known solution (X; KR,KI , δ) by constructing the solution curve to (4.23). The path-continuation can be performed by either simple or pseudo-arclength continuation. The former is known to fail at turning-point bifurcations, whereas the latter, which is computationally more expensive, does not. In the discussion below, details of these two methods are given with KI used as the continuation parameter but the methods are equivalent for the other parameters (δ, KR).

∗ n this is not to be confused with the Fourier coefﬁcients Gk used in the ﬁctitious time-scheme. †Unlike Yang & Keller [96], symmetry is not imposed on the solution Chapter 4. Numerical solution of the Dean equations for complex K 99

4.3.1.1 Simple continuation

0 Let X (KI ) denote the solution to (4.23) at some point (KR,KI, δ). The solution X(KI + 0 ds) at (KR,KI + ds,δ) is sought. The ﬁrst estimate of X(KI + ds) is X (KI). It is corrected using the Taylor expansion of X(KI + ds) as follows:

1 0 X (KI + ds)= X (KI )+ dsX˙ (KI ) , (4.26)

where the correction term X˙ denotes the derivative of X with respect to KI . The correction term X˙ (KI ) is constructed from the Taylor expansion of G:

G X˙ X0 = G, (4.27) X − − = X˙ = X0 G−1 G, (4.28) ⇒ − X where GX denotes the Jacobian of G with respect to X. The process is iterated to obtain estimates Xn, for n =1 ...m, until a sufﬁciently accurate solution is found. This scheme is depicted in Figure 4.2.

Figure 4.2: Depiction of simple continuation

The above technique converges well away from bifurcation points where GX is singular, 4.3 Path-continuation 100 such as occurs at fold points. The alternative method of pseudo-arclength continuation is an effective way to traverse these turning points. This is of speciﬁc interest here, as the results of Chapter 3 demonstrate that the singularity at iKc is of a simple fold type. In practice this method is adopted when simple continuation method takes too many steps to converge.

4.3.1.2 Pseudo-arclength continuation

In the pseudo-arclength continuation, rather than parameterising with respect to KI as in the simple continuation method, a new parameter, s, is introduced and solutions are sought to the augmented system:

G (X (s) ,KI (s) ,KR, δ)=0, (4.29) N (X (s) ,K (s) ,K , δ)= Re X˙ (s)[X(s) X (s )] + K˙ (s )[K (s) K (s )] ds , I R − 0 I 0 I − I 0 − = Re X˙ (s) ∆X + K˙ (s ) ∆K ds =0, (4.30) I 0 I − with the constraint: X˙ (s) 2 + K˙ (s) 2=1, (4.31) | | | I | ˙ dX(s) ˙ dKI (s) where X (s)= ds and KI (s)= ds . The second condition, N (X (s) ,KI (s) ,KR, δ)=

0, in the augmented system requires the point [X (s) ,KI (s)] to lie on the plane normal to the tangent (see the Figure 4.3). This is a geometric constraint, which is why the real part is considered in this complex system [37, 36]. The augmented system is nonsingular at regular points and at simple fold points. Herein, G (X (s) ,KI (s) ,KR, δ) will be denoted

G and N (X (s) ,KI (s) ,KR, δ)=0 by N.

The derivatives X˙ (s) and K˙ (s) are as found by the simple continuation technique from the previous step. Accordingly, the ﬁrst estimate of (X(s0 + ds),KI(s0 + ds)) is (X + dsX˙ K˙ I ,KI + dsK˙ I ). This estimate is corrected by Newton iteration, with the iterates n n−1 n n−1 X = X + ∆X and KI = KI + ∆KI . The correction terms ∆X and ∆KI are found as follows. The Taylor expansion of the augmented system gives:

G ∆X + G ∆K = G, (4.32) X KI I − Re X˙ ∆X + K˙ ∆K = N, (4.33) I I − Chapter 4. Numerical solution of the Dean equations for complex K 101

where GX denotes the Jacobian of G with respect to X and GKI denotes the Jacobian of G with respect to KI . This is solved by constructing ξ1 and ξ2 such that:

G ξ = G, (4.34) X 1 −

GX ξ2 = GKI , (4.35) and then forming

∆X = ξ1 + ∆KI ξ2, (4.36)

Re Xξ˙ 1 + N ∆K = . (4.37) I − Re Xξ˙ 2 + K˙I The scheme is depicted in Figure 4.3.

Figure 4.3: Depiction of pseudo-arclength continuation 4.3 Path-continuation 102

4.3.2 Results

The numerical scheme was run in the limit of small curvature, δ =0. The KRe = KIm =0 solutions w (r, θ)=1 r2 and ψ = 0 are used as initial estimates of the solution to 0 − 0 the system. Path-continuation was carried out in both KR and KI . The results have been presented for comparison with the techniques of Chapter 2 and Chapter 3 in those chapters.

Branch I was followed up to KR = 630 for (M, κ) = (200, 15). This value of KR is sufficiently large for the purposes of this study which required only an independent verification for the results of Chapter 2 and 3 for KR < 150. As expected, no bifurcations points were detected on this branch within this regime. The contour plots for KR = 1, 20 and 150 are shown in Figure 2.7 in Section 2.4.2, where they were compared with the Euler transform results. The flux ratio, total vorticity and outer wall shear stress are shown in Figure 3.4 in Section 3.2.1.2, where they were compared with the generalised Padéapproximants.

Path-continuation in KI has also proved successful. Branch I was continued for KI up to K = K . Branch II was followed up to K 0.83 for (M, κ) = (200, 15). Further I c I ≈ continuation on this branch proved difficult as neither the simple or the pseudo-arclength continuation method converge. In the latter case, this may result from solving the augmented system using (4.34) and (4.35) as GX may be nearly singular sufficiently close to the turning point at iKc. Branch III and IV have not been detected. The flux ratio, total vorticity and outer wall shear stress on these branches are shown in Figure 3.10 in Section 3.2.2.3 where they were compared with the generalised Padé approximants.

The contour plots for Branch I for K =0.40000i are shown in Figure 4.4. These contours take the form determined by the Dean series (1.16). As observed in Chapter 3, substituting K = iK, it follows that the real part of the solution corresponds to the even terms in the expansion and the imaginary part to the odd terms. In particular, from (2.8) and (2.9) the real parts of w (r, θ) and ψ (r, θ) are

∞ I2n−1 J2n Re(w (r, θ)) = ( 1)nK2n E cos(2iθ)r2j (4.38) − (2n),i,j n=0 i=0 j=0 and ∞ I2n−1 J2n−1 Re(ψ (r, θ)) = ( 1)nK2n C sin(2iθ)r2j (4.39) − (2n),i,j n=0 i=0 j=0 Chapter 4. Numerical solution of the Dean equations for complex K 103 and the imaginary parts are

∞ I2n+1−1 J2n+1 Im(w (r, θ)) = ( 1)nK2n+1 E sin((2i + 1)θ)r2j+1 (4.40) − (2n+1),i,j n=0 i=0 j=0 and

∞ I2n+1−1 J2n+1−1 Im(ψ (r, θ)) = ( 1)nK2n+1 C cos((2i + 1)θ)r2j+1. (4.41) − (2n+1),i,j n=0 i=0 j=0

Consequently, for small K

I0−1 J0 Re(w (r, θ)) E cos(2iθ)r2j + O(K2) (4.42) ≈ 0,i,j i=0 j=0 and I0−1 J0−1 Re(ψ (r, θ)) C sin(2iθ)r2j + O(K2) (4.43) ≈ 0,i,j i=0 j=0 and the imaginary parts are

I1−1 J1 Im(w (r, θ)) K E sin((2i + 1)θ)r2j+1 + O(K3) (4.44) ≈ − 1,i,j i=0 j=0 and I1−1 J1−1 Im(ψ (r, θ)) K C cos((2i + 1)θ)r2j+1 + O(K3). (4.45) ≈ − 1,i,j i=0 j=0 This behaviour certainly corresponds with that observed in Figure 4.4.

The contour plots for Branch II for K = 0.82921i are shown in Figure 4.5. It is clear that w, ψ and 2ψ are functions of r and θ with an indication of a singularity at the origin. ∇ The symmetry which makes the total vorticity in the upper half of the pipe imaginary and the ﬂux ratio real is also observed. The contour plots corroborate the assumption that there is no boundary layer structure present on this branch, as argued in Section 3.2.2.4 when considering the asymptotic behaviour of Branch II as K 0, and so K 0 on the I → → imaginary axis. 4.3 Path-continuation 104

0.9 0.01

0.8 0.008

0.006 0.7 0.004 0.6 0.002

I O 0.5 I O 0

−0.002 0.4 −0.004 0.3 −0.006

0.2 −0.008

−0.01 0.1

(a) NS real wK =0.2i (b) NS imaginary wK =0.2i

−4 x 10

8 0.08

6 0.06

4 0.04

2 0.02

I O 0 I O 0

−2 −0.02

−4 −0.04

−6 −0.06

−8 −0.08

2 0.03 1.5

0.02 1

0.01 0.5

I O 0 I O 0

−0.5 −0.01

−1 −0.02

−1.5 −0.03 −2

(e) NS real Ω K =0.2i (f) NS imaginary Ω K =0.2i

Figure 4.4: Contour plots of w, ψ and Ω on Branch I for K = 0.4i M = 200, κ = 15 Chapter 4. Numerical solution of the Dean equations for complex K 105

2.2 0.8

2 0.6

1.8 0.4 1.6 0.2 1.4

I O 1.2 I O 0

1 −0.2

0.8 −0.4 0.6 −0.6 0.4

0.2 −0.8

(a) NS real w for K =0.82921i (b) NS imaginary w for K =0.82921i

2 0.6

1.5 0.4 1

0.2 0.5

I O 0 I O 0

−0.5 −0.2

−1 −0.4 −1.5

−0.6 −2

25 50

20 40

15 30

10 20

5 10

I O 0 I O 0

−5 −10

−10 −20

−15 −30

−20 −40

−25 −50

(e) NS real Ω for K =0.82921i (f) NS imaginary Ω for K =0.82921i

Figure 4.5: Contour plots of w, ψ and Ω on Branch II for K = 0.82921i, M = 200, κ = 15 4.4 Summary and conclusion 106

4.4 Summary and conclusion

In this chapter two numerical schemes were developed to solve the Dean equations for complex K. The ﬁrst was based on the ﬁctitious time-stepping scheme of Siggers & Wa- ters [73] and the second on the path-continuation technique of Yang & Keller [96], where extension into the complex plane was achieved in light of Henderson’s work on complex bifurcation [37, 36]. Each approach provided new insight into the understanding of the Dean equations for imaginary K.

The time-stepping scheme successfully constructed Branch I for real and imaginary K. The results compared well with those of the generalised Pad´e approximants for the ﬂux ratio for imaginary K. The inability to detect Branch II-IV by this scheme suggests that these branches are unstable.

The path-continuation technique successfully constructed Branch I for real and imaginary K and detected Branch II for imaginary K for a finite number of steps. For real K, Branch I was followed up to K = 630 with no bifurcation detected. Further continuation of this branch may serve to test the conclusion of Yang & Keller who reported the possibility of an infinite number of 2n-vortex solutions [96], though this was not a specific objective of this project. For imaginary K the flux ratio, total vorticity and outer wall shear stress on Branch I and II compared well with those of the generalised Padéapproximants. Contour plots of the azimuthal velocity, streamfunction and vorticity were rendered on Branch I and II. For Branch I the imaginary and complex parts of these functions behaved according to the Dean series solution (1.16). The real part of azimuthal velocity behaved like w0 and its imaginary part like w1 defined in Chapter 2. Similarly, the real part of the streamfunction behaved like ψ2 and its imaginary part like ψ1.

The ﬂow on Branch II possess the symmetry postulated in Chapter 3, which makes the total vorticity in the upper half of the pipe imaginary and the ﬂux ratio real. The contour plots also corroborates the assumption of Section 3.2.2.4 that there is no boundary layer structure, further supporting the proposed asymptotic behaviour of Branch II as K 0 I → given in Chapter 3. Chapter 4. Numerical solution of the Dean equations for complex K 107

In conclusion, the numerical work herein has helped to corroborate the results of previous chapters. Notably, the poor convergence of the numerical scheme particularly on Branch II, serves to highlight the power of the generalised Pad´eapproximants of Chapter 3. This latter technique has successfully constructed complex solution to the Dean equations (1.12) and (1.13), without the need to solve the equations themselves and so avoids the convergence issues encountered in this chapter. 108 109

Chapter 5

Extended Stokes series: elliptic cross-section

In this chapter, Dean flow in pipes of elliptic cross-section of variable aspect ratio, λ, is considered by employing elliptical coordinates. In the limit of small and large aspect ratios asymptotic solutions are proposed. The equations are then solved using the extended Stokes series method. For specific λ the first 24 terms of the series are calculated. Whilst for various λ the series are found to have a finite radius of convergence limited by a conjugate pair of square-root singularities, for large λ more coefficients are required to find the analytic structure. Where calculated, the location of these singularities are found to vary with the aspect ratio. In particular, they are furthest away from the origin for small aspect ratios. The effect of the aspect ratio on the flow is investigated for small and suitably larger K still within the radius of convergence. It is found that the effect of the curvature of the pipe is diminished in the limit of small and large aspect ratios. A comparison of the computer-extended series with the asymptotic models and the earlier results of Cuming [19] and Srivastava [79] is also made.

5.1 Governing equations

First a curved pipe of elliptic cross-section with foci at L a, of small aspect ratio and ± radius of curvature L, is considered. The orthogonal elliptical coordinates (,θ,φ), with 5.1 Governing equations 110

=0...0 and θ =0...2π, are employed. These are shown in Figure 5.1, with :

x =(a cosh ()cos(θ)+ L)cos(φ) , y = a sinh () sin (θ) , z =(a cosh ()cos(θ)+ L) sin (φ) . (5.1)

The boundary at = 0, is an ellipse with semi-major axis, a cosh(0), on the x-axis and semi-minor axis, a sinh( ) on the y-axis. The corresponding aspect ratio λ = sinh(0) . As 0 cosh(0) , λ 1 and the cross-section is a circle (as shown in the dot dash line of Figure 0 → ∞ → 5.1). As 0, λ 0. 0 → → The full-derivation of the governing equations, analogous to the Dean’s derivation of the governing equations for pipes of circular cross-section, is given in Appendix E.1. Here, it suffices to comment that the streamfunction, ψ(, θ), which satisfies the continuity equation is defined as: ∂ψ ∂ψ = γu, = γv, (5.2) ∂θ ∂ − where γ = a2 sinh ()2 + sin (θ)2 . The length scales are non-dimensionalised by a cosh(0), the characteristic length of the ellipse in the x-direction. Accordingly the non-dimensionalised semi-minor axis varies like tanh(0). The azimuthal velocity w and the streamfunction in the cross-sectional plane ψ are non-dimensionalised analogously to the circle:

a2 sinh ()2 + sin (θ)2 ψ w γ2 = , ψ = and w = (5.3) 2 2 a cosh (0) ν ω0 where ω0 is the maximum down-pipe velocity in a straight pipe with elliptical cross-section driven by the same pressure gradient and ν is the viscosity.

Assuming that the curvature is small, the corresponding non-dimensionalised Dean equa- Chapter 5. Extended Stokes series: elliptic cross-section 111 tions are:

2 2 4 1 ∂ψ ∂ ψ ∂ψ ∂ ψ w cosh () sin (θ) ∂w sinh ()cos(θ) ∂w ψ = 2 ∇ ∇ 576K 2 + , ∇ γ ∂θ ∂ − ∂ ∂θ − γ cosh (0) ∂ cosh (0) ∂θ (5.4) 1 ∂ψ ∂w ∂ψ ∂w 1 1 2w = 2 + , (5.5) ∇ γ2 ∂θ ∂ − ∂ ∂θ − sinh2() cosh2() where

1 ∂2 ∂2 2 = + , (5.6) ∇ γ2 ∂2 ∂θ2 1 a cosh ( ) w 2 a cosh ( ) K = 0 0 0 . (5.7) 288 ν L

The corresponding no-slip boundary conditions are, using the deﬁnition of the streamfunciton (5.2), at = 0:

∂ψ w =0, =0, and ψ =0. (5.8) ∂

(a) Elliptic Coordinates 5.1 Governing equations 112

L-a L+a

(b) Geometry of the pipe

Figure 5.1: Coordinate system for curved pipe with elliptical cross-section with small λ. Dashed line is limit λ = 1

Pipes of elliptic cross-section with large aspect ratio (λ > 1) are considered by applying the following transformation to the above system:

π i + and a ia. (5.9) → 2 → −

Accordingly, the coordinates (5.1) become:

x =(a sinh ()cos(θ)+ L)cos(φ) , y = a cosh () sin (θ) , z =(a sinh ()cos(θ)+ L) sin (φ) . (5.10)

The boundary at = 0 is an ellipse with semi-major axis, a cosh(0), on the y-axis and semi-minor axis, a sinh( ), on the x-axis. The corresponding aspect ratio λ = cosh(0) 0 sinh(0) varies from 1 to as varies from to 0. Details of the corresponding non-dimensional ∞ 0 ∞ Dean equations for λ> 1 can be found in Appendix E.2.

In the remainder of the chapter the hat notation is dropped. The variables considered are Chapter 5. Extended Stokes series: elliptic cross-section 113 non-dimensional, unless stated otherwise.

5.2 Asymptotic behaviour for large and small λ

An asymptotic solution for small and large λ is presented. The following non-dimensional Cartesian coordinates rather than the elliptic coordinates are used:

x y x = and y = , (5.11) X X

where X is the lengthscale in the x-coordinate direction, with X = a cosh(0) for λ < 1 and X = a sinh(0) for λ> 1. In these coordinates, following Cuming [19] under suitable rescaling, the governing equations are, dropping the hat notation:

∂ψ ∂w ∂ψ ∂w 1 2w = 2 1+ , (5.12) ∇ ∂x ∂y − ∂y ∂x − λ2 ∂ψ ∂ 2ψ ∂ψ ∂ 2ψ ∂w 4ψ = ∇ ∇ 576Kw , (5.13) ∇ ∂x ∂y − ∂y ∂x − ∂y where ∂2 ∂2 2 = + . (5.14) ∇ ∂x2 ∂y2 The boundary conditions are

∂ψ ∂ψ w =0, =0, and =0, (5.15) ∂x ∂y

2 y2 on x + λ2 =1

Small λ. For small λ, ψ and w are expected to be independent of x near the centre of the pipe. Consequently, the derivatives with respect to x are zero and Dean equations (5.12) and (5.13) reduce to:

d2w 1 = 2 1+ , (5.16) dy2 − λ2 d4ψ K dw = w . (5.17) dy4 − 2 dy

1 As λ is small, it follows that the term λ2 dominates in the azimuthal momentum equation 5.2 Asymptotic behaviour for large and small λ 114

(5.16):

d2w 2 = , (5.18) dy2 −λ2 d4ψ K dw = w . (5.19) dy4 − 2 dy

These equations and the boundary conditions (5.15) are solved by:

y2 Kλ3 y2 2 y2 y w =1 and ψ = 1 5 . (5.20) − λ2 840 − λ2 − λ2 λ This solution is in accordance with Dean-Hele-Shaw ﬂow considered by Mestel & Zabiel- ski [58].

Large λ. For large λ, ψ and w are expected to be independent of y near the centre of the pipe . Consequently, the derivatives with respect to x are zero and Dean equations (5.12) and (5.13) reduce to:

d2w 1 = 2 1+ , (5.21) dx2 − λ2 d4ψ =0. (5.22) dx4

As λ is large:

d2w = 2, (5.23) dx2 − d4ψ =0. (5.24) dx4

These equations and the boundary conditions (5.15) are solved by:

w =1 x2 and ψ =0. (5.25) − Chapter 5. Extended Stokes series: elliptic cross-section 115

5.3 Computer-extended series

The Dean equations (5.4) and (5.5 ) are solved by regular asymptotic expansion in K, analogous to the pipe of circular cross-section considered in Chapter 2.

When K =0, the solution to equation (5.4) and (5.5 ) is:

4+ B A B A w (, θ)= w (, θ)= − + cos(2θ)cosh(2) − 0 4 4 A + B A + B cosh (2) cos (2θ) , − 4 − 4 ψ (, θ)= ψ0 (, θ) =0, (5.26) where: 1 1 A = 2 , B = 2 . cosh (0) sinh (0) When K is small, the solution to the Dean equations (5.4) and (5.5 ) is constructed as a regular perturbation in K of (5.26):

n w = wn (, θ) K , n ψ = ψn (, θ) K , 2ψ = Lψ (, θ) Kn. (5.27) ∇ n Substituting these series solutions into the Dean equations and collecting powers of K reduces the system to a sequence of successively more complicated linear problems which can be solved sequentially to ﬁnd the functions ψn (, θ), Lψn (, θ) and wn (, θ). Details can be found in Appendix E.3. By iteration, the following solution is found:

In In cosh(2j) cos(2iθ) n even w = e (A, B) , (5.28) n n,i,j  i=0 j=0 cosh((2j + 1)) cos((2i + 1)θ) n odd   Jn Jn sinh(2j) sin(2iθ) n even ψ = c (A, B) , (5.29) n n,i,j  i=0 j=0 sinh((2j + 1)) sin((2i + 1)θ) n odd   5.3 Computer-extended series 116

Ln Ln sinh(2j) sin(2iθ) n even 2ψ = d (A, B) , (5.30) ∇ n n,i,j  i=0 j=0 sinh((2j + 1)) sin((2i + 1)θ) n odd   7n+2 7n 7n−2 7n+1 7n−1 where for n even In = 2 , Jn = 2 , Ln = 2 and for n odd In = 2 , Jn = 2 , 7n−3 Ln = 2 .

The leading order contribution for the flow, w0, corresponds to that of axial Poiseuille flow in a straight pipe, whilst the first correction, given by ψ1, corresponds to Dean vortices in the cross-section.

The solution for ellipses with λ> 1 can be found by using transformation (5.9) as detailed in Appendix E.3.

∗ A MAPLE program has been written to calculate the coefﬁcients bn,i,j, cn,i,j and dn,i,j .

Although this code can calculate these coefﬁcients exactly as functions of 0, for speed particular values of 0 were chosen. The values of 0 considered and their corresponding aspect ratio are shown in Table 5.1. The ﬁrst 24 terms in the series solution have been calculated for each 0.

λ 0 λ 0 λ 0 0.1 0.10034 1.1 1.5223 2.0 0.54930 0.2 0.20273 1.2 1.1989 3.0 0.34658 0.3 0.30952 1.3 1.0184 4.0 0.25542 0.4 0.42365 1.4 0.89588 5.0 0.20274 0.5 0.54930 1.5 0.80472 6.0 0.16824 0.6 0.69315 1.6 0.73317 7.0 0.14383 0.7 0.86730 1.7 0.67496 8.0 0.12565 0.8 1.0986 1.8 0.62638 9.0 0.11157 0.9 1.4722 1.9 0.58504 10.0 0.10034 (a) λ< 1 (b) λ> 1 (using transformation (5.9))

Table 5.1: Values of 0 and λ for which the extended Stokes series has been calculated.

Specifically the flux ratio and total vorticity, shown below, are considered. Their derivation, from definitions 2.1.1 and 2.1.4 in Chapter 2, are given in Appendix E.4 and E.5 respectively.

∗Imposing the boundary conditions for this system involves solving a system of linear equations. It is this procedure which makes the program run slower than the one for a pipe with circular cross-section Chapter 5. Extended Stokes series: elliptic cross-section 117

The ﬂux down a straight pipe of elliptic cross-section:

π Qs (K)= 2 sinh (20) e0,0,0 (A, B) 2 cosh (0)

π sinh (40) + 0 + e0,0,1 (A, B) 2 cosh2 ( ) 4 0 π e0,1,1 (A, B) sinh (20) e0,1,0 (A, B) 0 . (5.31) − 2 cosh2 ( ) 2 − 0 As deﬁned in 2.1.1 with suitable coordinate transformation, the ﬂux ratio for λ< 1 is:

2n Q (K)= an (A, B) K (5.32) n where,

I(2n) 1 π sinh((2j + 2)0) sinh((2j 2)0) an = e2n,0,j (A, B) 2 + − Qc  2 cosh (0) (2j + 2) (2j 2) j=1 −  π sinh((4)0) +e2n,0,1 (A, B) + 0 2 cosh2 ( ) (4) 0 I n (2 ) π sinh((2j) ) π + e 0 r (A, B) − 2n,1,j 2 (2j) − 2n,1,0 2 0 j>0 2 cosh (0) 2 cosh (0)  π +e2n,0,0 (A, B) (sinh (20)) . (5.33) 2 cosh2 ( ) 0 As deﬁned in 2.1.4 with suitable coordinate transformation, the total vorticity for λ< 1 is:

2n+1 Ω(K)= bnK (5.34) n where,

J n J n λ (2 +1) (2 +1) ((2j + 1)2 (2i 1)2) b = c (A, B) − − (2 2cosh((2j + 1) )) . n Q (K) 2n+1,i,j (2j + 1)(2i + 1) − 0 s i=0 j=0 (5.35) The expressions for λ > 1 of the series solution of Q(K) and Ω(K) are found using transformation (5.9). Their structure is given in Appendix E.4. For the remainder of the chapter the labels (5.27), (5.32) and (5.34) will be used to refer to the series solution for all λ. 5.4 Radius of convergence 118

5.4 Radius of convergence

Domb-Sykes analysis was applied to the total vorticity and ﬂux ratio to determine the radius of convergence of the series solution (5.27). An overview of this technique was given in Chapter 2.

For most λ the series possess a ﬁnite radius of convergence limited by an imaginary conjugate pair of square-root singularity, iK , in the complex plane . The radius of convergence ± c and the location of the closest singularity varies with λ (see Table 5.2).

λ Kc λ Kc λ Kc 0.1 557.92 1.1 0.86276 2.0 0.53043 0.2 70.698 1.2 0.75670 3.0 0.61889 0.3 21.281 1.3 0.68252 4.0 0.81592 0.4 9.4021 1.4 0.63019 5.0 0.81816 0.5 5.1255 1.5 0.59326 6.0 X 0.6 3.1920 1.6 0.56752 7.0 X 0.7 2.1833 1.7 0.55010 8.0 X 0.8 1.6037 1.8 0.53898 9.0 X 0.9 1.2473 1.9 0.53278 10.0 X

(a) λ< 1 (b) λ> 1 (using transformation (5.9))

Table 5.2: Estimates of Kc for various λ. X denotes aspect ratios for which the Domb-Sykes plots show oscillations

The series solution of cross-sections with aspect ratio less than one have a very large radius of convergence. In this limit an asymptotic solution may be reached. The radius of convergence is smallest for λ near two and then continues to increase with λ. For λ =7, 8, 9, 10 the Domb-Sykes plots show oscillations which may correspond to fully complex singularities [89], but there is an indication that the radius of convergence increases with λ.

Studies of curved pipes of elliptical cross-section have shown that the critical point for the bifurcation from a two to a four-vortex solution varies with the aspect ratio of the cross- section [50]. Consequently, these singularities which correspond to a symmetry breaking bifurcation in the complex plane, as discussed in Chapter 3, should also vary with λ. Ana- lytic continuation by Euler transform was attempted. Subsequent analysis, for all λ posed similar challenges encountered for the circular pipe described in Chapter 2. Chapter 5. Extended Stokes series: elliptic cross-section 119

5.5 Generalised Pade´ approximants

A study of the lower order generalised Padéapproximants, d = 2, 3, indicates that the bifurcation diagram in the complex plane of K is similar for the pipes of circular cross- section discussed in Chapter 3. The results of the flux ratio for λ =0.8 and 1.8 are shown in Figure 5.2, using the same notation as in Chapter 3. These show that there is a bifurcation at iKc, with real flux ratio on Branch I and II, for Im(K) Kc.

1.4 1.3 λ =0.8 λ=1.8 1.35 1.25 1.3 Branch II 1.2 Branch III 1.25 ) K)

(K 1.2 1.15 Q Q(

1.15 1.1 1.1 1.05 1.05 Branch I Branch I 1 1 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

m(K) Im(K) (a) Q(K) λ =0.8 (b) Q(K) λ =1.8

1.1 1.07 λ=0.8 λ=1.8 1.06

1.095 1.05

1.04 Branch III 1.09

Q(K)) 1.03 (Q(K)) Branch III e Re( R 1.02 1.085 1.01

1 1.08 0.99 1.62 1.64 1.66 1.68 1.7 1.72 1.74 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Im(K) Im(K) (c) Re(Q(K)) λ =0.8 (d) Re(Q(K)) λ =1.8

0 0 λ=0.8 λ=1.8

-0.01 -0.02

-0.02 -0.04

-0.03

Im(Q(K)) Branch III Im(Q(K)) -0.06 -0.04 Branch III -0.08 -0.05

-0.06 -0.1 1.62 1.64 1.66 1.68 1.7 1.72 1.74 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Im(K) Im(K) (e) Im(Q(K)) λ =0.8 (f) Im(Q(K)) λ =1.8

Figure 5.2: Flux ratio on Branch I, II and III found by generalised Pad´eapproximant for λ = 0.8 and λ = 1.8 5.6 Discussion of the extended Stokes series 120

5.6 Discussion of the extended Stokes series

The flow given by the series solution (5.27) was examined for varying aspect ratio and Dean number. Herein, an overview of the findings are presented. First, the contour plots of the azimuthal velocity and streamfunction are discussed. The cross-sectional velocity for small λ and azimuthal velocity for large λ, given by the series expansion, are then compared with the asymptotic findings (5.20) and (5.25). Finally, variations of the total vorticity (5.34) and flux ratio (5.32) with λ are considered. These are compared with the previous results of Cuming [19] and Srivastava [79] respectively.

5.6.1 Contours of azimuthal velocity and streamfunction

1 The contours of the azimuthal velocity and streamfunction for λ = 0.8 for K = 576 and 200 576 are shown in Figure 5.3. They exhibit equivalent behaviour to pipes of circular cross- section discussed in Chapter 2. The streamfunction takes the form of two counter-rotating vortices. As K is increased the location of the maximal azimuthal velocity migrates to the outside bend.

I O I O

(a) Contour plot (b) Contour plot

1 Figure 5.3: Contour plot of azimuthal velocity and streamfunction for λ = 0.8 for K = 576 and 200 K = 576

The degree of the displacement of the maximal azimuthal velocity as K increases was Chapter 5. Extended Stokes series: elliptic cross-section 121

found to depend on λ. The location of the maximum azimuthal velocity xmax, has rescaled with the non-dimensional horizontal scale denoted x˜max. Accordingly, for λ< 1

x x˜ = max = x , (5.36) max 1 max and for λ> 1

x x˜ = max . (5.37) max λ

100 The value of x˜max for various λ at K = 576 is shown in Table 5.3. The results indicate that the displacement is reduced for values of λ away from λ = 1. For small λ, Cuming [19] found that the secondary flows u and v vary like λ2. For large λ, it was found that the secondary flows u and v vary like λ−2. It follows that for small and large aspect ratios the secondary flow is greatly reduced, leading to a decrease in the migration.

λ x˜max λ x˜max λ x˜max 0.1 4.4392e-5 1.1 2.1523e-2 2.0 9.9475e-2 0.2 5.5434e-4 1.2 3.4345e-2 3.0 1.1103e-1 0.3 2.0586e-3 1.3 4.6117e-2 4.0 1.0165e-1 0.4 4.6956e-3 1.4 5.6995e-2 5.0 9.0584e-2 0.5 8.1828e-3 1.5 6.6865e-2 6.0 8.0965e-2 0.6 1.1939e-2 1.6 7.5617e-2 7.0 7.2956e-2 0.7 1.5155e-2 1.7 8.3211e-2 8.0 6.6294e-2 0.8 1.6776e-2 1.8 8.9668e-2 9.0 6.0694e-2 0.9 1.5146e-2 1.9 9.5057e-2 10.0 5.5914e-2

(a) λ< 1 (b) λ> 1 (using transformation (5.9))

Table 5.3: Variation of xmax, the location of maximal azimuthal velocity as a ratio of the horizontal 100 distance, for various λ at K = 476 .

5.6.2 Cross-sectional velocities for small λ

For small λ, the cross-sectional velocity, u = cosh(0) ∂ψ , along the central-axis √(sinh2()+sin2(θ)) ∂ of the pipe (x =0) was examined. Figure 5.4 shows u along the central axis for various λ, where the y-axis has been rescaled by the non-dimensional aspect ratio λ for comparison. 5.6 Discussion of the extended Stokes series 122

In accordance with the contour plots discussed above, the maximum value of u along this axis is found to decrease with λ. As K is increased u becomes larger, as shown in Figure 5.4a. For relatively small λ the series solution is in agreement with the asymptotic model (5.20).

1 λ = 0.1 λ = 0.2 λ = 0.3 λ = 0.4 λ = 0.5 λ = 0.6 λ = 0.7 λ = 0.8 0.5 λ = 0.9 Asymptotic

λ 0 y/

-0.5

-1 -1.5e-02 -1.0e-02 -5.0e-03 0.0e+00 5.0e-03 1.0e-02 1.5e-02 u 1 (a) u for λ< 1 and K = 576

1 λ = 0.1 λ = 0.2 λ = 0.3 λ = 0.4 λ = 0.5 λ = 0.6 λ = 0.7 λ = 0.8 0.5 λ = 0.9 Asymptotic

λ 0 y/

-0.5

-1 -2 -1 0 1 2 u 200 (b) u for λ< 1 and K = 576

1 200 Figure 5.4: Secondary velocity u for λ< 1 and K = 576 and K = 576 Chapter 5. Extended Stokes series: elliptic cross-section 123

5.6.3 Azimuthal velocity for large λ

For large λ, the azimuthal velocity along the centre-line (y = 0) was examined. Figure 5.5 shows w along the horizontal-axis for various λ. In accordance with the contour plots discussed previously, the migration of the maximum azimuthal velocity is retarded for large λ. For relatively large λ, the series solution is in agreement with the asymptotic model (5.25).

1 λ = 2 λ = 3 λ = 4 λ = 5 0.8 λ = 6 λ = 7 λ = 8 λ = 9 0.6 λ = 10 Asymptotic w

0.4

0.2

0 -1.0e+00 -5.0e-01 0.0e+00 5.0e-01 1.0e+00 x 1 (a) w for λ< 1 and K = 576

1 λ = 2 λ = 3 λ = 4 λ = 5 0.8 λ = 6 λ = 7 λ = 8 λ = 9 0.6 λ = 10 Asymptotic w

0.4

0.2

0 -1.0e+00 -5.0e-01 0.0e+00 5.0e-01 1.0e+00 x 100 (b) w for λ< 1 and K = 576

100 Figure 5.5: Azimuthal velocity on the centre-line w for λ> 1 and K = 1 and K = 576 5.6 Discussion of the extended Stokes series 124

5.6.4 Total vorticity

In this section, the result of the total-vorticity for the extended Stokes series solution is compared with the results of Cuming [19]. In his work, Cuming calculated the Stokes series expansion to order K, a brief review of the details can be found in Appendix E.6.

The variation of the total vorticity (5.34) with λ is shown in Figure 5.6. It obtains a maximum for λ 1.5 and diminishes as λ 0 and λ . A log-log plot of the vorticity ≈ → → ∞ for small λ is shown in Figure 5.7a. It indicates that for small aspect ratios Ω = O(λ). Similarly, the log-log plot of the vorticity for large λ, shown in Figure 5.7b, indicates that for large aspect ratios Ω = O(λ−3). This is in agreement with Cuming’s results [19]. The total vorticity increases with the Dean number as for pipes of circular cross-section.

These results are compared with Cuming [19], under suitable rescaling, in Figure 5.6. They are found to be in close agreement for small Dean numbers, as shown in Figure 5.6a. For larger Dean numbers, the two solutions differ near λ = 1, as shown in Figure 5.6b. This difference is expected as Cuming’s solution corresponds to the series expansion solution truncated at order K.

Cuming [19] reported that his scaled total vorticity,

π Ω Ω= , (5.38) 2 Ω1 where Ω1 is the total vorticity for a pipe with circular cross-section, to be:

λ2 (125 + 310λ2 + 428λ4 + 82λ6 + 15λ8) Ω= . (5.39) (5+2λ2 + λ4)(35+84λ2 + 14λ4 + 20λ6 +3λ8) He reported, that this obtained a maximum at λ 2.2 which does not at appear to agree ≈ with his own plots of the total vorticity, nor with the calculations herein. This discrepancy was found to be an arithmetic error and the maximum of Ω, found by solving

dΩ d2Ω (5.40) =0, 2 =0,λ 0 , dλ dλ ≥ is obtained at λ 1.57. ≈ Similarly, the scaled total vorticity Ω is found to obtain a maximum at λ 1.57 which is ≈ Chapter 5. Extended Stokes series: elliptic cross-section 125 in agreement with the plots in Figure 5.6.

2.5e-03 ESS 2011 Cuming 1955

2.0e-03

1.5e-03 (K) Ω 1.0e-03

5.0e-04

0.0e+00 0 2 4 6 8 10 λ 1 (a) total vorticity for K = 576

3.0e-01 ESS 2011 Cuming 1955 2.5e-01

2.0e-01

(K) 1.5e-01 Ω

1.0e-01

5.0e-02

0.0e+00 0 2 4 6 8 10 λ 100 (b) total vorticity for K = 576

1 Figure 5.6: Comparing the total vorticity calculated by Cuming [19] and the ESS, for K = 576 and 100 K = 576 5.6 Discussion of the extended Stokes series 126

-1

-1.5

-2 (K) Ω

log -2.5

-3

ESS 2011 O(λ) -3.5 -2.5 -2 -1.5 -1 -0.5 0 logλ (a) log-log plot λ< 1

-6 ESS 2011 O(λ-3)

-7

-8 (K) Ω

log -9

-10

-11 0 0.5 1 1.5 2 2.5 3 logλ (b) log-log plot λ> 1

100 Figure 5.7: Log-log plot of the total vorticity for varying λ, K = 576 Chapter 5. Extended Stokes series: elliptic cross-section 127

5.6.5 Flux ratio

In thissection, the result of the ﬂux ratio for the extended Stokes series solution is compared with the results of Srivastava [79]. In his work, Srivastava extended the Stokes series expansion to order K2 in order to study the ﬂux ratio.

The variation of the flux ratio (5.32) with λ as found using the computer extended Stokes series is compared with the results of Srivastava [79] in Figure 5.8. They are found to be in close agreement for small Dean numbers, as shown in Figure 5.8a. However, for large Dean numbers the two solutions differ significantly for all aspect ratios as shown in Figure 5.8b. This difference is expected as Srivastava’s solution corresponds to the series expansion solution truncated at order K2. In agreement with the circular cross-section, the flux ratio decreases with the Dean number.

Notably, the results of this study indicate that the flux ratio obtains a minimum near λ ≈ 1.75. The existence of such a minimum flux ratio was not observed by Srivastava due to the aspect ratios he considered. However, this finding is in accordance with the observations regarding the total vorticity, which indicate that the influence of the curvature of the pipe is reduced for values of the aspect ratio away from λ =1.

The existence of a minimum flux ratio is of practical importance. For example, as discussed briefly in the introduction, curved pipes are generally found in heat exchange systems due to spatial constraints. In such systems, one of the adverse effects of the curvature of the pipe is a lossin pressure in the pipe, thatis a decrease in the flux ratio. The results discussed above indicate that this effect may be reduced by constructing pipes with aspect ratios away from λ 1.75, where the effect of the secondary flow is reduced. It seems likely that this ≈ behaviour would also extend to large Dean numbers. 5.6 Discussion of the extended Stokes series 128

1 1

0.99999995 0.99999995

0.9999999 0.9999999

0.99999985 0.99999985 Q(K)

0.9999998 0.9999998

0.99999975 0.99999975 ESS 2011 Srivastava 1980 0.9999997 0.9999997 0 2 4 6 8 10 λ 1 (a) Flux Ratio for K = 576

1 1

0.999995 0.9995 0.99999 0.999 0.999985

0.9985 0.99998 Q(K)

0.999975 0.998 0.99997 0.9975 0.999965 ESS 2011 Srivastava 1980 0.997 0.99996 0 2 4 6 8 10 λ 100 (b) Flux Ratio for K = 576

1 Figure 5.8: Comparing the ﬂux ratio calculated by Srivastava [79] and the ESS, for K = 576 and 100 K = 576 Chapter 5. Extended Stokes series: elliptic cross-section 129

5.7 Summary and conclusion

In this chapter Dean ﬂow in a weakly curved pipe of elliptic cross-section was considered. Asymptotic structures were proposed in the limit of small and large λ. With the small λ the asymptotic structure found to be in agreement with the work on Dean-Hele-Shaw ﬂow by Mestel & Zabielski [58].

Though the initial intention has been to construct the computer extended Stokes series for arbitrary aspect ratios, memory constraints of the program, meant that particular aspect ratios λ were examined by employing elliptic coordinates.

The Stokes series was computer-extended to the 24th term, (K24), for several values of λ. Domb-sykes analysis found the series to have a finite radius of convergence limited by an imaginary conjugate pair of square-root singularities. The radius of convergence was found to be largest for small values of λ. The radius of convergence was smallest for values λ 1.5. It was postulated that as λ 0 the radius of convergence becomes infinite as ≈ → an asymptotic solution is reached. Analytic continuation by Euler transform was found to have the same difficulty as the circular cross-section discussed in Chapter 2. Generalised Padéapproximants of low order indicate that the bifurcation diagram in the complex plane is similar to the circular cross-section. The large Dean number behaviour for this cross- section has yet to be studied.

The small Dean number solution was studied within the radius of convergences found. For small λ the secondary flow u on the central-axis was found to agree with the asymptotic structure proposed. Similarly, for large λ the azimuthal velocity on the horizontal-axis was found to agree with the asymptotic structure. For all λ the effect of the curvature of the pipe on the flow is as expected given the understanding of the curved pipe with circular cross- section. The flow consists of two counterrotating vortices in the upper and lower half of the pipe. As K is increased the maximum axial velocity migrates to the outside bend, the total vorticity increases, and the flux ratio decreases. The magnitude of this effect was found to decrease in the limit of small and large λ. In the limit of large λ this can be explained physically as the upper and lower walls which induce the secondary flow are further away.

The extended series solutions has provided minor numerical corrections to the work of Cuming [19], regarding the total vorticity, and Srivastava [79] regarding the flux ratio, in 5.7 Summary and conclusion 130 accordance with the truncation level of the Stokes series expansion. In the former case, this study concluded that for small λ, Ω O(λ) and for large λ Ω O(λ−3) , in accordance ∼ ∼ with the work of Cuming [19]. The total vorticity was found to reach a maximum for λ 1.5. Significantly, the flux ratio was minimum for λ 1.75. This hitherto unknown ≈ ≈ result, in addition to that of the total-vortitcity, suggests that in the construction of elliptical pipes of aspect ratios away from λ = 1.5 1.75 may reduce the effects of the secondary − flow in industrial systems. This is particularly advantageous in transport systems, where greater power is required to offset the pressure loss due to the curvature of the pipe. Though, as yet the robustness of this observation for large Dean number is not known.

In conclusion, this study suggests that the square-root singularity which limits the applicability of the Stokes series expansion, and the corresponding symmetry breaking bifurcation, is generic to Dean ﬂow. 131

Chapter 6

Final Conclusions

6.1 Summary of key results and possible extensions

The core of this thesis concerned the study of steady, fully-developed flows in weakly curved pipes of circular cross-section by the extended Stokes series method. The principal aims of this work were: (i) to address the discrepancy between the asymptotic behaviour of the flux ratio as claimed by Van Dyke [88] and boundary layer methods [7], and; (ii) to construct the multi-vortex solutions of the Dean equations reported by previous studies using generalised Padéapproximants.

The Stokes series for pipes of circular cross-section was computer-extended in the Dean number, K, up to order K196. Domb-Sykes analysis showed that the convergence of the series was limited by an imaginary conjugate pair of square-root singularities iK , as ± c reported by Van Dyke [88].

Calculations herein do not support Van Dyke’s conclusion that the flux ratio varies as K−1/10 for large K. The previously reported discrepancy between the extended Stokes series and boundary-layer theory is most likely a consequence of the use of the Euler transformation which necessarily maps large critical points of the solution close together, mak- ing the asymptotic behaviour difficult to determine. Furthermore, the series analytically continued by generalised Padéapproximants (Chapter 3) was found to be in close agreement with numerical results (Chapter 4) for large values of the Dean number at which they both differ from Van Dyke’s model (2.54). These findings suggest that the asymptotic be- 6.1 Summary of key results and possible extensions 132 haviour of the flux ratio may be as proposed by boundary-layer theory. However, such a conclusion must be deferred for a future study in which higher order generalised Padé approximants can be constructed.

Using the generalised Padéapproximants it was not possible to construct the multi-vortex solutions of the Dean equations reported by previous studies for real Dean number. The computation of further coefficients and construction of higher order approximants may allow further investigation of the analytic structure of the solution. However, its applicability may be limited if the point of bifurcation from the two to the four-vortex solution occurs at infinite Dean number. Such behaviour is exhibited in the stability analysis of straight-pipe flow, for which a subcritical bifurcation from infinite Reynolds number is responsible for the experimental observations.

The study of imaginary Dean number proved more successful in studying the bifurcation diagram. In Chapter 3 it was proposed that the singularityat iKc in some sense corresponds to a symmetry breaking bifurcation at which three new complex branches are found. Branch I corresponds to the extended Stokes series evaluated in the complex plane and Branch II-IV are previously unreported solutions. The nature of the singularity is supported by numerical results of the Dean equations for complex K found in Chapter 4. Branch I and II were constructed numerically and found to possess the proposed symmetry. It was postulated that on Branch II the azimuthal velocity behaves like K−1/2 and the streamfunction as K0 close to K =0. This was supported by the results from the generalised Pad´e approximants in Chapter 3.

The path-continuation technique adopted here may in the future be modiﬁed to construct

Branch III and IV from knowledge of the tangents of Branch I and II at iKc as described by Henderson [37]. Such an extension of this work would provide further understanding of the solution to the Dean equations in the complex plane.

The final part of the thesis concerned the study of pipes of elliptic cross-section. There were two principal aims of this section: (i) to investigate the small Dean number behaviour of this flow which has received little attention, and; (ii) to investigate the analytic structure of the extended Stokes series. The latter aim was motivated by the complex behaviour found for the circular cross-section and the question as to whether this was generic to Dean flow. Chapter 6. Final Conclusions 133

The Stokes series for pipes of elliptic cross-section with various aspect ratios, λ, was computer-extended up to order K24. For small Dean number, the solution provided numerical corrections to the previous work of Cuming [19] and Srivastava [79]. It was found that the effect of the curvature of the pipe was reduced for large and small aspect ratios. In particular, the total vorticity is maximised for λ 1.5, in accordance with Cuming [19] ≈ and the flux ratio is minimised for λ 1.75. This latter observation is a hitherto unre- ≈ ported result. It is yet to be seen whether this behaviour persists for high Dean number flow and may therefore be an area of future interest. In particular, the results suggest that in an industrial context it may be advantageous to construct pipes with a cross-section which becomes suitably elliptical at bends. The aspect ratio of this ellipse can be chosen to reduce the adverse effects of the secondary flow, such as curvature-induced pressure losses, or perhaps even enhance them, in order to increase particle residence times.

Interestingly, for all aspect ratios the convergence of the series solution is limited by an imaginary conjugate pair of square-root singularities iK (λ) which depend on the aspect ± c ratio. The largest radius of convergence occurs for the smallest aspect ratios. Furthermore, a preliminary study of the generalised Pad´eapproximants indicates that the bifurcation diagram for imaginary K, behaves in a similar manner to pipes of circular cross-section. Further coefﬁcients are required to construct the higher order approximants which will allow the bifurcation to be investigated more fully.

6.2 Conclusions and future work

The extended Stokes series falls within the broader class of computer extended series methods, which includes expansions in coordinate and time variables. These have been applied to several problems of compressible and viscous ﬂows [89].

In several cases it has met with a measure of success. Examples are the study of the boundary layer for steady, symmetrical ﬂow past a parabolic body by Van Dyke [86], and the problem of convection in a porous cavity by Mansour [52]. Notably, the results in this latter study agreed with boundary layer theory.

Yet there are some problems in which these techniques have led to controversial results, such as: subsonic flow past a circle [10, 34, 91] and laminar flow in a weakly curved pipe [88], a slowly rotating pipe [90] and a heated horizontal pipe [51]. The former controversy 6.2 Conclusions and future work 134 has been addressed. In particular, early application of the computer extended technique by Van Dyke & Guttmann [91] indicated that potential flow persisted beyond when the flow became sonic. A later reconsideration of this work by Guttmann & Thompson [34] showed that there was no evidence in the analysis of the series to support this claim - in agreement with Bollman’s earlier conclusion [10].

However, the controversies of the latter problems have yet to be successfully addressed. In particular, the extended Stokes series method applied to laminar ﬂow though a weakly curved pipe [88], a heated horizontal pipe [90] and a slowly rotating straight pipe [51] all conclude asymptotic behaviours which are not in agreement with boundary layer analysis or any other numerical or experimental result. Despite the success of this technique in other problems, this unresolved discrepancy ultimately lead to its general discredit. Nevertheless, in internal ﬂow problems such as these, where the boundary layer structure is unclear and where there is separation, this technique is theoretically an attractive alternative to the more traditional boundary layer methods.

Consequently, an assessment of the extended Stokes series method in fluid dynamics was a further objective of this work. In particular its vindication was sought by re-considering the Dean flow problem and applying the generalised Padéapproximants of Drazin & Tourigny [28] - the results from which are summarised above. Whilst the controversial asymptotic behaviour proposed by Van Dyke [88] has been discarded, the work primarily highlights the limitations of this technique in studying the high Reynolds number behaviour for this problem that was its principal attraction. Frustratingly, despite the vast number of terms calculated herein, only limited further understanding of the behaviour of the flow in this regime has been obtained. Therefore, with the present forms of analytic continuation, the application of the extended Stokes series method cannot be advocated in problems which are known to have a complex underlying structure. The insight gained in such cases is dwarfed by the results of standard numerical, experimental and boundary layer studies.

It is now left to conclude this discussion with more positive prospects, by turning to the understanding acquired regarding the square-root singularity at iK . Previously, the only ± c consideration given was to its removal as it was interpreted as a non-physical limitation of the Stokes series. Interestingly, this structure is found in the two additional internal flows discussed above as well as in several others including: flow inside a circle whose boundary Chapter 6. Final Conclusions 135 moves tangentially with a speed that varies with the circumference [18] and flow between infinite rotating disks [39]. It is possible that, as in the case of Dean flow, these singularities correspond to symmetry breaking bifurcations for the complex flow. The question arises whether these problems also admit a complex solution for which some component of the velocity tends to infinity as the Reynolds number tends to zero on the imaginary axis. This could be investigated by extending the series solution in each case to higher order and in- vestigating the bifurcation structure in the complex plane by means of the generalised Padé approximants. Alternatively, a numerical study of the complex solutions could be completed as in Chapter 4. If they are found to possess such behaviour for complex Reynolds number, this may suggest the existence of such solutions of the Navier-Stokes equations. In this way, the current study has uncovered a new and interesting aspect of Dean flow which may have ramifications in other such problems. This acknowledgement of the contribution of this work to the general understanding of fluid dynamics seems a fitting way with which to conclude. 136 137

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Appendix A

Introduction

A.1 Boundary layer asymptotics

For this discussion, the weakly curved pipe is considered.

Smith [76] assumed that the ﬂow was composed of an inviscid core surrounded by a thin boundary-layer, with characteristic length scale ǫ. In addition, he supposed that the azimuthal velocity w and streamfunction ψ scaled like W0 and Ψ0 respectively. Then, in the core ∂ 1 1, 1, (A.1) ∂r ∼ r ∼ whereas in the boundary ∂ 1 1 1 , . (A.2) ∂r ∼ ǫ r ∼ ǫ Therefore, for example, in the core the secondary velocity (u, v), from (1.8) in the small curvature limit, scales like

1 ∂ψ ∂ψ (u, v)= , (Ψ , Ψ ) (A.3) −r ∂θ ∂r ∼ 0 0 whereas in the boundary layer it scales like

1 ∂ψ ∂ψ Ψ Ψ (u, v)= , 0 , 0 . (A.4) −r ∂θ ∂r ∼ − ǫ ǫ

Estimates of the asymptotic scalings W0, Ψ0 and ǫ are now sought such that in the core Appendix A. Introduction 147 there is a balance between inertia forces and pressure gradients and in the boundary layer there is a balance between inertia, viscous and centrifugal forces.

In the core, from the Dean equation for vorticity (1.12) it follows that:

∂W ∂W 576KW sin θ 0 r cos θ 0 0. (A.5) 0 ∂θ − ∂r ∼ Using, (A.1)

∂W ∂W = sin θ 0 r cos θ 0 0. (A.6) ⇒ ∂θ − ∂r ∼ From (A.6), converting to cartesian coordinates (1.1), it follows that

∂W = 0 0. (A.7) ⇒ ∂y ∼

Equation (A.7), indicates that W0 is independent of y in the core. This agrees with the observation from numerical studies shown in Figure 1.6 (Chapter 1). Similarly, from the Dean equation in the azimuthal direction (1.13):

1 ∂Ψ ∂W ∂Ψ ∂W 0 0 0 0 4 (A.8) r ∂r ∂θ − ∂θ ∂r ∼ = Ψ W 1. (A.9) ⇒ 0 0 ∼

In the boundary layer, from the Dean equation for vorticity (1.12):

1 ∂Ψ ∂ ∂Ψ ∂ ∂W ∂W 4Ψ 0 0 2Ψ KW sin θ 0 r cos θ 0 . (A.10) ∇ 0 ∼ r ∂r ∂θ − ∂θ ∂r ∇ 0 ∼ 0 ∂θ − ∂r Using (A.2),

Ψ Ψ2 W 2K 0 0 0 (A.11) ǫ4 ∼ ǫ3 ∼ ǫ 1 1 = Ψ & W 2 . (A.12) ⇒ 0 ∼ ǫ 0 ∼ ǫ4K

Similarly, from the Dean equation in the azimuthal direction (1.13):

1 ∂Ψ ∂W ∂Ψ ∂W 2W 0 0 0 0 . (A.13) ∇ 0 ∼ r ∂r ∂θ − ∂θ ∂r A.1 Boundary layer asymptotics 148

Using (A.2),

W Ψ W 1 0 0 0 = Ψ , (A.14) ǫ2 ∼ ǫ ⇒ 0 ∼ ǫ

Equation (A.14) implies, from (A.9), that

W Ψ−1 ǫ. (A.15) 0 ∼ 0 ∼

Then, from (A.12)

1 ǫ2 (A.16) ∼ ǫ4K = ǫ K−1/6. (A.17) ⇒ ∼

Therefore, Ψ K1/6 and W K−1/6. Accordingly, the ﬂux ratio Q(K) W 0 ∼ 0 ∼ ∼ 0 ∼ K−1/6. 149

Appendix B

Extended Stokes series: circular cross-section

B.1 Expression for the recurrence relation

From identities (2.8) and (2.9), of solutions w(r, θ) and ψ(r, θ) to the Dean equations (1.12) and (1.13), the following identities are found:

In−1 Jn cos (2iθ) r2j n even ωn (r, θ)= En,i,j , (B.1)  2j+1 i=0 j=1 sin ((2i + 1) θ) r n odd   In−1 Jn−1 sin (2iθ) r2j n even ψn (r, θ)= Cn,i,j . (B.2)  2j+1 i=0 j=0 cos (((2i + 1) θ) r n odd  And so it follows that: 

In−1 Jn 2j−1 ∂ωn (r, θ) (2j)cos(2iθ) r n even = En,i,j , (B.3) ∂r  2j i=0 j=0 (2j + 1) sin ((2i + 1) θ) r n odd   In−1 Jn 2j ∂ωn (r, θ) (2i) sin (2iθ) r n even = En,i,j − , (B.4) ∂θ  2j+1 i=0 j=0 (2i + 1)cos((2i + 1) θ) r n odd   B.1 Expression for the recurrence relation 150

(2j) sin (2iθ) r2j−1 n even In−1 Jn−1 ∂ψn (r, θ)  = Cn,i,j  , (B.5) ∂r  i=0 j=0  (2j + 1)cos((2i + 1) θ) r2j n odd   In−1 Jn−1  2j ∂ψ (r, θ) (2i)cos(2iθ) r n even = Cn,i,j , (B.6) ∂θ  2j+1 i=0 j=0 (2i + 1) sin ((2i + 1) θ) r n odd −  In−1 Jn−1 ((2j)2 (2i)2) sin (2iθ) r2j−2 n even 2 ψn (r, θ)= Cn,i,j − , ∇  2 2 2j−1 i=0 j=0 ((2j + 1) (2i + 1) )cos((2i + 1) θ) r n odd  − (B.7)  2 In−1 Jn−1 2 2 2j−3 ∂ ψn (r, θ) (2j 2)((2j) (2i) ) sin (2iθ) r n even ∇ = Cn,i,j − − , ∂r  2 2 2j−2 i=0 j=0 (2j 1)((2j + 1) (2i + 1) )cos((2i + 1) θ) r n odd  − − (B.8)  2 In−1 Jn−1 2 2 2j−2 ∂ ψn (r, θ) (2i)((2j) (2i) )cos(2iθ) r n even ∇ = Cn,i,j − . ∂θ  2 2 2j+1 i=0 j=0 (2i + 1)((2j + 1) (2i + 1) ) sin ((2i + 1) θ) r n odd − − (B.9)  Appendix B. Extended Stokes series: circular cross-section 151 )] )] θ θ )] )] θ θ + 1) + 1) c c 1) 2 − + 1) +2 − c c a a 2 − +2 a a 2))cos((2 2))cos((2 − − d d 1))cos((2 − + 1))cos((2 c d + 1)(2 + 1)(2 )(2 )(2 b a a a (2 (2 − − ) )+(2 1)+(2 c c − + 1) d c + 1)(2 + 1)(2 )(2 )(2 b b b a )[((2 )[((2 )[((2 )[((2 2 2 2 2 ) ) c c (2 (2 + 1) + 1) c c − − 2 2 (2 (2 ) ) d d − − 2 2 ((2 ((2 3 3 − − d d + 1) + 1) +2 +2 d d b b 2 2 r r ((2 ((2 3 3 − − d d p,c,d p,c,d C C +2 +2 2 2 b b 2 2 r r ) m,a,b m,a,b C C p,c,d p,c,d r, θ 1 1 ( C C − − =0 =0 2 2 p p n d d J J 1 ψ 1 m,a,b m,a,b − − =0 =0 p p c c C C I I 1 1 1 1 and − − − − =0 =0 p p =0 =0 d d ) m m b b J J J J 1 1 1 1 − − =0 =0 − − p p r, θ c c =0 =0 I I ( m m a a I 1 1 I n } } − − n n =0 =0 ω m m b b = = J J p p 1 1 + + n n − − m m =0 =0 : : m m a a I I } } m,p m,p n n { { modd,peven modd,peven = = p p + − + + n n )= m m : : r, θ m,p m,p ( { { odd: meven,podd meven,podd n n − − ψ 4 B.1.1 Expanding For ∇ B.1 Expression for the recurrence relation 152 )] θ )] )] θ θ )] θ 1) + 1) c 1) + 1) − 2 c c − 2 2 − c 2 a − − a a − a cos ((2 − ) cos ((2 θ ) + cos((2 ) + cos((2 − θ θ ) θ 1) + 1) c + 1) − c c + 1) +2 c a +2 +2 a a +2 a )[cos ((2 )[cos ((2 c c + 1)[cos ((2 )[cos ((2 c c + − d d + − ( ( 1 1 d d ( ( − − d d +1 +1 d d +2 +2 b b 2 2 +2 +2 r r b b 2 2 r r y,c,d y,c,d E E y,c,d y,c,d 2 2 E E 2 2 x,a,b x,a,b E E x,a,b x,a,b E E y y =0 =0 J J d d y y =0 =0 J J 1 1 d d − − =0 =0 1 1 y y c c I I − − =0 =0 y y c c I I x x =0 =0 J J b b x x =0 =0 J J 1 1 b b − − =0 =0 1 1 x x a a I I − − =0 =0 x x } } n n a a I I n n } } +1= +1= y y n n +1= +1= + + y y x x : : n n + + x x : : x,y x,y xeven,y even xeven,y even { { x odd, y odd x odd, y odd x,y x,y { { − − + + Appendix B. Extended Stokes series: circular cross-section 153 ) ) θ θ 1) . + 1) − ) ) c θ θ a 2 +2 − + 1) + 1) a c c a 2 +2 − a c ))] sin ((2 ))] sin ((2 b b + 1)(2 + 1)(2 c c + 1))]sin((2 + 1))]sin((2 b b ((2 − )(2 )(2 c c ) )+((2 a a (2 − + 1)(2 + 1)(2 + 1) +1)+(2 d d a a [((2 [((2 )(2 )(2 1 1 d d − − d d +2 +2 [((2 [((2 b b 1 1 2 2 − − r r d d +2 +2 b b m,c,d m,c,d 2 2 r r C C 2 2 x,a,b x,a,b m,c,d m,c,d E E C C 2 2 1 1 − − =0 =0 x,a,b x,a,b m m d d J J E E 1 1 1 1 − − − − =0 =0 m =0 =0 c m c m m I I d d J J 1 1 x x =0 =0 J J − − b b =0 =0 m m c c 1 1 I I − − =0 =0 x x x x a a I I =0 =0 J J b b } } n n 1 1 = = x − − x =0 =0 x x a a + + I I n n m m } } n n : : = = x x m,x m,x + + { { modd,xeven modd,xeven n n m m : : − + )= m,x m,x { { meven,xodd meven,xodd r, θ + + ( n ω 2 ∇ B.1 Expression for the recurrence relation 154 )] θ )] θ + 2) ) c )] )] c θ θ 2 ) ) c c +2 − 2 a a +2 − a a + 1)) sin ((2 + 1)) sin ((2 2) sin ((2 2) sin ((2 c c − − d d )(2 )(2 + 1)(2 + 1)(2 a a b b (2 (2 − − ) )+(2 c c 1)+(2 1) )(2 )(2 − − b b d d )[((2 )[((2 2 2 ) ) c c + 1)(2 + 1)(2 (2 (2 a a − − 2 2 )[((2 )[((2 ) ) 2 2 d d ((2 ((2 4 4 + 1) + 1) − − c c d d (2 (2 +2 +2 b b 2 2 − − r r 2 2 p,c,d p,c,d + 1) + 1) C C 2 2 d d m,a,b m,a,b ((2 ((2 2 2 C C − − 1 1 d d − − =0 =0 p p +2 +2 d d b b J J 2 2 1 1 r r − − =0 =0 p p c c I I 1 1 p,c,d p,c,d − − C C =0 =0 m m 2 b b 2 J J 1 1 m,a,b m,a,b − − =0 =0 m m C a a C I I 1 1 } } n n − − =0 =0 p p = = d d J J p p 1 1 + + − − n n =0 =0 m m p p : : c c I I 1 1 m,p m,p − − { { =0 =0 meven,peven meven,peven m m b b J J + + 1 1 − − )= =0 =0 m m a a I I r, θ n n } } ( = = n p p + + ψ n n 4 m m : : even: ∇ n m,p m,p m odd, p odd m odd, p odd { { + − For Appendix B. Extended Stokes series: circular cross-section 155 )] θ )] )] . θ θ 2) )] ) θ c − ) 2 c c + 2) 2 2 − c 2 a − − a a − a sin ((2 sin ((2 ) + sin ((2 − − θ ) ) θ θ ) + sin ((2 ) θ a + 2) ) c c + 2) +2 c c +2 +2 a a +2 a )[sin ((2 )[sin ((2 c c + 1)[sin ((2 )[sin ((2 c c − + d d − + ( ( d d d d ( ( d d +2 +2 b b 2 2 +2 +2 r r b b 2 2 r r y,c,d y,c,d E E y,c,d y,c,d 2 2 E E 2 2 x,a,b x,a,b E E x,a,b x,a,b E E y y =0 =0 J J d d y y =0 =0 J J 1 1 d d − − =0 =0 1 1 y y c c I I − − =0 =0 y y c c I I x x =0 =0 J J b b x x =0 =0 1 1 J J b b − − =0 =0 1 1 x x a a I I − − =0 =0 x x n n a a } } I I n n } } +1= +1= y y n n +1= +1= + + y y x x : : n n + + x x : : x,y x,y xodd,y even xodd,y even { { x,y x,y xeven,y odd xeven,y odd { { − − − − B.1 Expression for the recurrence relation 156 . ) θ ) ) θ θ ) ) ) θ c c + 2) ) 2 c c 2 − +2 +2 − a a a a ))]cos((2 ))]cos((2 b b )(2 )(2 c c + 1))]cos((2 + 1))]cos((2 b b (2 − )+(2 ) a a + 1)(2 + 1)(2 c c )(2 )(2 d d (2 − [((2 [((2 2 2 − − d d + 1) +1)+(2 +2 +2 a a b b 2 2 r r m,c,d m,c,d + 1)(2 + 1)(2 C C d d 2 2 x,a,b x,a,b [((2 [((2 d d E E +2 +2 b b m m =0 =0 2 2 J J d d r r 1 1 − − =0 =0 m c m c m,c,d m,c,d I I C C 2 2 x x =0 =0 J J b b 1 x,a,b 1 x,a,b − − =0 =0 E E x x a a I I } } m m =0 =0 n n J J d d = = 1 1 x x − − + + =0 =0 n n m c m c m m I : I : x x =0 =0 m,x m,x J J b b { { meven,xeven meven,xeven 1 1 − − − + =0 =0 x x a a I I )= n n } } = = r, θ x x ( + + n n n m m : : ω 2 m,x m,x ∇ m odd, x odd m odd, x odd { { − + Appendix B. Extended Stokes series: circular cross-section 157

B.2 Coefﬁcients of the Euler transformed series

Table B.1 shows the first twelve coefficients for the Euler transformed series of the flux ratio, friction ratio, total vorticity and outer wall shear stress.

n n 2n+1 n n an (ǫ ) bn (ǫ ) dn ǫ 2 tn (ǫ 2 ) 0 1.0000000000 1.0000000000 3.9779317160e+00 -2.0000000000e+00 1 -3.1623448404e-02 3.1623448404e-02 1.3718461470e+00 -2.9005752090e-01 2 -1.8860302184e-02 1.9860344673e-02 8.4196226380e-01 -3.5720046560e-02 3 -1.3382351471e-02 1.4606831850e-02 6.1680191740e-01 -9.4595751130e-02 4 -1.0339385730e-02 1.1599072327e-02 4.9180096830e-01 -2.0195877990e-02 5 -8.4048423289e-03 9.6398794007e-03 4.1185995300e-01 -5.4175172060e-02 6 -7.0675438189e-03 8.2577596766e-03 3.5606961540e-01 -1.3136848450e-02 7 -6.0886465565e-03 7.2282681363e-03 3.1475950550e-01 -3.7054396670e-02 8 -5.3416059883e-03 6.4305401079e-03 2.8283672320e-01 -9.2435955120e-03 9 -4.7531274377e-03 5.7935547050e-03 2.5736016220e-01 -2.7688838390e-02 10 -4.2778297717e-03 5.2727661381e-03 2.3651012830e-01 -6.8231825090e-03 11 -3.8861171225e-03 4.8387804718e-03 2.1909879000e-01 -2.1822130360e-02 12 -3.5578678470e-03 4.4713977106e-03 2.0431682000e-01 -5.1957877580e-03

Table B.1: Coefficients for ǫ-series of the flux ratio (an), the friction ratio (bn), the total vorticity (dn) and the outer wall shear stress (tn) B.3 Revisiting Van Dyke’s work as K →∞ Van Dyke concluded that the Euler transform was successful in extending the validity of the series to all values of K. He also inferred that the flux ratio behaves like K−1/10 for large K. This is in opposition to the asymptotic theory which suggests that the flux ratio behaves like K−1/6 (as detailed in Appendix A.1).

The alternative techniques which Van Dyke used to discern the asymptotic structure of the Euler Transformed Dean series (discussed in section 2.3.2) are revisited in this section. Only the ﬂux and friction ratio are considered, with the analysis proving similar for the total vorticity and outer wall shear stress. The techniques which will be detailed all assume that the closest singularity in the ǫ-plane is at ǫ =1 and that the ﬂux and friction ratio take the form:

Q (ǫ)= A (1 ǫ)α (B.10) 1 − B.3 Revisiting Van Dyke’s work as K 158 →∞ and

Q (ǫ)= A (1 ǫ)−α . (B.11) 2 − It is argued that the difﬁculty in discerning the singularity structure at ǫ = 1, which corresponds to large K, indicates that this assumption is incorrect. One explanation is that there exists a cluster of singularities close to ǫ =1.

B.3.1 Completing the series

The method of completing the series [87] constructs estimates of the amplitudes A1 and A by ﬁtting successive terms in the Taylor expansion of A (1 ǫ)α and A (1 ǫ)−α to 2 1 − 2 − terms in the series Q (ǫ) and Q (ǫ) respectively for some suitably chosen α. The ‘correct’ 1 value of α is detected when the product of the estimates A1 and A2 behaves linearly in . n

2 α = 1/28 α 1.8 = 1/22 α = 1/20 α = 1/12 1.6 n=12 1.4

1.2 2 A 1 1 A

0.8

0.6

0.4

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 1/n

Figure B.1: Results from completing the series for the ﬂux and friction ratio

In Figure B.1 the product of the estimates for A1 and A2 for different values of α are shown.

For the ﬁrst twelve coefﬁcients available to Van Dyke A1A2 appears to extrapolate linearly 1 1 in n to unity for α = 20 . However when the higher order terms are considered this is no longer the case. In fact, no fraction value for α for which the product A1A2 behaved linearly or converged in any way to 1 could be found. This is strong evidence to suggest Appendix B. Extended Stokes series: circular cross-section 159 that Q (ǫ) and Q (ǫ) do not take the proposed singularity structure.

B.3.2 Comparison with ln(1 x)α − A sequence of estimates for α can be constructed by direct comparison of the series expansion of Q (ǫ) with the Taylor expansion of

ln [(1 x)α]= α ln(1 x). (B.12) − −

The sequence of estimates of the ﬁrst ﬁfteen terms in the ǫ- series are:

α =0.03162, 0.038721, 0.04197, 0.04348, 0.04649, 0.04694, 0.04729, 0.04756,... (B.13)

0.04777, 0.04793, 0.04806, 0.04817, 0.04825. (B.14)

1 In accordance with Van Dyke these are are ﬁtted to within 5 % by

1 1 α = 1 . (B.15) 20 − 2n However, the remaining estimates are not. For example, the estimates from the last ﬁve terms are:

α =0.04695, 0.04694, 0.04692, 0.04691, 0.04690. (B.16)

The convergence of these estimates has been tested, but has not produced any value of α which can be authenticated by any other technique. Furthermore the coefﬁcients do not behave monotonically which should be the case if Q(ǫ) had the assumed singularity structure.

B.3.3 Critical-point renormalisation

Critical-point renormalisation constructs an estimate of the difference between the exponents of two different functions known to be singular at the same point [42]. This involves n constructing a generating function n cnǫ whose exponent is one less than the difference of the original exponent. B.4 ‘Conﬂuent-singularity analysis’ 160

This technique is applied to the ﬂux and friction ratio. The coefﬁcients of the generating n function n cnǫ are an cn = (B.17) bn and its exponent is [42] 2α 1. (B.18) −

The Neville table analysis of the ﬁrst twelve coefﬁcients, yields the sequence of estimates [88]:

α =0.0555, 0.0550, 0.0535, 0.0523, 0.0512, 0.0503.

The last ﬁve available coefﬁcients yield the sequence of estimates:

α =0.06173, 0.06174, 0.06175, 0.06175. with a Neville table which does not behave monotonically. As stated previously, this is indicative of the generating function not taking the assumed form.

B.4 ‘Conﬂuent-singularity analysis’

In Chapter 2, section 2.3.2, the difficulty in discerning the singularity structure near ǫ = 1 of the Euler transformed Dean series was discussed. One possible explanation for this difficulty is the existence of a finite number of confluent singularities at ǫ =1, (K2 ). →∞ For example the flux ratio takes the following form:

m Q (ǫ)= A (1 ǫ)−αj . (B.19) j − j=1 Initially the simpler case of two such singularities, (m = 2) above, was considered

Q (ǫ)= A (1 ǫ)−α1 + A (1 ǫ)−α2 . (B.20) 1 − 2 −

Correction terms to the ﬂux ratio equations can be calculated in these cases [43]. Alter- Appendix B. Extended Stokes series: circular cross-section 161 natively, one could solve the simultaneous equations formed by four consecutive terms in the series expansion of (B.20), and look for solutions of (A1,α1, A2,α2) which are robust when the set of terms considered are changed. However, it is easier to calculate one of the exponents if the value of the other is known. Accordingly, the exponents α1 = 1/20 and α1 =1/12, corresponding to Van Dyke’s hypothesis and generally accepted boundary layer theory respectively, were considered in turn. However, no consistent structure was found. The analysis suffering the same ill-effects reported in Chapter 2.

As an alternative, the ‘Conﬂuent-singularity analysis’ [43] was considered. This involves constructing the auxiliary function G(s) as follows.

The variable ξ is introduced with ǫ = 1 e−ξ . Then Q (ξ) becomes: −

m m −αj ln(1−ǫ) αj ξ Q (ξ)= Aje = Aje . (B.21) j=1 j=1 The auxiliary function G(s) is then deﬁned as follows:

∞ G(ξ)= e−sQ (sξ) ds 0 ∞ m m ∞ −s αj ξs −s(1−αj ξ) = e Aje ds = Aj e ds 0 j=1 j=1 0 m ∞ e−s(1−αj ξ) = A j − 1 α ξ j=1 j 0 − m A = j . (B.22) 1 α ξ j=1 j − This is equivalent to expanding Q(ξ), in the variable ξ:

m αj ξ Q (ξ)= Aje j=1 m ∞ k k Aiα ξ = j . (B.23) k! j=1 k=0 B.5 Alternative extrapolation techniques 162

Then multiplying the kth coefﬁcient in the expansion by k!:

m ∞ k k Ajα ξ Q (ξ)= j (k!) k! j=1 k=0 m ∞ k k = Ajαj ξ . (B.24) j=1 k=0 and then summing over k to obtain G(ξ):

m A G (ξ)= j . (B.25) 1 α s j=1 j − G(ξ) therefore has a m poles at s = α−1 with residues A . The function G(s) was j − j analysed by means of Domb-Sykes analysis or linear Pad´eapproximants. The Domb- Sykes analysis showed that G(s) did not take this theoretical structure. This agrees with the possibility ǫ-series has a cluster of singularities close to ǫ = 1, but not identical to it, which may correspond to the critical points found by numerical studies [73].

B.5 Alternative extrapolation techniques

As observed in Chapter 2, though the Neville algorithm is the preferred extrapolation technique used in Domb-Sykes analysis there exists other extrapolation techniques which may overcome the correction terms due to conﬂuent singularitieswhich are suspected to exist for the Euler transformed series [26]. Furthermore, given that the Neville algorithm is irregular for the Euler transformed series it seemed that the application of alternative extrapolation technique was required. The extrapolation techniques considered were:

Wynn’s ǫ algorithm, • Brezinki’s θ algorithm, • Barber-Hamer algorithm, • Levin U-T transform and • Lubkin’s three-term transformation. • Appendix B. Extended Stokes series: circular cross-section 163

The code written to apply these algorithms can be found at the end of this section. These techniques were applied to the ratio of the coefficients of the Euler transformed flux ratio Q(ǫ), to find an estimate of the location of the nearest singularity and its critical exponent, without success.

In particular, in estimating the location of the nearest singularity Berzinki’s and Barber- Hamer’s algorithm proved unstable, whereas the other three suggested that there was a singularity close to ǫ =1 but where they converged, the estimates did not agree across the three algorithms.

The algorithm’s were also used calculate an estimate for the critical exponent under the assumption that the nearest singularity was at ǫ =1, that is they were applied to the sequence

aˆ 1 n , (B.26) − aˆ n−1 for n =1..m. Whilst Berzinki’s and Barber-Hamer’s algorithm also proved unstable in this case, the Levin U-T transform and Wynn’s algorithm appeared to both suggest the critical exponent was α 0.08. However, this estimate was not corroborated when considering ≈ the unbiased estimate of α found by applying the two techniques to the values [32]:

n(2 n) aˆn +(n 1)2 aˆn−1 aˆn− aˆn− − 1 − 2 , (B.27) aˆn−1 n aˆn (n 1) aˆn−1 − − aˆn−2 for n = 1..m. The unbiased estimates conclude that that α 0.11. Furtheremore, the ≈ − corresponding asymptote of these estimates of α do not ﬁt the ratio of the coefﬁcients of the Domb-Sykes plot. B.5 Alternative extrapolation techniques 164

WYNN'S EPS ALGORITHM: O WYNN dproc r T list , m T integer T Matrix local n, k, i, e; description "WYNN Algorithm to increase the rate of convergence of a sequence" for n from K1 to m C1 do e n K1 d 0 : od : for n from 0 to m K1 do e n 0 d evalf r nC1 : od : for k from 1 to m K1 do for i from 0 to m K1Kk do 1 e i k d evalf e iC1 k K2 C ; e iC1 k K1 Ke i k K1 od : od : return e; end proc : BREZINSKI'S THETA ALGORITHM: O BREZINSKI dproc r T list , m T integer T Matrix local n, k, i, θ; description "Brezinski's algorithm to increase the rate of convergence of a sequence" for i from 0 to m do θ i K1 d 0; od : for n from 0 to m K1 do θ n 0 d evalf r nC1 : od : for k from 1 to m K1 do if k mod 2 = 0 then k for i from 0 to m K1K3$ do 2 θ iC2 k K2 Kθ iC1 k K2 $ θ iC2 k K1 Kθ iC1 k K1 θ i k d θ iC1 k K2 C ; θ iC2 k K1 K2$θ iC1 k K1 Cθ i k K1 od : else k K1 for i from 0 to m K1K 3$ C1 do 2 1 θ i k d θ iC1 k K2 C ; θ iC1 k K1 Kθ i k K1 od : fi : od : return θ; end proc : BARBER-HAMER ALGORITHM: O BARHAM dproc r T list , m T integer T Matrix local n, k, i, bhs , bhe ; description "Barber-Hammer algorithm to increase the rate of convergence of a sequence" for i from 1 to m do bhs i 0 d evalf r i : od : for i from 1 to m do bhe i K1 d 0 : od : for k from 0 to m K1 do for i from k C1 to m K1Kk do 1 1 bhe i k dK $ 1K K1 k $bhe i kK1 C : 2 bhs iC1 k Kbhs i k od ; for i from k C2 to m K1Kk do 1 bhs i k C1 d bhs i k C : bhe i k Kbhe iK1 k od ; od ; return bhs ; end proc : Appendix B. Extended Stokes series: circular cross-section 165

LEVIN U-TANSFORM: O LEVUT dproc r T list , m T integer T Matrix local i, k, n, us , b, c, U; description "Levin u-transform to increase rate of convergence of a sequence" ; for i from 1 to m do us i d evalf r i : od : for n from 2 to m do for k from 0 to m do n k K 2$us n b n k d : us n Kus nK1 od : od : for n from 2 to m do for k from 0 to m do n k K 2 d c n k us n Kus nK1 : od : od : for n from 2 to m do b n 0 d U n 0 c n 0 : od ; m K1 for from to do k 1 trunc 2 for n from k C1 to m Kk do add binomial k, l $ K1 l$b nCk Kl k , l = 0 .. k U n k d evalf ; add binomial k, l $ K1 l$c nCk Kl k , l = 0 .. k od : od : return U; end proc : LUBKIN'S THREE-TERM TRANSFORMATION: O LUBKIN dproc r T list , m T integer T Matrix local k, i, n, ls ; description "Lubkin three-term transform to increase the rate of convergence of a sequence" ; for n from 0 to m K1 do ls n 0 d evalf r nC1 : od : m K1 for from to do k 1 trunc 3 for n from 3$k to m K1 do ls n k ls n k K1 ls nK1 k K1 d K $ ls n k K1 Kls nK1 k K1 2 ls nK1 k K1 Kls nK2 k K1 ls nK2 k K1 1 1 C K $ ls nK2 k K1 Kls nK3 k K1 ls n k K1 Kls nK1 k K1 2 ls nK1 k K1 Kls nK2 k K1 1 C ls nK2 k K1 Kls nK3 k K1 : od : od : return ls ; end proc : 166

Appendix C

Pade´ approximants

A brief overview of the theory of linear and quadratic Padéapproximants and results from their application to the flux ratio are detailed below. A more rigorous treatment of the theory of linear Padèapproximants can be found in [4] and [5]. They show limited success of analytic continuation and prompt the application of the more generalised form detailed in Chapter 3.

C.1 Linear Pade´ approximants

The [M, N] linear Pad´eapproximant of a function f(K), is the rational fraction of polynomials

f (K) a + a K + ... + a KM 0 0 1 M (C.1) [M, N]= = N , f1(K) b0 + b1K + ... + bN K where the coefficients a0...aM and b0...bN are chosen to make the first terms of the expansion of [M, N] agree with the first M + N +1 terms in the Taylor expansion of f(K) about K =0, i.e.∗:

f (K) f (K) f (K)= O KM+N+1 . (C.2) 1 − 0 ∗The linear equation from which the technique gets its name Appendix C. Pade´ approximants 167

Without loss of generality it is assumed that:

f1(0) = b0 =1. (C.3)

The system is well determined with M + N +1 equations for M + N + 1 unknowns

(M +1 from f0(K) and N from f1(K)). In practice, the coefficients a0...aM are found M+1 M+N first by equating the higher order terms (K ...K ). Then the coefficients of b0...bN are found by equating the remaining lower order terms.

The singularities of f(K) are represented by the zeros of f1(K). The accuracy of estimating the singularities in this way depends on the nature of the singularity. The zeros are generally close to actual poles, clustered near essential singularities and along certain cuts from branch points [26].

In the case of algebraic singularities such as (2.17), it is best to consider the Pad´eapproximants of the logarithm of the function:

d d α ln (f (K)) ln ((K K)α) − . (C.4) dK ∼ dK c − ∼ K K c − An estimate for α can be found by calculating the residue of the approximant at the estimate for Kc. This also applies to the singularity at inﬁnity, which will give the asymptotic behaviour of f(K). The diagonal Pad´eapproximants for which M = N are considered here as they are known to possess remarkable, though not yet fully understood, properties of analytic continuation [27].

C.1.1 Results

Figure C.1 shows the estimates of singularities by Linear Pad´eapproximants for the ﬂux 2 ratio. The approximants detect a singularity in agreement with Kc found by Domb-Sykes analysis, indicated by the solid line in Figure C.1. They also exhibit singularities along the branch cut in accordance with the square root singularity structure (2.37), indicated by the dashed res lines in Figure C.1. This was also found to be the case for the total vorticity and outer wall shear stress. C.1 Linear Pade´ approximants 168

−1 2 Kc −1.02

−1.04

−1.06

−1.08

−1.1

−1.12

−1.14

−1.16

eoo hdnmntrftheapproximant thedenominatorof Zerosof −1.18

−1.2 0 5 10 15 20 25 30 35 40 45 M Figure C.1: Estimate of singularities for the ﬂux ratio by linear Pad´eapproximants. indicates poles along the branch cut

Figure C.2 shows the estimates of the critical exponent of the singularity at infinity, α. The estimates exhibit the erratic convergence typical of Padé approximants. The estimates do not agree with boundary layer theory which suggests that Q(K) (K2)−1/12, Q(K) ∼ ∼ (K2)1/12, Ω(K) K1/3 and τ(K) K1 for large K. This is most marked for the outer ∼ ∼ wall shear stress τ(K). Notably, the estimates of the critical exponent for the flux ratio is in the vicinity of α = 1/20, the critical exponent suggested by Van Dyke. This is in agreement with Mansour [53].

For the current problem, one should be cautious of the results from this technique. Analysis in Chapter 2 suggests that the leading order singularity of the flux ratio (2.12) takes the additive singularity form (2.37). In the case of multiplicative and additive singularities such as this the linear Padéapproximants have been known to fail [4]. This is particularly true in the presence of a square-root singularity, the linear Padéapproximant adding poles along the branch cut. Appendix C. Pade´ approximants 169 45 40 50 1/12 1/20 35 40 30 25 30 M M 20 friction ratio 20 15 (b) outer wall shear stress 10 (d) 10 5 0 0 0 0

0.1 0.1

0.14 0.12 0.08 0.06 0.04 0.02 0.08 0.06 0.04 0.02

α α 45 45 40 40 −1/12 −1/20 35 35 of the ﬂux and friction ratio, total vorticity and α 30 30 25 25 M M 20 20 ﬂux ratio total vorticity (a) 15 15 (c) 10 10 5 5 0 0 0 0

−0.1

−0.02 −0.04 −0.06 −0.08 −0.12 −0.14 0.5 0.4 0.3 0.2 0.1

α α Plot of the estimates of critical exponent outer wall shear stress by linear Pad´eapproximants Figure C.2: C.2 Quadratic Pade´ approximants 170

C.2 Quadratic Pade´ approximants

From the analysis in Chapter 2 the leading singularity of the Dean series takes the form:

1 2 2 2 K + Kc g(K)+ h(K). (C.5) It therefore seems more sensible to consider a quadratic Pad´eapproximant rather than a linear one. This is constructed by solving the quadratic equation in f(K):

2 f2 (K) f (K)+ f1 (K) f (K)+ f0 (K)=0

f (K) f 2 (K) 4f (K) f (K) = f ± (K)=− 1 ± 1 − 2 0 , (C.6) ⇒ 2f2 (K) with

H n f0 (K)= f0,nK , (C.7) n=0 I n f1 (K)= f1,nK , (C.8) n=0 J n f2 (K)= f2,nK , (C.9) n=0 where f0(K) of order H, f1(K) of order I and f2(K) of order J are sought so that the N n truncated Taylor expansion f(K) = n=1 pnK is a solution to order H + I + J +1. As in the case of linear Pad´eapproximants, the diagonal approximants, H = I = J, are considered. Without loss of generality, it is assumed f2,0 =1. Furthermore, uniqueness at K =0 is imposed by taking f 2 f = 1,0 . (C.10) 0,0 4

C.2.1 Results

Table C.1 shows the estimates of iK by quadratic approximants for the flux ratio, total ± c vorticity and the outer wall shear stress. The estimates are in agreement with the Domb- Sykes analysis and linear Padéapproximants. Appendix C. Pade´ approximants 171 07501 77667 45086 45086 27317 50817 953706 445141 770276 249377 069161 174618 654816 341687 485211 648484 648484 018180 069717 orticity and outer wall shear stress c K 7362022627952698099919036592390787637404 1.01451655909025345020522269391 30518770051235120901 1.01664721756772127610326109799 24761406022865227149 1.01698827790960258533247426153 66548766037594444816 1.01698827790960258533247426153 32803596377597112781 1.01699350108032823540230188800 1.01699438012286113216062403086 58234276860901592560 0.97863843242658209467656703916 393537526432452562430827939938521766446145 1.0169944020557654939253076027 827825986463610832575 1.0169944015267313290827675896 827824158291188739038 1.0169944009109157081660182446 827824161164508675588 1.0169944009909871907355699023 827824160977519645628 1.0169944009954664097622008993 827824160962351601604 1.0169944009933336285866664801 827824160962471621811 1.0169944009934328161586198466 827824160962486781404 1.0169944009935458048069532790 827824160962486654655 1.0169944009935402911396908090 827824160962486653515 1.0169944009935402911396908090 827824160962486653508 1.0169944009935382664661416617 1.0169944009935382777283643893 using quadratic Padéapproximants of the flux ratio, total v c K Estimates of flux ratio total vorticity outer wall shear stress Table C.1: 45 1.01699880146682445559529264700886 1.016994923589942 1.01699446662457541247021704494647 1.0169944016692568572316776158280 1.016994386660847 8 1.016994400914366 1.01699440166925685723167761582809 1.016994400998501 1.016994400993630762029580228080210 1.0169944009935676527019812951931 1.016994400993642 1.016994400993567652701981295193111 1.016994400993537 12 1.01699440099354 1.0169944009935382793830167937408 1.016994400993538278468379540588513 1.01699440099353 14 1.01699440099353 1.0169944009935382782424969578454 1.016994400993538278241624579452215 1.01699440099353 1.01699440099353 1.01699440099353827824160983069041617 1.01699440099353 1.0169944009935382782416096311185 1.016994400993538278241609624852218 1.01699440099353 1.01699440099353 1.01699440099353827824160962486971920 1.01699440099353 1.0169944009935382782416096248665 1.016994400993538278241609624866521 1.01699440099353 1.01699440099353 1.0169944009935382782416096248665 1.01699440099353 d 3 1.0175287449179008569747741647765 1.017196938230642 C.3 Generalised Pade´ approximants: total vorticity Branch II 172

Figure C.3b shows the plots of the quadratic approximants of degree H = 20 of the flux ratio for real K. Both roots, f + and f − have a series of poles along a main branch which agree with the numerics of Chapter 4. All approximants exhibit this behaviour, though the location of the poles varies with degree H of the approximant. This behaviour is similar to the presence of poles along a branch cut for linear Padéapproximants. It suggests the possibility of a further singularity which the quadratic Padéapproximant has difficulty capturing. Figure C.3b shows the plots of the quadratic approximants H = 20 of the flux ratio for imaginary K. The f + approximant corresponds to the Dean series (1.16) evaluated for imaginary K. The f − approximant bifurcates at K2 and exhibits a pole − c near the origin. This behaviour is generic for approximants of any degree H, though the location of the poles varies with H.

The behaviour described above is similar for the total vorticity and outer wall shear stress. It suggests the presence of further singularities, not necessarily quadratic, and prompts the consideration of the generalised Pad´eapproximants discussed in Chapter 3.

C.3 Generalised Pade´ approximants: total vorticity Branch II

The approximants of Branch I-II of the total vorticity for imaginary K are shown in Figure C.4. On these branches the total vorticity is imaginary, though the ﬂow itself is complex. Branch I corresponds exactly to the Dean series solution (2.16).

The different order approximants of Branch II exhibit two behaviours. Approximants d = 2, 4, 5, 7, 8, and 12 cross the axes at the origin. Approximants d =3, 6, 9, 10 and 11 have a second turning point, which is closer to the K =0 axis for the higher order approximants. This behaviour is not in disagreement with the postulate that Ω(K) K0 as K 0 along ∼ → the imaginary axis, as proposed in section 3.2.2.4. Appendix C. Pade´ approximants 173

2 f+ f−

1.5

1 Q(K)

0.5

0 0 0.2 0.4 0.6 0.8 1 K2

(a) Quadratic Pad´eapproximant for K

2 f+ f−

1.5

1 Q(K)

0.5

0 −1 −0.8 −0.6 −0.4 −0.2 0 K2

(b) Quadratic Pad´eapproximant for imaginary K

Figure C.3: Plot of the quadratic Pad´eapproximants for the ﬂux ratio C.4 Generalised Pade´ approximants: path-continuation 174

1e+06

100000

10000 Branch II 1000 Ω )

( 100 d=2 d=3 Im d=4 10 d=5 d=6 1 d=7 d=8 d=9 0.1 d=10 Branch I d=11 0.01 d=12 1 0.8 0.6 0.4 0.2 0 Im(K) Figure C.4: Plot of the total vorticity on Branch I and II found using generalised Pad´eapproximants

C.4 Generalised Pade´ approximants: path-continuation

The generalised Pad´eapproximants deﬁned in Chapter 3 are solutions to the polynomial equation:

Fd (K,p)=0. (3.8)

These approximants were constructed using path-continuation techniques. This involves

ﬁnding a solution at (K0 +dK) from a known solution (K0,p0) by constructing the solution curve to (3.8) .

Initial solutions to the system were found using MAPLE †. The path-continuation method used depended on whether the solution found was a bifurcation point. In this problem three types of points were identiﬁed:

Regular points. ∂Fd =0 • ∂p Turning points. ∂Fd =0 and ∂Fd =0 (shown in Figure C.5) • ∂p ∂K 2 Transcritical points. ∂Fd = ∂Fd =0 and ∂ Fd =0. (shown in Figure C.6). • ∂p ∂K ∂p2

† Note from the deﬁnition of Fd the point (0, 0) is always a solution to the system Appendix C. Pade´ approximants 175

Figure C.5: Depiction of turning point bifurcation

Figure C.6: Depiction of transcritical bifurcation

At regular points path-continuation was achieved using simple continuation and at turning point bifurcations by pseudo-arclength continuation. When transcritical bifurcations were identiﬁed, simple continuation was used with an increased step-size in order to ‘jump’ the bifurcation point. At transcritical points, solutions on the secondary branch were later found using MAPLE. This secondary branch was then continued accordingly.

The two path-continuation techniques are as follows:

Simple continuation:

0 From an initial solution (K0,p ) of (3.8), a solution at K0 +dK is sought. The ﬁrst estimate 0 p = p(K0). It is corrected using the Taylor expansion of p(K0 + dK):

dp p(1) (K + dK)= p (K )+ dK . (C.11) 0 0 dK C.4 Generalised Pade´ approximants: path-continuation 176

dp 0 The correction term dK is found from the Taylor expansion of Fd(K + dK,p ):

∂ dp F K + dK,p0 p0 = F K ,p0 (C.12) ∂p d dK − − d 0 dp F(K ,p0) = = p0 d 0 . (C.13) ∂ 0 ⇒ dK − ∂p (Fd (K + dK,p ))

The process is iterated to obtain estimates pm, for m = 1 ... , until a sufﬁciently accurate solution p(K0 + dK) to (3.8) is found. This technique fails at turning point bifurcations.

Pseudo-arclength continuation:

In the pseudo-arclength continuation, a new arclength parameter, s, is introduced and solutions are sought to the augmented system:

Fd (K,p)=0, (C.14) N(K,p) =p ˙ (p(s) p(s )) + K˙ (K(s) K(s )) (s s ) − 0 − 0 − − 0 =p ˙∆p + K˙ ∆K (s s )=0, (C.15) − − 0 with the constraint: p˙ 2 + K˙ 2= ds2, (C.16) | | | | dp ˙ dK where p˙ = ds and K = ds . The second equations N(K,p)=0, requires the point (K(s),p(s)) to lie on the plane normal to the tangent. The system can also be solved using Newton’s method, by which

(F ) ∆p +(F ) ∆K = F , (C.17) d p d K − d p˙∆p + K˙ ∆K = N. (C.18) −

∂Fd ∂Fd ˙ where, (Fd)p = ∂p and (Fd)K = ∂K . For each step, the derivatives p˙ and K are found ﬁrst

ds K˙ = , (C.19) 1+ (Fd)K 2 | (Fd)p | (F ) p˙ = d K K.˙ (C.20) − (Fd)p Appendix C. Pade´ approximants 177

And the ﬁrst predictor is deﬁned as:

(1) K = K0 + K,˙ (C.21)

(1) p = p0 +p. ˙ (C.22)

The corrector step is then accordingly:

Fd N +p ˙ (F ) ∆K = − − d p , (C.23) K˙ +p ˙ (Fd)K − (Fd)p F (F ) ∆u = d + ∆K d K . (C.24) −(Fd)p − (Fd)p

So that subsequent corrector steps are deﬁned as:

(n+1) K = Kn + ∆K, (C.25)

(n+1) p = pn + ∆p. (C.26)

A new tangent is only found after convergence. For complex K, the above algorithm holds by taking the real part of N(K,p). 178

Appendix D

Numerical solution of the Dean equations for complex K

D.1 Path-continuation

The spectral coefﬁcients and ﬁnite difference equations of the Dean equations are ordered equivalently to Yang & Keller [96].

The spectral coefﬁcients (4.25) are ordered into the following vectors:

f0 = f0,0, g0 = g0,0, w0 = w0,0, (D.1) f T =(f ...f , 1 j M), (D.2) j −κ,j κ,j ≥ ≤ gT =(g ...g , 1 j M + 1), (D.3) j −κ,j κ,j ≥ ≤ wT =(w ...w , 1 j M). (D.4) j −κ,j κ,j ≥ ≤

And these vectors are ordered into the vector X

T X =(f0,g0,w0, f1,g1,w1,...,fM ,gM ,wM ,gM+1). (D.5)

The finite difference equations are ordered first in r-discretisation and then the spectral discretisation. That is for fixed j meshpoint the finite difference equations for κ k κ − ≤ ≤ are written in order. This is done for j =1, 2,...M and finally for j = M +1. The system Appendix D. Numerical solution of the Dean equations for complex K 179 is written in the vector equation form:

G (X; KRe,KIm, δ)=0. (D.6)

Here G (X; KRe,KIm, δ) has (3(2κ +1)+1) components, each being one of the difference equations. The advantage of constructing G (X; KRe,KIm, δ) in this manner is that its Jacobian matrix has a block-band structure shown in Figure D.1.

Figure D.1: Block matrix structure for Jacobian of Dean equations

Each square block is a matrix of order (3(2κ + 1))(3(2κ + 1)). There are M such rows of block matrices. This array of blocks is bordered by one row and column as shown.

All other elements of GX are zero. The sparse structure of the matrix is computationally advantageous. 180

Appendix E

Extended Stokes series: elliptic cross-section

E.1 Derivation of the Dean equations for λ< 1

The derivation of the Dean equations in elliptical coordinates is analogous to its derivation in polar coordinates, nevertheless it is given here for completeness.

The ﬂow is assumed incompressible, steady and fully developed (i.e. u, v and w but not the pressure are independent of φ). Accordingly, the continuity (1.2) and momentum equations (1.3) in elliptical coordinates (5.1) are:

1 ∂ (uγ) u ∂T 1 ∂ (vγ) v ∂T continuity: + + + =0, (E.1) γ2 ∂ γT ∂ γ2 ∂θ γT ∂θ ∂w u ∂w v a sinh ()cos(θ) a cosh () sin (θ) φ-momentum: + + uw vw (E.2) ∂ γ ∂θ γ γT − γT 1 ∂ p 1 ∂ 1 ∂(Tw) ∂ 1 ∂(Tw) = + + , −T ∂φ ρ γ2 ∂θ T ∂θ ∂ T ∂ (E.3) u ∂u a2 sin(θ) cos(θ) v ∂u -momentum: + vu + γ ∂ γ3 γ ∂θ a2 sinh () cosh () w2 v2 a sinh ()cos(θ) − γ3 − γT 1 ∂ p 1 ∂ T ∂(γv) ∂(γu) = , (E.4) −γ ∂ ρ − γT ∂θ γ2 ∂ − ∂θ Appendix E. Extended Stokes series: elliptic cross-section 181

u ∂v u2 vu θ-momentum: a2 sin (θ)cos(θ)+ a2 sinh () cosh () (E.5) γ ∂ − γ3 γ3 ∂v v w2 + + a cosh () sin (θ) ∂θ γ γT 1 ∂ p 1 ∂ T ∂(γv) ∂(γu) = , (E.6) −γ ∂θ ρ − γT −∂ γ2 ∂ − ∂θ where:

γ2 = a2 sinh ()2 + sin (θ)2 , T = (a cosh ()cos(θ)+ R).

a cosh(0) Assuming that the curvature of the pipe, R , is small the above equations reduce to:

1 ∂ (uγ) ∂ (vγ) continuity: + =0, (E.7) γ2 ∂ ∂θ ∂w u ∂w w 1 ∂ P 1 ∂2w ∂2w φ-momentum: + = + + , (E.8) ∂ γ ∂θ γ −R ∂φ ρ γ2 ∂θ2 ∂2 u ∂u a2 sin(θ)cos(θ) v ∂u -momentum: + vu + γ ∂ γ3 γ ∂θ a2 sinh () cosh () w2 v2 a sinh ()cos(θ) − γ3 − γR 1 ∂ p 1 ∂ 1 ∂(γv) ∂(γu) = , (E.9) −γ ∂ ρ − γ ∂θ γ2 ∂ − ∂θ u ∂v u2 vu θ-momentum: a2 sin (θ)cos(θ)+ a2 sinh () cosh () γ ∂ − γ3 γ3 ∂v v w2 + + a cosh () sin (θ) ∂θ γ γR 1 ∂ p 1 ∂ 1 ∂(γv) ∂(γu) = . (E.10) −γ ∂θ ρ − γ −∂ γ2 ∂ − ∂θ

p As u, v and w are assumed independent of φ, it follows from (E.8) that ρ must be of the form φf1 (, θ)+ f2 (, θ) and then from (E.9) and (E.10) that f1 (, θ) must be a constant. The axial pressure gradient is deﬁned as:

1 ∂ p G = , (E.11) −R ∂φ ρ ρ where G is a constant. E.1 Derivation of the Dean equations for λ< 1 182

The continuity equation (E.7) is satisﬁed by the streamfunction ψ(, θ) such that

∂ψ ∂ψ = γu, = γv. (E.12) ∂θ ∂ −

These identities are substituted into (E.8), (E.9) and (E.10) and the pressure terms are eliminated from (E.9) and (E.10) by cross-differentiation. Accordingly, the governing equations become:

1 ∂ψ ∂ 2ψ ∂ψ ∂ 2ψ ν 4ψ = ∇ ∇ (E.13) ∇ γ2 ∂θ ∂ − ∂ ∂θ 2wa cosh ( ) cosh () sin (θ) ∂w sinh ()cos(θ) ∂w 0 + , − Rγ2 cosh ( ) ∂ cosh ( ) ∂θ 0 0 1 ∂ψ ∂w ∂ψ ∂w ν 2w = + G, (E.14) ∇ γ2 ∂θ ∂ − ∂ ∂θ where 1 ∂2 ∂2 2 = + . (E.15) ∇ γ2 ∂2 ∂θ2 The following non-dimensional variables are introduced:

γ2 ψ w γ˜2 = , ψ˜ = , w˜ = , (E.16) 2 2 a cosh (0) ν ω0 where ω0, is the maximum down-pipe velcocity in a straight pipe with elliptical cross- section driven by the same pressure gradient and ν is the viscocity. The spatial non- dimensionalisation ﬁxes the semi-major axis of the system at 1, and the semi-minor axis varies as tanh( ). It follows that the limit is a curved pipe of circular cross- 0 0 → ∞ section with radius one. With this non-dimensionalisation, the system is reduced to (dropping the tilde notation):

1 ∂ψ ∂ 2ψ ∂ψ ∂ 2ψ 4ψ = ∇ ∇ ∇ γ2 ∂θ ∂ − ∂ ∂θ w cosh () sin (θ) ∂w sinh ()cos(θ) ∂w 576 K + , (E.17) − γ2 cosh ( ) ∂ cosh ( ) ∂θ 0 0 1 ∂ψ ∂w ∂ψ ∂w 2w = + C, (E.18) ∇ γ2 ∂θ ∂ − ∂ ∂θ

2 2 2 with C = Ga cosh (0) a constant, the dimensionless parameter K = a cosh(0)w0 a cosh(0) . νw0 ν 288R Appendix E. Extended Stokes series: elliptic cross-section 183

At K = 0, the equations are applicable to ﬂow in a straight-pipe of elliptic cross-section, so C = 2 1 + 1 : − sinh2() cosh2()

1 ∂ψ ∂ 2ψ ∂ψ ∂ 2ψ 4ψ = ∇ ∇ ∇ γ2 ∂θ ∂ − ∂ ∂θ w cosh () sin (θ) ∂w sinh ()cos(θ) ∂w 576 K + , (E.19) − γ2 cosh ( ) ∂ cosh ( ) ∂θ 0 0 1 ∂ψ ∂w ∂ψ ∂w 1 1 2w = 2 + . (E.20) ∇ γ2 ∂θ ∂ − ∂ ∂θ − sinh2() cosh2()

The corresponding no-slip boundary conditions (u,v,w) = (0, 0, 0) at = 0 are, using the deﬁnition of the streamfunciton (E.12)

∂ψ w =0, =0, and ψ =0. (E.21) ∂ E.2 Dean equations for λ> 1

Pipes of elliptic cross-section with λ> 1 can be considered by applying the transformation (5.9) to the Dean equations for pipes of elliptic cross-section with λ 1 (5.1) and (5.5) . ≤ Accordingly, the coordinates (5.1) become:

x =(a sinh ()cos(θ)+ L)cos(φ) , (E.22) y = a cosh () sin (θ) , (E.23) z =(a sinh ()cos(θ)+ L) sin (φ) , (E.24)

The boundary at = 0 is now an ellipse with the semi-major axis in the y-direction. The aspect ratio λ = cosh(0) . This transformation is depicted in Figure E.1. The boundary is sinh(0) at = 0. It is an ellipse with semi-minor axis, a sinh(0) on the x-axis and semi-major

cosh(0) axis, a cosh(0), on the y-axis . The corresponding aspect ratio λ = . As 0 , sinh(0) →∞ λ 1, and the cross-section is a circle (as shown in the dot dash line of Figure 5.1). As → 0, λ . 0 → →∞

The length scales are non-dimensionalised by a sinh(0), the characteristic length of the ellipse in the x-direction. Accordingly the non-dimensionalised semi-major axis varies like coth(0). The velocity components are non-dimensionalised in exactly the same manner E.3 Computer-extended series 184

L-a

L+a

Figure E.1: Coordinate system for curved pipe with elliptical cross-section with large λ. Dashed line is limit λ = 1 as the case λ> 1.

The corresponding non-dimensionalised Dean equations are therefore:

2 2 4 1 ∂ψ ∂ ψ ∂ψ ∂ ψ ψ = 2 ∇ ∇ ∇ γ ∂θ ∂ − ∂ ∂θ w sinh () sin (θ) ∂w cosh ()cos(θ) ∂w 576K + , (E.25) − γ2 sinh ( ) ∂ sinh ( ) ∂θ 0 0 1 ∂ψ ∂w ∂ψ ∂w 1 1 2w = +2 + , (E.26) ∇ γ2 ∂θ ∂ − ∂ ∂θ sinh2() cosh2() a2 sinh()2+sin(θ)2 2 2 ( ) 1 a sinh(0)w0 a sinh(0) where γ = 2 2 and K = . a sinh (0) 288 ν R The corresponding no-slip boundary conditions are the same as for λ< 1, (E.21).

E.3 Computer-extended series

Below, the derivation of the coefﬁcients of the computer-extended series (5.27) is given.

The series expansion (5.27) is substituted into the Dean equations for small aspect ratios (5.4) and (5.5). By collecting powers of K, the system is reduced to a sequence of successively more complicated linear problems: Appendix E. Extended Stokes series: elliptic cross-section 185

∞ ∂ψ (, θ) ∂ (Lψ (, θ)) ∂ψ (, θ) ∂ (Lψ (, θ)) γ2 2Lψ (, θ)= i j i j ∇ n ∂θ ∂ − ∂ ∂θ i,j:i+j=n ∞ cosh () sin (θ) ∂w (, θ) sinh ()cos(θ) ∂w (, θ) w (, θ) j + j , − i cosh ( ) ∂ cosh ( ) ∂θ i,j:i+j+1=n 0 0 ∂2ψ ∂2ψ γ2Lψ = n + n , n ∂2 ∂θ2 ∞ ∂ψ (, θ) ∂w (, θ) ∂ψ (, θ) ∂w (, θ) γ2 2w (, θ)= i j i j . (E.27) ∇ n ∂θ ∂ − ∂ ∂θ i,j:i+j=n Due to the the scaling term γ2 in the operator 2, the functions Lψ (, θ) are calculated ∇ n explicity. The equations are solved sequentially, with Lψ1 (, θ) considered ﬁrst:

B (3B + A) B ( 5B 16 + A) γ2 2Lψ (, θ) = sin (θ) sinh (5)+ − − sinh (3) ∇ 1 32 32 (A B) B ( 5B 16+ A) B + sin (3θ) − sinh (5) − − sinh () 32 − 32 B (B A) (3B + A) B + sin (5θ) − sinh (3) sinh () . (E.28) 32 − 32

The constants are dropped to ease explanation:

γ2 2Lψ (, θ) sin (θ) (sinh (5) , sinh (3)) ∇ 1 ∼ + sin (3θ) (sinh (5) , sinh ()) + sin (5θ) (sinh (3) , sinh ()) . (E.29)

For an arbitrary function f (, θ) = cosh((2j + 1) )cos((2i + 1) θ):

γ2 2f (, θ)= (2j + 1)2 (2i + 1)2 cosh ((2j + 1) )cos((2i + 1) θ) . (E.30) ∇ − E.3 Computer-extended series 186

It follows that:

Lψ (, θ) sin (θ) (sinh (5) , sinh (3) , sinh ()) 1 ∼ + sin (3θ) (sinh (5) , sinh (3) , sinh ()) + sin (5θ) (sinh (5) , sinh (3) , sinh ()) , (E.31) where the terms sin (θ) sinh (), sin (3θ) sinh (3) and sin (5θ) sinh (5) are the complementary functions. Also:

γ2 2ψ (, θ) γ2Lψ (, θ) ∇ 1 ∼ sin (θ) (sinh (7) , sinh (5) , sinh (3)) ∼ + sin (3θ) (sinh (7) , sinh (5) , sinh ()) + sin (5θ) (sinh (7) , sinh (3) , sinh ()) + sin (7θ) (sinh (5) , sinh (3) , sinh ()) . (E.32)

And so:

ψ (, θ) γ2Lψ (, θ) 1 ∼ sin (θ) (sinh (7) , sinh (5) , sinh (3) , sinh ()) ∼ + sin (3θ) (sinh (7) , sinh (5) , sinh (3) , sinh ()) + sin (5θ) (sinh (7) , sinh (5) , sinh (3) , sinh ()) + sin (7θ) (sinh (7) , sinh (5) , sinh (3) , sinh ()) , (E.33) where the terms sin (θ) sinh (), sin (3θ) sinh (3), sin (5θ) sinh (5) and sin (7θ) sinh (7) are complementary functions. Appendix E. Extended Stokes series: elliptic cross-section 187

Similarly for w (, θ):

γ2 2w (, θ) sin (θ) (sinh (9) , sinh (7) , sinh (5) , sinh (3)) ∇ 1 ∼ + sin (3θ) (sinh (9) , sinh (7) , sinh (5) , sinh ()) + sin (5θ) (sinh (9) , sinh (7) , sinh (3) , sinh ()) + sin (7θ) (sinh (9) , sinh (5) , sinh (3) , sinh ()) + sin (9θ) (sinh (7) , sinh (5) , sinh (3) , sinh ()) . (E.34)

So that:

w (, θ) sin (θ) (sinh (9) , sinh (7) , sinh (5) , sinh (3) , sinh ()) 1 ∼ + sin (3θ) (sinh (9) , sinh (7) , sinh (5) , sinh (3) , sinh ()) + sin (5θ) (sinh (9) , sinh (7) , sinh (5) , sinh (3) , sinh ()) + sin (7θ) (sinh (9) , sinh (7) , sinh (5) , sinh (3) , sinh ()) + sin (9θ) (sinh (9) , sinh (7) , sinh (5) , sinh (3) , sinh ()) , (E.35) where the terms sin (θ) sinh (), sin (3θ) sinh (3), sin (5θ) sinh (5), sin (7θ) sinh (7) and sin (9θ) sinh (9) are complementary functions.

By iteration the general structure of the Dean Series is found to be:

In In cosh(2j) cos(2iθ) n even w = e (A, B) , (E.36) n n,i,j  i=0 j=0 cosh((2j + 1)) cos((2i + 1)θ) n odd   Jn Jn sinh(2j) sin(2iθ) n even ψ = c (A, B) , (E.37) n n,i,j  i=0 j=0 sinh((2j + 1)) sin((2i + 1)θ) n odd   Ln Ln sinh(2j) sin(2iθ) n even 2ψ = d (A, B) , (E.38) ∇ n n,i,j  i=0 j=0 sinh((2j + 1)) sin((2i + 1)θ) n odd  7n+2 7n 7n−2 7n+1 7n−1 where for n even In = 2 , Jn =2 , Ln = 2 and for n odd In = 2 , Jn = 2 , 7n−3 Ln = 2 .

Using transformation (5.9), the solutions to equations (E.25) and (E.26) take the form: E.4 Calculation of the ﬂux ratio 188

In In cosh(2j) cos(2iθ) n even w˜ = e ( B, A) , (E.39) n n,i,j − −  i=0 j=0 sinh((2j + 1)) cos((2i + 1)θ) n odd  

Jn Jn sinh(2j) sin(2iθ) n even ψ˜ = c ( B, A) , (E.40) n n,i,j − −  i=0 j=0 cosh((2j + 1)) sin((2i + 1)θ) n odd  

Ln Ln sinh(2j) sin(2iθ) n even 2ψ˜ = d ( B, A) , (E.41) ∇ n n,i,j − −  i=0 j=0 cosh((2j + 1)) sin((2i + 1)θ) n odd  where: 

4 B + A B A w = − + cos(2θ)cosh(2) − 0 4 4 A + B A + B + cosh(2) + cos (2θ) , (E.42) 4 4 ψ =0. (E.43)

E.4 Calculation of the ﬂux ratio

The ﬂux of the ﬂow down a curved pipe with elliptic cross-section is given by:

0 2π 2 Qc (K)= γ w (K,,θ) d dθ. 0 0 Appendix E. Extended Stokes series: elliptic cross-section 189

Considering ﬁrst, the solutions for λ< 1:

Qc (K)= (E.44)

In In 0 2π n (cosh(2) cos(2θ)) cosh(2j) cos(2iθ) K en,i,j − 2 ddθ (E.45) n,even i=0 j=0 0 0 2 cosh (0) In In 0 2π n (cosh(2) cos(2θ)) cosh((2j + 1)) cos((2i + 1)θ) + K en,i,j − 2 ddθ i=0 j=0 0 0 2 cosh (0) n,odd (E.46)

In In 0 2π n 0 0 (cosh((2j + 2)) + cosh((2j 2))) cos(2iθ) = K en,i,j − (E.47) 4 cosh2 ( ) n,even i=0 j=0 0 (cos((2i 2)θ) + cos((2i + 2)θ)) cosh(2j) − 2 ddθ (E.48) − 4 cosh (0)

In π sinh((2j + 2) ) sinh((2j 2) ) = e (A, B) 0 + − 0 n,0,j 2 (2j + 2) (2j 2) n,even 2 cosh (0) j=1 − (E.49)

π sinh((4)0) +en,0,1 (A, B) + 0 (E.50) 2 cosh2 ( ) (4) 0 In π sinh((2j) ) π + e 0 e (A, B) (E.51) − n,1,j 2 (2j) − n,1,0 2 0 j>0 2 cosh (0) 2 cosh (0) π +en,0,0 (A, B) (sinh (20)) . (E.52) 2 cosh2 ( ) 0 E.4 Calculation of the ﬂux ratio 190

Applying transformation (5.9) the ﬂux in a curved pipe with λ> 1 is:

Jn π sinh((2j + 2) ) sinh((2j 2) ) Q (k)= e ( B, A) 0 + − 0 c n,0,j − − 2 (2j + 2) (2j 2) n,even 2 sinh (0) j=1 − (E.53)

π sinh((4)0) +en,0,1 ( B, A) + 0 + (E.54) − − 2 sinh2 ( ) (4) 0 Jn π sinh((2j) ) e ( B, A) 0 (E.55) n,1,j − − 2 (2j) j>0 2 sinh (0) π π +en,1,0 ( B, A) 0 + en,0,0 ( B, A) (sinh (20)) . − − 2 sinh2 ( ) − − 2 sinh2 ( ) 0 0 (E.56)

In both cases, the odd terms do not contribute to the ﬂux.

The ﬂux for the ﬂow down a straight pipe is:

For λ 1: ≤ π Qs (K)= 2 sinh (20) e0,0,0 (A, B) 2 cosh (0)

π sinh (40) + 0 + e0,0,1 (A, B) 2 cosh2 ( ) 4 0 π e0,1,1 (A, B) sinh (20) e0,1,0 (A, B) 0 . (E.57) − 2 cosh2 ( ) 2 − 0

For λ 1: ≥ π Qs (K)= 2 sinh (20) e0,0,0 ( B, A) 2 sinh (0) − −

π sinh (40) + 0 + e0,0,1 ( B, A) 2 sinh2 ( ) 4 − − 0 π e0,1,1 ( B, A) sinh (20) + − − + e0,1,0 ( B, A) 0 . (E.58) 2 sinh2 ( ) 2 − − 0 Appendix E. Extended Stokes series: elliptic cross-section 191

The ﬂux ratio as deﬁned in (2.1.1) is therefore:

Q (K) Q (K)= c = a (A, B) K2n, (E.59) Q (K) n s n where a depend on the coefﬁcients e (A, B) and e ( B, A) , where p is even. n p,i,j p,i,j − −

E.5 Calculation of the total vorticity

The total vorticity in the upper half of the plane:

π 0 2 2 2 Ωc (K)= γ ψ d dθ. (E.60) 0 − π ∇ 2 Considering ﬁrst, the solutions for λ< 1:

Ωc (K)=

Jn Jn π 0 2 n 2 2 K cn,i,j (A, B) γ sinh(2j) sin(2iθ) ddθ π ∇ n,even i=0 j=0 0 − 2 Jn Jn π 0 2 n 2 2 + K cn,i,j (A, B) γ sinh((2j + 1)) sin((2i + 1)θ) ddθ π ∇ i=0 j=0 0 − 2 n,odd Jn Jn π 0 2 n 2 2 = K cn,i,j (A, B) (2j) (2i) sinh(2j) sin(2iθ) ddθ 0 − π − n,even i=0 j=0 2 Jn Jn π 0 2 n 2 2 + K cn,i,j (A, B) (2j + 1) (2i + 1) sinh((2j + 1)) sin((2i + 1)θ) ddθ 0 − π − n,odd i=0 j=0 2 Jn Jn ((2j + 1)2 (2i 1)2) = Kn c (A, B) − − (2 2cosh((2j + 1) )) . − n,i,j (2j + 1)(2i + 1) − 0 i=0 j=0 n,odd (E.61)

Applying transformation (5.9) the total vorticity in a curved pipe with λ> 1 is:

Jn Jn ((2j + 1)2 (2i 1)2) Ω (K)= Kn c ( B, A) − − ( 2 sinh ((2j + 1) )) . c − n,i,j − − (2j + 1)(2i + 1) − 0 i=0 j=0 n,odd The even terms do not contribute to the total vorticity, so that the series expansion of Ω in E.6 Cuming’s scaling 192

K is odd. The normalised total vorticity (2.1.4):

Ω(K) = Ωc(K)λ = ∞ b K2n+1, (E.62) Qs(K) n n where bn depend on the coefﬁcients cn,i,j (A, B), where n is odd.

E.6 Cuming’s scaling

Cuming [19] studied steady, fully-developed ﬂow in toroidal pipes with elliptical cross- section for ﬁnite curvature using cartesian coordinates (x, y). Results for the weak curvature limit are presented below.

a x

Figure E.2: Cuming’s coordinate system for curved pipe with elliptic cross-section

He used the following spatial non-dimensionalisation:

y x z Y = , X = , Z = (E.63) λa a a and velocity components non-dimensionalised as follows:

ψ w Ψ= λ , W = , (E.64) ν w0

b where λ = a is the aspect ratio. He constructed solution for small K in accordance with Dean series for the circular cross-section, and found:

W =1 X2 Y 2, (E.65) − − 2 Ψ =576K 1 X2 Y 2 B + B X2 + B Y 2 Y, (E.66) − − 1 2 3 Appendix E. Extended Stokes series: elliptic cross-section 193 where

λ4 (375 + 820λ2 +1, 114λ4 + 212λ6 + 39λ8) B = , (E.67) 1 360(5+2λ2 + λ4)(35+84λ2 + 14λ4 + 20λ6 +3λ8) λ4 (75+2λ2 +3λ4) B = , (E.68) 2 −360(35 + 84λ2 + 14λ4 + 20λ6 +3λ8) λ4 (15 + 26λ2 + 39λ4) B = . (E.69) 3 −360(35 + 84λ2 + 14λ4 + 20λ6 +3λ8)

He found the total vorticity in the upper half of the pipe to be:

λ2 (125 + 310λ2 + 428λ4 + 82λ6 + 15λ8) Ω = 576K . (E.70) (5+2λ2 + λ4)(35+84λ2 + 14λ4 + 20λ6 +3λ8) In accordance with the normalisation of the total vorticity considered (2.1.4), and rescaling according to Ψ=ˆ λψˆ from (E.64) it follows that

π Ωλ = Ω (E.71) 2 2 λ (125 + 310λ2 + 428λ4 + 82λ6 + 15λ8) = Ω = 576K . (E.72) ⇒ π (5+2λ2 + λ4)(35+84λ2 + 14λ4 + 20λ6 +3λ8)

Accordingly, under the non-dimensionalisation and normalisation considered hereing, for small aspect ratios Ω= O(λ) and for large aspect ratios Ω= O(λ−3).