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Extended Stokes Series for Dean Flow in Weakly Curved Pipes

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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. Signed: 3 Copyright Copyright in text of this thesis rests with the Author. Copies (by any process) either in full, or of extracts, may be made only in accordance with instructions given by the Author and lodged in the doctorate thesis archive of the college central library. Details may be obtained from the Librarian. This page must form part of any such copies made. Further copies (by any process) of copies made in accordance with such instructions may not be made without the permission (in writing) of the Author. The ownership of any intellectual property rights which may be described in this thesis is vested in Imperial College, subject to any prior agreement to the contrary, and may not be made available for use by third parties without the written permission of the University, which will prescribe the terms and conditions of any such agreement. Further information on the conditions under which disclosures and exploitation may take place is available from the Imperial College registry. 4 5 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 prob- lem 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 imag- inary 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 LinearPad´eapproximants . 166 C.1.1 Results ................................167 C.2 QuadraticPad´eapproximants. .170 C.2.1 Results ................................170 C.3 Generalised Pad´eapproximants: total vorticity Branch II ..........172 C.4 Generalised Pad´eapproximants: 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 con- tours...................................... 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
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