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Numerical Investigation of the Thermal Interaction Between a Shape Memory Alloy and Low Reynolds Number Viscous Flow

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Numerical Investigation of the Thermal Interaction Between a Shape Memory Alloy and Low Reynolds Number Viscous Flow University of Alberta Numerical Investigation of the Thermal Interaction between a Shape Memory Alloy and Low Reynolds Number Viscous Flow Christopher Thomas Reaume O A thesis submitted to the Faculty of Craduate Studies and Research in partial hilfillment of the requirements for the degree of Master of Science. Department of Mechanicd Engineering Edmonton, Alberta Spring 2000 National Liitary Bibüothèque nationaie 1*1 ofCanada du Canada Acquisitions and Acquisitions et Bibliographie Se~kes senrices bibliographiques 385 wemngdon Sbwt 395. nie W~geon OttawuON K1AW ûüaviraON K1AûN4 Canada Canada The author has granted a non- L'auteur a accordé une licence non exclusive licence dowing the exc1usive pmettant à la National Li'bmy of Canada to BiblothQue nationale du Canada de reproduce, loan, distri'bute or seil reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la fonne de microfiche/nim, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droh d'autein qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or othemise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. A numerical scheme for the two dimensional rnodelling of the thermal interaction between a smart material and low Reynolds number cross flow has been developed. The phase change of the srnart material is modeled by the temperature dependence of the thermal conductivity and heat capacity of the solid. The change in these properties with temperature is shown to have a substan- tid egect on the temperature history of the solid. The thermal boundary between the fluid and solid is represented with a heat flux consehg boundary condition. This boundary condition represents the temperature dependent physics of the heat transfer at the surface. Results Frorn the two dimensiond scheme were compared to a one dimensional analysis of the average srnart materid temperature. This cornparison indicated that the one dimensional andysis is a poor indicator of the two dimemional results. This is due to the assumption in the one dimensiond analysis of constant heat flux wit h tirne. To rny parents, for always being there. First and foremost sincere thanks are extended to my supe~sor, Dr.Jeff Yokota. This work would not have been possible without his guidance and mentorship. Thanks are also extended to Dr.Bhattacharyya for lending his expertise in the area of shape memory alloys. Special thanks to everybody in the CFDAM lab. In addition, thanks to everyone fkom the lunch crew, anybody whoever tossed the frisbee around, went to the Plant for a few or just generdy made the time more enjoyable. 1 am &O grateful to the Natural Sciences and Engineering Re search Council of Canada for financial support of this research. 1 Introduction 1 1.1 Shape Meemory Alloys ...................... 1 1.2 Present State of Field ....................... 3 1.3 Convective Heat Transfer ..................... 4 2 Problem Description 7 2.1 SelectedApplication ....................... 7 2.2 Simon Nitinol n il ter@ ...................... 8 2.3 Insertion of Inferior Vena Cava Filter .............. 9 2.4 Computational Domain ...................... 10 Governing Equations 15 3.1 Generd Form of Equations .................... 15 3.2 Modification of Equations .................... 16 3.2.1 Boussinesq Approximation ................ 16 3.2.2 Application of Assumptions to Governing Equations . 18 3.2.3 Governing Equations in Fluid Region .......... 21 3.3 Modeling of Shape Memory Alloy Properties .......... 25 3.3.1 Property Model ...................... 25 3.3.2 Mathematical Representation of Phase ......... 28 3.3.3 Fùnctional Fit to Phase Equation ............ 31 3.3.4 Numerical Modeling of Thermal Conductivity ..... 35 3.3.5 Latent Energy of 'Itansformation ............ 38 3.3.6 Governing Equations in Solid Region .......... 41 3.4 Flow Decomposition ....................... 41 3.5 Coordinate Transformation .................... 42 3.6 Dimensional Analysis ....................... 45 4 Numerical Formulation 49 4.1 Solution Sequence ......................... 49 4.2 Convergence of Viscous Stream bction ............ 52 4.2.1 Approximate LU Factorization .............. 52 4.2.2 Multigrid Convergence Acceleration ........... 54 4.3 Mode1 Equation .......................... 56 4.4 Time Integration of Equations in Fluid Region ......... 57 4.5 Spatial Integration Scheme .................... 58 4.6 Interpolation of Convection Term ................ 60 4.7 Time Integation of Energy Equation in Solid ......... 62 4.8 Newton Iteration Solution of Energy Equation ......... 64 4.9 Numerical Stability ........................ 66 4.9.1 Single Step Semi-Implicit Time Advancing ....... 67 4.9.2 Multistage Convergence Iteration ............ 70 4.10 Boundary Conditions ....................... 73 4.10.1 Fluid Field Boundary Conditions ............ 73 410.2 ThedInteraction Boundary Condition ........ 76 5 Results 80 5.1 Values of Dimensiodess Groups ................. 80 5.1.1 Reestream Fiuid Velocity ................ 81 5.2 Numerical Domain ........................ 82 5.3 Development of Flow Field Periodic Solution .......... 86 5.4 Constant Material Properties in Solid Region .......... 93 5.4.1 Results from Two Dimensional Simulation ....... 93 5.4.2 Cornparison to One Dimensional Analysis ....... 96 5.5 Comparison between Smart W aterial and Constant Property Material .............................. 100 5.5.1 Results from Two Dimensional Simulation ....... 100 5.5.2 Cornparison to One Dimensionai Analysis .......106 5.6 Comparison of Smart Materials with Different Material Prop erties ................................ 110 6 Conclusions 126 2.1 Simon Nitinol Inferior Vena Cava Filter ........... 8 2.2 Inferior Vena Cava Viewed Along Axis ............ 12 2.3 Mapping of Cwed Plane onto Flat Space .......... 13 2.4 Domain of Problern ...................... 13 3-1 Displacive Phase Transformation between Austenite and Martensite ........................... 26 3.2 Cornparison of Approximate hction to Numerical Inte- gration ............................. 34 3.3 Thermal Conductivity for Both Material Phases ...... 37 3.4 Speciûc Interna1 Energy for Both Material Phases ..... 40 3.5 T'ransformation Erom (x?y) to (c. 7) Coordinate System . 43 4.1 Schematic of Six Level 'W' Cycle used in Mdtigrid Scheme 56 4.2 Error Amplification Factor for Wdtistage Solution of Con- duction Equation ........................ 72 4.3 Diagram of Fluid and Smart Material Boundary ...... 76 4.4 Boundq Cell ......................... 77 5.1 Supplied Grid in Fluid Portion of the Domain ........ 84 5.2 Developed Grid in Solid Portion of the Domain ....... 85 5.3(a) Vorticity Contours in Fluid Region. t = 0.004 ........ 87 5.3(b) Vorticity Contours in Fluid Region. t = 0.008 ........ 88 5.3 (c) Vorticity Contours in Fluid Region. t = 0.012 ........ 89 5.3(d) Vorticity Contows in Fluid Region. t = 0.016 ........ 90 5.3(e) Vorticity Contours in Fluid Region. t = 0.020 ........ 91 5.4 Average Vorticity History for Isothermal Simulation .... 92 5*5 Average Temperature History for Material with Constant Properties ........................... 94 Average Surface Heat Bansfer Coefficient History for Con- stant Property Material . 95 Comparison Between 1D and 2D Temperature Histories . 98 Time History of Biot Number for Constant Property Material 99 Average Temperature History for Smart Material and Con- stant Property Material . 102 Average Internai Energy History for Smart Material and Constant Property Materiai . 103 Average Surface Heat Transfer Coefficient for Smart Material 104 Cycle Averaged Surface Heat Transfer Coefficients . 105 Comparison Between ID and 2D Temperature History, Smart Material . 107 Comparison Between 1D and 2D Phase History . 108 Time History of Biot Number for Smart Materid . 109 Phase History for Srnart Materiais with DifIerent Transition Temperatures . 113 Phase History for Smart Matends with Different Latent Transformation Energy Amounts . 114 Temperature History for Srnart Materials with DifTerent La- tent Transformation Energy Amounts . 115 5.19(a) Phase Contours in Sm& Materiai 'A' for one Period of Vortex Shedding, t = 0.800 . 116 5.19(b) Phase Contours in Smart Material 'At for one Period of Vortex Shedding, t = 0.804 . 117 5.19(c) Phase Contours in Smart Material 'A' for one Period of Vortex Shedding, t = 0.808 . 118 5.19(d) Phase Contours in Smart Material 'A' for one Period of Vortex Shedding, t = 0.812 . 119 5.19(e) Phase Contours in Sm& Material 'A' for one Period of Vortex Shedding, t = 0.816 . 120 5.20(a) Phase Contours in Smart Material 'B' for one Period of Vortex Shedding, t = 0.020 . 121 5.20(b) Phase Contours in Smart Material 'B' for one Period of Vortex Shedding, t = 0.024 . 122 5.20(c) Phase Contours in Smart Material 'B' for one Period of Vortex Shedding, t = 0.028 . 123 5.20(d) Phase Contours in Smart Materiai 'B' for one Period of Vortex Shedding, t = 0.032 . 124 5.20(e) Phase Contours in Smart Material 'B' for one Period of Vortex Shedding, t = 0.036 . 125 5.1 Fluid Material Properties ..................... 80 5.2 Solid Material Properties ..................... 81 5.3 Dimensionless Groups ...................... 82 N OMENCLATURE LATIN SYMBOLS AI ........................... Austenite transfomation finish
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