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Numerical Analysis of Heat Conduction and Potential Flow Problems Over a Sphere and a 3- Dimensional Prolate Spheroidal Body

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Numerical Analysis of Heat Conduction and Potential Flow Problems Over a Sphere and a 3- Dimensional Prolate Spheroidal Body ISTANBUL TECHNICAL UNIVERSITY INSTITUTE OF SCIENCE AND TECHNOLOGY NUMERICAL ANALYSIS OF HEAT CONDUCTION AND POTENTIAL FLOW PROBLEMS OVER A SPHERE AND A 3- DIMENSIONAL PROLATE SPHEROIDAL BODY M.Sc. Thesis by Onur ÖLMEZ, B.Sc. Department: Ocean Engineering Programme: Ocean Engineering Supervisor : Assist. Prof. Şafak Nur Ertürk JUNE 2008 ISTANBUL TECHNICAL UNIVERSITY INSTITUTE OF SCIENCE AND TECHNOLOGY NUMERICAL ANALYSIS OF HEAT CONDUCTION AND POTENTIAL FLOW PROBLEMS OVER A SPHERE AND A 3- DIMENSIONAL PROLATE SPHEROIDAL BODY M.Sc. Thesis by Onur ÖLMEZ, B.Sc. (508051107) Date of submission : 21 April 2008 Date of defence examination: 10 June 2008 Supervisor (Chairman): Assist. Prof. Şafak Nur ERTÜRK (I.T.U) Members of the Examining Committee: Prof. Dr. Ömer GÖREN (I.T.U) Prof. Dr. Serdar BEJ İ (I.T.U) JUNE 2008 İSTANBUL TEKN İK ÜN İVERS İTES İ FEN B İLİMLER İ ENSTİTÜSÜ KÜRESEL VE 3 BOYUTLU PROLAT KÜRESEL GÖVDE ÇEVRES İNDE ISI TRANSFER İ VE POTANS İYEL AKI Ş PROBLEMLER İNİN NUMER İK ANAL İZİ YÜKSEK L İSANS TEZ İ Müh. Onur ÖLMEZ (508051107) Tezin Enstitüye Verildi ği Tarih : 21 Nisan 2008 Tezin Savunuldu ğu Tarih : 10 Haziran 2008 Tez Danı şmanı : Yrd. Doç. Dr. Şafak Nur ERTÜRK ( İ.T.Ü.) Di ğer Jüri Üyeleri: Prof. Dr. Ömer GÖREN ( İ.T.Ü.) Prof. Dr. Serdar BEJ İ ( İ.T.Ü.) HAZ İRAN 2008 ACKNOWLEDGEMENTS I would like to thank to my advisor Assist. Prof. Şafak Nur ERTÜRK who supported me to complete this study and illuminated me whenever I encountered with problems. I would also wish to thank to my friends for sophisticated discussions. Of course, special appreciation goes to my parents who provide me such a special life and never quit supporting me whatever I do. Onur ÖLMEZ May/2008 ii CONTENTS ACKNOWLEDGEMENT ii ABBREVIATIONS v LIST OF TABLES vii LIST OF FIGURES viii LIST OF SYMBOLS xi ÖZET xii SUMMARY xiii 1. INTRODUCTION 1 2. MATHEMATICAL MODELING AND NUMERICAL METHODS IN SPHERICALCOORDINATES 5 2.1 The Jakobi Iteration Method 7 2.2 Gauss Seidel Iteration Method 9 2.3 Successive Over Relaxation 10 2.4 The Alternating Direction Implicit (ADI) Method 11 3. MATHEMATICAL MODELLING IN PROLATE SPHERICAL COORDINATE SYSTEM 20 4. GRID GENERATION 23 4.1 General Methods 24 4.2 Forming Computational Domain 26 5. COMPUTATIONAL RESULTS 33 5.1 Solution of Heat Equation in Spherical Coordinates 33 5.2. Solution of Heat Equation in Prolate Spheroidal Coordinates 39 5.3. Solution of 3-D Potential Flow in Spherical Coordinates 43 5.4. Solution of 3-D Potential Flow in Prolate Spheroidal Coordinates 54 6. COMPARISOF RESULTS 64 6.1. In House Code versus Fluent and Symlab for a Sphere 64 6.2. In House Code versus Fluent and Symlab for Prolate Spheroidal Body 70 6.3 Comparison of In-House Code Results for Different Grid Densities 74 7.CONCLUSIONS 77 REFERENCES 79 APPENDICES 81 A Theoretical Pressure Coefficient Variation 81 iii B Node Number, Error, Computational Time Graphs 82 C Comparative Cp Presentation of Sphere Results 84 CURRICULUM VITAE 85 iv ABBREVIATIONS GRAD : Gradient ADI : Alternating Direction Implicit ADIE : ADI coefficient at geographic east ADIL : ADI coefficient at geographic lower position ADIN : ADI coefficient at geographic north ADIP : ADI coefficient at point ADIS : ADI coefficient at geographic south ADIU : ADI coefficient at geographic upper position ADIW : ADI coefficient at geographic west CFD : Computational Fluid Dynamics G-S : Gauss-Seidel GSE : Gauss-Seidel coefficient at geographic east GSL : Gauss-Seidel coefficient at geographic lower position GSN : Gauss-Seidel coefficient at geographic north GSP : Gauss-Seidel coefficient at point GSS : Gauss-Seidel coefficient at geographic south GSU : Gauss-Seidel coefficient at geographic upper position GSW : Gauss-Seidel coefficient at geographic west NR : Number of Nodes in r-ordinate NPHI : Number of Nodes in φ -ordinate NTHETA : Number of Nodes in θ-ordinate PE : Prolate Gauss-Seidel coefficient at geographic east PL : Prolate Gauss-Seidel coefficient at geographic lower position PN : Prolate Gauss-Seidel coefficient at geographic north PP : Prolate Gauss-Seidel coefficient at point PS : Prolate Gauss-Seidel coefficient at geographic south PU : Prolate Gauss-Seidel coefficient at geographic upper position PW : Prolate Gauss-Seidel coefficient at geographic west JE : Jakobi coefficient at geographic east JL : Jakobi coefficient at geographic lower position JN : Jakobi coefficient at geographic north JP : Jakobi coefficient at point JS : Jakobi coefficient at geographic south JU : Jakobi coefficient at geographic upper position JW : Jakobi coefficient at geographic west RANS : Reynolds-Avaraged Navier Stokes FDM : Finite Difference Method FEM : Finite Element Method BEM : Boundary Element Method FVM : Finite Volume Method Cp : Pressure Coefficient DDR : Double Data Rate v RAM : Random Access Memory AIAA : American Institute of Aeronautics and Astronautics DFVLR : Deutsche Forschungs- und Versuchsanstalt für Luft- und Raumfahrt (German Research & Development Institute for Air & Space Travel) 3-D : Three Dimentional 2-D : Two Dimentional PDE : Partial Differential Equation BC : Boundary Condition vi LIST OF TABLES Page Number Table 6.1 : Sphere Flow Results Comparison for Different Grid Densities..... 74 Table A.1 : Angle-Theoretical Pressure Coefficient Variation Datas………... 81 vii LIST OF FIGURES Page Number Figure 2.1 : 2-D View of the Coefficient Notation…………………………… 8 Figure 2.2 : Gauss-Seidel Iteration Schema……………………………..……. 9 Figure 2.3 : Schematical View of ADI x-direction sweep…………...……….. 12 Figure 2.4 : Schematical View of ADI x-direction sweep …..……………….. 14 Figure 2.5 : 2-D View of 7x3 Grid……………………………….…………... 17 Figure 4.1 : Transformation of coordinate systems...…………………………. 24 Figure 4.2 : Form-Fitted Grid Example (Blazek 2001. p.31).………………... 24 Figure 4.3 : Cartesian Grid Example (Blazek 2001. p.31)……………………. 25 Figure 4.4 : Spherical Coordinate’s Projection (Thomas 1998. p.534)...…….. 27 Figure 4.5 : 2-D View of Generated Fan……………………..………………. 28 Figure 4.6 : 3-D View of Generated Grid..…………………………………… 28 Figure 4.7 : Prolate Spheroidal Coordinate’s Projection (Moon 1988. p.31).... 29 Figure 4.8 : 2-D View of Generated Fan……………………..………………. 29 Figure 4.9 : 3-D View of Generated Grid..…………………………………… 30 Figure 4.10 : Form-Fitted Grid Around a Prolate Spheroidal Body.…..……..... 30 Figure 4.11 : Enlarged View of the Pole Region………………………………. 31 Figure 4.12 : Cross-Sectional View of the Prolate Spheroidal Body………….. 32 Figure 5.1 : Schematic 2-D View of Subsection 5.1...………………………... 35 Figure 5.2 : Algorithm for the Heat Conduction Problems……………….…... 36 Figure 5.3 : Grid View of Half Domain in Spherical Coordinates..………….. 37 Figure 5.4 : Different Aspect of Grid Domain in Spherical Coordinates…….. 37 Figure 5.5 : Half View of the Results with Grid in Spherical Coordinates..…. 38 Figure 5.6 : Another View of the Results without Grid in Spherical Coordinates...................................................................................... 38 Figure 5.7 : Schematic 2-D View of Subsection 5.2...………………………... 40 Figure 5.8 : Grid View of Half Domain in Prolate Spheroidal Coordinates..………………….………………………………….. 41 Figure 5.9 : Different Aspect of Grid Domain in Prolate Spheroidal Coordinate…...………………………………………………….... 41 Figure 5.10 : Half View of the Results with Grid in Prolate Spheroid Coordinates..…................................................................................ 42 Figure 5.11 : Another View of the Results without Grid in Prolate Spheroidal Coordinates...................................................................................... 42 Figure 5.12 : Schematic 2-D View of Subsection 5.3...………………………... 44 Figure 5.13 : Quarter Portion of a Half Sphere (Moon 1988. p.24)...…….….… 45 Figure 5.14 : Algorithm for Potential Flow Problems…....…………………….. 47 Figure 5.15 : 2-D Presentation of Velocity Potentials...………......…….……… 48 Figure 5.16 : 3-D Presentation of Velocity Potentials………….....…….……… 48 viii Figure 5.17 : Transformation from Spherical Coordinates to Cartesian Coordinates…………..…………………………………………… 49 Figure 5.18 : Transformation for r-direction (Thomas 1998. p.844)...….……… 50 Figure 5.19 : Velocity Distribution in the Domain……...…….......…….……… 51 Figure 5.20 : Another View ofVelocity Distribution…………......…….……… 51 Figure 5.21 : Velocity Distribution Contours Around a Sphere......…….……… 52 Figure 5.22 : Pressure Coefficient Around a Sphere…………….......….……… 53 Figure 5.23 : Schematic 2-D View of Subsection 5.4………….....…….……… 55 Figure 5.24 : 2-D Presentation of Velocity Potentials...………......…….……… 56 Figure 5.25 : 3-D Presentation of Velocity Potentials………….....…….……… 57 Figure 5.26 : Transformation from Prolate Spheroidal Coordinates to Cartesian Coordinates……………..…………………………...…….……… 58 Figure 5.27 : Transformation for η-direction………………...……...….……… 59 Figure 5.28 : Velocity Distribution in the Domain………...….......…….……… 60 Figure 5.29 : Another View of Velocity Distribution……...…......…….……… 61 Figure 5.30 : Velocity Distribution Contours Around a Prolate Spheroidal Body………..………………………………………………….…. 62 Figure 5.31 : Pressure Coefficient Around Sphere……………......…….……… 62 Figure 6.1 : Streamlines and Potential Lines for Inviscid Flow Past a Sphere (White 1995. p.538)……..……………………………………….. 64 Figure 6.2 : Vectorial Presentation of Velocity Results Around a Sphere……. 65 Figure 6.3 : Contoural Presentation of Velocity Results Around a Sphere…… 66 Figure 6.4 : Fluent Contoural Pressure Coefficient Distribution Results Around a Sphere……...…………………………………………..
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