AN EFFICIENT AND ROBUST ALGORITHM FOR INCOMPRESSIBLE FLOW AND ITS APPLICATION IN HEAT TRANSFER ENHANCEMENT CHENG YONGPAN NATIONAL UNIVERSITY OF SINGAPORE 2008 AN EFFICIENT AND ROBUST ALGORITHM FOR INCOMPRESSIBLE FLOW AND ITS APPLICATION IN HEAT TRANSFER ENHANCEMENT CHENG YONGPAN (B Eng., M Eng., Xian Jiaotong University, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2008 Acknowledgement I would like to express my deepest gratitude to my supervisors, Assoc Prof Lee Thong See and Assoc Prof Low Hong Tong for their continuous and invaluable support, supervision and encouragement Without their help, I cannot live through my Ph.D study and finish my thesis When I am going to finish my Ph.D thesis, I cannot help remembering my former supervisor in Xian Jiaotong University in China, the Academician of Chinese Academy of Science, Prof Tao Wen-quan, who led me to the field of Numerical Heat Transfer in 2001 His “hardworking, aggressive, practical and cooperative” attitude toward the research will benefit me for my whole life I would like to thank the kind colleagues in the Fluid Mechanics Laboratory, Shan Yongyuan, Sui Yi, Wu Jie, Chen Xiaobing, Wang Liping etc The discussion with them inspired me with many new ideas I would like to express my deepest love to my dear wife, Wang Wei, who gave me great love in my thesis preparation and defense; meanwhile, I would also like to express sincere thanks to my dear parents and sisters for their long-time love, support and understanding, which help me overcome the difficulty in the oversea life Finally, I would like to thank National University of Singapore for offering the research scholarship and valuable opportunity to pursue a Ph.D degree I Table of Contents Acknowledgement……………………………………………………………… I Table of Contents……………………………………………………………… II List of Figures……………………………………………………………….….VII List of Tables………………………………………………………………… XV Nomenclature……………………………………………………………… …XVI Abbreviations………………………………………………………………… XXI Summary………………………………………………………………… .XXIII Chapter Introduction and Literature Review……………………………… 1.1 Background………………………………………………………………… 1.2 Development in heat transfer enhancement in fin-and-tube heat exchangers 1.2.1 Recent development in experimental study…………………………… 1.2.1.1 Plain fin-and-tube heat exchanger…………………………………5 1.2.1.2 Wavy fin-and-tube heat exchanger………………………… ……8 1.2.2 Recent development in numerical study……………………………… 11 1.2.2.1 Plain fin-and-tube heat exchanger……………………………… 12 1.2.2.2 Wavy fin-and-tube heat exchanger……………………………….15 1.3 Development in numerical algorithms for incompressible flow…………… 16 1.3.1 Numerical algorithms on staggered grid……………………………….20 1.3.2 Numerical algorithms on collocated grid………………………………24 II 1.4 Objectives and significance of the study…………………………………… 29 1.5 Outline of the thesis………………………………………………………… 31 Chapter Grid Generation and Discretization of Governing Equations… 34 2.1 Requirement for grid…………………………………………………………34 2.2 Grid generation in two dimensions………………………………………… 37 2.3 Grid generation in three dimensions…………………………………………39 2.4 Discretization of governing equations……………………………………… 43 2.5 Implementation of high-order schemes………………………………………47 2.5.1 Normalized Variable and Space Formulation methodology…………… 47 2.5.2 Application of high-order schemes in equation discretization……… …49 Chapter CLEARER Algorithm on Staggered Grid……………………… 57 3.1 General review of SIMPLER algorithm on staggered grid……………………57 3.2 Mathematical formulation of CLEARER algorithm………………………… 61 3.3 Numerical validation and comparison with SIMPLER algorithm………… 65 3.3.1 Lid-driven flow in a square cavity…………………………………… 66 3.3.2 Lid-driven flow in a polar cavity…………………………………… ….68 3.4 Concluding remarks…………………………………………………… …….69 Chapter CLEARER Algorithm on Collocated Grid ……………………….81 4.1 General review of SIMPLER algorithm on collocated grid………………….81 III 4.2 Mathematical formulation of CLEARER algorithm…………………………87 4.2.1 Discussion on SIMPLER algorithm……………………………………87 4.2.2 Improved SIMPLER algorithm……………………………………… 88 4.2.3 Discussion on the second relaxation factor…………………………….90 4.2.4 Treatment of solid region in computational domain………………… 91 4.2.4.1 Treatment of temperature field………………………………… 92 4.2.4.2 Treatment of flow field………………………………………… 93 4.3 Numerical validation and comparison with SIMPLER algorithm………… 94 4.3.1 Lid-driven flow in a square cavity…………………………………… 96 4.3.2 Natural convection in a square cavity………………………………….97 4.3.3 Lid-driven flow in a polar cavity……………………………………….97 4.3.4 Natural convection in an annular enclosure……………………………98 4.4 Concluding remarks………………………………………………………….99 Chapter CLEARER Algorithm on Curvilinear Non-orthogonal Coordinates ………………………………………………………… 111 5.1 General review of SIMPLE algorithm on curvilinear non-orthogonal coordinates…………………………………………………………………….111 5.2 Mathematical formulation of CLEARER algorithm……………………… 116 5.2.1 The predictor step of CLEARER algorithm………………………… 116 5.2.2 The corrector step of CLEARER algorithm………………………… 122 5.2.3 Solution procedure of CLEARER algorithm…………………………123 IV 5.2.4 Discussion on the second relaxation factor ………………………….124 5.3 Numerical validation and comparison with SIMPLERM algorithm……… 126 5.3.1 Lid-driven flow in an inclined cavity…………………………………128 5.3.1.1 Lid-driven flow at Re=100…………………………………… 129 5.3.1.2 Lid-driven flow at Re=1000…………………………………….130 5.3.1.3 Lid-driven flow at Re=5000………………………………… …131 5.3.2 Natural convection in an inclined cavity…………………………… 132 5.3.3 Natural convection in an enclosure with eccentric cylinder and square duct………………………………………………………………………….133 5.4 Investigation of minimum intersection angle among grid lines to guarantee convergence…………………………………………………………………… 135 5.5 Concluding remarks……………………………………………………… 137 Chapter Extension of CLEARER Algorithm to 3D Curvilinear Non-orthogonal Coordinates……………………………………………… 158 6.1 Discretization of governing equations………………………………………158 6.2 Implementation of CLEARER algorithm………………………………… 162 6.2.1 The predictor step of CLEARER algorithm………………………… 162 6.2.2 The corrector step of CLEARER algorithm………………………… 167 6.3 Validation of CLEARER algorithm…………………………………………172 6.4 Concluding remarks……………………………………………………… 173 V Chapter Application of CLEARER Algorithm to Triangular Wavy Fin-and-Tube Heat Exchanger……………………………………………… 176 7.1 Physical model…………………………………………………………… 176 7.2 Mathematic description…………………………………………………… 177 7.2.1 Computational domain……………………………………………… 177 7.2.2 Boundary condition………………………………………………… 177 7.3 Brief introduction of field synergy principle……………………………… 178 7.4 Results and discussion………………………………………………………180 7.4.1 Mesh independence study…………………………………………….180 7.4.2 Influence of wavy angle………………………………………………181 7.4.3 Influence of fin pitch………………………………………………….182 7.4.4 Influence of tube diameter…………………………………………….184 7.4.5 Influence of wavy density…………………………………………….185 7.5 Concluding remarks……………………………………………………… 186 Chapter Conclusions and Recommendations………………………… … 196 8.1 Conclusions……………………………………………………………… …196 8.2 Recommendations for future work………………………………………… 198 References………………………………………………………….………… 199 List of Publications……………………………………………… …………….223 VI List of Figures Figure 1.1Heat transfer surface area density spectrum of heat exchanger surface33 Figure 1.2 Various enhanced heat transfer fins………………………………… 33 Figure 2.1 The relation between the physical domain and computational domain54 Figure 2.2 Grid generated in 2D complex enclosure…………………………… 54 Figure 2.3 Grid generated in 3D wavy fin-and-tube heat exchanger………… 54 Figure 2.4 Computational grid and the definition of parameters……………… 55 Figure 2.5 Original and normalized variable and profiles……………………… 56 Figure 2.6 Control volumes for two-dimensional problem………………………56 Figure 2.7 Treatment of boundary condition…………………………………… 56 Figure 3.1 Control volumes of staggered grid in 2D Cartesian coordinates…… 71 Figure 3.2 Lid-driven flow in a square cavity……………………………………71 Figure 3.3 Convergence histories of SIMPLER, CLEARER, CLEAR1 and CLEAR2………………………………………………………… 72 Figure 3.4 Accuracy test with fully developed flow in straight channel…… ……72 Figure 3.5 Comparison between predicted velocity distributions and benchmark solutions at Re=100…………………………………………………73 Figure 3.6 Comparison between predicted velocity distributions and benchmark solutions at Re=1000……………………………………………….73 Figure 3.7 Comparison between predicted velocity distributions and benchmark solutions at Re=5000…………………………………………………74 VII Figure 3.8 Comparison of iteration numbers among SIMPLER, CLEARER, CLEAR1 and CLEAR2 at Re=100…………………………………74 Figure 3.9 Comparison of iteration numbers among SIMPLER, CLEARER, CLEAR1 and CLEAR2 at Re=1000……………………………… 75 Figure 3.10 Comparison of iteration numbers among SIMPLER, CLEARER at Re=5000…………………………………………………………… 75 Figure 3.11 Comparison of iteration number ratio of CLEARER, CLEAR1, and CLEAR2 over SIMPLER at Re=100……………………………….76 Figure 3.12 Comparison of iteration number ratio of CLEARER, CLEAR1 and CLEAR2 over SIMPLER at Re=1000…………………………… 76 Figure 3.13 Comparison of iteration number ratio of CLEARER over SIMPLER at Re=5000……………………………………………….………….77 Figure 3.14 Lid-driven flow in polar cavity…………………………………… 77 Figure 3.15 Comparison of streamlines at Re=350………………………………78 Figure 3.16 Comparison of streamlines at Re=1000…………………………….78 Figure 3.17 Comparison of iteration numbers among SIMPLER, CLEARER, CLEAR1 and CLEAR2 at Re=350……………………………… 78 Figure 3.18 Comparison of iteration numbers among SIMPLER, CLEARER, CLEAR1 and CLEAR2 at Re=1000………………………………79 Figure 3.19 Comparison of iteration number ratio of CLEARER, CLEAR1 and CLEAR2 over SIMPLER at Re=350…………………………… 79 Figure 3.20 Comparison of iteration number ratio of CLEARER, CLEAR1 and VIII References Numerical 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transfer in a closed-ended horizontal concentric annulus with rotating inner heated wall The 4th International Conference on Computational Heat and Mass Transfer, Paris, France, pp.171-175, May 17-20, 2005 [P.N 0214944] Cheng Y.P., Lee T.S and Low H.T., The influence of strip arrangement on the performance of slit fin The Eleventh Asian Congress of Fluid Mechanics, Kuala Lumpur, Malaysia, pp.815-818, May 22-25, 2006.[P.N.0368654] Cheng Y P., Lee T.S., Low H.T and Tao W.Q., An efficient and robust SIMPLER-like numerical algorithm and its application, The Asian Symposium on Computational Heat Transfer and Fluid Flow, Xi’an, China, October 18-21, 2007 [P.N.0368658] Journal papers 1.Cheng Y P., Lee T.S and Low H.T., Numerical analysis of mixed convection in three-dimensional rectangular channel with flush-mounted heat sources based on field synergy principle International Journal for Numerical Methods in Fluids, vol.52, pp 987-1003, 2006 [P.N.: 0305521] 2.Cheng Y P., Lee T S., Low H T and Tao W Q., An efficient and robust numerical scheme for SIMPLER algorithm on non-orthogonal curvilinear coordinates, CLEARER, Numerical Heat Transfer, Part B, vol.51(5), pp 223 433-461, 2007 [P.N.: 0345670] Cheng Y P., Lee T S., Low H T and Tao W.Q., Improvement of SIMPLER algorithm for incompressible flow on collocated grid system Numerical Heat Transfer, Part B, vol.51(5), pp 463-486, 2007 [P.N.: 0322007] Lee T.S.，Cheng Y P and Low H T., Improvement of SIMPLER algorithm for incompressible flow on staggered grid system International Journal of Modern Physics C, vol.18(7), pp.1149-1155, 2007 [P.N.: 0345607] 5.Tao W.Q., Cheng Y P and Lee T.S., 3D numerical simulation on fluid flow and heat transfer characteristics in multistage heat exchanger with slit fins, Heat and Mass Transfer, vol.44, pp.125-136, 2007 [P.N.0368667] Tao W.Q., Cheng Y P and Lee T.S., The influence of strip location on the pressure drop and heat transfer performance of slotted fin, Numerical Heat Transfer, Part A, vol.52, pp.463-480, 2007 [P.N.: 0352393] Cheng Y P., Lee T S and Low H T., Numerical analysis of conjugate heat transfer in electronic cooling based on field synergy principle, Applied Thermal Engineering, vol.28, pp.1826-1833, 2008 [P.N.0368672] Cheng Y P., Lee T S and Low H T., Numerical analysis of periodically developed fluid flow and heat transfer characteristics in the triangular wavy fin-and-tube heat exchanger based on field synergy principle Numerical Heat Transfer, Part A, vol.53(8), pp.821-842, 2008 [P.N.0368679] Cheng Y P., Lee T S., Low H T and Sui Y., Implementation of CLEARER 224 algorithm on three-dimensional non-orthogonal curvilinear coordinates and its application Numerical Heat Transfer, Part B, vol.54(1), pp.62-83, 2008 10 Cheng Y P., Lee T S and Low H T., Numerical prediction of periodically developed fluid flow and heat transfer characteristics in the sinusoid wavy fin-and-tube heat exchanger Int J Numerical Methods for Heat and Fluid Flow, in press 11 Sui Y., Chew Y T., Roy P., Cheng Y P and Low H T., Dynamic motion of red blood cells in simple shear flow Physics of Fluids, in press 225 .. .AN EFFICIENT AND ROBUST ALGORITHM FOR INCOMPRESSIBLE FLOW AND ITS APPLICATION IN HEAT TRANSFER ENHANCEMENT CHENG YONGPAN (B Eng., M Eng., Xian Jiaotong University, China) A THESIS SUBMITTED FOR. .. airside convective heat transfer coefficient for the wet and frosted surfaces in heat exchangers with simultaneous heat and mass transfer 1.2.2.2 Wavy fin -and- tube heat exchanger Jang and Chen (1997)... performance for plain fin -and- tube heat exchanger in wet conditions Niederer (1986) performed experiments to investigate the frosting and defrosting effects on the heat transfer in heat exchangers