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Smoothed particle hydrodynamic (SPH) method is used to solve a variety of complex engineering problems. In the literature about SPH, there are two approaches to solving the pressure component of momentum conservation equation, namely incompressible SPH (ISPH) and weakly compressible SPH (WCSPH) methods. In this paper, we present a new comparative study of WCSPH (Lagrange), ISPH (Lagrange) and ISPH (Euler) methods, focusing on heat conduction issue by numerical solutions of natural convection in a square cavity. Temperature distributions, velocity distributions and Nusselt number distributions at different Rayleigh numbers (Ra=104, 105, 106) are provided in the paper. The quantitative comparisons of results show that WCSPH (Lagrange), ISPH (Lagrange) and ISPH (Euler) methods all perform very well at low Rayleigh number. And at high Rayleigh number, SPH (Lagrange) needs shifting particle technology to correct the distribution of particles, ISPH (Euler) performs best because of the motionless particles, WCSPH (Lagrange) performs better than ISPH (Lagrange).
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Keywords:
- smoothed particle hydrodynamics /
- incompressible smoothed particle hydrodynamics /
- shifting particle technology /
- natural convection
[1] Gingold R A, Monaghan J J 1977 Mon. Not. R. Astron. Soc. 181 375
[2] Lucy L B 1977 Astron. J. 82 1013
[3] Sun Z H, Han R J 2008 Chin. Phys. B 17 3185
[4] Zhang A M 2008 Chin. Phys. B 17 927
[5] Antoci C, Gallati M, Sibilla S 2007 Comput. Struct. 85 879
[6] Monaghan J J, Huppert H E, Worster M G 2005 J. Comput. Phys. 206 684
[7] Qiu L C 2013 Acta Phys. Sin. 62 124702 (in Chinese) [邱流潮 2013 物理学报 62 124702]
[8] Monaghan J J, Kocharyan A 1995 Comput. Phys. Commun. 87 225
[9] Shadloo M S, Yildiz M 2011 Int. J. Numer. Meth. Engng. 87 988
[10] Rook R A, Yildiz M, Dost S 2007 J. Numer. Heat Transfer B 51 1
[11] Lei J M, Huang C 2014 Acta Phys. Sin. 63 144702 (in Chinese) [雷娟棉, 黄灿 2014 物理学报 63 144702]
[12] Zhong C W, Xie J F, Zhuo C S, Xiong S W, Yin D C 2009 Chin. Phys. B 18 4083
[13] Cheng R J, Cheng Y M, Ge H X 2009 Chin. Phys. B 18 4059
[14] Wang J F, Sun F X, Cheng R J 2010 Chin. Phys. B 19 060201
[15] Zhang X, Liu Y, Ma S 2009 Adv. Mech. 39 1 (in Chinese) [张雄, 刘岩, 马上 2009 力学进展 39 1]
[16] Lee E S, Moulinec C, Xu R, Violeau D, Laurence D, Stansbyc P 2008 J. Comput. Phys. 227 8417
[17] Hughes J P, Graham D I 2010 J. Hydraul. Res. 48 105
[18] Shadloo M S, Zainali A, Yildiz M, Suleman A 2012 Int. J. Numer. Meth. Engng. 89 939
[19] Chen Z, Zong Z, Liu M B, Li H T 2013 Int. J. Numer. Meth. Fluids 73 813
[20] Zhou G Z, Ge W 2014 CIESC 65 1145 (in Chinese) [周光正, 葛蔚 2014 化工学报 65 1145]
[21] Li S W, Xiong L F 2007 Ind. Heat. 36 10 (in Chinese) [李世武, 熊莉芳 2007 工业加热 36 10]
[22] Hu J, Huang C, He J D 2014 Trans. Beijing Inst. Technol. 34 237 (in Chinese) [胡俊, 黄灿, 何建东 2014 北京理工大学学报 (自然科学版) 34 237]
[23] Danis M E, Orhan M, Ecder A 2013 Int. J. Comput. Fluid Dyn. 27 15
[24] Xu R, Stansby P, Laurence D 2009 J. Comput. Phys. 228 6703
[25] Liu G R, Liu M B 2003 Smoothed Particle Hydrodynamics: A Meshfree Particle Method (Singapore: World Scientific Publishing) pp33–56
[26] Dehnen W, Aly H 2012 Mon. R. Astron. Soc. 425 1068
[27] Cummins S J, Rudman M 1999 J. Comput. Phys. 152 584
[28] van der Vorst H A 1992 SIAM J. Sci. Stat. Comput. 13 631
[29] Sli E, Mayers D F 2003 An Introduction to Numerical Analysis (Cambridge university press) pp 325-329
[30] Monaghan J J 1994 J. Comput. Phys. 110 399
[31] Liu M B, Chang J Z 2010 Acta Phys. Sin. 59 3654 (in Chinese) [刘谋斌, 常建忠 2010 物理学报 59 3654]
[32] Shao S, Lo E Y M 2003 Adv. Water Res. 26 7
[33] Fu X J, Qiang H F, Yang Y C 2007 Adv. Mech. 37 375 (in Chinese) [傅学金, 强洪夫, 杨月诚 2007 力学进展 37 375]
[34] Szewc K, Pozorski J, Taniere A 2011 Int. J. Heat Mass Transfer 54 4807
[35] Chaniotis A K, Poulikakos D, Koumoutsakos P 2002 J. Comput. Phys. 182 67
[36] Cleary P W 1998 Appl. Math. Modell. 22 981
[37] Wan D C, Patnaik B S V, Wei G W 2001 Numer. Heat Transfer B: Fundam. 40 199
[38] Wakashima S, Saitoh T S 2004 Int. J. Heat Mass Transfer 47 853
[39] de Vahl Davis G 1983 Int. J. Numer. Meth. Fluids 3 249
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[1] Gingold R A, Monaghan J J 1977 Mon. Not. R. Astron. Soc. 181 375
[2] Lucy L B 1977 Astron. J. 82 1013
[3] Sun Z H, Han R J 2008 Chin. Phys. B 17 3185
[4] Zhang A M 2008 Chin. Phys. B 17 927
[5] Antoci C, Gallati M, Sibilla S 2007 Comput. Struct. 85 879
[6] Monaghan J J, Huppert H E, Worster M G 2005 J. Comput. Phys. 206 684
[7] Qiu L C 2013 Acta Phys. Sin. 62 124702 (in Chinese) [邱流潮 2013 物理学报 62 124702]
[8] Monaghan J J, Kocharyan A 1995 Comput. Phys. Commun. 87 225
[9] Shadloo M S, Yildiz M 2011 Int. J. Numer. Meth. Engng. 87 988
[10] Rook R A, Yildiz M, Dost S 2007 J. Numer. Heat Transfer B 51 1
[11] Lei J M, Huang C 2014 Acta Phys. Sin. 63 144702 (in Chinese) [雷娟棉, 黄灿 2014 物理学报 63 144702]
[12] Zhong C W, Xie J F, Zhuo C S, Xiong S W, Yin D C 2009 Chin. Phys. B 18 4083
[13] Cheng R J, Cheng Y M, Ge H X 2009 Chin. Phys. B 18 4059
[14] Wang J F, Sun F X, Cheng R J 2010 Chin. Phys. B 19 060201
[15] Zhang X, Liu Y, Ma S 2009 Adv. Mech. 39 1 (in Chinese) [张雄, 刘岩, 马上 2009 力学进展 39 1]
[16] Lee E S, Moulinec C, Xu R, Violeau D, Laurence D, Stansbyc P 2008 J. Comput. Phys. 227 8417
[17] Hughes J P, Graham D I 2010 J. Hydraul. Res. 48 105
[18] Shadloo M S, Zainali A, Yildiz M, Suleman A 2012 Int. J. Numer. Meth. Engng. 89 939
[19] Chen Z, Zong Z, Liu M B, Li H T 2013 Int. J. Numer. Meth. Fluids 73 813
[20] Zhou G Z, Ge W 2014 CIESC 65 1145 (in Chinese) [周光正, 葛蔚 2014 化工学报 65 1145]
[21] Li S W, Xiong L F 2007 Ind. Heat. 36 10 (in Chinese) [李世武, 熊莉芳 2007 工业加热 36 10]
[22] Hu J, Huang C, He J D 2014 Trans. Beijing Inst. Technol. 34 237 (in Chinese) [胡俊, 黄灿, 何建东 2014 北京理工大学学报 (自然科学版) 34 237]
[23] Danis M E, Orhan M, Ecder A 2013 Int. J. Comput. Fluid Dyn. 27 15
[24] Xu R, Stansby P, Laurence D 2009 J. Comput. Phys. 228 6703
[25] Liu G R, Liu M B 2003 Smoothed Particle Hydrodynamics: A Meshfree Particle Method (Singapore: World Scientific Publishing) pp33–56
[26] Dehnen W, Aly H 2012 Mon. R. Astron. Soc. 425 1068
[27] Cummins S J, Rudman M 1999 J. Comput. Phys. 152 584
[28] van der Vorst H A 1992 SIAM J. Sci. Stat. Comput. 13 631
[29] Sli E, Mayers D F 2003 An Introduction to Numerical Analysis (Cambridge university press) pp 325-329
[30] Monaghan J J 1994 J. Comput. Phys. 110 399
[31] Liu M B, Chang J Z 2010 Acta Phys. Sin. 59 3654 (in Chinese) [刘谋斌, 常建忠 2010 物理学报 59 3654]
[32] Shao S, Lo E Y M 2003 Adv. Water Res. 26 7
[33] Fu X J, Qiang H F, Yang Y C 2007 Adv. Mech. 37 375 (in Chinese) [傅学金, 强洪夫, 杨月诚 2007 力学进展 37 375]
[34] Szewc K, Pozorski J, Taniere A 2011 Int. J. Heat Mass Transfer 54 4807
[35] Chaniotis A K, Poulikakos D, Koumoutsakos P 2002 J. Comput. Phys. 182 67
[36] Cleary P W 1998 Appl. Math. Modell. 22 981
[37] Wan D C, Patnaik B S V, Wei G W 2001 Numer. Heat Transfer B: Fundam. 40 199
[38] Wakashima S, Saitoh T S 2004 Int. J. Heat Mass Transfer 47 853
[39] de Vahl Davis G 1983 Int. J. Numer. Meth. Fluids 3 249
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