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ISPH modeling of natural convection heat transfer with an analytical kernel renormalization factor

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Abstract

The objective of this study is to extend the attention of the incompressible smoothed particle hydrodynamics method (ISPH) in the heat transfer field. The ISPH method for the natural convection heat transfer under the Boussinesq approximation in various environments: pure-fluid, nanofluid, and non-Darcy porous medium is introduced. We adopted the improved analytical method for calculating the kernel renormalization factor and its gradient based on a quintic kernel function for the wall boundary treatment in the ISPH method. The proposed method requires no dummy particle layer to meet the impermeability condition and makes the heat flux over the wall boundary easy to implement. We performed four different numerical simulations of natural convection in cavities with increasing complexity in modeling and implementation: the natural convection in a square cavity with constant differentially heated wall temperature, natural convection with the heat flux from the bottom wall for a wide range of Rayleigh numbers, natural convection in a non-Darcy porous cavity fully filled with nanofluid in different flow regimes, and natural convection in a partially layered porous cavity. The results showed excellent agreement with results from literatures and the in-house P1–P1 finite element method code.

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References

  1. Lucy LB (1977) A numerical approach to the testing of the fission hypothesis. Astron J 82:1013–1024

    Article  ADS  Google Scholar 

  2. Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon Not R Astron Soc 181:375–389

    Article  ADS  MATH  Google Scholar 

  3. Koshizuka S, Oka Y (1996) Moving-particle semi-implicit method for fragmentation of incompressible fluid. Nucl Sci Eng 123:421–434

    Article  ADS  Google Scholar 

  4. Aly AM (2012) An Improved incompressible smoothed particle hydrodynamics to simulate fluid–soil–structure interactions. Kyushu University. https://wiki.manchester.ac.uk/s

  5. Amdahl DL (1993) Modeling aerodynamic problems using smoothed particle hydrodynamics (SPH). SAE Technical Paper, No. 932512. https://doi.org/10.4271/932512

  6. Hu XY, Adams NA (2006) A multi-phase SPH method for macroscopic and mesoscopic flows. J Comput Phys 213:844–861

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Monaghan JJ (1994) Simulating free surface flows with SPH. J Comput Phys 110:399–406

    Article  ADS  MATH  Google Scholar 

  8. Fourey G, Oger G, Touzé DL, Alessandrini B (2010) Violent fluid–structure interaction simulations using a coupled SPH/FEM method. In: IOP conference series: materials science and engineering, vol 10, p 012041

  9. Cummins SJ, Rudman M (1999) An SPH projection method. J Comput Phys 152:584–607

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Shao S, Lo EYM (2003) Incompressible SPH method for simulating newtonian and non-newtonian flows with a free surface. Adv Water Resour 26:787–800

    Article  ADS  Google Scholar 

  11. Khayyer A, Gotoh H, Shao SD (2008) Corrected incompressible SPH method for accurate water-surface tracking in breaking waves. Coast Eng 55:236–250

    Article  Google Scholar 

  12. Asai M, Sonoda AMA, Sakai YY (2012) A Stabilized incompressible SPH method by relaxing the density invariance condition. J Appl Math Mech 2012:24

    MATH  Google Scholar 

  13. Aly AM, Lee SW (2014) Numerical simulations of impact flows with incompressible smoothed particle hydrodynamics. J Mech Sci Technol 28:2179–2188

    Article  Google Scholar 

  14. Gotoh H, Khayyer A (2016) A Current achievements and future perspectives for projection-based particle methods with applications in ocean engineering. J Ocean Eng Mar Energy 2016:1–28

    Google Scholar 

  15. Shadloo MS, Oger G, Le Touzé D (2016) Smoothed particle hydrodynamics method for fluid flows, towards industrial applications: motivations, current state, and challenges. Comput Fluids 136:11–34

    Article  MathSciNet  Google Scholar 

  16. Basak T, Roy S, Thirumalesha C (2007) Finite element analysis of natural convection in a triangular enclosure: effects of various thermal boundary conditions. Chem Eng Sci 62:2623–2640

    Article  Google Scholar 

  17. Oztop HF, Varol Y, Pop I (2009) Investigation of natural convection in triangular enclosure filled with porous media saturated with water near 4°C. Energ Convers Manag 50:1473–1480

    Article  Google Scholar 

  18. Sieres J, Campo A, Ridouane EH, Fernándes-Seara J (2007) Effect of surface radiation on buoyant convection in vertical triangular cavities with variable aperture angles. Int J Heat Mass Transf 50:5139–5149

    Article  MATH  Google Scholar 

  19. Varol Y, Oztop HF, Pop I (2009) Entropy generation due to natural convection in non-uniformly heated porous isosceles triangular enclosures at different positions. Int J Heat Mass Transf 52:1193–1205

    Article  MATH  Google Scholar 

  20. Varol Y, Oztop HF, Pop I (2009) Natural convection in right-angle porous trapezoidal enclosure partially cooled from inclined wall. Int Commun Heat Mass 36:6–15

    Article  Google Scholar 

  21. Chamkha AJ, Ismael MA (2014) Natural convection in differentially heated partially porous layered cavities filled with a nanofluid. Numer Heat Transf Appl 65:1089–1113

    Article  ADS  Google Scholar 

  22. Chamkha AJ, Mansour MA, Ahmed SE (2010) Double-diffusive natural convection in inclined finned triangular porous enclosures in the presence of heat generation/absorption effects. Heat Mass Transf 46:757–768

    Article  ADS  Google Scholar 

  23. Aly AM (2017) Natural convection over circular cylinders in a porous enclosure filled with a nanofluid under thermo-diffusion effects. J Taiwan Inst Chem Eng 70:88–103

    Article  Google Scholar 

  24. Nguyen MT, Aly AM, Lee SW (2015) Natural convection in a non-Darcy porous cavity filled with cu–water nanofluid using the characteristic-based split procedure in finite-element method. Numer Heat Transf Appl 67:224–247

    Article  ADS  Google Scholar 

  25. Nithiarasu P, Sundararajan T, Seetharamu KN (1998) Finite element analysis of transient natural convection in an odd-shaped enclosure. Int J Numer Methods H 8:199–216

    Article  MATH  Google Scholar 

  26. Aly AM (2015) Modeling of multi-phase flows and natural convection in a square cavity using an incompressible smoothed particle hydrodynamics. Int J Numer Method Heat 25:513–533

    Article  MathSciNet  MATH  Google Scholar 

  27. Aly AM, Asai M (2015) Modelling of non-Darcy flows through porous media using extended incompressible smoothed particle hydrodynamics. Numer Heat Transf B Fundam 67:255–279

    Article  ADS  Google Scholar 

  28. Aly AM, Chamkha AJ, Lee SW, Al-Mudhaf A (2016) On mixed convection in an inclined lid-driven cavity with sinusoidal heated walls using the ISPH method. Int J Comput Therm 8:337–354

    Google Scholar 

  29. Leroy A, Violeau D, Ferrand M, Joly A (2015) Buoyancy modelling with incompressible SPH for laminar and turbulent flows. Int J Numer Methods Fluid 78:455–474

    Article  MathSciNet  Google Scholar 

  30. Cleary PW, Monaghan JJ (1999) Conduction modelling using smoothed particle hydrodynamics. J Comput Phys 148:227–264

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Chaniotis AK, Poulikakos D, Koumoutsakos P (2002) Remeshed smoothed particle hydrodynamics for the simulation of viscous and heat conducting flows. J Comput Phys 182:67–90

    Article  ADS  MATH  Google Scholar 

  32. Danis ME, Orhan M, Ecder A (2013) ISPH modelling of transient natural convection. Int J Comput Fluid D 27:15–31

    Article  MathSciNet  Google Scholar 

  33. Szewc K, Pozorski J, Taniére A (2011) Modeling of natural convection with smoothed particle hydrodynamics: non-Boussinesq formulation. Int J Heat Mass Transf 54:4807–4816

    Article  MATH  Google Scholar 

  34. Morris JP, Fox PJ, Zhu Y (1997) Modeling low Reynolds number incompressible flows using SPH. J Comput Phys 136:214–226

    Article  ADS  MATH  Google Scholar 

  35. Yildiz M, Rock RA, Suleman A (2009) SPH with the multiple boundary tangent method. Int J Numer Methods Eng 77:1416–1438

    Article  MathSciNet  MATH  Google Scholar 

  36. Leroy A, Violeau D, Ferrand M, Kassiotis C (2014) Unified semi-analytical wall boundary conditions applied to 2-D incompressible SPH. J Comput Phys 261:106–129

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. De Leffe M, Le Touze D, Alessandrini B (2009) Normal flux method at the boundary for SPH. 4th Int SPHERIC Workshop (SPHERIC 2009), Nantes, France

  38. Feldman J, Bonet J (2007) Dynamic refinement and boundary contact forces in SPH with applications in fluid flow problems. Int J Numer Methods Eng 72:295–324

    Article  MathSciNet  MATH  Google Scholar 

  39. Monaco AD, Manenti S, Gallati M, Sibilla S, Agate G, Guandalini R (2011) SPH modeling of solid boundaries through a semi-analytic approach. Eng Appl Comput Fluid 5:1–15

    Google Scholar 

  40. Ferrandd M, Laurence DR, Rogers BD, Violearu D, Kassiotis C (2013) Unified semi-analytical wall boundary conditions for inviscid, laminar or turbulent flows in the meshless SPH method. Int J Numer Methods Fluids 71:446–472

    Article  MathSciNet  Google Scholar 

  41. Mayrhofer A, Rogers BD, Violeau D, Ferrand M (2013) Investigation of wall bounded flows using SPH and the unified semi-analytical wall boundary conditions. Comput Phys Commun 184:2515–2527

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Macià F, González LM, Cercos-pita JL, Souto-iglesias A (2012) A boundary integral SPH formulation: consistency and applications to ISPH and WCSPH. Prog Theor Phys 128:439–462

    Article  ADS  MATH  Google Scholar 

  43. Ellero M, Serrano M, Espanol P (2007) Incompressible smoothed particle hydrodynamics. J Comput Phys 226(2):1731–1752

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Chorin AJ (1968) Numerical solution of the Navier–Stokes equations. Math Comput 22:745–762

    Article  MathSciNet  MATH  Google Scholar 

  45. Lind SJ, Xu R, Stansby PK, Rogers BD (2012) Incompressible smoothed particle hydrodynamics for free-surface flows: a generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves. J Comput Phys 231:1499–1523

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. Skillen A, Lind S, Stansby PK, Rogers BD (2013) Incompressible smoothed particle hydrodynamics (SPH) with reduced temporal noise and generalised Fickian smoothing applied to body–water slam and efficient wave–body interaction. Comput Method Appl M 265:163–173

    Article  MathSciNet  MATH  Google Scholar 

  47. Brookshaw L (1985) A method of calculating radiative heat diffusion in particle simulations. Proc Astron Soc Aust 6:207–210

    Article  ADS  Google Scholar 

  48. Kulasegaram S, Bonet J, Lewis WR, Profit M (2004) A variational formulation based contact algorithm for rigid boundaries in two-dimensional SPH applications. Comput Mech 33:316–325

    Article  MATH  Google Scholar 

  49. Davis GDV (1983) Natural convection of air in a square cavity: a bench mark numerical solution. Int J Numer Methods Fluids 3:249–264

    Article  MATH  Google Scholar 

  50. Nguyen MT, Aly AM, Lee SW (2016) Unsteady natural convection heat transfer in a nanofluid-filled square cavity with various heat source conditions. Adv Mech Eng 8(5):1–18

    Article  Google Scholar 

  51. Nithiarasu P, Seetharamu KN, Sundararajan T (1997) Natural convective heat transfer in a fluid saturated variable porosity medium. Int J Heat Mass Transf 40:3955–3967

    Article  MATH  Google Scholar 

  52. Maxwell J (1904) A treatise on electricity and magnetism, 2nd edn. Oxford University Press, Oxford

    MATH  Google Scholar 

  53. Brinkman HC (1952) The Viscosity of concentrated suspensions and solutions. J Chem Phys 20:571

    Article  ADS  Google Scholar 

  54. Beckermann C, Ramadhyani S, Viskanta R (1987) Natural convection flow and heat transfer between a fluid layer and a porous layer inside a rectangular enclosure. J Heat Transf 109:363–370

    Article  Google Scholar 

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Acknowledgements

This work was supported by the University of Ulsan, Korea.

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Authors and Affiliations

  1. Yonsei Cardiovascular Center, College of Medicine, Yonsei University, Seoul, South Korea

    Minh Tuan Nguyen

  2. Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt

    Abdelraheem M. Aly

  3. School of Mechanical Engineering, University of Ulsan, 93 Daehak-ro, Nam-gu, Ulsan, 680-749, South Korea

    Abdelraheem M. Aly & Sang-Wook Lee

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  1. Minh Tuan Nguyen
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Correspondence to Sang-Wook Lee.

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Nguyen, M.T., Aly, A.M. & Lee, SW. ISPH modeling of natural convection heat transfer with an analytical kernel renormalization factor. Meccanica 53, 2299–2318 (2018). https://doi.org/10.1007/s11012-018-0825-3

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