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A smoothed particle hydrodynamics approach for numerical simulation of nano-fluid flows

Application to forced convection heat transfer over a horizontal cylinder


Nano-fluidic flow and heat transfer around a horizontal cylinder at Reynolds numbers up to 250 are investigated by using weakly compressible smoothed particle hydrodynamics (WCSPH). To be able to simulate enhanced nanoparticle heat transfer, this manuscript describes for the first time a development that allows conductive and convective heat transfer to be modelled accurately for the Eckert problem using WCSPH. The simulations have been conducted for Pr = 0.01–40 with nanoparticle volumetric concentrations ranging from 0 to 4%. The velocity fields and the Nusselt profiles from the present simulations are in a good agreement with the experimental measurements. The results show that WCSPH is appropriate method for such numerical modelling. Additionally, the results of heat transfer characteristics of nano-fluid flow over a cylinder marked improvements comparing with the base fluids. This improvement is more evident in flows with higher Reynolds numbers and higher particle volume concentration.

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B :

Constant in equation of state

c o :

Artificial speed of sound

C p :

The specific heat at constant pressure

C v :

The specific heat at constant volume

D :

Cylinder diameter

e :

Energy of a given particle

f :

Arbitrarily function

h :

Smoothing length or support dimension

H :

Computational domain height

k :

Thermal conductivity of fluid

L :

Computational domain length

m :

Mass of a given particle

Nu :

Nusselt number

P :

Pressure acting on the particle

Pr :

Prandtl number

Q :

Heat flux terms in energy equation

r :

Distance between the centres of a couple of particles

r :

Position vector identifying the equilibrium state

Re :

Reynolds number

t :


T :


u :

Velocity component along x-axis

v :

Velocity component along y-axis

V :

Volume of the particle i

W :

Kernel or smoothing function

x :

The first Cartesian coordinate axis

y :

The second Cartesian coordinate axis

δ :

Dirac delta function

σ :

Stress tensor

μ :

Viscosity coefficient

ρ :

Mass density

ν :

Kinematic viscosity

γ :

Constant in state equation

φ :

Nanoparticle volume fraction

ε :

Parameter of artificial viscous heating


Value for reference condition


Cutting radius


Particle of interest


Neighbour particle






Solid nanoparticle


  1. Kramers HA. Heat transfer from spheres to flowing media. Physica. 1945;12:61–80.

    Article  Google Scholar 

  2. Perkins H. Forced convection heat transfer from a uniformly heated cylinder. J Heat Transf. 1962;84:257–63.

    Article  CAS  Google Scholar 

  3. Perkins H, Leppert G. Local heat-transfer coefficients on a uniformly heated cylinder. Int J Heat Mass Transf. 1964;7:143–58.

    Article  Google Scholar 

  4. Fand RM. Heat transfer by forced convection from a cylinder to water in crossflow. Int J Heat Mass Transf. 1965;8:995–1010.

    Article  CAS  Google Scholar 

  5. Zukauskas A, Ziugzda J. Heat transfer of a cylinder in crossflow. Washington: Hemishere Pub; 1985.

    Google Scholar 

  6. Whitaker S. Forced convection heat transfer calculations for flow in pipes, past flat plate, single cylinder, and for flow in packed beds and tube bundles. AIChE J. 1972;18:361–71.

    Article  CAS  Google Scholar 

  7. Churchill S, Bernstein M. A correlating equation for forced convection from gases and liquids to a circular cylinder in cross flow. J Heat Transf. 1977;99:300–6.

    Article  CAS  Google Scholar 

  8. Sanitjai S, Goldstein R. Heat transfer from a circular cylinder to mixtures of water and ethylene glycol. Int J Heat Mass Transf. 2004;47:4785–94.

    Article  CAS  Google Scholar 

  9. Sanitjai S, Goldstein RJ. Forced convection heat transfer from a circular cylinder in crossflow to air and liquids. Int J Heat Mass Transf. 2004;47:4795–805.

    Article  CAS  Google Scholar 

  10. Haeri S, Shrimpton J. On the application of immersed boundary, fictitious domain and body-conformal mesh methods to many particle multiphase flows. Int J Multiph Flow. 2012;40:38–55.

    Article  CAS  Google Scholar 

  11. Mittal R, Balachandar S. Effect of three-dimensionality on the lift and drag of nominally two-dimensional cylinders. Phys Fluids. 1995;7:1841–65.

    Article  CAS  Google Scholar 

  12. Williamson CHK. Vortex dynamics in the cylinder wake. Annu Rev Fluid Mech. 1996;28:477–539.

    Article  Google Scholar 

  13. Singh SP, Mittal S. Energy spectra of flow past a circular cylinder. Int J Comput Fluid Dyn. 2004;18:671–9.

    Article  Google Scholar 

  14. Sharma N, Patankar N. A fast computation technique for the direct numerical simulation of rigid particulate flows. J Comput Phys. 2005;205:439–57.

    Article  Google Scholar 

  15. Yu Z, Shao X. A direct-forcing fictitious domain method for particulate flows. J Comput Phys. 2007;227:292–314.

    Article  Google Scholar 

  16. Apte SV, Martin M, Patankar NA. A numerical method for fully resolved simulation (FRS) of rigid particle-flow interactions in complex flows. J Comput Phys. 2009;228:2712–38.

    Article  CAS  Google Scholar 

  17. Peskin CS. The immersed boundary method. Acta Numer. 2002;11:479–517.

    Article  Google Scholar 

  18. Griffith BE, Hornung RD, McQueen DM, Peskin CS. An adaptive, formally second order accurate version of the immersed boundary method. J Comput Phys. 2007;223:10–49.

    Article  Google Scholar 

  19. Roma A, Peskin C, Berger M. An adaptive version of the immersed boundary method. J Comput Phys. 1999;153:509–34.

    Article  Google Scholar 

  20. Luci L. A numerical approach to the testing of the fission hypothesis. Astron J. 1977;82:1013–24.

    Article  Google Scholar 

  21. Gingold RA, Monaghan JJ. Smoothed particle hydrodynamics: Theory and application to non-spherical stars. Sci Math MNRAS. 1977;181:375–89.

    Article  Google Scholar 

  22. Shadloo MS, Weiss R, Yildiz M, Dalrymple RA. Numerical simulations of the breaking and non-breaking long waves. Int J Offshore Polar Eng. 2015;25(1):1–7.

    Google Scholar 

  23. Shadloo MS, Le Touze D, Oger G. Mesh-free Lagrangian modelling of fast flow dynamics, Manuscript No.: 2015-TPC-0750. In: The 25th international ocean and polar engineering conference, June 21–26, 2015, Kona, Big Island.

  24. Shadloo MS, Yildiz M. Numerical modeling of Kelvin–Helmholtz instability using smoothed particle hydrodynamics method. Int J Numer Methods Eng. 2011;87:988–1006.

    Article  Google Scholar 

  25. Shadloo MS, Zainali A, Yildiz M. Simulation of single mode Rayleigh–Taylor instability by SPH method. Comput Mech. 2013;51:699–715.

    Article  Google Scholar 

  26. Fatehi R, Shadloo MS, Manzari MT. Numerical investigation of two-phase secondary Kelvin–Helmholtz instability. J Mech Eng Sci. 2014;228:1913–24.

    Article  Google Scholar 

  27. Shadloo MS, Rahmat A, Yildiz M. A smoothed particle hydrodynamics study on the electrohydrodynamic deformation of a droplet suspended in a neutrally buoyant Newtonian fluid. Comput Mech. 2013;52:693–707.

    Article  Google Scholar 

  28. Abdollahzadeh Jamalabadi MY, Ovosi M. Numerical simulation of interaction of a current with a circular cylinder near a rigid bed. J Appl Math Phys. 2016;4:398–411.

    Article  Google Scholar 

  29. Zainali A, Tofighi N, Shadloo MS, Yildiz M. Numerical investigation of Newtonian and non-Newtonian multiphase flows using ISPH method. Comput Methods Appl Mech Eng. 2013;254:99–113.

    Article  Google Scholar 

  30. Shamsoddini R, Sefid M, Fatehi R. Incompressible SPH modeling and analysis of non-Newtonian power-law fluids, mixing in a microchannel with an oscillating stirrer. J Mech Sci Technol. 2016;30(1):307–16.

    Article  Google Scholar 

  31. Esmaili Sikarudi MA, Nikseresht AH. Neumann and Robin boundary conditions for heat conduction modeling using smoothed particle hydrodynamics. Comput Phys Commun. 2016;198:1–11.

    Article  CAS  Google Scholar 

  32. Rook R, Yildiz M, Dost S. Modeling transient heat transfer using SPH and implicit time integration. Numer Heat Transf B. 2007;51:1–23.

    Article  CAS  Google Scholar 

  33. Devendiran DK, Amirtham VA. A review on preparation, characterization, properties and applications of nanofluids. Renew Sustain Energy Rev. 2016;60:21–40.

    Article  CAS  Google Scholar 

  34. Esfe MH, Karimipour A, Yan WM, Akbari M, Safaei MR, Dahari M. Experimental study on thermal conductivity of ethylene glycol based nanofluids containing Al2–O3 nanoparticles. Int J Heat Mass Transf. 2016;88:728–34.

    Article  CAS  Google Scholar 

  35. Esfe MH, Saedodin S, Akbari M, Karimipour A, Afrand M, Wongwises S, Safaei MR, Dahari M. Experimental investigation and development of new correlations for thermal conductivity of CuO/EG–water nanofluid. Int Commun Heat Mass Transf. 2015;65:47–51.

    Article  CAS  Google Scholar 

  36. Karimipour A, Esfe MH, Safaei MR, Semiromi DT, Jafari S, Kazi SN. Mixed convection of copper–water nanofluid in a shallow inclined lid driven cavity using the lattice Boltzmann method. Physica A. 2014;402:150–68.

    Article  CAS  Google Scholar 

  37. Mahian O, Kianifar A, Kalogirou SA, Pop I, Wongwises S. A review of the applications of nanofluids in solar energy. Int J Heat Mass Transf. 2013;57(2):582–94.

    Article  CAS  Google Scholar 

  38. Karimipour A, D’Orazio A, Shadloo MS. The effects of different nano particles of Al2 O3 and Ag on the MHD nano fluid flow and heat transfer in a microchannel including slip velocity and temperature jump. Phys E Low Dimens Syst Nanostruct. 2017;86:146–53.

    Article  CAS  Google Scholar 

  39. Rashidi MM, Nasiri M, Shadloo MS, Yang Z. Entropy generation in a circular tube heat exchanger using nanofluids: effects of different modeling approaches. Heat Transf Eng. 2017;38(9):853–66.

    Article  CAS  Google Scholar 

  40. Safaei MR, Gooarzi M, Akbari OA, Shadloo MS, Dahari M. Electronics cooling, chapter 6. In: Sohel Murshed SM, editor. Performance evaluation of nanofluids in an inclined ribbed microchannel for electronic cooling applications. ISBN 978-953-51-2406-1, Print ISBN 978-953-51-2405-4; Published: June 15, 2016.

  41. Loya A, Stair JL, Ren G. Simulation and experimental study of rheological properties of CeO2–water nanofluid. Int Nano Lett. 2015;5(1):1–7.

    Article  CAS  Google Scholar 

  42. Safaei MR, Shadloo MS, Goodarzi MS, Hadjadj A, Goshayeshi HR, Afrand M, Kazi SN. A survey on experimental and numerical studies of convection heat transfer of nanofluids inside closed conduits. Adv Mech Eng. 2016;8(10):1–14.

    Article  CAS  Google Scholar 

  43. Rea U, McKrell T, Hu L-W, Buongiorno J. Laminar convective heat transfer and viscous pressure loss of alumina–water and zirconia–water nanofluids. Int J Heat Mass Transf. 2009;52(7–8):2042–8.

    Article  CAS  Google Scholar 

  44. Zeinali Heris S, Etemad SG, Nasr Esfahany M. Experimental investigation of oxide nanofluids laminar flow convective heat transfer. Int Commun Heat Mass Transf. 2006;33(4):529–35.

    Article  CAS  Google Scholar 

  45. Ebrahimnia-Bajestan E, Niazmand H, Duangthongsuk W, Wongwises S, Renksizbulut M. Numerical investigation of effective parameters in convective heat transfer of nanofluids flowing under laminar flow regime. Int J Heat Mass Transf. 2011;54(19–20):4376–88.

    Article  CAS  Google Scholar 

  46. R Vignjevic, J Campbell, Review of development of the smooth particle hydrodynamics (SPH) method. In: Predictive modeling of dynamic processes. Springer; 2009, p 367–396.

  47. Batchelor GK. Introduction to fluid dynamics. Cambridge: Cambridge University Press; 1974.

    Google Scholar 

  48. Monaghan JJ. Simulating free surface flows with SPH. J Comput Phys. 1994;110:399–406.

    Article  Google Scholar 

  49. Monaghan JJ. Smoothed particle hydrodynamics. Annu Rev Astron Astrophys. 1992;30:543–74.

    Article  Google Scholar 

  50. Rahmat A, Tofighi N, Shadloo MS, Yildiz M. Numerical simulation of wall bounded and electrically excited Rayleigh Taylor instability using incompressible smoothed particle hydrodynamics. Colloids Surfaces A. 2014;460:60–70.

    Article  CAS  Google Scholar 

  51. Shadloo MS, Zainali A, Sadek SH, Yildiz M. Improved incompressible smoothed particle hydrodynamics method for simulating flow around bluff bodies. Comput Methods Appl Mech Eng. 2011;200(9):1008–20.

    Article  Google Scholar 

  52. Fatehi R, Manzari MT. Error estimation in smoothed particle hydrodynamics and a new scheme for second derivatives. Comput Math Appl. 2011;61(2):482–98.

    Article  Google Scholar 

  53. Liu GR, Liu MB. Smoothed particle hydrodynamics: a meshfree particle method. Singapore: World Scientific Publishing Co Pte Ltd; 2003. ISBN 978-981-238-456-0.

    Book  Google Scholar 

  54. Karimipour Arash. A novel case study for thermal radiation through a nanofluid as a semitransparent medium via discrete ordinates method to consider the absorption and scattering of nanoparticles along the radiation beams coupled with natural convection. Int Commun Heat Mass Transf. 2017;87:256–69.

    Article  CAS  Google Scholar 

  55. Ghasemi S, Karimipour A. Experimental investigation of the effects of temperature and mass fraction on the dynamic viscosity of CuO-paraffin nanofluid. Appl Therm Eng. 2018;128:189–97.

    Article  CAS  Google Scholar 

  56. Aramesh M, Pourfayaz F, Kasaeian A. Numerical investigation of the nanofluid effects on the heat extraction process of solar ponds in the transient step. Sol Energy. 2017;157:869–79.

    Article  CAS  Google Scholar 

  57. Kasaeian A, Pourfayaz F, Khodabandeh E, Yan WM. Experimental studies on the applications of PCMs and Nano-PCMs in buildings: a critical review. Energy Build. 2017;154:96–112.

    Article  Google Scholar 

  58. Eckert ERG, Soehngen E. Distribution of heat transfer coefficients around circular cylinders in cross flow at Reynolds numbers from 20 to 500. J Heat Transf. 1952;74:343–7.

    Google Scholar 

  59. Shadloo MS, Zainali A, Yildiz M, Suleman A. A robust weakly compressible SPH method and its comparison with an incompressible SPH. Int J Numer Method Eng. 2012;89(8):939–56.

    Article  Google Scholar 

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Correspondence to Mohammad Reza Safaei.

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Nasiri, H., Abdollahzadeh Jamalabadi, M.Y., Sadeghi, R. et al. A smoothed particle hydrodynamics approach for numerical simulation of nano-fluid flows. J Therm Anal Calorim 135, 1733–1741 (2019).

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  • Smoothed particle hydrodynamics (SPH)
  • Weakly compressible
  • Nanoparticles
  • Nano-fluid
  • Forced convection