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Lattice Boltzmann Simulation of MHD Rayleigh–Bénard Convection in Porous Media

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Abstract

Lattice Boltzmann method is used to investigate the Rayleigh–Bénard convection of magnetohydrodynamic fluid flow inside a rectangular cavity filled by porous media. The Brinkman–Forchheimer model is considered in the simulation to formulate a porous medium mathematically, and the multi-distribution function model is considered to include the magnetic field effect with different inclination angles. The water is considered as the working fluid, which is electrically conducting. A comprehensive analysis of the impact of governing dimensionless parameters is performed by varying Rayleigh (Ra), Hartmann (Ha), and Darcy (Da) numbers, porosity (\(\epsilon \)), and inclination angles (\(\phi \)) of the applied magnetic field. Numerical results are evaluated in the form of streamlines, isotherms, and the rate of heat transfer in terms of the local and average Nusselt number as well as the entropy generation due to the irreversibility of the fluid friction, temperature gradient, and magnetic field effects. The results imply that increasing Ha and decreasing Da reduce the rate of heat transfer. The average Bejan number \(\hbox {Be}_\mathrm{{avg}}\) increases for increasing the Hartmann number. On the other hand, augmenting Ra and \(\epsilon \) improves the heat transfer rate. It is also found that the change of the magnetic field inclination angle \(\phi \) changes the rate of heat transfer and entropy generation.

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Abbreviations

B :

Magnitude of the magnetic field

Be:

Bejan number

\(\hbox {Be}_\mathrm{{avg}}\) :

Average Bejan number

\(c_{i}\) :

Lattice speed

\(c_\mathrm{{s}}\) :

Speed of sound

Da:

Darcy number

\({\bar{e}}_{i}\) :

Discrete velocities

\(f_{i}\) :

Distribution function for flow fields

\(f^\mathrm{{eq}}_{i}\) :

Equilibrium distribution function

\({\bar{F}}_{i}\) :

Force terms

\(F^\mathrm{{M}}_{i}\) :

Force term for MHD

\(F^\mathrm{{P}}_{i}\) :

Force term for porous media

\(F^\mathrm{{b}}_{i}\) :

Buoyancy term

\(g_{i}\) :

Distribution function for temperature fields

\(g_{y}\) :

Gravitational force acting in y-direction

\(g^\mathrm{{eq}}_{i}\) :

Thermal equilibrium function

H :

Height of the cavity

Ha:

Hartmann number

K :

Permeability

m :

Lattice on the boundary

n :

Iteration index

Nu:

Nusselt number

\({\overline{\hbox {Nu}}}\) :

Average Nusselt number

Pr:

Prandtl number

Ra:

Rayleigh number

\(\hbox {Ra}_\mathrm{{c}}\) :

Critical Rayleigh number

\({\overline{S}}_\mathrm{{S}}\) :

Dum of irreversibilities

\({\overline{S}}_\mathrm{{F}}\) :

Irreversibilities due to fluid friction

\({\overline{S}}_\mathrm{{M}}\) :

Irreversibilities due to magnetic field

t :

Time

\(\Delta t\) :

Time interval

T :

Temperature

\(\Delta T\) :

Temperature difference

\(T_\mathrm{{c}}\) :

Cold temperature

\(T_\mathrm{{h}}\) :

Hot temperature

v :

Velocity component

\(\alpha \) :

Thermal diffusivity

\(\beta \) :

Thermal expansion

\(\Delta \) :

Differences

\(\epsilon \) :

Porosity

\(\mu \) :

Dynamic viscosity

\(\nu \) :

Kinematic viscosity

\(\rho \) :

Density of the fluid

\(\sigma \) :

Electrical conductivity

\(\phi \) :

Angle of inclination

\(\omega \) :

Weighting factor

avg:

Average

EM:

Empirical relation

i :

Lattice direction

s:

Sound

y :

y direction

b:

Buoyancy term

eq:

Equilibrium

M:

MHD

P:

Porous media

References

  1. Eikrem, G.O.; Imsland, L.; Foss, B.: Stabilization of gas lifted wells based on state estimation. IFAC Proc. Vol. 37(1), 323–328 (2004)

    Google Scholar 

  2. Hachem, E.; Rivaux, B.; Kloczko, T.; Digonnet, H.; Coupez, T.: Stabilized finite element method for incompressible flows with high Reynolds number. J. Comput. Phys. 229(23), 8643–8665 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Oreper, G.; Szekely, J.: The effect of an externally imposed magnetic field on buoyancy driven flow in a rectangular cavity. J. Cryst. Growth 64(3), 505–515 (1983)

    Google Scholar 

  4. El-Hakiem, M.: MHD oscillatory flow on free convection–radiation through a porous medium with constant suction velocity. J. Magn. Magn. Mater. 220(2–3), 271–276 (2000)

    Google Scholar 

  5. Muthuraj, R.; Srinivas, S.: A note on heat transfer to MHD oscillatory flow in an asymmetric wavy channel. Int. Commun. Heat Mass 37(9), 1255–1260 (2010)

    Google Scholar 

  6. Sheikholeslami, M.: Lattice Boltzmann method simulation for MHD non-Darcy nanofluid free convection. Phys. B Condens. Matter 516, 55–71 (2017)

    Google Scholar 

  7. Sheikholeslami, M.; Shehzad, S.; Li, Z.: Nanofluid heat transfer intensification in a permeable channel due to magnetic field using lattice Boltzmann method. Phys. B Condens. Matter 542, 51–58 (2018)

    Google Scholar 

  8. Nakagawa, Y.: An experiment on the inhibition of thermal convection by a magnetic field. Nature 175(4453), 417–419 (1955)

    Google Scholar 

  9. Zierep, J.: Rayleigh–Bénard convection with magnetic field. J. Theor. Appl. Mech. 30, 29–40 (2003)

    MATH  Google Scholar 

  10. Alchaar, S.; Vasseur, P.; Bilgen, E.: The effect of a magnetic field on natural convection in a shallow cavity heated from below. Chem. Eng. Commun. 134(1), 195–209 (1995)

    Google Scholar 

  11. Burr, U.; Müller, U.: Rayleigh–Bénard convection in liquid metal layers under the influence of a vertical magnetic field. Phys. Fluids 13(11), 3247–3257 (2001)

    MATH  Google Scholar 

  12. Pirmohammadi, M.; Ghassemi, M.: Effect of magnetic field on convection heat transfer inside a tilted square enclosure. Int. Commun. Heat Mass. 36(7), 776–780 (2009)

    Google Scholar 

  13. Sheikholeslami, M.; Gorji-Bandpy, M.; Vajravelu, K.: Lattice Boltzmann simulation of magnetohydrodynamic natural convection heat transfer of Al\(_{2}\)O\(_{3}\)–water nanofluid in a horizontal cylindrical enclosure with an inner triangular cylinder. Int. J. Heat Mass Transf. 80, 16–25 (2015)

    Google Scholar 

  14. Hasanuzzaman, M.; Öztop, H.F.; Rahman, M.; Rahim, N.; Saidur, R.; Varol, Y.: Magnetohydrodynamic natural convection in trapezoidal cavities. Int. Commun. Heat Mass. 39(9), 1384–1394 (2012)

    Google Scholar 

  15. Öztop, H.F.; Rahman, M.; Ahsan, A.; Hasanuzzaman, M.; Saidur, R.; Al-Salem, K.; Rahim, N.: MHD natural convection in an enclosure from two semi-circular heaters on the bottom wall. Int. J. Heat Mass Transf. 55(7–8), 1844–1854 (2012)

    Google Scholar 

  16. Grosan, T.; Revnic, C.; Pop, I.; Ingham, D.: Magnetic field and internal heat generation effects on the free convection in a rectangular cavity filled with a porous medium. Int. J. Heat Mass Transf. 52(5–6), 1525–1533 (2009)

    MATH  Google Scholar 

  17. Mahmud, S.; Fraser, R.A.: Magnetohydrodynamic free convection and entropy generation in a square porous cavity. Int. J. Heat Mass Transf. 47(14–16), 3245–3256 (2004)

    MATH  Google Scholar 

  18. Kuzhir, P.; Bossis, G.; Bashtovoi, V.; Volkova, O.: Flow of magnetorheological fluid through porous media. Eur. J. Mech. B Fluid. 22(4), 331–343 (2003)

    MATH  Google Scholar 

  19. McWhirter, J.D.; Crawford, M.E.; Klein, D.E.: Magnetohydrodynamic flows in porous media II: experimental results. Fusion Technol. 34(3P1), 187–197 (1998)

    Google Scholar 

  20. Khanafer, K.M.; Chamkha, A.J.: Hydromagnetic natural convection from an inclined porous square enclosure with heat generation. Numer. Heat Trans. A Appl. 33(8), 891–910 (1998)

    Google Scholar 

  21. Shan, X.: Simulation of Rayleigh–Bénard convection using a lattice Boltzmann method. Phys. Rev. E 55(3), 2780–2788 (1997)

    Google Scholar 

  22. Javed, T.; Mehmood, Z.; Abbas, Z.: Natural convection in square cavity filled with ferrofluid saturated porous medium in the presence of uniform magnetic field. Phys. B Condens. Matter. 506, 122–132 (2017)

    Google Scholar 

  23. Sheikholeslami, M.; Shamlooei, M.: Convective flow of nanofluid inside a lid driven porous cavity using CVFEM. Phys. B Condens. Matter. 521, 239–250 (2017)

    Google Scholar 

  24. Seta, T.; Takegoshi, E.; Okui, K.: Lattice Boltzmann simulation of natural convection in porous media. Math. Comput. Simul. 72(2–6), 195–200 (2006)

    MathSciNet  MATH  Google Scholar 

  25. Mohamad, A.A.: Lattice Boltzmann Method: Fundamentals and Engineering Applications with Computer Codes. Springer, London (2011)

    MATH  Google Scholar 

  26. Chen, S.; Doolen, G.D.: Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30(1), 329–364 (1998)

    MathSciNet  MATH  Google Scholar 

  27. Succi, S.: The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Numerical Mathematics and Scientific Computation. Clarendon Press, Oxford (2001)

    MATH  Google Scholar 

  28. Montgomery, D.; Doolen, G.D.: Magnetohydrodynamic cellular automata. Phys. Lett. A 120(5), 229–231 (1987)

    MATH  Google Scholar 

  29. Pattison, M.; Premnath, K.; Morley, N.; Abdou, M.: Progress in lattice Boltzmann methods for magnetohydrodynamic flows relevant to fusion applications. Fusion Eng. Des. 83(4), 557–572 (2008)

    Google Scholar 

  30. Hasanpour, A.; Farhadi, M.; Sedighi, K.; Ashorynejad, H.: Numerical study of Prandtl effect on MHD flow at a lid-driven porous cavity. Int. J. Numer. Methods Fluids. 70(7), 886–898 (2012)

    MathSciNet  MATH  Google Scholar 

  31. Molla, M.M.; Haque, M.J.; Khan, A.; Saha, S.C.: GPU accelerated multiple-relaxation-time lattice Boltzmann simulation of convective flows in a porous media. Front. Mech. Eng. 4, 1–15 (2018)

    Google Scholar 

  32. Wang, L.; Mi, J.; Guo, Z.: A modified lattice Bhatnagar–Gross–Krook model for convection heat transfer in porous media. Int. J. Heat Mass Transf. 94, 269–291 (2016)

    Google Scholar 

  33. Guo, Z.; Shu, C.: Lattice Boltzmann Method and Its Applications in Engineering, vol. 3. Advances in Computational Fluid DynamicsWorld Scientfic, New Jersey (2013)

    MATH  Google Scholar 

  34. Guo, Z.; Zhao, T.S.: A lattice Boltzmann model for convection heat transfer in porous media. Numer. Heat Trans. B Fundam. 47(2), 157–177 (2005)

    Google Scholar 

  35. Yuan, C.: Pore-scale modeling and hydromechanics of partially saturated granular materials. PhD thesis, University Grenoble Alpes, France (2016)

  36. Bear, J.: Dynamics of Fluids in Porous Media. Courier Corporation, New York (2013)

    MATH  Google Scholar 

  37. Mosthaf, K.: Modeling and analysis of coupled porous-medium and free flow with application to evaporation processes. PhD thesis, University of Stuttgart, Germany (2014)

  38. Sheikholeslami, M.; Gorji-Bandpy, M.; Ganji, D.: Lattice Boltzmann method for MHD natural convection heat transfer using nanofluid. Powder Technol. 254, 82–93 (2014)

    Google Scholar 

  39. Kao, P.-H.; Yang, R.-J.: Simulating oscillatory flows in Rayleigh–Bénard convection using the lattice Boltzmann method. Int. J. Heat Mass Transf. 50(17–18), 3315–3328 (2007)

    MATH  Google Scholar 

  40. Himika, T.A.; Hasan, M.F.; Molla, M.M.: Lattice Boltzmann simulation of airflow and mixed convection in a general ward of hospital. J. Comput. Eng. 2016, 1–15 (2016)

    Google Scholar 

  41. Kefayati, G.; Gorji, M.; Sajjadi, H.; Domiri Ganji, D.: Investigation of Prandtl number effect on natural convection MHD in an open cavity by lattice Boltzmann method. Eng. Comput. 30(1), 97–116 (2012)

    Google Scholar 

  42. Ergun, S.: Fluid flow through packed columns. Chem. Eng. Prog. 48, 89–94 (1952)

    Google Scholar 

  43. Hassan, S.; Himika, T.A.; Molla, M.M.; Hasan, M.F.: Lattice Boltzmann simulation of fluid flow and heat transfer through partially filled porous media. J. Comput. Eng. Phys. Model. 2(4), 21–30 (2019)

    Google Scholar 

  44. Perumal, D.A.; Yadav, A.K.: Computation of fluid flow in double sided cross-shaped lid-driven cavities using lattice Boltzmann method. Eur. J. Mech. B Fluid. 70, 46–72 (2018)

    MathSciNet  MATH  Google Scholar 

  45. Guo, Z.; Zhao, T.S.: Lattice Boltzmann model for incompressible flows through porous media. Phys. Rev. E. 66(3), 036304 (2002)

    Google Scholar 

  46. Hasan, M.F.; Ahmed Himika, T.; Molla, M.M.: Lattice Boltzmann simulation of airflow and heat transfer in a model ward of a hospital. J. Therm. Sci. Eng. Appl. 9(1), 011011 (2016)

    Google Scholar 

  47. Hasan, M.F., Himika, T.A., Molla, M.M.: Large-eddy simulation of airflow and heat transfer in a general ward of hospital. In: AIP Conference Proceedings, vol. 1754 of 1, Dhaka, Bangladesh (2016)

  48. Tang, G.; Tao, W.; He, Y.: Thermal boundary condition for the thermal lattice Boltzmann equation. Phys. Rev. E 72(1), 016703 (2005)

    Google Scholar 

  49. Clever, R.; Busse, F.: Transition to time-dependent convection. J. Fluid Mech. 65(4), 625–645 (1974)

    MATH  Google Scholar 

  50. He, X.; Chen, S.; Doolen, G.D.: A novel thermal model for the lattice Boltzmann method in incompressible limit. J. Comput. Phys. 146(1), 282–300 (1998)

    MathSciNet  MATH  Google Scholar 

  51. Demirel, Y.; Al-Ali, H.: Thermodynamic analysis of convective heat transfer in a packed duct with asymmetrical wall temperatures. Int. J. Heat Mass Transf. 40(5), 1145–1153 (1997)

    MATH  Google Scholar 

  52. Demirel, Y.: Irreversibility distribution for a pure convection case of the Smith-Hutton problem. Int. Commun. Heat Mass Transf. 25(5), 671–679 (1998)

    Google Scholar 

  53. Ilis, G.G.; Mobedi, M.; Sunden, B.: Effect of aspect ratio on entropy generation in a rectangular cavity with differentially heated vertical walls. Int. Commun. Heat Mass Transf. 35(6), 696–703 (2008)

    Google Scholar 

  54. Sivaraj, C.; Sheremet, M.: MHD natural convection and entropy generation of ferrofluids in a cavity with a non-uniformly heated horizontal plate. Int. J. Mech. Sci. 149, 326–337 (2018)

    Google Scholar 

  55. Nithiarasu, P.; Seetharamu, K.; Sundararajan, T.: Natural convective heat transfer in a fluid saturated variable porosity medium. Int. J. Heat Mass Transf. 40(16), 3955–3967 (1997)

    MATH  Google Scholar 

  56. Vafai, K. (ed.): Handbook of Porous Media, 3rd edn. CRC Press, Taylor & Francis Group, Boca Raton (2015)

    MATH  Google Scholar 

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Acknowledgements

This research is conducted by the financial support of North South University (NSU) as faculty research Grant “CTRG NSU-RP-18-067”.

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Authors and Affiliations

  1. Department of Mathematics and Physics, North South University, Dhaka, 1229, Bangladesh

    Taasnim Ahmed Himika, Sheikh Hassan & Md. Mamun Molla

  2. Department of Engineering and Physical Sciences, La Trobe University, Melbourne, Australia

    Md. Farhad Hasan

  3. Center for Applied Scientific Computing (CASC), North South University, Dhaka, 1229, Bangladesh

    Md. Mamun Molla

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  1. Taasnim Ahmed Himika
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Himika, T.A., Hassan, S., Hasan, M.F. et al. Lattice Boltzmann Simulation of MHD Rayleigh–Bénard Convection in Porous Media. Arab J Sci Eng 45, 9527–9547 (2020). https://doi.org/10.1007/s13369-020-04812-z

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