Document Type : Article

**Authors**

School of Mechanical Engineering, Sharif University of Technology, Azadi Avenue, Tehran, Iran

**Abstract**

In this study, turbulent natural convection in a square enclosure including one or four hot and cold bodies is numerically investigated in the range of Rayleigh numbers of . The shape of the internal bodies is square or rectangular with the same surface areas and different aspect ratios. In all cases, the horizontal walls of the enclosure are adiabatic and the vertical ones are isothermal. It is desired to investigate the influence of different shapes and arrangements of internal bodies on the heat transfer rate inside the enclosure with wide-ranging applications such as ventilation of buildings, electronic cooling and industrial coldbox packages. Governing equations including Reynolds-averaged-Navier-Stokes equations have been solved numerically with finite volume method and turbulence model in a staggered grid. The boundary condition for turbulence model is based on the standard wall function approach. Strongly implicit method is employed to solve the discretized systems of algebraic equations with a remarkable rate of convergence. The effects of several parameters such as distance between the bodies, aspect ratio and Rayleigh number on heat transfer rate have been investigated. The most change in heat transfer rate at high values of Rayleigh numbers is associated with alteration in distance between square bodies. Moreover, the horizontal installation of rectangular bodies with *h/w* = 1/3 is accompanied by a maximum reduction of heat transfer at low Rayleigh numbers. The present results have been compared with previous experimental and numerical works regarding enclosures with or without internal bodies and reasonable agreement is observed.

**Keywords**

**Main Subjects**

References:

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8. Ampofo, F. and Karayiannis, T.G. "Experimental benchmark data for turbulent natural convection in an air filled square cavity", Int. J. of Heat and Mass Transfer, 46, pp. 3551-3572 (2003). DOI:10.1016/S0017-9310(03)00147-9.

9. Salat, J., Xin, S., Joubert, P., Sergent, A., Penot, F., and Le Quere, P. "Experimental and numerical investigation of turbulent natural convection in a large air-filled cavity", Int. J. of Heat and Fluid Flow, 25, pp. 824-832 (2004). DOI: 10.1016/j.ijheatfluid

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10. De Vahl Davis, G. "Natural convection of air in a square cavity: a bench mark numerical solution", Int. J. for Numerical Methods in Fluids, 3, pp. 249-264 (1983). DOI: 10.1002/ d.1650030305.

11. Hortmann, M. Peric, M., and Scheuerer, G. "Finite volume multigrid prediction of laminar natural convection: Bench-mark solutions", Int. J. for Numererical Methods in Fluids., 11, pp. 189-207(1990). DOI:10.1002/ d.1650110206.

12. Le Quere, P. "Accurate solutions to the square thermally driven cavity at high Rayleigh number", Computers & Fluids, 20(1), pp. 29-41 (1991). DOI:10.1016/0045-7930(91)90025-D.

13. Phillips, T.N. "Natural convection in an enclosed cavity", J. of Computational Physics, 54(3), pp. 365- 381 (1984). DOI: 10.1016/0021-9991(84)90123-2.

14. Launder, B.E. and Spalding, D.B. "The numerical computation of turbulent flows", Computer Methods in Applied Mechanics and Engineering, 3, pp. 269-289 (1974). DOI: 10.1016/0045-7825(74)90029-2.

15. Ince, N.Z. and Launder, B.E. "On the computation of buoyancy-driven turbulent flows in rectangular enclosures", Int. J. of Heat and Fluid Flow, 10, pp. 110-117 (1989). DOI: 10.1016/0142-727X(89)90003-9.

16. Jones, W.P. and Launder, B.E. "The prediction of laminarization with a two-equation model of turbulence", Int. J. of Heat and Mass Transfer, 15, pp. 301-314 (1972). DOI: 10.1016/0017-9310(72)90076-2.

17. Henkes, R.A.W.M., Van Der Vlugt, F.F., and Hoogendoorn, C.J. "Natural-convection flow in a square cavity calculated with low-Reynolds-number turbulence models", Int. J. of Heat and Mass Transfer, 34 pp. 377-388(1991). DOI: 10.1016/0017-9310(91)90258-G.

18. Barakos, G. and Mitsoulis, E. "Natural convection flow in a square cavity revisited: laminar and turbulent models with wall functions", Int. J. for Numerical Methods in Fluids, 18, pp. 695-719 (1994). DOI:10.1002/ d.1650180705.

19. Chen, Q. "Comparison of different k " models for indoor air flow computations", Numer. Heat Transf. Part B Fundamental, 28, pp. 353-369 (1995). DOI:10.1080/10407799508928838.

20. Trias, F.X., Gorobets, A., Soria, M., and Oliva, A. "Direct numerical simulation of a differentially heated cavity of aspect ratio 4 with Rayleigh numbers up to 1011 - Part I: Numerical methods and time-averaged flow", Int. J. of Heat and Mass Transfer, 53, pp. 665-673(2010). DOI: 10.1016/j.ijheatmasstransfer.2009.10.026.

21. Trias, F.X., Gorobets, A., Soria, M., and Oliva, A. "Direct numerical simulation of a differentially heated cavity of aspect ratio 4 with Rayleigh numbers up to 1011 - Part II: Heat transfer and flow dynamics", Int. J. of Heat and Mass Transfer, 53, pp. 674-683 (2010). DOI: 10.1016/j.ijheatmasstransfer.2009.10.027.

22. Hsieh, K.J. and Lien, F.S. "Numerical modeling of buoyancy-driven turbulent flows in enclosures", Int. J. of Heat and Fluid Flow, 25, pp. 659-670 (2004). DOI: 10.1016/j.ijheat fluid flow.2003.11.023.

23. Hanjalic, K. and Vasic, S. "Computation of turbulent natural convection in rectangular enclosures with an algebraic flux model", Int. J. of Heat and Mass Transfer, 36, pp. 3603-3624 (1993). DOI: 10.1016/0017-9310(93)90178-9.

24. Dol, H.S., Hanjalic, K., and Kenjeres, S. "A comparative assessment of the second-moment differential and algebraic models in turbulent natural convection", Int. J. of Heat and Fluid Flow, 18, pp. 4-14 (1997). DOI:10.1016/S0142-727X(96)00149-X.

25. Craft, T.J., Gant, S.E., Gerasimov, A.V., Iacovides, H., and Launder, B.E. "Development and application of wall-function treatments for turbulent forced and mixed convection flows", Fluid Dynamics Research, 38, pp. 127-144 (2006). DOI: 10.1016/j. fluiddyn.2004.11.002.

26. Bairi, A., Zarco-pernia, E., and de Maria J.M.G. "A review on natural convection in enclosures for engineering applications, The particular case of the parallelogrammic diode cavity", Applied Thermal Engineering, 63, pp. 304-322 (2014). DOI: 10.1016/j.applthermaleng.2013.10.065.

27. Ho, C.J., Chang, W.S., and Wang, C.C. "Natural convection between two horizontal cylinders in an adiabatic circular enclosure", Transactions of ASME J. of Heat Transfer, 115. pp. 158-165 (1993). DOI:10.1115/1.2910642.

28. Ho, C.J., Cheng, Y.T., and Wang, C.C. "Natural convection between two horizontal cylinders inside a circular enclosure subjected to external convection", Int. J. of Heat and Fluid Flow, 15, pp. 299-306 (1994). DOI: 10.1016/0142-727X(94)90015-9.

29. Ha, M.Y, Jung, M.J., and Kim, Y.S. "Numerical study on transient heat transfer and fluid flow of natural convection in an enclosure with a heat generating conducting body, Numerical Heat Transfer, Part A, 35, pp. 415-433 (1999).

30. Ha, M.Y., Kim, I.K., Yoon, H.S., Yoon, K.S., Lee, J.R., Balachandar, S., and Chun, H.H. "Two- Dimensional and unsteady natural convection in a horizontal enclosure with a square body", Numerical Heat Transfer, Part A: Applications, 41, pp. 183-210 (2002). DOI: 10.1080/104077802317221393.

31. Oztop, H., Dagtekin, I., and Bahloul, A. "Comparison of position of a heated thin plate located in a cavity for natural convection", Int. Commun. Heat Mass Transfer, 31, pp. 121-132 (2004). DOI: 10.1016/S0735-1933(03)00207-0.

32. Oztop, H. and Bilgen, E. "Natural convection in differentially heated and partially divided square cavities with internal heat generation", Int. J. of Heat and Fluid Flow, 27, pp. 466-475 (2006). DOI: 10.1016/j.ijheatfluid flow.2005.11.003.

33. Kandaswamy, P., Lee, J., Abdul Hakeem, A.K., and Saravanan, S. "Effect of baffle-cavity ratios on buoyancy convection in a cavity with mutually orthogonal heated baffles", Int. J. of Heat and Mass Transf., 51, pp. 1830-1837 (2008). DOI: 10.1016/j.ijheatmasstransfer.2007.06.039.

34. Hakeem, A.K.A., Saravanan, S., and Kandaswamy, P. "Buoyancy convection in a square cavity with mutually orthogonal heat generating baffles", Int. J. of Heat and Fluid Flow., 29, pp. 1164-1173 (2008). DOI: 10.1016/j.ijheatfluid flow.2008.01.015.

35. Lee, J.M., Ha, M.Y., and Yoon, H.S. "Natural convection in a square enclosure with a circular cylinder at different horizontal and diagonal locations", Int. J. of Heat and Mass Transfer, 53, pp. 5905-5919 (2010). DOI: 10.1016/j.ijheatmasstransfer.2010.07.043.

36. Hussain, S.H. and Hussein, A.K. "Numerical investigation of natural convection phenomena in a uniformly heated circular cylinder immersed in square enclosure filled with air at different vertical locations", Int. Communications in Heat and Mass Transfer, 37, pp. 1115-1126 (2010). DOI: 10.1016/j.icheatmasstransfer.2010.05.016.

37. Bararnia, H., Soleimani, S., and Ganji, D.D. "Lattice Boltzmann simulation of natural convection around a horizontal elliptic cylinder inside a square enclosure", Int. Communications of Heat and Mass Transfer, 38, pp. 1436-1442 (2011). DOI:10.1016/j.icheatmasstransfer.2011.07.012.

38. Park, Y.G., Ha, M.Y., Choi, C., and Park, J. "Natural convection in a square enclosure with two inner circular cylinders positioned at different vertical locations", Int. J. of Heat and Mass Transfer, 77, pp. 501-518 (2014). DOI: 10.1016/j.ijheatmasstransfer.2014.05.041.

39. Garoosi, F., Bagheri, G., and Talebi, F. "Numerical simulation of natural convection of nano fluids in a square cavity with several pairs of heaters and coolers (HACs) inside", Int. J. of Heat and Mass Transfer, 67, pp. 362-376 (2013). DOI: 10.1016/j.ijheatmas stransfer.2013.08.034.

40. Garoosi, F. and Hoseininejad, F. "Numerical study of natural and mixed convection heat transfer between differentially heated cylinders in an adiabatic enclosure filled with nanofluid", J. of Molecular Liquids, 215, pp. 1-17 (2016). DOI: 10.1016/j.molliq.2015.12.016.

41. Patankar, S. and Spalding, D. "A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows", Int. J. of Heat and Mass Transfer, 15, pp. 1787-1806 (1972). DOI:10.1016/0017-9310(72)90054-3.

42. Stone, H.L. "Iterative solution of implicit approximations of multidimensional partial differential equations", SIAM J. Numerical Analysis, 5(3), pp. 530-558 (1968).

2. Saury, D., Rouger, N, Djanna, F., and Penot, F. "Natural convection in an air-filled cavity: Experimental results at large Rayleigh numbers", Int. Communications of Heat and Mass Transfer, 38, pp. 679-687 (2011). DOI: 10.1016/j.icheatmasstransfer.2011.03.019.

3. Dafa'Alla, A.A. and Betts, P.L. "Experimental study of turbulent natural convection in a tall air cavity", Exp. Heat Transfer, 9, pp. 165-194 (1996). DOI: 10.1080/08916159608946520.

4. Betts, P.L. and Bokhari, I.H. "Experiments on turbulent natural convection in an enclosed tall cavity", Int. J. of Heat and Fluid Flow, 21, pp. 675-683 (2000). DOI: 10.1016/S0142-727X(00)00033-3.

5. Kirkpatrick, A.T. and. Bohn, M. "An experimental investigation of mixed cavity natural convection in the high Rayleigh number regime", Int. J. of Heat and Mass Transfer, 29, pp. 69-82 (1986). DOI:10.1016/0017-9310(86)90035-9.

6. Tian, Y.S. and Karayiannis, T.G. "Low turbulence natural convection in an air filled square cavity", Part I, Int. J. of Heat and Mass Transfer, 43, pp. 849-866 (2000). DOI: 10.1016/S0017-9310(99)00199-4.

7. Tian, Y.S. and Karayiannis, T.G. "Low turbulence natural convection in an air filled square cavity", Part II, Int. J. of Heat and Mass Transfer, 43, pp. 867-884 (2000). DOI: 10.1016/S0017-9310(99)00200-8.

8. Ampofo, F. and Karayiannis, T.G. "Experimental benchmark data for turbulent natural convection in an air filled square cavity", Int. J. of Heat and Mass Transfer, 46, pp. 3551-3572 (2003). DOI:10.1016/S0017-9310(03)00147-9.

9. Salat, J., Xin, S., Joubert, P., Sergent, A., Penot, F., and Le Quere, P. "Experimental and numerical investigation of turbulent natural convection in a large air-filled cavity", Int. J. of Heat and Fluid Flow, 25, pp. 824-832 (2004). DOI: 10.1016/j.ijheatfluid

flow.2004.04.003.

10. De Vahl Davis, G. "Natural convection of air in a square cavity: a bench mark numerical solution", Int. J. for Numerical Methods in Fluids, 3, pp. 249-264 (1983). DOI: 10.1002/ d.1650030305.

11. Hortmann, M. Peric, M., and Scheuerer, G. "Finite volume multigrid prediction of laminar natural convection: Bench-mark solutions", Int. J. for Numererical Methods in Fluids., 11, pp. 189-207(1990). DOI:10.1002/ d.1650110206.

12. Le Quere, P. "Accurate solutions to the square thermally driven cavity at high Rayleigh number", Computers & Fluids, 20(1), pp. 29-41 (1991). DOI:10.1016/0045-7930(91)90025-D.

13. Phillips, T.N. "Natural convection in an enclosed cavity", J. of Computational Physics, 54(3), pp. 365- 381 (1984). DOI: 10.1016/0021-9991(84)90123-2.

14. Launder, B.E. and Spalding, D.B. "The numerical computation of turbulent flows", Computer Methods in Applied Mechanics and Engineering, 3, pp. 269-289 (1974). DOI: 10.1016/0045-7825(74)90029-2.

15. Ince, N.Z. and Launder, B.E. "On the computation of buoyancy-driven turbulent flows in rectangular enclosures", Int. J. of Heat and Fluid Flow, 10, pp. 110-117 (1989). DOI: 10.1016/0142-727X(89)90003-9.

16. Jones, W.P. and Launder, B.E. "The prediction of laminarization with a two-equation model of turbulence", Int. J. of Heat and Mass Transfer, 15, pp. 301-314 (1972). DOI: 10.1016/0017-9310(72)90076-2.

17. Henkes, R.A.W.M., Van Der Vlugt, F.F., and Hoogendoorn, C.J. "Natural-convection flow in a square cavity calculated with low-Reynolds-number turbulence models", Int. J. of Heat and Mass Transfer, 34 pp. 377-388(1991). DOI: 10.1016/0017-9310(91)90258-G.

18. Barakos, G. and Mitsoulis, E. "Natural convection flow in a square cavity revisited: laminar and turbulent models with wall functions", Int. J. for Numerical Methods in Fluids, 18, pp. 695-719 (1994). DOI:10.1002/ d.1650180705.

19. Chen, Q. "Comparison of different k " models for indoor air flow computations", Numer. Heat Transf. Part B Fundamental, 28, pp. 353-369 (1995). DOI:10.1080/10407799508928838.

20. Trias, F.X., Gorobets, A., Soria, M., and Oliva, A. "Direct numerical simulation of a differentially heated cavity of aspect ratio 4 with Rayleigh numbers up to 1011 - Part I: Numerical methods and time-averaged flow", Int. J. of Heat and Mass Transfer, 53, pp. 665-673(2010). DOI: 10.1016/j.ijheatmasstransfer.2009.10.026.

21. Trias, F.X., Gorobets, A., Soria, M., and Oliva, A. "Direct numerical simulation of a differentially heated cavity of aspect ratio 4 with Rayleigh numbers up to 1011 - Part II: Heat transfer and flow dynamics", Int. J. of Heat and Mass Transfer, 53, pp. 674-683 (2010). DOI: 10.1016/j.ijheatmasstransfer.2009.10.027.

22. Hsieh, K.J. and Lien, F.S. "Numerical modeling of buoyancy-driven turbulent flows in enclosures", Int. J. of Heat and Fluid Flow, 25, pp. 659-670 (2004). DOI: 10.1016/j.ijheat fluid flow.2003.11.023.

23. Hanjalic, K. and Vasic, S. "Computation of turbulent natural convection in rectangular enclosures with an algebraic flux model", Int. J. of Heat and Mass Transfer, 36, pp. 3603-3624 (1993). DOI: 10.1016/0017-9310(93)90178-9.

24. Dol, H.S., Hanjalic, K., and Kenjeres, S. "A comparative assessment of the second-moment differential and algebraic models in turbulent natural convection", Int. J. of Heat and Fluid Flow, 18, pp. 4-14 (1997). DOI:10.1016/S0142-727X(96)00149-X.

25. Craft, T.J., Gant, S.E., Gerasimov, A.V., Iacovides, H., and Launder, B.E. "Development and application of wall-function treatments for turbulent forced and mixed convection flows", Fluid Dynamics Research, 38, pp. 127-144 (2006). DOI: 10.1016/j. fluiddyn.2004.11.002.

26. Bairi, A., Zarco-pernia, E., and de Maria J.M.G. "A review on natural convection in enclosures for engineering applications, The particular case of the parallelogrammic diode cavity", Applied Thermal Engineering, 63, pp. 304-322 (2014). DOI: 10.1016/j.applthermaleng.2013.10.065.

27. Ho, C.J., Chang, W.S., and Wang, C.C. "Natural convection between two horizontal cylinders in an adiabatic circular enclosure", Transactions of ASME J. of Heat Transfer, 115. pp. 158-165 (1993). DOI:10.1115/1.2910642.

28. Ho, C.J., Cheng, Y.T., and Wang, C.C. "Natural convection between two horizontal cylinders inside a circular enclosure subjected to external convection", Int. J. of Heat and Fluid Flow, 15, pp. 299-306 (1994). DOI: 10.1016/0142-727X(94)90015-9.

29. Ha, M.Y, Jung, M.J., and Kim, Y.S. "Numerical study on transient heat transfer and fluid flow of natural convection in an enclosure with a heat generating conducting body, Numerical Heat Transfer, Part A, 35, pp. 415-433 (1999).

30. Ha, M.Y., Kim, I.K., Yoon, H.S., Yoon, K.S., Lee, J.R., Balachandar, S., and Chun, H.H. "Two- Dimensional and unsteady natural convection in a horizontal enclosure with a square body", Numerical Heat Transfer, Part A: Applications, 41, pp. 183-210 (2002). DOI: 10.1080/104077802317221393.

31. Oztop, H., Dagtekin, I., and Bahloul, A. "Comparison of position of a heated thin plate located in a cavity for natural convection", Int. Commun. Heat Mass Transfer, 31, pp. 121-132 (2004). DOI: 10.1016/S0735-1933(03)00207-0.

32. Oztop, H. and Bilgen, E. "Natural convection in differentially heated and partially divided square cavities with internal heat generation", Int. J. of Heat and Fluid Flow, 27, pp. 466-475 (2006). DOI: 10.1016/j.ijheatfluid flow.2005.11.003.

33. Kandaswamy, P., Lee, J., Abdul Hakeem, A.K., and Saravanan, S. "Effect of baffle-cavity ratios on buoyancy convection in a cavity with mutually orthogonal heated baffles", Int. J. of Heat and Mass Transf., 51, pp. 1830-1837 (2008). DOI: 10.1016/j.ijheatmasstransfer.2007.06.039.

34. Hakeem, A.K.A., Saravanan, S., and Kandaswamy, P. "Buoyancy convection in a square cavity with mutually orthogonal heat generating baffles", Int. J. of Heat and Fluid Flow., 29, pp. 1164-1173 (2008). DOI: 10.1016/j.ijheatfluid flow.2008.01.015.

35. Lee, J.M., Ha, M.Y., and Yoon, H.S. "Natural convection in a square enclosure with a circular cylinder at different horizontal and diagonal locations", Int. J. of Heat and Mass Transfer, 53, pp. 5905-5919 (2010). DOI: 10.1016/j.ijheatmasstransfer.2010.07.043.

36. Hussain, S.H. and Hussein, A.K. "Numerical investigation of natural convection phenomena in a uniformly heated circular cylinder immersed in square enclosure filled with air at different vertical locations", Int. Communications in Heat and Mass Transfer, 37, pp. 1115-1126 (2010). DOI: 10.1016/j.icheatmasstransfer.2010.05.016.

37. Bararnia, H., Soleimani, S., and Ganji, D.D. "Lattice Boltzmann simulation of natural convection around a horizontal elliptic cylinder inside a square enclosure", Int. Communications of Heat and Mass Transfer, 38, pp. 1436-1442 (2011). DOI:10.1016/j.icheatmasstransfer.2011.07.012.

38. Park, Y.G., Ha, M.Y., Choi, C., and Park, J. "Natural convection in a square enclosure with two inner circular cylinders positioned at different vertical locations", Int. J. of Heat and Mass Transfer, 77, pp. 501-518 (2014). DOI: 10.1016/j.ijheatmasstransfer.2014.05.041.

39. Garoosi, F., Bagheri, G., and Talebi, F. "Numerical simulation of natural convection of nano fluids in a square cavity with several pairs of heaters and coolers (HACs) inside", Int. J. of Heat and Mass Transfer, 67, pp. 362-376 (2013). DOI: 10.1016/j.ijheatmas stransfer.2013.08.034.

40. Garoosi, F. and Hoseininejad, F. "Numerical study of natural and mixed convection heat transfer between differentially heated cylinders in an adiabatic enclosure filled with nanofluid", J. of Molecular Liquids, 215, pp. 1-17 (2016). DOI: 10.1016/j.molliq.2015.12.016.

41. Patankar, S. and Spalding, D. "A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows", Int. J. of Heat and Mass Transfer, 15, pp. 1787-1806 (1972). DOI:10.1016/0017-9310(72)90054-3.

42. Stone, H.L. "Iterative solution of implicit approximations of multidimensional partial differential equations", SIAM J. Numerical Analysis, 5(3), pp. 530-558 (1968).

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