Vol. 25, No. 2 (2026), Fen26778 https://doi.org/10.24275/rmiq/Fen26778

Direct numerical simulation of the two-dimensional differentially heated square cavity at high Rayleigh numbers (Ra ≤ 1012)

 

Authors

F.I. Molina-Herrera, H. Montes-Rosales, N.E. Maldonado-Sierra, M.A. Sandoval-Hernández, G.M. Martinez-González, M.L. López-González, H. Jiménez-Islas

Abstract

This work presents a numerical study of natural convection in a 2-D differentially heated square cavity at high Rayleigh numbers (Ra ≤ 1012), using a dimensionless formulation of the incompressible Navier-Stokes equations. The problem was solved using the finite element method on an orthogonal quadrilateral mesh with boundary-layer-type refinement along the hot and cold vertical walls to resolve the steep thermal and hydrodynamic gradients near the boundaries. A systematic grid-independence study identified a mesh with 100 boundary-layer elements as sufficient to capture the relevant near-wall structure. Numerical consistency was assessed through macroscopic heat-balance verification, mesh-quality evaluation, and transient monitoring of the wall-integrated Nusselt number. Accuracy was further examined by comparison with classical and modern reference data available in the literature. The results show very good agreement with benchmark solutions across a wide range of Rayleigh numbers, and a consistent rise in heat transfer as Ra increases. In addition, a power-law regression of the computed data over 103 ≤ Ra ≤ 1011 predicts the Ra = 1012 result with excellent agreement. The steady-state and transient simulations converge to nearly identical Nusselt values at high Rayleigh numbers, confirming the robustness of the proposed numerical strategy. Overall, the adopted meshing approach provides stable and numerically consistent reference solutions for the classical 2-D differentially heated cavity benchmark at high Rayleigh numbers.

Keywords

Differentially heated square cavity; Natural convection; Rayleigh number; Finite element method; Boundary-layer mesh.

References
  • Abbassi, M.A., Safaei, M.R., Djebali, R., Guedri, K., Zeghmati, B., and Alrashed, A.A. (2018). LBM simulation of free convection in a nanofluid filled incinerator containing a hot block. Int. J. Mech. Sci. 144, 172–185. https://doi.org/10.1016/j.ijmecsci.2018.05.031
  • Barakos, G., Mitsoulis, E., and Assimacopoulos, D. (1994). Natural convection flow in a square cavity revisited: laminar and turbulent models with wall functions. Int. J. Numer. Methods Fluids 18(7), 695–719. https://doi.org/10.1002/fld.1650180705
  • Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (2009). Transport phenomena (Revised 2nd ed.). John Wiley and Sons, Inc., p. 928.
  • Cardona, L. M., Cortés-Rodríguez, M., & Castellanos-Galeano, F. J. (2023). Pineapple powder quality: Effect of the feedstock solution composition, homogenization and spray drying process. Revista Mexicana de Ingeniería Química, 22(2), Alim2993. https://doi.org/10.24275/rmiq/Alim2993
  • Castillo, P., and Henríquez, I. (2017). Assessment of time discretizations and LDG methods for diffusion and nonlinear reaction problems in 2D. Rev. Mex. Ing. Quim. 16(2), 541–554. https://www.redalyc.org/pdf/620/62052087019.pdf
  • Choi, S., and Kim, S. (2012). Turbulence modeling of natural convection in enclosures: a review. J. Mech. Sci. Technol. 26(1), 283–297. https://doi.org/10.1007/s12206-011-1037-0
  • Cintolesi, C., Petronio, A., and Armenio, V. (2015). Large eddy simulation of turbulent buoyant flow in a confined cavity with conjugate heat transfer. Phys. Fluids 27(9). https://doi.org/10.1063/1.4930815
  • Cortés-Avendaño, E. J., Castillo-Zamudio, R. I., Salgado-Cervantes, M. A., and García-Alvarado, M. A. (2023). Heat and mass transfer heterogeneous model applied for mathematical representation of Aloe vera extracts spray drying. Rev. Mex. Ing. Quim. 22(2), Alim237. https://doi.org/10.24275/rmiq/Alim237
  • De Vahl Davis, G. (1983). Natural convection of air in a square cavity: a benchmark numerical solution. Int. J. Numer. Methods Fluids 3(3), 249–264. https://doi.org/10.1002/fld.1650030305
  • Dixit, H., and Babu, V. (2006). Simulation of high Rayleigh number natural convection in a square cavity using the lattice Boltzmann method. Int. J. Heat Mass Transf. 49(3–4), 727–739. https://doi.org/10.1016/j.ijheatmasstransfer.2005.07.046
  • Dorfman, A., and Renner, Z. (2009). Conjugate problems in convective heat transfer: review. Math. Probl. Eng. Volume 2009, Article ID 927350. https://doi.org/10.1155/2009/927350
  • Duchaine, P. (2010). Experimental analysis of the dynamics of gaseous and two-phase counterflow flames submitted to upstream modulations [Doctoral dissertation, Ecole Centrale Paris]. HAL Archives Ouvertes. https://theses.hal.science/tel-00545418v1
  • El Hattab, M., and Lafdaili, Z. (2022). Turbulent natural convection heat transfer in a square cavity with nanofluids in presence of inclined magnetic field. Thermal Science 26(4A), 3201-3213. https://doi.org/10.2298/TSCI210825326E
  • Ghoben, Z.K., and Hussein, A.K. (2022). The natural convection inside a 3D triangular cross section cavity filled with nanofluid and included cylinder with different arrangements. Diagnostyka 23(2), 1–13. https://doi.org/10.29354/diag/149734
  • Goloviznin, V.M., Korotkin, I.A., and Finogenov, S.A. (2016). Parameter-free numerical method for modeling thermal convection in square cavities in a wide range of Rayleigh numbers. J. Appl. Mech. Tech. Phys. 57(7), 1159–1171. https://doi.org/10.1134/s0021894416070063
  • Henkes, R.A.W.M., and Hoogendoorn, C. (1989). Laminar natural convection boundary-layer flows along a heated vertical plate in a stratified environment. Int. J. Heat Mass Transf. 32(1), 147–155. https://doi.org/10.1016/0017-9310(89)90098-7
  • Hernández-López, I., Xamán, J., Álvarez, G., Chávez, Y., and Arce, J. (2016). Analysis of laminar and turbulent natural, mixed and forced convection in cavities by heatlines. Arch. Mech. 68(1), 27–53. https://doi.org/10.24423/aom.1905
  • Huang, Y., Ge, J., Chen, Y., and Zhang, C. (2023). Natural and forced convection heat transfer characteristics of single-phase immersion cooling systems for data centers. Int. J. Heat Mass Transf. 207, 124023. https://doi.org/10.1016/j.ijheatmasstransfer.2023.124023
  • Igci, A.A., and Arici, M.E. (2016). A comparative study of four low-Reynolds-number k-ε turbulence models for periodically fully developed duct flow and heat transfer. Numerical Heat Transfer, Part B: Fundamentals 69(3), 234–248. https://doi.org/10.1080/10407790.2015.1097141
  • Ji, Y. (2014). CFD modelling of natural convection in air cavities. CFD Lett. 6(1), 15–31. Retrieved from https://www.akademiabaru.com/doc/CFDLV6_N1_P15_31.pdf
  • Jiménez-Islas, H., Calderón-Ramírez, M., Navarrete-Bolaños, J. L., Botello-Álvarez, J. E., Martínez-González, G. M., & López-Isunza, F. (2009). Numerical study of natural convection in a 2-D square cavity with fluid-porous medium interface and heat generation. Revista Mexicana de Ingeniería Química, 8(2), 169–185. https://www.scielo.org.mx/pdf/rmiq/v8n2/v8n2a3.pdf
  • Jiménez-Islas, H., Calderón-Ramírez, M., Molina-Herrera, F., Martínez-González, G., Navarrete-Bolaños, J., and Castrejón-González, E. (2014). Low-RAM algorithm for solving 3-D natural convection problems using orthogonal collocation. Rev. Mex. Ing. Quim. 13(1), 259–277. https://www.redalyc.org/pdf/620/62031166020.pdf
  • Jones, W., and Launder, B. (1972). The prediction of laminarization with a two-equation model of turbulence. Int. J. Heat Mass Transf. 15(2), 301–314. https://doi.org/10.1016/0017-9310(72)90076-2
  • Markatos, N., and Pericleous, K. (1984). Laminar and turbulent natural convection in an enclosed cavity. Int. J. Heat Mass Transf. 27(5), 755–772. https://doi.org/10.1016/0017-9310(84)90145-5
  • Molina-Herrera, F.I., Quemada-Villagómez, L.I., Navarrete-Bolaños, J.L., and Jiménez-Islas, H. (2024). Comparative analysis of nondimensionalization approaches for solving the 2-D differentially heated cavity problem. Int. J. Numer. Methods Fluids 96(7), 1276–1303. https://doi.org/10.1002/fld.5285
  • Molina-Herrera, F.I., and Jiménez-Islas, H. (2025). Direct numerical simulation of the differentially heated cavity and comparison with the κ-ε model for high Rayleigh numbers. Modelling, 6(3), 66. https://doi.org/10.3390/modelling6030066
  • Orszag, S.A. (1970). Analytical theories of turbulence. J. Fluid Mech. 41(2), 363-386. https://doi.org/10.1017/S0022112070000642
  • Trias, F.X., Soria, M., Oliva, A., and Pérez-Segarra, C.D. (2007). Direct numerical simulations of two- and three-dimensional turbulent natural convection flows in a differentially heated cavity of aspect ratio 4. J. Fluid Mech. 586, 259–293. https://doi.org/10.1017/S0022112007006908
  • Trias, F.X., Gorobets, A., Soria, M., and Oliva, A. (2010). 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. Heat Mass Transf. 53(4), 665-673. https://doi.org/10.1016/j.ijheatmasstransfer.2009.10.026
  • Villa Ortiz, A., and Koloszar, L. (2023). RANS thermal modelling of a natural convection boundary layer at low Prandtl number. Comput. Fluids 254, 105809. https://doi.org/10.1016/j.compfluid.2023.105809
  • Welti-Chanes, J., Gómez-Palomares, O., Vergara-Balderas, F., & Maris-Alzamora, S. (2005). Engineering and transport phenomena applications to convective heat transfer study for canned foods. Rev. Mex. Ing. Quim.  4, 89–106. https://rmiq.org/iqfvp/Pdfs/Vol.%204,%20No.%201/7.pdf
  • Wilcox, D.C. (1993). Turbulence modeling for CFD. Retrieved from http://ci.nii.ac.jp/ncid/BA21991687
  • Weppe, A., Moreau, F., and Saury, D. (2022). Experimental investigation of a turbulent natural convection flow in a cubic cavity with an inner obstacle partially heated. Int. J. Heat Mass Transf. 194, 123052. https://doi.org/10.1016/j.ijheatmasstransfer.2022.123052
  • Xin, S., and Le Quéré, P. (1994). Numerical simulations of 2D turbulent natural convection in differentially heated cavities of aspect ratios 1 and 4. Fluid Mechanics and Its Applications, 423–434. https://doi.org/10.1007/978-94-011-1000-6_36
  • Zamora-Cisneros, D. U., & Ruiz Martínez, R. S. (2023). Simulation investigation of oxydehydrogenation of ethane in an industrial-scale turbulent fluidized bed over MoVTeNbO catalyst. Revista Mexicana de Ingeniería Química, 22(2), Cat2312. https://doi.org/10.24275/rmiq/Cat2312