An extension of Oberbeck–Boussinesq approximation for thermal convection problems

  • Bernardo Alan de Freitas Duarte
  • Rafael Romão da Silva Melo
  • Millena Martins Villar
  • Ricardo Serfaty
  • Aristeu da Silveira Neto
Technical Paper


An extension of the Oberbeck–Boussinesq approximation (OB) with expressive thermal effects is investigated to model specific mass with a temperature-dependent approach. An expansion of OB is proposed using a mathematical formulation with null velocity divergence in the continuity equation and variable specific mass in the momentum and energy equations (NOB). The NOB method has been previously tested in the literature, and the present paper aims to carry out a quantitative analysis between NOB and OB. The specific mass is calculated based on the temperature field and a divergence-free velocity is imposed on the NOB due to the small effects of variations of the specific mass in the continuity equation, as previously demonstrated by the OB. The purpose of the NOB is to overcome the OB’s restriction to single-phase flows, as well as to model the effects of specific mass variations more accurately, especially in problems with prominent thermal transfer effects, as that occur in the turbulent regime. Since the effects of a variable specific mass in the momentum and energy equations were taken into account by the NOB, it was expected that the thermal transfer results of the NOB would be improved compared with those of the OB. Then, three-dimensional natural convection simulations for a large range of Rayleigh numbers were performed using OB and NOB in a cubic cavity. Turbulence modeling was performed using Large Eddy Simulation (LES). Numerical results from both approaches were validated and all the results presented good quality. Thermal transfer was evaluated by the calculation of the Nusselt number at the heated wall and compared to experimental data from the literature. The OB and NOB provided adequate thermal transfer rate results; however, the differences between the literature and OB were higher than for NOB. Therefore, NOB is presented as a useful mathematical formulation for modeling incompressible flows with a temperature-dependent specific mass approach in single-phase or multiphase problems instead of solving the full compressible mathematical formulation. In addition, the differences between the literature and OB results increased as the Rayleigh number was raised, especially in the turbulent regime. The results of the NOB were closer to the literature than were those of the OB for the entire range of Rayleigh numbers tested. Therefore, the numerical results confirmed the higher accuracy of NOB compared to OB despite the NOB’s still being an approximate model. Lastly, flow visualization allowed the identification of coherent turbulent structures near the cavity walls in the simulation with Rayleigh number \(10^{10}\). The presence of hairpin and Tollmien–Schlichting instabilities revealed the importance of modeling the three-dimensional effects of natural convection in the turbulent regime.


Natural convection Variable specific mass Temperature-dependent properties Buoyancy Tollmien–Schlichting instability 



The authors gratefully acknowledge financial support from Petrobras, CNPQ, Fapemig and Capes. The authors are also grateful to the mechanical engineering graduate program from the Federal University of Uberlândia (UFU).


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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  • Bernardo Alan de Freitas Duarte
    • 1
  • Rafael Romão da Silva Melo
    • 1
  • Millena Martins Villar
    • 1
  • Ricardo Serfaty
    • 2
  • Aristeu da Silveira Neto
    • 1
  1. 1.Federal University of UberlândiaUberlândiaBrazil
  2. 2.CENPESRio de JaneiroBrazil

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