Natural Convection Study with Internal Heat Generation on Heat Transfer and Fluid Flow Within a Differentially Heated Square Cavity Filled with Different Working Fluids and Porous Media

  • Najib Hdhiri
  • Basma SouayehEmail author
  • Huda Alfannakh
  • Brahim Ben Beya


To study the intricate natural convection in square cavity filled with porous medium with an electrically conductive fluid in the presence of internal heat source, a numerical methodology based on the finite volume method and a full multigrid acceleration is utilized in this paper. The Darcy–Brinkman is adopted to model the fluid flow and energy transport equations in order to predict the heat transfer process in the porous medium. Numerical solutions are generated for representative combinations of the controlling Grashof number (103 ≤ Gr ≤ 106), the Prandtl number (0.015 ≤ Pr ≤ 0.054), and the Darcy number (10−5 ≤ Da ≤ 10−2). Typical sets of streamlines, isotherms, and average Nusselt number profiles are presented to analyze the flow patterns set up by the competition between homogenous and porous medium. It is revealed that average Nusselt number values are strongly affected by the increase of Prandtl number and the presence of homogeneous medium overestimates the rate of heat transfer better than the presence of a porous medium. Correlations of heat transfer rates in porous medium cases are established in the current investigation.


Natural convection Heat generation Porous medium Metal liquid Numerical simulation 



aspect ratio.


specific heat capacity (Jkg−1 K−1).


height of the enclosure (m).


thermal conductivity (Wm−1 K−1).


heat generation per unit volume (W m−3).


temperature (K).


x-velocity component (ms−1).


dimensionless X-velocity component, u H/ν.


y-velocity component (m s−1).


dimensionless Y-velocity component, v H/ν.


x-Cartesian coordinate (m).


dimensionless X-Cartesian coordinate, x/H.


y-Cartesian coordinate (m).


dimensionless Y-Cartesian coordinate, y/H.


reference temperature difference (K), TH-TC.


local Nusselt number.

\( \overline{Nu} \)

average Nusselt number.

\( \overline{N{u}_{corr}} \)

correlated average Nusselt number.


pressure (Nm2).


dimensionless pressure,pH20ν2.


Prandtl number, ν/α.


Grashof number, gβ ΔT H32.


External Rayleigh number, gβ ΔT H3/να.


Internal Rayleigh number, gβ Q H5/ναk.


The dimensionless heat generation/absorption parameter RaI/PrRaE.


cold wall temperature (K).


Hot wall temperature (K).


dimensional time (s).

Greek symbols


thermal diffusivity(m2 s−1).


thermal expansion coefficient (K−1).


difference value.


kinematic viscosity (m2 s−1).


dynamic viscosity, Ns/m2.


dimensionless temperature, (T-TC)/ΔT.


fluid density (kg/ m3).


dimensionless time, t ν/H2.


ratio of heat capacities.


permeability, m2.


Porosity values.


Darcy number, K/H2.


generic variable (U, V, P, or θ).


dimensionless stream-function.


standard deviation.



The authors would like to express their gratitude to King Faisal University, P.O. 380, Al-Ahsa-31982, Saudi Arabia, for providing the administrative and technical support.

Compliance with Ethical Standards

Conflict of Interest


Research Involving Humans and Animals Statement


Informed consent


Funding statement



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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Sciences of Tunis, Laboratory of fluid Mechanics, physics departmentUniversity of Tunis El ManarTunisTunisia
  2. 2.College of Science, Physics departmentKing Faisal UniversityAlahsaKingdom of Saudi Arabia

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