Effects of quadratic drag on natural convection in a tilted porous layer with uniform heat flux from the side Z. Alloui a,⇑ , R. Rebhi b , M. Mamou c , P. Vasseur d a Département de Génie Mécanique, Faculté de Technologie, Université Batna 2, 05000 Batna, Algeria b Laboratoire de Mécanique Physique et modélisation Mathématique (LMP2M), Université Yahia Fares de Médéa, Quartier Ain D’Heb, Algeria c Aerodynamics Laboratory, NRC Aerospace, National Research Council, Ottawa, Ontario K1A OR6, Canada d Ecole Polytechnique, Université de Montréal, C.P. 6079, Succ. «Centre Ville», Montréal, Québec H3C 3A7, Canada article info Article history: Received 25 February 2017 Received in revised form 7 June 2017 Accepted 4 July 2017 Keywords: Natural convection Tilted porous cavity Form drag effects Heat transfer abstract This paper reports a numerical study of natural convection in an inclined enclosure filled with a fluid- saturated porous medium. The Darcy-Dupuit model, which includes effects of flow form drag, is used to describe the flow in the porous layer. Thermal boundary conditions of the Neumann type are applied on the long side walls of the enclosure while the short ones are assumed adiabatic. The governing param- eters for the problem are the Rayleigh number R, inclination angle u, form drag parameter G and aspect ratio of the cavity A. A semi-analytical solution, valid for an infinite layer (A  1), is derived on the basis of the parallel flow approximation. It is demonstrated that both the inclination of the layer and the form drag parameter, have a strong influence on the strength of the natural convection heat transfer within the enclosure. The effect of form drag parameter on the existence of multiple steady state solutions, that are possible for an enclosure slightly inclined around the horizontal position, is investigated. For boundary layer flows in a vertical cavity it is demonstrated that, in the limit of Dupuit regime, the Nusselt number is given by Nu ¼ 0:556ðR=GÞ 1=4 . A good agreement is found between the predictions of the parallel flow approximation and the numerical results obtained by solving the full governing equations. Ó 2017 Elsevier Ltd. All rights reserved. 1. Introduction Over the past years considerable research efforts have been devoted to the study of natural convection in differentially heated cavities filled with a fluid saturated porous medium. Most studies on this topic are based on Darcy’s law which has the advantage of linearizing the momentum equation. However, this model is valid only when the order of magnitude of the pore size Reynolds num- ber is less than unity. It is thus valid for slow flows through porous media with low permeability (see for instance Bear [1]). Also, since Darcy’s law is of order one less than the Navier-Stokes equations it cannot account for the no-slip boundary condition on solid bound- aries. To overcome these limitations Brinkman [2] added a viscous like term in order to take into account the boundary frictional drag on solid walls for flows through medium of high permeability. Fur- thermore, Dupuit [3] and later Forchheimer [4] included a velocity- squared term in Darcy’s law to account for the form drag forces when the Reynolds number exceeds unity. The Brinkman-extended Darcy model was used first in the past by Katto and Masuaka [5], Walker and Homsy [6], Rudraiah et al. [7] and Vasseur et al. [8] to investigate the onset of motion in a horizontal layer heated from below and subjected to various ther- mal boundary conditions. The criterion for the onset of motion was predicted by these authors in terms of the Darcy number. In partic- ular it was demonstrated by Vasseur et al. [8] that the results of viscous fluid and the Darcy medium can be recovered in the limits of large and small Darcy numbers, respectively. The case of a ver- tical rectangular porous cavity, heated isothermally from the sides, has been considered by Chan et al. [9] on the basis of the Brinkman-extended Darcy model. Numerical results were obtained in terms of the Rayleigh and Darcy numbers and aspect ratio of the cavity. The results indicated that, when the Darcy number is suffi- ciently small, a good agreement with Darcy’s law is obtained. Tong and Subramanian [10] solved the boundary layer equations, using the modified Oseen method, to investigate natural convection in a cavity with the vertical walls maintained at constant temperatures. It was found that the flow field is governed by a given dimension- less parameter which depends upon the Rayleigh and Darcy num- bers and aspect ratio of the cavity. When the Darcy parameter is sufficiently small it was demonstrated that, except in a thin region http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.07.015 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail address: allouizineddine@gmail.com (Z. Alloui). International Journal of Heat and Mass Transfer 115 (2017) 314–325 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt