Effect of packing height and location of porous media on heat transfer in a cubical cavity_ Are extended Darcy simulations sufficient?

We numerically investigate natural convection in a bottom-heated top-cooled cavity, fully and partially filled with adiabatic spheres (with diameter-to-cavity-size ratio = d L / 0.2) arranged in a Simple Cubic Packing (SCP) configuration. We study the influence of packing height and location of porous media. We carry out the simulations using water as the working fluid with Prandtl number, = Pr 5.4 at Rayleigh number = × Ra 1.16 10 , 5 1.16 × 106 and 2.31 × 107. The applicability and suitability of Darcy-Forchheimer assumption to predict the global heat transfer is analysed by comparing it with the pore-structure resolved simulations. We found that the heat transfer in pore-structure resolved simulations is comparable to that in fluid-only cavities at high Rayleigh numbers, irrespective of the number of layers of packing and its location. Discrepancies in heat transfer between the Darcy-Forchheimer and the fully resolved simulations are observed when the porous medium is close to the isothermal wall and at high Ra, while it vanishes when the porous medium is away from the isothermal bottom


Introduction
Natural convective heat transfer in porous media packed cavities is of great importance in various engineering and real-life applications. It is an important mechanism in refrigeration devices Laguerre et al. (2008a,b), distribution transformers Torriano et al. (2018), nuclear waste disposal Shams et al. (2014), air dehumidifiers Fazilati et al. (2016), catalytic reactors Li et al. (2013); Thiagalingam et al. (2015), etc. It can also be of crucial importance in heat exchangers Boomsma et al. (2003); Missirlis et al. (2007) under safety mode operation, when the forced flow is blocked. In most of the applications mentioned above the domain may be partially filled Zhu et al. (2018) with materials and the relative position of these materials may vary (like food kept at the bottom or top of a refrigerator), which demands an in-depth understanding of the heat transfer mechanism in a partially filled porous media and its relative position.
Several experimental and numerical studies on natural convection in porous media are available in literature Nield et al. (2006). Most of the studies are on the influence of Rayleigh number Katto and Masuoka (1967), Prandtl number Catton and Jonsson (1987); Kladias and Prasad (1989), conductivity Zhao et al. (2005); Ataei- Dadavi et al. (2019b) and size of porous media Seki et al. (1978); Keene and Goldstein (2015); Nithiarasu et al. (1998). Studies also report that the heat transfer in fully packed porous media filled cavities asymptotically approach the heat transfer in a fluid-only cavity at high Rayleigh numbers Keene and Goldstein (2015); Ataei-Dadavi et al. (2019a). All these studies concentrate on the effect of the above mentioned parameters on fully packed cavities. However, the studies on the effect of location of the porous medium and its height are limited. Studies on partially filled porous media made of metal foam report a positive influence of porous media on heat transfer enhancement Kathare et al. (2008). The enhancement reported is mostly due to the increased conductivity of the medium. Experimental studies also report a non-monotonic change in heat transfer with packing height Prasad et al. (1991); Prasad (1993). Visualization from the top of the cavity hints at the channeling effect to be the reason for the nonmonotonic heat transfer variation.
A variety of different cases and conditions were tested using Darcy assumption and other more refined Darcy models Nield et al. (2006), with most of them being 2D simulations. Semi-analytical solution for 2dimensional Fahs et al. (2014Fahs et al. ( , 2015 and 3-dimensional T Shao et al. (2018) porous media are available for conditions at which Darcy assumption is valid. 3-dimensional numerical simulation of layered porous structure with Darcy assumption Guerrero-Martínez et al. (2017) discusses the influence of the thermal properties of the layer in heat transfer. However, at comparatively higher flow velocities, this linear relation is valid only at lower Reynolds number Nield et al. (2006). Extensive studies on porous-continuum representative volume (REV) based approach Vafai (1984); Nithiarasu et al. (1998) valid at higher flow velocities, with Darcy Basak et al. (2007); Das et al. (2017) and Darcy-Forchheimer Das et al. (2016) terms included are reported in literature. Heat transfer in partially filled cavities, obtained using REV approach with Darcy and extended Darcy assumptions show good agreement with experiments at low Ra Bagchi and Kulacki (2011);Poulikakos (1986). The simulations, however, deviate up to 15% from the experimental heat transfer measurements, when the Ra and Da are high, and also with the change in the location and height of the packing Su et al. (2015).
From our literature review, we find that the literature on the effect of pore-scale phenomena on heat transfer is mostly limited to fully packed cavities Karani and Huber (2017); Chakkingal et al. (2019) and almost no literature is available on the effect of location and height of packing Laguerre et al. (2008b). From our previous numerical Chakkingal et al. (2019Chakkingal et al. ( , 2020bChakkingal et al. ( , 2020a and experimental Ataei-Dadavi et al. (2019a, 2019b, 2020) studies on fully-packed porous media, we find a strong influence of pore-scale features on global and local heat transfer. Consequently, detailed pore-structure resolved simulations can give us a better understanding of the effect of height and location of the packing on heat transfer as well as the reason for deviation of REV approach based simulations from experiments. In our present work, we carry out 3D numerical simulations resolving the pore-structure and compare them to the Darcy-Forchheimer (D-F) simulations. We investigate the validity and applicability of the D-F assumption in partially filled cavities of different packing heights and locations, by comparing it with the pore-structure resolved simulations. We aim at providing an insight into the influence of local flow and thermal structures on the heat transfer in these cavities and explain the discrepancy in heat transfer between the pore-structure resolved and D-F simulations.

Geometry and boundary conditions
We study natural convection in a cubical L × L × L cavity filled with adiabatic spheres of diameter, = d L 0.2 at = × × Ra 1.16 10 , 1.16 10 5 6 and 2.3 × 10 7 . We use water ( = Pr 5.4) as the working fluid. The gravity, g acts along the negative Z-axis of the Cartesian coordinate system. From our previous experimental study on natural convection in fully-packed cavity Ataei-Dadavi et al. (2019a), we find that spherical beads of the dimensions mentioned above result in its influence on heat transfer at low Ra, while the effect reduces with an increase in Ra. The bottom and top walls of the cubical cavity are maintained at uniform temperatures T h and T c (T h > T c ) respectively. All the other vertical walls are adiabatic. Effects of packing height and packing location are investigated carrying out fully resolved numerical simulations and comparing with Darcy-Forchheimer simulations.
• Packing height:The domain is stacked with different number of horizontal layers ( Fig. 1  No-slip boundary condition is applied at all the walls for both fully resolved and D-F simulations, and also at the spherical surfaces in the fully resolved packed bed simulations. We ensure that βΔT ≪ 1 in all our simulations such that the Boussinesq approximation Gray and Giorgini (1976) is valid.

Governing equations and numerical method
The governing equations used for the simulations are split into two: For the fully resolved simulations using the Boussinesq approximation, we numerically solve the transient Navier-Stokes and thermal energy transport equations for Newtonian fluids: (1) To perform fully resolved numerical simulations in our complex geometry, we use Foam Extend 4.0 (a fork of OpenFOAM). We use unstructured polyhedral grids Zenklusen et al. (2014Zenklusen et al. ( , 2016 to carry out numerical simulations for the packed bed cavities and structured grids for the reference (fluid only) Rayleigh-Bénard convection simulations. The above set of equations, Eq. (1)-(3) are discretized and fully resolved numerical simulations are carried out using the standard solver buoy-antBoussinesqPisoFoam. The solver is validated by comparing the heat transfer obtained at different Ra in fluid-only cavities to that reported in the literature Ataei-Dadavi et al. (2019a); Chu and Goldstein (1973); Valencia et al. (2005). A detailed solver validation is also reported in Chakkingal et al. (2020c). The heat transfer in cavity fully-filled with adiabatic spherical beads (discussed later) at the highest Ra reported, is compared to that obtained with glass beads, experimentally Ataei-Dadavi et al. (2019a). At = × Ra 2.31 10 , 7 for a cavity fully-filled with adiabatic spherical beads we obtain a Nusselt number, = Nu 18.2. From experiments it is observed that, at the same Ra, in a cavity filled with glass beads of the same size and packing Ataei-Dadavi et al. (2019a) we obtain a Nusselt number = Nu 17.9 (interpolated from data). The heat transfer obtained numerically with adiabatic beads is comparable to that obtained with glass beads (of lower conductivity) experimentally. This confirms the suitability of the current solver for the pore-structure resolved simulations of the packed bed.
We use the numerical schemes as listed in Table 2 to solve the convective and diffusive termsWeller et al. (1998) and to handle the pressure-velocity-coupling at each time step Issa (1986). The adaptive time-stepping is carried out by limiting the maximum cell Courant number to 0.33.
For the D-F simulations in ANSYS-Fluent, the effect of porous medium is modelled by including the pressure drop in a porous medium due to the viscous and inertial effects as a source term S in the momentum equation, Eq. 5 Nithiarasu et al. (1998) where, • α eff is calculated using an effective thermal conductivity, s where λ f and λ s are the thermal conductivity of the fluid and solid (equal to zero in current simulations) respectively.
The source term, S in Eq. (5) is implemented as: For a packed bed cavity C 0 and C 1 are calculated using Ergun's equation as: where d and ϕ are the diameter of the spherical beads and local porosity, respectively. For each packing scheme the packed bed section is substituted by a Darcy equivalent, both in position/size and properties. The porosity, ϕ is calculated for the porous layer using the porous layer height, L H ( = L L L 0.2 , 0.4 H etc. for 1 Layer packing, 2 Layer packing etc. respectively) and the diameter of the sphere d as: where, V T porous and V T spheres is the total volume of the cavity of height L H and volume occupied by the spherical beads, respectively. The Eq. (4)-(7) in ANSYS Fluent user manual (when re-arranged) are the same as   Chakkingal, et al. International Journal of Heat and Fluid Flow 84 (2020) 108617 reported in Nithiarasu et al. (1998). The details of solver validation are reported in Appendix A. For the investigated d/L ratio and porosity, the permeability K is equal to × 1.05 10 6 resulting in a Da varying between × 2.62 10 3 and × 1.05 10 4 for packing height varying from 1 Layer to 5 Layers.
The governing equations in ANSYS-Fluent are solved using the numerical schemes listed in Table 3.
The time step size determined using for D-F simulations also satisfy the condition of maximum cell Courant number ≤ 0.33.

Mesh requirement
The pore-structure resolved (P-R) simulations are carried out using polyhedral mesh ( Fig. 2(b)), generated by converting a fine tetrahedral mesh ( Fig. 2(a)). The function polyDualMesh available in OpenFOAM is used to carry out this conversion. The number of mesh cells is decreased using this approach. For example, it reduces a 30 million cell tetrahedral mesh to a polyhedral mesh with around 6 million cells. Initial estimates of the grid requirement close to the wall and the bulk of the cavity are made based on the correlations provided in Shishkina et al. (2010) for Rayleigh-Bénard convection. Using the estimate for the highest = × Ra 2.3 10 7 as a base, a grid independence study is carried out using three different polyhedral meshes with 0.5 million, 4 million and 6 million cells for the cavity packed with 1 layer of spherical beads. The time-averaged integral Nusselt number (deviation less than 4%) and time-and plane-averaged non-dimensional temperature, θ m along the vertical direction (Fig. 3) are used to ensure that we get a grid-independent solution. Though the volume of the packed-cavity to be meshed decreases with an increase in the number of layers of spherical beads, we use~6 million polyhedral cells to account for the increase in curved surfaces in the domain.
For the fluid-only and Darcy-Forchheimer simulations, we use structured hexahedral grid cells, with a growth factor of 1.2 close to the horizontal isothermal walls. A grid independence study is carried out using 32 3 , 64 3 and 128 3 and 167 3 hexahedral cells. The deviation of time-and wall-averaged integral Nusselt number between 128 3 and 167 3 cells is less than 0.5% for = × Ra 2.3 10 7 . All the reported results are thus obtained with the 128 3 hexahedral cells.

Results and Discussion
The influence of Rayleigh number, packing height and location of packing of spherical beads on the flow and heat transfer is discussed in this section. In the following sections, the ANSYS Fluent simulations with Darcy and Forchheimer terms are referred to as Darcy-Forchheimer (D-F) simulations. All the D-F simulations are carried with grid-size required to fully resolve the flow in the porous media free region. In the porous region we find the flow velocities to be very low such that the Reynolds number calculated with maximum flow velocity as the velocity scale and square root of permeability K as the length scale is ≈ 3 within the porous region. Thus laminar models are used for the simulations.
The pore-scale resolved simulations are compared to D-F simulations to better understand the suitability and applicability of D-F assumptions to simulate the flow and heat transfer in fully and partially packed porous domains. The results obtained with D-F simulations are also reported in terms of intrinsic velocity u. All the results reported below are expressed in non-dimensional form. The instantaneous temperature is non-dimensionalised using the temperature difference be- The time-and wall-averaged global heat transfer is expressed in terms of Nusselt number defined as:   M. Chakkingal, et al. International Journal of Heat and Fluid Flow 84 (2020)

Influence of Ra and packing height
We compare the heat transfer and flow features in P-R cavities with 1-5 layers of spherical beads stacked from the bottom to the top. The results from P-R simulations are compared with the D-F simulation results with the same local Darcy number Da and ϕ. We also compare the heat transfer obtained with P-R and D-F simulations to that in a fluid-only filled cavity.
In Fig. 4 we plot the time-and plane-averaged (averaged along XY plane) non-dimensional temperature of pore-structure resolved (P-R) simulations with 1 Layer to 5 Layer of spherical beads. The vertical lines indicate the packing height, for e.g. = z L / 0.2 and = z L / 1 represent the height of 1Layer packing and 5 Layer packing, respectively. At low = × Ra 1.16 10 5 (Fig. 4(a)), the temperature distribution becomes conduction-dominated with the increase in the number of layers of spherical beads i.e. from 1Layer to 5Layer packing. The "close to linear" temperature in fully-packed cavity (5 Layer) indicates the absence of convection in the cavity. The flow resistance imposed by the adiabatic spherical beads results in this behaviour.
In contrast to the above, at = × Ra 2.3 10 7 (Fig. 4(b)) the slope of time-and plane-averaged temperature close to the wall ( = z L / 0 and = z L / 1) is comparable to that in a fluid-only filled cavity, indicating a convection dominated heat transfer. The presence of multi-layered adiabatic spherical beads results in an asymmetry in the temperature distribution, which vanishes in the cavity with 5 layers of packing.
From Fig. 5 we observe that the inclusion of spherical beads in the cavity, adversely affects the convective flow and thus the heat transfer is lower than in the fluid-only cavity, especially at low Ra. The nondimensional heat transfer Nu in P-R cavities and in cavities with the D-F assumption are comparable at lower Rayleigh numbers ( = × × Ra 1.16 10 , 1.16 10 5 6 ). At = × Ra 1.16 10 5 the heat transfer is much lower than in a fluid-only filled-cavity, and decreases with an increase in the number of layers of packing. At = × Ra 1.16 10 , 6 though the heat transfer is much lower than in fluid-only cavity, it is practically independent of the layers of packing (1 Layer-4 Layer). However, when the spherical beads are introduced close to the top cold wall (5 Layer), the heat transfer reduces further.
With the increase in Ra ( = × Ra 2.3 10 7 ), the heat transfer in P-R cavities are slightly lower than (1 Layer-3 Layer, 5 Layer) or comparable (4 Layer) to that in fluid-only filled cavities, while the heat transfer obtained from simulations with D-F assumption is observed to be much lower than in the fluid-only cavities. The results from the P-R simulations are qualitatively similar to that obtained experimentally for partially filled Prasad et al. (1991) and fully-filled cavities Ataei-Dadavi et al. (2019a). Unlike the P-R simulations, the D-F simulations cannot capture this behaviour and thus significantly under-predict the heat transfer.
To understand the difference in heat transfer (Nu) between P-R and D-F simulations we analyse the instantaneous local Nusselt number, local non-dimensional temperature and the local non-dimensional velocity magnitude in cavities with 1 Layer and 5 Layer packing, at = × Ra 2.3 10 7 .

Darcy-Forchheimer simulations
In simulations with D-F assumption, the local Nusselt number (Fig. 6) at the bottom hot wall significantly decreases with the increase in the height of packing. In the D-F cavity with porous layer equivalent to the height of 1 Layer of spherical bead packing ( Fig. 6(a)), the instantaneous heat transfer is high throughout the surface of the bottom wall. However, when the cavity is fully filled with porous media (Fig. 6(b)), the surface area with high local Nusselt number is smaller when compared to that in the cavity with 1 Layer of packing.
The reason for the reduction in heat transfer with the increase in porous layer height, becomes clear from the local non-dimensional instantaneous temperature (Fig. 7) and velocity magnitude (Fig. 8) along a plane ( = X L / 0.43) in the configurations with 1 Layer and 5 Layers of packing. From Fig. 7(a) we see that the cold thermal plumes erupting from the top cold wall moves downward (as indicated by the velocity vectors) towards the bottom wall. From the velocity vectors we see that the flow in the porous region is lower when compared to the fluid-only region above the porous layer ( Fig. 7(a)). The temperature at the interface of the fluid-only and porous region (Fig. 7(c)) is close to the cold fluid temperature. The fluid region above this interface thus behaves like a fluid only cavity.    M. Chakkingal, et al. International Journal of Heat and Fluid Flow 84 (2020) 108617 When the cavity is fully packed with porous media the asymmetry in temperature distribution (Fig. 7(b)) observed above, vanishes.The velocity vectors suggest the flow velocity to be much lower than that in the cavity with 1 Layer of packing ( Fig. 7(a)). The contours of nondimensional velocity magnitude confirm the reduction in flow velocity when the cavity is fully packed with porous media (Fig. 8(b)) when compared to the cavity with 1 Layer of packing ( Fig. 8(a)). From Fig. 7 and Fig. 8 we find that the convective contribution to the heat transfer in the cavity decreases when the cavity is fully packed. The fluid-only region in the partially filled cavity dictates the total heat transfer thus resulting in a higher heat transfer (Fig. 6(a)) than in the fully-packed cavities ( Fig. 6(b)).

Pore-structure resolved simulations
As observed in Fig. 5, the heat transfer obtained with pore-structure resolved simulations are comparable to that in fluid-only filled cavities and is higher than in D-F simulations, at high Ra. This is also evident from the local Nusselt number distribution of heat transfer at the bottom hot wall in P-R simulations, obtained in cavities with 1 Layer and 5 Layer of packing (Fig. 9). The local Nusselt number in P-R simulations ( Fig. 9) are higher than in D-F simulations (Fig. 6). Unlike the Nusselt number distribution with D-F simulations, the heat transfer in P-R simulations are high in the regions close to the pore-space and are close to 0 at regions very close to the point of contact of the spherical beads with the bottom wall. To explain this behaviour, we look at the local flow and temperature in the plane = X L / 0.43, where the spherical beads and the pore-space are comparable in size. In Fig. 10 we compare configurations with 1 Layer (Fig. 10(a)) and 5 Layer (Fig. 10(b)) of spherical bead packing, at = × Ra 2.3 10 7 . The instantaneous non-dimensional temperature contours also have the non-dimensional velocity vectors to give an indication of the direction and strength of the inplane (ZY plane) flow. At the top right corner of the temperature contours we have the non-dimensional out-of plane velocity (u * x ) contours. From the temperature contours we observe that the thermal plumes erupting from the isothermal walls are thinner than the pore-space size (discussed later in next section), resulting in a flow-dominated heat transfer. This results in the heat transfer in spherical bead filled cavities to be comparable to that in a fluid-only filled cavity. Unlike the cavity with 1 Layer packing in which the flow is 3-dimensional with high value of out of plane velocity u * x ( Fig. 10(a)), in the cavity with 5 Layer packing ( Fig. 10(b)), we observe a channeled (2D planar) flow with negligible out of plane velocity. The slight increase in heat transfer observed with 4 Layer packing (Fig. 5) could be due to the dominance of the channeled flow in which the thin hot and cold plumes move through the pore-space and directly impinge on the cold and hot walls. However, a slight reduction in heat transfer occurs when the 5 th layer of spherical beads is placed close to the cold wall. Losing the area for the eruption of cold plumes in the cavity with 5 layers of packing results in this reduction, the effect of which is expected to diminish at higher Ra Ataei-Dadavi et al. (2019a); Chakkingal et al. (2019).
To understand the local flow in P-R simulations, we look at the frequency spectra of thermal and kinetic energy at = × Ra 2.3 10 7 (of vertical velocity u z ) at the probe location ( = y L / 0.5, = x L / 0.6, = z L / 0.6) in cavities with 1 Layer, 3 Layer and 5 Layer packing. The probe is far from the spherical beads in cavity with 1Layer packing, close to the beads in 3 Layer packing and within the pore-space in 5Layer packing. We also compare the spectra with that in a fluid-only cavity (same probe location) to understand the influence of spherical beads on local flow and thermal features.
The frequency spectra of thermal and kinetic energy at z comparable to that reported in Zhou and Xia (2001) and Shang and Xia (2001), respectively. From the P-R simulations of the spherical bead filled cavities, we observe that the temporal flow and temperature fluctuations (Fig. 11) dies out with the addition of layers of spherical beads. When we increase the number of the packing layers, the flow is still turbulent and thus the energy spectra follow the turbulent scaling for 1 Layer and 3 Layer packing, while the flow becomes oscillatory in the cavity with 5 Layer packing.    Chakkingal, et al. International Journal of Heat and Fluid Flow 84 (2020) 108617 This indicates that the flow becomes less turbulent as the relative distance of the probe with the spherical beads decreases, because of the increased resistance imposed by the spheres.

Influence of shift of packing
In contrast to the cavities with spherical beads touching the hot wall (Fig. 5), the suppression in heat transfer is comparatively lower when the spherical beads are shifted away from the hot wall (Fig. 12), both in P-R and D-F simulations.

Pore-structure resolved simulations
With P-R simulations, at = × Ra 1.16 10 , 5 for both 1 Layer and 2 Layer cases (Fig. 12), when the packing is away from the wall i.e for 1L/ 2L Quarter/Middle (P-R), the suppression in heat transfer varies significantly from that in cavities with the spherical beads touching the wall 1L/2L Bottom (P-R), for e.g. Nu 1 Layer Middle > Nu 1 Layer Quarter > Nu 1 Layer Bottom . At higher Ra ( = × × Ra 1.6 10 , 2.3 10 6 7 ), the heat transfer in porous media filled cavities with the packing away from the isothermal walls becomes comparable to or even slightly higher than that in fluid-only filled cavities, unlike the cavities in which the spherical beads touch the hot wall (1L Bottom (P-R)). To understand this difference in heat transfer behaviour, we look at the thermal plume (boundary layer) thickness defined as: When the thermal plume size δ th is scaled with the diameter of the porespace d p (Fig. 13), we observe that the thermal plumes are thicker than the pore-space (δ th /d p > 1) at low Ra ( = × Ra 1.16 10 5 ). While with the increase in Ra, thermal plumes become comparable to pore-size ( = × Ra 1.16 10 6 ) and even gets smaller than the pore-size ( = × Ra 2.31 10 7 ). At the lowest Ra ( = × Ra 1.16 10 5 ), as the thermal plumes are thicker than the pore-space, the inclusion of the spherical beads leads to a considerable suppression in heat transfer as the thermal plumes cannot meander through the pore-space. The inclusion of spherical beads thus increases the resistance offered to the thermal plumes and even makes them thicker (Fig. 13).
A comparison of the cavities with 1 Layer of packing (Fig. 12) shows that the suppression in heat transfer is minimum when the porous layer is at the center of the cavity (1L Middle (P-R)). This indicates that the influence of the packing on heat transfer decreases with the increase in its distance from the hot isothermal wall. At = × Ra 1.16 10 , 6 as the thermal plume thickness of the fluid-only cavity is comparable to the pore-space (Fig. 13) and thus the reduction in heat transfer with the inclusion of spherical beads away from the isothermal walls is minimal (Fig. 12). This is also evident from the small change in the δ th /d p ratio in the cavities with spherical beads, when compared to the fluid-only filled cavity (Fig. 13). With the increase in Ra ( = × Ra 2.31 10 7 ), the thermal plumes become much thinner than the pore-space (Fig. 13). The inclusion of layers of packing irrespective of their location, thus does not significantly obstruct the flow as the plumes meander through the pore-space. Thus at high Ra, the heat transfer in cavities with spherical beads are comparable to that in a fluid-only cavity (Fig. 12).

Darcy-Forchheimer simulations
At the lowest = × Ra 1.16 10 , 5 when the porous media touches the bottom hot wall, the heat transfer with with D-F and P-R simulations are comparable, for e.g. Nu 1 Layer Bottom (D-F) ≈ Nu 1 Layer Bottom (P-R) (Fig. 12).
However, the heat transfer obtained with the D-F simulations are lower than the P-R simulations at higher Ra, for e.g. at = × Ra 2.31 10 , 7 Nu 1 Layer Bottom (D-F) < Nu 1 Layer Bottom (P-R). When the porous media is away from the isothermal hot wall, we observe a different behaviour.
• At = × Ra 1.16 10 , 5 Nu with D-F simulations are higher than with the P-R simulations (Fig. 12). This indicates that the flow within the pore-space of the spherical bead packing in P-R simulations is blocked as the thermal plumes are thicker than the pore-space (Fig. 13). While in the D-F simulations of the cavities with porous media not touching the isothermal walls, the convective flow is not fully choked as in P-R simulations, thus resulting in comparatively higher heat transfer (Fig. 12).
• At = × Ra 1.16 10 6 and = × Ra 2.31 10 , 7 the D-F and P-R simulations gives us comparable heat transfer, irrespective of the number of layers (Fig. 12).
We also observe that the resistance to the convective flow imposed by the porous media in D-F simulations is higher when it touches the isothermal wall when compared to the cavity with the porous media located away from it, thus resulting in lower heat transfer (Fig. 12), i.e. Nu 1 Layer Bottom (D-F) < Nu 1 Layer Quarter (D-F).  Chu and Goldstein (1973); Valencia et al. (2005). For naming convection of packing arrangement refer to Table 1. Fig. 13. Ratio of boundary layer thickness to pore-size at different Ra, for porescale resolved simulations. The boundary layer thickness of fluid-only cavity is also scaled with pore-size of the porous media filled-cavity for reference.
To understand the difference in the behaviour of the D-F simulations when the porous media touches the wall and when it is away from the wall, we look at the instantaneous non-dimensional velocity magnitude and temperature distribution along the plane, = X L / 0.43 in cavities with 2 Layer of porous media packing touching the isothermal wall ( Fig. 14(a1), Fig. 15(a1)) and away from it ( Fig. 14(b1), Fig. 15(b1)). We also compare it with the P-R simulations of the respective cavities ( Fig. 14(a2,b2), Fig. 15(a2,b2)).
Since the thermal boundary layer is thinner than the pore-space at = × Ra 2.31 10 7 (Fig. 13), the pore-space flow velocity in the P-R simulation of the cavity with 2 Layer packing touching the bottom wall ( Fig. 14(a2)), is less influenced by the spherical beads, resulting in hot thermal plumes at high velocity meandering through the pore-space ( Fig. 15(a2)). In the cavities with porous media touching the isothermal hot wall, the flow velocity in the porous region obtained with D-F simulation (Fig. 14(a1)) is much lower than with the P-R simulation (Fig. 14(a2)). We also observe that in the simulation with D-F assumption, hot thermal plumes of comparatively lower temperature (Fig. 15(a1)) erupt from the bottom hot wall. Unlike the D-F simulation, the temperature and flow velocity of the hot thermal plumes (erupting from the bottom wall) within the pore-space is higher in the P-R simulation ( Fig. 15(a2)). The combined effect of the thermal plume temperature and flow velocity results in lower convective heat transfer in the simulation with D-F assumption (Fig. 5).
However, in the D-F simulation with porous-media defined away from the wall (Fig. 14(b1)), the flow velocity close to the isothermal walls is less influenced by the porous media and is comparable to that in the P-R simulation (Fig. 14(b2)). Thus, comparable flow velocity close to the isothermal walls results in comparable heat transfer with D-F (2L Quarter (D-F)) and P-R (2L Quarter (P-R)) simulations (Fig. 12).

Summary and Conclusion
A detailed study on the validity and applicability of Darcy-Forchheimer assumption to simulate convective heat transfer in flow dominated porous media filled-cavity is reported. The Darcy-Forchheimer simulations of cavities with porous media touching the isothermal walls and away from them are compared with the porestructure resolved simulations at different Ra. Major findings of this work can be summarized as: (i) In cavities with porous media touching the isothermal wall, the heat transfer in simulations with Darcy-Forchheimer assumption is comparable to that in pore-structure resolved simulations at low Ra. With an increase in Ra the Darcy-Forchheimer simulations under-predict the heat transfer. At high Ra, the temperature and velocity of the thermal plumes impinging the hot wall are higher in the pore-structure resolved simulations resulting in heat transfer comparable to fluid-only cavities. (ii) In cavities with porous media not touching the isothermal wall, Darcy-Forchheimer simulations and pore-structure resolved simulations report comparable heat transfer irrespective of the number M. Chakkingal, et al. International Journal of Heat and Fluid Flow 84 (2020) 108617 of layers of packing and its location, for Ra ≥ 1.16 × 10 6 . However, at low = × Ra 1.16 10 5 D-F simulations over-predict the heat transfer. This shows that the laminar model for porous-media simulations predicts heat transfer comparable to the pore-structure resolved simulations, when the non-porous regions are fully resolved.
(iii) Heat transfer in a porous medium filled-cavity decreases when the porous medium is close to the isothermal walls. A slight enhancement in heat transfer with porous media over the fluid-only cavity occurs when the channeling effect dominates over the effect due to the reduction in the area available for plume impingement when the packings are away from the isothermal walls. (iv) From pore-structure resolved simulations, we observe that when the thermal boundary layer thickness (thermal plume size) is larger than the pore-space (of a pore-structure resolved cavity), the location and the number of layers of packing influence the heat transfer ( = × Ra 1.16 10 5 ) for all packing configurations. However, it is practically independent of the porous packing when the boundary layer thickness is considerably smaller than the pore-size and when the porous media does not touch the wall (( = × Ra 2.31 10 7 ). When the porous media touches the isothermal wall, the heat transfer is however reduced by the adiabatic spherical beads close to the wall. The reduction in heat transfer compared to fluid-only cavity is due to the comparatively lower flow velocity close to the wall.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:  15. Instantaneous non-dimensional temperature distribution and velocity vectors at = x L / 0.43, in a cavity with 2 layers of spherical beads at = × Ra 2.3 10 7 in Darcy-Forchheimer (a1,b1) and pore-scale resolved (a2,b2) simulations. (a1,a2) has porous media from = z L / 0 to 0.4 and (b1, b2) has porous media from = z L / 0.2 to 0.6. M. Chakkingal, et al. International Journal of Heat and Fluid Flow 84 (2020) 108617 To validate the Darcy-Forchheimer solver in ANSYS Fluent 2 , we carry out 2D steady-state simulation of a differentially heated L × L cavity filled with porous media for fluid with = Pr 1 at = Da 10 , = 0.6, and carry out with the results reported in Das et al. (2016). The simulations are carried out in a 80 × 80 hexahedral grids of equal size. The isotherm (Fig. A.16) obtained with different Da and Ra are comparable to that reported in Das et al. (2016). A plot of the temperature along the horizontal line at the mid-height of the cavity (Fig. A.17) shows us that the results obtained with ANSYS Fluent are comparable to that in literature.  = 0.6 using Darcy-Forchheimer assumption. The current results are compared to that reported in Das et al. (2016).