Introduction

Solid–fluid conjugate heat transfer (CHT) is commonly used in heat transfer in engineering devices, such as heat exchangers, combustion chambers, cooling of electronic devices, electronic equipment [1,2,3], the heat transport in microfuel cells and microchannels [4, 5]. In this context, many research topics have been published. In the area of industrial automobile, Ali et al. [6, 7] investigated experimentally the heat transfer inside a car radiator using metal oxide nanofluid. The use of nano-suspensions increases the heat transfer rate comparing with the water base fluid. The increase in the volume concentration enhances the heat transfer. The heat transfer rates are weakly dependent on the inlet nanofluid temperature.

Besides, numerical simulations of CHT problems are performed by many researchers using conventional numerical methods for solving the governing Navier–Stokes and energy equations. Finite volume method (FVM), finite difference method (FDM), and finite element method (FEM) have been successfully applied [8,9,10]. However, different material properties at solid–fluid interface caused some difficulties in simulation of heat transfer [11, 12]. In addition, the implementation of conjugate interface thermal boundary conditions is a challenging problem, particularly for complex geometries.

Moreover, with the growth of the capacity of computer machines, new numerical methods are emerging. This is typically the so-called lattice Boltzmann-type methods known by the acronym LBM or its ancestor lattice gas automata (LGA). Based on the evolution of particle distribution functions of discrete velocities, the LBM method with its four models (BGK, MRT, entropic and regularized LBM) has been developed rapidly in recent years and has received much attention in science and engineering as a powerful computational tool for solving a large class of problems. The LBM has been successfully used to solve a wide range of thermo-fluid problems [13,14,15,16]. LBM is based on mesoscopic scale, which has many advantages over conventional computational fluid dynamics (CFD) solvers.

Lu et al. [17] proposed FD-LBM in order to simulate CHT problems, in which the continuity of heat flux at the interface between two different media was obtained by modifying the streaming process to avoid exchanging information directly between different media because the conventional streaming process cannot keep heat flux continuous at the interface in CHT. The distribution functions at the interface can be obtained by coupling the interface conditions of temperature and heat flux with non-equilibrium extrapolation. Authors tested four cases and showed that simple difference method based on the coupling between two distribution functions has a good performance in simulating solid–fluid or solid–solid CHT problems. A promising difference method was proposed by Mohamad et al. [18]. The continuity of heat flux at the interface was ensured by coupling temperature difference with a new scaling law of energy distribution function. Koca et al. [19] used the Bejans’ heatline technique to analyze the effects of a differentially thickness change conductive partition on natural convection heat transfer in an enclosure. It was found that both heat transfer and flow strength strongly depend on the thermal conductivity ratio and Rayleigh number.

Karani and Huber [20] suggested using a source term to the evolution equation in such a way that temperature and heat flux continuities were satisfied at the interface. In their method, the interface was placed halfway between the boundary lattice nodes.

Benachour et al. [21] made a series of numerical simulations to develop a new correlation using the Lagrange polynomial interpolation method for high Rayleigh numbers. This method is used for predicting exchange coefficient and estimating Nusselt number to optimize the design of walls in buildings. Authors found that the heat transfer decreases with the increase in the wall thickness of the building material of the outer wall. The Nusselt number increases when the Rayleigh number increases.

Khatamifar et al. [22] made a series of numerical simulation for conjugate natural convection flow and heat transfer using the finite volume method in a partitioned differentially heated square cavity. They examined the effects of partition position and thickness on the unsteady conjugate natural convection heat transfer in a differentially heated partitioned enclosure over a wide range of Rayleigh numbers. The presence of the heat-conducting solid has significant effects on heat and flow characteristics in the cavity. In addition, the average Nusselt number is a decreasing function of partition thickness but is an increasing function of the Rayleigh number. Finally, the position of the heat-conducting solid has a negligible influence on the heat transfer rate for the whole range of Rayleigh number considered.

The numerical results of Ho and Yih [23] show that heat transfer in an air-filled partitioned rectangular cavity is considerably lower than that in a non-partitioned cavity. Indeed, the presence of a conducting partition in the cavity has a significant influence on the natural convection heat transfer characteristics across the cavity.

Saeid [24] used the finite volume method to simulate conjugate natural convection in two-dimensional vertical porous layer sandwiched between two equal-thickness walls. The numerical results indicated that the ratio of the wall thickness to its height, the wall to porous thermal conductivity ratio and the Rayleigh number have great effects on the heat transfer and fluid flow characteristics. It was found that as the average Nusselt number decreases the wall thickness increases. The strength of the circulation is much higher with thin walls and with high thermal conductivity ratio. The numerical results indicated that for small values of Rayleigh number, the average Nusselt number is approximately constant.

The earlier research in this area has been conducted by Wang et al. [25]. Authors proposed a CFD model for simulating the CHT at the fluid–solid interface using the LB method. The half-lattice technique was employed to model the interface. The model has given reasonable results for steady-state heat conduction problems.

A finite difference approach has been proposed by Seddiq et al. [26] to deal with the CHT problems, in which the continuous heat flux at the interface between two different media can be well guaranteed. However, this method is confined to solids with a square shape at steady state.

Souayeh et al. [27] investigated numerically the steady laminar natural convection in a water-filled 2D enclosure containing a rectangular conducting body. To study the impact on the flow structure and heat transfer characteristics, several parameters are highlighted such as Rayleigh number, solid–fluid thermal conductivity ratio and different locations of a conducting body. It was found that the location of the rectangular conducting body affects only the flow structure and has no effect on temperature field. By increasing the Rayleigh number, the gradient of thermal boundary layer becomes larger. As a consequence, the heat transfer rate increases. A comparative study was made between the two cases, vertical and horizontal location of the conducting bloc. It is found that the mean Nusselt number is higher for vertically positioned case.

Ben Nakhi and Chamkha [28] made a numerical investigation in which they studied the influence of length and inclination angle of heated thin fin attached in the hot wall of a differentially heated cavity. The FVM is used to obtain the numerical solution. They found that the considered parameters have a significant effect on the heat transfer. Besides, the heat transfers decrease when the length of the hot fin increases. Depending on the thin fin length, the behavior of the average Nusselt number changes when the inclination angle increases. In addition, the increase in Rayleigh numbers produces a significant augmentation in heat transfer. In fact, as well as the Rayleigh number, it is possible to enhance or reduce wall heat transfer by proper selection of both fin inclination angle and length. In another study, Ben Nakhi and Chamkha [29] studied numerically the CHT in a square cavity which has three thick-cooled thick walls and one thin-heated vertical wall with a heated inclined thin fin attached to its middle. They found that the thin fin inclination angle, length and the solid–fluid thermal conductivity ratio have a significant influence on the heat transfer rate. Besides, the control of the behavior of heat transfer through an enclosure is done by means of the selected parameters.

Ismael and Chamkha [30] investigate numerically the conjugate natural convection in a differentially square composite vertically layered cavity using the FDM. The cavity is subdivided from the left: a solid wall, a porous layer, and a nanofluid layer. It was found that, at low value of the porous medium permeability and when the porous layer thickness is greater than 0.5, the natural convection heat transfer was enhanced when the Rayleigh number is less than or equal to 104. The solid wall type is found to play a considerable role in the flow and heat transfer fields. It is also found that the conduction heat transfer within the solid wall is affected by the permeability of the porous layer.

Chamkha et al. [31] conducted a numerical study of mixed convection heat transfer in a heated square solid cylinder located at the center of a vented cavity filled with air. The FDM is used to solve the considering equations. They found that the heat transfer increases linearly with the Richardson number for all considered configurations. As the Richardson and Reynolds numbers increase, the heat transfer along the hot wall increases while the surface temperature decreases for the same parameters for all configurations. The authors found that, in terms of heat transfer, the performance of the CT configuration is more pronounced than the CC and CB configurations. The position of the inner square cylinder has an important role in the streamline and isotherm contour patterns.

Chamkha [32] performed a numerical study of CHT in porous cavity heated by triangular thick wall using the second-order central differences finite difference scheme with the successive over relaxation (SOR) method. It was found that the heat transfer and the circulation strength are the increasing functions with Rayleigh numbers. The increase in the solid wall thickness causes at the same time the augmentation of thermal resistance and increases the contact interface. When the thermal conductivity ratio is less than 0.1, the effect of Ra and D become more pronounced.

Chamkha and Ismael [33] made a numerical investigation of natural convection in a differentially heated and partially cavity filled with nanofluid. The obtained results show that the heat transfer is an increasing function of Ra at lower porous layer thicknesses. The use of nanofluid enhances the convective heat transfer, precisely at low permeability porous layer. It was found that the addition of the nanofluid accelerates the flow throughout the cavity. The convective heat transfer presents a maximum value corresponding to a critical porous layer thickness for a considered effective thermal conductivity. The aspect ratio A = 2 corresponds to the best value of convective heat transfer.

Ismael et al. [34] considered the case of Chamkha et al. [32], but they focus on the irreversibility of heat transfer and fluid friction process. The authors conclude that the considered nanofluid increases the global entropy generation. The global entropy generation rate is an increase function of conductivity solid wall. For low Ra, the rate of entropy generation increases with solid wall thickness for highly conductive wall. On the other hand, for high Ra, the maximum of heat transfer corresponds to a critical value of D = 0.7 for conductive solid wall less than 1 and the maximum irreversibility appears at D = 0.5.

The objective of the present contribution is to simulate laminar natural convection in a two-dimensional square cavity partitioned with a conducting solid.

The remainder of this paper is organized as follows: The section two is devoted to the numerical modeling where the mathematical formulation, boundary conditions and the LBM basis are presented. In the third section, the independence test and the code validation are performed. The fourth section is concerned with the results and discussion where the effect of the partition wall, the effect of wall thickness, the effects of partition position and Rayleigh number, the effect of thermal conductivity ratio on the dynamic and thermal structures and heat transfer rate measured using the Nusselt number are appraised. Concluding remarks are finally drawn in “Conclusions” section.

Numerical modeling

Problem statement

The geometry of the problem is shown in Fig. 1. It displays a two-dimensional partitioned square enclosure. In this configuration, the horizontal walls are adiabatic, while the vertical ones are maintained at constant but different temperatures so that the cavity is heated from the right vertical wall and cooled from the opposite left vertical one. A heat-conducting solid partition is placed in the cavity. All the walls are rigid, and no-slip BC is considered. The partition walls are heat conducting, with finite thermal conductivity ks [22]. The fluid flow is assumed to be laminar and incompressible, and the Boussinesq approximation is adopted for the variations of the density in the buoyancy term.

Fig. 1
figure 1

Physical domain and coordinate system

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Mathematical formulation and boundary conditions

The equations governing the dynamic and thermal fields are the Navier–Stokes coupled with the energy equations, given as follows:

Continuity equation:

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$$
(1)

Momentum equations:

$$\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = - \frac{1}{\rho }\frac{{\partial p^{*} }}{\partial x} + \upsilon \left( {\frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{\partial^{2} u}}{{\partial y^{2} }}} \right)$$
(2)
$$\frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} = - \frac{1}{\rho }\frac{{\partial p^{*} }}{\partial y} + \upsilon \left( {\frac{{\partial^{2} v}}{{\partial x^{2} }} + \frac{{\partial^{2} v}}{{\partial y^{2} }}} \right) + g\beta \left( {T_{\text{f}} - T_{\text{c}} } \right)$$
(3)

Energy equation for the fluid:

$$\frac{{\partial T_{\text{f}} }}{\partial t} + u\frac{{\partial T_{\text{f}} }}{\partial x} + v\frac{{\partial T_{\text{f}} }}{\partial y} = \alpha \left( {\frac{{\partial^{2} T_{\text{f}} }}{{\partial x^{2} }} + \frac{{\partial^{2} T_{\text{f}} }}{{\partial y^{2} }}} \right)$$
(4)

Energy equation for the solid:

$$\frac{{\partial^{2} T_{\text{s}} }}{{\partial x^{2} }} + \frac{{\partial^{2} T_{\text{s}} }}{{\partial y^{2} }} = 0$$
(5)

where p* = p + ρgy and p is the relative pressure.


Where Tc is the temperature of the right vertical wall of the cavity.

The following non-dimensional variables are introduced to proceed with the numerical solution of the above-mentioned equations:

$$X = \frac{x}{W},\quad Y = \frac{y}{W},\quad U = \frac{uW}{\alpha },\quad V = \frac{vW}{\alpha },p = \frac{{p^{*} W^{2} }}{{\rho \alpha^{2} }},\quad \theta = \frac{{T - T_{\text{c}} }}{{T_{{\text{h}}} - T_{{\text{c}}} }},\quad {\text{Ra}} = \frac{{g\beta \left( {T_{\text{h}} - T_{{\text{c}}} } \right)W^{3} }}{\alpha \upsilon },\quad \tau = \frac{t\alpha }{{W^{2} }},\quad \Pr = \frac{\upsilon }{\alpha }$$

The non-dimensionalized continuity, momentum and energy equations can be expressed as:

$$\frac{\partial U}{\partial X} + \frac{\partial V}{\partial Y} = 0$$
(6)
$$\frac{\partial U}{\partial \tau } + U\frac{\partial U}{\partial X} + V\frac{\partial U}{\partial Y} = - \frac{\partial P}{\partial X} + \Pr \left( {\frac{{\partial^{2} U}}{{\partial X^{2} }} + \frac{{\partial^{2} U}}{{\partial Y^{2} }}} \right)$$
(7)
$$\frac{\partial V}{\partial \tau } + U\frac{\partial V}{\partial X} + V\frac{\partial V}{\partial Y} = - \frac{\partial P}{\partial Y} + \Pr \left( {\frac{{\partial^{2} V}}{{\partial X^{2} }} + \frac{{\partial^{2} V}}{{\partial Y^{2} }}} \right) + RaPr\theta$$
(8)
$$\frac{{\partial \theta_{\text{f}} }}{\partial \tau } + U\frac{{\partial \theta_{\text{f}} }}{\partial X} + V\frac{{\partial \theta_{\text{f}} }}{\partial Y} = \frac{{\partial^{2} \theta_{\text{f}} }}{{\partial X^{2} }} + \frac{{\partial^{2} \theta_{\text{f}} }}{{\partial Y^{2} }}$$
(9)
$$\frac{{\partial^{2} \theta_{\text{s}} }}{{\partial X^{2} }} + \frac{{\partial^{2} \theta_{\text{s}} }}{{\partial Y^{2} }} = 0$$
(10)

The non-dimensional boundary conditions are defined as follows:

$$\left\{ {\begin{array}{*{20}l} {U = V = 0\quad {\text{at}}\,X = 0,W\quad {\text{and}}\quad Y = 0,W} \hfill \\ {\theta = 1\quad {\text{at}}\,X = 0\quad {\text{and}}\quad \theta = 0\,{\text{at}}\,X = W} \hfill \\ {\frac{\partial \theta }{\partial Y} = 0\quad {\text{at}}\,Y = 0,W} \hfill \\ {\left( {\frac{\partial \theta }{\partial X}} \right)_{\text{fluid}} = k_{\text{r}} \left( {\frac{\partial \theta }{\partial X}} \right)_{\text{solid}} \quad {\text{at}}\,X = X_{1} ,\,X_{2} } \hfill \\ \end{array} } \right.$$
(11)

where the thermal conductivity ratio is defined as kr = ks/kf.

The local Nusselt number evaluating the heat transfer rate at the active vertical walls is defined as follows:

$$Nu_{\text{local}} = \frac{\partial \theta }{\partial X}$$
(12)

The average Nusselt number \(\overline{Nu}\) is defined as follows:

$$\overline{Nu} \, = \int\limits_{0}^{1} {Nu_{\text{local}} {\text{ d}}Y} = \sum\limits_{0}^{n} {Nu_{{\text{i}}} }$$
(13)

Lattice Boltzmann method basis

The lattice Boltzmann equation (LBE) was originated from Ludwig Boltzmann’s kinetic theory of gases. It explains and predicts how the properties of atoms and molecules (microscopic properties) determine the phenomenological (macroscopic) properties of matter. The distribution function (probability of finding particles within a certain range of velocities at a certain range of locations at a given time) replaces tagging each particle, as in molecular dynamic simulations.

Flow and temperature fields

The distribution functions are calculated by solving the LBE, which is a special discretization of the kinetic Boltzmann equation:

$$\frac{{\partial f_{\text{k}} }}{\partial t} + c_{\text{k}} \nabla f_{\text{k}} = \frac{\Delta t}{{\tau_{\upsilon } }}\left( {f_{\text{k}}^{\text{eq}} - f_{\text{k}} } \right)$$
(14)

After introducing Bhatnagar–Gross–Krook (BGK) [35] approximation, the Boltzmann equation was reformulated consisting of two steps: collide and stream. The collision step is defined as follows:

$$f_{\text{k}} \left( {x + c_{\text{k}} \Delta t,t + \Delta t} \right) = f_{\text{k}} \left( {x,t} \right) + \frac{\Delta t}{{\tau_{\upsilon } }}\left[ {f_{\text{k}}^{\text{eq}} \left( {x,t} \right) - f_{\text{k}} \left( {x,t} \right)} \right]$$
(15)

The streaming step represents the advection of the fluid particles. This can be written as:

$$f_{\text{k}} \left( {x + c_{\text{k}} \Delta t,t + \Delta t} \right) = f_{\text{k}} \left( {x,t + \Delta t} \right)$$
(16)

The exchange of momentum and energy is achieved through particles collision and streaming. For non-isothermal problems, the LBM utilizes generally two distribution functions f and g, for the flow and temperature fields, respectively. In this study, the temperature field is simulated by D2Q9 new distribution function gi. The corresponding LBE without source/sink term is:

$$g_{\text{k}} \left( {x + c_{\text{k}} \Delta t,t + \Delta t} \right) = g_{\text{k}} \left( {x,t} \right) + \frac{\Delta t}{{\tau_{\alpha } }}\left[ {g_{\text{k}}^{\text{eq}} \left( {x,t} \right) - g_{\text{k}} \left( {x,t} \right)} \right]$$
(17)

The present study examines two-dimensional problem by square lattice with nine velocities (D2Q9 model). The velocity vectors of the D2Q9 model are shown in Fig. 2. For the D2Q9 model, the discrete velocities set ck are written as:

$$c_{\text{k}} = \left( {\begin{array}{*{20}c} {c_{\text{kx}} } \\ {c_{\text{ky}} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} 0 & 1 & 0 & { - 1} & 0 & 1 & { - 1} & { - 1} & 1 \\ 0 & 0 & 1 & 0 & { - 1} & 1 & 1 & { - 1} & { - 1} \\ \end{array} } \right)$$
(18)
Fig. 2
figure 2

Flow domain discretization using the D2Q9 model (left) and the nine-speed square lattice of the D2Q9 model (right)

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The distribution functions for the next time step are then computed with the post-streaming distribution functions and the equilibrium distribution functions. For the D2Q9 model, feq and geq are given by the following:

$$f_{\text{k}}^{\text{eq}} = \rho \omega_{\text{k}} \left[ {1 + 3\frac{{c_{\text{k}} .u_{\text{i}} }}{{c^{2} }} + \frac{{9\left( {c_{\text{k}} .u_{\text{i}} } \right)^{2} }}{{2c^{4} }} - \frac{{3u_{\text{i}}^{2} }}{{2c^{2} }}} \right]$$
(19)
$$g_{\text{k}}^{\text{eq}} = \theta \omega_{\text{k}} \left[ {1 + 3\frac{{c_{\text{k}} .u_{\text{i}} }}{{c^{2} }}} \right]$$
(20)

For simulation of heat transfer in the solid part, equilibrium distribution function is as follows:

$$g_{\text{k}}^{\text{eq}} = \theta \omega_{\text{k}}$$
(21)

where the values of ωk are as follows:

$$\omega_{0} = 4/9,\quad \omega_{{{\text{k}} = 1 - 4}} = 1/9,\quad \omega_{{{\text{k}} = 5 - 8}} = 1/36$$
(22)

The LB method is nowadays a powerful numerical technique, based on kinetic theory, for simulating fluid flow and heat transfer [36,37,38,39,40], and has many advantages in comparison with conventional CFD methods mentioned previously.

The dynamic and thermal macroscopic quantities are computed as:

$$\rho (x,t) = \sum\limits_{{{\text{k}} = 0}}^{8} {f_{\text{k}} }$$
(23)
$$\rho (x,t)u_{\text{i}} (x,t) = \sum\limits_{{{\text{k}} = 0}}^{8} {c_{\text{k}} f_{\text{k}} }$$
(24)
$$\theta (x,t) = \sum\limits_{{{\text{k}} = 0}}^{8} {g_{\text{k}} }$$
(25)

where the kinematic viscosity υ and the thermal diffusivity α are related to the flow and heat relaxation times as follows:

$$\upsilon = \left[ {\tau_{\upsilon } - \frac{1}{2}} \right]c_{\text{s}}^{2} \Delta t\quad \alpha = \left[ {\tau_{\alpha } - \frac{1}{2}} \right]c_{\text{s}}^{2} \Delta t$$
(26)

where cs is the lattice speed of sound and equals \(c_{\text{s}}^{2} = c/3\).

LBM boundary conditions treatment for fluid flow

One of the important and crucial issues in LBM simulation of flow and temperature is accurate implementation of the boundary conditions. Therefore, we need to determine appropriate equations for calculating those distribution functions f and g, for the flow and temperature at the boundaries for a given physical condition. The domain is divided into (n + 1) × (m + 1) nodes with n = m, so n2 square cells and will be confound in the following the node (xi, yi) and (i, j).

The incoming unknown distribution functions pointing to the fluid zone at boundary nodes must be specified. For the no-slip boundary condition, bounce-back boundary condition is applied on all solid boundaries. This means that incoming distribution functions are equal to outgoing ones after the collision [41]. For instance, for the east boundary, the following conditions are imposed:

$$f_{6} \left( {n,j} \right) = f_{8} \left( {n,j} \right),\quad f_{7} \left( {n,j} \right) = f_{5} \left( {n,j} \right),\quad f_{3} \left( {n,j} \right) = f_{1} \left( {n,j} \right)$$
(27)

LBM boundaries treatment of temperature field

Bounce-back boundary condition (adiabatic) is used on the north and the south boundaries. For instance, for the north boundary, the following conditions are imposed.

$$g_{7} \left( {i,m} \right) = g_{7} \left( {i,m - 1} \right),\quad g_{8} \left( {i,m} \right) = g_{8} \left( {i,m - 1} \right),\quad g_{8} \left( {i,m} \right) = g_{8} \left( {i,m - 1} \right)$$
(28)

The temperature at east and west walls is known, θ = 1 and θ = 0, respectively. Since we are using D2Q9, the unknowns for the west wall are g1, g5, g8 which are evaluated as:

$$\left\{ {\begin{array}{*{20}l} {g_{1} \left( {0,j} \right) = \theta_{h} \left( {\omega \left( 1 \right) + \omega \left( 3 \right)} \right) - g_{3} \left( {0,j} \right)} \hfill \\ {g_{5} \left( {0,j} \right) = \theta_{h} \left( {\omega \left( 5 \right) + \omega \left( 7 \right)} \right) - g_{7} \left( {0,j} \right)} \hfill \\ {g_{8} \left( {0,j} \right) = \theta_{h} \left( {\omega \left( 8 \right) + \omega \left( 6 \right)} \right) - g_{6} \left( {0,j} \right)} \hfill \\ \end{array} } \right.$$
(29)

The unknowns for the east wall are g3, g6, g7 which are evaluated as:

$$g_{3} \left( {n,j} \right) = - g_{1} \left( {n,j} \right),\quad g_{6} \left( {n,j} \right) = - g_{8} \left( {n,j} \right),\quad g_{7} \left( {n,j} \right) = - g_{5} \left( {n,j} \right)$$
(30)

LBM conjugate heat transfer treatment

In the present model, the temperature and the heat flux at the interface can be kept continuous naturally. Solving the last line of Eq. (11) for the temperature at the interface leads to:


At X = X1

$$\theta \left( {i,j} \right) = \frac{{\theta_{\text{f}} \left( {i - 1,j} \right) + k_{\text{r}} \times \theta_{\text{s}} \left( {i + 1,j} \right)}}{{1 + k_{\text{r}} }}$$
(31)

At X = X2

$$\theta \left( {i,j} \right) = \frac{{k_{\text{r}} \times \theta_{\text{s}} \left( {i - 1,j} \right) + \theta_{\text{f}} \left( {i + 1,j} \right)}}{{1 + k_{\text{r}} }}$$
(32)

Grid independence test and code validation

Enclosure without partition

The results presented in Table 1 show the maximum stream function magnitude and the average Nusselt number \(\overline{Nu}\) variations with different grid sizes for an enclosure without partition wall. For Ra = 105, we notice the convergence of the two physical quantities by increasing the mesh resolution. For the grid mesh 200 × 200 and 250 × 250, the change of the Nusselt number holds in the third and the change of the |ψ|max does not exceed 0.2%. We adopt 200 × 200 number of grid as sufficient to obtain the accurate results for the present problem for Ra ≤ 105.

Table 1 Average Nusselt numbers obtained with different grid meshes for Ra = 105
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The choice of this grid is confirmed by the horizontal velocity at the mid-width, the vertical velocity at the mid-height of the cavity and the temperature profile at the mid-width. To test the grid independence, numerical simulations were performed as shown in Fig. 3 for Ra = 105 using different numbers of grid. We notice that the horizontal velocity, the vertical velocity and the temperature profile are similar for fairly large meshes.

Fig. 3
figure 3

Horizontal velocity profile U(0.5, Y) (a), vertical velocity profile V(X, 0.5) (b), and temperature profile θ(X, 0.5) for different grid meshes

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In the following, we will adopt the grid resolutions 1002 for Ra = 103 and Ra = 104, 2002 for Ra = 105 and 2502 for Ra = 106. Table 2 shows a comparison of the average Nusselt numbers obtained with the present code and those available in the literature for a differentially heated cavity at different Rayleigh numbers. From this table, we noticed that the obtained results show a good agreement with the last findings. The deviation of the mean Nusselt number according to numerical studies does not exceed 1.35% for all Rayleigh numbers. The maximum deviation according to the experimental study carried out by Baϊri [42] is 6.59%.

Table 2 Comparison of the present and literature results at different Rayleigh numbers
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Enclosure with partition

A second test case of our model for the differentially heated cavity with CHT was performed in comparison with the present FE COMSOL results [52] for kr = 6.22 and Ra = 105. The temperature profiles and the local Nusselt number of the hot wall are depicted in Fig. 4. The results show an excellent agreement. The present and the COMSOL Nusselt numbers are 2.139 and 2.1142, respectively, thus a deviation close to 1.173%.

Fig. 4
figure 4

Temperature profile θ(X, 0.5) (a) and the local Nusselt number (b) compared with those obtained by COMSOL

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In order to test the effect of the grid mesh, we used the vertical velocity profile at the mid-height of the cavity for different numbers of grid (Fig. 5). This figure shows us that the velocity is the same even for fairly large meshes.

Fig. 5
figure 5

Vertical velocity profile V(X, 0.5) at different grid meshes for conjugate heat transfer

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The two used test cases with and without partition show the high correctness and credibility of the present code. The emerging results were compared with well-chosen high-resolution schemes of the FV, FE, FD and LB methods as well as experimental measurements, and an excellent agreement was obtained. The present code and grid resolution will be used in the following to analyze and discuss the convective flow and heat transfer inside partitioned square cavity.

Results and discussion

Numerical simulations are carried out for natural convection CHT for partitioned cavity. As mentioned above, the governing parameters are the thickness of the conducting body, Rayleigh numbers and thermal conductivity ratio. The results are shown in terms of streamlines, isotherms, the average and local Nusselt numbers. The air is chosen as a working fluid having Pr = 0.71.

Effect of the partition wall on heat transfer rate

The aim of this part is to compare the heat transfer in the two cases with and without conducting wall for Rayleigh numbers (103 ≤ Ra ≤ 106). Figure 6 shows that the conducting partition has a significant influence on the heat transfer intensity at the active wall. The heat transfer rate is considerably reduced in the case of partitioned cavity. By analyzing the figure, it is noticed that the local Nusselt number decreases at the active wall for all Rayleigh numbers.

Fig. 6
figure 6

Variation of the local Nusselt number in cases with and without partition for kr = 6.22, δ = 0.1

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The impact of the partition wall on the rate of heat transfer for different Rayleigh numbers is presented in Fig. 7. The figure shows that the average Nusselt number of the cavity with CHT is lower than that without partition wall. The Rayleigh number has a great influence on the augmentation of heat transfer in the cavity. Heat exchange in the cavity increases with the increase in Rayleigh number due to the augmentation of the heat intensity. By introducing the conducting partition wall, we observe that the Nusselt number value undergoes a substantial reduction. In the case with partition, the block plays a damping rule countering the convective currents inside the cavity by altering the dynamic behavior and enhancing the confinement and buoyancy stability.

Fig. 7
figure 7

Variation of the average Nusselt number at different Rayleigh numbers for a cavity with and without conducting wall for kr = 6.22, δ = 0.1

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Effect of wall thickness

The wall thickness effect on the flow structure and isotherms is depicted in Fig. 8 for kr = 6.22 and Ra = 105. It is observed that the thickness of partition does not have an important effect on the streamlines structure. As seen from the figure, two circulation cells are formed in each part of the enclosure and the flow structure are generally centro-symmetric. By increasing the thickness of the conducting body, the size of the formed circulation cells decreases and the streamlines become very tight. Both formed circulations rotate in a clockwise direction. As the wall thickness increases, the strength of the circulation within the cavity decreases and the absolute value of ψ decreases by increasing the wall thickness. As an example, |ψ|max = 6.264, 3.022 and 0. 0013 for, respectively, a wall thickness δ = 0.1, 0.4 and 0.95.

Fig. 8
figure 8

Streamlines at different partition thicknesses for Ra = 105, kr = 6.22 and Xs = 0.5

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For a wall thickness tending to the unity, the pressure inside each part of the enclosure increases because the distance between the active wall and the partition decreases. The confinement of the flow induces a strong dynamic and thermal coupling between the different boundaries of the enclosure.

Figure 9 shows the isotherms at different values of conducting body thickness. It can be seen clearly from that, by increasing the thickness value δ of the partition wall, the isotherms become parallel to both sides of the conducting body and active walls. For δ ≤ 0.5, the temperature field undergoes a deviation in the solid–fluid interface. For δ > 0.5, the isotherms become strictly parallel in the entire of the cavity. Obviously, as the wall thickness increases, the heat transfer mechanism changes from convection-dominant regime to conduction-dominant regime. By increasing the wall thickness δ, the intensity the temperature gradient becomes low and the convective motion is more dumped. This is more expressed at mid-height: the horizontal temperature gradient is non-constant at low δ (nonlinear variation); however, it is visibly constant at high values of δ, leading to a linear variation of X-coordinates.

Fig. 9
figure 9

Isotherms at different values of conducting body thickness for Ra = 105, kr = 6.22 and Xs = 0.5

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Figure 10 shows the temperature profiles θ(X, 0.5) presenting the transient evolution of the heat transfer in the domain. The temperature profiles present a centro-symmetric behavior (X = 0.5, Y = 0.5). By increasing the thickness δ, the temperature field follows a quasi-linear variation at mid-height and for high thickness values. This behavior is more expressed at low δ values. The temperature profile follows a linear variation when δ rises toward 1. Besides, the temperature profile decreases linearly in the wall and in each part of the cavity for δ ≥ 0.45.

Fig. 10
figure 10

Temperature profile θ(X, 0.5) at different values of conducting body thickness for Ra = 105, kr = 6.22 and Xs = 0.5

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Figure 11 depicts the variation of the local Nusselt number at different partition thicknesses δ. It is shown that the local Nusselt number Nu(y) on the active walls decreases with increasing vertical coordinate and with increasing partition thickness until a critical value is near 0.5. This is attributed to the partition dumping effect of the convective current from the hot to the cold wall. Since δ ≈ 0.5, as the wall thickness increases, the variation of the local Nusselt number becomes constant. The thermal communication is controlled by the conduction-dominant mode between the hot and cold walls which becomes more enhanced. This will increase the degree of heating and cooling on both walls.

Fig. 11
figure 11

Variation of the local Nusselt number at different partition thicknesses for Ra = 105, kr = 6.22 and Xs = 0.5

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Figure 12 shows the variation of the LB average Nusselt number at different wall thicknesses compared with the FE results obtained by COMSOL. A good agreement can be observed for the values obtained by the two approaches. A change in the isotherms (in temperature gradients) is noted. Such an effect has an impact on the average Nusselt number, which tends to decrease with the increase in the thickness of the wall. The partition exerts a lower driving force on each part of the enclosure and thus reduces the heat transfer for δ ≤ 0.45.

Fig. 12
figure 12

Average Nusselt number versus partition thickness for Ra = 105, kr = 6.22 and Xs = 0.5

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By increasing the thickness of the wall (δ ≥ 0.45), the distance between the partition and the active walls decreases to favor the confinement, which leads to an increased heat exchange with the partition. In addition, we observe an increase in the mean Nusselt number for δ ≥ 0.5. The convective currents become much accelerated. In addition, the rotational movement of the fluid adjacent to the hot and cold active walls improves the heat transfer through the enclosure.

Figure 13 shows the flow structure at different positions of the wall partition from the cavity for different Rayleigh numbers. It may be observed that by decreasing the distance between the partition and the hot wall at Ra = 103 and 104 a convective recirculation is formed in the big zone of the cavity. By increasing more the Ra number, we notice the formation of a convective small recirculation in the small zone of the cavity. This behavior is observed for Xs = 0.25 and Xs = 0.75. For Ra = 106 (Xs = 0.25 and 0.75), we notice the development of the flow and the formation of two convective recirculation in the big zone of the cavity. The flow becomes unstable.

Fig. 13
figure 13

Streamlines at different positions of conducting body: (left) Xs = 0.25, (middle) Xs = 0.5, (right) Xs = 0.75, for different Ra numbers, kr = 6.22 and δ = 0.1

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Effect of partition position and Rayleigh number

For Xs = 0.5, a formation of two symmetrical circular structures is marked. The strength of these recirculations is weakened by increasing the Ra number. For instance at Xs = 0.25, ψmin = − 0.545, − 3.397 and − 8. 071 for, respectively, Ra = 103, 104 and 105.

Figure 14 shows a succession of the isotherms contours at different positions of the conducting body for different Rayleigh numbers. The isotherms confirm the behavior of the flow structure. The isotherms are parallel near the smaller zone leading to a decrease in temperature gradient in this zone. With increasing Ra number, the heat transfer mechanism varies from conduction-dominant regime to convection-dominant regime.

Fig. 14
figure 14

Isotherms at different positions of partition: (left) Xs = 0.25, (middle) Xs = 0.5, (right) Xs = 0.75, for different Ra numbers, kr = 6.22 and δ = 0.1

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By analyzing the temperature profile at different positions of the wall partition and different Ra numbers (Fig. 15), the temperature decreases in the small zone of the cavity for Xs = 0.25 and Xs = 0.75 according to an affine line with a slope greater than that of the temperature in the solid conductor. For Xs = 0.25, the temperature presents a symmetrical character at mid-height and its profile is centro-symmetric of center (X = 0.5, Y = 0.5). At low Rayleigh numbers, the temperature decreases linearly. By increasing more the Rayleigh number, the temperature profile keeps the symmetrical character; however, it decreases linearly.

Fig. 15
figure 15

Temperature profile at different positions of partition: (left) Xs = 0.25, (middle) Xs = 0.5, (right) Xs = 0.75, for different Ra numbers, kr = 6.22 and δ = 0.1

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In Fig. 16, the local Nusselt number takes higher values in the leading edge of the hot surface (X = 0) and decreases by increasing the vertical coordinate Y. For Xs = 0.25, the local Nusselt number follows a symmetric behavior; this is explained by the symmetrical character of the convective flow. For Xs = 0.25, the heat transfer is more important at the cold wall than at the hot wall; indeed, the local Nusselt number at the cold wall is greater than that at the hot wall. At Ra = 106, the local Nusselt number is more important at the hot wall, this is due to the interaction between the two structures, and this last increases the local heat transfer. For Xs = 0.75, the local Nusselt number becomes more important at the hot wall execpted Ra = 106 the local Nusselt number becomes greater at the cold wall. This behavior is due to the unsteadiness. In addition, with decreasing partition position, the local Nusselt number increases. For instance, for Xs = 0.75 at Y = 0, the Nuavg is close to 11; the same is for Xs = 0.75. Besides, the Nusselt number increases with increasing Rayleigh number.

Fig. 16
figure 16

Local Nusselt number variation at hot and cold walls for different Rayleigh numbers and different positions for kr = 6.22 and δ = 0.1

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Figure 17 presents the variation of the average Nusselt number at different positions of the conducting solid for different Rayleigh numbers. The average Nusselt number increases with the partition. This is noticed for Ra = 104 and Ra = 105. For Ra = 103 and Ra = 106, the average Nusselt number is the same. Besides, the average Nusselt number increases with increasing Rayleigh number. The average Nusselt numbers at Xs = 0.25 and Xs = 0.75 are the same due to symmetry.

Fig. 17
figure 17

Variation of the average Nusselt number as a function of Rayleigh number at different positions of the wall for kr = 6.22 and δ = 0.1

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The effect of partition position on the average Nusselt number is shown in Fig. 18 for different values of Xs for Ra = 105 and Ra = 106. For Ra = 105, the heat transfer rate decreases with the partition position until a critical value close to 0.325 and rises slightly until Xs = 0.5. The critical position value decreases with the increase in Ra number and it is close to 0.2 for Ra = 106 where Nu = 3.766. For Xs ≥ 0.325, the average Nusselt number remains almost constant, indicating a negligible effect. This observed effect of the partition position was also reported by Khatamifar et al. [22] for 0.25 ≤ Xs ≤ 0.75 and Kahveci [53] where it has been found that the effect of the body position on the average Nusselt number is very weak.

Fig. 18
figure 18

Variation of the average Nusselt number as a function of the wall position for kr = 6.22 and δ = 0.1

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Effect of thermal conductivity ratio

Figure 19 shows the flow structure at different thermal conductivity ratios and different Rayleigh numbers. The variation of thermal conductivity has not an effect on the flow structure. The flow is unicellular on either sides of the partition, and diagonal symmetric streamlines are almost identical. By observing the values of the stream function, the flow keeps its symmetrical behavior.

Fig. 19
figure 19

Streamlines at different values of thermal conductivity and different Rayleigh numbers with Xs = 0.5 and δ = 0.1

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The isotherms at different thermal conductivity ratios are presented in Fig. 20. The contours of temperature are parallel at low values of kr and Ra. The heat transfer across the air cavities is evidently conductive dominated, as indicated by the uniformly distributed isotherms. By increasing the thermal conductivity ratio, the isotherms become vertical in the partition region. The existence of plateaus in the core regions may be observed. The isotherms in the cavity with CHT appear to be increasingly deformed and crowded into the vicinity of the solid wall.

Fig. 20
figure 20

Isotherms at different values of thermal conductivity and different Rayleigh numbers with Xs = 0.5 and δ = 0.1

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The average heat transfer rate across the cavity with CHT is the primary quantity of practical interest in this study. Figure 21 presents the variation of the average Nusselt number with the thermal conductivity ratio at different Rayleigh numbers. It can be found that for low Rayleigh numbers the heat transfer rate appears to be slightly affected by the increase in thermal conductivity ratio. For Ra = 103 and Ra = 104, the heat transfer rate is less affected by varying the thermal conductivity ratio. For Ra = 105 and Ra = 106, the average Nusselt number increases for kr < 6.22 and converges to constant value for high kr values.

Fig. 21
figure 21

Variation of the average Nusselt number at different values of thermal conductivity ratio for different Rayleigh numbers for Xs = 0.5 and δ = 0.1

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Heat transfer correlation

Several correlations are cited in the literature for the average Nusselt numbers for the two following cases:

  • Non-partitioned enclosure Karayiannis et al. [54] give a correlations as follows:

    $$\overline{Nu} = 0.155Ra^{0.29} \quad {\text{for}}\quad A_{\text{r}} = \frac{H}{W} = 1$$
    (33)
    $$\overline{Nu} = 0.247A_{\text{r}}^{ - 0.20} Ra^{0.26} \quad {\text{for}}\quad 2 \le A_{\text{r}} = \frac{H}{W} \le 10$$
    (34)

The present study is concerned with the case Ar = 1.

  • Cavity with conjugate heat transfer For this case, many correlations are proposed. For instance, the correlation proposed by Duxbury [55]:

    $$\overline{Nu} = 0.339A_{\text{r}}^{ - 0.25} Ra^{0.25} \left( {N + 1} \right)^{ - 1.25}$$
    (35)

    where N is the number of partitions inside the cavity.

Nishimura et al. [56] proposed a similar correlation, but he assumed a thin partition in the cavity.

$$\overline{Nu} = 0.29A_{\text{r}}^{ - 0.25} Ra^{0.25} \left( {N + 1} \right)^{ - 1} \quad {\text{For}}\quad A_{\text{r}} = \frac{H}{W} \ge 1$$
(36)

More recently, Souayeh et al. [27] developed the following correlation which incorporates the effect of Rayleigh number and the dimensionless conducting block location Xs.

$$\overline{Nu} {\text{ = A}}\left( {{\text{X}}_{\text{s}} } \right) + B\left( {{\text{X}}_{\text{s}} } \right) \times Ra^{{\frac{1}{2}}}$$
(37)

Based on the correlation of Karayiannis et al. [55], the mean Nusselt number obtained from aforementioned correlation with the numerical results for a non-partitioned enclosure is presented in Fig. 22. It is clear that Nu increases with increasing Ra. This figure approves a linear dependency of the variation of the average Nusselt number. A good agreement is observed in the figure.

Fig. 22
figure 22

Variation of the numerical and correlated mean Nusselt number with Rayleigh number for a non-partitioned enclosure kr = 6.22, Xs = 0.5 and δ = 0.1

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For the case of an enclosure with CHT and based on correlation of Duxbury [55] and Nishimura et al. [56], the comparison of the empirical correlation with the numerical average Nusselt number is depicted in Fig. 23. From the figure, it is seen that the average Nusselt number increases linearly with the Rayleigh number.

Fig. 23
figure 23

Variation of the numerical and correlated mean Nusselt number with Rayleigh number for a partitioned enclosure kr = 6.22, Xs = 0.5 and δ = 0.1

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Conclusions

In this paper, a numerical investigation of natural CHT in a differentially heated cavity divided by a conducting solid has been performed using LBM. The numerical simulation is carried over a wide range of the Rayleigh number 103–106. The effects of wall thickness, the wall position and Rayleigh number as well as the effect of the thermal conductivity ratio are studied. The distribution of streamlines, isotherms, local and the averaged Nusselt number is used in order to understand the behavior of the flow and the mode of heat transfer. The main conclusions of this work can be summarized as follows:

  • The presence of partition in the cavity has a significant influence on the natural convection heat transfer characteristics.

  • The rate of heat transfer decreases with increasing wall thickness for δ ≤ 0.5. When δ tends toward 1, the average Nusselt number increases.

  • The average Nusselt number is more affected by the position of the wall partition but is an increasing function of the Rayleigh number.

  • The variation of thermal conductivity ratio increases the rate of heat transfer.

The mainly studied thermal conductivity ratio is kr = 6.22. The air thermal conductivity at home temperature is close to 0.026. By multiplying the last value by 6.22, the solid thermal conductivity is close to 0.16. Such a value is characteristic of materials used in the building such as cellular concrete, wood, wood fiber-board and wood particle board, which presents a fallout of specific applications of partitioning of premises in the building.

Besides, it has been shown that the Lattice Boltzmann method is a powerful approach for investigating heat and fluid flow problems in multiphase media as well as for its simplicity of coding and its reasonable convergence CPU time in such stationary problems. In the following steps, MHD effects in microgeometries will be accounted for as an issue of important interest.