Darcy number effects on natural convection around a porous cylinder in L-shaped enclosure using Lattice Boltzmann method | Scientific Reports

Scientific Reports volume 15, Article number: 8448 (2025) Cite this article

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This study numerically evaluates fluid flow and natural convection heat transfer of a porous square cylinder in an L-shaped enclosure using the Lattice Boltzmann method. Three layouts along vertical and horizontal centrelines are explored, investigating the effects of Rayleigh number (Ra) (103 ≤ Ra ≤ 106), Darcy number (Da) (10−6 ≤ Da ≤ 10−2), and cylinder size. Results show that increasing Rayleigh numbers enhances heat transfer, with higher Mean Nusselt number (NuMean) values observed. Doubling the cylinder’s width at Da = 10−6 increases NuMean by 46.5%, and tripling the width results in a 118% enhancement. Higher Rayleigh values enhance buoyant forces’ intensity, improving heat transfer; for example, at Ra = 105 and Da = 10–2, there is a 42% increase in Nu mean for a cylinder with a 0.6 L side length compared to Ra = 103. Optimal cylinder orientation significantly maximizes convective heat transfer, especially at high Rayleigh and Darcy numbers. This study provides valuable insight for optimizing the orientation of electronic blocks in compact L-shaped enclosures for better thermal management.

Natural convection offers significant cost-effectiveness along with the benefits of simplicity, reliability, and durability, coupled with ease of maintenance and low energy consumption. Research on natural convection heat transfer has significantly advanced its application to various enclosed objects. These applications include solar collectors, drying technology, freezers, thermal storage, nuclear reactors, and geothermal reservoirs. Additionally, it extends to renewable heat waste disposal techniques, artificial bone manufacturing, and cooling and safety measures for electronic devices1,2,3. This numerical analysis primarily focuses on the free convection of heat within an enclosure that contains a porous cylinder positioned strategically in a designated area. The investigation is carried out by varying the size and position of these square blocks to observe the dissipation of heat within the enclosure.

Numerical simulation is employed to investigate the fluid flows driven by natural convection in enclosures, as these chambers often contain porous square cylinders and rely on natural convection for efficient heat transfer.

Various studies have already been made on various types of enclosures and provided solutions for both flow patterns and thermal fields but there is still a dearth of knowledge. Extensive research has given an accurate picture of the heat transfer and flow in structures constructed with several materials. These include heat blocks, baffles, fins, and other porous media such as nanofluids; various orientations of MHD; and so on, in various shapes and orientations4,5,6.

Vahl Davis’s innovative research has significantly advanced computational methods for solving free convection problems in square chambers with differentially combined horizontal walls7. His investigation revealed that stronger vortices would be formed within the chamber as the Rayleigh number increased. Moreover, Davis 'experiments also directly showed a positive correlation between Rayleigh numbers and maximum fluid velocity as well as heat transfer from the chamber. As a result, these discoveries have considerably enhanced our understanding of natural convection in enclosed spaces. This article follows up on de Vahl Davis’s pioneering work to make further progress in understanding the dynamics of free convection within differently connected square chambers.

Nithiarasu et al.8 studied the flow and heat transmission within an oddly shaped vertical container that was horizontally expanded. Various types of boundary conditions were examined to meet the actual application demand for irregular-shaped enclosures for cooling electronic equipment, thermal hydraulic analysis, solar collectors, ingot castings, etc. The study used a rigorous parametric analysis to assess the effects of geometry, Rayleigh number, Darcy number, and boundary condition specifics.

Chinnakotla et al.9, Angirasa et al.10, and Mahajan11 have conducted comprehensive research studies to examine the heat transfer and flow patterns within L-shaped enclosures under different conditions of boundary. The size, location, aspect ratio, and boundary condition of air-filled rectangular enclosures were analyzed for effects on natural convection by Chu et al.12. They determined that these factors are very essential parameters controlling temperature distribution as well as velocity profile.

There are several variables affecting the heat transfer phenomena in L-shaped enclosures. These include the spacing between walls, height of enclosure, angle of corners, and flow patterns at high Rayleigh numbers. Mahmud13 conducted a study on an L-shaped enclosure with different aspect ratios to investigate the flow pattern caused by buoyancy and its impact on heat transfer characteristics. Saidi and Karimi14 investigated the L-shaped enclosure geometry, which was adjusted by different cylindrical pin insertions and optimal angle inclinations and discovered a substantial influence on the rate of heat transfer coefficient owing to higher buoyancy forces.

The reason why people are paying more attention to L-shaped enclosures is that they are driven by their engineering design applications involving electronic packages, electrical equipment, and corners of buildings. Ruiz and Sparrow15 investigated heat transfer and flow phenomena in L-shaped corners as well as V-shaped ones. An enclosure of an L shape can also be described as a V-shaped one by turning the height and width 45° from horizontal. The research conducted by Nithiarasu and his colleagues related most directly to the exploration of an inverted L-shaped (Γ) enclosure on a vertical axis8.

The results set forth by Hasan and Baig16 for an irregularly shaped enclosure closely conform to the geometric characteristics of an inverted U-shaped enclosure.

Additionally, Mohebbi and Rashidi investigated the L-shaped enclosure using nanofluid as well as heat-blocking walls17. They showed that, for all the Rayleigh number values they tested, a larger volume fraction and internal porous square body vertical orientation leads to higher Nu numbers. In addition, they investigated the largest Nu number discovered when barriers and heat resources were positioned at the left wall of the lowest location.

Mojumder et al.18 investigated the influence of the Reynolds, Darcy, and Grashof numbers on temperature and velocity fields. When a greater Reynolds number, higher Darcy number, and a low Grashof number were applied to the vertical wall, the heat transfer rate increased, but the converse occurred on the horizontal wall.

Even though there have been numerous studies of natural convection heat transfer and flow in air-filled enclosures, the optimal conditions and maximum possible amount of heat transfer under those circumstances are unknown.

However, the literature review indicated that very few research had particularly studied the use of porous media to increase heat transfer rates within cavities. To the best of the authors’ knowledge, no researchers have looked at the impact of the dimensions and location of evenly heated porous square cylinders within variably heated L-shaped enclosures on convection that occurs naturally. Despite extensive research on natural convection heat transfer in various types of enclosures, there remains a significant gap in understanding how porous media can be strategically utilized to enhance thermal performance. This study tries to close that gap.

The primary goal of this research is to systematically examine how varying Rayleigh numbers (Ra), Darcy numbers (Da), sizes, and placements of porous square cylinders influence heat transfer and flow patterns within an L-shaped enclosure. The study enables determining the most effective placement of heated porous square cylinders of different sizes within an enclosure at different positions along the vertical and horizontal bottom-centre lines of the L-shaped enclosure for optimal heat dissipation. A unique aspect of this study is that the L-shaped enclosure introduces unique flow patterns due to its geometry. Additionally, through numerical simulations, the research clarifies the interactions between the hot and cold walls of the enclosure; thus, our findings could lead directly to improve design strategies for systems such as electronic cooling devices, solar collectors, or even advanced manufacturing processes. Moria Hazim studied the importance of the porous body in achieving maximum heat transfer rate via an L-shaped enclosure by altering the placements of the heating element and porous bodies attached to the wall cavity3. The novelty of this research lies in its investigation of the optimal positioning of the porous square cylinder to enhance heat transfer rate within L-shape enclosure.

The geometry under investigation is schematically shown in Fig. 1.

(a) Schematic illustration of the L-shaped cavity of the problem under consideration. (b) Various sizes of the porous square cylinder are considered.

The present model considered an L-shaped enclosure with left vertical and bottom horizontal walls equal in length and heated isothermally (Th) to induce buoyancy effect in the enclosure, and the inner walls of the right vertical and up horizontal are equal in length and cooled isothermally (Tc). The remaining walls are maintain adiabatic condition by thermal insulation with an aspect ratio of AR = w/L. It is assumed that the cavity under investigation is quite long in z-direction; thus, the variations of z are negligible. H = W is the length of the walls; L is the thickness of the rib and w is the porous square cylinder dimension which is placed at the top, bottom-centre, and bottom right of the L-shaped chamber at an equal distance in the enclosure. The insulated thickness of the enclosure is illustrated as L. It is assumed that the pores within the porous material are uniformly interconnected, rigid, and filled with working fluid. In order to examine the impact of varying the size and locations of the porous square cylinder inside an L-shaped enclosure for heat transfer improvement, the porous square cylinder with a dimension of \(w\) (0.2 L, 0.4 L, and 0.6 L) is located at an equal distance to the top, bottom-center and bottom-right of the thickness of the enclosure and moves along the vertical and right sides of the horizontal wall in the centreline of the cavity. The aspect ratio is described as:

The enclosure is filled with an incompressible fluid, which is commonly utilized for thermal transfer of heat in industrial applications. The immersed boundary technique is used to address the interior porous square cylinders, located at varied locations in both horizontal and vertical directions. Within the framework of Cartesian coordinates.

The protective container is filled with a liquid that is incompressible (Pr = 0.71). Fluid flow fields, isotherm patterns, and NuMean are displayed for Ra 103 to 105, Da from 10–6 ≤ Da ≤ 10–2, and porous thermal conductivity ratios of 0.1. Because there is only a tiny temperature differential between the hot walls and the inner porous walls, all thermophysical parameters are assumed to remain constant, with the exception of density changes and variations in buoyancy force, which are considered as a linear relationship with temperature. Radiation effects are ignored, and gravity is assumed to act downward along the y-axis. The study is based on the following assumptions.

Air, with constant properties as per Newtonian principles, is used as the working fluid.

The fluid flows steadily and it is incompressible in a two-dimensional manner.

Negligible radiation emanates from the heated cylinders.

The thermal conductivity ratio of the porous material and fluid is considered to be one.

Single-phase fluid moves through a porous medium made up of a uniform and isotropic medium.

The permeable cylinders are homogeneous in all directions and have a stable composition, allowing for the passage of a single-phase fluid.

Porosity and permeability values are consistent throughout this region.

The mathematical solution is achieved using the finite volume technique because the flow is regarded as laminar, stable, and two-dimensional, and the fluid inside the enclosure is considered to be Newtonian and incompressible, with minor effects from viscous dissipation. All solid barriers are considered to be stiff with no sliding circumstances. The flow and temperature field within the cavity with a porous square cylinder body are characterized as the transfer of heat by natural convection in two dimensions, using the continuity equation19:

Momentum equations:

Energy equation:

The derived governing equations above can be written in the form of non-dimensional as the following characteristic equations:

where x and y denote the dimensional coordinates, u and v stand for the dimensional velocity components in x and y direction respectively. The \({\rho }_{f}\), \({\alpha }_{f}\) & \({\alpha }_{s}\) are fluid density, and the thermal diffusivity of fluid and solid respectively. ν and β are the kinematic viscosity and coefficient of thermal expansion; g and μ are gravitational acceleration and dynamic viscosity respectively. \({C}_{1}\,\text{and}\, {C}_{2}\) are binary constants with a value of 0 at the fluid region and 1 at the permeable region. Dimensionless parameters can be defined as \(a Pr=\frac{\mu {C}_{p} }{{k}_{f}}\) is the Prandtl number \(, Ra= \frac{g\beta {\left({T}_{h}-{T}_{c} \right)H}^{3}}{v\alpha }\) Rayleigh number. (\({k}_{f}\)) is thermal properties of the fluid and (\({k}_{s}\)) is porous media assumed to be equal, hence \({k}_{f}={k}_{f}{=k}_{s}, \sigma =1\). The thermal conductivity ratio (\(\sigma\)) of fluid to the permeable body is 1 for indicating the flow effects, such as flow due to different porosity variations19. The effective thermal conductivity of a porous body can be stated by using Eq. (7) in the form of heat balancing between the fluid and porous media20

Lattice Boltzmann Methods (LBM) has gained prominence as a powerful computational fluid dynamics technique for simulating diverse fluid flow scenarios, particularly in heat transfer applications, distinguished by its efficiency and numerous advantages such as broad applicability, high precision, simplified algorithms, and discretization methods, along with compatibility for parallel processing. The concept is to imagine fluids as being made up of a large number of tiny particles that move randomly21,22. In this study, the Lattice Boltzmann method is employed with a simplified collision term that utilizes the Bhatnagar–Gross–Krook (BGK) scheme to solve numerical studies23,24,25. The key concept suggests that fluids can be visualized as a collection of numerous small particles moving randomly. The transfer of energy and momentum occurs through the flowing movement of particles and collisions akin to billiard interaction. In this approach, the domain is divided into discrete lattice points. These steps involve calculating the governing parameters such as density, temperature, and velocity. The arrangement of the lattice structure for D2Q9 employed in previous investigations26,27,28 is shown in Fig. 2. Figure 2a depicts the distribution functions of the bounce-back technique for all wall cavities at the boundary29 and (b) is the configuration of the lattice in D2Q9 Discretization two-dimensional nine-velocity model structure along the velocity vector30. The values for velocity in both x and y directions, along with temperature and density, are computed for all lattice points throughout the domain. Unlike nonlinear terms in the Navier–Stokes equation, the convective terms in the LB equations are linear. The probability distribution functions are influenced by factors like relaxation coefficient value and macroscopic property expressed in Lattice Boltzmann. The values are principally influenced by specific viscosity in fluid flow dynamics and thermal diffusivity in heat transfer calculations. As a result, for an incompressible non-isothermal issue, both critical factors fully control the flow and distribution functions fi and gi.

(a) Distribution functions at the boundary walls and (b) configuration of lattice in D2Q9 structure along the velocity vector.

In the standard procedure of the Lattice Boltzmann Method, there are two fundamental steps: collisions and streaming. Before the streaming process commences, collisions need to take place between particles at every node which can be given as23,26,27.

For flow field:

where, \({f}_{i}\) represents instantaneous particle motion velocity function. In the present study \({e}_{i}\) is lattice velocity vector in ith direction “e” is the speed magnitude lattice which can be expressed \(e=\frac{\Delta x}{\Delta t}=\frac{\Delta y}{\Delta t}\) when the lattice discretized uniformly, subscript “i” indicates the direction of the lattice, and \(\Delta t\) is lattice time step size (\(\Delta t=1)\). Fi represents the force term responsible for drag or viscous effects caused by the porous medium, while Fb considers both natural and forced convective effects. In this investigation, the LBM technique standard D2Q9 Lattice model, which is extensively used for solving fluid flow issues, is utilized to describe both flow and temperature in two dimensions. A particle lattice velocity vector. The particle lattice velocity vector \({e}_{i}\) (Fig. 2) is defined as23,29,31:

In Eq. (8), the term \(\tau\) is a factor of relaxation which is identical with the particles in all lattices. During each collision step, the particles relax toward a state of equilibrium. In the first term on the right side signifies a single relaxation time of BGK collision operators.

The fluid kinematic viscosity (\(v\)) is related to relaxation time for the flow field denoted as23:

where \({C}_{s}\) is speed of sound23 in lattices which is equal to: \({C}_{s}=\) \(\frac{c}{\sqrt{3}}=\frac{1}{\sqrt{3}}\) . The buoyancy force in natural convection Eq. (8), heat transfer problem is considered as the effect of external force \({F}_{i}\) is obtained by adding of force term including the effect of porosity of the permeable body given by23,26,27:

Darcy-Forchheimer force term, F is the magnitude of external force due to permeable regions both inertial and viscous that can expressed as32:

where porosity(ϵ), the permeability of the porous region (\(K)\), the kinematic viscosity of fluid (\(v)\) which is different from effective viscosity, \({C}_{F}\) is the Forchheimer non-dimensional inertial factor or drag coefficient given as \({C}_{F}=\frac{1.75}{\sqrt{150\times {\varepsilon }^{2}}}\), and \(\left|U\right|=\sqrt{{u}^{2}+{v}^{2}}\), \(u\) and \(v\) are velocity components of velocity U in the horizontal and vertical directions, with \(G\) representing gravitational body force.

In Eq. (8) is an equation relating the porosity distribution of local equilibrium function for natural convection heat transfer and flow field can be expressed as32:

The D2Q9 model has nine velocity vectors with central particle speed zero and individual directional component is associated with weighting factors values are \({\omega }_{0}=\frac{4}{9}\) for \(i=0,\) \({\omega }_{i}=\frac{1}{9}\) for \(i=\text{1,2},\text{3,4}\), and \({\omega }_{i}=\frac{1}{36}\) for \(i=\text{5,6},\text{7,8}\). At the meso-level, collisions occur and streaming takes place before applying suitable boundary conditions. This is followed by assessing the macroscopic properties. The total frequency distribution values at each node of the lattice determine the fluid density at that specific location. The density (ρ) and velocity (u) of the fluid can be calculated using specific equations based on these distribution function values. Finally, the summation of the distribution function at each lattice site can be used to extract macroscopically significant variables from the distribution function as follows23.

The variable V which expresses the auxiliary velocity due to the porous body can be defined as

In the porous region the actual velocity can be calculated by:

The parameters \({C}_{0}\) and \({C}_{1}\) in Eq. (16) can be determined by:

Density variations develop in the fluid, causing fluid motion caused by mass or gradients of temperature, also known as buoyant force, which is considered an external force. To solve the lattice Boltzmann equation using the Boussinesq approximation, an extra force term must be addressed. This additional force factor (Fb) is introduced into the collision equation to mimic free convection effects indicated in Eq. (8). It is approximated using the Boussinesq approximation.

where \({e}_{iy}\), \(\beta ,\) \({g}_{y}, \rho\), θ denotes the y-direction velocity vector, thermal expansion coefficient, gravitational acceleration, local density in the y-direction, and temperature respectively.

The LBM thermal field is described by the following equation33:

where \({g}_{i}\) represents instantaneous particle equilibrium distribution function with \({\tau }^{\prime}\) is the dimensionless relaxation time which is computed from thermal diffusivity (\(\alpha\)).

The thermal diffusivity (\(\alpha\)) is related to relaxation time for the temperature field is known as the Chapman–Enskog equation, that expresses thermal relaxation time (\({\tau }^{\prime}\)) denoted as23:

Equation (19), \({{g}_{i}}^{eq}\) shows the distribution of equilibrium function for natural convection heat transfer flow field can be written as32:

where \(\theta\) evaluates fluid temperature as:

In LBM simulation of flow, precise estimation of boundary conditions is critical for utilizing any numerical techniques. In the D2Q9 model, distribution functions in the inner position are undetermined at the lattice node positioned on the borders and change depending on the boundary conditions used. The temperature distribution function uses the reflection of bounce-back boundary conditions on all walls to calculate the flow field34. This condition states that after striking a wall, inward distribution equals outward distribution as it returns to the flow zone. Specifically, the following constraints are enforced for nodes located along the east boundary:

The thermal domain imposes a fixed-temperature boundary requirement on the heated and cooled walls, while the other walls use an adiabatic boundary condition. To demonstrate, for all nodes along the West of the border where an even temperature Tw prevails, special requirements are implemented31:

Regarding the temperature field, the bounce-back method is utilized to address the adiabatic boundary conditions of the enclosure. To illustrate, specific conditions are taken into account for all nodes situated along the adiabatic North boundary:

For instance, a temperature value of Th = 1 is specified on the West wall. Consequently, the distribution functions \({g}_{1}\), \({g}_{5}\), and \({g}_{8}\) that were previously unknown are calculated as shown in Fig. 2b: Boundary conditions utilized on enclosure walls when using LB methods are listed in Table 1.

The relevant boundary conditions are:

The Boussinesq approximation is utilized for natural convection. To ensure the computational code operates in an incompressible flow regime, the characteristic velocity must be significantly lower than the speed of sound of the fluid. Therefore, a Mach number less than 0.3 is recommended for simulations21. In this study, a fixed Mach number that belongs to an incompressible region of 0.1 was used across all cases considered21.

By keeping the Mach number, Prandtl number, and Rayleigh number constant, thermal diffusivity, and viscosity are determined based on these non-dimensional parameters’ definitions. In this research, the non-dimensional parameters employed include the Rayleigh number, Prandtl number, and Darcy number. The Ra is computed based on the number of lattice characteristic lengths. This can be written:

The enclosure length (L) is used in this study to represent the typical length. Porous medium is a fluid cell zone characteristic defined as a packed bed of spheres with porosity. The Carman-Kozeny equation is used to calculate the pressure decreases that occur when liquid passes through the densely packed surface of the sphere. The Darcy numbers are used to measure the non-dimensional permeability features of the porous medium and are described in terms of porosity by the Carman-Kozeny relation35.

In the Lattice Boltzmann Method, the numerical accuracy and stability of the solutions are mainly dependent on the number of lattices (N) on characteristics length (L).

Nusselt number is an important factor in analyzing the heat transfer by convection at a boundary; it is the ratio of heat transfer due to convection to heat transfer due to conduction.

The local Nusselt number (\({Nu}_{l})\) is computed for the hot wall located in the West/left vertical direction for the hot wall located in the South/ horizontal direction can be computed by replacing y in place of x as21,36:

The surface NuMean on the hot walls of the two-dimensional model was obtained as integrating the \({Nu}_{l}\) along the two hot walls23,29:

The current heat transfer investigation is carried out using in-house code based on the LB method for numerical simulations, employing collision equations with external force terms (Fi) (Eq. 11) across the computational domain to account for natural convection effects, including the Boussinesq approximation for the buoyant force term (Fb) throughout the domain due to gravity acceleration g. The initial velocities at all nodes were set to zero, while temperatures in heated regions were set to 1. Solution stability is contingent upon maintaining \(\sqrt{g\beta \Delta TL}\le 0.1\)24. Initialization involves formulating the initial velocity (v0), and LB kinematic viscosity based on characteristic length lattices (N) and calculating relaxation factors through the Chapman–Enskog relationship37.

Here, '\(n\)' denotes time steps, while '\(i\)' and '\(j\)' correspond to the positions of nodes. The velocities in the directions of x and y are indicated by \(u(i,j)\) and \(v(i,j)\), respectively, \(\theta (i,j)\) indicates the temperature value at that specific position.

In the present study, LB code is verified by evaluating the result of NuMean in free convection heat transfer from the solid hot wall of the enclosure and the porous square cylinder equipped at different places in an enclosure cavity. In every simulation, computations are iterated a significant number of times until the results converge to the accurate values. The iteration stops once the specified convergence criterion is met31:

To show the variation of different parameters (103 ≤ Ra ≤ 106) and porous square cylinder dimension (0.4 × L), the flow inside the enclosure is steady, two-dimensional, incompressible and laminar, and the working fluid is air (Pr = 0.71)38, the numerical validation is done as shown in Table 2 is a good agreement with the result of P. Nithiarasu39 investigated for L-shaped enclosure filled with working fluid air under various Rayleigh number. The successful code validation instills trust in the obtained results in this study. The NuMean calculated matches well with information from Table 2 found in the existing literature.

To check and evaluate the grid independence of the NuMean at the hot walls, five different mesh density combinations (100 × 100, 120 × 120, 140 × 140, 160 × 160, and 180 × 180) grid sizes for w = 0.2 L–0.6 L in the steps of 0.2 were performed. The size of each lattice is 1. A Darcy number 10–4 and Rayleigh number 105 are considered for the grid-independent test due to conduction being predominant at Ra = 104 while convection has a stronger impact at Rayleigh number 105 in heat transfer32. It is found that a mesh grid size of 160 × 160 ensures a grid-independent solution and the outcomes are compared with this finer mesh as Fig. 3b depicts. This was done to determine the grid size appropriate for natural convection inside the two-dimensional L-shaped enclosure. The slight differences in results using the current code and the literature shown in Tables 2 and 3 are due to discretization methods. The study conducted by P. Nithiarasu39 relies on L-shape enclosure for Rayleigh numbers (103 ≤ Ra ≤ 106), while Nithiarasu et al.19 and Moghimi et al.40 findings are for Darcy values of 10–6 and 10–2. It is found that the deviation in NuMean for 160 × 160 and 180 × 180 grids is always less than 0.50%. So that 160 × 160 mesh is considered for numerical simulation as shown in Fig. 3a. The current numerical study’s accuracy has been verified by comparing the streamline and temperature contours with those from a previous study by Kanna et al.36 and Kalteh et al.29.

Computational grids used for (a) Comparison of the Mean Nusselt number for different lattice sizes of a porous square cylinder with widths of 0.2 L, 0.4 L, and 0.6 L. (b) Uniform grid generation for the current numerical study.

In order to verify numerical simulation outcomes, comparisons were done between the current study and previous experimental or numerical investigations in the literature40. We further corroborated our findings using the results of Moghimi et al.40 and Nithiarasu et al.19 for convective heat transfer in a differentially heated and insulated wall cavity filled with porous media. The results in the porous cavity were compared to those of Moghimi et al.40, who used porous media centrally positioned inside the L shape cavity of the left wall at a constant high temperature, while the right wall was thermally insulated, and the other walls were cold. Nithiarasu et al.19 study a porous-filled hollow with insulated top and bottom walls and differentially heated side walls. Table 3 shows that the findings of this research are broadly consistent.

Additional validation shown in Figs. 4, 5 and 6 was conducted by examining flow lines and temperature contours in the current study with Kanna et al.36 and Kalteh et al.29 at Ra = 103, Ra = 104, and Ra = 105. The comparison in Figs. 4, 5 and 6, depicts that the result of these studies is very good match with the previously published results and strongly agreed with the present investigation of L-shape cavity.

Verification of isotherm and streamline contours of the present models with the findings of Kanna et al.36 for Ra = 103 (a,b).

Isotherms and streamline plots in the L-shape enclosure at Ra = 104 (a,b) with a result of Kale et al.29.

Comparison of isotherm and streamline contours for Ra = 105 (a,b) and Pr = 6.2 with results of Kanna et al.36 and Present work.

Another verification test was conducted to confirm \({Nu}_{l}\) in the L-shape enclosure at Ra = 103 and 104 with the simulation result of Naseri et al.22 heat distributions on the hot wall (AB and BC) at (a) Ra = 103 and (b) Ra = 104. The comparison of present result is strongly agreed with the literature. All suitable comparison confirmed that the findings in this study align with those from previous investigations (Fig. 7).

Comparison of present results with Naseri et al.22 simulation result of local Nusselt number distributions on the hot wall (AB and BC) at (a) Ra = 103 and (b) Ra = 104.

The study is performed numerically to examine how air flows and heat transfers within an enclosure containing vertically and horizontally equipped permeable square porous cylinder using LBM. The porous square cylinders have dimensions ranging from 0.2 L to 0.6 L. The study investigates the effects of the Darcy number, Rayleigh number, and porous square cylinder size on the heat transfer and flow with the following variations.

Darcy number (Da): 10–6, 10–4, and 10–2

Rayleigh number (Ra): 103, 104, and 105

Porosity (ε): 0.629, 0.8, and 0.993. These correspond values are Da = 10–6, 10–4, and 10–2, respectively and calculated as per Eq. (30).

The local Nusselt number is a dimensionless parameter measuring the ratio of convective to conductive heat transfer at any surface point. The \({Nu}_{l}\) on the heated wall of the enclosure is computed per Eq. (31). The changes in local Nusselt number for w = 0.2–0.6 L along the L shaped enclosure hot walls at Ra = 103, 104, and 105, Da = 10–6, 10–4 and 10–2 at different locations, top, (bottom-centre, and bottom-right) are analysed during the study. Nusselt at the vertical hot wall (AB) and horizontal hot wall (BC) are seen, thus the result of Nu is maximum when porous body placed at the bottom-centre irrespective of Darcy numbers and size of porous square cylinders. In Fig. 8a–d, for different position of the cylinder at Ra = 104 and 105 at Da = 10–6 and 10–2 for 0.6 L local Nusselt number are presented. It can be observed that the maximum value is found at bottom-centre position for the horizontal hot wall (BC), while the lowest value, nearly zero, is observed at the corners of the enclosure. Figure 8a–d depicts that the result of the maxima changes with the increases Rayleigh number of 105 and Darcy number 10–2 at bottom-centre position for 0.6L cylinder size. This is due to heated enclosure wall and porous square cylinder at the bottom-centre increases convective effect. The flow patterns shown by streamlines that reveal concentrated vortices near the lower right corner of the wall, where heat absorption is prominent with fluid moving toward surrounding boundaries; Additionally, isotherms are uniformly distribution along the bottom of the enclosure; heat transport is more elevated from bottom wall as well.

Local Nusselt number along enclosure hot wall, west wall (AB), and south wall (BC), for porous square cylinder 0.6 L size at Da = 10–6 and 10–2 when Ra = 104 (a,b) and Ra = 105(c,d).

As the size of the porous square cylinder varies from 0.2 to 0.6 L, changes in heat transfer due to the Darcy number (Da), and the Rayleigh (Ra) are observed. Figure 9a–c depicts how the location of the porous square cylinder and its size affect the NuMean on the enclosure walls at Rayleigh number 105. The remarkable impact of Ra, Da, porous square cylinder size, and porous square cylinder location can be noticed. The variation is shown in Figure 9a,b with different sizes and locations of the porous square cylinder within the L-shaped cavity. For all Ra, the NuMean value is highest when the porous square cylinder is positioned at the bottom centre of the enclosure. When the porous square cylinder is placed at the bottom-centre, NuMean is at its greatest due to the buoyant effects increase with Rayleigh number. This is more pronounced in isothermal contours presented in this paper. The porous square cylinder placed at the bottom-right wall of the enclosure to enhance heat transfer rates during natural convection effects is stronger.

Variations of Mean Nusselt number (NuMean) along the hot enclosure walls for different position and size of porous square cylinder in the cavity with darcy number and Ra = 105 for (a) 0.2 L, (b) 0.4 L, (c) 0.6 L.

Increasing the Ra, Da, and cylinder size enhances the heat transfer rate at all cylinder locations. Figure 10 shows that when the dimension of the porous square cylinders increases from 0.2 L to 0.6 L, there is a noticeable rise in the NuMean within the enclosure at constant Rayleigh number. Moreover, the magnitude of NuMean increases as the Darcy number and Rayleigh number increase, that leads to a strong buoyant force that causes convective effect. At Darcy numbers of 10–6 and 10–4 for a Rayleigh number of 103 and104 the NuMean has small variations; however, there is a significant increase observed at Darcy number 10–2. Doubling the width of the porous square cylinder at Da = 10–6 leads to approximately 46.5% rise in NuMean, while tripling it results in 118% heat transfer enhancement. As Ra increases to 104 and105, buoyant forces become more prominent, driving NuMean to higher values. For example, at Ra = 105 and Da = 10–2, NuMean increases by 42% for a porous square cylinder with side length 0.6L compared to lower Rayleigh values (103). Further investigation reveals that fluid circulation patterns vary depending on Rayleigh number: hotter fluid tends to circulate near top walls when Ra small but shifts towards sidewalls as Ra reach 105 leading to increased Nusselt numbers. The highest value for NuMean among all conditions occurs when the width of porous square cylinder equals 0.6L and Da equals 102 with Ra equalling 105. The larger size porous square cylinders limit space available for fluid recirculation inside L shaped enclosure; however, this limitation is counterbalanced by increased conductive effects alongside increases convective effects under high Ra conditions.

The mean Nusselt number of enclosure wall for various porous square cylinder size at bottom-centre of enclosure cavity for (Da = 10−6, 10−4, 10−2) for (a) Ra = 103, (b) Ra = 104, and (c) Ra = 105.

The heat transfer enhancement ratio (ER), which represents the relationship between the NuMean at different Darcy numbers compared to Da = 10−6, and is expressed as:

The effects of Darcy and Rayleigh number on the heat transfer enhancement ratio were also investigated in Fig. 11 With increasing Da, heat transfer augmentation becomes more pronounced as Ra increases.

Heat transfer enhancement ratio (ER) at different porous square cylinder sizes (0.2L ≤ w ≤ 0.6L) of Ra = 103, 104 and 105 at Da = 10–4 and 10–2.

Figure 14a–i show that how the streamline flow patterns changes when Ra, Da, position, and the size of the porous square cylinder change. Figure 12a–c show that the isotherm around the heated porous cylinder. Figures 12 and 14 display the isothermal lines and flow streamlines, respectively in an L-shaped enclosure containing porous square cylinder positioned at various locations. At low Ra, conductive mechanisms are dominant; introducing an internal porous square cylinder reduces overall heat transfer. As Ra increase and convective mechanisms become more significant, the conduction effects become negligible. Furthermore, with an increase in the Ra, the primary flow pattern elongates and becomes limited to vicinity of the walls. It’s important to note that the critical Ra at which both conductive and convective mechanisms have equal importance in determining overall energy transport within this system.

(a) Isotherms around porous square cylinder (i–ix) placed at top differentially heated L shape enclosure for different Rayleigh number and porous medium size at Da = 10–2. (b) Isotherms around porous square cylinder (i–ix) placed at bottom-centre differentially heated L shape enclosure for different Rayleigh number and porous medium size at Da = 10–2. (c) Isotherms around porous square cylinder (i–ix) placed at bottom-right differentially heated L shape enclosure for different Rayleigh number and porous medium size at Da = 10–2.

The influence of Darcy number on the Nusselt number is notable primarily at higher Rayleigh numbers, and this effect becomes more pronounced with larger Darcy numbers.41 For a Ra of 103 and porous size of 0.2 L, 0.4 L, and 0.6 L placed at the top, bottom-centre, and bottom-right positions, the isothermal lines exhibit a uniform decrease from the enclosure wall to the bottom-centre of the enclosure.

However, when increasing the Ra of porous square cylinder placed at bottom-centre to 105 as depicted in Fig. 12((b) iii, vi, and ix) it causes a shift in isothermal line flow towards the enclosure cold wall from its centreline due to the temperature radiant between the heated porous square cylinder and the enclosure walls. Due to density differences resulting from heat transfer within an enclosed space, warm fluid heated along the left wall rises towards the right wall, spreading over the top insulated surface before returning down adjacent L-shaped hot walls. Regardless of the specific Ra considered here, the temperature difference between left and right boundaries remains significant, while convection flows associated with specific Ra cannot change temperature gradient signs for different sizes and positions occupied by porous body. As indicated by the isotherm lines on Fig. 13, when the internal porous square cylinder is placed in different positions along the centreline, its temperature experiences a continuous increase from the left to the right in the hot wall. However, the temperature gradient range decreases as the Ra increases. As Ra increase, flow circulation near boundaries limited temperature gradients to those regions.

Effect of porous square cylinder size and location on isotherms for various Ra (\({10}^{3}\) and \({10}^{5}\)) at (a) top (b) bottom-centre, and (c) bottom-right in partially heated L shaped enclosure for \({Da=10}^{-2 }, \varepsilon =0.9\)

According to the isotherm lines depicted on Fig. 13, when the internal porous square cylinder is placed at various positions along the centreline, its temperature undergoes a gradual rise from left to right on the hot wall. However, as we observe higher Ra, there is a reduction in the extent of temperature variations. This is due to the circulation becoming near boundaries at high Ra, primarily limited temperature gradients within the adjacent boundary region. The calculated results are presented with isotherm and streamlined in the form of a graph.

In Fig. 12, at low Ra numbers (103, 104), the isotherm patterns exhibit a uniform structure parallel to the hot wall and cold walls, indicating predominant transfer of heat through conduction. The addition of a porous square cylinder doesn’t change the isotherm pattern but compresses it in in specific location, signifying enhanced heat transfer. This compression occurs as the hot wall moves closer to the cold wall due to the presence of a porous boundary.

The effect becomes more pronounced for porous square cylinder width 0.4 L and 0.6 L. A comparison of Ra numbers 103 and 104 porous body positioned at top, bottom-centre, and right-bottom reveals that changing the position of the porous has minimal impact beyond shifting where this compression in isotherms occurs.

Moreover, in Fig. 12, at high Ra number (105), significant changes and distortions occur in the isotherm patterns, indicating that heat transfer by convective is more dominant. A comparison between in case of high Ra number (105), porous body positioned at right bottom, has minimal impact on isotherm patterns; however, at top and bottom-centre, it disrupts the pattern, revealing that changing the position of the porous body has a high impact on where this isotherm occurs. Moreover, as seen in Fig. 12, with high Ra number (105) at the bottom-centre, lead to substantial alterations in isotherm configurations where compressed lines under the porous square cylinder to horizontal rightward become prominent. The impact of the Rayleigh number (Ra) and the position of a porous square cylinder on the formation of streamlines within an enclosure is illustrated in Fig. 14 for streamlines. At Ra = 103, one vortex forms when a porous square cylinder is placed at the top, two vortices emerge when it’s positioned at the bottom-centre, and three vortices appear when located at the bottom right side of the enclosure. In contrast, placing a porous square cylinder with w = 0.6L at its bottom-centre results in two vortices. The size and placement of this cylindrical structure significantly influence both streamlines and temperature distribution. As Ra increases to 104, one vortex forms with top placement, two with c bottom-centre positioning, three under bottom right location for a porous square cylinder; however, only two forms if it has dimensions w = 0.6L placed towards bottom right corner. Continuing to higher Ra values leads to more pronounced circulation zones until reaching Ra = 105 where three distinct circulation zones are observed occupying various parts within an enclosure as streamline patterns fill up entirely. At Ra = 105 specifically noted was that using a bottom-centre positioned porous square cylinder sized at 0.6L resulted in an arrangement featuring two streamline circulation zones filling up space within this configuration. Additionally, at Ra = 105, the size of the vortex greatly increased this indicates that natural convection is enhanced.

(a) Effect of porous square cylinder size on streamlines for Da (\({10}^{-6}, {10}^{-4},\) and \({10}^{-2}\)) at top in differentially heated L shaped enclosure at \({Ra=10}^{3 }, \varepsilon =0.9\). (b) Effect of porous square cylinder size on streamlines for Da (\({10}^{-6}, {10}^{-4},\) and \({10}^{-2}\)) at top in differentially heated L shaped enclosure at \({Ra=10}^{4 }, \varepsilon =0.9\). (c) Effect of porous square cylinder size on streamlines for Da (\({10}^{-6}, {10}^{-4},\) and \({10}^{-2}\)) at top in differentially heated L shaped enclosure at \({Ra=10}^{5 }, \varepsilon =0.9\). (d) Effect of porous square cylinder size on streamlines for Da (\({10}^{-6}, {10}^{-4},\) and \({10}^{-2}\)) at bottom-centre in differentially heated L shaped enclosure at \({Ra=10}^{3 }, \varepsilon =0.9\). (e) Effect of porous square cylinder size on streamlines for Da (\({10}^{-6}, {10}^{-4},\) and \({10}^{-2}\)) at bottom-centre in differentially heated L shaped enclosure at \({Ra=10}^{4 }, \varepsilon =0.9\). (f) Effect of porous square cylinder size on streamlines for Da (\({10}^{-6}, {10}^{-4},\) and \({10}^{-2}\)) at bottom-centre in differentially heated L shaped enclosure at \({Ra=10}^{5 }, \varepsilon =0.9\). (g) Effect of porous square cylinder size on streamlines for Da (\({10}^{-6}, {10}^{-4},\) and \({10}^{-2}\)) at bottom-right in differentially heated L shaped enclosure at \({Ra=10}^{3 }, \varepsilon =0.9\). (h) Effect of porous square cylinder size on streamlines for Da (\({10}^{-6}, {10}^{-4},\) and \({10}^{-2}\)) at bottom-right in differentially heated L shaped enclosure at \({Ra=10}^{4 }, \varepsilon =0.9\). (i) Effect of porous square cylinder size on streamlines for Da (\({10}^{-6}, {10}^{-4},\) and \({10}^{-2}\)) at bottom-right in differentially heated L shaped enclosure at \({Ra=10}^{5 }, \varepsilon =0.9\)

With an increase in Ra as depicted by Fig. 6, there’s a notable rise in streamlines along with new vortices formation within an L-shaped structure without a porous L shape enclosure showing in stagnant flow regions.

As demonstrated in (Fig. 14 Streamline and Fig. 12 isotherm), the internal porous square cylinder positioned in three different places (Top, bottom-centre and bottom-right) with three different sizes (0.2 L, 0.4 L, and 0.6 L), has a great effect on isotherm and streamlines flow in heat transfer.

The fluid does not enter the porous square cylinder and instead circulates around it at Da = 10−6. For Da = 10−6, two rotating eddies are evident for Ra = 103 and 104 when the cylinder is positioned at top, bottom-center, and bottom-right. However, for Ra = 105, three revolving eddies have developed as shown in Fig. 14. As the Darcy value increases to Da = 10−4, the porous square cylinder experiences less fluid penetration due to increased inertial resistance. At Da = 10−2, a significant volume of fluid penetrates the porosity square cylinder and reacts with the heated porous body, regardless of its size. The heated liquid flows back to the cool wall, suggesting increased heat dissipation. However, when Da = 10–2, heat transfer rates rise as a greater amount of fluid interacts with the heated porous square cylinders and flows towards the chilled enclosure wall. Figure 13 shows that fluctuations in the thermal field and temperature contours frequently travel towards the top boundary, whereas thermal lines tend to meet near the ceiling wall. With an increase in the Darcy number, the lone internal vortex shifts downward for the porous square cylinder positioned at the top for Ra = 103 and 104. For Ra = 105, the vortex moves to the cavity’s bottom-right side. At Da = 10−2, a thermal plume directed towards the top wall indicates the passage of lighter and warmer fluids upwards due to density differences. In Fig. 14, the square porous cylinder shifts to the bottom-center of the enclosure as the permeability level increases to Da = 10−2. For Ra = 103 and 104, the vortex region on the right side of the innermost eddy shifts towards the top, while for Ra = 105, the eddy shifts to the bottom-right side. The recirculating vortex in the bottom-right of the enclosure splits into two smaller vortices within and beside the porous body, while one massive vortex moves around and through the left vertical cavity of the enclosure as the square porous cylinder moves from the bottom-center to the bottom-right (Fig. 14). At Darcy number 10−2 and Ra = 105, the amount of fluid entering the cylinder rises. One large vortex and two smaller vortices are dense, suggesting strong convection heat transfer. At Da = 10−2 and Ra = 105, heat transfer rates rise across the cylinder (Figs. 12, 13, 14).

In the current study, LBM was used for computational modeling and simulation of two-dimensional natural convection in an L-shaped enclosure containing porous square cylinders at various places within the enclosure cavity. The governing equations were solved in a dimensionless manner using computational fluid dynamics (CFD). The aim of this study was to provide valuable insights for experimentalists looking to explore different combinations of orientation, size, and thermal properties of porous square cylinder in controlling fluid flow and thermal performance within enclosures at various Raleigh numbers (Ra) and Darcy numbers (Da). Our focus is on investigating how factors such as location, orientation, dimensions of porous square cylinders interact with different Ra and Da values. The fluid flow within an L-shaped cavity without a porous square cylinder was also simulated for isotherm, streamlines, and local Nu at various Ra, as shown in Figs. 4, 5, 6, and 7, to confirm the simulation’s accuracy with existing studies and to conduct a comparison for the L shape with a porous square cylinder. As a consequence, the influence of a porous square cylinder arrangement on width (w) and location (top, bottom-center, and bottom right) in different Rayleigh number (Ra) ranges was investigated. The results, which are displayed graphically, demonstrate that the most significant parameters in an L-shape enclosure are the Darcy number, Rayleigh number, size of the porous square cylinder, and its orientation. Furthermore, several streamlines, isotherm contours, local and NuMean were explored to compare various orientations of porous square cylinder in L-shaped cavity to optimize the location in order to increase convective heat. As a result, the following major implications may be drawn from the study findings:

Impact of Rayleigh number on heat transfer: As the Rayleigh number increases, the NuMean also increases, indicating a stronger convective heat transfer. However, for specific values of the Rayleigh number (Ra = 103 and 104), the NuMean does not change significantly and remains nearly constant. This indicates that conduction is the dominant method of heat transfer within the cavity during these ranges of Rayleigh numbers.

Significance of cylinder size: Among various parameters, the effect of porous square cylinder size is one of the most influential factors affecting convective heat transfer in an L-shaped enclosure compared to the other types. The large size of width (w = 0.6 L) are always more effective in increasing the overall Nusselt number value than the smaller size of width (w = 0.2 L).

Optimal positioning for enhanced performance: The result of this investigation shows that by changing porous square cylinder with the proper positioning at the bottom-centre are always best convective heat transfer performance than their corresponding configuration with the position of top and bottom right.

Effectiveness across scenarios: Using porous square cylinders significantly improves heat transfer performance across all tested scenarios; however, the NuMean remains relatively stable when Darcy numbers fall below Da = 10−5. This finding implies limited effectiveness for porous layers characterized by low permeability under these conditions.

Visual data representation considerations: While comprehensive visual data such as all isotherm contours and streamlines play an essential role in understanding complex flow patterns. All isotherm contours and streamlines were not presented here to save space.

The datasets used and analyzed during the current study are available from the corresponding author on request.

Enclosure aspect ratio

Binary constant

Non-dimensional Forchheimer term

Speed of sound

Darcy number

Discrete lattice velocity in i direction

Equilibrium temperature distribution function

Darcy–Forchheimer force term

Total force due to porous medium

Boussinesq force term

Particle distribution function along ith link direction

Temperature distribution function along ith link direction

Gravitational acceleration

Gravitational body force

Non-dimensional enclosure height

Thermal conductivity

Enclosure thickness, m

Local Nusselt number

Mean Nusselt number

Prandtl number

Rayleigh number

Porous square cylinder side-length

Non-dimensional enclosure length

Temperature

Velocity component along the x-direction

Non-dimensional velocity component along x-direction

Velocity component along the y-direction

Non-dimensional velocity component along y-direction

Solid to fluid conductivity ratio

Dynamic viscosity

Ratio of thermal conductivity

Kinematic viscosity

Thermal diffusivity

Volumetric isobaric coefficient of thermal expansion

Non-dimensional temperature \(\rho\) density

Cold

Fluid

Hot

Solid

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Department of Mechanical Engineering, College of Engineering and Technology, Wollega University, Nekemte, Ethiopia

Tadesse Beyene Hulle

Department of Mechanical Engineering, College of Engineering and Technology, Dambi Dollo University, Dambi Dollo, Ethiopia

Tadesse Beyene Hulle & Ramaswamy Krishnaraj

Center for Global Health Research, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai, India

Ramaswamy Krishnaraj

Department of Mechanical Engineering, Institute for Energy Studies, Anna University, Chennai, India

M. Venkata Ramanan

Department of Mechanical Engineering, ULTRA College of Engineering and Technology, Madurai, Tamil Nadu, 625 104, India

N. Nagaprasad

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Conceptualization, TB. H, R. K, M. VR and N. N.; Data curation, TB. H, R. K, M. VR and N. N.; Analysis and Validation, TB. H, R. K, M. VR and N. N.; Formal analysis, TB. H, R. K, M. VR and N. N; Investigation, TB. H, R. K, M. VR and N. N.; Methodology, TB. H, R. K, M. VR and N. N.; Project administration, K.R and M. VR. Resources TB. H, R. K, M. VR and N. N.; Software, TB. H, R. K, M. VR and N. N., Supervision, K.R and M. VR.; Validation, TB. H, R. K, M. VR and N. N.; Visualization, TB. H, R. K, M. VR and N. N.; Writing—original draft, TB. H, R. K, M. VR and N. N., Data Visualization, Editing and Rewriting, TB. H, R. K, M. VR and N. N.

Correspondence to Ramaswamy Krishnaraj.

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Hulle, T.B., Krishnaraj, R., Ramanan, M.V. et al. Darcy number effects on natural convection around a porous cylinder in L-shaped enclosure using Lattice Boltzmann method. Sci Rep 15, 8448 (2025). https://doi.org/10.1038/s41598-025-88548-6

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Received: 17 September 2024

Accepted: 29 January 2025

Published: 11 March 2025

DOI: https://doi.org/10.1038/s41598-025-88548-6

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