Natural convection in a square cavity filled with a porous medium: Effects of various thermal boundary conditions

Natural convection flows in a square cavity filled with a porous matrix has been studied numerically using penalty finite element method for uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls. Darcy–Forchheimer model is used to simulate the momentum transfer in the porous medium. The numerical procedure is adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra, 10³ ⩽ Ra ⩽ 10⁶, Darcy number Da, 10⁻⁵ ⩽ Da ⩽ 10⁻³, and Prandtl number Pr, 0.71 ⩽ Pr ⩽ 10) with respect to continuous and discontinuous thermal boundary conditions. Numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case for all Rayleigh numbers but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. It has been found that the heat transfer is primarily due to conduction for Da ⩽ 10⁻⁵ irrespective of Ra and Pr. The conductive heat transfer regime as a function of Ra has also been reported for Da ⩾ 10⁻⁴. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominated regimes the power law correlations between average Nusselt number and Rayleigh numbers are presented.     

Heat flow visualization for natural convection in rhombic enclosures due to isothermal and non-isothermal heating at the bottom wall

Analysis has been carried out for the energy distribution and thermal mixing in steady laminar natural convective flow through the rhombic enclosures with various inclination angles, φ for various industrial applications. Simulations are carried out for various regimes of Prandtl (Pr) and Rayleigh (Ra) numbers. Dimensionless streamfunctions and heatfunctions are used to visualize the flow and energy distribution, respectively. Multiple flow circulations are observed at Pr = 0.015 and 0.7 for all φs at Ra = 10⁵. On the other hand, two asymmetric flow circulation cells are found to occupy the entire cavity for φ = 75° at higher Pr (Pr = 7.2 and 1000) and Ra (Ra = 10⁵). Heatlines are found to be parallel circular arcs connecting the cold and hot walls for the conduction dominant heat transfer at Ra = 10³. The enhanced convective heat transfer is explained with dense heatlines and convective loop of heatlines at Ra = 10⁵. Heatlines clearly demonstrate that the left wall receives heat from the bottom wall as heatlines directly connect both the walls whereas the convective heat circulation cells play lead role to distribute the heat along the right wall, especially for smaller φs. On the other hand, the heat flow is evenly distributed to both side walls at higher φs via convection as well as direct conductive transport. Significant convective heat transfer from the bottom hot wall to the left cold wall occurs for φ = 30° cavity whereas the heat transfer to the right cold wall is maximum for φ = 75° irrespective of Pr. Average Nusselt number studies also show that φ = 30° cavity gives maximum heat transfer rate from the bottom to left wall irrespective of Pr in isothermal heating case. On the other hand, enhanced thermal mixing occurs at φ = 75° for both isothermal and non-isothermal heating strategies except at Pr = 0.015 in isothermal heating case.     

Steady natural convection flows in a square cavity with linearly heated side wall(s)

The present numerical study deals with natural convection flow in a closed square cavity when the bottom wall is uniformly heated and vertical wall(s) are linearly heated whereas the top wall is well insulated. Non-linear coupled PDEs governing the flow have been solved by penalty finite element method with bi-quadratic rectangular elements. Numerical results are obtained for various values of Rayleigh number (Ra) (10³ ⩽ Ra ⩽ 10⁵) and Prandtl number (Pr) (0.7 ⩽ Pr ⩽ 10). Results are presented in the form of streamlines, isotherm contours, local Nusselt number and the average Nusselt as a function of Rayleigh number.     

Turbulent Rayleigh–Bénard convection of water in cubical cavities: A numerical and experimental study

Experimental measurements and numerical simulations of natural convection in a cubical cavity heated from below and cooled from above are reported at turbulent Rayleigh numbers using water as a convective fluid (Pr = 6.0). Direct numerical simulations were carried out considering the Boussinesq approximation with a second-order finite volume code (10⁷ ⩽ Ra ⩽ 10⁸). The particle image velocimetry technique was used to measure the velocity field at Ra = 10⁷, Ra = 7 × 10⁷ and Ra = 10⁸ and there was general agreement between the predicted time averaged local velocities and those experimentally measured if the heat conduction through the sidewalls was considered in the simulations.     

Thermal lattice-BGK model based on large-eddy simulation of turbulent natural convection due to internal heat generation

To simulate turbulent convection at high Rayleigh number (Ra), we propose a new thermal lattice-BGK (LBGK) model based on large eddy simulation (LES). Two-dimensional numerical simulations of natural convection with internal heat generation in a square cavity were performed at Ra from 10⁶ to 10¹³ with Prandtl numbers (Pr) at 0.25 and 0.60. Simulation results indicate that our model is fit to simulate high Ra flow for its better numerical stability. At Ra = 10¹³, a global turbulent has occurred. With a further increase in Ra, the flow will arrive in a fully turbulence regime. The Nusselt–Rayleigh relationship is also discussed.     

Effects of thermal boundary conditions on natural convection flows within a square cavity

A numerical study to investigate the steady laminar natural convection flow in a square cavity with uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls has been performed. A penalty finite element method with bi-quadratic rectangular elements has been used to solve the governing mass, momentum and energy equations. The numerical procedure adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra, 10³ ⩽ Ra ⩽ 10⁵ and Prandtl number Pr, 0.7 ⩽ Pr ⩽ 10) with respect to continuous and discontinuous Dirichlet boundary conditions. Non-uniform heating of the bottom wall produces greater heat transfer rates at the center of the bottom wall than the uniform heating case for all Rayleigh numbers; however, average Nusselt numbers show overall lower heat transfer rates for the non-uniform heating case. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominated regimes, power law correlations between average Nusselt number and Rayleigh numbers are presented.     

Numerical investigation of several physical and geometric parameters in the natural convection into trapezoidal cavities

Natural convection in trapezoidal cavities, especially those with two internal baffles in conjunction with an insulated floor, inclined top surface, and isothermal left-heated and isothermal right-cooled vertical walls, has been investigated numerically using the Element based Finite Volume Method (EbFVM). In numerical simulations, the effect of three inclination angles of the upper surface as well as the effect of the Rayleigh number (Ra), the Prandtl number (Pr), and the baffle’s height (H_b) on the stream functions, temperature profiles, and local and average Nusselt numbers has been investigated. A parametric study was performed for a wide range of Ra numbers (10³ ? Ra ? 10⁶) H_b heights (H_b = H¹/3, 2H¹/3, and H¹), Pr numbers (Pr = 0.7, 10 and 130), and top angle (θ) ranges from 10 to 20. A correlation for the average Nusselt number in terms of Pr and Ra numbers, and the inclination of the upper surface of the cavity is proposed for each baffle height investigated.     

Numerical study of turbulent double-diffusive natural convection in a square cavity by LES-based lattice Boltzmann model

Turbulent double-diffusive natural convection in a square cavity represents numerous important problems in practice as well as in fundamental. However up to date the study on it is quite sparse and most previous studies just focus on laminar regime. To the best knowledge of the present authors, only several k–? models were developed to investigate turbulent double-diffusive convection and there is no attempt to use Large Eddy Simulation (LES). In order to deepen our knowledge on turbulent double-diffusive convection in a square cavity, we propose a novel LES-based lattice Boltzmann (LB) model to simulate such turbulent convectional flow. Previous LES-based LB models can be recovered from the present model. We find that the symmetry of the fluid circulation becomes broken since the Rayleigh number Ra = 10⁸, although the asymmetry is more clear when Ra ? 10¹⁰. More important, in the present study we find the power-law relationship among the Nusselt (Nu), the ratio of buoyancy forces (N) and the Rayleigh number (Ra) still exists in turbulent regime. The formula among them can be concluded as Nu = a × (Ra × ∣1 ? N∣)^b + c. The values of parameters a, b and c are given in this work.     

Analysis of Bejan’s heatlines on visualization of heat flow and thermal mixing in tilted square cavities

This article analyzes the detailed heat transfer phenomena during natural convection within tilted square cavities with isothermally cooled walls (BC and DA) and hot wall AB is parallel to the insulated wall CD. A penalty finite element analysis with bi-quadratic elements has been used to investigate the results in terms of streamlines, isotherms and heatlines. The present numerical procedure is performed over a wide range of parameters (10³ ? Ra ? 10⁵,0.015 ? Pr ? 1000,0° ? φ ? 90°). Secondary circulations cells are observed near corner regions of cavity for all φ’s at Pr = 0.015 with Ra = 10⁵. Two asymmetric flow circulation cells are found to occupy the entire cavity for φ = 15° at Pr = 0.7 and Pr = 1000 with Ra = 10⁵. Heatlines indicate that the cavity with inclination angle φ = 15° corresponds to large convective heat transfer from the wall AB to wall DA whereas the heat transfer to wall BC is maximum for φ = 75°. Heat transfer rates along the walls are obtained in terms of local and average Nusselt numbers and they are explained based on gradients of heatfunctions. Average Nusselt number distributions show that heat transfer rate along wall DA is larger for lower inclination angle (φ = 15°) whereas maximum heat transfer rate along wall BC occur for higher inclination angle (φ = 75°).     

Steady natural convection flow in a square cavity filled with a porous medium for linearly heated side wall(s)

In this paper natural convection flows in a square cavity filled with a porous matrix has been investigated numerically when the bottom wall is uniformly heated and vertical wall(s) are linearly heated whereas the top wall is well insulated. Darcy–Forchheimer model without the inertia term is used to simulate the momentum transfer in the porous medium. Penalty finite element method with bi-quadratic rectangular elements is used to solve the non-dimensional governing equations. Numerical results are presented for a range of parameters (Rayleigh number Ra, 10³ ⩽ Ra ⩽ 10⁶, Darcy number Da, 10⁻⁵ ⩽ Da ⩽ 10⁻³, and Prandtl number Pr, 0.2 ⩽ Pr ⩽ 100) in terms of stream functions and isotherm contours, and local and average Nusselt numbers.     

Numerical simulations of periodic flow oscillations in low Prandtl number fluids

The transition from steady to oscillatory flow for a very low Prandtl number fluid (Pr = 0.008) is computed for rectangular enclosures with aspect ratios (length/height) of 0.25, 0.4, 1.0, and 2.0 and are found to occur at Rayleigh numbers of 250,000, 130,000, 83,500, and 30,000 respectively. The structures of the oscillations are graphically depicted and are manifested in corner cells which dissipate into centered cells and then into opposite corner cells. A secondary flow transition is detected for a geometry with an aspect ratio of 1.0 at Ra = 1.2Ra_c2.     

Effect of a magnetic field on natural convection in an open cavity subjugated to water/alumina nanofluid using Lattice Boltzmann method

In this paper, the effect of a magnetic field on natural convection in an open enclosure which subjugated to water/alumina nanofluid using Lattice Boltzmann method has been investigated. The cavity is filled with water and nanoparticles of Al₂O₃ at the presence of a magnetic field. Calculations were performed for Rayleigh numbers (Ra = 10⁴–10⁶), volume fractions of nanoparticles (φ = 0,0.02,0.04 and .0.06) and Hartmann number (0 ≤ Ha ≤ 90) with interval 30 while the magnetic field is considered horizontally. Results show that the heat transfer decreases by the increment of Hartmann number for various Rayleigh numbers and volume fractions. The magnetic field augments the effect of nanoparticles at Rayleigh number of Ra = 10⁶ regularly. Just as the most effect of nanoparticles for Ra = 10⁴ is observed at Ha = 30, so the most influence of nanoparticles occurs at Ha = 60 for Ra = 10⁵.     

Natural convection effects on heat and mass transfer in a curvilinear triangular cavity

Finite element method is used in this study to analyze the effects of buoyancy ratio and Lewis number on heat and mass transfer in a triangular cavity with zig-zag shaped bottom wall. Buoyancy ratio is defined as the ratio of Grashof number of solutal and thermal. Inclined walls of the cavity have lower temperature and concentration according to zig-zag shaped bottom wall. Enclosed space consists mostly of an absorber plate and two inclined glass covers that form a cavity. Both high temperature and high concentrations are applied to bottom corrugated wall. Computations were done for different values of buoyancy ratio (?10 ? Br ? 10), Lewis number (0.1 ? Le ? 20) and thermal Rayleigh number (10⁴ ? Ra_T ? 10⁶). Streamlines, isotherms, iso-concentration, average Nusselt and Sherwood numbers are obtained. It is found that average Nusselt and Sherwood numbers increase by 89.18% and 101.91% respectively as Br increases from ?10 to 20 at Ra_T = 10⁶. Also, average Nusselt decreases by 16.22% and Sherwood numbers increases by 144.84% as Le increases from 0.1 to 20 at this Rayleigh number.     

Treatment of turbulent heat fluxes with the elliptic-blending second-moment closure for turbulent natural convection flows

In this paper a comparative study on the treatment of the turbulent heat fluxes with the elliptic-blending second-moment closure for natural convection flows is presented. Three different cases for treating the turbulent heat fluxes are considered. Those are the generalized gradient diffusion hypothesis (GGDH), the algebraic flux model (AFM) and the differential flux model (DFM). These models are implemented in a computer code especially designed for an evaluation of turbulent models. Calculations are performed for turbulent natural convection flows in an 1:5 rectangular cavity (Ra = 4.3 × 10¹⁰) and in a square cavity with conducting top and bottom walls (Ra = 1.58 × 10⁹). The calculated results are compared with the available experimental data. The results show that the GGDH, AFM and DFM models produce sufficiently accurate solutions for the turbulent natural convection in an 1:5 rectangular cavity where the strength of the thermal stratification is weak in a central region of the cavity. However, the GGDH model produces very erroneous solutions for the turbulent natural convection in a square cavity with conducting walls where the Rayleigh number is relatively small and the thermally stratified region is dominant. The AFM and DFM produce very accurate solutions for both cases without invoking any numerical problems.     

Flow visualization of natural convection in a tall,air-filled vertical cavity

Natural convection of air in a tall vertical cavity was studied using a smoke patterns and interferometry. Experiments covered Rayleigh numbers of 4850 < Ra < 54,800 and aspect ratio A ≈ 40. Secondary cells were noted at Ra as low as 6228. The flow was stable at Ra < 10⁴. As Ra exceeded 10⁴ the flow became irregular, the core flow became increasingly unsteady and 3-D motion became evident. Interferometry showed that most of the temperature drop exists in boundary layers near the walls. The core is well mixed and of relatively uniform temperature with little or no vertical stratification.     

Numerical simulation of liquid metals in differentially heated enclosure undergoing orthogonal rotation

In this work a numerical investigation has been performed to examine the characteristics of mixed convective heat transfer in square enclosures undergoing orthogonal rotation i.e. rotation axis and gravity axis are orthogonal to each other. A semi implicit finite difference code on a collocated grid is used to solve the momentum and energy equations subject to Boussinesq approximation. The study is carried out for a wide range of operating parameters such as Rayleigh number (Ra), Taylor number (Ta), Rotational Rayleigh number (Ra_w) for a fixed Prandtl number (Pr). The numerical experiments have been carried out for a fixed Pr = 0.01, Ra varies from 10⁵ to 10⁷ while Ta and Ra_w vary from almost 0 to 10⁹. Results reveal that significant increase or decrease in heat transfer rates can be achieved by the rotational effects, mainly influenced by centrifugal force.     

Laminar natural convection in a square cavity: Low Prandtl numbers and large density differences

Steady natural convection at low Prandtl numbers caused by large density differences in a square cavity heated through the side walls is investigated numerically and theoretically. An appropriate dimensionless parameter characterizing the density differences of the working fluid is identified by the Gay-Lussac number. The Boussinesq assumption is achieved when the Gay-Lussac number tends to zero. The Nusselt number is derived for the ranges in Rayleigh number 10 ? Ra ? 10⁸, in Prandtl number 0.0071 ? Pr ? 7.1 and in Gay-Lussac number 0 ? Ga < 2. The effects of the Rayleigh, Prandtl and Gay-Lussac numbers on the Nusselt number are discussed on physical grounds by means of a scale analysis. Finally, based on physical arguments, a heat transfer correlation is proposed, valid for all Prandtl and Gay-Lussac number ranges addressed.     

Laminar natural convection of non-Newtonian power-law fluids between concentric circular cylinders

The two-dimensional steady-state natural convection of power-law fluids is studied numerically between two concentric horizontal cylinders with different constant temperatures. The governing equations are discretized using finite volume technique based on second order upwind and are solved using the SIMPLE algorithm. The effects of Rayleigh number (10³ ≤ Ra ≤ 10⁵) and Prandtl number (10 ≤ Pr ≤ 10³) on the dimensionless velocity and temperature are investigated for both pseudoplastic and dilatant fluids. Also the mean Nusselt number for various values of governing parameters is obtained and discussed. The results indicate that with increasing the power-law index from 0.6 to 1.4, the mean Nusselt number decreases. In the best case among the range of parameters considered here the heat transfer rate for pseudo-plastic fluid (n = 0.6) is 170% higher than the Newtonian one and for dilatant fluid (n = 1.4) the heat transfer rate is 43% lower than the Newtonian fluid. So the pseudoplastic and dilatant fluids are more efficient than Newtonian fluids for cooling and insulating purposes, respectively. It is shown that as the Rayleigh number increases the cooling effect of pseudoplastic fluid and the insulating effect of dilatant fluid become more pronounced.     

Magnetic field and internal heat generation effects on the free convection in a rectangular cavity filled with a porous medium

A numerical investigation of the steady magnetohydrodynamics free convection in a rectangular cavity filled with a fluid-saturated porous medium and with internal heat generation has been performed. A uniform magnetic field, inclined at an angle γ with respect to the horizontal plane, is externally imposed. The values of the governing parameters are the inclined angle γ = 0, π/6, π/4 and π/2, Hartmann number Ha = 0, 1, 5, 10 and 50, Rayleigh number Ra = 10, 100, 10³ and 10⁵, and the aspect ratio a = 0.01, 0.2, 0.5 and 1 (square cavity). It is shown that the intensity of the core convection is considerably affected by the considered parameters. It is also found that the local Nusselt number Nu_Y decreases on the bottom wall as γ increases (magnetic field changes its direction from the horizontal to the vertical direction) and vice versa for the top wall of the cavity. The reported results are in good agreement with the available published work in the literature.