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.     

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.     

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 simulation of natural convection heat transfer in a porous cavity heated from below using a non-Darcian and thermal non-equilibrium model

The present paper investigates the numerical simulation of steady laminar incompressible natural convection heat transfer in an enclosed cavity that is filled with a fluid-saturated porous medium. The bottom wall is subjected to a relatively higher temperature than the top wall while the vertical walls are considered to be insulated. The flow field is modeled upon incorporating different non-Darcian effects, such as the convective term, Brinkman effect and Forchhiemer quadratic inertial effect. Moreover the two-equation model is used to separately account for the local fluid and solid temperatures. The numerical solution is obtained through the application of the finite volume method. The appraisals of the sought objectives are performed upon identifying key dimensionless groups of parameters. These dimensionless groups along with their operating domains are: Rayleigh number 1 ⩽ Ra ⩽ 400, Darcy number 10⁻⁴ ⩽ Da ⩽ 10⁻³, effective fluid-to-solid thermal conductivity ratio 0.1 ⩽ κ ⩽ 1.0, and the modified Biot number 1 ⩽ χ ⩽ 100. The non-Darcian effects are first examined over a broad range of Rayleigh number. Next, the implications of the group of parameters on the flow circulation intensity, local thermal non-equilibrium (LTNE) and average Nusselt number are highlighted and pertinent observations are documented.     

Entropy generation in natural convection in a symmetrically and uniformly heated vertical channel

In this study numerical predictions of local and global entropy generation rates in natural convection in air in a vertical channel symmetrically heated at uniform heat flux are reported. Results of entropy generation analysis are obtained by solving the entropy generation equation based on the velocity and temperature data. The analyzed regime is two-dimensional, laminar and steady state. The numerical procedure expands an existing computer code on natural convection in vertical channels. Results in terms of fields and profiles of local entropy generation, for various Rayleigh number, Ra, and aspect ratio values, L/b, are given. The distributions of local values show different behaviours for the different Ra values. A correlation between global entropy generation rates, Rayleigh number and aspect ratio is proposed in the ranges 10³ ⩽ Ra ⩽ 10⁶ and 5 ⩽ L/b ⩽ 20.     

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.     

Jet impingement interaction with cross flow in horizontal porous layer under thermal non-equilibrium conditions

In the present article the jet impingement cooling of heated portion of a horizontal surface immersed in a thermally non-equilibrium porous layer is considered for investigation numerically with the presence of a cross flow. The mathematical model is derived for steady, two-dimensional laminar flow based on Darcy model and two-energy equation for fluid and solid phases. A parametric study is carried out by varying the following parameters: cross flow to jet flow velocity ratio parameter (0 ⩽ M ⩽ 1); porosity scaled thermal conductivity ratio parameter (0.1 ⩽ K_r ⩽ 1000); heat transfer coefficient parameter (0.1 ⩽ H ⩽ 1000); Péclet number (1 ⩽ Pe ⩽ 1000) and Rayleigh number (10 ⩽ Ra ⩽ 100). The total average Nusselt number is defined based on the overall thermal conductivity, which is assumed to be the arithmetic mean of the porosity scaled thermal conductivity of the fluid and solid phases. The total average Nusselt number as well as the average Nusselt number for both fluid and solid phases is presented for different governing parameters. It is found that the presence of a weak cross flow in a jet impinging jet may degrade the heat transfer. The results show that the average Nusselt number calculated from the thermal equilibrium model are the maximum possible values and these values can be reproduced by large values of H × K_r.     

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.     

Natural convective flow in an inclined lid-driven enclosure with a heated thin plate in the middle

Natural convection heat transfer from a heated thin plate located in the middle of a lid-driven inclined square enclosure has been analyzed numerically. Left and right of the cavity are adiabatic, the two horizontal walls have constant temperature lower than the plate’s temperature. The study is formulated in terms of the vorticity-stream function procedure and numerical solution was performed using a fully higher-order compact (FHOC) finite difference scheme on the 9-point 2D stencil. Air was chosen as a working fluid (Pr = 0.71). Two cases are considered depending on the position of heated thin plate (Case I, horizontal position; Case II, vertical position). Governing parameters, which are effective on flow field and temperature distribution, are Rayleigh number values (Ra) ranging from 10³ to 10⁵ and inclination angles γ (0° ? γ < 360°). The fluid flow, heat transfer and heat transport characteristics were illustrated by streamlines, isotherms and Nusselt number (Nu). It is found that fluid flow and temperature fields strongly depend on Rayleigh numbers and inclination angles. Further, for the vertical located position of thin plate heat transfer becomes more enhanced with lower γ at various Rayleigh numbers.     

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.     

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.     

Natural convection heat transfer enhancement using adiabatic block: Optimal block size and Prandtl number effect

Numerical methods are used to solve the finite volume formulation of the two-dimensional mass, momentum and energy equations for steady-state natural convection inside a square enclosure. The enclosure consists of adiabatic horizontal walls and differentially heated vertical walls, but it also contains an adiabatic centrally-placed solid block. The aim of the study is to delineate the effect of such a block on the flow and temperature fields. The parametric study covers the range 10³ ⩽ Ra ⩽ 10⁶ and is done at three Pr namely, 0.071, 0.71 and 7.1. In addition the effect of increasing the size (characterized by the solidity Φ) of the adiabatic block is ascertained. It is found that the wall heat transfer increases, with increase in the Φ, until it reaches a critical value Φ = Φ_OPT, where the wall heat transfer attains its maximum. Further increases in the block size beyond Φ_OPT, reduces the wall heat transfer, for as the block size becomes larger than the conduction dominant core size it reduces the thermal mass of the convecting fluid. A steady-state heat transfer enhancement of 10% is observed for certain Ra and Pr values. Useful correlations predicting this optimum block size and the corresponding maximum heat transfer as a function of Ra and Pr are proposed; these predict within ±3%, the numerical results.     

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.     

Steady forced convection heat transfer from a heated circular cylinder to power-law fluids

Forced convection heat transfer to incompressible power-law fluids from a heated circular cylinder in the steady cross-flow regime has been investigated numerically by solving the momentum and thermal energy equations using a finite volume method and the QUICK scheme on a non-uniform Cartesian grid. The dependence of the average Nusselt number on the Reynolds number (5 ⩽ Re ⩽ 40), power-law index (0.6 ⩽ n ⩽ 2) and Prandtl number (1 ⩽ Pr ⩽ 1000) has been studied in detail. The numerical results are used to develop simple correlations as functions of the pertinent dimensionless variables. In addition to the average Nusselt number, the effects of Re, Pr and n on the local Nusselt number distribution have also been studied to provide further physical insights. The role of the two types of thermal boundary conditions, namely, constant temperature and uniform heat flux on the surface of the cylinder has also been presented.     

Investigation of natural circulation in cavities with uniform heat generation for different Prandtl number fluids

Natural convection in enclosures with uniform heat generation and isothermal side walls is studied here. For the rectangular enclosure, two-dimensional conservation equations are solved using SIMPLE algorithm. Parametric studies are conducted to examine the effects of orientation of the cavity, fluid properties (Pr number), and aspect ratio for Rayleigh numbers up to 10⁶. For a horizontal square cavity, the flow becomes periodically oscillating at Ra = 5 × 10⁴ and chaotic at Ra = 8 × 10⁵. With a slight increase in the inclination angle, the oscillations die and for inclination angles greater than 15⁰, the flow attain a steady state over a range of Ra. It is found that for tall cavities (aspect ratio > 1), the steady-state solution is obtained for all values of Ra considered here. However, for wide cavities (aspect ratio < 1), an oscillatory flow regime is observed. The maximum temperature within the cavity is calculated for the range of Ra, aspect ratio and Pr number. Correlations for the maximum cavity temperature is presented here. The values of critical Rayleigh number at which the convection sets in the rectangular cavity are also studied and two distinct criteria are determined to evaluate the critical Rayleigh number. Further, a three-dimensional simulation is performed for a cubic cavity. It is found that the steady state solutions are obtained for all Rayleigh number, except at Ra = 10⁶. This is in contrast to the predictions for a two-dimensional square cavity, which has an oscillatory zone from Ra = 5 × 10⁴ onwards.     

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.     

Natural convection and fluid flow in inclined enclosure with a corner heater

The phenomena of natural convection in an inclined square enclosure heated via corner heater have been studied numerically. Finite difference method is used for solving momentum and energy equations in the form of stream function–vorticity. One wall of the enclosure is isothermal but its temperature is colder than that of heaters while the remaining walls are adiabatic. The numerical procedure adopted in this analysis yields consistent performance over a wide range of parameters; Rayleigh number, Ra (10³ ? Ra ? 10⁶); Prandtl number, Pr (0.07 ? Pr ? 70); dimensionless lengths of heater in x and y directions (0.25 ? hx ? 0.75, 0.25 ? hy ? 0.75); and inclination angle, ? (0° ? ? ? 270°). It is observed that heat transfer is maximum or minimum depending on the inclination angle and depending on the length of the corner heaters. The effect of Prandtl number on mean Nusselt number is more significant for Pr < 1.     

Jet impingement cooling of a horizontal surface in an unconfined porous medium: Mixed convection regime

Two-dimensional slot jet impingement cooling of an isothermal horizontal surface immersed in an unconfined porous medium is simulated numerically to gain insight into thermal characteristics under mixed convection conditions with the limitation of the Darcy model. The jet direction is considered to be perpendicular from the top to the horizontal heated element; therefore, the jet flow and the buoyancy driven flow are in opposite directions. The results are presented in the mixed convection regime with wide ranges of the governing parameters: Péclet number (1 ? Pe ? 1000), Rayleigh number (10 ? Ra ? 100), half jet width (0.1 ? D ? 0.5), and the distance between the jet and the heated portion (0.1 ? H ? 1.0). It is found that the average Nusselt number increases with increase in either Rayleigh number or jet width for high values of Péclet number. The average Nusselt number also increases with decrease in the distance between the jet and the heated portion. It is shown that mixed convection mode can cause minimum average Nusselt number at two values of Péclet number and a maximum average Nusselt number occurs in between theses two Péclet numbers at higher Rayleigh number due to counteraction of jet flow against buoyancy driven flow. Hence careful consideration must be given while designing a system of jet impingement cooling through porous medium.     

Natural convection in a nanofluid-filled eccentric annulus with constant heat flux wall: A lattice Boltzmann study with immersed boundary method

A numerical investigation of natural convection in a Cu–water nanofluid-filled eccentric annulus with constant heat flux wall is presented. The governing equations of the flow and temperature fields are solved by lattice Boltzmann method (LBM), and the Dirichlet and Neumann boundary conditions are treated using the immersed boundary method (IBM). Influences of the Rayleigh number (10³ ≤ Ra ≤ 10⁷), eccentricity (ε = −0.625,0 and 0.625), nanoparticles volume fraction (0 ≤ ϕ ≤ 0.03) and radial ratio (rr = 2.33,2.6 and 3) on the streamlines, isotherms and Nusselt number are studied. It is found that the inclusion of the nanoparticles into pure fluid changes the flow pattern. And the Nusselt number has a positive relationship with nanoparticle volume fraction, Rayleigh number and radial ratio. Also, it can be confirmed that Nusselt number in the case with negative eccentricity (ε = −0.625) is larger than the others.     

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°).