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.     

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.     

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°).     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Natural convection in a rectangular enclosure containing an oval-shaped heat source and filled with Fe3O4/water nanofluid

This work concerns with the study of natural convection heat transfer in rectangular cavities with an inside oval-shaped heat source filled with Fe₃O₄/water nanofluid. The finite element method is employed to solve the governing equations for this problem. Average Nusselt numbers are presented for a wide range of Rayleigh number (10³ ≤ Ra ≤ 10⁵), volume fraction of nanoparticles (0 ≤ ϕ ≤ 14%), and four different size and shapes of the heat source. Depending on concentration of the nanoparticle, geometry of the heat source, and the value of Rayleigh number different behaviors are monitored for average Nusselt numbers. Configuration of the heat source dictates a significant change on the behavior of the average Nusselt number, while addition of the nanoparticles has a negative effect on the magnitude of Nusselt number for this problem.     

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.     

Heat transfer by free convection from the inside surface of the vertical and inclined elliptic tube

Free convection from the inside surface of vertical and inclined elliptic tubes of axis ratio (a:b) 2:1 with a uniformly heated surface (constant heat flux) is investigated experimentally. The effects of orientation angle (α) and inclination angle (ϕ) on the heat transfer coefficient were studied. The orientation angle (α) is varied from 0° (when the major axis is horizontal) to 90° (when the major axis is vertical) with steps of 15°. The inclination angle (ϕ) is measured from the horizontal and varied from 15° to 75° with steps of 15°. The vertical position is considered as a special case of the inclined case when ϕ = 90. The experiments covered a range of Rayleigh number, Ra from 2.6 × 10⁶ to 3.6 × 10⁷. The local and average Nusselt numbers are estimated for different orientation angles and inclination angles at different Rayleigh numbers. The results obtained showed that the local Nu increased with the increase of axial distance from the lower end of the elliptic tube until a maximum value near the upper end, and then, it gradually decreased. The average Nu increases with the increase of α or ϕ at the same Ra. The results obtained are correlated by dimensionless groups and with the available data of the inclined and vertical elliptic tubes.     

Comparative evaluation of the performance of sinuous symmetrically-heated natural convection channels

Buoyancy-driven flows established in vertical channels are investigated numerically. Two different sinuous morphologies for a vertical channel are proposed as alternatives to a typical (straight) vertical channel; one is a sinus-shaped channel, and the other is a three-segment channel. The objective is to carry out a comparative evaluation of the thermal and dynamic performance of the proposed configurations, comparing their performance with that corresponding to a typical vertical channel. The low-Reynolds k − ω turbulence model is employed to simulate the transitional or fully turbulent flow. The average Nusselt number and the dimensionless mass flow rate are obtained for Rayleigh numbers ranging 10² ≤ Ra_H ≤ 10¹⁵, with an aspect ratio of the channel equal to 0.1. Sizeable increases in the Nusselt number and the mass flow rate are found for the sinuous configurations, under given conditions.     

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.     

Role of finite element based grids and simulations on evaluation of Nusselt numbers for heatfunctions within square and triangular cavities involving multiple discrete heaters

Nusselt number is an important non-dimensional parameter which quantifies the heat transfer rate. Local Nusselt number is useful in predicting the heat transfer rate along the various hot and cold sections of the side walls in a discretely heated enclosed cavity. In addition, the overall heat balance in an enclosed cavity (total heat delivered by the hot isothermal walls should be equal to the total heat gained by the cold isothermal walls) can be validated via the average Nusselt numbers. Current finite element based simulations and post-processing have been carried out in order to analyze the influence of the multiple heaters on the Nusselt number along various sections (hot and cold) of the side walls in discretely heated square and triangular (design 1 and design 2) cavities. The working fluid is considered to be air (Pr = 0.7) and the numerical studies have been carried out for a large range of Rayleigh number (Ra = 10³–10⁵) for four different biquadratic elements (24 × 24, 28 × 28, 32 × 32 and 34 × 34). The current work also estimates the fractional error in the heat balance (ϵ) and it is clearly inferred that ϵ is comparatively lower for 34 × 34 biquadratic elements. Current work also reveals that the fractional error (ϵ) is mainly induced due to the sharp variations in the Nusselt number at the cold-hot junctions along the side walls. The present study also involves the detailed evaluation of the heatfunction (Π) expressions along the cold-hot junctions of the side walls. The computations of the heatfunctions are intrinsically related to the Nusselt numbers of the hot-cold junctions.     

The effect of inclination on impinging jets at small nozzle-to-plate spacing

The effect of inclination on heat transfer characteristics of an impinging slot air jet is experimentally investigated. The effects of inclination angle (0° ? θ ? 40°) and dimensionless pumping power on the Nusselt number are considered. The focus is on cases where the nozzle-to-plate spacing is equal to or less than one nozzle diameter (H/d_h ? 1.0). The results show that the heat transfer characteristics of small nozzle-to-plate spacings are significantly different from those of large nozzle-to-plate spacings. In the cases of fixed flow rate conditions, the impingement point and average Nusselt numbers at small nozzle-to-plate spacing (H/d_h ? 1.0) increase as the inclination angle increases due to an increase in the pumping power, while the impingement point and average Nusselt numbers at large nozzle-to-plate spacing (H/d_h > 1.0) decrease as the inclination angle increases due to momentum loss of the wall jet. In the cases of fixed pumping power conditions, the impingement point and average Nusselt numbers at both of small and large nozzle-to-plate spacings are independent of the inclination angle. Based on the experimental results, correlations for the impingement point and average Nusselt numbers of the impinging jet are suggested as a function of the pumping power alone.     

Numerical study on effect of vent locations on natural convection in an enclosure with an internal heat source

Natural convection is a widely studied phenomenon because of the extensive applications in cooling of large scale electrical and electronic equipments. The current study involves study of effect of vent locations on natural convection in enclosures with partial openings having an internal heat source. It involves the numerical simulation of 2D steady state natural convection in enclosure of different aspect ratios (H/W = 1, 2 and 3) for lower Rayleigh numbers (Ra_h = 10³, 10⁴ and 10⁵). Four different configurations have been considered based on the number and position of vents — same side (SS), diagonal side (DS), one inlet two outlets (1I2O) and two inlets one inlet (2I1O). The mass flow rate driven through the enclosure and the average Nusselt number over the heater surface for all the four configurations have been compared. It is found that the 2I1O configuration yielded better heat transfer rates of the four considered. It was found that the mass flow rates and Nu increased with increase in Ra_h and decrease in the aspect ratio.     

Simulation of turbulent natural convection in a porous cylindrical annulus using a macroscopic two-equation model

This work presents numerical computations for laminar and turbulent natural convection within a horizontal cylindrical annulus filled with a fluid saturated porous medium. Computations covered the range 25 < Ra_m < 500 and 3.2 × 10⁻⁴ > Da > 3.2 × 10⁻⁶ and made use of the finite volume method. The inner and outer walls are maintained at constant but different temperatures. The macroscopic k–ε turbulence model with wall function is used to handle turbulent flows in porous media. First, the turbulence model is switched off and the laminar branch of the solution is found when increasing the Rayleigh number, Ra_m. Subsequently, the turbulence model is included and calculations start at high Ra_m, merging to the laminar branch for a reducing Ra_m. This convergence of results as Ra_m decreases can be seen as an estimate of the so-called laminarization phenomenon. Here, a critical Rayleigh number was not identified and results indicated that when the porosity, Prandtl number, conductivity ratio between the fluid and the solid matrix and Ra_m are kept fixed, the lower the Darcy number, the higher is the difference of the average Nusselt number given by the laminar and turbulent models.