Investigation of a large top wall temperature on the natural convection plume along a heated vertical wall in a square cavity

The characteristics of the laminar natural convection in an air-filled square cavity heated and cooled on the side walls was studied for cases where the temperature of the top wall was significantly larger than the heated vertical wall. Experiments were performed for a horizontal Grashof number of 1.3 × 10⁸, and non-dimensional top wall temperatures from 1.4 to 2.3. The results show that the plume formed on the heated vertical wall separated from this wall before reaching the top wall. As a result, three different regions were observed in the cavity: a stratified core region, a buoyant plume region, and a highly stratified region above the plume after it had separated from the vertical wall. The highly stratified region above the plume became larger and more stable with an increase of the top wall temperature, stabilizing the motion of the plume across the cavity. The similarity solutions developed by Kulkarni et al. [A.K. Kulkarni, H.R. Jacobs, J.J. Hwang, Similarity solution for natural convection flow over an isothermal vertical wall immersed in thermally stratified medium, Int. J. Heat Mass Transfer 30 (1987) 691–698] to characterize the natural convection heat transfer along an isothermal single vertical plate did not agree with the results for the current measurements; however, the non-similarity model of Chen and Eichhorn [C.C. Chen, R. Eichhorn, Natural convection from a vertical surface to thermally stratified fluid, J. Heat Transfer 98 (1976) 446–451] was in good agreement over most of the wall. There were some discrepancies in the temperature distributions and the heat transfer characteristics, especially at y/H ? 0.8 due to the separated flow in this region.     

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.     

Natural convection in a partially open square cavity with internal heat source: An analysis of the opening mass flow

A steady buoyancy-driven flow of air in a partially open square 2D cavity with internal heat source, adiabatic bottom and top walls, and vertical walls maintained at different constant temperatures is investigated numerically in this work. A heat source with 1% of the cavity volume is present in the center of the bottom wall. The cold right wall contains a partial opening occupying 25%, 50% or 75% of the wall. The influence of the temperature gradient between the verticals walls was analyzed for Ra_e = 10³–10⁵, while the influence of the heat source was evaluated through the relation R = Ra_i/Ra_e, investigated at between 400 and 2000. Interesting results were obtained. For a low Rayleigh number, it is found that the isotherm plots are smooth and follow a parabolic shape indicating the dominance of the heat source. But as the Ra_e increases, the flow slowly becomes dominated by the temperature difference between the walls. It is also observed that multiple strong secondary circulations are formed for fluids with a small Ra_e whereas these features are absent at higher Ra_e. The comprehensive analysis is concluded with horizontal air velocity and temperature plots for the opening. The numerical results show a significant influence of the opening on the heat transfer in the cavity.     

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.     

Laminar natural convection in an air-filled square cavity with partitions on the top wall

The laminar natural convection in an air-filled square cavity with a partition on the top wall was experimentally investigated. Temperature measurements and flow visualizations were performed for cases with heated and cooled vertical walls (corresponding to a global Grashof number Gr_H of approximately 1.3 × 10⁸) and non-dimensional top wall temperatures θ_T of 0.56 (insulated) to 2.3. Experiments were performed with an aluminum partition with non-dimensional height H_P/H of 0.0625 and 0.125 attached to the top wall at x/H = 0.1, 0.2, 0.4 and 0.6. The blockage effect and/or the thermal effect of the partition resulted in changes to the temperature and flow fields, but were mainly limited to the vicinity of the partition. The partition on the heated top wall resulted in a recirculating flow between the partition and the heated vertical wall. For a given partition height, the structure of this recirculating flow was dependent on the partition location and θ_T. A thermal boundary layer developed along the rear surface of the partition due to the thermal effect of the partition. The ambient temperature outside the boundary layer and Nu near the corner region was affected by the partition height due to the change in the recirculating flow and due to the thermal effect on the rear surface of the partition.     

Effects of moving lid direction on MHD mixed convection in a linearly heated cavity

Effects of moving lid-direction on MHD mixed convection in a cavity with the bottom wall being linearly heated are analyzed using a numerical technique. Vertical walls of the enclosure are adiabatic and the sliding wall at the top has constant temperature. The lid moves in the negative and positive x-direction. Finite volume method has been used to solve the governing equations. Results are presented for different values of Hartmann number (0 ? Ha ? 30), Reynolds number (100 ? Re ? 1000) and Grashof number (10⁴ ? Gr ? 10⁶). It is found that direction of lid is more effective on heat transfer and fluid flow in the case of mixed convection than it is the case in forced convection. Heat transfer is also decreased with increasing of magnetic field for all studied parameters.     

MHD natural convection in an enclosure from two semi-circular heaters on the bottom wall

A computational study of the thermal and dynamical behavior of fluid in an enclosure with two isothermal semi-circular heaters is presented. The top wall and the flat surfaces on bottom wall are adiabatic while the vertical walls are kept at lower temperature than the semi-circular heaters. The radius of curvature of the semi-circular surfaces is chosen as one tenth of the cavity wall length. The governing equations are solved by the Galerkin weighted residual finite element method. The effect of magnetic field on the flow is another important parameter in this study. Numerical simulations were performed for several values of Rayleigh number (10³ ? Ra ? 10⁶), Hartmann number (0 ? Ha ? 50) and the distance between two semi-circular heaters (0.2 ? D ? 0.8). In all cases the Prandtl number is taken as 7. It is found that the distance between the semi-circular heaters is the most important parameter affecting the heat and fluid flow fields. In addition, Hartmann number was found to have an adverse affect on heat transfer.     

Visualization of natural convection heat transport using heatline method in porous non-isothermally heated triangular cavity

Natural convection heat transfer in a porous media filled and non-isothermally heated from the bottom wall of triangular enclosure is analyzed using finite difference technique. Darcy law was used to write equations of porous media. Dimensionless heatfunction was used to visualize the heat transport due to buoyancy forces. Three different boundary conditions were applied for the vertical and inclined boundaries of triangular enclosures as Case I; both vertical and inclined walls were isothermal, Case II; vertical wall was adiabatic and inclined one was isothermal, Case III; vertical wall is isothermal and inclined one is adiabatic. A cosine function was utilized to get non-isothermal wall condition. The study was performed for different aspect ratios (0.25 ? AR ? 1.0) and Darcy-modified Rayleigh numbers (100 ? Ra ? 1000). It was observed that heat transfer enhancement was formed when vertical and inclined walls were isothermal while bottom wall was at non-uniform temperature. Heat transfer from bottom wall did not vary when the value of aspect ratio was higher than 0.50. In addition, heatline visualization technique was a useful technique for non-isothermally heated and porous media filled triangular enclosures.     

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.     

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.     

Investigation of mixed convection heat transfer in a horizontal channel with discrete heat sources at the top and at the bottom

Mixed convection heat transfer from arrays of discrete heat sources inside a horizontal channel has been investigated experimentally. Each of the lower and upper surfaces of the channel was equipped with 8 × 4 flush mounted heat sources subjected to uniform heat flux. Sidewalls, lower and upper walls are insulated and adiabatic. The experimental parametric study was made for aspect ratios of AR = 2, 4 and 10, at various Reynolds and Grashof numbers. From the experimental measurements, row-average surface temperature and Nusselt number distributions of the discrete heat sources were obtained and effects of Reynolds and Grashof numbers on these numbers were investigated. From these results, the buoyancy affected secondary flow and the onset of instability have been discussed. Results show that top and bottom heater surface temperatures increase with increasing Grashof number. The top heater average-surface temperatures for AR = 2 are greater than those of bottom ones. For high values of Grashof numbers where natural convection is the dominant heat transfer regime (Gr¹/Re² ≫ 1), temperatures of top heaters can have much greater values. The variation of the row-average Nusselt numbers for the aspect ratio of AR = 4, show that with the increase in the buoyancy affected secondary flow and the onset of instability, values of Nusselt number level off and even rise as a result of heat transfer enhancement especially for low Reynolds numbers.     

Natural convection in an enclosure with a localized nonuniform heat source on the bottom wall

Natural convection in an air filled enclosure with a localized nonuniform heat source mounted centrally on the bottom wall is numerically investigated. The vertical walls are cooled while the top wall and the remaining portions of the bottom wall are insulated. The heat source is assumed to be isothermal with a linearly varying temperature. The governing equations were solved using finite volume method on a uniformly staggered grid system. The computational results are presented in the form of isotherm and streamline plots and Nusselt numbers. The effects of the source nonuniformity parameter, λ and the line source length, ε are investigated for the Grashof numbers Gr = 10⁶ and 10⁷. It is found that for Gr = 10⁶ nonuniform heating of the line source enhances the overall heat transfer rate markedly compared to uniform heating of the heat source whereas for Gr = 10⁷ its effect is marginal.     

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.     

Time evolution of double-diffusive convection in a vertical cylinder with radial temperature and axial solutal gradients

The characteristics of transient double-diffusive convection in a vertical cylinder are numerically simulated using a finite element method. Initially the fluid in the cavity is at uniform temperature and solute concentration, then constant temperature and solute concentration, which are lower than their initial values, are imposed along the sidewall and bottom wall, respectively. The time evolution of the double-diffusive convection is investigated for specific parameters, which are the Prandtl number, Pr = 7, the Lewis number, Le = 5, the thermal Grashof number, Gr_T = 10⁷, and the aspect ratio, A = 2, of the enclosure. The objective of the work is to identify the effect of the buoyancy ratio (the ratio of solutal Grashof to thermal Grashof numbers: N = Gr_S/Gr_T) on the evolution of the flow field, temperature and solute field in the cavity. It is found that initially the fluid near the bottom wall is squeezed by the cold flow from the sidewall, a crest of the solute field forms and then pushed to the symmetry line. In the case of N > 0, a domain with higher temperature and weak flow (dead region) forms on the bottom wall near the symmetry line, and the area of dead region increases when N varies from 0.5 to 1.5. More crests of the solute field are formed and the flow near the bottom wall fluctuates continuously for N < 0. The frequency of the fluctuation increases when N varies from −0.5 to −1.5. Corresponding to the variety of the thermal and solutal boundary layers, the average rates of heat transfer (Nu) at the sidewall remain almost unchanged while the average rates of mass transfer (Sh) at the bottom wall change much in the cases of N = 1, 0, −1.     

Vortex formation and heat transfer in turbulent flow past a transverse cavity with inclined frontal and rear walls

The process of vortex formation, distributions of pressure coefficients, and convective heat transfer in a turbulent flow past a cavity with a low aspect ratio and inclined frontal and rear walls were experimentally studied. The angle of wall inclination φ was varied in the interval from 30° to 90°. Visualization techniques were applied to trace the evolution of the flow with the angle φ as the transverse cavity became more open. Pressure fields in the longitudinal and transverse sections on the bottom wall of the cavity, and on its frontal and rear walls, were measured. The measured distributions of temperature in the longitudinal and transverse sections on the three heated walls, and the obtained thermographic fields over the whole heated surface, were used to calculate local and average heat-transfer coefficients. It is found that in the interval of wall inclination angles φ = 60–70° the flow in the cavity becomes unstable, with the primary vortex changing its structure from single-cellular to double-cellular. As a result, the distributions of static pressure and surface temperature across and along the cavity suffer dramatic changes. At smallest angles φ the flow re-attachment point gets displaced into the cavity to cause an abrupt growth of pressure and heat-transfer coefficients on the rear wall, which leads to a slight increase of the surface-mean pressure and heat transfer inside the cavity. At the angle of instability, φ = 60°, the local heat-transfer coefficient decreases markedly over the cavity span from the end faces of the cavity toward its center, and a most pronounced intensification of heat transfer is observed.     

Analysis of combined thermal and magnetic convection ferrofluid flow in a cavity

A numerical analysis of the magnetic gradient and thermal buoyancy induced cavity ferrofluid flow is conducted by a semi-implicit finite element method. The physical model for a square cavity containing two different temperature side walls and a magnet near bottom wall is described by mass, momentum and energy equations. Conditions for the fixed Prandtl number, Rayleigh number and different ferro-hydrodynamic interaction parameter are studied for 5 × 10⁸ ≤ β ≤ 1.6 × 10¹⁰. Results show the flow strength increases with the strengthening magnetic field. However, the side-wall heat transfer rate presents a decrease trend with the increase in magnetic field strength, for the magnet located near the bottom central area evokes the circulation to move toward the central portion. In summary, a proper choice of magnet strength and location can adjust the flow field and local heat transfer rate to fit the practical application.     

Experimental investigation of mixed convection heat transfer in a rectangular channel with discrete heat sources at the top and at the bottom

Mixed convection heat transfer in a top and bottom heated rectangular channel with discrete heat sources has been investigated experimentally for air. The lower and upper surfaces of the channel were equipped with 8 × 4 flush-mounted heat sources subjected to uniform heat flux. Sidewalls, the lower and upper walls were insulated and adiabatic. The experimental study was made for an aspect ratio of AR = 6, Reynolds numbers 955 ≤ Re_Dh ≤ 2220 and modified Grashof numbers Gr* = 1.7 × 10⁷ to 6.7 × 10⁷. From experimental measurements, surface temperature and Nusselt number distributions of the discrete heat sources were obtained for different Grashof numbers. Furthermore, Nusselt number distributions were calculated for different Reynolds numbers. Results show that surface temperatures increase with increasing Grashof number. The row-averaged Nusselt numbers first decrease with the row number and then, due to the increase in the buoyancy affected secondary flow and the onset of instability, they show an increase towards the exit as a result of heat transfer enhancement.