Laminar natural convection in inclined enclosures bounded by a solid wall

Laminar natural convection has been studied in enclosures bounded by a solid wall with its outer boundary at constant temperature while the opposing side has a constant heat flux. Two-dimensional equations of conservation of mass, momentum and energy, with the Boussinesq approximation are solved using a finite difference method. The numerical procedure adopted is based on the SIMPLER algorithm. Various parameters were: Rayleigh number (from 10³ to 10⁶), dimensionless conductivity of bounding wall (from 1 to 10) and dimensionless wall width (from 0.15 to 0.5), aspect ratio (from 0.5 to 1) and the inclination angle (from 30^° to 180^°). The results are reduced in terms of the normalized Nusselt number as a function of the Rayleigh number, and other dimensionless parameters. The isotherms and streamlines are produced for various Rayleigh numbers and geometrical conditions. It is found that the heat transfer is an increasing function of the Rayleigh number, wall to fluid conductivity ratio, enclosure aspect ratio and a decreasing function of the wall thickness. It passes from a maximum for the inclination angle of about 80^°.     

Laminar natural convection in a partially divided rectangular cavity at high Rayleigh number

Finite element predictions of two-dimensional laminar natural convection in a partially divided rectangular cavity at high Rayleigh number are presented. The walls are differentially heated, the horizontal surfaces are insulated and the cavity contains a partial vertical divider which is centrally located and whose height is varied. Detailed results are presented for an aluminium half-divider in water for Rayleigh number up to 10¹¹ and compared directly with recent experiments in a cavity of aspect ratio 1/2. The predicted flow and heat transfer are in good agreement with the measurements and confirm the existence of a high Rayleigh number regime with characteristic behaviour that differs significantly from that found at lower Rayleigh number. In addition, the effects of the divider height, the divider conductivity, the fluid Prandtl number and the cavity aspect ratio are studied. The results show that a direct simulation of the complex flow and heat transfer that occurs in partially divided cavities is possible for realistic physical conditions.     

Analytical and numerical study of natural convection heat transfer in an inclined composite enclosure

Laminar natural convection heat transfer in inclined fluid layers divided by a partition with finite thickness and conductivity is studied analytically and numerically. The governing equations for the fluid layers are solved analytically in the limit of a thin layered system with constant flux boundary conditions. The study covers of the range of Ra from 10³ to 10⁷, from 0° to 180° and the thermal conductivity ratio of partition to fluid ratioK from 10^–2 to 10⁶. The Prandtl number was 0.72 (for air). Results are obtained in terms of an overall Nusselt number as a function of Rayleigh number, angle of inclination of the system, mid layer thickness, and mid layer thermal conductivity. The critical Rayleigh number for the onset of convection in a bottom-heated horizontal system is predicted. The results are compared with the numerical results obtained by solving the complete system of governing equations, using SIMPLER method, as well as with the limiting cases in the literature.     

Natural convection in inclined rectangular enclosures with perfectly conducting fins attached on the heated wall

The natural convection heat transfer in inclined rectangular enclosures with perfectly conducting fins attached to the heated wall is numerically studied. The parameters governing this problem are the Rayleigh number (10²≤Ra≤2×10⁵), the aspect ratio of the enclosures (2.5≤A=H′/L′≤∞), the dimensionless lengths of the partitions (0≤B=?′/L′≤1), the aspect ratio of micro-cavities (A≤C=h′/L′≤0.33), the inclination angle (0≤φ≤60^°) and the Prandtl number (Pr=0.72). The results indicate that the heat transfer through the cover is considerably affected by the presence of the fins. At low Rayleigh numbers, the heat transfer regime is dominated by conduction. When B≈0.75 and C≈0.33, the heat transfer through the cold wall decreases considerably. This trend is enhanced when the enclosure is inclined. Useful engineering correlations are derived for practical applications.     

Natural convection flow of water near its density maximum in a rectangular enclosure having isothermal walls with heat generation

We consider unsteady laminar natural convection flow of water subject to density inversion in a rectangular cavity formed by isothermal vertical walls with internal heat generation. The top and bottom horizontal walls are considered to be adiabatic, whereas the temperature of the left vertical wall is assumed to be greater than that of the right vertical wall. The equations are non-dimensionalized and are solved numerically by an upwind finite difference method together with a successive over-relaxation (SOR) technique. The effects of both heat generation and variations in the aspect ratio on the streamlines, isotherms and the rate of heat transfer from the walls of the enclosure are presented. Investigations are performed for water taking Prandtl number to be Pr=11.58 and the Rayleigh number to be Ra=10⁵.     

Conjugate Natural Convection in a Porous Enclosure with Non-Uniform Heat Generation

Effects of a conductive wall on natural convection in a square porous enclosure having internal heating at a rate proportional to a power of temperature difference is studied numerically in this article. The horizontal heating is considered, where the vertical walls heated isothermally at different temperatures while the horizontal walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and finite difference method is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (0 ???Ra ???1000), the internal heating and the local exponent parameters (0 ????? ???5), (1 ????? ???3), the wall to porous thermal conductivity ratio (0.44 ???Kr ???9.9) and the ratio of wall thickness to its width (0.02 ???D ???0.5). The results are presented to show the effect of these parameters on the fluid flow and heat transfer characteristics. It is found a strong internal heating can generate significant maximum fluid temperature more than the conductive solid wall. Increasing value thermal conductivity ratio and/or decreasing the thickness of solid wall can increase the maximum fluid temperature. It is also found that at very low Rayleigh number, the heat transfer across the porous enclosure remain stable for any values of the thermal conductivity ratio.     

Conjugate heat transfer within a concentric annulus filled with micropolar fluid

This paper investigates numerically the conjugate heat transfer in an annulus between two concentric cylinders. The annulus contains micropolar fluid and is heated isothermally from its inner wall. The effect of Rayleigh number, thickness of inner wall, inner wall-fluid thermal conductivity ratio, and material parameters of micropolar fluid on heat transfer rate within the annulus has been investigated. The study has shown that for low Rayleigh number regimes and for thermal conductivity of the inner wall greater than that of the fluid, the increase of inner wall thickness increases the heat transfer rate through the annulus and vice versa. While for convection dominating regimes Ra ≥ 10⁴ the increase of inner wall thickness decreases the heat transfer rate. Moreover, the study has shown that for fixed geometrical and flow parameters the heat transfer decreases in case of micropolar fluids in comparison with that of Newtonian fluids.     

Conjugate natural convection in a square porous cavity

 Steady-state conjugate natural convection in a square cavity filled with a porous medium is studied numerically in this paper. The enclosure consists of two horizontal conductive walls of finite thickness and two vertical walls at different uniform temperatures. The focus is on the role of solid-fluid conductivity ratio, k, on the flow and heat transfer characteristics and the average Nusselt number, , over the vertical hot and cold walls of the cavity for a limited set of particular parameters. It was shown that the interface temperature, θ_w, along the top of the solid wall decreases with the increase in the wall conductivity k. Also, the values of decreases with the increase of the values of the parameter k. Comparison with known results from the open literature when the wall thickness of the horizontal solid walls is neglected (non-conjugate problem) is excellent. Received on 4 April 2000     

Natural convection in an inclined quadrantal cavity heated and cooled on adjacent walls

Natural convective flow and heat transfer in an inclined quadrantal cavity is studied experimentally and numerically. The particle tracing method is used to visualize the fluid motion in the enclosure. Numerical solutions are obtained via a commercial CFD package, Fluent. The working fluid is distilled water. The effects of the inclination angle, ? and the Rayleigh number, Ra on fluid flow and heat transfer are investigated for the range of angle of inclination between 0° ? ? ? 360°, and Ra from 10⁵ to 10⁷. It is disclosed that heat transfer changes dramatically according to the inclination angle which affects convection currents inside, i.e. flow physics inside. A fairly good agreement is observed between the experimental and numerical results.     

Lattice Boltzmann simulation of natural convection in open end cavity with inclined hot wall

Natural convection in an open end cavity with a hot inclined wall is simulated based on the lattice Boltzmann method (LBM). The physics of flow and energy transfer in open end cavities are addressed when the hot wall is inclined. The combination of the two topics (open cavity and inclined walls) is the main novelty of the present study. The effects of the angle of the hot inclined wall on the flow field and heat transfer are thoroughly investigated. The Prandtl number is fixed to 0.71 (air). The Rayleigh number and the angle of the hot inclined wall are varied in the range of 10⁴ to 10⁶ and 60? to 85?, respectively. The results are presented for two different aspect ratios, i.e., A = 1 and 2. The results obtained with the LBM are also compared with those of the finite volume method (FVM). The predicted results of the LBM conform to those of the FVM. The results show that by increasing the angle of the hot inclined wall and the aspect ratio of the cavity, the average Nusselt number decreases. The trend of the local Nusselt number on the inclined wall is also discussed.     

Natural convection in an inclined enclosure containing internal energy sources and cooled from below

Natural convection in an inclined enclosure from below and containing internally heated fluid has been investigated using a finite difference calculation procedure. Results have been obtained for Rayleigh number values up to 10⁶ and for inclination angles of 30 and 60°. For internal Rayleigh numbers that are much larger than the external Rayleigh number, the flow rises in the interior and moves down both the hot and cold walls. On the other hand, if the external Rayleigh number has a larger magnitude, the flow moves upwards along the hot surface and downwards along the cold surface. For the latter situation, the inner core is multicellular in nature at large external Rayleigh numbers. The average heat flux ratio along the cold surface (convective heat flux/corresponding conduction heat flux) increases with increasing external Rayleigh number and decreasing internal ratio is non-monotonic in nature. The heat flux ratio along both surfaces is observed to be strongly dependent on the inclination angle at high external Rayleigh numbers. A maximum in the local heat flux along the cold surface is obtained in the vicinity of x/L = 1 where hot fluid, either from the interior or directly from the opposite hot wall, meets the surface. Along the hot wall, a maximum in the heat flux ra flo     

Effect of Conduction in Bottom Wall on Darcy–Bénard Convection in a Porous Enclosure

Conjugate natural convection-conduction heat transfer in a square porous enclosure with a finite-wall thickness is studied numerically in this article. The bottom wall is heated and the upper wall is cooled while the verticals walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and the COMSOL Multiphysics software is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (100 ≤ Ra ≤ 1000), the wall to porous thermal conductivity ratio (0.44 ≤ K _r ≤ 9.90) and the ratio of wall thickness to its height (0.02 ≤ D ≤ 0.4). The results are presented to show the effect of these parameters on the heat transfer and fluid flow characteristics. It is found that the number of contrarotative cells and the strength circulation of each cell can be controlled by the thickness of the bottom wall, the thermal conductivity ratio and the Rayleigh number. It is also observed that increasing either the Rayleigh number or the thermal conductivity ratio or both, and decreasing the thickness of the bounded wall can increase the average Nusselt number for the porous enclosure.     

Natural convection in inclined two dimensional rectangular cavities

Steady two-dimensional natural convection in fluid filled cavities is numerically investigated. The channel is heated from below and cooled from the top with insulated side walls and the inclination angle is varied. The field equations for a Newtonian Boussinesq fluid are solved numerically for three cavity height based Rayleigh numbers, Ra = 10⁴, 10⁵ and 10⁶, and several aspect ratios. The calculations are in excellent agreement with previously published benchmark results. The effect of the inclination of the cavity to the horizontal with the angle varying from 0° to 180° and the effect of the startup conditions on the flow pattern, temperature distribution and the heat transfer rates have been investigated. Flow admits different configurations at different angles as the angle of inclination is increased depending on the initial conditions. Regardless of the initial conditions Nusselt number Nu exhibits discontinuities triggered by gradual transition from multiple cell to a single cell configuration. The critical angle of inclination at which the discontinuity occurs is strongly influenced by the assumed startup field. The hysteresis effect previously reported is not always present when the calculations are reversed from 90° to 0°. A comprehensive study of the flow structure, the Nu variation with varying angle of inclination, the effect of the initial conditions and the hysteresis effect are presented.     

A numerical study of mixed convection in a horizontal channel with a discrete heat source in an open cavity

This article aims to numerically investigate mixed convection heat transfer in a two-dimensional horizontal channel with an open cavity. A discrete heat source is considered to be located on one of the walls of the cavity. Three different heating modes are considered which relate to the location of the heat source on three different walls (left, right and bottom) of the cavity. The analysis is carried out for a range of Richardson numbers and cavity aspect ratios. The results show that there are noticeable differences among the three heating modes. When the heat source is located on the right wall, the cavity with an aspect ratio of two has the highest heat transfer rate compared to other cavity heating modes. Moreover, when the heat source is located on the bottom wall, the flow field in the cavity with an aspect ratio of two experiences a fluctuating behaviour for Richardson number of 10. The results also show that at a fixed value of Richardson number, all three different heating modes show noticeable improvements in the heat transfer mechanism as the cavity aspect ratio increases.     

Natural convection in a rectangular cavity with wall temperature decreasing at a uniform rate

The transient natural convection in a fluid contained in a rectangular enclosure, the wall of which is maintained at a uniform temperature which changes at a steady rate, is approached by a numerical method. Numerical solutions are obtained forPr=0.73, 7.3 and 73 and a range of Rayleigh numbersRa=10² ～ 10⁸. At relatively low Rayleigh numbers the flow is characterized by the development of double cells with flow up the center and down the sidewalk However it was found that an increase of the Rayleigh number leads to the development of strong secondary circulation on the axis of symmetry of the cavity near the top wall. Thus, as the Rayleigh number is increased the secondary cells grow in size. The effects of the secondary cells on the temperature field and heat transfer coefficients are discussed. Most results are obtained for the case of a square cavity (E=2) but the influence of the aspect ratio of the cavity is also studied forE=1 and 4.     

Natural convection in an inclined rectangular porous cavity with uniform heat flux from the side

The problem of natural convection in an inclined rectangular porous layer enclosure is studied numerically. The enclosure is heated from one side and cooled from the other by a constant heat flux while the two other walls are insulated. The effect of aspect ratio, inclination angle and Rayleigh number on heat transfer is studied. It is found that the enclosure orientation has a considerable effect on the heat transfer. The negative orientation sharply inhibits the convection and consequently the heat transfer and a positive orientation maximizes the energy transfer. The maximum temperature within the porous medium can be considerably higher than that induced by pure conduction when the cavity is negatively oriented. The peak of the average Nusselt number depends on the Rayleigh number and the aspect ratio. The heat transfer between the two thermally active boundaries is sensitive to the effect of aspect ratio. For an enclosure at high or low aspect ratio, the convection is considerably decreased and the heat transfer depends mainly on conduction.     

Lattice Boltzmann simulation of natural convection in nanofluid-filled 2D long enclosures at presence of magnetic field

In this paper, the effects of a magnetic field on natural convection flow in filled long enclosures with Cu/water nanofluid have been analyzed by lattice Boltzmann method. This study has been carried out for the pertinent parameters in the following ranges: the Rayleigh number of base fluid, Ra = 10³–10⁵, the volumetric fraction of nanoparticles between 0 and 6 %, the aspect ratio of the enclosure between A = 0.5 and 2. The Hartmann number has been varied from Ha = 0 to 90 with interval 30 while the magnetic field is considered at inclination angles of θ = 0°, 30°, 60° and 90°. Results show that the heat transfer decreases by the increment of Hartmann number for various Rayleigh numbers and the aspect ratios. Heat transfer decreases with the growth of the aspect ratio but this growth causes the effect of the nanoparticles to increase. The magnetic field augments the effect of the nanoparticles at high Rayleigh numbers (Ra = 10⁵). The effect of the nanoparticles rises for high Hartmann numbers when the aspect ratio increases. The rise in the magnetic field inclination improves heat transfer at aspect ratio of A = 0.5.     

Interferometric study of natural convection in rectangular air cavities of various orientations

The results of an experimental investigation into the temperature profiles and heat transfer associated with natural convection in rectangular air cavities are presented, the angle of inclination varying from 0 (heating at the bottom) to 180° (heating at the top). The range of Rayleigh numbers was R=2.68·10³–2.57·10⁵, and n=H/d=5.06–18.3. The investigation was carried out by an optical method, using an IZK-454 interferometer. For a horizontal orientation of the cavity the heat-transfer data satisfy the relation N=0.216 R^0.25, for a vertical orientation N=0.144 R^0.3h^–0.129, where N is the Nusselt number. In the region of an inclination of 30° the heat transfer passes through a maximum under all conditions studied.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 89–93, July–August, 1972.     

Conjugate natural convection from an array of discrete heat sources: part 2 — a numerical parametric study

Coupled conduction and natural convection transport within a discretely heated cavity have been investigated numerically. One vertical wall of the cavity is composed of discrete, isoflux heat sources mounted in a substrate of finite thermal conductivity. The opposite vertical wall and the horizontal walls are assumed to be isothermal and adiabatic, respectively. The governing steady-state partial differential equations for the fluid and solid region are solved simultaneously using a control volume formulation, coupled with an additive correction multigrid procedure that increases the convergence rate of the solution. The fluid Prandtl number and heater/fluid thermal conductivity ratio are fixed at 25 and 2350, respectively, corresponding to a dielectric fluid (FC-77) and heaters manufactured from silicon. With increasing modified Rayleigh number (10⁴ < Ra_Lz^* < 109), the cavity flow becomes more boundary layer-like along the vertical walls, and multiple fluid cells develop in the central region. Thermal spreading in the substrate increases with decreasing modified Rayleigh number and with increasing values of the substrate/fluid thermal conductivity ratio (10⁻¹ <- R_s ≤ 10³). For large R_s, the discrete heat sources lose their thermal identity, and the streamlines and isotherms resemble those associated with a differentially heated cavity. Thermal spreading in the substrate also has a significant effect on circulation in the cavity and on maximum surface temperatures.