Three-dimensional natural convection in an enclosure with a sphere at different vertical locations

Numerical calculations are carried out for the three-dimensional natural convection induced by a temperature difference between a cold outer cubic enclosure and a hot inner sphere. The immersed-boundary method (IBM) to model a sphere based on the finite volume method is used to study a three-dimensional natural convection for different Rayleigh numbers varying in the range of 10³–10⁶. This study investigates the effect of the inner sphere location on the heat transfer and fluid flow. The flow and thermal fields eventually reach the steady state for all Rayleigh numbers regardless of the sphere location. For Rayleigh numbers of 10⁵ and 10⁶, the variation of local Nusselt number of the sphere along the circumferential direction is large, showing the strong three dimensionality of the natural convection in the enclosure unlike to the cases of lower Rayleigh numbers of 10³ and 10⁴. For the highest Rayleigh number, the local peaks of the Nusselt number on the top wall of the enclosure shows the sinusoidal distribution along the circumferential direction. The flow and thermal fields, and the local and surface-averaged Nusselt numbers on the sphere and the enclosure are highlighted in detail.     

Effect of Prandtl Number on Free Convection from Two Cylinders in a Square Enclosure

This study explores the effect of Prandtl number on the laminar natural convection heat transfer to Newtonian fluids in a square enclosure consisting of one hot circular cylinder and one cold circular cylinder. The walls of the square enclosure are maintained isothermal and at the same temperature as the cold cylinder and the fluid medium. The governing partial differential equations have been solved numerically over the following ranges of conditions: Grashof number, 10 to 10⁵; Prandtl number, 0.7 to 100 (or the range of Rayleigh numbers as 7 to 10⁷); and relative positioning of the cylinders, ?0.25 to 0.25. However, the ratio of the radius of the cylinder to the side of the enclosure is held fixed at 0.2. Extensive results on the streamline and isotherm contours, the local Nusselt number distribution, and the average Nusselt number are discussed to delineate the influence of Grashof and Prandtl numbers on them for a given location with respect to the horizontal center line. The surface-averaged Nusselt number shows a positive dependence on Grashof and Prandtl numbers for a fixed location of the two cylinders. The heat transfer results have been correlated as a function of the Rayleigh number and geometric parameters, thereby enabling its prediction in a new application.     

Numerical simulation of natural convection in a horizontal enclosure with a heat-generating conducting body

The physical model considered here is a horizontal layer of fluid heated below and cold above with heat-generating conducting body placed at the center of the layer. The dimensionless thermal conductivities of body considered in the present study are 0.1, 1 and 50. The dimensionless temperature difference ratios considered are 0.0, 0.25, 2.5 and 25. Two-dimensional solution for unsteady natural convection is obtained using an accurate and efficient Chebyshev spectral methodology for variety of Rayleigh number from 10³ to 10⁶. Multi-domain technique is used to handle square-shaped heat-generating conducting body. The fluid flow, heat transfer and time- and surface-averaged Nusselt number are investigated for various ranges of Rayleigh number, thermal conductivity ratio and dimensionless temperature difference ratio. The results for the case of conducting body with heat generation are also compared to those without heat generation to see the effects of heat generation from the conducting body on the fluid flow and heat transfer in the enclosure.     

Conjugate natural convection heat transfer in an inclined square cavity containing a conducting block

The present work is concerned with computation of natural convection flow in a square enclosure with a centered internal conducting square block both of which are given an inclination angle. Finite volume method through the concepts of staggered grid and SIMPLE algorithm have been applied. Deferred QUICK scheme has been used to discretize the convective fluxes and central difference for diffusive fluxes. The problem of conjugate natural convection has been taken up for validating the code. The abrupt variation in the properties at the solid/fluid interface are taken care of with the harmonic mean formulation. Solution has been performed in the computational domain as a whole with proper treatment at the solid/fluid interface. Computations have been performed for Ra = 10³–10⁶, angle of inclination varying from 15° to 90° in steps of 15° and ratio of solid to fluid thermal conductivities of 0.2 and 5.0. Results are presented in terms of streamlines, isotherms, local and average Nusselt number.     

Conjugate turbulent natural convection with surface radiation in air filled rectangular enclosures

Conjugate turbulent natural convection and surface radiation in rectangular enclosures heated from below and cooled from other walls, typically encountered in Liquid Metal Fast Breeder Reactor (LMFBR) subsystems, have been investigated by a finite volume method for various aspect ratios. The formulation comprises the standard two equation k–ε turbulence model with physical boundary conditions (no wall functions), along with the Boussinesq approximation, for the flow and heat transfer. As far as radiation is concerned, the radiosity – irradiation formulation for a transparent fluid of Prandtl number 0.7 has been employed. The conjugate coupling on the walls has been handled by using a fin type formulation. The Rayleigh number based on the width of the enclosure is varied from 10⁸ to 10¹² and the aspect ratio is varied from 0.5 to 2.0. Detailed results including stream lines, temperature profiles, and convective, radiative and overall Nusselt numbers are presented. A correlation for the mean convection Nusselt number in terms of Rayleigh number and aspect ratio is proposed for design purposes. The influence of the wall emissivity and the external heat transfer coefficient on the heat transfer from the enclosure has also been investigated.     

Natural Convection in a Square Enclosure with Two Horizontal Cylinders

This study investigates natural convection in a cooled square enclosure with two inner heated circular cylinders with the same diameter. The centers of two equidiameter cylinders are placed at those of the lower and upper half of the enclosure, respectively. The immersed boundary method (IBM) to model the inner circular cylinders based on the finite volume method is used to study a two-dimensional natural convection for different Rayleigh numbers varying in the range of 10³ ≤ Ra ≤ 10⁵. The effect of the radius of inner circular cylinders in an enclosure on heat transfer and fluid flow at different Rayleigh numbers has been examined. As the Rayleigh number increases, the horizontal symmetry is broken and the asymmetry occurred from the smaller radius. As the radius decreases, the dependence of the convection on the Rayleigh number is considerable. The dependence of the Nusselt number on the radius and the Rayleigh number is presented.     

Effect of Inclination on Heat Transfer and Fluid Flow in a Finned Enclosure Filled with a Dielectric Liquid

The problem of two-dimensional natural convection flow of a dielectric fluid in a square inclined enclosure with a fin placed on the hot wall is investigated numerically. The fin thickness and length are 1/10 and 1/2 of the enclosure side, respectively. The Rayleigh number is varied from 10³ to 5 × 10⁵ and the solid to fluid thermal conductivity ratio is fixed at 10³. The enclosure tilt or inclination angle is varied from 0° to 90°. The streamlines and isotherms within the enclosure are produced and the heat transfer is calculated. It is found that for 2.5 × 10⁴ ≤ Ra ≤ 2.5 × 10⁵, the average Nusselt number is maximum when γ = 0° and minimum when γ = 90°. For Ra = 5 × 10⁵, the values of enclosure tilt angle for which the average Nusselt number is maximum or minimum are completely different due to the transition to unsteady state. In this case, the maximum heat transfer is obtained for γ = 60°, while the minimum heat transfer is predicted for γ = 0°. Monomial correlations relating the average Nusselt number with the different values of the Rayleigh number from 10⁴ to 10⁵ are determined for two different angles, γ = 0° and γ = 90°.     

Numerical Investigation of Unsteady Natural Convection in an Inclined Square Enclosure With Heat-Generating Porous Medium

The unsteady laminar natural convection in an inclined square enclosure with heat-generating porous medium whose heat varies by a cosine function is investigated by a thermal equilibrium model and the Brinkman–Darcy–Forchheimer model numerically, with the four cooled walls of closure as isothermal. The numerical code based on the finite-volume method has been validated by reference data before it was adopted. Influence of dimensionless frequency and inclination angle on heat transfer characteristics in a square enclosure, such as flow distribution, isotherm, averaged Nusselt number on each wall, and time-averaged Nusselt number, are discussed, with specified value for Rayleigh number = 10⁸, Darcy number = 10^?4, Prandtl number = 7, porosity = 0.4, and specific heat ratio = 1. It is found that when the internal heat source varies by cosine, the Nusselt numbers of the four walls oscillate with the same frequency as the internal heat source; however, phase difference occurs. Moreover, frequency has little impact on time-averaged Nusselt number of the four walls, which is different from the phenomenon discovered in natural convection with suitable periodic varying wall temperature boundary condition. Moreover, inclination angle plays an important role in the heat transfer characteristics of the walls studied.     

Investigation of buoyancy-driven flow and heat transfer in a trapezoidal cavity filled with water–Cu nanofluid

A numerical study is conducted to investigate the transport mechanism of free convection in a trapezoidal enclosure filled with water–Cu nanofluid. The horizontal walls of the enclosure are insulated while the inclined walls are kept at constant but different temperatures. The numerical approach is based on the finite element technique with Galerkin's weighted residual simulation. Solutions are obtained for a wide range of the aspect ratio (AR) and Prandtl number (Pr) with Rayleigh number (Ra = 10⁵) and solid volume fraction (? = 0.05). The streamlines, isotherm plots and the variation of the average Nusselt number at the left hot wall are presented and discussed. It is found that both AR and Pr affect the fluid flow and heat transfer in the enclosure. A correlation is also developed graphically for the average Nusselt number as a function of the Prandtl number as well as the cavity aspect ratio.     

Numerical simulation of natural convection heat transfer from a heated square cylinder in a square cavity filled with micropolar fluid

A numerical investigation has been performed to visualize the magnetohydrodynamic natural convective heat transfer from a heated square cylinder situated within a square enclosure subjected to nonuniform temperature distributions on the left wall. The flow inside the enclosure is unsteady, incompressible, and laminar and the working fluid is micropolar fluid with constant Prandtl number (Pr = 7). The governing equations of the flow problem are the conservation of mass, energy, and linear momentum, as well as the angular momentum equations. Governing equations formulated in dimensionless velocity and pressure form has been solved by Marker and Cell method with second-order accuracy finite difference scheme. Comprehensive verification of the utilized numerical method and mathematical model has shown a good agreement with numerical data of other authors. The results are discussed in terms of the distribution of streamlines and isotherms and surface-averaged Nusselt number, for combinations of Rayleigh number, Ra (10³–10⁶), Vortex viscosity parameter, K (0–5), and Ha parameter (0–50). It has been shown that an increase in the vortex viscosity parameter leads to attenuation of the convective flow and heat transfer inside the cavity.     

Lattice Boltzmann Method Based Simulation of Natural Convection in Partially Open Enclosure

A thermal lattice Boltzmann method‐based analysis was performed to numerically investigate the heat transfer by natural convection from an enclosure with a large vertical side opening. The height of the opening was less than the enclosure height and the vertical wall opposite to the opening was maintained at constant temperature. A parametric study was carried out for different values of Rayleigh number (Ra) ranging from 10³ to 10⁵ with air as the working fluid for three opening sizes and three opening locations. The Prandtl number was fixed at 0.71 and the enclosure aspect ratio was also fixed at 2 in all calculations. With Boussinesq approximation, the temperature distribution and stream functions in the enclosure were predicted. The profile of the normal velocity component at the opening location was determined. The opening size affects the stratification and recirculation pattern within the enclosure. The average Nusselt number at the heated wall was determined for all cases. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21110     

Laminar mixed convection in shallow inclined driven cavities with hot moving lid on top and cooled from bottom

Laminar mixed convective heat transfer in two-dimensional shallow rectangular driven cavities of aspect ratio 10 is studied numerically. The top moving lid of the cavity is at a higher temperature than the bottom wall. Computations are performed for Rayleigh numbers ranging from 10⁵ to 10⁷ keeping the Reynolds number fixed at 408.21, thus encompassing the dominating forced convection, mixed convection, and dominating natural convection flow regimes. The fluid Prandtl number is taken as 6 representing water. The effects of inclination of the cavity on the flow and thermal fields are investigated for inclination angles ranging from 0° to 30°. Interesting behaviours of the flow and thermal fields with increasing inclination are observed. The streamline and isotherm plots and the variation of the local and average Nusselt numbers at the hot and cold walls are presented. The average Nusselt number is found to increase with cavity inclination. The rate of increase of the average Nusselt number with cavity inclination is mild for dominating forced convection case while it is much steeper in dominating natural convection case.     

Natural convection in an enclosure with disconnected and conducting solid blocks

Natural convection through a rectangular enclosure filled with a fluid bathing several discrete, conducting and disconnected solid objects is common in many applications, such as environmental control (e.g., indoor buildings, storage), materials processing (e.g., drying), food processing (e.g., baking), and electronics (e.g., cabinets). The case of an enclosure heated from the side and containing equally spaced, conducting solid square blocks is investigated here by using a continuum model, which treats the fluid and solid constituents individually. The dispersive effect of the solid constituent is isolated by increasing the number of solid blocks (N) while reducing their size as to maintain their relative total volume constant. Results obtained for a wide Rayleigh number (Ra) range and several values of solid-to-fluid conductivity ratio (κ) show the strong hindrance effect of the blocks on the convection process to be dependent on a minimum number of blocks, N_min, for every Ra. An analytical expression predicting N_min is proposed and validated by the numerical results.     

Laminar Natural Convection of Bingham Fluids in Square Cross-Sectioned Cylindrical Annular Cavity with Differentially Heated Vertical Walls Subjected to Constant Heat Fluxes

Steady-state numerical simulations have been conducted to investigate natural convection of yield stress fluids obeying Bingham model in square cross-sectioned axisymmetric cylindrical annular enclosure with vertical walls subjected to constant heat fluxes for nominal Rayleigh number range of 10³ to 10⁶, nominal Prandtl number of 10 to 10³ for different values of internal cylinder radius. It is found that the mean Nusselt number on the inner periphery increases (decreases) with increasing nominal Rayleigh (Bingham) number due to strengthening (weakening) of thermal advection. However, the values of the mean Nusselt number on the inner periphery obtained for Bingham fluids are smaller than that obtained for Newtonian fluids for the same set of nominal Rayleigh and Prandtl numbers. The mean Nusselt number normalized by the corresponding value obtained for pure conductive transport increases with increasing internal radius before asymptotically approaching the mean Nusselt number for a square enclosure. This suggests that the ratio of the convective to the conductive transport strengthens with increasing cylinder radius in the cylindrical annular cavity. Detailed physical explanations have been provided for the effects of the aforementioned parameters on the mean Nusselt number on the inner periphery and correlations have been proposed for the mean Nusselt number on the inner periphery for both Newtonian and Bingham fluids.     

Conjugate natural convection in a fluid-saturated porous enclosure with two solid vertical partitions

Conjugate natural convection in a fluid-saturated square porous enclosure with two solid vertical partitions of finite and equal thickness equispaced from center of enclosure is investigated in this paper. The primary objective is to attenuate the Nusselt number (Nu) and hence the heat transfer rate across a differentially heated enclosure. Darcy's model is considered. Numerical computation is performed using successive accelerated replacement and explicit scheme. Partition ratio, partition length, thermal conductivity ratio, and modified Rayleigh number are the parameters under study. Fluid flow is analyzed by observing transient changes of streamlines and isotherms for partition length 0.3-1, thermal conductivity ratio 0.5-2, partition ratio 0.1-0.3 and modified Rayleigh number 100 and 1000 where partition ratio is the ratio of distance between center of enclosure and either of the partition center to the total length of the enclosure; while Nusselt number is calculated to estimate the heat transfer rate for each configuration. It is found that, employing a solid partition within the enclosure most definitely reduces the Nusselt number. The drop in Nusselt number is more for partition length 0-0.6 after which it does show a drop in Nu but only very subtle. Further, Nu is the least for partition ratio 0.2. Also, Nusselt number is proportional to thermal conductivity ratio which is the ratio of thermal conductivity of solid to porous medium.     

Experimental and numerical analysis of buoyancy-induced flow in inclined triangular enclosures

The buoyancy‐induced heat transfer and fluid flow in a triangular enclosure are investigated both numerically and experimentally. The enclosure is heated from one wall and the adjacent wall is insulated. Hypotenuse of the triangle is cooled isothermally. The numerical tests and experiments covered a range of Rayleigh number, Ra, from 1.5 × 10⁴ to 1.5 × 10⁵. The local and average Nusselt numbers are given for different orientation angles. A code was written based on finite difference method in Fortran platform to solve governing equations of natural convection. Experimental and numerical results show good agreement. It is observed that inclination angle can be used as a control parameter for heat transfer.     

Three-Dimensional Numerical Study of Natural Convective Heat Transfer of Liquid in a Cubic Enclosure

ABSTRACT

Steady-state laminar natural convection in a cubic enclosure with a cold vertical wall and two hot square heaters with constant temperature on the opposite wall is studied numerically. The enclosure is filled with various liquids. Three-dimensional Navier–Stokes Equations are solved by employing the SIMPLE algorithm. Computations are performed for a range of Rayleigh number from 10³ to 10⁷ while enclosure aspect ratio varies from 0.05 to 1.6. The effects of Rayleigh number, enclosure aspect ratio, and Prandtl number on heat transfer characteristics are studied in detail. The results show that the flow field is very complex and heat transfer from the two heaters is not the same. The effects of Prandtl number are negligible in the range from 5 to 140 with other parameters kept constant. This allows the use of liquids such as water for studying other dielectric liquids, provided the flow geometry and other nondimensional parameters are similar. The overall Nusselt number increases markedly with Rayleigh number. It is also affected by enclosure aspect ratio. It attains the maximum value when aspect ratio is in the range of 0.1–0.2 and decreases as enclosure aspect ratio varies from 0.2 to 1.6. Also, various settings of cooling face and arrangement of heaters are investigated, and the results show that they have considerable effects on heat transfer of both heaters.     

Unsteady natural convection within a differentially heated enclosure of sinusoidal corrugated side walls

Numerically investigation of natural convection within a differentially heated modified square enclosure with sinusoidally corrugated side walls has been performed for different values of Rayleigh number. The fluid inside the enclosure considered is air and is quiescent, initially. The top and bottom surfaces are flat and considered as adiabatic. Results reveal three main stages: an initial stage, a transitory or oscillatory stage and a steady stage for the development of natural convection flow inside the corrugated cavity. The numerical scheme is based on the finite element method adapted to triangular non-uniform mesh element by a non-linear parametric solution algorithm. Investigation has been performed for the Rayleigh number, Ra ranging from 10⁵ to 10⁸ with variation of corrugation amplitude and frequency. Constant physical properties for the fluid medium have been assumed except for the density where Boussinesq’s approximation has been considered. Results have been presented in terms of the isotherms, streamlines, temperature plots, average Nusselt numbers, traveling waves and thermal boundary layer thickness plots, temperature and velocity profiles. The effects of sudden differential heating and its consequent transient behavior on fluid flow and heat transfer characteristics have been observed for the range of governing parameters. The present results show that the transient phenomena are greatly influenced by the variation of the Rayleigh number with corrugation amplitude and frequency.     

Free Convection in Confined Power-Law Fluids From Two Side-by-Side Cylinders in a Square Enclosure

Laminar free convection heat transfer in power-law fluids from two side-by-side cylinders (one hot and one cold) confined in a square duct has been studied numerically in the two-dimensional flow regime. For a fixed value of the ratio of cylinder radius to size of the enclosure, the effect of geometrical placement of the cylinders is studied on the resulting velocity and temperature fields in the laminar free convection regime by considering six asymmetric locations of the two cylinders. In particular, extensive results reported herein span the range of conditions of Grashof number, 10 to 10⁵; Prandtl number, 0.7 to 100, thereby yielding the range of the Rayleigh number as 7 to 10⁷; power-law index, 0.3 to 1.8; and the relative positions (dimensionless) of the cylinders with respect to the centerline, –0.25 to 0.25. The heat transfer characteristics are analyzed in terms of the local Nusselt number along the surfaces of the two cylinders and the enclosure walls. Overall, the average Nusselt number shows a positive dependence on both the Grashof number and the Prandtl number irrespective of the values of power-law index and relative positioning of the cylinders. Also, all else being equal, shear-thinning fluid behavior promotes heat transfer with reference to that in Newtonian fluids. When the two cylinders are situated close to the bottom wall, the rate of heat transfer is augmented with reference to that for the symmetric positioning of the cylinders along the horizontal mid-plane of the enclosure. Conversely, heat transfer deteriorates as the cylinders are located above the centerline of the enclosure. The present numerical results have been consolidated via the use of a modified Rayleigh number, thereby enabling the estimation of the average Nusselt number in a new application.     

Effect of eccentricity and radius ratio on fluid flow and heat transfer inside an eccentric semicircular enclosure

latroductionNatural convechon heat transfer has gainedconsiderable attention because of itS' numerousaPPlications in the areas of energy conservation, coolingo f electrical and electronic components, design of solarcoil~, bed exchangers, and many others. Heattransfer inside annular space, air-filled cavity or annularsector has wide aPPlication in many engineeringProblems. In our earlier work["n, we have shown theeffeCt of eccentricity on heat transfer and flow field forradius ratio R'=2.0 f…