NUMERICAL EXPERIMENTS WITH FLUID CONVECTION IN TILTED NONRECTANGULAR ENCLOSURES

Numerical experiments were performed on an incompressible fluid contained in a tilted nonrectangular enclosure. Rayleigh numbers of l0²-l0⁵ and Prandtl numbers of 0.001-100 are considered. The wall angles are 22.5°, 45°, and 77.5° with aspect ratios of 3 and 6. Results indicate that the heat transfer and fluid motion within the enclosure are strong functions of Rayleigh number, Prandtl number, and orientation angle of the enclosure. For Rayleigh numbers greater than 1& and Prandtl numbers greater than 0.1, a minimum and a maximum mean Nusselt number occurred as the angle of orientation was increased from 0° to 360°. A transition in the mode of circulation occurred at the angles corresponding to the minimum or maximum rate of heat transfer.     

Effects of the aspect ratio on the optimal tilting angle for maximum convection heat transfer across air-filled rectangular enclosures differentially heated at sides

Natural convection in air-filled rectangular cavities inclined with respect to gravity, so that the heated wall is facing upwards, is studied numerically under the assumption of two-dimensional laminar flow. A computational code based on the SIMPLE-C algorithm is used for the solution of the system of the mass, momentum and energy transfer governing equations. Simulations are performed for height-to-width aspect ratios of the enclosure from 0.25 to 8, Rayleigh numbers based on the length of the heated and cooled walls from 10~2 to 10~7, and tilting angles of the enclosure from 0° to 75°. The existence of an optimal tilting angle is confirmed for any investigated configuration, at a location that increases as the Rayleigh number is decreased, and the height-to-width aspect ratio of the cavity are increased, unless the value of the Rayleigh number is that corresponding to the onset of convection or just higher. Dimensionless correlating equations are developed to predict the optimal tilting angle and the heat transfer performance of the enclosure.     

Numerical Studies on Natural Convection in a Trapezoidal Enclosure With Discrete Heating

Abstract

A steady state laminar natural convection flow in a trapezoidal enclosure with discretely heated bottom wall, adiabatic top wall, and constant temperature cold inclined walls is performed. The finite volume based commercial code “ANSYS-FLUENT” is used to investigate the influence of discrete heating on natural convection flows in a trapezoidal cavity. The numerical solution of the problem covers various Rayleigh numbers ranging from 10³ to 10⁶, non-dimensional heating length ranging from 0.2 to 0.8 and Prandtl number is 0.7. The performance of the present numerical approach is represented in the form of streamfunction, temperature profile and Nusselt number. Heat transfer increases with increase of Rayleigh numbers at the corners of the cavity for same heating length from center of the bottom wall. However, the heat transfer rate is less and almost constant for the Rayleigh numbers considered. It is found that the average Nusselt number monotonically increases with increase of Rayleigh number and length of heat source. The variation of local and average Nusselt numbers is more significant for larger length of heating than smaller one. The heat transfer correlations useful for practical design problems have been predicted.     

Convective transport in rectangular cavities partially heated at the bottom and cooled at one side

Laminar natural convection heat transfer inside air-filled, rectangular enclosures partially heated from below and cooled at one side is studied numerically. A computational code based on the SIMPLE-C algorithm is used for the solution of the system of the mass, momentum, and energy transfer governing equations. Simulations are performed for a complete range of heater size, for Rayleigh numbers based on the height of the cavity ranging from 10~3to 10~6, and for height-to-width aspect ratios of the cavity spanning from 0.25 to 4. It is found that the heat transfer rate increases with increasing the heater size and the Rayleigh number, while it decreases with increasing the aspect ratio of the cavity. Dimensionless heat transfer correlations are also proposed.     

Study of buoyancy-induced flows subjected to partially heated sources on the left and bottom walls in a square enclosure

In this paper, numerical simulations of laminar, steady, two-dimensional natural convection flows in a square enclosure with discrete heat sources on the left and bottom walls are presented using a finite-volume method. Two different orientated wall boundary conditions are designed to investigate the natural convection features. The computational results are expressed in the form of streamlines and isothermal lines for Rayleigh numbers ranging from 10² to 10⁷ in the cavity. In the course of study, a combination of third-order and exponential interpolating profile based on the convective boundedness criterion is proposed and tested against the partially heated cavity flow up to the highest Rayleigh number 10⁷. The effects of thermal strength and heating length on the hydrodynamic and thermal fields inside the enclosure are also presented. Numerical results indicate that the average Nusselt number increases as Rayleigh number increases for both cases. Moreover, it is seen that the effect of the heat transfer rate due to the heating strength on the left wall is different from the one on the bottom. For the heater size effect, it is observed that by increasing the length of heat source segment, the heat transfer rate is gradually increased for both cases.     

Numerical Simulation of Natural Convection in a Triangle Cavity Filled with Nanofluids Using Tiwari and Das' Model: Effects of Heat Flux

In this article, we present a fully higher‐order compact (FHOC) finite difference method to investigate the effects of heat flux on natural convection of nanofluids in a right‐angle triangle cavity, where the left vertical side is heated with constant heat flux both partially and throughout the entire wall, the inclined wall is cooled, and the rest of walls are kept adiabatic. The Darcy flow and the Tiwari and Das’ nanofluid models are considered. Investigations with four types of nanofluids were made for different values of Rayleigh numbers with the range of 100 ≤ Ra ≤ 50,000, size of heat flux as 0.1 ≤ ε ≤ 1.0, enclosure aspect ratio as 0.5 ≤ AR ≤ 2.0, and solid volume fraction parameter of nanofluids with the range of 0% ≤ ? ≤ 20%. Results show that the average heat transfer rate increases significantly as particle volume fraction and Rayleigh numbers increase, and the maximum value of average Nusselt number is obtained by decreasing the enclosure aspect ratio. The results also show that the average heat transfer decreases with an increase in the length of the heater. Furthermore, multiple correlations in terms of the Rayleigh numbers and the solid volume fraction of four types of nanoparticles have been established in a general form.     

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.     

Lattice Boltzmann simulation of natural convection and heat transfer from multiple heated blocks

This study is aimed to investigate the natural convection heat transfer from discrete heat sources (similar to heated microchips) using Bhatnagar‐Gross‐Krook lattice Boltzmann method via graphics process unit computing. The simulation is carried out separately for three and six heated blocks model for different Rayleigh numbers and fixed Prandtl number, $Pr=0.71$ (air). The uniformly heated blocks are placed at the bottom wall inside a rectangular enclosure. The enclosure is maintained by the cold temperature at its left and right walls. The top and bottom surface is maintained by adiabatic conditions apart from the regions where blocks are attached to the bottom wall. The numerical code is validated with the benchmark heat transfer problem of side‐heated square cavity as well as with an experimental study for one discrete heat source. The rate of heat transfer is presented in terms of the local Nusselt and average Nusselt number for each block. It is found that the heat transfer rate becomes maximized in the leftmost and rightmost blocks due to the adjacent cold walls. It is found that the number of blocks and their positions play a substantial role in determining their collective performance on the heat transfer rate.     

NATURAL CONVECTION IN OPEN CAVITIES AND SLOTS

The present work is concerned with natural convection from open cavities or heated plates attached with parallel vertical strips. The bottom of the cavity is heated, and the vertical walls are assumed adiabatic. Numerical results are presented for steady, laminar natural convection for the geometry described. Effects of Rayleigh numbers from 1 × 10 ³ to 1 × 10 ⁷ , inclination angles from 10^° to 90^°, and aspect ratios of 0.5, 1.0, and 2.0 are investigated for a fixed Prandtl number (0.7). It is found that the average Nusselt number is not very sensitive to the inclination angle. Flow becomes unstable at high Rayleigh numbers and at low inclination angles. Flow pattern and heat transfer results are presented and discussed.     

Effects of Heating Location and Size on Natural Convection in Partially Heated Open-Ended Enclosure by Using Lattice Boltzmann Method

Natural convection heat transfer in an square enclosure, consisting of a partially heated west wall with east end open to ambient, is investigated numerically by using an in-house computational fluid dynamics solver based on thermal lattice Boltzmann method. In particular, the influences of Rayleigh number (10³–10⁶), heating location (bottom, middle, and top) on west wall, and dimensionless heating length (0.25–0.75) on momentum and heat transfer characteristics of air are presented and discussed. The streamline patterns show bifurcation at the lowest Rayleigh number for bottom and middle heating, whereas at the highest Rayleigh number, all heating positions yield bifurcation and elongation of flow patterns with a secondary vortex near the lower side of open end. The middle and bottom heating locations show a linear increase in Nusselt number with heater size, whereas inverse dependence is seen for top heating. The maximum heat transfer is observed in the case of middle heating. As expected, average Nusselt number increased with increasing Rayleigh number. Finally, the functional dependence of the average Nusselt number on flow governing parameters (Rayleigh number and heating length) for different heating locations is presented as a simple predictive empirical relationship.     

Lattice Boltzmann Simulation of Natural Convection in a Differentially Heated Square Enclosure Containing Heat Generating Low‐Prandtl Number Fluid

Lattice Boltzmann simulations were conducted for the free convective flow of a low‐Prandtl number (Pr = 0.0321) fluid with internal heat generation in a square enclosure having adiabatic top and bottom walls and isothermal side walls. The problem of free convection with volumetric heat source has represented itself in connection with advanced engineering applications, such as water‐cooled lithium–lead breeder blankets for nuclear fusion reactors and liquid metal sources of spallation neutrons for subcritical fission systems. A single relaxation time (SRT) thermal lattice Boltzmann method (LBM) was employed. While applying SRT, a D2Q9 model was used to simulate the flow field and temperature field. Results have been obtained for various Rayleigh numbers characterizing internal and external heating from 10³ to 10⁶. Flow and temperature fields in terms of stream function and isotherms in the enclosure were predicted for these cases. The temperature of the fluid in the enclosure was found higher than the heated wall temperature at high values of internal Rayleigh numbers. The internal heat generation affected the rate of heat transfer significantly as two convection loops are observed in the enclosure. The average Nusselt number at the heated and cold wall was determined for all the cases.     

Conjugate Wall Conduction-Fluid Natural Convection in a Three-Dimensional Inclined Enclosure

Laminar conjugate conduction-natural convection heat transfer in a 3-D inclined cubic enclosure comprised of finite thickness conductive walls and central cavity is numerically investigated. The dimensionless governing equations describing the convective flow and wall heat conduction are solved by the high accuracy multidomain pseudospectral method. Computations are performed for different Rayleigh numbers (10³ ≤ Ra* ≤ 10⁶), thermal conductivity ratios (1 ≤ k ≤ 100), dimensionless wall thickness (0 ≤ s ≤ 0.25), and enclosure inclinations (?30° ≤ α ₁ ≤ 30°, 0° ≤ α ₂ ≤ 45°). The effects of the above controlling parameters on the heat transfer performances of the enclosure system are investigated in detail, with emphases on the variations of wall conduction and fluid convection heat transfer, and the interactive heat transfer conditions between solid walls and fluid in the central cavity. Numerical results reveal that the existence of enclosure walls reduces the temperature gradient across the cavity and alters the temperature distribution within the solid walls; thus, the fluid convection is complexly determined by the combined effects of k and s, and is greatly affected by enclosure inclinations at high Rayleigh numbers. Moreover, the temperature distributions and solid-fluid interactive heat transfer conditions are provided for further interpretation and demonstration of the effects of the solid walls.     

Acknowledgment

An experimental and numerical study has been carried out in order to investigate mixed and natural convection heat transfer in a two-dimensional enclosure. A discrete isothermal heat source is located at one of the vertical walls. Also, two ventilation ports are at the bottom and on top of the opposite wall. A forced flow condition was imposed by providing an inlet of air at the bottom port. A Mach–Zehnder interferometer was used to visualize the temperature field within the enclosure and to determine the local and average heat transfer characteristics of the heat source. Five heater positions on the vertical wall and different Rayleigh numbers (4.5 × 10⁵ to 1.15 × 10⁶) and Reynolds numbers (120 to 1600) were considered in the experiments. A finite volume code has been developed based on the SIMPLE algorithm and hybrid discretization scheme for the numerical study. It is observed that the interaction of natural convection with the forced flow leads to various flow fields depending on the Richardson number, Reynolds number and the heater position. Also, results show different trends for variation of the average Nusselt number with the heater position at low and high Reynolds numbers. An optimum position for the heat source, at which the maximum heat transfer is achieved, exists for high Reynolds numbers and has been found to be at the middle of the vertical wall.     

Multi-Prandtl correlating equations for free convection heat transfer from a horizontal tube of elliptic cross-section

Steady laminar free convection from a horizontal elliptic cylinder set in unbounded space is studied numerically under the assumption of uniform surface temperature. A specifically developed computer-code based on the SIMPLE-C algorithm is used for the solution of the mass, momentum and energy transfer governing equations. Simulations are performed for ratios between the minor and major axes of the elliptic cross-section of the cylinder in the range between 0.05 and 0.98, inclination angles of the major axis of the elliptic cross-section with respect to gravity in the range between 0° and 90°, Rayleigh numbers based on the major axis of the elliptic cross-section in the range between 10 and 10⁷, and Prandtl numbers in the range between 0.7 and 700. It is found that the heat transfer rate increases with increasing the Rayleigh and Prandtl numbers, while decreases with increasing the orientation angle of the cross-section of the cylinder, i.e., passing from the slender to the blunt configuration. In addition, a noteworthy fact is that in most cases the amount of heat exchanged at the cylinder surface has a peak at an optimum axis ratio which is practically independent of the Prandtl number, while may either increase or decrease with increasing the Rayleigh number depending on whether the orientation angle of the tube is above or below a critical value of approximately 67.5°. Dimensionless correlating equations are proposed both for the optimum axis ratio for maximum heat transfer and for the heat transfer rate from the cylinder surface to the undisturbed surrounding fluid reservoir.     

Three-dimensional numerical simulation of natural convection in an inclined liquid-filled enclosure with an array of discrete heaters

A three-dimensional (3-D) numerical model is developed to investigate the effects of inclination on the heat transport processes in a liquid-filled rectangular enclosure of finite size. The interactions of flows tangential and normal to the heater surfaces in an inclined enclosure cause a slight oscillation in Nusselt number. Inclination has little effects on the average heat transfer characteristics except configurations close to the conduction case with heaters on top leading to a sharp decrease in Nusselt number. The critical Rayleigh number of 1708 is also observed in the present study regardless of inclinations. Heat transfer from discrete heaters is non-uniform. Maximum Nusselt number occurs at the heater leading edge and decreases towards the trailing edge. Correlations for the row average Nusselt number vs. Rayleigh number at various inclination angles are presented. By taking the average of the discrete heater array, the numerical results compared well with the empirical correlations obtained from a single heater plate in literature.     

Flows and heat transfer of the transition to an unsteady state in a finned cavity for different Prandtl numbers

Flows and heat transfer of the transition to an unsteady state in a finned cavity are studied for Prandtl numbers (Pr) from 0.1 to 100 and Rayleigh numbers (Ra) from 10⁷ to 10¹⁰. Transient flows are described in the finned cavity. Critical Rayleigh numbers of the transition to an unsteady state are obtained for different Prandtl numbers and the relation between two dimensionless parameters is given. The spectral analysis is applied for the oscillations of unsteady flows and the dominant frequency dependent on governing parameters is presented. Heat transfer of the transition to an unsteady flow is quantified and the corresponding relations dependent on the Prandtl number and Rayleigh number are gained. It is demonstrated that the flow rate and the Nusselt number of the finned cavity significantly increase due to the presentence of the fin, which depend on the Prandtl number and the Rayleigh number.     

Laminar natural convection in an inclined complicated cavity with spatially variable wall temperature

Natural convection in two-dimensional enclosure with three flat and one wavy walls is numerically investigated. One wall is having a sinusoidal temperature profile. Other three walls including the wavy wall are maintained at constant cold temperature. This problem is solved by SIMPLE algorithm with deferred QUICK scheme in curvilinear co-ordinates. The tests were carried out for different inclination angles, amplitudes and Rayleigh numbers while the Prandtl number was kept constant. The geometrical configurations considered were namely one-, two- and three-undulations.The results obtained show that the angle of inclination affects the flow and heat transfer rate in the cavity. With increase in amplitude, the average Nusselt number on the wavy wall is appreciably high at low Rayleigh number. Increasing the number of undulations beyond two is not beneficial. The trend of local Nusselt number is wavy.     

Convective heat transfer combined with surface radiation in a rotating square cavity with a local heater

The effect of surface radiation on laminar natural convection in a rotating cavity with a discrete heater has been analyzed numerically. The enclosure is insulated at the bottom and top, heated by a constant temperature from the discrete heater located on the bottom wall, and cooled by a constant temperature from the side walls. Governing equations with corresponding initial and boundary conditions formulated in dimensionless stream function, vorticity, and temperature have been solved by finite difference method of the second-order accuracy. The effects of surface emissivity, Rayleigh number, and Taylor number on the fluid flow and heat transfer have been studied. Obtained results have revealed that rotation can be a very good control parameter for heat transfer and fluid flow.     

Analysis of Flow and Heat Transfer in a Parallelogram Non‐Uniformly Heated Enclosure Filled with Porous Medium

The flow and heat transfer in a parallelogram enclosure filled with a porous medium is analyzed numerically. The heated bottom wall has a sinusoidal temperature distribution and side walls cooled isothermally while the upper wall is well insulated. Dimensionless Darcy law and energy equations are solved using the finite difference method along with the corresponding boundary condition. Computations were carried out for four inclination angles of side walls (γ = 45°, 60°, 75°, 90°) with different Rayleigh numbers (100≤Ra≤1000) and their effects on the flow field and heat transfer are discussed. It is found that the inclination angle has a significant effect on flow pattern and heat transfer and an increase in the angle leads to a decrease in the strength of the right vortex. The study also revealed that as the Rayleigh number increases at γ = 45°, another (third) vortex develops along the left wall and its strength enhances with Rayleigh number. At the end, a correlation is extracted from the numerical data which represents the relation between the Nusselt number, inclination angle, and the Rayleigh number. © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res; 39(7): 497–506, 2010; Published online in Wiley Online Library ( wileyOnlinelibrary.com ). DOI 10.1002/htj.20312     

Laminar natural convection in an inclined complicated cavity with spatially variable wall temperature

Natural convection in two-dimensional enclosure with three flat and one wavy walls is numerically investigated. One wall is having a sinusoidal temperature profile. Other three walls including the wavy wall are maintained at constant cold temperature. This problem is solved by SIMPLE algorithm with deferred QUICK scheme in curvilinear co-ordinates. The tests were carried out for different inclination angles, amplitudes and Rayleigh numbers while the Prandtl number was kept constant. The geometrical configurations considered were namely one, two and three undulations.The results obtained show that the angle of inclination affects the flow and heat transfer rate in the cavity. With increase in amplitude, the average Nusselt number on the wavy wall is appreciably high at low Rayleigh number. Increasing the number of undulations beyond two is not beneficial. The trend of local Nusselt number is wavy.