UNSTEADY NATURAL CONVECTION OF AIR IN A TALL AXISYMMETRIC,NONISOTHERMAL ANNULUS

This article presents numerical predictions of axisymmetric natural convection within a tall annular cavity with an aspect ratio of 10 and a radius ratio of 0.6. The temperature of the bounding inner vertical cylinder is hot at its base and decreases linearly with height, while the outer cylinder is cold at its base and its temperature increases linearly with height. These boundary conditions promote a steady bi-cellular flow at low Rayleigh numbers. As the Rayleigh number is increased, the flow transitions to a time-varying state in which the interface between the two natural-convection cells starts to oscillate. If the Rayleigh number is further increased, hysteresis, subharmonics, multiple solutions, and a reverse transition back to a steady state are all predicted. Results predicting the onset of unsteady flow for the Cartesian case are also presented.     

A numerical study of natural convection in a horizontal enclosure with a conducting body

The physical model considered here is a horizontal layer of fluid heated below and cold above with a conducting body placed at the center of the layer. The body has dimensionless thermal conductivities of 0.1, 1 and 50. Two-dimensional solution for unsteady natural convection is obtained using an accurate and efficient Chebyshev spectral methodology for different Rayleigh numbers. Multi-domain technique is used to handle a square-shaped conducting body. The results for the case of a conducting body are also compared to those of adiabatic and neutral isothermal bodies. When the dimensionless thermal conductivity is 0.1, a pattern of fluid flow and isotherms and the corresponding surface- and time-averaged Nusselt number are almost the same as the case of an adiabatic body. When the dimensionless thermal conductivity is 50, a pattern of flow and isotherm and the corresponding surface- and time-averaged Nusselt number are similar to those of neutral isothermal body. The results for the case of dimensionless thermal conductivity of unity are also compared to those of pure natural convection.     

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.     

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.     

Transient natural convection in rectangular enclosures heated from one side and cooled from above

A numerical investigation into two-dimensional transient natural convection of single-phase fluids inside a completely filled square enclosure has been conducted for the Prandd numbers of 0.71 and 7.1, and the Rayleigh number range 10³–10⁷. The fluid is assumed to be initially at a uniform temperature and motionless. Then, at time zero, the flow is driven by instantaneously raising and lowering the temperatures at the left side and the top wall, respectively. Adiabatic boundary conditions are used at the remaining walls. The unsteady Navier-Stokes equations, governing the flow under Boussinesq approximation, are solved with the vorticity-stream function formulation using the finite difference method. The development of the flow and temperature fields following these temperature changes are determined numerically. The transient behaviour of the average Nusselt number at the hot wall is traced.     

A numerical study of natural convection in a square enclosure with a circular cylinder at different vertical locations

Numerical calculations are carried out for natural convection induced by a temperature difference between a cold outer square enclosure and a hot inner circular cylinder. A two-dimensional solution for unsteady natural convection is obtained, using the immersed boundary method (IBM) to model an inner circular cylinder based on the finite volume method for different Rayleigh numbers varying over the range of 10³–10⁶. The study goes further to investigate the effect of the inner cylinder location on the heat transfer and fluid flow. The location of the inner circular cylinder is changed vertically along the center-line of square enclosure. The number, size and formation of the cell strongly depend on the Rayleigh number and the position of the inner circular cylinder. The changes in heat transfer quantities have also been presented.     

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.     

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.     

NUMERICAL EVALUATION OF WEAKLY TURBULENT FLOW PATTERNS OF NATURAL CONVECTION IN A SQUARE ENCLOSURE WITH DIFFERENTIALLY HEATED SIDE WALLS

This article presents the results of numerical evaluation of weakly turbulent natural convection of air in a rectangular enclosure with differentially heated side walls and adiabatic horizontal walls. The turbulence in the natural convection was described by k–ε equations, which were solved by Strang splitting, while average thermal and fluid flow fields were described by statistically averaged equations, which were solved by the projection method PmIII. The combined application of projection method and the Strang splitting characterizes the numerical method in this study. Numerical results for Rayleigh number 1.58 × 10⁹ have revealed reasonable agreement with the existing experimental ones, with some discrepancy attributable to the adiabatic boundary conditions on the horizontal walls. The results are also in good agreement with some published numerical results, particularly at higher Rayleigh numbers. However, comparison with the latest experimental data reveals that the turbulent heat flux model is not quite capable of giving satisfactory temperature distribution.     

Suppressing natural convection in a differentially heated square cavity with an arc shaped baffle

The problem of laminar natural convection heat transfer in a square cavity with an adiabatic arc shaped baffle is analyzed in this paper. As boundary conditions of the cavity, two vertical opposite walls are kept at constant but different temperatures and the remaining two walls are kept thermally insulated. Results are presented for a range of Rayleigh numbers, arc lengths of the baffle, and shape parameters of the baffle. Finally, parametric results are presented in terms of isothermal lines and streamlines. It is identified that flow and thermal fields are modified by the blockage effect of the baffle. The degree of flow modification due to blockage is enhanced by increasing the shape parameter of the baffle.     

BUOYANT CONVECTION IN A CAVITY WITH A BAFFLE UNDER TIME-PERIODIC WALL TEMPERATURE

A numerical study is made of buoyant convection at high Rayleigh number in a square cavity that contains a horizontal baffle at midheight. The horizontal walls of the cavity are insulated. At the cold left vertical wall, the nondimensional temperature is constant θ = 0, and at the hot right vertical wall, the wall temperature is time periodic, θ     

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 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.     

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.     

Natural convection in partially divided square enclosures: Effects of thermal boundary conditions and thermal conductivity of the partitions

Natural convection in partitioned square enclosures filled with air is numerically studied, trying to characterize these enclosures mainly in what concerns its overall heat transfer performance. Two partitions of finite thickness are considered, placed in the enclosure following an ordered arrangement, which position, length and thermal conductivity are varied for some values of Rayleigh number and for different thermal boundary conditions. Study starts considering the simplest enclosures with two adiabatic partitions, after the more realistic enclosures of heat conductive walls and partitions are considered, and finally the even more realistic situation of enclosures with heat conductive partitions and walls subjected to cyclic thermal boundary conditions in the vertical direction is also considered. Position and length of the enclosures’ effects depend on the thermal boundary conditions prescribed for the enclosure, and different thermal boundary conditions (corresponding to the heating or cooling operations or seasons) are considered to capture this effect. Fluid flow field, thermal field and heat transfer are analyzed for some particular situations through the streamlines, isotherms, and heatlines. The overall thermal performance of the enclosure is analyzed through the overall Nusselt number, and many data are compactly presented for different placements and lengths of the partitions, for different thermal conductivity of the walls and partitions of the enclosure, for different Rayleigh numbers and for different thermal boundary conditions imposed to the enclosure. Considered boundary conditions and the enclosure walls and partitions of finite thickness and finite thermal conductivity are much more realistic conditions than simply the single cavity without walls and with perfectly adiabatic partitions usually considered in many studies of this kind.     

Combined Effect of Mixed Convection and Surface Radiation Heat Transfer for Thermally Developing Flow in Vertical Channels

Combined effect of laminar flow mixed convection and surface radiation heat transfer for thermally developing airflow in a vertical channel heated from a side has been experimentally examined with different thermal and geometric parameters. The channel boundary is made of two isothermal walls and two adiabatic walls, the isothermal parallel wall is heated uniformly and the opposite cold wall temperature is maintained equal to the inlet conditions. The heated wall temperature ranged from 55 to 100°C, Reynolds number ranged from 800 to 2900 and the heat flux was varied from 250 to 870 W/m². To cover the wide range of Reynolds numbers, two aspect ratios of square and rectangular section were used. Surface radiation from the internal walls is considered through two emissivities i.e. 0.05 and 0.85, to represent weak and strong radiation effects, respectively. From the experiments, surface temperature and Nusselt number distributions of convection and radiation heat transfer are obtained for different heat flux values. Flow structure inside the channel is visualized to observe the flow pattern. The results show the combined effect of laminar flow mixed convection and surface radiation on the total heat transfer rate within the channel. The accumulating buoyancy force and airflow moves together vertically in the upward direction to give significant heat transfer enhancement in the vertical orientation of the channel.     

Effect of Heat Transfer on Natural Convection in a Square Cavity With Two Source Pairs

In this paper, the influence of a small heating source, positioned in the lateral walls of a square cavity, is investigated. Numerical and experimental analyses are performed to investigate natural convection heat transfer in a square cavity heated by hot strips in the side walls. The H side square cavity is filled with air and heated by two hot strips with heights of H/4. The effect of placing the hot strips at two different positions is evaluated. The temperature distribution and the Nusselt numbers at different Rayleigh numbers are experimentally measured using both real-time and double-exposure holographic interferometry. The isothermal patterns obtained through the holographic interferometry are compared with the temperature and velocity fields from a numerical study performed using the finite-volume code Fluent.     

A Thermal Nonequilibrium Approach to Natural Convection in a Square Enclosure Due to the Partially Cooled Sidewalls of the Enclosure

This article presents a numerical investigation of steady non-Darcy natural convection heat transfer in a square cavity filled with a heat-generating porous medium with partial cooling using a local thermal nonequilibrium (LTNE) model. Five different partial cooling boundary conditions and the fully cooled boundary condition are investigated under LTNE and local thermal equilibrium (LTE). The cooling portions of the left and the right sidewalls of the cavity are maintained at temperature T ₀ while the enclosure's top and bottom walls, as well as the inactive parts of its sidewalls, are kept insulated. The simulation results show that the placement order of wall cooling has a significant effect on the flow pattern and heat transfer rate. Compared with the fully cooled wall, the partially cooled wall of the cavity yielded a higher local Nusselt number for both fluid and solid phases. Under the same boundary conditions, the LTNE and LTE models can demonstrate significant differences in flow patterns and temperature fields. The total heat transfer rate increases with both Darcy number and Rayleigh number. Enhancement of interphase heat transfer coefficient (H) reduces the impact of Darcy number on the heat transfer rate of a porous cavity. Also, the total heat transfer rate of the porous medium decreases steadily with thermal conductivity ratio γ and interphase heat transfer coefficient H.     

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.     

Isotherms in a horizontal differentially heated cavity at intermediate Rayleigh numbers

Rayleigh-Benard convection in a horizontal, differentially heated, high aspect ratio fluid layer is considered. Experiments with three Rayleigh numbers (13900, 34800 and 51800) corresponding to different flow regimes have been performed. The temperature field in the fluid has been recorded in the form of interferometric projections, the fringe patterns representing isotherms. For steady and quasi-steady flow, the three dimensional temperature field has been obtained using an iterative algebraic reconstruction technique. Results show the formation of longitudinal rolls in the cavity at a Rayleigh number of 13900. The flow is unsteady at a Rayleigh number of 34800, but a strong indication of the formation of cubic rolls is seen. At a Rayleigh number of 51800, the flow field is completely unsteady with no discernible pattern. The local ray-averaged Nusselt numbers at each of the heated and cooled walls have been determined at the two lower Rayleigh numbers. The systematic variation of Nusselt number over each of the solid surfaces confirms the flow models proposed at these Rayleigh numbers. The average Nusselt number qualitatively matches the published correlations.