SUMMARY OF BENCHMARK NUMERICAL STUDIES FOR 3-D NATURAL CONVECTION IN AN AIR-FILLED ENCLOSURE

A numerical benchmark study dealing with 3-D natural convection in an air-filled cavity oriented at three different angles was held at the International Center for Heat and Mass Transfer (ICHMT) 2nd International Symposium on Advances in Computational Heat Transfer (CHT'01), May 20 - 25, 2001, in Palm Cove, Australia. The benchmark problem derives from an experimental study conducted by Leong, Hollands, and Brung involving natural convection within a differentially heated cubical cavity oriented at three different angles (φ) and subjected to four different Rayleigh numbers ( Ra ), all in the range 10 5 h Ra h 10 8 . The cube sidewall temperature varied linearly with distance from the hot face. In view of the number of test cases reported and the relatively low experimental uncertainty (about 1% of measurement), the problem appeared to be well suited as a benchmark for numerical validation. Ten teams (sets of participants) submitted papers using various mesh sizes and computational techniques. Most teams reported the existence of oscillatory solutions for cases at high Ra where the heating was primarily from below. Computed Nusselt numbers agreed with measurements at Ra and at inclinations where the heating was at least partly from the side. However, no one set of participants produced Nusselt numbers that agreed with the experimental results for all test cases.     

Thermal transport analysis for natural convection in a porous corrugated rhombic enclosure

A numerical study of fluid flow and heat transfer, applying natural convection is carried out in a porous corrugated rhombic enclosure. A uniform heating source is applied from the bottom boundary wall while the inclined side walls are maintained to a constant cold temperature and the top corrugated wall is retained at insulated condition inside the enclosure. The heat transfer and flow features are presented for a wide spectrum of Rayleigh numbers (Ra), 10⁴ ≤ Ra ≤ 10⁶, and Darcy numbers (Da), 10^?3 ≤ Da ≤ 10^?2. The number of undulations (n) for the top and bottom walls have been varied from 1 to 13 keeping the amplitude of undulation fixed. It is revealed that the characteristics of heat transfer are conceivably modulated by changing the parameter of the undulation number on the enclosure walls, specifically at the bottom and top. The influencing control of n in altering the heat transfer rate is felt maximum on the left wall and minimum for the right wall, and there is a strong interplay between Ra and Da together with n on dictating the heat transfer characteristics. The critical value, where heat transfer rate is observed as maximum is at n = 11 and thereafter the values decrease.     

NUMERICAL STUDY OF DEVELOPING NATURAL LAMINAR CONVECTION IN A VERTICAL HYPERBOLIC DUCT OF A FIXED LENGTH AND WITH A CONSTANT WALL TEMPERATURE

The natural laminar convection in a vertical hyperbolic duct of a fixed length and with a constant wall temperature is numerically investigated. The governing equations are solved by a finite difference method. The results are obtained for the velocity, temperature, and pressure fields, and for the mean heat transfer coefficient. The numerical calculations are fulfilled for Rayleigh numbers ( Ra ) ranging from 5 to 3 · 104 and for the numerical eccentricity ranging from 5 to 100. The effects of the numerical eccentricity and Ra are examined and the results are compared with those of a cylindrical vertical duct. It was found that the flow fields and Nusselt number ( Nu ) are affected significantly at small values of the numerical eccentricity and Ra .     

A Differential Quadrature Solution of Natural Convection in an Enclosure with a Partial Partition

Natural convection in a partially divided enclosure has been examined numerically using a differential quadrature method. Governing equations for the flow and heat transfer have been constructed by the vorticity-stream function formulation and computational results have been obtained for the Rayleigh numbers, 10⁴, 10⁵, 10⁶, the locations of the partition, 0.2, 0.4, 0.6, 0.8, and the partition ratios, 0.25, 0.50, 0.75, 1.0. The results show that circulation strength, and therefore heat transfer, decreases considerably with increasing partition height especially for the higher values of the Rayleigh number. As the distance of the partition from the hot wall increases, first a decrease and then an increase is observed in the average Nusselt number for the low Rayleigh numbers and first an increase and then a decrease turns out for the high Rayleigh numbers. As for the average Nusselt number on the partition, first it shows a decreasing trend and then an increasing trend as the partition is distanced from the hot wall towards the cold wall. The increase in the partition height brings about significant increase in the average Nusselt number on the partition.     

Visualization of Heat Transport during Natural Convection Within Porous Triangular Cavities via Heatline Approach

In this article, natural convection in a porous triangular cavity has been analyzed. Bejan's heatlines concept has been used for visualization of heat transfer. Penalty finite-element method with biquadratic elements is used to solve the nondimensional governing equations for the triangular cavity involving hot inclined walls and cold top wall. The numerical solutions are studied in terms of isotherms, streamlines, heatlines, and local and average Nusselt numbers for a wide range of parameters Da (10^?5–10^?3), Pr (0.015–1000), and Ra (Ra = 10³–5 × 10⁵). For low Darcy number (Da = 10^?5), the heat transfer occurs due to conduction as the heatlines are smooth and orthogonal to the isotherms. As the Rayleigh number increases, conduction dominant mode changes into convection dominant mode for Da = 10^?3, and the critical Rayleigh number corresponding to the on-set of convection is obtained. Distribution of heatlines illustrate that most of the heat transport for a low Darcy number (Da = 10^?5) occurs from the top region of hot inclined walls to the cold top wall, whereas heat transfer is more from the bottom region of hot inclined walls to the cold top wall for a high Darcy number (Da = 10^?3). Interesting features of streamlines and heatlines are discussed for lower and higher Prandtl numbers. Heat transfer analysis is obtained in terms of local and average Nusselt numbers (Nu _l , Nu _t ) and the local and average Nusselt numbers are found to be correlated with heatline patterns within the cavity.     

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.     

Effects of Thermal Boundary Conditions on Entropy Generation during Natural Convection

A comprehensive numerical study on entropy generation during natural convection is studied in a square cavity subjected to a wide variety of thermal boundary conditions. Entropy generation terms involving thermal and velocity gradients are evaluated accurately based on the elemental basis set via the Galerkin finite element method. The thermal and fluid irreversibilities during the conduction and convection dominant regimes are analyzed in detail for various fluids (Pr = 0.026,988.24) within Ra = 10³–10⁵. Further, the effect of Ra on the total entropy generation and average Bejan number is discussed. It is observed that thermal boundary conditions significantly affect the thermal mixing, temperature uniformity, and the entropy generation in the cavity. A case where the bottom wall is hot isothermal with linearly cooled side walls and adiabatic top wall is found to result in high thermal mixing and a higher degree of temperature uniformity with minimum total entropy generation.     

Laminar Natural Convection of Bingham Fluids in a Square Enclosure with Vertical Walls Subjected to Constant Heat Flux

In this study, two-dimensional steady-state simulations of laminar natural convection in square enclosures with vertical sidewalls subjected to constant heat flux have been carried out, where the enclosures are considered to be completely filled with a yield-stress fluid obeying the Bingham model. Yield stress effects on heat and momentum transport are investigated for nominal values of Rayleigh number (Ra) in the range 10³–10⁶ and a Prandtl number (Pr) range of 0.1–100. It is found that the mean Nusselt number Nu increases with increasing values of Rayleigh number for both Newtonian and Bingham fluids. However, Nu values obtained for Bingham fluids are smaller than that obtained in the case of Newtonian fluids with the same nominal value of Rayleigh number Ra due to weakening of convective transport. The mean Nusselt number Nu in the case of Bingham fluids is found to decrease with increasing Bingham number, and for large values of Bingham number Bn, the value settles to unity (Nu = 1.0) as heat transfer takes place principally due to thermal conduction. The Nu values for the vertical walls subjected to constant heat flux are smaller than the corresponding values in the same configuration with constant vertical wall temperatures (for identical values of nominal Rayleigh, Prandtl, and Bingham numbers). However, the value of Bingham number at which Nu approaches to unity remains the same for both constant wall temperature and constant wall heat flux configurations. It is demonstrated that for small values of Bingham number Nu increases with increasing Prandtl number, but the opposite behavior occurs for large values of Bingham number. New correlations are proposed for the mean Nusselt number Nu for both Newtonian and Bingham fluids for square enclosures with vertical walls subjected to constant heat flux, which are shown to satisfactorily capture the correct qualitative and quantitative behavior of Nu in response to changes in Ra, Pr, and Bn.     

Heat transfer enhancement in cubical enclosures with vertical fins

Natural convection of air in a cubical enclosure with a thick partition fitted vertically on the hot wall is numerically investigated for Rayleigh numbers of 10³–10⁶. A three dimensional convective circulation is generated, in which the cold flow sweeps the fin faces and the hot wall, with low flow blockage. The combined contributions of these faces cause heat transfer enhancements over 40% at high Rayleigh numbers and thermal conductivity ratios (R_k). These enhancements significantly exceed the ones obtained with horizontal fins. Even low R_k values cause heat transfer enhancements, except at Ra = 10⁴.     

MULTIGRID COMPUTATION OF NATURAL CONVECTION IN ENCLOSURES WITH A CONDUCTIVE BAFFLE

Conjugate heat transfer has been investigated in a two-dimensional square enclosure with a conducting vertical baffle of finite thickness and varying height. The horizontal end walls are assumed to be adiabatic, and the vertical watts are at constant but different temperatures. Calculations are made by a staggered, finite volume multigrid procedure. The performance of the multigrid method in accelerating the convergence rate is remarkable by comparison with the usual iterative method. The influence of inserting a baffle into the buoyancy-driven square cavity on the heat transfer, as well as on the temperature distribution and velocity field, has been obtained for various Rayleigh numbers Ra, solid /fluid conductivity ratio k? dimensionless baffle height H, and baffle location L. Predicted flow patterns and isotherms indicate that the effect of inserting baffles of varying height upon the overall heat transfer and local temperature profiles in the cavity is limited, except when the height of baffle is large (H > 0.5). The effect of conductivity is also found to be marginal. However, both the height and the conductivity become very important when the baffle is located very near the hot wall or cold walls.     

ENCLOSED BUOYANT CONVECTION OF A VARIABLE-VISCOSITY FLUID UNDER TIME-PERIODIC THERMAL FORCING

Finite-volume numerical solutions are obtained for buoyant convection of a fluid with temperature-dependent viscosity in an enclosed space. At one vertical wall the temperature is constant, and at the other vertical wall the temperature is time-periodic. Solutions to the governing Navier-Stokes equations are acquired for a fixed Rayleigh number and over wide ranges of viscosity contrast, which measures the ratio of viscosities at the cold side wall and hot side wall. The effects of variable viscosity on the time-mean value as well as the amplitude of fluctuation of instantaneous heat transfer rate are delineated. The extensive results reveal the existence of resonance and show that resonance becomes more distinctive for large viscosity contrast in the case of the hot-wall temperature oscillation. As the viscosity contrast increases, the upward convective motion is invigorated during the relative heating phase due to the lowered viscosity in the thermal boundary layer near the hot vertical side wall. This is also reflected in the augmentation of the cycle-averaged heat transfer rate. When the temperature oscillation is imposed on the cold wall, the flow is less sensitive to the viscosity variation. Physical interpretations of the overall flow and heat transfer are offered.     

Natural convection in enclosure with heating and cooling by sinusoidal temperature profiles on one side

A numerical study has been carried out in rectangular enclosures, which have a vertical active wall with all the other walls insulated. The equally divided active sidewall is heated and cooled with sinusoidal temperature profiles. Two cases have been considered: the first is the lower part is heated while the upper part is cooled and the second, the upper part is heated and lower part is cooled. Steady state heat transfer by laminar natural convection has been studied by numerically solving equations of mass, momentum and energy, to determine the thermal penetration in the enclosures and heat transfer as a function of Rayleigh number, the aspect ratio and the position of side heating with respect to side cooling. Rayleigh number was varied from 10³ to 10⁶ and the aspect ratio from 0.2 to 5, and the results are presented in the form of streamlines and isotherms, local and average Nusselt number, and heat penetration length. It is found that the penetration approaches to 100% at high Rayleigh numbers when the lower part is heated while the higher part is cooled. In the case of the higher part is heated and the lower part is cooled, the penetration is limited to 70% passing through maxima at Rayleigh number below 10⁶.     

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     

Distribution of heat sources in vertical open channels with natural convection

In this paper we use the constructal method to determine the optimal distribution and sizes of discrete heat sources in a vertical open channel cooled by natural convection. Two classes of geometries are considered: (i) heat sources with fixed size and fixed heat flux, and (ii) single heat source with variable size and fixed total heat current. In both classes, the objective is the maximization of the global thermal conductance between the discretely heated wall and the cold fluid. This objective is equivalent to minimizing temperature of the hot spot that occurs at a point on the wall. The numerical results show that for low Rayleigh numbers (∼10²), the heat sources select as optimal location the inlet plane of the channel. For configuration (i), the optimal location changes as the Rayleigh number increases, and the last (downstream) heat source tends to migrate toward the exit plane, which results in a non-uniform distribution of heat sources on the wall. For configuration (ii) we also show that at low and moderate Rayleigh numbers (Ra_M ∼ 10² and 10³) the thermal performance is maximized when the heat source does not cover the entire wall. As the flow intensity increases, the optimal heat source size approaches the height of the wall. The importance to free the flow geometry to morph toward the configuration of minimal global resistance (maximal flow access) is also discussed.     

Lattice Boltzmann Simulation of Natural Convection in a Square Cavity with Linearly Heated Wall(s)

In this study, the lattice Boltzmann method is used in order to investigate the natural convection in a cavity with linearly heated wall(s). The bottom wall is heated uniformly and the vertical wall(s) are heated linearly, whereas the top wall is insulated. Investigation has been conducted for Rayleigh numbers of 10³ to 10⁵, while Prandtl number is varied from 0.7 to 10. The effects of an increase in Rayleigh number and Prandtl number on streamlines, isotherm counters, local Nusselt number and average Nusselt number are depicted. It has been observed that the average Nusselt number at the bottom wall augments with an increase in Prandtl number.     

A Study of Free Convection Induced by a Vertical Wavy Surface with Heat Flux in a Porous Enclosure

Free convection induced by a vertical wavy surface with uniform heat flux in a porous enclosure has been analyzed numerically using the finite element method (FEM). The flow and the convection process in the cavity is found to be sensitive to the flow parameter Rayleigh number (Ra), and geometrical parameters like wave amplitude (a), wave phase (φ), and number of waves (N) in the vertical dimension of the cavity. The study reveals that small sinusoidal drifts from the smoothness of a vertical wall with a phase angle of 60^o and high frequency enhances the free convection from a vertical wall with uniform heat flux.     

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.     

Simulation of high Rayleigh number natural convection in a square cavity using the lattice Boltzmann method

A thermal lattice Boltzmann method based on the BGK model has been used to simulate high Rayleigh number natural convection in a square cavity. The model uses the double populations approach to simulate hydrodynamic and thermal fields. The traditional lattice Boltzmann method on a uniform grid has unreasonably high grid requirements at higher Rayleigh numbers which renders the method impractical. In this work, the interpolation supplemented lattice Boltzmann method has been utilized. This is shown to be effective even at high Rayleigh numbers. Numerical results are presented for natural convection in a square cavity with insulated horizontal walls and isothermal vertical walls maintained at different temperatures. Very fine grids (wall y⁺ < 0.3) have been used for the higher Rayleigh number simulations. A universal structure is shown to exist in the mean velocity turbulent boundary layer profile for y⁺ < 10. This agrees extremely well with previously reported experimental data. The numerical results (for Rayleigh numbers up to 10¹⁰) are in very good agreement with the benchmark results available in the literature. The highlight of the calculations is that no turbulence model has been employed.     

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.