Numerical Simulation of Natural Convection and Radiation Heat Transfer in Two-Dimensional Enclosure on Hybrid Grids

Natural convection and radiation heat transfer interaction commonly exists in engineering problems, and a numerical method for combined natural convection and radiation heat transfer is very important in practical engineering applications. In this article, the finite-volume method (FVM) for radiation is formulated and implemented in the fluid flow solver GTEA on hybrid grids. For comparison and validation, three test cases, an equilateral triangular enclosure and a square enclosure with/without baffles, are chosen. Then, natural convections in a cavity with/without baffles are simulated with the present FVM to take into account the radiation heat transfer effects. All the results obtained by the presented FVM agree very well with the exact solutions as well as results obtained by the zone method. Natural convection under low gravity is researched with Gr = 0.7 and 700, and the radiation effects on the temperature distribution are also studied with variation conduction-radiation numbers, Nr = 0.06, 0.1, and 0.15. It is found that the solutions are sensitive to the conduction-radiation number but do not change very much with the Grashof number.     

Influences of a confined elliptic cylinder at different aspect ratios and inclinations on the laminar natural and mixed convection flows

Numerical investigations were carried out for natural and mixed convection within domains with stationary and rotating complex geometry by using an immersed-boundary method. The method was first validated with flows induced by natural convection in the annulus between concentric circular cylinder and square enclosure, and the grid-function convergence tests were also examined. Natural convection induced by isothermally elliptic cylinder was further investigated for different Rayleigh numbers within the range of 10⁴–10⁶ and the influence of the outer enclosure was also considered. The parameters investigated in the study included Rayleigh number, axis ratio and inclination angle of the elliptic cross-section. Local and average heat transfer characteristics were fully studied around the surfaces of both inner cylinder and outer enclosure. Finally, mixed convection in a square enclosure with an active rotating elliptic cylinder was considered and the heat transfer quantities of the system were obtained for different rotating speeds.     

Unstructured Polygonal Finite-Volume Solutions of Radiative Heat Transfer in a Complex Axisymmetric Enclosure

Radiative heat transfer in a complex axisymmetric enclosure with participating medium is investigated by using the finite-volume method (FVM). In particular, an implementation of the unstructured polygonal meshes is adopted by connecting each center point of the unstructured triangular meshes rather than joining the centroids of the triangular elements to the midpoints of the corresponding sides to form a polygonal element. Also, typical considerations regarding application of the present polygonal mesh system to axisymmetric radiation are discussed. After a mathematical formulation and corresponding discretization equation for the radiative transfer equation (RTE) are derived, the final discretization equation is introduced with the conventional finite-volume approach by using the directional weights. For validation and comparison, three test examples with complex axisymmetric geometries have been accomplished. The present study shows that not only is the method flexible in treating radiative problems with complex geometries, it is also accurate and efficient for the analysis of axisymmetric radiative heat transfer.     

Numerical Method on Natural Convection and Radiation Heat Transfer with an Isotropic Scattering Medium

The finite-volume method (FVM) for radiation heat transfer with a nonscattering medium is extended to an isotropic scattering medium, and this method is implemented in the fluid flow solver GTEA on hybrid grids. For comparison and validation, three test cases, a semicircle enclosure with a hole, a rhombic enclosure, and a square cavity, are chosen. All the results obtained by the present FVM agree very well with the numerical solutions in the references. Furthermore, the effects of the extinction coefficient and scattering albedo on the flow and temperature distribution are studied numerically in the cavity based on present approach. As the extinction coefficient increases from 0.2 to 5, the temperature gradient adjacent to the hot and cold walls gradually decreases at Ra = 10⁵, however, the temperature profiles become similar at Ra = 10⁶. For Ra = 10⁵, 10⁶, the scattering albedo affects the structures of the isotherm and streamline to some extent. As the scattering albedo increases, the convection heat transfer in the middle region of the hot wall increases, but the radiation heat transfer and the total radiation heat transfer along the hot wall decrease.     

NUMERICAL ANALYSIS OF NATURAL CONVECTIVE AND RADIATIVE HEAT TRANSFER IN AN ARBITRARILY SHAPED ENCLOSURE

The flow and thermal characteristics of the interactions of natural convection and radiation in an enclosure containing circular ducts are analyzed numerically. For calculation of flow fields, the SIMPLE algorithm originally developed in Cartesian coordinates is extended and modified to apply to the curvilinear coordinates system. The radiation part of the problem for an arbitrarily shaped domain is solved by using the finite volume method (FVM). Cartesian velocity components are used as the dependent variables in momentum equations, and a nonstaggered grid system is employed. The flow and thermal fields for an irregular geometry are investigated for the variation of such parameters as scattering albedo, optical thickness, and Planck number. The test problem is compared with both the exact solutions and the discrete ordinates method solution. The results show that the FVM is an effective method to predict radiative heat transfer processes in irregular geometries and that the change of optical thickness has more effect than that of scattering albedo on flow and thermal fields.     

Analysis of Conduction and Radiation Heat Transfer in a 2-D Cylindrical Medium Using the Modified Discrete Ordinate Method and the Lattice Boltzmann Method

This article deals with the analysis of radiative transport with and without conduction in a finite concentric cylindrical enclosure containing absorbing, emitting, and scattering medium. Isothermal medium as the radiation source confined between the cold cylinders and a nonisothermal medium with the inner cylinder as the radiation source are the two nonradiative and radiative equilibrium problems. They involve only calculation of radiative information. In the third problem, a conducting-radiating medium is thermally perturbed by raising the temperature of the inner cylinder. In all problems, radiative information is computed using the modified discrete ordinate method (MDOM), and in the third problem, the lattice Boltzmann method (LBM) is used to formulate and solve the energy equation. Depending on the problems, effects of various parameters such as the extinction coefficient, the scattering albedo, the boundary emissivity, the conduction-radiation parameter, and the radius ratio are studied on temperature and heat flux distributions. The MDOM and the LBM-MDOM results are compared with those available in the literature. To further establish the accuracy of the MDOM and the LBM-MDOM results, in all problems, comparisons are made with the results obtained from the finite volume method (FVM) and the finite difference method-FVM approach, in which FVM provides the radiative information. The selection of the FDM-FVM for the third problem is also with the objective that for this problem, not much work is reported in which the FVM is used to calculate the radiative information. MDOM and LBM-MDOM results are found to compare well with those available in the literature, and in all cases they are in excellent agreement with FVM and FDM-FVM approaches.     

Development of a finite element radiation model applied to two‐dimensional participating media

A finite element method (FEM) for radiative heat transfer has been developed and it is applied to 2D problems with unstructured meshes. The present work provides a solution for temperature distribution in a rectangular enclosure with black or gray walls containing an absorbing, emitting, isotropically scattering medium. Compared with the results available from Monte Carlo simulation and finite volume method (FVM), the present FEM can predict the radiative heat transfer accurately. © 2005 Wiley Periodicals, Inc. Heat Trans Asian Res, 34(6): 386–395, 2005; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20076     

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.     

Heat transfer regulation in a rectangular enclosure using a rotating plate

The effects of an inner rotating plate with horizontal axis on the heat transfer in a differentially heated vertical enclosure were investigated experimentally. The aspect ratio of the enclosure height/width was 1 throughout the experiments. An acrylic plate with a small thermal conductivity was installed horizontally at the center of the square enclosure, and was rotated at various speeds for normal and reverse rotations by using the motor attached outside of the enclosure. Purified water was used as the working fluid. The flow pattern was sketched by a visualization experiment using aluminum powder. The heat transfer results were also compared with those from a previous paper on a rotating cylinder. It is clarified here that the heat transfer rate of the enclosure depends largely on the parameter Gr_w/Re_ω², and is characterized by three regions. The heat transfer rate of the enclosure with a rotating plate is somewhat larger than that of a rotating cylinder in the forced convection region. The rotating plate used here will be useful for regulation of wide‐ranging heat transfer. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(4): 342–353, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10099     

Steady mixed convection in a differentially heated square enclosure with an active rotating circular cylinder

Mixed convection in a square enclosure with a rotating cylinder centered within it is numerically studied. Depending on the rotation of the cylinder, the natural and forced convection effects can be combined or opposite. However, due to the complex flow structure assumed by the flow, in some conditions the combined effects do not lead to the maximum overall heat transfer crossing the enclosure. The rotating cylinder participates on both the conductive and convective heat transfer processes, and exchanges heat with the fluid naturally, without imposition of a thermal condition at its surface. It is explored the influence of the cylinder through its radius, rotating velocity, thermal conductivity and thermal capacity on the resulting mixed convection problem. Thermal field is visualized using the isotherms, the flow structure is visualized through the streamlines, and the heat transfer process is visualized through the heatlines. For the first time, these visualization tools are applied to a moving solid. The overall thermal performance of the enclosure is analyzed through the overall Nusselt number. For a better understanding of the participation of the cylinder on the heat transfer process, the local heat exchanged between the cylinder surface and the fluid is also analyzed. Results clearly show how the rotating cylinder affects the thermal performance of the enclosure, and how the thermophysical properties of the cylinder are important on the overall heat transfer process across the enclosure.     

A Meshless Local Radial Basis Function Method for Two-Dimensional Incompressible Navier-Stokes Equations

A meshless local radial basis function method is developed for two-dimensional incompressible Navier-Stokes equations. The distributed nodes used to store the variables are obtained by the philosophy of an unstructured mesh, which results in two main advantages of the method. One is that the unstructured nodes generation in the computational domain is quite simple, without much concern about the mesh quality; the other is that the localization of the obtained collocations for the discretization of equations is performed conveniently with the supporting nodes. The algebraic system is solved by a semi-implicit pseudo-time method, in which the convective and source terms are explicitly marched by the Runge-Kutta method, and the diffusive terms are implicitly solved. The proposed method is validated by several benchmark problems, including natural convection in a square cavity, the lid-driven cavity flow, and the natural convection in a square cavity containing a circular cylinder, and very good agreement with the existing results are obtained.     

Laminar combined magnetoconvection in a wavy enclosure with the effect of heat conducting cylinder

In the current work, a numerical study of the flow characteristics on combined magnetoconvection in a lid-driven square enclosure, differentially heated, is carried out. This problem is solved by using finite element method of the partial differential equations, which are the heat transfer and stream function in Cartesian coordinates. The tests are performed for different solid–fluid thermal conductivity ratio, cylinder location and Richardson number while the Prandtl number, Reynolds number, magnetic and Joule heating parameters are kept constant. One geometrical configuration is used namely two undulations. The outcome obtained shows that the heat conducting inner square cylinder affects the flow and the heat transfer rate in the enclosure. The trend of the local heat transfer is found to follow a wavy pattern. Results are presented in terms of streamlines, isotherms, average Nusselt number at the heated wavy wall, average temperature of the fluid in the enclosure and dimensionless temperature at the cylinder center for different combinations of the governing parameters.     

Heat transfer in an inclined enclosure with an inner rotating plate

The effects of an inner rotating plate on the heat transfer in a differentially heated inclined enclosure were investigated experimentally. The aspect ratio of the enclosure (height/width) was 1 throughout the experiments. An acrylic plate with a small thermal conductivity was installed horizontally at the center of the square enclosure, and was rotated at various speeds by using a motor attached outside of the enclosure. The inclination angle of the enclosure was varied from –90° to 90°. Purified water was used for the working fluid. The flow pattern was sketched by a visualization experiment using aluminum powder. The heat transfer enhancement can be clearly seen for the inclined enclosure with the hot wall downward facing. The rotating plate used here is useful for the regulation of a wide‐ranging heat transfer rate. © 2001 Scripta Technica, Heat Trans Asian Res, 30(4): 331–340, 2001     

Analysis of Conduction and Radiation Heat Transfer in a Differentially Heated 2‐D Square Enclosure

Radiative heat transfer with and without conduction in a differentially heated 2‐D square enclosure is analyzed. The enclosure with diffuse gray boundaries contains radiating and/or conducting gray homogeneous medium. Radiatively, the medium is absorbing, emitting and scattering. On the south boundary, four types of discrete heated regions, viz., the full boundary, the left one‐third, left two third and middle one third, are considered. In the absence of conduction, distributions of heat flux along the south boundary are studied for the effect of extinction coefficient. In the presence of conduction, distributions of radiation, conduction and total heat fluxes along the south boundary are analyzed for the effects of extinction coefficient, scattering albedo, conduction–radiation parameter, and south boundary emissivity. Effects of these parameters on centerline temperature distribution are also studied. To assess the performance of three commonly used radiative transfer methods, in all cases, the radiative transfer equation is solved using the discrete ordinate method (DOM), the conventional discrete ordinate method (CDOM) and the finite volume method (FVM). In the combined mode problem, with volumetric radiative information known from one of the three methods, viz., DOM, CDOM, and FVM, the energy equation is solved using the finite difference method (FDM). In all cases, the results from FDM‐DOM, FDM‐CDOM, and FDM‐FVM are in good agreement. Computationally, all three sets of methods are equally efficient.     

Numerical investigation of natural convection phenomena in a uniformly heated circular cylinder immersed in square enclosure filled with air at different vertical locations

In this model, a numerical study of two dimensional steady natural convection is performed for a uniform heat source applied on the inner circular cylinder in a square air (Pr = 0.7) filled enclosure in which all boundaries are assumed to be isothermal (at a constant low temperature). The developed mathematical model is governed by the coupled equations of continuity, momentum and energy and is solved by finite volume method. The effects of vertical cylinder locations and Rayleigh numbers on fluid flow and heat transfer performance are investigated. Rayleigh number is varied from 10³ to 10⁶ and the location of the inner cylinder is changed vertically along the centerline of the enclosure from − 0.25 L to 0.25 L upward and downward, respectively. It is found that at small Rayleigh numbers does not have much influence on the flow field while at high Rayleigh numbers have considerable effect on the flow pattern. In addition, the numerical solutions yield a two cellular flow field between the inner cylinder and the enclosure. Also, the total average Nusselt number behaves nonlinearly as a function of locations. Results are presented in terms of the streamlines, isotherms, local and average Nusselt numbers. Detailed results of the numerical has been compared with literature ones, and it gives a reliable agreement.     

Meshless method based on the local weak-forms for steady-state heat conduction problems

In this article, the meshless local Petrov–Galerkin (MLPG) method is applied to compute two steady-state heat conduction problems of irregular complex domain in 2D space. The essential boundary condition is enforced by the transformation method, and the MLS method is used for interpolation schemes. A numerical example that has analytical solution shows the present method can obtain desired accuracy and efficiency. Two cases in engineering with irregular boundary are computed to validate the approach by comparing the present method with the finite volume method (FVM) solutions obtained from a commercial CFD package FLUENT 6.3. The results show that the present method is in good agreement with FVM. It is expected that MLPG method (which is a truly meshless) is very promising in solving engineering heat conduction problems within irregular domains.     

Application of the lattice Boltzmann method for solving the energy equation of a 2-D transient conduction-radiation problem

A lattice Boltzmann method (LBM) is used to solve the energy equation in a test problem involving thermal radiation and to thus investigate the suitability of scalar diffusion LBM for a new class of problems. The problem chosen is transient conductive and radiative heat transfer in a 2-D rectangular enclosure filled with an optically absorbing, emitting and scattering medium. The energy equation of the problem is solved alternatively with a previously used finite volume method (FVM) and with the LBM, while the radiative transfer equation is solved in both cases using the collapsed dimension method. In a parametric study on the effects of the conduction-radiation parameter, extinction coefficient, scattering albedo, and enclosure aspect ratio, FVM and LBM are compared in each case. It is found that, for given level of accuracy, LBM converges in fewer iterations to the steady-state solution, independent of the influence of radiation. On the other hand, the computational cost per iteration is higher for LBM than for the FVM for a simple grid. For coupled radiation-diffusion, the LBM is faster than the FVM because the radiative transfer computation is more time-consuming than that of diffusion.     

Natural convection between a circular enclosure and an elliptic cylinder using Control Volume based Finite Element Method

In this study natural convection heat transfer in a cold outer circular enclosure containing a hot inner elliptic cylinder is investigated numerically using the Control Volume based Finite Element Method (CVFEM). Both of the circular enclosure and the inner cylinder are maintained at constant temperatures with air filled inside the enclosure. The governing equations are used in their vorticity stream function form to simulate the fluid flow and heat transfer. The numerical calculations are performed for various Rayleigh numbers, the inclination angle of the enclosure and different sizes of inner cylinder. The results show that streamlines, isotherms, and the number, size and formation of the cells inside the enclosure are strongly depend on these parameters which considerably enhance the heat transfer rate.     

USE OF GCG METHODS FOR THE EFFICIENT SOLUTION OF MATRIX PROBLEMS ARISING FROM THE FVM FORMULATION OF RADIATIVE TRANSFER

Preconditioned generalized conjugate gradient (GCG) iterative methods are applied to the solution of large, sparse, and unsymmetric linear algebraic equations resulting from the application of the finite-volume method to the problem of radiative heat transfer in an absorbing, emitting, and scattering gray medium, with the boundary surfaces reflecting radiation in both diffuse and specular regimes. The governing radiative transfer equation, which is a complicated integro-differential equation, has been discretized using the S N finite-volume method (FVM). Different variants of GCG methods have been tested on a problem of 2-D radiation in a cylinder, and efficiencies of the methods have been compared. Numerical results indicate that preconditioning suggested in the article dramatically improves the performance of the GCG methods. Results on test problems based on S 8 FVM agree well with exact results reported in the literature.     

A numerical study of laminar natural convective heat transfer around a horizontal cylinder inside a concentric air-filled triangular enclosure

A numerical investigation of steady-state laminar natural convective heat transfer around a horizontal cylinder to its concentric triangular enclosure was carried out. The enclosure was filled with air and both the inner and outer cylinders were maintained at uniform temperatures. The buoyancy effect was modeled by applying the Boussinesq approximation of density to the momentum equation and the governing equations were iteratively solved using the control volume approach. The effects of the Rayleigh number and the aspect ratio were examined. Flow and thermal fields were exhibited by means of streamlines and isotherms, respectively. Variations of the maximum value of the dimensionless stream function and the local and average Nusselt numbers were also presented. The average Nusselt number was correlated to the Rayleigh number based on curve-fitting for each aspect ratio. At the highest Rayleigh number studied, the effects of different inclination angles of the enclosure and various cross-section geometries of the inner cylinder were investigated. The computed results indicated that at constant aspect ratio, both the inclination angle and cross-section geometry have insignificant effects on the overall heat transfer rates though the flow patterns are significantly modified.