Local RBF meshless scheme for coupled radiative and conductive heat transfer

ABSTRACT

A local radial basis function meshless (LRBFM) method is developed to solve coupled radiative and conductive heat transfer problems in multidimensional participating media, in which compact support radial basis functions (RBFs) augmented on a polynomial basis are employed to construct the trial function, and the radiative transfer equation (RTE) and energy conservation equation are discretized directly at nodes by the collocation method. LRBFM belongs to a class of truly meshless methods which require no mesh or grid, and can be readily implemented in a set of uniform or irregular node distributions with no node connectivity. Performances of the LRBFM is compared to numerical results reported in the literature via a variety of coupled radiative and conductive heat transfer problems in 1D and 2D geometries. It is demonstrated that the local radial basis function meshless method provides high accuracy and great efficiency to solve coupled radiative and conductive heat transfer problems in multidimensional participating media with uniform and irregular node distribution, especially for coupled heat transfer problems in irregular geometry with Cartesian coordinates. In addition, it is extremely simple to implement.     

Global Versus Localized RBF Meshless Methods for Solving Incompressible Fluid Flow with Heat Transfer

Global and localized radial basis function (RBF) meshless methods are compared for solving viscous incompressible fluid flow with heat transfer using structured multiquadratic RBFs. In the global approach, the collocation is made globally over the whole domain, so the size of the discretization matrices scales as the number of the nodes in the domain. The localized meshless method uses a local collocation defined over a set of overlapping domains of influence. Only small systems of linear equations need to be solved for each node. The computational effort thus grows linearly with the number of nodes—the localized approach is slightly more expensive on serial processors, but is highly parallelizable. Numerical results are presented for three benchmark problems—the lid-driven cavity, natural convection within an enclosure, and forced convective flow over a backward-facing step—and results are compared with the finite-element method (FEM) and experimental data.     

The meshless analog equation method for solving heat transfer to molten polymer flow in tubes

In this paper, we proposed a meshless analog equation method (MAEM) to solve a heat transfer problem of molten polymer flow, which is considered to be a generalized Newtonian viscous flow. The MAEM, free from mesh generation and numerical integration, is a powerful meshless method. The numerical solutions are expressed by a linear combination of the derived radial basis functions (RBFs). This paper considers two different viscosity models for the molten polymer; one is temperature-independent power-law model and the other is temperature-dependent power-law model. The viscous dissipation term is included in the energy equation to capture the relevant physical phenomena. From the comparisons of numerical simulation, the meshless solutions are in good agreement with some analytical solutions and other finite element solutions. Moreover, the MAEM uses much less CPU-time and computer memory to simulate molten polymer flows. Therefore, it is believed that the RBF-based meshless method of the MAEM is a promising and flexible numerical scheme for molten polymer flow simulation.     

Benchmark solutions for natural convection in a cubic cavity using the high-order time-space method

Benchmark numerical solutions for a three-dimensional natural convection heat transfer problem in a cubical cavity are presented in this paper. The 3-D cavity has two differentially heated and isothermal vertical walls and also four adiabatic walls. The computations are conducted for three Rayleigh numbers of 10⁴, 10⁵ and 10⁶. The filled fluid is with air and the Prandtl number is fixed at 0.71. The computed results are efficiently obtained by using the time-space method, which was proposed by Saitoh (1991) as a highly efficient and fast solver for general heat transfer and fluid flow problems. In our computations, the high-accuracy finite differences of a fourth-order were employed for the spatial discretization of governing equations and boundary conditions. In addition the third-order backward finite difference was used in timewise discretization. The resultant converged flow and temperature characteristics are also presented. The spatial grid dependency of the solutions was examined on a uniform grid. In addition, the grid-independent benchmark solutions were obtained by Richardson extrapolation for three cases. The present benchmark solutions will be useful for checking the performance and accuracy of any numerical methodologies.     

Investigation of the use of radial basis functions in local collocation method for solving diffusion problems

Multidimensional diffusion problems can be solved by a local collocation method using radial basis functions. This method is computationally efficient because it is meshless and yields a sparse system of algebraic equations. Three types of radial basis functions—the generalized multiquadric function, the Gaussian function, and the generalized thin-plate spline function—are considered as interpolation functions in this method. Analysis of truncation error indicates that the use of the generalized multiquadric function or the Gaussian function produces satisfactory results, but the generalized thin-plate spline function is not a good interpolation function.     

Local RBF-DQ method for two-dimensional transient heat conduction problems

The meshless local radial basis function-based differential quadrature (RBF-DQ) method is applied on two-dimensional heat conduction for different irregular geometries. This method is the combination of differential quadrature approximation of derivatives and function approximation of radial basis function. Four different geometries with regular and irregular boundaries are considered, and numerical results are compared with those gained by finite element (FE) solution achieved by COMSOL commercial code. Outcomes prove that current technique is in very good agreement with FEM and this fact that RBF-DQ method is an accurate and flexible method in solution of heat conduction problems.     

Determination of convective heat transfer for fenestration with between-the-glass louvered shades

In previous work, a two-dimensional steady laminar natural convection model of a window cavity with between-panes louvers (i.e., slats) was developed by approximating the system as a vertical cavity with isothermal walls at different temperatures, and with rotatable baffles located midway between the walls. The baffles were set to a third temperature so that night-time and day-time conditions, and the effects of low emissivity coatings (low-e), could be considered. It was found that the system is suited to a traditional one-dimensional analysis. A novel approach that allows the use of standard vertical cavity convection correlations and a modified cavity half-width is described, and a cavity modification factor, n^*, is presented. Finally, the n^* factor and vertical cavity convection correlation are joined with a longwave radiant model, and the results are compared to experimental results. The models show good agreement with experiments.     

Comparative Study on Accuracy and Solution Cost of the First/Second-Order Radiative Transfer Equations Using the Meshless Method

The direct collocation meshless (DCM) method is applied to solve and evaluate the performance of the second-order radiative transfer equation (SORTE) proposed by Zhao and Liu (Numer. Heat Transfer B, vol. 51, pp. 391–409, 2007). The SORTE transforms the original first-order radiative transfer equation (FORTE) into a form similar to a diffusion equation, so no additional artificial diffusion or upwinding treatment is needed in the numerical discretization for stabilization. In order to investigate the accuracy and cost of the direct collocation meshless method based on the SORTE, two typical radiative transfer problems are considered. These cases are also solved by the DCM approach and the least-squares collocation meshless (LSCM) approach based on the FORTE. Numerical results show that the DCM approach based on the SORTE is more accurate and stable than the DCM approach and the LSCM approach based on the FORTE. The convergence rate of the SORTE-based methods with increase of collocation point number is faster than that of the FORTE-based methods. For obtaining the same target accuracy, the DCM approach based on the SORTE is more efficient than the other two meshless methods based on the FORTE. In addition, the DCM approach based on the SORTE also exhibits higher accuracy in solving radiative transfer problems with complex geometries or discontinuous temperature distributions along the boundary.     

Two-level meshless local Petrov Galerkin method for multi-dimensional nonlinear convection–diffusion equation based on radial basis function

Abstract

In this article, a two-level meshless local Petrov Galerkin method (MLPG) is proposed to analyze nonlinear convection–diffusion equation based on the radial basis function (RBF) collocation method. Two-level method is employed to save the computation time, solving one small linearized convection–diffusion equation in a coarse nodal distribution by Newton iterative scheme and then processing one linear equation on a fine nodal distribution. The convergence analysis for the MLPG method and the two-level method is proven by applying the RBF interpolation method. Finally, the numerical examples provide a sufficient support and show that the proposed method is efficient for solving nonlinear convection–diffusion equation.     

A new meshless “fragile points method” and a local variational iteration method for general transient heat conduction in anisotropic nonhomogeneous media. Part II: Validation and discussion

Abstract

In the first part of this two-paper series, a new computational approach is presented for analyzing transient heat conduction problems in anisotropic nonhomogeneous media. The approach consists of a truly meshless Fragile Points Method (FPM) being utilized for spatial discretization, and a Local Variational Iteration (LVI) scheme for time discretization. In the present article, extensive numerical results are provided as validations, followed by a discussion on the recommended computational parameters. The FPM?+?LVIM approach shows its capability in solving 2?D and 3?D transient heat transfer problems in complex geometries with mixed boundary conditions, including preexisting cracks. Both functionally graded materials and composite materials are considered. It is shown that, with appropriate computational parameters, the FPM?+?LVIM approach is not only accurate, but also efficient, and has reliable stability under relatively large time intervals.     

Natural convection of a binary mixture confined in a slightly inclined tall enclosure

This paper presents an analytical and numerical study of natural convection of a binary mixture confined in a tall enclosure, slightly inclined about the gravity field. The cavity is heated from the bottom by a constant heat flux while the long side walls are impermeable and adiabatic. Both double-diffusive convection and Soret-induced convection are considered. The basis of the analytical approximation is an assumption of parallel flow over a large portion of the layer. The existence of multiple steady states, for small enough inclinations around the vertical plane, is demonstrated. Numerical confirmation of the stable analytical results is also presented.     

Jet Emerging from a Nozzle and Impinging on a Conical Cavity: Influence of Nozzle and Cavity Geometric Configurations

A jet emerging from a nozzle with different cone angles and impinging onto a conical cavity with different depths and diameters is considered. The flow simulation is extended to include a jet emerging from a pipe and impinging onto the cavity for the comparison. The Reynolds stress turbulence model is incorporated to account for the turbulence. The control volume approach is used to discretize the governing equations of flow and heat transfer. It is found that the flow structure above the cavity differs significantly due to radial expansion of the flow emerging from the nozzle. This modifies the flow structure in the cavity, particularly for a large diameter. The influence of the nozzle cone angle on the heat transfer coefficient and the shear stress along the cavity wall is more pronounced for a large diameter cavity.     

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.     

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.     

Meshless local Petrov Galerkin method for 2D/3D nonlinear convection–diffusion equations based on LS-RBF-PUM

In this article, we propose a meshless local Petrov Galerkin (MLPG) method based on least square radial basis function partition of unity method (LS-RBF-PUM), which is applied to the nonlinear convection–diffusion equations. The proposed method is not sensitive to the node layout, and has good stability and flexibility to complex domain. In order to treat nonlinear term, Picard iterative scheme is employed to confirm the convergence of iterative process. Error estimates are derived by the radial basis function interpolation method and convergence rate is proven to be second order. Numerical examples are performed for the nonlinear convection–diffusion equations in two and three space dimensions (2D/3D), which not only supports the theoretical results but also finds out superconvergence of third order.     

Numerical study of mixed convection in a ventilated square enclosure with the lattice Boltzmann method

In this article, numerical study of heat transfer by convection in a square cavity was investigated. The vertical walls of the cavity are differentially heated and the horizontal walls are considered adiabatic. A ventilation jet is created by a fan placed in the cavity. A lattice Boltzmann model for incompressible flow equation is used to simulate the problem. A parametric study was performed presenting the influence of Reynolds number (20 ≤ Re?≤?500), Rayleigh number (10≤Ra?≤?10⁺⁶), and fan position (0.2?≤?L_F≤0.8). It has been observed that heat transfer rate increases with the Reynolds number increasing and it is maximal for the L_F=0.2.     

Effect of anisotropy on the development of convective boundary layer flow in porous media

The effects of anisotropy on the development of thermal boundary layer flow in a rectangular porous cavity is studied. The side walls of the cavity are respectively heated and cooled isothermally. Top and bottom walls are insulated. The porous medium is anisotropic both in permeability and thermal conductivity with its principal axes oriented in a direction that is oblique to the gravity vector. Scale analysis is applied to predict the orders of magnitude involved in the boundary layer regime. In the large Rayleigh number limit, the governing boundary layer equations are solved in closed form, using an intergral approach. A finite difference method is used to obtain numerical solutions of the full governing equations. The effects of the anisotropy in permeability and thermal conductivity on the development of free convective boundary layer flow are found to be significant.     

Experimental validation of the coupled fluid flow, heat transfer and electromagnetic numerical model of the medium-power dry-type electrical transformer

This paper presents experimental validation of a numerical model of coupled processes within a three-phase medium-power dry-type electrical transformer. The analysis carried out employed a multi-disciplinary approach involving heat, fluid flow and electromagnetics. The thermal and fluid flow analysis was coupled with an electromagnetic model in order to examine the specific power losses within the coils and the core. The thermal boundary conditions, i.e. the local and temperature-dependent heat fluxes, were computed by considering a numerical model of the surrounding internal and external air. Moreover, separate numerical and analytical models were considered in order to obtain the anisotropic thermal conductivities for different types of coils and also for laminated cores. To validate the numerical model, experimental transformer temperature tests in the short-circuit, open-circuit, and under nominal parameters according to the current European Standards for dry-type transformers were performed. During the tests, temperatures were measured at selected points on elements of the transformer using thermocouples and thermometers, while on the external tank walls an infrared thermography was employed. The obtained numerical results showed that the prediction of the temperature distribution within the analyzed transformers and their surroundings was very accurate.     

A numerical study of three-dimensional laminar mixed convection past an open cavity

A numerical study has been carried out to analyze the effects of mixed convective flow over a three-dimensional cavity that lies at the bottom of a horizontal channel. The vertical walls of the cavity are isothermal and all other walls are adiabatic. The cavity is assumed to be cubic in geometry and the flow is laminar and incompressible. A direct numerical simulation is undertaken to investigate the flow structure, the heat transfer characteristics and the complex interaction between the induced stream flow at ambient temperature and the buoyancy-induced flow from the heated wall over a wide range of the Grashof number (10³–10⁶) and two Reynolds numbers Re = 100 and 1000. The computed thermal and flow fields are displayed and discussed in terms of the velocity fields, streamlines, the temperature distribution and the averaged Nusselt number at the heated and cooled walls. It is found that the flow becomes stable at moderate Grashof number and exhibit a three-dimensional structure, while for both high Reynolds and Grashof numbers the mixed convection effects come into play, push the recirculating zone further upstream and the flow becomes unsteady with Kelvin–Helmholtz instabilities at the shear layer.     

WALL HEAT CONDUCTION EFFECT ON NATURAL CONVECTION IN AN ENCLOSURE FILLED WITH A NON-DARCIAN POROUS MEDIUM

ABSTRACT

A numerical analysis has been made of the conjugate natural convection in a rectangular enclosure filled with a fluid-saturated porous medium and surrounded with four solid walls. The conductance of the walls is assumed to be much greater than that of the cavity filled with a porous medium. The main objective was to investigate the influences of the ratio of thermal conductivity of the wall to that of the fluid-porous matrix composite, the Darcy-modified Rayleigh number, the Prandtl number, and the aspect ratio. The streamlines and isotherms are presented; also, the local and average Nusselt numbers are presented along the interface between walls and cavity. A non-Darcian model was employed and the numerical method was SIMPLE-C. The numerical results indicate that the wall heat conduction effects decrease the heat transfer rate. When the wall heat conduction is considered, the greater the conductance of the solid walls surrounding the cavity, the greater is the rate of heat transfer.