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.     

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.     

Inverse Geometry Design of Radiating Enclosure Filled with Participating Media Using Meshless Method

A new inverse geometry design methodology is presented in this work for designing a two-dimensional radiating enclosure filled with participating media to meet the pre-specified radiative heat flux distribution on a designed boundary wall. Akima cubic interpolation is employed to approximate the shape of the unknown design surface and transform the continuous geometry shape design to the discrete points' position design. To avoid the tedious remeshing of the variable computational domain in the inverse geometry design processes, the direct collocation meshless method is adopted to solve the radiative transfer problem in the enclosure. The geometry shape of the design surface is optimized using the conjugate gradient method, and the zeroth order regularization method is chosen to stabilize the inverse solutions. A test example is taken to verify the new method presented in this work. The inverse design results show that pre-specified design requirement on the boundary wall can be successfully obtained using the new methodology.     

Meshless method for solving radiative transfer problems in complex two-dimensional and three-dimensional geometries

A meshless method is presented to solve the radiative transfer equation in complex 2D and 3D geometries. In order to avoid numerical oscillations, the even parity formulation of the discrete ordinates method is used. A moving least squares approximation meshless method is used to solve the second order partial differential equations. Prediction results of radiative heat transfer problems obtained by the proposed method are compared with reference in order to assess the correctness of the present method.     

Meshless local Petrov–Galerkin approach for coupled radiative and conductive heat transfer

A meshless local Petrov–Galerkin approach is employed for solving the coupled radiative and conductive heat transfer in absorbing, emitting and scattering media. The meshless local Petrov–Galerkin approach with upwind scheme for radiative transfer is based on the discrete ordinate equations. The moving least square approximation is used to construct the shape function. Three particular test cases for coupled radiative and conductive heat transfer are examined to verify this new approximate method. The dimensionless temperatures and the dimensionless heat fluxes are obtained. The results are compared with the other benchmark approximate solutions. By comparison, the results show that the meshless local Petrov–Galerkin approach has a good accuracy in solving the coupled radiative and conductive heat transfer in absorbing, emitting and scattering media.     

Chebyshev collocation spectral method for one-dimensional radiative heat transfer in graded index media

Chebyshev spectral collocation method based on discrete ordinates equation is developed to solve radiative transfer problems in a one-dimensional absorbing, emitting and scattering semitransparent slab with spatially variable refractive index. For radiative transfer equation, the angular domain is discretized by discrete ordinates method, and the spatial domain is discretized by Chebyshev collocation spectral method. Due to the exponential convergence of spectral methods, a very high accuracy can be obtained even using few nodes for present problems. Numerical results by the Chebyshev collocation spectral-discrete ordinates method (SP-DOM) are compared with those available data in references. Effects of refractive index gradient on radiative intensity are studied for space dependent scattering media. The results show that SP-DOM has a good accuracy and efficiency for solving radiative heat transfer problems in even spatially varying absorbing, emitting, scattering, and graded index media.     

Meshless analysis of the substrate temperature in plasma spraying process

The substrate temperature plays a very important role in coating formation and its quality during the thermal spraying. Heating effect of the plasma and particle flux on the substrate is explored in detail in terms of different spraying distances using the meshless local Petrov–Galerkin method (MLPG). Based on this approach, a 3D transient heat transfer model is derived rigorously, in which the moving least-squares (MLS) method is introduced to construct the shape functions. A quartic spline function is selected as the weight function of the MLS scheme and also the test function for the discretized weak form, in which the penalty technique is used to treat the essential boundary conditions. For comparison, the finite element method (FEM) is also adopted to solve the same problem. It is found that the computed temperature is in very good agreement with the empirical data and better than that obtained using FEM, which validates the meshless formulation. Both numerical and experimental results indicate that the spraying distance has a crucial influence on heating effect of the plasma jet and particle flux onto the substrate.     

Meshless method for radiation heat transfer in graded index medium

Because ray goes along a curved path determined by the Fermat principle, curved ray tracing is very difficult and complex in graded index media. To avoid the difficult and complex computation of curved ray trajectories, a meshless local Petrov–Galerkin approach based on discrete-ordinate equations is developed to solve the radiative transfer problem in multi-dimensional absorbing–emitting–scattering semitransparent graded index media. A moving least square approximation is used to construct the shape function. Two particular test problems in radiative transfer are taken as examples to verify this meshless approach. The predicted temperature distributions and the dimensionless radiative heat fluxes are determined by the proposed method and compared with the other benchmark approximate solutions. The results show that the meshless local Petrov–Galerkin approach based on discrete-ordinate equations has a good accuracy in solving the radiative transfer problems in absorbing–emitting–scattering semitransparent graded index media.     

Meshless Method for Solving Transient Radiative and Conductive Heat Transfer in Two-Dimensional Complex Geometries

A diffuse approximation meshless method (DAM) is employed to solve the transient radiative and conductive heat transfer problem in a semitransparent medium enclosed in 2-D complex geometries. The computational spatial domain is discretized by a set of nodes scattered in the domain and boundary without information on the relationship between them. The meshless method for radiative transfer equation is based on the even-parity formulation of the discrete ordinates method without any form of upwinding. Results of dimensionless temperature distribution at different dimensionless times are obtained and validated with other benchmark approximate solutions in order to illustrate the performance of the proposed method.     

Discontinuous Galerkin finite element method for radiative heat transfer in two-dimensional media with inner obstacles

A discontinuous Galerkin finite element method (DGFEM) with unstructured meshes is presented to solve the radiative transfer equation (RTE) in two-dimensional media with inner obstacles. The computation domain is discretized into a tessellation of unstructured elements and the elements are assumed to be discontinuous on the inner-element boundaries. The shape functions are constructed on each element and the continuity of the computation domain is maintained by modeling an up-winding numerical flux across the inner boundaries, which makes the DGFEM suitable and numerical stable for radiative transfer problems involved with strong non-uniformity and discontinuity induced by ray effects. The DGFEM discretization for RTE is presented and the accuracy of DGFEM is verified. Radiative transfer problems in square and irregular media with inner obstacles are investigated, the influence of medium parameters and the obstacle shielding effects are discussed.     

Numerical predictions of two-dimensional conduction,convection, and radiation heat transfer. II. Validation

This paper presents several test problems that were used to validate the formulation and implementation of a CVFEM for combined-mode heat transfer in participating media. The objective here is to demonstrate that the proposed CVFEM can be used to solve combined modes of heat transfer in media that emit, absorb, and scatter radiant energy in regularly- and irregularly-shaped geometries. The paper first assesses briefly the CVFEM for the solution of convection–diffusion problems, then attention is focused on radiative heat transfer problems. Finally, several combined mode problems are investigated. Results show that the proposed method adequately solves the governing equations (energy and RTE): as the solutions compare favorably with those obtained with other acknowledged methods or with analytical solutions, when available. The proposed CVFEM could, however, be improved to broaden its scope of application and enhance its numerical efficiency. In a near future, the method will be combined with a code already established for fluid flow calculations.     

A Consistent Interpolation Function for the Solution of Radiative Transfer on Triangular Meshes. I—Comprehensive Formulation

This article presents a first-order skewed upwinding procedure for application to discretization numerical methods in the context of radiative transfer involving gray participating media. This scheme: (1) yields fast convergence of the algorithm; (2) inherently precludes the possibility of computing negative coefficients to the discretized algebraic equations; (3) reduces false scattering (diffusion); (4) is relatively insensitive to grid orientation; and (5) produces solutions completely free from undesirable oscillations. Theses attributes render the scheme attractive, especially in the context of combined modes of heat transfer and fluid flow problems for which computational time is a major concern. The suggested scheme has been validated by application to several basic test problems discussed in a companion article.     

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.     

Solution of incompressible fluid flow problems with heat transfer by means of an efficient RBF-FD meshless approach

The localized radial basis function collocation meshless method (LRBFCMM), also known as radial basis function generated finite differences (RBF-FD) meshless method, is employed to solve time-dependent, two-dimensional (2D) incompressible fluid flow problems with heat transfer using multiquadric RBFs. A projection approach is employed to decouple the continuity and momentum equations for which a fully implicit scheme is adopted for the time integration. The node distributions are characterized by non-Cartesian node arrangements and large sizes, i.e., in the order of 10⁵ nodes, while nodal refinement is employed where large gradients are expected, i.e., near the walls. Particular attention is given to the accurate and efficient solution of unsteady flows at high Reynolds or Rayleigh numbers, in order to assess the capability of this specific meshless approach to deal with practical problems. Three benchmark test cases are considered: a lid-driven cavity, a differentially heated cavity and a flow past a circular cylinder between parallel walls. The obtained numerical results compare very favorably with literature references for each of the considered cases. It is concluded that the presented numerical approach can be employed for the efficient simulation of fluid-flow problems of engineering relevance over complex-shaped domains.     

The Combined Radiative Integral Equations and Finite-Element Method for Radiation in Anisotropic Scattering Media

A combined procedure of the radiative integral equation and finite-element method (IEFEM) is proposed for handling radiative heat transfer in linearly anisotropic scattering media. The IEFEM can eliminate the angular discretization and easily handle irregular geometries. The present work provides a solution of radiative transfer in rectangular and irregular quadrilateral enclosures containing participating media. The influences of emissivities, albedos, and anisotropy on the boundary fluxes or incident intensity have been analyzed. Compared with the results in published references, the present IEFEM has no limitation to geometry and can predict the radiative heat transfer in linearly anisotropic scattering media accurately.     

First-Order and Second-Order Meshless Formulations of the Radiative Transfer Equation: A Comparative Study

In this article we compare the first- and second-order formulations of the radiative transfer equation (RTE) by using a diffuse approximation meshless (DAM) method. These formulations include the first-order radiative transfer equation (FORTE), the second-order radiative transfer equation (SORTE), and the even parity method. All of them are solved without upwinding treatment. In order to investigate the accuracy and cost, different radiative transfer problems in complex 2-D and 3-D geometries are considered. Prediction results obtained by the proposed methods are compared with other references in order to illustrate the performance of different formulations and of the meshless method.     

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.     

Nongray inverse boundary design problems in enclosures at radiative equilibrium

In this paper, an inverse analysis is used to find an appropriate heat flux distribution over the heater surface of radiant enclosures, filled with nongray media at radiative equilibrium from the knowledge of desired (prespecified) temperature and heat flux distributions over the given design surface. Regular and irregular 2D enclosures filled with nongray combustive gas products are considered. Radiation is considered the dominant mode of heat transfer and the medium temperature is obtained from the energy equation. To evaluate the nongray behavior of the participating gases properly, the spectral‐line weighted‐sum‐of‐gray‐gases (SLW) model with updated correlations is used. The dependence of absorption coefficients and the weights of the SLW model on the temperature of the medium makes the inverse problem nonlinear and difficult to handle. Here, the inverse problem is formulated as an optimization problem and the Levenberg‐Marquardt method has been used to solve it. The finite volume method is exploited for the discretization of the energy equation and the spatial discretization of the radiative transfer equation (RTE). The discrete ordinates method (TN quadrature) is used for the angular discretization of RTE. Five test cases, including homogeneous and inhomogeneous media, are investigated to prove the ability of the present methodology for achieving the desired conditions.     

Thermal radiation in layered participating cylindrical media using an integral equation method

Radiative transfer in a layered cylindrical medium is analyzed using a newly developed S-type integral equation of transfer in terms of incident radiation and net radiative heat flux. The physical systems in hollow and solid cylindrical geometry are modeled as nonhomogeneous participating media with stepwise variable properties. Thus, the necessity to establish the equation of transfer for each layer and combine these layers through the intensity continuity on interface is avoided. Numerical predictions from a collocation method are presented to illustrate the effects of radiation properties on the radiative transfer for the systems considered. Comparison of the results with other available data in literature shows that highly accurate results are obtainable by current method with simplicity.     

A High-Order-Accurate GPU-Based Radiative Transfer Equation Solver for Combustion and Propulsion Applications

In this article we present a high-order-accurate solver for the radiative transfer equation (RTE) which uses the discontinuous Galerkin (DG) method and is designed for graphics processing units (GPUs). The compact nature of the high-order DG method enhances scalability, particularly on GPUs. High-order spatial accuracy can be used to reduce discretization errors on a given computational mesh, and can also reduce the mesh size needed to achieve a desired error tolerance. Computational efficiency is a key concern in solutions to radiative heat transfer problems, due to potentially large problem sizes created by (a) the presence of participating nongray media in a full-spectrum analysis, (b) the need to resolve a large number of angular directions and spatial extent of the domain for an accurate solution, and (c) potentially large variations in material and flow properties in the domain. We present here a simulation strategy, as well as a set of physical models, accompanied by a number of case studies, demonstrating the accuracy and superior performance in terms of computational efficiency of this approach.