Finite-Element Solution of Integral Equations Arising in Radiative Heat Transfer and Laminar Boundary-Layer Theory

Abstract

Finite-element solutions of a number of one-dimensional linear and nonlinear (nonsingular) integral equations arising in radiative heat transfer and boundarylayer theory are presented.     

Comparison of Quadrature Schemes in DOM for Anisotropic Scattering Radiative Transfer Analysis

The commonly implemented level-symmetric S _N quadrature set for the discrete-ordinates method suffers from a limitation in discrete direction number to avoid physically unrealistic weighting factors. This limitation can have an adverse impact for determining radiative transfer, as directional discretization results in angular false scattering errors due to distortion of the scattering phase function in addition to the ray effect. To combat this limitation, several higher-order quadrature schemes with no directional limitation have been developed. Here, four higher-order quadrature sets (Legendre-equal weight, Legendre-Chebyshev, triangle tessellation, and spherical ring approximation) are implemented for determination of radiative transfer in a 3-D cubic enclosure containing participating media. Heat fluxes obtained at low direction number are compared to the S _N quadrature and Monte Carlo predictions to gauge and compare quadrature accuracy. Investigation into the reduction/elimination of angular false scattering with increase in direction number, including heat flux accuracy with respect to Monte Carlo and computational efficiency, is presented. It is found that while the higher-order quadrature sets are able to effectively minimize angular false scattering, the number of directions required is extremely large, and thus it is more computationally efficient to implement proper phase-function normalization to obtain accurate results.     

Improved Treatment of Anisotropic Scattering in Radiation Transfer Analysis Using the Finite Volume Method

Discretization of the integral anisotropic-scattering term in the equation of radiative transfer will result in two kinds of numerical errors: alterations in scattered energy and asymmetry factor. Though quadrature flexibility with large angular directions and further solid-angle splitting in the finite volume method (FVM) allow for reduction/minimization of these errors, computational efficiency is adversely impacted. A phase-function normalization technique to get rid of these errors is simpler and is applied to the three-dimensional (3-D) FVM for the first time to improve anisotropic radiation transfer computation accuracy and efficiency. FVM results are compared to Monte Carlo and discrete-ordinates method predictions of radiative heat transfer in a cubic enclosure housing a highly anisotropic participating medium. It is found that the FVM results generated using the normalization technique conform accurately to the results of the other two methods with little impact on computational efficiency.     

蒙特卡洛法求二维矩形散射性介质内的辐射传递

引入辐射传递因子RDij的概念，由于该因子与温度的关系较小，因此可以将蒙特卡洛法模拟与温度场的迭代求解分开来进行。建立任意几何形状条件下蒙特卡洛法求解辐射传递因子的计算模型，并对二维矩形黑体及灰体壁面条件下的纯散射灰介质内辐射传递问题进行了模拟计算。其中，黑体壁面计算结果与文献[5,6]的结果吻合较好。另外，计算了二维矩形灰体壁面、灰体吸收散射性介质内的温度场及壁面热流分布，其结果可以供比较参考。     

Second-Order Radiative Transfer Equation and Its Properties of Numerical Solution Using the Finite-Element Method

The original radiative transfer equation is a first-order integrodifferential equation, which can be taken as a convection-dominated equation. The presence of the convection term may cause nonphysical oscillation of solutions. This type of instability can occur in many numerical methods, including the finite-difference method and the finite-element method, if no special stability treatment is used. To overcome this problem, a second-order radiative transfer equation is derived, which is a diffusion-type equation similar to the heat conduction equation for an anisotropic medium. The consistency of the second-order radiative transfer equation with the original radiative transfer equation is demonstrated. The perturbation characteristics of error are analyzed and compared for both the first- and second-order equations. Good numerical properties are found for the second-order radiative transfer equation. To show the properties of the numerical solution, the standard Galerkin finite-element method is employed to solve the second-order radiative transfer equation. Four test problems are taken as examples to check the numerical properties of the second-order radiative transfer equation. The results show that the standard Galerkin finite-element solution of the second-order radiative transfer equation is numerically stable, efficient, and accurate.     

各向异性散射介质的辐射传热分析

利用Ｍｏｎｔｅ－ＣａｒｌｏＺｏｎｅ相结合的数值计算方法（简称ＭＣＺ方法）分析各向同性和各向异性散射介质的辐射换热。为了便于对照,本文选取了一维平板系统,利用编程序对各向同性散射吸收介质和线性相函数各向异性纯散射介质的半球透射率和半球反射率以及线性相函数各向异性散射吸收介质的平板中辐射传热分别进行了计算,获得了较好的结论。     

Nodal Integral Method Using Quadrilateral Elements for Transport Equations: Part 2-Navier-Stokes Equations

A numerical scheme for the Navier-Stokes equations in an irregular shaped domain using the nodal integral method (NIM) is developed. A mapping similar to the convection-diffusion article (Part 1) is used for the Navier-Stokes equations using the NIM in an arbitrary-shaped domain. Use of a recently developed pressure correction-based iterative scheme for the NIM ensures that the final solution satisfies the continuity condition. Lid-driven and buoyancy-driven flows in a skewed cavity are used as two test cases for comparison and verification. From the detailed comparative study of the results obtained by the NIM and very-fine-grid results (using finite-volume or similar approaches) of previous studies, it is established that this NIM-based scheme for the Navier-Stokes equation retains its capability to produce accurate results in comparatively much coarser grids, even with nonorthogonal grids     

Evaluation of the FTn Finite Volume Method for Transient Radiative Transfer in Anisotropically Scattering Medium

In this paper, we formulated, applied, and tested the FTn Finite Volume Method (FTn FVM) for transient radiative transfer in three-dimensional absorbing, emitting, and anisotropically scattering medium. Both the STEP and the Curved-Line Advection Method (CLAM) are introduced for spatial discretization of the transient radiative transfer equation. The results show that FTn FVM reduces largely the ray effects and it is more accurate than the standard FVM. Also, using both STEP and CLAM schemes, FTn FVM has smaller convergence time than the standard FVM for all cases. On the contrary, the STEP scheme is faster than the CLAM scheme but it has less accuracy. Then, the effects of optical thickness, scattering albedo, and anisotropy factor on incident radiation and radiative flux are presented and discussed.     

The Improved Direct Collocation Meshless Approach for Radiative Heat Transfer in Participating Media

The moving least-squares (MLS) direct collocation meshless method (DCM) is an effective numerical scheme for solving the radiative heat transfer in participating media. In this method the trial function is constructed by a MLS approximation and the radiative transfer equation (RTE) is discretized directly at nodes by collocation. The main drawback of this method is that, like most of the other numerical methods, the solution to the RTE by the DCM also suffers much from nonphysical oscillations in some cases caused by the convection-dominated property of the RTE. To overcome the numerical oscillations, special stabilization techniques are usually adopted, which increases the complexity and computation time of problem. In the present work a new scheme based on the outflow-boundary intensity interpolation correction is proposed that can easily ensure a large reduction in numerical oscillations of results without any complex stabilization technique. Adaptive support domain technique is also adopted, and the size of the support domain of each evaluated point changes with the density of nodes with irregular distribution. Five cases are studied to illustrate the numerical performance of these improvements. The numerical results compare well with the benchmark approximate solutions, and it is shown that the improved moving least-square direct collocation meshless method (iDCM) is easily implemented, efficient, of high accuracy, and excellent stability, to solve radiative heat transfer in homogeneous participating media.     

Improved Finite Volume Method for Three-Dimensional Radiative Heat Transfer in Complex Enclosures Containing Homogenous and Inhomogeneous Participating Media

The paper presents a modified finite volume method for the solution of the radiative transport equation, which implements the FTn angular discretization along with the bounded high-resolution curved line advection method to alleviate ray effect and false scattering, respectively, and consequently improve the accuracy of the final results. Using the blocked-off-region procedure, the present formulation is capable of treating blockage effects caused by inner/outer obstructing bodies. The developed methodology based on the combination of the above methods is evaluated against five three-dimensional test cases considering either homogenous or inhomogeneous participating media. For all cases, the predictions reveal the mitigation of false scattering and ray effects consequently the improvement of accuracy, employing this model for solving radiation heat transfer in industrial applications. In industrial application, the radiative heat transfer problem is solved for a unity boiler furnace where an inhomogeneous medium is assumed. The effects of the scattering albedo, walls emissivity and walls temperature are investigated.     

Non－Darcian and Anisotropic Anisotropic Effects on Natural Convection in Horizontal Porous Media Enclosure

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Improved Iterative Convergence of the Finite-Element Method in the Solution of the Second-Order Radiative Transfer Equation by Modified Diffusion Synthetic Acceleration

A modified diffusion synthetic acceleration model is developed to improve the convergence of the second-order radiative transfer equation (SORTE). The model considers the influence of a diffusely reflecting boundary and adopts the P₁ diffusion approximation to rectify the scattering source term of the SORTE and the reflecting term of the boundary condition. It can remarkably reduce the computational time in the solution of the SORTE, especially when the participating medium with intensive diffusely reflecting walls is simulated. The accuracy of the model is verified by comparing the results with those of the Monte Carlo method and the finite-element method without any accelerative technique. The effects of the emissivity of walls, the optical thickness, and the scattering albedo on the convergence are also investigated. The results indicate that the accuracy of the proposed model is reliable and its accelerative effect is significant for optically thick and scattering-dominated media with intensive diffusely reflecting walls.     

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.     

炉内辐射换热过程的有限体积法

简要分析了含吸收散射性介质的三维空腔内辐射传递方程的有限体积法求解过程,应用该方法对四角切圆炉膛内的辐射换热过程进行模拟计算,得出了炉膛内温度分布,并将计算结果与实测值进行了比较。通过数值计算表明：有限体积法计算速度快,对不规则边界适应性强,具有很高的工程可用性。     

Finite-Element Discretized Symplectic Method for Steady-State Heat Conduction with Singularities in Composite Structures

A new-finite element discretized symplectic method for solving the steady-state heat conduction problem with singularities in composite structures is presented. The model with a singularity is divided into two regions, near and far fields, and meshed using conventional finite elements. In the near field, the temperature and heat flux densities are expanded in exact symplectic eigensolutions. After a matrix transformation, the unknowns in the near field are transformed to coefficients of the symplectic series, while those in the far field are as usual. The exact local solutions for temperature and heat flux densities are obtained simultaneously without any post-processing.     

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.     

用共轭梯度法与蒙特卡洛法求解三维系统的辐射热负荷反问题

采用蒙特卡洛法与共轭梯度法确定三维矩形炉膛中燃烧器或加热表面的温度分与热负荷分布。讨论了测量误差对计算结果的影响,结果表明：当测量值不含误差时,用共轭梯并法计算得到的加热表面的温度值和热流量值与其真实值吻合很好。实验测量误差对计算精度的影响与加热表面及受热表面之间的辐射热交换系数有关,辐射热交换系数越大,测量误差对计算结果的影响越大。     

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.     

Comparison of the Discrete-Ordinates Method and the Finite-Volume Method for Steady-State and Ultrafast Radiative Transfer Analysis in Cylindrical Coordinates

The time-dependent equation of radiative transfer is solved for an axisymmetric cylindrical medium using both the discrete-ordinates method and the finite-volume method. Steady and transient flux profiles are determined for absorbing and scattering media. Results for each solution method are compared and shown for various grid numbers, scattering albedos, and optical thicknesses. A comparison of computational time and memory usage between the methods is presented. It is found that the finite-volume method uses more memory and has a longer convergence time than the discrete-ordinates method for all cases, due to the difference in angular treatment.