Discrete ordinates interpolation method incorporated into a flow and energy solver for solution of combined heat transfer problems

Combined conduction-radiation and natural convection-radiation in two-dimensional enclosures containing gray absorbing/emitting medium are numerically investigated. The discrete ordinates interpolation method (DOIM) is used to solve the radiative transfer equation (RTE). It is incorporated into a commercial software (FLUENT^®) by using user-defined function (UDF) to be used in a finite volume-based code for fluid flow computation. Two issues are critically examined: accuracy and versatility. Cases of combined conduction-radiation are considered first and the results are compared with other benchmark solutions to validate the accuracy. Additional problems are also tested to verify the capability of handling unstructured grid system and irregular geometry. Combined natural convection-radiation problem is then examined varying the optical thickness. The radiation effect is investigated through the profiles of velocity, temperature distributions and streamlines. The results are compared with discrete ordinates (DO) solutions, Rosseland solutions and P1 solutions which are offered by FLUENT^® package. The accuracy and other numerical characteristics of DOIM are scrutinized. The DOIM shows very successful results from the viewpoint of accuracy and grid compatibility. It is proved to be a reliable future numerical tool for combined heat transfer problems in engineering applications.     

Discrete ordinates interpolation method for radiative heat transfer problems in three-dimensional enclosures filled with non-gray or scattering medium

The discrete ordinates interpolation method (DOIM) is applied to three groups of problems of radiative heat transfer in three-dimensional rectangular enclosures containing non-gray or scattering medium. The original DOIM is first extended to a gray gas model using a new geometric interpolation scheme. It is applied to participating media for different scattering phase functions and optical thicknesses. For the non-gray gas model, the DOIM coupled with the narrow band-based weighted-sum-of-gray-gases (WSGG) model is developed. A few test problems with real gases such as pure H₂O and a mixture of CO₂, H₂O and N₂ are taken. The wall heat flux is calculated and compared with the exact solutions or reference values. All results of test problems are found to be reliable in this study. The DOIM closely reproduces the Monte Carlo reference solutions for different scattering phase functions and optical thicknesses. The non-gray gas results are compared with reference calculations based on the statistical narrow band model and they also show good agreements. The DOIM shows a remarkable merit in the computation time and the grid compatibility, to prove its usefulness for engineering applications.     

A least square finite element formulation of the collimated irradiation in frequency domain for optical tomography applications

This paper presents an extension of the least square finite element formulation associated to the discrete ordinates method to solve collimated irradiation problems in frequency domain. The features of the method are shown with a separation of the intensity into its collimated and scattered parts for a better handling of discontinuities due to the boundary conditions of Dirichlet type used in optical tomography applications. Numerical tests are used to gauge the accuracy of the model in both isotropic and anisotropic scattering media, with and without frequency modulation. The results show that the method is accurate compared to some reference solutions.     

Least-squares collocation meshless approach for radiative heat transfer in absorbing and scattering media

A least-squares collocation meshless method is employed for solving the radiative heat transfer in absorbing, emitting and scattering media. The least-squares collocation meshless method for radiative transfer is based on the discrete ordinates equation. A moving least-squares approximation is applied to construct the trial functions. Except for the collocation points which are used to construct the trial functions, a number of auxiliary points are also adopted to form the total residuals of the problem. The least-squares technique is used to obtain the solution of the problem by minimizing the summation of residuals of all collocation and auxiliary points. Three numerical examples are studied to illustrate the performance of this new solution method. The numerical results are compared with the other benchmark approximate solutions. By comparison, the results show that the least-squares collocation meshless method is efficient, accurate and stable, and can be used for solving the radiative heat transfer in absorbing, emitting and scattering media.     

Least-squares finite element formulations for one-dimensional radiative transfer

We present least-squares-based finite element formulations for the numerical solution of the radiative transfer equation in its first-order primitive variable form. The use of least-squares principles leads to a variational unconstrained minimization problem in a setting of residual minimization. In addition, the resulting linear algebraic problem will always have a symmetric positive definite coefficient matrix, allowing the use of robust and fast iterative methods for its solution. We consider space-angle coupled and decoupled formulations. In the coupled formulation, the space-angle dependency is represented by two-dimensional finite element expansions and the least-squares functional minimized in the continuous space-angle domain. In the decoupled formulation the angular domain is represented by discrete ordinates, the spatial dependence represented by one-dimensional finite element expansions, and the least-squares functional minimized continuously in space domain and at discrete locations in the angle domain. Numerical examples are presented to demonstrate the merits of the formulations in slab geometry, for absorbing, emitting, anisotropically scattering mediums, allowing for spatially varying absorption and scattering coefficients. For smooth solutions in space-angle domain, exponentially fast decay of error measures is demonstrated as the p-level of the finite element expansions is increased. The formulations represent attractive alternatives to weak form Galerkin finite element formulations, typically applied to the more complicated second-order even- and odd-parity forms of the radiative transfer equation.     

Nonaxisymmetric radiative transfer in inhomogeneous cylindrical media with anisotropic scattering

In this paper, the control volume finite element method (CVFEM) is applied for the first time to solve nonaxisymmetric radiative transfer in inhomogeneous, emitting, absorbing and anisotropic scattering cylindrical media. Mathematical formulations as well as numerical implementation are given and the final discretized equations are based on similar meshes used for convective and conductive heat transfer in computational fluid dynamic analysis. In order to test the efficiency of the developed method, four nonaxisymmetric problems have been examined. Also, the grid dependence and the false scattering of the CVFEM are investigated and compared with the finite volume method and the discrete ordinates interpolation method.     

Radiative transfer in finite cylindrical media using transformed integral equations

A modified direct integration method is presented to solve three-dimensional radiative transfer in emitting, absorbing and linear-anisotropic scattering finite cylindrical media. This scheme effectively avoids an integral singularity in the coupled Fredholm type integral equations of radiative transfer. The scheme leads to faster and more accurate results, which are needed in combined mode and non-gray problems. The calculated incident radiation and heat fluxes agree well with published results by discrete ordinates method. Using the transformed integral equations, the effects of boundary emission and reflection can also be easily handled.     

Modified discrete ordinates solution of radiative transfer in two-dimensional rectangular enclosures

A modified discrete ordinates solution is developed for radiative transfer in a two-dimensional rectangular enclosure which contains an absorbing, emitting and isotropically scattering medium. Uniform and non-uniform diffuse loadings on the top boundary are considered. The intensity is broken into direct and diffuse components. The direct component is determined analytically, and the diffuse transport equation is solved numerically by conventional discrete ordinates procedure. Results are presented for various aspect ratios, media extinction properties, and loading parameters. The standard discrete ordinates solution exhibits anomalies in the flux leaving the bottom which are attributed to ‘ray-effects’. Large errors are observed when the aspect ratio is small. Numerical results obtained using the modified discrete ordinates scheme compare well with benchmark solutions and show no anomalies in the bottom flux.     

Étude tridimensionnelle du transfert radiatif dans un milieu semi-transparent diffusant anisotrope par la méthode des transferts discrets modifiée

The development of the modified discrete transfer method (MDT) in a three-dimensional rectangular configuration allowed us to simulate the thermal behaviour of a semi-transparent, grey, absorbing emitting and anisotropically scattering medium at the radiative equilibrium. An internal source distributes heat uniformly in the medium while the walls of the enclosure that surround it, opaque, grey, diffuse for emission and reflection, are submitted to prescribed temperatures. A linear variation law of the temperature, as well as the scattered radiation intensity, within a grid cell associated with the direction set of the discrete ordinates method has been adopted. A grid close enough to each inner wall was necessary for a better estimation of the incident flux near the singularities of the considered system. These global improvements led to a new version of the stable MDT method, as accurate as the zonal method and as flexible as the discrete ordinates one.     

Prediction of radiative heat transfer in 3D complex geometries using the unstructured control volume finite element method

In this paper, a 3D algorithm for the treatment of radiative heat transfer in emitting, absorbing, and scattering media is developed. The numerical approach is based on the utilization of the unstructured control volume finite element method (CVFEM) which, to the knowledge of the authors, is applied for the first time to simulate radiative heat transfer in participated media confined in 3D complex geometries. This simulation makes simultaneously the use of the merits of both the finite element method and the control volume method. Unstructured 3D triangular element grids are employed in the spatial discretization and azimuthal discretization strategy is employed in the angular discretization. The general discretization equation is presented and solved by the conditioned conjugate gradient squared method (CCGS). In order to test the efficiency of the developed method, several 3D complex geometries including a hexahedral enclosure, a 3D equilateral triangular enclosure, a 3D L-shaped enclosure and 3D elliptical enclosure are examined. The results are compared with the exact solutions or published references and the accuracy obtained in each case is shown to be highly satisfactory. Moreover, this approach required a less CPU time and iterations compared with those of even parity formulation of the discrete ordinates method.     

Analysis of collimated radiation in participating media using the discrete transfer method

A general formulation of the discrete transfer method is provided to analyze radiative heat transfer problems in a participating medium subjected to collimated radiation. The formulation is validated by considering 1-D planar absorbing, emitting and anisotropically scattering gray medium in radiative equilibrium. Anisotropy of the medium is approximated by linear anisotropic phase function. For the purpose of comparison, the problem is also solved analytically. Results are obtained for different angles of incidence of the collimated radiation. At a given angle of incidence, results are obtained for forward, isotropic and backward scattering situations. Heat flux results are compared over a wide range of values of the extinction coefficient. Emissive power distributions in the medium are also obtained for some cases. The discrete transfer method results are found to compare very well with the analytic results.     

一维瞬态辐射传输的有限元法模拟

随着超短脉冲激光的快速发展,吸收散射性介质内的瞬态辐射传输引起了人们的广泛关注.本文基于离散坐标法和最小二乘有限元法(LSFEM),提出了模拟多维吸收散射性介质内瞬态辐射传输的数值模型.该模型有效地克服了在标准Galerkin有限元法(GFEM)中发生的伪振荡现象,在时间步长较大的情况下仍然可以得到光滑无振荡的解.而且,最小二乘法产生的求解系数矩阵是对称正定的,与GFEM中的系数矩阵相比,仅需要存储一半的非零系数,可以应用许多高效的迭代求解方法进行求解.为了检验模型,本文研究了一维吸收散射性介质内瞬态辐射传输问题,其结果与蒙特卡洛法(MCM)和积分模型法(IE)的结果进行了比较,结果证实:本文的方法可以精确、高效地模拟参与性介质内的瞬态辐射传输.     

P1 approximation,MDA and IDA for the simulation of radiative transfer in a 3D geometry for an absorbing scattering medium

A numerical study has been conducted using the differential approximation and two modified versions of this method, known as MDA and IDA. The methods have been applied on a 3D geometry containing a participating, absorbing, emitting and anisotropically scattering medium. Comparisons have been carried out with some cases of the literature. It has been observed that the IDA can significantly improve the solution, where the classical P1 approximation is not able to reproduce accurately the actual flux or flux divergence in some of studied configurations. The MDA does not really improve the solution in the purely absorbing cases and when radiation is mainly coming from the medium. In other situations (scattering media, relevant influence of the boundaries), the MDA is more accurate than the P1 but the IDA still performs better, with uncertainties around a few percents for the literature cases studied in the present work.     

Elliptic PDE formulation and boundary conditions of the spherical harmonics method of arbitrary order for general three-dimensional geometries

The inherent complexity of the radiative transfer equation makes the exact treatment of radiative heat transfer impossible even for idealized situations and simple boundary conditions. Therefore, a wide variety of efficient solution methods have been developed for the RTE. Among these solution methods the spherical harmonics method, the moment method, and the discrete ordinates method provide means to obtain higher-order approximate solutions to the equation of radiative transfer. Although the assembly of the governing equations for the spherical harmonics method requires tedious algebra, their final form promises great accuracy for any given order, since it is a spectral method (rather than finite difference/finite volume in the case of discrete ordinates). In this study, a new methodology outlined in a previous paper on the spherical harmonics method (P_N) is further developed. The new methodology employs successive elimination of spherical harmonic tensors, thus reducing the number of first-order partial differential equations needed to be solved simultaneously by previous P_N approximations (=(N+1)²). The result is a relatively small set (=N(N+1)/2) of second-order, elliptic partial differential equations, which can be solved with standard PDE solution packages. General boundary conditions and supplementary conditions using rotation of spherical harmonics in terms of local coordinates are formulated for the general P_N approximation for arbitrary three-dimensional geometries. Accuracy of the P_N approximation can be further improved by applying the “modified differential approximation” approach first developed for the P₁-approximation. Numerical computations are carried out with the P₃ approximation for several new two-dimensional problems with emitting, absorbing, and scattering media. Results are compared to Monte Carlo solutions and discrete ordinates simulations and a discussion of ray effects and false scattering is provided.     

Radiative transfer in absorbing, emitting and linearly anisotropic-scattering inhomogeneous cylindrical medium

Radiative integral transfer equations for one-dimensional solid cylinder with absorbing, emitting and linearly anisotropic-scattering inhomogeneous medium were derived by Abulwafa et al. (JQSRT 62 (1999) 755). The anisotropic terms in the integral equations and their results for anisotropic benchmark problems (JQSRT 66 (2000) 487) are inaccurate. In this study, the integral equations for absorbing, emitting and linearly anisotropic-scattering medium are rederived, and the integral equations for one-dimensional solid cylindrical medium are solved. The results are compared with selected cases using the discrete ordinates S₁₆ and the exact solutions available in the literature.     

Parallel Jacobian-free Newton Krylov solution of the discrete ordinates method with flux limiters for 3D radiative transfer

The present study introduces a parallel Jacobian-free Newton Krylov (JFNK) general minimal residual (GMRES) solution for the discretized radiative transfer equation (RTE) in 3D, absorbing, emitting and scattering media. For the angular and spatial discretization of the RTE, the discrete ordinates method (DOM) and the finite volume method (FVM) including flux limiters are employed, respectively. Instead of forming and storing a large Jacobian matrix, JFNK methods allow for large memory savings as the required Jacobian-vector products are rather approximated by semiexact and numerical formulations, for which convergence and computational times are presented. Parallelization of the GMRES solution is introduced in a combined memory-shared/memory-distributed formulation that takes advantage of the fact that only large vector arrays remain in the JFNK process. Results are presented for 3D test cases including a simple homogeneous, isotropic medium and a more complex non-homogeneous, non-isothermal, absorbing–emitting and anisotropic scattering medium with collimated intensities. Additionally, convergence and stability of Gram–Schmidt and Householder orthogonalizations for the Arnoldi process in the parallel GMRES algorithms are discussed and analyzed. Overall, the introduction of JFNK methods results in a parallel, yet scalable to the tested 2048 processors, and memory affordable solution to 3D radiative transfer problems without compromising the accuracy and convergence of a Newton-like solution.     

求解辐射传递方程的离散坐标法

阐述了含吸收散射性介质三维空腔内辐射传递方程的离散坐标解法。讨论了入射散射项积分格式的选取,以及假散射和射线效应对解精度的影响。对三维矩形炉膛内辐射传递过程进行了数值模拟,并与区域法和离散传递法进行比较。比较结果表明离散坐标法具有较好的精度,是目前燃烧室内辐射传热过程数值模拟的一种较好的方法。     

吸收散射性三维矩形介质内辐射源项的反问题

提出了一种由边界出射辐射强度反演吸收散射性三维矩形介质内辐射源项分布的方法。该方法是在辐射传递方程离散坐标近似的基础上,用求目标函数极小值的共轭梯度法进行反演计算。通过对介质辐射特性、光学厚度等参数对反演精度影响的分析,结果表明,即使存在测量误差,本文所提出的方法可较精确地反演辐射源项。     

Simulation of the infrared radiation characteristics of high-temperature exhaust plume including particles using the backward Monte Carlo method

The analysis of the infrared radiation characteristics of high-temperature free-stream flow including particles is very significant for the field of target detection, combustion diagnosis and temperature measurement of flame. In this paper, the infrared radiation characteristics of high-temperature free-stream flow are calculated and analyzed using the backward Monte Carlo method, considering the effect of the directional radiation heat flux due to the particle scattering and the different boundary conditions. The calculation results of emitting, absorbing and anisotropically scattering media are compared with the forward Monte Carlo and finite-volume methods results, which shows the superiority on computational efficiency with the backward Monte Carlo method.     

矩形介质内辐射换热的有限元法

利用有限单元法离散求解辐射传递方程和能量控制方程.分别计算了边界为黑体和灰体条件下矩形吸收、发射、各向同性散射介质内的平均入射强度和温度分布,并同蒙特卡罗法(M-C法)计算结果进行了比较.