Multidimensional radiative transfer in absorbing,emitting, and linearly anisotropic scattering cylindrical medium with space-dependent properties

The integral form of three-dimensional radiative transfer equation for an absorbing, emitting, and linear-anisotropic scattering medium with space-dependent properties is formulated. A product-integration method is subsequently applied to develop a numerical scheme for solving the corresponding integral transfer equations in a two-dimensional, axisymmetric and nonhomogeneous medium subjected to externally incident radiation or bounded by emitting and diffusely-reflecting walls. The numerical solutions for cases of constant, continuous, and stepwise variations of scattering albedo are presented to illustrate its accuracy and flexibility, and validated by comparing with results available in the literature.     

A general integration method for radiative transfer in 3-D non-homogeneous cylindrical media with anisotropic scattering

A general set of integral equations is presented to solve 3-D radiative heat transfer problems in emitting, absorbing and linear anisotropic scattering finite hollow or solid cylinders with non-homogeneous media. By tracing a ray to compute the intensity,it is much easier to handle the spatial change properties including extinction coefficient. Both the continuous change property and step-change property are dealt with without difficulties. The solid angle integration in getting the incident radiation and heat fluxes is represented by the bounding surface integration. In order to avoid the singularity problem near the bounding surface, the surface integrations are transformed to new modified integral equations by mathematical methods. By doing so, we get more flexible general integral equations applicable to all cases (3-D solid cylinders, 3-D hollow cylinders, finite cylinders or infinite cylinders). This scheme has been verified by comparing the results with published data in the literature. It is believed that this method will be useful in combined radiation and convection heat transfer problems.     

Tau method approximation for radiative transfer problems in a slab medium

An approximate method for solving the radiative transfer equation in a slab medium with an isotropic scattering is proposed. The method is based upon constructing the double Legendre series to approximate the required solution using Legendre tau method. The differential and integral expressions which arise in the radiative transfer equation are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. Numerical examples are included to demonstrate the validity and applicability of the method and a comparison is made with existing results.     

Finite element method for the two-dimensional atmospheric radiative transfer

The finite element method is applied to the solution of the two-dimensional atmospheric radiative transfer. The analysis is mainly focussed on the derivation of the cell or element equation. The Galerkin method and several hybrid methods using the integral and finite difference form of the radiative transfer equation are employed to obtain the cell equation. The assembled system of equations relating the radiances at the lower and upper boundary of the domain is solved by a direct method.     

A comparison of differential and integral equations of radiative transfer

The equations of radiative transfer and of statistical equilibrium of a two-level atom are solved by means of differential and integral equations for a one-dimensional medium. The numerical solutions are compared to the analytic solution. It is found that the integral equation for piecewise quadratic source functions gives more accurate results than does the differential equation.     

A method of cauchy integral equation for non-coherent transfer in half-space

A technique is presented which allows easy construction of solutions for various half-space problems arising in non-coherent radiative transfer with complete redistribution. By use of an inverse Laplace transform method, Wiener-Hopf integral equations are reduced to Cauchy-type singular integral equations. The factorization technique used by Case and Zweifel for coherent scattering can then be carried over to non-coherent transfer. The method is applied to the inhomogeneous integral equation for the source function of a two-level atom, previously solved by Ivanov. It is also applied to the conservative, homogeneous case and to singular Wiener-Hopf equations arising from asymptotic expansions in the limit of vanishing probability of collisional destruction ?. Consequences for the scaling laws in a finite slab are examined in a companion paper.     

High-Precision Direct Method for the Radiative Transfer Problems

It is the main aim of this paper to investigate the numerical methods of the radiative transfer equation.Using the five-point formula to approximate the differential part and the Simpson formula to substitute for integral part respectively, a new high-precision numerical scheme, which has 4-order local truncation error, is obtained. Subsequently,a numerical example for radiative transfer equation is carried out, and the calculation results show that the new numerical scheme is more accurate.     

Spatial differencing of the discrete ordinate equations in optically thick media

We present a modification to the linear characteristic method used to finite difference the discrete ordinate (S-N) equations. This modification is designed to produce the correct diffusion limit in the radiative transfer context in the limit of optically thick mesh cells. Numerical tests suggest that this finite difference method is a practical calculational scheme for radiative transfer problems.     

辐射输运菱形差分SN方程的扩散综合加速方法

讨论辐射输运菱形差分S_N方程的源迭代加速方法,在一维情形下给出与输运算子离散相容的线性多频灰体加速计算格式.数值算例表明该加速算法是健壮有效的.     

热辐射输运问题的隐式蒙特卡罗方法求解

热辐射与物质相互作用及辐射光子在物质中的传输是惯性约束聚变研究中的重要课题. 介绍了基于隐式蒙特卡罗方法的辐射输运方程,在该方程的积分-微分形式基础上,推导了利于蒙特卡罗方法模拟的等价的积分输运方程;基于积分方程设计数值模拟流程,编写三维蒙特卡罗数值模拟程序;针对热辐射输运典型问题及benchmark问题开展了数值实验,计算结果验证了方法的适应性及程序的正确性. 关键词： 热辐射 惯性约束聚变 输运方程 隐式蒙特卡罗     

An Hermitian method for the solution of radiative transfer problems

A new method for the solution of the radiative transfer equation is developed. The resulting system, like the ordinary difference equations is tridiagonal in form and easy to use. Numerical examples are given which show the superiority of the new technique to previous differential and integral equation methods.     

Coupled conductive-radiative heat transfer problem for two-layer slab

The coupled conductive radiative transfer problem in two homogeneous layers slab of anisotropic scattering with specularly reflecting boundaries has been considered. A Galerkin-iterative technique is used to solve the coupled conductive radiative heat equations in integral forms for the two layers. Numerical results are obtained for the temperature, the conductive, radiative and the total heat fluxes for the two homogeneous layers with isotropic and anisotropic scattering. The calculations are also carried out for homogeneous plane parallel medium with anisotropic scattering which show good agreement with the published calculations.     

Thermal analysis for transient radiative cooling of a conducting semitransparent layer of ceramic in high-temperature applications

A method is developed for obtaining transient temperature distribution in a cooling semitransparent layer of ceramic. The layer is emitting, absorbing, isotropically scattering and heat conducting with a refractive index ranging from 1 to 2. The solution involves solving simultaneously the energy equation and the integral equation for the radiative flux gradient. The energy equation is solved using an implicit finite volume scheme and the integral equation of radiative heat transfer is solved using the singularity technique and Gaussian integration. The effects of scattering are investigated. It is shown that scattering has a significant effect on the transient temperature distribution and the transient mean temperature of the layer.     

辐射与物质强耦合问题的二维输运数值模拟

分析确定性输运数值方法模拟辐射与物质强耦合问题的难点,给出在任意四边形网格上求解该类问题的简单隅角平衡算法,并应用灰体输运综合加速算法提高输运方程的源迭代收敛速度,最后给出数值例子.     

Theoretical study of combined radiative conductive and convective heat transfer in semi-transparent porous medium in a spherical enclosure

A new method for the solution of the radiative transfer equation in spherical media based on a modified discrete ordinates method is extended to study radiative, conductive and convective heat transfer in a semi-transparent scattering porous medium. The set of differential equations is solved using the fourth-order Runge-Kutta method. Various results are obtained for the case of combined radiative and conductive heat transfer, as well as for the interaction of those modes with convection. The effects of some radiative properties of the medium on the heat transfer rate are examined.     

Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere-ocean system

A vector radiative transfer model has been developed for a coupled atmosphere-ocean system. The radiative transfer scheme is based on the discrete ordinate and matrix operator methods. The reflection/transmission matrices and source vectors are obtained for each atmospheric or oceanic layer through the discrete ordinate solution. The vertically inhomogeneous system is constructed using the matrix operator method, which combines the radiative interaction between the layers. This radiative transfer scheme is flexible for a vertically inhomogeneous system including the oceanic layers as well as the ocean surface. Compared with the benchmark results, the computational error attributable to the radiative transfer scheme has been less than 0.1% in the case of eight discrete ordinate directions. Furthermore, increasing the number of discrete ordinate directions has produced computations with higher accuracy. Based on our radiative transfer scheme, simulations of sun glint radiation have been presented for wavelengths of 670 nm and 1.6 μm. Results of simulations have shown reasonable characteristics of the sun glint radiation such as the strongly peaked, but slightly smoothed radiation by the rough ocean surface and depolarization through multiple scattering by the aerosol-loaded atmosphere. The radiative transfer scheme of this paper has been implemented to the numerical model named Pstar as one of the OpenCLASTR/STAR radiative transfer code systems, which are widely applied to many radiative transfer problems, including the polarization effect.     

Half-Moment Closure for Radiative Transfer Equations

In this paper a moment method for radiative transfer equations is considered which has been developed and investigated using different approaches. Problems appearing for this moment system for boundary value problems using Maxwell-type boundary conditions are described. A new method based on the consideration of positive and negative half fluxes is developed and shown to overcome the above problems. Moreover, a numerical scheme and numerical results for the new moment system are presented.     

Application of Synthetic Kernel (SKN) method to participating linearly anisotropically scattering solid spherical medium

The Synthetic Kernel (SK_N) method is applied to a solid spherical absorbing, emitting and linearly anisotropically scattering homogeneous and inhomogeneous medium. The SK_N method relies on approximating the integral transfer kernels by Synthetic Kernels. The radiative integral transfer equation is then reducible to a set of coupled second-order differential equations. The SK_N method, which uses Gauss quadratures, is tested against integral equation and the discrete-ordinates S₈ solutions for various optical radius and scattering albedo variations.     

Spectral collocation method with a flexible angular discretization scheme for radiative transfer in multi-layer graded index medium

The spectral collocation method (SCM) is employed to solve the radiative transfer in multi-layer semitransparent medium with graded index. A new flexible angular discretization scheme is employed to discretize the solid angle domain freely to overcome the limit of the number of discrete radiative direction when adopting traditional S_N discrete ordinate scheme. Three radial basis function interpolation approaches, named as multi-quadric (MQ), inverse multi-quadric (IMQ) and inverse quadratic (IQ) interpolation, are employed to couple the radiative intensity at the interface between two adjacent layers and numerical experiments show that MQ interpolation has the highest accuracy and best stability. Variable radiative transfer problems in double-layer semitransparent media with different thermophysical properties are investigated and the influence of these thermophysical properties on the radiative transfer procedure in double-layer semitransparent media is also analyzed. All the simulated results show that the present SCM with the new angular discretization scheme can predict the radiative transfer in multi-layer semitransparent medium with graded index efficiently and accurately.     

Thermoelastic stress in thin large-area windows grown from melt by the Stepanov method

The mathematical model for thermoelastic stress in a thin large-area window grown from melt by the Stepanov method is considered. The distribution and magnitude of stress are analyzed depending on the height of the radiative shield and on the distance between the shield and growing window. The model includes the heat-conduction equation and the integral equations relating the thermal radiation fluxes and the temperatures of the surfaces involved in heat transfer.