Analytical solutions of the simplified spherical harmonics equations

We derived analytical solutions of the simplified spherical harmonics equations, an approximation of the radiative transfer equation, for infinitely extended scattering media. The derived equations are simple (sum of exponential functions) and quickly evaluated. We compared the solutions with Monte Carlo simulations in the steady-state and time domains and found much better agreement compared to solutions of the diffusion equation, especially for large absorption coefficients, short time values, and small distances from the source.     

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.     

球谐函数法求解辐射传输方程的假散射和射线效应

假散射和射线效应是辐射传输方程近似解法中出现的特有误差.从辐射传输方程的近似求解过程出发,在定性分析的基础上构造物理模型,通过数值模拟研究球谐函数法(P₁和P₃近似法)的假散射和射线效应.构造激光平行入射和倾斜入射二维半透明介质的物理模型,通过内部温度场的分布特征研究假散射.构造顶部侧面保持高温而其余侧面保持低温的二维半透明介质方案,通过对比底面边界净热流密度分析射线效应.计算结果表明球谐函数法中同时存在假散射和射线效应,P₃近似比P₁近似减小了射线效应.同时,球谐函数法的射线效应随光学厚度的增加而减小.     

Radiative transfer in three-dimensional rectangular enclosures containing inhomogeneous,anisotropically scattering media

Radiative transfer in a three-dimensional rectangular enclosure containing radiatively participating gases and particles is studied using the first- and third-order spherical harmonics approximations. Inhomogeneities in the radiative properties of the medium, as well as in the radiation characteristics of the boundaries, are allowed for. The scattering phase function is represented by the delta-Eddington approximation, and it is assumed to be a function of the location in order to account for density variation of the particles in the medium. Numerical solutions of the model equations are obtained using a finite difference scheme. For the purpose of validating the P₃-approximation, the results are compared with those based on Hottel's zonal method.     

The direct solution of the radiative transfer equation for optically thick atmospheres

A simple improvement of a previous direct method of solution of the spherical harmonics approximation to the radiative transfer equation results in higher computational efficiencies by factors of 2–4; these higher efficiencies are particularly important to shorten the lengthy computations required in optically thick nonhomogeneous atmospheres.     

An efficient diffusion approximation for 3D radiative transfer parameterization: application to cloudy atmospheres

The three-dimensional (3D) diffusion radiative transfer equation, which utilizes a four-term spherical harmonics expansion for the scattering phase function and intensity, has been efficiently solved by using the full multigrid numerical method. This approach can simulate the transfer of solar and thermal infrared radiation in inhomogeneous cloudy conditions with different boundary conditions and sharp boundary discontinuity. The correlated k-distribution method is used in this model for incorporation of the gaseous absorption in multiple-scattering atmospheres for the calculation of broadband fluxes and heating rates in the solar and infrared spectra. Comparison of the results computed from this approach with those computed from plane-parallel and 3D Monte Carlo models shows excellent agreement. This 3D radiative transfer approach is well suited for radiation parameterization involving 3D and inhomogeneous clouds in climate models.     

Numerical study of heat transfer in an optically thick semi-transparent spherical porous medium

In this paper, heat transfer by simultaneous convection, conduction and radiation in a semi-transparent spherical porous medium is investigated. The ROSSELAND approximation is adopted to take account of radiation in the heat transfer rate. The routine used here to solve the set of differential equations is taken from the IMSL MATH/LIBRARY. Various results are obtained for the dimensionless temperature profiles in the solid and fluid phases, the radiative, conductive, convective and total heat fluxes. The effects of some radiative properties of the medium on the heat transfer rate are examined.     

多维梯度折射率介质内辐射传递的扩散近似

推导多维梯度折射率介质内稳态辐射传递的扩散近似方程.使用有限元法对扩散近似进行离散和求解,利用两个二维半透明介质的稳态辐射传递问题验证该扩散近似的精度及适用性.算例考虑介质为均匀折射率及梯度折射率两种情况.利用扩散近似分别求解辐射平衡时的边界热流、介质内温度场分布,并与辐射传递方程的求解结果进行对比分析.结果表明:介质折射率变化、散射特性、光学厚度及散射反照率均直接影响扩散近似的精度;在光学厚及强散射条件下,该扩散近似可以作为一种快速算法应用于梯度折射率介质稳态辐射传递的求解.     

A simplified implementation of the discrete-ordinates method for a class of problems in radiative transfer with polarization

A simplified implementation of the analytical discrete ordinates (ADO) method in radiative transfer with polarization is presented in this work. The class of problems that can be solved with the simplified ADO approach consists of problems defined in plane-parallel geometry and driven by external illumination in the form of obliquely incident parallel rays. Numerical results of benchmark quality are tabulated for the albedo problem with polarization and Lambert reflection. The new results improve on a tabulation made available in a previous work by the authors that was based on the (less accurate) spherical harmonics method.     

Radiative transfer in a nongray spherical layer: Simplified rectangular model

The problem of radiative transfer in a nongray, absorbing-emitting spherical layer is investigated. The absorption coefficient is assumed to be only a function of frequency, i.e. k_v = α(v)k, and the function α(v) is allowed only two values, zero or unity (simplified rectangular model). The nongray radiative transfer problem is reduced to a gray solution without the use of any approximation (such asthe Plank or Rosseland means) for an isothermal layer and for radiative equilibr     

Towards linearization of atmospheric radiative transfer in spherical geometry

We present a general approach for the linearization of radiative transfer in a spherical planetary atmosphere. The approach is based on the forward-adjoint perturbation theory. In the first part we develop the theoretical background for a linearization of radiative transfer in spherical geometry. Using an operator formulation of radiative transfer allows one to derive the linearization principles in a universally valid notation. The application of the derived principles is demonstrated for a radiative transfer problem in simplified spherical geometry in the second part of this paper. Here, we calculate the derivatives of the radiance at the top of the atmosphere with respect to the absorption properties of a trace gas species in the case of a nadir-viewing satellite instrument.     

Time-dependent equation for the intensity in the diffusion limit using a higher-order angular expansion

The first two terms in the spherical-harmonic expansion (the P(1) approximation) of the radiative transfer equation yield the diffusion equation. This approximation applies to multiple scattering and results in a solution for the energy density, the gradient of which is proportional to the light intensity. In this work a higher-order spherical-harmonic expansion of the radiative transfer equation is developed. This equation applies to the radiant intensity rather than the energy density. The equation can be decomposed into two terms: a propagator term obtained from the determinant of the coupled equations describing the individual components of the intensity, and a mixing matrix that describes the cross coupling between different orders of the expansion. Using the Fourier transform, an approximation based on expanding at small wave vectors k leads to an equation similar to the diffusion equation. The equation is expected to predict the intensity for multiple scattering at earlier times and shorter distances than the diffusion equation can. The notion of an equivalent wave field is introduced.     

Analysis of combined conduction and radiation heat transfer in presence of participating medium by the development of hybrid method

The current study addresses the mathematical modeling aspects of coupled conductive and radiative heat transfer in the presence of absorbing, emitting and isotropic scattering gray medium within two-dimensional square enclosure. A blended method where the concepts of modified differential approximation employed by combining discrete ordinate method and spherical harmonics method, has been developed for modeling the radiative transport equation. The gray participating medium is bounded by isothermal walls of two-dimensional enclosure which are considered to be opaque, diffuse and gray. The effect of various influencing parameters i.e., radiation-conduction parameter, surface emissivity, single scattering albedo and optical thickness has been illustrated. The adaptability of the present method has also been addressed.     

High fidelity radiative heat transfer models for high-pressure laminar hydrogen–air diffusion flames

Radiative heat transfer is studied numerically for high-pressure laminar H₂–air jet diffusion flames, with pressure ranging from 1 to 30 bar. Water vapour is assumed to be the only radiatively participating species. Two different radiation models are employed, the first being the full spectrum k-distribution model together with conventional Radiative Transfer Equation (RTE) solvers. Narrowband k-distributions of water vapour are calculated and databased from the HITEMP 2010 database, which claims to retain accuracy up to 4000 K. The full-spectrum k-distributions are assembled from their narrowband counterparts to yield high accuracy with little additional computational cost. The RTE is solved using various spherical harmonics methods, such as P₁, simplified P₃ (SP₃) and simplified P₅ (SP₅). The resulting partial differential equations as well as other transport equations in the laminar diffusion flames are discretized with the finite-volume method in OpenFOAM^®. The second radiation model is a Photon Monte Carlo (PMC) method coupled with a line-by-line spectral model. The PMC absorption coefficient database is derived from the same spectroscopy database as the k-distribution methods. A time blending scheme is used to reduce PMC calculations at each time step. Differential diffusion effects, which are important in laminar hydrogen flames, are also included in the scalar transport equations. It was found that the optically thin approximation overpredicts radiative heat loss at elevated pressures. Peak flame temperature is less affected by radiation because of faster chemical reactions at high pressures. Significant cooling effects are observed at downstream locations. As pressure increases, the performance of RTE models starts to deviate due to increased optical thickness. SP_N models perform only marginally better than P₁ because P₁ is adequate except at very high pressure.     

A direct solution of the radiative transfer equation: Application to Rayleigh and Mie atmospheres

A direct method is given for the solution of the spherical harmonics approximation to the equation of radiative transfer in plane-parallel atmospheres. Although the method is formulated theoretically for non-homogeneous atmospheres with an arbitrary phase function, at present it has only been implemented for homogeneous atmospheres. Test computations performed for Rayleigh and Mie scattering phase functions show that the direct method is unconditionally stable and solves efficiently problems both for optically thin and very thick atmospheres. Timing comparisons with the method of Chandrasekhar for Rayleigh atmospheres and with an integral-equation iterative method for Mie atmospheres are quite favorable to the proposed method.     

Tensorielle Kugelflächenfunktionsentwicklung des Stoßintegrals für ein schwach ionisiertes Plasma II

A spherical harmonic expansion of the deduced approximation [1] of the collision integral is obtained by expanding the electron distribution in spherical harmonics of the electron velocity. General spherical harmonic tensors are used. From the expansion a hierarchy of partial differential equations for the expansion tensors follows. This hierarchy is equivalent to the Boltzmann-equation. In this paper the calculations are performed up to linear terms of the approximation.     

Radiative transfer in a spherical, emitting, absorbing and anisotropically scattering medium

The atmospheres of planets (including Earth) and the outer layers of stars have often been treated in radiative transfer as plane-parallel media, instead of spherical shells, which can lead to inaccuracy, e.g. limb darkening. We give an exact solution of the radiative transfer specific intensity at all points and directions in a finite spherical medium having arbitrary radial spectral distribution of: source (temperature), absorption, emission and anisotropic scattering. The power and efficiency of the method stems from the spherical numerical gridding used to discretize the transfer equations prior to matrix solution: the wanted ray and the rays which scatter into it both have the same physico-geometric structure. Very good agreement is found with an isotropic astrophysical benchmark [Avrett EH, Loeser R. Methods in radiative transfer. In: Kalkofen W, editor. Cambridge: Cambridge University Press; 1984. pp. 341-79]. We introduce a specimen arbitrary forward- side-back phase scattering function for future comparisons. Our method directly and exactly addresses spherical symmetry with anisotropic scattering, and could be used to study the Earth's climate, nuclear power (neutron diffusion) and the astrophysics of stars and planets.     

辐射传输方程的三阶球谐展开(P3)近似的有限元法求解

从辐射传输方程出发,应用球谐函数方法对辐射率进行多项式展开,并利用球谐函数的正交性和递推性,在二维笛卡儿坐标下,导出了三阶展开的球谐函数微分方程组（P3近似）,改进了以往成像文献中忽略各项异性因子的P3近似,并用有限元方法对二维圆域均匀和非均匀两种情况做了数值模拟.与漫射近似模型相比较,P3近似能更准确地描述光源附近及吸收较强情况下边界的光辐射分布情况.     

Comparison of radiative transfer approximations for a highly forward scattering planar medium

We examine critically the accuracy of the two-flux, spherical harmonics and discrete ordinates methods for predicting radiative transfer in a planar, highly-forward scattering and absorbing medium. Numerical results for the radiative fluxes show that the two-flux and P₃-approximations yield accurate results compared to solutions based on the F_N-method. Indeed, these approximate methods are relatively simple and have potential for generalization to predict radiative transfer in multidimensional systems, as long as an appropriate simplification of the phase function is utilized.     

Boundary conditions for differential approximations

Low order spherical harmonic (P-N) approximations are applied to a radiative transfer Marshak wave problem. A modified Milne boundary condition is developed for the P-2 approximation, similar to one suggested earlier for the P-1 approximation. Comparison with exact Monte Carlo results suggests that this modified P-2 method may be an accurate and generally applicable differential approximation to the equation of transfer. The Monte Carlo results presented should be useful for testing other approximate formulations of radiative transfer and validating time dependent numerical solution methods for the equation of transfer.