Two-dimensional isotropic scattering in a semi-infinite medium

Exact results are presented for the source function, radiative flux, and intensity at the boundary of a two-dimensional, isotropically scattering, semi-infinite medium subjected to collimated or diffuse radiation. The spatial distributions of incident radiation considered are (1) cosine-varying, (2) semi-infinite step, (3) step at the origin and (4) finite strip. Two-dimensional effects are most pronounced at large albedos.     

Single and double scattering approximations for a two-dimensional cylindrical medium

Large spatial frequency expansions for the source function, radiative flux, and intensity are obtained for an isotropically scattering finite two-dimensional medium exposed to collimated radiation. With these expansions, the single and double scattering results are obtained which are valid at small optical distances away from the incident radiation. Results are presented for a circular disk, exponential distribution and a Gaussian distribution of incident radiation.     

紫外波段多分散系气溶胶散射相函数随机抽样方法研究

基于电磁散射与辐射传输中的基本理论,对紫外波段霾尺度范围内满足特定分布的多种气溶胶粒子的散射相函数进行了研究.提出了一种直接随机抽样拟合散射相函数的方法.比较了H-G相函数、改进的H-G相函数及随机抽样拟合的相函数与多分散系Mie相函数的偏离程度.数值计算了不同相函数拟合方法对应气溶胶的传输特性.计算结果表明,相函数的准确模拟计算对于蒙特卡罗方法等辐射传输问题的解决具有十分重要的意义.     

散射相函数对一维介质内辐射传递的影响规律

采用有限体积法研究了一维线性各向异性散射介质内散射相函数对辐射传递的影响规律.经与理论解、辐射元法、蒙特卡洛法计算结果比较表明,有限体积法的计算结果更可靠,且不同散射相函数的辐射换热系统中,其无因次热流之比与光学厚度之间存在某种单调变化的函数关系,利用该函数关系可以检验模型的准确度.     

Three-dimensional transient radiative transfer modeling using the finite-volume method

This article presents a finite-volume method to calculate transient radiative transfer in a three-dimensional enclosure. The fully implicit scheme is used to discretize the transient term. The step and CLAM spatial differencing schemes are used in this study. The procedure is validated using available published results. The ability of the present formulation in modeling an absorbing, emitting and isotropically scattering medium is examined using heat fluxes and incident radiation. Ray effect and false scattering are discussed.     

Non-Fourier heat conduction with radiation in an absorbing, emitting, and isotropically scattering medium

This article numerically analyses the combined conductive and radiative heat transfer in an absorbing, emitting, and isotropically scattering medium. The non-Fourier heat conduction equation, which includes the time lag between heat flux and the temperature gradient, is used to model the conductive heat transfer in the medium. It predicts that a temperature disturbance will propagate as a wave at finite speed. The radiative heat transfer is solved using the P₃ approximation method. In addition, the MacCormack's explicit predictor-corrector scheme is used to solve the non-Fourier problem. The effects of radiation including single scattering albedo, conduction-to-radiation parameter, and optical thickness of the medium on the transient and steady state temperature distributions are investigated in detail. Analysis results indicate that the internal radiation in the medium significantly influences the wave nature. The thermal wave nature in the combined non-Fourier heat conduction with radiation is more obvious for large values of conduction-to-radiation parameter, small values of optical thickness and higher scattering medium. The results from non-Fourier-effect equation are also compared to those obtained from the Fourier equation. Non-Fourier effect becomes insignificant as either time increases or the effect of radiation increases.     

Light scattering in disperse layers with a high concentration of optically soft particles

A model describing the propagation of radiation in disperse close-packed layers of optically soft particles is proposed. The model is based on the radiative transfer equation used for describing a light field in the multiple scattering regime and on the interference approximation for describing collective scattering effects; it also takes into account the Fresnel reflection at the layer boundaries. To describe the single-scattering characteristics, the Mie formulas are used. The model allows one to consider scattering in layers of any optical thickness. Calculations of the angular structure of the radiation leaving the layer and estimates of the errors of the calculations were performed. The possibility of forming a maximum at small scattering angles in media with correlated particles is shown. The theoretical results are in good agreement with the available experimental data.     

Transient radiative transfer in irregular two-dimensional geometries

This article presents a finite-volume method to calculate transient radiative transfer in two-dimensional irregularly shaped enclosures. The fully implicit scheme is used to discretize the transient term. The step and CLAM spatial differencing schemes are used in this study. The procedure is validated with available published results. The capabilities of the procedure are then examined using two additional test problems. The ability of the present formulation in modeling absorbing, emitting and anisotropically scattering media is examined using wall heat fluxes and incident radiation.     

The non-equilibrium Marshak wave problem

An analytic solution to a particular Marshak wave problem is given. The radiative transfer model used is the gray, non-equilibrium diffusion approximation which allows the radiation and material fields to be out of equilibrium. This solution should be useful as a reference problem for validating time-dependent radiative transfer computer codes, as well as investigating the convergence, as a function of space and time step size, for such codes. The coupling of the radiation field to the material field in a multigroup code, a difficult numerical problem, can also be tested against this solution. Typical numerical results are given for surface quantities, integral quantities, and the distribution of radiative energy and material temperature as a function of space and time.     

NEW APPROXIMATE PHASE FUNCTIONS FOR SCATTERING OF UNPOLARIZED LIGHT BY DIELECTRIC PARTICLES

We present two new phase functions, one for particles small compared to the wavelength of the scattering radiation and the other for particles large compared to the wavelength of the scattering radiation. These phase functions have been validated for the case of Mie scatterers. For small particles, the results of the new phase function are found to be identical with the Mie results. For large particles, comparison with the Mie results show that the phase function presented here is an extremely good approximation to the Mie phase function. We believe that these phase functions can be expediently used in problems relating to solutions of the radiative transfer equations.     

Radiative transfer in a two-dimensional rectangular medium exposed to diffuse radiation

The integral equation for radiative transfer in a two-dimensional rectangular scattering medium exposed to diffuse radiation is solved numerically by removing the singularity. This method yielded accurate results except at very large optical thicknesses. Graphical and tabular results for the source function, flux, and intensity are presented. The source function is also calculated using the first term of a Taylor series expansion. The Taylor series is fairly accurate for small optical thicknesses and columnar geometries. A method is presented for extending these results to the problem of a strongly anisotropic scattering phase function which is made up of a spike in the forward direction superimposed on an isotropic phase function.     

Time-dependent radiative transfer using Chapman-Enskog maximum entropy method

The time-dependent problems of radiative transfer involve a coupling between radiation and material energy fields and are nonlinear because of proposed temperature dependence of the medium characteristics in semi-infinite medium with Rayleigh anisotropic scattering. By means of the limited flux, Chapman-Enskog and maximum entropy technique the time-dependent radiative transfer equation has been solved explicitly. The maximum entropy method is used to solve the resulting differential equation for radiative energy density. The calculations are carried out for temperature (normalized dimensionless) Θ(x,τ), radiative energy density and net flux with Rayleigh and anisotropic scattering for different space at different times.     

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.     

Three-dimensional vector radiative transfer in a semi-infinite, Rayleigh scattering medium exposed to spatially varying polarized radiation: generalized reflection matrix

Three-dimensional vector radiative transfer in a semi-infinite medium exposed to spatially varying, polarized radiation is studied. The problem is to determine the generalized reflection matrix for a multiple scattering medium characterized by a 4×4 scattering matrix. A double integral transform is used to convert the three-dimensional vector radiative transfer equation to a one-dimensional form, and a modified Ambarzumian's method is then applied to derive a nonlinear integral equation for the generalized reflection matrix. The spatially varying backscattered radiation for an arbitrarily polarized incident beam can be found from the generalized reflection matrix. For Rayleigh scattering and normal incidence and emergence, the generalized reflection matrix is shown to have five non-zero elements. Benchmark results for these five elements are presented and compared to asymptotic results. When the incident radiation is polarized, the vector approach used in this study correctly predicts three-dimensional behavior, while the scalar approach does not. When the incident radiation is unpolarized, both the vector and scalar approaches predict a two-dimensional distribution of the intensity, but the error in the scalar prediction can be as high as 20%.     

Radiative properties of densely packed spheres in semitransparent media: A new geometric optics approach

This contribution presents a new Ray-tracing method for calculating effective radiative properties of densely packed spheres in non-absorbing or semitransparent host medium. The method is restricted to the geometric optic objects and neglects the wave effects. The effective radiative properties such as the absorption and scattering coefficients, and phase function are retrieved from the calculation of mean-free paths of scattering and absorption, and the angular scattering probability of radiation propagating in the dispersed medium. The model accounts for the two geometric effects called here as non-point scattering and ray transportation effects. The successful comparison of the current model with data of radiative properties and transmittances of particle beds in a non-absorbing medium reported in the literature confirm its suitability. It is shown that: (i) for opaque or absorbing particles (not systematically opaque), the non-point scattering is the dominant geometric effects whereas both non-point scattering and ray transportation effects occur for weakly absorbing and transparent particles. In the later cases, these two geometric effects oppose and may cancel out. This may explain why the Independent scattering theory works well for packed of quasi-transparent particles; (ii) the non-point scattering and ray transportation effects can be captured through the scattering and absorption coefficients while using the classical form of phase function. This enables using the standard radiative transfer equation (RTE); (iii) the surrounding medium absorption can be accounted for without any homogenization rule. It contributes to increasing the effective absorption coefficient of the composite medium as expected but, at the same time, it reduces the particle extinction; and (iv) the current transfer calculation predicts remarkably the results of direct Monte Carlo (MC) simulation. This study tends therefore to confirm that the RTE can be applied to densely packed media by using effective radiative properties.     

A single-scattering approximation for infrared radiative transfer in limb geometry in the Martian atmosphere

We present a single-scattering approximation for infrared radiative transfer in limb geometry in the Martian atmosphere. It is based on the assumption that the upwelling internal radiation field is dominated by a surface with a uniform brightness temperature. It allows the calculation of the scattering source function for individual aerosol types, mixtures of aerosol types, and mixtures of gas and aerosol. The approximation can be applied in a Curtis-Godson radiative transfer code and is used for operational retrievals from Mars Climate Sounder measurements. Radiance comparisons with a multiple scattering model show good agreement in the mid- and far-infrared although the approximate model tends to underestimate the radiances in realistic conditions of the Martian atmosphere. Relative radiance differences are found to be about 2% in the lowermost atmosphere, increasing to ∼10% in the middle atmosphere of Mars. The increasing differences with altitude are mostly due to the increasing contribution to limb radiance of scattering relative to emission at the colder, higher atmospheric levels. This effect becomes smaller toward longer wavelengths at typical Martian temperatures. The relative radiance differences are expected to produce systematic errors of similar magnitude in retrieved opacity profiles.     

Time-dependent polarized radiation transfer in semi-infinite binary Markovian media

The problem of time-dependent radiation transfer in a semi-infinite plane-parallel random medium with Rayleigh scattering phase function including polarization is considered. The random medium is assumed to consist of two immiscible mixed materials with specular reflecting boundary. The mixing statistics of the two components of the medium is described by the two-state homogeneous Markovian statistics. A formalism, developed to treat radiative transfer in statistical mixtures, is used to obtain the ensemble-averaged solution. Two different weight functions are used to obtain the numerical results for the ensemble-average for reflectivity, radiant energy, and net flux of the medium at any time.     

Diffuse reflectance of TiO2 pigmented paints: Spectral dependence of the average pathlength parameter and the forward scattering ratio

Diffuse reflectance spectra of paint coatings with different pigment concentrations, normally illuminated with unpolarized radiation, have been measured. A four-flux radiative transfer approach is used to model the diffuse reflectance of TiO₂ (rutile) pigmented coatings through the solar spectral range. The spectral dependence of the average pathlength parameter and of the forward scattering ratio for diffuse radiation, are explicitly incorporated into this four-flux model from two novel approximations. The size distribution of the pigments has been taken into account to obtain the averages of the four-flux parameters: scattering and absorption cross sections, forward scattering ratios for collimated and isotropic diffuse radiation, and coefficients involved in the expansion of the single particle phase function in terms of Legendre polynomials.     

Pomraning-Eddington approximation for radiative transfer in a spherical turbid medium

Abstract

The Pomraning-Eddington approximation is used to solve the radiative transfer problem for anisotropic scattering in a spherical homogeneous turbid medium with diffuse and specular reflecting boundaries. This approximation replaces the radiative transfer integro-differential equation by a second-order differential equation which has an analytical solution in terms of the modified Bessel function. Here, we calculate the partial heat flux at the boundary of anisotropic scattering on a homogeneous solid sphere. The calculations are carried out for spherical media of radii 0.1, 1.0 and 10 mfp and for scattering albedos between 0.1 and 1.0. In addition, the calculations are given for media with transparent, diffuse reflecting and diffuse and specular reflecting boundaries. Two different weight functions are used to verify the boundary conditions. Our results are compared with those given by the Galerkin technique and show greater accuracy for thick and highly scattering media.     

Metallic foams: Radiative properties/comparison between different models

The aim of this study is to determine the radiative properties, which are the extinction coefficient, the scattering albedo and the scattering phase function, of highly porous open-cell aluminium foam, using more-or-less simple predictive models, and to compare all these models. The radiative properties are predicted using geometric optics laws to model the interaction of radiation with the particles forming the foam. Moreover, the particles forming the foam are large compared with the considered wavelength and are supposed to be sufficiently distant from each other to scatter radiation independently. Thus, the radiative characteristics of the foam can be determined by adding the contributions of each particle. A particular attention is paid on microstructure analysis and modelling. We considered different kinds of cell shapes and struts cross-section, using microscopic and tomographic analysis. Furthermore, a new phase function modelling is presented. Finally, we compare the results of each method with the radiative properties obtained from experimental measurements of directional and hemispherical transmittances and hemispherical reflectance.