Radiative transfer in inhomogeneous stratified scattering media with use of the auxiliary function method (original) (raw)
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We present an analysis of a number of different approximations for the diffuse reflectance (spherical and plane albedo) of a semi-infinite, unbounded, plane-parallel, and optically homogeneous layer. The maximally wide optical conditions (from full absorption to full scattering and from fully forward to fully backward scattering) at collimated, diffuse, and combined illumination conditions were considered. The approximations were analyzed from the point of view of their physical limitations and compared to the numerical radiative transfer solutions, whenever it was possible. The main factors impacting the spherical and plane albedo were revealed for the known and unknown scattering phase functions. The main criterion for inclusion of the models in analysis was the possibility of practical use, i.e., approximations were well parameterized and only included easily measured or estimated parameters. We give a detailed analysis of errors for different models. An algorithm for recalculation of results under combined (direct and diffuse) illumination also has been developed.
International Journal of Heat and Mass Transfer, 2012
In this work, we derive the integral equations of radiative transfer in terms of intensity moments for radiative transfer in an anisotropically scattering slab with a spatially varying refractive index (VRI). The integral equations are solved by the Nyström method. We apply this method to study radiative heat transfer in a cold slab with higher-degree anisotropic scattering and linearly VRI. The slab lays on an opaque substratum. The refractive index may have a jump at the interface between the surroundings and the slab, while the interface between the slab and the substratum is assumed to be non-reflecting. To exemplify the application of the integral formulation, we consider the case with irradiation from external source in the surroundings and the case with an emitting substratum. We also solve the problems by the Monte Carlo method (MCM). The hemispherical reflectance and transmittance of the slabs obtained by solving integral equations are in excellent agreement with those obtained by the MCM. A positive gradient of refractive index (n 0) enhances forward radiative transfer, and so the dimensionless radiative heat flux increases with the increase of n 0 for the cases with irradiation from the surroundings. Effects of the optical thickness, the scattering albedo and the scattering phase function are also investigated.
Applied Mathematics and Computation, 2005
We consider the radiation transfer problem in the discrete-ordinate, plane-parallel approach. We introduce two benchmark problems with exact known solutions and show that for strongly nonhomogeneous media the homogeneous layers approximation can lead to errors of 10% in the estimation of the intensity. We propose and validate a general purpose numerical method that transforming the two-boundary problem into an initial boundary problem, using an adaptative step integration and an interpolation of the local optical properties, can improve the accuracy of the solution up to two orders of magnitude. This is furthermore of interest for practical applications, such as atmospheric radiation transfer, where the scattering and absorbing properties of the media vary strongly with height and are only known, based on measurements or models, at certain discrete points.
Generalized method for evaluating scattering parameters used in radiative transfer models
Journal of The Optical Society of America A-optics Image Science and Vision, 1997
The effective scattering and absorption coefficients used to describe the optical properties of particulate materials in radiative transfer models are determined by the average path-length parameter of the diffuse radiation, as well as by the fraction of energy that each particle scatters into the forward and backward hemispheres relative to the direction of the impinging radiation. Until now, there were no well-established methods to calculate these parameters. We have devised an approach for evaluating average path-length parameters and forward-scattering ratios for both forward and backward diffuse radiation intensities. Single-scattering processes are described by Lorenz-Mie theory, and multiple-scattering effects have been taken into account by a generalization of Hartel theory. As a consequence of the formalism, the Kubelka-Munk scattering and absorption coefficients are explicitly related to average path-length parameters and forward-scattering ratios. These parameters display an optical depth dependence, characterized by values smoothly increasing or decreasing from the perpendicularly illuminated interface and saturation values at large optical depths.
Radiative transfer in a rectangular anisotropically scattering medium exposed to diffuse radiation
Journal of Quantitative Spectroscopy and Radiative Transfer, 1990
Linearly anisotropic scattering in a two-dimensional rectangular medium exposed to diffuse radiation is studied using three methods. A standard differential approximation and Olfe's modified differential approximation are presented, and an exact integral formulation in terms of moments of intensity is developed. A collocation method is extended to the solutions of the 2-D integral equations. For the purpose of validating the method, the results of the integral formulation are compared with the existing solutions of 2-D isotropic scattering. We then use the solutions of the integral formulation as a benchmark to examine the accuracy of the two differential approximations. Comparisons of the results derived from the three methods show that both differential approximations work well for radiative transfer in an optically-thick medium, while the modified differential approximation is superior to the standard differential approximation for radiative transfer around the boundaries and for radiative transfer in an optically-thin medium. However, the modified differential approximation understimates the total intensity around the center of a medium. Graphic results are also reported to show the effects of anisotropic scattering, optical sizes and scattering albedos.
A Discrete Transfer Method for Radiative Transfer through Anisotropically Scattering Media
American Journal of Heat and Mass Transfer, 2016
A discrete transfer formulation for analyzing radiative heat transfer through participating medium with highly anisotropic scattering phase function is presented. This formulation used the finite volume method to evaluate integral of the radiative heat transfer problem. Hence, participating media anisotropic scattering phase function along a centre scattered sub-solid angle of the discrete transfer direction is defined by an average on the incident sub-solid angle. Numerical simulations on the efficiency of the proposed discrete transfer formulation for one-dimensional radiative heat transfer through participating, and anisotropic scattering medium under diffuse incidence is carried out. Results indicate that present discrete transfer radiative heat fluxes, transmittance and reflectance predictions have accuracy comparable to available analytical and numerical literature benchmark solutions. Excellent agreements are noted for different anisotropic scattering phase functions.
Radiative transfer in a spherical inhomogeneous medium with anisotropic scattering
Journal of Quantitative Spectroscopy and Radiative Transfer, 1991
The radialIke heal flu\: a~ the boundary oi a sphere comalnmp an mternal energ! source and subject to general boundary condnlons IS obtained m lerms ol' the albedo of the corresponding source-free problem wh lsolropic boundaq condIllon. The relallons obiamed apply IO the general case of amsotroplc scatiennp m an inhomogeneous medium. The adkanlage of these relalions IS the result of the l-act thal there is no need LO obtain a particular solutlon l-or specilicd mlernal sources Therefore. calculations can be done easily ior a non-uniform source dislrlbulion The phase funcrlon 1s approalmaled b> usmg a linear anisotropIc relallon. The linear coefficient IS laken IO Ix the sum of Ihe coefficients of the Legendre expansion of the phase funcuon. The resullmg relalions are used IO calculate the partial heal flux and emiswlt> for a gl\en mlernal enerp) source and mhomogeneous medium I
Journal of The Atmospheric Sciences - J ATMOS SCI, 2006
The multiresolution radiative transfer equations of Part I of this paper are solved numerically for the case of inhomogeneous model clouds using Meyer's basis functions. After analyzing the properties of Meyer's connection coefficients and effective coupling operators (ECOs) for two examples of extinction functions, the present approach is validated by comparisons with Spherical Harmonic Discrete Ordinate Method (SHDOM) and Monte Carlo codes, and a preliminary analysis of the local-scale coupling between the cloud inhomogeneities and the radiance fields is presented. It is demonstrated that the contribution of subpixel-scale cloud inhomogeneities to pixel-scale radiation fields may be very important and that it varies considerably as a function of local cloud inhomogeneities.
International Communications in Heat and Mass Transfer, 2013
In this paper, an investigation of radiative transfer in a rectangular medium with one-dimensional or twodimensional graded index is presented. The integral equations of intensity moments are derived and then the cases of a cold medium exposed to diffuse irradiation at the left boundary are solved by the Nyström method. The results obtained by solving integral equations are in excellent agreement with those obtained by the Monte Carlo method and the discrete ordinates method. For the case with an increasing refractive index in the direction to the right boundary, the distribution of refractive index enhances the radiation in the direction to the right, and so the half-range flux toward the right boundary increases with the increase of the gradient of refractive index. Besides, the half-range flux toward the right boundary decreases as the position considered approaches the top or bottom boundary.