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For a four-stream approximation of the kinetic model of radiative transfer with isotropic scattering, a numerical scheme endowed with both truly 2D well-balanced and diffusive asymptotic-preserving properties is derived, in the same spirit as what was done in [L. Gosse and G. Toscani, C. R. Math. Acad. Sci. Paris, 334 (2002), pp. 337-342] in the 1D case. Building on former results of Birkhoff and Abu-Shumays [J. Math. Anal. Appl., 28 (1969), pp. 211-221], it is possible to express 2D kinetic steady-states by means of harmonic polynomials, and this allows one to build a scattering S-matrix yielding a time-marching scheme. Such an S-matrix can be decomposed, as in [L. Gosse and N. Vauchelet, Numer. Math., 141 (2019), pp. 627-680], so as to deduce another scheme, well-suited for a diffusive approximation of the kinetic model, for which rigorous convergence can be proved. Challenging benchmarks are also displayed on coarse grids.

We compute the exact solutions of the radiative transfer equation in two dimensions for isotropic scattering. The intensity and the radiance are given in the space-time domain for isotropic and unidirectional sources. These analytical results are compared to Monte-Carlo simulations in four particular situations., Comment: 6 pages, 5 figures

Here we present results of Monte Carlo simulations employing an effective numerical approach, based upon a description of radiative transfer in terms of the Bethe-Salpeter equation. We develop a simulation algorithm which presents a stochastic calculation of the multi-fold spatial integrals of the Bethe-Salpeter iterative solution. The algorithm is a modification of the widely known MCLM method with a signal registration recipe resulting in a significant reducing of sampling volume. For isotropic scattering the obtained results exhibit an excellent agreement with the known theoretical solution. We simulate backscattering from layers of different thickness for the Henyey-Greenstein or Rayleigh-Gans scattering patterns.

Highly precise numerical solutions to the radiative transfer equation with polarization present a special challenge. Here, we establish a precise numerical solution to the radiative transfer equation with combined Rayleigh and isotropic scattering in a 1D-slab medium with simple polarization. The 2- Stokes vector solution for the fully discretized radiative transfer equation in space and direction derives from the method of doubling and adding enhanced through convergence acceleration. Updates to benchmark solutions found in the literature to 7 places for reflectance and transmittance as well as for angular flux follow. Finally, we conclude with the numerical solution in a partially randomly absorbing heterogeneous medium., Comment: 44 pages 9 Tables 4 Figures

The double interval spherical harmonic method introduced effectively by Wilson and Sen has already been used by Ghosh and Karanjai to solve the equation of radiative transfer in coherent isotropic scattering atmosphere, originally developed by Woolley and Stibbs. The same method has been successfully used in this paper to solve the equation of transfer for coherent anisotropic scattering.

Through using the Monte Carlo method to calculate the fraction distributions of the emission scattered within an emitting, absorbing and scattering gray medium with a vacuum or a cold black wall, a new solution for the angular distributions of radiative intensity in the medium is presented. The angular distributions of intensity in one or more points, or even everywhere, can be calculated from the given distributions of temperature and radiative properties in the medium without the requirement of radiative equilibrium. For the emitting, absorbing, isotropic scattering, parallel-plane, gray media, some important relationships between the new and the formal solutions of the radiative transfer equation are delivered. Validation results for the angular distributions of radiative intensity got for the medium with uniformly specified distributions of temperature and radiative property and optical thickness less than 2.0 demonstrate that the new solution agrees well with the formal solution in the same condition. The work presented in this paper extends the application of the common Monte Carlo methods. In the recent future, more complicated parallel-plane cases will be studied by the same idea, especially with different kinds of wall conditions and/or anisotropic scattering.

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.

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 a Galerkin-type 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.

The discrete-ordinates method is used to develop a solution to a basic polarization problem in radiative transfer. In particular, a solution for the coupled I and Q components of the Stokes vector is developed for a polarization model based on a mixture of Rayleigh and isotropic scattering. The solution is evaluated for the case of a finite plane-parallel layer that has a polarized beam incident on one surface and which has Lambertian reflection on the other surface.

The time-independent linear transport problem in a stochastic finite-plane medium with linear anisotropic scattering is considered. The medium is assumed to consist of two randomly mixed immiscible fluids, with the mixing statistics described as a two-state homogeneous Markov process. The Pomraning–Eddington approach is used to obtain an explicit solution to the problem in the deterministic case. A formalism, developed to treat radiative transfer in statistical mixtures, is used to obtain the ensemble-averaged solution for the problem under consideration. In the case of isotropic scattering, explicit analytic results for reflectivity and transmissivity, which show a good agreement with Monte Carlo benchmark results, are given. Results for reflectivity and transmissivity in the case of linear anisotropic scattering are also given.

A Cartesian-based method is proposed for combustion-radiation problems in gray enclosures having one ordinate symmetry and containing absorbing-emitting media with isotropic scattering. This method collapses the 3-D radiative information of the problem into its 2-D solution plane and is shown to be applicable for all ranges of optical thicknesses, including the optically very thin case, while maintaining near-analytic accuracy over most of this range. To accomplish this task, the collapsed dimension method makes use of effective intensity rays, each of which contains the information of the conglomerate of all the conventional intensity rays focused on the same point and contained in a plane that is both inclined at the same angle as the ray and is also normal to the solution plane. This novel procedure thus eliminates the use of solid angles and reduces both the complexity and the computational expense involved in the solution of such problems. The method is shown to be applicable to gray wall problems en...

The method of computing the radiation field in an infinite circular cylinder proposed in Part I is now applied to the case of isotropic scattering with sources on the boundary and axis of the cylinder, as well as for a uniform distribution of sources inside the cylinder. For the simplest kernel we obtain exact solutions of the basic integral equation in explicit form. For scattering in a spectral line with complete frequency redistribution and a power absorption profile we develop an asymptotic theory for the case when the optical radius of the cylinder is large. We solve the asymptotic equations for the basic characteristics of the scattering in closed form for conservative scattering and find its asymptotics. We obtain estimates of the mean number of scatterings with a layered source, and also the mean and variance of the number of scatterings with a uniform source distribution.

When waves are radiated from a point source in a random medium, single scattering provides a good description of the propagation characteristics at small distances from the source and at a short lapse time from the source origin time; however, multiple scattering dominates over single scattering as travel distance or lapse time increases.

An approximate form for theH-function for isotropic scattering developed by Karanjai and Sen (1971) has been extended to the case of four-term scattering indicatrix.

9th International Conference on Advanced Computational Methods in Heat Transfer -- JUL, 2006 -- -- Ashurst, ENGLAND, WOS: 000239953800011, The SKN, (Synthetic Kernel) approximation is applied to a two-dimensional homogeneous cylindrical participating medium with isotropic scattering. The SKN equations are tested against benchmark problems consisting of cold homogeneous participating medium. The solutions are compared with those obtained by various methods available in the literature. The SK3 approximation results for geometries of very large and very small aspect ratios are in excellent agreement with those of benchmark solutions. The SK3 solutions with moderate aspect ratios are accurate within several percent., Wessex Inst Technol, Lund Univ Technol, Dev Heat Transfer Book Series

An approximate form for the H -function for isotropic scattering developed by Karanjai and Sen has been extended to anisotropic scatering with a characteristic function of the form ψ ( μ ) = w 1 + w 2 μ 2 + w 3 μ 4 . Our method is applicable to w (1 + x cos θ ), Rayleigh's phase function, 1 + x 1 P 1 (cos θ ) + w 2 P 2 (cos θ ), and similar phase functions.

A procedure for calculating heat-transfer characteristics in absorbing and isotropically scattering media is developed and tested by applying it to internally fired furnaces.

THE problem of the radiation field in a plane-parallel multilayer system, whose outer boundary is irradiated by parallel rays, is considered. The case of non-isotropic radiation is studied. A formal solution of the problem is given, and using this, the spatial-angular distribution function of the intensity of the diffused radiation is constructed by applying the method of discrete ordinates. Consider a multilayer plane-parallel system, consisting of N layers in contact with one another which are irradiated by a parallel pencil of rays incident at some angle to the outward normal of the system. We assume that the boundaries of the layers are known and that the source functions are defined differently in each layer. It is required to determine the radiative field in any region of the system and find the law of the angular distribution of the outgoing radiation.

A linear problem of radiative transfer in the spectral line in a plane- parallel layer of finite thickness, under general assumptions on the function of redistribution in frequency, independent of the angle of scattering, is considered. The solution of the system of integral equations obtained is reduced to the solution of certain functional equations of Volterra type. Some calculations are carried out in the case of Doppler broadening of the line. (auth)

The kernel of the integral equation for the source function in a three-dimensional homogeneous atmosphere possesses the properties of a Green's function. These properties are used to transform the integral equation into a singular integral equation for the kernel. The particular case of a homogeneous plane parallel atmosphere is discussed and a solution to the kernel equation is obtained at all points of the atmosphere.

Integral forms of transport equations first appeared in radiative transfer theory at the beginning of this century. If one considers radiative transfer with isotropic scattering in a layer of finite optical thickness τ, the boundary value problem may be written (cf. Section IX.1)

A six-flux theory is derived for the diffuse reflectance and fluorescence of mixed powder samples of various shades of gray. One component of the powder is assumed to be fluorescent, with nonoverlapping excitation and emission bands. For symmetrical normal illumination and isotropic scattering, the six-flux reflectance reduces exactly to the Kubelka-Munk (two-flux) expression. Measurements of the fluorescence of a series of silica-silicon carbide-calcium tungstate mixed powder samples, which were found to vary over a range of 60:1, correlate well with the six-flux theory over the full range, whereas a two-flux theory is satisfactory only at high sample reflectances.

Time dependent radiative transfer equation for plane-parallel isotropic scattering medium in first Gaussian approximation