The influence of anisotropic scattering on the radiative intensity in a gray, plane-parallel medium calculated by the DRESOR method

Even though there have been many ways to treat complex anisotropic scattering problems, in most of the cases only the radiation flux or its dimensionless data were provided, and radiative intensity with high directional resolution could merely be seen. In this paper, a comprehensive formulation for the DRESOR method was proposed to deal with the anisotropic scattering, emitting, absorbing, plane-parallel media with different boundary conditions. The method was validated by the data from literature and the integral formulation of RTE. The DRESOR value plays an important role in the DRESOR method, and how it is determined by the anisotropic scattering was demonstrated by some typical results. The intensities with high directional resolution at any point can be given by the present method. It was found that the scattering phase function has little effect on the intensity for thin optical thickness, for example, 0.1. And there is the largest boundary intensity for the medium with the largest forward scattering capability, and the smallest one with the largest backward scattering capability. An attractive phenomenon was observed that the scattering of the medium makes the intensity at boundary can not reach the blackbody emission capability with the same temperature, even if the optical thickness tends to very large. It was also revealed that the scattering of the medium does not mean it cannot alter the magnitude of the energy; actually, stronger scattering causes the energy to have more chance to be absorbed by the medium, and indirectly changes the energy magnitude in the medium. Finally, it is easy to deduce all the associated quantities such as the radiation flux, the incident radiation and the heat source from the intensity, just as done in literature.     

Lorenz-Mie Theory for Spheres Immersed in an absorbing host medium

Light scattering by particles is often used to determine velocities or concentrations of particles in gaseous or liquid streams. Within the Lorenz-Mie theory, light scattering is well understood both for a single compact spherical particle and a single multilayered particle in a non-absorbing surrounding medium. However, in some cases of practical importance the Lorenz-Mie theory in its present form may fail to describe the scattering because the host medium is absorbing (e.g. water droplets in oil). In this case, a new treatment of the scattering theory is required. In previous work, solutions were obtained in the far-field of the scattering sphere. In this paper, a rigorous solution is derived from the calculation of the total absorption rate of the particle in the host medium, which is valid for all distances from the surface of the encapsulated particle. It is shown that it is necessary to consider finite sizes R of the integrating sphere when dealing with absorbing host media. Cross-sections are defined which are characteristic quantities not only for the particle, depending on the size of a conceptual sphere around the scatterer and the imaginary part of the refractive index of the host medium. The results obtained are discussed for the case of non-absorbing host media and in the far-field approximation. Some numerical examples are given which are also related to experimental results.     

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.     

A multi-dimensional differential approximation for absorbing/emitting and anisotropically scattering media with collimated irradiation

By considering the intensity within a medium to consist of a collimated and a fairly diffuse part, the overall problem of radiative transfer is reduced to two simpler ones: first the collimated intensity is obtained (equivalent in complexity to a nonscattering medium); for the evaluation of the diffuse part of the radiation (due to emission and scattering), a new differential approximation has been developed. To demonstrate the accuracy and simplicity of the present method, two sample cases are presented for which some exact solutions can be found in the literature: results are presented (i) for cosine-varying irradiation incident upon a two-dimensional, isotropically scattering slab and (ii) for irradiation with a Gaussian intensity distribution of a two-dimensional, anisotropically scattering semi-infinite cylindrical medium.     

Radiative heat transfer in a planar medium with anisotropic scattering and directional boundaries

Radiation heat transfer in an absorbing, emitting and scattering medium has been the subject of many previous investigations. Most solutions are numerically complex and the existing analytical solutions are restricted in application by the simplifying assumptions involved. A plane-parallel medium is considered which scatters anisotropically. The boundaries are considered to be specular reflectors, as predicted by Fresnel's relations, while the diffusely incident radiation is refracted according to Snell's law. The emission is restricted to a medium with a uniform temperature distribution. Approximate closed-form solutions for the radiative heat flux and incident intensity are presented for dielectric layers and linear anisotropic scattering. Numerical results are also presented and show that the effects of directional boundaries, anisotropic scattering, scattering albedo and optical depth are accurately predicted by the approximate solution.     

Propagation of a light beam in an absorbing medium with large-scale inhomogeneities

A method of solving the radiative transfer equation is proposed; it enables one to take into account the influence of absorption on the angular and spatial distributions of radiation under conditions of sharply anisotropic multiple scattering. For phase functions that decrease with an increase in the scattering angle by the power law, the total flux attenuation and profiles of the angular and spatial distributions in a strongly absorbing medium are studied. The obtained analytical dependences exhibit a good agreement with results of numerical solution of the radiative transfer equation.     

Analysis of radiative transport in a cylindrical enclosure—An application of the modified discrete ordinate method

Application of the modified discrete ordinate method (MDOM) proposed by Mishra et al. [Mishra SC, Roy HK, Misra N. Discrete ordinate method with a new and simple quadrature scheme. J Quant Spectrosc Radiat Transfer 2006;101:249-262.] has been extended for calculation of volumetric radiative information in a cylindrical enclosure. Radiatively, the medium inside a diffuse gray 1-D concentric cylinder is absorbing, emitting and scattering. Three types of problems, viz., an isothermal medium representing non-radiative equilibrium case, a non-isothermal medium representing radiative equilibrium situation and the case of a combined mode conduction and radiation heat transfer have been used to test the robustness of the MDOM. Temperature/emissive power and heat flux/energy flow rate distributions in the medium have been found for the effects of various parameters like the extinction coefficient, the scattering albedo, the boundary emissivity and the conduction-radiation parameter. To check the accuracy of the results of the MDOM, results have been compared with those available in the literature and also by obtaining the radiative information using the finite volume method. MDOM has been found to provide accurate results.     

Time-dependent radiation transfer with Rayleigh scattering in finite slab media

The time-dependent radiation transfer equation in a finite plane geometry with Rayleigh scattering is studied. The traveling wave transformation is used to obtain the corresponding stationary-like equation. Pomraning-Eddington approximation is then used to find the solution. Numerical results for reflectivity at the left boundary and transmissivity from the right boundary are presented at different times. The medium is assumed to have specular-reflecting boundaries with angular-dependent externally incident flux. Two different weight functions are introduced to force the boundary conditions to fulfill.     

Parallel Jacobian-free Newton Krylov solution of the discrete ordinates method with flux limiters for 3D radiative transfer

The present study introduces a parallel Jacobian-free Newton Krylov (JFNK) general minimal residual (GMRES) solution for the discretized radiative transfer equation (RTE) in 3D, absorbing, emitting and scattering media. For the angular and spatial discretization of the RTE, the discrete ordinates method (DOM) and the finite volume method (FVM) including flux limiters are employed, respectively. Instead of forming and storing a large Jacobian matrix, JFNK methods allow for large memory savings as the required Jacobian-vector products are rather approximated by semiexact and numerical formulations, for which convergence and computational times are presented. Parallelization of the GMRES solution is introduced in a combined memory-shared/memory-distributed formulation that takes advantage of the fact that only large vector arrays remain in the JFNK process. Results are presented for 3D test cases including a simple homogeneous, isotropic medium and a more complex non-homogeneous, non-isothermal, absorbing–emitting and anisotropic scattering medium with collimated intensities. Additionally, convergence and stability of Gram–Schmidt and Householder orthogonalizations for the Arnoldi process in the parallel GMRES algorithms are discussed and analyzed. Overall, the introduction of JFNK methods results in a parallel, yet scalable to the tested 2048 processors, and memory affordable solution to 3D radiative transfer problems without compromising the accuracy and convergence of a Newton-like solution.     

Resonance scattering of canonical elastic shells in absorbing fluid medium

Resonance scattering of elastic spherical shell and cylindrical shell while the surrounding fluid medium has absorption is studied. The normal mode solution derived using exact elastic theory and the separation of variables is still applicable. However, the scattering form function has to be modified for the absorbing medium, otherwise the unreasonable result would be obtained. The backscattering form function in the absorbing medium is redefined, and the form function of elastic spherical and cylindrical shell with vacuum or solid matter filled is calculated in various absorption conditions. The results show that the absorption of surrounding fluid leads to notable attenuation of the coincidence resonances in the mid-frequency, but it has a little influence on the low-frequency resonance scattering induced by the filler inside the shell.     

The use of the galerkin method for radiation transfer in an anisotropically scattering slab with reflecting boundaries

Radiation transfer in an absorbing, emitting, anisotropically scattering, plane-parallel medium with diffusely reflecting boundaries is solved by application of the Galerkin method. With this approach, the radiation heat flux, angular distribution of radiation intensity, and the divergence of the radiation heat flux anywhere in the medium can be determined highly accurately. For optical thicknesses up to about 10, exact results are also readily obtainable if sufficient number of terms are considered in the expansion. Numerical results are presented for representative cases.     

Interior radiances in optically deep absorbing media—II Rayleigh scattering

The interior radiances are calculated within an optically deep absorbing medium scattering according to the Rayleigh phase function. The accuracy of the matrix operator method is improved by many orders of magnitude through the use of accurate starting values obtained by the Runge-Kutta method rather than from the single scattering approximation. The radiance and flux are given for a range of solar zenith angles and for single scattering albedos of 1, 0.99, 0.9, 0.5 and 0.1. The development of the asymptotic angular distribution of the radiance is illustrated. It is shown that this asymptotic distribution is probably physically unobservable when ω₀ < 0.8, since the flux is less than 10^-8 of its original value at the beginning of the asymptotic region. The ratio of the upward to downward flux is calculated and is shown to be remarkably constant within the medium except very close to the boundaries. The heating rate within the medium is found to be very nearly proportional to the downward flux, except near the boundaries. When the single scattering albedo is small, a number of examples illustrate the significant contribution of the direct solar flux to the total flux even at great optical depths within the medium. The total downward flux decreases exponentially with optical depth away from boundaries when the single scattering albedo is greater than or equal to 0.9; when it is less than or equal to 0.5 only an approximate exponential fit can be obtained within the region accessible to experimental observation.     

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

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.     

Transient radiative transfer in participating media with pulse-laser irradiation—an approximate Galerkin solution

An assessment is made of the Galerkin technique as an effective method of solution for transient radiative transfer problems in participating media. A one-dimensional absorbing and isotropically scattering plane-parallel gray medium irradiated with a short-pulse laser on one of its boundaries is considered for the application of the method. The medium is non-emitting and the boundaries are non-reflecting and non-refracting. In the integral formulation of the problem for the source function, the time-wise variation of the radiation intensity at any point and in any direction in the medium is assumed to be the same as the time-wise variation of the average intensity at the same point as an approximation for the application of the method. The transient transmittance and reflectance of the medium are evaluated for various values of the optical thickness, scattering albedo and pulse duration. The results are in agreement with those available in the literature. It is demonstrated that the method is relatively simple to implement and yields accurate results.     

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.     

Radiation transport and internal reflection in a sphere

We construct an integral equation for the flux intensity in a scattering and absorbing medium using the integro-differential form of the radiative transfer equation in a sphere. The sphere is uniformly irradiated by an external source of arbitrary angular distribution. The Fresnel boundary conditions, which incorporate reflection and refraction, are used. For the special cases of a non-scattering medium, and in the limit of an optically transparent medium, we obtain exact solutions for specular and diffuse refection. Some numerical examples are given which give qualitative agreement with some recent work of Tian and Chiu (JQSRT, 2005).     

Scattering amplitude of absorbing and nonabsorbing spheroidal particles in the WKB approximation

This work is devoted to a theoretical study of scattering of light by absorbing and nonabsorbing oriented spheroidal particles in the Wentzel-Kramers-Brillouin (WKB) approximation. Within the framework of the scattering theory, we investigate the form factor and the scattering amplitude for this approximation. The Rayleigh-Gans-Debye theory (RGD), the diffraction approximation (DA), and the anomalous diffraction (AD) are treated as particular cases for nonabsorbing spheroids. To illustrate our formalism, we analyze some numerical examples.     

Differential approximations for transient radiative transfer through a participating medium exposed to collimated irradiation

First, we apply the modified differential approximation (MDA) suggested by Chandrasekhar to transient radiative transfer in a scattering planar medium exposed to collimated pulse irradiation. Next, a hybrid method of the P_1/3 approximation suggested by Olson and the MDA is developed. The hybrid method may be referred to as the modified P_1/3 approximation (MP_1/3A) and is also applied to the same example. Comparisons of the results obtained by solving the MDA, the MP_1/3A and the exact integral equation are made. The comparisons show that the temporal distribution of the transmissivity obtained by the MDA contains a small protuberance or an abrupt slope change, which decreases with the decrease of the scattering albedo. The results obtained by the MP_1/3A are more accurate than those obtained by the MDA for most of the cases considered, because the MP_1/3A corrects the propagation speed of the transmitted radiation.     

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.     

Improving the Born approximation for the scattering of ultrasound in elastic solids

It has recently been demonstrated that the Born approximation for predicting the scattering response of flaws can be improved through the use of simple modifications called the "doubly distorted Born approximation". In this paper the doubly distorted Born approximation itself is modified with phase and amplitude corrections that further improve the Born scattering results for isotropic elastic media. The reliability of this new modification of the Born approximation has been evaluated by comparison with the exact solution for spherical inclusions obtained with the method of separation of variables. Unlike the ordinary Born approximation which works well only for very weak scattering inclusions, our modification of the doubly distorted Born approximation gives improved scattering results for both weak and strong scattering inclusions.