Radiative transfer theory with time delay for the effect of pulse entrapping in a resonant random medium: General transfer equation and point-like scatterer model

Abstract

A pulse propagation of a vector electromagnetic wave field in a discrete random medium under the condition of Mie resonant scattering is considered on the basis of the Bethe–Salpeter equation in the two-frequency domain in the form of an exact kinetic equation which takes into account the energy accumulation inside scatterers. The kinetic equation is simplified using the transverse field and far wave zone approximations which give a new general tensor radiative transfer equation with strong time delay by resonant scattering. This new general radiative transfer equation, being specified in terms of the low-density limit and the resonant point-like scatterer model, takes the form of a new tensor radiative transfer equation with three Lorentzian time-delay kernels by resonant scattering. In contrast to the known phenomenological scalar Sobolev equation with one Lorentzian time-delay kernel, the derived radiative transfer equation does take into account effects of (i) the radiation polarization, (ii) the energy accumulation inside scatterers, (iii) the time delay in three terms, namely in terms with the Rayleigh phase tensor, the extinction coefficient and a coefficient of the energy accumulation inside scatterers, respectively (i.e. not only in a term with the Rayleigh phase tensor). It is worth noting that the derived radiative transfer equation is coordinated with Poynting's theorem for non-stationary radiation, unlike the Sobolev equation. The derived radiative transfer equation is applied to study the Compton–Milne effect of a pulse entrapping by its diffuse reflection from the semi-infinite random medium when the pulse, while propagating in the medium, spends most of its time inside scatterers. This specific albedo problem for the derived radiative transfer equation is resolved in scalar approximation using a version of the time-dependent invariance principle. In fact, the scattering function of the diffusely reflected pulse is expressed in terms of a generalized time-dependent Chandrasekhar H-function which satisfies a governing nonlinear integral equation. Simple analytic asymptotics are obtained for the scattering function of the front and the back parts of the diffusely reflected Dirac delta function incident pulse, depending on time, the angle of reflection, the mean free time, the microscopic time delay and a parameter of the energy accumulation inside scatterers. These asymptotics show quantitatively how the rate of increase of the front part and the rate of decrease of the rear part of the diffusely reflected pulse become slower with transition from the regime of conventional radiative transfer to that of pulse entrapping in the resonant random medium.     

Diffusion approximations for the scattering of resonance-line photons: Interior and boundary layer solutions

Resonance-line scattering in static low density media with large optical thickness has a diffusive behavior in both space and frequency because photons belonging to the Lorentzian wings of the line may be scattered almost monochromatically a very large number of times. This diffusive behavior holds on frequency scales and spatial scales, χ_c and τ_c, much larger than the scales associated with one elementary scattering of a wing-photon.A method developed for diffusion approximations in neutron transport theory, suitably generalized to handle diffusion in frequency space, is applied to the case of conservative scattering in a bounded medium with interior sources and zero incoming radiation. The method is to separate the line radiation field into an interior part and a boundary layer part which goes to zero in the interior. Each part is expanded in terms of a small parameter ?, which is the ratio of the mean free-path at frequency χ_c to the characteristic spatial scale τ_c.It is shown that the leading term in the interior asymptotic expansion is isotropic, zero on the boundary, and obeys a space and frequency diffusion equation. In the boundary-layer expansion, the leading term is of order ? and is a solution to a monochromatic transfer equation in a semi-infinite, plane-parallel medium. The emergent radiation field is shown to be of order ? and proportional to the gradient of the interior solution at the boundary. Its angular dependence, in the case of isotropic scattering in the atom frame, is given by the Ambartsoumian H-function. A comparison is presented between numerical solutions of the full transfer equation and asymptotic solutions. Non-conservative scattering and time-dependent problems are briefly 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.     

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.     

Exploiting the linearity of radiative transfer

A standard problem in radiative transfer is finding the external and internal radiative fields produced by uniform, parallel rays illuminating the top of a one-dimensional, scattering and absorbing medium of finite optical thickness. This problem has been solved in several ways with various physical restrictions. One approach is by finding the source function that represents the rate of production of scattered radiation per unit volume per unit solid angle at each point in the medium. The present paper develops and uses the idea that the standard source function is an influence function for a given medium. The linearity of radiative transfer is then used to find certain general source functions in terms of the standard one. The usefulness of the above concept is demonstrated by the following four problems: (1) derivation of Chandrasekhar's four principles of invariance from the radiative transfer equation, (2) derivation of the equations governing Chandrasekhar's X- and Y- functions without using the invariance principles or resolvent kernels, (3) finding the source function for a medium with a Lambert's-law bottom, and (4) finding the source function for a medium with a bottom that is a perfect specular reflector.     

Radiative transfer in one-dimensional hollow sphere with anisotropic scattering and variable medium properties

The effects of variable medium properties on radiation transfer in participating and anisotropically scattering one-dimensional spherical medium were investigated by Tsai et al. (JQSRT 42(3) (1989) 187). The discrete ordinates method solutions they provided for hollow spherical medium cases are incorrect. The correct DOM S₈ and the integral transfer equation solutions are provided.     

Application of Synthetic Kernel (SKN) method to participating linearly anisotropically scattering solid spherical medium

The Synthetic Kernel (SK_N) method is applied to a solid spherical absorbing, emitting and linearly anisotropically scattering homogeneous and inhomogeneous medium. The SK_N method relies on approximating the integral transfer kernels by Synthetic Kernels. The radiative integral transfer equation is then reducible to a set of coupled second-order differential equations. The SK_N method, which uses Gauss quadratures, is tested against integral equation and the discrete-ordinates S₈ solutions for various optical radius and scattering albedo variations.     

Simultaneous solution of Kompaneets equation and radiative transfer equation in the photon energy range 1-125 keV

Radiative transfer equation in plane parallel geometry and Kompaneets equation is solved simultaneously to obtain theoretical spectrum of 1-125 keV photon energy range. Diffuse radiation field are calculated using time-independent radiative transfer equation in plane parallel geometry, which is developed using discrete space theory (DST) of radiative transfer in a homogeneous medium for different optical depths. We assumed free-free emission and absorption and emission due to electron gas to be operating in the medium. The three terms n, n² and (∂n/∂x_k) where n is photon phase density and x_k=(hν/kT_e), in Kompaneets equation and those due to free-free emission are utilized to calculate the change in the photon phase density in a hot electron gas. Two types of incident radiation are considered: (1) isotropic radiation with the modified black body radiation I^MB[1] and (2) anisotropic radiation which is angle dependent. The emergent radiation at τ=0 and reflected radiation τ=τ_max are calculated by using the diffuse radiation from the medium. The emergent and reflected radiation contain the free-free emission and emission from the hot electron gas. Kompaneets equation gives the changes in photon phase densities in different types of media. Although the initial spectrum is angle dependent, the Kompaneets equation gives a spectrum which is angle independent after several Compton scattering times.     

Numerical results for the thermal scattering functions

A recent formulation in radiative transfer defined the thermal scattering functions that characterize radiative transfer from a general, plane-parallel, finite medium driven solely by an internal distribution of thermal sources. Exiting diffuse intensities are expressed as space convolutions of the thermal scattering functions with any thermal source distribution. A parametric study is presented to obtain the basic structure of these scattering functions. The independent variables of these azimuthally independent functions are the direction consine μ and source location t, while the parameters are the single scattering albedo ω, total optical depth t₀, and the asymmetry factor g in the Henyey-Greenstein phase function. The basic functional trends are discussed using various parametric plots, and selected tabular results are given to allow numerical checks. The computational method is invariant imbedding. As a particular application, these functions are used in the following companion paper to obtain exiting intensities from inhomogeneous and nonisothermal media.     

Relativistic charged particles in the field of an electromagnetic plane wave in a medium

Solutions of the Klein-Gordon and Dirac equation for a charged particle in the field of an electromagnetic plane wave in a medium with a constant refractive index n are discussed. Generally, for n² < 1, spontaneous pair creation from the vacuum, and for n² > 1, energy bands are observed. The interplay of Compton and Cherenkov scattering is discussed. Some doubts are formulated as to the physical relevance of calculating pair creation in a homogeneous electric field as it is usually done.     

Theory of transfer with delay for trapping of nonstationary acoustic radiation in a resonant randomly inhomogeneous medium

The propagation of a quasimonochromatic wave packet of acoustic radiation in a discrete randomly-inhomogeneous medium under the condition that the carrier frequency of the packet is close to the resonance frequency of Mie scattering by an isolated scatterer is studied. The two-frequency Bethe-Salpeter equation in the form of an exact kinetic equation that takes account of the accumulation of the acoustic energy of the radiation inside the scatterers is taken as the initial equation. This kinetic equation is simplified by using the model of resonant point scatterers, the approximation of low scatterer density, and the Fraunhofer approximation in the theory of multiple scattering of waves. This leads to a new transport equation for nonstationary radiation with three Lorentzian delay kernels. In contrast to the well-known Sobolev radiative transfer equation with one Lorentzian delay kernel, the new transfer equation takes account of the accumulation of radiation energy inside the scatterers and is consistent with the Poynting theorem for nonstationary acoustic radiation. The transfer equation obtained with three Lorentzian delay kernels is used to study the Compton-Milne effect—trapping of a pulse of acoustic radiation diffusely reflected from a semi-infinite resonant randomly-inhomogeneous medium, when the pulse can spend most of its propagation time in the medium being “trapped” inside the scatterers. This specific albedo problem for the transfer equation obtained is solved by applying a generalized nonstationary invariance principle. As a result, the function describing the scattering of a diffusely reflected pulse can be expressed in terms of a generalized nonstationary Chandrasekhar H-function, satisfying a nonlinear integral equation. Simple analytical asymptotic expressions are found for the scattering function for the leading and trailing edges of a diffusely reflected δ-pulse as functions of time, the reflection angle, the mean scattering time of the radiation, the elementary delay time, and the parameter describing the accumulation of radiation energy inside the scatterers. These asymptotic expressions demonstrate quantitatively the retardation of the growth of the leading edge and the retardation of the decay of the trailing edge of a diffusely reflected δ-pulse when the conventional radiative transfer regime goes over to a regime of radiation trapping in a resonant randomly-inhomogeneous medium. Zh. éksp. Teor. Fiz. 113, 432–444 (February 1998)     

Radiative and conductive heat transfer in a nongrey semitransparent medium. Application to fire protection curtains

This paper deals with heat transfer in nongrey media which scatter, absorb and emit radiation. Considering a two dimensional geometry, radiative and conductive phenomena through the medium have been taken into account. The radiative part of the problem was solved using the discrete ordinate method with classical S_n quadratures. The absorption and scattering coefficients involved in the radiative transfer equation (RTE) were obtained from the Mie theory. Conduction inside the medium was linked to the RTE through the energy conservation. Validation of the model has been achieved with several simulation of water spray curtains used as fire protection walls.     

Probe into the reflection from GaP nanoparticles via different solutions of radiative transfer equation

The reflection and absorption spectra of gallium phosphide (GaP) nanoparticles were measured. The radiative transfer equation (RTE) for the medium with scattering and absorption is solved by three different solutions. The ratio of the absorption and scattering coefficients (E _a/E _s) of the GaP nanoparticles layer is calculated from the reflection spectrum via the three solutions, respectively, and the result derived with the three-flux model is closest to the exact solution given by Giovanelli. The E _a/E _s curves all exhibit the energy band gaps of GaP nanoparticles, which are consistent with the absorption spectrum measurement. The shape of the reflection spectrum is mainly determined by the absorption, and the scattering only influences its intensity. The energy band structure of the powder sample plays an important role in the reflection phenomenon, and the reflectance data can be used for quantitative analyses.     

Inverse solutions to radiative-transfer problems with partially transparent boundaries and diffuse reflection

Analytical techniques are used to solve a class of inverse radiative-transfer problems relevant to finite and semi-infinite plane-parallel media. While the assumption of isotropic scattering is made, diffuse reflection is allowed at the surface, for the semi-infinite case, and at both surfaces for the case of a finite layer. For the general case based on a semi-infinite medium, a cubic algebraic equation is used to define the basic result, but for the specific case of a semi-infinite medium illuminated by a constant incident distribution of radiation, very simple exact expressions are developed for the albedo for single scattering ? and the coefficient for diffuse reflection ρ. Analytical results are also developed (again in terms of a cubic algebraic equation) for the case of a finite layer with equal reflection coefficients relevant to the two surfaces. For the general case of a finite layer with unequal reflection coefficients, two specific formulations are given. The first algorithm is based on a system of three quadratic algebraic equations for the two reflection coefficients ρ₁ and ρ₂ and the single-scattering albedo ?. Secondly, an elimination between these three algebraic equations is carried out to yield two coupled algebraic equations for ρ₁ and ρ₂ plus an explicit expression for ? in terms of ρ₁ and ρ₂. In addition, an exact expression for τ₀, the optical thickness of the finite layer, is developed in terms of ?, ρ₁ and ρ₂. As is typical with the considered class of inverse problems in radiative transfer, all surface quantities are either specified or considered available from experimental measurements. All basic results are tested numerically.     

A multiperipheral model with continued cross channel unitarity

We derive, by using a spectral representation in momentum transfer, t, an integral equation, similar in structure to a multipheral equation, with continued cross channel unitarity, for the absorptive part for a composite particle scattering amplitude from a Bethe-Salpeter equation describing composite particle scattering in the s channel. At high energy in the t channel, the equation becomes homogeneous and has a Reggeized solution. We indicate how this equation may be solved using determinental techniques. We also show how the composite particle amplitude resulting from the original equation may be used to construct production and three body amplitudes. We also infer the possibility of studying, using the amplitude from the cross channel problem, the effect of extra unitarity on Reggeon-Reggeon-particle vertices.     

Numerical prediction of heat transfer by natural convection and radiation in an enclosure filled with an isotropic scattering medium

This paper deals with the numerical solution for natural convection and volumetric radiation in an isotropic scattering medium within a heated square cavity using a hybrid thermal lattice Boltzmann method (HTLBM). The multiple relaxation time lattice Boltzmann method (MRT-LBM) has been coupled to the finite difference method (FDM) to solve momentum and energy equations, while the discrete ordinates method (DOM) has been adopted to solve the radiative transfer equation (RTE) using the S8 quadrature. Based on these approaches, the effects of various influencing parameters such as the Rayleigh number (Ra), the wall emissivity (ε_ι), the Planck number (Pl), and the scattering albedo (ω), have been considered. The results presented in terms of isotherms, streamlines and averaged Nusselt number, show that in absence of radiation, the temperature and the flow fields are centro-symmetrics and the cavity core is thermally stratified. However, radiation causes an overall increase in the temperature and velocity gradients along both thermally active walls. The maximum heat transfer rate is obtained when the surfaces of the enclosure walls are regarded as blackbodies. It is also seen that the scattering medium can generate a multicellular flow.     

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.     

Complete matrix solution of radiative transfer equation for PILE of horizontally homogeneous slabs

This article covers the analytical solution of the discretized radiative transfer equation in the matrix form. The equation is discretized according to the discrete ordinates method. The solution is based on the representation of the light field in a scattering medium as a superposition of an anisotropic and a smooth regular parts. The first of them is calculated analytically using the smoothness of the solution angular spectrum. The regular part is obtained from a radiative transfer equation boundary problem with the anisotropic part as a source function by discrete ordinates method with a scaling transformation and a matrix-operator method applied. There is no limitation of the scattering law in a medium.     

Coupled radiative and conductive heat transfer in a non-grey absorbing and emitting semitransparent media under collimated radiation

This paper deals with heat transfer in non-grey semitransparent two-dimensional sample. Considering an homogeneous purely absorbing medium, we calculated the temperature field and heat fluxes of a material irradiated under a specific direction. Coupled radiative and conductive heat transfer were considered. The radiative heat transfer equation (RTE) was solved using a S₈ quadrature and a discrete ordinate method. Reflection and absorption coefficients of the medium were calculated with the silica optical properties. The conduction inside the medium was linked to the RTE through the energy conservation. Validation of the model and two original cases are also presented.     

On long-wave sound scattering by a Rankine vortex: Non-resonant and resonant cases

The well-known two-dimensional problem of sound scattering by a Rankine vortex at small Mach number M is considered. Despite its long history, the solutions obtained by many authors still are not free from serious objections. The common approach to the problem consists in the transformation of governing equations to the d’Alembert equation with right-hand part. It was recently shown [I.V. Belyaev, V.F. Kopiev, On the problem formulation of sound scattering by cylindrical vortex, Acoustical Physics 54(5) (2008) 603-614] that due to the slow decay of the mean velocity field at infinity the convective equation with nonuniform coefficients instead of the d’Alembert equation should be considered, and the incident wave should be excited by a point source placed at a large but finite distance from the vortex instead of specifying an incident plane wave (which is not a solution of the governing equations).Here we use the new formulation of Belyaev and Kopiev to obtain the correct solution for the problem of non-resonant sound scattering, to second order in Mach number M. The partial harmonic expansion approach and the method of matched asymptotic expansions are employed. The scattered field in the region far outside the vortex is determined as the solution of the convective wave equation, and van Dyke's matching principle is used to match the fields inside and outside the vortical region. Finally, resonant scattering is also considered; an O(M²) result is found that unifies earlier solutions in the literature. These problems are considered for the first time.