Differential approximations for transient radiative transfer in refractive planar media with pulse irradiation

Transient radiative transfer in an anisotropically scattering refractive planar medium with pulse irradiation is solved by various approximation methods, such as P?1, P?1 parabolic, P_1/3 and two-flux. The time-resolved transmittance and reflectance are calculated for various radiative parameters, and are compared with those obtained by the discrete ordinate method (DOM). Among the approximation methods considered, the P_1/3 approximation is the better one, because its results are in overall good agreement with those obtained by the more rigorous DOM, except the transmittance around the peak for neither thin nor very thick slabs. It is found that the curved paths of radiation and the internal reflection of the back scattered radiation enhance the effect of scattering.     

A semi-analytical numerical method for time-dependent radiative transfer problems in slab geometry with coherent isotropic scattering

We describe a semi-analytical numerical method for coherent isotropic scattering time-dependent radiative transfer problems in slab geometry. This numerical method is based on a combination of two classes of numerical methods: the spectral methods and the Laplace transform (LTS_N) methods applied to the radiative transfer equation in the discrete ordinates (S_N) formulation. The basic idea is to use the essence of the spectral methods and expand the intensity of radiation in a truncated series of Laguerre polynomials in the time variable and then solve recursively the resulting set of “time-independent” S_N problems by using the LTS_N method. We show some numerical experiments for a typical model problem.     

Evaluation of solution methods for radiative heat transfer in gaseous oxy-fuel combustion environments

The exact solution to radiative heat transfer in combusting flows is not possible analytically due to the complex nature of the integro-differential radiative transfer equation (RTE). Many different approximate solution methods for the solution of the RTE in multi-dimensional problems are available. In this paper, two of the principal methods, the spherical harmonics (P₁) and the discrete ordinates method (DOM) are used to calculate radiation. The radiative properties of the gases are calculated using a non-gray gas full spectrum k-distribution method and a gray method. Analysis of the effects of numerical quadrature in the DOM and its effect on computation time is performed. Results of different radiative property methods are compared with benchmark statistical narrow band (SNB) data for both cases that simulate air combustion and oxy-fuel combustion. For both cases, results of the non-gray full spectrum k-distribution method are in good agreement with the SNB data. In the case of oxy-fuel simulations with high partial pressures of carbon dioxide, use of gray method for the radiative properties may cause errors and should be avoided.     

Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere-ocean system

A vector radiative transfer model has been developed for a coupled atmosphere-ocean system. The radiative transfer scheme is based on the discrete ordinate and matrix operator methods. The reflection/transmission matrices and source vectors are obtained for each atmospheric or oceanic layer through the discrete ordinate solution. The vertically inhomogeneous system is constructed using the matrix operator method, which combines the radiative interaction between the layers. This radiative transfer scheme is flexible for a vertically inhomogeneous system including the oceanic layers as well as the ocean surface. Compared with the benchmark results, the computational error attributable to the radiative transfer scheme has been less than 0.1% in the case of eight discrete ordinate directions. Furthermore, increasing the number of discrete ordinate directions has produced computations with higher accuracy. Based on our radiative transfer scheme, simulations of sun glint radiation have been presented for wavelengths of 670 nm and 1.6 μm. Results of simulations have shown reasonable characteristics of the sun glint radiation such as the strongly peaked, but slightly smoothed radiation by the rough ocean surface and depolarization through multiple scattering by the aerosol-loaded atmosphere. The radiative transfer scheme of this paper has been implemented to the numerical model named Pstar as one of the OpenCLASTR/STAR radiative transfer code systems, which are widely applied to many radiative transfer problems, including the polarization effect.     

Radiative absorption in an infinitely long hollow cylinder with Fresnel surfaces

Radiation absorption in an infinitely long hollow cylinder with Fresnel surfaces is studied using the ray tracing method. It is found that the inner boundary can be modeled as a total reflective surface for the infinitely long hollow cylinder. Radiative absorption of hollow cylinders with Fresnel surfaces is compared to diffusive surfaces predicted by the finite volume method. Effects of refractive index, optical thickness and hole size on radiative absorption are studied. Abrupt changes in radiative absorption near τ_r/τ_Ro=1/n are observed for hollow cylinders with Fresnel surfaces. It is because the Fresnel relation predicts a critical angle at . This trend is not observed in diffusive surfaces. Refractive index and optical thickness are two competing factors that govern the radiative absorption. Higher refractive index drives higher absorption close to the inner surface, while higher optical thickness yields higher absorption near the outer surface. The results of this study can also serve as benchmark solutions for modeling radiative heat transfer in hollow cylinders with Fresnel surfaces. It is also found that the directional or hemispherical emittance can be calculated without solving the radiative transfer equation in the media when the temperature variation in the media is small.     

Experimental and theoretical studies on high-temperature thermal properties of fibrous insulation

In the present paper, an experimental apparatus has been developed to measure heat transfer through high-alumina fibrous insulation for thermal protection system. Effective thermal conductivities of the fibrous insulation were measured over a wide range of temperature (300-973 K) and pressure (10⁻²-10⁵ Pa) using the developed apparatus. The specific heat and the transmittance spectra in the wavelength range of 2.5-25 μm were also measured. The spectral extinction coefficients and Rosseland mean extinction coefficients were obtained from transmittance data at various temperatures to investigate the radiative heat transfer in fibrous insulation. A one-dimensional finite volume numerical model combined radiation and conduction heat transfer was developed to predict the behavior of the effective thermal conductivity of the fibrous insulation at various temperatures and pressures. The two-flux approximation was used to model the radiation heat transfer through the insulation. The experimentally measured specific heat and Rosseland mean extinction coefficients were used in the numerical heat transfer model to calculate the effective thermal conductivity. The average deviation between the numerical results for different values of albedo of scattering and the experimental results was investigated. The numerical results for ω=1 and experimental data were compared. It was found that the calculated values corresponded with the experimental values within an average of 13.5 percent. Numerical results were consistent with experimental results through the environmental conditions under examination.     

Elliptic PDE formulation and boundary conditions of the spherical harmonics method of arbitrary order for general three-dimensional geometries

The inherent complexity of the radiative transfer equation makes the exact treatment of radiative heat transfer impossible even for idealized situations and simple boundary conditions. Therefore, a wide variety of efficient solution methods have been developed for the RTE. Among these solution methods the spherical harmonics method, the moment method, and the discrete ordinates method provide means to obtain higher-order approximate solutions to the equation of radiative transfer. Although the assembly of the governing equations for the spherical harmonics method requires tedious algebra, their final form promises great accuracy for any given order, since it is a spectral method (rather than finite difference/finite volume in the case of discrete ordinates). In this study, a new methodology outlined in a previous paper on the spherical harmonics method (P_N) is further developed. The new methodology employs successive elimination of spherical harmonic tensors, thus reducing the number of first-order partial differential equations needed to be solved simultaneously by previous P_N approximations (=(N+1)²). The result is a relatively small set (=N(N+1)/2) of second-order, elliptic partial differential equations, which can be solved with standard PDE solution packages. General boundary conditions and supplementary conditions using rotation of spherical harmonics in terms of local coordinates are formulated for the general P_N approximation for arbitrary three-dimensional geometries. Accuracy of the P_N approximation can be further improved by applying the “modified differential approximation” approach first developed for the P₁-approximation. Numerical computations are carried out with the P₃ approximation for several new two-dimensional problems with emitting, absorbing, and scattering media. Results are compared to Monte Carlo solutions and discrete ordinates simulations and a discussion of ray effects and false scattering is provided.     

Radiative transfer in a nongray spherical layer: Simplified rectangular model

The problem of radiative transfer in a nongray, absorbing-emitting spherical layer is investigated. The absorption coefficient is assumed to be only a function of frequency, i.e. k_v = α(v)k, and the function α(v) is allowed only two values, zero or unity (simplified rectangular model). The nongray radiative transfer problem is reduced to a gray solution without the use of any approximation (such asthe Plank or Rosseland means) for an isothermal layer and for radiative equilibr     

Spectral collocation method with a flexible angular discretization scheme for radiative transfer in multi-layer graded index medium

The spectral collocation method (SCM) is employed to solve the radiative transfer in multi-layer semitransparent medium with graded index. A new flexible angular discretization scheme is employed to discretize the solid angle domain freely to overcome the limit of the number of discrete radiative direction when adopting traditional S_N discrete ordinate scheme. Three radial basis function interpolation approaches, named as multi-quadric (MQ), inverse multi-quadric (IMQ) and inverse quadratic (IQ) interpolation, are employed to couple the radiative intensity at the interface between two adjacent layers and numerical experiments show that MQ interpolation has the highest accuracy and best stability. Variable radiative transfer problems in double-layer semitransparent media with different thermophysical properties are investigated and the influence of these thermophysical properties on the radiative transfer procedure in double-layer semitransparent media is also analyzed. All the simulated results show that the present SCM with the new angular discretization scheme can predict the radiative transfer in multi-layer semitransparent medium with graded index efficiently and accurately.     

Comments on the simulation of radiative transfer in laser compression experiments

The problems of calculating the radiative transfer for spherically symmetric laser irradiated microspheres are analyzed. The methods presently used to perform the simulation are outlined. A discussion of time dependent simulations is presented to clarify the differences in results which one expects between ab initio coupled radiative transfer—hydrodynamics and semi-empirical approaches.     

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.     

Radiative transfer in an isotropically scattering two-region slab with reflecting boundaries

The problem of radiative heat transfer in an absorbing, emitting, isotropically scattering two-layer slab with diffusely and specularly reflecting boundaries is solved by the F_N method and results are presented for the transmissivity and reflectivity of the slab.     

The use of the itFN method for radiative transfer problems with reflective boundary conditions

The F_N method is used to compute the net radiative heat flux relevant to radiative transfer in an anisotropically scattering, plane-parallel medium with specularly and diffusely reflecting boundaries.     

Radiative transfer calculations in spheres and cylinders

Integral transformation techniques and the F_N method are used to solve, for the case of isotropic scattering, radiative transfer problems in spherical and cylindrical geometries. Numerical results accurate to five or six significant figures are given for selected cases basic to problems with internal heat generation and emitting and diffusely reflecting surfaces.     

A numerical model for multigroup radiation hydrodynamics

We present in this paper a multigroup model for radiation hydrodynamics to account for variations of the gas opacity as a function of frequency. The entropy closure model (M₁) is applied to multigroup radiation transfer in a radiation hydrodynamics code. In difference from the previous grey model, we are able to reproduce the crucial effects of frequency-variable gas opacities, a situation omnipresent in physics and astrophysics. We also account for the energy exchange between neighbouring groups which is important in flows with strong velocity divergence. These terms were computed using a finite volume method in the frequency domain. The radiative transfer aspect of the method was first tested separately for global consistency (reversion to grey model) and against a well-established kinetic model through Marshak wave tests with frequency-dependent opacities. Very good agreement between the multigroup M₁ and kinetic models was observed in all tests. The successful coupling of the multigroup radiative transfer to the hydrodynamics was then confirmed through a second series of tests. Finally, the model was linked to a database of opacities for a Xe gas in order to simulate realistic multigroup radiative shocks in Xe. The differences with the previous grey models are discussed.     

On the numerical characteristics of an inverse solution for three-term radiative transfer

Certain numerical characteristics of an inverse formulation for three-term scattering radiative transfer are investigated. Specifically, approximate solutions to the direct problem are constructed by the F_N and Monte Carlo methods, allowing approximation of the various surface angular moments and related quantities needed for the inverse calculation. Several numerical schemes are employed in order of demonstrate the computational characteristics for some specific phase functions. The numerical results indicate that the single-scatter albedo can be calculated fairly consistently and accurately, but higher order coefficients of the scattering law are more difficult to obtain by this method.     

Numerical solution of radiative and conductive heat transfer in concentric spherical and cylindrical media

A new technique is presented to improve the performance of the discrete ordinates method when solving the coupled conduction-radiation problems in spherical and cylindrical media. In this approach the angular derivative term of the discretized one-dimensional radiative transfer equation is derived from an expansion of the radiative intensity on the basis of Chebyshev polynomials. The set of resulting differential equations, obtained by the application of the S_N method, is numerically solved using the boundary value problem with the finite difference algorithm. Results are presented for the different independent parameters. Numerical results obtained using the Chebyshev transform method compare well with the benchmark approximate solutions. Moreover, the new technique can easily be applied to higher-order S_N calculations.     

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.     

Resistance-network representation of radiative heat transfer with particulate scattering

The resistance-network representation for radiative heat transfer is developed for a planar absorbing-scattering medium on the basis of the two-flux model and the linear anisotropic scattering model. Particular attention is given to the scattering effect due to particulates such as flame soot or smoke particles. Limiting relations for various radiative regimes are derived and the physical significance of the resistances are discussed. An illustrative example is presented for thermal radiation from a smoke layer. Extension to two-phase dispersed systems is also demonstrated.     

Combining the independent pixel and point-spread function approaches to simulate the actinic radiation field in moderately inhomogeneous 3D cloudy media

A fast method is presented for gaining 3D actinic flux density fields, F_act, in clouds employing the Independent Pixel Approximation (IPA) with a parameterized horizontal photon transport to imitate radiative smoothing effects. For 3D clouds the IPA is an efficient method to simulate radiative transfer, but it suffers from the neglect of horizontal photon fluxes leading to significant errors (up to locally 30% in the present study). Consequently, the resulting actinic flux density fields exhibit an unrealistically rough and rugged structure. In this study, the radiative smoothing is approximated by applying a physically based smoothing algorithm to the calculated IPA actinic flux field.