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.     

The Pomraning-Eddington approximation to diffusion of light in turbid materials

The source-free diffusion problem of light in turbid media with generalized boundary conditions is considered. The intensity of light is considered as a sum of collimated and diffused radiance. In this way the problem is transformed to a source problem with a collimated source (problem 1). This problem is solved in terms of the corresponding source-free problem of simple boundary conditions (problem 2). The Pomraning-Eddington method is used to solve problem 2. Two coupled first-order differential equations are obtained involving the energy density and the radiation net flux. Weight functions are introduced in order to force the boundary conditions to be fulfilled. Numerical results are given and compared with previous calculations. The calculations show that the accuracy depends on the choice of the weight function.     

Sparse tensor spherical harmonics approximation in radiative transfer

The stationary monochromatic radiative transfer equation is a partial differential transport equation stated on a five-dimensional phase space. To obtain a well-posed problem, boundary conditions have to be prescribed on the inflow part of the domain boundary.     

Tau method approximation for radiative transfer problems in a slab medium

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.     

Equivalence of MTF of a turbid medium and radiative transfer field

The equivalence of the modulation transfer function (MTF) of a turbid medium and the transmitted radiance from the medium under isotropic diffuse illumination is demonstrated. MTF of a turbid medium can be fully evaluated by numerically solving a radiative transfer problem in a plane parallel medium. MTF for a homogenous single layer turbid medium is investigated as illustration. General features of the MTF in the low and high spatial frequency domains are provided through their dependence on optical thickness, single scattering albedo, asymmetrical factor, and phase function type.     

Heat transfer in a spherical turbid medium with conduction and radiation

The heat transfer through a spherical media with conduction and radiation is considered. The medium is considered to be turbid and anisotropically scattering with diffusely reflecting boundaries of constant temperatures. The radiative transfer problem is solved using the Galerkin method. An iterative method is used to solve the nonlinear relation between the radiative transfer equation and the conductive energy equation. Calculations are carried out and compared for a homogeneous, isotropically scattering medium with isothermal, transparent boundaries. The results show good agreement with previous work. Calculations are carried out for inhomogeneous media with isotropic, and forward and backward anisotropic scattering. The boundaries of the media are considered to be isothermal and may be transparent or diffusely reflecting boundaries. The calculations are used to study the effects of the single scattering albedo, the anisotropic scattering parameter, the conduction-radiation parameter and the heat source.     

Comparison between radiative transfer theory and the simplified spherical harmonics approximation for a semi-infinite geometry

In this study, the third-order simplified spherical harmonics equations (SP3), an approximation of the radiative transfer equation, are solved for a semi-infinite geometry considering the exact simplified spherical harmonics boundary conditions. The obtained Green's function is compared to radiative transfer calculations and the diffusion theory. In general, it is shown that the SP3 equations provide better results than the diffusion approximation in media with high absorption coefficient values but no improvement is found for small distances to the source.     

The differential approximation for radiative transfer in an emitting,absorbing and anisotropically scattering medium

The validity of the well-established differential approximation in radiative transfer is extended to include linear-anisotropic scattering. Sample results demonstrate the accuracy of the method.     

On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering

The authors developed a numerical method of the boundary-value problem solution in the vectorial radiative transfer theory applicable to the turbid media with an arbitrary three-dimensional geometry. The method is based on the solution representation as the sum of an anisotropic part that contains all the singularities of the exact solution and a smooth regular part. The regular part of the solution could be found numerically by the finite element method that enables to extend the approach to the arbitrary medium geometry. The anisotropic part of the solution is determined analytically by the special form of the small-angle approximation. The method development is performed by the examples of the boundary-value problems for the plane unidirectional and point isotropic sources in a turbid medium slab.     

Pomraning-Eddington approximation for solving the Spencer-Lewis equation

By means of Pomraning-Eddington approximation and maximum entropy method the Spencer-Lewis equation in an infinite medium has been solved explicitly. The behaviour of the approximate solution for the total electron density are shown graphically, and compared with that obtained by using the flux-limited approach. The results reported in this article provide further evidence of the usefulness of both Pomraning-Eddington and flux-limited. Knowing the electron density distribution allows us to calculate directly some physical quantity, such as the reflection function and energy deposition profile.     

Theoretical study of combined radiative conductive and convective heat transfer in semi-transparent porous medium in a spherical enclosure

A new method for the solution of the radiative transfer equation in spherical media based on a modified discrete ordinates method is extended to study radiative, conductive and convective heat transfer in a semi-transparent scattering porous medium. The set of differential equations is solved using the fourth-order Runge-Kutta method. Various results are obtained for the case of combined radiative and conductive heat transfer, as well as for the interaction of those modes with convection. The effects of some radiative properties of the medium on the heat transfer rate are examined.     

Two-dimensional radiative transfer in a finite scattering planar medium

The source function, radiative flux, and intensity at the boundaries are calculated for a two-dimensional, scattering, finite medium subjected to collimated radiation. The scattering phase function is composed of a spike in the forward direction super-imposed on an isotropic background. Exact radiative transfer theory is used to formulate the problem and Ambarzumian's method is used to obtain results. Using the principle of superposition, the results for any step variation in incident radiation are expressed in terms of universal functions for the semi-infinite step case. Two-dimensional effects are most pronounced at large optical thicknesses and albedos.     

Nongray radiative transfer in a semi-infinite medium: H-function

Numerical results are presented for the dimensionless emissive power at the boundary of a nongray semi-infinite medium in radiative equilibrium. The adsorption coefficient of an array of equal intensity, nonoverlapping bands or lines. Specifically, the rectangular, triangular, exponential, Doppler and Lorentz profiles are considered.     

Simulating radiative transfer with filtered spherical harmonics

This Letter presents a novel application of filters to the spherical harmonics (PN) expansion for radiative transfer problems in the high-energy-density regime. The filter, which is based on non-oscillatory spherical splines, preserves both the equilibrium diffusion limit and formal convergence properties of the unfiltered expansion. While the method requires further mathematical justification and computational studies, preliminary results demonstrate that solutions to the filtered PN equations are (1) more robust and less oscillatory than standard PN solutions and (2) more accurate than discrete ordinates solutions of comparable order. The filtered P₇ solution demonstrates comparable accuracy to an implicit Monte Carlo solution for a benchmark hohlraum problem. Given the benefits of this method we believe it will enable more routine use of high-fidelity radiation-hydrodynamics calculations in the simulation of physical systems.     

Non-grey radiative heat transfer in the picket-fence approximation

The radiative heat transfer between parallel plates separated by an absorbing gas with a “picket-fence” absorption coefficient is treated by a variational technique. For the heat transfer, a simple rational algebraic expression is obtained and the variational results are found in excellent agreement with the numerically “exact” results reported recently by Reithet al.(⁸) An expression for the extrapolation distance is also deduced and the results are compared with the values reported earlier by Bond and Siewert.(⁵)     

A direct method for the integration of the equation of radiative transfer in a turbid atmosphere

In this paper we apply a numerical method on the equation of radiative transfer in a turbid atmosphere. The solution is obtained by means of direct integration of the equation of radiative transfer without any circuitous series development.     

Robust and accurate filtered spherical harmonics expansions for radiative transfer

We present a novel application of filters to the spherical harmonics (P_N) expansion for radiative transfer problems in the high-energy-density regime. The filter we use is based on non-oscillatory spherical splines and a filter strength chosen to (i) preserve the equilibrium diffusion limit and (ii) vanish as the expansion order tends to infinity. Our implementation is based on modified equations that are derived by applying the filter after every time step in a simple first-order time integration scheme. The method is readily applied to existing codes that solve the P_N equations. Numerical results demonstrate that the solution to the filtered P_N equations are (i) more robust and less oscillatory than standard P_N solutions and (ii) more accurate than discrete ordinates solutions of comparable order. In particular, the filtered P₇ solution demonstrates comparable accuracy to an implicit Monte Carlo solution for a benchmark hohlraum problem in 2D Cartesian geometry.     

Towards linearization of atmospheric radiative transfer in spherical geometry

We present a general approach for the linearization of radiative transfer in a spherical planetary atmosphere. The approach is based on the forward-adjoint perturbation theory. In the first part we develop the theoretical background for a linearization of radiative transfer in spherical geometry. Using an operator formulation of radiative transfer allows one to derive the linearization principles in a universally valid notation. The application of the derived principles is demonstrated for a radiative transfer problem in simplified spherical geometry in the second part of this paper. Here, we calculate the derivatives of the radiance at the top of the atmosphere with respect to the absorption properties of a trace gas species in the case of a nadir-viewing satellite instrument.