Two-dimensional Radiative Transfer Equation Research Articles

On the application of the ADO method to the solution of two-dimensional radiative transfer problems in anisotropic scattering media

In this work, the Analytical Discrete Ordinates (ADO) method is applied to the solution of the discrete ordinates approximation of the two-dimensional radiative transfer equation in Cartesian geometry. The phase function is expanded in terms of Legendre polynomials up to arbitrary order L to model higher-order anisotropy. The discrete ordinates equations are transversally integrated over regions of the domain, yielding one-dimensional equations for average angular intensities in x and y-directions. The ADO method is then applied to the one-dimensional equations, with approximations for the unknown intensities on the contours of the regions, to derive explicit solutions for the spatial variables. This approach, based on the use of the ADO method along with the nodal technique, is usually referred to as ADO-Nodal. The scheme does not use sweeping. Linear systems have to be solved to fully establish the general solutions. Here, using a recent rearrangement of those linear systems, numerical results are provided for more refined domain partitions, into regions, than previously. Higher-order quadrature schemes are also explored. Comparisons with few results available in the literature reinforce the good performance of the method. A detailed analysis is carried out to contribute in the direction of obtaining reference solutions for the class of problems investigated. Furthermore, numerical results for different optical thicknesses and section average intensities are included, that according to our knowledge it has not been presented before in the literature.

Energy density distribution of light inside two-dimensional randomly scattering slabs illuminated by a plane wave

In this study, light scattering inside random two-dimensional media, illuminated by an incident plane wave, is simulated by numerically solving Maxwell’s equations. The results are compared to the corresponding, newly derived analytical solution of the radiative transfer equation (RTE) for a laterally infinitely extended slab. The two-dimensional scattering medium consists of a random distribution of circular scatterers with well-defined optical properties and area fraction. The ensemble-averaged energy density calculated from the electromagnetic fields obtained from Maxwell’s equations is compared with the energy density obtained from the two-dimensional radiative transfer equation. Simulations for different area fractions show good agreement of the energy densities of the two theories for small area fractions ( < 0.2) of the scatterers. For low concentrations, three different slab thicknesses are examined. The ensemble-averaged energy density calculated with Maxwell’s equations show good agreement for all three slab thicknesses compared with the energy density obtained with the RTE. The results of this study can be used to calculate the average energy density distribution of light inside slabs of fiber-like structures with fast calculations of the presented analytical RTE solution.

Errata: Applying a new computational method for biological tissue optics based on the time-dependent two-dimensional radiative transfer equation

Errata: Applying a new computational method for biological tissue optics based on the time-dependent two-dimensional radiative transfer equation

Applying a new computational method for biological tissue optics based on the time-dependent two-dimensional radiative transfer equation

Optical tomography is a medical imaging technique based on light propagation in the near infrared (NIR) part of the spectrum. We present a new way of predicting the short-pulsed NIR light propagation using a time-dependent two-dimensional-global radiative transfer equation in an absorbing and strongly anisotropically scattering medium. A cell-vertex finite-volume method is proposed for the discretization of the spatial domain. The closure relation based on the exponential scheme and linear interpolations was applied for the first time in the context of time-dependent radiative heat transfer problems. Details are given about the application of the original method on unstructured triangular meshes. The angular space (4πSr) is uniformly subdivided into discrete directions and a finite-differences discretization of the time domain is used. Numerical simulations for media with physical properties analogous to healthy and metastatic human liver subjected to a collimated short-pulsed NIR light are presented and discussed. As expected, discrepancies between the two kinds of tissues were found. In particular, the level of light flux was found to be weaker (inside the medium and at boundaries) in the healthy medium than in the metastatic one.

Green's functions for the two-dimensional radiative transfer equation in bounded media

In this study, the two-dimensional radiative transfer equation is solved considering the exact boundary conditions for describing the propagation of particles in bounded anisotropically scattering media. The obtained Green's functions have an analytical dependence on the spatial and angular variables and, therefore, can be evaluated rapidly and implemented without any computational complexities. The Monte Carlo method was used for a successful verification of the derived solutions.

Analytical approach for solving the radiative transfer equation in two-dimensional layered media

This study presents an analytical approach for obtaining Green's function of the two-dimensional radiative transfer equation to the boundary-value problem of a layered medium. A conventional Fourier transform and a modified Fourier series which is defined in a rotated reference frame are applied to derive an analytical solution of the radiance in the transformed space. The Monte Carlo method was used for a successful validation of the derived solutions.

Radiative transfer in two-dimensional infinitely extended scattering media

In this study, Green’s function of the two-dimensional radiative transfer equation is derived for an infinitely extended anisotropically scattering medium, which is illuminated by a unidirectional source distribution. In the steady-state domain, the final results, which are based on eigenvalues and eigenvectors, are given analytically apart from the eigenvalues. For the time-dependent case an additional numerical inverse Fourier transform is required. The obtained solutions were successfully validated with another exact analytical solution in the time domain for isotropically scattering and with the Monte Carlo method for anisotropically scattering media.

Effects of Radiative Transfer Modeling on Transient Temperature Distribution in Semitransparent Glass Rod

This paper presents a method of modeling the radiative energy transfer that takes place during the transient of joining two concentric, semitransparent glass cylinders. Specifically, we predict the two-dimensional transient temperature and heat flux distributions to a ramp input which advances the cylinders into a furnace at high temperature. In this paper, we discretize the fully conservative form of two-dimensional Radiative Transfer Equation (RTE) in both curvilinear and cylindrical coordinate systems so that it can be used for arbitrary axisymmetric cylindrical geometry. We compute the transient temperature field using both the Discrete Ordinate Method (DOM) and the widely used Rosseland’s approximation. The comparison shows that Rosseland’s approximation fails badly near the gap inside the glass media and when the radiative heat flux is dominant at short wavelengths where the spectral absorption coefficient is relatively small. Most prior studies of optical fiber drawing processes at the melting point (generally used Myers’ two-step band model at room temperature) neglect the effects of the spectral absorption coefficient at short wavelengths λ&lt;3μm. In this study, we suggest a modified band model that includes the glass absorption coefficient in the short-wavelength band. Our results show that although the spectral absorption coefficient at short wavelengths is relatively small, its effects on the temperature and heat flux are considerable.

Structure and Instabilities of an Irradiated Viscous Protoplanetary Disk

We investigate the structure and instabilities of a protoplanetary disk that is heated by a viscous process in itself and by its central star. The disk is set to rotate with Keplerian velocity and has the surface density distribution of a minimum-mass solar nebula. We assume vertical hydrostatic equilibrium and radiative equilibrium at each point and solve the two-dimensional radiative transfer equation by means of the short characteristics method in spherical coordinates in order to determine the disk structure. Our calculation shows that at the outer region of the disk, with a distance from the central star of x > 1 AU, the radiative heating from the inner disk dominates the viscous heating, even near the midplane. This is because of the high temperature distribution in the optically thin surface layer and the relatively high disk height (z∞ ~ 0.7x at x ~ 1 AU) as a consequence of the irradiation from the inner hot region of the disk. In addition, we examine the convective and the magnetorotational instabilities of the disk. We find that the whole disk is convectively stable, since the dusty region is not heated by the viscous dissipation from the midplane, but by the radial radiative heating. On the other hand, almost all the disk is magnetorotationally unstable, except for the region near the equatorial plane of 2 AU < x < 10 AU. Finally, we discuss the growth and the size distribution of dust particles in the disk, which suggest that there exist centimeter-sized particles in the surface layer, namely, in the exposed region of the disk.