L. B. Barichello

A discrete-ordinates solution for a non-grey model with complete frequency redistribution

Abstract The discrete-ordinates method is used to develop a solution to a class of non-grey problems in the theory of radiative transfer. The model considered allows for scattering with complete frequency redistribution (completely non-coherent scattering) and continuum absorption. In addition to a general formulation for semi-infinite and finite plane-parallel media, specific computations, for both the Doppler and the Lorentz profiles of the line-scattering coefficient, are discussed in regard to a half-space application concerning a linearly varying Planck function and also in regard to a basic problem from which, except for the conservative case, the classical X and Y functions can be extracted.

Particular solutions for the discrete-ordinates method

Abstract A full-range orthogonality relation is developed and used to construct the infinite-medium Green’s function for a general form of the discrete-ordinates approximation to the transport equation in plane geometry. The Green’s function is then used to define a particular solution that is required in the solution of inhomogeneous versions of the discrete-ordinates equations.

The temperature-jump problem in rarefied-gas dynamics

An analytical version of the discrete-ordinates method is used here to solve the classical temperature-jump problem based on the BGK model in rarefied-gas dynamics. In addition to a complete development of the discrete-ordinates method for the application considered, the computational algorithm is implemented to yield very accurate results for the temperature jump and the complete temperature and density distributions in the gas. The algorithm is easy to use, and the developed code runs typically in less than a second on a 400 MHz Pentium-based PC.

The temperature-jump problem for a variable collision frequency model

An analytical version of the discrete-ordinates method is used here in the field of rarefied-gas dynamics to solve a version of the temperature-jump problem that is based on a linearized, variable collision frequency model of the Boltzmann equation. In addition to a complete development of the discrete-ordinates method for the application considered, the computational algorithm is implemented to yield accurate numerical results for three specific cases: the classical BGK model, the Williams model (the collision frequency is proportional to the magnitude of the velocity), and the rigid-sphere model.

Some comments on modeling the linearized Boltzmann equation

Abstract Some exact solutions of the homogeneous and the inhomogeneous linearized Boltzmann equation (LBE) for rigid-sphere collisions are used to define two model equations in the general area of rarefied-gas dynamics. These equations are obtained from a systematic development of two synthetic scattering kernels that yield model equations that have as exact solutions certain known exact solutions of the homogeneous and of the inhomogeneous LBE. The first model established is defined in terms of the collisional invariants and the Chapman–Enskog integral equations for viscosity and for heat conduction. An extended model is defined also in terms of the collisional invariants and the Chapman–Enskog functions for viscosity and heat conduction, but the first and second Burnett functions are also included in the model. The variable collision frequency or generalized BGK model is also obtained as a special case. In addition, the exact mean-free paths defined, for rigid-sphere collisions and the LBE, in terms of viscosity or heat conduction are employed to define approximations of these quantities that are consistent with the use of the variable collision frequency model.

An analytical solution for the multigroup slab geometry discrete ordinates problems

Abstract In this work an analytical formulation is presented for the solution of the multigroup discrete ordinates problem in planar geometry with isotropic scattering. The approach is based on the application of the Laplace transform to the discrete ordinates equations and analytical inversion. Numerical results are presented for a test problem.

Solution of the three-dimensional one-group discrete ordinates problems by the LTSn method

Abstract In this work, the LTS n formulation is extended to the three-dimensional discrete ordinates problems in cartesian geometry. Numerical simulations are presented for problems with cylindrical geometry.

A SPHERICAL-HARMONICS SOLUTION FOR RADIATIVE-TRANSFER PROBLEMS WITH REFLECTING BOUNDARIES AND INTERNAL SOURCES

Abstract The spherical-harmonics method, including some recent improvements, is used to establish the complete solution for a general problem concerning radiative transfer in a plane-parallel medium. An L-th order Legendre expansion of the phase function is allowed, internal sources and reflecting boundaries are included in the model, and since a non-normally incident beam is impinging on one surface, all components in a Fourier decomposition of the intensity are required in the solution. Numerical results for two test problems are reported.

An analytical discrete-ordinates solution for dual-mode heat transfer in a cylinder

Abstract A modern analytical version of the discrete-ordinates method is used along with Hermite cubic splines and Newtons method to solve a class of coupled nonlinear radiation–conduction heat-transfer problems in a solid cylinder. Computational details of the solution are discussed, and the algorithm is implemented to establish high-quality results for various data sets which include some difficult cases.

On inverse boundary-condition problems in radiative transfer

Abstract Some elementary computations are reported to suggest that a certain type of inverse boundary-condition problem in radiative transfer can, in some cases, be solved quite simply.