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Results for “Radiative transfer equation”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 361 records · Page 20
All sourcesNASA NTRSDOE OSTI
NASA NTRS2021

Fitting Strategies of Accretion Column Models and Application to the Broadband Spectrum of Cen X-3

Due to the complexity of modeling the radiative transfer inside the accretion columns of neutron star binaries, their X-ray spectra are still commonly described with phenomenological models, e.g., a cut off power law. While the behavior of these models is well understood and they allow for a comparison of different sources and studying source behavior, the extent to which the underlying physics can be derived from the model parameters is very limited. During recent years, several physically motivated spectral models have been developed to overcome these limitations. Their application, however, is generally computationally much more expensive and they require a high number of parameters which are difficult to constrain. In particular, Becker & Wolff(2007) presented an analytical solution to the radiative transfer equation inside the accretion column assuming a velocity profile that is linear in the optical depth. An implementation of this model that is both fast and accurate enough to be fitted to observed spectra is available in XSPEC. The main difficulty of this implementation is that some solutions violate energy conservation and therefore have to be rejected by the user. We propose a novel fitting strategy that ensures energy conservation during theχ2-minimization which simplifies the application of the model considerably. We demonstrate this approach as well as a study of possible parameter degeneracies with a comprehensive Markov-Chain Monte Carlo analysis of the complete parameter space for a combined NuSTAR and Swift/XRT data-set of Cen X-3.The derived accretion-flow structure features a small column radius of∼63 m and a spectrum dominated by bulk-Comptonization of bremsstrahlung seed photons, in agreement with previous studies

Philipp Thalhammer↗
NASA NTRS1982

Applicaton of radiative transfer theory to microwave transmission medium calibrations

Precise determinations of the transmission medium loss and noise temperature contribution which are important to the performance characterization of low noise microwave receiving systems and thermal noise standards are discussed. Tropospheric loss is frequently inferred from microwave radiometer noise temperature measurements. Interpretation of these measurements requires an inversion of the radiative transfer integral equation. This is inconvenient even with computer techniques. Solutions of a rapidly convergent power series of the radiative transfer equations are presented. This solution is applicable to a low loss medium with either uniform or nonuniform loss distributions. A four layer atmosphere model is investigated to demonstrate the accuracy of the solution relative to the model. Applications include thermal noise standards and single- and dual-frequency water radiometers.

Stelzried, C. T.↗
NASA NTRS1976

A new formulation for anisotropic radiative transfer problems. I - Solution with a variational technique

The equations of radiative transfer in anisotropically scattering media are reformulated as linear operator equations in a single independent variable. The resulting equations are suitable for solution by a variety of standard mathematical techniques. The operators appearing in the resulting equations are in general nonsymmetric; however, it is shown that every bounded linear operator equation can be embedded in a symmetric linear operator equation and a variational solution can be obtained in a straightforward way. For purposes of demonstration, a Rayleigh-Ritz variational method is applied to three problems involving simple phase functions. It is to be noted that the variational technique demonstrated is of general applicability and permits simple solutions for a wide range of otherwise difficult mathematical problems in physics.

Cheyney, H., III↗
NASA NTRS1991

Thermal protection for hypervelocity flight in earth's atmosphere by use of radiation backscattering ablating materials

A heat-shield-material response code predicting the transient performance of a material subject to the combined convective and radiative heating associated with the hypervelocity flight is developed. The code is dynamically interactive to the heating from a transient flow field, including the effects of material ablation on flow field behavior. It accomodates finite time variable material thickness, internal material phase change, wavelength-dependent radiative properties, and temperature-dependent thermal, physical, and radiative properties. The equations of radiative transfer are solved with the material and are coupled to the transfer energy equation containing the radiative flux divergence in addition to the usual energy terms.

Howe, John T.↗
NASA NTRS2006

Heat Transfer In High-Temperature Multilayer Insulation

The combined radiation/conduction heat transfer in high-temperature multilayer insulations for typical reentry of reusable launch vehicles from low Earth orbit was investigated experimentally and numerically. The high-temperature multilayer insulation investigated consisted of gold-coated reflective foils separated by alumina fibrous insulation spacers. The steady-state heat transfer through four multilayer insulation configurations was investigated experimentally over the temperature range of 300-1300 K and environmental pressure range of 1.33 10(exp -5)-101.32 kPa. It was shown that including the reflective foils reduced the effective thermal conductivity compared to fibrous insulation sample at 1.5 times the density of the multilayer sample. A finite volume numerical model was developed to solve the governing combined radiation/conduction heat transfer equations. The radiation heat transfer in the fibrous insulation spacers was modeled using the modified two-flux approximation assuming anisotropic scattering and gray medium. The numerical model was validated by comparison with steady-state experimental data. The root mean square deviation between the predicted and measured effective thermal conductivity of the samples was 9.5%.

Daryabeigi, Kamran↗
NASA NTRS2018

Overview of Methods for Deriving the Radiative Transfer Theory from the Maxwell Equations. I: Approach Based on the Far-Field Foldy Equations

In this paper, we revisit, with further enhancements and clarifications, the self-consistent first-principles approach developed previously for deriving the vector radiative transfer theory for a discrete random medium with a sparse concentration of particles. We specifically consider the case of a plane-parallel particulate layer embedded in an otherwise homogeneous unbounded medium. The solution method is based on the far-field Foldy equations, an order-of-scattering expansion for the total field derived under the Twersky approximation, the computation of the coherent field by assuming that the positions of the particles are uncorrelated, and the ladder approximation for the coherency dyadic. The latter yields an integral equation for the diffuse specific coherency dyadic, defined through an angular spectrum repre- sentation for the coherency dyadic, which in turn, gives the vector radiative transfer equation for the diffuse specific intensity column vector. We analyze specifically the computation of the coherent field for inhomogeneous particulate media and multiple species of particles, the continuous extension of the far-field representation to the near field, the Foldy approximation, and the Foldy integral equation for the coherent field. Finally, we discuss the transition from the vector to the scalar radiative transfer equation.

Doicu, Adrian↗
NASA NTRS1986

Radiative transfer in a sphere illuminated by a parallel beam - An integral equation approach

The problem of multiple scattering of nonpolarized light in a planetary body of arbitrary shape illuminated by a parallel beam is formulated using the integral equation approach. There exists a simple functional whose stationarity condition is equivalent to solving the equation of radiative transfer and whose value at the stationary point is proportional to the differential cross section. The analysis reveals a direct relation between the microscopic symmetry of the phase function for each scattering event and the macroscopic symmetry of the differential cross section for the entire planetary body, and the interconnection of these symmetry relations and the variational principle. The case of a homogeneous sphere containing isotropic scatterers is investigated in detail. It is shown that the solution can be expanded in a multipole series such that the general spherical problem is reduced to solving a set of decoupled integral equations in one dimension. Computations have been performed for a range of parameters of interest, and illustrative examples of applications to planetary problems as provided.

Shia, R.-L.↗
NASA NTRS1973

Matrix operator theory of radiative transfer. I - Rayleigh scattering.

An entirely rigorous method for the solution of the equations for radiative transfer based on the matrix operator theory is reviewed. The advantages of the present method are: (1) all orders of the reflection and transmission matrices are calculated at once; (2) layers of any thickness may be combined, so that a realistic model of the atmosphere can be developed from any arbitrary number of layers, each with different properties and thicknesses; (3) calculations can readily be made for large optical depths and with highly anisotropic phase functions; (4) results are obtained for any desired value of the surface albedo including the value unity and for a large number of polar and azimuthal angles; (5) all fundamental equations can be interpreted immediately in terms of the physical interactions appropriate to the problem; and (6) both upward and downward radiance can be calculated at interior points from relatively simple expressions.

Plass, G. N.↗
NASA NTRS1974

Efficient stream distributions in radiative transfer theory

This paper discusses a new, computationally-efficient method for approximating the integro-differential equation of radiative transfer with a finite set of coupled differential equations for discrete streams. The method uses recommended spatial distributions of streams that are quite different from those typically used in that they are based on the symmetry of several regular Platonic solids. To facilitate the use of such distributions, an explicit, one-parameter relationship between the physical radiance and the abstract stream is formulated. The parameter is used to determine the minimum number of streams required in the radiative transfer model. Accuracy and computational efficiency are shown to be served best by choosing a stream distribution that is invariant to a large number of three space rotations. For various values of the above-mentioned parameter, the resulting recommended stream distribution is shown to be more computationally efficient than more conventional stream distributions. Finally, the incorporation of polarization in the stream definition is described.

Whitney, C.↗
NASA NTRS1991

An analytic solution of the transient behavior of backscattering thermal protective coatings exposed to combined radiative and convective heating

An analytic solution of the material response to combined radiative and convective heating is presented. The solution includes the equations of radiative transfer (within the material), coupled to a transient energy equation which contains both radiative and convective terms. The analysis allows for unlimited spectral detail, but assumes that within the range of applicability, the various material properties do not vary significantly with temperature. Also, to facilitate development of the analytic solution, it is assumed that scattering within the material dominates absorption, and the material exposed surface does not ablate. The exposed surface boundary condition includes convective heating and spectral radiation, some of which is absorbed by the surface and some which penetrates the surface.

Cornelison, Charles J.↗
NASA NTRS1972

Implications of a quadratic stream definition in radiative transfer theory.

An explicit definition of the radiation-stream concept is stated and applied to approximate the integro-differential equation of radiative transfer with a set of twelve coupled differential equations. Computational efficiency is enhanced by distributing the corresponding streams in three-dimensional space in a totally symmetric way. Polarization is then incorporated in this model. A computer program based on the model is briefly compared with a Monte Carlo program for simulation of horizon scans of the earth's atmosphere. It is found to be considerably faster.

Whitney, C.↗
NASA NTRS1976

A tensor formulation of the equation of transfer for spherically symmetric flows

A tensor formulation of the equation of radiative transfer is derived in a seven-dimensional Riemannian space such that the resulting equation constitutes a divergence in any coordinate system. After being transformed to a spherically symmetric comoving coordinate system, the transfer equation contains partial derivatives in angle and frequency, as well as optical depth due to the effects of aberration and the Doppler shift. However, by virtue of the divergence form of this equation, the divergence theorem may be applied to yield a numerical differencing scheme which is expected to be stable and to conserve luminosity. It is shown that the equation of transfer derived by this method in a Lagrangian coordinate system may be reduced to that given by Castor (1972), although it is, of course, desirable to leave the equation in divergence form.

Haisch, B. M.↗
DOE OSTI2024

A reduced-order model for nonlinear radiative transfer problems based on moment equations and POD-Petrov-Galerkin projection of the normalized Boltzmann transport equation

A data-driven projection-based reduced-order model (ROM) for nonlinear thermal radiative transfer (TRT) problems is presented. The TRT ROM is formulated by (i) a hierarchy of low-order quasidiffusion (aka variable Eddington factor) equations for moments of the radiation intensity and (ii) the normalized Boltzmann transport equation (BTE). The multilevel system of moment equations is derived by projection of the BTE onto a sequence of subspaces which represent elements of the phase space of the problem. Exact closure for the moment equations is provided by the Eddington tensor. A Petrov-Galerkin (PG) projection of the normalized BTE is formulated using a proper orthogonal decomposition (POD) basis representing the normalized radiation intensity over the whole phase space and time. The Eddington tensor linearly depends on the solution of the normalized BTE. By linear superposition of the POD basis functions, a low-rank expansion of the Eddington tensor is constructed with coefficients defined by the PG projected normalized BTE. The material energy balance (MEB) equation is coupled with the effective gray low-order equations which exist on the same dimensional scale as the MEB equation. The resulting TRT ROM is structure and asymptotic preserving. A detailed analysis of the ROM is performed on the classical Fleck-Cummings (F-C) TRT multigroup test problem in 2D geometry. Numerical results are presented to demonstrate the ROM's effectiveness in the simulation of radiation wave phenomena. Importantly, the ROM is shown to produce solutions with sufficiently high accuracy while using low-rank approximation of the normalized BTE solution. Essential physical characteristics of supersonic radiation wave are preserved in the ROM solutions.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
NASA NTRS1991

Fragmentation in stellar collapse

A three dimensional Cartesian code was developed to model molecular cloud collapse and protostellar formation. The basic Eulerian scheme employs spatially second-order accurate, finite difference methods to advance the fluid variables in time. The hydrodynamic equations governing the collapse include the effects of self-gravity, rotation, and radiative transfer. Supplementary equations include Poisson's equation which relates the gravitational potential to the density distribution and a mean intensity equation derived with the Eddington approximation for radiative transfer. The gas pressures and internal energies are determined from equations of state derived for a molecular cloud composed of hydrogen, helium, and heavier elements.

Myhill, E. M.↗
NASA NTRS1985

Radiative Amplification of Acoustic Waves in Hot Stars

The discovery of broad P Cygni profiles in early type stars and the detection of X-rays emitted from the envelopes of these stars made it clear, that a considerable amount of mechanical energy has to be present in massive stars. An attack on the problem, which has proven successful when applied to late type stars is proposed. It is possible that acoustic waves form out of random fluctuations, amplify by absorbing momentum from stellar radiation field, steepen into shock waves and dissipate. A stellar atmosphere was constructed, and sinusoidal small amplitude perturbations of specified Mach number and period at the inner boundary was introduced. The partial differential equations of hydrodynamics and the equations of radiation transfer for grey matter were solved numerically. The equation of motion was augmented by a term which describes the absorption of momentum from the radiation field in the continuum and in lines, including the Doppler effect and allows for the treatment of a large number of lines in the radiative acceleration term.

Wolf, B. E.↗
DOE OSTI2023

Data-driven, structure-preserving approximations to entropy-based moment closures for kinetic equations

In this study, we present a data-driven approach for approximating entropy-based closures of moment systems from kinetic equations. The proposed closure learns the entropy function by fitting the map between the moments and the entropy of the moment system, and thus does not depend on the spacetime discretization of the moment system or specific problem configurations such as initial and boundary conditions. With convex and C 2 approximations, this data-driven closure inherits several structural properties from entropy-based closures, such as entropy dissipation, hyperbolicity, and H-Theorem. We construct convex approximations to the Maxwell–Boltzmann entropy using convex splines and neural networks, test them on the plane source benchmark problem for linear transport in slab geometry, and compare the results to the standard, entropy-based systems which solve a convex optimization problem to find the closure. Numerical results indicate that these data-driven closures provide accurate solutions in much less computation time than that required by the optimization routine.

97 MATHEMATICS AND COMPUTING↗