The vectorial radiative transfer equation problem in the small angle modification of the spherical harmonics method with the determination of the solution smooth part

2006 ◽  
Author(s):  
V. P. Budak ◽  
S. V. Korkin
Keyword(s):  
Radiative Transfer ◽  
Spherical Harmonics ◽  
Small Angle ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Equation Problem ◽  
Smooth Part

Solution of the Radiative Transfer Equation in Three-Dimensional Participating Media Using a Hybrid Discrete Ordinates: Spherical Harmonics Method

2012 ◽  
Vol 134 (11) ◽  
Author(s):  
Maathangi Sankar ◽  
Sandip Mazumder
Keyword(s):  
Monte Carlo ◽  
Radiative Transfer ◽  
Spherical Harmonics ◽  
Hybrid Method ◽  
Transfer Equation ◽  
Large Range ◽  
Radiative Transfer Equation ◽  
Three Dimensional ◽  
Discrete Ordinates ◽  
Medium Component

In this article, a new hybrid solution to the radiative transfer equation (RTE) is proposed. Following the modified differential approximation (MDA), the radiation intensity is first split into two components: a “wall” component, and a “medium” component. Traditionally, the wall component is determined using a viewfactor-based surface-to-surface exchange formulation, while the medium component is determined by invoking the first-order spherical harmonics (P1) approximation. Recent studies have shown that although the MDA approach is accurate over a large range of optical thicknesses, it is prohibitive for complex three-dimensional geometry with obstructions, both from a computational efficiency as well as memory standpoint. The inefficiency stems from the use of the viewfactor-based approach for determination of the wall-emitted component. In this work, instead, the wall component is determined directly using the control angle discrete ordinates method (CADOM). The new hybrid method was validated for both two-dimensional (2D) and three-dimensional (3D) geometries against benchmark Monte Carlo results for gray media in which the optical thickness was varied over a large range. In all cases, the accuracy of the hybrid method was found to be within a few percent of Monte Carlo results, and comparable to the solutions of the RTE obtained directly using CADOM. Finally, the new hybrid method was explored for 3D nongray media in the presence of reflecting walls and various scattering albedos. As a noteworthy advantage, irrespective of the conditions used, it was always found to be computationally more efficient than standalone CADOM and up to 15 times more efficient than standalone CADOM for optically thick media with strong scattering.

scholarly journals Combining magnetohydrostatic constraints with Stokes profiles inversions

2019 ◽  
Vol 632 ◽  
pp. A111 ◽  
Author(s):  
J. M. Borrero ◽  
A. Pastor Yabar ◽  
M. Rempel ◽  
B. Ruiz Cobo
Keyword(s):  
Magnetic Field ◽  
Radiative Transfer ◽  
Boundary Conditions ◽  
Optical Depth ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Three Dimensional ◽  
Gas Pressure ◽  
Order Of Magnitude

Context. Inversion codes for the polarized radiative transfer equation, when applied to spectropolarimetric observations (i.e., Stokes vector) in spectral lines, can be used to infer the temperature T, line-of-sight velocity vlos, and magnetic field B as a function of the continuum optical-depth τc. However, they do not directly provide the gas pressure Pg or density ρ. In order to obtain these latter parameters, inversion codes rely instead on the assumption of hydrostatic equilibrium (HE) in addition to the equation of state (EOS). Unfortunately, the assumption of HE is rather unrealistic across magnetic field lines, causing estimations of Pg and ρ to be unreliable. This is because the role of the Lorentz force, among other factors, is neglected. Unreliable gas pressure and density also translate into an inaccurate conversion from optical depth τc to geometrical height z. Aims. We aim at improving the determination of the gas pressure and density via the application of magnetohydrostatic (MHS) equilibrium instead of HE. Methods. We develop a method to solve the momentum equation under MHS equilibrium (i.e., taking the Lorentz force into account) in three dimensions. The method is based on the iterative solution of a Poisson-like equation. Considering the gas pressure Pg and density ρ from three-dimensional magnetohydrodynamic (MHD) simulations of sunspots as a benchmark, we compare the results from the application of HE and MHS equilibrium using boundary conditions with different degrees of realism. Employing boundary conditions that can be applied to actual observations, we find that HE retrieves the gas pressure and density with an error smaller than one order of magnitude (compared to the MHD values) in only about 47% of the grid points in the three-dimensional domain. Moreover, the inferred values are within a factor of two of the MHD values in only about 23% of the domain. This translates into an error of about 160 − 200 km in the determination of the z − τc conversion (i.e., Wilson depression). On the other hand, the application of MHS equilibrium with similar boundary conditions allows determination of Pg and ρ with an error smaller than an order of magnitude in 84% of the domain. The inferred values are within a factor of two in more than 55% of the domain. In this latter case, the z − τc conversion is obtained with an accuracy of 30 − 70 km. Inaccuracies are due in equal part to deviations from MHS equilibrium and to inaccuracies in the boundary conditions. Results. Compared to HE, our new method, based on MHS equilibrium, significantly improves the reliability in the determination of the density, gas pressure, and conversion between geometrical height z and continuum optical depth τc. This method could be used in conjunction with the inversion of the radiative transfer equation for polarized light in order to determine the thermodynamic, kinematic, and magnetic parameters of the solar atmosphere.

ΕΞΕΤΑΣΗ ΦΑΙΝΟΜΕΝΩΝ ΘΕΡΜΙΚΗΣ ΑΚΤΙΝΟΒΟΛΙΑΣ ΣΕ ΕΣΤΙΕΣ ΣΤΕΡΕΟΥ ΚΑΥΣΙΜΟΥ

1998 ◽  
Author(s):  
Ιωάννης Μαράκης
Keyword(s):  
Heat Transfer ◽  
Radiative Transfer ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Operating Conditions ◽  
Radiative Properties ◽  
Band Model ◽  
Combustion Chambers ◽  
Practical Applications

THEMATIC AREA OF THIS THESIS IS THE HEAT TRANSFER IN COMBUSTION CHAMBERS. THE ORIGINALITY ITEMS ARE CONCERNED WITH THE DEVELOPMENT OF ACCURATE METHODS BOTH FOR THE CALCULATION OF THE FLUE GAS AND COMBUSTION PARTICLE RADIATIVE PROPERTIES, AS WELL AS THE SOLUTION OF THE RADIATIVE TRANSFER EQUATION IN FURNACE - LIKE ENCLOSURES. SPECIFICALLY, THIS WORK CONTRIBUTES TO THE EXACT DETERMINATION OF THE INFLUENCE THAT THE TEMPERATURE AND PRESSURE OPERATING CONDITIONS HAVE ON THE RADIATIVE FLUXES AND SOURCE TERMS, THE LATTER BEING THE NET THERMAL ENERGY EMITTED OR ABSORBED PER UNIT VOLUME. THE THESIS INCLUDES THE DEVELOPMENT OF TWO METHODS FOR THE SOLUTION OF THE RADIATIVE TRANSFER EQUATION (A MONTE CARLO VARIANT AND A NEW INTEGRAL METHOD NAMED DIRECT NUMERICAL INTEGRATION),TWO STATISTICAL NARROW BAND AND A WIDE BAND MODEL FOR THE CALCULATION OF THE NON - GRAY GAS SPECTRAL TRANSMISSIVITY, AN ALGORITHM BASED ON MIE THEORY FOR THE DETERMINATION OF THE ABSORPTION AND SCATTERING COEFFICIENTS, THE PHASE FUNCTION AND THE ASYMMETRY PARAMETER OF COAL, CHAR, FLY - ASH AND SOOT PARTICLES AND CORRELATIONS FOR THE RESPECTIVE SPECTRAL OPTICAL PROPERTIES. THE EXACT SOLUTION OF THE THERMAL RADIATION TRANSFER HAS SIGNIFICANT PRACTICAL APPLICATIONS, SUCH AS: 1) DESIGN OF COMBUSTION CHAMBERS AND HEAT TRANSFER SURFACES, 2) DETERMINATION OF THE RADIATIVE FLUX AT THE BOUNDARIES OF A GIVEN GEOMETRY (ABSTRACT TRUNCATED)

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