Rarefied gas flow between parallel plates based on the discrete ordinate method.
Rarefied gas flow between parallel plates based on discrete ordinate method
GAS FLOW↗
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Modifications Of Discrete Ordinate Method For Computations With High Scattering Anisotropy: Comparative Analysis
A numerical accuracy analysis of the radiative transfer equation (RTE) solution based on separation of the diffuse light field into anisotropic and smooth parts is presented. The analysis uses three different algorithms based on the discrete ordinate method (DOM). Two methods, DOMAS and DOM2+, that do not use the truncation of the phase function, are compared against the TMS-method. DOMAS and DOM2+ use the Small-Angle Modification of RTE and the single scattering term, respectively, as an anisotropic part. The TMS method uses Delta-M method for truncation of the phase function along with the single scattering correction. For reference, a standard discrete ordinate method, DOM, is also included in analysis. The obtained results for cases with high scattering anisotropy show that at low number of streams (16, 32) only DOMAS provides an accurate solution in the aureole area. Outside of the aureole, the convergence and accuracy of DOMAS, and TMS is found to be approximately similar: DOMAS was found more accurate in cases with coarse aerosol and liquid water cloud models, except low optical depth, while the TMS showed better results in case of ice cloud.
Korkin, Sergey V.↗
Some uses of discrete ordinates codes for shielding analysis at NASA, Lewis Research Center
Discrete ordinates codes used to determine secondary gamma sources in reactor assemblies and shields
Lahti, G. P.↗
Discrete ordinate theory of radiative transfer. 2: Scattering from maritime haze
Discrete ordinate theory was used to calculate the reflected and transmitted radiance of photons which have interacted with plane parallel maritime haze layers. The results are presented for three solar zenith angles, three values of the surface albedo, and a range of optical thicknesses from very thin to very thick. The diffuse flux at the lower boundary and the cloud albedo were tabulated. The forward peak and other features in the single scattered phase function caused the radiance in many cases to be very different from that for Rayleigh scattering. The variation of the radiance with both the zenith or nadir angle and the azimuthal angle is more marked, and the relative limb darkening under very thick layers is greater, for haze than for Rayleigh scattering. The downward diffuse flux at the lower boundary for A = O is always greater and the cloud albedo is always less for haze than for Rayleigh layers.
Kattawar, G. W.↗
The discrete ordinate method for unsteady linearized Boltzmann Bhatnagar-Gross-Krook equation.
Discrete ordinate method for unsteady linearized Boltzmann Bhatnagar-Gross-Krook equation of rarefied gas flow
KROOK EQUATION↗
An evaluation of three spatial differencing schemes for the discrete ordinates method in participating media
Three popular spatial differencing schemes for the discrete ordinates method are examined for two-dimensional Cartesian coordinates system. These are a positive, the step, and the diamond schemes. Contrary to the common belief that negative intensities will not occur when fine spatial discretization is used with the diamond scheme, under certain conditions, the diamond scheme will produce negative intensities irrespective of the number of control volums employed. The positive scheme can produce physically unrealistic trends. The diamond and positive schemes are also capable of producing physically unrealistic overshoots. In absorbing-emitting or absorbing-emitting-scattering media, grid refinement can result in negative intensities when the diamond or positive scheme is used.
Chai, John C.↗
The discrete ordinate method for the linearized boundary value problems in kinetic theory of gases.
Discrete ordinate method for boundary value problems in gas dynamics using differential equations yields solutions to Couette flow
BOUNDARY VALUE↗
Comparison of spherical harmonics method and discrete ordinates method for radiative transfer in a turbulent jet flame
Here, in this study, we systematically compared the accuracy and computational cost of two popular solution methods for the radiative transfer equation (RTE): the spherical harmonics method (P N ) and the discrete ordinates method (DOM). We first investigated convergence characteristics of different orders of P N and DOM in a series of 1D homogeneous configurations with varying optical thicknesses. Both solvers perform better for optically thicker cases. The accuracy of P N methods increases with its order, , but the gain in accuracy reduces with the increase in , i.e., improvement of P 7 over P 5 is less than that of P 3 over P 1 . This decreasing trend becomes more prominent as the optical thickness decreases. On the other hand, DOM’s accuracy increases almost linearly with the increase in the number of ordinates (or polar angles in this study) in all cases. While comparing the directional profile of radiative intensity, both solvers perform better when the radiative intensity is more isotropic. These solvers were then connected with a full spectrum k-distribution (FSK) spectral model and used to perform radiation-coupled simulations of a turbulent jet flame in an axi-symmetric cylindrical domain. Results obtained from P 1 to P 7 approximations for P N , and 2 x 4, 4 x 4, 4 x 8, 8 x 8 finite angles for DOM are compared with that from an optically thin model, and a reference solution from line-by-line (LBL) photon Monte Carlo (PMC) method. The choice of radiation solver shows a noticeable impact on the temperature distribution of the flame. The P N solvers lead to slightly higher radiant fractions and the DOM solvers lead to slightly lower radiant fractions than the PMC benchmark solution. Finally, the computational costs of each of these solvers are also reported and an intermittent evaluation / time blending scheme to improve the computational efficiency of radiation solvers in radiation-coupled simulations are also demonstrated.
42 ENGINEERING↗
Finite element discrete ordinates method for radiative transfer in non-rotationally invariant scattering media - Application to the leaf canopy problem
A finite element discrete ordinates method for solving the radiative transfer equation in nonrotationally invariant scattering media has been applied to the lead-canopy problem, and results are presented on the cross sections and the reflection functions. The method is based on a unique implementation of the Galerkin integral law formulation of the transport equation. For both near-normal and grazing incidences, the transfer functions of leaf canopies are found to be strongly anisotropic, with relatively more scattered flux in the vertical directions. It is suggested that the assumption of isotropic scattering in leaf canopies is not valid.
Myneni, Ranga B.↗
Gamma-Weighted Discrete Ordinate Two-Stream Approximation for Computation of Domain Averaged Solar Irradiance
An algorithm is developed for the gamma-weighted discrete ordinate two-stream approximation that computes profiles of domain-averaged shortwave irradiances for horizontally inhomogeneous cloudy atmospheres. The algorithm assumes that frequency distributions of cloud optical depth at unresolved scales can be represented by a gamma distribution though it neglects net horizontal transport of radiation. This algorithm is an alternative to the one used in earlier studies that adopted the adding method. At present, only overcast cloudy layers are permitted.
Kato, S.↗
Application of the method of discrete ordinates to the solution of the transport equation in the case of an inhomogeneous planetary atmosphere
A version of the method of discrete ordinate (MDO) which can be used for an optically thick inhomogeneous atmosphere was developed. The MDO may be implemented as an algorithm simpler than those proposed earlier. The algorithm is the same for each azimuth dependent component of intensity. The MDO can determine the scattered radiation field over its entire length along the vertical. It is shown the MDO, when used for homogeneous atmospheres is in agreement with the results obtained by other methods.
Zasova, L. V.↗
General Purpose Fortran Program for Discrete-Ordinate-Method Radiative Transfer in Scattering and Emitting Layered Media: An Update of DISORT
This update reports a state-of-the-art discrete ordinate algorithm for monochromatic unpolarized radiative transfer in non-isothermal, vertically inhomogeneous, but horizontally homogeneous media. The physical processes included are Planckian thermal emission, scattering with arbitrary phase function, absorption, and surface bidirectional reflection. The system may be driven by parallel or isotropic diffuse radiation incident at the top boundary, as well as by internal thermal sources and thermal emission from the boundaries. Radiances, fluxes, and mean intensities are returned at user-specified angles and levels. DISORT has enjoyed considerable popularity in the atmospheric science and other communities since its introduction in 1988. Several new DISORT features are described in this update: intensity correction algorithms designed to compensate for the 8-M forward-peak scaling and obtain accurate intensities even in low orders of approximation; a more general surface bidirectional reflection option; and an exponential-linear approximation of the Planck function allowing more accurate solutions in the presence of large temperature gradients. DISORT has been designed to be an exemplar of good scientific software as well as a program of intrinsic utility. An extraordinary effort has been made to make it numerically well-conditioned, error-resistant, and user-friendly, and to take advantage of robust existing software tools. A thorough test suite is provided to verify the program both against published results, and for consistency where there are no published results. This careful attention to software design has been just as important in DISORT's popularity as its powerful algorithmic content.
Tsay, Si-Chee↗
A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogeneous, emitting and scattering atmosphere. I - Theory. II - Application
A theory is developed for discretizing the vector integro-differential radiative transfer equation including both solar and thermal radiation. A complete solution and boundary equations are obtained using the discrete-ordinate method. An efficient numerical procedure is presented for calculating the phase matrix and achieving computational stability. With natural light used as a beam source, the Stokes parameters from the model proposed here are compared with the analytical solutions of Chandrasekhar (1960) for a Rayleigh scattering atmosphere. The model is then applied to microwave frequencies with a thermal source, and the brightness temperatures are compared with those from Stamnes'(1988) radiative transfer model.
Weng, Fuzhong↗
Discrete ordinate solution of the radiative transfer equation in the 'polarization normal wave representation'
The transfer equations for normal waves in finite, inhomogeneous and plane-parallel magnetoactive media are solved using the discrete ordinate method. The physical process of absorption, emission, and multiple scattering are accounted for, and the medium may be forced both at the top and bottom boundary by anisotropic radiation as well as by internal anisotropic sources. The computational procedure is numerically stable for arbitrarily large optical depths, and the computer time is independent of optical thickness.
Kylling, A.↗
Thermal radiation analysis of a pressurized oxy-coal combustor with discrete ordinate model
Concerns over climate change have led to numerous efforts in developing low-carbon energy technologies. Pressurized oxy-combustion (POC) is a promising candidate to reduce carbon emissions in power generation. Pressure strongly impacts thermal radiation, leading to a substantial difference in heat transfer between POC and conventional atmospheric pressure combustion. A good understanding of thermal radiation in POC is needed to aid new combustor development. The present work is a step in this direction, initiating a systematic analysis of thermal radiation and heat transfer in a pilot-scale POC combustor, which has been developed at Washington University in St. Louis (WUSTL). In a POC process, pulverized coal is burned under elevated pressure and O2-CO2 environment. Few researchers have investigated the thermal radiation of pressurized oxy-combustion. This work adopted the discrete ordinate model to simulate the radiation of a 1-D cylinder with predetermined conditions (temperature, pressure, gas compositions) to predict the influence of radiative properties of flue gas on the radiation in the combustor.
Li, Lei↗
Transport Corrections and Sensitivities in the Discrete-Ordinates Method
Not Available
42 ENGINEERING↗
Meshless discretization of the discrete-ordinates transport equation with integration based on Voronoi cells
The time-dependent, gray, linear radiation transport equation is discretized using the meshless local Petrov-Galerkin method with reproducing kernels. The integration is performed using a Voronoi tessellation, which creates a partition of unity that only depends on the position and extent of the kernels. The resolution of the integration automatically follows the particles and requires no manual adjustment. The discretization includes streamline-upwind Petrov-Galerkin stabilization to prevent oscillations and improve numerical conditioning. The angular quadrature is selectively refineable to increase angular resolution in chosen directions. The time discretization is done using backward Euler. The transport solve for each direction and the solve for the scattering source are both done using Krylov iterative methods. The results indicate first-order convergence in time and second-order convergence in space for linear reproducing kernels.
97 MATHEMATICS AND COMPUTING↗
Iterative Discrete Ordinates Solution of the Equation for the Surface-Reflected Radiance
This paper presents a new method of numerical solution of the integral equation for the radiance reflected from an anisotropic surface. The equation relates the radiance at the surface level with BRDF and solutions of the standard radiative transfer problems for a slab with no reflection on its surfaces. It is also shown that the kernel of the equation satisfies the condition of the existence of a unique solution and the convergence of the successive approximations to that solution. The developed method features two basic steps: discretization on a 2D quadrature, and solving the resulting system of algebraic equations with successive over-relaxation method based on the Gauss-Seidel iterative process. Presented numerical examples show good coincidence between the surface-reflected radiance obtained with DISORT and the proposed method. Analysis of contributions of the direct and diffuse (but not yet reflected) parts of the downward radiance to the total solution is performed. Together, they represent a very good initial guess for the iterative process. This fact ensures fast convergence. The numerical evidence is given that the fastest convergence occurs with the relaxation parameter of 1 (no relaxation). An integral equation for BRDF is derived as inversion of the original equation. The potential of this new equation for BRDF retrievals is analyzed. The approach is found not viable as the BRDF equation appears to be an ill-posed problem, and it requires knowledge the surface-reflected radiance on the entire domain of both Sun and viewing zenith angles.
Alexander Radkevich↗