Numerical simulation of radiative heat transfer from non-gray gases in three-dimensional enclosures

The discrete ordinates and the discrete transfer methods are applied to the numerical simulation of radiative heat transfer from non-gray gases in three-dimensional enclosures. Several gas radiative property models are used, namely the correlated k-distribution (CK), the spectral line-based weighted-sum-of-gray-gases (SLW) and the weighted-sum-of-gray-gases (WSGG) methods. The results are compared with recently published accurate calculations based on the statistical narrow band model. The WSGG model is computationally efficient, but often yields relatively large errors. It should be used only if moderate accuracy is sufficient. The SLW model is the best alternative regarding the compromise between accuracy and numerical efficiency. However, an optimization of the coefficients of the model is essential to reduce the computational requirements, especially in the case of gas mixtures. The CK model is the most accurate of the methods evaluated here, but too time consuming for engineering applications, although recent developments may partly overcome this shortcoming.     

A multi-spectral reordering technique for the full spectrum SLMB modeling of radiative heat transfer in nonuniform gaseous mixtures

A Multi-Spectral Reordering (MSR) scheme is introduced to improve the performances of the Spectral-Line Moment-Based (SLMB) modeling for the handling of full spectrum radiative heat transfer calculations in nonuniform media. Using this simultaneous reordering of the spectrum for several thermophysical conditions together with employing approximate formulations to evaluate path-averaged transmission functions for nonisothermal and nonhomogenous gaseous paths, a novel full spectrum gas radiation modeling method in nonuniform gaseous mixtures is constituted. The method is presented in details as well as the building of associated databases for CO₂ and H₂O at atmospheric pressure and for the temperature range of 300-2700 K. The new model is validated against line-by-line reference computations for a series of existing benchmarks and for a flame configuration. The MSR-SLMB modeling is shown to perform accurately and better than the standard SLMB one, while involving reasonable additional computational costs.     

Fast approximate technique for the cumulative wavenumber model to modeling radiative transfer in a mixture of real gas media

In the cumulative wavenumber (CW) model, the total range of the absorption cross-section Cη is subdivided into the supplementary absorption cross-section of gray gases C_j, j=1,…,n, where n is the number of gray gases; and the wavenumber region is subdivided into intervals Δ_i=[η_i−1, η_i], i=1, 2,…,p, where p is the number of intervals. The intersection of the two spectral subdivisions is used to define the modeling of the fractional gray gas D_ij. In the CW model, we solve the radiative transfer equation (RTE) in every subinterval D_ij; then it is necessary to solve n x p times the spectral form of the RTE for complete spectral integration. In this work, the CW model is used with a numerical approximation technique based on additive properties of radiative intensity to reduce the solution of RTE to n new fractional gray gas D_j for complete spectral integration. The CW model was first coupled with the discrete ordinates method and the accuracy of the simplified technique and the algorithm was first examined for one-dimensional homogeneous media; results are compared with line-by-line calculations and it is found that the CW model with the simplified technique is exact for the homogeneous media examined. Also, the fast approach is tested in the diffuse reflecting boundaries case. The CW model is implemented in a bi-dimensional enclosure containing real gases in isothermal cases. Afterwards, this approximate technique is extended to non-isothermal and non-homogeneous cases; the results are compared with line-by-line calculations taken from literature and good agreement was found. The results obtained using the acceleration technique for the CW model agree with the results of original CW model. With this acceleration technique the CPU time decreases p times. Spectral database HITRAN and HITEMP are used to obtain the molecular absorption spectrum of the gases.     

Discrete ordinates interpolation method for radiative heat transfer problems in three-dimensional enclosures filled with non-gray or scattering medium

The discrete ordinates interpolation method (DOIM) is applied to three groups of problems of radiative heat transfer in three-dimensional rectangular enclosures containing non-gray or scattering medium. The original DOIM is first extended to a gray gas model using a new geometric interpolation scheme. It is applied to participating media for different scattering phase functions and optical thicknesses. For the non-gray gas model, the DOIM coupled with the narrow band-based weighted-sum-of-gray-gases (WSGG) model is developed. A few test problems with real gases such as pure H₂O and a mixture of CO₂, H₂O and N₂ are taken. The wall heat flux is calculated and compared with the exact solutions or reference values. All results of test problems are found to be reliable in this study. The DOIM closely reproduces the Monte Carlo reference solutions for different scattering phase functions and optical thicknesses. The non-gray gas results are compared with reference calculations based on the statistical narrow band model and they also show good agreements. The DOIM shows a remarkable merit in the computation time and the grid compatibility, to prove its usefulness for engineering applications.     

The method of lines solution of discrete ordinates method for radiative heat transfer in cylindrical enclosures

A radiation code based on method of lines solution of discrete ordinates method for radiative heat transfer in axisymmetric cylindrical enclosures containing absorbing-emitting medium was developed and tested for predictive accuracy by applying it to (i) test problems with black and grey walls (ii) a gas turbine combustor simulator enclosing a non-homogeneous absorbing-emitting medium and benchmarking its steady-state predictions against exact solutions and measurements. Comparisons show that it provides accurate solutions for radiative heat fluxes and can be used with confidence in conjunction with CFD codes based on the same approach.     

Discrete ordinates solution of coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation

The coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation is solved. The collimated and diffuse components problems are treated separately. The solution for diffuse radiation is obtained by using a composite discrete ordinates method and includes the development of adaptive directional quadratures to overcome the difficulties usually encountered at the interfaces. The complete radiation numerical model is validated against the predictions obtained by using the Monte Carlo method.     

On the use of a meshless method for solving radiative transfer with the discrete ordinates formulations

A meshless method is presented for solving the radiative transfer equation in the discrete ordinates approach. It is shown that the primitive variables formulation is unstable for low values of the absorption coefficient while the even parity formulation is always stable and accurate.     

Assessment of the re-ordered wide band model for non-grey radiative transfer calculations in 3D enclosures

The objective of the present study is to evaluate variations of the re-ordered wide band model for non-grey radiative transfer calculations in 3D enclosures using the discrete ordinates method. First, the performance of various angular and spatial discretisation schemes of the discrete ordinates method is investigated. Then, several formulations, averaging procedures, and scaling methods of the re-ordered wide band model are tested, and the results are validated against those of a statistical narrow band model. The grey gases formulation using three optimised absorption coefficient is found to be the most efficient method.     

Application of multidimensional scheme and the discrete ordinate method to radiative heat transfer in a two-dimensional enclosure with diffusely emitting and reflecting boundary walls

Radiative heat transfer is the dominant mode of heat transfer in many engineering problems, including combustion chambers, space, greenhouses, rocket plume sensing, among others. The aim of this study is to develop an efficient method capable of eliminating ray effects in complex 2D situations and to use the developed code for other problems including combined conduction and convection in connection with CFD codes. A complete genuinely multidimensional discretization in two-dimensional discrete ordinates method is formulated to solve radiative heat transfer in a rectangular enclosure composed of diffusely emitting and reflecting boundaries and containing homogeneous media that absorbs, emits and scatters radiation. A new genuinely multidimensional differencing scheme is used to solve the radiative transfer equation with S₄, S₆, S₈, T₆, T₇, T₈ and T₉ angular quadrature schemes. Different cases are analyzed and the results are compared when possible with those obtained by others researchers.     

Finite element formulation of the first- and second-order discrete ordinates equations for radiative heat transfer calculation in three-dimensional participating media

Two finite element methods (FEMs), FEDOM1 and FEDOM2 (standing for the first and the second finite element discrete ordinates methods, respectively), are formulated and numerically tested. The reference second-order discrete equation is modified in its scattering terms and is applied to the problems of absorbing/emitting and anisotropically scattering media by using the FEM. Numerical features of the developed FEMs are compared with one of the discrete ordinates interpolation method (DOIM), which uses a finite difference scheme. Prediction results of radiative heat transfer by these two FEMs are compared with reference solutions and verified in three-dimensional enclosures containing participating media. The results of FEDOM1 and FEDOM2 agree well with exact solutions for the problem of absorbing/emitting medium with various range of optical thickness. Generally, the two FEMs show more accurate results than DOIM. And FEDOM1 shows more accurate results than FEDOM2 in most of the test problems. Both of the developed FEMs show reasonable results compared with published Monte Carlo solutions for the tested absorbing/emitting and anisotropically scattering media. Although the FEDOM2 is not as accurate as the FEDOM1, it shows its own advantages that it reduces CPU time and memory space of dependent variable to half.     

Unsteady coupling of Navier-Stokes and radiative heat transfer solvers applied to an anisothermal multicomponent turbulent channel flow

Direct numerical simulations (DNS) of an anisothermal reacting turbulent channel flow with and without radiative source terms have been performed to study the influence of the radiative heat transfer on the optically non-homogeneous boundary layer structure. A methodology for the study of the emitting/absorbing turbulent boundary layer (TBL) is presented. Details on the coupling strategy and the parallelization techniques are exposed. An analysis of the first order statistics is then carried out. It is shown that, in the studied configuration, the global structure of the thermal boundary layer is not significantly modified by radiation. However, the radiative transfer mechanism is not negligible and contributes to the heat losses at the walls. The classical law-of-the-wall for temperature can thus be improved for RANS/LES simulations taking into account the radiative contribution.     

Analysis of the radiative transfer equation with highly assymetric phase function

This paper considers a scalar radiative transfer problem with high scattering anisotropy. Two computational methods are presented based on decomposition of the diffuse light field into a regular and anisotropic part. The first algorithm (DOMAS) singles out the anisotropic radiance in the forward scattering peak using the Small-Angle Modification of RTE. The second algorithm (DOM2+) separates the single scattering radiance as an anisotropic part, which largely defines the fine detail of the total radiance in the backscattering directions. In both cases, the anisotropic part is represented analytically. With anisotropy subtraction, the regular part of the signal, which requires a numerical solution, is essentially smoothed as a function of angles. Further, the transport equation is obtained for the regular part that contains an additional source function from the anisotropic part of the signal. This equation is solved with the discrete ordinates method. A conducted numerical analysis of this work showed that algorithm DOMAS has a strong advantage as compared to the standard discrete ordinates method for simulation of the radiance transmission, and DOM2+ is the best of the three for the reflection computations. Both algorithms offer at least a factor of three acceleration of convergence of the azimuthal series for highly anisotropic phase functions.     

Simultaneous derivation of intensities and weighting functions in a general pseudo-spherical discrete ordinate radiative transfer treatment

The retrieval of atmospheric constituents from measurements of backscattered light requires a radiative transfer forward model that can simulate both intensities and weighting functions (partial derivatives of intensity with respect to atmospheric parameters being retrieved). The radiative transfer equation is solved in a multi-layer multiply-scattering atmosphere using the discrete ordinate method. In an earlier paper dealing with the upwelling top-of-the-atmosphere radiation field, it was shown that a full internal perturbation analysis of the plane-parallel discrete ordinate solution leads in a natural way to the simultaneous generation of analytically-derived weighting functions with respect to a wide range of atmospheric variables. In the present paper, a more direct approach is used to evaluate explicitly all partial derivatives of the intensity field. A generalization of the post-processing function is developed for the derivation of weighting functions at arbitrary optical depth and stream angles for both upwelling and downwelling directions. Further, a complete treatment is given for the pseudo-spherical approximation of the direct beam attenuation; this is an important extension to the range of viewing geometries encountered in practical radiative transfer applications. The numerical model LIDORT developed for this work is able to generate intensities and weighting functions for a wide range of retrieval scenarios, in addition to the passive remote sensing application from space. We present a number of examples in an atmosphere with O₃ absorption in the UV, for satellite (upwelling radiation) and ground-based (downwelling radiation) applications. In particular, we examine the effect of various pseudo-spherical parameterizations on backscatter intensities and weighting functions with respect to O₃ volume mixing ratio. In addition, the use of layer-integrated multiple scatter output from the model is shown to be important for satellite instruments with wide-angle off-nadir viewing geometries.     

The method of lines solution of discrete ordinates method for non-grey media

A radiation code based on method of lines (MOL) solution of discrete ordinates method (DOM) for radiative heat transfer in non-grey absorbing-emitting media was developed by incorporation of a gas spectral radiative property model, namely wide band correlated-k (WBCK) model, which is compatible with MOL solution of DOM. Predictive accuracy of the code was evaluated by applying it to 1-D parallel plate and 2-D axisymmetric cylindrical enclosure problems containing absorbing-emitting medium and benchmarking its predictions against line-by-line solutions available in the literature. Comparisons reveal that MOL solution of DOM with WBCK model produces accurate results for radiative heat fluxes and source terms and can be used with confidence in conjunction with computational fluid dynamics codes based on the same approach.     

The k-moment method for the narrow band modeling of radiative properties of nonuniform gaseous media

Moments of the absorption coefficient distribution function are used for the derivation of statistical narrow band (SNB) model parameters of nonuniform optical paths in gases. The approach yields approximations for the path-averaged first- and second-order k-moments from which equivalent SNB parameters are determined in the frame of the Malkmus model. The approach is assessed through comparisons with LBL data. The nonuniform approximation is shown to enable the computation of transmissivities and radiation intensities with accuracy similar to or higher than those achieved by the Curtis-Godson one.     

Development and comparison of different spatial numerical schemes for the radiative transfer equation resolution using three-dimensional unstructured meshes

In the present work four different spatial numerical schemes have been developed with the aim of reducing the false-scattering of the numerical solutions obtained with the discrete ordinates (DOM) and the finite volume (FVM) methods. These schemes have been designed specifically for unstructured meshes by means of the extrapolation of nodal values of intensity on the studied radiative direction. The schemes have been tested and compared in several 3D benchmark test cases using both structured orthogonal and unstructured grids.     

MOL solution of DOM for transient radiative transfer in 3-D scattering media

A methodology based on the method of lines solution of discrete ordinates method for solution of the 3-D transient radiative transfer equation is introduced. The method is applied to the prediction of transient and steady state transmittances in a cubical enclosure containing purely scattering medium and validated against Monte Carlo solutions from the literature. The flexibility of the method for implementation of linear spatial differencing schemes, flux limiters and weighted essentially non-oscillatory methods is demonstrated. Van Leer flux limiter is found to provide stable, accurate and efficient solutions.     

Numerical solution of radiative and conductive heat transfer in concentric spherical and cylindrical media

A new technique is presented to improve the performance of the discrete ordinates method when solving the coupled conduction-radiation problems in spherical and cylindrical media. In this approach the angular derivative term of the discretized one-dimensional radiative transfer equation is derived from an expansion of the radiative intensity on the basis of Chebyshev polynomials. The set of resulting differential equations, obtained by the application of the S_N method, is numerically solved using the boundary value problem with the finite difference algorithm. Results are presented for the different independent parameters. Numerical results obtained using the Chebyshev transform method compare well with the benchmark approximate solutions. Moreover, the new technique can easily be applied to higher-order S_N calculations.     

The sensitivity of radiative transfer calculations to the changes in the HITRAN database from 1982 to 2004

Over the last quarter century, improvements in the determination of the spectroscopic characteristics of the infrared-active trace species have enhanced our ability to retrieve quantitative distributions of temperatures, clouds, and abundances for various trace species within the Earth's atmosphere. These improvements have also allowed for refinements in the estimates of climatic effects attributed to changes in the Earth's atmospheric composition. Modeling efforts, however, have frequently experienced significant delays in assimilating improved spectroscopic information. Such is the case for highly parameterized models, where considerable effort is typically required to incorporate any revisions. Thus, a line-by-line radiative transfer model has been used to investigate the magnitude of the effects resulting from modifications to the spectroscopic information. Calculations from this line-by-line model have demonstrated that recent modifications to the HITRAN (High Resolution Transmission) line parameters, the continuum formulation, and the CO₂ line-mixing formulation can significantly affect the interpretation of the high spectral resolution radiance and brightness temperature retrievals. For certain moderate-resolution satellite remote sensing channels, modifications to these spectroscopic parameters and formulations have shown the capacity to induce changes in the calculated radiances equivalent to brightness temperature differences of 1-2 K. Model calculations have further shown that modifications of the spectroscopic characteristics tend to have a modest effect on the determination of spectrally integrated radiances, fluxes, and radiative forcing estimates, with the largest differences being of order 1 W m⁻² for the total thermal infrared fluxes, and of order 2-3% of the calculated radiative forcing at the tropopause attributed to the combined doubling of CO₂, N₂O, and CH₄. The results from this investigation are intended to function as a guide to differentiate between cases where older parameterizations provide acceptable results, within specified accuracy bounds, and cases where upgrades to the latest spectroscopic database are necessary.     

An improved treatment of overlapping absorption bands based on the correlated k distribution model for thermal infrared radiative transfer calculations

This paper discusses several schemes for handling gaseous overlapping bands in the context of the correlated k distribution model (CKD). Commonly used methods are generally based on certain spectral correlation assumptions; thus they are either less accurate or less efficient and rarely apply to all overlapping bands. We propose a new treatment, which we developed from the traditional absorber amount weighted scheme and improved for application to various bands. This approach is quite efficient for treating the gaseous mixture as if it were a “single gas.” Numerical experiments demonstrate that the new scheme achieves high accuracy with a fast operating speed. To validate the new scheme, we conducted spectrally integrated calculations and sensitivity experiments in the thermal infrared region. Compared to line-by-line integration results, errors in cooling rates were less than 0.2 K/day below 70 Km and rose to 1 K/day from above 70 Km up to 100 Km; flux differences did not exceed 0.8 W/m² at any altitude. Changes in CO₂ and H₂O concentrations slightly influenced the accuracy of the results.