Curved ray tracing method for one-dimensional radiative transfer in the linear-anisotropic scattering medium with graded index

The curved ray tracing method (CRT) is extended to radiative transfer in the linear-anisotropic scattering medium with graded index from non-scattering medium. In this paper, the CRT is presented to solve one-dimensional radiative transfer in the linear-anisotropic scattering gray medium with a linear refractive index and two black boundaries. The predicted temperature distributions and radiative heat flux at radiative equilibrium are determined by the proposed method, and numerical results are compared with the data in references. The results show that the CRT has a good accuracy for radiative transfer in the linear-anisotropic scattering medium with graded index and the dimensionless emissive power and dimensionless radiative heat flux depend on the dimensionless refractive index gradient. It can also be seen that the dimensionless refractive index gradient has important effects on the temperature discontinuity at the boundaries.     

Temperature field of radiative equilibrium in a two-dimensional graded index medium with gray boundaries

In this paper, a numerical method is presented for the study of the radiative transfer in a two-dimensional graded index semitransparent medium with diffuse gray boundaries. The numerical method is a combination of the linear refractive index bar model, the discrete curved ray-tracing technique and the pseudo source adding method (LRIB-CRTP). In the traditional ray-tracing technique, it is difficult to deal with the diffuse gray boundary while solving the radiative transfer. Using the pseudo source adding method, the diffuse gray boundary of the medium can be treated as a black boundary. We have also studied the radiative equilibrium temperature field of the medium and analyzed the influence of some parameters involved. The results show that the directional discrete number is important for the medium having a large absorption coefficient. The results also show that the refractive index distribution greatly influences the temperature field, whereas the linear absorption coefficient distribution has little influence on the temperature field.     

Discontinuous finite element method for radiative heat transfer in semitransparent graded index medium

To avoid the complicated and time-consuming computation of curved ray trajectories, a discontinuous finite element method based on discrete ordinate equation is extended to solve the radiative transfer problem in a multi-dimensional semitransparent graded index medium. Two cases of radiative heat transfer in two-dimensional rectangular gray semitransparent graded index medium enclosed by opaque boundary are examined to verify this discontinuous finite element method. Special layered and radial graded index distributions are considered. The predicted dimensionless net radiative heat fluxes and dimensionless temperature distributions are determined by the discontinuous finite element method and compared with the results obtained by the curved Monte Carlo method in references. The results show that the discontinuous finite element method has a good accuracy in solving the multi-dimensional radiative transfer problem in a semitransparent graded index medium.     

Benchmark numerical solutions for radiative heat transfer in two-dimensional medium with graded index distribution

In graded index media, the ray goes along a curved path determined by Fermat principle. Generally, the curved ray trajectory in graded index media is a complex implicit function, and the curved ray tracing is very difficult and complex. Only for some special refractive index distributions, the curved ray trajectory can be expressed as a simple explicit function. Two important examples are the layered and the radial graded index distributions. In this paper, the radiative heat transfer problems in two-dimensional square semitransparent with layered and radial graded index distributions are analyzed. After deduction of the ray trajectory, the radiative heat transfer problems are solved by using the Monte Carlo curved ray-tracing method. Some numerical solutions of dimensionless net radiative heat flux and medium temperature are tabulated as the benchmark solutions for the future development of approximation techniques for multi-dimensional radiative heat transfer in graded index media.     

Finite element approach for radiative transfer in multi-layer graded index cylindrical medium with Fresnel surfaces

Because the optical plane defined by the incidence and reflection direction at a cylindrical surface has a complicated relation with the local azimuthal angle and zenith angle in the traditional cylindrical coordinate system, it is difficult to deal with the specular reflective boundary condition in the solution of the traditional radiative transfer equation for cylindrical system. In this paper, a new radiative transfer equation for graded index medium in cylindrical system (RTEGCN) is derived based on a newly defined cylindrical coordinate system. In this new cylindrical coordinate system, the optical plane defined by the incidence and reflection direction is just the isometric plane of the local azimuthal angle, which facilitates the RTEGCN in dealing with cylindrical specular reflective boundaries. A least squares finite element method (LSFEM) is developed for solving radiative transfer in single and multi-layer cylindrical medium based on the discrete ordinates form of the RTEGCN. For multi-layer cylindrical medium, a radial basis function interpolation method is proposed to couple the radiative intensity at the interface between two adjacent layers. Various radiative transfer problems in both single and multi-layer cylindrical medium are tested. The results show that the present finite element approach has good accuracy to predict the radiative heat transfer in multi-layer cylindrical medium with Fresnel surfaces.     

Finite element method for modeling radiative transfer in semitransparent graded index cylindrical medium

Both Galerkin finite element method (GFEM) and least squares finite element method (LSFEM) are developed and their performances are compared for solving the radiative transfer equation of graded index medium in cylindrical coordinate system (RTEGC). The angular redistribution term of the RTEGC is discretized by finite difference approach and after angular discretization the RTEGC is formulated into a discrete-ordinates form, which is then discretized based on Galerkin or least squares finite element approach. To overcome the RTEGC-led numerical singularity at the origin of cylindrical coordinate system, a pole condition is proposed as a special mathematical boundary condition. Compared with the GFEM, the LSFEM has very good numerical properties and can effectively mitigate the nonphysical oscillation appeared in the GFEM solutions. Various problems of both axisymmetry and nonaxisymmetry, and with medium of uniform refractive index distribution or graded refractive index distribution are tested. The results show that both the finite element approaches have good accuracy to predict the radiative heat transfer in semitransparent graded index cylindrical medium, while the LSFEM has better numerical stability.     

Modified finite volume method applied to radiative transfer in 2D complex geometries and graded index media

A modified finite volume method with unstructured triangular meshes is proposed to solve the RTE in 2D complex geometries and for graded index media. In such media, the RTE has an additional term corresponding to “angular redistribution”. This term is due to the change in the orientation of the direction of propagation for the radiation along curved optical paths. Some benchmark cases applied to a slab (1D) and a square cavity (2D) with linear and nonlinear refractive graded index are used to validate the new method. New results are presented for a disk with radial graded index.     

Finite element method for radiation heat transfer in multi-dimensional graded index medium

In graded index medium, ray goes along a curved path determined by Fermat principle, and curved ray-tracing is very difficult and complex. To avoid the complicated and time-consuming computation of curved ray trajectories, a finite element method based on discrete ordinate equation is developed to solve the radiative transfer problem in a multi-dimensional semitransparent graded index medium. Two particular test problems of radiative transfer are taken as examples to verify this finite element method. The predicted dimensionless net radiative heat fluxes are determined by the proposed method and compared with the results obtained by finite volume method. The results show that the finite element method presented in this paper has a good accuracy in solving the multi-dimensional radiative transfer problem in semitransparent graded index medium.     

Least-squares finite element method for radiation heat transfer in graded index medium

To avoid the complicated and time-consuming computation of curved ray trajectories, a least-squares finite element method based on discrete ordinate equation is extended to solve the radiative transfer problem in a multi-dimensional semitransparent graded index medium. Four cases of radiative heat transfer are examined to verify this least-squares finite element method. Linear and nonlinear graded index are considered. The predicted dimensionless net radiative heat fluxes are determined by the least-squares finite element method and compared with the results obtained by other methods. The results show that the least-squares finite element method is stable and has a good accuracy in solving the multi-dimensional radiative transfer problem in a semitransparent graded index medium, while the Galerkin finite element method sometimes suffers from nonphysical oscillations.     

Spectral collocation method with a flexible angular discretization scheme for radiative transfer in multi-layer graded index medium

The spectral collocation method (SCM) is employed to solve the radiative transfer in multi-layer semitransparent medium with graded index. A new flexible angular discretization scheme is employed to discretize the solid angle domain freely to overcome the limit of the number of discrete radiative direction when adopting traditional S_N discrete ordinate scheme. Three radial basis function interpolation approaches, named as multi-quadric (MQ), inverse multi-quadric (IMQ) and inverse quadratic (IQ) interpolation, are employed to couple the radiative intensity at the interface between two adjacent layers and numerical experiments show that MQ interpolation has the highest accuracy and best stability. Variable radiative transfer problems in double-layer semitransparent media with different thermophysical properties are investigated and the influence of these thermophysical properties on the radiative transfer procedure in double-layer semitransparent media is also analyzed. All the simulated results show that the present SCM with the new angular discretization scheme can predict the radiative transfer in multi-layer semitransparent medium with graded index efficiently and accurately.     

Meshless approach for coupled radiative and conductive heat transfer in one-dimensional graded index medium

A meshless local Petrov-Galerkin (MLPG) approach is employed for solving the coupled radiative and conductive heat transfer in a one-dimensional slab with graded index media. The angular distribution term in discrete ordinate equation of radiative transfer within a one-dimensional graded index slab is discretized by a step scheme, and the meshless approach for radiative transfer is based on the discrete ordinate equation. A moving least-squares approximation is used to construct the shape function. Two particular test cases for coupled radiative and conductive heat transfer within a one-dimensional graded index slab are examined to verify this new approximate method. The temperatures and the radiative heat fluxes are obtained. The results are compared with the other benchmark approximate solutions. By comparison, the results show that the MLPG approach has a good accuracy in solving the coupled radiative and conductive heat transfer in one-dimensional graded index media.     

Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems

In graded index medium, the ray goes along a curved path determined by Fermat principle, and the curved ray-tracing is very difficult and complex. To avoid the complicated and time-consuming computation of curved ray trajectory, the methods not based on ray-tracing technique need to be developed for the solution of radiative transfer in graded index medium. For this purpose, in this paper the streaming operator along a curved ray trajectory in original radiative transfer equation for graded index medium is transformed and expressed in spatial and angular ordinates and the radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems are derived. The conservative and the non-conservative forms of radiative transfer equation for three-dimensional graded index medium are given, which can be used as base equations to develop the numerical simulation methods, such as finite volume method, discrete ordinates method, and finite element method, for radiative transfer in graded index medium in cylindrical and spherical coordinate systems.     

The DRESOR method for radiative heat transfer in a one-dimensional medium with variable refractive index

This paper extends the DRESOR (Distribution of Ratios of Energy Scattered by the medium Or Reflected by the boundary surface) method to radiative transfer in a variable refractive index medium. In this method, the intensity is obtained from the source term along the curved integration paths determined only by the variable refractive index, and the DRESOR values are calculated by the Monte Carlo method in which the propagation of the energy bundles are affected by Snell's law. With given temperatures on the black boundaries of a one-dimensional medium, the temperature distribution inside the medium with a variable scattering property is calculated under the condition of radiative equilibrium. It is shown that the DRESOR method has a good accuracy in the cases studied. For an isotropic-scattering medium with the same optical thickness, the scattering albedo has no effect on the temperature distribution, which can be obtained from the general equations and can be seen as an extension of what exists for a constant refractive index; however, the different refractive index causes obvious changes in the temperatures inside the medium. The effect of anisotropic scattering on the temperature distribution cannot be ignored, although it is still weaker than the effect caused by variation in the refractive index.     

The Iterative-DRESOR method to solve radiative transfer in a plane-parallel, anisotropic scattering medium with specular-diffuse boundaries

Using the intensity with high directional resolution obtained by the Basic-DRESOR method as an initial guess, which is substituted into the integrated radiative transfer equation (IRTE), an iterative algorithm is proposed, called the Iterative-DRESOR method. This method can reduce the error levels of the intensity from several percent using the Basic-DRESOR method to a level of less than 1.0×10⁻⁶ with acceptable computation costs. The method is also validated against the exact heat flux in literature in some cases. It further clarifies some uncertain results for the reflectance in a pure, linearly anisotropic scattering medium with specular-diffuse boundaries. The directional distributions of intensity are obviously influenced by the reflecting modes of the boundary, especially in the zone near the boundary. The reflecting mode of an emitting boundary has little effect on the transmittance or reflectance. The reflecting mode of a non-emitting boundary also has little effect on the transmittance, but it obviously influences the reflectance. The difference between the reflectance for specular and diffuse boundaries increases at first, and then decreases, as the optical thickness of the medium increases. The difference will decrease as the scattering albedo of the medium increases, and it is negligible when the medium is pure scattering. The effect of the scattering phase function of the medium on the difference can also not be ignored. The Iterative-DRESOR method is expected to strengthen the capability of the Monte Carlo method to produce accurate results and to validate the results of other methods to solve RTE.     

Transient radiation and conduction heat transfer inside a plane-parallel participating gray medium with boundaries having different reflecting characteristics

A hybrid ray-tracing method is developed for the solution to the radiative transfer in a plane-parallel participating medium having one specular surface and another diffuse surface. By this method, radiative transfer coefficients (RTCs) for specular–diffuse (S–D) surfaces are deduced. The medium surfaces are considered to be semitransparent. The effects of convection–radiation parameter, conduction–radiation parameter and refractive index on transient coupled heat transfer are investigated. Results show that the temperature curves of the medium having S–D surfaces is higher than those of the medium having S–S surfaces (two specular surfaces); the total heat flux at steady state for the S–D surfaces is lower than that for the S–S surfaces.     

Discrete transfer method applied to radiative transfer in a variable refractive index semitransparent medium

Application of the discrete transfer method (DTM) has been extended to the analysis of radiative heat transfer in a variable refractive index participating medium. To validate the DTM formulation, radiative heat transfer in an absorbing, emitting and isotropically scattering planar medium was considered. The participating medium was assumed to be in radiative equilibrium. For both constant and variable refractive indices of the medium, the DTM results were compared with those available in the literature. The DTM was found to provide accurate results.     

Three-dimensional radiative transfer in an anisotropically scattering, semi-infinite medium: generalized reflection function

Three-dimensional radiative transfer in an anisotropic scattering medium exposed to spatially varying, collimated radiation is studied. The generalized reflection function for a semi-infinite medium with a very general scattering phase function is the focus of this investigation. An integral transform is used to reduce the three-dimensional transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to formulate a nonlinear integral equation for the generalized reflection function. The integration is over both the polar and azimuthal angles; hence, the integral equation is said to be in the double-integral form. The double-integral, reflection function formulation can handle a variety of anisotropic phase functions and does not require an expansion of the phase function in a Legendre polynomial series. Complicated kernel transformations of previous single-integral studies are eliminated. Single and double scattering approximations are developed. Numerical results are presented for a Rayleigh phase function to illustrate the computational characteristics of the method and are compared to results obtained with the single-integral method. Agreement between the two approaches is excellent; however, as the transform variable increases beyond five the number of quadrature points required for the double-integral method to produce accurate solutions significantly increases. A new interpolation scheme produces accurate results when the transform variable is large.     

Three-dimensional radiative transfer in an isotropically scattering, semi-infinite medium: generalized reflection function

The focus of this study is the generalized reflection function of multidimensional radiative transfer. The physical situation considered is spatially varying, collimated radiation incident on the upper boundary of an isotropically scattering, semi-infinite medium. An integral transform is used to reduce the three-dimensional transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to formulate a nonlinear integral equation for the generalized reflection function. The resulting equation is said to be in double-integral form because the integration is over both angular variables. Computational issues associated with this generalized reflection function formulation are investigated. The source function and reflection function formulations are compared, and the relative merits of the two approaches are discussed.     

Three-dimensional radiative transfer in an anisotropically scattering, plane-parallel medium: generalized reflection and transmission functions

The problem of spatially varying, collimated radiation incident on an anisotropically scattering, plane-parallel medium is considered. A very general phase function is allowed. An integral transform is used to reduce the three-dimensional radiative transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to derive nonlinear integral and integro-differential equations for the generalized reflection and transmission functions. The integration is over the polar and azimuthal angles—this formulation is referred to as the double-integral formulation. The integral equations are used to illustrate symmetry relationships and to obtain single- and double-scattering approximations. The generalized reflection and transmission functions are important in the construction of the solutions to many multidimensional problems. Coupled integral equations for the interior and emergent intensities are developed and, for the case of two identical homogeneous layers, used to formulate a doubling procedure. Results for an isotropic and Rayleigh scattering medium are presented to illustrate the computational characteristics of the formulation.     

The influence of anisotropic scattering on the radiative intensity in a gray, plane-parallel medium calculated by the DRESOR method

Even though there have been many ways to treat complex anisotropic scattering problems, in most of the cases only the radiation flux or its dimensionless data were provided, and radiative intensity with high directional resolution could merely be seen. In this paper, a comprehensive formulation for the DRESOR method was proposed to deal with the anisotropic scattering, emitting, absorbing, plane-parallel media with different boundary conditions. The method was validated by the data from literature and the integral formulation of RTE. The DRESOR value plays an important role in the DRESOR method, and how it is determined by the anisotropic scattering was demonstrated by some typical results. The intensities with high directional resolution at any point can be given by the present method. It was found that the scattering phase function has little effect on the intensity for thin optical thickness, for example, 0.1. And there is the largest boundary intensity for the medium with the largest forward scattering capability, and the smallest one with the largest backward scattering capability. An attractive phenomenon was observed that the scattering of the medium makes the intensity at boundary can not reach the blackbody emission capability with the same temperature, even if the optical thickness tends to very large. It was also revealed that the scattering of the medium does not mean it cannot alter the magnitude of the energy; actually, stronger scattering causes the energy to have more chance to be absorbed by the medium, and indirectly changes the energy magnitude in the medium. Finally, it is easy to deduce all the associated quantities such as the radiation flux, the incident radiation and the heat source from the intensity, just as done in literature.