The SKN Approximation for Solving Radiative Transfer Problems in Absorbing, Emitting, and Isotropically Scattering Plane-Parallel Medium: Part 1

2002 ◽  
Vol 124 (4) ◽  
pp. 674-684 ◽  
Author(s):  
Zekeriya Altac¸
Keyword(s):  
Integral Equation ◽  
Radiative Transfer ◽  
Differential Equations ◽  
Order Approximation ◽  
High Order ◽  
High Order Approximation ◽  
One Dimensional ◽  
Second Order Differential Equations ◽  
Plane Parallel ◽  
Transfer Problems

A high order approximation, the SKN method—a mnemonic for synthetic kernel—is proposed for solving radiative transfer problems in participating medium. The method relies on approximating the integral transfer kernel by a sum of exponential kernels. The radiative integral equation is then reducible to a set of coupled second-order differential equations. The method is tested for one-dimensional plane-parallel participating medium. Three quadrature sets are proposed for the method, and the convergence of the method with the proposed sets is explored. The SKN solutions are compared with the exact, PN, and SN solutions. The SK1 and SK2 approximations using quadrature Set-2 possess the capability of solving radiative transfer problems in optically thin systems.

The SKN Approximation for Solving Radiative Transfer Problems in Absorbing, Emitting, and Linearly Anisotropically Scattering Plane-Parallel Medium: Part 2

2002 ◽  
Vol 124 (4) ◽  
pp. 685-695 ◽  
Author(s):  
Zekeriya Altac¸
Keyword(s):  
Radiative Transfer ◽  
Incident Energy ◽  
Radiative Heat ◽  
Isotropic Scattering ◽  
Benchmark Problems ◽  
Scattering Medium ◽  
Second Order Differential Equations ◽  
Plane Parallel ◽  
Kernel Approximation ◽  
Transfer Problems

The SKN (Synthetic Kernel) approximation is proposed for solving radiative transfer problems in linearly anisotropically scattering homogeneous and inhomogeneous participating plane-parallel medium. The radiative integral equations for the incident energy and the radiative heat flux using synthetic kernels are reduced to a set of coupled second-order differential equations for which proper boundary conditions are established. Performance of the three quadrature sets proposed for isotropic scattering medium are further tested for linearly anisotropically scattering medium. The method and its convergence with respect to the proposed quadrature sets are explored by comparing the results of benchmark problems using the exact, P11, and S128 solutions. The SKN method yields excellent results even for low orders using appropriate quadrature set.

scholarly journals Radiative transfer in Rayleigh–Taylor unstable ICF pellets

1990 ◽  
Vol 8 (4) ◽  
pp. 741-751 ◽  
Author(s):  
G. C. Pomraning
Keyword(s):  
Integral Equation ◽  
Radiative Transfer ◽  
Differential Equations ◽  
Markov Model ◽  
Master Equation ◽  
Energy Flow ◽  
Fluid Mixing ◽  
Renewal Processes ◽  
Integral Equation Formulation ◽  
Statistical Effects

A formulation of radiative transfer is discussed describing energy flow in a turbulent mixture in the vicinity of a Rayleigh–Taylor unstable interface, as might be extant in an ICF pellet. Included in this discussion are (1) the method of smoothing and the Liouville master equation approaches in the case of Markovian statistics as the description of the fluid mixing, (2) the use of asymptotics to derive various limiting descriptions of the Markov model, (3) the use of the theory of alternating renewal processes to obtain an integral equation formulation for non-Markovian statistics, and (4) the reduction of this non-Markovian integral formulation to integro-differential equations of the generic transport form, with statistical effects represented by pseudoscattering terms.

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