scholarly journals Formulações espectrais para solução de problemas de transporte de partículas

2020 ◽  
Vol 42 ◽  
pp. e39
Author(s):  
Liliane Basso Barichello ◽  
Patricia Rodrigues Fortes ◽  
Camila Becker Picoloto ◽  
Mariza De Camargo
Keyword(s):  
Gas Dynamics ◽  
Eigenvalue Problems ◽  
Rarefied Gas ◽  
Neutron Transport ◽  
Linear Boltzmann Equation ◽  
Discrete Ordinates ◽  
Deterministic Approach ◽  
Relevant Feature ◽  
Integral Term ◽  
Reduced Order

In this work, a deterministic approach to the solution of the integro differential linear Boltzmann equation, in one and twodimensional media, is presented. Explicit solutions, in terms of the spatial variables, for the discrete ordinates approximation of the original model are obtained from a spectral formulation. A relevant feature of the methodology is the reduced order of the eigenvalue problems, which is given as half of the number of the discrete directions. Such aspect, along with the use of arbitrary quadrature schemes to represent the integral term of the equation, are fundamental to provide fast, concise and accurate solutions to several problems of interest. In particular, the solutions here are derived for models associated with two different applications: rarefied gas dynamics and neutron transport. Mathematical aspects present in the solution of the two models are emphasized and extensions of the formulation to other applications are commented and referenced.

Kramers' problem for a variable collision frequency model

2001 ◽  
Vol 12 (2) ◽  
pp. 179-191 ◽  
Author(s):  
C. E. SIEWERT
Keyword(s):  
Gas Dynamics ◽  
Rarefied Gas ◽  
Collision Frequency ◽  
Discrete Ordinates ◽  
Rigid Sphere ◽  
Sphere Model ◽  
Bgk Model ◽  
Frequency Model ◽  
Change Of Variables ◽  
Kramers Problem

The often-studied problem known as Kramers' problem, in the general area of rarefied-gas dynamics, is investigated in terms of a linearized, variable collision frequency model of the Boltzmann equation. A convenient change of variables is used to reduce the general case considered to a canonical form that is well suited for analysis by analytical and/or numerical methods. While the general formulation developed is valid for an unspecified collision frequency, a recently developed version of the discrete-ordinates method is used to compute the viscous-slip coefficient and the velocity defect in the Knudsen layer for three specific cases: the classical BGK model, the Williams model (the collision frequency is proportional to the magnitude of the velocity) and the rigid-sphere model.

The temperature-jump problem in rarefied-gas dynamics

2000 ◽  
Vol 11 (4) ◽  
pp. 353-364 ◽  
Author(s):  
L. B. BARICHELLO ◽  
C. E. SIEWERT
Keyword(s):  
Gas Dynamics ◽  
Computational Algorithm ◽  
Temperature Jump ◽  
Rarefied Gas ◽  
Rarefied Gas Dynamics ◽  
Discrete Ordinates ◽  
Jump Problem ◽  
Bgk Model ◽  
Discrete Ordinates Method ◽  
Density Distributions

An analytical version of the discrete-ordinates method is used here to solve the classical temperature-jump problem based on the BGK model in rarefied-gas dynamics. In addition to a complete development of the discrete-ordinates method for the application considered, the computational algorithm is implemented to yield very accurate results for the temperature jump and the complete temperature and density distributions in the gas. The algorithm is easy to use, and the developed code runs typically in less than a second on a 400 MHz Pentium-based PC.

scholarly journals A Discrete-Ordinates Solution for the Strong Evaporation Problem in Rarefied Gas Dynamics

2021 ◽  
Vol 22 (2) ◽  
pp. 179-199
Author(s):  
C. S. Scherer
Keyword(s):  
Gas Dynamics ◽  
Rarefied Gas ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Rarefied Gas Dynamics ◽  
Form Solution ◽  
Discrete Ordinates ◽  
Bgk Model ◽  
Complete Development ◽  
Strong Evaporation

In this work we solve the nonlinear strong evaporation problem in rarefied gas dynamics. The analysis is based on the BGK model, with three dimensional velocity vector, derived from the Boltzmann equation. We present the complete development of a closed form solution for evaluating density, velocity, temperature perturbations and the heat flux of a gas. Numerical results are presented and discussed.  

Solution and Study of the Two-Dimensional Nodal Neutron Transport Equation

2002 ◽  
Author(s):  
Ruben Panta Pazos ◽  
Marco Tullio de Vilhena ◽  
Eliete Biasotto Hauser
Keyword(s):  
Transport Equation ◽  
Truncation Error ◽  
Neutron Transport ◽  
Discrete Ordinates ◽  
Angular Variable ◽  
Two Dimensional ◽  
Neutron Transport Equation ◽  
Discrete Approximations ◽  
Integral Term ◽  
The Laplace Transform

In the last decade Vilhena and coworkers10 reported an analytical solution to the two-dimensional nodal discrete-ordinates approximations of the neutron transport equation in a convex domain. The key feature of these works was the application of the combined collocation method of the angular variable and nodal approach in the spatial variables. By nodal approach we mean the transverse integration of the SN equations. This procedure leads to a set of one-dimensional SN equations for the average angular fluxes in the variables x and y. These equations were solved by the old version of the LTSN method9, which consists in the application of the Laplace transform to the set of nodal SN equations and solution of the resulting linear system by symbolic computation. It is important to recall that this procedure allow us to increase N the order of SN up to 16. To overcome this drawback we step forward performing a spectral painstaking analysis of the nodal SN equations for N up to 16 and we begin the convergence of the SN nodal equations defining an error for the angular flux and estimating the error in terms of the truncation error of the quadrature approximations of the integral term. Furthermore, we compare numerical results of this approach with those of other techniques used to solve the two-dimensional discrete approximations of the neutron transport equation6.

scholarly journals Neutron angular flux reconstruction in slab geometry using multigroup discrete ordinates transport models

2021 ◽  
Vol 8 (3A) ◽  
Author(s):  
Hermes Alves Filho
Keyword(s):  
Neutron Transport ◽  
Isotropic Scattering ◽  
Computational Time ◽  
Discrete Ordinates ◽  
Neutron Transport Equation ◽  
Slab Geometry ◽  
Neutron Shielding ◽  
Integral Term ◽  
Spatial Domains ◽  
Neutron Propagation

In this article, we present an application of the coarse-mesh Deterministic Spectral Method (SDM) to generate multigroup angular fluxes in one-dimensional spatial domains using the neutron transport stationary equation, in the formulation of discrete ordinates (SN), considering isotropic scattering source. After obtaining the analytical solution of the SN equations, we replace the integral term of the scattering source in the original neutron transport equation. Thus, we obtain analytically two expressions for angular fluxes in the multigroup formulation, considering the neutron propagation in the positive ( ) and negative ( ) directions, presenting a meaningful reduction in the computational time of simulations of typical neutron shielding problems.

scholarly journals A Revisit to CMFD Schemes: Fourier Analysis and Enhancement

Energies ◽  
2021 ◽  
Vol 14 (2) ◽  
pp. 424
Author(s):  
Dean Wang ◽  
Zuolong Zhu
Keyword(s):  
Fourier Analysis ◽  
Eigenvalue Problems ◽  
Neutron Transport ◽  
Coarse Mesh ◽  
Successive Overrelaxation ◽  
Computational Cell ◽  
Artificial Diffusion ◽  
Acceleration Method ◽  
The Stability ◽  
Partial Current

The coarse-mesh finite difference (CMFD) scheme is a very effective nonlinear diffusion acceleration method for neutron transport calculations. CMFD can become unstable and fail to converge when the computational cell optical thickness is relatively large in k-eigenvalue problems or diffusive fixed-source problems. Some variants and fixups have been developed to enhance the stability of CMFD, including the partial current-based CMFD (pCMFD), optimally diffusive CMFD (odCMFD), and linear prolongation-based CMFD (lpCMFD). Linearized Fourier analysis has proven to be a very reliable and accurate tool to investigate the convergence rate and stability of such coupled high-order transport/low-order diffusion iterative schemes. It is shown in this paper that the use of different transport solvers in Fourier analysis may have some potential implications on the development of stabilizing techniques, which is exemplified by the odCMFD scheme. A modification to the artificial diffusion coefficients of odCMFD is proposed to improve its stability. In addition, two explicit expressions are presented to calculate local optimal successive overrelaxation (SOR) factors for lpCMFD to further enhance its acceleration performance for fixed-source problems and k-eigenvalue problems, respectively.

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