Nodal Methods for Problems in Fluid Mechanics and Neutron Transport (navier-Stokes Equation, Boussinesq, Lid-Driven Cavity, Natural Thermal Convection, Diffusion Synthetic Acceleration)
Azmy, Yousry Youssef
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https://hdl.handle.net/2142/70902
Description
Title
Nodal Methods for Problems in Fluid Mechanics and Neutron Transport (navier-Stokes Equation, Boussinesq, Lid-Driven Cavity, Natural Thermal Convection, Diffusion Synthetic Acceleration)
Author(s)
Azmy, Yousry Youssef
Issue Date
1985
Department of Study
Nuclear Engineering
Discipline
Nuclear Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Engineering, Nuclear
Abstract
A new high-accuracy, coarse-mesh, nodal integral approach is developed for the efficient numerical solution of linear partial differential equations. It is shown that various special cases of this general nodal integral approach correspond to several high-efficiency nodal methods developed recently for the numerical solution of neutron diffusion and neutron transport problems.
The new approach is extended to the nonlinear Navier-Stokes equations of fluid mechanics; its extension to these equations leads to a new computational method, the nodal integral method which is implemented for the numerical solution of these equations. Application to several test problems demonstrates the superior computational efficiency of this new method over previously developed methods. The solutions obtained for several driven cavity problems are compared with the available experimental data and are shown to be in very good agreement with experiment. Additional comparisons also show that the coarse-mesh, nodal integral method results agree very well with the results of definitive ultra-fine-mesh, finite-difference calculations for the driven cavity problem up to fairly high Reynolds numbers. A solution continuation study for the closed driven cavity problem is done to identify singular points and to continue solutions through them for the very coarse meshes for which they exist. The study uncovers the origin of some spurious solutions to this problem which had been reported earlier and attributed to other causes. These final results underscore the value of using continuation methods when solutions are sought for problems with high Reynolds numbers.
A second new nodal integral method for solving natural thermal convection problems described by the nonlinear Boussinesq equations also is developed and tested. Results of the test problems are compared to the analytic solution, and to h('2)-extrapolated finite-difference solutions. Our study shows that our new method is extremely accurate even on very coarse meshes.
A two-dimensional Diffusion Synthetic Acceleration method for the Discrete Nodal Transport Method is developed, implemented, and tested, along with its one-dimensional analogue. The DSA is shown to be very effective in reducing the number of iterations and the CPU time required for convergence over unaccelerated schemes especially for highly-scattering, large systems. The DSA also is shown to be superior to coarse mesh rebalance and to be more predictable for a wider class of problems.
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