Evidence of nonuniqueness and oscillatory solutions in computational fluid mechanics
Abstract
We will review some of our recent experiences in computing solutions for nonlinear fluids in relatively simple, twodimensional geometries. The purpose of this discussion will be to display by example some of the interesting but difficult questions that arise when illbehaved solutions are obtained numerically. We will consider two examples. As the first example, we will consider a nonlinear elastic (compressible) fluid with chemical reactions and discuss solutions for detonation and detonation failure in a twodimesnisonal cylinder. In this case, the numerical algorithm utilizes a finitedifference method with artifical viscosity (von NeumanRichtmyer method) and leads to two, distinctly different, stable solutions depending on the time step criterion used. The second example to be considered involves the convection of a viscous fluid in a rectangular container as a result of an exothermic polymerization reaction. A solidification front develops near the top of the container and propagates down through the fluid, changing the aspect ratio of the region ahead of the front. Using a Galerkinbased finite element method, a numerical solution of the partial differential equations is obtained which tracks the front and correctly predicts the fluid temperatures near the walls. However, the solution also exhibits oscillatory behavior with regard to the number of cells in the fluid ahead of the front and in the strength of the cell. More definitive experiments and analysis are required to determine whether this oscillatory phenomena is a numerical artifact or a physical reality.
 Publication:

NASA STI/Recon Technical Report N
 Pub Date:
 1985
 Bibcode:
 1985STIN...8630111N
 Keywords:

 Computational Fluid Dynamics;
 Finite Difference Theory;
 Finite Element Method;
 Fluid Mechanics;
 Oscillations;
 Partial Differential Equations;
 Polymerization;
 Prediction Analysis Techniques;
 Algorithms;
 Chemical Reactions;
 Compressible Flow;
 Detonation;
 Elastic Properties;
 Nonlinearity;
 Solidification;
 Viscosity;
 Fluid Mechanics and Heat Transfer