POTENTIALITIES AND LIMITATIONS OF MIXING SIMULATIONS

As numerical simulations in mixing become pervasive, an analysis of er rors becomes crucial. Purposely discretized examples with exact analyt ical solutions provide a reference point from which to judge the sound ness of numerical solutions. Three types of errors are identified and examined: discretization, time integration, and round-off with emphasi s on the first two. Theoretical derivations and numerical examples for 2-D, steady (regular) and time-periodic (chaotic) flows indicate that errors, in general behave as material lines. In regular flows, their magnitude increases, on the average, with at most t(2) while in chaoti c flows it increases exponentially. Errors tend to align with the dire ction of the streamlines in regular flows and with manifolds in chaoti c flows. As a result, even though exact and calculated trajectories di verge exponentially fast in chaotic flows, overall mixing patterns are reproduced, at least qualitatively, even when the velocity field is c alculated using coarse meshes. For example, approximate velocity field s do reproduce qualitatively the main features of a line as it is defo rmed by the flow although the error in its length may be more than 100 %. It is concluded that accurate quantitative information such as the location of periodic points or the length of a deformed line, can be o btained from numerical simulations. However, robust application of sta ndard numerical analysis tools, such as mesh refinement, is necessary, which, in turn, can lend to nearly prohibitive computational costs.