Potentialities and limitations of mixing simulations

As numerical simulations in mixing become pervasive, an analysis of errors becomes crucial. Purposely discretized examples with exact analytical solutions provide a reference point from which to judge the soundness of numerical solutions. Three types of errors are identified and examined: discretization, time integration, and round‐off, with emphasis 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 t², while in chaotic flows it increases exponentially. Errors tend to align with the direction of the streamlines in regular flows and with manifolds in chaotic flows. As a result, even though exact and calculated trajectories diverge exponentially fast in chaotic flows, overall mixing patterns are reproduced, at least qualitatively, even when the velocity field is calculated using the main features of a line as it is deformed 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 obtained from numerical simulations. However, robust application of standard numerical analysis tools, such as mesh refinement, is necessary, which, in turn, can lead to nearly prohibitive computational costs.