DIMACS TR: 99-55

Apparent Motion in Spatial Time-varying Data: A Variational Approach to Pointwise Tracking of Coherent Structures in Computational Fluid Dynamics

Authors: Vadim Mottl, Alexander Blinov, Norman Zabusky and Ilya Muchnik


By a massive time-varying data set is meant an experimentally acquired or mathematically simulated function of spatial coordinates and time which is to be analyzed with the purpose of studying behavior of the respective distributed dynamical system. We address here primarily problems of computational fluid dynamics that are concerned with the necessity to visualize and study overwhelming time-varying data sets generated by numerical unsteady simulations. Of a major interest is time tracking of persistent coherent structures of the fluid flow that use to arise, move in space for a time while, fade, and be replaced by new pattern building up from their remains. The spatial motion of persistent patterns in a time-varying data set is called here apparent motion because it does not coincide, in the general case, with mechanical motion of any physical particles. With the purpose of tracking coherent structures, it is proposed to evaluate the apparent motion in terms of apparent velocity of space points. The apparent velocity is to be estimated in the form of a time-varying vector field as result of processing the original time-varying data set or immediately in the course of numerical simulating the respective physical reality as solution of an additional system of partial differential equations. In both versions, we lean upon the variational approach to the analysis of massive data sets and estimate the vector field of apparent velocity as time-varying function that delivers the minimum value to a quadratic functional. In experiments with simulated two-dimensional time-varying spatial dynamics of the fluid's density and vorticity in shock wave interaction with an elliptical bubble, the estimated apparent velocity provided long-term pointwise tracking of persistent coherent structures.

Paper Available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1999/99-55.ps.gz
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