## 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

**
ABSTRACT
**

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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