Coarse analysis of multiscale systems: diffuser flows, charged particle motion, and connections to averaging theory

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Jimmy Fung
PhD Dissertation, Caltech Aeronautics, May 2005

We describe a technique for the efficient computation of the dominant-scale dynamics of a fluid system when only a high-fidelity simulation is available. Such a technique is desirable when governing equations for the dominant scales are unavailable, when model reduction is impractical, or when the original high-fidelity computation is expensive. We adopt the coarse analysis framework proposed by I. G. Kevrekidis (Comm. Math. Sci. 2003), where a computational superstructure is designed to use short-time, high-fidelity simulations to extract the dominant features for a multi- scale system. We apply this technique to compute the dominant features of the compressible flow through a planar diffuser. We apply the proper orthogonal decomposition to classify the dominant and subdominant scales of diffuser flows. We derive a suitable coarse pro jective Adams-Bashforth time integration routine and apply it to compute averaged diffuser flows. The results include accu- rate tracking of the dominant-scale dynamics for a range of parameter values for the computational superstructure. These results demonstrate that coarse analysis methods are useful for solving fluid flow problems of a multiscale nature.

In order to elucidate the behavior of coarse analysis techniques, we make comparisons to averaging theory. To this end, we derive governing equations for the average motion of charged particles in a magnetic field in a number of different settings. First, we apply a novel procedure, inspired by WKB theory and Whitham averaging, to average the variational principle. The resulting equations are equivalent to the guiding center equations for charged particle motion; this marks an instance where averaging and variational principles commute. Secondly, we apply Lagrangian averaging techniques, previously applied in fluid mechanics, to derive averaged equations. Making comparisons to the WKB/Whitham-style derivation allows for the necessary closure of the Lagrangian averaging formulation. We also discuss the Hamiltonian setting and show that averaged Hamiltonian systems may be derivable using concepts from coarse analysis. Finally, we apply a prototypical coarse analysis procedure to the system of charged particles and generate tra jectories that resemble guiding center tra jectories. We make connections to perturbation theory to derive guidelines for the design of coarse analysis techniques and comment on the prototypical coarse analysis application.