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Speaker : Troy Smoth

Control & Dynamical Systems, Caltech

Title: LOW DIMENSIONAL MODELS OF PLANE COUETTE FLOW USING THE PROPER ORTHOGONAL DECOMPOSITION

Abstract:

This talk describes efforts to model plane Couette flow in a minimal flow unit -- a domain whose spanwise and streamwise extent is just sufficient to maintain turbulence -- by expanding the velocity field as a sum of optimal modes calculated via proper orthogonal decomposition from numerical data. Ordinary differential equations are obtained by Galerkin projection of the Navier-Stokes equations onto these modes. The focus will be on two models consisting of only a few ordinary differential equations, the dynamics of which convincingly approximate the dynamics of plane Couette flow. The "core" of these models is the quadratic normal form describing the generic interaction of Fourier modes of wavenumbers 0, 1 and 2 under the symmetry group O(2) of rotations and reflections. Several types of interesting dynamics for this system, which is a generalisation of the 1:2 modal interaction studied by Armbruster, Guckenheimer, Holmes and others, will also be discussed.