A model for the unstable manifold of the bursting behavior in the 2D Navier-Stokes flow

Quasi-periodic and bursting behaviors of the two- dimensional (2D) Navier S tokes flow are analyzed. The tools used are the proper orthogonal decomposi tion (POD) method and the artificial neural network (ANN) method. The POD i s used to extract coherent structures and prominent features from PDE simul ations of quasi-periodic regime and bursting regime. Eigenfunctions of the two regimes were related by the symmetries of the 2D Navier-Stokes equation s. Three eigenfunctions that represent the dynamics of the quasi-periodic r egime and two eigenfunctions associated with the unstable manifold of the b ursting regime were derived. Calculations of the POD eigenfunctions are per formed on the Fourier amplitudes in comoving frame. Inverse Fourier transfo rm is applied to represent the POD eigenfunctions in both streamfunction an d vorticity formulations so that the number of relevant eigenfunctions for streamfunction and vorticity data is the same. Projection onto the two eige nfunctions associated with the unstable manifold reduces the data to two ti me series. Processing these time series through an ANN results in low-dimen sional model describing the unstable manifold of the bursting regime that c an be used to predict the onset of burst.