A Model for the Unstable Manifold of the Bursting Behavior in the 2D Navier--Stokes Flow

Abstract

Quasi-periodic and bursting behaviors of the two-dimensional (2D) Navier--Stokes flow are analyzed. The tools used are the proper orthogonal decomposition (POD) method and the artificial neural network (ANN) method. The POD is used to extract coherent structures and prominent features from PDE simulations of a quasi-periodic regime and a bursting regime. Eigenfunctions of the two regimes were related by the symmetries of the 2D Navier--Stokes equations. Three eigenfunctions that represent the dynamics of the quasi-periodic regime and two eigenfunctions associated with the unstable manifold of the bursting regime were derived. Calculations of the POD eigenfunctions are performed on the Fourier amplitudes in a comoving frame. Inverse Fourier transform is applied to represent the POD eigenfunctions in both streamfunction and vorticity formulations so that the number of relevant eigenfunctions for streamfunction and vorticity data is the same. Projection onto the two eigenfunctions associated with the unstable manifold reduces the data to two time series. Processing these time series through an ANN results in a low-dimensional model describing the unstable manifold of the bursting regime that can be used to predict the onset of a burst.