From snapshots to manifolds - a tale of shear flows

We propose a novel nonlinear manifold learning from snapshot data and demonstrate its superiority over proper orthogonal decomposition (POD) for shedding-dominated shear flows. Key enablers are isometric feature mapping, Isomap, as encoder and, K-nearest neighbours (KNN) algorithm as decoder. The proposed technique is applied to numerical and experimental datasets including the fluidic pinball, a swirling jet and the wake behind a couple of tandem cylinders. Analysing the fluidic pinball, the manifold is able to describe the pitchfork bifurcation and the chaotic regime with only three feature coordinates. These coordinates are linked to the vortex-shedding phases and the force coefficients. The manifold coordinates of the swirling jet are comparable to the POD mode amplitudes, yet allow for a more distinct and less noise-sensitive manifold identification. A similar observation is made for the wake of two tandem cylinders. The tandem cylinders are aligned and located at a streamwise distance which corresponds to the transition between the single bluff body and the reattachment regimes of vortex shedding. Isomap unveils these two shedding regimes while the Lissajous plot of the first two POD mode amplitudes features a single circle. The reconstruction error of the manifold model is small compared with the fluctuation level, indicating that the low embedding dimensions contain the coherent structure dynamics. The proposed Isomap-KNN manifold learner is expected to be of great importance in estimation, dynamic modelling and control for a large range of configurations with dominant coherent structures.

Automated summary

We propose a novel nonlinear manifold learning method using Isomap as encoder and K-nearest neighbours algorithm as decoder, and demonstrate its superiority over POD for shedding-dominated shear flows. The method is applied to numerical and experimental datasets including fluidic pinball, swirling jet and wake behind tandem cylinders, and is able to describe the bifurcation, chaotic regime and shedding phases of the flow. The reconstruction error of the manifold model is small, indicating that the low embedding dimensions contain the coherent structure dynamics.