Actuation manifold from snapshot data

TL;DR

This work tackles the challenge of obtaining a low-dimensional, control-oriented representation of flows under multiple actuations. It introduces an actuation manifold learned with ISOMAP as the encoder and a neural-network plus $k$NN decoder, applied to the fluidic pinball at $Re = 30$, yielding a 5D embedding with small representation error. Key findings show that the latent coordinates align with physically meaningful actuation mechanisms (boat-tailing, Magnus effect, forward stagnation point) and the wake dynamics (amplitude and phase of vortex shedding), enabling accurate full-state flow estimation from few sensors (cosine similarity near 1). The approach has broad implications for estimation and control of actuation-driven flows, offering a data-driven, interpretable framework that can be extended to other multi-input flow scenarios.

Abstract

We propose a data-driven methodology to learn a low-dimensional manifold of controlled flows. The starting point is resolving snapshot flow data for a representative ensemble of actuations. Key enablers for the actuation manifold are isometric mapping as encoder and a combination of a neural network and a k-nearest-neighbour interpolation as decoder. This methodology is tested for the fluidic pinball, a cluster of three parallel cylinders perpendicular to the oncoming uniform flow. The centres of these cylinders are the vertices of an equilateral triangle pointing upstream. The flow is manipulated by constant rotation of the cylinders, i.e. described by three actuation parameters. The Reynolds number based on a cylinder diameter is chosen to be 30. The unforced flow yields statistically symmetric periodic shedding represented by a one-dimensional limit cycle. The proposed methodology yields a five-dimensional manifold describing a wide range of dynamics with small representation error. Interestingly, the manifold coordinates automatically unveil physically meaningful parameters. Two of them describe the downstream periodic vortex shedding. The other three describe the near-field actuation, i.e. the strength of boat-tailing, the Magnus effect and forward stagnation point. The manifold is shown to be a key enabler for control-oriented flow estimation.

Figures (7)

• Figure 1: Illustration of the methodology for actuation-manifold learning and full-state estimation. The diagram highlights key steps, from flow data collection to data-driven actuation manifold discovery (upper section). A neural network, incorporating actuation parameters ($p_1$, $p_2$ and $p_3$) and sensor information ($s_1$ and $s_2$), determines ISOMAP coordinates ($\gamma_1, \gamma_2, \ldots, \gamma_n$) and a $k$NN decoder is used for the full-state flow reconstruction.
• Figure 2: (a) The Frobenius norm of the geodesic distance matrix plotted against the number of neighbours employed in Floyd's algorithm. (b) The ratio between the connected snapshots and the total number of snapshots as $k_e$ increases. Results of the manifold obtained for $k_e = 40$ are presented in panels (c) and (d). The former illustrates the residual variance of the first $10$ ISOMAP coordinates, whereas the latter showcases all possible manifold sections identified by the first five coordinates.
• Figure 3: Three-dimensional projections of the manifold colour-coded with actuation parameters and forces coefficients. The first (boat-tailing) and second (Magnus) actuation parameters are plotted against lift and drag coefficients to understand physical control mechanisms.
• Figure 4: At the top, representation of the manifold section identified by the coordinates $\gamma_1$ and $\gamma_2$ colour coded with $b_1$ (left), $b_2$ (centre) and $b_3$ (right). At the bottom, a diagram explaining how the coordinate $\gamma_2$ provides indications of horizontal symmetry in the flow field.
• Figure 5: LIC representations of the normalised actuation modes. The shadowed contour represents the local velocity magnitude of the pseudomodes.