We propose the first least-order Galerkin model of an incompressible flow undergoing two successive supercritical bifurcations of Hopf and pitchfork type. A key enabler is a mean-field consideration exploiting the symmetry of the mean flow and the asymmetry of the fluctuation. These symmetries generalize mean-field theory, e.g. no assumption of slow growth-rate is needed. The resulting 5-dimensional Galerkin model successfully describes the phenomenogram of the fluidic pinball, a two-dimensional wake flow around a cluster of three equidistantly spaced cylinders. The corresponding transition scenario is shown to undergo two successive supercritical bifurcations, namely a Hopf and a pitchfork bifurcations on the way to chaos. The generalized mean-field Galerkin methodology may be employed to describe other transition scenarios. †

We propose a self-supervised cluster-based hierarchical reduced-order modelling methodology to model and analyse the complex dynamics arising from a sequence of bifurcations for a two-dimensional incompressible flow of the fluidic pinball. The hierarchy is guided by a triple decomposition separating a slowly varying base flow, dominant shedding and secondary flow structures. All these flow components are kinematically resolved by a hierarchy of clusters. The transition dynamics between these clusters is described by a directed network, called the cluster-based hierarchical network model (HiCNM). Three consecutive Reynolds number regimes for different dynamics are considered: (i) periodic shedding at $Re=80$ , (ii) quasi-periodic shedding at $Re=105$ and (iii) chaotic shedding at $Re=130$ , involving three unstable fixed points, three limit cycles, two quasi-periodic attractors and a chaotic attractor. The HiCNM enables identification of the dynamics between multiple invariant sets in a self-supervised manner. Both the global trends and the local structures during the transition are well resolved by a moderate number of hierarchical clusters. The proposed HiCNM provides a visual representation of transient and post-transient, multi-frequency, multi-attractor behaviour and may automate the identification and analysis of complex dynamics with multiple scales and multiple invariant sets.

The aim of our work is to advance a self-learning, model-free control method to tame complex nonlinear flows-building on the pioneering work of Dracopoulous [1]. The cornerstone is the formulation of the control problem as a function optimization problem. The control law is derived by solving a nonsmooth optimization problem thanks to an artificial intelligence technique, genetic programming (GP). Metaparameters optimization of the algorithm and complexity penalization have been our main contribution and have been tested on a cluster of three equidistant cylinders immersed in a incoming flow, the fluidic pinball. The means of control is the independent rotation of the cylinders. GP derived a control law associated to each cylinder in order to minimize the net drag power and managed to outperform past open-loop studies with a 46.0 % net drag power reduction by combining two strategies from literature. This success of MIMO control including sensor history is promising for exploring even more complex dynamics.

The fluidic pinball has been recently proposed as an attractive and effective flow configuration for exploring machine learning fluid flow control. In this contribution, we focus on the route to chaos in this system without actuation, as the Reynolds number is smoothly increased. It was found to be of the Newhouse-Ruelle-Takens kind, with a secondary pitchfork bifurcation that breaks the symmetry of the mean flow field on the route to quasi-periodicity.

Abstract

Coal will still be China’s basic energy for quite a long time. With the increase of mining depth, gas content and pressure also increase. The problems of gas emission and overrun affect the safety and efficient production of coal resource to a certain extent. In this work, the field test of gas drainage borehole peeping and trajectory measurement in coal seam of Changling coal mine are carried out. These coal seams include C5b coal seam, upper adjacent C5a coal seam, C6a coal seams, C6c in lower adjacent strata, and C5b coal seam in high-level borehole. The view of gas drainage borehole peeping and trajectory measurement in the working seam, upper adjacent layer, lower adjacent layer, and high position are obtained. It is found that the hole collapses at the position of about 20 m in both adjacent strata and high-level boreholes, and there are a lot of cracks in the high-level boreholes before 12 m. The deviation distance of high-level borehole is large, and the actual vertical deviation of upper adjacent layer is small. Finally, the strategies to prevent the deviation of drilling construction are put forward. It includes four aspects: ensuring the reliability of drilling equipment, reasonably controlling the drilling length, standardizing the drilling, and reasonably selecting the drilling process parameters.

We propose a cluster-based control (CBC) strategy for model-free feedback drag reduction with multiple actuators and full-state feedback. CBC consists of three steps. First, the input of the feedback law is clustered from unforced flow data. Second, the feedback law is interpolated with actuation commands associated with the cluster centroids. Thus,centroids and these actuation commands facilitate a low-dimensional parameterization of the feedback law. Third, the centroid-based actuation commands are optimized, e.g., with a downhill simplex method. This framework generalizes the feature-based CBC from Nair et al. ["Cluster-based feedback control of turbulent post-stall separated flows," J. Fluid Mech. 875, 345-375 (2019)] in three aspects. First, the control law input is the velocity field. Second, the control law output commands multiple actuators here. Third, a reformulation of the downhill simplex method allows parallelizing the simulations, thus accelerating the computation threefold. Full-state CBC is demonstrated on a multiple-input configuration, the so-called fluidic pinball in three flow regimes, including symmetric periodic at Re = 30, asymmetricperiodic at Re = 100, and chaotic vortex shedding at Re = 150. The net drag reductions for the three cases amount to 33.06%, 24.15%, and 12.23%, respectively. CBC shows distinct advantages for robustness control at different flow conditions. The full-state CBC further reveals the evolution of the control flow associated with the centroids, which contributes to the physical interpretation of the feedback control process.

We propose a novel trajectory-optimized Cluster-based Network Model (tCNM) for nonlinear model order reduction from time-resolved data following Li et al. ["Cluster-based network model," J. Fluid Mech. 906, A21 (2021)] and improving the accuracy for a given number of centroids. The starting point is k-means++ clustering which minimizes the representation error of the snapshots by their closest centroids. The dynamics is presented by 'flights' between the centroids. The proposed trajectory-optimized clustering aims to reduce the kinematic representation error further by shifting the centroids closer to the snapshot trajectory and refining state propagation with trajectory support points. Thus, curved trajectories are better resolved. The resulting tCNM is demonstrated for the sphere wake for three flow regimes, including the periodic, quasi-periodic, and chaotic dynamics. The representation error of tCNM is 5 times smaller as compared to the approximation by the closest centroid. Thus, the error at the same level as Proper Orthogonal Decomposition (POD) of same order. Yet, tCNM has distinct advantages over POD modeling: it is human interpretable by representing dynamics by a handful of coherent structures and their transitions; it shows robust dynamics by design, i.e., stable long-time behavior; and its development is fully automatable, i.e., it does not require tuneable auxiliary closure and other models.