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We mitigate vortex-induced vibrations of a square cylinder at a Reynolds number of 100 using deep reinforcement learning (DRL)-based active flow control (AFC). The proposed method exploits the powerful nonlinear and high-dimensional problem-solving capabilities of DRL, overcoming limitations of linear and model-based control approaches. Three positions of jet actuators including the front, the middle, and the back of the cylinder sides were tested. The DRL agent as a controller is able to optimize the velocity of the jets to minimize drag and lift coefficients and refine the control strategy. The results show that a significant reduction in vibration amplitude of 86%, 79%, and 96% is achieved for the three different positions of the jet actuators, respectively. The DRL-based AFC method is robust under various reduced velocities. This study successfully demonstrates the potential of DRL-based AFC method in mitigating flow-induced instabilities.

We present the first machine-learned multiple-input multiple-output aerodynamic feedback control under varying operating conditions. Closed-loop control is relevant to many fluid dynamic applications ranging from gust mitigation to drag reduction. Existing machine learning control investigations have been mainly applied under steady conditions. The current study leverages gradient-enriched machine learning control (Cornejo Maceda et al. in J Fluid Mech 917:A42, 2021) to identify optimal control laws under unsteady conditions. The approach is exemplified on a coupled oscillator system with unsteady coupling and demonstrated for a generic truck model undergoing a yawing maneuver. Key enablers of the experiment are a rich set of pneumatic actuators and pressure sensors. The results demonstrate the method’s capabilities in identifying an efficient forcing for control under dynamically changing conditions. This automated and generalizable closed-loop control strategy complements and expands the machine learning control field and promises a new fast-track avenue to efficiently control a broader set of fluid flow problems.

We propose the first machine-learned control-oriented flow estimation for multiple-input multiple-output plants. Starting point is constant actuation with open-loop actuation commands leading to a database with simultaneously recorded actuation commands, sensor signals and flow fields. A key enabler is an estimator input vector comprising sensor signals and actuation commands. The mapping from the sensor signals and actuation commands to the flow fields is realized in an analytically simple, data-centric and general nonlinear approach. The analytically simple estimator generalizes Linear Stochastic Estimation (LSE) for actuation commands. The data-centric approach yields flow fields from estimator inputs by interpolating from the database -- similar to Loiseau et al. (2018) for unforced flow. The interpolation is performed with k Nearest Neighbors (kNN). The general global nonlinear mapping from inputs to flow fields is obtained from a Deep Neural Network (DNN) via an iterative training approach. The estimator comparison is performed for the fluidic pinball plant, which is a multiple-input, multiple-output wake control benchmark (Deng et al. 2020) featuring rich dynamics under steady controls. We conclude that the machine learning methods clearly outperform the linear model. The performance of kNN and DNN estimators are comparable for periodic dynamics. Yet, DNN performs consistently better when the flow is chaotic. Moreover, a thorough comparison regarding to the complexity, computational cost, and prediction accuracy is presented to demonstrate the relative merits of each estimator. The proposed method can be generalized for closed-loop flow control plants., 34 pages, 27 figures, 4 tables

Flow, turbulence and combustion (2020). doi:10.1007/s10494-020-00110-8, Published by Springer Science + Business Media B.V., Dordrecht [u.a.]

International audience

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, asymmetric periodic 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 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.

We stabilize an open cavity flow experiment to 1% of its original fluctuation level. For the first time, a multi-modal feedback control is automatically learned for this configuration. The key enabler is automatic in-situ optimization of control laws with machine learning augmented by a gradient descent algorithm, named gradient-enriched machine learning control (Cornejo Maceda et al. 2021, gMLC). gMLC is shown to learn one order of magnitude faster than MLC (Duriez et al. 2017, MLC). The physical interpretation of the feedback mechanism is assisted by a novel cluster-based control law visualization for flow dynamics and corresponding actuation commands. Starting point of the control experiment are two unforced open cavity benchmark configurations: a narrow-bandwidth regime with a single dominant frequency and a mode-switching regime where two frequencies compete. The feedback control commands the DBD actuator located at the leading edge. The flow is monitored by a downstream hot-wire sensor over the trailing edge. The feedback law is optimized with respect to the monitored fluctuation level. As reference, the self-oscillations of the mixing layer are mitigated with steady actuation. Then, a feedback controller is optimized with gMLC. As expected, feedback control outperforms steady actuation by achieving both, a better amplitude reduction and a significantly smaller actuation power, about about 1% of the actuation energy required for similarly effective steady forcing. Intriguingly, optimized laws learned for one regime performs well for the other untested regime as well. The proposed control strategy can be expected to be applicable for many other shear flow experiments., Comment: 48 pages, 25 figures

Deep learning has been widely utilized to accurately estimate the flow state from the sparse sensor measurements. Yet there is still a lack of understanding of the actual dimension of this type of regression problem. In this study, we propose an Autoencoder (AE) based estimation method to tackle the control-oriented, sensor-based flow estimation problem. This method encodes the input information into a few-nodes bottleneck, then decodes the compressed information to the estimated the flow state. This network stands for the least order representation of the input-output relationship. We choose the fluidic pinball as the benchmark problem which contains multiple-input and multiple-output. The rotations of three cylinders in the flow generates rich flow physics. Consequently, strong non-linearity arises between input measurements and output flow state. For fair comparison the Deep Neural Network (DNN) is also employed to highlight the influence of the encoding-decoding process in AE. The results show that when the flow is periodic, the input vector can be effectively encoded into a five-dimensional subspace without a significant loss of estimation accuracy. However, the five-dimensional encoding will eliminate partial information from the input vector when the forced flow becomes chaotic, resulting in a lower estimation accuracy than DNN.

Generically, a local bifurcation only affects a single solution branch. However, branches that are quite different may nonetheless share certain eigenvectors and eigenvalues, leading to coincident bifurcations. For the fluidic pinball, two supercritical pitchfork bifurcations, of the equilibrium and the periodic solutions, occur at nearly the same Reynolds number. The mechanism of this kind of non-generic coincidence is modelled and explained.

We present the first general metric for attractor overlap (MAO) facilitating an unsupervised comparison of flow data sets. The starting point is two or more attractors, i.e., ensembles of states representing different operating conditions. The proposed metric generalizes the standard Hilbert-space distance between two snapshots to snapshot ensembles of two attractors. A reduced-order analysis for big data and many attractors is enabled by coarse-graining the snapshots into representative clusters with corresponding centroids and population probabilities. For a large number of attractors, MAO is augmented by proximity maps for the snapshots, the centroids, and the attractors, giving scientifically interpretable visual access to the closeness of the states. The coherent structures belonging to the overlap and disjoint states between these attractors are distilled by few representative centroids. We employ MAO for two quite different actuated flow configurations: (1) a two-dimensional wake of the fluidic pinball with vortices in a narrow frequency range and (2) three-dimensional wall turbulence with broadband frequency spectrum manipulated by spanwise traveling transversal surface waves. MAO compares and classifies these actuated flows in agreement with physical intuition. For instance, the first feature coordinate of the attractor proximity map correlates with drag for the fluidic pinball and for the turbulent boundary layer. MAO has a large spectrum of potential applications ranging from a quantitative comparison between numerical simulations and experimental particle-image velocimetry data to the analysis of simulations representing a myriad of different operating conditions., Comment: 33 pages, 20 figures

We propose an open-source Python platform for applications of deep reinforcement learning (DRL) in fluid mechanics. DRL has been widely used in optimizing decision making in nonlinear and high-dimensional problems. Here, an agent maximizes a cumulative reward by learning a feedback policy by acting in an environment. In control theory terms, the cumulative reward would correspond to the cost function, the agent to the actuator, the environment to the measured signals, and the learned policy to the feedback law. Thus, DRL assumes an interactive environment or, equivalently, a control plant. The setup of a numerical simulation plant with DRL is challenging and time-consuming. In this work, a novel Python platform, namely DRLinFluids, is developed for this purpose, with DRL for flow control and optimization problems in fluid mechanics. The simulations employ OpenFOAM as a popular, flexible Navier–Stokes solver in industry and academia, and Tensorforce or Tianshou as widely used versatile DRL packages. The reliability and efficiency of DRLinFluids are demonstrated for two wake stabilization benchmark problems. DRLinFluids significantly reduces the application effort of DRL in fluid mechanics, and it is expected to greatly accelerate academic and industrial applications.

We propose an aerodynamic force model associated with a Galerkin model for the unforced fluidic pinball, the two-dimensional flow around three equal cylinders with one radius distance to each other. The starting point is a Galerkin model of a bluff-body flow. The force on this body is derived as a constant-linear-quadratic function of the mode amplitudes from first principles following the pioneering work of Noca (On the evaluation of time-dependent fluid-dynamic forces on bluff bodies. PhD thesis, California Institute of Technology, 1997), Noca et al. (J. Fluids Struct., vol. 13, issue 5, 1999, pp. 551–578) and Liang & Dong (Virtual force measurement of POD modes for a flat plate in low Reynolds number flows. AIAA Paper 2014-0054). The force model is simplified for the mean-field model of the unforced fluidic pinball (Deng et al., J. Fluid Mech., vol. 884, 2020, p. A37) using symmetry properties and sparse calibration. The model is successfully applied to transient and post-transient dynamics in different Reynolds number regimes: the periodic vortex shedding after the Hopf bifurcation and the asymmetric vortex shedding after the pitchfork bifurcation comprising six different Navier–Stokes solutions. We foresee many applications of the Galerkin force model for other bluff bodies and flow control.

A generally applicable robust data-driven network modeling strategy offers rapid means to predict and control complex systems., We propose a universal method for data-driven modeling of complex nonlinear dynamics from time-resolved snapshot data without prior knowledge. Complex nonlinear dynamics govern many fields of science and engineering. Data-driven dynamic modeling often assumes a low-dimensional subspace or manifold for the state. We liberate ourselves from this assumption by proposing cluster-based network modeling (CNM) bridging machine learning, network science, and statistical physics. CNM describes short- and long-term behavior and is fully automatable, as it does not rely on application-specific knowledge. CNM is demonstrated for the Lorenz attractor, ECG heartbeat signals, Kolmogorov flow, and a high-dimensional actuated turbulent boundary layer. Even the notoriously difficult modeling benchmark of rare events in the Kolmogorov flow is solved. This automatable universal data-driven representation of complex nonlinear dynamics complements and expands network connectivity science and promises new fast-track avenues to understand, estimate, predict, and control complex systems in all scientific fields.

The transfer of internal energy fluctuation is numerically investigated for the stationary compressible isotropic turbulence in vibrational non-equilibrium with large-scale thermal forcing. We observe the spectra of velocity, solenoidal pressure component, density and temperatures all exhibiting the scaling in the inertial range. The Helmholtz decomposition results reveal that the solenoidal velocity component predominates over the dilatational component. Fluctuations of the solenoidal velocity and pressure components are nearly insensitive to the turbulent Mach number and vibrational relaxation, while those of the dilatational velocity and pressure components are closely related to them. In addition, the weak and strong acoustic equilibrium hypotheses are verified. On global average, the dissipation of translational–rotational energy fluctuation stems mainly from the thermal conduction and vibrational relaxation, while effects of the dilatation and viscosity are negligible. For the vibrational energy fluctuation, the dilatation effect is insignificant, while the dissipation due to the thermal conduction is roughly equivalent to the production owing to the vibrational relaxation. The cascades of translational–rotational and vibrational energy fluctuations are mainly dominated by the solenoidal component of filtered velocity. The direct subgrid-scale (SGS) fluxes of translational–rotational and vibrational energy fluctuations due to the dilatational component of filtered velocity in the compression region are balanced by the reverse SGS fluxes in the expansion region. On the other hand, the dependencies of the SGS fluxes due to the solenoidal component of filtered velocity on the local compressibility are closely related to the relaxation effect. The sensibility gradually disappears as the relaxation effect weakens.