Historically, investigations on gust encounters have been limited to thin airfoils. In this work, we examine vortex-gust encounters by a family of airfoils at a chord-based Reynolds number italic Re Subscript c Baseline equals 100 Re c = 100 ${\textit{Re}}_c=100$ , which includes variations in the gust ratio, initial gust position, gust radius, angle of attack, airfoil thickness and airfoil camber. We examine differences in the flow fields, lift-element distributions and aerodynamic responses across several airfoil–gust interactions. We observe a large deviation of the flow fields and aerodynamic responses with respect to the baseline flows for increasing gust ratios and gust sizes. The initial position of the vortex gust influences the magnitude of the velocity gradients observed near the leading edge, effectively heightening or mitigating the amplitude of the lift response. Moreover, the lift fluctuation increases with the angle of attack until it flattens around 10 Superscript ring 10 ∘ $10^\circ$ , reminiscent of an unsteady stall-like regime. Furthermore, we report a decrease in the amplitude of the gust-induced lift fluctuations for thicker airfoils, which we attribute to a decrease in the vorticity production levels from the leading edge, as well as a spatial redirection of the transient aerodynamic forces, which shifts the response in favour of drag fluctuations. The exploration of a sensitive subset of the parameter space uncovers relevant trends, shedding light on regions that have received limited attention in past studies, with special focus on the influence of airfoil geometry.

The fluid dynamics community has increasingly adopted machine learning to analyze, model, predict, and control a wide range of flows. These methods offer powerful computational capabilities for regression, compression, and optimization. In some cases, machine learning has even outperformed traditional approaches. However, many fluid mechanics problems remain beyond the reach of current machine learning techniques. As the field moves from its current state toward a more mature paradigm, this article offers a critical assessment of the key challenges that must be addressed. Tackling these technical issues will not only deepen our understanding of flow physics but also expand the applicability of machine learning beyond fundamental research. We also highlight the importance of community-maintained datasets and open-source code repositories to accelerate progress in this area. Furthermore, the future success of machine learning in fluid dynamics will depend on effective training-not only for the next generation of researchers but also for established fluid mechanicians adapting to this evolving landscape. Data-driven fluid dynamics is in its critical transitional state over the next few years to shape its future. This perspective article aims to spark discussions and encourage collaborative efforts to advance the integration of machine learning in fluid dynamics.

We present a machine-learning method for data-driven synchronization of rhythmic spatiotemporal patterns in reaction-diffusion systems. Extending the phase autoencoder [Yawata et al., Chaos 34, 063111 (2024)1054-150010.1063/5.0205718] for low-dimensional oscillators, we develop a framework to map high-dimensional field variables of the reaction-diffusion system to low-dimensional latent variables characterizing the asymptotic phase and amplitudes of the field variables. This yields a reduced phase description of the limit cycle underlying the rhythmic spatiotemporal dynamics in a data-driven manner. We propose a method to drive the system along the tangential direction of the limit cycle, enabling phase control without inducing amplitude deviations. With examples of 1D oscillating spots and 2D spiral waves in the FitzHugh-Nagumo reaction-diffusion system, we show that the method achieves rapid synchronization in both reference-based and coupling-based settings. These results demonstrate the potential of data-driven phase description based on the phase autoencoder for synchronization of high-dimensional spatiotemporal dynamics.

Biological flyers and swimmers navigate in unsteady wake flows using limited sensory abilities and actuation energies. Understanding how vortical structures can be leveraged for energy-efficient navigation in unsteady flows is beneficial in developing autonomous navigation for small-scale aerial and marine vehicles. Such vehicles are typically operated with constrained onboard actuation and sensing capabilities, making energy-efficient trajectory planning critically important. This study finds that trajectory planners can leverage three-dimensionality appearing in a complex unsteady wake for efficient navigation using limited flowfield information. This is revealed with comprehensive investigations by finite-horizon model-predictive control for trajectory planning of a swimmer behind a cylinder wake at Reynolds number of 300. The navigation performance of three-dimensional cases is compared with scenarios in a two-dimensional (2-D) wake. The underactuated swimmer is able to reach the target by leveraging the background flow when the prediction horizon exceeds one-tenth of the wake-shedding period, demonstrating that navigation is feasible with limited information about the flowfield. Further, we identify that the swimmer can leverage the secondary transverse vortical structures to reach the target faster than is achievable navigating in a 2-D wake.

This study seeks a low-rank representation of turbulent flow data obtained from multiple sources. To uncover such a representation, we consider finding a finite-dimensional manifold that captures underlying turbulent flow structures and characteristics. While nonlinear machine-learning techniques can be considered to seek a low-order manifold from flow field data, there exists an infinite number of transformations between data-driven low-order representations, causing difficulty in understanding turbulent flows on a manifold. Finding a manifold that captures turbulence characteristics becomes further challenging when considering multi-source data together due to the presence of inherent noise or uncertainties and the difference in the spatiotemporal length scale resolved in flow snapshots, which depends on approaches in collecting data. With an example of numerical and experimental data sets of transitional turbulent boundary layers, this study considers an observable-augmented nonlinear autoencoder-based compression, enabling data-driven feature extraction with prior knowledge of turbulence. We show that it is possible to find a low-rank subspace that not only captures structural features of flows across the Reynolds number but also distinguishes the data source. Along with machine-learning-based super-resolution, we further argue that the present manifold can be used to validate the outcome of modern data-driven techniques when training and evaluating across data sets collected through different techniques. The current approach could serve as a foundation for a range of analyses including reduced-complexity modelling and state estimation with multi-source turbulent flow data.

This study uncovers low-dimensional latent space representations of plunging airfoil wakes with nonlinear unsupervised machine learning. The dataset is prepared by direct numerical simulations of flow over a NACA0012 airfoil undergoing sinusoidal plunging oscillation at Re=100 across a range of plunging amplitude, reduced frequency, and angle of attack. In the present analysis, the flowfield data is first compressed using a drag-augmented autoencoder, which includes a drag-estimation subnetwork from the latent variables to capture the relationship between vortical structures and wake-related properties such as thrust generation. Using the current method, the diverse wake data can be represented only in three-dimensional coordinates. We find that for the wake dynamics, atomic orbit-like distributions for each wake type are extracted in the three-dimensional space. This enables the classification of different wake patterns in a data-driven manner with the characterization of physical parameters of the plunging airfoil geometrically in each orbit. Moreover, the low-order coordinates can be estimated only from sparse pressure sensors on the airfoil surface, allowing sensor-input-based unsupervised wake characterization. This low-dimensional representation achieved by nonlinear machine learning offers possibilities for low-dimensional modeling and control in feature space.

A wide range of techniques exist for extracting the dominant flow dynamics and features about steady, or periodic base flows. However, there have been limited efforts in extracting the dominant dynamics about unsteady, aperiodic base flow. These flows appear in many applications such as when there is a sudden change in the flow rate through a pipe, when an airfoil experiences stall, or when a vortex forms. For these unsteady flows, it is valuable to know not only the dynamics of the base flow but also the features that form around this base flow. Here, we discuss the current state of research on extracting important flow structures and their dynamics in such cases with time-varying base flows. In particular, we consider data-driven decompositions, operator-based methods, causality analysis, and some other approaches. We also offer an outlook and call attention to key areas that require future efforts.

The transition to electric vehicles is driving a fundamental shift in the automobile design process. Changes in constraints afforded by the absence of a combustion engine create new opportunities for modifying vehicle geometries. Current approaches to optimizing vehicle aerodynamics require a vast amount of computational studies and physical experiments, which are expensive when performing parameter sweeps over conceivable geometric configurations, suggesting the need for more efficient surrogate models to assist analysis. Here we analyze a dataset of industry-quality automobile geometries with their associated aerodynamic performance obtained from experimentally validated, high-fidelity large-eddy simulations. We show that a relationship between these geometries and their respective aerodynamics can be extracted in a low-dimensional manner by leveraging a nonlinear autoencoder which is simultaneously trained to estimate the drag coefficient from the latent variables. We perform aerodynamic design optimization of vehicle designs by making use of the learned aerodynamic relationship in the low-order space obtained by the model. We demonstrate that the aerodynamic trends for the geometries produced from the optimization process show agreement with validation simulations. The findings of this work demonstrate the application of data-driven approaches to the analysis and design of vehicles in a production environment.

We present a data-driven feedforward control to attenuate large transient lift experienced by an airfoil disturbed by an extreme level of discrete vortex gust. The current analysis uses a nonlinear machine-learning technique to compress the high-dimensional flow dynamics onto a low-dimensional manifold. While the interaction dynamics between the airfoil and extreme vortex gust are parametrized by its size, gust ratio and position, the wake responses are well captured on this simple manifold. The effect of extreme vortex disturbance about the undisturbed baseline flows can be extracted in a physically interpretable manner. Furthermore, we call on phase-amplitude reduction to model and control the complex nonlinear extreme aerodynamic flows. The present phase-amplitude reduction model reveals the sensitivity of the dynamical system in terms of the phase shift and amplitude change induced by external forcing with respect to the baseline periodic orbit. By performing the phase-amplitude analysis for a latent dynamical model identified by sparse regression, the sensitivity functions of low-dimensionalized aerodynamic flows for both phase and amplitude are derived. With the phase and amplitude sensitivity functions, optimal forcing can be determined to quickly suppress the effect of extreme vortex gusts towards the undisturbed states in a low-order space. The present optimal flow modification built upon the machine-learned low-dimensional subspace quickly alleviates the impact of transient vortex gusts for a variety of extreme aerodynamic scenarios, providing a potential foundation for flight of small-scale air vehicles in adverse atmospheric conditions.

Modern machine-learning techniques are generally considered data-hungry. However, this may not be the case for turbulence as each of its snapshots can hold more information than a single data file in general machine-learning settings. This study asks the question of whether nonlinear machine-learning techniques can effectively extract physical insights even from as little as a single snapshot of turbulent flow. As an example, we consider machine-learning-based super-resolution analysis that reconstructs a high-resolution field from low-resolution data for two examples of two-dimensional isotropic turbulence and three-dimensional turbulent channel flow. First, we reveal that a carefully designed machine-learning model trained with flow tiles sampled from only a single snapshot can reconstruct vortical structures across a range of Reynolds numbers for two-dimensional decaying turbulence. Successful flow reconstruction indicates that nonlinear machine-learning techniques can leverage scale-invariance properties to learn turbulent flows. We also show that training data of turbulent flows can be cleverly collected from a single snapshot by considering characteristics of rotation and shear tensors. Second, we perform the single-snapshot super-resolution analysis for turbulent channel flow, showing that it is possible to extract physical insights from a single flow snapshot even with inhomogeneity. The present findings suggest that embedding prior knowledge in designing a model and collecting data is important for a range of data-driven analyses for turbulent flows. More broadly, this work hopes to stop machine-learning practitioners from being wasteful with turbulent flow data.

Estimation of aerodynamic loads is a significant challenge in complex gusty environments due to the associated complexities of flow separation and strong nonlinearities. In this study, we explore the practical feasibility of multilayer perceptron (MLP) for estimating aerodynamic loads in gusts, when confounded by noisy and spatially distributed sparse surface pressure measurements. As a demonstration, a nonslender delta wing experiencing various gusts with different initial and final conditions is considered. Time-resolved lift and drag, and spatially distributed sparse surface pressure measurements are collected in a towing-tank facility. The nonlinear MLP model is used to estimate gust scenarios that are unseen in training progress. A filtering process allows us to examine the fluctuation of the dynamic response from the pressure measurements on the MLP. Estimation results show that the MLP model is able to capture the relationship between surface pressure and aerodynamic loads with a minimum quantity of learning samples, delivering accurate estimations, despite the slightly large errors for the cases at the boundary of the datasets. The results also indicate that the dynamic response of the pressure measurements has an influence on the learning of MLP. We further utilize gradient maps to perform a sensitivity analysis, so as to evaluate the contribution of the pressure data to the estimation of gust loads. This study reveals the significant contribution of the sensors located near the leading edge and at the nose of the delta wing. Our findings suggest the potential for an efficient sensor deployment strategy in data-driven aerodynamic load estimation.

At high incidence, low-aspect-ratio wings present a unique set of aerodynamic characteristics, including flow separation, vortex shedding and unsteady force production. Furthermore, low-aspect-ratio wings exhibit a highly impactful tip vortex, which introduces strong spanwise gradients into an already complex flow. In this work, we explore the interaction between leading-edge flow separation and a strong, persistent tip vortex over a Reynolds number range of $600 \leq Re \leq 10{\,}000$ . In performing this study, we aim to bridge the insight gained from existing low-Reynolds-number studies of separated flow on finite wings ( $Re \approx 10^2$ ) and turbulent flows at higher Reynolds numbers ( $Re \approx 10^4$ ). Our study suggests two primary effects of the Reynolds number. First, we observe a break from periodicity, along with a dramatic increase in the intensity and concentration of small-scale eddies, as we shift from $Re = 600$ to $Re = 2500$ . Second, we observe that many of our flow diagnostics, including the time-averaged aerodynamic force, exhibit reduced sensitivity to Reynolds number beyond $Re = 2500$ , an observation attributed to the stabilising impact of the wing tip vortex. This latter point illustrates the manner by which the tip vortex drives flow over low-aspect-ratio wings, and provides insight into how our existing understanding of this flow field may be adjusted for higher-Reynolds-number applications.

We perform direct numerical simulations of actively controlled laminar separated wakes around low-aspect-ratio wings with two primary goals: (i) reducing the size of the separation bubble and (ii) attenuating the wing tip vortex. Instead of preventing separation, we modify the three-dimensional (3-D) dynamics to exploit wake vortices for aerodynamic enhancements. A direct wake modification is considered using optimal harmonic forcing modes from triglobal resolvent analysis. For this study, we consider wings at angles of attack of $14^\circ$ and $22^\circ$ , taper ratios $0.27$ and $1$ , and leading edge sweep angles of $0^\circ$ and $30^\circ$ , at a mean-chord-based Reynolds number of $600$ . The wakes behind these wings exhibit 3-D reversed-flow bubble and large-scale vortical structures. For tapered swept wings, the diversity of wake vortices increases substantially, posing a challenge for flow control. To achieve the first control objective for an untapered unswept wing, root-based actuation at the shedding frequency is introduced to reduce the reversed-flow bubble size by taking advantage of the wake vortices to significantly enhance the aerodynamic performance of the wing. For both untapered and tapered swept wings, root-based actuation modifies the stalled flow, reduces the reversed-flow region and enhances aerodynamic performance by increasing the root contribution to lift. For the goal of controlling the tip vortex, we demonstrate the effectiveness of actuation with high-frequency perturbations near the tip. This study shows how insights from resolvent analysis for unsteady actuation can enable global modification of 3-D separated wakes and achieve improved aerodynamics of wings.

Nonlinear machine learning for turbulent flows can exhibit robust performance even outside the range of training data. This is achieved when machine-learning models can accommodate scale-invariant characteristics of turbulent flow structures. This study presents a data-driven approach to reveal scale-invariant vortical structures across Reynolds numbers that provide insights for supporting nonlinear machine-learning-based studies of turbulent flows. To uncover conditions for which nonlinear models are likely to perform well, we use a Buckingham-Pi-based sparse nonlinear scaling to find the influence of the Pi groups on the turbulent flow data. We consider nonlinear scalings of the invariants of the velocity gradient tensor for an example of three-dimensional decaying isotropic turbulence. The present scaling not only enables the identification of vortical structures that are interpolatory and extrapolatory for the given flow field data but also captures non-equilibrium effects of the energy cascade. As a demonstration, the present findings are applied to machine-learning-based super-resolution analysis of three-dimensional isotropic turbulence. We show that machine-learning models reconstruct vortical structures well in the interpolatory space with reduced performance in the extrapolatory space revealed by the nonlinearly scaled invariants. The present approach enables us to depart from labelling turbulent flow data with a single parameter of Reynolds number and comprehensively examine the flow field to support training and testing of nonlinear machine-learning techniques.

Analyzing the perturbation dynamics is challenging for unsteady base flows due to their transient dynamics and the high degree of freedom. We leverage a linear reduced-order method based on the optimally time-dependent (OTD) bases to trace the dominant amplified directions of vortex-airfoil interaction. We consider complex transient dynamics with a positive vortical gust advecting over a NACA 0012 airfoil at the angle of incidence of 12 degrees. The flow around the airfoil is at a chord-length-based Reynolds number of 400. This gust incites vortex shedding above the airfoil and wake fluctuations behind the airfoil trailing edge (TE). We reveal the transitions of the most amplified region from the leading edge region to the negative leading-edge vortex, with a secondary amplified direction due to the wake deformation behind the airfoil TE. This study provides physical insights into the time-evolving perturbation amplification of transient, non-periodic, and nonlinear flows.

We present a phase autoencoder that encodes the asymptotic phase of a limit-cycle oscillator, a fundamental quantity characterizing its synchronization dynamics. This autoencoder is trained in such a way that its latent variables directly represent the asymptotic phase of the oscillator. The trained autoencoder can perform two functions without relying on the mathematical model of the oscillator: first, it can evaluate the asymptotic phase and the phase sensitivity function of the oscillator; second, it can reconstruct the oscillator state on the limit cycle in the original space from the phase value as an input. Using several examples of limit-cycle oscillators, we demonstrate that the asymptotic phase and the phase sensitivity function can be estimated only from time-series data by the trained autoencoder. We also present a simple method for globally synchronizing two oscillators as an application of the trained autoencoder.

We investigate the effect of three-dimensionality on the synchronisation characteristics of the wake behind an oscillating circular cylinder at ${\textit {Re}} = 300$. Cylinder oscillations in rotation, transverse translation and streamwise translation are considered. We utilise phase-reduction analysis, which quantifies the phase-sensitivity function of periodic flows, to examine the synchronisation properties. Here, we present an ensemble-based framework for phase-reduction analysis to handle three-dimensional wakes that are not perfectly time-periodic. Based on the phase-sensitivity functions, synchronisability to three types of cylinder oscillations is evaluated. In spite of similar trends, we find that phase-sensitivity functions involving three-dimensional wakes are lower in magnitude compared with those of two-dimensional wakes, which leads to narrower conditions for synchronisation to weak cylinder oscillations. We unveil that the difference between the phase-sensitivity functions of two- and three-dimensional flows is strongly correlated to the amplitude variation of the three-dimensional flow by the cylinder motions. This finding reveals that the cylinder motion modifies the three-dimensionality of the wake as well as the phase of vortex shedding, which leads to reduced phase modulation. The synchronisation conditions of three-dimensional wakes, predicted by phase-reduction analysis, agree with the identification by parametric studies using direct numerical simulations for forced oscillations with small amplitudes. This study presents the potential capability of phase-reduction to study synchronisation characteristics of complex flows.

Large amplitude gust encounters exhibit a range of separated flow phenomena, making them difficult to characterize using the traditional tools of aerodynamics. In this work, we propose a dynamical systems approach to gust encounters, viewing the flow as a cycle (or a closed trajectory) in state space. We posit that the topology of this cycle, or its shape and structure, provides a compact description of the flow, and can be used to identify coordinates in which the dynamics evolve in a simple, intuitive way. To demonstrate this idea, we consider flowfield measurements of a transverse gust encounter. For each case in the dataset, we characterize the full-state dynamics of the flow using persistent homology, a tool that identifies holes in point cloud data, and transform the dynamics to a reduced-order space using a nonlinear autoencoder. Critically, we constrain the autoencoder such that it preserves topologically relevant features of the original dynamics, or those features identified by persistent homology. Using this approach, we are able to transform six separate gust encounters to a three-dimensional latent space, in which each gust encounter reduces to a simple circle, and from which the original flow can be reconstructed. This result shows that topology can guide the creation of low-dimensional state representations for strong transverse gust encounters, a crucial step toward the modeling and control of airfoil-gust interactions.