We find the optimally time-dependent (OTD) orthogonal modes about a time-varying flow generated by a strong gust vortex impacting a NACA 0012 airfoil. This OTD analysis reveals the amplification characteristics of perturbations about the unsteady base flow and their amplified spatiotemporal structures that evolve over time. We consider four time-varying laminar base flows in which a vortex with a strength corresponding to the gust ratio $G$ of $\{-1,-0.5,0.5,1\}$ impinges on the leading edge of the airfoil at an angle of attack of $12^\circ$. In these cases, the impingement of the strong gust vortex causes massive separation and the generation of large-scale vortices around the airfoil within two convective time units. As these flow structures develop around the airfoil on a short time scale, the airfoil experiences large transient vortical lift variations in the positive and negative directions that are approximately five to ten times larger than the baseline lift. The highly unsteady nature of these vortex–airfoil interactions necessitates an advanced analytical technique capable of capturing the transient perturbation dynamics. For each of the considered gust ratios, the OTD analysis identifies the most amplified region to perturbations, the location of which changes as the wake evolves differently. For interactions between a moderate positive vortex gust ($G=0.5$) and the airfoil, the area where perturbations are amplified transitions from the leading-edge vortex (LEV) sheet to the forming LEV. Later, this most amplified structure becomes supported in the airfoil wake directly behind the trailing edge. In contrast, a strong vortex gust ($G=\pm 1$) encountered by the airfoil shows the most amplified OTD mode to appear around the core of the shed vortices. This study provides an analysis technique and fundamental insights into the broader family of unsteady aerodynamic problems.

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.

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.

We study the effect of acceleration and deceleration on the stability of channel flows. To do so, we derive an exact solution for laminar profiles of channel flows with an arbitrary, time-varying wall motion and pressure gradient. This solution then allows us to investigate the stability of any unsteady channel flow. In particular, we restrict our investigation to the non-normal growth of perturbations about time-varying base flows with exponentially decaying acceleration and deceleration, with comparisons to growth about a constant base flow (i.e. the time-invariant simple shear or parabolic profile). We apply this acceleration and deceleration through the velocity of the walls and through the flow rate. For accelerating base flows, perturbations never grow larger than perturbations about a constant base flow, while decelerating flows show massive amplification of perturbations – at a Reynolds number of $500$, properly timed perturbations about the decelerating base flow grow $ {O}(10^5)$ times larger than perturbations grow about a constant base flow. This amplification increases as we raise the rate of deceleration and the Reynolds number. We find that this amplification arises due to a transition from spanwise perturbations leading to the largest amplification to streamwise perturbations leading to the largest amplification that only occurs in the decelerating base flow. By evolving the optimal perturbations through the linearized equations of motion, we reveal that the decelerating base flow achieves this massive amplification through the Orr mechanism, or the down-gradient Reynolds stress mechanism, which accelerating and constant base flows cannot maintain.

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.

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.

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.

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.

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 towards the modelling and control of aerofoil–gust interactions.

While tapered swept wings are widely used, the influence of taper on their post-stall wake characteristics remains largely unexplored. To address this issue, we conduct an extensive study using direct numerical simulations to characterize the wing taper and sweep effects on laminar separated wakes. We analyse flows behind NACA 0015 cross-sectional profile wings at post-stall angles of attack $\alpha =14^\circ$–$22^\circ$ with taper ratios $\lambda =0.27$–$1$, leading-edge sweep angles $0^\circ$–$50^\circ$ and semi aspect ratios $sAR =1$ and $2$ at a mean-chord-based Reynolds number of $600$. Tapered wings have smaller tip chord length, which generates a weaker tip vortex, and attenuates inboard downwash. This results in the development of unsteadiness over a large portion of the wingspan at high angles of attack. For tapered wings with backward-swept leading edges, unsteadiness emerges near the wing tip. On the other hand, wings with forward-swept trailing edges are shown to concentrate wake-shedding structures near the wing root. For highly swept untapered wings, the wake is steady, while unsteady shedding vortices appear near the tip for tapered wings with high leading-edge sweep angles. For such wings, larger wake oscillations emerge near the root as the taper ratio decreases. While the combination of taper and sweep increases flow unsteadiness, we find that tapered swept wings have more enhanced aerodynamic performance than untapered and unswept wings, exhibiting higher time-averaged lift and lift-to-drag ratio. The current findings shed light on the fundamental aspects of flow separation over tapered wings in the absence of turbulent flow effects.

We obtain an optimal actuation waveform for fast synchronization of periodic airfoil wakes through the phase reduction approach. Using the phase reduction approach for periodic wake flows, the spatial sensitivity fields with respect to the phase of the vortex shedding are obtained. The phase sensitivity fields can uncover the synchronization properties in the presence of periodic actuation. This study seeks a periodic actuation waveform using phase-based analysis to minimize the time for synchronization to modify the wake shedding frequency of NACA0012 airfoil wakes. This fast synchronization waveform is obtained theoretically from the phase sensitivity function by casting an optimization problem. The obtained optimal actuation waveform becomes increasingly non-sinusoidal for higher angles of attack. Actuation based on the obtained waveform achieves rapid synchronization within as low as two vortex shedding cycles irrespective of the forcing frequency, whereas traditional sinusoidal actuation requires ${O}(10)$ shedding cycles. Further, we analyse the influence of actuation frequency on the vortex shedding and the aerodynamic coefficients using force-element analysis. The present analysis provides an efficient way to modify the vortex lock-on properties in a transient manner with applications to fluid–structure interactions and unsteady flow control.

Tonal noise emitted from the trailing edge of an airfoil is considered using modal analysis techniques to investigate secondary quadrupole tones. We examine the origin of quadrupole sound generated from two-dimensional unsteady laminar flow over a NACA0012 airfoil. In this paper, we consider two flow configurations at Mach numbers of $M_\infty = 0.1$ and 0.05 that lead to different acoustic characteristics: the former has a significant high-frequency quadrupole noise source, whereas the latter does not. We use vortex sound theory, dynamic mode decomposition (DMD), and resolvent analysis to analyze the sound source. First, we employ DMD modes to reveal that the quadrupole sound is only observed in the higher Mach number case. Next, the vortex dynamics in the vicinity of the trailing edge are studied to identify the origin of quadrupole sound. It is found that the quadrupole sound is caused by vortex shedding in the vicinity of the trailing edge. The complex vortex interaction between both sides of the airfoil strengthens the quadrupole source in the higher Mach number case, while it is negligible in the lower one. Furthermore, we perform resolvent analysis to examine the vortex generation over the airfoil. The resolvent mode indicates that the interaction between the vortices on both sides of the airfoil causes a multi-scale vortex structure on the suction-side wall.

Through triglobal resolvent analysis, we reveal the effects of wing tip and sweep angle on laminar separated wakes over swept wings. For the present study, we consider wings with semi-aspect ratios from $1$ to $4$, sweep angles from $0^\circ$ to $45^\circ$ and angles of attack of $20^\circ$ and $30^\circ$ at a chord-based Reynolds number of $400$ and a Mach number of $0.1$. Using direct numerical simulations, we observe that unswept wings develop vortex shedding near the wing root with a quasi-steady tip vortex. For swept wings, vortex shedding is seen near the wing tip for low sweep angles, while the wakes are steady for wings with high sweep angles. To gain further insights into the mechanisms of flow unsteadiness, triglobal resolvent analysis is used to identify the optimal spatial input–output mode pairs and the associated gains over a range of frequencies. The three-dimensional forcing and response modes reveal that harmonic fluctuations are directed towards the root for unswept wings and towards the wing tip for swept wings. The overlapping region of the forcing–response mode pairs uncovers triglobal resolvent wavemakers associated with self-sustained unsteady wakes of swept wings. Furthermore, we show that for low-aspect-ratio wings optimal perturbations develop globally over the entire wingspan. The present study uncovers physical insights on the effects of tip and sweep on the growth of optimal harmonic perturbations and the wake dynamics of separated flows over swept wings.

We reveal the effects of sweep on the wake dynamics around NACA 0015 wings at high angles of attack using direct numerical simulations and resolvent analysis. The influence of sweep on the wake dynamics is considered for sweep angles from $0^\circ$ to $45^\circ$ and angles of attack from $16^\circ$ to $30^\circ$ for a spanwise periodic wing at a chord-based Reynolds number of $400$ and a Mach number of $0.1$. Wing sweep affects the wake dynamics, especially in terms of stability and spanwise fluctuations with implications on the development of three-dimensional (3-D) wakes. We observe that wing sweep attenuates spanwise fluctuations. Even as the sweep angle influences the wake, force and pressure coefficients can be collapsed for low angles of attack when examined in wall-normal and wingspan-normal independent flow components. Some small deviations at high sweep and incidence angles are attributed to vortical wake structures that impose secondary aerodynamic loads, revealed through the force element analysis. Furthermore, we conduct global resolvent analysis to uncover oblique modes with high disturbance amplification. The resolvent analysis also reveals the presence of wavemakers in the shear-dominated region associated with the emergence of 3-D wakes at high angles of attack. For flows at high sweep angles, the optimal convection speed of the response modes is shown to be faster than the optimal wavemakers speed suggesting a mechanism for the attenuation of perturbations. The present findings serve as a fundamental stepping stone to understanding separated flows at higher Reynolds numbers.

We consider the use of sparsity-promoting norms in obtaining localised forcing structures from resolvent analysis. By formulating the optimal forcing problem as a Riemannian optimisation, we are able to maximise cost functionals whilst maintaining a unit-energy forcing. Taking the cost functional to be the energy norm of the driven response results in a traditional resolvent analysis and is solvable by a singular value decomposition (SVD). By modifying this cost functional with the $L_1$-norm, we target spatially localised structures that provide an efficient amplification in the energy of the response. We showcase this optimisation procedure on two flows: plane Poiseuille flow at Reynolds number $Re=4000$, and turbulent flow past a NACA 0012 aerofoil at $Re=23\,000$. In both cases, the optimisation yields sparse forcing modes that maintain important features of the structures arising from an SVD in order to provide a gain in energy. These results showcase the benefits of utilising a sparsity-promoting resolvent formulation to uncover sparse forcings, specifically with a view to using them as actuation locations for flow control.

Linear modal instabilities of flow over untapered wings with aspect ratios $AR=4$ and 8, based on the NACA 0015 profile, have been investigated numerically over a range of angles of attack, $\alpha$, and angles of sweep, $\varLambda$, at chord Reynolds numbers $100\le Re\le 400$. Laminar base flows have been generated using direct numerical simulation and selective frequency damping, as appropriate. Several families of unstable three-dimensional linear global (TriGlobal) eigenmodes have been identified and their dependence on geometric parameters has been examined in detail at $Re=400$. The leading global mode A is associated with the peak recirculation in the three-dimensional laminar separation bubble formed on the wing and becomes unstable when recirculation reaches $\textit {O}(10\,\%)$. On unswept wings, this mode peaks in the midspan region of the wake and moves towards the wing tip with increasing $\varLambda$, following the displacement of peak recirculation; its linear amplification leads to wake unsteadiness. Additional amplified modes exist at nearly the same and higher frequencies compared to mode A. The critical $Re$ has been identified and it is shown that amplification increases with increasing sweep, up to $\varLambda \approx 10^\circ$. At higher $\varLambda$, all global modes become less amplified and are ultimately stable at $\varLambda =30^\circ$. An increase in amplification of the leading mode with sweep was not observed over the $AR=4$ wing, where tip vortex effects were shown to dominate.

A large eddy simulation is performed to study secondary tones generated by a NACA0012 airfoil at angle of attack of $\alpha = 3^{\circ}$ with freestream Mach number of $M_{\infty} = 0.3$ and Reynolds number of $Re = 5 \times 10^4$. Laminar separation bubbles are observed over the suction side and near the trailing edge, on the pressure side. Vortex shedding occurs aft of the suction side separation bubble, and vortex interaction results in merging or bursting such that coherent structures or turbulent packets are advected towards the trailing edge. This mechanism modulates the amplitude of the incident pressure signal, leading to different levels of noise emission. Despite the intermittent occurrence of laminar–turbulent transition, the noise spectrum depicts a main tone with multiple equidistant secondary tones. To understand the role of flow instabilities on the tones, the linearised Navier–Stokes equations are examined in their operator form through biglobal stability and resolvent analyses, and by time evolution of disturbances using a matrix-free method. These linear global analyses reveal amplification of disturbances over the suction side separation bubble. Spanwise-averaged pressure fluctuations elucidate aspects of the acoustic feedback loop mechanism in the nonlinear solutions. This feedback process is self-sustained by acoustic waves radiated from the trailing edge, which reach the most sensitive flow location near the leading edge, as identified by the resolvent analysis. Flow disturbances arising from secondary diffraction and phase interference among the most unstable frequencies computed in the eigenspectrum are shown to have an important role in the feedback loop.

Phase-reduction analysis captures the linear phase dynamics with respect to a limit cycle subjected to weak external forcing. We apply this technique to study the phase dynamics of the self-sustained oscillations produced by a Rijke tube undergoing thermoacoustic instability. Through the phase-reduction formulation, we are able to reduce these dynamics to a scalar equation for the phase, which allows us to efficiently determine the synchronisation properties of the system. For the thermoacoustic system, we find the conditions for which $m:n$ frequency locking occurs, which sheds light on the mechanisms behind asynchronous and synchronous quenching. We also reveal the optimal placement of pressure actuators that provide the most efficient route to synchronisation.

Unsteady bluff-body flows exhibit dominant oscillatory behaviour owing to periodic vortex shedding. The ability to manipulate this vortex shedding is critical to improving the aerodynamic performance of bodies in a flow. This goal requires a precise understanding of how the perturbations affect the asymptotic behaviour of the oscillatory flow and of the ability to control transient dynamics. In this work, we develop an energy-efficient flow-control strategy to alter the oscillation phase of time-periodic fluid flows rapidly. First, we perform a phase-sensitivity analysis to construct a reduced-order model for the response of the flow oscillation to impulsive control inputs at various phases. Next, we introduce a real-time optimal phase-control strategy based on the phase-sensitivity function obtained by solving the associated Euler–Lagrange equations as a two-point boundary-value problem. Our approach is demonstrated for the incompressible laminar flow past a circular cylinder and an airfoil. We show the effectiveness of phase control with different actuation inputs, including blowing and rotary control. Moreover, our control approach is a sensor-based approach without the need for access to high-dimensional measurements of the entire flow field.

We use resolvent analysis to develop a physics-based, open-loop, unsteady control strategy to attenuate pressure fluctuations in turbulent flow over a rectangular cavity with a length-to-depth ratio of $6$ at a Mach number of $1.4$ and a Reynolds number based on cavity depth of $10\,000$. Large-eddy simulations (LES) of the baseline uncontrolled flow reveal the dominance of Rossiter modes II and IV that generate high-amplitude unsteadiness via trailing-edge impingement and oblique shock waves that obstruct the free stream. To suppress the oscillations, we introduce three-dimensional unsteady blowing along the cavity leading edge. We leverage resolvent analysis as a linear model with respect to the baseline flow to guide the selections of the optimal spanwise wavenumber and frequency of the unsteady actuation input for a fixed momentum coefficient of 0.02. Instead of choosing the most amplified resolvent forcing modes, we seek a disturbance that yields sustained amplification of the primary response mode-based kinetic energy distribution over the entire cavity length. This necessary but not sufficient guideline for effective mean flow modification is evaluated using LES of the controlled cavity flows. The most effective control case reduces the pressure root mean square level up to $52\,\%$ along cavity walls relative to the baseline and is approximately twice that achievable by comparable steady blowing. Dynamic mode decomposition on the controlled flows confirms that the optimal actuation input indeed suppresses the formation of the large-scale Rossiter modes. It is expected that the present flow control guideline derived from resolvent analysis will also be applicable at higher Reynolds numbers with the aid of physical insights and further validation.