We use video footage of a water-tunnel experiment to construct a 2-D reduced-order model of the flapping dynamics of an inverted flag in uniform flow. The model is obtained as the reduced dynamics on a 2-D attracting spectral submanifold (SSM) that emanates from the two slowest modes of the unstable fixed point of the flag. Beyond an unstable fixed point and a limit cycle expected from observations, our SSM-reduced model also confirms the existence of two unstable fixed points for the flag, which were found by previous studies. Importantly, the model correctly reconstructs the dynamics from a small number of general trajectories and no further information on the system. In the chaotic flapping regime, we construct a 4-D SSM-reduced model that captures the system's chaotic attractor.

The aeroacoustics of a boundary layer ingesting (BLI) ducted fan is investigated experimentally. The study examines a ducted fan immersed in an adverse streamwise pressure gradient turbulent boundary layer developed over a curved wall. Aeroacoustics measurements indicate that the noise from the BLI ducted fan results from a complex interaction among the fan, duct and the incoming boundary layer. The fundamental mechanisms of noise generation are explained using a general source separation strategy. A detailed noise comparison is made at varying fan rotational speeds and across a wide range of axial inflow velocities. In a low thrust regime, the noise is found to be driven by the fan loading, coupled with duct acoustics and the haystacking phenomenon. In a high thrust regime, the contribution from duct acoustics diminishes, and the noise is predominantly driven by the fan loading coupled with the haystacking phenomenon.

This paper presents a novel Hamiltonian formulation of the isotropic Navier–Stokes problem based on a minimum-action principle derived from the principle of least squares. This formulation uses the velocities $u_{i}(x_{j},t)$ and pressure $p(x_{j},t)$ as the field quantities to be varied, along with canonically conjugate momenta deduced from the analysis. From these, a conserved Hamiltonian functional $H^{*}$ satisfying Hamilton's canonical equations is constructed, and the associated Hamilton–Jacobi equation is formulated for both compressible and incompressible flows. This Hamilton–Jacobi equation reduces the problem of finding four separate field quantities ($u_{i}$,$p$) to that of finding a single scalar functional in those fields – Hamilton's principal functional ${S}^{*}[u_{i},p,t]$. Moreover, the transformation theory of Hamilton and Jacobi now provides a prescribed recipe for solving the Navier–Stokes problem: find ${S}^{*}$. If an analytical expression for ${S}^{*}$ can be obtained, it will lead via canonical transformation to a new set of fields which are simply equal to their initial values, giving analytical expressions for the original velocity and pressure fields. Failing that, if one can only show that a complete solution to this Hamilton–Jacobi equation does or does not exist, that will also resolve the question of existence of solutions. The method employed here is not specific to the Navier–Stokes problem or even to classical mechanics, and can be applied to any traditionally non-Hamiltonian problem.

Large eddy simulation of flow past a cricket ball with its seam at $30^\circ$ to the free stream is carried out for $5 \times 10^4 \le Re \le 4.5 \times 10^5$. Three regimes of flow are identified on the basis of the time-averaged swing force coefficient ($\bar {C}_Z$) – no swing (NS), conventional swing (CS, $\bar {C}_Z>0$) and reverse swing (RS, $\bar {C}_Z<0$). The effect of seam on the boundary layer is investigated. Contrary to the popular belief, the boundary layer does not transition to a turbulent state in the initial stages of CS. The seam energizes the laminar boundary layer and delays its separation. The delay is significantly larger in a region near the poles, whose extent increases with an increase in $Re$ causing $\bar {C}_Z$ to increase. Here $\bar {C}_Z$ assumes a near constant value in the later stage of CS. The boundary layer transitions to a turbulent state via formation of a laminar separation bubble (LSB) in the equatorial region and directly, without a LSB, in the polar region. The extent of the LSB shrinks while the region of direct transition near the poles increases with an increase in $Re$. A LSB forms on the non-seam side of the ball in the RS regime. A secondary vortex is observed in the wake bubble. While it exists on the non-seam side for the entire range of $Re$ considered, the mixing in the flow introduced by the seam causes it to disappear beyond a certain $Re$ on the seam side. The pressure difference between the seam and non-seam sides sets up wing-tip-like vortices. Their polarity reverses with the switch from the CS to RS regime.

We study theoretically and experimentally pressure-driven flow between a flat wall and a parallel corrugated wall, a design used widely in microfluidics for low-Reynolds-number mixing and particle separation. In contrast to previous work, which focuses on recirculating helicoidal flows along the microfluidic channel that result from its confining lateral walls, we study the three-dimensional pressure and flow fields and trajectories of tracer particles at the scale of each corrugation. Employing a perturbation approach for small surface roughness, we find that anisotropic pressure gradients generated by the surface corrugations, which are tilted with respect to the applied pressure gradient, drive transverse flows. We measure experimentally the flow fields using particle image velocimetry and quantify the effect of the ratio of the surface wavelength to the channel height on the transverse flows. Further, we track tracer particles moving near the surface structures and observe three-dimensional skewed helical trajectories. Projecting the helical motion to two dimensions reveals oscillatory near-surface motion with an overall drift along the surface corrugations, reminiscent of earlier experimental observations and independent of the secondary helical flows that are induced by confining lateral walls. Finally, we quantify the hydrodynamically induced drift transverse to the mean flow direction as a function of distance to the surface and the wavelength of the surface corrugations.

Recent microfluidic experiments have evidenced complex spatio-temporal fluctuations in low-Reynolds-number flows of polymer solutions through lattices of obstacles. However, understanding the nonlinear physics of such systems remains a challenge. Here, we use high performance simulations to study viscoelastic flows through a hexagonal lattice of cylindrical obstacles. We find that structures of localized polymer stress – in particular birefringent strands – control the stability and the dynamics. We first show that, at steady state, strands act as a web of sticky flow barriers that induce channelization, multistability and hysteresis. We then demonstrate that a spontaneous destabilization of the strands drives the transition to unsteady flow with regimes of self-sustained oscillations, travelling waves and strand pulsations. We further show that these pulsations, which result from the destabilization of envelope patterns of stress with strands wrapped around multiple obstacles, are integral to the transition towards elastic turbulence in our two-dimensional simulations. Our study provides a new perspective on the role of birefringent strands and a framework for understanding experimental observations. We anticipate that it is an important step towards unifying existing interpretations of the nonlinear physics of viscoelastic flows through complex structures.

The effects of surfactants on a mechanically generated plunging breaker are studied experimentally in a laboratory wave tank. Waves are generated using a dispersively focused wave packet with a characteristic wavelength of $\lambda _0 = 1.18$ m. Experiments are performed with two sets of surfactant solutions. In the first set, increasing amounts of the soluble surfactant Triton X-100 are mixed into the tank water, while in the second set filtered tap water is left undisturbed in the tank for wait times ranging from 15 min to 21 h. Increasing Triton X-100 concentrations and longer wait times lead to surfactant-induced changes in the dynamic properties of the free surface in the tank. It is found that low surface concentrations of surfactants can dramatically change the wave breaking process by changing the shape of the jet and breaking up the entrained air cavity at the time of jet impact. Direct numerical simulations (DNS) of plunging breakers with constant surface tension are used to show that there is significant compression of the free surface near the plunging jet tip and dilatation elsewhere. To explore the effect of this compression/dilatation, the surface tension isotherm is measured in all experimental cases. The effects of surfactants on the plunging jet are shown to be primarily controlled by the surface tension gradient ($\Delta \mathcal {E}$) while the ambient surface tension of the undisturbed wave tank ($\sigma _0$) plays a secondary role.

To date, the predominant means for computing the probability density function (p.d.f.) for the free surface elevation of a nonlinear, irregular water wave field, free of assumptions involving narrow-bandedness and small directionality, is the approximate Gram–Charlier series solution of Longuet-Higgins (J. Fluid Mech., vol. 17, 1963, pp. 459–480, hereafter LH63). In this paper we re-visit the derivation of this p.d.f. to second order in the wave steepness, utilizing both moment and cumulant generating functions. We show that LH63's approximate solution based on the cumulant generating function, in fact, matches that derived from the moment generating function. Moreover, through a change of variables coupled with complex analysis, it is shown that the approximation employed by LH63 is unnecessary, and the second-order p.d.f. stemming from the cumulant generating function can be represented exactly in terms of the Airy function. The new second-order p.d.f. predicts increased probability of extreme positive surface elevations typical of e.g. rogue waves, relative to both second- and third-order solutions of LH63. This heavy positive tail is inherent, and is explained through comparison of the asymptotic limits of the p.d.f.s for large surface elevations. A semi-theoretical method is also proposed for remedying non-physical spurious oscillations that arise in the negative tail, based on the envelope of the Airy function with negative arguments. This modified negative tail is valid for irregular wave fields having skewness less than or equal to 0.2. The new p.d.f.s are compared against those based on data sets generated from second-order irregular wave theory as well as a fully nonlinear, spectrally accurate numerical wave model. Good accuracy is collectively demonstrated for directionally spread irregular seas in both finite and infinite water depths for a range of directional spreading.

In this paper we revive a special, less-common, variational principle in analytical mechanics (Hertz’ principle of least curvature) to develop a novel variational analogue of Euler's equations for the dynamics of an ideal fluid. The new variational formulation is fundamentally different from those formulations based on Hamilton's principle of least action. Using this new variational formulation, we generalize the century-old problem of the flow over a two-dimensional body; we developed a variational closure condition that is, unlike the Kutta condition, derived from first principles. The developed variational principle reduces to the classical Kutta–Zhukovsky condition in the special case of a sharp-edged airfoil, which challenges the accepted wisdom about the Kutta condition being a manifestation of viscous effects. Rather, we found that it represents conservation of momentum. Moreover, the developed variational principle provides, for the first time, a theoretical model for lift over smooth shapes without sharp edges where the Kutta condition is not applicable. We discuss how this fundamental divergence from current theory can explain discrepancies in computational studies and experiments with superfluids.

Falling paper flutters and tumbles through air, whereas a paper airplane glides smoothly if its leading edge is appropriately weighted. We investigate this transformation from ‘plain paper’ to ‘paper plane’ through experiments, aerodynamic modelling and free flight simulations of thin plates with differing centre of mass (CoM) locations. Periodic modes such as fluttering, tumbling and bounding give way to steady gliding and then downward diving as the CoM is increasingly displaced towards one edge. To explain these observations, we formulate a quasi-steady aerodynamic model whose force and torque coefficients are informed by experimental measurements. The dependencies on angle of attack reflect the transition from attached to separated flow and a dynamic centre of pressure, effects that prove critical to reproducing the observed motions of paper planes in air and plates in water. Because the model successfully accounts for unsteady and steady flight modes, it may be usefully applied to further problems involving actuated motions, feedback control and interactions with ambient flows.

It has been commonly observed on open waters that ducklings/goslings follow their mothers in a highly organized formation. The questions arise: (1) why are they swimming in formation? (2) what is the best swimming formation? (3) how much energy can be preserved by each individual in formation swimming? To address these questions, we established a simplified mathematical and numerical model and calculated the wave drag on a group of waterfowl in a swimming formation. We observed two new and interesting findings: wave-riding and wave-passing. By riding the waves generated by a mother duck, a trailing duckling can obtain a significant wave-drag reduction. When a duckling swims at the ‘sweet point’ behind its mother, a destructive wave interference phenomenon occurs and the wave drag of the duckling turns positive, pushing the duckling forward. More interestingly, this wave-riding benefit could be sustained by the rest of the ducklings in a single-file line formation. Starting from the third one in a queue, the wave drag of individuals gradually tended towards zero, and a delicate dynamic equilibrium was achieved. Each individual under that equilibrium acted as a wave passer, passing the waves’ energy to its trailing one without any energy losses. Wave-riding and wave-passing are probably the principal reasons for the evolution of swimming formation by waterfowl. This study is the first to reveal the reasons why the formation movement of waterfowl can preserve individuals’ energy expenditure. Our calculations provide new insights into the mechanisms of formation swimming.

We present a new data reconstruction method with supervised machine learning techniques inspired by super resolution and inbetweening to recover high-resolution turbulent flows from grossly coarse flow data in space and time. For the present machine-learning-based data reconstruction, we use the downsampled skip-connection/multiscale model based on a convolutional neural network, incorporating the multiscale nature of fluid flows into its network structure. As an initial example, the model is applied to the two-dimensional cylinder wake at $Re_D = 100$. The reconstructed flow fields by the present method show great agreement with the reference data obtained by direct numerical simulation. Next, we apply the current model to a two-dimensional decaying homogeneous isotropic turbulence. The machine-learned model is able to track the decaying evolution from spatial and temporal coarse input data. The proposed concept is further applied to a complex turbulent channel flow over a three-dimensional domain at $Re_{\tau }=180$. The present model reconstructs high-resolved turbulent flows from very coarse input data in space, and also reproduces the temporal evolution for appropriately chosen time interval. The dependence on the number of training snapshots and duration between the first and last frames based on a temporal two-point correlation coefficient are also assessed to reveal the capability and robustness of spatio-temporal super resolution reconstruction. These results suggest that the present method can perform a range of flow reconstructions in support of computational and experimental efforts.

Flow physics plays a key role in nearly every facet of the COVID-19 pandemic. This includes the generation and aerosolization of virus-laden respiratory droplets from a host, its airborne dispersion and deposition on surfaces, as well as the subsequent inhalation of these bioaerosols by unsuspecting recipients. Fluid dynamics is also key to preventative measures such as the use of face masks, hand washing, ventilation of indoor environments and even social distancing. This article summarizes what we know and, more importantly, what we need to learn about the science underlying these issues so that we are better prepared to tackle the next outbreak of COVID-19 or a similar disease.

Surfactants – molecules and particles that preferentially adsorb to fluid interfaces – play a ubiquitous role in the fluids of industry, of nature and of life. Since most surfactants cannot be seen directly, their behaviour must be inferred from their impact on observed flows, like the buoyant rise of a bubble, or the thickness of a coating film. In so doing, however, a difficulty arises: physically distinct surfactant processes can affect measurable flows in qualitatively identical ways, raising the spectre of confusion or even misinterpretation. This Perspective describes, in one coherent piece, both the equilibrium properties and dynamic processes of surfactants, to better enable the fluid mechanics community to understand, interpret and design surfactant/fluid systems. Specifically, we treat the equilibrium thermodynamics of surfactants at interfaces, including surface pressure, isotherms of soluble and insoluble surfactants and surface dilatational moduli (Gibbs and Marangoni). We describe surfactant dynamics in fluid systems, including surfactant transport and interfacial stress boundary conditions, the competition between surface diffusion, advection and adsorption/desorption, Marangoni stresses and flows and surface-excess rheology. We discuss paradigmatic problems from fluid mechanics that are impacted by surfactants, including translating drops and bubbles, surfactant adsorption to clean and oscillating interfaces; capillary wave damping, thin-film dynamics, foam drainage and the dynamics of particles and probes at surfactant-laden interfaces. Finally, we discuss the additional richness and complexity that frequently arise in ‘real’ surfactants, including phase transitions, phase coexistence and polycrystalline phases within surfactant monolayers, and their impact on non-Newtonian surface rheology.