We report a new interferometric method to study the interactions between two gas bubbles undergoing small-amplitude oscillations in a liquid, based on the extension of a previously developed one-bubble set-up. Nanometric oscillations of millimetre-sized supported bubbles are excited acoustically; the response of each bubble is recorded interferometrically, as a function of the mutual distance (from quasi-contact to greater than the bubbles radii). The interferometric nature of the technique and the resonant nature of the vibration modes enable the accurate measurement of the amplitude (with sub-nanometric sensitivity), frequency and mutual phase of oscillation, whose variations over the bubble–bubble distance range allow the interactions to be probed. The bubbles oscillate at the same frequencies, exhibiting a low-frequency, in-phase and a high-frequency, out-of-phase resonance peak, whose separation is a function of distance, in good agreement with the theory for free interacting bubbles. The technique, here demonstrated for the volume modes of air bubbles in water, can be extended to other gas–liquid and liquid–liquid interfaces, bare or adsorbate-covered, as well as to shape oscillations.

Flow control for turbulent skin-friction drag reduction is applied to a transonic airfoil to improve its aerodynamic performance. The study is based on direct numerical simulations (with up to 1.8 billion cells) of the compressible turbulent flow around a supercritical airfoil, at Reynolds and Mach numbers of $Re_\infty = 3 \times 10^{5}$ and $M_\infty =0.7$. Control via spanwise forcing is applied over a fraction of the suction side of the airfoil. Besides locally reducing friction, the control modifies the shock wave and significantly improves the aerodynamic efficiency of the airfoil by increasing lift and decreasing drag. Hence, the airfoil can achieve the required lift at a lower angle of attack and with a lower drag. Estimates at the aircraft level indicate that substantial savings are possible; when control is active, its energy cost becomes negligible thanks to the small application area. We suggest that skin-friction drag reduction should be considered not only as a goal, but also as a tool to improve the global aerodynamics of complex flows.

The rise of a single bubble confined between two vertical plates is investigated over a wide range of Reynolds numbers. In particular, we focus on the evolution of the bubble speed, aspect ratio and drag coefficient during the transition from the viscous to the inertial regime. For sufficiently large bubbles, a simple model based on power balance captures the transition for the bubble velocity and matches all the experimental data despite strong time variations of bubble aspect ratio at large Reynolds numbers. Surprisingly, bubbles in the viscous regime systematically exhibit an ellipse elongated along its direction of motion while bubbles in the inertia-dominated regime are always flattened perpendicularly to it.

We study the kinematics, dynamics and flow fields generated by an oscillating, compliant membrane hydrofoil extracting energy from a uniform water stream at a chord-based Reynolds number $Re \approx 3 \times 10^4$. Hydrodynamic forces during the foil's motion cause the membrane to dynamically morph its shape, effectively increasing the camber during the oscillation cycle. The membrane's deflection is modelled using the Young–Laplace equation, with pressure term approximated from thin-airfoil theory. Simultaneous tracking of the membrane deformation and the surrounding flow field using laser profiling and particle image velocimetry, respectively, reveals the role of dynamic cambering in stabilizing the leading-edge vortices on the membrane. In this regime of operation, we obtain up to 160 % higher power extraction when compared to a rigid, symmetric hydrofoil. The present work provides a demonstration of how passive compliance of soft materials interacting with fluids may be exploited in tidal and fluvial energy extraction.

We report non-monotonic wettability effects on displacement efficiency in heterogeneous porous structures at the post-breakthrough stage, in contrast to the monotonic ones in homogeneous porous structures. Experiments on designed microfluidic chips show that there exists a critical wettability to attain the highest efficiency of displacement in the porous matrix structure combined with a preferential flow pathway, while a stronger wettability of the displacing fluid leads to a higher displacement efficiency on the same matrix structure only. The porous structure with or without a preferential flow pathway results in totally different topological characteristics of phase distribution during displacement. Pore-scale mechanisms are identified to elucidate the formation of this non-monotonic wettability rule: cooperative pore filling under weakly water-wet conditions yields the best displacement; corner flow under strongly water-wet conditions and Haines events under strongly oil-wet conditions decrease the displacement efficiency. The pore-scale findings may provide unique insights into the joint effects of both wettability and flow heterogeneity on fluid displacement in porous media.

We revisit the theoretical solutions of the problem of the reflected flow from a vertical obstacle encountered by a current generated by dambreak of a Cartesian long reservoir over a dry bottom and with tailwaters. Previous investigations derived accurate solutions by quite complex manipulation of the balances along the characteristics, supported by numerical solutions, of the shallow-water equations (a significant extension is the recent paper Hogg & Skevington, Q. J. Mech. Appl. Maths, vol. 74, no. 4, 2021, pp. 441–465). Here we developed a simplified model, based on the major assumption that the fluid between the obstacle (wall) and the reflected jump is stagnant. This allows the solution of the problem by a straightforward numerical integration of one initial-value ordinary differential equation. The model provides the position, height and speed of the jump as functions of time. For long times a simple analytical approximation is also available. The model points out clear-cut effects of the presence of the tailwaters. The model has been validated by comparisons with exact solutions of the shallow-water equations of the recent study of Hogg & Skevington (Q. J. Mech. Appl. Maths, vol. 74, no. 4, 2021, pp. 441–465) (obtained by significantly more complex hodograph-plane methods). In all the tested cases the agreement is good, for long periods of time. This model provides reliable insights and fast quantitative predictions recommended for use in research and engineering problems where a fair approximation is sufficient for the application.

The disturbance flow field for a Mach $6$ flat-plate boundary layer flow with a wall-to-free-stream temperature ratio of $0.5$ is studied using direct numerical simulation (DNS), linear stability theory (LST) and biorthogonal decomposition. In the second-mode instability region, the DNS flow field can be reconstructed using a single LST mode, namely $F^+_1$. However, when the supersonic mode emerges, none of the discrete modes nor the continuous branches alone can precisely match the DNS data. A superposition of the pair of discrete supersonic modes $F^\pm _1$ and the slow acoustic continuous spectrum is required to reproduce the disturbance amplitude distributions and the amplification rates observed in the DNS. Another finding is that the supersonic mode $F^-_1$, which should be decaying following LST, in actuality is amplified based on the projected DNS data.

Dynamic stall on airfoils negatively impacts their aerodynamic performance and can lead to structural damage. Accurate prediction and modelling of the dynamic stall loads are crucial for a more robust design of wings and blades that operate under unsteady conditions susceptible to dynamic stall and for widening the range of operation of these lifting surfaces. Many dynamic stall models rely on empirical parameters that need to be obtained from experimental or numerical data which limits their generalisability. Here, we introduce physically derived time scales to replace the empirical parameters in the Goman–Khrabrov dynamic stall model. The physics-based time constants correspond to the dynamic stall delay and the decay of post-stall load fluctuations. The dynamic stall delay is largely independent of the type of motion, the Reynolds number and the airfoil geometry, and is described as a function of a normalised instantaneous pitch rate. The post-stall decay is independent of the motion kinematics and is related to the Strouhal number of the post-stall vortex shedding. The general validity of our physics-based time constants is demonstrated using three sets of experimental dynamic stall data covering various airfoil profiles, Reynolds numbers varying from 75 000 to 1 000 000, and sinusoidal and ramp-up pitching motions. The use of physics-based time constants generalises the Goman–Khrabrov dynamic stall model and extends its range of application.

The inner–outer interaction model (Marusic et al., Science, vol. 329, 2010, pp. 193–196) and the attached-eddy model (Townsend, Cambridge University Press, 1976) are two fundamental models describing the multiscale turbulence interactions and the organization of energy-containing motions in the logarithmic region of high-Reynolds-number wall-bounded turbulence, respectively. In this paper, by coupling the additive description with the attached-eddy model, the generation process of streamwise wall-shear fluctuations, resulting from wall-attached eddies, is portrayed. Then, by resorting to the inner–outer interaction model, the streamwise wall-shear stress fluctuations generated by attached eddies in a turbulent channel flow are isolated. Direct comparison between the statistics from these two models demonstrates that they are consistent with and complement each other. Meanwhile, we further show that the superpositions of attached eddies follow an additive process strictly by verifying the validity of the strong and extended self-similarity. Moreover, we propose a Gaussian model to characterize the instantaneous distribution of streamwise wall-shear stress, resulting from the attached-eddy superpositions. These findings are important for developing an advanced reduced-order wall model.

The composition of the Earth's mantle, built by impacts of metallic core planetesimals, depends on the time offered to metal–silicate equilibration (concentration equalization by mixing) as the impactor fragments settle down. Smaller fragments equilibrate while bigger ones remain segregated and accumulate in the – therefore iron rich – Earth's core. To understand the primary instability setting the interpenetration depth and wavelength between the impactor and the mantle, Lherm et al. (J. Fluid Mech., vol. 937, 2022, A20) analyse the intermingled phenomena occurring when a drop impacts a substrate of a pool of another liquid, digging a crater. The density difference between the phases triggers a Rayleigh–Taylor instability when the drop is denser than the pool. The overall planet chemical composition relies on a tradeoff between fragmentation and mixing; this study provides elements to decipher their relative importance.

There are numerous ways to control objects in the Stokes regime, with microscale examples ranging from the use of optical tweezers to the application of external magnetic fields. In contrast, there are relatively few explorations of theoretical controllability, which investigate whether or not refined and precise control is indeed possible in a given system. In this work, seeking to highlight the utility and broad applicability of such rigorous analysis, we recount and illustrate key concepts of geometric control theory in the context of multiple particles in Stokesian fluids interacting with each other, such that they may be readily and widely applied in this largely unexplored fluid-dynamical setting. Motivated both by experimental and abstract questions of control, we exemplify these techniques by explicit and detailed application to multiple problems concerning the control of two particles, such as the motion of tracers in flow and the guidance of one sphere by another. Further, we showcase how this analysis of controllability can directly lead to the construction of schemes for control, in addition to facilitating explorations of mechanical efficiency and contributing to our overall understanding of non-local hydrodynamic interactions in the Stokes limit.

Machine Learning (ML) is becoming increasingly popular in fluid dynamics. Powerful ML algorithms such as neural networks or ensemble methods are notoriously difficult to interpret. Here, we introduce the novel Shapley additive explanations (SHAP) algorithm (Lundberg & Lee, Advances in Neural Information Processing Systems, 2017, pp. 4765–4774), a game-theoretic approach that explains the output of a given ML model in the fluid dynamics context. We give a proof of concept concerning SHAP as an explainable artificial intelligence method providing useful and human-interpretable insight for fluid dynamics. To show that the feature importance ranking provided by SHAP can be interpreted physically, we first consider data from an established low-dimensional model based on the self-sustaining process (SSP) in wall-bounded shear flows, where each data feature has a clear physical and dynamical interpretation in terms of known representative features of the near-wall dynamics, i.e. streamwise vortices, streaks and linear streak instabilities. SHAP determines consistently that only the laminar profile, the streamwise vortex and a specific streak instability play a major role in the prediction. We demonstrate that the method can be applied to larger fluid dynamics datasets by a SHAP evaluation on plane Couette flow in a minimal flow unit focussing on the relevance of streaks and their instabilities for the prediction of relaminarisation events. Here, we find that the prediction is based on proxies for streak modulations corresponding to linear streak instabilities within the SSP. That is, the SHAP analysis suggests that the break-up of the self-sustaining cycle is connected with a suppression of streak instabilities.

To investigate the transport of heavy small particles (inertial particles) in high-Reynolds- number wall turbulence, we conduct direct numerical simulations of inertial particles in turbulent channel flow at the friction Reynolds number $Re_\tau =1000$. In the statistically steady state, particles distribute inhomogeneously; particles with different relaxation times form voids with different sizes in the bulk of the flow, whereas they form streak-like clusters with different widths along the streamwise direction. To explore the origin of the multiscale voids and clusters of particles, we identify objectively the axes of tubular vortices and the spines of streaks with different sizes. These identifications enable us to show quantitatively that (i) vortices sweep out the particles of the relaxation time comparable with their turnover time, irrespective of their size and existing height, and that, among these particles, (ii) those swept out by wall-detached tubular vortices form clusters isotropically around them, whereas (iii) those swept out by wall-attached vortices are attracted by a nearby low-speed streak. These explain the reason why the multiscale clusterings are well described in terms of the local Stokes number defined by the turnover time of multiscale vortices. Furthermore, these descriptions of the particle transport give us a clear view to understand velocity statistics and wall-deposition mechanism of inertial particles in high-Reynolds-number wall turbulence.

The bistable fluttering response of heavy inverted flags with different aspect ratios ($AR$) is investigated to determine how the vortical structures affect the intermittent vibration response of the flag. A heavy inverted flag in a uniform flow may exhibit several response modes; amongst them are three major modes that occur over an extended velocity range: stationary, large-scale periodic oscillation and one-sided deflected modes. Significant hysteretic bistability is observed at the transition between these modes for all $AR$, which is notably different from the conventional flag vibration with a fixed leading edge and free trailing edge where no hysteresis is observed at the lower $AR$ limit ($AR<1$). The difference is associated with the distinct roles of vortices around the flag. Experiments with flags made of spring steel are conducted in a wind tunnel, where the flow speed is steadily increased and later decreased to obtain different oscillatory modes of the heavy inverted flags. The experimental results are used to validate the numerical model of the same problem. It is found that different critical velocities exist for increasing and decreasing flow velocities, and there is a sustained hysteresis for all $AR$ controlled by the initiation threshold and growth of the leading-edge and side-edge vortices. The effect of the vortices in the bistable oscillation regime is quantified by formulating a modal force partitioning approach. It is shown that $AR$ can significantly alter the static and dynamic vortex interaction with the flexible plate, thereby changing the flag's hysteresis behaviour and bistable response.

We consider the time evolution in two spatial dimensions of a double vorticity layer consisting of two contiguous, infinite material fluid strips, each with uniform but generally differing vorticity, embedded in an otherwise infinite, irrotational, inviscid incompressible fluid. The potential application is to the wake dynamics formed by two boundary layers separating from a splitter plate. A thin-layer approximation is constructed where each layer thickness, measured normal to the common centre curve, is small in comparison with the local radius of curvature of the centre curve. The three-curve equations of contour dynamics that fully describe the double-layer dynamics are expanded in the small thickness parameter. At leading order, closed nonlinear initial-value evolution equations are obtained that describe the motion of the centre curve together with the time and spatial variation of each layer thickness. In the special case where the layer vorticities are equal, these equations reduce to the single-layer equation of Moore (Stud. Appl. Math., vol. 58, 1978, pp. 119–140). Analysis of the linear stability of the first-order equations to small-amplitude perturbations shows Kelvin–Helmholtz instability when the far-field fluid velocities on either side of the double layer are unequal. Equal velocities define a circulation-free double vorticity layer, for which solution of the initial-value problem using the Laplace transform reveals a double pole in transform space leading to linear algebraic growth in general, but there is a class of interesting initial conditions with no linear growth. This is shown to agree with the long-wavelength limit of the full linearized, three-curve stability equations.

In neutrally stratified shallow water, full-depth Langmuir cells (LCs) can interact with the turbulent benthic boundary layer and, thus, influence bottom wall shear stresses. In this paper the impacts of full-depth LCs on the streamwise and spanwise wall shear stresses are systematically studied using the database obtained from wall-resolved large-eddy simulation of shallow-water Langmuir turbulence. Analyses focus on the instantaneous wall shear stress fluctuations and the joint probability density functions between the stress fluctuations and the LCs parts of the velocity fluctuations, which show that the linear superimposition effect and nonlinear modulation effect of LCs are responsible for the spanwise organized distribution of wall shear stress fluctuations. Compared with the statistics in pure shear-driven turbulence without LCs, the mean square values of wall shear stress fluctuations in shallow-water Langmuir turbulence are enhanced by the strong linear superimposition effect of LCs, while the skewness and kurtosis are reduced by the combination of the linear superimposition effect and nonlinear modulation effect of LCs. Based on the scalings of these effects, a new predictive model of wall shear stress fluctuations is proposed for shallow-water Langmuir turbulence. The proposed model can predict the spatial distribution and statistics of wall shear stress fluctuations using the LCs parts of velocity fluctuations measured above the water bottom. Owing to the persistence of the spanwise inhomogeneity of wall shear stresses induced by full-depth LCs, the new predictive model will be useful for improving the wall-layer modelling for shallow-water Langmuir turbulent flows.

We derive analytical solutions for hydrodynamic sources and sinks to granular temperature in moderately dense suspensions of elastic particles at finite Reynolds numbers. Modelling the neighbour-induced drag disturbances with a Langevin equation allows an exact solution for the joint fluctuating acceleration–velocity distribution function $P(v^{\prime },a^{\prime };t)$. Quadrant-conditioned covariance integrals of $P(v^{\prime },a^{\prime };t)$ yield the hydrodynamic source and sink that dictate the evolution of granular temperature that can be used in Eulerian two-fluid models. Analytical predictions agree with benchmark data from particle-resolved direct numerical simulations and show promise as a general theory from gas–solid to bubbly flows.

Using an improved numerical code we investigate the creation and evolution of quantum knots and links as defects of the Gross–Pitaevskii equation. The particular constraints put on quantum hydrodynamics make this an ideal context for application of geometric and topological methods to investigate dynamical properties. Evolutionary processes are classified into three generic scenarios representing (i) direct topological cascade and collapse, (ii) structural and topological cycles, and (iii) inverse topological cascade of complex structures. Several examples and test cases are studied; the head-on collision of quantum vortex rings and the creation of a trefoil knot from initially unlinked, unknotted loops are realized for the first time. Each type of scenario is studied by carrying out a detailed evaluation of fundamental geometric and dynamical properties associated with evolution. Direct topological cascade that governs the decay of complex structures to small-scale vortex rings is identified by writhe measures, while picks of total curvature are found to provide a clear signature of reconnection events. We demonstrate that isophase minimal surfaces spanning knots and links have a privileged role in the decay process by detecting surface energy relaxation of complex structures. Minimal surfaces are shown to be critical markers for energy and prove to be appropriate detectors for the evolution of complex systems.

Buoyancy-driven exchange flow, i.e. drainage of a heavy colloidal suspension by an immiscible light pure fluid in a vertical duct, is studied analytically. A lubrication model is developed for two-dimensional channel and axisymmetric pipe geometries. The outcome transport equations are solved numerically to capture the effect of important governing dimensionless parameters: Péclet number, precursor film thickness, Reynolds number, mixtures’ viscosity ratio, initial volume fraction of hard-sphere particles, particles concentration within the precursor film, and capillary number. Recent study of colloidal film flow over a flat surface reveals interface thinning over time due to diffusion impact on suspension viscosity. However, the current model suggests that there is no such thinning phenomenon observed for colloidal flows within confined geometry unless in the absence of surface tension. The extent of the mixed particle zone along the duct grows as the Péclet number is decreased. Precursor film thickness is found to decrease the capillary ridge height forming at heavy suspension front. Either an increase in the Reynolds number or a decrease in light-to-heavy-fluid viscosity ratio extends the interpenetration rate of mixtures. As the bulk volume fraction of particles is increased close to 50 %, interesting secondary fronts emerge within the flow that coincide with the range where the diffusion coefficient is minimized. The heavy suspension layer drains slightly faster when the gradient in particle concentration between the bulk and precursor film regions is maximum. Finally, the non-monotonic impact of surface tension on the extent of mixtures’ exchange zone is uncovered via simulations conducted at various capillary numbers.

We investigate the effects of helical swimmer shape (i.e. helical pitch angle and tail thickness) on swimming dynamics in a constant viscosity viscoelastic (Boger) fluid via a combination of particle tracking velocimetry, particle image velocimetry and three-dimensional simulations of the finitely extensible nonlinear elastic model with Peterlin approximation (FENE-P). The 3D-printed helical swimmer is actuated in a magnetic field using a custom-built rotating Helmholtz coil. Our results indicate that increasing the swimmer tail thickness and pitch angle enhances the normalized swimming speed (i.e. ratio of swimming speed in the Boger fluid to that of the Newtonian fluid). Strikingly, unlike the Newtonian fluid, the viscoelastic flow around the swimmer is characterized by formation of a front–back flow asymmetry that is characterized by a strong negative wake downstream of the swimmer's body. Evidently, the strength of the negative wake is inversely proportional to the normalized swimming speed. Three-dimensional simulations of the swimmer with the FENE-P model with conditions that match those of experiments, confirm formation of a similar front–back flow asymmetry around the swimmer. Finally, by developing an approximate force balance in the streamwise direction, we show that the contribution of polymer stresses in the interior region of the helix may provide a mechanism for swimming enhancement or diminution in the viscoelastic fluid.