In gas evolving electrolysis, bubbles grow at electrodes due to a diffusive influx from oversaturation generated locally in the electrolyte by the electrode reaction. When considering electrodes of micrometre size resembling catalytic islands, direct numerical simulations show that bubbles may approach dynamic equilibrium states at which they neither grow nor shrink. These are found in under- and saturated bulk electrolytes during both pinning and expanding wetting regimes of the bubbles. The equilibrium is based on the balance of local influx near the bubble foot and global outflux. To identify the parameter regions of bubble growth, dissolution and dynamic equilibrium by analytical means, we extend the solution of Zhang & Lohse (2023) J. Fluid Mech. 975, R3, by taking into account modified gas fluxes across the bubble interface, that result from a non-uniform distribution of dissolved gas. The Damköhler numbers at equilibrium are found to range from small to intermediate values. Unlike pinned nano-bubbles studied earlier, for micrometre-sized bubbles the Laplace pressure plays only a minor role. With respect to the stability of the dynamic equilibrium states, we extend the methodology of Lohse & Zhang (2015a) Phys. Rev. E 91 (3), 031003(R), by additionally taking into account the electrode reaction. Under contact line pinning, the equilibrium states are found to be stable for flat nano-bubbles and for micro-bubbles in general. For unpinned bubbles, the equilibrium states are always stable. Finally, we draw conclusions on how to possibly enhance the efficiency of electrolysis.

Turbulent emulsions are ubiquitous in chemical engineering, food processing, pharmaceuticals and other fields. However, our experimental understanding of this area remains limited due to the multiscale nature of turbulent flow and the presence of extensive interfaces, which pose significant challenges to optical measurements. In this study, we address these challenges by precisely matching the refractive indices of the continuous and dispersed phases, enabling us to measure local velocity information at high volume fractions. The emulsion is generated in a turbulent Taylor–Couette flow, with velocity measured at two radial locations: near the inner cylinder (boundary layer) and in the middle gap (bulk region). Near the inner cylinder, the presence of droplets suppresses the emission of angular velocity plumes, which reduces the mean azimuthal velocity and its root mean squared fluctuation. The former effect leads to a higher angular velocity gradient in the boundary layer, resulting in greater global drag on the system. In the bulk region, although droplets suppress turbulence fluctuations, they enhance the cross-correlation between azimuthal and radial velocities, leaving the angular velocity flux contributed by the turbulent flow nearly unchanged. In both locations, droplets suppress turbulence at scales larger than the average droplet diameter and increase the intermittency of velocity increments. However, the effects of the droplets are more pronounced near the inner cylinder than in the bulk, likely because droplets fragment in the boundary layer but are less prone to break up in the bulk. Our study provides experimental insights into how dispersed droplets modulate global drag, coherent structures and the multiscale characteristics of turbulent flow.

Combined theoretical and quantitative experimental study of resonant internal standing waves in a pycnocline between two miscible liquids in a narrow rectangular basin is presented. The waves are excited by a cylinder that harmonically oscillates in the vertical direction. A linear theoretical model describing the internal wave structure that accounts for pycnocline thickness, the finite wavemaker size and dissipation is developed. Separate series of measurements were performed using shadowgraphy and time-resolved particle image velocimetry. Accurate density profile measurements were carried out to monitor the variation of the pycnocline parameters in the course of the experiments; these measurements were used as the input parameters for the model simulations. The detected broadening of the pycnocline is attributed mainly to the presence of the waves and leads to the variation of the wave structure. The complex spatio-temporal structure of the observed internal wavefield was elucidated by carrying band-pass filtering in the temporal domain. The experiments demonstrate the coexistence of multiple spatial modes at the forcing frequency as well as the presence of the internal wave system at the second harmonic of the forcing frequency. The results of the theoretical model are in good agreement with the experiments.

A spherical vesicle is made up of a liquid core bounded by a semi-permeable membrane that is impermeable to solute molecules. When placed in an externally imposed gradient of solute concentration, the osmotic pressure jump across the membrane results in an inward trans-membrane solvent flux at the solute-depleted side of the vesicle, and and outward flux in its solute-enriched side. As a result, a freely suspended vesicle drifts down the concentration gradient, a phenomenon known as osmophoresis. An experimental study of lipid vesicles observed drift velocities that are more than three orders of magnitude larger than the linearised non-equilibrium prediction (Nardi et al., Phys. Rev. Lett., vol. 82, 1999, pp. 5168–5171). Inspired by this study, we analyse osmophoresis of a vesicle in close proximity to an impermeable wall, where the vesicle–wall separation $a\delta$ is small compared with the vesicle radius $a$. Due to intensification of the solute concentration gradient in the narrow gap between the membrane and the wall, the ‘osmophoretic’ force and torque on a stationary vesicle scale as an irrational power, $1/\sqrt {2}-1\ (\approx -0.29289\ldots )$, of $\delta$. Both the rectilinear velocity $\mathcal V$ and the angular velocity $\unicode {x1D6FA}$ of a freely suspended vesicle scale as the ratio of that power to $\ln \delta$. In contrast to the classical problem of sedimentation parallel to a wall, where the ratio $a\unicode {x1D6FA}/\mathcal V$ approaches $1/4$ as $\delta \to 0$, here the ratio approaches unity, as though the vesicle performs pure rigid-body rolling without slippage. Our approximations are in excellent agreement with hitherto unexplained numerical computations in the literature.

The attached-eddy model (AEM) predicts that the mean streamwise velocity and streamwise velocity variance profiles follow a logarithmic shape, while the vertical velocity variance remains invariant with height in the overlap region of high Reynolds number wall-bounded turbulent flows. Moreover, the AEM coefficients are presumed to attain asymptotically constant values at very high Reynolds numbers. Here, the AEM predictions are examined using sonic anemometer measurements in the near-neutral atmospheric surface layer, with a focus on the logarithmic behaviour of the streamwise velocity variance. Utilizing an extensive 210-day dataset collected from a 62 m meteorological tower located in the Eastern Snake River Plain, Idaho, USA, the inertial sublayer is first identified by analysing the measured momentum flux and mean velocity profiles. The logarithmic behaviour of the streamwise velocity variance and the associated ‘$-1$’ scaling of the streamwise velocity energy spectra are then investigated. The findings indicate that the Townsend–Perry coefficient ($A_1$) is influenced by mild non-stationarity that manifests itself as a Reynolds number dependence. After excluding non-stationary runs, and requiring the bulk Reynolds number defined using the atmospheric boundary layer height to be larger than $4 \times 10^{7}$, the inferred $A_1$ converges to values ranging between 1 and 1.25, consistent with laboratory experiments. Furthermore, nine benchmark cases selected through a restrictive quality control reveal a close relation between the ‘$-1$’ scaling in the streamwise velocity energy spectrum and the logarithmic behaviour of streamwise velocity variance. However, additional data are required to determine whether the plateau value of the pre-multiplied streamwise velocity energy spectrum is identical to $A_1$.

This study presents a novel investigation into the vortex dynamics of flow around a near-wall rectangular cylinder based on direct numerical simulation at $Re=1000$, marking the first in-depth exploration of these phenomena. By varying aspect ratios ($L/D = 5$, $10$, $15$) and gap ratios ($G/D = 0.1$, $0.3$, $0.9$), the study reveals the vortex dynamics influenced by the near-wall effect, considering the incoming laminar boundary layer flow. Both $L/D$ and $G/D$ significantly influence vortex dynamics, leading to behaviours not observed in previous bluff body flows. As $G/D$ increases, the streamwise scale of the upper leading edge (ULE) recirculation grows, delaying flow reattachment. At smaller $G/D$, lower leading edge (LLE) recirculation is suppressed, with upper Kelvin–Helmholtz vortices merging to form the ULE vortex, followed by instability, differing from conventional flow dynamics. Larger $G/D$ promotes the formation of an LLE shear layer. An intriguing finding at $L/D = 5$ and $G/D = 0.1$ is the backward flow of fluid from the downstream region to the upper side of the cylinder. At $G/D = 0.3$, double-trailing-edge vortices emerge for larger $L/D$, with two distinct flow behaviours associated with two interactions between gap flow and wall recirculation. These interactions lead to different multiple flow separations. For $G/D = 0.9$, the secondary vortex (SV) from the plate wall induces the formation of a tertiary vortex from the lower side of the cylinder. Double-SVs are observed at $L/D = 5$. Frequency locking is observed in most cases, but is suppressed at $L/D = 10$ and $G/D = 0.9$, where competing shedding modes lead to two distinct evolutions of the SV.

The transition to chaos in the subcritical regime of counter-rotating Taylor–Couette flow is investigated using a minimal periodic domain capable of sustaining coherent structures. Following a Feigenbaum cascade, the dynamics is found to be remarkably well approximated by a simple discrete map that admits rigorous proof of its chaotic nature. The chaotic set that arises for the map features densely distributed periodic points that are in one-to-one correspondence with unstable periodic orbits (UPOs) of the Navier–Stokes system. This supports the increasingly accepted view that UPOs may serve as the backbone of turbulence and, indeed, we demonstrate that it is possible to reconstruct every statistical property of chaotic fluid flow from UPOs.

Undulatory swimming is among the most common swimming forms found in nature across various length scales. In this study, we analyse the inertial effects of both the fluid and the swimmer on the transient motion of undulatory swimming using Taylor’s waving sheet model. We derive the transient velocity of the sheet for combined longitudinal and transverse waves in the Laplace domain, identifying three contributions to the velocity: the ‘slip’ velocity, fluid convection and a hydrodynamic force contribution. By numerically inverting the Laplace transform, we obtain the time history of the velocity for swimmers with varying swimming parameters and initial configurations. The acceleration performance of two types of swimmers is analysed by considering three dimensionless parameters: the acceleration rate $1/T$, sheet mass $M$, and Reynolds number $Re$, representing the effects of unsteady, convective and swimmer inertia, respectively. Under a relatively strong inertia effect, the start-up time scales as $\sim TM^2\,Re$ and $\sim TM^2$ for longitudinal and transverse waving sheets, respectively. Under weak inertia effects, the start-up time approximately reaches a constant for longitudinal waves, while it scales as $\sim T$ for transverse waves. Additionally, the transverse waving may induce a velocity overshoot, and enhances the burst swimming performance.

In rotating convection, analysis of heat transfer reveals a distinct shift in behaviour as the system transitions from a steep scaling regime near the onset of convection to a shallower scaling at higher Rayleigh numbers ($Ra$), irrespective of whether the top and bottom plates have stress-free, no-slip or no boundaries (homogeneous convection). However, while most research on this transition focuses on no-slip boundary conditions, geophysical and astrophysical flows commonly involve stress-free and homogeneous convection models as well. This study delves into the transition from the rapidly rotating regime to the non-rotating one with both stress-free and homogeneous models, leveraging direct numerical simulations (DNS) and existing literature data. Our findings unveil that for stress-free boundary conditions, the transitional Rayleigh number ($Ra_T$) exhibits a relationship $Ra_T\sim Ek^{-12/7}$, whereas for homogeneous rotating convection, $Ra_T\sim Ek^{-2}\, Pr$, where $Ek$ denotes the Ekman number, and $Pr$ denotes the Prandtl number. Both of these relationships align with the data obtained through DNS.

Head-on collisions between elliptic vortex rings (EVRs) and walls were studied experimentally using planar laser-induced fluorescence visualisations and time-resolved particle image velocimetry. Aspect ratios of $AR=2$ and 4 EVRs at a Reynolds number of $Re=4000$ were used. Collision locations were based on four key axis-switching stages of freely translating EVRs, which would shed light upon how axis-switching behaviour and aspect ratio variations affect the collision outcomes. Results show that non-uniform circumferential induced velocities in both colliding EVRs produce different behaviours along major and minor planes, where vortex-stretching/compression and hence circumferential flows play key roles in the vortex dynamics. Non-uniform formations of secondary/tertiary EVRs also lead to varied entanglements around the primary EVRs. As such, secondary vortex rings form vortex loops that may congregate along the collision axis, depending on the exact collision location. Vortex-core trajectories show the net primary/secondary vortex-core movements result from a balance between EVR diameter expansion due to collision and EVR segment motions associated with the axis-switching stage at the point of collision. Confinement effects are also observed to dominate over aspect ratio effects when the collision occurs closest to the orifice. While increasing the aspect ratio leads to different vortex-stretching/compression behaviour and more varied vortex-core trajectories due to the greater non-uniform induced velocities, they could still be understood by the preceding interpretations. Finally, three-dimensional vortex flows are reconstructed based on the experimental results to further explain the flow mechanisms.

A reactive control strategy is implemented to attenuate the streaks formed on a wing boundary layer due to free-stream turbulence (FST). Numerical simulations are performed on a section of a NACA0008 profile, considering its leading edge, while forced by FST with turbulence intensities of 0.5 % and 2.5 %. The controller is composed of localised sensors and actuators, with the control law consisting of a linear quadratic Gaussian regulator designed on a reduced-order model based only on the impulse responses of the system. Three configurations are evaluated by considering three different numbers of sensors/actuators along the spanwise direction. It is found that all configurations are effective in damping the streaks inside the boundary layer, whose effect is sustained downstream of the objective function location. However, distinct behaviours are observed when comparing the capability of the controllers with delay transition, where the best performance is attained for the case with larger number of sensors/actuators. This is attributed to the effectiveness of the controller in damping the streaks that will later break down, which in this case are associated with relatively short spanwise wavelength. This observation is confirmed by analysing the stability of the flow before the appearance of turbulent spots. Our results suggest that for an effective transition delay, efforts should not only be put into control of streaks with average spanwise wavelength, but also in the short spanwise wavelength associated with breakdown.

We investigate the deformation, dynamics and rheology of a single and a suspension of elastic capsules in inertial shear flow using high-fidelity particle-resolved simulations. For a single capsule in the shear flow, we elucidate the interplay of flow inertia and viscosity ratio, revealing the mechanism behind the stretching of capsule surface during tank-treading motion and the sign changes in normal stress differences with increasing inertia. When examining capsule suspensions, we thoroughly discuss the impact of volume fraction on average deformation, diffusion and rheology. Notably, we observe the formation of bridge structures due to hydrodynamic interactions, which enhance the inhomogeneity of the microstructure and alter the surface stress distribution within the suspension. We identify a critical Reynolds number range that marks the transition of capsule diffusion from non-inertial to inertial regimes. Furthermore, we reveal close connections between the behaviour of individual capsules and dense suspensions, particularly regarding capsule deformation and dynamics. Additionally, we propose multiple new empirical correlations for predicting the deformation factor of a single capsule and the relative viscosity of the suspension. These findings provide valuable insights into the complex behaviour of elastic capsules in inertial flows, informing the design of more accurate and efficient inertial microfluidic systems.

Sedimenting flows occur in a range of society-critical systems, such as circulating fluidised bed reactors and pyroclastic density currents (PDCs), the most hazardous volcanic process. In these systems, mass loading is sufficiently high ($\gg \mathcal {O}(1)$) and momentum coupling between the phases gives rise to mesoscale behaviour, such as formation of coherent structures capable of generating and sustaining turbulence in the carrier phase and directly impacting large-scale quantities of interest, such as settling time. While contemporary work has explored the physical processes underpinning these multiphase phenomena for monodispersed particles, polydispersed behaviour has been largely understudied. Since all real-world flows are polydisperse, understanding the role of polydispersity in gas–solid systems is critical for informing closures that are accurate and robust. This work characterises the sedimentation behaviour of two polydispersed gas–solid flows, with properties of the particles sampled from historical PDC ejecta. Highly resolved data at two volume fractions (1 % and 10 %) are collected using an EulerLagrange framework and is compared with monodisperse configurations of particles with diameters equivalent to the arithmetic mean of the polydisperse configurations. From these data, we find that polydispersity has an important impact on cluster formation and structure and that this is most pronounced for dilute flows. At higher volume fraction, the effect of polydispersity is reduced. We also propose a new metric for predicting the degree of clustering, termed ‘surface loading’, and a model for the coefficient of drag that accurately captures the settling velocity observed in the high-fidelity data.

Experiments are conducted over smooth and rough walls to explore the influence of pressure-gradient histories on skin friction and mean flow of turbulent boundary layers. Different pressure-gradient histories are imposed on the boundary layer through an aerofoil mounted in the free stream. Hot-wire measurements are taken at different free-stream velocities downstream of the aerofoil where the flow has locally recovered to zero pressure gradient but retains the history effects. Direct skin friction measurements are also made using oil film interferometry for smooth walls and a floating-element drag balance for rough walls. The friction Reynolds number, $Re_\tau$, varies between $3000$ and $27\,000$, depending both on the surface conditions and the free-stream velocity ensuring sufficient scale separation. Results align with previous findings, showing that adverse pressure gradients just upstream of the measurement location increase wake strength and reduce the local skin friction while favourable pressure gradients suppress the wake and increase skin friction. The roughness length scale, $y_0$, remains constant across different pressure-gradient histories for rough wall boundary layers. Inspired by previous works, a new correlation is proposed to infer skin friction based on the mean flow. The difference in skin friction by matching the turbulence profiles and flow structure between an arbitrary pressure-gradient history and zero pressure-gradient condition can be predicted using only the local wake strength parameter ($\Pi$), and the variations in wake strength for different histories are related to a weighted integral of the pressure-gradient history normalised by local quantities. This allows us to develop a general correlation that can be used to infer skin friction for turbulent boundary layers experiencing arbitrary pressure-gradient histories.

The quasi-steady shock refraction at a diffusive air–SF$_6$ interface (fast–slow type) is investigated numerically and theoretically. A new refraction pattern where both shock and expansion waves are simultaneously present in the reflected waves (named RRR-E) is first observed at the diffusive interface. The new refraction pattern is a regular pattern that is not expected to occur in classical shock refraction at a sharp fast–slow interface. Through the shock polar method, continuous refraction processes occur within the diffusion layer to satisfy the kinematic relationship between the reflected wave and the transmitted shock, which results in the RRR-E formation. Subsequently, the conditions for the RRR-E occurrence are obtained theoretically and verified numerically. In the phase diagram of the refraction patterns, the presence of RRR-E results in the transition boundaries of different refraction patterns at the sharp fast–slow interface no longer being valid. Specifically, the appearance of RRR-E delays the Mach reflection refraction (MRR) process, which is of great significance for the design of scramjet engines.

Geophysical and astrophysical fluid flows are typically driven by buoyancy and strongly constrained at large scales by planetary rotation. Rapidly rotating Rayleigh–Bénard convection (RRRBC) provides a paradigm for experiments and direct numerical simulations (DNS) of such flows, but the accessible parameter space remains restricted to moderately fast rotation rates (Ekman numbers ${ {Ek}} \gtrsim 10^{-8}$), while realistic ${Ek}$ for geo- and astrophysical applications are orders of magnitude smaller. On the other hand, previously derived reduced equations of motion describing the leading-order behaviour in the limit of very rapid rotation ($ {Ek}\to 0$) cannot capture finite rotation effects, and the physically most relevant part of parameter space with small but finite ${Ek}$ has remained elusive. Here, we employ the rescaled rapidly rotating incompressible Navier–Stokes equations (RRRiNSE) – a reformulation of the Navier–Stokes–Boussinesq equations informed by the scalings valid for ${Ek}\to 0$, recently introduced by Julien et al. (2024) – to provide full DNS of RRRBC at unprecedented rotation strengths down to $ {Ek}=10^{-15}$ and below, revealing the disappearance of cyclone–anticyclone asymmetry at previously unattainable Ekman numbers (${Ek}\approx 10^{-9}$). We also identify an overshoot in the heat transport as ${Ek}$ is varied at fixed $\widetilde { {Ra}} \equiv {Ra}{Ek}^{4/3}$, where $Ra$ is the Rayleigh number, associated with dissipation due to ageostrophic motions in the boundary layers. The simulations validate theoretical predictions based on thermal boundary layer theory for RRRBC and show that the solutions of RRRiNSE agree with the reduced equations at very small ${Ek}$. These results represent a first foray into the vast, largely unexplored parameter space of very rapidly rotating convection rendered accessible by RRRiNSE.