We study numerically a sequence of eddies in two-dimensional electrohydrodynamics (EHD) flows of a dielectric liquid, driven by an electric potential difference between a hyperbolic blade electrode and a flat plate electrode (or the blade–plate configuration). The electrically driven flow impinges on the plate to generate vortices, which resemble Moffatt eddies (Moffatt, J. Fluid Mech., vol. 18, 1964, pp. 1–18). Such a phenomenon in EHD was first reported in the experimental work of Perri et al. (J. Fluid Mech., vol. 900, 2020, A12). We conduct direct numerical simulations of the EHD flow with three Moffatt-type eddies in a large computational domain at moderate electric Rayleigh numbers ($T$, quantifying the strength of the electric field). The ratios of size and intensity of the adjacent eddies are examined, and they can be compared favourably to the theoretical prediction of Moffatt; interestingly, the quantitative comparison is remarkably accurate for the two eddies in the far field. Our investigation also shows that a larger $T$ strengthens the vortex intensity, and a stronger charge diffusion effect enlarges the vortex size. A sufficiently large $T$ can further result in an oscillating flow, consistent with the experimental observation. In addition, a global stability analysis of the steady blade–plate EHD flow is conducted. The global mode is characterised in detail at different values of $T$. When $T$ is large, the confinement effect of the geometry in the centre region may lead to an increased oscillation frequency. This work contributes to the quantitative characterisation of the Moffatt-type eddies in EHD flows.

The curvature instability of thin vortex rings is a parametric instability discovered from short-wavelength analysis by Hattori & Fukumoto (Phys. Fluids, vol. 15, 2003, pp. 3151–3163). A full-wavelength analysis using normal modes then followed in Fukumoto & Hattori (J. Fluid Mech., vol. 526, 2005, pp. 77–115). The present work extends these results to the case with different densities inside and outside the vortex core in the presence of surface tension. The maximum growth rate and the instability half-bandwidth are calculated from the dispersion relation given by the resonance between two Kelvin waves of $m$ and $m+1$, where $m$ is the azimuthal wavenumber. The result shows that vortex rings are unstable to resonant waves in the presence of density and surface tension. The curvature instability for the principal modes is enhanced by density variations in the small axial wavenumber regime, while the asymptote for short wavelengths is close to the constant density case. The effect of surface tension is marginal. The growth rates of non-principal modes are also examined, and long waves are most unstable.

Many geological layers include cross bedding, which leads to different values for permeability along and across the bedding planes. We explore how such cross bedding impacts buoyancy-driven flow through an inclined aquifer. For each bedding angle and ratio of the permeability along and across the bedding, a free buoyancy-driven plume rises at a particular angle to the horizontal. If the angle of inclination of the aquifer to the horizontal is smaller than this angle, then the plume rises along the upper boundary, otherwise, somewhat surprisingly, the buoyant plume rises along the lower boundary of the aquifer. We present new laboratory experiments to support these predictions. We also test a model for the effective permeabilities which control the speed and the rate of spread of the plume along one or other boundary of the aquifer. We consider the impact of our results for modelling geological storage of ${\rm CO}_2$ or aquifer thermal energy storage.

This experimental study investigated the control induced by a spanwise surface wire on a rigid circular cylinder undergoing vortex-induced vibration (VIV) under the conditions of low mass damping in the lower synchronization branch. Being motivated by the idea of VIV-based energy harvesting from ocean and river flows, this elastically mounted cylinder was immersed in a water channel, leaving a free end at its bottom spanwise end, while the free water surface bounded its top. The cylinder was constrained to vibrate in the cross-stream direction. The wire diameter was 6.25 % of the cylinder diameter. Experimental research was conducted by attaching this large-scale wire along the span of the cylinder at various angular positions ranging from 0° to 180° (with respect to the most upstream point of the cylinder) at a fixed Reynolds number of 104 (based on the cylinder diameter). Simultaneous to measuring the trajectory of the cylinder motion via a laser distance sensor, the instantaneous velocity field in the near wake of the cylinder was obtained using particle image velocimetry. Several VIV response categories were identified depending on the angular position of the wire, which led to the classification of distinct angular ranges for the wire application. Associated with the structural vibrations in these categories, different vortex-formation modes induced by the wire were revealed. For specific wire positions, decreases of up to 98 % and increases of up to 102 % were identified in the oscillation amplitude of the cylinder compared with the amplitude of the clean cylinder under similar conditions.

A rotating disk is the canonical experiment for measuring surface reaction rates in geochemical and electrochemical systems. Using the similarity solution for laminar flow around an infinite disk, the mass transfer coefficient can be simply related to the intrinsic reaction rate at the surface. However, measurements of mass transfer rates use a finite-size disk within a larger container of solution; here the flow is no longer strictly laminar, but there must always be some recirculation. Our interest was initially in the assumption of a uniform radial concentration field, how this breaks down near the perimeter of the disk, and what effect that might have on the measured mass transfer rates. However, our numerical simulations suggest that the flow around a finite-size disk becomes time dependent at Reynolds number ($Re$) below 1000, which is much smaller than the typical values in mass-transfer measurements (${Re} \sim 10^4$). We observe the formation of coherent structures in the flow, which suggest a non-uniform mass transfer at the disk surface. The rotating-disk flow follows a similar sequence of instabilities to the Taylor–Couette flow: a centrifugal instability leading an axisymmetric, time-invariant flow, followed by a Hopf bifurcation to a time-periodic flow. To minimise the possibility that our results are a numerical artefact, we have also simulated the instability in the stationary boundary layer of a rotor–stator flow, comparing with self-similar solutions at low ${Re}$ and with spectral methods near the critical Reynolds number.

Fluid entering the periphery of a steadily rotating cylindrical tank exits through an off-axis drain hole, located in the tank's base at the half-radius. Experiments show that, though a concentrated vortex forms over the drain, it soon advects around the tank in what is at first a circular path. Though inviscid vortex dynamics predicts continued motion, our experiments show that the vortex moves inwards from the predicted circular path, finally coming to rest at approximately $50^{\circ }$ from the drain. In this final state, the vorticity is concentrated in a thin shear layer bounding an irrotational core, which passes over the drain. The broadening of the vortex structure and eventual steady-state formation are believed to be due to the growing boundary layer on the outer wall.

This empirical study aims to characterize the dynamical behaviour of sprays using time-series analysis of the size–velocity data acquired using a phase Doppler particle analyser. The prime motivation of this analysis is to capture the spatio-temporal correlations using time-series modelling paradigms that provide valuable new insights into spray dynamics. As a first step, we study long-held assumptions, especially on stationarity and time unsteadiness. We show that air-blast sprays have increased drop size as well as velocity ordering near the edge of the spray. Analysis of the inter-particle time of the droplets shows non-Poisson behaviour where droplets that are closely spaced in time are also closely spaced in the size and velocity coordinates. Temporal auto-correlation and partial auto-correlation calculations reveal the presence of inherent correlated features in the spray. This correlation is stronger and short lived in an air-blast spray and weaker but more persistent in a pressure swirl spray. These correlations render the probability density function (p.d.f.) estimate obtained from standard methods inaccurate; therefore, we propose a technically correct way of estimating the p.d.f. using a suitable downsampling and averaging method. Statistical analysis of residuals (from appropriate autoregressive integrated moving average time-series models) uncovers an interesting feature of spray data pertaining to heteroskedasticity (stochastically changing variance) of the diameter series. In order to account for heteroskedasticity, appropriate generalized autoregressive conditional heteroskedasticity models are developed. Finally, we present a utilitarian view of these results as an empirically consistent boundary condition implementation tool for computational fluid dynamics (CFD).

We investigate jumping of sessile droplets from a solid surface in ambient oil using modulated electrowetting actuation. We focus on the case in which the electrowetting effect is activated to cause droplet spreading and then deactivated exactly at the moment the droplet reaches its maximum deformation. By systematically varying the control parameters such as the droplet radius, liquid viscosity and applied voltage, we provide detailed characterisation of the resulting behaviours including a comprehensive phase diagram separating detachment from non-detachment behaviours, as well as how the detach velocity and detach time, i.e. duration leading to detachment, depend on the control parameters. We then construct a theoretical model predicting the detachment condition using energy conservation principles. We finally validate our theoretical analysis by experimental data obtained in the explored ranges of the control parameters.

The scaling and mechanism of the propagation speed of turbulent fronts in pipe flow with the Reynolds number has been a long-standing problem in the past decades. Here, we derive an explicit scaling law for the upstream front speed, which approaches a power-law scaling at high Reynolds numbers, and we explain the underlying mechanism. Our data show that the average wall distance of low-speed streaks at the tip of the upstream front, where transition occurs, appears to be constant in local wall units in the wide bulk-Reynolds-number range investigated, between 5000 and 60 000. By further assuming that the axial propagation of velocity fluctuations at the front tip, resulting from streak instabilities, is dominated by the advection of the local mean flow, the front speed can be derived as an explicit function of the Reynolds number. The derived formula agrees well with the speed measured by front tracking. Our finding reveals a relationship between the structure and speed of a front, which enables a close approximation to be obtained of the front speed based on a single velocity field without having to track the front over time.

Extensive studies have focused on the self-propulsion of a droplet in a viscous environment driven by the Marangoni effect in the absence of inertial effects. In order to capture the influence of inertia on the self-propulsion of a droplet, we use the singular perturbation solution for small but finite Reynolds number ($Re$) flow past a spherical droplet with inhomogeneous surface tension. We calculate the swimming speed and the corresponding flow fields generated by the droplet in an axisymmetric unbounded medium at $O(Re^2)$. The present results reveal how the choice of the stress parameter $\sigma$, which is the ratio of the first two modes of the induced stress field, distinguishes between the different swimming styles, and determines the role of inertia on the swimming speed, energy expenditure and swimming efficiency of the droplet. Inertia enhances the swimming speed and the associated swimming efficiency of the droplet by abating the energy expenditure. It is striking to observe how a droplet swimmer with $\sigma <0$ has a competitive advantage over a rigid squirmer with an equivalent surface activity due to the existence of an internal flow. We independently treat the potential influence of the viscosity ratio on the swimming properties of the droplet at finite $Re$. Additionally, using linear stability analysis, we provide insights into the stability of the estimated migration velocity at $O(Re)$. We argue that the droplet achieves a distinct stable equilibrium velocity, which occurs due to the inertial effect of the surrounding medium.

This study introduces an analytical solution for the laminar entrance flow in circular pipes, aiming to confirm the occurrence of velocity overshoot. Velocity overshoot is characterised by the maximum axial velocity appearing near the pipe wall instead of the central axis. Similar to the previous studies, the analytical solution is derived from the parabolised Navier–Stokes equation; however, the specific approach used in linearising the momentum equation has not been attempted before. The accuracy of this analytical solution has been verified through a comprehensive comparison with various published experimental data. The existence of velocity overshoot at a short distance from the inlet, which is evident in numerous numerical calculations based on the full Navier–Stokes equations and corroborated by recent magnetic resonance (MR) velocimetry experiments, is identified analytically for the first time. The parabolised Navier–Stokes equation has inherent self-similarity with respect to the Reynolds number, implying that $Re$ is incorporated into the dimensionless variables rather than serving as an independent flow parameter. According to both MR velocimetry measurements and the present analytical solution, the self-similarity is not valid immediately following the pipe inlet, and this becomes more evident as $Re$ decreases; hence, the analytical solution derived from the parabolised Navier–Stokes equation cannot accurately predict the evolution of the velocity profile within this region near the pipe inlet.

Even without breaking or wind influence, ocean surface waves are observed to produce turbulence in the water, possibly influencing ocean surface dynamics and air–sea interactions. Based on the water-side free-surface simulations, recent studies suggest that such turbulence is produced through the interaction between the waves and the near-surface Eulerian current associated with the viscous attenuation of waves. To clarify the dynamical role of the air–water interface in the turbulence production, the attenuating interfacial gravity waves were simulated directly using a newly developed two-phase wave-resolving numerical model. The air–water coupling enhanced the wave energy dissipation through the formation of a strong shear at the air-side viscous boundary layer. This led to an enhancement of the wave-to-current momentum transfer and the formation of the down-wave Eulerian mean sheared current, which is favourable for the CL2 instability responsible for the production of Langmuir circulations. As a result, the water-side turbulence grew stronger compared with the corresponding free surface (water-only) wave-resolving simulation. The evolution of the wave-averaged field was well reproduced with the Craik–Leibovich equation with the upper boundary condition provided with the virtual wave stress based on linear theory. The wave energy dissipation by air–water coupling plays a significant role in the quantitative understanding of the wave-induced turbulence at the laboratory and field scales.

We study the effect of geometrical confinement on thermal convection by laboratory experiments and direct numerical simulations using Hele-Shaw geometries (typically the gap-to-height aspect ratio $0.12$) for the Prandtl number $Pr \geq 40$ and the Rayleigh number $Ra \leq 6 \times 10^7$. Under such strong unidirectional confinement, the convective flows are forced to squeeze within the narrow gap and exhibit unique spatiotemporal signatures, which contrast those in unconfined systems. With the increase of $Ra$, we identify that the system experiences five convective regimes that can be classified from two aspects, time dependency and flow dimensionality: (I) quasi-two-dimensional (quasi-2-D) steady flow; (II) quasi-2-D flow with oscillatory corner rolls; (III) three-dimensional (3-D) flow with oscillatory corner rolls; (IV) 3-D steady flow; and (V) 3-D time-dependent motion of plumes around sidewalls. Notably, unsteadiness does not emerge globally, but is localised near the sidewalls as oscillatory corner rolls, resulting in the regime transitions happening in a quasi-steady manner. We confirm that these regime transitions show less dependence on both $Pr$ and the other (wider) horizontal scale of the geometry. Moreover, we find that a recently proposed criterion ‘degree of confinement’ (Noto et al., Proc. Natl Acad. Sci. USA, vol. 121, issue 28, 2024, e2403699121) successfully explains the emergence of 3-D structures, expanding its applicable range to smaller $Ra$. This study deepens the comprehension of the thermal convection emerging in tight geometries, impacting across disciplines, such as Earth and planetary science, and thermal engineering.

We study the spreading of Newtonian viscous (aqueous glycerin solution) and viscoelastic (aqueous polymer solution) drops on solid substrates with different wettabilities. For drops of the same zero-shear viscosity, we find in the early stages of spreading that viscoelastic drops (i) spread faster and (ii) their contact radius shows a different power law vs time than Newtonian drops. We argue that the effect of viscoelasticity is only observable for experimental time scales of the order of or larger than the internal relaxation time of the viscoelastic polymer solution. We attribute this behaviour to the shear thinning of the viscoelastic polymer solution. When approaching the contact line, the shear rate increases and the steady-state viscosity of the viscoelastic drop is lower than that of the Newtonian drop. We support our experimental findings with a simple (first-order) perturbation model that qualitatively agrees with our findings.

The scaling relations mapping the turbulence statistics in compressible turbulent boundary layers (TBLs) onto their incompressible counterparts are of fundamental significance for turbulence modelling, such as the Morkovin scaling for velocity fields, while for pressure fluctuation fields, a corresponding scaling relation is currently absent. In this work, the underlying scaling relations of pressure fluctuations about Mach number ($M$) contained in their generation mechanisms are explored by analysing a series of direct numerical simulation data of compressible TBLs over a wide Mach number range $(0.5\leq M \leq 8.0)$. Based on the governing equation of pressure fluctuations, they are decomposed into components according to the properties of source terms. It is notable that the intensity of the compressible component, predominantly originating from the acoustic mode, obeys a monotonic distribution about the Mach number and wall distance; further, the intensity of the rest of the pressure components, which are mainly generated by the vorticity mode, demonstrates a uniform distribution consistent with its incompressible counterpart. Moreover, the coupling between the two components is negligibly weak. Based on the scaling relations, semiempirical models for the fluctuation intensity of both pressure and its components are constructed. Hence, a mapping relation is obtained that the profiles of pressure fluctuation intensities in compressible TBLs can be mapped onto their incompressible counterparts by removing the contribution from the acoustic mode, which can be provided by the model. The intrinsic scaling relation can provide some basic insight for pressure fluctuation modelling.