A specific family of spanwise-localised invariant solutions of plane Couette flow exhibits homoclinic snaking, a process by which spatially localised invariant solutions of a nonlinear partial differential equation smoothly grow additional structure at their fronts while undergoing a sequence of saddle-node bifurcations. Homoclinic snaking is well understood in the context of simpler pattern-forming systems such as the one-dimensional Swift–Hohenberg equation with cubic-quintic nonlinearity. The Swift–Hohenberg solutions closely resemble the snaking solutions of plane Couette flow, yet this remarkable resemblance and the mechanisms supporting homoclinic snaking within the three-dimensional Navier–Stokes equations remain to be fully understood. Studies of Swift–Hohenberg revealed the central importance of discrete symmetries for homoclinic snaking to be supported by an equation. We therefore study the structural stability of the characteristic snakes-and-ladders structure associated with homoclinic snaking in three-dimensional plane Couette flow for flow modifications that break symmetries of the flow. We demonstrate that wall-normal suction modifies the bifurcation structure of three-dimensional plane Couette solutions in the same way a symmetry-breaking quadratic term modifies solutions of the one-dimensional Swift–Hohenberg equation. These modifications are related to the breaking of the discrete rotational symmetry. At large amplitudes of the symmetry-breaking wall-normal suction the connected snakes-and-ladders structure is destroyed. Previously unknown solution branches are created and can be parametrically continued to vanishing suction. This yields new localised solutions of plane Couette flow that exist in a wide range of Reynolds numbers.

The effect of turbulence on snow precipitation is not incorporated into present weather forecasting models. Here we show evidence that turbulence is in fact a key influence on both fall speed and spatial distribution of settling snow. We consider three snowfall events under vastly different levels of atmospheric turbulence. We characterize the size and morphology of the snow particles, and we simultaneously image their velocity, acceleration and relative concentration over vertical planes approximately $30\ \textrm {m}^2$ in area. We find that turbulence-driven settling enhancement explains otherwise contradictory trends between the particle size and velocity. The estimates of the Stokes number and the correlation between vertical velocity and local concentration are consistent with the view that the enhanced settling is rooted in the preferential sweeping mechanism. When the snow vertical velocity is large compared to the characteristic turbulence velocity, the crossing trajectories effect results in strong accelerations. When the conditions of preferential sweeping are met, the concentration field is highly non-uniform and clustering appears over a wide range of scales. These clusters, identified for the first time in a naturally occurring flow, display the signature features seen in canonical settings: power-law size distribution, fractal-like shape, vertical elongation and large fall speed that increases with the cluster size. These findings demonstrate that the fundamental phenomenology of particle-laden turbulence can be leveraged towards a better predictive understanding of snow precipitation and ground snow accumulation. They also demonstrate how environmental flows can be used to investigate dispersed multiphase flows at Reynolds numbers not accessible in laboratory experiments or numerical simulations.

We numerically investigate the flow structure of periodic steady water waves of fixed relative mass flux propagating on rotational flows with piece-wise constant vorticity. We show that, for wave solutions along the global bifurcation diagram, the stagnation point can first occur internally or at the bottom, and then again occurs at the crest with further increase in wave height. We observe that the bifurcation diagram has a new branch which is not connected to the trivial solution. Furthermore, we present an in-depth discussion of the results on pressure distributions and particle trajectories beneath large-amplitude steady waves near stagnation. We also expand on previous results concerning the amplitude and mass flux of steady water waves travelling on rotational flows with discontinuous vorticity.

A numerical study of the $d$-dimensional eddy damped quasi-normal Markovian equations is performed to investigate the dependence on spatial dimension of homogeneous isotropic fluid turbulence. Relationships between structure functions and energy and transfer spectra are derived for the $d$-dimensional case. Additionally, an equation for the $d$-dimensional enstrophy analogue is derived and related to the velocity derivative skewness. Comparisons are made to recent four-dimensional direct numerical simulation results. Measured energy spectra show a magnified bottleneck effect which grows with dimension whilst transfer spectra show a varying peak in the nonlinear energy transfer as the dimension is increased. These results are consistent with an increased forward energy transfer at higher dimensions, further evidenced by measurements of a larger asymptotic dissipation rate with growing dimension. The enstrophy production term, related to the velocity derivative skewness, is seen to reach a maximum at around five dimensions and may reach zero in the limit of infinite dimensions, raising interesting questions about the nature of turbulence in this limit.

The generation of a magnetic field in an electrically conducting fluid generally involves the complicated nonlinear interaction of flow turbulence, rotation and field. This dynamo process is of great importance in geophysics, planetary science and astrophysics, since magnetic fields are known to play a key role in the dynamics of these systems. This paper gives an introduction to dynamo theory for the fluid dynamicist. It proceeds by laying the groundwork, introducing the equations and techniques that are at the heart of dynamo theory, before presenting some simple dynamo solutions. The problems currently exercising dynamo theorists are then introduced, along with the attempts to make progress. The paper concludes with the argument that progress in dynamo theory will be made in the future by utilising and advancing some of the current breakthroughs in neutral fluid turbulence such as those in transition, self-sustaining processes, turbulence/mean-flow interaction, statistical and data-driven methods and maintenance and loss of balance.

Micro- and nanodroplets have many important applications such as in drug delivery, liquid–liquid extraction, nanomaterial synthesis and cosmetics. A commonly used method to generate a large number of micro- or nanodroplets in one simple step is solvent exchange (also called nanoprecipitation), in which a good solvent of the droplet phase is displaced by a poor one, generating an oversaturation pulse that leads to droplet nucleation. Despite its crucial importance, the droplet growth resulting from the oversaturation pulse in this ternary system is still poorly understood. We experimentally and theoretically study this growth in Hele-Shaw-like channels by measuring the total volume of the oil droplets that nucleate out of it. In order to prevent the oversaturated oil from exiting the channel, we decorated some of the channels with a porous region in the middle. Solvent exchange is performed with various solution compositions, flow rates and channel geometries, and the measured droplets volume is found to increase with the Péclet number, $Pe$, with an approximate effective power law $V\propto Pe^{0.50}$. A theoretical model is developed to account for this finding. With this model we can indeed explain the $V\propto Pe^{1/2}$ scaling, including the prefactor, which can collapse all data of the ‘porous’ channels onto one universal curve, irrespective of channel geometry and composition of the mixtures. Our work provides a macroscopic approach to this bottom-up method of droplet generation and may guide further studies on oversaturation and nucleation in ternary systems.

When a water droplet is deposited within a sparsely miscible liquid medium, such as certain oils, the droplet surprisingly vanishes, even in a confined geometry. Such a phenomenon has crucial consequences for multiphase flows in which confined nano- and/or picolitre droplets are considered. We report here experiments of microdroplet dissolution in microchannels that reveal an enhancement of the shrinkage of confined water microdroplets in oil due to the permeability of the walls – made of polydimethilsiloxane (PDMS) – and a delay when collective effects are present. The system is first modelled assuming that the dissolution of the droplet in its surrounding liquid follows the Epstein–Plesset solution of the diffusion equation. The dissolution of small isolated droplets can indeed be described by this solution of the diffusion equation, while the vanishing of droplets larger than a certain critical value and those closer to other droplets requires numerical simulations. Experimental measurements and simulations compare well only when the boundary conditions of the confined system, the neighbouring droplets and, interestingly, the evaporative water vapour flux through the PDMS are all taken into account in the numerical model. Our results thus reveal the important role of the water solubility in oil and, most remarkably, of the water vapour transport through permeable walls.

We report on turbulent thermal convection experiments in a rotating cylinder with a diameter ($D$) to height ($H$) aspect ratio of $\varGamma =D/H=0.5$. Using nitrogen and pressurised sulphur hexafluoride we cover Rayleigh numbers (Ra) from $8\times 10^{9}$ to $8\times 10^{14}$ at Prandtl numbers $0.72\lesssim {\textit {Pr}}\lesssim 0.94$. For these Ra we measure the global vertical heat flux (i.e. the Nusselt number – Nu), as well as statistical quantities of local temperature measurements, as a function of the rotation rate, i.e. the inverse Rossby number – 1/Ro. In contrast to measurements in fluids with a higher Pr we do not find a heat transport enhancement, but only a decrease of Nu with increasing 1/Ro. When normalised with Nu(0) for the non-rotating system, data for all different Ra collapse and, for sufficiently large 1/Ro, follow a power law ${\textit {Nu}}/{\textit {Nu}}_0\propto (1/{\textit {Ro}})^{-0.43}$. Furthermore, we find that both the heat transport and the temperature field qualitatively change at rotation rates $1/{\textit {Ro}}^*_1=0.8$ and $1/{\textit {Ro}}^*_2=4$. We interpret the first transition at $1/{\textit {Ro}}^*_1$ as change from a large-scale circulation roll to the recently discovered boundary zonal flow (BZF). The second transition at rotation rate $1/{\textit {Ro}}^*_2$ is not associated with a change of the flow morphology, but is rather the rotation rate for which the BZF is at its maximum. For faster rotation the vertical transport of warm and cold fluid near the sidewall within the BZF decreases and hence so does Nu.

In this paper we experimentally study the transitional range of Reynolds numbers in plane Couette–Poiseuille flow, focusing our attention on the localized turbulent structures triggered by a strong impulsive jet and the large-scale flow generated around these structures. We present a detailed investigation of the large-scale flow and show how its amplitude depends on Reynolds number and amplitude perturbation. In addition, we characterize the initial dynamics of the localized turbulent spot, which includes the coupling between the small and large scales, as well as the dependence of the advection speed on the large-scale flow generated around the spot. Finally, we provide the first experimental measurements of the large-scale flow around an oblique turbulent band.

The energetically independent linear wave and geostrophic (vortex) solutions are shown to be a complete basis for velocity and density variables $(u,v,w,\rho )$ in a rotating non-hydrostatic Boussinesq fluid with arbitrary stratification and non-periodic vertical boundaries. This work extends the familiar wave-vortex decomposition for triply periodic domains with constant stratification. As a consequence of the decomposition, the fluid can be unambiguously separated into decoupled linear wave and geostrophic components at each instant in time, without the need for temporal filtering. The fluid can then be diagnosed for temporal changes in wave and geostrophic coefficients at each unique wavenumber and mode, including those that inevitably occur due to nonlinear interactions. We demonstrate that this methodology can be used to determine which physical interactions cause the transfer of energy between modes by projecting the nonlinear equations of motion onto the wave-vortex basis. In the particular example given, we show that an eddy in geostrophic balance superimposed with inertial oscillations at the surface transfers energy from the inertial oscillations to internal gravity wave modes. This approach can be applied more generally to determine which mechanisms are involved in energy transfers between wave and vortices, including their respective scales. Finally, we show that the nonlinear equations of motion expressed in a wave-vortex basis are computationally efficient for certain problems. In cases where stratification profiles vary strongly with depth, this approach may be an attractive alternative to traditional spectral models for rotating Boussinesq flow.

Heat and mass transport are generally closely correlated to momentum transport in shear flows. This so-called Reynolds analogy between advective heat or mass transport and momentum transport hinders efficiency improvements in engineering heat and mass transfer applications. I show through direct numerical simulations that in plane Couette and Taylor–Couette flow, rotation can strongly influence wall-to-wall passive tracer transport and make it much faster than momentum transport, clearly in violation of the Reynolds analogy. This difference between passive tracer transport, representative of heat/mass transport, and momentum transport is observed in steady flows with large counter-rotating vortices at low Reynolds numbers as well as in fully turbulent flows at higher Reynolds numbers. It is especially large near the neutral (Rayleigh's) stability limit. The rotation-induced Coriolis force strongly damps the streamwise/azimuthal velocity fluctuations when this limit is approached, while tracer fluctuations are much less affected. Accordingly, momentum transport is much more reduced than tracer transport, showing that the Coriolis force breaks the Reynolds analogy. At higher Reynolds numbers, this strong advective transport dissimilarity is accompanied by approximate limit cycle dynamics with intense low-frequency bursts of turbulence when approaching the neutral stability limit. The study demonstrates that simple body forces can cause clear dissimilarities between heat/mass and momentum transport in shear flows.

A Mach-10 hypersonic boundary layer of air overriding a cold, isothermal, non-catalytic flat wall, and with a stagnation enthalpy of $21.6\ \textrm {MJ}\ \textrm {kg}^{-1}$, is analysed using direct numerical simulations. The calculations include multicomponent transport, equilibrium vibrational excitation and chemical kinetics for air dissociation. The initially laminar boundary layer undergoes transition to turbulence by the resonance of a two-dimensional mode injected by a suction-and-blowing boundary condition imposed over a narrow spanwise porous strip. The ensuing turbulent boundary layer has a momentum Reynolds number of 3826 near the outflow of the computational domain. The relatively low temperature of the free stream renders the air chemically frozen there. However, the high temperatures generated within the boundary layer by viscous aerodynamic heating, peaking at a wall-normal distance $y^\star \simeq 10\text {--}20$ in semi-local viscous units, lead to air dissociation in under-equilibrium amounts equivalent to 4 %–7 % on a molar basis of atomic oxygen, along with smaller concentrations of nitric oxide, which is mainly produced by the Zel'dovich mechanism, and of atomic nitrogen, the latter being mostly in steady state. A statistical analysis of the results is provided, including the streamwise evolution of (a) the skin friction coefficient and dimensionless wall heat flux; (b) the mean profiles of temperature, velocity, density, molar fractions, chemical production rates and chemical heat-release rate; (c) the Reynolds stresses and root-mean-squares of the fluctuations of temperature, density, pressure, molar fractions and chemical heat-release rate; and (d) the temperature/velocity and mass-fraction/velocity correlations.

We report on experiments of the quasi-static growth and detachment of air bubbles in water from a superhydrophobic substrate, overcoming the maximum size limitation of conventional injectors due to the Rayleigh–Taylor instability. The observations are in good agreement with a hydrostatic model, demonstrating that bubbles grow through a sequence of quasi-equilibrium states. Our experiments corroborate the theoretical prediction of a maximum bubble volume of approximately $6.04 {\rm \pi}$ and a critical base radius of $3.22$, both numbers in units of the capillary length (Michael & Williams, Proc. R. Soc. Lond. A. vol. 351, 1976, pp. 117–127). This maximum is also reached when bubbles grow in an unbounded, ideally non-wetting surface, establishing the ultimate size limit of quasi-static bubble formation.

A millimetre-size superhydrophobic sphere impacting on the free surface of a quiescent bath can be propelled back into the air by capillary effects and dynamic fluid forces, whilst transferring part of its energy to the fluid. We report the findings of a thorough investigation of this phenomenon, involving different approaches. Over the range from minimum impact velocities required to produce rebounds to impact velocities that cause the sinking of the solid sphere, we focus on the dependence of the coefficient of restitution, contact time and maximum surface deflection on the different physical parameters of the problem. Experiments, simulations and asymptotic analysis reveal trends in the rebound metrics, uncover new phenomena at both ends of the Weber number spectrum, and collapse the data. Direct numerical simulations using a pseudo-solid sphere successfully reproduce experimental data whilst also providing insight into flow quantities that are challenging to determine from experiments. A model based on matching the motion of a perfectly hydrophobic impactor to a linearised fluid free surface is validated against direct numerical simulations and used in the low-Weber-number regime. The hierarchical and cross-validated models in this study allow us to explore the entirety of our target parameter space within a challenging multi-scale system.

We experimentally investigate the effect of geometrical anisotropy for buoyant spheroidal particles rising in a still fluid. All other parameters, such as the Galileo number (the ratio of gravitational to viscous forces) $Ga \approx 6000$, the ratio of the particle to fluid density $\varGamma \approx 0.53$ and the dimensionless moment of inertia $\boldsymbol{\mathsf{I}}^*= \boldsymbol{\mathsf{I}}_p/\boldsymbol{\mathsf{I}}_f$ (with $\boldsymbol{\mathsf{I}}_p$ being the moment of inertia of the particle and $\boldsymbol{\mathsf{I}}_f$ that of the fluid in an equivalent volume), are kept constant. The geometrical aspect ratio of the spheroids, $\chi$ , is varied systematically from $\chi = 0.2$ (oblate) to 5 (prolate). Based on tracking all degrees of particle motion, we identify six regimes characterised by distinct rise dynamics. Firstly, for $0.83 \le \chi \le 1.20$, increased rotational dynamics are observed and the particle flips over semi-regularly in a ‘tumbling’-like motion. Secondly, for oblate particles with $0.29 \le \chi \le 0.75$, planar regular ‘zig–zag’ motion is observed, where the drag coefficient is independent of $\chi$. Thirdly, for the most extreme oblate geometries ($\chi \le 0.25$), a ‘flutter’-like behaviour is found, characterised by precession of the oscillation plane and an increase in the drag coefficient. For prolate geometries, we observed two coexisting oscillation modes that contribute to complex trajectories: the first is related to oscillations of the pointing vector and the second corresponds to a motion perpendicular to the particle's symmetry axis. We identify a ‘longitudinal’ regime ($1.33 \le \chi \le 2.5$), where both modes are active and a different one, the ‘broadside’-regime ($3 \le \chi \le 4$), where only the second mode is present. Remarkably, for the most prolate particles ($\chi = 5$), we observe an entirely different ‘helical’ rise with completely unique features.

This paper investigates a long-standing question about the effect of surface roughness on turbulent flow: What is the equivalent roughness sand-grain height for a given roughness topography? Deep neural network (DNN) and Gaussian process regression (GPR) machine learning approaches are used to develop a high-fidelity prediction approach of the Nikuradse equivalent sand-grain height $k_s$ for turbulent flows over a wide variety of different rough surfaces. To this end, 45 surface geometries were generated and the flow over them simulated at ${Re}_\tau =1000$ using direct numerical simulations. These surface geometries differed significantly in moments of surface height fluctuations, effective slope, average inclination, porosity and degree of randomness. Thirty of these surfaces were considered fully rough, and they were supplemented with experimental data for fully rough flows over 15 more surfaces available from previous studies. The DNN and GPR methods predicted $k_s$ with an average error of less than 10 % and a maximum error of less than 30 %, which appears to be significantly more accurate than existing prediction formulae. They also identified the surface porosity and the effective slope of roughness in the spanwise direction as important factors in drag prediction.

Froth flotation by small air bubbles has been traditionally used in industry to capture fine minerals and other hydrophobic particles. This method, however, is not efficient for capturing very small particles. The present work is motivated by a new agglomeration process that overcomes this lack of efficiency. It consists of mixing a particle suspension and saltwater-filled droplets covered with semi-permeable oil layers. This paper investigates the two-particle dynamics of a solid particle and a semi-permeable spherical drop that expands due to osmosis in an external, pure extensional flow field. A dimensionless engulfment parameter measures the relative effects of droplet growth and convective flow. The computational results from numerical integration determine a transient collision efficiency, which describes the influence of hydrodynamic interactions and osmotic flow on particle capture. The results show that drop expansion, which decays slowly with time, greatly increases particle capture rates, especially for small particles. Moreover, as the engulfment parameter increases, there is a transition from flow-dominated capture to expansion-dominated capture. For the case of a non-expanding droplet, we provide a numerical solution for the transient pair distribution function, which enables us to explain the transient particle-capture rate in terms of the microstructure of the suspension. Furthermore, we derive an analytical expression for the initial collision efficiency at zero times, which agrees with our numerical data. The numerical results for non-expanding droplets at long times show increasing collision efficiency as the permeability increases and when the size ratio is near unity, in agreement with previous steady-state calculations.

The leapfrogging of coaxial vortex rings is a famous effect which has been noticed since the times of Helmholtz. Recent advances in ultra-cold atomic gases show that the effect can now be studied in quantum fluids. The strong confinement which characterises these systems motivates the study of leapfrogging of vortices within narrow channels. Using the two-dimensional point vortex model, we show that in the constrained geometry of a two-dimensional channel the dynamics is richer than in an unbounded domain: alongside the known regimes of standard leapfrogging and the absence of it, we identify new regimes of image-driven leapfrogging and periodic orbits. Moreover, by solving the Gross–Pitaevskii equation for a Bose–Einstein condensate, we show that all four regimes exist for quantum vortices too. Finally, we discuss the differences between classical and quantum vortex leapfrogging which appear when the quantum healing length becomes significant compared to the vortex separation or the channel size, and when, due to high velocity, compressibility effects in the condensate becomes significant.

The transient cavity dynamics during water entry of a heavy, non-rotating sphere impacting a rotating pool of liquid is studied experimentally, numerically and theoretically. We show that the pool rotation advances the transition of the cavity type – from deep seal to surface seal – marked by a reduction in the transitional Froude number. The role of the dimensionless rotational number $\mathcal {S} \equiv \omega R_0/U_0$ on the transient cavity dynamics is unveiled, where $R_0$ is the sphere radius, $\omega$ the angular speed of the liquid and $U_0$ the impact velocity. The rotating background liquid has two discernible effects on the cavity evolution. Firstly, an increase in the underwater pressure field due to centripetal effects; and secondly, a reduction in the pressure of airflow in the cavity neck near the water surface. The non-dimensional pinch-off time of the deep seal shows a robust $1/2$ power-law dependence on the Froude number, but with a reducing prefactor for increasing $\omega$. Our findings reveal that the effects of a rotating background liquid on the water entry can be traced back to the subtle differences in the initial stage splash and the near-surface cavity dynamics.

Studies on the evaporation of multicomponent droplets have revealed complex and important physical mechanisms, induced by preferential phase change or mediated by external vapour sources, e.g. occurrence of density-driven flows, phase separation, transient Marangoni flow and solutal effects, etc. With the addition of hygroscopic salts, the adhesive property of the droplet can be tuned, and the direction of water vapour mass flux reversed. This paper focuses on the dynamics of hygroscopic aqueous solution droplets, and analyses the interplay between different physical processes. Specifically, a lubrication-type model is established with the assumption of a precursor film in front of the three-phase contact line, which indicates qualitative agreement with our experimental results, quantitatively with respect to the initial spreading rates and qualitatively with respect to the overall behaviour. We derive the expression of absorptive mass flux combining the balance of chemical potential across the solution–air interface and the Hertz–Knudsen equation. Depending on the droplet state and the ambient condition, evaporation or vapour absorption occurs. The evaporative/absorptive mass flux varies both spatially and temporally as the droplet approaches equilibrium. It is demonstrated that the dominating mechanisms, i.e. capillary, thermal Marangoni and solutal Marangoni, compete with each other, and lead to diverse droplet dynamics at different stages of evaporation or vapour absorption. The findings shed light on the physical processes within droplets with both positive and negative interfacial mass fluxes, and provide rational explanations for the experimental observations.