We study the mixing processes inside a forced fountain using data from direct numerical simulation. The outer boundary of the fountain with the ambient is a turbulent/non-turbulent interface. Inside the fountain, two internal boundaries, both turbulent/turbulent interfaces, are identified: (i) the classical boundary between upflow and downflow which is composed of the loci of points of zero mean vertical velocity; and (ii) the streamline that separates the mean flow emitted by the source from the entrained fluid from the ambient (the separatrix). We show that entrainment due to turbulent fluxes across the internal boundary is at least as important as that by the mean flow. However, entrainment by the turbulence behaves substantively differently from that by the mean flow and cannot be modelled using the same assumptions. This presents a challenge for existing models of turbulent fountains and other environmental flows that evolve inside turbulent environments.

The interaction of supercritical turbulent flows with granular sediment beds is challenging to study both experimentally and numerically. This challenging task has hampered advances in understanding antidunes, the most characteristic bedform of supercritical flows. This article presents the first numerical attempt to simulate upstream-migrating antidunes with geometrically resolved particles and a liquid–gas interface. Our simulations provide data at a resolution higher than laboratory experiments, and they can therefore provide new insights into the mechanisms of antidune migration and contribute to a deeper understanding of the underlying physics. To manage the simulations’ computational costs and physical complexity, we employ the cumulant lattice Boltzmann method in conjunction with a discrete element method for particle interactions, as well as a volume-of-fluid scheme to track the deformable free surface of the fluid. By reproducing two flow configurations of previous experiments (Pascal et al., Earth Surf. Process. Landf., vol. 46, issue 9, 2021, pp. 1750–1765), we demonstrate that our approach is robust and accurately predicts the antidunes’ amplitude, wavelength and celerity. Furthermore, the simulated wall shear stress, a key parameter governing sediment transport, is in excellent agreement with the experimental measurements. The highly resolved data of fluid and particle motion from our simulation approach open new perspectives for detailed studies of morphodynamics in shallow supercritical flows.

Understanding how turbulence enhances irreversible scalar mixing in density-stratified fluids is a central problem in geophysical fluid dynamics. While isotropic overturning regions are commonly the focus of mixing analyses, we here investigate whether significant mixing may arise in anisotropic statically stable regions of the flow. Focusing on a single forced direct numerical simulation of stratified turbulence, we analyse spatial correlations between the vertical density gradient $\partial \rho /\partial z$ and the dissipation rates of kinetic energy $\epsilon$ and scalar variance $\chi$, the latter quantifying scalar mixing. The domain is characterized by relatively well-mixed density layers separated by sharp stable interfaces that are correlated with high vertical shear. While static instability is most prevalent within the mixed layers, much of the scalar mixing is localized to the intervening interfaces, a phenomenon not apparent if considering local static instability or $\epsilon$ alone. While the majority of the domain is characterized by the canonical flux coefficient $\varGamma \equiv \chi /\epsilon =0.2$, often assumed in ocean mixing parametrizations, extreme values of $\chi$ within the statically stable interfaces, associated with elevated $\varGamma$, strongly skew the bulk statistics. Our findings suggest that current parametrizations of turbulent mixing may be biased by undersampling, such that the most common, but not necessarily the most significant, mixing events are overweighted. Having focused here on a single simulation of stratified turbulence, it is hoped that our results motivate a broader investigation into the role played by stable density interfaces in mixing, across a wider range of parameters and forcing schemes representative of ocean turbulence.

Direct numerical simulation data obtained from four pairs of turbulent, lean hydrogen–air, complex-chemistry flames are analysed to explore the influence of molecular diffusion on flame surface density, displacement speed $S_d$ and the flame surface density transport equation terms. Each pair involves (i) a flame where mixture-averaged molecular diffusivities are adopted and Lewis number $Le$ is significantly less than unity and (ii) an equidiffusive flame where all molecular diffusivities are set equal to molecular heat diffusivity of the mixture and $Le=1$, with other things being equal. Reported results show that significantly higher turbulent burning rates simulated in the former flames result mainly from an increase in the local fuel consumption rate, whereas an increase in flame surface area plays a secondary role, especially in more intense turbulence. The rate increase stems from (i) an increase in the peak local fuel consumption rate and (ii) an increase in a width of a zone where the rate is significant. The latter phenomenon is of more importance in richer flames and both phenomena are most pronounced in the vicinity of the flame leading edges, thus indicating a crucial role played by the leading edge of a premixed turbulent flame in its propagation. Moreover, mean displacement speed differs significantly from the laminar flame speed even in the equidiffusive flames, varies substantially across flame brush and may be negative at the leading edges of highly turbulent flames.

This study investigates the dynamics of the simultaneous impact of two droplets on a dry substrate. We develop a new micro-controlled droplet generator that releases two equally sized water droplets simultaneously on-demand, with no trailing droplets. The impact Weber number, based on impact droplet size and velocity, and the inter-droplet spacing relative to the impact droplet size are varied in the ranges of 54 to 155 and 1.32 to 2.25, respectively, leading to the strong interaction of the spreading lamellae that form a central uprising sheet, which eventually deposits or breaks into tiny droplets. We analyse the impact processes for both deposition and splashing of the uprising sheets. Simultaneous high-speed imaging from two orthogonal views of the droplet impacts quantifies the three-dimensional structure of the sheet morphology, including the temporal evolution of the rim-bounded ‘semilunar’ shape, surface waves, rim corrugations and finger formation, and deposition or splashing of the liquid sheet. The characteristics of the sheet surface waves and the rim instabilities are quantified. Novel scaling is developed for the maximum sheet height, sheet width and thickness, which considers the geometrical constraints and mass balance of the interacting lamellae to describe the temporal evolution of a ‘semilunar’ uprising sheet and is in good agreement with the measurements. The uprising sheet splashing generates larger droplets than those from splashing of single-droplet impacts, and it occurs due to the end-pinching of sheet fingers and at conditions that single-droplet impacts lead only to liquid deposition.

The effect of surfactants on the flow characteristics during rapid drop formation in a microchannel is investigated using high-speed imaging, micro-particle image velocimetry and numerical simulations; the latter are performed using a three- dimensional multiphase solver that accounts for the transport of soluble surfactants in the bulk and at the interface. Drops are generated in a flow-focusing microchannel, using silicone oil ($4.6$ mPa s) as the continuous phase and a 52 % w/w glycerol solution as the dispersed phase. A non-ionic surfactant (Triton X-100) is dissolved in the dispersed phase at concentrations below and above the critical micelle concentration. Good agreement is found between experimental and numerical data for the drop size, drop formation time and circulation patterns. The results reveal strong circulation patterns in the forming drop in the absence of surfactants, whose intensity decreases with increasing surfactant concentration. The surfactant concentration profiles in the bulk and at the interface are shown for all stages of drop formation. The surfactant interfacial concentration is large at the front and the back of the forming drop, while the neck region is almost surfactant free. Marangoni stresses develop away from the neck, contributing to changes in the velocity profile inside the drop.

In a dilute suspension where sinking spheroids or motile gyrotactic micro-organisms are modelled as orientable and negatively buoyant particles, we have found analytical solutions to their steady distributions under any arbitrary continuous vertical shear flow. The two-way coupling between their distribution and the vertical flow is nonlinear, enabling the uniform base state to bifurcate into a structure reminiscent of the streamers in settling spheroid suspensions and gyrotactic plumes. This bifurcation depends on a single parameter that is proportional to the average number of particles on a horizontal cross-section. In a three-dimensional axisymmetric system, the plume structure blows up when the parameter is above a threshold. We discuss how this singularity is analogous to the chemotactic collapse of a Keller–Segel model, and the significance that this analogy entails.

The root systems of mangroves, a tree species found in intertidal tropical and subtropical coastal zones, provide a natural barrier that dissipates wave energy effectively and reduces sediment erosion. Here, we use a combination of experiments and numerical simulations to examine the wake and drag characteristics of porous arrays of cylinders, which serve as simplified models of mangrove root networks. Optimal arrangements of the arrays are obtained by coupling Navier–Stokes simulations with a multi-objective optimization algorithm, which seeks configurations that minimize wake enstrophy and maximize drag on the porous structure. These optimal configurations are investigated using particle image velocimetry, and the internal and external flows around the porous arrays are analysed using a combination of proper orthogonal decomposition and Lagrangian particle tracking. Large variations in drag and enstrophy are observed by varying the relative positions of the cylinders, which indicates that the geometrical arrangement of porous arrays plays a prominent role in determining wake and drag characteristics. A sensitivity analysis suggests that enstrophy is more sensitive than drag to specific cylinder placement, and depends on distinctive flow patterns that develop in the interior due to interactions among neighbouring cylinders. Arrays with higher drag involve a combination of larger projected frontal area and minimal flux through the interior, leading to increased wake enstrophy, which is unfavourable for particle deposition and erosion. Based on the analysis of characteristics associated with the optimal arrays, several manually designed arrays are tested, and they display the expected behaviour with regard to drag and enstrophy.

We revisit the classical problem of calculating the pancake-like shape of a sessile drop at large Bond numbers. Starting from a formulation where drop volume and contact angle are prescribed, we develop an asymptotic scheme which systematically produces approximations to the two key pancake parameters, height and radius. The scheme is based on asymptotic matching of a ‘flat region’ where capillarity is negligible and an ‘edge region’ near the contact line. Major simplifications follow from the distinction between algebraically and exponentially small terms, together with the use of two exact integral relations. The first represents a force balance in the vertical direction. The second, which can be interpreted as a radial force balance on the drop edge (up to exponentially small terms), generalises an approximate force balance used in classical treatments. The resulting approximations for the geometric pancake parameters, which go beyond known leading-order results, are compared with numerical calculations tailored to the pancake limit. These, in turn, are facilitated by an asymptotic approximation for the exponentially small apex curvature, which we obtain using a Wentzel–Kramers–Brillouin method. We also consider the comparable two-dimensional problem, where similar integral balances explicitly determine the pancake parameters in closed form up to an exponentially small error.

Thin-film flow down a fibre exhibits rich dynamics and is relevant to applications such as desalination, fibre coating and fog harvesting. These flows are subject to instabilities that result in dynamic bead-on-fibre patterns. We perform an experimental study of shear-thinning flow down fibres using 20 different xanthan gum solutions as our working liquid. The bead-on-fibre morphology can be oriented either symmetrically or asymmetrically on the fibre, and this depends upon the surface tension, fibre diameter and liquid rheology, as defined by the Ostwald power-law index. For highly shear-thinning liquids, it is possible for the pattern to be complex and exhibit simultaneously both asymmetric large beads and symmetric small beads in the isolated and convective flow regimes. We quantify the transition between flow regimes and bead dynamics for the asymmetric morphology, and compare with Newtonian flow, as it depends upon the experimental parameters. Finally, the dimensionless bead frequency is shown to scale with the Bond number for all of our experimental data (symmetric and asymmetric).

The dynamics of thin, non-circular droplets evaporating in the diffusion-limited regime is examined. The challenging non-rectilinear mixed boundary problem this poses is solved using a novel asymptotic approach and an asymptotic expansion for the evaporative flux from the free surface of the droplet is found. While theoretically valid only for droplets that are close to circular, it is demonstrated that the methodology can successfully be applied to droplets with a wide variety of footprint shapes, including polygons and highly non-convex domains. As our solution for the flux fundamentally represents a novel result in potential theory, the applications are numerous, as the mixed boundary value problem arises in fields as diverse as electrostatics and contact mechanics. Here, we demonstrate the practicality of our result by considering the analytically tractable case of deposition of solute from large droplets in detail, including a matched asymptotic analysis to resolve the pressure, streamlines and deposition up to second order.

We study the effect of insoluble surfactants on the impact of surfactant-free droplets onto surfactant-laden thin liquid films via a fully three-dimensional direct numerical simulation approach that employs a hybrid interface-tracking/level-set method, and by taking into account surfactant-induced Marangoni stresses due to gradients in interfacial surfactant concentration. Our numerical predictions for the temporal evolution of the surfactant-free crown are validated against the experimental work by Che & Matar (Langmuir, vol. 33, 2017, pp. 12140–12148). We focus on the ‘crown-splash regime’, and we observe that the crown dynamics evolves through various stages: from the growth of linear modes (through a Rayleigh–Plateau instability) to the development of nonlinearities leading to primary and secondary breakup events (through droplet shedding modulated by an end-pinching mechanism). We show that the addition of surfactants does not affect the wave selection via the Rayleigh–Plateau instability. However, the presence of surfactants plays a key role in the late stages of the dynamics as soon as the ligaments are driven out from the rim. Surfactant-induced Marangoni stresses delay the end-pinching mechanisms to result in longer ligaments prior to their capillary singularity. Our results indicate that Marangoni stresses bridge the gap between adjacent protrusions promoting the adjacent protrusions' collision and the merging of ligaments. Finally, we demonstrate that the addition of surfactants leads to surface rigidification and consequently to the retardation of the flow dynamics.

Although natural and industrial flows often transport and mix multiple scalars, relatively few studies of turbulent multi-scalar mixing have been undertaken. In the present work, a novel three-wire thermal-anemometry-based probe – capable of simultaneously measuring velocity, helium concentration and temperature – is used to investigate the evolution of multiple scalars ($\phi _1$, $\phi _2$, $\phi _3$) and velocity in turbulent coaxial jets. The jets consist of (i) a centre jet containing a mixture of helium and air ($\phi _1$), (ii) an annular jet containing pure (unheated) air ($\phi _2$) and (iii) a coflow of (pure) heated air ($\phi _3$). Axial measurements are made at three different momentum flux ratios ($M=0.77, 2.1, 4.2$). Increasing $M$ was observed to result in complex, competing effects. Larger momentum flux ratios cause the potential core of the centre jet to decrease in size, greater scalar fluctuations and more rapid correlation of $\phi _1$ and $\phi _2$. However, at the same time, certain statistics, including those describing the velocity field, evolve more slowly. Moreover, the flow near the beginning of the fully merged region appears to be less mixed at higher values of $M$. The present work finally demonstrates that differences can be observed in the evolution and mixing of coaxial jets between those in which $M<1$ and those in which $M>1$, thereby presenting an opportunity by which the mixing process in coaxial jets may be controlled.

Supraglacial lakes play a central role in storing melt water, enhancing surface melt and ultimately in driving ice flow and ice shelf melt through injecting water into the subglacial environment and through facilitating fracturing. Here, we develop a model for the drainage of supraglacial lakes through the dissipation-driven incision of a surface channel. The model consists of the St Venant equations for flow in the channel, fed by an upstream lake reservoir, coupled with an equation for the evolution of channel elevation due to advection, uplift and downward melting. After reduction to a ‘stream power’-type hyperbolic model, we show that lake drainage occurs above a critical rate of water supply to the lake due to the backward migration of a shock that incises the lake seal. The critical water supply rate depends on advection velocity and uplift (or more precisely, drawdown downstream of the lake) as well as model parameters such as channel wall roughness and the parameters defining the relationship between channel cross-section and wetted perimeter. Once lake drainage does occur, it can either continue until the lake is empty, or terminate early, leading to oscillatory cycles of lake filling and draining, with the latter favoured by large lake volumes and relatively small water supply rates.

We propose a compressible multi-scale model that (i) captures the dynamics of both large vapour cavities (resolved vapour) and micro-bubbles (unresolved vapour), and (ii) accounts for medium compressibility. The vapour mass, momentum and energy in the compressible homogeneous mixture equations are explicitly decomposed into constituent resolved and unresolved components that are independently treated. The homogeneous mixture of liquid and resolved vapour is tracked as a continuum in an Eulerian sense. The unresolved vapour terms are expressed in terms of subgrid bubble velocities and radii that are tracked in a Lagrangian sense using a novel ‘$kR$-$RP$ equation’ (k, constant multiple; R, bubble size; RP, Rayleigh-Plesset). The $kR$-$RP$ equation is formally derived in terms of the pressure at a finite distance ($kR$) from the bubble while accounting for the effects of neighbouring bubbles; $p(kR)$ may therefore be either a near-field or far-field pressure. The equation exactly recovers the classical Rayleigh–Plesset and Keller–Miksis equations in the limits that $k$ and $c$ (speed of sound) become very large. Also, the results are independent of $k$ for a single bubble for all $k$, and for multiple bubbles when $kR < d$ (where $d$ denotes separation distance). Numerical results show this robustness of the model to the choice of $k$, which can be different for each bubble. The multi-scale model is validated for the collapse of a single resolved/unresolved bubble. Its ability to capture inter-bubble interactions is demonstrated for multiple bubbles exposed to an acoustic pulse. The model is then applied to a problem where resolved and unresolved bubbles co-exist. Finally, it is validated using a cluster of $1200$ bubbles exposed to a strong acoustic pulse. The results show the impact of the bubble cluster on the transmitted and reflected waves and the shielding effect where bubbles at the edge of the cluster shield the interior bubbles by dampening the incident acoustic wave.

The large-scale pulsations of shock-induced separation with length scale that significantly exceeds the incoming boundary layer thickness are investigated. The shock–boundary layer interaction (SBLI) unit is generated by an inward-turning axisymmetric compression ramp at an inflow Mach number of 2.5. A substantial region surrounding the centre azimuth exhibited mean and dynamic flow features that are consistent with two-dimensional separation. Two-dimensional highly resolved maps of surface pressure field are obtained using fast-response pressure-sensitive paint fluorescence imaging at 40 kHz repetition rate. The measurement domain covered significant regions of the incoming boundary layer through the relaxing boundary layer downstream of the reattachment as well as over 25 boundary layer thicknesses in the azimuthal direction. These measurements provide new insights into the spanwise coupling of the SBLI unit in addition to its inherent dynamics. The power spectral density (PSD) of the centreline pressure exhibits very good agreement with theoretical models and complementary measurements using fast-response pressure transducers, which served to validate the pressure field measurements. Detailed examination of the PSD reveals strong agreement with the literature, which includes the peak Strouhal number of the separation and reattachment shock motions as well as the downward frequency shift along the separation bubble. Furthermore, the pressure fluctuation maps reveal streamwise-elongated structures just downstream of the ramp leading edge that persist well downstream of the reattachment. A time sequence of conditional average pressure fluctuation maps is constructed surrounding isolated pressure excursions in the intermittent region. This sequence, along with two-point cross correlation analysis, provides critical information about the flow processes that drive the separation bubble pulsations in the SBLI units with large separation scales. Overall, the imbalance in the mass within the separation bubble appears to be the critical mechanism that drives the separation bubble pulsations. Furthermore, the pressure perturbations originating at azimuthally offset locations are also observed to influence the separation bubble dynamics.

Subaqueous vortex ripples in equilibrium are characterized by their unique geometry and dimensions. Motivated by the recent direct numerical simulation study of oscillatory turbulent flow over a wavy bottom by Önder & Yuan (J. Fluid Mech., vol. 858, 2019, pp. 264–314), the objective of this study is to further investigate the fluid dynamical controls that determine the distinctive equilibrium dimensions of vortex ripples. We use direct numerical simulations to investigate the differences in flow kinetics between sinusoidal oscillatory flow over equilibrium and out-of-equilibrium vortex ripples. In comparison with the equilibrium case, the spanwise coherent vortices, the averaged bottom shear stress on overlying flow and the shear stress distribution on the ripple surface are identified as the key fluid dynamical controls on equilibrium dimensions. Based on these controls, we propose mechanisms in the selection of vortex ripple dimensions. We observe that the flow adjusts in such a way that the interaction between overlying flow and vortex ripples tends to generate the strongest coherent vortices while the ripple surface (or overlying flow) experiences the smallest shear stress averaged over ripple wavelength during the selection process. Through a triple decomposition of the flow, the component of the ripple-induced fluctuation is found to dictate these fluid dynamical controls, which implies that this component plays an important role in the evolution of vortex ripples.

This paper presents a novel theoretical framework in the Hamiltonian theory of nonlinear surface gravity waves. The envelope of surface elevation and the velocity potential on the free water surface are introduced in the framework, which are shown to be a new pair of canonical variables. Using the two envelopes as the main unknowns, coupled envelope evolution equations (CEEEs) are derived based on a perturbation expansion. Similar to the high-order spectral method, the CEEEs can be derived up to arbitrary order in wave steepness. In contrast, they have a temporal scale as slow as the rate of change of a wave spectrum and allow for the wave fields prescribed on a computational (spatial) domain with a much larger size and with spacing longer than the characteristic wavelength at no expense of accuracy and numerical efficiency. The energy balance equation is derived based on the CEEEs. The nonlinear terms in the CEEEs are in a form of the separation of wave harmonics, due to which an individual term is shown to have clear physical meanings in terms of whether or not it is able to force free waves that obey the dispersion relation. Both the nonlinear terms that can only lead to the forcing of bound waves and those that are capable of forcing free waves are demonstrated, in the case of the latter through the analysis of the quartet and quintet resonant interactions of linear waves. The relations between the CEEEs and two other existing theoretical frameworks are established, including the theory for a train of Stokes waves up to second order in wave steepness (Fenton, ASCE J. Waterway Port Coastal Ocean Engng, vol. 111, issue 2, 1985, pp. 216–234) and a semi-analytical framework for three-dimensional weakly nonlinear surface waves with arbitrary bandwidth and large directional spreading by Li & Li (Phys. Fluids, vol. 33, issue 7, 2021, 076609).

We consider the multidirectional swelling and drying of hydrogels formed from super-absorbent polymers and water, focusing on the elastic deformation caused by differential swelling. By modelling hydrogels as instantaneously incompressible, linear-elastic materials and considering situations in which there can be large isotropic strains (arising from swelling) while deviatoric strains remain small, it is possible to describe accurately a wider range of gel states than traditional linear elastic theories allow. An equation is derived relating the displacement field to the polymer fraction in such hydrogels, permitting the shape of the swelling gel to be determined as it evolves in time, using the formulation of Part 1 to find the local polymer fraction. We discuss the boundary conditions to be applied at the surfaces of a gel, both on the bulk elastic stress and on the pervadic (pore) pressure in the interstices. Similarities between the equation for the displacements and the equations of classical plate theory are investigated by considering a model problem of a slender cylinder with its base immersed in water drying by evaporation into the surrounding air. In this problem, there is differential drying along the axis of the cylinder, as the base remains swollen while the top dries. The results of our displacement formulation agree qualitatively with experiments that we have conducted, and provide a physical interpretation of the forced biharmonic equation describing the displacement field.

Reinforcement learning is applied to the development of control strategies in order to reduce skin friction drag in a fully developed turbulent channel flow at a low Reynolds number. Motivated by the so-called opposition control (Choi et al., J. Fluid Mech., vol. 253, 1993, pp. 509–543), in which a control input is applied so as to cancel the wall-normal velocity fluctuation on a detection plane at a certain distance from the wall, we consider wall blowing and suction as a control input, and its spatial distribution is determined by the instantaneous streamwise and wall-normal velocity fluctuations at distance 15 wall units above the wall. A deep neural network is used to express the nonlinear relationship between the sensing information and the control input, and it is trained so as to maximize the expected long-term reward, i.e. drag reduction. When only the wall-normal velocity fluctuation is measured and a linear network is used, the present framework reproduces successfully the optimal linear weight for the opposition control reported in a previous study (Chung & Talha, Phys. Fluids, vol. 23, 2011, 025102). In contrast, when a nonlinear network is used, more complex control strategies based on the instantaneous streamwise and wall-normal velocity fluctuations are obtained. Specifically, the obtained control strategies switch abruptly between strong wall blowing and suction for downwelling of a high-speed fluid towards the wall and upwelling of a low-speed fluid away from the wall, respectively. Extracting key features from the obtained policies allows us to develop novel control strategies leading to drag reduction rates as high as 37 %, which is higher than the 23 % achieved by the conventional opposition control at the same Reynolds number. Finding such an effective and nonlinear control policy is quite difficult by relying solely on human insights. The present results indicate that reinforcement learning can be a novel framework for the development of effective control strategies through systematic learning based on a large number of trials.