Bubble–particle collisions in turbulence are central to a variety of processes such as froth flotation. Despite their importance, details of the collision process have not received much attention yet. This is compounded by the sometimes counter-intuitive behaviour of bubbles and particles in turbulence, as exemplified by the fact that they segregate in space. Although bubble–particle relative behaviour is fundamentally different from that of identical particles, the existing theoretical models are nearly all extensions of theories for particle–particle collisions in turbulence. The adequacy of these theories has yet to be assessed as appropriate data remain scarce to date. In this investigation, we study the geometric collision rate by means of direct numerical simulations of bubble–particle collisions in homogeneous isotropic turbulence using the point-particle approach over a range of the relevant parameters, including the Stokes and Reynolds numbers. We analyse the spatial distribution of bubble and particles, and quantify to what extent their segregation reduces the collision rate. This effect is countered by increased approach velocities for bubble–particle compared to monodisperse pairs, which we relate to the difference in how bubbles and particles respond to fluid accelerations. We found that in the investigated parameter range, these collision statistics are not altered significantly by the inclusion of a lift force or different drag parametrisations, or when assuming infinite particle density. Furthermore, we critically examine existing models and discuss inconsistencies therein that contribute to the discrepancy.

The presence of very-large-scale motions in wall-bounded turbulent flows is commonly associated with their footprint in the form of the superposition of the large scales at the wall and the additional amplitude modulation of small-scale near-wall turbulence. These two phenomena are currently understood to be interlinked, with the superposed large-scale velocity gradient causing the modulation of small-scale activity in the proximity of the wall. To challenge this idea, we devise a numerical strategy that selectively suppresses either superposition or amplitude modulation, in an effort to isolate and study the remaining phenomenon. Results from our direct numerical simulations indicate that a positive correlation between the amplitude of the small scales in the near-wall region and the large-scale signal in the outer flow persists even when near-wall large-scale motions are suppressed – i.e. in absence of superposition. Clearly, this kind of correlation cannot be caused by the near-wall large-scale velocity or its gradients, as both are absent. Conversely, when modulation is blocked, the near-wall footprints of the large scales seem to disappear. This study has been carried out on channel flows at friction Reynolds number $Re_\tau =1000$ in both standard simulation domains and minimal streamwise units (MSUs), where the streamwise fluctuation energy is enhanced. The consistency of the results obtained by the two approaches suggests that MSUs can capture correctly this kind of scale interaction at a much reduced cost.

The dynamics of hydrogen bubbles produced by water electrolysis in an acidic electrolyte is studied using electrochemical and optical methods. A defined cyclic modulation of the electric potential is applied at a microelectrode to produce pairs of interacting H$_2$ bubbles in a controlled manner. Three scenarios of interactions are identified and studied systematically. The most prominent one consists of a sudden reversal in the motion of the first detached bubble, its return to the electrode, and finally its coalescence with the second bubble. Attested by Toepler's schlieren technique, an explanation of contactless motion reversal is provided by the competition between buoyancy and thermocapillary effects.

We study the linear stability of bubbles in a capillary tube under external flow. Yu et al. (J. Fluid Mech., vol. 911, 2021, pp. 1–19) showed that a rich variety of bubble dynamics occurs when a downward external flow is applied, opposing the buoyancy-driven ascent of the bubble. They found experimentally and numerically the existence of two branches of solutions that overlap over a finite range of the capillary number of the downward external flow in cases where the Reynolds number is small and the Bond number is larger than the critical value for which the bubble can rise spontaneously (Bretherton, J. Fluid Mech., vol. 10, issue 2, 1961, pp. 166–188). Furthermore, inertialess, symmetry-breaking steady-state shapes were found as the bubble transits near the tipping points of the solution branches. In this work, using steady axisymmetric simulations, we show that the reported multiplicity of solutions can be described using bifurcation diagrams with three branches of steady axisymmetric solutions and two limit points. The linear global stability analysis of the different branches of the stationary axisymmetric solutions demonstrates that the symmetry breaking is due to the development of three-dimensional instabilities with azimuthal wavenumber $|m|=1$.

The transition to dripping in the gravity-driven flow of a liquid film under an inclined plate is investigated at zero Reynolds number. Computations are carried out on a periodic domain assuming either a fixed fluid volume or a fixed flow rate for a hierarchy of models: two lubrication models with either linearised curvature or full curvature (the LCM and FCM, respectively), and the full equations of Stokes flow. Of particular interest is the breakdown of travelling-wave solutions as the plate inclination angle is increased. For any fixed volume, the LCM reaches the horizontal state where it attains a cosine-shaped profile. For sufficiently small volume, the FCM and Stokes solutions attain a weak Young–Laplace equilibrium profile, the approach to which is described by an asymptotic analysis generalising that of Kalliadasis & Chang (J. Fluid Mech., vol. 261, 1994, pp. 135–168) for the LCM. For large volumes, the bifurcation curves for the FCM and Stokes model have a turning point so that the fully inverted state is never reached. For fixed flow rate, the LCM blows up at a critical angle that is well predicted by asymptotic analysis. The bifurcation curve for the FCM either has a turning point or else reaches a point at which the surface profile has an infinite slope singularity, indicating the onset of multi-valuedness. The latter is confirmed by the Stokes model, which can be continued to obtain overturning surface profiles. Overall, the thin-film models either provide an accurate prediction for dripping onset or else supply an upper bound on the critical inclination angle.

High-speed water entry is a transient hydrodynamic process that is accompanied by strongly compressible flow, free surface splash, cavity evolution and other nonlinear hydrodynamic phenomena. To address these problems, a novel fluid–structure interaction (FSI) scheme based on the immersed boundary method is proposed which is suitable for strongly compressible multiphase flows. In this scheme, considering the multiphase interfaces at the immersed boundary, an improved immersed boundary method for effectively suppressing the non-physical force oscillation is proposed. Additionally, a quaternion-based six degrees of freedom motion system is used to describe rigid body motion, and the multiphase flow Eulerian finite element method is applied as the fluid solver. Using analytical solutions, experimental data and literature data, the accuracy and robustness of the FSI scheme are validated. Finally, the high-speed water entry of the slender body with different noses is investigated, and the hydrodynamic loads including the axial and normal drag forces and the bending moment are extensively discussed. The hydrodynamic load and motion trajectory are determined by the nose configuration. The tail slamming phenomenon is the primary focus, and it is revealed that its formation is primarily related to the pitch moment formed at the stage of crossing the free surface. Tail slamming also causes violent impact loads, especially bending moments, which may cause slender projectiles to break off. Finally, to combine the features of the flat and hemispherical noses, the water entry of the projectile with a truncated hemispherical nose is simulated and discussed.

Rain cleans pollution out of the atmosphere, as droplets collide with airborne particles during free fall. Such particle–droplet collisions have been presumed to be capture events, but the details of such collisions lack thorough investigation. We show that rain droplets and pollution particles interact through multiple collision behaviours, including captures on the droplet surface, cavity-forming droplet entries and ricochets. Rain drop diameter and free fall velocity, in addition to pollution particle density, size, wettability and droplet impact location, determine which capture or escape behaviour occurs. Our findings reveal that rain does not capture all airborne pollutants equally even upon collision, and certain pollutants prove more difficult to remove from the air than others. Consequently, we must account for both rain and pollution characteristics to understand pollution capture mechanisms and pollution fluxes in the environment.

The incompressible motion of viscoplastic fluid between two semi-infinite rigid plates, hinged at their ends and rotating towards one another at constant angular velocity, generates self-similar flow fields because there is no externally imposed length scale in the absence of inertia. The magnitude of the strain rate scales with the angular velocity of the plates and the dimensionless deviatoric stresses are functions only of the polar angle and a dimensionless measure of the yield stress; they are independent of the radial distance from the corner. These flows feature unyielded regions adjacent to the boundaries for sufficiently large angles between the plates. Moreover, when the dimensionless yield stress is large, there are viscoplastic boundary layers that are attached to the boundary or the plug, the asymptotic structures of which are constructed.

When an aqueous drop contacts an immiscible oil film, it displays complex interfacial dynamics. When the spreading factor is positive, upon contact, the oil spreads onto the drop's liquid–air interface, first forming a liquid bridge whose curvature drives an apparent drop spreading motion and later engulfs the drop. We study this flow using both three-phase lattice Boltzmann simulations based on the conservative phase field model, and experiments. Inertially and viscously limited dynamics are explored using the Ohnesorge number $Oh$ and the ratio between the film height $H$ and the initial drop radius $R$. Both regimes show that the radial growth of the liquid bridge $r$ is fairly insensitive to the film height $H$, and scales with time $T$ as $r\sim T^{1/2}$ for $Oh\ll 1$, and as $r\sim T^{2/5}$ for $Oh\gg 1$. For $Oh\gg 1$, we show experimentally that this immiscible liquid bridge growth is analogous with the miscible drop–film coalescence case. Contrary to the growth of the liquid bridge, however, we find that the late-time engulfment dynamics and final interface profiles are significantly affected by the ratio $H/R$.

A droplet charged above the Rayleigh limit is unstable. In the resulting dynamical process, referred to as a Coulomb explosion, smaller droplets with higher charge-to-mass ratios are ejected, reducing the charge of the parent droplet below the Rayleigh limit. Furthermore, if the droplet is sufficiently small, the electric field on its surface can promote ion field emission. Ion emission can lower the charge of a spherical droplet below its Rayleigh limit, keeping it stable, or reduce the charge of a deforming droplet, changing its dynamics and potentially preventing the Coulomb explosion. This article develops a continuum phase field electrohydrodynamic model to study the interplay between Coulomb explosions and ion emission, using charged nanodroplets of the ionic liquid 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide (EMI-Im) as a case study. In small droplets (diameter $D \lesssim 20$ nm for EMI-Im), the electric field is strong enough to emit ions in the early phase of the droplet's evolution, suppressing the Coulomb explosion. For $20 \lesssim D \lesssim 45$ nm, the electric field on the EMI-Im droplet may not promote significant ion emission; however, as the unstable droplet becomes ellipsoidal, ions are emitted from its vertices, ultimately suppressing the Coulomb explosion while shedding $20\unicode{x2013}40\,\%$ of the initial charge. For larger EMI-Im droplets, $45 \lesssim D \lesssim 100$ nm, the evolution typical of a Coulomb explosion is observed, accompanied by ion emission which is however insufficient to prevent the Coulomb explosion. Ion emission and the smaller progeny droplets account for $24\,\%$ and $16\,\%$ of the initial charge, respectively.

This study presents the spreading of a single filament of a yield stress (viscoplastic) fluid extruded onto a pre-wetted solid surface. The filaments spread laterally under surface tension forces until they reach a final equilibrium shape when the yield stress dominates. We use a simple experimental set-up to print the filaments on a moving surface and measure their final width using optical coherence tomography. Additionally, we present a scaling law for the final width and determine the corresponding prefactor using asymptotic analysis. We then analyse the level of agreement between the theory and experiments, and discuss the possible origins of discrepancies. The process studied here has applications in extrusion-based thermoplastic and bio-three-dimensional printing.

The Lighthill–Panton and Lyman–Huggins interpretations of vorticity dynamics are extended to the dynamics of enstrophy. There exist two competing definitions of the vorticity current tensor, which describes the flow rate of vorticity in the fluid interior, and the corresponding boundary vorticity flux, which represents the local vorticity creation rate on a boundary. It is demonstrated that each definition of the vorticity current tensor leads to a consistent set of definitions for the enstrophy current, boundary enstrophy flux and the enstrophy dissipation term. This leads to two alternative interpretations of vorticity and enstrophy dynamics: the Lighthill–Panton and Lyman–Huggins interpretations. Although the kinematic evolution of the vorticity and enstrophy fields are the same under each set of definitions, the dynamical interpretation of the motion generally differs. For example, we consider the Stokes flow over a rotating sphere, and find that the flow approaches a steady state where, under the Lyman–Huggins interpretation, there is no enstrophy creation or dissipation. Under the Lighthill–Panton interpretation, however, the steady-state flow features a balance between the continuous generation and subsequent dissipation of enstrophy. Moreover, the Lyman–Huggins interpretation has previously been shown to offer several benefits in understanding the dynamics of vorticity, and therefore it is beneficial to extend this interpretation to the dynamics of enstrophy. For example, the Lyman–Huggins interpretation allows the creation of vorticity, and therefore enstrophy, to be interpreted as an inviscid process, due to the relative acceleration between the fluid and the boundary.

The spontaneous emergence of structure is a ubiquitous process observed in fluid and plasma turbulence. These structures typically manifest as flows which remain coherent over a range of spatial and temporal scales, resisting statistically homogeneous description. This work conducts a computational and theoretical study of coherence in turbulent flows in the stochastically forced barotropic $\beta$-plane quasi-geostrophic equations. These equations serve as a prototypical two-dimensional model for turbulent flows in Jovian atmospheres, and can also be extended to study flows in magnetically confined fusion plasmas. First, analysis of direct numerical simulations demonstrates that a significant fraction of the flow energy is organized into coherent large-scale Rossby wave eigenmodes, comparable with the total energy in the zonal flows. A characterization is given for Rossby wave eigenmodes as nearly integrable perturbations to zonal flow Lagrangian trajectories, linking finite-dimensional deterministic Hamiltonian chaos in the plane to a laminar-to-turbulent flow transition. Poincaré section analysis reveals that Lagrangian flows induced by the zonal flows plus large-scale waves exhibit localized chaotic regions bounded by invariant tori, manifesting as Rossby wave breaking in the absence of critical layers. It is argued that the surviving invariant tori organize the large-scale flows into a single temporally and zonally varying laminar flow, suggesting a form of self-organization and wave stability that can account for the resilience of the observed large-amplitude Rossby waves.

This work deals with the investigation and modelling of wall pressure fluctuations induced by a supersonic jet over a tangential flat plate. The analysis is performed at several nozzle pressure ratios around the nozzle design Mach number, including slightly over-expanded and under-expanded conditions, and for different radial positions of the rigid plate. Pitot measurements and flow visualizations through the background oriented schlieren technique provided a general overview of the aerodynamic interactions between the jet flow and the plate at the different regimes and configurations. Wall pressure fluctuations were measured using a couple of piezoelectric pressure transducers flush mounted over the plate surface. The spectral analysis has been carried out to clarify the effect of the plate position on the single and multivariate wall pressure statistics, including the screech tone amplitude. The experimental dataset is used to assess and validate a surrogate model based on artificial neural networks. Sound pressure levels and coherence functions are modelled by means of a single fully connected network, built on the basis of a recently implemented fully deterministic topology optimization algorithm. The metamodel uncertainty is also quantified using the spatial correlation function. It is shown that the flow behaviour as well as the screech and broadband noise signatures are significantly influenced by the presence of the plate, and the effects on spectral quantities are correctly reproduced by the proposed data-driven model that provides predictions in agreement with the available data.

When a liquid drop falls on a solid substrate, the air layer between them delays the occurrence of liquid–solid contact. For impacts on smooth substrates, the air film can even prevent wetting, allowing the drop to bounce off with dynamics identical to that observed for impacts on superamphiphobic materials. In this paper, we investigate similar bouncing phenomena, occurring on viscous liquid films, that mimic atomically smooth substrates, with the goal to probe their effective repellency. We elucidate the mechanisms associated with the bouncing to non-bouncing (floating) transition using experiments, simulations, and a minimal model that predicts the main characteristics of drop impact, the contact time and the coefficient of restitution. In the case of highly viscous or very thin films, the impact dynamics is not affected by the presence of the viscous film. Within this substrate-independent limit, bouncing is suppressed once the drop viscosity exceeds a critical value, as on superamphiphobic substrates. For thicker or less viscous films, both the drop and film properties influence the rebound dynamics and conspire to inhibit bouncing above a critical film thickness. This substrate-dependent regime also admits a limit, for low-viscosity drops, in which the film properties alone determine the limits of repellency.

Non-wetting substrates allow impacting liquid drops to spread, recoil and take-off, provided they are not too heavy (Biance et al., J. Fluid Mech., vol. 554, 2006, pp. 47–66) or too viscous (Jha et al., Soft Matt., vol. 16, no. 31, 2020, pp. 7270–7273). In this article, using direct numerical simulations with the volume of fluid method, we investigate how viscous stresses and gravity oppose capillarity to inhibit drop rebound. Close to the bouncing to non-bouncing transition, we evidence that the initial spreading stage can be decoupled from the later retraction and take-off, allowing us to understand the rebound as a process converting the surface energy of the spread liquid into kinetic energy. Drawing an analogy with coalescence-induced jumping, we propose a criterion for the transition from the bouncing to the non-bouncing regime, namely by the condition ${Oh}_{c} + {Bo}_{c} \sim 1$, where ${Oh}_{c}$ and ${Bo}_{c}$ are the Ohnesorge number and Bond number at the transition, respectively. This criterion is in excellent agreement with the numerical results. We also elucidate the mechanisms of bouncing inhibition in the heavy and viscous drop limiting regimes by calculating the energy budgets and relating them to the drop's shape and internal flow.

The purpose of the paper is to identify Mach-number effects on pressure fluctuations $p'$ in compressible turbulent plane channel flow. We use data from a specifically constructed $(Re_{\tau ^\star },\bar {M}_{{CL}_x})$-matrix direct numerical simulation (DNS) database, with systematic variation of the centreline streamwise Mach number $0.32\leqslant \bar {M}_{{CL}_x}\leqslant 2.49$ and of the HCB (Huang et al., J. Fluid Mech., vol. 305, 1995, pp. 185–218) friction Reynolds number $66\leqslant Re_{\tau ^\star }\lessapprox 1000$. Strong $\bar {M}_{{CL}_x}$ effects (enhanced by the increasingly cold-wall condition) appear for $\bar {M}_{{CL}_x}\gtrapprox 2$, for all $Re_{\tau ^\star }$, very close to the wall ($y^\star \lessapprox 15$). Compared with incompressible flow at the same $Re_{\tau ^\star }$, the wall root-mean-square $[p'_{rms}]^+_w$ (in wall-units, i.e. scaled by the average wall shear stress $\bar {\tau }_w$) strongly increases with $\bar {M}_{{CL}_x}$. In contrast, the peak level across the channel, $[p'_{rms}]^+_{PEAK}$, slightly decreases with increasing $\bar {M}_{{CL}_x}$. In order to study the near-wall coherent structures we introduce a new wall-distance-independent non-local system of units, based for all $y$ on wall friction and the extreme values of density and dynamic viscosity, namely, for cold walls $\{\bar {\tau }_w,\min _y\bar {\rho },\max _y\bar {\mu }\}$. The average spanwise distance between streaks, scaled by this length-unit, is nearly independent of $\bar {M}_{{CL}_x}$ at constant $Re_{\tau ^\star }$. Using the in-plane (parallel to the wall) Laplacian $\nabla ^2_{xz}p'$ we find that the $(+/-)\text {-}p'$ wave-packet-like structures appearing inside the low-speed streaks ($y^\star \lessapprox 15$) with increasing $\bar {M}_{{CL}_x}\gtrapprox 2$ are part of a more complex wave system with spanwise extent over several streaks, whose spatial density decreases rapidly with decreasing $\bar {M}_{{CL}_x}$ or increasing $y^\star$. These $p'$ wave packets appear to be collocated with strong $(+/-)$-$v'$ events and could be responsible for compensating towards 0 the negative incompressible-flow correlation coefficient $c_{p'v'}$, with increasing $\bar {M}_{{CL}_x}$ very near the wall.

The scattering of three-dimensional inertia-gravity waves by a turbulent geostrophic flow leads to the redistribution of their action through what is approximately a diffusion process in wavevector space. The corresponding diffusivity tensor was obtained by Kafiabad et al. (J. Fluid Mech., vol. 869, 2019, R7) under the assumption of a time-independent geostrophic flow. We relax this assumption to examine how the weak diffusion of wave action across constant-frequency cones that results from the slow time dependence of the geostrophic flow affects the distribution of wave energy. We find that the stationary wave-energy spectrum that arises from a single-frequency wave forcing is localised within a thin boundary layer around the constant-frequency cone, with a thickness controlled by the acceleration spectrum of the geostrophic flow. We obtain an explicit analytic formula for the wave-energy spectrum which shows good agreement with the results of a high-resolution simulation of the Boussinesq equations.

The rebound of droplets impacting a deep fluid bath is studied both experimentally and theoretically. Millimetric drops are generated using a piezoelectric droplet-on-demand generator and normally impact a bath of the same fluid. Measurements of the droplet trajectory and other rebound metrics are compared directly with the predictions of a linear quasipotential model, as well as fully resolved direct numerical simulations of the unsteady Navier–Stokes equations. Both models resolve the time-dependent bath and droplet shapes in addition to the droplet trajectory. In the quasipotential model, the droplet and bath shape are decomposed using orthogonal function decompositions leading to two sets of coupled damped linear harmonic oscillator equations solved using an implicit numerical method. The underdamped dynamics of the drop are directly coupled to the response of the bath through a single-point kinematic match condition which we demonstrate to be an effective and efficient model in our parameter regime of interest. Starting from the inertio-capillary limit in which both gravitational and viscous effects are negligible, increases in gravity or viscosity lead to a decrease in the coefficient of restitution and an increase in the contact time. The inertio-capillary limit defines an upper bound on the possible coefficient of restitution for droplet–bath impact, depending only on the Weber number. The quasipotential model is able to rationalize historical experimental measurements for the coefficient of restitution, first presented by Jayaratne & Mason (Proc. R. Soc. Lond. A, vol. 280, issue 1383, 1964, pp. 545–565).