A numerical investigation was conducted on $Re_{\varGamma _{0}}=3000$ vortex rings colliding with wall-mounted hemispheres to study how their relative sizes affect the resulting vortex dynamics and structures. The hemisphere to vortex ring diameter ratio ranges from $D/d=0.5$ to $D/d=2$. Secondary/tertiary vortex rings are observed to result from hemispheric surface boundary layer separations rather than wall boundary layer separations as the diameter ratio increases. While those for $D/d\leq 1$ hemispheres can be attributed to sequential hemispheric and wall boundary layer separations, the primary vortex ring produces a series of secondary/tertiary vortex rings only along the $D/d=2$ hemispheric surface. This indicates that the presence of the wall makes little difference when the hemisphere is sufficiently large. On top of comparing vortex ring circulations and translational velocities between hemisphere and flat-wall based collisions, present collision outcomes have also been compared with those predicted by specific discharge velocity models. Additionally, comparisons of vortex core trajectories and vortex ring formation locations with earlier cylindrical convex surface based collisions provide more clarity on differences between two- and three-dimensional convex surfaces. Finally, vortex flow models are presented to account for the significantly different flow behaviour as the hemisphere size varies. Specifically, the vortex flow model for the $D/d=2$ hemisphere hypothesizes that the recurring tertiary vortex ring formations cease only when the primary vortex ring slows down sufficiently for the last tertiary vortex ring to entangle with it and render it incoherent. Until that happens, the primary vortex ring will continue to induce more tertiary vortex rings to form, with potential implications for heat/mass transfer optimizations.

Large-amplitude gust encounters exhibit a range of separated flow phenomena, making them difficult to characterize using the traditional tools of aerodynamics. In this work, we propose a dynamical systems approach to gust encounters, viewing the flow as a cycle (or a closed trajectory) in state space. We posit that the topology of this cycle, or its shape and structure, provides a compact description of the flow, and can be used to identify coordinates in which the dynamics evolve in a simple, intuitive way. To demonstrate this idea, we consider flowfield measurements of a transverse gust encounter. For each case in the dataset, we characterize the full-state dynamics of the flow using persistent homology, a tool that identifies holes in point cloud data, and transform the dynamics to a reduced-order space using a nonlinear autoencoder. Critically, we constrain the autoencoder such that it preserves topologically relevant features of the original dynamics, or those features identified by persistent homology. Using this approach, we are able to transform six separate gust encounters to a three-dimensional latent space, in which each gust encounter reduces to a simple circle, and from which the original flow can be reconstructed. This result shows that topology can guide the creation of low-dimensional state representations for strong transverse gust encounters, a crucial step towards the modelling and control of aerofoil–gust interactions.

The first stages of the path instability phenomenon affecting the buoyancy-driven motion of gas bubbles rising in weakly or moderately viscous liquids are examined using a recently developed numerical approach designed to assess the global linear stability of incompressible flows involving freely evolving interfaces. Predictions for the critical bubble size and frequency of the most unstable mode are found to agree well with reference data obtained in ultrapure water and in several silicone oils. By varying the bubble size, stability diagrams are built for several specific fluids, revealing three distinct regimes with different bifurcation sequences. The spatial structure of the unstable modes is analysed, together with the variations of the bubble shape, position and orientation. For this purpose, displacements of the bubble surface are split into rigid-body components and volume-preserving deformations, allowing us to determine how the relative magnitude of the latter varies with the fluid properties and bubble size. Predictions obtained with freely deformable bubbles are compared with those found by maintaining the bubble shape determined in the base state frozen during the stability analysis. This comparison reveals that deformations leave the phenomenology of the first bifurcations unchanged in low-viscosity fluids, especially water. Hence, in such fluids, bubbles behave essentially as freely moving rigid bodies submitted to constant-force and zero-torque constraints, at the surface of which the fluid obeys a shear-free condition. In contrast, deformations change the nature of the primary bifurcation in oils slightly more viscous than water, whereas, somewhat surprisingly, they leave the near-threshold phenomenology unchanged in more viscous oils.

We report the first experiments on hydrodynamic instabilities of a single-mode light/heavy interface driven by co-directional rarefaction and shock waves. The experiments are conducted in a specially designed rarefaction-shock tube that enables the decoupling of interfacial instabilities caused by these co-directional waves. After the impacts of rarefaction and shock waves, the interface evolution transitions into Richtmyer–Meshkov unstable states from Rayleigh–Taylor (RT) stable states, which is different from the finding in the previous case with counter-directional rarefaction and shock waves. A scaling method is proposed, which effectively collapses the RT stable perturbation growths. An analytical theory for predicting the time-dependent acceleration and density induced by rarefaction waves is established. Based on the analytical theory, the model proposed by Mikaelian (Phys. Fluids, vol. 21, 2009, p. 024103) is revised to provide a good description of the dimensionless RT stable behaviour. Before the shock arrival, the unequal interface velocities, caused by rarefaction-induced uneven vorticity, result in a V-shape-like interface. The linear growth rate of the amplitude is insensitive to the pre-shock interface shape, and can be well predicted by the linear superposition of growth rates induced by rarefaction and shock waves. The nonlinear growth rate is higher than that of a pure single-mode case, which can be predicted by the nonlinear models (Sadot et al., Phys. Rev. Lett., vol. 80, 1998, pp. 1654–1657; Dimonte & Ramaprabhu, Phys. Fluids, vol. 22, 2010, p. 014104).

The secondary motion caused by turbulence anisotropy is one of the crucial factors for determining the size of corner-flow separation in a side-wall interference flow field. Therefore, through a wall-resolved large-eddy simulation (LES) of a side-wall interference flow field, this study investigates the effects of the secondary motion on the corner-flow separation and explores the turbulence modelling that can reproduce the secondary flow motion. The momentum transport analysis using the LES results shows that the secondary vortex has twofold effects on delaying the corner-flow separation: the convective transport of the streamwise momentum towards the corner, and the enhanced production of turbulence by increasing the shear. Also, the vorticity transport analysis reconfirms that the secondary motion is caused primarily by turbulence anisotropy in the outer layer of the turbulent boundary layer. Furthermore, a quadratic constitutive relation (QCR) is proposed based on the analysis of the relationship between the Reynolds stress and velocity gradient. The proposed QCR consists of two quadratic terms and three constant parameters. The a priori analysis using the LES data shows that the proposed QCR represents the anisotropy of the Reynolds stress overall better than the existing QCR. Reynolds-averaged Navier–Stokes simulation using the proposed QCR with the Spalart–Allmaras turbulence model shows improvements in the prediction of the corner-flow separation compared to the results obtained by the existing QCR with the same turbulence model.

A comprehensive direct numerical simulation of electroconvection (EC) turbulence caused by strong unipolar charge injection in a two-dimensional cavity is performed. The EC turbulence has strong fluctuations and intermittency in the closed cavity. Several dominant large-scale structures are found, including two vertical main rolls and a single primary roll. The flow mode significantly influences the charge transport efficiency. A nearly $Ne \sim T^{1/2}$ scaling stage is observed, and the optimal $Ne$ increment is related to the mode with two vertical rolls, while the single roll mode decreases the charge transport efficiency. As the flow strength increases, EC turbulence transitions from an electric force-dominated mode to an inertia-dominated mode. The former utilizes the Coulomb force more effectively and allocates more energy to convection. The vertical mean profiles of charge, electric field and energy budget provide intuitive information on the spatial energy distribution. With the aid of the energy-box technique, a detailed energy transport evolution is illustrated with changing electric Rayleigh numbers. This exploration of EC turbulence can help explain more complicated electrokinetic turbulence mechanisms and the successful utilization of Fourier mode decomposition and energy-box techniques is expected to benefit future EC studies.

Plane unsteady potential flows of an ideal incompressible fluid with a free boundary are considered in the absence of external forces and surface tension. Examples of exact solutions in situations where the entire boundary of the domain occupied by the fluid is completely free are constructed. There may be polar singularities of the complex velocity function inside the fluid, which corresponds to the presence of a source or a sink there.

Direct numerical simulations have been conducted to investigate turbulent Rayleigh– Bénard convection (RBC) of liquid metal in a cuboid vessel with aspect ratio $\varGamma =5$ under an imposed horizontal magnetic field. Flows with Prandtl number $Pr=0.033$, Rayleigh numbers ranging up to $Ra\leq 10^{7}$, and Chandrasekhar numbers up to $Q\leq 9 \times 10^6$ are considered. For weak magnetic fields, our findings reveal that a previously undiscovered decreasing region precedes the enhancement of heat transfer and kinetic energy. For moderate magnetic fields, we have reproduced the reversals of the large-scale flow, which are considered a reorganization process of the roll-like structures that were reported experimentally by Yanagisawa et al. (Phys. Rev. E, vol. 83, 2011, 036307). Nevertheless, the proposed approach of skewed-varicose instability has been substantiated as insufficient to elucidate fundamentally the phenomenon of flow reversal, an occurrence bearing a striking resemblance to the large-scale intermittency observed in magnetic channel flows. As we increase the magnetic field strength further, we observe that the energy dissipation of the system comes primarily from the viscous dissipation within the boundary layer. Consequently, the dependence of Reynolds number $Re$ on $Q$ approaches a scaling as $Re\,Pr/Ra^{2/3} \sim Q^{-1/3}$. At the same time, we find the law for the cutoff frequency that separates large quasi-two-dimensional scales from small three-dimensional ones in RBC flow, which scales with the interaction parameter as ${\sim }N^{1/3}$.

The characterisation and the modelling of air concentration distributions in self-aerated free-surface flows has been subject to sustained research interest since the 1970s. Recently, a novel two-state formulation of the structure of a self-aerated flow was proposed by Kramer & Valero (J. Fluid Mech., vol. 966, 2023, A37), which physically explains the air concentration through the weak interaction of two canonical flow momentum layers, comprising a turbulent boundary layer and a turbulent wavy layer (TWL). The TWL was modelled using a Gaussian error function, assuming that the most dominant contribution are wave troughs. Here, it is shown that air bubbles form an integral part of the TWL, and its formulation is expanded by adopting a superposition principle of entrapped air (waves) and entrained air (bubbles). Combining the superposition principle with the two-state formulation, an expression for the depth-averaged (mean) air concentration is derived, which allows us to quantify the contribution of different physical mechanisms to the mean air concentration. Overall, the presented concepts help to uncover new flow physics, thereby contributing fundamentally to our understanding of self-aerated flows.

In numerical studies of thermal convection that includes a layer of lighter surface fluid, the light fluid naturally forms clusters that bulge downward at downwelling sites. A curious result is that in some cases, the clusters have maximum bulging downward near the sides of the cluster instead of a single bulge downward centred above the downwelling. The fluid mechanics leading to this ‘double bulge’ formation is analysed. To accomplish this, a simplified model replaces the thermally driven convection cells with driving cells with a fixed speed. Adding a layer of dense fluid on the bottom to the previous configuration leads to bulges along the top and bottom. More importantly, this allows a new scaling that reduces the number of governing parameters from four to three and even to two in this study. The mechanism for the double bulges comes from buoyancy of the clusters. This produces localized vorticity at the sides of the cluster that has the opposite sign of the driving cells. When this vorticity is approximately the same order of magnitude as the driving cell vorticity, a divergence in the middle of each cluster leads to the double bulges. The effect can be so great that the underlying flow cells are tilted so that vertical motion is reversed under the middle of each bulge.

Floating objects will drift due to the action of surface gravity waves. This drift will depart from that of a perfect Lagrangian tracer due to both viscous effects (non-potential flow) and wave–body interaction (potential flow). We examine the drift of freely floating objects in regular (non-breaking) deep-water wave fields for object sizes that are large enough to cause significant diffraction. Systematic numerical simulations are performed using a hybrid numerical solver, qaleFOAM, which deals with both viscosity and wave–body interaction. For very small objects, the model predicts a wave-induced drift equal to the Stokes drift. For larger objects, the drift is generally greater and increases with object size (we examine object sizes up to $10\,\%$ of the wavelength). The effects of different shapes, sizes and submergence depths and steepnesses are examined. Furthermore, we derive a ‘diffraction-modified Stokes drift’ akin to Stokes (Trans. Camb. Phil. Soc., vol. 8, 1847, pp. 411–455), but based on the combination of incident, diffracted and radiated wave fields, which are based on potential-flow theory and obtained using the boundary element method. This diffraction-modified Stokes drift explains both qualitatively and quantitatively the increase in drift. Generally, round objects do not diffract the wave field significantly and do not experience a significant drift enhancement as a result. For box-shape objects, drift enhancement is greater for larger objects with greater submergence depths (we report an increase of $92\,\%$ for simulations without viscosity and $113\,\%$ with viscosity for a round-cornered box whose size is $10\,\%$ of the wavelength). We identify the specific standing wave pattern that arises near the object because of diffraction as the main cause of the enhanced drift. Viscosity plays a small positive role in the enhanced drift behaviour of large objects, increasing the drift further by approximately $20\,\%$.

A collection of microswimmers immersed in an incompressible fluid is characterised by strong interactions due to the long-range nature of the hydrodynamic fields generated by individual organisms. As a result, suspensions of rear-actuated ‘pusher’ swimmers such as bacteria exhibit a collective motion state often referred to as ‘bacterial turbulence’, characterised by large-scale chaotic flows. The onset of collective motion in pusher suspensions is classically understood within the framework of mean-field kinetic theories for dipolar swimmers. In bulk two and three dimensions, the theory predicts that the instability leading to bacterial turbulence is due to mutual swimmer reorientation and sets in at the largest length scale available to the suspension. Here, we construct a similar kinetic theory for the case of a dipolar microswimmer suspension restricted to a two-dimensional plane embedded in a three-dimensional incompressible fluid. This setting qualitatively mimics the effect of swimming close to a two-dimensional interface. We show that the in-plane flow fields are effectively compressible in spite of the incompressibility of the three-dimensional bulk fluid, and that microswimmers on average act as sources (pushers) or sinks (pullers). We analyse the stability of the homogeneous and isotropic state, and find two types of instability that are qualitatively different from the bulk, three-dimensional case: first, we show that the analogue of the orientational pusher instability leading to bacterial turbulence in bulk systems instead occurs at the smallest length scale available to the system. Second, an instability associated with density variations arises in puller suspensions as a generic consequence of the effective in-plane compressibility. Given these qualitative differences with respect to the standard bulk setting, we conclude that confinement can have a crucial role in determining the collective behaviour of microswimmer suspensions.

In sandstorms and thunderclouds, turbulence-induced collisions between solid particles and ice crystals lead to inevitable triboelectrification. The charge segregation is usually size dependent, with small particles charged negatively and large particles charged positively. In this work, we perform numerical simulations to study the influence of charge segregation on the dynamics of bidispersed inertial particles in turbulence. Direct numerical simulations of homogeneous isotropic turbulence are performed with the Taylor Reynolds number ${Re}_{\lambda }=147.5$, while particles are subjected to both electrostatic interactions and fluid drag, with Stokes numbers of 1 and 10 for small and large particles, respectively. Coulomb repulsion/attraction is shown to effectively inhibit/enhance particle clustering within a short range. Besides, the mean relative velocity between same-size particles is found to rise as the particle charge increases because of the exclusion of low-velocity pairs, while the relative velocity between different-size particles is almost unaffected, emphasizing the dominant roles of differential inertia. The mean Coulomb-turbulence parameter, ${Ct}_0$, is then defined to characterize the competition between the Coulomb potential energy and the mean relative kinetic energy. In addition, a model is proposed to quantify the rate at which charged particles approach each other and to capture the transition of the particle relative motion from the turbulence-dominated regime to the electrostatic-dominated regime. Finally, the probability distribution function of the approach rate between particle pairs is examined, and its dependence on the Coulomb force is further discussed using the extended Coulomb-turbulence parameter.

The dipole–multipole transition in rapidly rotating dynamos is investigated through the analysis of forced magnetohydrodynamic waves in an unstably stratified fluid. The focus of this study is on the inertia-free limit applicable to planetary cores, where the Rossby number is small not only on the core depth but also on the length scale of columnar convection. By progressively increasing the buoyant forcing in a linear magnetoconvection model, the slow magnetic–Archimedean–Coriolis (MAC) waves are significantly attenuated so that their kinetic helicity decreases to zero; the fast MAC wave helicity, on the other hand, is practically unaffected. In turn, polarity reversals in low-inertia spherical dynamos are shown to occur when the slow MAC waves disappear under strong forcing. Two dynamically similar regimes are identified – the suppression of slow waves in a strongly forced dynamo and the excitation of slow waves in a moderately forced dynamo starting from a small seed field. While the former regime results in polarity reversals, the latter regime produces the axial dipole from a chaotic multipolar state. For either polarity transition, a local Rayleigh number based on the mean wavenumber of the energy-containing scales bears the same linear relationship with the square of the peak magnetic field measured at the transition. The self-similarity of the dipole–multipole transition can place a constraint on the Rayleigh number for polarity reversals in the Earth.

The Reynolds number dependent flow resistance of heterogeneous rough surfaces is largely unknown at present. The present work provides novel reference data for spanwise-alternating sandpaper strips as one idealised case of a heterogeneous rough surface. Experimental data are presented and analysed in direct comparison with drag measurements of homogeneous sandpaper surfaces and numerical simulations. Based on the homogeneous roughness data, the related challenges and sensitivities for the evaluation of roughness functions from experiments and simulations are discussed. A hydraulic channel height is suggested as an alternative measure for the drag impact of rough surfaces in internal flows. For the investigated heterogeneous roughness, it is found that turbulent flow does not exhibit a fully rough flow behaviour, indicating that the assignment of an equivalent sand grain height as commonly applied for homogeneous roughness is not possible. A prediction of the drag behaviour of rough strips based on an average between rough and smooth drag curves appears promising, but requires further refinement to capture the impact of turbulent secondary flows and spatial transients linking smooth and rough surface parts. While turbulent secondary flow induced by the roughness strips yield significant spanwise variation of the mean velocity profile for the investigated rough strips, we show that the spanwise averaged velocity profiles collapse reasonably well with a smooth or homogeneous rough wall flow. This allows to extract a global roughness function from the spanwise averaged flow field in good agreement with the one deduced from global pressure drop measurements.

The Zakharov equation describes the evolution of weakly nonlinear surface gravity waves for arbitrary spectral shape. For deep-water waves, results from the Zakharov equation are well established. However, for two-dimensional propagation, in intermediate and shallow water, there are problems related to the treatment of apparent singularities in the contribution of the wave-induced set-up to the evolution of the surface gravity waves. More specifically, the kernel in the integral term is characterized by a regular and an apparent singular contribution. Here, we show that the Davey–Stewartson equation can be directly derived from the Zakharov equation, also in the shallow water limit. This result provides guidance on how to treat the singular contribution to the evolution of the action variable. A relevant result that is obtained is that the growth rate obtained from the stability analysis of a plane wave in shallow water does not depend on the singular part of the kernel of the Zakharov equation.

For the dispersion of soluble matter in solvent flowing through a tube as investigated originally by G.I. Taylor, a streamwise dispersion theory is developed from a Lagrangian perspective for the whole process with multi-scale effects. By means of a convected coordinate system to decouple convection from diffusion, a diffusion-type governing equation is presented to reflect superposable diffusion processes with a multi-scale time-dependent anisotropic diffusivity tensor. A short-time benchmark, complementing the existing Taylor–Aris solution, is obtained to reveal novel statistical and physical features of mean concentration for an initial phase with isotropic molecular diffusion. For long times, effective streamwise diffusion prevails asymptotically corresponding to the overall enhanced diffusion in Taylor's classical theory. By inverse integral expansions of local concentration moments, a general streamwise dispersion model is devised to match the short- and long-time asymptotic solutions. Analytical solutions are provided for most typical cases of point and area sources in a Poiseuille tube flow, predicting persistent long tails and skewed platforms. The theoretical findings are substantiated through Monte Carlo simulations, from the initial release to the Taylor dispersion regime. Asymmetries of concentration distribution in a circular tube are certified as originated from (a) initial non-uniformity, (b) unidirectional flow convection, and (c) non-penetration boundary effect. Peculiar peaks in the concentration cloud, enhanced streamwise dispersivity and asymmetric collective phenomena of concentration distributions are illustrated heuristically and characterised to depict the non-equilibrium dispersion. The streamwise perspective could advance our understanding of macro-transport processes of both passive solutes and active suspensions.

The axisymmetric problem of a conical shell impact onto an inviscid and incompressible liquid of infinite depth is studied. The shell is thin, and its deadrise angle is small. The problem is inertia dominated. Gravity, surface tension and viscous effects are not taken into account. The hydrodynamic loads acting on the shell and the shell displacements are determined at the same time. The model by Scolan (J. Sound Vib., vol. 277, issue 1–2, 2004, pp. 163–203) is used to find the flow and hydrodynamic pressure caused by the shell impact. This model is based on the Wagner theory of water impact, which was generalised to axisymmetric problems of hydroelastic slamming. Dry and wet modes of the conical shell, as well as the corresponding frequencies, are calculated. It is shown that a conical shell can be approximated by a circular plate only for a very small deadrise angle. Deflections and strains in the conical shell during the impact stage, when the wetted part of the shell increases at high rate, as well as the hydrodynamic loads, are determined and analysed.

We report numerical simulations confirming the predictions in Gordillo & Riboux (J. Fluid Mech., vol. 941, 2022, A10), where we elucidated the lubrication mechanism by which a drop of a low-viscosity liquid impacting over a smooth solid substrate skates over a thin gas film that prevents contact with the wall. Moreover, with the purpose of explaining the so-called lift-off mechanism reported in Kolinski et al. (Phys. Rev. Lett., vol. 112, issue 13, 2014, 134501), we extend our previous findings and derive expressions for the time-varying thickness of the gas layer at the region where the distance to the wall is minimum, finding good agreement with the numerical results. In addition, we report that our predictions for the minimum thickness of the gas film separating a falling drop from a wall at room temperature follow closely the experimental values when gas kinetic effects are retained in the analysis, and also report that the analogous equation for the minimum thickness of the vapour layer formed after a drop impacts a superheated wall predicts well the experimental measurements.

The rupture of the thin film at the top of a bubble at a liquid–gas interface leads to an axisymmetric collapse of the bubble cavity. We present scaling laws for such a cavity collapse, established from experiments conducted with bubbles spanning a wide range of Bond (${10^{-3}< Bo\leq 1}$) and Ohnesorge numbers (${10^{-3}< Oh<10^{-1}}$), defined with the bubble radius $R$. The cavity collapse is a capillary-driven process, with a dependency on viscosity and gravity, affecting respectively, precursory capillary waves on the cavity boundary and the static bubble shape. The collapse is characterised by the normal interface velocity ($U_n$) and by the tangential wave propagation velocity of the kink ($U_t$), defined by the intersection of the concave cavity boundary formed after the rupture of the thin film with the convex boundary of the bubble cavity. During the collapse, $U_t$ remains constant and is shown to be $U_t=4.5U_c{\mathcal {W}}_R$, where $U_c$ is the capillary velocity and ${\mathcal {W}}_R(Oh,Bo)={(1-\sqrt {Oh {\mathscr {L}}} )^{-1/2}}$ is the wave resistance factor due to the precursory capillary waves, with $\mathscr {L}(Bo)$ being the path correction of the kink motion. The movement of the kink in the normal direction is part of the inward shrinkage of the whole cavity due to the sudden reduction of gas pressure inside the bubble cavity after the thin film rupture. This normal velocity is shown to scale as $U_c$ in the equatorial plane, while at the bottom of the cavity $\bar {U}_{nb}=U_c(Z_c/R)({\mathcal {W}_R}/ {\mathscr {L}})$, where $Z_c(Bo)$ is the static cavity depth. The filling rate of the cavity, which remains a constant throughout the collapse, is shown to be entirely determined by the shrinking velocity and scales as ${Q_T\simeq 2{\rm \pi} R Z_c U_c}$. From $Q_T$ we recover the jet velocity scaling, thereby relating the cavity collapse with the jet velocity scaling.