A droplet impacting a deep fluid bath is as common as rain over the ocean. If the impact is sufficiently gentle, the mediating air layer remains intact, and the droplet may rebound completely from the interface. In this work, we experimentally investigate the role of translational bath motion on the bouncing to coalescence transition. Over a range of parameters, we find that the relative bath motion systematically decreases the normal Weber number required to transition from bouncing to merging. Direct numerical simulations demonstrate that the depression created during impact combined with the translational motion of the bath enhances the air-layer drainage on the upstream side of the droplet, ultimately favouring coalescence. A simple geometric argument is presented that rationalises the collapse of the experimental threshold data, extending what is known for the case of axisymmetric normal impacts to the more general three-dimensional scenario of interest herein.

In this paper, a two-component discrete Boltzmann method is used to investigate the influence of the relaxation time on the two-dimensional compressible Kelvin–Helmholtz instability under dual-mode perturbations. The evolution characteristics of density gradient, vorticity, mixing entropy, Knudsen number (Kn), and thermodynamic non-equilibrium (TNE) effects are analysed. The results reveal that increasing relaxation time enhances diffusive and dissipative effects, leading to smoother interfaces, weaker vortex structures and suppressed instability growth. The global density gradient and vorticity intensity decrease accordingly. Mixing entropy analysis shows that larger relaxation times promote early mixing through diffusion, while smaller ones enhance late-stage mixing via vortex-induced convection. The Kn and TNE counters exhibit similar spatial and temporal variations, both effectively capturing the interface dynamics and deviations from local equilibrium. Both their magnitudes and the area over which they are spatially distributed increase with relaxation time, reflecting enhanced non-equilibrium effects. Besides, the global Kn and average TNE intensity initially rise, then decline, increase again and finally decrease, increasing with the relaxation time. These are jointly driven by the competitive physical mechanisms of the interface stretching, vortex merging and diffusion mechanisms. The findings provide a theoretical foundation for further exploration of non-equilibrium processes in complex fluid systems.

We investigate the interaction between two equally signed neutral vortices, namely vortices with a vanishing area integral of vorticity in inviscid non–divergent two-dimensional flows or a vanishing volume integral of potential vorticity anomaly in three-dimensional quasi-geostrophic (QG) flows. The vortices have a continuous (potential) vorticity distribution, and are linear combinations of appropriately normalised cylindrical (or spherical) Bessel functions of order 0, truncated at a zero of the Bessel function of order 1. Some pairs of neutral vortices reach an oscillating near-equilibrium state, attracting and repelling each other as a result of the exchange of small amounts of vorticity in their peripheries. This vorticity exchange generates a dipolar moment within each vortex which separates the vortices slightly, whereas the subsequent radial redistribution of the vorticity causes the vortices to come back closer again. The interaction is slower and weaker in three-dimensional QG flows, as the potential vorticity exchange primarily takes place close to the horizontal mid-plane of the vortices. These results have been corroborated using two radically different numerical models, namely a pseudo-spectral model and a high-resolution contour-advection model, both in two and in three dimensions. The observed oscillation mechanism could explain the persistence of geophysical vortices under the influence of other vortices and their ability to form stable vortex structures without experiencing vortex merging.

This study explores the effect of friction Reynolds number ($\textit{Re}_\tau \approx 3000$–$13{\,}000$) on secondary flows in three-dimensional turbulent boundary layers induced by spanwise surface heterogeneity. Using a combination of floating-element drag balance and high-resolution hot-wire anemometry, we examine how varying spanwise spacing ($S/\delta$ where $\delta$ is the boundary layer thickness defined as the distance from the wall where streamwise mean velocity $U = 0.99U_\infty$) influences frictional drag, turbulence intensity, spectral energy distribution and the organisation of coherent structures. The results reveal that secondary flows modulate turbulence differently depending on $S/\delta$, with strong near-wall effects at $S/\delta \lt 1$ and outer-layer modulation at $S/\delta \gtrsim 1$. A robust spectral signature of secondary flows peaking at $\lambda _x \approx 3\delta$ and $y \approx 0.5\delta$ emerges across all cases. This peak coexists with, or suppresses, very large-scale motions (VLSMs), depending on flow region and spacing. While VLSMs are suppressed in low-momentum pathways, they gradually recover in high-momentum pathways at higher $S/\delta$ and $\textit{Re}_\tau$. These findings offer insights into the interplay between fluctuations caused by secondary motions and boundary layer structures at high Reynolds numbers.

The dynamics of turbulent Rayleigh–Taylor (RT) mixing layers is investigated across a broad range of Atwood and Reynolds numbers using the statistically stationary RT flow configuration – a computational framework that enables simulation of a minimal flow unit for RT flows at reduced cost. Normalizations are developed for all dominant non-transport terms in the continuity, mixed mass and turbulent kinetic energy budgets in terms of the input parameters: the mixing layer height $h$, gravitational acceleration $g$ and fluid densities $\rho _H$ and $\rho _L$. Most normalized quantities collapse well across the parameter space. In some cases, variations in the Atwood number $A$ (or the density ratio $R$) lead to consistent integral magnitudes but spatially shifted profiles. These shifts are primarily related to a division by density and are similarly observed in the analytical solution of the one-dimensional variable-density diffusion problem. The analysis introduces a reference density for the mixed mass, examines trends in Favre-averaged statistics and derives a scaling law for the growth rate of the mixing layer. For height definitions encompassing the full extent of the layer, the conventional growth parameter, $\alpha =\dot {h}^2/4Agh$, varies with Atwood number. Our analysis leads to an alternative formulation using an effective Atwood number, $A^*= (\ln R)/2$, that is consistent with the scaling proposed by Belen’kii & Fradkin (1965 Trudy FIAN 29, 207–238). Applying this $A^*$ scaling to existing RT data, the corresponding growth parameter, $\alpha ^*=\dot {h}^2/4A^*gh$, remains nearly constant across all Atwood numbers considered, offering a unified scaling for variable-density RT flows.

We study the dynamics of inertial particles in turbulence using datasets obtained from both direct numerical simulations and laboratory experiments of turbulent swirling flows. By analysing time series of particle velocity increments at different scales, we show that their evolution is consistent with a Markov process across the inertial range. This Markovian character enables a coarse-grained description of particle dynamics through a Fokker–Planck equation, from which we can extract drift and diffusion coefficients directly from the data. The inferred coefficients reveal scale-dependent relaxation and noise amplitudes, indicative of inertial filtering and intermittency effects. Beyond the kinematic description, we analyse the thermodynamic properties of particle trajectories by computing the trajectory-dependent entropy production. We show that the statistics of entropy fluctuations satisfy both the integral fluctuation theorem and, under certain conditions, the detailed fluctuation theorem. These results establish a quantitative bridge between stochastic thermodynamics and particle-laden flows, and open the door to modelling turbulent transport using effective stochastic theories constrained by data and physical consistency.

This study reveals the presence of frequency components lower than the wake vortex-shedding frequency within the classical Mode A. Primaries are one-third of the vortex-shedding frequency and frequencies corresponding to recirculation bubble pumping, as previously studied. However, when the spanwise domain size $L_z$ in numerical simulations is sufficiently large, their interaction relationship becomes obscured. To clarify the interaction relationship, we introduce a process in which distinct frequency components gradually emerge by starting with a small spanwise domain of $3.3D$ and then increasing it to $4.7D$, where $D$ represents the diameter of the cylinder. At $L_z \leqslant 3.5D$, only the vortex-shedding frequency harmonics are present. One-third of the vortex-shedding frequency component appeared in $L_z \geqslant 3.6D$. Bispectral mode decomposition and energy transfer analysis reveal that the difference interaction between the one-third-shedding frequency and the vortex-shedding frequency component transfers energy to another low-frequency component. The recirculation bubble pumping is evident in the flow fields $L_z \geqslant 3.8D$. The frequency components after this emergence are not only the harmonics of the lowest-frequency component, and the periodic nature is disrupted, which is marked as a quasi-periodic state. Nonlinear interactions between the lowest-frequency component corresponding to recirculation bubble pumping, primary frequency components such as wake vortex shedding, and approximately one-third of the vortex-shedding frequency complicate the temporal behaviour of the flow field. Utilising the constraint of the spanwise domain size, our approach effectively reveals the interaction relationship among frequency components inherent in a flow field with several coherent spectral components.

We present a theoretical framework for modelling a plane-strain hydraulic fracture propagating in a poroelastic rock in the toughness-dominated regime. The formulation explicitly incorporates two-dimensional (2-D) pore-pressure diffusion, thereby generalising the classical Carter leak-off model, which can be interpreted as the limiting case of one-dimensional (1-D) diffusion. The poroelastic response is captured by superposing pore pressure and backstress contributions from a spatial and temporal distribution of instantaneous point sources along the extending fracture. A scaling analysis reveals the existence of a class of large-time, self-similar solutions for which the fracture length grows as $\ell \sim t^{1/2}$, with a prefactor function of a dimensionless injection rate $\mathcal{I}$ and a poroelastic stress coefficient $\eta$. The injection rate $\mathcal{I}$ emerges as the dominant controlling parameter. Asymptotic analysis provides large-time closed-form solutions in the limits of both large and small $\mathcal{I}$, which show excellent agreement with full numerical simulations. For large $\mathcal{I}$, diffusion reduces to 1-D and the solution converges to the classical toughness- and leak-off-dominated solution governed by Carter’s law. For small $\mathcal{I}$, fracture growth is instead controlled by pseudo-steady (2-D) diffusion. The transition from 2-D to 1-D diffusion is characterised by an increase in the fracture length prefactor and a reduction in leak-off. The poroelastic coefficient $\eta$ acts to shorten and narrow the fracture while increasing both leak-off and driving pressure. This framework delineates the transition between 2-D and 1-D diffusion and establishes quantitative conditions under which Carter’s law remains valid in the large-time limit.

We propose a novel stability criterion for incompressible shear flows by combining input–output analysis and the small gain theorem. The criterion yields an explicit threshold on the magnitude of velocity perturbations about a given base flow that guarantees stability. If this threshold is crossed – either due to non-modal growth, exponential growth or a bypass transition scenario – our analysis predicts a loss of stability that may lead to transition to turbulence. We consider three approximated models for nonlinearity: unstructured, structured with non-repeated blocks and structured with repeated blocks. We show that the imposed threshold obtained by these three methods complies with a hierarchical relationship, where the unstructured case is the most conservative, imposing the lowest bound on disturbance magnitude. We apply this approach to three canonical and well-studied base flows: Couette, plane Poiseuille and Blasius. For these three base flows, we compare our results with experiments, direct numerical simulation results, non-modal nonlinear stability results and linear stability theory (LST). In the limit of infinitesimally small perturbation magnitude, our stability criterion for the unstructured case recovers the results of LST. For finite perturbations, the structured cases that account for nonlinear interactions provided stability thresholds that are consistent with experimental observations and simulation results of transition at both subcritical and post-critical Reynolds numbers for the considered base flows in our study. In particular, we utilise our stability criterion to demonstrate that Couette flow can become unstable and transition can be triggered at different Reynolds numbers, which is consistent with past experimental observations.

We present a three-dimensional numerical study of the splashing dynamics of non-spherical droplets impacting a quiescent liquid film, covering a wide range of aspect ratios ($A_r$) and Weber numbers ($ \textit{We}$). The simulations reveal distinct impact dynamics, such as spreading, splashing type-1, splashing type-2 and canopy formation, which are delineated in a regime map constructed in the $A_r$–$ \textit{We}$ parameter space. Our results demonstrate that droplet morphology during the impact significantly influences crown evolution and splash initiation, with oblate drops promoting finger growth and fragmentation due to enhanced rim deceleration, while prolate drops tend to form canopies. We observe that the hole instability, which becomes more prominent at higher Weber numbers, arises from lamella rupture in the thinnest region of the film, located just beneath the crown rim. A linear stability analysis, supplemented by the temporal evolution of the crown obtained from the numerical simulations, adequately predicts the number of fingers formed along the crown rim by accounting for both Rayleigh–Plateau (RP) and Rayleigh–Taylor (RT) instabilities. The theoretical analysis demonstrates the dominant role of the RP instability in determining the number and wavelength of early undulations, with the RT instability serving to amplify the growth rate of the disturbances. Our findings highlight the critical role of the droplet shape in splash dynamics, which is relevant to a range of applications involving droplet impact.

The current work analyses the onset characteristics of buoyancy and thermocapillary-driven instabilities in two-layer binary fluid systems near their upper critical solution temperature (UCST). To account for the non-trivial thickness of the fluids’ interface and the temperature-dependent solubility in such regimes, the present analysis utilises the phase-field approach with a modified free-energy expression. The spatial discretisation of the field variables is carried out here using the spectral collocation approach with a suitable grid mapping strategy to accurately evaluate the field gradients around the diffuse-interface region. The results reveal that in the case of pure buoyancy-driven (Rayleigh–Bénard) convection, the parametric range for oscillatory onset is found to shrink when the system approaches the UCST, as the increased solubility results in less favourable conditions for oscillatory onset. The marginal stability curves of different fluid combinations considered here exhibit unique drift patterns based on their thermo-physical and transport properties. For systems with added thermocapillarity effects (Rayleigh–Bénard–Marangoni convection), the changing solubilities and the interfacial thickness, like the interfacial tension, exhibit a dual role that results in system-specific expansion/shrinkage of the parametric space for oscillatory flow onset.