The two-dimensional to three-dimensional wake transition of a circular cylinder in a sinusoidal oscillatory flow arises from the Honji instability at a critical Keulegan–Carpenter number (denoted $\textit{KC}_{cr}$) with a corresponding critical spanwise wavelength (denoted $\lambda _{cr}$) for a given Stokes number (denoted $\beta$) larger than approximately 50. However, significant discrepancies in the $\textit{KC}_{cr}$ and $\lambda _{cr}$ values exist among the theoretical predictions by Hall (J. Fluid Mech., vol. 146, 1984, pp. 347–367), empirical formulae by Sarpkaya (J. Fluid Mech., vol. 457, 2002, pp. 157–180) and other experimental and numerical results in the literature. These long-standing discrepancies are addressed in this study, and new equations for $\textit{KC}_{cr}$ and $\lambda _{cr}$ are proposed for $\beta = 55$–$10^{6}$. The present $\textit{KC}_{cr}$ and $\lambda _{cr}$ values agree well with the Floquet analysis results of Elston et al. (J. Fluid Mech., vol. 550, 2006, pp. 359–389) for $\beta \sim 50$–$100$, and asymptotically converge to theoretical predictions by Hall (1984) as $\beta \to \infty$, but deviate significantly from the empirical formulae by Sarpkaya (2002). The underlying physical mechanisms for these deviations are elucidated. In addition, we reproduce the quasi-coherent structure (QCS) numerically for the first time, and demonstrate that the QCS observed by Sarpkaya (2002), where transient Honji vortices become pronounced near peak flow velocities but diminish during deceleration, is physically induced by ambient disturbances inevitably contained in physical experiments, such that $\textit{KC}_{cr}$ given by Sarpkaya (2002) is specific to the level of disturbance in his experimental setting and is somewhat arbitrary.

Liquid sheets arise in curtain coating, polymer processing and sprays. When a fluid is ejected from a die (nozzle) to form a liquid sheet, its cross-section is rectangular albeit for the two rounded ends. The latter retract due to surface tension. The retraction dynamics is also affected by stresses owing to bulk rheology, which may be viscous and/or viscoelastic in nature, and surface rheology, which may be due to the presence of surface-active agents. We analyse theoretically and numerically the retraction dynamics of highly viscous Newtonian liquid sheets when surface viscous stresses are present. While it has been shown recently that viscoelasticity increases retraction rate, it is demonstrated that surface viscosity operates synergistically with bulk viscosity to decrease retraction rate. As the two surfaces of a retracting sheet remain flat outside of the two tip regions, an exact analytical solution is obtained for the transient sheet thickness in terms of the Lambert W function. An asymptotic solution for sheet thickness, valid for early times, is also obtained and shown to agree well with the analytical solution and simulations. An energy analysis is performed to rationalise that at early times, the rate of energy dissipation due to the action of surface viscous stresses can be dominant in slowing retraction, but it can wane in importance and be overtaken at large times by the rate at which energy is dissipated due to the action of bulk viscous stresses.

In the article, the unsteady flow phenomenon of self-excited and forced oscillations in a rectangular diverging isolator is studied by using large eddy simulation, and the shock train region is analysed particularly. Self-excited oscillations are analysed under four pressure ratios, with pressure statistically processed to reveal shock train oscillation characteristics. The interference factors of the external environment on the unsteady flow in the isolator are investigated, and the function of upstream disturbance and downstream disturbance on the shock train oscillation is studied. Both disturbance types show that pressure amplitude increases oscillation amplitude, while frequency variations have opposing effects. Due to mismatched response speeds, low-frequency disturbances intensify oscillations, whereas high-frequency ones suppress them. The difference is that the pressure frequency excitation upstream is transmitted along the flow direction and directly acts on the shock train in the trough period of each unsteady pressure transformation, which intensifies the negative effect on shock train oscillations. The downstream disturbance arrives at the shock train region after passing through the complex flow coupling in the mixing region. The superposition of the external pressure excitation frequency and the mixing region makes the response of the shock train slower leading to a weakening effect of the shock train oscillation. Moreover, the unsteady flow develops in the mixing district and transmits upstream, and the inhibition effect is stronger than that of upstream pressure frequency excitation.

The description of riblets and other drag-reducing devices has long used the concept of longitudinal and transverse protrusion heights, both as a means to predict the drag reduction itself and as equivalent boundary conditions to simplify numerical simulations by transferring the effect of riblets onto a flat virtual boundary. The limitation of this idea is that it stems from a first-order approximation in the riblet-size parameter $s^+$, and as a consequence it cannot predict other than a linear dependence of drag reduction upon $s^+$; in other words, the initial slope of the drag-reduction curve. Here the concept is extended to a full asymptotic expansion using matched asymptotics, which consistently provides higher-order protrusion coefficients and higher-order equivalent boundary conditions on a virtual flat surface. While the majority of this expansion, though nonlinear in $s^+$, remains linear in velocity, and therefore we shall not directly address the shape of the drag-reduction curve, this procedure will also allow us to explore the way nonlinearities of the Navier–Stokes equations first enter the $s^+$ expansion, with somewhat surprising negative results.

The accuracy obtained with computational fluid dynamics and process simulations of flotation critically depends on the quality and robustness of the underlying models for the non-resolved subprocesses. An important issue in flotation is the collision between particles and air bubbles. Many models have been developed, but their accuracy for applications in flotation is limited. In particular, the significant size difference between particles and bubbles and their intricate coupling to the turbulent flow field pose severe challenges. The present paper first reviews presently employed collision models, highlighting their advantages and disadvantages when applied to flotation. On this basis, the `integrated multisize collision model’ (IMSC) is proposed. After a detailed evaluation, it combines existing approaches from various sources and introduces new developments designed to address present shortcomings. The model is validated by own direct numerical simulation data as well as data from the literature. It is shown that, overall, the IMSC provides better predictions for the collision rate in typical flotation conditions than presently employed collision models and covers the entire parameter range of the flotation process very well. Using the available data, some of the underlying modelling assumptions are validated. Finally, a comprehensive overview of the model is provided for further use in Euler–Euler frameworks or process simulations also beyond flotation.

Turbulent flows, despite their apparent randomness, exhibit coherent structures that underpin their dynamics. Proper orthogonal decomposition (POD) has been widely used to extract these structures from experimental data. Periodic features such as vortex shedding can appear as POD mode pairs in strongly periodic flows, but detecting propagating structures in more complex flows is challenging. Hilbert proper orthogonal decomposition (HPOD) addresses this by applying POD to the analytic signal of the turbulent fluctuations, which yields complex modes with a $\pi /2$ phase shift between the real and imaginary components. These modes capture propagating structures effectively but introduce spectral leakage from the Hilbert transform used to derive the analytic signal. The current work investigates the relationship between the modes of the POD and those of the HPOD on the velocity fluctuations in the wake of a sphere. By comparing their outputs, POD mode pairs that correspond to the same propagating structures revealed by HPOD are identified. Furthermore, this study explores whether computing the analytic signal of the POD modes can replicate the HPOD modes, offering a more computationally efficient method for determining the pairs of POD modes that represent propagating structures. The results show that the pairs of POD modes identified by the HPOD can be determined more efficiently using the Hilbert transform directly on the POD modes. This method enhances the interpretive power of POD, enabling more detailed analysis of the turbulent dynamics without the need to compute the analytic signal of the entire turbulent fluctuation data.

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.

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.

Slender fibres, including textile-derived microplastics, are abundant in aquatic environments and often extend beyond the Kolmogorov length scale. While breakup at dissipative scales has been characterised by velocity-gradient statistics, no closure existed for inertial-range spans where eddy turnover sets the clock. Here we develop a turbulence-informed kinetic theory of fibre fragmentation bridging turbulence forcing and slender-beam mechanics. First, we derive a load-to-curvature mapping showing that spanwise forcing generates peak bending moments scaling as $\sim U_L L^2$, with $U_L$ the velocity increment across fibre length $L$. Second, we construct a breakup hazard $h(L)$ from curvature-threshold exceedances over eddy-time blocks, which identifies a turbulence-defined critical span $\ell _c$. For $L\gt \ell _c$, breakup is eddy-time-limited, $h(L)=O(\bar \varepsilon ^{1/3}L^{-2/3})$ with $\bar \varepsilon$ the mean turbulent energy dissipation rate, whereas for $L\lt \ell _c$, it is a rare-event process with $h(L)\propto L^{5/3+\alpha }$, $\alpha$ denoting the small correction from intermittency. Embedding this hazard in a self-similar binary kernel yields a closed population-balance equation for the fragment distribution $n(L,t)$ with sources and sinks. The framework produces explicit predictions: intermittency-corrected curvature scalings, critical spans set by material and flow parameters, start-up and halving times linked to surf-zone conditions and scaling profiles in the cascade. The steady-state bulk distribution on the subcritical branch, with vertical removal induced by horizontal convergence, follows $n(L)\propto L^{-8/3-\alpha }\simeq L^{-2.7}$, in striking agreement with the mean slope $\simeq -2.68$ observed for environmental microfibres in recent surveys. The reported variability of slopes is naturally explained in our framework by the coexistence of supercritical and subcritical branches together with $L$-dependent removal-driven sinks.

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.

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.

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.

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.

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.

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.