This study investigates droplet impact on elastic plates using a two-phase lattice Boltzmann method in both two-dimensional (2-D) and three-dimensional (3-D) configurations, with a focus on rebound dynamics and contact time. The 2-D simulations reveal three distinct rebound modes – conventional bounce, early bounce and rim rising – driven by fluid–structure interaction. Among them, the early bounce mode uniquely achieves a significant reduction in contact time, occurring only at moderate plate oscillation frequency. Momentum analysis shows a non-monotonic relationship between vertical momentum transfer and rebound efficiency: increased momentum does not necessarily promote rebound if it concentrates in a central jet, which contributes minimally to lift-off. This introduces a novel rebound mechanism governed by momentum distribution morphology rather than total magnitude. A theoretical model treating the droplet–plate system as coupled oscillators is developed to predict contact time in the early bounce regime, showing good agreement with numerical results. The mechanism and model are further validated through fully 3-D simulations, confirming the robustness of the findings.

Plane unsteady potential flows of an ideal incompressible fluid with a free boundary are considered in the absence of external forces and surface tension. At the initial time, the flow occupy a wedge with an angle at the apex. For different initial flow velocities and values of the angle at the vertex, a family of exact solutions is found. A method for finding solutions based on reducing the boundary-value problems to systems of ordinary differential equations.

Hysteresis in the transition between regular reflection (RR) and Mach reflection (MR) has been predicted theoretically and numerically for decades, yet successful experimental demonstrations have remained limited to wedge-angle-variation-induced hysteresis. This work presents the first successful experimental demonstration of Mach-number-variation-induced hysteresis. Utilising a newly developed continuously variable Mach 5–8 wind-tunnel nozzle, Mach-sweep experiments were conducted on a pair of wedges at three different angles ($25^{\circ }$, $27^{\circ }$ and $28^{\circ }$). A stable RR was first established at Mach 7 within the dual-solution domain for each angle, and then the Mach number was decreased to 5. For the $27^{\circ }$ and $28^{\circ }$ cases, transition from RR to MR was observed at Mach 5.3 and 5.9, respectively, during the downward Mach sweep, and the MR state persisted throughout the upward sweep back to Mach 7. During the $25^{\circ }$ case, a stable RR was maintained throughout the entire Mach sweep, prompting further experiments into the effect of free-stream disturbances on the stability of the RR state. Preliminary results revealed a free-stream-disturbance-induced hysteresis and that the RR state is metastable with potential stochastic behaviour.

A heaving and pitching wing encountering effective angle-of-attack perturbations at the Reynolds numbers of 2000 and 20 000 is numerically studied by using an immersed boundary–lattice Boltzmann method. The perturbations are introduced as an abrupt heaving or pitching motion superposed on the baseline motion. It is found that the lift increment scales with the increase in the perturbation effective angle of attack, especially during the heaving perturbation. The pitching perturbation is more likely to disrupt this scaling due to the transition of the leading-edge vortex (LEV) detachment mechanism, where the detachment mechanism of the LEV transitions from bluff-body shedding dominant to vorticity layer eruption dominant. Despite the same variation in the effective angle of attack for the heaving and pitching perturbations, vorticity layer eruption is more likely to occur under the fast pitching perturbation. When the Reynolds number is increased to 20 000, the time histories of aerodynamic force are similar to those at the Reynolds number of 2000. Moreover, the boundary layer under the LEV is more resistant to the adverse pressure gradient, leading to greater variability in vorticity layer eruption.

Transonic buffet is a complex and strongly nonlinear unstable flow sensitive to variations in the incoming flow state. This poses great challenges for establishing accurate-enough reduced-order models, limiting the application of model-based control strategies in transonic buffet control problems. To address these challenges, this paper presents a time-variant modelling approach that incorporates rolling sampling, recursive parameter updating and inner iteration strategies under dynamic incoming flow conditions. The results demonstrate that this method successfully overcomes the difficulty in designing appropriate training signals and obtaining unstable steady base flow. Additionally, it improves the global predictive capability and identification efficiency of linear models for nonlinear flow-system responses by more than one order of magnitude. Furthermore, two adaptive control strategies – minimum variance control and generalised predictive control – are validated as effective based on the time-variant reduced-order model through numerical simulations of the transonic buffet flow over the NACA 0012 aerofoil. The adaptive controllers effectively regulate the unstable eigenvalues of the flow system, achieving the desired control outcomes. They ensure that the shock wave buffet phenomenon does not recur after control is applied, and that the actuator deflection, specifically the trailing-edge flap, returns to zero. Moreover, the control results further confirm the global instability essence of transonic buffet flow from a control perspective, thereby deepening the cognition of this nonlinear unstable flow.

The low Reynolds number solution of the wind–wave interaction problem is found in Cimarelli et al. (2023 J. Fluid Mech. vol. 956, A13), to be characterised by a skewed pattern of small-elevation waves on the bottom of a turbulent wind where drag reduction is caused by a wave-induced Stokes sublayer. The inhomogeneous, anisotropic and multiscale phenomena at the basis of this interesting solution are analysed here by means of the generalised Kolmogorov equation. It is found that the large and coherent structures populating the wind are the result of an upward shift of the self-sustaining production mechanisms of turbulence and of intense reverse energy cascade phenomena. The upward shift of production and the intensification of the reverse cascade are recognised to be the result of a periodically distributed pumping of scale energy induced by the pressure field associated with the wave-induced Stokes sublayer. The low dissipative nature of the wind–wave interface region is also investigated and is found to be related to a layering effect generated by the simultaneous presence of wave-induced pressure fluctuations and of wind-induced velocity fluctuations that interact with each other in an incoherent manner. Finally, the theoretical framework provided by the generalised Kolmogorov equation is also used to rigorously define two relevant cross-over scales for the filtering formalism, the shear scale identifying the energy-containing motion and the split energy cascade scale identifying the cross-over between forward and backward cascades. Well-defined quantitative criteria for the definition of spatial resolution and for the selection of turbulence closures in coarse-grained approaches to the wind–wave problem are provided.

We explore the fundamental flow structure of temporally evolving inclined gravity currents with direct numerical simulations. A velocity maximum naturally divides the current into inner and outer shear layers, which are weakly coupled by momentum and buoyancy exchanges on time scales that are much longer than the typical time scale characterising either layer. The outer layer evolves to a self-similar state and can be described by theory developed for a current on a free-slip slope (Van Reeuwijk et al. 2019, J. Fluid Mech., vol. 873, pp. 786–815) when expressed in terms of outer-layer properties. The inner layer evolves to a quasi-steady state and is essentially unstratified for shallow slopes, with flow statistics that are virtually indistinguishable from fully developed open channel flow. We present the classic buoyancy–drag force balance proposed by Ellison & Turner (1959, J. Fluid Mech., vol. 6, pp. 423–448) for each layer, and find that buoyancy forces in the outer layer balance entrainment drag, while buoyancy forces in the inner layer balance wall friction drag. Using scaling laws within each layer and a matching condition at the velocity maximum, the entire flow system can be solved as a function of the slope angle, in good agreement with the simulation data. We further derive an entrainment law from the solution, which exhibits relatively high accuracy across a wide range of Richardson numbers, and provides new insights into the long runout of oceanographic gravity currents on mild slopes.

Understanding the flow behaviour of wet granular materials is essential for comprehending the dynamics of numerous geological and physical phenomena, but remains a significant challenge, especially the transition of these flow regimes. In this study, we perform a series of rotating drum experiments to systematically investigate the dynamic observables and flow regimes of wet mono-dispersed particles. Two typical continuous flows including rolling and cascading regimes are identified and analysed, concentrating on the impact of fluid density and rotation speed. The probability density functions of surface angles, $\theta _{\textit{top}}$ and $\theta _{\textit{lo}w\textit{er}}$, reveal distinct patterns for these two flow regimes. A morphological parameter thus proposed, termed angle divergence, is used to characterise the rolling–cascading regime transition quantitatively. By integrating quantitative observables, we construct the flow phase diagram and flow curve to delineate the transition rules governing these regimes. Notably, the resulting nonlinear phase boundary demonstrates that higher fluid densities significantly enhance the likelihood of the system transitioning into the cascading regime. This finding is further supported by corresponding variations in flow fluctuations. Our results provide new insights into the fundamental dynamics of wet granular matter, offering valuable implications for understanding the complex rheology of underwater landslides and related phenomena.

A spherical cap, lined internally with a surfactant-laden liquid film, is studied numerically as a model of lung alveoli. Large-amplitude oscillations are considered (deep breathing), which may lead to collapse of the surfactant monolayer during compression, with formation of a sub-surface reservoir that replenishes the monolayer during re-expansion. Independent conservation equations are satisfied for the monolayer and the total surface concentration of surfactant and a novel kinetic expression is introduced to model the two-way internal transport with the reservoir. Marangoni stresses, which drive shearing flow, are not significantly hindered by the collapse of the monolayer, unless the latter is singularly stiff. However, volumetric flow rate and wall shear stress exhibit abrupt changes with monolayer collapse, mainly because of the strong modification of capillary stresses. These changes induce complex temporal variability in the epithelial shear, a condition known to stimulate enhanced surfactant secretion. The effect may counterbalance the predicted increase with amplitude in surfactant drift from the alveolar opening, thereby contributing to homeostasis. Nano-particles deposited on the liquid layer are slowly transported by the flow towards the alveolar rim, with exit half-time in order-of-magnitude agreement with in vivo data. Thus, Marangoni stresses are proposed as a key mechanism of alveolar clearance. Both particle displacement speed and surfactant drift from the alveoli are found to vary with solubility, with the former increasing monotonically and the latter exhibiting maximum at intermediate solubilities.

Surface roughness of fairly small (micron-sized) height is known to influence significantly three-dimensional boundary-layer transition. In this paper, we investigate this sensitive effect from the viewpoint that roughness alters the base flow thereby inducing new instabilities. We consider distributed roughness in the form of a wavy wall with its height being taken to be of $\mathit{O} (R^{-1/3 } \delta ^{\ast })$, where the Reynolds number $R$ is defined using the local boundary-layer thickness $\delta ^{\ast }$. Despite having a height much smaller than $\delta ^{\ast }$, the roughness is high enough to induce nonlinear responses. The roughness-distorted boundary-layer flow is characterised by a wall layer (WL) – a thin layer adjacent to the surface – the main layer and a critical layer (CL) – the vicinity of a special position at which a singularity of the Rayleigh equation occurs. The widths of both the WL and CL are of $\mathit{O} (R^{-1/3} \delta ^{\ast })$. Surface roughness alters the base flow significantly, leading to $\mathit{O} (1)$ vorticity distortions in these layers. We show for the first time that the nonlinearly distorted flows in these layers support small-scale local instabilities due to the roughness-induced $\mathit{O} (1)$ vorticities. Two types of modes, CL and WL modes, are identified. The CL modes have short wavelengths and high frequencies, with the spatial and temporal instabilities being governed by essentially the same equation. Thus, we focus on the former, which can be formulated as a linear generalised eigenvalue problem. The WL modes have short wavelengths but $\mathit{O} (1)$ frequencies. The temporal WL mode is governed by a linear eigenvalue problem similar to that for the CL modes, while the spatial WL mode is described by a nonlinear eigenvalue problem. The onset of these small-scale fluctuations could form a crucial step in the transition to turbulence.

The Richtmyer–Meshkov instability (RMI) develops when a planar shock front hits a rippled contact surface separating two different fluids. After the incident shock refraction, a transmitted shock is always formed and another shock or a rarefaction is reflected back. The pressure/entropy/vorticity fields generated by the rippled wavefronts are responsible of the generation of hydrodynamic perturbations in both fluids. In linear theory, the contact surface ripple reaches an asymptotic normal velocity which is dependent on the incident shock Mach number, fluid density ratio and compressibilities. In this work we only deal with the situations in which a shock is reflected. Our main goal is to show an explicit, closed form expression of the asymptotic linear velocity of the corrugation at the contact surface, valid for arbitrary Mach number, fluid compressibilities and pre-shock density ratio. An explicit analytical formula (closed form expression) is presented that works quite well in both limits: weak and strong incident shocks. The new formula is obtained by approximating the contact surface by a rigid piston. This work is a natural continuation of J. G. Wouchuk (2001 Phys. Rev. E vol. 63, p. 056303) and J. G. Wouchuk (2025 Phys. Rev. E vol. 111, p. 035102). It is shown here that a rigid piston approximation (RPA) works quite well in the general case, giving reasonable agreement with existing simulations, previous analytical models and experiments. An estimate of the relative error incurred because of the RPA is shown as a function of the incident shock Mach number $M_i$ and ratio of $\gamma $ values at the contact surface. The limits of validity of this approximation are also discussed. The calculations shown here have been done with the scientific software Mathematica. The files used to do these calculations can be retrieved as Supplemental Files to this article.

We present a new solution to the nonlinear shallow water equations (NSWEs) and show that it accurately predicts the swash flow due to obliquely approaching bores in large-scale wave basin experiments. The solution is based on an application of Snell’s law of refraction in settings where the bore approach angle $\theta$ is small. We therefore use the weakly two-dimensional NSWEs (Ryrie 1983 J. Fluid Mech. 129, 193), where the cross-shore dynamics are independent of, and act as a forcing to, the alongshore dynamics. Using a known solution to the cross-shore dynamics (Antuono 2010 J. Fluid Mech. 658, 166), we solve for the alongshore flow using the method of characteristics and show that it differs from previous solutions. Since the cross-shore solution assumes a constant forward-moving characteristic variable, $\alpha$, we call our solution the ‘small-$\theta$, constant-$\alpha$’ solution. We test our solution in large-scale experiments with data from 16 wave cases, including both normally and obliquely incident waves generated using the wall reflection method. We measure water depths and fluid velocities using in situ sensors within the surf and swash zones, and track shoreline motion using quantitative imaging. The data show that the basic assumptions of the theory (Snell’s law of refraction and constant-$\alpha$) are satisfied and that our solution accurately predicts the swash flow. In particular, the data agrees well with our expression for the time-averaged alongshore velocity, which is expected to improve predictions of alongshore transport at coastlines.

Mixing and heat transfer rates are typically enhanced in high-pressure transcritical turbulent flow regimes. This is largely due to the rapid variation of thermophysical properties near the pseudo-boiling region, which can significantly amplify velocity fluctuations and promote flow destabilisation. The stability conditions are influenced by the presence of baroclinic torque, primarily driven by steep, localised density gradients across the pseudo-boiling line; an effect intensified by differentially heated wall boundaries. As a result, enstrophy levels increase compared with equivalent low-pressure systems, and flow dynamics diverge from those of classical wall-bounded turbulence. In this study the dynamic equilibrium of these instabilities is systematically analysed using linear stability theory. It is shown that under isothermal wall transcritical conditions, the nonlinear thermodynamics near the pseudo-boiling region favour destabilisation more readily than in subcritical or supercritical states; though this typically requires high-Mach-number regimes. The destabilisation is further intensified in non-isothermal wall configurations, even at low Brinkman and significantly low Mach numbers. In particular, the sensitivity of neutral curves to Brinkman number variations, along with the modal and non-modal perturbation profiles of hydrodynamic and thermodynamic modes, offer preliminary insight into the conditions driving early destabilisation. Notably, a non-isothermal set-up (where walls are held at different temperatures) is found to be a necessary condition for triggering destabilisation in low-Mach, low-Reynolds-number regimes. For the same Brinkman number, such configurations accelerate destabilisation and enhance algebraic growth compared with isothermal wall cases. As a consequence, high-pressure transcritical flows exhibit increased kinetic energy budgets, driven by elevated production rates and reduced viscous dissipation.

The Boltzmann kinetic equation is considered to compute the transport coefficients associated with the mass flux of intruders in a granular gas. Intruders and granular gas are immersed in a gas of elastic hard spheres (molecular gas). We assume that the granular particles are sufficiently rarefied so that the state of the molecular gas is not affected by the presence of the granular gas. Thus, the gas of elastic hard spheres can be considered as a thermostat (or bath) at a fixed temperature $T_g$. In the absence of spatial gradients, the system achieves a steady state where the temperature of the granular gas $T$ differs from that of the intruders $T_0$ (energy non-equipartition). Approximate theoretical predictions for the temperature ratio $T_0/T_g$ and the kurtosis $c_0$ associated with the intruders compare very well with Monte Carlo simulations for conditions of practical interest. For states close to the steady homogeneous state, the Boltzmann equation for the intruders is solved by means of the Chapman–Enskog method to first order in the spatial gradients. As expected, the diffusion transport coefficients are given in terms of the solutions of a set of coupled linear integral equations which are approximately solved by considering the first Sonine approximation. In dimensionless form, the transport coefficients are nonlinear functions of the mass and diameter ratios, the coefficients of restitution and the (reduced) bath temperature. Interestingly, previous results derived from a suspension model based on an effective fluid–solid interaction force are recovered when $m/m_g\to \infty$ and $m_0/m_g\to \infty$, where $m$, $m_0$ and $m_g$ are the masses of the granular particles, intruders and molecular gas particles, respectively. Finally, as an application of our results, thermal diffusion segregation is exhaustively analysed.

Linear-stability modelling suggests that all sufficiently large riblets promote maximally growing spanwise rollers (García-Mayoral & Jiménez 2011 J. Fluid Mech. vol. 678, 317–347), yet direct numerical simulations (DNS) have shown that this is not the case (Endrikat et al. 2021 J. Fluid Mech. vol. 913, A37) some riblet shapes do not form spanwise rollers at all. Thus, the drag-reduction breakdown across all riblet shapes cannot be solely attributed to maximally growing spanwise rollers, prompting a reappraisal of the modelling. In this paper, comparing DNS data with riblet-resolving linear-stability predictions shows that the spanwise rollers are actually marginal modes, not maximally growing instabilities. This riblet-resolved linear analysis also predicts that not all riblet shapes promote spanwise rollers, in agreement with DNS, and unlike earlier linear-stability modelling, which relied on a one-dimensional (1-D) mean flow and on an over-simplified effective wall-admittance boundary condition. These riblet-resolved calculations further inform how to capture the effect of the riblet shape in a 1D model. Once captured, predictions with an effective boundary condition match riblet-resolved results, but still do not indicate what features of the riblet geometry promote the roller instability. Thus, the wall admittance is measured near the riblet crests, in both the riblet-resolved linear analysis and DNS, to show that the in-groove dynamics is dominated by a balance between the overlying pressure and unsteady inertia, and not viscous diffusion, as previously assumed. This pressure–unsteady-inertia balance sets the linear scaling of the wall admittance with riblet size, as observed in DNS, and is a key factor in setting the streamwise wavelength of the spanwise rollers. Furthermore, modelling this pressure–unsteady-inertia balance in the wall admittance reveals the role of riblet slenderness in promoting spanwise rollers, which provides the missing link in previous correlations between the riblet geometry and the presence or lack of rollers.

We investigate Lighthill’s proposed turbulent mechanism for near-wall concentration of spanwise vorticity by calculating mean flows conditioned on motion away from or toward the wall in an (friction Reynolds number) ${\textit{Re}}_\tau =1000$ database of plane-parallel channel flow. Our results corroborate Lighthill’s proposal throughout the entire logarithmic layer, but extended by counter-flows that help explain anti-correlation of vorticity transport by advection and by stretching/tilting. We present evidence also for Lighthill’s hypothesis that the vorticity transport in the log layer is a ‘cascade process’ through a scale hierarchy of eddies, with intense competition between transport outward from and inward to the wall. Townsend’s model of attached eddies of hairpin-vortex type accounts for half of the vorticity cascade, whereas we identify necklace type or ’shawl vortices’ that envelop turbulent sweeps as supplying the other half.