The coalescence and breakup of drops are classic examples of flows that feature singularities. The behaviour of viscoelastic fluids near these singularities is particularly intriguing – not only because of their added complexity, but also due to the unexpected responses they often exhibit. In particular, experiments have shown that the coalescence of viscoelastic sessile drops can differ significantly from that of their Newtonian counterparts, sometimes resulting in a sharply distorted interface. However, the mechanisms driving these differences in dynamics, as well as the potential influence of the contact angle are not fully known. Here, we study two different flow regimes effectively induced by varying the contact angle and demonstrate how that leads to markedly different coalescence behaviours. We show that the coalescence dynamics is effectively unaltered by viscoelasticity at small contact angles. The Deborah number, which is the ratio of the relaxation time of the polymer to the time scale of the background flow, scales as $\theta ^3$ for $\theta \ll 1$, thus rationalising the near-Newtonian response. On the other hand, it has been shown previously that viscoelasticity dramatically alters the shape of the interface during coalescence at large contact angles. We study this large contact angle limit using two-dimensional numerical simulations of the equation of motion. We show that the departure of the coalescence dynamics from the Newtonian case is a function of the Deborah number and the elastocapillary number, which is the ratio between the shear modulus of the polymer solution and the characteristic stress in the fluid.

Boundary-layer instability and transition control have drawn extensive attention from the hypersonic community. The acoustic metasurface has become a promising passive control method, owing to its straightforward implementation and lack of requirement for external energy input. Currently, the effects of the acoustic metasurface on the early and late transitional stages remain evidently less understood than the linear instability stage. In this study, the transitional stage of a flat-plate boundary layer at Mach 6 is investigated, with a particular emphasis on the nonlinear mode–mode interaction. The acoustic metasurface is modelled by the well-validated time-domain impedance boundary condition. First, the resolvent analysis is performed to obtain the optimal disturbances, which reports two peaks corresponding to the oblique first mode and the planar Mack second mode. These two most amplified responses are regarded as the dominant primary instabilities that trigger the transition. Subsequently, both optimal forcings are introduced upstream in the direct numerical simulation, which leads to pronounced detuned modes before breakdown. The takeaway is that the location of the acoustic metasurface is significant in minimising skin friction and delaying transition onset simultaneously. The bispectral mode decomposition results reveal the dominant energy-transfer routine along the streamwise direction – from primary modes to low-frequency detuned modes. By employing the acoustic metasurface, the nonlinear triadic interaction between high- and low-frequency primary modes is effectively suppressed, ultimately delaying transition onset, whereas the late interaction related to lower-frequency detuned modes is reinforced, promoting the late skin friction. The placement of the metasurface in the linearly unstable region of the second mode delays the transition, which is due to the suppressed streak in the oblique breakdown scenario. However, in the late stage of the transition, the acoustic metasurface induces an undesirable increment of skin friction overshoot due to the augmented shear-induced dissipation work, which mainly arises from reinforced detuned modes related to the combination resonance. Meanwhile, by restricting the location of the metasurface upstream of the overshoot region, this undesirable augmentation of skin friction can be eliminated. As a result, the reasonable placement of the metasurface is crucial to damping the early instability while causing less negative impacts on the late transitional stage.

This work investigates the weakly nonlinear dynamics of internal shear layers and the mean zonal flow induced by the longitudinal libration of an inner core within a spherical shell. Building on the work of He et al. (2022 J. Fluid Mech., vol. 939, p. A3), which focused on linear dynamics, we adopt a similar set-up to explore the nonlinear regime using both asymptotic theory and numerical computations, with Ekman numbers as low as $E=10^{-10}$. A specific forcing frequency of $\widehat {\omega }=\sqrt {2}\widehat {\varOmega }$, where $\widehat {\varOmega }$ denotes the rotation rate, is introduced to generate a closed rectangular path of characteristics for the inertial wave beam generated at the critical latitude. Our approach extends previous results by Le Dizès (2020 J. Fluid Mech., vol. 899, p. A21) and reveals that nonlinear interactions are predominantly localised around regions where the wave beam reflects on the boundary. We derive specific scaling laws governing the nonlinear interactions: the width of the interaction region scales as $E^{1/3}$ and the amplitude of the resulting mean zonal flow scales as $E^{1/6}$ in general. However, near the rotation axis, where the singularity of the self-similar solution becomes more pronounced, the amplitude exhibits a scaling of $E^{-1/2}$. In addition, our study also examines the nonlinear interactions of beams that are governed by different scaling laws. Through comparison with numerical results, we validate the theoretical predictions of the asymptotic framework, observing good agreement as the Ekman number decreases.

In this study, we revisit the validity of eddy viscosity models for predicting wave-induced airflow disturbances over ocean surface waves. We first derive a turbulence curvilinear model for the phase-averaged Navier–Stokes equations, extending the work of Cao, Deng & Shen (2020 J. Fluid Mech. 901, A27), by incorporating turbulence stress terms previously neglected in the linearised viscous curvilinear model. To verify our formulation, we perform a priori tests by numerically solving the model using mean wind and turbulence stress profiles from large-eddy simulations (LES) of airflow over waves across various wave ages. Results show that including turbulence stress terms improves wave-induced airflow predictions compared with the previous viscous curvilinear model. We further show that using a standard mixing-length eddy viscosity yields inaccurate predictions at certain wave ages, as it fails to capture wave-induced turbulence, which fundamentally differs from mean shear-driven turbulence. The LES data show that accurate representations of wave-induced stresses require a complex-valued eddy viscosity. The maximum magnitude of this eddy viscosity scales as $\sim \!u_\tau \zeta _{\textit{inner}}$, where $u_\tau$ is the friction velocity and $\zeta _{\textit{inner}}$ is the inner-layer thickness, the height at which the eddy-turnover time matches the wave advection time scale. This scaling aligns with the prediction by Belcher & Hunt (1993 J. Fluid Mech. 251, 109–148). Overall, the findings demonstrate that traditional eddy viscosity models are inadequate for capturing wave-induced turbulence. More sophisticated turbulence models are essential for the accurate prediction of airflow disturbances and form drag in wind–wave interaction models.

An experimental and computational analysis of a wing tip at moderate angle of attack highlights the leading role of the wing-tip vortex wandering along the direction grazing the wing-tip corner in generating far-field noise. The cases of Reynolds numbers $ \textit{Re}_c=0.6\times 10^6$ and $1.0\times 10^6$ at angle of attack $\alpha =10^\circ$ are presented. The vorticity field shows the existence of a system of three wing-tip vortices that co-rotate to form a helical structure. The vortices have wandering motions that develop as they travel downstream. Surface pressure measurements indicate the unsteadiness in the primary vortex to be coherent at a chord-based Strouhal number $ \textit{St}_c\approx 9$. The coherence between the surface pressure fluctuations and the far-field noise is the highest at the primary vortex crossover from the tip surface to the suction surface, which also occurs at $ \textit{St}_c\approx 9$. This is supported by computational results, where the crossover position on the wing surface experiences local maxima of pressure fluctuations at $ \textit{St}_c=9$, and the dilatation shows a wavefront emanating from the vortex crossover location. Given the downstream convection of the unsteadiness along the primary vortex, the crossover is suggested to be converting the pressure fluctuations in the vortex to acoustic waves rather than being a source of a new spectral feature. The causality correlation calculated between the surface pressure and the proper orthogonal decomposition modes of the flow field identifies the vortex kinematic modes that contribute the most to the surface pressure fluctuations at the vortex crossover.

We study the surfing motion of an active particle along a planar interface, separating a semi-infinite layer of gas from a deep layer of liquid. The interface-trapped particle self-propels, thanks to an uneven distribution of surface tension in its immediate vicinity, which itself results from a non-uniform release of an active agent from the particle’s surface. We use the reciprocal theorem in conjunction with singular perturbation expansions to calculate the leading-order contributions to the propulsion speed of the surfer due to the advective transport of mass and momentum when the Péclet and Reynolds numbers (denoted by $\textit{Pe}$ and $\textit{Re}$, respectively) are small but finite. Assuming that the surface tension varies linearly with the concentration of the agent with a slope of negative $\alpha$, we show, perhaps unexpectedly, that the normalised speed for a purely translating (but otherwise arbitrarily shaped) particle, independent of the agent discharge mechanism, can be expressed as $\mathscr{U} = 1 + \mathscr{A} ( 2 \textit{Pe} \ln \textit{Pe} + \textit{Re} \ln \textit{Re} ) + \mathscr{O}(\textit{Pe}) + \mathscr{O}(\textit{Re})$, where the prefactor $\mathscr{A}$ is positive for negative $\alpha$ and vice versa. For reference, the self-propulsion speed of autophoretic Janus spheres varies with $\textit{Pe}$ as $\mathscr{U} = 1 + \mathscr{B} \, \textit{Pe} + {\cdots}$, where $\mathscr{B}$ is positive when the mobility coefficient of the particle is negative and vice versa. Also, the speed of spherical squirmers changes with $\textit{Re}$ as $\mathscr{U} = 1 + \mathscr{C} \, \textit{Re} + \mathscr{O}(\textit{Re})^2$, with $\mathscr{C}$ being positive for pushers and negative for pullers. Our asymptotic formula reveals that the speed of a Marangoni surfer is a non-monotonic function of the Péclet and Reynolds numbers, hinting at the existence of optimal values for both $\textit{Pe}$ and $\textit{Re}$. The information contained within the multiplier $\mathscr{A}$ also offers guidance for customising the shape of the surfer, as well as the release rate and configuration of the agent, to enhance the self-surfing performance. Our general theoretical analysis is complemented by detailed numerical simulations for a representative spherical surfer. These simulations confirm our theoretical predictions and shed light on the effects of intermediate and large values of $\textit{Pe}$ and $\textit{Re}$ on the performance of Marangoni surfers.

Fluid flows around hypersonic vehicles experience chemical non-equilibrium effects at extreme temperature conditions. Reynolds-averaged Navier–Stokes (RANS) equations are primarily used to simulate turbulent external flow at full vehicle scales. However, the turbulent closure of near-wall reactions related to gas dissociation is omitted in practice because it remains unknown how to close the associated mean reaction rate, despite research efforts in this direction for more than a decade. This paper aims to discover an appropriate turbulent closure strategy of the involved finite-rate dissociative reaction through direct numerical simulation of a hypersonic turbulent boundary at Mach 9.2 with an isothermal cold wall surface, computed using Park’s five-species air dissociation model. Three sets of calculations are conducted, including two sets with non-catalytic and catalytic wall surface conditions, and one set without chemical reaction. Results show that the involved endothermic reaction mainly affects the magnitude of mean temperature and its fluctuations, whereas it has a relatively slight influence on the velocity and wall surface statistics. Turbulence-chemistry interaction is analysed within the same probability density function (PDF) framework as Wang & Xu (2024 J. Fluid Mech. vol. 998, A1), which considers temperature and species compositions in sample space. We find that modelling only the PDF of temperature, with simple knowledge of the mean species concentrations, is sufficient to reasonably well close the turbulent reaction rates and heat absorption rates, except for quantitative errors in the reaction rate of atomic nitrogen. This finding avoids the need for a more complex multivariable PDF in closure and also eliminates the requirement to model species fluctuations in RANS. Assuming a log-normal distribution for temperature provides better results, owing to the strongly skewed temperature distribution near the wall surface. The dependence and sensitivity of the single model parameter, temperature skewness, are further investigated. It is shown that the accuracy of closure result is not highly sensitive to the exact skewness value, as long as a negative one within a relatively wide range is selected. The developed closure model is applied to a wall model with species balance equations, showing significant improvement over the laminar closure, while further closure modelling efforts in the atomic nitrogen are still needed to improve computation robustness.

We study natural convection in porous media using a lattice Boltzmann method that recovers the incompressible Navier–Stokes–Fourier dynamics. The porous structure consists of a staggered two-dimensional cylinder array with half-cylinders at the walls, forming a Darcy continuum at the domain scale. Hydrodynamic reference simulations reveal distinct flow regimes: laminar (Darcy), steady inertial (Forchheimer) and vortex shedding. We then analyse the effects of porosity and solid-to-fluid conductivity ratio ($k_s/k_{\!f}$) on natural convection. At low porosity ($\varphi = 33\,\%$), convection is highly sensitive to thermal coupling, particularly for insulating solids, whereas conductive matrices buffer this effect through lateral diffusion. Increasing porosity ($\varphi = 43\,\%$) smooths the transition as solid and fluid phases become more balanced. Across the explored range, two inertial regimes emerge governed by plume-scale confinement. The transition from Darcy to inertia-driven convection begins once the dynamics resembles the Forchheimer regime of the reference simulations. Based on our data, the system is governed by the confinement parameter $\varLambda$, which relates the plume-neck width, equivalent to the thermal boundary-layer thickness, to the pore scale: for $\varLambda \gtrsim 1$, the dynamics follows Forchheimer scaling, while for $\varLambda \lt 1/2$ it shifts toward Rayleigh–Bénard behaviour. Comparison with experimental data shows the same trend: the nominal Darcy–Rayleigh-to-porous-Prandtl ratio, $Ra^*/\textit{Pr}_{\!p} \approx 1$, holds for $\varLambda \gt 10$, but weaker confinement causes earlier departure. Finally, we revise benchmark Nusselt numbers for a cavity with square obstacles, showing that the reference by Merrikh & Lage (2005 Intl J. Heat Transfer 48(7), 1361–1372) misrepresents trends due to improper normalisation.

The instabilities of a floating droplet under the action of an inclined temperature gradient in the presence of the spatial modulation of the transverse temperature gradient are investigated. The problem is studied numerically in the framework of the slender droplet approximation and the precursor model. It is shown that the spatial modulation of the transverse component of the Marangoni number is accompanied by the change of the droplet shape and can lead to development of periodic oscillations. In the definite region of parameters, quasi-periodic oscillations accompanied by the creation of pulsating satellites have been obtained. The separation and the recombination of the ‘main’ droplet with the satellites have been observed.

Lagrangian transit times on basin to planetary scales are controlled by the interplay of multiscale processes. The primary advective time scale is set by throughflow currents, such as interhemispheric western boundary currents. Dispersion by mesoscale eddies introduces fluctuations that erase memory and enhance dispersion, widening the transit-time distribution. The tortuous paths of Lagrangian parcels, particularly within ocean gyres, significantly enhance dispersion beyond the levels attributed to mesoscale eddies alone. Additionally, trapping by ocean gyres leads to multimodal distributions of Lagrangian transit times. These processes are illustrated in three complementary contexts: eddy-permitting ocean state estimates, simplified spatially extended three-dimensional flows and diffusively coupled two-dimensional pipe models.

Bounds on turbulent averages in shear flows can be derived from the Navier–Stokes equations by a mathematical approach called the background method. Bounds that are optimal within this method can be computed at each Reynolds number $ \textit{Re}$ by numerically optimising subject to a spectral constraint, which requires a quadratic integral to be non-negative for all possible velocity fields. Past authors have eased computations by enforcing the spectral constraint only for streamwise-invariant (2.5-D) velocity fields, assuming this gives the same result as enforcing it for three-dimensional (3-D) fields. Here, we compute optimal bounds over 2.5-D fields and then verify, without doing computations over 3-D fields, that the bounds indeed apply to 3-D flows. One way is to directly check that an optimiser computed using 2.5-D fields satisfies the spectral constraint for all 3-D fields. A second way uses a criterion we derive that is based on a theorem of Busse (1972 Arch. Ration. Mech. Anal., vol. 47, pp. 28–35) for energy stability analysis of models with certain symmetry. The advantage of checking this criterion, as opposed to directly checking the 3-D constraint, is lower computational cost and natural extrapolation of the criterion to large $ \textit{Re}$. We compute optimal upper bounds on friction coefficients for the wall-bounded Kolmogorov flow known as Waleffe flow and for plane Couette flow. This requires lower bounds on dissipation in the first model and upper bounds in the second. For Waleffe flow, all bounds computed using 2.5-D fields satisfy our criterion, so they hold for 3-D flows. For Couette flow, where bounds have been previously computed using 2.5-D fields by Plasting & Kerswell (2003 J. Fluid Mech., vol. 477, pp. 363–379), our criterion holds only up to moderate $ \textit{Re}$, so at larger $ \textit{Re}$ we directly verify the 3-D spectral constraint. Over the $ \textit{Re}$ range of our computations, this confirms the assumption by Plasting & Kerswell that their bounds hold for 3-D flows.