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.

The dynamic behaviours of an axisymmetric ferrofluid jet, surrounded by a non-magnetisable and immiscible fluid of equal density, are investigated from both asymptotic and numerical perspectives. This two-layer system consists of incompressible, inviscid fluids that flow irrotationally within each layer. Based on the expansions of the axisymmetric Dirichlet–Neumann operators developed by Xu & Wang (2025 J. Fluid Mech., vol. 1002, p. A23), strongly nonlinear longwave models – without assuming small wave amplitudes – are derived in various limits from the magnetised Euler equations within the Hamiltonian framework. In the supercritical regime, where the magnetic field is strong enough to completely suppress the Rayleigh–Plateau instability, these models show good agreement with the full Euler equations for monotonic solitary waves. This is particularly true concerning wave profiles and speed–energy bifurcations, even when the wave trough approaches the rigid bottom. Thus, these models overcome the limitations of the cubic full-dispersion model proposed in previous studies. An analytic criterion related to wave energy for the stability exchange of axisymmetric interfacial solitary waves under longitudinal perturbations is established for the full Euler equations. Guided by this criterion, the dynamic evolution of unstable solitary waves is then numerically solved using the derived strongly nonlinear equations. In the subcritical regime, the flow experiences the Rayleigh–Plateau instability. The phenomenon of singularity is examined in a configuration where the thickness of the outer layer is infinite, employing a newly proposed model that incorporates a non-local operator. It is demonstrated that infinite-slope singularities arise before pinching for most initial conditions; however, pinching may occur for sufficiently small initial amplitudes.

We consider the flow of a viscous fluid through a two-dimensional symmetric cross-slot geometry with sharp corners. The problem is analysed using the unified transform method in the complex plane, providing a quasi-analytical solution that can be used to compute all the physical quantities of interest. This study is a novel application of this method to a complicated geometry featuring multiple sharp corner singularities and multiple inlets and outlets. Our approach offers the advantage of resolving unbounded domains, as well as providing quantities of interest, such as the velocity and stress profiles, and the Couette pressure correction, from the solution of low-order linear systems. Our results agree well with the existing literature, which has largely used truncated bounded geometries with rounded or curved corners.

Pre-existing bubbles in the water play a critical role in influencing the impact pressure characteristics during the wedge water entry. This study experimentally and analytically investigates the effect of aeration on water-entry impact. A series of controlled drop tests were conducted using a wedge with a 20° deadrise angle at varying impact velocities and void fractions. Four classical pure water impact models (the Zhao & Faltinsen model (ZFM), original Logvinovich model (OLM), modified Logvinovich model (MLM) and generalised Wagner model (GWM)) were extended to account for the effect of aeration. These modifications accounted for compressibility effects, the time-dependent void fraction, three-dimensional flow corrections and area-averaged pressure calculations, resulting in four modified models (M-ZFM, M-OLM, M-MLM and M-GWM). This marks the first systematic theoretical extension of multiple classical water-entry models to aerated conditions. The proposed models demonstrated good agreement with experimental results, with the M-MLM providing accurate peak pressure predictions and M-GWM performing best in capturing the post-peak behaviours. The results indicated that the expansion velocity of the wetted surface varied spatially and closely matched the M-ZFM predictions. While the peak pressures decreased by up to 32.8 % in highly aerated water, the prolonged impact durations led to a comparable or slightly increased pressure impulse than that in pure water. This finding suggests that prolonged lower-magnitude impacts in aerated water may pose a greater risk to structural safety than short-duration high-magnitude impacts. These contributions offer new physical insight and validated tools relevant to marine engineering design in aerated environments.

We present a framework to calculate the scale-resolved turbulent Prandtl number ${\textit{Pr}}_t$ for the well-mixed and highly inertial bulk of a turbulent Rayleigh–Bénard mesoscale convection layer at a molecular Prandtl number of ${\textit{Pr}}=10^{-3}$. It builds on Kolmogorov’s refined similarity hypothesis of homogeneous isotropic fluid and passive scalar turbulence, based on log–normally distributed amplitudes of kinetic energy and scalar dissipation rates that are coarse-grained over variable scales $r$ in the inertial subrange. Our definitions of turbulent (or eddy) viscosity and diffusivity do not rely on mean gradient-based Boussinesq closures of Reynolds stresses and convective heat fluxes. Such gradients are practically absent or indefinite in the bulk. The present study is based on direct numerical simulation of plane-layer convection at an aspect ratio of $\varGamma =25$ for Rayleigh numbers $10^5\leqslant Ra\leqslant 10^7$. We find that the turbulent Prandtl number is effectively up to four orders of magnitude larger than the molecular one, ${\textit{Pr}}_t\sim 10$. This holds particularly for the upper end of the inertial subrange, where the eddy diffusivity exceeds the molecular value, $\kappa _e(r)\gt \kappa$. Highly inertial low-Prandtl-number convection becomes effectively a higher-Prandtl-number turbulent flow, when turbulent mixing processes on scales that reach into the inertial range are included. This might have some relevance for prominent low-Prandtl-number applications, such as solar convection.