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.

Bypass transition, momentum and passive scalar transports in an initially laminar low Reynolds number channel flow with a specific roughness morphology are investigated by direct numerical simulations. The roughness elements are square bars of large heights $k$. Turbulence cannot be triggered in an initially laminar flow without external noise, when the bars extend the entire width of the channel. A staggered configuration is necessary to break up the spanwise symmetry, in which case a pseudo-fully developed rough regime sets up and self-sustains near and below the subcritical Reynolds number. The critical parameter is the shift $s$ between two consecutive staggered bars spanning half the width of the channel. A small shift $s/k$ is enough to trigger the turbulent field. Momentum and scalar fields are analysed for different $s/k$ configurations. The Townsend similarity hypothesis postulating that the outer layer is insensitive to the roughness effects, and that the rough- and smooth-wall statistics collapse in the outer layer, holds well for the momentum field despite the large roughness heights. A particular attention is paid to the deviation of the scalar statistics from the Townsend hypothesis. There is a dissimilarity between the fluctuating temperature and the velocity fields. The Reynolds analogy does not hold stricto sensu. Wake-induced terms determined through the double-averaging procedure play an important role in the rough sublayer. For instance, a significative production of the fluctuating spanwise velocity intensity, which is absent in the canonical flow, appears as a wake-induced term at small shifts. This is solely due to the imposed spanwise asymmetry. The nature, the generation and the self-sustaining mechanisms of the coherent structures near and between the roughness elements are analysed in detail in different configurations. There is a substantial increase of the Nusselt number at particularly low Reynolds numbers.

We derive equations for three-dimensional internal wave beams propagating over a uniform slope in a uniformly stratified fluid. Using small-amplitude expansions, linear solutions for internal waves are obtained under weakly viscous conditions. Furthermore, a set of equations is constructed for the Lagrangian mean flow induced by the weakly nonlinear internal waves, providing the corresponding Lagrangian mean flow solutions within the boundary layer. The momentum equations of the Lagrangian mean velocity show that the Lagrangian mean flow is driven by the internal wave-induced body force, with its barotropic component related to the pressure gradient force, and its baroclinic component influenced by both viscosity and buoyancy. The Lagrangian-averaged buoyancy equation demonstrates that only the horizontal velocity for the Lagrangian mean flow exists throughout a vertically stable stratification region. This study emphasises the potential role of the Lagrangian mean flow in transporting time-averaged potential vorticity and solves for the Lagrangian mean flow in the inviscid region via mean potential vorticity conservation. The main results of the internal waves and the Lagrangian mean flow are visualised, revealing that the range of the boundary layer is related to the Reynolds number and that the intensity of the Lagrangian mean flow within the boundary layer is affected by the incidence angle and the reflection obliqueness. Theoretical analysis is provided to explain these phenomena.

Reverse osmosis (RO) is an efficient desalination approach, but the widely used solution-diffusion model was challenged for failing to explain field-dependent permeabilities, particularly when the continuum theory may break down in Ångström scale. Here we developed a non-equilibrium statistical theory, supported by molecular dynamics simulations that captures the field-dependent water and ion permeabilities through a single Ångström-scale channel. Surprisingly, our simulation reveals a counterintuitive negative differential flow resistance (NDFR) effect, where the flow velocity decreases with increasing pressure. This phenomenon arises from ion trapping at the nanotube entrance, caused by dielectric and dehydration barriers and hydrodynamic friction. The NDFR effect significantly reduces water permeability and may be a predominant factor constraining the selectivity-permeability trade-off in RO. Our statistical theory is based on a bidirectional escape framework that predicts the pressure- and size-dependent permeabilities and explains the NDFR effect. Our findings offer molecular-level insights into RO and can be extended to broader transport phenomena in confined systems.

The hydrodynamics of wetting involves a singularity of viscous stress, and its microscopic regularisation ultimately determines the speed at which contact lines move over a surface. In a recent paper, Luo & Gao (J. Fluid Mech., vol. 1019, 2025, A52) explore a new analytical solution, based on which they construct a model for ‘slippery wedge flow’. This lucid approach provides an accurate description of viscous wetting flows in the presence of slip, without the usual restriction to small contact angles, and offers a quantitative multiscale formalism for slippery contact lines.

We present a novel approach to harness the oscillation energy from cilia in chaotic flow to enhance scalar transport, addressing limitations of the laminar boundary layer. In contrast to the scallop theorem, where reciprocal motion yields negligible transport, coordinated rigid cilium oscillations in chaotic flow trigger boundary-layer resonance, significantly boosting scalar transport at specific frequencies. Under relatively high rigidity, the cilia undergo only small elastic deformations at the driving frequency, and their strokes remain nearly time symmetric. Nevertheless, unlike the classical expectation that reciprocal motion yields negligible transport, coordinated rigid cilium oscillations in chaotic flow trigger boundary-layer resonance, producing a sharp, frequency-selective boost in transport. At low to medium frequencies, cilium-driven fluid displacement enhances transport via vertical mixing. Above a critical frequency, rapid cilium motion induces unstable shear flow, generating coherent vortical structures that amplify mixing in chaotic flow regimes. These vortices, which interact with the inherent coherent structures of the chaotic flow, dramatically improve the efficiency of transport. Our findings reveal a dynamic coupling between cilium-driven resonance and chaotic flow coherent structures, providing a paradigm for optimising transport in thermal systems through active flow control.

Solidification of droplets is of great importance to various technological applications, drawing considerable attention from scientists aiming to unravel the fundamental physical mechanisms. In the case of multicomponent droplets undergoing solidification, the emergence of concentration gradients may trigger significant interfacial flows that dominate the freezing dynamics. Here, we experimentally investigate the fascinating interfacial freezing dynamics of supercooled ethanol–water droplets, accompanied with the migration and growth of massive ice particles. We reveal that this unique freezing dynamics is driven by solidification-induced solutal Marangoni flow within the droplets. Our model, which incorporates the temperature- and concentration-dependent properties of the ethanol–water mixture, quantitatively predicts both the migration velocity and the growth rate of the ice particles. The former is determined by the solutal Marangoni flow velocity, while the latter is governed by a balance between the latent heat release and the enhanced thermal dissipation by the Marangoni flow. Moreover, we show that the final wrapping state of droplets can be modulated by the concentration of ethanol. Our findings may pave the way for novel insights into the physicochemical hydrodynamics of multicomponent liquids undergoing phase transitions.

For decades, it has been established that there are two distinct types of instability waves leading to rotating stall in compressors, known as modes and spikes. Modal-type stall inception can be explained by conventional stability theory; however, spike-type instabilities are inherently nonlinear, whose exploration requires a different theoretical approach. For this problem, a two-dimensional point vortex instability model is developed in this paper. This simple model represents a cascade of blades by a row of bound vortices and large-scale shed vortices by point vortices. It assumes that lift on an overloaded blade abruptly drops as local incidence exceeds a critical value, analogous to leading edge stall of an isolated aerofoil, such that local cascade characteristic can be expressed as a discontinuous function. The nonlinearity thus introduced precludes the possibility of modal-type inception. As the results show, a localised stall cell will be formed in the cascade once a local perturbation triggers a discontinuous drop in blade loading, which is bounded by the stall and starting vortices shed respectively from the stalling and unstalling blades. Accordingly, a spike appears in the calculated velocity or pressure trace, directly growing into rotating stall. With this model, the experimentally observed features of spike stall are qualitatively reproduced. Moreover, the temporal variation of the stall cell size is predicted for the first time, showing qualitative agreement with existing experiments. Finally, a new prediction is made that the spike amplitude increases approximately linearly with time, in contrast to the exponential growth of linear modes.

We introduce a novel experimental approach for measuring Onsager coefficients in steady-state multiphase flow through porous media, leveraging the fluctuation–dissipation theorem to analyse saturation fluctuations. This method provides a new tool for probing transport properties in porous media, which could aid in the characterisation of key macroscopic coefficients such as relative permeability. The experimental set-up consists of a steady-state flow system in which two incompressible fluids are simultaneously injected into a modified Hele-Shaw cell, allowing direct visualisation of the dynamics through optical imaging. By computing the temporal correlations of saturation fluctuations, we extract Onsager coefficients that govern the coupling between phase fluxes. Additionally, we have performed a statistical analysis of the fluctuations in the derivative of saturation under different flow conditions. This analysis reveals that while the fluctuations follow Gaussian statistics up to 2–3 standard deviations, they exhibit heavy tails beyond this range. This work provides an experimental foundation for recent theoretical developments in the extention of non-equilibrium thermodynamics to multiphase porous media flows. By linking microscopic fluctuations to macroscopic transport behaviour, our approach offers a new perspective that may complement existing techniques in the study of multiphase flow, making it relevant to both statistical physics and the broader fluid mechanics community.

The objective of this work is to investigate the unexplored laminar-to-turbulent transition of a heated flat-plate boundary layer with a fluid at supercritical pressure. Two temperature ranges are considered: a subcritical case, where the fluid remains entirely in the liquid-like regime, and a transcritical case, where the pseudo-critical (Widom) line is crossed and pseudo-boiling occurs. Fully compressible direct numerical simulations are used to study (i) the linear and nonlinear instabilities, (ii) the breakdown to turbulence, and (iii) the fully developed turbulent boundary layer. In the transcritical regime, two-dimensional forcing generates not only a train of billow-like structures around the Widom line, resembling Kelvin–Helmholtz instability, but also near-wall travelling regions of flow reversal. These spanwise-oriented billows dominate the early nonlinear stage. When high-amplitude subharmonic three-dimensional forcing is applied, staggered $\Lambda$-vortices emerge more abruptly than in the subcritical case. However, unlike the classic H-type breakdown under zero pressure gradient observed in ideal-gas and subcritical regimes, the H-type breakdown is triggered by strong shear layers caused by flow reversals – similar to that observed in adverse pressure gradient boundary layers. Without oblique wave forcing, transition is only slightly delayed and follows a naturally selected fundamental breakdown (K-type) scenario. Hence in the transcritical regime, it is possible to trigger nonlinearities and achieve transition to turbulence relatively early using only a single two-dimensional wave that strongly amplifies background noise. In the fully turbulent region, we demonstrate that variable-property scaling accurately predicts turbulent skin-friction and heat-transfer coefficients.

The present work aims at exploring the scale-by-scale kinetic energy exchanges in multiphase turbulence. For this purpose, we derive the Kármán–Howarth–Monin equation which accounts for the variations of density and viscosity across the two phases together with the effect of surface tension. We consider both conventional and phase conditional averaging operators. This framework is applied to numerical data from detailed simulations of forced homogeneous and isotropic turbulence covering different values for the liquid volume fraction, the liquid–gas density ratio, the Reynolds number and the Weber number. We confirm the existence of an additional transfer term due to surface tension. Part of the kinetic energy injected at large scales is transferred into kinetic energy at smaller scales by classical nonlinear transport while another part is transferred to surface energy before being released back into kinetic energy, but at smaller scales. The overall kinetic energy transfer rate is larger than in single-phase flows. Kinetic energy budgets conditioned in a given phase show that the scale-by-scale transport of turbulent kinetic energy due to pressure is a gain (loss) of kinetic energy for the lighter (heavier) phase. Its contribution can be dominant when the gas volume fraction becomes small or when the density ratio increases. Building on previous work, we hypothesise the existence of a pivotal scale above which kinetic energy is stored into surface deformation and below which the kinetic energy is released by interface restoration. Some phenomenological predictions for this scale are discussed.

The linear Faraday instability of a viscous liquid film on a vibrating substrate is analysed. The importance is in the first step in applications for ultrasonic liquid-film destabilisation. The equations of motion are linearised and solved for a liquid film with constant thickness vibrating in a direction normal to its interface with an ambient gaseous medium treated as dynamically inert. Motivated by empirical evidence and the weakly nonlinear analysis of Miles (J. Fluid Mech., vol. 248, 1993, pp. 671–683), we choose an ansatz that the free liquid-film surface forms a square-wave pattern with the same wavenumbers in the two horizontal directions. The result of the stability analysis is a complex rate factor in the time dependency of the film surface deformation caused by the vibrations at a given excitation frequency and vibration amplitude. The analysis allows Hopf bifurcations in the liquid-film behaviour to be identified. Regimes of the deformation wavenumber and the vibration amplitude characterised by unstable film behaviour are found. Inside the regimes, states with given values of the deformation growth rate are identified. The influence of all the governing parameters, such as the vibration amplitude and frequency, the deformation wavenumber and the liquid material properties, on the liquid-film stability is quantified. Non-dimensional relations for vibration amplitudes characteristic for changing stability behaviour are presented.

The turbulent evolution of the shallow water system exhibits asymmetry in vorticity. This emergent phenomenon can be classified as ‘balanced’, that is, it is not due to the inertial-gravity-wave modes. The quasi-geostrophic (QG) system, the canonical model for balanced motion, has a symmetric evolution of vorticity, thus misses this phenomenon. Here, we present a next-order-in-Rossby extension of QG, $\textrm {QG}^{+1}$, in the shallow water context. We recapitulate the derivation of the model in one-layer shallow water grounded in physical principles and provide a new formulation using ‘potentials’. Then, the multi-layer extension of the shallow water quasi-geostrophic equation ($\textrm {SWQG}^{+1}$) model is formulated for the first time. The $\textrm {SWQG}^{+1}$ system is still balanced in the sense that there is only one prognostic variable, potential vorticity (PV), and all other variables are diagnosed from PV. It filters out inertial-gravity waves by design. This feature is attractive for modelling the dynamics of balanced motions that dominate transport in geophysical systems. The diagnostic relations connect ageostrophic physical variables and extend the massively useful geostrophic balance. Simulations of these systems in classical set-ups provide evidence that $\textrm {SWQG}^{+1}$ captures the vorticity asymmetry in the shallow water system. Simulations of freely decaying turbulence in one layer show that $\textrm {SWQG}^{+1}$ can capture the negatively skewed vorticity, and simulations of the nonlinear evolution of a baroclinically unstable jet show that it can capture vorticity asymmetry and finite divergence of strain-driven fronts.