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.

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.