Cerebrospinal fluid (CSF) is a Newtonian fluid that bathes the brain and spinal cord, and oscillates in response to the physiological periodic changes in brain volume, of which the cardiac cycle is a major driver. Understanding this motion is essential for clarifying its contribution to solute transport, waste clearance and drug delivery. In this work, we study oscillatory and steady streaming flow in the cranial subarachnoid space using a lubrication-based theoretical framework. The model represents the cranial CSF compartment as a thin fluid layer bounded internally by the brain surface and externally by the dura, driven by time-dependent brain surface displacements. We first derive simplified governing equations for flow over an arbitrary smooth sphere-like brain surface and obtain analytical solutions for an idealised spherical geometry with uniform displacements. We then incorporate realistic displacement fields reconstructed from magnetic resonance imaging (MRI) measurements in healthy subjects and solve the reduced equations numerically. The results show that oscillatory forcing produces a steady streaming component that may enhance solute transport compared with diffusion alone. This work provides a mechanistic description of the flow generated by physiological brain motion and highlights the potential presence of steady streaming in cranial subarachnoid fluid dynamics.

We investigate the propagation of counter-rotating convection rolls in Rayleigh–Bénard convection initiated locally in a quiescent fluid layer under supercritical conditions. The velocity of the front separating quiescent fluid from the forming convection rolls, and the wavenumber of the convection rolls remaining behind the front, are explored. We numerically investigate fronts of forming convection rolls over five orders of magnitude of the reduced Rayleigh number, $\epsilon$, in two- and three-dimensional domains, for a broad range of boundary conditions, and for different front initiation approaches. In all cases, the front velocity increases as $\epsilon ^{1/2}$ with increasing $\epsilon$ for $\epsilon \lesssim 1$ in agreement with predictions using the amplitude equation. The amplitude equation description of the front velocity remains accurate for $\epsilon \lesssim 10$ except when the Prandtl number is large which yields a velocity that is faster than predicted for a fluid layer far from threshold. The wavenumber of the convection rolls increases linearly with $\epsilon$ in agreement with the wavenumber that maximises the growth rate of perturbations in the linear regime. Farther from onset, the wavenumber growth transitions to a reduced scaling of $\epsilon ^{1/4}$ in agreement with predictions using the Swift–Hohenberg equation in the large-$\epsilon$ limit. The scalings describing the wavenumber variation with $\epsilon$ are independent of the domain geometry, boundary conditions and front initiation method. However, the front-selected wavenumber at criticality does not equal the critical wavenumber of the bulk instability, in general, and depends significantly upon these details. We compare our results with experimental measurements where possible.

We investigate how negatively buoyant inertial spheres settle in progressive surface gravity waves, focusing on the phase-dependent modulation of the particle–fluid relative motion. Starting from a Maxey–Riley framework with a quasi-steady nonlinear-drag closure, we formulate a phase-resolved equation of motion in terms of the slip velocity and obtain a perturbation solution in the limit of small wave amplitude. The leading state corresponds to the quiescent-water terminal slip, the first-order response yields a dominant harmonic modulation induced by the wave kinematics and second-order terms introduce higher harmonics arising from higher-order interactions. A key element is that, under nonlinear drag with finite particle Reynolds number, the particle relaxation time and hence the Stokes number depend on the instantaneous slip. This dependence is retained consistently in the perturbation expansion, allowing the vertical-slip phase at finite Stokes number to be predicted without the additional ordering adopted in previous work. We test predictions using wave flume experiments with simultaneous particle tracking velocimetry/particle image velocimetry, which provide slip estimates along particle trajectories and phase-binned ensemble statistics. The analysis first assesses whether particles attain a terminal settling regime within the observation window, then quantifies the phase-dependent oscillation of the horizontal and vertical slip. For cases that attain the terminal regime, the perturbation solution reproduces the amplitude and the phase of the dominant slip oscillation. However, cases that remain in a wave-acceleration-dominated relaxation regime show settling hindrance near the water surface; the wave-averaged vertical slip approaches the quiescent terminal velocity with depth, and no clear settling enhancement is observed.

We present a phase-based framework for reducing the pressure fluctuations within a spanwise-periodic supersonic turbulent cavity flow with an incoming free-stream Mach number of 1.4 and a depth-based Reynolds number of $10^4$. Open cavity flows exhibit large fluctuations due to the feedback between the shear-layer instabilities and the acoustic field. The dominant flow physics includes the formation, convection and impingement of large-scale spanwise-oriented vortical structures into the aft wall. We implement a phase-reduction approach to identify the flow response about this time-varying convective process by defining a phase variable using dynamic mode decomposition. Three-dimensional impulse perturbations with various spanwise actuation wavenumbers, $\beta$, are introduced from the cavity leading edge to characterise the phase advancement or delay of convection. We perform open-loop flow control through unsteady blowing using actuation frequencies slightly different from the vortex-convection frequency to disrupt the feedback loop. After designing a phase-sensitivity-based actuation waveform optimised for quick flow modification, we investigate the speed of fluctuation reduction and compare it with a sinusoidal waveform actuation. At a spanwise actuation wavenumber of $\beta =2\pi$, both perform similarly, achieving a 46 % reduction of pressure fluctuations within five convective times. At $\beta =\pi$, the phase-sensitivity-based actuation performs better by achieving a 40 % reduction compared with the 30 % with a sinusoidal waveform within six convective times. The present method explicitly targets the temporal dynamics to achieve rapid suppression within a few convective times. This study shows the potential of phase-based analysis for timing-based (event-based) flow control and rapid flow modification of unsteady turbulent flows.

We introduce a modal representation for Lagrangian trajectories in turbulence, termed Lagrangian proper orthogonal decomposition. An ensemble of particle trajectories is used to construct velocity time series, which are normalized independently for each trajectory to isolate fluctuations. Principal component analysis is then applied to the resulting dataset, with temporal instances defining the feature space. The method is tested on trajectories from both direct numerical simulations of homogeneous isotropic turbulence and a three-dimensional particle-tracking experiment, showing that the leading modes exhibit similar structures and energy distributions in both cases. Truncated reconstructions are obtained by combining modes and coefficients, rescaling the fluctuations and integrating in time. For trajectories of the order of the integral time scale, single-particle dispersion and curvature statistics are accurately reproduced using a limited number of modes ($\sim O(10)$), whereas capturing the tails of acceleration distributions and higher-order Lagrangian structure functions requires a larger set ($\approx$ 40–80). Longer trajectories require progressively more modes for accurate reconstruction. These results suggest a possible route to data-driven generation of synthetic particle trajectories via stochastic sampling of the modal Lagrangian dynamics.

We explore thermo-viscous instabilities in an idealised setting motivated by ice-sheet flow, modelled as a shallow layer of fluid with a temperature-dependent viscosity flowing down an inclined plane and subject to a constant geothermal heat flux. In this setting, multiple steady states are possible for a given slope, and we examine their linear stability to general wave-like perturbations in both the streamwise and transverse directions. We confirm that the lower and upper branches are stable with respect to perturbations that are spatially uniform in the two horizontal directions, but the intermediate state is unstable. For finite-wavenumber perturbations and with heat diffusion into the underlying bedrock, we find that all three branches are unstable. The lower and upper branches exhibit a relatively weak instability with respect to waves with finite streamwise wavelength and long transverse wavelengths, caused by diffusion of heat into the bedrock. Across all branches, the strongest instability occurs when the transverse wavelength is infinite and streamwise wavelengths are order one. Along the intermediate branch, modes remain unstable for all transverse wavenumbers, and growth rates do not decay in the limit of short transverse wavelengths. Motivated by this observation, we also briefly examine a simplified nonlinear problem for transverse perturbations, in order to clarify the implications of this missing short-wavelength cutoff.

We use direct numerical simulations (DNSs) of columnar Taylor–Green vortex arrays under stable density stratification to characterise the route to, and the properties of, layered anisotropic stratified turbulence (LAST). Building on the linear stability analysis of Guo et al. (2024 J. Fluid Mech., vol. 997, A34) and the convection-driven turbulence transition reported by the same authors (2025 J. Fluid Mech., vol. 1016, A44), we extend the DNS database to Froude numbers $\textit {Fr}\leqslant 0.25$ and Reynolds numbers $\textit {Re}$ up to $3200$. In this strongly stratified regime, the vortex array breaks down through a shear-driven path: vertical shear layers that form spontaneously in the zigzag-deformed base flow destabilise locally and transition to LAST, in contrast with the convection-driven path identified at $\textit {Fr}\geqslant 0.5$. Four flow regimes are charted, two of which follow the buoyancy-driven scaling $\ell _v\propto U_h/N$: LAST and a layered anisotropic viscously affected flow (LAVAF) regime, where $\ell _v$ is the vertical integral scale, $U_h$ the horizontal velocity and $N$ the buoyancy frequency. In both regimes, the ratio of buoyancy scale $\ell _b=2\pi U_h/N$ to $\ell _v$ plateaus near $0.74$ at peak dissipation. Layered anisotropic stratified turbulence is distinguished from LAVAF by the buoyancy Reynolds number $\textit {Re}_b$ exceeding unity, so that the Ozmidov and Thorpe scales both rise above the Kolmogorov scale and small-scale overturns develop under weaker viscous damping. Across LAST, the cumulative mixing efficiency is $\varGamma \in [0.24,0.49]$ and the horizontal Froude number $\textit {Fr}_h=U_h/(N\ell _h)$ at peak dissipation is of order $0.01$, where $\ell _h$ is the horizontal integral scale. In terms of turbulence-based parameters, LAST is bounded by $\textit {Fr}_h\lt 0.031$–$0.044$ and $\textit {Re}_b\gtrsim 1$; via empirical fits, these map to base-flow cutoffs $\textit {Fr}\lt 0.26$–$0.37$ and $\textit {Re}\,\textit {Fr}^{5/3}\gtrsim 110$.

Minimising bacterial surface adhesion and subsequent biofilm formation in industrial and medical settings requires understanding how bacteria are transported and adhere to complex surface geometries in the presence of non-uniform flow. In this paper, we consider the transport of a dilute suspension of motile bacteria through a corrugated two-dimensional channel with perfectly adhesive walls. Considering flows much faster than the bacterial swimming speed, we asymptotically analyse the diffusive boundary layer that forms using a curvilinear coordinate system based on the fluid streamfunction. We present a similarity solution to the diffusivity-varying diffusion-type equation that arises, from which we derive an analytical expression for the bacterial adhesion rate as a function of surface arclength and the spatially varying wall shear rate. Our model predicts that bacterial adhesion becomes localised on curved surfaces, with bacteria showing preferential adhesion to wall ‘peaks’ at lower shear rates and preferential adhesion to wall ‘valleys’ at higher shear rates. More broadly, our results highlight how spatially varying flows generated by complex geometries can lead to localised bacterial adhesion, with potential implications for both enhancing and minimising biofilm formation.

Analysing the dynamics of phase-changing liquid films is essential for enhancing the performance of thermal-management systems. Still, direct simulation of the full governing equations is computationally expensive. To circumvent this limitation, I derived a weighted-integral boundary-layer (WIBL) model under long-wave assumptions, weak evaporation and strong surface tension, also accounting for variable-substrate heating. In the linear regime, the WIBL reproduces the growth rates and cutoff wavenumbers of unstable modes with significantly higher accuracy than commonly used Benney-type models for Reynolds number ${{\textit{Re}}}\lt 40$, and with accuracy comparable to that of the Orr–Sommerfeld equations. The linear analysis further reveals a threshold separating streamwise- and spanwise-dominated instabilities in hanging films, arising from the competition between the Kapitza and Rayleigh–Taylor mechanisms; the WIBL accurately predicts this threshold for small ${\textit{Re}}$ and inclination angles. In the nonlinear regime, with space- and time-varying substrate heating, the WIBL model captures the evolution of the free-surface thickness and temperature with an accuracy of approximately 6 % relative to the original Navier–Stokes equations. Three-dimensional simulations show that a condensing film undergoes dry out due to Kapitza instability, whereas unsteady substrate heating promotes spanwise momentum spreading, modifies wave dynamics, and prevents dry out. The WIBL model provides a good level of accuracy at a low computational cost, enabling extensive parametric studies, nonlinear-stability analyses and the design of optimal substrate-heating control strategies.

We investigate start-up flow from an initially quiescent state in three ordered sphere packings (hexagonal close-packed, face-centred cubic, body-centred cubic) driven by a constant intrinsic pressure gradient using direct numerical simulation. The main finding is that the first nonlinear departure from the initial linear regime occurs at a characteristic inviscid time $\tau _{{inv}}=\sqrt {\rho d/\vert \boldsymbol{\nabla }\langle p\rangle _{\textit{i}}\vert }$ (where $\rho$, $d$, $\boldsymbol{\nabla }\langle p\rangle _{\textit{i}}$ correspond to fluid density, sphere diameter, intrinsic pressure gradient), rather than at a critical instantaneous Reynolds number. At early times, the flow consists of an irrotational outer motion and thin Stokes-type boundary layers near the solid surfaces; nonlinear effects emerge when the convective term in these boundary layers becomes comparable to the other leading-order terms. At the pore scale, the transient evolution is characterised by the growth of thin vorticity layers on the sphere surfaces, their large-scale migration into the pore space after times of order $\tau _{{inv}}$ and the formation of inertial cores. Despite geometric differences, these processes occur in a remarkably similar sequence across all three packings. Vorticity magnitude exhibits laminar boundary-layer scaling with the governing dimensionless parameter Hagen number, whilst in the body-centred cubic sphere pack case a transition towards turbulent-type scaling is observed. These results establish $\tau _{{inv}}$ as a unifying measure for the onset of nonlinearity in strongly accelerated porous media flows, with direct implications for the modelling of unsteady transport in natural and engineered systems.

Stalagmites grow from the repeated impact of supersaturated water drops onto a cave floor. Existing theoretical descriptions typically model the water supply as a continuous thin film driven by a steady flux. Consequently, they neglect the discrete nature of individual drops and fail to predict a non-zero minimal stalagmite radius. Here, we demonstrate that the coupled hydrodynamics and precipitation kinetics of a single drop spreading as a thin viscous film on a stalagmite tip select a specific minimal radius. Using the similarity solution for an axisymmetric gravity current of fixed volume, coupled with precipitation kinetics, we derive an explicit expression for this lower bound in terms of drop volume, fluid properties and reaction rate. For typical cave parameters, the predicted radius is of the order of two centimetres, in quantitative agreement with the smallest stalagmites observed in the field. Our results provide a falsifiable constraint on stalagmite morphodynamics and offer a prototype for scale selection in reactive gravity currents generated by discrete sources.

Fragmentation of slender fibres by turbulence controls length distributions in many natural and industrial suspensions, yet most theories assume a prescribed carrier flow, and treat fibres as dilute and dynamically passive. Here, we develop a self-consistent kinetic description for fragmentation in the opposite limit of sufficiently dense fibre suspensions that back-react on the turbulence. Starting from an Euler–Bernoulli model in the quasi-static regime, we show that collective resistive-force drag defines a cut-off span $L_c$ inside the inertial range, marking the crossover from a Kolmogorov-like inertial range with an effective linear friction (for scales above $L_c$) to a fibre-dominated frictional regime with $E(k)\propto k^{-3}$ for scales below $L_c$. In this regime, the curvature amplitude of an almost straight fibre rises steeply with its span, so that the supercritical breaking hazard loses its dependence on length, and approaches a plateau controlled solely by the fibre loading. Embedding this hazard into a scale-invariant binary fragmentation equation, adopted here as a minimal quasi-static closure, we obtain, without adjustable parameters, a corresponding asymptotic steady length spectrum $n(L)\propto L^{-2}$ in the fibre-dominated regime, in contrast with the shallower $L^{-4/3}$ law predicted in the Kolmogorov-like regime. The crossover between these two spectra is set self-consistently by the same length density that fixes $L_c$ and the onset of the frictional modification of the cascade, thus linking turbulence modification, curvature statistics and kinetic fragmentation in a single coupled framework.

Secondary droplets produced by interactions between falling fluid drops and a liquid pool play a significant role in engineering applications and geophysical processes in nature. This study uses direct numerical simulations to investigate the dynamics of secondary droplets generated by raindrop–liquid pool interactions. The raindrop parameters feature realistic speed ${\sim} 7\ \mathrm{m \ s}^{-1}$, effective diameters ${\sim} 1$–$4\ \mathrm{mm}$, and surface tension values ranging from $25\,\%$ to twice the typical air–water interface value. The numerical configurations include both a single raindrop and two raindrops separated by distances between two and four times the raindrop diameter. The secondary droplet size distribution $N_d$ is found to scale with the droplet radius $r_s$ as $N_d(r_s)\propto r_s^{-5/2}$, with additional dependencies on surface tension and raindrop diameter. When normalised according to this new scaling law, the droplet size distribution obtained from simulations with different parameter values collapses onto a single curve. Analysis of the impact morphology reveals distinct stages of raindrop interactions, and identifies the formation and break-up of a central liquid film. Spatial and temporal analyses of the secondary droplets show that raindrop interaction can influence both the percentage of droplets captured by the cavity and the duration over which they re-merge with the pool. These behaviours arise from the combined effects of differences in the birth times of secondary droplets of various sizes and aerodynamic forcing associated with the cavity airflow.

In Hass & Lele (2026 J. Fluid Mech. 1037, A20), we looked at the statistical structure (self-similarity) and entrainment rate of a shear-free turbulent layer bounded by a net zero momentum source layer above and a stable density interface below and found agreement with oscillating grid turbulence experiments. However, interfacial waves supported by the stable stratification make quantification of variances in the near-interfacial region ambiguous. In the present paper we propose a decomposition of the fluctuating fields based on eigenmodes of the linearized equations. Evidence that a linear model is appropriate is provided by the wavenumber–frequency spectra for low Froude number simulations. Using the decomposition, a detailed analysis of the wave–turbulence statistics, including budgets of kinetic and potential energy, is conducted. An energy conversion pathway is uncovered consistent with the internal wave-breaking entrainment mechanism proposed by Hannoun & List (1988 J. Fluid Mech. 189, 211–234), Fernando & Hunt (1997 J. Fluid Mech. 347, 197–234), and McGrath et al. (1997 J. Fluid Mech. 347, 235–261). A locally defined mixing or flux coefficient $\varGamma =\chi /\epsilon$ scales with the (local) turbulent Froude number in the same way as in homogeneous turbulence simulations where transport is identically zero. Finally, we connect our spatially inhomogeneous simulations to the strongly stratified turbulence theory demonstrating that the parameter regime required by the theory is, in principle, accessible for sufficiently energetic forcing, but is not realized in our simulations.

In the ocean, internal wave beams are generated by the barotropic tide passing over steep topography. Observations suggest that these wave beams are dissipated as they propagate into the upper ocean. This differs from idealised numerical simulations of wave propagation, in which the wave beams often reflect off the surface. Inspired by this discrepancy, we hypothesise that wave beams in the ocean can be absorbed by stratified turbulence, inhibiting their reflection. In this idealised study we perform three-dimensional direct numerical simulations, where stratified turbulence is generated in the top half of the domain and forcing is applied near the bottom, creating upward-propagating waves with fixed frequency that travel toward the turbulent layer. We apply dynamic mode decomposition to extract the structure of disturbances at the wave-forcing frequency and then use Hilbert transforms to separate upward from downward travelling disturbances. Thus we diagnose wave reflection, should it occur. We compare results for cases in which only turbulence is present with cases in which waves are forced with different frequencies and amplitudes. Although wave reflection is observed for strongly non-hydrostatic waves, sufficiently low-frequency waves are absorbed by the turbulent layer.

The generation of renewable power from ocean waves at a commercial scale requires arrays of wave energy converters (WECs) to be designed for efficient power capture over broad frequency bands and a wide range of incident wave directions. Here, array designs are developed in a three-dimensional model based on linear potential-flow theory, by arranging heaving-buoy-type WECs into rows, and spatially grading the resonant properties of successive rows via linear spring–damper power take-off (PTO) mechanisms. The grading leverages theories for rainbow reflection and rainbow absorption, and involves analyses of Bloch waves and pole–zero pairs in complex frequency space to inform the array layout and choice of PTO parameters. The spacing within an array alters the resonant properties of the rows and the behaviour of higher-order passbands, which has implications for the efficiency of an array. Strategies are developed to grade arrays composed of rows with infinitely many WECs, for efficient, broadband absorption over a broad target frequency range, and average absorptions exceeding $90\,\%$ are achieved over a wide range of wave directions. The strategies are applied to finite arrays, which demonstrate broadband absorption in directionally spread irregular sea states, and are shown to outperform the absorption of uniform arrays across a wide range of peak periods.

We study the onset of two-dimensional secondary shear instability (SSI) in the braid regions connecting primary Kelvin–Helmholtz billows in stratified shear flows. While strain induced by the billows stabilises the braids, it also compresses their tilted isopycnals, enhancing baroclinic shear that enables rapid perturbation growth. By modifying the classical analysis of Corcos & Sherman (1976 J. Fluid Mech. 73, 241–264) in braid-aligned coordinates and adding an instability criterion based on the ratio of strain rate to shear, we develop an inviscid, time-dependent model for the braid evolution and the onset of SSI. We show that the criterion for instability can be achieved significantly earlier than the saturation of the primary billow at sufficiently high initial Richardson number $ \textit{Ri}$, because increased stratification slows the rate of billow growth relative to the rate of baroclinic shear production. Two-dimensional direct numerical simulations up to Reynolds numbers $ \textit{Re}=10^7$ quantify the role of viscosity. At high $ \textit{Re}$, we find that SSI indeed develops early in the braid, as predicted by the inviscid model, while the primary billow is still growing and before viscosity slows braid thinning. These results provide a mechanistic explanation for field observations of braid-dominated mixing and suggest that, at geophysically relevant $ \textit{Ri}$ and $ \textit{Re}$, SSI can control the three-dimensional transition and ensuing mixing by preceding and pre-empting both vortex pairing instabilities and secondary convective instabilities in the billow core.

Aerodynamic interactions between closely spaced lifting bodies are fundamental to many natural and engineered systems, including flapping insect wings, multi-rotor aerial vehicles and vertical-axis wind turbines. During operation, these systems undergo periodic counter-translating interactions (cross-over) in which adjacent wings or blades translate past each other in close proximity, producing strong unsteady coupling and transient load excursions. This study investigates the canonical form of this problem: two counter-translating lifting bodies in a hydrodynamic towing facility to isolate the fluid mechanics of a single cross-over event. Time-resolved load and flow-field measurements reveal that the unsteady forces arise from a combination of induced effects and transient pressure fields imposed by the mutual interaction between the wings. The resulting disturbance behaves as a self-induced gust, generated collectively by the coupled system rather than externally imposed. Comparison with linear unsteady aerodynamic models demonstrates that the measured response most closely follows Miles’s travelling-gust formulation. This connection establishes a continuum between externally imposed and self-induced gust interactions. A parametric analysis further identifies the primary non-dimensional groups governing the phenomenon, Reynolds number, aerofoil thickness, bound circulation and inter-wing spacing, and quantifies their influence on the transient lift response. Together, these results reveal a new regime of multi-body unsteady aerodynamics, where the interplay between circulatory and non-circulatory effects govern the short-time dynamics of interacting lifting surfaces.