We study the transport and deposition of inhaled aerosols in a mid-generation, mucus-lined lung airway, with the aim of understanding if and how airborne particles can avoid the mucus and deposit on the airway wall – an outcome that is harmful in case of allergens and pathogens, but beneficial in case of aerosolised drugs. We adopt the weighted-residual integral boundary-layer model of Dietze and Ruyer-Quil (J. Fluid Mech. 762, 2015, 68–109, to describe the dynamics of the mucus–air interface, as well as the flow in both phases. The transport of mucus induced by wall-attached cilia is also considered, via a coarse-grained boundary condition at the base of the mucus. We show that the capillary-driven Rayleigh–Plateau instability plays an important role in particle deposition by drawing the mucus into large annular humps and leaving substantial areas of the wall exposed to particles. We find, counter-intuitively, that these mucus-depleted zones enlarge on increasing the mucus volume fraction. Our simulations are eased by the fact that the effects of cilia and air turn out to be rather simple: the long-term interface profile is slowly translated by cilia and is unaffected by the laminar airflow. The streamlines of the airflow, though, are strongly modified by the non-uniform mucus film, and this has important implications for aerosol entrapment. Particles spanning a range of sizes (0.1–50 microns) are modelled using the Maxey–Riley equation, augmented with Brownian forces. We find a non-monotonic dependence of deposition on size. Small particles diffuse across streamlines due to Brownian motion, while large particles are thrown off streamlines by inertial forces – particularly when air flows past mucus humps. Intermediate-sized particles are tracer-like and deposit the least. Remarkably, increasing the mucus volume need not increase entrapment: the effect depends on particle size, because more mucus produces not only deeper humps that intercept inertial particles, but also larger depleted zones that enable diffusive particles to deposit on the wall.

The interface shape near a moving contact line is described by the Cox–Voinov theory, which contains a constant term that is not trivially obtained. In this work, an approximate expression of this term in explicit form is derived under the condition of a Navier slip. Introducing the approximation of a local slippery wedge flow, we first propose a novel form of the generalised lubrication equation. A matched asymptotic analysis of this equation yields the Cox–Voinov relation with the constant term expressed in elementary functions. For various viscosity ratios and contact angles, the theoretical predictions are rigorously validated against full numerical solutions of the Stokes equations and available asymptotic results.

We experimentally investigate the rotational dynamics of neutrally buoyant flat bodies of revolution (spheroids, disks and rings with different cross-sectional shapes) in shear flows. In the Stokes regime, the axis of revolution of these rigid particles moves in one of a family of closed periodic Jeffery orbits. Inertia is able to lift the orbit degeneracy and induces drift among several rotations towards limiting stable orbits. Furthermore, permanent alignment can be achieved for disks and rings with triangular cross-sectional shapes, provided the inertia is sufficiently high. The bifurcations between the different dynamics are compared with those predicted by small-inertia asymptotic theories and numerical simulations.

Compressible jets impinging on a perpendicular surface can produce high-intensity, discrete-frequency tones. The character of these tones is a function of nozzle shape, jet Mach number, impingement-plate geometry, and the distance between nozzle and plate. Though it has long been recognised that these tones are associated with a resonance cycle, the exact mechanism by which they are generated has remained a topic of some debate. In this work, we present evidence for a number of distinct tone-generation mechanisms, reconciling some of the different findings of prior authors. We demonstrate that the upstream-propagating waves that close resonance can be confined within the jet, or external to it. These waves can be either weak and relatively linear, or strong and nonlinear from their inception. The waves can undergo coalescence or merging, and in some configurations, pairs of waves rather than singletons appear. We discuss both historical and new evidence for multiple distinct processes by which upstream-propagating waves are produced: direct vortex sound, shock leakage, wall-jet-boundary fluctuations, and wall-jet shocklets. We link these various mechanisms to the disparate collection of upstream-propagating waves observed in the data. We also demonstrate that multiple mechanisms can be provoked by a single vortex, providing an explanation as to why sometimes pairs of waves or merging waves are observed. Through this body of work, we demonstrate that rather than being in opposition, the various pieces of past research on this topic were simply identifying different mechanisms that can support resonance.

In rotating fluids, the viscous smoothing of inviscid singular inertial waves leads to the formation of internal shear layers. In previous works, we analysed the internal shear layers excited by a viscous forcing (longitudinal libration) in a spherical shell geometry (He et al., 2022 J. Fluid Mech. 939, A3; He et al., 2023 J. Fluid Mech. 974, A3). We now consider the stronger inviscid forcing corresponding to the vertical oscillation of the inner boundary. We limit our analysis to two-dimensional geometries but examine three different configurations: freely propagating wave beams in an unbounded domain and two wave patterns (a periodic orbit and an attractor) in a cylindrical shell geometry. The asymptotic structures of the internal shear layers are assumed to follow the similarity solution of Moore & Saffman (1969 Phil. Trans. R. Soc. Lond. A, 264, 597–634) in the small viscous limit. The two undefined parameters of the similarity solution (singularity strength and amplitude) are derived by asymptotically matching the similarity solution with the inviscid solution. For each case, the derivation of the latter is achieved either through separation of variables combined with analytical continuation or the method of characteristics. Global inviscid solutions, when obtained, closely match numerical solutions for small Ekman numbers far from the critical lines, while viscous asymptotic solutions show excellent performance near those lines. The amplitude scalings of the internal shear layers excited by an inviscid forcing are found to be divergent as the Ekman number $E$ decreases, specifically $O(E^{-1/6})$ for the critical-point singularity and $O(E^{-1/3})$ for attractors, in contrast to the convergent scalings found for a viscous forcing.

The physical fidelity of turbulence models can benefit from a partial resolution of fluctuations, but doing so often comes with an increase in computational cost. To explore this trade-off in the context of wall-bounded flows, this paper introduces a framework for turbulence-resolving integral simulations (TRIS) with the goal of efficiently resolving the largest motions using a two-dimensional, three-component representation of the flow defined by instantaneous wall-normal integrals of velocity and pressure. Self-sustaining turbulence with qualitatively realistic large-scale structures is demonstrated for TRIS on an open-channel (half-channel) flow configuration using moment-of-momentum integral equations derived from Navier–Stokes with relatively simple closure approximations. Evidence from direct numerical simulations (DNS) suggests that TRIS can theoretically resolve $35\,\%{-}40\,\%$ of the turbulent skin friction enhancement for friction Reynolds numbers between $180$ and $5200$, without a noticeable decrease or increase as a function of Reynolds number. The current implementation of TRIS can match this resolution while simulating one flow through time in ${\sim}1$ minute on a single processor, even for very large Reynolds numbers. The framework facilitates a detailed apples-to-apples comparison of predicted statistics against data from DNS. Comparisons at friction Reynolds numbers of $395$ and $590$ show that TRIS generates a relatively accurate representation of the flow, while highlighting discrepancies that demonstrate a need for improving the closure models. The present results for open-channel flow represent a proof of concept for TRIS as a new approach for wall-bounded turbulence modelling, motivating extension to more general flow configurations such as boundary layers on immersed objects.

Experiments are conducted over smooth and rough walls to explore the influence of pressure-gradient histories on skin friction and mean flow of turbulent boundary layers. Different pressure-gradient histories are imposed on the boundary layer through an aerofoil mounted in the free stream. Hot-wire measurements are taken at different free-stream velocities downstream of the aerofoil where the flow has locally recovered to zero pressure gradient but retains the history effects. Direct skin friction measurements are also made using oil film interferometry for smooth walls and a floating-element drag balance for rough walls. The friction Reynolds number, $Re_\tau$, varies between $3000$ and $27\,000$, depending both on the surface conditions and the free-stream velocity ensuring sufficient scale separation. Results align with previous findings, showing that adverse pressure gradients just upstream of the measurement location increase wake strength and reduce the local skin friction while favourable pressure gradients suppress the wake and increase skin friction. The roughness length scale, $y_0$, remains constant across different pressure-gradient histories for rough wall boundary layers. Inspired by previous works, a new correlation is proposed to infer skin friction based on the mean flow. The difference in skin friction by matching the turbulence profiles and flow structure between an arbitrary pressure-gradient history and zero pressure-gradient condition can be predicted using only the local wake strength parameter ($\Pi$), and the variations in wake strength for different histories are related to a weighted integral of the pressure-gradient history normalised by local quantities. This allows us to develop a general correlation that can be used to infer skin friction for turbulent boundary layers experiencing arbitrary pressure-gradient histories.

Recent studies focusing on the response of turbulent boundary layers (TBLs) to a step change in roughness have provided insight into the scaling and characterisation of TBLs and the development of the internal layer. Although various step-change combinations have been investigated, ranging from smooth-to-rough to rough-to-smooth, the minimum required roughness fetch length over which the TBL returns to its homogeneously rough behaviour remains unclear. Moreover, the relationship between a finite- and infinite-fetch roughness function (and the equivalent sand-grain roughness) is also unknown. In this study, we determine the minimum ‘equilibrium fetch length’ for a TBL developing over a smooth-to-rough step change as well as the expected error in local skin friction if the fetch length is under this minimum threshold. An experimental study is carried out where the flow is initially developed over a smooth wall, and then a step change is introduced using patches of P24 sandpaper. Twelve roughness fetch lengths are tested in this study, systematically increasing from $L = 1\delta _2$ up to $L = 39\delta _2$ (where L is the roughness fetch length and $\delta _2$ is the TBL thickness of the longest fetch case), measured over a range of Reynolds numbers ($4\times 10^3 \leqslant Re_\tau \leqslant 2\times 10^4$). Results show that the minimum fetch length needed to achieve full equilibrium recovery is around $20\delta _2$. Furthermore, we observe that the local friction coefficient, $C_{\! f}$, recovers to within 10 % of its recovered value for fetch lengths $\geqslant 10\delta _2$. This information allows us to incorporate the effects of roughness fetch length on the skin friction and roughness function.

We study the near-wall behaviour of pressure spectra and associated variances in canonical wall-bounded flows, with a special focus on pipe flow. Analysis of the pressure spectra reveals the universality of small and large scales, supporting the establishment of $ k^{-1}$ spectral layers as predicted by fundamental physical theories. However, this universality does not extend to the velocity spectra (Pirozzoli, J. Fluid Mech., vol. 989, 2024, A5), which show a lack of universality at the large-scale end and systematic deviations from the $ k^{-1}$ behaviour. We attribute this fundamental difference to the limited influence of direct viscous effects on pressure, with implied large differences in the near-wall behaviour. Consequently, the inner-scaled pressure variances continue to increase logarithmically with the friction Reynolds number as we also infer from a refined version of the attached-eddy model, while the growth of the velocity variance tends to saturate. Extrapolated distributions of the pressure variance at extremely high Reynolds numbers are inferred.

Geophysical flows are typically composed of wave and mean motions with a wide range of overlapping temporal scales, making separation between the two types of motion in wave-resolving numerical simulations challenging. Lagrangian filtering – whereby a temporal filter is applied in the frame of the flow – is an effective way to overcome this challenge, allowing clean separation of waves from mean flow based on frequency separation in a Lagrangian frame. Previous implementations of Lagrangian filtering have used particle tracking approaches, which are subject to large memory requirements or difficulties with particle clustering. Kafiabad & Vanneste (2023, Computing Lagrangian means, J. Fluid Mech., vol. 960, A36) recently proposed a novel method for finding Lagrangian means without particle tracking by solving a set of partial differential equations alongside the governing equations of the flow. In this work, we adapt the approach of Kafiabad & Vanneste to develop a flexible, on-the-fly, partial differential equation-based method for Lagrangian filtering using arbitrary convolutional filters. We present several different wave–mean decompositions, demonstrating that our Lagrangian methods are capable of recovering a clean wave field from a nonlinear simulation of geostrophic turbulence interacting with Poincaré waves.

Evolution of solitary waves and an undular bore intruding through an abrupt transition from a wide basin into a narrow channel with opposing current is investigated. The laboratory experiments are performed in a wave tank that is crafted to achieve a steady and symmetrical shallow-water jet in the basin. The channel has a breadth comparable to the wave lengths, and the flow has Froude number approximately 0.1. The opposing current amplifies and slows the incoming waves on the jet in the basin, but the propagation speed is faster than the local Doppler effect of the current due to the influence of the wave propagating in the flank of the jet. At the channel mouth, the wave amplitude is enhanced due to the waveform altered by the current in the basin, although the amplification in the upstream channel is similar with and without the current. The longer incident waves have greater amplification into the channel. The leading wave of the undular bore is impacted by the opposing flow and transition similarly to the solitary waves. In contrast, the subsequent waves of the undular bore have a complex phase interference on the jet that causes disconnection in the lateral wave formation across the breadth of the jet. At the transition, the subsequent waves exhibit greater amplification than the leading one due to accumulated wave energy at the channel mouth. The intrusion of the undular bore against the current further enhances a rise in mean water level in the channel.

Ocean turbulence at meso- and submesocales affects the propagation of surface waves through refraction and scattering, inducing spatial modulations in significant wave height (SWH). We develop a theoretical framework that relates these modulations to the current that induces them. We exploit the asymptotic smallness of the ratio of typical current speed to wave group speed to derive a linear map – the U2H map – between surface current velocity and SWH anomaly. The U2H map is a convolution, non-local in space, expressible as a product in Fourier space by a factor independent of the magnitude of the wavenumber vector. Analytic expressions of the U2H map show how the SWH responds differently to the vortical and divergent parts of the current, and how the anisotropy of the wave spectrum is key to large current-induced SWH anomalies. We implement the U2H map numerically and test its predictions against WAVEWATCH III numerical simulations for both idealised and realistic current configurations.

Understanding the mechanism of hydrodynamic cloud cavitation is crucial to reducing noise, vibration and wear. Recent studies have clarified the physics of two distinct formation mechanisms of cloud cavitation. Ganesh et al. (J. Fluid Mech., vol. 802, 2016, pp. 37–78) identified the propagation of bubbly shockwaves as a cloud detachment mechanism. Pelz et al. (J. Fluid Mech., vol. 817, 2017, pp. 439–454) explained the influence of Reynolds number and cavitation number on asymptotic growth of the cavity sheet and its periodic shedding caused by re-entrant flow. In this paper the two mechanisms are set in relation to each other. For this, we show firstly that the transition from re-entrant flow to shockwave-driven cloud cavitation is given by a kinematic condition, namely the asymptotic sheet length equates to the chord length, $\hat {a}=L$. For $\hat {a}>L$ shockwave-driven cloud cavitation dominates. For $\hat {a}< L$ re-entrant flow-driven cloud cavitation dominates. As the cavitation number decreases, the closure region of the cavity sheet reaches the trailing edge of the hydrofoil, identifying the trailing edge as a trigger for condensation shockwaves, particularly as re-entrant flow-driven cavitation diminishes. Since the sheet length is an implicit function of the cavitation number, the kinematic condition $\hat {a}/L=1$ results in a critical cavitation number ${\sigma _\mathrm {II,III}}$ that is calculated analytically and validated by experiments. Secondly, we derive the relationship between the Strouhal number and the asymptotic sheet length for re-entrant flow-driven cloud cavitation. The model presented here is thoroughly validated by experiments.

A liquid drop impacting a rigid substrate undergoes deformation and spreading due to normal reaction forces, which are counteracted by surface tension. On a non-wetting substrate, the drop subsequently retracts and takes off. Our recent work (Zhang et al., Phys. Rev. Lett., vol. 129, 2022, 104501) revealed two peaks in the temporal evolution of the normal force $F(t)$ – one at impact and another at jump-off. The second peak coincides with a Worthington jet formation, which vanishes at high viscosities due to increased viscous dissipation affecting flow focusing. In this article, using experiments, direct numerical simulations and scaling arguments, we characterize both the peak amplitude $F_1$ at impact and the one at takeoff ($F_2$) and elucidate their dependency on the control parameters: the Weber number $We$ (dimensionless impact kinetic energy) and the Ohnesorge number $Oh$ (dimensionless viscosity). The first peak amplitude $F_1$ and the time $t_1$ to reach it depend on inertial time scales for low viscosity liquids, remaining nearly constant for viscosities up to 100 times that of water. For high viscosity liquids, we balance the rate of change in kinetic energy with viscous dissipation to obtain new scaling laws: $F_1/F_\rho \sim \sqrt {Oh}$ and $t_1/\tau _\rho \sim 1/\sqrt {Oh}$, where $F_\rho$ and $\tau _\rho$ are the inertial force and time scales, respectively, which are consistent with our data. The time $t_2$ at which the amplitude $F_2$ appears is set by the inertiocapillary time scale $\tau _\gamma$, independent of both the viscosity and the impact velocity of the drop. However, these properties dictate the magnitude of this amplitude.

Surface roughness significantly modifies the liquid film thickness entrained when dip coating a solid surface, particularly at low coating velocity. Using a homogenization approach, we present a predictive model for determining the liquid film thickness coated on a rough plate. A homogenized boundary condition at an equivalent flat surface is used to model the rough boundary, accounting for both flow through the rough texture layer, through an interface permeability term, and slip at the equivalent surface. While the slip term accounts for tangential velocity induced by viscous shear stress, accurately predicting the film thickness requires the interface permeability term to account for additional tangential flow driven by pressure gradients along the interface. We find that a greater degree of slip and interface permeability signifies less viscous stress that would promote deposition, thus reducing the amount of free film coated above the textures. The model is found to be in good agreement with experimental measurements, and requires no fitting parameters. Furthermore, our model may be applied to arbitrary periodic roughness patterns, opening the door to flexible characterization of surfaces found in natural and industrial coating processes.

We introduce a novel recursive procedure to a neural-network-based subgrid-scale (NN-based SGS) model for large eddy simulation (LES) of high-Reynolds-number turbulent flow. This process is designed to allow an SGS model to be applicable to a hierarchy of different grid sizes without requiring expensive filtered direct numerical simulation (DNS) data: (1) train an NN-based SGS model with filtered DNS data at a low Reynolds number; (2) apply the trained SGS model to LES at a higher Reynolds number; (3) update this SGS model with training data augmented with filtered LES (fLES) data, accommodating coarser filter size; (4) apply the updated NN to LES at a further higher Reynolds number; (5) go back to Step (3) until a target (very coarse) filter size divided by the Kolmogorov length scale is reached. We also construct an NN-based SGS model using a dual NN architecture whose outputs are the SGS normal stresses for one NN and the SGS shear stresses for the other NN. The input is composed of the velocity gradient tensor and grid size. Furthermore, for the application of an NN-based SGS model trained with one flow to another flow, we modify the NN by eliminating bias and introducing a leaky rectified linear unit function as an activation function. The present recursive SGS model is applied to forced homogeneous isotropic turbulence (FHIT) and successfully predicts FHIT at high Reynolds numbers. The present model trained from FHIT is also applied to decaying homogeneous isotropic turbulence and shows an excellent prediction performance.

We provide scaling relations for the Nusselt number $Nu$ and the friction coefficient $C_{S}$ in sheared Rayleigh–Bénard convection, i.e. in Rayleigh–Bénard flow with Couette- or Poiseuille-type shear forcing, by extending the Grossmann & Lohse (J. Fluid Mech., vol. 407, 2000, pp. 27–56, Phys. Rev. Lett., vol. 86, 2001, pp. 3316–3319, Phys. Rev. E, vol. 66, 2002, 016305, Phys. Fluids, vol. 16, 2004, pp. 4462–4472) theory to sheared thermal convection. The control parameters for these systems are the Rayleigh number $Ra$, the Prandtl number $Pr$ and the Reynolds number $Re_S$ that characterises the strength of the imposed shear. By direct numerical simulations and theoretical considerations, we show that, in turbulent Rayleigh–Bénard convection, the friction coefficients associated with the applied shear and the shear generated by the large-scale convection rolls are both well described by Prandtl's (Ergeb. Aerodyn. Vers. Gött., vol. 4, 1932, pp. 18–29) logarithmic friction law, suggesting some kind of universality between purely shear-driven flows and thermal convection. These scaling relations hold well for $10^6 \leq Ra \leq 10^8$, $0.5 \leq Pr \leq 5.0$, and $0 \leq Re_S \leq 10^4$.

This study explores precession-driven flows in a non-axisymmetric ellipsoid spinning around its medium axis. Using an experimental approach, we focus on two aspects of the flow: the base flow of uniform vorticity and the development of fluid instabilities. In contrast to a preceding paper (J. Fluid. Mech., vol. 932, 2022, A24), where the ellipsoid rotated around its shortest axis, we do not observe bi-stability or hysteresis of the base flow, but a continuous transition from small to large differential rotation and tilt of the fluid rotation axis. We then use the model developed by Noir & Cébron (J. Fluid. Mech., vol. 737, 2013, pp. 412–439) to numerically determine regions in the parameter space of axial and equatorial deformations for which bi-stability may exist. Concerning fluid instabilities, we use three independent observations to track the onset of both boundary layer and parametric instabilities. Our results clearly show the presence of a parametric instability, yet the exact nature of the underlying mechanism (conical shear layer instability, shear instability and elliptical instability) is not unambiguously identified. A coexisting boundary layer instability, although unlikely, cannot be ruled out based on our experimental data. To make further progress on this topic, a new generation of experiments at significantly lower Ekman numbers (ratio of rotation and viscous time scales) is clearly needed.

Turbulent shear flows driven by a combination of a pressure gradient and buoyancy forcing are investigated using direct numerical simulations. Specifically, we consider the set-up of a differentially heated vertical channel subject to a Poiseuille-like horizontal pressure gradient. We explore the response of the system to its three control parameters: the Grashof number $Gr$, the Prandtl number $Pr$, and the Reynolds number $Re$ of the pressure-driven flow. From these input parameters, the relative strength of buoyancy driving to the pressure gradient can be quantified by the Richardson number $Ri=Gr/Re^2$. We compare the response of the mixed vertical convection configuration to that of mixed Rayleigh–Bénard convection, and find a nearly identical behaviour, including an increase in wall friction at higher $Gr$, and a drop in the heat flux relative to natural convection for $Ri=O(1)$. This closely matched response is despite vastly different flow structures in the systems. No large-scale organisation is visible in visualisations of mixed vertical convection – an observation that is confirmed quantitatively by spectral analysis. This analysis, combined with a statistical description of the wall heat flux, highlights how moderate shear suppresses the growth of small-scale plumes and reduces the likelihood of extreme events in the local wall heat flux. Vice versa, starting from a pure shear flow, the addition of thermal driving enhances the drag due to the emission of thermal plumes.

We present evidence for layered anisotropic stratified turbulence (LAST) and mixing produced in a freely evolving uniformly stratified shear layer where the direction of shear is orthogonal to gravity. As originally reported by Basak & Sarkar (J. Fluid. Mech., vol. 568, 2006, pp. 19–54), such a flow develops a rich three-dimensional structure in the form of interlocking columnar vortices formed by horizontal shear instability that remain coherent at large scales due to the stabilising vertical stratification. Here, we modify the initial velocity field by introducing additional small-amplitude vertical perturbations designed to be representative of pre-existing horizontal layers often observed in strongly stratified ocean environments. This reveals a novel finite amplitude, non-normal mode growth mechanism through which the vertical shear between layers may be rapidly amplified by its interaction with the horizontal shear layer prior to the growth of shear instability, leading to a rapid turbulent transition instigated by the subsequent interaction of the layers with the emerging columnar vortices. Through a consideration of relevant flow statistics and associated dimensionless parameters, we demonstrate that turbulence can enter the LAST regime, thereby indicating a generic mechanism leading to the transient development of regions of strongly stratified turbulence in the ocean. We discuss the properties of mixing and the parameterisation of mixing efficiency in terms of the relationship between turbulent length scales in the flow, in particular highlighting links to models based on the classical vertical shear instability paradigm typically associated with more weakly stratified flows that produce isolated turbulent ‘patches’.