A better understanding of the fluid dynamics of disease transmission by disintegrated respiratory droplets has been the focus of great attention since the recent outbreak of COVID-19. In particular, human respiratory activities such as coughing, sneezing and even talking and eating expel a large amount of pathogen-laden droplets. Particularly, during eating or drinking, the physical properties of saliva can be changed. In this study, we investigate the atomization morphology of expelled artificial saliva mixtures from the perspective of varying fluid physical properties, specifically surface tension and dynamic viscosity. Using high-speed shadowgraph experiments on artificial saliva, we visualize and analyse the disintegration of saliva liquid sheets into ligaments and droplets. We find that the viscosity and surface tension affect the droplet size formed from expelled saliva and follow scaling laws that have been previously observed and predicted for constant shear viscosity. We conclude that the changes in physical properties of saliva induced by eating and drinking tend to favour the formation of smaller droplets during sneezing or coughing, which could drive the airborne transmission pathway of pathogens. Furthermore, we derive a theoretical model based on scaling arguments that shows the breakup time of ligaments produced from the artificial saliva mixtures is dependent on the capillary number.

We conduct three-dimensional direct numerical simulations to investigate the mixing, entrainment and energy budgets of gravity currents emerging from two-layer stratified locks. Depending on the density and layer thickness ratios, we find that either the upper layer or lower layer fluid can propagate faster, and that the density structure of the overall gravity current can range from strongly stratified to near-complete mixing. We furthermore observe that intermediate values of the density ratio can maximise mixing between the gravity current layers. Based on the vorticity budget, we propose a theoretical model for predicting the overall gravity current height, along with the front velocity of the two layers, for situations in which the lower layer moves faster than the upper layer. The model identifies the role of the height and thickness ratios in determining the velocity structure of the current, and it clarifies the dynamics of the ambient counter-current. A detailed analysis of the energy budget quantifies the conversion of potential into kinetic energy as a function of the governing parameters, along with the energy transfer between the different layers of the gravity current and the ambient fluid. Depending on the values of the density and layer thickness ratios, we find that the lower lock layer can gain or lose energy, whereas the upper layer always loses energy.

We study the dynamics of capillary waves at the interface of a two-layer stratified turbulent channel flow. We use a combined pseudo-spectral/phase field method to solve for the turbulent flow in the two liquid layers and to track the dynamics of the liquid–liquid interface. The two liquid layers have same thickness and same density, but different viscosity. We vary the viscosity of the upper layer (two different values) to mimic a stratified oil–water flow. This allows us to study the interplay between inertial, viscous and surface tension forces in the absence of gravity. In the present set-up, waves are naturally forced by turbulence over a broad range of scales, from the larger scales, whose size is of order of the system scale, down to the smaller dissipative scales. After an initial transient, we observe the emergence of a stationary capillary wave regime, which we study by means of temporal and spatial spectra. The computed frequency and wavenumber power spectra of wave elevation are in line with previous experimental findings and can be explained in the frame of the weak wave turbulence theory. Finally, we show that the dispersion relation, which gives the frequency ($\omega$) as a function of the wavenumber ($k$), is in good agreement with the well-established theoretical prediction, $\omega (k) \sim k^{3/2}$.

This paper addresses the viscous flow developing about an array of equally spaced identical circular cylinders aligned with an incompressible fluid stream whose velocity oscillates periodically in time. The focus of the analysis is on harmonically oscillating flows with stroke lengths that are comparable to or smaller than the cylinder radius, such that the flow remains two-dimensional, time-periodic and symmetric with respect to the centreline. Specific consideration is given to the limit of asymptotically small stroke lengths, in which the flow is harmonic at leading order, with the first-order corrections exhibiting a steady-streaming component, which is computed here along with the accompanying Stokes drift. As in the familiar case of oscillating flow over a single cylinder, for small stroke lengths, the associated time-averaged Lagrangian velocity field, given by the sum of the steady-streaming and Stokes-drift components, displays recirculating vortices, which are quantified for different values of the two relevant controlling parameters, namely, the Womersley number and the ratio of the inter-cylinder distance to the cylinder radius. Comparisons with results of direct numerical simulations indicate that the description of the Lagrangian mean flow for infinitesimally small values of the stroke length remains reasonably accurate even when the stroke length is comparable to the cylinder radius. The numerical integrations are also used to quantify the streamwise flow rate induced by the presence of the cylinder array in cases where the periodic surrounding motion is driven by an anharmonic pressure gradient, a problem of interest in connection with the oscillating flow of cerebrospinal fluid around the nerve roots located along the spinal canal.

This paper presents an experimental study on how both variable solid volume fractions and aspect ratios (length/width) of a centre-channel rectangular porous patch under aligned configuration of rigid and emergent stems impact the flow behaviour and wake structure. This study forms an essential extension to the existing fundamentals and knowledge on this topic. Through rigorous experimental tests by velocity measurement and dye visualization, the aspect ratio, rarely addressed before, is confirmed to play a critical role. Vortex street, unable to be triggered under a low solid volume fraction, however, can be generated by elongating the patch (increasing the aspect ratio). The key reason is that patch elongation promotes the generation of the wake vortex street by producing a relatively high transverse velocity gradient in the wake region. Meanwhile, Kelvin–Helmholtz vortex streets are triggered along the two patch lateral edges, re-increasing the in-patch velocity and imposing contributions to the wake vortex streets generation. By scaling the characteristic velocity (at the wake vortex initiation position and patch trailing edge) and solid volume fraction with the patch aspect ratio, three non-dimensional threshold maps can be established to express the combined effects of the solid volume fraction and aspect ratio on the initiation of the wake vortex street. They could be alternatively used for theoretical analysis and implementation on wake formation and structure subject to parameter availability.

Extensive studies of the hydraulics of pipes have focused on limiting cases, such as fully-developed laminar or turbulent flow through long conduits and the accelerating flow through an orifice, for which there exist laws relating pressure drop and flow rate. We carry out experiments on smooth, circular pipes for dimensions and flow rates that interrogate intermediate conditions between the well-studied limits. Organizing this information in terms of dimensionless friction factor, Reynolds number and pipe aspect ratio yields a surface $f_D(Re,\alpha )$ that is shown to match the three laws associated with developed laminar, developed turbulent, and orifice flows. While each law fails outside its applicable range of $(Re,\alpha )$, we present a hybrid theoretical–empirical model that includes inlet, development and transition effects, and that proves accurate to approximately 10 % over wide ranges of $Re$ and $\alpha$. We also present simple formulas for the boundaries between the three hydraulic regimes, which intersect at a triple point. Measurements show that sipping through a straw is an everyday example of such intermediate conditions not accounted for by existing laws but described accurately by our model. More generally, our findings provide formulas for predicting frictional resistance for intermediate-$Re$ flows through finite-length pipes.

A model based on a convolutional neural network (CNN) is designed to reconstruct the three-dimensional turbulent flows beneath a free surface using surface measurements, including the surface elevation and surface velocity. Trained on datasets obtained from the direct numerical simulation of turbulent open-channel flows with a deformable free surface, the proposed model can accurately reconstruct the near-surface flow field and capture the characteristic large-scale flow structures away from the surface. The reconstruction performance of the model, measured by metrics such as the normalised mean squared reconstruction errors and scale-specific errors, is considerably better than that of the traditional linear stochastic estimation (LSE) method. We further analyse the saliency maps of the CNN model and the kernels of the LSE model and obtain insights into how the two models utilise surface features to reconstruct subsurface flows. The importance of different surface variables is analysed based on the saliency map of the CNN, which reveals knowledge about the surface–subsurface relations. The CNN is also shown to have a good generalisation capability with respect to the Froude number if a model trained for a flow with a high Froude number is applied to predict flows with lower Froude numbers. The results presented in this work indicate that the CNN is effective regarding the detection of subsurface flow structures and by interpreting the surface–subsurface relations underlying the reconstruction model, the CNN can be a promising tool for assisting with the physical understanding of free-surface turbulence.

Our focus concerns the turbulent convective flow above a uniformly heated high-aspect-ratio rectangular plate on $z=0$. Prior to this study, classic plume theory could not be applied directly as the Richardson number is ill-defined at the plate. Guided by observation, conservation equations are posed for the near-plate region where the attached buoyant flow is predominantly horizontal. Analytical solutions under the Boussinesq approximation reveal this to be a dynamically invariant region where the attached ‘plumes’ grow linearly toward the plate centreline, their merger forming an ‘apparent’ source for the vertical plume above. Coupling predictions with data from flow visualisation and temperature measurement, we deduce the half-width $0.72b_0$, height ${z=0.46b_0}$ and finite Richardson number $({\sim }18)$ of an apparent source from which plume theory can be applied to model the plume above a heated plate of width $2b_0$. Finally, practical implications of this advancement to the analytical theory of turbulent plumes are noted.

This serial work presents a linear-time-invariance (LTI) notion to the Koopman analysis, finding consistent and physically meaningful Koopman modes and addressing a long-standing problem of fluid mechanics: deterministically relating the fluid excitations and corresponding structure reactions. Part 1 (Li et al., Phys. Fluids, vol. 34, no. 12, p. 125136) developed the Koopman-LTI architecture and applied it to a pedagogical prism wake. By a systematic analytical procedure, the Koopman-LTI generated sampling-independent linear models that captured all the recurring dynamics embedded in the input data, finding six corresponding, orthogonal, and in-synch fluid–structure mechanisms. This Part 2 analyses the six modal duplets to underpin their physical implications, providing a phenomenological analysis of the subcritical prism wake. Visualizing the newly proposed dynamic Koopman modes, results show that two mechanisms at St1 = 0.1242 and St5 = 0.0497 describe shear layer dynamics, the associated Bérnard–Kármán shedding and turbulence production, which together overwhelm the upstream and crosswind walls by instigating a reattachment-type of reaction. The on-wind walls’ dynamical similarity renders them a spectrally unified fluid–structure interface. Another four harmonic counterparts, namely the subharmonic at St7 = 0.0683, the second harmonic at St3 = 0.2422, and two ultra-harmonics at St7 = 0.1739 and St13 = 0.1935, govern the downstream wall. Finally, this work discovered the vortex breathing phenomenon, describing the constant energy exchange in the wake's circulation-entrainment-deposition processes. With the Koopman-LTI, one may pinpoint the exact excitations responsible for a specific structure reaction, benefiting future investigations into fluid–structure interactions and nonlinear, stochastic systems.

We use small-amplitude inviscid theory to study the swimming performance of a flexible flapping plate with time-varying flexibility. The stiffness of the plate oscillates at twice the frequency of the kinematics in order to maintain a symmetric motion. Plates with constant and time-periodic stiffness are compared over a range of mean plate stiffnesses, oscillating stiffness amplitudes and oscillating stiffness phases for isolated heaving, isolated pitching and combined leading-edge kinematics. We find that there is a profound impact of oscillating stiffness on the thrust, with a lesser impact on propulsive efficiency. Thrust improvements of up to 35 % relative to a constant-stiffness plate are observed. For large enough frequencies and amplitudes of the stiffness oscillation, instabilities emerge. The unstable regions may confer enhanced propulsive performance; this hypothesis must be verified via experiments or nonlinear simulations.

The potential of frequency-tuned surfaces as a passive control strategy for reducing drag in wall-bounded turbulent flows is investigated using resolvent analysis. These surfaces are considered to have geometries with impedances that permit transpiration and/or slip at the wall in response to wall pressure and/or shear and are tuned to target the dynamically important structures of wall turbulence. It is shown that wall impedance can suppress the modes resembling the near-wall cycle and the very-large-scale motions and the Reynolds stress contribution of these modes. Suppression of the near-wall cycle requires a more reactive impedance. In addition to these dynamically important modes, the effect of wall impedance across the spectral space is analysed by considering varying mode speeds and wavelengths. It is shown that the materials designed for suppression of the near-wall modes lead to gain reduction over a wide range across the spectral space. Furthermore, a wall with only shear-driven impedance is found to suppress turbulent structures over a wider range in spectral space, leading to an overall turbulent drag reduction. Most importantly, the present analysis shows that the drag-reducing impedance is non-unique and the control performance is not sensitive to variations of the surface impedance within a favourable range. This implies that specific frequency bandwidths can be targeted with periodic material design.

Biologically inspired finlet treatments have been shown to effectively reduce the trailing-edge noise of a flat plate and hence are a viable noise-suppression technology for engineering applications. The present work performs a thorough experimental investigation on the near-field dynamics of finlet surface treatments applied to a flat plate. To examine the underlying noise-reduction mechanism, the manipulated flow field is analysed using data from detailed static, unsteady wall-pressure as well as velocity measurements and their correlations. Specifically, the densely populated dynamic transducers allow for the tracking of the turbulent boundary-layer development from upstream to the wake of the finlet-treated area (see supplementary movies), which elucidates the formation of ‘finlet-induced turbulence’ through flow–finlet interaction. Associated turbulence structures are found to further develop within the treated area and structures shed from the top of the finlets are observed to mix and merge with the turbulence being channelled through the space between the finlets in the finlet wake. While the mixing process increases the spanwise turbulence length scale, it significantly attenuates the unsteady wall-pressure fluctuation at the trailing edge and thus leads to broadband reduction of the trailing-edge noise. Moreover, it corroborates the findings of earlier studies suggesting that there exists an optimal distance between finlets and trailing-edge where the mixing effects are most beneficial.

Collision of two counterflowing gravity currents of equal densities and heights was investigated by means of three-dimensional high-resolution simulations with the goal of understanding the flow structures and energetics in the collision region in more detail. The lifetime of collision is approximately $3 \tilde {H}/\tilde {u}_f$, where $\tilde {H}$ is the depth of heavy and ambient fluids, and $\tilde {u}_f$ is the front velocity of the approaching gravity currents, and the lifetime of collision can be divided into three phases. During Phase I, $-0.2 \leqslant (\tilde {t}-\tilde {t}_c) \tilde {u}_f/\tilde {H} \leqslant 0.5$, where $\tilde {t}$ is the time, and $\tilde {t}_c$ is the time instance at which the two colliding gravity currents have fully osculated, geometric distortions of the gravity current fronts result in stretching of pre-existing vorticity in the wall-normal direction inside the fronts, and an array of vertical vortices extending throughout the updraught fluid column develop along the interface separating the two colliding gravity currents. The array of vertical vortices is responsible for the mixing between the heavy fluids of the two colliding gravity currents and for the production of turbulent kinetic energy in the collision region. The presence of the top boundary deflects the updraughts into the horizontal direction, and a number of horizontal streamwise vortices are generated close to the top boundary. During Phase II, $0.5 \leqslant (\tilde {t}-\tilde {t}_c) \tilde {u}_f/\tilde {H} \leqslant 1.2$, the horizontal streamwise vortices close to the top boundary induce turbulent buoyancy flux and break up into smaller structures. While the production of turbulent kinetic energy weakens, the rate of transfer of energy to turbulent flow due to turbulent buoyancy flux reaches its maximum and becomes the primary supply in the turbulent kinetic energy in Phase II. During Phase III, $1.2 \leqslant (\tilde {t}-\tilde {t}_c) \tilde {u}_f/\tilde {H} \leqslant 2.8$, the collided fluid slumps away from the collision region, while the production of turbulent kinetic energy, turbulent buoyancy flux and dissipation of energy attenuate. From the point of view of energetics, the production of turbulent kinetic energy and turbulent buoyancy flux transfers energy away from the mean flow to the turbulent flow during the collision. Our study complements previous experimental investigations on the collision of gravity currents in that the flow structures, spatial distribution and temporal evolution of the mean flow and turbulent flow characteristics in the collision region are presented clearly. It is our understanding that such complete information on the energy budgets in the collision region can be difficult to attain in laboratory experiments.

The events following the 15 January 2022 explosions of the Hunga Tonga-Hunga Ha'apai volcano highlighted the need for a better understanding of ocean-atmosphere interactions when large amounts of energy are locally injected into one (or both). Starting from the compressible Euler equations, a two-way coupled (TWC) system is derived governing the long-wave behaviour of the ocean and atmosphere under isentropic constraint. Bathymetry and topography are accounted for along with three-dimensional atmospheric non-uniformities through their depth average over a spherical shell. A linear analysis, yielding two pairs of gravito-acoustic waves, offers explanations for phenomena observed during the Tonga event. A continuous transcritical regime (in terms of water depth) is identified as the source of large wave generation in deep water bodies, removing the singularity-driven Proudman-type resonance observed in one-way coupled models. The refractive properties, governing the interaction of the atmospheric wave with step changes in water depth, are derived to comment on mode-to-mode energy transfer. Two-dimensional global simulations modelling the propagation of the atmospheric wave (under realistic conditions on the day) and its worldwide effect on oceans are presented. Local maxima of water-height disturbance in the farfield from the volcano, linked to the atmospheric wave deformation (in agreement with observations), are identified, emphasising the importance of the TWC model for any daylong predictions. The proposed framework can be extended to include additional layers and physics, e.g. ocean and atmosphere stratification. With the aim of contributing to warning system improvement, the code necessary to simulate the event with the proposed model is made available.

The effects of perturbation-based active flow control on supersonic rectangular twin jets (SRTJ) over a wide range of nozzle pressure ratios (NPR = 2.77 to 6.7, corresponding to fully expanded Mach numbers Mj = 1.3 to 1.9) were investigated. The aspect ratio and design Mach number for the bi-conic, converging-diverging nozzles were 2 and 1.5, respectively. The flow and acoustic fields of SRTJ are known to couple, often generating high near-field (NF) pressure fluctuations and elevated far-field (FF) noise levels. Large-scale structures (LSS), or equivalently instability waves or wave packets, are responsible for mixing noise, broadband shock-associated noise, screech and coupling. The primary objective of this research was to manipulate the development of LSS in this complex flow to better understand and mitigate their effects. The organization and passage frequency of the LSS were altered by excitation of instabilities over a wide range of frequencies and modes. Key findings include: (1) the screech mode of each jet was flapping along its minor axis; (2) the jets coupled, out-of-phase primarily in overexpanded cases and in-phase primarily in underexpanded cases, along the minor axis of the SRTJ; (3) coupling has significant effects on the NF pressure fluctuations, but only minor effect on the FF noise; (4) standing waves were observed only on the minor axis plane of the SRTJ; (5) altering or suppressing coupling can significantly reduce NF pressure fluctuations; (6) two high-frequency excitation methods proved effective in reducing the FF noise; and (7) nonlinear interactions between the screech tones and excitation input were observed in controlled cases in which screech was only partially suppressed.

In the fully rough regime, proposed models predict a scaling for a roughness heat-transfer coefficient, e.g. the roughness Stanton number ${St}_k \sim (k^+)^{-p} {Pr}^{-m}$ where the exponent values $p$ and $m$ are model dependent, giving diverse predictions. Here, $k^+$ is the roughness Reynolds number and ${Pr}$ is the Prandtl number. To clarify this ambiguity, we conduct direct numerical simulations of forced convection over a three-dimensional sinusoidal surface spanning $k^+ = 5.5$–$111$ for Prandtl numbers ${Pr} = 0.5$, 1.0 and 2.0. These unprecedented parameter ranges are reached by employing minimal channels, which resolve the roughness sublayer at an affordable cost. We focus on the fully rough phenomenologies, which fall into two groups: $p=1/2$ (Owen & Thomson, J. Fluid Mech., vol. 15, issue 3, 1963, pp. 321–334; Yaglom & Kader, J. Fluid Mech., vol. 62, issue 3, 1974, pp. 601–623) and $p=1/4$ (Brutsaert, Water Resour. Res., vol. 11, issue 4, 1975b, pp. 543–550). Although we find the mean heat transfer favours the $p=1/4$ scaling, the Prandtl–Blasius boundary-layer ideas associated with the Reynolds–Chilton–Colburn analogy that underpin the $p=1/2$ can remain an apt description of the flow locally in regions exposed to high shear. Sheltered regions, meanwhile, violate this behaviour and are instead dominated by reversed flow, where no clear correlation between heat and momentum transfer is evident. The overall picture of fully rough heat transfer is then not encapsulated by one singular mechanism or phenomenology, but rather an ensemble of different behaviours locally. The implications of the approach to a Reynolds-analogy-like behaviour locally on bulk measures of the Nusselt and Stanton numbers are also examined, with evidence pointing to the onset of a regime transition at even-higher Reynolds numbers.

We describe a theoretical and experimental study of an axisymmetric viscous gravity current with a constant flux, confined to the space between two horizontal parallel plates. The effect of confinement results in two regions of flow: an inner region where the fluid is in contact with both plates and an outer annular region where the fluid forms a gravity current along the lower plate. We present a simple theoretical model that describes the flow dynamics by a single dimensionless parameter $J$, which is the ratio of the characteristic height of an unconfined gravity current to the height of the confined space. Theoretical height profiles display the same characteristics as unconfined gravity currents until $J \approx 0.48$, where a rapid change in behaviour occurs as confinement comes into effect. For larger values of $J$, the confined viscous gravity current gradually tends to Hele-Shaw flow, with the transition essentially complete by $J \approx 2$. We compare the findings from our theoretical model with the results of a series of experiments using golden syrup with various fluxes and gap spacings. Although the data aligns with the major aspects of the model, it is clear that other physics is at play and a single non-dimensional parameter is not sufficient to capture the flow behaviour fully. We speculate on the factors absent in our model that may be responsible for this mismatch.

Undulating and breaking bores are generated in the laboratory using a programmable long-stroke wavemaker. By changing the stroke length and the speed of the wavemaker, both non-decaying and decaying bores are generated and studied. Bore strength, height and duration are measured and compared with the solutions derived by using the method of characteristics, with excellent agreement. The measurements for inundation depth, runup height and flood duration are checked with the formulas presented in Barranco & Liu (J. Fluid Mech., vol. 915, 2021). The comparisons show that the formulas are also accurate for the non-decaying bores generated by the wavemaker. The maximum inundation depth predicted by the formula for zero bore length at the beach toe agrees with the laboratory observations for decaying bores. Using a high-speed particle image velocimetry system, the ensemble-averaged velocities and fluctuating velocities under undulating bores and breaking bores are measured in constant water depth and in the vicinity of the still water shoreline. Detailed analyses of the velocity fields are presented and discussed. For the undulating bore a long quiescent flood duration is observed, while for the breaking bore the up-rush flow changes into down-rush flow almost linearly.

Power-law shear-thinning fluid motions induced by a translating spherical bubble with sinusoidal oscillation at a high frequency are numerically studied. We focus on reducing the time-averaged drag force $D$ on the bubble owing to the oscillation-enhanced shear-thinning effect. Under the assumption of negligible convection, the unsteady Stokes equation is directly solved in a finite-difference manner over a wide parameter space of the dimensionless oscillation amplitude $A$ (corresponding to the oscillatory-to-translational velocity ratio) and the power-law index $n$ of the viscosity. The results show that, for small amplitude ($A\ll 1$), the drag reduction ratio $1-D/D_0$ (here, $D_0$ is $D$ with no oscillation) is proportional to $A^2$. In contrast, for large amplitude ($A \gg 1$), the drag ratio $D/D_0$ is proportional to $A^{n-1}$, revealing a power-law behaviour. In the case of $A \gg 1$ for a strong shear-thinning fluid with small $n$, the square of the vorticity over the entire domain is much smaller than that of the shear rate, and thus the bulk flow may be regarded as irrotational. To provide a fundamental perspective on the drag reduction mechanism, a theoretical model is proposed based on potential theory, and demonstrated to well capture the power-law relation between $D/D_0$ and $A$.

Bubble–particle collisions in turbulence are central to a variety of processes such as froth flotation. Despite their importance, details of the collision process have not received much attention yet. This is compounded by the sometimes counter-intuitive behaviour of bubbles and particles in turbulence, as exemplified by the fact that they segregate in space. Although bubble–particle relative behaviour is fundamentally different from that of identical particles, the existing theoretical models are nearly all extensions of theories for particle–particle collisions in turbulence. The adequacy of these theories has yet to be assessed as appropriate data remain scarce to date. In this investigation, we study the geometric collision rate by means of direct numerical simulations of bubble–particle collisions in homogeneous isotropic turbulence using the point-particle approach over a range of the relevant parameters, including the Stokes and Reynolds numbers. We analyse the spatial distribution of bubble and particles, and quantify to what extent their segregation reduces the collision rate. This effect is countered by increased approach velocities for bubble–particle compared to monodisperse pairs, which we relate to the difference in how bubbles and particles respond to fluid accelerations. We found that in the investigated parameter range, these collision statistics are not altered significantly by the inclusion of a lift force or different drag parametrisations, or when assuming infinite particle density. Furthermore, we critically examine existing models and discuss inconsistencies therein that contribute to the discrepancy.