A slowly varying modes solution of Wentzel–Kramers–Brillouin type is derived for the problem of sound propagation in a slowly varying two-dimensional duct with homentropic inviscid sheared mean flow and acoustically lined walls of slowly varying impedance. The modal shape function and axial wavenumber are described by the Pridmore-Brown eigenvalue equation. The slowly varying modal amplitude is determined in the usual way by an equation resulting from a solvability condition. For a general mean flow, this equation can be solved in the form of an incomplete adiabatic invariant. Due to conservation of specific mean vorticity along streamlines, two simplifications prove possible for a linearly sheared mean flow: (i) an analytically exact approximation for the mean flow, and (ii) a complete adiabatic invariant for the acoustics. For this last configuration some example cases are evaluated numerically, where the Pridmore-Brown eigenvalue problem is solved by a Galerkin projection combined with an efficient nonlinear iteration.

We propose an automatable data-driven methodology for robust nonlinear reduced-order modelling from time-resolved snapshot data. In the kinematical coarse-graining, the snapshots are clustered into a few centroids representing the whole ensemble. The dynamics is conceptualized as a directed network, where the centroids represent nodes and the directed edges denote possible finite-time transitions. The transition probabilities and times are inferred from the snapshot data. The resulting cluster-based network model constitutes a deterministic–stochastic grey-box model resolving the coherent-structure evolution. This model is motivated by limit-cycle dynamics, illustrated for the chaotic Lorenz attractor and successfully demonstrated for the laminar two-dimensional mixing layer featuring Kelvin–Helmholtz vortices and vortex pairing, and for an actuated turbulent boundary layer with complex dynamics. Cluster-based network modelling opens a promising new avenue with unique advantages over other model-order reductions based on clustering or proper orthogonal decomposition.

Heterogeneous nucleation is the most effective mechanism for the inception of phase transformation. Solid walls and impurities act as a catalyst for the formation of a new thermodynamic phase by reducing the activation energy required for a phase change, hence enhancing nucleation. The formation of vapour bubbles close to solid, ideally flat, walls is addressed here by exploiting a mesoscale description that couples diffuse interface modelling of the two-phase vapour–liquid system with fluctuating hydrodynamics, extending previous work by the authors on homogeneous nucleation. The technical focus of this work is to directly account for hydrophobic or hydrophilic walls through appropriate boundary conditions compliant with the fluctuation–dissipation balance, a crucial point in the context of fluctuating hydrodynamics theory. This methodology provides access to the complete dynamics of the nucleation process, from the inception of multiple bubbles up to their long-time macroscopic expansion, on time and spatial scales unaffordable by standard techniques for nucleation, such as molecular dynamics. The analysis mainly focuses on the effect of wall wettability on the nucleation rate, and, albeit qualitatively in agreement with classical nucleation theory predictions, it reveals several discrepancies to be ascribed to layering effects in the liquid close to the boundary and to bubble–bubble interactions. In particular, it is found that, close to moderately hydrophilic surfaces, the most probable nucleation events occur away from the wall through a homogeneous mechanism.

Our study concerns the propagation of acoustic waves through a thin screen made of a periodic arrangement of air bubbles in water. The bubbles are oscillators of the Minnaert type whose dynamics is modified by the containment. This nonlinear dynamics is obtained in the time domain using asymptotic analysis and a homogenization technique involving three scales, those being the scale of a bubble, that of the array and eventually that of the wavelength. The resulting effective model is set in the water (the screen has disappeared) and it encapsulates the effect of the screen in a jump of the normal acoustic velocity. The jump is linked to the continuous version of the bubble radius which satisfies an equation of the Rayleigh–Plesset type. This allows us to highlight two important effects. Firstly, a bubble within the array has a much larger radiative damping than an isolated bubble. Secondly it perceives a pressure which differs from the acoustic pressure imposed by the source due to bubble–bubble interactions; it results in a term of mass correction deduced from the Green's function for a Laplace problem which accounts for the bubble arrangement. Our findings are exemplified by numerical experiments of the scattering of a short pulse in the linear and nonlinear regimes.

Stability analysis of a liquid layer subjected to an oblique temperature gradient (OTG) is carried out. The general linear stability analysis reveals a stabilization effect of the imposed horizontal component (horizontal temperature gradient, HTG) of the OTG on the long-wave instabilities introduced by the vertical component (vertical temperature gradient, VTG) of the OTG. This stabilization is due to the VTG induced by the prescribed HTG, which counteracts the imposed VTG. The induced VTG arises due to the presence of advection of the energy. As a result of the stabilization, the long-wave mode forms an island of instability in the $\eta$–$Ma_c$ plane, where $\eta$ and $Ma_c$ are the ratio of the strength of the imposed HTG to imposed VTG components of the OTG, and the critical Marangoni number, respectively. However, for sufficiently high $\eta$, a new class of modes emerge with the critical Marangoni number scaling as $Ma_c \sim 1/\eta$. These modes originate as a result of the interaction between the thermocapillary flow caused by the imposed HTG on the one hand, and the VTG on the other, and remain the dominant modes of instability at higher $\eta$. The long-wave analysis is carried out and, in its framework, the nonlinear evolution equation is derived, and, based on it, linear and weakly nonlinear analyses are performed. An increase in $\eta$ changes the type of bifurcation from subcritical to supercritical. The numerical solution of the evolution equation around the critical parameter values validates the predictions of the weakly nonlinear analysis. The present study illustrates a possible use of imposing the HTG to prevent dry-spot formation and rupture of the film caused by the imposed VTG.

We propose an image-based flow decomposition developed from the two-dimensional (2-D) tensor empirical wavelet transform (EWT) (Gilles, IEEE Trans. Signal Process., vol. 61, 2013, pp. 3999–4010). The idea is to decompose the instantaneous flow data, or their visualisation, adaptively according to the averaged Fourier supports for the identification of spatially localised structures. The resulting EWT modes stand for the decomposed flows, and each accounts for part of the spectrum, illustrating fluid physics with different scales superimposed in the original flow. With the proposed method, decomposition of an instantaneous three-dimensional (3-D) flow becomes feasible without resorting to its time series. Examples first focus on the interaction between a jet plume and 2-D wake, where only experimental visualisations are available. The proposed method is capable of separating the jet/wake flows and their instabilities. Then the decomposition is applied to an early-stage boundary layer transition, where direct numerical simulations provided a full dataset. The tested inputs are the 3-D flow data and their visualisation using streamwise velocity and $\lambda _{2}$ vortex identification criterion. With both types of inputs, EWT modes robustly extract the streamwise-elongated streaks, multiple secondary instabilities and helical vortex filaments. Results from 2-D stability analysis justify the EWT modes that represent the streak instabilities. In contrast to proper orthogonal decomposition or dynamic modal decomposition that extract spatial modes according to energy or frequency, EWT provides a new strategy for decomposing an instantaneous flow from its spatial scales.

The motion of a single spherical particle in a two-sided lid-driven cavity is investigated experimentally. The flow in which the particle moves is created by two facing cavity sidewalls which move with equal velocity in opposite directions. For a long cavity with width-to-height cross-sectional aspect ratio $\varGamma =W/H=1.6$ the flow field at Reynolds number ${Re}=400$ consists of steady spatially periodic three-dimensional convection cells. Nearly neutrally buoyant particles with radius in units of $H$ ranging from $1.1\times 10^{-2}$ to $7.1\times 10^{-2}$ are found to be attracted to periodic or quasi-periodic orbits in close vicinity of Kolmogorov–Arnold–Moser (KAM) tori of the unperturbed flow. Like the KAM tori the attractors of neutrally buoyant particles arise in mirror-symmetric pairs within each convection cell. The particle attractors are created by a dissipative effect in the dynamical system describing the particle motion which arises when the finite-size particle closely passes the moving walls. When the particle density deviates from that of the fluid, inertial attractors arise whose symmetry is broken by buoyancy, and other periodic attractors are created which do not have KAM tori as counterparts.

This paper considers the effects of smoothly varying chordwise porosity of a finite perforated plate on turbulence–aerofoil interaction noise. The aeroacoustic model is made possible through the use of a novel Mathieu function collocation method, rather than a traditional Wiener–Hopf approach which would be unable to deal with chordwise-varying quantities. The main focus is on two bio-inspired porosity distributions, modelled from air flow resistance data obtained from the wings of barn owls (tyto alba) and common buzzards (buteo buteo). Trailing-edge noise is much reduced for the owl-like distribution, but, perhaps surprisingly, so too is leading-edge noise, despite both wings having similar porosity values at the leading edge. A general monotonic variation is then considered indicating that there may indeed be a significant acoustic impact of how the porosity is distributed along the whole chord of the plate, not just its values at the scattering edges. Through this investigation, it is found that a plate whose porosity continuously decreases from the trailing edge to a zero-porosity leading edge can, in fact, generate lower levels of trailing-edge noise than a plate whose porosity remains constant at the trailing-edge value.

Fluid flows coupled by a deformable solid layer (DS) may be found in both natural and industrial settings. To analyse the stability of such systems, we consider plane Couette flows coupled by a DS. The shear stress exerted by the flowing fluids on the DS leads to its deformation, coupling the motion in each fluid. We demonstrate that, as a consequence of the coupling and irrespective of the thickness of the DS, a viscous instability exists, which can lead to absolute instability at sufficiently high dimensionless speed of the lower plate. The predicted instability apparently arises due to the energy exchange between the fluids and DS via the tangential velocities at the interface. A linear elasticity model for the DS is employed, removing the non-physical growth rate predicted by previous models. A spatio-temporal analysis further reveals the existence of an absolute instability. Insight on this instability could be potentially utilised in manipulating mixing in microfluidic systems and membrane separation as well as for understanding morphologies evolving during membrane formation via interfacial polymerisation.

We study the spreading of viscous fluid injected under an elastic sheet, which is driven by gravity and by elastic bending and tension forces and resisted by viscous forces. The injected fluid forms a large blister and spreads outwards analogously to a viscous gravity current or a capillary droplet. The relative strengths of the three driving forces are determined by how the horizontal length scales of the system compare with three key transition length scales. Bending is dominant on small length scales, tension is dominant on intermediate length scales and gravity is dominant on large length scales. We show how to use the method of matched asymptotic expansions to predict the spreading rate and thickness profile of the blister of fluid in the seven possible asymptotic regimes, for both two-dimensional and axisymmetric geometries. Consideration of different physical effects at the fluid front increases the number of regimes yet further.

The small-scale physics within the first centimetres above the wavy air–sea interface are the gateway for transfers of momentum and scalars between the atmosphere and the ocean. We present an experimental investigation of the surface wind stress over laboratory wind-generated waves. Measurements were performed at the University of Delaware's large wind-wave-current facility using a recently developed state-of-the-art wind-wave imaging system. The system was deployed at a fetch of 22.7 m, with wind speeds from 2.19 to $16.63\ \textrm {m}\ \textrm {s}^{-1}$. Airflow velocity fields were acquired using particle image velocimetry above the wind waves down to $100\ \mathrm {\mu }\textrm {m}$ above the surface, and wave profiles were detected using laser-induced fluorescence. The airflow intermittently separates downwind of wave crests, starting at wind speeds as low as $2.19\ \textrm {m}\ \textrm {s}^{-1}$. Such events are accompanied by a dramatic drop in tangential viscous stress past the wave's crest, and a gradual regeneration of the viscous sublayer upon the following (downwind) crest. This contrasts with non-airflow separating waves, where the surface viscous stress drop is less significant. Airflow separation becomes increasingly dominant with increasing wind speed and wave slope $a k$ (where $a$ and $k$ are peak wave amplitude and wavenumber, respectively). At the highest wind speed ($16.63\ \textrm {m}\ \textrm {s}^{-1}$), airflow separation occurs over nearly 100 % of the wave crests. The total air–water momentum flux is partitioned between viscous stress and form drag at the interface. Viscous stress (respectively form drag) dominates at low (respectively high) wave slopes. Tangential viscous forcing makes a minor contribution (${\sim }3\,\%$) to wave growth.

Entropy perturbations generate sound when accelerated/decelerated by a non-uniform flow. Current analytical models provide a good prediction of this entropy noise when the flow cross-sectional area changes are gradual, as is the case for nozzle flows. However, they typically rely on quasi-1-D and isentropic assumptions, and their predictions differ significantly from experimental measurements when sudden area increases are involved. This work uses a theoretical approach to quantitatively identify the main mechanisms responsible for the mismatch. A new form of the acoustic analogy is derived in which the entropy-related source terms are systematically identified for the first time. The theory includes three-dimensional and non-isentropic effects. The approach is applied to the flow through a sudden area expansion, for which the large-scale flow separation creates a recirculation zone. The derived acoustic analogy is simplified for low Mach numbers and frequencies, and solved using a Green's function method. The results provide the first quantitative evidence that the presence and spatial extent of the recirculation zone, rather than the flow non-isentropicity, is the dominant factor causing deviation from predictions from quasi-1-D, isentropic theory.

Water particle orbits are key elements in the Lagrange wave formulation. The stochastic Miche implementation of the Lagrange model is a linear Gaussian two-dimensional or three-dimensional space–time model which exhibits typical nonlinear wave characteristics when transformed to Eulerian coordinates. This paper investigates the statistical relation between the degree of front–back asymmetry of individual waves and the orbit orientation for the particle located at the wave maximum at the point of observation. It is shown that, in the Lagrangian model with statistical front–back symmetry, for individual waves there is a clear connexion between the degree of individual wave asymmetry and the orientation of the randomly deformed elliptic orbit: a steep front correlates with upward tilt, a steep back is correlated with a downward tilt. This holds both for waves observed in time and in space, and the dependence is stronger for large amplitude waves than for smaller ones. The dependence is strongly dependent on the depth and on the significant steepness and spectral width. Inclusion of the average Stokes drift has a moderate effect on the dependence. For models with forced front–back asymmetry there is both a systematic dependence and a statistical correlation between asymmetry and tilt; for large amplitude waves the systematic relation dominates. The conclusions are based on Fourier simulations of Gauss–Lagrange waves of first and second order with a wind–sea Pierson–Moskowitz spectrum and a narrow swell JONSWAP spectrum.

Rayleigh–Bénard (RB) convection with free-slip plates and horizontally periodic boundary conditions is investigated using direct numerical simulations. Two configurations are considered, one is two-dimensional (2-D) RB convection and the other one three-dimensional (3-D) RB convection with a rotating axis parallel to the plate, which for strong rotation mimics 2-D RB convection. For the 2-D simulations, we explore the parameter range of Rayleigh numbers $Ra$ from $10^{7}$ to $10^{9}$ and Prandtl numbers $Pr$ from $1$ to $100$. The effect of the width-to-height aspect ratio $\varGamma$ is investigated for $1\leqslant \varGamma \leqslant 128$. We show that zonal flow, which was observed, for example, by Goluskin et al. (J. Fluid. Mech., vol. 759, 2014, pp. 360–385) for $\varGamma =2$, is only stable when $\varGamma$ is smaller than a critical value, which depends on $Ra$ and $Pr$. The regime in which only zonal flow can exist is called the first regime in this study. With increasing $\varGamma$, we find a second regime in which both zonal flow and different convection roll states can be statistically stable. For even larger $\varGamma$, in a third regime, only convection roll states are statistically stable and zonal flow is not sustained. How many convection rolls form (or in other words, what the mean aspect ratio of an individual roll is), depends on the initial conditions and on $Ra$ and $Pr$. For instance, for $Ra=10^{8}$ and $Pr=10$, the aspect ratio $\varGamma _r$ of an individual, statistically stable convection roll can vary in a large range between $16/11$ and $64$. A convection roll with a large aspect ratio of $\varGamma _r = 64$, or more generally already with $\varGamma _r \gg 10$, can be seen as ‘localized’ zonal flow, and indeed carries over various properties of the global zonal flow. For the 3-D simulations, we fix $Ra=10^{7}$ and $Pr=0.71$, and compare the flow for $\varGamma =8$ and $\varGamma = 16$. We first show that with increasing rotation rate both the flow structures and global quantities like the Nusselt number $Nu$ and the Reynolds number $Re$ increasingly behave like in the 2-D case. We then demonstrate that with increasing aspect ratio $\varGamma$, zonal flow, which was observed for small $\varGamma =2{\rm \pi}$ by von Hardenberg et al. (Phys. Rev. Lett., vol. 15, 2015, 134501), completely disappears for $\varGamma =16$. For such large $\varGamma$, only convection roll states are statistically stable. In-between, here for medium aspect ratio $\varGamma = 8$, the convection roll state and the zonal flow state are both statistically stable. What state is taken depends on the initial conditions, similarly as we found for the 2-D case.

We investigate experimentally the effect of double diffusion on the dynamics of initially neutrally buoyant warm and salty turbulent jets discharged horizontally into stationary cooler freshwater ambient. Jets over a range of source Reynolds numbers and source temperature/salinity combinations are examined. In all cases, we observed sinking jet trajectories and the formation of salt fingers along the lower surface of the jet. Increasing the source concentration of both scalar properties led to more pronounced jet sinking trajectories, and to a reduction in the distance between the source and the onset point of salt fingers, demonstrating the significance of the double-diffusive processes. We propose that is it the differential double-diffusive fluxes across the jet–ambient turbulent/non-turbulent interfaces that causes the build-up of negative buoyancy and hence the sinking motion. In addition, we make predictions on the onset point of the salt fingers based on the balance between diffusive processes and the jet entrainment, and compare them with the experimental observations.

We study the deformation and dewetting of liquid films under impinging gas jets using experimental, analytical and numerical techniques. We first derive a reduced-order model (a thin-film equation) based on the long-wave assumption and on appropriate decoupling of the gas problem from that for the liquid. The model not only provides insight into relevant flow regimes, but is also used in conjunction with experimental data to guide more computationally prohibitive direct numerical simulations of the full governing equations. A unique feature of our modelling solution is the use of an efficient iterative procedure in order to update the interfacial deformation based on stresses originating from computational data. We show that both gas normal and tangential stresses are equally important for achieving accurate predictions. The interplay between these techniques allows us to study previously unreported flow features. These include finite-size effects of the host geometry, with consequences for flow and vortex formation inside the liquid, as well as the specific individual contributions from the non-trivial gas flow components on interfacial deformation. Dewetting phenomena are found to depend on either a dominant gas flow or contact line motion, with the observed behaviour (including healing effects) being explained using a bifurcation diagram of steady-state solutions in the absence of the gas flow.

Taylor–Couette (TC) flow is the shear-driven flow between two coaxial independently rotating cylinders. In recent years, high-fidelity simulations and experiments revealed the shape of the streamwise and angular velocity profiles up to very high Reynolds numbers. However, due to curvature effects, so far no theory has been able to correctly describe the turbulent streamwise velocity profile for a given radius ratio, as the classical Prandtl–von Kármán logarithmic law for turbulent boundary layers over a flat surface at most fits in a limited spatial region. Here, we address this deficiency by applying the idea of a Monin–Obukhov curvature length to turbulent TC flow. This length separates the flow regions where the production of turbulent kinetic energy is governed by pure shear from that where it acts in combination with the curvature of the streamlines. We demonstrate that for all Reynolds numbers and radius ratios, the mean streamwise and angular velocity profiles collapse according to this separation. We then develop the functional form of the velocity profile. Finally, using the newly developed angular velocity profiles, we show that these lead to an alternative constant in the model proposed by Cheng et al. (J. Fluid Mech., vol. 890, 2020, A17) for the dependence of the torque on the Reynolds number, or, in other words, of the generalized Nusselt number (i.e. the dimensionless angular velocity transport) on the Taylor number.

We study the axisymmetric interaction of two chemically active Janus particles. By relying on the linearity of the field equations and symmetry arguments, we derive a generic solution for the relative velocity of the particles. We show that, regardless of the chemical properties of the system, the relative velocity can be written as a linear summation of geometrical functions which only depend on the gap size between the particles. We evaluate these functions via an exact approach which accounts for the full chemical and hydrodynamic interactions. Using the obtained solution, we expose the role of each compartment in the relative motion, and also discuss the contribution of different interactions. We then show that the dynamical system describing the relative motion of two Janus particles can have up to three fixed points. These fixed points can be stable or unstable, indicating that a system of two Janus particles can exhibit a variety of non-trivial behaviours depending on their initial gap size, and their chemical properties. We also look at the specific case of Janus particles in which one compartment is inert, and present regime diagrams for their relative behaviour in the activity–mobility parameter space.

The aim of this paper is to provide a complete description of the bifurcation scenario of a uniform flow past a rotating circular cylinder up to $Re = 200$. Linear stability theory is used to depict the neutral curves and analyse the arising unstable global modes. Three codimension-two bifurcation points are identified, namely a Takens–Bogdanov, a cusp and generalised Hopf, which are closely related to qualitative changes in orbit dynamics. The occurrence of the cusp and Takens–Bogdanov bifurcations for very close parameters (corresponding to an imperfect codimension-three bifurcation) is shown to be responsible for the existence of multiple steady states, as already observed in previous studies. Two bistability regions are identified, the first with two stable fixed points and the second with a fixed point and a cycle. The presence of homoclinic and heteroclinic orbits, which are classical in the presence of Takens–Bogdanov bifurcations, is confirmed by direct numerical simulations. Finally, a weakly nonlinear analysis is performed in the neighbourhood of the generalised Hopf, showing that above this point the Hopf bifurcation is subcritical, leading to a third range of bistability characterised by both a stable fixed point and a stable cycle.

Extensive studies have focused on the self-propulsion of a droplet in a viscous environment driven by the Marangoni effect in the absence of inertial effects. In order to capture the influence of inertia on the self-propulsion of a droplet, we use the singular perturbation solution for small but finite Reynolds number ($Re$) flow past a spherical droplet with inhomogeneous surface tension. We calculate the swimming speed and the corresponding flow fields generated by the droplet in an axisymmetric unbounded medium at $O(Re^2)$. The present results reveal how the choice of the stress parameter $\sigma$, which is the ratio of the first two modes of the induced stress field, distinguishes between the different swimming styles, and determines the role of inertia on the swimming speed, energy expenditure and swimming efficiency of the droplet. Inertia enhances the swimming speed and the associated swimming efficiency of the droplet by abating the energy expenditure. It is striking to observe how a droplet swimmer with $\sigma <0$ has a competitive advantage over a rigid squirmer with an equivalent surface activity due to the existence of an internal flow. We independently treat the potential influence of the viscosity ratio on the swimming properties of the droplet at finite $Re$. Additionally, using linear stability analysis, we provide insights into the stability of the estimated migration velocity at $O(Re)$. We argue that the droplet achieves a distinct stable equilibrium velocity, which occurs due to the inertial effect of the surrounding medium.