The classical model of evaporation of liquids hinges on Maxwell's assumption that the air near the liquid's surface is saturated. It allows one to find the evaporative flux without considering the interface separating liquid and air. Maxwell's hypothesis is based on an implicit assumption that the vapour-emission capacity of the interface exceeds the throughput of air (i.e. its ability to pass the vapour on to infinity). If this is indeed so, then the air adjacent to the liquid would get quickly saturated, justifying Maxwell's hypothesis. In the present paper, the so-called diffuse-interface model is used to account for the interfacial physics and thus derive a generalised version of Maxwell's boundary condition for the near-interface vapour density. It is then applied to a spherical drop floating in air. It turns out that the vapour-emission capacity of the interface exceeds the throughput of air only if the drop's radius is $r_{d}\gtrsim 10\ \mathrm {\mu } \mathrm {m}$, but for $r_{d}\approx 2\ \mathrm {\mu } {\rm m}$, the two are comparable. For $r_{d} \lesssim 1\ \mathrm {\mu } {\rm m}$, evaporation is interface-driven, and the resulting evaporation rate is noticeably smaller than that predicted by the classical model.

In this paper, we study the aeroacoustic instability which occurs in a deep axisymmetric cavity in a turbulent pipe flow. This phenomenon is the axisymmetric counterpart of the classical whistling of a rectangular deep cavity subject to a grazing flow. The whistling of such axisymmetric cavity originates from the interaction of the coherent fluctuations of the vorticity at the cavity's opening with one of its trapped azimuthal or radial acoustic modes. We focus here on the situation involving the first pure azimuthal mode, which is trapped in the cavity. As a consequence of the rotational symmetry of the configuration, azimuthal modes are actually pairs of degenerate eigenmodes, or almost degenerate in the presence of small asymmetries. Therefore, the aeroacoustic instabilities exhibit more complex mechanisms than in the case of a rectangular deep cavity. In particular, we show that self-sustained spinning modes induce a symmetry breaking of the mean flow and we will elucidate the details of this phenomenon. To that end, simultaneous acoustic and time-resolved stereoscopic particle image velocimetry (PIV) measurements are performed. They reveal that when large-amplitude aeroacoustic waves spin around the cavity, a quasi-steady mean flow starts whirling slowly in the opposite direction to the wave propagation. A linear perturbation analysis around an axisymmetric mean flow confirms the experimental observations: although the incoming pipe flow is not swirling, the hydrodynamic component of the aeroacoustic wave induces such whirling motion of the mean flow because of the forcing from the steady part of the coherent Reynolds stress tensor.

The whistling induced by a low-Mach turbulent flow through a deep axisymmetric cavity in a duct is investigated theoretically and experimentally. The experiments include acoustic measurements and stereoscopic particle image velocimetry (PIV). The paper focuses on the effect of a mean swirl on the dynamics of the azimuthal aeroacoustic modes. The mean swirl in the cavity has two origins: one component is imposed by a controlled tangential air injection upstream of the cavity, and the other component spontaneously arises under the action of the self-sustained azimuthal aeroacoustic mode, as explained in the companion paper, Part 1 (Faure-Beaulieu, Xiong, Pedergnana & Noiray, J. Fluid Mech., vol. 971, 2023, A21). Experiments show that the dynamics of the aeroacoustic wave is influenced by the imposed swirl. In particular, the spinning wave propagating against the swirl is promoted. To explain this, a linear perturbation analysis is performed around an incompressible mean swirling flow obtained from RANS simulations. It reveals that the dominant shear layer modes of azimuthal order 1 and −1 involved in the whistling phenomenon are helical modes winding respectively with and against the swirl, and spinning respectively in counterswirl and co-swirl directions. The counterswirl hydrodynamic mode is the least damped of the two, which is in agreement with the experimental observations. Finally, a low-order model based on the wave equation is derived. With only a few parameters, it fully reproduces the experimental observations for a wide range of imposed swirl intensity in the duct flow, and it allows us to disentangle the mechanisms responsible for this complex aeroacoustic instability.

Pore-resolved direct numerical simulations are performed to investigate the interactions between streamflow turbulence and groundwater flow through a randomly packed porous sediment bed for three permeability Reynolds numbers, $Re_K=2.56$, 5.17 and 8.94, representative of natural stream or river systems. Time–space averaging is used to quantify the Reynolds stress, form-induced stress, mean flow and shear penetration depths, and mixing length at the sediment–water interface (SWI). The mean flow and shear penetration depths increase with $Re_K$ and are found to be nonlinear functions of non-dimensional permeability. The peaks and significant values of the Reynolds stresses, form-induced stresses, and pressure variations are shown to occur in the top layer of the bed, which is also confirmed by conducting simulations of just the top layer as roughness elements over an impermeable wall. The probability distribution functions (p.d.f.s) of normalized local bed stress are found to collapse for all Reynolds numbers, and their root-mean-square fluctuations are assumed to follow logarithmic correlations. The fluctuations in local bed stress and resultant drag and lift forces on sediment grains are mainly a result of the top layer; their p.d.f.s are symmetric with heavy tails, and can be well represented by a non-Gaussian model fit. The bed stress statistics and the pressure data at the SWI potentially can be used in providing better boundary conditions in modelling of incipient motion and reach-scale transport in the hyporheic zone.

To assess how the presence of surfactant in lung airways alters the flow of mucus that leads to plug formation and airway closure, we investigate the effect of insoluble surfactant on the instability of a viscoplastic liquid coating the interior of a cylindrical tube. Evolution equations for the layer thickness using thin-film and long-wave approximations are derived that incorporate yield-stress effects and capillary and Marangoni forces. Using numerical simulations and asymptotic analysis of the thin-film system, we quantify how the presence of surfactant slows growth of the Rayleigh–Plateau instability, increases the size of initial perturbation required to trigger instability and decreases the final peak height of the layer. When the surfactant strength is large, the thin-film dynamics coincide with the dynamics of a surfactant-free layer but with time slowed by a factor of four and the capillary Bingham number, a parameter proportional to the yield stress, exactly doubled. By solving the long-wave equations numerically, we quantify how increasing surfactant strength can increase the critical layer thickness for plug formation to occur and delay plugging. The previously established effect of the yield stress in suppressing plug formation (Shemilt et al., J. Fluid Mech., vol. 944, 2022, A22) is shown to be amplified by introducing surfactant. We discuss the implications of these results for understanding the impact of surfactant deficiency and increased mucus yield stress in obstructive lung diseases.

We report an experimental study of the motion of a clapping body consisting of two flat plates pivoted at the leading edge by a torsion spring. Clapping motion and forward propulsion of the body are initiated by the sudden release of the plates, initially held apart at an angle $2\theta _o$. Results are presented for the clapping and forward motions, and for the wake flow field for 24 cases, where depth-to-length ratio ($d^* = 1.5,1\text { and }0.5$), spring stiffness per unit depth ($Kt$), body mass ($m_b$) and initial separation angle ($2\theta _o = 45^{\circ }\text { and }60^{\circ }$) are varied. The body initially accelerates rapidly forward, then slowly retards to nearly zero velocity. Whereas the acceleration phase involves a complex interaction between plate and fluid motions, the retardation phase is simply fluid dynamic drag slowing the body. The wake consists of either a single axis-switching elliptical vortex loop (for $d^* = 1\text { and }1.5$) or multiple vortex loops (for $d^* = 0.5$). The body motion is nearly independent of $d^*$ and most affected by variation in $\theta _o$ and $Kt$. Using conservation of linear momentum and conversion of spring strain energy into kinetic energy in the fluid and body, we obtain a relation for the translation velocity of the body in terms of the various parameters. Approximately 80 % of the initial stored energy is transferred to the fluid, only 20 % to the body. The experimentally obtained cost of transport lies between 2 and $8\ \mathrm {J}\ \mathrm {kg}^{-1}\ \mathrm {m}^{-1}$.

We present a statistical characterization of the interaction between a planar shock and a finite-diameter, cylindrical column of dense gas based on three-dimensional, large-eddy simulation results. In the simulation, the column of gas is initially inclined at an angle $\alpha _0$ with respect to the shock plane. Effects of the initial column angle on the mixing characteristics are examined at Mach number 2.0 for column incline angles $1^\circ$, $5^\circ$, $10^\circ$ and $30^\circ$. Mean velocity profiles show that the column angle affects the gas velocity components in the vertical plane, but not in the spanwise direction. The gas undergoes higher initial upward acceleration at larger initial column incline angles. With time, the gas motion tends to become one-dimensional in the streamwise direction. Initially, velocity fluctuations are most intense within the interior of the column, but concentrate near the column leading edge over time. At high wavenumbers $\kappa$, the turbulent kinetic energy spectra follow a power-law scaling of $\kappa ^{-1}$. The structure functions of the mass fraction do not clearly demonstrate power-law scaling except at early times for $\alpha _0=30^\circ$, manifesting overall trends very similar to those observed in earlier experiments. Probability distributions of the mass fraction show independence of the mean and the standard deviation of the mixed gas on $\alpha _0$. The column angle was also found to have little effect on the mixing efficiency characterized by the molecular mixedness. Velocity components in the streamwise and transverse directions tend towards a bimodal distribution for larger $\alpha _{0}$.

Eigenvectors of the observability and controllability Gramians represent responsive and receptive flow structures that enjoy a well-established connection to resolvent forcing and response modes. However, whereas resolvent modes have demonstrated great potential to guide sensor and actuator placement, observability and controllability modes have been leveraged exclusively in the context of model reduction via input and output projections. In this work, we introduce interpolatory, rather than orthogonal, input and output projections, that can be leveraged for sensor and actuator placement and open-loop control design. An interpolatory projector is an oblique projector with the property of preserving certain entries in the vector being projected. We review the connection between the resolvent operator and the Gramians, and present several numerical examples where we perform both orthogonal and interpolatory input and output projections onto the dominant forcing and response subspaces. Input projections are used to identify dynamically relevant disturbances, place sensors to measure disturbances, and place actuators for feedforward control in the linearized Ginzburg–Landau equation. Output projections are used to identify coherent structures and place sensors aiming at state reconstruction in the turbulent flow in a minimal channel at $Re_{\tau }=185$. The framework does not require data snapshots and relies only on knowledge of the steady or mean flow.

The shock wave–turbulent boundary layer interaction over a compression corner is studied using global stability analysis (GSA) and resolvent analysis based on a separation of scales between the low-frequency, large-scale motions and the turbulent fluctuations. The GSA identifies a leading stationary mode, which becomes globally unstable as the ramp angle is beyond a critical value. For globally stable flows, the resolvent analysis captures two-dimensional and three-dimensional local maxima in optimal gain, both of which are due to modal resonance between the forcing and the leading global mode. Notably, the frequency-premultiplied optimal gain associated with two-dimensional disturbances peaks at a low frequency. For different interaction strengths, the peak frequencies collapse onto a universal value of 0.015 when non-dimensionalized using the length of the separation region and the free-stream velocity. A numerical simulation perturbed with the corresponding optimal forcing reveals that the response is in the form of a back-and-forth shock motion.

The model eukaryotic microalgae Chlamydomonas reinhardtii is well known for its ability to generate bioconvection flows that are associated to intricate concentration patterns. Recently, it was demonstrated that the propensity of these algae to move toward a light source – a phenomenon termed phototaxis – can be exploited to locally concentrate micro-organisms and induce (photo)-bioconvection in algal suspensions by inducing a localised excess of density. In the present study we show experimentally that a cell population in a thin liquid layer self-organises in the presence of a heterogeneous light field and displays remarkable symmetry-breaking instabilities that are ruled by both the width of the light beam and the photo-bioconvection Rayleigh number. Beside circular stable states, fingers, dendrites and wave instabilities are reported, quantified and classified in a general phase diagram. Next, we use lubrication theory to develop an asymptotic model for bioconvection in a thin liquid layer, that includes the influences of both gyrotaxis and phototaxis. We obtain a single nonlinear anisotropic diffusion–drift equation describing the spatiotemporal evolution of the depth-averaged algal population. Analytical and numerical solutions are presented and show a very good agreement with the experimental results. In particular, we show that the dendrite instability arises as a result of a subtle coupling between the nonlinearity of the phototactic response, the gyrotactic effect and the self-induced bioconvective flows. Such complex flow fields might find applications in photo-bioreactors, through the efficient stirring of the harvested biomass.

The linear and nonlinear stability of two concentric jets separated by a duct wall is analysed by means of global linear stability and weakly nonlinear analysis. Three governing parameters are considered, the Reynolds number based on the inner jet, the inner-to-outer jet velocity ratio ($\delta _u$) and the length of the duct wall ($L$) separating the jet streams. Global linear stability analysis demonstrates the existence of unsteady modes of inherent convective nature, and symmetry-breaking modes that lead to a new non-axisymmetric steady state with a single or double helix. Additionally, we highlight the existence of multiple steady states, as a result of a series of saddle-node bifurcations and its connection to the changes in the topology of the flow. The neutral lines of stability have been computed for inner-to-outer velocity ratios within the range $0 < \delta _u < 2$ and duct wall distances in the interval $0.5 < L < 4$. They reveal the existence of hysteresis, and mode switching between two symmetry-breaking modes with azimuthal wavenumbers $1:2$. Finally, the mode interaction is analysed, highlighting the presence of travelling waves emerging from the resonant interaction of the two steady states, and the existence of robust heteroclinic cycles that are asymptotically stable.

An experimental study on the aerobreakup of a wall-attached droplet induced by a shock wave is carried out in this work. The shock wave and the post-wave air stream are generated by a shock tube facility, and the dynamics of droplet breakup are captured through high-speed imaging from the side, the oblique front and the back views, respectively. The wall-attached droplets with different wettabilities are considered, including the hydrophobic, hemispherical and hydrophilic ones. For the convenience of analysis, droplets with different wettabilities are unified with the same equivalent radius and, thus, the same Weber number and Reynolds number. Different from the evolution of a free spherical droplet, it is observed that the hydrophobic droplet can develop ‘peak’ structures on the windward and leeward sides, respectively; while the hemispherical and hydrophilic droplets leave a liquid film on the wall. The occurrence of ‘peak’ structures is caused by the local standing vortex on the droplet windward and leeward sides, and the residual of a liquid film on a solid wall is related to the boundary layer near the wall. Through comparing the height, width and advancing distance of droplet deformation between the spherical droplet and the wall-attached droplets with different wettabilities, it is found that the wall will inhibit the deformation and fragmentation of droplets on the flow direction. The ‘peak’ and film structures are responsible for the deviation of the dimensionless width between spherical and wall-attached droplets. The ‘lip’ structure is related to the recirculation flow at the rear of the droplet, which causes the reverse of velocity on the surface. The radius of the ‘lip’ decreases with the decrease of contact angle. The Rayleigh–Taylor (R–T) and Kelvin–Helmholtz (K–H) instabilities of wall-attached and spherical droplets are also studied by experiments and theoretical analysis. With the decrease of contact angle, the development of the K–H instability hardly changes, while the R–T instability results in a smaller wavelength of perturbation on the windward side. This study is expected to give some guidance for regulating the aerodynamic fragmentation of droplets in engineering applications.

A thin liquid droplet spreads on a soft viscoelastic substrate with arbitrary rheology. Lubrication theory is applied to the governing field equations in the liquid and solid domains, which are coupled through the free boundary at the solid–liquid interface, to derive a set of reduced equations that describe the spreading dynamics. Fourier transform techniques and the finite difference method are used to construct a solution for the dynamic liquid–gas and solid–liquid interface shapes, as well as the macroscopic contact angle. Substrate properties affect the spreading dynamics through the contact angle and internal droplet flow fields, and these mechanisms are revealed. Increased substrate softness increases the spreading rate, whereas increased viscoelasticity decreases the spreading rate. For the case of a purely elastic substrate, the spreading power-law exponent recovers Tanner's law in the rigid limit and increases with substrate softness.

We present numerical and analytical predictions of mucociliary clearance based on the continuum description of a viscoelastic mucus film, where momentum transfer from the beating cilia is represented via a Navier-slip boundary condition introduced by Bottier et al. (PLoS Comput. Biol., vol. 13, issue 7, 2017a, e1005552). Mucus viscoelasticity is represented via the Oldroyd-B model, where the relaxation time and the viscosity ratio have been fitted to experimental data for the storage and loss moduli of different types of real mucus, ranging from healthy to diseased conditions. We solve numerically the fully nonlinear governing equations for inertialess flow, and develop analytical solutions via asymptotic expansion in two limits: (i) weak viscoelasticity, i.e. low Deborah number; (ii) low cilia beat amplitude (CBA). All our approaches predict a drop in the mucus flow rate in relation to the Newtonian reference value, as the cilia beat frequency is increased. This relative drop increases with decreasing CBA and slip length. In diseased conditions, e.g. mucus properties characteristic of cystic fibrosis, the drop reaches 30 % in the physiological frequency range. In the case of healthy mucus, no significant drop is observed, even at very high frequency. This contrasts with the deterioration of microorganism propulsion predicted by the low-amplitude theory of Lauga (Phys. Fluids, vol. 19, issue 8, 2007, 083104), and is due to the larger beat amplitude and slip length associated with mucociliary clearance. In the physiological range of the cilia beat frequency, the low-amplitude prediction is accurate for both healthy and diseased conditions. Finally, we find that shear-thinning, modelled via a multi-mode Giesekus model, does not significantly alter our weakly viscoelastic and low-amplitude predictions based on the Oldroyd-B model.

Droplet coalescence is a common phenomenon and plays an important role in multidisciplinary applications. Previous studies mainly consider the coalescence of miscible liquids, even though the coalescence of immiscible droplets on a solid surface is a common process. In this study, we explore the coalescence of two immiscible droplets on a partial wetting surface experimentally and theoretically. We find that the coalescence process can be divided into three stages based on the time scales and force interactions involved, namely (I) the growth of a liquid bridge, (II) the oscillation of the coalescing sessile droplet and (III) the formation of a partially engulfed compound sessile droplet and the subsequent retraction. In stage I, the immiscible interface is found not to affect the scaling of the temporal evolution of the liquid bridge, which follows the same 2/3 power law as that of miscible droplets. In stage II, by developing a new capillary time scale considering both surface and interfacial tensions, we show that the interfacial tension between the two immiscible liquids functions as a non-negligible resistance to the oscillation which decreases the oscillation periods. In stage III, a modified Ohnesorge number is developed to characterize the visco-capillary and inertia-capillary time scales involved during the displacement of water by oil; a new model based on energy balance is proposed to analyse the maximum retraction velocity, highlighting that the viscous resistance is concentrated in a region close to the contact line.

The transfer of momentum between the atmosphere and oceanic motions affected by the Earth's rotation occurs through the thin surface Ekman layer. The exchange depends on the surface wind stress, which produces the Ekman pumping of fluid to the ocean upper layer. The Ekman pumping mainly depends on: (i) the curl of the wind stress and (ii) the advection of vorticity due to the Ekman transport. The wind stress is usually parametrised as a quadratic function of the relative speed between the wind and the ocean currents, providing a feedback mechanism between the two fluids. Under steady and spatially uniform wind conditions over mesoscale vortices, the first mechanism generates vertical motions that induce the vortex decay (top drag), while the second promotes the horizontal advection of vorticity in the Ekman transport direction. This study examines the nonlinear effects of both mechanisms in cyclonic and anticyclonic vortices. The analyses consist of simple analytical approximations and nonlinear numerical simulations of quasi-two-dimensional vortices. When considering only the top drag mechanism, it is found that anticyclones decay faster than cyclones. By considering only the vorticity-advection effect, the vortices acquire horizontal momentum and drift; furthermore, anticyclones are reinforced while cyclones are weakened. The joint action of both mechanisms and the possible consequences on vertical transport properties are also discussed.

The near wake of a hemisphere immersed in a laminar boundary layer is studied utilizing time-resolved tomographic particle image velocimetry (TPIV). Focus is placed on the three-dimensional vortical structures and the formation details of hairpin vortices before the onset of transition. The three-dimensional instantaneous pressure field of the hemisphere wake is reconstructed for better understanding the flow mechanism. Experiments are carried out with Reynolds number $Re_{r}=1370$, based on the hemisphere radius $R$. Features of periodicity of the near wake are analysed using proper orthogonal decomposition and Fourier transformation. The velocity fluctuation in the wall-normal direction is shown to be crucial to the formation of hairpin vortices in the near wake. By investigating the transport of mass and vorticity, and the correlation between pressure and hairpin vortex strength, the formation mechanism is revealed clearly. Specifically, the main hairpin vortices (MHVs) are formed within the reaction of outer high-speed flow and near-wall flow. The formation of the head portion is followed by the leg portion formation. The shedding of the MHVs is highly correlated with the pressure, as well as the pressure gradient in the wall-normal direction. For the side hairpin vortices (SHVs), the leg portion is formed first, followed by the generation of the head portion thanks to induction of the re-oriented standing vortices. The generation of the SHVs can be regarded as the downstream bridging of the standing vortices, similar to the generation of hairpin vortices due to the connection of streamwise vortices in turbulent boundary layers.

We study the effect of a confined turbulent counter-current gas flow on the waviness of a weakly inclined falling liquid film. Our study is centred on experiments in a channel of 13 mm height, using water and air, where we have successively increased the counter-current gas flow rate until flooding. Computations with a new low-dimensional model and linear stability calculations are used to elucidate the linear and nonlinear wave dynamics. We find that the gas pressure gradient plays an important role in countering the stabilizing effect of the tangential gas shear stress at the liquid–gas interface. At very low inclination angles, the latter effect dominates and can suppress the long-wave Kapitza instability unconditionally. By contrast, for non-negligible inclination, the gas effect is linearly destabilizing, amplifies the height of nonlinear Kapitza waves, and exacerbates coalescence-induced formation of large-amplitude tsunami waves. Kapitza waves do not undergo any catastrophic transformation when the counter-current gas flow rate is increased beyond the absolute instability (AI) limit. On the contrary, we find that AI is an effective linear wave selection mechanism in a noise-driven wave evolution scenario, leading to highly regular downward-travelling nonlinear wave trains, which preclude coalescence events. In our experiments, where Kapitza waves develop in a protected region before coming into contact with the gas, flooding is eventually caused far beyond the AI limit by upward-travelling short-wave ripples. Based on our linear stability calculations for arbitrary wavenumbers, we have uncovered a new short-wave interfacial instability mode with negative linear wave speed, causing these ripples.

Correct prediction of particle transport by surface waves is crucial in many practical applications such as search and rescue or salvage operations and pollution tracking and clean-up efforts. Recent results by Deike et al. (J. Fluid Mech., vol. 829, 2017, pp. 364–391) and Pizzo et al. (J. Phys. Oceanogr., vol. 49, no. 4, 2019, pp. 983–992) have indicated transport by deep-water breaking waves is enhanced compared with non-breaking waves. To model particle transport in irregular waves, some of which break, we develop a stochastic differential equation describing both mean particle transport and its uncertainty. The equation combines a Brownian motion, which captures non-breaking drift-diffusion effects, and a compound Poisson process, which captures jumps in particle positions due to breaking. From the corresponding Fokker–Planck equation for the evolution of the probability density function for particle position, we obtain closed-form expressions for its first three moments. We corroborate these predictions with new experiments, in which we track large numbers of particles in irregular breaking waves. For breaking and non-breaking wave fields, our experiments confirm that the variance of the particle position grows linearly with time, in accordance with Taylor's single-particle dispersion theory. For wave fields that include breaking, the compound Poisson process increases the linear growth rate of the mean and variance and introduces a finite skewness of the particle position distribution.

This study examines the effects of surface topography on the flow and performance of a self-propelled swimming (SPS) body. We consider a thin flat plate with an egg-carton roughness texture undergoing prescribed undulatory swimming kinematics at Strouhal number $0.3$ and tail amplitude to length ratio $0.1$; we use plate Reynolds numbers $Re=6$, 12 and $24\times 10^3$, and focus on $12\,000$. As the roughness wavelength is decreased, we find that the undulation wave speed must be increased to overcome the additional drag from the roughness and maintain SPS. Correspondingly, the extra wave speed raises the power required to maintain SPS, making the swimmer less efficient. To decouple the roughness and the kinematics, we compare the rough plates to equivalent smooth cases by matching the kinematic conditions. We find that all but the longest roughness wavelengths reduce the required swimming power and the unsteady amplitude of the forces when compared to a smooth plate undergoing identical kinematics. Additionally, roughness can enhance flow enstrophy by up to 116 % compared to the smooth cases without a corresponding spike in forces; this suggests that the increased mixing is not due to increased vorticity production at the wall. Instead, the enstrophy is found to peak strongly when the roughness wavelength is approximately twice the boundary layer thickness over the $Re$ range, indicating the roughness induces large-scale secondary flow structures that extend to the edge of the boundary layer. This study reveals the nonlinear interaction between roughness and kinematics beyond a simple increase or decrease in drag, illustrating that roughness studies on static shapes do not transfer directly to unsteady swimmers.