The interplay between chemical reaction and substrate deformation is discussed by adapting Ranz's formulation for scalar mixing to the case of a reactive mixture between segregated reactants, initially separated by an interface whose thickness may not be vanishingly small. Experiments in a simple shear flow demonstrate the existence of three regimes depending on the Damköhler number $Da=t_s/t_c$ where $t_s$ is the mixing time of the interface width and $t_c$ is the chemical time. Instead of treating explicitly the chemical cross-term, we rationalize these different regimes by globalizing it as a production term involving a flux which depends on the rate at which the reaction zone is fed by the reactants, a formulation valid for $Da>1$. For $Da<1$, the reactants interpenetrate before they react, giving rise to a ‘diffusio-chemical’ regime where chemical production occurs within a substrate whose width is controlled by molecular diffusion.

The thermal resistance associated with the interface between a solid and a liquid is analysed from an atomistic point of view. Partial evaluation of the associated Green–Kubo integral elucidates the various factors governing heat transport across the interface and leads to a quantitative model for the thermal resistance in terms of atomistic-level system parameters. The model is validated using molecular dynamics simulations.

Multiphase flow in porous materials is conventionally described by an empirical extension to Darcy's law, which assumes that the pressure gradient is proportional to the flow rate. Through a series of two-phase flow experiments, we demonstrate that even when capillary forces are dominant at the pore scale, there is a nonlinear intermittent flow regime with a power-law dependence between pressure gradient and flow rate. Energy balance is used to predict accurately the start of the intermittent regime in hydrophobic porous media. The pore-scale explanation of the behaviour based on the periodic filling of critical flow pathways is confirmed through 3D micron-resolution X-ray imaging.

Results from seven direct and large-eddy simulations of gravity currents on slopes ranging from 0.14$^{\circ }$ to 2.86$^{\circ }$ that span from the subcritical to the supercritical regime are studied. By considering a long domain, attention is focused on the near-self-similar state approached by these currents far downstream. In the self-similar limit, the various shape factors, local Richardson number, entrainment coefficient, velocity scale and basal drag coefficient reach a constant value, while the current height, volume and momentum fluxes continue to increase linearly. Their dependence on slope is presented.

The evaporation of multiple sessile droplets is both scientifically interesting and practically important, occurring in many natural and industrial applications. Although there are simple analytic expressions to predict evaporation rates of single droplets, there are no such frameworks for general configurations of droplets of arbitrary size, contact angle or spacing. However, a recent theoretical contribution by Masoud, Howell & Stone (J. Fluid Mech., vol. 927, 2021, R4) shows how considerable insight can be gained into the evaporation of arbitrary configurations of droplets without having either to obtain the solution for the concentration of vapour in the atmosphere or to perform direct numerical simulations of the full problem. The theoretical predictions show excellent agreement with simulations for all configurations, only deviating by 25 % for the most confined droplets.

Ice breaking has become one of the main problems faced by ships and other equipment operating in an ice-covered water region. New methods are always being pursued and studied to improve ice-breaking capabilities and efficiencies. Based on the strong damage capability, a high-speed water jet impact is proposed to be used to break an ice plate in contact with water. A series of experiments of water jet impacting ice were performed in a transparent water tank, where the water jets at tens of metres per second were generated by a home-made device and circular ice plates of various thicknesses and scales were produced in a cold room. The entire evolution of the water jet and ice was recorded by two high-speed cameras from the top and front views simultaneously. The focus was the responses of the ice plate, such as crack development and breakup, under the high-speed water jet loads, which involved compressible pressure ${P_1}$ and incompressible pressure ${P_2}$. According to the main cause and crack development sequence, it was found that the damage of the ice could be roughly divided into five patterns. On this basis, the effects of water jet strength, ice thickness, ice plate size and boundary conditions were also investigated. Experiments validated the ice-breaking capability of the high-speed water jet, which could be a new auxiliary ice-breaking method in the future.

Experimental studies of the influence of fluid–structure interaction on cloud cavitation about a stiff stainless steel (SS) and a flexible composite (CF) hydrofoil have been presented in Parts I (Smith et al., J. Fluid Mech., vol. 896, 2020a, p. A1) and II (Smith et al., J. Fluid Mech., vol. 897, 2020b, p. A28). This work further analyses the data and complements the measurements with reduced-order model predictions to explain the complex response. A two degrees-of-freedom steady-state model is used to explain why the tip bending and twisting deformations are much higher for the CF hydrofoil, while the hydrodynamic load coefficients are very similar. A one degree-of-freedom dynamic model, which considers the spanwise bending deflection only, is used to capture the dynamic response of both hydrofoils. Peaks in the frequency response spectrum are observed at the re-entrant jet-driven and shock-wave-driven cavity shedding frequencies, system bending frequency and heterodyne frequencies caused by the mixing of the two cavity shedding frequencies. The predictions capture the increase of the mean system bending frequency and wider bandwidth of frequency modulation with decreasing cavitation number. The results show that, in general, the amplitude of the deformation fluctuation is higher, but the amplitude of the load fluctuation is lower for the CF hydrofoil compared with the SS hydrofoil. Significant dynamic load amplification is observed at subharmonic lock-in when the shock-wave-driven cavity shedding frequency matches with the nearest subharmonic of the system bending frequency of the CF hydrofoil. Both measurements and predictions show an absence of dynamic load amplification at primary lock-in because of the low intensity of cavity load fluctuations with high cavitation number.

We study the development of perturbations in a submerged air jet with a round cross-section and a long laminar region (five jet diameters) at a Reynolds number of 5400 by both inviscid linear stability theory and experiments. The theoretical analysis shows that there are two modes of growing axisymmetric perturbations, which are generated by three generalized inflection points of the jet's velocity profile. To validate the results of linear stability theory, we conduct experiments with controlled axisymmetric perturbations to the jet. The characteristics of growing waves are obtained by visualization, thermoanemometer measurements and correlation analysis. Experimentally measured wavelengths, growth rates and spatial distributions of velocity fluctuations for both growing modes are in good agreement with theoretical calculations. Therefore, it is demonstrated that small perturbations to the laminar jet closely follow the predictions of inviscid linear stability theory.

When a liquid slams into a solid, the intermediate gas is squeezed out at a speed that diverges when approaching the moment of impact. Although there is mounting experimental evidence that instabilities form on the liquid interface during such an event, understanding of the nature of these instabilities is limited. This study therefore addresses the stability of a liquid–gas interface with surface tension, subject to a diverging flow in the gas phase, where the liquid and the gas phase are both represented as potential fluids. We perform a Kelvin–Helmholtz-type linear modal stability analysis of the surface to obtain an amplitude equation that is subsequently analysed in detail and applied to two cases of interest for impact problems, namely, the parallel impact of a wave onto a vertical wall, and the impact of a horizontal plate onto a liquid surface. In both cases we find that long wavelengths are stabilised considerably in comparison with what may be expected based upon classical knowledge of the stability of interfaces subject to a constant gas flow. In the former case, this leads to the prediction of a marginally stable wavelength that is completely absent in the classical analysis. For the latter we find much resemblance to the classical case, with the connotation that the instability is suppressed for smaller disk sizes. The study ends with a discussion of the influence of gas viscosity and gas compressibility on the respective stability diagrams.

Experiments are conducted to explore the rolling of a cylinder over a pool of viscous fluid. The speed, width and loading of the cylinder are varied along with the initial depth and length of the viscous pool. Depending on the conditions, the cylinder will either ride on a lubrication film or remain in solid contact with the underlying substrate. For the former situation, a lubrication theory is presented that describes the pressure underneath the cylinder and the thickness of the film. The theory approximates the flow by the one-dimensional Reynolds equation with the addition of one term, with an adjustable parameter, to account for the flux of fluid to the cylinder sides. Once this parameter is calibrated against experiment, the theory predicts peak lubrication pressures, gap sizes and film thicknesses to within approximately ten per cent. For lubricated rolling, the film splits evenly between the cylinder and substrate downstream of the nip. The printer's instability arises during the splitting process, patterning the residual fluid films on the substrate and cylinder. If the pool length is less than the cylinder circumference, the fluid adhering to the cylinder is rotated back into contact with the substrate, and when there is sufficient adhered fluid a lubrication film forms that can again be modelled by the theory. Conversely, if there is insufficient adhered fluid, no contiguous lubrication film is formed; instead, the pattern from the printer's instability ‘prints’ from the cylinder to the substrate.

The direct numerical simulation of a non-equilibrium turbulent heat transfer case is performed in a channel flow, where non-equilibrium is induced by a step change in surface temperature. The domain is thus made of two parts in the streamwise direction. Upstream, the flow is turbulent, homogeneous in temperature and the channel walls are adiabatic. The inflow conditions are extracted from a recycling plane located further downstream, so that a fully developed turbulent adiabatic flow reaches the second part. In the domain located downstream, isothermal boundary conditions are prescribed at the walls. The boundary layer, initially at equilibrium, is perturbed by the abrupt change of boundary conditions, and a non-equilibrium transient phase is observed until, further downstream, the flow reaches a new equilibrium state, presenting a fully developed thermal boundary layer. The work aims at identifying the non-equilibrium effects that are expected to be encountered in comparable flows, while providing the means to understand them. In particular, the study allows for the identification of an inner region of the developing boundary layer where several quantities are at equilibrium. Other quantities, instead, exhibit a behaviour of their own, especially in proximity to the leading edge. The analysis is supported by mean and root-mean-square profiles of temperature and velocity, as well as by budgets of first- and second-order moment balance equations for the enthalpy and momentum turbulent fields.

Capillary instability of a two-layer liquid film lining a rigid tube is studied computationally as a model for liquid plug formation and closure of human airways. The two-layer liquid consists of a serous layer, also called the periciliary liquid layer, at the inner side and a mucus layer at the outer side. Together, they form the airway surface liquid lining the airway wall and surrounding an air core. Liquid plug formation occurs due to Plateau–Rayleigh instability when the liquid film thickness exceeds a critical value. Numerical simulations are performed for the entire closure process, including the pre- and post-coalescence phases. The mechanical stresses and their gradients on the airway wall are investigated for physiologically relevant ranges of the mucus-to-serous thickness ratio, the viscosity ratio, and the air–mucus and serous–mucus surface tensions encompassing healthy and pathological conditions of a typical adult human lung. The growth rate of the two-layer model is found to be higher in comparison with a one-layer equivalent configuration. This leads to a much sooner closure in the two-layer model than that in the corresponding one-layer model. Moreover, it is found that the serous layer generally provides an effective protection to the pulmonary epithelium against high shear stress excursions and their gradients. A linear stability analysis is also performed, and the results are found to be in good qualitative agreement with the simulations. Finally, a secondary coalescence that may occur during the post-closure phase is investigated.

The Poiseuille flow (centreline velocity $U_c$) of a fluid (kinematic viscosity $\nu$) past a circular cylinder (radius $R$) in a Hele-Shaw cell (height $2h$) is traditionally characterised by a Stokes flow ($\varLambda =(U_cR/\nu )(h/R)^2 \ll 1$) through a thin gap ($\epsilon =h/R \ll 1$). In this work we use asymptotic methods and direct numerical simulations to explore the parameter space $\varLambda$–$\epsilon$ when these conditions are not met. Starting with the Navier–Stokes equations and increasing $\varLambda$ (which corresponds to increasing inertial effects), four successive regimes are identified, namely the linear regime, nonlinear regimes I and II in the boundary layer (the ‘ inner’ region) and a nonlinear regime III in both the inner and outer region. Flow phenomena are studied with extensive comparisons made between reduced calculations, direct numerical simulations and previous analytical work. For $\epsilon =0.01$, the limiting condition for a steady flow as $\varLambda$ is increased is the instability of the Poiseuille flow. However, for larger $\epsilon$, this limit is at a much higher $\varLambda$, resulting in a laminar separation bubble, of size ${O}(h)$, forming for a certain range of $\epsilon$ at the back of the cylinder, where the azimuthal location was dependent on $\epsilon$. As $\epsilon$ is increased to approximately 0.5, the secondary flow becomes increasingly confined adjacent to the sidewalls. The results of the analysis and numerical simulations are summarised in a plot of the parameter space $\varLambda$–$\epsilon$.

The centrifugal spinning method is a recently invented technique to extrude polymer melts/solutions into ultra-fine nanofibres. Here, we present a superior integrated string-based mathematical model, to quantify the nanofibre fabrication performance in the centrifugal spinning process. Our model enables us to analyse the critical flow parameters covering an extensive range, by incorporating the angular momentum equations, the Giesekus viscoelastic constitutive model, the air-to-fibre drag effects and the energy equation into the string model equations. Using the model, we can analyse the dynamic behaviour of polymer melt/solution jets through the dimensionless flow parameters, namely, the Rossby ($Rb$), Reynolds ($Re$), Weissenberg ($Wi$), Weber ($We$), Froude ($Fr$), air Péclet ($Pe^*$) and air Reynolds ($Re^*$) numbers as well as the viscosity ratio ($\delta _s$), corresponding to rotational, inertial, viscous, viscoelastic, surface tension, gravitational, air thermal diffusivity, aerodynamic and viscosity ratio effects. We find that the nonlinear rheology remarkably affects the fibre trajectory, radius and normal stresses. Increasing $Wi$ leads to a thicker fibre, whereas increasing $\delta _s$ shows an opposite trend. In addition, by increasing $Wi$, the fibre curvature is enhanced, causing the fibre to spiral closer to the rotation centre.

A simulation of stably stratified plane Poiseuille flow at a moderate Reynolds number ($\textit {Re}_\tau = 550$) and Richardson number ($\textit {Ri}_\tau = 480$) is presented. For the first time, the dynamics in the channel core are shown to be described as a series of internal waves that approximately obey a linear wave dispersion relationship. For a given streamwise wavenumber $k_x$ there are two internal wave solutions, a dominant low frequency mode and a weaker-amplitude high-frequency mode, respectively corresponding to ‘backward’ and ‘forward’ propagating internal waves relative to the mean flow. Analysis of linearised equations shows that the dominant low-frequency mode appears to arise due to a particularly sensitive response of the mean flow profiles to incoherent forcing. Instantaneous visualisations reveal that hairpin vortices dominate the outer region of the channel flow, neighbouring the buoyancy dominated channel core. These hairpins are fundamentally different from those observed in canonical unstratified boundary layer flows, as they arise via quasi-linear local processes far from the wall, governed by background shear. Outer region ejection events are common and can be induced by high amplitude waves. Ejected hairpins are transported into the channel core, in turn ‘ringing’ the prevailing strong buoyancy gradient and thus generating high-amplitude internal waves, high dissipation and wave breaking, induced by spanwise vortex stretching and baroclinic vorticity generation. Such spontaneous and sustained generation of quasi-linear internal waves by wall-bounded sheared turbulence may provide novel idealised solutions for, and insight into, large-scale turbulent mixing in a wide range of environmental and industrial flows.

Despite its pivotal role in the classic theory of turbulent line plumes, there has been no consensus on the value of the entrainment coefficient $\alpha$ suitable for a pure plume. Reported measurements vary by 100 %, from $\alpha =0.1$ to $\alpha =0.2$, hindering the predictive capabilities of plume theory. Following our theoretical developments, measurements of plume entrainment using a new approach and a rigorous assessment of reported values for $\alpha$, we conclude that ${\alpha =0.11\pm 15\,\%}$ should be adopted as the consensus value. Our theoretical framework demonstrates how $\alpha$ is determined from underlying plume measurements, and places an emphasis on the link between measurement uncertainty and uncertainty in $\alpha$. This framework inspired our experimental design, intentionally conceived to precisely determine $\alpha$. From measurements of the plume scalar width and the entrainment velocity outside the plume, we determine that $\alpha =0.108\pm 2\,\%$ ($95\,\%$ confidence interval). Complementing our experiments is an evaluation of the historical data which, after we explain why some reported values of $\alpha$ are erroneous, supports the range $0.095\lesssim \alpha \lesssim 0.13$. The proposed consensus value thus represents both our precisely determined value and the variation in the published data. The significance of a consensus value for $\alpha$ can be summarised as follows: (i) it enhances confidence in the application of plume theory to practical situations and (ii) it permits more detailed comparison of entrainment between pure line plumes and related turbulent flows, including forced and lazy line plumes and wall plumes.

We use spectral proper orthogonal decomposition (SPOD) to extract and analyse coherent structures in the turbulent wake of a disk at Reynolds number $ {\textit {Re}} = 5 \times 10^{4}$ and Froude numbers $ {\textit {Fr}} = 2$, 10. We find that the SPOD eigenspectra of both wakes exhibit a low-rank behaviour and the relative contribution of low-rank modes to total fluctuation energy increases with $x/D$. The vortex shedding (VS) mechanism, which corresponds to $ {\textit {St}} \approx 0.11 - 0.13$ in both wakes, is active and dominant throughout the domain in both wakes. The continual downstream decay of the SPOD eigenspectrum peak at the VS mode, which is a prominent feature of the unstratified wake, is inhibited by buoyancy, particularly for $ {\textit {Fr}} = 2$. The energy at and near the VS frequency is found to appear in the outer region of the wake when the downstream distance exceeds $Nt = Nx/U = 6 - 8$. Visualizations show that unsteady internal gravity waves (IGWs) emerge at the same $Nt = 6 - 8$. A causal link between the VS mechanism and the unsteady IGW generation is also established using the SPOD-based reconstruction and analysis of the pressure transport term. These IGWs are also picked up in SPOD analysis as a structural change in the shape of the leading SPOD eigenmode. The $ {\textit {Fr}} = 2$ wake shows layering in the wake core at $Nt > 15$ which is captured by the leading SPOD eigenmodes of the VS frequency at downstream locations $x/D > 30$. The VS mode of the $ {\textit {Fr}} = 2$ wake is streamwise coherent, consisting of $V$-shaped structures at $x/D \gtrsim 30$. Overall, we find that the coherence of wakes, initiated by the VS mode at the body, is prolonged by buoyancy to far downstream. Also, this coherence is spatially modified by buoyancy into horizontal layers and IGWs. Low-order truncations of SPOD modes are shown to efficiently reconstruct important second-order statistics.

We report a systematic study of the collective effect of thermal plumes on the probability density function (p.d.f.) $P(\delta T)$ of temperature fluctuations $\delta T(t)$ in turbulent Rayleigh–Bénard convection. By decomposing $\delta T(t)$ into four basic fluctuation modes associated with single and multiple warm and cold plumes and a turbulent background, we derive an analytic form of $P(\delta T)$ based on the convolutions of the five independent modes. To test the derived form of $P(\delta T)$ in the multiple-plume regions, where the thermal plumes are heavily populated, we conduct time series measurements of temperature fluctuations in two convection cells; one is a vertical thin disk and the other is an upright cylinder of aspect ratio unity. For a given normalized position in most regions of the convection cell, all of the measured p.d.f.s $P(\delta T)$ for different Rayleigh numbers fall onto a single master curve, once $\delta T$ is normalized by its root-mean-square (r.m.s.) value $\sigma _T$. It is found that the measured $P(\delta T/\sigma _T)$ at different locations along the symmetric horizontal and vertical axes of the convection cells can all be well described by the derived form of $P(\delta T/\sigma _T)$. The fitted values of the parameters associated with the number of plumes in multiple plume clusters and their relative strengths and degrees of intermittency are closely linked to the spatial distribution of thermal plumes and local dynamics of the large-scale circulation in a closed convection cell. Our work thus provides a unified theoretical approach for understanding scalar p.d.f.s in a turbulent field, which is very useful not only for the present study but also for the study of many turbulent mixing problems of practical interest.

The generation of liquid jets and drops using tightly focused femtosecond laser pulses near a liquid–air interface is a convenient contactless solution for printing functional materials as well as bio-materials. Jets and drops emerge following the nucleation of a cavitation bubble in the liquid bulk by a laser-induced plasma. During the initial expansion of the bubble, a thin and fast jet is produced at the liquid surface. Moments later a second thick and slow jet emanates from the surface when the bubble has nearly deflated. Despite potential applications, little is known about the mechanism behind this complex phenomenology. Here, experiments and simulations are used to investigate this two-jet process. Counter-intuitively, the second jet is not the result of bubble expansion, as with the first jet, but originates from the secondary flows induced by the bubble dynamics. Our study links the second jet properties to the control parameters of the problem and establishes a phase diagram for its emergence.

We perform interface-resolved simulations of finite-size evaporating droplets in weakly compressible homogeneous shear turbulence. The study is conducted by varying three dimensionless physical parameters: the initial gas temperature over the critical temperature $T_{g,0}/T_c$, the initial droplet diameter over the Kolmogorov scale $d_0/\eta$ and the surface tension, i.e. the shear-based Weber number, $We_{\mathcal {S}}$. For the smallest $We_{\mathcal {S}}$, we first discuss the impact on the evaporation rate of the three thermodynamic models employed to evaluate the gas thermophysical properties: a constant property model and two variable-properties approaches where either the gas density or all the gas properties are allowed to vary. Taking this last approach as reference, the model assuming constant gas properties and evaluated with the ‘1/3’ rule is shown to predict the evaporation rate better than the model where the only variable property is the gas density. Moreover, we observe that the well-known Frössling/Ranz-Marshall correlation underpredicts the Sherwood number at low temperatures, $T_{g,0}/T_c=0.75$. Next, we show that the ratio between the actual evaporation rate in turbulence and the one computed in stagnant conditions is always much higher than one for weakly deformable droplets: it decreases with $T_{g,0}/T_c$ without approaching unity at the highest $T_{g,0}/T_c$ considered. This suggests an evaporation enhancement due to turbulence also in conditions typical of combustion applications. Finally, we examine the overall evaporation rate and the local interfacial mass flux at higher $We_{\mathcal {S}}$, showing a positive correlation between evaporation rate and interfacial curvature, especially at the lowest $T_{g,0}/T_c$.