We develop perturbation theories to describe the flow dynamics of a droplet with a thin layer of insoluble surfactant whose mechanics are described by interfacial viscosity, i.e. a Boussinesq–Scriven constitutive law. The theories quantify droplet deformation in the limit of small capillary number, large viscosity ratio, or large shear Boussinesq number, to a sufficient level of approximation where one can extract information about nonlinear rheology and droplet breakup. In the first part of this manuscript, we quantify the Taylor deformation parameter and inclination angle in shear and extensional flows, developing expressions that resolve discrepancies between current analytical theories and boundary element simulations. Interestingly, the theories we develop appear to accurately describe the inclination angle of a clean droplet over a wider range of viscosity ratios and capillary numbers than previous works. In the second part of the manuscript, we calculate how interfacial viscosity alters the extra stress of a dilute suspension of droplets, in particular the shear stress, normal stress differences, shear thinning and extensional thickening. The normal stresses are intimately related to the lateral migration of droplets in wall-bound shear flow, and we explore the influence of interfacial viscosity on this phenomenon. We conclude by discussing how one can use these theories to describe droplet breakup, and how one can incorporate additional effects into the perturbation theories such as viscoelastic membranes and/or Marangoni flows.

The role of the external boundary conditions of the nozzle surface on the azimuthal mode selection of impinging supersonic jets is demonstrated for the first time. Jets emanating from thin- and infinite-lipped nozzles at a nozzle pressure ratio of $3.4$ and plate spacing of $5.0D$, where $D$ is the nozzle exit diameter, are investigated using high resolution particle image velocimetry (PIV) and acoustic measurements. Proper orthogonal decomposition is applied to the PIV fields and a difference in dominant instability mode is found. To investigate possible explanations for the change in instability mode, additional nozzle external boundary conditions are investigated, including the addition of acoustic dampening foam. A difference in acoustic feedback path is suggested to be the cause for the change in dominant azimuthal modes between the flows. This is due to the thin-lip case containing a feedback path that is concluded to be closed exclusively by a reflection from the nozzle base surface, rather than directly to the nozzle lip. The ability of the flow to form a feedback path that maximises the impingement tone gain is discussed with consideration of the numerous acoustic feedback paths possible for the given nozzle external boundary conditions.

Inertial particles in turbulent flows are characterised by preferential concentration and segregation and, at sufficient mass loading, dense particle clusters may spontaneously arise due to momentum coupling between the phases. These clusters, in turn, can generate and sustain turbulence in the fluid phase, which we refer to as cluster-induced turbulence (CIT). In the present work, we tackle the problem of developing a framework for the stochastic modelling of moderately dense particle-laden flows, based on a Lagrangian probability-density-function formalism. This framework includes the Eulerian approach, and hence can be useful also for the development of two-fluid models. A rigorous formalism and a general model have been put forward focusing, in particular, on the two ingredients that are key in moderately dense flows, namely, two-way coupling in the carrier phase, and the decomposition of the particle-phase velocity into its spatially correlated and uncorrelated components. Specifically, this last contribution allows us to identify in the stochastic model the contributions due to the correlated fluctuating energy and to the granular temperature of the particle phase, which determine the time scale for particle–particle collisions. The model is then validated and assessed against direct-numerical-simulation data for homogeneous configurations of increasing difficulty: (i) homogeneous isotropic turbulence, (ii) decaying and shear turbulence and (iii) CIT.

Most terrestrial sand seas form at ‘horse’ latitudes, where the wind direction exhibits seasonal variation. Here, we extend the two-dimensional linear stability analysis of a flat sand bed associated with a unidirectional wind to the three-dimensional case in order to account for multidirectional wind regimes. Focusing on the simplest case of bidirectional flow regimes, we show that the transition from transverse to oblique or longitudinal patterns is controlled by the transport ratio and the divergence angle between the two flows. Our predictions agree with previous results for dune orientation, and also provide a wider range of possible alignments depending on flow strength, especially when the two winds are perpendicular, at which the transition occurs. This analysis also predicts the selected pattern wavelength, which either decreases close to the transition angle for strong winds, due to a geometric effect, or increases at low winds, when the bed slope affects the transport. This theoretical analysis is complemented by analogous subaqueous experiments, where bedforms are submitted to alternate water flows. For transverse bedforms, the experimental data validate the model at strong flows, providing evidence for the predicted geometric effect, but also for the increase of the wavelength close to the transport threshold. For longitudinal bedforms, a discrepancy is observed, which we interpret as the sign of enhanced nonlinearities induced by the development of slip faces when the flow alternately blows on both sides of the dune.

The fluid flow and mass transfer characteristics in two-dimensional hydrodynamic focusing are theoretically investigated by considering different physical properties for the sample and sheath flows. Adopting a single-domain formulation, which assigns the region variable physical properties, three-dimensional analytical solutions are obtained for species transport under hydrodynamically fully developed conditions. In addition, simplified analytical solutions are derived assuming a uniform velocity field appropriate to electrokinetic focusing. The results show that the normalized overall mean velocity is an increasing function of the height to width ratio and a decreasing function of the sheath to sample viscosity ratio. The dependence of this normalized mean velocity on the sheath to sample flow-rate ratio is, however, non-monotonic: it grows with the flow-rate ratio when the sample fluid is more viscous than the sheath fluid, whereas the opposite is true when the sheath fluid is more viscous. Moreover, although an increase in either the viscosity or flow-rate ratios results in creating a smaller value of the normalized focused width, varying the channel aspect ratio may lead to either thinner or thicker focused regions, depending on the viscosity ratio. The inspection of the mass transport characteristics reveals that only the viscosity ratio and the Péclet number can significantly alter the mixing length. Surprisingly, the minimum mixing length in the presence of significant axial diffusion is achieved for a single-phase flow. Finally, the dimensionless mixing length is reduced by increasing the Péclet number, although the changes are negligible when this parameter is above 10. This threshold is only true when the Péclet number is calculated based on the higher of the sample and sheath diffusion coefficients.

We present a new approach to model coastal waves in the shoaling and surf zones. The model can be described as a depth-averaged large-eddy simulation model with a cutoff in the inertial subrange. The large-scale turbulence is explicitly resolved through an extra variable called enstrophy while the small-scale turbulence is modelled with a turbulent-viscosity hypothesis. The equations are derived by averaging the mass, momentum and kinetic energy equations assuming a shallow-water flow, a negligible bottom shear stress and a weakly turbulent flow assumption which is not restrictive in practice. The model is fully nonlinear and has the same dispersive properties as the Green–Naghdi equations. It is validated by numerical tests and by comparison with experimental results of the literature on the propagation of a one-dimensional solitary wave over a mild sloping beach. The wave breaking is characterized by a sudden increase of the enstrophy which allows us to propose a breaking criterion based on the new concept of virtual enstrophy. The model features three empirical parameters. The first one governs the turbulent dissipation and was found to be a constant. The eddy viscosity is determined by a turbulent Reynolds number depending only on the bottom slope. The third parameter defines the breaking criterion and depends only on the wave initial nonlinearity. These dependences give a predictive character to the model which is suitable for further developments.

A computational study was performed both of a single agglomerate and of the collision of two agglomerates in a shear flow. The agglomerates were extracted from a direct numerical simulation of a turbulent agglomeration process, and had the loosely packed fractal structure typical of agglomerate structures formed in turbulent agglomeration processes. The computation was performed using a discrete-element method for adhesive particles with four-way coupling, accounting both for forces between the fluid and the particles (and vice versa) as well as force transmission directly between particles via particle collisions. In addition to understanding and characterizing the particle dynamics, the study focused on illuminating the fluid flow field induced by the agglomerate in the presence of a background shear and the effect of collisions on this particle-induced flow. Perhaps the most interesting result of the current work was the observation that the flow field induced by a particle agglomerate rotating in a shear flow has the form of two tilted vortex rings with opposite-sign circulation. These rings are surrounded by a sea of stretched vorticity from the background shear flow. The agglomerate rotates in the shear flow, but at a slower rate than the ambient fluid elements. In the computations with two colliding agglomerates, we observed cases resulting in agglomerate merger, bouncing and fragmentation. However, the bouncing cases were all observed to also result in an exchange of particles between the two colliding agglomerates, so that they were influenced both by elastic rebound of the agglomerate structures as well as by tearing away of particulate matter between the agglomerates. Overall, the problems of agglomerate–flow interaction and of the collision of two agglomerates in a shear flow are considerably richer in physical phenomena and more complex than can be described by the common approximation that represents each agglomerate by an ‘equivalent sphere’.

The dynamics of $A+B\rightarrow C$ fronts is analysed numerically in a radial geometry. We are interested to understand miscible fingering instabilities when the simple chemical reaction changes the viscosity of the fluid locally and a non-monotonic viscosity profile with a global maximum or minimum is formed. We consider viscosity-matched reactants $A$ and $B$ generating a product $C$ having different viscosity than the reactants. Depending on the effect of $C$ on the viscosity relative to the reactants, different viscous fingering (VF) patterns are captured which are in good qualitative agreement with the existing radial experiments. We have found that, for a given chemical reaction rate, an unfavourable viscosity contrast is not always sufficient to trigger the instability. For every fixed Péclet number ($Pe$), these effects of chemical reaction on VF are summarized in the Damköhler number ($Da$) $-$ the log-mobility ratio ($R_{c}$) parameter space that exhibits a stable region separating two unstable regions corresponding to the cases of more and less viscous product. Fixing $Pe$, we determine $Da$-dependent critical log-mobility ratios $R_{c}^{+}$ and $R_{c}^{-}$ such that no VF is observable whenever $R_{c}^{-}\leqslant R_{c}\leqslant R_{c}^{+}$. The effect of geometry is observable on the onset of instability, where we obtain significant differences from existing results in the rectilinear geometry.

Turbulent mixing of heat and momentum in the stably-stratified ocean interior occurs in discrete events driven by vertical variations of the horizontal velocity. Typically, these events have been modelled assuming an initially laminar stratified shear flow which develops wavelike instabilities, becomes fully turbulent, and then relaminarizes into a stable state. However, in the real ocean there is always some level of turbulence left over from previous events. Using direct numerical simulations, we show that the evolution of a stably-stratified shear layer may be significantly modified by pre-existing turbulence. The classical billow structure associated with Kelvin–Helmholtz instability is suppressed and eventually eliminated as the strength of the initial turbulence is increased. A corresponding energetics analysis shows that potential energy changes and dissipation of kinetic energy depend non-monotonically on initial turbulence strength, with the largest effects when initial turbulence is present but insufficient to prevent billow formation. The mixing efficiency decreases with increasing initial turbulence amplitude as the development of the Kelvin–Helmholtz billow, with its large pre-turbulent mixing efficiency, is arrested.

We experimentally investigate particle migration in a non-Brownian suspension sheared in a Taylor–Couette configuration and in the limit of vanishing Reynolds number. Highly resolved index-matching techniques are used to measure the local particulate volume fraction. In this wide-gap Taylor–Couette configuration, we find that for a large range of bulk volume fraction, $\unicode[STIX]{x1D719}_{b}\in [20\,\%{-}50\,\%]$, the fully developed concentration profiles are well predicted by the suspension balance model of Nott & Brady (J. Fluid Mech., vol. 275, 1994, pp. 157–199). Moreover, we provide systematic measurements of the migration strain scale and of the migration amplitude which highlight the limits of the suspension balance model predictions.

A theoretical model is developed which illustrates the dynamics of layering instability, frequently realized in ocean regions with active fingering convection. Thermohaline layering is driven by the interplay between large-scale stratification and primary double-diffusive instabilities operating at the microscale – temporal and spatial scales set by molecular dissipation. This interaction is described by a combination of direct numerical simulations and an asymptotic multiscale model. The multiscale theory is used to formulate explicit and dynamically consistent flux laws, which can be readily implemented in large-scale analytical and numerical models. Most previous theoretical investigations of thermohaline layering were based on the flux-gradient model, which assumes that the vertical transport of density components is uniquely determined by their local background gradients. The key deficiency of this approach is that layering instabilities predicted by the flux-gradient model have unbounded growth rates at high wavenumbers. The resulting ultraviolet catastrophe precludes the analysis of such basic properties of layering instability as its preferred wavelength or the maximal growth rate. The multiscale model, on the other hand, incorporates hyperdiffusion terms that stabilize short layering modes. Overall, the presented theory carries the triple advantage of (i) offering an explicit description of the interaction between microstructure and layering modes, (ii) taking into account the influence of non-uniform stratification on microstructure-driven mixing, and (iii) avoiding unphysical behaviour of the flux-gradient laws at small scales. While the multiscale approach to the parametrization of time-dependent small-scale processes is illustrated here on the example of fingering convection, we expect the proposed technique to be readily adaptable to a wide range of applications.

The onset of interfacial instability in two-fluid systems using a viscous, leaky dielectric model is studied. The instability arises as a result of resonance between the parametric frequency of an imposed electric field and the system’s natural frequency. In addition to a rigorous model that uses Floquet instability analysis, where both viscous and charge effects are considered, this study also provides convincing validating experiments. In other results, it is shown that (a) the imposition of a periodic electrostatic potential acts to counter gravity and this countering effect becomes more effective if a DC voltage is also added, (b) a critical DC voltage exists at which the interface becomes unstable such that no parametric frequency is required to completely destabilize the interface and (c) the leaky dielectric model approaches a model for a perfect dielectric/perfect conductor pair as the conductivity ratio becomes large. It is also shown via experiments that parametric resonant instability using electrostatic forcing may be reliably used to estimate interfacial tension to sufficient accuracy.

The interaction of a thin viscous film with an elastic sheet results in coupling of pressure and deformation, which can be utilized as an actuation mechanism for surface deformations in a wide range of applications, including microfluidics, optics and soft robotics. Implementation of such configurations inherently takes place over finite domains and often requires some pre-stretching of the sheet. Under the assumptions of strong pre-stretching and small deformations of the lubricated elastic sheet, we use the linearized Reynolds and Föppl–von Kármán equations to derive closed-form analytical solutions describing the deformation in a finite domain due to external forces, accounting for both bending and tension effects. We provide a closed-form solution for the case of a square-shaped actuation region and present the effect of pre-stretching on the dynamics of the deformation. We further present the dependence of the deformation magnitude and time scale on the spatial wavenumber, as well as the transition between stretching- and bending-dominant regimes. We also demonstrate the effect of spatial discretization of the forcing (representing practical actuation elements) on the achievable resolution of the deformation. Extending the problem to an axisymmetric domain, we investigate the effects arising from nonlinearity of the Reynolds and Föppl–von Kármán equations and present the deformation behaviour as it becomes comparable to the initial film thickness and dependent on the induced tension. These results set the theoretical foundation for implementation of microfluidic soft actuators based on elastohydrodynanmics.

During filtration in reverse osmosis membranes (ROM), the system performance is dramatically affected by membrane fouling which causes a significant decrease in permeate flux as well as an increase in the energy input required to operate the system. In this work, we develop a model, able to dynamically capture foulant evolution, that couples the transient Navier–Stokes and the advection–diffusion equations, with an adsorption–desorption equation for the foulant accumulation. The model is validated against unsteady measurements of permeate flux as well as steady-state spatial fouling patterns. For a straight channel, we derive a universal scaling relationship between the Sherwood and Bejan numbers, i.e. the dimensionless permeate flux through the membrane and the pressure drop along the channel, respectively, and generalize this result to membranes subject to morphological and/or topological modifications, i.e. whose shape (wiggliness) or surface roughness is altered from the rectangular and flat reference case. We demonstrate that a universal scaling can be identified through the definition of a modified Reynolds number, $Re^{\star }$, that accounts for the additional length scales introduced by the membrane modifications, and a membrane performance index, $\unicode[STIX]{x1D709}$, an aggregate efficiency measure with respect to both clean permeate flux and energy input required to operate the system. Our numerical simulations demonstrate that ‘wiggly’ membranes outperform ‘rough’ membranes for smaller values of $Re^{\star }$, while the trend is reversed at higher $Re^{\star }$. The proposed approach is able to quantitatively investigate, optimize and guide the design of both morphologically and topologically altered membranes under the same framework, while providing insights into the physical mechanisms controlling the overall system performance.

Lattice Boltzmann direct numerical simulation of turbulent open-channel flows over randomly distributed hemispheres at $Re_{\unicode[STIX]{x1D70F}}=600$ is carried out to reveal the influence of roughness parameters related to a probability density function of rough-surface elevation on turbulence by analysing the spatial and Reynolds- (double-) averaged Navier–Stokes equation. This study specifically concentrates on the influence of the root-mean-square roughness and the skewness, and profiles of turbulence statistics are compared by introducing an effective wall-normal distance defined as a wall-normal integrated plane porosity. The effective distance can completely collapse the total shear stress outside the roughness sublayer, and thus the similarity of the streamwise mean velocity is clearer by introducing the effective distance. In order to examine the influence of the root-mean-square roughness and the skewness on dynamical effects that contribute to an increase in the skin friction coefficient, the triple-integrated double-averaged Navier–Stokes equation is analysed. The main contributors to the skin friction coefficient are found to be turbulence and drag force. The turbulence contribution increases with the root-mean-square roughness and/or the skewness. The drag force contribution, on the other hand, increases in particular with the root-mean-square roughness whereas an increase in the skewness does not increase the drag force contribution because it does not necessarily increase the surface area of the roughness elements. The contribution of the mean velocity dispersion induced by spatial inhomogeneity of the rough surfaces substantially increases with the root-mean-square roughness. A linear correlation is confirmed between the root-mean-square roughness and the equivalent roughness while the equivalent roughness monotonically increases with the skewness. A new correlation function based on the root-mean-square roughness and the skewness is developed with the available experimental and direct numerical simulation data, and it is confirmed that the developed correlation reasonably predicts the equivalent roughness of various types of real rough surfaces.

We present a study of absolute and convective instabilities in electrohydrodynamic flow subjected to a Poiseuille flow (EHD-Poiseuille). The electric field is imposed on two infinite flat plates filled with a non-conducting dielectric fluid with unipolar ion injection. Mathematically, the dispersion relation of the linearised problem is studied based on the asymptotic response of an impulse disturbance imposed on the base EHD-Poiseuille flow. Transverse, longitudinal and oblique rolls are investigated to identify the saddle point satisfying the pinching condition in the corresponding complex wavenumber space. It is found that when the ratio of Coulomb force to viscous force increases, the transverse rolls can transit from convective instability to absolute instability. The ratio of hydrodynamic mobility to electric mobility, which exerts negligible effect on the linear stability criterion when the cross-flow is small, has significant influence on the convective–absolute instability transition, especially when the ratio is small. As we change the value of the mobility ratio, a saddle point shift phenomenon occurs in the case of transverse rolls. The unstable longitudinal rolls are convectively unstable as long as there is a cross-flow, a result which is deduced from a one-mode Galerkin approximation. Longitudinal rolls have a larger growth rate than transverse rolls except for a small cross-flow. Finally, regarding the oblique rolls, a numerical search for the saddle point simultaneously in the complex streamwise and transverse wavenumber spaces always yields an absolute transverse wavenumber of zero, implying that oblique rolls give way to transverse rolls when the flow is unstable.

Secondary flows can develop in turbulent boundary layers that grow over surfaces with spanwise inhomogeneities. In this article, we demonstrate the formation of secondary flows in both experimental and numerical tests and dissect the instantaneous structure and topology of these secondary motions. We show that the formation of secondary flows is not very sensitive to the Reynolds number range investigated, and direct numerical simulations and experiments produce similar results in the mean flow as well as the dispersive and turbulent stress distributions. The numerical methods capture time-resolved features of the instantaneous flow and provide insight into the near-wall flow structures, that were previously obscured in the experimental measurements. Proper orthogonal decomposition was shown to capture the essence of the secondary flows in relatively few modes and to be useful as a filter to analyse the instantaneous flow patterns. The secondary flows are found to create extended regions of high Reynolds stress away from the wall that comprise predominantly sweeps similar to what one would expect to see near the wall and which are comparable in magnitude to the near-wall stress. Analysis of the instantaneous flow patterns reveals that the secondary flows are the result of a non-homogeneous distribution of mid-size vortices.

We present a theoretical framework to characterize the steady and unsteady aeroelastic behaviour of compliant membrane wings under different conditions. We develop an analytic model based on thin airfoil theory coupled with a membrane equation. Adopting a numerical solution to the model equations, we study the effects of wing compliance, inertia and flapping kinematics on aerodynamic performance. The effects of added mass and fluid damping on a flapping membrane are quantified using a simple damped oscillator model. As the flapping frequency is increased, membranes go through a transition from thrust to drag around the resonant frequency, and this transition is earlier for more compliant membranes. The wake also undergoes a transition from a reverse von Kármán wake to a traditional von Kármán wake. The wake transition frequency is predicted to be higher than the thrust–drag transition frequency for highly compliant wings.

Geophysical turbulence is observed to self-organize into large-scale flows such as zonal jets and coherent vortices. Previous studies of barotropic $\unicode[STIX]{x1D6FD}$-plane turbulence have shown that coherent flows emerge from a background of homogeneous turbulence as a bifurcation when the turbulence intensity increases. The emergence of large-scale flows has been attributed to a new type of collective, symmetry-breaking instability of the statistical state dynamics of the turbulent flow. In this work, we extend the analysis to stratified flows and investigate turbulent self-organization in a two-layer fluid without any imposed mean north–south thermal gradient and with turbulence supported by an external random stirring. We use a second-order closure of the statistical state dynamics, that is termed S3T, with an appropriate averaging ansatz that allows the identification of statistical turbulent equilibria and their structural stability. The bifurcation of the statistically homogeneous equilibrium state to inhomogeneous equilibrium states comprising zonal jets and/or large-scale waves when the energy input rate of the excitation passes a critical threshold is analytically studied. Our theory predicts that there is a large bias towards the emergence of barotropic flows. If the scale of excitation is of the order of (or larger than) the deformation radius, the large-scale structures are barotropic. Mixed barotropic–baroclinic states with jets and/or waves arise when the excitation is at scales shorter than the deformation radius with the baroclinic component of the flow being subdominant for low energy input rates and insignificant for higher energy input rates. The predictions of the S3T theory are compared with nonlinear simulations. The theory is found to accurately predict both the critical transition parameters and the scales of the emergent structures but underestimates their amplitude.

A self-similar flow generated by water entry of an expanding two-dimensional smooth and curved body is studied based on the incompressible velocity potential theory, with gravity and surface tension effects being ignored. At each expansion speed, mathematical solutions for detached flow with a splash jet and attached flow with jet leaning on the body surface are obtained, both of which have been found possible in experiment, corresponding to hydrophobic and hydrophilic bodies, respectively. The problem is solved using the integral hodograph method, which converts the governing Laplace equations into two integral equations along the half real and imaginary axes of a parameter plane. For the detached flow, the conditions for continuity and finite spatial derivative of the velocity at the point of flow departing form the body surface are imposed. It is found that the Brillouin–Villat criterion for flow detachment of steady flow is also met in this self-similar flow, which requires the curvatures of the free surface and the body surface to be the same at the detachment point. Solutions for the detached flow have been obtained in the whole range of the expansion speeds, from zero to infinity, relative to the water entry speed. For the attached flow there is a minimal expansion speed below which no solution cannot be obtained. Detailed results in terms of pressure distribution, free surface shape and streamlines and tip angle of the jet are presented. It is revealed that when solutions for both detached and attached flows exist, the pressure distributions on the cylinder surface are almost the same up to the point near the jet root. Beyond that point, the pressure relative to the ambient one drops to zero at the detachment point in the former, while it drops below zero in the jet attached on the body and then returns to zero at the contact point in the latter.