Taylor–Couette flow is inevitably associated with the visually appealing toroidal vortices, waves, and spirals that are instigated by linear instability. The linearly stable regimes, however, pose a new challenge: do they undergo transition to turbulence and if so, what is its mechanism? Maretzke et al. (J. Fluid Mech., vol. 742, 2014, pp. 254–290) begin to address this question by determining the transient growth over the entire parameter space. They find that in the quasi-Keplerian regime, the optimal perturbations take the form of Taylor columns and that the maximum energy achieved depends only on the shear.

In this paper, the transient settling dynamics of a spherical particle sedimenting in a linearly stratified fluid is investigated by performing fully resolved direct numerical simulations. The settling behaviour is quantified for different values of Reynolds, Froude and Prandtl numbers. It is demonstrated that the transient settling dynamics is correlated to the induced Lagrangian drift of flow around the settling particle. A simplified model is provided to predict the maximum velocity of the settling particle in linearly stratified fluids. The peak velocity can be followed by the oscillation of the settling velocity and the particle can even reverse its direction of motion before reaching to its neutrally buoyant level. The frequency of oscillation of settling velocity scales with the Brunt–Väisälä frequency and the motion of the particle can lead to the formation of secondary and tertiary vortices following the primary vortex.

Double-diffusive flux-gradient laws are commonly used to describe the development of large-scale structures driven by salt fingers – thermohaline staircases, collective instability waves and intrusions. The flux-gradient model assumes that the vertical transport is uniquely determined by the local background temperature and salinity gradients. While flux-gradient laws adequately capture mixing characteristics on scales that greatly exceed those of primary double-diffusive instabilities, their accuracy rapidly deteriorates when the scale separation between primary and secondary instabilities is reduced. This study examines conditions for the breakdown of the flux-gradient laws using a combination of analytical arguments and direct numerical simulations. The applicability (failure) of the flux-gradient laws at large (small) scales is illustrated through the example of layering instability, which results in the spontaneous formation of thermohaline staircases from uniform temperature and salinity gradients. Our inquiry is focused on the properties of the ‘point-of-failure’ scale ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H_{pof}$) at which the vertical transport becomes significantly affected by the non-uniformity of the background stratification. It is hypothesized that $H_{pof} $ can control some key characteristics of secondary double-diffusive phenomena, such as the thickness of high-gradient interfaces in thermohaline staircases. A more general parametrization of the vertical transport – the flux-gradient-aberrancy law – is proposed, which includes the selective damping of relatively short wavelengths that are inadequately represented by the flux-gradient models. The new formulation is free from the unphysical behaviour of the flux-gradient laws at small scales (e.g. the ultraviolet catastrophe) and can be readily implemented in theoretical and large-scale numerical models of double-diffusive convection.

The wake behind a cube with a face normal to the flow was investigated experimentally in a water tunnel using laser induced fluorescence (LIF) visualisation and particle image velocimetry (PIV) techniques. Measurements were carried out for moderate Reynolds numbers between 100 and 400 and in this range a sequence of two flow bifurcations was confirmed. Values for both onsets were determined in the framework of Landau’s instability model. The measured longitudinal vorticity was separated into three components corresponding to each of the identified regimes. It was shown that the vorticity associated with a basic flow regime originates from corners of the bluff body, in contrast to the two other contributions which are related to instability effects. The present experimental results are compared with numerical simulation carried out earlier by Saha (Phys. Fluids, vol. 16, 2004, pp. 1630–1646).

The relationship between nonlinear equilibrium solutions of the full Navier–Stokes equations and the high-Reynolds-number asymptotic vortex–wave interaction (VWI) theory developed for general shear flows by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666) is investigated. Using plane Couette flow as a prototype shear flow, we show that all solutions having $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}O(1)$ wavenumbers converge to VWI states with increasing Reynolds number. The converged results here uncover an upper branch of VWI solutions missing from the calculations of Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). For small values of the streamwise wavenumber, the converged lower-branch solutions take on the long-wavelength state of Deguchi, Hall & Walton (J. Fluid Mech., vol. 721, 2013, pp. 58–85) while the upper-branch solutions are found to be quite distinct, with new states associated with instabilities of jet-like structures playing the dominant role. Between these long-wavelength states, a complex ‘snaking’ behaviour of solution branches is observed. The snaking behaviour leads to complex ‘entangled’ states involving the long-wavelength states and the VWI states. The entangled states exhibit different-scale fluid motions typical of those found in shear flows.

A method for generating controllable two-dimensional velocity fluctuations using two pitching foils was derived theoretically in a previous companion paper. The present work describes the experimental implementation of the method. The experiments are carried out in a re-circulating water channel optimised to provide low turbulence intensity in the incoming flow. Velocities are measured using an acoustic Doppler velocimeter (ADV). The pitching motions of the foils are position-controlled using a closed-loop control system. Two velocity fluctuation patterns are investigated. They consist of a combination of sinusoidal components. Theoretical predictions and experimental measurements are compared in the time and frequency domain. Although some discrepancies are observed, the agreement is generally good and therefore validates the theoretical method for the conditions investigated.

The paper describes a process which allows a vertical circular cylinder subject to plane monochromatic surface gravity waves to appear invisible to the far-field observer. This is achieved by surrounding the cylinder with an annular region of variable bathymetry. Two approaches are taken to investigate this effect. First a mild-slope approximation is applied to the governing linearised three-dimensional water wave equations to formulate a depth-averaged two-dimensional wave equation with varying wavenumber over the variable bathmetry. This is then solved by formulating a domain integral equation, solved numerically by discretisation. For a given set of geometrical and wave parameters, the bathymetry is selected by a numerical optimisation process and it is shown that the scattering cross-section is reduced towards zero with increasing refinement of the bathymetry. A fully three-dimensional boundary-element method, based on the WAMIT solver (see www.wamit.com) but adapted here to allow for depressions in the bed, is used to assess the accuracy of the mild-slope results and then further numerically optimise the bathymetry towards a cloaking structure. Numerical results provide strong evidence that perfect cloaking is possible for the fully three-dimensional problem. One practical application of the results is that cloaking implies a reduced mean drift force on the cylinder.

When a flexible vesicle is placed in an extensional flow (planar or uniaxial), it undergoes two unique sets of shape transitions that to the best of the authors’ knowledge have not been observed for droplets. At intermediate reduced volumes (i.e. intermediate particle aspect ratio) and high extension rates, the vesicle stretches into an asymmetric dumbbell separated by a long, cylindrical thread. At low reduced volumes (i.e. high particle aspect ratio), the vesicle extends symmetrically without bound, in a manner similar to the breakup of liquid droplets. During this ‘burst’ phase, ‘pearling’ occasionally occurs, where the vesicle develops a series of periodic beads in its central neck. In this paper, we describe the physical mechanisms behind these seemingly unrelated instabilities by solving the Stokes flow equations around a single, fluid-filled particle whose interfacial dynamics is governed by a Helfrich energy (i.e. the membranes are inextensible with bending resistance). By examining the linear stability of the steady-state shapes, we determine that vesicles are destabilized by curvature changes on its interface, similar to the Rayleigh–Plateau phenomenon. This result suggests that the vesicle’s initial geometry plays a large role in its shape transitions under tension. The stability criteria calculated by our simulations and scaling analyses agree well with available experiments. We hope that this work will lend insight into the stretching dynamics of other types of biological particles with nearly incompressible membranes, such as cells.

The numerical simulation of a flow through a duct requires an externally specified forcing that makes the fluid flow against viscous friction. To this end, it is customary to enforce a constant value for either the flow rate (CFR) or the pressure gradient (CPG). When comparing a laminar duct flow before and after a geometrical modification that induces a change of the viscous drag, both approaches lead to a change of the power input across the comparison. Similarly, when carrying out direct numerical simulation or large-eddy simulation of unsteady turbulent flows, the power input is not constant over time. Carrying out a simulation at constant power input (CPI) is thus a further physically sound option, that becomes particularly appealing in the context of flow control, where a comparison between control-on and control-off conditions has to be made. We describe how to carry out a CPI simulation, and start with defining a new power-related Reynolds number, whose velocity scale is the bulk flow that can be attained with a given pumping power in the laminar regime. Under the CPI condition, we derive a relation that is equivalent to the Fukagata–Iwamoto–Kasagi relation valid for CFR (and to its extension valid for CPG), that presents the additional advantage of naturally including the required control power. The implementation of the CPI approach is then exemplified in the standard case of a plane turbulent channel flow, and then further applied to a flow control case, where a spanwise-oscillating wall is used for skin-friction drag reduction. For this low-Reynolds-number flow, using 90 % of the available power for the pumping system and the remaining 10 % for the control system is found to be the optimum share that yields the largest increase of the flow rate above the reference case where 100 % of the power goes to the pump.

Closed-form solutions describing the behaviour of two-dimensional planar turbulent rising plumes and fountains from horizontal planar area and line sources in unconfined quiescent environments of uniform density are proposed. Extending the analysis on axisymmetric releases by van den Bremer & Hunt (J. Fluid Mech., vol. 644, 2010, pp. 165–192) to planar releases, the local flux balance parameter $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\varGamma =\varGamma (z)$ is instrumental in describing the bulk behaviour of steady Boussinesq and non-Boussinesq planar plumes and the initial rise behaviour of Boussinesq planar fountains as a function of height $z$. Expressions for the asymptotic virtual source correction are developed and the results elucidated by ‘scale diagrams’ (cf. Morton & Middleton, J. Fluid Mech., vol. 58, 1973, pp. 165–176) showing certain characteristic heights for different source conditions. These diagrams capture all the different manifestations of plume behaviour, encompassing fountains, jets, source-momentum-dominated or ‘forced’ plumes, pure plumes and source-buoyancy-dominated or ‘lazy’ plumes, and their associated key features. Other flow features identified include a gravity-driven deceleration regime and a mixing-driven regime for forced fountains. Deceleration in lazy fountains is purely gravity-driven. The results can be shown to be valid for both Boussinesq and non-Boussinesq plumes (but not for non-Boussinesq fountains) thus resulting in universal solutions valid for both cases provided the entrainment velocity is unaffected by non-Boussinesq effects. This paper presents and explores these universal solutions. An accompanying paper (van den Bremer & Hunt, J. Fluid Mech., vol. 750, 2014, pp. 245–258) examines the implications for non-Boussinesq plumes. The existing solutions of Lee & Emmons (J. Fluid Mech., vol. 11, 1961, pp. 353–368) generalized herein are valid for a constant entrainment coefficient $\alpha $. New results for an entrainment coefficient that varies linearly with $\varGamma (z)$ and thus captures experimental values far more realistically are presented for forced plumes.

In an accompanying paper (van den Bremer & Hunt, J. Fluid Mech., vol. 750, 2014, pp. 210–244) closed-form solutions, describing the behaviour of two-dimensional planar turbulent rising plumes from horizontal planar area and line sources in unconfined quiescent environments of uniform density, that are universally applicable to Boussinesq and non-Boussinesq plumes, are proposed. This universality relies on an entrainment velocity unmodified by non-Boussinesq effects, an assumption that is derived in the literature based on similarity arguments and is, in fact, in contradiction with the axisymmetric case, in which entrainment is modified by non-Boussinesq effects. Exploring these solutions, we show that a non-Boussinesq plume model predicts exactly the same behaviour with height for a pure plume as would a Boussinesq model, whereas the effects on forced and lazy plumes are opposing. Non-intuitively, the non-Boussinesq model predicts larger fluxes of volume and mass for lazy plumes, but smaller fluxes for forced plumes at any given height compared to the Boussinesq model. This raises significant questions regarding the validity of the unmodified entrainment model for planar non-Boussinesq plumes based on similarity arguments and calls for detailed experiments to resolve this debate.

The excursion number, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}Ex = U_0/\varOmega l$, is a parameter that characterizes the ratio of streamwise fluid advection during a tidal oscillation of amplitude $U_0$ and frequency $\varOmega $ to the streamwise topographic length scale $l$. Direct numerical simulations are performed to study how internal gravity waves and turbulence change when $Ex$ is varied from a low value (typical of a ridge in the deep ocean) to a value of unity (corresponding to energetic tides over a small topographic feature). An isolated obstacle having a smoothed triangular shape and 20 % of the streamwise length at critical slope is considered. With increasing values of $Ex$, the near field of the internal waves loses its beam-like character, the wave response becomes asymmetric with respect to the ridge centre, and transient lee waves form. Analysis of the baroclinic energy balance shows significant reduction in the radiated wave flux in the cases with higher $Ex$ owing to a substantial rise in advection and baroclinic dissipation as well as a decrease in conversion. Turbulence changes qualitatively with increasing $Ex$. In the situation with $Ex \sim 0.1$, turbulence is intensified at the near-critical regions of the slope, and is also significant in the internal wave beams above the ridge where there is intensified shear. At $Ex = O(1)$, the transient lee waves overturn adjacent to the ridge flanks and, owing to convective instability, buoyancy acts as a source for turbulent kinetic energy. The size of the turbulent overturns has a non-monotonic dependence on excursion number: the largest overturns, as tall as twice the obstacle height, occur in the $Ex = 0.4$ case, but there is a substantial decrease of overturn size at larger values of $Ex$ simulated here.

The theory to predict the performance and loads on rotors (propellers, screws, windmills) has a history of more than a century. Apart from modern computational fluid dynamics and vortex panel models taking the true blade geometry into account, most other models proceed from an infinitely thin actuator disc or line. These models assume an externally defined force field distributed at the disc or line, representing the loads on the real rotor. Given this force field, the flow is solved by momentum balances or by the equations of motion. The use of external force fields was discussed in textbooks of the first decades of the 20th century, but has received little attention since then. This paper investigates the higher-order effect of adding thickness to the actuator disc or changing the actuator line to a blade with cross-sectional dimensions. For the generation of a Rankine vortex by a force field acting on an actuator disc with thickness, an exact solution has been found in which not only the thrust and torque determine the flow, but also a radial force. This force is conservative, in contrast to the other force components. For rotor blades, a conservative normal and radial force acting on the chordwise bound vorticity is present. This explains the experimentally observed inboard motion of the tip vortex of model wind turbine rotors before the wake induction field drives it outboard. Simulations by computational fluid mechanics and a vortex panel code reproduce the inboard motion, but an actuator line analysis, in which the chordwise vorticity is absent, does not. The conservative load is only $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1\mbox{--}2\, \%$ of the thrust on the entire blade but ${\approx }10\, \%$ of the thrust at the tip ($r/R>0.9$). Conservative forces at the disc and rotor blade vanish for vanishing disc thickness or blade cross-section, so play no role in any of the infinitely thin actuator disc or line methods. However, if higher-order effects of non-zero dimensions are to be modelled, the conservative force field has to be included.

Active control of a turbulent boundary layer has been experimentally investigated with a view to reducing the skin-friction drag and gaining some insight into the mechanism that leads to drag reduction. A spanwise-aligned array of piezo-ceramic actuators was employed to generate a transverse travelling wave along the wall surface, with a specified phase shift between adjacent actuators. Local skin-friction drag exhibits a strong dependence on control parameters, including the wavelength, amplitude and frequency of the oscillation. A maximum drag reduction of 50 % has been achieved at 17 wall units downstream of the actuators. The near-wall flow structure under control, measured using smoke–wire flow visualization, hot-wire and particle image velocimetry techniques, is compared with that without control. The data have been carefully analysed using techniques such as streak detection, power spectra and conditional averaging based on the variable-interval time-average detection. All the results point to a pronounced change in the organization of the perturbed boundary layer. It is proposed that the actuation-induced wave generates a layer of highly regularized streamwise vortices, which acts as a barrier between the large-scale coherent structures and the wall, thus interfering with the turbulence production cycle and contributing partially to the drag reduction. Associated with the generation of regularized vortices is a significant increase, in the near-wall region, of the mean energy dissipation rate, as inferred from a substantial decrease in the Taylor microscale. This increase also contributes to the drag reduction. The scaling of the drag reduction is also examined empirically, providing valuable insight into the active control of drag reduction.

Bubble oscillations play a crucial role in ultrasonic cleaning, a process by which micro- and nanoscale contaminant particles are removed from solid surfaces, such as semiconductor wafers, photomasks and membranes. Although it is well known that the ultrasonic cleaning may damage the functional patterns of ever-shrinking size in current manufacturing technology while removing dust and debris, the mechanisms leading to such damage have been elusive. Here we report observations of the dynamics of bubbles that yield microstructure damage under a continuous ultrasonic field via high-speed imaging. We find that the bubble behaviour can be classified into four types, namely volume oscillation, shape oscillation, splitting and chaotic oscillation, depending on the acoustic pressure and bubble size. This allows us to construct a regime map that can predict the bubble behaviour near a wall based on the experimental parameters. Our visualization experiments reveal that damage of microwalls and microcantilevers arises due to either splitting small bubbles or chaotically oscillating large bubbles in the ultrasonic field, with the forces generated by them quantitatively measured.

The free interface separating an exterior, viscous fluid from an intrusive conduit of buoyant, less viscous fluid is known to support strongly nonlinear solitary waves due to a balance between viscosity-induced dispersion and buoyancy-induced nonlinearity. The overtaking, pairwise interaction of weakly nonlinear solitary waves has been classified theoretically for the Korteweg–de Vries equation and experimentally in the context of shallow water waves, but a theoretical and experimental classification of strongly nonlinear solitary wave interactions is lacking. The interactions of large amplitude solitary waves in viscous fluid conduits, a model physical system for the study of one-dimensional, truly dissipationless, dispersive nonlinear waves, are classified. Using a combined numerical and experimental approach, three classes of nonlinear interaction behaviour are identified: purely bimodal, purely unimodal, and a mixed type. The magnitude of the dispersive radiation due to solitary wave interactions is quantified numerically and observed to be beyond the sensitivity of our experiments, suggesting that conduit solitary waves behave as ‘physical solitons’. Experimental data are shown to be in excellent agreement with numerical simulations of the reduced model. Experimental movies are available with the online version of the paper.

The steady, two-dimensional transition criteria between regular and Mach reflection are well established. Little has been done on the dynamic effect on transition due to a rapidly rotating wedge. Results from experiments and computations done on steady and unsteady shock wave transition from Mach reflection to regular reflection, MR $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\rightarrow $ RR, are described. The measured motion and the initial shock incidence was used to mimic the experiment with a two-dimensional numerical code. The maximum rotation speed achieved at transition was approximately $2500^\circ \ {\mathrm{s}}^{-1}$. Rapid wedge rotation was shown to have a significant measurable effect on transition. The code was also applied to the dependence of dynamic MR $\rightarrow $ RR transition on other variables in the parameter space. These include rotation about the leading or trailing edge, initial incidence and rotation speed at two free-stream conditions. Impulsively started rotation in these cases was used with the rotation specified by $M_E = \omega c/a_{\infty }$ where $\omega $ is constant angular velocity (negative anticlockwise), $c$ the distance from the edge considered to the pivot point and $a_{\infty }$ the free-stream sound speed. For the Mach numbers and range of rotation speeds tested, both the wedge and shock angle at transition decreased with increased rotation speed. The sensitivity of the transition angle to changing the rotation point from the trailing edge to the experimental model pivot point was investigated briefly at a free-stream Mach number of $M = 2.98$ with $M_E = -0.1$. The wedge angle at transition increased by 1.5° and the shock angle at transition decreased by 1.5°, a significant variation. The effect of the initial incidence was also investigated. By reducing the initial wedge angle from 24.5 to 23.5° for these initial conditions the shock angle at transition decreased by approximately 1.8°, also a marked sensitivity. The flow field development for impulsive rotation about the wedge trailing and leading edges at $M = 1.93$ for $M_E = -0.075$ was analysed in some detail. The flow field development is very sensitive to the rotation centre, more especially at large rotation rates. Four phases of the Mach stem development were identified in both cases. For rotation about the wedge leading edge the Mach stem height remains constant until the expansion waves arrive at the triple point. This is followed by an increase in Mach stem height, which then remains constant for a short period after which it decreases until transition to RR. For rotation about the wedge trailing edge the impulsive start generates a disturbance on the incident wave which propagates down the wave, through the triple point and down the Mach stem. The stem height is constant until the arrival of the incident wave disturbance. This causes a sudden decrease in Mach stem height. Subsequently, the Mach stem height remains constant for a short time, before it decreases until transition to RR. Similar effects in the variation of stem height with wedge angle occur at the higher Mach number of 2.98. It was demonstrated that MR can be maintained for a while at zero wedge incidence with a sufficiently large rotation rate of $M_E = -0.1$, with $M=1.93$, for both leading and trailing edge pivot points.

The propagation of axisymmetric solitary waves on the surface of an otherwise cylindrical ferrofluid jet subjected to a magnetic field is investigated. An azimuthal magnetic field is generated by an electric current flowing along a stationary metal rod which is mounted along the axis of the moving jet. A numerical method is used to compute fully nonlinear travelling solitary waves, and the predictions of elevation waves and depression waves made by Rannacher and Engel (New J. Phys., vol. 8, 2006, pp. 108–123) using a weakly nonlinear theory are confirmed in the appropriate ranges of the magnetic Bond number. New nonlinear branches of solitary wave solutions are identified. As the Bond number is varied, the solitary wave profiles may approach a limiting configuration with a trapped toroidal-shaped bubble, or they may approach a static wave (i.e. one with zero phase speed). For a sufficiently large axial rod, the limiting profile may exhibit a cusp.

A novel concept for simulation of turbulent mixing, termed hierarchical parcel swapping (HiPS), was recently proposed. The method involves either a parameterized representation of the turbulent flow or a more self-contained flow simulation. As a step toward turbulent mixing applications, the latter formulation is used for the first numerical demonstration of model performance. Owing to its suitability for this purpose and its role as a canonical benchmark, channel flow is the target application. Despite its idealized representation of this flow, HiPS is shown to capture salient features of the flow with a notable degree of quantitative accuracy. The implications of this finding with regard to flow physics and with regard to the applicability of HiPS to other problems are discussed.

We consider the linear and nonlinear stability of two-phase density-matched but viscosity-contrasted fluids subject to laminar Poiseuille flow in a channel, paying particular attention to the formation of three-dimensional waves. A combination of Orr–Sommerfeld–Squire analysis (both modal and non-modal) with direct numerical simulation of the three-dimensional two-phase Navier–Stokes equations is used. For the parameter regimes under consideration, under linear theory, the most unstable waves are two-dimensional. Nevertheless, we demonstrate several mechanisms whereby three-dimensional waves enter the system, and dominate at late time. There exists a direct route, whereby three-dimensional waves are amplified by the standard linear mechanism; for certain parameter classes, such waves grow at a rate less than but comparable to that of the most dangerous two-dimensional mode. Additionally, there is a weakly nonlinear route, whereby a purely spanwise wave grows according to transient linear theory and subsequently couples to a streamwise mode in weakly nonlinear fashion. Consideration is also given to the ultimate state of these waves: persistent three-dimensional nonlinear waves are stretched and distorted by the base flow, thereby producing regimes of ligaments, ‘sheets’ or ‘interfacial turbulence’. Depending on the parameter regime, these regimes are observed either in isolation, or acting together.