We investigate the evolution and stability of a wetting viscous fluid layer flowing down the surface of a cylinder, and surrounded by a conductive gas. The inner cylinder is an electrode kept at constant voltage, and a second, concentric electrode encloses the system whose potential is allowed to vary spatially. This induces electrostatic forces at the interface in competition with surface tension and viscous stresses. Asymptotic methods are used to derive a long-wave axisymmetric model governing the interfacial position and charge density. The resulting system of equations is investigated both analytically and numerically to determine its stability characteristics in the linear and nonlinear regimes.

The linear stability of a rotating contained perfect gas flowing axially is shown to depend on a ‘centrifugal Richardson number’ when the ratio of the axial flow to the peripheral velocity, and the ratio of the peripheral velocity to the sound speed are both small. The Boussinesq approximation is not made. In the limit of infinite sound speed the known incompressible result (unstable, Pedley 1968) is reproduced. Comparison with computational results indicates that the asymptotic theory is pessimistic.

The current work investigates how turbulence affects the mass transfer rate between inertial particles and fluid in a dilute, polydisperse particle system. Direct numerical simulations are performed in which all scales of turbulence are fully resolved and particles are represented in a Lagrangian reference frame. The results show that, similarly to a monodisperse system, the mass transfer rate between particles and fluid decreases as a result of particle clustering. This occurs when the flow time scale (based on the turbulence integral scale) is long relative to the chemical time scale, and is strongest when the particle time scale is one order of magnitude smaller than the flow time scale (i.e. the Stokes number is around 0.1). It is also found that for larger solid mass fractions, the clustering of the heavier particles is enhanced by the effect of drag force from the particles on the fluid (momentum back-reactions or two-way coupling). In particular, when two-way coupling is accounted for, locations of particles of different sizes are much more correlated, which leads to a stronger effect of clustering, and thus a greater reduction of the particle–fluid mass transfer rate.

Laboratory experiments with a bottom hinged surface-piercing cylinder, exposed to irregular deep water waves, are used to investigate high-frequency forcing. The focus is on the secondary load cycle, a strongly nonlinear phenomenon regarding the wave load on a vertical cylinder, first identified by Grue et al. (1993 Preprint Series. Mechanics and Applied Mathematics, pp. 1–30. University of Oslo, available at http://urn.nb.no/URN:NBN:no-52740; 1994 Ninth International Workshop on Water Waves and Floating Bodies (ed. M. Ohkusu), pp. 77–81, available at http://iwwwfb.org). For a total of 2166 single wave events, the force above $3\unicode[STIX]{x1D714}$ (where $\unicode[STIX]{x1D714}$ is the governing wave frequency) is used to identify and split the strongly nonlinear forces into two peaks: a high-frequency peak closely correlated in time with the wave crest when the total load is positive and a high-frequency peak defining the secondary load cycle which occurs close in time to the wave zero downcrossing when the total load is negative. The two peaks are studied by regression analysis as a function of either the Keulegan–Carpenter number ($KC$) or the Froude number ($Fr$). Regarding the secondary load cycle, the best correlation is found with $Fr$. The speed of the travelling edge of the undisturbed wave approximates the fluid velocity. A threshold value separating between small and large forces is found for $KC\sim 4$–5, indicating effects of flow separation. Alternatively, the threshold occurs for $Fr\sim 0.3$–0.4, indicating local wave effects at the scale of the cylinder diameter. The findings suggest that both effects are present and important.

An analysis of the quasi-steady streaming of the liquid in a vertically vibrated horizontal soap film is reported. The air around the soap film is seen to play a variety of roles: it transmits normal and tangential oscillatory stresses to the film, damps out Marangoni waves, and forces non-oscillatory deflection of the film and tangential motion of the liquid. Non-oscillatory volume forcing originating inside the liquid is also analysed. This forcing dominates the quasi-steady streaming when the excitation frequency is close to the eigenfrequency of a Marangoni mode of the soap film, while both volume forcing in the liquid and surface forcing of the gas on the liquid are important when no Marangoni mode resonates. Different manners by which the combined forcings can induce quasi-steady streaming motion are discussed and some numerical simulations of the quasi-steady liquid flow are presented.

When a vertical liquid jet impacts on a solid horizontal surface, the first expectation is to have a circular hydraulic jump. However, in some conditions, for highly viscous fluids, the transition from supercritical to subcritical flow occurs with non-circular shapes such as polygons. Indeed, a quick rotational wave appears on the circular jump before the formation of a polygonal form, which may be related to the Rayleigh–Plateau instability. In this paper, stable polygonal jumps are studied to complete this research. The region of stability is defined for polygonal jumps, and the dependence of this region on the flow governing dimensionless groups is determined experimentally. The results confirm the multistability (hysteresis) of the polygonal jumps, and imply that polygonal jumps with different corner numbers can be created in a certain parameter regime. The size and curvature of the sides of the polygons due to variations of flow rate and downstream obstacle height are also investigated. In addition to the stable ones, our experiments reveal a new type of polygonal jump that has an unstable structure and displays a rotational behaviour with a constant angular velocity, which we call it, ‘rotational hydraulic jump’. It is observed that the angular velocity of this kind of jump depends on the jet flow rate, jet radius and downstream height of the jump. Our observations suggest that the nature of the rotational jump is some kind of surface wave along the jump in clockwise or anticlockwise direction. It seems that the rotational jump has a flow structure that is the same as a type IIb jump. The jump dimensions are studied; the inscribed and circumscribed circular radii of each polygon are measured in order to compare the various polygons together and to find a mean jump radius to compare with Watson’s theory.

Although potential flow, including viscous dissipation, explains quite well the flow around individual bubbles of about 1mm radius rising in water, and e.g. predicts their drag quite accurately, this model cannot explain the homogeneous rise of a bubbly suspension. From numerical and analytical work it follows that eventually all bubbles cluster together. On the other hand it has been shown that velocity fluctuations of the bubbles of sufficient intensity, expressed in terms of a critical (pseudo) temperature, prevents clustering.

Bubbles with radius above 0.8 mm rising in water perform zigzag or spiralling motions. Recently experimental and numerical work has made it clear that such bubbles have a wake behind them consisting of twin vortical threads carrying vorticity of opposite sign in the direction of motion. It is the purpose of this contribution to make an estimate of the velocity fluctuations induced by these trailing vortices in neighbouring bubbles. To this end the two-threaded wake is represented as a horseshoe vortex similar to the wake behind an airfoil. A pair of bubbles is considered and first the velocity induced by the horseshoe vortex behind one of the pair at the centre of the other is calculated. After this the force exerted on the latter based on the induced velocity and on the relative velocity of the bubbles, due to hydrodynamic interaction is calculated. Then the motion of one bubble in the pair is analysed under the influence both of this force and the hydrodynamic forces already there in the absence of the horseshoe vortex. Using these results and appropriate averaging, an estimate is made of the intensity of the velocity fluctuations of bubbles, and the corresponding temperature.

We present the first general metric for attractor overlap (MAO) facilitating an unsupervised comparison of flow data sets. The starting point is two or more attractors, i.e. ensembles of states representing different operating conditions. The proposed metric generalizes the standard Hilbert-space distance between two snapshot-to-snapshot ensembles of two attractors. A reduced-order analysis for big data and many attractors is enabled by coarse graining the snapshots into representative clusters with corresponding centroids and population probabilities. For a large number of attractors, MAO is augmented by proximity maps for the snapshots, the centroids and the attractors, giving scientifically interpretable visual access to the closeness of the states. The coherent structures belonging to the overlap and disjoint states between these attractors are distilled by a few representative centroids. We employ MAO for two quite different actuated flow configurations: a two-dimensional wake with vortices in a narrow frequency range and three-dimensional wall turbulence with a broadband spectrum. In the first application, seven control laws are applied to the fluidic pinball, i.e. the two-dimensional flow around three circular cylinders whose centres form an equilateral triangle pointing in the upstream direction. These seven operating conditions comprise unforced shedding, boat tailing, base bleed, high- and low-frequency forcing as well as two opposing Magnus effects. In the second example, MAO is applied to three-dimensional simulation data from an open-loop drag reduction study of a turbulent boundary layer. The actuation mechanisms of 38 spanwise travelling transversal surface waves are investigated. MAO compares and classifies these actuated flows in agreement with physical intuition. For instance, the first feature coordinate of the attractor proximity map correlates with drag for the fluidic pinball and for the turbulent boundary layer. MAO has a large spectrum of potential applications ranging from a quantitative comparison between numerical simulations and experimental particle-image velocimetry data to the analysis of simulations representing a myriad of different operating conditions.

We consider the linear temporal stability of a Couette flow of a Maxwell gas within the gap between a rotating inner cylinder and a concentric stationary outer cylinder both maintained at the same temperature. The neutral curve is obtained for arbitrary Mach (Ma) and arbitrarily small Knudsen (Kn) numbers by use of a ‘slip-flow’ continuum model and is verified via comparison to direct simulation Monte Carlo results. At subsonic rotation speeds we find, for the radial ratios considered here, that the neutral curve nearly coincides with the constant-Reynolds-number curve pertaining to the critical value for the onset of instability in the corresponding incompressible-flow problem. With increasing Mach number, transition is deferred to larger Reynolds numbers. It is remarkable that for a fixed Reynolds number, instability is always eventually suppressed beyond some supersonic rotation speed. To clarify this we examine the variation with increasing (Ma) of the reference Couette flow and analyse the narrow-gap limit of the compressible TC problem. The results of these suggest that, as in the incompressible problem, the onset of instability at supersonic speeds is still essentially determined through the balance of inertial and viscous-dissipative effects. Suppression of instability is brought about by increased rates of dissipation associated with the elevated bulk-fluid temperatures occurring at supersonic speeds. A useful approximation is obtained for the neutral curve throughout the entire range of Mach numbers by an adaptation of the familiar incompressible stability criteria with the critical Reynolds (or Taylor) numbers now based on average fluid properties. The narrow-gap analysis further indicates that the resulting approximate neutral curve obtained in the (Ma, Kn) plane consists of two branches: (i) the subsonic part corresponding to a constant ratio (Ma/Kn) (i.e. a constant critical Reynolds number) and (ii) a supersonic branch which at large Ma values corresponds to a constant product Ma Kn. Finally, our analysis helps to resolve some conflicting views in the literature regarding apparently destabilizing compressibility effects.

A linearized analysis of the two-dimensional double vortex sheet model of a jet shows that inviscid jet instabilities occur over a wide range of frequencies at all jet Mach numbers. No particular frequency for maximum growth rate exists unless finite shear layer thickness effects are considered. It is suggested that the model describes the essential characteristics of a real jet disturbed by long wavelength perturbations. The idea is advanced that the jet flow constitutes a broad band amplifier of high gain. Disturbances can grow rapidly to a size when nonlinear effects bring about significant interaction with the mean flow. By seeding the jet with disturbances of a type that are highly amplified it is argued that gross features of the flow may be affected and that the jet may be rendered less noisy at high Mach number. It is argued that some of these ideas are supported by the observation that a supersonic jet diffuses at an unusually rapid rate when subject to the oscillatory condition known as ‘screech’.

A theoretical and numerical investigation of two classes of models for pressure-driven core–annular flow is presented. Both classes, referred to as ‘long-wave’ and ‘thin-film’ models, may be derived from a unified perspective using long-wave asymptotics, but are distinct from one another in the role played by the curved tube geometry with respect to the planar (limiting) case. Analytical and numerical techniques are used to show and quantify the significant differences between the behaviour of solutions to both model types. Temporal linear stability analysis of the constant solution is carried out first to pinpoint with closed-form mathematical expressions the different dynamical regimes associated with absolute or convective instabilities. Numerical simulations for the models are then performed and qualitative differences in the evolution of the free surface are explored. Mathematically, different levels of asymptotic accuracy are found to result in different regularizing properties affecting the long-time behaviour of generic numerical solutions. Travelling wave solutions are also studied, and qualitative differences in the topology of streamline patterns describing the flow of the film in a moving reference frame are discussed. These topological differences allow for further classification of the models. In particular, a transition from a regime in which waves trap a fluid core to one where waves travel faster than any parcel of the underlying fluid is documented for a variant of the primary model. In the corresponding thin-film model, no such transition is found to occur. The source of these differences is examined, and a comparison of the results with those of related models in the literature is given. A brief discussion of the merits of each class of models concludes this study.

This paper examines the unsteady lift, drag and moment coefficients experienced by a thin airfoil in high-amplitude pitch ramp motion. Experiments have been carried out in a wind tunnel at moderate Reynolds number ( $Re\approx 1.45\times 10^{4}$ ), using a rigid flat-plate model. Forces and moments have been measured for reduced pitch rates ranging from 0.01 to 0.18, four maximum pitch angles ( $30^{\circ },45^{\circ },60^{\circ },90^{\circ }$ ) and different pivot axis locations between the leading and the trailing edge. Results confirm that for reduced pitch rates lower than 0.03, the unsteady aerodynamics is limited to a stall delay effect. For higher pitch rates, the unsteady response is dominated by a buildup of the circulation, which increases with the pitch rate and the absolute distance between the pivot axis and the $3/4$ -chord location. This circulatory effect induces an overshoot in the normal force and moment coefficients, which is slightly reduced for a flat plate with a finite aspect ratio close to 8 in comparison with the two-dimensional configuration. A new time-dependent model has been tested for both the normal force and moment coefficients. It is mainly based on the superposition of step responses, using the Wagner function and a time-varying input that accounts for the nonlinear variation of the steady aerodynamics, the pivot point location and an additional circulation which depends on the pitch rate. When compared with experiments, it gives satisfactory results for $0^{\circ }$ to $90^{\circ }$ pitch ramp motion and captures the main effect of reduced pitch rate and pivot point location.

Receptivity of a laminar boundary layer to the interaction of time-harmonic free-stream disturbances with a three-dimensional roughness element is studied. The three-dimensional nonlinear triple–deck equations are solved numerically to provide the basic steady-state motion. At high Reynolds numbers, the governing equations for the unsteady motion are the unsteady linearized three-dimensional triple-deck equations. These equations can only be solved numerically. In the absence of any roughness element, the free-stream disturbances, to the first order, produce the classical Stokes flow, in the thin Stokes layer near the wall (on the order of our lower deck). However, with the introduction of a small three-dimensional roughness element, the interaction between the hump and the Stokes flow introduces a spectrum of all spatial disturbances inside the boundary layer. For supercritical values of the scaled Strouhal number, S0 > 2, these Tollmien–Schlichting waves are amplified in a wedge-shaped region, 15° to 18° to the basic-flow direction, extending downstream of the hump. The amplification rate approaches a value slightly higher than that of two-dimensional Tollmien–Schlichting waves, as calculated by the linearized analysis, far downstream of the roughness element.

This paper presents an approximate method for solving Oseen's linearized equations for a two-dimensional steady flow of incompressible viscous fluid past arbitrary cylindrical bodies at low Reynolds numbers. The formulation is based on a discrete singularity method with a least squares criterion for satisfying the no-slip boundary condition. That is, sets of Oseenlets, sinks, sources and vortices are discretely distributed in the interior of the body, and then the least squares criterion attempts to minimize the integrated squares of velocities along the body contour, thus leading to a system of simultaneous algebraic equations. Complex-variable arithmetic, usually available on modern computers, makes the computation algorithm very simple. Furthermore, the method is applicable to cases that cannot be solved by classical analytical approaches. As examples of application, we computed the forces acting on a single circular cylinder, two circular cylinders of equal radius separated by a distance, an inclined elliptic cylinder and an inclined square cylinder all of which are immersed in uniform flow fields. The computed results agree very well with those of classical analytical methods.

We describe the long-term dynamics of sustained stratified shear flows in the laboratory. The stratified inclined duct (SID) experiment sets up a two-layer exchange flow in an inclined duct connecting two reservoirs containing salt solutions of different densities. This flow is primarily characterised by two non-dimensional parameters: the tilt angle of the duct with respect to the horizontal, $\unicode[STIX]{x1D703}$ (a few degrees at most), and the Reynolds number $Re$, an input parameter based on the density difference driving the flow. The flow can be sustained with constant forcing over arbitrarily long times and exhibits a wealth of dynamical behaviours representative of geophysically relevant sustained stratified shear flows. Varying $\unicode[STIX]{x1D703}$ and $Re$ leads to four qualitatively different regimes: laminar flow; mostly laminar flow with finite-amplitude, travelling Holmboe waves; spatio-temporally intermittent turbulence with substantial interfacial mixing; and sustained, vigorous interfacial turbulence (Meyer & Linden, J. Fluid Mech., vol. 753, 2014, pp. 242–253). We seek to explain the scaling of the transitions between flow regimes in the two-dimensional plane of input parameters $(\unicode[STIX]{x1D703},Re)$. We improve upon previous studies of this problem by providing a firm physical basis and non-dimensional scaling laws that are mutually consistent and in good agreement with the empirical transition curves we inferred from 360 experiments spanning $\unicode[STIX]{x1D703}\in [-1^{\circ },6^{\circ }]$ and $Re\in [300,5000]$. To do so, we employ state-of-the-art simultaneous volumetric measurements of the density field and the three-component velocity field, and analyse these experimental data using time- and volume-averaged potential and kinetic energy budgets. We show that regime transitions are caused by an increase in the non-dimensional time- and volume-averaged kinetic energy dissipation within the duct, which scales with $\unicode[STIX]{x1D703}Re$ at high enough angles. As the power input scaling with $\unicode[STIX]{x1D703}Re$ is increased above zero, the two-dimensional, parallel-flow dissipation (power output) increases to close the budget through an increase in the magnitude of the exchange flow, incidentally triggering Holmboe waves above a certain threshold in interfacial shear. However, once the hydraulic limit of two-layer exchange flows is reached, two-dimensional dissipation plateaus and three-dimensional dissipation at small scales (turbulence) takes over, at first intermittently, and then steadily, in order to close the budget and follow the $\unicode[STIX]{x1D703}Re$ scaling. This general understanding of regime transitions and energetics in the SID experiment may serve as a basis for the study of more complex sustained stratified shear flows found in the natural environment.