In a recent paper, Deguchi & Hall (J. Fluid Mech., vol. 752, 2014a, pp. 602–625) described a new kind of exact coherent structure which sits at the edge of an asymptotic suction boundary layer at high values of the Reynolds number $Re$. At a distance $\ln Re$ from the wall, the structure is driven by the fully nonlinear interaction of tiny rolls, waves and streaks convected downstream at almost the free-stream speed. The interaction problem satisfies the unit-Reynolds-number three-dimensional Navier–Stokes equations and is localized in a layer of the same depth as the unperturbed boundary layer. Here, we show that the interaction problem is generic to any boundary layer that approaches its free-stream form through an exponentially small correction. It is shown that away from the layer where it is generated the induced roll–streak flow is dominated by non-parallel effects which now play a major role in the streamwise evolution of the structure. The similarity with the parallel boundary layer case is restricted only to the layer where it is generated. It is shown that non-parallel effects cause the structure to persist only over intervals of finite length in any growing boundary layer and lead to a flow structure reminiscent of turbulent boundary layer simulations. The results found shed light on a possible mechanism to couple near-wall streaks with coherent structures located towards the edge of a turbulent boundary layer. Some discussion of how the mechanism adapts to a three-dimensional base flow is given.

We perform a linearized local stability analysis for short-wavelength perturbations of a circular Couette flow with a radial temperature gradient. Axisymmetric and non-axisymmetric perturbations are considered and both the thermal diffusivity and the kinematic viscosity of the fluid are taken into account. The effect of asymmetry of the heating both on centrifugally unstable flows and on the onset of instabilities of centrifugally stable flows, including flows with a Keplerian shear profile, is thoroughly investigated. It is found that an inward temperature gradient destabilizes the Rayleigh-stable flow either via Hopf bifurcation if the liquid is a very good heat conductor or via steady state bifurcation if viscosity prevails over the thermal conductance.

It is shown how a two-dimensional surface wave can be either totally reflected or totally transmitted in the presence of two parallel vertical barriers each containing a small gap. Total transmission of a surface wave past obstacles has been known to occur in many situations in water-wave theory, but total reflexion is a comparatively new phenomenon which could be of practical use in the design of breakwaters.

Shear flow along a slotted plate can give rise to highly coherent, small-scale oscillations within each slot; the dimensionless frequency is an order of magnitude lower than the frequency of the classical Kelvin–Helmholtz instability. Within a given slot, the unsteady development of a swirl pattern, as well as the flow into and out of the slot, is compatible with the evolution of a large-scale structure, which shows increased agglomeration of vorticity along the surface of the plate. Moreover, the streamwise phase shift of the unsteady flow rates through the slots is in accord with the phase shift of the velocity fluctuation of the downstream propagating disturbance along either side of the plate. This phase shift is associated with development of the large-scale structure, and corresponds to a dimensionless frequency that scales on the plate length, as verified by experiments over ranges of length and inflow velocity. The value of this dimensionless frequency is close to that in the absence of the slotted plate i.e. the Kelvin–Helmholtz instability of the free shear layer.

Volumetric expansion driven drainage flow of a viscous compressible fluid from a small capillary with a sealed end is studied in the low Mach number limit using the linearized compressible Navier–Stokes equations with no-slip condition. Density relaxation, oscillation and decay as well as the velocity field are investigated in detail. It is shown that fluid drainage is controlled by the slow decay of the standing acoustic wave inside the capillary; and the acoustic wave retards the density diffusion by reducing the diffusion coefficient of the density envelope equation by one half. Remarkably the no-slip flow exhibits a slip-like mass flow rate. The period-averaged mass flow rate at the exit (drainage rate) is found proportional to the fluid’s kinematic viscosity via the density diffusion coefficient and the average drainage speed is independent of the capillary radius. These findings are valid for arbitrarily small capillaries as long as the continuum assumption holds and they are in stark contrast to the classical lubrication based theory. Generalization to a capillary with a sound absorbing end is achieved by a simple model. The reported results offer new insights to the nature of slow viscous compressible flows in very small capillaries.

Electrochemical measurements of the local wall shear were made along the inner bend of a curved pipe in the entry region, to examine the nature of the secondary-flow boundary-layer collision. Analysis by Stewartson, Cebeci & Chang (1980) predicts a vanishing of the wall shear at the collision point, but without flow reversal. The experimental results are in good agreement with the analytical predictions upstream of the collision point, and its location appears to be predicted accurately. However, downstream of the collision point agreement is poor.

The transformation of a time-dependent pressure pulse imposed on the front face of a rigid porous medium sample, mounted in a tunnel, through the sample and a fixed-volume air gap between the rear face of the sample and the end wall of a tunnel is studied both experimentally and analytically. In the experiments, rigid porous samples that are placed at various distances from a shock tube end wall are subjected to the impingement of shock waves. The pressure buildup behind the porous sample is monitored and compared with the pressure imposed at the front face of the porous sample. The shock tube is fitted with a short driver section in order to generate blast-like decaying pressure profiles, which continue to decay after the initial shock impingement. In this scenario, the measured pressure profile at the end wall, which is affected by the properties of the porous medium and the size of the air gap separating its rear face and the shock tube end wall, is significantly different from the pressure profile imposed on the front face of the porous sample. The mechanism governing the pressure transformation provided by the porous medium is attributed to a selective filtration process that attenuates the pressure changes associated with high frequencies. The results of the present study are also analysed in conjunction with previously published analytical and numerical models to achieve a broader understanding of the physical mechanisms affecting the pressure buildup.

An experimental study of local temperature statistics in turbulent thermal convection is presented. The emissions of plumes and plume clusters are detected by an array of thermistors embedded in the top and bottom plates of a 1 m diameter convection cell. We found that the product STST′ of the temperature skewness ST and the skewness of the temperature time derivative ST′ from the embedded thermistors may be used as a measure of the intensity of plume emissions and that STST′ exhibits a pattern that corresponds well to the orientation of the large-scale circulation in the convecting flow. This is despite the fact that the temperature distribution across the plates is highly uniform, as indicated by the mean temperature of the embedded thermistors. By comparing the spatial distributions of STST′ and of the RMS temperature σ, we further find that the maximum temperature fluctuations take place in regions dominated by plume mixing instead of regions of plume emission. It is also found that temperature fluctuations inside the conducting plates have the same statistical and scaling properties as those in the cell centre.

The series of papers on the flow dynamics due to wave-induced macrovortices is completed with a statistical analysis of the mixing of the shallow flows occurring around submerged structures used for coastal protection. This is investigated with specific focus on the role played by large-scale horizontal eddies shed in coastal areas by waves breaking corresponding to topographic features like submerged breakwaters. As in Part 2, conditions due to isolated or arrays of breakwaters are studied. Analysis of particle statistics is used to determine both the features of the induced quasi-two-dimensional flow and to derive general properties. In particular three distinct regimes are found to characterize the flow evolution. Asymptotic regimes for small and large times share in any of the features of typical ‘ballistic’ and ‘Brownian’ regimes. Focus is mainly placed on properties of the ‘intermediate time’ regime which are seen to depend on the chosen topographic configuration. In agreement with the deterministic results of Part 2, we find that, because of an intense longshore current, an isolated breakwater induces a larger dispersion than that due to an array of breakwaters, characterized by a rip current. Moreover, for the same topography, the diffusivity grows with the local wavelength. Comparison with field data suggests that results of scaled-down laboratory experiments reproduce well natural mixing conditions. A simple formulation of absolute diffusivity, to be used in practical applications related to environmental quality management, is, finally, proposed.

A variational formulation is presented for a class of magnetohydrodynamic (MHD) channel flow problems. This formulation yields solutions for the fluid velocity and the induced electric potential in a channel with a uniform transverse static magnetic field. The channel cross-section is constant but arbitrary, and the channel walls can be either insulators or conductors with finite electrical conductivity. Electric currents are permitted to enter and leave the channel walls so that the solutions are suitable for MHD generator and pump applications. An example of a square channel with conducting walls is solved as an illustration.

Some properties of trapped oscillations in an inviscid rotating fluid were studied by Stewartson (1971, 1972). In this paper the modifications necessary to allow for a small viscosity in the fluid are discussed. It is found that concentrated disturbances of the kind used in ray theory are only possible if the rays are thought of as emanating from two sources outside the fluid and at opposite ends of the ray. Otherwise the ray must have a lateral tail, which vitiates the notion of a ray. The double-source character of rays is preserved at reflexions except when at glancing incidence. In this case a substantial part of the ray strength is lost and may give rise to a lateral tail if uncompensated. The focusing of the rays at the limit cycle is also discussed and shown to lead to a catastropic loss of energy, so that a free oscillation if set up rapidly decays. A simple forced oscillation is considered and shown to lead to weak resonances in the bands of trapped oscillations.

We consider a model for the solidification of an ideal ternary alloy in a mushy layer that incorporates the effects of thermal and solutal diffusion, convection and solidification. Our results reveal that although the temperature and solute fields are constrained to the liquidus surface of the phase diagram, the system still admits double-diffusive modes of instability. Additionally, modes of instability exist even in situations in which the thermal and solute fields are each individually stable from a static point of view. We identify these instabilities for a general model in which the base-state solution and its linear stability are computed numerically. We then highlight these instabilities in a much simpler model that admits an analytical solution.

A reacting flow free of molecular transport exhibits noteworthy behaviour in the neighbourhood of a blunt, symmetrical stagnation point. A local analytical study using the Lighthill-Freeman gas model reveals the basic structure of such a flow. Chemical activity is found to affect some, but not all, of the local characteristics of the flow. Unaffected are the pressure and velocity fields near the stagnation point, where the pressure varies quadratically and the velocity varies linearly as in an inert flow. In addition, the stagnation point is found to be in chemical equilibrium for all non-zero reaction rates. On the other hand the density, temperature, and concentration fields are affected by the non-equilibrium reactions. The extent of this effect can be predicted on the basis of a reaction parameter that measures the rate of reaction in terms of the velocity gradient at the stagnation point. A rapidly reacting flow (with reaction parameter greater than unity) approaches the stagnation point with vanishing gradients of density and temperature, whereas a slowly reacting flow approaches with infinite gradients. The flow field is represented mathematically by functions that are regular along the body but non-analytic in the normal direction. Numerical computations support the validity of the local closed-form solution, and provide information on the local effects of the chemical history of the flow.

An experimental investigation to establish the maximum rise height zm attained by a finite volume of fluid forced impulsively vertically upwards against its buoyancy into quiescent surroundings of uniform density is described. In the absence of a density contrast, the release propagates as a vortex ring and the vertical trajectory is limited by viscous effects. On increasing the source density of the release, gravitational effects limit the trajectory and a maximum rise height zm is reached. For these negatively buoyant releases, the dependence of zm on the length L of the column of ejected fluid, nozzle diameter D (= 2r0), dispensing time and source reduced gravity is determined by injecting saline solution into a fresh-water environment. For 3.4 ≲ L/D ≲ 9.0, zm/r0 is shown to scale on the source parameter η = Fr(L/D), a product of the source Froude number Fr and the aspect ratio L/D for the finite-volume release. Our results show that the morphology of the cap that develops above the source and the vortical motion induced within are sensitively dependent on the source conditions. Moreover, three rise-height regimes are identified: ‘weak-fountain-transition’, ‘vorticity-development’ and ‘forced-release’ regimes, each with a distinct morphology and dependence of dimensionless rise height on η.

An estimate is obtained of the heat transfer from a constant-temperature hot wire in a two-dimensional low Reynolds number flow (R [Lt ] 1). The flow has a small sinusoidally fluctuating velocity superimposed on the mean velocity. For the fluctuating components it is shown that at low frequencies the heat transfer is in phase with the velocity, the magnitude of the heat transfer being given by the result for steady heat transfer, but at high frequencies the heat transfer lags behind the velocity, the relative magnitude of the heat transfer decreasing as (frequency)−1.

A linear analysis of the perturbations associated with the jet flow, $\overline{u}(y) = {\rm sech}\,y$, of an inviscid compressible fluid is considered. Some subsonic stable solutions, not associated with stability boundaries, are first determined. Then a subsonic neutral solution is found and used as an aid in determining stability boundaries of the symmetric and antisymmetric disturbance modes. Numerical methods are also used to determine instability characteristics, including the Reynolds stress distributions. Comparisons are made with previous results obtained for the hyperbolic-tangent velocity profile and with unstable characteristics of the Bickley jet.

This paper considers the role of long finite-amplitude Rossby waves in determining the evolution of flow along a rapidly rotating channel with an uneven floor. The Rossby waves travel on a potential vorticity interface in a channel with a cross-channel step change in depth, where step position varies slowly along the channel. A nonlinear wave equation is derived describing the evolution of the potential vorticity interface. To leading order this is the hydraulic equation derived by Haynes, Johnson & Hurst (1993). Dispersion appears at the next order. Various solution regimes are identified. As well as slowly varying hydraulic solutions, two further types of steady solutions appear: approach-controlled flows and twin supercritical leaps. Both these solutions are characterized by leaps between supercritical branches of the hydraulic function. It is shown how the position and size of these ‘supercritical leaps’ can be determined within the context of hydraulic theory. To fully resolve the internal structure of these leaps dispersive effects must be included and leaps are shown to correspond to kink soliton solutions of the steady unforced problem. It is also shown that increasing dispersion (decreasing topographic length scale) causes the loss of the subcritical solution branch in some subcritical flows. The only candidate for a steady solution in these regimes is then an approach-controlled flow. Integrations of initial value problems show that in general flows evolve towards the dispersive form of the solution predicted by hydraulic theory, at least near the topographic perturbation. However, in those subcritical flows where sufficiently large dispersion causes the subcritical branch to disappear, unsteady integrations evolve to approach-controlled flows even when the dispersion is sufficiently small that the subcritical branch still exists.

In order to understand the dynamics of pancake shaped vortices in stably stratified fluids, we perform a linear stability analysis of an axisymmetric vortex with Gaussian angular velocity in both the radial and axial directions with an aspect ratio of ${\it\alpha}$. The results are compared to those for a columnar vortex (${\it\alpha}=\infty$) in order to identify the instabilities. Centrifugal instability occurs when $\mathscr{R}>c(m)$ where $\mathscr{R}=ReF_{h}^{2}$ is the buoyancy Reynolds number, $F_{h}$ the Froude number, $Re$ the Reynolds number and $c(m)$ a constant which differs for the three unstable azimuthal wavenumbers $m=0,1,2$. The maximum growth rate depends mostly on $\mathscr{R}$ and is almost independent of the aspect ratio ${\it\alpha}$. For sufficiently large buoyancy Reynolds number, the axisymmetric mode is the most unstable centrifugal mode whereas for moderate $\mathscr{R}$, the mode $m=1$ is the most unstable. Shear instability for $m=2$ develops only when $F_{h}\leqslant 0.5{\it\alpha}$. By considering the characteristics of shear instability for a columnar vortex with the same parameters, this condition is shown to be such that the vortex is taller than the minimum wavelength of shear instability in the columnar case. For larger Froude number $F_{h}\geqslant 1.5{\it\alpha}$, the isopycnals overturn and gravitational instability can operate. Just below this threshold, the azimuthal wavenumbers $m=1,2,3$ are unstable to baroclinic instability. A simple model shows that baroclinic instability develops only above a critical vertical Froude number $F_{h}/{\it\alpha}$ because of confinement effects.

The neutralization of a dilute spray of electrically charged droplets by ions of the opposite polarity generated by a corona discharge at a wire ring is analysed numerically. A Lagrangian description of the spray and Eulerian descriptions of the gas and the ions are used to deal with this two-way coupled problem. A model of the corona consisting of a line of charge and a distribution of ion sources is proposed. In the configuration that is analysed, neutralization usually begins at the shroud of the spray and extends to inner regions when the corona current increases. The number density of droplets is large at the shroud due to neutralized droplets that are no longer pushed by the electric field. These droplets can be dragged towards a collector surface by a weak forced flow that overcomes the ionic wind due to the force of the ions on the gas. The fraction of the spray charge that is neutralized increases with the corona current, but the value of this current required for full neutralization is several times larger than the inlet electric current of the spray owing to loss of ions to the boundaries of the system. The electric field induced by the charge of the droplets opposes the field due to the voltage applied between the wire ring and the extractor through which the droplets are injected, and thus reduces the threshold voltage of the corona and significantly affects its current–voltage characteristic, which may become multivalued. In turn, the electric field due to the applied voltage and the space charge of the ions affects the shape of the spray and the velocity of the droplets.

This paper describes the results of examining the influence of radiative heat transfer on turbulent natural convection above fires in an atmosphere of constant potential temperature, under both the ‘opaque’ and ‘transparent’ approximations. It turns out that on the basis of the over-all approximations introduced in this investigation, the former case reduces to that of no radiative transfer. For the latter case, the initial fire size, the energy release rate (initial temperature difference) and the absorption coefficient have been regarded as independent parameters. The solution curves presented cover a range over which these parameters are expected to vary in practice.