A closed flow circuit, comprising a smooth horizontal plastic pipe transporting a dilute suspension of plastic particles in water, was modified to produce a satisfactory environment for the detection of mean particle location across a fixed plane normal to the mean turbulent flow direction. It was established that mean concentration in the vicinity of the vertical diameter was a function of the vertical co-ordinate only, and this result enabled the particle diffusivity to be evaluated for a core region of the cross-section. For a particle relative density in the range 1.035 to 1.126 and a variety of flows, a suitable expression for the dimensionless particle diffusivity is A (1 - BW/u) SN, where u is the friction velocity, W the fall velocity for a particle in still water, S the relative density of the suspended material, and A, B and N are constants.

An analytical and experimental investigation is made of the compression wave generated when a train enters a tunnel fitted with a long, uniform hood with a rectangular window. The window is situated at the junction of the hood and tunnel, which are taken to have the same uniform cross-sectional area. An understanding of the mechanics of this canonical configuration is important for the design of tunnel entrance hoods for new high-speed trains, with speeds in excess of 300 km h$^{-1}$. The compression wave is formed in two stages: as the train nose enters the hood and as it passes the window. The elevated pressure within the hood produces a flow of air from the window in the form of a high-speed jet, whose inertia generates an additional rise in pressure that propagates into the tunnel as a localized pulse. Multiple reflections from the window and the hood portal cause the temporary trapping of wave energy within the hood (prior to its radiation into the tunnel). All of these aspects of the flow are modelled analytically and the results are found to be in good accord with new model-scale measurements and flow visualization studies reported in this paper.

A recent hydrodynamic theory of liquid slippage on a solid substrate (Kirkinis & Davis, Phys. Rev. Lett., vol. 110, 2013, 234503) gives rise to a sequence of eddies (Moffatt vortices) that co-move with a moving contact line (CL) in a liquid wedge. The presence of these vortices is established through secular equations that depend on the dynamic contact angle $\alpha $ and capillary number Ca. The limiting case $\alpha \rightarrow 0$ is associated with the appearance of such vortices in a channel. The vortices are generated by the relative motion of the interfaces, which in turn is due to the motion of the CL. This effect has yet to be observed in experiment.

A soft elastohydrodynamic lubrication model is formulated for deformable roll coating involving two contra-rotating rolls, one rigid and the other covered with a compliant layer. Included is a finite-strip model (FSM) for the deformation of the layer and a lubrication model with suitable boundary conditions for the motion of the fluid. The scope of the analysis is restricted to Newtonian fluids, linear elasticity/viscoelasticity and equal roll speeds, with application to the industrially relevant highly loaded or ‘negative gap’ regime. Predictions are presented for coated film thickness, inter-roll thickness, meniscus location, pressure and layer deformation as the control parameters – load (gap), elasticity, layer thickness and capillary number, $\hbox{\it Ca}$ – are varied. There are four main results:

1. Hookean spring models are shown to be unable to model effectively the deformation of a compliant layer when Poisson's ratio $\nu\rightarrow 0.5$. In particular, they fail to predict the swelling of the layer at the edge of the contact region which increases as $\nu\rightarrow 0.5$; they also fail to locate accurately the position of the meniscus, $X_M$, and to identify the presence, close to the meniscus, of a ‘nib’ (constriction in gap thickness) and associated magnification of the sub-ambient pressure loop.

2. Scaling arguments suggest that layer thickness and elasticity may have similar effects on the field variables. It is shown that for positive gaps this is true, whereas for negative gaps they have similar effects on the pressure profile and flow rate yet quite different effects on layer swelling (deformation at the edge of the contact region) and different effects on $X_M$.

3. For negative gaps and $\hbox{\it Ca}\,{\sim}\,O(1)$, the effect of varying either viscosity or speed and hence $\hbox{\it Ca}$ is to significantly alter both the coating thickness and $X_M$. This is contrary to the case of fixed-gap rigid roll coating.

4. Comparison between theoretical predictions and experimental data shows quantitive agreement in the case of $X_M$ and qualitive agreement for flow rate. It is shown that this difference in the latter case may be due to viscoelastic effects in the compliant layer.

The transient process by which an incompressible dissipative rotating stratified fluid adjusts to a small change in the rotation rate of its container is examined theoretically. The aim is to clarify the effects of the imposed density stratification and of the boundary condition specified for the density perturbation on the behaviour of the fluid, particularly during the time span when the adjustment is performed in a homogeneous fluid. For a weakly stratified fluid in a cylinder, it is shown how these two factors govern the nature and intensity of boundary layers on the vertical wall which close the secondary meridional circulation generated by Ekman layers along the horizontal boundaries. For a more strongly stratified fluid, the usefulness and importance of potential vorticity conservation in determining the quasi-steady motion is verified, and a calculation for a spherical container demonstrates some new features that arise only when the container boundaries are not normal or parallel to the rotation axis. It is shown that experimental results of Holton (1965) are in less good agreement with predictions of the linear theory than had been previously indicated.

Bulk-coupled instability at the interface between miscible fluids which have identical mechanical properties but disparate electrical conductivities and are stressed by an equilibrium normal electric field is studied experimentally and theoretically. Observations of fluid motions permit measurement of an instability wavenumber and growth rate.

A model with a layer of diffusive conductivity distribution coupled to uniform bounding half-spaces is developed. Numerical integration of linearized electromechanical equations leads to a set of growing eigenfrequencies with corresponding eigenmodes, pure real for small wavenumber and complex (propagating) for large wavenumber.

The fastest growing wavenumber and corresponding growth rate are characterized as a function of the conductivity ratio and a time-constant ratio, which reflects the importance of inertial and viscous effects. In the viscous-dominated limit, the description agrees with a corresponding surface-coupled theory. The model is evaluated for parameters corresponding to experimental observations.

Experiments are presented in which axisymmetric internal waves are generated by an oscillating sphere moving vertically in a stably stratified salt solution. The Reynolds numbers for the sphere based on the diameter and the mean velocity are between 10 and 200. Lighthill's theory for dispersive waves is used to calculate the phase configuration of the internal waves. The agreement between experiment and theory is reasonably good.

The stability and dynamics of two-dimensional gravity–capillary solitary waves in deep water within the fully nonlinear water-wave equations are numerically studied. It is well known that there are two families of symmetric gravity–capillary solitary waves – depression waves and elevation waves – bifurcating from infinitesimal periodic waves at the minimum of the phase speed. The stability of both branches was previously examined by Calvo & Akylas (J. Fluid Mech., vol. 452, 2002, pp. 123–143) by means of a numerical spectral analysis. Their results show that the depression solitary waves with single-valued profiles are stable, while the elevation branch experiences a stability exchange at a turning point on the speed–amplitude curve. In the present paper, we provide numerical evidence that the depression solitary waves with an overhanging structure are also stable. On the other hand, Dias et al. (Eur. J. Mech. B, vol. 15, 1996, pp. 17–36) numerically traced the elevation branch and discovered that its speed–amplitude bifurcation curve features a ‘snake-like’ behaviour with many turning points, whereas Calvo & Akylas (J. Fluid Mech., vol. 452, 2002, pp. 123–143) only considered the stability exchange near the first turning point. Our results reveal that the stability exchange occurs again near the second turning point. A branch of asymmetric solitary waves is also considered and found to be unstable, even when the wave profile consists of a depression wave and a stable elevation one. The excitation of stable gravity–capillary solitary waves is carried out via direct numerical simulations. In particular, the stable elevation waves, which feature two troughs connected by a small dimple, can be excited by moving two fully localised, well-separated pressures on the free surface with the speed slightly below the phase speed minimum and removing the pressures simultaneously after a period of time.

The diffusion of heat injected from a line source into turbulence generated by regular and fractal grids with the same solidity and inlet velocity was investigated experimentally with particular interest in the effects of grid geometry and relative location of the source on the width of the thermal plume and the mixing efficiency. These grids included one fractal square grid (FSG) and three regular square grids with mesh sizes that were comparable to the first (RG160), second (RG80) and fourth (RG18) iterations of the fractal grid. The heated line source was inserted on the centre plane of the grids, spanning the entire width of the wind tunnel at either of two downstream locations, an upstream location or a location nearly coincident with a grid. It was found that, in all cases examined, RG160 produced the strongest diffusion of the thermal plume and the highest level of scalar mixing. These observations were consistent with the evolution of the corresponding turbulent diffusivities, which, according to Taylor’s theory of diffusion, are the product of the transverse turbulence intensity and the integral length scale. We argue that to maximise scalar diffusion and mixing of a scalar released from a concentrated source inside a duct, one should prefer a regular grid over a fractal square grid; we also recommend the use of a grid with a mesh size roughly equal to half the height of the duct and placed at approximately one duct height upstream of the source.

A general numerical method of characteristics applicable to problems in magneto-fluid dynamics as well as ordinary fluid dynamics is described. The method can be applied to unsteady three-dimensional flows of chemically reacting, non-equilibrium, multi-component media. Dissipative phenomena must be neglected in order to make the governing equations of change hyperbolic, because the method can be applied only to quasi-linear, hyperbolic, partial differential equations. Practical restrictions on computation time usually require unsteady problems to be limited to cases with short transient times although theoretically the method applies to all unsteady flows. In steady flow the local velocity must be greater than the largest local wave speed. The characteristic and compatibility equations are derived for the most general case of magnetofluid dynamics. A new finite-difference network and its corresponding equations are developed similarly. Specialization of the general method to consider simpler problems is outlined. Preliminary numerical results of calculations using the method are presented. The practicality and feasibility of utilizing the general numerical method of characteristics on presently available, electronic digital computers is evaluated in the light of recent experience in calculating multi-dimensional flows with the method.

We present theoretical, numerical and experimental studies of the release of a finite volume of fluid instantaneously from an edge of a rectangular domain for high Reynolds number flows. For the cases we considered, the results indicate that approximately half of the initial volume exits during an early adjustment period. Then, the inertial gravity current reaches a self-similar phase during which approximately 40 % of its volume drains and its height decreases as $\unicode[STIX]{x1D70F}^{-2}$, where $\unicode[STIX]{x1D70F}$ is a dimensionless time that is derived with the typical gravity wave speed and the horizontal length of the domain. Based on scaling arguments, we reduce the shallow-water partial differential equations into two nonlinear ordinary differential equations (representing the continuity and momentum equations), which are solved analytically by imposing a zero velocity boundary condition at the closed end wall and a critical Froude number condition at the open edge. The solutions are in good agreement with the performed experiments and direct numerical simulations for various geometries, densities and viscosities. This study provides new insights into the dynamical behaviour of a fluid draining from an edge in the inertial regime. The solutions may be useful for environmental, geophysical and engineering applications such as open channel flows, ventilations and dam-break problems.

The effects of sinusoidal protrusions on steady laminar free convection between vertical walls is investigated in this paper. Numerical results are presented for various values of the size and spacing of the protrusions. In particular optimum values are found which yield maximum wall heat-transfer coefficients.

This paper represents the second part of a study of semi-geostrophic (SG) geophysical fluid dynamics. SG dynamics shares certain attractive properties with the better known and more widely used quasi-geostrophic (QG) model, but is also a good prototype for balanced models that are more accurate than QG dynamics. The development of such balanced models is an area of great current interest. The goal of the present work is to extend a central body of QG theory, concerning the evolution of disturbances to prescribed basic states, to SG dynamics. Part 1 was based on the pseudomomentum; Part 2 is based on the pseudoenergy.

A pseudoenergy invariant is a conserved quantity, of second order in disturbance amplitude relative to a prescribed steady basic state, which is related to the time symmetry of the system. We derive such an invariant for the semi-geostrophic equations, and use it to obtain: (i) a linear stability theorem analogous to Arnol'd's ‘first theorem’; and (ii) a small-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit. The results are analogous to their quasi-geostrophic forms, and reduce to those forms in the limit of small Rossby number.

The results are derived for both the f-plane Boussinesq form of semi-geostrophic dynamics, and its extension to β-plane compressible flow by Magnusdottir & Schubert. Novel features particular to semi-geostrophic dynamics include apparently unnoticed lateral boundary stability criteria. Unlike the boundary stability criteria found in the first part of this study, however, these boundary criteria do not necessarily preclude the construction of provably stable basic states.

The interior semi-geostrophic dynamics has an underlying Hamiltonian structure, which guarantees that symmetries in the system correspond naturally to the system's invariants. This is an important motivation for the theoretical approach used in this study. The connection between symmetries and conservation laws is made explicit using Noether's theorem applied to the Eulerian form of the Hamiltonian description of the interior dynamics.

A model is presented of the two-dimensional boundary-layer and interior flow in a rectangular box resulting from the application of a quadratic temperature variation on its lower surface. The other walls are insulating. It is shown that a similarity form exists for the narrow, thermocline-like layer near the lower surface, and that this can satisfy all known consistency conditions with the interior, together with either laminar or turbulent side-wall regions. The interior temperature and Nusselt number are shown to be insensitive to Prandtl number, and to be primarily functions of the horizontal Rayleigh number RaL.

Specifically, the interior temperature, relative to the coldest applied value, is 60% of the total applied temperature range. The Nusselt number is predicted to vary as $0.26 Ra^{\frac{1}{5}}_L$ for a box with unit aspect ratio. The dynamics of the side-wall region, and the details of the imposed temperature variation appear to be unimportant in determining the overall buoyancy exchange. The solutions are compared with numerical and observational results, and generally good agreement is found.

We developed a numerical method for the set of equations governing fully compressible convection in the limit of infinite Prandtl numbers. Reduced models have also been analysed, such as the anelastic approximation and the anelastic liquid approximation. The tests of our numerical schemes against self-consistent criteria have shown that our numerical simulations are consistent from the point of view of energy dissipation, heat transfer and entropy budget. The equation of state of an ideal gas has been considered in this work. Specific effects arising because of the compressibility of the fluid are studied, like the scaling of viscous dissipation and the scaling of the heat flux contribution due to the mechanical power exerted by viscous forces. We analysed the solutions obtained with each model (fully compressible model, anelastic and anelastic liquid approximations) in a wide range of dimensionless parameters and determined the errors induced by each approximation with respect to the fully compressible solutions. Based on a rationale on the development of the thermal boundary layers, we can explain reasonably well the differences between the fully compressible and anelastic models, in terms of both the heat transfer and viscous dissipation dependence on compressibility. This could be mostly an effect of density variations on thermal diffusivity. Based on the different forms of entropy balance between exact and anelastic models, we find that a necessary condition for convergence of the anelastic results to the exact solutions is that the product $\unicode[STIX]{x1D716}q$ must be small compared to unity, where $\unicode[STIX]{x1D716}$ is the ratio of the superadiabatic temperature difference to the adiabatic difference, and $q$ is the ratio of the superadiabatic heat flux to the heat flux conducted along the adiabat. The same condition seems also to be associated with a convergence of the computed heat fluxes. Concerning the anelastic liquid approximation, we confirm previous estimates by Anufriev et al. (Phys. Earth Planet. Inter., vol. 152, 2005, pp. 163–190) and find that its results become generally close to those of the fully compressible model when $\unicode[STIX]{x1D6FC}T{\mathcal{D}}$ is small compared to unity, where $\unicode[STIX]{x1D6FC}$ is the isobaric thermal expansion coefficient, $T$ is the temperature (here $\unicode[STIX]{x1D6FC}T=1$ for an ideal gas) and ${\mathcal{D}}$ is the dissipation number.

The distribution of kinetic helicity in a dipolar planetary dynamo is central to the success of that dynamo. Motivated by the helicity distributions observed in numerical simulations of the Earth’s dynamo, we consider the relationship between the kinetic helicity, $h=\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{u}$, and the buoyancy field that acts as a source of helicity, where $\boldsymbol{u}$ is velocity. We show that, in the absence of a magnetic field, helicity evolves in accordance with the equation $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}t=-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{F}+S_{h}$, where the flux, $\boldsymbol{F}$, represents the transport of helicity by inertial waves, and the helicity source, $S_{h}$, involves the product of the buoyancy and the velocity fields. In the numerical simulations it is observed that the helicity outside the tangent cylinder is predominantly negative in the north and positive in the south, a feature which the authors had previously attributed to the transport of helicity by waves (Davidson & Ranjan, Geophys. J. Intl, vol. 202, 2015, pp. 1646–1662). It is also observed that there is a strong spatial correlation between the distribution of $h$ and of $S_{h}$, with $S_{h}$ also predominantly negative in the north and positive in the south. This correlation tentatively suggests that it is the in situ generation of helicity by buoyancy that establishes the distribution of $h$ outside the tangent cylinder, rather than the dispersal of helicity by waves, as had been previously argued by the authors. However, although $h$ and $S_{h}$ are strongly correlated, there is no such correlation between $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}t$ and $S_{h}$, as might be expected if the distribution of $h$ were established by an in situ generation mechanism. We explain these various observations by showing that inertial waves interact with the buoyancy field in such a way as to induce a source $S_{h}$ which has the same sign as the helicity in the local wave flux, and that the sign of $h$ is simply determined by the direction of that flux. We conclude that the observed distributions of $h$ and $S_{h}$ outside the tangent cylinder are consistent with the transport of helicity by waves.

This paper presents numerical simulations of two-dimensional incompressible flow around two circular cylinders in relative motion, which may result in impact. Viscous flow computations are carried out using a streamfunction–vorticity method for two equal-diameter cylinders undergoing a two-dimensional impact in otherwise stationary fluid and for cases of similar impact of two cylinders in a steady incident flow. These results are supported by potential flow calculations carried out using a Möbius conformal transformation and infinite arrays of image singularities. The inviscid flow results are compared with other published work and show that the inviscid forces induced on the cylinders have an inverse square root singularity with respect to the time to impact. All impacts considered in this paper result from steady motion of the cylinders along the line joining their centres.

Turbulent plumes, which are seen in a wide number of industrial and natural flows, have been extensively studied; however, very little attention has been paid to plumes which have an internal mechanism for changing buoyancy. Such plumes arise in e.g. industrial chimneys, where species can react and change the density of the plume material. These plumes with chemical reaction are the focus of this study. An integral model describing the behaviour of a plume undergoing a second-order chemical reaction between a component in the plume (A) and a component in the surrounding fluid (B), which alters the buoyancy flux, is considered. The behaviour of a reactive plume is shown to depend on four dimensionless groups: the volume and momentum fluxes at the source, the parameter ϵ which indicates the additional buoyancy flux generated by the reaction and γ which is a dimensionless rate of depletion of species B. Additionally, approximate analytical solutions are sought for a reactive plume rising from a point source of buoyancy when species B is in great excess. These analytical results show excellent agreement with numerical simulations. It is also shown that the behaviour of a reactive plume in the far field is equivalent to an inert plume issuing from a virtual source downstream of the real source, and the dependence of the location of the virtual source on ϵ and γ is discussed. The effects of varying the volume flux at the source and the Morton source parameter Γ0 are further investigated by solving the full governing equations numerically. These solutions indicate that ϵ is important in determining the buoyancy generated by the reaction, and the length scale over which this reaction occurs depends on γ when γ > 1. It is also shown that when the dimensionless buoyancy ϵ < − 1, the reaction can cause the plume to collapse.

In this paper the problem of interaction of an oblique blast wave encountering a small bend along a plane wall is studied for the case when relative outflow from the reflected shock is supersonic. A theoretical solution is developed completely with the help of conformal transformations and complex variable techniques. Numerical results showing the pressure distribution along the wall have been obtained for two incident shock strengths.