We address two aspects of the influence of double diffusion on convection from point buoyancy sources (plumes). First we consider double-diffusive convection acting in conjunction with buoyancy from an isolated source in an unconfined environment, and we study the problem of when fluxes of properties occur in the opposite direction to that of the initial buoyancy. A simple dimensional analysis is in good agreement with the experimental data for these ‘counterbuoyant’ fluxes of properties. Secondly, we consider a double-diffusive plume discharging into a confined environment. The plume now does not always reach the bottom (or top) of the confined environment, but can spread out into the environment at intermediate depths. Double-diffusive interfaces form, and we describe their development and the evolution of the density profile in the confined environment. A quantitative measure of the importance of double-diffusive convection in relation to the normal filling-box mechanism is obtained in the experiments by using the measured density profile, and this correlates well with a theoretically derived parameter Bρ.

An experimental investigation of resonant standing water waves in a rectangular tank with a corrugated bottom is reported. The study was stimulated by the theory of Howard & Yu (J. Fluid Mech., vol. 593, 2007, pp. 209–234) predicting the existence of normal modes that can be significantly affected by Bragg reflection/scattering. As a result, the amplitude of the standing waves (normal modes) varies exponentially along the entire length of the tank, or from the centre out in each direction, depending on the phase of the corrugations at the tank endwalls. Experiments were conducted in a 5 m tank fitted with a sinusoidal bottom with one adjustable endwall. Waves were excited by small-amplitude sinusoidal horizontal movement of the tank using an electrical motor drive system. Simultaneous time-series data of standing oscillations were recorded at well-separated positions along the tank to measure the growth in amplitude. Waveforms over a section of the tank were filmed through the transparent acrylic walls. Except for very shallow depths and near the tank endwalls, the experimental measurements of resonant frequencies, mean wavelengths, free-surface waveforms and amplitude growth are found in essential agreement with the Bragg resonant normal mode theory.

The classical plume model due to Morton, Taylor & Turner (Proc. R. Soc. Lond. A, vol. 234, 1956, pp. 1–23) is re-cast in terms of the non-dimensional plume radius, the plume ‘laziness’ defined as the squared ratio of the source radius and the jet length, and the buoyancy flux. It is shown that many of the key results of this classical model can then be read straight from the equations without recourse to solving them. Based on this observation, derivative models that consider plumes propagating through stratified environments or undergoing chemical reactions are similarly re-cast. We show again that key results can be read straight from the governing equations and results that have previously only been demonstrated numerically can be found analytically. In particular, we unify two previously distinct models that consider plumes propagating through stable and unstable stratified environments whose stratification has a power-law dependence on height. We present analytical solutions for the range of stratification power-law decay rates for which straight-sided plumes are possible. This result unifies the sets of solutions by Batchelor (Q. J. R. Meteorol. Soc., vol. 80, 1954, pp. 339–358) and Caulfield & Woods (J. Fluid Mech., vol. 360, 1998, pp. 229–248). We are able to explain the unstable behaviour previously found when the power lies in the range (−4, −8/3). Finally we show that this method also has limited advantages when applied to plumes with unsteady source conditions.

Analytical solutions for the quasi-one-dimensional flow of a gas not in thermodynamic equilibrium are presented for two distinct types of rate equation, namely, the linear rate equation which governs vibrational relaxation, and the nonlinear rate equation which governs dissociation. The solutions are derived for the case when, to a first approximation, the rate equation is uncoupled from the remaining flow equations.

There are, in general, three distinct regions in non-equilibrium nozzle flow. First, a so-called near equilibrium region where a perturbation solution is expected to hold. This region is followed by a narrow transition-layer in which there is a rapid departure from equilibrium. Finally, downstream of this layer, the energy in the lagging mode tends asymptotically to some constant ‘frozenout’ value. The solutions applicable to each of these three regions are derived for both rate equations, the boundary conditions for the transition-layer solution and the asymptotic solution are obtained from appropriate matching procedures.

In particular the structures of the asymptotic solutions are discussed. Several approximate methods for determining the asymptotic frozen level of theenergy in the lagging mode have been proposed in the literature. For the present case, when there is only a small amount of energy in the lagging mode, it is shown that none of these approximate methods is mathematically correct.

Consider an infinite two-dimensional acoustic waveguide containing a long rectangular obstacle placed symmetrically with respect to the centreline. We search for trapped modes, i.e. modes of oscillation at particular frequencies which decay down the waveguide. We provide analytic estimates for trapped mode frequencies and prove that the number of trapped modes is asymptotically proportional to the length of the obstacle.

We investigate the merger of two co-rotating geophysical vortices at finite Rossby and Froude number. The initial conditions consist of two uniform potential vorticity vortices in near-equilibrium and in a nearly ‘balanced’ state (i.e. with negligible emission of inertia–gravity wave radiation). We determine the critical merger distance between the two vortices. This distance is found to increase with the magnitude of the Rossby number: intense cyclones or intense anticyclones are able to merge from further apart compared to weaker cyclones and anticyclones. Note that the Froude number is proportional to the Rossby number for the near-equilibrium initial conditions considered. The critical merging distance also depends on the sign of the potential vorticity anomaly, which is positive for ‘cyclones’ and negative for ‘anticyclones’. We show that ageostrophic motions occurring at finite Rossby number tend to draw cyclones together but draw anticyclones apart. On the other hand, we show that anticyclones tend to deform more, in particular when subject to vertical shear (as when the vortices are vertically offset). These two effects compete. Overall, nearly aligned cyclones tend to merge from further apart than their anticyclonic counterparts, while vertically offset anticyclones merge from further apart than cyclones.

Experiments are described in which the populations of metastable levels in ionizing argon are measured through spatially resolved hook interferometry. The results are compared with the present model for shock-induced ionization and a recently proposed mechanism to explain observed flow instabilities. It is found that the experimental measurements support the presently accepted model, which states that electron–atom collisions play the dominant role in the excitation process, but contradicts recent proposals which predict a rapid build-up of anomalously high metastable populations through atom–atom collisions.

The stability of an infinite layer of fluid of uniform thickness at rest with two horizontal boundaries is investigated when the difference between the temperatures at the top and bottom boundaries has a component which is a fluctuating periodic function of time in addition to a constant part. When both boundaries are free small fluctuations have a stabilizing effect on the layer, while large fluctuations tend to make it less stable, consistently with the numerical results of Yih & Li (1972); the effect is attributable entirely to the variation of the temperature gradient with time. An approximate relation between the mean Rayleigh number and the amplitude of the fluctuations is found which separates stable situations from unstable ones. This is compared with the criteria deduced by Homsy (1974) using energy arguments for disturbances of any amplitude.

Experiments are described in which a body was towed at large Richardson numbers through a uniformly stratified fluid, with particular attention being given to the parameter range in which diffusive effects are likely to play an important role. When the Richardson number is only moderately large (i.e. less than about 106) it appears that a linear viscous diffusive theory fails to model the experimental observations, but at larger Richardson numbers it seems that such a theory is a strong candidate for modelling the physical situation. At Richardson numbers of about 103 a non-diffusive viscous model (see the theories of Graebel 1969; Janowitz 1971) appears to give a fairly good description of the experimental results, except for some discrepancies between the theories and the observed flows in the wake downstream from the body. However, the conditions under which the experiments were carried out were not completely favourable for the application of these theories.

A brief survey and new interpretations of some of the work on the rotatingfluid counterparts of the present experiments are also given. In view of the theoretical similarities between the two situations we have tried to assess and compare (where possible) the results of experimental investigations in each field and we feel that the observations in the rotating-fluid experiments are in good qualitative agreement with the present results. No measurements of the drag acting on the body were made in the present experiments but this has been done carefully in the rotating-fluid experiments, and the results of those measurements are in poor agreement with the theoretical predictions. Because of the close similarities between the axisymmetric model for viscous rotating fluids (cf. Moore & Saffman 1969) and the two-dimensional model for diffusive stratified fluids (Freund & Meyer 1972), an explanation for the discrepancy should be sought before the linear viscous diffusive model can be applied with any confidence to the flow of stratified fluids at very large Richardson numbers.

The work that follows considers the velocity profiles within the boundary layer at the wall of an arbitrarily converging funnel. The occurrence of super-velocities, i.e. components of velocity within the boundary layer exceeding their corresponding free stream component, is investigated and the relevance of such a phenomenon to the efficiency of discharge discussed.

We employ the moderate-Péclet-number macroscale model developed in part 2 of this sequence (Schnitzer et al., J. Fluid Mech., vol. 704, 2012, pp. 109–136) towards the calculation of electroviscous forces on charged solid particles engendered by an imposed relative motion between these particles and the electrolyte solution in which they are suspended. In particular, we are interested in the kinematic irreversibility of these forces, stemming from the diffusio-osmotic slip which accompanies the salt-concentration polarisation induced by that imposed motion. We illustrate the electroviscous irreversibility using two prototypic problems, one involving side-by-side sedimentation of two spherical particles, and the other involving a force-free spherical particle suspended in the vicinity of a planar wall and exposed to a simple shear flow. We focus on the pertinent limit of near-contact configurations, where use of lubrication approximations provides closed-form expressions for the leading-order lateral repulsion. In this approximation scheme, the need to solve the advection–diffusion equation governing the salt-concentration polarisation is circumvented.

A theory is developed for the speed and structure of steady-state non-dissipative gravity currents in rotating channels. The theory is an extension of that of Benjamin (1968) for non-rotating gravity currents, and in a similar way makes use of the steady-state and perfect-fluid (incompressible, inviscid and immiscible) approximations, and supposes the existence of a hydrostatic ‘control point’ in the current some distance away from the nose. The model allows for fully non-hydrostatic and ageostrophic motion in a control volume V ahead of the control point, with the solution being determined by the requirements, consistent with the perfect-fluid approximation, of energy and momentum conservation in V, as expressed by Bernoulli's theorem and a generalized flow-force balance. The governing parameter in the problem, which expresses the strength of the background rotation, is the ratio W = B/R, where B is the channel width and R = (g′H)1/2/f is the internal Rossby radius of deformation based on the total depth of the ambient fluid H. Analytic solutions are determined for the particular case of zero front-relative flow within the gravity current. For each value of W there is a unique non-dissipative two-layer solution, and a non-dissipative one-layer solution which is specified by the value of the wall-depth h0. In the two-layer case, the non-dimensional propagation speed c = cf(g′H)−1/2 increases smoothly from the non-rotating value of 0.5 as W increases, asymptoting to unity for W → ∞. The gravity current separates from the left-hand wall of the channel at W = 0.67 and thereafter has decreasing width. The depth of the current at the right-hand wall, h0, increases, reaching the full depth at W = 1.90, after which point the interface outcrops on both the upper and lower boundaries, with the distance over which the interface slopes being 0.881R. In the one-layer case, the wall-depth based propagation speed Froude number c0 = cf(g′h0)−1/2 = 21/2, as in the non-rotating one-layer case. The current separates from the left-hand wall of the channel at W0 ≡ B/R0 = 2−1/2, and thereafter has width 2−1/2R0, where R0 = (g′h0)1/2/f is the wall-depth based deformation radius.

We examine the flows induced by horizontally modulated, vertically confined (or guided), internal wavepackets in a stratified, Boussinesq fluid. The wavepacket induces both an Eulerian flow and a Stokes drift, which together determine the Lagrangian transport of passive tracers. We derive equations describing the wave-induced flows in arbitrary stable stratification and consider four special cases: a two-layer fluid, symmetric and asymmetric piecewise constant (‘top-hat’) stratification and, more representative of the ocean, exponential stratification. In a two-layer fluid, the Stokes drift is positive everywhere with the peak value at the interface, whereas the Eulerian flow is negative and uniform with depth for long groups. Combined, the net depth-integrated Lagrangian transport is zero. If one layer is shallower than the other, the wave-averaged interface displaces into that layer making the Eulerian flow in that layer more negative and the Eulerian flow in the opposite layer more positive so that the depth-integrated Eulerian transports are offset by the same amount in each layer. By contrast, in continuous stratification the depth-integrated transport due to the Stokes drift and Eulerian flow are each zero, but the Eulerian flow is singular if the horizontal phase speed of the induced flow equals the group velocity of the wavepacket, giving rise to a single resonance in uniform stratification (McIntyre, J. Fluid Mech., vol. 60, 1973, pp. 801–811). In top-hat stratification, this single resonance disappears, being replaced by multiple resonances occurring when the horizontal group velocity of the wavepacket matches the horizontal phase speed of higher-order modes. Furthermore, if the stratification is not vertically symmetric, then the Eulerian induced flow varies as the inverse squared horizontal wavenumber for shallow waves, the same as for the asymmetric two-layer case. This ‘infrared catastrophe’ also occurs in the case of exponential stratification suggesting significant backward near-surface transport by the Eulerian induced flow for modulated oceanic internal modes. Numerical simulations are performed confirming these theoretical predictions.

Self-similar turbulent vortex rings are investigated theoretically in the framework of the semi-empirical turbulence theory for the modified Helmholtz equation. The velocity and vorticity fields are established, as well as the transport of passive admixture by turbulent vortex rings. Turbulent vortex rings of propellant gases originating from the muzzle of a gun after a gunshot are an important phenomenon to consider in crime scene reconstruction. In this work, it is shown that this has a significant repercussion on the outcome of backward blood spatter resulting from a gunshot. Turbulent vortex rings of propellant gases skew the distribution of bloodstains on the ground and can either propel blood droplets further from the target, or even turn them backwards towards the target. This is revealed through the final bloodstain locations and the respective distributions of the number of stains and their area as a function of distance from the target for two different shooter-to-target distances. An image of the propagating muzzle gases after bullet ejection is overlaid with the predicted flow field, which reveals satisfactory agreement. Gunshot residue is an important factor in determining the events of a violent crime due to a gunshot and are considered to be entrained and transported by the propellant gases. The self-similar solutions for the flow, vorticity and concentration of gunpowder particles are predicted and the results are shown to be within the measured range of a limited set of experimental data.

An upper bound on the rate of vertical heat transport is established in three dimensions for stress-free velocity boundary conditions on horizontally periodic plates. A variation of the background method is implemented that allows negative values of the quadratic form to yield ‘small’ ($O\left(1/ \mathit{Pr}\right)$) corrections to the subsequent bound. For large (but finite) Prandtl numbers this bound is an improvement over the ‘ultimate’ $R{a}^{1/ 2} $ scaling and, in the limit of infinite $Pr$, agrees with the bound of $R{a}^{5/ 12} $ recently derived in that limit for stress-free boundaries.

Spin-up in a rectangular container with a free surface is investigated numerically and experimentally. In the formulation of two-dimensional numerical computation, we use a potential-like function in addition to the stream function to deal with the first-order Ekman pumping model. It is shown that our numerical results are in good agreement with those obtained by the experiment when either the leading-order or first-order pumping model is used. On the other hand, when no pumping effect is considered the numerical results show, except in the initial development, a considerable discrepancy from those of the experiment. Our attention in this study is focused on clarifying the physical mechanism of cyclonic vortex merging. At low Reynolds numbers and/or liquid depths the Ekman pumping damps the vortical flows fast, resulting in non-merging. At moderate Reynolds numbers, it enhances merging because the cyclonic vortices expand due to the Ekman pumping. We discuss the influence of various parameters, including Reynolds number, Rossby number, and dimensionless liquid depth, on the evolution of the vortical flows.

The behaviour of the interface between two immiscible fluids of different density and viscosity in a centrifuge is investigated both analytically and experimentally. The study includes the interfacial shape and behaviour during both steady and transient operation of the centrifuge. The interface shape is investigated for rigid-body rotation and during slow steady acceleration or deceleration of the centrifuge. The dependence of the interface shape on time is investigated during a rapid spin-up to a slightly different value of the angular velocity, which then remains constant. Some aspects of interface stability are also reported.

Numerical solutions of finite-difference approximations to the Navier–Stokes equations have been obtained for the axisymmetric motion of a Boussinesq liquid in a rigidly bounded rotating annulus. For most of the cases studied, a temperature difference is maintained between the top and bottom surfaces such that essentially a basic stable density stratification is imposed on the fluid. The side walls are thermally insulated and the motion is driven by a differential rotation of the top surface. Approximate steady-state solutions are obtained for various values of the Rossby number ε and the stratification parameter S = N2/Ω2, where N is the Brunt–Väisälä frequency and Ω is the rotational frequency. The changes in the flow field with the variation of these parameters is studied. Particular attention is given to an investigation of the meridional, or up welling, circulation and its dependence on the stratification parameter. The effects on the flow of different boundary conditions, such as an applied stress driving, specified temperature at the side walls and an applied heat flux at the top and bottom surfaces, are also investigated.