Diapycnal mixing plays a key role in the thermohaline circulation of the deep ocean. Field observations have suggested that this mixing is intensified over the rough topography along the boundaries of the ocean. In this study, we experimentally explore the transport of salinity and tracer in a horizontally stirred, stratified fluid with a steady vertical buoyancy flux. The mechanical mixing occurs either uniformly across the tank or near a sidewall. To compare uniform and boundary mixing, first we explore the steady state dynamics and find that, in both cases, the surfaces of constant density are horizontal. With uniform mixing, vertical transport of buoyancy occurs uniformly throughout the tank while, with boundary mixing, transport is confined to the turbulent region and the interior space remains quiescent and does not mix or have any net movement. We then explore the transient evolution of the stratification when the source of buoyancy is removed from the base of the system. In the boundary mixing case, the resulting divergence of the turbulent diffusive flux in the boundary region leads to a reduction in the buoyancy of the fluid in the boundary region, and a net upflow develops in the boundary region. In turn, this drives a downwelling in the interior. Vertical gradients in the rate of downwelling lead to stretching of isopycnals and, together, these processes enable the interior stratification to evolve. The experiments highlight that, with boundary mixing, the main transport of buoyancy occurs near the boundaries even though the interior is stratified; independent measurements of tracer mixing in the interior show that this fluid may be quiescent and stratified even though there is a large flux being transported in the boundary region. This has important implications for the interpretation of mixing data in the ocean. In a companion paper, (Li and Woods Part 2), we explore the interaction of a net upwelling with such boundary mixing.

Many geological layers include cross bedding, which leads to different values for permeability along and across the bedding planes. We explore how such cross bedding impacts buoyancy-driven flow through an inclined aquifer. For each bedding angle and ratio of the permeability along and across the bedding, a free buoyancy-driven plume rises at a particular angle to the horizontal. If the angle of inclination of the aquifer to the horizontal is smaller than this angle, then the plume rises along the upper boundary, otherwise, somewhat surprisingly, the buoyant plume rises along the lower boundary of the aquifer. We present new laboratory experiments to support these predictions. We also test a model for the effective permeabilities which control the speed and the rate of spread of the plume along one or other boundary of the aquifer. We consider the impact of our results for modelling geological storage of ${\rm CO}_2$ or aquifer thermal energy storage.

A series of laboratory experiments are reported in which a continuous stream of bubbles rise from a small source at the base of a tank of water. Using different nozzles, bubble sizes $d$ ranging from 1.2 to 11.6 mm were produced for a number of gas volume fluxes, $Q_b$, ranging between 1.1 and $21.1\times 10^{-6}\ {\rm m}^3\ {\rm s}^{-1}$. Within a small distance from the source, the slip speed of these bubbles exceeds the speed of the equivalent single-phase plume with the same buoyancy flux, leading to formation of what we refer to as the ‘slip plume’ regime. Through a combination of high-speed photography, coupled with flow visualisation in the plume and the ambient fluid using dye, we find that the bubble speed and the fluid speed remain nearly constant with height, with the maximum fluid speed being of order $0.30\pm 0.03$ of the bubble speed. Using the filling box method, we also find that the net fluid volume flux in the slip plume increases linearly with distance from the source at a rate $Q_l = \lambda Bz/v_s^2$, where $B$ is the buoyancy flux of the gas, $v_s$ the rise speed of the gas bubbles, $z$ the distance above the source and $\lambda$ is a constant related to the dimensionless volume of fluid in the wake of each bubble. This slip-dominated flow regime can be understood in terms of kinetic energy imparted to the fluid as the bubbles rise and release potential energy. Further experiments with particle-laden plumes illustrate similar scalings for the volume flux in a particle-driven slip plume once the slip speed of the particles exceeds the bulk speed of the equivalent single-phase buoyant plume with the same buoyancy flux. Near the source the slip speed may be smaller than the plume speed, and the flow follows the classical model for a turbulent buoyant plume, with the transition to the slip regime occurring at a distance $z^*\approx (32\pm 5)\lambda ^{3/2} B/v_s^3$ from the source, where the dimensionless parameter $\lambda$ relates to the dimensionless volume of the fluid wake, which we find varies with the Reynolds number of the particles.

We present new experiments of particle-laden turbulent fountains in a uniform horizontal crossflow, $u_a$, with momentum flux, $M_0$, and buoyancy flux, $B_0$. We use the ratio, $P$, of the crossflow speed to the characteristic fountain speed, $M_0^{-1/4}|B_0|^{1/2}$, and the ratio $U$, of the Stokes fall speed of the particles, $v_s$, to the characteristic fountain speed, to characterise the dynamics of a particle fountain in a crossflow. We find that the dynamics of these particle fountains can be categorised into three distinct regimes. In regime I when the fall speed of the particles is small in comparison with the characteristic fountain speed ($U\ll 1$), the particles remain well-coupled to the fountain fluid and the flow essentially behaves as a single-phase fountain in a crossflow. In the transitional regime II ($0.1< U<1$), when the fall speed of particles is comparable to the characteristic fountain speed, we observe some particles separating from the fountain fluid during the descent of the flow which leaves some fluid neutrally buoyant. As $U>1$ (regime III), we observe particles separating from the fountain as it rises from the source. We measure the average dispersal distance of the particles and the speed of the descending particles as a function of $U$ and $P$ and compare these results with models of a single-phase fountain in a crossflow. We build a regime diagram to describe the effect of $U$ and $P$ on the flow dynamics and consider our work in the context of deep-submarine volcanic eruptions.

We explore the motion of an axisymmetric gravity current in an anisotropic porous medium in which the horizontal permeability is larger than the vertical permeability. It is well known that the classical axisymmetric gravity current supplied by a constant point source of fluid has an unphysical singularity near the origin. We address this by considering a pressure-dominated region near the origin which allows for vertical flow from the source, such that the current remains of finite depth, whilst beyond this region the flow is gravity dominated. At early times the inner pressure-driven region controls the spreading of the current, but at late times the inner region occupies a progressively smaller fraction of the current such that the radius increases as ${\sim }t^{3/7}$, while the depth near the origin increases approximately as ${\sim }t^{1/7}$. The presence of anisotropy highlights this phenomenon, since the vertical permeability maintains an effect on the flow at late times through the pressure-driven flow near the origin. Using these results we provide some quantitative insights into the dominant dynamics which controls CO$_2$ migration through permeable aquifers, as occurs in the context of carbon capture and storage.

We present a series of experiments to illustrate the dynamics of positively or negatively buoyant particle-laden plumes in a cross-flow, with relevance for the discharge of sediment into the ocean during deep-sea mining operations. In an unstratified ambient fluid, our experiments identify three different regimes, corresponding to (i) a dense particle-laden plume, host to relatively dense saline fluid, in which the particles separate from the descending plume as the flow speed falls below the particle settling speed; (ii) a dense particle-laden plume, host to buoyant fluid, in which the fluid gradually rises from the sinking plume of particles, to form a second rising plume of source fluid; and (iii) a buoyant particle-laden plume, host to buoyant fluid, which rises from the discharge pipe, and from which particles gradually sediment. Classical models of single-phase plumes describe the initial motion of the plumes in cases (i) and (iii), but as the flow speed falls below the particle fall speed, sedimentation leads to a change in the averaged buoyancy, and, hence, the plume speed. Our data also suggest that the sedimentation leads to a reduction in the rate of entrainment of ambient fluid, compared with the classical single-phase plumes. We also show that with a density stratified ambient fluid, the stratification may arrest the plume prior to significant particle sedimentation, and in this case, the plume tends to spread downstream at the level of neutral buoyancy where particle sedimentation proceeds. The bulk density of the residual plume fluid may then remain intermediate between the density of the upper and lower layer fluid, or may become less dense than the upper layer fluid, in which case, following sedimentation, the plume fluid rises through the upper layer. While the dynamics of deep-sea mining plumes in the ocean are more complex, for example, owing to background turbulence and mixing, the results of our new laboratory experiments highlight the range of flow processes which may influence the initial dispersion and sedimentation of particles in such plumes following release into the water, depending on the initial conditions, the ambient density and the particle fall speed. We also discuss the relevance of our work in the context of ash dispersal by volcanic plumes.

We explore the interior stratification that develops in a reservoir of fluid in which there is a localised source of destabilising (positive) buoyancy and a distributed source of stabilising (negative) buoyancy at the surface of the fluid, combined with an interior turbulent diffusivity. The source of destabilising buoyancy forms a descending turbulent plume that entrains ambient fluid, and this, coupled with the interior turbulent diffusivity, leads to mixing of the interior fluid. When the ratio $Pe$ of the plume source volume flux to the diffusive flux across the whole reservoir is large, the reservoir is well mixed except for a narrow surface boundary layer, while for smaller $Pe$, the reservoir is more stratified, consistent with the results of Hughes et al. (J. Fluid Mech., vol. 581, 2007, pp. 251–276) and Hughes & Griffiths (Annu. Rev. Fluid Mech., vol. 40, issue 1, 2008, pp. 185–208). We explore the evolution of the system when the source buoyancy flux varies in time, exploring both sudden changes and oscillatory changes in the forcing. Provided that the time scale of the change in the forcing is comparable to or smaller than the response time of the reservoir, as the source buoyancy flux decreases the plume may become trapped in the upper part of the reservoir, leading to a much shallower mixed zone. However, in the case of a source whose strength oscillates in time, as the source buoyancy flux increases, again the deeper water becomes involved in the mixing and circulation. We discuss briefly how these results may provide insight into the impact of changes in the forcing on mixing in deep ocean waters.

We present new experiments of particle-driven turbulent plumes issuing from a constant source of dense particle-laden fluid, with buoyancy flux, $B$, in a uniform horizontal current, $u$. Experiments show that a turbulent, well-mixed plume develops, in which the downward vertical speed $w$ decreases with depth $z$ according to $w = 0.76 (B/uz)^{1/2}$ while the horizontal speed rapidly asymptotes to the current speed $u$, provided that the Stokes settling speed of the particles $v<0.92 w$. For $v > 0.92 w$, the particles separate from the plume fluid, and their depth $z$ increases according to the simple sedimentation trajectory $\textrm {d}z/{\textrm {d}\kern0.7pt x} = v/u$. As the particles sediment, they form clusters of particles, which lead to fluctuations in the particle load with position, but do not appear to change the time-average sedimentation speed. We explore the impact of these results for deep-sea mining, in which the fate of the plume water as well as the particles is key for assessing potential environmental impacts.

We analyse the stability of the interface between two immiscible fluids both flowing in the horizontal direction in a thin cell with vertically varying gap width. The dispersion relation for the growth rate of each mode is derived. The stability is approximately determined by the sign of a simple expression, which incorporates the density difference between the fluids and the effect of surface tension in the along- and cross-cell directions. The latter arises from the varying channel width: if the non-wetting fluid is in the thinner part of the channel, the interface is unstable as it will preferentially migrate into the thicker part. The density difference may suppress or complement this effect. The system is always stable to sufficiently large wavenumbers owing to the along-flow component of surface tension.

We consider the release of preferentially wetting fluid in a laterally extensive V-shaped channel initially filled with a second fluid, presenting solutions for the initial exchange flow and the late time spreading of the wetting fluid along the narrow part of the channel. We also show that, if there is a buoyancy force acting in the cross-channel direction, the early time exchange flow depends on the Bond number, and the intermediate time slumping flow may initially be dominated by buoyancy, but at long times becomes controlled by capillarity. Where there is an along-channel component of gravity we show that the flow spreads out downslope, with capillarity controlling the structure of the nose. We then consider the case where the channel is connected to a reservoir of wetting fluid at constant pressure. We show that, depending on this pressure, either a zero flux exchange flow develops, or a net inflow through the whole width of the channel develops, as in the classical Washburn, Lucas, Bell and Cameron capillary imbibition flow. We show these flows are analogous to the classical model for one-dimensional capillary driven flows in porous media, with the current width in the channel corresponding to the saturation in the pore space.

We present a series of experiments to explore the dynamics of particle-laden fountains rising through a stratified environment with zero buoyancy flux at the source. We find that the ratio $U$ between the particle sedimentation speed $V_s$ and the characteristic fountain velocity $(M_0N^{2})^{1/4}$, where $M_0$ is the initial momentum flux and $N$ the frequency of the ambient stratification, has a profound effect on the structure of the fountain and the dispersal of the particles. In a mono-disperse particle fountain, when the settling speed of the particles is small in comparison to the characteristic fountain speed ($U\ll 1$) the flow initially behaves in an analogous fashion to a single-phase fountain, forming an intrusion at a height of approximately 0.5 of the maximum fountain height. As the fluid–particle mixture spreads out, the particles gradually sediment to the tank floor. The intruding fluid subsequently rises and forms a new intrusion at its neutral buoyancy height. Some of the particles are carried up from the original intrusion with the rising fluid. This leads to the formation of a sedimenting column of particles with a characteristic radius. We observe a transition in the behaviour of the particle fountains in the vicinity of $U\sim 0.1$, with the particles now separating from the fluid near the top of the fountain. The separation of the particles leads to a reduction in the steady-state height of the particle-laden fountain, while the fluid in the fountain continues upwards until reaching its neutral buoyancy height and forming an intrusion above the fountain top. We compare the experimental data with two models of turbulent fountains to help rationalise the trends observed as a function of the dimensionless fall speed $U$. We briefly consider the dynamics of poly-disperse particle fountains and relate their dynamics to the regimes observed in their mono-disperse counterparts. We discuss the implications of this work for the dispersal of different sized particles from submarine volcanic eruptions.

The post-injection migration of a plume of CO$_{2}$ through an inclined, confined porous layer across which the permeability varies is studied theoretically. We derive a sharp-interface lubrication model which accounts for the capillary trapping of CO$_{2}$ at the receding edge of the plume. Eventually the CO$_{2}$ becomes entirely trapped in the pore throats, and the final run-out distance is a key quantity for determining storage security and efficiency. We deploy asymptotic approximations to show that when the CO$_{2}$ plume migrates a long distance relative to its initial length, the run-out distance is approximately proportional to the permeability at the top of the layer. The permeability structure away from the top boundary is important at early and intermediate times. Dissolution of the CO$_{2}$ and three-dimensional effects are incorporated, which demonstrate that the influence of heterogeneity is quite general. The initial distribution of the CO$_{2}$ at the end of the injection phase also influences the post-injection spreading and trapping. At low injection rates, the CO$_{2}$ remains near the top of the layer so that the end-of-injection plume shape has a small aspect ratio leading to a relatively large run-out length. At very high fluxes, the end-of-injection shape and hence the final run-out distance becomes nearly independent of the injection flux because the role of buoyancy becomes negligible during injection. Our results illustrate the strong control of reservoir heterogeneity on the sweep efficiency of a CO$_{2}$ plume and hence the storage capacity. In many situations, it may be possible to increase the storage volume by injecting at a higher rate.

In recent years, a variety of efforts have been made in political science to enable, encourage, or require scholars to be more open and explicit about the bases of their empirical claims and, in turn, make those claims more readily evaluable by others. While qualitative scholars have long taken an interest in making their research open, reflexive, and systematic, the recent push for overarching transparency norms and requirements has provoked serious concern within qualitative research communities and raised fundamental questions about the meaning, value, costs, and intellectual relevance of transparency for qualitative inquiry. In this Perspectives Reflection, we crystallize the central findings of a three-year deliberative process—the Qualitative Transparency Deliberations (QTD)—involving hundreds of political scientists in a broad discussion of these issues. Following an overview of the process and the key insights that emerged, we present summaries of the QTD Working Groups’ final reports. Drawing on a series of public, online conversations that unfolded at www.qualtd.net, the reports unpack transparency’s promise, practicalities, risks, and limitations in relation to different qualitative methodologies, forms of evidence, and research contexts. Taken as a whole, these reports—the full versions of which can be found in the Supplementary Materials—offer practical guidance to scholars designing and implementing qualitative research, and to editors, reviewers, and funders seeking to develop criteria of evaluation that are appropriate—as understood by relevant research communities—to the forms of inquiry being assessed. We dedicate this Reflection to the memory of our coauthor and QTD working group leader Kendra Koivu.1

We study the longitudinal dispersion of a passive tracer by a two-dimensional pressure-driven flow through a layer of heterogeneous porous rock which is bounded above and below by impermeable seal rock. We assume there are localised regions of different permeability at random vertical positions within an otherwise uniform layer. This leads to a Fickian-type dispersion (Eames & Bush, Proc. R. Soc. Lond. A, vol. 455, issue 1990, 1999, pp. 3665–3686) and a shear near the boundaries (Rabinovich et al., Phys. Rev. E, vol. 86, issue 4, 2012, 046601). We illustrate this effect in a long two-dimensional layer, consisting of $(a)$ long and thin lenses and $(b)$ elliptical lenses, using expressions for depth-dependent mean speed and dispersivity to derive a depth-averaged transport equation for the flow. We derive asymptotic solutions for the layer-averaged dispersal of tracer, showing that at long times the spreading of the tracer is controlled by the shear. We discuss the importance of our results for the interpretation of tracer dispersal tests in layered permeable rocks.

We report a series of experiments in which a cylinder, with a vertical axis, is moved back and forth along a long narrow channel containing fresh water at Reynolds numbers $Re=3220\text {--}13\,102$. We examine the mixing of a cloud of dye along the channel by the oscillatory motion of the cylinder. Using light attenuation techniques to measure the time evolution of the concentration of dye along the channel, we find that at early times the concentration profile collapses to a Gaussian profile with dispersivity, $D=(2.4\pm 0.5) fdW$, where $f$ is the frequency of the cylinder oscillation, $d$ is the diameter of the cylinder and $W$ is the width of the channel, respectively. For times much longer than $L^2/D$, with $L$ being the length of the channel, the concentration becomes progressively more uniform over the whole length of the channel, and we show that the long-time non-uniform component decays with time dependence $\exp (- 4{\rm \pi} ^2 Dt / L^2)$. We consider the implications of these experiments for the dispersal of viral aerosols along poorly ventilated corridors, with implications for infection transmission in hospitals and public buildings.

We study the longitudinal spreading of a passive tracer by a two-dimensional pressure-driven flow through a composite layer of porous rock which is bounded above and below by impermeable seal rock. We focus on the flow across the interface between two neighbouring zones of the rock. First, we show that, with isotropic permeability, if the interface between the two zones is tilted relative to the boundaries, then this results in a difference in travel times across the formation which in turns leads to a net shear flow. We explore the strength of this shear as a function of (a) the permeability ratio across the interface, and (b) the interface angle. Second, we show that if one zone of the rock is cross-bedded, then with uniform flow, the pressure gradient is directed at an angle to the boundary. As a result, there is a transition zone across the interface, which again leads to a net shear, even if the interface is normal to the boundaries of the layer. We explore the competition between these effects, showing how they may combine constructively to produce a larger shear, or may negate one another, reducing or reversing the sign of the shear, depending on the angle of the interface, the degree of anisotropy and the change in effective downstream permeability across the interface. We discuss some of the implications of this shear for modelling flow in such composite rocks.

We consider the injection of a buoyant low viscosity fluid into an aquifer saturated with a higher viscosity fluid. The nose region of the flow, where the thickness of the injected fluid is less than the thickness of the aquifer, grows in proportion to time and as a result fluid continually migrates further into the nose where it has a progressively smaller vertical extent. We explore how the flow in the nose influences the migration of a pulse of tracer. The growth of the nose stretches a pulse of tracer of initial length, $L_0$, longitudinally to have a length proportional to $L_0 (T/T_E)^{1/2}$, where $T_E$ is the nose entry time. Diffusion acts at the same rate and the combination of the two processes results in an tracer spreading longitudinally with a length proportional to $(D T\, \log T)^{1/2}$ at long times after entering the nose, where D is the coefficient of diffusion. The results are generalised to consider the case in which the permeability in the aquifer varies with depth. At early times, the tracer is sheared. As the tracer migrates into continually thinner regions of the growing nose, the permeability contrast sampled by the tracer rapidly decays. The role of the shear becomes dominated by the stretching of the nose and ultimately the late-time behaviour is as in a uniform aquifer. However, the effective pulse length of the tracer upon asymptoting to the stretching regime is now given by $L_0={\rm \Delta} U T_E$, where ${\rm \Delta} U$ is the magnitude of the shear. The spreading in the stretching regime then has a length scale of ${\rm \Delta} U (T_E T)^{1/2}$, which may be much faster than in the case of a uniform aquifer. If the diffusion is sufficiently fast, there may be an intermediate regime in which Taylor dispersion is important prior to the stretching dominating.

Many permeable aquifers have vertical variations in the permeability, which are correlated over long lateral distances. When a buoyant high viscosity fluid is injected into a permeable aquifer and displaces a less viscous fluid, the interface that develops has finite extent and travels with constant speed. The effect of the permeability variations leads to fluid in the high permeability regions travelling into the front, where it migrates into the lower permeability part. Subsequently, it lags progressively further behind the advancing front. We explore the influence of this cycling on the dispersion of a tracer or additive in the fluid to determine its distribution within the current as a function of time. At early times tracer is sheared owing to the vertically varying permeability. At later times, cross-aquifer diffusion homogenises the tracer distribution, which spreads longitudinally at a faster rate than by diffusion alone in this shear dispersion regime. Eventually, tracer reaches the front of the current. This acts as a no-flux boundary and the concentration profile transitions to a half-Gaussian, with the maximum concentration at the front. The centre of mass of the tracer spreads backwards relative to the fixed nose at a rate proportional to $(DT)^{1/2}$, where T is time and D is the diffusion coefficient. The initial release of tracer may not be vertically uniform owing to the heterogeneity and we show that this can lead to the centre of mass of tracer initially advancing faster than the mean flow. Although our model is highly idealised, it illustrates how, owing to a vertical variation of permeability, it is possible for a finite pulse of tracer or chemical added after the start of injection to reach the front with important implications for tracer tests and strategies for enhanced recovery.

We present an experimental investigation of steady particle-driven gravity currents with Reynolds numbers in the range 500–1600, and with the ratio of the initial current speed to the fall speed of the particles, $S=u_{0}/u_{fall}$, in the range $5<S<160$. We identify three regimes: (i) For $S<10$, the particles settle close to the source at a velocity corresponding to their fall speed, consistent with the observation of sedimenting fronts in classical settling column experiments. (ii) In the range $10<S<40$, a steady gravity current develops within the tank. The experiments show that the depth of the gravity current gradually decreases away from the source and dye added to the source liquid appears above the gravity current along its entire length, suggesting that there is a sedimentation front, so that the volume and momentum fluxes of the current gradually decrease with distance from the source. We find that as $S$ increases, the descent speed of the sedimentation front decreases relative to the fall speed of the particles, and the run-out length of the gravity current increases. We note that the density of the interstitial fluid corresponds to the density of the ambient fluid, so that any reduction in buoyancy of the gravity current is attributed to the sedimentation of particles on the floor of the tank and we do not observe lofting of the interstitial fluid. (iii) For $40<S<160$, the gravity currents reach the end of our experimental tank and we no longer observe a sedimentation front. For these experiments, it appears that the entrainment at the top of the current begins to match the sedimentation and so the current depth does not change significantly over the scale of the tank, but a larger scale experimental system would be needed to explore the full run-out behaviour for these larger values of $S$. For the intermediate case, $10<S<40$, we develop a model for the conservation of volume, momentum and buoyancy fluxes in the current, accounting for the sedimentation front and the release of fluid at the top surface of the gravity current, and we compare this with our new experimental data.