When a flat plate is withdrawn from a liquid pool, a liquid film is deposited on the plate. This simple process is called dip coating. In the case of vertically upward withdrawal, gravity competes with the surface tension and viscous drag, and the balance between those determine the meniscus shape and hence the film thickness. Most of the previous studies on dip coating assumed that the pool is sufficiently large so that the stationary container wall does not affect the film thickness. However, the cases where the stationary wall affects the entrained film have not been examined thoroughly so far. In this confined dip coating, the film thickness deviates from that of unconfined dip coating under the same conditions such as the withdrawal speed and the physical properties of the liquid. The meniscus in a confined pool is more curved than that in an unconfined pool owing to wetting on the stationary wall, which is parallel to the plate. Besides, a channel between the moving plate and the stationary wall appears; therefore, the flow inside the channel should be included in an analysis of confined dip coating. In the present study, we analyse the mechanism that determines the film thickness, both theoretically and numerically.

In the formulation of stability of fluid flow through channels and tubes with deformable walls, while the fluid is naturally treated in an Eulerian framework, the solid can be treated either in a Lagrangian or Eulerian framework. A consistent formulation, then, should yield results that are independent of the chosen framework. Previous studies have demonstrated this consistency for the stability of plane Couette flow past a deformable solid layer modelled as a neo-Hookean solid, in the creeping-flow limit. However, a similar exercise carried out in the creeping-flow limit for the stability of pressure-driven flow in a neo-Hookean tube shows that while the flow is stable in the Lagrangian formulation, it is unstable in the existing Eulerian formulation. The present work resolves this discrepancy by presenting consistent Lagrangian and Eulerian frameworks for performing stability analyses in flow through deformable tubes and channels. The resolution is achieved by making important modifications to the Lagrangian formulation to make it fundamentally consistent, as well as by proposing a proper formulation for the neo-Hookean constitutive relation in the Eulerian framework. In the neo-Hookean model, the Cauchy stress tensor in the solid is proportional to the Finger tensor. We demonstrate that the neo-Hookean constitutive model within the Eulerian formulation used in the previous studies is a special case of the Mooney–Rivlin solid, with the Cauchy stress tensor being proportional to the inverse of the Finger tensor unlike in a true neo-Hookean solid. Remarkably, for plane Couette flow subjected to two-dimensional perturbations, there is perfect agreement between the results obtained using earlier Eulerian and Lagrangian formulations despite the crucial difference in the constitutive relation owing to the rather simple kinematics of the base state. However, the consequences are drastic for pressure-driven flow in a tube even for axisymmetric disturbances. We propose a consistent neo-Hookean constitutive relation in the Eulerian framework, which yields results that are in perfect agreement with the results from the Lagrangian formulation for both plane Couette and tube flows at arbitrary Reynolds number. The present study thus provides an unambiguous formulation for carrying out stability analyses in flow through deformable channels and tubes. We further show that unlike plane Couette flow and Hagen–Poiseuille flow in rigid-walled conduits where there is a remarkable similarity in the linear stability characteristics between these two flows, the stability behaviour for these two flows is very different when the walls are deformable. The instability of plane Couette flow past a deformable wall is very robust and is not sensitive to the constitutive nature of the solid, but the stability of pressure-driven flow in a deformable tube is rather sensitive to the constitutive nature of the deformable solid, especially at low Reynolds number.

Fluidisation is the process by which the weight of a bed of particles is supported by a gas flow passing through it from below. When fluidised materials flow down an incline, the dynamics of the motion differs from their non-fluidised counterparts because the granular agitation is no longer required to support the weight of the flowing layer. Instead, the weight is borne by the imposed gas flow and this leads to a greatly increased flow mobility. In this paper, a framework is developed to model this two-phase motion by incorporating a kinetic theory description for the particulate stresses generated by the flow. In addition to calculating numerical solutions for fully developed flows, it is shown that for sufficiently thick flows there is often a local balance between the production and dissipation of the granular temperature. This phenomenon permits an asymptotic reduction of the full governing equations and the identification of a simple state in which the volume fraction of the flow is uniform. The results of the model are compared with new experimental measurements of the internal velocity profiles of steady granular flows down slopes. The distance covered with time by unsteady granular flows down slopes and along horizontal surfaces and their shapes are also measured and compared with theoretical predictions developed for flows that are thin relative to their streamwise extent. For the horizontal flows, it was found that resistance from the sidewalls was required in addition to basal resistance to capture accurately the unsteady evolution of the front position and the depth of the current and for situations in which sidewall drag dominates, similarity solutions are found for the experimentally measured motion.

This paper concerns feedback stabilization of point-vortex equilibria above an inclined thin plate and a three-plate configuration known as the Kasper wing in the presence of an oncoming uniform flow. The flow is assumed to be potential and is modelled by the two-dimensional incompressible Euler equations. Actuation has the form of blowing and suction localized on the main plate and is represented in terms of a sink–source singularity, whereas measurement of pressure across the plate serves as system output. We focus on point-vortex equilibria forming a one-parameter family with locus approaching the trailing edge of the main plate and show that these equilibria are either unstable or neutrally stable. Using methods of linear control theory we find that the system dynamics linearized around these equilibria is both controllable and observable for almost all actuator and sensor locations. The design of the feedback control is based on the linear–quadratic–Gaussian (LQG) compensator. Computational results demonstrate the effectiveness of this control and the key finding of this study is that Kasper wing configurations are in general not only more controllable than their single-plate counterparts, but also exhibit larger basins of attraction under LQG feedback control. The feedback control is then applied to systems with additional perturbations added to the flow in the form of random fluctuations of the angle of attack and a vorticity shedding mechanism. Another important observation is that, in the presence of these additional perturbations, the control remains robust, provided the system does not deviate too far from its original state. Furthermore, except in a few isolated cases, introducing a vorticity-shedding mechanism enhanced the effectiveness of the control. Physical interpretation is provided for the results of the controllability and observability analysis as well as the response of the feedback control to different perturbations.

We study the fluid–structure interaction (FSI) of a compliant panel with developing Blasius boundary-layer flow. The linearised Navier–Stokes equations in velocity–vorticity form are solved using a Helmholtz decomposition coupled with the dynamics of a plate-spring compliant panel couched in finite-difference form. The FSI system is written as an eigenvalue problem and the various flow- and wall-based instabilities are analysed. It is shown that global temporal instability can occur through the interaction of travelling wave flutter (TWF) with a structural mode or as a resonance between Tollmien–Schlichting wave (TSW) instability and discrete structural modes of the compliant panel. The former is independent of compliant panel length and upstream inflow disturbances while the specific behaviour arising from the latter phenomenon is dependent upon the frequency of a disturbance introduced upstream of the compliant panel. The inclusion of axial displacements in the wall model does not lead to any further global instabilities. The dependence of instability-onset Reynolds numbers with structural stiffness and damping for the global modes is quantified. It is also shown that the TWF-based global instability is stabilised as the boundary layer progresses downstream while the TSW-based global instability exhibits discrete resonance-type behaviour as Reynolds number increases. At sufficiently high Reynolds numbers, a globally unstable divergence instability is identified when the wavelength of its wall-based mode is longer than that of the least stable TSW mode. Finally, a non-modal analysis reveals a high level of transient growth when the flow interacts with a compliant panel which has structural properties capable of reducing TSW growth but which is prone to global instability through wall-based modes.

We investigate theoretically the statics and dynamics of bubbles in fully confined liquids, i.e. in liquids surrounded by solid walls in all directions of space. This situation is found in various natural and technological contexts (geological fluid inclusions, plant cells and vessels, soil tensiometers, etc.), where such bubbles can pre-exist in the trapped liquid or appear by nucleation (cavitation). We focus on volumetric deformations and first establish the potential energy of fully confined bubbles as a function of their radius, including contributions from gas compressibility, surface tension, liquid compressibility and elastic deformation of the surrounding solid. We evaluate how the Blake threshold of unstable bubble growth is modified by confinement and we also obtain an original bubble stability phase diagram with a regime of liquid superstability (spontaneous bubble collapse) for strong confinements. We then calculate the liquid velocity field associated with radial deformations of the bubble and strain in the solid, and we predict large deviations in the kinematics compared to bubbles in extended liquids. Finally, we derive the equations governing the natural oscillation dynamics of fully confined bubbles, extending Minnaert’s formula and the Rayleigh–Plesset equation, and we show that the compressibility of the liquid as well as the elasticity of the walls can result in ultra-fast bubble radial oscillations and unusually quick damping. We find excellent agreement between the predictions of our model and recent experimental results.

Unstable equilibrium solutions in a homogeneous shear flow with sinuous (streamwise-shift-reflection and spanwise-shift-rotation) symmetry are numerically found in large-eddy simulations (LES) with no kinetic viscosity. The small-scale properties are determined by the mixing length scale $l_{S}$ used to define eddy viscosity, and the large-scale motion is induced by the mean shear at the integral scale, which is limited by the spanwise box dimension $L_{z}$. The fraction $R_{S}=L_{z}/l_{S}$, which plays the role of a Reynolds number, is used as a numerical continuation parameter. It is shown that equilibrium solutions appear by a saddle-node bifurcation as $R_{S}$ increases, and that the flow structures resemble those in plane Couette flow with the same sinuous symmetry. The vortical structures of both lower- and upper-branch solutions become spontaneously localised in the vertical direction. The lower-branch solution is an edge state at low $R_{S}$, and takes the form of a thin critical layer as $R_{S}$ increases, as in the asymptotic theory of generic shear flow at high Reynolds numbers. On the other hand, the upper-branch solutions are characterised by a tall velocity streak with multiscale multiple vortical structures. At the higher end of $R_{S}$, an incipient multiscale structure is found. The LES turbulence occasionally visits vertically localised states whose vortical structure resembles the present vertically localised LES equilibria.

We experimentally investigate scale-to-scale anisotropy from the integral to the dissipative scales in homogeneous turbulence. We employ an apparatus in which two facing arrays of randomly actuated air jets generate turbulence with negligible mean flow and shear, over a volume several times larger than the energy-containing eddy size. The Reynolds number based on the Taylor microscale is varied in the range $Re_{\unicode[STIX]{x1D706}}\approx 300{-}500$, while the axial-to-radial ratio of the root mean square velocity fluctuations ranges between 1.38 and 1.72. Two velocity components are measured by particle image velocimetry at varying resolutions, capturing from the integral to the Kolmogorov scales and yielding statistics up to sixth order. Over the inertial range, the scaling exponents of the velocity structure functions are found to differ not only between the longitudinal and transverse components, but also between the axial and radial directions of separation. At the dissipative scales, the moments of the velocity gradients indicate that departure from isotropy is, at the present Reynolds numbers, significant and more pronounced for stronger large-scale anisotropy. The generalized flatness factors of the longitudinal velocity differences tend towards isotropy as the separation is reduced from the inertial to the near-dissipative scales (down to about $10\unicode[STIX]{x1D702}$, $\unicode[STIX]{x1D702}$ being the Kolmogorov length), but become more anisotropic for even smaller scales which are characterized by high intermittency. At the large scales, the direction of turbulence forcing is associated with a larger integral length, defined as the distance over which the velocity component in a given direction is spatially correlated. Because of anisotropy, the definition of the integral length is not trivial and may lead to dissimilar conclusions on the qualitative behaviour of the large scales and on the quantitative values of the normalized dissipation. Alternative definitions of these quantities are proposed to account for the anisotropy. Overall, these results highlight the importance of evaluating both the different velocity components and the different spatial directions across all scales of the flow.

We investigate the flow and turbulence structure in front of a cylinder mounted on a flat plate by a combined study using highly resolved large-eddy simulation and particle image velocimetry. The Reynolds number based on the bulk velocity and cylinder diameter is $Re_{D}=39\,000$. As the cylinder is placed in an open channel, we take special care to simulate open-channel flow as the inflow condition, including secondary flows that match the inflow in the experiment. Due to the high numerical resolution, subgrid contributions to the Reynolds stresses are negligible and the modelled dissipation plays a minor role in major parts of the flow field. The accordance of the experimental and numerical results is good. The shear in the approach flow creates a vertical pressure gradient, inducing a downflow in the cylinder front. This downflow, when deflected in the upstream direction at the bottom plate, gives rise to a so-called horseshoe vortex system. The most upstream point of flow reversal at the wall is found to be a stagnation point which appears as a sink instead of a separation point in the symmetry plane in front of the cylinder. The wall shear stress is largest between the main (horseshoe) vortex and the cylinder, and seems to be mainly governed by the strong downflow in front of the cylinder as turbulent stresses are small in this region. Due to a strong acceleration along the streamlines, a region of relatively small turbulent kinetic energy is found between the horseshoe vortex and the cylinder. When passing under the horseshoe vortex, the upstream-directed jet formed by the deflected downflow undergoes a deceleration which gives rise to a strong production of turbulent kinetic energy. We find that pressure transport of turbulent kinetic energy is important for the initiation of the large production rates by increasing the turbulence level in the upstream jet near the wall. The distribution of the dissipation of turbulent kinetic energy is similar to that of the turbulent kinetic energy. Large values of dissipation occur around the centre of the horseshoe vortex and near the wall in the region where the jet decelerates. While the small scales are nearly isotropic in the horseshoe vortex centre, they are anistotropic near the wall. This can be explained by a vertical flapping of the upstream-directed jet. The distribution and level of dissipation, turbulent and pressure transport of turbulent kinetic energy are of crucial interest to turbulence modelling in the Reynolds-averaged context. To the best of our knowledge, this is the first time that these terms have been documented in this kind of flow.

First-principle-based prediction of mean-flow quantities of wall-bounded turbulent flows (channel, pipe and turbulent boundary layer (TBL)) is of great importance from both physics and engineering standpoints. Here we present a symmetry-based approach which yields analytical expressions for the mean-velocity profile (MVP) from a Lie-group analysis. After verifying the dilatation-group invariance of the Reynolds averaged Navier–Stokes (RANS) equation in the presence of a wall, we depart from previous Lie-group studies of wall turbulence by selecting a stress length function as a similarity variable. We argue that this stress length function characterizes the symmetry property of wall flows having a simple dilatation-invariant form. Three kinds of (local) invariant forms of the length function are postulated, a combination of which yields a multi-layer formula giving its distribution in the entire flow region normal to the wall and hence also the MVP, using the mean-momentum equation. In particular, based on this multi-layer formula, we obtain analytical expressions for the (universal) wall function and separate wake functions for pipe and channel, which are validated by data from direct numerical simulations (DNS). In conclusion, an analytical expression for the entire MVP of wall turbulence, beyond the log law or power law, is developed in this paper and the theory can be used to describe the mean turbulent kinetic-energy distribution, as well as a variety of boundary conditions such as pressure gradient, wall roughness, buoyancy, etc. where the dilatation-group invariance is valid in the wall-normal direction.

We present an effective reduced-order model (ROM) technique to couple an incompressible flow with a transversely vibrating bluff body in a state-space format. The ROM of the unsteady wake flow is based on the Navier–Stokes equations and is constructed by means of an eigensystem realization algorithm (ERA). We investigate the underlying mechanism of vortex-induced vibration (VIV) of a circular cylinder at low Reynolds number via linear stability analysis. To understand the frequency lock-in mechanism and self-sustained VIV phenomenon, a systematic analysis is performed by examining the eigenvalue trajectories of the ERA-based ROM for a range of reduced oscillation frequency $(F_{s})$, while maintaining fixed values of the Reynolds number ($Re$) and mass ratio ($m^{\ast }$). The effects of the Reynolds number $Re$, the mass ratio $m^{\ast }$ and the rounding of a square cylinder are examined to generalize the proposed ERA-based ROM for the VIV lock-in analysis. The considered cylinder configurations are a basic square with sharp corners, a circle and three intermediate rounded squares, which are created by varying a single rounding parameter. The results show that the two frequency lock-in regimes, the so-called resonance and flutter, only exist when certain conditions are satisfied, and the regimes have a strong dependence on the shape of the bluff body, the Reynolds number and the mass ratio. In addition, the frequency lock-in during VIV of a square cylinder is found to be dominated by the resonance regime, without any coupled-mode flutter at low Reynolds number. To further discern the influence of geometry on the VIV lock-in mechanism, we consider the smooth curve geometry of an ellipse and two sharp corner geometries of forward triangle and diamond-shaped bluff bodies. While the ellipse and diamond geometries exhibit the flutter and mixed resonance–flutter regimes, the forward triangle undergoes only the flutter-induced lock-in for $30\leqslant Re\leqslant 100$ at $m^{\ast }=10$. In the case of the forward triangle configuration, the ERA-based ROM accurately predicts the low-frequency galloping instability. We observe a kink in the amplitude response associated with 1:3 synchronization, whereby the forward triangular body oscillates at a single dominant frequency but the lift force has a frequency component at three times the body oscillation frequency. Finally, we present a stability phase diagram to summarize the VIV lock-in regimes of the five smooth-curve- and sharp-corner-based bluff bodies. These findings attempt to generalize our understanding of the VIV lock-in mechanism for bluff bodies at low Reynolds number. The proposed ERA-based ROM is found to be accurate, efficient and easy to use for the linear stability analysis of VIV, and it can have a profound impact on the development of control strategies for nonlinear vortex shedding and VIV.

We present an active feedback blowing and suction (AFBS) procedure via model reduction for unsteady wake flow and the vortex-induced vibration (VIV) of circular cylinders. The reduced-order model (ROM) for the AFBS procedure is developed by the eigensystem realization algorithm (ERA), which provides a low-order representation of the unsteady flow dynamics in the neighbourhood of the equilibrium steady state. The actuation is considered via vertical suction and a blowing jet at the porous surface of a circular cylinder with a body-mounted force sensor. While the optimal gain is obtained using a linear quadratic regulator (LQR), Kalman filtering is employed to estimate the approximate state vector. The feedback control system shifts the unstable eigenvalues of the wake flow and the VIV system to the left half-complex-plane, and subsequently results in suppression of the vortex street and the VIV in elastically mounted structures. The resulting controller designed by a linear low-order approximation is able to suppress the nonlinear saturated state of wake vortex shedding from the circular cylinder. A systematic linear ROM-based stability analysis is performed to understand the eigenvalue distribution for the flow past stationary and elastically mounted circular cylinders. The results from the ROM analysis are consistent with those obtained from full nonlinear fluid–structure interaction simulations, thereby confirming the validity of the proposed ROM-based AFBS procedure. A sensitivity study on the number of suction/blowing actuators, the angular arrangement of actuators and the combined versus independent control architectures has been performed for the flow past a stationary circular cylinder. Overall, the proposed control concept based on the ERA-based ROM and the LQR algorithm is found to be effective in suppressing the vortex street and the VIV for a range of reduced velocities and mass ratios.

Using a one-layer quasi-geostrophic model, we study the effect of random monoscale topography on forced beta-plane turbulence. The forcing is a uniform steady wind stress that produces both a uniform large-scale zonal flow $U(t)$ and smaller-scale macroturbulence characterized by standing and transient eddies. The large-scale flow $U$ is retarded by a combination of Ekman drag and the domain-averaged topographic form stress produced by the eddies. The topographic form stress typically balances most of the applied wind stress, while the Ekman drag provides all of the energy dissipation required to balance the wind work. A collection of statistically equilibrated numerical solutions delineate the main flow regimes and the dependence of the time average of $U$ on parameters such as the planetary potential vorticity (PV) gradient $\unicode[STIX]{x1D6FD}$ and the statistical properties of the topography. We obtain asymptotic scaling laws for the strength of the large-scale flow $U$ in the limiting cases of weak and strong forcing. If $\unicode[STIX]{x1D6FD}$ is significantly smaller than the topographic PV gradient, the flow consists of stagnant pools attached to pockets of closed geostrophic contours. The stagnant dead zones are bordered by jets and the flow through the domain is concentrated into a narrow channel of open geostrophic contours. In most of the domain, the flow is weak and thus the large-scale flow $U$ is an unoccupied mean. If $\unicode[STIX]{x1D6FD}$ is comparable to, or larger than, the topographic PV gradient, then all geostrophic contours are open and the flow is uniformly distributed throughout the domain. In this open-contour case, there is an ‘eddy saturation’ regime in which $U$ is insensitive to large changes in the wind stress. We show that eddy saturation requires strong transient eddies that act effectively as PV diffusion. This PV diffusion does not alter the kinetic energy of the standing eddies, but it does increase the topographic form stress by enhancing the correlation between the topographic slope and the standing-eddy pressure field. Using bounds based on the energy and enstrophy power integrals, we show that as the strength of the wind stress increases, the flow transitions from a regime in which the form stress balances most of the wind stress to a regime in which the form stress is very small and large transport ensues.

Experiments are presented that demonstrate how liquid-infused surfaces can reduce turbulent drag significantly in Taylor–Couette flow. The test liquid was water, and the test surface was composed of square microscopic grooves measuring $100~\unicode[STIX]{x03BC}\text{m}$ to $800~\unicode[STIX]{x03BC}\text{m}$, filled with alkane liquids with viscosities from 0.3 to 1.4 times that of water. We achieve drag reduction exceeding 35 %, four times higher than previously reported for liquid-infused surfaces in turbulent flow. The level of drag reduction increased with viscosity ratio, groove width, fluid area fraction and Reynolds number. The optimum groove width was given by $w^{+}\approx 35$.

A micro-continuum approach is proposed to simulate the dissolution of solid minerals at the pore scale under single-phase flow conditions. The approach employs a Darcy–Brinkman–Stokes formulation and locally averaged conservation laws combined with immersed boundary conditions for the chemical reaction at the solid surface. The methodology compares well with the arbitrary-Lagrangian–Eulerian technique. The simulation framework is validated using an experimental microfluidic device to image the dissolution of a single calcite crystal. The evolution of the calcite crystal during the acidizing process is analysed and related to the flow conditions. Macroscopic laws for the dissolution rate are proposed by upscaling the pore-scale simulations. Finally, the emergence of wormholes during the injection of acid in a two-dimensional domain of calcite grains is discussed based on pore-scale simulations.

The propulsive performance of a pair of tandem flapping foils is sensitively dependent on the spacing and phasing between them. Large increases in thrust and efficiency of the hind foil are possible, but the mechanisms governing these enhancements remain largely unresolved. Two-dimensional numerical simulations of tandem and single foils oscillating in heave and pitch at a Reynolds number of 7000 are performed over a broad and dense parameter space, allowing the effects of inter-foil spacing ($S$) and phasing ($\unicode[STIX]{x1D711}$) to be investigated over a range of non-dimensional frequencies (or Strouhal number, $St$). Results indicate that the hind foil can produce from no thrust to twice the thrust of a single foil depending on its spacing and phasing with respect to the fore foil, which is consistent with previous studies that were carried out over a limited parameter space. Examination of instantaneous flow fields indicates that high thrust occurs when the hind foil weaves between the vortices that have been shed by the fore foil, and low thrust occurs when the hind foil intercepts these vortices. Contours of high thrust and minimal thrust appear as inclined bands in the $S-\unicode[STIX]{x1D711}$ parameter space and this behaviour is apparent over the entire range of Strouhal numbers considered $(0.2\leqslant St\leqslant 0.5)$. A novel quasi-steady model that utilises kinematics of a virtual hind foil together with data obtained from simulations of a single flapping foil shows that performance augmentation is primarily determined through modification of the instantaneous angle of attack of the hind foil by the vortex street established by the fore foil. This simple model provides estimates of thrust and efficiency for the hind foil, which is consistent with data obtained through full simulations. The limitations of the virtual hind foil method and its physical significance is also discussed.

Dynamics and mixing of a variable-density turbulent flow subject to an externally imposed acceleration field in the zero-Mach-number limit are studied in a series of direct numerical simulations. The flow configuration studied consists of alternating slabs of high- and low-density fluid in a triply periodic domain. Density ratios in the range of $1.05\leqslant R\equiv \unicode[STIX]{x1D70C}_{1}/\unicode[STIX]{x1D70C}_{2}\leqslant 10$ are investigated. The flow produces temporally evolving shear layers. A perpendicular density–pressure gradient is maintained in the mean as the flow evolves, with multi-scale baroclinic torques generated in the turbulent flow that ensues. For all density ratios studied, the simulations attain Reynolds numbers at the beginning of the fully developed turbulence regime. An empirical relation for the convection velocity predicts the observed entrainment-ratio and dominant mixed-fluid composition statistics. Two mixing-layer temporal evolution regimes are identified: an initial diffusion-dominated regime with a growth rate ${\sim}t^{1/2}$ followed by a turbulence-dominated regime with a growth rate ${\sim}t^{3}$. In the turbulent regime, composition probability density functions within the shear layers exhibit a slightly tilted (‘non-marching’) hump, corresponding to the most probable mole fraction. The shear layers preferentially entrain low-density fluid by volume at all density ratios, which is reflected in the mixed-fluid composition.

This paper presents fundamental analysis of the injection and release of fluid into porous media or geological reservoirs saturated by a different fluid undergoing a background flow, and tests the predictions using analogue laboratory experiments. The study reveals new results important for an understanding of the transport of hazardous contaminants through aquifers and the long-term fate of carbon dioxide ($\text{CO}_{2}$) in geological $\text{CO}_{2}$ sequestration. Using numerical and asymptotic analysis, we describe a variety of flow regimes that arise, and demonstrate an almost instantaneous control of injected fluid by the far field conditions in geological reservoirs. For a continuous input, the flow develops a horizontal interface between the injected and ambient fluids. The background flow thereby effectively caps the height of the injected fluid into a shallower region of vertical confinement. For a released parcel of fluid, gravitational spreading is found to become negligible after a short time. A dominant control of the interface by the background pressure gradient arises, and stems from the different velocities at which it drives the injected and ambient fluids individually. Similarity solutions describing these dynamics show that the parcel approaches a slender triangular profile that grows horizontally as $t^{1/2}$, where $t$ is time, a rate faster than relaxation under gravity. Shock layers develop at the front or back of the parcel, depending on whether it is more or less viscous than the ambient fluid. New analytical results describing the long-term effects of residual trapping due to capillary retention are developed, which yield explicit predictions for the time and length scales on which a parcel of $\text{CO}_{2}$ becomes retained. We end by applying our results to geological contexts, concluding that even slight background motion can have considerable implications for long-term transport through the subsurface.

This paper discusses the fundamental mechanisms of vortex–droplet interactions leading to flow distortion, droplet dispersion and breakup in a complex swirling gas flow field. In particular, the way in which the location of droplet injection determines the degree of inhomogeneous dispersion and breakup modes has been elucidated in detail using high-fidelity laser diagnostics. The droplets are injected as monodispersed streams at various spatial locations such as the vortex breakdown bubble and the shear layers (inner and outer) exhibited by the swirling flow. Simultaneous time-resolved particle image velocimetry ($3500~\text{frames}~\text{s}^{-1}$) and high-speed shadowgraphy measurements are employed to delineate the two-phase interaction dynamics. These measurements have been used to evaluate the fluctuations in instantaneous circulation strength $\unicode[STIX]{x1D6E4}^{\prime }$ caused by the flow field eddies and the resultant angular dispersion in the droplet trajectories $\unicode[STIX]{x1D703}^{\prime }$. The droplet–flow interactions show two-way coupling at low momentum ratios ($MR$) and strong one-way coupling at high momentum ratios. The gas phase flow field is globally altered at low airflow rates (low $MR$) due to impact of droplets with the vortex core. The flow perturbation is found to be minimal and mainly local at high airflow rates (high $MR$). Spectral coherence analysis is carried out to understand the correlation between eddy circulation strength $\unicode[STIX]{x1D6E4}^{\prime }$ and droplet dispersion $\unicode[STIX]{x1D703}^{\prime }$. The droplet dispersion shows strong coherence with the flow in certain frequency bands. Subsequently, proper orthogonal decomposition (POD) is implemented to elucidate the governing instability mechanism and frequency signatures associated with the turbulent coherent structures. The POD results suggest dominance of the Kelvin–Helmholtz (KH) instability mode (axial and azimuthal shear). The frequency range pertaining to high coherence between dispersion and circulation shows good agreement with KH instability quantified from POD analysis. The droplets injected at the inner shear layer (ISL) and outer shear layer (OSL) show different interaction dynamics. For instance, droplet dispersion at the OSL exhibits secondary frequency (shedding mode) coupling in addition to the KH mode, whereas ISL injection couples only in a single narrow frequency band (i.e. KH mode). Further, high-speed shadow imaging ($7500~\text{frames}~\text{s}^{-1}$) is employed to visualize the breakup dynamics of the droplets. The effect of coherent structures on the droplet breakup modes is shown as a function of the Weber number ($We$) defined based on the circulation strength. The wide fluctuations caused in the instantaneous circulation strength lead to different breakup modes (bag, multimodal, shear thinning, catastrophic) even for fixed airflow rates. These fluctuations also lead to inhomogeneous spatial dispersion of the droplets in the swirling gas flow field. We are able to present the dispersion contours in terms of the Stokes number and a spatial homogeneity parameter. In essence, the dispersion inhomogeneity is found to be a strong function of the injection location, the phase relationship with the eddies and the momentum ratio ($MR$).

When a liquid drop is placed on a highly superheated surface, it can be levitated by its own vapour. This remarkable phenomenon is referred to as the Leidenfrost effect. The thermally insulating vapour film results in a severe reduction of the heat transfer rate compared to experiments at lower surface temperatures, where the drop is in direct contact with the solid surface. A commonly made assumption is that this solid surface is isothermal, which is at least questionable for materials of low thermal conductivity, resulting in an overestimation of the surface temperature and heat transfer for such systems. Here we aim to obtain more quantitative insight into how surface cooling affects the Leidenfrost effect. We develop a technique based on Mach–Zehnder interferometry to investigate the surface cooling of a quartz plate by a Leidenfrost drop. The three-dimensional plate temperature field is reconstructed from interferometric data by an Abel inversion method using a basis function expansion of the underlying temperature field. By this method we are able to quantitatively measure the local cooling inside the plate, which can be as strong as 80 K. We develop a numerical model which shows good agreement with experiments and enables extending the analysis beyond the experimental parameter space. Based on the numerical and experimental results we quantify the effect of surface cooling on the Leidenfrost phenomenon. By focusing on the role of the solid surface we provide new insights into the Leidenfrost effect and demonstrate how to adjust current models to account for non-isothermal solids and use previously obtained isothermal scaling laws for the neck thickness and evaporation rate.