Rotating flow in which the angular velocity decreases outward while the angular momentum increases is known as ‘quasi-Keplerian’. Despite the general tendency of shear flow to break down into turbulence, this type of flow seems to maintain stability at very large Reynolds number, even when nonlinearly perturbed, a behaviour that strongly influences our understanding of astrophysical accretion discs. Investigating these flows in the laboratory is difficult because secondary Ekman flows, caused by the retaining Couette cylinders, can become turbulent on their own. A recent high Reynolds number numerical study by Lopez & Avila (J. Fluid Mech., vol. 817, 2017, pp. 21–34) reconciles apparently discrepant laboratory experiments by confirming that this secondary flow recedes toward the axial boundaries of the container as the Reynolds number is increased, a result that enhances our understanding of nonlinear quasi-Keplerian flow stability.

A three-dimensional direct numerical simulation is conducted for a temporally evolving planar jet of n-heptane at a pressure of 40 atmospheres and in a coflow of air at 1100 K. At these conditions, n-heptane exhibits a two-stage ignition due to low- and high-temperature chemistry, which is reproduced by the global chemical model used in this study. The results show that ignition occurs in several overlapping stages and multiple modes of combustion are present. Low-temperature chemistry precedes the formation of multiple spatially localised high-temperature chemistry autoignition events, referred to as ‘kernels’. These kernels form within the shear layer and core of the jet at compositions with short homogeneous ignition delay times and in locations experiencing low scalar dissipation rates. An analysis of the kernel histories shows that the ignition delay time is correlated with the mixing rate history and that the ignition kernels tend to form in vortically dominated regions of the domain, as corroborated by an analysis of the topology of the velocity gradient tensor. Once ignited, the kernels grow rapidly and establish edge flames where they envelop the stoichiometric isosurface. A combination of kernel formation (autoignition) and the growth of existing burning surface (via edge-flame propagation) contributes to the overall ignition process. An analysis of propagation speeds evaluated on the burning surface suggests that although the edge-flame speed is promoted by the autoignitive conditions due to an increase in the local laminar flame speed, edge-flame propagation of existing burning surfaces (triggered initially by isolated autoignition kernels) is the dominant ignition mode in the present configuration.

Locomotion of microswimmers near an interface has attracted recent attention and has several applications related to synthetic swimmers and microorganisms. In this work, we study the motion of a model swimmer called the ‘squirmer’ with an arbitrary time-dependent swimming gait near a weakly deforming interface. We first obtain an exact solution of the governing equations for the motion of the swimmer near a plane interface using the bipolar coordinate method, and then an approximate solution using the method of reflections. We thereby derive the velocity of a swimmer due to small interface deformations using the domain perturbation method and Lorentz reciprocal theorem. We use our solution to study the dynamics of a swimmer with steady, as well as time-reversible, squirming gaits. The long-time dynamics of a time-reversible swimmer is such that it either moves towards or away from the interface. Thus, we divide its phase space into regions of attraction (repulsion) towards (from) the interface. The long-time orientation of a time-reversible swimmer that is moving towards the interface depends on its initial orientation. Additionally, we find that the method of reflections is accurate to $O(1)$ distances of the swimmer from the interface.

Here we show that the distribution of energy of internal gravity waves over a patch of seabed corrugations strongly depends on the distance of the patch to adjacent seafloor features located downstream of the patch. Specifically, we consider the steady state energy distribution due to an incident internal wave arriving at a patch of seabed ripples neighbouring (i) another patch of ripples (i.e. a second patch) and (ii) a vertical wall. Seabed undulations with dominant wavenumber twice as large as overpassing internal waves reflect back part of the energy of the incident internal waves (Bragg reflection) and allow the rest of the energy to transmit downstream. In the presence of a neighbouring topography on the downstream side, the transmitted energy from the patch may reflect back; partially if the downstream topography is another set of seabed ripples or fully if it is a vertical wall. The reflected wave from the downstream topography is again reflected back by the patch of ripples through the same mechanism. This consecutive reflection goes on indefinitely, leading to a complex interaction pattern including constructive and destructive interference of multiply reflected waves as well as an interplay between higher mode internal waves resonated over the topography. We show here that when steady state is reached both the qualitative and quantitative behaviour of the energy distribution over the patch is a strong function of the distance between the patch and the downstream topography: it can increase or decrease exponentially fast along the patch or stay (nearly) unchanged. As a result, for instance, the local energy density in the water column can become an order of magnitude larger in certain areas merely based on where the downstream topography is. This may result in the formation of steep waves in specific areas of the ocean, leading to breaking and enhanced mixing. At a particular distance, the wall or the second patch may also result in a complete disappearance of the trace of the seabed undulations on the upstream and the downstream wave field.

The linear stability of three-dimensional vortices in rotating, stratified flows has been studied by analysing the non-hydrostatic inviscid Boussinesq equations. We have focused on a widely used model of geophysical and astrophysical vortices, which assumes an axisymmetric Gaussian structure for pressure anomalies in the horizontal and vertical directions. For a range of Rossby numbers ($-0.5<Ro<0.5$) and Burger numbers ($0.02<Bu<2.3$) relevant to observed long-lived vortices, the growth rate and spatial structure of the most unstable eigenmodes have been numerically calculated and presented as a function of $Ro{-}Bu$. We have found neutrally stable vortices only over a small region of the $Ro{-}Bu$ parameter space: cyclones with $Ro\sim 0.02{-}0.05$ and $Bu\sim 0.85{-}0.95$. However, we have also found that anticyclones in general have slower growth rates compared to cyclones. In particular, the growth rate of the most unstable eigenmode for anticyclones in a large region of the parameter space (e.g. $Ro<0$ and $0.5\lesssim Bu\lesssim 1.3$) is slower than 50 turnaround times of the vortex (which often corresponds to several years for ocean eddies). For cyclones, the region with such slow growth rates is confined to $0<Ro<0.1$ and $0.5\lesssim Bu\lesssim 1.3$. While most calculations have been done for $f/\bar{N}=0.1$ (where $f$ and $\bar{N}$ are the Coriolis and background Brunt–Väisälä frequencies), we have numerically verified and explained analytically, using non-dimensionalized equations, the insensitivity of the results to reducing $f/\bar{N}$ to the more ocean-relevant value of 0.01. The results of our stability analysis of Gaussian vortices both support and contradict the findings of earlier studies with QG or multilayer models or with other families of vortices. The results of this paper provide a stepping stone to study the more complicated problems of the stability of geophysical (e.g. those in the atmospheres of giant planets) and astrophysical vortices (in accretion disks).

The polymer coil–stretch mechanism in turbulent drag reducing flows is analysed using direct numerical simulations of viscoelastic finitely extensible nonlinear elastic fluids with the Peterlin approximation. The study is carried out taking into account low and high drag reduction regimes. The polymer stretching and the alignment between the conformation tensor and other relevant entities are investigated using statistical and tensor analysis. The significant alignment between the former and the velocity fluctuations product tensor indicates that the initial polymer stretching due to the mean shear is increased by the flow stress fluctuations, providing a supplementary polymer extension. In addition, interactions between the turbulence and the polymer are evaluated from an instantaneous turbulent energy exchange perspective by considering streamwise work fluctuating terms in elliptical and hyperbolic flow regions separately. Near the wall, polymers not only release energy to the streaks, but also to the elliptical (or vortical) and hyperbolic (or extensional) structures. However, polymers can also be dragged around near-wall vortices, passing through hyperbolic regions and experiencing a significant straining within both these turbulent structures and storing their energy. Hence, polymers weaken elliptical and hyperbolic structures leading to a tendency toward relaminarization of the flow. Polymer release of energy occurs primarily in the streamwise direction, which is in agreement with the enhanced streamwise velocity fluctuation observed in drag reducing flows. A detailed polymer coil–stretch mechanism is provided.

This article presents a reconstruction of the unsteady behaviour of a round jet at a Reynolds number equal to 3300, from the sole knowledge of the time-averaged flow field and one pointwise unsteady measurement. The reconstruction approach is an application of the work of Beneddine et al. (J. Fluid Mech., vol. 798, 2016, pp. 485–504) and relies on the computation of the dominant resolvent modes of the flow, using a parabolised stability equations analysis. To validate the procedure, the unsteady velocity field of the jet has been characterised by time-resolved particle image velocimetry (TR-PIV), yielding an experimental reference. We first show that the dominant resolvent modes are proportional to the experimental Fourier modes, as predicted by Beneddine et al. (J. Fluid Mech., vol. 798, 2016, pp. 485–504). From these results, it is then possible to fully reconstruct the unsteady velocity and pressure fluctuation fields, yielding a flow field that displays good agreement with the experimental reference. Finally, it is found that the robustness of the reconstruction mainly depends on the location of the pointwise unsteady measurement, which should be within energetic regions of the flow, and this robustness as well as the quality of the reconstruction can be greatly improved by considering a few pointwise measurements instead of a single one. The effects of other experimental parameters on the reconstruction, such as the size of the interrogation window used for the TR-PIV processing and the accuracy of the positioning of the sensors, are also investigated in this paper.

Near-field acoustic measurements and time-resolved schlieren visualisations are performed on 10 round jets with the aim of analysing the different parts of the feedback loop related to the screech phenomenon in a systematic fashion. The ideally expanded Mach number of the studied jets ranges from $M_{j}=1.07$ to $M_{j}=1.50$. The single source of screech acoustic waves is found at the fourth shock tip for A1 and A2 modes, and at either the third or the fourth shock tip for the B mode, depending on the Mach number. The phase of the screech cycle is measured throughout schlieren visualisations in the shear layer from the nozzle to the source. Estimates of the convective velocities are deduced for each case, and a trend for the convective velocity to grow with the axial distance is pointed out. These results are used together with source localisation deduced from a two-microphone survey to determine the number of screech periods contained in a screech loop. For the A1 and B modes, four periods are contained in a loop for cases in which the radiating shock is the fourth, and three periods when the radiating shock tip is the third, whereas the loop of the A2 mode contains five periods.

In this paper, we address the linear stability analysis of a confined potential vortex with a free surface. This particular flow has been recently used by Tophøj et al. (Phys. Rev. Lett., vol. 110(19), 2013, article 194502) as a model for the swirling flow of fluid in an open cylindrical container, driven by rotating the bottom plate (the rotating bottom experiment) to explain the so-called rotating polygons instability (Vatistas J. Fluid Mech., vol. 217, 1990, pp. 241–248; Jansson et al., Phys. Rev. Lett., vol. 96, 2006, article 174502) in terms of surface wave interactions leading to resonance. Global linear stability results are complemented by a Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) analysis in the shallow-water limit as well as new experimental observations. It is found that global stability results predict additional resonances that cannot be captured by the simple wave coupling model presented in Tophøj et al. (2013). Both the main resonances (thought to be at the root of the rotating polygons) and these secondary resonances are interpreted in terms of over-reflection phenomena by the WKBJ analysis. Finally, we provide experimental evidence for a secondary resonance supporting the numerical and theoretical analysis presented. These different methods and observations allow to support the unstable wave coupling mechanism as the physical process at the origin of the polygonal patterns observed in free-surface rotating flows.

In this work, an exact Euler solution is derived under the fundamental contingencies of axisymmetric, steady, rotational, incompressible, single-phase, non-reactive and inviscid fluid, which also stand behind the ubiquitously used mean flow profile named ‘Taylor–Culick.’ In comparison with the latter, which proves to be complex lamellar, the present model is derived in the context of a Trkalian flow field, and hence is capable of generating a non-zero swirl component that increases linearly in the streamwise direction. This enables us to provide an essential mathematical representation that is appropriate for flow configurations where the bulk gaseous motion is driven to swirl. From a procedural standpoint, the new Trkalian solution is deduced directly from the Bragg–Hawthorne equation, which has been repeatedly shown to possess sufficient latitude to reproduce several existing profiles such as Taylor–Culick’s as special cases. Throughout this study, the fundamental properties of the present model are considered and discussed in the light of existing flow approximations. Consistent with the original Taylor–Culick mean flow motion, the Trkalian velocity is seen to exhibit both axial and tangential components that increase linearly with the distance from the headwall, and a radial component that remains axially invariant. Furthermore, the Trkalian model is shown to form a subset of the Beltramian class of solutions for which the velocity and vorticity vectors are not only parallel but also directly proportional. This characteristic feature is interesting, as it stands in sharp contrast to the complex-lamellar nature of the Taylor–Culick motion, where the velocity and vorticity vectors remain orthogonal. By way of verification, a numerical simulation is carried out using a finite-volume solver, thus leading to a favourable agreement between theoretical and numerical predictions.

We present a theoretical study of nonlinear effects that result from modal interactions in internal waves in a non-uniformly stratified finite-depth fluid with background rotation. A linear wave field containing modes $m$ and $n$ (of horizontal wavenumbers $k_{m}$ and $k_{n}$) at a fixed frequency $\unicode[STIX]{x1D714}$ results in two different terms in the steady-state weakly nonlinear solution: (i) a superharmonic wave of frequency $2\unicode[STIX]{x1D714}$, horizontal wavenumber $k_{m}+k_{n}$ and a vertical structure $\bar{h}_{mn}(z)$ and (ii) a time-independent term (Eulerian mean flow) with horizontal wavenumber $k_{m}-k_{n}$. For some $(m,n)$, $\bar{h}_{mn}(z)$ is infinitely large along specific curves on the $(\unicode[STIX]{x1D714}/N_{0},f/\unicode[STIX]{x1D714})$ plane, where $N_{0}$ and $f$ are the deep ocean stratification and the Coriolis frequency, respectively; these curves are referred to as divergence curves in the rest of this paper. In uniform stratifications, a unique divergence curve occurs on the $(\unicode[STIX]{x1D714}/N_{0},f/\unicode[STIX]{x1D714})$ plane for those $(m,n\neq m)$ that satisfy $(m/3)<n<(3m)$. In the presence of a pycnocline (whose strength is quantified by the maximum stratification $N_{max}$), divergence curves occur for several more modal interactions than those for a uniform stratification; furthermore, a given $(m,n)$ interaction can result in multiple divergence curves on the $(\unicode[STIX]{x1D714}/N_{0},f/\unicode[STIX]{x1D714})$ plane for a fixed $N_{max}/N_{0}$. Nearby high-mode interactions in a uniform stratification and any modal interaction in a non-uniform stratification with a sufficiently strong pycnocline are shown to result in near-horizontal divergence curves around $f/\unicode[STIX]{x1D714}\approx 1$, thus implying that strong nonlinear effects often occur as a result of interaction within triads containing two different modes at the near-inertial frequency. Notably, self-interaction of certain modes in a non-uniform stratification results in one or more divergence curves on the $(\unicode[STIX]{x1D714}/N_{0},f/\unicode[STIX]{x1D714})$ plane, thus suggesting that even arbitrarily small-amplitude individual modes cannot remain linear in a non-uniform stratification. We show that internal wave resonant triads containing modes $m$ and $n$ at frequency $\unicode[STIX]{x1D714}$ occur along the divergence curves, and their existence is guaranteed upon the satisfaction of two different criteria: (i) the horizontal component of the standard triadic resonance criterion $\boldsymbol{k}_{1}+\boldsymbol{k}_{2}+\boldsymbol{k}_{3}=0$ and (ii) a non-orthogonality criterion. For uniform stratifications, criterion (ii) reduces to the vertical component of the standard triadic resonance criterion. For non-uniform stratifications, criterion (ii) seems to be always satisfied whenever criterion (i) is satisfied, thus significantly increasing the number of modal interactions that result in strong nonlinear effects irrespective of the wave amplitudes. We then adapt our theoretical framework to identify resonant triads and hence provide insights into the generation of higher harmonics in two different oceanic scenarios: (i) low-mode internal tide propagating over small- or large-scale topography and (ii) an internal wave beam incident on a pycnocline in the upper ocean, for which our results are in qualitative agreement with the numerical study of Diamessis et al. (Dynam. Atmos. Oceans., vol. 66, 2014, pp. 110–137).

Transonic buffet is a phenomenon of aerodynamic instability with shock wave motions which occurs at certain combinations of Mach number and mean angle of attack, and which limits the aircraft flight envelope. The objective of this study is to develop a modelling method for unstable flow with oscillating shock waves and moving boundaries, and to perform model-based feedback control of the two-dimensional buffet flow by means of trailing-edge flap oscillations. System identification based on the ARX algorithm is first used to derive a linear model of the input–output dynamics between the flap rotation (the control input) and the lift and pitching moment coefficients (system outputs). The model features a pair of unstable complex-conjugate poles at the characteristic buffet frequency. An appropriate reduced-order model (ROM) with a lower dimension is further obtained by a balanced truncation method that keeps the pair of unstable poles in the unstable subspace but truncates the dynamics in the stable subspace. Based on this balanced ROM, two kinds of feedback control are designed by pole assignment and linear quadratic methods respectively. These independent designs, however, result in similar suboptimal static output feedback control laws. When introduced in numerical simulations, they are both able to completely suppress the buffet instability. Furthermore, the resulting controllers are even able to stabilize buffet flows with nonlinear disturbances and in off-design flow conditions, thus implying their robustness. The analysis of the feedback control laws indicates that parameters (frequency and phase) corresponding to the ‘anti-resonance’ of the linear input–output model are vital for optimal control. The best performance is obtained when the control operates close to the ‘anti-resonance’, which is supported by the optimal frequency and the phase of the open-loop control as well as by the optimal phase of the closed-loop control.

The K-type and H-type transitions of a natural convection boundary layer of a fluid of Prandtl number 7 adjacent to an isothermally heated vertical surface are investigated by means of three-dimensional direct numerical simulation (DNS). These two types of transitions refer to different flow features at the transitional stage from laminar to turbulence caused by two different types of perturbations. To excite the K-type transition, superimposed Tollmien–Schlichting (TS) and oblique waves of the same frequency are introduced into the boundary layer. It is found that a three-layer longitudinal vortex structure is present in the boundary layer undergoing the K-type transition. The typical aligned $\wedge$-shaped vortices characterizing the K-type transition are observed for the first time in pure natural convection boundary layers. For exciting the H-type transition, superimposed TS and oblique waves of different frequencies, with the frequency of the oblique waves being half of the frequency of the TS waves, are introduced into the boundary layer. Unlike the three-layer longitudinal vortex structure observed in the K-type transition, a double-layer longitudinal vortex structure is observed in the boundary layer undergoing the H-type transition. The successively staggered $\wedge$-shaped vortices characterizing the H-type transition are also observed in the downstream boundary layer. The staggered pattern of $\wedge$-shaped vortices is considered to be caused by temporal modulation of the TS and oblique waves. Interestingly the flow structures of both the K-type and H-type transitions observed in the natural convection boundary layer are qualitatively similar to those observed in Blasius boundary layers. However, an analysis of turbulence energy production suggests that the turbulence energy production by buoyancy rather than Reynolds stresses dominates the K-type and H-type transitions. In contrast, the turbulence energy production by Reynolds stresses is the only factor contributing to the transition in Blasius boundary layers.

The accuracy and performance of entropic multi-relaxation time lattice Boltzmann models are assessed for transitional flows of engineering interest. A simulation of the flow over a low-Reynolds-number $SD7003$ airfoil at $Re=6\times 10^{4}$, at an angle of attack $\unicode[STIX]{x1D6FC}=4^{\circ }$, is performed and thoroughly compared to available numerical and experimental data. In order to include blockage and curvature effects, simulations of the flow in a low-pressure turbine passage composed of $T106$ blade profiles, at a chord Reynolds number of $Re=6\times 10^{4}$ or $Re=1.48\times 10^{5}$, for different free-stream turbulence intensities are presented. Using a multi-domain grid refinement strategy in combination with Grad’s boundary conditions yields good agreement for all simulations. The results demonstrate that the entropic lattice Boltzmann model is a viable, parameter-free alternative to modelling approaches such as large-eddy simulations with similar resolution requirements.

A basic framework characterising the interaction between aquatic flows and permeable sediment beds is presented here. Through the permeability Reynolds number ($Re_{K}=\sqrt{K}u_{\ast }/\unicode[STIX]{x1D708}$, where $K$ is the sediment permeability, $u_{\ast }$ is the shear velocity and $\unicode[STIX]{x1D708}$ is the fluid viscosity), the framework unifies two classical flow typologies, namely impermeable boundary layer flows ($Re_{K}\ll 1$) and highly permeable canopy flows ($Re_{K}\gg 1$). Within this range, the sediment–water interface (SWI) is identified as a transitional region, with $Re_{K}$ in aquatic systems typically $O(0.001{-}10)$. As the sediments obstruct conventional measurement techniques, experimental observations of interfacial hydrodynamics remain extremely rare. The use of refractive index matching here allows measurement of the mean and turbulent flow across the SWI and thus direct validation of the proposed framework. This study demonstrates a strong relationship between the structure of the mean and turbulent flow at the SWI and $Re_{K}$. Hydrodynamic characteristics, such as the interfacial turbulent shear stress, velocity, turbulence intensities and turbulence anisotropy tend towards those observed in flows over impermeable boundaries as $Re_{K}\rightarrow 0$ and towards those seen in flows over highly permeable boundaries as $Re_{K}\rightarrow \infty$. A value of $Re_{K}\approx 1{-}2$ is seen to be an important threshold, above which the turbulent stress starts to dominate the fluid shear stress at the SWI, the penetration depths of turbulence and the mean flow into the sediment bed are comparable and similarity relationships developed for highly permeable boundaries hold. These results are used to provide a new perspective on the development of interfacial transport models at the SWI.

Extending previous linear stability analyses of the instabilities developing in permeable Taylor–Couette–Poiseuille flows where axial and radial throughflows are superimposed on the usual Taylor–Couette flow, we further examine the linear behaviour and expand the analysis to consider the weakly nonlinear behaviour of convective-type instabilities by means of the derivation of the fifth-order amplitude equation together with direct numerical simulations. Special attention is paid to the influence of the radius ratio $\unicode[STIX]{x1D702}=r_{in}/r_{out}$, and particularly to wide gaps (small $\unicode[STIX]{x1D702}$) and how they magnify the effects of the radial flow. The instabilities take the form of pairs of counter-rotating toroidal vortices superseded by helical ones as the axial flow is increased. Increasing the radial inflow draws these vortices near the inner cylinder, where they shrink relative to the annular gap, when this gap is wide. Strong axial and radial flows in a narrow annular gap lead to a very large azimuthal wavenumber with steeply sloped helical vortices. Strong radial outflow in a wide annular gap results in very large helical vortices. The analytical and numerical saturated vortices match quite well. In addition, radial inflows or outflows can turn the usually supercritical bifurcation from laminar to vortical flow into a subcritical one. The radial flow above which this change occurs decreases as the radius ratio $\unicode[STIX]{x1D702}$ decreases. A practical motivation for this weakly nonlinear analysis is found in modelling dynamic filtration devices, which rely on vortical instabilities to reduce the processes of accumulation on their membranes.

This paper is concerned with rapid-distortion theory on transversely sheared mean flows which (among other things) can be used to analyse the unsteady motion resulting from the interaction of a turbulent shear flow with a solid surface. It extends previous analyses of Goldstein et al. (J. Fluid Mech., vol. 736, 2013a, pp. 532–569; NASA/TM-2013-217862, 2013b) which showed that the unsteady motion is completely determined by specifying two arbitrary convected quantities. The present paper uses a pair of previously derived conservation laws to derive upstream boundary conditions that relate these quantities to experimentally measurable flow variables. The result is dependent on the imposition of causality on an intermediate variable that appears in the conservation laws. Goldstein et al. (2013a) related the convected quantities to the physical flow variables at the location of the interaction, but the results were not generic and hard to reconcile with experiment. That problem does not occur in the present formulation, which leads to a much simpler and more natural result than the one given in Goldstein et al. (2013a). We also show that the present formalism yields better predictions of the sound radiation produced by the interaction of a two-dimensional jet with the downstream edge of a flat plate than the Goldstein et al. (2013a) result. The role of causality is also discussed.

We conduct numerical simulations using the Eulerian–Lagrangian approach to investigate the formation of the leaking, finger, and stable-settling modes in convective sedimentation when a sediment-laden fluid layer descends through a sharply stratified ambient flow. We show that the temporal evolution of the sedimentation process for the leaking mode can be divided into three stages, including (in temporal order) Rayleigh–Taylor instability, convection, and leaking stages. The presence of sheet-like descending plumes of suspended particles is an important characteristic of the leaking mode, which marks the existence of the leaking stage. For larger particles, the motion is more dominated by gravitational settling and less affected by buoyancy-induced flow motion. The resulting lack of the leaking stage for the larger-particle case leads to persistent finger-like plumes of suspended particles, known as the finger mode. The stable-settling mode occurs when the particles are large and the concentration is dilute such that flow motion due to Rayleigh–Taylor instability has no effect on the particle motion, and the convective motion of suspended particles is insignificant. For the third stage of the leaking mode, which is also the final stationary state, we derive the criterion for the occurrence of the leaking pattern from a scaling argument of the viscous boundary layer. The criterion is further confirmed by the present simulation results and previous laboratory experiments. Through analysis of the energy budget and the vertical flux, we show that although the settling of individual particles is accelerated, the presence of the sheet-like descending plumes in the leaking mode does not contribute to an efficient settling enhancement compared with the finger mode and the Rayleigh–Taylor instability, i.e., the cases with no background stratification. This implies a negative effect on the settling enhancement for small suspended particles when a stable background density stratification exists. In addition, simulations using the equilibrium Eulerian description for the suspended particles are also conducted to examine the difference between the present Lagrangian particle approach and the conventional Eulerian model.

The aim of this work is to better understand fluid displacement mechanisms at the pore scale in relation to capillary-filling rules. Using specifically designed micro-models we investigate the role of pore body shape on fluid displacement during drainage and imbibition via quasi-static and spontaneous experiments at ambient conditions. The experimental results are directly compared to lattice Boltzmann (LB) simulations. The critical pore-filling pressures for the quasi-static experiments agree well with those predicted by the Young–Laplace equation and follow the expected filling events. However, the spontaneous imbibition experimental results differ from those predicted by the Young–Laplace equation; instead of entering the narrowest available downstream throat the wetting phase enters an adjacent throat first. Thus, pore geometry plays a vital role as it becomes the main deciding factor in the displacement pathways. Current pore network models used to predict displacement at the field scale may need to be revised as they currently use the filling rules proposed by Lenormand et al. (J. Fluid Mech., vol. 135, 1983, pp. 337–353). Energy balance arguments are particularly insightful in understanding the aspects affecting capillary-filling rules. Moreover, simulation results on spontaneous imbibition, in excellent agreement with theoretical predictions, reveal that the capillary number itself is not sufficient to characterise the two phase flow. The Ohnesorge number, which gives the relative importance of viscous forces over inertial and capillary forces, is required to fully describe the fluid flow, along with the viscosity ratio.

We consider the dynamics of an axisymmetric, partially wetting droplet of a one-component volatile liquid. The droplet is supported on a smooth superheated substrate and evaporates into a pure vapour atmosphere. In this process, we take the liquid properties to be constant and assume that the vapour phase has poor thermal conductivity and small dynamic viscosity so that we may decouple its dynamics from the dynamics of the liquid phase. This leads to a so-called ‘one-sided’ lubrication-type model for the evolution of the droplet thickness, which accounts for the effects of evaporation, capillarity, gravity, slip and kinetic resistance to evaporation. By asymptotically matching the flow near the contact line region and the bulk of the droplet in the limit of a small slip length and commensurably small evaporation and kinetic resistance effects, we obtain coupled evolution equations for the droplet radius and volume. The predictions of our asymptotic analysis, which also include an estimate of the evaporation time, are found to be in excellent agreement with numerical simulations of the governing lubrication model for a broad range of parameter regimes.