We present a theory that quantifies the interplay between intrapore and interpore flow variabilities and their impact on hydrodynamic dispersion. The theory reveals that porous media with varying levels of structural disorder exhibit notable differences in interpore flow variability, characterised by the flux-weighted probability density function (PDF), $\hat {\psi }_\tau (\tau ) \sim \tau ^{-\theta -2}$, for advection times $\tau$ through conduits. These differences result in varying relative strengths of interpore and intrapore flow variabilities, leading to distinct scaling behaviours of the hydrodynamic dispersion coefficient $D_L$, normalised by the molecular diffusion coefficient $D_m$, with respect to the Péclet number $Pe$. Specifically, when $\hat {\psi }_\tau (\tau )$ exhibits a broad distribution of $\tau$ with $\theta$ in the range of $(0, 1)$, the dispersion undergoes a transition from power-law scaling, $D_L/D_m \sim Pe^{2-\theta }$, to linear scaling, $D_L/D_m \sim Pe$, and eventually to logarithmic scaling, $D_L/D_m \sim Pe\ln (Pe)$, as $Pe$ increases. Conversely, when $\tau$ is narrowly distributed or when $\theta$ exceeds 1, dispersion consistently follows a logarithmic scaling, $D_L/D_m \sim Pe\ln (Pe)$. The power-law and linear scaling occur when interpore variability predominates over intrapore variability, while logarithmic scaling arises under the opposite condition. These theoretical predictions are supported by experimental data and network simulations across a broad spectrum of porous media.

We investigate the concentration fluctuations of passive scalar plumes emitted from small, localised (point-like) steady sources in a neutrally stratified turbulent boundary layer over a rough wall. The study utilises high-resolution large-eddy simulations for sources of varying sizes and heights. The numerical results, which show good agreement with wind-tunnel studies, are used to estimate statistical indicators of the concentration field, including spectra and moments up to the fourth order. These allow us to elucidate the mechanisms responsible for the production, transport and dissipation of concentration fluctuations, with a focus on the very near field, where the skewness is found to have negative values – an aspect not previously highlighted. The gamma probability density function is confirmed to be a robust model for the one-point concentration at sufficiently large distances from the source. However, for ground-level releases in a well-defined area around the plume centreline, the Gaussian distribution is found to be a better statistical model. As recently demonstrated by laboratory results, for elevated releases, the peak and shape of the pre-multiplied scalar spectra are confirmed to be independent of the crosswind location for a given downwind distance. Using a stochastic model and theoretical arguments, we demonstrate that this is due to the concentration spectra being directly shaped by the transverse and vertical velocity components governing the meandering of the plume. Finally, we investigate the intermittency factor, i.e. the probability of non-zero concentration, and analyse its variability depending on the thresholds adopted for its definition.

We report on Lagrangian statistics of turbulent Rayleigh–Bénard convection under very different conditions. For this, we conducted particle tracking experiments in a $H=1.1$-m-high cylinder of aspect ratio $\varGamma =1$ filled with air (Pr = 0.7), as well as in two rectangular cells of heights $H=0.02$ m ($\varGamma =16$) and $H=0.04$ m ($\varGamma =8$) filled with water (Pr = 7.0), covering Rayleigh numbers in the range $10^6\le {\textit {Ra}}\le 1.6\times 10^9$. Using the Shake-The-Box algorithm, we have tracked up to 500 000 neutrally buoyant particles over several hundred free-fall times for each set of control parameters. We find the Reynolds number to scale at small Ra (large Pr) as $ {\textit{Re}} \propto {\textit{Ra}}^{0.6}$. Further, the averaged horizontal particle displacement is found to be universal and exhibits a ballistic regime at small times and a diffusive regime at larger times, for sufficiently large $\varGamma$. The diffusive regime occurs for time lags larger than $\tau _{co}$, which is the time scale related to the decay of the velocity autocorrelation. Compensated as $\tau _{co} {\textit {Pr}}^{-0.3}$, this time scale is universal and rather independent of $ {\textit {Ra}}$ and $\varGamma$. We have also investigated the Lagrangian velocity structure function $S^2_i(\tau )$, which is dominated by viscous effects for times smaller than the Kolmogorov time $\tau _\eta$ and hence $S^2_i\propto \tau ^2$. For larger times we find a novel scaling for the different components with exponents smaller than what is expected in the inertial range of homogeneous isotropic turbulence without buoyancy. Studying particle-pair dispersion, we find a Batchelor scaling (${\propto }\,t^2$) on small time scales, diffusive scaling (${\propto }\,t$) on large time scales and Richardson-like scaling (${\propto }\,t^3$) for intermediate time scales.

Diffusion-driven flow is a boundary layer flow arising from the interplay of gravity and diffusion in density-stratified fluids when a gravitational field is non-parallel to an impermeable solid boundary. This study investigates diffusion-driven flow within a nonlinearly density-stratified fluid confined between two tilted parallel walls. We introduce an asymptotic expansion inspired by the centre manifold theory, where quantities are expanded in terms of derivatives of the cross-sectional averaged stratified scalar (such as salinity or temperature). This technique provides accurate approximations for velocity, density and pressure fields. Furthermore, we derive an evolution equation describing the cross-sectional averaged stratified scalar. This equation takes the form of the traditional diffusion equation but replaces the constant diffusion coefficient with a positive-definite function dependent on the solution's derivative. Numerical simulations validate the accuracy of our approximations. Our investigation of the effective equation reveals that the density profile depends on a non-dimensional parameter denoted as $\gamma$ representing the flow strength. In the large $\gamma$ limit, the system is approximated by a diffusion process with an augmented diffusion coefficient of $1+\cot ^{2}\theta$, where $\theta$ signifies the inclination angle of the channel domain. This parameter regime is where diffusion-driven flow exhibits its strongest mixing ability. Conversely, in the small $\gamma$ regime, the density field behaves like pure diffusion with distorted isopycnals. Lastly, we show that the classical thin film equation aligns with the results obtained using the proposed expansion in the small $\gamma$ regime but fails to accurately describe the dynamics of the density field for large $\gamma$.

Droplet clustering in sprays refers to the dynamic evolution of highly concentrated regions due to the preferential accumulation of the polydisperse droplets in the turbulent airflow entrained by the spray. In the current study, we aim to experimentally investigate the collective vaporization of the droplets in droplet clusters in an air-assisted acetone spray characterized by the Group number, $G$. The magnitude of $G$ depends on the cluster length scale and interdroplet spacing, and it indicates the vaporization mode that may vary from the isolated mode ($G \ll 1$) to external group mode ($G \gg 1$). The droplet measurements were obtained under atmospheric conditions at different axial and radial locations within the spray. Application of the Voronoi analysis to particle image velocimetry images of the spray droplets facilitated the identification and characterization of the droplet clusters, which allowed the measurement of $G$ for each cluster. The results highlighted that multiscale clustering of the evaporating droplets leads to multimode group evaporation of the clusters (characterized by a wide range of $G$: 0.001–10). The trend of interdroplet spacing versus cluster area allowed the classification of the droplet clusters into small-scale clusters (which are of the order of the Kolmogorov length scale) and large-scale clusters (that scale with the large-scale turbulent eddies), that are found to exhibit distinct group evaporation behaviour. A theoretical model is invoked to correlate $G$ with the droplet evaporation rate for individual clusters, and some interesting observations are identified, which are explained in the paper.

Cystic and alveolar echinococcosis are considered the second and third most significant foodborne parasitic diseases worldwide. The microscopic eggs excreted in the feces of the definitive host are the only source of contamination for intermediate and dead-end hosts, including humans. However, estimating the respective contribution of the environment, fomites, animals or food in the transmission of Echinococcus eggs is still challenging. Echinococcus granulosus and E. multilocularis seem to have a similar survival capacity regarding temperature under laboratory conditions. In addition, field experiments have reported that the eggs can survive several weeks to years outdoors, with confirmation of the relative susceptibility of Echinococcus eggs to desiccation. Bad weather (such as rain and wind), invertebrates and birds help scatter Echinococcus eggs in the environment and may thus impact human exposure. Contamination of food and the environment by taeniid eggs has been the subject of renewed interest in the past decade. Various matrices from endemic regions have been found to be contaminated by Echinococcus eggs. These include water, soil, vegetables and berries, with heterogeneous rates highlighting the need to acquire more robust data so as to obtain an accurate assessment of the risk of human infection. In this context, it is essential to use efficient methods of detection and to develop methods for evaluating the viability of eggs in the environment and food.

We experimentally investigate the flow through a hollow cube, with an indoor ground-level passive scalar source, immersed in a rough-wall turbulent boundary layer inside a water tunnel. The focus is on characterizing scalar transport within the cube, through simultaneous scalar and flow measurements using planar laser-induced fluorescence and particle image velocimetry. To understand the role of window positioning, three cube configurations, labelled as ‘centre’, ‘up-down’ and ‘down-up’, distinguished by window positions at the upstream and downstream ends, are studied. Varying window position alters the flow characteristics within the cube, resulting in differences in scalar concentration and distribution. The steady-state concentration is highest for ‘centre’, followed by ‘up-down’ and ‘down-up’ configurations. Regarding the scalar distribution, ‘centre’ showed accumulation near the top and bottom walls, while ‘up-down’ and ‘down-up’ exhibited scalar buildup in the lower and upper half of the cube, respectively. The flow patterns and scalar transport mechanisms remained consistent across different Reynolds numbers ($Re=U_{Ref}H/\nu = 20\ 000$, 35 000, 50 000) for each configuration; $U_{Ref}=$ incoming flow velocity at cube height ($H$), and $\nu =\,$ kinematic viscosity of water. The analysis is extended by revising the classical box model, accounting for practical complexities such as non-perfect mixing. Our results can help better understand and model indoor–outdoor pollutant exchange in complex urban environments.

The coupling between advection and diffusion in position space can often lead to enhanced mass transport compared with diffusion without flow. An important framework used to characterize the long-time diffusive transport in position space is the generalized Taylor dispersion theory. In contrast, the dynamics and transport in orientation space remains less developed. In this work we develop a rotational Taylor dispersion theory that characterizes the long-time orientational transport of a spheroidal particle in linear flows that is constrained to rotate in the velocity-gradient plane. Similar to Taylor dispersion in position space, the orientational distribution of axisymmetric particles in linear flows at long times satisfies an effective advection–diffusion equation in orientation space. Using this framework, we then calculate the long-time average angular velocity and dispersion coefficient for both simple shear and extensional flows. Analytic expressions for the transport coefficients are derived in several asymptotic limits including nearly spherical particles, weak flow and strong flow. Our analysis shows that at long times the effective rotational dispersion is enhanced in simple shear and suppressed in extensional flow. The asymptotic solutions agree with full numerical solutions of the derived macrotransport equations and results from Brownian dynamics simulations. Our results show that the interplay between flow-induced rotations and Brownian diffusion can fundamentally change the long-time transport dynamics.

We report on an experimental study in which Lagrangian tracking is applied to millions of microscopic particles floating on the free surface of turbulent water. We leverage a large jet-stirred zero-mean-flow apparatus, where the Reynolds number is sufficiently high for an inertial range to emerge while the surface deformation remains minimal. Two-point statistics reveal specific features of the flow, deviating from the classic description derived for incompressible turbulence. The magnitude of the relative velocity is strongly intermittent, especially at small separations, leading to anomalous scaling of the second-order structure functions in the dissipative range. This is driven by the divergent component of the flow, leading to fast approaching/separation rates of nearby particles. The Lagrangian relative velocity shows strong persistence of the initial state, such that the ballistic pair separation extends to the inertial range of time delays. Based on these observations, we propose a classification of particle pairs based on their initial separation rate. When this is much smaller than the relative velocity prescribed by inertial scaling (which is the case for the majority of the observed particle pairs), the relative velocity transitions to a diffusive growth and the Richardson–Obukhov super-diffusive dispersion is recovered.

It is known that the dispersion of colloidal particles in porous media is determined by medium structure, pore-scale flow variability and diffusion. However, much less is known about how diffusiophoresis, that is, the motion of colloidal particles along salt gradients, impacts large-scale particle dispersion in porous media. To shed light on this question, we perform detailed pore-scale simulations of fluid flow, solute transport and diffusiophoretic particle transport in a two-dimensional hyper-uniform porous medium. Particles and solute are initially uniformly distributed throughout the medium. The medium is flushed at constant flow rate, and particle breakthrough curves are recorded at the outlet to assess the macroscopic effects of diffusiophoresis. Particle breakthrough curves show non-Fickian behaviour manifested by strong tailing that is controlled by the diffusiophoretic mobility. Although diffusiophoresis is a short-time, microscopic phenomenon owing to the fast attenuation of salt gradients, it governs macroscopic colloid dispersion through the partitioning of particles into transmitting and dead-end pores. We quantify these behaviours by an upscaled analytical model that describes both the retention and release of colloids in dead-end pores and the observed long-time tailings. Our results suggest that diffusiophoresis is an efficient tool to control particle dispersion and filtration through porous media.

Low Stokes number particles at dilute concentrations in turbulent flows can reasonably be approximated as passive scalars. The added presence of a drift velocity due to buoyancy or gravity when considering the transport of such passive scalars can reduce the turbulent dispersion of the scalar via a diminution of the eddy diffusivity. In this work, we propose a model to describe this decay and use a recently developed technique to accurately and efficiently measure the eddy diffusivity using Eulerian fields and quantities. We then show a correspondence between this method and standard Lagrangian definitions of diffusivity and collect data across a range of drift velocities and Reynolds numbers. The proposed model agrees with data from these direct numerical simulations, offers some improvement to previous models in describing other computational and experimental data and satisfies theoretical constraints that are independent of Reynolds number.

The dispersion of clays is of great importance in determining various soil properties such as hydraulic conductivity. A procedure which involves fixing followed by embedding of clay particles in an epoxy resin is described. This procedure enables the observation of cross sections of clay tactoids under a transmittance electron microscope, and the determination of the number of plates per tactoid. The use of the procedure for the determination of the relation between the exchangeable sodium percentage (ESP) and tactoid size in suspensions of a Na/Ca bentonite system is presented. It was demonstrated that even at ESP 5 significant dispersion already occurs, the average number of plates per tactoid being 6.6 as compared to 16.1 at ESP 0.

Utilizing the discrete element method and the pore network model, we numerically investigate the impact of compaction on the longitudinal dispersion coefficient of porous media. Notably, the dispersion coefficient exhibits a non-monotonic dependence on the degree of compaction, which is distinguished by the presence of three distinct regimes in the variation of dispersion coefficient. The non-monotonic variation of dispersion coefficient is attributed to the disparate effect of compaction on dispersion mechanisms. Specifically, the porous medium tightens with an increasing pressure load, reducing the effect of molecular diffusion that primarily governs at small Péclet numbers. On the other hand, heightened pressure loads enhance the heterogeneity of pore structures, resulting in increased disorder and a higher proportion of stagnant zones within porous media flow. These enhancements further strengthen mechanical dispersion and hold-up dispersion, respectively, both acting at higher Péclet numbers. It is crucial to highlight that hold-up dispersion is induced by the low-velocity regions in porous media flow, which differ fundamentally from zero-velocity regions (such as dead-ends or the interior of permeable grains) as described by the classical theory of dispersion. The competition between weakened molecular diffusion and enhanced hold-up dispersion and mechanical dispersion, together with the shift in the dominance of dispersion mechanisms across various Péclet numbers, results in multiple regimes in the variation of dispersion coefficients. Our study provides unique insights into structural design and modulation of the dispersion coefficient of porous materials.

We consider the process of convective dissolution in a homogeneous and isotropic porous medium. The flow is unstable due to the presence of a solute that induces a density difference responsible for driving the flow. The mixing dynamics is thus driven by a Rayleigh–Taylor instability at the pore scale. We investigate the flow at the scale of the pores using Hele-Shaw type experiment with bead packs, two-dimensional direct numerical simulations and physical models. Experiments and simulations have been specifically designed to mimic the same flow conditions, namely matching porosities, high Schmidt numbers and linear dependency of fluid density with solute concentration. In addition, the solid obstacles of the medium are impermeable to fluid and solute. We characterise the evolution of the flow via the mixing length, which quantifies the extension of the mixing region and grows linearly in time. The flow structure, analysed via the centreline mean wavelength, is observed to grow in agreement with theoretical predictions. Finally, we analyse the dissolution dynamics of the system, quantified through the mean scalar dissipation, and three mixing regimes are observed. Initially, the evolution is controlled by diffusion, which produces solute mixing across the initial horizontal interface. Then, when the interfacial diffusive layer is sufficiently thick, it becomes unstable, forming finger-like structures and driving the system into a convection-dominated phase. Finally, when the fingers have grown sufficiently to touch the horizontal boundaries of the domain, the mixing reduces dramatically due to the absence of fresh unmixed fluid. With the aid of simple physical models, we explain the physics of the results obtained numerically and experimentally. The solute evolution presents a self-similar behaviour, and it is controlled by different length scales in each stage of the mixing process, namely the length scale of diffusion, the pore size and the domain height.

Shear-induced migration of elongated micro-swimmers exhibiting anisotropic Brownian diffusion at a population scale is investigated analytically in this work. We analyse the steady motion of confined ellipsoidal micro-swimmers subject to coupled diffusion in a general setting within a continuum homogenisation framework, as an extension of existing studies on macro-transport processes, by allowing for the direct coupling of convection and diffusion in local and global spaces. The analytical solutions are validated successfully by comparison with numerical results from Monte Carlo simulations. Subsequently, we demonstrate from the probability perspective that symmetric actuation does not yield net vertical polarisation in a horizontal flow, unless non-spherical shapes, external fields or direct coupling effects are harnessed to generate steady locomotion. Coupled diffusivities modify remarkably the drift velocity and vertical migration of motile micro-swimmers exposed to fluid shear. The interplay between stochastic swimming and preferential alignment could explain the diverse concentration and orientation distributions, including rheological formations of depletion layers, centreline focusing and surface accumulation. Results of the analytical study shed light on unravelling peculiar self-propulsion strategies and dispersion dynamics in active-matter systems, with implications for various transport problems arising from the fluctuating shape, size and other external or inter-particle interactions of swimmers in confined environments.

Advective–diffusive transport in Poiseuille flow through a channel with partially absorbing walls is a classical problem with applications to a broad range of natural and engineered scenarios, ranging from solute and heat transport in porous and fractured media to absorption in biological systems and chromatography. We study this problem from the perspective of transverse distributions of surviving mass and velocity, which are a central ingredient of recent stochastic models of transport based on the sampling of local flow velocities along trajectories. We show that these distributions tend to asymptotic equilibria for large times and travel distances, and derive rigorous explicit expressions for arbitrary reaction rate. We find that the equality of flux-weighted and breakthrough distributions that holds for conservative transport breaks in the presence of reaction, and that the average velocity of the scalar plume is no longer fully characterized by the transverse distribution of flow velocities sampled at a given time.