We analyse the process of convective mixing in two-dimensional, homogeneous and isotropic porous media with dispersion. We considered a Rayleigh–Taylor instability in which the presence of a solute produces density differences driving the flow. The effect of dispersion is modelled using an anisotropic Fickian dispersion tensor (Bear, J. Geophys. Res., vol. 66, 1961, pp. 1185–1197). In addition to molecular diffusion ($D_m^*$), the solute is redistributed by an additional spreading, in longitudinal and transverse flow directions, which is quantified by the coefficients $D_l^*$ and $D_t^*$, respectively, and it is produced by the presence of the pores. The flow is controlled by three dimensionless parameters: the Rayleigh–Darcy number $\textit{Ra}$, defining the relative strength of convection and diffusion, and the dispersion parameters $r=D_l^*/D_t^*$ and $\varDelta =D_m^*/D_t^*$. With the aid of numerical Darcy simulations, we investigate the mixing dynamics without and with dispersion. We find that in the absence of dispersion ($\varDelta \to \infty$) the dynamics is self-similar and independent of $\textit{Ra}$, and the flow evolves following several regimes, which we analyse. Then we analyse the effect of dispersion on the flow evolution for a fixed value of the Rayleigh–Darcy number ($\textit{Ra}=10^4$). A detailed analysis of the molecular and dispersive components of the mean scalar dissipation reveals a complex interplay between flow structures and solute mixing. We find that the dispersion parameters $r$ and $\varDelta$ affect the formation of fingers and their dynamics: the lower the value of $\varDelta$ (or the larger the value of $r$), the wider, more convoluted and diffused the fingers. We also find that for strong anisotropy, $r=O(10)$, the role of $\varDelta$ is crucial: except for the intermediate phases of the flow dynamics, dispersive flows show more efficient (or at least comparable) mixing than in non-dispersive systems. Finally, we look at the effect of the anisotropy ratio $r$, and we find that it produces only second-order effects, with relevant changes limited to the intermediate phase of the flow evolution, where it appears that the mixing is more efficient for small values of anisotropy. The proposed theoretical framework, in combination with pore-scale simulations and bead packs experiments, can be used to validate and improve current dispersion models to obtain more reliable estimates of solute transport and spreading in buoyancy-driven subsurface flows.

Hierarchical parcel swapping (HiPS) is a multiscale stochastic model of turbulent mixing based on a binary tree. Length scales decrease geometrically with increasing tree level, and corresponding time scales follow inertial range scaling. Turbulent eddies are represented by swapping subtrees. Lowest-level swaps change fluid parcel pairings, with new pairings instantly mixed. This formulation suitable for unity Schmidt number $Sc$ is extended to non-unity $Sc$. For high $Sc$, the tree is extended to the Batchelor level, assigning the same time scale (governing the rate of swap occurrences) to the added levels as the time scale at the base of the $Sc=3$ tree. For low $Sc$, a swap at the Obukhov–Corrsin level mixes all parcels within corresponding subtrees. Well-defined model analogues of turbulent diffusivity, and mean scalar-variance production and dissipation rates are identified. Simulations idealising stationary homogeneous turbulence with an imposed scalar gradient reproduce various statistical properties of viscous-range and inertial-range pair dispersion, and of the scalar power spectrum in the inertial-advective, inertial-diffusive and viscous-advective regimes. The viscous-range probability density functions of pair separation and scalar dissipation agree with applicable theory, including the stretched-exponential tail shape associated with viscous-range scalar intermittency. Previous observation of that tail shape for $Sc=1$, heretofore not modelled or explained, is reproduced. Comparisons to direct numerical simulation allow evaluation of empirical coefficients, facilitating quantitative applications. Parcel-pair mixing is a common mixing treatment, e.g. in subgrid closures for coarse-grained flow simulation, so HiPS can improve model physics simply by smarter (yet nearly cost-free) selection of pairs to be mixed.

Motivated by the need for a better understanding of marine plastic transport, we experimentally investigate finite-size particles floating in free-surface turbulence. Using particle tracking velocimetry, we study the motion of spheres and discs along the quasi-flat free-surface above homogeneous isotropic grid turbulence in open channel flows. The focus is on the effect of the particle diameter, which varies from the Kolmogorov scale to the integral scale of the turbulence. We find that particles of size up to approximately one-tenth of the integral scale display motion statistics indistinguishable from surface flow tracers. For larger sizes, the particle fluctuating energy and acceleration variance decrease, the correlation times of their velocity and acceleration increase, and the particle diffusivity is weakly dependent on their diameter. Unlike in three-dimensional turbulence, the acceleration of finite-size floating particles becomes less intermittent with increasing size, recovering a Gaussian distribution for diameters in the inertial subrange. These results are used to assess the applicability of two distinct frameworks: temporal filtering and spatial filtering. Neglecting preferential sampling and assuming an empirical linear relation between the particle size and its response time, the temporal filtering approach is found to correctly predict the main trends, though with quantitative discrepancies. However, the spatial filtering approach, based on the spatial autocorrelation of the free-surface turbulence, accurately reproduces the decay of the fluctuating energy with increasing diameter. Although the scale separation is limited, power-law scaling relations for the particle acceleration variance based on spatial filtering are compatible with the observations.

We examine the dispersion of prolate spheroidal microswimmers in pressure-driven channel flow, with the emphasis on a novel anomalous scaling regime. When time scales corresponding to swimmer orientation relaxation, and diffusion in the gradient and flow directions, are all well separated, a multiple scales analysis leads to a closed form expression for the shear-enhanced diffusivity, $D_{\it{eff}}$, governing the long-time spread of the swimmer population along the flow (longitudinal) direction. This allows one to organize the different $D_{\it{eff}}$-scaling regimes as a function of the rotary Péclet number (${\it{{\it{Pe}}}}_r)$, where the latter parameter measures the relative importance of shear-induced rotation and relaxation of the swimmer orientation due to rotary diffusion. For large ${\it{{\it{Pe}}}}_r$, $D_{\it{eff}}$ scales as $O({\it{{\it{Pe}}}}_r^4D_t)$ for $1 \leqslant \kappa \lesssim 2$, and as $O({\it{{\it{Pe}}}}_r^{ {10}/{3}}D_t)$ for $\kappa = \infty$, with $D_t$ being the intrinsic translational diffusivity of the swimmer arising from a combination of swimming and rotary diffusion, and $\kappa$ being the swimmer aspect ratio; $\kappa = 1$ for spherical swimmers. For $2 \lesssim \kappa \lt \infty$, the swimmers collapse onto the centreline with increasing ${\it{{\it{Pe}}}}_r$, leading to an anomalously reduced longitudinal diffusivity of $O({\it{{\it{Pe}}}}_r^{5-C(\kappa )}D_t)$. Here, $C(\kappa )\!\gt \!1$ characterizes the algebraic decay of swimmer concentration outside an $O({\it{{\it{Pe}}}}_r^{-1})$ central core, with the anomalous exponent $(5-C)$ governed by large velocity variations occasionally sampled by swimmers outside this core. Here, $C(\kappa )\gt 5$ for $\kappa \gtrsim 10$, leading to $D_{\it{eff}}$ eventually decreasing with increasing ${\it{{\it{Pe}}}}_r$, in turn implying a flow-independent maximum, at a finite ${\it{{\it{Pe}}}}_r$, for the rate of slender swimmer dispersion.

Dispersion of microswimmers is widespread in environmental and biomedical applications. In the category of continuum modelling, the present study investigates the dispersion of microswimmers in a confined unidirectional flow under a diffuse reflection boundary condition, instead of the specular reflection and the Robin boundary conditions prevailing in existing studies. By the moment analysis based on the Smoluchowski equation, the asymptotic and transient solutions are directly obtained, as validated against random walk simulations, to illustrate the effects of mean flow velocity, swimming velocity and gyrotaxis on the migration and distribution patterns of elongated microswimmers. Under the diffuse reflection boundary condition, microswimmers are found more likely to exhibit M-shaped low-shear trapping and even pronounced centreline aggregation, and elongated shape affects depletion at the centreline. Along the flow direction, they readily form unimodal distributions oriented downstream, resulting in prominent downstream migration. Near the centreline, the migration is almost entirely downstream, while upstream and vertical migrations are confined near the boundaries. When the mean flow velocity and swimming velocity are comparable, the system undergoes a temporal transition from M-shaped low-shear trapping to M-shaped high-shear trapping and ultimately to centreline aggregation. The downstream migration continuously strengthens over time, while the upstream first strengthens and then weakens. Moreover, the coupling between swimming-induced diffusion and convective dispersion leads to non-monotonic, fluctuating trends in both drift velocity and dispersivity over time. These results contribute to a deeper understanding of the underlying mechanisms governing the locomotion and control of natural and synthetic microswimmers.

We study the behaviour of shallow water waves propagating over bathymetry that varies periodically in one direction and is constant in the other. Plane waves travelling along the constant direction are known to evolve into solitary waves, due to an effective dispersion. We apply multiple-scale perturbation theory to derive an effective constant-coefficient system of equations, showing that the transversely averaged wave approximately satisfies a Boussinesq-type equation, while the lateral variation in the wave is related to certain integral functions of the bathymetry. Thus the homogenized equations not only accurately describe these waves but also predict their full two-dimensional shape in some detail. Numerical experiments confirm the good agreement between the effective equations and the variable-bathymetry shallow water equations.

The dispersion behaviour of solutes in flow is crucial to the design of chemical separation systems and microfluidics devices. These systems often rely on coupled electroosmotic and pressure-driven flows to transport and separate chemical species, making the transient dispersive behaviour of solutes highly relevant. However, previous studies of Taylor dispersion in coupled electroosmotic and pressure-driven flows focused on the long-term dispersive behaviour and the associated analyses cannot capture the transient behaviour of solute. Further, the radial distribution of solute has not been analysed. In the current study, we analyse the Taylor dispersion for coupled electroosmotic and pressure-driven flows across all time regimes, assuming a low zeta potential (electric potential at the shear plane), the Debye–Hückel approximation and a finite electric double layer thickness. We first derive analytical expressions for the effective dispersion coefficient in the long-time regime. We also derive an unsteady, two-dimensional (radial and axial) solute concentration field applicable in the latter regime. We next apply Aris’ method of moments to characterise the unsteady propagation of the mean axial position and the unsteady growth of the variance of the solute zone in all time regimes. We benchmark our predictions with Brownian dynamics simulations across a wide and relevant dynamical regime, including various time scales. Lastly, we derive expressions for the optimal relative magnitudes of electroosmotic versus pressure-driven flow and the optimum Péclet number to minimise dispersion across all time scales. These findings offer valuable insights for the design of chemical separation systems, including the optimisation of capillary electrophoresis devices and electrokinetic microchannels and nanochannels.

Dispersion in spatio-temporal random flows is dominated by the competition between spatial and temporal velocity resets along particle paths. This competition admits a range of normal and anomalous dispersion behaviours characterised by the Kubo number, which compares the relative strength of spatial and temporal velocity resets. To shed light on these behaviours, we develop a Lagrangian stochastic approach for particle motion in spatio-temporally fluctuating flow fields. For space–time separable flows, particle motion is mapped onto a continuous time random walk (CTRW) for steady flow in warped time, which enables the upscaling and prediction of the large-scale dispersion behaviour. For non-separable flows, we measure Lagrangian velocities in terms of a new sampling variable, the average number of velocity transitions (both temporal and spatial) along pathlines, which renders the velocity series Markovian. Based on this, we derive a Lagrangian stochastic model that represents particle motion as a coupled space–time random walk, that is, a CTRW for which the space and time increments are intrinsically coupled. This approach sheds light on the fundamental mechanisms of particle motion in space–time variable flows, and allows for its systematic quantification. Furthermore, these results indicate that alternative strategies for the analysis of Lagrangian velocity data using new sampling variables may facilitate the identification of (hidden) Markov models, and enable the development of reduced-order models for otherwise complex particle dynamics.

In soft porous media, deformation drives solute transport via the intrinsic coupling between flow of the fluid and rearrangement of the pore structure. Solute transport driven by periodic loading, in particular, can be of great relevance in applications including the geomechanics of contaminants in the subsurface and the biomechanics of nutrient transport in living tissues and scaffolds for tissue engineering. However, the basic features of this process have not previously been systematically investigated. Here, we fill this hole in the context of a one-dimensional model problem. We do so by expanding the results from a companion study, in which we explored the poromechanics of periodic deformations, by introducing and analysing the impact of the resulting fluid and solid motion on solute transport. We first characterise the independent roles of the three main mechanisms of solute transport in porous media – advection, molecular diffusion and hydrodynamic dispersion – by examining their impacts on the solute concentration profile during one loading cycle. We next explore the impact of the transport parameters, showing how these alter the relative importance of diffusion and dispersion. We then explore the loading parameters by considering a range of loading periods – from slow to fast, relative to the poroelastic time scale – and amplitudes – from infinitesimal to large. We show that solute spreading over several loading cycles increases monotonically with amplitude, but is maximised for intermediate periods because of the increasing poromechanical localisation of the flow and deformation near the permeable boundary as the period decreases.

Many particles, whether passive or active, possess elongated shapes. When these particles settle or swim in shear flows, they often form regions of accumulation and depletion. Additionally, the density contrast between the particles and the fluid can further alter the flow by increasing the local suspension density, resulting in a two-way buoyancy–flow coupling mechanism. This study investigates the buoyancy–flow coupled dispersion of active spheroids, examining the effects of elongation, orientation-dependent settling and gyrotaxis in a vertical pipe subjected to either downwards or upwards discharge. While the concentration and velocity profiles of passive settling spheroids and spherical gyrotactic swimmers can be analysed similarly to a recent study, notable differences in dispersion characteristics emerge due to different streamline-crossing mechanisms. For suspensions of elongated swimmers, the interplay between orientation-dependent settling, gyrotaxis-induced accumulation and shear-induced trapping results in distinct concentration and velocity distributions compared to those of neutrally buoyant particles and extremely dilute suspensions with negligible coupling effect. These differences further impact drift velocity, dispersivity, and the time elapsed to steady dispersion under varying flow rates. Interestingly, low-shear trapping of non-settling elongated swimmers around the centreline, commonly observed in planar Poiseuille flow, is absent in the vertical pipe due to the change of confinement from reflectional to rotational symmetry. However, elongated settling swimmers show a non-trivial concentration response to strong downwelling discharge. This phenomenon, linked to the centreline accumulation of passive settling spheroids, bears similarities to low-shear trapping observed in planar Poiseuille flow.

We conduct a large scale experiment to investigate peer effects in computer assisted learning (CAL). Identification relies on three levels of randomization. We find an average 0.17 standard deviation improvement in math scores among primary school students. This average effect is the same for students treated individually or in pairs, implying that peer effects double the learning benefit from a given equipment. Among paired students, poor performers benefit more from CAL when paired with good performers and vice versa. Average performers benefit equally irrespective of who they are paired with. This suggests that the treatment is dominated by knowledge exchange between peers. We also find that CAL treatment reduces the dispersion in math scores and that the beneficial effects of CAL can be strengthened if weak students are systematically paired with strong students.

Several ways of using the traditional analysis of variance to test heterogeneity of spread in factorial designs with equal or unequal n are compared using both theoretical and Monte Carlo results. Two types of spread variables, (1) the jackknife pseudovalues of s2 and (2) the absolute deviations from the cell median, are shown to be robust and relatively powerful. These variables seem to be generally superior to the Z-variance and Box-Scheffé procedures.

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 $\gamma$.