Dispersion in turbulent flows is of broad interest in engineering and environmental processes, particularly for rivers, lakes and oceanic water bodies. Based on our streamwise dispersion model grounded in a Lagrangian perspective of convection–diffusion dynamics (Guan & Chen, 2024, J. Fluid Mech., vol. 980, A33), this work presents a comprehensive solution that consistently unifies dispersion across the Reynolds number spectrum, bridging laminar and turbulent regimes. The streamwise dispersion mechanism is general across time scales, yet its statistical behaviour cannot be fully described using conventional coarse-grained moments averaged over cross-sections. While classical drift–diffusion models that are effective for long-time asymptotics fail to capture the turbulent dynamics of the pre-asymptotic phase, our analytical model enables a complete spatio-temporal characterisation of concentration, and reveals how local statistics evolve towards their asymptotic, coarse-grained limits. Through asymptotic expansions and eigenfunction analysis, we quantify the time-dependent behaviour of phenomenological dispersion coefficients, and distinguish between local and mean statistics, which diverge significantly during the pre-asymptotic phase. The early regime exhibits robust features, including an overshoot in local dispersivity, asymmetric long tails in mean concentration, and island-shaped solute accumulation near the free surface. Three regimes are identified in the evolution of the local concentration: (i) an initially uniform line source, (ii) a transitional logarithmic profile shaped by vertical shear, and (iii) an emergent Gaussian dispersion regime approaching vertical uniformity. Comparisons of both local and mean concentration demonstrate quantitative agreement with finite difference and Monte Carlo simulations across all regimes. These findings clarify the interplay between shear and turbulent diffusion, laying a foundation for addressing more intricate and physically significant transport problems.

Because of the high dimensionality and geometric complexity of the circular-pipe problem, formulating and implementing boundary conditions are challenging, and most existing theoretical studies either neglect boundary effects or impose purely specular-reflection boundary conditions. To address this gap, we devise and explore an analytical model for microswimmer dispersion in a cylindrical pipe flow under a diffuse-reflection boundary condition, extending our earlier studies (Jiang & Chen, J. Fluid Mech., vol. 899, 2020, A18; Zeng et al., J. Fluid Mech., 1018, 2025, A27). We derive a well-posed Laplacian eigenvalue problem under diffuse reflection and obtain a complete basis formed by products of Bessel functions and spherical harmonics. The moment equations are solved by the Galerkin spectral method, and the computation is simplified by decomposing the operator and basis functions, together with an analytical treatment of the orientational integrals. The study follows the entire transport process by examining the local and radial distributions, the drift velocity and the dispersivity, and we assess the effects of key parameters with comparisons to the specular reflection conditions. Our results show that diffuse reflection drives microswimmers away from the wall more efficiently and promotes downstream alignment and cross-stream migration. When swimming is strong, non-gyrotactic microswimmers can develop centre accumulation, whereas gyrotaxis promotes near-wall accumulation that counteracts the effect of diffuse reflection, in contrast to classical behaviour. Distinct mechanisms dominate different stages of the transient evolution, leading to different temporal trends in the radial distribution and dispersivity. Overall, diffuse reflection yields a larger drift velocity and a smaller dispersivity, while both gyrotaxis and elongation increase dispersivity.

This study investigates the influence of wind tunnel ground conditions (stationary/moving) on flow topology and passive scalar dispersion in the wake of the Ahmed body with rear slant angles, $\phi$ = 25$^\circ$ and 40$^\circ$. We implement field measurements of both velocity and scalar concentrations in the wake, for both the ground conditions, within the same experimental set-up, allowing for structural correlation between wake topology and scalar dispersion. Particle image velocimetry measurements reveal the existence of a third spanwise vortex (vortex G) near the stationary wind tunnel ground, due to the floor boundary layer, for both of the Ahmed bodies ($\phi$ = 25$^\circ$, 40$^\circ$). Concentration field measurements performed using quantitative smoke visualisation show higher scalar dispersion in the wake of both Ahmed bodies for the stationary ground condition. Comparing the velocity and concentration fields further identifies vortex G as the primary physical driver for the enhanced vertical dispersion of the scalar, observed in stationary ground conditions. To quantify the dispersion and characterise these effects, we introduce dispersion parameters, such as non-dimensional dispersion ($\mathscr{D}$) and dispersion length scales ($\mathscr{L}_y, \mathscr{L}_z$). These parameters confirm that, while lateral dispersion remains relatively insensitive to wind tunnel ground conditions, the presence of vortex G in stationary ground conditions leads to an overestimation of vertical dispersion by up to $\approx$29 % ($\phi$ = 25$^\circ$) and $\approx$49 % ($\phi$ = 40$^\circ$). This study quantifies the overestimated dispersion, identifies the vortical structures responsible for scalar redistribution, provides physical insight into the wake dispersion phenomenon and highlights the importance of correct wind tunnel ground conditions in the vehicle wake dispersion studies.

The superlinear scaling relationship between the hydrodynamic dispersion coefficient and the Péclet number in porous media has been widely acknowledged. Nevertheless, the mechanisms driving this behaviour remain inadequately understood. In this work, we investigate the mechanism responsible for this superlinear scaling using a Lagrangian framework that combines a statistical model, which links the global probability density function of tracer transition time to flow variability in porous media, with a continuous time random walk framework. Our analysis reveals that the intra-pore and inter-pore flow variabilities are the primary sources responsible for the superlinear scaling, with their relative significance characterised by a structure-specific parameter, $\chi$. Specifically, the inter-pore flow variability dominates when $\chi \gt 1$, while the intra-pore variability prevails for $0\lt \chi \lt 1$. The parameter $\chi$ is derived exclusively from the statistical distributions of pore-throat radius, length and orientation angle, which can be readily obtained from structural characterisation techniques such as X-ray computed tomography imaging. These theoretical predictions are validated through extensive numerical simulations on tube networks with substantial structural variation. This study resolves discrepancies in previous studies regarding the mechanisms of superlinear scaling in hydrodynamic dispersion and offers valuable insights into modulate dispersion and mixing in porous media.

The dispersion phenomenon of mass and heat transport in oscillatory flows has wide applications in environmental, physiological and microfluidic flows. The method of concentration moments is a powerful theoretical framework for analysing transport characteristics and is well developed for steady flows: general solution expressions of moments have been profoundly derived by Barton (J. Fluid Mech. 126, 1983, 205–218). However, it was thought that these expressions could not be directly applied to unsteady flows. Prior studies needed to re-solve the governing equations of moments from scratch, encountering the complication induced by the time-periodic velocity, leaving higher-order statistics like skewness and kurtosis analytically intractable except for specific cases. This work proposes a novel approach based on a two-time-variable extension to tackle these challenges. By introducing an auxiliary time variable, referred to as oscillation time to characterise the inherent oscillation in the dispersion due to the oscillating flow, the transport problem is extended to a two-time-variable system with a ‘steady’ flow term. This enables the direct use of Barton’s expressions and thus avoids the prior complication. This approach not only offers an intuitive physical perspective for the influence of the velocity oscillation, but also clarifies the solution structure of concentration moments. As a preliminary verification, we examine the transport problem in an oscillatory Couette flow. The analytical solution agrees well with the numerical result by Brownian dynamics simulations. The effects of the point-source release and the phase shift of velocity on the transport characteristics are investigated. By extending the classic steady-flow solution to the time-dependent flows, this work provides a versatile framework for transient dispersion analysis, enhancing predictions in oscillatory transport problems.

Predicting and controlling the transport of colloids in porous media is essential for a broad range of applications, from drug delivery to contaminant remediation. Chemical gradients are ubiquitous in these environments, arising from reactions, precipitation/dissolution or salinity contrasts, and can drive particle motion via diffusiophoresis. Yet our current understanding mostly comes from idealised settings with sharply imposed solute gradients, whereas in porous media, flow disorder enhances solute dispersion, and leads to diffuse solute fronts. This raises a central question: Does front dispersion suppress diffusiophoretic migration of colloids in dead-end pores, rendering the effect negligible at larger scales? We address this question using an idealised one-dimensional dead-end geometry. We derive an analytical model for the spatio-temporal evolution of colloids subjected to slowly varying solute fronts and validate it with numerical simulations and microfluidic experiments. Counterintuitively, we find that diffuseness of the solute front enhances removal from dead-end pores: although smoothing reduces instantaneous gradient magnitude, it extends the temporal extent of phoretic forcing, yielding a larger cumulative drift and higher clearance efficiency than sharp fronts. Our results highlight that solute dispersion does not weaken the phoretic migration of colloids from dead-end pores, pointing to the potential relevance of diffusiophoresis at larger scales, with implications for filtration, remediation and targeted delivery in porous media.

Non-Newtonian fluid flow in porous media results in spatially varying viscosity, driven by flow–pore–geometry interactions, potentially leading to non-monotonic dispersion. In this work, using high-resolution micro-particle image velocimetry, we present a direct experimental observation of shear-viscosity-distribution-dependent transport with non-Newtonian fluid flows in porous media. We experimentally investigate dispersion in porous media in a microfluidic chip featuring a physical rock geometry, comparing a shear-thinning, non-Newtonian fluid with its Newtonian analogue at various Péclet numbers. We demonstrate that, in the absence of advective fluxes driven by elastic instabilities, non-Newtonian fluid flows at either extreme of the shear-dependent viscosity ($\eta _0,\eta _{\infty }$) converge to the Newtonian analogue. In contrast, for flows between these extremes, the non-Newtonian velocity fields are broadly distributed along the streamline curvature, leading to a larger enhancement in dispersion.

We introduce a description of passive scalar transport based on a (deterministic and hyperbolic) Liouville master equation. Defining a noise term based on time-independent random coefficients, instead of time-dependent stochastic processes, we circumvent the use of stochastic calculus to capture the one-point space–time statistics of solute particles in Lagrangian form deterministically. To find the proper noise term, we solve a closure problem for the first two moments locally in a streamline coordinate system, such that averaging the Liouville equation over the coefficients leads to the Fokker–Planck equation of solute particle locations. This description can be used to trace solute plumes of arbitrary shape, for any Péclet number, and in arbitrarily defined grids, thanks to the time reversibility of hyperbolic systems. In addition to grid flexibility, this approach offers some computational advantages as compared with particle tracking algorithms and grid-based partial differential equation solvers, including reduced computational cost, no Monte-Carlo-type sampling and unconditional stability. We reproduce known analytical results for the case of simple shear flow and extend the description of mixing in a vortex model to consider diffusion radially and nonlinearities in the flow, which govern the long time decay of the maximum concentration. Finally, we validate our formulation by comparing it with Monte Carlo particle tracking simulations in a heterogeneous flow field at the Darcy (continuum) scale.

Lagrangian transit times on basin to planetary scales are controlled by the interplay of multiscale processes. The primary advective time scale is set by throughflow currents, such as interhemispheric western boundary currents. Dispersion by mesoscale eddies introduces fluctuations that erase memory and enhance dispersion, widening the transit-time distribution. The tortuous paths of Lagrangian parcels, particularly within ocean gyres, significantly enhance dispersion beyond the levels attributed to mesoscale eddies alone. Additionally, trapping by ocean gyres leads to multimodal distributions of Lagrangian transit times. These processes are illustrated in three complementary contexts: eddy-permitting ocean state estimates, simplified spatially extended three-dimensional flows and diffusively coupled two-dimensional pipe models.

The dispersion of solutes has been extensively studied due to its important applications in microfluidic devices for mixing, separation and other related processes. Solute dispersion in fluids can be analysed over multiple time scales; however, Taylor dispersion specifically addresses long-term behaviour, which is primarily influenced by advective dispersion. This study investigates Taylor–Aris dispersion in a viscoelastic fluid flowing through axisymmetric channels of arbitrary shape. The fluid’s rheology is described using the simplified Phan-Thien–Tanner (sPTT) model. Although the channel walls are axisymmetric, they can adopt any geometry, provided they maintain small axial slopes. Drawing inspiration from the work of Chang & Santiago (2023 J. Fluid Mech. vol. 976, p. A30) on Newtonian fluids, we have developed a governing equation for solute dynamics that accounts for the combined effects of fluid viscoelasticity, molecular diffusivity and channel geometry. This equation is expressed using key dimensionless parameters: the Weissenberg number, the Péclet number and a shape-dependent dimensionless function. Solving this model allows us to analyse the temporal evolution of the solute distribution, including its mean and variance. Our analysis shows that viscoelasticity significantly decreases the effective solute diffusivity compared with that observed in a Newtonian fluid. Additionally, we have identified a specific combination of parameters that results in zero or negative transient growth of the variance. This finding is illustrated in a phase diagram and provides a means for transient control over dispersion. We validated our results against Brownian dynamics simulations and previous literature, highlighting potential applications for the design and optimisation of microfluidic devices.

Statistical regularities can be acquired from usage. To examine language speakers’ statistical metacognition about multiword expressions (MWEs), we collected ratings for frequency, dispersion, and directional association strength of English binomials from L1, advanced and intermediate L2 speakers. Mixed-effects modeling showed all speakers had limited speaker-to-corpus consistency but significant sensitivity to statistical regularities of language, supporting usage-based (Gries & Ellis, 2015) and statistical learning theories (Christiansen, 2019). Their statistical metacognition was also shaped by word-level cues, consistent with dual-route model (Carrol & Conklin, 2014). Despite similarities, frequency metacognition showed the strongest speaker-to-corpus consistency, while dispersion metacognition was the hardest to develop. Advanced L2 speakers showed the greatest speaker-to-corpus consistency and sensitivity, while lower-proficiency speakers relied more on word-level cues in metacognitive judgments, supporting the shallow-structure hypothesis (Clahsen & Felser, 2006). Overall, L1 and L2 speakers develop diverse statistical metacognition, with L2 speakers not necessarily inferior, suggesting that statistical metacognition is not solely shaped by usage-based experience.

This work investigates the long-time asymptotic behaviour of a diffusing passive scalar advected by fluid flow in a straight channel with a periodically varying cross-section. The goal is to derive an asymptotic expansion for the scalar field and estimate the time scale over which this expansion remains valid, thereby generalising Taylor dispersion theory to periodically modulated channels. By reformulating the eigenvalue problem for the advection–diffusion operator on a unit cell using a Floquet–Bloch-type eigenfunction expansion, we extend the classical Fourier integral of the flat channel problem to a periodic setting, yielding an integral representation of the scalar field. This representation reveals a slow manifold that governs the algebraically decaying dynamics, while the difference between the scalar field and the slow manifold decays exponentially in time. Building on this, we derive a long-time asymptotic expansion of the scalar field. We show that the validity time scale of the expansion is determined by the real part of the eigenvalues of a modified advection–diffusion operator, which depends solely on the flow and geometry within a single unit cell. This framework offers a rigorous and systematic method for estimating mixing time scales in channels with complex geometries. We show that non-flat channel boundaries tend to increase the time scale, while transverse velocity components tend to decrease it. The approach developed here is broadly applicable and can be extended to derive long-time asymptotics for other systems with periodic coefficients or periodic microstructures.

In this paper, we study experimentally the dispersion of colloids in a two-dimensional, time-independent, Rayleigh–Bénard flow in the presence of salt gradients. Due to the additional scalar, the colloids do not follow exactly the Eulerian flow field, but have a (small) extra velocity $\boldsymbol{v}_{{dp}} = D_{{dp}}\, \boldsymbol{\nabla }\log C_s$, where $D_{{dp}}$ is the phoretic constant, and $C_s$ is the salt concentration. Such a configuration is motivated by the theoretical work by Volk et al. (2022, J. Fluid Mech., vol. 948, A42), which predicted enhanced transport or blockage in a stationary cellular flow depending on the value of a blockage coefficient. By means of high dynamical range light-induced fluorescence, we study the evolution of the colloids concentration field at large Péclet number. We find good agreement with the theoretical work, although a number of hypotheses are not satisfied, as the experiment is non-homogeneous in space, and intrinsically transient. In particular, we observe enhanced transport when salt and colloids are injected at both ends of the Rayleigh–Bénard chamber, and blockage when colloids and salt are injected together and phoretic effects are strong enough.

The integration of electro-osmotic effect to the underlying flow enhances solute dispersion precision in microfluidic systems, which is crucial for applications such as drug delivery and on-chip fluidic functionalities. We investigate, in this study, the solute dispersion characteristics of couple-stress fluids in a two-dimensional microchannel configuration under the combined effects of electro-osmotic actuation and applied pressure gradients. We consider both homogeneous and heterogeneous reactions in the present analysis. Couple-stress fluids, which account for additional stresses due to the presence of the microstructures in the fluids, offer a more accurate model to describe the rheological behaviour of biofluids. While previous studies have addressed longitudinal Gaussianity and transverse uniformity of solute distribution, we focus uniquely in this endeavour on longitudinal uniformity. Using Mei’s multiscale homogenisation technique, we solve a two-dimensional convection–diffusion model, extending it to third-order approximation to analyse the dispersion coefficient, concentration profiles, and variation rates of concentration within microchannel flow. Results show that forcing and couple-stress parameters enhance the gradients of the longitudinal variation rate, while boundary absorption reduces this variation rate near the walls. The couple-stress parameter exhibits dual behaviour: initially, it enhances solute dispersion, but beyond a certain value of couple-stress parameter $B_{cr}$ (which depends on forcing comparison and the Debye–Hückel parameter), it reduces dispersion. In the absence of pressure, solute distribution remains longitudinally uniform. However, as the pressure gradient increases, concentration levels drop sharply, and the distribution shifts to a parabolic profile, underscoring the significant influence of pressure on flow behaviour in electro-osmotic flow.

Mass dispersion in oscillatory flows is closely tied to various environmental and biological processes, differing markedly from dispersion in steady flows due to the periodic expansion and contraction of particle patches. In this study, we investigate the Taylor–Aris dispersion of active particles in laminar oscillatory flows between parallel plates. Two complementary approaches are employed: a two-time-variable expansion of the Smoluchowski equation is used to facilitate Aris’ method of moments for the pre-asymptotic dispersion, while the generalised Taylor dispersion theory is extended to capture phase-dependent periodic drift and dispersivity in the long-time asymptotic limit. Applying both frameworks, we find that spherical non-gyrotactic swimmers can exhibit greater or lesser diffusivity than passive solutes in purely oscillatory flows, depending on the oscillation frequency. This behaviour arises primarily from the disruption of cross-streamline migration governed by Jeffery orbits. When a steady component is superimposed, oscillation induces a non-monotonic dual effect on diffusivity. We further examine two well-studied shear-related accumulation mechanisms, arising from gyrotaxis and elongation. Although these accumulation effects are less pronounced than in steady flows due to flow unsteadiness, gyrotactic swimmers respond more strongly to the unsteady shear profile, significantly modifying their drift and dispersivity. This work offers new insights into the dispersion of active particles in oscillatory flows, and also provides a foundation for studying periodic active dispersion beyond the oscillatory flow, such as periodic variations in shape and swimming speed.

In this work the fascinating dynamics of a two-layered channel flow characterised by the dispersion in composite media within its layers is investigated in depth. The top layer comprises of a fluid zone that allows the fluid to travel along its surface easily (with relatively higher velocity), while the bottom layer is packed with porous media. The primary objective of this research is to do an in-depth investigation of the complex two-dimensional concentration distribution of a passive solute discharged from the inflow region. A multi-scale perturbation analysis approach has been implemented to address the system’s inherent complexity. This accurate determination of the dispersion coefficient, mean concentration distribution and two-dimensional concentration distribution is accomplished deftly using Mei’s homogenisation approach up to second-order approximation, which satisfactorily capture the minor variations in the solute dynamics also. The influence of various flow and porous media elements on these basic parameters is thoroughly investigated, expanding our comprehension of the complex interaction between flow dynamics and porous media’s properties. The effect of Darcy number and the ratio of two viscosities ($M$) on the dispersion coefficient depends on the height of the porous layer. As the Péclet number ratio increases, the dispersion coefficient experiences a concurrent increase, resulting in a decline in the concentration peak. The results of the analytical studies have also been compared with those results obtained using a purely computational method to establish the validity of our studies. Both the sets of results show quite good agreement with each other. In this study, alternate flow models have been used for the porous region, and the outcomes are compared to determine which approach yields more suitable results under different conditions.

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.