We investigate the dynamics of heavy inertial particles in a flow field due to an isolated, non-axisymmetric vortex. For our study, we consider a canonical elliptical vortex – the Kirchhoff vortex and its strained variant, the Kida vortex. Contrary to the anticipated centrifugal dispersion of inertial particles, which is typical in open vortical flows, we observe the clustering of particles around co-rotating attractors near the Kirchhoff vortex due to its non-axisymmetric nature. We analyse the inertia-modified stability characteristics of the fixed points, highlighting how some of the fixed points migrate in physical space, collide and then annihilate with increasing particle inertia. The introduction of external straining, the Kida vortex being an example, introduces chaotic tracer transport. Using a Melnikov analysis, we show that particle inertia and external straining can compete, where chaotic transport can be suppressed beyond a critical value of particle inertia.

In the absence of large-scale coherent structures, a widely used statistical theory of two-dimensional turbulence developed by Kraichnan, Leith, and Batchelor (KLB) predicts a power-law scaling for the energy, $E(k)\propto k^\alpha$ with an integral exponent $\alpha ={-3}$, in the inertial range associated with the direct cascade. A power-law scaling is also observed in the presence of coherent structures, but the scaling exponent becomes fractal and often differs substantially from the value predicted by the KLB theory. Here we present a dynamical theory that sheds new light on the relationship between the spatial and temporal structure of the large-scale flow and the scaling of small-scale structures representing filamentary vorticity. Specifically, we find hyperbolic regions of the large-scale flow to play a key role in the flux of enstrophy between scales. Small-scale vorticity in these regions can be described by dynamically self-similar solutions of the Euler equation, which explains the power-law scaling. Furthermore, we find that correlations between different hyperbolic regions are responsible for the emergence of fractal scaling exponents.

Mixing describes the process by which solutes evolve from an initial heterogeneous state to uniformity under the stirring action of a fluid flow. Fluid stretching forms thin scalar lamellae that coalesce due to molecular diffusion. Owing to the linearity of the advection–diffusion equation, coalescence can be envisioned as an aggregation process. Here, we demonstrate that in smooth two-dimensional chaotic flows, mixing obeys a correlated aggregation process, where the spatial distribution of the number of lamellae in aggregates is highly correlated with their elongation, and is set by the fractal properties of the advected material lines. We show that the presence of correlations makes mixing less efficient than a completely random aggregation process because lamellae with similar elongations and scalar levels tend to remain isolated from each other. We show that correlated aggregation is uniquely determined by a single exponent that quantifies the effective number of random aggregation events. These findings expand aggregation theories to a larger class of systems, which have relevance to various fundamental and applied mixing problems.

Our understanding of the material organization of complex fluid flows has benefited recently from mathematical developments in the theory of objective coherent structures. These methods have provided a wealth of approaches that identify transport barriers in three-dimensional (3-D) turbulent flows. Specifically, theoretical advances have been incorporated into numerical algorithms that extract the most influential advective, diffusive and active barriers to transport from data sets in a frame-indifferent fashion. To date, however, there has been very limited investigation into these objectively defined transport barriers in 3-D unsteady flows with complicated spatiotemporal dynamics. Similarly, no systematic comparison of advective, diffusive and active barriers has been carried out in a 3-D flow with both thermally driven and mechanically modified structures. In our study, we utilize simulations of turbulent rotating Rayleigh–Bénard convection to uncover the interplay between advective transport barriers (Lagrangian coherent structures), material barriers to diffusive heat transport, and objective Eulerian barriers to momentum transport. For a range of (inverse) Rossby numbers, we identify each type of barrier and find intriguing relationships between momentum and heat transport that can be related to changes in the relative influence of mechanical and thermal forces. Further connections between bulk behaviours and structure-specific behaviours are also developed.

The motion of a finite-size particle in the cuboidal lid-driven cavity flow is investigated experimentally for Reynolds numbers $100$ and $200$ for which the flow is steady. These steady three-dimensional flows exhibit chaotic and regular streamlines, where the latter are confined to Kolmogorov–Arnold–Moser (KAM) tori. The interaction between the moving wall and the particle creates global particle attractors. For neutrally buoyant particles, these attractors are periodic or quasi-periodic, strongly attracting and located in or near KAM tori of the flow. As the density mismatch between particle and fluid increases, buoyancy and inertia become important, and the attractors evolve from those for neutrally buoyant particles, changing their shape, position and attraction rates.

This work describes the application of a technique that extracts branched manifolds from time series to study numerically generated fluid particle behaviour in the wake past a cylinder performing a rotary oscillation at low Reynolds numbers, and compares it with the results obtained for a paradigmatic analytical model of Lagrangian motion: the driven double gyre. The approach does not require prior knowledge of the underlying equations defining the dataset. The time series taken as input corresponds to the evolution of a position coordinate of an individual fluid particle. A delay embedding is used to reconstruct the dynamics in phase space, and a cell complex is built to characterize the topology of the embedding. Fluid particles are said to belong to the same topological class when the Betti numbers, orientability chains and weak boundaries of the associated cell complexes coincide. Topological colouring consists of labelling or ‘colouring’ advected particles with the topological class obtained in their finite-time analyses. The results suggest that topological colouring can be used to distinguish between regions of the flow where trajectories exhibit different finite-time dynamics.

In a sheared suspension, the chaotic motion of the particles disperses the suspending liquid and drives an exponential and broadly distributed growth of the elongation of its material lines. This paper addresses experimentally and theoretically the consequences of this complex advection on the mixing of an initially segregated blob of diffusive dye, at large Péclet number and down to the finest scales of mixing. As the suspension is sheared, the combined action of the mean and fluctuating components of the flow stretches and folds the blob into a multi-scale lamellar structure. At short time, overlaps between the lamellae are scarce and the probability distribution concentration can be understood from the local stretching statistics only. At intermediate times, dispersion limits the volume within which lamellae deploy; overlaps become abundant and their statistics determine the concentration distribution. At longer times, the dye becomes homogeneous at the particle scale and the concentration distribution is set by dispersion only. Predictions for both the concentration distribution and the transitions between these successive stages of mixing are provided and compared to the experimental results.

In the Stokesian limit, the streamline topology around a single neutrally buoyant sphere is identical to the topology of pair-sphere pathlines, both in an ambient simple shear flow. In both cases there are fore–aft symmetric open and closed trajectories spatially demarcated by an axisymmetric separatrix surface. We show that the topology of the fluid pathlines around a neutrally buoyant freely rotating spheroid, in simple shear flow, is profoundly different, and will have a crucial bearing on transport from such particles in shearing flows. An inertialess non-Brownian spheroid in a simple shear flow rotates indefinitely in any one of a one-parameter family of Jeffery orbits. The parameter is the orbit constant $C$, with $C = 0$ and $C = \infty$ denoting the limiting cases of a spinning (log-rolling) spheroid, and a spheroid tumbling in the flow–gradient plane, respectively. The streamline pattern around a spinning spheroid is qualitatively identical to that around a sphere regardless of its aspect ratio. For a spheroid in any orbit other than the spinning one ($C >0$), the velocity field being time dependent in all such cases, the fluid pathlines may be divided into two categories. Pathlines in the first category extend from upstream to downstream infinity without ever crossing the flow axis; unlike the spinning case, these pathlines are fore–aft asymmetric, suffering a net displacement in both the gradient and vorticity directions. The second category includes primarily those pathlines that loop around the spheroid, and to a lesser extent those that cross the flow axis, without looping around the spheroid, reversing direction in the process. The residence time, in the neighbourhood of the spheroid, is a smooth function of upstream conditions for pathlines belonging to the first category. In contrast, the number of loops, and thence, the residence time associated with pathlines in the second category, is extremely sensitive to upstream conditions. Plots reveal a fractal structure with singularities distributed on a Cantor-like set, suggesting the existence of a chaotic saddle in the vicinity of the spheroid.

The motion of a single spherical particle in a two-sided lid-driven cavity is investigated experimentally. The flow in which the particle moves is created by two facing cavity sidewalls which move with equal velocity in opposite directions. For a long cavity with width-to-height cross-sectional aspect ratio $\varGamma =W/H=1.6$ the flow field at Reynolds number ${Re}=400$ consists of steady spatially periodic three-dimensional convection cells. Nearly neutrally buoyant particles with radius in units of $H$ ranging from $1.1\times 10^{-2}$ to $7.1\times 10^{-2}$ are found to be attracted to periodic or quasi-periodic orbits in close vicinity of Kolmogorov–Arnold–Moser (KAM) tori of the unperturbed flow. Like the KAM tori the attractors of neutrally buoyant particles arise in mirror-symmetric pairs within each convection cell. The particle attractors are created by a dissipative effect in the dynamical system describing the particle motion which arises when the finite-size particle closely passes the moving walls. When the particle density deviates from that of the fluid, inertial attractors arise whose symmetry is broken by buoyancy, and other periodic attractors are created which do not have KAM tori as counterparts.

Inertial particles (i.e. with mass and of finite size) immersed in a fluid in motion are unable to adapt their velocities to the carrying flow and thus they have been the subject of much interest in fluid mechanics. In this paper we consider an ocean setting with inertial particles elastically connected forming a network that floats at the interface with the atmosphere. The network evolves according to a recently derived and validated Maxey–Riley equation for inertial particle motion in the ocean. We rigorously show that, under sufficiently calm wind conditions, rotationally coherent quasigeostrophic vortices (which have material boundaries that resist outward filamentation) always possess finite-time attractors for elastic networks if they are anticyclonic, while if they are cyclonic provided that the networks are sufficiently stiff. This result is supported numerically under more general wind conditions and, most importantly, is consistent with observations of rafts of pelagic Sargassum, for which the elastic inertial networks represent a minimal model. Furthermore, our finding provides an effective mechanism for the long range transport of Sargassum, and thus for its connectivity between accumulation regions and remote sources.

We consider the numerical solution of competitive exothermic and endothermic reactions in the presence of a chaotic advection flow. The resulting behaviour is characterized by a strong dependence on the competitive reaction history. The burnt temperature is not immediately connected to simple enthalpy calculations, so there is a subtlety in the interplay between the major parameters, notably the Damköhler number, the ratio of the heats of exothermic and endothermic reactions, as well as the ratio of their respective activation energies. This paper seeks to explore the way these parameters affect the steady states of these reaction fronts and their stability.

Using index matching and particle tracking, we measure the three-dimensional velocity field in an isotropic porous medium composed of randomly packed solid spheres. This high-resolution experimental dataset provides new insights into the dynamics of dispersion and stretching in porous media. Dynamic-range velocity measurements indicate that the distribution of the velocity magnitude, $U$, is flat at low velocity (probability density function $(U)\propto U^{0}$). While such a distribution should lead to a persistent anomalous dispersion process for advected non-diffusive point particles, we show that the dispersion of non-diffusive tracers nonetheless becomes Fickian beyond a time set by the smallest effective velocity of the tracers. We derive expressions for the onset time of the Fickian regime and the longitudinal and transverse dispersion coefficients as a function of the velocity field properties. The experimental velocity field is also used to study, by numerical advection, the stretching histories of fluid material lines. The mean and the variance of the line elongations are found to grow exponentially in time and the distribution of elongation is log-normal. These results confirm the chaotic nature of advection within three-dimensional porous media. By providing the laws of dispersion and stretching, the present study opens the way to a complete description of mixing in porous media.

We study the joint mixing of colloids and salt released together in a stagnation point or in a globally chaotic flow. In the presence of salt inhomogeneities, the mixing time is strongly modified depending on the sign of the diffusiophoretic coefficient $D_{dp}$. Mixing is delayed when $D_{dp}>0$ (salt-attracting configuration), or faster when $D_{dp}<0$ (salt-repelling configuration). In both configurations, as for molecular diffusion alone, large scales are barely affected in the dilating direction while the Batchelor scale for the colloids, $\ell _{c,diff}$, is strongly modified by diffusiophoresis. We propose here to measure a global effect of diffusiophoresis in the mixing process through an effective Péclet number built on this modified Batchelor scale. Whilst this small scale is obtained analytically for the stagnation point, in the case of chaotic advection, we derive it using the equation of gradients of concentration, following Raynal & Gence (Intl J. Heat Mass Transfer, vol. 40 (14), 1997, pp. 3267–3273). Comparing to numerical simulations, we show that the mixing time can be predicted by using the same function as in absence of salt, but as a function of the effective Péclet numbers computed for each configuration. The approach is shown to be valid when the ratio $D_{dp}^{2}/D_{s}D_{c}\gg 1$, where $D_{c}$ and $D_{s}$ are the diffusivities of the colloids and salt.

Application of an electric field across the pressure-driven stratified flow of a pair of miscible fluids inside a microchannel manifests interesting electrohydrodynamic (EHD) instabilities. Experiments uncover distinctive instability regimes with an increase in electric field Rayleigh number ($Ra^{\unicode[STIX]{x1D713}}$) – a linear-onset regime, a time-periodic nonlinear regime analogous to the von Kármán vortex street in the downstream and a regime with coherent flow patterns. The experiments also reveal that such linear and nonlinear instabilities can be stimulated non-invasively in a microchannel to mix or de-mix fluids simply by turning the electric field on or off, indicating the suitability of the process for on-demand micromixing. The characteristics of these instabilities have been theoretically investigated with the help of an Orr–Sommerfeld framework, which discloses the possibility of five distinctive finite-wavenumber modes for the instability. The EHD stresses originating due to the application of electric field stimulate a pair of shorter-wavelength electric field modes beyond a critical value of $Ra^{\unicode[STIX]{x1D713}}$. Increase in the levels of charge injection and EHD stresses lower the critical $Ra^{\unicode[STIX]{x1D713}}$ of these modes. The relatively longer-wavelength viscous mode is found to appear when the viscosity stratification between the fluid layers is high. Beyond a threshold Schmidt number ($Sc$), a diffusive mode is also found to appear near the mixed interfacial region. A thinner interface between the fluids at a higher $Sc$ helps this mode to behave as the interfacial mode of immiscible fluids. Contrast of ionic mobility in the fluids leads to the appearance of the K-mode of instability at much shorter wavelengths. The reported phenomena can be of significance in the domains of microscale mixing, pumping, heat exchange, mass transfer and reaction engineering.

This paper uses compressible flow simulation to analyse the hyperbolicity, shadowing directions and sensitivities of a weakly turbulent three-dimensional cylinder flow at Reynolds number 525 and Mach number 0.1. By computing the first 40 covariant Lyapunov vectors (CLVs), we find that unstable CLVs are active in the near-wake region, whereas stable CLVs are active in the far-wake region. This phenomenon is related to hyperbolicity since it shows that CLVs point to different directions; it also suggests that for open flows there is a large fraction of CLVs that are stable. However, due to the extra neutral CLV and the occasional tangencies between CLVs, our system is not uniform hyperbolic. By the non-intrusive least-squares shadowing (NILSS) algorithm, we compute shadowing directions and sensitivities of long-time-averaged objectives. Our results suggest that shadowing methods may be valid for general chaotic fluid problems.

The classical Prandtl–Batchelor theorem (Prandtl, Proc. Intl Mathematical Congress, Heidelberg, 1904, pp. 484–491; Batchelor, J. Fluid Mech., vol. 1 (02), 1956, pp. 177–190) states that in the regions of steady 2D flow where viscous forces are small and streamlines are closed, the vorticity is constant. In this paper, we extend this theorem to recirculating flows with quasiperiodic time dependence using ergodic and geometric analysis of Lagrangian dynamics. In particular, we show that 2D quasiperiodic viscous flows, in the limit of zero viscosity, cannot converge to recirculating inviscid flows with non-uniform vorticity distribution. A corollary of this result is that if the vorticity contours form a family of closed curves in a quasiperiodic viscous flow, then at the limit of zero viscosity, vorticity is constant in the area enclosed by those curves at all times.

Lagrangian data assimilation is a complex problem in oceanic and atmospheric modelling. Tracking drifters in large-scale geophysical flows can involve uncertainty in drifter location, complex inertial effects and other factors which make comparing them to simulated Lagrangian trajectories from numerical models extremely challenging. Temporal and spatial discretisation, factors necessary in modelling large scale flows, also contribute to separation between real and simulated drifter trajectories. The chaotic advection inherent in these turbulent flows tends to separate even closely spaced tracer particles, making error metrics based solely on drifter displacements unsuitable for estimating model parameters. We propose to instead use error in the coherent structure colouring (CSC) field to assess model skill. The CSC field provides a spatial representation of the underlying coherent patterns in the flow, and we show that it is a more robust metric for assessing model accuracy. Through the use of two test cases, one considering spatial uncertainty in particle initialisation, and one examining the influence of stochastic error along a trajectory and temporal discretisation, we show that error in the coherent structure colouring field can be used to accurately determine single or multiple simultaneously unknown model parameters, whereas a conventional error metric based on error in drifter displacement fails. Because the CSC field enhances the difference in error between correct and incorrect model parameters, error minima in model parameter sweeps become more distinct. The effectiveness and robustness of this method for single and multi-parameter estimation in analytical flows suggest that Lagrangian data assimilation for real oceanic and atmospheric models would benefit from a similar approach.

The deformation of elementary fluid volumes by velocity gradients is a key process for scalar mixing, chemical reactions and biological processes in flows. Whilst fluid deformation in unsteady, turbulent flow has gained much attention over the past half-century, deformation in steady random flows with complex structure – such as flow through heterogeneous porous media – has received significantly less attention. In contrast to turbulent flow, the steady nature of these flows constrains fluid deformation to be anisotropic with respect to the fluid velocity, with significant implications for e.g. longitudinal and transverse mixing and dispersion. In this study we derive an ab initio coupled continuous-time random walk (CTRW) model of fluid deformation in random steady three-dimensional flow that is based upon a streamline coordinate transform which renders the velocity gradient and fluid deformation tensors upper triangular. We apply this coupled CTRW model to several model flows and find that these exhibit a remarkably simple deformation structure in the streamline coordinate frame, facilitating solution of the stochastic deformation tensor components. These results show that the evolution of longitudinal and transverse fluid deformation for chaotic flows is governed by both the Lyapunov exponent and power-law exponent of the velocity probability distribution function at small velocities, whereas algebraic deformation in non-chaotic flows arises from the intermittency of shear events following similar dynamics as that for steady two-dimensional flow.

Taylor–Couette flows have been widely studied in part due to the enhanced mixing performance from the variety of hydrodynamic flow states accessible. These process improvements have been demonstrated despite the traditionally limited injection mechanisms from the complexity of the Taylor–Couette geometry. In this study, using a newly designed, modified Taylor–Couette cell, axial mass transport behaviour is experimentally determined over two orders of magnitude of Reynolds number. Four different flow states, including laminar and turbulent Taylor vortex flows and laminar and turbulent wavy vortex flows, were studied. Using flow visualization techniques, the measured dispersion coefficient was found to increase with increasing $Re$, and a single, unified regression is found for all vortices studied. In addition to mass transport, the vortex structures’ stability to radial injection is also quantified. A dimensionless stability criterion, the ratio of injection to diffusion time scales, was found to capture the conditions under which vortex structures are stable to injection. Using the stability criterion, global and transitional stability regions are identified as a function of Reynolds number, $Re$.