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.

Surfactant transport is central to a diverse range of natural phenomena with numerous practical applications in physics and engineering. Surprisingly, this process remains relatively poorly understood at the molecular scale. Here, we use non-equilibrium molecular dynamics (NEMD) simulations to study the spreading of sodium dodecyl sulphate on a thin film of liquid water. The molecular form of the control volume is extended to a coordinate system moving with the liquid–vapour interface to track surfactant spreading. We use this to compare the NEMD results to the continuum description of surfactant transport on an interface. By including the molecular details in the continuum model, we establish that the transport equation preserves substantial accuracy in capturing the underlying physics. Moreover, the relative importance of the different mechanisms involved in the transport process is identified. Consequently, we derive a novel exact molecular equation for surfactant transport along a deforming surface. Close agreement between the two conceptually different approaches, i.e. NEMD simulations and the numerical solution of the continuum equation, is found as measured by the surfactant concentration profiles, and the time dependence of the so-called spreading length. The current study focuses on a relatively simple specific solvent–surfactant system, and the observed agreement with the continuum model may not arise for more complicated industrially relevant surfactants and anti-foaming agents. In such cases, the continuum approach may fail to predict accompanying phase transitions, which can still be captured through the NEMD framework.

We investigate the emergent three-dimensional (3-D) dynamics of a rapidly yawing spheroidal swimmer interacting with a viscous shear flow. We show that the rapid yawing generates non-axisymmetric emergent effects, with the active swimmer behaving as an effective passive particle with two orthogonal planes of symmetry. We also demonstrate that this effective asymmetry generated by the rapid yawing can cause chaotic behaviour in the emergent dynamics, in stark contrast to the emergent dynamics generated by rapidly rotating spheroids, which are equivalent to those of effective passive spheroids. In general, we find that the shape of the equivalent effective particle under rapid yawing is different to the average shape of the active particle. Moreover, despite having two planes of symmetry, the equivalent passive particle is not an ellipsoid in general, except for specific scenarios in which the effective shape is a spheroid. In these scenarios, we calculate analytically the equivalent aspect ratio of the effective spheroid. We use a multiple scales analysis for systems to derive the emergent swimmer behaviour, which requires solving a non-autonomous nonlinear 3-D dynamical system, and we validate our analysis via comparison to numerical simulations.

Direct numerical simulations in a low-curvature viscoelastic turbulent Taylor vortex flow, with Reynolds numbers ranging from 1500 to 8000 and maximum chain extensibility ($L$) from 50 to 200, reveal a maximum drag reduction (MDR) asymptote. Compared with the classical MDR observed in planar wall-bounded shear flows, that is, drag reduction (DR) is $\sim -80\, \%$, this MDR state achieves only moderate levels of DR ($\sim -60\,\%$). This is due to the existence of large-scale structures (LSSs). A careful examination of the flow structures reveals that the polymer–turbulence interaction suppresses small-scale vortices and stabilizes the LSSs. These structural changes in turn lead to a reduction of Reynolds stress, and consequently to a DR flow state. Although Reynolds stress does not vanish as observed in classical MDR states, the small-scale vortices that heavily populate the near-wall region are also almost completely eliminated in this flow state. Concurrently, significant polymer stresses develop as a consequence of the interaction between polymer chains and LSSs that partially offset the magnitude of DR, leading to MDR asymptotes with moderate levels of DR. Moreover, we demonstrate that polymer deformation, i.e. deviation from the equilibrium state, is directly correlated with the LSSs dynamics, while the polymer deformation fluctuation displays a universal property in the MDR state. Hence, it is not surprising that the extent of DR exhibits a non-monotonic dependence on the maximum chain extensibility. Specifically, the variation in $L$ alters the incoherent and coherent angular momentum transport by small- and large-scale flow structures, respectively. To that end, the most DR flow state occurs at a moderate value $L=100$. Overall, this study further supports the universal property of polymer-induced asymptotic states in wall-bounded turbulence and paves the way for mechanistic understanding of drag modification that arises from the interaction of polymers with small- and large-scale flow structures.

We propose a hybrid numerical model for the collective motion of fish groups, which integrates an agent-based model with a computational fluid dynamics (CFD)-based model. In the agent-based model, the fish group is represented by self-propelled particles (SPPs), incorporating social forces with local interactive rules. The CFD-based model treats the fish body with an undulated filament that responds to the hydrodynamic forces imposed by the surrounding fluid flow. These two models are coupled using a central pattern generator controller. We test this hybrid model with groups of 30–50 individuals. The results show that the group exhibits various collective behaviours, including tight schooling, sparse schooling and milling patterns, by adjusting the coefficients in the SPP model. Due to the hydrodynamic interactions, particularly with the obstacle avoidance model, both the individuals’ and the group’s mean speed fluctuate, differentiating it from traditional SPP models that typically consider volumeless particles. More interestingly, our findings indicate that fish benefit from collective motion in terms of energetic consumption in both schooling and milling patterns. It is important to note that the swimming fish are actuated using a very simple mechanism without any optimisation strategy. An additional study investigates the effects of the Reynolds number, demonstrating the capability of the current hybrid model to account for fish groups of varying body lengths or swimming speeds. Future applications of this model are promising, offering potential insights into the energetic advantages of collective motion in large-scale fish groups.

The flow over a cambered NACA 65(1)–412 airfoil at $Re\,=\,2\times 10^4$ is described based on a high-order direct numerical simulation. Simulations are run over a range of angles of attack, $\alpha$, where a number of instabilities in the unsteady, three-dimensional flow field are identified. The balance and competing effects of these instabilities are responsible for significant and abrupt (with respect to $\alpha$) changes in flow regime, with measurable consequences in time-averaged, integrated force coefficients, and in the far-wake footprint. At low $\alpha$, the flow is strongly influenced by vortex roll-up from the pressure side at the trailing edge. The interaction of this large-scale structure with shear and three-dimensional modal instabilities in the separated shear layer and associated wake region on the suction side, explains the transitions and bifurcations of the the flow states as $\alpha$ increases. The transition from a separation at low $\alpha$ to reattachment and establishment of a laminar separation bubble at the trailing edge at critical $\alpha$ is driven by instabilities within the separated shear layer that are absent at lower angles. Instabilities of different wavelengths are then shown to pave the path to turbulence in the near wake.

The temporal characteristics of fully localised turbulent bands in transitional channel flow remains unclear due to the difficulty in resolving the large length and time scales involved. Here, we tackle this problem by performing statistical lifetime studies in sufficiently large computational domains. The results show signs of stochastic memoryless decay of a fully localised band, suggesting a chaotic-saddle behaviour of the entire band as a coherent entity. Although the mean lifetime of a turbulent band was reported to increase with the band length, our data suggest that it saturates at a certain length. This saturation results in a characteristic lifetime for a fully developed band with a changing length due to the intermittent chipping and decay of turbulence at the upstream end. This memoryless behaviour is observed down to Reynolds number $Re=630$ in our study and we propose that the onset of the memoryless behaviour is in the range of $Re\simeq 620{-}630$. Our data also show that the time it takes for a perturbed flow to enter the saddle, i.e. to start behaving memorylessly, can be thousands of convective time units, which is comparable to the maximum achievable observation time in existing channel set-ups and may pose difficulties for experiments.

In this study, we tackle the challenge of inferring the initial conditions of a Rayleigh–Taylor mixing zone for modelling purposes by analysing zero-dimensional (0-D) turbulent quantities measured at an unspecified time. This approach assesses the extent to which 0-D observations retain the memory of the flow, evaluating their effectiveness in determining initial conditions and, consequently, in predicting the flow’s evolution. To this end, we generated a comprehensive dataset of direct numerical simulations, focusing on miscible fluids with low density contrasts. The initial interface deformations in these simulations are characterised by an annular spectrum parametrised by four non-dimensional numbers. To study the sensitivity of 0-D turbulent quantities to initial perturbation distributions, we developed a surrogate model using a physics-informed neural network (PINN). This model enables computation of the Sobol indices for the turbulent quantities, disentangling the effects of the initial parameters on the growth of the mixing layer. Within a Bayesian framework, we employ a Markov chain Monte Carlo (MCMC) method to determine the posterior distributions of initial conditions and time, given various state variables. This analysis sheds light on inertial and diffusive trajectories, as well as the progressive loss of initial conditions memory during the transition to turbulence. Furthermore, it identifies which turbulent quantities serve as better predictors of Rayleigh–Taylor mixing zone dynamics by more effectively retaining the memory of the flow. By inferring initial conditions and forward propagating the maximum a posteriori (MAP) estimate, we propose a strategy for modelling the Rayleigh–Taylor transition to turbulence.

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.

The interaction between porous structures and flows with mean and oscillatory components has many applications in fluid dynamics. One such application is the hydrodynamic forces on offshore jacket structures from waves and current, which have been shown to give a significant blockage effect, leading to a reduction in drag forces. To better understand this, we derived analytical expressions that describe the effect of current on drag forces from large waves, and conducted experiments that measured forces on a model jacket in collinear waves and currents. We utilised symmetry and phase-inversion techniques, relying on the underlying physics of wave structure interaction, to separate Morison drag and inertia-type forces and to decompose these forces into their respective frequency harmonics. We find that the odd harmonics of the drag force mostly contain the loads from waves, while even harmonics vary much more rapidly with the current speed flowing through the jacket. At the time of peak force, these current speeds were estimated to be 40 % of the undisturbed current and 50 % of the industry-standard estimates, a result that has significant implications for design and re-assessment of jackets. At times away from the peak force, when there are no waves and only current, the blockage effects are reduced. Hence, the variation in blocked current speeds appears to occur on a relatively fast time scale similar to the compact wave envelope. These findings may be generalisable to any jacket-type structure in flows with mean and high Keulegan–Carpenter number oscillatory components.

Turbidity currents, which are stratified, sediment-laden bottom flows in the ocean or lakes, can run out for hundreds or thousands of kilometres in submarine channels without losing their stratified structure. Here, we derive a layer-averaged, two-layer model for turbidity currents, specifically designed to capture long-runout. A number of previous models have captured runout of only tens of kilometres, beyond which thickening of the flows becomes excessive, and the models without a lateral overspill mechanism fail. In our framework, a lower layer containing nearly all the sediment is a faster, gravity-driven flow that propels an upper layer, where sediment concentration is nearly zero. The thickness of the lower layer is controlled by competition between interfacial water entrainment due to turbulent mixing and water detrainment due to sediment settling at the interface. The detrainment mechanism, first identified in experiments, is the key feature that prevents excessive thickening of the lower layer and allows long-runout. Under normal flow conditions, we obtain an exact solution to the two-layer formulation, revealing a constant velocity and a constant thickening rate in each of the two layers. Numerical simulations applied

to gradually varied flows on both constant and exponentially declining bed slopes, with boundary conditions mimicking field observations, show that the predicted lower layer thickness after 200 km flow propagation compares with observed submarine channel depths, whereas previous models overestimate this thickness three- to fourfold. This formulation opens new avenues for modelling the fluid mechanics and morphodynamics of long-runout turbidity currents in the submarine setting.

We investigate the effects of bottom roughness on bottom boundary-layer (BBL) instability beneath internal solitary waves (ISWs) of depression. Applying both two-dimensional (2-D) numerical simulations and linear stability theory, an extensive parametric study explores the effect of the Reynolds number, pressure gradient, roughness (periodic bump) height $h_b$ and roughness wavelength $\lambda _b$ on BBL instability. The simulations show that small $h_b$, comparable to that of laboratory-flume materials ($\sim$100 times less than the thickness of the viscous sublayer $\delta _v$), can destabilize the BBL and trigger vortex shedding at critical Reynolds numbers much lower than what occurs for numerically smooth surfaces. We identify two mechanisms of vortex shedding, depending on $h_b/\delta _v$. For $h_b/\delta _v \gtrapprox 1$, vortices are forced directly by local flow separation in the lee of each bump. Conversely, for $h_b/\delta _v \lessapprox 10^{-1}$ the roughness seeds perturbations in the BBL, which are amplified by the BBL flow. Roughness wavelengths close to those associated with the most unstable BBL mode, as predicted by linear instability theory, are preferentially amplified. This resonant amplification nature of the BBL flow, beneath ISWs, is consistent with what occurs in a BBL driven by surface solitary waves and by periodic monochromatic waves. Using the $N$-factor method for Tollmien–Schlichting waves, we propose an analogy between the roughness height and seed noise required to trigger instability. Including surface roughness, or more generally an appropriate level of seed noise, reconciles the discrepancies between the vortex-shedding threshold observed in the laboratory versus that predicted by otherwise smooth-bottomed 2-D spectral simulations.

Double-diffusive convection can arise when the fluid density is set by multiple species which diffuse at different rates. Different flow regimes are possible depending on the distribution of the diffusing species, including salt fingering and diffusive convection. Flows arising from diffusive convection commonly exhibit step-like density profiles with sharp density interfaces separated by well-mixed layers. The formation of density layers is also seen in stratified turbulence, where a framework based on sorted density coordinates (Winters & D’Asaro 1996 J. Fluid Mech. 317, 179–193) has been used to diagnose layer formation (Zhou et al. 2017 J. Fluid Mech. 823, 198–229; Taylor & Zhou 2017 J. Fluid Mech. 823, R5). In this framework, the evolution of the sorted density profile can be expressed solely in terms of the eddy diffusivity, $\kappa _e$. Here, we use the same framework to diagnose layer formation in two-dimensional simulations of double-diffusive convection. We show that $\kappa _e$ is negative everywhere, with the antidiffusive effect strongest in ‘well-mixed’ layers where a positive diffusion coefficient may be expected. By considering a decomposition of the budget of the square of the Brunt-Väisälä frequency $\partial N^2_*/\partial t$, we demonstrate that the density layers are maintained by fundamentally different processes than in single-component stratified turbulence. In more complicated flows where stratified turbulence and double-diffusive convection can coexist, this framework could provide a method to distinguish between the mechanisms responsible for generating density layers.

We study time-dependent density segregation of granular mixtures flowing over an inclined plane. Discrete element method (DEM) simulations in a periodic box are performed for granular mixtures of same size and different density particles flowing under the influence of gravity. In addition, a continuum model is developed to solve the momentum balance equations along with the species transport equation by accounting for the inter-coupling of segregation and rheology. The particle force-based density segregation theory has been used along with the $\mu {-}I$ rheology to predict evolution of flow properties with time for binary and multicomponent mixtures. The effect of particle arrangements on the transient evolution of flow properties for three different initial configurations is investigated using both continuum and DEM simulations. Continuum predictions for various flow properties of interest such as species concentration, velocity, pressure and shear stress at different time instants are compared with DEM simulations. The results from the discrete and continuum models are found to be in good agreement with each other for well-mixed and heavy-near-base initial configurations. Kinetic theory-based predictions of segregation evolution, however, show good quantitative agreement only for the heavy-near-base configuration with a much slower evolution for the well-mixed case. Interestingly, the continuum model is unable to predict the flow evolution for the light-near-base initial configuration. DEM simulations reveal the presence of an instability driven, quick segregation for this configuration which is not predicted by the one-dimensional model and requires generalisation to three dimensions.

We report an anomalous capillary phenomenon that reverses typical capillary trapping via nanoparticle suspension and leads to a counterintuitive self-removal of non-aqueous fluid from dead-end structures under weakly hydrophilic conditions. Fluid interfacial energy drives the trapped liquid out by multiscale surfaces: the nanoscopic structure formed by nanoparticle adsorption transfers the molecular-level adsorption film to hydrodynamic film by capillary condensation, and maintains its robust connectivity, then the capillary pressure gradient in the dead-end structures drives trapped fluid motion out of the pore continuously. The developed mathematical models agree well with the measured evolution dynamics of the released fluid. This reversing capillary trapping phenomenon via nanoparticle suspension can be a general event in a random porous medium and could dramatically increase displacement efficiency. Our findings have implications for manipulating capillary pressure gradient direction via nanoparticle suspensions to trap or release the trapped fluid from complex geometries, especially for site-specific delivery, self-cleaning, or self-recover systems.

A purely elastic linear instability was recently reported for viscoelastic plane Poiseuille flow in the limit of ultra-dilute (solvent to solution viscosity ratio $\beta \gt 0.99$), highly elastic (Weissenberg number $W \sim 1000$) polymer solutions, within the framework of the Oldroyd-B model (Khalid et al., Phys. Rev. Lett., vol. 127, 2021, pp. 134–502). This is the first instance of a purely elastic instability in a strictly rectilinear shearing flow, with the phase speed of the unstable ‘centre mode’ being close to the base-state maximum velocity at the channel centreline. Subsequently, Buza, Page and Kerswell (J. Fluid Mech., vol. 940, 2022, A11) have shown, using the FENE-P model, that the centre-mode instability persists down to moderate elasticities ($W \sim O (100)$), the reduction in threshold evidently due to the finite extensibility of the polymer molecules. In this work, we augment this latter finding and provide a comprehensive account of the effect of finite extensibility on the centre-mode instability in viscoelastic channel flow, using the FENE-P and FENE-CR models, in both the absence and presence of fluid inertia. In both these models, finite extensibility causes a decrease in the polymer relaxation time at high shear rates, and the resulting weakening of elastic stresses would seem to indicate a stabilising effect. The latter trend has been demonstrated by earlier analyses of hoop-stress-driven instabilities in curvilinear flows, and is indeed borne out for the FENE-CR case, where finite extensibility has a largely stabilising influence on the centre-mode instability. In stark contrast, for the FENE-P model, finite extensibility plays a dual role – a stabilising one at lower values of the elasticity number $E$, but, surprisingly, a destabilising one at higher $E$ values. Further, the centre-mode instability is predicted over a significantly larger domain of the $Re$–$E$–$\beta$ parameter space, compared to the Oldroyd-B model, making it more amenable to experimental observations.

The study examines supersonic square jets in a twin nozzle configuration with the aim of identifying and characterising emergent instability modes during overexpanded operation. Unlike screeching rectangular jets that undergo strong fluctuations normal to the wider jet dimension, the equilateral nature of the exit geometry in square nozzles leads to multiple instability states dictated by shock–turbulence interactions and nozzle operating conditions. Furthermore, strong coupling modes between the jets were identified that led to either phase locked or out of phase interactions of the inner shear layers. Results from experimental studies were examined using spatial and temporal decomposition techniques based on spectral methods to identify the resultants from triadic shock–turbulence interactions. The primary instability mode across both operating conditions were driven by optimal interactions while the harmonics were found to be associated with the suboptimal shock–turbulence interactions.