We study the longitudinal dispersion of a passive tracer by a two-dimensional pressure-driven flow through a layer of heterogeneous porous rock which is bounded above and below by impermeable seal rock. We assume there are localised regions of different permeability at random vertical positions within an otherwise uniform layer. This leads to a Fickian-type dispersion (Eames & Bush, Proc. R. Soc. Lond. A, vol. 455, issue 1990, 1999, pp. 3665–3686) and a shear near the boundaries (Rabinovich et al., Phys. Rev. E, vol. 86, issue 4, 2012, 046601). We illustrate this effect in a long two-dimensional layer, consisting of $(a)$ long and thin lenses and $(b)$ elliptical lenses, using expressions for depth-dependent mean speed and dispersivity to derive a depth-averaged transport equation for the flow. We derive asymptotic solutions for the layer-averaged dispersal of tracer, showing that at long times the spreading of the tracer is controlled by the shear. We discuss the importance of our results for the interpretation of tracer dispersal tests in layered permeable rocks.

The development of laminar thermals and the formation of buoyant vortex rings in thermals are studied by performing direct numerical simulations. The formation number of buoyant vortex rings in thermals is also analysed. We find that the development of thermals can be classified into three modes: the starting vortex ring dominated mode; the mode with the occurrence of a secondary vortex ring with breakup; and the mode with the occurrence of a secondary vortex ring without breakup. For the latter two modes, owing to the stretching of the thermal cap, the fluid at the leading edge rolls up, and a secondary vortex ring occurs, grows and replaces the starting vortex ring. The boundary of non-occurrence and occurrence of the secondary vortex ring is determined in a space of Richardson number (Ri) and injection duration ($t_{i}$). The final mode occurs only in a small region. For $Ri<0.6$, the secondary vortex ring does not occur even for very long injection duration. The effective Rayleigh number ($Ra_{m}$) is proposed to accommodate the cases $Ri>0.7$ and $t_{i}<5$, with $Ra_{m}$ larger than the critical value (approximates to 1. 95 × 105) for the occurrence of the secondary vortex ring. The formation number of buoyant vortex rings in thermals is beyond the universal formation number of 4 for non-buoyant vortex rings, and increases with the increase of the Richardson number and the injection duration. The switching between the thermal modes by changing the Richardson number and the injection duration has no significant effect on the value of the formation number.

When two swimming cells collide, some ciliates show biological reactions such as avoidance and escape on sensing mechanical agitation through membrane ion channels. However, there is still no clear understanding of the mechanosensing mechanism, despite it being proposed more than 50 years ago. Moreover, it is largely unknown how the deformability of cells affects hydrodynamic interactions between microswimmers. In this study, we investigate hydrodynamic interactions between two deformable microswimmers and quantify the membrane tension during the interaction. A ciliate was modelled as a capsule with a hyperelastic membrane, and thrust generated by the cilia was modelled as torque distributed above the cell body. The model we use is the simplest we could think of that can swim, deform and exert membrane tension, such that hydrodynamic interactions can be analysed non-trivially. The results showed that softer swimmers induced smaller membrane tension during the interaction, and tended to have smaller scattering angles after the interaction. In the case of chasing interactions, swimmers attracted each other, which was not observed in non-deformable velocity squirmer models. Finally, we discuss the biological implications of these results, comparing them to those of our former experiments using Paramecium. This study is useful for understanding the effect of swimmer deformability in hydrodynamic interactions, and provides a theoretical basis for investigating a mechanical picture of the biological reactions of ciliates.

Inspired by the intermittent locomotion of fish schools, numerical simulations are performed with two self-propelled flexible fins in a side-by-side configuration with anti-phase oscillation actuated by laterally constrained heaving motions. For an intermittent swimming gait, one type of the half-tail-beating mode (HT mode) and two types of multiple-tail-beating modes coasting at the smallest (MTS mode) and largest (MTL mode) lateral gap distances are applied. Similar to the continuous-tail-beating mode (CT mode), equilibrium lateral gap distances between two fins with HT and MTL modes exist, whereas two fins with MTS mode do not maintain a lateral equilibrium state. Although the cycle-averaged lateral force acting on two fins with CT and MTL modes is mostly determined by an outward deflected jet and enhanced positive pressure between two fins, an added-mass lateral force related to an asymmetric flapping kinematics by passive flexibility also plays an important role in MTL mode to achieve a stable state with a lateral gap distance smaller than that in CT mode. When the cruising speed or the cycle-averaged input power is identical in a stable state, the cost of transport (COT) for two fins with MTL mode is smaller than that with CT mode due to not only a benefit from the intermittent swimming gait but also an enhanced schooling benefit with a small equilibrium lateral gap distance. The COT for two fins with CT mode is reduced further when the bending rigidity increases, whereas it is opposite with MTL mode.

Orbits tracked by ellipsoids immersed in inviscid and viscous environments are studied by means of Kirchhoff's equations and high resolution numerical simulations using a variant of the immersed boundary method. We explore the consequences of Kozlov and Onishchenko's theorem of non-integrability of Kirchhoff's equations to show how the fraction of phase space in chaotic orbits is sensitively determined by the body shape, fluid/solid density ratio and the fraction of initial energy in rotational motion. We show how the added mass tensor of the system is an important player in both viscous and inviscid flow, in causing chaos in a triaxial ellipsoid while acting to suppress it in a spheroid. We identify a new integral of motion for a spheroid in inviscid fluid: one component of the generalised angular momentum. A spheroid, which can never execute chaotic dynamics in inviscid flow, is shown to display chaos in viscous flow due to irregular vortex shedding. But the dynamics of the spheroid is restricted whether in viscous or in inviscid flow, unlike in the triaxial ellipsoid, due to our extra integral of motion.

What is the most robust way to communicate flow trajectories? To answer this question, we employ two neural networks to respectively deconstruct (the encoder) and reconstruct (the decoder) trajectories, where information is passed between the two networks through a low-dimensional latent space in a set-up known as an autoencoder. To ensure that their communications are robust, we add noise to the coded information passed through this latent space. In the low-noise limit the latent space structures are non-spatial in nature, resembling modes of a principle component analysis (PCA). However, as the signal-to-noise ratio is decreased, we uncover Lagrangian coherent structures (LCS) as the most compact representations which still allow the decoder to accurately reconstruct trajectories. This relationship offers increased interpretability to both PCA and LCS analysis, and helps to bridge the gap between two methods of flow analysis.

The present work describes the vortex–vortex interactions observed during laboratory experiments, where a single regular water wave is allowed to travel over a discontinuous rigid bed promoting the generation of both near-bed and surface vortices. While near-bed vortices are generated by the flow separation occurring at the bed discontinuity, surface vortices are induced by the wave breaking in conjunction with a breaking-induced jet. A ‘backward breaking’ (previously observed in the case of solitary waves) occurs at the air–water interface downstream of the discontinuity and generates a surface anticlockwise vortex that interacts with the near-bed clockwise vortex. With the vortex–vortex interaction influenced by many physical mechanisms, a point-vortex model, by which vortices evolve under both self-advection (in relation to both free surface and seabed) and mutual interaction, has been implemented to separately investigate the vortex- and wave-induced dynamics. The available data indicate that both self-advection and mutual interaction are the governing mechanisms for the downward motion of the surface vortex, with the effect of the breaking-induced jet being negligible. The same two mechanisms, combined with the mean flow, are responsible for the almost horizontal and oscillating path of the near-bed vortex. The investigation of the vortex paths allow us to group the performed tests into three distinct classes, each characterized by a specific range of wave nonlinearity. The time evolution of the main variables characterizing the vortices (e.g. circulation, kinetic energy, enstrophy, radius) and their maximum values increase with the wave nonlinearity, such dependences being described by synthetic best-fit formulas.

When an object is accelerated in a fluid, a primary vortex is formed through the roll-up of a shear layer. This primary vortex does not grow indefinitely and will reach a limiting size and strength. Additional vorticity beyond the critical limit will end up in a trailing shear layer and accumulate into secondary vortices. The secondary vortices are typically considerably smaller than the primary vortex. In this paper, we focus on the formation, shedding and trajectory of secondary vortices generated by a rotating rectangular plate in a quiescent fluid using time-resolved particle image velocimetry. The Reynolds number $\textit {Re}$ based on the maximum rotational velocity of the plate and the distance between the centre of rotation and the tip of the plate is varied from 840 to 11 150. At low $\textit {Re}$, the shear layer is a continuous uninterrupted layer of vorticity that rolls up into a single coherent primary vortex. At $\textit {Re} = 1955$, the shear layer becomes unstable and secondary vortices emerge and subsequently move away from the tip of the plate. For $\textit {Re} > 4000$, secondary vortices are discretely released from the plate tip and are not generated from the stretching of an unstable shear layer. First, we demonstrate that the roll-up of the shear layer, the trajectory of the primary vortex and the path of secondary vortices can be predicted by a modified Kaden spiral for the entire $\textit {Re}$ range considered. Second, the timing of the secondary vortex shedding is analysed using the swirling strength criterion. The separation time of each secondary vortex is identified as a local maximum in the temporal evolution of the average swirling strength close to the plate tip. The time interval between the release of successive secondary vortices is not constant during the rotation but increases the more vortices have been shed. The shedding time interval also increases with decreasing Reynolds number. The increased time interval under both conditions is due to a reduced circulation feeding rate.

A wide class of problems for free-surface gravity waves fall into a weakly dispersive regime, in which wavelength is large compared to water depth, and wave phase speed differs by a small amount from the speed $c_{0}=\sqrt{gh}$ of shallow-water waves. The resulting problem is treated naturally using Taylor series expansions of dependent variables in the vertical coordinate, leading to a class of models that are collectively referred to here as Boussinesq-type models. Madsen & Fuhrman (J. Fluid Mech., vol. 889, 2020, A38) have recently shown that certain members of this broad class of models are subject to a high-wavenumber instability, which can grow rapidly when the elevation of the wave trough is sufficiently depressed below the mean water surface. This newly revealed instability may provide an explanation for the modelling community’s frequent observations of noisy behaviour in Boussinesq-type model calculations.

We study the mixing processes inside a forced fountain using data from direct numerical simulation. The outer boundary of the fountain with the ambient is a turbulent/non-turbulent interface. Inside the fountain, two internal boundaries, both turbulent/turbulent interfaces, are identified: (i) the classical boundary between upflow and downflow which is composed of the loci of points of zero mean vertical velocity; and (ii) the streamline that separates the mean flow emitted by the source from the entrained fluid from the ambient (the separatrix). We show that entrainment due to turbulent fluxes across the internal boundary is at least as important as that by the mean flow. However, entrainment by the turbulence behaves substantively differently from that by the mean flow and cannot be modelled using the same assumptions. This presents a challenge for existing models of turbulent fountains and other environmental flows that evolve inside turbulent environments.

For turbulent bubbly flows, multi-phase simulations resolving both the liquid and bubbles are prohibitively expensive in the context of different natural phenomena. One example is breaking waves, where bubbles strongly influence wave impact loads, acoustic emissions and atmospheric-ocean transfer, but detailed simulations in all but the simplest settings are infeasible. An alternative approach is to resolve only large scales, and model small-scale bubbles adopting sub-resolution closures. Here, we introduce a large eddy simulation smoothed particle hydrodynamics (SPH) scheme for simulations of bubbly flows. The continuous liquid phase is resolved with a semi-implicit isothermally compressible SPH framework. This is coupled with a discrete Lagrangian bubble model. Bubbles and liquid interact via exchanges of volume and momentum, through turbulent closures, bubble breakup and entrainment, and free-surface interaction models. By representing bubbles as individual particles, they can be tracked over their lifetimes, allowing closure models for sub-resolution fluctuations, bubble deformation, breakup and free-surface interaction in integral form, accounting for the finite time scales over which these events occur. We investigate two flows: bubble plumes and breaking waves, and find close quantitative agreement with published experimental and numerical data. In particular, for plunging breaking waves, our framework accurately predicts the Hinze scale, bubble size distribution, and growth rate of the entrained bubble population. This is the first coupling of an SPH framework with a discrete bubble model, with potential for cost-effective simulations of wave–structure interactions and more accurate predictions of wave impact loads.

The present paper uses the detailed flow data produced by direct numerical simulation (DNS) of a three-dimensional, spatially developing plane free shear layer to assess several commonly used turbulence models in compressible flows. The free shear layer is generated by two parallel streams separated by a splitter plate, with a naturally developing inflow condition. The DNS is conducted using a high-order discontinuous spectral element method (DSEM) for various convective Mach numbers. The DNS results are employed to provide insights into turbulence modelling. The analyses show that with the knowledge of the Reynolds velocity fluctuations and averages, the considered strong Reynolds analogy models can accurately predict temperature fluctuations and Favre velocity averages, while the extended strong Reynolds analogy models can correctly estimate the Favre velocity fluctuations and the Favre shear stress. The pressure–dilatation correlation and dilatational dissipation models overestimate the corresponding DNS results, especially with high compressibility. The pressure–strain correlation models perform excellently for most pressure–strain correlation components, while the compressibility modification model gives poor predictions. The results of an a priori test for subgrid-scale (SGS) models are also reported. The scale similarity and gradient models, which are non-eddy viscosity models, can accurately reproduce SGS stresses in terms of structure and magnitude. The dynamic Smagorinsky model, an eddy viscosity model but based on the scale similarity concept, shows acceptable correlation coefficients between the DNS and modelled SGS stresses. Finally, the Smagorinsky model, a purely dissipative model, yields low correlation coefficients and unacceptable accumulated errors.

We characterise the incompressible turbulence cascade in terms of the concurrent inter-scale and inter-space exchanges of the scale-by-scale energy, helicity and enstrophy. The governing equations for the scale-by-scale helicity and enstrophy are derived in a similar fashion to that of the second order structure function following Hill (J. Fluid Mech., vol. 468, 2002, pp. 317–326). We examine the instantaneous dynamics, applying these equations to forced periodic turbulence and a von Kármán flow focusing on scales in the dissipative range $r=2.5\eta$, the near-dissipative range $r=0.5\lambda$ and the onset inertial range $r=\lambda$ (where $\eta$ and $\lambda$ are the Kolmogorov and Taylor length scales, respectively). The signature of the random sweeping effect is observed in all three individual budgets and between the energy and enstrophy transfers. As in the energy cascade, the anti correlation of the pressure transport and non-linear transfer is identified also in the helicity cascade. Owing to its lack of positive definiteness, the helicity transfers are found to be decorrelated from the others. However a connection between the energy cascade and helicity is identified kinematically. This connection reveals the large-scale sweeping motions are a key element in the overall energy cascade and underpins previous observations of large-scale intermittency. Taken together, this work extends a classic framework to gain novel insight on turbulence dynamics that underlay the statistically steady state, and demonstrates how transfers are interconnected.

We study the inverse flux of waves in one of the simplest geophysical fluid dynamics models: one-dimensional rotating shallow water equations. Based on direct numerical integration of the governing equations, we find that waves injected at small scales get transferred upscale predominantly via resonant quartic interactions between wave modes. The waves’ upscale transfer is non-local and involves turbulent transfer between disparate scales of the flow. Our analysis reveals that the upscale transfer of waves is extremely intermittent and is a result of localized-in-time bursts in wave action flux. These intermittent events of flux bursts lead to shallower waves’ spectrum and relatively higher amplitude wave fields in physical space. On examining statistics of the flow fields, we find that low-energy high wavenumbers more or less comply with the assumptions used in wave turbulence theory, such as uniformly distributed wave phases and the Gaussian distribution of fields, while non-uniform distribution of wave phases and non-Gaussian statistics dominate at large scales or low wavenumbers that contain a major share of the flow energy. Our findings point out that the one-dimensional rotating shallow water equations, despite being a simple geophysical fluid dynamic model, harbour complex and intricate features associated with the upscale transfer of waves that have not been recognized in the past.