The swift deformations of flagella and cilia are crucial for locomotion and fluid transport on the micron scale. Most hydrodynamic models of flagellar and ciliary flows assume the zero Reynolds number limit and model the flow using Stokes equations. Recent work has demonstrated that this quasi-steady approximation breaks down at increasing distances from the cilia. Here, we use optical tweezer-based velocimetry to measure the flow velocity with high temporal accuracy, and to reconstruct the entire unsteady flow field around beating cilia. We report both the steady and the unsteady component of the ciliary flow and compare them with the solutions to both the Stokes and the Navier–Stokes equations. Our experimental measurements of the velocity and vorticity fields are in agreement with the numerical solution to the Navier–Stokes equations and show significant differences with the solution to the Stokes equations. We characterize the phase difference between the flow oscillations and the oscillations of the ciliary motion and evidence a significant anisotropic phase lag. We show that this phase lag presents the spatiotemporal characteristics of the unsteady Stokes equations and that the flow field around beating cilia is well represented by the fundamental solution to the unsteady Stokes equations: the oscillet.

We report the results of an experimental investigation into the decay of turbulence in plane Couette–Poiseuille flow using ‘quench’ experiments where the flow laminarises after a sudden reduction in Reynolds number $Re$. Specifically, we study the velocity field in the streamwise–spanwise plane. We show that the spanwise velocity containing rolls decays faster than the streamwise velocity, which displays elongated regions of higher or lower velocity called streaks. At final Reynolds numbers above $425$, the decay of streaks displays two stages: first a slow decay when rolls are present and secondly a more rapid decay of streaks alone. The difference in behaviour results from the regeneration of streaks by rolls, called the lift-up effect. We define the turbulent fraction as the portion of the flow containing turbulence and this is estimated by thresholding the spanwise velocity component. It decreases linearly with time in the whole range of final $Re$. The corresponding decay slope increases linearly with final $Re$. The extrapolated value at which this decay slope vanishes is $Re_{a_z}\approx 656\pm 10$, close to $Re_g\approx 670$ at which turbulence is self-sustained. The decay of the energy computed from the spanwise velocity component is found to be exponential. The corresponding decay rate increases linearly with $Re$, with an extrapolated vanishing value at $Re_{A_z}\approx 688\pm 10$. This value is also close to the value at which the turbulence is self-sustained, showing that valuable information on the transition can be obtained over a wide range of $Re$.

Based on the linear shallow-water equations, new analytical solutions are derived for trapped waves over a ridge with a hyperbolic-cosine squared cross-sectional profile which may be used to idealize many real-world ocean ridges. In the new analytical formulation, the free surface of the trapped waves is described using the combination of the first and second kinds of the associated Legendre functions, which is further analysed to reveal the existence of both symmetrical and anti-symmetrical trapped waves on the ridge under consideration. New algebraic equations are also derived to depict the wave dispersion relationships, allowing explicit quantification of their sensitivity to the topographic profile. Furthermore, a ray-tracing method is applied to interpret the propagation paths of trapped waves over the ridge and better understand the excitation mechanisms. Finally, an extensively validated Boussinesq wave model is used to conduct numerical experiments for trapped waves induced by tsunamis. The numerical predictions are consistent with the new analytical solutions, which effectively confirms the validity of the new analytical framework for trapped waves over a more general type of oceanic ridges.

Synthetic jet actuators remain coveted components in flow control applications as the convection of vortex rings induces a redistribution of the momentum in a boundary layer although the net mass flux remains zero. The ability to predict the trajectory of these vortical structures, and subsequently the jet, is critical for efficient and targeted usage of such actuators. In this investigation, a synthetic jet is issued into a zero pressure gradient turbulent boundary layer from rectangular orifices with aspect ratios 3, 6 and 12 over a range of actuation frequencies and velocity ratios, with the flow field captured through particle image velocimetry measurements. An assessment of the trajectories culminates in scaling characteristics which encapsulate the aspect ratio, the momentum ratio between the jet and the cross-flow and the Strouhal number. However, the scaling mechanism is found to be sensitive to constraints which include the interaction between successive vortex pairs and the various regions of the flow field where the degree of interaction of the jet with the cross-flow changes. By redefining the threshold between the near-field, transitioning and far-field regions and accounting for vortex interaction, we observe the relevant non-dimensional parameters describing the scaling to vary. Additionally, together with cross-flow properties, variation in the orifice dimensions and actuation parameters enable the exact form of the Strouhal number for each combination of constraints to be determined. Under conditions of no vortex interaction, synthetic jets are found to follow identical trajectories provided the ratio of total momentum between the jet and the cross-flow remains the same, irrespective of the actuation frequency or Reynolds number of the incoming flow.

Free-falling objects impacting onto water pools experience a very high initial impact force, greatest at the moment when breaking through the free surface. Many have intuitively wondered whether throwing another object in front of an important object (like oneself) before impacting the water surface may reduce this high impact force. Here, we test this idea experimentally by allowing two spheres to consecutively enter the water and measuring the forces on the trailing sphere. We find that the impact acceleration reduction on the trailing sphere depends on the dynamics of the cavity created by the first sphere and the relative timing of the second sphere impact. These combined effects are captured by the non-dimensional ‘Matryoshka’ number, which classifies the observed phenomena into four major regimes. In three of these regimes, we find that the impact acceleration on the second sphere is reduced by up to 78 % relative to impact on a quiescent water surface. Surprisingly, in one of the regimes the force on the trailing sphere is dramatically increased by more than 400 % in the worst case observed. We explain how the various stages of cavity evolution result in the observed alterations in impact force in this multi-body water entry problem.

Two mean-field models for polyelectrolytes in simultaneous electric field and pressure-driven flow field were developed and compared. The models predict migration perpendicular to the anti-parallel or parallel fields, where the migration is caused by electrohydrodynamic interactions calculated using either a short- or long-range approximation. Inputs for the mean-field models were determined from Brownian dynamics simulations in a simple shear flow. Both models qualitatively reproduce experimental observations of DNA focusing as reported in previous publications. Specifically, it is observed that combination of the shear and electric fields leads to polyelectrolyte motion in the direction transverse to the flow and electric field direction, which in turn leads to concentration of the polyelectrolyte in the centre of a microfluidic channel. Furthermore, both models predict that there is an optimal strength of electric field that leads to the narrowest distribution profile of the polyelectrolyte in the centre of the channel. The analysis suggests that this is due to dispersion induced by the electrohydrodynamic interactions. However, quantitative disagreement between the model predictions and the experimental data indicates that further progress in the model development is needed.

The evolution of the boundary layer vortex sheet on a rotating and translating accelerating circular cylinder at Reynolds numbers of 10 000 and 20 000 is investigated using planar particle image velocimetry. The vortex sheet is decomposed into contributions resulting from translation and rotation as well as from local and far-field vorticity. Their individual development is explored to understand the overall time history of the boundary layer as well as its evolution at the unsteady separation point. The boundary layer vortex sheet distribution changes considerably throughout the motion as well as between different kinematic cases. The same is observed for the vortex sheet strength at the unsteady separation point. A non-dimensional parameter is proposed which removes the effect of rotation rate, instantaneous velocity and shed vorticity accumulating in the far field. It was found that this was successful at collapsing the vortex sheet strength at the unsteady separation point during cylinder motion as well as for the individual kinematic test cases investigated. This confirms that cylinder kinematics and far-field vorticity are driving factors contributing to the development of the unsteady boundary layer and its strength at the separation point.

Elastohydrodynamic lubrication, or simply soft lubrication, refers to the motion of deformable objects near a boundary lubricated by a fluid, and is one of the key physical mechanisms to minimise friction and wear in natural and engineered systems. Hence, it is of particular interest to relate the thickness of the lubricant layer to the entrainment (sliding/rolling) velocity, the mechanical loading exerted onto the contacting elements and the properties of the elastic boundary. In this work, we provide an overview of the various regimes of soft lubrication for two-dimensional cylinders in lubricated contact with compliant walls. We discuss the limits of small and large entrainment velocity, which are equivalent to large and small elastic deformations, as the cylinder moves near thick or thin elastic layers. The analysis focusses on thin elastic coatings, both compressible and incompressible, for which analytical scaling laws are not yet available in the regime of large deformations. By analysing the elastohydrodynamic boundary layers that appear at the edge of the contact, we establish the missing scaling laws – including prefactors. As such, we offer a rather complete overview of the physically relevant limits of soft lubrication.

Stereoscopic particle image velocimetry measurements of open-channel flows over streamwise-orientated triangular-shaped ridges were used to explore interactions between ridge-induced secondary currents (SCs) and turbulence. Terms in the double-averaged (in space and in time) momentum and energy conservation equations were analysed for a range of ridge spacings ($s$) between 0.4 and 4.0 flow depths ($H$). The double-averaged equations neatly partition momentum and energy fluxes into turbulence and SC contributions, making them well suited to this study. The obtained data indicate that for a range of $s/H$ between 0.4 and 2.0, the normalised momentum and energy fluxes due to SCs approximately collapse when plotted as functions of $(z-d)/s$, where $z$ is the vertical coordinate and $d$ a constant that aligns the elevations of SC cell centres. The SCs controlled the shape of the mean velocity distribution with the vertical gradient of the double-averaged streamwise velocity found to be inversely proportional to $s$ near the elevations of SC cell centres. Partitioning the total kinetic energy into double-mean (DMKE), dispersive (DKE) and turbulent (SATKE) components and considering the balance equation for each component indicated that at the elevation of SC cell centres the production rate of SATKE via exchange with DMKE was comparable in magnitude to the production rate via exchange with DKE (due to SCs). For all ridge spacings, SATKE was reduced compared to a no-ridge benchmark case due to suppression of very-large-scale turbulent motions by the SCs. Finally, it is demonstrated that energy is supplied to SCs by turbulence.

Geometric focusing of monochromatic internal waves is a well-known linear mechanism that provides a pathway to smaller length scales and hence energy dissipation, especially when repeated reflection from inclined boundaries focuses these waves onto internal wave attractors (IWAs). While simple attractors have been well documented both theoretically and in idealized laboratory settings, the small aspect ratio and complex shape of the ocean has made it difficult to detect the narrow frequency bands where attractors may form. The greatest obstruction however is the restricted spatiotemporal resolution of measurements of the ocean's internal wave field. Moored instruments provide a detailed temporal but poor spatial resolution, except along line-segments spanning part of the water column only. The ability to measure a fully resolved two-dimensional section through a three-dimensional field, as in present-day laboratory experiments, is missing in the ocean. This prohibits the search for IWAs in ocean data, especially if this concerns a multi-frequency wave field. Mimicking an ocean observation set-up, our experimental study aims at detecting IWA signatures from laboratory observations along a vertical line-segment. We perform small and large amplitude forcing experiments at a fixed wavenumber with both monochromatic and broadband frequency components. We examine energy intensification at dissipative length scales at frequencies known to support attractors. Importantly, our laboratory experiments indicate that even in the presence of multiple forcing frequencies, or for large-amplitude monochromatic forcing (where beams become unstable to Triadic Resonant Instability), focusing reflections due to attractors are the predominant mechanism for transferring energy to dissipative scales.

We propose a model and numerical method for the propulsion of rectangular and rhombic lattices of flapping plates at $O$(10–100) Reynolds numbers in incompressible flow. The numerical method uses an adaptive mesh to mitigate singularities at the plates’ edges. We establish convergence rates and find good numerical accuracy in a test problem: Laplace's equation in the region exterior to a plate. We then use the method to establish benchmark results for a single flapping plate, including vortex wake characteristics and Froude efficiency over ranges of flapping amplitude, frequency and Strouhal number. As a prelude to a study of propulsive efficiency in Part 2 (J. Fluid Mech., vol. 915, 2021, A21), we study a key ingredient: the time-averaged input power in lattices of plates. Scaling laws for the mean input power are estimated in the limits of small and large streamwise spacings, using steady flow models with small-gap and Poiseuille-like flows between the plates respectively in the two limits. For both lattice types, the mean input power saturates as the lateral spacing becomes large (and thrust occurs). At small lateral spacings, the rhombic lattices’ input power becomes much larger when the plates overlap. The time-averaged input power in flapping lattices agrees qualitatively with the steady models.

We present a numerical study of spatially quasi-periodic travelling waves on the surface of an ideal fluid of infinite depth. This is a generalization of the classic Wilton ripple problem to the case when the ratio of wavenumbers satisfying the dispersion relation is irrational. We propose a conformal mapping formulation of the water wave equations that employs a quasi-periodic variant of the Hilbert transform to compute the normal velocity of the fluid from its velocity potential on the free surface. We develop a Fourier pseudo-spectral discretization of the travelling water wave equations in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on the torus. This leads to an overdetermined nonlinear least-squares problem that we solve using a variant of the Levenberg–Marquardt method. We investigate various properties of quasi-periodic travelling waves, including Fourier resonances, time evolution in conformal space on the torus, asymmetric wave crests, capillary wave patterns that change from one gravity wave trough to the next without repeating and the dependence of wave speed and surface tension on the amplitude parameters that describe a two-parameter family of waves.

We study liquid migration in partly saturated shear bands of granular media where liquid is transported away from the shear-band centre. Earlier studies show that the liquid migration can be modelled as a diffusive process with a shear-rate-dependent diffusion coefficient. Here, we apply this model in a two-dimensional Cartesian split-bottom shear cell with one wide, steady shear band. Initially, a high liquid concentration peak develops at the edges of the shear band, which propagates away from the shear band, splitting the shear cell into a liquid-depleted shear-band region and an outer region not yet affected by the liquid migration. Assuming the liquid transport in the vertical direction is negligible, we simplify the liquid migration model to a one-dimensional form evolving over time. By coordinate transformation, an analytically solvable drift-diffusion model is obtained for liquid migration from the simplified model. From here, we obtain analytical solutions for the liquid concentration as a function of space and time. The significance of the mechanisms is studied in terms of the local Péclet number. While drift enhances drying of the shear band and accumulates the liquid in the peak, diffusion shifts the liquid further away from the shear band. To validate the model, we predict numerically the trajectory of the liquid concentration peak from the continuum model and compare with discrete particle method (DPM) simulations. Our continuum model results give a perfect qualitative and an approximate quantitative agreement with the overall results predicted from the DPM model.

We study propulsion of rectangular and rhombic lattices of flapping plates at $O$(10–100) Reynolds numbers in incompressible flow. The fluid dynamics often converges to time periodic in 5–30 flapping periods, facilitating accurate computations of time-averaged thrust force and input power. We classify the propulsive performances of the lattices and the periodicities of the flows with respect to flapping amplitude and frequency, horizontal and vertical spacings between plates, and oncoming flow velocity. Non-periodic states are most common at small streamwise spacing, large lateral spacing and large Reynolds number. Lattices that are closely spaced in the streamwise direction produce intense vortex dipoles between adjacent plates. The flows transition sharply from drag- to thrust-producing as these dipoles switch from upstream to downstream orientations at critical flow speeds. Near these transitions the flows pass through a variety of periodic and non-periodic states, with and without up–down symmetry, and multiple stable self-propelled speeds can occur. As the streamwise spacing increases (and with large lateral spacing), the plates may shed vortex streets that impinge on downstream neighbours. The most efficient streamwise spacing increases with flapping amplitude. With small lateral spacing, the rectangular lattices have Poiseuille-type flows that yield net drag, while the rhombic lattices may shed vortices and generate net thrust, sometimes with relatively high efficiency. As lateral spacing increases to one plate length and beyond, the rectangular lattices begin to shed vortices and generate thrust, eventually with efficiencies similar to the rhombic lattices', as the two types of flows converge. At $Re = 70$, the lattices’ maximum Froude efficiencies are approximately twice those of an isolated plate (only considering nearly periodic lattice flows). As $Re$ decreases, the lattices’ efficiency advantage increases further.

Selective heating of a microparticle surface had been observed to cause its autonomous movement in a fluid medium due to self-generated temperature gradients. Here, we theoretically investigate the response of such an auto-thermophoretic particle near an isothermal planar wall. We derive an exact solution of the energy equation and employ the Lorentz reciprocal theorem to obtain the translational and rotational swimming velocities in the creeping-flow limit. We report fixed points for vertical movement of the micromotor for its specific orientations relative to the wall. The critical wall gap for fixed points shows unique non-monotonic dependence on the metallic coating coverage on the particle. Also, the micromotor trajectories can be switched either from wall-bound sliding or stationary state to escape from the near-wall zone by tuning the particle and the surrounding fluid pair thermal conductivity contrast. The scenario holds several exclusive distinguishing features from the otherwise extensively studied self-diffusiophoresis phenomenon near an inert wall, despite obvious analogies in the respective constitutive laws relating the fluxes with the gradients of the concerned forcing parameters. The most contrasting locomotion is the ability of a self-thermophoretic micromotor with a large heated cap to migrate towards the wall even if it is initially directed away from the wall. During the stationary states of swimming, the cold portion on the micromotor surface faces away from the wall under all conditions. Such unique aspects hold the potential of being harnessed in practice towards achieving intricate control over the autonomous motion of microparticles in thermally regulated fluidic environments.

It is known that the pitchfork bifurcation of Kelvin–Helmholtz instability occurring at minimum gradient Richardson number $Ri_m \simeq 1/4$ in viscous stratified shear flows can be subcritical or supercritical depending on the value of the Prandtl number, $Pr$. Here, we study stratified shear flow restricted to two dimensions at finite Reynolds number, continuously forced to have a constant background density gradient and a hyperbolic tangent shear profile, corresponding to the ‘Drazin model’ base flow. Bifurcation diagrams are produced for fluids with $Pr=0.7$ (typical for air), 3 and $7$ (typical for water). For $Pr=3$ and $7$, steady billow-like solutions are found to exist for strongly stable stratification of $Ri_m$ beyond $1/2$. Interestingly, these solutions are not a direct product of a Kelvin–Helmholtz instability, having half the wavelength of the linear instability, and arising through a superharmonic bifurcation. These short-wavelength states can be tracked down to at least $Pr \approx 2.3$ and act as instigators of complex dynamics, even in strongly stratified flows. Direct numerical simulations of forced and unforced two-dimensional flows are performed, which support the results of the bifurcation analyses. Perturbations are observed to grow approximately exponentially from random initial conditions where no modal instability is predicted by a linear stability analysis.

This paper analyses the turbulent energy cascade from the perspective of statistical mechanics, and relates interscale energy fluxes to statistical irreversibility and information entropy production. The microscopical reversibility of the energy cascade is tested by constructing a reversible three-dimensional turbulent system using a dynamic model for the sub-grid stresses. This system, when reversed in time, develops a sustained inverse cascade towards the large scales, evidencing that the characterisation of the inertial energy cascade must consider the possibility of an inverse regime. This experiment is used to study the origin of statistical irreversibility and the prevalence of direct over inverse energy cascades in isotropic turbulence. Statistical irreversibility, a property of statistical ensembles in phase space related to entropy production, is connected to the dynamics of the energy cascade in physical space by considering the space locality of the energy fluxes and their relation to the local structure of the flow. A mechanism to explain the probabilistic prevalence of direct energy transfer is proposed based on the dynamics of the rate-of-strain tensor, which is identified as the most important source of statistical irreversibility in the energy cascade.

An informal introduction is provided to a range of topics in fluid dynamics having a topological character. These topics include flows with boundary singularities, Lagrangian chaos, frozen-in fields, magnetohydrodynamic analogies, fast- and slow-dynamo mechanisms, magnetic relaxation, minimum-energy states, knotted flux tubes, vortex reconnection and the finite-time singularity problem. The paper concludes with a number of open questions concerning the above topics.

The description of the feedback mechanism is one of the crucial points in the understanding of self-sustained flow oscillations, such as the edge tone or the cavity tone. In this paper a new model for the edge tone, using linear adjoint modes is proposed for low speeds, when acoustic effects are negligible. The two crucial points of the model are (i) a new relationship between the jet oscillation and the generated vortex at the edge tip and (ii) a new model for the effect of the generated vortex on the oscillation. The frequency corresponding to the optimum open-loop gain of the feedback mechanism was compared with experimental data and good agreement was found.

We introduce a network (graph) theoretic community-based framework to extract vortical structures that serve the role of connectors for the vortical interactions in two- and three-dimensional isotropic turbulence. The present framework represents the vortical interactions on a network, where the vortical elements are viewed as the nodes and the vortical interactions are regarded as edges weighted by induced velocity. We identify closely interacting vortical elements as vortical network communities through community detection algorithms. The inter- and intra-community interactions are used to identify the communities which have the strongest and weakest interactions amongst them. These vortical communities are referred to as the connector and peripheral communities, respectively. We demonstrate the influence of the network-based structures to modify the dynamics of a collection of discrete point vortices. Taking advantage of the strong inter-community interactions, connector community can significantly modify the collective dynamics of vortices through the application of multiple impulse perturbations. We then apply the community-based framework to extract influential structures in isotropic turbulence. The connector and peripheral communities extracted from turbulent flows resemble shear-layer and vortex-core-like structures, respectively. The influence of the connector structures on the flow field and their neighbouring vortical structures is analysed by adding impulse perturbations to the connectors in direct numerical simulations. The findings are compared with the cases of perturbing the strongest vortex tube and shear-layer regions. We reveal that perturbing the connector structures enhances local turbulent mixing beyond what is achieved by the other cases.