Families of exact coherent states in elliptical pipe flow obtained from the travelling-wave solutions in circular pipe flow by a continuation approach are found. Results are given in a regime of the aspect ratio $A$ in which the laminar flow is linearly stable. The results suggest the possibility of two distinct classes of solutions of elliptical travelling waves at higher values of $A$: (i) rotationally symmetric centre-mode states that collapse towards the pipe centre and (ii) rotationally asymmetric vortex–wave interaction states with additional mirror symmetry exhibiting organization of the waves around a critical layer. These are the first calculations of three-dimensional travelling waves in elliptical pipes. Investigation of these states has the potential to provide fresh insight into the relationship between exact coherent structures in Poiseuille flow in pipes and channels.

We study the effect of random perturbations of body positions in large uniformly spaced arrays of axisymmetric wave energy converters (WECs). We perform systematic computational simulations of ensembles of randomized array configurations that are obtained by introducing zero-mean position perturbations (characterized by randomness parameter $\varepsilon$) to line arrays of uniform spacing $d$. Of special interest are the conditions under which these randomized arrays can extract more energy than the underlying uniform arrays. We find that random WEC arrays achieve substantial energy extraction gains over the same number of isolated bodies in monochromatic and irregular incident seas. Introducing $\varepsilon >0$ acts to smooth out the uneven performance of uniformly spaced arrays over varying incident wavenumber. In the low-scattering regime, the standard deviation of array gain grows with the square of wavenumber, for which we provide a theoretical explanation. We show that the uniform line array with spacing optimized for a given incident spectrum generally outperforms randomized arrays of any mean $d$ and $\varepsilon$ in that spectrum, and we offer a heuristic explanation. This holds for a wide range of incident spectra.

Rotations of spheroidal particles immersed in turbulent flows reflect the combined effects of fluid strain and vorticity, as well as the time history of these quantities along the particle's trajectory. Conversely, particle rotation statistics in turbulence provide a way to characterise the Lagrangian properties of velocity gradients. Particle rotations are also important for a range of environmental and industrial processes where particles of various shapes and sizes are immersed in a turbulent flow. In this study, we investigate the rotations of inertialess spheroidal particles that follow Lagrangian fluid trajectories. We perform direct numerical simulations (DNS) of homogeneous isotropic turbulence and investigate the dynamics of different particle shapes at different scales in turbulence using a filtering approach. We find that the mean-square particle angular velocity is nearly independent of particle shape across all scales from the Kolmogorov scale to the integral scale. The particle shape does determine the relative split between different modes of rotation (spinning vs tumbling), but this split is also almost independent of the filter scale suggesting a Lagrangian scale-invariance in velocity gradients. We show how the split between spinning and tumbling can be quantitatively related to the particle's alignment with respect to the fluid vorticity.

We study the convective and absolute forms of azimuthal magnetorotational instability (AMRI) in a cylindrical Taylor–Couette (TC) flow with an imposed azimuthal magnetic field. We show that the domain of the convective AMRI is wider than that of the absolute AMRI. Actually, it is the absolute instability which is the most relevant and important for magnetic TC flow experiments. The absolute AMRI, unlike the convective one, stays in the device, displaying a sustained growth that can be experimentally detected. We also study the global AMRI in a TC flow of finite height using direct numerical simulation and find that its emerging butterfly-type structure – a spatio-temporal variation in the form of axially upward and downward travelling waves – is in a very good agreement with the linear analysis, which indicates the presence of two dominant absolute AMRI modes in the flow giving rise to this global butterfly pattern.

The ability of microswimmers to deploy optimal propulsion strategies is of paramount importance for their locomotory performance and survival at low Reynolds numbers. Although for perfectly spherical swimmers minimum dissipation requires a neutral-type swimming, any departure from the spherical shape may lead the swimmer to adopt a new propulsion strategy, namely those of puller- or pusher-type swimming. In this study, by using the minimum dissipation theorem for microswimmers, we determine the flow field of an optimal nearly spherical swimmer, and show that indeed depending on the shape profile, the optimal swimmer can be a puller, pusher or neutral. Using an asymptotic approach, we find that amongst all the modes of the shape function, only the third mode determines, to leading order, the swimming type of the optimal swimmer.

Shedding of vortices can be observed in the wake of a fixed cylinder at Reynolds numbers larger than $Re=47$. This might give the impression that a vortex-induced vibration (VIV), which occurs when the frequency of vortex shedding in the wake of a flexibly mounted cylinder synchronizes with the natural frequency of the structure, could be observed only at Reynolds numbers larger than $Re=47$. Recent numerical simulations and theoretical work, however, have shown that it is possible to observe VIV at subcritical Reynolds numbers, i.e. Reynolds numbers smaller than $Re=47$. In these studies, a VIV has been observed numerically at Reynolds numbers as low as $Re=22$. In the present work, the first experimental evidence of VIV at subcritical Reynolds number is presented. We have designed and built an experimental set-up that makes it possible to conduct VIV experiments at subcritical Reynolds numbers, and at a constant Reynolds number over the entire lock-in range (i.e. the range for which oscillations are observed). Using this experimental set-up, we have confirmed experimentally that VIV can indeed be observed at subcritical Reynolds numbers, by observing VIV at Reynolds numbers as low as $Re=19$. We have observed subcritical VIV both when the Reynolds number stays constant over the entire lock-in range, and when the Reynolds number increases with increasing reduced velocity, while staying within the subcritical range.

We study experimentally the primary instability of a visco-plastic film flow down an inclined plane. The experimental set-up is a channel with a varying slope angle, in which a permanent flow of a Herschel–Bulkley fluid (carbopol or kaolin) is established. Controlled perturbations are imposed at the entrance of the channel to generate surface waves, and their downstream evolution is observed with a laser sensor system measuring the local fluid thickness. Growth rates and cutoff frequencies are obtained after processing the thickness signal, and experimental critical Reynolds and Bingham numbers are deduced. We find that the experimental stability map obtained is well described by the pseudo-plug model of Balmforth & Liu (J. Fluid Mech., vol, 519, 2004, pp. 33–54), a model obtained after neglecting the film thickness compared with its length. This is not the case for dispersion effects (growth rates, cutoff frequencies, phase speeds), for which a more accurate model is needed.

We prove a new rigorous bound for the mean convective heat transport $\langle w T \rangle$, where $w$ and $T$ are the non-dimensional vertical velocity and temperature, in internally heated convection between an insulating lower boundary and an upper boundary with a fixed heat flux. The quantity $\langle wT \rangle$ is equal to half the ratio of convective to conductive vertical heat transport, and also to $\frac 12$ plus the mean temperature difference between the top and bottom boundaries. An analytical application of the background method based on the construction of a quadratic auxiliary function yields $\langle w T \rangle \leq \frac {1}{2}(\frac {1}{2}+ \frac {1}{\sqrt {3}} ) - 1.6552\, {\textit {R}}^{-(1/3)}$ uniformly in the Prandtl number, where R is the non-dimensional control parameter measuring the strength of the internal heating. Numerical optimisation of the auxiliary function suggests that the asymptotic value of this bound and the $-1/3$ exponent are optimal within our bounding framework. This new result halves the best existing (uniform in $ {\textit {R}}$) bound (Goluskin, Internally Heated Convection and Rayleigh–Bénard Convection, Springer, 2016, table 1.2), and its dependence on $ {\textit {R}}$ is consistent with previous conjectures and heuristic scaling arguments. Contrary to physical intuition, however, it does not rule out a mean heat transport larger than $\frac 12$ at high $ {\textit {R}}$, which corresponds to the top boundary being hotter than the bottom one on average.

This paper addresses peristaltic flow induced in a non-axisymmetric annular tube by a periodic small-amplitude wave of arbitrary shape propagating axially along its inner surface, assumed to be a circular cylinder. The study is motivated by recent in vivo experimental observations pertaining to the flow of cerebrospinal fluid along the perivascular spaces of cerebral arteries. The analysis employs the lubrication approximation, describing low-Reynolds-number peristaltic flow in the long-wavelength approximation. Closed-form analytic expressions are derived for the average pumping rate in infinitely long tubes and also in tubes of finite length. Consideration is also given to the transverse motion arising in non-axisymmetric tubes. For small-amplitude waves, the solution is reduced to the integration of a parameter-free Stokes-flow problem, which is solved for relevant cross-sectional shapes, with closed-form analytical results derived for thin canals.

Oscillatory flows have the potential to overcome long-standing limitations encountered when using steady flows for inertial focusing due to low particle Reynolds numbers. The periodic displacement generated by oscillatory flow increases the total path length travelled by a suspended particle with no net displacement within a channel or the need for increased channel length. The effects of unsteady inertial forces on inertial focusing, however, have not been thoroughly examined. Here, we present a combined theoretical and experimental study on the effect of Womersley number on inertial focusing in planar pulsatile flows. The migration velocity for a small and weakly inertial particle was evaluated for different oscillation frequencies using the method of matched asymptotics. Using experiments in a custom-built microchannel, we show that oscillatory flows are remarkably efficient for inertial focusing, even at low particle Reynolds numbers. For appropriately selected oscillation amplitude and frequency, inertial focusing is achieved in only a fraction of the channel length (1 % to 10 %) compared to what would be required in a steady flow. We show that the Womersley number influences the equilibrium focusing position and the overall focusing efficiency. In fact, above a critical Womersley number, inertial focusing does not occur despite increasing particle Reynolds number. Lastly, the application of oscillatory inertial focusing for the direct measurement of particle migration velocity is demonstrated.

We identify an exceptional point in the numerical stability map of a combustion experiment operated with laminar premixed flames. A low-order model of the experiment allows one to investigate the interplay between the system modes and an exceptional point. The latter is located in the unstable region of the complex eigenvalue plane. Under gradual variation of the operating parameters, the branch-switching characteristic of the exceptional point can facilitate abrupt changes between strong instability and stable operation. The results also indicate that intrinsic thermoacoustic feedback plays an important role in the stability characteristics of this experiment.

Gas volumes with an equivalent spherical diameter greater than that of the tube in which they rise by buoyancy – so-called Taylor bubbles – assume a characteristic bullet shape and ascend with a velocity that is dependent primarily on the tube diameter, and also on the physical properties of the liquid, but, remarkably, is mostly independent of the gas volume. The requirement of liquid volume conservation suggests a plausible explanation of this paradoxical feature in that the space vacated under the rising bubble must be replenished by the liquid film falling along the bubble surface. It is demonstrated by numerical means that, by limiting the bubble diameter, a cylindrical ‘cage’ of thin vertical rods coaxial with the tube permits the flow rate of this film to increase, with the result that the bubble is able to ascend with a significantly higher velocity than in an empty tube with the same diameter.

Owing to the Taylor–Proudman theorem, it is generally believed that rotating flows should have two-dimensional dynamics for rapid background rotation. Yet, we show that two infinitely long counter-rotating vertical vortices always remain unstable to three-dimensional perturbations even for large rotation rate about the vertical axis. The dominant instability consists of quasi-antisymmetric displacements of the two vortices. Its growth rate is independent of the rotation rate when it is sufficiently large, while the most amplified vertical wavelength scales like the rotation rate. Direct numerical simulations show that the instability leads ultimately to a full three-dimensional breakdown of the vortex pair. In other words, a two-dimensional vortex pair will spontaneously develop three-dimensional variations even in the limit of infinite rotation rate but the wavelength will tend to infinity. This implies that the Taylor–Proudman theorem can be strictly valid only in vertically bounded flows. The scaling for the wavelength is next generalized by showing that the typical vertical scale in rapidly rotating unbounded flows is $L_v \sim L_h/Ro$, where $L_h$ is the horizontal scale and $Ro$ the Rossby number. This scaling law is shown to derive from a self-similarity of the Navier–Stokes equations for $Ro \ll 1$. The resulting reduced equations are identical to those obtained first by Julien et al. (Theor. Comput. Fluid Dyn., vol. 11, issue 3–4, 1998, pp. 251–261) and Nazarenko & Schekochihin (J. Fluid Mech., vol. 677, 2011, pp. 134–153) by means of multiscale analyses in the cases of rotating convection or turbulence. The self-similarity demonstrated herein suggests that these reduced equations are valid for any rapidly rotating unbounded flow.

Tsunami waves induced by landslides are a threat to human activities and safety along coastal areas. In this paper, we characterize experimentally the waves generated by the gravity-driven collapse of a dry granular column into water. Three nonlinear wave regimes are identified depending on the Froude number ${Fr}_f$ based on the ratio of the velocity of the advancing granular front and the velocity of linear gravity waves in shallow water: transient bores for large ${Fr}_f$, solitary waves for intermediate values of ${Fr}_f$, and nonlinear transition waves at small ${Fr}_f$. The wave amplitude relative to the water depth increases with ${Fr}_f$ in the three regimes but with different nonlinear scalings, and the relative wavelength is an increasing or decreasing function of ${Fr}_f$ depending on the wave regime. Two of these wave regimes are rationalized by considering that the advancing granular front acts as a vertical piston pushing the water, while the last one is found to be a transition from shallow- to deep-water conditions. The present modelling contributes to a better understanding of the rich hydrodynamics of the generated waves, with coastal risk assessment as practical applications.

Abrupt depth transitions (ADTs) have recently been identified as potential causes of ‘rogue’ ocean waves. When stationary and (close-to-) normally distributed waves travel into shallower water over an ADT, distinct spatially localized peaks in the probability of extreme waves occur. These peaks have been predicted numerically, observed experimentally, but not explained theoretically. Providing this theoretical explanation using a leading-order-physics-based statistical model, we show, by comparing to new experiments and numerical simulations, that the peaks arise from the interaction between linear free and second-order bound waves, also present in the absence of the ADT, and new second-order free waves generated due to the ADT.

An analytical solution of the impulsive impact of a cylindrical body submerged below a calm water surface is obtained by solving a free boundary problem. The shape of the cross-section of the body is arbitrary. The integral hodograph method is applied to derive the complex velocity potential defined in a parameter plane. The boundary-value problem is reduced to a Fredholm integral equation of the first kind in the velocity magnitude on the free surface. The velocity field, the impulsive pressure on the body surface and the added mass are determined in a wide range of depths of submergence for various cross-sectional shapes, such as a flat plate, a circular cylinder and a rectangle.

Turbulent convection plays a crucial role in many natural environments and engineering applications. One of the most fundamental questions is how the heat flux depends on the thermal driving and fluid property. It has been proposed that when the fluid layer experiences extremely strong thermal driving, the boundary layers will become fully turbulent and the so-called ultimate regime will emerge. In this work we proposed a numerical experiment in which the thermal boundary layer can be spatially decoupled with the viscous one. We demonstrate that, once the thermal boundary layer is fully decoupled from the viscous boundary layer and locates entirely inside the turbulent bulk, the scaling laws corresponding to the ultimate regime can be obtained, namely $Nu \sim Ra^{1/2}Pr^{1/2}$ and $Re \sim Ra^{1/2} Pr^{-1/2}$ with $Nu$ being the Nusselt number, $Re$ the Reynolds number, $Ra$ the Rayleigh number and $Pr$ the Prandtl number, respectively. Therefore, our results support the physical conjecture of the ultimate regime for the turbulent convection.

The effect of a counter-current gas flow on the linear stability of an inclined falling liquid film switches from destabilizing to stabilizing, as the flow confinement is increased. We confront this linear effect with the response of nonlinear surface waves resulting from long-wave interfacial instability. For the strongest confinement studied, the gas flow damps both the linear growth rate and the amplitude of nonlinear travelling waves, and this holds for waves of the most-amplified frequency and for low-frequency solitary waves. In the latter case, waves are shaped into elongated humps with a flat top that resist secondary instabilities. For intermediate confinement, the linear and nonlinear responses are opposed and can be non-monotonic. The linear growth rate of the most-amplified waves first decreases and then increases as the gas velocity is increased, whereas their nonlinear amplitude is first amplified and then damped. Conversely, solitary waves are amplified linearly but damped nonlinearly. For the weakest confinement, solitary waves are prone to two secondary instability modes that are not observed in unconfined falling films. The first involves waves of diminishing amplitude slipstreaming towards their growing leading neighbours. The second causes wave splitting events that lead to a train of smaller, shorter waves.

Scaling laws for the thrust production and power consumption of a purely pitching hydrofoil in ground effect are presented. For the first time, ground-effect scaling laws based on physical insights capture the propulsive performance over a wide range of biologically relevant Strouhal numbers, dimensionless amplitudes and dimensionless ground distances. This is achieved by advancing previous scaling laws (Moored & Quinn (AIAA J., 2018, pp. 1–15)) with physics-driven modifications to the added mass and circulatory forces to account for ground distance variations. The key physics introduced are the increase in the added mass of a foil near the ground and the reduction in the influence of a wake-vortex system due to the influence of its image system. The scaling laws are found to be in good agreement with new inviscid simulations and viscous experiments, and can be used to accelerate the design of bio-inspired hydrofoils that oscillate near a ground plane or two out-of-phase foils in a side-by-side arrangement.

A Boussinesq fluid inside a stably thermally stratified square container whose walls are inclined ${45}^{\circ }$ with respect to gravity, with two opposite walls kept at constant temperatures and the other two insulated is nearly isothermal in the regions above and below the horizontal diagonal. The flow is concentrated in the wall boundary layers and a shear layer centred about the horizontal diagonal. The equilibrium is maintained by the balance between dissipation in the shear and boundary layers, the heat fluxes at the constant temperature walls, and the induced flow resulting from the no-flux condition at the inclined insulated walls. The dynamical response of the fluid to vertical oscillations of the container is studied over a range of forcing frequencies. For a small forcing amplitude and below a viscosity-dependent cutoff forcing frequency, this response exhibits a modal cellular structure localized about the shear layer. With increasing forcing amplitude, the response experiences instabilities, studied here numerically at a forcing frequency above the cutoff frequency, that are similar to those encountered in the Faraday wave problem, such as parametric subharmonic instability, triadic resonance and resonant collapse.