The effects of the slope of an ice–seawater interface on the mechanisms and rate of ablation of the ice by natural convection are examined using turbulence-resolving simulations. Solutions are obtained for ice slopes $\unicode[STIX]{x1D703}=2^{\circ }{-}90^{\circ }$, at a fixed ambient salinity and temperature, chosen to represent common Antarctic ocean conditions. For laminar boundary layers the ablation rate decreases with height, whereas in the turbulent regime the ablation rate is found to be height independent. The simulated laminar ablation rates scale with $(\sin \unicode[STIX]{x1D703})^{1/4}$, whereas in the turbulent regime it follows a $(\sin \unicode[STIX]{x1D703})^{2/3}$ scaling, both consistent with the theoretical predictions developed here. The reduction in the ablation rate with shallower slopes arises as a result of the development of stable density stratification beneath the ice face, which reduces turbulent buoyancy fluxes to the ice. The turbulent kinetic energy budget of the flow shows that, for very steep slopes, both buoyancy and shear production are drivers of turbulence, whereas for shallower slopes shear production becomes the dominant mechanism for sustaining turbulence in the convective boundary layer.

Sea ripples are small-scale bedforms which originate from the interaction of an oscillatory flow with an erodible sand bed. The phenomenon of sea ripple formation is investigated by means of direct numerical simulation in which the sediment bed is represented by a large number of fully resolved spherical grains (i.e. the flow around each individual particle is accounted for). Two sets of parameter values (differing in the amplitude and frequency of fluid oscillations, among other quantities) are adopted which are motivated by laboratory experiments on the formation of laminar rolling-grain ripples. The knowledge of the origin of ripples is presently enriched by insights and by providing fluid- and sediment-related quantities that are difficult to obtain in the laboratory (e.g. particle forces, statistics of particle motion, bed shear stress). In particular, detailed analysis of flow and sediment bed evolution has confirmed that ripple wavelength is determined by the action of steady recirculating cells which tend to accumulate sediment grains into ripple crests. The ripple amplitude is observed to grow exponentially, consistent with established linear stability analysis theories. Particles at the bed surface exhibit two kinds of motion depending on their position with respect to the recirculating cells: particles at ripple crests are significantly faster and show larger excursions than those lying in ripple troughs. In analogy with the segregation phenomenon of polydisperse sediments, the non-uniform distribution of the velocity field promotes the formation of ripples. The wider the gap between the excursion of fast and slow particles, the larger the resulting growth rate of the ripples. Finally, it is revealed that, in the absence of turbulence, the sediment flow rate is driven by both the bed shear stress and the wave-induced pressure gradient, the dominance of each depending on the phase of the oscillation period. In phases of maximum bed shear stress, the sediment flow rate correlates more with the Shields number while the pressure gradient tends to drive sediment bed motion during phases of minimum bed shear stress.

A fluid flowing over a granular bed can move its superficial grains, and eventually deform it by erosion and deposition. This coupling generates a beautiful variety of patterns, such as ripples, bars and streamwise streaks. Here, we investigate the latter, sometimes called ‘sand ridges’ or ‘sand ribbons’. We perturb a sediment bed with sinusoidal streaks, the crests of which are aligned with the flow. We find that, when their wavelength is much larger than the flow depth, bedload diffusion brings mobile grains from troughs, where they are more numerous, to crests. Surprisingly, gravity can only counter this destabilising mechanism when sediment transport is intense enough. Relaxing the long-wavelength approximation, we find that the cross-stream diffusion of momentum mitigates the influence of the bed perturbation on the flow, and even reverses it for short wavelengths. Viscosity thus opposes the diffusion of entrained grains to select the most unstable wavelength. This instability might turn single-thread alluvial rivers into braided channels.

Unsteadiness in separated shock–boundary layer interactions have been previously analysed in order to propose a scenario of entrainment–discharge as the origin of unsteadiness. It was assumed that the fluid in the separated zone is entrained by the free shear layer formed at its edge, and that this layer follows the properties of the canonical mixing layer. This last point is addressed by reanalysing the velocity measurements in an oblique shock reflection at a nominal Mach number of 2.3 and for two cases of flow deviation ($8^{\circ }$ and $9.5^{\circ }$). The rate of spatial growth of this layer is evaluated from the spatial growth of the turbulent stress profiles. Moreover, the entrainment velocity at the edge of the layer is determined from the mean velocity profiles. It is shown that the values of turbulent shear stress, spreading rate and entrainment velocity are consistent, and that they follow the classical laws for turbulent transport in compressible shear layers. Moreover, the measurements suggest that the vertical normal stress is sensitive to compressibility, so that the anisotropy of turbulence is affected by high Mach numbers. Dimensional considerations proposed by Brown & Roshko (J. Fluid Mech., vol. 64, 1974, 775–781) are reformulated to explain this observed trend.

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

Steady and unsteady linearised flow past a submerged source are studied in the small-surface-tension limit, in the absence of gravitational effects. The free-surface capillary waves generated are exponentially small in the surface tension, and are determined using the theory of exponential asymptotics. In the steady problem, capillary waves are found to extend upstream from the source, switching on across curves on the free surface known as Stokes lines. Asymptotic predictions are compared with computational solutions for the position of the free surface. In the unsteady problem, transient effects cause the solution to display more complicated asymptotic behaviour, such as higher-order Stokes lines. The theory of exponential asymptotics is applied to show how the capillary waves evolve over time, and eventually tend to the steady solution.

We study the evolution of three-dimensional (3-D), small-scale, small-amplitude perturbations on a plane internal gravity wave using the local stability approach. The plane internal wave is characterised by its non-dimensional amplitude, $A$, and the angle the group velocity vector makes with gravity, $\unicode[STIX]{x1D6F7}$. For a given $(A,\unicode[STIX]{x1D6F7})$, the local stability equations are solved on the periodic fluid particle trajectories to obtain growth rates for all two-dimensional (2-D) and 3-D perturbation wave vectors. For small $A$, the local stability approach recovers previous results of 2-D parametric subharmonic instability (PSI) while offering new insights into 3-D PSI. Higher-order triadic resonances, and associated deviations from them, are also observed at small $A$. Moreover, for small $A$, purely transverse instabilities resulting from parametric resonance are shown to occur at select values of $\unicode[STIX]{x1D6F7}$. The possibility of a non-resonant instability mechanism for transverse perturbations at finite $A$ allows us to derive a heuristic, modified gravitational instability criterion. We then study the extension of small $A$ to finite $A$ internal wave instabilities, where we recover and build upon existing knowledge of small-scale, small-amplitude internal wave instabilities. Four distinct regions of the $(A,\unicode[STIX]{x1D6F7})$-plane based on the dominant instability modes are identified: 2-D PSI, 3-D oblique, quasi-2-D shear-aligned, and 3-D transverse. Our study demonstrates the local stability approach as a physically insightful and computationally efficient tool, with potentially broad utility for studies that are based on other theoretical approaches and numerical simulations of small-scale instabilities of internal waves in various settings.

The flow of a viscous gravity current over a lubricating layer of fluid is modelled using lubrication theory. We study the case of an axisymmetric current with constant influx which allows for a similarity solution, which depends on three parameters: a non-dimensional influx rate ${\mathcal{Q}}$; a viscosity ratio $m$ between the lower and upper layer fluid; and a relative density difference $\unicode[STIX]{x1D700}$. The limit of equal densities $\unicode[STIX]{x1D700}=0$ is singular, as the interfacial evolution equation changes nature from parabolic to hyperbolic. Theoretical analysis of this limit reveals that a discontinuity, or shock, in the interfacial height forms above a critical viscosity ratio $m_{crit}=3/2$, i.e. for a sufficiently less viscous upper-layer fluid. The physical mechanism for shock formation is described, which is based on advective steepening of the interface between the two fluids and relies on the lack of a contribution to the pressure gradient from the interfacial slope for equal-density fluids. In the limit of small but non-zero density differences, local travelling-wave solutions are found which regularise the singular structure of a potential shock and lead to a constraint on the possible shock heights in the form of an Oleinik entropy condition. Calculation of a simplified time-dependent system reveals the appropriate boundary conditions for the late-time similarity solution, which includes a shock at the nose of the current for $m>3/2$. The numerically calculated similarity solutions compare well to experimental measurements with respect to the predictions of self-similarity, the radial extent and the self-similar top-surface shapes of the current.

In the present study, we experimentally investigate water surface deformation due to the impact of rounded cylindrical projectiles with different aspect ratios (1.0–8.0). The subsequent jet and splash formation is closely related to the dynamics of an underwater cavity. To control the cavity formation, two kinds of surface conditions (smooth and rough) are applied to the front parts of the projectiles, and two impact speeds are considered. The Froude, Reynolds and Weber numbers are in the ranges of 32–90, $5\times 10^{4}{-}8.4\times 10^{4}$ and 1700–5000, respectively. When the front is smooth, the water film rises up along the body surface immediately after impact, and the temporal variation of its height is analytically estimated. The film converges at the rear pole to create an apex jet at lower aspect ratios and simply rises up and falls with the body at higher aspect ratios. The jets could be further distinguished as thin and thick jets, whose breakdown is found to be a function of the viscous force and surface tension, i.e. the Ohnesorge number. On the other hand, when the front is rough, the water film cannot rise up along the body surface, and instead early separation occurs to make the splash above a free surface. The splash size is quantified to assess the effects of the aspect ratio and impact speed. Upon splash formation, a cavity is created under the free surface, which emanates from the nose of the projectile. As the body sinks, the cavity pinch-off occurs due to the imbalance between the hydrostatic pressure and air pressure inside the cavity. At higher aspect ratios, cavity pinch-off occurs on the side wall of the projectile and leaves a portion of the cavity bubble on it. When the surface is smooth, no underwater cavity forms. Finally, we compare the hydrodynamic force acting on the sinking bodies with and without cavity formation, based on the underwater trajectory of each projectile. It is found that the underwater cavity reduces the drag force on the sinking body when it fully encapsulates the body; however, if the air bubbles are partially attached to the body after pinch-off, they tend to detach irregularly or impose additional drag on the body.

Steady detonation in multi-dimensional flow is controlled by the chemical energy release that occurs in a subsonic elliptic flow region known as the detonation driving zone (DDZ). It is the region encompassing the detonation shock and sonic flow locus (in the frame of the detonation shock). A detonation that is strongly confined by material surrounding the explosive has the shock and sonic locus separated at the material interface. Information about the material boundary is traditionally believed to influence the DDZ structure via the subsonic flow on the boundary ahead of the sonic locus. A detonation that is weakly confined has the detonation shock and sonic locus intersecting at the material boundary. The sonic nature of the flow at the intersection point on the boundary is believed to isolate the DDZ structure from the material properties of the confinement. In this study, we examine the paths of characteristics propagating information about the confinement through the supersonic hyperbolic flow region that exists beyond the sonic locus, and determine whether these paths may impinge on the sonic locus and consequently influence the DDZ structure. Our configuration consists of a solid wall boundary deflected through a specified angle on detonation shock arrival, so that the streamline turning angle of the wall at the explosive edge is unambiguously defined. By varying the wall deflection angle from small through large values, we can systematically capture the evolution of the DDZ structure and the characteristic flow regions that influence its structure for strongly to weakly confined detonations. In all strong and weak confinement cases examined, we find that a subset of characteristics from the supersonic flow regions always impinge on the sonic locus. Limiting characteristics are identified that define the boundary between characteristics that impinge on the sonic surface and those that propagate information downstream of the sonic surface. In combination with an oblique-shock polar analysis, we show that the effects on the DDZ of characteristic impingement can be significant.

The present study investigates the puffing behaviour of planar buoyant plumes by employing linear BiGlobal stability analysis and experiments. The BiGlobal instability characteristics of two-dimensional plumes have been explored using stability analysis and compared with the puffing behaviour of both rectangular plumes and square plumes obtained from experiments. In the parameter space investigated, which spans a Richardson number range $0.03<Ri<960$, instability analysis reveals that planar plumes exhibit BiGlobal instability only for varicose perturbations, while they remain stable for sinuous perturbations. The BiGlobal frequency and growth rates of the unstable varicose mode are used to obtain Strouhal number correlation and stability curves. An investigation into the effect of the spanwise wavenumber on BiGlobal instability indicates that planar plumes are more unstable to two-dimensional perturbations than to three-dimensional perturbations. An increase in the spanwise wavenumber tends to stabilize planar plumes without affecting their oscillation frequencies. Experiments suggest that the puffing frequencies in rectangular plumes closely follow the power law obtained from two-dimensional instability analysis while exhibiting a weaker dependence on inlet aspect ratio. To further explore the effect of aspect ratio on puffing behaviour, experiments have been carried out in plumes of aspect ratio 1, i.e. square plumes. Square plumes are found to be more stable and to exhibit higher puffing frequencies than rectangular plumes. The reasons for these differences in puffing dynamics between rectangular and square plumes have been explored from the phase-locked streamwise and spanwise flow visualizations. In addition to puffing, spanwise visualizations in both rectangular and square plumes show the presence of secondary flows at their corners, similar to their constant-density jet counterparts. Finally, from experiments, we deduced a new universal puffing frequency correlation with the hydraulic diameter as the length scale which eliminates the aspect ratio dependence, and is valid for both square and low-aspect-ratio rectangular plumes.

Recent experimental observations (Kühnen et al., Nat. Phys., vol. 14, 2018b, pp. 386–390) have shown that flattening a turbulent streamwise velocity profile in pipe flow destabilises the turbulence so that the flow relaminarises. We show that a similar phenomenon exists for laminar pipe flow profiles in the sense that the nonlinear stability of the laminar state is enhanced as the profile becomes more flattened. The flattening of the laminar base profile is produced by an artificial localised body force designed to mimic an obstacle used in the experiments of Kühnen et al. (Flow Turbul. Combust., vol. 100, 2018a, pp. 919–943) and the nonlinear stability measured by the size of the energy of the initial perturbations needed to trigger transition. Significant drag reduction is also observed for the turbulent flow when triggered by sufficiently large disturbances. In order to make the nonlinear stability computations more efficient, we examine how indicative the minimal seed – the disturbance of smallest energy for transition – is in measuring transition thresholds. We first show that the minimal seed is relatively robust to base profile changes and spectral filtering. We then compare the (unforced) transition behaviour of the minimal seed with several forms of randomised initial conditions in the range of Reynolds numbers $Re=2400$–$10\,000$ and find that the energy of the minimal seed after the Orr and oblique phases of its evolution is close to that of a critical localised random disturbance. In this sense, the minimal seed at the end of the oblique phase can be regarded as a good proxy for typical disturbances (here taken to be the localised random ones) and is thus used as initial condition in the simulations with the body force. The enhanced nonlinear stability and drag reduction predicted in the present study are an encouraging first step in modelling the experiments of Kühnen et al. and should motivate future developments to fully exploit the benefits of this promising direction for flow control.

Considering two-dimensional potential ideal flow with a free surface and finite depth, we study the dynamics of small-amplitude and short-wavelength wavetrains propagating in the background of a steepening nonlinear wave. This can be seen as a model for small ripples developing on the slopes of breaking waves in the surf zone. Using the concept of wave action as an adiabatic invariant, we derive an explicit asymptotic expression for the change of ripple steepness. Through this expression, nonlinear effects are described using the intrinsic frequency and intrinsic gravity along Lagrangian (material) trajectories on a free surface. We show that strong compression near the tip on the wave leads to an explosive ripple instability. This instability may play an important role in the understanding of fragmentation and whitecapping at the surface of breaking waves. Analytical results are confirmed by numerical simulations using a potential theory model.

The problem of determining the threshold of motion of a sediment particle resting on the bed of an open channel has historically been dominated by an approach based on the time–space-averaged bed shear stress (i.e. Shields criterion). Recently, experimental studies have promoted an alternative approach to predict the dislodgement threshold, which is based on the impulse of the flow-induced force. Nonetheless, theoretical analyses accompanying these studies result in complex expressions that fail to provide a direct estimate of said impulse threshold. We employ the work–energy principle to derive a prediction of the fundamental impulse threshold that the destabilising hydrodynamic force must overcome in order to achieve full particle dislodgement. For the bed configuration studied, which is composed of spheres, the proposed expression depends on the mobile particle’s size and mass, and shows excellent agreement with experimental observations previously published. The derivation presented in this paper may thus represent a robust theoretical framework that aids in the reinterpretation of existing data, as well as in the design of future experiments aimed at analysing the importance of hydrodynamic impulse as a criterion for prediction of particle dislodgement.

We study the flow induced by eye rotations in the anterior chamber (AC) of the eye, the region between the cornea and the iris. We model the geometry of the AC as a thin domain sitting on the surface of a sphere, and study both the simpler case of a constant-height domain as well as a more realistic AC shape. We model eye rotations as harmonic in time with prescribed frequency $\unicode[STIX]{x1D714}_{f}$ and amplitude $\unicode[STIX]{x1D6FD}$, and use lubrication theory to simplify the governing equations. We write the equations in a reference frame moving with the domain and show that fluid motion is governed by three dimensionless parameters: the aspect ratio $\unicode[STIX]{x1D716}$ of the AC, the angular amplitude $\unicode[STIX]{x1D6FD}$ and the Womersley number $\unicode[STIX]{x1D6FC}$. We simplify the equations under the physiologically realistic assumptions that $\unicode[STIX]{x1D716}$ is small and $\unicode[STIX]{x1D6FC}$ large, leading to a linear system that can be decomposed into three harmonics: a dominant frequency component, with frequency $\unicode[STIX]{x1D714}_{f}$, and a steady streaming component and a third component with frequency $2\unicode[STIX]{x1D714}_{f}$. We solve the problem analytically for the constant-height domain and numerically as the solution of ordinary differential equations in the more realistic geometry. Both the primary flow and the steady streaming are shown to have a highly three-dimensional structure, which has not been highlighted in previous numerical works. We show that the steady streaming is particularly relevant from the clinical point of view, as it induces fluid mixing in the AC. Furthermore, the steady flow component is the dominant mixing mechanism during the night, when the thermal flow induced by temperature variations across the AC is suppressed.

We investigate theoretically and experimentally the stability of two interlaced helical vortices with respect to displacement perturbations having wavelengths that are large compared to the size of the vortex cores. First, existing theoretical results are recalled and applied to the present configuration. Various modes of unstable perturbations, involving different phase relationships between the two vortices, are identified and their growth rates are calculated. They lead to a local pairing of neighbouring helix loops, or to a global pairing with one helix expanding and the other one contracting. A relation is established between this instability and the three-dimensional pairing of arrays of straight parallel vortices, and a striking quantitative agreement concerning the growth rates and frequencies is found. This shows that the local pairing of vortices is the driving mechanism behind the instability of the helix system. Second, an experimental study designed to observe these instabilities in a real flow is presented. Two helical vortices are generated by a two-bladed rotor in a water channel and characterised through dye visualisations and particle image velocimetry measurements. Unstable displacement modes are triggered individually, either by varying the rotation frequency of the rotor, or by imposing a small rotor eccentricity. The observed unstable mode structure, and the corresponding growth rates obtained from advanced processing of visualisation sequences, are in good agreement with theoretical predictions. The nonlinear late stages of the instability are also documented experimentally. Whereas local pairing leads to strong deformations and subsequent breakup of the vortices, global pairing results in a leapfrogging phenomenon, which temporarily restores the initial double-helix geometry, in agreement with recent observations from numerical simulations.

The forces on an object impacting the water are extreme in the early moments of water entry and can cause structural damage to biological and man-made bodies alike. These early-time forces arise largely from added mass, peaking when the submergence is much less than one body length. We experimentally investigate a means of reducing impact forces on a rigid sphere by placing the sphere inside a jet of water so that the jet strikes the quiescent water surface prior to entry of the sphere into the pool. The water jet accelerates the pool liquid and forms a cavity into which a sphere falls. Through on-board accelerometer measurements and high-speed imaging, we quantify the force reduction compared to the case of a sphere entering a quiescent pool. Finally, we find the emergence of a critical jet volume required to maximize force reduction; the critical volume is rationalized using scaling arguments informed by near-surface particle image velocimetry (PIV) data.

Motivated by the success of wavepackets in modelling the noise from subsonic and perfectly expanded supersonic jets, we apply the wavepacket model to imperfectly expanded supersonic jets. Recent studies with subsonic jets have demonstrated the importance of capturing the ‘jitter’ of wavepackets in order to correctly predict the intensity of far-field sound. Wavepacket jitter may be statistically represented using a two-point coherence function; accurate prediction of noise requires identification of this coherence function. Following the analysis of Cavalieri & Agarwal (J. Fluid Mech., vol. 748, 2014. pp. 399–415), we extend their methodology to model the acoustic sources of broadband shock-associated noise in imperfectly expanded supersonic jets using cross-spectral densities of the turbulent and shock-cell quantities. The aim is to determine the relationship between wavepacket coherence-decay and far-field broadband shock-associated noise, using the model as a vehicle to explore the flow mechanisms at work. Unlike the subsonic case where inclusion of coherence decay amplifies the sound pressure level over the whole acoustic spectrum, we find that it does not play such a critical role in determining the peak sound amplitude for shock-cell noise. When higher-order shock-cell modes are used to reconstruct the acoustic spectrum at higher frequencies, however, the inclusion of a jittering wavepacket is necessary. These results suggest that the requirement for coherence decay identified in prior broadband shock-associated noise (BBSAN) models is in reality the statistical signature of jittering wavepackets. The results from this modelling approach suggest that nonlinear jittering effects of wavepackets need to be included in dynamic models for broadband shock-associated noise.

Turbulent separation bubbles over and behind a two-dimensional forward–backward-facing step submerged in a deep turbulent boundary layer are investigated using a time-resolved particle image velocimetry. The Reynolds number based on the step height and free-stream velocity is 12 300, and the ratio of the streamwise length to the height of the step is 2.36. The upstream turbulent boundary layer thickness is 4.8 times the step height to ensure a strong interaction of the upstream turbulence structures with the separated shear layers over and behind the step. The velocity measurements were performed in streamwise–vertical planes at the channel mid-span and streamwise–spanwise planes at various vertical distances from the wall. The unsteady characteristics of the separation bubbles and their associated turbulence structures are studied using a variety of techniques including linear stochastic estimation, proper orthogonal decomposition and variable-interval time averaging. The results indicate that the low-frequency flapping motion of the separation bubble over the step is induced by the oncoming large-scale alternating low- and high-velocity streaky structures. Dual separation bubbles appear periodically over the step at a higher frequency than the flapping motion, and are attributed to the inherent instability in the rear part of the mean separation bubble. The separation bubble behind the step exhibits a flapping motion at the same frequency as the separation bubble over the step, but with a distinct phase delay. At instances when an enlarged separation bubble is formed in front of the step, a pair of vertical counter-rotating vortices is formed in the immediate vicinity of the leading edge.

Diverse organisms that swim and fly in the inertial regime use the flapping or pumping of flexible appendages and cavities to propel themselves through a fluid. It has long been postulated that the speed and efficiency of locomotion are optimized by oscillating these appendages at their frequency of free vibration. In jellyfish swimming, a significant contribution to locomotory efficiency has been attributed to the effects passive energy recapture, whereby the bell is passively propelled through the fluid through its interaction with stopping vortex rings formed during each expansion of the bell. In this paper, we investigate the interplay between resonance and passive energy recapture using a three-dimensional implementation of the immersed boundary method to solve the fluid–structure interaction of an elastic oblate jellyfish bell propelling itself through a viscous fluid. The motion is generated through a fixed duration application of active tension to the bell margin, which mimics the action of the coronal swimming muscles. The pulsing frequency is then varied by altering the length of time between the application of applied tension. We find that the swimming speed is maximized when the bell is driven at its resonant frequency. However, the cost of transport is maximized by driving the bell at lower frequencies whereby the jellyfish passively coasts between active contractions through its interaction with the stopping vortex ring. Furthermore, the thrust generated by passive energy recapture was found to be dependent on the elastic properties of the jellyfish bell.