Few fluid phenomena are as beautiful, fragile and ephemeral as the crown splash that is created by the impact of an object on a liquid. The crown-shaped phenomenon and the physics behind it have mesmerised and intrigued scientists for over a century, and still the scientific world has not yet uncovered all of the secrets of the splash. This is exemplified in a particularly striking manner in Marston et al. (J. Fluid Mech., vol. 794, 2016, pp. 506–529) where a 6 m tall vacuum chamber is employed to study the splash formed upon impact of a sphere onto a deep liquid pool, at both atmospheric and reduced ambient pressures. They shed light into the classical problem of the surface seal and study the buckling of the splash. With an almost magical touch they devise a method to create a splash without the liquid and the sphere ever coming into contact. The images that accompany the paper – taken with state-of-the-art high-speed cameras – are as stunning as the physics that is uncovered in them.

The atmospheric ventilation of a surface-piercing hydrofoil is examined in a series of towing-tank experiments, performed on a vertically cantilevered hydrofoil with an immersed free tip. The results of the experiments expand upon previous studies by contributing towards a comprehensive understanding of the topology, formation and elimination of ventilated flows at low-to-moderate Froude and Reynolds numbers. Fully wetted, fully ventilated and partially ventilated flow regimes are identified, and their stability regions are presented in parametric space. The stability of partially and fully ventilated regimes is related to the angle of the re-entrant jet, leading to a set of criteria for identifying flow regimes in a laboratory environment. The stability region of fully wetted flow overlaps those of partially and fully ventilated flows, forming bi-stable regions where hysteresis occurs. Ventilation transition mechanisms are classified as formation and elimination mechanisms, which separate the three steady flow regimes from one another. Ventilation formation requires air ingress into separated flow at sub-atmospheric pressure from a continuously available air source. Ventilation washout is caused by upstream flow of the re-entrant jet. The boundary denoting washout of fully ventilated flow is expressed as a semi-theoretical scaling relation, which captures past and present experimental data well across a wide range of Froude and Reynolds numbers.

Hemiwicking refers to the spreading of a liquid on a rough hydrophilic surface driven by capillarity. Here, we construct scaling laws to predict the velocity of hemiwicking on a rough substrate and experimentally corroborate them with various arrangements and dimensions of micropillar arrays. At the macroscopic scale, where the wetting front appears parallel to the free surface of the reservoir, the wicking distance is shown to grow diffusively, i.e. like $t^{1/2}$ with $t$ being time. We show that our model is consistent with pillar arrays of a wide range of pitch-to-height ratios, either square or skewed. At the microscopic scale, where the meniscus extension from individual pillars at the wetting front is considered, the extension distance begins to grow like $t$ but the spreading slows down to behave like $t^{1/3}$ when the meniscus is far from the pillar. Our microscopic flow modelling allows us to find pillar spacing conditions under which the assumption of densely spaced pillars is valid.

We propose a hierarchy of low-dimensional proper orthogonal decomposition (POD) models for the transient and post-transient flow around a high-lift airfoil with unsteady Coanda blowing over the trailing edge. The modal expansion comprises actuation modes as a lifting method for wall actuation following Graham et al. (Intl J. Numer. Meth. Engng, vol. 44 (7), 1999, pp. 945–972) and Kasnakoğlu et al. (Intl J. Control, vol. 81 (9), 2008, pp. 1475–1492). A novel element is separate actuation modes for different frequencies. The structure of the dynamic model rests on a Galerkin projection using the Navier–Stokes equations, simplifying mean-field considerations, and a stochastic term representing the background turbulence. The model parameters are identified with a data assimilation (4D-Var) method. We propose a model hierarchy from a linear oscillator explaining the suppression of vortex shedding by blowing to a fully nonlinear model resolving unactuated and actuated transients with steady and high-frequency modulation of blowing. The models’ accuracy is assessed through the mode amplitudes and an estimator for the lift coefficient. The robustness of the model is physically justified, and then observed for the training and the validation dataset.

It is shown that the formal expression for the effective viscosity of a dilute suspension of arbitrary-shaped particles in Poiseuille flow contains a novel quadrupole term, besides the expected stresslet. This term becomes important for a very confined geometry. For a high-frequency flow field (in the sense used in Feuillebois et al. (J. Fluid Mech., vol. 764, 2015, pp. 133–147), the suspension rheology is Newtonian at first order in volume fraction. The effective viscosity is calculated for suspensions of $N$-bead rods and of prolate spheroids with the same length, volume and aspect ratio (up to 6), entrained by the Poiseuille flow between two infinite parallel flat hard walls. The numerical computations, based on solving the Stokes equations, indicate that the quadrupole term gives a significant positive contribution to the intrinsic viscosity $[{\it\mu}]$ if the distance between the walls is less than ten times the particle width, or less. It is found that the intrinsic viscosity in bounded Poiseuille flow is generally smaller than the corresponding value in unbounded flow, except for extremely narrow gaps when it becomes larger because of lubrication effects. The intrinsic viscosity is at a minimum for a gap between walls of the order of 1.5–2 particle width. For spheroids, the intrinsic viscosity is generally smaller than for chains of beads with the same aspect ratio, but when normalized by its value in the bulk, the results are qualitatively the same. Therefore, a rigid chain of beads can serve as a simple model of an orthotropic particle with a more complicated shape. The important conclusion is that the intrinsic viscosity in shear flow is larger than in the Poiseuille flow between two walls, and the difference is significant even for relatively wide channels, e.g. three times wider than the particle length. For such confined geometries, the hydrodynamic interactions with the walls are significant and should be taken into account.

A detailed numerical study of the influence of dense gas effects on the large-scale dynamics of decaying homogeneous isotropic turbulence is carried out by using the van der Waals gas model. More specifically, we focus on dense gases of the Bethe–Zel’dovich–Thompson type, which may exhibit non-classical nonlinearities in the transonic and supersonic flow regimes, under suitable thermodynamic conditions. The simulations are based on the inviscid conservation equations, solved by means of a ninth-order numerical scheme. The simulations rely on the numerical viscosity of the scheme to dissipate energy at the finest scales, while leaving the larger scales mostly unaffected. The results are systematically compared with those obtained for a perfect gas. Dense gas effects are found to have a significant influence on the time evolution of the average and root mean square (r.m.s.) of the thermodynamic properties for flows characterized by sufficiently high initial turbulent Mach numbers (above 0.5), whereas the influence on kinematic properties, such as the kinetic energy and the vorticity, are smaller. However, the flow dilatational behaviour is very different, due to the non-classical variation of the speed of sound in flow regions where the dense gas is characterized by a value of the fundamental derivative of the gas dynamics (a measure of the variation of the speed of sound in isentropic compressions) smaller than one or even negative. The most significant differences between the perfect and the dense gas case are found for the repartition of dilatation levels in the flow field. For the perfect gas, strong compressions occupy a much larger volume fraction than expansion regions, leading to probability distributions of the velocity divergence highly skewed toward negative values. For the dense gas, the volume fractions occupied by strong expansion and compression regions are much more balanced; moreover, strong expansion regions are characterized by sheet-like structures, unlike the perfect gas which exhibits tubular structures. In strong compression regions, where compression shocklets may occur, both the dense and the perfect gas exhibit sheet-like structures. This suggests the possibility that expansion eddy shocklets may appear in the dense gas. This hypothesis is also supported by the fact that, in dense gas, vorticity is created with equal probability in strong compression and expansion regions, whereas for a perfect gas, vorticity is more likely to be created in the strong compression ones.

The term ‘history effect’ refers to the contribution of any past mass transfer events between a gas bubble and its liquid surroundings towards the current diffusion-driven growth or dissolution dynamics of that same bubble. The history effect arises from the (non-instantaneous) development of the dissolved gas concentration boundary layer in the liquid in response to changes in the concentration at the bubble interface caused, for instance, by variations of the ambient pressure in time. Essentially, the history effect amounts to the acknowledgement that at any given time the mass flux across the bubble is conditioned by the preceding time history of the concentration at the bubble boundary. Considering the canonical problem of an isolated spherical bubble at rest, we show that the contribution of the history effect in the current interfacial concentration gradient is fully contained within a memory integral of the interface concentration. Retaining this integral term, we formulate a governing differential equation for the bubble dynamics, analogous to the well-known Epstein–Plesset solution. Our equation does not make use of the quasi-static radius approximation. An analytical solution is presented for the case of multiple step-like jumps in pressure. The nature and relevance of the history effect is then assessed through illustrative examples. Finally, we investigate the role of the history effect in rectified diffusion for a bubble that pulsates under harmonic pressure forcing in the non-inertial, isothermal regime.

The gravito-inertial waves propagating over a shellular baroclinic flow inside a rotating spherical shell are analysed using the Boussinesq approximation. The wave properties are examined by computing paths of characteristics in the non-dissipative limit, and by solving the full dissipative eigenvalue problem using a high-resolution spectral method. Gravito-inertial waves are found to obey a mixed-type second-order operator and to be often focused around short-period attractors of characteristics or trapped in a wedge formed by turning surfaces and boundaries. We also find eigenmodes that show a weak dependence with respect to viscosity and heat diffusion just like truly regular modes. Some axisymmetric modes are found unstable and likely destabilized by baroclinic instabilities. Similarly, some non-axisymmetric modes that meet a critical layer (or corotation resonance) can turn unstable at sufficiently low diffusivities. In all cases, the instability is driven by the differential rotation. For many modes of the spectrum, neat power laws are found for the dependence of the damping rates with diffusion coefficients, but the theoretical explanation for the exponent values remains elusive in general. The eigenvalue spectrum turns out to be very rich and complex, which lets us suppose an even richer and more complex spectrum for rotating stars or planets that own a differential rotation driven by baroclinicity.

For the first time, we used computer simulations to study lift forces on two static disks placed side-by-side within a two-dimensional granular flow and found them to be either repulsive or attractive depending on the flow velocity and separation between the disks. Our simulations results reveal that differences in the flow velocity between the disks and outside of that region are closely correlated with the lift force. We propose an empirical function for the lift force based on this correlation and our dimensional analysis. The specific region where the measured velocity exhibits this correlation suggests that attractive lift is not a Bernoulli-like effect. Instead, we speculate that it might be explained by a force balance based on Coulomb’s theory of passive failure in a Mohr–Coulomb material. Our results confirm that repulsive lift is due to the jamming of particles flowing between the disks.

Various microfluidic systems, such as chemical and biological lab-on-a-chip devices, involve motion of multiple droplets within an immersing fluid in shallow microchannels. Modelling the dynamics of such systems requires calculation of the forces of interaction between the moving droplets. These forces are commonly approximated by superposition of dipole solutions, which requires an assumption of sufficiently large distance between the droplets. In this work we obtain exact solutions (in the Hele-Shaw limit) for two moving droplets, and a droplet within a droplet, located within a moving immersing fluid, without limitation on the distance between the droplets. This is achieved by solution of the pressure field in a bipolar coordinate system and calculation of the force in Cartesian coordinates. Our results are compared with numerical computations, experimental data and the existing dipole-based models. We utilize the results to calculate the dynamics of a droplet within a droplet, and of two close droplets, located within an immersing fluid with oscillating speed. Overall, the obtained results establish the solid base for the rather important future extensions for modelling the complex, long-range interdroplet interactions in the limit of dense droplet media.

Steady nozzle flows of Bethe–Zel’dovich–Thompson fluids – substances exhibiting non-classical gasdynamic behaviour in a finite vapour-phase thermodynamic region in close proximity to the liquid–vapour saturation curve – are examined. Non-classical flow features include rarefaction shock waves, shock waves with either upstream or downstream sonic states and split shocks. Exact solutions for a mono-component single-phase fluid expanding from a reservoir into a stationary atmosphere through a conventional converging–diverging nozzle are determined within the quasi-one-dimensional inviscid flow approximation. The novel analytical approach makes it possible to elucidate the connection between the adiabatic, possibly non-isentropic flow field and the underlying local isentropic-flow features, including the possible qualitative alterations in passing through shock waves. Contrary to previous predictions based on isentropic-flow inspection, shock disintegration is found to occur also from reservoir states corresponding to a single sonic point. The global layout of the flow configurations produced by a monotonic decrease in the ambient pressure, namely the functioning regime, is examined for reservoir conditions resulting in single-phase flows. Accordingly, a classification of steady nozzle flows into 10 different functioning regimes is proposed. Flow conditions determining the transition between the different classes of flow are investigated and each functioning regime is associated with the corresponding thermodynamic region of reservoir states.

We perform a comprehensive characterization of the propulsive performance of a thrust generating pitching foil over a wide range of Reynolds ($10\leqslant Re\leqslant 2000$) and Strouhal ($St$) numbers using a high-resolution viscous vortex particle method. For a given $Re$, we show that the mean thrust coefficient $\overline{C_{T}}$ increases monotonically with $St$, exhibiting a sharp rise as the location of the inception of the wake asymmetry shifts towards the trailing edge. As a result, the propulsive efficiency too rises steeply before attaining a maximum and eventually declining at an asymptotic rate that is consistent with the inertial scalings of $St^{2}$ for $\overline{C_{T}}$ and $St^{3}$ for the mean power coefficient, with the latter scaling holding, quite remarkably, over the entire range of $Re$. We find the existence of a sharp increase in the peak propulsive efficiency ${\it\eta}_{max}$ (at a given $Re$) in the $Re$ range of 50 to approximately 1000, with ${\it\eta}_{max}$ increasing rapidly from about 1.7 % to the saturated asymptotic value of approximately $16\,\%$. The $St$ at which ${\it\eta}_{max}$ is attained decreases progressively with $Re$ towards an asymptotic limit of $0.45$ and always exceeds the one for transition from a reverse von Kármán to a deflected wake. Moreover, the drag-to-thrust transition occurs at a Strouhal number $St_{tr}$ that exceeds the one for von Kármán to reverse von Kármán transition. The $St_{tr}$ and the corresponding power coefficient $\overline{C_{p,}}_{tr}$ are found to be remarkably consistent with the simple scaling relationships $St_{tr}\sim Re^{-0.37}$ and $\overline{C_{p,}}_{tr}\sim Re^{-1.12}$ that are derived from a balance of the thrust generated from the pitching motion and the drag force arising out of viscous resistance to the foil motion. The fact that the peak propulsive efficiency degrades appreciably only below $Re\approx 10^{3}$ establishes a sharp lower threshold for energetically efficient thrust generation from a pitching foil. Our findings should be generalizable to other thrust-producing flapping foil configurations and should aid in establishing the link between wake patterns and energetic cost of thrust production in similar systems.

We use the adjoint method to compute sensitivity maps for the limit-cycle frequency and amplitude of the Bénard–von Kármán vortex street in the wake of a circular cylinder. The sensitivity analysis is performed in the frame of the semi-linear self-consistent model recently introduced by Mantič et al. (Phys. Rev. Lett., vol. 113, 2014, 084501), which allows us to describe accurately the effect of the control on the mean flow, but also on the finite-amplitude fluctuation that couples back nonlinearly onto the mean flow via the formation of Reynolds stress. The sensitivity is computed with respect to arbitrary steady and synchronous time-harmonic body forces. For a small amplitude of the control, the theoretical variations of the limit-cycle frequency predict well those of the controlled flow, as obtained from either self-consistent modelling or direct numerical simulation of the Navier–Stokes equations. This is not the case if the variations are computed in the simpler mean flow approach overlooking the coupling between the mean and fluctuating components of the flow perturbation induced by the control. The variations of the limit-cycle amplitude (that falls out the scope of the mean flow approach) are also correctly predicted, meaning that the approach can serve as a relevant and systematic guideline to control strongly unstable flows exhibiting non-small, finite amplitudes of oscillation. As an illustration, we apply the method to control by means of a small secondary control cylinder and discuss the obtained results in the light of the seminal experiments of Strykowski & Sreenivasan (J. Fluid Mech., vol. 218, 1990, pp. 71–107).

The diffusion of heat injected from a line source into turbulence generated by regular and fractal grids with the same solidity and inlet velocity was investigated experimentally with particular interest in the effects of grid geometry and relative location of the source on the width of the thermal plume and the mixing efficiency. These grids included one fractal square grid (FSG) and three regular square grids with mesh sizes that were comparable to the first (RG160), second (RG80) and fourth (RG18) iterations of the fractal grid. The heated line source was inserted on the centre plane of the grids, spanning the entire width of the wind tunnel at either of two downstream locations, an upstream location or a location nearly coincident with a grid. It was found that, in all cases examined, RG160 produced the strongest diffusion of the thermal plume and the highest level of scalar mixing. These observations were consistent with the evolution of the corresponding turbulent diffusivities, which, according to Taylor’s theory of diffusion, are the product of the transverse turbulence intensity and the integral length scale. We argue that to maximise scalar diffusion and mixing of a scalar released from a concentrated source inside a duct, one should prefer a regular grid over a fractal square grid; we also recommend the use of a grid with a mesh size roughly equal to half the height of the duct and placed at approximately one duct height upstream of the source.

Large-eddy simulation data for a Mach 1.3 round jet are decomposed into acoustic, hydrodynamic and thermal components using Doak’s momentum potential theory. The decomposed fields are then analysed to examine the properties of each mode and their dynamics based on the transport equation for the total fluctuating enthalpy. The solenoidal fluctuations highlight hydrodynamic components of the jet and capture the shear layer growth and breakdown process. The acoustic mode exhibits a jittering coherent wavepacket structure in the turbulent region and consequent highly directional downstream radiation. The expected radial decay rates, $r^{-6}$ for hydrodynamic and $r^{-2}$ for acoustic, are recovered and closely follow the universal radiation spectra in the sideline and downstream directions. The scalogram of the acoustic mode in the near-acoustic-field region is consistent with that of the pressure perturbation signal in the acoustic-frequency range, but effectively removes the hydrodynamic and thermal content. The time-resolved and mean behaviour of terms in the total fluctuating enthalpy equation is analysed in detail. A large-scale intermittent event in the near-acoustic field is shown to be associated with an intrusion of vortices from the shear layer into the core of the jet. Acoustic sources are created when the resulting negative fluctuations in the solenoidal component interact with positive fluctuations in the Coriolis acceleration term. The latter are associated with regions of high vorticity on the inner side of the shear layer. In contrast, sinks result from the interaction of solenoidal momentum fluctuations with positive entropy gradients along entrainment streaks.

Kinetic theory provides a rigorous foundation to explore the unsteady (oscillatory) flow of a dilute gas, which is often generated by nanomechanical devices. Recently, formal asymptotic analyses of unsteady (oscillatory) flows at small Knudsen numbers have been derived from the linearised Boltzmann–Bhatnagar–Gross–Krook (Boltzmann–BGK) equation, in both the low- and high-frequency limits (Nassios & Sader, J. Fluid Mech., vol. 708, 2012, pp. 197–249 and vol. 729, 2013, pp. 1–46; Takata & Hattori, J. Stat. Phys., vol. 147, 2012, pp. 1182–1215). These asymptotic theories predict that unsteadiness can couple strongly with heat transport to dramatically modify the overall gas flow. Here, we study the gas flow generated between two parallel plane walls whose temperatures vary sinusoidally in time. Predictions of the asymptotic theories are compared to direct numerical solutions, which are valid for all Knudsen numbers and normalised frequencies. Excellent agreement is observed, providing the first numerical validation of the asymptotic theories. The asymptotic analyses also provide critical insight into the physical mechanisms underlying these flow phenomena, establishing that mass conservation (not momentum or energy) drives the flows – this explains the identical results obtained using different previous theoretical treatments of these linear thermal flows. This study highlights the unique gas flows that can be generated under oscillatory non-isothermal conditions and the importance of both numerical and asymptotic analyses in explaining the underlying mechanisms.

We present a three-dimensional computational study of fully deformable red blood cells of the biconcave resting shape subject to sinusoidally oscillating shear flow. A comprehensive analysis of the cell dynamics and deformation response is considered over a wide range of flow frequency, shear rate amplitude and viscosity ratio. We observe that the cell exhibits either a periodic motion or a chaotic motion. In the periodic motion, the cell reverses its orientation either by passing through the flow direction (horizontal axis) or by passing through the flow gradient (vertical axis). The chaotic dynamics is characterized by a non-periodic sequence of horizontal and vertical reversals. The study provides the first conclusive evidence of the chaotic dynamics of fully deformable cells in oscillating flow using a deterministic numerical model without the introduction of any stochastic noise. In certain regimes of the periodic motion, the initial conditions are completely forgotten and the cells become entrained in the same sequence of horizontal reversals. We show that chaos is only possible in certain frequency bands when the cell membrane can rotate by a certain amount, allowing the cells to swing near the maximum shear rate. As such, the bifurcation between the horizontal and vertical attractors in phase space always occurs via a swinging inflection. While the reversal sequence evolves in an unpredictable way in the chaotic regime, we find a novel result that there exists a critical inclination angle at the instant of flow reversal which determines whether a vertical or horizontal reversal takes place, and is independent of the flow frequency. The chaotic dynamics, however, occurs at a viscosity ratio less than the physiological values. We further show that the cell shape in oscillatory shear at large amplitude exhibits a remarkable departure from the biconcave shape, and that the deformation is significantly greater than that in steady shear flow. A large compression of the cells occurs during the reversals which leads to over/undershoots in the deformation parameter. We show that due to the large deformation experienced by the cells, the regions of chaos in parameter space diminish and eventually disappear at high shear rate, in contradiction to the prediction of reduced-order models. While the findings bolster support for reduced-order models at low shear rate, they also underscore the important role that the cell deformation plays in large-amplitude oscillatory flows.

We study the transient dynamics of a viscous liquid contained in a narrow gap between a rigid surface and a parallel elastic plate. The elastic plate is deformed due to an externally applied time-varying pressure field. We model the flow field via the lubrication approximation and the plate deformation by the Kirchhoff–Love plate theory. We obtain a self-similarity solution for the case of an external point force acting on the elastic plate. The pressure and deformation field during and after the application of the external force are derived and presented by closed-form expressions. We examine a distributed external pressure, spatially uniform and linearly increasing with time, acting on the elastic plate over a finite region and during a finite time period, similar to the viscous–elastic interaction time-scale. The interaction between elasticity and viscosity is shown to reduce by an order of magnitude the pressure within the Hele-Shaw cell compared with the externally applied pressure. The results thus suggest that elastic Hele-Shaw configurations may be used to achieve significant impact mitigation.

We use direct numerical simulations to study the effect of thermal boundary conditions on developing turbulent pipe flows with fluids at supercritical pressure. The Reynolds number based on pipe diameter and friction velocity at the inlet is $Re_{{\it\tau}0}=360$ and Prandtl number at the inlet is $Pr_{0}=3.19$. The thermodynamic conditions are chosen such that the temperature range within the flow domain incorporates the pseudo-critical point where large variations in thermophysical properties occur. Two different thermal wall boundary conditions are studied: one that permits temperature fluctuations and one that does not allow temperature fluctuations at the wall (equivalent to cases where the thermal effusivity ratio approaches infinity and zero, respectively). Unlike for turbulent flows with constant thermophysical properties and Prandtl numbers above unity – where the effusivity ratio has a negligible influence on heat transfer – supercritical fluids shows a strong dependency on the effusivity ratio. We observe a reduction of 7 % in Nusselt number when the temperature fluctuations at the wall are suppressed. On the other hand, if temperature fluctuations are permitted, large property variations are induced that consequently cause an increase of wall-normal velocity fluctuations very close to the wall and thus an increased overall heat flux and skin friction.

The Stokes wave is a finite-amplitude periodic gravity wave propagating with constant velocity in an inviscid fluid. The complex analytical structure of the Stokes wave is analysed using a conformal mapping of a free fluid surface of the Stokes wave onto the real axis with the fluid domain mapped onto the lower complex half-plane. There is one square root branch point per spatial period of the Stokes wave located in the upper complex half-plane at a distance $v_{c}$ from the real axis. The increase of Stokes wave height results in $v_{c}$ approaching zero with the limiting Stokes wave formation at $v_{c}=0$. The limiting Stokes wave has a $2/3$ power-law singularity forming a $2{\rm\pi}/3$ radians angle on the crest which is qualitatively different from the square root singularity valid for arbitrary small but non-zero $v_{c}$, making the limit of zero $v_{c}$ highly non-trivial. That limit is addressed by crossing a branch cut of a square root into the second and subsequently higher sheets of the Riemann surface to find coupled square root singularities at distances $\pm v_{c}$ from the real axis at each sheet. The number of sheets is infinite and the analytical continuation of the Stokes wave into all of these sheets is found together with the series expansion in half-integer powers at singular points within each sheet. It is conjectured that a non-limiting Stokes wave at the leading order consists of an infinite number of nested square root singularities which also implies the existence in the third and higher sheets of additional square root singularities away from the real and imaginary axes. These nested square roots form a $2/3$ power-law singularity of the limiting Stokes wave as $v_{c}$ vanishes.