Reversed von Kármán streets are responsible for a velocity surplus in the wake of flapping foils, indicating the onset of thrust generation. However, the wake pattern cannot be predicted based solely on the flapping peak-to-peak amplitude $A$ and frequency $f$ because the transition also depends sensitively on other details of the kinematics. In this work we replace $A$ with the cycle-averaged swept trajectory ${\mathcal{T}}$ of the foil chordline. Two-dimensional simulations are performed for pure heave, pure pitch and a variety of heave-to-pitch coupling. In a phase space of dimensionless ${\mathcal{T}}-f$ we show that the drag-to-thrust wake transition of all tested modes occurs for a modified Strouhal $St_{{\mathcal{T}}}\rightarrow 1$. Physically, the product ${\mathcal{T}}f$ expresses the induced velocity of the foil and indicates that propulsive jets occur when this velocity exceeds $U_{\infty }$. The new metric offers a unique insight into the thrust-producing strategies of biological swimmers and flyers alike, as it directly connects the wake development to the chosen kinematics, enabling a self-similar characterisation of flapping foil propulsion.

The flow of fluid through a pipe has been instrumental in illuminating the subcritical route to turbulence typical of many wall-bounded shear flows. Especially important in this process are the turbulent–laminar fronts that separate the turbulent and laminar flow. Four years ago Michael Graham (Nature, vol. 526, 2015, p. 508) wrote a commentary entitled ‘Turbulence spreads like wildfire’, which is a picturesque but also accurate characterisation of the way turbulence spreads through laminar flow in a straight pipe. In this spirit, the recent article by Rinaldi et al. (J. Fluid Mech., vol. 866, 2019, pp. 487–502) shows that turbulent wildfires are substantially tamed in bent pipes. These authors find that even at modest pipe curvature, the characteristic strong turbulent–laminar fronts of straight pipe flow vanish. As a result, the propagation of turbulent structures is modified and there are hints that the route to turbulence is fundamentally altered.

Capillary rise of a liquid displacing gas is analysed for both open and closed capillaries. We include menisci mass and hysteresis, and show that oscillations due to inertia are muted by friction at the advancing meniscus. From single-phase numerical computations in a no-slip/slip capillary, we quantify losses due to entry, flow development, meniscus slip, exit and acceleration of fluid within the reservoir. For closed capillaries, determining viscous drag due to gas requires inclusion of compressibility, and solving a moving boundary problem. This solution is derived through perturbation expansion with respect to two different small parameters for obtaining pressure above the liquid meniscus. Our rise predictions spanning a large range of experimental conditions and fluids for both open and closed capillaries match the data. The experimental data confirm the adequacy of the theoretically constructed dimensionless groups for predicting oscillatory behaviour.

Conceptually, tidal rivers are seen as narrow channels along which the cross-section geometry remains constant and the bed is horizontal. As tidal waves propagate along such a channel, they decrease exponentially in height. The more rapid the decrease, the stronger the river flow. Near the coast, the tidally averaged width and depth change little throughout the year, even if the river discharge varies strongly between the seasons. However, further upstream, the water depth varies considerably with the river discharge. Recent observations from the Kapuas River, Indonesia, show that the water surface forms a backwater profile when the river flow is low. In this case, the depth converges, i.e. it gradually decreases between the river mouth and the point where the bed reaches sea level. This effect distinctly influences how tidal waves propagate up river so that their wave height does not decrease exponentially any more. We present a theoretical analysis of this phenomenon, which reveals several so far overlooked aspects of river tides. These aspects are particularly relevant to low river flow. Along the downstream part of the tidal river, depth convergence counteracts frictional damping so that the tidal range is higher than expected. Along the upstream parts of the tidal river, the low depth increases the damping so that the tide more rapidly attenuates. The point where the bed reaches sea level effectively limits the tidal intrusion, which carries over to the overtide and the subtidal water level set-up.

Mack-mode waves are measured in a hypersonic boundary layer using high-frequency focusing schlieren deflectometry. Experiments are performed using a $5^{\circ }$ flared cone at $0^{\circ }$ angle of attack in the low-disturbance Mach 6 Quiet Tunnel at Texas A&M University across a free-stream unit Reynolds number range of $7.8\times 10^{6}~\text{m}^{-1}\leqslant \mathit{Re}^{\prime }\leqslant 11.0\times 10^{6}~\text{m}^{-1}$. The high-frequency response of the measurement system allows harmonics and other nonlinear behaviour to be measured. Mack-mode waves and several harmonics are clearly observed at a frequency of $f_{0}\approx 250~\text{kHz}$. Bispectral analysis is used to show that these waves undergo several quadratic phase-coupled sum and difference interactions with themselves to produce harmonics, as well interact with a relatively low-frequency wave that results in amplitude modulation. Bispectral analysis is used to highlight these interactions.

Liquid films on wettable solid surfaces can be disturbed to dewet when low surface tension liquids or surfactants are added because the surface tension difference gives rise to stresses on the film interface. Here we consider an alcohol drop placed above a thin aqueous film, which punctures a hole in the film starting from underneath the alcohol drop. Such film dewetting is attributed to the Marangoni effects caused by the spatial gradient of alcohol vapour concentration. We measure the liquid–gas interfacial tension of aqueous liquids rapidly responding to the surrounding isopropyl alcohol vapour concentration, and observe evolution of the film morphology consisting of central hole, fringe film, thinning region and bulk. We construct scaling laws to predict the dewetting rates of the film by considering the Marangoni stress, viscous shear stress and evaporation. It is shown that our experiments are consistent with our theory.

An analytical investigation is performed to understand the lock-in phenomenon, observed in vortex shedding combustors. Several aeroengine afterburners and ramjets use a bluff body to stabilize the flame. The bluff body sheds vortices. During the occurrence of high-amplitude combustion instability, the frequency of vortex shedding locks in to the frequency of the chamber acoustic field. This phenomenon is termed vortex-acoustic lock-in. In general, there is a two-way coupling between the vortex shedding process and the acoustic field, making analytical investigation difficult. Since the frequency of the latter remains largely unaltered, performing an investigation to study the response of vortex shedding to external excitation not only allows one to gain insights, but also make the problem analytically tractable. We begin with a lower-order model available in the literature to describe the vortex shedding process in non-reacting flows, arising from sharp corners in the presence of upstream velocity excitation. The continuous time domain model is transformed to a discrete map, which connects the time instances of two successive vortex shedding events. The frequency and amplitude of excitation are varied to study the instantaneous vortex shedding time period, as the response of the system. In the absence of forcing, the iterates of the map form a period-1 solution with the frequency equalling the natural vortex shedding frequency. On increasing the amplitude of excitation, quasi-periodic behaviour of the iterates is observed, followed by a period-1 lock-in solution, where vortex shedding occurs at the excitation frequency. On further increasing the amplitude, de-lock-in occurs. From the map, an analytical solution is extracted, which represents the lock-in state. The condition and thereby the region in the frequency–amplitude parameter space where a general $p:1$ lock-in occurs is then identified. Several important analytical expressions, such as for (1) critical threshold frequency above which lock-in occurs, (2) boundary of lock-in region in the parameter space, that are of direct importance to the design of quieter combustors are obtained. The study also identifies the transition of higher-order $p:1$ to $1:1$ lock-in state, through a series of lock-in and de-lock-in steps, whose occurrence could be verified from future experiments.

We develop a general model to describe a network of interconnected thin viscous sheets, or viscidas, which evolve under the action of surface tension. A junction between two viscidas is analysed by considering a single viscida containing a smoothed corner, where the centreline angle changes rapidly, and then considering the limit as the smoothing tends to zero. The analysis is generalized to derive a simple model for the behaviour at a junction between an arbitrary number of viscidas, which is then coupled to the governing equation for each viscida. We thus obtain a general theory, consisting of $N$ partial differential equations and $3J$ algebraic conservation laws, for a system of $N$ viscidas connected at $J$ junctions. This approach provides a framework to understand the fabrication of microstructured optical fibres containing closely spaced holes separated by interconnected thin viscous struts. We show sample solutions for simple networks with $J=2$ and $N=2$ or 3. We also demonstrate that there is no uniquely defined junction model to describe interconnections between viscidas of different thicknesses.

Flows in channels exposed to spatially distributed heating were investigated. Such flows are of interest as theoretical analyses suggest that heating leads to the reduction of pressure losses. A special apparatus providing the means for the creation of well-controlled spatially periodic heating with the desired intensity as well as precise control of the flow rate in flows with small Reynolds numbers was constructed. The apparatus works with air and provides optical access to the flow interior. The relevant theory has been generalized to handle the temperature fields measured in the experiments. The experiments were carried out for Reynolds numbers $Re<20$ and at a single Rayleigh number based on the peak-to-peak temperature difference and channel half-height of $Ra_{p}=3500$. Flow visualization and particle image velocimetry measurements demonstrated the formation of two-dimensional steady rolls whose size was dictated by $Re$, with the largest rolls observed for the smallest $Re$ and the roll size being gradually reduced as $Re$ increased until their complete elimination at the largest $Re$ used in the experiment. An excellent agreement between the theoretically and experimentally determined complex flow fields was found. Wall shear stresses extracted from the velocity measurements agree with their theoretical counterparts within the expected accuracy. The agreement between the experimental and theoretical velocity fields and their unique relation with the corresponding pressure fields indirectly verify the heating-induced pressure-gradient-reducing effect.

A novel approach to identify internal interfacial layers, or IILs, in wall-bounded turbulent flows is proposed. Using a fuzzy cluster method (FCM) on the streamwise velocity component, a unique and unambiguous grouping of the uniform momentum zones (UMZs) is achieved, thus allowing the identification of the IILs. The approach overcomes some of the key limitations of the histogram-based IIL identification methods. The method is insensitive to the streamwise domain length, can be used on inhomogeneous grids, uses all the available flow field data, is trivially extended to three dimensions and does not need user-defined parameters (e.g. number of bins) other than the number of zones. The number of zones for a given snapshot can be automatically determined by an a priori algorithm based on a kernel density estimation algorithm, or KDE. This automated approach is applied to compute the average number of UMZs as a function of Reynolds number $Re_{\unicode[STIX]{x1D70F}}$ in turbulent channel flows in several numerical simulations. This systematic approach reveals a dependence of the Reynolds number on the average number of UMZs in the channel flow; this supports previously reported observations in the boundary layer. The fuzzy clustering approach is applied to the turbulent boundary layer (experimental, planar particle image velocimetry) and channel flow (numerical, direct numerical simulation) at varying Reynolds numbers. The interfacial layers are characterized by a strong concentration of spanwise vorticity, with the outer-most layer located at the upper edge of the log layer. The three-dimensional interface identification reveals a streak-like organization. The large-scale motion (LSM) at the outer region of the channel flow boundary layer modulates the outer IIL. The corrugations of the outer IIL are aligned with the LSM and the conditional correlation of the inner and outer IIL height shows that extreme near-wall events leave their mark on the outer IIL corrugations.

Fluid deformable surfaces are ubiquitous in cell and tissue biology, including lipid bilayers, the actomyosin cortex or epithelial cell sheets. These interfaces exhibit a complex interplay between elasticity, low Reynolds number interfacial hydrodynamics, chemistry and geometry, and govern important biological processes such as cellular traffic, division, migration or tissue morphogenesis. To address the modelling challenges posed by this class of problems, in which interfacial phenomena tightly interact with the shape and dynamics of the surface, we develop a general continuum mechanics and computational framework for fluid deformable surfaces. The dual solid–fluid nature of fluid deformable surfaces challenges classical Lagrangian or Eulerian descriptions of deforming bodies. Here, we extend the notion of arbitrarily Lagrangian–Eulerian (ALE) formulations, well-established for bulk media, to deforming surfaces. To systematically develop models for fluid deformable surfaces, which consistently treat all couplings between fields and geometry, we follow a nonlinear Onsager formalism according to which the dynamics minimizes a Rayleighian functional where dissipation, power input and energy release rate compete. Finally, we propose new computational methods, which build on Onsager’s formalism and our ALE formulation, to deal with the resulting stiff system of higher-order partial differential equations. We apply our theoretical and computational methodology to classical models for lipid bilayers and the cell cortex. The methods developed here allow us to formulate/simulate these models in their full three-dimensional generality, accounting for finite curvatures and finite shape changes.

The flow rate of a pressure-driven liquid through a microchannel may be enhanced by texturing its no-slip boundaries with grooves aligned with the flow. In such cases, the grooves may contain vapour and/or an inert gas and the liquid is trapped in the Cassie state, resulting in (apparent) slip. The flow-rate enhancement is of benefit to different applications including the increase of throughput of a liquid in a lab-on-a-chip, and the reduction of thermal resistance associated with liquid metal cooling of microelectronics. At any given cross-section, the meniscus takes the approximate shape of a circular arc whose curvature is determined by the pressure difference across it. Hence, it typically protrudes into the grooves near the inlet of a microchannel and is gradually drawn into the microchannel as it is traversed and the liquid pressure decreases. For sufficiently large Reynolds numbers, the variation of the meniscus shape and hence the flow geometry necessitates the inclusion of inertial (non-parallel) flow effects. We capture them for a slender microchannel, where our small parameter is the ratio of ridge pitch-to-microchannel height, and order-one Reynolds numbers. This is done by using a hybrid analytical–numerical method to resolve the nonlinear three-dimensional (3-D) problem as a sequence of two-dimensional (2-D) linear ones in the microchannel cross-section, allied with non-local conditions that determine the slowly varying pressure distribution at leading and first orders. When the pressure difference across the microchannel is constrained by the advancing contact angle of the liquid on the ridges and its surface tension (which is high for liquid metals), inertial effects can significantly reduce the flow rate for realistic parameter values. For example, when the solid fraction of the ridges is 0.1, the microchannel height-to-(half) ridge pitch ratio is 6, the Reynolds number of the flow is 1 and the small parameter is 0.1, they reduce the flow rate of a liquid metal (Galinstan) by approximately 50 %. Conversely, for sufficiently large microchannel heights, they enhance it. Physical explanations of both of these phenomena are given.

Thin liquid films sitting on a heated solid substrate and surrounded by a colder ambient gas phase are strongly affected by surface-shear stresses induced by surface tension and temperature gradients, as well as by viscous and capillary forces. The temperature dependence of surface tension may lead to thinning of liquid-film depressions promoting instability which takes place when a critical temperature difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D717}_{cr}$ between the substrate and the ambient gas phase is exceeded. In this article we show theoretically that viscous heating, previously neglected in related literature, may delay or suppress the thermocapillary instability and leads to film healing. The viscous heating effect, by inhibiting heat transfer, prevents the system from reaching the critical value $\unicode[STIX]{x0394}\unicode[STIX]{x1D717}_{cr}$ required to bring about instability. As a consequence, the system remains within the stability region, suppressing film rupture. The presence of the viscous heating effect leads to a persistent circulating motion of two counter-rotating vortices lying diametrically opposite to a depression of the liquid–gas interface reducing the wavelength of disturbances to one half of its initial value. This effect has yet to be observed in experiment.

We present a mesoscale kinetic model for multicomponent flows, augmented with a short range forcing term, aimed at describing the combined effect of surface tension and near-contact interactions operating at the fluid interface level. Such a mesoscale approach is shown to (i) accurately capture the complex dynamics of bouncing colliding droplets for different values of the main governing parameters, (ii) predict quantitatively the effective viscosity of dense emulsions in micro-channels and (iii) simulate the formation of the so-called soft flowing crystals in microfluidic focusers.

This paper investigates the influence of three-dimensional effects on the transfer function of a rectangular-section body in turbulent flow. The dimensionless factor $\unicode[STIX]{x1D713}$, as derived by Li et al. (J. Fluid Mech., vol. 847, 2018, pp. 768–785), is adapted to evaluate this influence. The calculation of $\unicode[STIX]{x1D713}$ requires the spanwise influence term. For this purpose, an adapted form of the lift coherence function is derived, enabling the use of the measured lift coherence for the estimation of the spanwise influence term. Three rectangular models with different cross-sections (chord-to-depth ratios of 3, 5 and 10) are chosen for testing, and a NACA 0015 airfoil model is tested for comparison. Using the measured spanwise influence terms, the dimensionless factors of these models are then numerically calculated under different ratios of the turbulent integral scale to the chord $\unicode[STIX]{x1D6FE}$ and aspect ratios $\unicode[STIX]{x1D703}$. It is shown that the dimensionless factors of the rectangular models increase as $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D703}$ increase, which are similar to the dimensionless factor of the airfoil model. If $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D703}$ have suitable values, the strip theory could be applicable to the rectangular-section body. It is also found that the dimensionless factors of all the rectangular models are larger than the dimensionless factor of the airfoil model under the same parameters. The smaller the chord-to-depth ratio is, the larger the dimensionless factor is. Using the strip theory to calculate the lift response of the rectangular-section body may provide more accurate estimation. Additionally, the one-wavenumber transfer functions of these models are determined under the consideration of the three-dimensional effects. The results show that the experimental transfer functions of the rectangular models cannot be captured by the Sears function. They are larger than the Sears function at lower frequencies, while falling at a faster rate as the frequency increases. For bluff bodies with separated flow, the modified transfer function presented here appears to be an appropriate approach.

We present experimental observations of the velocity and spatial distribution of inertial particles dispersed in turbulent downward flow through a vertical channel at friction Reynolds numbers $\mathit{Re}_{\unicode[STIX]{x1D70F}}=235$ and 335. The working fluid is air laden with size-selected glass microspheres, having Stokes numbers $St=\mathit{O}(10)$ and $\mathit{O}(100)$ when based on the Kolmogorov and viscous time scales, respectively. Cases at solid volume fractions $\unicode[STIX]{x1D719}_{v}=3\times 10^{-6}$ and $5\times 10^{-5}$ are considered. In the more dilute regime, the particle concentration profile shows near-wall and centreline maxima compatible with a turbophoretic drift down the gradient of turbulence intensity; the particles travel at speed similar to that of the unladen flow except in the near-wall region; and their velocity fluctuations generally follow the unladen flow level over the channel core, exceeding it in the near-wall region. The denser regime presents substantial differences in all measured statistics: the near-wall concentration peak is much more pronounced, while the centreline maximum is absent; the mean particle velocity decreases over the logarithmic and buffer layers; and particle velocity fluctuations and deposition velocities are enhanced. An analysis of the spatial distributions of particle positions and velocities reveals different behaviours in the core and near-wall regions. In the channel core, dense clusters form which are somewhat elongated, tend to be preferentially aligned with the vertical/streamwise direction and travel faster than the less concentrated particles. In the near-wall region, the particles arrange in highly elongated streaks associated with negative streamwise velocity fluctuations, several channel heights in length and spaced by $\mathit{O}(100)$ wall units, supporting the view that these are coupled to fluid low-speed streaks typical of wall turbulence. The particle velocity fields contain a significant component of random uncorrelated motion, more prominent for higher $St$ and in the near-wall region.

Surfactant-laden liquid plug propagation and rupture occurring in lower lung airways are studied computationally using a front-tracking method. The plug is driven by an applied constant pressure in a rigid axisymmetric tube whose inner surface is coated by a thin liquid film. The evolution equations of the interfacial and bulk surfactant concentrations coupled with the incompressible Navier–Stokes equations are solved in the front-tracking framework. The numerical method is first validated for a surfactant-free case and the results are found to be in good agreement with the earlier simulations of Fujioka et al. (Phys. Fluids, vol. 20, 2008, 062104) and Hassan et al. (Intl J. Numer. Meth. Fluids, vol. 67, 2011, pp. 1373–1392). Then extensive simulations are performed to investigate the effects of surfactant on the mechanical stresses that could be injurious to epithelial cells, such as pressure and shear stress. It is found that the liquid plug ruptures violently to induce large pressure and shear stress on airway walls and even a tiny amount of surfactant significantly reduces the pressure and shear stress and thus improves cell survivability. However, addition of surfactant also delays the plug rupture and thus airway reopening.

In this paper, a finite difference scheme on non-uniform staggered grids is proposed for wormhole propagation with the Darcy–Brinkman–Forchheimer framework in porous media by introducing an auxiliary flux variable to guarantee full mass conservation. Error estimates for the pressure, velocity, porosity, concentration and auxiliary flux with second-order superconvergence in different discrete norms are established rigorously and carefully on non-uniform grids. We also obtain second-order superconvergence for some terms of the $H^{1}$ norm of the velocity on non-uniform grids. Finally, some numerical experiments are presented to verify the theoretical analysis and effectiveness of the proposed scheme.

This paper examines the recovery of the wall-shear stress of a turbulent boundary layer that has undergone a sudden transition from a rough to a smooth surface. Early work of Antonia & Luxton (J. Fluid Mech., vol. 53, 1972, pp. 737–757) questioned the reliability of standard smooth-wall methods for measuring wall-shear stress in such conditions, and subsequent studies show significant disagreement depending on the approach used to determine the wall-shear stress downstream. Here we address this by utilising a collection of experimental databases at $Re_{\unicode[STIX]{x1D70F}}\approx 4100$ that have access to both ‘direct’ and ‘indirect’ measures of the wall-shear stress to understand the recovery to equilibrium conditions of the new surface. Our results reveal that the viscous region ($z^{+}\lesssim 4$) recovers almost immediately to an equilibrium state with the new wall conditions; however, the buffer region and beyond takes several boundary layer thicknesses before recovering to equilibrium conditions, which is longer than previously thought. A unique direct numerical simulation database of a wall-bounded flow with a rough-to-smooth wall transition is employed to confirm these findings. In doing so, we present evidence that any estimate of the wall-shear stress from the mean velocity profile in the buffer region or further away from the wall tends to underestimate its magnitude in the near vicinity of the rough-to-smooth transition, and this is likely to be partly responsible for the large scatter of recovery lengths to equilibrium conditions reported in the literature. Our results also reveal that smaller energetic scales in the near-wall region recover to an equilibrium state associated with the new wall conditions within one boundary layer thickness downstream of the transition, while larger energetic scales exhibit an over-energised state for several boundary layer thicknesses downstream of the transition. Based on these observations, an alternative approach to estimating the wall-shear stress from the premultiplied energy spectrum is proposed.

We perform Lagrangian investigations of the dynamics of velocity gradients in compressible decaying turbulence. Specifically, we examine the evolution of the invariants of the velocity-gradient tensor. We employ well-resolved direct numerical simulations over a range of Mach number along with a Lagrangian particle tracker to examine trajectories of fluid particles in the space of the invariants of the velocity gradient tensor. This allows us to accurately measure the lifetimes of major topologies of compressible turbulence and provide an explanation of why some selective topologies tend to exist longer than the others. Further, the influence of dilatation on the lifetime of various topologies is examined. Finally, we explain why the so-called conditional mean trajectories (CMT) used previously by several researchers fail to predict the lifetime of topologies accurately.