Flapping wings are important in many biological and bioinspired systems. Here, we investigate the fluid mechanics of flapping wings that possess a single flexible hinge allowing passive wing pitch rotation under load. We perform experiments on an insect-scale (${\approx}1$  cm wing span) robotic flapper and compare the results with a quasi-steady dynamical model and a coupled fluid–structure computational fluid dynamics model. In experiments we measure the time varying kinematics, lift force and two-dimensional velocity fields of the induced flow from particle image velocimetry. We find that increasing hinge stiffness leads to advanced wing pitching, which is beneficial towards lift force production. The classical quasi-steady model gives an accurate prediction of passive wing pitching if the relative phase difference between the wing stroke and the pitch kinematics, ${\it\delta}$, is small. However, the quasi-steady model cannot account for the effect of ${\it\delta}$ on leading edge vortex (LEV) growth and lift generation. We further explore the relationships between LEV, lift force, drag force and wing kinematics through experiments and numerical simulations. We show that the wing kinematics and flapping efficiency depend on the stiffness of a passive compliant hinge. Our dual approach of running at-scale experiments and numerical simulations gives useful guidelines for choosing wing hinge stiffnesses that lead to efficient flapping.

Theory and experiments are reported that explore the behaviour of the Rayleigh–Taylor instability initiated with a diffuse interface. Experiments are performed in which an interface between two gases of differing density is made unstable by acceleration generated by a rarefaction wave. Well-controlled, diffuse, two-dimensional and three-dimensional, single-mode perturbations are generated by oscillating the gases either side to side, or vertically for the three-dimensional perturbations. The puncturing of a diaphragm separating a vacuum tank beneath the test section generates a rarefaction wave that travels upwards and accelerates the interface downwards. This rarefaction wave generates a large, but non-constant, acceleration of the order of $1000g_{0}$, where $g_{0}$ is the acceleration due to gravity. Initial interface thicknesses are measured using a Rayleigh scattering diagnostic and the instability is visualized using planar laser-induced Mie scattering. Growth rates agree well with theoretical values, and with the inviscid, dynamic diffusion model of Duff et al. (Phys. Fluids, vol. 5, 1962, pp. 417–425) when diffusion thickness is accounted for, and the acceleration is weighted using inviscid Rayleigh–Taylor theory. The linear stability formulation of Chandrasekhar (Proc. Camb. Phil. Soc., vol. 51, 1955, pp. 162–178) is solved numerically with an error function diffusion profile using the Riccati method. This technique exhibits good agreement with the dynamic diffusion model of Duff et al. for small wavenumbers, but produces larger growth rates for large-wavenumber perturbations. Asymptotic analysis shows a $1/k^{2}$ decay in growth rates as $k\rightarrow \infty$ for large-wavenumber perturbations.

Spectral transfer processes in homogeneous magnetohydrodynamic (MHD) turbulence are investigated analytically by decomposition of the velocity and magnetic fields in Fourier space into helical modes. Steady solutions of the dynamical system which governs the evolution of the helical modes are determined, and a stability analysis of these solutions is carried out. The interpretation of the analysis is that unstable solutions lead to energy transfer between the interacting modes while stable solutions do not. From this, a dependence of possible interscale energy and helicity transfers on the helicities of the interacting modes is derived. As expected from the inverse cascade of magnetic helicity in 3-D MHD turbulence, mode interactions with like helicities lead to transfer of energy and magnetic helicity to smaller wavenumbers. However, some interactions of modes with unlike helicities also contribute to an inverse energy transfer. As such, an inverse energy cascade for non-helical magnetic fields is shown to be possible. Furthermore, it is found that high values of the cross-helicity may have an asymmetric effect on forward and reverse transfer of energy, where forward transfer is more quenched in regions of high cross-helicity than reverse transfer. This conforms with recent observations of solar wind turbulence. For specific helical interactions the relation to dynamo action is established. The present analysis provides new theoretical insights into physical processes where inverse cascade and dynamo action are involved, such as the evolution of cosmological and astrophysical magnetic fields and laboratory plasmas.

Strongly nonlinear three-dimensional interactions between a roll–streak structure and a Tollmien–Schlichting wave in plane Poiseuille flow are considered in this study. Equations governing the interaction at high Reynolds number originally derived by Bennett et al. (J. Fluid Mech., vol. 223, 1991, pp. 475–495) are solved numerically. Travelling wave states bifurcating from the lower branch linear neutral point are tracked to finite amplitudes, where they are observed to localize in the spanwise direction. The nature of the localization is analysed in detail near the relevant spanwise locations, revealing the presence of a singularity which slowly develops in the governing interaction equations as the amplitude of the motion is increased. Comparisons with the full Navier–Stokes equations demonstrate that the finite Reynolds number solutions gradually approach the numerical asymptotic solutions with increasing Reynolds number.

Flow in a horizontal channel exposed to external heating which results in sinusoidal temperature variations along the upper and lower walls with a phase shift between them has been studied using a combination of analytical and numerical methods. The most intense convection is observed when the upper and lower hot spots are located above each other. It has been demonstrated that the heating results in a significant reduction of the pressure gradient required to drive the flow when compared to a similar flow in an isothermal channel. The drag reduction is associated with the formation of separation bubbles which insulate the stream from direct contact with the bounding walls. The fluid inside of the bubbles rotates due to horizontal density gradients, which further reduces the required pressure gradient. The magnitude of the drag reduction depends on the phase shift between the heating patterns and can increase by up to threefold when compared to the drag reduction which can be achieved by heating only one wall. A detailed analysis of the associated heat fluxes has been presented.

The interaction between the large and the small scales of turbulence is investigated in a mixing layer, at a Reynolds number based on the Taylor microscale ($Re_{{\it\lambda}}$) of $250$, via direct numerical simulations. The analysis is performed in physical space, and the local vorticity root-mean-square (r.m.s.) is taken as a measure of the small-scale activity. It is found that positive large-scale velocity fluctuations correspond to large vorticity r.m.s. on the low-speed side of the mixing layer, whereas, they correspond to low vorticity r.m.s. on the high-speed side. The relationship between large and small scales thus depends on position if the vorticity r.m.s. is correlated with the large-scale velocity fluctuations. On the contrary, the correlation coefficient is nearly constant throughout the mixing layer and close to unity if the vorticity r.m.s. is correlated with the large-scale velocity gradients. Therefore, the small-scale activity appears closely related to large-scale gradients, while the correlation between the small-scale activity and the large-scale velocity fluctuations is shown to reflect a property of the large scales. Furthermore, the vorticity from unfiltered (small scales) and from low pass filtered (large scales) velocity fields tend to be aligned when examined within vortical tubes. These results provide evidence for the so-called ‘scale invariance’ (Meneveau & Katz, Annu. Rev. Fluid Mech., vol. 32, 2000, pp. 1–32), and suggest that some of the large-scale characteristics are not lost at the small scales, at least at the Reynolds number achieved in the present simulation.

We conduct experiments in a turbulent bubbly flow to study the nature of the transition between the classical $-5/3$ energy spectrum scaling for a single-phase turbulent flow and the $-3$ scaling for a swarm of bubbles rising in a quiescent liquid and of bubble-dominated turbulence. The bubblance parameter (Lance & Bataille J. Fluid Mech., vol. 222, 1991, pp. 95–118; Rensen et al., J. Fluid Mech., vol. 538, 2005, pp. 153–187), which measures the ratio of the bubble-induced kinetic energy to the kinetic energy induced by the turbulent liquid fluctuations before bubble injection, is often used to characterise bubbly flow. We vary the bubblance parameter from $b=\infty$ (pseudoturbulence) to $b=0$ (single-phase flow) over 2–3 orders of magnitude (0.01–5) to study its effect on the turbulent energy spectrum and fluctuations in liquid velocity. The probability density functions (PDFs) of the fluctuations in liquid velocity show deviations from the Gaussian profile for $b>0$, i.e. when bubbles are present in the system. The PDFs are asymmetric with higher probability in the positive tails. The energy spectra are found to follow the $-3$ scaling at length scales smaller than the size of the bubbles for bubbly flows. This $-3$ spectrum scaling holds not only in the well-established case of pseudoturbulence, but surprisingly in all cases where bubbles are present in the system ($b>0$). Therefore, it is a generic feature of turbulent bubbly flows, and the bubblance parameter is probably not a suitable parameter to characterise the energy spectrum in bubbly turbulent flows. The physical reason is that the energy input by the bubbles passes over only to higher wavenumbers, and the energy production due to the bubbles can be directly balanced by the viscous dissipation in the bubble wakes as suggested by Lance & Bataille (1991). In addition, we provide an alternative explanation by balancing the energy production of the bubbles with viscous dissipation in the Fourier space.

In spanwise rotating plane Couette flow (RPCF) a secondary flow dominated by three-dimensional roll-cell structures develops. At high enough rotation rates the flow exhibits a state of zero absolute vorticity at the centre of the channel, as described by Suryadi et al. (Phys. Rev. E, vol. 89, 2014, 033003). They suggested that the zero-absolute-vorticity state is caused by the secondary flow motion of the coherent roll-cell structure induced by the Coriolis force. In the present study we focus on the momentum transport caused by the roll-cell structure of laminar RPCF in order to further understand how the zero-absolute-vorticity state is maintained by the coherent roll cells. The flow is studied through stereoscopic particle image velocimetry measurements, which allow both the Reynolds shear stress and the wall shear stress to be quantified and used as measures of the momentum transport across the channel. Various types of roll-cell structures at different system rotation rates and the momentum transport induced by them are investigated, and the processes in which the momentum is transported in the wall-normal direction are discussed based on a displaced-particle argument as well as the production of the Reynolds stresses. It is shown that the wall-normal fluid motion driven by secondary flow of the roll-cell structure induces two different effects on the mean flow which conflict each other, the momentum transport in the wall-normal direction and the Coriolis acceleration, and the zero-absolute-vorticity state is a stable state where these two effects cancel each other.

In data-based control design, system-identification techniques are used to extract low-dimensional representations of the input–output map between actuators and sensors from observed data signals. Under realistic conditions, noise in the signals is present and is expected to influence the identified system representation. For the subsequent design of the controller, it is important to gauge the sensitivity of the system representation to noise in the observed data; this information will impact the robustness of the controller and influence the stability margins for a closed-loop configuration. Commonly, full Monte Carlo analysis has been used to quantify the effect of data noise on the system identification and control design, but in fluid systems, this approach is often prohibitively expensive, due to the high dimensionality of the data input space, for both numerical simulations and physical experiments. Instead, we present a framework for the estimation of statistical properties of identified system representations given an uncertainty in the processed data. Our approach consists of a perturbative method, relating noise in the data to identified system parameters, which is followed by a Monte Carlo technique to propagate uncertainties in the system parameters to error bounds in Nyquist and Bode plots. This hybrid approach combines accuracy, by treating the system-identification part perturbatively, and computational efficiency, by applying Monte Carlo techniques to the low-dimensional input space of the control design and performance/stability evaluation part. This combination makes the proposed technique affordable and efficient even for large-scale flow-control problems. The ARMarkov/LS identification procedure has been chosen as a representative system-identification technique to illustrate this framework and to obtain error bounds on the identified system parameters based on the signal-to-noise ratio of the input–output data sequence. The procedure is illustrated on the control design for flow over an idealized aerofoil with a trailing-edge splitter plate.

The virial corrections to short-time self- and collective diffusion coefficients as well as the effective viscosity are calculated for suspensions of hard spheres with the same radii and constant (blocked within the particle) magnetization modelled by a point dipole. Analytic, integral formulae derived from basic principles of statistical mechanics are provided for both cases – in the absence and in the presence of an external magnetic field. In the former case the diffusion and viscosity coefficients are evaluated numerically as a function of the strength of magnetic interactions between the particles and it is reported that the translational collective diffusion coefficient is significantly greater than all the other coefficients. In the presence of an external magnetic field the coefficients become anisotropic and are evaluated in the asymptotic regime of weak interparticle magnetic interactions.

A series of laboratory experiments was conducted to study the formation of internal boluses through the run up of periodic internal wave trains on a uniform slope/shelf topography in a two-layer stratified fluid system. In the experiments, the forcing parameters of the incident waves (wave amplitude and frequency) are varied for constant slope angle and layer depths. Simultaneous particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) measurements are used to calculate high resolution, two-dimensional velocity and density fields. Over the range of wave forcing conditions, four bolus formation types were observed: backward overturning into a coherent bolus, top breaking into a turbulent bolus, top breaking into a turbulent surge and forward breaking into a turbulent surge. Wave forcing parameters, including a wave Froude number $Fr$, a wave Reynolds number $Re$ and a wave steepness parameter $ka_{0}$, are used to relate initial wave forcing to a dominant bolus formation mechanism. Bolus characteristics, including the bolus propagation speed and turbulent components, are also related to wave forcing. Results indicate that for $Fr>0.20$ and $ka_{0}>0.40$, the generated boluses become more turbulent in nature. As wave forcing continues to increase further, boluses are no longer able to form.

In this paper, we have found two new nonlinear travelling wave solutions in pipe flows. We investigate possible asymptotic structures at large Reynolds number $R$ when wavenumber is independent of $R$ and identify numerically calculated solutions as finite $R$ realizations of a nonlinear viscous core (NVC) state that collapses towards the pipe centre with increasing $R$ at a rate $R^{-1/4}$. We also identify previous numerically calculated states as finite $R$ realizations of a vortex wave interacting (VWI) state with an asymptotic structure similar to the ones in channel flows studied earlier by Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). In addition, asymptotics suggests the possibility of a VWI state that collapses towards the pipe centre like $R^{-1/6}$, though this remains to be confirmed numerically.

We consider theoretically and experimentally the propagation in porous media of variable-density gravity currents containing a stably stratified density field, with most previous studies of gravity currents having focused on cases of uniform density. New thin-layer equations are developed to describe stably stratified fluid flows in which the density field is materially advected with the flow. Similarity solutions describing both the fixed-volume release of a distributed density stratification and the continuous input of fluid containing a distribution of densities are obtained. The results indicate that the density distribution of the stratification significantly influences the vertical structure of the gravity current. When more mass is distributed into lighter densities, it is found that the shape of the current changes from the convex shape familiar from studies of the uniform-density case to a concave shape in which lighter fluid accumulates primarily vertically above the origin of the current. For a constant-volume release, the density contours stratify horizontally, a simplification which is used to develop analytical solutions. For currents introduced continuously, the horizontal velocity varies with vertical position, a feature which does not apply to uniform-density gravity currents in porous media. Despite significant effects on vertical structure, the density distribution has almost no effect on overall horizontal propagation, for a given total mass. Good agreement with data from a laboratory study confirms the predictions of the model.

We investigate the entrainment of liquid films on a partially wetting plate vertically withdrawn from a reservoir of viscous liquid using a combination of diffuse-interface numerical simulation and lubrication analysis. So far available theoretical investigations were commonly conducted by focusing on separate parameter regions, and a complete description of the flow regimes with increasing plate speed is still missing. By solving the full Stokes equations, we present a complete scenario of film transition in the presence of moving contact line. With increasing plate speed, we identify numerically four successive flow regimes in terms of the interfacial morphologies: (1) a stationary meniscus, (2) a speed-independent thick film connected to the liquid bath through a stationary dimple, (3) coexistence of a thick film and the classical Landau–Levich–Derjaguin (LLD) film connected by a propagating capillary shock and (4) a film with a monotonically varying thickness. The characteristics of the film profiles in different regions of the interfaces are analysed with lubrication theory as applicable, and satisfactory agreements with the numerical results are obtained. In particular, we confirm that the onset of film deposition occurs at a vanishing apparent contact angle, consistent with the predictions of lubrication theory. Numerical results suggest that the critical capillary number for the onset of film deposition is smaller than that for the onset of LLD film despite the fact that it is higher than the experimentally observed one, showing that the thick film can be realized in the two-dimensional model. We also demonstrated that the LLD film is triggered by the bifurcation of the stationary dimple, which is found to admit multiple branches of stable and unstable solutions.

In this paper, the stability of a plane, unidirectional Couette flow of a dense granular material is investigated via the means of a normal mode stability analysis. Our studies are based on a continuum mechanical model for the flows of interest coupled with the constitutive expressions for the normal and the shear stresses of the granular material induced by the ${\it\mu}(I)$-rheology. According to our analysis, both the Couette gap and the wall velocity play a destabilizing role in the flows of interest as opposed to the initial concentration that acts as stabilizer. For sufficiently high Couette gaps and wall velocities, unstable modes are recovered. The predicted instability manifests itself through shear-induced dilatancy that, in turn, engenders particle migration and the formation of bulbs, similar to the ones that have been acquired through molecular dynamics simulations.

The scattering of sound by a finite rigid plate with a finite poroelastic extension interacting with an unsteady acoustic source is investigated to determine the effects of porosity, elasticity and the length of the extension when compared to a purely rigid plate. The problem is solved using the Wiener–Hopf technique, and an approximate Wiener–Hopf factorisation process is implemented to yield reliable far-field results quickly. Importantly, finite chord-length effects are taken into account, principally the interaction of a rigid leading-edge acoustic field with a poroelastic trailing-edge acoustic field. The model presented discusses how the poroelastic trailing-edge property of owls’ wings could inspire quieter aeroacoustic designs in bladed systems such as wind turbines, and provides a framework for analysing the potential noise reduction of these designs.

Quantitative mathematical models describe planar, spontaneous, reaction wave propagation (Zeldovich, Combust. Flame, vol. 39, 1980, pp. 211–214) in a finite hot spot volume of reactive gas. The results describe the complete thermomechanical response of the gas to a one-step, high-activation-energy exothermic reaction initiated by a tiny initial temperature non-uniformity in a gas at rest with uniform pressure. Initially, the complete conservation equations, including all transport terms, are non-dimensionalized to identify parameters that quantify the impact of viscosity, conduction and diffusion. The results demonstrate unequivocally that transport terms are tiny relative to all other terms in the equations, given the relevant time and length scales. The asymptotic analyses, based on the reactive Euler equations, describe both induction and post-induction period models for a fast heat release rate (induction time scale short compared to the acoustic time of the spot), as well as a modest heat release rate (induction time scale equivalent to the acoustic time). Analytical results are obtained for the fast heating rate problem and emphasize the physics of near constant-volume heating during the induction period. Weak hot spot expansion is the source of fluid expelled from the original finite volume and is a ‘piston-effect’ source of acoustic mechanical disturbances beyond the spot. The post-induction period is characterized by the explosive appearance of an ephemeral, spatially uniform high-temperature, high-pressure spot embedded in a cold, low-pressure environment. In analogy with a shock tube the subsequent expansion process occurs on the acoustic time scale of the spot and will be the source of shocks propagating beyond the spot. The modest heating rate induction period is characterized by weakly compressible phenomena that can be described by a novel system of linear wave equations for the temperature, pressure and induced velocity perturbations driven by nonlinear chemical heating, which provides physical insights difficult to obtain from the more familiar ‘Clarke equation’. When the heating rate is modest, reaction terms in the post-induction period Euler equations exhibit a form of singular behaviour in the high-activation-energy limit, implying the need to use a nonlinear exponential scaling for time and space, developed originally to describe spatially uniform thermal explosions (Kassoy, Q. J. Mech. Appl. Maths, vol. 30, 1977, pp. 71–89). Here again the result will be the explosive appearance of an ephemeral spatially uniform high-temperature, high-pressure hot spot. These results demonstrate that an initially weak temperature non-uniformity in a finite hot spot can be the source of acoustic and shock wave mechanical disturbances in the gas beyond the spot that may be related to rocket engine instability and engine knock.

The behaviour of a single droplet in an immiscible external fluid, submitted to shear flow is investigated using numerical simulations. The surface of the droplet is modelled by a Boussinesq–Scriven constitutive law involving the interfacial viscosities and a constant surface tension. A numerical method using Loop subdivision surfaces to represent droplet interface is introduced. This method couples boundary element method for fluid flows and finite element method to take into account the stresses due to the surface dilational and shear viscosities and surface tension. Validation of the numerical scheme with respect to previous analytic and computational work is provided, with particular attention to the viscosity contrast and the shear and dilational viscosities characterized both by a Boussinesq number $B_{q}$. Then, influence of equal surface viscosities on steady-state characteristics of a droplet in shear flow are studied, considering both small and large deformations and for a large range of bulk viscosity contrast. We find that small deformation analysis is surprisingly predictive at moderate and high surface viscosities. Equal surface viscosities decrease the Taylor deformation parameter and tank-treading angle and also strongly modify the dynamics of the droplet: when the Boussinesq number (surface viscosity) is large relative to the capillary number (surface tension), the droplet displays damped oscillations prior to steady-state tank-treading, reminiscent from the behaviour at large viscosity contrast. In the limit of infinite capillary number $Ca$, such oscillations are permanent. The influence of surface viscosities on breakup is also investigated, and results show that the critical capillary number is increased. A diagram $(B_{q};Ca)$ of breakup is established with the same inner and outer bulk viscosities. Additionally, the separate roles of shear and dilational surface viscosity are also elucidated, extending results from small deformation analysis. Indeed, shear (dilational) surface viscosity increases (decreases) the stability of drops to breakup under shear flow. The steady-state deformation (Taylor parameter) varies nonlinearly with each Boussinesq number or a linear combination of both Boussinesq numbers. Finally, the study shows that for certain combinations of shear and dilational viscosities, drop deformation for a given capillary number is the same as in the case of a clean surface while the inclination angle varies.

We examine the dynamics of a thin viscous liquid film on the outer surface of a solid sphere rotating around its vertical axis in the presence of gravity. An asymptotic model describing the evolution of the film thickness is derived in the rotating frame based on the lubrication approximation. The model includes the centrifugal and gravity forces and the stabilizing effect of surface tension. Depending on the values of the parameters, the problem admits different types of steady states: one with a uniformly positive film thickness, or those with one or two dry zones on the sphere. We prove that all steady states are energy minimizers and hence global attractors for axisymmetric states.

We compute the equilibrium contact angles for an evaporating droplet whose contact line lies over a solid wedge. The stability of the liquid interface is also considered and an integro-differential equation for small perturbations is deduced. The analysis of this equation yields criteria for stability and instability of the contact line, where the instability represents transition from the pinned to unpinned contact line representative of stick–slip motion.