It was thought that the wing hinge position can be tuned to stabilize an uncontrolled fly. However here, our Floquet stability analysis shows that the hinge position has a weak dependence on the flight stability. As long as the hinge position is within the fly’s body length, both hovering and ascending flight are unstable. Instead, there is an optimal hinge position, $h^{\ast }$, at which the ascending speed is maximized. $h^{\ast }$ is approximately half way between the centre of mass and the top of the body. We show that the optimal $h^{\ast }$ is associated with the anti-resonance of the body–wing coupling, and is independent of the stroke amplitude. At $h^{\ast }$, the torque due to wing inertia nearly cancels the torque due to aerodynamic lift, minimizing the body oscillation thus maximizing the upward force. Our analysis using a simplified model of two coupled masses further predicts, $h^{\ast }=(m_{t}/2m_{w})(g/\unicode[STIX]{x1D714}^{2})$. These results suggest that the ascending speed, in addition to energetics and stability, is a trait that insects are likely to optimize.

The present study focuses on the behaviour of fully resolved oblate spheroids in turbulent channel flows using the lattice Boltzmann method (LBM). Mean and maximum drag reductions of 1.3 % and 4.4 %, respectively, are observed at a solid-phase volume fraction of 10 % by using oblate spheroids of aspect ratio $\unicode[STIX]{x1D706}=1/3$ and equivalent diameter $D_{eq}/H=1/6.5$. The behaviour of oblate spheroids and spheres are found to be different in the near-wall region, where oblate spheroids, in contrast to spheres, do not augment the near-wall turbulence. Strong reduction of spanwise and wall-normal fluid velocity fluctuations in the buffer layer by using oblate spheroids is observed. It is also seen that reduction of Reynolds shear stress cannot solely guarantee the occurrence of drag reduction, as spherical particles increase the drag in spite of the reduction in Reynolds shear stress. Moreover, in drag-reduced particle-laden flows, although the spanwise spacing of streaks increases, vortices are stronger and smaller. Thus, in addition to the streak cycle of turbulence regeneration, the vortices generated by particles contribute to higher vortex strengths. By examining the role of the oblate spheroid concentration, it is found to mainly influence the streak spacing, with a minor effect on the strength of vortices. Drag reduction in oblate-spheroid-laden flows is thus attributed to the reduction of fluid velocity fluctuations in the transverse directions and enhancement of the spanwise streak spacing. Higher vortex strength has less effect on the initiation of drag reduction but can influence the level of drag reduction and explains the lower drag-reduction effect of rigid particles compared to polymers. By quadrant analysis, sweeps and ejections are seen to have less contribution to the Reynolds shear stress by using oblate spheroids. With respect to the particle data, oblate spheroids tend to orient with their symmetry axis mainly along the wall-normal direction. Very close to the wall, spanwise orientation is dominant and wall-normal alignment of the symmetry axis is reduced. Furthermore, oblate spheroids move faster than the fluid, before reaching a similar streamwise velocity in the channel centre. In contrast, spheres move slower than the fluid in the buffer region, as well as showing a local concentration peak near the wall. It is also seen that the angular velocity of oblate spheroids decays very rapidly whereas spheres have a higher rotation rate which shows a smoother reduction towards the centre.

The Richtmyer–Meshkov instability (RMI) is experimentally investigated in a vertical shock tube using a broadband initial condition imposed on an interface between a helium–acetone mixture and argon (Atwood number $A\approx 0.7$). In the present work, a shear layer is introduced at the interface to serve as a statistically repeatable, broadband initial condition to the RMI, and the density interface is accelerated by either an $M=1.6$ or $M=2.2$ planar shock wave. The development of the ensuing mixing layer is investigated using simultaneous planar laser-induced fluorescence (PLIF) and particle image velocimetry (PIV). PLIF images are processed to reveal the light-gas mole fraction, while PIV particle image pairs yield corresponding two-component planar velocity results. Field structure and distribution are explored through probability density functions (PDFs), and a decomposition is performed on concentration and velocity results to obtain a mean flow field and define fluctuations. Simultaneous concentration and velocity field measurements allow – for the first time in this regime – experimentally determined turbulence quantities such as Reynolds stresses, turbulent mass-flux velocities and turbulent kinetic energy to be obtained. We show that by the latest times the mixing layer has passed the turbulent threshold, and there is evidence of turbulent mixing occurring sooner for the higher Mach number case. Interface measurements show nonlinear growth with a power-law fit to the thickness data, and that integral measurements of mixing layer thickness are proportional to threshold measurements. Spectral analysis demonstrates the emergence of an inertial range with a slope ${\sim}k^{-5/3}$ when considering both density and velocity effects in planar turbulent kinetic energy (TKE) measurements.

Steady states of inviscid, compressible and axisymmetric swirling flows of a single-phase, inert, thermodynamically supercritical fluid in a rotating, finite-length, straight, long circular pipe are studied. The fluid thermodynamic behaviour is modelled by the van der Waals equation of state. A nonlinear partial differential equation for the solution of the flow streamfunction is derived from the fluid equations of motion in terms of the inlet flow specific total enthalpy, specific entropy and circulation functions. This equation reflects the complicated, nonlinear thermo-physical interactions in the flows, specifically when the inlet state temperature and density profiles vary around the critical thermodynamic point, flow compressibility is significant and the inlet swirl ratio is high. Several types of solutions of the resulting nonlinear ordinary differential equation for the axially independent case describe the flow outlet state when the pipe is sufficiently long. The approach is applied to an inlet flow described by a solid-body rotation with uniform profiles of the axial velocity and temperature. The solutions are used to form the bifurcation diagrams of steady compressible flows of real fluids as the inlet swirl level and the centreline inlet density are increased at a fixed inlet Mach number and temperature. Focus is on heavy-molecule fluids with low values of $R/C_{v}$. Computed results provide theoretical predictions of the critical swirl levels for the exchange of stability of the columnar state and for the appearance of non-columnar states and of vortex breakdown states as a function of inlet centreline density. The difference in the dynamical behaviour between that of a calorically perfect gas and of a real gas is explored. The analysis sheds new fundamental light on the complex dynamics of high-Reynolds-number, compressible, subsonic swirling flows of real gases.

Gas turbine engines are prone to the phenomenon of thermoacoustic instability, which is highly detrimental to their components. Recently, in turbulent combustors, it was observed that the transition to thermoacoustic instability occurs through an intermediate state, known as intermittency, where the system exhibits epochs of ordered behaviour, randomly appearing amidst disordered dynamics. We investigate the onset of intermittency and the ensuing self-organization in the reactive flow field, which, under certain conditions, could result in the transition to thermoacoustic instability. We characterize this transition from a state of disordered and incoherent dynamics to a state of ordered and coherent dynamics as pattern formation in the turbulent combustor, utilizing high-speed flame images representing the distribution of the local heat release rate fluctuations, flow field measurements (two-dimensional particle image velocimetry), unsteady pressure and global heat release rate signals. Separately, through planar Mie scattering images using oil droplets, the collective behaviour of small scale vortices interacting and resulting in the emergence of large scale coherent structures is illustrated. We show the emergence of spatial patterns using statistical tools used to study transitions in other pattern forming systems. In this paper, we propose that the intertwined and highly intricate interactions between the wide spatio-temporal scales in the flame, the flow and the acoustics are through pattern formation.

The acoustics of a straight annular lined duct containing a swirling mean flow is considered. The classical Ingard–Myers impedance boundary condition is shown not to be correct for swirling flow. By considering behaviour within the thin boundary layers at the duct walls, the correct impedance boundary condition for an infinitely thin boundary layer with swirl is derived, which reduces to the Ingard–Myers condition when the swirl is set to zero. The correct boundary condition contains a spring-like term due to centrifugal acceleration at the walls, and consequently has a different sign at the inner (hub) and outer (tip) walls. Examples are given for mean flows relevant to the interstage region of aeroengines. Surface waves in swirling flows are also considered, and are shown to obey a more complicated dispersion relation than for non-swirling flows. The stability of the surface waves is also investigated, and as in the non-swirling case, one unstable surface wave per wall is found.

A qualitative explanation for the scaling of energy dissipation by high-Reynolds-number fluid flows in contact with solid obstacles is proposed in the light of recent mathematical and numerical results. Asymptotic analysis suggests that it is governed by a fast, small-scale Rayleigh–Tollmien–Schlichting instability with an unstable range whose lower and upper bounds scale as $Re^{3/8}$ and $Re^{1/2}$, respectively. By linear superposition, the unstable modes induce a boundary vorticity flux of order $Re^{1}$, a key ingredient in detachment and drag generation according to a theorem of Kato. These predictions are confirmed by numerically solving the Navier–Stokes equations in a two-dimensional periodic channel discretized using compact finite differences in the wall-normal direction, and a spectral scheme in the wall-parallel direction.

Understanding and predicting dense granular flows is of importance in geology and industrial applications. Still, most theoretical work has been limited to flows and packings composed of discs or spheres, a narrow subset of all possible packings. To advance our understanding of more realistic flows we here study the granular rheology of ellipses in steady-state flow with a focus on the effects of elongation and interparticle friction. We carry out novel numerical simulations of amorphous granular flows in a shear cell under confining pressure, at constant shear rate and at various aspect ratios. Both frictionless and frictional particles are considered. The various rheological curves follow the semi-empirical constitutive relations previously found for granular flows composed of discs or spheres. At the shear jamming point one finds well-defined packings, all characterised by their own set of critical parameters such as critical packing fraction, effective friction, etc. Packings composed of frictionless or almost frictionless particles are found to have a non-monotonic dependence of the macroscopic friction but a monotonic increase in packing fraction as the aspect ratio increases. For packings composed of particles with high interparticle friction the reverse is found. While frictionless packings are found to be hypostatic (except in the disc limit) frictional packings are remarkably close to the isostaticity point of having three contacts per particle. Both frictional and frictionless packings are found to have an increasing nematic ordering as the aspect ratio increases. The onset of a rolling, rather than sliding, motion between very frictional particles diminish this nematic ordering substantially. These findings put new and previously unknown bounds on the packing ratios and yield criteria for these amorphous packings at shear jamming.

Variations in the local stability of the flow in a Taylor–Couette cell can be imposed by adding an axial Poiseuille flow and a radial flow associated with one or both of the cylinders being permeable. At a given rotation rate of the inner cylinder, this results in adjacent regions of the flow that can be simultaneously stable, convectively unstable, and absolutely unstable, making this system fit for studying global modes of instability. To this end, building on the existing stability analysis in absolute modes developing over axially invariant base flows, we consider the case of axially varying base flows in systems for which the outer cylinder is impermeable, and the inner cylinder is a weakly permeable membrane through which the radial flow is governed by Darcy’s law. The frameworks of linear and nonlinear global modes are used to describe the instabilities and assess the results of direct numerical simulations using a dedicated pseudospectral method. Three different axially evolving set-ups are considered. In the first, fluid injection occurs along the full inner cylinder. In the second, fluid extraction occurs along the full inner cylinder. Besides its fundamental interest, this set-up is relevant to filtration devices. In the third, fluid flux through the inner cylinder evolves from extraction to injection as cross-flow reversal occurs. In agreement with the global mode analyses, the numerical simulations develop centrifugal instabilities above the predicted critical rotation rates and downstream of the predicted axial locations. The global mode analyses do not fully explain, however, that the instabilities observed in the numerical simulations take the form of axial stacks of wavepackets characterized by jumps of the temporal frequency.

Stability and nonlinear analyses were employed to study symmetric and asymmetric steady flows through a straight channel with a smooth constriction with 50 % occlusion. Linear stability analysis was carried out to determine the unstable regions and the critical Reynolds numbers for the primary and secondary global instabilities. The primary bifurcation demonstrated an intricate aspect: the three-dimensional modes transfer their energy to the two-dimensional mode, which causes a symmetry breaking of the flow. This behaviour could be observed for Reynolds number lower than the critical, showing that this primary bifurcation is hysteretical. The secondary bifurcation also presented subcritical behaviour, exhibiting a pitchfork diagram with a large hysteretic curve. Given the subcritical character of both bifurcations, the relevance of non-normal growth in these flows were assessed, so that their convective mechanisms were exhaustively investigated. In addition, we could show that, for the secondary instability, optimal initial disturbances with large enough initial energy were able to promote a subcritical nonlinear saturation for a Reynolds number lower than the critical. The physical mechanism behind the transition process occurred by nonlinear interaction between the two- and three-dimensional modes, which established oscillatory behaviour, moreover, this energy exchange between the modes led the flow to the nonlinear saturated state. Therefore, the two-dimensional modes play a key role in the primary and secondary bifurcations of this system.

The sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for binary fluids with moving contact lines are studied by asymptotic analysis and numerical simulations. The effects of the mobility number as well as a phenomenological relaxation parameter on the boundary condition are considered. In asymptotic analysis, we consider both the cases that the mobility number is proportional to the Cahn number and the square of the Cahn number, and derive the sharp-interface limits for several set-ups of the boundary relaxation parameter. It is shown that the sharp-interface limit of the phase-field model is the standard two-phase incompressible Navier–Stokes equations coupled with several different slip boundary conditions. Numerical results are consistent with the analysis results and also illustrate the different convergence rates of the sharp-interface limits for different scalings of the two parameters.

We present the results of an experimental and theoretical investigation into the influence of proximate boundaries on the motion of an rotationally oscillating sphere in a viscous fluid. The angular oscillations of the sphere are controlled using the magnetic torque generated by a spatially uniform, oscillatory magnetic field which interacts with a small magnet embedded within the sphere. We study the motion of the sphere in the vicinity of stationary walls that are parallel and perpendicular to the rotational axis of the sphere, and near a second passive sphere that is non-magnetic and free to move. We find that rigid boundaries introduce viscous resistance to motion that acts to suppress the oscillations of the driven sphere. The amount of viscous resistance depends on the orientation of the wall with respect to the axis of rotation of the oscillating sphere. A passive sphere also introduces viscous resistance to motion, but for this case the rotational oscillations of the active sphere establish a standing wave that imparts vorticity to the fluid and induces oscillations of the passive sphere. The standing wave is analogous to the case of an oscillating plate in a viscous fluid; the amplitude of the wave decays exponentially with radial distance from the surface of the oscillating sphere. The standing wave introduces a phase lag between the motion of the active sphere and the response of the passive sphere which increases linearly with separation distance.

The effects of triboelectricity in a small-scale fluidized bed of polyethylene particles were investigated by imaging the particle layer in the vicinity of the column wall and by measuring the pressure drop across the bed. The average charge on the particles was altered by changing the relative humidity of the gas. A triboelectric charging model coupled with a computational fluid dynamics–discrete element method (CFD-DEM) model was utilized to simulate gas–particle flow in the bed. The electrostatic forces were evaluated based on a particle–particle particle–mesh method, accounting for the surface charge on the insulating walls. It was found that simulations with fixed and uniform charge distribution among the particles capture remarkably well both the agglomeration of the particles on the wall and the associated decrease in the pressure drop across the bed. With a dynamic tribocharging model, the charging rate had to be accelerated to render the computations affordable. Such simulations with an artificial acceleration significantly over-predict charge segregation and the wall becomes rapidly sheeted with a single layer of strongly charged particles.

Streaks are a common feature of disturbed boundary-layer flows. They play a central role in transient growth mechanisms and are a building block of self-sustained structures. Most theoretical work has focused on streaks that are periodic in the spanwise direction, but in this work we consider a single spatially localised streak embedded into a Blasius boundary layer. For small streak amplitudes, we show the perturbation can be described in terms of a set of eigenmodes that correspond to an isolated streak/roll structure. These modes are new, and arise from a bi-global eigenvalue calculation; they decay algebraically downstream and may be viewed as the natural three-dimensional extension of the classical two-dimensional Libby & Fox (J. Fluid Mech., vol. 17 (3), 1963, pp. 433–449) solutions. Despite their bi-global nature, we show that a subset of these eigenmodes (including the slowest decaying) is fundamentally related to the solutions first presented by Luchini (J. Fluid Mech., vol. 327, 1996, pp. 101–116), as derived for spanwise-periodic disturbances (at small spanwise wavenumber). This surprising connection is made by an analysis of the far-field decay of the bi-global state. We also address the fully non-parallel downstream development of nonlinear streaks, confirming that the aforementioned eigenmodes are recovered as the streak/roll decays downstream. Encouraging comparisons are made with available experimental data.

The effects of shear on double-diffusive fingering and on the settling-driven instability are assessed by means of a transient growth analysis. Employing Kelvin waves within a linearized framework allows for the consideration of time-dependent waveforms in uniform shear. In this way, the effects of boundaries and of shear-driven instability modes can be eliminated, so that the influence of shear on the double-diffusive and settling-driven instabilities can be analysed in isolation. Shear is seen to dampen both instabilities, which is consistent with previous findings by other authors. The shear damping is more pronounced for parameter values that produce larger unsheared growth. These trends can be explained in terms of instantaneous linear stability results for the unsheared case. For both double-diffusive and settling-driven instabilities, low Prandtl number ($Pr$) values result in less damping and an increased importance of the Orr mechanism, for which a quantitative scaling law is obtained.

Axisymmetric boundary layers are studied using integral analysis of the governing equations for axial flow over a circular cylinder. The analysis includes the effect of pressure gradient and focuses on the effect of transverse curvature on boundary layer parameters such as shape factor ($H$) and skin-friction coefficient ($C_{f}$), defined as $H=\unicode[STIX]{x1D6FF}^{\ast }/\unicode[STIX]{x1D703}$ and $C_{f}=\unicode[STIX]{x1D70F}_{w}/(0.5\unicode[STIX]{x1D70C}U_{e}^{2})$ respectively, where $\unicode[STIX]{x1D6FF}^{\ast }$ is displacement thickness, $\unicode[STIX]{x1D703}$ is momentum thickness, $\unicode[STIX]{x1D70F}_{w}$ is the shear stress at the wall, $\unicode[STIX]{x1D70C}$ is density and $U_{e}$ is the streamwise velocity at the edge of the boundary layer. Relations are obtained relating the mean wall-normal velocity at the edge of the boundary layer ($V_{e}$) and $C_{f}$ to the boundary layer and pressure gradient parameters. The analytical relations reduce to established results for planar boundary layers in the limit of infinite radius of curvature. The relations are used to obtain $C_{f}$ which shows good agreement with the data reported in the literature. The analytical results are used to discuss different flow regimes of axisymmetric boundary layers in the presence of pressure gradients.

The values of the highest Lyapunov exponent (HLE) $\unicode[STIX]{x1D706}_{1}$ for turbulent flow in a plane channel at Reynolds numbers up to $Re_{\unicode[STIX]{x1D70F}}=586$ are determined. The instantaneous and statistical properties of the corresponding leading Lyapunov vector (LLV) are investigated. The LLV is calculated by numerical solution of the Navier–Stokes equations linearized about the non-stationary base solution corresponding to the developed turbulent flow. The base turbulent flow is calculated in parallel with the calculation of the evolution of the perturbations. For arbitrary initial conditions, the regime of exponential growth ${\sim}\exp (\unicode[STIX]{x1D706}_{1}t)$ which corresponds to the approaching of the perturbation to the LLV is achieved already at $t^{+}<50$. It is found that the HLE increases with increasing Reynolds number from $\unicode[STIX]{x1D706}_{1}^{+}\approx 0.021$ at $Re_{\unicode[STIX]{x1D70F}}=180$ to $\unicode[STIX]{x1D706}_{1}^{+}\approx 0.026$ at $Re_{\unicode[STIX]{x1D70F}}=586$. The LLV structures are concentrated mainly in a region of the buffer layer and are manifested in the form of spots of increased fluctuation intensity localized both in time and space. The root-mean-square (r.m.s.) profiles of the velocity and vorticity intensities in the LLV are qualitatively close to the corresponding profiles in the base flow with artificially removed near-wall streaks. The difference is the larger concentration of LLV perturbations in the vicinity of the buffer layer and a relatively larger (by approximately 80 %) amplitude of the vorticity pulsations. Based on the energy spectra of velocity and vorticity pulsations, the integral spatial scales of the LLV structures are determined. It is found that LLV structures are on average twice narrower and twice shorter than the corresponding structures of the base flow. The contribution of each of the terms entering into the expression for the production of the perturbation kinetic energy is determined. It is shown that the process of perturbation development is essentially dictated by the inhomogeneity of the base flow, as well as by the presence of transversal motion in it. Neglecting of these factors leads to a significant underestimation of the perturbation growth rate. The presence of near-wall streaks in the base flow, on the contrary, does not play a significant role in the development of the LLV perturbations. Artificial removal of streaks from the base flow does not change the character of the perturbation growth.

This paper focuses on the characteristics of the regime C flow (Tatsuno & Bearman, J. Fluid Mech., vol. 211, 1990, pp. 157–182) around an oscillating circular cylinder in still water. The regime C flow is characterised by the formation of large-scale vortex cores arranged as opposed von Kármán vortex streets, resulting from a regular switching of vortex shedding directions with respect to the axis of oscillation. Both Floquet analysis and direct numerical simulations (DNS) are performed to investigate the two- (2-D) and three-dimensional (3-D) instabilities. The present study reveals that the low-wavenumber 3-D instability can emerge slightly before the 2-D instability in regime C. In total, five spanwise vortex modes were identified: (i) standing-wave pattern, S-mode; (ii) travelling-wave pattern, T-mode; (iii) mixed ST-mode; (iv) X-type vortex pattern, X-mode; and (v) U-type vortex pattern, U-mode. The modal analysis conducted in this study demonstrates that the vortex patterns and the corresponding spatial and temporal modulations of the dynamic loads of the S-, T- and mixed ST-modes are mainly induced by the 3-D instability of a single wavenumber. The characteristics of the X-mode are due to the superposition of the 3-D instabilities of multiple wavenumbers. The U-mode is dominated by a 2-D instability and its interaction with 3-D instabilities. The domain size dependence study demonstrates that the regime C flow is very sensitive to the spanwise length of the computational domain. The subcritical nature of the regime C flow is responsible for the discrepancy in the marginal stability curves obtained by independent Floquet stability analysis, DNS and physical experiments.

The unsteady aerodynamic and aerothermal phenomena resulting from the interaction between a shock-induced supersonic boundary-layer flow and a wall-mounted cylinder are investigated. Experiments were conducted in a shock tube at three different post-shock unit Reynolds numbers and a single Mach number to investigate the effects of differing ratios of inviscid and viscous temporal scales on the flow development. Two cylinder heights were studied: ‘large’ and ‘small’ protuberances based on calculated boundary-layer thicknesses. Heat-flux measurements on the shock-tube wall were performed using an ultra-fast-response temperature sensitive paint and verified by independent thermocouple measurements. High-speed schlieren provided visualizations of the inviscid flow phenomena. The unsteady shock-wave/boundary-layer interaction ahead of the cylinder resulted in high transient heat loading on the wall and caused transition to turbulence of the incoming laminar boundary layer. Once this incoming boundary layer had naturally transitioned, the region of enhanced heat flux collapsed back towards the cylinder; during this process, heat transfer in the immediate wake increased significantly. The overall heat flux upstream of the cylinder was higher for the large protuberance, whereas the downstream heat flux was generally higher for the small protuberance. In the case of the large protuberance, the viscous scaling appeared to best collapse the upstream heat-flux development for the three different unit Reynolds numbers, though the agreement downstream was less satisfactory. Neither the viscous nor the inviscid scaling appeared to adequately collapse the development for the small protuberance.

We explore the rotational stability of hovering flight in an idealized two-dimensional model. Our model is motivated by an experimental pyramid-shaped object (Weathers et al., J. Fluid Mech, vol. 650, 2010, pp. 415–425; Liu et al., Phys. Rev. Lett., vol. 108, 2012, 068103) and a computational $\wedge$-shaped analogue (Huang et al., Phys. Fluids, vol. 27 (6), 2015, 061706; Huang et al., J. Fluid Mech., vol. 804, 2016, pp. 531–549) hovering passively in oscillating airflows; both systems have been shown to maintain rotational balance during free flight. Here, we attach the $\wedge$-shaped flyer at its apex in oscillating flow, allowing it to rotate freely akin to a pendulum. We use computational vortex sheet methods and we develop a quasi-steady point-force model to analyse the rotational dynamics of the flyer. We find that the flyer exhibits stable concave-down ($\wedge$) and concave-up ($\vee$) behaviour. Importantly, the down and up configurations are bistable and co-exist for a range of background flow properties. We explain the aerodynamic origin of this bistability and compare it to the inertia-induced stability of an inverted pendulum oscillating at its base. We then allow the flyer to flap passively by introducing a rotational spring at its apex. For stiff springs, flexibility diminishes upward stability but as stiffness decreases, a new transition to upward stability is induced by flapping. We conclude by commenting on the implications of these findings for biological and man-made aircraft.