This work is focused on the development of an analytical model for the axisymmetric deformation of droplets with a given velocity lag in an outer flow. This leads either to oscillations around a deformed equilibrium shape or to strong deformations which may ultimately lead to their breakup when the equilibrium shape loses its stability. To obtain an evolution equation for the droplet shape evolution, it is first supposed that the droplet deforms like a spheroid. Then, a balance between kinetic energy of deformation, surface energy creation, external pressure work and viscous dissipation within the droplet is written. This is close to the approach developed in the classical Taylor analogy breakup or droplet deformation and breakup model. The main originality of the present modelling is that pressure work can be computed at any time, assuming an outer potential flow. Pressure is assumed to work on either the whole droplet or only on the forward half of the droplet due to flow separation. Results of this model are then compared with reported droplet oscillation frequencies. For suddenly accelerated droplets, oscillations are present in the liquid–liquid metal case and successfully compared with some new direct numerical simulation results. Comparison is also successful when compared with previous results on droplet oscillating at terminal falling velocity, whether in the liquid–liquid case or for falling raindrop. Lastly, comparisons are made with previously published secondary fragmentation experiments and direct numerical simulations whether in a shock tube or for droplets falling in a cross-flow.

A joint experimental and theoretical study is made of instability mechanisms of locally confined internal gravity wave beams in a stratified fluid. Using as forcing a horizontal cylinder that is oscillated harmonically in the direction of beam propagation makes it possible to generate coherent finite-amplitude internal wave beams whose spatial profile comprises no more than a single wavelength. For forcing amplitude above a certain threshold depending on the driving frequency, such thin wave beams are observed to undergo an instability that involves two subharmonic perturbations with wavepacket-like spatial structure. Although it bears resemblance to the triadic resonant instability (TRI) of small-amplitude sinusoidal waves, the present instability cannot be predicted by TRI theory as the primary wave is not nearly monochromatic, but instead contains broadband wavenumber spectrum. In contrast, the experimental observations are in good agreement with the predictions of a formal linear stability analysis based on Floquet theory. Finally, experimental evidence is presented that transverse beam variations induce a horizontal mean flow of the streaming type and greatly subdue the instability.

The dynamics of Reynolds stresses and turbulent kinetic energy (TKE) in Langmuir turbulence are analysed using data of large-eddy simulations with the wave phase resolved. It is found that the streamwise and spanwise Reynolds normal stresses and the Reynolds shear stress vary appreciably with the wave phase, while the vertical normal stress is only weakly dependent on the wave phase. Budget analyses indicate that the production due to wave straining and the effects associated with turbulence pressure are the dominant mechanisms for the wave-phase variation of Reynolds stresses. The accumulative effect of wave–turbulence interactions on TKE is then investigated using the Lagrangian average. It is discovered that the energy transfer from wave to turbulence is contributed by two mechanisms. The first mechanism is the turbulence production by the Lagrangian mean wave shearing and the mean shear stress, which is consistent with the traditional wave-phase-averaged model. The second mechanism, which is not accounted for in previous studies, is the correlation between the wave-phase variation of the Reynolds shear stress and the wave orbital shearing. A model is proposed for the second mechanism. Comparison of the frequency spectrum with Craik–Leibovich simulation results shows that the correlation effect can affect the turbulence fluctuations at time scales around the wave period, indicating the importance of this effect on Reynolds stresses and TKE.

This work presents a detailed analysis of the flow structures relevant to extreme wall shear stress events for turbulent pipe flow direct numerical simulation data at a friction Reynolds number $\textit {Re}_{\tau} \approx 1000$. The results reveal that extreme positive wall-friction events are located below an intense sweep (Q4) event originated from a strong quasi-streamwise vortex at the buffer region. This vortex transports high streamwise momentum from the overlap and the outer layers towards the wall, giving rise to a high-speed streak within the inner region. This vortical structure also relates to regions with extreme wall-normal velocity. Consequently, the conditional fields of turbulence production and viscous dissipation exhibit peaks whose magnitudes are approximately 25 times higher than the ensemble mean quantities in the vicinity of the extreme positive events. An analysis of the turbulent inertia force reveals that the energetic quasi-streamwise vortex acts as an essential source of momentum at the near-wall region. Similarly, extremely rare backflow events are studied. An examination of the wall-normal vorticity and velocity vector fields shows an identifiable oblique vortical structure along with two other large-scale roll modes. These counter-rotating motions contribute to the formation of backflow events by transporting streamwise momentum from the inner to the outer region, creating a large-scale meandering low-speed streak. It is found that extreme events are clustered below large-scale structures of positive streamwise momentum that interact with near-wall low-speed streaks, related to regions densely populated with vortical structures. Finally, a three-dimensional model is proposed to conceptualise the flow dynamics associated with extreme events.

We study singular jets from the collapse of drop-impact craters, when the drop and pool are of different immiscible liquids. The fastest jets emerge from a dimple at the bottom of the rebounding crater, when no bubble is pinched off. The parameter space is considerably more complex than for identical liquids, revealing intricate compound-dimple shapes. In contrast to the universal capillary–inertial drop pinch-off regime, where the neck radius scales as $R\sim t^{2/3}$, for a purely inertial air dimple the collapse has $R \sim t^{1/2}$. The bottom dimple dynamics is not self-similar but possesses memory effects, being sensitive to initial and boundary conditions. Sequence of capillary waves can therefore mould the air dimple into different collapse shapes, such as bamboo-like and telescopic forms. The finest jets are only $12\ \mathrm {\mu }\textrm {m}$ in diameter and the normalized jetting speeds are up to one order of magnitude larger than for jets from bursting bubbles. We study the cross-over between the two power laws approaching the singularity. The singular jets show the earliest cross-over into the inertial regime. The fastest jets can pinch off a toroidal micro-bubble from the cusp at the base of the jet.

The breakup of water droplets when exposed to high-speed gas flows is investigated using both high-magnification shadowgraphy experiments as well as fully three-dimensional numerical simulations, which account for viscous as well as capillary effects. After thorough validation of the simulations with respect to the experiments, we elucidate the ligament formation process and the effect of surface tension. By Fourier decomposition of the flow field, we observe the development of specific azimuthal modes, which destabilize the liquid sheet surrounding the droplet. Eventually, the liquid sheet is ruptured, which leads to the formation of ligaments. We further observe the ligament formation and shedding to be a recurrent process. While the first ligament shedding weakly depends on the Weber number, subsequent shedding processes seem to be driven primarily by inertia and the vortex shedding in the wake of the deformed droplet.

The dynamics of coherent structures forming in a turbulent shallow mixing layer (ML) between two parallel streams advancing in a constant-depth, open channel is investigated using three-dimensional, time-accurate simulations. The large channel length to flow depth ratio ($L_{x}/H = 400\text{--}800$) allows characterization of the spatial evolution of shallow MLs until the mean velocity difference between the two streams becomes less than 3% of the initial value at the end of the splitter plate. Away from the ML origin, the dynamics and coherence of the Kelvin–Helmholtz (KH) billows are affected by the destabilizing effect of the mean shear between the two streams and by the stabilizing effect of the bed friction. A linear decay of the entrainment coefficient α with the bed-friction factor, S, applies only over the region where merging of neighbouring KH billows is still observed (transition to quasi-equilibrium regime). At larger distances from the origin, where the billows are severely stretched in the streamwise direction before being destroyed, the rates of increase of the ML width, δ, and centreline shift, lcy, become very small and α is exponentially decaying with increasing S toward zero (quasi-equilibrium regime). During the initial stages of the quasi-equilibrium regime where the KH vortices are severely stretched, the ML assumes an undulatory shape in horizontal planes. New relationships are proposed to characterize the downstream variation of the non-dimensional ML width and centreline shift over the transition and quasi-equilibrium regimes. During the transition to equilibrium regime, the ML boundary on the fast-stream side remains close to vertical, while that on the slow-stream side becomes strongly tilted. The ML boundary on the slow-stream side becomes again close to vertical once the quasi-equilibrium regime is reached. During the transition to the equilibrium regime, the passage of the KH billows and the generation of streamwise cells of secondary flow generate regions of high instantaneous bed shear stress, such that the region where the erosive capacity of the flow peaks does not correspond to the fast stream. The paper also investigates the effects of flow shallowness and initial velocity ratio between the two streams on the turbulent kinetic energy inside the ML, the depth-averaged lateral momentum fluxes, the passage frequency and size of the KH billows and the wavelength and period of the undulatory motions of the ML during the early stages of the quasi-equilibrium regime.

This study presents a class of steadily translating two-dimensional approximately circular vortices. The proposed solutions take the form of a superposition of two nearly concentric vorticity patches with zero net vorticity. Exact quasi-monopolar solutions of this type are found for propagation speeds that are much less than typical azimuthal velocities. While all V-states obtained are shown to be formally unstable, a large subset of configurations is characterized by very low growth rates. Fully nonlinear simulations reveal that such nearly stable eddies can propagate large distances from the point of origin while maintaining their structure and intensity. Therefore, the proposed solutions can serve as models of geophysical vortices that are known to drift relative to the ambient fluid, often exhibiting remarkable longevity and resilience to external perturbations.

We study the pattern formation of a thin film flowing under an inclined planar substrate. The phenomenon is studied in the context of the Rayleigh–Taylor instability using the lubrication equation. Inspired by experimental observations, we numerically study the thin film response to a streamwise-invariant sinusoidal initial condition. The numerical response shows the emergence of predominant streamwise-aligned structures, modulated along the direction perpendicular to the flow, called rivulets. Oscillations of the thickness profile along the streamwise direction do not grow significantly when the inclination is very large or the liquid layer very thin. However, for small inclinations or thick films, streamwise perturbations grow on rivulets. A secondary stability analysis of one-dimensional and steady rivulets reveals a strong stabilization mechanism for large inclinations or very thin films. The theoretical results are compared with experimental measurements of the streamwise oscillations of the rivulet profile, showing a good agreement. The emergence of rivulets is investigated by studying the impulse response. Both the experimental observation and the numerical simulation show a marked anisotropy favouring streamwise-aligned structures. A weakly nonlinear model is proposed to rationalize the levelling of all but streamwise-aligned structures.

One of the successful feedback controls for skin-friction drag reduction designed by Choi et al. (J. Fluid Mech., vol. 262, 1994, pp. 75–110), called ‘opposition control’, has a limitation in application because the sensors need to be placed slightly away from the wall, i.e. at $y^+ = 10$, and measure the instantaneous wall-normal velocity. In the present study we train convolutional neural networks using the database of uncontrolled turbulent channel flow at $Re_{\tau } = 178$ to extract the spatial distributions of the wall shear stresses and pressure that closely represent the wall-normal velocity at $y^+ = 10$. The correlations between the predicted wall-normal velocities at $y^+ = 10$ from the wall-variable distributions and true ones are very high, and they are 0.92, 0.96 and 0.96 for the streamwise and spanwise wall shear stresses and pressure, respectively. We perform feedback controls of turbulent channel flow with instantaneous blowing and suction determined by the trained convolutional neural networks from the measured wall-variable distributions. The predicted wall-normal velocities during the controls have higher energy at small to intermediate scales than the true ones, which degrades the control performance in skin-friction drag reduction. By applying a low-pass filter to the predicted wall-normal velocities to remove those scales, we reduce skin-friction drag by up to 18 % whose amount is comparable to that by opposition control. The convolutional neural networks trained at $Re_{\tau } = 178$ are also applied to a higher Reynolds number flow ($Re_{\tau } = 578$), and provide a successful skin-friction drag reduction of 15 %.

The equations governing the evolution of quantum vortex defects subject to twist are derived in standard hydrodynamic form. Vortex defects emerge as solutions of the Gross–Pitaevskii equation, that by Madelung transformation admits a hydrodynamic description. Here, we consider a vortex defect subject to superposed twist due to the rotation of the phase of the wave function. We prove that, when twist is present, the corresponding Hamiltonian is non-Hermitian and determine the effect of twist on the energy expectation value of the system. We show how twist diffusion may trigger linear instability, a property directly related to the non-Hermiticity of the Hamiltonian. We derive the correct continuity equation and, by applying defect theory, we obtain the correct momentum equation. Finally, by coupling twist kinematics and vortex dynamics we determine the full set of hydrodynamic equations governing quantum vortex evolution subject to twist.

A new methodology to accurately and efficiently examine the radio frequency blackout phenomenon during the hypersonic reentry process is introduced and validated. The current state-of-the-art thermochemical modelling of $\textrm {CO}_2$ flows is reviewed and one-dimensional stagnation line studies are performed in order to determine a suitable chemical mechanism for the electron density modelling. Hypersonic computational fluid dynamics (CFD) simulations are performed with a simplified chemical model including only neutral species, in order to calculate the flow field surrounding the ExoMars Schiapparelli module in flight conditions. A novel decoupled CFD approach is then applied where the calculation of the electron density is performed separately using a computationally inexpensive Lagrangian approach. Subsequently, a ray tracing algorithm is applied in order to model the propagation of electromagnetic waves in the wake flow past the ExoMars vehicle accounting for collisions between electrons and gas particles. The numerical results of the proposed novel approach for blackout analysis consisting of CFD, Lagrangian and ray tracing algorithms are in good agreement with the flight data.

Ruelle predicted that the maximal amplification of perturbations in homogeneous isotropic turbulence is exponential $\exp ({\sigma \sqrt {Re} \,t})$ (where $\sigma \sqrt {Re}$ is the maximal Lyapunov exponent). In our earlier works, we predicted that the maximal amplification of perturbations in fully developed turbulence is faster than exponential and is given by $\exp ({\sigma \sqrt {Re} \sqrt {t} +\sigma _1 t})$ where $\sigma \sqrt {Re} \sqrt {t}$ is much larger than $\sigma \sqrt {Re} \, t$ for small $t$. That is, we predicted superfast initial amplification of perturbations. Built upon our earlier numerical verification of our prediction, here, we conduct a large numerical verification with resolution up to $2048^3$ and Reynolds number up to $6210$. Our direct numerical simulation here confirms our analytical prediction. Our numerical simulation also demonstrates that such superfast amplification of perturbations leads to superfast nonlinear saturation. We conclude that such superfast amplification and superfast nonlinear saturation of ever existing perturbations suggest a mechanism for the generation, development and persistence of fully developed turbulence.

Extensive studies have focused on the self-propulsion of a droplet in a viscous environment driven by the Marangoni effect in the absence of inertial effects. In order to capture the influence of inertia on the self-propulsion of a droplet, we use the singular perturbation solution for small but finite Reynolds number ($Re$) flow past a spherical droplet with inhomogeneous surface tension. We calculate the swimming speed and the corresponding flow fields generated by the droplet in an axisymmetric unbounded medium at $O(Re^2)$. The present results reveal how the choice of the stress parameter $\sigma$, which is the ratio of the first two modes of the induced stress field, distinguishes between the different swimming styles, and determines the role of inertia on the swimming speed, energy expenditure and swimming efficiency of the droplet. Inertia enhances the swimming speed and the associated swimming efficiency of the droplet by abating the energy expenditure. It is striking to observe how a droplet swimmer with $\sigma <0$ has a competitive advantage over a rigid squirmer with an equivalent surface activity due to the existence of an internal flow. We independently treat the potential influence of the viscosity ratio on the swimming properties of the droplet at finite $Re$. Additionally, using linear stability analysis, we provide insights into the stability of the estimated migration velocity at $O(Re)$. We argue that the droplet achieves a distinct stable equilibrium velocity, which occurs due to the inertial effect of the surrounding medium.

Flow pulsatility is ubiquitous in biological and biomedical fluid dynamics. Cardiovascular flow, the most well-studied pulsatile flow, has benefited from decades of fundamental studies of the temporal influences of basic parameter changes in simplified models. A similar approach is employed herein on a relatively unstudied, canonical flow configuration. Using experiments and simulations, we examine highly pulsatile flow over a surface-mounted bluff body. The pulsatile (i.e. with no flow reversal) inflow waveform is sinusoidal, and the inflow pulsation frequency is varied from low-frequency, quasi-steady pulsation to high-frequency pulsatility. A wake regime map encompassing the range of pulsation frequency is created, and a mechanistic explanation of the regimes observed is put forth. Finally, we introduce a non-dimensional parameter applicable to pulsatile flows and point out remarkable similarity to the formation time parameter associated with vortex ring generation, including a similar critical value, despite substantive differences in the flow configurations.

A wall-resolved large-eddy simulation of a symmetric Joukowski aerofoil with a 12 % thickness at a Reynolds number of 250 000, a Mach number of 0.4 and zero incidence angle is performed in order to investigate the effect of using a serrated trailing edge on the acoustic feedback event that generates a tonal noise. The acoustic feedback is investigated in detail to emphasise the interaction between the upstream travelling acoustic expansion wave and the laminar separation bubble. The simulation shows that the serrated trailing edges may result in a significant reduction of the tonal noise. This paper provides detailed investigations into the noise reduction mechanisms. The main finding is that the presence of a serrated trailing edge decreases the amplitude of the acoustic source pressure in the transitional region and gives rise to destructive phase interference in the wall pressure fluctuations in the vicinity of the trailing edge which weakens the acoustic feedback loop.

The Ginzburg–Landau equation (GLE) can phenomenologically model several key features of non-equilibrium systems including those in fluid mechanics. Its validity in real flows, however, remains questionable. Here, we show that the linear GLE can be formulated such that it has the same Wentzel–Kramers–Brillouin (WKB) approximation as for the linear global stability problem in open shear flows. We use the GLE to model the linear global modes of three different wakes and find that it can accurately capture the linear growth rate and frequency to first order in the WKB approximation. Furthermore, we find that it can also provide the shapes of the direct and adjoint eigenvectors and the regions of maximal structural sensitivity. The proposed model requires only the basic flow as input, but gives robust predictions and is computationally inexpensive. As well as opening up new possibilities for GLE-based control strategies, the proposed model makes accurate stability calculations possible, even for some computationally intractable open shear flows.

Interaction between body motion and fluid motion is considered inside a nonlinear viscous wall layer, with this unsteady two-way coupling leading to impact of the body on the wall. The present paper involves a reduced system analysis which is shown to be consistent with computational solutions from direct numerical simulations for a basic flat-plate shape presented in an allied paper (Palmer & Smith, J. Fluid Mech., 2020). The occurrence of impact depends mainly on fluid parameters and initial conditions. The body considered is translating upstream or downstream relative to the wall. Subsequent analysis focusses on the unusual nature of the impact at the leading edge. The impacting flow structure is found to have two nonlinear viscous–inviscid regions lying on either side of a small viscous region. The flow properties in the regions dictate the lift and torque which drive the body towards the wall. Pronounced flow separations are common as the impact then cuts off the mass flux in the gap between the body and the wall; here, a nonlinear similarity solution sheds extra light on the separations. Comparisons are made between results from direct simulations and asymptotics at increased flow rate.

Reconfiguration events in turbulent mixed convection, i.e. the superposition of thermal and forced flow contributions, at the two different Richardson numbers $Ri=1.5$ and $Ri=3.7$ and similar Rayleigh numbers of $Ra \approx 10^8$ are investigated with tomographic particle image velocimetry in combination with local temperature measurements. For both cases, the three-dimensional velocity fields reflect diagonally aligned large-scale circulations (LSC) switching their alignment by rotating their axes around a pivot located at the centre of the LSC, while the temperatures perform a translation movement of the structures in agreement with earlier temperature-based investigations. For the high $Ri$ case, the switching process of the observed spontaneous reconfigurations is induced by a reversing thermal flow contribution while the forced flow contribution is constant. Furthermore, it is shown that a secondary roll structure, which drives the reconfiguration process in Rayleigh–Bénard convection, also exists in mixed convection. However, in the latter, the flow reversals are triggered by different structures which accumulate and release their kinetic energy according to a proper orthogonal decomposition analysis. In contrast, for the low $Ri$ case, the structure formation during continuous reconfigurations is governed by a Taylor- or Görtler-type instability. This means that the forced convection substantially affects the reconfiguration mechanism of these structures. Therefore, the reconfigurations cannot be described by a simple superposition of structures associated with the two flow contributions as for the high $Ri$.

We investigate theoretically, on the basis of the steady Stokes equations for a viscous incompressible fluid, the flow induced by a stokeslet located on the centre axis of two coaxially positioned rigid disks. The stokeslet is directed along the centre axis. No-slip boundary conditions are assumed to hold at the surfaces of the disks. We perform the calculation of the associated Green's function in large parts analytically, reducing the spatial evaluation of the flow field to one-dimensional integrations amenable to numerical treatment. To this end, we formulate the solution of the hydrodynamic problem for the viscous flow surrounding the two disks as a mixed boundary-value problem, which we then reduce to a system of four dual integral equations. We show the existence of viscous toroidal eddies arising in the fluid domain bounded by the two disks, manifested in the plane containing the centre axis through adjacent counter-rotating eddies. Additionally, we probe the effect of the confining disks on the slow dynamics of a point-like particle by evaluating the hydrodynamic mobility function associated with axial motion. Thereupon, we assess the appropriateness of the commonly employed superposition approximation and discuss its validity and applicability as a function of the geometrical properties of the system. Additionally, we complement our semi-analytical approach by finite-element computer simulations, which reveals a good agreement. Our results may find applications in guiding the design of microparticle-based sensing devices and electrokinetic transport in small-scale capacitors.