In this article the erratic coupling that can occur in screeching supersonic twin jets is characterised. Non-stationary acoustic analysis is used to investigate the temporal behaviour of the coupling phenomena. The results show that where the phase between the jets is time varying, the screech tone experiences interruptions. The interruptions are either correlated and experienced by both jets or are anti-correlated and only by one. During the anti-correlated interruption, the uninterrupted jet screeches as an isolated jet. The instantaneous velocity field shows that for the majority of snapshots during an acoustic interruption, the jets do not exhibit a coupled oscillation. When the jets are uninterrupted, they are oscillating in either a coupled symmetric or anti-symmetric mode. This behaviour manifests at a condition between two operating points characterised by different coupling modes. It suggests the interruptions arise due to a competition between two global modes of the flow. Despite the existence of multiple acoustic tones in the region where these modes are competing, analysis of the individual jets reveals energetic structures with only a single wavelength. It is found that jets whose own oscillation is characterised by a single wavelength can, through coupling either symmetrically or anti-symmetrically about their symmetry plane, produce different acoustic tones. These findings are consistent across three experimental facilities. The observed modes are a function of the jet spacing and nozzle pressure, therefore future studies investigating other spacings must recharacterise the encountered coupled modes. This article provides the signatures to characterise the behaviour for future studies.

We experimentally investigate the breakup mechanisms and probability of Hinze-scale bubbles in turbulence. The Hinze scale is defined as the critical bubble size based on the critical mean Weber number, across which the bubble breakup probability was believed to have an abrupt transition from being dominated by turbulence stresses to being suppressed completely by the surface tension. In this work, to quantify the breakup probability of bubbles with sizes close to the Hinze scale and to examine different breakup mechanisms, both bubbles and their surrounding tracer particles were simultaneously tracked. From the experimental results, two Weber numbers, one calculated from the slip velocity between the two phases and the other acquired from local velocity gradients, are separated and fitted with models that can be linked back to turbulence characteristics. Moreover, we also provide an empirical model to link bubble deformation to the two Weber numbers by extending the relationship obtained from potential flow theory. The proposed relationship between bubble aspect ratio and the Weber numbers seems to work consistently well for a range of bubble sizes. Furthermore, the time traces of bubble aspect ratio and the two Weber numbers are connected using the linear forced oscillator model. Finally, having access to the distributions of these two Weber numbers provides a unique way to extract the breakup probability of bubbles with sizes close to the Hinze scale.

We prove a duality between the functional forms of the Faxén formulas associated with a particle of a given shape and material composition and the corresponding singularity solutions for the velocity disturbances induced by that particle, and extend it to the case of systems with coupled transport processes, enabling the solution of a large family of problems via Faxén methods. Prior approaches to constructing proofs of duality of Faxén formulas and Stokes-flow singularities relied on knowledge of all boundary conditions on all particle surfaces, viz. the Lorentz reciprocal theorem approach. We recognized that, in order to bypass the complexity of boundary conditions one can instead invoke energy methods that give reciprocity between operators rather than between specific stress and velocity fields. We derive reciprocal relations between operators, from which we demonstrate that the Faxén/singularity duality is a consequence of a generalized reciprocal relation between conjugate thermodynamic variables. We use our reciprocal relations to derive expressions for the hydrodynamic force on a sphere of arbitrary composition, the hydrodynamic stresslet exerted on a deformable droplet in an arbitrary velocity field, the phoretic force exerted on a rigid particle in the thin double-layer limit in response to arbitrary externally imposed field and the total stresslet on a charged particle in an arbitrary velocity field, i.e. an electroviscous Faxén law.

This paper presents a detailed investigation of the role played by the wave steepness in connection with the statistical properties of the surface elevation and fluid kinematics in irregular, directionally spread, deep-water wave fields initially given by a JONSWAP spectrum. Using ensembles of large wave fields obtained from fully nonlinear simulations, we first consider the statistical properties of the surface elevation. In that connection we determine the probability density functions (PDFs) of the surface and crest elevations for wave fields of relatively small to unprecedentedly large steepness, and compare them with theoretical results from the literature in order to establish the latter's accuracy. We then consider certain statistical aspects of the fluid kinematics found at the surface and of the fluid kinematics accompanying large crests, which to our knowledge marks the first investigation of these properties in the literature. We first determine the PDFs of the horizontal fluid velocities and accelerations as well as the vertical fluid acceleration at the surface. Next, we investigate the joint PDF of the surface elevation and each of the velocities and accelerations at the surface, and use it to determine the surface elevations for which the velocities and accelerations at the surface are large. We then present an analysis of the largest fluid velocities and accelerations found in the vicinity of large crests, and compute the PDFs of these quantities. Finally, we consider the PDFs of the location at which the largest velocities and accelerations occur relative to the crest.

Expressing the evolution equations for the filtered velocity gradient tensor (FVGT) in the strain-rate eigenframe provides an insightful way to disentangle and understand various processes such as strain self-amplification, vortex stretching and tilting, and to consider their properties at different scales in the flow. Using data from direct numerical simulation of the forced Navier–Stokes equation, we consider the relative importance of local and non-local terms in the FVGT eigenframe equations across the scales using statistical analysis. The eigenframe rotation rate (that drives vorticity tilting) is shown to exhibit highly non-Gaussian fluctuations even at large scales due to kinematic effects, but dynamically, the anisotropic pressure Hessian plays a key role. The anisotropic pressure Hessian conditioned on the eigenvalues and principal vorticity components exhibits highly nonlinear behaviour at low values of normalized local gradients, with important modelling implications. We derive a generalization of the Lumley triangle that allows us to show that the pressure Hessian has a preference for two-component axisymmetric configurations at small scales, with a transition to a more isotropic state at larger scales. Correlations between the sub-grid stress and other terms in the eigenframe equations are considered, highlighting the coupling between the sub-grid and nonlinear amplification terms, with the sub-grid term playing an important role in regularizing the system. These results provide useful guidelines for improving Lagrangian models of the FVGT, since current models fail to capture a number of subtle features observed in our results.

Centrifugal instability of the boundary layer is known to induce spiral vortices over a rotating slender cone that is facing an axial inflow. This paper shows how a deviation from the symmetry of such axial inflow affects the boundary layer instability over a rotating slender cone with half-angle $\psi =15^\circ$. The spiral vortices are experimentally detected using their thermal footprint on the cone surface for both axial and non-axial inflow conditions. In axial inflow, the onset and growth of the spiral vortices are governed by the local rotational speed ratio $S$ and Reynolds number $Re_l$ in agreement with the literature. During their growth, the spiral vortices significantly affect the mean velocity field as they entrain and bring high-momentum flow closer to the wall. It is found that the centrifugal instability induces these spiral vortices in non-axial inflow as well; however, the asymmetry of the non-axial inflow inhibits the initial growth of the spiral vortices, and they appear at higher local rotational speed ratio and Reynolds number, where the azimuthal variations in the instability characteristics (azimuthal number $n$ and vortex angle $\phi$) are low.

This article presents a computational approach for determining the optimal slip velocities on any given shape of an axisymmetric micro-swimmer suspended in a viscous fluid. The objective is to minimize the power loss to maintain a target swimming speed, or equivalently to maximize the efficiency of the micro-swimmer. Owing to the linearity of the Stokes equations governing the fluid motion, we show that this PDE-constrained optimization problem reduces to a simpler quadratic optimization problem, whose solution is found using a high-order accurate boundary integral method. We consider various families of shapes parameterized by the reduced volume and compute their swimming efficiency. Among those, prolate spheroids were found to be the most efficient micro-swimmer shapes for a given reduced volume. We propose a simple shape-based scalar metric that can determine whether the optimal slip on a given shape makes it a pusher, a puller or a neutral swimmer.

In this work we propose and demonstrate a Fresnel-lens-inspired method to focus multiple laser-induced shock waves through time-delay superposition at arbitrary locations. While the principle works for any geometry, we demonstrate that this method already achieves focusing with two pairs of photoacoustic shock wave emitters located on a line centred around the acoustic axis ($z$) in a quasi-two-dimensional liquid geometry. Each emitter pair is created by focusing one laser pulse simultaneously at two spots with a spatial light modulator at $z=0\ \mathrm {\mu }$m with $y=\pm 145\ \mathrm {\mu }$m and $y=\pm 75\ \mathrm {\mu }$m. The delays between the emitters necessary to vary the location of the focus from $z\approx 0$ to ${\sim }206\ \mathrm {\mu }$m are $35$ and $0$ ns, respectively. We find that the location of constructive superposition is significantly closer to the origin than what would be expected for linear waves in homogeneous media. This is confirmed with simulations using an Euler solver that shows the importance of finite-amplitude effects. The simulated dynamics are in reasonable agreement with our measurements. Finally, pressure gains at various locations along the acoustic axis are tested with the response of gaseous microbubbles acting as pressure probes. The measurements agree with calculated pressure ratios at different positions.

We investigate the role of linear mechanisms in the emergence of nonlinear horizontal self-propelled states of a heaving foil in a quiescent fluid. Two states are analysed: a periodic state of unidirectional motion and a quasi-periodic state of slow back and forth motion around a mean horizontal position. The states emergence is explained through a fluid–solid Floquet stability analysis of the non-propulsive symmetric base solution. Unlike a purely hydrodynamic analysis, our analysis accurately determines the locomotion states onset. An unstable synchronous mode is found when the unidirectional propulsive solution is observed. The obtained mode has a propulsive character, featuring a mean horizontal velocity and an asymmetric flow that generates a horizontal force accelerating the foil. An unstable asynchronous mode, also featuring flow asymmetry and a non-zero velocity, is found when the back and forth state is observed. Its associated complex multiplier introduces a slow modulation of the flapping period, agreeing with the quasi-periodic nature of the back and forth regime. The temporal evolution of this perturbation shows how the horizontal force exerted by the flow is alternatively propulsive or resistive over a slow period. For both modes, an analysis of the velocity and force perturbation time-averaged over the flapping period is used to establish physical instability criteria. The behaviour for a large solid-to-fluid density ratio of the modes is thus analysed. The asynchronous fluid–solid mode converges towards the purely hydrodynamic one, whereas the synchronous mode becomes marginally unstable in our analysis not converging to the purely hydrodynamic analysis where it is never destabilised.

Recent attempts to use deep learning for super-resolution reconstruction of turbulent flows have used supervised learning, which requires paired data for training. This limitation hinders more practical applications of super-resolution reconstruction. Therefore, we present an unsupervised learning model that adopts a cycle-consistent generative adversarial network (CycleGAN) that can be trained with unpaired turbulence data for super-resolution reconstruction. Our model is validated using three examples: (i) recovering the original flow field from filtered data using direct numerical simulation (DNS) of homogeneous isotropic turbulence; (ii) reconstructing full-resolution fields using partially measured data from the DNS of turbulent channel flows; and (iii) generating a DNS-resolution flow field from large-eddy simulation (LES) data for turbulent channel flows. In examples (i) and (ii), for which paired data are available for supervised learning, our unsupervised model demonstrates qualitatively and quantitatively similar performance as that of the best supervised learning model. More importantly, in example (iii), where supervised learning is impossible, our model successfully reconstructs the high-resolution flow field of statistical DNS quality from the LES data. Furthermore, we find that the present model has almost universal applicability to all values of Reynolds numbers within the tested range. This demonstrates that unsupervised learning of turbulence data is indeed possible, opening a new door for the wide application of super-resolution reconstruction of turbulent fields.

Viscoelastic liquids are usually blends of a polymeric solute and a Newtonian solvent. In the presence of a temperature gradient, stratification of these solutes can take place via the Soret effect. Here, we investigate the classical Marangoni instability problem for a thin viscoelastic film considering this binary aspect of the liquid. The film, bounded above by a deformable free surface, is subjected to heating from below by a solid substrate. Linear stability analysis performed numerically for perturbations of finite wavelength (short-wave perturbations) reveals that both monotonic and oscillatory instabilities can emerge in this system. In the presence of Soret diffusion, the interaction between thermocapillary and solutocapillary forces is found to give rise to two different oscillatory instabilities, of which one mode was overlooked previously, even for the Newtonian binary mixtures. As a principal result of this work, we provide a complete picture of the susceptibility to different instability modes based on model parameter values. Finally, an approximate model is developed under the framework of long-wave analysis, which can qualitatively depict the stability behaviour of the system without numerically solving the problem.

This paper investigates pre- and post-reconnection dynamics of an unperturbed trefoil knotted vortex for circulation-based Reynolds numbers $Re_\varGamma = 2\times 10^3$ and $6\times 10^3$ by means of direct numerical simulations based on an adaptive mesh refinement framework. Companion coherent-vorticity preserving large-eddy simulations are also carried out on a uniform Cartesian grid. The complete vortex structure and flow evolution are simulated, including reconnection and subsequent separation into a smaller and a larger vortex ring, and the resulting helicity dynamics. The self-advection velocity before reconnection is found to scale with inviscid parameters. The reconnection process, however, occurs earlier (and more rapidly) in the higher Reynolds number case due to higher induced velocities associated with a thinner vortex core. The vortex propagation velocities after reconnection and separation are also affected by viscous effects, more prominently for the smaller vortex ring; the larger one is shown to carry the bulk of the helicity and enstrophy after reconnection. The domain integrated, or total helicity, $H(t)$, does not significantly change up until reconnection, at which point it varies abruptly due to the rapid dissipation of helicity caused by super-helicity hotspots localized at the reconnection sites. The total helicity dissipation rate predicted by the large-eddy simulation agrees reasonably well with the direct numerical simulation results, with a significant contribution from the modelled subgrid-scale stresses. On the other hand, variations in the vortex centreline helicity, $H_C(t)$, and the vortex-tube-integrated helicity $H_V(t)$ are less sensitive to the reconnection process. Periodic vortex bursting events are also observed and are shown to be due to converging axial flow velocities in the detached vortex rings at later stages of evolution.

We study the effect of the Coriolis force on centrifugal buoyancy-driven convection in a rotating cylindrical shell with inner cold wall and outer hot wall. This is done by performing direct numerical simulations for increasing inverse Rossby number $Ro^{-1}$ from zero (no Coriolis force) to $20$ (very large Coriolis force) and for Rayleigh number $Ra$ from $10^{7}$ to $10^{10}$ and Prandtl number $Pr = 0.7$, corresponding to air. We invoke the thin-shell limit, which neglects the curvature and radial variations of the centripetal acceleration. As $Ro^{-1}$ increases from zero, the system forms an azimuthal bidirectional wind that reaches its maximum momentum at an optimal $Ro^{-1}_{opt}$, associated with a maximal skin-friction coefficient $C_f$ and a minimal Nusselt number $Nu$. Just beyond $Ro^{-1}_{opt}$, the wind weakens and an axial, quasi-two-dimensional cyclone, corotating with the system, begins to form. A local ‘turbulence’ inverse Rossby number (non-dimensionalised by the eddy turnover time) determines the onset of cyclone formation for all $Ra$, when its value reaches approximately $4$. At $Ro^{-1} \gg Ro^{-1}_{opt}$, the system falls into the geostrophic regime with a sudden drop in $Nu$. The bidirectional wind for $Ro^{-1} \le Ro^{-1}_{opt}$ is a feature of this system, as it hastens the boundary layer transition from laminar to turbulent, towards the ultimate regime. We see the onset of this transition at $Ra=10^{10}$ and $Ro^{-1}\simeq Ro^{-1}_{opt}$, although the mean flow profile has not yet fully collapsed on the Prandtl–von Kármán (logarithmic) law.

We present a numerical study of the flow states and reversals of the large-scale circulation (LSC) in a two-dimensional circular Rayleigh–Bénard cell. Long-time direct numerical simulations are carried out in the Rayleigh number ($Ra$) range $10^{7} \leqslant Ra \leqslant 10^{8}$ and Prandtl number ($Pr$) range $2.0 \leqslant Pr \leqslant 20.0$. We found that a new, long-lived, chaotic flow state exists, in addition to the commonly observed circulation states (the LSC in the clockwise and counterclockwise directions). The circulation states consist of one primary roll in the middle and two secondary rolls near the top and bottom circular walls. The primary roll becomes stronger and larger, while the two secondary rolls diminish, with increasing $Ra$. Our results suggest that the reversal of the LSC is accompanied by the secondary rolls growing, breaking the primary roll and then connecting to form a new primary roll with reversed direction. We mapped out the phase diagram of the existence of the LSC and the reversal in the $Ra\text{--}Pr$ space, which reveals that the flow is in the circulation states when $Ra$ is large and $Pr$ is small. The reversal of the LSC can only occur in a limited $Pr$ range. The phase diagram can be understood in terms of competition between the thermal and viscous diffusions. We also found that the internal flow states manifested themselves into global properties such as Nusselt and Reynolds numbers.

Direct numerical simulations at $Re=200$ have been conducted of the flow past rows of tandem cylinders. It is shown that when the pitch between the two upstream cylinders is large, the wake downstream is characterised by a two-row vortex structure. Placing a third body on the wake centreline in the majority of this two-row structure has basically no impact both upstream and downstream – the third body is cloaked. However, a region is identified where the placement of a body suppresses vortex shedding from the first cylinder and the two-row structure is destroyed, globally broadcasting the presence of the third body. The effect is shown to occur for different third-body shapes. To understand the existence of this broadcasting region, local instability analysis is conducted which shows the majority of the two-row structure to be convectively unstable, with only a small region adjacent to the rear of the second cylinder that is absolutely unstable. This suggests only bodies placed close to the second body will trigger the global change, and this is supported by a global sensitivity analysis and observation from the simulations. However, neither the local analysis nor the global sensitivity analysis explains the presence of a lower limit for the third-body position that will trigger a global change. However the simulation results clearly show that a third body placed very close to the second body does not trigger this change.

We investigated the linear stability of pipe flow with anisotropic slip length at the wall by considering streamwise and azimuthal slip separately as the limiting cases. Our numerical analysis shows that streamwise slip renders the flow less stable but does not cause instability. The exponential decay rate of the least stable mode appears to be $\propto Re^{-1}$ when the Reynolds number is sufficiently large. Azimuthal slip can cause linear instability if the slip length is sufficiently large. The critical Reynolds number can be reduced to a few hundred given large slip lengths. In addition to numerical calculations, we present a proof of the linear stability of the flow to three-dimensional yet streamwise-independent disturbances for arbitrary Reynolds number and slip length, as an alternative to the usual energy analysis. Meanwhile we derived analytical solutions to the eigenvalue and eigenvector, and explained the structure of the spectrum and the dependence of the leading eigenvalue on the slip length. The scaling of the exponential decay rate of streamwise independent modes is shown to be rigorously $\propto Re^{-1}$. Our non-modal analysis shows that overall streamwise slip reduces the non-modal growth, and azimuthal slip has the opposite effect. Nevertheless, both slip cases still give the $Re^2$-scaling of the maximum non-modal growth and the most amplified disturbances are still streamwise rolls, which are qualitatively the same as in the no-slip case.

A swirling wake flow submitted to an adverse pressure gradient is studied by bifurcation analysis, modal analysis and direct numerical simulations. In contrast to experiments in diverging tubes, the adverse pressure gradient is imposed by the presence of a downstream axisymmetric obstacle centred on the vortex axis. Different adverse pressure gradients are investigated by modifying the obstacle radius, which results in the deceleration of the vortex axial velocity. Hence, vortex breakdown occurs for a sufficiently large pressure gradient. We observe a spiral vortex breakdown type without any recirculation bubble, which contrasts with classical spiral vortex breakdown developing in the bubble wake. A weakly nonlinear analysis is performed to characterize this self-sustained instability. The resulting Landau equation reveals the sub-critical character of this Hopf bifurcation, highlighting a sub-critical vortex breakdown. In addition, the stabilization mechanism of this spiral vortex breakdown caused by an off-centre displacement of the downstream axisymmetric obstacle is investigated by direct numerical simulations. Nonlinear dynamics, such as a quasi-periodic state, is observed as a consequence of nonlinear interactions between the spiral vortex breakdown and the misalignment of the obstacle before the stabilization occurs.

In turbulent wall sheared thermal convection, there are three different flow regimes, depending on the relative relevance of thermal forcing and wall shear. In this paper, we report the results of direct numerical simulations of such sheared Rayleigh–Bénard convection, at fixed Rayleigh number $Ra=10^{6}$, varying the wall Reynolds number in the range $0 \leqslant Re_w \leqslant 4000$ and Prandtl number $0.22 \leqslant Pr \leqslant 4.6$, extending our prior work by Blass et al. (J. Fluid Mech., vol. 897, 2020, A22), where $Pr$ was kept constant at unity and the thermal forcing ($Ra$) varied. We cover a wide span of bulk Richardson numbers $0.014 \leqslant Ri \leqslant 100$ and show that the Prandtl number strongly influences the morphology and dynamics of the flow structures. In particular, at fixed $Ra$ and $Re_w$, a high Prandtl number causes stronger momentum transport from the walls and therefore yields a greater impact of the wall shear on the flow structures, resulting in an increased effect of $Re_w$ on the Nusselt number. Furthermore, we analyse the thermal and kinetic boundary layer thicknesses and relate their behaviour to the resulting flow regimes. For the largest shear rates and $Pr$ numbers, we observe the emergence of a Prandtl–von Kármán log layer, signalling the onset of turbulent dynamics in the boundary layer.

We consider the time-dependent response of a gravitationally thinning inviscid liquid sheet (a coating curtain) leaving a vertical slot to sinusoidal ambient pressure disturbances. The theoretical investigation employs the hyperbolic partial differential equation developed by Weinstein et al. (Phys. Fluids, vol. 9, issue 12, 1997, pp. 3625–3636). The response of the curtain is characterized by the slot Weber number, $W_{e_0} = \rho q V/2\sigma$, where $V$ is the speed of the curtain at the slot, $q$ is the volumetric flow rate per unit width, $\sigma$ is the surface tension and $\rho$ is the fluid density. Flow disturbances travel along characteristics with speeds relative to the curtain of $\pm \sqrt {uV/W_{e_0}}$, where $u = \sqrt {V^{2} + 2gx}$ is the curtain speed at a distance $x$ downstream from the slot. Here g is the acceleration of gravity. When the flow is subcritical ($W_{e_0} < 1$), upstream travelling disturbances near the slot affect the curtain centreline, and the slope of the curtain centreline at the slot oscillates with an amplitude that is a function of $W_{e_0}$. In contrast, all disturbances travel downstream in supercritical curtains ($W_{e_0} > 1$) and the slope of the curtain at the slot is vertical. Here, we specifically examine the curtain response under supercritical and subcritical flow conditions near $W_{e_0} = 1$ to deduce whether there is a substantial change in the overall shape and magnitude of the curtain responses. Despite the local differences in the curtain solution near the slot, we find that subcritical and supercritical curtains have similar responses for all imposed sinusoidal frequencies.

We elucidate the physics underlying the birth, evolution and breakup of ligaments on a rim bounding an unsteady liquid sheet. This rim destabilizes into corrugations that can grow into ligaments, which in turn, break into secondary droplets via end-pinching. Combining experiments and theory, we show that not all corrugations can grow into ligaments. The number of corrugations is captured by linear instability coupled with nonlinear rim thickness self-adjustment ($\text {Bond number} = 1 \text { criterion}$, Wang et al. (Phys. Rev. Lett., vol. 120, 2018, 204503)) and scales as $N_c \sim We^{3/4}$ with Weber number, $We$. The number of ligaments scales as $N_{\ell } \sim We^{3/8}$. The growth of a ligament is governed by the competition between the constraint imposed by the geometry of the local rim–ligament junction; the local force balance including the fictitious force from the continuously decelerating rim; and the global rim mass conservation constraint. The temporal evolution of the average width of ligaments is predicted. Key to understanding the ligament population, a minimum distance between two corrugations is required to enable their actual transition into ligaments. By predicting this minimal distance, we derive the evolution of the number of ligaments. We show that droplets are shed, one at a time, following a chaotic dripping end-pinching regime independent of $We$. Finally, the number of droplets shed per unit of time decreases over time and scales as $S_d \sim We^{3/4}$; while the volume shed per unit of time increases over time and is independent of $We$. Theoretical predictions are validated without fitting parameters.