Wake interaction of two rotationally oscillating cylinders in side-by-side configuration is studied experimentally at a Reynolds number of 150. Five spacing ratios, $T/D$ (ratio of centre-to-centre spacing to cylinder diameter), are considered, namely, 1.4, 1.8, 2.5, 4.0 and 7.5. Both in-phase and antiphase forcing are investigated. Oscillation amplitude is varied from ${\rm \pi} /8$ to ${\rm \pi}$, and forcing frequency, $FR$ (ratio of the oscillation frequency to the vortex-shedding frequency of a stationary cylinder) is varied from 0 to 5. The experimental investigation is done using laser-induced fluorescence, hot film anemometry and particle-image velocimetry (PIV). The interaction between the two cylinders under forcing results in new wake modes and vortex structures and a comprehensive study from the wake visualisations is conducted. Quantitative results are presented in terms of streamwise and cross-stream mean velocity profiles, centreline velocity recovery, peak velocity deficit, wake width, fluctuation intensity, circulation, vorticity contours and drag coefficient. The magnitude of streamwise velocity deficit and cross-stream velocity variation is strongly affected by the presence of second cylinder. The recirculation region behind the cylinders is found to extend further downstream with increase in the forcing. Scaling analysis is carried out to express the peak velocity deficit variation with forcing. It is observed that the relative strength of the vortices shed from inner and outer shear layers depends on the phase of oscillation. An experimental set-up for direct force measurement is designed and the drag force acting on the oscillating cylinders assembly is directly measured and the effect of forcing on the variation of ${C}_{d}$ is studied. An estimate for drag coefficient is also made from the PIV data following a detailed control volume analysis. It is observed that the set of forcing parameters that correspond to maximum and minimum drag also yield extrema in the values of circulation and fluctuation intensity.

Turbulent shear flows are abundant in geophysical and astrophysical systems and in engineering-technology applications. They are often riddled with large-scale secondary flows that drastically modify the characteristics of the primary stream, preventing or enhancing mixing, mass and heat transfer. Using experiments and numerical simulations, we study the possibility of modifying these secondary flows by using superhydrophobic surface treatments that reduce the local shear. We focus on the canonical problem of Taylor–Couette flow, the flow between two coaxial and independently rotating cylinders, which has robust secondary structures called Taylor rolls that persist even at significant levels of turbulence. We generate these structures by rotating only the inner cylinder of the system, and show that an axially spaced superhydrophobic treatment can weaken the rolls through a mismatching surface heterogeneity, as long as the roll size can be fixed. The minimum hydrophobicity of the treatment required for this flow control is rationalized, and its effectiveness beyond the Reynolds numbers studied here is also discussed.

A resolvent-based model was used to predict the phase-difference profile between velocity and stress coherent motions measured in a high Reynolds number channel flow as a proxy for predicting large- and small-scale turbulent interactions. The resolvent model is based on the transfer-function approach for scale interactions in wall turbulence proposed in Jacobi et al. (J. Fluid Mech., vol 914, 2021, pp. 1–27), but incorporates a quasi-empirical weighting scheme to construct composite mode shapes that represent the realistic dispersion of convection velocities associated with the large scales of turbulence. The weighting scheme was derived from the observed similarity between the spectral region where the resolvent operator is low rank and the streamwise spectral energy density of wall-bounded turbulence, and was found to be superior to both single-convection velocity models and models based on linearly weighted modes, when compared with cross-spectral phase calculations from a channel flow computation. The ability to predict the phase relationship between large-scale coherent motions and their associated stress fluctuations allows for refining and extending resolvent-based models of turbulence to describe small-scale features of wall-bounded flows.

A steady granular flow experiment was performed in a confined annular shear cell to examine how the wall friction coefficient $\mu _w$ degrades from the intrinsic sliding friction coefficient $f$ between the grains and the container wall. Two existing models are invoked to examine the decay trend of $\mu _w/f$ in view of the ratio of shear velocity to the square root of granular temperature $\chi$ (Artoni & Richard, Phys. Rev. Lett., vol. 115, 2015, 158001) and the ratio of grain angular and slip velocities $\varOmega$ (Yang & Huang, Granul. Matt., vol. 18, issue 4, 2016, p. 77), respectively. As both models correlate $\mu _w/f$ to different flow properties, a hidden relation is speculated between $\chi$ and $\varOmega$, or equivalently, between the granular temperature and the grain rotation speed. We used experiment data to confirm and reveal this hidden relation. From there, a unified $\mu _w/f-\chi$ model is proposed with physical meanings for the model coefficients and to show general agreement with the measured trend. Hence we may conclude that both the fluctuations in grain translations and their mean rotation are the crucial yet equivalent mechanisms to degrade $\mu _w/f$.

We present experimental evidence that the superstructures in turbulent boundary layers comprise smaller, geometrically self-similar coherent motions. The evidence comes from identifying and analysing instantaneous superstructures from large-scale particle image velocimetry datasets acquired at high Reynolds numbers, capable of capturing streamwise elongated motions extending up to 12 times the boundary layer thickness. Given the challenge in identifying the constituent motions of the superstructures based on streamwise velocity signatures, a new approach is adopted that analyses the wall-normal velocity fluctuations within these very long motions, which reveals the constituent motions unambiguously. The conditional streamwise energy spectra of the Reynolds shear stress and the wall-normal fluctuations, corresponding exclusively to the superstructure region, are found to exhibit the well-known distance-from-the-wall scaling in the intermediate-scale range. It suggests that geometrically self-similar motions are the constituent motions of these very-large-scale structures. Investigation of the spatial organization of the wall-normal momentum-carrying eddies, within the superstructures, also lends empirical support to the concatenation hypothesis for the formation of these structures. The association between the superstructures and self-similar motions is reaffirmed on comparing the vertical coherence of the Reynolds-shear-stress-carrying motions, by computing conditionally averaged two-point correlations, which are found to match with the mean correlations. The mean vertical coherence of these motions, investigated for the log region across three decades of Reynolds numbers, exhibits a unique distance-from-the-wall scaling invariant with Reynolds number. The findings support modelling of these dynamically significant motions via data-driven coherent structure-based models.

A simple efficient method is presented for the determination of the intrinsic transmissivity tensor, as well as the intrinsic correction tensors at successive orders in the dimensionless slip parameter, that predicts the effective transmissivity tensorial coefficient for steady, one-phase, isothermal, creeping flow of a Newtonian fluid with slip boundary condition in a rough fracture. It is demonstrated that the solution of the first $N$ ancillary closure problems provides the slip correction tensors up to the $2N-1$ order, hence reducing the computational requirements by a factor of ${\sim }2$ compared with the conventional approach. In particular, the first-order correction tensor (i.e. a Klinkenberg-like tensor) can be obtained by solving the closure problem required for the computation of the intrinsic transmissivity tensor. In addition, symmetry and definiteness (positiveness or negativeness) properties of the individual tensors are analysed. It is shown that a Padé approximant, built on the correction tensors at the first three orders, outperforms the predictions for the effective transmissivity tensor. The new approach is illustrated and validated with numerical examples on model rough fractures.

The disturbance flow field in a hypersonic boundary layer excited by particle impingement was investigated with a focus on the first stage of the laminar-to-turbulent transition process, namely the receptivity process. A previously validated direct numerical simulation approach adopting disturbance flow tracking is used to simulate the particle-induced transition process. Particle impingement generates a highly complex disturbance flow field that can be characterised by a wide range of frequencies and wavenumbers. After providing some insight about the spectral characteristics of the disturbance flow field in the frequency and wavenumber domains, biorthogonal decomposition is employed to reveal the composition of the disturbance flow field consisting of different continuous and discrete eigenmodes that are triggered through particle impingement. The disturbance flow characteristics for different frequency and wavenumber pairs are discussed where large contributions in the disturbance flow spectrum are observed in the vicinity of the impingement location. A significant amount of the disturbance energy is diverted into the free stream leading to large coefficients of projection for the slow and fast acoustic branches while contributions to the entropy and vorticity branches are negligible. In addition to the continuous acoustic spectra, the first-, second- and other higher-order Mack modes are activated and provide large contributions to the disturbance flow field inside the boundary layer. Finally, it is demonstrated that the disturbance flow field in the vicinity of the impingement location can be reconstructed with a maximum relative error of $2.3\,\%$ by employing a theoretical biorthogonal eigenfunction system expansion and by considering contributions from fast and slow acoustic waves and at most four discrete modes only.

New exact solutions are presented to the problem of steadily travelling water waves with vorticity wherein a submerged von Kármán point vortex street cotravels with a wave on a linear shear current. Surface tension and gravity are ignored. The work generalizes an earlier study by Crowdy & Nelson (Phys. Fluids, vol. 22, 2010, 096601) who found analytical solutions for a single point vortex row cotravelling with a water wave in a linear shear current. The main theoretical tool is the Schwarz function of the wave, and the work builds on a novel framework set out recently by Crowdy (J. Fluid Mech., vol. 954, 2022, A47). Conformal mapping theory is used to construct Schwarz functions with the requisite properties and to parametrize the waveform. A two-parameter family of solutions is found by solving a pair of nonlinear algebraic equations. This system of equations has intriguing properties: indeed, it is degenerate, which radically reduces the number of possible solutions, although the space of physically admissible equilibria is still found to be rich and diverse. For inline vortex streets, where the two vortex rows are aligned vertically, there is generally a single physically admissible solution. However, for staggered streets, where the two vortex rows are offset horizontally, certain parameter regimes produce multiple solutions. An important outcome of the work is that while only degenerate von Kármán point vortex streets can exist in an unbounded simple shear current, a broad array of such equilibria is possible in a shear current beneath a cotravelling wave on a free surface.

We derive explicit formulae for the mean profiles of passive scalars (either temperature or concentration of a diffusing substance), and their respective wall fluxes (either heat or mass fluxes), in forced turbulent convection, as a function of the Reynolds and Prandtl numbers. Direct numerical simulation data for turbulent flow within a smooth straight pipe of circular cross-section, at friction Reynolds number ${{Re}}_{\tau }=1140$, in the range of Prandtl numbers from ${{Pr}}=0.00625$ to ${{Pr}}=16$, are used to infer the proper analytical form of the eddy diffusivity. This is leveraged to derive accurate predictive formulae for the mean passive scalar profiles, and for the corresponding logarithmic offset function. Asymptotic scaling laws result for the thickness of the conductive (diffusive) layer, and for the Nusselt number, which significantly extend the predictive envelope of classical formulae.

The influence of surface roughness on transition to turbulence in a Mach 4.5 boundary layer is studied using direct numerical simulations. Transition is initiated by the nonlinearly most dangerous inflow disturbance, which causes the earliest possible breakdown on a flat plate for the prescribed inflow energy and Mach number. This disturbance primarily comprises two normal second-mode instability waves and an oblique first mode. When localized roughness is introduced, its shape and location relative to the synchronization points of the inflow waves are confirmed to have a clear impact on the amplification of the second-mode instabilities. The change in modal amplification coincides with the change in the height of the near-wall region where the instability wave speed is supersonic relative to the mean flow; the net effect of a protruding roughness is destabilizing when placed upstream of the synchronization point and stabilizing when placed downstream. Assessment of the effect of the roughness location is followed by an optimization of the roughness height, abruptness and width with the objective of achieving maximum transition delay. The optimization is performed using an ensemble-variational (EnVar) approach, while the location of the roughness is fixed upstream of the synchronization points of the two second-mode waves. The optimal roughness disrupts the phase of the near-wall pressure waves, suppresses the amplification of the primary instability waves and mitigates the nonlinear interactions that lead to breakdown to turbulence. The outcome is a sustained non-turbulent flow throughout the computational domain.

It is known that the disintegration of vertical liquid curtains (sheets) is affected crucially by the amplification of free edge holes forming inside the curtain. This paper aims to investigate the influence of the hole expansion dynamics, driven by the so-called rim retraction, on the breakup of a liquid curtain, in both supercritical (Weber number $We > 1$) and subcritical ($We < 1$) conditions. The analysis is based on three-dimensional direct numerical simulations. For a selected supercritical configuration, the steady flow topology is first analysed. The investigation reveals the classic triangular shape regime of the steady curtain, due to the surface-tension-induced borders retraction towards its centre plane. The unsteady dynamics is then investigated as the curtain response to a hole perturbation introduced artificially in the steady flow configuration. The hole evolution determines a rim retraction phenomenon inside the curtain, which is influenced by both capillary and gravity forces. In supercritical conditions, the hole does not influence the curtain flow dynamics in the long-time limit. By reducing the Weber number slightly under the critical threshold ($We=1$), the initial amplification rate of the hole area increases, due to the stronger retraction effect of surface tension acting on the hole rims. The free hole expansion in fully subcritical conditions ($We < 1$) is investigated finally by simulating an edge-free curtain flow. As $We$ decreases progressively, the hole expands while it is convected downstream by gravity acceleration. In the range $0.4< We<1$, the subcritical curtain returns to the intact unperturbed configuration after the hole expulsion at the downstream outflow. For $We<0.4$, the surface tension force becomes strong enough to reverse the gravitational motion of the hole top point, which retracts upstream towards the sheet inlet section while expanding along the lateral directions. This last phenomenon causes finally the breakup of the curtain, which results in a columnar regime strictly resembling similar experimental findings of the literature.

In this work we present experiments and simulations on the nucleation and successive dynamics of laser-induced bubbles inside liquid droplets in free-fall motion, i.e. a case where the bubbles are subjected to the influence of a free boundary in all directions. Within this spherical millimetric droplet, we have investigated the nucleation of secondary bubbles induced by the rarefaction wave that is produced when the shock wave emitted by the laser-induced plasma reflects at the drop surface. Interestingly, three-dimensional clusters of cavitation bubbles are observed. Their shape is compared with the negative pressure distribution computed with a computational fluid dynamics model and allows us to estimate a cavitation threshold value. In particular, we observed that the focusing of the waves in the vicinity of the free surface can give rise to explosive cavitation events that end up in fast liquid ejections. High-speed recordings of the drop/bubble dynamics are complemented by the velocity and pressure fields simulated for the same initial conditions. The effect of the proximity of a curved free surface on the jetting dynamics of the bubbles was qualitatively assessed by classifying the cavitation events using a non-dimensional stand-off parameter ${\Upsilon\hskip -1,05em -\,}$ that depends on the drop size, the bubble maximum radius and the relative position of the bubble inside the drop. Additionally, we studied the role of the drop's curvature by implementing a structural similarity algorithm to compare cases with bubbles produced near a flat surface to the bubbles inside the drop. Interestingly, this quantitative comparison method indicated the existence of equivalent stand-off distances at which bubbles influenced by different boundaries behave in a very similar way. The oscillation of the laser-induced bubbles promotes the onset of Rayleigh–Taylor and Rayleigh–Plateau instabilities, observed on the drop's surface. This phenomenon was studied by varying the ratio of the maximum radii of the bubble and the drop. The specific mechanisms leading to the destabilisation of the droplet surface were identified.

We study the gravity-induced collisions of charged spheres of dielectric materials dispersed in a gaseous medium. When the gap thickness between the surfaces of two spheres is shorter than the mean free path of the surrounding fluid medium, continuum assumptions for the hydrodynamics interactions are no longer valid, and the non-continuum lubrication interactions result in surface-to-surface contact in finite time. Two like-charged dielectric spheres attract each other at close separations for a wide range of size and charge ratio values. We use trajectory analysis to calculate the collision rate and, thus, explore the role of electrostatic interactions in the collision dynamics of a pair of like-charged dielectric spheres. We present the modifications of pair trajectories due to electrostatic forces and show how collision efficiencies vary with the non-dimensional parameter capturing the relative strength of the electrostatic force to gravity as well as the charge ratio and size ratio.

In direct numerical simulations (DNS) of turbulent Couette flow, the observation has been made that the long streamwise rolls increase in length with the Reynolds number (Lee & Moser, J. Fluid Mech., vol. 842, 2018, pp. 128–145). To understand this, we employ both linear stability theory and its extension to resolvent analysis. For this, we emphasise the high Reynolds number ($Re \rightarrow \infty$) and small streamwise wavenumbers ($\alpha \rightarrow 0$) limit, imposing the distinguished limit $Re_{\alpha }=Re \, \alpha = O(1)$. We find that in case of linear stability theory, $Re_{\alpha }$ acts as a global invariant in the resulting eigenvalue problem, while in case of resolvent analysis, $Re_{\alpha }$ acts as a local invariant in the behaviour of the energy of the system characterised through the first singular value $\sigma _1$ of the resolvent operator within the investigated asymptotic limit. In order to obtain constant streamwise structures for increasing Reynolds numbers, the respective streamwise wavenumber has to decrease, which verifies the observations from DNS studies of an increasing length of the streamwise structures with the Reynolds number. In linear stability theory, a parameter reduction is achieved for the above asymptotic limit, resulting in the modified Orr–Sommerfeld and Squire equations being dependent only on $Re_{\alpha }$. The behaviour of both the coherent structures obtained from linear stability theory and resolvent analysis are compared with each other and show similar behaviours over $Re_{\alpha }$.

Understanding the dynamics of staircases in salt fingering convection presents a long-standing theoretical challenge to fluid dynamicists. Although there has been significant progress, particularly through numerical simulations, there are a number of conflicting theoretical explanations as to the driving mechanism underlying staircase formation. The Phillips effect proposes that layering in stirred stratified flow is due to an antidiffusive process, and it has been suggested that this mechanism may also be responsible for salt fingering staircases. However, the details of this process, as well as mathematical models to predict the evolution and merger dynamics of staircases, have yet to be established. We generalise the theory of the Phillips effect to a three-component system (e.g. temperature, salinity, energy) and demonstrate a regularised nonlinear model of layering based on mixing length parametrisations. The model predicts both the inception of layering and its long-term evolution through mergers, while generalising, and remaining consistent with, previous results for double-diffusive layering based on flux ratios. Our model of salt fingering is formulated using spatial averaging processes, and closed by a mixing length parametrised in terms of the kinetic energy and the ratio of the temperature and salt gradients. The model predicts a layering instability for a bounded range of parameter values in the salt fingering regime. Nonlinear solutions show that an initially unstable linear buoyancy gradient develops into layers, which proceed to merge through a process of stronger interfaces growing at the expense of weaker ones. Our results indicate that these mergers are responsible for the characteristic increase of buoyancy flux through thermohaline staircases.

The statistics of breaking wave fields are characterised within a novel multi-layer framework, which generalises the single-layer Saint-Venant system into a multi-layer and non-hydrostatic formulation of the Navier–Stokes equations. We simulate an ensemble of phase-resolved surface wave fields in physical space, where strong nonlinearities, including directional wave breaking and the subsequent highly rotational flow motion, are modelled, without surface overturning. We extract the kinematics of wave breaking by identifying breaking fronts and their speed, for freely evolving wave fields initialised with typical wind wave spectra. The $\varLambda (c)$ distribution, defined as the length of breaking fronts (per unit area) moving with speed $c$ to $c+{\rm d}c$ following Phillips (J. Fluid Mech., vol. 156, 1985, pp. 505–531), is reported for a broad range of conditions. We recover the $\varLambda (c) \propto c^{-6}$ scaling without wind forcing for sufficiently steep wave fields. A scaling of $\varLambda (c)$ based solely on the root-mean-square slope and peak wave phase speed is shown to describe the modelled breaking distributions well. The modelled breaking distributions are in good agreement with field measurements and the proposed scaling can be applied successfully to the observational data sets. The present work paves the way for simulations of the turbulent upper ocean directly coupled to a realistic breaking wave dynamics, including Langmuir turbulence, and other sub-mesoscale processes.

The Perron–Frobenius operator (PFO) is adapted from dynamical-system theory to the study of turbulent channel flow. It is shown that, as long as the analysis is restricted to the system attractor, the PFO can be used to differentiate causality and coherence from simple correlation without performing interventional experiments, and that the key difficulty remains the collection of enough data to populate the operator matrix. This is alleviated by limiting the analysis to two-dimensional projections of the phase space, and developing a series of indicators to choose the best parameter pairs from a large number of possibilities. The techniques thus developed are applied to the study of bursting in the inertial layer of the channel, with emphasis on the process by which bursts are reinitiated after they have decayed. Conditional averaging over phase-space trajectories suggested by the PFO shows, somewhat counter-intuitively, that a key ingredient for the burst recovery is the development of a low-shear region near the wall, overlaid by a lifted shear layer. This is confirmed by a computational experiment in which the control of the mean velocity profile by the turbulence fluctuations is artificially relaxed. The behaviour of the mean velocity profile is thus modified, but the association of low wall shear with the initiation of the bursts is maintained.

The effect of surface vibrations on the pressure-gradient-driven flows in channels has been studied. The analysis considered monochromatic waves and laminar flows. The effectiveness of the vibrations was gauged by determining the pressure gradient correction required to maintain the same flow rate as without vibrations. Waves propagating upstream always increase pressure losses. Flow response to waves propagating downstream is more complex and changes as a function of the flow Reynolds number. Such waves reduce losses if the Reynolds number $Re <\ \sim\!\!100$, but these waves must be sufficiently fast to reduce pressure losses for larger Re values. In general, the supercritical waves, i.e. waves faster than the reference flow, reduce pressure losses with the magnitude of reduction increasing monotonically with the wave phase speed and wavenumber. The need for an external pressure gradient is eliminated if sufficiently short and fast waves are used. Generally, the subcritical waves, i.e. waves with velocities similar to the reference flow, increase pressure losses. This increase changes somewhat irregularly as a function of the wave phase speed and wavenumber forming local maxima and minima. These waves can reduce pressure losses only if the Reynolds number becomes large enough. It is shown that subcritical waves with very small amplitudes but matching the natural flow frequencies produce significant pressure losses.

Separate and joint droplets, clusters, and voids characteristics of sprays injected in a turbulent co-flow are investigated experimentally. Simultaneous Mie scattering and interferometric laser imaging for droplet sizing along with separate hotwire anemometry are performed. The turbulent co-flow characteristics are adjusted using zero, one or two perforated plates. The Taylor-length-scale-based Reynolds number varies from 10 to 38, and the Stokes number estimated based on the Kolmogorov time scale varies from 3 to 25. The results show that the mean length scale of the clusters normalized by the Kolmogorov length scale varies linearly with the Stokes number. However, the mean void length scale is of the order of the integral length scale. It is shown that the number density of the droplets inside the clusters is approximately 7 times larger than that in the voids. The ratios of the droplets number densities in the clusters and voids to the total number density are independent of the test conditions and equal 5.5 and 0.8, respectively. The joint probability density function of the droplets diameter and clusters area shows that the droplets with the most probable diameter are found in the majority of the clusters. It is argued that intensifying the turbulence broadens the range of turbulent eddy size in the co-flow which allows for accommodating droplets with a broad range of diameters in the clusters. The results are of significance for engineering applications that aim to modify the clustering characteristics of large-Stokes-number droplets sprayed into turbulent co-flows.

We present measurements of turbulent drag reduction (DR) in boundary layers at high friction Reynolds numbers in the range of $4500 \le Re_\tau \le 15\ 000$. The efficacy of the approach, using streamwise travelling waves of spanwise wall oscillations, is studied for two actuation regimes: (i) inner-scaled actuation (ISA), as investigated in Part 1 of this study, which targets the relatively high-frequency structures of the near-wall cycle, and (ii) outer-scaled actuation (OSA), which was recently presented by Marusic et al. (Nat. Commun., vol. 12, 2021) for high-$Re_\tau$ flows, targeting the lower-frequency, outer-scale motions. Multiple experimental techniques were used, including a floating-element balance to directly measure the skin-friction drag force, hot-wire anemometry to acquire long-time fluctuating velocity and wall-shear stress, and stereoscopic particle image velocimetry to measure the turbulence statistics of all three velocity components across the boundary layer. Under the ISA pathway, DR of up to 25 % was achieved, but mostly with net power saving (NPS) losses due to the high-input power cost associated with the high-frequency actuation. The low-frequency OSA pathway, however, with its lower input power requirements, was found to consistently result in positive NPS of 5–10 % for moderate DRs of 5–15 %. The results suggest that OSA is an attractive pathway for energy-efficient DR in high-Reynolds-number applications.