In this study, changes in the mean flow of a compressible turbulent boundary layer spatially evolving from low to ‘moderate’ Reynolds numbers are examined. All discussions are based on literature data and a direct numerical simulation (DNS) of a supersonic boundary layer specifically designed to be effectively free of spurious inflow effects in the range $4000 \lessapprox Re_\theta \lessapprox 5000$, which enables discussion of sensitive properties such as the turbulent wake. Most noticeably, the DNS data show the formation of a distinct ‘bend’ in the friction coefficient distribution reflected in sudden deviation from established low-Reynolds-number correlations. As will be shown, the bend is related to the surprisingly abrupt saturation of the turbulent wake, marking the change from low- to moderate-Reynolds-number behaviour; in previous studies, this trend was potentially obscured by data scatter in experiments and/or insufficient domain length in DNS. Moreover, the influence of the wake saturation on the formation of the early logarithmic overlap layer is assessed, which, if fully developed, leads to the onset of high-Reynolds-number behaviour further downstream.

We simulate thermal convection in a two-dimensional square box using the no-slip condition on all boundaries, and isothermal bottom and top walls, and adiabatic sidewalls. We choose 0.1 and 1 for the Prandtl number $Pr$ and vary the Rayleigh number $Ra$ between $10^6$ and $10^{12}$. We particularly study the temporal evolution of integral transport quantities towards their steady states. Perhaps not surprisingly, the velocity field evolves more slowly than the thermal field, and its steady state – which is nominal in the sense that large-amplitude low-frequency oscillations persist around plausible averages – is reached exponentially. We study these oscillation characteristics. The transient time for the velocity field to achieve its nominal steady state increases almost linearly with the Reynolds number. For large $Ra$, the Reynolds number itself scales almost as $Ra^{2/3}\, Pr^{-1}$, and the Nusselt number as $Ra^{2/7}$.

In rotating fluids, the viscous smoothing of inviscid singular inertial waves leads to the formation of internal shear layers. In previous works, we analysed the internal shear layers excited by a viscous forcing (longitudinal libration) in a spherical shell geometry (He et al., 2022 J. Fluid Mech. 939, A3; He et al., 2023 J. Fluid Mech. 974, A3). We now consider the stronger inviscid forcing corresponding to the vertical oscillation of the inner boundary. We limit our analysis to two-dimensional geometries but examine three different configurations: freely propagating wave beams in an unbounded domain and two wave patterns (a periodic orbit and an attractor) in a cylindrical shell geometry. The asymptotic structures of the internal shear layers are assumed to follow the similarity solution of Moore & Saffman (1969 Phil. Trans. R. Soc. Lond. A, 264, 597–634) in the small viscous limit. The two undefined parameters of the similarity solution (singularity strength and amplitude) are derived by asymptotically matching the similarity solution with the inviscid solution. For each case, the derivation of the latter is achieved either through separation of variables combined with analytical continuation or the method of characteristics. Global inviscid solutions, when obtained, closely match numerical solutions for small Ekman numbers far from the critical lines, while viscous asymptotic solutions show excellent performance near those lines. The amplitude scalings of the internal shear layers excited by an inviscid forcing are found to be divergent as the Ekman number $E$ decreases, specifically $O(E^{-1/6})$ for the critical-point singularity and $O(E^{-1/3})$ for attractors, in contrast to the convergent scalings found for a viscous forcing.

The wake merging of two side-by-side porous discs with varying disc spacing is investigated experimentally in a wind tunnel. Two disc designs used in the literature are employed: a non-uniform disc and a mesh disc. Hot-wire anemometry is utilised to acquire two spanwise profiles at 8 and 30 disc diameters downstream and along the centreline between the dual-disc configuration up to 40 diameters downstream. The spanwise Castaing parameter profiles confirm the appearance of rings of internal intermittency at the outermost parts of the wakes. These rings are the first feature to interact between the discs. After this point, the turbulence develops to a state whereby an inertial range is observable in the spectra. Farther downstream, the internal intermittency shows the classical features of homogeneous, isotropic turbulence. These events are repeatable and occur in the same order for both types of porous discs. This robustness allows us to develop a general map of the merging of the two wakes.

Modelling the nonlinear forcing is critical for linear models based on resolvent or input–output analyses. For compressible wall-bounded turbulence, little is known on what the real forcing looks like due to limited data, so the prediction agrees more qualitatively than quantitatively with direct numerical simulations (DNSs). Here, we present detailed forcing statistics of stochastic linear models, derived from elaborate DNS datasets for channel flows with bulk Mach number reaching 3. These statistics directly explain the success and failure of current models and provide guidance for further improvements. The benchmark linearised Navier–Stokes (LNS) and eLNS models are considered; the latter is assisted by eddy-viscosity-related terms. First, we prove the self-consistency of the models by using DNS-computed forcing as the input. Second, we present the spectral distributions of the forcing and its components. Third, we quantify the acoustic components, absent in incompressible cases, within the linear models. We reveal that the LNS forcing can exhibit relatively high coherence and low rank, very different from the modelled diagonal full-rank forcing. The eddy-viscosity-related term is not partial modelling of the LNS forcing; contrarily, the former is much larger than the latter, serving to disrupt the low-rank feature, enhance diagonal dominance and increase robustness across scales. The scales narrow in either horizontal direction are most susceptible to acoustic modes, while the others are little affected (${\lt}2\,\%$ in energy). Furthermore, the extended strong Reynolds analogy is assessed in predicting the density and temperature components.

We study buoyant miscible injections of dense viscoplastic fluids into lighter Newtonian fluids in inclined closed-end pipes, at the high-Péclet-number regime. We integrate experiments involving camera imaging and ultrasound Doppler velocimetry, and computational fluid dynamics simulations, to provide a detailed analysis of interfacial dynamics, flow phases/regimes, velocity field, yielded and unyielded zones, and interfacial arrest mechanisms. The flow dynamics is governed by Reynolds ($Re$), Froude ($Fr$) and Bingham ($B$) numbers, the viscosity ratio ($M$), inclination angle ($\beta$), or their combinations, such as $\chi \equiv 2Re/Fr^2$. As the interface evolves, our results reveal a transition from an inertial-dominated phase, characterised by linear front advancement at the injection velocity, to a viscoplastic-dominated phase, marked by deceleration and eventual interfacial arrest governed by the yield stress. The critical transition length between these phases $(\mathcal{L} \approx 1.26 Fr^{0.14})$ is determined by a balance between inertial and buoyant stresses. Experimental findings confirm buoyancy-driven slumping in our flows, consistent with the theoretical yield number criterion ($Y \equiv B/\chi$), with maximum interfacial arrest lengths scaling as $L_s \sim 1/Y$. These results also classify arrested and unhalted interfacial flow regimes on a plane involving ${\chi \cos (\beta )}/{B}$ and $Y$. Furthermore, we demonstrate that the interfacial arrest mechanism arises from interactions between buoyancy, rheology and geometry, as diminishing shear stresses promote unyielded zone expansion near the interface, progressively encompassing the viscoplastic layer and halting flow when stresses fall below the yield stress.

A systematic study is conducted both experimentally and theoretically on the wake-induced vibration of an inelastic or zero structural stiffness cylinder placed behind a perfectly elastic or rigid cylinder. The mass ratio m* of the inelastic cylinder is 11.1. The spacing ratio L/D is 2.0–6.0, where L is the distance between centers of the two cylinders, and D is the cylinder diameter. The range of Reynolds number Re is 1.97 × 103–1.18 × 104. It has been found that the inelastic cylinder becomes aerodynamically elastic because the cylinder and the fluctuating wake interact, inducing an effective stiffness and thus giving rise to an aeroelastic natural frequency. This frequency depends on the added mass, fluid damping and flow-induced stiffness and is always smaller than the vortex shedding frequency, irrespective of Re and L/D. The wake-induced vibration of the inelastic cylinder may be divided into a desynchronisation branch and a galloping branch. The vibration amplitude jumps greatly at the transition from desynchronisation to galloping for L/D = 2.0–4.5 but not so for L/D = 5.0–6.0. The flow-induced stiffness is linearly correlated with Re, generally higher in the reattachment regime than in the coshedding regime and smaller in galloping than in desynchronisation. Other aspects of the inelastic cylinder are also investigated in detail, including the dependence on Re of the Strouhal numbers, hydrodynamic forces, phase lag between lift and displacement and flow characteristics.

The nucleation of bubbles on rough substrates has been widely investigated in various applications such as electrolysis processes and fluid transportation in pipelines. However, the microscopic mechanisms underlying surface bubble nucleation are not fully understood. Using molecular dynamics simulations, we evaluate the probability of surface bubble nucleation, quantified by the magnitude of the nucleation threshold. Bubble nucleation preferentially occurs at the solid interfaces containing nanoscale defects or wells (nanowells), where reduced nucleation thresholds are observed. For the gas-entrapped nanowell, as the nanowell width decreases, the threshold of bubble nucleation around the nanowell gradually increases, eventually approaching a critical value close to that of a smooth surface. This results from a decrease in the amount of entrapped gas that promotes bubble nucleation, and the entrapped gas eventually converges to a critical state as the width decreases. For the liquid-filled nanowell, bubble nucleation initiates from the inner corner of the large nanowell. As the nanowell width decreases, the threshold is first kept constant and then decreases. This results from a decrease in the amount of filled liquid that inhibits bubble nucleation and from the enhanced confinement effect of the inner wall on the filled liquid as the width decreases. In this work, we propose a multiscale model integrating classical nucleation theory, van der Waals fluid theory and statistical mechanics to describe the relationship between nucleation threshold and nanowell width. Eventually, a unified phase diagram of bubble nucleation at the rough interface is summarised, offering fundamental insights for integrated system design.

This study presents an experimental investigation on the drag reduction (DR) over air-fed hydrophobic surfaces (AFHS) with longitudinal grooves in a turbulent boundary layer (TBL). The AFHS, designed with longitudinal grooves and air supplement channels, enables active maintenance and reversible restoration of the plastron in TBL. The shear stress sensor, particle image velocimetry (PIV) and interfacial visualization are applied for simultaneous measurement of the skin friction drag, TBL velocity profiles and plastron coverage. The AFHS demonstrated the ability to control plastron shape and enhance its sustainability with friction Reynolds numbers up to 1723. Drag reductions ranging from 14.8–35.8 % are obtained over the AFHS. At same designed air fraction, the AFHS exhibits higher DR than the conventional hydrophobic surface. By minimizing influences of the degradation of plastron coverage and the shape, the monotonic increase in DR and slip velocity with Reynolds number is confirmed, which corroborates trends from direct numerical simulations. Turbulence statistics measured by PIV reveal an apparent decrease in near-wall viscous shear stress, and corresponding slip velocities both in the viscous sublayer and log-law region. The Reynolds shear stress and streamwise velocity fluctuations over the AFHS are larger than those over a smooth wall, where near-wall vortex cores of the AFHS are found to be shifted 10 % towards the wall. This study presents the first simultaneous experimental quantification of skin friction, plastron coverage and turbulence statistics under sustained plastron conditions in TBL. The results demonstrate the efficacy of the plastron control strategy on hydrophobic surfaces and address a critical gap in validating numerical predictions for turbulent flows in practical applications.

We investigate the energy transfer from the mean profile to velocity fluctuations in channel flow by calculating nonlinear optimal disturbances, i.e. the initial condition of a given finite energy that achieves the highest possible energy growth during a given fixed time horizon. It is found that for a large range of time horizons and initial disturbance energies, the nonlinear optimal exhibits streak spacing and amplitude consistent with direct numerical simulation (DNS) at least at ${Re}_\tau = 180$, which suggests that they isolate the relevant physical mechanisms that sustain turbulence. Moreover, the time horizon necessary for a nonlinear disturbance to outperform a linear optimal is consistent with previous DNS-based estimates using eddy turnover time, which offers a new perspective on how some turbulent time scales are determined.

This paper focuses on the concept of delaying laminar–turbulent transition in hypersonic boundary layers by stabilising fundamental resonance (FR), a key nonlinear mechanism in which finite-amplitude Mack modes support the rapid growth of oblique perturbations. As a pioneering demonstration of this control strategy, we introduce surface heating applied exclusively during the nonlinear phase. Unlike traditional control methods that target the linear phase, the suppressive effect of surface heating on secondary instability modes during FR is evident across various Reynolds numbers, wall temperatures and fundamental frequencies, as confirmed by direct numerical simulations (DNS) and secondary instability analyses (SIA). To gain deeper insights into this control concept, an asymptotic analysis is conducted, revealing an almost linear relationship between the suppression effect and the heating intensity. The asymptotic predictions align overall with the DNS and SIA calculations. The asymptotic theory reveals that the suppression effect of FR is primarily influenced by modifications to the fundamental-mode profile, while mean-flow distortion has a comparatively modest yet opposing impact on this process. This research presents a promising approach to controlling transition considering the nonlinear evolution of boundary-layer perturbations, demonstrating advantages over conventional methods that are sensitive to frequency variations.

Eddies within the meso/submeso-scale range are prevalent throughout the Arctic Ocean, playing a pivotal role in regulating the freshwater budget, heat transfer and sea ice transport. While observations have suggested a strong connection between the dynamics of sea ice and the underlying turbulent flows, quantifying this relationship remains an ambitious task due to the challenges of acquiring concurrent sea ice and ocean measurements. Recently, an innovative study using a unique algorithm to track sea ice floes showed that ice floes can be used as vorticity-meters of the ocean. Here, we present a numerical and analytical evaluation of this result by estimating the kinematic link between free-drifting ice floes and underlying ocean eddies using idealised vortex models. These analyses are expanded to explore local eddies in quasi-geostrophic turbulence, providing a more realistic representation of eddies in the Arctic Ocean. We find that in both flow fields, the relationship between floe rotation rates and ocean vorticity depends on the relative size of the ice floe to the eddy. As the floe size approaches and exceeds the eddy size, the floe rotation rates depart from half of the ocean vorticity. Finally, the effects of ice floe thickness, atmospheric winds and floe collisions on floe rotations are investigated. The derived relations and floe statistics set the foundation for leveraging remote sensing observations of floe motions to characterise eddy vorticity at small to moderate scales. This innovative approach opens new possibilities for quantifying Arctic Ocean eddy characteristics, providing valuable inputs for more accurate climate projections.

Depth-averaged systems of equations describing the motion of fluid–sediment mixtures have been widely adopted by scientists in pursuit of models that can predict the paths of dangerous overland flows of debris. As models have become increasingly sophisticated, many have been developed from a multi-phase perspective in which separate, but mutually coupled sets of equations govern the evolution of different components of the mixture. However, this creates the opportunity for the existence of pathological instabilities stemming from resonant interactions between the phases. With reference to the most popular approaches, analyses of two- and three-phase models are performed, which demonstrate that they are more often than not ill posed as initial-value problems over physically relevant parameter regimes – an issue which renders them unsuitable for scientific applications. Additionally, a general framework for detecting ill posedness in models with any number of phases is developed. This is used to show that small diffusive terms in the equations for momentum transport, which are sometimes neglected, can reliably eliminate this issue. Conditions are derived for the regularisation of models in this way, but they are typically not met by multi-phase models that feature diffusive terms.

Flapping-based propulsive systems rely on fluid–structure interactions to produce thrust. At intermediate and high Reynolds numbers, vortex formation and organisation in the wake of such systems are crucial for the generation of a propulsive force. In this work, we experimentally investigate the wake produced by a tethered robotic fish immersed in a water tunnel. By systematically varying the amplitude and frequency of the fish tail as well as the free stream speed, we are able to observe and characterise different vortex streets as a function of the Strouhal number. The produced wakes are three-dimensional and exhibit a classical V-shape, mainly with two oblique trains of vortex rings convecting outward. Using two-dimensional particle image velocimetry in the mid-span plane behind the fish and through extensive data processing of the velocity and vorticity fields, we demonstrate the strong couplings at place between vortex dynamics, thrust production and wake structure. The main results are twofold. First, by accounting for the obliqueness of the vortex trains, we quantify in experiments the evolution of vortex velocity components in both streamwise and transverse directions. We also measure key geometrical and dynamical properties such as wake angle, vortex ring orientation, diameter and vorticity. Remarkably, all of these quantities collapse onto master curves that also encompass data from previous studies. Second, we develop a quasi-two-dimensional model that incorporates both components of the momentum balance equation and introduces an effective spanwise thickness of the wake structure. This additional dimension, which scales with the physical thickness of the fish, captures the fine features of the three-dimensional wake. The model successfully explains the experimental master curves and highlights the links between vortex dynamics, thrust and wake geometry. Together, this framework offers a comprehensive understanding of the influence of the Strouhal number, providing universal insights relevant for both biological locomotion and bio-inspired propulsion systems.

We study experimentally the starting vortices shed by airfoils accelerating uniformly from rest in superfluid helium-4 (He II). The vortices behave apparently as if they were moving in a classical Newtonian fluid, such as air or water. Specifically, the starting vortex positions obtained from the experimental data are found to be very close to those computed numerically in a Newtonian fluid, at sufficiently small times, when self-similar behaviour is expected to occur, and for Reynolds numbers ranging between approximately $5 \times 10^2$ and $5 \times 10^5$. The result indicates neatly that turbulent flows of He II can be very similar to classical flows of Newtonian fluids, when thermal effects can be neglected and at sufficiently large flow scales, i.e. the study demonstrates that He II could also be employed to study classical Newtonian flows.

Breaking wave impacts on rigid structures have been extensively studied, yet the role of structural elasticity in shaping the impact and response remains insufficiently understood. In this study, we experimentally investigate the hydroelastic behaviour of a vertical cantilever plate subjected to multimodal solitary breaking wave impacts. The plate is mounted near the still water level on a 1 : 10 sloping beach, and the wave height-to-depth ratio ($H/h$) is varied from 0.15 to 0.40 to systematically control the impact type from non-breaking to highly aerated wave impacts. We show that aeration significantly affects hydroelastic impacts. The spatio-temporal extent of the impact pressure on the elastic plate increases with air entrapment, while the peak pressure becomes highly sensitive as the wave approaches the flip-through regime. Pressure oscillations associated with bubble formation induce high-frequency structural vibrations, particularly under low-aeration conditions. Furthermore, we find that the elasticity has a limited effect on the peak pressure, impact duration and impulse, but increases the maximum quasi-hydrostatic force on the plate for the scenarios investigated. Following the impact, two distinct free-top deflections are identified, i.e. a deflection $\Delta x_{\textit{imp}}$ with high acceleration induced by the impact pressure and a deflection $\Delta x_{{hp}}$ with high magnitude caused by the maximum quasi-hydrostatic pressure. These deflections scale with the Cauchy number as $\Delta x_{\textit{imp}}/l \sim Ca_{\textit{imp}}/6$ and $\Delta x_{{hp}}/l \sim Ca_{{hp}}/12$ (where l is the plate length), exhibiting parabolic and linear trends with $H/h$, respectively. This work presents a benchmark dataset and introduces a predictive law for structural deflection, providing practical insights into hydroelastic effects across various impact regimes.

The effect of Stokes number on turbulence modulation in particle-laden channel flow is investigated through four-way coupled point-particle direct numerical simulations, with the mass loading fixed at 0.6 and the friction Stokes number $St^+$ varying from 3 to 300. A full transition pathway is observed, from a drag-enhanced to a drag-reduced regime, eventually approaching the single-phase state as $St^+$ increases towards 300. A set of transport equations for the particle phase is derived analytically to characterise the interphase coupling, within the framework of the point-based statistical description of particle-laden turbulence. By virtue of this, two dominant mechanisms are identified and quantitatively characterised: a positive, particle-induced extra transport that decreases monotonically with increasing $St^+$, and a negative, particle-induced extra dissipation that varies non-monotonically with $St^+$. The coupling of these two mechanisms leads to a direct contribution of the particle phase to the shear stress balance, the turbulent kinetic energy budgets and the Reynolds stress budgets. Consequently, as $St^+$ increases, the self-sustaining cycle of near-wall turbulence transitions from being augmented to being suppressed and, eventually, returns to the single-phase state. This gives rise to an indirect effect, manifested as a non-monotonic modulation of Reynolds shear stress and turbulence production rate. Taken together, complex interplays between particle-modified turbulent transport, particle-induced extra transport and extra dissipation are analysed and summarised, providing a holistic physical picture composed of consistent interpretations of turbulence modulation induced by small heavy particles.

Compressible jets impinging on a perpendicular surface can produce high-intensity, discrete-frequency tones. The character of these tones is a function of nozzle shape, jet Mach number, impingement-plate geometry, and the distance between nozzle and plate. Though it has long been recognised that these tones are associated with a resonance cycle, the exact mechanism by which they are generated has remained a topic of some debate. In this work, we present evidence for a number of distinct tone-generation mechanisms, reconciling some of the different findings of prior authors. We demonstrate that the upstream-propagating waves that close resonance can be confined within the jet, or external to it. These waves can be either weak and relatively linear, or strong and nonlinear from their inception. The waves can undergo coalescence or merging, and in some configurations, pairs of waves rather than singletons appear. We discuss both historical and new evidence for multiple distinct processes by which upstream-propagating waves are produced: direct vortex sound, shock leakage, wall-jet-boundary fluctuations, and wall-jet shocklets. We link these various mechanisms to the disparate collection of upstream-propagating waves observed in the data. We also demonstrate that multiple mechanisms can be provoked by a single vortex, providing an explanation as to why sometimes pairs of waves or merging waves are observed. Through this body of work, we demonstrate that rather than being in opposition, the various pieces of past research on this topic were simply identifying different mechanisms that can support resonance.

Dot array deposition through electrohydrodynamic (EHD) printing is widely used for high resolution and material utilization advantages. However, the conventional printing method is subject to a printing frequency limit known as the capillary frequency of the meniscus oscillation, where the jet directly contacts the substrate. This makes the printing frequency of EHD printing maintain at a low level and that is difficult to improve. In this work, a method for high-frequency EHD printing through continuous pinch-off is proposed. The characteristic frequency is broken through. A model is established to reveal the printing mechanism by combining the Poisson–Nernst–Planck equation and the phase field method. The unreal charge leakage is prevented by constructing a transition function for the fluid’s properties. The stability of the Taylor cone’s deformation and the droplets’ generation is studied. The measurement criterion for printing frequency is determined. The suitable printing height that can prevent the jet from directly contacting the substrate is obtained by investigating its influence on the printing states and frequency. The phase diagram considering the liquid’s conductivity and viscosity is presented to distinguish whether the printing is based on the end-pinching or Rayleigh–Plateau instability. The influence of the conductivity, viscosity, flow rate and printing voltage on the printing frequencies is studied quantitatively. Finally, scaling laws for printing frequency are proposed by theoretical analyses and summarizing the numerical data. This work could be beneficial for further enhancing the printing frequency of EHD printing.