The shape of a free-surface slump of viscoplastic material supported by an oblique barrier on an inclined plane is investigated theoretically and experimentally. The barrier is sufficiently tall that it is not surmounted by the viscoplastic fluid, and a focus of this study is the largest volume of rigid viscoplastic fluid that can be supported upstream of it. A lubrication model is integrated numerically to determine the transient flow as the maximal rigid shape is approached. Away from the region supported by the barrier, the viscoplastic layer attains a uniform thickness in which the gravitational stresses are in balance with the yield stress of the material. However, closer to the barrier, the layer thickens and the barrier bears the additional gravitational loading. An exact solution for the rigid shape of the viscoplastic material is constructed from the steady force balance and computed by integrating Charpit’s equations along characteristics that emanate from the barrier wall. The characteristics represent the late-time streamlines of the flow as it approaches the rigid shape. The exact solution depends on a single dimensionless group, which incorporates the slope inclination, the barrier width and the fluid’s yield stress. It is shown that the shape is insensitive to the transient flow from which it originates. The force exerted by the slump is calculated for different barrier shapes. The results of new laboratory experiments are reported; these show that although convergence to the final rigid state is slow, there is good agreement with the experimental measurements at long times.

This study examines the similarity properties of hypersonic turbulent boundary layers using direct numerical simulations within a two-species mixture, composed of molecular and atomic oxygen. A dissociation–recombination mechanism is considered, at varying reaction rates. The results show that while the hydrodynamic field remains largely unaffected by changes in reaction rates, temperature profiles are slightly altered, with faster reactions leading to lower temperature peaks. The chemical mechanisms significantly influence the wall heat flux, with frozen chemistry overestimating the flux. The reference simulations are compared with companion calculations, where chemical reactions are activated downstream within the fully turbulent region. These calculations represent set-ups in which the computational domain effectively starts with an inflow in a fully turbulent state, where hydrodynamic and thermal quantities are accurately described at the boundary and the chemical inflow profile is derived from a frozen-chemistry assumption. In this set-up, chemical source terms rapidly relax towards the baseline downstream of the chemistry activation location. This behaviour is due to an approximate global self-similarity shown by the chemical species transport in the fully turbulent region. Unlike laminar boundary layers where streamwise fluxes are relevant, source terms are balanced only by wall-normal transport in the turbulent region. A chemical relaxation length scale is introduced to collapse the results of all mechanisms.

Accurately predicting the mean flow properties of wall-bounded turbulence is essential for both fundamental research and engineering applications. In atmospheric boundary layers, the mean flow within the surface layer is typically described by Monin–Obukhov similarity theory (MOST). However, beyond the surface layer, MOST no longer applies as the Coriolis effect becomes significant. To address this issue, this study introduces a novel analytical model for the mean turbulent momentum fluxes and geostrophic wind deficits in nocturnal stable atmospheric boundary layers (NSBLs), which are stably stratified near the surface and transition to neutrally stratified flow above. The model solutions are derived from the Ekman equations using the eddy viscosity approach and a new parametrisation of the flux Richardson number. The solutions show that the geostrophic wind deficits scale with $u_*^2/(hf)$, where $u_*$ is the friction velocity, $h$ is the boundary layer height, and $f$ is the Coriolis parameter. The model’s predictions align closely with recent large-eddy simulation studies, confirming the model’s accuracy. Combined with the geostrophic drag law, the model can reliably predict the wind speed profile above the surface layer of NSBLs. This work marks a significant step in modelling atmospheric turbulence and its fundamental dynamics.

The reactivity of transverse waves in detonations of methane, oxygen and nitrogen are experimentally assessed using MHz rate schlieren and chemiluminescence imaging. In these highly unstable mixtures, the mode of wave propagation is more complex than what is described by the cellular instability model that is conventionally used for weakly unstable mixtures. Behind the low-speed leading shock in unstable waves, the processed gas remains essentially unreacted until transverse waves reach this region. In highly unstable waves, the transverse waves have a range of reactivity, that is rates of reaction in the flow immediately behind the wave. In this study, we present examples of transverse waves for near-limit detonations and analyse four cases in detail. In some cases, these waves appear to be essentially non-reactive or cause very slow reaction. In other cases, the transverse waves can be highly reactive. In the most extreme example, the transverse wave is propagating at the Chapman–Jouguet speed with a small reaction zone, i.e. a transverse detonation. A reactive oblique shock model is used to approximate the triple-point configuration of this case as a double-Mach reflection, which shows good agreement with the images. The reaction evolution along path lines is analysed using detailed reaction mechanisms and considerations about flow-field unsteadiness. Length scales of the energy release and expansion processes within the reaction zone region are used to explain the observed modes of wave propagation and interaction.

Buoyancy-driven exchange flows in geophysical contexts often exhibit significant interfacial turbulence leading to a partially mixed intermediate layer between two counterflowing layers. In this paper we perform a three-layer hydraulic analysis of such flows, highlighting the dynamical importance of the middle mixed layer. Our analysis is based on the viscous, shallow water, Boussinesq equations and includes the effects of mixing as a non-hydrostatic pressure forcing. We demonstrate the superior predictive accuracy of three-layer hydraulics over the more classical two-layer approach by applying it to direct numerical simulation data in stratified inclined duct exchange flows where turbulence is controlled by a modest slope of the duct. The three-layer model predicts a region bounded by two control points in the middle of the duct, linked to the onset of instability and turbulence, whereas a two-layer model only predicts one control point. We show that the nonlinear characteristics of the three-layer model correspond to linear long waves perturbing a three-layer mean flow. We also provide the first evidence of long-wave resonance, as well as resonance between long and short waves, and their connection to turbulence. These results challenge current parameterisations for turbulent transport, which typically overlook long waves and internal hydraulics induced by streamwise variations of the flow.

We address the Reynolds number dependence of the turbulent skin-friction drag reduction induced by streamwise-travelling waves of spanwise wall oscillations. The study relies on direct numerical simulations of drag-reduced flows in a plane open channel at friction Reynolds numbers in the range $1000 \leqslant Re_\tau \leqslant 6000$, which is the widest range considered so far in simulations with spanwise forcing. Our results corroborate the validity of the predictive model proposed by Gatti & Quadrio (J. Fluid Mech. vol. 802, 2016, pp. 553–558): regardless of the control parameters, the drag reduction decreases monotonically with $Re$ at a rate that depends on the drag reduction itself and on the skin-friction of the uncontrolled flow. We do not find evidence in support of the results of Marusic et al. (Nat. Commun. vol. 12, no. 1, 2021, pp. 5805), which instead report by experiments an increase of the drag reduction with $Re$ in turbulent boundary layers, for control parameters that target low-frequency, outer-scaled motions. Possible explanations for this discrepancy are provided, including obvious differences between open channel flows and boundary layers, and possible limitations of laboratory experiments.

By incorporating leading-edge (L-E) protuberances inspired by humpback whale flippers, this study enhances hydrodynamic performance, mitigates cavitation effects and develops efficient models to minimise noise emissions in aquatic systems. Experimental and numerical simulations are conducted on four semi-elliptical NACA 16020 three-dimensional (3-D) hydrofoils, including a baseline hydrofoil and three modified versions featuring sinusoidal L-E alterations. These alterations encompass amplitudes of 2 %, wavelengths of 8.33 % and 4.1667 % of the mean chord length (C), and wavenumbers of 12 and 6. Experimental analysis encompassing both cavitational and non-cavitational regimes at varying attack angles revealed significant relationships between the hydrodynamic performance and partial sheet cavitation. Hydrodynamic force analysis shows that hydrofoils with L-E protuberances generate elevated lift at moderate and high angles of attack (AOA) in cavitating and non-cavitating conditions. Under lower-severity cavitating conditions, models with L-E protuberances exhibit no significant reduction in sound pressure level. In contrast, at higher severity, the presence of L-E protuberances effectively reduces the flow-induced noise, with partial cavities covering 30 %–50 % of the chord. Numerical simulations were conducted to investigate the turbulent kinetic energy (TKE) distribution and the presence of counter-rotating vortices on each protuberance. The results reveal a significantly enhanced TKE around the trough area and the presence of counter-rotating vortices at each protuberance peak. The more realistic asymmetric design performed better than the other modifications regarding hydrodynamic force, whereas the symmetric model with wavelengths of 8.33 % excelled at cavitation and noise suppression. Therefore, this study offers promising avenues for advancing hydrofoil design in diverse engineering domains.

We investigate the effects of external harmonic forcing on flow through a duct with square cross-section containing two circular orifice plates – a double-orifice cavity – at an operating condition where self-sustained limit cycle oscillations are observed. When the oscillatory flow is periodically forced at a frequency $f_f$ near its natural frequency $f_n$ ($0.9\leqslant f_f /f_n \leqslant 1.1$), it undergoes lock-in and amplitude suppression through synchronous quenching. We observe phase-drifting (or phase-slipping) prior to lock-in that happens via a saddle-node bifurcation. However, when the flow system is forced far from its natural frequency ($0.8\leqslant f_f /f_n\leqslant 0.9$ and $1.1\leqslant f_f /f_n\leqslant 1.4$) lock-in happens via asynchronous quenching through a Neimark–Sacker bifurcation (torus death). In asynchronous quenching, phase-drifting and phase-trapping are observed before lock-in. An asymmetry is present in the synchronization map on forcing either side of the natural frequency, which becomes more pronounced in the asynchronous quenching regime. There is also an observed saturation of the synchronization map for $f_f/f_n\gt 1$ over the range of frequencies explored. Subharmonic synchronization or $1:2$ lock-in with period-two oscillations is also observed when the system is forced near $f_n/2$ ($ 0.49 \leqslant f_f /f_n \leqslant 0.51$). The route to lock-in consists of a three frequency regime where subharmonics of the forcing frequency ($f_f/2$ and $f_f/3$) play an important role in the dynamics. The transition from $1:1$ to $1:2$ lock-in occurs via a de-lock-in regime ($ 0.55 \leqslant f_f /f_n \leqslant 0.65$), where a lock-in boundary is present; i.e. the system delocks after lock-in if the amplitude is raised beyond a critical value. The de-lock-in regime is also characterized by a nonlinear phase drift after de-lock-in and a significant jump in the forcing amplitude for lock-in for $f_f/f_n=0.6$. Amplification is observed for $f_f/f_n\gt 1$ and also in the $1:2$ lock-in and de-lock-in regimes where the total signal power exceeds the unforced system’s power for small increases in forcing amplitude after lock-in. Based on these results, we identify the asynchronous quenching regime for $f_f/f_n\lt 1$ as the optimal frequency range where active control is most effective. Finally, we introduce a reduced-order phenomenological model based on vortex–acoustic interaction dynamics from first principles. The model correctly identifies the four regimes, their dynamics leading to lock-in, and asymmetry and saturation in the synchronization map.

The evaporation of liquid from within a porous medium is a complicated process involving coupled capillary flow, vapour diffusion and phase change. Different drying behaviour is observed at different stages during the process. Initially, liquid is drawn to the surface by capillary forces, where it evaporates at a near constant rate; thereafter, a drying front recedes into the material, with a slower net evaporation rate. Modelling drying porous media accurately is challenging due to the multitude of relevant spatial and temporal scales, and the large number of constitutive laws required for model closure. Key aspects of the drying process, including the net evaporation rate and the time of the sudden transition between stages, are not well understood or reliably predicted. We derive simplified mathematical models for both stages of this drying process by systematically reducing an averaged continuum multi-phase flow model, using the method of matched asymptotic expansions, in the physically relevant limit of slow vapour diffusion relative to the local evaporation rate (the large-Péclet-number limit). By solving our reduced models, we compute the evolving net evaporation rate, fluid fluxes and saturation profiles, and estimate the transition time to be when the initial constant-rate-period model ceases to be valid. We additionally characterise properties of the constitutive laws that affect the qualitative drying behaviour: the model is shown to exhibit a receding-front period only if the relative permeability for the liquid phase decays sufficiently quickly relative to the blow up in the capillary pressure as the liquid saturation decreases.

Following Scott & Cambon (2024 J. Fluid Mech. vol. 979, A17), henceforth referred to as [I], a spectral approach is used and the flow is expressed as a sum of normal modes, which are of two types: inertial/gravity waves and non-propagating (NP) modes. It was shown in [I] that, for weak (small Rossby or Froude number) turbulence, the NP component of the flow decouples from the waves at leading order and here we focus on the NP part alone. It is demonstrated that the evolution equations of the NP component are equivalent to the three-dimensional, quasi-geostrophic (QG) approximation of geophysical fluid dynamics. For QG turbulence, the seminal paper of Charney (1971 J. Atmos. Sci. vol. 28, pp. 1087–1095), referred to as [II], concluded that, as for two-dimensional turbulence, the energy cascade for QG turbulence should go from smaller to larger scales and that the inertial-range spectrum at wavenumber $k$ should behave as $k^{-3}$. He also proposed that the energy distribution in spectral space is isotropic if the vertical wavenumber is appropriately scaled and deduced a principle of equipartition in which the average kinetic energy is twice the potential one. We use Charney’s transformation of spectral coordinates to effectively eliminate the parameter $\beta =2{\varOmega} /N$, where ${\varOmega}$ is the rotation rate and $N$ the Brunt–Vaisala frequency, and give results of numerical calculations concerning the energy distribution. The results mostly agree with [II] at large enough times, although they do not support Charney isotropy. They further suggest self-similarity of the time evolution of the three-dimensional energy distribution in spectral space away from the vertical axis.

This paper presents an experimental application of reactive control to jet installation noise based on destructive interference. The work is motivated by the success of previous studies in applying this control approach to mixing layers (Sasaki et al. Theor. 2018b Comput. Fluid Dyn. 32, 765–788), boundary layers (Brito et al. 2021 Exp. Fluids 62, 1–13; Audiffred et al. 2023 Phys. Rev. Fluids 8, 073902), flow over a backward-facing step (Martini et al. 2022 J. Fluid Mech. 937, A19) and, more recently, to turbulent jets (Maia et al. 2021 Phys. Rev. Fluids 6, 123901; Maia et al. 2022 Phys. Rev. Fluids 7, 033903; Audiffred et al. 2024b J. Fluid Mech. 994, A15). We exploit the fact that jet–surface interaction noise is underpinned by wavepackets that can be modelled in a linear framework and develop a linear control strategy where piezoelectric actuators situated at the edge of a scattering surface are driven in real time by sensor measurements in the near field of the jet, the objective being to reduce noise radiated in the acoustic field. The control mechanism involves imposition of an anti-dipole at the trailing edge to cancel the scattering dipole that arises due to an incident wavepacket perturbation. We explore two different control strategies: (i) the inverse feed-forward approach, where causality is imposed by truncating the control kernel, and (ii) the Wiener–Hopf approach, where causality is optimally enforced in building the control kernel. We show that the Wiener–Hopf approach has better performance than that obtained using the truncated inverse feed-forward kernel. We also explore different positions of the near-field sensors and show that control performance is better for sensors installed for streamwise positions downstream in the jet plume, where the signature of hydrodynamic wavepacket is better captured by the sensors. Broadband noise reductions of up to 50 % are achieved.

Several million years of natural evolution have endowed marine animals with high flexibility and mobility. A key factor in this achievement is their ability to modulate stiffness during swimming. However, an unresolved puzzle remains regarding how muscles modulate stiffness, and the implications of this capability for achieving high swimming efficiency. Inspired by this, we proposed a self-propulsor model that employs a parabolic stiffness-tuning strategy, emulating the muscle tensioning observed in biological counterparts. Furthermore, efforts have been directed towards developing the nonlinear vortex sheet method, specifically designed to address nonlinear fluid–structure coupling problems. This work aims to analyse how and why nonlinear tunable stiffness influences swimming performance. Numerical results demonstrate that swimmers with nonlinear tunable stiffness can double their speed and efficiency across nearly the entire frequency range. Additionally, our findings reveal that high-efficiency biomimetic propulsion originates from snap-through instability, which facilitates the emergence of quasi-quadrilateral swimming patterns and enhances vortex strength. Moreover, this study examines the influence of nonlinear stiffness on swimming performance, providing valuable insights into the optimisation of next-generation, high-performance, fish-inspired robotic systems.

We study the mixing of passive scalars in a velocity field generated by selected-eddy simulations (SES), an approach where only a randomly selected subset of spectrally distributed modes obey Navier–Stokes dynamics. The Taylor Reynolds number varies from 140 to 400 and the Schmidt number ($Sc$) varies from 0.25 to 1. By comparing the results with direct numerical simulations (DNS), we show that most statistics are captured with as low as $0.5\,\%$ of Navier–Stokes modes in the velocity field. This includes scalar gradients, spectra, structure functions and their departures from classical scaling due to intermittency. The results suggest that all modes need not be resolved to accurately capture turbulent mixing for $Sc\leqslant 1$ scalars.

Understanding how bubbles on a substrate respond to ultrasound is crucial for applications from industrial cleaning to biomedical treatments. Under ultrasonic excitation, bubbles can undergo shape deformations due to Faraday instability, periodically producing high-speed jets that may cause damage. While recent studies have begun to elucidate this behaviour for free bubbles, the dynamics of wall-attached bubbles is still largely unexplored. In particular, the selection and evolution of non-spherical modes in these bounded systems have not previously been resolved in three dimensions, and the resulting jetting dynamics has yet to be compared with that observed in free bubbles. In this study, we investigate individual micrometric air bubbles in contact with a rigid substrate and subjected to ultrasound. We introduce a novel dual-view imaging technique that combines top-view bright-field microscopy with side-view phase-contrast X-ray imaging, enabling visualisation of bubble shape evolution from two orthogonal perspectives. This set-up reveals the progression of bubble shape through four distinct dynamic regimes: purely spherical oscillations, onset of harmonic axisymmetric meniscus waves, emergence of half-harmonic axisymmetric Faraday waves and the superposition of half-harmonic sectoral Faraday waves. This stepwise evolution contrasts with the behaviour of free bubbles, which exhibit their ultimate Faraday wave pattern immediately upon instability onset. For the substrate chosen, the resulting shape-mode spectrum appears to be degenerate and exhibits a continuous range of shape mode degrees, in line with our theoretical predictions derived from kinematic arguments. While free bubbles also display a degenerate spectrum, their shape mode degrees remain discrete, constrained by the bubble spherical periodicity. Experimentally measured ultrasound pressure thresholds for the onset of Faraday instability agree well with classical interface stability theory, modified to incorporate the effects of a rigid boundary. Complementary three-dimensional boundary element simulations of bubble shape evolution align closely with experimental observations, validating this method’s predictive capability. Finally, we determine the acceleration threshold at which shape mode lobes initiate cyclic jetting. Unlike free bubbles, jetting in wall-attached bubbles consistently emerges from the side not restricted by the substrate.

We focus on the wake of a cylinder placed in uniform flow and forced to rotate periodically at subcritical Reynolds numbers, i.e. for Reynolds numbers smaller than 47 calculated based on the incoming flow velocity and the cylinder diameter, where vortices are not shed in the wake of a fixed cylinder. We show that in the near wake, the imposed periodic rotation causes the Föppl vortices (the symmetric steady vortices that are formed right behind a fixed cylinder within the Reynolds number range of $5\lt {Re}\lt 47$) to appear only momentarily during each rotation cycle until they disappear at higher rotation rates. In the far wake, vortices can be induced for certain values of rotation rate, $\alpha$, and rotation frequency, $f$. The shedding of these vortices in the wake results in a periodic lift force that acts on the cylinder. We have defined a new parameter $\omega /(f\alpha )\equiv 1/F$, where $\omega$ is the angular velocity of the cylinder, which is significant in describing the system. For any values of angular velocity and the frequency of change in the rotation direction, the wake pattern remains the same if the value of $1/F$ stays constant. Subsequently, the fluctuating lift coefficient and the average drag coefficient peak at the same value of $1/F$ for any value of $\omega /f\equiv \alpha /F$. The Reynolds number for the onset of shedding decreases with increasing rotation rate at a constant $\alpha /F$. We have observed shedding at Reynolds numbers as low as ${Re}=1$ for higher rotation rates.

Axisymmetric turbulent boundary layers are of great significance in industry and the fluid dynamics community. In this paper, direct numerical simulations of an axially developing axisymmetric turbulent boundary layer along a slender cylinder are performed. Periodical suction and blowing perturbation are used to trigger the transition from laminar inflow to turbulent flow downstream, resulting in the boundary layer thickness varying from 7 to 13 times the cylinder radius, and the friction Reynolds number varying from 300 to 510. Turbulence statistics including wall friction coefficient, mean velocity profile and Reynolds stresses are obtained. The turbulence intensities are weakened compared with the planar turbulent layer, and the inter-component energy transfer is also inhibited. A curvature-weighted transformation is proposed, and the transformed Reynolds stresses and mean velocity deficit collapse well with the planar case in the near-wall region. The velocity streaks and vortical structures are explored. The wall-normal variation of the mean spanwise spacing of low-speed streaks is greatly influenced by the cylindrical geometry. Quasi-streamwise vortices dominate the near-wall region, and the arch vortices are prevalent in the outer region. The prograde hairpin vortices can be commonly observed.

A rotating detonation combustor exhibits corotating $N$-wave modes with $N$ detonation waves propagating in the same direction. These modes and their responses to ignition conditions and disturbances were studied using a surrogate model. Through numerical continuation, a mode curve (MC) is obtained, depicting the relationship between the wave speed of the one-wave mode and a defined baseline of the combustor circumference ($L_{{base}}$) under fixed equation parameters, limited by deflagration and flow choking. The modes’ existence is confirmed by the equivalence between a one-wave mode within a combustor with circumference $L_{{base}}$/$N$ on the MC and an $N$-wave mode in an $L_{{base}}$ combustor. The stability, measured by the real part of the eigenvalue from linear stability analysis (LSA), revealed the dynamic properties. When multiple stable modes exist under the same parameters, ignition conditions with a spatial period of $L_{{base}}$/$N$ are more likely to form $N$-wave modes. An unstable evolution in formed modes, occurs in the dynamics from stable to unstable modes through saddle-node bifurcation and Hopf bifurcation induced by parameter perturbations and from unstable to stable modes induced by state disturbances. Eigenmodes from LSA reveal mechanisms of the unstable evolution, including the effect of secondary deflagration in the unstable one-wave mode and competitive interaction between detonation waves in the unstable multiwave mode, crucial for the combustor to mode transition.